diff --git "a/random/test.jsonl" "b/random/test.jsonl"
new file mode 100644--- /dev/null
+++ "b/random/test.jsonl"
@@ -0,0 +1,2000 @@
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TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nhb : ∃ C, ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\n⊢ ∃ r,\n    (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),\n        l.MemBaseSet I c₁ (r c₁) π₁ →\n          π₁.IsPartition →\n            l.MemBaseSet I c₂ (r c₂) π₂ →\n              π₂.IsPartition →\n                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε","tactic":"refine integrable_iff_cauchy_basis.2 fun ε ε0 ↦ ?_","premises":[{"full_name":"BoxIntegral.integrable_iff_cauchy_basis","def_path":"Mathlib/Analysis/BoxIntegral/Basic.lean","def_pos":[202,8],"def_end_pos":[202,35]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]}]},{"state_before":"ι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nhb : ∃ C, ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\n⊢ ∃ r,\n    (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),\n        l.MemBaseSet I c₁ (r c₁) π₁ →\n          π₁.IsPartition →\n            l.MemBaseSet I c₂ (r c₂) π₂ →\n              π₂.IsPartition →\n                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε","state_after":"case intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nhb : ∃ C, ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\n⊢ ∃ r,\n    (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),\n        l.MemBaseSet I c₁ (r c₁) π₁ →\n          π₁.IsPartition →\n            l.MemBaseSet I c₂ (r c₂) π₂ →\n              π₂.IsPartition →\n                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε","tactic":"rcases exists_pos_mul_lt ε0 (2 * μ.toBoxAdditive I) with ⟨ε₁, ε₁0, hε₁⟩","premises":[{"full_name":"MeasureTheory.Measure.toBoxAdditive","def_path":"Mathlib/Analysis/BoxIntegral/Partition/Measure.lean","def_pos":[93,4],"def_end_pos":[93,17]},{"full_name":"exists_pos_mul_lt","def_path":"Mathlib/Algebra/Order/Field/Basic.lean","def_pos":[404,8],"def_end_pos":[404,25]}]},{"state_before":"case intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nhb : ∃ C, ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\n⊢ ∃ r,\n    (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),\n        l.MemBaseSet I c₁ (r c₁) π₁ →\n          π₁.IsPartition →\n            l.MemBaseSet I c₂ (r c₂) π₂ →\n              π₂.IsPartition →\n                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε","state_after":"case intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\n⊢ ∃ r,\n    (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),\n        l.MemBaseSet I c₁ (r c₁) π₁ →\n          π₁.IsPartition →\n            l.MemBaseSet I c₂ (r c₂) π₂ →\n              π₂.IsPartition →\n                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε","tactic":"rcases hb with ⟨C, hC⟩","premises":[]},{"state_before":"case intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\n⊢ ∃ r,\n    (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),\n        l.MemBaseSet I c₁ (r c₁) π₁ →\n          π₁.IsPartition →\n            l.MemBaseSet I c₂ (r c₂) π₂ →\n              π₂.IsPartition →\n                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε","state_after":"case intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\n⊢ ∃ r,\n    (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),\n        l.MemBaseSet I c₁ (r c₁) π₁ →\n          π₁.IsPartition →\n            l.MemBaseSet I c₂ (r c₂) π₂ →\n              π₂.IsPartition →\n                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε","tactic":"have C0 : 0 ≤ C := by\n    obtain ⟨x, hx⟩ := BoxIntegral.Box.nonempty_coe I\n    exact le_trans (norm_nonneg (f x)) <| hC x (I.coe_subset_Icc hx)","premises":[{"full_name":"BoxIntegral.Box.coe_subset_Icc","def_path":"Mathlib/Analysis/BoxIntegral/Box/Basic.lean","def_pos":[213,8],"def_end_pos":[213,22]},{"full_name":"BoxIntegral.Box.nonempty_coe","def_path":"Mathlib/Analysis/BoxIntegral/Box/Basic.lean","def_pos":[127,8],"def_end_pos":[127,20]},{"full_name":"le_trans","def_path":"Mathlib/Order/Defs.lean","def_pos":[48,8],"def_end_pos":[48,16]},{"full_name":"norm_nonneg","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[401,29],"def_end_pos":[401,40]}]},{"state_before":"case intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\n⊢ ∃ r,\n    (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),\n        l.MemBaseSet I c₁ (r c₁) π₁ →\n          π₁.IsPartition →\n            l.MemBaseSet I c₂ (r c₂) π₂ →\n              π₂.IsPartition →\n                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε","state_after":"case intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\n⊢ ∃ r,\n    (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),\n        l.MemBaseSet I c₁ (r c₁) π₁ →\n          π₁.IsPartition →\n            l.MemBaseSet I c₂ (r c₂) π₂ →\n              π₂.IsPartition →\n                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε","tactic":"rcases exists_pos_mul_lt ε0 (4 * C) with ⟨ε₂, ε₂0, hε₂⟩","premises":[{"full_name":"exists_pos_mul_lt","def_path":"Mathlib/Algebra/Order/Field/Basic.lean","def_pos":[404,8],"def_end_pos":[404,25]}]},{"state_before":"case intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf��� g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\n⊢ ∃ r,\n    (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),\n        l.MemBaseSet I c₁ (r c₁) π₁ →\n          π₁.IsPartition →\n            l.MemBaseSet I c₂ (r c₂) π₂ →\n              π₂.IsPartition →\n                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε","state_after":"case intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\n⊢ ∃ r,\n    (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),\n        l.MemBaseSet I c₁ (r c₁) π₁ →\n          π₁.IsPartition →\n            l.MemBaseSet I c₂ (r c₂) π₂ →\n              π₂.IsPartition →\n                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε","tactic":"have ε₂0' : ENNReal.ofReal ε₂ ≠ 0 := ne_of_gt <| ofReal_pos.2 ε₂0","premises":[{"full_name":"ENNReal.ofReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[187,28],"def_end_pos":[187,34]},{"full_name":"ENNReal.ofReal_pos","def_path":"Mathlib/Data/ENNReal/Real.lean","def_pos":[192,8],"def_end_pos":[192,18]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"ne_of_gt","def_path":"Mathlib/Order/Defs.lean","def_pos":[85,8],"def_end_pos":[85,16]}]},{"state_before":"case intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\n⊢ ∃ r,\n    (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),\n        l.MemBaseSet I c₁ (r c₁) π₁ →\n          π₁.IsPartition →\n            l.MemBaseSet I c₂ (r c₂) π₂ →\n              π₂.IsPartition →\n                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε","state_after":"case intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\n⊢ ∃ r,\n    (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),\n        l.MemBaseSet I c₁ (r c₁) π₁ →\n          π₁.IsPartition →\n            l.MemBaseSet I c₂ (r c₂) π₂ →\n              π₂.IsPartition →\n                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε","tactic":"let D := { x ∈ Box.Icc I | ¬ ContinuousWithinAt f (Box.Icc I) x }","premises":[{"full_name":"And","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[516,10],"def_end_pos":[516,13]},{"full_name":"BoxIntegral.Box.Icc","def_path":"Mathlib/Analysis/BoxIntegral/Box/Basic.lean","def_pos":[185,14],"def_end_pos":[185,17]},{"full_name":"ContinuousWithinAt","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[158,4],"def_end_pos":[158,22]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Not","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[224,4],"def_end_pos":[224,7]},{"full_name":"setOf","def_path":"Mathlib/Init/Set.lean","def_pos":[56,4],"def_end_pos":[56,9]}]},{"state_before":"case intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\n⊢ ∃ r,\n    (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),\n        l.MemBaseSet I c₁ (r c₁) π₁ →\n          π₁.IsPartition →\n            l.MemBaseSet I c₂ (r c₂) π₂ →\n              π₂.IsPartition →\n                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε","state_after":"case intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\n⊢ ∃ r,\n    (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),\n        l.MemBaseSet I c₁ (r c₁) π₁ →\n          π₁.IsPartition →\n            l.MemBaseSet I c₂ (r c₂) π₂ →\n              π₂.IsPartition →\n                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε","tactic":"let μ' := μ.restrict (Box.Icc I)","premises":[{"full_name":"BoxIntegral.Box.Icc","def_path":"Mathlib/Analysis/BoxIntegral/Box/Basic.lean","def_pos":[185,14],"def_end_pos":[185,17]},{"full_name":"MeasureTheory.Measure.restrict","def_path":"Mathlib/MeasureTheory/Measure/Restrict.lean","def_pos":[43,18],"def_end_pos":[43,26]}]},{"state_before":"case intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\n⊢ ∃ r,\n    (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),\n        l.MemBaseSet I c₁ (r c₁) π₁ →\n          π₁.IsPartition →\n            l.MemBaseSet I c₂ (r c₂) π₂ →\n              π₂.IsPartition →\n                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε","state_after":"case intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\n⊢ ∃ r,\n    (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),\n        l.MemBaseSet I c₁ (r c₁) π₁ →\n          π₁.IsPartition →\n            l.MemBaseSet I c₂ (r c₂) π₂ →\n              π₂.IsPartition →\n                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε","tactic":"have μ'D : μ' D = 0 := by\n    rcases eventually_iff_exists_mem.1 hc with ⟨V, ae, hV⟩\n    exact eq_of_le_of_not_lt (mem_ae_iff.1 ae ▸ (μ'.mono <| fun x h xV ↦ h.2 (hV x xV))) not_lt_zero","premises":[{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"ENNReal.not_lt_zero","def_path":"Mathlib/Data/ENNReal/Operations.lean","def_pos":[113,8],"def_end_pos":[113,19]},{"full_name":"Filter.eventually_iff_exists_mem","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[990,8],"def_end_pos":[990,33]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"MeasureTheory.OuterMeasure.mono","def_path":"Mathlib/MeasureTheory/OuterMeasure/Defs.lean","def_pos":[46,12],"def_end_pos":[46,16]},{"full_name":"MeasureTheory.mem_ae_iff","def_path":"Mathlib/MeasureTheory/OuterMeasure/AE.lean","def_pos":[70,8],"def_end_pos":[70,18]},{"full_name":"eq_of_le_of_not_lt","def_path":"Mathlib/Order/Basic.lean","def_pos":[345,8],"def_end_pos":[345,26]}]},{"state_before":"case intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\n⊢ ∃ r,\n    (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),\n        l.MemBaseSet I c₁ (r c₁) π₁ →\n          π₁.IsPartition →\n            l.MemBaseSet I c₂ (r c₂) π₂ →\n              π₂.IsPartition →\n                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε","state_after":"case intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < μ' D + ENNReal.ofReal ε₂\n⊢ ∃ r,\n    (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),\n        l.MemBaseSet I c₁ (r c₁) π₁ →\n          π₁.IsPartition →\n            l.MemBaseSet I c₂ (r c₂) π₂ →\n              π₂.IsPartition →\n                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε","tactic":"obtain ⟨U, UD, Uopen, hU⟩ := Set.exists_isOpen_lt_add D (show μ' D ≠ ⊤ by simp [μ'D]) ε₂0'","premises":[{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Set.exists_isOpen_lt_add","def_path":"Mathlib/MeasureTheory/Measure/Regular.lean","def_pos":[340,8],"def_end_pos":[340,39]},{"full_name":"Top.top","def_path":"Mathlib/Order/Notation.lean","def_pos":[94,2],"def_end_pos":[94,5]}]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < μ' D + ENNReal.ofReal ε₂\n⊢ ∃ r,\n    (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),\n        l.MemBaseSet I c₁ (r c₁) π₁ →\n          π₁.IsPartition →\n            l.MemBaseSet I c₂ (r c₂) π₂ →\n              π₂.IsPartition →\n                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε","state_after":"case intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\n⊢ ∃ r,\n    (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),\n        l.MemBaseSet I c₁ (r c₁) π₁ →\n          π₁.IsPartition →\n            l.MemBaseSet I c₂ (r c₂) π₂ →\n              π₂.IsPartition →\n                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε","tactic":"rw [μ'D, zero_add] at hU","premises":[{"full_name":"zero_add","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[408,2],"def_end_pos":[408,13]}]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\n⊢ ∃ r,\n    (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),\n        l.MemBaseSet I c₁ (r c₁) π₁ →\n          π₁.IsPartition →\n            l.MemBaseSet I c₂ (r c₂) π₂ →\n              π₂.IsPartition →\n                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε","state_after":"case intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\ncomp : IsCompact (Box.Icc I \\ U)\n⊢ ∃ r,\n    (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),\n        l.MemBaseSet I c₁ (r c₁) π₁ →\n          π₁.IsPartition →\n            l.MemBaseSet I c₂ (r c₂) π₂ →\n              π₂.IsPartition →\n                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε","tactic":"have comp : IsCompact (Box.Icc I \\ U) :=\n    I.isCompact_Icc.of_isClosed_subset (I.isCompact_Icc.isClosed.sdiff Uopen) Set.diff_subset","premises":[{"full_name":"BoxIntegral.Box.Icc","def_path":"Mathlib/Analysis/BoxIntegral/Box/Basic.lean","def_pos":[185,14],"def_end_pos":[185,17]},{"full_name":"BoxIntegral.Box.isCompact_Icc","def_path":"Mathlib/Analysis/BoxIntegral/Box/Basic.lean","def_pos":[198,18],"def_end_pos":[198,31]},{"full_name":"IsClosed.sdiff","def_path":"Mathlib/Topology/Basic.lean","def_pos":[186,8],"def_end_pos":[186,22]},{"full_name":"IsCompact","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[254,4],"def_end_pos":[254,13]},{"full_name":"IsCompact.isClosed","def_path":"Mathlib/Topology/Separation.lean","def_pos":[1757,8],"def_end_pos":[1757,26]},{"full_name":"IsCompact.of_isClosed_subset","def_path":"Mathlib/Topology/Compactness/Compact.lean","def_pos":[93,8],"def_end_pos":[93,36]},{"full_name":"SDiff.sdiff","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[415,2],"def_end_pos":[415,7]},{"full_name":"Set.diff_subset","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1432,8],"def_end_pos":[1432,19]}]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\ncomp : IsCompact (Box.Icc I \\ U)\n⊢ ∃ r,\n    (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),\n        l.MemBaseSet I c₁ (r c₁) π₁ →\n          π₁.IsPartition →\n            l.MemBaseSet I c₂ (r c₂) π₂ →\n              π₂.IsPartition →\n                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε","state_after":"case intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\ncomp : IsCompact (Box.Icc I \\ U)\nthis : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁\n⊢ ∃ r,\n    (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),\n        l.MemBaseSet I c₁ (r c₁) π₁ →\n          π₁.IsPartition →\n            l.MemBaseSet I c₂ (r c₂) π₂ →\n              π₂.IsPartition →\n                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε","tactic":"have : ∀ x ∈ (Box.Icc I \\ U), oscillationWithin f (Box.Icc I) x < (ENNReal.ofReal ε₁) := by\n    intro x hx\n    suffices oscillationWithin f (Box.Icc I) x = 0 by rw [this]; exact ofReal_pos.2 ε₁0\n    simpa [OscillationWithin.eq_zero_iff_continuousWithinAt, D, hx.1] using hx.2 ∘ (fun a ↦ UD a)","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"BoxIntegral.Box.Icc","def_path":"Mathlib/Analysis/BoxIntegral/Box/Basic.lean","def_pos":[185,14],"def_end_pos":[185,17]},{"full_name":"ENNReal.ofReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[187,28],"def_end_pos":[187,34]},{"full_name":"ENNReal.ofReal_pos","def_path":"Mathlib/Data/ENNReal/Real.lean","def_pos":[192,8],"def_end_pos":[192,18]},{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"OscillationWithin.eq_zero_iff_continuousWithinAt","def_path":"Mathlib/Analysis/Oscillation.lean","def_pos":[74,8],"def_end_pos":[74,38]},{"full_name":"SDiff.sdiff","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[415,2],"def_end_pos":[415,7]},{"full_name":"oscillationWithin","def_path":"Mathlib/Analysis/Oscillation.lean","def_pos":[37,18],"def_end_pos":[37,35]}]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\ncomp : IsCompact (Box.Icc I \\ U)\nthis : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁\n⊢ ∃ r,\n    (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),\n        l.MemBaseSet I c₁ (r c₁) π₁ →\n          π₁.IsPartition →\n            l.MemBaseSet I c₂ (r c₂) π₂ →\n              π₂.IsPartition →\n                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε","state_after":"case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\ncomp : IsCompact (Box.Icc I \\ U)\nthis : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁\nr : ℝ\nr0 : r > 0\nhr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁\n⊢ ∃ r,\n    (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),\n        l.MemBaseSet I c₁ (r c₁) π₁ →\n          π₁.IsPartition →\n            l.MemBaseSet I c₂ (r c₂) π₂ →\n              π₂.IsPartition →\n                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε","tactic":"rcases comp.uniform_oscillationWithin this with ⟨r, r0, hr⟩","premises":[{"full_name":"IsCompact.uniform_oscillationWithin","def_path":"Mathlib/Analysis/Oscillation.lean","def_pos":[101,8],"def_end_pos":[101,33]}]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\ncomp : IsCompact (Box.Icc I \\ U)\nthis : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁\nr : ℝ\nr0 : r > 0\nhr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁\n⊢ ∃ r,\n    (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),\n        l.MemBaseSet I c₁ (r c₁) π₁ →\n          π₁.IsPartition →\n            l.MemBaseSet I c₂ (r c₂) π₂ →\n              π₂.IsPartition →\n                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε","state_after":"case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\ncomp : IsCompact (Box.Icc I \\ U)\nthis : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁\nr : ℝ\nr0 : r > 0\nhr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁\nc₁ c₂ : ℝ≥0\nπ₁ π₂ : TaggedPrepartition I\nh₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁\nh₁p : π₁.IsPartition\nh₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂\nh₂p : π₂.IsPartition\n⊢ dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε","tactic":"refine ⟨fun _ _ ↦ ⟨r / 2, half_pos r0⟩, fun _ _ _ ↦ rfl, fun c₁ c₂ π₁ π₂ h₁ h₁p h₂ h₂p ↦ ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"half_pos","def_path":"Mathlib/Algebra/Order/Field/Basic.lean","def_pos":[351,8],"def_end_pos":[351,16]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\ncomp : IsCompact (Box.Icc I \\ U)\nthis : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁\nr : ℝ\nr0 : r > 0\nhr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁\nc₁ c₂ : ℝ≥0\nπ₁ π₂ : TaggedPrepartition I\nh₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁\nh₁p : π₁.IsPartition\nh₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂\nh₂p : π₂.IsPartition\n⊢ dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε","state_after":"case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε��\ncomp : IsCompact (Box.Icc I \\ U)\nthis : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁\nr : ℝ\nr0 : r > 0\nhr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁\nc₁ c₂ : ℝ≥0\nπ₁ π₂ : TaggedPrepartition I\nh₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁\nh₁p : π₁.IsPartition\nh₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂\nh₂p : π₂.IsPartition\n⊢ ‖∑ x ∈ (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes,\n        μ.toBoxAdditive x •\n          (f ((π₁.infPrepartition π₂.toPrepartition).tag x) - f ((π₂.infPrepartition π₁.toPrepartition).tag x))‖ ≤\n    ε","tactic":"simp only [dist_eq_norm, integralSum_sub_partitions _ _ h₁p h₂p, toSMul_apply, ← smul_sub]","premises":[{"full_name":"BoxIntegral.BoxAdditiveMap.toSMul_apply","def_path":"Mathlib/Analysis/BoxIntegral/Partition/Additive.lean","def_pos":[174,8],"def_end_pos":[174,20]},{"full_name":"BoxIntegral.integralSum_sub_partitions","def_path":"Mathlib/Analysis/BoxIntegral/Basic.lean","def_pos":[109,8],"def_end_pos":[109,34]},{"full_name":"smul_sub","def_path":"Mathlib/Algebra/GroupWithZero/Action/Defs.lean","def_pos":[279,8],"def_end_pos":[279,16]}]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\ncomp : IsCompact (Box.Icc I \\ U)\nthis : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁\nr : ℝ\nr0 : r > 0\nhr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁\nc₁ c₂ : ℝ≥0\nπ₁ π₂ : TaggedPrepartition I\nh₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁\nh₁p : π₁.IsPartition\nh₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂\nh₂p : π₂.IsPartition\n⊢ ‖∑ x ∈ (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes,\n        μ.toBoxAdditive x •\n          (f ((π₁.infPrepartition π₂.toPrepartition).tag x) - f ((π₂.infPrepartition π₁.toPrepartition).tag x))‖ ≤\n    ε","state_after":"case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\ncomp : IsCompact (Box.Icc I \\ U)\nthis : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁\nr : ℝ\nr0 : r > 0\nhr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁\nc₁ c₂ : ℝ≥0\nπ₁ π₂ : TaggedPrepartition I\nh₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁\nh₁p : π₁.IsPartition\nh₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂\nh₂p : π₂.IsPartition\nμI : μ ↑I < ⊤\n⊢ ‖∑ x ∈ (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes,\n        μ.toBoxAdditive x •\n          (f ((π₁.infPrepartition π₂.toPrepartition).tag x) - f ((π₂.infPrepartition π₁.toPrepartition).tag x))‖ ≤\n    ε","tactic":"have μI : μ I < ⊤ := lt_of_le_of_lt (μ.mono I.coe_subset_Icc) I.isCompact_Icc.measure_lt_top","premises":[{"full_name":"BoxIntegral.Box.coe_subset_Icc","def_path":"Mathlib/Analysis/BoxIntegral/Box/Basic.lean","def_pos":[213,8],"def_end_pos":[213,22]},{"full_name":"BoxIntegral.Box.isCompact_Icc","def_path":"Mathlib/Analysis/BoxIntegral/Box/Basic.lean","def_pos":[198,18],"def_end_pos":[198,31]},{"full_name":"IsCompact.measure_lt_top","def_path":"Mathlib/MeasureTheory/Measure/Typeclasses.lean","def_pos":[1124,8],"def_end_pos":[1124,39]},{"full_name":"MeasureTheory.OuterMeasure.mono","def_path":"Mathlib/MeasureTheory/OuterMeasure/Defs.lean","def_pos":[46,12],"def_end_pos":[46,16]},{"full_name":"Top.top","def_path":"Mathlib/Order/Notation.lean","def_pos":[94,2],"def_end_pos":[94,5]},{"full_name":"lt_of_le_of_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[99,8],"def_end_pos":[99,22]}]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\ncomp : IsCompact (Box.Icc I \\ U)\nthis : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁\nr : ℝ\nr0 : r > 0\nhr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁\nc₁ c₂ : ℝ≥0\nπ₁ π₂ : TaggedPrepartition I\nh₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁\nh₁p : π₁.IsPartition\nh₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂\nh₂p : π₂.IsPartition\nμI : μ ↑I < ⊤\n⊢ ‖∑ x ∈ (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes,\n        μ.toBoxAdditive x •\n          (f ((π₁.infPrepartition π₂.toPrepartition).tag x) - f ((π₂.infPrepartition π₁.toPrepartition).tag x))‖ ≤\n    ε","state_after":"case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\ncomp : IsCompact (Box.Icc I \\ U)\nthis : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁\nr : ℝ\nr0 : r > 0\nhr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁\nc₁ c₂ : ℝ≥0\nπ₁ π₂ : TaggedPrepartition I\nh₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁\nh₁p : π₁.IsPartition\nh₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂\nh₂p : π₂.IsPartition\nμI : μ ↑I < ⊤\nt₁ : Box ι → ι → ℝ := fun J => (π₁.infPrepartition π₂.toPrepartition).tag J\n⊢ ‖∑ x ∈ (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes,\n        μ.toBoxAdditive x •\n          (f ((π₁.infPrepartition π₂.toPrepartition).tag x) - f ((π₂.infPrepartition π₁.toPrepartition).tag x))‖ ≤\n    ε","tactic":"let t₁ (J : Box ι) : ℝⁿ := (π₁.infPrepartition π₂.toPrepartition).tag J","premises":[{"full_name":"BoxIntegral.Box","def_path":"Mathlib/Analysis/BoxIntegral/Box/Basic.lean","def_pos":[73,10],"def_end_pos":[73,13]},{"full_name":"BoxIntegral.TaggedPrepartition.infPrepartition","def_path":"Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean","def_pos":[178,4],"def_end_pos":[178,19]},{"full_name":"BoxIntegral.TaggedPrepartition.tag","def_path":"Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean","def_pos":[43,2],"def_end_pos":[43,5]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]}]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\ncomp : IsCompact (Box.Icc I \\ U)\nthis : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁\nr : ℝ\nr0 : r > 0\nhr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁\nc₁ c₂ : ℝ≥0\nπ₁ π₂ : TaggedPrepartition I\nh₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁\nh₁p : π₁.IsPartition\nh₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂\nh₂p : π₂.IsPartition\nμI : μ ↑I < ⊤\nt₁ : Box ι → ι → ℝ := fun J => (π₁.infPrepartition π₂.toPrepartition).tag J\n⊢ ‖∑ x ∈ (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes,\n        μ.toBoxAdditive x •\n          (f ((π₁.infPrepartition π₂.toPrepartition).tag x) - f ((π₂.infPrepartition π₁.toPrepartition).tag x))‖ ≤\n    ε","state_after":"case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\ncomp : IsCompact (Box.Icc I \\ U)\nthis : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁\nr : ℝ\nr0 : r > 0\nhr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁\nc₁ c₂ : ℝ≥0\nπ₁ π₂ : TaggedPrepartition I\nh₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁\nh₁p : π₁.IsPartition\nh₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂\nh₂p : π₂.IsPartition\nμI : μ ↑I < ⊤\nt₁ : Box ι → ι → ℝ := fun J => (π₁.infPrepartition π₂.toPrepartition).tag J\nt₂ : Box ι → ι → ℝ := fun J => (π₂.infPrepartition π₁.toPrepartition).tag J\n⊢ ‖∑ x ∈ (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes,\n        μ.toBoxAdditive x •\n          (f ((π₁.infPrepartition π₂.toPrepartition).tag x) - f ((π₂.infPrepartition π₁.toPrepartition).tag x))‖ ≤\n    ε","tactic":"let t₂ (J : Box ι) : ℝⁿ := (π₂.infPrepartition π₁.toPrepartition).tag J","premises":[{"full_name":"BoxIntegral.Box","def_path":"Mathlib/Analysis/BoxIntegral/Box/Basic.lean","def_pos":[73,10],"def_end_pos":[73,13]},{"full_name":"BoxIntegral.TaggedPrepartition.infPrepartition","def_path":"Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean","def_pos":[178,4],"def_end_pos":[178,19]},{"full_name":"BoxIntegral.TaggedPrepartition.tag","def_path":"Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean","def_pos":[43,2],"def_end_pos":[43,5]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]}]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\ncomp : IsCompact (Box.Icc I \\ U)\nthis : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁\nr : ℝ\nr0 : r > 0\nhr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁\nc₁ c₂ : ℝ≥0\nπ₁ π₂ : TaggedPrepartition I\nh₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁\nh₁p : π₁.IsPartition\nh₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂\nh₂p : π₂.IsPartition\nμI : μ ↑I < ⊤\nt₁ : Box ι → ι → ℝ := fun J => (π₁.infPrepartition π₂.toPrepartition).tag J\nt₂ : Box ι → ι → ℝ := fun J => (π₂.infPrepartition π₁.toPrepartition).tag J\n⊢ ‖∑ x ∈ (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes,\n        μ.toBoxAdditive x •\n          (f ((π₁.infPrepartition π₂.toPrepartition).tag x) - f ((π₂.infPrepartition π₁.toPrepartition).tag x))‖ ≤\n    ε","state_after":"case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\ncomp : IsCompact (Box.Icc I \\ U)\nthis : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁\nr : ℝ\nr0 : r > 0\nhr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁\nc₁ c₂ : ℝ≥0\nπ₁ π₂ : TaggedPrepartition I\nh₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁\nh₁p : π₁.IsPartition\nh₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂\nh₂p : π₂.IsPartition\nμI : μ ↑I < ⊤\nt₁ : Box ι → ι → ℝ := fun J => (π₁.infPrepartition π₂.toPrepartition).tag J\nt₂ : Box ι → ι → ℝ := fun J => (π₂.infPrepartition π₁.toPrepartition).tag J\nB : Finset (Box ι) := (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes\n⊢ ‖∑ x ∈ (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes,\n        μ.toBoxAdditive x •\n          (f ((π₁.infPrepartition π₂.toPrepartition).tag x) - f ((π₂.infPrepartition π₁.toPrepartition).tag x))‖ ≤\n    ε","tactic":"let B := (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes","premises":[{"full_name":"BoxIntegral.Prepartition.boxes","def_path":"Mathlib/Analysis/BoxIntegral/Partition/Basic.lean","def_pos":[55,2],"def_end_pos":[55,7]},{"full_name":"Inf.inf","def_path":"Mathlib/Order/Notation.lean","def_pos":[53,2],"def_end_pos":[53,5]}]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\ncomp : IsCompact (Box.Icc I \\ U)\nthis : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁\nr : ℝ\nr0 : r > 0\nhr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁\nc₁ c₂ : ℝ≥0\nπ₁ π₂ : TaggedPrepartition I\nh₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁\nh₁p : π₁.IsPartition\nh₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂\nh₂p : π₂.IsPartition\nμI : μ ↑I < ⊤\nt₁ : Box ι → ι → ℝ := fun J => (π₁.infPrepartition π₂.toPrepartition).tag J\nt₂ : Box ι → ι → ℝ := fun J => (π₂.infPrepartition π₁.toPrepartition).tag J\nB : Finset (Box ι) := (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes\n⊢ ‖∑ x ∈ (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes,\n        μ.toBoxAdditive x •\n          (f ((π₁.infPrepartition π₂.toPrepartition).tag x) - f ((π₂.infPrepartition π₁.toPrepartition).tag x))‖ ≤\n    ε","state_after":"case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\ncomp : IsCompact (Box.Icc I \\ U)\nthis : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁\nr : ℝ\nr0 : r > 0\nhr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁\nc₁ c₂ : ℝ≥0\nπ₁ π₂ : TaggedPrepartition I\nh₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁\nh₁p : π₁.IsPartition\nh₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂\nh₂p : π₂.IsPartition\nμI : μ ↑I < ⊤\nt₁ : Box ι → ι → ℝ := fun J => (π₁.infPrepartition π₂.toPrepartition).tag J\nt₂ : Box ι → ι → ℝ := fun J => (π₂.infPrepartition π₁.toPrepartition).tag J\nB : Finset (Box ι) := (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes\nB' : Finset (Box ι) := Finset.filter (fun J => ↑J ⊆ U) B\n⊢ ‖∑ x ∈ (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes,\n        μ.toBoxAdditive x •\n          (f ((π₁.infPrepartition π₂.toPrepartition).tag x) - f ((π₂.infPrepartition π₁.toPrepartition).tag x))‖ ≤\n    ε","tactic":"let B' := B.filter (fun J ↦ J.toSet ⊆ U)","premises":[{"full_name":"BoxIntegral.Box.toSet","def_path":"Mathlib/Analysis/BoxIntegral/Box/Basic.lean","def_pos":[101,4],"def_end_pos":[101,9]},{"full_name":"Finset.filter","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2144,4],"def_end_pos":[2144,10]},{"full_name":"HasSubset.Subset","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[384,2],"def_end_pos":[384,8]}]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\ncomp : IsCompact (Box.Icc I \\ U)\nthis : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁\nr : ℝ\nr0 : r > 0\nhr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁\nc₁ c₂ : ℝ≥0\nπ₁ π₂ : TaggedPrepartition I\nh₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁\nh₁p : π₁.IsPartition\nh₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂\nh₂p : π₂.IsPartition\nμI : μ ↑I < ⊤\nt₁ : Box ι → ι → ℝ := fun J => (π₁.infPrepartition π₂.toPrepartition).tag J\nt₂ : Box ι → ι → ℝ := fun J => (π₂.infPrepartition π₁.toPrepartition).tag J\nB : Finset (Box ι) := (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes\nB' : Finset (Box ι) := Finset.filter (fun J => ↑J ⊆ U) B\n⊢ ‖∑ x ∈ (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes,\n        μ.toBoxAdditive x •\n          (f ((π₁.infPrepartition π₂.toPrepartition).tag x) - f ((π₂.infPrepartition π₁.toPrepartition).tag x))‖ ≤\n    ε","state_after":"case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\ncomp : IsCompact (Box.Icc I \\ U)\nthis : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁\nr : ℝ\nr0 : r > 0\nhr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁\nc₁ c₂ : ℝ≥0\nπ₁ π₂ : TaggedPrepartition I\nh₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁\nh₁p : π₁.IsPartition\nh₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂\nh₂p : π₂.IsPartition\nμI : μ ↑I < ⊤\nt₁ : Box ι → ι → ℝ := fun J => (π₁.infPrepartition π₂.toPrepartition).tag J\nt₂ : Box ι → ι → ℝ := fun J => (π₂.infPrepartition π₁.toPrepartition).tag J\nB : Finset (Box ι) := (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes\nB' : Finset (Box ι) := Finset.filter (fun J => ↑J ⊆ U) B\nhB' : B' ⊆ B\n⊢ ‖∑ x ∈ (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes,\n        μ.toBoxAdditive x •\n          (f ((π₁.infPrepartition π₂.toPrepartition).tag x) - f ((π₂.infPrepartition π₁.toPrepartition).tag x))‖ ≤\n    ε","tactic":"have hB' : B' ⊆ B := B.filter_subset (fun J ↦ J.toSet ⊆ U)","premises":[{"full_name":"BoxIntegral.Box.toSet","def_path":"Mathlib/Analysis/BoxIntegral/Box/Basic.lean","def_pos":[101,4],"def_end_pos":[101,9]},{"full_name":"Finset.filter_subset","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2152,8],"def_end_pos":[2152,21]},{"full_name":"HasSubset.Subset","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[384,2],"def_end_pos":[384,8]}]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\ncomp : IsCompact (Box.Icc I \\ U)\nthis : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁\nr : ℝ\nr0 : r > 0\nhr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁\nc₁ c₂ : ℝ≥0\nπ₁ π₂ : TaggedPrepartition I\nh₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁\nh₁p : π₁.IsPartition\nh₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂\nh₂p : π₂.IsPartition\nμI : μ ↑I < ⊤\nt₁ : Box ι → ι → ℝ := fun J => (π₁.infPrepartition π₂.toPrepartition).tag J\nt₂ : Box ι → ι → ℝ := fun J => (π₂.infPrepartition π₁.toPrepartition).tag J\nB : Finset (Box ι) := (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes\nB' : Finset (Box ι) := Finset.filter (fun J => ↑J ⊆ U) B\nhB' : B' ⊆ B\n⊢ ‖∑ x ∈ (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes,\n        μ.toBoxAdditive x •\n          (f ((π₁.infPrepartition π₂.toPrepartition).tag x) - f ((π₂.infPrepartition π₁.toPrepartition).tag x))‖ ≤\n    ε","state_after":"case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\ncomp : IsCompact (Box.Icc I \\ U)\nthis : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁\nr : ℝ\nr0 : r > 0\nhr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁\nc₁ c₂ : ℝ≥0\nπ₁ π₂ : TaggedPrepartition I\nh₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁\nh₁p : π₁.IsPartition\nh₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂\nh₂p : π₂.IsPartition\nμI : μ ↑I < ⊤\nt₁ : Box ι → ι → ℝ := fun J => (π₁.infPrepartition π₂.toPrepartition).tag J\nt₂ : Box ι → ι → ℝ := fun J => (π₂.infPrepartition π₁.toPrepartition).tag J\nB : Finset (Box ι) := (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes\nB' : Finset (Box ι) := Finset.filter (fun J => ↑J ⊆ U) B\nhB' : B' ⊆ B\nμJ_ne_top : ∀ J ∈ B, μ ↑J ≠ ⊤\n⊢ ‖∑ x ∈ (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes,\n        μ.toBoxAdditive x •\n          (f ((π₁.infPrepartition π₂.toPrepartition).tag x) - f ((π₂.infPrepartition π₁.toPrepartition).tag x))‖ ≤\n    ε","tactic":"have μJ_ne_top : ∀ J ∈ B, μ J ≠ ⊤ :=\n    fun J hJ ↦ lt_top_iff_ne_top.1 <| lt_of_le_of_lt (μ.mono (Prepartition.le_of_mem' _ J hJ)) μI","premises":[{"full_name":"BoxIntegral.Prepartition.le_of_mem'","def_path":"Mathlib/Analysis/BoxIntegral/Partition/Basic.lean","def_pos":[57,2],"def_end_pos":[57,12]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"MeasureTheory.OuterMeasure.mono","def_path":"Mathlib/MeasureTheory/OuterMeasure/Defs.lean","def_pos":[46,12],"def_end_pos":[46,16]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Top.top","def_path":"Mathlib/Order/Notation.lean","def_pos":[94,2],"def_end_pos":[94,5]},{"full_name":"lt_of_le_of_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[99,8],"def_end_pos":[99,22]},{"full_name":"lt_top_iff_ne_top","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[122,8],"def_end_pos":[122,25]}]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\ncomp : IsCompact (Box.Icc I \\ U)\nthis : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁\nr : ℝ\nr0 : r > 0\nhr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁\nc₁ c₂ : ℝ≥0\nπ₁ π₂ : TaggedPrepartition I\nh₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁\nh₁p : π₁.IsPartition\nh₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂\nh₂p : π₂.IsPartition\nμI : μ ↑I < ⊤\nt₁ : Box ι → ι → ℝ := fun J => (π₁.infPrepartition π₂.toPrepartition).tag J\nt₂ : Box ι → ι → ℝ := fun J => (π₂.infPrepartition π₁.toPrepartition).tag J\nB : Finset (Box ι) := (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes\nB' : Finset (Box ι) := Finset.filter (fun J => ↑J ⊆ U) B\nhB' : B' ⊆ B\nμJ_ne_top : ∀ J ∈ B, μ ↑J ≠ ⊤\n⊢ ‖∑ x ∈ (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes,\n        μ.toBoxAdditive x •\n          (f ((π₁.infPrepartition π₂.toPrepartition).tag x) - f ((π₂.infPrepartition π₁.toPrepartition).tag x))‖ ≤\n    ε","state_after":"case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\ncomp : IsCompact (Box.Icc I \\ U)\nthis : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁\nr : ℝ\nr0 : r > 0\nhr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁\nc₁ c₂ : ℝ≥0\nπ₁ π₂ : TaggedPrepartition I\nh₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁\nh₁p : π₁.IsPartition\nh₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂\nh₂p : π₂.IsPartition\nμI : μ ↑I < ⊤\nt₁ : Box ι → ι → ℝ := fun J => (π₁.infPrepartition π₂.toPrepartition).tag J\nt₂ : Box ι → ι → ℝ := fun J => (π₂.infPrepartition π₁.toPrepartition).tag J\nB : Finset (Box ι) := (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes\nB' : Finset (Box ι) := Finset.filter (fun J => ↑J ⊆ U) B\nhB' : B' ⊆ B\nμJ_ne_top : ∀ J ∈ B, μ ↑J ≠ ⊤\nun : ∀ S ⊆ B, ⋃ J ∈ S, ↑J ⊆ ↑I\n⊢ ‖∑ x ∈ (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes,\n        μ.toBoxAdditive x •\n          (f ((π₁.infPrepartition π₂.toPrepartition).tag x) - f ((π₂.infPrepartition π₁.toPrepartition).tag x))‖ ≤\n    ε","tactic":"have un : ∀ S ⊆ B, ⋃ J ∈ S, J.toSet ⊆ I.toSet :=\n    fun S hS ↦ iUnion_subset_iff.2 (fun J ↦ iUnion_subset_iff.2 fun hJ ↦ le_of_mem' _ J (hS hJ))","premises":[{"full_name":"BoxIntegral.Box.toSet","def_path":"Mathlib/Analysis/BoxIntegral/Box/Basic.lean","def_pos":[101,4],"def_end_pos":[101,9]},{"full_name":"BoxIntegral.Prepartition.le_of_mem'","def_path":"Mathlib/Analysis/BoxIntegral/Partition/Basic.lean","def_pos":[57,2],"def_end_pos":[57,12]},{"full_name":"HasSubset.Subset","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[384,2],"def_end_pos":[384,8]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Set.iUnion","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[178,4],"def_end_pos":[178,10]},{"full_name":"Set.iUnion_subset_iff","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[230,8],"def_end_pos":[230,25]}]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\ncomp : IsCompact (Box.Icc I \\ U)\nthis : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁\nr : ℝ\nr0 : r > 0\nhr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁\nc₁ c₂ : ℝ≥0\nπ₁ π₂ : TaggedPrepartition I\nh₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁\nh₁p : π₁.IsPartition\nh₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂\nh₂p : π₂.IsPartition\nμI : μ ↑I < ⊤\nt₁ : Box ι → ι → ℝ := fun J => (π₁.infPrepartition π₂.toPrepartition).tag J\nt₂ : Box ι → ι → ℝ := fun J => (π₂.infPrepartition π₁.toPrepartition).tag J\nB : Finset (Box ι) := (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes\nB' : Finset (Box ι) := Finset.filter (fun J => ↑J ⊆ U) B\nhB' : B' ⊆ B\nμJ_ne_top : ∀ J ∈ B, μ ↑J ≠ ⊤\nun : ∀ S ⊆ B, ⋃ J ∈ S, ↑J ⊆ ↑I\n⊢ ‖∑ x ∈ (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes,\n        μ.toBoxAdditive x •\n          (f ((π₁.infPrepartition π₂.toPrepartition).tag x) - f ((π₂.infPrepartition π₁.toPrepartition).tag x))‖ ≤\n    ε","state_after":"case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\ncomp : IsCompact (Box.Icc I \\ U)\nthis : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁\nr : ℝ\nr0 : r > 0\nhr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁\nc₁ c₂ : ℝ≥0\nπ₁ π₂ : TaggedPrepartition I\nh₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁\nh₁p : π₁.IsPartition\nh₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂\nh₂p : π₂.IsPartition\nμI : μ ↑I < ⊤\nt₁ : Box ι → ι → ℝ := fun J => (π₁.infPrepartition π₂.toPrepartition).tag J\nt₂ : Box ι → ι → ℝ := fun J => (π₂.infPrepartition π₁.toPrepartition).tag J\nB : Finset (Box ι) := (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes\nB' : Finset (Box ι) := Finset.filter (fun J => ↑J ⊆ U) B\nhB' : B' ⊆ B\nμJ_ne_top : ∀ J ∈ B, μ ↑J ≠ ⊤\nun : ∀ S ⊆ B, ⋃ J ∈ S, ↑J ⊆ ↑I\n⊢ ‖∑ x ∈ B \\ B',\n          μ.toBoxAdditive x •\n            (f ((π₁.infPrepartition π₂.toPrepartition).tag x) - f ((π₂.infPrepartition π₁.toPrepartition).tag x)) +\n        ∑ x ∈ B',\n          μ.toBoxAdditive x •\n            (f ((π₁.infPrepartition π₂.toPrepartition).tag x) - f ((π₂.infPrepartition π₁.toPrepartition).tag x))‖ ≤\n    ε / 2 + ε / 2","tactic":"rw [← sum_sdiff hB', ← add_halves ε]","premises":[{"full_name":"Finset.sum_sdiff","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[509,2],"def_end_pos":[509,13]},{"full_name":"add_halves","def_path":"Mathlib/Algebra/CharZero/Lemmas.lean","def_pos":[115,8],"def_end_pos":[115,18]}]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\ncomp : IsCompact (Box.Icc I \\ U)\nthis : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁\nr : ℝ\nr0 : r > 0\nhr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁\nc₁ c₂ : ℝ≥0\nπ₁ π₂ : TaggedPrepartition I\nh₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁\nh₁p : π₁.IsPartition\nh₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂\nh₂p : π₂.IsPartition\nμI : μ ↑I < ⊤\nt₁ : Box ι → ι → ℝ := fun J => (π₁.infPrepartition π₂.toPrepartition).tag J\nt₂ : Box ι → ι → ℝ := fun J => (π₂.infPrepartition π₁.toPrepartition).tag J\nB : Finset (Box ι) := (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes\nB' : Finset (Box ι) := Finset.filter (fun J => ↑J ⊆ U) B\nhB' : B' ⊆ B\nμJ_ne_top : ∀ J ∈ B, μ ↑J ≠ ⊤\nun : ∀ S ⊆ B, ⋃ J ∈ S, ↑J ⊆ ↑I\n⊢ ‖∑ x ∈ B \\ B',\n          μ.toBoxAdditive x •\n            (f ((π₁.infPrepartition π₂.toPrepartition).tag x) - f ((π₂.infPrepartition π₁.toPrepartition).tag x)) +\n        ∑ x ∈ B',\n          μ.toBoxAdditive x •\n            (f ((π₁.infPrepartition π₂.toPrepartition).tag x) - f ((π₂.infPrepartition π₁.toPrepartition).tag x))‖ ≤\n    ε / 2 + ε / 2","state_after":"ι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\ncomp : IsCompact (Box.Icc I \\ U)\nthis : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁\nr : ℝ\nr0 : r > 0\nhr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁\nc₁ c₂ : ℝ≥0\nπ₁ π₂ : TaggedPrepartition I\nh₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁\nh₁p : π₁.IsPartition\nh₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂\nh₂p : π₂.IsPartition\nμI : μ ↑I < ⊤\nt₁ : Box ι → ι → ℝ := fun J => (π₁.infPrepartition π₂.toPrepartition).tag J\nt₂ : Box ι → ι → ℝ := fun J => (π₂.infPrepartition π₁.toPrepartition).tag J\nB : Finset (Box ι) := (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes\nB' : Finset (Box ι) := Finset.filter (fun J => ↑J ⊆ U) B\nhB' : B' ⊆ B\nμJ_ne_top : ∀ J ∈ B, μ ↑J ≠ ⊤\nun : ∀ S ⊆ B, ⋃ J ∈ S, ↑J ⊆ ↑I\n⊢ ‖∑ x ∈ B \\ B',\n        μ.toBoxAdditive x •\n          (f ((π₁.infPrepartition π₂.toPrepartition).tag x) - f ((π₂.infPrepartition π₁.toPrepartition).tag x))‖ ≤\n    ε / 2\n\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\ncomp : IsCompact (Box.Icc I \\ U)\nthis : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁\nr : ℝ\nr0 : r > 0\nhr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁\nc₁ c₂ : ℝ≥0\nπ₁ π₂ : TaggedPrepartition I\nh₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁\nh₁p : π₁.IsPartition\nh₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂\nh₂p : π₂.IsPartition\nμI : μ ↑I < ⊤\nt₁ : Box ι → ι → ℝ := fun J => (π₁.infPrepartition π₂.toPrepartition).tag J\nt₂ : Box ι → ι → ℝ := fun J => (π₂.infPrepartition π₁.toPrepartition).tag J\nB : Finset (Box ι) := (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes\nB' : Finset (Box ι) := Finset.filter (fun J => ↑J ⊆ U) B\nhB' : B' ⊆ B\nμJ_ne_top : ∀ J ∈ B, μ ↑J ≠ ⊤\nun : ∀ S ⊆ B, ⋃ J ∈ S, ↑J ⊆ ↑I\n⊢ ‖∑ x ∈ B',\n        μ.toBoxAdditive x •\n          (f ((π₁.infPrepartition π₂.toPrepartition).tag x) - f ((π₂.infPrepartition π₁.toPrepartition).tag x))‖ ≤\n    ε / 2","tactic":"apply le_trans (norm_add_le _ _) (add_le_add ?_ ?_)","premises":[{"full_name":"add_le_add","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[182,31],"def_end_pos":[182,41]},{"full_name":"le_trans","def_path":"Mathlib/Order/Defs.lean","def_pos":[48,8],"def_end_pos":[48,16]},{"full_name":"norm_add_le","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[385,14],"def_end_pos":[385,25]}]}]}
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+{"url":"Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean","commit":"","full_name":"AlgebraicGeometry.ProjectiveSpectrum.Proj.toOpen_toSpec_val_c_app","start":[674,0],"end":[678,88],"file_path":"Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean","tactics":[{"state_before":"R : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : A\nU : (Opens ↑↑(Spec A⁰_ f).toPresheafedSpace)ᵒᵖ\n⊢ StructureSheaf.toOpen (A⁰_ f) (unop U) ≫ (toSpec 𝒜 f).val.c.app U =\n    StructureSheaf.toOpen (A⁰_ f) (unop U) ≫\n      ((ΓSpec.locallyRingedSpaceAdjunction.homEquiv (Proj.restrict ⋯) (op (CommRingCat.of (A⁰_ f))))\n                (awayToΓ 𝒜 f).op).val.c.app\n        U","state_after":"no goals","tactic":"congr","premises":[]}]}
+{"url":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","commit":"","full_name":"List.cons_union","start":[919,0],"end":[920,74],"file_path":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : BEq α\na : α\nl₁ l₂ : List α\n⊢ a :: l₁ ∪ l₂ = List.insert a (l₁ ∪ l₂)","state_after":"no goals","tactic":"simp [List.union_def, foldr]","premises":[{"full_name":"List.foldr","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[416,18],"def_end_pos":[416,23]},{"full_name":"List.union_def","def_path":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","def_pos":[915,8],"def_end_pos":[915,17]}]}]}
+{"url":"Mathlib/Geometry/Manifold/IntegralCurve.lean","commit":"","full_name":"isIntegralCurveOn_comp_mul_ne_zero","start":[252,0],"end":[259,91],"file_path":"Mathlib/Geometry/Manifold/IntegralCurve.lean","tactics":[{"state_before":"E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\nH : Type u_2\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : SmoothManifoldWithCorners I M\nγ γ' : ℝ → M\nv : (x : M) → TangentSpace I x\ns s' : Set ℝ\nt₀ a : ℝ\nha : a ≠ 0\n⊢ IsIntegralCurveOn γ v s ↔ IsIntegralCurveOn (γ ∘ fun x => x * a) (a • v) {t | t * a ∈ s}","state_after":"E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\nH : Type u_2\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : SmoothManifoldWithCorners I M\nγ γ' : ℝ → M\nv : (x : M) → TangentSpace I x\ns s' : Set ℝ\nt₀ a : ℝ\nha : a ≠ 0\nhγ : IsIntegralCurveOn (γ ∘ fun x => x * a) (a • v) {t | t * a ∈ s}\n⊢ IsIntegralCurveOn γ v s","tactic":"refine ⟨fun hγ ↦ hγ.comp_mul a, fun hγ ↦ ?_⟩","premises":[{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"IsIntegralCurveOn.comp_mul","def_path":"Mathlib/Geometry/Manifold/IntegralCurve.lean","def_pos":[244,6],"def_end_pos":[244,32]}]},{"state_before":"E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\nH : Type u_2\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : SmoothManifoldWithCorners I M\nγ γ' : ℝ → M\nv : (x : M) → TangentSpace I x\ns s' : Set ℝ\nt₀ a : ℝ\nha : a ≠ 0\nhγ : IsIntegralCurveOn (γ ∘ fun x => x * a) (a • v) {t | t * a ∈ s}\n⊢ IsIntegralCurveOn γ v s","state_after":"case h.e'_11\nE : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\nH : Type u_2\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : SmoothManifoldWithCorners I M\nγ γ' : ℝ → M\nv : (x : M) → TangentSpace I x\ns s' : Set ℝ\nt₀ a : ℝ\nha : a ≠ 0\nhγ : IsIntegralCurveOn (γ ∘ fun x => x * a) (a • v) {t | t * a ∈ s}\n⊢ γ = (γ ∘ fun x => x * a) ∘ fun x => x * a⁻¹\n\ncase h.e'_12\nE : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\nH : Type u_2\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : SmoothManifoldWithCorners I M\nγ γ' : ℝ → M\nv : (x : M) → TangentSpace I x\ns s' : Set ℝ\nt₀ a : ℝ\nha : a ≠ 0\nhγ : IsIntegralCurveOn (γ ∘ fun x => x * a) (a • v) {t | t * a ∈ s}\n⊢ v = a⁻¹ • a • v\n\ncase h.e'_13\nE : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\nH : Type u_2\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : SmoothManifoldWithCorners I M\nγ γ' : ℝ → M\nv : (x : M) → TangentSpace I x\ns s' : Set ℝ\nt₀ a : ℝ\nha : a ≠ 0\nhγ : IsIntegralCurveOn (γ ∘ fun x => x * a) (a • v) {t | t * a ∈ s}\n⊢ s = {t | t * a⁻¹ ∈ {t | t * a ∈ s}}","tactic":"convert hγ.comp_mul a⁻¹","premises":[{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"IsIntegralCurveOn.comp_mul","def_path":"Mathlib/Geometry/Manifold/IntegralCurve.lean","def_pos":[244,6],"def_end_pos":[244,32]}]}]}
+{"url":"Mathlib/CategoryTheory/Idempotents/Karoubi.lean","commit":"","full_name":"CategoryTheory.Idempotents.Karoubi.decomp_p","start":[265,0],"end":[267,73],"file_path":"Mathlib/CategoryTheory/Idempotents/Karoubi.lean","tactics":[{"state_before":"C : Type u_1\ninst✝ : Category.{u_2, u_1} C\nP : Karoubi C\n⊢ (toKaroubi C).map P.p = P.decompId_p ≫ P.decompId_i","state_after":"case h\nC : Type u_1\ninst✝ : Category.{u_2, u_1} C\nP : Karoubi C\n⊢ ((toKaroubi C).map P.p).f = (P.decompId_p ≫ P.decompId_i).f","tactic":"ext","premises":[]},{"state_before":"case h\nC : Type u_1\ninst✝ : Category.{u_2, u_1} C\nP : Karoubi C\n⊢ ((toKaroubi C).map P.p).f = (P.decompId_p ≫ P.decompId_i).f","state_after":"no goals","tactic":"simp only [comp_f, decompId_p_f, decompId_i_f, P.idem, toKaroubi_map_f]","premises":[{"full_name":"CategoryTheory.Idempotents.Karoubi.comp_f","def_path":"Mathlib/CategoryTheory/Idempotents/Karoubi.lean","def_pos":[110,8],"def_end_pos":[110,14]},{"full_name":"CategoryTheory.Idempotents.Karoubi.decompId_i_f","def_path":"Mathlib/CategoryTheory/Idempotents/Karoubi.lean","def_pos":[249,2],"def_end_pos":[249,7]},{"full_name":"CategoryTheory.Idempotents.Karoubi.decompId_p_f","def_path":"Mathlib/CategoryTheory/Idempotents/Karoubi.lean","def_pos":[254,2],"def_end_pos":[254,7]},{"full_name":"CategoryTheory.Idempotents.Karoubi.idem","def_path":"Mathlib/CategoryTheory/Idempotents/Karoubi.lean","def_pos":[47,2],"def_end_pos":[47,6]},{"full_name":"CategoryTheory.Idempotents.toKaroubi_map_f","def_path":"Mathlib/CategoryTheory/Idempotents/Karoubi.lean","def_pos":[139,2],"def_end_pos":[139,7]}]}]}
+{"url":"Mathlib/LinearAlgebra/Matrix/Circulant.lean","commit":"","full_name":"Matrix.Fin.circulant_injective","start":[61,0],"end":[63,39],"file_path":"Mathlib/LinearAlgebra/Matrix/Circulant.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nm : Type u_3\nn : Type u_4\nR : Type u_5\n⊢ Injective fun v => circulant v","state_after":"no goals","tactic":"simp [Injective]","premises":[{"full_name":"Function.Injective","def_path":"Mathlib/Logic/Function/Defs.lean","def_pos":[101,4],"def_end_pos":[101,13]}]}]}
+{"url":"Mathlib/CategoryTheory/Monoidal/Internal/FunctorCategory.lean","commit":"","full_name":"CategoryTheory.Monoidal.ComonFunctorCategoryEquivalence.functorObj_obj_X","start":[125,0],"end":[142,54],"file_path":"Mathlib/CategoryTheory/Monoidal/Internal/FunctorCategory.lean","tactics":[{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : MonoidalCategory D\nA : Comon_ (C ⥤ D)\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\n⊢ A.X.map f ≫\n      ((fun X =>\n            { X := A.X.obj X, counit := A.counit.app X, comul := A.comul.app X, counit_comul := ⋯, comul_counit := ⋯,\n              comul_assoc := ⋯ })\n          Y✝).counit =\n    ((fun X =>\n          { X := A.X.obj X, counit := A.counit.app X, comul := A.comul.app X, counit_comul := ⋯, comul_counit := ⋯,\n            comul_assoc := ⋯ })\n        X✝).counit","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : MonoidalCategory D\nA : Comon_ (C ⥤ D)\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\n⊢ A.X.map f ≫ A.counit.app Y✝ = A.counit.app X✝","tactic":"dsimp","premises":[]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : MonoidalCategory D\nA : Comon_ (C ⥤ D)\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\n⊢ A.X.map f ≫ A.counit.app Y✝ = A.counit.app X✝","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : MonoidalCategory D\nA : Comon_ (C ⥤ D)\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\n⊢ A.counit.app X✝ ≫ 𝟙 (𝟙_ D) = A.counit.app X✝","tactic":"rw [A.counit.naturality, tensorUnit_map]","premises":[{"full_name":"CategoryTheory.Monoidal.tensorUnit_map","def_path":"Mathlib/CategoryTheory/Monoidal/FunctorCategory.lean","def_pos":[93,8],"def_end_pos":[93,22]},{"full_name":"CategoryTheory.NatTrans.naturality","def_path":"Mathlib/CategoryTheory/NatTrans.lean","def_pos":[50,2],"def_end_pos":[50,12]},{"full_name":"Comon_.counit","def_path":"Mathlib/CategoryTheory/Monoidal/Comon_.lean","def_pos":[42,2],"def_end_pos":[42,8]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : MonoidalCategory D\nA : Comon_ (C ⥤ D)\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\n⊢ A.counit.app X✝ ≫ 𝟙 (𝟙_ D) = A.counit.app X✝","state_after":"no goals","tactic":"rw [Category.comp_id]","premises":[{"full_name":"CategoryTheory.Category.comp_id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[153,2],"def_end_pos":[153,9]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : MonoidalCategory D\nA : Comon_ (C ⥤ D)\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\n⊢ A.X.map f ≫\n      ((fun X =>\n            { X := A.X.obj X, counit := A.counit.app X, comul := A.comul.app X, counit_comul := ⋯, comul_counit := ⋯,\n              comul_assoc := ⋯ })\n          Y✝).comul =\n    ((fun X =>\n            { X := A.X.obj X, counit := A.counit.app X, comul := A.comul.app X, counit_comul := ⋯, comul_counit := ⋯,\n              comul_assoc := ⋯ })\n          X✝).comul ≫\n      (A.X.map f ⊗ A.X.map f)","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : MonoidalCategory D\nA : Comon_ (C ⥤ D)\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\n⊢ A.X.map f ≫ A.comul.app Y✝ = A.comul.app X✝ ≫ (A.X.map f ⊗ A.X.map f)","tactic":"dsimp","premises":[]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : MonoidalCategory D\nA : Comon_ (C ⥤ D)\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\n⊢ A.X.map f ≫ A.comul.app Y✝ = A.comul.app X✝ ≫ (A.X.map f ⊗ A.X.map f)","state_after":"no goals","tactic":"rw [A.comul.naturality, tensorObj_map]","premises":[{"full_name":"CategoryTheory.Monoidal.tensorObj_map","def_path":"Mathlib/CategoryTheory/Monoidal/FunctorCategory.lean","def_pos":[101,8],"def_end_pos":[101,21]},{"full_name":"CategoryTheory.NatTrans.naturality","def_path":"Mathlib/CategoryTheory/NatTrans.lean","def_pos":[50,2],"def_end_pos":[50,12]},{"full_name":"Comon_.comul","def_path":"Mathlib/CategoryTheory/Monoidal/Comon_.lean","def_pos":[44,2],"def_end_pos":[44,7]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : MonoidalCategory D\nA : Comon_ (C ⥤ D)\nX : C\n⊢ {\n          obj := fun X =>\n            { X := A.X.obj X, counit := A.counit.app X, comul := A.comul.app X, counit_comul := ⋯, comul_counit := ⋯,\n              comul_assoc := ⋯ },\n          map := fun {X Y} f => { hom := A.X.map f, hom_counit := ⋯, hom_comul := ⋯ } }.map\n      (𝟙 X) =\n    𝟙\n      ({\n            obj := fun X =>\n              { X := A.X.obj X, counit := A.counit.app X, comul := A.comul.app X, counit_comul := ⋯, comul_counit := ⋯,\n                comul_assoc := ⋯ },\n            map := fun {X Y} f => { hom := A.X.map f, hom_counit := ⋯, hom_comul := ⋯ } }.obj\n        X)","state_after":"case w\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : MonoidalCategory D\nA : Comon_ (C ⥤ D)\nX : C\n⊢ ({\n            obj := fun X =>\n              { X := A.X.obj X, counit := A.counit.app X, comul := A.comul.app X, counit_comul := ⋯, comul_counit := ⋯,\n                comul_assoc := ⋯ },\n            map := fun {X Y} f => { hom := A.X.map f, hom_counit := ⋯, hom_comul := ⋯ } }.map\n        (𝟙 X)).hom =\n    (𝟙\n        ({\n              obj := fun X =>\n                { X := A.X.obj X, counit := A.counit.app X, comul := A.comul.app X, counit_comul := ⋯,\n                  comul_counit := ⋯, comul_assoc := ⋯ },\n              map := fun {X Y} f => { hom := A.X.map f, hom_counit := ⋯, hom_comul := ⋯ } }.obj\n          X)).hom","tactic":"ext","premises":[]},{"state_before":"case w\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : MonoidalCategory D\nA : Comon_ (C ⥤ D)\nX : C\n⊢ ({\n            obj := fun X =>\n              { X := A.X.obj X, counit := A.counit.app X, comul := A.comul.app X, counit_comul := ⋯, comul_counit := ⋯,\n                comul_assoc := ⋯ },\n            map := fun {X Y} f => { hom := A.X.map f, hom_counit := ⋯, hom_comul := ⋯ } }.map\n        (𝟙 X)).hom =\n    (𝟙\n        ({\n              obj := fun X =>\n                { X := A.X.obj X, counit := A.counit.app X, comul := A.comul.app X, counit_comul := ⋯,\n                  comul_counit := ⋯, comul_assoc := ⋯ },\n              map := fun {X Y} f => { hom := A.X.map f, hom_counit := ⋯, hom_comul := ⋯ } }.obj\n          X)).hom","state_after":"case w\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : MonoidalCategory D\nA : Comon_ (C ⥤ D)\nX : C\n⊢ A.X.map (𝟙 X) = 𝟙 (A.X.obj X)","tactic":"dsimp","premises":[]},{"state_before":"case w\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : MonoidalCategory D\nA : Comon_ (C ⥤ D)\nX : C\n⊢ A.X.map (𝟙 X) = 𝟙 (A.X.obj X)","state_after":"no goals","tactic":"rw [CategoryTheory.Functor.map_id]","premises":[{"full_name":"CategoryTheory.Functor.map_id","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[39,2],"def_end_pos":[39,8]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : MonoidalCategory D\nA : Comon_ (C ⥤ D)\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\n⊢ {\n          obj := fun X =>\n            { X := A.X.obj X, counit := A.counit.app X, comul := A.comul.app X, counit_comul := ⋯, comul_counit := ⋯,\n              comul_assoc := ⋯ },\n          map := fun {X Y} f => { hom := A.X.map f, hom_counit := ⋯, hom_comul := ⋯ } }.map\n      (f ≫ g) =\n    {\n            obj := fun X =>\n              { X := A.X.obj X, counit := A.counit.app X, comul := A.comul.app X, counit_comul := ⋯, comul_counit := ⋯,\n                comul_assoc := ⋯ },\n            map := fun {X Y} f => { hom := A.X.map f, hom_counit := ⋯, hom_comul := ⋯ } }.map\n        f ≫\n      {\n            obj := fun X =>\n              { X := A.X.obj X, counit := A.counit.app X, comul := A.comul.app X, counit_comul := ⋯, comul_counit := ⋯,\n                comul_assoc := ⋯ },\n            map := fun {X Y} f => { hom := A.X.map f, hom_counit := ⋯, hom_comul := ⋯ } }.map\n        g","state_after":"case w\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : MonoidalCategory D\nA : Comon_ (C ⥤ D)\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\n⊢ ({\n            obj := fun X =>\n              { X := A.X.obj X, counit := A.counit.app X, comul := A.comul.app X, counit_comul := ⋯, comul_counit := ⋯,\n                comul_assoc := ⋯ },\n            map := fun {X Y} f => { hom := A.X.map f, hom_counit := ⋯, hom_comul := ⋯ } }.map\n        (f ≫ g)).hom =\n    ({\n              obj := fun X =>\n                { X := A.X.obj X, counit := A.counit.app X, comul := A.comul.app X, counit_comul := ⋯,\n                  comul_counit := ⋯, comul_assoc := ⋯ },\n              map := fun {X Y} f => { hom := A.X.map f, hom_counit := ⋯, hom_comul := ⋯ } }.map\n          f ≫\n        {\n              obj := fun X =>\n                { X := A.X.obj X, counit := A.counit.app X, comul := A.comul.app X, counit_comul := ⋯,\n                  comul_counit := ⋯, comul_assoc := ⋯ },\n              map := fun {X Y} f => { hom := A.X.map f, hom_counit := ⋯, hom_comul := ⋯ } }.map\n          g).hom","tactic":"ext","premises":[]},{"state_before":"case w\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : MonoidalCategory D\nA : Comon_ (C ⥤ D)\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\n⊢ ({\n            obj := fun X =>\n              { X := A.X.obj X, counit := A.counit.app X, comul := A.comul.app X, counit_comul := ⋯, comul_counit := ⋯,\n                comul_assoc := ⋯ },\n            map := fun {X Y} f => { hom := A.X.map f, hom_counit := ⋯, hom_comul := ⋯ } }.map\n        (f ≫ g)).hom =\n    ({\n              obj := fun X =>\n                { X := A.X.obj X, counit := A.counit.app X, comul := A.comul.app X, counit_comul := ⋯,\n                  comul_counit := ⋯, comul_assoc := ⋯ },\n              map := fun {X Y} f => { hom := A.X.map f, hom_counit := ⋯, hom_comul := ⋯ } }.map\n          f ≫\n        {\n              obj := fun X =>\n                { X := A.X.obj X, counit := A.counit.app X, comul := A.comul.app X, counit_comul := ⋯,\n                  comul_counit := ⋯, comul_assoc := ⋯ },\n              map := fun {X Y} f => { hom := A.X.map f, hom_counit := ⋯, hom_comul := ⋯ } }.map\n          g).hom","state_after":"case w\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : MonoidalCategory D\nA : Comon_ (C ⥤ D)\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\n⊢ A.X.map (f ≫ g) = A.X.map f ≫ A.X.map g","tactic":"dsimp","premises":[]},{"state_before":"case w\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : MonoidalCategory D\nA : Comon_ (C ⥤ D)\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\n⊢ A.X.map (f ≫ g) = A.X.map f ≫ A.X.map g","state_after":"no goals","tactic":"rw [Functor.map_comp]","premises":[{"full_name":"CategoryTheory.Functor.map_comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[41,2],"def_end_pos":[41,10]}]}]}
+{"url":"Mathlib/CategoryTheory/Monoidal/Mon_.lean","commit":"","full_name":"Mon_.mul_braiding","start":[534,0],"end":[555,28],"file_path":"Mathlib/CategoryTheory/Monoidal/Mon_.lean","tactics":[{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : SymmetricCategory C\nX Y : Mon_ C\n⊢ (X ⊗ Y).mul ≫ (β_ X.X Y.X).hom = ((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫ (Y ⊗ X).mul","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : SymmetricCategory C\nX Y : Mon_ C\n⊢ (α_ X.X Y.X (X.X ⊗ Y.X)).hom ≫\n      X.X ◁ (α_ Y.X X.X Y.X).inv ≫\n        X.X ◁ (β_ Y.X X.X).hom ▷ Y.X ≫\n          X.X ◁ (β_ X.X Y.X).hom ▷ Y.X ≫\n            (α_ X.X (Y.X ⊗ X.X) Y.X).inv ≫\n              (α_ X.X Y.X X.X).inv ▷ Y.X ≫\n                (β_ X.X Y.X).hom ▷ X.X ▷ Y.X ≫\n                  (α_ (Y.X ⊗ X.X) X.X Y.X).hom ≫\n                    (α_ Y.X X.X (X.X ⊗ Y.X)).hom ≫\n                      Y.X ◁ X.X ◁ (β_ X.X Y.X).hom ≫\n                        Y.X ◁ (α_ X.X Y.X X.X).inv ≫\n                          Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫\n                            Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul) =\n    ((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫\n      (α_ Y.X X.X (Y.X ⊗ X.X)).hom ≫\n        Y.X ◁ (α_ X.X Y.X X.X).inv ≫\n          Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫ Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul)","tactic":"simp only [monMonoidalStruct_tensorObj_X, tensor_mul, tensor_μ, Category.assoc,\n    BraidedCategory.braiding_naturality, BraidedCategory.braiding_tensor_right,\n    BraidedCategory.braiding_tensor_left, comp_whiskerRight, whisker_assoc,\n    MonoidalCategory.whiskerLeft_comp, pentagon_assoc, pentagon_inv_hom_hom_hom_inv_assoc,\n    Iso.inv_hom_id_assoc, whiskerLeft_hom_inv_assoc]","premises":[{"full_name":"CategoryTheory.BraidedCategory.braiding_naturality","def_path":"Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean","def_pos":[121,8],"def_end_pos":[121,27]},{"full_name":"CategoryTheory.BraidedCategory.braiding_tensor_left","def_path":"Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean","def_pos":[89,8],"def_end_pos":[89,28]},{"full_name":"CategoryTheory.BraidedCategory.braiding_tensor_right","def_path":"Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean","def_pos":[98,8],"def_end_pos":[98,29]},{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.MonoidalCategory.comp_whiskerRight","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[249,8],"def_end_pos":[249,25]},{"full_name":"CategoryTheory.MonoidalCategory.whiskerLeft_comp","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[232,8],"def_end_pos":[232,24]},{"full_name":"CategoryTheory.MonoidalCategory.whisker_assoc","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[266,8],"def_end_pos":[266,21]},{"full_name":"CategoryTheory.tensor_μ","def_path":"Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean","def_pos":[508,4],"def_end_pos":[508,12]},{"full_name":"Mon_.monMonoidalStruct_tensorObj_X","def_path":"Mathlib/CategoryTheory/Monoidal/Mon_.lean","def_pos":[416,8],"def_end_pos":[416,19]},{"full_name":"Mon_.tensor_mul","def_path":"Mathlib/CategoryTheory/Monoidal/Mon_.lean","def_pos":[493,8],"def_end_pos":[493,18]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : SymmetricCategory C\nX Y : Mon_ C\n⊢ (α_ X.X Y.X (X.X ⊗ Y.X)).hom ≫\n      X.X ◁ (α_ Y.X X.X Y.X).inv ≫\n        X.X ◁ (β_ Y.X X.X).hom ▷ Y.X ≫\n          X.X ◁ (β_ X.X Y.X).hom ▷ Y.X ≫\n            (α_ X.X (Y.X ⊗ X.X) Y.X).inv ≫\n              (α_ X.X Y.X X.X).inv ▷ Y.X ≫\n                (β_ X.X Y.X).hom ▷ X.X ▷ Y.X ≫\n                  (α_ (Y.X ⊗ X.X) X.X Y.X).hom ≫\n                    (α_ Y.X X.X (X.X ⊗ Y.X)).hom ≫\n                      Y.X ◁ X.X ◁ (β_ X.X Y.X).hom ≫\n                        Y.X ◁ (α_ X.X Y.X X.X).inv ≫\n                          Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫\n                            Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul) =\n    ((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫\n      (α_ Y.X X.X (Y.X ⊗ X.X)).hom ≫\n        Y.X ◁ (α_ X.X Y.X X.X).inv ≫\n          Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫ Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul)","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : SymmetricCategory C\nX Y : Mon_ C\n⊢ (α_ X.X Y.X (X.X ⊗ Y.X)).hom ≫\n      X.X ◁ (α_ Y.X X.X Y.X).inv ≫\n        ((((((((((X.X ◁ 𝟙 (Y.X ⊗ X.X) ▷ Y.X ≫ (α_ X.X (Y.X ⊗ X.X) Y.X).inv) ≫ (α_ X.X Y.X X.X).inv ▷ Y.X) ≫\n                          (β_ X.X Y.X).hom ▷ X.X ▷ Y.X) ≫\n                        (α_ (Y.X ⊗ X.X) X.X Y.X).hom) ≫\n                      (α_ Y.X X.X (X.X ⊗ Y.X)).hom) ≫\n                    Y.X ◁ X.X ◁ (β_ X.X Y.X).hom) ≫\n                  Y.X ◁ (α_ X.X Y.X X.X).inv) ≫\n                Y.X ◁ (β_ X.X Y.X).hom ▷ X.X) ≫\n              Y.X ◁ (α_ Y.X X.X X.X).hom) ≫\n            (α_ Y.X Y.X (X.X ⊗ X.X)).inv) ≫\n          (Y.mul ⊗ X.mul) =\n    ((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫\n      (α_ Y.X X.X (Y.X ⊗ X.X)).hom ≫\n        Y.X ◁ (α_ X.X Y.X X.X).inv ≫\n          Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫ Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul)","tactic":"slice_lhs 3 4 =>\n    -- We use symmetry here:\n    rw [← MonoidalCategory.whiskerLeft_comp, ← comp_whiskerRight, SymmetricCategory.symmetry]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.MonoidalCategory.comp_whiskerRight","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[249,8],"def_end_pos":[249,25]},{"full_name":"CategoryTheory.MonoidalCategory.whiskerLeft_comp","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[232,8],"def_end_pos":[232,24]},{"full_name":"CategoryTheory.SymmetricCategory.symmetry","def_path":"Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean","def_pos":[358,2],"def_end_pos":[358,10]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : SymmetricCategory C\nX Y : Mon_ C\n⊢ (α_ X.X Y.X (X.X ⊗ Y.X)).hom ≫\n      X.X ◁ (α_ Y.X X.X Y.X).inv ≫\n        ((((((((((X.X ◁ 𝟙 (Y.X ⊗ X.X) ▷ Y.X ≫ (α_ X.X (Y.X ⊗ X.X) Y.X).inv) ≫ (α_ X.X Y.X X.X).inv ▷ Y.X) ≫\n                          (β_ X.X Y.X).hom ▷ X.X ▷ Y.X) ≫\n                        (α_ (Y.X ⊗ X.X) X.X Y.X).hom) ≫\n                      (α_ Y.X X.X (X.X ⊗ Y.X)).hom) ≫\n                    Y.X ◁ X.X ◁ (β_ X.X Y.X).hom) ≫\n                  Y.X ◁ (α_ X.X Y.X X.X).inv) ≫\n                Y.X ◁ (β_ X.X Y.X).hom ▷ X.X) ≫\n              Y.X ◁ (α_ Y.X X.X X.X).hom) ≫\n            (α_ Y.X Y.X (X.X ⊗ X.X)).inv) ≫\n          (Y.mul ⊗ X.mul) =\n    ((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫\n      (α_ Y.X X.X (Y.X ⊗ X.X)).hom ≫\n        Y.X ◁ (α_ X.X Y.X X.X).inv ≫\n          Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫ Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul)","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : SymmetricCategory C\nX Y : Mon_ C\n⊢ (α_ (X.X ⊗ Y.X) X.X Y.X).inv ≫\n      (β_ X.X Y.X).hom ▷ X.X ▷ Y.X ≫\n        (α_ (Y.X ⊗ X.X) X.X Y.X).hom ≫\n          (α_ Y.X X.X (X.X ⊗ Y.X)).hom ≫\n            Y.X ◁ X.X ◁ (β_ X.X Y.X).hom ≫\n              Y.X ◁ (α_ X.X Y.X X.X).inv ≫\n                Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫\n                  Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul) =\n    ((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫\n      (α_ Y.X X.X (Y.X ⊗ X.X)).hom ≫\n        Y.X ◁ (α_ X.X Y.X X.X).inv ≫\n          Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫ Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul)","tactic":"simp only [id_whiskerRight, MonoidalCategory.whiskerLeft_id, Category.id_comp, Category.assoc,\n    pentagon_inv_assoc, Iso.hom_inv_id_assoc]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Category.id_comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[151,2],"def_end_pos":[151,9]},{"full_name":"CategoryTheory.MonoidalCategory.id_whiskerRight","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[172,2],"def_end_pos":[172,17]},{"full_name":"CategoryTheory.MonoidalCategory.whiskerLeft_id","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[170,2],"def_end_pos":[170,16]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : SymmetricCategory C\nX Y : Mon_ C\n⊢ (α_ (X.X ⊗ Y.X) X.X Y.X).inv ≫\n      (β_ X.X Y.X).hom ▷ X.X ▷ Y.X ≫\n        (α_ (Y.X ⊗ X.X) X.X Y.X).hom ≫\n          (α_ Y.X X.X (X.X ⊗ Y.X)).hom ≫\n            Y.X ◁ X.X ◁ (β_ X.X Y.X).hom ≫\n              Y.X ◁ (α_ X.X Y.X X.X).inv ≫\n                Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫\n                  Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul) =\n    ((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫\n      (α_ Y.X X.X (Y.X ⊗ X.X)).hom ≫\n        Y.X ◁ (α_ X.X Y.X X.X).inv ≫\n          Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫ Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul)","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : SymmetricCategory C\nX Y : Mon_ C\n⊢ (((((((((β_ X.X Y.X).hom ▷ (X.X ⊗ Y.X) ≫ (α_ (Y.X ⊗ X.X) X.X Y.X).inv) ≫ (α_ (Y.X ⊗ X.X) X.X Y.X).hom) ≫\n                  (α_ Y.X X.X (X.X ⊗ Y.X)).hom) ≫\n                Y.X ◁ X.X ◁ (β_ X.X Y.X).hom) ≫\n              Y.X ◁ (α_ X.X Y.X X.X).inv) ≫\n            Y.X ◁ (β_ X.X Y.X).hom ▷ X.X) ≫\n          Y.X ◁ (α_ Y.X X.X X.X).hom) ≫\n        (α_ Y.X Y.X (X.X ⊗ X.X)).inv) ≫\n      (Y.mul ⊗ X.mul) =\n    ((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫\n      (α_ Y.X X.X (Y.X ⊗ X.X)).hom ≫\n        Y.X ◁ (α_ X.X Y.X X.X).inv ≫\n          Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫ Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul)","tactic":"slice_lhs 1 2 =>\n    rw [← associator_inv_naturality_left]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.MonoidalCategory.associator_inv_naturality_left","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[456,8],"def_end_pos":[456,38]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : SymmetricCategory C\nX Y : Mon_ C\n⊢ (((((((((β_ X.X Y.X).hom ▷ (X.X ⊗ Y.X) ≫ (α_ (Y.X ⊗ X.X) X.X Y.X).inv) ≫ (α_ (Y.X ⊗ X.X) X.X Y.X).hom) ≫\n                  (α_ Y.X X.X (X.X ⊗ Y.X)).hom) ≫\n                Y.X ◁ X.X ◁ (β_ X.X Y.X).hom) ≫\n              Y.X ◁ (α_ X.X Y.X X.X).inv) ≫\n            Y.X ◁ (β_ X.X Y.X).hom ▷ X.X) ≫\n          Y.X ◁ (α_ Y.X X.X X.X).hom) ≫\n        (α_ Y.X Y.X (X.X ⊗ X.X)).inv) ≫\n      (Y.mul ⊗ X.mul) =\n    ((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫\n      (α_ Y.X X.X (Y.X ⊗ X.X)).hom ≫\n        Y.X ◁ (α_ X.X Y.X X.X).inv ≫\n          Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫ Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul)","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : SymmetricCategory C\nX Y : Mon_ C\n⊢ (β_ X.X Y.X).hom ▷ (X.X ⊗ Y.X) ≫\n      ((((((𝟙 ((Y.X ⊗ X.X) ⊗ X.X ⊗ Y.X) ≫ (α_ Y.X X.X (X.X ⊗ Y.X)).hom) ≫ Y.X ◁ X.X ◁ (β_ X.X Y.X).hom) ≫\n                Y.X ◁ (α_ X.X Y.X X.X).inv) ≫\n              Y.X ◁ (β_ X.X Y.X).hom ▷ X.X) ≫\n            Y.X ◁ (α_ Y.X X.X X.X).hom) ≫\n          (α_ Y.X Y.X (X.X ⊗ X.X)).inv) ≫\n        (Y.mul ⊗ X.mul) =\n    ((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫\n      (α_ Y.X X.X (Y.X ⊗ X.X)).hom ≫\n        Y.X ◁ (α_ X.X Y.X X.X).inv ≫\n          Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫ Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul)","tactic":"slice_lhs 2 3 =>\n    rw [Iso.inv_hom_id]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Iso.inv_hom_id","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[58,2],"def_end_pos":[58,12]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : SymmetricCategory C\nX Y : Mon_ C\n⊢ (β_ X.X Y.X).hom ▷ (X.X ⊗ Y.X) ≫\n      ((((((𝟙 ((Y.X ⊗ X.X) ⊗ X.X ⊗ Y.X) ≫ (α_ Y.X X.X (X.X ⊗ Y.X)).hom) ≫ Y.X ◁ X.X ◁ (β_ X.X Y.X).hom) ≫\n                Y.X ◁ (α_ X.X Y.X X.X).inv) ≫\n              Y.X ◁ (β_ X.X Y.X).hom ▷ X.X) ≫\n            Y.X ◁ (α_ Y.X X.X X.X).hom) ≫\n          (α_ Y.X Y.X (X.X ⊗ X.X)).inv) ≫\n        (Y.mul ⊗ X.mul) =\n    ((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫\n      (α_ Y.X X.X (Y.X ⊗ X.X)).hom ≫\n        Y.X ◁ (α_ X.X Y.X X.X).inv ≫\n          Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫ Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul)","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : SymmetricCategory C\nX Y : Mon_ C\n⊢ (β_ X.X Y.X).hom ▷ (X.X ⊗ Y.X) ≫\n      ((((((α_ Y.X X.X (X.X ⊗ Y.X)).hom ≫ Y.X ◁ X.X ◁ (β_ X.X Y.X).hom) ≫ Y.X ◁ (α_ X.X Y.X X.X).inv) ≫\n              Y.X ◁ (β_ X.X Y.X).hom ▷ X.X) ≫\n            Y.X ◁ (α_ Y.X X.X X.X).hom) ≫\n          (α_ Y.X Y.X (X.X ⊗ X.X)).inv) ≫\n        (Y.mul ⊗ X.mul) =\n    ((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫\n      (α_ Y.X X.X (Y.X ⊗ X.X)).hom ≫\n        Y.X ◁ (α_ X.X Y.X X.X).inv ≫\n          Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫ Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul)","tactic":"rw [Category.id_comp]","premises":[{"full_name":"CategoryTheory.Category.id_comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[151,2],"def_end_pos":[151,9]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : SymmetricCategory C\nX Y : Mon_ C\n⊢ (β_ X.X Y.X).hom ▷ (X.X ⊗ Y.X) ≫\n      ((((((α_ Y.X X.X (X.X ⊗ Y.X)).hom ≫ Y.X ◁ X.X ◁ (β_ X.X Y.X).hom) ≫ Y.X ◁ (α_ X.X Y.X X.X).inv) ≫\n              Y.X ◁ (β_ X.X Y.X).hom ▷ X.X) ≫\n            Y.X ◁ (α_ Y.X X.X X.X).hom) ≫\n          (α_ Y.X Y.X (X.X ⊗ X.X)).inv) ≫\n        (Y.mul ⊗ X.mul) =\n    ((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫\n      (α_ Y.X X.X (Y.X ⊗ X.X)).hom ≫\n        Y.X ◁ (α_ X.X Y.X X.X).inv ≫\n          Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫ Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul)","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : SymmetricCategory C\nX Y : Mon_ C\n⊢ (β_ X.X Y.X).hom ▷ (X.X ⊗ Y.X) ≫\n      ((((((Y.X ⊗ X.X) ◁ (β_ X.X Y.X).hom ≫ (α_ Y.X X.X (Y.X ⊗ X.X)).hom) ≫ Y.X ◁ (α_ X.X Y.X X.X).inv) ≫\n              Y.X ◁ (β_ X.X Y.X).hom ▷ X.X) ≫\n            Y.X ◁ (α_ Y.X X.X X.X).hom) ≫\n          (α_ Y.X Y.X (X.X ⊗ X.X)).inv) ≫\n        (Y.mul ⊗ X.mul) =\n    ((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫\n      (α_ Y.X X.X (Y.X ⊗ X.X)).hom ≫\n        Y.X ◁ (α_ X.X Y.X X.X).inv ≫\n          Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫ Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul)","tactic":"slice_lhs 2 3 =>\n    rw [← associator_naturality_right]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.MonoidalCategory.associator_naturality_right","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[476,8],"def_end_pos":[476,35]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : SymmetricCategory C\nX Y : Mon_ C\n⊢ (β_ X.X Y.X).hom ▷ (X.X ⊗ Y.X) ≫\n      ((((((Y.X ⊗ X.X) ◁ (β_ X.X Y.X).hom ≫ (α_ Y.X X.X (Y.X ⊗ X.X)).hom) ≫ Y.X ◁ (α_ X.X Y.X X.X).inv) ≫\n              Y.X ◁ (β_ X.X Y.X).hom ▷ X.X) ≫\n            Y.X ◁ (α_ Y.X X.X X.X).hom) ≫\n          (α_ Y.X Y.X (X.X ⊗ X.X)).inv) ≫\n        (Y.mul ⊗ X.mul) =\n    ((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫\n      (α_ Y.X X.X (Y.X ⊗ X.X)).hom ≫\n        Y.X ◁ (α_ X.X Y.X X.X).inv ≫\n          Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫ Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul)","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : SymmetricCategory C\nX Y : Mon_ C\n⊢ (((((((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫ (α_ Y.X X.X (Y.X ⊗ X.X)).hom) ≫ Y.X ◁ (α_ X.X Y.X X.X).inv) ≫\n            Y.X ◁ (β_ X.X Y.X).hom ▷ X.X) ≫\n          Y.X ◁ (α_ Y.X X.X X.X).hom) ≫\n        (α_ Y.X Y.X (X.X ⊗ X.X)).inv) ≫\n      (Y.mul ⊗ X.mul) =\n    ((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫\n      (α_ Y.X X.X (Y.X ⊗ X.X)).hom ≫\n        Y.X ◁ (α_ X.X Y.X X.X).inv ≫\n          Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫ Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul)","tactic":"slice_lhs 1 2 =>\n    rw [← tensorHom_def]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.MonoidalCategory.tensorHom_def","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[157,2],"def_end_pos":[157,15]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : SymmetricCategory C\nX Y : Mon_ C\n⊢ (((((((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫ (α_ Y.X X.X (Y.X ⊗ X.X)).hom) ≫ Y.X ◁ (α_ X.X Y.X X.X).inv) ≫\n            Y.X ◁ (β_ X.X Y.X).hom ▷ X.X) ≫\n          Y.X ◁ (α_ Y.X X.X X.X).hom) ≫\n        (α_ Y.X Y.X (X.X ⊗ X.X)).inv) ≫\n      (Y.mul ⊗ X.mul) =\n    ((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫\n      (α_ Y.X X.X (Y.X ⊗ X.X)).hom ≫\n        Y.X ◁ (α_ X.X Y.X X.X).inv ≫\n          Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫ Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul)","state_after":"no goals","tactic":"simp only [Category.assoc]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]}]}]}
+{"url":"Mathlib/Data/Complex/Exponential.lean","commit":"","full_name":"Real.tanh_eq_sinh_div_cosh","start":[892,0],"end":[893,49],"file_path":"Mathlib/Data/Complex/Exponential.lean","tactics":[{"state_before":"x y : ℝ\n⊢ ↑(tanh x) = ↑(sinh x / cosh x)","state_after":"no goals","tactic":"simp [tanh_eq_sinh_div_cosh]","premises":[{"full_name":"Complex.tanh_eq_sinh_div_cosh","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[329,8],"def_end_pos":[329,29]}]}]}
+{"url":"Mathlib/Data/Nat/Bitwise.lean","commit":"","full_name":"Nat.land_assoc","start":[333,0],"end":[333,90],"file_path":"Mathlib/Data/Nat/Bitwise.lean","tactics":[{"state_before":"n m k : ℕ\n⊢ n &&& m &&& k = n &&& (m &&& k)","state_after":"no goals","tactic":"bitwise_assoc_tac","premises":[{"full_name":"Bool.and_assoc","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[221,8],"def_end_pos":[221,22]},{"full_name":"Bool.bne_eq_xor","def_path":"Mathlib/Data/Bool/Basic.lean","def_pos":[177,8],"def_end_pos":[177,18]},{"full_name":"Bool.or_assoc","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[224,8],"def_end_pos":[224,21]},{"full_name":"Nat.binaryRec","def_path":"Mathlib/Data/Nat/Bits.lean","def_pos":[162,4],"def_end_pos":[162,13]}]}]}
+{"url":"Mathlib/SetTheory/Game/Birthday.lean","commit":"","full_name":"SetTheory.PGame.neg_birthday_le","start":[127,0],"end":[128,64],"file_path":"Mathlib/SetTheory/Game/Birthday.lean","tactics":[{"state_before":"a b x : PGame\n⊢ -x.birthday.toPGame ≤ x","state_after":"no goals","tactic":"simpa only [neg_birthday, ← neg_le_iff] using le_birthday (-x)","premises":[{"full_name":"SetTheory.PGame.le_birthday","def_path":"Mathlib/SetTheory/Game/Birthday.lean","def_pos":[118,8],"def_end_pos":[118,19]},{"full_name":"SetTheory.PGame.neg_birthday","def_path":"Mathlib/SetTheory/Game/Birthday.lean","def_pos":[103,8],"def_end_pos":[103,20]},{"full_name":"SetTheory.PGame.neg_le_iff","def_path":"Mathlib/SetTheory/Game/PGame.lean","def_pos":[1208,8],"def_end_pos":[1208,18]}]}]}
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+{"url":"Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean","commit":"","full_name":"uniformCauchySeqOnFilter_of_fderiv","start":[106,0],"end":[161,92],"file_path":"Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean","tactics":[{"state_before":"ι : Type u_1\nl : Filter ι\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝³ : RCLike 𝕜\ninst✝² : NormedSpace 𝕜 E\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : ι → E → G\ng : E → G\nf' : ι → E → E →L[𝕜] G\ng' : E → E →L[𝕜] G\nx : E\nhf' : UniformCauchySeqOnFilter f' l (𝓝 x)\nhf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2\nhfg : Cauchy (map (fun n => f n x) l)\n⊢ UniformCauchySeqOnFilter f l (𝓝 x)","state_after":"ι : Type u_1\nl : Filter ι\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝³ : RCLike 𝕜\ninst✝² : NormedSpace 𝕜 E\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : ι → E → G\ng : E → G\nf' : ι → E → E →L[𝕜] G\ng' : E → E →L[𝕜] G\nx : E\nhf' : UniformCauchySeqOnFilter f' l (𝓝 x)\nhf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2\nhfg : Cauchy (map (fun n => f n x) l)\nthis : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E\n⊢ UniformCauchySeqOnFilter f l (𝓝 x)","tactic":"letI : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 _","premises":[{"full_name":"NormedSpace","def_path":"Mathlib/Analysis/Normed/Module/Basic.lean","def_pos":[43,6],"def_end_pos":[43,17]},{"full_name":"NormedSpace.restrictScalars","def_path":"Mathlib/Analysis/Normed/Module/Basic.lean","def_pos":[457,4],"def_end_pos":[457,31]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]}]},{"state_before":"ι : Type u_1\nl : Filter ι\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝³ : RCLike 𝕜\ninst✝² : NormedSpace 𝕜 E\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : ι → E → G\ng : E → G\nf' : ι → E → E →L[𝕜] G\ng' : E → E →L[𝕜] G\nx : E\nhf' : UniformCauchySeqOnFilter f' l (𝓝 x)\nhf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2\nhfg : Cauchy (map (fun n => f n x) l)\nthis : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E\n⊢ UniformCauchySeqOnFilter f l (𝓝 x)","state_after":"ι : Type u_1\nl : Filter ι\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝³ : RCLike 𝕜\ninst✝² : NormedSpace 𝕜 E\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : ι → E → G\ng : E → G\nf' : ι → E → E →L[𝕜] G\ng' : E → E →L[𝕜] G\nx : E\nhf' : TendstoUniformlyOnFilter (fun n z => f' n.1 z - f' n.2 z) 0 (l ×ˢ l) (𝓝 x)\nhf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2\nhfg : Cauchy (map (fun n => f n x) l)\nthis : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E\n⊢ TendstoUniformlyOnFilter (fun n z => f n.1 z - f n.2 z) 0 (l ×ˢ l) (𝓝 x)","tactic":"rw [SeminormedAddGroup.uniformCauchySeqOnFilter_iff_tendstoUniformlyOnFilter_zero] at hf' ⊢","premises":[{"full_name":"SeminormedAddGroup.uniformCauchySeqOnFilter_iff_tendstoUniformlyOnFilter_zero","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[864,2],"def_end_pos":[864,13]}]},{"state_before":"ι : Type u_1\nl : Filter ι\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝³ : RCLike 𝕜\ninst✝² : NormedSpace 𝕜 E\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : ι → E → G\ng : E → G\nf' : ι → E → E →L[𝕜] G\ng' : E → E →L[𝕜] G\nx : E\nhf' : TendstoUniformlyOnFilter (fun n z => f' n.1 z - f' n.2 z) 0 (l ×ˢ l) (𝓝 x)\nhf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2\nhfg : Cauchy (map (fun n => f n x) l)\nthis : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E\n⊢ TendstoUniformlyOnFilter (fun n z => f n.1 z - f n.2 z) 0 (l ×ˢ l) (𝓝 x)","state_after":"ι : Type u_1\nl : Filter ι\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝³ : RCLike 𝕜\ninst✝² : NormedSpace 𝕜 E\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : ι → E → G\ng : E → G\nf' : ι → E → E →L[𝕜] G\ng' : E → E →L[𝕜] G\nx : E\nhf' : TendstoUniformlyOnFilter (fun n z => f' n.1 z - f' n.2 z) 0 (l ×ˢ l) (𝓝 x)\nhf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2\nhfg : Cauchy (map (fun n => f n x) l)\nthis : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E\n⊢ TendstoUniformlyOnFilter (fun n z => f n.1 z - f n.2 z - (f n.1 x - f n.2 x)) 0 (l ×ˢ l) (𝓝 x) ∧\n    TendstoUniformlyOnFilter (fun n x_1 => f n.1 x - f n.2 x) 0 (l ×ˢ l) (𝓝 x)","tactic":"suffices\n    TendstoUniformlyOnFilter (fun (n : ι × ι) (z : E) => f n.1 z - f n.2 z - (f n.1 x - f n.2 x)) 0\n        (l ×ˢ l) (𝓝 x) ∧\n      TendstoUniformlyOnFilter (fun (n : ι × ι) (_ : E) => f n.1 x - f n.2 x) 0 (l ×ˢ l) (𝓝 x) by\n    have := this.1.add this.2\n    rw [add_zero] at this\n    exact this.congr (by simp)","premises":[{"full_name":"And","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[516,10],"def_end_pos":[516,13]},{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"Prod","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[479,10],"def_end_pos":[479,14]},{"full_name":"Prod.fst","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[483,2],"def_end_pos":[483,5]},{"full_name":"Prod.snd","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[485,2],"def_end_pos":[485,5]},{"full_name":"TendstoUniformlyOnFilter","def_path":"Mathlib/Topology/UniformSpace/UniformConvergence.lean","def_pos":[82,4],"def_end_pos":[82,28]},{"full_name":"TendstoUniformlyOnFilter.add","def_path":"Mathlib/Topology/Algebra/UniformGroup.lean","def_pos":[393,2],"def_end_pos":[393,13]},{"full_name":"TendstoUniformlyOnFilter.congr","def_path":"Mathlib/Topology/UniformSpace/UniformConvergence.lean","def_pos":[184,8],"def_end_pos":[184,38]},{"full_name":"add_zero","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[412,2],"def_end_pos":[412,13]},{"full_name":"nhds","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[113,16],"def_end_pos":[113,20]}]},{"state_before":"ι : Type u_1\nl : Filter ι\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝³ : RCLike 𝕜\ninst✝² : NormedSpace 𝕜 E\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : ι → E → G\ng : E → G\nf' : ι → E → E →L[𝕜] G\ng' : E → E →L[𝕜] G\nx : E\nhf' : TendstoUniformlyOnFilter (fun n z => f' n.1 z - f' n.2 z) 0 (l ×ˢ l) (𝓝 x)\nhf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2\nhfg : Cauchy (map (fun n => f n x) l)\nthis : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E\n⊢ TendstoUniformlyOnFilter (fun n z => f n.1 z - f n.2 z - (f n.1 x - f n.2 x)) 0 (l ×ˢ l) (𝓝 x) ∧\n    TendstoUniformlyOnFilter (fun n x_1 => f n.1 x - f n.2 x) 0 (l ×ˢ l) (𝓝 x)","state_after":"case left\nι : Type u_1\nl : Filter ι\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝³ : RCLike 𝕜\ninst✝² : NormedSpace 𝕜 E\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : ι → E → G\ng : E → G\nf' : ι → E → E →L[𝕜] G\ng' : E → E →L[𝕜] G\nx : E\nhf' : TendstoUniformlyOnFilter (fun n z => f' n.1 z - f' n.2 z) 0 (l ×ˢ l) (𝓝 x)\nhf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2\nhfg : Cauchy (map (fun n => f n x) l)\nthis : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E\n⊢ TendstoUniformlyOnFilter (fun n z => f n.1 z - f n.2 z - (f n.1 x - f n.2 x)) 0 (l ×ˢ l) (𝓝 x)\n\ncase right\nι : Type u_1\nl : Filter ι\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝³ : RCLike 𝕜\ninst✝² : NormedSpace 𝕜 E\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : ι → E → G\ng : E → G\nf' : ι → E → E →L[𝕜] G\ng' : E → E →L[𝕜] G\nx : E\nhf' : TendstoUniformlyOnFilter (fun n z => f' n.1 z - f' n.2 z) 0 (l ×ˢ l) (𝓝 x)\nhf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2\nhfg : Cauchy (map (fun n => f n x) l)\nthis : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E\n⊢ TendstoUniformlyOnFilter (fun n x_1 => f n.1 x - f n.2 x) 0 (l ×ˢ l) (𝓝 x)","tactic":"constructor","premises":[]}]}
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+{"url":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","commit":"","full_name":"MeasureTheory.exists_measurable_le_lintegral_eq","start":[164,0],"end":[180,58],"file_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\n⊢ ∃ g, Measurable g ∧ g ≤ f ∧ ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), g a ∂μ","state_after":"case inl\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nh₀ : ∫⁻ (a : α), f a ∂μ = 0\n⊢ ∃ g, Measurable g ∧ g ≤ f ∧ ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), g a ∂μ\n\ncase inr\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nh₀ : ∫⁻ (a : α), f a ∂μ ≠ 0\n⊢ ∃ g, Measurable g ∧ g ≤ f ∧ ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), g a ∂μ","tactic":"rcases eq_or_ne (∫⁻ a, f a ∂μ) 0 with h₀ | h₀","premises":[{"full_name":"MeasureTheory.lintegral","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[59,16],"def_end_pos":[59,25]},{"full_name":"eq_or_ne","def_path":"Mathlib/Logic/Basic.lean","def_pos":[167,8],"def_end_pos":[167,16]}]},{"state_before":"case inr\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nh₀ : ∫⁻ (a : α), f a ∂μ ≠ 0\n⊢ ∃ g, Measurable g ∧ g ≤ f ∧ ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), g a ∂μ","state_after":"case inr.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nh₀ : ∫⁻ (a : α), f a ∂μ ≠ 0\nL : ℕ → ℝ≥0∞\nleft✝ : StrictMono L\nhLf : ∀ (n : ℕ), L n ∈ Ioo ⊥ (∫⁻ (a : α), f a ∂μ)\nhL_tendsto : Tendsto L atTop (𝓝 (∫⁻ (a : α), f a ∂μ))\n⊢ ∃ g, Measurable g ∧ g ≤ f ∧ ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), g a ∂μ","tactic":"rcases exists_seq_strictMono_tendsto' h₀.bot_lt with ⟨L, _, hLf, hL_tendsto⟩","premises":[{"full_name":"Ne.bot_lt","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[301,8],"def_end_pos":[301,17]},{"full_name":"exists_seq_strictMono_tendsto'","def_path":"Mathlib/Topology/Order/IsLUB.lean","def_pos":[169,8],"def_end_pos":[169,38]}]},{"state_before":"case inr.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nh₀ : ∫⁻ (a : α), f a ∂μ ≠ 0\nL : ℕ → ℝ≥0∞\nleft✝ : StrictMono L\nhLf : ∀ (n : ℕ), L n ∈ Ioo ⊥ (∫⁻ (a : α), f a ∂μ)\nhL_tendsto : Tendsto L atTop (𝓝 (∫⁻ (a : α), f a ∂μ))\n⊢ ∃ g, Measurable g ∧ g ≤ f ∧ ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), g a ∂μ","state_after":"case inr.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nh₀ : ∫⁻ (a : α), f a ∂μ ≠ 0\nL : ℕ → ℝ≥0∞\nleft✝ : StrictMono L\nhLf : ∀ (n : ℕ), L n ∈ Ioo ⊥ (∫⁻ (a : α), f a ∂μ)\nhL_tendsto : Tendsto L atTop (𝓝 (∫⁻ (a : α), f a ∂μ))\nthis : ∀ (n : ℕ), ∃ g, Measurable g ∧ g ≤ f ∧ L n < ∫⁻ (a : α), g a ∂μ\n⊢ ∃ g, Measurable g ∧ g ≤ f ∧ ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), g a ∂μ","tactic":"have : ∀ n, ∃ g : α → ℝ≥0∞, Measurable g ∧ g ≤ f ∧ L n < ∫⁻ a, g a ∂μ := by\n    intro n\n    simpa only [← iSup_lintegral_measurable_le_eq_lintegral f, lt_iSup_iff, exists_prop] using\n      (hLf n).2","premises":[{"full_name":"And","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[516,10],"def_end_pos":[516,13]},{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"ENNReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[96,4],"def_end_pos":[96,11]},{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"Measurable","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","def_pos":[479,4],"def_end_pos":[479,14]},{"full_name":"MeasureTheory.iSup_lintegral_measurable_le_eq_lintegral","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[106,8],"def_end_pos":[106,49]},{"full_name":"MeasureTheory.lintegral","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[59,16],"def_end_pos":[59,25]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]},{"full_name":"lt_iSup_iff","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[485,8],"def_end_pos":[485,19]}]},{"state_before":"case inr.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nh₀ : ∫⁻ (a : α), f a ∂μ ≠ 0\nL : ℕ → ℝ≥0∞\nleft✝ : StrictMono L\nhLf : ∀ (n : ℕ), L n ∈ Ioo ⊥ (∫⁻ (a : α), f a ∂μ)\nhL_tendsto : Tendsto L atTop (𝓝 (∫⁻ (a : α), f a ∂μ))\nthis : ∀ (n : ℕ), ∃ g, Measurable g ∧ g ≤ f ∧ L n < ∫⁻ (a : α), g a ∂μ\n⊢ ∃ g, Measurable g ∧ g ≤ f ∧ ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), g a ∂μ","state_after":"case inr.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nh₀ : ∫⁻ (a : α), f a ∂μ ≠ 0\nL : ℕ → ℝ≥0∞\nleft✝ : StrictMono L\nhLf : ∀ (n : ℕ), L n ∈ Ioo ⊥ (∫⁻ (a : α), f a ∂μ)\nhL_tendsto : Tendsto L atTop (𝓝 (∫⁻ (a : α), f a ∂μ))\ng : ℕ → α → ℝ≥0∞\nhgm : ∀ (n : ℕ), Measurable (g n)\nhgf : ∀ (n : ℕ), g n ≤ f\nhLg : ∀ (n : ℕ), L n < ∫⁻ (a : α), g n a ∂μ\n⊢ ∃ g, Measurable g ∧ g ≤ f ∧ ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), g a ∂μ","tactic":"choose g hgm hgf hLg using this","premises":[]},{"state_before":"case inr.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nh₀ : ∫⁻ (a : α), f a ∂μ ≠ 0\nL : ℕ → ℝ≥0∞\nleft✝ : StrictMono L\nhLf : ∀ (n : ℕ), L n ∈ Ioo ⊥ (∫⁻ (a : α), f a ∂μ)\nhL_tendsto : Tendsto L atTop (𝓝 (∫⁻ (a : α), f a ∂μ))\ng : ℕ → α → ℝ≥0∞\nhgm : ∀ (n : ℕ), Measurable (g n)\nhgf : ∀ (n : ℕ), g n ≤ f\nhLg : ∀ (n : ℕ), L n < ∫⁻ (a : α), g n a ∂μ\n⊢ ∃ g, Measurable g ∧ g ≤ f ∧ ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), g a ∂μ","state_after":"case inr.intro.intro.intro.refine_1\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nh₀ : ∫⁻ (a : α), f a ∂μ ≠ 0\nL : ℕ → ℝ≥0∞\nleft✝ : StrictMono L\nhLf : ∀ (n : ℕ), L n ∈ Ioo ⊥ (∫⁻ (a : α), f a ∂μ)\nhL_tendsto : Tendsto L atTop (𝓝 (∫⁻ (a : α), f a ∂μ))\ng : ℕ → α → ℝ≥0∞\nhgm : ∀ (n : ℕ), Measurable (g n)\nhgf : ∀ (n : ℕ), g n ≤ f\nhLg : ∀ (n : ℕ), L n < ∫⁻ (a : α), g n a ∂μ\n⊢ ∫⁻ (a : α), f a ∂μ ≤ ∫⁻ (a : α), (fun x => ⨆ n, g n x) a ∂μ\n\ncase inr.intro.intro.intro.refine_2\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nh₀ : ∫⁻ (a : α), f a ∂μ ≠ 0\nL : ℕ → ℝ≥0∞\nleft✝ : StrictMono L\nhLf : ∀ (n : ℕ), L n ∈ Ioo ⊥ (∫⁻ (a : α), f a ∂μ)\nhL_tendsto : Tendsto L atTop (𝓝 (∫⁻ (a : α), f a ∂μ))\ng : ℕ → α → ℝ≥0∞\nhgm : ∀ (n : ℕ), Measurable (g n)\nhgf : ∀ (n : ℕ), g n ≤ f\nhLg : ∀ (n : ℕ), L n < ∫⁻ (a : α), g n a ∂μ\n⊢ ∫⁻ (a : α), (fun x => ⨆ n, g n x) a ∂μ ≤ ∫⁻ (a : α), f a ∂μ","tactic":"refine\n    ⟨fun x => ⨆ n, g n x, measurable_iSup hgm, fun x => iSup_le fun n => hgf n x, le_antisymm ?_ ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"iSup","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[56,4],"def_end_pos":[56,8]},{"full_name":"iSup_le","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[661,8],"def_end_pos":[661,15]},{"full_name":"le_antisymm","def_path":"Mathlib/Order/Defs.lean","def_pos":[156,8],"def_end_pos":[156,19]},{"full_name":"measurable_iSup","def_path":"Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean","def_pos":[725,8],"def_end_pos":[725,23]}]}]}
+{"url":"Mathlib/Order/CompleteLattice.lean","commit":"","full_name":"iSup_sup","start":[1031,0],"end":[1032,30],"file_path":"Mathlib/Order/CompleteLattice.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nβ₂ : Type u_3\nγ : Type u_4\nι : Sort u_5\nι' : Sort u_6\nκ : ι → Sort u_7\nκ' : ι' → Sort u_8\ninst✝¹ : CompleteLattice α\nf✝ g s t : ι → α\na✝ b : α\ninst✝ : Nonempty ι\nf : ι → α\na : α\n⊢ (⨆ x, f x) ⊔ a = ⨆ x, f x ⊔ a","state_after":"no goals","tactic":"rw [iSup_sup_eq, iSup_const]","premises":[{"full_name":"iSup_const","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[842,8],"def_end_pos":[842,18]},{"full_name":"iSup_sup_eq","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[1000,8],"def_end_pos":[1000,19]}]}]}
+{"url":"Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean","commit":"","full_name":"LinearMap.ker_eq_bot_iff_range_eq_top","start":[537,0],"end":[539,57],"file_path":"Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean","tactics":[{"state_before":"K : Type u\nV : Type v\ninst✝⁵ : DivisionRing K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nV₂ : Type v'\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\ninst✝ : FiniteDimensional K V\nf : V →ₗ[K] V\n⊢ ker f = ⊥ ↔ range f = ⊤","state_after":"no goals","tactic":"rw [range_eq_top, ker_eq_bot, injective_iff_surjective]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"LinearMap.injective_iff_surjective","def_path":"Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean","def_pos":[522,8],"def_end_pos":[522,32]},{"full_name":"LinearMap.ker_eq_bot","def_path":"Mathlib/Algebra/Module/Submodule/Ker.lean","def_pos":[185,8],"def_end_pos":[185,18]},{"full_name":"LinearMap.range_eq_top","def_path":"Mathlib/Algebra/Module/Submodule/Range.lean","def_pos":[89,8],"def_end_pos":[89,20]}]}]}
+{"url":"Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean","commit":"","full_name":"Matrix.isRepresentation.toEnd_surjective","start":[187,0],"end":[191,14],"file_path":"Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean","tactics":[{"state_before":"ι : Type u_1\ninst✝⁴ : Fintype ι\nM : Type u_2\ninst✝³ : AddCommGroup M\nR : Type u_3\ninst✝² : CommRing R\ninst✝¹ : Module R M\nI : Ideal R\nb : ι → M\nhb : Submodule.span R (Set.range b) = ⊤\ninst✝ : DecidableEq ι\n⊢ Function.Surjective ⇑(toEnd R b hb)","state_after":"ι : Type u_1\ninst✝⁴ : Fintype ι\nM : Type u_2\ninst✝³ : AddCommGroup M\nR : Type u_3\ninst✝² : CommRing R\ninst✝¹ : Module R M\nI : Ideal R\nb : ι → M\nhb : Submodule.span R (Set.range b) = ⊤\ninst✝ : DecidableEq ι\nf : Module.End R M\n⊢ ∃ a, (toEnd R b hb) a = f","tactic":"intro f","premises":[]},{"state_before":"ι : Type u_1\ninst✝⁴ : Fintype ι\nM : Type u_2\ninst✝³ : AddCommGroup M\nR : Type u_3\ninst✝² : CommRing R\ninst✝¹ : Module R M\nI : Ideal R\nb : ι → M\nhb : Submodule.span R (Set.range b) = ⊤\ninst✝ : DecidableEq ι\nf : Module.End R M\n⊢ ∃ a, (toEnd R b hb) a = f","state_after":"case intro.intro\nι : Type u_1\ninst✝⁴ : Fintype ι\nM✝ : Type u_2\ninst✝³ : AddCommGroup M✝\nR : Type u_3\ninst✝² : CommRing R\ninst✝¹ : Module R M✝\nI : Ideal R\nb : ι → M✝\nhb : Submodule.span R (Set.range b) = ⊤\ninst✝ : DecidableEq ι\nf : Module.End R M✝\nM : ↥(isRepresentation R b)\ne : (toEnd R b hb) M = f\n⊢ ∃ a, (toEnd R b hb) a = f","tactic":"obtain ⟨M, e, -⟩ := Matrix.isRepresentation.toEnd_exists_mem_ideal R b hb f ⊤ (by simp)","premises":[{"full_name":"Matrix.isRepresentation.toEnd_exists_mem_ideal","def_path":"Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean","def_pos":[169,8],"def_end_pos":[169,54]},{"full_name":"Top.top","def_path":"Mathlib/Order/Notation.lean","def_pos":[94,2],"def_end_pos":[94,5]}]},{"state_before":"case intro.intro\nι : Type u_1\ninst✝⁴ : Fintype ι\nM✝ : Type u_2\ninst✝³ : AddCommGroup M✝\nR : Type u_3\ninst✝² : CommRing R\ninst✝¹ : Module R M✝\nI : Ideal R\nb : ι → M✝\nhb : Submodule.span R (Set.range b) = ⊤\ninst✝ : DecidableEq ι\nf : Module.End R M✝\nM : ↥(isRepresentation R b)\ne : (toEnd R b hb) M = f\n⊢ ∃ a, (toEnd R b hb) a = f","state_after":"no goals","tactic":"exact ⟨M, e⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]}]}
+{"url":"Mathlib/LinearAlgebra/Matrix/DotProduct.lean","commit":"","full_name":"Matrix.dotProduct_eq","start":[50,0],"end":[52,80],"file_path":"Mathlib/LinearAlgebra/Matrix/DotProduct.lean","tactics":[{"state_before":"m : Type u_1\nn : Type u_2\np : Type u_3\nR : Type u_4\ninst✝¹ : Semiring R\ninst✝ : Fintype n\nv w : n → R\nh : ∀ (u : n → R), v ⬝ᵥ u = w ⬝ᵥ u\n⊢ v = w","state_after":"case h\nm : Type u_1\nn : Type u_2\np : Type u_3\nR : Type u_4\ninst✝¹ : Semiring R\ninst✝ : Fintype n\nv w : n → R\nh : ∀ (u : n → R), v ⬝ᵥ u = w ⬝ᵥ u\nx : n\n⊢ v x = w x","tactic":"funext x","premises":[{"full_name":"funext","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1817,8],"def_end_pos":[1817,14]}]},{"state_before":"case h\nm : Type u_1\nn : Type u_2\np : Type u_3\nR : Type u_4\ninst✝¹ : Semiring R\ninst✝ : Fintype n\nv w : n → R\nh : ∀ (u : n → R), v ⬝ᵥ u = w ⬝ᵥ u\nx : n\n⊢ v x = w x","state_after":"no goals","tactic":"classical rw [← dotProduct_stdBasis_one v x, ← dotProduct_stdBasis_one w x, h]","premises":[{"full_name":"Matrix.dotProduct_stdBasis_one","def_path":"Mathlib/LinearAlgebra/Matrix/DotProduct.lean","def_pos":[46,8],"def_end_pos":[46,31]}]}]}
+{"url":"Mathlib/NumberTheory/NumberField/Embeddings.lean","commit":"","full_name":"NumberField.InfinitePlace.IsUnramified.comap_algHom","start":[735,0],"end":[738,39],"file_path":"Mathlib/NumberTheory/NumberField/Embeddings.lean","tactics":[{"state_before":"k : Type u_1\ninst✝⁷ : Field k\nK : Type u_2\ninst✝⁶ : Field K\nF : Type u_3\ninst✝⁵ : Field F\ninst✝⁴ : NumberField K\ninst✝³ : Algebra k K\ninst✝² : Algebra k F\ninst✝¹ : Algebra K F\ninst✝ : IsScalarTower k K F\nσ : K ≃ₐ[k] K\nw✝ : InfinitePlace K\nw : InfinitePlace F\nh : IsUnramified k w\nf : K →ₐ[k] F\n⊢ IsUnramified k (w.comap ↑f)","state_after":"k : Type u_1\ninst✝⁷ : Field k\nK : Type u_2\ninst✝⁶ : Field K\nF : Type u_3\ninst✝⁵ : Field F\ninst✝⁴ : NumberField K\ninst✝³ : Algebra k K\ninst✝² : Algebra k F\ninst✝¹ : Algebra K F\ninst✝ : IsScalarTower k K F\nσ : K ≃ₐ[k] K\nw✝ : InfinitePlace K\nw : InfinitePlace F\nh : IsUnramified k w\nf : K →ₐ[k] F\n⊢ (w.comap ↑f).mult ≤ w.mult","tactic":"rw [InfinitePlace.isUnramified_iff_mult_le, ← InfinitePlace.comap_comp, f.comp_algebraMap, h.eq]","premises":[{"full_name":"AlgHom.comp_algebraMap","def_path":"Mathlib/Algebra/Algebra/Hom.lean","def_pos":[204,8],"def_end_pos":[204,23]},{"full_name":"NumberField.InfinitePlace.IsUnramified.eq","def_path":"Mathlib/NumberTheory/NumberField/Embeddings.lean","def_pos":[728,6],"def_end_pos":[728,21]},{"full_name":"NumberField.InfinitePlace.comap_comp","def_path":"Mathlib/NumberTheory/NumberField/Embeddings.lean","def_pos":[607,6],"def_end_pos":[607,16]},{"full_name":"NumberField.InfinitePlace.isUnramified_iff_mult_le","def_path":"Mathlib/NumberTheory/NumberField/Embeddings.lean","def_pos":[730,6],"def_end_pos":[730,30]}]},{"state_before":"k : Type u_1\ninst✝⁷ : Field k\nK : Type u_2\ninst✝⁶ : Field K\nF : Type u_3\ninst✝⁵ : Field F\ninst✝⁴ : NumberField K\ninst✝³ : Algebra k K\ninst✝² : Algebra k F\ninst✝¹ : Algebra K F\ninst✝ : IsScalarTower k K F\nσ : K ≃ₐ[k] K\nw✝ : InfinitePlace K\nw : InfinitePlace F\nh : IsUnramified k w\nf : K →ₐ[k] F\n⊢ (w.comap ↑f).mult ≤ w.mult","state_after":"no goals","tactic":"exact InfinitePlace.mult_comap_le _ _","premises":[{"full_name":"NumberField.InfinitePlace.mult_comap_le","def_path":"Mathlib/NumberTheory/NumberField/Embeddings.lean","def_pos":[626,6],"def_end_pos":[626,19]}]}]}
+{"url":"Mathlib/CategoryTheory/Limits/Shapes/Terminal.lean","commit":"","full_name":"CategoryTheory.Limits.asEmptyCone_π_app","start":[29,0],"end":[34,29],"file_path":"Mathlib/CategoryTheory/Limits/Shapes/Terminal.lean","tactics":[{"state_before":"C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX : C\n⊢ (X_1 : Discrete PEmpty.{1}) → ((Functor.const (Discrete PEmpty.{1})).obj X).obj X_1 ⟶ (Functor.empty C).obj X_1","state_after":"no goals","tactic":"aesop_cat","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]}]}]}
+{"url":"Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean","commit":"","full_name":"MeasurableEmbedding.measurableSet_preimage","start":[89,0],"end":[91,62],"file_path":"Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns✝ t u : Set α\nmα : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf : α → β\ng : β → γ\nhf : MeasurableEmbedding f\ns : Set β\n⊢ MeasurableSet (f ⁻¹' s) ↔ MeasurableSet (s ∩ range f)","state_after":"no goals","tactic":"rw [← image_preimage_eq_inter_range, hf.measurableSet_image]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"MeasurableEmbedding.measurableSet_image","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean","def_pos":[67,8],"def_end_pos":[67,27]},{"full_name":"Set.image_preimage_eq_inter_range","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[666,8],"def_end_pos":[666,37]}]}]}
+{"url":"Mathlib/RingTheory/Valuation/Basic.lean","commit":"","full_name":"Valuation.comap_supp","start":[565,0],"end":[567,85],"file_path":"Mathlib/RingTheory/Valuation/Basic.lean","tactics":[{"state_before":"K : Type u_1\nF : Type u_2\nR : Type u_3\ninst✝⁵ : DivisionRing K\nΓ₀ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝⁴ : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝³ : CommRing R\ninst✝² : LinearOrderedCommMonoidWithZero Γ₀\ninst✝¹ : LinearOrderedCommMonoidWithZero Γ'₀\nv : Valuation R Γ₀\nS : Type u_7\ninst✝ : CommRing S\nf : S →+* R\nx : S\n⊢ x ∈ (comap f v).supp ↔ x ∈ Ideal.comap f v.supp","state_after":"no goals","tactic":"rw [mem_supp_iff, Ideal.mem_comap, mem_supp_iff, comap_apply]","premises":[{"full_name":"Ideal.mem_comap","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[66,8],"def_end_pos":[66,17]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Valuation.comap_apply","def_path":"Mathlib/RingTheory/Valuation/Basic.lean","def_pos":[227,8],"def_end_pos":[227,19]},{"full_name":"Valuation.mem_supp_iff","def_path":"Mathlib/RingTheory/Valuation/Basic.lean","def_pos":[540,8],"def_end_pos":[540,20]}]}]}
+{"url":"Mathlib/Probability/Kernel/RadonNikodym.lean","commit":"","full_name":"ProbabilityTheory.Kernel.measure_mutuallySingularSetSlice","start":[217,0],"end":[235,11],"file_path":"Mathlib/Probability/Kernel/RadonNikodym.lean","tactics":[{"state_before":"α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ✝ η✝ : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\n⊢ (η a) (κ.mutuallySingularSetSlice η a) = 0","state_after":"α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ✝ η✝ : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\nh_coe : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b\n⊢ (η a) (κ.mutuallySingularSetSlice η a) = 0","tactic":"have h_coe : ∀ b, (Real.toNNReal b : ℝ��0∞) = ENNReal.ofReal b := fun _ ↦ rfl","premises":[{"full_name":"ENNReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[96,4],"def_end_pos":[96,11]},{"full_name":"ENNReal.ofReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[187,28],"def_end_pos":[187,34]},{"full_name":"Real.toNNReal","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[110,18],"def_end_pos":[110,38]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ✝ η✝ : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\nh_coe : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b\n⊢ (η a) (κ.mutuallySingularSetSlice η a) = 0","state_after":"α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ✝ η✝ : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\nh_coe : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b\n⊢ (((κ + η).withDensity fun a x => ↑(1 - κ.rnDerivAux (κ + η) a x).toNNReal) a) {x | 1 ≤ κ.rnDerivAux (κ + η) a x} = 0","tactic":"suffices withDensity (κ + η) (fun a x ↦ Real.toNNReal\n      (1 - rnDerivAux κ (κ + η) a x)) a {x | 1 ≤ rnDerivAux κ (κ + η) a x} = 0 by\n    rwa [withDensity_one_sub_rnDerivAux κ η] at this","premises":[{"full_name":"ProbabilityTheory.Kernel.rnDerivAux","def_path":"Mathlib/Probability/Kernel/RadonNikodym.lean","def_pos":[88,4],"def_end_pos":[88,14]},{"full_name":"ProbabilityTheory.Kernel.withDensity","def_path":"Mathlib/Probability/Kernel/WithDensity.lean","def_pos":[44,18],"def_end_pos":[44,29]},{"full_name":"ProbabilityTheory.Kernel.withDensity_one_sub_rnDerivAux","def_path":"Mathlib/Probability/Kernel/RadonNikodym.lean","def_pos":[164,6],"def_end_pos":[164,36]},{"full_name":"Real.toNNReal","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[110,18],"def_end_pos":[110,38]},{"full_name":"setOf","def_path":"Mathlib/Init/Set.lean","def_pos":[56,4],"def_end_pos":[56,9]}]},{"state_before":"α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ✝ η✝ : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\nh_coe : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b\n⊢ (((κ + η).withDensity fun a x => ↑(1 - κ.rnDerivAux (κ + η) a x).toNNReal) a) {x | 1 ≤ κ.rnDerivAux (κ + η) a x} = 0","state_after":"α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ✝ η✝ : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\nh_coe : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b\n⊢ (((κ + η).withDensity fun a x => ENNReal.ofReal (1 - κ.rnDerivAux (κ + η) a x)) a)\n      {x | 1 ≤ κ.rnDerivAux (κ + η) a x} =\n    0","tactic":"simp_rw [h_coe]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]}]},{"state_before":"α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ✝ η✝ : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\nh_coe : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b\n⊢ (((κ + η).withDensity fun a x => ENNReal.ofReal (1 - κ.rnDerivAux (κ + η) a x)) a)\n      {x | 1 ≤ κ.rnDerivAux (κ + η) a x} =\n    0","state_after":"α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ✝ η✝ : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\nh_coe : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b\n⊢ ∀ᵐ (x : γ) ∂(κ + η) a, x ∈ {x | 1 ≤ κ.rnDerivAux (κ + η) a x} → ENNReal.ofReal (1 - κ.rnDerivAux (κ + η) a x) = 0 x\n\nα : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ✝ η✝ : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\nh_coe : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b\n⊢ MeasurableSet {x | ENNReal.ofReal (1 - κ.rnDerivAux (κ + η) a x) = 0 x}\n\nα : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ✝ η✝ : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\nh_coe : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b\n⊢ Measurable fun b => ENNReal.ofReal (1 - κ.rnDerivAux (κ + η) a b)\n\ncase hf\nα : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ✝ η✝ : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\nh_coe : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b\n⊢ Measurable (Function.uncurry fun a x => ENNReal.ofReal (1 - κ.rnDerivAux (κ + η) a x))","tactic":"rw [Kernel.withDensity_apply', lintegral_eq_zero_iff, EventuallyEq, ae_restrict_iff]","premises":[{"full_name":"Filter.EventuallyEq","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1260,4],"def_end_pos":[1260,16]},{"full_name":"MeasureTheory.ae_restrict_iff","def_path":"Mathlib/MeasureTheory/Measure/Restrict.lean","def_pos":[533,8],"def_end_pos":[533,23]},{"full_name":"MeasureTheory.lintegral_eq_zero_iff","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[921,8],"def_end_pos":[921,29]},{"full_name":"ProbabilityTheory.Kernel.withDensity_apply'","def_path":"Mathlib/Probability/Kernel/WithDensity.lean","def_pos":[63,18],"def_end_pos":[63,36]}]},{"state_before":"α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ✝ η✝ : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\nh_coe : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b\n⊢ ∀ᵐ (x : γ) ∂(κ + η) a, x ∈ {x | 1 ≤ κ.rnDerivAux (κ + η) a x} → ENNReal.ofReal (1 - κ.rnDerivAux (κ + η) a x) = 0 x\n\nα : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ✝ η✝ : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\nh_coe : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b\n⊢ MeasurableSet {x | ENNReal.ofReal (1 - κ.rnDerivAux (κ + η) a x) = 0 x}\n\nα : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ✝ η✝ : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\nh_coe : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b\n⊢ Measurable fun b => ENNReal.ofReal (1 - κ.rnDerivAux (κ + η) a b)\n\ncase hf\nα : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ✝ η✝ : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\nh_coe : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b\n⊢ Measurable (Function.uncurry fun a x => ENNReal.ofReal (1 - 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κ.rnDerivAux (κ + η) a x))\n\nα : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ✝ η✝ : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\nh_coe : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b\n⊢ ∀ᵐ (x : γ) ∂(κ + η) a, x ∈ {x | 1 ≤ κ.rnDerivAux (κ + η) a x} → ENNReal.ofReal (1 - κ.rnDerivAux (κ + η) a x) = 0 x","tactic":"rotate_left","premises":[]},{"state_before":"α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ✝ η✝ : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\nh_coe : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b\n⊢ ∀ᵐ (x : γ) ∂(κ + η) a, x ∈ {x | 1 ≤ κ.rnDerivAux (κ + η) a x} → ENNReal.ofReal (1 - κ.rnDerivAux (κ + η) a x) = 0 x","state_after":"α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ✝ η✝ : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\nh_coe : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b\nx : γ\nhx : x ∈ {x | 1 ≤ κ.rnDerivAux (κ + η) a x}\n⊢ ENNReal.ofReal (1 - 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+{"url":"Mathlib/Combinatorics/Additive/FreimanHom.lean","commit":"","full_name":"AddEquivClass.isAddFreimanIso","start":[193,0],"end":[197,76],"file_path":"Mathlib/Combinatorics/Additive/FreimanHom.lean","tactics":[{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝⁴ : CommMonoid α\ninst✝³ : CommMonoid β\ninst✝² : CommMonoid γ\nA A₁ A₂ : Set α\nB B₁ B₂ : Set β\nC : Set γ\nf✝ f₁ f₂ : α → β\ng : β → γ\nm n : ℕ\ninst✝¹ : EquivLike F α β\ninst✝ : MulEquivClass F α β\nf : F\nhfAB : BijOn (⇑f) A B\ns t : Multiset α\nx✝³ : ∀ ⦃x : α⦄, x ∈ s → x ∈ A\nx✝² : ∀ ⦃x : α⦄, x ∈ t → x ∈ A\nx✝¹ : card s = n\nx✝ : card t = n\n⊢ (map (⇑f) s).prod = (map (⇑f) t).prod ↔ s.prod = t.prod","state_after":"no goals","tactic":"rw [← map_multiset_prod, ← map_multiset_prod, EquivLike.apply_eq_iff_eq]","premises":[{"full_name":"EquivLike.apply_eq_iff_eq","def_path":"Mathlib/Data/FunLike/Equiv.lean","def_pos":[174,8],"def_end_pos":[174,23]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"map_multiset_prod","def_path":"Mathlib/Algebra/BigOperators/Group/Multiset.lean","def_pos":[194,6],"def_end_pos":[194,30]}]}]}
+{"url":"Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean","commit":"","full_name":"CategoryTheory.leftUnitor_monoidal","start":[614,0],"end":[627,11],"file_path":"Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean","tactics":[{"state_before":"C : Type u₁\ninst✝⁸ : Category.{v₁, u₁} C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category.{v₂, u₂} D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category.{v₃, u₃} E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ : C\n⊢ (λ_ X₁).hom ⊗ (λ_ X₂).hom = tensor_μ C (𝟙_ C, X₁) (𝟙_ C, X₂) ≫ (λ_ (𝟙_ C)).hom ▷ (X₁ ⊗ X₂) ≫ (λ_ (X₁ ⊗ X₂)).hom","state_after":"C : Type u₁\ninst✝⁸ : Category.{v₁, u₁} C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category.{v₂, u₂} D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category.{v₃, u₃} E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ : C\n⊢ (λ_ X₁).hom ⊗ (λ_ X₂).hom =\n    ((α_ (𝟙_ C) X₁ (𝟙_ C ⊗ X₂)).hom ≫\n        𝟙_ C ◁ (α_ X₁ (𝟙_ C) X₂).inv ≫\n          𝟙_ C ◁ (β_ X₁ (𝟙_ C)).hom ▷ X₂ ≫ 𝟙_ C ◁ (α_ (𝟙_ C) X₁ X₂).hom ≫ (α_ (𝟙_ C) (𝟙_ C) (X₁ ⊗ X₂)).inv) ≫\n      (λ_ (𝟙_ C)).hom ▷ (X₁ ⊗ X₂) ≫ (λ_ (X₁ ⊗ X₂)).hom","tactic":"dsimp only [tensor_μ]","premises":[{"full_name":"CategoryTheory.tensor_μ","def_path":"Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean","def_pos":[508,4],"def_end_pos":[508,12]}]},{"state_before":"C : Type u₁\ninst✝⁸ : Category.{v₁, u₁} C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category.{v₂, u₂} D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category.{v₃, u₃} E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ : C\n⊢ (λ_ X₁).hom ⊗ (λ_ X₂).hom =\n    ((α_ (𝟙_ C) X₁ (𝟙_ C ⊗ X₂)).hom ≫\n        𝟙_ C ◁ (α_ X₁ (𝟙_ C) X₂).inv ≫\n          𝟙_ C ◁ (β_ X₁ (𝟙_ C)).hom ▷ X₂ ≫ 𝟙_ C ◁ (α_ (𝟙_ C) X₁ X₂).hom ≫ (α_ (𝟙_ C) (𝟙_ C) (X₁ ⊗ X₂)).inv) ≫\n      (λ_ (𝟙_ C)).hom ▷ (X₁ ⊗ X₂) ≫ (λ_ (X₁ ⊗ X₂)).hom","state_after":"C : Type u₁\ninst✝⁸ : Category.{v₁, u₁} C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category.{v₂, u₂} D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category.{v₃, u₃} E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ : C\nthis :\n  (λ_ X₁).hom ⊗ (λ_ X₂).hom =\n    (α_ (𝟙_ C) X₁ (𝟙_ C ⊗ X₂)).hom ≫ 𝟙_ C ◁ (α_ X₁ (𝟙_ C) X₂).inv ≫ (λ_ ((X₁ ⊗ 𝟙_ C) ⊗ X₂)).hom ≫ (ρ_ X₁).hom ▷ X₂\n⊢ (λ_ X₁).hom ⊗ (λ_ X₂).hom =\n    ((α_ (𝟙_ C) X₁ (𝟙_ C ⊗ X₂)).hom ≫\n        𝟙_ C ◁ (α_ X₁ (𝟙_ C) X₂).inv ≫\n          𝟙_ C ◁ (β_ X₁ (𝟙_ C)).hom ▷ X₂ ≫ 𝟙_ C ◁ (α_ (𝟙_ C) X₁ X₂).hom ≫ (α_ (𝟙_ C) (𝟙_ C) (X₁ ⊗ X₂)).inv) ≫\n      (λ_ (𝟙_ C)).hom ▷ (X₁ ⊗ X₂) ≫ (λ_ (X₁ ⊗ X₂)).hom","tactic":"have :\n    (λ_ X₁).hom ⊗ (λ_ X₂).hom =\n      (α_ (𝟙_ C) X₁ (𝟙_ C ⊗ X₂)).hom ≫\n        (𝟙_ C ◁ (α_ X₁ (𝟙_ C) X₂).inv) ≫ (λ_ ((X₁ ⊗ 𝟙_ C) ⊗ X₂)).hom ≫ ((ρ_ X₁).hom ▷ X₂) := by\n    coherence","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"CategoryTheory.BicategoricalCoherence.hom","def_path":"Mathlib/Tactic/CategoryTheory/BicategoricalComp.lean","def_pos":[29,2],"def_end_pos":[29,5]},{"full_name":"CategoryTheory.Bicategory.comp_whiskerLeft","def_path":"Mathlib/CategoryTheory/Bicategory/Basic.lean","def_pos":[81,2],"def_end_pos":[81,18]},{"full_name":"CategoryTheory.Bicategory.comp_whiskerRight","def_path":"Mathlib/CategoryTheory/Bicategory/Basic.lean","def_pos":[89,2],"def_end_pos":[89,19]},{"full_name":"CategoryTheory.Bicategory.id_whiskerLeft","def_path":"Mathlib/CategoryTheory/Bicategory/Basic.lean","def_pos":[77,2],"def_end_pos":[77,16]},{"full_name":"CategoryTheory.Bicategory.id_whiskerRight","def_path":"Mathlib/CategoryTheory/Bicategory/Basic.lean","def_pos":[87,2],"def_end_pos":[87,17]},{"full_name":"CategoryTheory.Bicategory.whiskerLeft_comp","def_path":"Mathlib/CategoryTheory/Bicategory/Basic.lean","def_pos":[73,2],"def_end_pos":[73,18]},{"full_name":"CategoryTheory.Bicategory.whiskerLeft_id","def_path":"Mathlib/CategoryTheory/Bicategory/Basic.lean","def_pos":[71,2],"def_end_pos":[71,16]},{"full_name":"CategoryTheory.Bicategory.whiskerRight_comp","def_path":"Mathlib/CategoryTheory/Bicategory/Basic.lean","def_pos":[97,2],"def_end_pos":[97,19]},{"full_name":"CategoryTheory.Bicategory.whiskerRight_id","def_path":"Mathlib/CategoryTheory/Bicategory/Basic.lean","def_pos":[93,2],"def_end_pos":[93,17]},{"full_name":"CategoryTheory.Bicategory.whisker_assoc","def_path":"Mathlib/CategoryTheory/Bicategory/Basic.lean","def_pos":[103,2],"def_end_pos":[103,15]},{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.CategoryStruct.comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]},{"full_name":"CategoryTheory.FreeMonoidalCategory.projectMap","def_path":"Mathlib/CategoryTheory/Monoidal/Free/Basic.lean","def_pos":[312,4],"def_end_pos":[312,14]},{"full_name":"CategoryTheory.Iso.hom","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[51,2],"def_end_pos":[51,5]},{"full_name":"CategoryTheory.Iso.inv","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[53,2],"def_end_pos":[53,5]},{"full_name":"CategoryTheory.MonoidalCategory.comp_whiskerRight","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[249,8],"def_end_pos":[249,25]},{"full_name":"CategoryTheory.MonoidalCategory.id_tensorHom","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[222,8],"def_end_pos":[222,20]},{"full_name":"CategoryTheory.MonoidalCategory.id_whiskerLeft","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[237,8],"def_end_pos":[237,22]},{"full_name":"CategoryTheory.MonoidalCategory.id_whiskerRight","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[172,2],"def_end_pos":[172,17]},{"full_name":"CategoryTheory.MonoidalCategory.tensorHom_def","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[157,2],"def_end_pos":[157,15]},{"full_name":"CategoryTheory.MonoidalCategory.tensorHom_id","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[227,8],"def_end_pos":[227,20]},{"full_name":"CategoryTheory.MonoidalCategory.tensor_whiskerLeft","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[242,8],"def_end_pos":[242,26]},{"full_name":"CategoryTheory.MonoidalCategory.whiskerLeft_comp","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[232,8],"def_end_pos":[232,24]},{"full_name":"CategoryTheory.MonoidalCategory.whiskerLeft_id","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[170,2],"def_end_pos":[170,16]},{"full_name":"CategoryTheory.MonoidalCategory.whiskerRight_id","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[254,8],"def_end_pos":[254,23]},{"full_name":"CategoryTheory.MonoidalCategory.whiskerRight_tensor","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[259,8],"def_end_pos":[259,27]},{"full_name":"CategoryTheory.MonoidalCategory.whisker_assoc","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[266,8],"def_end_pos":[266,21]},{"full_name":"CategoryTheory.MonoidalCategoryStruct.associator","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[94,2],"def_end_pos":[94,12]},{"full_name":"CategoryTheory.MonoidalCategoryStruct.leftUnitor","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[96,2],"def_end_pos":[96,12]},{"full_name":"CategoryTheory.MonoidalCategoryStruct.rightUnitor","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[98,2],"def_end_pos":[98,13]},{"full_name":"CategoryTheory.MonoidalCategoryStruct.tensorHom","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[89,2],"def_end_pos":[89,11]},{"full_name":"CategoryTheory.MonoidalCategoryStruct.tensorObj","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[82,2],"def_end_pos":[82,11]},{"full_name":"CategoryTheory.MonoidalCategoryStruct.tensorUnit","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[92,2],"def_end_pos":[92,12]},{"full_name":"CategoryTheory.MonoidalCategoryStruct.whiskerLeft","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[84,2],"def_end_pos":[84,13]},{"full_name":"CategoryTheory.MonoidalCategoryStruct.whiskerRight","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[86,2],"def_end_pos":[86,14]},{"full_name":"CategoryTheory.bicategoricalComp","def_path":"Mathlib/Tactic/CategoryTheory/BicategoricalComp.lean","def_pos":[47,4],"def_end_pos":[47,21]},{"full_name":"CategoryTheory.monoidalComp","def_path":"Mathlib/Tactic/CategoryTheory/MonoidalComp.lean","def_pos":[67,4],"def_end_pos":[67,16]},{"full_name":"Mathlib.Tactic.Coherence.LiftHom.lift","def_path":"Mathlib/Tactic/CategoryTheory/Coherence.lean","def_pos":[54,12],"def_end_pos":[54,16]},{"full_name":"id","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[33,14],"def_end_pos":[33,16]}]},{"state_before":"C : Type u₁\ninst✝⁸ : Category.{v₁, u₁} C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category.{v₂, u₂} D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category.{v₃, u₃} E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ : C\nthis :\n  (λ_ X₁).hom ⊗ (λ_ X₂).hom =\n    (α_ (𝟙_ C) X₁ (𝟙_ C ⊗ X₂)).hom ≫ 𝟙_ C ◁ (α_ X₁ (𝟙_ C) X₂).inv ≫ (λ_ ((X₁ ⊗ 𝟙_ C) ⊗ X₂)).hom ≫ (ρ_ X₁).hom ▷ X₂\n⊢ (λ_ X₁).hom ⊗ (λ_ X₂).hom =\n    ((α_ (𝟙_ C) X₁ (𝟙_ C ⊗ X₂)).hom ≫\n        𝟙_ C ◁ (α_ X₁ (𝟙_ C) X₂).inv ≫\n          𝟙_ C ◁ (β_ X₁ (𝟙_ C)).hom ▷ X₂ ≫ 𝟙_ C ◁ (α_ (𝟙_ C) X₁ X₂).hom ≫ (α_ (𝟙_ C) (𝟙_ C) (X₁ ⊗ X₂)).inv) ≫\n      (λ_ (𝟙_ C)).hom ▷ (X₁ ⊗ X₂) ≫ (λ_ (X₁ ⊗ X₂)).hom","state_after":"C : Type u₁\ninst✝⁸ : Category.{v₁, u₁} C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category.{v₂, u₂} D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category.{v₃, u₃} E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ : C\nthis :\n  (λ_ X₁).hom ⊗ (λ_ X₂).hom =\n    (α_ (𝟙_ C) X₁ (𝟙_ C ⊗ X₂)).hom ≫ 𝟙_ C ◁ (α_ X₁ (𝟙_ C) X₂).inv ≫ (λ_ ((X₁ ⊗ 𝟙_ C) ⊗ X₂)).hom ≫ (ρ_ X₁).hom ▷ X₂\n⊢ (α_ (𝟙_ C) X₁ (𝟙_ C ⊗ X₂)).hom ≫ 𝟙_ C ◁ (α_ X₁ (𝟙_ C) X₂).inv ≫ (λ_ ((X₁ ⊗ 𝟙_ C) ⊗ X₂)).hom ≫ (ρ_ X₁).hom ▷ X₂ =\n    ((α_ (𝟙_ C) X₁ (𝟙_ C ⊗ X₂)).hom ≫\n        𝟙_ C ◁ (α_ X₁ (𝟙_ C) X₂).inv ≫\n          𝟙_ C ◁ (β_ X₁ (𝟙_ C)).hom ▷ X₂ ≫ 𝟙_ C ◁ (α_ (𝟙_ C) X₁ X₂).hom ≫ (α_ (𝟙_ C) (𝟙_ C) (X₁ ⊗ X₂)).inv) ≫\n      (λ_ (𝟙_ C)).hom ▷ (X₁ ⊗ X₂) ≫ (λ_ (X₁ ⊗ X₂)).hom","tactic":"rw [this]","premises":[]},{"state_before":"C : Type u₁\ninst✝⁸ : Category.{v₁, u₁} C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category.{v₂, u₂} D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category.{v₃, u₃} E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ : C\nthis :\n  (λ_ X₁).hom ⊗ (λ_ X₂).hom =\n    (α_ (𝟙_ C) X₁ (𝟙_ C ⊗ X₂)).hom ≫ 𝟙_ C ◁ (α_ X₁ (𝟙_ C) X₂).inv ≫ (λ_ ((X₁ ⊗ 𝟙_ C) ⊗ X₂)).hom ≫ (ρ_ X₁).hom ▷ X₂\n⊢ (α_ (𝟙_ C) X₁ (𝟙_ C ⊗ X₂)).hom ≫ 𝟙_ C ◁ (α_ X₁ (𝟙_ C) X₂).inv ≫ (λ_ ((X₁ ⊗ 𝟙_ C) ⊗ X₂)).hom ≫ (ρ_ X₁).hom ▷ X₂ =\n    ((α_ (𝟙_ C) X₁ (𝟙_ C ⊗ X₂)).hom ≫\n        𝟙_ C ◁ (α_ X₁ (𝟙_ C) X₂).inv ≫\n          𝟙_ C ◁ (β_ X₁ (𝟙_ C)).hom ▷ X₂ ≫ 𝟙_ C ◁ (α_ (𝟙_ C) X₁ X₂).hom ≫ (α_ (𝟙_ C) (𝟙_ C) (X₁ ⊗ X₂)).inv) ≫\n      (λ_ (𝟙_ C)).hom ▷ (X₁ ⊗ X₂) ≫ (λ_ (X₁ ⊗ X₂)).hom","state_after":"C : Type u₁\ninst✝⁸ : Category.{v₁, u₁} C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category.{v₂, u₂} D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category.{v₃, u₃} E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ : C\n⊢ (α_ (𝟙_ C) X₁ (𝟙_ C ⊗ X₂)).hom ≫ 𝟙_ C ◁ (α_ X₁ (𝟙_ C) X₂).inv ≫ (λ_ ((X₁ ⊗ 𝟙_ C) ⊗ X₂)).hom ≫ (ρ_ X₁).hom ▷ X₂ =\n    ((α_ (𝟙_ C) X₁ (𝟙_ C ⊗ X₂)).hom ≫\n        𝟙_ C ◁ (α_ X₁ (𝟙_ C) X₂).inv ≫\n          𝟙_ C ◁ (β_ X₁ (𝟙_ C)).hom ▷ X₂ ≫ 𝟙_ C ◁ (α_ (𝟙_ C) X₁ X₂).hom ≫ (α_ (𝟙_ C) (𝟙_ C) (X₁ ⊗ X₂)).inv) ≫\n      (λ_ (𝟙_ C)).hom ▷ (X₁ ⊗ X₂) ≫ (λ_ (X₁ ⊗ X₂)).hom","tactic":"clear this","premises":[]},{"state_before":"C : Type u₁\ninst✝⁸ : Category.{v₁, u₁} C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category.{v₂, u₂} D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category.{v₃, u₃} E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ : C\n⊢ (α_ (𝟙_ C) X₁ (𝟙_ C ⊗ X₂)).hom ≫ 𝟙_ C ◁ (α_ X₁ (𝟙_ C) X₂).inv ≫ (λ_ ((X₁ ⊗ 𝟙_ C) ⊗ X₂)).hom ≫ (ρ_ X₁).hom ▷ X₂ =\n    ((α_ (𝟙_ C) X₁ (𝟙_ C ⊗ X₂)).hom ≫\n        𝟙_ C ◁ (α_ X₁ (𝟙_ C) X₂).inv ≫\n          𝟙_ C ◁ (β_ X₁ (𝟙_ C)).hom ▷ X₂ ≫ 𝟙_ C ◁ (α_ (𝟙_ C) X₁ X₂).hom ≫ (α_ (𝟙_ C) (𝟙_ C) (X₁ ⊗ X₂)).inv) ≫\n      (λ_ (𝟙_ C)).hom ▷ (X₁ ⊗ X₂) ≫ (λ_ (X₁ ⊗ X₂)).hom","state_after":"C : Type u₁\ninst✝⁸ : Category.{v₁, u₁} C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : 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+{"url":"Mathlib/CategoryTheory/WithTerminal.lean","commit":"","full_name":"CategoryTheory.WithTerminal.lift_map_liftStar","start":[302,0],"end":[307,5],"file_path":"Mathlib/CategoryTheory/WithTerminal.lean","tactics":[{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u_1\ninst✝ : Category.{u_2, u_1} D\nZ : D\nF : C ⥤ D\nM : (x : C) → F.obj x ⟶ Z\nhM : ∀ (x y : C) (f : x ⟶ y), F.map f ≫ M y = M x\nx : C\n⊢ (lift F M hM).map (starTerminal.from (incl.obj x)) ≫ (liftStar F M hM).hom = (inclLift F M hM).hom.app x ≫ M x","state_after":"C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u_1\ninst✝ : Category.{u_2, u_1} D\nZ : D\nF : C ⥤ D\nM : (x : C) → F.obj x ⟶ Z\nhM : ∀ (x y : C) (f : x ⟶ y), F.map f ≫ M y = M x\nx : C\n⊢ (lift F M hM).map (starTerminal.from (incl.obj x)) = M x","tactic":"erw [Category.id_comp, Category.comp_id]","premises":[{"full_name":"CategoryTheory.Category.comp_id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[153,2],"def_end_pos":[153,9]},{"full_name":"CategoryTheory.Category.id_comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[151,2],"def_end_pos":[151,9]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]}]},{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u_1\ninst✝ : Category.{u_2, u_1} D\nZ : D\nF : C ⥤ D\nM : (x : C) → F.obj x ⟶ Z\nhM : ∀ (x y : C) (f : x ⟶ y), F.map f ≫ M y = M x\nx : C\n⊢ (lift F M hM).map (starTerminal.from (incl.obj x)) = M x","state_after":"no goals","tactic":"rfl","premises":[]}]}
+{"url":"Mathlib/CategoryTheory/Monoidal/Functor.lean","commit":"","full_name":"CategoryTheory.LaxMonoidalFunctor.associativity_inv","start":[172,0],"end":[177,32],"file_path":"Mathlib/CategoryTheory/Monoidal/Functor.lean","tactics":[{"state_before":"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nX Y Z : C\n⊢ F.obj X ◁ F.μ Y Z ≫ F.μ X (Y ⊗ Z) ≫ F.map (α_ X Y Z).inv =\n    (α_ (F.obj X) (F.obj Y) (F.obj Z)).inv ≫ F.μ X Y ▷ F.obj Z ≫ F.μ (X ⊗ Y) Z","state_after":"no goals","tactic":"rw [Iso.eq_inv_comp, ← F.associativity_assoc, ← F.toFunctor.map_comp, Iso.hom_inv_id,\n    F.toFunctor.map_id, comp_id]","premises":[{"full_name":"CategoryTheory.Category.comp_id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[153,2],"def_end_pos":[153,9]},{"full_name":"CategoryTheory.Functor.map_comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[41,2],"def_end_pos":[41,10]},{"full_name":"CategoryTheory.Functor.map_id","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[39,2],"def_end_pos":[39,8]},{"full_name":"CategoryTheory.Iso.eq_inv_comp","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[181,8],"def_end_pos":[181,19]},{"full_name":"CategoryTheory.Iso.hom_inv_id","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[55,2],"def_end_pos":[55,12]}]}]}
+{"url":"Mathlib/Algebra/Group/Units.lean","commit":"","full_name":"IsAddUnit.add_eq_add_of_sub_eq_sub","start":[892,0],"end":[895,91],"file_path":"Mathlib/Algebra/Group/Units.lean","tactics":[{"state_before":"α : Type u\nM : Type u_1\nN : Type u_2\ninst✝ : DivisionCommMonoid α\na✝ b c✝ d : α\nhb : IsUnit b\nhd : IsUnit d\na c : α\nh : a / b = c / d\n⊢ a * d = c * b","state_after":"no goals","tactic":"rw [← mul_one a, ← hb.div_self, ← mul_comm_div, h, div_mul_eq_mul_div, hd.div_mul_cancel]","premises":[{"full_name":"IsUnit.div_mul_cancel","def_path":"Mathlib/Algebra/Group/Units.lean","def_pos":[805,16],"def_end_pos":[805,30]},{"full_name":"IsUnit.div_self","def_path":"Mathlib/Algebra/Group/Units.lean","def_pos":[770,16],"def_end_pos":[770,24]},{"full_name":"div_mul_eq_mul_div","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[558,8],"def_end_pos":[558,26]},{"full_name":"mul_comm_div","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[564,8],"def_end_pos":[564,20]},{"full_name":"mul_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[413,8],"def_end_pos":[413,15]}]}]}
+{"url":"Mathlib/Probability/Distributions/Gaussian.lean","commit":"","full_name":"ProbabilityTheory.integrable_gaussianPDFReal","start":[75,0],"end":[94,38],"file_path":"Mathlib/Probability/Distributions/Gaussian.lean","tactics":[{"state_before":"μ : ℝ\nv : ℝ≥0\n⊢ Integrable (gaussianPDFReal μ v) ℙ","state_after":"μ : ℝ\nv : ℝ���0\n⊢ Integrable (fun x => (√(2 * π * ↑v))⁻¹ * rexp (-(x - μ) ^ 2 / (2 * ↑v))) ℙ","tactic":"rw [gaussianPDFReal_def]","premises":[{"full_name":"ProbabilityTheory.gaussianPDFReal_def","def_path":"Mathlib/Probability/Distributions/Gaussian.lean","def_pos":[47,6],"def_end_pos":[47,25]}]},{"state_before":"μ : ℝ\nv : ℝ≥0\n⊢ Integrable (fun x => (√(2 * π * ↑v))⁻¹ * rexp (-(x - μ) ^ 2 / (2 * ↑v))) ℙ","state_after":"case pos\nμ : ℝ\nv : ℝ≥0\nhv : v = 0\n⊢ Integrable (fun x => (√(2 * π * ↑v))⁻¹ * rexp (-(x - μ) ^ 2 / (2 * ↑v))) ℙ\n\ncase neg\nμ : ℝ\nv : ℝ≥0\nhv : ¬v = 0\n⊢ Integrable (fun x => (√(2 * π * ↑v))⁻¹ * rexp (-(x - μ) ^ 2 / (2 * ↑v))) ℙ","tactic":"by_cases hv : v = 0","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case neg\nμ : ℝ\nv : ℝ≥0\nhv : ¬v = 0\n⊢ Integrable (fun x => (√(2 * π * ↑v))⁻¹ * rexp (-(x - μ) ^ 2 / (2 * ↑v))) ℙ","state_after":"case neg\nμ : ℝ\nv : ℝ≥0\nhv : ¬v = 0\ng : ℝ → ℝ := fun x => (√(2 * π * ↑v))⁻¹ * rexp (-x ^ 2 / (2 * ↑v))\n⊢ Integrable (fun x => (√(2 * π * ↑v))⁻¹ * rexp (-(x - μ) ^ 2 / (2 * ↑v))) ℙ","tactic":"let g : ℝ → ℝ := fun x ↦ (√(2 * π * v))⁻¹ * rexp (- x ^ 2 / (2 * v))","premises":[{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Real.exp","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[102,11],"def_end_pos":[102,14]},{"full_name":"Real.pi","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","def_pos":[119,28],"def_end_pos":[119,30]},{"full_name":"Real.sqrt","def_path":"Mathlib/Data/Real/Sqrt.lean","def_pos":[109,18],"def_end_pos":[109,22]}]},{"state_before":"case neg\nμ : ℝ\nv : ℝ≥0\nhv : ¬v = 0\ng : ℝ → ℝ := fun x => (√(2 * π * ↑v))⁻¹ * rexp (-x ^ 2 / (2 * ↑v))\n⊢ Integrable (fun x => (√(2 * π * ↑v))⁻¹ * rexp (-(x - μ) ^ 2 / (2 * ↑v))) ℙ","state_after":"case neg\nμ : ℝ\nv : ℝ≥0\nhv : ¬v = 0\ng : ℝ → ℝ := fun x => (√(2 * π * ↑v))⁻¹ * rexp (-x ^ 2 / (2 * ↑v))\nhg : Integrable g ℙ\n⊢ Integrable (fun x => (√(2 * π * ↑v))⁻¹ * rexp (-(x - μ) ^ 2 / (2 * ↑v))) ℙ","tactic":"have hg : Integrable g := by\n    suffices g = fun x ↦ (√(2 * π * v))⁻¹ * rexp (- (2 * v)⁻¹ * x ^ 2) by\n      rw [this]\n      refine (integrable_exp_neg_mul_sq ?_).const_mul (√(2 * π * v))⁻¹\n      simp [lt_of_le_of_ne (zero_le _) (Ne.symm hv)]\n    ext x\n    simp only [g, zero_lt_two, mul_nonneg_iff_of_pos_left, NNReal.zero_le_coe, Real.sqrt_mul',\n      mul_inv_rev, NNReal.coe_mul, NNReal.coe_inv, NNReal.coe_ofNat, neg_mul, mul_eq_mul_left_iff,\n      Real.exp_eq_exp, mul_eq_zero, inv_eq_zero, Real.sqrt_eq_zero, NNReal.coe_eq_zero, hv,\n      false_or]\n    rw [mul_comm]\n    left\n    field_simp","premises":[{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"MeasureTheory.Integrable","def_path":"Mathlib/MeasureTheory/Function/L1Space.lean","def_pos":[389,4],"def_end_pos":[389,14]},{"full_name":"MeasureTheory.Integrable.const_mul","def_path":"Mathlib/MeasureTheory/Function/L1Space.lean","def_pos":[1080,8],"def_end_pos":[1080,28]},{"full_name":"NNReal.coe_eq_zero","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[183,25],"def_end_pos":[183,36]},{"full_name":"NNReal.coe_inv","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[168,18],"def_end_pos":[168,25]},{"full_name":"NNReal.coe_mul","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[164,18],"def_end_pos":[164,25]},{"full_name":"NNReal.coe_ofNat","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[311,18],"def_end_pos":[311,27]},{"full_name":"NNReal.zero_le_coe","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[539,8],"def_end_pos":[539,19]},{"full_name":"Ne.symm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[704,8],"def_end_pos":[704,15]},{"full_name":"Real.exp","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[102,11],"def_end_pos":[102,14]},{"full_name":"Real.exp_eq_exp","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[1024,8],"def_end_pos":[1024,18]},{"full_name":"Real.pi","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","def_pos":[119,28],"def_end_pos":[119,30]},{"full_name":"Real.sqrt","def_path":"Mathlib/Data/Real/Sqrt.lean","def_pos":[109,18],"def_end_pos":[109,22]},{"full_name":"Real.sqrt_eq_zero","def_path":"Mathlib/Data/Real/Sqrt.lean","def_pos":[251,8],"def_end_pos":[251,20]},{"full_name":"Real.sqrt_mul'","def_path":"Mathlib/Data/Real/Sqrt.lean","def_pos":[321,8],"def_end_pos":[321,17]},{"full_name":"false_or","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[122,16],"def_end_pos":[122,24]},{"full_name":"integrable_exp_neg_mul_sq","def_path":"Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean","def_pos":[124,8],"def_end_pos":[124,33]},{"full_name":"inv_eq_zero","def_path":"Mathlib/Algebra/GroupWithZero/Basic.lean","def_pos":[353,8],"def_end_pos":[353,19]},{"full_name":"lt_of_le_of_ne","def_path":"Mathlib/Order/Defs.lean","def_pos":[164,8],"def_end_pos":[164,22]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]},{"full_name":"mul_eq_mul_left_iff","def_path":"Mathlib/Algebra/GroupWithZero/Basic.lean","def_pos":[197,8],"def_end_pos":[197,27]},{"full_name":"mul_eq_zero","def_path":"Mathlib/Algebra/GroupWithZero/Defs.lean","def_pos":[248,8],"def_end_pos":[248,19]},{"full_name":"mul_inv_rev","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[982,8],"def_end_pos":[982,19]},{"full_name":"mul_nonneg_iff_of_pos_left","def_path":"Mathlib/Algebra/Order/Ring/Unbundled/Basic.lean","def_pos":[628,8],"def_end_pos":[628,34]},{"full_name":"neg_mul","def_path":"Mathlib/Algebra/Ring/Defs.lean","def_pos":[268,8],"def_end_pos":[268,15]},{"full_name":"zero_le","def_path":"Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean","def_pos":[105,29],"def_end_pos":[105,36]},{"full_name":"zero_lt_two","def_path":"Mathlib/Algebra/Order/Monoid/NatCast.lean","def_pos":[62,14],"def_end_pos":[62,25]}]},{"state_before":"case neg\nμ : ℝ\nv : ℝ≥0\nhv : ¬v = 0\ng : ℝ → ℝ := fun x => (√(2 * π * ↑v))⁻¹ * rexp (-x ^ 2 / (2 * ↑v))\nhg : Integrable g ℙ\n⊢ Integrable (fun x => (√(2 * π * ↑v))⁻¹ * rexp (-(x - μ) ^ 2 / (2 * ↑v))) ℙ","state_after":"no goals","tactic":"exact Integrable.comp_sub_right hg μ","premises":[{"full_name":"MeasureTheory.Integrable.comp_sub_right","def_path":"Mathlib/MeasureTheory/Group/Integral.lean","def_pos":[103,2],"def_end_pos":[103,13]}]}]}
+{"url":"Mathlib/CategoryTheory/Dialectica/Monoidal.lean","commit":"","full_name":"CategoryTheory.Dial.associator_hom_F","start":[67,0],"end":[71,71],"file_path":"Mathlib/CategoryTheory/Dialectica/Monoidal.lean","tactics":[{"state_before":"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasFiniteProducts C\ninst✝ : HasPullbacks C\nX Y Z : Dial C\n⊢ ((X.tensorObj Y).tensorObj Z).rel =\n    (Subobject.pullback (prod.map (prod.associator X.src Y.src Z.src).hom (prod.associator X.tgt Y.tgt Z.tgt).hom)).obj\n      (X.tensorObj (Y.tensorObj Z)).rel","state_after":"no goals","tactic":"simp [Subobject.inf_pullback, ← Subobject.pullback_comp, inf_assoc]","premises":[{"full_name":"CategoryTheory.Subobject.inf_pullback","def_path":"Mathlib/CategoryTheory/Subobject/Lattice.lean","def_pos":[424,8],"def_end_pos":[424,20]},{"full_name":"CategoryTheory.Subobject.pullback_comp","def_path":"Mathlib/CategoryTheory/Subobject/Basic.lean","def_pos":[498,8],"def_end_pos":[498,21]},{"full_name":"inf_assoc","def_path":"Mathlib/Order/Lattice.lean","def_pos":[389,8],"def_end_pos":[389,17]}]}]}
+{"url":"Mathlib/Data/Set/Lattice.lean","commit":"","full_name":"iSup_iUnion","start":[1902,0],"end":[1904,35],"file_path":"Mathlib/Data/Set/Lattice.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nι' : Sort u_5\nι₂ : Sort u_6\nκ : ι → Sort u_7\nκ₁ : ι → Sort u_8\nκ₂ : ι → Sort u_9\nκ' : ι' → Sort u_10\ninst✝ : CompleteLattice β\ns : ι → Set α\nf : α → β\n⊢ ⨆ a ∈ ⋃ i, s i, f a = ⨆ i, ⨆ a ∈ s i, f a","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nι' : Sort u_5\nι₂ : Sort u_6\nκ : ι → Sort u_7\nκ₁ : ι → Sort u_8\nκ₂ : ι → Sort u_9\nκ' : ι' → Sort u_10\ninst✝ : CompleteLattice β\ns : ι → Set α\nf : α → β\n⊢ ⨆ a ∈ ⋃ i, s i, f a = ⨆ j, ⨆ i, ⨆ (_ : j ∈ s i), f j","tactic":"rw [iSup_comm]","premises":[{"full_name":"iSup_comm","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[921,8],"def_end_pos":[921,17]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nι' : Sort u_5\nι₂ : Sort u_6\nκ : ι → Sort u_7\nκ₁ : ι → Sort u_8\nκ₂ : ι → Sort u_9\nκ' : ι' → Sort u_10\ninst✝ : CompleteLattice β\ns : ι → Set α\nf : α → β\n⊢ ⨆ a ∈ ⋃ i, s i, f a = ⨆ j, ⨆ i, ⨆ (_ : j ∈ s i), f j","state_after":"no goals","tactic":"simp_rw [mem_iUnion, iSup_exists]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Set.mem_iUnion","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[254,8],"def_end_pos":[254,18]},{"full_name":"iSup_exists","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[1076,8],"def_end_pos":[1076,19]}]}]}
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+{"url":"Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean","commit":"","full_name":"AffineSubspace.parallel_iff_direction_eq_and_eq_bot_iff_eq_bot","start":[1656,0],"end":[1674,27],"file_path":"Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean","tactics":[{"state_before":"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns₁ s₂ : AffineSubspace k P\n⊢ s₁ ∥ s₂ ↔ s₁.direction = s₂.direction ∧ (s₁ = ⊥ ↔ s₂ = ⊥)","state_after":"case refine_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns₁ s₂ : AffineSubspace k P\nh : s₁ ∥ s₂\n⊢ s₁ = ⊥ → s₂ = ⊥\n\ncase refine_2\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns₁ s₂ : AffineSubspace k P\nh : s₁ ∥ s₂\n⊢ s₂ = ⊥ → s₁ = ⊥\n\ncase refine_3\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns₁ s₂ : AffineSubspace k P\nh : s₁.direction = s₂.direction ∧ (s₁ = ⊥ ↔ s₂ = ⊥)\n⊢ s₁ ∥ s₂","tactic":"refine ⟨fun h => ⟨h.direction_eq, ?_, ?_⟩, fun h => ?_⟩","premises":[{"full_name":"AffineSubspace.Parallel.direction_eq","def_path":"Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean","def_pos":[1641,8],"def_end_pos":[1641,29]},{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]}]}]}
+{"url":"Mathlib/RingTheory/WittVector/Identities.lean","commit":"","full_name":"WittVector.iterate_verschiebung_mul_coeff","start":[168,0],"end":[181,39],"file_path":"Mathlib/RingTheory/WittVector/Identities.lean","tactics":[{"state_before":"p : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CharP R p\nx y : 𝕎 R\ni j : ℕ\n⊢ ((⇑verschiebung)^[i] x * (⇑verschiebung)^[j] y).coeff (i + j) = x.coeff 0 ^ p ^ j * y.coeff 0 ^ p ^ i","state_after":"case calc_1\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CharP R p\nx y : 𝕎 R\ni j : ℕ\n⊢ ((⇑verschiebung)^[i] x * (⇑verschiebung)^[j] y).coeff (i + j) =\n    ((⇑verschiebung)^[i + j] ((⇑frobenius)^[j] x * (⇑frobenius)^[i] y)).coeff (i + j)\n\ncase calc_2\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CharP R p\nx y : 𝕎 R\ni j : ℕ\n⊢ ((⇑verschiebung)^[i + j] ((⇑frobenius)^[j] x * (⇑frobenius)^[i] y)).coeff (i + j) =\n    ((⇑frobenius)^[j] x * (⇑frobenius)^[i] y).coeff 0\n\ncase calc_3\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CharP R p\nx y : 𝕎 R\ni j : ℕ\n⊢ ((⇑frobenius)^[j] x * (⇑frobenius)^[i] y).coeff 0 = ((⇑frobenius)^[j] x).coeff 0 * ((⇑frobenius)^[i] y).coeff 0\n\ncase calc_4\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CharP R p\nx y : 𝕎 R\ni j : ℕ\n⊢ ((⇑frobenius)^[j] x).coeff 0 * ((⇑frobenius)^[i] y).coeff 0 = x.coeff 0 ^ p ^ j * y.coeff 0 ^ p ^ i","tactic":"calc\n    _ = (verschiebung^[i + j] (frobenius^[j] x * frobenius^[i] y)).coeff (i + j) := ?_\n    _ = (frobenius^[j] x * frobenius^[i] y).coeff 0 := ?_\n    _ = (frobenius^[j] x).coeff 0 * (frobenius^[i] y).coeff 0 := ?_\n    _ = _ := ?_","premises":[{"full_name":"Nat.iterate","def_path":"Mathlib/Logic/Function/Iterate.lean","def_pos":[36,4],"def_end_pos":[36,15]},{"full_name":"WittVector.coeff","def_path":"Mathlib/RingTheory/WittVector/Defs.lean","def_pos":[53,2],"def_end_pos":[53,7]},{"full_name":"WittVector.frobenius","def_path":"Mathlib/RingTheory/WittVector/Frobenius.lean","def_pos":[233,4],"def_end_pos":[233,13]},{"full_name":"WittVector.verschiebung","def_path":"Mathlib/RingTheory/WittVector/Verschiebung.lean","def_pos":[106,18],"def_end_pos":[106,30]}]}]}
+{"url":"Mathlib/LinearAlgebra/Lagrange.lean","commit":"","full_name":"Lagrange.eval_basis_not_at_node","start":[593,0],"end":[597,75],"file_path":"Mathlib/LinearAlgebra/Lagrange.lean","tactics":[{"state_before":"F : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns : Finset ι\nv r : ι → F\ni : ι\nx : F\nhi : i ∈ s\nhxi : x ≠ v i\n⊢ eval x (Lagrange.basis s v i) = eval x (nodal s v) * (nodalWeight s v i * (x - v i)⁻¹)","state_after":"no goals","tactic":"rw [mul_comm, basis_eq_prod_sub_inv_mul_nodal_div hi, eval_mul, eval_C, ←\n    nodal_erase_eq_nodal_div hi, eval_nodal, eval_nodal, mul_assoc, ← mul_prod_erase _ _ hi, ←\n    mul_assoc (x - v i)⁻¹, inv_mul_cancel (sub_ne_zero_of_ne hxi), one_mul]","premises":[{"full_name":"Finset.mul_prod_erase","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[1656,8],"def_end_pos":[1656,22]},{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"Lagrange.basis_eq_prod_sub_inv_mul_nodal_div","def_path":"Mathlib/LinearAlgebra/Lagrange.lean","def_pos":[588,8],"def_end_pos":[588,43]},{"full_name":"Lagrange.eval_nodal","def_path":"Mathlib/LinearAlgebra/Lagrange.lean","def_pos":[491,8],"def_end_pos":[491,18]},{"full_name":"Lagrange.nodal_erase_eq_nodal_div","def_path":"Mathlib/LinearAlgebra/Lagrange.lean","def_pos":[564,8],"def_end_pos":[564,32]},{"full_name":"Polynomial.eval_C","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[308,8],"def_end_pos":[308,14]},{"full_name":"Polynomial.eval_mul","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[938,8],"def_end_pos":[938,16]},{"full_name":"inv_mul_cancel","def_path":"Mathlib/Algebra/GroupWithZero/NeZero.lean","def_pos":[50,8],"def_end_pos":[50,22]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]},{"full_name":"one_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[409,8],"def_end_pos":[409,15]},{"full_name":"sub_ne_zero_of_ne","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[387,2],"def_end_pos":[387,13]}]}]}
+{"url":".lake/packages/batteries/Batteries/Data/List/Basic.lean","commit":"","full_name":"List.zipWithLeft_eq_zipWithLeftTR","start":[1123,0],"end":[1129,26],"file_path":".lake/packages/batteries/Batteries/Data/List/Basic.lean","tactics":[{"state_before":"⊢ @zipWithLeft = @zipWithLeftTR","state_after":"case h.h.h.h.h.h\nα : Type u_3\nβ : Type u_2\nγ : Type u_1\nf : α → Option β → γ\nas : List α\nbs : List β\n⊢ zipWithLeft f as bs = zipWithLeftTR f as bs","tactic":"funext α β γ f as bs","premises":[{"full_name":"funext","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1817,8],"def_end_pos":[1817,14]}]},{"state_before":"case h.h.h.h.h.h\nα : Type u_3\nβ : Type u_2\nγ : Type u_1\nf : α → Option β → γ\nas : List α\nbs : List β\n⊢ zipWithLeft f as bs = zipWithLeftTR f as bs","state_after":"case h.h.h.h.h.h\nα : Type u_3\nβ : Type u_2\nγ : Type u_1\nf : α → Option β → γ\nas : List α\nbs : List β\n⊢ zipWithLeft f as bs = zipWithLeftTR.go f as bs #[]","tactic":"simp [zipWithLeftTR]","premises":[{"full_name":"List.zipWithLeftTR","def_path":".lake/packages/batteries/Batteries/Data/List/Basic.lean","def_pos":[1115,14],"def_end_pos":[1115,27]}]},{"state_before":"α : Type u_3\nβ : Type u_2\nγ : Type u_1\nf : α → Option β → γ\nas : List α\nbs✝ : List β\nacc : Array γ\nbs : List β\n⊢ zipWithLeftTR.go f [] bs acc = acc.toList ++ zipWithLeft f [] bs","state_after":"no goals","tactic":"simp [zipWithLeftTR.go]","premises":[{"full_name":"List.zipWithLeftTR.go","def_path":".lake/packages/batteries/Batteries/Data/List/Basic.lean","def_pos":[1118,2],"def_end_pos":[1118,4]}]},{"state_before":"α : Type u_3\nβ : Type u_2\nγ : Type u_1\nf : α → Option β → γ\nas : List α\nbs : List β\nacc : Array γ\nhead✝ : α\ntail✝ : List α\n⊢ zipWithLeftTR.go f (head✝ :: tail✝) [] acc = acc.toList ++ zipWithLeft f (head✝ :: tail✝) []","state_after":"no goals","tactic":"simp [zipWithLeftTR.go, Array.foldl_data_eq_map]","premises":[{"full_name":"Array.foldl_data_eq_map","def_path":".lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean","def_pos":[188,8],"def_end_pos":[188,25]},{"full_name":"List.zipWithLeftTR.go","def_path":".lake/packages/batteries/Batteries/Data/List/Basic.lean","def_pos":[1118,2],"def_end_pos":[1118,4]}]},{"state_before":"α : Type u_3\nβ : Type u_2\nγ : Type u_1\nf : α → Option β → γ\nas✝ : List α\nbs✝ : List β\nacc : Array γ\na : α\nas : List α\nb : β\nbs : List β\n⊢ zipWithLeftTR.go f (a :: as) (b :: bs) acc = acc.toList ++ zipWithLeft f (a :: as) (b :: bs)","state_after":"no goals","tactic":"simp [zipWithLeftTR.go, go _ as bs]","premises":[{"full_name":"List.zipWithLeftTR.go","def_path":".lake/packages/batteries/Batteries/Data/List/Basic.lean","def_pos":[1118,2],"def_end_pos":[1118,4]}]},{"state_before":"case h.h.h.h.h.h\nα : Type u_3\nβ : Type u_2\nγ : Type u_1\nf : α → Option β → γ\nas : List α\nbs : List β\n⊢ zipWithLeft f as bs = zipWithLeftTR.go f as bs #[]","state_after":"no goals","tactic":"simp [zipWithLeftTR, go]","premises":[{"full_name":"List.zipWithLeftTR","def_path":".lake/packages/batteries/Batteries/Data/List/Basic.lean","def_pos":[1115,14],"def_end_pos":[1115,27]}]}]}
+{"url":"Mathlib/CategoryTheory/Filtered/Final.lean","commit":"","full_name":"CategoryTheory.isFiltered_structuredArrow_of_isFiltered_of_exists","start":[56,0],"end":[72,45],"file_path":"Mathlib/CategoryTheory/Filtered/Final.lean","tactics":[{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝ : IsFilteredOrEmpty C\nh₁ : ∀ (d : D), ∃ c, Nonempty (d ⟶ F.obj c)\nh₂ : ∀ {d : D} {c : C} (s s' : d ⟶ F.obj c), ∃ c' t, s ≫ F.map t = s' ≫ F.map t\nd : D\n⊢ IsFiltered (StructuredArrow d F)","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝ : IsFilteredOrEmpty C\nh₁ : ∀ (d : D), ∃ c, Nonempty (d ⟶ F.obj c)\nh₂ : ∀ {d : D} {c : C} (s s' : d ⟶ F.obj c), ∃ c' t, s ≫ F.map t = s' ≫ F.map t\nd : D\nthis : Nonempty (StructuredArrow d F)\n⊢ IsFiltered (StructuredArrow d F)","tactic":"have : Nonempty (StructuredArrow d F) := by\n    obtain ⟨c, ⟨f⟩⟩ := h₁ d\n    exact ⟨.mk f⟩","premises":[{"full_name":"CategoryTheory.StructuredArrow","def_path":"Mathlib/CategoryTheory/Comma/StructuredArrow.lean","def_pos":[39,4],"def_end_pos":[39,19]},{"full_name":"CategoryTheory.StructuredArrow.mk","def_path":"Mathlib/CategoryTheory/Comma/StructuredArrow.lean","def_pos":[65,4],"def_end_pos":[65,6]},{"full_name":"Nonempty","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[709,16],"def_end_pos":[709,24]},{"full_name":"Nonempty.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[711,4],"def_end_pos":[711,9]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝ : IsFilteredOrEmpty C\nh₁ : ∀ (d : D), ∃ c, Nonempty (d ⟶ F.obj c)\nh₂ : ∀ {d : D} {c : C} (s s' : d ⟶ F.obj c), ∃ c' t, s ≫ F.map t = s' ≫ F.map t\nd : D\nthis : Nonempty (StructuredArrow d F)\n⊢ IsFiltered (StructuredArrow d F)","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝ : IsFilteredOrEmpty C\nh₁ : ∀ (d : D), ∃ c, Nonempty (d ⟶ F.obj c)\nh₂ : ∀ {d : D} {c : C} (s s' : d ⟶ F.obj c), ∃ c' t, s ≫ F.map t = s' ≫ F.map t\nd : D\nthis : Nonempty (StructuredArrow d F)\n⊢ IsFilteredOrEmpty (StructuredArrow d F)","tactic":"suffices IsFilteredOrEmpty (StructuredArrow d F) from IsFiltered.mk","premises":[{"full_name":"CategoryTheory.IsFilteredOrEmpty","def_path":"Mathlib/CategoryTheory/Filtered/Basic.lean","def_pos":[74,6],"def_end_pos":[74,23]},{"full_name":"CategoryTheory.StructuredArrow","def_path":"Mathlib/CategoryTheory/Comma/StructuredArrow.lean","def_pos":[39,4],"def_end_pos":[39,19]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝ : IsFilteredOrEmpty C\nh₁ : ∀ (d : D), ∃ c, Nonempty (d ⟶ F.obj c)\nh₂ : ∀ {d : D} {c : C} (s s' : d ⟶ F.obj c), ∃ c' t, s ≫ F.map t = s' ≫ F.map t\nd : D\nthis : Nonempty (StructuredArrow d F)\n⊢ IsFilteredOrEmpty (StructuredArrow d F)","state_after":"case refine_1\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝ : IsFilteredOrEmpty C\nh₁ : ∀ (d : D), ∃ c, Nonempty (d ⟶ F.obj c)\nh₂ : ∀ {d : D} {c : C} (s s' : d ⟶ F.obj c), ∃ c' t, s ≫ F.map t = s' ≫ F.map t\nd : D\nthis : Nonempty (StructuredArrow d F)\nf g : StructuredArrow d F\n⊢ ∃ Z x x, True\n\ncase refine_2\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝ : IsFilteredOrEmpty C\nh₁ : ∀ (d : D), ∃ c, Nonempty (d ⟶ F.obj c)\nh₂ : ∀ {d : D} {c : C} (s s' : d ⟶ F.obj c), ∃ c' t, s ≫ F.map t = s' ≫ F.map t\nd : D\nthis : Nonempty (StructuredArrow d F)\nf g : StructuredArrow d F\nη μ : f ⟶ g\n⊢ ∃ Z h, η ≫ h = μ ≫ h","tactic":"refine ⟨fun f g => ?_, fun f g η μ => ?_⟩","premises":[]}]}
+{"url":"Mathlib/RingTheory/Localization/NumDen.lean","commit":"","full_name":"IsFractionRing.exists_reduced_fraction","start":[35,0],"end":[45,57],"file_path":"Mathlib/RingTheory/Localization/NumDen.lean","tactics":[{"state_before":"R : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommRing P\nA : Type u_4\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\n⊢ ∃ a b, IsRelPrime a ↑b ∧ mk' K a b = x","state_after":"case intro.mk.intro\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommRing P\nA : Type u_4\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nb : A\nb_nonzero : b ∈ nonZeroDivisors A\na : A\nhab : (algebraMap A K) a = ↑⟨b, b_nonzero⟩ • x\n⊢ ∃ a b, IsRelPrime a ↑b ∧ mk' K a b = x","tactic":"obtain ⟨⟨b, b_nonzero⟩, a, hab⟩ := exists_integer_multiple (nonZeroDivisors A) x","premises":[{"full_name":"IsLocalization.exists_integer_multiple","def_path":"Mathlib/RingTheory/Localization/Integer.lean","def_pos":[76,8],"def_end_pos":[76,31]},{"full_name":"nonZeroDivisors","def_path":"Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean","def_pos":[84,4],"def_end_pos":[84,19]}]},{"state_before":"case intro.mk.intro\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommRing P\nA : Type u_4\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nb : A\nb_nonzero : b ∈ nonZeroDivisors A\na : A\nhab : (algebraMap A K) a = ↑⟨b, b_nonzero⟩ • x\n⊢ ∃ a b, IsRelPrime a ↑b ∧ mk' K a b = x","state_after":"case intro.mk.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommRing P\nA : Type u_4\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\na' b' c' : A\nno_factor : IsRelPrime a' b'\nb_nonzero : c' * b' ∈ nonZeroDivisors A\nhab : (algebraMap A K) (c' * a') = ↑⟨c' * b', b_nonzero⟩ • x\n⊢ ∃ a b, IsRelPrime a ↑b ∧ mk' K a b = x","tactic":"obtain ⟨a', b', c', no_factor, rfl, rfl⟩ :=\n    UniqueFactorizationMonoid.exists_reduced_factors' a b\n      (mem_nonZeroDivisors_iff_ne_zero.mp b_nonzero)","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"UniqueFactorizationMonoid.exists_reduced_factors'","def_path":"Mathlib/RingTheory/UniqueFactorizationDomain.lean","def_pos":[894,8],"def_end_pos":[894,31]},{"full_name":"mem_nonZeroDivisors_iff_ne_zero","def_path":"Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean","def_pos":[199,8],"def_end_pos":[199,39]}]},{"state_before":"case intro.mk.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommRing P\nA : Type u_4\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\na' b' c' : A\nno_factor : IsRelPrime a' b'\nb_nonzero : c' * b' ∈ nonZeroDivisors A\nhab : (algebraMap A K) (c' * a') = ↑⟨c' * b', b_nonzero⟩ • x\n⊢ ∃ a b, IsRelPrime a ↑b ∧ mk' K a b = x","state_after":"case intro.mk.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommRing P\nA : Type u_4\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\na' b' c' : A\nno_factor : IsRelPrime a' b'\nb_nonzero : c' * b' ∈ nonZeroDivisors A\nhab : (algebraMap A K) (c' * a') = ↑⟨c' * b', b_nonzero⟩ • x\nleft✝ : c' ∈ nonZeroDivisors A\nb'_nonzero : b' ∈ nonZeroDivisors A\n⊢ ∃ a b, IsRelPrime a ↑b ∧ mk' K a b = x","tactic":"obtain ⟨_, b'_nonzero⟩ := mul_mem_nonZeroDivisors.mp b_nonzero","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"mul_mem_nonZeroDivisors","def_path":"Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean","def_pos":[169,8],"def_end_pos":[169,31]}]},{"state_before":"case intro.mk.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommRing P\nA : Type u_4\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\na' b' c' : A\nno_factor : IsRelPrime a' b'\nb_nonzero : c' * b' ∈ nonZeroDivisors A\nhab : (algebraMap A K) (c' * a') = ↑⟨c' * b', b_nonzero⟩ • x\nleft✝ : c' ∈ nonZeroDivisors A\nb'_nonzero : b' ∈ nonZeroDivisors A\n⊢ ∃ a b, IsRelPrime a ↑b ∧ mk' K a b = x","state_after":"case intro.mk.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommRing P\nA : Type u_4\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\na' b' c' : A\nno_factor : IsRelPrime a' b'\nb_nonzero : c' * b' ∈ nonZeroDivisors A\nhab : (algebraMap A K) (c' * a') = ↑⟨c' * b', b_nonzero⟩ • x\nleft✝ : c' ∈ nonZeroDivisors A\nb'_nonzero : b' ∈ nonZeroDivisors A\n⊢ mk' K a' ⟨b', b'_nonzero⟩ = x","tactic":"refine ⟨a', ⟨b', b'_nonzero⟩, no_factor, ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]},{"state_before":"case intro.mk.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommRing P\nA : Type u_4\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\na' b' c' : A\nno_factor : IsRelPrime a' b'\nb_nonzero : c' * b' ∈ nonZeroDivisors A\nhab : (algebraMap A K) (c' * a') = ↑⟨c' * b', b_nonzero⟩ • x\nleft✝ : c' ∈ nonZeroDivisors A\nb'_nonzero : b' ∈ nonZeroDivisors A\n⊢ mk' K a' ⟨b', b'_nonzero⟩ = x","state_after":"case intro.mk.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommRing P\nA : Type u_4\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\na' b' c' : A\nno_factor : IsRelPrime a' b'\nb_nonzero : c' * b' ∈ nonZeroDivisors A\nhab : (algebraMap A K) (c' * a') = ↑⟨c' * b', b_nonzero⟩ • x\nleft✝ : c' ∈ nonZeroDivisors A\nb'_nonzero : b' ∈ nonZeroDivisors A\n⊢ (algebraMap A K) (c' * b') * mk' K a' ⟨b', b'_nonzero⟩ = (algebraMap A K) (c' * b') * x","tactic":"refine mul_left_cancel₀ (IsFractionRing.to_map_ne_zero_of_mem_nonZeroDivisors b_nonzero) ?_","premises":[{"full_name":"IsFractionRing.to_map_ne_zero_of_mem_nonZeroDivisors","def_path":"Mathlib/RingTheory/Localization/FractionRing.lean","def_pos":[90,18],"def_end_pos":[90,55]},{"full_name":"mul_left_cancel₀","def_path":"Mathlib/Algebra/GroupWithZero/Defs.lean","def_pos":[48,8],"def_end_pos":[48,24]}]},{"state_before":"case intro.mk.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommRing P\nA : Type u_4\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\na' b' c' : A\nno_factor : IsRelPrime a' b'\nb_nonzero : c' * b' ∈ nonZeroDivisors A\nhab : (algebraMap A K) (c' * a') = ↑⟨c' * b', b_nonzero⟩ • x\nleft✝ : c' ∈ nonZeroDivisors A\nb'_nonzero : b' ∈ nonZeroDivisors A\n⊢ (algebraMap A K) (c' * b') * mk' K a' ⟨b', b'_nonzero⟩ = (algebraMap A K) (c' * b') * x","state_after":"case intro.mk.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommRing P\nA : Type u_4\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\na' b' c' : A\nno_factor : IsRelPrime a' b'\nb_nonzero : c' * b' ∈ nonZeroDivisors A\nleft✝ : c' ∈ nonZeroDivisors A\nb'_nonzero : b' ∈ nonZeroDivisors A\nhab : (algebraMap A K) c' * (algebraMap A K) a' = (algebraMap A K) c' * (algebraMap A K) b' * x\n⊢ (algebraMap A K) c' * (algebraMap A K) b' * mk' K a' ⟨b', b'_nonzero⟩ = (algebraMap A K) c' * (algebraMap A K) b' * x","tactic":"simp only [Subtype.coe_mk, RingHom.map_mul, Algebra.smul_def] at *","premises":[{"full_name":"Algebra.smul_def","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[270,8],"def_end_pos":[270,16]},{"full_name":"RingHom.map_mul","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[484,18],"def_end_pos":[484,25]},{"full_name":"Subtype.coe_mk","def_path":"Mathlib/Data/Subtype.lean","def_pos":[86,8],"def_end_pos":[86,14]}]},{"state_before":"case intro.mk.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommRing P\nA : Type u_4\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\na' b' c' : A\nno_factor : IsRelPrime a' b'\nb_nonzero : c' * b' ∈ nonZeroDivisors A\nleft✝ : c' ∈ nonZeroDivisors A\nb'_nonzero : b' ∈ nonZeroDivisors A\nhab : (algebraMap A K) c' * (algebraMap A K) a' = (algebraMap A K) c' * (algebraMap A K) b' * x\n⊢ (algebraMap A K) c' * (algebraMap A K) b' * mk' K a' ⟨b', b'_nonzero⟩ = (algebraMap A K) c' * (algebraMap A K) b' * x","state_after":"no goals","tactic":"erw [← hab, mul_assoc, mk'_spec' _ a' ⟨b', b'_nonzero⟩]","premises":[{"full_name":"IsLocalization.mk'_spec'","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[246,8],"def_end_pos":[246,17]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]}]}]}
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+{"url":"Mathlib/Probability/ProbabilityMassFunction/Monad.lean","commit":"","full_name":"PMF.bindOnSupport_eq_zero_iff","start":[227,0],"end":[231,52],"file_path":"Mathlib/Probability/ProbabilityMassFunction/Monad.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\np : PMF α\nf : (a : α) → a ∈ p.support → PMF β\nb : β\n⊢ (p.bindOnSupport f) b = 0 ↔ ∀ (a : α) (ha : p a ≠ 0), (f a ha) b = 0","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\np : PMF α\nf : (a : α) → a ∈ p.support → PMF β\nb : β\n⊢ (∀ (i : α), ¬p i = 0 → (if h : p i = 0 then 0 else (f i h) b) = 0) ↔ ∀ (a : α) (ha : p a ≠ 0), (f a ha) b = 0","tactic":"simp only [bindOnSupport_apply, ENNReal.tsum_eq_zero, mul_eq_zero, or_iff_not_imp_left]","premises":[{"full_name":"Classical.or_iff_not_imp_left","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[147,8],"def_end_pos":[147,27]},{"full_name":"ENNReal.tsum_eq_zero","def_path":"Mathlib/Topology/Instances/ENNReal.lean","def_pos":[778,18],"def_end_pos":[778,30]},{"full_name":"PMF.bindOnSupport_apply","def_path":"Mathlib/Probability/ProbabilityMassFunction/Monad.lean","def_pos":[196,8],"def_end_pos":[196,27]},{"full_name":"mul_eq_zero","def_path":"Mathlib/Algebra/GroupWithZero/Defs.lean","def_pos":[248,8],"def_end_pos":[248,19]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\np : PMF α\nf : (a : α) → a ∈ p.support → PMF β\nb : β\n⊢ (∀ (i : α), ¬p i = 0 → (if h : p i = 0 then 0 else (f i h) b) = 0) ↔ ∀ (a : α) (ha : p a ≠ 0), (f a ha) b = 0","state_after":"no goals","tactic":"exact ⟨fun h a ha => Trans.trans (dif_neg ha).symm (h a ha),\n    fun h a ha => Trans.trans (dif_neg ha) (h a ha)⟩","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"Trans.trans","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1163,2],"def_end_pos":[1163,7]},{"full_name":"dif_neg","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[954,8],"def_end_pos":[954,15]}]}]}
+{"url":"Mathlib/Analysis/MeanInequalities.lean","commit":"","full_name":"Real.Lp_add_le_hasSum_of_nonneg","start":[758,0],"end":[778,29],"file_path":"Mathlib/Analysis/MeanInequalities.lean","tactics":[{"state_before":"ι : Type u\ns : Finset ι\nf g : ι → ℝ\np q : ℝ\nhp : 1 ≤ p\nhf : ∀ (i : ι), 0 ≤ f i\nhg : ∀ (i : ι), 0 ≤ g i\nA B : ℝ\nhA : 0 ≤ A\nhB : 0 ≤ B\nhfA : HasSum (fun i => f i ^ p) (A ^ p)\nhgB : HasSum (fun i => g i ^ p) (B ^ p)\n⊢ ∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p)","state_after":"case intro\nι : Type u\ns : Finset ι\ng : ι → ℝ\np q : ℝ\nhp : 1 ≤ p\nhg : ∀ (i : ι), 0 ≤ g i\nA B : ℝ\nhA : 0 ≤ A\nhB : 0 ≤ B\nhgB : HasSum (fun i => g i ^ p) (B ^ p)\nf : ι → ℝ≥0\nhfA : HasSum (fun i => (fun i => ↑(f i)) i ^ p) (A ^ p)\n⊢ ∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => ((fun i => ↑(f i)) i + g i) ^ p) (C ^ p)","tactic":"lift f to ι → ℝ≥0 using hf","premises":[{"full_name":"NNReal","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[60,4],"def_end_pos":[60,10]}]},{"state_before":"case intro\nι : Type u\ns : Finset ι\ng : ι → ℝ\np q : ℝ\nhp : 1 ≤ p\nhg : ∀ (i : ι), 0 ≤ g i\nA B : ℝ\nhA : 0 ≤ A\nhB : 0 ≤ B\nhgB : HasSum (fun i => g i ^ p) (B ^ p)\nf : ι → ℝ≥0\nhfA : HasSum (fun i => (fun i => ↑(f i)) i ^ p) (A ^ p)\n⊢ ∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => ((fun i => ↑(f i)) i + g i) ^ p) (C ^ p)","state_after":"case intro.intro\nι : Type u\ns : Finset ι\np q : ℝ\nhp : 1 ≤ p\nA B : ℝ\nhA : 0 ≤ A\nhB : 0 ≤ B\nf : ι → ℝ≥0\nhfA : HasSum (fun i => (fun i => ↑(f i)) i ^ p) (A ^ p)\ng : ι → ℝ≥0\nhgB : HasSum (fun i => (fun i => ↑(g i)) i ^ p) (B ^ p)\n⊢ ∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => ((fun i => ↑(f i)) i + (fun i => ↑(g i)) i) ^ p) (C ^ p)","tactic":"lift g to ι → ℝ≥0 using hg","premises":[{"full_name":"NNReal","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[60,4],"def_end_pos":[60,10]}]},{"state_before":"case intro.intro\nι : Type u\ns : Finset ι\np q : ℝ\nhp : 1 ≤ p\nA B : ℝ\nhA : 0 ≤ A\nhB : 0 ≤ B\nf : ι → ℝ≥0\nhfA : HasSum (fun i => (fun i => ↑(f i)) i ^ p) (A ^ p)\ng : ι → ℝ≥0\nhgB : HasSum (fun i => (fun i => ↑(g i)) i ^ p) (B ^ p)\n⊢ ∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => ((fun i => ↑(f i)) i + (fun i => ↑(g i)) i) ^ p) (C ^ p)","state_after":"case intro.intro.intro\nι : Type u\ns : Finset ι\np q : ℝ\nhp : 1 ≤ p\nB : ℝ\nhB : 0 ≤ B\nf g : ι → ℝ≥0\nhgB : HasSum (fun i => (fun i => ↑(g i)) i ^ p) (B ^ p)\nA : ℝ≥0\nhfA : HasSum (fun i => (fun i => ↑(f i)) i ^ p) (↑A ^ p)\n⊢ ∃ C, 0 ≤ C ∧ C ≤ ↑A + B ∧ HasSum (fun i => ((fun i => ↑(f i)) i + (fun i => ↑(g i)) i) ^ p) (C ^ p)","tactic":"lift A to ℝ≥0 using hA","premises":[{"full_name":"NNReal","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[60,4],"def_end_pos":[60,10]}]},{"state_before":"case intro.intro.intro\nι : Type u\ns : Finset ι\np q : ℝ\nhp : 1 ≤ p\nB : ℝ\nhB : 0 ≤ B\nf g : ι → ��≥0\nhgB : HasSum (fun i => (fun i => ↑(g i)) i ^ p) (B ^ p)\nA : ℝ≥0\nhfA : HasSum (fun i => (fun i => ↑(f i)) i ^ p) (↑A ^ p)\n⊢ ∃ C, 0 ≤ C ∧ C ≤ ↑A + B ∧ HasSum (fun i => ((fun i => ↑(f i)) i + (fun i => ↑(g i)) i) ^ p) (C ^ p)","state_after":"case intro.intro.intro.intro\nι : Type u\ns : Finset ι\np q : ℝ\nhp : 1 ≤ p\nf g : ι → ℝ≥0\nA : ℝ≥0\nhfA : HasSum (fun i => (fun i => ↑(f i)) i ^ p) (↑A ^ p)\nB : ℝ≥0\nhgB : HasSum (fun i => (fun i => ↑(g i)) i ^ p) (↑B ^ p)\n⊢ ∃ C, 0 ≤ C ∧ C ≤ ↑A + ↑B ∧ HasSum (fun i => ((fun i => ↑(f i)) i + (fun i => ↑(g i)) i) ^ p) (C ^ p)","tactic":"lift B to ℝ≥0 using hB","premises":[{"full_name":"NNReal","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[60,4],"def_end_pos":[60,10]}]},{"state_before":"case intro.intro.intro.intro\nι : Type u\ns : Finset ι\np q : ℝ\nhp : 1 ≤ p\nf g : ι → ℝ≥0\nA : ℝ≥0\nhfA : HasSum (fun i => (fun i => ↑(f i)) i ^ p) (↑A ^ p)\nB : ℝ≥0\nhgB : HasSum (fun i => (fun i => ↑(g i)) i ^ p) (↑B ^ p)\n⊢ ∃ C, 0 ≤ C ∧ C ≤ ↑A + ↑B ∧ HasSum (fun i => ((fun i => ↑(f i)) i + (fun i => ↑(g i)) i) ^ p) (C ^ p)","state_after":"case intro.intro.intro.intro\nι : Type u\ns : Finset ι\np q : ℝ\nhp : 1 ≤ p\nf g : ι → ℝ≥0\nA : ℝ≥0\nhfA : HasSum (fun i => ↑(f i) ^ p) (↑A ^ p)\nB : ℝ≥0\nhgB : HasSum (fun i => ↑(g i) ^ p) (↑B ^ p)\n⊢ ∃ C, 0 ≤ C ∧ C ≤ ↑A + ↑B ∧ HasSum (fun i => ((fun i => ↑(f i)) i + (fun i => ↑(g i)) i) ^ p) (C ^ p)","tactic":"beta_reduce at hfA hgB","premises":[]},{"state_before":"case intro.intro.intro.intro\nι : Type u\ns : Finset ι\np q : ℝ\nhp : 1 ≤ p\nf g : ι → ℝ≥0\nA : ℝ≥0\nhfA : HasSum (fun i => ↑(f i) ^ p) (↑A ^ p)\nB : ℝ≥0\nhgB : HasSum (fun i => ↑(g i) ^ p) (↑B ^ p)\n⊢ ∃ C, 0 ≤ C ∧ C ≤ ↑A + ↑B ∧ HasSum (fun i => ((fun i => ↑(f i)) i + (fun i => ↑(g i)) i) ^ p) (C ^ p)","state_after":"case intro.intro.intro.intro\nι : Type u\ns : Finset ι\np q : ℝ\nhp : 1 ≤ p\nf g : ι → ℝ≥0\nA B : ℝ≥0\nhfA : HasSum (fun a => f a ^ p) (A ^ p)\nhgB : HasSum (fun a => g a ^ p) (B ^ p)\n⊢ ∃ C, 0 ≤ C ∧ C ≤ ↑A + ↑B ∧ HasSum (fun i => ((fun i => ↑(f i)) i + (fun i => ↑(g i)) i) ^ p) (C ^ p)","tactic":"norm_cast at hfA hgB","premises":[]},{"state_before":"case intro.intro.intro.intro\nι : Type u\ns : Finset ι\np q : ℝ\nhp : 1 ≤ p\nf g : ι → ℝ≥0\nA B : ℝ≥0\nhfA : HasSum (fun a => f a ^ p) (A ^ p)\nhgB : HasSum (fun a => g a ^ p) (B ^ p)\n⊢ ∃ C, 0 ≤ C ∧ C ≤ ↑A + ↑B ∧ HasSum (fun i => ((fun i => ↑(f i)) i + (fun i => ↑(g i)) i) ^ p) (C ^ p)","state_after":"case intro.intro.intro.intro.intro.intro\nι : Type u\ns : Finset ι\np q : ℝ\nhp : 1 ≤ p\nf g : ι → ℝ≥0\nA B : ℝ≥0\nhfA : HasSum (fun a => f a ^ p) (A ^ p)\nhgB : HasSum (fun a => g a ^ p) (B ^ p)\nC : ℝ≥0\nhC₁ : C ≤ A + B\nhC₂ : HasSum (fun i => (f i + g i) ^ p) (C ^ p)\n⊢ ∃ C, 0 ≤ C ∧ C ≤ ↑A + ↑B ∧ HasSum (fun i => ((fun i => ↑(f i)) i + (fun i => ↑(g i)) i) ^ p) (C ^ p)","tactic":"obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB","premises":[{"full_name":"NNReal.Lp_add_le_hasSum","def_path":"Mathlib/Analysis/MeanInequalities.lean","def_pos":[595,8],"def_end_pos":[595,24]}]},{"state_before":"case intro.intro.intro.intro.intro.intro\nι : Type u\ns : Finset ι\np q : ℝ\nhp : 1 ≤ p\nf g : ι → ℝ≥0\nA B : ℝ≥0\nhfA : HasSum (fun a => f a ^ p) (A ^ p)\nhgB : HasSum (fun a => g a ^ p) (B ^ p)\nC : ℝ≥0\nhC₁ : C ≤ A + B\nhC₂ : HasSum (fun i => (f i + g i) ^ p) (C ^ p)\n⊢ ∃ C, 0 ≤ C ∧ C ≤ ↑A + ↑B ∧ HasSum (fun i => ((fun i => ↑(f i)) i + (fun i => ↑(g i)) i) ^ p) (C ^ p)","state_after":"case h\nι : Type u\ns : Finset ι\np q : ℝ\nhp : 1 ≤ p\nf g : ι → ℝ≥0\nA B : ℝ≥0\nhfA : HasSum (fun a => f a ^ p) (A ^ p)\nhgB : HasSum (fun a => g a ^ p) (B ^ p)\nC : ℝ≥0\nhC₁ : C ≤ A + B\nhC₂ : HasSum (fun i => (f i + g i) ^ p) (C ^ p)\n⊢ 0 ≤ ↑C ∧ ↑C ≤ ↑A + ↑B ∧ HasSum (fun i => ((fun i => ↑(f i)) i + (fun i => ↑(g i)) i) ^ p) (↑C ^ p)","tactic":"use C","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]}]},{"state_before":"case h\nι : Type u\ns : Finset ι\np q : ℝ\nhp : 1 ≤ p\nf g : ι → ℝ≥0\nA B : ℝ≥0\nhfA : HasSum (fun a => f a ^ p) (A ^ p)\nhgB : HasSum (fun a => g a ^ p) (B ^ p)\nC : ℝ≥0\nhC₁ : C ≤ A + B\nhC₂ : HasSum (fun i => (f i + g i) ^ p) (C ^ p)\n⊢ 0 ≤ ↑C ∧ ↑C ≤ ↑A + ↑B ∧ HasSum (fun i => ((fun i => ↑(f i)) i + (fun i => ↑(g i)) i) ^ p) (↑C ^ p)","state_after":"case h\nι : Type u\ns : Finset ι\np q : ℝ\nhp : 1 ≤ p\nf g : ι → ℝ≥0\nA B : ℝ≥0\nhfA : HasSum (fun a => f a ^ p) (A ^ p)\nhgB : HasSum (fun a => g a ^ p) (B ^ p)\nC : ℝ≥0\nhC₁ : C ≤ A + B\nhC₂ : HasSum (fun i => (f i + g i) ^ p) (C ^ p)\n⊢ 0 ≤ ↑C ∧ ↑C ≤ ↑A + ↑B ∧ HasSum (fun i => (↑(f i) + ↑(g i)) ^ p) (↑C ^ p)","tactic":"beta_reduce","premises":[]},{"state_before":"case h\nι : Type u\ns : Finset ι\np q : ℝ\nhp : 1 ≤ p\nf g : ι → ℝ≥0\nA B : ℝ≥0\nhfA : HasSum (fun a => f a ^ p) (A ^ p)\nhgB : HasSum (fun a => g a ^ p) (B ^ p)\nC : ℝ≥0\nhC₁ : C ≤ A + B\nhC₂ : HasSum (fun i => (f i + g i) ^ p) (C ^ p)\n⊢ 0 ≤ ↑C ∧ ↑C ≤ ↑A + ↑B ∧ HasSum (fun i => (↑(f i) + ↑(g i)) ^ p) (↑C ^ p)","state_after":"case h\nι : Type u\ns : Finset ι\np q : ℝ\nhp : 1 ≤ p\nf g : ι → ℝ≥0\nA B : ℝ≥0\nhfA : HasSum (fun a => f a ^ p) (A ^ p)\nhgB : HasSum (fun a => g a ^ p) (B ^ p)\nC : ℝ≥0\nhC₁ : C ≤ A + B\nhC₂ : HasSum (fun i => (f i + g i) ^ p) (C ^ p)\n⊢ 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun a => (f a + g a) ^ p) (C ^ p)","tactic":"norm_cast","premises":[]},{"state_before":"case h\nι : Type u\ns : Finset ι\np q : ℝ\nhp : 1 ≤ p\nf g : ι → ℝ≥0\nA B : ℝ≥0\nhfA : HasSum (fun a => f a ^ p) (A ^ p)\nhgB : HasSum (fun a => g a ^ p) (B ^ p)\nC : ℝ≥0\nhC₁ : C ≤ A + B\nhC₂ : HasSum (fun i => (f i + g i) ^ p) (C ^ p)\n⊢ 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun a => (f a + g a) ^ p) (C ^ p)","state_after":"no goals","tactic":"exact ⟨zero_le _, hC₁, hC₂⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"zero_le","def_path":"Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean","def_pos":[105,29],"def_end_pos":[105,36]}]}]}
+{"url":"Mathlib/CategoryTheory/Limits/Final.lean","commit":"","full_name":"CategoryTheory.Functor.final_of_adjunction","start":[119,0],"end":[130,100],"file_path":"Mathlib/CategoryTheory/Limits/Final.lean","tactics":[{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\nadj : L ⊣ R\nc : C\nu : StructuredArrow c R := StructuredArrow.mk (adj.unit.app c)\nf g : StructuredArrow c R\n⊢ u.hom ≫ R.map ((adj.homEquiv c f.right).symm f.hom) = f.hom","state_after":"no goals","tactic":"simp [u]","premises":[]},{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\nadj : L ⊣ R\nc : C\nu : StructuredArrow c R := StructuredArrow.mk (adj.unit.app c)\nf g : StructuredArrow c R\n⊢ u.hom ≫ R.map ((adj.homEquiv c g.right).symm g.hom) = g.hom","state_after":"no goals","tactic":"simp [u]","premises":[]}]}
+{"url":"Mathlib/Algebra/Lie/Weights/Basic.lean","commit":"","full_name":"LieModule.mem_weightSpaceOf","start":[162,0],"end":[164,22],"file_path":"Mathlib/Algebra/Lie/Weights/Basic.lean","tactics":[{"state_before":"K : Type u_1\nR : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\ninst✝⁴ : LieAlgebra.IsNilpotent R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nχ : R\nx : L\nm : M\n⊢ m ∈ weightSpaceOf M χ x ↔ ∃ k, (((toEnd R L M) x - χ • 1) ^ k) m = 0","state_after":"no goals","tactic":"simp [weightSpaceOf]","premises":[{"full_name":"LieModule.weightSpaceOf","def_path":"Mathlib/Algebra/Lie/Weights/Basic.lean","def_pos":[149,4],"def_end_pos":[149,17]}]}]}
+{"url":"Mathlib/RingTheory/PowerSeries/WellKnown.lean","commit":"","full_name":"PowerSeries.mk_one_pow_eq_mk_choose_add","start":[82,0],"end":[97,23],"file_path":"Mathlib/RingTheory/PowerSeries/WellKnown.lean","tactics":[{"state_before":"S : Type u_1\ninst✝ : CommRing S\nd : ℕ\n⊢ mk 1 ^ (d + 1) = mk fun n => ↑((d + n).choose d)","state_after":"no goals","tactic":"induction d with\n  | zero => ext; simp\n  | succ d hd =>\n      ext n\n      rw [pow_add, hd, pow_one, mul_comm, coeff_mul]\n      simp_rw [coeff_mk, Pi.one_apply, one_mul]\n      norm_cast\n      rw [Finset.sum_antidiagonal_choose_add, ← Nat.choose_succ_succ, Nat.succ_eq_add_one,\n        add_right_comm]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Finset.sum_antidiagonal_choose_add","def_path":"Mathlib/Data/Nat/Choose/Sum.lean","def_pos":[217,8],"def_end_pos":[217,35]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Nat.choose_succ_succ","def_path":"Mathlib/Data/Nat/Choose/Basic.lean","def_pos":[57,8],"def_end_pos":[57,24]},{"full_name":"Nat.succ_eq_add_one","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[154,16],"def_end_pos":[154,31]},{"full_name":"Pi.one_apply","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[60,8],"def_end_pos":[60,17]},{"full_name":"PowerSeries.coeff_mk","def_path":"Mathlib/RingTheory/PowerSeries/Basic.lean","def_pos":[170,8],"def_end_pos":[170,16]},{"full_name":"PowerSeries.coeff_mul","def_path":"Mathlib/RingTheory/PowerSeries/Basic.lean","def_pos":[282,8],"def_end_pos":[282,17]},{"full_name":"add_right_comm","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[155,2],"def_end_pos":[155,13]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]},{"full_name":"one_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[409,8],"def_end_pos":[409,15]},{"full_name":"pow_add","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[598,6],"def_end_pos":[598,13]},{"full_name":"pow_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[571,6],"def_end_pos":[571,13]}]}]}
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1)","state_after":"case refine_1\np : ℕ+\nk : ℕ\nK : Type u\ninst✝² : Field K\ninst✝¹ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {2 ^ (k + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ ↑(2 ^ (k + 1))\nthis : NumberField K := numberField {2 ^ (k + 1)} ℚ K\nh : hζ.toInteger - 1 = 0\n⊢ False\n\ncase refine_2\np : ℕ+\nk : ℕ\nK : Type u\ninst✝² : Field K\ninst✝¹ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {2 ^ (k + 1)} ℚ K\nh�� : IsPrimitiveRoot ζ ↑(2 ^ (k + 1))\nthis : NumberField K := numberField {2 ^ (k + 1)} ℚ K\n⊢ Irreducible (Ideal.absNorm (Ideal.span {hζ.toInteger - 1}))","tactic":"refine Ideal.prime_of_irreducible_absNorm_span (fun h ↦ ?_) ?_","premises":[{"full_name":"Ideal.prime_of_irreducible_absNorm_span","def_path":"Mathlib/RingTheory/Ideal/Norm.lean","def_pos":[370,8],"def_end_pos":[370,41]}]},{"state_before":"case refine_2\np : ℕ+\nk : ℕ\nK : Type u\ninst✝² : Field K\ninst✝¹ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {2 ^ (k + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ ↑(2 ^ (k + 1))\nthis : NumberField K := numberField {2 ^ (k + 1)} ℚ K\n⊢ Irreducible (Ideal.absNorm (Ideal.span {hζ.toInteger - 1}))","state_after":"case refine_2\np : ℕ+\nk : ℕ\nK : Type u\ninst✝² : Field K\ninst✝¹ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {2 ^ (k + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ ↑(2 ^ (k + 1))\nthis : NumberField K := numberField {2 ^ (k + 1)} ℚ K\n⊢ Prime ((Algebra.norm ℤ) (hζ.toInteger - 1))","tactic":"rw [Nat.irreducible_iff_prime, Ideal.absNorm_span_singleton, ← Nat.prime_iff,\n    ← Int.prime_iff_natAbs_prime]","premises":[{"full_name":"Ideal.absNorm_span_singleton","def_path":"Mathlib/RingTheory/Ideal/Norm.lean","def_pos":[310,8],"def_end_pos":[310,30]},{"full_name":"Int.prime_iff_natAbs_prime","def_path":"Mathlib/RingTheory/Int/Basic.lean","def_pos":[115,8],"def_end_pos":[115,34]},{"full_name":"Nat.irreducible_iff_prime","def_path":"Mathlib/Data/Nat/Prime/Defs.lean","def_pos":[419,8],"def_end_pos":[419,29]},{"full_name":"Nat.prime_iff","def_path":"Mathlib/Data/Nat/Prime/Defs.lean","def_pos":[414,8],"def_end_pos":[414,17]}]},{"state_before":"case refine_2\np : ℕ+\nk : ℕ\nK : Type u\ninst✝² : Field K\ninst✝¹ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {2 ^ (k + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ ↑(2 ^ (k + 1))\nthis : NumberField K := numberField {2 ^ (k + 1)} ℚ K\n⊢ Prime ((Algebra.norm ℤ) (hζ.toInteger - 1))","state_after":"case refine_2.zero\np : ℕ+\nK : Type u\ninst✝² : Field K\ninst✝¹ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {2 ^ (0 + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ ↑(2 ^ (0 + 1))\nthis : NumberField K := numberField {2 ^ (0 + 1)} ℚ K\n⊢ Prime ((Algebra.norm ℤ) (hζ.toInteger - 1))\n\ncase refine_2.succ\np : ℕ+\nK : Type u\ninst✝² : Field K\ninst✝¹ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\nn✝ : ℕ\ninst✝ : IsCyclotomicExtension {2 ^ (n✝ + 1 + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ ↑(2 ^ (n✝ + 1 + 1))\nthis : NumberField K := numberField {2 ^ (n✝ + 1 + 1)} ℚ K\n⊢ Prime ((Algebra.norm ℤ) (hζ.toInteger - 1))","tactic":"cases k","premises":[]},{"state_before":"case refine_2.succ\np : ℕ+\nK : Type u\ninst✝² : Field K\ninst✝¹ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\nn✝ : ℕ\ninst✝ : IsCyclotomicExtension {2 ^ (n✝ + 1 + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ ↑(2 ^ (n✝ + 1 + 1))\nthis : NumberField K := numberField {2 ^ (n✝ + 1 + 1)} ℚ K\n⊢ Prime ((Algebra.norm ℤ) (hζ.toInteger - 1))","state_after":"case h.e'_3\np : ℕ+\nK : Type u\ninst✝² : Field K\ninst✝¹ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\nn✝ : ℕ\ninst✝ : IsCyclotomicExtension {2 ^ (n✝ + 1 + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ ↑(2 ^ (n✝ + 1 + 1))\nthis : NumberField K := numberField {2 ^ (n✝ + 1 + 1)} ℚ K\n⊢ (Algebra.norm ℤ) (hζ.toInteger - 1) = 2","tactic":"convert Int.prime_two","premises":[{"full_name":"Int.prime_two","def_path":"Mathlib/Data/Nat/Prime/Basic.lean","def_pos":[325,8],"def_end_pos":[325,17]}]},{"state_before":"case h.e'_3\np : ℕ+\nK : Type u\ninst✝² : Field K\ninst✝¹ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\nn✝ : ℕ\ninst✝ : IsCyclotomicExtension {2 ^ (n✝ + 1 + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ ↑(2 ^ (n✝ + 1 + 1))\nthis : NumberField K := numberField {2 ^ (n✝ + 1 + 1)} ℚ K\n⊢ (Algebra.norm ℤ) (hζ.toInteger - 1) = 2","state_after":"case h.e'_3.a\np : ℕ+\nK : Type u\ninst✝² : Field K\ninst✝¹ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\nn✝ : ℕ\ninst✝ : IsCyclotomicExtension {2 ^ (n✝ + 1 + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ ↑(2 ^ (n✝ + 1 + 1))\nthis : NumberField K := numberField {2 ^ (n✝ + 1 + 1)} ℚ K\n⊢ (algebraMap ℤ ℚ) ((Algebra.norm ℤ) (hζ.toInteger - 1)) = (algebraMap ℤ ℚ) 2","tactic":"apply RingHom.injective_int (algebraMap ℤ ℚ)","premises":[{"full_name":"Int","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean","def_pos":[40,10],"def_end_pos":[40,13]},{"full_name":"Rat","def_path":".lake/packages/batteries/Batteries/Data/Rat/Basic.lean","def_pos":[17,10],"def_end_pos":[17,13]},{"full_name":"RingHom.injective_int","def_path":"Mathlib/Data/Int/CharZero.lean","def_pos":[36,8],"def_end_pos":[36,29]},{"full_name":"algebraMap","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[108,4],"def_end_pos":[108,14]}]},{"state_before":"case h.e'_3.a\np : ℕ+\nK : Type u\ninst✝² : Field K\ninst✝¹ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\nn✝ : ℕ\ninst✝ : IsCyclotomicExtension {2 ^ (n✝ + 1 + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ ↑(2 ^ (n✝ + 1 + 1))\nthis : NumberField K := numberField {2 ^ (n✝ + 1 + 1)} ℚ K\n⊢ (algebraMap ℤ ℚ) ((Algebra.norm ℤ) (hζ.toInteger - 1)) = (algebraMap ℤ ℚ) 2","state_after":"case h.e'_3.a\np : ℕ+\nK : Type u\ninst✝² : Field K\ninst✝¹ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\nn✝ : ℕ\ninst✝ : IsCyclotomicExtension {2 ^ (n✝ + 1 + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ ↑(2 ^ (n✝ + 1 + 1))\nthis : NumberField K := numberField {2 ^ (n✝ + 1 + 1)} ℚ K\n⊢ (Algebra.norm ℚ) ((algebraMap (𝓞 K) K) (hζ.toInteger - 1)) = (algebraMap ℤ ℚ) 2","tactic":"rw [← Algebra.norm_localization (Sₘ := K) ℤ (nonZeroDivisors ℤ)]","premises":[{"full_name":"Algebra.norm_localization","def_path":"Mathlib/RingTheory/Localization/NormTrace.lean","def_pos":[59,8],"def_end_pos":[59,33]},{"full_name":"Int","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean","def_pos":[40,10],"def_end_pos":[40,13]},{"full_name":"nonZeroDivisors","def_path":"Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean","def_pos":[84,4],"def_end_pos":[84,19]}]},{"state_before":"case h.e'_3.a\np : ℕ+\nK : Type u\ninst✝² : Field K\ninst✝¹ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\nn✝ : ℕ\ninst✝ : IsCyclotomicExtension {2 ^ (n✝ + 1 + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ ↑(2 ^ (n✝ + 1 + 1))\nthis : NumberField K := numberField {2 ^ (n✝ + 1 + 1)} ℚ K\n⊢ (Algebra.norm ℚ) ((algebraMap (𝓞 K) K) (hζ.toInteger - 1)) = (algebraMap ℤ ℚ) 2","state_after":"case h.e'_3.a\np : ℕ+\nK : Type u\ninst✝² : Field K\ninst✝¹ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\nn✝ : ℕ\ninst✝ : IsCyclotomicExtension {2 ^ (n✝ + 1 + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ ↑(2 ^ (n✝ + 1 + 1))\nthis : NumberField K := numberField {2 ^ (n✝ + 1 + 1)} ℚ K\n⊢ (Algebra.norm ℚ) ((algebraMap (𝓞 K) K) (hζ.toInteger - 1)) = (Int.castRingHom ℚ) 2","tactic":"simp only [PNat.pow_coe, id.map_eq_id, RingHomCompTriple.comp_eq, RingHom.coe_coe,\n    Subalgebra.coe_val, algebraMap_int_eq, map_natCast]","premises":[{"full_name":"Algebra.id.map_eq_id","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[361,8],"def_end_pos":[361,17]},{"full_name":"PNat.pow_coe","def_path":"Mathlib/Data/PNat/Basic.lean","def_pos":[226,8],"def_end_pos":[226,15]},{"full_name":"RingHom.coe_coe","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[392,8],"def_end_pos":[392,15]},{"full_name":"RingHomCompTriple.comp_eq","def_path":"Mathlib/Algebra/Ring/CompTypeclasses.lean","def_pos":[62,2],"def_end_pos":[62,9]},{"full_name":"Subalgebra.coe_val","def_path":"Mathlib/Algebra/Algebra/Subalgebra/Basic.lean","def_pos":[339,8],"def_end_pos":[339,15]},{"full_name":"algebraMap_int_eq","def_path":"Mathlib/Algebra/Algebra/Basic.lean","def_pos":[254,8],"def_end_pos":[254,25]},{"full_name":"map_natCast","def_path":"Mathlib/Data/Nat/Cast/Basic.lean","def_pos":[163,8],"def_end_pos":[163,19]}]},{"state_before":"case h.e'_3.a\np : ℕ+\nK : Type u\ninst✝² : Field K\ninst✝¹ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\nn✝ : ℕ\ninst✝ : IsCyclotomicExtension {2 ^ (n✝ + 1 + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ ↑(2 ^ (n✝ + 1 + 1))\nthis : NumberField K := numberField {2 ^ (n✝ + 1 + 1)} ℚ K\n⊢ (Algebra.norm ℚ) ((algebraMap (𝓞 K) K) (hζ.toInteger - 1)) = (Int.castRingHom ℚ) 2","state_after":"no goals","tactic":"exact hζ.norm_sub_one_two Nat.AtLeastTwo.prop (cyclotomic.irreducible_rat (by simp))","premises":[{"full_name":"IsPrimitiveRoot.norm_sub_one_two","def_path":"Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean","def_pos":[498,8],"def_end_pos":[498,24]},{"full_name":"Nat.AtLeastTwo.prop","def_path":"Mathlib/Data/Nat/Cast/Defs.lean","def_pos":[37,2],"def_end_pos":[37,6]},{"full_name":"Polynomial.cyclotomic.irreducible_rat","def_path":"Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean","def_pos":[190,8],"def_end_pos":[190,34]}]}]}
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p.2","premises":[{"full_name":"Continuous","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[141,10],"def_end_pos":[141,20]},{"full_name":"Prod","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[479,10],"def_end_pos":[479,14]},{"full_name":"Prod.fst","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[483,2],"def_end_pos":[483,5]},{"full_name":"Prod.snd","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[485,2],"def_end_pos":[485,5]}]},{"state_before":"X✝ : Type u\nY✝ : Type v\nZ✝ : Type u_1\nW : Type u_2\nε : Type u_3\nζ : Type u_4\ninst✝⁵ : TopologicalSpace X✝\ninst✝⁴ : TopologicalSpace Y✝\ninst✝³ : TopologicalSpace Z✝\ninst✝² : TopologicalSpace W\ninst✝¹ : TopologicalSpace ε\ninst✝ : TopologicalSpace ζ\nX : Type u_5\nY : Type u_6\nZ : Type u_7\nf : X → Y → Z\nta1 ta2 : TopologicalSpace X\ntb1 tb2 : TopologicalSpace Y\ntc1 : TopologicalSpace Z\nh : Continuous fun p => f p.1 p.2\n⊢ Continuous fun p => f p.1 p.2","state_after":"X✝ : Type u\nY✝ : Type v\nZ✝ : Type u_1\nW : Type u_2\nε : Type u_3\nζ : Type u_4\ninst✝⁵ : TopologicalSpace X✝\ninst✝⁴ : TopologicalSpace Y✝\ninst✝³ : TopologicalSpace Z✝\ninst✝² : TopologicalSpace W\ninst✝¹ : TopologicalSpace ε\ninst✝ : TopologicalSpace ζ\nX : Type u_5\nY : Type u_6\nZ : Type u_7\nf : X → Y → Z\nta1 ta2 : TopologicalSpace X\ntb1 tb2 : TopologicalSpace Y\ntc1 : TopologicalSpace Z\nh : Continuous fun p => f p.1 p.2\nha : Continuous id\n⊢ Continuous fun p => f p.1 p.2","tactic":"have ha := @continuous_inf_dom_right _ _ id ta1 ta2 ta2 (@continuous_id _ (id _))","premises":[{"full_name":"continuous_id","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1382,8],"def_end_pos":[1382,21]},{"full_name":"continuous_inf_dom_right","def_path":"Mathlib/Topology/Order.lean","def_pos":[684,8],"def_end_pos":[684,32]},{"full_name":"id","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[33,14],"def_end_pos":[33,16]}]},{"state_before":"X✝ : Type u\nY✝ : Type v\nZ✝ : Type u_1\nW : Type u_2\nε : Type u_3\nζ : Type u_4\ninst✝⁵ : TopologicalSpace X✝\ninst✝⁴ : TopologicalSpace Y✝\ninst✝³ : TopologicalSpace Z✝\ninst✝² : TopologicalSpace W\ninst✝¹ : TopologicalSpace ε\ninst✝ : TopologicalSpace ζ\nX : Type u_5\nY : Type u_6\nZ : Type u_7\nf : X → Y → Z\nta1 ta2 : TopologicalSpace X\ntb1 tb2 : TopologicalSpace Y\ntc1 : TopologicalSpace Z\nh : Continuous fun p => f p.1 p.2\nha : Continuous id\n⊢ Continuous fun p => f p.1 p.2","state_after":"X✝ : Type u\nY✝ : Type v\nZ✝ : Type u_1\nW : Type u_2\nε : Type u_3\nζ : Type u_4\ninst✝⁵ : TopologicalSpace X✝\ninst✝⁴ : TopologicalSpace Y✝\ninst✝³ : TopologicalSpace Z✝\ninst✝² : TopologicalSpace W\ninst✝¹ : TopologicalSpace ε\ninst✝ : TopologicalSpace ζ\nX : Type u_5\nY : Type u_6\nZ : Type u_7\nf : X → Y → Z\nta1 ta2 : TopologicalSpace X\ntb1 tb2 : TopologicalSpace Y\ntc1 : TopologicalSpace Z\nh : Continuous fun p => f p.1 p.2\nha : Continuous id\nhb : Continuous id\n⊢ Continuous fun p => f p.1 p.2","tactic":"have hb := @continuous_inf_dom_right _ _ id tb1 tb2 tb2 (@continuous_id _ (id _))","premises":[{"full_name":"continuous_id","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1382,8],"def_end_pos":[1382,21]},{"full_name":"continuous_inf_dom_right","def_path":"Mathlib/Topology/Order.lean","def_pos":[684,8],"def_end_pos":[684,32]},{"full_name":"id","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[33,14],"def_end_pos":[33,16]}]},{"state_before":"X✝ : Type u\nY✝ : Type v\nZ✝ : Type u_1\nW : Type u_2\nε : Type u_3\nζ : Type u_4\ninst✝⁵ : TopologicalSpace X✝\ninst✝⁴ : TopologicalSpace Y✝\ninst✝³ : TopologicalSpace Z✝\ninst✝² : TopologicalSpace W\ninst✝¹ : TopologicalSpace ε\ninst✝ : TopologicalSpace ζ\nX : Type u_5\nY : Type u_6\nZ : Type u_7\nf : X → Y → Z\nta1 ta2 : TopologicalSpace X\ntb1 tb2 : TopologicalSpace Y\ntc1 : TopologicalSpace Z\nh : Continuous fun p => f p.1 p.2\nha : Continuous id\nhb : Continuous id\n⊢ Continuous fun p => f p.1 p.2","state_after":"X✝ : Type u\nY✝ : Type v\nZ✝ : Type u_1\nW : Type u_2\nε : Type u_3\nζ : Type u_4\ninst✝⁵ : TopologicalSpace X✝\ninst✝⁴ : TopologicalSpace Y✝\ninst✝³ : TopologicalSpace Z✝\ninst✝² : TopologicalSpace W\ninst✝¹ : TopologicalSpace ε\ninst✝ : TopologicalSpace ζ\nX : Type u_5\nY : Type u_6\nZ : Type u_7\nf : X → Y → Z\nta1 ta2 : TopologicalSpace X\ntb1 tb2 : TopologicalSpace Y\ntc1 : TopologicalSpace Z\nh : Continuous fun p => f p.1 p.2\nha : Continuous id\nhb : Continuous id\nh_continuous_id : Continuous fun p => (id p.1, id p.2)\n⊢ Continuous fun p => f p.1 p.2","tactic":"have h_continuous_id := @Continuous.prod_map _ _ _ _ ta2 tb2 (ta1 ⊓ ta2) (tb1 ⊓ tb2) _ _ ha hb","premises":[{"full_name":"Continuous.prod_map","def_path":"Mathlib/Topology/Constructions.lean","def_pos":[403,8],"def_end_pos":[403,27]},{"full_name":"Inf.inf","def_path":"Mathlib/Order/Notation.lean","def_pos":[53,2],"def_end_pos":[53,5]}]},{"state_before":"X✝ : Type u\nY✝ : Type v\nZ✝ : Type u_1\nW : Type u_2\nε : Type u_3\nζ : Type u_4\ninst✝⁵ : TopologicalSpace X✝\ninst✝⁴ : TopologicalSpace Y✝\ninst✝³ : TopologicalSpace Z✝\ninst✝² : TopologicalSpace W\ninst✝¹ : TopologicalSpace ε\ninst✝ : TopologicalSpace ζ\nX : Type u_5\nY : Type u_6\nZ : Type u_7\nf : X → Y → Z\nta1 ta2 : TopologicalSpace X\ntb1 tb2 : TopologicalSpace Y\ntc1 : TopologicalSpace Z\nh : Continuous fun p => f p.1 p.2\nha : Continuous id\nhb : Continuous id\nh_continuous_id : Continuous fun p => (id p.1, id p.2)\n⊢ Continuous fun p => f p.1 p.2","state_after":"no goals","tactic":"exact @Continuous.comp _ _ _ (id _) (id _) _ _ _ h h_continuous_id","premises":[{"full_name":"Continuous.comp","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1389,8],"def_end_pos":[1389,23]},{"full_name":"id","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[33,14],"def_end_pos":[33,16]}]}]}
+{"url":"Mathlib/Algebra/Group/Equiv/Basic.lean","commit":"","full_name":"MulEquiv.comp_right_injective","start":[454,0],"end":[456,94],"file_path":"Mathlib/Algebra/Group/Equiv/Basic.lean","tactics":[{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nA : Type u_4\nB : Type u_5\nM : Type u_6\nN : Type u_7\nP : Type u_8\nQ : Type u_9\nG : Type u_10\nH : Type u_11\ninst✝³ : EquivLike F α β\ninst✝² : MulOneClass M\ninst✝¹ : MulOneClass N\ninst✝ : MulOneClass P\ne : M ≃* N\nf : P →* M\n⊢ (↑e.symm).comp ((↑e).comp f) = f","state_after":"no goals","tactic":"simp [← MonoidHom.comp_assoc]","premises":[{"full_name":"MonoidHom.comp_assoc","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[732,8],"def_end_pos":[732,28]}]}]}
+{"url":"Mathlib/GroupTheory/FreeGroup/Basic.lean","commit":"","full_name":"FreeGroup.Red.Step.append_left_iff","start":[157,0],"end":[160,66],"file_path":"Mathlib/GroupTheory/FreeGroup/Basic.lean","tactics":[{"state_before":"α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\n⊢ Step ([] ++ L₁) ([] ++ L₂) ↔ Step L₁ L₂","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\np : α × Bool\nl : List (α × Bool)\n⊢ Step (p :: l ++ L₁) (p :: l ++ L₂) ↔ Step L₁ L₂","state_after":"no goals","tactic":"simp [Step.append_left_iff l, Step.cons_cons_iff]","premises":[{"full_name":"FreeGroup.Red.Step.cons_cons_iff","def_path":"Mathlib/GroupTheory/FreeGroup/Basic.lean","def_pos":[154,8],"def_end_pos":[154,26]}]}]}
+{"url":".lake/packages/batteries/Batteries/Data/Array/Lemmas.lean","commit":"","full_name":"Array.size_filter_le","start":[88,0],"end":[91,29],"file_path":".lake/packages/batteries/Batteries/Data/Array/Lemmas.lean","tactics":[{"state_before":"α : Type u_1\np : α → Bool\nl : Array α\n⊢ (filter p l 0).size ≤ l.size","state_after":"α : Type u_1\np : α → Bool\nl : Array α\n⊢ (List.filter p l.data).length ≤ l.data.length","tactic":"simp only [← data_length, filter_data]","premises":[{"full_name":"Array.data_length","def_path":".lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean","def_pos":[29,16],"def_end_pos":[29,27]},{"full_name":"Array.filter_data","def_path":".lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean","def_pos":[719,16],"def_end_pos":[719,27]}]},{"state_before":"α : Type u_1\np : α → Bool\nl : Array α\n⊢ (List.filter p l.data).length ≤ l.data.length","state_after":"no goals","tactic":"apply List.length_filter_le","premises":[{"full_name":"List.length_filter_le","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean","def_pos":[861,8],"def_end_pos":[861,24]}]}]}
+{"url":"Mathlib/Data/Finsupp/Defs.lean","commit":"","full_name":"Finsupp.support_onFinset","start":[649,0],"end":[652,25],"file_path":"Mathlib/Data/Finsupp/Defs.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst✝¹ : Zero M\ninst✝ : DecidableEq M\ns : Finset α\nf : α → M\nhf : ∀ (a : α), f a ≠ 0 → a ∈ s\n⊢ (onFinset s f hf).support = filter (fun a => f a ≠ 0) s","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst✝¹ : Zero M\ninst✝ : DecidableEq M\ns : Finset α\nf : α → M\nhf : ∀ (a : α), f a ≠ 0 → a ∈ s\n⊢ filter (fun x => ¬f x = 0) s = filter (fun a => ¬f a = 0) s","tactic":"dsimp [onFinset]","premises":[{"full_name":"Finsupp.onFinset","def_path":"Mathlib/Data/Finsupp/Defs.lean","def_pos":[626,4],"def_end_pos":[626,12]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst✝¹ : Zero M\ninst✝ : DecidableEq M\ns : Finset α\nf : α → M\nhf : ∀ (a : α), f a ≠ 0 → a ∈ s\n⊢ filter (fun x => ¬f x = 0) s = filter (fun a => ¬f a = 0) s","state_after":"no goals","tactic":"congr","premises":[]}]}
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+{"url":"Mathlib/Analysis/Analytic/Inverse.lean","commit":"","full_name":"FormalMultilinearSeries.leftInv_removeZero","start":[72,0],"end":[87,18],"file_path":"Mathlib/Analysis/Analytic/Inverse.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\ni : E ≃L[𝕜] F\n⊢ p.removeZero.leftInv i = p.leftInv i","state_after":"case h\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\ni : E ≃L[𝕜] F\nn : ℕ\n⊢ p.removeZero.leftInv i n = p.leftInv i n","tactic":"ext1 n","premises":[]},{"state_before":"case h\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\ni : E ≃L[𝕜] F\nn : ℕ\n⊢ p.removeZero.leftInv i n = p.leftInv i n","state_after":"case h.H\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\ni : E ≃L[𝕜] F\nn : ℕ\nIH : ∀ m < n, p.removeZero.leftInv i m = p.leftInv i m\n⊢ p.removeZero.leftInv i n = p.leftInv i n","tactic":"induction' n using Nat.strongRec' with n IH","premises":[{"full_name":"Nat.strongRec'","def_path":"Mathlib/Data/Nat/Defs.lean","def_pos":[811,14],"def_end_pos":[811,24]}]},{"state_before":"case h.H\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\ni : E ≃L[𝕜] F\nn : ℕ\nIH : ∀ m < n, p.removeZero.leftInv i m = p.leftInv i m\n⊢ p.removeZero.leftInv i n = p.leftInv i n","state_after":"no goals","tactic":"match n with\n  | 0 => simp -- if one replaces `simp` with `refl`, the proof times out in the kernel.\n  | 1 => simp -- TODO: why?\n  | n + 2 =>\n    simp only [leftInv, neg_inj]\n    refine Finset.sum_congr rfl fun c cuniv => ?_\n    rcases c with ⟨c, hc⟩\n    ext v\n    dsimp\n    simp [IH _ hc]","premises":[{"full_name":"Finset.sum_congr","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[380,2],"def_end_pos":[380,13]},{"full_name":"FormalMultilinearSeries.leftInv","def_path":"Mathlib/Analysis/Analytic/Inverse.lean","def_pos":[55,18],"def_end_pos":[55,25]},{"full_name":"neg_inj","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[294,2],"def_end_pos":[294,13]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
+{"url":"Mathlib/Topology/Algebra/Monoid.lean","commit":"","full_name":"tendsto_list_sum","start":[465,0],"end":[475,73],"file_path":"Mathlib/Topology/Algebra/Monoid.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nM : Type u_3\nN : Type u_4\nX : Type u_5\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : Monoid M\ninst✝ : ContinuousMul M\nf : ι → α → M\nx : Filter α\na : ι → M\nx✝ : ∀ i ∈ [], Tendsto (f i) x (𝓝 (a i))\n⊢ Tendsto (fun b => (List.map (fun c => f c b) []).prod) x (𝓝 (List.map a []).prod)","state_after":"no goals","tactic":"simp [tendsto_const_nhds]","premises":[{"full_name":"tendsto_const_nhds","def_path":"Mathlib/Topology/Basic.lean","def_pos":[844,8],"def_end_pos":[844,26]}]},{"state_before":"ι : Type u_1\nα : Type u_2\nM : Type u_3\nN : Type u_4\nX : Type u_5\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : Monoid M\ninst✝ : ContinuousMul M\nf✝ : ι → α → M\nx : Filter α\na : ι → M\nf : ι\nl : List ι\nh : ∀ i ∈ f :: l, Tendsto (f✝ i) x (𝓝 (a i))\n⊢ Tendsto (fun b => (List.map (fun c => f✝ c b) (f :: l)).prod) x (𝓝 (List.map a (f :: l)).prod)","state_after":"ι : Type u_1\nα : Type u_2\nM : Type u_3\nN : Type u_4\nX : Type u_5\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : Monoid M\ninst✝ : ContinuousMul M\nf✝ : ι → α → M\nx : Filter α\na : ι → M\nf : ι\nl : List ι\nh : ∀ i ∈ f :: l, Tendsto (f✝ i) x (𝓝 (a i))\n⊢ Tendsto (fun b => f✝ f b * (List.map (fun c => f✝ c b) l).prod) x (𝓝 (a f * (List.map a l).prod))","tactic":"simp only [List.map_cons, List.prod_cons]","premises":[{"full_name":"List.map_cons","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[366,16],"def_end_pos":[366,24]},{"full_name":"List.prod_cons","def_path":"Mathlib/Algebra/BigOperators/Group/List.lean","def_pos":[85,8],"def_end_pos":[85,17]}]},{"state_before":"ι : Type u_1\nα : Type u_2\nM : Type u_3\nN : Type u_4\nX : Type u_5\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : Monoid M\ninst✝ : ContinuousMul M\nf✝ : ι → α → M\nx : Filter α\na : ι → M\nf : ι\nl : List ι\nh : ∀ i ∈ f :: l, Tendsto (f✝ i) x (𝓝 (a i))\n⊢ Tendsto (fun b => f✝ f b * (List.map (fun c => f✝ c b) l).prod) x (𝓝 (a f * (List.map a l).prod))","state_after":"no goals","tactic":"exact\n      (h f (List.mem_cons_self _ _)).mul\n        (tendsto_list_prod l fun c hc => h c (List.mem_cons_of_mem _ hc))","premises":[{"full_name":"Filter.Tendsto.mul","def_path":"Mathlib/Topology/Algebra/Monoid.lean","def_pos":[116,8],"def_end_pos":[116,26]},{"full_name":"List.mem_cons_of_mem","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean","def_pos":[291,8],"def_end_pos":[291,23]},{"full_name":"List.mem_cons_self","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean","def_pos":[289,8],"def_end_pos":[289,21]}]}]}
+{"url":"Mathlib/FieldTheory/Adjoin.lean","commit":"","full_name":"IntermediateField.lift_top","start":[414,0],"end":[416,83],"file_path":"Mathlib/FieldTheory/Adjoin.lean","tactics":[{"state_before":"F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : Set E\nK : IntermediateField F E\n⊢ lift ⊤ = K","state_after":"no goals","tactic":"rw [lift, ← AlgHom.fieldRange_eq_map, fieldRange_val]","premises":[{"full_name":"AlgHom.fieldRange_eq_map","def_path":"Mathlib/FieldTheory/Adjoin.lean","def_pos":[256,8],"def_end_pos":[256,39]},{"full_name":"IntermediateField.fieldRange_val","def_path":"Mathlib/FieldTheory/IntermediateField.lean","def_pos":[459,8],"def_end_pos":[459,22]},{"full_name":"IntermediateField.lift","def_path":"Mathlib/FieldTheory/IntermediateField.lean","def_pos":[547,4],"def_end_pos":[547,8]}]}]}
+{"url":"Mathlib/Combinatorics/Enumerative/Composition.lean","commit":"","full_name":"Composition.single_embedding","start":[504,0],"end":[508,6],"file_path":"Mathlib/Combinatorics/Enumerative/Composition.lean","tactics":[{"state_before":"n✝ : ℕ\nc : Composition n✝\nn : ℕ\nh : 0 < n\ni : Fin n\n⊢ ((single n h).embedding 0) i = i","state_after":"case h\nn✝ : ℕ\nc : Composition n✝\nn : ℕ\nh : 0 < n\ni : Fin n\n⊢ ↑(((single n h).embedding 0) i) = ↑i","tactic":"ext","premises":[]},{"state_before":"case h\nn✝ : ℕ\nc : Composition n✝\nn : ℕ\nh : 0 < n\ni : Fin n\n⊢ ↑(((single n h).embedding 0) i) = ↑i","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean","commit":"","full_name":"Orientation.two_zsmul_oangle_neg_right","start":[197,0],"end":[205,37],"file_path":"Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean","tactics":[{"state_before":"V : Type u_1\nV' : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : NormedAddCommGroup V'\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : InnerProductSpace ℝ V'\ninst✝¹ : Fact (finrank ℝ V = 2)\ninst✝ : Fact (finrank ℝ V' = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\n⊢ 2 • o.oangle x (-y) = 2 • o.oangle x y","state_after":"case pos\nV : Type u_1\nV' : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : NormedAddCommGroup V'\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : InnerProductSpace ℝ V'\ninst✝¹ : Fact (finrank ℝ V = 2)\ninst✝ : Fact (finrank ℝ V' = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nhx : x = 0\n⊢ 2 • o.oangle x (-y) = 2 • o.oangle x y\n\ncase neg\nV : Type u_1\nV' : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : NormedAddCommGroup V'\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : InnerProductSpace ℝ V'\ninst✝¹ : Fact (finrank ℝ V = 2)\ninst✝ : Fact (finrank ℝ V' = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nhx : ¬x = 0\n⊢ 2 • o.oangle x (-y) = 2 • o.oangle x y","tactic":"by_cases hx : x = 0","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
+{"url":"Mathlib/RingTheory/Valuation/ValuationSubring.lean","commit":"","full_name":"ValuationSubring.primeSpectrumOrderEquiv_symm_apply_asIdeal_carrier","start":[337,0],"end":[348,52],"file_path":"Mathlib/RingTheory/Valuation/ValuationSubring.lean","tactics":[{"state_before":"K : Type u\ninst✝ : Field K\nA : ValuationSubring K\na✝ b✝ : (PrimeSpectrum ↥A)ᵒᵈ\nh : __src✝ a✝ ≤ __src✝ b✝\n⊢ a✝ ≤ b✝","state_after":"K : Type u\ninst✝ : Field K\nA : ValuationSubring K\na✝ b✝ : (PrimeSpectrum ↥A)ᵒᵈ\nh : A.primeSpectrumEquiv a✝ ≤ A.primeSpectrumEquiv b✝\n⊢ a✝ ≤ b✝","tactic":"dsimp at h","premises":[]},{"state_before":"K : Type u\ninst✝ : Field K\nA : ValuationSubring K\na✝ b✝ : (PrimeSpectrum ↥A)ᵒᵈ\nh : A.primeSpectrumEquiv a✝ ≤ A.primeSpectrumEquiv b✝\n⊢ a✝ ≤ b✝","state_after":"case refine_3\nK : Type u\ninst✝ : Field K\nA : ValuationSubring K\na✝ b✝ : (PrimeSpectrum ↥A)ᵒᵈ\nh : A.primeSpectrumEquiv a✝ ≤ A.primeSpectrumEquiv b✝\nthis : A.idealOfLE ↑(A.primeSpectrumEquiv b✝) ?refine_2 ≤ A.idealOfLE ↑(A.primeSpectrumEquiv a✝) ?refine_1\n⊢ a✝ ≤ b✝\n\ncase refine_1\nK : Type u\ninst✝ : Field K\nA : ValuationSubring K\na✝ b✝ : (PrimeSpectrum ↥A)ᵒᵈ\nh : A.primeSpectrumEquiv a✝ ≤ A.primeSpectrumEquiv b✝\n⊢ A ≤ ↑(A.primeSpectrumEquiv a✝)\n\ncase refine_2\nK : Type u\ninst✝ : Field K\nA : ValuationSubring K\na✝ b✝ : (PrimeSpectrum ↥A)ᵒᵈ\nh : A.primeSpectrumEquiv a✝ ≤ A.primeSpectrumEquiv b✝\n⊢ A ≤ ↑(A.primeSpectrumEquiv b✝)","tactic":"have := idealOfLE_le_of_le A _ _ ?_ ?_ h","premises":[{"full_name":"ValuationSubring.idealOfLE_le_of_le","def_path":"Mathlib/RingTheory/Valuation/ValuationSubring.lean","def_pos":[321,8],"def_end_pos":[321,26]}]},{"state_before":"case refine_3\nK : Type u\ninst✝ : Field K\nA : ValuationSubring K\na✝ b✝ : (PrimeSpectrum ↥A)ᵒᵈ\nh : A.primeSpectrumEquiv a✝ ≤ A.primeSpectrumEquiv b✝\nthis : A.idealOfLE ↑(A.primeSpectrumEquiv b✝) ?refine_2 ≤ A.idealOfLE ↑(A.primeSpectrumEquiv a✝) ?refine_1\n⊢ a✝ ≤ b✝\n\ncase refine_1\nK : Type u\ninst✝ : Field K\nA : ValuationSubring K\na✝ b✝ : (PrimeSpectrum ↥A)ᵒᵈ\nh : A.primeSpectrumEquiv a✝ ≤ A.primeSpectrumEquiv b✝\n⊢ A ≤ ↑(A.primeSpectrumEquiv a✝)\n\ncase refine_2\nK : Type u\ninst✝ : Field K\nA : ValuationSubring K\na✝ b✝ : (PrimeSpectrum ↥A)ᵒᵈ\nh : A.primeSpectrumEquiv a✝ ≤ A.primeSpectrumEquiv b✝\n⊢ A ≤ ↑(A.primeSpectrumEquiv b✝)","state_after":"case refine_3\nK : Type u\ninst✝ : Field K\nA : ValuationSubring K\na✝ b✝ : (PrimeSpectrum ↥A)ᵒᵈ\nh : A.primeSpectrumEquiv a✝ ≤ A.primeSpectrumEquiv b✝\nthis : b✝.asIdeal ≤ a✝.asIdeal\n⊢ a✝ ≤ b✝\n\ncase refine_1\nK : Type u\ninst✝ : Field K\nA : ValuationSubring K\na✝ b✝ : (PrimeSpectrum ↥A)ᵒᵈ\nh : A.primeSpectrumEquiv a✝ ≤ A.primeSpectrumEquiv b✝\n⊢ A ≤ ↑(A.primeSpectrumEquiv a✝)\n\ncase refine_2\nK : Type u\ninst✝ : Field K\nA : ValuationSubring K\na✝ b✝ : (PrimeSpectrum ↥A)ᵒᵈ\nh : A.primeSpectrumEquiv a✝ ≤ A.primeSpectrumEquiv b✝\n⊢ A ≤ ↑(A.primeSpectrumEquiv b✝)","tactic":"iterate 2 erw [idealOfLE_ofPrime] at this","premises":[{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"ValuationSubring.idealOfLE_ofPrime","def_path":"Mathlib/RingTheory/Valuation/ValuationSubring.lean","def_pos":[290,8],"def_end_pos":[290,25]}]},{"state_before":"case refine_1\nK : Type u\ninst✝ : Field K\nA : ValuationSubring K\na✝ b✝ : (PrimeSpectrum ↥A)ᵒᵈ\nh : A.primeSpectrumEquiv a✝ ≤ A.primeSpectrumEquiv b✝\n⊢ A ≤ ↑(A.primeSpectrumEquiv a✝)\n\ncase refine_2\nK : Type u\ninst✝ : Field K\nA : ValuationSubring K\na✝ b✝ : (PrimeSpectrum ↥A)ᵒᵈ\nh : A.primeSpectrumEquiv a✝ ≤ A.primeSpectrumEquiv b✝\n⊢ A ≤ ↑(A.primeSpectrumEquiv b✝)","state_after":"no goals","tactic":"all_goals exact le_ofPrime A (PrimeSpectrum.asIdeal _)","premises":[{"full_name":"PrimeSpectrum.asIdeal","def_path":"Mathlib/RingTheory/PrimeSpectrum.lean","def_pos":[61,2],"def_end_pos":[61,9]},{"full_name":"ValuationSubring.le_ofPrime","def_path":"Mathlib/RingTheory/Valuation/ValuationSubring.lean","def_pos":[281,8],"def_end_pos":[281,18]}]},{"state_before":"K : Type u\ninst✝ : Field K\nA : ValuationSubring K\na✝ b✝ : (PrimeSpectrum ↥A)ᵒᵈ\nh : a✝ ≤ b✝\n⊢ __src✝ a✝ ≤ __src✝ b✝","state_after":"case h\nK : Type u\ninst✝ : Field K\nA : ValuationSubring K\na✝ b✝ : (PrimeSpectrum ↥A)ᵒᵈ\nh : a✝ ≤ b✝\n⊢ b✝.asIdeal ≤ a✝.asIdeal","tactic":"apply ofPrime_le_of_le","premises":[{"full_name":"ValuationSubring.ofPrime_le_of_le","def_path":"Mathlib/RingTheory/Valuation/ValuationSubring.lean","def_pos":[318,8],"def_end_pos":[318,24]}]},{"state_before":"case h\nK : Type u\ninst✝ : Field K\nA : ValuationSubring K\na✝ b✝ : (PrimeSpectrum ↥A)ᵒᵈ\nh : a✝ ≤ b✝\n⊢ b✝.asIdeal ≤ a✝.asIdeal","state_after":"no goals","tactic":"exact h","premises":[]}]}
+{"url":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","commit":"","full_name":"WeierstrassCurve.Jacobian.dblU_smul","start":[567,0],"end":[569,7],"file_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","tactics":[{"state_before":"R : Type u\ninst✝¹ : CommRing R\nW' : Jacobian R\nF : Type v\ninst✝ : Field F\nW : Jacobian F\nP : Fin 3 → R\nu : R\n⊢ W'.dblU (u • P) = u ^ 4 * W'.dblU P","state_after":"R : Type u\ninst✝¹ : CommRing R\nW' : Jacobian R\nF : Type v\ninst✝ : Field F\nW : Jacobian F\nP : Fin 3 → R\nu : R\n⊢ W'.a₁ * (u ^ 3 * P y) * (u * P z) -\n      (3 * (u ^ 2 * P x) ^ 2 + 2 * W'.a₂ * (u ^ 2 * P x) * (u * P z) ^ 2 + W'.a₄ * (u * P z) ^ 4) =\n    u ^ 4 * (W'.a₁ * P y * P z - (3 * P x ^ 2 + 2 * W'.a₂ * P x * P z ^ 2 + W'.a₄ * P z ^ 4))","tactic":"simp only [dblU_eq, smul_fin3_ext]","premises":[{"full_name":"WeierstrassCurve.Jacobian.dblU_eq","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","def_pos":[563,6],"def_end_pos":[563,13]},{"full_name":"WeierstrassCurve.Jacobian.smul_fin3_ext","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","def_pos":[137,6],"def_end_pos":[137,19]}]},{"state_before":"R : Type u\ninst✝¹ : CommRing R\nW' : Jacobian R\nF : Type v\ninst✝ : Field F\nW : Jacobian F\nP : Fin 3 → R\nu : R\n⊢ W'.a₁ * (u ^ 3 * P y) * (u * P z) -\n      (3 * (u ^ 2 * P x) ^ 2 + 2 * W'.a₂ * (u ^ 2 * P x) * (u * P z) ^ 2 + W'.a₄ * (u * P z) ^ 4) =\n    u ^ 4 * (W'.a₁ * P y * P z - (3 * P x ^ 2 + 2 * W'.a₂ * P x * P z ^ 2 + W'.a₄ * P z ^ 4))","state_after":"no goals","tactic":"ring1","premises":[]}]}
+{"url":"Mathlib/Data/Real/Irrational.lean","commit":"","full_name":"irrational_sqrt_natCast_iff","start":[122,0],"end":[124,21],"file_path":"Mathlib/Data/Real/Irrational.lean","tactics":[{"state_before":"n : ℕ\n⊢ Irrational √↑n ↔ ¬IsSquare n","state_after":"no goals","tactic":"rw [← Rat.isSquare_natCast_iff, ← irrational_sqrt_ratCast_iff_of_nonneg n.cast_nonneg,\n    Rat.cast_natCast]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Nat.cast_nonneg","def_path":"Mathlib/Data/Nat/Cast/Order/Ring.lean","def_pos":[29,8],"def_end_pos":[29,19]},{"full_name":"Rat.cast_natCast","def_path":"Mathlib/Data/Rat/Cast/Defs.lean","def_pos":[116,8],"def_end_pos":[116,20]},{"full_name":"Rat.isSquare_natCast_iff","def_path":"Mathlib/Data/Rat/Lemmas.lean","def_pos":[112,8],"def_end_pos":[112,28]},{"full_name":"irrational_sqrt_ratCast_iff_of_nonneg","def_path":"Mathlib/Data/Real/Irrational.lean","def_pos":[96,8],"def_end_pos":[96,45]}]}]}
+{"url":"Mathlib/Data/Matroid/Restrict.lean","commit":"","full_name":"Matroid.Indep.indep_restriction","start":[354,0],"end":[355,39],"file_path":"Mathlib/Data/Matroid/Restrict.lean","tactics":[{"state_before":"α : Type u_1\nM : Matroid α\nR I J X Y : Set α\nN : Matroid α\nhI : M.Indep I\nhNM : N ≤r M\nhIN : I ⊆ N.E\n⊢ N.Indep I","state_after":"case intro.intro\nα : Type u_1\nM : Matroid α\nR✝ I J X Y : Set α\nhI : M.Indep I\nR : Set α\nhIN : I ⊆ (M ↾ R).E\n⊢ (M ↾ R).Indep I","tactic":"obtain ⟨R, -, rfl⟩ := hNM","premises":[]},{"state_before":"case intro.intro\nα : Type u_1\nM : Matroid α\nR✝ I J X Y : Set α\nhI : M.Indep I\nR : Set α\nhIN : I ⊆ (M ↾ R).E\n⊢ (M ↾ R).Indep I","state_after":"no goals","tactic":"simpa [hI]","premises":[]}]}
+{"url":"Mathlib/Algebra/Polynomial/Eval.lean","commit":"","full_name":"Polynomial.eval₂_X","start":[63,0],"end":[64,59],"file_path":"Mathlib/Algebra/Polynomial/Eval.lean","tactics":[{"state_before":"R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\nx : S\n⊢ eval₂ f x X = x","state_after":"no goals","tactic":"simp [eval₂_eq_sum]","premises":[{"full_name":"Polynomial.eval₂_eq_sum","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[44,8],"def_end_pos":[44,20]}]}]}
+{"url":"Mathlib/RingTheory/Coprime/Basic.lean","commit":"","full_name":"IsCoprime.mul_add_right_right","start":[304,0],"end":[306,31],"file_path":"Mathlib/RingTheory/Coprime/Basic.lean","tactics":[{"state_before":"R : Type u\ninst✝ : CommRing R\nx y : R\nh : IsCoprime x y\nz : R\n⊢ IsCoprime x (z * x + y)","state_after":"R : Type u\ninst✝ : CommRing R\nx y : R\nh : IsCoprime x y\nz : R\n⊢ IsCoprime x (y + z * x)","tactic":"rw [add_comm]","premises":[{"full_name":"add_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[293,2],"def_end_pos":[293,13]}]},{"state_before":"R : Type u\ninst✝ : CommRing R\nx y : R\nh : IsCoprime x y\nz : R\n⊢ IsCoprime x (y + z * x)","state_after":"no goals","tactic":"exact h.add_mul_right_right z","premises":[{"full_name":"IsCoprime.add_mul_right_right","def_path":"Mathlib/RingTheory/Coprime/Basic.lean","def_pos":[288,8],"def_end_pos":[288,27]}]}]}
+{"url":"Mathlib/Algebra/Homology/ShortComplex/Exact.lean","commit":"","full_name":"CategoryTheory.ShortComplex.Splitting.ext_r","start":[518,0],"end":[526,5],"file_path":"Mathlib/Algebra/Homology/ShortComplex/Exact.lean","tactics":[{"state_before":"C : Type u_1\nD : Type u_2\ninst✝³ : Category.{u_3, u_1} C\ninst✝² : Category.{?u.82092, u_2} D\ninst✝¹ : Preadditive C\ninst✝ : Preadditive D\nS : ShortComplex C\ns s' : S.Splitting\nh : s.r = s'.r\n⊢ s = s'","state_after":"C : Type u_1\nD : Type u_2\ninst✝³ : Category.{u_3, u_1} C\ninst✝² : Category.{?u.82092, u_2} D\ninst✝¹ : Preadditive C\ninst✝ : Preadditive D\nS : ShortComplex C\ns s' : S.Splitting\nh : s.r = s'.r\nthis : Epi S.g\n⊢ s = s'","tactic":"have := s.epi_g","premises":[{"full_name":"CategoryTheory.ShortComplex.Splitting.epi_g","def_path":"Mathlib/Algebra/Homology/ShortComplex/Exact.lean","def_pos":[508,6],"def_end_pos":[508,11]}]},{"state_before":"C : Type u_1\nD : Type u_2\ninst✝³ : Category.{u_3, u_1} C\ninst✝² : Category.{?u.82092, u_2} D\ninst✝¹ : Preadditive C\ninst✝ : Preadditive D\nS : ShortComplex C\ns s' : S.Splitting\nh : s.r = s'.r\nthis : Epi S.g\n⊢ s = s'","state_after":"C : Type u_1\nD : Type u_2\ninst✝³ : Category.{u_3, u_1} C\ninst✝² : Category.{?u.82092, u_2} D\ninst✝¹ : Preadditive C\ninst✝ : Preadditive D\nS : ShortComplex C\ns s' : S.Splitting\nh : s.r = s'.r\nthis : Epi S.g\neq : s.r ≫ S.f + S.g ≫ s.s = 𝟙 S.X₂\n⊢ s = s'","tactic":"have eq := s.id","premises":[{"full_name":"CategoryTheory.ShortComplex.Splitting.id","def_path":"Mathlib/Algebra/Homology/ShortComplex/Exact.lean","def_pos":[480,2],"def_end_pos":[480,4]}]},{"state_before":"C : Type u_1\nD : Type u_2\ninst✝³ : Category.{u_3, u_1} C\ninst✝² : Category.{?u.82092, u_2} D\ninst✝¹ : Preadditive C\ninst✝ : Preadditive D\nS : ShortComplex C\ns s' : S.Splitting\nh : s.r = s'.r\nthis : Epi S.g\neq : s.r ≫ S.f + S.g ≫ s.s = 𝟙 S.X₂\n⊢ s = s'","state_after":"C : Type u_1\nD : Type u_2\ninst✝³ : Category.{u_3, u_1} C\ninst✝² : Category.{?u.82092, u_2} D\ninst✝¹ : Preadditive C\ninst✝ : Preadditive D\nS : ShortComplex C\ns s' : S.Splitting\nh : s.r = s'.r\nthis : Epi S.g\neq : s.s = s'.s\n⊢ s = s'","tactic":"rw [← s'.id, h, add_right_inj, cancel_epi S.g] at eq","premises":[{"full_name":"CategoryTheory.ShortComplex.Splitting.id","def_path":"Mathlib/Algebra/Homology/ShortComplex/Exact.lean","def_pos":[480,2],"def_end_pos":[480,4]},{"full_name":"CategoryTheory.ShortComplex.g","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[39,2],"def_end_pos":[39,3]},{"full_name":"CategoryTheory.cancel_epi","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[260,8],"def_end_pos":[260,18]},{"full_name":"add_right_inj","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[63,2],"def_end_pos":[63,13]}]},{"state_before":"C : Type u_1\nD : Type u_2\ninst✝³ : Category.{u_3, u_1} C\ninst✝² : Category.{?u.82092, u_2} D\ninst✝¹ : Preadditive C\ninst✝ : Preadditive D\nS : ShortComplex C\ns s' : S.Splitting\nh : s.r = s'.r\nthis : Epi S.g\neq : s.s = s'.s\n⊢ s = s'","state_after":"case mk\nC : Type u_1\nD : Type u_2\ninst✝³ : Category.{u_3, u_1} C\ninst✝² : Category.{?u.82092, u_2} D\ninst✝¹ : Preadditive C\ninst✝ : Preadditive D\nS : ShortComplex C\ns' : S.Splitting\nthis : Epi S.g\nr✝ : S.X₂ ⟶ S.X₁\ns✝ : S.X₃ ⟶ S.X₂\nf_r✝ : S.f ≫ r✝ = 𝟙 S.X₁\ns_g✝ : s✝ ≫ S.g = 𝟙 S.X₃\nid✝ : r✝ ≫ S.f + S.g ≫ s✝ = 𝟙 S.X₂\nh : { r := r✝, s := s✝, f_r := f_r✝, s_g := s_g✝, id := id✝ }.r = s'.r\neq : { r := r✝, s := s✝, f_r := f_r✝, s_g := s_g✝, id := id✝ }.s = s'.s\n⊢ { r := r✝, s := s✝, f_r := f_r✝, s_g := s_g✝, id := id✝ } = s'","tactic":"cases s","premises":[]},{"state_before":"case mk\nC : Type u_1\nD : Type u_2\ninst✝³ : Category.{u_3, u_1} C\ninst✝² : Category.{?u.82092, u_2} D\ninst✝¹ : Preadditive C\ninst✝ : Preadditive D\nS : ShortComplex C\ns' : S.Splitting\nthis : Epi S.g\nr✝ : S.X₂ ⟶ S.X₁\ns✝ : S.X₃ ⟶ S.X₂\nf_r✝ : S.f ≫ r✝ = 𝟙 S.X₁\ns_g✝ : s✝ ≫ S.g = 𝟙 S.X₃\nid✝ : r✝ ≫ S.f + S.g ≫ s✝ = 𝟙 S.X₂\nh : { r := r✝, s := s✝, f_r := f_r✝, s_g := s_g✝, id := id✝ }.r = s'.r\neq : { r := r✝, s := s✝, f_r := f_r✝, s_g := s_g✝, id := id✝ }.s = s'.s\n⊢ { r := r✝, s := s✝, f_r := f_r✝, s_g := s_g✝, id := id✝ } = s'","state_after":"case mk.mk\nC : Type u_1\nD : Type u_2\ninst✝³ : Category.{u_3, u_1} C\ninst✝² : Category.{?u.82092, u_2} D\ninst✝¹ : Preadditive C\ninst✝ : Preadditive D\nS : ShortComplex C\nthis : Epi S.g\nr✝¹ : S.X₂ ⟶ S.X₁\ns✝¹ : S.X₃ ⟶ S.X₂\nf_r✝¹ : S.f ≫ r✝¹ = 𝟙 S.X₁\ns_g✝¹ : s✝¹ ≫ S.g = 𝟙 S.X₃\nid✝¹ : r✝¹ ≫ S.f + S.g ≫ s✝¹ = 𝟙 S.X₂\nr✝ : S.X₂ ⟶ S.X₁\ns✝ : S.X₃ ⟶ S.X₂\nf_r✝ : S.f ≫ r✝ = 𝟙 S.X₁\ns_g✝ : s✝ ≫ S.g = 𝟙 S.X₃\nid✝ : r✝ ≫ S.f + S.g ≫ s✝ = 𝟙 S.X₂\nh :\n  { r := r✝¹, s := s✝¹, f_r := f_r✝¹, s_g := s_g✝¹, id := id✝¹ }.r =\n    { r := r✝, s := s✝, f_r := f_r✝, s_g := s_g✝, id := id✝ }.r\neq :\n  { r := r✝¹, s := s✝¹, f_r := f_r✝¹, s_g := s_g✝¹, id := id✝¹ }.s =\n    { r := r✝, s := s✝, f_r := f_r✝, s_g := s_g✝, id := id✝ }.s\n⊢ { r := r✝¹, s := s✝¹, f_r := f_r✝¹, s_g := s_g✝¹, id := id✝¹ } =\n    { r := r✝, s := s✝, f_r := f_r✝, s_g := s_g✝, id := id✝ }","tactic":"cases s'","premises":[]},{"state_before":"case mk.mk\nC : Type u_1\nD : Type u_2\ninst✝³ : Category.{u_3, u_1} C\ninst✝² : Category.{?u.82092, u_2} D\ninst✝¹ : Preadditive C\ninst✝ : Preadditive D\nS : ShortComplex C\nthis : Epi S.g\nr✝¹ : S.X₂ ⟶ S.X₁\ns✝¹ : S.X₃ ⟶ S.X₂\nf_r✝¹ : S.f ≫ r✝¹ = 𝟙 S.X₁\ns_g✝¹ : s✝¹ ≫ S.g = 𝟙 S.X₃\nid✝¹ : r✝¹ ≫ S.f + S.g ≫ s✝¹ = 𝟙 S.X₂\nr✝ : S.X₂ ⟶ S.X₁\ns✝ : S.X₃ ⟶ S.X₂\nf_r✝ : S.f ≫ r✝ = 𝟙 S.X₁\ns_g✝ : s✝ ≫ S.g = 𝟙 S.X₃\nid✝ : r✝ ≫ S.f + S.g ≫ s✝ = 𝟙 S.X₂\nh :\n  { r := r✝¹, s := s✝¹, f_r := f_r✝¹, s_g := s_g✝¹, id := id✝¹ }.r =\n    { r := r✝, s := s✝, f_r := f_r✝, s_g := s_g✝, id := id✝ }.r\neq :\n  { r := r✝¹, s := s✝¹, f_r := f_r✝¹, s_g := s_g✝¹, id := id✝¹ }.s =\n    { r := r✝, s := s✝, f_r := f_r✝, s_g := s_g✝, id := id✝ }.s\n⊢ { r := r✝¹, s := s✝¹, f_r := f_r✝¹, s_g := s_g✝¹, id := id✝¹ } =\n    { r := r✝, s := s✝, f_r := f_r✝, s_g := s_g✝, id := id✝ }","state_after":"case mk.mk\nC : Type u_1\nD : Type u_2\ninst✝³ : Category.{u_3, u_1} C\ninst✝² : Category.{?u.82092, u_2} D\ninst✝¹ : Preadditive C\ninst✝ : Preadditive D\nS : ShortComplex C\nthis : Epi S.g\nr✝¹ : S.X₂ ⟶ S.X₁\ns✝ : S.X₃ ⟶ S.X₂\nf_r✝¹ : S.f ≫ r✝¹ = 𝟙 S.X₁\ns_g✝¹ : s✝ ≫ S.g = 𝟙 S.X₃\nid✝¹ : r✝¹ ≫ S.f + S.g ≫ s✝ = 𝟙 S.X₂\nr✝ : S.X₂ ⟶ S.X₁\nf_r✝ : S.f ≫ r✝ = 𝟙 S.X₁\ns_g✝ : { r := r✝¹, s := s✝, f_r := f_r✝¹, s_g := s_g✝¹, id := id✝¹ }.s ≫ S.g = 𝟙 S.X₃\nid✝ : r✝ ≫ S.f + S.g ≫ { r := r✝¹, s := s✝, f_r := f_r✝¹, s_g := s_g✝¹, id := id✝¹ }.s = 𝟙 S.X₂\nh :\n  { r := r✝¹, s := s✝, f_r := f_r✝¹, s_g := s_g✝¹, id := id✝¹ }.r =\n    { r := r✝, s := { r := r✝¹, s := s✝, f_r := f_r✝¹, s_g := s_g✝¹, id := id✝¹ }.s, f_r := f_r✝, s_g := s_g✝,\n        id := id✝ }.r\n⊢ { r := r✝¹, s := s✝, f_r := f_r✝¹, s_g := s_g✝¹, id := id✝¹ } =\n    { r := r✝, s := { r := r✝¹, s := s✝, f_r := f_r✝¹, s_g := s_g✝¹, id := id✝¹ }.s, f_r := f_r✝, s_g := s_g✝,\n      id := id✝ }","tactic":"obtain rfl := eq","premises":[]},{"state_before":"case mk.mk\nC : Type u_1\nD : Type u_2\ninst✝³ : Category.{u_3, u_1} C\ninst✝² : Category.{?u.82092, u_2} D\ninst✝¹ : Preadditive C\ninst✝ : Preadditive D\nS : ShortComplex C\nthis : Epi S.g\nr✝¹ : S.X₂ ⟶ S.X₁\ns✝ : S.X₃ ⟶ S.X₂\nf_r✝¹ : S.f ≫ r✝¹ = 𝟙 S.X₁\ns_g✝¹ : s✝ ≫ S.g = 𝟙 S.X₃\nid✝¹ : r✝¹ ≫ S.f + S.g ≫ s✝ = 𝟙 S.X₂\nr✝ : S.X₂ ⟶ S.X₁\nf_r✝ : S.f ≫ r✝ = 𝟙 S.X₁\ns_g✝ : { r := r✝¹, s := s✝, f_r := f_r✝¹, s_g := s_g✝¹, id := id✝¹ }.s ≫ S.g = 𝟙 S.X₃\nid✝ : r✝ ≫ S.f + S.g ≫ { r := r✝¹, s := s✝, f_r := f_r✝¹, s_g := s_g✝¹, id := id✝¹ }.s = 𝟙 S.X₂\nh :\n  { r := r✝¹, s := s✝, f_r := f_r✝¹, s_g := s_g✝¹, id := id✝¹ }.r =\n    { r := r✝, s := { r := r✝¹, s := s✝, f_r := f_r✝¹, s_g := s_g✝¹, id := id✝¹ }.s, f_r := f_r✝, s_g := s_g✝,\n        id := id✝ }.r\n⊢ { r := r✝¹, s := s✝, f_r := f_r✝¹, s_g := s_g✝¹, id := id✝¹ } =\n    { r := r✝, s := { r := r✝¹, s := s✝, f_r := f_r✝¹, s_g := s_g✝¹, id := id✝¹ }.s, f_r := f_r✝, s_g := s_g✝,\n      id := id✝ }","state_after":"case mk.mk\nC : Type u_1\nD : Type u_2\ninst✝³ : Category.{u_3, u_1} C\ninst✝² : Category.{?u.82092, u_2} D\ninst✝¹ : Preadditive C\ninst✝ : Preadditive D\nS : ShortComplex C\nthis : Epi S.g\nr✝ : S.X₂ ⟶ S.X₁\ns✝ : S.X₃ ⟶ S.X₂\nf_r✝¹ : S.f ≫ r✝ = 𝟙 S.X₁\ns_g✝¹ : s✝ ≫ S.g = 𝟙 S.X₃\nid✝¹ : r✝ ≫ S.f + S.g ≫ s✝ = 𝟙 S.X₂\ns_g✝ : { r := r✝, s := s✝, f_r := f_r✝¹, s_g := s_g✝¹, id := id✝¹ }.s ≫ S.g = 𝟙 S.X₃\nf_r✝ : S.f ≫ { r := r✝, s := s✝, f_r := f_r✝¹, s_g := s_g✝¹, id := id✝¹ }.r = 𝟙 S.X₁\nid✝ :\n  { r := r✝, s := s✝, f_r := f_r✝¹, s_g := s_g✝¹, id := id✝¹ }.r ≫ S.f +\n      S.g ≫ { r := r✝, s := s✝, f_r := f_r✝¹, s_g := s_g✝¹, id := id✝¹ }.s =\n    𝟙 S.X₂\n⊢ { r := r✝, s := s✝, f_r := f_r✝¹, s_g := s_g✝¹, id := id✝¹ } =\n    { r := { r := r✝, s := s✝, f_r := f_r✝¹, s_g := s_g✝¹, id := id✝¹ }.r,\n      s := { r := r✝, s := s✝, f_r := f_r✝¹, s_g := s_g✝¹, id := id✝¹ }.s, f_r := f_r✝, s_g := s_g✝, id := id✝ }","tactic":"obtain rfl := h","premises":[]},{"state_before":"case mk.mk\nC : Type u_1\nD : Type u_2\ninst✝³ : Category.{u_3, u_1} C\ninst✝² : Category.{?u.82092, u_2} D\ninst✝¹ : Preadditive C\ninst✝ : Preadditive D\nS : ShortComplex C\nthis : Epi S.g\nr✝ : S.X₂ ⟶ S.X₁\ns✝ : S.X₃ ⟶ S.X₂\nf_r✝¹ : S.f ≫ r✝ = 𝟙 S.X₁\ns_g✝¹ : s✝ ≫ S.g = 𝟙 S.X₃\nid✝¹ : r✝ ≫ S.f + S.g ≫ s✝ = 𝟙 S.X₂\ns_g✝ : { r := r✝, s := s✝, f_r := f_r✝¹, s_g := s_g✝¹, id := id✝¹ }.s ≫ S.g = 𝟙 S.X₃\nf_r✝ : S.f ≫ { r := r✝, s := s✝, f_r := f_r✝¹, s_g := s_g✝¹, id := id✝¹ }.r = 𝟙 S.X₁\nid✝ :\n  { r := r✝, s := s✝, f_r := f_r✝¹, s_g := s_g✝¹, id := id✝¹ }.r ≫ S.f +\n      S.g ≫ { r := r✝, s := s✝, f_r := f_r✝¹, s_g := s_g✝¹, id := id✝¹ }.s =\n    𝟙 S.X₂\n⊢ { r := r✝, s := s✝, f_r := f_r✝¹, s_g := s_g✝¹, id := id✝¹ } =\n    { r := { r := r✝, s := s✝, f_r := f_r✝¹, s_g := s_g✝¹, id := id✝¹ }.r,\n      s := { r := r✝, s := s✝, f_r := f_r✝¹, s_g := s_g✝¹, id := id✝¹ }.s, f_r := f_r✝, s_g := s_g✝, id := id✝ }","state_after":"no goals","tactic":"rfl","premises":[]}]}
+{"url":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","commit":"","full_name":"MeasureTheory.tendsto_setLIntegral_zero","start":[493,0],"end":[502,36],"file_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nι : Type u_5\nf : α → ℝ≥0∞\nh : ∫⁻ (x : α), f x ∂μ ≠ ⊤\nl : Filter ι\ns : ι → Set α\nhl : Tendsto (⇑μ ∘ s) l (𝓝 0)\n⊢ Tendsto (fun i => ∫⁻ (x : α) in s i, f x ∂μ) l (𝓝 0)","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nι : Type u_5\nf : α → ℝ≥0∞\nh : ∫⁻ (x : α), f x ∂μ ≠ ⊤\nl : Filter ι\ns : ι → Set α\nhl : ∀ (i : ℝ≥0∞), 0 < i → ∀ᶠ (a : ι) in l, (⇑μ ∘ s) a < i\n⊢ ∀ (i : ℝ≥0∞), 0 < i → ∀ᶠ (a : ι) in l, ∫⁻ (x : α) in s a, f x ∂μ < i","tactic":"simp only [ENNReal.nhds_zero, tendsto_iInf, tendsto_principal, mem_Iio,\n    ← pos_iff_ne_zero] at hl ⊢","premises":[{"full_name":"ENNReal.nhds_zero","def_path":"Mathlib/Topology/Instances/ENNReal.lean","def_pos":[177,8],"def_end_pos":[177,17]},{"full_name":"Filter.tendsto_iInf","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2729,8],"def_end_pos":[2729,20]},{"full_name":"Filter.tendsto_principal","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2761,16],"def_end_pos":[2761,33]},{"full_name":"Set.mem_Iio","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[106,8],"def_end_pos":[106,15]},{"full_name":"pos_iff_ne_zero","def_path":"Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean","def_pos":[123,2],"def_end_pos":[123,13]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nι : Type u_5\nf : α → ℝ≥0∞\nh : ∫⁻ (x : α), f x ∂μ ≠ ⊤\nl : Filter ι\ns : ι → Set α\nhl : ∀ (i : ℝ≥0∞), 0 < i → ∀ᶠ (a : ι) in l, (⇑μ ∘ s) a < i\n⊢ ∀ (i : ℝ≥0∞), 0 < i → ∀ᶠ (a : ι) in l, ∫⁻ (x : α) in s a, f x ∂μ < i","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nι : Type u_5\nf : α → ℝ≥0∞\nh : ∫⁻ (x : α), f x ∂μ ≠ ⊤\nl : Filter ι\ns : ι → Set α\nhl : ∀ (i : ℝ≥0∞), 0 < i → ∀ᶠ (a : ι) in l, (⇑μ ∘ s) a < i\nε : ℝ≥0∞\nε0 : 0 < ε\n⊢ ∀ᶠ (a : ι) in l, ∫⁻ (x : α) in s a, f x ∂μ < ε","tactic":"intro ε ε0","premises":[]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nι : Type u_5\nf : α → ℝ≥0∞\nh : ∫⁻ (x : α), f x ∂μ ≠ ⊤\nl : Filter ι\ns : ι → Set α\nhl : ∀ (i : ℝ≥0∞), 0 < i → ∀ᶠ (a : ι) in l, (⇑μ ∘ s) a < i\nε : ℝ≥0∞\nε0 : 0 < ε\n⊢ ∀ᶠ (a : ι) in l, ∫⁻ (x : α) in s a, f x ∂μ < ε","state_after":"case intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ✝ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nι : Type u_5\nf : α → ℝ≥0∞\nh : ∫⁻ (x : α), f x ∂μ ≠ ⊤\nl : Filter ι\ns : ι → Set α\nhl : ∀ (i : ℝ≥0∞), 0 < i → ∀ᶠ (a : ι) in l, (⇑μ ∘ s) a < i\nε : ℝ≥0∞\nε0 : 0 < ε\nδ : ℝ≥0∞\nδ0 : δ > 0\nhδ : ∀ (s : Set α), μ s < δ → ∫⁻ (x : α) in s, f x ∂μ < ε\n⊢ ∀ᶠ (a : ι) in l, ∫⁻ (x : α) in s a, f x ∂μ < ε","tactic":"rcases exists_pos_setLIntegral_lt_of_measure_lt h ε0.ne' with ⟨δ, δ0, hδ⟩","premises":[{"full_name":"LT.lt.ne'","def_path":"Mathlib/Order/Basic.lean","def_pos":[267,8],"def_end_pos":[267,11]},{"full_name":"MeasureTheory.exists_pos_setLIntegral_lt_of_measure_lt","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[459,8],"def_end_pos":[459,48]}]},{"state_before":"case intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ✝ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nι : Type u_5\nf : α → ℝ≥0∞\nh : ∫⁻ (x : α), f x ∂μ ≠ ⊤\nl : Filter ι\ns : ι → Set α\nhl : ∀ (i : ℝ≥0∞), 0 < i → ∀ᶠ (a : ι) in l, (⇑μ ∘ s) a < i\nε : ℝ≥0∞\nε0 : 0 < ε\nδ : ℝ≥0∞\nδ0 : δ > 0\nhδ : ∀ (s : Set α), μ s < δ → ∫⁻ (x : α) in s, f x ∂μ < ε\n⊢ ∀ᶠ (a : ι) in l, ∫⁻ (x : α) in s a, f x ∂μ < ε","state_after":"no goals","tactic":"exact (hl δ δ0).mono fun i => hδ _","premises":[{"full_name":"Filter.Eventually.mono","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1002,8],"def_end_pos":[1002,23]}]}]}
+{"url":"Mathlib/Algebra/Order/Field/Basic.lean","commit":"","full_name":"one_div_lt_one_div_of_lt","start":[321,0],"end":[322,69],"file_path":"Mathlib/Algebra/Order/Field/Basic.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : LinearOrderedSemifield α\na b c d e : α\nm n : ℤ\nha : 0 < a\nh : a < b\n⊢ 1 / b < 1 / a","state_after":"no goals","tactic":"rwa [lt_div_iff' ha, ← div_eq_mul_one_div, div_lt_one (ha.trans h)]","premises":[{"full_name":"div_eq_mul_one_div","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[345,8],"def_end_pos":[345,26]},{"full_name":"div_lt_one","def_path":"Mathlib/Algebra/Order/Field/Basic.lean","def_pos":[303,8],"def_end_pos":[303,18]},{"full_name":"lt_div_iff'","def_path":"Mathlib/Algebra/Order/Field/Basic.lean","def_pos":[73,8],"def_end_pos":[73,19]}]}]}
+{"url":"Mathlib/Algebra/Order/Rearrangement.lean","commit":"","full_name":"MonovaryOn.sum_comp_perm_smul_eq_sum_smul_iff","start":[153,0],"end":[171,32],"file_path":"Mathlib/Algebra/Order/Rearrangement.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : LinearOrderedRing α\ninst✝² : LinearOrderedAddCommGroup β\ninst✝¹ : Module α β\ninst✝ : OrderedSMul α β\ns : Finset ι\nσ : Perm ι\nf : ι → α\ng : ι → β\nhfg : MonovaryOn f g ↑s\nhσ : {x | σ x ≠ x} ⊆ ↑s\n⊢ ∑ i ∈ s, f (σ i) • g i = ∑ i ∈ s, f i • g i ↔ MonovaryOn (f ∘ ⇑σ) g ↑s","state_after":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : LinearOrderedRing α\ninst✝² : LinearOrderedAddCommGroup β\ninst✝¹ : Module α β\ninst✝ : OrderedSMul α β\ns : Finset ι\nσ : Perm ι\nf : ι → α\ng : ι → β\nhfg : MonovaryOn f g ↑s\nhσ : {x | σ x ≠ x} ⊆ ↑s\nhσinv : {x | σ⁻¹ x ≠ x} ⊆ ↑s\n⊢ ∑ i ∈ s, f (σ i) • g i = ∑ i ∈ s, f i • g i ↔ MonovaryOn (f ∘ ⇑σ) g ↑s","tactic":"have hσinv : { x | σ⁻¹ x ≠ x } ⊆ s := (set_support_inv_eq _).subset.trans hσ","premises":[{"full_name":"Eq.subset","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[180,8],"def_end_pos":[180,17]},{"full_name":"Equiv.Perm.set_support_inv_eq","def_path":"Mathlib/GroupTheory/Perm/Support.lean","def_pos":[233,8],"def_end_pos":[233,26]},{"full_name":"HasSubset.Subset","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[384,2],"def_end_pos":[384,8]},{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"setOf","def_path":"Mathlib/Init/Set.lean","def_pos":[56,4],"def_end_pos":[56,9]}]},{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : LinearOrderedRing α\ninst✝² : LinearOrderedAddCommGroup β\ninst✝¹ : Module α β\ninst✝ : OrderedSMul α β\ns : Finset ι\nσ : Perm ι\nf : ι → α\ng : ι → β\nhfg : MonovaryOn f g ↑s\nhσ : {x | σ x ≠ x} ⊆ ↑s\nhσinv : {x | σ⁻¹ x ≠ x} ⊆ ↑s\n⊢ ∑ i ∈ s, f (σ i) • g i = ∑ i ∈ s, f i • g i ↔ MonovaryOn (f ∘ ⇑σ) g ↑s","state_after":"case refine_1\nι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : LinearOrderedRing α\ninst✝² : LinearOrderedAddCommGroup β\ninst✝¹ : Module α β\ninst✝ : OrderedSMul α β\ns : Finset ι\nσ : Perm ι\nf : ι → α\ng : ι → β\nhfg : MonovaryOn f g ↑s\nhσ : {x | σ x ≠ x} ⊆ ↑s\nhσinv : {x | σ⁻¹ x ≠ x} ⊆ ↑s\n⊢ ∑ i ∈ s, f (σ i) • g i = ∑ i ∈ s, f i • g i ↔ ∑ i ∈ s, f i • g (σ⁻¹ i) = ∑ i ∈ s, f i • g i\n\ncase refine_2\nι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : LinearOrderedRing α\ninst✝² : LinearOrderedAddCommGroup β\ninst✝¹ : Module α β\ninst✝ : OrderedSMul α β\ns : Finset ι\nσ : Perm ι\nf : ι → α\ng : ι → β\nhfg : MonovaryOn f g ↑s\nhσ : {x | σ x ≠ x} ⊆ ↑s\nhσinv : {x | σ⁻¹ x ≠ x} ⊆ ↑s\nh : MonovaryOn f (g ∘ ⇑σ⁻¹) ↑s\n⊢ MonovaryOn (f ∘ ⇑σ) g ↑s\n\ncase refine_3\nι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : LinearOrderedRing α\ninst✝² : LinearOrderedAddCommGroup β\ninst✝¹ : Module α β\ninst✝ : OrderedSMul α β\ns : Finset ι\nσ : Perm ι\nf : ι → α\ng : ι → β\nhfg : MonovaryOn f g ↑s\nhσ : {x | σ x ≠ x} ⊆ ↑s\nhσinv : {x | σ⁻¹ x ≠ x} ⊆ ↑s\nh : MonovaryOn (f ∘ ⇑σ) g ↑s\n⊢ MonovaryOn f (g ∘ ⇑σ⁻¹) ↑s","tactic":"refine (Iff.trans ?_ <| hfg.sum_smul_comp_perm_eq_sum_smul_iff hσinv).trans\n    ⟨fun h ↦ ?_, fun h ↦ ?_⟩","premises":[{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"Iff.trans","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[803,8],"def_end_pos":[803,17]},{"full_name":"MonovaryOn.sum_smul_comp_perm_eq_sum_smul_iff","def_path":"Mathlib/Algebra/Order/Rearrangement.lean","def_pos":[111,8],"def_end_pos":[111,53]}]}]}
+{"url":"Mathlib/Dynamics/BirkhoffSum/NormedSpace.lean","commit":"","full_name":"dist_birkhoffAverage_birkhoffAverage","start":[53,0],"end":[56,52],"file_path":"Mathlib/Dynamics/BirkhoffSum/NormedSpace.lean","tactics":[{"state_before":"α : Type u_1\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝² : RCLike 𝕜\ninst✝¹ : Module 𝕜 E\ninst✝ : BoundedSMul 𝕜 E\nf : α → α\ng : α → E\nn : ℕ\nx y : α\n⊢ dist (birkhoffAverage 𝕜 f g n x) (birkhoffAverage 𝕜 f g n y) = dist (birkhoffSum f g n x) (birkhoffSum f g n y) / ↑n","state_after":"no goals","tactic":"simp [birkhoffAverage, dist_smul₀, div_eq_inv_mul]","premises":[{"full_name":"birkhoffAverage","def_path":"Mathlib/Dynamics/BirkhoffSum/Average.lean","def_pos":[42,4],"def_end_pos":[42,19]},{"full_name":"dist_smul₀","def_path":"Mathlib/Analysis/Normed/MulAction.lean","def_pos":[98,8],"def_end_pos":[98,18]},{"full_name":"div_eq_inv_mul","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[522,8],"def_end_pos":[522,22]}]}]}
+{"url":"Mathlib/Computability/DFA.lean","commit":"","full_name":"DFA.accepts_comap","start":[197,0],"end":[203,20],"file_path":"Mathlib/Computability/DFA.lean","tactics":[{"state_before":"α : Type u\nσ : Type v\nM : DFA α σ\nα' : Type u_1\nσ' : Type u_2\nf : α' → α\n⊢ (comap f M).accepts = List.map f ⁻¹' M.accepts","state_after":"case h\nα : Type u\nσ : Type v\nM : DFA α σ\nα' : Type u_1\nσ' : Type u_2\nf : α' → α\nx : List α'\n⊢ x ∈ (comap f M).accepts ↔ x ∈ List.map f ⁻¹' M.accepts","tactic":"ext x","premises":[]},{"state_before":"case h\nα : Type u\nσ : Type v\nM : DFA α σ\nα' : Type u_1\nσ' : Type u_2\nf : α' → α\nx : List α'\n⊢ x ∈ (comap f M).accepts ↔ x ∈ List.map f ⁻¹' M.accepts","state_after":"case h\nα : Type u\nσ : Type v\nM : DFA α σ\nα' : Type u_1\nσ' : Type u_2\nf : α' → α\nx : List α'\n⊢ x ∈ (comap f M).accepts ↔ M.eval (List.map f x) ∈ M.accept","tactic":"conv =>\n    rhs\n    rw [Set.mem_preimage, mem_accepts]","premises":[{"full_name":"DFA.mem_accepts","def_path":"Mathlib/Computability/DFA.lean","def_pos":[102,8],"def_end_pos":[102,19]},{"full_name":"Set.mem_preimage","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[112,8],"def_end_pos":[112,20]}]},{"state_before":"case h\nα : Type u\nσ : Type v\nM : DFA α σ\nα' : Type u_1\nσ' : Type u_2\nf : α' → α\nx : List α'\n⊢ x ∈ (comap f M).accepts ↔ M.eval (List.map f x) ∈ M.accept","state_after":"no goals","tactic":"simp [mem_accepts]","premises":[{"full_name":"DFA.mem_accepts","def_path":"Mathlib/Computability/DFA.lean","def_pos":[102,8],"def_end_pos":[102,19]}]}]}
+{"url":"Mathlib/Data/Multiset/FinsetOps.lean","commit":"","full_name":"Multiset.mem_ndinter","start":[210,0],"end":[212,28],"file_path":"Mathlib/Data/Multiset/FinsetOps.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : DecidableEq α\ns✝ s t : Multiset α\na : α\n⊢ a ∈ s.ndinter t ↔ a ∈ s ∧ a ∈ t","state_after":"no goals","tactic":"simp [ndinter, mem_filter]","premises":[{"full_name":"Multiset.mem_filter","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[1723,8],"def_end_pos":[1723,18]},{"full_name":"Multiset.ndinter","def_path":"Mathlib/Data/Multiset/FinsetOps.lean","def_pos":[190,4],"def_end_pos":[190,11]}]}]}
+{"url":"Mathlib/Combinatorics/SimpleGraph/Operations.lean","commit":"","full_name":"SimpleGraph.card_edgeFinset_sup_edge","start":[177,0],"end":[179,44],"file_path":"Mathlib/Combinatorics/SimpleGraph/Operations.lean","tactics":[{"state_before":"V : Type u_1\ninst✝³ : DecidableEq V\nG : SimpleGraph V\ns t : V\ninst✝² : Fintype V\ninst✝¹ : DecidableRel G.Adj\ninst✝ : Fintype ↑(G ⊔ edge s t).edgeSet\nhn : ¬G.Adj s t\nh : s ≠ t\n⊢ (G ⊔ edge s t).edgeFinset.card = G.edgeFinset.card + 1","state_after":"no goals","tactic":"rw [G.edgeFinset_sup_edge hn h, card_cons]","premises":[{"full_name":"Finset.card_cons","def_path":"Mathlib/Data/Finset/Card.lean","def_pos":[84,8],"def_end_pos":[84,17]},{"full_name":"SimpleGraph.edgeFinset_sup_edge","def_path":"Mathlib/Combinatorics/SimpleGraph/Operations.lean","def_pos":[171,8],"def_end_pos":[171,27]}]}]}
+{"url":"Mathlib/Algebra/Homology/Homotopy.lean","commit":"","full_name":"prevD_comp_right","start":[95,0],"end":[99,27],"file_path":"Mathlib/Algebra/Homology/Homotopy.lean","tactics":[{"state_before":"ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf✝ g✝ : C ⟶ D\nh k : D ⟶ E\ni : ι\nf : (i j : ι) → C.X i ⟶ D.X j\ng : D ⟶ E\nj : ι\n⊢ ((prevD j) fun i j => f i j ≫ g.f j) = (prevD j) f ≫ g.f j","state_after":"ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf✝ g✝ : C ⟶ D\nh k : D ⟶ E\ni : ι\nf : (i j : ι) → C.X i ⟶ D.X j\ng : D ⟶ E\nj : ι\n⊢ (f j (c.prev j) ≫ g.f (c.prev j)) ≫ E.d (c.prev j) j = (f j (c.prev j) ≫ D.d (c.prev j) j) ≫ g.f j","tactic":"dsimp [prevD]","premises":[{"full_name":"prevD","def_path":"Mathlib/Algebra/Homology/Homotopy.lean","def_pos":[67,4],"def_end_pos":[67,9]}]},{"state_before":"ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf✝ g✝ : C ⟶ D\nh k : D ⟶ E\ni : ι\nf : (i j : ι) → C.X i ⟶ D.X j\ng : D ⟶ E\nj : ι\n⊢ (f j (c.prev j) ≫ g.f (c.prev j)) ≫ E.d (c.prev j) j = (f j (c.prev j) ≫ D.d (c.prev j) j) ≫ g.f j","state_after":"no goals","tactic":"simp only [assoc, g.comm]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"HomologicalComplex.Hom.comm","def_path":"Mathlib/Algebra/Homology/HomologicalComplex.lean","def_pos":[220,8],"def_end_pos":[220,16]}]}]}
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+{"url":"Mathlib/Combinatorics/Hindman.lean","commit":"","full_name":"Hindman.FS.add","start":[100,0],"end":[117,32],"file_path":"Mathlib/Combinatorics/Hindman.lean","tactics":[{"state_before":"M : Type u_1\ninst✝ : Semigroup M\na : Stream' M\nm : M\nhm : m ∈ FP a\n⊢ ∃ n, ∀ m' ∈ FP (Stream'.drop n a), m * m' ∈ FP a","state_after":"case head\nM : Type u_1\ninst✝ : Semigroup M\na✝ : Stream' M\nm : M\na : Stream' M\n⊢ ∃ n, ∀ m' ∈ FP (Stream'.drop n a), a.head * m' ∈ FP a\n\ncase tail\nM : Type u_1\ninst✝ : Semigroup M\na✝ : Stream' M\nm✝ : M\na : Stream' M\nm : M\nhm : FP a.tail m\nih : ∃ n, ∀ m' ∈ FP (Stream'.drop n a.tail), m * m' ∈ FP a.tail\n⊢ ∃ n, ∀ m' ∈ FP (Stream'.drop n a), m * m' ∈ FP a\n\ncase cons\nM : Type u_1\ninst✝ : Semigroup M\na✝ : Stream' M\nm✝ : M\na : Stream' M\nm : M\nhm : FP a.tail m\nih : ∃ n, ∀ m' ∈ FP (Stream'.drop n a.tail), m * m' ∈ FP a.tail\n⊢ ∃ n, ∀ m' ∈ FP (Stream'.drop n a), a.head * m * m' ∈ FP a","tactic":"induction' hm with a a m hm ih a m hm ih","premises":[]}]}
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+{"url":"Mathlib/CategoryTheory/Localization/Bousfield.lean","commit":"","full_name":"CategoryTheory.Localization.LeftBousfield.W_isoClosure","start":[61,0],"end":[74,76],"file_path":"Mathlib/CategoryTheory/Localization/Bousfield.lean","tactics":[{"state_before":"C : Type u_1\nD : Type u_2\ninst✝¹ : Category.{u_3, u_1} C\ninst✝ : Category.{?u.563, u_2} D\nP : C → Prop\n⊢ W (isoClosure P) = W P","state_after":"case h\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{u_3, u_1} C\ninst✝ : Category.{?u.563, u_2} D\nP : C → Prop\nX Y : C\nf : X ⟶ Y\n⊢ W (isoClosure P) f ↔ W P f","tactic":"ext X Y f","premises":[]},{"state_before":"case h\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{u_3, u_1} C\ninst✝ : Category.{?u.563, u_2} D\nP : C → Prop\nX Y : C\nf : X ⟶ Y\n⊢ W (isoClosure P) f ↔ W P f","state_after":"case h.mp\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{u_3, u_1} C\ninst✝ : Category.{?u.563, u_2} D\nP : C → Prop\nX Y : C\nf : X ⟶ Y\n⊢ W (isoClosure P) f → W P f\n\ncase h.mpr\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{u_3, u_1} C\ninst✝ : Category.{?u.563, u_2} D\nP : C → Prop\nX Y : C\nf : X ⟶ Y\n⊢ W P f → W (isoClosure P) f","tactic":"constructor","premises":[]}]}
+{"url":"Mathlib/CategoryTheory/Preadditive/Biproducts.lean","commit":"","full_name":"CategoryTheory.Limits.IsBilimit.binary_total","start":[310,0],"end":[312,59],"file_path":"Mathlib/CategoryTheory/Preadditive/Biproducts.lean","tactics":[{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nX Y : C\nb : BinaryBicone X Y\ni : b.IsBilimit\nj : Discrete WalkingPair\n⊢ (b.fst ≫ b.inl + b.snd ≫ b.inr) ≫ b.toCone.π.app j = 𝟙 b.pt ≫ b.toCone.π.app j","state_after":"no goals","tactic":"rcases j with ⟨⟨⟩⟩ <;> simp","premises":[]}]}
+{"url":"Mathlib/Data/Finset/Basic.lean","commit":"","full_name":"Finset.disjoint_filter","start":[2250,0],"end":[2252,73],"file_path":"Mathlib/Data/Finset/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\np✝ q✝ : α → Prop\ninst✝³ : DecidablePred p✝\ninst✝² : DecidablePred q✝\ns✝ t s : Finset α\np q : α → Prop\ninst✝¹ : DecidablePred p\ninst✝ : DecidablePred q\n⊢ _root_.Disjoint (filter p s) (filter q s) ↔ ∀ x ∈ s, p x → ¬q x","state_after":"no goals","tactic":"constructor <;> simp (config := { contextual := true }) [disjoint_left]","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Finset.disjoint_left","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[814,8],"def_end_pos":[814,21]}]}]}
+{"url":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","commit":"","full_name":"_private.Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.0.WeierstrassCurve.Jacobian.toAffine_addX_of_eq","start":[653,0],"end":[657,7],"file_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","tactics":[{"state_before":"R : Type u\ninst✝¹ : CommRing R\nW' : Jacobian R\nF : Type v\ninst✝ : Field F\nW : Jacobian F\nP : Fin 3 → F\nn d : F\nhPz : P z ≠ 0\nhd : d ≠ 0\n⊢ W.toAffine.addX (P x / P z ^ 2) (P x / P z ^ 2) (-n / (P z * d)) =\n    (n ^ 2 - W.a₁ * n * P z * d - W.a₂ * P z ^ 2 * d ^ 2 - 2 * P x * d ^ 2) / (P z * d) ^ 2","state_after":"R : Type u\ninst✝¹ : CommRing R\nW' : Jacobian R\nF : Type v\ninst✝ : Field F\nW : Jacobian F\nP : Fin 3 → F\nn d : F\nhPz : P z ≠ 0\nhd : d ≠ 0\n⊢ (((n ^ 2 * (P z * d) + -(W.toAffine.a₁ * n * (P z * d) ^ 2) - (P z * d) ^ 2 * (P z * d) * W.toAffine.a₂) * P z ^ 2 -\n            (P z * d) ^ 2 * (P z * d) * P x) *\n          P z ^ 2 -\n        (P z * d) ^ 2 * (P z * d) * P z ^ 2 * P x) *\n      (P z * d) ^ 2 =\n    (n ^ 2 - W.a₁ * n * P z * d - W.a₂ * P z ^ 2 * d ^ 2 - 2 * P x * d ^ 2) *\n      ((P z * d) ^ 2 * (P z * d) * P z ^ 2 * P z ^ 2)","tactic":"field_simp [mul_ne_zero hPz hd]","premises":[{"full_name":"mul_ne_zero","def_path":"Mathlib/Algebra/GroupWithZero/Basic.lean","def_pos":[80,8],"def_end_pos":[80,19]}]},{"state_before":"R : Type u\ninst✝¹ : CommRing R\nW' : Jacobian R\nF : Type v\ninst✝ : Field F\nW : Jacobian F\nP : Fin 3 → F\nn d : F\nhPz : P z ≠ 0\nhd : d ≠ 0\n⊢ (((n ^ 2 * (P z * d) + -(W.toAffine.a₁ * n * (P z * d) ^ 2) - (P z * d) ^ 2 * (P z * d) * W.toAffine.a₂) * P z ^ 2 -\n            (P z * d) ^ 2 * (P z * d) * P x) *\n          P z ^ 2 -\n        (P z * d) ^ 2 * (P z * d) * P z ^ 2 * P x) *\n      (P z * d) ^ 2 =\n    (n ^ 2 - W.a₁ * n * P z * d - W.a₂ * P z ^ 2 * d ^ 2 - 2 * P x * d ^ 2) *\n      ((P z * d) ^ 2 * (P z * d) * P z ^ 2 * P z ^ 2)","state_after":"no goals","tactic":"ring1","premises":[]}]}
+{"url":"Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean","commit":"","full_name":"MeasureTheory.SimpleFunc.integrable_approxOn_range","start":[247,0],"end":[250,58],"file_path":"Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nE : Type u_4\nF : Type u_5\n𝕜 : Type u_6\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : BorelSpace E\nf : β → E\nμ : Measure β\nfmeas : Measurable f\ninst✝ : SeparableSpace ↑(Set.range f ∪ {0})\nhf : Integrable f μ\nn : ℕ\n⊢ 0 ∈ Set.range f ∪ {0}","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/RingTheory/Ideal/QuotientOperations.lean","commit":"","full_name":"Ideal.quotientEquivAlgOfEq_symm","start":[608,0],"end":[612,5],"file_path":"Mathlib/RingTheory/Ideal/QuotientOperations.lean","tactics":[{"state_before":"R : Type u\nS : Type v\nF : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : Semiring S\nR₁ : Type u_1\nR₂ : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁴ : CommSemiring R₁\ninst✝³ : CommSemiring R₂\ninst✝² : CommRing A\ninst✝¹ : Algebra R₁ A\ninst✝ : Algebra R₂ A\nI J : Ideal A\nh : I = J\n⊢ (quotientEquivAlgOfEq R₁ h).symm = quotientEquivAlgOfEq R₁ ⋯","state_after":"case h\nR : Type u\nS : Type v\nF : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : Semiring S\nR₁ : Type u_1\nR₂ : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁴ : CommSemiring R₁\ninst✝³ : CommSemiring R₂\ninst✝² : CommRing A\ninst✝¹ : Algebra R₁ A\ninst✝ : Algebra R₂ A\nI J : Ideal A\nh : I = J\na✝ : A ⧸ J\n⊢ (quotientEquivAlgOfEq R₁ h).symm a✝ = (quotientEquivAlgOfEq R₁ ⋯) a✝","tactic":"ext","premises":[]},{"state_before":"case h\nR : Type u\nS : Type v\nF : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : Semiring S\nR₁ : Type u_1\nR₂ : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁴ : CommSemiring R₁\ninst✝³ : CommSemiring R₂\ninst✝² : CommRing A\ninst✝¹ : Algebra R₁ A\ninst✝ : Algebra R₂ A\nI J : Ideal A\nh : I = J\na✝ : A ⧸ J\n⊢ (quotientEquivAlgOfEq R₁ h).symm a✝ = (quotientEquivAlgOfEq R₁ ⋯) a✝","state_after":"no goals","tactic":"rfl","premises":[]}]}
+{"url":"Mathlib/Analysis/InnerProductSpace/Projection.lean","commit":"","full_name":"orthogonalProjection_singleton","start":[598,0],"end":[609,42],"file_path":"Mathlib/Analysis/InnerProductSpace/Projection.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : _root_.RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : HasOrthogonalProjection K\nv w : E\n⊢ ↑((orthogonalProjection (Submodule.span 𝕜 {v})) w) = (⟪v, w⟫_𝕜 / ↑(‖v‖ ^ 2)) • v","state_after":"case pos\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : _root_.RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : HasOrthogonalProjection K\nv w : E\nhv : v = 0\n⊢ ↑((orthogonalProjection (Submodule.span 𝕜 {v})) w) = (⟪v, w⟫_𝕜 / ↑(‖v‖ ^ 2)) • v\n\ncase neg\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : _root_.RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : HasOrthogonalProjection K\nv w : E\nhv : ¬v = 0\n⊢ ↑((orthogonalProjection (Submodule.span 𝕜 {v})) w) = (⟪v, w⟫_𝕜 / ↑(‖v‖ ^ 2)) • v","tactic":"by_cases hv : v = 0","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case neg\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : _root_.RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : HasOrthogonalProjection K\nv w : E\nhv : ¬v = 0\n⊢ ↑((orthogonalProjection (Submodule.span 𝕜 {v})) w) = (⟪v, w⟫_𝕜 / ↑(‖v‖ ^ 2)) • v","state_after":"case neg\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : _root_.RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : HasOrthogonalProjection K\nv w : E\nhv : ¬v = 0\nhv' : ‖v‖ ≠ 0\n⊢ ↑((orthogonalProjection (Submodule.span 𝕜 {v})) w) = (⟪v, w⟫_𝕜 / ↑(‖v‖ ^ 2)) • v","tactic":"have hv' : ‖v‖ ≠ 0 := ne_of_gt (norm_pos_iff.mpr hv)","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Norm.norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[59,2],"def_end_pos":[59,6]},{"full_name":"ne_of_gt","def_path":"Mathlib/Order/Defs.lean","def_pos":[85,8],"def_end_pos":[85,16]},{"full_name":"norm_pos_iff","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[1216,29],"def_end_pos":[1216,41]}]},{"state_before":"case neg\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : _root_.RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : HasOrthogonalProjection K\nv w : E\nhv : ¬v = 0\nhv' : ‖v‖ ≠ 0\n⊢ ↑((orthogonalProjection (Submodule.span 𝕜 {v})) w) = (⟪v, w⟫_𝕜 / ↑(‖v‖ ^ 2)) • v","state_after":"case neg\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : _root_.RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : HasOrthogonalProjection K\nv w : E\nhv : ¬v = 0\nhv' : ‖v‖ ≠ 0\nkey :\n  ((↑(‖v‖ ^ 2))⁻¹ * ↑(‖v‖ ^ 2)) • ↑((orthogonalProjection (Submodule.span 𝕜 {v})) w) = ((↑(‖v‖ ^ 2))⁻¹ * ⟪v, w⟫_𝕜) • v\n⊢ ↑((orthogonalProjection (Submodule.span 𝕜 {v})) w) = (⟪v, w⟫_𝕜 / ↑(‖v‖ ^ 2)) • v","tactic":"have key :\n    (((‖v‖ ^ 2 : ℝ) : 𝕜)⁻¹ * ((‖v‖ ^ 2 : ℝ) : 𝕜)) • ((orthogonalProjection (𝕜 ∙ v) w) : E) =\n      (((‖v‖ ^ 2 : ℝ) : 𝕜)⁻¹ * ⟪v, w⟫) • v := by\n    simp [mul_smul, smul_orthogonalProjection_singleton 𝕜 w, -map_pow]","premises":[{"full_name":"Inner.inner","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[79,2],"def_end_pos":[79,7]},{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"MulAction.mul_smul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[99,2],"def_end_pos":[99,10]},{"full_name":"Norm.norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[59,2],"def_end_pos":[59,6]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Singleton.singleton","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[467,2],"def_end_pos":[467,11]},{"full_name":"Submodule.span","def_path":"Mathlib/LinearAlgebra/Span.lean","def_pos":[49,4],"def_end_pos":[49,8]},{"full_name":"map_pow","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[423,8],"def_end_pos":[423,15]},{"full_name":"orthogonalProjection","def_path":"Mathlib/Analysis/InnerProductSpace/Projection.lean","def_pos":[439,4],"def_end_pos":[439,24]},{"full_name":"smul_orthogonalProjection_singleton","def_path":"Mathlib/Analysis/InnerProductSpace/Projection.lean","def_pos":[588,8],"def_end_pos":[588,43]}]},{"state_before":"case neg\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : _root_.RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : HasOrthogonalProjection K\nv w : E\nhv : ¬v = 0\nhv' : ‖v‖ ≠ 0\nkey :\n  ((↑(‖v‖ ^ 2))⁻¹ * ↑(‖v‖ ^ 2)) • ↑((orthogonalProjection (Submodule.span 𝕜 {v})) w) = ((↑(‖v‖ ^ 2))⁻¹ * ⟪v, w⟫_𝕜) • v\n⊢ ↑((orthogonalProjection (Submodule.span 𝕜 {v})) w) = (⟪v, w⟫_𝕜 / ↑(‖v‖ ^ 2)) • v","state_after":"no goals","tactic":"convert key using 1 <;> field_simp [hv']","premises":[]}]}
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+{"url":"Mathlib/Algebra/Group/Subgroup/Basic.lean","commit":"","full_name":"AddSubgroup.iSup_eq_closure","start":[984,0],"end":[986,86],"file_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","tactics":[{"state_before":"G : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst✝³ : Group G\ninst✝² : Group G'\ninst✝¹ : Group G''\nA : Type u_4\ninst✝ : AddGroup A\nH K : Subgroup G\nk : Set G\nι : Sort u_5\np : ι → Subgroup G\n⊢ ⨆ i, p i = closure (⋃ i, ↑(p i))","state_after":"no goals","tactic":"simp_rw [closure_iUnion, closure_eq]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Subgroup.closure_eq","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[958,8],"def_end_pos":[958,18]},{"full_name":"Subgroup.closure_iUnion","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[978,8],"def_end_pos":[978,22]}]}]}
+{"url":"Mathlib/Logic/Function/Basic.lean","commit":"","full_name":"Function.extend_injective","start":[641,0],"end":[646,9],"file_path":"Mathlib/Logic/Function/Basic.lean","tactics":[{"state_before":"α : Sort u_1\nβ : Sort u_2\nγ : Sort u_3\nf : α → β\nhf : Injective f\ne' : β → γ\n⊢ Injective fun g => extend f g e'","state_after":"α : Sort u_1\nβ : Sort u_2\nγ : Sort u_3\nf : α → β\nhf : Injective f\ne' : β → γ\ng₁ g₂ : α → γ\nhg : (fun g => extend f g e') g₁ = (fun g => extend f g e') g₂\n⊢ g₁ = g₂","tactic":"intro g₁ g₂ hg","premises":[]},{"state_before":"α : Sort u_1\nβ : Sort u_2\nγ : Sort u_3\nf : α → β\nhf : Injective f\ne' : β → γ\ng₁ g₂ : α → γ\nhg : (fun g => extend f g e') g₁ = (fun g => extend f g e') g₂\n⊢ g₁ = g₂","state_after":"α : Sort u_1\nβ : Sort u_2\nγ : Sort u_3\nf : α → β\nhf : Injective f\ne' : β → γ\ng₁ g₂ : α → γ\nhg : (fun g => extend f g e') g₁ = (fun g => extend f g e') g₂\nx : α\n⊢ g₁ x = g₂ x","tactic":"refine funext fun x ↦ ?_","premises":[{"full_name":"funext","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1817,8],"def_end_pos":[1817,14]}]},{"state_before":"α : Sort u_1\nβ : Sort u_2\nγ : Sort u_3\nf : α → β\nhf : Injective f\ne' : β → γ\ng₁ g₂ : α → γ\nhg : (fun g => extend f g e') g₁ = (fun g => extend f g e') g₂\nx : α\n⊢ g₁ x = g₂ x","state_after":"α : Sort u_1\nβ : Sort u_2\nγ : Sort u_3\nf : α → β\nhf : Injective f\ne' : β → γ\ng₁ g₂ : α → γ\nhg : (fun g => extend f g e') g₁ = (fun g => extend f g e') g₂\nx : α\nH : (fun g => extend f g e') g₁ (f x) = (fun g => extend f g e') g₂ (f x)\n⊢ g₁ x = g₂ x","tactic":"have H := congr_fun hg (f x)","premises":[]},{"state_before":"α : Sort u_1\nβ : Sort u_2\nγ : Sort u_3\nf : α → β\nhf : Injective f\ne' : β → γ\ng₁ g₂ : α → γ\nhg : (fun g => extend f g e') g₁ = (fun g => extend f g e') g₂\nx : α\nH : (fun g => extend f g e') g₁ (f x) = (fun g => extend f g e') g₂ (f x)\n⊢ g₁ x = g₂ x","state_after":"α : Sort u_1\nβ : Sort u_2\nγ : Sort u_3\nf : α → β\nhf : Injective f\ne' : β → γ\ng₁ g₂ : α → γ\nhg : (fun g => extend f g e') g₁ = (fun g => extend f g e') g₂\nx : α\nH : g₁ x = g₂ x\n⊢ g₁ x = g₂ x","tactic":"simp only [hf.extend_apply] at H","premises":[{"full_name":"Function.Injective.extend_apply","def_path":"Mathlib/Logic/Function/Basic.lean","def_pos":[623,8],"def_end_pos":[623,30]}]},{"state_before":"α : Sort u_1\nβ : Sort u_2\nγ : Sort u_3\nf : α → β\nhf : Injective f\ne' : β → γ\ng₁ g₂ : α → γ\nhg : (fun g => extend f g e') g₁ = (fun g => extend f g e') g₂\nx : α\nH : g₁ x = g₂ x\n⊢ g₁ x = g₂ x","state_after":"no goals","tactic":"exact H","premises":[]}]}
+{"url":"Mathlib/Data/Nat/GCD/Basic.lean","commit":"","full_name":"Nat.coprime_add_mul_right_left","start":[177,0],"end":[179,47],"file_path":"Mathlib/Data/Nat/GCD/Basic.lean","tactics":[{"state_before":"m n k : ℕ\n⊢ (m + k * n).Coprime n ↔ m.Coprime n","state_after":"no goals","tactic":"rw [Coprime, Coprime, gcd_add_mul_right_left]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Nat.Coprime","def_path":".lake/packages/batteries/Batteries/Data/Nat/Gcd.lean","def_pos":[20,17],"def_end_pos":[20,24]},{"full_name":"Nat.gcd_add_mul_right_left","def_path":"Mathlib/Data/Nat/GCD/Basic.lean","def_pos":[47,8],"def_end_pos":[47,30]}]}]}
+{"url":"Mathlib/Data/Finset/Image.lean","commit":"","full_name":"Finset.subtype_map","start":[637,0],"end":[643,79],"file_path":"Mathlib/Data/Finset/Image.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\np : α → Prop\ninst✝ : DecidablePred p\ns : Finset α\n⊢ map (Embedding.subtype p) (Finset.subtype p s) = filter p s","state_after":"case a\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\np : α → Prop\ninst✝ : DecidablePred p\ns : Finset α\nx : α\n⊢ x ∈ map (Embedding.subtype p) (Finset.subtype p s) ↔ x ∈ filter p s","tactic":"ext x","premises":[]},{"state_before":"case a\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\np : α → Prop\ninst✝ : DecidablePred p\ns : Finset α\nx : α\n⊢ x ∈ map (Embedding.subtype p) (Finset.subtype p s) ↔ x ∈ filter p s","state_after":"no goals","tactic":"simp [@and_comm _ (_ = _), @and_left_comm _ (_ = _), @and_comm (p x) (x ∈ s)]","premises":[{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"and_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[819,8],"def_end_pos":[819,16]},{"full_name":"and_left_comm","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[56,8],"def_end_pos":[56,21]}]}]}
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1‖ < i} → (homothety x x_1) y ∈ s\n\ncase inr\nα : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst✝⁸ : SeminormedAddCommGroup V\ninst✝⁷ : PseudoMetricSpace P\ninst✝⁶ : NormedAddTorsor V P\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : MetricSpace Q\ninst✝³ : NormedAddTorsor W Q\n𝕜 : Type u_6\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 V\ninst✝ : NormedSpace 𝕜 W\nx : Q\ns : Set Q\ny : Q\nhy : y ∈ interior s\nh : y ≠ x\n⊢ ∃ i, 0 < i ∧ ∀ ⦃x_1 : 𝕜⦄, x_1 ∈ {y | ‖y - 1‖ < i} → (homothety x x_1) y ∈ s","tactic":"rcases eq_or_ne y x with h | h","premises":[{"full_name":"eq_or_ne","def_path":"Mathlib/Logic/Basic.lean","def_pos":[167,8],"def_end_pos":[167,16]}]},{"state_before":"case inr\nα : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst✝⁸ : SeminormedAddCommGroup V\ninst✝⁷ : PseudoMetricSpace P\ninst✝⁶ : NormedAddTorsor V P\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : MetricSpace Q\ninst✝³ : NormedAddTorsor W Q\n𝕜 : Type u_6\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 V\ninst✝ : NormedSpace 𝕜 W\nx : Q\ns : Set Q\ny : Q\nhy : y ∈ interior s\nh : y ≠ x\n⊢ ∃ i, 0 < i ∧ ∀ ⦃x_1 : 𝕜⦄, x_1 ∈ {y | ‖y - 1‖ < i} → (homothety x x_1) y ∈ s","state_after":"case inr\nα : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst✝⁸ : SeminormedAddCommGroup V\ninst✝⁷ : PseudoMetricSpace P\ninst✝⁶ : NormedAddTorsor V P\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : MetricSpace Q\ninst✝³ : NormedAddTorsor W Q\n𝕜 : Type u_6\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 V\ninst✝ : NormedSpace 𝕜 W\nx : Q\ns : Set Q\ny : Q\nhy : y ∈ interior s\nh : y ≠ x\nhxy : 0 < ‖y -ᵥ x‖\n⊢ ∃ i, 0 < i ∧ ∀ ⦃x_1 : 𝕜⦄, x_1 ∈ {y | ‖y - 1‖ < i} → (homothety x x_1) y ∈ s","tactic":"have hxy : 0 < ‖y -ᵥ x‖ := by rwa [norm_pos_iff, vsub_ne_zero]","premises":[{"full_name":"Norm.norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[59,2],"def_end_pos":[59,6]},{"full_name":"VSub.vsub","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[94,2],"def_end_pos":[94,6]},{"full_name":"norm_pos_iff","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[1216,29],"def_end_pos":[1216,41]},{"full_name":"vsub_ne_zero","def_path":"Mathlib/Algebra/AddTorsor.lean","def_pos":[126,8],"def_end_pos":[126,20]}]},{"state_before":"case inr\nα : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst✝⁸ : SeminormedAddCommGroup V\ninst✝⁷ : PseudoMetricSpace P\ninst✝⁶ : NormedAddTorsor V P\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : MetricSpace Q\ninst✝³ : NormedAddTorsor W Q\n𝕜 : Type u_6\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 V\ninst✝ : NormedSpace 𝕜 W\nx : Q\ns : Set Q\ny : Q\nhy : y ∈ interior s\nh : y ≠ x\nhxy : 0 < ‖y -ᵥ x‖\n⊢ ∃ i, 0 < i ∧ ∀ ⦃x_1 : 𝕜⦄, x_1 ∈ {y | ‖y - 1‖ < i} → (homothety x x_1) y ∈ s","state_after":"case inr.intro.intro.intro\nα : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst✝⁸ : SeminormedAddCommGroup V\ninst✝⁷ : PseudoMetricSpace P\ninst✝⁶ : NormedAddTorsor V P\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : MetricSpace Q\ninst✝³ : NormedAddTorsor W Q\n𝕜 : Type u_6\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 V\ninst✝ : NormedSpace 𝕜 W\nx : Q\ns : Set Q\ny : Q\nhy : y ∈ interior s\nh : y ≠ x\nhxy : 0 < ‖y -ᵥ x‖\nu : Set Q\nhu₁ : u ⊆ s\nhu₂ : IsOpen u\nhu₃ : y ∈ u\n⊢ ∃ i, 0 < i ∧ ∀ ⦃x_1 : 𝕜⦄, x_1 ∈ {y | ‖y - 1‖ < i} → (homothety x x_1) y ∈ s","tactic":"obtain ⟨u, hu₁, hu₂, hu₃⟩ := mem_interior.mp hy","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"mem_interior","def_path":"Mathlib/Topology/Basic.lean","def_pos":[215,8],"def_end_pos":[215,20]}]},{"state_before":"case inr.intro.intro.intro\nα : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst✝⁸ : SeminormedAddCommGroup V\ninst✝⁷ : PseudoMetricSpace P\ninst✝⁶ : NormedAddTorsor V P\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : MetricSpace Q\ninst✝³ : NormedAddTorsor W Q\n𝕜 : Type u_6\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 V\ninst✝ : NormedSpace 𝕜 W\nx : Q\ns : Set Q\ny : Q\nhy : y ∈ interior s\nh : y ≠ x\nhxy : 0 < ‖y -ᵥ x‖\nu : Set Q\nhu₁ : u ⊆ s\nhu₂ : IsOpen u\nhu₃ : y ∈ u\n⊢ ∃ i, 0 < i ∧ ∀ ⦃x_1 : 𝕜⦄, x_1 ∈ {y | ‖y - 1‖ < i} → (homothety x x_1) y ∈ s","state_after":"case inr.intro.intro.intro.intro.intro\nα : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst✝⁸ : SeminormedAddCommGroup V\ninst✝⁷ : PseudoMetricSpace P\ninst✝⁶ : NormedAddTorsor V P\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : MetricSpace Q\ninst✝³ : NormedAddTorsor W Q\n𝕜 : Type u_6\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 V\ninst✝ : NormedSpace 𝕜 W\nx : Q\ns : Set Q\ny : Q\nhy : y ∈ interior s\nh : y ≠ x\nhxy : 0 < ‖y -ᵥ x‖\nu : Set Q\nhu₁ : u ⊆ s\nhu₂ : IsOpen u\nhu₃ : y ∈ u\nε : ℝ\nhε : ε > 0\nhyε : Metric.ball y ε ⊆ u\n⊢ ∃ i, 0 < i ∧ ∀ ⦃x_1 : 𝕜⦄, x_1 ∈ {y | ‖y - 1‖ < i} → (homothety x x_1) y ∈ s","tactic":"obtain ⟨ε, hε, hyε⟩ := Metric.isOpen_iff.mp hu₂ y hu₃","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Metric.isOpen_iff","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[857,8],"def_end_pos":[857,18]}]},{"state_before":"case inr.intro.intro.intro.intro.intro\nα : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst✝⁸ : SeminormedAddCommGroup V\ninst✝⁷ : PseudoMetricSpace P\ninst✝⁶ : NormedAddTorsor V P\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : MetricSpace Q\ninst✝³ : NormedAddTorsor W Q\n𝕜 : Type u_6\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 V\ninst✝ : NormedSpace 𝕜 W\nx : Q\ns : Set Q\ny : Q\nhy : y ∈ interior s\nh : y ≠ x\nhxy : 0 < ‖y -ᵥ x‖\nu : Set Q\nhu₁ : u ⊆ s\nhu₂ : IsOpen u\nhu₃ : y ∈ u\nε : ℝ\nhε : ε > 0\nhyε : Metric.ball y ε ⊆ u\n⊢ ∃ i, 0 < i ∧ ∀ ⦃x_1 : 𝕜⦄, x_1 ∈ {y | ‖y - 1‖ < i} → (homothety x x_1) y ∈ s","state_after":"case inr.intro.intro.intro.intro.intro\nα : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst✝⁸ : SeminormedAddCommGroup V\ninst✝⁷ : PseudoMetricSpace P\ninst✝⁶ : NormedAddTorsor V P\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : MetricSpace Q\ninst✝³ : NormedAddTorsor W Q\n𝕜 : Type u_6\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 V\ninst✝ : NormedSpace 𝕜 W\nx : Q\ns : Set Q\ny : Q\nhy : y ∈ interior s\nh : y ≠ x\nhxy : 0 < ‖y -ᵥ x‖\nu : Set Q\nhu₁ : u ⊆ s\nhu₂ : IsOpen u\nhu₃ : y ∈ u\nε : ℝ\nhε : ε > 0\nhyε : Metric.ball y ε ⊆ u\nδ : 𝕜\nhδ : ‖δ - 1‖ < ε / ‖y -ᵥ x‖\n⊢ (homothety x δ) y ∈ Metric.ball y ε","tactic":"refine ⟨ε / ‖y -ᵥ x‖, div_pos hε hxy, fun δ (hδ : ‖δ - 1‖ < ε / ‖y -ᵥ x‖) => hu₁ (hyε ?_)⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Norm.norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[59,2],"def_end_pos":[59,6]},{"full_name":"VSub.vsub","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[94,2],"def_end_pos":[94,6]},{"full_name":"div_pos","def_path":"Mathlib/Algebra/Order/Field/Unbundled/Basic.lean","def_pos":[45,6],"def_end_pos":[45,13]}]},{"state_before":"case inr.intro.intro.intro.intro.intro\nα : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst✝⁸ : SeminormedAddCommGroup V\ninst✝⁷ : PseudoMetricSpace P\ninst✝⁶ : NormedAddTorsor V P\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : MetricSpace Q\ninst✝³ : NormedAddTorsor W Q\n𝕜 : Type u_6\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 V\ninst✝ : NormedSpace 𝕜 W\nx : Q\ns : Set Q\ny : Q\nhy : y ∈ interior s\nh : y ≠ x\nhxy : 0 < ‖y -ᵥ x‖\nu : Set Q\nhu₁ : u ⊆ s\nhu₂ : IsOpen u\nhu₃ : y ∈ u\nε : ℝ\nhε : ε > 0\nhyε : Metric.ball y ε ⊆ u\nδ : 𝕜\nhδ : ‖δ - 1‖ < ε / ‖y -ᵥ x‖\n⊢ (homothety x δ) y ∈ Metric.ball y ε","state_after":"case inr.intro.intro.intro.intro.intro\nα : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst✝⁸ : SeminormedAddCommGroup V\ninst✝⁷ : PseudoMetricSpace P\ninst✝⁶ : NormedAddTorsor V P\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : MetricSpace Q\ninst✝³ : NormedAddTorsor W Q\n𝕜 : Type u_6\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 V\ninst✝ : NormedSpace 𝕜 W\nx : Q\ns : Set Q\ny : Q\nhy : y ∈ interior s\nh : y ≠ x\nhxy : 0 < ‖y -ᵥ x‖\nu : Set Q\nhu₁ : u ⊆ s\nhu₂ : IsOpen u\nhu₃ : y ∈ u\nε : ℝ\nhε : ε > 0\nhyε : Metric.ball y ε ⊆ u\nδ : 𝕜\nhδ : ‖δ • (y -ᵥ x) - (y -ᵥ x)‖ < ε\n⊢ (homothety x δ) y ∈ Metric.ball y ε","tactic":"rw [lt_div_iff hxy, ← norm_smul, sub_smul, one_smul] at hδ","premises":[{"full_name":"lt_div_iff","def_path":"Mathlib/Algebra/Order/Field/Basic.lean","def_pos":[70,8],"def_end_pos":[70,18]},{"full_name":"norm_smul","def_path":"Mathlib/Analysis/Normed/MulAction.lean","def_pos":[79,8],"def_end_pos":[79,17]},{"full_name":"one_smul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[379,6],"def_end_pos":[379,14]},{"full_name":"sub_smul","def_path":"Mathlib/Algebra/Module/Defs.lean","def_pos":[245,8],"def_end_pos":[245,16]}]},{"state_before":"case inr.intro.intro.intro.intro.intro\nα : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst✝⁸ : SeminormedAddCommGroup V\ninst✝⁷ : PseudoMetricSpace P\ninst✝⁶ : NormedAddTorsor V P\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : MetricSpace Q\ninst✝³ : NormedAddTorsor W Q\n𝕜 : Type u_6\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 V\ninst✝ : NormedSpace 𝕜 W\nx : Q\ns : Set Q\ny : Q\nhy : y ∈ interior s\nh : y ≠ x\nhxy : 0 < ‖y -ᵥ x‖\nu : Set Q\nhu₁ : u ⊆ s\nhu₂ : IsOpen u\nhu₃ : y ∈ u\nε : ℝ\nhε : ε > 0\nhyε : Metric.ball y ε ⊆ u\nδ : 𝕜\nhδ : ‖δ • (y -ᵥ x) - (y -ᵥ x)‖ < ε\n⊢ (homothety x δ) y ∈ Metric.ball y ε","state_after":"no goals","tactic":"rwa [homothety_apply, Metric.mem_ball, dist_eq_norm_vsub W, vadd_vsub_eq_sub_vsub]","premises":[{"full_name":"AffineMap.homothety_apply","def_path":"Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean","def_pos":[768,8],"def_end_pos":[768,23]},{"full_name":"Metric.mem_ball","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[354,8],"def_end_pos":[354,16]},{"full_name":"dist_eq_norm_vsub","def_path":"Mathlib/Analysis/Normed/Group/AddTorsor.lean","def_pos":[68,8],"def_end_pos":[68,25]},{"full_name":"vadd_vsub_eq_sub_vsub","def_path":"Mathlib/Algebra/AddTorsor.lean","def_pos":[142,8],"def_end_pos":[142,29]}]}]}
+{"url":"Mathlib/Data/Fintype/Card.lean","commit":"","full_name":"Finset.exists_superset_card_eq","start":[677,0],"end":[680,94],"file_path":"Mathlib/Data/Fintype/Card.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : Fintype α\nn : ℕ\ns : Finset α\nhsn : s.card ≤ n\nhnα : n ≤ Fintype.card α\n⊢ ∃ t, s ⊆ t ∧ t.card = n","state_after":"no goals","tactic":"simpa using exists_subsuperset_card_eq s.subset_univ hsn hnα","premises":[{"full_name":"Finset.exists_subsuperset_card_eq","def_path":"Mathlib/Data/Finset/Card.lean","def_pos":[554,6],"def_end_pos":[554,32]},{"full_name":"Finset.subset_univ","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[120,8],"def_end_pos":[120,19]}]}]}
+{"url":"Mathlib/RingTheory/Localization/LocalizationLocalization.lean","commit":"","full_name":"IsFractionRing.isFractionRing_of_isDomain_of_isLocalization","start":[264,0],"end":[275,32],"file_path":"Mathlib/RingTheory/Localization/LocalizationLocalization.lean","tactics":[{"state_before":"R : Type u_1\ninst✝¹⁰ : CommRing R\nM : Submonoid R\nS✝ : Type u_2\ninst✝⁹ : CommRing S✝\ninst✝⁸ : IsDomain R\nS : Type u_3\nT : Type u_4\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsFractionRing R T\n⊢ IsFractionRing S T","state_after":"R : Type u_1\ninst✝¹⁰ : CommRing R\nM : Submonoid R\nS✝ : Type u_2\ninst✝⁹ : CommRing S✝\ninst✝⁸ : IsDomain R\nS : Type u_3\nT : Type u_4\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsFractionRing R T\nthis : Nontrivial T\n⊢ IsFractionRing S T","tactic":"haveI := IsFractionRing.nontrivial R T","premises":[{"full_name":"IsFractionRing.nontrivial","def_path":"Mathlib/RingTheory/Localization/FractionRing.lean","def_pos":[246,18],"def_end_pos":[246,28]}]},{"state_before":"R : Type u_1\ninst✝¹⁰ : CommRing R\nM : Submonoid R\nS✝ : Type u_2\ninst✝⁹ : CommRing S✝\ninst✝⁸ : IsDomain R\nS : Type u_3\nT : Type u_4\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsFractionRing R T\nthis : Nontrivial T\n⊢ IsFractionRing S T","state_after":"R : Type u_1\ninst✝¹⁰ : CommRing R\nM : Submonoid R\nS✝ : Type u_2\ninst✝⁹ : CommRing S✝\ninst✝⁸ : IsDomain R\nS : Type u_3\nT : Type u_4\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝�� : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsFractionRing R 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TopologicalSpace A\ninst✝¹ : (i : I) → TopologicalSpace (X i)\ninst✝ : (i : I) → TopologicalSpace (Y i)\nx✝ : C(A, (i : I) → X i)\n⊢ pi ((fun f i => (eval i).comp f) x✝) = x✝","state_after":"case h.h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : TopologicalSpace β\ninst✝⁴ : TopologicalSpace γ\ninst✝³ : TopologicalSpace δ\nf g : C(α, β)\nI : Type u_5\nA : Type u_6\nX : I → Type u_7\nY : I → Type u_8\ninst✝² : TopologicalSpace A\ninst✝¹ : (i : I) → TopologicalSpace (X i)\ninst✝ : (i : I) → TopologicalSpace (Y i)\nx✝¹ : C(A, (i : I) → X i)\na✝ : A\nx✝ : I\n⊢ (pi ((fun f i => (eval i).comp f) x✝¹)) a✝ x✝ = x✝¹ a✝ x✝","tactic":"ext","premises":[]},{"state_before":"case h.h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : TopologicalSpace β\ninst✝⁴ : TopologicalSpace γ\ninst✝³ : TopologicalSpace δ\nf g : C(α, β)\nI : Type u_5\nA : Type u_6\nX : I → Type u_7\nY : I → Type u_8\ninst✝² : TopologicalSpace A\ninst✝¹ : (i : I) → TopologicalSpace (X i)\ninst✝ : (i : I) → TopologicalSpace (Y i)\nx✝¹ : C(A, (i : I) → X i)\na✝ : A\nx✝ : I\n⊢ (pi ((fun f i => (eval i).comp f) x✝¹)) a✝ x✝ = x✝¹ a✝ x✝","state_after":"no goals","tactic":"simp [pi]","premises":[{"full_name":"ContinuousMap.pi","def_path":"Mathlib/Topology/ContinuousFunction/Basic.lean","def_pos":[324,4],"def_end_pos":[324,6]}]}]}
+{"url":"Mathlib/Order/Filter/Prod.lean","commit":"","full_name":"Filter.map_prod_map_const_id_principal_coprod_principal","start":[500,0],"end":[516,23],"file_path":"Mathlib/Order/Filter/Prod.lean","tactics":[{"state_before":"α✝ : Type u_1\nβ✝ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι✝ : Sort u_5\nf : Filter α✝\ng : Filter β✝\nα : Type u_6\nβ : Type u_7\nι : Type u_8\na : α\nb : β\ni : ι\n⊢ map (Prod.map (fun x => b) id) ((𝓟 {a}).coprod (𝓟 {i})) = 𝓟 ({b} ×ˢ univ)","state_after":"α✝ : Type u_1\nβ✝ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι✝ : Sort u_5\nf : Filter α✝\ng : Filter β✝\nα : Type u_6\nβ : Type u_7\nι : Type u_8\na : α\nb : β\ni : ι\n⊢ 𝓟 (Prod.map (fun x => b) id '' ({a}ᶜ ×ˢ {i}ᶜ)ᶜ) = 𝓟 ({b} ×ˢ univ)","tactic":"rw [principal_coprod_principal, map_principal]","premises":[{"full_name":"Filter.map_principal","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1644,8],"def_end_pos":[1644,21]},{"full_name":"Filter.principal_coprod_principal","def_path":"Mathlib/Order/Filter/Prod.lean","def_pos":[474,8],"def_end_pos":[474,34]}]},{"state_before":"α✝ : Type u_1\nβ✝ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι✝ : Sort u_5\nf : Filter α✝\ng : Filter β✝\nα : Type u_6\nβ : Type u_7\nι : Type u_8\na : α\nb : β\ni : ι\n⊢ 𝓟 (Prod.map (fun x => b) id '' ({a}ᶜ ×ˢ {i}ᶜ)ᶜ) = 𝓟 ({b} ×ˢ univ)","state_after":"case e_s\nα✝ : Type u_1\nβ✝ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι✝ : Sort u_5\nf : Filter α✝\ng : Filter β✝\nα : Type u_6\nβ : Type u_7\nι : Type u_8\na : α\nb : β\ni : ι\n⊢ Prod.map (fun x => b) id '' ({a}ᶜ ×ˢ {i}ᶜ)ᶜ = {b} ×ˢ univ","tactic":"congr","premises":[]},{"state_before":"case e_s\nα✝ : Type u_1\nβ✝ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι✝ : Sort u_5\nf : Filter α✝\ng : Filter β✝\nα : Type u_6\nβ : Type u_7\nι : Type u_8\na : α\nb : β\ni : ι\n⊢ Prod.map (fun x => b) id '' ({a}ᶜ ×ˢ {i}ᶜ)ᶜ = {b} ×ˢ univ","state_after":"case e_s.h.mk\nα✝ : Type u_1\nβ✝ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι✝ : Sort u_5\nf : Filter α✝\ng : Filter β✝\nα : Type u_6\nβ : Type u_7\nι : Type u_8\na : α\nb : β\ni : ι\nb' : β\ni' : ι\n⊢ (b', i') ∈ Prod.map (fun x => b) id '' ({a}ᶜ ×ˢ {i}ᶜ)ᶜ ↔ (b', i') ∈ {b} ×ˢ univ","tactic":"ext ⟨b', i'⟩","premises":[]},{"state_before":"case e_s.h.mk\nα✝ : Type u_1\nβ✝ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι✝ : Sort u_5\nf : Filter α✝\ng : Filter β✝\nα : Type u_6\nβ : Type u_7\nι : Type u_8\na : α\nb : β\ni : ι\nb' : β\ni' : ι\n⊢ (b', i') ∈ Prod.map (fun x => b) id '' ({a}ᶜ ×ˢ {i}ᶜ)ᶜ ↔ (b', i') ∈ {b} ×ˢ univ","state_after":"case e_s.h.mk.mp\nα✝ : Type u_1\nβ✝ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι✝ : Sort u_5\nf : Filter α✝\ng : Filter β✝\nα : Type u_6\nβ : Type u_7\nι : Type u_8\na : α\nb : β\ni : ι\nb' : β\ni' : ι\n⊢ (b', i') ∈ Prod.map (fun x => b) id '' ({a}ᶜ ×ˢ {i}ᶜ)ᶜ → (b', i') ∈ {b} ×ˢ univ\n\ncase e_s.h.mk.mpr\nα✝ : Type u_1\nβ✝ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι✝ : Sort u_5\nf : Filter α✝\ng : Filter β✝\nα : Type u_6\nβ : Type u_7\nι : Type u_8\na : α\nb : β\ni : ι\nb' : β\ni' : ι\n⊢ (b', i') ∈ {b} ×ˢ univ → (b', i') ∈ Prod.map (fun x => b) id '' ({a}ᶜ ×ˢ {i}ᶜ)ᶜ","tactic":"constructor","premises":[]}]}
+{"url":"Mathlib/Topology/Sheaves/Sheafify.lean","commit":"","full_name":"TopCat.Presheaf.stalkToFiber_surjective","start":[82,0],"end":[89,63],"file_path":"Mathlib/Topology/Sheaves/Sheafify.lean","tactics":[{"state_before":"X : TopCat\nF : Presheaf (Type v) X\nx : ↑X\n⊢ Function.Surjective (F.stalkToFiber x)","state_after":"case w\nX : TopCat\nF : Presheaf (Type v) X\nx : ↑X\n⊢ ∀ (t : F.stalk x), ∃ U f, ∃ (_ : (Sheafify.isLocallyGerm F).pred f), f ⟨x, ⋯⟩ = t","tactic":"apply TopCat.stalkToFiber_surjective","premises":[{"full_name":"TopCat.stalkToFiber_surjective","def_path":"Mathlib/Topology/Sheaves/LocalPredicate.lean","def_pos":[229,8],"def_end_pos":[229,31]}]},{"state_before":"case w\nX : TopCat\nF : Presheaf (Type v) X\nx : ↑X\n⊢ ∀ (t : F.stalk x), ∃ U f, ∃ (_ : (Sheafify.isLocallyGerm F).pred f), f ⟨x, ⋯⟩ = t","state_after":"case w\nX : TopCat\nF : Presheaf (Type v) X\nx : ↑X\nt : F.stalk x\n⊢ ∃ U f, ∃ (_ : (Sheafify.isLocallyGerm F).pred f), f ⟨x, ⋯⟩ = t","tactic":"intro t","premises":[]},{"state_before":"case w\nX : TopCat\nF : Presheaf (Type v) X\nx : ↑X\nt : F.stalk x\n⊢ ∃ U f, ∃ (_ : (Sheafify.isLocallyGerm F).pred f), f ⟨x, ⋯⟩ = t","state_after":"case w.intro.intro.intro\nX : TopCat\nF : Presheaf (Type v) X\nx : ↑X\nU : Opens ↑X\nm : x ∈ U\ns : (forget (Type v)).obj (F.obj (op U))\n⊢ ∃ U_1 f, ∃ (_ : (Sheafify.isLocallyGerm F).pred f), f ⟨x, ⋯⟩ = (F.germ ⟨x, m⟩) s","tactic":"obtain ⟨U, m, s, rfl⟩ := F.germ_exist _ t","premises":[{"full_name":"TopCat.Presheaf.germ_exist","def_path":"Mathlib/Topology/Sheaves/Stalks.lean","def_pos":[402,8],"def_end_pos":[402,18]}]},{"state_before":"case w.intro.intro.intro\nX : TopCat\nF : Presheaf (Type v) X\nx : ↑X\nU : Opens ↑X\nm : x ∈ U\ns : (forget (Type v)).obj (F.obj (op U))\n⊢ ∃ U_1 f, ∃ (_ : (Sheafify.isLocallyGerm F).pred f), f ⟨x, ⋯⟩ = (F.germ ⟨x, m⟩) s","state_after":"case h\nX : TopCat\nF : Presheaf (Type v) X\nx : ↑X\nU : Opens ↑X\nm : x ∈ U\ns : (forget (Type v)).obj (F.obj (op U))\n⊢ ∃ f, ∃ (_ : (Sheafify.isLocallyGerm F).pred f), f ⟨x, ⋯⟩ = (F.germ ⟨x, m⟩) s","tactic":"use ⟨U, m⟩","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]}]},{"state_before":"case h\nX : TopCat\nF : Presheaf (Type v) X\nx : ↑X\nU : Opens ↑X\nm : x ∈ U\ns : (forget (Type v)).obj (F.obj (op U))\n⊢ ∃ f, ∃ (_ : (Sheafify.isLocallyGerm F).pred f), f ⟨x, ⋯⟩ = (F.germ ⟨x, m⟩) s","state_after":"case h.w\nX : TopCat\nF : Presheaf (Type v) X\nx : ↑X\nU : Opens ↑X\nm : x ∈ U\ns : (forget (Type v)).obj (F.obj (op U))\n⊢ (y : ↥{ obj := U, property := m }.obj) → F.stalk ↑y\n\ncase h.h\nX : TopCat\nF : Presheaf (Type v) X\nx : ↑X\nU : Opens ↑X\nm : x ∈ U\ns : (forget (Type v)).obj (F.obj (op U))\n⊢ ∃ (_ : (Sheafify.isLocallyGerm F).pred ?h.w), ?h.w ⟨x, ⋯⟩ = (F.germ ⟨x, m⟩) s","tactic":"fconstructor","premises":[]}]}
+{"url":"Mathlib/CategoryTheory/Limits/Shapes/Pullback/Cospan.lean","commit":"","full_name":"CategoryTheory.Limits.cospanExt_inv_app_left","start":[362,0],"end":[364,90],"file_path":"Mathlib/CategoryTheory/Limits/Shapes/Pullback/Cospan.lean","tactics":[{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nX Y Z X' Y' Z' : C\niX : X ≅ X'\niY : Y ≅ Y'\niZ : Z ≅ Z'\nf : X ⟶ Z\ng : Y ⟶ Z\nf' : X' ⟶ Z'\ng' : Y' ⟶ Z'\nwf : iX.hom ≫ f' = f ≫ iZ.hom\nwg : iY.hom ≫ g' = g ≫ iZ.hom\n⊢ (cospanExt iX iY iZ wf wg).inv.app WalkingCospan.left = iX.inv","state_after":"no goals","tactic":"dsimp [cospanExt]","premises":[{"full_name":"CategoryTheory.Limits.cospanExt","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Pullback/Cospan.lean","def_pos":[330,4],"def_end_pos":[330,13]}]}]}
+{"url":"Mathlib/Algebra/Group/Basic.lean","commit":"","full_name":"sub_sub","start":[542,0],"end":[543,52],"file_path":"Mathlib/Algebra/Group/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nG : Type u_3\nM : Type u_4\ninst✝ : DivisionCommMonoid α\na b c d : α\n⊢ a / b / c = a / (b * c)","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/Algebra/Group/Pi/Basic.lean","commit":"","full_name":"Sum.elim_single_zero","start":[450,0],"end":[453,62],"file_path":"Mathlib/Algebra/Group/Pi/Basic.lean","tactics":[{"state_before":"I : Type u\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : I → Type v₁\ng : I → Type v₂\nh : I → Type v₃\nx y : (i : I) → f i\ni✝ : I\na a' : α → γ\nb b' : β → γ\ninst✝² : DecidableEq α\ninst✝¹ : DecidableEq β\ninst✝ : One γ\ni : α\nc : γ\n⊢ Sum.elim (Pi.mulSingle i c) 1 = Pi.mulSingle (inl i) c","state_after":"no goals","tactic":"simp only [Pi.mulSingle, Sum.elim_update_left, elim_one_one]","premises":[{"full_name":"Pi.mulSingle","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[264,4],"def_end_pos":[264,13]},{"full_name":"Sum.elim_one_one","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[447,8],"def_end_pos":[447,20]},{"full_name":"Sum.elim_update_left","def_path":"Mathlib/Data/Sum/Basic.lean","def_pos":[222,8],"def_end_pos":[222,24]}]}]}
+{"url":"Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean","commit":"","full_name":"MeasureTheory.ae_eq_zero_of_forall_setIntegral_isClosed_eq_zero","start":[583,0],"end":[600,13],"file_path":"Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean","tactics":[{"state_before":"α : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\nμ✝ : Measure α\ns t : Set α\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_3\ninst✝² : TopologicalSpace β\ninst✝¹ : MeasurableSpace β\ninst✝ : BorelSpace β\nμ : Measure β\nf : β → E\nhf : Integrable f μ\nh'f : ∀ (s : Set β), IsClosed s → ∫ (x : β) in s, f x ∂μ = 0\n⊢ f =ᶠ[ae μ] 0","state_after":"α : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\nμ✝ : Measure α\ns t : Set α\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_3\ninst✝² : TopologicalSpace β\ninst✝¹ : MeasurableSpace β\ninst✝ : BorelSpace β\nμ : Measure β\nf : β → E\nhf : Integrable f μ\nh'f : ∀ (s : Set β), IsClosed s → ∫ (x : β) in s, f x ∂μ = 0\n⊢ ∀ (s : Set β), MeasurableSet s → ∫ (x : β) in s, f x ∂μ = 0","tactic":"suffices ∀ s, MeasurableSet s → ∫ x in s, f x ∂μ = 0 from\n    hf.ae_eq_zero_of_forall_setIntegral_eq_zero (fun s hs _ ↦ this s hs)","premises":[{"full_name":"MeasurableSet","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","def_pos":[61,4],"def_end_pos":[61,17]},{"full_name":"MeasureTheory.Integrable.ae_eq_zero_of_forall_setIntegral_eq_zero","def_path":"Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean","def_pos":[550,8],"def_end_pos":[550,59]},{"full_name":"MeasureTheory.Measure.restrict","def_path":"Mathlib/MeasureTheory/Measure/Restrict.lean","def_pos":[43,18],"def_end_pos":[43,26]},{"full_name":"MeasureTheory.integral","def_path":"Mathlib/MeasureTheory/Integral/Bochner.lean","def_pos":[714,16],"def_end_pos":[714,24]}]},{"state_before":"α : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\nμ✝ : Measure α\ns t : Set α\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_3\ninst✝² : TopologicalSpace β\ninst✝¹ : MeasurableSpace β\ninst✝ : BorelSpace β\nμ : Measure β\nf : β → E\nhf : Integrable f μ\nh'f : ∀ (s : Set β), IsClosed s → ∫ (x : β) in s, f x ∂μ = 0\n⊢ ∀ (s : Set β), MeasurableSet s → ∫ (x : β) in s, f x ∂μ = 0","state_after":"α : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\nμ✝ : Measure α\ns t : Set α\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_3\ninst✝² : TopologicalSpace β\ninst✝¹ : MeasurableSpace β\ninst✝ : BorelSpace β\nμ : Measure β\nf : β → E\nhf : Integrable f μ\nh'f : ∀ (s : Set β), IsClosed s → ∫ (x : β) in s, f x ∂μ = 0\nA : ∀ (t : Set β), MeasurableSet t → ∫ (x : β) in t, f x ∂μ = 0 → ∫ (x : β) in tᶜ, f x ∂μ = 0\n⊢ ∀ (s : Set β), MeasurableSet s → ∫ (x : β) in s, f x ∂μ = 0","tactic":"have A : ∀ (t : Set β), MeasurableSet t → ∫ (x : β) in t, f x ∂μ = 0\n      → ∫ (x : β) in tᶜ, f x ∂μ = 0 := by\n    intro t t_meas ht\n    have I : ∫ x, f x ∂μ = 0 := by rw [← integral_univ]; exact h'f _ isClosed_univ\n    simpa [ht, I] using integral_add_compl t_meas hf","premises":[{"full_name":"HasCompl.compl","def_path":"Mathlib/Order/Notation.lean","def_pos":[34,2],"def_end_pos":[34,7]},{"full_name":"MeasurableSet","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","def_pos":[61,4],"def_end_pos":[61,17]},{"full_name":"MeasureTheory.Measure.restrict","def_path":"Mathlib/MeasureTheory/Measure/Restrict.lean","def_pos":[43,18],"def_end_pos":[43,26]},{"full_name":"MeasureTheory.integral","def_path":"Mathlib/MeasureTheory/Integral/Bochner.lean","def_pos":[714,16],"def_end_pos":[714,24]},{"full_name":"MeasureTheory.integral_add_compl","def_path":"Mathlib/MeasureTheory/Integral/SetIntegral.lean","def_pos":[159,8],"def_end_pos":[159,26]},{"full_name":"MeasureTheory.integral_univ","def_path":"Mathlib/MeasureTheory/Integral/SetIntegral.lean","def_pos":[151,8],"def_end_pos":[151,21]},{"full_name":"Set","def_path":"Mathlib/Init/Set.lean","def_pos":[53,4],"def_end_pos":[53,7]},{"full_name":"isClosed_univ","def_path":"Mathlib/Topology/Basic.lean","def_pos":[157,16],"def_end_pos":[157,29]}]},{"state_before":"α : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\nμ✝ : Measure α\ns t : Set α\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_3\ninst✝² : TopologicalSpace β\ninst✝¹ : MeasurableSpace β\ninst✝ : BorelSpace β\nμ : Measure β\nf : β → E\nhf : Integrable f μ\nh'f : ∀ (s : Set β), IsClosed s → ∫ (x : β) in s, f x ∂μ = 0\nA : ∀ (t : Set β), MeasurableSet t → ∫ (x : β) in t, f x ∂μ = 0 → ∫ (x : β) in tᶜ, f x ∂μ = 0\n⊢ ∀ (s : Set β), MeasurableSet s → ∫ (x : β) in s, f x ∂μ = 0","state_after":"α : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\nμ✝ : Measure α\ns✝ t : Set α\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_3\ninst✝² : TopologicalSpace β\ninst✝¹ : MeasurableSpace β\ninst✝ : BorelSpace β\nμ : Measure β\nf : β → E\nhf : Integrable f μ\nh'f : ∀ (s : Set β), IsClosed s → ∫ (x : β) in s, f x ∂μ = 0\nA : ∀ (t : Set β), MeasurableSet t → ∫ (x : β) in t, f x ∂μ = 0 → ∫ (x : β) in tᶜ, f x ∂μ = 0\ns : Set β\nhs : MeasurableSet s\n⊢ ∫ (x : β) in s, f x ∂μ = 0","tactic":"intro s hs","premises":[]},{"state_before":"α : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\nμ✝ : Measure α\ns✝ t : Set α\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_3\ninst✝² : TopologicalSpace β\ninst✝¹ : MeasurableSpace β\ninst✝ : BorelSpace β\nμ : Measure β\nf : β → E\nhf : Integrable f μ\nh'f : ∀ (s : Set β), IsClosed s → ∫ (x : β) in s, f x ∂μ = 0\nA : ∀ (t : Set β), MeasurableSet t → ∫ (x : β) in t, f x ∂μ = 0 → ∫ (x : β) in tᶜ, f x ∂μ = 0\ns : Set β\nhs : MeasurableSet s\n⊢ ∫ (x : β) in s, f x ∂μ = 0","state_after":"case refine_1\nα : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\nμ✝ : Measure α\ns✝ t : Set α\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_3\ninst✝² : TopologicalSpace β\ninst✝¹ : MeasurableSpace β\ninst✝ : BorelSpace β\nμ : Measure β\nf : β → E\nhf : Integrable f μ\nh'f : ∀ (s : Set β), IsClosed s → ∫ (x : β) in s, f x ∂μ = 0\nA : ∀ (t : Set β), MeasurableSet t → ∫ (x : β) in t, f x ∂μ = 0 → ∫ (x : β) in tᶜ, f x ∂μ = 0\ns : Set β\nhs : MeasurableSet s\nU : Set β\nhU : IsOpen U\n⊢ ∫ (x : β) in U, f x ∂μ = 0\n\ncase refine_2\nα : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\nμ✝ : Measure α\ns✝ t : Set α\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_3\ninst✝² : TopologicalSpace β\ninst✝¹ : MeasurableSpace β\ninst✝ : BorelSpace β\nμ : Measure β\nf : β → E\nhf : Integrable f μ\nh'f : ∀ (s : Set β), IsClosed s → ∫ (x : β) in s, f x ∂μ = 0\nA : ∀ (t : Set β), MeasurableSet t → ∫ (x : β) in t, f x ∂μ = 0 → ∫ (x : β) in tᶜ, f x ∂μ = 0\ns : Set β\nhs : MeasurableSet s\ng : ℕ → Set β\ng_disj : Pairwise (Disjoint on g)\ng_meas : ∀ (i : ℕ), MeasurableSet (g i)\nhg : ∀ (i : ℕ), ∫ (x : β) in g i, f x ∂μ = 0\n⊢ ∫ (x : β) in ⋃ i, g i, f x ∂μ = 0","tactic":"refine MeasurableSet.induction_on_open (fun U hU ↦ ?_) A (fun g g_disj g_meas hg ↦ ?_) hs","premises":[{"full_name":"MeasurableSet.induction_on_open","def_path":"Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean","def_pos":[222,8],"def_end_pos":[222,39]}]}]}
+{"url":"Mathlib/LinearAlgebra/Matrix/GeneralLinearGroup/Card.lean","commit":"","full_name":"Matrix.card_GL_field","start":[83,0],"end":[90,88],"file_path":"Mathlib/LinearAlgebra/Matrix/GeneralLinearGroup/Card.lean","tactics":[{"state_before":"𝔽 : Type u_1\ninst✝¹ : Field 𝔽\ninst✝ : Fintype 𝔽\nn : ℕ\n⊢ Nat.card (GL (Fin n) 𝔽) = ∏ i : Fin n, (q ^ n - q ^ ↑i)","state_after":"case inl\n𝔽 : Type u_1\ninst✝¹ : Field 𝔽\ninst✝ : Fintype 𝔽\n⊢ Nat.card (GL (Fin 0) 𝔽) = ∏ i : Fin 0, (q ^ 0 - q ^ ↑i)\n\ncase inr\n𝔽 : Type u_1\ninst✝¹ : Field 𝔽\ninst✝ : Fintype 𝔽\nn : ℕ\nhn : n > 0\n⊢ Nat.card (GL (Fin n) 𝔽) = ∏ i : Fin n, (q ^ n - q ^ ↑i)","tactic":"rcases Nat.eq_zero_or_pos n with rfl | hn","premises":[{"full_name":"Nat.eq_zero_or_pos","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[350,8],"def_end_pos":[350,22]}]}]}
+{"url":"Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean","commit":"","full_name":"CliffordAlgebra.submodule_comap_mul_reverse","start":[251,0],"end":[255,71],"file_path":"Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean","tactics":[{"state_before":"R : Type u_1\ninst✝² : CommRing R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\np q : Submodule R (CliffordAlgebra Q)\n⊢ Submodule.comap reverse (p * q) = Submodule.comap reverse q * Submodule.comap reverse p","state_after":"no goals","tactic":"simp_rw [← submodule_map_reverse_eq_comap, submodule_map_mul_reverse]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"CliffordAlgebra.submodule_map_mul_reverse","def_path":"Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean","def_pos":[245,8],"def_end_pos":[245,33]},{"full_name":"CliffordAlgebra.submodule_map_reverse_eq_comap","def_path":"Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean","def_pos":[225,8],"def_end_pos":[225,38]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]}]}]}
+{"url":"Mathlib/Algebra/Star/NonUnitalSubalgebra.lean","commit":"","full_name":"NonUnitalStarSubalgebra.iSupLift_mk","start":[1008,0],"end":[1013,25],"file_path":"Mathlib/Algebra/Star/NonUnitalSubalgebra.lean","tactics":[{"state_before":"F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝¹⁷ : CommSemiring R\ninst✝¹⁶ : StarRing R\ninst✝¹⁵ : NonUnitalSemiring A\ninst✝¹⁴ : StarRing A\ninst✝¹³ : Module R A\ninst✝¹² : IsScalarTower R A A\ninst✝¹¹ : SMulCommClass R A A\ninst✝¹⁰ : StarModule R A\ninst✝⁹ : NonUnitalSemiring B\ninst✝⁸ : StarRing B\ninst✝⁷ : Module R B\ninst✝⁶ : IsScalarTower R B B\ninst✝⁵ : SMulCommClass R B B\ninst✝⁴ : StarModule R B\ninst✝³ : FunLike F A B\ninst✝² : NonUnitalAlgHomClass F R A B\ninst✝¹ : NonUnitalStarAlgHomClass F R A B\nS : NonUnitalStarSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalStarSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →⋆ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)\nT : NonUnitalStarSubalgebra R A\nhT : T = iSup K\ni : ι\nx : ↥(K i)\nhx : ↑x ∈ T\n⊢ (iSupLift K dir f hf T hT) ⟨↑x, hx⟩ = (f i) x","state_after":"F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝¹⁷ : CommSemiring R\ninst✝¹⁶ : StarRing R\ninst✝¹⁵ : NonUnitalSemiring A\ninst✝¹⁴ : StarRing A\ninst✝¹³ : Module R A\ninst✝¹² : IsScalarTower R A A\ninst✝¹¹ : SMulCommClass R A A\ninst✝¹⁰ : StarModule R A\ninst✝⁹ : NonUnitalSemiring B\ninst✝⁸ : StarRing B\ninst✝⁷ : Module R B\ninst✝⁶ : IsScalarTower R B B\ninst✝⁵ : SMulCommClass R B B\ninst✝⁴ : StarModule R B\ninst✝³ : FunLike F A B\ninst✝² : NonUnitalAlgHomClass F R A B\ninst✝¹ : NonUnitalStarAlgHomClass F R A B\nS : NonUnitalStarSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalStarSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →⋆ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)\ni : ι\nx : ↥(K i)\nhx : ↑x ∈ iSup K\n⊢ (iSupLift K dir f hf (iSup K) ⋯) ⟨↑x, hx⟩ = (f i) x","tactic":"subst hT","premises":[]},{"state_before":"F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝¹⁷ : CommSemiring R\ninst✝¹⁶ : StarRing R\ninst✝¹⁵ : NonUnitalSemiring A\ninst✝¹⁴ : StarRing A\ninst✝¹³ : Module R A\ninst✝¹² : IsScalarTower R A A\ninst✝¹¹ : SMulCommClass R A A\ninst✝¹⁰ : StarModule R A\ninst✝⁹ : NonUnitalSemiring B\ninst✝⁸ : StarRing B\ninst✝⁷ : Module R B\ninst✝⁶ : IsScalarTower R B B\ninst✝⁵ : SMulCommClass R B B\ninst✝⁴ : StarModule R B\ninst✝³ : FunLike F A B\ninst✝² : NonUnitalAlgHomClass F R A B\ninst✝¹ : NonUnitalStarAlgHomClass F R A B\nS : NonUnitalStarSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalStarSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →⋆ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)\ni : ι\nx : ↥(K i)\nhx : ↑x ∈ iSup K\n⊢ (iSupLift K dir f hf (iSup K) ⋯) ⟨↑x, hx⟩ = (f i) x","state_after":"F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝¹⁷ : CommSemiring R\ninst✝¹⁶ : StarRing R\ninst✝¹⁵ : NonUnitalSemiring A\ninst✝¹⁴ : StarRing A\ninst✝¹³ : Module R A\ninst✝¹² : IsScalarTower R A A\ninst✝¹¹ : SMulCommClass R A A\ninst✝¹⁰ : StarModule R A\ninst✝⁹ : NonUnitalSemiring B\ninst✝⁸ : StarRing B\ninst✝⁷ : Module R B\ninst✝⁶ : IsScalarTower R B B\ninst✝⁵ : SMulCommClass R B B\ninst✝⁴ : StarModule R B\ninst✝³ : FunLike F A B\ninst✝² : NonUnitalAlgHomClass F R A B\ninst✝¹ : NonUnitalStarAlgHomClass F R A B\nS : NonUnitalStarSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalStarSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →⋆ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)\ni : ι\nx : ↥(K i)\nhx : ↑x ∈ iSup K\n⊢ Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x) ⋯ ↑(iSup K) ⋯ ⟨↑x, hx⟩ = (f i) x","tactic":"dsimp 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i)\nhx : ↑x ∈ iSup K\n⊢ Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x) ⋯ ↑(iSup K) ⋯ ⟨↑x, hx⟩ = (f i) x","state_after":"no goals","tactic":"apply Set.iUnionLift_mk","premises":[{"full_name":"Set.iUnionLift_mk","def_path":"Mathlib/Data/Set/UnionLift.lean","def_pos":[62,8],"def_end_pos":[62,21]}]}]}
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+{"url":"Mathlib/MeasureTheory/Measure/Sub.lean","commit":"","full_name":"MeasureTheory.Measure.sub_apply","start":[59,0],"end":[87,39],"file_path":"Mathlib/MeasureTheory/Measure/Sub.lean","tactics":[{"state_before":"α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ns : Set α\ninst✝ : IsFiniteMeasure ν\nh₁ : MeasurableSet s\nh₂ : ν ≤ μ\nmeasure_sub : Measure α := ofMeasurable (fun t x => μ t - ν t) ⋯ ⋯\n⊢ (μ - ν) s = μ s - ν s","state_after":"α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ns : Set α\ninst✝ : IsFiniteMeasure ν\nh₁ : MeasurableSet s\nh₂ : ν ≤ μ\nmeasure_sub : Measure α := ofMeasurable (fun t x => μ t - ν t) ⋯ ⋯\nh_measure_sub_add : ν + measure_sub = μ\n⊢ (μ - ν) s = μ s - ν s","tactic":"have h_measure_sub_add : ν + measure_sub = μ := by\n    ext1 t h_t_measurable_set\n    simp only [Pi.add_apply, coe_add]\n    rw [MeasureTheory.Measure.ofMeasurable_apply _ h_t_measurable_set, add_comm,\n      tsub_add_cancel_of_le (h₂ t)]","premises":[{"full_name":"MeasureTheory.Measure.coe_add","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","def_pos":[781,8],"def_end_pos":[781,15]},{"full_name":"MeasureTheory.Measure.ofMeasurable_apply","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[119,8],"def_end_pos":[119,26]},{"full_name":"Pi.add_apply","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[81,2],"def_end_pos":[81,13]},{"full_name":"add_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[293,2],"def_end_pos":[293,13]},{"full_name":"tsub_add_cancel_of_le","def_path":"Mathlib/Algebra/Order/Sub/Canonical.lean","def_pos":[28,8],"def_end_pos":[28,29]}]},{"state_before":"α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ns : Set α\ninst✝ : IsFiniteMeasure ν\nh₁ : MeasurableSet s\nh₂ : ν ≤ μ\nmeasure_sub : Measure α := ofMeasurable (fun t x => μ t - ν t) ⋯ ⋯\nh_measure_sub_add : ν + measure_sub = μ\nh_measure_sub_eq : μ - ν = measure_sub\n⊢ (μ - ν) s = μ s - ν s","state_after":"α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ns : Set α\ninst✝ : IsFiniteMeasure ν\nh₁ : MeasurableSet s\nh₂ : ν ≤ μ\nmeasure_sub : Measure α := ofMeasurable (fun t x => μ t - ν t) ⋯ ⋯\nh_measure_sub_add : ν + measure_sub = μ\nh_measure_sub_eq : μ - ν = measure_sub\n⊢ measure_sub s = μ s - ν s","tactic":"rw [h_measure_sub_eq]","premises":[]},{"state_before":"α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ns : Set α\ninst✝ : IsFiniteMeasure ν\nh₁ : MeasurableSet s\nh₂ : ν ≤ μ\nmeasure_sub : Measure α := ofMeasurable (fun t x => μ t - ν t) ⋯ ⋯\nh_measure_sub_add : ν + measure_sub = μ\nh_measure_sub_eq : μ - ν = measure_sub\n⊢ measure_sub s = μ s - ν s","state_after":"no goals","tactic":"apply Measure.ofMeasurable_apply _ h₁","premises":[{"full_name":"MeasureTheory.Measure.ofMeasurable_apply","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[119,8],"def_end_pos":[119,26]}]}]}
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+{"url":"Mathlib/Algebra/BigOperators/Group/Finset.lean","commit":"","full_name":"Finset.sum_multiset_map_count","start":[1373,0],"end":[1377,32],"file_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","tactics":[{"state_before":"�� : Type u_1\nκ : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝² : CommMonoid β\ninst✝¹ : DecidableEq α\ns : Multiset α\nM : Type u_6\ninst✝ : CommMonoid M\nf : α → M\n⊢ (Multiset.map f s).prod = ∏ m ∈ s.toFinset, f m ^ Multiset.count m s","state_after":"ι : Type u_1\nκ : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝² : CommMonoid β\ninst✝¹ : DecidableEq α\ns : Multiset α\nM : Type u_6\ninst✝ : CommMonoid M\nf : α → M\nl : List α\n⊢ (Multiset.map f (Quot.mk Setoid.r l)).prod =\n    ∏ m ∈ Multiset.toFinset (Quot.mk Setoid.r l), f m ^ Multiset.count m (Quot.mk Setoid.r l)","tactic":"refine Quot.induction_on s fun l => ?_","premises":[{"full_name":"Quot.induction_on","def_path":"Mathlib/Data/Quot.lean","def_pos":[39,18],"def_end_pos":[39,30]}]},{"state_before":"ι : Type u_1\nκ : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝² : CommMonoid β\ninst✝¹ : DecidableEq α\ns : Multiset α\nM : Type u_6\ninst✝ : CommMonoid M\nf : α → M\nl : List α\n⊢ (Multiset.map f (Quot.mk Setoid.r l)).prod =\n    ∏ m ∈ Multiset.toFinset (Quot.mk Setoid.r l), f m ^ Multiset.count m (Quot.mk Setoid.r l)","state_after":"no goals","tactic":"simp [prod_list_map_count l f]","premises":[{"full_name":"Finset.prod_list_map_count","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[1338,8],"def_end_pos":[1338,27]}]}]}
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+{"url":"Mathlib/Analysis/Normed/Lp/lpSpace.lean","commit":"","full_name":"memℓp_gen'","start":[107,0],"end":[117,12],"file_path":"Mathlib/Analysis/Normed/Lp/lpSpace.lean","tactics":[{"state_before":"α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nC : ℝ\nf : (i : α) → E i\nhf : ∀ (s : Finset α), ∑ i ∈ s, ‖f i‖ ^ p.toReal ≤ C\n⊢ Memℓp f p","state_after":"case hf\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nC : ℝ\nf : (i : α) → E i\nhf : ∀ (s : Finset α), ∑ i ∈ s, ‖f i‖ ^ p.toReal ≤ C\n⊢ Summable fun i => ‖f i‖ ^ p.toReal","tactic":"apply memℓp_gen","premises":[{"full_name":"memℓp_gen","def_path":"Mathlib/Analysis/Normed/Lp/lpSpace.lean","def_pos":[97,8],"def_end_pos":[97,17]}]},{"state_before":"case hf\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nC : ℝ\nf : (i : α) → E i\nhf : ∀ (s : Finset α), ∑ i ∈ s, ‖f i‖ ^ p.toReal ≤ C\n⊢ Summable fun i => ‖f i‖ ^ p.toReal","state_after":"case h\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nC : ℝ\nf : (i : α) → E i\nhf : ∀ (s : Finset α), ∑ i ∈ s, ‖f i‖ ^ p.toReal ≤ C\n⊢ HasSum (fun i => ‖f i‖ ^ p.toReal) (⨆ s, ∑ i ∈ s, ‖f i‖ ^ p.toReal)","tactic":"use ⨆ s : Finset α, ∑ i ∈ s, ‖f i‖ ^ p.toReal","premises":[{"full_name":"ENNReal.toReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[184,14],"def_end_pos":[184,20]},{"full_name":"Finset","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[133,10],"def_end_pos":[133,16]},{"full_name":"Finset.sum","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[53,2],"def_end_pos":[53,13]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Norm.norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[59,2],"def_end_pos":[59,6]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]},{"full_name":"iSup","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[56,4],"def_end_pos":[56,8]}]},{"state_before":"case h\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nC : ℝ\nf : (i : α) → E i\nhf : ∀ (s : Finset α), ∑ i ∈ s, ‖f i‖ ^ p.toReal ≤ C\n⊢ HasSum (fun i => ‖f i‖ ^ p.toReal) (⨆ s, ∑ i ∈ s, ‖f i‖ ^ p.toReal)","state_after":"case h.h\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nC : ℝ\nf : (i : α) → E i\nhf : ∀ (s : Finset α), ∑ i ∈ s, ‖f i‖ ^ p.toReal ≤ C\n⊢ ∀ (i : α), 0 ≤ ‖f i‖ ^ p.toReal\n\ncase h.hf\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nC : ℝ\nf : (i : α) → E i\nhf : ∀ (s : Finset α), ∑ i ∈ s, ‖f i‖ ^ p.toReal ≤ C\n⊢ IsLUB (Set.range fun s => ∑ i ∈ s, ‖f i‖ ^ p.toReal) (⨆ s, ∑ i ∈ s, ‖f i‖ ^ p.toReal)","tactic":"apply hasSum_of_isLUB_of_nonneg","premises":[{"full_name":"hasSum_of_isLUB_of_nonneg","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Order.lean","def_pos":[242,2],"def_end_pos":[242,13]}]},{"state_before":"case h.hf\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nC : ℝ\nf : (i : α) → E i\nhf : ∀ (s : Finset α), ∑ i ∈ s, ‖f i‖ ^ p.toReal ≤ C\n⊢ IsLUB (Set.range fun s => ∑ i ∈ s, ‖f i‖ ^ p.toReal) (⨆ s, ∑ i ∈ s, ‖f i‖ ^ p.toReal)","state_after":"case h.hf.H\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nC : ℝ\nf : (i : α) → E i\nhf : ∀ (s : Finset α), ∑ i ∈ s, ‖f i‖ ^ p.toReal ≤ C\n⊢ BddAbove (Set.range fun s => ∑ i ∈ s, ‖f i‖ ^ p.toReal)","tactic":"apply isLUB_ciSup","premises":[{"full_name":"isLUB_ciSup","def_path":"Mathlib/Order/ConditionallyCompleteLattice/Basic.lean","def_pos":[464,8],"def_end_pos":[464,19]}]},{"state_before":"case h.hf.H\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nC : ℝ\nf : (i : α) → E i\nhf : ∀ (s : Finset α), ∑ i ∈ s, ‖f i‖ ^ p.toReal ≤ C\n⊢ BddAbove (Set.range fun s => ∑ i ∈ s, ‖f i‖ ^ p.toReal)","state_after":"case h\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nC : ℝ\nf : (i : α) → E i\nhf : ∀ (s : Finset α), ∑ i ∈ s, ‖f i‖ ^ p.toReal ≤ C\n⊢ C ∈ upperBounds (Set.range fun s => ∑ i ∈ s, ‖f i‖ ^ p.toReal)","tactic":"use C","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]}]},{"state_before":"case h\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nC : ℝ\nf : (i : α) → E i\nhf : ∀ (s : Finset α), ∑ i ∈ s, ‖f i‖ ^ p.toReal ≤ C\n⊢ C ∈ upperBounds (Set.range fun s => ∑ i ∈ s, ‖f i‖ ^ p.toReal)","state_after":"case h.intro\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nC : ℝ\nf : (i : α) → E i\nhf : ∀ (s : Finset α), ∑ i ∈ s, ‖f i‖ ^ p.toReal ≤ C\ns : Finset α\n⊢ (fun s => ∑ i ∈ s, ‖f i‖ ^ p.toReal) s ≤ C","tactic":"rintro - ⟨s, rfl⟩","premises":[]},{"state_before":"case h.intro\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nC : ℝ\nf : (i : α) → E i\nhf : ∀ (s : Finset α), ∑ i ∈ s, ‖f i‖ ^ p.toReal ≤ C\ns : Finset α\n⊢ (fun s => ∑ i ∈ s, ‖f i‖ ^ p.toReal) s ≤ C","state_after":"no goals","tactic":"exact hf s","premises":[]}]}
+{"url":"Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean","commit":"","full_name":"CategoryTheory.Subgroupoid.IsNormal.generatedNormal_le","start":[355,0],"end":[364,17],"file_path":"Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean","tactics":[{"state_before":"C : Type u\ninst✝ : Groupoid C\nS✝ : Subgroupoid C\nX : (c d : C) → Set (c ⟶ d)\nS : Subgroupoid C\nSn : S.IsNormal\n⊢ generatedNormal X ≤ S ↔ ∀ (c d : C), X c d ⊆ S.arrows c d","state_after":"case mp\nC : Type u\ninst✝ : Groupoid C\nS✝ : Subgroupoid C\nX : (c d : C) → Set (c ⟶ d)\nS : Subgroupoid C\nSn : S.IsNormal\n⊢ generatedNormal X ≤ S → ∀ (c d : C), X c d ⊆ S.arrows c d\n\ncase mpr\nC : Type u\ninst✝ : Groupoid C\nS✝ : Subgroupoid C\nX : (c d : C) → Set (c ⟶ d)\nS : Subgroupoid C\nSn : S.IsNormal\n⊢ (∀ (c d : C), X c d ⊆ S.arrows c d) → generatedNormal X ≤ S","tactic":"constructor","premises":[]}]}
+{"url":"Mathlib/Topology/Homotopy/HomotopyGroup.lean","commit":"","full_name":"GenLoop.copy_eq","start":[124,0],"end":[126,21],"file_path":"Mathlib/Topology/Homotopy/HomotopyGroup.lean","tactics":[{"state_before":"N : Type u_1\nX : Type u_2\ninst✝ : TopologicalSpace X\nx : X\nf : ↑(Ω^ N X x)\ng : (N → ↑I) → X\nh : g = ⇑f\n⊢ copy f g h = f","state_after":"case H\nN : Type u_1\nX : Type u_2\ninst✝ : TopologicalSpace X\nx✝ : X\nf : ↑(Ω^ N X x✝)\ng : (N → ↑I) → X\nh : g = ⇑f\nx : N → ↑I\n⊢ (copy f g h) x = f x","tactic":"ext x","premises":[]},{"state_before":"case H\nN : Type u_1\nX : Type u_2\ninst✝ : TopologicalSpace X\nx✝ : X\nf : ↑(Ω^ N X x✝)\ng : (N → ↑I) → X\nh : g = ⇑f\nx : N → ↑I\n⊢ (copy f g h) x = f x","state_after":"no goals","tactic":"exact congr_fun h x","premises":[]}]}
+{"url":"Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean","commit":"","full_name":"EuclideanGeometry.tan_oangle_right_of_oangle_eq_pi_div_two","start":[607,0],"end":[612,85],"file_path":"Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean","tactics":[{"state_before":"V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Module.Oriented ℝ V (Fin 2)\np₁ p₂ p₃ : P\nh : ∡ p₁ p₂ p₃ = ↑(π / 2)\n⊢ (∡ p₂ p₃ p₁).tan = dist p₁ p₂ / dist p₃ p₂","state_after":"V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Module.Oriented ℝ V (Fin 2)\np₁ p₂ p₃ : P\nh : ∡ p₁ p₂ p₃ = ↑(π / 2)\nhs : (∡ p₂ p₃ p₁).sign = 1\n⊢ (∡ p₂ p₃ p₁).tan = dist p₁ p₂ / dist p₃ p₂","tactic":"have hs : (∡ p₂ p₃ p₁).sign = 1 := by rw [oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two]","premises":[{"full_name":"EuclideanGeometry.oangle","def_path":"Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean","def_pos":[40,4],"def_end_pos":[40,10]},{"full_name":"EuclideanGeometry.oangle_rotate_sign","def_path":"Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean","def_pos":[406,8],"def_end_pos":[406,26]},{"full_name":"Real.Angle.sign","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean","def_pos":[712,4],"def_end_pos":[712,8]},{"full_name":"Real.Angle.sign_coe_pi_div_two","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean","def_pos":[812,8],"def_end_pos":[812,27]}]},{"state_before":"V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Module.Oriented ℝ V (Fin 2)\np₁ p₂ p₃ : P\nh : ∡ p₁ p₂ p₃ = ↑(π / 2)\nhs : (∡ p₂ p₃ p₁).sign = 1\n⊢ (∡ p₂ p₃ p₁).tan = dist p₁ p₂ / dist p₃ p₂","state_after":"no goals","tactic":"rw [oangle_eq_angle_of_sign_eq_one hs, Real.Angle.tan_coe,\n    tan_angle_of_angle_eq_pi_div_two (angle_eq_pi_div_two_of_oangle_eq_pi_div_two h)]","premises":[{"full_name":"EuclideanGeometry.angle_eq_pi_div_two_of_oangle_eq_pi_div_two","def_path":"Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean","def_pos":[363,8],"def_end_pos":[363,51]},{"full_name":"EuclideanGeometry.oangle_eq_angle_of_sign_eq_one","def_path":"Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean","def_pos":[342,8],"def_end_pos":[342,38]},{"full_name":"EuclideanGeometry.tan_angle_of_angle_eq_pi_div_two","def_path":"Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean","def_pos":[401,8],"def_end_pos":[401,40]},{"full_name":"Real.Angle.tan_coe","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean","def_pos":[655,8],"def_end_pos":[655,15]}]}]}
+{"url":"Mathlib/Order/WellFoundedSet.lean","commit":"","full_name":"Set.IsWF.isPWO","start":[484,0],"end":[489,34],"file_path":"Mathlib/Order/WellFoundedSet.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nπ : ι → Type u_5\ninst✝ : LinearOrder α\ns : Set α\nhs : s.IsWF\n⊢ s.IsPWO","state_after":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nπ : ι → Type u_5\ninst✝ : LinearOrder α\ns : Set α\nhs : s.IsWF\nf : ℕ → α\nhf : ∀ (n : ℕ), f n ∈ s\n⊢ ∃ m n, m < n ∧ (fun x x_1 => x ≤ x_1) (f m) (f n)","tactic":"intro f hf","premises":[]},{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nπ : ι → Type u_5\ninst✝ : LinearOrder α\ns : Set α\nhs : s.IsWF\nf : ℕ → α\nhf : ∀ (n : ℕ), f n ∈ s\n⊢ ∃ m n, m < n ∧ (fun x x_1 => x ≤ x_1) (f m) (f n)","state_after":"case intro\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nπ : ι → Type u_5\ninst✝ : LinearOrder α\ns : Set α\nhs : s.IsWF\nf : ℕ → ↑s\n⊢ ∃ m n, m < n ∧ (fun x x_1 => x ≤ x_1) ((fun i => ↑(f i)) m) ((fun i => ↑(f i)) n)","tactic":"lift f to ℕ → s using hf","premises":[{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]}]},{"state_before":"case intro\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nπ : ι → Type u_5\ninst✝ : LinearOrder α\ns : Set α\nhs : s.IsWF\nf : ℕ → ↑s\n⊢ ∃ m n, m < n ∧ (fun x x_1 => x ≤ x_1) ((fun i => ↑(f i)) m) ((fun i => ↑(f i)) n)","state_after":"case intro.intro.intro.intro\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nπ : ι → Type u_5\ninst✝ : LinearOrder α\ns : Set α\nhs : s.IsWF\nf : ℕ → ↑s\nm : ℕ\nhm : ∀ x ∈ range f, ¬(fun x x_1 => x < x_1) ↑x ↑(f m)\n⊢ ∃ m n, m < n ∧ (fun x x_1 => x ≤ x_1) ((fun i => ↑(f i)) m) ((fun i => ↑(f i)) n)","tactic":"rcases hs.has_min (range f) (range_nonempty _) with ⟨_, ⟨m, rfl⟩, hm⟩","premises":[{"full_name":"Set.range","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[144,4],"def_end_pos":[144,9]},{"full_name":"Set.range_nonempty","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[641,8],"def_end_pos":[641,22]},{"full_name":"WellFounded.has_min","def_path":"Mathlib/Order/WellFounded.lean","def_pos":[45,8],"def_end_pos":[45,15]}]},{"state_before":"case intro.intro.intro.intro\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nπ : ι → Type u_5\ninst✝ : LinearOrder α\ns : Set α\nhs : s.IsWF\nf : ℕ → ↑s\nm : ℕ\nhm : ∀ x ∈ range f, ¬(fun x x_1 => x < x_1) ↑x ↑(f m)\n⊢ ∃ m n, m < n ∧ (fun x x_1 => x ≤ x_1) ((fun i => ↑(f i)) m) ((fun i => ↑(f i)) n)","state_after":"case intro.intro.intro.intro\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nπ : ι → Type u_5\ninst✝ : LinearOrder α\ns : Set α\nhs : s.IsWF\nf : ℕ → ↑s\nm : ℕ\nhm : ∀ (i : ℕ), ↑(f m) ≤ ↑(f i)\n⊢ ∃ m n, m < n ∧ (fun x x_1 => x ≤ x_1) ((fun i => ↑(f i)) m) ((fun i => ↑(f i)) n)","tactic":"simp only [forall_mem_range, not_lt] at hm","premises":[{"full_name":"Set.forall_mem_range","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[564,8],"def_end_pos":[564,24]},{"full_name":"not_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[312,8],"def_end_pos":[312,14]}]},{"state_before":"case intro.intro.intro.intro\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nπ : ι → Type u_5\ninst✝ : LinearOrder α\ns : Set α\nhs : s.IsWF\nf : ℕ → ↑s\nm : ℕ\nhm : ∀ (i : ℕ), ↑(f m) ≤ ↑(f i)\n⊢ ∃ m n, m < n ∧ (fun x x_1 => x ≤ x_1) ((fun i => ↑(f i)) m) ((fun i => ↑(f i)) n)","state_after":"no goals","tactic":"exact ⟨m, m + 1, by omega, hm _⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]}]}
+{"url":"Mathlib/Algebra/Group/Units.lean","commit":"","full_name":"IsUnit.div_div_cancel_left","start":[907,0],"end":[909,80],"file_path":"Mathlib/Algebra/Group/Units.lean","tactics":[{"state_before":"α : Type u\nM : Type u_1\nN : Type u_2\ninst✝ : DivisionCommMonoid α\na b c d : α\nh : IsUnit a\n⊢ a / b / a = b⁻¹","state_after":"no goals","tactic":"rw [div_eq_mul_inv, div_eq_mul_inv, mul_right_comm, h.mul_inv_cancel, one_mul]","premises":[{"full_name":"IsUnit.mul_inv_cancel","def_path":"Mathlib/Algebra/Group/Units.lean","def_pos":[736,18],"def_end_pos":[736,32]},{"full_name":"div_eq_mul_inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[907,8],"def_end_pos":[907,22]},{"full_name":"mul_right_comm","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[156,8],"def_end_pos":[156,22]},{"full_name":"one_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[409,8],"def_end_pos":[409,15]}]}]}
+{"url":"Mathlib/MeasureTheory/Integral/Bochner.lean","commit":"","full_name":"MeasureTheory.integral_smul","start":[823,0],"end":[829,23],"file_path":"Mathlib/MeasureTheory/Integral/Bochner.lean","tactics":[{"state_before":"α : Type u_1\nE : Type u_2\nF : Type u_3\n𝕜 : Type u_4\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\nhE : CompleteSpace E\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : SMulCommClass ℝ 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : CompleteSpace F\nG : Type u_5\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace ℝ G\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedSpace 𝕜 G\ninst✝ : SMulCommClass ℝ ���� G\nc : 𝕜\nf : α → G\n⊢ ∫ (a : α), c • f a ∂μ = c • ∫ (a : α), f a ∂μ","state_after":"case pos\nα : Type u_1\nE : Type u_2\nF : Type u_3\n𝕜 : Type u_4\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\nhE : CompleteSpace E\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : SMulCommClass ℝ 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : CompleteSpace F\nG : Type u_5\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace ℝ G\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedSpace 𝕜 G\ninst✝ : SMulCommClass ℝ 𝕜 G\nc : 𝕜\nf : α → G\nhG : CompleteSpace G\n⊢ ∫ (a : α), c • f a ∂μ = c • ∫ (a : α), f a ∂μ\n\ncase neg\nα : Type u_1\nE : Type u_2\nF : Type u_3\n𝕜 : Type u_4\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\nhE : CompleteSpace E\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : SMulCommClass ℝ 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : CompleteSpace F\nG : Type u_5\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace ℝ G\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedSpace 𝕜 G\ninst✝ : SMulCommClass ℝ 𝕜 G\nc : 𝕜\nf : α → G\nhG : ¬CompleteSpace G\n⊢ ∫ (a : α), c • f a ∂μ = c • ∫ (a : α), f a ∂μ","tactic":"by_cases hG : CompleteSpace G","premises":[{"full_name":"CompleteSpace","def_path":"Mathlib/Topology/UniformSpace/Cauchy.lean","def_pos":[360,6],"def_end_pos":[360,19]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
+{"url":"Mathlib/Tactic/Rify.lean","commit":"","full_name":"Mathlib.Tactic.Rify.ratCast_lt","start":[79,0],"end":[79,79],"file_path":"Mathlib/Tactic/Rify.lean","tactics":[{"state_before":"a b : ℚ\n⊢ a < b ↔ ↑a < ↑b","state_after":"no goals","tactic":"simp","premises":[]}]}
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+{"url":"Mathlib/CategoryTheory/EssentiallySmall.lean","commit":"","full_name":"CategoryTheory.essentiallySmall_congr","start":[64,0],"end":[70,42],"file_path":"Mathlib/CategoryTheory/EssentiallySmall.lean","tactics":[{"state_before":"C✝ : Type u\ninst✝² : Category.{v, u} C✝\nC : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\ne : C ≌ D\n⊢ EssentiallySmall.{w, v, u} C ↔ EssentiallySmall.{w, v', u'} D","state_after":"case mp\nC✝ : Type u\ninst✝² : Category.{v, u} C✝\nC : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\ne : C ≌ D\n⊢ EssentiallySmall.{w, v, u} C → EssentiallySmall.{w, v', u'} D\n\ncase mpr\nC✝ : Type u\ninst✝² : Category.{v, u} C✝\nC : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\ne : C ≌ D\n⊢ EssentiallySmall.{w, v', u'} D → EssentiallySmall.{w, v, u} C","tactic":"fconstructor","premises":[]}]}
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+{"url":"Mathlib/Probability/Kernel/Composition.lean","commit":"","full_name":"ProbabilityTheory.Kernel.ae_null_of_compProd_null","start":[290,0],"end":[298,53],"file_path":"Mathlib/Probability/Kernel/Composition.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_4\nmγ : MeasurableSpace γ\ns : Set (β × γ)\nκ : Kernel α β\ninst✝¹ : IsSFiniteKernel κ\nη : Kernel (α × β) γ\ninst✝ : IsSFiniteKernel η\na : α\nh : ((κ ⊗ₖ η) a) s = 0\n⊢ (fun b => (η (a, b)) (Prod.mk b ⁻¹' s)) =ᶠ[ae (κ a)] 0","state_after":"case intro.intro.intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_4\nmγ : MeasurableSpace γ\ns : Set (β × γ)\nκ : Kernel α β\ninst✝¹ : IsSFiniteKernel κ\nη : Kernel (α × β) γ\ninst✝ : IsSFiniteKernel η\na : α\nh : ((κ ⊗ₖ η) a) s = 0\nt : Set (β × γ)\nhst : s ⊆ t\nmt : MeasurableSet t\nht : ((κ ⊗ₖ η) a) t = 0\n⊢ (fun b => (η (a, b)) (Prod.mk b ⁻¹' s)) =ᶠ[ae (κ a)] 0","tactic":"obtain ⟨t, hst, mt, ht⟩ := exists_measurable_superset_of_null h","premises":[{"full_name":"MeasureTheory.exists_measurable_superset_of_null","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[204,8],"def_end_pos":[204,42]}]},{"state_before":"case intro.intro.intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_4\nmγ : MeasurableSpace γ\ns : Set (β × γ)\nκ : Kernel α β\ninst✝¹ : IsSFiniteKernel κ\nη : Kernel (α × β) γ\ninst✝ : IsSFiniteKernel η\na : α\nh : ((κ ⊗ₖ η) a) s = 0\nt : Set (β × γ)\nhst : s ⊆ t\nmt : MeasurableSet t\nht : ((κ ⊗ₖ η) a) t = 0\n⊢ (fun b => (η (a, b)) (Prod.mk b ⁻¹' s)) =ᶠ[ae (κ a)] 0","state_after":"case intro.intro.intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_4\nmγ : MeasurableSpace γ\ns : Set (β × γ)\nκ : Kernel α β\ninst✝¹ : IsSFiniteKernel κ\nη : Kernel (α × β) γ\ninst✝ : IsSFiniteKernel η\na : α\nh : ((κ ⊗ₖ η) a) s = 0\nt : Set (β × γ)\nhst : s ⊆ t\nmt : MeasurableSet t\nht : (fun b => (η (a, b)) (Prod.mk b ⁻¹' t)) =ᶠ[ae (κ a)] 0\n⊢ (fun b => (η (a, b)) (Prod.mk b ⁻¹' s)) =ᶠ[ae (κ a)] 0","tactic":"simp_rw [compProd_null a mt] at ht","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"ProbabilityTheory.Kernel.compProd_null","def_path":"Mathlib/Probability/Kernel/Composition.lean","def_pos":[284,8],"def_end_pos":[284,21]}]},{"state_before":"case intro.intro.intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_4\nmγ : MeasurableSpace γ\ns : Set (β × γ)\nκ : Kernel α β\ninst✝¹ : IsSFiniteKernel κ\nη : Kernel (α × β) γ\ninst✝ : IsSFiniteKernel η\na : α\nh : ((κ ⊗ₖ η) a) s = 0\nt : Set (β × γ)\nhst : s ⊆ t\nmt : MeasurableSet t\nht : (fun b => (η (a, b)) (Prod.mk b ⁻¹' t)) =ᶠ[ae (κ a)] 0\n⊢ (fun b => (η (a, b)) (Prod.mk b ⁻¹' s)) =ᶠ[ae (κ a)] 0","state_after":"case intro.intro.intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_4\nmγ : MeasurableSpace γ\ns : Set (β × γ)\nκ : Kernel α β\ninst✝¹ : IsSFiniteKernel κ\nη : Kernel (α × β) γ\ninst✝ : IsSFiniteKernel η\na : α\nh : ((κ ⊗ₖ η) a) s = 0\nt : Set (β × γ)\nhst : s ⊆ t\nmt : MeasurableSet t\nht : (fun b => (η (a, b)) (Prod.mk b ⁻¹' t)) =ᶠ[ae (κ a)] 0\n⊢ (fun b => (η (a, b)) (Prod.mk b ⁻¹' s)) ≤ᶠ[ae (κ a)] 0 ∧ 0 ≤ᶠ[ae (κ a)] fun b => (η (a, b)) (Prod.mk b ⁻¹' s)","tactic":"rw [Filter.eventuallyLE_antisymm_iff]","premises":[{"full_name":"Filter.eventuallyLE_antisymm_iff","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1493,8],"def_end_pos":[1493,33]}]},{"state_before":"case intro.intro.intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_4\nmγ : MeasurableSpace γ\ns : Set (β × γ)\nκ : Kernel α β\ninst✝¹ : IsSFiniteKernel κ\nη : Kernel (α × β) γ\ninst✝ : IsSFiniteKernel η\na : α\nh : ((κ ⊗ₖ η) a) s = 0\nt : Set (β × γ)\nhst : s ⊆ t\nmt : MeasurableSet t\nht : (fun b => (η (a, b)) (Prod.mk b ⁻¹' t)) =ᶠ[ae (κ a)] 0\n⊢ (fun b => (η (a, b)) (Prod.mk b ⁻¹' s)) ≤ᶠ[ae (κ a)] 0 ∧ 0 ≤ᶠ[ae (κ a)] fun b => (η (a, b)) (Prod.mk b ⁻¹' s)","state_after":"no goals","tactic":"exact\n    ⟨Filter.EventuallyLE.trans_eq\n        (Filter.eventually_of_forall fun x => (measure_mono (Set.preimage_mono hst) : _)) ht,\n      Filter.eventually_of_forall fun x => zero_le _⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Filter.EventuallyLE.trans_eq","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1479,8],"def_end_pos":[1479,29]},{"full_name":"Filter.eventually_of_forall","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[979,8],"def_end_pos":[979,28]},{"full_name":"MeasureTheory.measure_mono","def_path":"Mathlib/MeasureTheory/OuterMeasure/Basic.lean","def_pos":[49,8],"def_end_pos":[49,20]},{"full_name":"Set.preimage_mono","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[56,8],"def_end_pos":[56,21]},{"full_name":"zero_le","def_path":"Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean","def_pos":[105,29],"def_end_pos":[105,36]}]}]}
+{"url":"Mathlib/Order/Interval/Set/OrdConnectedComponent.lean","commit":"","full_name":"Set.ordConnectedComponent_eq_empty","start":[53,0],"end":[55,66],"file_path":"Mathlib/Order/Interval/Set/OrdConnectedComponent.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : LinearOrder α\ns t : Set α\nx y z : α\n⊢ s.ordConnectedComponent x = ∅ ↔ x ∉ s","state_after":"no goals","tactic":"rw [← not_nonempty_iff_eq_empty, nonempty_ordConnectedComponent]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Set.nonempty_ordConnectedComponent","def_path":"Mathlib/Order/Interval/Set/OrdConnectedComponent.lean","def_pos":[50,8],"def_end_pos":[50,38]},{"full_name":"Set.not_nonempty_iff_eq_empty","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[481,8],"def_end_pos":[481,33]}]}]}
+{"url":"Mathlib/LinearAlgebra/Trace.lean","commit":"","full_name":"LinearMap.trace_eq_matrix_trace","start":[85,0],"end":[88,41],"file_path":"Mathlib/LinearAlgebra/Trace.lean","tactics":[{"state_before":"R : Type u\ninst✝⁶ : CommSemiring R\nM : Type v\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nι : Type w\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\nκ : Type u_1\ninst✝¹ : DecidableEq κ\ninst✝ : Fintype κ\nb : Basis ι R M\nc : Basis κ R M\nf : M →ₗ[R] M\n⊢ (trace R M) f = ((toMatrix b b) f).trace","state_after":"no goals","tactic":"rw [trace_eq_matrix_trace_of_finset R b.reindexFinsetRange, ← traceAux_def, ← traceAux_def,\n    traceAux_eq R b b.reindexFinsetRange]","premises":[{"full_name":"Basis.reindexFinsetRange","def_path":"Mathlib/LinearAlgebra/Basis.lean","def_pos":[496,4],"def_end_pos":[496,22]},{"full_name":"LinearMap.traceAux_def","def_path":"Mathlib/LinearAlgebra/Trace.lean","def_pos":[47,8],"def_end_pos":[47,20]},{"full_name":"LinearMap.traceAux_eq","def_path":"Mathlib/LinearAlgebra/Trace.lean","def_pos":[51,8],"def_end_pos":[51,19]},{"full_name":"LinearMap.trace_eq_matrix_trace_of_finset","def_path":"Mathlib/LinearAlgebra/Trace.lean","def_pos":[78,8],"def_end_pos":[78,39]}]}]}
+{"url":"Mathlib/CategoryTheory/Limits/IsLimit.lean","commit":"","full_name":"CategoryTheory.Limits.IsColimit.coconePointsIsoOfNatIso_hom_desc","start":[745,0],"end":[749,21],"file_path":"Mathlib/CategoryTheory/Limits/IsLimit.lean","tactics":[{"state_before":"J : Type u₁\ninst✝² : Category.{v₁, u₁} J\nK : Type u₂\ninst✝¹ : Category.{v₂, u₂} K\nC : Type u₃\ninst✝ : Category.{v₃, u₃} C\nF✝ : J ⥤ C\nt✝ : Cocone F✝\nF G : J ⥤ C\ns : Cocone F\nr t : Cocone G\nP : IsColimit s\nQ : IsColimit t\nw : F ≅ G\n⊢ ∀ (j : J), s.ι.app j ≫ (P.coconePointsIsoOfNatIso Q w).hom ≫ Q.desc r = s.ι.app j ≫ P.map r w.hom","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/Analysis/Normed/Group/Basic.lean","commit":"","full_name":"comap_norm_nhdsWithin_Ioi_zero","start":[1272,0],"end":[1274,69],"file_path":"Mathlib/Analysis/Normed/Group/Basic.lean","tactics":[{"state_before":"𝓕 : Type u_1\n𝕜 : Type u_2\nα : Type u_3\nι : Type u_4\nκ : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst✝¹ : NormedGroup E\ninst✝ : NormedGroup F\na b : E\n⊢ comap norm (𝓝[>] 0) = 𝓝[≠] 1","state_after":"no goals","tactic":"simp [nhdsWithin, comap_norm_nhds_one, Set.preimage, Set.compl_def]","premises":[{"full_name":"Set.compl_def","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1290,8],"def_end_pos":[1290,17]},{"full_name":"Set.preimage","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[106,4],"def_end_pos":[106,12]},{"full_name":"comap_norm_nhds_one","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[721,8],"def_end_pos":[721,27]},{"full_name":"nhdsWithin","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[121,4],"def_end_pos":[121,14]}]}]}
+{"url":"Mathlib/Algebra/Order/Floor/Div.lean","commit":"","full_name":"ceilDiv_one","start":[149,0],"end":[150,56],"file_path":"Mathlib/Algebra/Order/Floor/Div.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝⁴ : OrderedSemiring α\ninst✝³ : OrderedAddCommMonoid β\ninst✝² : MulActionWithZero α β\ninst✝¹ : CeilDiv α β\na : α\ninst✝ : Nontrivial α\nb c : β\n⊢ b ⌈/⌉ 1 ≤ c ↔ b ≤ c","state_after":"no goals","tactic":"simp [zero_lt_one' α]","premises":[{"full_name":"zero_lt_one'","def_path":"Mathlib/Algebra/Order/ZeroLEOne.lean","def_pos":[39,6],"def_end_pos":[39,18]}]}]}
+{"url":"Mathlib/Data/Set/Pointwise/Interval.lean","commit":"","full_name":"Set.preimage_mul_const_Icc_of_neg","start":[579,0],"end":[581,98],"file_path":"Mathlib/Data/Set/Pointwise/Interval.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : LinearOrderedField α\na✝ a b c : α\nh : c < 0\n⊢ (fun x => x * c) ⁻¹' Icc a b = Icc (b / c) (a / c)","state_after":"no goals","tactic":"simp [← Ici_inter_Iic, h, inter_comm]","premises":[{"full_name":"Set.Ici_inter_Iic","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[506,8],"def_end_pos":[506,21]},{"full_name":"Set.inter_comm","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[742,8],"def_end_pos":[742,18]}]}]}
+{"url":"Mathlib/CategoryTheory/Triangulated/Opposite.lean","commit":"","full_name":"CategoryTheory.Pretriangulated.TriangleOpEquivalence.counitIso_inv_app_hom₁","start":[253,0],"end":[267,18],"file_path":"Mathlib/CategoryTheory/Triangulated/Opposite.lean","tactics":[{"state_before":"C : Type u_1\ninst✝¹ : Category.{?u.192952, u_1} C\ninst✝ : HasShift C ℤ\nT : Triangle Cᵒᵖ\n⊢ (inverse C ⋙ functor C).obj T ≅ (𝟭 (Triangle Cᵒᵖ)).obj T","state_after":"case refine_1\nC : Type u_1\ninst✝¹ : Category.{?u.192952, u_1} C\ninst✝ : HasShift C ℤ\nT : Triangle Cᵒᵖ\n⊢ ((inverse C ⋙ functor C).obj T).mor₁ ≫ (Iso.refl ((inverse C ⋙ functor C).obj T).obj₂).hom =\n    (Iso.refl ((inverse C ⋙ functor C).obj T).obj₁).hom ≫ ((𝟭 (Triangle Cᵒᵖ)).obj T).mor₁\n\ncase refine_2\nC : Type u_1\ninst✝¹ : Category.{?u.192952, u_1} C\ninst✝ : HasShift C ℤ\nT : Triangle Cᵒᵖ\n⊢ ((inverse C ⋙ functor C).obj T).mor₂ ≫ (Iso.refl ((inverse C ⋙ functor C).obj T).obj₃).hom =\n    (Iso.refl ((inverse C ⋙ functor C).obj T).obj₂).hom ≫ ((𝟭 (Triangle Cᵒᵖ)).obj T).mor₂\n\ncase refine_3\nC : Type u_1\ninst✝¹ : Category.{?u.192952, u_1} C\ninst✝ : HasShift C ℤ\nT : Triangle Cᵒᵖ\n⊢ ((inverse C ⋙ functor C).obj T).mor₃ ≫ (shiftFunctor Cᵒᵖ 1).map (Iso.refl ((inverse C ⋙ functor C).obj T).obj₁).hom =\n    (Iso.refl ((inverse C ⋙ functor C).obj T).obj₃).hom ≫ ((𝟭 (Triangle Cᵒᵖ)).obj T).mor₃","tactic":"refine Triangle.isoMk _ _ (Iso.refl _) (Iso.refl _) (Iso.refl _) ?_ ?_ ?_","premises":[{"full_name":"CategoryTheory.Iso.refl","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[114,4],"def_end_pos":[114,8]},{"full_name":"CategoryTheory.Pretriangulated.Triangle.isoMk","def_path":"Mathlib/CategoryTheory/Triangulated/Basic.lean","def_pos":[185,4],"def_end_pos":[185,18]}]},{"state_before":"C : Type u_1\ninst✝¹ : Category.{?u.192952, u_1} C\ninst✝ : HasShift C ℤ\n⊢ ∀ {X Y : Triangle Cᵒᵖ} (f : X ⟶ Y),\n    (inverse C ⋙ functor C).map f ≫\n        ((fun T =>\n              ((inverse C ⋙ functor C).obj T).isoMk ((𝟭 (Triangle Cᵒᵖ)).obj T)\n                (Iso.refl ((inverse C ⋙ functor C).obj T).obj₁) (Iso.refl ((inverse C ⋙ functor C).obj T).obj₂)\n                (Iso.refl ((inverse C ⋙ functor C).obj T).obj₃) ⋯ ⋯ ⋯)\n            Y).hom =\n      ((fun T =>\n              ((inverse C ⋙ functor C).obj T).isoMk ((𝟭 (Triangle Cᵒᵖ)).obj T)\n                (Iso.refl ((inverse C ⋙ functor C).obj T).obj₁) (Iso.refl ((inverse C ⋙ functor C).obj T).obj₂)\n                (Iso.refl ((inverse C ⋙ functor C).obj T).obj₃) ⋯ ⋯ ⋯)\n            X).hom ≫\n        (𝟭 (Triangle Cᵒᵖ)).map f","state_after":"no goals","tactic":"aesop_cat","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]}]}]}
+{"url":"Mathlib/Data/List/Basic.lean","commit":"","full_name":"List.disjoint_pmap","start":[2745,0],"end":[2754,47],"file_path":"Mathlib/Data/List/Basic.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\np : α → Prop\nf : (a : α) → p a → β\ns t : List α\nhs : ∀ (a : α), a ∈ s → p a\nht : ∀ (a : α), a ∈ t → p a\nhf : ∀ (a a' : α) (ha : p a) (ha' : p a'), f a ha = f a' ha' → a = a'\nh : s.Disjoint t\n⊢ (pmap f s hs).Disjoint (pmap f t ht)","state_after":"ι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\np : α → Prop\nf : (a : α) → p a → β\ns t : List α\nhs : ∀ (a : α), a ∈ s → p a\nht : ∀ (a : α), a ∈ t → p a\nhf : ∀ (a a' : α) (ha : p a) (ha' : p a'), f a ha = f a' ha' → a = a'\nh : s.Disjoint t\n⊢ ∀ ⦃a : β⦄, (∃ a_1 h, f a_1 ⋯ = a) → (∃ a_2 h, f a_2 ⋯ = a) → False","tactic":"simp only [Disjoint, mem_pmap]","premises":[{"full_name":"List.Disjoint","def_path":".lake/packages/batteries/Batteries/Data/List/Basic.lean","def_pos":[773,4],"def_end_pos":[773,12]},{"full_name":"List.mem_pmap","def_path":"Mathlib/Data/List/Basic.lean","def_pos":[1917,8],"def_end_pos":[1917,16]}]},{"state_before":"ι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\np : α → Prop\nf : (a : α) → p a → β\ns t : List α\nhs : ∀ (a : α), a ∈ s → p a\nht : ∀ (a : α), a ∈ t → p a\nhf : ∀ (a a' : α) (ha : p a) (ha' : p a'), f a ha = f a' ha' → a = a'\nh : s.Disjoint t\n⊢ ∀ ⦃a : β⦄, (∃ a_1 h, f a_1 ⋯ = a) → (∃ a_2 h, f a_2 ⋯ = a) → False","state_after":"case intro.intro.intro.intro\nι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\np : α → Prop\nf : (a : α) → p a → β\ns t : List α\nhs : ∀ (a : α), a ∈ s → p a\nht : ∀ (a : α), a ∈ t → p a\nhf : ∀ (a a' : α) (ha : p a) (ha' : p a'), f a ha = f a' ha' → a = a'\nh : s.Disjoint t\na : α\nha : a ∈ s\na' : α\nha' : a' ∈ t\nha'' : f a' ⋯ = f a ⋯\n⊢ False","tactic":"rintro b ⟨a, ha, rfl⟩ ⟨a', ha', ha''⟩","premises":[]},{"state_before":"case intro.intro.intro.intro\nι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\np : α → Prop\nf : (a : α) → p a → β\ns t : List α\nhs : ∀ (a : α), a ∈ s → p a\nht : ∀ (a : α), a ∈ t → p a\nhf : ∀ (a a' : α) (ha : p a) (ha' : p a'), f a ha = f a' ha' → a = a'\nh : s.Disjoint t\na : α\nha : a ∈ s\na' : α\nha' : a' ∈ t\nha'' : f a' ⋯ = f a ⋯\n⊢ False","state_after":"case intro.intro.intro.intro\nι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\np : α → Prop\nf : (a : α) → p a → β\ns t : List α\nhs : ∀ (a : α), a ∈ s → p a\nht : ∀ (a : α), a ∈ t → p a\nhf : ∀ (a a' : α) (ha : p a) (ha' : p a'), f a ha = f a' ha' → a = a'\nh : s.Disjoint t\na : α\nha : a ∈ s\na' : α\nha' : a' ∈ t\nha'' : f a' ⋯ = f a ⋯\n⊢ a ∈ t","tactic":"apply h ha","premises":[]},{"state_before":"case intro.intro.intro.intro\nι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\np : α → Prop\nf : (a : α) → p a → β\ns t : List α\nhs : ∀ (a : α), a ∈ s → p a\nht : ∀ (a : α), a ∈ t → p a\nhf : ∀ (a a' : α) (ha : p a) (ha' : p a'), f a ha = f a' ha' → a = a'\nh : s.Disjoint t\na : α\nha : a ∈ s\na' : α\nha' : a' ∈ t\nha'' : f a' ⋯ = f a ⋯\n⊢ a ∈ t","state_after":"no goals","tactic":"rwa [hf a a' (hs a ha) (ht a' ha') ha''.symm]","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]}]}]}
+{"url":"Mathlib/GroupTheory/Congruence/Basic.lean","commit":"","full_name":"AddCon.quotientKerEquivOfRightInverse_apply","start":[898,0],"end":[908,61],"file_path":"Mathlib/GroupTheory/Congruence/Basic.lean","tactics":[{"state_before":"M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝² : MulOneClass M\ninst✝¹ : MulOneClass N\ninst✝ : MulOneClass P\nc : Con M\nx✝ y : M\nf✝ f : M →* P\ng : P → M\nhf : Function.RightInverse g ⇑f\nx : (ker f).Quotient\n⊢ (kerLift f) ((toQuotient ∘ g) ((kerLift f) x)) = (kerLift f) x","state_after":"no goals","tactic":"rw [Function.comp_apply, kerLift_mk, hf]","premises":[{"full_name":"Con.kerLift_mk","def_path":"Mathlib/GroupTheory/Congruence/Basic.lean","def_pos":[848,8],"def_end_pos":[848,18]},{"full_name":"Function.comp_apply","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[35,16],"def_end_pos":[35,35]}]},{"state_before":"M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝² : MulOneClass M\ninst✝¹ : MulOneClass N\ninst✝ : MulOneClass P\nc : Con M\nx✝ y : M\nf✝ f : M →* P\ng : P → M\nhf : Function.RightInverse g ⇑f\nx : P\n⊢ (kerLift f) ((toQuotient ∘ g) x) = x","state_after":"M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝² : MulOneClass M\ninst✝¹ : MulOneClass N\ninst✝ : MulOneClass P\nc : Con M\nx✝ y : M\nf✝ f : M →* P\ng : P → M\nhf : Function.RightInverse g ⇑f\nx : P\n⊢ (kerLift f) ((toQuotient ∘ g) x) = f (g x)","tactic":"(conv_rhs => rw [← hf x])","premises":[]},{"state_before":"M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝² : MulOneClass M\ninst✝¹ : MulOneClass N\ninst✝ : MulOneClass P\nc : Con M\nx✝ y : M\nf✝ f : M →* P\ng : P → M\nhf : Function.RightInverse g ⇑f\nx : P\n⊢ (kerLift f) ((toQuotient ∘ g) x) = f (g x)","state_after":"no goals","tactic":"rfl","premises":[]}]}
+{"url":"Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean","commit":"","full_name":"pow_div_pow_eventuallyEq_atBot","start":[45,0],"end":[49,24],"file_path":"Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\np q : ℕ\n⊢ (fun x => x ^ p / x ^ q) =ᶠ[atBot] fun x => x ^ (↑p - ↑q)","state_after":"𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\np q : ℕ\n⊢ ∀ x < 0, (fun x => x ^ p / x ^ q) x = (fun x => x ^ (↑p - ↑q)) x","tactic":"apply (eventually_lt_atBot (0 : 𝕜)).mono fun x hx => _","premises":[{"full_name":"Filter.Eventually.mono","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1002,8],"def_end_pos":[1002,23]},{"full_name":"Filter.eventually_lt_atBot","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[192,8],"def_end_pos":[192,27]}]},{"state_before":"𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\np q : ℕ\n⊢ ∀ x < 0, (fun x => x ^ p / x ^ q) x = (fun x => x ^ (↑p - ↑q)) x","state_after":"𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\np q : ℕ\nx : 𝕜\nhx : x < 0\n⊢ (fun x => x ^ p / x ^ q) x = (fun x => x ^ (↑p - ↑q)) x","tactic":"intro x hx","premises":[]},{"state_before":"𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\np q : ℕ\nx : 𝕜\nhx : x < 0\n⊢ (fun x => x ^ p / x ^ q) x = (fun x => x ^ (↑p - ↑q)) x","state_after":"no goals","tactic":"simp [zpow_sub₀ hx.ne]","premises":[{"full_name":"zpow_sub₀","def_path":"Mathlib/Algebra/GroupWithZero/Units/Basic.lean","def_pos":[355,6],"def_end_pos":[355,15]}]}]}
+{"url":"Mathlib/CategoryTheory/Preadditive/AdditiveFunctor.lean","commit":"","full_name":"CategoryTheory.Functor.additive_of_full_essSurj_comp","start":[102,0],"end":[114,18],"file_path":"Mathlib/CategoryTheory/Preadditive/AdditiveFunctor.lean","tactics":[{"state_before":"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁹ : Category.{u_4, u_1} C\ninst✝⁸ : Category.{u_5, u_2} D\ninst✝⁷ : Category.{u_6, u_3} E\ninst✝⁶ : Preadditive C\ninst✝⁵ : Preadditive D\ninst✝⁴ : Preadditive E\nF : C ⥤ D\ninst✝³ : F.Additive\ninst✝² : F.Full\ninst✝¹ : F.EssSurj\nG : D ⥤ E\ninst✝ : (F ⋙ G).Additive\nX Y : D\nf g : X ⟶ Y\n⊢ G.map (f + g) = G.map f + G.map g","state_after":"case intro\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁹ : Category.{u_4, u_1} C\ninst✝⁸ : Category.{u_5, u_2} D\ninst✝⁷ : Category.{u_6, u_3} E\ninst✝⁶ : Preadditive C\ninst✝⁵ : Preadditive D\ninst✝⁴ : Preadditive E\nF : C ⥤ D\ninst✝³ : F.Additive\ninst✝² : F.Full\ninst✝¹ : F.EssSurj\nG : D ⥤ E\ninst✝ : (F ⋙ G).Additive\nX Y : D\nf g : X ⟶ Y\nf' : F.objPreimage X ⟶ F.objPreimage Y\nhf' : F.map f' = (F.objObjPreimageIso X).hom ≫ f ≫ (F.objObjPreimageIso Y).inv\n⊢ G.map (f + g) = G.map f + G.map g","tactic":"obtain ⟨f', hf'⟩ := F.map_surjective ((F.objObjPreimageIso X).hom ≫ f ≫\n      (F.objObjPreimageIso Y).inv)","premises":[{"full_name":"CategoryTheory.CategoryStruct.comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]},{"full_name":"CategoryTheory.Functor.map_surjective","def_path":"Mathlib/CategoryTheory/Functor/FullyFaithful.lean","def_pos":[69,8],"def_end_pos":[69,22]},{"full_name":"CategoryTheory.Functor.objObjPreimageIso","def_path":"Mathlib/CategoryTheory/EssentialImage.lean","def_pos":[136,4],"def_end_pos":[136,21]},{"full_name":"CategoryTheory.Iso.hom","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[51,2],"def_end_pos":[51,5]},{"full_name":"CategoryTheory.Iso.inv","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[53,2],"def_end_pos":[53,5]}]},{"state_before":"case intro\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁹ : Category.{u_4, u_1} C\ninst✝⁸ : Category.{u_5, u_2} D\ninst✝⁷ : Category.{u_6, u_3} E\ninst✝⁶ : Preadditive C\ninst✝⁵ : Preadditive D\ninst✝⁴ : Preadditive E\nF : C ⥤ D\ninst✝³ : F.Additive\ninst✝² : F.Full\ninst✝¹ : F.EssSurj\nG : D ⥤ E\ninst✝ : (F ⋙ G).Additive\nX Y : D\nf g : X ⟶ Y\nf' : F.objPreimage X ⟶ F.objPreimage Y\nhf' : F.map f' = (F.objObjPreimageIso X).hom ≫ f ≫ (F.objObjPreimageIso Y).inv\n⊢ G.map (f + g) = G.map f + G.map g","state_after":"case intro.intro\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁹ : Category.{u_4, u_1} C\ninst✝⁸ : Category.{u_5, u_2} D\ninst✝⁷ : Category.{u_6, u_3} E\ninst✝⁶ : Preadditive C\ninst✝⁵ : Preadditive D\ninst✝⁴ : Preadditive E\nF : C ⥤ D\ninst✝³ : F.Additive\ninst✝² : F.Full\ninst✝¹ : F.EssSurj\nG : D ⥤ E\ninst✝ : (F ⋙ G).Additive\nX Y : D\nf g : X ⟶ Y\nf' : F.objPreimage X ⟶ F.objPreimage Y\nhf' : F.map f' = (F.objObjPreimageIso X).hom ≫ f ≫ (F.objObjPreimageIso Y).inv\ng' : F.objPreimage X ⟶ F.objPreimage Y\nhg' : F.map g' = (F.objObjPreimageIso X).hom ≫ g ≫ (F.objObjPreimageIso Y).inv\n⊢ G.map (f + g) = G.map f + G.map g","tactic":"obtain ⟨g', hg'⟩ := F.map_surjective ((F.objObjPreimageIso X).hom ≫ g ≫\n      (F.objObjPreimageIso Y).inv)","premises":[{"full_name":"CategoryTheory.CategoryStruct.comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]},{"full_name":"CategoryTheory.Functor.map_surjective","def_path":"Mathlib/CategoryTheory/Functor/FullyFaithful.lean","def_pos":[69,8],"def_end_pos":[69,22]},{"full_name":"CategoryTheory.Functor.objObjPreimageIso","def_path":"Mathlib/CategoryTheory/EssentialImage.lean","def_pos":[136,4],"def_end_pos":[136,21]},{"full_name":"CategoryTheory.Iso.hom","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[51,2],"def_end_pos":[51,5]},{"full_name":"CategoryTheory.Iso.inv","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[53,2],"def_end_pos":[53,5]}]},{"state_before":"case intro.intro\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁹ : Category.{u_4, u_1} C\ninst✝⁸ : Category.{u_5, u_2} D\ninst✝⁷ : Category.{u_6, u_3} E\ninst✝⁶ : Preadditive C\ninst✝⁵ : Preadditive D\ninst✝⁴ : Preadditive E\nF : C ⥤ D\ninst✝³ : F.Additive\ninst✝² : F.Full\ninst✝¹ : F.EssSurj\nG : D ⥤ E\ninst✝ : (F ⋙ G).Additive\nX Y : D\nf g : X ⟶ Y\nf' : F.objPreimage X ⟶ F.objPreimage Y\nhf' : F.map f' = (F.objObjPreimageIso X).hom ≫ f ≫ (F.objObjPreimageIso Y).inv\ng' : F.objPreimage X ⟶ F.objPreimage Y\nhg' : F.map g' = (F.objObjPreimageIso X).hom ≫ g ≫ (F.objObjPreimageIso Y).inv\n⊢ G.map (f + g) = G.map f + G.map g","state_after":"case intro.intro\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁹ : Category.{u_4, u_1} C\ninst✝⁸ : Category.{u_5, u_2} D\ninst✝⁷ : Category.{u_6, u_3} E\ninst✝⁶ : Preadditive C\ninst✝⁵ : Preadditive D\ninst✝⁴ : Preadditive E\nF : C ⥤ D\ninst✝³ : F.Additive\ninst✝² : F.Full\ninst✝¹ : F.EssSurj\nG : D ⥤ E\ninst✝ : (F ⋙ G).Additive\nX Y : D\nf g : X ⟶ Y\nf' : F.objPreimage X ⟶ F.objPreimage Y\nhf' : F.map f' = (F.objObjPreimageIso X).hom ≫ f ≫ (F.objObjPreimageIso Y).inv\ng' : F.objPreimage X ⟶ F.objPreimage Y\nhg' : F.map g' = (F.objObjPreimageIso X).hom ≫ g ≫ (F.objObjPreimageIso Y).inv\n⊢ G.map\n      ((F.objObjPreimageIso X).hom ≫ f ≫ (F.objObjPreimageIso Y).inv +\n        (F.objObjPreimageIso X).hom ≫ g ≫ (F.objObjPreimageIso Y).inv) =\n    G.map ((F.objObjPreimageIso X).hom ≫ f ≫ (F.objObjPreimageIso Y).inv) +\n      G.map ((F.objObjPreimageIso X).hom ≫ g ≫ (F.objObjPreimageIso Y).inv)","tactic":"simp only [← cancel_mono (G.map (F.objObjPreimageIso Y).inv),\n      ← cancel_epi (G.map (F.objObjPreimageIso X).hom),\n      Preadditive.add_comp, Preadditive.comp_add, ← Functor.map_comp]","premises":[{"full_name":"CategoryTheory.Functor.map_comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[41,2],"def_end_pos":[41,10]},{"full_name":"CategoryTheory.Functor.objObjPreimageIso","def_path":"Mathlib/CategoryTheory/EssentialImage.lean","def_pos":[136,4],"def_end_pos":[136,21]},{"full_name":"CategoryTheory.Iso.hom","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[51,2],"def_end_pos":[51,5]},{"full_name":"CategoryTheory.Iso.inv","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[53,2],"def_end_pos":[53,5]},{"full_name":"CategoryTheory.Preadditive.add_comp","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[58,2],"def_end_pos":[58,10]},{"full_name":"CategoryTheory.Preadditive.comp_add","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[60,2],"def_end_pos":[60,10]},{"full_name":"CategoryTheory.cancel_epi","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[260,8],"def_end_pos":[260,18]},{"full_name":"CategoryTheory.cancel_mono","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[263,8],"def_end_pos":[263,19]},{"full_name":"Prefunctor.map","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[57,2],"def_end_pos":[57,5]}]},{"state_before":"case intro.intro\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁹ : Category.{u_4, u_1} C\ninst✝⁸ : Category.{u_5, u_2} D\ninst✝⁷ : Category.{u_6, u_3} E\ninst✝⁶ : Preadditive C\ninst✝⁵ : Preadditive D\ninst✝⁴ : Preadditive E\nF : C ⥤ D\ninst✝³ : F.Additive\ninst✝² : F.Full\ninst✝¹ : F.EssSurj\nG : D ⥤ E\ninst✝ : (F ⋙ G).Additive\nX Y : D\nf g : X ⟶ Y\nf' : F.objPreimage X ⟶ F.objPreimage Y\nhf' : F.map f' = (F.objObjPreimageIso X).hom ≫ f ≫ (F.objObjPreimageIso Y).inv\ng' : F.objPreimage X ⟶ F.objPreimage Y\nhg' : F.map g' = (F.objObjPreimageIso X).hom ≫ g ≫ (F.objObjPreimageIso Y).inv\n⊢ G.map\n      ((F.objObjPreimageIso X).hom ≫ f ≫ (F.objObjPreimageIso Y).inv +\n        (F.objObjPreimageIso X).hom ≫ g ≫ (F.objObjPreimageIso Y).inv) =\n    G.map ((F.objObjPreimageIso X).hom ≫ f ≫ (F.objObjPreimageIso Y).inv) +\n      G.map ((F.objObjPreimageIso X).hom ≫ g ≫ (F.objObjPreimageIso Y).inv)","state_after":"case intro.intro\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁹ : Category.{u_4, u_1} C\ninst✝⁸ : Category.{u_5, u_2} D\ninst✝⁷ : Category.{u_6, u_3} E\ninst✝⁶ : Preadditive C\ninst✝⁵ : Preadditive D\ninst✝⁴ : Preadditive E\nF : C ⥤ D\ninst✝³ : F.Additive\ninst✝² : F.Full\ninst✝¹ : F.EssSurj\nG : D ⥤ E\ninst✝ : (F ⋙ G).Additive\nX Y : D\nf g : X ⟶ Y\nf' : F.objPreimage X ⟶ F.objPreimage Y\nhf' : F.map f' = (F.objObjPreimageIso X).hom ≫ f ≫ (F.objObjPreimageIso Y).inv\ng' : F.objPreimage X ⟶ F.objPreimage Y\nhg' : F.map g' = (F.objObjPreimageIso X).hom ≫ g ≫ (F.objObjPreimageIso Y).inv\n⊢ G.map (F.map f' + F.map g') = (F ⋙ G).map (f' + g')","tactic":"erw [← hf', ← hg', ← (F ⋙ G).map_add]","premises":[{"full_name":"CategoryTheory.Functor.comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[100,4],"def_end_pos":[100,8]},{"full_name":"CategoryTheory.Functor.map_add","def_path":"Mathlib/CategoryTheory/Preadditive/AdditiveFunctor.lean","def_pos":[52,8],"def_end_pos":[52,15]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]}]},{"state_before":"case intro.intro\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁹ : Category.{u_4, u_1} C\ninst✝⁸ : Category.{u_5, u_2} D\ninst✝⁷ : Category.{u_6, u_3} E\ninst✝⁶ : Preadditive C\ninst✝⁵ : Preadditive D\ninst✝⁴ : Preadditive E\nF : C ⥤ D\ninst✝³ : F.Additive\ninst✝² : F.Full\ninst✝¹ : F.EssSurj\nG : D ⥤ E\ninst✝ : (F ⋙ G).Additive\nX Y : D\nf g : X ⟶ Y\nf' : F.objPreimage X ⟶ F.objPreimage Y\nhf' : F.map f' = (F.objObjPreimageIso X).hom ≫ f ≫ (F.objObjPreimageIso Y).inv\ng' : F.objPreimage X ⟶ F.objPreimage Y\nhg' : F.map g' = (F.objObjPreimageIso X).hom ≫ g ≫ (F.objObjPreimageIso Y).inv\n⊢ G.map (F.map f' + F.map g') = (F ⋙ G).map (f' + g')","state_after":"case intro.intro\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁹ : Category.{u_4, u_1} C\ninst✝⁸ : Category.{u_5, u_2} D\ninst✝⁷ : Category.{u_6, u_3} E\ninst✝⁶ : Preadditive C\ninst✝⁵ : Preadditive D\ninst✝⁴ : Preadditive E\nF : C ⥤ D\ninst✝³ : F.Additive\ninst✝² : F.Full\ninst✝¹ : F.EssSurj\nG : D ⥤ E\ninst✝ : (F ⋙ G).Additive\nX Y : D\nf g : X ⟶ Y\nf' : F.objPreimage X ⟶ F.objPreimage Y\nhf' : F.map f' = (F.objObjPreimageIso X).hom ≫ f ≫ (F.objObjPreimageIso Y).inv\ng' : F.objPreimage X ⟶ F.objPreimage Y\nhg' : F.map g' = (F.objObjPreimageIso X).hom ≫ g ≫ (F.objObjPreimageIso Y).inv\n⊢ G.map (F.map f' + F.map g') = G.map (F.map (f' + g'))","tactic":"dsimp","premises":[]},{"state_before":"case intro.intro\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁹ : Category.{u_4, u_1} C\ninst✝⁸ : Category.{u_5, u_2} D\ninst✝⁷ : Category.{u_6, u_3} E\ninst✝⁶ : Preadditive C\ninst✝⁵ : Preadditive D\ninst✝⁴ : Preadditive E\nF : C ⥤ D\ninst✝³ : F.Additive\ninst✝² : F.Full\ninst✝¹ : F.EssSurj\nG : D ⥤ E\ninst✝ : (F ⋙ G).Additive\nX Y : D\nf g : X ⟶ Y\nf' : F.objPreimage X ⟶ F.objPreimage Y\nhf' : F.map f' = (F.objObjPreimageIso X).hom ≫ f ≫ (F.objObjPreimageIso Y).inv\ng' : F.objPreimage X ⟶ F.objPreimage Y\nhg' : F.map g' = (F.objObjPreimageIso X).hom ≫ g ≫ (F.objObjPreimageIso Y).inv\n⊢ G.map (F.map f' + F.map g') = G.map (F.map (f' + g'))","state_after":"no goals","tactic":"rw [F.map_add]","premises":[{"full_name":"CategoryTheory.Functor.map_add","def_path":"Mathlib/CategoryTheory/Preadditive/AdditiveFunctor.lean","def_pos":[52,8],"def_end_pos":[52,15]}]}]}
+{"url":"Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean","commit":"","full_name":"CategoryTheory.Pretriangulated.comp_distTriang_mor_zero₁₂","start":[112,0],"end":[125,70],"file_path":"Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean","tactics":[{"state_before":"C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : Preadditive C\ninst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive\nhC : Pretriangulated C\nT : Triangle C\nH : T ∈ distinguishedTriangles\n⊢ T.mor₁ ≫ T.mor₂ = 0","state_after":"case intro\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : Preadditive C\ninst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive\nhC : Pretriangulated C\nT : Triangle C\nH : T ∈ distinguishedTriangles\nc : (contractibleTriangle T.obj₁).obj₃ ⟶ T.obj₃\nhc :\n  (contractibleTriangle T.obj₁).mor₂ ≫ c = T.mor₁ ≫ T.mor₂ ∧\n    (contractibleTriangle T.obj₁).mor₃ ≫ (shiftFunctor C 1).map (𝟙 T.obj₁) = c ≫ T.mor₃\n⊢ T.mor₁ ≫ T.mor₂ = 0","tactic":"obtain ⟨c, hc⟩ :=\n    complete_distinguished_triangle_morphism _ _ (contractible_distinguished T.obj₁) H (𝟙 T.obj₁)\n      T.mor₁ rfl","premises":[{"full_name":"CategoryTheory.CategoryStruct.id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[87,2],"def_end_pos":[87,4]},{"full_name":"CategoryTheory.Pretriangulated.Triangle.mor₁","def_path":"Mathlib/CategoryTheory/Triangulated/Basic.lean","def_pos":[48,2],"def_end_pos":[48,6]},{"full_name":"CategoryTheory.Pretriangulated.Triangle.obj₁","def_path":"Mathlib/CategoryTheory/Triangulated/Basic.lean","def_pos":[42,2],"def_end_pos":[42,6]},{"full_name":"CategoryTheory.Pretriangulated.complete_distinguished_triangle_morphism","def_path":"Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean","def_pos":[78,2],"def_end_pos":[78,42]},{"full_name":"CategoryTheory.Pretriangulated.contractible_distinguished","def_path":"Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean","def_pos":[68,2],"def_end_pos":[68,28]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"case intro\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : Preadditive C\ninst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive\nhC : Pretriangulated C\nT : Triangle C\nH : T ∈ distinguishedTriangles\nc : (contractibleTriangle T.obj₁).obj₃ ⟶ T.obj₃\nhc :\n  (contractibleTriangle T.obj₁).mor₂ ≫ c = T.mor₁ ≫ T.mor₂ ∧\n    (contractibleTriangle T.obj₁).mor₃ ≫ (shiftFunctor C 1).map (𝟙 T.obj₁) = c ≫ T.mor₃\n⊢ T.mor₁ ≫ T.mor₂ = 0","state_after":"no goals","tactic":"simpa only [contractibleTriangle_mor₂, zero_comp] using hc.left.symm","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"CategoryTheory.Limits.zero_comp","def_path":"Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean","def_pos":[66,8],"def_end_pos":[66,17]},{"full_name":"CategoryTheory.Pretriangulated.contractibleTriangle_mor₂","def_path":"Mathlib/CategoryTheory/Triangulated/Basic.lean","def_pos":[79,2],"def_end_pos":[79,8]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]}]}]}
+{"url":"Mathlib/Computability/TuringMachine.lean","commit":"","full_name":"Turing.TM0to1.tr_respects","start":[1825,0],"end":[1846,9],"file_path":"Mathlib/Computability/TuringMachine.lean","tactics":[{"state_before":"Γ : Type u_1\ninst✝¹ : Inhabited Γ\nΛ : Type u_2\ninst✝ : Inhabited Λ\nM : TM0.Machine Γ Λ\nx✝ : Cfg₀\nq : Λ\nT : Tape Γ\n⊢ FRespects (TM1.step (tr M)) (trCfg M) (trCfg M { q := q, Tape := T }) (TM0.step M { q := q, Tape := T })","state_after":"case none\nΓ : Type u_1\ninst✝¹ : Inhabited Γ\nΛ : Type u_2\ninst✝ : Inhabited Λ\nM : TM0.Machine Γ Λ\nx✝ : Cfg₀\nq : Λ\nT : Tape Γ\ne : M q T.head = none\n⊢ FRespects (TM1.step (tr M)) (trCfg M) (trCfg M { q := q, Tape := T }) (TM0.step M { q := q, Tape := T })\n\ncase some\nΓ : Type u_1\ninst✝¹ : Inhabited Γ\nΛ : Type u_2\ninst✝ : Inhabited Λ\nM : TM0.Machine Γ Λ\nx✝ : Cfg₀\nq : Λ\nT : Tape Γ\nval : Λ × TM0.Stmt Γ\ne : M q T.head = some val\n⊢ FRespects (TM1.step (tr M)) (trCfg M) (trCfg M { q := q, Tape := T }) (TM0.step M { q := q, Tape := T })","tactic":"cases' e : M q T.1 with val","premises":[{"full_name":"Turing.Tape.head","def_path":"Mathlib/Computability/TuringMachine.lean","def_pos":[453,2],"def_end_pos":[453,6]}]},{"state_before":"case some\nΓ : Type u_1\ninst✝¹ : Inhabited Γ\nΛ : Type u_2\ninst✝ : Inhabited Λ\nM : TM0.Machine Γ Λ\nx✝ : Cfg₀\nq : Λ\nT : Tape Γ\nval : Λ × TM0.Stmt Γ\ne : M q T.head = some val\n⊢ FRespects (TM1.step (tr M)) (trCfg M) (trCfg M { q := q, Tape := T }) (TM0.step M { q := q, Tape := T })","state_after":"case some.mk\nΓ : Type u_1\ninst✝¹ : Inhabited Γ\nΛ : Type u_2\ninst✝ : Inhabited Λ\nM : TM0.Machine Γ Λ\nx✝ : Cfg₀\nq : Λ\nT : Tape Γ\nq' : Λ\ns : TM0.Stmt Γ\ne : M q T.head = some (q', s)\n⊢ FRespects (TM1.step (tr M)) (trCfg M) (trCfg M { q := q, Tape := T }) (TM0.step M { q := q, Tape := T })","tactic":"cases' val with q' s","premises":[]},{"state_before":"case some.mk\nΓ : Type u_1\ninst✝¹ : Inhabited Γ\nΛ : Type u_2\ninst✝ : Inhabited Λ\nM : TM0.Machine Γ Λ\nx✝ : Cfg₀\nq : Λ\nT : Tape Γ\nq' : Λ\ns : TM0.Stmt Γ\ne : M q T.head = some (q', s)\n⊢ FRespects (TM1.step (tr M)) (trCfg M) (trCfg M { q := q, Tape := T }) (TM0.step M { q := q, Tape := T })","state_after":"case some.mk\nΓ : Type u_1\ninst✝¹ : Inhabited Γ\nΛ : Type u_2\ninst✝ : Inhabited Λ\nM : TM0.Machine Γ Λ\nx✝ : Cfg₀\nq : Λ\nT : Tape Γ\nq' : Λ\ns : TM0.Stmt Γ\ne : M q T.head = some (q', s)\n⊢ Reaches₁ (TM1.step (tr M)) { l := some (Λ'.normal q), var := (), Tape := T }\n    {\n      l :=\n        match\n          match\n            M q'\n              (match s with\n                | TM0.Stmt.move d => Tape.move d T\n                | TM0.Stmt.write a => Tape.write a T).head with\n          | some val => true\n          | none => false with\n        | true => some (Λ'.normal q')\n        | false => none,\n      var := (),\n      Tape :=\n        match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T }","tactic":"simp only [FRespects, TM0.step, trCfg, e, Option.isSome, cond, Option.map_some']","premises":[{"full_name":"Option.isSome","def_path":".lake/packages/lean4/src/lean/Init/Data/Option/Basic.lean","def_pos":[25,14],"def_end_pos":[25,20]},{"full_name":"Option.map_some'","def_path":".lake/packages/lean4/src/lean/Init/Data/Option/Basic.lean","def_pos":[115,16],"def_end_pos":[115,25]},{"full_name":"Turing.FRespects","def_path":"Mathlib/Computability/TuringMachine.lean","def_pos":[848,4],"def_end_pos":[848,13]},{"full_name":"Turing.TM0.step","def_path":"Mathlib/Computability/TuringMachine.lean","def_pos":[957,4],"def_end_pos":[957,8]},{"full_name":"Turing.TM0to1.trCfg","def_path":"Mathlib/Computability/TuringMachine.lean","def_pos":[1822,4],"def_end_pos":[1822,9]},{"full_name":"cond","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1006,20],"def_end_pos":[1006,24]}]},{"state_before":"case some.mk\nΓ : Type u_1\ninst✝¹ : Inhabited Γ\nΛ : Type u_2\ninst✝ : Inhabited Λ\nM : TM0.Machine Γ Λ\nx✝ : Cfg₀\nq : Λ\nT : Tape Γ\nq' : Λ\ns : TM0.Stmt Γ\ne : M q T.head = some (q', s)\n⊢ Reaches₁ (TM1.step (tr M)) { l := some (Λ'.normal q), var := (), Tape := T }\n    {\n      l :=\n        match\n          match\n            M q'\n              (match s with\n                | TM0.Stmt.move d => Tape.move d T\n                | TM0.Stmt.write a => Tape.write a T).head with\n          | some val => true\n          | none => false with\n        | true => some (Λ'.normal q')\n        | false => none,\n      var := (),\n      Tape :=\n        match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T }","state_after":"case some.mk\nΓ : Type u_1\ninst✝¹ : Inhabited Γ\nΛ : Type u_2\ninst✝ : Inhabited Λ\nM : TM0.Machine Γ Λ\nx✝ : Cfg₀\nq : Λ\nT : Tape Γ\nq' : Λ\ns : TM0.Stmt Γ\n⊢ M q T.head = some (q', s) →\n    Reaches₁ (TM1.step (tr M)) { l := some (Λ'.normal q), var := (), Tape := T }\n      {\n        l :=\n          match\n            match\n              M q'\n                (match s with\n                  | TM0.Stmt.move d => Tape.move d T\n                  | TM0.Stmt.write a => Tape.write a T).head with\n            | some val => true\n            | none => false with\n          | true => some (Λ'.normal q')\n          | false => none,\n        var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }","tactic":"revert e","premises":[]},{"state_before":"case some.mk\nΓ : Type u_1\ninst✝¹ : Inhabited Γ\nΛ : Type u_2\ninst✝ : Inhabited Λ\nM : TM0.Machine Γ Λ\nx✝ : Cfg₀\nq : Λ\nT : Tape Γ\nq' : Λ\ns : TM0.Stmt Γ\n⊢ M q T.head = some (q', s) →\n    Reaches₁ (TM1.step (tr M)) { l := some (Λ'.normal q), var := (), Tape := T }\n      {\n        l :=\n          match\n            match\n              M q'\n                (match s with\n                  | TM0.Stmt.move d => Tape.move d T\n                  | TM0.Stmt.write a => Tape.write a T).head with\n            | some val => true\n            | none => false with\n          | true => some (Λ'.normal q')\n          | false => none,\n        var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }","state_after":"case some.mk\nΓ : Type u_1\ninst✝¹ : Inhabited Γ\nΛ : Type u_2\ninst✝ : Inhabited Λ\nM : TM0.Machine Γ Λ\nx✝ : Cfg₀\nq : Λ\nT : Tape Γ\nq' : Λ\ns : TM0.Stmt Γ\nthis :\n  TM1.step (tr M) { l := some (Λ'.act s q'), var := (), Tape := T } =\n    some\n      { l := some (Λ'.normal q'), var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }\n⊢ M q T.head = some (q', s) →\n    Reaches₁ (TM1.step (tr M)) { l := some (Λ'.normal q), var := (), Tape := T }\n      {\n        l :=\n          match\n            match\n              M q'\n                (match s with\n                  | TM0.Stmt.move d => Tape.move d T\n                  | TM0.Stmt.write a => Tape.write a T).head with\n            | some val => true\n            | none => false with\n          | true => some (Λ'.normal q')\n          | false => none,\n        var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }","tactic":"have : TM1.step (tr M) ⟨some (Λ'.act s q'), (), T⟩ = some ⟨some (Λ'.normal q'), (), match s with\n        | TM0.Stmt.move d => T.move d\n        | TM0.Stmt.write a => T.write a⟩ := by\n      cases' s with d a <;> rfl","premises":[{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]},{"full_name":"Turing.TM0.Stmt.move","def_path":"Mathlib/Computability/TuringMachine.lean","def_pos":[912,4],"def_end_pos":[912,8]},{"full_name":"Turing.TM0.Stmt.write","def_path":"Mathlib/Computability/TuringMachine.lean","def_pos":[913,4],"def_end_pos":[913,9]},{"full_name":"Turing.TM0to1.tr","def_path":"Mathlib/Computability/TuringMachine.lean","def_pos":[1812,4],"def_end_pos":[1812,6]},{"full_name":"Turing.TM0to1.Λ'.act","def_path":"Mathlib/Computability/TuringMachine.lean","def_pos":[1794,4],"def_end_pos":[1794,7]},{"full_name":"Turing.TM0to1.Λ'.normal","def_path":"Mathlib/Computability/TuringMachine.lean","def_pos":[1793,4],"def_end_pos":[1793,10]},{"full_name":"Turing.TM1.step","def_path":"Mathlib/Computability/TuringMachine.lean","def_pos":[1152,4],"def_end_pos":[1152,8]},{"full_name":"Turing.Tape.move","def_path":"Mathlib/Computability/TuringMachine.lean","def_pos":[479,4],"def_end_pos":[479,13]},{"full_name":"Turing.Tape.write","def_path":"Mathlib/Computability/TuringMachine.lean","def_pos":[588,4],"def_end_pos":[588,14]},{"full_name":"Unit.unit","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[148,24],"def_end_pos":[148,33]}]},{"state_before":"case some.mk\nΓ : Type u_1\ninst✝¹ : Inhabited Γ\nΛ : Type u_2\ninst✝ : Inhabited Λ\nM : TM0.Machine Γ Λ\nx✝ : Cfg₀\nq : Λ\nT : Tape Γ\nq' : Λ\ns : TM0.Stmt Γ\nthis :\n  TM1.step (tr M) { l := some (Λ'.act s q'), var := (), Tape := T } =\n    some\n      { l := some (Λ'.normal q'), var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }\n⊢ M q T.head = some (q', s) →\n    Reaches₁ (TM1.step (tr M)) { l := some (Λ'.normal q), var := (), Tape := T }\n      {\n        l :=\n          match\n            match\n              M q'\n                (match s with\n                  | TM0.Stmt.move d => Tape.move d T\n                  | TM0.Stmt.write a => Tape.write a T).head with\n            | some val => true\n            | none => false with\n          | true => some (Λ'.normal q')\n          | false => none,\n        var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }","state_after":"case some.mk\nΓ : Type u_1\ninst✝¹ : Inhabited Γ\nΛ : Type u_2\ninst✝ : Inhabited Λ\nM : TM0.Machine Γ Λ\nx✝ : Cfg₀\nq : Λ\nT : Tape Γ\nq' : Λ\ns : TM0.Stmt Γ\nthis :\n  TM1.step (tr M) { l := some (Λ'.act s q'), var := (), Tape := T } =\n    some\n      { l := some (Λ'.normal q'), var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }\ne : M q T.head = some (q', s)\n⊢ Reaches₁ (TM1.step (tr M)) { l := some (Λ'.normal q), var := (), Tape := T }\n    {\n      l :=\n        match\n          match\n            M q'\n              (match s with\n                | TM0.Stmt.move d => Tape.move d T\n                | TM0.Stmt.write a => Tape.write a T).head with\n          | some val => true\n          | none => false with\n        | true => some (Λ'.normal q')\n        | false => none,\n      var := (),\n      Tape :=\n        match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T }","tactic":"intro e","premises":[]},{"state_before":"case some.mk\nΓ : Type u_1\ninst✝¹ : Inhabited Γ\nΛ : Type u_2\ninst✝ : Inhabited Λ\nM : TM0.Machine Γ Λ\nx✝ : Cfg₀\nq : Λ\nT : Tape Γ\nq' : Λ\ns : TM0.Stmt Γ\nthis :\n  TM1.step (tr M) { l := some (Λ'.act s q'), var := (), Tape := T } =\n    some\n      { l := some (Λ'.normal q'), var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }\ne : M q T.head = some (q', s)\n⊢ Reaches₁ (TM1.step (tr M)) { l := some (Λ'.normal q), var := (), Tape := T }\n    {\n      l :=\n        match\n          match\n            M q'\n              (match s with\n                | TM0.Stmt.move d => Tape.move d T\n                | TM0.Stmt.write a => Tape.write a T).head with\n          | some val => true\n          | none => false with\n        | true => some (Λ'.normal q')\n        | false => none,\n      var := (),\n      Tape :=\n        match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T }","state_after":"case some.mk.refine_1\nΓ : Type u_1\ninst✝¹ : Inhabited Γ\nΛ : Type u_2\ninst✝ : Inhabited Λ\nM : TM0.Machine Γ Λ\nx✝ : Cfg₀\nq : Λ\nT : Tape Γ\nq' : Λ\ns : TM0.Stmt Γ\nthis :\n  TM1.step (tr M) { l := some (Λ'.act s q'), var := (), Tape := T } =\n    some\n      { l := some (Λ'.normal q'), var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }\ne : M q T.head = some (q', s)\n⊢ { l := some (Λ'.act s q'), var := (), Tape := T } ∈ TM1.step (tr M) { l := some (Λ'.normal q), var := (), Tape := T }\n\ncase some.mk.refine_2\nΓ : Type u_1\ninst✝¹ : Inhabited Γ\nΛ : Type u_2\ninst✝ : Inhabited Λ\nM : TM0.Machine Γ Λ\nx✝ : Cfg₀\nq : Λ\nT : Tape Γ\nq' : Λ\ns : TM0.Stmt Γ\nthis :\n  TM1.step (tr M) { l := some (Λ'.act s q'), var := (), Tape := T } =\n    some\n      { l := some (Λ'.normal q'), var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }\ne : M q T.head = some (q', s)\n⊢ ReflTransGen (fun a b => b ∈ TM1.step (tr M) a)\n    { l := some (Λ'.normal q'), var := (),\n      Tape :=\n        match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T }\n    {\n      l :=\n        match\n          match\n            M q'\n              (match s with\n                | TM0.Stmt.move d => Tape.move d T\n                | TM0.Stmt.write a => Tape.write a T).head with\n          | some val => true\n          | none => false with\n        | true => some (Λ'.normal q')\n        | false => none,\n      var := (),\n      Tape :=\n        match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T }","tactic":"refine TransGen.head ?_ (TransGen.head' this ?_)","premises":[{"full_name":"Relation.TransGen.head","def_path":"Mathlib/Logic/Relation.lean","def_pos":[357,8],"def_end_pos":[357,12]},{"full_name":"Relation.TransGen.head'","def_path":"Mathlib/Logic/Relation.lean","def_pos":[349,8],"def_end_pos":[349,13]}]},{"state_before":"case some.mk.refine_2\nΓ : Type u_1\ninst✝¹ : Inhabited Γ\nΛ : Type u_2\ninst✝ : Inhabited Λ\nM : TM0.Machine Γ Λ\nx✝ : Cfg₀\nq : Λ\nT : Tape Γ\nq' : Λ\ns : TM0.Stmt Γ\nthis :\n  TM1.step (tr M) { l := some (Λ'.act s q'), var := (), Tape := T } =\n    some\n      { l := some (Λ'.normal q'), var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }\ne : M q T.head = some (q', s)\n⊢ ReflTransGen (fun a b => b ∈ TM1.step (tr M) a)\n    { l := some (Λ'.normal q'), var := (),\n      Tape :=\n        match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T }\n    {\n      l :=\n        match\n          match\n            M q'\n              (match s with\n                | TM0.Stmt.move d => Tape.move d T\n                | TM0.Stmt.write a => Tape.write a T).head with\n          | some val => true\n          | none => false with\n        | true => some (Λ'.normal q')\n        | false => none,\n      var := (),\n      Tape :=\n        match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T }","state_after":"case some.mk.refine_2.none\nΓ : Type u_1\ninst✝¹ : Inhabited Γ\nΛ : Type u_2\ninst✝ : Inhabited Λ\nM : TM0.Machine Γ Λ\nx✝ : Cfg₀\nq : Λ\nT : Tape Γ\nq' : Λ\ns : TM0.Stmt Γ\nthis :\n  TM1.step (tr M) { l := some (Λ'.act s q'), var := (), Tape := T } =\n    some\n      { l := some (Λ'.normal q'), var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }\ne : M q T.head = some (q', s)\ne' :\n  M q'\n      (match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T).head =\n    none\n⊢ ReflTransGen (fun a b => b ∈ TM1.step (tr M) a)\n    { l := some (Λ'.normal q'), var := (),\n      Tape :=\n        match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T }\n    {\n      l :=\n        match\n          match none with\n          | some val => true\n          | none => false with\n        | true => some (Λ'.normal q')\n        | false => none,\n      var := (),\n      Tape :=\n        match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T }\n\ncase some.mk.refine_2.some\nΓ : Type u_1\ninst✝¹ : Inhabited Γ\nΛ : Type u_2\ninst✝ : Inhabited Λ\nM : TM0.Machine Γ Λ\nx✝ : Cfg₀\nq : Λ\nT : Tape Γ\nq' : Λ\ns : TM0.Stmt Γ\nthis :\n  TM1.step (tr M) { l := some (Λ'.act s q'), var := (), Tape := T } =\n    some\n      { l := some (Λ'.normal q'), var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }\ne : M q T.head = some (q', s)\nval✝ : Λ × TM0.Stmt Γ\ne' :\n  M q'\n      (match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T).head =\n    some val✝\n⊢ ReflTransGen (fun a b => b ∈ TM1.step (tr M) a)\n    { l := some (Λ'.normal q'), var := (),\n      Tape :=\n        match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T }\n    {\n      l :=\n        match\n          match some val✝ with\n          | some val => true\n          | none => false with\n        | true => some (Λ'.normal q')\n        | false => none,\n      var := (),\n      Tape :=\n        match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T }","tactic":"cases e' : M q' _","premises":[]}]}
+{"url":"Mathlib/Algebra/Group/Basic.lean","commit":"","full_name":"sub_nsmul","start":[792,0],"end":[794,65],"file_path":"Mathlib/Algebra/Group/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nG : Type u_3\nM : Type u_4\ninst✝ : Group G\na✝ b c d : G\nn✝ : ℤ\na : G\nm n : ℕ\nh : n ≤ m\n⊢ a ^ (m - n) * a ^ n = a ^ m","state_after":"no goals","tactic":"rw [← pow_add, Nat.sub_add_cancel h]","premises":[{"full_name":"Nat.sub_add_cancel","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[893,26],"def_end_pos":[893,40]},{"full_name":"pow_add","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[598,6],"def_end_pos":[598,13]}]}]}
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+{"url":"Mathlib/Algebra/Homology/ShortComplex/ShortExact.lean","commit":"","full_name":"CategoryTheory.ShortComplex.shortExact_iff_of_iso","start":[56,0],"end":[59,34],"file_path":"Mathlib/Algebra/Homology/ShortComplex/ShortExact.lean","tactics":[{"state_before":"C : Type u_1\nD : Type u_2\ninst✝³ : Category.{u_3, u_1} C\ninst✝² : Category.{?u.3513, u_2} D\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasZeroMorphisms D\nS S₁ S₂ : ShortComplex C\ne : S₁ ≅ S₂\n⊢ S₁.ShortExact ↔ S₂.ShortExact","state_after":"case mp\nC : Type u_1\nD : Type u_2\ninst✝³ : Category.{u_3, u_1} C\ninst✝² : Category.{?u.3513, u_2} D\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasZeroMorphisms D\nS S₁ S₂ : ShortComplex C\ne : S₁ ≅ S₂\n⊢ S₁.ShortExact → S₂.ShortExact\n\ncase mpr\nC : Type u_1\nD : Type u_2\ninst✝³ : Category.{u_3, u_1} C\ninst✝² : Category.{?u.3513, u_2} D\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasZeroMorphisms D\nS S₁ S₂ : ShortComplex C\ne : S₁ ≅ S₂\n⊢ S₂.ShortExact → S₁.ShortExact","tactic":"constructor","premises":[]}]}
+{"url":"Mathlib/Data/Set/Prod.lean","commit":"","full_name":"Set.pi_nonempty_iff","start":[640,0],"end":[641,39],"file_path":"Mathlib/Data/Set/Prod.lean","tactics":[{"state_before":"ι : Type u_1\nα : ι → Type u_2\nβ : ι → Type u_3\ns s₁ s₂ : Set ι\nt t₁ t₂ : (i : ι) → Set (α i)\ni : ι\n⊢ (s.pi t).Nonempty ↔ ∀ (i : ι), ∃ x, i ∈ s → x ∈ t i","state_after":"no goals","tactic":"simp [Classical.skolem, Set.Nonempty]","premises":[{"full_name":"Classical.skolem","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[116,8],"def_end_pos":[116,14]},{"full_name":"Set.Nonempty","def_path":"Mathlib/Init/Set.lean","def_pos":[222,14],"def_end_pos":[222,22]}]}]}
+{"url":"Mathlib/RingTheory/DiscreteValuationRing/Basic.lean","commit":"","full_name":"DiscreteValuationRing.HasUnitMulPowIrreducibleFactorization.of_ufd_of_unique_irreducible","start":[212,0],"end":[230,26],"file_path":"Mathlib/RingTheory/DiscreteValuationRing/Basic.lean","tactics":[{"state_before":"R : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₁ : ∃ p, Irreducible p\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\n⊢ HasUnitMulPowIrreducibleFactorization R","state_after":"case intro\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\n⊢ HasUnitMulPowIrreducibleFactorization R","tactic":"obtain ⟨p, hp⟩ := h₁","premises":[]},{"state_before":"case intro\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\n⊢ HasUnitMulPowIrreducibleFactorization R","state_after":"case intro\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\n⊢ ∀ {x : R}, x ≠ 0 → ∃ n, Associated (p ^ n) x","tactic":"refine ⟨p, hp, ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]},{"state_before":"case intro\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\n⊢ ∀ {x : R}, x ≠ 0 → ∃ n, Associated (p ^ n) x","state_after":"case intro\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\n⊢ ∃ n, Associated (p ^ n) x","tactic":"intro x hx","premises":[]},{"state_before":"case intro\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\n⊢ ∃ n, Associated (p ^ n) x","state_after":"case intro.intro\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\nfx : Multiset R\nhfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x\n⊢ ∃ n, Associated (p ^ n) x","tactic":"cases' WfDvdMonoid.exists_factors x hx with fx hfx","premises":[{"full_name":"WfDvdMonoid.exists_factors","def_path":"Mathlib/RingTheory/UniqueFactorizationDomain.lean","def_pos":[95,8],"def_end_pos":[95,22]}]},{"state_before":"case intro.intro\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\nfx : Multiset R\nhfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x\n⊢ ∃ n, Associated (p ^ n) x","state_after":"case intro.intro\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\nfx : Multiset R\nhfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x\n⊢ Associated (p ^ Multiset.card fx) x","tactic":"refine ⟨Multiset.card fx, ?_⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Multiset.card","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[659,4],"def_end_pos":[659,8]}]},{"state_before":"case intro.intro\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\nfx : Multiset R\nhfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x\n⊢ Associated (p ^ Multiset.card fx) x","state_after":"case intro.intro\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\nfx : Multiset R\nhfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x\nH : Associated fx.prod x\n⊢ Associated (p ^ Multiset.card fx) x","tactic":"have H := hfx.2","premises":[{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]}]},{"state_before":"case intro.intro\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\nfx : Multiset R\nhfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x\nH : Associated fx.prod x\n⊢ Associated (p ^ Multiset.card fx) x","state_after":"case intro.intro\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\nfx : Multiset R\nhfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x\nH : Associates.mk fx.prod = Associates.mk x\n⊢ Associates.mk (p ^ Multiset.card fx) = Associates.mk x","tactic":"rw [← Associates.mk_eq_mk_iff_associated] at H ⊢","premises":[{"full_name":"Associates.mk_eq_mk_iff_associated","def_path":"Mathlib/Algebra/Associated/Basic.lean","def_pos":[743,8],"def_end_pos":[743,31]}]},{"state_before":"case intro.intro\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\nfx : Multiset R\nhfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x\nH : Associates.mk fx.prod = Associates.mk x\n⊢ Associates.mk (p ^ Multiset.card fx) = Associates.mk x","state_after":"case intro.intro\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\nfx : Multiset R\nhfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x\nH : Associates.mk fx.prod = Associates.mk x\n⊢ (Multiset.replicate (Multiset.card fx) (Associates.mk p)).prod = (Multiset.map Associates.mk fx).prod","tactic":"rw [← H, ← Associates.prod_mk, Associates.mk_pow, ← 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u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\nfx : Multiset R\nhfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x\nH : Associates.mk fx.prod = Associates.mk x\n⊢ Multiset.replicate (Multiset.card fx) (Associates.mk p) = Multiset.map Associates.mk fx","tactic":"congr 1","premises":[]},{"state_before":"case intro.intro.e_a\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\nfx : Multiset R\nhfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x\nH : Associates.mk fx.prod = Associates.mk x\n⊢ Multiset.replicate (Multiset.card fx) (Associates.mk p) = Multiset.map Associates.mk 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F U).obj c).pt ≫\n      ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj c).π.app WalkingParallelPair.zero =\n    ((coneEquivInverse F U ⋙ coneEquivFunctor F U).obj c).π.app WalkingParallelPair.zero\n\ncase one\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasProducts C\nX : TopCat\nF : Presheaf C X\nι : Type v'\nU : ι → Opens ↑X\nc : Cone (SheafConditionEqualizerProducts.diagram F U)\n⊢ 𝟙 ((coneEquivInverse F U ⋙ coneEquivFunctor F U).obj c).pt ≫\n      ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj c).π.app WalkingParallelPair.one =\n    ((coneEquivInverse F U ⋙ coneEquivFunctor F U).obj c).π.app WalkingParallelPair.one","tactic":"rintro ⟨_ | _⟩","premises":[]},{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasProducts C\nX : TopCat\nF : Presheaf C X\nι : Type v'\nU : ι → Opens ↑X\nc : Cone (SheafConditionEqualizerProducts.diagram F U)\n⊢ ∀ (j : WalkingParallelPair),\n    𝟙 ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj c).pt ≫\n        ((coneEquivInverse F U ⋙ coneEquivFunctor F U).obj c).π.app j =\n      ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj c).π.app j","state_after":"case zero\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasProducts C\nX : TopCat\nF : Presheaf C X\nι : Type v'\nU : ι → Opens ↑X\nc : Cone (SheafConditionEqualizerProducts.diagram F U)\n⊢ 𝟙 ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj c).pt ≫\n      ((coneEquivInverse F U ⋙ coneEquivFunctor F U).obj c).π.app WalkingParallelPair.zero =\n    ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj c).π.app WalkingParallelPair.zero\n\ncase one\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasProducts C\nX : TopCat\nF : Presheaf C X\nι : Type v'\nU : ι → Opens ↑X\nc : Cone (SheafConditionEqualizerProducts.diagram F U)\n⊢ 𝟙 ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj c).pt ≫\n      ((coneEquivInverse F U ⋙ coneEquivFunctor F U).obj c).π.app WalkingParallelPair.one =\n    ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj c).π.app WalkingParallelPair.one","tactic":"rintro ⟨_ | _⟩","premises":[]},{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasProducts C\nX : TopCat\nF : Presheaf C X\nι : Type v'\nU : ι → Opens ↑X\nc d : Cone (SheafConditionEqualizerProducts.diagram F U)\nf : c ⟶ d\n⊢ (coneEquivInverse F U ⋙ coneEquivFunctor F U).map f ≫\n      ((fun c =>\n            { hom := { hom := 𝟙 ((coneEquivInverse F U ⋙ coneEquivFunctor F U).obj c).pt, w := ⋯ },\n              inv := { hom := 𝟙 ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj c).pt, w := ⋯ },\n              hom_inv_id := ⋯, inv_hom_id := ⋯ })\n          d).hom =\n    ((fun c =>\n            { hom := { hom := 𝟙 ((coneEquivInverse F U ⋙ coneEquivFunctor F U).obj c).pt, w := ⋯ },\n              inv := { hom := 𝟙 ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj c).pt, w := ⋯ },\n              hom_inv_id := ⋯, inv_hom_id := ⋯ })\n          c).hom ≫\n      (𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).map f","state_after":"case w\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasProducts C\nX : TopCat\nF : Presheaf C X\nι : Type v'\nU : ι → Opens ↑X\nc d : Cone (SheafConditionEqualizerProducts.diagram F U)\nf : c ⟶ d\n⊢ ((coneEquivInverse F U ⋙ coneEquivFunctor F U).map f ≫\n        ((fun c =>\n              { hom := { hom := 𝟙 ((coneEquivInverse F U ⋙ coneEquivFunctor F U).obj c).pt, w := ⋯ },\n                inv := { hom := 𝟙 ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj c).pt, w := ⋯ },\n                hom_inv_id := ⋯, inv_hom_id := ⋯ })\n            d).hom).hom =\n    (((fun c =>\n              { hom := { hom := 𝟙 ((coneEquivInverse F U ⋙ coneEquivFunctor F U).obj c).pt, w := ⋯ },\n                inv := { hom := 𝟙 ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj c).pt, w := ⋯ },\n                hom_inv_id := ⋯, inv_hom_id := ⋯ })\n            c).hom ≫\n        (𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).map f).hom","tactic":"ext","premises":[]},{"state_before":"case w\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasProducts C\nX : TopCat\nF : Presheaf C X\nι : Type v'\nU : ι → Opens ↑X\nc d : Cone (SheafConditionEqualizerProducts.diagram F U)\nf : c ⟶ d\n⊢ ((coneEquivInverse F U ⋙ coneEquivFunctor F U).map f ≫\n        ((fun c =>\n              { hom := { hom := 𝟙 ((coneEquivInverse F U ⋙ coneEquivFunctor F U).obj c).pt, w := ⋯ },\n                inv := { hom := 𝟙 ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj c).pt, w := ⋯ },\n                hom_inv_id := ⋯, inv_hom_id := ⋯ })\n            d).hom).hom =\n    (((fun c =>\n              { hom := { hom := 𝟙 ((coneEquivInverse F U ⋙ coneEquivFunctor F U).obj c).pt, w := ⋯ },\n                inv := { hom := 𝟙 ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj c).pt, w := ⋯ },\n                hom_inv_id := ⋯, inv_hom_id := ⋯ })\n            c).hom ≫\n        (𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).map f).hom","state_after":"case w\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasProducts C\nX : TopCat\nF : Presheaf C X\nι : Type v'\nU : ι → Opens ↑X\nc d : Cone (SheafConditionEqualizerProducts.diagram F U)\nf : c ⟶ d\n⊢ f.hom ≫ 𝟙 d.pt = 𝟙 c.pt ≫ f.hom","tactic":"dsimp","premises":[]},{"state_before":"case w\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasProducts C\nX : TopCat\nF : Presheaf C X\nι : Type v'\nU : ι → Opens ↑X\nc d : Cone (SheafConditionEqualizerProducts.diagram F U)\nf : c ⟶ d\n⊢ f.hom ≫ 𝟙 d.pt = 𝟙 c.pt ≫ f.hom","state_after":"no goals","tactic":"simp only [Category.comp_id, Category.id_comp]","premises":[{"full_name":"CategoryTheory.Category.comp_id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[153,2],"def_end_pos":[153,9]},{"full_name":"CategoryTheory.Category.id_comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[151,2],"def_end_pos":[151,9]}]}]}
+{"url":"Mathlib/Order/Atoms.lean","commit":"","full_name":"CompleteLattice.isStronglyCoatomic","start":[958,0],"end":[963,40],"file_path":"Mathlib/Order/Atoms.lean","tactics":[{"state_before":"ι : Sort u_1\nα : Type u_2\nβ : Type u_3\ninst✝² : CompleteLattice α\ninst✝¹ : IsLowerModularLattice α\ninst✝ : IsCoatomistic α\n⊢ IsStronglyCoatomic α","state_after":"ι : Sort u_1\nα : Type u_2\nβ : Type u_3\ninst✝² : CompleteLattice α\ninst✝¹ : IsLowerModularLattice α\ninst✝ : IsCoatomistic α\n⊢ IsStronglyAtomic αᵒᵈ","tactic":"rw [← isStronglyAtomic_dual_iff_is_stronglyCoatomic]","premises":[{"full_name":"isStronglyAtomic_dual_iff_is_stronglyCoatomic","def_path":"Mathlib/Order/Atoms.lean","def_pos":[336,8],"def_end_pos":[336,53]}]},{"state_before":"ι : Sort u_1\nα : Type u_2\nβ : Type u_3\ninst✝² : CompleteLattice α\ninst✝¹ : IsLowerModularLattice α\ninst✝ : IsCoatomistic α\n⊢ IsStronglyAtomic αᵒᵈ","state_after":"no goals","tactic":"exact CompleteLattice.isStronglyAtomic","premises":[{"full_name":"CompleteLattice.isStronglyAtomic","def_path":"Mathlib/Order/Atoms.lean","def_pos":[946,8],"def_end_pos":[946,40]}]}]}
+{"url":"Mathlib/LinearAlgebra/Finsupp.lean","commit":"","full_name":"Finsupp.supported_empty","start":[345,0],"end":[347,86],"file_path":"Mathlib/LinearAlgebra/Finsupp.lean","tactics":[{"state_before":"α : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nR : Type u_5\nS : Type u_6\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nl : α →₀ M\nh : l ∈ supported M R ∅\n⊢ l = 0","state_after":"case h\nα : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nR : Type u_5\nS : Type u_6\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nl : α →₀ M\nh : l ∈ supported M R ∅\na✝ : α\n⊢ l a✝ = 0 a✝","tactic":"ext","premises":[]},{"state_before":"case h\nα : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nR : Type u_5\nS : Type u_6\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nl : α →₀ M\nh : l ∈ supported M R ∅\na✝ : α\n⊢ l a✝ = 0 a✝","state_after":"no goals","tactic":"simp_all [mem_supported']","premises":[{"full_name":"Finsupp.mem_supported'","def_path":"Mathlib/LinearAlgebra/Finsupp.lean","def_pos":[286,8],"def_end_pos":[286,22]}]}]}
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+{"url":"Mathlib/CategoryTheory/Sites/Sieves.lean","commit":"","full_name":"CategoryTheory.Sieve.sSup_apply","start":[342,0],"end":[345,31],"file_path":"Mathlib/CategoryTheory/Sites/Sieves.lean","tactics":[{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y✝ Z : C\nf✝ : Y✝ ⟶ X\nS R : Sieve X\nSs : Set (Sieve X)\nY : C\nf : Y ⟶ X\n⊢ (sSup Ss).arrows f ↔ ∃ S, ∃ (_ : S ∈ Ss), S.arrows f","state_after":"no goals","tactic":"simp [sSup, Sieve.sup, setOf]","premises":[{"full_name":"CategoryTheory.Sieve.sup","def_path":"Mathlib/CategoryTheory/Sites/Sieves.lean","def_pos":[276,14],"def_end_pos":[276,17]},{"full_name":"SupSet.sSup","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[42,2],"def_end_pos":[42,6]},{"full_name":"setOf","def_path":"Mathlib/Init/Set.lean","def_pos":[56,4],"def_end_pos":[56,9]}]}]}
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+{"url":"Mathlib/Order/Filter/CardinalInter.lean","commit":"","full_name":"Filter.eventually_cardinal_ball","start":[107,0],"end":[111,30],"file_path":"Mathlib/Order/Filter/CardinalInter.lean","tactics":[{"state_before":"ι α β : Type u\nc : Cardinal.{u}\nl : Filter α\ninst✝ : CardinalInterFilter l c\nS : Set ι\nhS : #↑S < c\np : α → (i : ι) → i ∈ S → Prop\n⊢ (∀ᶠ (x : α) in l, ∀ (i : ι) (hi : i ∈ S), p x i hi) ↔ ∀ (i : ι) (hi : i ∈ S), ∀ᶠ (x : α) in l, p x i hi","state_after":"ι α β : Type u\nc : Cardinal.{u}\nl : Filter α\ninst✝ : CardinalInterFilter l c\nS : Set ι\nhS : #↑S < c\np : α → (i : ι) → i ∈ S → Prop\n⊢ ⋂ i, ⋂ (i_1 : i ∈ S), {x | p x i i_1} ∈ l ↔ ∀ (i : ι) (hi : i ∈ S), {x | p x i hi} ∈ l","tactic":"simp only [Filter.Eventually, setOf_forall]","premises":[{"full_name":"Filter.Eventually","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[948,14],"def_end_pos":[948,24]},{"full_name":"Set.setOf_forall","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[210,8],"def_end_pos":[210,20]}]},{"state_before":"ι α β : Type u\nc : Cardinal.{u}\nl : Filter α\ninst✝ : CardinalInterFilter l c\nS : Set ι\nhS : #↑S < c\np : α → (i : ι) → i ∈ S → Prop\n⊢ ⋂ i, ⋂ (i_1 : i ∈ S), {x | p x i i_1} ∈ l ↔ ∀ (i : ι) (hi : i ∈ S), {x | p x i hi} ∈ l","state_after":"no goals","tactic":"exact cardinal_bInter_mem hS","premises":[{"full_name":"Filter.cardinal_bInter_mem","def_path":"Mathlib/Order/Filter/CardinalInter.lean","def_pos":[96,8],"def_end_pos":[96,27]}]}]}
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s₂.tail\n⊢ s₁ ⋈ s₂ = s₁.head :: s₂.head :: t","tactic":"show s₁ ⋈ s₂ = head s₁::head s₂::t","premises":[{"full_name":"Stream'.cons","def_path":"Mathlib/Data/Stream/Defs.lean","def_pos":[25,4],"def_end_pos":[25,8]},{"full_name":"Stream'.head","def_path":"Mathlib/Data/Stream/Defs.lean","def_pos":[35,7],"def_end_pos":[35,11]},{"full_name":"Stream'.interleave","def_path":"Mathlib/Data/Stream/Defs.lean","def_pos":[90,4],"def_end_pos":[90,14]}]},{"state_before":"α : Type u\nβ : Type v\nδ : Type w\ns₁ s₂ : Stream' α\nt : Stream' α := s₁.tail ⋈ s₂.tail\n⊢ s₁ ⋈ s₂ = s₁.head :: s₂.head :: t","state_after":"α : Type u\nβ : Type v\nδ : Type w\ns₁ s₂ : Stream' α\nt : Stream' α := s₁.tail ⋈ s₂.tail\n⊢ (corecOn (s₁, s₂)\n      (fun x =>\n        match x with\n        | (s₁, snd) => s₁.head)\n      fun x =>\n      match x with\n      | (s₁, s₂) => (s₂, s₁.tail)) =\n    s₁.head :: s₂.head :: t","tactic":"unfold interleave","premises":[{"full_name":"Stream'.interleave","def_path":"Mathlib/Data/Stream/Defs.lean","def_pos":[90,4],"def_end_pos":[90,14]}]},{"state_before":"α : Type u\nβ : Type v\nδ : Type w\ns₁ s₂ : Stream' α\nt : Stream' α := s₁.tail ⋈ s₂.tail\n⊢ (corecOn (s₁, s₂)\n      (fun x =>\n        match x with\n        | (s₁, snd) => s₁.head)\n      fun x =>\n      match x with\n      | (s₁, s₂) => (s₂, s₁.tail)) =\n    s₁.head :: s₂.head :: t","state_after":"α : Type u\nβ : Type v\nδ : Type w\ns₁ s₂ : Stream' α\nt : Stream' α := s₁.tail ⋈ s₂.tail\n⊢ corec\n      (fun x =>\n        match x with\n        | (s₁, snd) => s₁.head)\n      (fun x =>\n        match x with\n        | (s₁, s₂) => (s₂, s₁.tail))\n      (s₁, s₂) =\n    s₁.head :: s₂.head :: t","tactic":"unfold corecOn","premises":[{"full_name":"Stream'.corecOn","def_path":"Mathlib/Data/Stream/Defs.lean","def_pos":[75,4],"def_end_pos":[75,11]}]},{"state_before":"α : Type u\nβ : Type v\nδ : Type w\ns₁ s₂ : Stream' α\nt : Stream' α := s₁.tail ⋈ s₂.tail\n⊢ corec\n      (fun x =>\n        match x with\n        | (s₁, snd) => s₁.head)\n      (fun x =>\n        match x with\n        | (s₁, s₂) => (s₂, s₁.tail))\n      (s₁, s₂) =\n    s₁.head :: s₂.head :: t","state_after":"α : Type u\nβ : Type v\nδ : Type w\ns₁ s₂ : Stream' α\nt : Stream' α := s₁.tail ⋈ s₂.tail\n⊢ (match (s₁, s₂) with\n      | (s₁, snd) => s₁.head) ::\n      corec\n        (fun x =>\n          match x with\n          | (s₁, snd) => s₁.head)\n        (fun x =>\n          match x with\n          | (s₁, s₂) => (s₂, s₁.tail))\n        (match (s₁, s₂) with\n        | (s₁, s₂) => (s₂, s₁.tail)) =\n    s₁.head :: s₂.head :: t","tactic":"rw [corec_eq]","premises":[{"full_name":"Stream'.corec_eq","def_path":"Mathlib/Data/Stream/Init.lean","def_pos":[315,8],"def_end_pos":[315,16]}]},{"state_before":"α : Type u\nβ : Type v\nδ : Type w\ns₁ s₂ : Stream' α\nt : Stream' α := s₁.tail ⋈ s₂.tail\n⊢ (match (s₁, s₂) with\n      | (s₁, snd) => s₁.head) ::\n      corec\n        (fun x =>\n          match x with\n          | (s₁, snd) => s₁.head)\n        (fun x =>\n          match x with\n          | (s₁, s₂) => (s₂, s₁.tail))\n        (match (s₁, s₂) with\n        | (s₁, s₂) => (s₂, s₁.tail)) =\n    s₁.head :: s₂.head :: t","state_after":"α : Type u\nβ : Type v\nδ : Type w\ns₁ s₂ : Stream' α\nt : Stream' α := s₁.tail ⋈ s₂.tail\n⊢ s₁.head :: corec (fun x => x.1.head) (fun x => (x.2, x.1.tail)) (s₂, s₁.tail) = s₁.head :: s₂.head :: t","tactic":"dsimp","premises":[]},{"state_before":"α : Type u\nβ : Type v\nδ : Type w\ns₁ s₂ : Stream' α\nt : Stream' α := s₁.tail ⋈ s₂.tail\n⊢ s₁.head :: corec (fun x => x.1.head) (fun x => (x.2, x.1.tail)) (s₂, s₁.tail) = s₁.head :: s₂.head :: t","state_after":"α : Type u\nβ : Type v\nδ : Type w\ns₁ s₂ : Stream' α\nt : Stream' α := s₁.tail ⋈ s₂.tail\n⊢ s₁.head ::\n      (s₂, s₁.tail).1.head ::\n        corec (fun x => x.1.head) (fun x => (x.2, x.1.tail)) ((s₂, s₁.tail).2, (s₂, s₁.tail).1.tail) =\n    s₁.head :: s₂.head :: t","tactic":"rw [corec_eq]","premises":[{"full_name":"Stream'.corec_eq","def_path":"Mathlib/Data/Stream/Init.lean","def_pos":[315,8],"def_end_pos":[315,16]}]},{"state_before":"α : Type u\nβ : Type v\nδ : Type w\ns₁ s₂ : Stream' α\nt : Stream' α := s₁.tail ⋈ s₂.tail\n⊢ s₁.head ::\n      (s₂, s₁.tail).1.head ::\n        corec (fun x => x.1.head) (fun x => (x.2, x.1.tail)) ((s₂, s₁.tail).2, (s₂, s₁.tail).1.tail) =\n    s₁.head :: s₂.head :: t","state_after":"no goals","tactic":"rfl","premises":[]}]}
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f))","tactic":"intro a f' f ih","premises":[]},{"state_before":"n : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα✝ α : TypeVec.{u} n\np : Fix F α → Prop\nh : ∀ (x : F (α ::: Fix F α)), LiftP (α.PredLast p) x → p (mk x)\nx : (P F).W α\na : (P F).A\nf' : (P F).drop.B a ⟹ α\nf : (P F).last.B a → (P F).W α\nih : ∀ (i : (P F).last.B a), p (Quot.mk Setoid.r (f i))\n⊢ p (Quot.mk Setoid.r ((P F).wMk a f' f))","state_after":"n : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα✝ α : TypeVec.{u} n\np : Fix F α → Prop\nh : ∀ (x : F (α ::: Fix F α)), LiftP (α.PredLast p) x → p (mk x)\nx : (P F).W α\na : (P F).A\nf' : (P F).drop.B a ⟹ α\nf : (P F).last.B a → (P F).W α\nih : ∀ (i : (P F).last.B a), p (Quot.mk Setoid.r (f i))\n⊢ p ⟦(P F).wMk a f' f⟧","tactic":"change p ⟦q.P.wMk a f' f⟧","premises":[{"full_name":"MvPFunctor.wMk","def_path":"Mathlib/Data/PFunctor/Multivariate/W.lean","def_pos":[153,4],"def_end_pos":[153,7]},{"full_name":"MvQPF.P","def_path":"Mathlib/Data/QPF/Multivariate/Basic.lean","def_pos":[87,2],"def_end_pos":[87,3]},{"full_name":"Quotient.mk","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1592,14],"def_end_pos":[1592,16]}]},{"state_before":"n : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα✝ α : TypeVec.{u} n\np : Fix F α → Prop\nh : ∀ (x : F (α ::: Fix F α)), LiftP (α.PredLast p) x → p (mk x)\nx : (P F).W α\na : (P F).A\nf' : (P F).drop.B a ⟹ α\nf : (P F).last.B a → (P F).W α\nih : ∀ (i : (P F).last.B a), p (Quot.mk Setoid.r (f i))\n⊢ p ⟦(P F).wMk a f' f⟧","state_after":"n : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα✝ α : TypeVec.{u} n\np : Fix F α → Prop\nh : ∀ (x : F (α ::: Fix F α)), LiftP (α.PredLast p) x → p (mk x)\nx : (P F).W α\na : (P F).A\nf' : (P F).drop.B a ⟹ α\nf : (P F).last.B a → (P F).W α\nih : ∀ (i : (P F).last.B a), p (Quot.mk Setoid.r (f i))\n⊢ p (mk (abs ⟨a, (P F).appendContents f' fun x => ⟦f x⟧⟩))","tactic":"rw [← Fix.ind_aux a f' f]","premises":[{"full_name":"MvQPF.Fix.ind_aux","def_path":"Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean","def_pos":[205,8],"def_end_pos":[205,19]}]},{"state_before":"n : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα✝ α : TypeVec.{u} n\np : Fix F α → Prop\nh : ∀ (x : F (α ::: Fix F α)), LiftP (α.PredLast p) x → p (mk x)\nx : (P F).W α\na : (P F).A\nf' : (P F).drop.B a ⟹ α\nf : (P F).last.B a → (P F).W α\nih : ∀ (i : (P F).last.B a), p (Quot.mk Setoid.r (f i))\n⊢ p (mk (abs ⟨a, (P F).appendContents f' fun x => ⟦f x⟧⟩))","state_after":"case a\nn : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα✝ α : TypeVec.{u} n\np : Fix F α → Prop\nh : ∀ (x : F (α ::: Fix F α)), LiftP (α.PredLast p) x → p (mk x)\nx : (P F).W α\na : (P F).A\nf' : (P F).drop.B a ⟹ α\nf : (P F).last.B a → (P F).W α\nih : ∀ (i : (P F).last.B a), p (Quot.mk Setoid.r (f i))\n⊢ LiftP (α.PredLast p) (abs ⟨a, (P F).appendContents f' fun x => ⟦f x⟧⟩)","tactic":"apply h","premises":[]},{"state_before":"case a\nn : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα✝ α : TypeVec.{u} n\np : Fix F α → Prop\nh : ∀ (x : F (α ::: Fix F α)), LiftP (α.PredLast p) x → p (mk x)\nx : (P F).W α\na : (P F).A\nf' : (P F).drop.B a ⟹ α\nf : (P F).last.B a → (P F).W α\nih : ∀ (i : (P F).last.B a), p (Quot.mk Setoid.r (f i))\n⊢ LiftP (α.PredLast p) (abs ⟨a, (P F).appendContents f' fun x => ⟦f x⟧⟩)","state_after":"case a\nn : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα✝ α : TypeVec.{u} n\np : Fix F α → Prop\nh : ∀ (x : F (α ::: Fix F α)), LiftP (α.PredLast p) x → p (mk x)\nx : (P F).W α\na : (P F).A\nf' : (P F).drop.B a ⟹ α\nf : (P F).last.B a → (P F).W α\nih : ∀ (i : (P F).last.B a), p (Quot.mk Setoid.r (f i))\n⊢ ∃ a_1 f_1,\n    abs ⟨a, (P F).appendContents f' fun x => ⟦f x⟧⟩ = abs ⟨a_1, f_1⟩ ∧\n      ∀ (i : Fin2 (n + 1)) (j : (P F).B a_1 i), α.PredLast p (f_1 i j)","tactic":"rw [MvQPF.liftP_iff]","premises":[{"full_name":"MvQPF.liftP_iff","def_path":"Mathlib/Data/QPF/Multivariate/Basic.lean","def_pos":[119,8],"def_end_pos":[119,17]}]},{"state_before":"case a\nn : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα✝ α : TypeVec.{u} n\np : Fix F α → Prop\nh : ∀ (x : F (α ::: Fix F α)), LiftP (α.PredLast p) x → p (mk x)\nx : (P F).W α\na : (P F).A\nf' : (P F).drop.B a ⟹ α\nf : (P F).last.B a → (P F).W α\nih : ∀ (i : (P F).last.B a), p (Quot.mk Setoid.r (f i))\n⊢ ∃ a_1 f_1,\n    abs ⟨a, (P F).appendContents f' fun x => ⟦f x⟧⟩ = abs ⟨a_1, f_1⟩ ∧\n      ∀ (i : Fin2 (n + 1)) (j : (P F).B a_1 i), α.PredLast p (f_1 i j)","state_after":"case a\nn : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα✝ α : TypeVec.{u} n\np : Fix F α → Prop\nh : ∀ (x : F (α ::: Fix F α)), LiftP (α.PredLast p) x → p (mk x)\nx : (P F).W α\na : (P F).A\nf' : (P F).drop.B a ⟹ α\nf : (P F).last.B a → (P F).W α\nih : ∀ (i : (P F).last.B a), p (Quot.mk Setoid.r (f i))\n⊢ ∀ (i : Fin2 (n + 1)) (j : (P F).B a i), α.PredLast p ((P F).appendContents f' (fun x => ⟦f x⟧) i j)","tactic":"refine ⟨_, _, rfl, ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"case a\nn : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα✝ α : TypeVec.{u} n\np : Fix F α → Prop\nh : ∀ (x : F (α ::: Fix F α)), LiftP (α.PredLast p) x → p (mk x)\nx : (P F).W α\na : (P F).A\nf' : (P F).drop.B a ⟹ α\nf : (P F).last.B a → (P F).W α\nih : ∀ (i : (P F).last.B a), p (Quot.mk Setoid.r (f i))\n⊢ ∀ (i : Fin2 (n + 1)) (j : (P F).B a i), α.PredLast p ((P F).appendContents f' (fun x => ⟦f x⟧) i j)","state_after":"case a\nn : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα✝ α : TypeVec.{u} n\np : Fix F α → Prop\nh : ∀ (x : F (α ::: Fix F α)), LiftP (α.PredLast p) x → p (mk x)\nx : (P F).W α\na : (P F).A\nf' : (P F).drop.B a ⟹ α\nf : (P F).last.B a → (P F).W α\nih : ∀ (i : (P F).last.B a), p (Quot.mk Setoid.r (f i))\ni : Fin2 (n + 1)\nj : (P F).B a i\n⊢ α.PredLast p ((P F).appendContents f' (fun x => ⟦f x⟧) i j)","tactic":"intro i j","premises":[]},{"state_before":"case a\nn : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα✝ α : TypeVec.{u} n\np : Fix F α → Prop\nh : ∀ (x : F (α ::: Fix F α)), LiftP (α.PredLast p) x → p (mk x)\nx : (P F).W α\na : (P F).A\nf' : (P F).drop.B a ⟹ α\nf : (P F).last.B a → (P F).W α\nih : ∀ (i : (P F).last.B a), p (Quot.mk Setoid.r (f i))\ni : Fin2 (n + 1)\nj : (P F).B a i\n⊢ α.PredLast p ((P F).appendContents f' (fun x => ⟦f x⟧) i j)","state_after":"case a.fz\nn : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα✝ α : TypeVec.{u} n\np : Fix F α → Prop\nh : ∀ (x : F (α ::: Fix F α)), LiftP (α.PredLast p) x → p (mk x)\nx : (P F).W α\na : (P F).A\nf' : (P F).drop.B a ⟹ α\nf : (P F).last.B a → (P F).W α\nih : ∀ (i : (P F).last.B a), p (Quot.mk Setoid.r (f i))\nj : (P F).B a Fin2.fz\n⊢ α.PredLast p ((P F).appendContents f' (fun x => ⟦f x⟧) Fin2.fz j)\n\ncase a.fs\nn : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα✝ α : TypeVec.{u} n\np : Fix F α → Prop\nh : ∀ (x : F (α ::: Fix F α)), LiftP (α.PredLast p) x → p (mk x)\nx : (P F).W α\na : (P F).A\nf' : (P F).drop.B a ⟹ α\nf : (P F).last.B a → (P F).W α\nih : ∀ (i : (P F).last.B a), p (Quot.mk Setoid.r (f i))\na✝ : Fin2 n\nj : (P F).B a a✝.fs\n⊢ α.PredLast p ((P F).appendContents f' (fun x => ⟦f x⟧) a✝.fs j)","tactic":"cases i","premises":[]}]}
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+{"url":"Mathlib/Algebra/Module/Torsion.lean","commit":"","full_name":"Submodule.exists_isTorsionBy","start":[798,0],"end":[811,79],"file_path":"Mathlib/Algebra/Module/Torsion.lean","tactics":[{"state_before":"R : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : (x : M) → Decidable (x = 0)\np : R\nhM : IsTorsion' M ↥(Submonoid.powers p)\nd : ℕ\nhd : d ≠ 0\ns : Fin d → M\nhs : span R (Set.range s) = ⊤\n⊢ ∃ j, IsTorsionBy R M (p ^ pOrder hM (s j))","state_after":"R : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : (x : M) → Decidable (x = 0)\np : R\nhM : IsTorsion' M ↥(Submonoid.powers p)\nd : ℕ\nhd : d ≠ 0\ns : Fin d → M\nhs : span R (Set.range s) = ⊤\noj : Option (Fin d) := List.argmax (fun i => pOrder hM (s i)) (List.finRange d)\n⊢ ∃ j, IsTorsionBy R M (p ^ pOrder hM (s j))","tactic":"let oj := List.argmax (fun i => pOrder hM <| s i) (List.finRange d)","premises":[{"full_name":"List.argmax","def_path":"Mathlib/Data/List/MinMax.lean","def_pos":[96,4],"def_end_pos":[96,10]},{"full_name":"List.finRange","def_path":"Mathlib/Data/List/Range.lean","def_pos":[87,4],"def_end_pos":[87,12]},{"full_name":"Submodule.pOrder","def_path":"Mathlib/Algebra/Module/Torsion.lean","def_pos":[785,4],"def_end_pos":[785,10]}]},{"state_before":"R : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : (x : M) → Decidable (x = 0)\np : R\nhM : IsTorsion' M ↥(Submonoid.powers p)\nd : ℕ\nhd : d ≠ 0\ns : Fin d → M\nhs : span R (Set.range s) = ⊤\noj : Option (Fin d) := List.argmax (fun i => pOrder hM (s i)) (List.finRange d)\n⊢ ∃ j, IsTorsionBy R M (p ^ pOrder hM (s j))","state_after":"R : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : (x : M) → Decidable (x = 0)\np : R\nhM : IsTorsion' M ↥(Submonoid.powers p)\nd : ℕ\nhd : d ≠ 0\ns : Fin d → M\nhs : span R (Set.range s) = ⊤\noj : Option (Fin d) := List.argmax (fun i => pOrder hM (s i)) (List.finRange d)\nhoj : oj.isSome = true\n⊢ ∃ j, IsTorsionBy R M (p ^ pOrder hM (s j))","tactic":"have hoj : oj.isSome :=\n    Option.ne_none_iff_isSome.mp fun eq_none =>\n      hd <| List.finRange_eq_nil.mp <| List.argmax_eq_none.mp eq_none","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"List.argmax_eq_none","def_path":"Mathlib/Data/List/MinMax.lean","def_pos":[147,8],"def_end_pos":[147,22]},{"full_name":"List.finRange_eq_nil","def_path":"Mathlib/Data/List/Range.lean","def_pos":[109,8],"def_end_pos":[109,23]},{"full_name":"Option.isSome","def_path":".lake/packages/lean4/src/lean/Init/Data/Option/Basic.lean","def_pos":[25,14],"def_end_pos":[25,20]},{"full_name":"Option.ne_none_iff_isSome","def_path":".lake/packages/lean4/src/lean/Init/Data/Option/Lemmas.lean","def_pos":[77,8],"def_end_pos":[77,26]}]},{"state_before":"R : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : (x : M) → Decidable (x = 0)\np : R\nhM : IsTorsion' M ↥(Submonoid.powers p)\nd : ℕ\nhd : d ≠ 0\ns : Fin d → M\nhs : span R (Set.range s) = ⊤\noj : Option (Fin d) := List.argmax (fun i => pOrder hM (s i)) (List.finRange d)\nhoj : oj.isSome = true\n⊢ ∃ j, IsTorsionBy R M (p ^ pOrder hM (s j))","state_after":"case h\nR : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : (x : M) → Decidable (x = 0)\np : R\nhM : IsTorsion' M ↥(Submonoid.powers p)\nd : ℕ\nhd : d ≠ 0\ns : Fin d → M\nhs : span R (Set.range s) = ⊤\noj : Option (Fin d) := List.argmax (fun i => pOrder hM (s i)) (List.finRange d)\nhoj : oj.isSome = true\n⊢ IsTorsionBy R M (p ^ pOrder hM (s (oj.get hoj)))","tactic":"use Option.get _ hoj","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Option.get","def_path":".lake/packages/lean4/src/lean/Init/Data/Option/Basic.lean","def_pos":[127,14],"def_end_pos":[127,17]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]}]},{"state_before":"case h\nR : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : (x : M) → Decidable (x = 0)\np : R\nhM : IsTorsion' M ↥(Submonoid.powers p)\nd : ℕ\nhd : d ≠ 0\ns : Fin d → M\nhs : span R (Set.range s) = ⊤\noj : Option (Fin d) := List.argmax (fun i => pOrder hM (s i)) (List.finRange d)\nhoj : oj.isSome = true\n⊢ IsTorsionBy R M (p ^ pOrder hM (s (oj.get hoj)))","state_after":"case h\nR : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : (x : M) → Decidable (x = 0)\np : R\nhM : IsTorsion' M ↥(Submonoid.powers p)\nd : ℕ\nhd : d ≠ 0\ns : Fin d → M\nhs : span R (Set.range s) = ⊤\noj : Option (Fin d) := List.argmax (fun i => pOrder hM (s i)) (List.finRange d)\nhoj : oj.isSome = true\n⊢ ∀ (y : Fin d), s y ∈ ↑(torsionBy R M (p ^ pOrder hM (s (oj.get hoj))))","tactic":"rw [isTorsionBy_iff_torsionBy_eq_top, eq_top_iff, ← hs, Submodule.span_le,\n    Set.range_subset_iff]","premises":[{"full_name":"Module.isTorsionBy_iff_torsionBy_eq_top","def_path":"Mathlib/Algebra/Module/Torsion.lean","def_pos":[305,8],"def_end_pos":[305,40]},{"full_name":"Set.range_subset_iff","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[623,8],"def_end_pos":[623,24]},{"full_name":"Submodule.span_le","def_path":"Mathlib/LinearAlgebra/Span.lean","def_pos":[74,8],"def_end_pos":[74,15]},{"full_name":"eq_top_iff","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[116,8],"def_end_pos":[116,18]}]},{"state_before":"case h\nR : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : (x : M) → Decidable (x = 0)\np : R\nhM : IsTorsion' M ↥(Submonoid.powers p)\nd : ℕ\nhd : d ≠ 0\ns : Fin d → M\nhs : span R (Set.range s) = ⊤\noj : Option (Fin d) := List.argmax (fun i => pOrder hM (s i)) (List.finRange d)\nhoj : oj.isSome = true\n⊢ ∀ (y : Fin d), s y ∈ ↑(torsionBy R M (p ^ pOrder hM (s (oj.get hoj))))","state_after":"case h\nR : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : (x : M) → Decidable (x = 0)\np : R\nhM : IsTorsion' M ↥(Submonoid.powers p)\nd : ℕ\nhd : d ≠ 0\ns : Fin d → M\nhs : span R (Set.range s) = ⊤\noj : Option (Fin d) := List.argmax (fun i => pOrder hM (s i)) (List.finRange d)\nhoj : oj.isSome = true\ni : Fin d\n⊢ s i ∈ ↑(torsionBy R M (p ^ pOrder hM (s (oj.get hoj))))","tactic":"intro i","premises":[]},{"state_before":"case h\nR : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : (x : M) → Decidable (x = 0)\np : R\nhM : IsTorsion' M ↥(Submonoid.powers p)\nd : ℕ\nhd : d ≠ 0\ns : Fin d → M\nhs : span R (Set.range s) = ⊤\noj : Option (Fin d) := List.argmax (fun i => pOrder hM (s i)) (List.finRange d)\nhoj : oj.isSome = true\ni : Fin d\n⊢ s i ∈ ↑(torsionBy R M (p ^ pOrder hM (s (oj.get hoj))))","state_after":"case h\nR : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : (x : M) → Decidable (x = 0)\np : R\nhM : IsTorsion' M ↥(Submonoid.powers p)\nd : ℕ\nhd : d ≠ 0\ns : Fin d → M\nhs : span R (Set.range s) = ⊤\noj : Option (Fin d) := List.argmax (fun i => pOrder hM (s i)) (List.finRange d)\nhoj : oj.isSome = true\ni : Fin d\n⊢ p ^ pOrder hM (s (oj.get hoj)) • s i = 0","tactic":"change (p ^ pOrder hM (s (Option.get oj hoj))) • s i = 0","premises":[{"full_name":"Option.get","def_path":".lake/packages/lean4/src/lean/Init/Data/Option/Basic.lean","def_pos":[127,14],"def_end_pos":[127,17]},{"full_name":"Submodule.pOrder","def_path":"Mathlib/Algebra/Module/Torsion.lean","def_pos":[785,4],"def_end_pos":[785,10]}]},{"state_before":"case h\nR : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : (x : M) → Decidable (x = 0)\np : R\nhM : IsTorsion' M ↥(Submonoid.powers p)\nd : ℕ\nhd : d ≠ 0\ns : Fin d → M\nhs : span R (Set.range s) = ⊤\noj : Option (Fin d) := List.argmax (fun i => pOrder hM (s i)) (List.finRange d)\nhoj : oj.isSome = true\ni : Fin d\n⊢ p ^ pOrder hM (s (oj.get hoj)) • s i = 0","state_after":"case h\nR : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : (x : M) → Decidable (x = 0)\np : R\nhM : IsTorsion' M ↥(Submonoid.powers p)\nd : ℕ\nhd : d ≠ 0\ns : Fin d → M\nhs : span R (Set.range s) = ⊤\noj : Option (Fin d) := List.argmax (fun i => pOrder hM (s i)) (List.finRange d)\nhoj : oj.isSome = true\ni : Fin d\nthis : pOrder hM (s i) ≤ pOrder hM (s (oj.get hoj))\n⊢ p ^ pOrder hM (s (oj.get hoj)) • s i = 0","tactic":"have : pOrder hM (s i) ≤ pOrder hM (s <| Option.get _ hoj) :=\n    List.le_of_mem_argmax (List.mem_finRange i) (Option.get_mem hoj)","premises":[{"full_name":"List.le_of_mem_argmax","def_path":"Mathlib/Data/List/MinMax.lean","def_pos":[159,8],"def_end_pos":[159,24]},{"full_name":"List.mem_finRange","def_path":"Mathlib/Data/List/Range.lean","def_pos":[95,8],"def_end_pos":[95,20]},{"full_name":"Option.get","def_path":".lake/packages/lean4/src/lean/Init/Data/Option/Basic.lean","def_pos":[127,14],"def_end_pos":[127,17]},{"full_name":"Option.get_mem","def_path":".lake/packages/lean4/src/lean/Init/Data/Option/Lemmas.lean","def_pos":[25,8],"def_end_pos":[25,15]},{"full_name":"Submodule.pOrder","def_path":"Mathlib/Algebra/Module/Torsion.lean","def_pos":[785,4],"def_end_pos":[785,10]}]},{"state_before":"case h\nR : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : (x : M) → Decidable (x = 0)\np : R\nhM : IsTorsion' M ↥(Submonoid.powers p)\nd : ℕ\nhd : d ≠ 0\ns : Fin d → M\nhs : span R (Set.range s) = ⊤\noj : Option (Fin d) := List.argmax (fun i => pOrder hM (s i)) (List.finRange d)\nhoj : oj.isSome = true\ni : Fin d\nthis : pOrder hM (s i) ≤ pOrder hM (s (oj.get hoj))\n⊢ p ^ pOrder hM (s (oj.get hoj)) • s i = 0","state_after":"no goals","tactic":"rw [← Nat.sub_add_cancel this, pow_add, mul_smul, pow_pOrder_smul, smul_zero]","premises":[{"full_name":"MulAction.mul_smul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[99,2],"def_end_pos":[99,10]},{"full_name":"Nat.sub_add_cancel","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[893,26],"def_end_pos":[893,40]},{"full_name":"Submodule.pow_pOrder_smul","def_path":"Mathlib/Algebra/Module/Torsion.lean","def_pos":[790,8],"def_end_pos":[790,23]},{"full_name":"pow_add","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[598,6],"def_end_pos":[598,13]},{"full_name":"smul_zero","def_path":"Mathlib/Algebra/GroupWithZero/Action/Defs.lean","def_pos":[63,8],"def_end_pos":[63,17]}]}]}
+{"url":"Mathlib/Data/NNReal/Basic.lean","commit":"","full_name":"NNReal.coe_sub","start":[177,0],"end":[179,69],"file_path":"Mathlib/Data/NNReal/Basic.lean","tactics":[{"state_before":"r₁ r₂ : ℝ≥0\nh : r₂ ≤ r₁\n⊢ ↑r₂ ≤ ↑r₁ - 0","state_after":"no goals","tactic":"simp [show (r₂ : ℝ) ≤ r₁ from h]","premises":[{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]}]}]}
+{"url":"Mathlib/RingTheory/Algebraic.lean","commit":"","full_name":"inv_eq_of_root_of_coeff_zero_ne_zero","start":[412,0],"end":[420,55],"file_path":"Mathlib/RingTheory/Algebraic.lean","tactics":[{"state_before":"R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_3\nL : Type u_4\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : L\np : K[X]\naeval_eq : (aeval x) p = 0\ncoeff_zero_ne : p.coeff 0 ≠ 0\n⊢ x⁻¹ = -((aeval x) p.divX / (algebraMap K L) (p.coeff 0))","state_after":"case h.e'_3\nR : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_3\nL : Type u_4\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : L\np : K[X]\naeval_eq : (aeval x) p = 0\ncoeff_zero_ne : p.coeff 0 ≠ 0\n⊢ -((aeval x) p.divX / (algebraMap K L) (p.coeff 0)) = (aeval x) p.divX / ((aeval x) p - (algebraMap K L) (p.coeff 0))\n\ncase convert_2\nR : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_3\nL : Type u_4\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : L\np : K[X]\naeval_eq : (aeval x) p = 0\ncoeff_zero_ne : p.coeff 0 ≠ 0\nh : (aeval x) p.divX = 0\n⊢ (algebraMap K L) (p.coeff 0) = (algebraMap K L) 0","tactic":"convert inv_eq_of_aeval_divX_ne_zero (p := p) (L := L)\n    (mt (fun h => (algebraMap K L).injective ?_) coeff_zero_ne) using 1","premises":[{"full_name":"RingHom.injective","def_path":"Mathlib/Algebra/Field/Basic.lean","def_pos":[214,18],"def_end_pos":[214,27]},{"full_name":"algebraMap","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[108,4],"def_end_pos":[108,14]},{"full_name":"inv_eq_of_aeval_divX_ne_zero","def_path":"Mathlib/RingTheory/Algebraic.lean","def_pos":[405,8],"def_end_pos":[405,36]},{"full_name":"mt","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[647,8],"def_end_pos":[647,10]}]},{"state_before":"case convert_2\nR : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_3\nL : Type u_4\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : L\np : K[X]\naeval_eq : (aeval x) p = 0\ncoeff_zero_ne : p.coeff 0 ≠ 0\nh : (aeval x) p.divX = 0\n⊢ (algebraMap K L) (p.coeff 0) = (algebraMap K L) 0","state_after":"case convert_2\nR : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_3\nL : Type u_4\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : L\np : K[X]\naeval_eq : (aeval x) p = 0\ncoeff_zero_ne : p.coeff 0 ≠ 0\nh : (aeval x) p.divX = 0\n⊢ (algebraMap K L) (p.coeff 0) = 0","tactic":"rw [RingHom.map_zero]","premises":[{"full_name":"RingHom.map_zero","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[472,18],"def_end_pos":[472,26]}]},{"state_before":"case convert_2\nR : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_3\nL : Type u_4\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : L\np : K[X]\naeval_eq : (aeval x) p = 0\ncoeff_zero_ne : p.coeff 0 ≠ 0\nh : (aeval x) p.divX = 0\n⊢ (algebraMap K L) (p.coeff 0) = 0","state_after":"case h.e'_2\nR : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_3\nL : Type u_4\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : L\np : K[X]\naeval_eq : (aeval x) p = 0\ncoeff_zero_ne : p.coeff 0 ≠ 0\nh : (aeval x) p.divX = 0\n⊢ (algebraMap K L) (p.coeff 0) = (aeval x) p","tactic":"convert aeval_eq","premises":[]},{"state_before":"case h.e'_2\nR : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_3\nL : Type u_4\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : L\np : K[X]\naeval_eq : (aeval x) p = 0\ncoeff_zero_ne : p.coeff 0 ≠ 0\nh : (aeval x) p.divX = 0\n⊢ (algebraMap K L) (p.coeff 0) = (aeval x) p","state_after":"case h.e'_2\nR : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_3\nL : Type u_4\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : L\np : K[X]\naeval_eq : (aeval x) p = 0\ncoeff_zero_ne : p.coeff 0 ≠ 0\nh : (aeval x) p.divX = 0\n⊢ (algebraMap K L) (p.coeff 0) = (aeval x) (p.divX * X + C (p.coeff 0))","tactic":"conv_rhs => rw [← divX_mul_X_add p]","premises":[{"full_name":"Polynomial.divX_mul_X_add","def_path":"Mathlib/Algebra/Polynomial/Inductions.lean","def_pos":[44,8],"def_end_pos":[44,22]}]},{"state_before":"case h.e'_2\nR : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_3\nL : Type u_4\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : L\np : K[X]\naeval_eq : (aeval x) p = 0\ncoeff_zero_ne : p.coeff 0 ≠ 0\nh : (aeval x) p.divX = 0\n⊢ (algebraMap K L) (p.coeff 0) = (aeval x) (p.divX * X + C (p.coeff 0))","state_after":"no goals","tactic":"rw [map_add, map_mul, h, zero_mul, zero_add, aeval_C]","premises":[{"full_name":"MulZeroClass.zero_mul","def_path":"Mathlib/Algebra/GroupWithZero/Defs.lean","def_pos":[35,2],"def_end_pos":[35,10]},{"full_name":"Polynomial.aeval_C","def_path":"Mathlib/Algebra/Polynomial/AlgebraMap.lean","def_pos":[254,8],"def_end_pos":[254,15]},{"full_name":"map_add","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[280,2],"def_end_pos":[280,13]},{"full_name":"map_mul","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[281,8],"def_end_pos":[281,15]},{"full_name":"zero_add","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[408,2],"def_end_pos":[408,13]}]}]}
+{"url":"Mathlib/Order/Interval/Set/WithBotTop.lean","commit":"","full_name":"WithTop.preimage_coe_Ioo_top","start":[70,0],"end":[72,24],"file_path":"Mathlib/Order/Interval/Set/WithBotTop.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : Preorder α\na b : α\n⊢ some ⁻¹' Ioo ↑a ⊤ = Ioi a","state_after":"no goals","tactic":"simp [← Ioi_inter_Iio]","premises":[{"full_name":"Set.Ioi_inter_Iio","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[515,8],"def_end_pos":[515,21]}]}]}
+{"url":"Mathlib/CategoryTheory/Preadditive/Mat.lean","commit":"","full_name":"CategoryTheory.Mat.id_apply_self","start":[521,0],"end":[522,94],"file_path":"Mathlib/CategoryTheory/Preadditive/Mat.lean","tactics":[{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nR : Type u\ninst✝ : Semiring R\nM : Mat R\ni : ↑M\n⊢ 𝟙 M i i = 1","state_after":"no goals","tactic":"simp [id_apply]","premises":[{"full_name":"CategoryTheory.Mat.id_apply","def_path":"Mathlib/CategoryTheory/Preadditive/Mat.lean","def_pos":[518,8],"def_end_pos":[518,16]}]}]}
+{"url":"Mathlib/Analysis/SpecialFunctions/Log/Base.lean","commit":"","full_name":"Real.logb_neg_eq_logb","start":[60,0],"end":[62,45],"file_path":"Mathlib/Analysis/SpecialFunctions/Log/Base.lean","tactics":[{"state_before":"b x✝ y x : ℝ\n⊢ logb b (-x) = logb b x","state_after":"no goals","tactic":"rw [← logb_abs x, ← logb_abs (-x), abs_neg]","premises":[{"full_name":"Real.logb_abs","def_path":"Mathlib/Analysis/SpecialFunctions/Log/Base.lean","def_pos":[58,8],"def_end_pos":[58,16]},{"full_name":"abs_neg","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[70,2],"def_end_pos":[70,13]}]}]}
+{"url":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","commit":"","full_name":"MeasureTheory.lintegral_finset_sum'","start":[653,0],"end":[660,42],"file_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\ns : Finset β\nf : β → α → ℝ≥0∞\nhf : ∀ b ∈ s, AEMeasurable (f b) μ\n⊢ ∫⁻ (a : α), ∑ b ∈ s, f b a ∂μ = ∑ b ∈ s, ∫⁻ (a : α), f b a ∂μ","state_after":"case empty\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nf : β → α → ℝ≥0∞\nhf : ∀ b ∈ ∅, AEMeasurable (f b) μ\n⊢ ∫⁻ (a : α), ∑ b ∈ ∅, f b a ∂μ = ∑ b ∈ ∅, ∫⁻ (a : α), f b a ∂μ\n\ncase insert\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nf : β → α → ℝ≥0∞\na : β\ns : Finset β\nhas : a ∉ s\nih : (∀ b ∈ s, AEMeasurable (f b) μ) → ∫⁻ (a : α), ∑ b ∈ s, f b a ∂μ = ∑ b ∈ s, ∫⁻ (a : α), f b a ∂μ\nhf : ∀ b ∈ insert a s, AEMeasurable (f b) μ\n⊢ ∫⁻ (a_1 : α), ∑ b ∈ insert a s, f b a_1 ∂μ = ∑ b ∈ insert a s, ∫⁻ (a : α), f b a ∂μ","tactic":"induction' s using Finset.induction_on with a s has ih","premises":[{"full_name":"Finset.induction_on","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[1076,18],"def_end_pos":[1076,30]}]}]}
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+{"url":"Mathlib/Data/Nat/Digits.lean","commit":"","full_name":"Nat.eleven_dvd_of_palindrome","start":[703,0],"end":[710,35],"file_path":"Mathlib/Data/Nat/Digits.lean","tactics":[{"state_before":"n : ℕ\np : (digits 10 n).Palindrome\nh : Even (digits 10 n).length\n⊢ 11 ∣ n","state_after":"n : ℕ\np : (digits 10 n).Palindrome\nh : Even (digits 10 n).length\ndig : List ℤ := List.map (fun n => ↑n) (digits 10 n)\n⊢ 11 ∣ n","tactic":"let dig := (digits 10 n).map fun n : ℕ => (n : ℤ)","premises":[{"full_name":"Int","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean","def_pos":[40,10],"def_end_pos":[40,13]},{"full_name":"List.map","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[361,18],"def_end_pos":[361,21]},{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]},{"full_name":"Nat.digits","def_path":"Mathlib/Data/Nat/Digits.lean","def_pos":[76,4],"def_end_pos":[76,10]}]},{"state_before":"n : ℕ\np : (digits 10 n).Palindrome\nh : Even (digits 10 n).length\ndig : List ℤ := List.map (fun n => ↑n) (digits 10 n)\n⊢ 11 ∣ n","state_after":"n : ℕ\np : (digits 10 n).Palindrome\ndig : List ℤ := List.map (fun n => ↑n) (digits 10 n)\nh : Even dig.length\n⊢ 11 ∣ n","tactic":"replace h : Even dig.length := by rwa [List.length_map]","premises":[{"full_name":"Even","def_path":"Mathlib/Algebra/Group/Even.lean","def_pos":[44,2],"def_end_pos":[44,13]},{"full_name":"List.length","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2316,4],"def_end_pos":[2316,15]},{"full_name":"List.length_map","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean","def_pos":[725,16],"def_end_pos":[725,26]}]},{"state_before":"n : ℕ\np : (digits 10 n).Palindrome\ndig : List ℤ := List.map (fun n => ↑n) (digits 10 n)\nh : Even dig.length\n⊢ 11 ∣ n","state_after":"n : ℕ\np : (digits 10 n).Palindrome\ndig : List ℤ := List.map (fun n => ↑n) (digits 10 n)\nh : Even dig.length\n⊢ dig.alternatingSum = 0","tactic":"refine eleven_dvd_iff.2 ⟨0, (?_ : dig.alternatingSum = 0)⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"List.alternatingSum","def_path":"Mathlib/Algebra/BigOperators/Group/List.lean","def_pos":[41,4],"def_end_pos":[41,18]},{"full_name":"Nat.eleven_dvd_iff","def_path":"Mathlib/Data/Nat/Digits.lean","def_pos":[697,8],"def_end_pos":[697,22]}]},{"state_before":"n : ℕ\np : (digits 10 n).Palindrome\ndig : List ℤ := List.map (fun n => ↑n) (digits 10 n)\nh : Even dig.length\n⊢ dig.alternatingSum = 0","state_after":"n : ℕ\np : (digits 10 n).Palindrome\ndig : List ℤ := List.map (fun n => ↑n) (digits 10 n)\nh : Even dig.length\nthis : dig.reverse.alternatingSum = (-1) ^ (dig.length + 1) • dig.alternatingSum\n⊢ dig.alternatingSum = 0","tactic":"have := dig.alternatingSum_reverse","premises":[{"full_name":"List.alternatingSum_reverse","def_path":"Mathlib/Algebra/BigOperators/Group/List.lean","def_pos":[720,2],"def_end_pos":[720,13]}]},{"state_before":"n : ℕ\np : (digits 10 n).Palindrome\ndig : List ℤ := List.map (fun n => ↑n) (digits 10 n)\nh : Even dig.length\nthis : dig.reverse.alternatingSum = (-1) ^ (dig.length + 1) • dig.alternatingSum\n⊢ dig.alternatingSum = 0","state_after":"n : ℕ\np : (digits 10 n).Palindrome\ndig : List ℤ := List.map (fun n => ↑n) (digits 10 n)\nh : Even dig.length\nthis : (List.map (fun n => ↑n) (digits 10 n)).alternatingSum = -dig.alternatingSum\n⊢ dig.alternatingSum = 0","tactic":"rw [(p.map _).reverse_eq, _root_.pow_succ', h.neg_one_pow, mul_one, neg_one_zsmul] at this","premises":[{"full_name":"Even.neg_one_pow","def_path":"Mathlib/Algebra/Ring/Parity.lean","def_pos":[43,6],"def_end_pos":[43,22]},{"full_name":"List.Palindrome.map","def_path":"Mathlib/Data/List/Palindrome.lean","def_pos":[66,18],"def_end_pos":[66,21]},{"full_name":"List.Palindrome.reverse_eq","def_path":"Mathlib/Data/List/Palindrome.lean","def_pos":[47,8],"def_end_pos":[47,18]},{"full_name":"mul_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[413,8],"def_end_pos":[413,15]},{"full_name":"neg_one_zsmul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[917,14],"def_end_pos":[917,27]},{"full_name":"pow_succ'","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[573,33],"def_end_pos":[573,42]}]},{"state_before":"n : ℕ\np : (digits 10 n).Palindrome\ndig : List ℤ := List.map (fun n => ↑n) (digits 10 n)\nh : Even dig.length\nthis : (List.map (fun n => ↑n) (digits 10 n)).alternatingSum = -dig.alternatingSum\n⊢ dig.alternatingSum = 0","state_after":"no goals","tactic":"exact eq_zero_of_neg_eq this.symm","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"eq_zero_of_neg_eq","def_path":"Mathlib/Algebra/Order/Group/Defs.lean","def_pos":[114,14],"def_end_pos":[114,31]}]}]}
+{"url":"Mathlib/Data/Matrix/Basic.lean","commit":"","full_name":"Matrix.diagonal_neg","start":[408,0],"end":[413,10],"file_path":"Mathlib/Data/Matrix/Basic.lean","tactics":[{"state_before":"l : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nm' : o → Type u_5\nn' : o → Type u_6\nR : Type u_7\nS : Type u_8\nα : Type v\nβ : Type w\nγ : Type u_9\ninst✝¹ : DecidableEq n\ninst✝ : NegZeroClass α\nd : n → α\n⊢ -diagonal d = diagonal fun i => -d i","state_after":"case a\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nm' : o → Type u_5\nn' : o → Type u_6\nR : Type u_7\nS : Type u_8\nα : Type v\nβ : Type w\nγ : Type u_9\ninst✝¹ : DecidableEq n\ninst✝ : NegZeroClass α\nd : n → α\ni j : n\n⊢ (-diagonal d) i j = diagonal (fun i => -d i) i j","tactic":"ext i j","premises":[]},{"state_before":"case a\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nm' : o → Type u_5\nn' : o → Type u_6\nR : Type u_7\nS : Type u_8\nα : Type v\nβ : Type w\nγ : Type u_9\ninst✝¹ : DecidableEq n\ninst✝ : NegZeroClass α\nd : n → α\ni j : n\n⊢ (-diagonal d) i j = diagonal (fun i => -d i) i j","state_after":"no goals","tactic":"by_cases h : i = j <;>\n  simp [h]","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
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+{"url":"Mathlib/Algebra/BigOperators/Finprod.lean","commit":"","full_name":"finprod_mem_pair","start":[760,0],"end":[764,32],"file_path":"Mathlib/Algebra/BigOperators/Finprod.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g : α → M\na b : α\ns t : Set α\nh : a ≠ b\n⊢ ∏ᶠ (i : α) (_ : i ∈ {a, b}), f i = f a * f b","state_after":"case h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g : α → M\na b : α\ns t : Set α\nh : a ≠ b\n⊢ a ∉ {b}\n\ncase hs\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g : α → M\na b : α\ns t : Set α\nh : a ≠ b\n⊢ {b}.Finite","tactic":"rw [finprod_mem_insert, finprod_mem_singleton]","premises":[{"full_name":"finprod_mem_insert","def_path":"Mathlib/Algebra/BigOperators/Finprod.lean","def_pos":[728,8],"def_end_pos":[728,26]},{"full_name":"finprod_mem_singleton","def_path":"Mathlib/Algebra/BigOperators/Finprod.lean","def_pos":[702,8],"def_end_pos":[702,29]}]},{"state_before":"case h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g : α → M\na b : α\ns t : Set α\nh : a ≠ b\n⊢ a ∉ {b}\n\ncase hs\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g : α → M\na b : α\ns t : Set α\nh : a ≠ b\n⊢ {b}.Finite","state_after":"no goals","tactic":"exacts [h, finite_singleton b]","premises":[{"full_name":"Set.finite_singleton","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[725,8],"def_end_pos":[725,24]}]}]}
+{"url":"Mathlib/SetTheory/Cardinal/Basic.lean","commit":"","full_name":"Cardinal.lt_one_iff_zero","start":[1306,0],"end":[1308,38],"file_path":"Mathlib/SetTheory/Cardinal/Basic.lean","tactics":[{"state_before":"α β : Type u\nc : Cardinal.{u_1}\n⊢ c < 1 ↔ c = 0","state_after":"no goals","tactic":"simpa using lt_succ_bot_iff (a := c)","premises":[{"full_name":"Order.lt_succ_bot_iff","def_path":"Mathlib/Order/SuccPred/Basic.lean","def_pos":[482,8],"def_end_pos":[482,23]}]}]}
+{"url":"Mathlib/Analysis/Normed/Group/Basic.lean","commit":"","full_name":"norm_sub_eq_zero_iff","start":[1224,0],"end":[1225,88],"file_path":"Mathlib/Analysis/Normed/Group/Basic.lean","tactics":[{"state_before":"𝓕 : Type u_1\n𝕜 : Type u_2\nα : Type u_3\nι : Type u_4\nκ : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst✝¹ : NormedGroup E\ninst✝ : NormedGroup F\na b : E\n⊢ ‖a / b‖ = 0 ↔ a = b","state_after":"no goals","tactic":"rw [norm_eq_zero'', div_eq_one]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"div_eq_one","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[739,8],"def_end_pos":[739,18]},{"full_name":"norm_eq_zero''","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[1209,8],"def_end_pos":[1209,22]}]}]}
+{"url":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","commit":"","full_name":"MeasurableSpace.measurableSpace_iSup_eq","start":[462,0],"end":[466,5],"file_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort u_6\ns t u : Set α\nm : ι → MeasurableSpace α\n⊢ ⨆ n, m n = generateFrom {s | ∃ n, MeasurableSet s}","state_after":"case h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort u_6\ns✝ t u : Set α\nm : ι → MeasurableSpace α\ns : Set α\n⊢ MeasurableSet s ↔ MeasurableSet s","tactic":"ext s","premises":[]},{"state_before":"case h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort u_6\ns✝ t u : Set α\nm : ι → MeasurableSpace α\ns : Set α\n⊢ MeasurableSet s ↔ MeasurableSet s","state_after":"case h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort u_6\ns✝ t u : Set α\nm : ι → MeasurableSpace α\ns : Set α\n⊢ GenerateMeasurable {s | ∃ i, MeasurableSet s} s ↔ MeasurableSet s","tactic":"rw [measurableSet_iSup]","premises":[{"full_name":"MeasurableSpace.measurableSet_iSup","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","def_pos":[458,8],"def_end_pos":[458,26]}]},{"state_before":"case h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort u_6\ns✝ t u : Set α\nm : ι → MeasurableSpace α\ns : Set α\n⊢ GenerateMeasurable {s | ∃ i, MeasurableSet s} s ↔ MeasurableSet s","state_after":"no goals","tactic":"rfl","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]}]}]}
+{"url":"Mathlib/Geometry/Euclidean/Angle/Sphere.lean","commit":"","full_name":"Orientation.oangle_eq_two_zsmul_oangle_sub_of_norm_eq","start":[28,0],"end":[46,88],"file_path":"Mathlib/Geometry/Euclidean/Angle/Sphere.lean","tactics":[{"state_before":"V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y z : V\nhxyne : x ≠ y\nhxzne : x ≠ z\nhxy : ‖x‖ = ‖y‖\nhxz : ‖x‖ = ‖z‖\n⊢ o.oangle y z = 2 • o.oangle (y - x) (z - x)","state_after":"V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y z : V\nhxyne : x ≠ y\nhxzne : x ≠ z\nhxy : ‖x‖ = ‖y‖\nhxz : ‖x‖ = ‖z‖\nhy : y ≠ 0\n⊢ o.oangle y z = 2 • o.oangle (y - x) (z - x)","tactic":"have hy : y ≠ 0 := by\n    rintro rfl\n    rw [norm_zero, norm_eq_zero] at hxy\n    exact hxyne hxy","premises":[{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"norm_eq_zero","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[1208,29],"def_end_pos":[1208,41]},{"full_name":"norm_zero","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[436,29],"def_end_pos":[436,38]}]},{"state_before":"V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y z : V\nhxyne : x ≠ y\nhxzne : x ≠ z\nhxy : ‖x‖ = ‖y‖\nhxz : ‖x‖ = ‖z‖\nhy : y ≠ 0\n⊢ o.oangle y z = 2 • o.oangle (y - x) (z - x)","state_after":"V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y z : V\nhxyne : x ≠ y\nhxzne : x ≠ z\nhxy : ‖x‖ = ‖y‖\nhxz : ‖x‖ = ‖z‖\nhy : y ≠ 0\nhx : x ≠ 0\n⊢ o.oangle y z = 2 • o.oangle (y - x) (z - x)","tactic":"have hx : x ≠ 0 := norm_ne_zero_iff.1 (hxy.symm ▸ norm_ne_zero_iff.2 hy)","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"norm_ne_zero_iff","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[1212,14],"def_end_pos":[1212,30]}]},{"state_before":"V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y z : V\nhxyne : x ≠ y\nhxzne : x ≠ z\nhxy : ‖x‖ = ‖y‖\nhxz : ‖x‖ = ‖z‖\nhy : y ≠ 0\nhx : x ≠ 0\n⊢ o.oangle y z = 2 • o.oangle (y - x) (z - x)","state_after":"V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y z : V\nhxyne : x ≠ y\nhxzne : x ≠ z\nhxy : ‖x‖ = ‖y‖\nhxz : ‖x‖ = ‖z‖\nhy : y ≠ 0\nhx : x ≠ 0\nhz : z ≠ 0\n⊢ o.oangle y z = 2 • o.oangle (y - x) (z - x)","tactic":"have hz : z ≠ 0 := norm_ne_zero_iff.1 (hxz ▸ norm_ne_zero_iff.2 hx)","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"norm_ne_zero_iff","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[1212,14],"def_end_pos":[1212,30]}]},{"state_before":"V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y z : V\nhxyne : x ≠ y\nhxzne : x ≠ z\nhxy : ‖x‖ = ‖y‖\nhxz : ‖x‖ = ‖z‖\nhy : y ≠ 0\nhx : x ≠ 0\nhz : z ≠ 0\n⊢ o.oangle y z = 2 • o.oangle (y - x) (z - x)","state_after":"no goals","tactic":"calc\n    o.oangle y z = o.oangle x z - o.oangle x y := (o.oangle_sub_left hx hy hz).symm\n    _ = π - (2 : ℤ) • o.oangle (x - z) x - (π - (2 : ℤ) • o.oangle (x - y) x) := by\n      rw [o.oangle_eq_pi_sub_two_zsmul_oangle_sub_of_norm_eq hxzne.symm hxz.symm,\n        o.oangle_eq_pi_sub_two_zsmul_oangle_sub_of_norm_eq hxyne.symm hxy.symm]\n    _ = (2 : ℤ) • (o.oangle (x - y) x - o.oangle (x - z) x) := by abel\n    _ = (2 : ℤ) • o.oangle (x - y) (x - z) := by\n      rw [o.oangle_sub_right (sub_ne_zero_of_ne hxyne) (sub_ne_zero_of_ne hxzne) hx]\n    _ = (2 : ℤ) • o.oangle (y - x) (z - x) := by rw [← oangle_neg_neg, neg_sub, neg_sub]","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Int","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean","def_pos":[40,10],"def_end_pos":[40,13]},{"full_name":"Ne.symm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[704,8],"def_end_pos":[704,15]},{"full_name":"Orientation.oangle","def_path":"Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean","def_pos":[51,4],"def_end_pos":[51,10]},{"full_name":"Orientation.oangle_eq_pi_sub_two_zsmul_oangle_sub_of_norm_eq","def_path":"Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean","def_pos":[503,8],"def_end_pos":[503,56]},{"full_name":"Orientation.oangle_neg_neg","def_path":"Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean","def_pos":[209,8],"def_end_pos":[209,22]},{"full_name":"Orientation.oangle_sub_left","def_path":"Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean","def_pos":[463,8],"def_end_pos":[463,23]},{"full_name":"Orientation.oangle_sub_right","def_path":"Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean","def_pos":[470,8],"def_end_pos":[470,24]},{"full_name":"Real.pi","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","def_pos":[119,28],"def_end_pos":[119,30]},{"full_name":"neg_sub","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[399,2],"def_end_pos":[399,13]},{"full_name":"sub_ne_zero_of_ne","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[387,2],"def_end_pos":[387,13]}]}]}
+{"url":"Mathlib/Algebra/Order/Monoid/Unbundled/WithTop.lean","commit":"","full_name":"WithTop.le_of_add_le_add_right","start":[187,0],"end":[194,67],"file_path":"Mathlib/Algebra/Order/Monoid/Unbundled/WithTop.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\ninst✝² : Add α\na b c d : WithTop α\nx y : α\ninst✝¹ : LE α\ninst✝ : ContravariantClass α α (swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nha : a ≠ ⊤\nh : b + a ≤ c + a\n⊢ b ≤ c","state_after":"case intro\nα : Type u\nβ : Type v\ninst✝² : Add α\nb c d : WithTop α\nx y : α\ninst✝¹ : LE α\ninst✝ : ContravariantClass α α (swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na : α\nh : b + ↑a ≤ c + ↑a\n⊢ b ≤ c","tactic":"lift a to α using ha","premises":[]},{"state_before":"case intro\nα : Type u\nβ : Type v\ninst✝² : Add α\nb c d : WithTop α\nx y : α\ninst✝¹ : LE α\ninst✝ : ContravariantClass α α (swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na : α\nh : b + ↑a ≤ c + ↑a\n⊢ b ≤ c","state_after":"case intro.top\nα : Type u\nβ : Type v\ninst✝² : Add α\nb d : WithTop α\nx y : α\ninst✝¹ : LE α\ninst✝ : ContravariantClass α α (swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na : α\nh : b + ↑a ≤ ⊤ + ↑a\n⊢ b ≤ ⊤\n\ncase intro.coe\nα : Type u\nβ : Type v\ninst✝² : Add α\nb d : WithTop α\nx y : α\ninst✝¹ : LE α\ninst✝ : ContravariantClass α α (swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na a✝ : α\nh : b + ↑a ≤ ↑a✝ + ↑a\n⊢ b ≤ ↑a✝","tactic":"cases c","premises":[]}]}
+{"url":"Mathlib/Algebra/MvPolynomial/PDeriv.lean","commit":"","full_name":"MvPolynomial.pderiv_pow","start":[103,0],"end":[105,71],"file_path":"Mathlib/Algebra/MvPolynomial/PDeriv.lean","tactics":[{"state_before":"R : Type u\nσ : Type v\na a' a₁ a₂ : R\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\ni : σ\nf : MvPolynomial σ R\nn : ℕ\n⊢ (pderiv i) (f ^ n) = ↑n * f ^ (n - 1) * (pderiv i) f","state_after":"no goals","tactic":"rw [(pderiv i).leibniz_pow f n, nsmul_eq_mul, smul_eq_mul, mul_assoc]","premises":[{"full_name":"Derivation.leibniz_pow","def_path":"Mathlib/RingTheory/Derivation/Basic.lean","def_pos":[136,8],"def_end_pos":[136,19]},{"full_name":"MvPolynomial.pderiv","def_path":"Mathlib/Algebra/MvPolynomial/PDeriv.lean","def_pos":[57,4],"def_end_pos":[57,10]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]},{"full_name":"nsmul_eq_mul","def_path":"Mathlib/Data/Nat/Cast/Basic.lean","def_pos":[71,14],"def_end_pos":[71,33]},{"full_name":"smul_eq_mul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[79,6],"def_end_pos":[79,17]}]}]}
+{"url":"Mathlib/Data/Complex/Exponential.lean","commit":"","full_name":"Complex.cos_neg","start":[454,0],"end":[455,86],"file_path":"Mathlib/Data/Complex/Exponential.lean","tactics":[{"state_before":"x y : ℂ\n⊢ cos (-x) = cos x","state_after":"no goals","tactic":"simp [cos, sub_eq_add_neg, exp_neg, add_comm]","premises":[{"full_name":"Complex.cos","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[60,4],"def_end_pos":[60,7]},{"full_name":"Complex.exp_neg","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[214,8],"def_end_pos":[214,15]},{"full_name":"add_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[293,2],"def_end_pos":[293,13]},{"full_name":"sub_eq_add_neg","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[905,2],"def_end_pos":[905,13]}]}]}
+{"url":"Mathlib/Combinatorics/Quiver/Cast.lean","commit":"","full_name":"Quiver.eq_nil_of_length_zero","start":[128,0],"end":[132,54],"file_path":"Mathlib/Combinatorics/Quiver/Cast.lean","tactics":[{"state_before":"U : Type u_1\ninst✝ : Quiver U\nu v : U\np : Path u v\nhzero : p.length = 0\n⊢ Path.cast ⋯ ⋯ p = nil","state_after":"case nil\nU : Type u_1\ninst✝ : Quiver U\nu : U\nhzero : nil.length = 0\n⊢ Path.cast ⋯ ⋯ nil = nil\n\ncase cons\nU : Type u_1\ninst✝ : Quiver U\nu v b✝ : U\na✝¹ : Path u b✝\na✝ : b✝ ⟶ v\nhzero : (a✝¹.cons a✝).length = 0\n⊢ Path.cast ⋯ ⋯ (a✝¹.cons a✝) = nil","tactic":"cases p","premises":[]}]}
+{"url":"Mathlib/Algebra/Order/Group/MinMax.lean","commit":"","full_name":"abs_max_sub_max_le_abs","start":[88,0],"end":[90,40],"file_path":"Mathlib/Algebra/Order/Group/MinMax.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\na✝ b✝ c✝ a b c : α\n⊢ |max a c - max b c| ≤ |a - b|","state_after":"no goals","tactic":"simpa only [sub_self, abs_zero, max_eq_left (abs_nonneg (a - b))]\n    using abs_max_sub_max_le_max a c b c","premises":[{"full_name":"abs_max_sub_max_le_max","def_path":"Mathlib/Algebra/Order/Group/MinMax.lean","def_pos":[78,8],"def_end_pos":[78,30]},{"full_name":"abs_nonneg","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[95,29],"def_end_pos":[95,39]},{"full_name":"abs_zero","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[91,2],"def_end_pos":[91,13]},{"full_name":"max_eq_left","def_path":"Mathlib/Init/Order/LinearOrder.lean","def_pos":[118,8],"def_end_pos":[118,19]},{"full_name":"sub_self","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[695,29],"def_end_pos":[695,37]}]}]}
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+{"url":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","commit":"","full_name":"Real.rpow_sub_nat'","start":[395,0],"end":[396,35],"file_path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","tactics":[{"state_before":"x y z : ℝ\nn : ℕ\nhx : 0 ≤ x\nh : y - ↑n ≠ 0\n⊢ x ^ (y - ↑n) = x ^ y / x ^ n","state_after":"no goals","tactic":"rw [rpow_sub' hx h, rpow_natCast]","premises":[{"full_name":"Real.rpow_natCast","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","def_pos":[63,8],"def_end_pos":[63,20]},{"full_name":"Real.rpow_sub'","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","def_pos":[224,8],"def_end_pos":[224,17]}]}]}
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+{"url":"Mathlib/MeasureTheory/Integral/MeanInequalities.lean","commit":"","full_name":"ENNReal.lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_ne_top","start":[104,0],"end":[125,86],"file_path":"Mathlib/MeasureTheory/Integral/MeanInequalities.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\np q : ℝ\nhpq : p.IsConjExponent q\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhf_nontop : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤\nhg_nontop : ∫⁻ (a : α), g a ^ q ∂μ ≠ ⊤\nhf_nonzero : ∫⁻ (a : α), f a ^ p ∂μ ≠ 0\nhg_nonzero : ∫⁻ (a : α), g a ^ q ∂μ ≠ 0\n⊢ ∫⁻ (a : α), (f * g) a ∂μ ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), g a ^ q ∂μ) ^ (1 / q)","state_after":"α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\np q : ℝ\nhpq : p.IsConjExponent q\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhf_nontop : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤\nhg_nontop : ∫⁻ (a : α), g a ^ q ∂μ ≠ ⊤\nhf_nonzero : ∫⁻ (a : α), f a ^ p ∂μ ≠ 0\nhg_nonzero : ∫⁻ (a : α), g a ^ q ∂μ ≠ 0\nnpf : ℝ≥0∞ := (∫⁻ (c : α), f c ^ p ∂μ) ^ (1 / p)\n⊢ ∫⁻ (a : α), (f * g) a ∂μ ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), g a ^ q ∂μ) ^ (1 / q)","tactic":"let npf := (∫⁻ c : α, f c ^ p ∂μ) ^ (1 / p)","premises":[{"full_name":"MeasureTheory.lintegral","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[59,16],"def_end_pos":[59,25]}]},{"state_before":"α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\np q : ℝ\nhpq : p.IsConjExponent q\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhf_nontop : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤\nhg_nontop : ∫⁻ (a : α), g a ^ q ∂μ ≠ ⊤\nhf_nonzero : ∫⁻ (a : α), f a ^ p ∂μ ≠ 0\nhg_nonzero : ∫⁻ (a : α), g a ^ q ∂μ ≠ 0\nnpf : ℝ≥0∞ := (∫⁻ (c : α), f c ^ p ∂μ) ^ (1 / p)\n⊢ ∫⁻ (a : α), (f * g) a ∂μ ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), g a ^ q ∂μ) ^ (1 / q)","state_after":"α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\np q : ℝ\nhpq : p.IsConjExponent q\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhf_nontop : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤\nhg_nontop : ∫⁻ (a : α), g a ^ q ∂μ ≠ ⊤\nhf_nonzero : ∫⁻ (a : α), f a ^ p ∂μ ≠ 0\nhg_nonzero : ∫⁻ (a : α), g a ^ q ∂μ ≠ 0\nnpf : ℝ≥0∞ := (∫⁻ (c : α), f c ^ p ∂μ) ^ (1 / p)\nnqg : ℝ≥0∞ := (∫⁻ (c : α), g c ^ q ∂μ) ^ (1 / q)\n⊢ ∫⁻ (a : α), (f * g) a ∂μ ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), g a ^ q ∂μ) ^ (1 / q)","tactic":"let nqg := (∫⁻ c : α, g c ^ q ∂μ) ^ (1 / q)","premises":[{"full_name":"MeasureTheory.lintegral","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[59,16],"def_end_pos":[59,25]}]},{"state_before":"α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\np q : ℝ\nhpq : p.IsConjExponent q\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhf_nontop : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤\nhg_nontop : ∫⁻ (a : α), g a ^ q ∂μ ≠ ⊤\nhf_nonzero : ∫⁻ (a : α), f a ^ p ∂μ ≠ 0\nhg_nonzero : ∫⁻ (a : α), g a ^ q ∂μ ≠ 0\nnpf : ℝ≥0∞ := (∫⁻ (c : α), f c ^ p ∂μ) ^ (1 / p)\nnqg : ℝ≥0∞ := (∫⁻ (c : α), g c ^ q ∂μ) ^ (1 / q)\n⊢ ∫⁻ (a : α), (f * g) a ∂μ ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), g a ^ q ∂μ) ^ (1 / q)","state_after":"no goals","tactic":"calc\n    (∫⁻ a : α, (f * g) a ∂μ) =\n        ∫⁻ a : α, (funMulInvSnorm f p μ * funMulInvSnorm g q μ) a * (npf * nqg) ∂μ := by\n      refine lintegral_congr fun a => ?_\n      rw [Pi.mul_apply, fun_eq_funMulInvSnorm_mul_eLpNorm f hf_nonzero hf_nontop,\n        fun_eq_funMulInvSnorm_mul_eLpNorm g hg_nonzero hg_nontop, Pi.mul_apply]\n      ring\n    _ ≤ npf * nqg := by\n      rw [lintegral_mul_const' (npf * nqg) _\n          (by simp [npf, nqg, hf_nontop, hg_nontop, hf_nonzero, hg_nonzero, ENNReal.mul_eq_top])]\n      refine mul_le_of_le_one_left' ?_\n      have hf1 := lintegral_rpow_funMulInvSnorm_eq_one hpq.pos hf_nonzero hf_nontop\n      have hg1 := lintegral_rpow_funMulInvSnorm_eq_one hpq.symm.pos hg_nonzero hg_nontop\n      exact lintegral_mul_le_one_of_lintegral_rpow_eq_one hpq (hf.mul_const _) hf1 hg1","premises":[{"full_name":"AEMeasurable.mul_const","def_path":"Mathlib/MeasureTheory/Group/Arithmetic.lean","def_pos":[109,8],"def_end_pos":[109,30]},{"full_name":"ENNReal.funMulInvSnorm","def_path":"Mathlib/MeasureTheory/Integral/MeanInequalities.lean","def_pos":[79,4],"def_end_pos":[79,18]},{"full_name":"ENNReal.fun_eq_funMulInvSnorm_mul_eLpNorm","def_path":"Mathlib/MeasureTheory/Integral/MeanInequalities.lean","def_pos":[82,8],"def_end_pos":[82,41]},{"full_name":"ENNReal.lintegral_mul_le_one_of_lintegral_rpow_eq_one","def_path":"Mathlib/MeasureTheory/Integral/MeanInequalities.lean","def_pos":[63,8],"def_end_pos":[63,53]},{"full_name":"ENNReal.lintegral_rpow_funMulInvSnorm_eq_one","def_path":"Mathlib/MeasureTheory/Integral/MeanInequalities.lean","def_pos":[97,8],"def_end_pos":[97,44]},{"full_name":"ENNReal.mul_eq_top","def_path":"Mathlib/Data/ENNReal/Operations.lean","def_pos":[186,8],"def_end_pos":[186,18]},{"full_name":"MeasureTheory.lintegral","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[59,16],"def_end_pos":[59,25]},{"full_name":"MeasureTheory.lintegral_congr","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[312,8],"def_end_pos":[312,23]},{"full_name":"MeasureTheory.lintegral_mul_const'","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[727,8],"def_end_pos":[727,28]},{"full_name":"Pi.mul_apply","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[82,8],"def_end_pos":[82,17]},{"full_name":"Real.IsConjExponent.pos","def_path":"Mathlib/Data/Real/ConjExponents.lean","def_pos":[53,8],"def_end_pos":[53,11]},{"full_name":"Real.IsConjExponent.symm","def_path":"Mathlib/Data/Real/ConjExponents.lean","def_pos":[89,24],"def_end_pos":[89,28]},{"full_name":"mul_le_of_le_one_left'","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[353,8],"def_end_pos":[353,30]}]}]}
+{"url":"Mathlib/Topology/MetricSpace/ThickenedIndicator.lean","commit":"","full_name":"tendsto_mulIndicator_cthickening_mulIndicator_closure","start":[285,0],"end":[295,26],"file_path":"Mathlib/Topology/MetricSpace/ThickenedIndicator.lean","tactics":[{"state_before":"α : Type u_1\ninst✝² : PseudoEMetricSpace α\nβ : Type u_2\ninst✝¹ : One β\ninst✝ : TopologicalSpace β\nf : α → β\nE : Set α\n⊢ Tendsto (fun δ => (cthickening δ E).mulIndicator f) (𝓝 0) (𝓝 ((closure E).mulIndicator f))","state_after":"α : Type u_1\ninst✝² : PseudoEMetricSpace α\nβ : Type u_2\ninst✝¹ : One β\ninst✝ : TopologicalSpace β\nf : α → β\nE : Set α\n⊢ ∀ (x : α), Tendsto (fun i => (cthickening i E).mulIndicator f x) (𝓝 0) (𝓝 ((closure E).mulIndicator f x))","tactic":"rw [tendsto_pi_nhds]","premises":[{"full_name":"tendsto_pi_nhds","def_path":"Mathlib/Topology/Constructions.lean","def_pos":[1159,8],"def_end_pos":[1159,23]}]},{"state_before":"α : Type u_1\ninst✝² : PseudoEMetricSpace α\nβ : Type u_2\ninst✝¹ : One β\ninst✝ : TopologicalSpace β\nf : α → β\nE : Set α\n⊢ ∀ (x : α), Tendsto (fun i => (cthickening i E).mulIndicator f x) (𝓝 0) (𝓝 ((closure E).mulIndicator f x))","state_after":"α : Type u_1\ninst✝² : PseudoEMetricSpace α\nβ : Type u_2\ninst✝¹ : One β\ninst✝ : TopologicalSpace β\nf : α → β\nE : Set α\nx : α\n⊢ Tendsto (fun i => (cthickening i E).mulIndicator f x) (𝓝 0) (𝓝 ((closure E).mulIndicator f x))","tactic":"intro x","premises":[]},{"state_before":"α : Type u_1\ninst✝² : PseudoEMetricSpace α\nβ : Type u_2\ninst✝¹ : One β\ninst✝ : TopologicalSpace β\nf : α → β\nE : Set α\nx : α\n⊢ Tendsto (fun i => (cthickening i E).mulIndicator f x) (𝓝 0) (𝓝 ((closure E).mulIndicator f x))","state_after":"α : Type u_1\ninst✝² : PseudoEMetricSpace α\nβ : Type u_2\ninst✝¹ : One β\ninst✝ : TopologicalSpace β\nf : α → β\nE : Set α\nx : α\n⊢ Tendsto (fun x_1 => (closure E).mulIndicator f x) (𝓝 0) (𝓝 ((closure E).mulIndicator f x))","tactic":"rw [tendsto_congr' (mulIndicator_cthickening_eventually_eq_mulIndicator_closure f E x)]","premises":[{"full_name":"Filter.tendsto_congr'","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2631,8],"def_end_pos":[2631,22]},{"full_name":"mulIndicator_cthickening_eventually_eq_mulIndicator_closure","def_path":"Mathlib/Topology/MetricSpace/ThickenedIndicator.lean","def_pos":[261,6],"def_end_pos":[261,65]}]},{"state_before":"α : Type u_1\ninst✝² : PseudoEMetricSpace α\nβ : Type u_2\ninst✝¹ : One β\ninst✝ : TopologicalSpace β\nf : α → β\nE : Set α\nx : α\n⊢ Tendsto (fun x_1 => (closure E).mulIndicator f x) (𝓝 0) (𝓝 ((closure E).mulIndicator f x))","state_after":"no goals","tactic":"apply tendsto_const_nhds","premises":[{"full_name":"tendsto_const_nhds","def_path":"Mathlib/Topology/Basic.lean","def_pos":[844,8],"def_end_pos":[844,26]}]}]}
+{"url":"Mathlib/ModelTheory/Semantics.lean","commit":"","full_name":"FirstOrder.Language.Formula.realize_graph","start":[564,0],"end":[568,14],"file_path":"Mathlib/ModelTheory/Semantics.lean","tactics":[{"state_before":"L : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst✝² : L.Structure M\ninst✝¹ : L.Structure N\ninst✝ : L.Structure P\nα : Type u'\nβ : Type v'\nγ : Type u_3\nn : ℕ\nφ ψ : L.Formula α\nv : α → M\nf : L.Functions n\nx : Fin n → M\ny : M\n⊢ (graph f).Realize (cons y x) ↔ funMap f x = y","state_after":"L : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst✝² : L.Structure M\ninst✝¹ : L.Structure N\ninst✝ : L.Structure P\nα : Type u'\nβ : Type v'\nγ : Type u_3\nn : ℕ\nφ ψ : L.Formula α\nv : α → M\nf : L.Functions n\nx : Fin n → M\ny : M\n⊢ (y = funMap f fun i => x i) ↔ funMap f x = y","tactic":"simp only [Formula.graph, Term.realize, realize_equal, Fin.cons_zero, Fin.cons_succ]","premises":[{"full_name":"Fin.cons_succ","def_path":"Mathlib/Data/Fin/Tuple/Basic.lean","def_pos":[115,8],"def_end_pos":[115,17]},{"full_name":"Fin.cons_zero","def_path":"Mathlib/Data/Fin/Tuple/Basic.lean","def_pos":[118,8],"def_end_pos":[118,17]},{"full_name":"FirstOrder.Language.Formula.graph","def_path":"Mathlib/ModelTheory/Syntax.lean","def_pos":[719,4],"def_end_pos":[719,9]},{"full_name":"FirstOrder.Language.Formula.realize_equal","def_path":"Mathlib/ModelTheory/Semantics.lean","def_pos":[561,8],"def_end_pos":[561,21]},{"full_name":"FirstOrder.Language.Term.realize","def_path":"Mathlib/ModelTheory/Semantics.lean","def_pos":[67,4],"def_end_pos":[67,11]}]},{"state_before":"L : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst✝² : L.Structure M\ninst✝¹ : L.Structure N\ninst✝ : L.Structure P\nα : Type u'\nβ : Type v'\nγ : Type u_3\nn : ℕ\nφ ψ : L.Formula α\nv : α → M\nf : L.Functions n\nx : Fin n → M\ny : M\n⊢ (y = funMap f fun i => x i) ↔ funMap f x = y","state_after":"no goals","tactic":"rw [eq_comm]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"eq_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[811,8],"def_end_pos":[811,15]}]}]}
+{"url":"Mathlib/Data/Nat/Nth.lean","commit":"","full_name":"Nat.range_nth_of_infinite","start":[146,0],"end":[150,50],"file_path":"Mathlib/Data/Nat/Nth.lean","tactics":[{"state_before":"p : ℕ → Prop\nhf : (setOf p).Infinite\n⊢ Set.range (nth p) = setOf p","state_after":"p : ℕ → Prop\nhf : (setOf p).Infinite\n⊢ Set.range (Subtype.val ∘ ⇑(Subtype.orderIsoOfNat (setOf p))) = setOf p","tactic":"rw [nth_eq_orderIsoOfNat hf]","premises":[{"full_name":"Nat.nth_eq_orderIsoOfNat","def_path":"Mathlib/Data/Nat/Nth.lean","def_pos":[126,8],"def_end_pos":[126,28]}]},{"state_before":"p : ℕ → Prop\nhf : (setOf p).Infinite\n⊢ Set.range (Subtype.val ∘ ⇑(Subtype.orderIsoOfNat (setOf p))) = setOf p","state_after":"p : ℕ → Prop\nhf : (setOf p).Infinite\nthis : Infinite ↑(setOf p)\n⊢ Set.range (Subtype.val ∘ ⇑(Subtype.orderIsoOfNat (setOf p))) = setOf p","tactic":"haveI := hf.to_subtype","premises":[]},{"state_before":"p : ℕ → Prop\nhf : (setOf p).Infinite\nthis : Infinite ↑(setOf p)\n⊢ Set.range (Subtype.val ∘ ⇑(Subtype.orderIsoOfNat (setOf p))) = setOf p","state_after":"no goals","tactic":"classical exact Nat.Subtype.coe_comp_ofNat_range","premises":[{"full_name":"Nat.Subtype.coe_comp_ofNat_range","def_path":"Mathlib/Logic/Denumerable.lean","def_pos":[273,8],"def_end_pos":[273,28]}]}]}
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f.hom","premises":[{"full_name":"CategoryTheory.Comma.hom","def_path":"Mathlib/CategoryTheory/Comma/Basic.lean","def_pos":[64,2],"def_end_pos":[64,5]},{"full_name":"CategoryTheory.CostructuredArrow.w","def_path":"Mathlib/CategoryTheory/Comma/StructuredArrow.lean","def_pos":[431,8],"def_end_pos":[431,9]}]},{"state_before":"C₁ : Type u₁\nC₂ : Type u₂\nC₃ : Type u₃\nC₄ : Type u₄\ninst✝³ : Category.{v₁, u₁} C₁\ninst✝² : Category.{v₂, u₂} C₂\ninst✝¹ : Category.{v₃, u₃} C₃\ninst✝ : Category.{v₄, u₄} C₄\nT : C₁ ⥤ C₂\nL : C₁ ⥤ C₃\nR : C₂ ⥤ C₄\nB : C₃ ⥤ C₄\nw : TwoSquare T L R B\nX₂ : C₂\nX₃ : C₃\ng : R.obj X₂ ⟶ B.obj X₃\nf₁ f₂ : w.StructuredArrowRightwards g\nφ : f₁ ⟶ f₂\n⊢ ((fun f => CostructuredArrow.mk (StructuredArrow.homMk f.right.hom ⋯)) f₁).left.hom ≫ T.map φ.right.left =\n    ((fun f => CostructuredArrow.mk (StructuredArrow.homMk f.right.hom ⋯)) f₂).left.hom","state_after":"C₁ : Type u₁\nC₂ : Type u₂\nC₃ : Type u₃\nC₄ : Type u₄\ninst✝³ : Category.{v₁, u₁} C₁\ninst✝² : Category.{v₂, u₂} 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+{"url":"Mathlib/Order/Bounds/Basic.lean","commit":"","full_name":"BddAbove.union","start":[347,0],"end":[353,69],"file_path":"Mathlib/Order/Bounds/Basic.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ns✝ t✝ : Set α\na b : α\ninst✝ : IsDirected α fun x x_1 => x ≤ x_1\ns t : Set α\n⊢ BddAbove s → BddAbove t → BddAbove (s ∪ t)","state_after":"case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ns✝ t✝ : Set α\na✝ b✝ : α\ninst✝ : IsDirected α fun x x_1 => x ≤ x_1\ns t : Set α\na : α\nha : a ∈ upperBounds s\nb : α\nhb : b ∈ upperBounds t\n⊢ BddAbove (s ∪ t)","tactic":"rintro ⟨a, ha⟩ ⟨b, hb⟩","premises":[]},{"state_before":"case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ns✝ t✝ : Set α\na✝ b✝ : α\ninst✝ : IsDirected α fun x x_1 => x ≤ x_1\ns t : Set α\na : α\nha : a ∈ upperBounds s\nb : α\nhb : b ∈ upperBounds t\n⊢ BddAbove (s ∪ t)","state_after":"case intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ns✝ t✝ : Set α\na✝ b✝ : α\ninst✝ : IsDirected α fun x x_1 => x ≤ x_1\ns t : Set α\na : α\nha : a ∈ upperBounds s\nb : α\nhb : b ∈ upperBounds t\nc : α\nhca : a ≤ c\nhcb : b ≤ c\n⊢ BddAbove (s ∪ t)","tactic":"obtain ⟨c, hca, hcb⟩ := exists_ge_ge a b","premises":[{"full_name":"exists_ge_ge","def_path":"Mathlib/Order/Directed.lean","def_pos":[157,8],"def_end_pos":[157,20]}]},{"state_before":"case intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ns✝ t✝ : Set α\na✝ b✝ : α\ninst✝ : IsDirected α fun x x_1 => x ≤ x_1\ns t : Set α\na : α\nha : a ∈ upperBounds s\nb : α\nhb : b ∈ upperBounds t\nc : α\nhca : a ≤ c\nhcb : b ≤ c\n⊢ BddAbove (s ∪ t)","state_after":"case intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ns✝ t✝ : Set α\na✝ b✝ : α\ninst✝ : IsDirected α fun x x_1 => x ≤ x_1\ns t : Set α\na : α\nha : a ∈ upperBounds s\nb : α\nhb : b ∈ upperBounds t\nc : α\nhca : a ≤ c\nhcb : b ≤ c\n⊢ (upperBounds s ∩ upperBounds t).Nonempty","tactic":"rw [BddAbove, upperBounds_union]","premises":[{"full_name":"BddAbove","def_path":"Mathlib/Order/Bounds/Basic.lean","def_pos":[52,4],"def_end_pos":[52,12]},{"full_name":"upperBounds_union","def_path":"Mathlib/Order/Bounds/Basic.lean","def_pos":[305,8],"def_end_pos":[305,25]}]},{"state_before":"case intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ns✝ t✝ : Set α\na✝ b✝ : α\ninst✝ : IsDirected α fun x x_1 => x ≤ x_1\ns t : Set α\na : α\nha : a ∈ upperBounds s\nb : α\nhb : b ∈ upperBounds t\nc : α\nhca : a ≤ c\nhcb : b ≤ c\n⊢ (upperBounds s ∩ upperBounds t).Nonempty","state_after":"no goals","tactic":"exact ⟨c, upperBounds_mono_mem hca ha, upperBounds_mono_mem hcb hb⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"upperBounds_mono_mem","def_path":"Mathlib/Order/Bounds/Basic.lean","def_pos":[177,8],"def_end_pos":[177,28]}]}]}
+{"url":"Mathlib/Analysis/Normed/Operator/Banach.lean","commit":"","full_name":"ContinuousLinearMap.coe_ofSeqClosedGraph","start":[508,0],"end":[512,5],"file_path":"Mathlib/Analysis/Normed/Operator/Banach.lean","tactics":[{"state_before":"𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹⁴ : NontriviallyNormedField 𝕜\ninst✝¹³ : NontriviallyNormedField 𝕜'\nσ : 𝕜 →+* 𝕜'\nσ' : 𝕜' →+* 𝕜\ninst✝¹² : RingHomInvPair σ σ'\ninst✝¹¹ : RingHomInvPair σ' σ\ninst✝¹⁰ : RingHomIsometric σ\ninst✝⁹ : RingHomIsometric σ'\nE : Type u_3\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF✝ : Type u_4\ninst✝⁶ : NormedAddCommGroup F✝\ninst✝⁵ : NormedSpace 𝕜' F✝\nf : E →SL[σ] F✝\ninst✝⁴ : CompleteSpace F✝\ninst✝³ : CompleteSpace E\nF : Type u_5\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\ng : E →ₗ[𝕜] F\nhg : ∀ (u : ℕ → E) (x : E) (y : F), Tendsto u atTop (𝓝 x) → Tendsto (⇑g ∘ u) atTop (𝓝 y) → y = g x\n⊢ ↑(ofSeqClosedGraph hg) = g","state_after":"case h\n𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹⁴ : NontriviallyNormedField 𝕜\ninst✝¹³ : NontriviallyNormedField 𝕜'\nσ : 𝕜 →+* 𝕜'\nσ' : 𝕜' →+* 𝕜\ninst✝¹² : RingHomInvPair σ σ'\ninst✝¹¹ : RingHomInvPair σ' σ\ninst✝¹⁰ : RingHomIsometric σ\ninst✝⁹ : RingHomIsometric σ'\nE : Type u_3\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF✝ : Type u_4\ninst✝⁶ : NormedAddCommGroup F✝\ninst✝⁵ : NormedSpace 𝕜' F✝\nf : E →SL[σ] F✝\ninst✝⁴ : CompleteSpace F✝\ninst✝³ : CompleteSpace E\nF : Type u_5\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\ng : E →ₗ[𝕜] F\nhg : ∀ (u : ℕ → E) (x : E) (y : F), Tendsto u atTop (𝓝 x) → Tendsto (⇑g ∘ u) atTop (𝓝 y) → y = g x\nx✝ : E\n⊢ ↑(ofSeqClosedGraph hg) x✝ = g x✝","tactic":"ext","premises":[]},{"state_before":"case h\n𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹⁴ : NontriviallyNormedField 𝕜\ninst✝¹³ : NontriviallyNormedField 𝕜'\nσ : 𝕜 →+* 𝕜'\nσ' : 𝕜' →+* 𝕜\ninst✝¹² : RingHomInvPair σ σ'\ninst✝¹¹ : RingHomInvPair σ' σ\ninst✝¹⁰ : RingHomIsometric σ\ninst✝⁹ : RingHomIsometric σ'\nE : Type u_3\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF✝ : Type u_4\ninst✝⁶ : NormedAddCommGroup F✝\ninst✝⁵ : NormedSpace 𝕜' F✝\nf : E →SL[σ] F✝\ninst✝⁴ : CompleteSpace F✝\ninst✝³ : CompleteSpace E\nF : Type u_5\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\ng : E →ₗ[𝕜] F\nhg : ∀ (u : ℕ → E) (x : E) (y : F), Tendsto u atTop (𝓝 x) → Tendsto (⇑g ∘ u) atTop (𝓝 y) → y = g x\nx✝ : E\n⊢ ↑(ofSeqClosedGraph hg) x✝ = g x✝","state_after":"no goals","tactic":"rfl","premises":[]}]}
+{"url":"Mathlib/Combinatorics/SimpleGraph/Finite.lean","commit":"","full_name":"SimpleGraph.degree_compl","start":[214,0],"end":[218,94],"file_path":"Mathlib/Combinatorics/SimpleGraph/Finite.lean","tactics":[{"state_before":"V : Type u_1\nG : SimpleGraph V\ne : Sym2 V\nv : V\ninst✝² : Fintype ↑(G.neighborSet v)\ninst✝¹ : Fintype ↑(Gᶜ.neighborSet v)\ninst✝ : Fintype V\n⊢ Gᶜ.degree v = Fintype.card V - 1 - G.degree v","state_after":"no goals","tactic":"classical\n    rw [← card_neighborSet_union_compl_neighborSet G v, Set.toFinset_union]\n    simp [card_union_of_disjoint (Set.disjoint_toFinset.mpr (compl_neighborSet_disjoint G v))]","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Set.disjoint_toFinset","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[597,8],"def_end_pos":[597,25]},{"full_name":"Set.toFinset_union","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[610,8],"def_end_pos":[610,22]},{"full_name":"SimpleGraph.card_neighborSet_union_compl_neighborSet","def_path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","def_pos":[707,8],"def_end_pos":[707,48]},{"full_name":"SimpleGraph.compl_neighborSet_disjoint","def_path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","def_pos":[693,8],"def_end_pos":[693,34]}]}]}
+{"url":"Mathlib/MeasureTheory/Function/UniformIntegrable.lean","commit":"","full_name":"MeasureTheory.Memℒp.eLpNorm_indicator_le","start":[411,0],"end":[421,37],"file_path":"Mathlib/MeasureTheory/Function/UniformIntegrable.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nhf : Memℒp f p μ\nε : ℝ\nhε : 0 < ε\n⊢ ∃ δ,\n    ∃ (_ : 0 < δ),\n      ∀ (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator f) p μ ≤ ENNReal.ofReal ε","state_after":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nhf : Memℒp f p μ\nε : ℝ\nhε : 0 < ε\nhℒp : Memℒp f p μ\n⊢ ∃ δ,\n    ∃ (_ : 0 < δ),\n      ∀ (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator f) p μ ≤ ENNReal.ofReal ε","tactic":"have hℒp := hf","premises":[]},{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nhf : Memℒp f p μ\nε : ℝ\nhε : 0 < ε\nhℒp : Memℒp f p μ\n⊢ ∃ δ,\n    ∃ (_ : 0 < δ),\n      ∀ (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator f) p μ ≤ ENNReal.ofReal ε","state_after":"case intro.intro.intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nhℒp : Memℒp f p μ\nright✝ : eLpNorm f p μ < ⊤\nf' : α → β\nhf' : StronglyMeasurable f'\nheq : f =ᶠ[ae μ] f'\n⊢ ∃ δ,\n    ∃ (_ : 0 < δ),\n      ∀ (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator f) p μ ≤ ENNReal.ofReal ε","tactic":"obtain ⟨⟨f', hf', heq⟩, _⟩ := hf","premises":[]},{"state_before":"case intro.intro.intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nhℒp : Memℒp f p μ\nright✝ : eLpNorm f p μ < ⊤\nf' : α → β\nhf' : StronglyMeasurable f'\nheq : f =ᶠ[ae μ] f'\n⊢ ∃ δ,\n    ∃ (_ : 0 < δ),\n      ∀ (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator f) p μ ≤ ENNReal.ofReal ε","state_after":"case intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nhℒp : Memℒp f p μ\nright✝ : eLpNorm f p μ < ⊤\nf' : α → β\nhf' : StronglyMeasurable f'\nheq : f =ᶠ[ae μ] f'\nδ : ℝ\nhδpos : 0 < δ\nhδ : ∀ (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator f') p μ ≤ ENNReal.ofReal ε\n⊢ ∃ δ,\n    ∃ (_ : 0 < δ),\n      ∀ (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator f) p μ ≤ ENNReal.ofReal ε","tactic":"obtain ⟨δ, hδpos, hδ⟩ := (hℒp.ae_eq heq).eLpNorm_indicator_le_of_meas hp_one hp_top hf' hε","premises":[{"full_name":"MeasureTheory.Memℒp.ae_eq","def_path":"Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean","def_pos":[692,8],"def_end_pos":[692,19]},{"full_name":"MeasureTheory.Memℒp.eLpNorm_indicator_le_of_meas","def_path":"Mathlib/MeasureTheory/Function/UniformIntegrable.lean","def_pos":[398,8],"def_end_pos":[398,42]}]},{"state_before":"case intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nhℒp : Memℒp f p μ\nright✝ : eLpNorm f p μ < ⊤\nf' : α → β\nhf' : StronglyMeasurable f'\nheq : f =ᶠ[ae μ] f'\nδ : ℝ\nhδpos : 0 < δ\nhδ : ∀ (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator f') p μ ≤ ENNReal.ofReal ε\n⊢ ∃ δ,\n    ∃ (_ : 0 < δ),\n      ∀ (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator f) p μ ≤ ENNReal.ofReal ε","state_after":"case intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nhℒp : Memℒp f p μ\nright✝ : eLpNorm f p μ < ⊤\nf' : α → β\nhf' : StronglyMeasurable f'\nheq : f =ᶠ[ae μ] f'\nδ : ℝ\nhδpos : 0 < δ\nhδ : ∀ (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator f') p μ ≤ ENNReal.ofReal ε\ns : Set α\nhs : MeasurableSet s\nhμs : μ s ≤ ENNReal.ofReal δ\n⊢ eLpNorm (s.indicator f) p μ ≤ ENNReal.ofReal ε","tactic":"refine ⟨δ, hδpos, fun s hs hμs => ?_⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]},{"state_before":"case intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nhℒp : Memℒp f p μ\nright✝ : eLpNorm f p μ < ⊤\nf' : α → β\nhf' : StronglyMeasurable f'\nheq : f =ᶠ[ae μ] f'\nδ : ℝ\nhδpos : 0 < δ\nhδ : ∀ (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator f') p μ ≤ ENNReal.ofReal ε\ns : Set α\nhs : MeasurableSet s\nhμs : μ s ≤ ENNReal.ofReal δ\n⊢ eLpNorm (s.indicator f) p μ ≤ ENNReal.ofReal ε","state_after":"case h.e'_3\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nhℒp : Memℒp f p μ\nright✝ : eLpNorm f p μ < ⊤\nf' : α → β\nhf' : StronglyMeasurable f'\nheq : f =ᶠ[ae μ] f'\nδ : ℝ\nhδpos : 0 < δ\nhδ : ∀ (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator f') p μ ≤ ENNReal.ofReal ε\ns : Set α\nhs : MeasurableSet s\nhμs : μ s ≤ ENNReal.ofReal δ\n⊢ eLpNorm (s.indicator f) p μ = eLpNorm (s.indicator f') p μ","tactic":"convert hδ s hs hμs using 1","premises":[]},{"state_before":"case h.e'_3\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nhℒp : Memℒp f p μ\nright✝ : eLpNorm f p μ < ⊤\nf' : α → β\nhf' : StronglyMeasurable f'\nheq : f =ᶠ[ae μ] f'\nδ : ℝ\nhδpos : 0 < δ\nhδ : ∀ (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator f') p μ ≤ ENNReal.ofReal ε\ns : Set α\nhs : MeasurableSet s\nhμs : μ s ≤ ENNReal.ofReal δ\n⊢ eLpNorm (s.indicator f) p μ = eLpNorm (s.indicator f') p μ","state_after":"case h.e'_3\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nhℒp : Memℒp f p μ\nright✝ : eLpNorm f p μ < ⊤\nf' : α → β\nhf' : StronglyMeasurable f'\nheq : f =ᶠ[ae μ] f'\nδ : ℝ\nhδpos : 0 < δ\nhδ : ∀ (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator f') p μ ≤ ENNReal.ofReal ε\ns : Set α\nhs : MeasurableSet s\nhμs : μ s ≤ ENNReal.ofReal δ\n⊢ eLpNorm f p (μ.restrict s) = eLpNorm f' p (μ.restrict s)","tactic":"rw [eLpNorm_indicator_eq_eLpNorm_restrict hs, eLpNorm_indicator_eq_eLpNorm_restrict hs]","premises":[{"full_name":"MeasureTheory.eLpNorm_indicator_eq_eLpNorm_restrict","def_path":"Mathlib/MeasureTheory/Function/LpSpace.lean","def_pos":[612,8],"def_end_pos":[612,45]}]},{"state_before":"case h.e'_3\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nhℒp : Memℒp f p μ\nright✝ : eLpNorm f p μ < ⊤\nf' : α → β\nhf' : StronglyMeasurable f'\nheq : f =ᶠ[ae μ] f'\nδ : ℝ\nhδpos : 0 < δ\nhδ : ∀ (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator f') p μ ≤ ENNReal.ofReal ε\ns : Set α\nhs : MeasurableSet s\nhμs : μ s ≤ ENNReal.ofReal δ\n⊢ eLpNorm f p (μ.restrict s) = eLpNorm f' p (μ.restrict s)","state_after":"no goals","tactic":"exact eLpNorm_congr_ae heq.restrict","premises":[{"full_name":"Filter.EventuallyEq.restrict","def_path":"Mathlib/MeasureTheory/Measure/Restrict.lean","def_pos":[551,8],"def_end_pos":[551,43]},{"full_name":"MeasureTheory.eLpNorm_congr_ae","def_path":"Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean","def_pos":[683,8],"def_end_pos":[683,24]}]}]}
+{"url":"Mathlib/Analysis/SpecificLimits/Normed.lean","commit":"","full_name":"Antitone.cauchySeq_series_mul_of_tendsto_zero_of_bounded","start":[638,0],"end":[647,6],"file_path":"Mathlib/Analysis/SpecificLimits/Normed.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nE : Type u_4\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nb : ℝ\nf : ℕ → ℝ\nz : ℕ → E\nhfa : Antitone f\nhf0 : Tendsto f atTop (𝓝 0)\nhzb : ∀ (n : ℕ), ‖∑ i ∈ Finset.range n, z i‖ ≤ b\n⊢ CauchySeq fun n => ∑ i ∈ Finset.range n, f i • z i","state_after":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nE : Type u_4\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nb : ℝ\nf : ℕ → ℝ\nz : ℕ → E\nhfa : Antitone f\nhf0 : Tendsto f atTop (𝓝 0)\nhzb : ∀ (n : ℕ), ‖∑ i ∈ Finset.range n, z i‖ ≤ b\nhfa' : Monotone fun n => -f n\n⊢ CauchySeq fun n => ∑ i ∈ Finset.range n, f i • z i","tactic":"have hfa' : Monotone fun n ↦ -f n := fun _ _ hab ↦ neg_le_neg <| hfa hab","premises":[{"full_name":"Monotone","def_path":"Mathlib/Order/Monotone/Basic.lean","def_pos":[76,4],"def_end_pos":[76,12]},{"full_name":"neg_le_neg","def_path":"Mathlib/Algebra/Order/Group/Defs.lean","def_pos":[172,31],"def_end_pos":[172,41]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nE : Type u_4\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nb : ℝ\nf : ℕ → ℝ\nz : ℕ → E\nhfa : Antitone f\nhf0 : Tendsto f atTop (𝓝 0)\nhzb : ∀ (n : ℕ), ‖∑ i ∈ Finset.range n, z i‖ ≤ b\nhfa' : Monotone fun n => -f n\n⊢ CauchySeq fun n => ∑ i ∈ Finset.range n, f i • z i","state_after":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nE : Type u_4\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nb : ℝ\nf : ℕ → ℝ\nz : ℕ → E\nhfa : Antitone f\nhf0 : Tendsto f atTop (𝓝 0)\nhzb : ∀ (n : ℕ), ‖∑ i ∈ Finset.range n, z i‖ ≤ b\nhfa' : Monotone fun n => -f n\nhf0' : Tendsto (fun n => -f n) atTop (𝓝 0)\n⊢ CauchySeq fun n => ∑ i ∈ Finset.range n, f i • z i","tactic":"have hf0' : Tendsto (fun n ↦ -f n) atTop (𝓝 0) := by\n    convert hf0.neg\n    norm_num","premises":[{"full_name":"Filter.Tendsto","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2567,4],"def_end_pos":[2567,11]},{"full_name":"Filter.Tendsto.neg","def_path":"Mathlib/Topology/Algebra/Group/Basic.lean","def_pos":[214,2],"def_end_pos":[214,13]},{"full_name":"Filter.atTop","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[40,4],"def_end_pos":[40,9]},{"full_name":"nhds","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[113,16],"def_end_pos":[113,20]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nE : Type u_4\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nb : ℝ\nf : ℕ → ℝ\nz : ℕ → E\nhfa : Antitone f\nhf0 : Tendsto f atTop (𝓝 0)\nhzb : ∀ (n : ℕ), ‖∑ i ∈ Finset.range n, z i‖ ≤ b\nhfa' : Monotone fun n => -f n\nhf0' : Tendsto (fun n => -f n) atTop (𝓝 0)\n⊢ CauchySeq fun n => ∑ i ∈ Finset.range n, f i • z i","state_after":"case h.e'_5.h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nE : Type u_4\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nb : ℝ\nf : ℕ → ℝ\nz : ℕ → E\nhfa : Antitone f\nhf0 : Tendsto f atTop (𝓝 0)\nhzb : ∀ (n : ℕ), ‖∑ i ∈ Finset.range n, z i‖ ≤ b\nhfa' : Monotone fun n => -f n\nhf0' : Tendsto (fun n => -f n) atTop (𝓝 0)\nx✝ : ℕ\n⊢ ∑ i ∈ Finset.range x✝, f i • z i = (-fun n => ∑ i ∈ Finset.range n, -f i • z i) x✝","tactic":"convert (hfa'.cauchySeq_series_mul_of_tendsto_zero_of_bounded hf0' hzb).neg","premises":[{"full_name":"CauchySeq.neg","def_path":"Mathlib/Topology/Algebra/UniformGroup.lean","def_pos":[378,2],"def_end_pos":[378,13]},{"full_name":"Monotone.cauchySeq_series_mul_of_tendsto_zero_of_bounded","def_path":"Mathlib/Analysis/SpecificLimits/Normed.lean","def_pos":[620,8],"def_end_pos":[620,64]}]},{"state_before":"case h.e'_5.h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nE : Type u_4\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nb : ℝ\nf : ℕ → ℝ\nz : ℕ → E\nhfa : Antitone f\nhf0 : Tendsto f atTop (𝓝 0)\nhzb : ∀ (n : ℕ), ‖∑ i ∈ Finset.range n, z i‖ ≤ b\nhfa' : Monotone fun n => -f n\nhf0' : Tendsto (fun n => -f n) atTop (𝓝 0)\nx✝ : ℕ\n⊢ ∑ i ∈ Finset.range x✝, f i • z i = (-fun n => ∑ i ∈ Finset.range n, -f i • z i) x✝","state_after":"no goals","tactic":"simp","premises":[]}]}
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+{"url":"Mathlib/Deprecated/Subgroup.lean","commit":"","full_name":"Group.exists_list_of_mem_closure","start":[452,0],"end":[466,87],"file_path":"Mathlib/Deprecated/Subgroup.lean","tactics":[{"state_before":"G : Type u_1\nH : Type u_2\nA : Type u_3\na✝ a₁ a₂ b c : G\ninst✝ : Group G\ns✝ s : Set G\na : G\nh : a ∈ closure s\nx✝² : G\nx✝¹ : InClosure s x✝²\nx✝ : ∃ l, (∀ x ∈ l, x ∈ s ∨ x⁻¹ ∈ s) ∧ l.prod = x✝²\nL : List G\nHL1 : ∀ x ∈ L, x ∈ s ∨ x⁻¹ ∈ s\nHL2 : L.prod = x✝²\nx : G\nhx : x ∈ List.map Inv.inv L.reverse\ny : G\nhy1 : y ∈ L.reverse\nhy2 : y⁻¹ = x\n⊢ y ∈ s → y⁻¹⁻¹ ∈ s","state_after":"G : Type u_1\nH : Type u_2\nA : Type u_3\na✝ a₁ a₂ b c : G\ninst✝ : Group G\ns✝ s : Set G\na : G\nh : a ∈ closure s\nx✝² : G\nx✝¹ : InClosure s x✝²\nx✝ : ∃ l, (∀ x ∈ l, x ∈ s ∨ x⁻¹ ∈ s) ∧ l.prod = x✝²\nL : List G\nHL1 : ∀ x ∈ L, x ∈ s ∨ x⁻¹ ∈ s\nHL2 : L.prod = x✝²\nx : G\nhx : x ∈ List.map Inv.inv L.reverse\ny : G\nhy1 : y ∈ L.reverse\nhy2 : y⁻¹ = x\n⊢ y ∈ s → y ∈ s","tactic":"rw [inv_inv]","premises":[{"full_name":"inv_inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[734,8],"def_end_pos":[734,15]}]},{"state_before":"G : Type u_1\nH : Type u_2\nA : Type u_3\na✝ a₁ a₂ b c : G\ninst✝ : Group G\ns✝ s : Set G\na : G\nh : a ∈ closure s\nx✝² : G\nx✝¹ : InClosure s x✝²\nx✝ : ∃ l, (∀ x ∈ l, x ∈ s ∨ x⁻¹ ∈ s) ∧ l.prod = x✝²\nL : List G\nHL1 : ∀ x ∈ L, x ∈ s ∨ x⁻¹ ∈ s\nHL2 : L.prod = x✝²\nx : G\nhx : x ∈ List.map Inv.inv L.reverse\ny : G\nhy1 : y ∈ L.reverse\nhy2 : y⁻¹ = x\n⊢ y ∈ s → y ∈ s","state_after":"no goals","tactic":"exact id","premises":[{"full_name":"id","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[33,14],"def_end_pos":[33,16]}]},{"state_before":"G : Type u_1\nH : Type u_2\nA : Type u_3\na✝ a₁ a₂ b c : G\ninst✝ : Group G\ns✝ s : Set G\na : G\nh : a ∈ closure s\nx : G\nx✝¹ : InClosure s x\nx✝ : ∃ l, (∀ x ∈ l, x ∈ s ∨ x⁻¹ ∈ s) ∧ l.prod = x\nL : List G\nHL1 : ∀ x ∈ L, x ∈ s ∨ x⁻¹ ∈ s\nHL2 : L.prod = x\nhd : G\ntl : List G\nih : (List.map Inv.inv tl.reverse).prod = tl.prod⁻¹\n⊢ (List.map Inv.inv (hd :: tl).reverse).prod = (hd :: tl).prod⁻¹","state_after":"no goals","tactic":"rw [List.reverse_cons, List.map_append, List.prod_append, ih, List.map_singleton,\n              List.prod_cons, List.prod_nil, mul_one, List.prod_cons, mul_inv_rev]","premises":[{"full_name":"List.map_append","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean","def_pos":[839,16],"def_end_pos":[839,26]},{"full_name":"List.map_singleton","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean","def_pos":[723,8],"def_end_pos":[723,21]},{"full_name":"List.prod_append","def_path":"Mathlib/Algebra/BigOperators/Group/List.lean","def_pos":[102,8],"def_end_pos":[102,19]},{"full_name":"List.prod_cons","def_path":"Mathlib/Algebra/BigOperators/Group/List.lean","def_pos":[85,8],"def_end_pos":[85,17]},{"full_name":"List.prod_nil","def_path":"Mathlib/Algebra/BigOperators/Group/List.lean","def_pos":[60,8],"def_end_pos":[60,16]},{"full_name":"List.reverse_cons","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[522,16],"def_end_pos":[522,28]},{"full_name":"mul_inv_rev","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[982,8],"def_end_pos":[982,19]},{"full_name":"mul_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[413,8],"def_end_pos":[413,15]}]},{"state_before":"G : Type u_1\nH : Type u_2\nA : Type u_3\na✝ a₁ a₂ b c : G\ninst✝ : Group G\ns✝ s : Set G\na : G\nh : a ∈ closure s\nx y : G\nx✝³ : InClosure s x\nx✝² : InClosure s y\nx✝¹ : ∃ l, (∀ x ∈ l, x �� s ∨ x⁻¹ ∈ s) ∧ l.prod = x\nx✝ : ∃ l, (∀ x ∈ l, x ∈ s ∨ x⁻¹ ∈ s) ∧ l.prod = y\nL1 : List G\nHL1 : ∀ x ∈ L1, x ∈ s ∨ x⁻¹ ∈ s\nHL2 : L1.prod = x\nL2 : List G\nHL3 : ∀ x ∈ L2, x ∈ s ∨ x⁻¹ ∈ s\nHL4 : L2.prod = y\n⊢ (L1 ++ L2).prod = x * y","state_after":"no goals","tactic":"rw [List.prod_append, HL2, HL4]","premises":[{"full_name":"List.prod_append","def_path":"Mathlib/Algebra/BigOperators/Group/List.lean","def_pos":[102,8],"def_end_pos":[102,19]}]}]}
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* r b)\n\ncase neg\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ a ∈ t, r a ≤ R\nτ : ℝ\nhτ : 3 < τ\nh✝ : t.Nonempty\nht : ¬∀ a ∈ t, r a < 0\n⊢ ∃ u ⊆ t,\n    (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧\n      ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)","tactic":"by_cases ht : ∀ a ∈ t, r a < 0","premises":[{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case neg\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ a ∈ t, r a ≤ R\nτ : ℝ\nhτ : 3 < τ\nh✝ : t.Nonempty\nht : ¬∀ a ∈ t, r a < 0\n⊢ ∃ u ⊆ t,\n    (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧\n      ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)","state_after":"case neg\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ a ∈ t, r a ≤ R\nτ : ℝ\nhτ : 3 < τ\nh✝ : t.Nonempty\nht : ∃ a ∈ t, 0 ≤ r a\n⊢ ∃ u ⊆ t,\n    (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧\n      ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)","tactic":"push_neg at ht","premises":[]},{"state_before":"case neg\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ a ∈ t, r a ≤ R\nτ : ℝ\nhτ : 3 < τ\nh✝ : t.Nonempty\nht : ∃ a ∈ t, 0 ≤ r a\n⊢ ∃ u ⊆ t,\n    (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧\n      ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)","state_after":"case neg\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ a ∈ t, r a ≤ R\nτ : ℝ\nhτ : 3 < τ\nh✝ : t.Nonempty\nht : ∃ a ∈ t, 0 ≤ r a\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\n⊢ ∃ u ⊆ t,\n    (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧\n      ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)","tactic":"let t' := { a ∈ t | 0 ≤ r a }","premises":[{"full_name":"And","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[516,10],"def_end_pos":[516,13]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"setOf","def_path":"Mathlib/Init/Set.lean","def_pos":[56,4],"def_end_pos":[56,9]}]},{"state_before":"case neg\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ a ∈ t, r a ≤ R\nτ : ℝ\nhτ : 3 < τ\nh✝ : t.Nonempty\nht : ∃ a ∈ t, 0 ≤ r a\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\n⊢ ∃ u ⊆ t,\n    (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧\n      ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)","state_after":"case neg.intro.intro.intro\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ a ∈ t, r a ≤ R\nτ : ℝ\nhτ : 3 < τ\nh✝ : t.Nonempty\nht : ∃ a ∈ t, 0 ≤ r a\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : u.PairwiseDisjoint fun a => closedBall (x a) (r a)\nhu : ∀ a ∈ t', ∃ b ∈ u, (closedBall (x a) (r a) ∩ closedBall (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b\n⊢ ∃ u ⊆ t,\n    (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧\n      ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)","tactic":"rcases exists_disjoint_subfamily_covering_enlargment (fun a => closedBall (x a) (r a)) t' r\n      ((τ - 1) / 2) (by linarith) (fun a ha => ha.2) R (fun a ha => hr a ha.1) fun a ha =>\n      ⟨x a, mem_closedBall_self ha.2⟩ with\n    ⟨u, ut', u_disj, hu⟩","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Metric.closedBall","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[402,4],"def_end_pos":[402,14]},{"full_name":"Metric.mem_closedBall_self","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[432,8],"def_end_pos":[432,27]},{"full_name":"Vitali.exists_disjoint_subfamily_covering_enlargment","def_path":"Mathlib/MeasureTheory/Covering/Vitali.lean","def_pos":[56,8],"def_end_pos":[56,53]}]},{"state_before":"case neg.intro.intro.intro\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ a ∈ t, r a ≤ R\nτ : ℝ\nhτ : 3 < τ\nh✝ : t.Nonempty\nht : ∃ a ∈ t, 0 ≤ r a\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : u.PairwiseDisjoint fun a => closedBall (x a) (r a)\nhu : ∀ a ∈ t', ∃ b ∈ u, (closedBall (x a) (r a) ∩ closedBall (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b\n⊢ ∃ u ⊆ t,\n    (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧\n      ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)","state_after":"case neg.intro.intro.intro\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ a ∈ t, r a ≤ R\nτ : ℝ\nhτ : 3 < τ\nh✝ : t.Nonempty\nht : ∃ a ∈ t, 0 ≤ r a\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : u.PairwiseDisjoint fun a => closedBall (x a) (r a)\nhu : ∀ a ∈ t', ∃ b ∈ u, (closedBall (x a) (r a) ∩ closedBall (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b\nA : ∀ a ∈ t', ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)\n⊢ ∃ u ⊆ t,\n    (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧\n      ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)","tactic":"have A : ∀ a ∈ t', ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b) := by\n    intro a ha\n    rcases hu a ha with ⟨b, bu, hb, rb⟩\n    refine ⟨b, bu, ?_⟩\n    have : dist (x a) (x b) ≤ r a + r b := dist_le_add_of_nonempty_closedBall_inter_closedBall hb\n    apply closedBall_subset_closedBall'\n    linarith","premises":[{"full_name":"And","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[516,10],"def_end_pos":[516,13]},{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Dist.dist","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[80,2],"def_end_pos":[80,6]},{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"HasSubset.Subset","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[384,2],"def_end_pos":[384,8]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Metric.closedBall","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[402,4],"def_end_pos":[402,14]},{"full_name":"Metric.closedBall_subset_closedBall'","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[513,8],"def_end_pos":[513,37]},{"full_name":"Metric.dist_le_add_of_nonempty_closedBall_inter_closedBall","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[530,8],"def_end_pos":[530,59]}]},{"state_before":"case neg.intro.intro.intro\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ a ∈ t, r a ≤ R\nτ : ℝ\nhτ : 3 < τ\nh✝ : t.Nonempty\nht : ∃ a ∈ t, 0 ≤ r a\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : u.PairwiseDisjoint fun a => closedBall (x a) (r a)\nhu : ∀ a ∈ t', ∃ b ∈ u, (closedBall (x a) (r a) ∩ closedBall (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b\nA : ∀ a ∈ t', ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)\n⊢ ∃ u ⊆ t,\n    (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧\n      ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)","state_after":"case neg.intro.intro.intro\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ a ∈ t, r a ≤ R\nτ : ℝ\nhτ : 3 < τ\nh✝ : t.Nonempty\nht : ∃ a ∈ t, 0 ≤ r a\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : u.PairwiseDisjoint fun a => closedBall (x a) (r a)\nhu : ∀ a ∈ t', ∃ b ∈ u, (closedBall (x a) (r a) ∩ closedBall (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b\nA : ∀ a ∈ t', ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)\na : ι\nha : a ∈ t\n⊢ ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)","tactic":"refine ⟨u, ut'.trans fun a ha => ha.1, u_disj, fun a ha => ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]},{"state_before":"case neg.intro.intro.intro\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ a ∈ t, r a ≤ R\nτ : ℝ\nhτ : 3 < τ\nh✝ : t.Nonempty\nht : ∃ a ∈ t, 0 ≤ r a\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : u.PairwiseDisjoint fun a => closedBall (x a) (r a)\nhu : ∀ a ∈ t', ∃ b ∈ u, (closedBall (x a) (r a) ∩ closedBall (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b\nA : ∀ a ∈ t', ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)\na : ι\nha : a ∈ t\n⊢ ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)","state_after":"case neg.intro.intro.intro.inl\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ a ∈ t, r a ≤ R\nτ : ℝ\nhτ : 3 < τ\nh✝ : t.Nonempty\nht : ∃ a ∈ t, 0 ≤ r a\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : u.PairwiseDisjoint fun a => closedBall (x a) (r a)\nhu : ∀ a ∈ t', ∃ b ∈ u, (closedBall (x a) (r a) ∩ closedBall (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b\nA : ∀ a ∈ t', ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)\na : ι\nha : a ∈ t\nh'a : 0 ≤ r a\n⊢ ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)\n\ncase neg.intro.intro.intro.inr\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ a ∈ t, r a ≤ R\nτ : ℝ\nhτ : 3 < τ\nh✝ : t.Nonempty\nht : ∃ a ∈ t, 0 ≤ r a\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : u.PairwiseDisjoint fun a => closedBall (x a) (r a)\nhu : ∀ a ∈ t', ∃ b ∈ u, (closedBall (x a) (r a) ∩ closedBall (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b\nA : ∀ a ∈ t', ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)\na : ι\nha : a ∈ t\nh'a : r a < 0\n⊢ ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)","tactic":"rcases le_or_lt 0 (r a) with (h'a | h'a)","premises":[{"full_name":"le_or_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[290,8],"def_end_pos":[290,16]}]}]}
+{"url":"Mathlib/Algebra/Lie/IdealOperations.lean","commit":"","full_name":"LieIdeal.comap_bracket_incl","start":[294,0],"end":[302,32],"file_path":"Mathlib/Algebra/Lie/IdealOperations.lean","tactics":[{"state_before":"R : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nI₁ I₂ : LieIdeal R L\n⊢ ⁅comap I.incl I₁, comap I.incl I₂⁆ = comap I.incl ⁅I ⊓ I₁, I ⊓ I₂⁆","state_after":"R : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nI₁ I₂ : LieIdeal R L\n⊢ ⁅comap I.incl I₁, comap I.incl I₂⁆ = comap I.incl ⁅I.incl.idealRange ⊓ I₁, I.incl.idealRange ⊓ I₂⁆","tactic":"conv_rhs =>\n    congr\n    next => skip\n    rw [← I.incl_idealRange]","premises":[{"full_name":"LieIdeal.incl_idealRange","def_path":"Mathlib/Algebra/Lie/Submodule.lean","def_pos":[1209,8],"def_end_pos":[1209,23]}]},{"state_before":"R : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nI₁ I₂ : LieIdeal R L\n⊢ ⁅comap I.incl I₁, comap I.incl I₂⁆ = comap I.incl ⁅I.incl.idealRange ⊓ I₁, I.incl.idealRange ⊓ I₂⁆","state_after":"R : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nI₁ I₂ : LieIdeal R L\n⊢ ⁅comap I.incl I₁, comap I.incl I₂⁆ = ⁅comap I.incl I₁, comap I.incl I₂⁆ ⊔ I.incl.ker\n\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nI₁ I₂ : LieIdeal R L\n⊢ I.incl.IsIdealMorphism","tactic":"rw [comap_bracket_eq]","premises":[{"full_name":"LieIdeal.comap_bracket_eq","def_path":"Mathlib/Algebra/Lie/IdealOperations.lean","def_pos":[268,8],"def_end_pos":[268,24]}]}]}
+{"url":"Mathlib/Order/LiminfLimsup.lean","commit":"","full_name":"Filter.blimsup_sup_not","start":[1012,0],"end":[1014,53],"file_path":"Mathlib/Order/LiminfLimsup.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Type u_5\ninst✝ : CompleteDistribLattice α\nf : Filter β\np q : β → Prop\nu : β → α\n⊢ (blimsup u f p ⊔ blimsup u f fun x => ¬p x) = limsup u f","state_after":"no goals","tactic":"simp_rw [← blimsup_or_eq_sup, or_not, blimsup_true]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Filter.blimsup_or_eq_sup","def_path":"Mathlib/Order/LiminfLimsup.lean","def_pos":[1005,8],"def_end_pos":[1005,25]},{"full_name":"Filter.blimsup_true","def_path":"Mathlib/Order/LiminfLimsup.lean","def_pos":[489,8],"def_end_pos":[489,20]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"or_not","def_path":"Mathlib/Logic/Basic.lean","def_pos":[159,8],"def_end_pos":[159,14]}]}]}
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+{"url":"Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean","commit":"","full_name":"CategoryTheory.Presieve.FamilyOfElements.isCompatible_map_smul","start":[87,0],"end":[113,83],"file_path":"Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean","tactics":[{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\n⊢ ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp))).Compatible","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\n⊢ (A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) =\n    (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)","tactic":"intro Y₁ Y₂ Z g₁ g₂ f₁ f₂ h₁ h₂ fac","premises":[]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\n⊢ (A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) =\n    (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\n⊢ (A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) =\n    (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)","tactic":"let a₁ := r₀ f₁ h₁","premises":[]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\n⊢ (A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) =\n    (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\nb₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁\n⊢ (A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) =\n    (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)","tactic":"let b₁ := m₀ f₁ h₁","premises":[]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\nb₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁\n⊢ (A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) =\n    (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\nb₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁\na₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂\n⊢ (A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) =\n    (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)","tactic":"let a₂ := r₀ f₂ h₂","premises":[]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\nb₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁\na₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂\n⊢ (A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) =\n    (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\nb₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁\na₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂\nb₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂\n⊢ (A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) =\n    (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)","tactic":"let b₂ := m₀ f₂ h₂","premises":[]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\nb₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁\na₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂\nb₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂\n⊢ (A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) =\n    (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\nb₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁\na₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂\nb₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂\na₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁\n⊢ (A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) =\n    (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)","tactic":"let a₀ := R₀.map g₁.op a₁","premises":[{"full_name":"Prefunctor.map","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[57,2],"def_end_pos":[57,5]},{"full_name":"Quiver.Hom.op","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[136,4],"def_end_pos":[136,10]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\nb₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁\na₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂\nb₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂\na₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁\n⊢ (A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) =\n    (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\nb₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁\na₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂\nb₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂\na₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁\nb₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁\n⊢ (A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) =\n    (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)","tactic":"let b₀ := M₀.map g₁.op b₁","premises":[{"full_name":"PresheafOfModules.map","def_path":"Mathlib/Algebra/Category/ModuleCat/Presheaf.lean","def_pos":[62,4],"def_end_pos":[62,7]},{"full_name":"Quiver.Hom.op","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[136,4],"def_end_pos":[136,10]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\nb₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁\na₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂\nb₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂\na₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁\nb₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁\n⊢ (A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) =\n    (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\nb₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁\na₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂\nb₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂\na₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁\nb₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁\nha₁ : (α.app (Opposite.op Y₁)) a₁ = (R.map f₁.op) r\n⊢ (A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) =\n    (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)","tactic":"have ha₁ : (α.app (Opposite.op Y₁)) a₁ = (R.map f₁.op) r := (hr₀ f₁ h₁).symm","premises":[{"full_name":"CategoryTheory.NatTrans.app","def_path":"Mathlib/CategoryTheory/NatTrans.lean","def_pos":[48,2],"def_end_pos":[48,5]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Opposite.op","def_path":"Mathlib/Data/Opposite.lean","def_pos":[35,2],"def_end_pos":[35,4]},{"full_name":"Prefunctor.map","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[57,2],"def_end_pos":[57,5]},{"full_name":"Quiver.Hom.op","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[136,4],"def_end_pos":[136,10]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\nb₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁\na₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂\nb₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂\na₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁\nb₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁\nha₁ : (α.app (Opposite.op Y₁)) a₁ = (R.map f₁.op) r\n⊢ (A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) =\n    (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\nb₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁\na₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂\nb₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂\na₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁\nb₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁\nha₁ : (α.app (Opposite.op Y₁)) a₁ = (R.map f₁.op) r\nha₂ : (α.app (Opposite.op Y₂)) a₂ = (R.map f₂.op) r\n⊢ (A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) =\n    (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)","tactic":"have ha₂ : (α.app (Opposite.op Y₂)) a₂ = (R.map f₂.op) r := (hr₀ f₂ h₂).symm","premises":[{"full_name":"CategoryTheory.NatTrans.app","def_path":"Mathlib/CategoryTheory/NatTrans.lean","def_pos":[48,2],"def_end_pos":[48,5]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Opposite.op","def_path":"Mathlib/Data/Opposite.lean","def_pos":[35,2],"def_end_pos":[35,4]},{"full_name":"Prefunctor.map","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[57,2],"def_end_pos":[57,5]},{"full_name":"Quiver.Hom.op","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[136,4],"def_end_pos":[136,10]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\nb₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁\na₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂\nb₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂\na₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁\nb₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁\nha₁ : (α.app (Opposite.op Y₁)) a₁ = (R.map f₁.op) r\nha₂ : (α.app (Opposite.op Y₂)) a₂ = (R.map f₂.op) r\n⊢ (A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) =\n    (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\nb₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁\na₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂\nb₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂\na₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁\nb₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁\nha₁ : (α.app (Opposite.op Y₁)) a₁ = (R.map f₁.op) r\nha₂ : (α.app (Opposite.op Y₂)) a₂ = (R.map f₂.op) r\nhb₁ : (φ.app (Opposite.op Y₁)) b₁ = (A.map f₁.op) m\n⊢ (A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) =\n    (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)","tactic":"have hb₁ : (φ.app (Opposite.op Y₁)) b₁ = (A.map f₁.op) m := (hm₀ f₁ h₁).symm","premises":[{"full_name":"CategoryTheory.NatTrans.app","def_path":"Mathlib/CategoryTheory/NatTrans.lean","def_pos":[48,2],"def_end_pos":[48,5]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Opposite.op","def_path":"Mathlib/Data/Opposite.lean","def_pos":[35,2],"def_end_pos":[35,4]},{"full_name":"Prefunctor.map","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[57,2],"def_end_pos":[57,5]},{"full_name":"Quiver.Hom.op","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[136,4],"def_end_pos":[136,10]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\nb₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁\na₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂\nb₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂\na₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁\nb₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁\nha₁ : (α.app (Opposite.op Y₁)) a₁ = (R.map f₁.op) r\nha₂ : (α.app (Opposite.op Y₂)) a₂ = (R.map f₂.op) r\nhb₁ : (φ.app (Opposite.op Y₁)) b₁ = (A.map f₁.op) m\n⊢ (A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) =\n    (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\nb₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁\na₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂\nb₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂\na₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁\nb₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁\nha₁ : (α.app (Opposite.op Y₁)) a₁ = (R.map f₁.op) r\nha₂ : (α.app (Opposite.op Y₂)) a₂ = (R.map f₂.op) r\nhb₁ : (φ.app (Opposite.op Y₁)) b₁ = (A.map f₁.op) m\nhb₂ : (φ.app (Opposite.op Y₂)) b₂ = (A.map f₂.op) m\n⊢ (A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) =\n    (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)","tactic":"have hb₂ : (φ.app (Opposite.op Y₂)) b₂ = (A.map f₂.op) m := (hm₀ f₂ h₂).symm","premises":[{"full_name":"CategoryTheory.NatTrans.app","def_path":"Mathlib/CategoryTheory/NatTrans.lean","def_pos":[48,2],"def_end_pos":[48,5]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Opposite.op","def_path":"Mathlib/Data/Opposite.lean","def_pos":[35,2],"def_end_pos":[35,4]},{"full_name":"Prefunctor.map","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[57,2],"def_end_pos":[57,5]},{"full_name":"Quiver.Hom.op","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[136,4],"def_end_pos":[136,10]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\nb₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁\na₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂\nb₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂\na₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁\nb₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁\nha₁ : (α.app (Opposite.op Y₁)) a₁ = (R.map f₁.op) r\nha₂ : (α.app (Opposite.op Y₂)) a₂ = (R.map f₂.op) r\nhb₁ : (φ.app (Opposite.op Y₁)) b₁ = (A.map f₁.op) m\nhb₂ : (φ.app (Opposite.op Y₂)) b₂ = (A.map f₂.op) m\n⊢ (A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) =\n    (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\nb₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁\na₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂\nb₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂\na₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁\nb₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁\nha₁ : (α.app (Opposite.op Y₁)) a₁ = (R.map f₁.op) r\nha₂ : (α.app (Opposite.op Y₂)) a₂ = (R.map f₂.op) r\nhb₁ : (φ.app (Opposite.op Y₁)) b₁ = (A.map f₁.op) m\nhb₂ : (φ.app (Opposite.op Y₂)) b₂ = (A.map f₂.op) m\nha₀ : (α.app (Opposite.op Z)) a₀ = (R.map (f₁.op ≫ g₁.op)) r\n⊢ (A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) =\n    (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)","tactic":"have ha₀ : (α.app (Opposite.op Z)) a₀ = (R.map (f₁.op ≫ g₁.op)) r := by\n    dsimp [a₀]\n    rw [NatTrans.naturality_apply, ha₁, Functor.map_comp, comp_apply]","premises":[{"full_name":"CategoryTheory.CategoryStruct.comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]},{"full_name":"CategoryTheory.Functor.map_comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[41,2],"def_end_pos":[41,10]},{"full_name":"CategoryTheory.NatTrans.app","def_path":"Mathlib/CategoryTheory/NatTrans.lean","def_pos":[48,2],"def_end_pos":[48,5]},{"full_name":"CategoryTheory.NatTrans.naturality_apply","def_path":"Mathlib/CategoryTheory/ConcreteCategory/Basic.lean","def_pos":[216,6],"def_end_pos":[216,31]},{"full_name":"CategoryTheory.comp_apply","def_path":"Mathlib/CategoryTheory/ConcreteCategory/Basic.lean","def_pos":[120,16],"def_end_pos":[120,26]},{"full_name":"Opposite.op","def_path":"Mathlib/Data/Opposite.lean","def_pos":[35,2],"def_end_pos":[35,4]},{"full_name":"Prefunctor.map","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[57,2],"def_end_pos":[57,5]},{"full_name":"Quiver.Hom.op","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[136,4],"def_end_pos":[136,10]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\nb₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁\na₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂\nb₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂\na₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁\nb₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁\nha₁ : (α.app (Opposite.op Y₁)) a₁ = (R.map f₁.op) r\nha₂ : (α.app (Opposite.op Y₂)) a₂ = (R.map f₂.op) r\nhb₁ : (φ.app (Opposite.op Y₁)) b₁ = (A.map f₁.op) m\nhb₂ : (φ.app (Opposite.op Y₂)) b₂ = (A.map f₂.op) m\nha₀ : (α.app (Opposite.op Z)) a₀ = (R.map (f₁.op ≫ g₁.op)) r\n⊢ (A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) =\n    (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\nb₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁\na₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂\nb₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂\na₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁\nb₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁\nha₁ : (α.app (Opposite.op Y₁)) a₁ = (R.map f₁.op) r\nha₂ : (α.app (Opposite.op Y₂)) a₂ = (R.map f₂.op) r\nhb₁ : (φ.app (Opposite.op Y₁)) b₁ = (A.map f₁.op) m\nhb₂ : (φ.app (Opposite.op Y₂)) b₂ = (A.map f₂.op) m\nha₀ : (α.app (Opposite.op Z)) a₀ = (R.map (f₁.op ≫ g₁.op)) r\nhb₀ : (φ.app (Opposite.op Z)) b₀ = (A.map (f₁.op ≫ g₁.op)) m\n⊢ (A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) =\n    (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)","tactic":"have hb₀ : (φ.app (Opposite.op Z)) b₀ = (A.map (f₁.op ≫ g₁.op)) m := by\n    dsimp [b₀]\n    erw [NatTrans.naturality_apply, hb₁, Functor.map_comp, comp_apply]\n    rfl","premises":[{"full_name":"CategoryTheory.CategoryStruct.comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]},{"full_name":"CategoryTheory.Functor.map_comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[41,2],"def_end_pos":[41,10]},{"full_name":"CategoryTheory.NatTrans.app","def_path":"Mathlib/CategoryTheory/NatTrans.lean","def_pos":[48,2],"def_end_pos":[48,5]},{"full_name":"CategoryTheory.NatTrans.naturality_apply","def_path":"Mathlib/CategoryTheory/ConcreteCategory/Basic.lean","def_pos":[216,6],"def_end_pos":[216,31]},{"full_name":"CategoryTheory.comp_apply","def_path":"Mathlib/CategoryTheory/ConcreteCategory/Basic.lean","def_pos":[120,16],"def_end_pos":[120,26]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Opposite.op","def_path":"Mathlib/Data/Opposite.lean","def_pos":[35,2],"def_end_pos":[35,4]},{"full_name":"Prefunctor.map","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[57,2],"def_end_pos":[57,5]},{"full_name":"Quiver.Hom.op","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[136,4],"def_end_pos":[136,10]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\nb₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁\na₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂\nb₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂\na₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁\nb₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁\nha₁ : (α.app (Opposite.op Y₁)) a₁ = (R.map f₁.op) r\nha₂ : (α.app (Opposite.op Y₂)) a₂ = (R.map f₂.op) r\nhb₁ : (φ.app (Opposite.op Y₁)) b₁ = (A.map f₁.op) m\nhb₂ : (φ.app (Opposite.op Y₂)) b₂ = (A.map f₂.op) m\nha₀ : (α.app (Opposite.op Z)) a₀ = (R.map (f₁.op ≫ g₁.op)) r\nhb₀ : (φ.app (Opposite.op Z)) b₀ = (A.map (f₁.op ≫ g₁.op)) m\n⊢ (A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) =\n    (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\nb₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁\na₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂\nb₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂\na₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁\nb₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁\nha₁ : (α.app (Opposite.op Y₁)) a₁ = (R.map f₁.op) r\nha₂ : (α.app (Opposite.op Y₂)) a₂ = (R.map f₂.op) r\nhb₁ : (φ.app (Opposite.op Y₁)) b₁ = (A.map f₁.op) m\nhb₂ : (φ.app (Opposite.op Y₂)) b₂ = (A.map f₂.op) m\nha₀ : (α.app (Opposite.op Z)) a₀ = (R.map (f₁.op ≫ g₁.op)) r\nhb₀ : (φ.app (Opposite.op Z)) b₀ = (A.map (f₁.op ≫ g₁.op)) m\nha₀' : (α.app (Opposite.op Z)) a₀ = (R.map (f₂.op ≫ g₂.op)) r\n⊢ (A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) =\n    (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)","tactic":"have ha₀' : (α.app (Opposite.op Z)) a₀ = (R.map (f₂.op ≫ g₂.op)) r := by\n    rw [ha₀, ← op_comp, fac, op_comp]","premises":[{"full_name":"CategoryTheory.CategoryStruct.comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]},{"full_name":"CategoryTheory.NatTrans.app","def_path":"Mathlib/CategoryTheory/NatTrans.lean","def_pos":[48,2],"def_end_pos":[48,5]},{"full_name":"CategoryTheory.op_comp","def_path":"Mathlib/CategoryTheory/Opposites.lean","def_pos":[72,8],"def_end_pos":[72,15]},{"full_name":"Opposite.op","def_path":"Mathlib/Data/Opposite.lean","def_pos":[35,2],"def_end_pos":[35,4]},{"full_name":"Prefunctor.map","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[57,2],"def_end_pos":[57,5]},{"full_name":"Quiver.Hom.op","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[136,4],"def_end_pos":[136,10]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\nb₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁\na₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂\nb₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂\na₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁\nb₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁\nha₁ : (α.app (Opposite.op Y₁)) a₁ = (R.map f₁.op) r\nha₂ : (α.app (Opposite.op Y₂)) a₂ = (R.map f₂.op) r\nhb₁ : (φ.app (Opposite.op Y₁)) b₁ = (A.map f₁.op) m\nhb₂ : (φ.app (Opposite.op Y₂)) b₂ = (A.map f₂.op) m\nha₀ : (α.app (Opposite.op Z)) a₀ = (R.map (f₁.op ≫ g₁.op)) r\nhb₀ : (φ.app (Opposite.op Z)) b₀ = (A.map (f₁.op ≫ g₁.op)) m\nha₀' : (α.app (Opposite.op Z)) a₀ = (R.map (f₂.op ≫ g₂.op)) r\n⊢ (A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) =\n    (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\nb₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁\na₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂\nb₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂\na₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁\nb₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁\nha₁ : (α.app (Opposite.op Y₁)) a₁ = (R.map f₁.op) r\nha₂ : (α.app (Opposite.op Y₂)) a₂ = (R.map f₂.op) r\nhb₁ : (φ.app (Opposite.op Y₁)) b₁ = (A.map f₁.op) m\nhb₂ : (φ.app (Opposite.op Y₂)) b₂ = (A.map f₂.op) m\nha₀ : (α.app (Opposite.op Z)) a₀ = (R.map (f₁.op ≫ g₁.op)) r\nhb₀ : (φ.app (Opposite.op Z)) b₀ = (A.map (f₁.op ≫ g₁.op)) m\nha₀' : (α.app (Opposite.op Z)) a₀ = (R.map (f₂.op ≫ g₂.op)) r\nhb₀' : (φ.app (Opposite.op Z)) b₀ = (A.map (f₂.op ≫ g₂.op)) m\n⊢ (A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) =\n    (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)","tactic":"have hb₀' : (φ.app (Opposite.op Z)) b₀ = (A.map (f₂.op ≫ g₂.op)) m := by\n    rw [hb₀, ← op_comp, fac, op_comp]","premises":[{"full_name":"CategoryTheory.CategoryStruct.comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]},{"full_name":"CategoryTheory.NatTrans.app","def_path":"Mathlib/CategoryTheory/NatTrans.lean","def_pos":[48,2],"def_end_pos":[48,5]},{"full_name":"CategoryTheory.op_comp","def_path":"Mathlib/CategoryTheory/Opposites.lean","def_pos":[72,8],"def_end_pos":[72,15]},{"full_name":"Opposite.op","def_path":"Mathlib/Data/Opposite.lean","def_pos":[35,2],"def_end_pos":[35,4]},{"full_name":"Prefunctor.map","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[57,2],"def_end_pos":[57,5]},{"full_name":"Quiver.Hom.op","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[136,4],"def_end_pos":[136,10]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\nb₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁\na₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂\nb₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂\na₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁\nb₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁\nha₁ : (α.app (Opposite.op Y₁)) a₁ = (R.map f₁.op) r\nha₂ : (α.app (Opposite.op Y₂)) a₂ = (R.map f₂.op) r\nhb₁ : (φ.app (Opposite.op Y₁)) b₁ = (A.map f₁.op) m\nhb₂ : (φ.app (Opposite.op Y₂)) b₂ = (A.map f₂.op) m\nha₀ : (α.app (Opposite.op Z)) a₀ = (R.map (f₁.op ≫ g₁.op)) r\nhb₀ : (φ.app (Opposite.op Z)) b₀ = (A.map (f₁.op ≫ g₁.op)) m\nha₀' : (α.app (Opposite.op Z)) a₀ = (R.map (f₂.op ≫ g₂.op)) r\nhb₀' : (φ.app (Opposite.op Z)) b₀ = (A.map (f₂.op ≫ g₂.op)) m\n⊢ (A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) =\n    (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\nb₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁\na₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂\nb₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂\na₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁\nb₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁\nha₁ : (α.app (Opposite.op Y₁)) a₁ = (R.map f₁.op) r\nha₂ : (α.app (Opposite.op Y₂)) a₂ = (R.map f₂.op) r\nhb₁ : (φ.app (Opposite.op Y₁)) b₁ = (A.map f₁.op) m\nhb₂ : (φ.app (Opposite.op Y₂)) b₂ = (A.map f₂.op) m\nha₀ : (α.app (Opposite.op Z)) a₀ = (R.map (f₁.op ≫ g₁.op)) r\nhb₀ : (φ.app (Opposite.op Z)) b₀ = (A.map (f₁.op ≫ g₁.op)) m\nha₀' : (α.app (Opposite.op Z)) a₀ = (R.map (f₂.op ≫ g₂.op)) r\nhb₀' : (φ.app (Opposite.op Z)) b₀ = (A.map (f₂.op ≫ g₂.op)) m\n⊢ (A.map g₁.op) ((φ.app (Opposite.op Y₁)) (r₀.smul m₀ f₁ h₁)) =\n    (A.map g₂.op) ((φ.app (Opposite.op Y₂)) (r₀.smul m₀ f₂ h₂))","tactic":"dsimp","premises":[]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\nb₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁\na₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂\nb₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂\na₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁\nb₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁\nha₁ : (α.app (Opposite.op Y₁)) a₁ = (R.map f₁.op) r\nha₂ : (α.app (Opposite.op Y₂)) a₂ = (R.map f₂.op) r\nhb₁ : (φ.app (Opposite.op Y₁)) b₁ = (A.map f₁.op) m\nhb₂ : (φ.app (Opposite.op Y₂)) b₂ = (A.map f₂.op) m\nha₀ : (α.app (Opposite.op Z)) a₀ = (R.map (f₁.op ≫ g₁.op)) r\nhb₀ : (φ.app (Opposite.op Z)) b₀ = (A.map (f₁.op ≫ g₁.op)) m\nha₀' : (α.app (Opposite.op Z)) a₀ = (R.map (f₂.op ≫ g₂.op)) r\nhb₀' : (φ.app (Opposite.op Z)) b₀ = (A.map (f₂.op ≫ g₂.op)) m\n⊢ (A.map g₁.op) ((φ.app (Opposite.op Y₁)) (r₀.smul m₀ f₁ h₁)) =\n    (A.map g₂.op) ((φ.app (Opposite.op Y₂)) (r₀.smul m₀ f₂ h₂))","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\nb₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁\na₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂\nb₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂\na₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁\nb₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁\nha₁ : (α.app (Opposite.op Y₁)) a₁ = (R.map f₁.op) r\nha₂ : (α.app (Opposite.op Y₂)) a₂ = (R.map f₂.op) r\nhb₁ : (φ.app (Opposite.op Y₁)) b₁ = (A.map f₁.op) m\nhb₂ : (φ.app (Opposite.op Y₂)) b₂ = (A.map f₂.op) m\nha₀ : (α.app (Opposite.op Z)) a₀ = (R.map (f₁.op ≫ g₁.op)) r\nhb₀ : (φ.app (Opposite.op Z)) b₀ = (A.map (f₁.op ≫ g₁.op)) m\nha₀' : (α.app (Opposite.op Z)) a₀ = (R.map (f₂.op ≫ g₂.op)) r\nhb₀' : (φ.app (Opposite.op Z)) b₀ = (A.map (f₂.op ≫ g₂.op)) m\n⊢ (φ.app (Opposite.op Z)) ((M₀.presheaf.map g₁.op) (r₀.smul m₀ f₁ h₁)) =\n    (φ.app (Opposite.op Z)) ((M₀.presheaf.map g₂.op) (r₀.smul m₀ f₂ h₂))","tactic":"erw [← NatTrans.naturality_apply, ← NatTrans.naturality_apply]","premises":[{"full_name":"CategoryTheory.NatTrans.naturality_apply","def_path":"Mathlib/CategoryTheory/ConcreteCategory/Basic.lean","def_pos":[216,6],"def_end_pos":[216,31]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\nb₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁\na₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂\nb₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂\na₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁\nb₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁\nha₁ : (α.app (Opposite.op Y₁)) a₁ = (R.map f₁.op) r\nha₂ : (α.app (Opposite.op Y₂)) a₂ = (R.map f₂.op) r\nhb₁ : (φ.app (Opposite.op Y₁)) b₁ = (A.map f₁.op) m\nhb₂ : (φ.app (Opposite.op Y₂)) b₂ = (A.map f₂.op) m\nha₀ : (α.app (Opposite.op Z)) a₀ = (R.map (f₁.op ≫ g₁.op)) r\nhb₀ : (φ.app (Opposite.op Z)) b₀ = (A.map (f₁.op ≫ g₁.op)) m\nha₀' : (α.app (Opposite.op Z)) a₀ = (R.map (f₂.op ≫ g₂.op)) r\nhb₀' : (φ.app (Opposite.op Z)) b₀ = (A.map (f₂.op ≫ g₂.op)) m\n⊢ (φ.app (Opposite.op Z)) ((M₀.presheaf.map g₁.op) (r₀.smul m₀ f₁ h₁)) =\n    (φ.app (Opposite.op Z)) ((M₀.presheaf.map g₂.op) (r₀.smul m₀ f₂ h₂))","state_after":"no goals","tactic":"exact (isCompatible_map_smul_aux α φ hA r m f₁ g₁ a₁ a₀ b₁ b₀ ha₁ ha₀ hb₁ hb₀).trans\n    (isCompatible_map_smul_aux α φ hA r m f₂ g₂ a₂ a₀ b₂ b₀ ha₂ ha₀' hb₂ hb₀').symm","premises":[{"full_name":"CategoryTheory.Presieve.FamilyOfElements.isCompatible_map_smul_aux","def_path":"Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean","def_pos":[74,6],"def_end_pos":[74,31]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Eq.trans","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[335,8],"def_end_pos":[335,16]}]}]}
+{"url":"Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean","commit":"","full_name":"Polynomial.unique_int_coeff_of_cycl","start":[214,0],"end":[222,15],"file_path":"Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean","tactics":[{"state_before":"K✝ : Type u_1\ninst✝³ : Field K✝\nK : Type u_2\ninst✝² : CommRing K\ninst✝¹ : IsDomain K\ninst✝ : CharZero K\nζ : K\nn : ℕ+\nh : IsPrimitiveRoot ζ ↑n\n⊢ ∃! P, map (Int.castRingHom K) P = cyclotomic' (↑n) K","state_after":"case intro\nK✝ : Type u_1\ninst✝³ : Field K✝\nK : Type u_2\ninst✝² : CommRing K\ninst✝¹ : IsDomain K\ninst✝ : CharZero K\nζ : K\nn : ℕ+\nh : IsPrimitiveRoot ζ ↑n\nP : ℤ[X]\nhP : map (Int.castRingHom K) P = cyclotomic' (↑n) K ∧ P.degree = (cyclotomic' (↑n) K).degree ∧ P.Monic\n⊢ ∃! P, map (Int.castRingHom K) P = cyclotomic' (↑n) K","tactic":"obtain ⟨P, hP⟩ := int_coeff_of_cyclotomic' h","premises":[{"full_name":"Polynomial.int_coeff_of_cyclotomic'","def_path":"Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean","def_pos":[180,8],"def_end_pos":[180,32]}]},{"state_before":"case intro\nK✝ : Type u_1\ninst✝³ : Field K✝\nK : Type u_2\ninst✝² : CommRing K\ninst✝¹ : IsDomain K\ninst✝ : CharZero K\nζ : K\nn : ℕ+\nh : IsPrimitiveRoot ζ ↑n\nP : ℤ[X]\nhP : map (Int.castRingHom K) P = cyclotomic' (↑n) K ∧ P.degree = (cyclotomic' (↑n) K).degree ∧ P.Monic\n⊢ ∃! P, map (Int.castRingHom K) P = cyclotomic' (↑n) K","state_after":"case intro\nK✝ : Type u_1\ninst✝³ : Field K✝\nK : Type u_2\ninst✝² : CommRing K\ninst✝¹ : IsDomain K\ninst✝ : CharZero K\nζ : K\nn : ℕ+\nh : IsPrimitiveRoot ζ ↑n\nP : ℤ[X]\nhP : map (Int.castRingHom K) P = cyclotomic' (↑n) K ∧ P.degree = (cyclotomic' (↑n) K).degree ∧ P.Monic\nQ : ℤ[X]\nhQ : (fun P => map (Int.castRingHom K) P = cyclotomic' (↑n) K) Q\n⊢ Q = P","tactic":"refine ⟨P, hP.1, fun Q hQ => ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]},{"state_before":"case intro\nK✝ : Type u_1\ninst✝³ : Field K✝\nK : Type u_2\ninst✝² : CommRing K\ninst✝¹ : IsDomain K\ninst✝ : CharZero K\nζ : K\nn : ℕ+\nh : IsPrimitiveRoot ζ ↑n\nP : ℤ[X]\nhP : map (Int.castRingHom K) P = cyclotomic' (↑n) K ∧ P.degree = (cyclotomic' (↑n) K).degree ∧ P.Monic\nQ : ℤ[X]\nhQ : (fun P => map (Int.castRingHom K) P = cyclotomic' (↑n) K) Q\n⊢ Q = P","state_after":"case intro.a\nK✝ : Type u_1\ninst✝³ : Field K✝\nK : Type u_2\ninst✝² : CommRing K\ninst✝¹ : IsDomain K\ninst✝ : CharZero K\nζ : K\nn : ℕ+\nh : IsPrimitiveRoot ζ ↑n\nP : ℤ[X]\nhP : map (Int.castRingHom K) P = cyclotomic' (↑n) K ∧ P.degree = (cyclotomic' (↑n) K).degree ∧ P.Monic\nQ : ℤ[X]\nhQ : (fun P => map (Int.castRingHom K) P = cyclotomic' (↑n) K) Q\n⊢ map (Int.castRingHom K) Q = map (Int.castRingHom K) P","tactic":"apply map_injective (Int.castRingHom K) Int.cast_injective","premises":[{"full_name":"Int.castRingHom","def_path":"Mathlib/Data/Int/Cast/Lemmas.lean","def_pos":[84,4],"def_end_pos":[84,15]},{"full_name":"Int.cast_injective","def_path":"Mathlib/Data/Int/Cast/Lemmas.lean","def_pos":[69,6],"def_end_pos":[69,20]},{"full_name":"Polynomial.map_injective","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[708,8],"def_end_pos":[708,21]}]},{"state_before":"case intro.a\nK✝ : Type u_1\ninst✝³ : Field K✝\nK : Type u_2\ninst✝² : CommRing K\ninst✝¹ : IsDomain K\ninst✝ : CharZero K\nζ : K\nn : ℕ+\nh : IsPrimitiveRoot ζ ↑n\nP : ℤ[X]\nhP : map (Int.castRingHom K) P = cyclotomic' (↑n) K ∧ P.degree = (cyclotomic' (↑n) K).degree ∧ P.Monic\nQ : ℤ[X]\nhQ : (fun P => map (Int.castRingHom K) P = cyclotomic' (↑n) K) Q\n⊢ map (Int.castRingHom K) Q = map (Int.castRingHom K) P","state_after":"no goals","tactic":"rw [hP.1, hQ]","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]}]}]}
+{"url":"Mathlib/Data/Real/EReal.lean","commit":"","full_name":"EReal.top_mul_of_neg","start":[1036,0],"end":[1038,24],"file_path":"Mathlib/Data/Real/EReal.lean","tactics":[{"state_before":"x : EReal\nh : x < 0\n⊢ ⊤ * x = ⊥","state_after":"x : EReal\nh : x < 0\n⊢ x * ⊤ = ⊥","tactic":"rw [EReal.mul_comm]","premises":[{"full_name":"EReal.mul_comm","def_path":"Mathlib/Data/Real/EReal.lean","def_pos":[200,18],"def_end_pos":[200,26]}]},{"state_before":"x : EReal\nh : x < 0\n⊢ x * ⊤ = ⊥","state_after":"no goals","tactic":"exact mul_top_of_neg h","premises":[{"full_name":"EReal.mul_top_of_neg","def_path":"Mathlib/Data/Real/EReal.lean","def_pos":[1017,6],"def_end_pos":[1017,20]}]}]}
+{"url":"Mathlib/Analysis/BoxIntegral/Basic.lean","commit":"","full_name":"BoxIntegral.integralSum_neg","start":[133,0],"end":[136,80],"file_path":"Mathlib/Analysis/BoxIntegral/Basic.lean","tactics":[{"state_before":"ι : Type u\nE : Type v\nF : Type w\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nI J : Box ι\nπ✝ : TaggedPrepartition I\nf : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\nπ : TaggedPrepartition I\n⊢ integralSum (-f) vol π = -integralSum f vol π","state_after":"no goals","tactic":"simp only [integralSum, Pi.neg_apply, (vol _).map_neg, Finset.sum_neg_distrib]","premises":[{"full_name":"BoxIntegral.integralSum","def_path":"Mathlib/Analysis/BoxIntegral/Basic.lean","def_pos":[78,4],"def_end_pos":[78,15]},{"full_name":"ContinuousLinearMap.map_neg","def_path":"Mathlib/Topology/Algebra/Module/Basic.lean","def_pos":[1209,18],"def_end_pos":[1209,25]},{"full_name":"Finset.sum_neg_distrib","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[1803,2],"def_end_pos":[1803,13]},{"full_name":"Pi.neg_apply","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[125,2],"def_end_pos":[125,13]}]}]}
+{"url":"Mathlib/Algebra/RingQuot.lean","commit":"","full_name":"RingQuot.liftAlgHom_def","start":[591,0],"end":[607,72],"file_path":"Mathlib/Algebra/RingQuot.lean","tactics":[{"state_before":"R : Type uR\ninst✝⁶ : Semiring R\nS : Type uS\ninst✝⁵ : CommSemiring S\nT : Type uT\nA : Type uA\ninst✝⁴ : Semiring A\ninst✝³ : Algebra S A\nr : R → R → Prop\ninst✝² : Semiring T\nB : Type u₄\ninst✝¹ : Semiring B\ninst✝ : Algebra S B\ns : A → A → Prop\nf : { f // ∀ ⦃x y : A⦄, s x y → f x = f y }\n⊢ (fun F => ⟨F.comp (mkAlgHom S s), ⋯⟩) ((fun f' => preLiftAlgHom S ⋯) f) = f","state_after":"case a.H\nR : Type uR\ninst✝⁶ : Semiring R\nS : Type uS\ninst✝⁵ : CommSemiring S\nT : Type uT\nA : Type uA\ninst✝⁴ : Semiring A\ninst✝³ : Algebra S A\nr : R → R → Prop\ninst✝² : Semiring T\nB : Type u₄\ninst✝¹ : Semiring B\ninst✝ : Algebra S B\ns : A → A → Prop\nf : { f // ∀ ⦃x y : A⦄, s x y → f x = f y }\nx✝ : A\n⊢ ↑((fun F => ⟨F.comp (mkAlgHom S s), ⋯⟩) ((fun f' => preLiftAlgHom S ⋯) f)) x✝ = ↑f x✝","tactic":"ext","premises":[]},{"state_before":"case a.H\nR : Type uR\ninst✝⁶ : Semiring R\nS : Type uS\ninst✝⁵ : CommSemiring S\nT : Type uT\nA : Type uA\ninst✝⁴ : Semiring A\ninst✝³ : Algebra S A\nr : R → R → Prop\ninst✝² : Semiring T\nB : Type u₄\ninst✝¹ : Semiring B\ninst✝ : Algebra S B\ns : A → A → Prop\nf : { f // ∀ ⦃x y : A⦄, s x y → f x = f y }\nx✝ : A\n⊢ ↑((fun F => ⟨F.comp (mkAlgHom S s), ⋯⟩) ((fun f' => preLiftAlgHom S ⋯) f)) x✝ = ↑f x✝","state_after":"no goals","tactic":"simp only [preLiftAlgHom_def, mkAlgHom_def, mkRingHom_def, RingHom.toMonoidHom_eq_coe,\n                 RingHom.coe_monoidHom_mk, AlgHom.coe_comp, AlgHom.coe_mk, RingHom.coe_mk,\n                 MonoidHom.coe_mk, OneHom.coe_mk, 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0‖) =O[atTop] fun x => x ^ r","state_after":"a : UnitAddCircle\nr v : ℝ\nhv : 0 < v\nhv' : (fun x => ‖sinKernel a x‖) =O[atTop] fun x => rexp (-v * x)\n⊢ (fun x => ‖sinKernel a x‖) =O[atTop] fun x => x ^ r","tactic":"simp_rw [Function.comp_def, sub_zero, norm_real]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Complex.norm_real","def_path":"Mathlib/Analysis/Complex/Basic.lean","def_pos":[141,8],"def_end_pos":[141,17]},{"full_name":"Function.comp_def","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[37,8],"def_end_pos":[37,25]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"sub_zero","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[353,2],"def_end_pos":[353,13]}]},{"state_before":"a : UnitAddCircle\nr v : ℝ\nhv : 0 < v\nhv' : (fun x => ‖sinKernel a x‖) =O[atTop] fun x => rexp (-v * x)\n⊢ (fun x => ‖sinKernel a x‖) =O[atTop] fun x => x ^ r","state_after":"no goals","tactic":"exact hv'.trans (isLittleO_exp_neg_mul_rpow_atTop hv _).isBigO","premises":[{"full_name":"Asymptotics.IsBigO.trans","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[431,8],"def_end_pos":[431,20]},{"full_name":"Asymptotics.IsLittleO.isBigO","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[193,8],"def_end_pos":[193,24]},{"full_name":"isLittleO_exp_neg_mul_rpow_atTop","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean","def_pos":[304,8],"def_end_pos":[304,40]}]},{"state_before":"a : UnitAddCircle\nx : ℝ\nhx : x ∈ Ioi 0\n⊢ (ofReal' ∘ oddKernel a) (1 / x) = (1 * ↑(x ^ (3 / 2))) • (ofReal' ∘ sinKernel a) x","state_after":"no goals","tactic":"simp_rw [Function.comp_apply, one_mul, smul_eq_mul, ← ofReal_mul,\n    oddKernel_functional_equation a, one_div x, one_div x⁻¹, inv_rpow (le_of_lt hx), one_div,\n    inv_inv]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Complex.ofReal_mul","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[214,8],"def_end_pos":[214,18]},{"full_name":"Function.comp_apply","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[35,16],"def_end_pos":[35,35]},{"full_name":"HurwitzZeta.oddKernel_functional_equation","def_path":"Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean","def_pos":[184,6],"def_end_pos":[184,35]},{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Real.inv_rpow","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","def_pos":[430,8],"def_end_pos":[430,16]},{"full_name":"inv_inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[734,8],"def_end_pos":[734,15]},{"full_name":"le_of_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[89,8],"def_end_pos":[89,16]},{"full_name":"one_div","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[338,8],"def_end_pos":[338,15]},{"full_name":"one_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[409,8],"def_end_pos":[409,15]},{"full_name":"smul_eq_mul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[79,6],"def_end_pos":[79,17]}]}]}
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u_7\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup L\ninst✝¹⁰ : Module R L\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\nφ : L →ₗ[R] Module.End R M\ninst✝⁷ : Fintype ι\ninst✝⁶ : Fintype ι'\ninst✝⁵ : Fintype ιM\ninst✝⁴ : DecidableEq ι\ninst✝³ : DecidableEq ι'\ninst✝² : DecidableEq ιM\nb : Basis ι R L\nbₘ : Basis ιM R M\nA : Type u_8\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\n⊢ Polynomial.map (↑(bind₁ (toMvPolynomial (basis A b) (basis A bₘ).end (tensorProduct R A M M ∘ₗ baseChange A φ))))\n      (charpoly.univ A ιM) =\n    Polynomial.map (MvPolynomial.map (algebraMap R A))\n      (Polynomial.map (↑(bind₁ (toMvPolynomial b bₘ.end φ))) (charpoly.univ R ιM))","tactic":"simp only [polyCharpolyAux]","premises":[{"full_name":"LinearMap.polyCharpolyAux","def_path":"Mathlib/Algebra/Module/LinearMap/Polynomial.lean","def_pos":[237,4],"def_end_pos":[237,19]}]},{"state_before":"R : Type u_1\nL : Type u_2\nM : Type u_3\nn : Type u_4\nι : Type u_5\nι' : Type u_6\nιM : Type u_7\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup L\ninst✝¹⁰ : Module R L\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\nφ : L →ₗ[R] Module.End R M\ninst✝⁷ : Fintype ι\ninst✝⁶ : Fintype ι'\ninst✝⁵ : Fintype ιM\ninst✝⁴ : DecidableEq ι\ninst✝³ : DecidableEq ι'\ninst✝² : DecidableEq ιM\nb : Basis ι R L\nbₘ : Basis ιM R M\nA : Type u_8\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\n⊢ Polynomial.map (↑(bind₁ (toMvPolynomial (basis A b) (basis A bₘ).end (tensorProduct R A M M ∘ₗ baseChange A φ))))\n      (charpoly.univ A ιM) =\n    Polynomial.map (MvPolynomial.map (algebraMap R A))\n      (Polynomial.map (↑(bind₁ (toMvPolynomial b bₘ.end φ))) (charpoly.univ R ιM))","state_after":"R : Type u_1\nL : Type u_2\nM : Type u_3\nn : Type u_4\nι : Type u_5\nι' : Type u_6\nιM : Type u_7\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup L\ninst✝¹⁰ : Module R L\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\nφ : L →ₗ[R] Module.End R M\ninst✝⁷ : Fintype ι\ninst✝⁶ : Fintype ι'\ninst✝⁵ : Fintype ιM\ninst✝⁴ : DecidableEq ι\ninst✝³ : DecidableEq ι'\ninst✝² : DecidableEq ιM\nb : Basis ι R L\nbₘ : Basis ιM R M\nA : Type u_8\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\n⊢ Polynomial.map (↑(bind₁ (toMvPolynomial (basis A b) (basis A bₘ).end (tensorProduct R A M M ∘ₗ baseChange A φ))))\n      (Polynomial.map (MvPolynomial.map (algebraMap R A)) (charpoly.univ R ιM)) =\n    Polynomial.map (MvPolynomial.map (algebraMap R A))\n      (Polynomial.map (↑(bind₁ (toMvPolynomial b bₘ.end φ))) (charpoly.univ R ιM))","tactic":"rw [← charpoly.univ_map_map _ (algebraMap R A)]","premises":[{"full_name":"Matrix.charpoly.univ_map_map","def_path":"Mathlib/LinearAlgebra/Matrix/Charpoly/Univ.lean","def_pos":[60,6],"def_end_pos":[60,18]},{"full_name":"algebraMap","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[108,4],"def_end_pos":[108,14]}]},{"state_before":"R : Type u_1\nL : Type u_2\nM : Type u_3\nn : Type u_4\nι : Type u_5\nι' : Type u_6\nιM : Type u_7\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup L\ninst✝¹⁰ : Module R L\ninst✝⁹ 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ι'\ninst✝² : DecidableEq ιM\nb : Basis ι R L\nbₘ : Basis ιM R M\nA : Type u_8\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\n⊢ Polynomial.map\n      ((↑(bind₁ (toMvPolynomial (basis A b) (basis A bₘ).end (tensorProduct R A M M ∘ₗ baseChange A φ)))).comp\n        (MvPolynomial.map (algebraMap R A)))\n      (charpoly.univ R ιM) =\n    Polynomial.map ((MvPolynomial.map (algebraMap R A)).comp ↑(bind₁ (toMvPolynomial b bₘ.end φ))) (charpoly.univ R ιM)","tactic":"simp only [Polynomial.map_map]","premises":[{"full_name":"Polynomial.map_map","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[684,8],"def_end_pos":[684,15]}]},{"state_before":"R : Type u_1\nL : Type u_2\nM : Type u_3\nn : Type u_4\nι : Type u_5\nι' : Type u_6\nιM : Type u_7\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup L\ninst✝¹⁰ : Module R L\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\nφ : L →ₗ[R] Module.End R M\ninst✝⁷ : Fintype ι\ninst✝⁶ : Fintype ι'\ninst✝⁵ : Fintype ιM\ninst✝⁴ : DecidableEq ι\ninst✝³ : DecidableEq ι'\ninst✝² : DecidableEq ιM\nb : Basis ι R L\nbₘ : Basis ιM R M\nA : Type u_8\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\n⊢ Polynomial.map\n      ((↑(bind₁ (toMvPolynomial (basis A b) (basis A bₘ).end (tensorProduct R A M M ∘ₗ baseChange A φ)))).comp\n        (MvPolynomial.map (algebraMap R A)))\n      (charpoly.univ R ιM) =\n    Polynomial.map ((MvPolynomial.map (algebraMap R A)).comp ↑(bind₁ (toMvPolynomial b bₘ.end φ))) (charpoly.univ R ιM)","state_after":"case e_f\nR : Type u_1\nL : Type u_2\nM : Type u_3\nn : Type u_4\nι : Type u_5\nι' : Type u_6\nιM : Type u_7\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup L\ninst✝¹⁰ : Module R L\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\nφ : L →ₗ[R] Module.End R M\ninst✝⁷ : Fintype ι\ninst✝⁶ : Fintype ι'\ninst✝⁵ : Fintype ιM\ninst✝⁴ : DecidableEq ι\ninst✝³ : DecidableEq ι'\ninst✝² : DecidableEq ιM\nb : Basis ι R L\nbₘ : Basis ιM R M\nA : Type u_8\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\n⊢ (↑(bind₁ (toMvPolynomial (basis A b) (basis A bₘ).end (tensorProduct R A M M ∘ₗ baseChange A φ)))).comp\n      (MvPolynomial.map (algebraMap R A)) =\n    (MvPolynomial.map (algebraMap R A)).comp ↑(bind₁ (toMvPolynomial b bₘ.end φ))","tactic":"congr 1","premises":[]},{"state_before":"case e_f\nR : Type u_1\nL : Type u_2\nM : Type u_3\nn : Type u_4\nι : Type u_5\nι' : Type u_6\nιM : Type u_7\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup L\ninst✝¹⁰ : Module R L\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\nφ : L →ₗ[R] Module.End R M\ninst✝⁷ : Fintype ι\ninst✝⁶ : Fintype ι'\ninst✝⁵ : Fintype ιM\ninst✝⁴ : DecidableEq ι\ninst✝³ : DecidableEq ι'\ninst✝² : DecidableEq ιM\nb : Basis ι R L\nbₘ : Basis ιM R M\nA : Type u_8\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\n⊢ (↑(bind₁ (toMvPolynomial (basis A b) (basis A bₘ).end (tensorProduct R A M M ∘ₗ baseChange A φ)))).comp\n      (MvPolynomial.map (algebraMap R A)) =\n    (MvPolynomial.map (algebraMap R A)).comp ↑(bind₁ (toMvPolynomial b bₘ.end φ))","state_after":"case e_f.hC\nR : Type u_1\nL : Type u_2\nM : Type u_3\nn : Type u_4\nι : Type u_5\nι' : Type u_6\nιM : Type u_7\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup L\ninst✝¹⁰ : Module R L\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\nφ : L →ₗ[R] Module.End R M\ninst✝⁷ : Fintype ι\ninst✝⁶ : Fintype ι'\ninst✝⁵ : Fintype ιM\ninst✝⁴ : DecidableEq ι\ninst✝³ : DecidableEq ι'\ninst✝² : DecidableEq ιM\nb : Basis ι R L\nbₘ : Basis ιM R M\nA : Type u_8\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\n⊢ ∀ (r : R),\n    ((↑(bind₁ (toMvPolynomial (basis A b) (basis A bₘ).end (tensorProduct R A M M ∘ₗ baseChange A φ)))).comp\n          (MvPolynomial.map (algebraMap R A)))\n        (C r) =\n      ((MvPolynomial.map (algebraMap R A)).comp ↑(bind₁ (toMvPolynomial b bₘ.end φ))) (C r)\n\ncase e_f.hX\nR : Type u_1\nL : Type u_2\nM : Type u_3\nn : Type u_4\nι : Type u_5\nι' : Type u_6\nιM : Type u_7\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup L\ninst✝¹⁰ : Module R L\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\nφ : L →ₗ[R] Module.End R M\ninst✝⁷ : Fintype ι\ninst✝⁶ : Fintype ι'\ninst✝⁵ : Fintype ιM\ninst✝⁴ : DecidableEq ι\ninst✝³ : DecidableEq ι'\ninst✝² : DecidableEq ιM\nb : Basis ι R L\nbₘ : Basis ιM R M\nA : Type u_8\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\n⊢ ∀ (i : ιM × ιM),\n    ((↑(bind₁ (toMvPolynomial (basis A b) (basis A bₘ).end (tensorProduct R A M M ∘ₗ baseChange A φ)))).comp\n          (MvPolynomial.map (algebraMap R A)))\n        (X i) =\n      ((MvPolynomial.map (algebraMap R A)).comp ↑(bind₁ (toMvPolynomial b bₘ.end φ))) (X i)","tactic":"apply ringHom_ext","premises":[{"full_name":"MvPolynomial.ringHom_ext","def_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","def_pos":[403,8],"def_end_pos":[403,19]}]}]}
+{"url":"Mathlib/Algebra/Order/Ring/Unbundled/Basic.lean","commit":"","full_name":"lt_mul_of_lt_one_left","start":[502,0],"end":[506,57],"file_path":"Mathlib/Algebra/Order/Ring/Unbundled/Basic.lean","tactics":[{"state_before":"α : Type u\nβ : Type u_1\ninst✝⁵ : Semiring α\ninst✝⁴ : PartialOrder α\na b c d : α\ninst✝³ : ExistsAddOfLE α\ninst✝² : MulPosStrictMono α\ninst✝¹ : CovariantClass α α (swap fun x x_1 => x + x_1) fun x x_1 => x < x_1\ninst✝ : ContravariantClass α α (swap fun x x_1 => x + x_1) fun x x_1 => x < x_1\nhb : b < 0\nh : a < 1\n⊢ b < a * b","state_after":"no goals","tactic":"simpa only [one_mul] using mul_lt_mul_of_neg_right h hb","premises":[{"full_name":"mul_lt_mul_of_neg_right","def_path":"Mathlib/Algebra/Order/Ring/Unbundled/Basic.lean","def_pos":[487,8],"def_end_pos":[487,31]},{"full_name":"one_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[409,8],"def_end_pos":[409,15]}]}]}
+{"url":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","commit":"","full_name":"Asymptotics.IsBigOWith.prod_left_same","start":[826,0],"end":[828,71],"file_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n𝕜 : Type u_15\n𝕜' : Type u_16\ninst✝¹³ : Norm E\ninst✝¹² : Norm F\ninst✝¹¹ : Norm G\ninst✝¹⁰ : SeminormedAddCommGroup E'\ninst✝⁹ : SeminormedAddCommGroup F'\ninst✝⁸ : SeminormedAddCommGroup G'\ninst✝⁷ : NormedAddCommGroup E''\ninst✝⁶ : NormedAddCommGroup F''\ninst✝⁵ : NormedAddCommGroup G''\ninst✝⁴ : SeminormedRing R\ninst✝³ : SeminormedAddGroup E'''\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedDivisionRing 𝕜\ninst✝ : NormedDivisionRing 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nhf : IsBigOWith c l f' k'\nhg : IsBigOWith c l g' k'\n⊢ IsBigOWith c l (fun x => (f' x, g' x)) k'","state_after":"α : Type u_1\nβ : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n𝕜 : Type u_15\n𝕜' : Type u_16\ninst✝¹³ : Norm E\ninst✝¹² : Norm F\ninst✝¹¹ : Norm G\ninst✝¹⁰ : SeminormedAddCommGroup E'\ninst✝⁹ : SeminormedAddCommGroup F'\ninst✝⁸ : SeminormedAddCommGroup G'\ninst✝⁷ : NormedAddCommGroup E''\ninst✝⁶ : NormedAddCommGroup F''\ninst✝⁵ : NormedAddCommGroup G''\ninst✝⁴ : SeminormedRing R\ninst✝³ : SeminormedAddGroup E'''\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedDivisionRing 𝕜\ninst✝ : NormedDivisionRing 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nhf : ∀ᶠ (x : α) in l, ‖f' x‖ ≤ c * ‖k' x‖\nhg : ∀ᶠ (x : α) in l, ‖g' x‖ ≤ c * ‖k' x‖\n⊢ ∀ᶠ (x : α) in l, ‖(f' x, g' x)‖ ≤ c * ‖k' x‖","tactic":"rw [isBigOWith_iff] at *","premises":[{"full_name":"Asymptotics.isBigOWith_iff","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[85,8],"def_end_pos":[85,22]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n𝕜 : Type u_15\n𝕜' : Type u_16\ninst✝¹³ : Norm E\ninst✝¹² : Norm F\ninst✝¹¹ : Norm G\ninst✝¹⁰ : SeminormedAddCommGroup E'\ninst✝⁹ : SeminormedAddCommGroup F'\ninst✝⁸ : SeminormedAddCommGroup G'\ninst✝⁷ : NormedAddCommGroup E''\ninst✝⁶ : NormedAddCommGroup F''\ninst✝⁵ : NormedAddCommGroup G''\ninst✝⁴ : SeminormedRing R\ninst✝³ : SeminormedAddGroup E'''\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedDivisionRing 𝕜\ninst✝ : NormedDivisionRing 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nhf : ∀ᶠ (x : α) in l, ‖f' x‖ ≤ c * ‖k' x‖\nhg : ∀ᶠ (x : α) in l, ‖g' x‖ ≤ c * ‖k' x‖\n⊢ ∀ᶠ (x : α) in l, ‖(f' x, g' x)‖ ≤ c * ‖k' x‖","state_after":"no goals","tactic":"filter_upwards [hf, hg] with x using max_le","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Filter.mp_mem","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[159,8],"def_end_pos":[159,14]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]},{"full_name":"max_le","def_path":"Mathlib/Init/Order/LinearOrder.lean","def_pos":[66,8],"def_end_pos":[66,14]}]}]}
+{"url":"Mathlib/Data/Set/Card.lean","commit":"","full_name":"Set.Finite.exists_bijOn_of_encard_eq","start":[444,0],"end":[449,45],"file_path":"Mathlib/Data/Set/Card.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ns✝ t✝ s : Set α\nt : Set β\nf : α → β\ninst✝ : Nonempty β\nhs : s.Finite\nh : s.encard = t.encard\n⊢ ∃ f, BijOn f s t","state_after":"case intro.intro\nα : Type u_1\nβ : Type u_2\ns✝ t✝ s : Set α\nt : Set β\nf✝ : α → β\ninst✝ : Nonempty β\nhs : s.Finite\nh : s.encard = t.encard\nf : α → β\nhf : s ⊆ f ⁻¹' t\nhinj : InjOn f s\n⊢ ∃ f, BijOn f s t","tactic":"obtain ⟨f, hf, hinj⟩ := hs.exists_injOn_of_encard_le h.le","premises":[{"full_name":"Set.Finite.exists_injOn_of_encard_le","def_path":"Mathlib/Data/Set/Card.lean","def_pos":[417,8],"def_end_pos":[417,40]}]},{"state_before":"case intro.intro\nα : Type u_1\nβ : Type u_2\ns✝ t✝ s : Set α\nt : Set β\nf✝ : α → β\ninst✝ : Nonempty β\nhs : s.Finite\nh : s.encard = t.encard\nf : α → β\nhf : s ⊆ f ⁻¹' t\nhinj : InjOn f s\n⊢ ∃ f, BijOn f s t","state_after":"case h\nα : Type u_1\nβ : Type u_2\ns✝ t✝ s : Set α\nt : Set β\nf✝ : α → β\ninst✝ : Nonempty β\nhs : s.Finite\nh : s.encard = t.encard\nf : α → β\nhf : s ⊆ f ⁻¹' t\nhinj : InjOn f s\n⊢ BijOn f s t","tactic":"use f","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]}]},{"state_before":"case h\nα : Type u_1\nβ : Type u_2\ns✝ t✝ s : Set α\nt : Set β\nf✝ : α → β\ninst✝ : Nonempty β\nhs : s.Finite\nh : s.encard = t.encard\nf : α → β\nhf : s ⊆ f ⁻¹' t\nhinj : InjOn f s\n⊢ BijOn f s t","state_after":"case h.e'_5\nα : Type u_1\nβ : Type u_2\ns✝ t✝ s : Set α\nt : Set β\nf✝ : α → β\ninst✝ : Nonempty β\nhs : s.Finite\nh : s.encard = t.encard\nf : α → β\nhf : s ⊆ f ⁻¹' t\nhinj : InjOn f s\n⊢ t = f '' s","tactic":"convert hinj.bijOn_image","premises":[{"full_name":"Set.InjOn.bijOn_image","def_path":"Mathlib/Data/Set/Function.lean","def_pos":[919,8],"def_end_pos":[919,25]}]},{"state_before":"case h.e'_5\nα : Type u_1\nβ : Type u_2\ns✝ t✝ s : Set α\nt : Set β\nf✝ : α → β\ninst✝ : Nonempty β\nhs : s.Finite\nh : s.encard = t.encard\nf : α → β\nhf : s ⊆ f ⁻¹' t\nhinj : InjOn f s\n⊢ t = f '' s","state_after":"no goals","tactic":"rw [(hs.image f).eq_of_subset_of_encard_le' (image_subset_iff.mpr hf)\n    (h.symm.trans hinj.encard_image.symm).le]","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Eq.trans","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[335,8],"def_end_pos":[335,16]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Set.Finite.eq_of_subset_of_encard_le'","def_path":"Mathlib/Data/Set/Card.lean","def_pos":[208,8],"def_end_pos":[208,41]},{"full_name":"Set.Finite.image","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[735,8],"def_end_pos":[735,20]},{"full_name":"Set.InjOn.encard_image","def_path":"Mathlib/Data/Set/Card.lean","def_pos":[380,8],"def_end_pos":[380,26]},{"full_name":"Set.image_subset_iff","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[407,8],"def_end_pos":[407,24]}]}]}
+{"url":"Mathlib/Algebra/ContinuedFractions/ConvergentsEquiv.lean","commit":"","full_name":"GenContFract.squashGCF_nth_of_lt","start":[206,0],"end":[209,73],"file_path":"Mathlib/Algebra/ContinuedFractions/ConvergentsEquiv.lean","tactics":[{"state_before":"K : Type u_1\nn : ℕ\ng : GenContFract K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\nm : ℕ\nm_lt_n : m < n\n⊢ (g.squashGCF (n + 1)).s.get? m = g.s.get? m","state_after":"no goals","tactic":"simp only [squashGCF, squashSeq_nth_of_lt m_lt_n, Nat.add_eq, add_zero]","premises":[{"full_name":"GenContFract.squashGCF","def_path":"Mathlib/Algebra/ContinuedFractions/ConvergentsEquiv.lean","def_pos":[183,4],"def_end_pos":[183,13]},{"full_name":"GenContFract.squashSeq_nth_of_lt","def_path":"Mathlib/Algebra/ContinuedFractions/ConvergentsEquiv.lean","def_pos":[116,8],"def_end_pos":[116,27]},{"full_name":"Nat.add_eq","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[107,16],"def_end_pos":[107,22]},{"full_name":"add_zero","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[412,2],"def_end_pos":[412,13]}]}]}
+{"url":"Mathlib/Data/Nat/PartENat.lean","commit":"","full_name":"PartENat.ofENat_lt","start":[642,0],"end":[645,57],"file_path":"Mathlib/Data/Nat/PartENat.lean","tactics":[{"state_before":"x y : ℕ∞\n⊢ ↑x < ↑y ↔ x < y","state_after":"no goals","tactic":"classical\n  rw [← toWithTop_lt, toWithTop_ofENat, toWithTop_ofENat]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"PartENat.toWithTop_lt","def_path":"Mathlib/Data/Nat/PartENat.lean","def_pos":[591,8],"def_end_pos":[591,20]},{"full_name":"PartENat.toWithTop_ofENat","def_path":"Mathlib/Data/Nat/PartENat.lean","def_pos":[628,8],"def_end_pos":[628,24]}]}]}
+{"url":"Mathlib/Data/Int/Order/Lemmas.lean","commit":"","full_name":"Int.eq_zero_of_abs_lt_dvd","start":[40,0],"end":[46,41],"file_path":"Mathlib/Data/Int/Order/Lemmas.lean","tactics":[{"state_before":"a b : ℤ\nn : ℕ\nm x : ℤ\nh1 : m ∣ x\nh2 : |x| < m\n⊢ x = 0","state_after":"case inl\na b : ℤ\nn : ℕ\nx : ℤ\nh1 : 0 ∣ x\nh2 : |x| < 0\n⊢ x = 0\n\ncase inr\na b : ℤ\nn : ℕ\nm x : ℤ\nh1 : m ∣ x\nh2 : |x| < m\nhm : m ≠ 0\n⊢ x = 0","tactic":"obtain rfl | hm := eq_or_ne m 0","premises":[{"full_name":"eq_or_ne","def_path":"Mathlib/Logic/Basic.lean","def_pos":[167,8],"def_end_pos":[167,16]}]},{"state_before":"case inr\na b : ℤ\nn : ℕ\nm x : ℤ\nh1 : m ∣ x\nh2 : |x| < m\nhm : m ≠ 0\n⊢ x = 0","state_after":"case inr.intro\na b : ℤ\nn : ℕ\nm : ℤ\nhm : m ≠ 0\nd : ℤ\nh2 : |m * d| < m\n⊢ m * d = 0","tactic":"rcases h1 with ⟨d, rfl⟩","premises":[]},{"state_before":"case inr.intro\na b : ℤ\nn : ℕ\nm : ℤ\nhm : m ≠ 0\nd : ℤ\nh2 : |m * d| < m\n⊢ m * d = 0","state_after":"case inr.intro.h\na b : ℤ\nn : ℕ\nm : ℤ\nhm : m ≠ 0\nd : ℤ\nh2 : |m * d| < m\n⊢ d = 0","tactic":"apply mul_eq_zero_of_right","premises":[{"full_name":"mul_eq_zero_of_right","def_path":"Mathlib/Algebra/GroupWithZero/Defs.lean","def_pos":[241,8],"def_end_pos":[241,28]}]},{"state_before":"case inr.intro.h\na b : ℤ\nn : ℕ\nm : ℤ\nhm : m ≠ 0\nd : ℤ\nh2 : |m * d| < m\n⊢ d = 0","state_after":"case inr.intro.h\na b : ℤ\nn : ℕ\nm : ℤ\nhm : m ≠ 0\nd : ℤ\nh2 : |m * d| < m\n⊢ |m * d| < |m|","tactic":"rw [← abs_lt_one_iff, ← mul_lt_iff_lt_one_right (abs_pos.mpr hm), ← abs_mul]","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Int.abs_lt_one_iff","def_path":"Mathlib/Algebra/Order/Group/Int.lean","def_pos":[77,8],"def_end_pos":[77,22]},{"full_name":"abs_mul","def_path":"Mathlib/Algebra/Order/Ring/Abs.lean","def_pos":[42,6],"def_end_pos":[42,13]},{"full_name":"abs_pos","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[208,29],"def_end_pos":[208,36]},{"full_name":"mul_lt_iff_lt_one_right","def_path":"Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean","def_pos":[635,8],"def_end_pos":[635,31]}]},{"state_before":"case inr.intro.h\na b : ℤ\nn : ℕ\nm : ℤ\nhm : m ≠ 0\nd : ℤ\nh2 : |m * d| < m\n⊢ |m * d| < |m|","state_after":"no goals","tactic":"exact lt_of_lt_of_le h2 (le_abs_self m)","premises":[{"full_name":"le_abs_self","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[63,2],"def_end_pos":[63,13]},{"full_name":"lt_of_lt_of_le","def_path":"Mathlib/Order/Defs.lean","def_pos":[93,8],"def_end_pos":[93,22]}]}]}
+{"url":"Mathlib/MeasureTheory/Group/Prod.lean","commit":"","full_name":"MeasureTheory.lintegral_lintegral_add_neg","start":[182,0],"end":[195,20],"file_path":"Mathlib/MeasureTheory/Group/Prod.lean","tactics":[{"state_before":"G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SFinite ν\ninst✝³ : SFinite μ\ns : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : μ.IsMulLeftInvariant\ninst✝ : ν.IsMulLeftInvariant\nf : G → G → ℝ≥0∞\nhf : AEMeasurable (uncurry f) (μ.prod ν)\n⊢ ∫⁻ (x : G), ∫⁻ (y : G), f (y * x) x⁻¹ ∂ν ∂μ = ∫⁻ (x : G), ∫⁻ (y : G), f x y ∂ν ∂μ","state_after":"G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SFinite ν\ninst✝³ : SFinite μ\ns : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : μ.IsMulLeftInvariant\ninst✝ : ν.IsMulLeftInvariant\nf : G → G → ℝ≥0∞\nhf : AEMeasurable (uncurry f) (μ.prod ν)\nh : Measurable fun z => (z.2 * z.1, z.1⁻¹)\n⊢ ∫⁻ (x : G), ∫⁻ (y : G), f (y * x) x⁻¹ ∂ν ∂μ = ∫⁻ (x : G), ∫⁻ (y : G), f x y ∂ν ∂μ","tactic":"have h : Measurable fun z : G × G => (z.2 * z.1, z.1⁻¹) :=\n    (measurable_snd.mul measurable_fst).prod_mk measurable_fst.inv","premises":[{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"Measurable","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","def_pos":[479,4],"def_end_pos":[479,14]},{"full_name":"Measurable.inv","def_path":"Mathlib/MeasureTheory/Group/Arithmetic.lean","def_pos":[381,8],"def_end_pos":[381,22]},{"full_name":"Measurable.mul","def_path":"Mathlib/MeasureTheory/Group/Arithmetic.lean","def_pos":[119,8],"def_end_pos":[119,22]},{"full_name":"Measurable.prod_mk","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Basic.lean","def_pos":[645,8],"def_end_pos":[645,26]},{"full_name":"Prod","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[479,10],"def_end_pos":[479,14]},{"full_name":"Prod.fst","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[483,2],"def_end_pos":[483,5]},{"full_name":"Prod.mk","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[481,2],"def_end_pos":[481,4]},{"full_name":"Prod.snd","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[485,2],"def_end_pos":[485,5]},{"full_name":"measurable_fst","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Basic.lean","def_pos":[613,8],"def_end_pos":[613,22]},{"full_name":"measurable_snd","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Basic.lean","def_pos":[618,8],"def_end_pos":[618,22]}]},{"state_before":"G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SFinite ν\ninst✝³ : SFinite μ\ns : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : μ.IsMulLeftInvariant\ninst✝ : ν.IsMulLeftInvariant\nf : G → G → ℝ≥0∞\nhf : AEMeasurable (uncurry f) (μ.prod ν)\nh : Measurable fun z => (z.2 * z.1, z.1⁻¹)\n⊢ ∫⁻ (x : G), ∫⁻ (y : G), f (y * x) x⁻¹ ∂ν ∂μ = ∫⁻ (x : G), ∫⁻ (y : G), f x y ∂ν ∂μ","state_after":"G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SFinite ν\ninst✝³ : SFinite μ\ns : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : μ.IsMulLeftInvariant\ninst✝ : ν.IsMulLeftInvariant\nf : G → G → ℝ≥0∞\nhf : AEMeasurable (uncurry f) (μ.prod ν)\nh : Measurable fun z => (z.2 * z.1, z.1⁻¹)\nh2f : AEMeasurable (uncurry fun x y => f (y * x) x⁻¹) (μ.prod ν)\n⊢ ∫⁻ (x : G), ∫⁻ (y : G), f (y * x) x⁻¹ ∂ν ∂μ = ∫⁻ (x : G), ∫⁻ (y : G), f x y ∂ν ∂μ","tactic":"have h2f : AEMeasurable (uncurry fun x y => f (y * x) x⁻¹) (μ.prod ν) :=\n    hf.comp_quasiMeasurePreserving (measurePreserving_mul_prod_inv μ ν).quasiMeasurePreserving","premises":[{"full_name":"AEMeasurable","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[376,4],"def_end_pos":[376,16]},{"full_name":"AEMeasurable.comp_quasiMeasurePreserving","def_path":"Mathlib/MeasureTheory/Measure/AEMeasurable.lean","def_pos":[161,8],"def_end_pos":[161,35]},{"full_name":"Function.uncurry","def_path":"Mathlib/Logic/Function/Defs.lean","def_pos":[189,4],"def_end_pos":[189,11]},{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"MeasureTheory.Measure.prod","def_path":"Mathlib/MeasureTheory/Constructions/Prod/Basic.lean","def_pos":[308,26],"def_end_pos":[308,30]},{"full_name":"MeasureTheory.MeasurePreserving.quasiMeasurePreserving","def_path":"Mathlib/Dynamics/Ergodic/MeasurePreserving.lean","def_pos":[84,18],"def_end_pos":[84,40]},{"full_name":"MeasureTheory.measurePreserving_mul_prod_inv","def_path":"Mathlib/MeasureTheory/Group/Prod.lean","def_pos":[135,8],"def_end_pos":[135,38]}]},{"state_before":"G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SFinite ν\ninst✝³ : SFinite μ\ns : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : μ.IsMulLeftInvariant\ninst✝ : ν.IsMulLeftInvariant\nf : G → G → ℝ≥0∞\nhf : AEMeasurable (uncurry f) (μ.prod ν)\nh : Measurable fun z => (z.2 * z.1, z.1⁻¹)\nh2f : AEMeasurable (uncurry fun x y => f (y * x) x⁻¹) (μ.prod ν)\n⊢ ∫⁻ (x : G), ∫⁻ (y : G), f (y * x) x⁻¹ ∂ν ∂μ = ∫⁻ (x : G), ∫⁻ (y : G), f x y ∂ν ∂μ","state_after":"G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SFinite ν\ninst✝³ : SFinite μ\ns : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : μ.IsMulLeftInvariant\ninst✝ : ν.IsMulLeftInvariant\nf : G → G → ℝ≥0∞\nhf : AEMeasurable (uncurry f) (μ.prod ν)\nh : Measurable fun z => (z.2 * z.1, z.1⁻¹)\nh2f : AEMeasurable (uncurry fun x y => f (y * x) x⁻¹) (μ.prod ν)\n⊢ ∫⁻ (z : G × G), f (z.2 * z.1) z.1⁻¹ ∂μ.prod ν = ∫⁻ (z : G × G), f z.1 z.2 ∂μ.prod ν","tactic":"simp_rw [lintegral_lintegral h2f, lintegral_lintegral hf]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"MeasureTheory.lintegral_lintegral","def_path":"Mathlib/MeasureTheory/Constructions/Prod/Basic.lean","def_pos":[927,8],"def_end_pos":[927,27]}]},{"state_before":"G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SFinite ν\ninst✝³ : SFinite μ\ns : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : μ.IsMulLeftInvariant\ninst✝ : ν.IsMulLeftInvariant\nf : G → G → ℝ≥0∞\nhf : AEMeasurable (uncurry f) (μ.prod ν)\nh : Measurable fun z => (z.2 * z.1, z.1⁻¹)\nh2f : AEMeasurable (uncurry fun x y => f (y * x) x⁻¹) (μ.prod ν)\n⊢ ∫⁻ (z : G × G), f (z.2 * z.1) z.1⁻¹ ∂μ.prod ν = ∫⁻ (z : G × G), f z.1 z.2 ∂μ.prod ν","state_after":"G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SFinite ν\ninst✝³ : SFinite μ\ns : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : μ.IsMulLeftInvariant\ninst✝ : ν.IsMulLeftInvariant\nf : G → G → ℝ≥0∞\nhf : AEMeasurable (uncurry f) (μ.prod ν)\nh : Measurable fun z => (z.2 * z.1, z.1⁻¹)\nh2f : AEMeasurable (uncurry fun x y => f (y * x) x⁻¹) (μ.prod ν)\n⊢ ∫⁻ (z : G × G), f (z.2 * z.1) z.1⁻¹ ∂μ.prod ν =\n    ∫⁻ (z : G × G), f z.1 z.2 ∂map (fun z => (z.2 * z.1, z.1⁻¹)) (μ.prod ν)","tactic":"conv_rhs => rw [← (measurePreserving_mul_prod_inv μ ν).map_eq]","premises":[{"full_name":"MeasureTheory.MeasurePreserving.map_eq","def_path":"Mathlib/Dynamics/Ergodic/MeasurePreserving.lean","def_pos":[43,12],"def_end_pos":[43,18]},{"full_name":"MeasureTheory.measurePreserving_mul_prod_inv","def_path":"Mathlib/MeasureTheory/Group/Prod.lean","def_pos":[135,8],"def_end_pos":[135,38]}]},{"state_before":"G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SFinite ν\ninst✝³ : SFinite μ\ns : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : μ.IsMulLeftInvariant\ninst✝ : ν.IsMulLeftInvariant\nf : G → G → ℝ≥0∞\nhf : AEMeasurable (uncurry f) (μ.prod ν)\nh : Measurable fun z => (z.2 * z.1, z.1⁻¹)\nh2f : AEMeasurable (uncurry fun x y => f (y * x) x⁻¹) (μ.prod ν)\n⊢ ∫⁻ (z : G × G), f (z.2 * z.1) z.1⁻¹ ∂μ.prod ν =\n    ∫⁻ (z : G × G), f z.1 z.2 ∂map (fun z => (z.2 * z.1, z.1⁻¹)) (μ.prod ν)","state_after":"G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SFinite ν\ninst✝³ : SFinite μ\ns : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : μ.IsMulLeftInvariant\ninst✝ : ν.IsMulLeftInvariant\nf : G → G → ℝ≥0∞\nhf : AEMeasurable (uncurry f) (μ.prod ν)\nh : Measurable fun z => (z.2 * z.1, z.1⁻¹)\nh2f : AEMeasurable (uncurry fun x y => f (y * x) x⁻¹) (μ.prod ν)\n⊢ ∫⁻ (z : G × G), f z.1 z.2 ∂map (fun z => (z.2 * z.1, z.1⁻¹)) (μ.prod ν) =\n    ∫⁻ (z : G × G), f (z.2 * z.1) z.1⁻¹ ∂μ.prod ν","tactic":"symm","premises":[]},{"state_before":"G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SFinite ν\ninst✝³ : SFinite μ\ns : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : μ.IsMulLeftInvariant\ninst✝ : ν.IsMulLeftInvariant\nf : G → G → ℝ≥0∞\nhf : AEMeasurable (uncurry f) (μ.prod ν)\nh : Measurable fun z => (z.2 * z.1, z.1⁻¹)\nh2f : AEMeasurable (uncurry fun x y => f (y * x) x⁻¹) (μ.prod ν)\n⊢ ∫⁻ (z : G × G), f z.1 z.2 ∂map (fun z => (z.2 * z.1, z.1⁻¹)) (μ.prod ν) =\n    ∫⁻ (z : G × G), f (z.2 * z.1) z.1⁻¹ ∂μ.prod ν","state_after":"no goals","tactic":"exact\n    lintegral_map' (hf.mono' (measurePreserving_mul_prod_inv μ ν).map_eq.absolutelyContinuous)\n      h.aemeasurable","premises":[{"full_name":"AEMeasurable.mono'","def_path":"Mathlib/MeasureTheory/Measure/AEMeasurable.lean","def_pos":[62,18],"def_end_pos":[62,23]},{"full_name":"Measurable.aemeasurable","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[380,8],"def_end_pos":[380,31]},{"full_name":"MeasureTheory.MeasurePreserving.map_eq","def_path":"Mathlib/Dynamics/Ergodic/MeasurePreserving.lean","def_pos":[43,12],"def_end_pos":[43,18]},{"full_name":"MeasureTheory.lintegral_map'","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[1364,8],"def_end_pos":[1364,22]},{"full_name":"MeasureTheory.measurePreserving_mul_prod_inv","def_path":"Mathlib/MeasureTheory/Group/Prod.lean","def_pos":[135,8],"def_end_pos":[135,38]}]}]}
+{"url":"Mathlib/GroupTheory/GroupAction/Hom.lean","commit":"","full_name":"DistribMulActionHom.id_comp","start":[509,0],"end":[511,43],"file_path":"Mathlib/GroupTheory/GroupAction/Hom.lean","tactics":[{"state_before":"M : Type u_1\ninst✝¹⁵ : Monoid M\nN : Type u_2\ninst✝¹⁴ : Monoid N\nP : Type u_3\ninst✝¹³ : Monoid P\nφ : M →* N\nφ' : N →* M\nψ : N →* P\nχ : M →* P\nA : Type u_4\ninst✝¹² : AddMonoid A\ninst✝¹¹ : DistribMulAction M A\nB : Type u_5\ninst✝¹⁰ : AddMonoid B\ninst✝⁹ : DistribMulAction N B\nB₁ : Type u_6\ninst✝⁸ : AddMonoid B₁\ninst✝⁷ : DistribMulAction M B₁\nC : Type u_7\ninst✝⁶ : AddMonoid C\ninst✝⁵ : DistribMulAction P C\nA' : Type u_8\ninst✝⁴ : AddGroup A'\ninst✝³ : DistribMulAction M A'\nB' : Type u_9\ninst✝² : AddGroup B'\ninst✝¹ : DistribMulAction N B'\nF : Type u_10\ninst✝ : FunLike F A B\nf : A →ₑ+[φ] B\nx : A\n⊢ ((DistribMulActionHom.id N).comp f) x = f x","state_after":"no goals","tactic":"rw [comp_apply, id_apply]","premises":[{"full_name":"DistribMulActionHom.comp_apply","def_path":"Mathlib/GroupTheory/GroupAction/Hom.lean","def_pos":[505,8],"def_end_pos":[505,18]},{"full_name":"DistribMulActionHom.id_apply","def_path":"Mathlib/GroupTheory/GroupAction/Hom.lean","def_pos":[468,8],"def_end_pos":[468,16]}]}]}
+{"url":"Mathlib/GroupTheory/Perm/Support.lean","commit":"","full_name":"Equiv.Perm.support_subtype_perm","start":[574,0],"end":[579,24],"file_path":"Mathlib/GroupTheory/Perm/Support.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : DecidableEq α\ns : Finset α\nf : Perm α\nh : ∀ (x : α), x ∈ s ↔ f x ∈ s\n⊢ (f.subtypePerm h).support = filter (fun x => decide (f ↑x ≠ ↑x) = true) s.attach","state_after":"case a\nα : Type u_1\ninst✝ : DecidableEq α\ns : Finset α\nf : Perm α\nh : ∀ (x : α), x ∈ s ↔ f x ∈ s\na✝ : { x // x ∈ s }\n⊢ a✝ ∈ (f.subtypePerm h).support ↔ a✝ ∈ filter (fun x => decide (f ↑x ≠ ↑x) = true) s.attach","tactic":"ext","premises":[]},{"state_before":"case a\nα : Type u_1\ninst✝ : DecidableEq α\ns : Finset α\nf : Perm α\nh : ∀ (x : α), x ∈ s ↔ f x ∈ s\na✝ : { x // x ∈ s }\n⊢ a✝ ∈ (f.subtypePerm h).support ↔ a✝ ∈ filter (fun x => decide (f ↑x ≠ ↑x) = true) s.attach","state_after":"no goals","tactic":"simp [Subtype.ext_iff]","premises":[{"full_name":"Subtype.ext_iff","def_path":"Mathlib/Data/Subtype.lean","def_pos":[62,18],"def_end_pos":[62,25]}]}]}
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+{"url":".lake/packages/batteries/Batteries/Classes/Order.lean","commit":"","full_name":"Ordering.swap_inj","start":[14,0],"end":[15,55],"file_path":".lake/packages/batteries/Batteries/Classes/Order.lean","tactics":[{"state_before":"o₁ o₂ : Ordering\nh : o₁.swap = o₂.swap\n⊢ o₁ = o₂","state_after":"no goals","tactic":"simpa using congrArg swap h","premises":[{"full_name":"Ordering.swap","def_path":".lake/packages/lean4/src/lean/Init/Data/Ord.lean","def_pos":[20,4],"def_end_pos":[20,8]},{"full_name":"congrArg","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[362,8],"def_end_pos":[362,16]}]}]}
+{"url":"Mathlib/Data/Matrix/Basic.lean","commit":"","full_name":"Matrix.comp_equiv_symm_dotProduct","start":[721,0],"end":[725,88],"file_path":"Mathlib/Data/Matrix/Basic.lean","tactics":[{"state_before":"l : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nm' : o → Type u_5\nn' : o → Type u_6\nR : Type u_7\nS : Type u_8\nα : Type v\nβ : Type w\nγ : Type u_9\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : NonUnitalNonAssocSemiring α\nu v w : m → α\nx y : n → α\ne : m ≃ n\nx✝¹ : m\nx✝ : x✝¹ ∈ Finset.univ\n⊢ (u ∘ ⇑e.symm) (e x✝¹) * x (e x✝¹) = u x✝¹ * (x ∘ ⇑e) x✝¹","state_after":"no goals","tactic":"simp only [Function.comp, Equiv.symm_apply_apply]","premises":[{"full_name":"Equiv.symm_apply_apply","def_path":"Mathlib/Logic/Equiv/Defs.lean","def_pos":[241,16],"def_end_pos":[241,32]},{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]}]}]}
+{"url":"Mathlib/RingTheory/Localization/Integral.lean","commit":"","full_name":"isIntegral_localization","start":[202,0],"end":[222,91],"file_path":"Mathlib/RingTheory/Localization/Integral.lean","tactics":[{"state_before":"R : Type u_1\ninst✝¹⁰ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\nP : Type u_3\ninst✝⁷ : CommRing P\nRₘ : Type u_4\nSₘ : Type u_5\ninst✝⁶ : CommRing Rₘ\ninst✝⁵ : CommRing Sₘ\ninst✝⁴ : Algebra R Rₘ\ninst✝³ : IsLocalization M Rₘ\ninst✝² : Algebra S Sₘ\ninst✝¹ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ\ninst✝ : Algebra.IsIntegral R S\n⊢ (IsLocalization.map Sₘ (algebraMap R S) ⋯).IsIntegral","state_after":"R : Type u_1\ninst✝¹⁰ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\nP : Type u_3\ninst✝⁷ : CommRing P\nRₘ : Type u_4\nSₘ : Type u_5\ninst✝⁶ : CommRing Rₘ\ninst✝⁵ : CommRing Sₘ\ninst✝⁴ : Algebra R Rₘ\ninst✝³ : IsLocalization M Rₘ\ninst✝² : Algebra S Sₘ\ninst✝¹ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ\ninst✝ : Algebra.IsIntegral R S\nx : Sₘ\n⊢ (IsLocalization.map Sₘ (algebraMap R S) ⋯).IsIntegralElem x","tactic":"intro x","premises":[]},{"state_before":"R : Type u_1\ninst✝¹⁰ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\nP : Type u_3\ninst✝⁷ : CommRing P\nRₘ : Type u_4\nSₘ : Type u_5\ninst✝⁶ : CommRing Rₘ\ninst✝⁵ : CommRing Sₘ\ninst✝⁴ : Algebra R Rₘ\ninst✝³ : IsLocalization M Rₘ\ninst✝² : Algebra S Sₘ\ninst✝¹ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ\ninst✝ : Algebra.IsIntegral R S\nx : Sₘ\n⊢ (IsLocalization.map Sₘ (algebraMap R S) ⋯).IsIntegralElem x","state_after":"case intro.mk.mk\nR : Type u_1\ninst✝¹⁰ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\nP : Type u_3\ninst✝⁷ : CommRing P\nRₘ : Type u_4\nSₘ : Type u_5\ninst✝⁶ : CommRing Rₘ\ninst✝⁵ : CommRing Sₘ\ninst✝⁴ : Algebra R Rₘ\ninst✝³ : IsLocalization M Rₘ\ninst✝² : Algebra S Sₘ\ninst✝¹ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ\ninst✝ : Algebra.IsIntegral R S\nx : Sₘ\ns u : S\nhu : u ∈ Algebra.algebraMapSubmonoid S M\nhx : x * (algebraMap S Sₘ) ↑(s, ⟨u, hu⟩).2 = (algebraMap S Sₘ) (s, ⟨u, hu⟩).1\n⊢ (IsLocalization.map Sₘ (algebraMap R S) ⋯).IsIntegralElem x","tactic":"obtain ⟨⟨s, ⟨u, hu⟩⟩, hx⟩ := surj (Algebra.algebraMapSubmonoid S M) x","premises":[{"full_name":"Algebra.algebraMapSubmonoid","def_path":"Mathlib/Algebra/Algebra/Basic.lean","def_pos":[108,4],"def_end_pos":[108,23]},{"full_name":"IsLocalization.surj","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[120,8],"def_end_pos":[120,12]}]},{"state_before":"case intro.mk.mk\nR : Type u_1\ninst✝¹⁰ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\nP : Type u_3\ninst✝⁷ : CommRing P\nRₘ : Type u_4\nSₘ : Type u_5\ninst✝⁶ : CommRing Rₘ\ninst✝⁵ : CommRing Sₘ\ninst✝⁴ : Algebra R Rₘ\ninst✝³ : IsLocalization M Rₘ\ninst✝² : Algebra S Sₘ\ninst✝¹ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ\ninst✝ : Algebra.IsIntegral R S\nx : Sₘ\ns u : S\nhu : u ∈ Algebra.algebraMapSubmonoid S M\nhx : x * (algebraMap S Sₘ) ↑(s, ⟨u, hu⟩).2 = (algebraMap S Sₘ) (s, ⟨u, hu⟩).1\n⊢ (IsLocalization.map Sₘ (algebraMap R S) ⋯).IsIntegralElem x","state_after":"case intro.mk.mk.intro\nR : Type u_1\ninst✝¹⁰ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\nP : Type u_3\ninst✝⁷ : CommRing P\nRₘ : Type u_4\nSₘ : Type u_5\ninst✝⁶ : CommRing Rₘ\ninst✝⁵ : CommRing Sₘ\ninst✝⁴ : Algebra R Rₘ\ninst✝³ : IsLocalization M Rₘ\ninst✝² : Algebra S Sₘ\ninst✝¹ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ\ninst✝ : Algebra.IsIntegral R S\nx : Sₘ\ns u : S\nv : R\nhv : v ∈ ↑M ∧ (algebraMap R S) v = u\nhx : x * (algebraMap S Sₘ) ↑(s, ⟨u, ⋯⟩).2 = (algebraMap S Sₘ) (s, ⟨u, ⋯⟩).1\n⊢ (IsLocalization.map Sₘ (algebraMap R S) ⋯).IsIntegralElem x","tactic":"obtain ⟨v, hv⟩ := hu","premises":[]},{"state_before":"case intro.mk.mk.intro\nR : Type u_1\ninst✝¹⁰ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\nP : Type u_3\ninst✝⁷ : CommRing P\nRₘ : Type u_4\nSₘ : Type u_5\ninst✝⁶ : CommRing Rₘ\ninst✝⁵ : CommRing Sₘ\ninst✝⁴ : Algebra R Rₘ\ninst✝³ : IsLocalization M Rₘ\ninst✝² : Algebra S Sₘ\ninst✝¹ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ\ninst✝ : Algebra.IsIntegral R S\nx : Sₘ\ns u : S\nv : R\nhv : v ∈ ↑M ∧ (algebraMap R S) v = u\nhx : x * (algebraMap S Sₘ) ↑(s, ⟨u, ⋯⟩).2 = (algebraMap S Sₘ) (s, ⟨u, ⋯⟩).1\n⊢ (IsLocalization.map Sₘ (algebraMap R S) ⋯).IsIntegralElem x","state_after":"case intro.mk.mk.intro.intro\nR : Type u_1\ninst✝¹⁰ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\nP : Type u_3\ninst✝⁷ : CommRing P\nRₘ : Type u_4\nSₘ : Type u_5\ninst✝⁶ : CommRing Rₘ\ninst✝⁵ : CommRing Sₘ\ninst✝⁴ : Algebra R Rₘ\ninst✝³ : IsLocalization M Rₘ\ninst✝² : Algebra S Sₘ\ninst✝¹ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ\ninst✝ : Algebra.IsIntegral R S\nx : Sₘ\ns u : S\nv : R\nhv : v ∈ ↑M ∧ (algebraMap R S) v = u\nhx : x * (algebraMap S Sₘ) ↑(s, ⟨u, ⋯⟩).2 = (algebraMap S Sₘ) (s, ⟨u, ⋯⟩).1\nv' : Rₘ\nhv' : v' * (algebraMap R Rₘ) ↑⟨v, ⋯⟩ = 1\n⊢ (IsLocalization.map Sₘ (algebraMap R S) ⋯).IsIntegralElem x","tactic":"obtain ⟨v', hv'⟩ := isUnit_iff_exists_inv'.1 (map_units Rₘ ⟨v, hv.1⟩)","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"IsLocalization.map_units","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[115,8],"def_end_pos":[115,17]},{"full_name":"isUnit_iff_exists_inv'","def_path":"Mathlib/Algebra/Group/Units.lean","def_pos":[602,8],"def_end_pos":[602,30]}]},{"state_before":"case intro.mk.mk.intro.intro\nR : Type u_1\ninst✝¹⁰ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\nP : Type u_3\ninst✝⁷ : CommRing P\nRₘ : Type u_4\nSₘ : Type u_5\ninst✝⁶ : CommRing Rₘ\ninst✝⁵ : CommRing Sₘ\ninst✝⁴ : Algebra R Rₘ\ninst✝³ : IsLocalization M Rₘ\ninst✝² : Algebra S Sₘ\ninst✝¹ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ\ninst✝ : Algebra.IsIntegral R S\nx : Sₘ\ns u : S\nv : R\nhv : v ∈ ↑M ∧ (algebraMap R S) v = u\nhx : x * (algebraMap S Sₘ) ↑(s, ⟨u, ⋯⟩).2 = (algebraMap S Sₘ) (s, ⟨u, ⋯⟩).1\nv' : Rₘ\nhv' : v' * (algebraMap R Rₘ) ↑⟨v, ⋯⟩ = 1\n⊢ (IsLocalization.map Sₘ (algebraMap R S) ⋯).IsIntegralElem x","state_after":"case intro.mk.mk.intro.intro.refine_1\nR : Type u_1\ninst✝¹⁰ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\nP : Type u_3\ninst✝⁷ : CommRing P\nRₘ : Type u_4\nSₘ : Type u_5\ninst✝⁶ : CommRing Rₘ\ninst✝⁵ : CommRing Sₘ\ninst✝⁴ : Algebra R Rₘ\ninst✝³ : IsLocalization M Rₘ\ninst✝² : Algebra S Sₘ\ninst✝¹ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ\ninst✝ : Algebra.IsIntegral R S\nx : Sₘ\ns u : S\nv : R\nhv : v ∈ ↑M ∧ (algebraMap R S) v = u\nhx : x * (algebraMap S Sₘ) ↑(s, ⟨u, ⋯⟩).2 = (algebraMap S Sₘ) (s, ⟨u, ⋯⟩).1\nv' : Rₘ\nhv' : v' * (algebraMap R Rₘ) ↑⟨v, ⋯⟩ = 1\n⊢ (algebraMap Rₘ Sₘ) v' * (algebraMap S Sₘ) u = 1\n\ncase intro.mk.mk.intro.intro.refine_2\nR : Type u_1\ninst✝¹⁰ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\nP : Type u_3\ninst✝⁷ : CommRing P\nRₘ : Type u_4\nSₘ : Type u_5\ninst✝⁶ : CommRing Rₘ\ninst✝⁵ : CommRing Sₘ\ninst✝⁴ : Algebra R Rₘ\ninst✝³ : IsLocalization M Rₘ\ninst✝² : Algebra S Sₘ\ninst✝¹ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ\ninst✝ : Algebra.IsIntegral R S\nx : Sₘ\ns u : S\nv : R\nhv : v ∈ ↑M ∧ (algebraMap R S) v = u\nhx : x * (algebraMap S Sₘ) ↑(s, ⟨u, ⋯⟩).2 = (algebraMap S Sₘ) (s, ⟨u, ⋯⟩).1\nv' : Rₘ\nhv' : v' * (algebraMap R Rₘ) ↑⟨v, ⋯⟩ = 1\n⊢ IsIntegral Rₘ (x * (algebraMap S Sₘ) u)","tactic":"refine @IsIntegral.of_mul_unit Rₘ _ _ _ (localizationAlgebra M S) x (algebraMap S Sₘ u) v' ?_ ?_","premises":[{"full_name":"IsIntegral.of_mul_unit","def_path":"Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean","def_pos":[260,8],"def_end_pos":[260,30]},{"full_name":"algebraMap","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[108,4],"def_end_pos":[108,14]},{"full_name":"localizationAlgebra","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[1081,18],"def_end_pos":[1081,37]}]}]}
+{"url":"Mathlib/GroupTheory/OrderOfElement.lean","commit":"","full_name":"image_range_orderOf","start":[946,0],"end":[950,75],"file_path":"Mathlib/GroupTheory/OrderOfElement.lean","tactics":[{"state_before":"G : Type u_1\nH : Type u_2\nA : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝² : Group G\ninst✝¹ : Fintype G\nx : G\nn : ℕ\ninst✝ : DecidableEq G\n⊢ Finset.image (fun i => x ^ i) (Finset.range (orderOf x)) = (↑(zpowers x)).toFinset","state_after":"case a\nG : Type u_1\nH : Type u_2\nA : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝² : Group G\ninst✝¹ : Fintype G\nx✝ : G\nn : ℕ\ninst✝ : DecidableEq G\nx : G\n⊢ x ∈ Finset.image (fun i => x✝ ^ i) (Finset.range (orderOf x✝)) ↔ x ∈ (↑(zpowers x✝)).toFinset","tactic":"ext x","premises":[]},{"state_before":"case a\nG : Type u_1\nH : Type u_2\nA : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝² : Group G\ninst✝¹ : Fintype G\nx✝ : G\nn : ℕ\ninst✝ : DecidableEq G\nx : G\n⊢ x ∈ Finset.image (fun i => x✝ ^ i) (Finset.range (orderOf x✝)) ↔ x ∈ (↑(zpowers x✝)).toFinset","state_after":"no goals","tactic":"rw [Set.mem_toFinset, SetLike.mem_coe, mem_zpowers_iff_mem_range_orderOf]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Set.mem_toFinset","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[540,8],"def_end_pos":[540,20]},{"full_name":"SetLike.mem_coe","def_path":"Mathlib/Data/SetLike/Basic.lean","def_pos":[168,8],"def_end_pos":[168,15]},{"full_name":"mem_zpowers_iff_mem_range_orderOf","def_path":"Mathlib/GroupTheory/OrderOfElement.lean","def_pos":[775,6],"def_end_pos":[775,39]}]}]}
+{"url":"Mathlib/Analysis/Complex/Angle.lean","commit":"","full_name":"Complex.angle_mul_left","start":[49,0],"end":[51,48],"file_path":"Mathlib/Analysis/Complex/Angle.lean","tactics":[{"state_before":"a x✝ y✝ : ℂ\nha : a ≠ 0\nx y : ℂ\n⊢ angle (a * x) (a * y) = angle x y","state_after":"no goals","tactic":"obtain rfl | hx := eq_or_ne x 0 <;> obtain rfl | hy := eq_or_ne y 0 <;>\n    simp [angle_eq_abs_arg, mul_div_mul_left, *]","premises":[{"full_name":"Complex.angle_eq_abs_arg","def_path":"Mathlib/Analysis/Complex/Angle.lean","def_pos":[37,6],"def_end_pos":[37,22]},{"full_name":"eq_or_ne","def_path":"Mathlib/Logic/Basic.lean","def_pos":[167,8],"def_end_pos":[167,16]},{"full_name":"mul_div_mul_left","def_path":"Mathlib/Algebra/GroupWithZero/Units/Basic.lean","def_pos":[427,6],"def_end_pos":[427,22]}]}]}
+{"url":"Mathlib/Algebra/Associated/Basic.lean","commit":"","full_name":"Associated.refl","start":[364,0],"end":[366,14],"file_path":"Mathlib/Algebra/Associated/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝ : Monoid α\nx : α\n⊢ x * ↑1 = x","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/Order/Interval/Set/OrdConnected.lean","commit":"","full_name":"Set.ordConnected_iff_uIcc_subset_right","start":[309,0],"end":[311,59],"file_path":"Mathlib/Order/Interval/Set/OrdConnected.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : LinearOrder α\ns : Set α\nx : α\nhx : x ∈ s\n⊢ s.OrdConnected ↔ ∀ ⦃y : α⦄, y ∈ s → [[y, x]] ⊆ s","state_after":"no goals","tactic":"simp_rw [ordConnected_iff_uIcc_subset_left hx, uIcc_comm]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Set.ordConnected_iff_uIcc_subset_left","def_path":"Mathlib/Order/Interval/Set/OrdConnected.lean","def_pos":[305,8],"def_end_pos":[305,41]},{"full_name":"Set.uIcc_comm","def_path":"Mathlib/Order/Interval/Set/UnorderedInterval.lean","def_pos":[72,6],"def_end_pos":[72,15]}]}]}
+{"url":"Mathlib/Topology/Algebra/Module/Basic.lean","commit":"","full_name":"ContinuousLinearEquiv.image_symm_eq_preimage","start":[1932,0],"end":[1933,74],"file_path":"Mathlib/Topology/Algebra/Module/Basic.lean","tactics":[{"state_before":"R₁ : Type u_1\nR₂ : Type u_2\nR₃ : Type u_3\ninst✝²⁴ : Semiring R₁\ninst✝²³ : Semiring R₂\ninst✝²² : Semiring R₃\nσ₁₂ : R₁ →+* R₂\nσ₂₁ : R₂ →+* R₁\ninst✝²¹ : RingHomInvPair σ₁₂ σ₂₁\ninst✝²⁰ : RingHomInvPair σ₂₁ σ₁₂\nσ₂₃ : R₂ →+* R₃\nσ₃₂ : R₃ →+* R₂\ninst✝¹⁹ : RingHomInvPair σ₂₃ σ₃₂\ninst✝¹⁸ : RingHomInvPair σ₃₂ σ₂₃\nσ₁₃ : R₁ →+* R₃\nσ₃₁ : R₃ →+* R₁\ninst✝¹⁷ : RingHomInvPair σ₁₃ σ₃₁\ninst✝¹⁶ : RingHomInvPair σ₃₁ σ₁₃\ninst✝¹⁵ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹⁴ : RingHomCompTriple σ₃₂ σ₂₁ σ₃₁\nM₁ : Type u_4\ninst✝¹³ : TopologicalSpace M₁\ninst✝¹² : AddCommMonoid M₁\nM'₁ : Type u_5\ninst✝¹¹ : TopologicalSpace M'₁\ninst✝¹⁰ : AddCommMonoid M'₁\nM₂ : Type u_6\ninst✝⁹ : TopologicalSpace M₂\ninst✝⁸ : AddCommMonoid M₂\nM₃ : Type u_7\ninst✝⁷ : TopologicalSpace M₃\ninst✝⁶ : AddCommMonoid M₃\nM₄ : Type u_8\ninst✝⁵ : TopologicalSpace M₄\ninst✝⁴ : AddCommMonoid M₄\ninst✝³ : Module R₁ M₁\ninst✝² : Module R₁ M'₁\ninst✝¹ : Module R₂ M₂\ninst✝ : Module R₃ M₃\ne : M₁ ≃SL[σ₁₂] M₂\ns : Set M₂\n⊢ ⇑e.symm '' s = ⇑e ⁻¹' s","state_after":"no goals","tactic":"rw [e.symm.image_eq_preimage, e.symm_symm]","premises":[{"full_name":"ContinuousLinearEquiv.image_eq_preimage","def_path":"Mathlib/Topology/Algebra/Module/Basic.lean","def_pos":[1929,18],"def_end_pos":[1929,35]},{"full_name":"ContinuousLinearEquiv.symm","def_path":"Mathlib/Topology/Algebra/Module/Basic.lean","def_pos":[1773,14],"def_end_pos":[1773,18]},{"full_name":"ContinuousLinearEquiv.symm_symm","def_path":"Mathlib/Topology/Algebra/Module/Basic.lean","def_pos":[1912,8],"def_end_pos":[1912,17]}]}]}
+{"url":"Mathlib/Data/Nat/Factorization/PrimePow.lean","commit":"","full_name":"isPrimePow_iff_factorization_eq_single","start":[36,0],"end":[49,88],"file_path":"Mathlib/Data/Nat/Factorization/PrimePow.lean","tactics":[{"state_before":"R : Type u_1\ninst✝ : CommMonoidWithZero R\nn✝ p : R\nk n : ℕ\n⊢ IsPrimePow n ↔ ∃ p k, 0 < k ∧ n.factorization = Finsupp.single p k","state_after":"R : Type u_1\ninst✝ : CommMonoidWithZero R\nn✝ p : R\nk n : ℕ\n⊢ (∃ p k, Nat.Prime p ∧ 0 < k ∧ p ^ k = n) ↔ ∃ p k, 0 < k ∧ n.factorization = Finsupp.single p k","tactic":"rw [isPrimePow_nat_iff]","premises":[{"full_name":"isPrimePow_nat_iff","def_path":"Mathlib/Algebra/IsPrimePow.lean","def_pos":[63,8],"def_end_pos":[63,26]}]},{"state_before":"R : Type u_1\ninst✝ : CommMonoidWithZero R\nn✝ p : R\nk n : ℕ\n⊢ (∃ p k, Nat.Prime p ∧ 0 < k ∧ p ^ k = n) ↔ ∃ p k, 0 < k ∧ n.factorization = Finsupp.single p k","state_after":"R : Type u_1\ninst✝ : CommMonoidWithZero R\nn✝ p✝ : R\nk✝ n p k : ℕ\n⊢ Nat.Prime p ∧ 0 < k ∧ p ^ k = n ↔ 0 < k ∧ n.factorization = Finsupp.single p k","tactic":"refine exists₂_congr fun p k => ?_","premises":[{"full_name":"exists₂_congr","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[219,8],"def_end_pos":[219,21]}]},{"state_before":"R : Type u_1\ninst✝ : CommMonoidWithZero R\nn✝ p✝ : R\nk✝ n p k : ℕ\n⊢ Nat.Prime p ∧ 0 < k ∧ p ^ k = n ↔ 0 < k ∧ n.factorization = Finsupp.single p k","state_after":"case mp\nR : Type u_1\ninst✝ : CommMonoidWithZero R\nn✝ p✝ : R\nk✝ n p k : ℕ\n⊢ Nat.Prime p ∧ 0 < k ∧ p ^ k = n → 0 < k ∧ n.factorization = Finsupp.single p k\n\ncase mpr\nR : Type u_1\ninst✝ : CommMonoidWithZero R\nn✝ p✝ : R\nk✝ n p k : ℕ\n⊢ 0 < k ∧ n.factorization = Finsupp.single p k → Nat.Prime p ∧ 0 < k ∧ p ^ k = n","tactic":"constructor","premises":[]}]}
+{"url":"Mathlib/CategoryTheory/Triangulated/Triangulated.lean","commit":"","full_name":"CategoryTheory.IsTriangulated.mk'","start":[201,0],"end":[218,86],"file_path":"Mathlib/CategoryTheory/Triangulated/Triangulated.lean","tactics":[{"state_before":"C : Type u_1\ninst✝⁵ : Category.{u_2, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nh :\n  ∀ ⦃X₁' X₂' X₃' : C⦄ (u₁₂' : X₁' ⟶ X₂') (u₂₃' : X₂' ⟶ X₃'),\n    ∃ X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ u₁₂ u₂₃ e₁ e₂ e₃,\n      ∃ (_ : u₁₂' ≫ e₂.hom = e₁.hom ≫ u₁₂) (_ : u₂₃' ≫ e₃.hom = e₂.hom ≫ u₂₃),\n        ∃ v₁₂ w₁₂,\n          ∃ (h₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ∈ distinguishedTriangles),\n            ∃ v₂₃ w₂₃,\n              ∃ (h₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ∈ distinguishedTriangles),\n                ∃ v₁₃ w₁₃,\n                  ∃ (h₁₃ : Triangle.mk (u₁₂ ≫ u₂₃) v₁₃ w₁₃ ∈ distinguishedTriangles),\n                    Nonempty (Octahedron ⋯ h₁₂ h₂₃ h₁₃)\nX₁' X₂' X₃' Z₁₂' Z₂₃' Z₁₃' : C\nu₁₂' : X₁' ⟶ X₂'\nu₂₃' : X₂' ⟶ X₃'\nu₁₃' : X₁' ⟶ X₃'\ncomm' : u₁₂' ≫ u₂₃' = u₁₃'\nv₁₂' : X₂' ⟶ Z₁₂'\nw₁₂' : Z₁₂' ⟶ (shiftFunctor C 1).obj X₁'\nh₁₂' : Triangle.mk u₁₂' v₁₂' w₁₂' ∈ distinguishedTriangles\nv₂₃' : X₃' ⟶ Z₂₃'\nw₂₃' : Z₂₃' ⟶ (shiftFunctor C 1).obj X₂'\nh₂₃' : Triangle.mk u₂₃' v₂₃' w₂₃' ∈ distinguishedTriangles\nv₁₃' : X₃' ⟶ Z₁₃'\nw₁₃' : Z₁₃' ⟶ (shiftFunctor C 1).obj X₁'\nh₁₃' : Triangle.mk u₁₃' v₁₃' w₁₃' ∈ distinguishedTriangles\n⊢ Nonempty (Octahedron comm' h₁₂' h₂₃' h₁₃')","state_after":"case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nC : Type u_1\ninst✝⁵ : Category.{u_2, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nh :\n  ∀ ⦃X₁' X₂' X₃' : C⦄ (u₁₂' : X₁' ⟶ X₂') (u₂₃' : X₂' ⟶ X₃'),\n    ∃ X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ u₁₂ u₂₃ e₁ e₂ e₃,\n      ∃ (_ : u₁₂' ≫ e₂.hom = e₁.hom ≫ u₁₂) (_ : u₂₃' ≫ e₃.hom = e₂.hom ≫ u₂₃),\n        ∃ v₁₂ w₁₂,\n          ∃ (h₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ∈ distinguishedTriangles),\n            ∃ v₂₃ w₂₃,\n              ∃ (h₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ∈ distinguishedTriangles),\n                ∃ v₁₃ w₁₃,\n                  ∃ (h₁₃ : Triangle.mk (u₁₂ ≫ u₂₃) v₁₃ w₁₃ ∈ distinguishedTriangles),\n                    Nonempty (Octahedron ⋯ h₁₂ h₂₃ h₁₃)\nX₁' X₂' X₃' Z₁₂' Z₂₃' Z₁₃' : C\nu₁₂' : X₁' ⟶ X₂'\nu₂₃' : X₂' ⟶ X₃'\nu₁₃' : X₁' ⟶ X₃'\ncomm' : u₁₂' ≫ u₂₃' = u₁₃'\nv₁₂' : X₂' ⟶ Z₁₂'\nw₁₂' : Z₁₂' ⟶ (shiftFunctor C 1).obj X₁'\nh₁₂' : Triangle.mk u₁₂' v₁₂' w₁₂' ∈ distinguishedTriangles\nv₂₃' : X₃' ⟶ Z₂₃'\nw₂₃' : Z₂₃' ⟶ (shiftFunctor C 1).obj X₂'\nh₂₃' : Triangle.mk u₂₃' v₂₃' w₂₃' ∈ distinguishedTriangles\nv₁₃' : X₃' ⟶ Z₁₃'\nw₁₃' : Z₁₃' ⟶ (shiftFunctor C 1).obj X₁'\nh₁₃' : Triangle.mk u₁₃' v₁₃' w₁₃' ∈ distinguishedTriangles\nX₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C\nu₁₂ : X₁ ⟶ X₂\nu₂₃ : X₂ ⟶ X₃\ne₁ : X₁' ≅ X₁\ne₂ : X₂' ≅ X₂\ne₃ : X₃' ≅ X₃\ncomm₁₂ : u₁₂' ≫ e₂.hom = e₁.hom ≫ u₁₂\ncomm₂₃ : u₂₃' ≫ e₃.hom = e₂.hom ≫ u₂₃\nv₁₂ : X₂ ⟶ Z₁₂\nw₁₂ : Z₁₂ ⟶ (shiftFunctor C 1).obj X₁\nh₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ∈ distinguishedTriangles\nv₂₃ : X₃ ⟶ Z₂₃\nw₂₃ : Z₂₃ ⟶ (shiftFunctor C 1).obj X₂\nh₂₃ : Triangle.mk u₂₃ v₂₃ w��₃ ∈ distinguishedTriangles\nv₁₃ : X₃ ⟶ Z₁₃\nw₁₃ : Z₁₃ ⟶ (shiftFunctor C 1).obj X₁\nh₁₃ : Triangle.mk (u₁₂ ≫ u₂₃) v₁₃ w₁₃ ∈ distinguishedTriangles\nH : Nonempty (Octahedron ⋯ h₁₂ h₂₃ h₁₃)\n⊢ Nonempty (Octahedron comm' h₁₂' h₂₃' h₁₃')","tactic":"obtain ⟨X₁, X₂, X₃, Z₁₂, Z₂₃, Z₁₃, u₁₂, u₂₃, e₁, e₂, e₃, comm₁₂, comm₂₃,\n      v₁₂, w₁₂, h₁₂, v₂₃, w₂₃, h₂₃, v₁₃, w₁₃, h₁₃, H⟩ := h u₁₂' u₂₃'","premises":[]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nC : Type u_1\ninst✝⁵ : Category.{u_2, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nh :\n  ∀ ⦃X₁' X₂' X₃' : C⦄ (u₁₂' : X₁' ⟶ X₂') (u₂₃' : X₂' ⟶ X₃'),\n    ∃ X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ u₁₂ u₂₃ e₁ e₂ e₃,\n      ∃ (_ : u₁₂' ≫ e₂.hom = e₁.hom ≫ u₁₂) (_ : u₂₃' ≫ e₃.hom = e₂.hom ≫ u₂₃),\n        ∃ v₁₂ w₁₂,\n          ∃ (h₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ∈ distinguishedTriangles),\n            ∃ v₂₃ w₂₃,\n              ∃ (h₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ∈ distinguishedTriangles),\n                ∃ v₁₃ w₁₃,\n                  ∃ (h₁₃ : Triangle.mk (u₁₂ ≫ u₂₃) v₁₃ w₁₃ ∈ distinguishedTriangles),\n                    Nonempty (Octahedron ⋯ h₁₂ h₂₃ h₁₃)\nX₁' X₂' X₃' Z₁₂' Z₂₃' Z₁₃' : C\nu₁₂' : X₁' ⟶ X₂'\nu₂₃' : X₂' ⟶ X₃'\nu₁₃' : X₁' ⟶ X₃'\ncomm' : u₁₂' ≫ u₂₃' = u₁₃'\nv₁₂' : X₂' ⟶ Z₁₂'\nw₁₂' : Z₁₂' ⟶ (shiftFunctor C 1).obj X₁'\nh₁₂' : Triangle.mk u₁₂' v₁₂' w₁₂' ∈ distinguishedTriangles\nv₂₃' : X₃' ⟶ Z₂₃'\nw₂₃' : Z₂₃' ⟶ (shiftFunctor C 1).obj X₂'\nh₂₃' : Triangle.mk u₂₃' v₂₃' w₂₃' ∈ distinguishedTriangles\nv₁₃' : X₃' ⟶ Z₁₃'\nw₁₃' : Z₁₃' ⟶ (shiftFunctor C 1).obj X₁'\nh₁₃' : Triangle.mk u₁₃' v₁₃' w₁₃' ∈ distinguishedTriangles\nX₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C\nu₁₂ : X₁ ⟶ X₂\nu₂₃ : X₂ ⟶ X₃\ne₁ : X₁' ≅ X₁\ne₂ : X₂' ≅ X₂\ne₃ : X₃' ≅ X₃\ncomm₁₂ : u₁₂' ≫ e₂.hom = e₁.hom ≫ u₁₂\ncomm₂₃ : u₂₃' ≫ e₃.hom = e₂.hom ≫ u₂₃\nv₁₂ : X₂ ⟶ Z₁₂\nw₁₂ : Z₁₂ ⟶ (shiftFunctor C 1).obj X₁\nh₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ∈ distinguishedTriangles\nv₂₃ : X₃ ⟶ Z₂₃\nw₂₃ : Z₂₃ ⟶ (shiftFunctor C 1).obj X₂\nh₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ∈ distinguishedTriangles\nv₁₃ : X₃ ⟶ Z₁₃\nw₁₃ : Z₁₃ ⟶ (shiftFunctor C 1).obj X₁\nh₁₃ : Triangle.mk (u₁₂ ≫ u₂₃) v₁₃ w₁₃ ∈ distinguishedTriangles\nH : Nonempty (Octahedron ⋯ h₁₂ h₂₃ h₁₃)\n⊢ Nonempty (Octahedron comm' h₁₂' h₂₃' h₁₃')","state_after":"no goals","tactic":"exact ⟨Octahedron.ofIso u₁₂' u₂₃' u₁₃' comm' h₁₂' h₂₃' h₁₃'\n      u₁₂ u₂₃ _ rfl e₁ e₂ e₃ comm₁₂ comm₂₃ v₁₂ w₁₂ h₁₂ v₂₃ w₂₃ h₂₃ v₁₃ w₁₃ h₁₃ H.some⟩","premises":[{"full_name":"CategoryTheory.Triangulated.Octahedron.ofIso","def_path":"Mathlib/CategoryTheory/Triangulated/Triangulated.lean","def_pos":[106,4],"def_end_pos":[106,9]},{"full_name":"Nonempty.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[711,4],"def_end_pos":[711,9]},{"full_name":"Nonempty.some","def_path":"Mathlib/Logic/Nonempty.lean","def_pos":[81,31],"def_end_pos":[81,44]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
+{"url":"Mathlib/Order/SuccPred/Basic.lean","commit":"","full_name":"Order.succ_lt_succ_iff","start":[328,0],"end":[328,61],"file_path":"Mathlib/Order/SuccPred/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝² : Preorder α\ninst✝¹ : SuccOrder α\na b : α\ninst✝ : NoMaxOrder α\n⊢ succ a < succ b ↔ a < b","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/Data/Set/Pointwise/Basic.lean","commit":"","full_name":"Set.neg_singleton","start":[227,0],"end":[228,95],"file_path":"Mathlib/Data/Set/Pointwise/Basic.lean","tactics":[{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝ : InvolutiveInv α\ns t : Set α\na✝ a : α\n⊢ {a}⁻¹ = {a⁻¹}","state_after":"no goals","tactic":"rw [← image_inv, image_singleton]","premises":[{"full_name":"Set.image_inv","def_path":"Mathlib/Data/Set/Pointwise/Basic.lean","def_pos":[208,8],"def_end_pos":[208,17]},{"full_name":"Set.image_singleton","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[286,8],"def_end_pos":[286,23]}]}]}
+{"url":"Mathlib/Probability/Kernel/CondDistrib.lean","commit":"","full_name":"ProbabilityTheory.condDistrib_ae_eq_of_measure_eq_compProd","start":[110,0],"end":[122,16],"file_path":"Mathlib/Probability/Kernel/CondDistrib.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nΩ : Type u_3\nF : Type u_4\ninst✝⁵ : MeasurableSpace Ω\ninst✝⁴ : StandardBorelSpace Ω\ninst✝³ : Nonempty Ω\ninst✝² : NormedAddCommGroup F\nmα : MeasurableSpace α\nμ : Measure α\ninst✝¹ : IsFiniteMeasure μ\nX : α → β\nY : α → Ω\nmβ : MeasurableSpace β\ns : Set Ω\nt : Set β\nf : β × Ω → F\nhX : Measurable X\nhY : Measurable Y\nκ : Kernel β Ω\ninst✝ : IsFiniteKernel κ\nhκ : Measure.map (fun x => (X x, Y x)) μ = Measure.map X μ ⊗ₘ κ\n⊢ ∀ᵐ (x : β) ∂Measure.map X μ, κ x = (condDistrib Y X μ) x","state_after":"α : Type u_1\nβ : Type u_2\nΩ : Type u_3\nF : Type u_4\ninst✝⁵ : MeasurableSpace Ω\ninst✝⁴ : StandardBorelSpace Ω\ninst✝³ : Nonempty Ω\ninst✝² : NormedAddCommGroup F\nmα : MeasurableSpace α\nμ : Measure α\ninst✝¹ : IsFiniteMeasure μ\nX : α → β\nY : α → Ω\nmβ : MeasurableSpace β\ns : Set Ω\nt : Set β\nf : β × Ω → F\nhX : Measurable X\nhY : Measurable Y\nκ : Kernel β Ω\ninst✝ : IsFiniteKernel κ\nhκ : Measure.map (fun x => (X x, Y x)) μ = Measure.map X μ ⊗ₘ κ\nheq : Measure.map X μ = (Measure.map (fun x => (X x, Y x)) μ).fst\n⊢ ∀ᵐ (x : β) ∂Measure.map X μ, κ x = (condDistrib Y X μ) x","tactic":"have heq : μ.map X = (μ.map (fun x ↦ (X x, Y x))).fst := by\n    ext s hs\n    rw [Measure.map_apply hX hs, Measure.fst_apply hs, Measure.map_apply]\n    exacts [rfl, Measurable.prod hX hY, measurable_fst hs]","premises":[{"full_name":"Measurable.prod","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Basic.lean","def_pos":[633,8],"def_end_pos":[633,23]},{"full_name":"MeasureTheory.Measure.fst","def_path":"Mathlib/MeasureTheory/Constructions/Prod/Basic.lean","def_pos":[956,18],"def_end_pos":[956,21]},{"full_name":"MeasureTheory.Measure.fst_apply","def_path":"Mathlib/MeasureTheory/Constructions/Prod/Basic.lean","def_pos":[959,8],"def_end_pos":[959,17]},{"full_name":"MeasureTheory.Measure.map","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","def_pos":[1090,16],"def_end_pos":[1090,19]},{"full_name":"MeasureTheory.Measure.map_apply","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","def_pos":[1159,8],"def_end_pos":[1159,17]},{"full_name":"Prod.mk","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[481,2],"def_end_pos":[481,4]},{"full_name":"measurable_fst","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Basic.lean","def_pos":[613,8],"def_end_pos":[613,22]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nΩ : Type u_3\nF : Type u_4\ninst✝⁵ : MeasurableSpace Ω\ninst✝⁴ : StandardBorelSpace Ω\ninst✝³ : Nonempty Ω\ninst✝² : NormedAddCommGroup F\nmα : MeasurableSpace α\nμ : Measure α\ninst✝¹ : IsFiniteMeasure μ\nX : α → β\nY : α → Ω\nmβ : MeasurableSpace β\ns : Set Ω\nt : Set β\nf : β × Ω → F\nhX : Measurable X\nhY : Measurable Y\nκ : Kernel β Ω\ninst✝ : IsFiniteKernel κ\nhκ : Measure.map (fun x => (X x, Y x)) μ = Measure.map X μ ⊗ₘ κ\nheq : Measure.map X μ = (Measure.map (fun x => (X x, Y x)) μ).fst\n⊢ ∀ᵐ (x : β) ∂Measure.map X μ, κ x = (condDistrib Y X μ) x","state_after":"α : Type u_1\nβ : Type u_2\nΩ : Type u_3\nF : Type u_4\ninst✝⁵ : MeasurableSpace Ω\ninst✝⁴ : StandardBorelSpace Ω\ninst✝³ : Nonempty Ω\ninst✝² : NormedAddCommGroup F\nmα : MeasurableSpace α\nμ : Measure α\ninst✝¹ : IsFiniteMeasure μ\nX : α → β\nY : α → Ω\nmβ : MeasurableSpace β\ns : Set Ω\nt : Set β\nf : β × Ω → F\nhX : Measurable X\nhY : Measurable Y\nκ : Kernel β Ω\ninst✝ : IsFiniteKernel κ\nhκ : Measure.map (fun x => (X x, Y x)) μ = Measure.map X μ ⊗ₘ κ\nheq : Measure.map X μ = (Measure.map (fun x => (X x, Y x)) μ).fst\n⊢ ∀ᵐ (x : β) ∂(Measure.map (fun x => (X x, Y x)) μ).fst, κ x = (Measure.map (fun a => (X a, Y a)) μ).condKernel x","tactic":"rw [heq, condDistrib]","premises":[{"full_name":"ProbabilityTheory.condDistrib","def_path":"Mathlib/Probability/Kernel/CondDistrib.lean","def_pos":[60,30],"def_end_pos":[60,41]},{"full_name":"ProbabilityTheory.condDistrib_def","def_path":"Mathlib/Probability/Kernel/CondDistrib.lean","def_pos":[60,30],"def_end_pos":[60,41]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nΩ : Type u_3\nF : Type u_4\ninst✝⁵ : MeasurableSpace Ω\ninst✝⁴ : StandardBorelSpace Ω\ninst✝³ : Nonempty Ω\ninst✝² : NormedAddCommGroup F\nmα : MeasurableSpace α\nμ : Measure α\ninst✝¹ : IsFiniteMeasure μ\nX : α → β\nY : α → Ω\nmβ : MeasurableSpace β\ns : Set Ω\nt : Set β\nf : β × Ω → F\nhX : Measurable X\nhY : Measurable Y\nκ : Kernel β Ω\ninst✝ : IsFiniteKernel κ\nhκ : Measure.map (fun x => (X x, Y x)) μ = Measure.map X μ ⊗ₘ κ\nheq : Measure.map X μ = (Measure.map (fun x => (X x, Y x)) μ).fst\n⊢ ∀ᵐ (x : β) ∂(Measure.map (fun x => (X x, Y x)) μ).fst, κ x = (Measure.map (fun a => (X a, Y a)) μ).condKernel x","state_after":"α : Type u_1\nβ : Type u_2\nΩ : Type u_3\nF : Type u_4\ninst✝⁵ : MeasurableSpace Ω\ninst✝⁴ : StandardBorelSpace Ω\ninst✝³ : Nonempty Ω\ninst✝² : NormedAddCommGroup F\nmα : MeasurableSpace α\nμ : Measure α\ninst✝¹ : IsFiniteMeasure μ\nX : α → β\nY : α → Ω\nmβ : MeasurableSpace β\ns : Set Ω\nt : Set β\nf : β × Ω → F\nhX : Measurable X\nhY : Measurable Y\nκ : Kernel β Ω\ninst✝ : IsFiniteKernel κ\nhκ : Measure.map (fun x => (X x, Y x)) μ = Measure.map X μ ⊗ₘ κ\nheq : Measure.map X μ = (Measure.map (fun x => (X x, Y x)) μ).fst\n⊢ Measure.map (fun a => (X a, Y a)) μ = (Measure.map (fun a => (X a, Y a)) μ).fst ⊗ₘ κ","tactic":"refine eq_condKernel_of_measure_eq_compProd _ ?_","premises":[{"full_name":"ProbabilityTheory.eq_condKernel_of_measure_eq_compProd","def_path":"Mathlib/Probability/Kernel/Disintegration/Unique.lean","def_pos":[81,8],"def_end_pos":[81,44]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nΩ : Type u_3\nF : Type u_4\ninst✝⁵ : MeasurableSpace Ω\ninst✝⁴ : StandardBorelSpace Ω\ninst✝³ : Nonempty Ω\ninst✝² : NormedAddCommGroup F\nmα : MeasurableSpace α\nμ : Measure α\ninst✝¹ : IsFiniteMeasure μ\nX : α → β\nY : α → Ω\nmβ : MeasurableSpace β\ns : Set Ω\nt : Set β\nf : β × Ω → F\nhX : Measurable X\nhY : Measurable Y\nκ : Kernel β Ω\ninst✝ : IsFiniteKernel κ\nhκ : Measure.map (fun x => (X x, Y x)) μ = Measure.map X μ ⊗ₘ κ\nheq : Measure.map X μ = (Measure.map (fun x => (X x, Y x)) μ).fst\n⊢ Measure.map (fun a => (X a, Y a)) μ = (Measure.map (fun a => (X a, Y a)) μ).fst ⊗ₘ κ","state_after":"case h.e'_3.h.e'_5\nα : Type u_1\nβ : Type u_2\nΩ : Type u_3\nF : Type u_4\ninst✝⁵ : MeasurableSpace Ω\ninst✝⁴ : StandardBorelSpace Ω\ninst✝³ : Nonempty Ω\ninst✝² : NormedAddCommGroup F\nmα : MeasurableSpace α\nμ : Measure α\ninst✝¹ : IsFiniteMeasure μ\nX : α → β\nY : α → Ω\nmβ : MeasurableSpace β\ns : Set Ω\nt : Set β\nf : β × Ω → F\nhX : Measurable X\nhY : Measurable Y\nκ : Kernel β Ω\ninst✝ : IsFiniteKernel κ\nhκ : Measure.map (fun x => (X x, Y x)) μ = Measure.map X μ ⊗ₘ κ\nheq : Measure.map X μ = (Measure.map (fun x => (X x, Y x)) μ).fst\n⊢ (Measure.map (fun a => (X a, Y a)) μ).fst = Measure.map X μ","tactic":"convert hκ","premises":[]},{"state_before":"case h.e'_3.h.e'_5\nα : Type u_1\nβ : Type u_2\nΩ : Type u_3\nF : Type u_4\ninst✝⁵ : MeasurableSpace Ω\ninst✝⁴ : StandardBorelSpace Ω\ninst✝³ : Nonempty Ω\ninst✝² : NormedAddCommGroup F\nmα : MeasurableSpace α\nμ : Measure α\ninst✝¹ : IsFiniteMeasure μ\nX : α → β\nY : α → Ω\nmβ : MeasurableSpace β\ns : Set Ω\nt : Set β\nf : β × Ω → F\nhX : Measurable X\nhY : Measurable Y\nκ : Kernel β Ω\ninst✝ : IsFiniteKernel κ\nhκ : Measure.map (fun x => (X x, Y x)) μ = Measure.map X μ ⊗ₘ κ\nheq : Measure.map X μ = (Measure.map (fun x => (X x, Y x)) μ).fst\n⊢ (Measure.map (fun a => (X a, Y a)) μ).fst = Measure.map X μ","state_after":"no goals","tactic":"exact heq.symm","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]}]}]}
+{"url":"Mathlib/LinearAlgebra/Dual.lean","commit":"","full_name":"LinearEquiv.dualMap_refl","start":[227,0],"end":[231,5],"file_path":"Mathlib/LinearAlgebra/Dual.lean","tactics":[{"state_before":"R : Type u\ninst✝⁴ : CommSemiring R\nM₁ : Type v\nM₂ : Type v'\ninst✝³ : AddCommMonoid M₁\ninst✝² : Module R M₁\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R M₂\n⊢ (refl R M₁).dualMap = refl R (Dual R M₁)","state_after":"case h.h\nR : Type u\ninst✝⁴ : CommSemiring R\nM₁ : Type v\nM₂ : Type v'\ninst✝³ : AddCommMonoid M₁\ninst✝² : Module R M₁\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R M₂\nx✝¹ : Dual R M₁\nx✝ : M₁\n⊢ ((refl R M₁).dualMap x✝¹) x✝ = ((refl R (Dual R M₁)) x✝¹) x✝","tactic":"ext","premises":[]},{"state_before":"case h.h\nR : Type u\ninst✝⁴ : CommSemiring R\nM₁ : Type v\nM₂ : Type v'\ninst✝³ : AddCommMonoid M₁\ninst✝² : Module R M₁\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R M₂\nx✝¹ : Dual R M₁\nx✝ : M₁\n⊢ ((refl R M₁).dualMap x✝¹) x✝ = ((refl R (Dual R M₁)) x✝¹) x✝","state_after":"no goals","tactic":"rfl","premises":[]}]}
+{"url":"Mathlib/RepresentationTheory/Basic.lean","commit":"","full_name":"Representation.ofMulAction_self_smul_eq_mul","start":[325,0],"end":[340,7],"file_path":"Mathlib/RepresentationTheory/Basic.lean","tactics":[{"state_before":"k : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝³ : CommSemiring k\ninst✝² : Group G\ninst✝¹ : AddCommMonoid V\ninst✝ : Module k V\nρ : Representation k G V\nx : MonoidAlgebra k G\ny : (ofMulAction k G G).asModule\ng : G\n⊢ (fun z => z • y = z * y) ((of k G) g)","state_after":"k : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝³ : CommSemiring k\ninst✝² : Group G\ninst✝¹ : AddCommMonoid V\ninst✝ : Module k V\nρ : Representation k G V\nx : MonoidAlgebra k G\ny : (ofMulAction k G G).asModule\ng : G\n⊢ ((ofMulAction k G G).asAlgebraHom ((of k G) g)) y = (of k G) g * y","tactic":"show asAlgebraHom (ofMulAction k G G) _ _ = _","premises":[{"full_name":"Representation.asAlgebraHom","def_path":"Mathlib/RepresentationTheory/Basic.lean","def_pos":[91,18],"def_end_pos":[91,30]},{"full_name":"Representation.ofMulAction","def_path":"Mathlib/RepresentationTheory/Basic.lean","def_pos":[251,18],"def_end_pos":[251,29]}]},{"state_before":"k : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝³ : CommSemiring k\ninst✝² : Group G\ninst✝¹ : AddCommMonoid V\ninst✝ : Module k V\nρ : Representation k G V\nx : MonoidAlgebra k G\ny : (ofMulAction k G G).asModule\ng : G\n⊢ ((ofMulAction k G G).asAlgebraHom ((of k G) g)) y = (of k G) g * y","state_after":"case h\nk : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝³ : CommSemiring k\ninst✝² : Group G\ninst✝¹ : AddCommMonoid V\ninst✝ : Module k V\nρ : Representation k G V\nx : MonoidAlgebra k G\ny : (ofMulAction k G G).asModule\ng a✝ : G\n⊢ (((ofMulAction k G G).asAlgebraHom ((of k G) g)) y) a✝ = ((of k G) g * y) a✝","tactic":"ext","premises":[]},{"state_before":"case h\nk : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝³ : CommSemiring k\ninst✝² : Group G\ninst✝¹ : AddCommMonoid V\ninst✝ : Module k V\nρ : Representation k G V\nx : MonoidAlgebra k G\ny : (ofMulAction k G G).asModule\ng a✝ : G\n⊢ (((ofMulAction k G G).asAlgebraHom ((of k G) g)) y) a✝ = ((of k G) g * y) a✝","state_after":"case h\nk : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝³ : CommSemiring k\ninst✝² : Group G\ninst✝¹ : AddCommMonoid V\ninst✝ : Module k V\nρ : Representation k G V\nx : MonoidAlgebra k G\ny : (ofMulAction k G G).asModule\ng a✝ : G\n⊢ y (g⁻¹ * a✝) = (MonoidAlgebra.single g 1 * y) a✝","tactic":"simp only [MonoidAlgebra.of_apply, asAlgebraHom_single, one_smul,\n        ofMulAction_apply, smul_eq_mul]","premises":[{"full_name":"MonoidAlgebra.of_apply","def_path":"Mathlib/Algebra/MonoidAlgebra/Basic.lean","def_pos":[472,2],"def_end_pos":[472,7]},{"full_name":"Representation.asAlgebraHom_single","def_path":"Mathlib/RepresentationTheory/Basic.lean","def_pos":[98,8],"def_end_pos":[98,27]},{"full_name":"Representation.ofMulAction_apply","def_path":"Mathlib/RepresentationTheory/Basic.lean","def_pos":[310,8],"def_end_pos":[310,25]},{"full_name":"one_smul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[379,6],"def_end_pos":[379,14]},{"full_name":"smul_eq_mul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[79,6],"def_end_pos":[79,17]}]},{"state_before":"case h\nk : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝³ : CommSemiring k\ninst✝² : Group G\ninst✝¹ : AddCommMonoid V\ninst✝ : Module k V\nρ : Representation k G V\nx : MonoidAlgebra k G\ny : (ofMulAction k G G).asModule\ng a✝ : G\n⊢ y (g⁻¹ * a✝) = (MonoidAlgebra.single g 1 * y) a✝","state_after":"no goals","tactic":"rw [MonoidAlgebra.single_mul_apply, one_mul]","premises":[{"full_name":"MonoidAlgebra.single_mul_apply","def_path":"Mathlib/Algebra/MonoidAlgebra/Basic.lean","def_pos":[1006,8],"def_end_pos":[1006,24]},{"full_name":"one_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[409,8],"def_end_pos":[409,15]}]},{"state_before":"k : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝³ : CommSemiring k\ninst✝² : Group G\ninst✝¹ : AddCommMonoid V\ninst✝ : Module k V\nρ : Representation k G V\nx✝ : MonoidAlgebra k G\ny✝ : (ofMulAction k G G).asModule\nx y : MonoidAlgebra k G\nhx : (fun z => z • y✝ = z * y✝) x\nhy : (fun z => z • y✝ = z * y✝) y\n⊢ (fun z => z • y✝ = z * y✝) (x + y)","state_after":"no goals","tactic":"simp only [hx, hy, add_mul, add_smul]","premises":[{"full_name":"add_smul","def_path":"Mathlib/Algebra/Module/Defs.lean","def_pos":[87,8],"def_end_pos":[87,16]}]},{"state_before":"k : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝³ : CommSemiring k\ninst✝² : Group G\ninst✝¹ : AddCommMonoid V\ninst✝ : Module k V\nρ : Representation k G V\nx✝ : MonoidAlgebra k G\ny : (ofMulAction k G G).asModule\nr : k\nx : MonoidAlgebra k G\nhx : (fun z => z • y = z * y) x\n⊢ (fun z => z • y = z * y) (r • x)","state_after":"k : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝³ : CommSemiring k\ninst✝² : Group G\ninst✝¹ : AddCommMonoid V\ninst✝ : Module k V\nρ : Representation k G V\nx✝ : MonoidAlgebra k G\ny : (ofMulAction k G G).asModule\nr : k\nx : MonoidAlgebra k G\nhx : (fun z => z • y = z * y) x\n⊢ ((ofMulAction k G G).asAlgebraHom (r • x)) y = r • x * y","tactic":"show asAlgebraHom (ofMulAction k G G) _ _ = _","premises":[{"full_name":"Representation.asAlgebraHom","def_path":"Mathlib/RepresentationTheory/Basic.lean","def_pos":[91,18],"def_end_pos":[91,30]},{"full_name":"Representation.ofMulAction","def_path":"Mathlib/RepresentationTheory/Basic.lean","def_pos":[251,18],"def_end_pos":[251,29]}]},{"state_before":"k : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝³ : CommSemiring k\ninst✝² : Group G\ninst✝¹ : AddCommMonoid V\ninst✝ : Module k V\nρ : Representation k G V\nx✝ : MonoidAlgebra k G\ny : (ofMulAction k G G).asModule\nr : k\nx : MonoidAlgebra k G\nhx : (fun z => z • y = z * y) x\n⊢ ((ofMulAction k G G).asAlgebraHom (r • x)) y = r • x * y","state_after":"k : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝³ : CommSemiring k\ninst✝² : Group G\ninst✝¹ : AddCommMonoid V\ninst✝ : Module k V\nρ : Representation k G V\nx✝ : MonoidAlgebra k G\ny : (ofMulAction k G G).asModule\nr : k\nx : MonoidAlgebra k G\nhx : (fun z => z • y = z * y) x\n⊢ r • ((ofMulAction k G G).asAlgebraHom x) y = r • (x * y)","tactic":"simp only [map_smul, smul_apply, Algebra.smul_mul_assoc]","premises":[{"full_name":"Algebra.smul_mul_assoc","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[320,18],"def_end_pos":[320,32]},{"full_name":"LinearMap.smul_apply","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[748,8],"def_end_pos":[748,18]},{"full_name":"map_smul","def_path":"Mathlib/GroupTheory/GroupAction/Hom.lean","def_pos":[108,8],"def_end_pos":[108,16]}]},{"state_before":"k : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝³ : CommSemiring k\ninst✝² : Group G\ninst✝¹ : AddCommMonoid V\ninst✝ : Module k V\nρ : Representation k G V\nx✝ : MonoidAlgebra k G\ny : (ofMulAction k G G).asModule\nr : k\nx : MonoidAlgebra k G\nhx : (fun z => z • y = z * y) x\n⊢ r • ((ofMulAction k G G).asAlgebraHom x) y = r • (x * y)","state_after":"k : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝³ : CommSemiring k\ninst✝² : Group G\ninst✝¹ : AddCommMonoid V\ninst✝ : Module k V\nρ : Representation k G V\nx✝ : MonoidAlgebra k G\ny : (ofMulAction k G G).asModule\nr : k\nx : MonoidAlgebra k G\nhx : (fun z => z • y = z * y) x\n⊢ r • ((ofMulAction k G G).asAlgebraHom x) y = r • x • y","tactic":"rw [← hx]","premises":[]},{"state_before":"k : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝³ : CommSemiring k\ninst✝² : Group G\ninst✝¹ : AddCommMonoid V\ninst✝ : Module k V\nρ : Representation k G V\nx✝ : MonoidAlgebra k G\ny : (ofMulAction k G G).asModule\nr : k\nx : MonoidAlgebra k G\nhx : (fun z => z • y = z * y) x\n⊢ r • ((ofMulAction k G G).asAlgebraHom x) y = r • x • y","state_after":"no goals","tactic":"rfl","premises":[]}]}
+{"url":"Mathlib/RingTheory/TwoSidedIdeal/Lattice.lean","commit":"","full_name":"TwoSidedIdeal.iSup_ringCon","start":[81,0],"end":[83,50],"file_path":"Mathlib/RingTheory/TwoSidedIdeal/Lattice.lean","tactics":[{"state_before":"R : Type u_1\ninst✝ : NonUnitalNonAssocRing R\nι : Type u_2\nI : ι → TwoSidedIdeal R\n⊢ (⨆ i, I i).ringCon = ⨆ i, (I i).ringCon","state_after":"R : Type u_1\ninst✝ : NonUnitalNonAssocRing R\nι : Type u_2\nI : ι → TwoSidedIdeal R\n⊢ sSup (ringCon '' Set.range fun i => I i) = sSup (Set.range fun i => (I i).ringCon)","tactic":"simp only [iSup, sSup_ringCon]","premises":[{"full_name":"TwoSidedIdeal.sSup_ringCon","def_path":"Mathlib/RingTheory/TwoSidedIdeal/Lattice.lean","def_pos":[78,6],"def_end_pos":[78,18]},{"full_name":"iSup","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[56,4],"def_end_pos":[56,8]}]},{"state_before":"R : Type u_1\ninst✝ : NonUnitalNonAssocRing R\nι : Type u_2\nI : ι → TwoSidedIdeal R\n⊢ sSup (ringCon '' Set.range fun i => I i) = sSup (Set.range fun i => (I i).ringCon)","state_after":"case e_a\nR : Type u_1\ninst✝ : NonUnitalNonAssocRing R\nι : Type u_2\nI : ι → TwoSidedIdeal R\n⊢ (ringCon '' Set.range fun i => I i) = Set.range fun i => (I i).ringCon","tactic":"congr","premises":[]},{"state_before":"case e_a\nR : Type u_1\ninst✝ : NonUnitalNonAssocRing R\nι : Type u_2\nI : ι → TwoSidedIdeal R\n⊢ (ringCon '' Set.range fun i => I i) = Set.range fun i => (I i).ringCon","state_after":"case e_a.h\nR : Type u_1\ninst✝ : NonUnitalNonAssocRing R\nι : Type u_2\nI : ι → TwoSidedIdeal R\nx✝ : RingCon R\n⊢ (x✝ ∈ ringCon '' Set.range fun i => I i) ↔ x✝ ∈ Set.range fun i => (I i).ringCon","tactic":"ext","premises":[]},{"state_before":"case e_a.h\nR : Type u_1\ninst✝ : NonUnitalNonAssocRing R\nι : Type u_2\nI : ι → TwoSidedIdeal R\nx✝ : RingCon R\n⊢ (x✝ ∈ ringCon '' Set.range fun i => I i) ↔ x✝ ∈ Set.range fun i => (I i).ringCon","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/Algebra/Polynomial/Roots.lean","commit":"","full_name":"Polynomial.roots_prod_X_sub_C","start":[225,0],"end":[229,62],"file_path":"Mathlib/Algebra/Polynomial/Roots.lean","tactics":[{"state_before":"R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np q : R[X]\ns : Finset R\n⊢ (∏ a ∈ s, (X - C a)).roots = s.val","state_after":"R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\ninst�� : IsDomain R\np q : R[X]\ns : Finset R\n⊢ (s.val.bind fun i => (X - C i).roots) = s.val\n\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np q : R[X]\ns : Finset R\n⊢ ∏ a ∈ s, (X - C a) ≠ 0","tactic":"apply (roots_prod (fun a => X - C a) s ?_).trans","premises":[{"full_name":"Eq.trans","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[335,8],"def_end_pos":[335,16]},{"full_name":"Polynomial.C","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[426,4],"def_end_pos":[426,5]},{"full_name":"Polynomial.X","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[474,4],"def_end_pos":[474,5]},{"full_name":"Polynomial.roots_prod","def_path":"Mathlib/Algebra/Polynomial/Roots.lean","def_pos":[201,8],"def_end_pos":[201,18]}]}]}
+{"url":"Mathlib/Analysis/Calculus/FDeriv/Analytic.lean","commit":"","full_name":"ContinuousMultilinearMap.hasFDerivAt","start":[327,0],"end":[330,15],"file_path":"Mathlib/Analysis/Calculus/FDeriv/Analytic.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE✝ : Type u\ninst✝⁷ : NormedAddCommGroup E✝\ninst✝⁶ : NormedSpace 𝕜 E✝\nF : Type v\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nι : Type u_2\nE : ι → Type u_3\ninst✝³ : (i : ι) → NormedAddCommGroup (E i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹ : Fintype ι\nf : ContinuousMultilinearMap 𝕜 E F\nn : ℕ∞\nx : (i : ι) → E i\ninst✝ : DecidableEq ι\n⊢ HasFDerivAt (⇑f) (f.linearDeriv x) x","state_after":"𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE✝ : Type u\ninst✝⁷ : NormedAddCommGroup E✝\ninst✝⁶ : NormedSpace 𝕜 E✝\nF : Type v\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nι : Type u_2\nE : ι → Type u_3\ninst✝³ : (i : ι) → NormedAddCommGroup (E i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹ : Fintype ι\nf : ContinuousMultilinearMap 𝕜 E F\nn : ℕ∞\nx : (i : ι) → E i\ninst✝ : DecidableEq ι\n⊢ HasFDerivAt (⇑f) ((continuousMultilinearCurryFin1 𝕜 ((i : ι) → E i) F) (f.toFormalMultilinearSeries.changeOrigin x 1))\n    x","tactic":"rw [← changeOrigin_toFormalMultilinearSeries]","premises":[{"full_name":"ContinuousMultilinearMap.changeOrigin_toFormalMultilinearSeries","def_path":"Mathlib/Analysis/Calculus/FDeriv/Analytic.lean","def_pos":[293,8],"def_end_pos":[293,46]}]},{"state_before":"𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE✝ : Type u\ninst✝⁷ : NormedAddCommGroup E✝\ninst✝⁶ : NormedSpace 𝕜 E✝\nF : Type v\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nι : Type u_2\nE : ι → Type u_3\ninst✝³ : (i : ι) → NormedAddCommGroup (E i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹ : Fintype ι\nf : ContinuousMultilinearMap 𝕜 E F\nn : ℕ∞\nx : (i : ι) → E i\ninst✝ : DecidableEq ι\n⊢ HasFDerivAt (⇑f) ((continuousMultilinearCurryFin1 𝕜 ((i : ι) → E i) F) (f.toFormalMultilinearSeries.changeOrigin x 1))\n    x","state_after":"case h.e'_11\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE✝ : Type u\ninst✝⁷ : NormedAddCommGroup E✝\ninst✝⁶ : NormedSpace 𝕜 E✝\nF : Type v\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nι : Type u_2\nE : ι → Type u_3\ninst✝³ : (i : ι) → NormedAddCommGroup (E i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹ : Fintype ι\nf : ContinuousMultilinearMap 𝕜 E F\nn : ℕ∞\nx : (i : ι) → E i\ninst✝ : DecidableEq ι\n⊢ x = 0 + x","tactic":"convert f.hasFiniteFPowerSeriesOnBall.hasFDerivAt (y := x) ENNReal.coe_lt_top","premises":[{"full_name":"ContinuousMultilinearMap.hasFiniteFPowerSeriesOnBall","def_path":"Mathlib/Analysis/Calculus/FDeriv/Analytic.lean","def_pos":[277,18],"def_end_pos":[277,45]},{"full_name":"ENNReal.coe_lt_top","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[284,16],"def_end_pos":[284,26]},{"full_name":"HasFiniteFPowerSeriesOnBall.hasFDerivAt","def_path":"Mathlib/Analysis/Calculus/FDeriv/Analytic.lean","def_pos":[171,8],"def_end_pos":[171,47]}]},{"state_before":"case h.e'_11\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE✝ : Type u\ninst✝⁷ : NormedAddCommGroup E✝\ninst✝⁶ : NormedSpace 𝕜 E✝\nF : Type v\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nι : Type u_2\nE : ι → Type u_3\ninst✝³ : (i : ι) → NormedAddCommGroup (E i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹ : Fintype ι\nf : ContinuousMultilinearMap 𝕜 E F\nn : ℕ∞\nx : (i : ι) → E i\ninst✝ : DecidableEq ι\n⊢ x = 0 + x","state_after":"no goals","tactic":"rw [zero_add]","premises":[{"full_name":"zero_add","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[408,2],"def_end_pos":[408,13]}]}]}
+{"url":"Mathlib/Order/SuccPred/Limit.lean","commit":"","full_name":"SuccOrder.limitRecOn_succ","start":[180,0],"end":[189,64],"file_path":"Mathlib/Order/SuccPred/Limit.lean","tactics":[{"state_before":"α : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\na b : α\nC : α → Sort u_2\ninst✝ : WellFoundedLT α\nH_succ : (a : α) → ¬IsMax a → C a → C (succ a)\nH_lim : (a : α) → IsSuccLimit a → ((b : α) → b < a → C b) → C a\nha : ¬IsMax a\n⊢ SuccOrder.limitRecOn (succ a) H_succ H_lim = H_succ a ha (SuccOrder.limitRecOn a H_succ H_lim)","state_after":"α : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\na b : α\nC : α → Sort u_2\ninst✝ : WellFoundedLT α\nH_succ : (a : α) → ¬IsMax a → C a → C (succ a)\nH_lim : (a : α) → IsSuccLimit a → ((b : α) → b < a → C b) → C a\nha : ¬IsMax a\nh : ¬IsSuccLimit (succ a)\n⊢ SuccOrder.limitRecOn (succ a) H_succ H_lim = H_succ a ha (SuccOrder.limitRecOn a H_succ H_lim)","tactic":"have h := not_isSuccLimit_succ_of_not_isMax ha","premises":[{"full_name":"Order.not_isSuccLimit_succ_of_not_isMax","def_path":"Mathlib/Order/SuccPred/Limit.lean","def_pos":[67,8],"def_end_pos":[67,41]}]},{"state_before":"α : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\na b : α\nC : α → Sort u_2\ninst✝ : WellFoundedLT α\nH_succ : (a : α) → ¬IsMax a → C a → C (succ a)\nH_lim : (a : α) → IsSuccLimit a → ((b : α) → b < a → C b) → C a\nha : ¬IsMax a\nh : ¬IsSuccLimit (succ a)\n⊢ SuccOrder.limitRecOn (succ a) H_succ H_lim = H_succ a ha (SuccOrder.limitRecOn a H_succ H_lim)","state_after":"α : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\na b : α\nC : α → Sort u_2\ninst✝ : WellFoundedLT α\nH_succ : (a : α) → ¬IsMax a → C a → C (succ a)\nH_lim : (a : α) → IsSuccLimit a → ((b : α) → b < a → C b) → C a\nha : ¬IsMax a\nh : ¬IsSuccLimit (succ a)\n⊢ (let x := Classical.indefiniteDescription (fun x => ¬IsMax x ∧ succ x = succ a) ⋯;\n    ⋯ ▸\n      H_succ ↑x ⋯\n        ((fun y x =>\n            ⋯.fix\n              (fun a IH =>\n                if h : IsSuccLimit a then H_lim a h IH\n                else\n                  let x := Classical.indefiniteDescription (fun x => ¬IsMax x ∧ succ x = a) ⋯;\n                  ⋯ ▸ H_succ ↑x ⋯ (IH ↑x ⋯))\n              y)\n          ↑x ⋯)) =\n    H_succ a ha (SuccOrder.limitRecOn a H_succ H_lim)","tactic":"rw [SuccOrder.limitRecOn, WellFounded.fix_eq, dif_neg h]","premises":[{"full_name":"SuccOrder.limitRecOn","def_path":"Mathlib/Order/SuccPred/Limit.lean","def_pos":[173,34],"def_end_pos":[173,61]},{"full_name":"WellFounded.fix_eq","def_path":".lake/packages/lean4/src/lean/Init/WF.lean","def_pos":[94,8],"def_end_pos":[94,14]},{"full_name":"dif_neg","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[954,8],"def_end_pos":[954,15]}]},{"state_before":"α : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\na b : α\nC : α → Sort u_2\ninst✝ : WellFoundedLT α\nH_succ : (a : α) → ¬IsMax a → C a → C (succ a)\nH_lim : (a : α) → IsSuccLimit a → ((b : α) → b < a → C b) → C a\nha : ¬IsMax a\nh : ¬IsSuccLimit (succ a)\n⊢ (let x := Classical.indefiniteDescription (fun x => ¬IsMax x ∧ succ x = succ a) ⋯;\n    ⋯ ▸\n      H_succ ↑x ⋯\n        ((fun y x =>\n            ⋯.fix\n              (fun a IH =>\n                if h : IsSuccLimit a then H_lim a h IH\n                else\n                  let x := Classical.indefiniteDescription (fun x => ¬IsMax x ∧ succ x = a) ⋯;\n                  ⋯ ▸ H_succ ↑x ⋯ (IH ↑x ⋯))\n              y)\n          ↑x ⋯)) =\n    H_succ a ha (SuccOrder.limitRecOn a H_succ H_lim)","state_after":"α : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\na b : α\nC : α → Sort u_2\ninst✝ : WellFoundedLT α\nH_succ : (a : α) → ¬IsMax a → C a → C (succ a)\nH_lim : (a : α) → IsSuccLimit a → ((b : α) → b < a → C b) → C a\nha : ¬IsMax a\nh : ¬IsSuccLimit (succ a)\nthis :\n  ∀ {b c : α} {hb : ¬IsMax b} {hc : ¬IsMax c} {x : (a : α) → C a} (h : b = c), ⋯ ▸ H_succ b hb (x b) = H_succ c hc (x c)\n⊢ (let x := Classical.indefiniteDescription (fun x => ¬IsMax x ∧ succ x = succ a) ⋯;\n    ⋯ ▸\n      H_succ ↑x ⋯\n        ((fun y x =>\n            ⋯.fix\n              (fun a IH =>\n                if h : IsSuccLimit a then H_lim a h IH\n                else\n                  let x := Classical.indefiniteDescription (fun x => ¬IsMax x ∧ succ x = a) ⋯;\n                  ⋯ ▸ H_succ ↑x ⋯ (IH ↑x ⋯))\n              y)\n          ↑x ⋯)) =\n    H_succ a ha (SuccOrder.limitRecOn a H_succ H_lim)","tactic":"have {b c hb hc} {x : ∀ a, C a} (h : b = c) :\n    congr_arg succ h ▸ H_succ b hb (x b) = H_succ c hc (x c) := by subst h; rfl","premises":[{"full_name":"Order.succ","def_path":"Mathlib/Order/SuccPred/Basic.lean","def_pos":[203,4],"def_end_pos":[203,8]}]},{"state_before":"α : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\na b : α\nC : α → Sort u_2\ninst✝ : WellFoundedLT α\nH_succ : (a : α) → ¬IsMax a → C a → C (succ a)\nH_lim : (a : α) → IsSuccLimit a → ((b : α) → b < a → C b) → C a\nha : ¬IsMax a\nh : ¬IsSuccLimit (succ a)\nthis :\n  ∀ {b c : α} {hb : ¬IsMax b} {hc : ¬IsMax c} {x : (a : α) → C a} (h : b = c), ⋯ ▸ H_succ b hb (x b) = H_succ c hc (x c)\n⊢ (let x := Classical.indefiniteDescription (fun x => ¬IsMax x ∧ succ x = succ a) ⋯;\n    ⋯ ▸\n      H_succ ↑x ⋯\n        ((fun y x =>\n            ⋯.fix\n              (fun a IH =>\n                if h : IsSuccLimit a then H_lim a h IH\n                else\n                  let x := Classical.indefiniteDescription (fun x => ¬IsMax x ∧ succ x = a) ⋯;\n                  ⋯ ▸ H_succ ↑x ⋯ (IH ↑x ⋯))\n              y)\n          ↑x ⋯)) =\n    H_succ a ha (SuccOrder.limitRecOn a H_succ H_lim)","state_after":"α : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\na b : α\nC : α → Sort u_2\ninst✝ : WellFoundedLT α\nH_succ : (a : α) → ¬IsMax a → C a → C (succ a)\nH_lim : (a : α) → IsSuccLimit a → ((b : α) → b < a → C b) → C a\nha : ¬IsMax a\nh : ¬IsSuccLimit (succ a)\nthis :\n  ∀ {b c : α} {hb : ¬IsMax b} {hc : ¬IsMax c} {x : (a : α) → C a} (h : b = c), ⋯ ▸ H_succ b hb (x b) = H_succ c hc (x c)\nx : { x // ¬IsMax x ∧ succ x = succ a } := Classical.indefiniteDescription (fun x => ¬IsMax x ∧ succ x = succ a) ⋯\n⊢ (let x := Classical.indefiniteDescription (fun x => ¬IsMax x ∧ succ x = succ a) ⋯;\n    ⋯ ▸\n      H_succ ↑x ⋯\n        ((fun y x =>\n            ⋯.fix\n              (fun a IH =>\n                if h : IsSuccLimit a then H_lim a h IH\n                else\n                  let x := Classical.indefiniteDescription (fun x => ¬IsMax x ∧ succ x = a) ⋯;\n                  ⋯ ▸ H_succ ↑x ⋯ (IH ↑x ⋯))\n              y)\n          ↑x ⋯)) =\n    H_succ a ha (SuccOrder.limitRecOn a H_succ H_lim)","tactic":"let x := Classical.indefiniteDescription _ (not_isSuccLimit_iff.mp h)","premises":[{"full_name":"Classical.indefiniteDescription","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[15,18],"def_end_pos":[15,39]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Order.not_isSuccLimit_iff","def_path":"Mathlib/Order/SuccPred/Limit.lean","def_pos":[111,8],"def_end_pos":[111,27]}]},{"state_before":"α : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\na b : α\nC : α → Sort u_2\ninst✝ : WellFoundedLT α\nH_succ : (a : α) → ¬IsMax a → C a → C (succ a)\nH_lim : (a : α) → IsSuccLimit a → ((b : α) → b < a → C b) → C a\nha : ¬IsMax a\nh : ¬IsSuccLimit (succ a)\nthis :\n  ∀ {b c : α} {hb : ¬IsMax b} {hc : ¬IsMax c} {x : (a : α) → C a} (h : b = c), ⋯ ▸ H_succ b hb (x b) = H_succ c hc (x c)\nx : { x // ¬IsMax x ∧ succ x = succ a } := Classical.indefiniteDescription (fun x => ¬IsMax x ∧ succ x = succ a) ⋯\n⊢ (let x := Classical.indefiniteDescription (fun x => ¬IsMax x ∧ succ x = succ a) ⋯;\n    ⋯ ▸\n      H_succ ↑x ⋯\n        ((fun y x =>\n            ⋯.fix\n              (fun a IH =>\n                if h : IsSuccLimit a then H_lim a h IH\n                else\n                  let x := Classical.indefiniteDescription (fun x => ¬IsMax x ∧ succ x = a) ⋯;\n                  ⋯ ▸ H_succ ↑x ⋯ (IH ↑x ⋯))\n              y)\n          ↑x ⋯)) =\n    H_succ a ha (SuccOrder.limitRecOn a H_succ H_lim)","state_after":"no goals","tactic":"exact this ((succ_eq_succ_iff_of_not_isMax x.2.1 ha).mp x.2.2)","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Order.succ_eq_succ_iff_of_not_isMax","def_path":"Mathlib/Order/SuccPred/Basic.lean","def_pos":[377,8],"def_end_pos":[377,37]},{"full_name":"Subtype.property","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[590,2],"def_end_pos":[590,10]}]}]}
+{"url":"Mathlib/FieldTheory/Adjoin.lean","commit":"","full_name":"IntermediateField.map_comap_eq_self","start":[1411,0],"end":[1413,54],"file_path":"Mathlib/FieldTheory/Adjoin.lean","tactics":[{"state_before":"K : Type u_1\nL : Type u_2\nL' : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Field L'\ninst✝¹ : Algebra K L\ninst✝ : Algebra K L'\nf : L →ₐ[K] L'\nS : IntermediateField K L'\nh : S ≤ f.fieldRange\n⊢ map f (comap f S) = S","state_after":"no goals","tactic":"simpa only [inf_of_le_left h] using map_comap_eq f S","premises":[{"full_name":"IntermediateField.map_comap_eq","def_path":"Mathlib/FieldTheory/Adjoin.lean","def_pos":[1407,8],"def_end_pos":[1407,20]}]}]}
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+{"url":"Mathlib/Data/Set/Image.lean","commit":"","full_name":"Set.preimage_subset_preimage_iff","start":[699,0],"end":[706,20],"file_path":"Mathlib/Data/Set/Image.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nι' : Sort u_5\nf✝ : ι → α\ns✝ t✝ s t : Set α\nf : β → α\nhs : s ⊆ range f\n⊢ f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t","state_after":"case mp\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nι' : Sort u_5\nf✝ : ι → α\ns✝ t✝ s t : Set α\nf : β → α\nhs : s ⊆ range f\n⊢ f ⁻¹' s ⊆ f ⁻¹' t → s ⊆ t\n\ncase mpr\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nι' : Sort u_5\nf✝ : ι → α\ns✝ t✝ s t : Set α\nf : β → α\nhs : s ⊆ range f\n⊢ s ⊆ t → f ⁻¹' s ⊆ f ⁻¹' t","tactic":"constructor","premises":[]},{"state_before":"case mpr\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nι' : Sort u_5\nf✝ : ι → α\ns✝ t✝ s t : Set α\nf : β → α\nhs : s ⊆ range f\n⊢ s ⊆ t → f ⁻¹' s ⊆ f ⁻¹' t","state_after":"case mpr\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nι' : Sort u_5\nf✝ : ι → α\ns✝ t✝ s t : Set α\nf : β → α\nhs : s ⊆ range f\nh : s ⊆ t\nx : β\n⊢ x ∈ f ⁻¹' s → x ∈ f ⁻¹' t","tactic":"intro h x","premises":[]},{"state_before":"case mpr\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nι' : Sort u_5\nf✝ : ι → α\ns✝ t✝ s t : Set α\nf : β → α\nhs : s ⊆ range f\nh : s ⊆ t\nx : β\n⊢ x ∈ f ⁻¹' s → x ∈ f ⁻¹' t","state_after":"no goals","tactic":"apply h","premises":[]}]}
+{"url":"Mathlib/Algebra/Group/Subgroup/MulOpposite.lean","commit":"","full_name":"Subgroup.op_closure","start":[146,0],"end":[150,42],"file_path":"Mathlib/Algebra/Group/Subgroup/MulOpposite.lean","tactics":[{"state_before":"ι : Sort u_1\nG : Type u_2\ninst✝ : Group G\ns : Set G\n⊢ (closure s).op = closure (MulOpposite.unop ⁻¹' s)","state_after":"ι : Sort u_1\nG : Type u_2\ninst✝ : Group G\ns : Set G\n⊢ sInf {a | s ⊆ MulOpposite.op ⁻¹' ↑a} = sInf {K | MulOpposite.unop ⁻¹' s ⊆ ↑K}","tactic":"simp_rw [closure, op_sInf, Set.preimage_setOf_eq, Subgroup.unop_coe]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Set.preimage_setOf_eq","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[92,8],"def_end_pos":[92,25]},{"full_name":"Subgroup.closure","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[841,4],"def_end_pos":[841,11]},{"full_name":"Subgroup.op_sInf","def_path":"Mathlib/Algebra/Group/Subgroup/MulOpposite.lean","def_pos":[125,8],"def_end_pos":[125,15]},{"full_name":"Subgroup.unop_coe","def_path":"Mathlib/Algebra/Group/Subgroup/MulOpposite.lean","def_pos":[40,23],"def_end_pos":[40,28]}]},{"state_before":"ι : Sort u_1\nG : Type u_2\ninst✝ : Group G\ns : Set G\n⊢ sInf {a | s ⊆ MulOpposite.op ⁻¹' ↑a} = sInf {K | MulOpposite.unop ⁻¹' s ⊆ ↑K}","state_after":"case e_a.h\nι : Sort u_1\nG : Type u_2\ninst✝ : Group G\ns : Set G\na : Subgroup Gᵐᵒᵖ\n⊢ a ∈ {a | s ⊆ MulOpposite.op ⁻¹' ↑a} ↔ a ∈ {K | MulOpposite.unop ⁻¹' s ⊆ ↑K}","tactic":"congr with a","premises":[]},{"state_before":"case e_a.h\nι : Sort u_1\nG : Type u_2\ninst✝ : Group G\ns : Set G\na : Subgroup Gᵐᵒᵖ\n⊢ a ∈ {a | s ⊆ MulOpposite.op ⁻¹' ↑a} ↔ a ∈ {K | MulOpposite.unop ⁻¹' s ⊆ ↑K}","state_after":"no goals","tactic":"exact MulOpposite.unop_surjective.forall","premises":[{"full_name":"Function.Surjective.forall","def_path":"Mathlib/Logic/Function/Basic.lean","def_pos":[164,18],"def_end_pos":[164,35]},{"full_name":"MulOpposite.unop_surjective","def_path":"Mathlib/Algebra/Opposites.lean","def_pos":[129,8],"def_end_pos":[129,23]}]}]}
+{"url":"Mathlib/FieldTheory/Adjoin.lean","commit":"","full_name":"minpoly.algEquiv_apply","start":[1358,0],"end":[1364,53],"file_path":"Mathlib/FieldTheory/Adjoin.lean","tactics":[{"state_before":"K : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx y : L\nhx : IsAlgebraic K x\nh_mp : minpoly K x = minpoly K y\n⊢ (algEquiv hx h_mp) (AdjoinSimple.gen K x) = AdjoinSimple.gen K y","state_after":"K : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx y : L\nhx : IsAlgebraic K x\nh_mp : minpoly K x = minpoly K y\nhy : IsAlgebraic K y\n⊢ (algEquiv hx h_mp) (AdjoinSimple.gen K x) = AdjoinSimple.gen K y","tactic":"have hy : IsAlgebraic K y := ⟨minpoly K x, ne_zero hx.isIntegral, (h_mp ▸ aeval _ _)⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"IsAlgebraic","def_path":"Mathlib/RingTheory/Algebraic.lean","def_pos":[30,4],"def_end_pos":[30,15]},{"full_name":"minpoly","def_path":"Mathlib/FieldTheory/Minpoly/Basic.lean","def_pos":[36,18],"def_end_pos":[36,25]},{"full_name":"minpoly.aeval","def_path":"Mathlib/FieldTheory/Minpoly/Basic.lean","def_pos":[79,8],"def_end_pos":[79,13]},{"full_name":"minpoly.ne_zero","def_path":"Mathlib/FieldTheory/Minpoly/Basic.lean","def_pos":[55,8],"def_end_pos":[55,15]}]},{"state_before":"K : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx y : L\nhx : IsAlgebraic K x\nh_mp : minpoly K x = minpoly K y\nhy : IsAlgebraic K y\n⊢ (algEquiv hx h_mp) (AdjoinSimple.gen K x) = AdjoinSimple.gen K y","state_after":"no goals","tactic":"rw [algEquiv, trans_apply, ← adjoinRootEquivAdjoin_apply_root K hx.isIntegral,\n    symm_apply_apply, trans_apply, AdjoinRoot.algEquivOfEq_apply_root,\n    adjoinRootEquivAdjoin_apply_root K hy.isIntegral]","premises":[{"full_name":"AdjoinRoot.algEquivOfEq_apply_root","def_path":"Mathlib/FieldTheory/Adjoin.lean","def_pos":[1303,8],"def_end_pos":[1303,31]},{"full_name":"AlgEquiv.symm_apply_apply","def_path":"Mathlib/Algebra/Algebra/Equiv.lean","def_pos":[357,8],"def_end_pos":[357,24]},{"full_name":"AlgEquiv.trans_apply","def_path":"Mathlib/Algebra/Algebra/Equiv.lean","def_pos":[370,8],"def_end_pos":[370,19]},{"full_name":"IntermediateField.adjoinRootEquivAdjoin_apply_root","def_path":"Mathlib/FieldTheory/Adjoin.lean","def_pos":[997,8],"def_end_pos":[997,40]},{"full_name":"minpoly.algEquiv","def_path":"Mathlib/FieldTheory/Adjoin.lean","def_pos":[1351,18],"def_end_pos":[1351,26]}]}]}
+{"url":"Mathlib/Order/Basic.lean","commit":"","full_name":"min_def_lt","start":[928,0],"end":[930,20],"file_path":"Mathlib/Order/Basic.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\nπ : ι → Type u_4\ninst✝ : LinearOrder α\np : α → Prop\nx✝ y✝ x y : α\n⊢ min x y = if x < y then x else y","state_after":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\nπ : ι → Type u_4\ninst✝ : LinearOrder α\np : α → Prop\nx✝ y✝ x y : α\n⊢ (if ¬y ≤ x then x else y) = if x < y then x else y","tactic":"rw [min_comm, min_def, ← ite_not]","premises":[{"full_name":"ite_not","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[169,16],"def_end_pos":[169,23]},{"full_name":"min_comm","def_path":"Mathlib/Init/Order/LinearOrder.lean","def_pos":[76,8],"def_end_pos":[76,16]},{"full_name":"min_def","def_path":"Mathlib/Init/Order/LinearOrder.lean","def_pos":[30,8],"def_end_pos":[30,15]}]},{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\nπ : ι → Type u_4\ninst✝ : LinearOrder α\np : α → Prop\nx✝ y✝ x y : α\n⊢ (if ¬y ≤ x then x else y) = if x < y then x else y","state_after":"no goals","tactic":"simp only [not_le]","premises":[{"full_name":"not_le","def_path":"Mathlib/Order/Defs.lean","def_pos":[316,8],"def_end_pos":[316,14]}]}]}
+{"url":"Mathlib/SetTheory/Ordinal/Arithmetic.lean","commit":"","full_name":"Ordinal.IsNormal.sup","start":[1115,0],"end":[1119,23],"file_path":"Mathlib/SetTheory/Ordinal/Arithmetic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nf : Ordinal.{max u v} → Ordinal.{max u w}\nH : IsNormal f\nι : Type u\ng : ι → Ordinal.{max u v}\ninst✝ : Nonempty ι\na : Ordinal.{max u w}\n⊢ f (Ordinal.sup g) ≤ a ↔ Ordinal.sup (f ∘ g) ≤ a","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nf : Ordinal.{max u v} → Ordinal.{max u w}\nH : IsNormal f\nι : Type u\ng : ι → Ordinal.{max u v}\ninst✝ : Nonempty ι\na : Ordinal.{max u w}\n⊢ f (Ordinal.sup g) ≤ a ↔ ∀ (i : ι), (f ∘ g) i ≤ a","tactic":"rw [sup_le_iff]","premises":[{"full_name":"Ordinal.sup_le_iff","def_path":"Mathlib/SetTheory/Ordinal/Arithmetic.lean","def_pos":[1087,8],"def_end_pos":[1087,18]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nf : Ordinal.{max u v} → Ordinal.{max u w}\nH : IsNormal f\nι : Type u\ng : ι → Ordinal.{max u v}\ninst✝ : Nonempty ι\na : Ordinal.{max u w}\n⊢ f (Ordinal.sup g) ≤ a ↔ ∀ (i : ι), (f ∘ g) i ≤ a","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nf : Ordinal.{max u v} → Ordinal.{max u w}\nH : IsNormal f\nι : Type u\ng : ι → Ordinal.{max u v}\ninst✝ : Nonempty ι\na : Ordinal.{max u w}\n⊢ f (Ordinal.sup g) ≤ a ↔ ∀ (i : ι), f (g i) ≤ a","tactic":"simp only [comp]","premises":[{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nf : Ordinal.{max u v} → Ordinal.{max u w}\nH : IsNormal f\nι : Type u\ng : ι → Ordinal.{max u v}\ninst✝ : Nonempty ι\na : Ordinal.{max u w}\n⊢ f (Ordinal.sup g) ≤ a ↔ ∀ (i : ι), f (g i) ≤ a","state_after":"no goals","tactic":"rw [H.le_set' Set.univ Set.univ_nonempty g] <;>\n      simp [sup_le_iff]","premises":[{"full_name":"Ordinal.IsNormal.le_set'","def_path":"Mathlib/SetTheory/Ordinal/Arithmetic.lean","def_pos":[399,8],"def_end_pos":[399,24]},{"full_name":"Ordinal.sup_le_iff","def_path":"Mathlib/SetTheory/Ordinal/Arithmetic.lean","def_pos":[1087,8],"def_end_pos":[1087,18]},{"full_name":"Set.univ","def_path":"Mathlib/Init/Set.lean","def_pos":[157,4],"def_end_pos":[157,8]},{"full_name":"Set.univ_nonempty","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[425,8],"def_end_pos":[425,21]}]}]}
+{"url":"Mathlib/Algebra/Star/Free.lean","commit":"","full_name":"FreeAlgebra.star_algebraMap","start":[66,0],"end":[68,24],"file_path":"Mathlib/Algebra/Star/Free.lean","tactics":[{"state_before":"R : Type u_1\ninst✝ : CommSemiring R\nX : Type u_2\nr : R\n⊢ star ((algebraMap R (FreeAlgebra R X)) r) = (algebraMap R (FreeAlgebra R X)) r","state_after":"no goals","tactic":"simp [star, Star.star]","premises":[{"full_name":"Star.star","def_path":"Mathlib/Algebra/Star/Basic.lean","def_pos":[46,2],"def_end_pos":[46,6]}]}]}
+{"url":"Mathlib/Data/Finset/NAry.lean","commit":"","full_name":"Finset.card_image₂_singleton_right","start":[198,0],"end":[199,98],"file_path":"Mathlib/Data/Finset/NAry.lean","tactics":[{"state_before":"α : Type u_1\nα' : Type u_2\nβ : Type u_3\nβ' : Type u_4\nγ : Type u_5\nγ' : Type u_6\nδ : Type u_7\nδ' : Type u_8\nε : Type u_9\nε' : Type u_10\nζ : Type u_11\nζ' : Type u_12\nν : Type u_13\ninst✝⁷ : DecidableEq α'\ninst✝⁶ : DecidableEq β'\ninst✝⁵ : DecidableEq γ\ninst✝⁴ : DecidableEq γ'\ninst✝³ : DecidableEq δ\ninst✝² : DecidableEq δ'\ninst✝¹ : DecidableEq ε\ninst✝ : DecidableEq ε'\nf f' : α → β → γ\ng g' : α → β → γ → δ\ns s' : Finset α\nt t' : Finset β\nu u' : Finset γ\na a' : α\nb b' : β\nc : γ\nhf : Injective fun a => f a b\n⊢ (image₂ f s {b}).card = s.card","state_after":"no goals","tactic":"rw [image₂_singleton_right, card_image_of_injective _ hf]","premises":[{"full_name":"Finset.card_image_of_injective","def_path":"Mathlib/Data/Finset/Card.lean","def_pos":[246,8],"def_end_pos":[246,31]},{"full_name":"Finset.image₂_singleton_right","def_path":"Mathlib/Data/Finset/NAry.lean","def_pos":[128,8],"def_end_pos":[128,30]}]}]}
+{"url":"Mathlib/CategoryTheory/Localization/HomEquiv.lean","commit":"","full_name":"CategoryTheory.LocalizerMorphism.id_homMap","start":[86,0],"end":[89,62],"file_path":"Mathlib/CategoryTheory/Localization/HomEquiv.lean","tactics":[{"state_before":"C : Type u_1\nC₁ : Type u_2\nC₂ : Type u_3\nC₃ : Type u_4\nD₁ : Type u_5\nD₂ : Type u_6\nD₃ : Type u_7\ninst✝⁹ : Category.{?u.24683, u_1} C\ninst✝⁸ : Category.{u_9, u_2} C₁\ninst✝⁷ : Category.{?u.24691, u_3} C₂\ninst✝⁶ : Category.{?u.24695, u_4} C₃\ninst✝⁵ : Category.{u_8, u_5} D₁\ninst✝⁴ : Category.{?u.24703, u_6} D₂\ninst✝³ : Category.{?u.24707, u_7} D₃\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nW₃ : MorphismProperty C₃\nΦ : LocalizerMorphism W₁ W₂\nΨ : LocalizerMorphism W₂ W₃\nL₁ : C₁ ⥤ D₁\ninst✝² : L₁.IsLocalization W₁\nL₂ : C₂ ⥤ D₂\ninst✝¹ : L₂.IsLocalization W₂\nL₃ : C₃ ⥤ D₃\ninst✝ : L₃.IsLocalization W₃\nX Y Z : C₁\nf : L₁.obj X ⟶ L₁.obj Y\n⊢ (id W₁).homMap L₁ L₁ f = f","state_after":"no goals","tactic":"simpa using (id W₁).homMap_apply L₁ L₁ (𝟭 D₁) (Iso.refl _) f","premises":[{"full_name":"CategoryTheory.Functor.id","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[72,14],"def_end_pos":[72,16]},{"full_name":"CategoryTheory.Iso.refl","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[114,4],"def_end_pos":[114,8]},{"full_name":"CategoryTheory.LocalizerMorphism.homMap_apply","def_path":"Mathlib/CategoryTheory/Localization/HomEquiv.lean","def_pos":[70,6],"def_end_pos":[70,18]},{"full_name":"CategoryTheory.LocalizerMorphism.id","def_path":"Mathlib/CategoryTheory/Localization/LocalizerMorphism.lean","def_pos":[51,4],"def_end_pos":[51,6]}]}]}
+{"url":"Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean","commit":"","full_name":"CategoryTheory.regularTopology.isLocallySurjective_iff","start":[40,0],"end":[54,24],"file_path":"Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean","tactics":[{"state_before":"C : Type u_1\nD : Type u_2\ninst✝³ : Category.{u_3, u_1} C\ninst✝² : Category.{u_4, u_2} D\ninst✝¹ : ConcreteCategory D\ninst✝ : Preregular C\nF G : Cᵒᵖ ⥤ D\nf : F ⟶ G\n⊢ Presheaf.IsLocallySurjective (regularTopology C) f ↔\n    ∀ (X : C) (y : (forget D).obj (G.obj (op X))),\n      ∃ X' φ, ∃ (_ : EffectiveEpi φ), ∃ x, (f.app (op X')) x = (G.map (op φ)) y","state_after":"case mp\nC : Type u_1\nD : Type u_2\ninst✝³ : Category.{u_3, u_1} C\ninst✝² : Category.{u_4, u_2} D\ninst✝¹ : ConcreteCategory D\ninst✝ : Preregular C\nF G : Cᵒᵖ ⥤ D\nf : F ⟶ G\n⊢ Presheaf.IsLocallySurjective (regularTopology C) f →\n    ∀ (X : C) (y : (forget D).obj (G.obj (op X))),\n      ∃ X' φ, ∃ (_ : EffectiveEpi φ), ∃ x, (f.app (op X')) x = (G.map (op φ)) y\n\ncase mpr\nC : Type u_1\nD : Type u_2\ninst✝³ : Category.{u_3, u_1} C\ninst✝² : Category.{u_4, u_2} D\ninst✝¹ : ConcreteCategory D\ninst✝ : Preregular C\nF G : Cᵒᵖ ⥤ D\nf : F ⟶ G\n⊢ (∀ (X : C) (y : (forget D).obj (G.obj (op X))),\n      ∃ X' φ, ∃ (_ : EffectiveEpi φ), ∃ x, (f.app (op X')) x = (G.map (op φ)) y) →\n    Presheaf.IsLocallySurjective (regularTopology C) f","tactic":"constructor","premises":[]}]}
+{"url":"Mathlib/Analysis/Convex/Star.lean","commit":"","full_name":"starConvex_compl_Iic","start":[342,0],"end":[352,12],"file_path":"Mathlib/Analysis/Convex/Star.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : OrderedRing 𝕜\ninst✝² : OrderedAddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : OrderedSMul 𝕜 E\nx y : E\nh : x < y\n⊢ StarConvex 𝕜 y (Iic x)ᶜ","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : OrderedRing 𝕜\ninst✝² : OrderedAddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : OrderedSMul 𝕜 E\nx y : E\nh : x < y\nz : E\nhz : z ∈ (Iic x)ᶜ\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ a • y + b • z ∈ (Iic x)ᶜ","tactic":"refine (starConvex_iff_forall_pos <| by simp [h.not_le]).mpr fun z hz a b ha hb hab ↦ ?_","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"starConvex_iff_forall_pos","def_path":"Mathlib/Analysis/Convex/Star.lean","def_pos":[147,8],"def_end_pos":[147,33]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : OrderedRing 𝕜\ninst✝² : OrderedAddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : OrderedSMul 𝕜 E\nx y : E\nh : x < y\nz : E\nhz : z ∈ (Iic x)ᶜ\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ a • y + b • z ∈ (Iic x)ᶜ","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : OrderedRing 𝕜\ninst✝² : OrderedAddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : OrderedSMul 𝕜 E\nx y : E\nh : x < y\nz : E\nhz : ¬z ≤ x\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ ¬a • y + b • z ≤ x","tactic":"rw [mem_compl_iff, mem_Iic] at hz ⊢","premises":[{"full_name":"Set.mem_Iic","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[114,8],"def_end_pos":[114,15]},{"full_name":"Set.mem_compl_iff","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[84,16],"def_end_pos":[84,29]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : OrderedRing 𝕜\ninst✝² : OrderedAddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : OrderedSMul 𝕜 E\nx y : E\nh : x < y\nz : E\nhz : ¬z ≤ x\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ ¬a • y + b • z ≤ x","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : OrderedRing 𝕜\ninst✝² : OrderedAddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : OrderedSMul 𝕜 E\nx y : E\nh : x < y\nz : E\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhz : a • y + b • z ≤ x\n⊢ z ≤ x","tactic":"contrapose! hz","premises":[{"full_name":"Mathlib.Tactic.Contrapose.mtr","def_path":"Mathlib/Tactic/Contrapose.lean","def_pos":[24,6],"def_end_pos":[24,9]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : OrderedRing 𝕜\ninst✝² : OrderedAddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : OrderedSMul 𝕜 E\nx y : E\nh : x < y\nz : E\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhz : a • y + b • z ≤ x\n⊢ z ≤ x","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : OrderedRing 𝕜\ninst✝² : OrderedAddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : OrderedSMul 𝕜 E\nx y : E\nh : x < y\nz : E\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhz : a • y + b • z ≤ x\n⊢ b • z < b • x","tactic":"refine (lt_of_smul_lt_smul_of_nonneg_left ?_ hb.le).le","premises":[]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : OrderedRing 𝕜\ninst✝² : OrderedAddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : OrderedSMul 𝕜 E\nx y : E\nh : x < y\nz : E\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhz : a • y + b • z ≤ x\n⊢ b • z < b • x","state_after":"no goals","tactic":"calc\n    b • z ≤ (a + b) • x - a • y := by rwa [le_sub_iff_add_le', hab, one_smul]\n    _ < b • x := by\n      rw [add_smul, sub_lt_iff_lt_add']\n      gcongr","premises":[{"full_name":"add_smul","def_path":"Mathlib/Algebra/Module/Defs.lean","def_pos":[87,8],"def_end_pos":[87,16]},{"full_name":"le_sub_iff_add_le'","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Basic.lean","def_pos":[545,2],"def_end_pos":[545,13]},{"full_name":"one_smul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[379,6],"def_end_pos":[379,14]},{"full_name":"sub_lt_iff_lt_add'","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Basic.lean","def_pos":[670,2],"def_end_pos":[670,13]}]}]}
+{"url":"Mathlib/LinearAlgebra/Ray.lean","commit":"","full_name":"units_smul_eq_neg_iff","start":[504,0],"end":[507,12],"file_path":"Mathlib/LinearAlgebra/Ray.lean","tactics":[{"state_before":"R : Type u_1\ninst✝³ : LinearOrderedCommRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : NoZeroSMulDivisors R M\nu : Rˣ\nv : Module.Ray R M\n⊢ u • v = -v ↔ ↑u < 0","state_after":"no goals","tactic":"rw [← neg_inj, neg_neg, ← Module.Ray.neg_units_smul, units_smul_eq_self_iff, Units.val_neg,\n    neg_pos]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Module.Ray.neg_units_smul","def_path":"Mathlib/LinearAlgebra/Ray.lean","def_pos":[418,8],"def_end_pos":[418,22]},{"full_name":"Units.val_neg","def_path":"Mathlib/Algebra/Ring/Units.lean","def_pos":[35,18],"def_end_pos":[35,25]},{"full_name":"neg_inj","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[294,2],"def_end_pos":[294,13]},{"full_name":"neg_neg","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[733,2],"def_end_pos":[733,13]},{"full_name":"units_smul_eq_self_iff","def_path":"Mathlib/LinearAlgebra/Ray.lean","def_pos":[500,8],"def_end_pos":[500,30]}]}]}
+{"url":"Mathlib/Algebra/Group/Subgroup/Pointwise.lean","commit":"","full_name":"AddSubgroup.add_inf_assoc","start":[209,0],"end":[221,36],"file_path":"Mathlib/Algebra/Group/Subgroup/Pointwise.lean","tactics":[{"state_before":"α : Type u_1\nG : Type u_2\nA✝ : Type u_3\nS : Type u_4\ninst✝¹ : Group G\ninst✝ : AddGroup A✝\ns : Set G\nA B C : Subgroup G\nh : A ≤ C\n⊢ ↑A * ↑(B ⊓ C) = ↑A * ↑B ∩ ↑C","state_after":"case h\nα : Type u_1\nG : Type u_2\nA✝ : Type u_3\nS : Type u_4\ninst✝¹ : Group G\ninst✝ : AddGroup A✝\ns : Set G\nA B C : Subgroup G\nh : A ≤ C\nx✝ : G\n⊢ x✝ ∈ ↑A * ↑(B ⊓ C) ↔ x✝ ∈ ↑A * ↑B ∩ ↑C","tactic":"ext","premises":[]},{"state_before":"case h\nα : Type u_1\nG : Type u_2\nA✝ : Type u_3\nS : Type u_4\ninst✝¹ : Group G\ninst✝ : AddGroup A✝\ns : Set G\nA B C : Subgroup G\nh : A ≤ C\nx✝ : G\n⊢ x✝ ∈ ↑A * ↑(B ⊓ C) ↔ x✝ ∈ ↑A * ↑B ∩ ↑C","state_after":"case h\nα : Type u_1\nG : Type u_2\nA✝ : Type u_3\nS : Type u_4\ninst✝¹ : Group G\ninst✝ : AddGroup A✝\ns : Set G\nA B C : Subgroup G\nh : A ≤ C\nx✝ : G\n⊢ (∃ x ∈ ↑A, ∃ y, (y ∈ ↑B ∧ y ∈ ↑C) ∧ x * y = x✝) ↔ (∃ x ∈ ↑A, ∃ y ∈ ↑B, x * y = x✝) ∧ x✝ ∈ ↑C","tactic":"simp only [coe_inf, Set.mem_mul, Set.mem_inter_iff]","premises":[{"full_name":"Set.mem_inter_iff","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[718,8],"def_end_pos":[718,21]},{"full_name":"Set.mem_mul","def_path":"Mathlib/Data/Set/Pointwise/Basic.lean","def_pos":[273,8],"def_end_pos":[273,15]},{"full_name":"Subgroup.coe_inf","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[746,8],"def_end_pos":[746,15]}]},{"state_before":"case h\nα : Type u_1\nG : Type u_2\nA✝ : Type u_3\nS : Type u_4\ninst✝¹ : Group G\ninst✝ : AddGroup A✝\ns : Set G\nA B C : Subgroup G\nh : A ≤ C\nx✝ : G\n⊢ (∃ x ∈ ↑A, ∃ y, (y ∈ ↑B ∧ y ∈ ↑C) ∧ x * y = x✝) ↔ (∃ x ∈ ↑A, ∃ y ∈ ↑B, x * y = x✝) ∧ x✝ ∈ ↑C","state_after":"case h.mp\nα : Type u_1\nG : Type u_2\nA✝ : Type u_3\nS : Type u_4\ninst✝¹ : Group G\ninst✝ : AddGroup A✝\ns : Set G\nA B C : Subgroup G\nh : A ≤ C\nx✝ : G\n⊢ (∃ x ∈ ↑A, ∃ y, (y ∈ ↑B ∧ y ∈ ↑C) ∧ x * y = x✝) → (∃ x ∈ ↑A, ∃ y ∈ ↑B, x * y = x✝) ∧ x✝ ∈ ↑C\n\ncase h.mpr\nα : Type u_1\nG : Type u_2\nA✝ : Type u_3\nS : Type u_4\ninst✝¹ : Group G\ninst✝ : AddGroup A✝\ns : Set G\nA B C : Subgroup G\nh : A ≤ C\nx✝ : G\n⊢ (∃ x ∈ ↑A, ∃ y ∈ ↑B, x * y = x✝) ∧ x✝ ∈ ↑C → ∃ x ∈ ↑A, ∃ y, (y ∈ ↑B ∧ y ∈ ↑C) ∧ x * y = x✝","tactic":"constructor","premises":[]},{"state_before":"case h.mpr\nα : Type u_1\nG : Type u_2\nA✝ : Type u_3\nS : Type u_4\ninst✝¹ : Group G\ninst✝ : AddGroup A✝\ns : Set G\nA B C : Subgroup G\nh : A ≤ C\nx✝ : G\n⊢ (∃ x ∈ ↑A, ∃ y ∈ ↑B, x * y = x✝) ∧ x✝ ∈ ↑C → ∃ x ∈ ↑A, ∃ y, (y ∈ ↑B ∧ y ∈ ↑C) ∧ x * y = x✝","state_after":"case h.mpr.intro.intro.intro.intro.intro\nα : Type u_1\nG : Type u_2\nA✝ : Type u_3\nS : Type u_4\ninst✝¹ : Group G\ninst✝ : AddGroup A✝\ns : Set G\nA B C : Subgroup G\nh : A ≤ C\ny : G\nhy : y ∈ ↑A\nz : G\nhz : z ∈ ↑B\nhyz : y * z ∈ ↑C\n⊢ ∃ x ∈ ↑A, ∃ y_1, (y_1 ∈ ↑B ∧ y_1 ∈ ↑C) ∧ x * y_1 = y * z","tactic":"rintro ⟨⟨y, hy, z, hz, rfl⟩, hyz⟩","premises":[]},{"state_before":"case h.mpr.intro.intro.intro.intro.intro\nα : Type u_1\nG : Type u_2\nA✝ : Type u_3\nS : Type u_4\ninst✝¹ : Group G\ninst✝ : AddGroup A✝\ns : Set G\nA B C : Subgroup G\nh : A ≤ C\ny : G\nhy : y ∈ ↑A\nz : G\nhz : z ∈ ↑B\nhyz : y * z ∈ ↑C\n⊢ ∃ x ∈ ↑A, ∃ y_1, (y_1 ∈ ↑B ∧ y_1 ∈ ↑C) ∧ x * y_1 = y * z","state_after":"case h.mpr.intro.intro.intro.intro.intro\nα : Type u_1\nG : Type u_2\nA✝ : Type u_3\nS : Type u_4\ninst✝¹ : Group G\ninst✝ : AddGroup A✝\ns : Set G\nA B C : Subgroup G\nh : A ≤ C\ny : G\nhy : y ∈ ↑A\nz : G\nhz : z ∈ ↑B\nhyz : y * z ∈ ↑C\n⊢ z ∈ ↑C","tactic":"refine ⟨y, hy, z, ⟨hz, ?_⟩, rfl⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"case h.mpr.intro.intro.intro.intro.intro\nα : Type u_1\nG : Type u_2\nA✝ : Type u_3\nS : Type u_4\ninst✝¹ : Group G\ninst✝ : AddGroup A✝\ns : Set G\nA B C : Subgroup G\nh : A ≤ C\ny : G\nhy : y ∈ ↑A\nz : G\nhz : z ∈ ↑B\nhyz : y * z ∈ ↑C\n⊢ z ∈ ↑C","state_after":"case h.mpr.intro.intro.intro.intro.intro\nα : Type u_1\nG : Type u_2\nA✝ : Type u_3\nS : Type u_4\ninst✝¹ : Group G\ninst✝ : AddGroup A✝\ns : Set G\nA B C : Subgroup G\nh : A ≤ C\ny : G\nhy : y ∈ ↑A\nz : G\nhz : z ∈ ↑B\nhyz : y * z ∈ ↑C\n⊢ y⁻¹ * (y * z) ∈ C","tactic":"suffices y⁻¹ * (y * z) ∈ C by simpa","premises":[{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]}]},{"state_before":"case h.mpr.intro.intro.intro.intro.intro\nα : Type u_1\nG : Type u_2\nA✝ : Type u_3\nS : Type u_4\ninst✝¹ : Group G\ninst✝ : AddGroup A✝\ns : Set G\nA B C : Subgroup G\nh : A ≤ C\ny : G\nhy : y ∈ ↑A\nz : G\nhz : z ∈ ↑B\nhyz : y * z ∈ ↑C\n⊢ y⁻¹ * (y * z) ∈ C","state_after":"no goals","tactic":"exact mul_mem (inv_mem (h hy)) hyz","premises":[{"full_name":"InvMemClass.inv_mem","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[97,2],"def_end_pos":[97,9]},{"full_name":"MulMemClass.mul_mem","def_path":"Mathlib/Algebra/Group/Subsemigroup/Basic.lean","def_pos":[62,2],"def_end_pos":[62,9]}]}]}
+{"url":"Mathlib/CategoryTheory/Closed/Cartesian.lean","commit":"","full_name":"CategoryTheory.CartesianClosed.uncurry_id_eq_ev","start":[213,0],"end":[214,42],"file_path":"Mathlib/CategoryTheory/Closed/Cartesian.lean","tactics":[{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\nA✝ B X✝ X' Y Y' Z : C\ninst✝² : HasFiniteProducts C\ninst✝¹ : Exponentiable A✝\nA X : C\ninst✝ : Exponentiable A\n⊢ uncurry (𝟙 (A ⟹ X)) = (exp.ev A).app X","state_after":"no goals","tactic":"rw [uncurry_eq, prod.map_id_id, id_comp]","premises":[{"full_name":"CategoryTheory.CartesianClosed.uncurry_eq","def_path":"Mathlib/CategoryTheory/Closed/Cartesian.lean","def_pos":[207,8],"def_end_pos":[207,18]},{"full_name":"CategoryTheory.Category.id_comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[151,2],"def_end_pos":[151,9]},{"full_name":"CategoryTheory.Limits.prod.map_id_id","def_path":"Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean","def_pos":[628,8],"def_end_pos":[628,22]}]}]}
+{"url":"Mathlib/Order/JordanHolder.lean","commit":"","full_name":"CompositionSeries.Equivalent.snoc","start":[291,0],"end":[303,62],"file_path":"Mathlib/Order/JordanHolder.lean","tactics":[{"state_before":"X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ : CompositionSeries X\nx₁ x₂ : X\nhsat₁ : IsMaximal (last s₁) x₁\nhsat₂ : IsMaximal (last s₂) x₂\nhequiv : s₁.Equivalent s₂\nhlast : Iso (last s₁, x₁) (last s₂, x₂)\ne : Fin s₁.length.succ ≃ Fin s₂.length.succ :=\n  Trans.trans (Trans.trans finSuccEquivLast (Functor.mapEquiv Option (Exists.choose hequiv))) finSuccEquivLast.symm\ni : Fin (snoc s₁ x₁ hsat₁).length\n⊢ Iso ((snoc s₁ x₁ hsat₁).toFun i.castSucc, (snoc s₁ x₁ hsat₁).toFun i.succ)\n    ((snoc s₂ x₂ hsat₂).toFun (e i).castSucc, (snoc s₂ x₂ hsat₂).toFun (e i).succ)","state_after":"case refine_1\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ : CompositionSeries X\nx₁ x₂ : X\nhsat₁ : IsMaximal (last s₁) x₁\nhsat₂ : IsMaximal (last s₂) x₂\nhequiv : s₁.Equivalent s₂\nhlast : Iso (last s₁, x₁) (last s₂, x₂)\ne : Fin s₁.length.succ ≃ Fin s₂.length.succ :=\n  Trans.trans (Trans.trans finSuccEquivLast (Functor.mapEquiv Option (Exists.choose hequiv))) finSuccEquivLast.symm\ni : Fin (snoc s₁ x₁ hsat₁).length\n⊢ Iso\n    ((snoc s₁ x₁ hsat₁).toFun (Fin.last (s₁.length + (RelSeries.singleton IsMaximal x₁).length)).castSucc,\n      (snoc s₁ x₁ hsat₁).toFun (Fin.last (s₁.length + (RelSeries.singleton IsMaximal x₁).length)).succ)\n    ((snoc s₂ x₂ hsat₂).toFun (e (Fin.last (s₁.length + (RelSeries.singleton IsMaximal x₁).length))).castSucc,\n      (snoc s₂ x₂ hsat₂).toFun (e (Fin.last (s₁.length + (RelSeries.singleton IsMaximal x₁).length))).succ)\n\ncase refine_2\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ : CompositionSeries X\nx₁ x₂ : X\nhsat₁ : IsMaximal (last s₁) x₁\nhsat₂ : IsMaximal (last s₂) x₂\nhequiv : s₁.Equivalent s₂\nhlast : Iso (last s₁, x₁) (last s₂, x₂)\ne : Fin s₁.length.succ ≃ Fin s₂.length.succ :=\n  Trans.trans (Trans.trans finSuccEquivLast (Functor.mapEquiv Option (Exists.choose hequiv))) finSuccEquivLast.symm\ni : Fin (snoc s₁ x₁ hsat₁).length\n⊢ ∀ (i : Fin (s₁.length + (RelSeries.singleton IsMaximal x₁).length)),\n    Iso ((snoc s₁ x₁ hsat₁).toFun i.castSucc.castSucc, (snoc s₁ x₁ hsat₁).toFun i.castSucc.succ)\n      ((snoc s₂ x₂ hsat₂).toFun (e i.castSucc).castSucc, (snoc s₂ x₂ hsat₂).toFun (e i.castSucc).succ)","tactic":"refine Fin.lastCases ?_ ?_ i","premises":[{"full_name":"Fin.lastCases","def_path":".lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean","def_pos":[718,20],"def_end_pos":[718,29]}]}]}
+{"url":"Mathlib/MeasureTheory/Integral/SetIntegral.lean","commit":"","full_name":"MeasureTheory.setIntegral_nonpos_ae","start":[847,0],"end":[849,81],"file_path":"Mathlib/MeasureTheory/Integral/SetIntegral.lean","tactics":[{"state_before":"X : Type u_1\nY : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝ : MeasurableSpace X\nμ : Measure X\nf : X → ℝ\ns : Set X\nhs : MeasurableSet s\nhf : ∀ᵐ (x : X) ∂μ, x ∈ s → f x ≤ 0\n⊢ f ≤ᶠ[ae (μ.restrict s)] 0","state_after":"no goals","tactic":"rwa [EventuallyLE, ae_restrict_iff' hs]","premises":[{"full_name":"Filter.EventuallyLE","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1434,4],"def_end_pos":[1434,16]},{"full_name":"MeasureTheory.ae_restrict_iff'","def_path":"Mathlib/MeasureTheory/Measure/Restrict.lean","def_pos":[547,8],"def_end_pos":[547,24]}]}]}
+{"url":"Mathlib/RingTheory/Multiplicity.lean","commit":"","full_name":"multiplicity.finite_nat_iff","start":[249,0],"end":[264,39],"file_path":"Mathlib/RingTheory/Multiplicity.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝³ : Monoid α\ninst✝² : Monoid β\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\na b : ℕ\n⊢ Finite a b ↔ a ≠ 1 ∧ 0 < b","state_after":"α : Type u_1\nβ : Type u_2\ninst✝³ : Monoid α\ninst✝² : Monoid β\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\na b : ℕ\n⊢ (∀ (n : ℕ), a ^ n ∣ b) ↔ a = 1 ∨ b = 0","tactic":"rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne, Classical.not_not, not_lt,\n    Nat.le_zero]","premises":[{"full_name":"Classical.not_not","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[135,16],"def_end_pos":[135,23]},{"full_name":"Nat.le_zero","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[592,8],"def_end_pos":[592,15]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"multiplicity.not_finite_iff_forall","def_path":"Mathlib/RingTheory/Multiplicity.lean","def_pos":[70,8],"def_end_pos":[70,29]},{"full_name":"not_and_or","def_path":"Mathlib/Logic/Basic.lean","def_pos":[339,8],"def_end_pos":[339,18]},{"full_name":"not_iff_not","def_path":"Mathlib/Logic/Basic.lean","def_pos":[319,8],"def_end_pos":[319,19]},{"full_name":"not_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[312,8],"def_end_pos":[312,14]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝³ : Monoid α\ninst✝² : Monoid β\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\na b : ℕ\n⊢ (∀ (n : ℕ), a ^ n ∣ b) ↔ a = 1 ∨ b = 0","state_after":"no goals","tactic":"exact\n    ⟨fun h =>\n      or_iff_not_imp_right.2 fun hb =>\n        have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1\n        Classical.by_contradiction fun ha1 : a ≠ 1 =>\n          have ha_gt_one : 1 < a :=\n            lt_of_not_ge fun _ =>\n              match a with\n              | 0 => ha rfl\n              | 1 => ha1 rfl\n              | b+2 => by omega\n          not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b),\n      fun h => by cases h <;> simp [*]⟩","premises":[{"full_name":"Classical.or_iff_not_imp_right","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[148,8],"def_end_pos":[148,28]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Nat.le_of_dvd","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Dvd.lean","def_pos":[46,8],"def_end_pos":[46,17]},{"full_name":"Nat.lt_pow_self","def_path":"Mathlib/Data/Nat/Defs.lean","def_pos":[617,6],"def_end_pos":[617,17]},{"full_name":"Nat.pos_of_ne_zero","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[354,18],"def_end_pos":[354,32]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"lt_of_not_ge","def_path":"Mathlib/Order/Defs.lean","def_pos":[284,8],"def_end_pos":[284,20]},{"full_name":"not_lt_of_ge","def_path":"Mathlib/Order/Defs.lean","def_pos":[125,8],"def_end_pos":[125,20]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]},{"full_name":"zero_dvd_iff","def_path":"Mathlib/Algebra/GroupWithZero/Divisibility.lean","def_pos":[31,8],"def_end_pos":[31,20]}]}]}
+{"url":"Mathlib/RingTheory/MvPowerSeries/Inverse.lean","commit":"","full_name":"MvPowerSeries.invOfUnit_eq'","start":[233,0],"end":[239,14],"file_path":"Mathlib/RingTheory/MvPowerSeries/Inverse.lean","tactics":[{"state_before":"σ : Type u_1\nR : Type u_2\nk : Type u_3\ninst✝ : Field k\nφ : MvPowerSeries σ k\nu : kˣ\nh : (constantCoeff σ k) φ = ↑u\n⊢ φ.invOfUnit u = φ⁻¹","state_after":"σ : Type u_1\nR : Type u_2\nk : Type u_3\ninst✝ : Field k\nφ : MvPowerSeries σ k\nu : kˣ\nh : (constantCoeff σ k) φ = ↑u\n⊢ φ.invOfUnit u = φ.invOfUnit (Units.mk0 ((constantCoeff σ k) φ) ⋯)","tactic":"rw [← invOfUnit_eq φ (h.symm ▸ u.ne_zero)]","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"MvPowerSeries.invOfUnit_eq","def_path":"Mathlib/RingTheory/MvPowerSeries/Inverse.lean","def_pos":[229,8],"def_end_pos":[229,20]},{"full_name":"Units.ne_zero","def_path":"Mathlib/Algebra/GroupWithZero/Units/Basic.lean","def_pos":[33,8],"def_end_pos":[33,15]}]},{"state_before":"σ : Type u_1\nR : Type u_2\nk : Type u_3\ninst✝ : Field k\nφ : MvPowerSeries σ k\nu : kˣ\nh : (constantCoeff σ k) φ = ↑u\n⊢ φ.invOfUnit u = φ.invOfUnit (Units.mk0 ((constantCoeff σ k) φ) ⋯)","state_after":"σ : Type u_1\nR : Type u_2\nk : Type u_3\ninst✝ : Field k\nφ : MvPowerSeries σ k\nu : kˣ\nh : (constantCoeff σ k) φ = ↑u\n⊢ u = Units.mk0 ((constantCoeff σ k) φ) ⋯","tactic":"apply congrArg (invOfUnit φ)","premises":[{"full_name":"MvPowerSeries.invOfUnit","def_path":"Mathlib/RingTheory/MvPowerSeries/Inverse.lean","def_pos":[83,4],"def_end_pos":[83,13]},{"full_name":"congrArg","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[362,8],"def_end_pos":[362,16]}]},{"state_before":"σ : Type u_1\nR : Type u_2\nk : Type u_3\ninst✝ : Field k\nφ : MvPowerSeries σ k\nu : kˣ\nh : (constantCoeff σ k) φ = ↑u\n⊢ u = Units.mk0 ((constantCoeff σ k) φ) ⋯","state_after":"σ : Type u_1\nR : Type u_2\nk : Type u_3\ninst✝ : Field k\nφ : MvPowerSeries σ k\nu : kˣ\nh : (constantCoeff σ k) φ = ↑u\n⊢ ↑u = ↑(Units.mk0 ((constantCoeff σ k) φ) ⋯)","tactic":"rw [Units.ext_iff]","premises":[{"full_name":"Units.ext_iff","def_path":"Mathlib/Algebra/Group/Units.lean","def_pos":[137,18],"def_end_pos":[137,25]}]},{"state_before":"σ : Type u_1\nR : Type u_2\nk : Type u_3\ninst✝ : Field k\nφ : MvPowerSeries σ k\nu : kˣ\nh : (constantCoeff σ k) φ = ↑u\n⊢ ↑u = ↑(Units.mk0 ((constantCoeff σ k) φ) ⋯)","state_after":"no goals","tactic":"exact h.symm","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]}]}]}
+{"url":"Mathlib/Combinatorics/HalesJewett.lean","commit":"","full_name":"Combinatorics.Subspace.apply_inr","start":[132,0],"end":[132,84],"file_path":"Mathlib/Combinatorics/HalesJewett.lean","tactics":[{"state_before":"η✝ : Type u_1\nα✝ : Type u_2\nι✝ : Type u_3\nκ✝ : Type u_4\nη : Type u_5\nα : Type u_6\nι : Type u_7\nκ : Type u_8\nl : Subspace η α ι\nx : η → α\ni : ι\na : α\ne : η\nh : l.idxFun i = Sum.inr e\n⊢ ↑l x i = x e","state_after":"no goals","tactic":"simp [apply_def, h]","premises":[{"full_name":"Combinatorics.Subspace.apply_def","def_path":"Mathlib/Combinatorics/HalesJewett.lean","def_pos":[130,6],"def_end_pos":[130,15]}]}]}
+{"url":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","commit":"","full_name":"WeierstrassCurve.Jacobian.negDblY_of_Z_ne_zero","start":[695,0],"end":[701,86],"file_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","tactics":[{"state_before":"R : Type u\ninst✝¹ : CommRing R\nW' : Jacobian R\nF : Type v\ninst✝ : Field F\nW : Jacobian F\nP Q : Fin 3 → F\nhP : W.Equation P\nhQ : W.Equation Q\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhx : P x * Q z ^ 2 = Q x * P z ^ 2\nhy : P y * Q z ^ 3 ≠ W.negY Q * P z ^ 3\n⊢ W.negDblY P / W.dblZ P ^ 3 =\n    W.toAffine.negAddY (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3)\n      (W.toAffine.slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3))","state_after":"no goals","tactic":"rw [negDblY, dblX, toAffine_slope_of_eq hP hQ hPz hQz hx hy, dblZ, ← (X_eq_iff hPz hQz).mp hx,\n    toAffine_negAddY_of_eq hPz <| sub_ne_zero_of_ne <| Y_ne_negY_of_Y_ne' hP hQ hx hy]","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"WeierstrassCurve.Jacobian.X_eq_iff","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","def_pos":[217,6],"def_end_pos":[217,14]},{"full_name":"WeierstrassCurve.Jacobian.Y_ne_negY_of_Y_ne'","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","def_pos":[531,6],"def_end_pos":[531,24]},{"full_name":"WeierstrassCurve.Jacobian.dblX","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","def_pos":[634,18],"def_end_pos":[634,22]},{"full_name":"WeierstrassCurve.Jacobian.dblZ","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","def_pos":[587,4],"def_end_pos":[587,8]},{"full_name":"WeierstrassCurve.Jacobian.negDblY","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","def_pos":[668,18],"def_end_pos":[668,25]},{"full_name":"_private.Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.0.WeierstrassCurve.Jacobian.toAffine_negAddY_of_eq","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","def_pos":[687,14],"def_end_pos":[687,36]},{"full_name":"_private.Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.0.WeierstrassCurve.Jacobian.toAffine_slope_of_eq","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","def_pos":[621,14],"def_end_pos":[621,34]},{"full_name":"sub_ne_zero_of_ne","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[387,2],"def_end_pos":[387,13]}]}]}
+{"url":"Mathlib/Order/CompactlyGenerated/Basic.lean","commit":"","full_name":"sSup_compact_le_eq","start":[322,0],"end":[326,95],"file_path":"Mathlib/Order/CompactlyGenerated/Basic.lean","tactics":[{"state_before":"ι : Sort u_1\nα : Type u_2\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b✝ : α\ns : Set α\nb : α\n⊢ sSup {c | CompleteLattice.IsCompactElement c ∧ c ≤ b} = b","state_after":"case intro.intro\nι : Sort u_1\nα : Type u_2\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns✝ s : Set α\nhs : ∀ x ∈ s, CompleteLattice.IsCompactElement x\n⊢ sSup {c | CompleteLattice.IsCompactElement c ∧ c ≤ sSup s} = sSup s","tactic":"rcases IsCompactlyGenerated.exists_sSup_eq b with ⟨s, hs, rfl⟩","premises":[{"full_name":"IsCompactlyGenerated.exists_sSup_eq","def_path":"Mathlib/Order/CompactlyGenerated/Basic.lean","def_pos":[316,2],"def_end_pos":[316,16]}]},{"state_before":"case intro.intro\nι : Sort u_1\nα : Type u_2\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns✝ s : Set α\nhs : ∀ x ∈ s, CompleteLattice.IsCompactElement x\n⊢ sSup {c | CompleteLattice.IsCompactElement c ∧ c ≤ sSup s} = sSup s","state_after":"no goals","tactic":"exact le_antisymm (sSup_le fun c hc => hc.2) (sSup_le_sSup fun c cs => ⟨hs c cs, le_sSup cs⟩)","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"le_antisymm","def_path":"Mathlib/Order/Defs.lean","def_pos":[156,8],"def_end_pos":[156,19]},{"full_name":"le_sSup","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[77,8],"def_end_pos":[77,15]},{"full_name":"sSup_le","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[80,8],"def_end_pos":[80,15]},{"full_name":"sSup_le_sSup","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[95,8],"def_end_pos":[95,20]}]}]}
+{"url":"Mathlib/LinearAlgebra/AffineSpace/Ordered.lean","commit":"","full_name":"lineMap_mono_right","start":[58,0],"end":[60,61],"file_path":"Mathlib/LinearAlgebra/AffineSpace/Ordered.lean","tactics":[{"state_before":"k : Type u_1\nE : Type u_2\nPE : Type u_3\ninst✝³ : OrderedRing k\ninst✝² : OrderedAddCommGroup E\ninst✝¹ : Module k E\ninst✝ : OrderedSMul k E\na a' b b' : E\nr r' : k\nhb : b ≤ b'\nhr : 0 ≤ r\n⊢ (lineMap a b) r ≤ (lineMap a b') r","state_after":"k : Type u_1\nE : Type u_2\nPE : Type u_3\ninst✝³ : OrderedRing k\ninst✝² : OrderedAddCommGroup E\ninst✝¹ : Module k E\ninst✝ : OrderedSMul k E\na a' b b' : E\nr r' : k\nhb : b ≤ b'\nhr : 0 ≤ r\n⊢ (1 - r) • a + r • b ≤ (1 - r) • a + r • b'","tactic":"simp only [lineMap_apply_module]","premises":[{"full_name":"AffineMap.lineMap_apply_module","def_path":"Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean","def_pos":[463,8],"def_end_pos":[463,28]}]},{"state_before":"k : Type u_1\nE : Type u_2\nPE : Type u_3\ninst✝³ : OrderedRing k\ninst✝² : OrderedAddCommGroup E\ninst✝¹ : Module k E\ninst✝ : OrderedSMul k E\na a' b b' : E\nr r' : k\nhb : b ≤ b'\nhr : 0 ≤ r\n⊢ (1 - r) • a + r • b ≤ (1 - r) • a + r • b'","state_after":"no goals","tactic":"exact add_le_add_left (smul_le_smul_of_nonneg_left hb hr) _","premises":[{"full_name":"add_le_add_left","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[61,31],"def_end_pos":[61,46]},{"full_name":"smul_le_smul_of_nonneg_left","def_path":"Mathlib/Algebra/Order/Module/Defs.lean","def_pos":[272,16],"def_end_pos":[272,43]}]}]}
+{"url":"Mathlib/Topology/MetricSpace/HausdorffDistance.lean","commit":"","full_name":"Metric.infEdist_eq_top_iff","start":[455,0],"end":[456,94],"file_path":"Mathlib/Topology/MetricSpace/HausdorffDistance.lean","tactics":[{"state_before":"ι : Sort u_1\nα : Type u\nβ : Type v\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\ns t u : Set α\nx y : α\nΦ : α → β\n⊢ infEdist x s = ⊤ ↔ s = ∅","state_after":"no goals","tactic":"rcases s.eq_empty_or_nonempty with rfl | hs <;> simp [*, Nonempty.ne_empty, infEdist_ne_top]","premises":[{"full_name":"Metric.infEdist_ne_top","def_path":"Mathlib/Topology/MetricSpace/HausdorffDistance.lean","def_pos":[450,8],"def_end_pos":[450,23]},{"full_name":"Set.eq_empty_or_nonempty","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[506,8],"def_end_pos":[506,28]}]}]}
+{"url":"Mathlib/RingTheory/Polynomial/Cyclotomic/Expand.lean","commit":"","full_name":"Polynomial.cyclotomic_mul_prime_dvd_eq_pow","start":[130,0],"end":[139,62],"file_path":"Mathlib/RingTheory/Polynomial/Cyclotomic/Expand.lean","tactics":[{"state_before":"R : Type u_1\np n : ℕ\nhp : Fact (Nat.Prime p)\ninst✝¹ : Ring R\ninst✝ : CharP R p\nhn : p ∣ n\n⊢ cyclotomic (n * p) R = cyclotomic n R ^ p","state_after":"R : Type u_1\np n : ℕ\nhp : Fact (Nat.Prime p)\ninst✝¹ : Ring R\ninst✝ : CharP R p\nhn : p ∣ n\nthis : Algebra (ZMod p) R := ZMod.algebra R p\n⊢ cyclotomic (n * p) R = cyclotomic n R ^ p","tactic":"letI : Algebra (ZMod p) R := ZMod.algebra _ _","premises":[{"full_name":"Algebra","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[100,6],"def_end_pos":[100,13]},{"full_name":"ZMod","def_path":"Mathlib/Data/ZMod/Defs.lean","def_pos":[89,4],"def_end_pos":[89,8]},{"full_name":"ZMod.algebra","def_path":"Mathlib/Data/ZMod/Algebra.lean","def_pos":[42,7],"def_end_pos":[42,14]}]},{"state_before":"R : Type u_1\np n : ℕ\nhp : Fact (Nat.Prime p)\ninst✝¹ : Ring R\ninst✝ : CharP R p\nhn : p ∣ n\nthis : Algebra (ZMod p) R := ZMod.algebra R p\n⊢ cyclotomic (n * p) R = cyclotomic n R ^ p","state_after":"R : Type u_1\np n : ℕ\nhp : Fact (Nat.Prime p)\ninst✝¹ : Ring R\ninst✝ : CharP R p\nhn : p ∣ n\nthis : Algebra (ZMod p) R := ZMod.algebra R p\n⊢ cyclotomic (n * p) (ZMod p) = cyclotomic n (ZMod p) ^ p","tactic":"suffices cyclotomic (n * p) (ZMod p) = cyclotomic n (ZMod p) ^ p by\n    rw [← map_cyclotomic _ (algebraMap (ZMod p) R), ← map_cyclotomic _ (algebraMap (ZMod p) R),\n      this, Polynomial.map_pow]","premises":[{"full_name":"Polynomial.cyclotomic","def_path":"Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean","def_pos":[231,4],"def_end_pos":[231,14]},{"full_name":"Polynomial.map_cyclotomic","def_path":"Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean","def_pos":[264,8],"def_end_pos":[264,22]},{"full_name":"Polynomial.map_pow","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[783,18],"def_end_pos":[783,25]},{"full_name":"ZMod","def_path":"Mathlib/Data/ZMod/Defs.lean","def_pos":[89,4],"def_end_pos":[89,8]},{"full_name":"algebraMap","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[108,4],"def_end_pos":[108,14]}]},{"state_before":"R : Type u_1\np n : ℕ\nhp : Fact (Nat.Prime p)\ninst✝¹ : Ring R\ninst✝ : CharP R p\nhn : p ∣ n\nthis : Algebra (ZMod p) R := ZMod.algebra R p\n⊢ cyclotomic (n * p) (ZMod p) = cyclotomic n (ZMod p) ^ p","state_after":"no goals","tactic":"rw [← ZMod.expand_card, ← map_cyclotomic_int n, ← map_expand,\n    cyclotomic_expand_eq_cyclotomic hp.out hn, map_cyclotomic]","premises":[{"full_name":"Fact.out","def_path":"Mathlib/Logic/Basic.lean","def_pos":[92,2],"def_end_pos":[92,5]},{"full_name":"Polynomial.cyclotomic_expand_eq_cyclotomic","def_path":"Mathlib/RingTheory/Polynomial/Cyclotomic/Expand.lean","def_pos":[75,8],"def_end_pos":[75,39]},{"full_name":"Polynomial.map_cyclotomic","def_path":"Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean","def_pos":[264,8],"def_end_pos":[264,22]},{"full_name":"Polynomial.map_cyclotomic_int","def_path":"Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean","def_pos":[242,8],"def_end_pos":[242,26]},{"full_name":"Polynomial.map_expand","def_path":"Mathlib/Algebra/Polynomial/Expand.lean","def_pos":[158,8],"def_end_pos":[158,18]},{"full_name":"ZMod.expand_card","def_path":"Mathlib/FieldTheory/Finite/Basic.lean","def_pos":[512,8],"def_end_pos":[512,19]}]}]}
+{"url":"Mathlib/Tactic/ReduceModChar.lean","commit":"","full_name":"Tactic.ReduceModChar.CharP.intCast_eq_mod","start":[47,0],"end":[51,50],"file_path":"Mathlib/Tactic/ReduceModChar.lean","tactics":[{"state_before":"u : Level\nR : Type u_1\ninst✝¹ : Ring R\np : ℕ\ninst✝ : CharP R p\nk : ℤ\n⊢ ↑k = ↑(k % ↑p)","state_after":"no goals","tactic":"calc\n    (k : R) = ↑(k % p + p * (k / p)) := by rw [Int.emod_add_ediv]\n    _ = ↑(k % p) := by simp [CharP.cast_eq_zero R]","premises":[{"full_name":"CharP.cast_eq_zero","def_path":"Mathlib/Algebra/CharP/Defs.lean","def_pos":[56,14],"def_end_pos":[56,26]},{"full_name":"Int.emod_add_ediv","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean","def_pos":[201,8],"def_end_pos":[201,21]}]}]}
+{"url":"Mathlib/Analysis/BoundedVariation.lean","commit":"","full_name":"eVariationOn.add_le_union","start":[365,0],"end":[433,51],"file_path":"Mathlib/Analysis/BoundedVariation.lean","tactics":[{"state_before":"α : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns t : Set α\nh : ∀ x ∈ s, ∀ y ∈ t, x ≤ y\n⊢ eVariationOn f s + eVariationOn f t ≤ eVariationOn f (s ∪ t)","state_after":"case pos\nα : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns t : Set α\nh : ∀ x ∈ s, ∀ y ∈ t, x ≤ y\nhs : s = ∅\n⊢ eVariationOn f s + eVariationOn f t ≤ eVariationOn f (s ∪ t)\n\ncase neg\nα : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns t : Set α\nh : ∀ x ∈ s, ∀ y ∈ t, x ≤ y\nhs : ¬s = ∅\n⊢ eVariationOn f s + eVariationOn f t ≤ eVariationOn f (s ∪ t)","tactic":"by_cases hs : s = ∅","premises":[{"full_name":"EmptyCollection.emptyCollection","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[447,2],"def_end_pos":[447,17]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case neg\nα : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns t : Set α\nh : ∀ x ∈ s, ∀ y ∈ t, x ≤ y\nhs : ¬s = ∅\n⊢ eVariationOn f s + eVariationOn f t ≤ eVariationOn f (s ∪ t)","state_after":"case neg\nα : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns t : Set α\nh : ∀ x ∈ s, ∀ y ∈ t, x ≤ y\nhs : ¬s = ∅\nthis : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }\n⊢ eVariationOn f s + eVariationOn f t ≤ eVariationOn f (s ∪ t)","tactic":"have : Nonempty { u // Monotone u ∧ ∀ i : ℕ, u i ∈ s } :=\n    nonempty_monotone_mem (nonempty_iff_ne_empty.2 hs)","premises":[{"full_name":"And","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[516,10],"def_end_pos":[516,13]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Monotone","def_path":"Mathlib/Order/Monotone/Basic.lean","def_pos":[76,4],"def_end_pos":[76,12]},{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]},{"full_name":"Nonempty","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[709,16],"def_end_pos":[709,24]},{"full_name":"Set.nonempty_iff_ne_empty","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[485,8],"def_end_pos":[485,29]},{"full_name":"Subtype","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[583,10],"def_end_pos":[583,17]},{"full_name":"eVariationOn.nonempty_monotone_mem","def_path":"Mathlib/Analysis/BoundedVariation.lean","def_pos":[78,8],"def_end_pos":[78,29]}]},{"state_before":"case neg\nα : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns t : Set α\nh : ∀ x ∈ s, ∀ y ∈ t, x ≤ y\nhs : ¬s = ∅\nthis : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }\n⊢ eVariationOn f s + eVariationOn f t ≤ eVariationOn f (s ∪ t)","state_after":"case pos\nα : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns t : Set α\nh : ∀ x ∈ s, ∀ y ∈ t, x ≤ y\nhs : ¬s = ∅\nthis : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }\nht : t = ∅\n⊢ eVariationOn f s + eVariationOn f t ≤ eVariationOn f (s ∪ t)\n\ncase neg\nα : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns t : Set α\nh : ∀ x ∈ s, ∀ y ∈ t, x ≤ y\nhs : ¬s = ∅\nthis : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }\nht : ¬t = ∅\n⊢ eVariationOn f s + eVariationOn f t ≤ eVariationOn f (s ∪ t)","tactic":"by_cases ht : t = ∅","premises":[{"full_name":"EmptyCollection.emptyCollection","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[447,2],"def_end_pos":[447,17]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case neg\nα : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns t : Set α\nh : ∀ x ∈ s, ∀ y ∈ t, x ≤ y\nhs : ¬s = ∅\nthis : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }\nht : ¬t = ∅\n⊢ eVariationOn f s + eVariationOn f t ≤ eVariationOn f (s ∪ t)","state_after":"case neg\nα : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns t : Set α\nh : ∀ x ∈ s, ∀ y ∈ t, x ≤ y\nhs : ¬s = ∅\nthis✝ : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }\nht : ¬t = ∅\nthis : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ t }\n⊢ eVariationOn f s + eVariationOn f t ≤ eVariationOn f (s ∪ t)","tactic":"have : Nonempty { u // Monotone u ∧ ∀ i : ℕ, u i ∈ t } :=\n    nonempty_monotone_mem (nonempty_iff_ne_empty.2 ht)","premises":[{"full_name":"And","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[516,10],"def_end_pos":[516,13]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Monotone","def_path":"Mathlib/Order/Monotone/Basic.lean","def_pos":[76,4],"def_end_pos":[76,12]},{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]},{"full_name":"Nonempty","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[709,16],"def_end_pos":[709,24]},{"full_name":"Set.nonempty_iff_ne_empty","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[485,8],"def_end_pos":[485,29]},{"full_name":"Subtype","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[583,10],"def_end_pos":[583,17]},{"full_name":"eVariationOn.nonempty_monotone_mem","def_path":"Mathlib/Analysis/BoundedVariation.lean","def_pos":[78,8],"def_end_pos":[78,29]}]},{"state_before":"case neg\nα : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns t : Set α\nh : ∀ x ∈ s, ∀ y ∈ t, x ≤ y\nhs : ¬s = ∅\nthis✝ : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }\nht : ¬t = ∅\nthis : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ t }\n⊢ eVariationOn f s + eVariationOn f t ≤ eVariationOn f (s ∪ t)","state_after":"case neg\nα : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns t : Set α\nh : ∀ x ∈ s, ∀ y ∈ t, x ≤ y\nhs : ¬s = ∅\nthis✝ : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }\nht : ¬t = ∅\nthis : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ t }\n⊢ ∀ (i : ℕ × { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }) (j : ℕ × { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ t }),\n    ∑ i_1 ∈ Finset.range i.1, edist (f (↑i.2 (i_1 + 1))) (f (↑i.2 i_1)) +\n        ∑ i ∈ Finset.range j.1, edist (f (↑j.2 (i + 1))) (f (↑j.2 i)) ≤\n      eVariationOn f (s ∪ t)","tactic":"refine ENNReal.iSup_add_iSup_le ?_","premises":[{"full_name":"ENNReal.iSup_add_iSup_le","def_path":"Mathlib/Topology/Instances/ENNReal.lean","def_pos":[524,8],"def_end_pos":[524,24]}]},{"state_before":"case neg\nα : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns t : Set α\nh : ∀ x ∈ s, ∀ y ∈ t, x ≤ y\nhs : ¬s = ∅\nthis✝ : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }\nht : ¬t = ∅\nthis : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ t }\n⊢ ∀ (i : ℕ × { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }) (j : ℕ × { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ t }),\n    ∑ i_1 ∈ Finset.range i.1, edist (f (↑i.2 (i_1 + 1))) (f (↑i.2 i_1)) +\n        ∑ i ∈ Finset.range j.1, edist (f (↑j.2 (i + 1))) (f (↑j.2 i)) ≤\n      eVariationOn f (s ∪ t)","state_after":"case neg.mk.mk.intro.mk.mk.intro\nα : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns t : Set α\nh : ∀ x ∈ s, ∀ y ∈ t, x ≤ y\nhs : ¬s = ∅\nthis✝ : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }\nht : ¬t = ∅\nthis : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ t }\nn : ℕ\nu : ℕ → α\nhu : Monotone u\nus : ∀ (i : ℕ), u i ∈ s\nm : ℕ\nv : ℕ → α\nhv : Monotone v\nvt : ∀ (i : ℕ), v i ∈ t\n⊢ ∑ i ∈ Finset.range (n, ⟨u, ⋯⟩).1, edist (f (↑(n, ⟨u, ⋯⟩).2 (i + 1))) (f (↑(n, ⟨u, ⋯⟩).2 i)) +\n      ∑ i ∈ Finset.range (m, ⟨v, ⋯⟩).1, edist (f (↑(m, ⟨v, ⋯⟩).2 (i + 1))) (f (↑(m, ⟨v, ⋯⟩).2 i)) ≤\n    eVariationOn f (s ∪ t)","tactic":"rintro ⟨n, ⟨u, hu, us⟩⟩ ⟨m, ⟨v, hv, vt⟩⟩","premises":[]},{"state_before":"case neg.mk.mk.intro.mk.mk.intro\nα : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns t : Set α\nh : ∀ x ∈ s, ∀ y ∈ t, x ≤ y\nhs : ¬s = ∅\nthis✝ : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }\nht : ¬t = ∅\nthis : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ t }\nn : ℕ\nu : ℕ → α\nhu : Monotone u\nus : ∀ (i : ℕ), u i ∈ s\nm : ℕ\nv : ℕ → α\nhv : Monotone v\nvt : ∀ (i : ℕ), v i ∈ t\n⊢ ∑ i ∈ Finset.range (n, ⟨u, ⋯⟩).1, edist (f (↑(n, ⟨u, ⋯⟩).2 (i + 1))) (f (↑(n, ⟨u, ⋯⟩).2 i)) +\n      ∑ i ∈ Finset.range (m, ⟨v, ⋯⟩).1, edist (f (↑(m, ⟨v, ⋯⟩).2 (i + 1))) (f (↑(m, ⟨v, ⋯⟩).2 i)) ≤\n    eVariationOn f (s ∪ t)","state_after":"case neg.mk.mk.intro.mk.mk.intro\nα : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns t : Set α\nh : ∀ x ∈ s, ∀ y ∈ t, x ≤ y\nhs : ¬s = ∅\nthis✝ : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }\nht : ¬t = ∅\nthis : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ t }\nn : ℕ\nu : ℕ → α\nhu : Monotone u\nus : ∀ (i : ℕ), u i ∈ s\nm : ℕ\nv : ℕ → α\nhv : Monotone v\nvt : ∀ (i : ℕ), v i ∈ t\nw : ℕ → α := fun i => if i ≤ n then u i else v (i - (n + 1))\n⊢ ∑ i ∈ Finset.range (n, ⟨u, ⋯⟩).1, edist (f (↑(n, ⟨u, ⋯⟩).2 (i + 1))) (f (↑(n, ⟨u, ⋯⟩).2 i)) +\n      ∑ i �� Finset.range (m, ⟨v, ⋯⟩).1, edist (f (↑(m, ⟨v, ⋯⟩).2 (i + 1))) (f (↑(m, ⟨v, ⋯⟩).2 i)) ≤\n    eVariationOn f (s ∪ t)","tactic":"let w i := if i ≤ n then u i else v (i - (n + 1))","premises":[{"full_name":"ite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[970,20],"def_end_pos":[970,23]}]}]}
+{"url":".lake/packages/batteries/Batteries/Data/String/Lemmas.lean","commit":"","full_name":"String.join_eq.go","start":[448,0],"end":[451,45],"file_path":".lake/packages/batteries/Batteries/Data/String/Lemmas.lean","tactics":[{"state_before":"ss : List String\nx✝ : List Char\n⊢ List.foldl (fun x x_1 => x ++ x_1) { data := x✝ } [] = { data := x✝ ++ (List.map data []).join }","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"ss✝ : List String\ns : List Char\nss : List String\nx✝ : List Char\n⊢ { data := x✝ ++ s ++ (List.map data ss).join } = { data := x✝ ++ (List.map data ({ data := s } :: ss)).join }","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/Order/ModularLattice.lean","commit":"","full_name":"wellFounded_lt_exact_sequence","start":[221,0],"end":[237,36],"file_path":"Mathlib/Order/ModularLattice.lean","tactics":[{"state_before":"α : Type u_1\ninst✝³ : Lattice α\ninst✝² : IsModularLattice α\nx y z : α\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : PartialOrder β\ninst✝ : Preorder γ\nh₁ : WellFounded fun x x_1 => x < x_1\nh₂ : WellFounded fun x x_1 => x < x_1\nK : α\nf₁ : β → α\nf₂ : α → β\ng₁ : γ → α\ng₂ : α → γ\ngci : GaloisCoinsertion f₁ f₂\ngi : GaloisInsertion g₂ g₁\nhf : ∀ (a : α), f₁ (f₂ a) = a ⊓ K\nhg : ∀ (a : α), g₁ (g₂ a) = a ⊔ K\nA B : α\nhAB : A < B\n⊢ Prod.Lex (fun x x_1 => x < x_1) (fun x x_1 => x < x_1) (f₂ A, g₂ A) (f₂ B, g₂ B)","state_after":"α : Type u_1\ninst✝³ : Lattice α\ninst✝² : IsModularLattice α\nx y z : α\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : PartialOrder β\ninst✝ : Preorder γ\nh₁ : WellFounded fun x x_1 => x < x_1\nh₂ : WellFounded fun x x_1 => x < x_1\nK : α\nf₁ : β → α\nf₂ : α → β\ng₁ : γ → α\ng₂ : α → γ\ngci : GaloisCoinsertion f₁ f₂\ngi : GaloisInsertion g₂ g₁\nhf : ∀ (a : α), f₁ (f₂ a) = a ⊓ K\nhg : ∀ (a : α), g₁ (g₂ a) = a ⊔ K\nA B : α\nhAB : A < B\n⊢ A ⊓ K ≤ B ⊓ K ∧ ¬B ⊓ K ≤ A ⊓ K ∨ (A ⊓ K ≤ B ⊓ K ∧ B ⊓ K ≤ A ⊓ K) ∧ A ⊔ K ≤ B ⊔ K ∧ ¬B ⊔ K ≤ A ⊔ K","tactic":"simp only [Prod.lex_def, lt_iff_le_not_le, ← gci.l_le_l_iff, ← gi.u_le_u_iff, hf, hg,\n          le_antisymm_iff]","premises":[{"full_name":"GaloisCoinsertion.l_le_l_iff","def_path":"Mathlib/Order/GaloisConnection.lean","def_pos":[728,8],"def_end_pos":[728,18]},{"full_name":"GaloisInsertion.u_le_u_iff","def_path":"Mathlib/Order/GaloisConnection.lean","def_pos":[521,8],"def_end_pos":[521,18]},{"full_name":"Prod.lex_def","def_path":".lake/packages/lean4/src/lean/Init/WF.lean","def_pos":[228,8],"def_end_pos":[228,15]},{"full_name":"le_antisymm_iff","def_path":"Mathlib/Order/Defs.lean","def_pos":[161,8],"def_end_pos":[161,23]},{"full_name":"lt_iff_le_not_le","def_path":"Mathlib/Order/Defs.lean","def_pos":[51,8],"def_end_pos":[51,24]}]},{"state_before":"α : Type u_1\ninst✝³ : Lattice α\ninst✝² : IsModularLattice α\nx y z : α\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : PartialOrder β\ninst✝ : Preorder γ\nh₁ : WellFounded fun x x_1 => x < x_1\nh₂ : WellFounded fun x x_1 => x < x_1\nK : α\nf₁ : β → α\nf₂ : α → β\ng₁ : γ → α\ng₂ : α → γ\ngci : GaloisCoinsertion f₁ f₂\ngi : GaloisInsertion g₂ g₁\nhf : ∀ (a : α), f₁ (f₂ a) = a ⊓ K\nhg : ∀ (a : α), g₁ (g₂ a) = a ⊔ K\nA B : α\nhAB : A < B\n⊢ A ⊓ K ≤ B ⊓ K ∧ ¬B ⊓ K ≤ A ⊓ K ∨ (A ⊓ K ≤ B ⊓ K ∧ B ⊓ K ≤ A ⊓ K) ∧ A ⊔ K ≤ B ⊔ K ∧ ¬B ⊔ K ≤ A ⊔ K","state_after":"α : Type u_1\ninst✝³ : Lattice α\ninst✝² : IsModularLattice α\nx y z : α\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : PartialOrder β\ninst✝ : Preorder γ\nh₁ : WellFounded fun x x_1 => x < x_1\nh₂ : WellFounded fun x x_1 => x < x_1\nK : α\nf₁ : β → α\nf₂ : α → β\ng₁ : γ → α\ng₂ : α → γ\ngci : GaloisCoinsertion f₁ f₂\ngi : GaloisInsertion g₂ g₁\nhf : ∀ (a : α), f₁ (f₂ a) = a ⊓ K\nhg : ∀ (a : α), g₁ (g₂ a) = a ⊔ K\nA B : α\nhAB : A < B\n⊢ A ⊓ K < B ⊓ K ∨ A ⊓ K = B ⊓ K ∧ A ⊔ K < B ⊔ K","tactic":"simp only [gci.l_le_l_iff, gi.u_le_u_iff, ← lt_iff_le_not_le, ← le_antisymm_iff]","premises":[{"full_name":"GaloisCoinsertion.l_le_l_iff","def_path":"Mathlib/Order/GaloisConnection.lean","def_pos":[728,8],"def_end_pos":[728,18]},{"full_name":"GaloisInsertion.u_le_u_iff","def_path":"Mathlib/Order/GaloisConnection.lean","def_pos":[521,8],"def_end_pos":[521,18]},{"full_name":"le_antisymm_iff","def_path":"Mathlib/Order/Defs.lean","def_pos":[161,8],"def_end_pos":[161,23]},{"full_name":"lt_iff_le_not_le","def_path":"Mathlib/Order/Defs.lean","def_pos":[51,8],"def_end_pos":[51,24]}]},{"state_before":"α : Type u_1\ninst✝³ : Lattice α\ninst✝² : IsModularLattice α\nx y z : α\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : PartialOrder β\ninst✝ : Preorder γ\nh₁ : WellFounded fun x x_1 => x < x_1\nh₂ : WellFounded fun x x_1 => x < x_1\nK : α\nf₁ : β → α\nf₂ : α → β\ng₁ : γ → α\ng₂ : α → γ\ngci : GaloisCoinsertion f₁ f₂\ngi : GaloisInsertion g₂ g₁\nhf : ∀ (a : α), f₁ (f₂ a) = a ⊓ K\nhg : ∀ (a : α), g₁ (g₂ a) = a ⊔ K\nA B : α\nhAB : A < B\n⊢ A ⊓ K < B ⊓ K ∨ A ⊓ K = B ⊓ K ∧ A ⊔ K < B ⊔ K","state_after":"case inl\nα : Type u_1\ninst✝³ : Lattice α\ninst✝² : IsModularLattice α\nx y z : α\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : PartialOrder β\ninst✝ : Preorder γ\nh₁ : WellFounded fun x x_1 => x < x_1\nh₂ : WellFounded fun x x_1 => x < x_1\nK : α\nf₁ : β → α\nf₂ : α → β\ng₁ : γ → α\ng₂ : α → γ\ngci : GaloisCoinsertion f₁ f₂\ngi : GaloisInsertion g₂ g₁\nhf : ∀ (a : α), f₁ (f₂ a) = a ⊓ K\nhg : ∀ (a : α), g₁ (g₂ a) = a ⊔ K\nA B : α\nhAB : A < B\nh : A ⊓ K < B ⊓ K\n⊢ A ⊓ K < B ⊓ K ∨ A ⊓ K = B ⊓ K ∧ A ⊔ K < B ⊔ K\n\ncase inr\nα : Type u_1\ninst✝³ : Lattice α\ninst✝² : IsModularLattice α\nx y z : α\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : PartialOrder β\ninst✝ : Preorder γ\nh₁ : WellFounded fun x x_1 => x < x_1\nh₂ : WellFounded fun x x_1 => x < x_1\nK : α\nf₁ : β → α\nf₂ : α → β\ng₁ : γ → α\ng₂ : α → γ\ngci : GaloisCoinsertion f₁ f₂\ngi : GaloisInsertion g₂ g₁\nhf : ∀ (a : α), f₁ (f₂ a) = a ⊓ K\nhg : ∀ (a : α), g₁ (g₂ a) = a ⊔ K\nA B : α\nhAB : A < B\nh : A ⊓ K = B ⊓ K\n⊢ A ⊓ K < B ⊓ K ∨ A ⊓ K = B ⊓ K ∧ A ⊔ K < B ⊔ K","tactic":"rcases lt_or_eq_of_le (inf_le_inf_right K (le_of_lt hAB)) with h | h","premises":[{"full_name":"inf_le_inf_right","def_path":"Mathlib/Order/Lattice.lean","def_pos":[374,8],"def_end_pos":[374,24]},{"full_name":"le_of_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[89,8],"def_end_pos":[89,16]},{"full_name":"lt_or_eq_of_le","def_path":"Mathlib/Order/Defs.lean","def_pos":[191,8],"def_end_pos":[191,22]}]}]}
+{"url":"Mathlib/NumberTheory/Cyclotomic/Rat.lean","commit":"","full_name":"IsPrimitiveRoot.zeta_sub_one_prime","start":[324,0],"end":[330,46],"file_path":"Mathlib/NumberTheory/Cyclotomic/Rat.lean","tactics":[{"state_before":"p : ℕ+\nk : ℕ\nK : Type u\ninst✝² : Field K\ninst✝¹ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\n⊢ Prime (hζ.toInteger - 1)","state_after":"case pos\np : ℕ+\nk : ℕ\nK : Type u\ninst✝² : Field K\ninst✝¹ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhtwo : p = 2\n⊢ Prime (hζ.toInteger - 1)\n\ncase neg\np : ℕ+\nk : ℕ\nK : Type u\ninst✝² : Field K\ninst✝¹ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhtwo : ¬p = 2\n⊢ Prime (hζ.toInteger - 1)","tactic":"by_cases htwo : p = 2","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
+{"url":"Mathlib/CategoryTheory/Adjunction/Over.lean","commit":"","full_name":"CategoryTheory.Under.mapPushoutAdj_counit_app","start":[137,0],"end":[157,3],"file_path":"Mathlib/CategoryTheory/Adjunction/Over.lean","tactics":[{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\nX✝ : C\ninst✝ : HasPushouts C\nX Y : C\nf : X ⟶ Y\nx : Under X\ny : Under Y\nu : (pushout f).obj x ⟶ y\n⊢ x.hom ≫ pushout.inl x.hom f ≫ u.right = ((map f).obj y).hom","state_after":"C : Type u\ninst✝¹ : Category.{v, u} C\nX✝ : C\ninst✝ : HasPushouts C\nX Y : C\nf : X ⟶ Y\nx : Under X\ny : Under Y\nu : (pushout f).obj x ⟶ y\n⊢ x.hom ≫ pushout.inl x.hom f ≫ u.right = f ≫ y.hom","tactic":"simp only [map_obj_hom]","premises":[{"full_name":"CategoryTheory.Under.map_obj_hom","def_path":"Mathlib/CategoryTheory/Comma/Over.lean","def_pos":[461,8],"def_end_pos":[461,19]}]},{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\nX✝ : C\ninst✝ : HasPushouts C\nX Y : C\nf : X ⟶ Y\nx : Under X\ny : Under Y\nu : (pushout f).obj x ⟶ y\n⊢ x.hom ≫ pushout.inl x.hom f ≫ u.right = f ≫ y.hom","state_after":"C : Type u\ninst✝¹ : Category.{v, u} C\nX✝ : C\ninst✝ : HasPushouts C\nX Y : C\nf : X ⟶ Y\nx : Under X\ny : Under Y\nu : (pushout f).obj x ⟶ y\n⊢ x.hom ≫ pushout.inl x.hom f ≫ u.right = f ≫ ((pushout f).obj x).hom ≫ u.right","tactic":"rw [← Under.w u]","premises":[{"full_name":"CategoryTheory.Under.w","def_path":"Mathlib/CategoryTheory/Comma/Over.lean","def_pos":[391,8],"def_end_pos":[391,9]}]},{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\nX✝ : C\ninst✝ : HasPushouts C\nX Y : C\nf : X ��� Y\nx : Under X\ny : Under Y\nu : (pushout f).obj x ⟶ y\n⊢ x.hom ≫ pushout.inl x.hom f ≫ u.right = f ≫ ((pushout f).obj x).hom ≫ u.right","state_after":"C : Type u\ninst✝¹ : Category.{v, u} C\nX✝ : C\ninst✝ : HasPushouts C\nX Y : C\nf : X ⟶ Y\nx : Under X\ny : Under Y\nu : (pushout f).obj x ⟶ y\n⊢ x.hom ≫ pushout.inl x.hom f ≫ u.right = f ≫ pushout.inr x.hom f ≫ u.right","tactic":"simp only [Functor.const_obj_obj, map_obj_right, Functor.id_obj, pushout_obj, mk_right,\n          mk_hom]","premises":[{"full_name":"CategoryTheory.Functor.const_obj_obj","def_path":"Mathlib/CategoryTheory/Functor/Const.lean","def_pos":[32,2],"def_end_pos":[32,7]},{"full_name":"CategoryTheory.Functor.id_obj","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[85,8],"def_end_pos":[85,14]},{"full_name":"CategoryTheory.Under.map_obj_right","def_path":"Mathlib/CategoryTheory/Comma/Over.lean","def_pos":[457,8],"def_end_pos":[457,21]},{"full_name":"CategoryTheory.Under.mk_hom","def_path":"Mathlib/CategoryTheory/Comma/Over.lean","def_pos":[394,15],"def_end_pos":[394,18]},{"full_name":"CategoryTheory.Under.mk_right","def_path":"Mathlib/CategoryTheory/Comma/Over.lean","def_pos":[394,9],"def_end_pos":[394,14]},{"full_name":"CategoryTheory.Under.pushout_obj","def_path":"Mathlib/CategoryTheory/Adjunction/Over.lean","def_pos":[130,2],"def_end_pos":[130,7]}]},{"state_before":"C : Type u\ninst✝¹ 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+{"url":"Mathlib/MeasureTheory/Measure/Haar/Basic.lean","commit":"","full_name":"MeasureTheory.Measure.haar.chaar_mem_clPrehaar","start":[354,0],"end":[358,14],"file_path":"Mathlib/MeasureTheory/Measure/Haar/Basic.lean","tactics":[{"state_before":"G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₀ : PositiveCompacts G\nV : OpenNhdsOf 1\n⊢ chaar K₀ ∈ clPrehaar (↑K₀) V","state_after":"G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₀ : PositiveCompacts G\nV : OpenNhdsOf 1\nthis : Classical.choose ⋯ ∈ ⋂ V, clPrehaar (↑K₀) V\n⊢ chaar K₀ ∈ clPrehaar (↑K₀) V","tactic":"have := (Classical.choose_spec (nonempty_iInter_clPrehaar K₀)).2","premises":[{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"Classical.choose_spec","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[28,8],"def_end_pos":[28,19]},{"full_name":"MeasureTheory.Measure.haar.nonempty_iInter_clPrehaar","def_path":"Mathlib/MeasureTheory/Measure/Haar/Basic.lean","def_pos":[321,8],"def_end_pos":[321,33]}]},{"state_before":"G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₀ : PositiveCompacts G\nV : OpenNhdsOf 1\nthis : Classical.choose ⋯ ∈ ⋂ V, clPrehaar (↑K₀) V\n⊢ chaar K₀ ∈ clPrehaar (↑K₀) V","state_after":"G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₀ : PositiveCompacts G\nV : OpenNhdsOf 1\nthis : ∀ (i : OpenNhdsOf 1), Classical.choose ⋯ ∈ clPrehaar (↑K₀) i\n⊢ chaar K₀ ∈ clPrehaar (↑K₀) V","tactic":"rw [mem_iInter] at this","premises":[{"full_name":"Set.mem_iInter","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[259,8],"def_end_pos":[259,18]}]},{"state_before":"G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₀ : PositiveCompacts G\nV : OpenNhdsOf 1\nthis : ∀ (i : OpenNhdsOf 1), Classical.choose ⋯ ∈ clPrehaar (↑K₀) i\n⊢ chaar K₀ ∈ clPrehaar (↑K₀) V","state_after":"no goals","tactic":"exact this V","premises":[]}]}
+{"url":"Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean","commit":"","full_name":"Matrix.charpoly_degree_eq_dim","start":[89,0],"end":[112,9],"file_path":"Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean","tactics":[{"state_before":"R : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM✝ : Matrix n n R\ninst✝ : Nontrivial R\nM : Matrix n n R\n⊢ M.charpoly.degree = ↑(Fintype.card n)","state_after":"case pos\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM✝ : Matrix n n R\ninst✝ : Nontrivial R\nM : Matrix n n R\nh : Fintype.card n = 0\n⊢ M.charpoly.degree = ↑(Fintype.card n)\n\ncase neg\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM✝ : Matrix n n R\ninst✝ : Nontrivial R\nM : Matrix n n R\nh : ¬Fintype.card n = 0\n⊢ M.charpoly.degree = ↑(Fintype.card n)","tactic":"by_cases h : Fintype.card n = 0","premises":[{"full_name":"Fintype.card","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[62,4],"def_end_pos":[62,8]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case neg\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM✝ : Matrix n n R\ninst✝ : Nontrivial R\nM : Matrix n n R\nh : ¬Fintype.card n = 0\n⊢ M.charpoly.degree = ↑(Fintype.card n)","state_after":"case neg\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM✝ : Matrix n n R\ninst✝ : Nontrivial R\nM : Matrix n n R\nh : ¬Fintype.card n = 0\n⊢ (M.charpoly - ∏ i : n, (X - C (M i i)) + ∏ i : n, (X - C (M i i))).degree = ↑(Fintype.card n)","tactic":"rw [← sub_add_cancel M.charpoly (∏ i : n, (X - C (M i i)))]","premises":[{"full_name":"Finset","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[133,10],"def_end_pos":[133,16]},{"full_name":"Finset.prod","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[58,14],"def_end_pos":[58,18]},{"full_name":"Finset.univ","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[67,4],"def_end_pos":[67,8]},{"full_name":"Matrix.charpoly","def_path":"Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean","def_pos":[90,4],"def_end_pos":[90,12]},{"full_name":"Polynomial.C","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[426,4],"def_end_pos":[426,5]},{"full_name":"Polynomial.X","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[474,4],"def_end_pos":[474,5]},{"full_name":"sub_add_cancel","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[691,2],"def_end_pos":[691,13]}]},{"state_before":"case neg\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM✝ : Matrix n n R\ninst✝ : Nontrivial R\nM : Matrix n n R\nh : ¬Fintype.card n = 0\nh1 : (∏ i : n, (X - C (M i i))).degree = ↑(Fintype.card n)\n⊢ (M.charpoly - ∏ i : n, (X - C (M i i)) + ∏ i : n, (X - C (M i i))).degree = ↑(Fintype.card n)","state_after":"case neg\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM✝ : Matrix n n R\ninst✝ : Nontrivial R\nM : Matrix n n R\nh : ¬Fintype.card n = 0\nh1 : (∏ i : n, (X - C (M i i))).degree = ↑(Fintype.card n)\n⊢ (∏ i : n, (X - C (M i i))).degree = ↑(Fintype.card n)\n\ncase neg\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM✝ : Matrix n n R\ninst✝ : Nontrivial R\nM : Matrix n n R\nh : ¬Fintype.card n = 0\nh1 : (∏ i : n, (X - C (M i i))).degree = ↑(Fintype.card n)\n⊢ (M.charpoly - ∏ i : n, (X - C (M i i))).degree < (∏ i : n, (X - C (M i i))).degree","tactic":"rw [degree_add_eq_right_of_degree_lt]","premises":[{"full_name":"Polynomial.degree_add_eq_right_of_degree_lt","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[613,8],"def_end_pos":[613,40]}]},{"state_before":"case neg\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM✝ : Matrix n n R\ninst✝ : Nontrivial R\nM : Matrix n n R\nh : ¬Fintype.card n = 0\nh1 : (∏ i : n, (X - C (M i i))).degree = ↑(Fintype.card n)\n⊢ (M.charpoly - ∏ i : n, (X - C (M i i))).degree < (∏ i : n, (X - C (M i i))).degree","state_after":"case neg\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM✝ : Matrix n n R\ninst✝ : Nontrivial R\nM : Matrix n n R\nh : ¬Fintype.card n = 0\nh1 : (∏ i : n, (X - C (M i i))).degree = ↑(Fintype.card n)\n⊢ (M.charpoly - ∏ i : n, (X - C (M i i))).degree < ↑(Fintype.card n)","tactic":"rw [h1]","premises":[]},{"state_before":"case neg\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM✝ : Matrix n n R\ninst✝ : Nontrivial R\nM : Matrix n n R\nh : ¬Fintype.card n = 0\nh1 : (∏ i : n, (X - C (M i i))).degree = ↑(Fintype.card n)\n⊢ (M.charpoly - ∏ i : n, (X - C (M i i))).degree < ↑(Fintype.card n)","state_after":"case neg\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM✝ : Matrix n n R\ninst✝ : Nontrivial R\nM : Matrix n n R\nh : ¬Fintype.card n = 0\nh1 : (∏ i : n, (X - C (M i i))).degree = ↑(Fintype.card n)\n⊢ ↑(Fintype.card n - 1) < ↑(Fintype.card n)","tactic":"apply lt_trans (charpoly_sub_diagonal_degree_lt M)","premises":[{"full_name":"Matrix.charpoly_sub_diagonal_degree_lt","def_path":"Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean","def_pos":[57,8],"def_end_pos":[57,39]},{"full_name":"lt_trans","def_path":"Mathlib/Order/Defs.lean","def_pos":[74,8],"def_end_pos":[74,16]}]},{"state_before":"case neg\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM✝ : Matrix n n R\ninst✝ : Nontrivial R\nM : Matrix n n R\nh : ¬Fintype.card n = 0\nh1 : (∏ i : n, (X - C (M i i))).degree = ↑(Fintype.card n)\n⊢ ↑(Fintype.card n - 1) < ↑(Fintype.card n)","state_after":"case neg\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM✝ : Matrix n n R\ninst✝ : Nontrivial R\nM : Matrix n n R\nh : ¬Fintype.card n = 0\nh1 : (∏ i : n, (X - C (M i i))).degree = ↑(Fintype.card n)\n⊢ Fintype.card n - 1 < Fintype.card n","tactic":"rw [Nat.cast_lt]","premises":[{"full_name":"Nat.cast_lt","def_path":"Mathlib/Data/Nat/Cast/Order/Basic.lean","def_pos":[82,8],"def_end_pos":[82,15]}]},{"state_before":"case neg\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM✝ : Matrix n n R\ninst✝ : Nontrivial R\nM : Matrix n n R\nh : ¬Fintype.card n = 0\nh1 : (∏ i : n, (X - C (M i i))).degree = ↑(Fintype.card n)\n⊢ Fintype.card n - 1 < Fintype.card n","state_after":"case neg\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM✝ : Matrix n n R\ninst✝ : Nontrivial R\nM : Matrix n n R\nh : ¬Fintype.card n = 0\nh1 : (∏ i : n, (X - C (M i i))).degree = ↑(Fintype.card n)\n⊢ (Fintype.card n).pred < Fintype.card n","tactic":"rw [← Nat.pred_eq_sub_one]","premises":[{"full_name":"Nat.pred_eq_sub_one","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[840,16],"def_end_pos":[840,31]}]},{"state_before":"case neg\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM✝ : Matrix n n R\ninst✝ : Nontrivial R\nM : Matrix n n R\nh : ¬Fintype.card n = 0\nh1 : (∏ i : n, (X - C (M i i))).degree = ↑(Fintype.card n)\n⊢ (Fintype.card n).pred < Fintype.card n","state_after":"case neg.a\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM✝ : Matrix n n R\ninst✝ : Nontrivial R\nM : Matrix n n R\nh : ¬Fintype.card n = 0\nh1 : (∏ i : n, (X - C (M i i))).degree = ↑(Fintype.card n)\n⊢ Fintype.card n ≠ 0","tactic":"apply Nat.pred_lt","premises":[{"full_name":"Nat.pred_lt","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[285,8],"def_end_pos":[285,15]}]},{"state_before":"case neg.a\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM✝ : Matrix n n R\ninst✝ : Nontrivial R\nM : Matrix n n R\nh : ¬Fintype.card n = 0\nh1 : (∏ i : n, (X - C (M i i))).degree = ↑(Fintype.card n)\n⊢ Fintype.card n ≠ 0","state_after":"no goals","tactic":"apply h","premises":[]}]}
+{"url":"Mathlib/Analysis/NormedSpace/Pointwise.lean","commit":"","full_name":"cthickening_thickening","start":[263,0],"end":[268,29],"file_path":"Mathlib/Analysis/NormedSpace/Pointwise.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : NormedField 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace ℝ E\nx✝ y z : E\nδ ε : ℝ\nhε : 0 ≤ ε\nhδ : 0 < δ\ns : Set E\nx : E\n⊢ x ∈ cthickening (ε + δ) s → x ∈ cthickening ε (thickening δ s)","state_after":"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : NormedField 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace ℝ E\nx✝ y z : E\nδ ε : ℝ\nhε : 0 ≤ ε\nhδ : 0 < δ\ns : Set E\nx : E\n⊢ infEdist x s ≤ ENNReal.ofReal ε + ENNReal.ofReal δ → infEdist x s - ENNReal.ofReal δ ≤ ENNReal.ofReal ε","tactic":"simp_rw [mem_cthickening_iff, ENNReal.ofReal_add hε hδ.le, infEdist_thickening hδ]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"ENNReal.ofReal_add","def_path":"Mathlib/Data/ENNReal/Real.lean","def_pos":[59,8],"def_end_pos":[59,18]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Metric.mem_cthickening_iff","def_path":"Mathlib/Topology/MetricSpace/Thickening.lean","def_pos":[181,8],"def_end_pos":[181,27]},{"full_name":"infEdist_thickening","def_path":"Mathlib/Analysis/NormedSpace/Pointwise.lean","def_pos":[231,8],"def_end_pos":[231,27]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : NormedField 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace ℝ E\nx✝ y z : E\nδ ε : ℝ\nhε : 0 ≤ ε\nhδ : 0 < δ\ns : Set E\nx : E\n⊢ infEdist x s ≤ ENNReal.ofReal ε + ENNReal.ofReal δ → infEdist x s - ENNReal.ofReal δ ≤ ENNReal.ofReal ε","state_after":"no goals","tactic":"exact tsub_le_iff_right.2","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"tsub_le_iff_right","def_path":"Mathlib/Algebra/Order/Sub/Defs.lean","def_pos":[59,8],"def_end_pos":[59,25]}]}]}
+{"url":"Mathlib/FieldTheory/Galois.lean","commit":"","full_name":"IsGalois.is_separable_splitting_field","start":[289,0],"end":[297,59],"file_path":"Mathlib/FieldTheory/Galois.lean","tactics":[{"state_before":"F : Type u_1\ninst✝⁴ : Field F\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : FiniteDimensional F E\ninst✝ : IsGalois F E\n⊢ ∃ p, p.Separable ∧ Polynomial.IsSplittingField F E p","state_after":"case intro\nF : Type u_1\ninst✝⁴ : Field F\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : FiniteDimensional F E\ninst✝ : IsGalois F E\nα : E\nh1 : F⟮α⟯ = ⊤\n⊢ ∃ p, p.Separable ∧ Polynomial.IsSplittingField F E p","tactic":"cases' Field.exists_primitive_element F E with α h1","premises":[{"full_name":"Field.exists_primitive_element","def_path":"Mathlib/FieldTheory/PrimitiveElement.lean","def_pos":[208,8],"def_end_pos":[208,32]}]},{"state_before":"case intro\nF : Type u_1\ninst✝⁴ : Field F\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : FiniteDimensional F E\ninst✝ : IsGalois F E\nα : E\nh1 : F⟮α⟯ = ⊤\n⊢ ∃ p, p.Separable ∧ Polynomial.IsSplittingField F E p","state_after":"case adjoin_rootSet'\nF : Type u_1\ninst✝⁴ : Field F\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : FiniteDimensional F E\ninst✝ : IsGalois F E\nα : E\nh1 : F⟮α⟯ = ⊤\n⊢ Algebra.adjoin F ((minpoly F α).rootSet E) = ⊤","tactic":"use minpoly F α, separable F α, IsGalois.splits F α","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"IsGalois.separable","def_path":"Mathlib/FieldTheory/Galois.lean","def_pos":[72,8],"def_end_pos":[72,17]},{"full_name":"IsGalois.splits","def_path":"Mathlib/FieldTheory/Galois.lean","def_pos":[75,8],"def_end_pos":[75,14]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]},{"full_name":"minpoly","def_path":"Mathlib/FieldTheory/Minpoly/Basic.lean","def_pos":[36,18],"def_end_pos":[36,25]}]},{"state_before":"case adjoin_rootSet'\nF : Type u_1\ninst✝⁴ : Field F\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : FiniteDimensional F E\ninst✝ : IsGalois F E\nα : E\nh1 : F⟮α⟯ = ⊤\n⊢ Algebra.adjoin F ((minpoly F α).rootSet E) = ⊤","state_after":"case adjoin_rootSet'\nF : Type u_1\ninst✝⁴ : Field F\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : FiniteDimensional F E\ninst✝ : IsGalois F E\nα : E\nh1 : F⟮α⟯ = ⊤\n⊢ F⟮α⟯.toSubalgebra ≤ Algebra.adjoin F ((minpoly F α).rootSet E)","tactic":"rw [eq_top_iff, ← IntermediateField.top_toSubalgebra, ← h1]","premises":[{"full_name":"IntermediateField.top_toSubalgebra","def_path":"Mathlib/FieldTheory/Adjoin.lean","def_pos":[118,8],"def_end_pos":[118,24]},{"full_name":"eq_top_iff","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[116,8],"def_end_pos":[116,18]}]},{"state_before":"case adjoin_rootSet'\nF : Type u_1\ninst✝⁴ : Field F\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : FiniteDimensional F E\ninst✝ : IsGalois F E\nα : E\nh1 : F⟮α⟯ = ⊤\n⊢ F⟮α⟯.toSubalgebra ≤ Algebra.adjoin F ((minpoly F α).rootSet E)","state_after":"case adjoin_rootSet'\nF : Type u_1\ninst✝⁴ : Field F\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : FiniteDimensional F E\ninst✝ : IsGalois F E\nα : E\nh1 : F⟮α⟯ = ⊤\n⊢ Algebra.adjoin F {α} ≤ Algebra.adjoin F ((minpoly F α).rootSet E)","tactic":"rw [IntermediateField.adjoin_simple_toSubalgebra_of_integral (integral F α)]","premises":[{"full_name":"IntermediateField.adjoin_simple_toSubalgebra_of_integral","def_path":"Mathlib/FieldTheory/Adjoin.lean","def_pos":[564,8],"def_end_pos":[564,46]},{"full_name":"IsGalois.integral","def_path":"Mathlib/FieldTheory/Galois.lean","def_pos":[69,8],"def_end_pos":[69,16]}]},{"state_before":"case adjoin_rootSet'\nF : Type u_1\ninst✝⁴ : Field F\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : FiniteDimensional F E\ninst✝ : IsGalois F E\nα : E\nh1 : F⟮α⟯ = ⊤\n⊢ Algebra.adjoin F {α} ≤ Algebra.adjoin F ((minpoly F α).rootSet E)","state_after":"case adjoin_rootSet'.H\nF : Type u_1\ninst✝⁴ : Field F\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : FiniteDimensional F E\ninst✝ : IsGalois F E\nα : E\nh1 : F⟮α⟯ = ⊤\n⊢ {α} ⊆ (minpoly F α).rootSet E","tactic":"apply Algebra.adjoin_mono","premises":[{"full_name":"Algebra.adjoin_mono","def_path":"Mathlib/RingTheory/Adjoin/Basic.lean","def_pos":[56,8],"def_end_pos":[56,19]}]},{"state_before":"case adjoin_rootSet'.H\nF : Type u_1\ninst✝⁴ : Field F\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : FiniteDimensional F E\ninst✝ : IsGalois F E\nα : E\nh1 : F⟮α⟯ = ⊤\n⊢ {α} ⊆ (minpoly F α).rootSet E","state_after":"case adjoin_rootSet'.H\nF : Type u_1\ninst✝⁴ : Field F\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : FiniteDimensional F E\ninst✝ : IsGalois F E\nα : E\nh1 : F⟮α⟯ = ⊤\n⊢ minpoly F α ≠ 0 ∧ (Polynomial.aeval α) (minpoly F α) = 0","tactic":"rw [Set.singleton_subset_iff, Polynomial.mem_rootSet]","premises":[{"full_name":"Polynomial.mem_rootSet","def_path":"Mathlib/Algebra/Polynomial/Roots.lean","def_pos":[496,8],"def_end_pos":[496,19]},{"full_name":"Set.singleton_subset_iff","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1047,8],"def_end_pos":[1047,28]}]},{"state_before":"case adjoin_rootSet'.H\nF : Type u_1\ninst✝⁴ : Field F\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : FiniteDimensional F E\ninst✝ : IsGalois F E\nα : E\nh1 : F⟮α⟯ = ⊤\n⊢ minpoly F α ≠ 0 ∧ (Polynomial.aeval α) (minpoly F α) = 0","state_after":"no goals","tactic":"exact ⟨minpoly.ne_zero (integral F α), minpoly.aeval _ _⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"IsGalois.integral","def_path":"Mathlib/FieldTheory/Galois.lean","def_pos":[69,8],"def_end_pos":[69,16]},{"full_name":"minpoly.aeval","def_path":"Mathlib/FieldTheory/Minpoly/Basic.lean","def_pos":[79,8],"def_end_pos":[79,13]},{"full_name":"minpoly.ne_zero","def_path":"Mathlib/FieldTheory/Minpoly/Basic.lean","def_pos":[55,8],"def_end_pos":[55,15]}]}]}
+{"url":"Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean","commit":"","full_name":"MeasureTheory.eLpNorm'_const","start":[347,0],"end":[353,53],"file_path":"Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean","tactics":[{"state_before":"α : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nc : F\nhq_pos : 0 < q\n⊢ eLpNorm' (fun x => c) q μ = ↑‖c‖₊ * μ Set.univ ^ (1 / q)","state_after":"α : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nc : F\nhq_pos : 0 < q\n⊢ (↑‖c‖₊ ^ q) ^ (1 / q) * μ Set.univ ^ (1 / q) = ↑‖c‖₊ * μ Set.univ ^ (1 / q)","tactic":"rw [eLpNorm', lintegral_const, ENNReal.mul_rpow_of_nonneg _ _ (by simp [hq_pos.le] : 0 ≤ 1 / q)]","premises":[{"full_name":"ENNReal.mul_rpow_of_nonneg","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean","def_pos":[588,8],"def_end_pos":[588,26]},{"full_name":"MeasureTheory.eLpNorm'","def_path":"Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean","def_pos":[69,4],"def_end_pos":[69,12]},{"full_name":"MeasureTheory.lintegral_const","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[126,8],"def_end_pos":[126,23]}]},{"state_before":"α : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nc : F\nhq_pos : 0 < q\n⊢ (↑‖c‖₊ ^ q) ^ (1 / q) * μ Set.univ ^ (1 / q) = ↑‖c‖₊ * μ Set.univ ^ (1 / q)","state_after":"case e_a\nα : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nc : F\nhq_pos : 0 < q\n⊢ (↑‖c‖₊ ^ q) ^ (1 / q) = ↑‖c‖₊","tactic":"congr","premises":[]},{"state_before":"case e_a\nα : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nc : F\nhq_pos : 0 < q\n⊢ (↑‖c‖₊ ^ q) ^ (1 / q) = ↑‖c‖₊","state_after":"case e_a\nα : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nc : F\nhq_pos : 0 < q\n⊢ ↑‖c‖₊ ^ (q * (1 / q)) = ↑‖c‖₊","tactic":"rw [← ENNReal.rpow_mul]","premises":[{"full_name":"ENNReal.rpow_mul","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean","def_pos":[514,8],"def_end_pos":[514,16]}]},{"state_before":"case e_a\nα : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nc : F\nhq_pos : 0 < q\n⊢ ↑‖c‖₊ ^ (q * (1 / q)) = ↑‖c‖₊","state_after":"case e_a\nα : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nc : F\nhq_pos : 0 < q\n⊢ q * (1 / q) = 1","tactic":"suffices hq_cancel : q * (1 / q) = 1 by rw [hq_cancel, ENNReal.rpow_one]","premises":[{"full_name":"ENNReal.rpow_one","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean","def_pos":[428,8],"def_end_pos":[428,16]}]},{"state_before":"case e_a\nα : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nc : F\nhq_pos : 0 < q\n⊢ q * (1 / q) = 1","state_after":"no goals","tactic":"rw [one_div, mul_inv_cancel (ne_of_lt hq_pos).symm]","premises":[{"full_name":"Ne.symm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[704,8],"def_end_pos":[704,15]},{"full_name":"mul_inv_cancel","def_path":"Mathlib/Algebra/GroupWithZero/Defs.lean","def_pos":[195,14],"def_end_pos":[195,28]},{"full_name":"ne_of_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[83,8],"def_end_pos":[83,16]},{"full_name":"one_div","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[338,8],"def_end_pos":[338,15]}]}]}
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+{"url":"Mathlib/Algebra/BigOperators/Group/Finset.lean","commit":"","full_name":"Finset.prod_fiberwise_of_maps_to","start":[686,0],"end":[689,59],"file_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","tactics":[{"state_before":"ι✝ : Type u_1\nκ✝ : Type u_2\nα✝ : Type u_3\nβ : Type u_4\nγ : Type u_5\ns✝ s₁ s₂ : Finset α✝\na : α✝\nf✝¹ g✝¹ : α✝ → β\ninst✝² : CommMonoid β\nι : Type u_6\nκ : Type u_7\nα : Type u_8\ninst✝¹ : CommMonoid α\ns : Finset ι\nt : Finset κ\nf✝ : ι → α\ng✝ : κ → α\ninst✝ : DecidableEq κ\ng : ι → κ\nh : ∀ i ∈ s, g i ∈ t\nf : ι → α\n⊢ ∏ j ∈ t, ∏ i ∈ filter (fun i => g i = j) s, f i = ∏ i ∈ s, f i","state_after":"no goals","tactic":"rw [← prod_disjiUnion, disjiUnion_filter_eq_of_maps_to h]","premises":[{"full_name":"Finset.disjiUnion_filter_eq_of_maps_to","def_path":"Mathlib/Data/Finset/Union.lean","def_pos":[97,6],"def_end_pos":[97,37]},{"full_name":"Finset.prod_disjiUnion","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[398,8],"def_end_pos":[398,23]}]}]}
+{"url":"Mathlib/FieldTheory/Adjoin.lean","commit":"","full_name":"IntermediateField.finrank_bot","start":[884,0],"end":[885,93],"file_path":"Mathlib/FieldTheory/Adjoin.lean","tactics":[{"state_before":"F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nα : E\nS : Set E\nK L : IntermediateField F E\n⊢ finrank F ↥⊥ = 1","state_after":"no goals","tactic":"rw [finrank_eq_one_iff]","premises":[{"full_name":"IntermediateField.finrank_eq_one_iff","def_path":"Mathlib/FieldTheory/Adjoin.lean","def_pos":[877,8],"def_end_pos":[877,26]}]}]}
+{"url":"Mathlib/Data/Set/NAry.lean","commit":"","full_name":"Set.image_prod","start":[67,0],"end":[68,33],"file_path":"Mathlib/Data/Set/NAry.lean","tactics":[{"state_before":"α : Type u_1\nα' : Type u_2\nβ : Type u_3\nβ' : Type u_4\nγ : Type u_5\nγ' : Type u_6\nδ : Type u_7\nδ' : Type u_8\nε : Type u_9\nε' : Type u_10\nζ : Type u_11\nζ' : Type u_12\nν : Type u_13\nf f' : α → β → γ\ng g' : α �� β → γ → δ\ns s' : Set α\nt t' : Set β\nu u' : Set γ\nv : Set δ\na a' : α\nb b' : β\nc c' : γ\nd d' : δ\nx✝ : γ\n⊢ x✝ ∈ (fun x => f x.1 x.2) '' s ×ˢ t ↔ x✝ ∈ image2 f s t","state_after":"no goals","tactic":"simp [and_assoc]","premises":[{"full_name":"and_assoc","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[148,8],"def_end_pos":[148,17]}]}]}
+{"url":"Mathlib/Analysis/Analytic/Basic.lean","commit":"","full_name":"FormalMultilinearSeries.changeOrigin_eval","start":[1222,0],"end":[1260,72],"file_path":"Mathlib/Analysis/Analytic/Basic.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\ninst✝ : CompleteSpace F\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nr R : ℝ≥0\nh : ↑‖x‖₊ + ↑‖y‖₊ < p.radius\n⊢ (p.changeOrigin x).sum y = p.sum (x + y)","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\ninst✝ : CompleteSpace F\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nr R : ℝ≥0\nh : ↑‖x‖₊ + ↑‖y‖₊ < p.radius\nradius_pos : 0 < p.radius\n⊢ (p.changeOrigin x).sum y = p.sum (x + y)","tactic":"have radius_pos : 0 < p.radius := lt_of_le_of_lt (zero_le _) h","premises":[{"full_name":"FormalMultilinearSeries.radius","def_path":"Mathlib/Analysis/Analytic/Basic.lean","def_pos":[125,4],"def_end_pos":[125,10]},{"full_name":"lt_of_le_of_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[99,8],"def_end_pos":[99,22]},{"full_name":"zero_le","def_path":"Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean","def_pos":[105,29],"def_end_pos":[105,36]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\ninst✝ : CompleteSpace F\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nr R : ℝ≥0\nh : ↑‖x‖₊ + ↑‖y‖₊ < p.radius\nradius_pos : 0 < p.radius\n⊢ (p.changeOrigin x).sum y = p.sum (x + y)","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\ninst✝ : CompleteSpace F\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nr R : ℝ≥0\nh : ↑‖x‖₊ + ↑‖y‖₊ < p.radius\nradius_pos : 0 < p.radius\nx_mem_ball : x ∈ EMetric.ball 0 p.radius\n⊢ (p.changeOrigin x).sum y = p.sum (x + y)","tactic":"have x_mem_ball : x ∈ EMetric.ball (0 : E) p.radius :=\n    mem_emetric_ball_zero_iff.2 ((le_add_right le_rfl).trans_lt h)","premises":[{"full_name":"EMetric.ball","def_path":"Mathlib/Topology/EMetricSpace/Basic.lean","def_pos":[484,4],"def_end_pos":[484,8]},{"full_name":"FormalMultilinearSeries.radius","def_path":"Mathlib/Analysis/Analytic/Basic.lean","def_pos":[125,4],"def_end_pos":[125,10]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"le_add_right","def_path":"Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean","def_pos":[147,2],"def_end_pos":[147,13]},{"full_name":"le_rfl","def_path":"Mathlib/Order/Defs.lean","def_pos":[43,8],"def_end_pos":[43,14]},{"full_name":"mem_emetric_ball_zero_iff","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[704,2],"def_end_pos":[704,13]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\ninst✝ : CompleteSpace F\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nr R : ℝ≥0\nh : ↑‖x‖₊ + ↑‖y‖₊ < p.radius\nradius_pos : 0 < p.radius\nx_mem_ball : x ∈ EMetric.ball 0 p.radius\n⊢ (p.changeOrigin x).sum y = p.sum (x + y)","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace ��� G\ninst✝ : CompleteSpace F\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nr R : ℝ≥0\nh : ↑‖x‖₊ + ↑‖y‖₊ < p.radius\nradius_pos : 0 < p.radius\nx_mem_ball : x ∈ EMetric.ball 0 p.radius\ny_mem_ball : y ∈ EMetric.ball 0 (p.changeOrigin x).radius\n⊢ (p.changeOrigin x).sum y = p.sum (x + y)","tactic":"have y_mem_ball : y ∈ EMetric.ball (0 : E) (p.changeOrigin x).radius := by\n    refine mem_emetric_ball_zero_iff.2 (lt_of_lt_of_le ?_ p.changeOrigin_radius)\n    rwa [lt_tsub_iff_right, add_comm]","premises":[{"full_name":"EMetric.ball","def_path":"Mathlib/Topology/EMetricSpace/Basic.lean","def_pos":[484,4],"def_end_pos":[484,8]},{"full_name":"FormalMultilinearSeries.changeOrigin","def_path":"Mathlib/Analysis/Analytic/Basic.lean","def_pos":[1096,4],"def_end_pos":[1096,16]},{"full_name":"FormalMultilinearSeries.changeOrigin_radius","def_path":"Mathlib/Analysis/Analytic/Basic.lean","def_pos":[1191,8],"def_end_pos":[1191,27]},{"full_name":"FormalMultilinearSeries.radius","def_path":"Mathlib/Analysis/Analytic/Basic.lean","def_pos":[125,4],"def_end_pos":[125,10]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"add_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[293,2],"def_end_pos":[293,13]},{"full_name":"lt_of_lt_of_le","def_path":"Mathlib/Order/Defs.lean","def_pos":[93,8],"def_end_pos":[93,22]},{"full_name":"lt_tsub_iff_right","def_path":"Mathlib/Algebra/Order/Sub/Defs.lean","def_pos":[357,8],"def_end_pos":[357,25]},{"full_name":"mem_emetric_ball_zero_iff","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[704,2],"def_end_pos":[704,13]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\ninst✝ : CompleteSpace F\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nr R : ℝ≥0\nh : ↑‖x‖₊ + ↑‖y‖₊ < p.radius\nradius_pos : 0 < p.radius\nx_mem_ball : x ∈ EMetric.ball 0 p.radius\ny_mem_ball : y ∈ EMetric.ball 0 (p.changeOrigin x).radius\n⊢ (p.changeOrigin x).sum y = p.sum (x + y)","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\ninst✝ : CompleteSpace F\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nr R : ℝ≥0\nh : ↑‖x‖₊ + ↑‖y‖₊ < p.radius\nradius_pos : 0 < p.radius\nx_mem_ball : x ∈ EMetric.ball 0 p.radius\ny_mem_ball : y ∈ EMetric.ball 0 (p.changeOrigin x).radius\nx_add_y_mem_ball : x + y ∈ EMetric.ball 0 p.radius\n⊢ (p.changeOrigin x).sum y = p.sum (x + y)","tactic":"have x_add_y_mem_ball : x + y ∈ EMetric.ball (0 : E) p.radius := by\n    refine mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt ?_ h)\n    exact mod_cast nnnorm_add_le x y","premises":[{"full_name":"EMetric.ball","def_path":"Mathlib/Topology/EMetricSpace/Basic.lean","def_pos":[484,4],"def_end_pos":[484,8]},{"full_name":"FormalMultilinearSeries.radius","def_path":"Mathlib/Analysis/Analytic/Basic.lean","def_pos":[125,4],"def_end_pos":[125,10]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"lt_of_le_of_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[99,8],"def_end_pos":[99,22]},{"full_name":"mem_emetric_ball_zero_iff","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[704,2],"def_end_pos":[704,13]},{"full_name":"nnnorm_add_le","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[643,14],"def_end_pos":[643,27]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\ninst✝ : CompleteSpace F\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nr R : ℝ≥0\nh : ↑‖x‖₊ + ↑‖y‖₊ < p.radius\nradius_pos : 0 < p.radius\nx_mem_ball : x ∈ EMetric.ball 0 p.radius\ny_mem_ball : y ∈ EMetric.ball 0 (p.changeOrigin x).radius\nx_add_y_mem_ball : x + y ∈ EMetric.ball 0 p.radius\n⊢ (p.changeOrigin x).sum y = p.sum (x + y)","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\ninst✝ : CompleteSpace F\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nr R : ℝ≥0\nh : ↑‖x‖₊ + ↑‖y‖₊ < p.radius\nradius_pos : 0 < p.radius\nx_mem_ball : x ∈ EMetric.ball 0 p.radius\ny_mem_ball : y ∈ EMetric.ball 0 (p.changeOrigin x).radius\nx_add_y_mem_ball : x + y ∈ EMetric.ball 0 p.radius\nf : (k : ℕ) × (l : ℕ) × { s // s.card = l } → F :=\n  fun s => ((p.changeOriginSeriesTerm s.fst s.snd.fst ↑s.snd.snd ⋯) fun x_1 => x) fun x => y\n⊢ (p.changeOrigin x).sum y = p.sum (x + y)","tactic":"set f : (Σ k l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) → F := fun s =>\n    p.changeOriginSeriesTerm s.1 s.2.1 s.2.2 s.2.2.2 (fun _ => x) fun _ => y","premises":[{"full_name":"Fin","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1831,10],"def_end_pos":[1831,13]},{"full_name":"Finset","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[133,10],"def_end_pos":[133,16]},{"full_name":"Finset.card","def_path":"Mathlib/Data/Finset/Card.lean","def_pos":[40,4],"def_end_pos":[40,8]},{"full_name":"FormalMultilinearSeries.changeOriginSeriesTerm","def_path":"Mathlib/Analysis/Analytic/Basic.lean","def_pos":[1042,4],"def_end_pos":[1042,26]},{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]},{"full_name":"Sigma","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[169,10],"def_end_pos":[169,15]},{"full_name":"Sigma.fst","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[175,2],"def_end_pos":[175,5]},{"full_name":"Sigma.snd","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[177,2],"def_end_pos":[177,5]},{"full_name":"Subtype","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[583,10],"def_end_pos":[583,17]},{"full_name":"Subtype.property","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[590,2],"def_end_pos":[590,10]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\ninst✝ : CompleteSpace F\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nr R : ℝ≥0\nh : ↑‖x‖₊ + ↑‖y‖₊ < p.radius\nradius_pos : 0 < p.radius\nx_mem_ball : x ∈ EMetric.ball 0 p.radius\ny_mem_ball : y ∈ EMetric.ball 0 (p.changeOrigin x).radius\nx_add_y_mem_ball : x + y ∈ EMetric.ball 0 p.radius\nf : (k : ℕ) × (l : ℕ) × { s // s.card = l } → F :=\n  fun s => ((p.changeOriginSeriesTerm s.fst s.snd.fst ↑s.snd.snd ⋯) fun x_1 => x) fun x => y\n⊢ (p.changeOrigin x).sum y = p.sum (x + y)","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\ninst✝ : CompleteSpace F\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nr R : ℝ≥0\nh : ↑‖x‖₊ + ↑‖y‖₊ < p.radius\nradius_pos : 0 < p.radius\nx_mem_ball : x ∈ EMetric.ball 0 p.radius\ny_mem_ball : y ∈ EMetric.ball 0 (p.changeOrigin x).radius\nx_add_y_mem_ball : x + y ∈ EMetric.ball 0 p.radius\nf : (k : ℕ) × (l : ℕ) × { s // s.card = l } → F :=\n  fun s => ((p.changeOriginSeriesTerm s.fst s.snd.fst ↑s.snd.snd ⋯) fun x_1 => x) fun x => y\nhsf : Summable f\n⊢ (p.changeOrigin x).sum y = p.sum (x + y)","tactic":"have hsf : Summable f := by\n    refine .of_nnnorm_bounded _ (p.changeOriginSeries_summable_aux₁ h) ?_\n    rintro ⟨k, l, s, hs⟩\n    dsimp only [Subtype.coe_mk]\n    exact p.nnnorm_changeOriginSeriesTerm_apply_le _ _ _ _ _ _","premises":[{"full_name":"FormalMultilinearSeries.changeOriginSeries_summable_aux₁","def_path":"Mathlib/Analysis/Analytic/Basic.lean","def_pos":[1142,8],"def_end_pos":[1142,40]},{"full_name":"FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm_apply_le","def_path":"Mathlib/Analysis/Analytic/Basic.lean","def_pos":[1064,8],"def_end_pos":[1064,46]},{"full_name":"Subtype.coe_mk","def_path":"Mathlib/Data/Subtype.lean","def_pos":[86,8],"def_end_pos":[86,14]},{"full_name":"Summable","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Defs.lean","def_pos":[88,2],"def_end_pos":[88,13]},{"full_name":"Summable.of_nnnorm_bounded","def_path":"Mathlib/Analysis/Normed/Group/InfiniteSum.lean","def_pos":[157,8],"def_end_pos":[157,34]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\ninst✝ : CompleteSpace F\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nr R : ℝ≥0\nh : ↑‖x‖₊ + ↑‖y‖₊ < p.radius\nradius_pos : 0 < p.radius\nx_mem_ball : x ∈ EMetric.ball 0 p.radius\ny_mem_ball : y ∈ EMetric.ball 0 (p.changeOrigin x).radius\nx_add_y_mem_ball : x + y ∈ EMetric.ball 0 p.radius\nf : (k : ℕ) × (l : ℕ) × { s // s.card = l } → F :=\n  fun s => ((p.changeOriginSeriesTerm s.fst s.snd.fst ↑s.snd.snd ⋯) fun x_1 => x) fun x => y\nhsf : Summable f\nhf : HasSum f ((p.changeOrigin x).sum y)\n⊢ (p.changeOrigin x).sum y = p.sum (x + y)","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\ninst✝ : CompleteSpace F\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nr R : ℝ≥0\nh : ↑‖x‖₊ + ↑‖y‖₊ < p.radius\nradius_pos : 0 < p.radius\nx_mem_ball : x ∈ EMetric.ball 0 p.radius\ny_mem_ball : y ∈ EMetric.ball 0 (p.changeOrigin x).radius\nx_add_y_mem_ball : x + y ∈ EMetric.ball 0 p.radius\nf : (k : ℕ) × (l : ℕ) × { s // s.card = l } → F :=\n  fun s => ((p.changeOriginSeriesTerm s.fst s.snd.fst ↑s.snd.snd ⋯) fun x_1 => x) fun x => y\nhsf : Summable f\nhf : HasSum f ((p.changeOrigin x).sum y)\n⊢ HasSum (f ∘ ⇑changeOriginIndexEquiv.symm) (p.sum (x + y))","tactic":"refine hf.unique (changeOriginIndexEquiv.symm.hasSum_iff.1 ?_)","premises":[{"full_name":"Equiv.hasSum_iff","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Basic.lean","def_pos":[143,2],"def_end_pos":[143,13]},{"full_name":"Equiv.symm","def_path":"Mathlib/Logic/Equiv/Defs.lean","def_pos":[146,14],"def_end_pos":[146,18]},{"full_name":"FormalMultilinearSeries.changeOriginIndexEquiv","def_path":"Mathlib/Analysis/Analytic/Basic.lean","def_pos":[1108,4],"def_end_pos":[1108,26]},{"full_name":"HasSum.unique","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Defs.lean","def_pos":[159,2],"def_end_pos":[159,13]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\ninst✝ : CompleteSpace F\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nr R : ℝ≥0\nh : ↑‖x‖₊ + ↑‖y‖₊ < p.radius\nradius_pos : 0 < p.radius\nx_mem_ball : x ∈ EMetric.ball 0 p.radius\ny_mem_ball : y ∈ EMetric.ball 0 (p.changeOrigin x).radius\nx_add_y_mem_ball : x + y ∈ EMetric.ball 0 p.radius\nf : (k : ℕ) × (l : ℕ) × { s // s.card = l } → F :=\n  fun s => ((p.changeOriginSeriesTerm s.fst s.snd.fst ↑s.snd.snd ⋯) fun x_1 => x) fun x => y\nhsf : Summable f\nhf : HasSum f ((p.changeOrigin x).sum y)\n⊢ HasSum (f ∘ ⇑changeOriginIndexEquiv.symm) (p.sum (x + y))","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\ninst✝ : CompleteSpace F\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nr R : ℝ≥0\nh : ↑‖x‖₊ + ↑‖y‖₊ < p.radius\nradius_pos : 0 < p.radius\nx_mem_ball : x ∈ EMetric.ball 0 p.radius\ny_mem_ball : y ∈ EMetric.ball 0 (p.changeOrigin x).radius\nx_add_y_mem_ball : x + y ∈ EMetric.ball 0 p.radius\nf : (k : ℕ) × (l : ℕ) × { s // s.card = l } → F :=\n  fun s => ((p.changeOriginSeriesTerm s.fst s.snd.fst ↑s.snd.snd ⋯) fun x_1 => x) fun x => y\nhsf : Summable f\nhf : HasSum f ((p.changeOrigin x).sum y)\nn : ℕ\n⊢ HasSum (fun c => (f ∘ ⇑changeOriginIndexEquiv.symm) ⟨n, c⟩) ((p n) fun x_1 => x + y)","tactic":"refine HasSum.sigma_of_hasSum\n    (p.hasSum x_add_y_mem_ball) (fun n => ?_) (changeOriginIndexEquiv.symm.summable_iff.2 hsf)","premises":[{"full_name":"Equiv.summable_iff","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Basic.lean","def_pos":[152,2],"def_end_pos":[152,13]},{"full_name":"Equiv.symm","def_path":"Mathlib/Logic/Equiv/Defs.lean","def_pos":[146,14],"def_end_pos":[146,18]},{"full_name":"FormalMultilinearSeries.changeOriginIndexEquiv","def_path":"Mathlib/Analysis/Analytic/Basic.lean","def_pos":[1108,4],"def_end_pos":[1108,26]},{"full_name":"FormalMultilinearSeries.hasSum","def_path":"Mathlib/Analysis/Analytic/Basic.lean","def_pos":[325,18],"def_end_pos":[325,24]},{"full_name":"HasSum.sigma_of_hasSum","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean","def_pos":[118,2],"def_end_pos":[118,13]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\ninst✝ : CompleteSpace F\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nr R : ℝ≥0\nh : ↑‖x‖₊ + ↑‖y‖₊ < p.radius\nradius_pos : 0 < p.radius\nx_mem_ball : x ∈ EMetric.ball 0 p.radius\ny_mem_ball : y ∈ EMetric.ball 0 (p.changeOrigin x).radius\nx_add_y_mem_ball : x + y ∈ EMetric.ball 0 p.radius\nf : (k : ℕ) × (l : ℕ) × { s // s.card = l } → F :=\n  fun s => ((p.changeOriginSeriesTerm s.fst s.snd.fst ↑s.snd.snd ⋯) fun x_1 => x) fun x => y\nhsf : Summable f\nhf : HasSum f ((p.changeOrigin x).sum y)\nn : ℕ\n⊢ HasSum (fun c => (f ∘ ⇑changeOriginIndexEquiv.symm) ⟨n, c⟩) ((p n) fun x_1 => x + y)","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\ninst✝ : CompleteSpace F\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nr R : ℝ≥0\nh : ↑‖x‖₊ + ↑‖y‖₊ < p.radius\nradius_pos : 0 < p.radius\nx_mem_ball : x ∈ EMetric.ball 0 p.radius\ny_mem_ball : y ∈ EMetric.ball 0 (p.changeOrigin x).radius\nx_add_y_mem_ball : x + y ∈ EMetric.ball 0 p.radius\nf : (k : ℕ) × (l : ℕ) × { s // s.card = l } → F :=\n  fun s => ((p.changeOriginSeriesTerm s.fst s.snd.fst ↑s.snd.snd ⋯) fun x_1 => x) fun x => y\nhsf : Summable f\nhf : HasSum f ((p.changeOrigin x).sum y)\nn : ℕ\n⊢ HasSum (fun c => (f ∘ ⇑changeOriginIndexEquiv.symm) ⟨n, c⟩)\n    (∑ s : Finset (Fin n), (p n) (s.piecewise (fun x_1 => x) fun x => y))","tactic":"erw [(p n).map_add_univ (fun _ => x) fun _ => y]","premises":[{"full_name":"ContinuousMultilinearMap.map_add_univ","def_path":"Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean","def_pos":[374,8],"def_end_pos":[374,20]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\ninst✝ : CompleteSpace F\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nr R : ℝ≥0\nh : ↑‖x‖₊ + ↑‖y‖₊ < p.radius\nradius_pos : 0 < p.radius\nx_mem_ball : x ∈ EMetric.ball 0 p.radius\ny_mem_ball : y ∈ EMetric.ball 0 (p.changeOrigin x).radius\nx_add_y_mem_ball : x + y ∈ EMetric.ball 0 p.radius\nf : (k : ℕ) × (l : ℕ) × { s // s.card = l } → F :=\n  fun s => ((p.changeOriginSeriesTerm s.fst s.snd.fst ↑s.snd.snd ⋯) fun x_1 => x) fun x => y\nhsf : Summable f\nhf : HasSum f ((p.changeOrigin x).sum y)\nn : ℕ\n⊢ HasSum (fun c => (f ∘ ⇑changeOriginIndexEquiv.symm) ⟨n, c⟩)\n    (∑ s : Finset (Fin n), (p n) (s.piecewise (fun x_1 => x) fun x => y))","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\ninst✝ : CompleteSpace F\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nr R : ℝ≥0\nh : ↑‖x‖₊ + ↑‖y‖₊ < p.radius\nradius_pos : 0 < p.radius\nx_mem_ball : x ∈ EMetric.ball 0 p.radius\ny_mem_ball : y ∈ EMetric.ball 0 (p.changeOrigin x).radius\nx_add_y_mem_ball : x + y ∈ EMetric.ball 0 p.radius\nf : (k : ℕ) × (l : ℕ) × { s // s.card = l } → F :=\n  fun s => ((p.changeOriginSeriesTerm s.fst s.snd.fst ↑s.snd.snd ⋯) fun x_1 => x) fun x => y\nhsf : Summable f\nhf : HasSum f ((p.changeOrigin x).sum y)\nn : ℕ\n⊢ HasSum (fun c => (f ∘ ⇑changeOriginIndexEquiv.symm) ⟨n, c⟩)\n    (∑ x_1 : Finset (Fin n),\n      ((p.changeOriginSeriesTerm (changeOriginIndexEquiv.symm ⟨n, x_1⟩).fst\n            (changeOriginIndexEquiv.symm ⟨n, x_1⟩).snd.fst ↑(changeOriginIndexEquiv.symm ⟨n, x_1⟩).snd.snd ⋯)\n          fun x_2 => x)\n        fun x => y)","tactic":"simp_rw [← changeOriginSeriesTerm_changeOriginIndexEquiv_symm]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"FormalMultilinearSeries.changeOriginSeriesTerm_changeOriginIndexEquiv_symm","def_path":"Mathlib/Analysis/Analytic/Basic.lean","def_pos":[1132,6],"def_end_pos":[1132,56]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\ninst✝ : CompleteSpace F\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nr R : ℝ≥0\nh : ↑‖x‖₊ + ↑‖y‖₊ < p.radius\nradius_pos : 0 < p.radius\nx_mem_ball : x ∈ EMetric.ball 0 p.radius\ny_mem_ball : y ∈ EMetric.ball 0 (p.changeOrigin x).radius\nx_add_y_mem_ball : x + y ∈ EMetric.ball 0 p.radius\nf : (k : ℕ) × (l : ℕ) × { s // s.card = l } → F :=\n  fun s => ((p.changeOriginSeriesTerm s.fst s.snd.fst ↑s.snd.snd ⋯) fun x_1 => x) fun x => y\nhsf : Summable f\nhf : HasSum f ((p.changeOrigin x).sum y)\nn : ℕ\n⊢ HasSum (fun c => (f ∘ ⇑changeOriginIndexEquiv.symm) ⟨n, c⟩)\n    (∑ x_1 : Finset (Fin n),\n      ((p.changeOriginSeriesTerm (changeOriginIndexEquiv.symm ⟨n, x_1⟩).fst\n            (changeOriginIndexEquiv.symm ⟨n, x_1⟩).snd.fst ↑(changeOriginIndexEquiv.symm ⟨n, x_1⟩).snd.snd ⋯)\n          fun x_2 => x)\n        fun x => y)","state_after":"no goals","tactic":"exact hasSum_fintype (fun c => f (changeOriginIndexEquiv.symm ⟨n, c⟩))","premises":[{"full_name":"Equiv.symm","def_path":"Mathlib/Logic/Equiv/Defs.lean","def_pos":[146,14],"def_end_pos":[146,18]},{"full_name":"FormalMultilinearSeries.changeOriginIndexEquiv","def_path":"Mathlib/Analysis/Analytic/Basic.lean","def_pos":[1108,4],"def_end_pos":[1108,26]},{"full_name":"Sigma.mk","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[173,2],"def_end_pos":[173,4]},{"full_name":"hasSum_fintype","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Defs.lean","def_pos":[131,2],"def_end_pos":[131,13]}]}]}
+{"url":"Mathlib/Data/Set/Lattice.lean","commit":"","full_name":"Set.nonempty_sInter","start":[1024,0],"end":[1026,51],"file_path":"Mathlib/Data/Set/Lattice.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nι' : Sort u_5\nι₂ : Sort u_6\nκ : ι → Sort u_7\nκ₁ : ι → Sort u_8\nκ₂ : ι → Sort u_9\nκ' : ι' → Sort u_10\nc : Set (Set α)\n⊢ (⋂₀ c).Nonempty ↔ ∃ a, ∀ b ∈ c, a ∈ b","state_after":"no goals","tactic":"simp [nonempty_iff_ne_empty, sInter_eq_empty_iff]","premises":[{"full_name":"Set.nonempty_iff_ne_empty","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[485,8],"def_end_pos":[485,29]},{"full_name":"Set.sInter_eq_empty_iff","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[1008,8],"def_end_pos":[1008,27]}]}]}
+{"url":"Mathlib/Algebra/Polynomial/Eval.lean","commit":"","full_name":"Polynomial.eval₂_ofFinsupp","start":[144,0],"end":[147,5],"file_path":"Mathlib/Algebra/Polynomial/Eval.lean","tactics":[{"state_before":"R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝² : Semiring R\np✝ q r : R[X]\ninst✝¹ : Semiring S\nf✝ : R →+* S\nx✝ : S\ninst✝ : Semiring T\nf : R →+* S\nx : S\np : R[ℕ]\n⊢ eval₂ f x { toFinsupp := p } = (liftNC ↑f ⇑((powersHom S) x)) p","state_after":"R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝² : Semiring R\np✝ q r : R[X]\ninst✝¹ : Semiring S\nf✝ : R →+* S\nx✝ : S\ninst✝ : Semiring T\nf : R →+* S\nx : S\np : R[ℕ]\n⊢ ∑ x_1 ∈ p.support, f (p x_1) * x ^ x_1 = (liftNC ↑f ⇑((powersHom S) x)) p","tactic":"simp only [eval₂_eq_sum, sum, toFinsupp_sum, support, coeff]","premises":[{"full_name":"Polynomial.coeff","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[557,4],"def_end_pos":[557,9]},{"full_name":"Polynomial.eval₂_eq_sum","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[44,8],"def_end_pos":[44,20]},{"full_name":"Polynomial.sum","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[812,4],"def_end_pos":[812,7]},{"full_name":"Polynomial.support","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[348,4],"def_end_pos":[348,11]},{"full_name":"Polynomial.toFinsupp_sum","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[339,8],"def_end_pos":[339,21]}]},{"state_before":"R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝² : Semiring R\np✝ q r : R[X]\ninst✝¹ : Semiring S\nf✝ : R →+* S\nx✝ : S\ninst✝ : Semiring T\nf : R →+* S\nx : S\np : R[ℕ]\n⊢ ∑ x_1 ∈ p.support, f (p x_1) * x ^ x_1 = (liftNC ↑f ⇑((powersHom S) x)) p","state_after":"no goals","tactic":"rfl","premises":[]}]}
+{"url":"Mathlib/Order/Filter/Basic.lean","commit":"","full_name":"Filter.bind_mono","start":[2543,0],"end":[2548,50],"file_path":"Mathlib/Order/Filter/Basic.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nδ : Type u_1\nι : Sort x\nl f₁ f₂ : Filter α\ng₁ g₂ : α → Filter β\nhf : f₁ ≤ f₂\nhg : g₁ ≤ᶠ[f₁] g₂\n⊢ f₁.bind g₁ ≤ f₂.bind g₂","state_after":"α : Type u\nβ : Type v\nγ : Type w\nδ : Type u_1\nι : Sort x\nl f₁ f₂ : Filter α\ng₁ g₂ : α → Filter β\nhf : f₁ ≤ f₂\nhg : g₁ ≤ᶠ[f₁] g₂\ns : Set β\nhs : s ∈ (map g₂ f₁).join\n⊢ s ∈ f₁.bind g₁","tactic":"refine le_trans (fun s hs => ?_) (join_mono <| map_mono hf)","premises":[{"full_name":"Filter.join_mono","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[942,8],"def_end_pos":[942,17]},{"full_name":"Filter.map_mono","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2040,8],"def_end_pos":[2040,16]},{"full_name":"le_trans","def_path":"Mathlib/Order/Defs.lean","def_pos":[48,8],"def_end_pos":[48,16]}]},{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nδ : Type u_1\nι : Sort x\nl f₁ f₂ : Filter α\ng₁ g₂ : α → Filter β\nhf : f₁ ≤ f₂\nhg : g₁ ≤ᶠ[f₁] g₂\ns : Set β\nhs : s ∈ (map g₂ f₁).join\n⊢ s ∈ f₁.bind g₁","state_after":"α : Type u\nβ : Type v\nγ : Type w\nδ : Type u_1\nι : Sort x\nl f₁ f₂ : Filter α\ng₁ g₂ : α → Filter β\nhf : f₁ ≤ f₂\nhg : g₁ ≤ᶠ[f₁] g₂\ns : Set β\nhs : g₂ ⁻¹' {t | s ∈ t} ∈ f₁\n⊢ {a | s ∈ g₁ a} ∈ f₁","tactic":"simp only [mem_join, mem_bind', mem_map] at hs ⊢","premises":[{"full_name":"Filter.mem_bind'","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2527,8],"def_end_pos":[2527,17]},{"full_name":"Filter.mem_join","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[296,8],"def_end_pos":[296,16]},{"full_name":"Filter.mem_map","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1658,8],"def_end_pos":[1658,15]}]},{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nδ : Type u_1\nι : Sort x\nl f₁ f₂ : Filter α\ng₁ g₂ : α → Filter β\nhf : f₁ ≤ f₂\nhg : g₁ ≤ᶠ[f₁] g₂\ns : Set β\nhs : g₂ ⁻¹' {t | s ∈ t} ∈ f₁\n⊢ {a | s ∈ g₁ a} ∈ f₁","state_after":"no goals","tactic":"filter_upwards [hg, hs] with _ hx hs using hx hs","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Filter.mp_mem","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[159,8],"def_end_pos":[159,14]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]}]}]}
+{"url":"Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean","commit":"","full_name":"Real.dist_le_of_mem_Icc_01","start":[34,0],"end":[35,80],"file_path":"Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\ninst✝ : PseudoMetricSpace α\nx y : ℝ\nhx : x ∈ Icc 0 1\nhy : y ∈ Icc 0 1\n⊢ dist x y ≤ 1","state_after":"no goals","tactic":"simpa only [sub_zero] using Real.dist_le_of_mem_Icc hx hy","premises":[{"full_name":"Real.dist_le_of_mem_Icc","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean","def_pos":[29,6],"def_end_pos":[29,29]},{"full_name":"sub_zero","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[353,2],"def_end_pos":[353,13]}]}]}
+{"url":"Mathlib/Topology/Semicontinuous.lean","commit":"","full_name":"lowerSemicontinuous_iff_isClosed_epigraph","start":[314,0],"end":[327,60],"file_path":"Mathlib/Topology/Semicontinuous.lean","tactics":[{"state_before":"α : Type u_1\ninst✝⁵ : TopologicalSpace α\nβ : Type u_2\ninst✝⁴ : Preorder β\nf✝ g : α → β\nx : α\ns t : Set α\ny z : β\nγ : Type u_3\ninst✝³ : CompleteLinearOrder γ\ninst✝² : DenselyOrdered γ\ninst✝¹ : TopologicalSpace γ\ninst✝ : OrderTopology γ\nf : α → γ\n⊢ LowerSemicontinuous f ↔ IsClosed {p | f p.1 ≤ p.2}","state_after":"case mp\nα : Type u_1\ninst✝⁵ : TopologicalSpace α\nβ : Type u_2\ninst✝⁴ : Preorder β\nf✝ g : α → β\nx : α\ns t : Set α\ny z : β\nγ : Type u_3\ninst✝³ : CompleteLinearOrder γ\ninst✝² : DenselyOrdered γ\ninst✝¹ : TopologicalSpace γ\ninst✝ : OrderTopology γ\nf : α → γ\n⊢ LowerSemicontinuous f → IsClosed {p | f p.1 ≤ p.2}\n\ncase mpr\nα : Type u_1\ninst✝⁵ : TopologicalSpace α\nβ : Type u_2\ninst✝⁴ : Preorder β\nf✝ g : α → β\nx : α\ns t : Set α\ny z : β\nγ : Type u_3\ninst✝³ : CompleteLinearOrder γ\ninst✝² : DenselyOrdered γ\ninst✝¹ : TopologicalSpace γ\ninst✝ : OrderTopology γ\nf : α → γ\n⊢ IsClosed {p | f p.1 ≤ p.2} → LowerSemicontinuous f","tactic":"constructor","premises":[]}]}
+{"url":"Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean","commit":"","full_name":"Polynomial.isRoot_cyclotomic_iff","start":[94,0],"end":[99,27],"file_path":"Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean","tactics":[{"state_before":"R : Type u_1\ninst✝² : CommRing R\nn : ℕ\ninst✝¹ : IsDomain R\ninst✝ : NeZero ↑n\nμ : R\n⊢ (cyclotomic n R).IsRoot μ ↔ IsPrimitiveRoot μ n","state_after":"R : Type u_1\ninst✝² : CommRing R\nn : ℕ\ninst✝¹ : IsDomain R\ninst✝ : NeZero ↑n\nμ : R\nhf : Function.Injective ⇑(algebraMap R (FractionRing R))\n⊢ (cyclotomic n R).IsRoot μ ↔ IsPrimitiveRoot μ n","tactic":"have hf : Function.Injective _ := IsFractionRing.injective R (FractionRing R)","premises":[{"full_name":"FractionRing","def_path":"Mathlib/RingTheory/Localization/FractionRing.lean","def_pos":[266,7],"def_end_pos":[266,19]},{"full_name":"Function.Injective","def_path":"Mathlib/Logic/Function/Defs.lean","def_pos":[101,4],"def_end_pos":[101,13]},{"full_name":"IsFractionRing.injective","def_path":"Mathlib/RingTheory/Localization/FractionRing.lean","def_pos":[77,18],"def_end_pos":[77,27]}]},{"state_before":"R : Type u_1\ninst✝² : CommRing R\nn : ℕ\ninst✝¹ : IsDomain R\ninst✝ : NeZero ↑n\nμ : R\nhf : Function.Injective ⇑(algebraMap R (FractionRing R))\n⊢ (cyclotomic n R).IsRoot μ ↔ IsPrimitiveRoot μ n","state_after":"R : Type u_1\ninst✝² : CommRing R\nn : ℕ\ninst✝¹ : IsDomain R\ninst✝ : NeZero ↑n\nμ : R\nhf : Function.Injective ⇑(algebraMap R (FractionRing R))\nthis : NeZero ↑n\n⊢ (cyclotomic n R).IsRoot μ ↔ IsPrimitiveRoot μ n","tactic":"haveI : NeZero (n : FractionRing R) := NeZero.nat_of_injective hf","premises":[{"full_name":"FractionRing","def_path":"Mathlib/RingTheory/Localization/FractionRing.lean","def_pos":[266,7],"def_end_pos":[266,19]},{"full_name":"NeZero","def_path":"Mathlib/Algebra/NeZero.lean","def_pos":[23,6],"def_end_pos":[23,12]},{"full_name":"NeZero.nat_of_injective","def_path":"Mathlib/Data/Nat/Cast/Order/Basic.lean","def_pos":[169,8],"def_end_pos":[169,31]}]},{"state_before":"R : Type u_1\ninst✝² : CommRing R\nn : ℕ\ninst✝¹ : IsDomain R\ninst✝ : NeZero ↑n\nμ : R\nhf : Function.Injective ⇑(algebraMap R (FractionRing R))\nthis : NeZero ↑n\n⊢ (cyclotomic n R).IsRoot μ ↔ IsPrimitiveRoot μ n","state_after":"no goals","tactic":"rw [← isRoot_map_iff hf, ← IsPrimitiveRoot.map_iff_of_injective hf, map_cyclotomic, ←\n    isRoot_cyclotomic_iff']","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"IsPrimitiveRoot.map_iff_of_injective","def_path":"Mathlib/RingTheory/RootsOfUnity/Basic.lean","def_pos":[488,8],"def_end_pos":[488,28]},{"full_name":"Polynomial.isRoot_map_iff","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[1072,8],"def_end_pos":[1072,22]},{"full_name":"Polynomial.map_cyclotomic","def_path":"Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean","def_pos":[264,8],"def_end_pos":[264,22]},{"full_name":"_private.Mathlib.RingTheory.Polynomial.Cyclotomic.Roots.0.Polynomial.isRoot_cyclotomic_iff'","def_path":"Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean","def_pos":[65,16],"def_end_pos":[65,38]}]}]}
+{"url":"Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean","commit":"","full_name":"TensorProduct.gradedMul_assoc","start":[231,0],"end":[246,6],"file_path":"Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean","tactics":[{"state_before":"R : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹¹ : CommSemiring ι\ninst✝¹⁰ : Module ι (Additive ℤˣ)\ninst✝⁹ : DecidableEq ι\n𝒜 : ι → Type u_5\nℬ : ι → Type u_6\ninst✝⁸ : CommRing R\ninst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)\ninst✝⁶ : (i : ι) → AddCommGroup (ℬ i)\ninst✝⁵ : (i : ι) → Module R (𝒜 i)\ninst✝⁴ : (i : ι) → Module R (ℬ i)\ninst✝³ : DirectSum.GRing 𝒜\ninst✝² : DirectSum.GRing ℬ\ninst✝¹ : DirectSum.GAlgebra R 𝒜\ninst✝ : DirectSum.GAlgebra R ℬ\nx y z : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ\n⊢ ((gradedMul R 𝒜 ℬ) (((gradedMul R 𝒜 ℬ) x) y)) z = ((gradedMul R 𝒜 ℬ) x) (((gradedMul R 𝒜 ℬ) y) z)","state_after":"R : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹¹ : CommSemiring ι\ninst✝¹⁰ : Module ι (Additive ℤˣ)\ninst✝⁹ : DecidableEq ι\n𝒜 : ι → Type u_5\nℬ : ι → Type u_6\ninst✝⁸ : CommRing R\ninst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)\ninst✝⁶ : (i : ι) → AddCommGroup (ℬ i)\ninst✝⁵ : (i : ι) → Module R (𝒜 i)\ninst✝⁴ : (i : ι) → Module R (ℬ i)\ninst✝³ : DirectSum.GRing 𝒜\ninst✝² : DirectSum.GRing ℬ\ninst✝¹ : DirectSum.GAlgebra R 𝒜\ninst✝ : DirectSum.GAlgebra R ℬ\nx y z : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ\nmA : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ :=\n  gradedMul R 𝒜 ℬ\n⊢ ((gradedMul R 𝒜 ℬ) (((gradedMul R 𝒜 ℬ) x) y)) z = ((gradedMul R 𝒜 ℬ) x) (((gradedMul R 𝒜 ℬ) y) z)","tactic":"let mA := gradedMul R 𝒜 ℬ","premises":[{"full_name":"TensorProduct.gradedMul","def_path":"Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean","def_pos":[173,30],"def_end_pos":[173,39]}]},{"state_before":"R : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹¹ : CommSemiring ι\ninst✝¹⁰ : Module ι (Additive ℤˣ)\ninst✝⁹ : DecidableEq ι\n𝒜 : ι → Type u_5\nℬ : ι → Type u_6\ninst✝⁸ : CommRing R\ninst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)\ninst✝⁶ : (i : ι) → AddCommGroup (ℬ i)\ninst✝⁵ : (i : ι) → Module R (𝒜 i)\ninst✝⁴ : (i : ι) → Module R (ℬ i)\ninst✝³ : DirectSum.GRing 𝒜\ninst✝² : DirectSum.GRing ℬ\ninst✝¹ : DirectSum.GAlgebra R 𝒜\ninst✝ : DirectSum.GAlgebra R ℬ\nx y z : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ\nmA : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ :=\n  gradedMul R 𝒜 ℬ\n⊢ ((gradedMul R 𝒜 ℬ) (((gradedMul R 𝒜 ℬ) x) y)) z = ((gradedMul R 𝒜 ℬ) x) (((gradedMul R 𝒜 ℬ) y) z)","state_after":"R : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹¹ : CommSemiring ι\ninst✝¹⁰ : Module ι (Additive ℤˣ)\ninst✝⁹ : DecidableEq ι\n𝒜 : ι → Type u_5\nℬ : ι → Type u_6\ninst✝⁸ : CommRing R\ninst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)\ninst✝⁶ : (i : ι) → AddCommGroup (ℬ i)\ninst✝⁵ : (i : ι) → Module R (𝒜 i)\ninst✝⁴ : (i : ι) → Module R (ℬ i)\ninst✝³ : DirectSum.GRing 𝒜\ninst✝² : DirectSum.GRing ℬ\ninst✝¹ : DirectSum.GAlgebra R 𝒜\ninst✝ : DirectSum.GAlgebra R ℬ\nx y z : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ\nmA : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ :=\n  gradedMul R 𝒜 ℬ\n⊢ (LinearMap.llcomp R (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ) (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ)\n          (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ))\n        mA ∘ₗ\n      mA =\n    (((LinearMap.llcomp R (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ)\n            (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R]\n              DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ)\n            (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R]\n              DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ))\n          LinearMap.lflip)\n        ((LinearMap.llcomp R (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ) (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ)\n              (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ))\n            mA.flip ∘ₗ\n          mA)).flip","tactic":"suffices LinearMap.llcomp R _ _ _ mA ∘ₗ mA =\n      (LinearMap.llcomp R _ _ _ LinearMap.lflip <| LinearMap.llcomp R _ _ _ mA.flip ∘ₗ mA).flip by\n    exact DFunLike.congr_fun (DFunLike.congr_fun (DFunLike.congr_fun this x) y) z","premises":[{"full_name":"DFunLike.congr_fun","def_path":"Mathlib/Data/FunLike/Basic.lean","def_pos":[199,18],"def_end_pos":[199,27]},{"full_name":"LinearMap.comp","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[489,4],"def_end_pos":[489,8]},{"full_name":"LinearMap.flip","def_path":"Mathlib/LinearAlgebra/BilinearMap.lean","def_pos":[105,4],"def_end_pos":[105,8]},{"full_name":"LinearMap.lflip","def_path":"Mathlib/LinearAlgebra/BilinearMap.lean","def_pos":[234,4],"def_end_pos":[234,9]},{"full_name":"LinearMap.llcomp","def_path":"Mathlib/LinearAlgebra/BilinearMap.lean","def_pos":[273,4],"def_end_pos":[273,10]},{"full_name":"RingHom.id","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[538,4],"def_end_pos":[538,6]}]},{"state_before":"R : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹¹ : CommSemiring ι\ninst✝¹⁰ : Module ι (Additive ℤˣ)\ninst✝⁹ : DecidableEq ι\n𝒜 : ι → Type u_5\nℬ : ι → Type u_6\ninst✝⁸ : CommRing R\ninst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)\ninst✝⁶ : (i : ι) → AddCommGroup (ℬ i)\ninst✝⁵ : (i : ι) → Module R (𝒜 i)\ninst✝⁴ : (i : ι) → Module R (ℬ i)\ninst✝³ : DirectSum.GRing 𝒜\ninst✝² : DirectSum.GRing ℬ\ninst✝¹ : DirectSum.GAlgebra R 𝒜\ninst✝ : DirectSum.GAlgebra R ℬ\nx y z : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ\nmA : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ :=\n  gradedMul R 𝒜 ℬ\n⊢ (LinearMap.llcomp R (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ) (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ)\n          (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ))\n        mA ∘ₗ\n      mA =\n    (((LinearMap.llcomp R (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ)\n            (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R]\n              DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ)\n            (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R]\n              DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ))\n          LinearMap.lflip)\n        ((LinearMap.llcomp R (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ) (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ)\n              (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ))\n            mA.flip ∘ₗ\n          mA)).flip","state_after":"case a.H.h.H.h.a.H.h.H.h.a.H.h.H.h\nR : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹¹ : CommSemiring ι\ninst✝¹⁰ : Module ι (Additive ℤˣ)\ninst✝⁹ : DecidableEq ι\n𝒜 : ι → Type u_5\nℬ : ι → Type u_6\ninst✝⁸ : CommRing R\ninst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)\ninst✝⁶ : (i : ι) → AddCommGroup (ℬ i)\ninst✝⁵ : (i : ι) → Module R (𝒜 i)\ninst✝⁴ : (i : ι) → Module R (ℬ i)\ninst✝³ : DirectSum.GRing 𝒜\ninst✝² : DirectSum.GRing ℬ\ninst✝¹ : DirectSum.GAlgebra R 𝒜\ninst✝ : DirectSum.GAlgebra R ℬ\nx y z : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ\nmA : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ :=\n  gradedMul R 𝒜 ℬ\nixa : ι\nxa : 𝒜 ixa\nixb : ι\nxb : ℬ ixb\niya : ι\nya : 𝒜 iya\niyb : ι\nyb : ℬ iyb\niza : ι\nza : 𝒜 iza\nizb : ι\nzb : ℬ izb\n⊢ ((AlgebraTensorModule.curry\n              (((AlgebraTensorModule.curry\n                        (((AlgebraTensorModule.curry\n                                  ((LinearMap.llcomp R (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ)\n                                        (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ)\n                                        (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ))\n                                      mA ∘ₗ\n                                    mA) ∘ₗ\n                                lof R ι 𝒜 ixa)\n                              xa ∘ₗ\n                            lof R ι ℬ ixb)\n                          xb) ∘ₗ\n                      lof R ι 𝒜 iya)\n                    ya ∘ₗ\n                  lof R ι ℬ iyb)\n                yb) ∘ₗ\n            lof R ι 𝒜 iza)\n          za ∘ₗ\n        lof R ι ℬ izb)\n      zb =\n    ((AlgebraTensorModule.curry\n              (((AlgebraTensorModule.curry\n                        (((AlgebraTensorModule.curry\n                                  (((LinearMap.llcomp R (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ)\n                                          (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R]\n                                            DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ)\n                                          (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R]\n                                            DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ))\n                                        LinearMap.lflip)\n                                      ((LinearMap.llcomp R (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ)\n                                            (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ)\n                                            (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ))\n                                          mA.flip ∘ₗ\n                                        mA)).flip ∘ₗ\n                                lof R ι 𝒜 ixa)\n                              xa ∘ₗ\n                            lof R ι ℬ ixb)\n                          xb) ∘ₗ\n                      lof R ι 𝒜 iya)\n                    ya ∘ₗ\n                  lof R ι ℬ iyb)\n                yb) ∘ₗ\n            lof R ι 𝒜 iza)\n          za ∘ₗ\n        lof R ι ℬ izb)\n      zb","tactic":"ext ixa xa ixb xb iya ya iyb yb iza za izb zb","premises":[]},{"state_before":"case a.H.h.H.h.a.H.h.H.h.a.H.h.H.h\nR : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹¹ : CommSemiring ι\ninst✝¹⁰ : Module ι (Additive ℤˣ)\ninst✝⁹ : DecidableEq ι\n𝒜 : ι → Type u_5\nℬ : ι → Type u_6\ninst✝⁸ : CommRing R\ninst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)\ninst✝⁶ : (i : ι) → AddCommGroup (ℬ i)\ninst✝⁵ : (i : ι) → Module R (𝒜 i)\ninst✝⁴ : (i : ι) → Module R (ℬ i)\ninst✝³ : DirectSum.GRing 𝒜\ninst✝² : DirectSum.GRing ℬ\ninst✝¹ : DirectSum.GAlgebra R 𝒜\ninst✝ : DirectSum.GAlgebra R ℬ\nx y z : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ\nmA : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ :=\n  gradedMul R 𝒜 ℬ\nixa : ι\nxa : 𝒜 ixa\nixb : ι\nxb : ℬ ixb\niya : ι\nya : 𝒜 iya\niyb : ι\nyb : ℬ iyb\niza : ι\nza : 𝒜 iza\nizb : ι\nzb : ℬ izb\n⊢ ((AlgebraTensorModule.curry\n              (((AlgebraTensorModule.curry\n                        (((AlgebraTensorModule.curry\n                                  ((LinearMap.llcomp R (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ)\n                                        (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ)\n                                        (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ))\n                                      mA ∘ₗ\n                                    mA) ∘ₗ\n                                lof R ι 𝒜 ixa)\n                              xa ∘ₗ\n                            lof R ι ℬ ixb)\n                          xb) ∘ₗ\n                      lof R ι 𝒜 iya)\n                    ya ∘ₗ\n                  lof R ι ℬ iyb)\n                yb) ∘ₗ\n            lof R ι 𝒜 iza)\n          za ∘ₗ\n        lof R ι ℬ izb)\n      zb =\n    ((AlgebraTensorModule.curry\n              (((AlgebraTensorModule.curry\n                        (((AlgebraTensorModule.curry\n                                  (((LinearMap.llcomp R (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ)\n                                          (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R]\n                                            DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ)\n                                          (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R]\n                                            DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ))\n                                        LinearMap.lflip)\n                                      ((LinearMap.llcomp R (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ)\n                                            (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ)\n                                            (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ))\n                                          mA.flip ∘ₗ\n                                        mA)).flip ∘ₗ\n                                lof R ι 𝒜 ixa)\n                              xa ∘ₗ\n                            lof R ι ℬ ixb)\n                          xb) ∘ₗ\n                      lof R ι 𝒜 iya)\n                    ya ∘ₗ\n                  lof R ι ℬ iyb)\n                yb) ∘ₗ\n            lof R ι 𝒜 iza)\n          za ∘ₗ\n        lof R ι ℬ izb)\n      zb","state_after":"case a.H.h.H.h.a.H.h.H.h.a.H.h.H.h\nR : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹¹ : CommSemiring ι\ninst✝¹⁰ : Module ι (Additive ℤˣ)\ninst✝⁹ : DecidableEq ι\n𝒜 : ι → Type u_5\nℬ : ι → Type u_6\ninst✝⁸ : CommRing R\ninst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)\ninst✝⁶ : (i : ι) → AddCommGroup (ℬ i)\ninst✝⁵ : (i : ι) → Module R (𝒜 i)\ninst✝⁴ : (i : ι) → Module R (ℬ i)\ninst✝³ : DirectSum.GRing 𝒜\ninst✝² : DirectSum.GRing ℬ\ninst✝¹ : DirectSum.GAlgebra R 𝒜\ninst✝ : DirectSum.GAlgebra R ℬ\nx y z : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ\nmA : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ :=\n  gradedMul R 𝒜 ℬ\nixa : ι\nxa : 𝒜 ixa\nixb : ι\nxb : ℬ ixb\niya : ι\nya : 𝒜 iya\niyb : ι\nyb : ℬ iyb\niza : ι\nza : 𝒜 iza\nizb : ι\nzb : ℬ izb\n⊢ ((gradedMul R 𝒜 ℬ)\n        (((gradedMul R 𝒜 ℬ) ((lof R ι 𝒜 ixa) xa ⊗ₜ[R] (lof R ι ℬ ixb) xb))\n          ((lof R ι 𝒜 iya) ya ⊗ₜ[R] (lof R ι ℬ iyb) yb)))\n      ((lof R ι 𝒜 iza) za ⊗ₜ[R] (lof R ι ℬ izb) zb) =\n    ((gradedMul R 𝒜 ℬ) ((lof R ι 𝒜 ixa) xa ⊗ₜ[R] (lof R ι ℬ ixb) xb))\n      (((gradedMul R 𝒜 ℬ) ((lof R ι 𝒜 iya) ya ⊗ₜ[R] (lof R ι ℬ iyb) yb)) ((lof R ι 𝒜 iza) za ⊗ₜ[R] (lof R ι ℬ izb) zb))","tactic":"dsimp [mA]","premises":[]},{"state_before":"case a.H.h.H.h.a.H.h.H.h.a.H.h.H.h\nR : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹¹ : CommSemiring ι\ninst✝¹⁰ : Module ι (Additive ℤˣ)\ninst✝⁹ : DecidableEq ι\n𝒜 : ι → Type u_5\nℬ : ι → Type u_6\ninst✝⁸ : CommRing R\ninst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)\ninst✝⁶ : (i : ι) → AddCommGroup (ℬ i)\ninst✝⁵ : (i : ι) → Module R (𝒜 i)\ninst✝⁴ : (i : ι) → Module R (ℬ i)\ninst✝³ : DirectSum.GRing 𝒜\ninst✝² : DirectSum.GRing ℬ\ninst✝¹ : DirectSum.GAlgebra R 𝒜\ninst✝ : DirectSum.GAlgebra R ℬ\nx y z : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ\nmA : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ :=\n  gradedMul R 𝒜 ℬ\nixa : ι\nxa : 𝒜 ixa\nixb : ι\nxb : ℬ ixb\niya : ι\nya : 𝒜 iya\niyb : ι\nyb : ℬ iyb\niza : ι\nza : 𝒜 iza\nizb : ι\nzb : ℬ izb\n⊢ ((gradedMul R 𝒜 ℬ)\n        (((gradedMul R 𝒜 ℬ) ((lof R ι 𝒜 ixa) xa ⊗ₜ[R] (lof R ι ℬ ixb) xb))\n          ((lof R ι 𝒜 iya) ya ⊗ₜ[R] (lof R ι ℬ iyb) yb)))\n      ((lof R ι 𝒜 iza) za ⊗ₜ[R] (lof R ι ℬ izb) zb) =\n    ((gradedMul R 𝒜 ℬ) ((lof R ι 𝒜 ixa) xa ⊗ₜ[R] (lof R ι ℬ ixb) xb))\n      (((gradedMul R 𝒜 ℬ) ((lof R ι 𝒜 iya) ya ⊗ₜ[R] (lof R ι ℬ iyb) yb)) ((lof R ι 𝒜 iza) za ⊗ₜ[R] (lof R ι ℬ izb) zb))","state_after":"case a.H.h.H.h.a.H.h.H.h.a.H.h.H.h\nR : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹¹ : CommSemiring ι\ninst✝¹⁰ : Module ι (Additive ℤˣ)\ninst✝⁹ : DecidableEq ι\n𝒜 : ι → Type u_5\nℬ : ι → Type u_6\ninst✝⁸ : CommRing R\ninst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)\ninst✝⁶ : (i : ι) → AddCommGroup (ℬ i)\ninst✝⁵ : (i : ι) → Module R (𝒜 i)\ninst✝⁴ : (i : ι) → Module R (ℬ i)\ninst✝³ : DirectSum.GRing 𝒜\ninst✝² : DirectSum.GRing ℬ\ninst✝¹ : DirectSum.GAlgebra R 𝒜\ninst✝ : DirectSum.GAlgebra R ℬ\nx y z : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ\nmA : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ :=\n  gradedMul R 𝒜 ℬ\nixa : ι\nxa : 𝒜 ixa\nixb : ι\nxb : ℬ ixb\niya : ι\nya : 𝒜 iya\niyb : ι\nyb : ℬ iyb\niza : ι\nza : 𝒜 iza\nizb : ι\nzb : ℬ izb\n⊢ ↑↑((-1) ^ (ixb * iya)) •\n      (-1) ^ ((ixb + iyb) * iza) •\n        ((lof R ι 𝒜 ixa) xa * ((lof R ι 𝒜 iya) ya * (lof R ι 𝒜 iza) za)) ⊗ₜ[R]\n          ((lof R ι ℬ ixb) xb * ((lof R ι ℬ iyb) yb * (lof R ι ℬ izb) zb)) =\n    ↑↑((-1) ^ (iyb * iza)) •\n      (-1) ^ (ixb * (iya + iza)) •\n        ((lof R ι 𝒜 ixa) xa * ((lof R ι 𝒜 iya) ya * (lof R ι 𝒜 iza) za)) ⊗ₜ[R]\n          ((lof R ι ℬ ixb) xb * ((lof R ι ℬ iyb) yb * (lof R ι ℬ izb) zb))","tactic":"simp_rw [tmul_of_gradedMul_of_tmul, Units.smul_def, ← Int.cast_smul_eq_nsmul R,\n    LinearMap.map_smul₂, LinearMap.map_smul, DirectSum.lof_eq_of, DirectSum.of_mul_of,\n    ← DirectSum.lof_eq_of R, tmul_of_gradedMul_of_tmul, DirectSum.lof_eq_of, ← DirectSum.of_mul_of,\n    ← DirectSum.lof_eq_of R, mul_assoc]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"DirectSum.lof_eq_of","def_path":"Mathlib/Algebra/DirectSum/Module.lean","def_pos":[66,8],"def_end_pos":[66,17]},{"full_name":"DirectSum.of_mul_of","def_path":"Mathlib/Algebra/DirectSum/Ring.lean","def_pos":[206,8],"def_end_pos":[206,17]},{"full_name":"Int.cast_smul_eq_nsmul","def_path":"Mathlib/Algebra/Module/Defs.lean","def_pos":[373,6],"def_end_pos":[373,28]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"LinearMap.map_smul","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[348,18],"def_end_pos":[348,26]},{"full_name":"LinearMap.map_smul₂","def_path":"Mathlib/LinearAlgebra/BilinearMap.lean","def_pos":[139,8],"def_end_pos":[139,17]},{"full_name":"TensorProduct.tmul_of_gradedMul_of_tmul","def_path":"Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean","def_pos":[183,8],"def_end_pos":[183,33]},{"full_name":"Units.smul_def","def_path":"Mathlib/Algebra/Group/Action/Units.lean","def_pos":[30,21],"def_end_pos":[30,29]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]}]},{"state_before":"case a.H.h.H.h.a.H.h.H.h.a.H.h.H.h\nR : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹¹ : CommSemiring ι\ninst✝¹⁰ : Module ι (Additive ℤˣ)\ninst✝⁹ : DecidableEq ι\n𝒜 : ι → Type u_5\nℬ : ι → Type u_6\ninst✝⁸ : CommRing R\ninst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)\ninst✝⁶ : (i : ι) → AddCommGroup (ℬ i)\ninst✝⁵ : (i : ι) → Module R (𝒜 i)\ninst✝⁴ : (i : ι) → Module R (ℬ i)\ninst✝³ : DirectSum.GRing 𝒜\ninst✝² : DirectSum.GRing ℬ\ninst✝¹ : DirectSum.GAlgebra R 𝒜\ninst✝ : DirectSum.GAlgebra R ℬ\nx y z : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ\nmA : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ :=\n  gradedMul R 𝒜 ℬ\nixa : ι\nxa : 𝒜 ixa\nixb : ι\nxb : ℬ ixb\niya : ι\nya : 𝒜 iya\niyb : ι\nyb : ℬ iyb\niza : ι\nza : 𝒜 iza\nizb : ι\nzb : ℬ izb\n⊢ ↑↑((-1) ^ (ixb * iya)) •\n      (-1) ^ ((ixb + iyb) * iza) •\n        ((lof R ι 𝒜 ixa) xa * ((lof R ι 𝒜 iya) ya * (lof R ι 𝒜 iza) za)) ⊗ₜ[R]\n          ((lof R ι ℬ ixb) xb * ((lof R ι ℬ iyb) yb * (lof R ι ℬ izb) zb)) =\n    ↑↑((-1) ^ (iyb * iza)) •\n      (-1) ^ (ixb * (iya + iza)) •\n        ((lof R ι 𝒜 ixa) xa * ((lof R ι 𝒜 iya) ya * (lof R ι 𝒜 iza) za)) ⊗ₜ[R]\n          ((lof R ι ℬ ixb) xb * ((lof R ι ℬ iyb) yb * (lof R ι ℬ izb) zb))","state_after":"case a.H.h.H.h.a.H.h.H.h.a.H.h.H.h\nR : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹¹ : CommSemiring ι\ninst✝¹⁰ : Module ι (Additive ℤˣ)\ninst✝⁹ : DecidableEq ι\n𝒜 : ι → Type u_5\nℬ : ι → Type u_6\ninst✝⁸ : CommRing R\ninst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)\ninst✝⁶ : (i : ι) → AddCommGroup (ℬ i)\ninst✝⁵ : (i : ι) → Module R (𝒜 i)\ninst✝⁴ : (i : ι) → Module R (ℬ i)\ninst✝³ : DirectSum.GRing 𝒜\ninst✝² : DirectSum.GRing ℬ\ninst✝¹ : DirectSum.GAlgebra R 𝒜\ninst✝ : DirectSum.GAlgebra R ℬ\nx y z : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ\nmA : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ :=\n  gradedMul R 𝒜 ℬ\nixa : ι\nxa : 𝒜 ixa\nixb : ι\nxb : ℬ ixb\niya : ι\nya : 𝒜 iya\niyb : ι\nyb : ℬ iyb\niza : ι\nza : 𝒜 iza\nizb : ι\nzb : ℬ izb\n⊢ (-1) ^ (ixb * iya + (ixb * iza + iyb * iza)) •\n      ((lof R ι 𝒜 ixa) xa * ((lof R ι 𝒜 iya) ya * (lof R ι 𝒜 iza) za)) ⊗ₜ[R]\n        ((lof R ι ℬ ixb) xb * ((lof R ι ℬ iyb) yb * (lof R ι ℬ izb) zb)) =\n    (-1) ^ (iyb * iza + (ixb * iya + ixb * iza)) •\n      ((lof R ι 𝒜 ixa) xa * ((lof R ι 𝒜 iya) ya * (lof R ι 𝒜 iza) za)) ⊗ₜ[R]\n        ((lof R ι ℬ ixb) xb * ((lof R ι ℬ iyb) yb * (lof R ι ℬ izb) zb))","tactic":"simp_rw [Int.cast_smul_eq_nsmul R, ← Units.smul_def, smul_smul, ← uzpow_add, add_mul, mul_add]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Int.cast_smul_eq_nsmul","def_path":"Mathlib/Algebra/Module/Defs.lean","def_pos":[373,6],"def_end_pos":[373,28]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Units.smul_def","def_path":"Mathlib/Algebra/Group/Action/Units.lean","def_pos":[30,21],"def_end_pos":[30,29]},{"full_name":"smul_smul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[374,6],"def_end_pos":[374,15]},{"full_name":"uzpow_add","def_path":"Mathlib/Data/ZMod/IntUnitsPower.lean","def_pos":[94,6],"def_end_pos":[94,15]}]},{"state_before":"case a.H.h.H.h.a.H.h.H.h.a.H.h.H.h\nR : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹¹ : CommSemiring ι\ninst✝¹⁰ : Module ι (Additive ℤˣ)\ninst✝⁹ : DecidableEq ι\n𝒜 : ι → Type u_5\nℬ : ι → Type u_6\ninst✝⁸ : CommRing R\ninst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)\ninst✝⁶ : (i : ι) → AddCommGroup (ℬ i)\ninst✝⁵ : (i : ι) → Module R (𝒜 i)\ninst✝⁴ : (i : ι) → Module R (ℬ i)\ninst✝³ : DirectSum.GRing 𝒜\ninst✝² : DirectSum.GRing ℬ\ninst✝¹ : DirectSum.GAlgebra R 𝒜\ninst✝ : DirectSum.GAlgebra R ℬ\nx y z : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ\nmA : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ :=\n  gradedMul R 𝒜 ℬ\nixa : ι\nxa : 𝒜 ixa\nixb : ι\nxb : ℬ ixb\niya : ι\nya : 𝒜 iya\niyb : ι\nyb : ℬ iyb\niza : ι\nza : 𝒜 iza\nizb : ι\nzb : ℬ izb\n⊢ (-1) ^ (ixb * iya + (ixb * iza + iyb * iza)) •\n      ((lof R ι 𝒜 ixa) xa * ((lof R ι 𝒜 iya) ya * (lof R ι 𝒜 iza) za)) ⊗ₜ[R]\n        ((lof R ι ℬ ixb) xb * ((lof R ι ℬ iyb) yb * (lof R ι ℬ izb) zb)) =\n    (-1) ^ (iyb * iza + (ixb * iya + ixb * iza)) •\n      ((lof R ι 𝒜 ixa) xa * ((lof R ι 𝒜 iya) ya * (lof R ι 𝒜 iza) za)) ⊗ₜ[R]\n        ((lof R ι ℬ ixb) xb * ((lof R ι ℬ iyb) yb * (lof R 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a.H.h.H.h.a.H.h.H.h.a.H.h.H.h.e_a.e_a\nR : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹¹ : CommSemiring ι\ninst✝¹⁰ : Module ι (Additive ℤˣ)\ninst✝⁹ : DecidableEq ι\n𝒜 : ι → Type u_5\nℬ : ι → Type u_6\ninst✝⁸ : CommRing R\ninst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)\ninst✝⁶ : (i : ι) → AddCommGroup (ℬ i)\ninst✝⁵ : (i : ι) → Module R (𝒜 i)\ninst✝⁴ : (i : ι) → Module R (ℬ i)\ninst✝³ : DirectSum.GRing 𝒜\ninst✝² : DirectSum.GRing ℬ\ninst✝¹ : DirectSum.GAlgebra R 𝒜\ninst✝ : DirectSum.GAlgebra R ℬ\nx y z : DirectSum ι 𝒜 ⊗[R] DirectSum �� ℬ\nmA : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ :=\n  gradedMul R 𝒜 ℬ\nixa : ι\nxa : 𝒜 ixa\nixb : ι\nxb : ℬ ixb\niya : ι\nya : 𝒜 iya\niyb : ι\nyb : ℬ iyb\niza : ι\nza : 𝒜 iza\nizb : ι\nzb : ℬ izb\n⊢ ixb * iya + (ixb * iza + iyb * iza) = iyb * iza + (ixb * iya + ixb * iza)","state_after":"no goals","tactic":"abel","premises":[]}]}
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X₂)).hom","state_after":"no goals","tactic":"simpa using tensor_left_unitality C X₁ X₂","premises":[{"full_name":"CategoryTheory.tensor_left_unitality","def_path":"Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean","def_pos":[544,8],"def_end_pos":[544,29]}]},{"state_before":"C : Type u₁\ninst✝⁸ : Category.{v₁, u₁} C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category.{v₂, u₂} D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category.{v₃, u₃} E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nx✝ : C × C\nX₁ X₂ : C\n⊢ (ρ_ (__src✝.obj (X₁, X₂))).hom =\n    __src✝.obj (X₁, X₂) ◁ (λ_ (𝟙_ C)).inv ≫ tensor_μ C (X₁, X₂) (𝟙_ (C × C)) ≫ __src✝.map (ρ_ (X₁, X₂)).hom","state_after":"no goals","tactic":"simpa using tensor_right_unitality C X₁ 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MonoidalCategory E\ninst✝ : BraidedCategory E\n⊢ ∀ (X Y : C × C),\n    IsIso\n      ((α_ X.1 X.2 (Y.1 ⊗ Y.2)).hom ≫\n        X.1 ◁ (α_ X.2 Y.1 Y.2).inv ≫\n          X.1 ◁ (β_ X.2 Y.1).hom ▷ Y.2 ≫ X.1 ◁ (α_ Y.1 X.2 Y.2).hom ≫ (α_ X.1 Y.1 (X.2 ⊗ Y.2)).inv)","tactic":"dsimp [tensor_μ]","premises":[{"full_name":"CategoryTheory.tensor_μ","def_path":"Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean","def_pos":[508,4],"def_end_pos":[508,12]}]},{"state_before":"C : Type u₁\ninst✝⁸ : Category.{v₁, u₁} C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category.{v₂, u₂} D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category.{v₃, u₃} E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\n⊢ ∀ (X Y : C × C),\n    IsIso\n      ((α_ X.1 X.2 (Y.1 ⊗ Y.2)).hom ≫\n        X.1 ◁ (α_ X.2 Y.1 Y.2).inv ≫\n          X.1 ◁ (β_ X.2 Y.1).hom ▷ Y.2 ≫ X.1 ◁ (α_ Y.1 X.2 Y.2).hom ≫ (α_ X.1 Y.1 (X.2 ⊗ Y.2)).inv)","state_after":"no 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+{"url":"Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean","commit":"","full_name":"MeasureTheory.exists_upperSemicontinuous_le_integral_le","start":[404,0],"end":[434,37],"file_path":"Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean","tactics":[{"state_before":"α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : μ.WeaklyRegular\nf : α → ℝ≥0\nfint : Integrable (fun x => ↑(f x)) μ\nε : ℝ\nεpos : 0 < ε\n⊢ ∃ g,\n    (∀ (x : α), g x ≤ f x) ∧\n      UpperSemicontinuous g ∧ Integrable (fun x => ↑(g x)) μ ∧ ∫ (x : α), ↑(f x) ∂μ - ε ≤ ∫ (x : α), ↑(g x) ∂μ","state_after":"case intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : μ.WeaklyRegular\nf : α → ℝ≥0\nfint : Integrable (fun x => ↑(f x)) μ\nε : ℝ≥0\nεpos : 0 < ↑ε\n⊢ ∃ g,\n    (∀ (x : α), g x ≤ f x) ∧\n      UpperSemicontinuous g ∧ Integrable (fun x => ↑(g x)) μ ∧ ∫ (x : α), ↑(f x) ∂μ - ↑ε ≤ ∫ (x : α), ↑(g x) ∂μ","tactic":"lift ε to ℝ≥0 using εpos.le","premises":[{"full_name":"NNReal","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[60,4],"def_end_pos":[60,10]}]},{"state_before":"case intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : μ.WeaklyRegular\nf : α → ℝ≥0\nfint : Integrable (fun x => ↑(f x)) μ\nε : ℝ≥0\nεpos : 0 < ↑ε\n⊢ ∃ g,\n    (∀ (x : α), g x ≤ f x) ∧\n      UpperSemicontinuous g ∧ Integrable (fun x => ↑(g x)) μ ∧ ∫ (x : α), ↑(f x) ∂μ - ↑ε ≤ ∫ (x : α), ↑(g x) ∂μ","state_after":"case intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : μ.WeaklyRegular\nf : α → ℝ≥0\nfint : Integrable (fun x => ↑(f x)) μ\nε : ℝ≥0\nεpos : 0 < ε\n⊢ ∃ g,\n    (∀ (x : α), g x ≤ f x) ∧\n      UpperSemicontinuous g ∧ Integrable (fun x => ↑(g x)) μ ∧ ∫ (x : α), ↑(f x) ∂μ - ↑ε ≤ ∫ (x : α), ↑(g x) ∂μ","tactic":"NNReal.coe_pos,","premises":[{"full_name":"NNReal.coe_pos","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[329,25],"def_end_pos":[329,32]}]},{"state_before":"case intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : μ.WeaklyRegular\nf : α → ℝ≥0\nfint : Integrable (fun x => ↑(f x)) μ\nε : ℝ≥0\nεpos : 0 < ε\n⊢ ∃ g,\n    (∀ (x : α), g x ≤ f x) ∧\n      UpperSemicontinuous g ∧ Integrable (fun x => ↑(g x)) μ ∧ ∫ (x : α), ↑(f x) ∂μ - ↑ε ≤ ∫ (x : α), ↑(g x) ∂μ","state_after":"case intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : μ.WeaklyRegular\nf : α → ��≥0\nfint : Integrable (fun x => ↑(f x)) μ\nε : ℝ≥0\nεpos : 0 < ↑ε\n⊢ ∃ g,\n    (∀ (x : α), g x ≤ f x) ∧\n      UpperSemicontinuous g ∧ Integrable (fun x => ↑(g x)) μ ∧ ∫ (x : α), ↑(f x) ∂μ - ↑ε ≤ ∫ (x : α), ↑(g x) ∂μ","tactic":"← ENNReal.coe_pos","premises":[{"full_name":"ENNReal.coe_pos","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[338,27],"def_end_pos":[338,34]}]},{"state_before":"case intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : μ.WeaklyRegular\nf : α → ℝ≥0\nfint : Integrable (fun x => ↑(f x)) μ\nε : ℝ≥0\nεpos : 0 < ↑ε\n⊢ ∃ g,\n    (∀ (x : α), g x ≤ f x) ∧\n      UpperSemicontinuous g ∧ Integrable (fun x => ↑(g x)) μ ∧ ∫ (x : α), ↑(f x) ∂μ - ↑ε ≤ ∫ (x : α), ↑(g x) ∂μ","state_after":"case intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : μ.WeaklyRegular\nf : α → ℝ≥0\nfint : Integrable (fun x => ↑(f x)) μ\nε : ℝ≥0\nεpos : 0 < ↑ε\nIf : ∫⁻ (x : α), ↑(f x) ∂μ < ⊤\n⊢ ∃ g,\n    (∀ (x : α), g x ≤ f x) ∧\n      UpperSemicontinuous g ∧ Integrable (fun x => ↑(g x)) μ ∧ ∫ (x : α), ↑(f x) ∂μ - ↑ε ≤ ∫ (x : α), ↑(g x) ∂μ","tactic":"have If : (∫⁻ x, f x ∂μ) < ∞ := hasFiniteIntegral_iff_ofNNReal.1 fint.hasFiniteIntegral","premises":[{"full_name":"ENNReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[96,4],"def_end_pos":[96,11]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"MeasureTheory.Integrable.hasFiniteIntegral","def_path":"Mathlib/MeasureTheory/Function/L1Space.lean","def_pos":[403,8],"def_end_pos":[403,36]},{"full_name":"MeasureTheory.hasFiniteIntegral_iff_ofNNReal","def_path":"Mathlib/MeasureTheory/Function/L1Space.lean","def_pos":[116,8],"def_end_pos":[116,38]},{"full_name":"MeasureTheory.lintegral","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[59,16],"def_end_pos":[59,25]},{"full_name":"Top.top","def_path":"Mathlib/Order/Notation.lean","def_pos":[94,2],"def_end_pos":[94,5]}]},{"state_before":"case intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : μ.WeaklyRegular\nf : α → ℝ≥0\nfint : Integrable (fun x => ↑(f x)) μ\nε : ℝ≥0\nεpos : 0 < ↑ε\nIf : ∫⁻ (x : α), ↑(f x) ∂μ < ⊤\n⊢ ∃ g,\n    (∀ (x : α), g x ≤ f x) ∧\n      UpperSemicontinuous g ∧ Integrable (fun x => ↑(g x)) μ ∧ ∫ (x : α), ↑(f x) ∂μ - ↑ε ≤ ∫ (x : α), ↑(g x) ∂μ","state_after":"case intro.intro.intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : μ.WeaklyRegular\nf : α → ℝ≥0\nfint : Integrable (fun x => ↑(f x)) μ\nε : ℝ≥0\nεpos : 0 < ↑ε\nIf : ∫⁻ (x : α), ↑(f x) ∂μ < ⊤\ng : α → ℝ≥0\ngf : ∀ (x : α), g x ≤ f x\ngcont : UpperSemicontinuous g\ngint : ∫⁻ (x : α), ↑(f x) ∂μ ≤ ∫⁻ (x : α), ↑(g x) ∂μ + ↑ε\n⊢ ∃ g,\n    (∀ (x : α), g x ≤ f x) ∧\n      UpperSemicontinuous g ∧ Integrable (fun x => ↑(g x)) μ ∧ ∫ (x : α), ↑(f x) ∂μ - ↑ε ≤ ∫ (x : α), ↑(g x) ∂μ","tactic":"rcases exists_upperSemicontinuous_le_lintegral_le f If.ne εpos.ne' with ⟨g, gf, gcont, gint⟩","premises":[{"full_name":"LT.lt.ne'","def_path":"Mathlib/Order/Basic.lean","def_pos":[267,8],"def_end_pos":[267,11]},{"full_name":"MeasureTheory.exists_upperSemicontinuous_le_lintegral_le","def_path":"Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean","def_pos":[375,8],"def_end_pos":[375,50]}]},{"state_before":"case intro.intro.intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : μ.WeaklyRegular\nf : α → ℝ≥0\nfint : Integrable (fun x => ↑(f x)) μ\nε : ℝ≥0\nεpos : 0 < ↑ε\nIf : ∫⁻ (x : α), ↑(f x) ∂μ < ⊤\ng : α → ℝ≥0\ngf : ∀ (x : α), g x ≤ f x\ngcont : UpperSemicontinuous g\ngint : ∫⁻ (x : α), ↑(f x) ∂μ ≤ ∫⁻ (x : α), ↑(g x) ∂μ + ↑ε\n⊢ ∃ g,\n    (∀ (x : α), g x ≤ f x) ∧\n      UpperSemicontinuous g ∧ Integrable (fun x => ↑(g x)) μ ∧ ∫ (x : α), ↑(f x) ∂μ - ↑ε ≤ ∫ (x : α), ↑(g x) ∂μ","state_after":"case intro.intro.intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : μ.WeaklyRegular\nf : α → ℝ≥0\nfint : Integrable (fun x => ↑(f x)) μ\nε : ℝ≥0\nεpos : 0 < ↑ε\nIf : ∫⁻ (x : α), ↑(f x) ∂μ < ⊤\ng : α → ℝ≥0\ngf : ∀ (x : α), g x ≤ f x\ngcont : UpperSemicontinuous g\ngint : ∫⁻ (x : α), ↑(f x) ∂μ ≤ ∫⁻ (x : α), ↑(g x) ∂μ + ↑ε\nIg : ∫⁻ (x : α), ↑(g x) ∂μ < ⊤\n⊢ ∃ g,\n    (∀ (x : α), g x ≤ f x) ∧\n      UpperSemicontinuous g ∧ Integrable (fun x => ↑(g x)) μ ∧ ∫ (x : α), ↑(f x) ∂μ - ↑ε ≤ ∫ (x : α), ↑(g x) ∂μ","tactic":"have Ig : (∫⁻ x, g x ∂μ) < ∞ := by\n    refine lt_of_le_of_lt (lintegral_mono fun x => ?_) If\n    simpa using gf x","premises":[{"full_name":"ENNReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[96,4],"def_end_pos":[96,11]},{"full_name":"MeasureTheory.lintegral","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[59,16],"def_end_pos":[59,25]},{"full_name":"MeasureTheory.lintegral_mono","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[95,8],"def_end_pos":[95,22]},{"full_name":"Top.top","def_path":"Mathlib/Order/Notation.lean","def_pos":[94,2],"def_end_pos":[94,5]},{"full_name":"lt_of_le_of_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[99,8],"def_end_pos":[99,22]}]},{"state_before":"case intro.intro.intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : μ.WeaklyRegular\nf : α → ℝ≥0\nfint : Integrable (fun x => ↑(f x)) μ\nε : ℝ≥0\nεpos : 0 < ↑ε\nIf : ∫⁻ (x : α), ↑(f x) ∂μ < ⊤\ng : α → ℝ≥0\ngf : ∀ (x : α), g x ≤ f x\ngcont : UpperSemicontinuous g\ngint : ∫⁻ (x : α), ↑(f x) ∂μ ≤ ∫⁻ (x : α), ↑(g x) ∂μ + ↑ε\nIg : ∫⁻ (x : α), ↑(g x) ∂μ < ⊤\n⊢ ∃ g,\n    (∀ (x : α), g x ≤ f x) ∧\n      UpperSemicontinuous g ∧ Integrable (fun x => ↑(g x)) μ ∧ ∫ (x : α), ↑(f x) ∂μ - ↑ε ≤ ∫ (x : α), ↑(g x) ∂μ","state_after":"case intro.intro.intro.intro.refine_1\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : μ.WeaklyRegular\nf : α → ℝ≥0\nfint : Integrable (fun x => ↑(f x)) μ\nε : ℝ≥0\nεpos : 0 < ↑ε\nIf : ∫⁻ (x : α), ↑(f x) ∂μ < ⊤\ng : α → ℝ≥0\ngf : ∀ (x : α), g x ≤ f x\ngcont : UpperSemicontinuous g\ngint : ∫⁻ (x : α), ↑(f x) ∂μ ≤ ∫⁻ (x : α), ↑(g x) ∂μ + ↑ε\nIg : ∫⁻ (x : α), ↑(g x) ∂μ < ⊤\n⊢ Integrable (fun x => ↑(g x)) μ\n\ncase intro.intro.intro.intro.refine_2\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : μ.WeaklyRegular\nf : α → ℝ≥0\nfint : Integrable (fun x => ↑(f x)) μ\nε : ℝ≥0\nεpos : 0 < ↑ε\nIf : ∫⁻ (x : α), ↑(f x) ∂μ < ⊤\ng : α → ℝ≥0\ngf : ∀ (x : α), g x ≤ f x\ngcont : UpperSemicontinuous g\ngint : ∫⁻ (x : α), ↑(f x) ∂μ ≤ ∫⁻ (x : α), ↑(g x) ∂μ + ↑ε\nIg : ∫⁻ (x : α), ↑(g x) ∂μ < ⊤\n⊢ ∫ (x : α), ↑(f x) ∂μ - ↑ε ≤ ∫ (x : α), ↑(g x) ∂μ","tactic":"refine ⟨g, gf, gcont, ?_, ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]}]}
+{"url":"Mathlib/GroupTheory/Perm/Cycle/Factors.lean","commit":"","full_name":"Equiv.Perm.SameCycle.exists_pow_eq","start":[270,0],"end":[286,86],"file_path":"Mathlib/GroupTheory/Perm/Cycle/Factors.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\nf✝ g : Perm α\nx y : α\ninst✝³ : DecidableRel f✝.SameCycle\ninst✝² : DecidableRel g.SameCycle\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf : Perm α\nh : f.SameCycle x y\n⊢ ∃ i, 0 < i ∧ i ≤ (f.cycleOf x).support.card + 1 ∧ (f ^ i) x = y","state_after":"case pos\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nf✝ g : Perm α\nx y : α\ninst✝³ : DecidableRel f✝.SameCycle\ninst✝² : DecidableRel g.SameCycle\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf : Perm α\nh : f.SameCycle x y\nhx : x ∈ f.support\n⊢ ∃ i, 0 < i ∧ i ≤ (f.cycleOf x).support.card + 1 ∧ (f ^ i) x = y\n\ncase neg\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nf✝ g : Perm α\nx y : α\ninst✝³ : DecidableRel f✝.SameCycle\ninst✝² : DecidableRel g.SameCycle\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf : Perm α\nh : f.SameCycle x y\nhx : x ∉ f.support\n⊢ ∃ i, 0 < i ∧ i ≤ (f.cycleOf x).support.card + 1 ∧ (f ^ i) x = y","tactic":"by_cases hx : x ∈ f.support","premises":[{"full_name":"Equiv.Perm.support","def_path":"Mathlib/GroupTheory/Perm/Support.lean","def_pos":[267,4],"def_end_pos":[267,11]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
+{"url":"Mathlib/CategoryTheory/Localization/Adjunction.lean","commit":"","full_name":"CategoryTheory.Adjunction.localization_counit_app","start":[120,0],"end":[126,26],"file_path":"Mathlib/CategoryTheory/Localization/Adjunction.lean","tactics":[{"state_before":"C₁ : Type u_1\nC₂ : Type u_2\nD₁ : Type u_3\nD₂ : Type u_4\ninst✝⁷ : Category.{u_8, u_1} C₁\ninst✝⁶ : Category.{u_7, u_2} C₂\ninst✝⁵ : Category.{u_6, u_3} D₁\ninst✝⁴ : Category.{u_5, u_4} D₂\nG : C₁ ⥤ C₂\nF : C₂ ⥤ C₁\nadj : G ⊣ F\nL₁ : C₁ ⥤ D₁\nW₁ : MorphismProperty C₁\ninst✝³ : L₁.IsLocalization W₁\nL₂ : C₂ ⥤ D₂\nW₂ : MorphismProperty C₂\ninst✝² : L₂.IsLocalization W₂\nG' : D₁ ⥤ D₂\nF' : D₂ ⥤ D₁\ninst✝¹ : CatCommSq G L₁ L₂ G'\ninst✝ : CatCommSq F L₂ L₁ F'\nX₂ : C₂\n⊢ (adj.localization L₁ W₁ L₂ W₂ G' F').counit.app (L₂.obj X₂) =\n    G'.map ((CatCommSq.iso F L₂ L₁ F').inv.app X₂) ≫\n      (CatCommSq.iso G L₁ L₂ G').inv.app (F.obj X₂) ≫ L₂.map (adj.counit.app X₂)","state_after":"no goals","tactic":"apply Localization.η_app","premises":[{"full_name":"CategoryTheory.Adjunction.Localization.η_app","def_path":"Mathlib/CategoryTheory/Localization/Adjunction.lean","def_pos":[65,6],"def_end_pos":[65,11]}]}]}
+{"url":"Mathlib/Analysis/SumOverResidueClass.lean","commit":"","full_name":"not_summable_of_antitone_of_neg","start":[47,0],"end":[57,65],"file_path":"Mathlib/Analysis/SumOverResidueClass.lean","tactics":[{"state_before":"f : ℕ → ℝ\nhf : Antitone f\nn : ℕ\nhn : f n < 0\n⊢ ¬Summable f","state_after":"f : ℕ → ℝ\nhf : Antitone f\nn : ℕ\nhn : f n < 0\nhs : Summable f\n⊢ False","tactic":"intro hs","premises":[]},{"state_before":"f : ℕ → ℝ\nhf : Antitone f\nn : ℕ\nhn : f n < 0\nhs : Summable f\n⊢ False","state_after":"f : ℕ → ℝ\nhf : Antitone f\nn : ℕ\nhn : f n < 0\nhs : Summable f\nthis : Filter.Tendsto f Filter.atTop (nhds 0)\n⊢ False","tactic":"have := hs.tendsto_atTop_zero","premises":[{"full_name":"Summable.tendsto_atTop_zero","def_path":"Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean","def_pos":[292,2],"def_end_pos":[292,13]}]},{"state_before":"f : ℕ → ℝ\nhf : Antitone f\nn : ℕ\nhn : f n < 0\nhs : Summable f\nthis : Filter.Tendsto f Filter.atTop (nhds 0)\n⊢ False","state_after":"f : ℕ → ℝ\nhf : Antitone f\nn : ℕ\nhn : f n < 0\nhs : Summable f\nthis : ∀ ε > 0, ∃ N, ∀ n ≥ N, |f n| < ε\n⊢ False","tactic":"simp only [Metric.tendsto_atTop, dist_zero_right, Real.norm_eq_abs] at this","premises":[{"full_name":"Metric.tendsto_atTop","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[932,8],"def_end_pos":[932,21]},{"full_name":"Real.norm_eq_abs","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[1132,8],"def_end_pos":[1132,19]},{"full_name":"dist_zero_right","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[360,2],"def_end_pos":[360,13]}]},{"state_before":"f : ℕ → ℝ\nhf : Antitone f\nn : ℕ\nhn : f n < 0\nhs : Summable f\nthis : ∀ ε > 0, ∃ N, ∀ n ≥ N, |f n| < ε\n⊢ False","state_after":"case intro\nf : ℕ → ℝ\nhf : Antitone f\nn : ℕ\nhn : f n < 0\nhs : Summable f\nthis : ∀ ε > 0, ∃ N, ∀ n ≥ N, |f n| < ε\nN : ℕ\nhN : ∀ n_1 ≥ N, |f n_1| < |f n|\n⊢ False","tactic":"obtain ⟨N, hN⟩ := this (|f n|) (abs_pos_of_neg hn)","premises":[{"full_name":"abs","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[33,2],"def_end_pos":[33,13]},{"full_name":"abs_pos_of_neg","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[217,14],"def_end_pos":[217,28]}]},{"state_before":"case intro\nf : ℕ → ℝ\nhf : Antitone f\nn : ℕ\nhn : f n < 0\nhs : Summable f\nthis : ∀ ε > 0, ∃ N, ∀ n ≥ N, |f n| < ε\nN : ℕ\nhN : ∀ n_1 ≥ N, |f n_1| < |f n|\n⊢ False","state_after":"case intro\nf : ℕ → ℝ\nhf : Antitone f\nn : ℕ\nhn : f n < 0\nhs : Summable f\nthis : ∀ ε > 0, ∃ N, ∀ n ≥ N, |f n| < ε\nN : ℕ\nhN : |f (max n N)| < |f n|\n⊢ False","tactic":"specialize hN (max n N) (n.le_max_right N)","premises":[{"full_name":"Max.max","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1129,2],"def_end_pos":[1129,5]},{"full_name":"Nat.le_max_right","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/MinMax.lean","def_pos":[55,18],"def_end_pos":[55,30]}]},{"state_before":"case intro\nf : ℕ → ℝ\nhf : Antitone f\nn : ℕ\nhn : f n < 0\nhs : Summable f\nthis : ∀ ε > 0, ∃ N, ∀ n ≥ N, |f n| < ε\nN : ℕ\nhN : |f (max n N)| < |f n|\n⊢ False","state_after":"case intro\nf : ℕ → ℝ\nhf : Antitone f\nn : ℕ\nhn : f n < 0\nhs : Summable f\nthis : ∀ ε > 0, ∃ N, ∀ n ≥ N, |f n| < ε\nN : ℕ\nhN : ¬False\n⊢ |f n| ≤ |f (max n N)|","tactic":"contrapose! hN","premises":[{"full_name":"Mathlib.Tactic.Contrapose.mtr","def_path":"Mathlib/Tactic/Contrapose.lean","def_pos":[24,6],"def_end_pos":[24,9]}]},{"state_before":"case intro\nf : ℕ → ℝ\nhf : Antitone f\nn : ℕ\nhn : f n < 0\nhs : Summable f\nthis : ∀ ε > 0, ∃ N, ∀ n ≥ N, |f n| < ε\nN : ℕ\nhN : ¬False\n⊢ |f n| ≤ |f (max n N)|","state_after":"case intro\nf : ℕ → ℝ\nhf : Antitone f\nn : ℕ\nhn : f n < 0\nhs : Summable f\nthis : ∀ ε > 0, ∃ N, ∀ n ≥ N, |f n| < ε\nN : ℕ\n⊢ |f n| ≤ |f (max n N)|","tactic":"clear hN","premises":[]},{"state_before":"case intro\nf : ℕ → ℝ\nhf : Antitone f\nn : ℕ\nhn : f n < 0\nhs : Summable f\nthis : ∀ ε > 0, ∃ N, ∀ n ≥ N, |f n| < ε\nN : ℕ\n⊢ |f n| ≤ |f (max n N)|","state_after":"case intro\nf : ℕ → ℝ\nhf : Antitone f\nn : ℕ\nhn : f n < 0\nhs : Summable f\nthis : ∀ ε > 0, ∃ N, ∀ n ≥ N, |f n| < ε\nN : ℕ\nH : f (max n N) ≤ f n\n⊢ |f n| ≤ |f (max n N)|","tactic":"have H : f (max n N) ≤ f n := hf (n.le_max_left N)","premises":[{"full_name":"Max.max","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1129,2],"def_end_pos":[1129,5]},{"full_name":"Nat.le_max_left","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/MinMax.lean","def_pos":[53,18],"def_end_pos":[53,29]}]},{"state_before":"case intro\nf : ℕ → ℝ\nhf : Antitone f\nn : ℕ\nhn : f n < 0\nhs : Summable f\nthis : ∀ ε > 0, ∃ N, ∀ n ≥ N, |f n| < ε\nN : ℕ\nH : f (max n N) ≤ f n\n⊢ |f n| ≤ |f (max n N)|","state_after":"no goals","tactic":"rwa [abs_of_neg hn, abs_of_neg (H.trans_lt hn), neg_le_neg_iff]","premises":[{"full_name":"abs_of_neg","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[84,2],"def_end_pos":[84,13]},{"full_name":"neg_le_neg_iff","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Basic.lean","def_pos":[216,2],"def_end_pos":[216,13]}]}]}
+{"url":"Mathlib/CategoryTheory/Generator.lean","commit":"","full_name":"CategoryTheory.isDetector_iff_reflectsIsomorphisms_coyoneda_obj","start":[547,0],"end":[556,66],"file_path":"Mathlib/CategoryTheory/Generator.lean","tactics":[{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nG : C\n⊢ IsDetector G ↔ (coyoneda.obj (op G)).ReflectsIsomorphisms","state_after":"case refine_1\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nG : C\nhG : IsDetector G\nA✝ B✝ : C\nf : A✝ ⟶ B✝\nhf : IsIso ((coyoneda.obj (op G)).map f)\nh : G ⟶ B✝\n⊢ ∃! h', h' ≫ f = h\n\ncase refine_2\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nG : C\nh : (coyoneda.obj (op G)).ReflectsIsomorphisms\nX Y : C\nf : X ⟶ Y\nhf : ∀ (h : G ⟶ Y), ∃! h', h' ≫ f = h\n⊢ IsIso f","tactic":"refine\n    ⟨fun hG => ⟨fun f hf => hG.def _ fun h => ?_⟩, fun h =>\n      (isDetector_def _).2 fun X Y f hf => ?_⟩","premises":[{"full_name":"CategoryTheory.IsDetector.def","def_path":"Mathlib/CategoryTheory/Generator.lean","def_pos":[427,8],"def_end_pos":[427,22]},{"full_name":"CategoryTheory.isDetector_def","def_path":"Mathlib/CategoryTheory/Generator.lean","def_pos":[419,8],"def_end_pos":[419,22]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]}]}]}
+{"url":"Mathlib/RingTheory/Ideal/QuotientOperations.lean","commit":"","full_name":"Ideal.ker_quotient_lift","start":[128,0],"end":[143,17],"file_path":"Mathlib/RingTheory/Ideal/QuotientOperations.lean","tactics":[{"state_before":"R : Type u\nS : Type v\nF : Type w\ninst✝¹ : CommRing R\ninst✝ : Semiring S\nI : Ideal R\nf : R →+* S\nH : I ≤ ker f\n⊢ ker (Quotient.lift I f H) = map (Quotient.mk I) (ker f)","state_after":"case h\nR : Type u\nS : Type v\nF : Type w\ninst✝¹ : CommRing R\ninst✝ : Semiring S\nI : Ideal R\nf : R →+* S\nH : I ≤ ker f\n⊢ ∀ (x : R ⧸ I), x ∈ ker (Quotient.lift I f H) ↔ x ∈ map (Quotient.mk I) (ker f)","tactic":"apply Ideal.ext","premises":[{"full_name":"Ideal.ext","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[69,8],"def_end_pos":[69,11]}]},{"state_before":"case h\nR : Type u\nS : Type v\nF : Type w\ninst✝¹ : CommRing R\ninst✝ : Semiring S\nI : Ideal R\nf : R →+* S\nH : I ≤ ker f\n⊢ ∀ (x : R ⧸ I), x ∈ ker (Quotient.lift I f H) ↔ x ∈ map (Quotient.mk I) (ker f)","state_after":"case h\nR : Type u\nS : Type v\nF : Type w\ninst✝¹ : CommRing R\ninst✝ : Semiring S\nI : Ideal R\nf : R →+* S\nH : I ≤ ker f\nx : R ⧸ I\n⊢ x ∈ ker (Quotient.lift I f H) ↔ x ∈ map (Quotient.mk I) (ker f)","tactic":"intro x","premises":[]},{"state_before":"case h\nR : Type u\nS : Type v\nF : Type w\ninst✝¹ : CommRing R\ninst✝ : Semiring S\nI : Ideal R\nf : R →+* S\nH : I ≤ ker f\nx : R ⧸ I\n⊢ x ∈ ker (Quotient.lift I f H) ↔ x ∈ map (Quotient.mk I) (ker f)","state_after":"case h.mp\nR : Type u\nS : Type v\nF : Type w\ninst✝¹ : CommRing R\ninst✝ : Semiring S\nI : Ideal R\nf : R →+* S\nH : I ≤ ker f\nx : R ⧸ I\n⊢ x ∈ ker (Quotient.lift I f H) → x ∈ map (Quotient.mk I) (ker f)\n\ncase h.mpr\nR : Type u\nS : Type v\nF : Type w\ninst✝¹ : CommRing R\ninst✝ : Semiring S\nI : Ideal R\nf : R →+* S\nH : I ≤ ker f\nx : R ⧸ I\n⊢ x ∈ map (Quotient.mk I) (ker f) → x ∈ ker (Quotient.lift I f H)","tactic":"constructor","premises":[]}]}
+{"url":"Mathlib/Data/Nat/Prime/Basic.lean","commit":"","full_name":"Nat.Prime.dvd_factorial","start":[199,0],"end":[205,93],"file_path":"Mathlib/Data/Nat/Prime/Basic.lean","tactics":[{"state_before":"n✝ n p : ℕ\nhp : Prime p\n⊢ p ∣ (n + 1)! ↔ p ≤ n + 1","state_after":"n✝ n p : ℕ\nhp : Prime p\n⊢ p ∣ n + 1 ∨ p ≤ n ↔ p ≤ n + 1","tactic":"rw [factorial_succ, hp.dvd_mul, Prime.dvd_factorial hp]","premises":[{"full_name":"Nat.Prime.dvd_mul","def_path":"Mathlib/Data/Nat/Prime/Defs.lean","def_pos":[410,8],"def_end_pos":[410,21]},{"full_name":"Nat.factorial_succ","def_path":"Mathlib/Data/Nat/Factorial/Basic.lean","def_pos":[43,8],"def_end_pos":[43,22]}]},{"state_before":"n✝ n p : ℕ\nhp : Prime p\n⊢ p ∣ n + 1 ∨ p ≤ n ↔ p ≤ n + 1","state_after":"no goals","tactic":"exact\n      ⟨fun h => h.elim (le_of_dvd (succ_pos _)) le_succ_of_le, fun h =>\n        (_root_.lt_or_eq_of_le h).elim (Or.inr ∘ le_of_lt_succ) fun h => Or.inl <| by rw [h]⟩","premises":[{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"Nat.le_of_dvd","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Dvd.lean","def_pos":[46,8],"def_end_pos":[46,17]},{"full_name":"Nat.le_of_lt_succ","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1716,8],"def_end_pos":[1716,25]},{"full_name":"Nat.le_succ_of_le","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1687,8],"def_end_pos":[1687,25]},{"full_name":"Nat.succ_pos","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1693,8],"def_end_pos":[1693,20]},{"full_name":"Or.elim","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[550,8],"def_end_pos":[550,15]},{"full_name":"Or.inl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[534,4],"def_end_pos":[534,7]},{"full_name":"Or.inr","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[536,4],"def_end_pos":[536,7]},{"full_name":"lt_or_eq_of_le","def_path":"Mathlib/Order/Defs.lean","def_pos":[191,8],"def_end_pos":[191,22]}]}]}
+{"url":"Mathlib/GroupTheory/MonoidLocalization/Basic.lean","commit":"","full_name":"AddSubmonoid.LocalizationMap.mk'_add_eq_mk'_of_add","start":[639,0],"end":[641,38],"file_path":"Mathlib/GroupTheory/MonoidLocalization/Basic.lean","tactics":[{"state_before":"M : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type u_3\ninst✝ : CommMonoid P\nf : S.LocalizationMap N\nx₁ x₂ : M\ny : ↥S\n⊢ f.mk' x₂ y * f.toMap x₁ = f.mk' (x₁ * x₂) y","state_after":"no goals","tactic":"rw [mul_comm, mul_mk'_eq_mk'_of_mul]","premises":[{"full_name":"Submonoid.LocalizationMap.mul_mk'_eq_mk'_of_mul","def_path":"Mathlib/GroupTheory/MonoidLocalization/Basic.lean","def_pos":[636,8],"def_end_pos":[636,29]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]}]}]}
+{"url":"Mathlib/Topology/MetricSpace/Isometry.lean","commit":"","full_name":"IsometryEquiv.ediam_preimage","start":[392,0],"end":[394,32],"file_path":"Mathlib/Topology/MetricSpace/Isometry.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\nh : α ≃ᵢ β\ns : Set β\n⊢ EMetric.diam (⇑h ⁻¹' s) = EMetric.diam s","state_after":"no goals","tactic":"rw [← image_symm, ediam_image]","premises":[{"full_name":"IsometryEquiv.ediam_image","def_path":"Mathlib/Topology/MetricSpace/Isometry.lean","def_pos":[305,8],"def_end_pos":[305,19]},{"full_name":"IsometryEquiv.image_symm","def_path":"Mathlib/Topology/MetricSpace/Isometry.lean","def_pos":[378,8],"def_end_pos":[378,18]}]}]}
+{"url":"Mathlib/CategoryTheory/Idempotents/KaroubiKaroubi.lean","commit":"","full_name":"CategoryTheory.Idempotents.KaroubiKaroubi.inverse_map_f","start":[38,0],"end":[42,60],"file_path":"Mathlib/CategoryTheory/Idempotents/KaroubiKaroubi.lean","tactics":[{"state_before":"C : Type u_1\ninst✝ : Category.{?u.1672, u_1} C\nP : Karoubi (Karoubi C)\n⊢ P.p.f ≫ P.p.f = P.p.f","state_after":"no goals","tactic":"simpa only [hom_ext_iff] using P.idem","premises":[{"full_name":"CategoryTheory.Idempotents.Karoubi.hom_ext_iff","def_path":"Mathlib/CategoryTheory/Idempotents/Karoubi.lean","def_pos":[98,8],"def_end_pos":[98,19]},{"full_name":"CategoryTheory.Idempotents.Karoubi.idem","def_path":"Mathlib/CategoryTheory/Idempotents/Karoubi.lean","def_pos":[47,2],"def_end_pos":[47,6]}]},{"state_before":"C : Type u_1\ninst✝ : Category.{?u.1672, u_1} C\nX✝ Y✝ : Karoubi (Karoubi C)\nf : X✝ ⟶ Y✝\n⊢ f.f.f =\n    ((fun P => { X := P.X.X, p := P.p.f, idem := ⋯ }) X✝).p ≫\n      f.f.f ≫ ((fun P => { X := P.X.X, p := P.p.f, idem := ⋯ }) Y✝).p","state_after":"no goals","tactic":"simpa only [hom_ext_iff] using f.comm","premises":[{"full_name":"CategoryTheory.Idempotents.Karoubi.Hom.comm","def_path":"Mathlib/CategoryTheory/Idempotents/Karoubi.lean","def_pos":[73,2],"def_end_pos":[73,6]},{"full_name":"CategoryTheory.Idempotents.Karoubi.hom_ext_iff","def_path":"Mathlib/CategoryTheory/Idempotents/Karoubi.lean","def_pos":[98,8],"def_end_pos":[98,19]}]}]}
+{"url":"Mathlib/Data/Nat/Fib/Basic.lean","commit":"","full_name":"Nat.fib_add","start":[147,0],"end":[154,8],"file_path":"Mathlib/Data/Nat/Fib/Basic.lean","tactics":[{"state_before":"m n : ℕ\n⊢ fib (m + n + 1) = fib m * fib n + fib (m + 1) * fib (n + 1)","state_after":"case zero\nm : ℕ\n⊢ fib (m + 0 + 1) = fib m * fib 0 + fib (m + 1) * fib (0 + 1)\n\ncase succ\nn : ℕ\nih : ∀ (m : ℕ), fib (m + n + 1) = fib m * fib n + fib (m + 1) * fib (n + 1)\nm : ℕ\n⊢ fib (m + (n + 1) + 1) = fib m * fib (n + 1) + fib (m + 1) * fib (n + 1 + 1)","tactic":"induction' n with n ih generalizing m","premises":[]}]}
+{"url":"Mathlib/RingTheory/Polynomial/Quotient.lean","commit":"","full_name":"MvPolynomial.quotientEquivQuotientMvPolynomial_rightInverse","start":[210,0],"end":[230,93],"file_path":"Mathlib/RingTheory/Polynomial/Quotient.lean","tactics":[{"state_before":"R : Type u_1\nσ : Type u_2\ninst✝ : CommRing R\nr : R\nI : Ideal R\n⊢ Function.RightInverse\n    (eval₂ (Ideal.Quotient.lift I ((Ideal.Quotient.mk (Ideal.map C I)).comp C) ⋯) fun i =>\n      (Ideal.Quotient.mk (Ideal.map C I)) (X i))\n    ⇑(Ideal.Quotient.lift (Ideal.map C I) (eval₂Hom (C.comp (Ideal.Quotient.mk I)) X) ⋯)","state_after":"R : Type u_1\nσ : Type u_2\ninst✝ : CommRing R\nr : R\nI : Ideal R\nf : MvPolynomial σ (R ⧸ I)\n⊢ (Ideal.Quotient.lift (Ideal.map C I) (eval₂Hom (C.comp (Ideal.Quotient.mk I)) X) ⋯)\n      (eval₂ (Ideal.Quotient.lift I ((Ideal.Quotient.mk (Ideal.map C I)).comp C) ⋯)\n        (fun i => (Ideal.Quotient.mk (Ideal.map C I)) (X i)) f) =\n    f","tactic":"intro f","premises":[]},{"state_before":"R : Type u_1\nσ : Type u_2\ninst✝ : CommRing R\nr : R\nI : Ideal R\nf : MvPolynomial σ (R ⧸ I)\n⊢ (Ideal.Quotient.lift (Ideal.map C I) (eval₂Hom (C.comp (Ideal.Quotient.mk I)) X) ⋯)\n      (eval₂ (Ideal.Quotient.lift I ((Ideal.Quotient.mk (Ideal.map C I)).comp C) ⋯)\n        (fun i => (Ideal.Quotient.mk (Ideal.map C I)) (X i)) f) =\n    f","state_after":"case h_C\nR : Type u_1\nσ : Type u_2\ninst✝ : CommRing R\nr : R\nI : Ideal R\nf : MvPolynomial σ (R ⧸ I)\n⊢ ∀ (a : R ⧸ I),\n    (Ideal.Quotient.lift (Ideal.map C I) (eval₂Hom (C.comp (Ideal.Quotient.mk I)) X) ⋯)\n        (eval₂ (Ideal.Quotient.lift I ((Ideal.Quotient.mk (Ideal.map C I)).comp C) ⋯)\n          (fun i => (Ideal.Quotient.mk (Ideal.map C I)) (X i)) (C a)) =\n      C a\n\ncase h_add\nR : Type u_1\nσ : Type u_2\ninst✝ : CommRing R\nr : R\nI : Ideal R\nf : MvPolynomial σ (R ⧸ I)\n⊢ ∀ (p q : MvPolynomial σ (R ⧸ I)),\n    (Ideal.Quotient.lift (Ideal.map C I) (eval₂Hom (C.comp (Ideal.Quotient.mk I)) X) ⋯)\n          (eval₂ (Ideal.Quotient.lift I ((Ideal.Quotient.mk (Ideal.map C I)).comp C) ⋯)\n            (fun i => (Ideal.Quotient.mk (Ideal.map C I)) (X i)) p) =\n        p →\n      (Ideal.Quotient.lift (Ideal.map C I) (eval₂Hom (C.comp (Ideal.Quotient.mk I)) X) ⋯)\n            (eval₂ (Ideal.Quotient.lift I ((Ideal.Quotient.mk (Ideal.map C I)).comp C) ⋯)\n              (fun i => (Ideal.Quotient.mk (Ideal.map C I)) (X i)) q) =\n          q →\n        (Ideal.Quotient.lift (Ideal.map C I) (eval₂Hom (C.comp (Ideal.Quotient.mk I)) X) ⋯)\n            (eval₂ (Ideal.Quotient.lift I ((Ideal.Quotient.mk (Ideal.map C I)).comp C) ⋯)\n              (fun i => (Ideal.Quotient.mk (Ideal.map C I)) (X i)) (p + q)) =\n          p + q\n\ncase h_X\nR : Type u_1\nσ : Type u_2\ninst✝ : CommRing R\nr : R\nI : Ideal R\nf : MvPolynomial σ (R ⧸ I)\n⊢ ∀ (p : MvPolynomial σ (R ⧸ I)) (n : σ),\n    (Ideal.Quotient.lift (Ideal.map C I) (eval₂Hom (C.comp (Ideal.Quotient.mk I)) X) ⋯)\n          (eval₂ (Ideal.Quotient.lift I ((Ideal.Quotient.mk (Ideal.map C I)).comp C) ⋯)\n            (fun i => (Ideal.Quotient.mk (Ideal.map C I)) (X i)) p) =\n        p →\n      (Ideal.Quotient.lift (Ideal.map C I) (eval₂Hom (C.comp (Ideal.Quotient.mk I)) X) ⋯)\n          (eval₂ (Ideal.Quotient.lift I ((Ideal.Quotient.mk (Ideal.map C I)).comp C) ⋯)\n            (fun i => (Ideal.Quotient.mk (Ideal.map C I)) (X i)) (p * X n)) =\n        p * X n","tactic":"apply induction_on f","premises":[{"full_name":"MvPolynomial.induction_on","def_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","def_pos":[399,8],"def_end_pos":[399,20]}]}]}
+{"url":"Mathlib/Data/Nat/PartENat.lean","commit":"","full_name":"PartENat.withTopEquiv_symm_ofNat","start":[703,0],"end":[705,6],"file_path":"Mathlib/Data/Nat/PartENat.lean","tactics":[{"state_before":"n : ℕ\ninst✝ : n.AtLeastTwo\n⊢ withTopEquiv.symm (OfNat.ofNat n) = OfNat.ofNat n","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/CategoryTheory/Functor/KanExtension/Basic.lean","commit":"","full_name":"CategoryTheory.Functor.isLeftKanExtension_iff_precomp","start":[428,0],"end":[437,68],"file_path":"Mathlib/CategoryTheory/Functor/KanExtension/Basic.lean","tactics":[{"state_before":"C : Type u_1\nC' : Type u_2\nH : Type u_3\nH' : Type u_4\nD : Type u_5\nD' : Type u_6\ninst✝⁶ : Category.{u_8, u_1} C\ninst✝⁵ : Category.{u_10, u_2} C'\ninst✝⁴ : Category.{u_7, u_3} H\ninst✝³ : Category.{?u.142795, u_4} H'\ninst✝² : Category.{u_9, u_5} D\ninst✝¹ : Category.{?u.142803, u_6} D'\nL : C ⥤ D\nF : C ⥤ H\nF' : D ⥤ H\nG : C' ⥤ C\ninst✝ : G.IsEquivalence\nα : F ⟶ L ⋙ F'\n⊢ F'.IsLeftKanExtension α ↔ F'.IsLeftKanExtension (whiskerLeft G α ≫ (G.associator L F').inv)","state_after":"C : Type u_1\nC' : Type u_2\nH : Type u_3\nH' : Type u_4\nD : Type u_5\nD' : Type u_6\ninst✝⁶ : Category.{u_8, u_1} C\ninst✝⁵ : Category.{u_10, u_2} C'\ninst✝⁴ : Category.{u_7, u_3} H\ninst✝³ : Category.{?u.142795, u_4} H'\ninst✝² : Category.{u_9, u_5} D\ninst✝¹ : Category.{?u.142803, u_6} D'\nL : C ⥤ D\nF : C ⥤ H\nF' : D ⥤ H\nG : C' ⥤ C\ninst✝ : G.IsEquivalence\nα : F ⟶ L ⋙ F'\neq : StructuredArrow.IsUniversal (LeftExtension.mk F' α) ≃\n  StructuredArrow.IsUniversal (LeftExtension.mk F' (whiskerLeft G α ≫ (G.associator L F').inv)) :=\n  (LeftExtension.isUniversalPrecompEquiv L F G (LeftExtension.mk F' α)).trans\n    (IsInitial.equivOfIso\n      (StructuredArrow.isoMk (Iso.refl ((LeftExtension.precomp L F G).obj (LeftExtension.mk F' α)).right) ⋯))\n⊢ F'.IsLeftKanExtension α ↔ F'.IsLeftKanExtension (whiskerLeft G α ≫ (G.associator L F').inv)","tactic":"let eq : (LeftExtension.mk _ α).IsUniversal ≃ (LeftExtension.mk _\n      (whiskerLeft G α ≫ (Functor.associator _ _ _).inv)).IsUniversal :=\n    (LeftExtension.isUniversalPrecompEquiv L F G _).trans\n    (IsInitial.equivOfIso (StructuredArrow.isoMk (Iso.refl _)))","premises":[{"full_name":"CategoryTheory.CategoryStruct.comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]},{"full_name":"CategoryTheory.Functor.LeftExtension.isUniversalPrecompEquiv","def_path":"Mathlib/CategoryTheory/Functor/KanExtension/Basic.lean","def_pos":[416,18],"def_end_pos":[416,55]},{"full_name":"CategoryTheory.Functor.LeftExtension.mk","def_path":"Mathlib/CategoryTheory/Functor/KanExtension/Basic.lean","def_pos":[60,4],"def_end_pos":[60,20]},{"full_name":"CategoryTheory.Functor.associator","def_path":"Mathlib/CategoryTheory/Whiskering.lean","def_pos":[242,4],"def_end_pos":[242,14]},{"full_name":"CategoryTheory.Iso.inv","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[53,2],"def_end_pos":[53,5]},{"full_name":"CategoryTheory.Iso.refl","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[114,4],"def_end_pos":[114,8]},{"full_name":"CategoryTheory.Limits.IsInitial.equivOfIso","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Terminal.lean","def_pos":[135,4],"def_end_pos":[135,24]},{"full_name":"CategoryTheory.StructuredArrow.IsUniversal","def_path":"Mathlib/CategoryTheory/Comma/StructuredArrow.lean","def_pos":[344,7],"def_end_pos":[344,18]},{"full_name":"CategoryTheory.StructuredArrow.isoMk","def_path":"Mathlib/CategoryTheory/Comma/StructuredArrow.lean","def_pos":[161,4],"def_end_pos":[161,9]},{"full_name":"CategoryTheory.whiskerLeft","def_path":"Mathlib/CategoryTheory/Whiskering.lean","def_pos":[45,4],"def_end_pos":[45,15]},{"full_name":"Equiv","def_path":"Mathlib/Logic/Equiv/Defs.lean","def_pos":[61,10],"def_end_pos":[61,15]},{"full_name":"Equiv.trans","def_path":"Mathlib/Logic/Equiv/Defs.lean","def_pos":[162,14],"def_end_pos":[162,19]}]},{"state_before":"C : Type u_1\nC' : Type u_2\nH : Type u_3\nH' : Type u_4\nD : Type u_5\nD' : Type u_6\ninst✝⁶ : Category.{u_8, u_1} C\ninst✝⁵ : Category.{u_10, u_2} C'\ninst✝⁴ : Category.{u_7, u_3} H\ninst✝³ : Category.{?u.142795, u_4} H'\ninst✝² : Category.{u_9, u_5} D\ninst✝¹ : Category.{?u.142803, u_6} D'\nL : C ⥤ D\nF : C ⥤ H\nF' : D ⥤ H\nG : C' ⥤ C\ninst✝ : G.IsEquivalence\nα : F ⟶ L ⋙ F'\neq : StructuredArrow.IsUniversal (LeftExtension.mk F' α) ≃\n  StructuredArrow.IsUniversal (LeftExtension.mk F' (whiskerLeft G α ≫ (G.associator L F').inv)) :=\n  (LeftExtension.isUniversalPrecompEquiv L F G (LeftExtension.mk F' α)).trans\n    (IsInitial.equivOfIso\n      (StructuredArrow.isoMk (Iso.refl ((LeftExtension.precomp L F G).obj (LeftExtension.mk F' α)).right) ⋯))\n⊢ F'.IsLeftKanExtension α ↔ F'.IsLeftKanExtension (whiskerLeft G α ≫ (G.associator L F').inv)","state_after":"case mp\nC : Type u_1\nC' : Type u_2\nH : Type u_3\nH' : Type u_4\nD : Type u_5\nD' : Type u_6\ninst✝⁶ : Category.{u_8, u_1} C\ninst✝⁵ : Category.{u_10, u_2} C'\ninst✝⁴ : Category.{u_7, u_3} H\ninst✝³ : Category.{?u.142795, u_4} H'\ninst✝² : Category.{u_9, u_5} D\ninst✝¹ : Category.{?u.142803, u_6} D'\nL : C ⥤ D\nF : C ⥤ H\nF' : D ⥤ H\nG : C' ⥤ C\ninst✝ : G.IsEquivalence\nα : F ⟶ L ⋙ F'\neq : StructuredArrow.IsUniversal (LeftExtension.mk F' α) ≃\n  StructuredArrow.IsUniversal (LeftExtension.mk F' (whiskerLeft G α ≫ (G.associator L F').inv)) :=\n  (LeftExtension.isUniversalPrecompEquiv L F G (LeftExtension.mk F' α)).trans\n    (IsInitial.equivOfIso\n      (StructuredArrow.isoMk (Iso.refl ((LeftExtension.precomp L F G).obj (LeftExtension.mk F' α)).right) ⋯))\n⊢ F'.IsLeftKanExtension α → F'.IsLeftKanExtension (whiskerLeft G α ≫ (G.associator L F').inv)\n\ncase mpr\nC : Type u_1\nC' : Type u_2\nH : Type u_3\nH' : Type u_4\nD : Type u_5\nD' : Type u_6\ninst✝⁶ : Category.{u_8, u_1} C\ninst✝⁵ : Category.{u_10, u_2} C'\ninst✝⁴ : Category.{u_7, u_3} H\ninst✝³ : Category.{?u.142795, u_4} H'\ninst✝² : Category.{u_9, u_5} D\ninst✝¹ : Category.{?u.142803, u_6} D'\nL : C ⥤ D\nF : C ⥤ H\nF' : D ⥤ H\nG : C' ⥤ C\ninst✝ : G.IsEquivalence\nα : F ⟶ L ⋙ F'\neq : StructuredArrow.IsUniversal (LeftExtension.mk F' α) ≃\n  StructuredArrow.IsUniversal (LeftExtension.mk F' (whiskerLeft G α ≫ (G.associator L F').inv)) :=\n  (LeftExtension.isUniversalPrecompEquiv L F G (LeftExtension.mk F' α)).trans\n    (IsInitial.equivOfIso\n      (StructuredArrow.isoMk (Iso.refl ((LeftExtension.precomp L F G).obj (LeftExtension.mk F' α)).right) ⋯))\n⊢ F'.IsLeftKanExtension (whiskerLeft G α ≫ (G.associator L F').inv) → F'.IsLeftKanExtension α","tactic":"constructor","premises":[]}]}
+{"url":"Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean","commit":"","full_name":"IsIntegralClosure.mk'_algebraMap","start":[488,0],"end":[492,88],"file_path":"Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean","tactics":[{"state_before":"R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing A\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : Algebra A B\ninst✝² : IsIntegralClosure A R B\ninst✝¹ : Algebra R A\ninst✝ : IsScalarTower R A B\nx : R\nh : optParam (IsIntegral R ((algebraMap R B) x)) ⋯\n⊢ (algebraMap A B) (mk' A ((algebraMap R B) x) h) = (algebraMap A B) ((algebraMap R A) x)","state_after":"no goals","tactic":"rw [algebraMap_mk', ← IsScalarTower.algebraMap_apply]","premises":[{"full_name":"IsIntegralClosure.algebraMap_mk'","def_path":"Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean","def_pos":[467,8],"def_end_pos":[467,22]},{"full_name":"IsScalarTower.algebraMap_apply","def_path":"Mathlib/Algebra/Algebra/Tower.lean","def_pos":[122,8],"def_end_pos":[122,24]}]}]}
+{"url":"Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean","commit":"","full_name":"CochainComplex.HomComplex.Cochain.leftShift_leftUnshift","start":[139,0],"end":[145,55],"file_path":"Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean","tactics":[{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nK L M : CochainComplex C ℤ\nn✝ : ℤ\nγ✝ γ₁ γ₂ : Cochain K L n✝\na n' : ℤ\nγ : Cochain ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj K) L n'\nn : ℤ\nhn' : n + a = n'\n⊢ (γ.leftUnshift n hn').leftShift a n' hn' = γ","state_after":"case h\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nK L M : CochainComplex C ℤ\nn✝ : ℤ\nγ✝ γ₁ γ₂ : Cochain K L n✝\na n' : ℤ\nγ : Cochain ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj K) L n'\nn : ℤ\nhn' : n + a = n'\np q : ℤ\nhpq : p + n' = q\n⊢ ((γ.leftUnshift n hn').leftShift a n' hn').v p q hpq = γ.v p q hpq","tactic":"ext p q hpq","premises":[]},{"state_before":"case h\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nK L M : CochainComplex C ℤ\nn✝ : ℤ\nγ✝ γ₁ γ₂ : Cochain K L n✝\na n' : ℤ\nγ : Cochain ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj K) L n'\nn : ℤ\nhn' : n + a = n'\np q : ℤ\nhpq : p + n' = q\n⊢ ((γ.leftUnshift n hn').leftShift a n' hn').v p q hpq = γ.v p q hpq","state_after":"no goals","tactic":"rw [(γ.leftUnshift n hn').leftShift_v a n' hn' p q hpq (q-n) (by omega),\n    γ.leftUnshift_v n hn' (q-n) q (by omega) p hpq, Linear.comp_units_smul, smul_smul,\n    Iso.hom_inv_id_assoc, Int.units_mul_self, one_smul]","premises":[{"full_name":"CategoryTheory.Linear.comp_units_smul","def_path":"Mathlib/CategoryTheory/Linear/Basic.lean","def_pos":[169,6],"def_end_pos":[169,21]},{"full_name":"CochainComplex.HomComplex.Cochain.leftShift_v","def_path":"Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean","def_pos":[61,6],"def_end_pos":[61,17]},{"full_name":"CochainComplex.HomComplex.Cochain.leftUnshift","def_path":"Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean","def_pos":[85,4],"def_end_pos":[85,15]},{"full_name":"CochainComplex.HomComplex.Cochain.leftUnshift_v","def_path":"Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean","def_pos":[90,6],"def_end_pos":[90,19]},{"full_name":"Int.units_mul_self","def_path":"Mathlib/Data/Int/Order/Units.lean","def_pos":[27,8],"def_end_pos":[27,22]},{"full_name":"one_smul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[379,6],"def_end_pos":[379,14]},{"full_name":"smul_smul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[374,6],"def_end_pos":[374,15]}]}]}
+{"url":".lake/packages/batteries/Batteries/Data/UInt.lean","commit":"","full_name":"USize.size_eq","start":[211,0],"end":[213,42],"file_path":".lake/packages/batteries/Batteries/Data/UInt.lean","tactics":[{"state_before":"⊢ size = 2 ^ System.Platform.numBits","state_after":"this : 1 ≤ 2 ^ System.Platform.numBits\n⊢ size = 2 ^ System.Platform.numBits","tactic":"have : 1 ≤ 2 ^ System.Platform.numBits := Nat.succ_le_of_lt (Nat.two_pow_pos _)","premises":[{"full_name":"Nat.succ_le_of_lt","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[348,8],"def_end_pos":[348,21]},{"full_name":"Nat.two_pow_pos","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Power2.lean","def_pos":[11,18],"def_end_pos":[11,29]},{"full_name":"System.Platform.numBits","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1820,4],"def_end_pos":[1820,27]}]},{"state_before":"this : 1 ≤ 2 ^ System.Platform.numBits\n⊢ size = 2 ^ System.Platform.numBits","state_after":"no goals","tactic":"rw [USize.size, Nat.sub_add_cancel this]","premises":[{"full_name":"Nat.sub_add_cancel","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[893,26],"def_end_pos":[893,40]},{"full_name":"USize.size","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2078,7],"def_end_pos":[2078,17]}]}]}
+{"url":"Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean","commit":"","full_name":"AlgebraicTopology.DoldKan.Γ₀.Obj.map_on_summand₀","start":[174,0],"end":[187,48],"file_path":"Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean","tactics":[{"state_before":"C : Type u_1\ninst✝⁴ : Category.{u_2, u_1} C\ninst✝³ : Preadditive C\nK K' : ChainComplex C ℕ\nf : K ⟶ K'\nΔ✝ Δ'✝ Δ''✝ : SimplexCategory\ninst✝² : HasFiniteCoproducts C\nΔ Δ' : SimplexCategoryᵒᵖ\nA : Splitting.IndexSet Δ\nθ : Δ ⟶ Δ'\nΔ'' : SimplexCategory\ne : unop Δ' ⟶ Δ''\ni : Δ'' ⟶ unop A.fst\ninst✝¹ : Epi e\ninst✝ : Mono i\nfac : e ≫ i = θ.unop ≫ A.e\n⊢ Sigma.ι (summand K Δ) A ≫ map K θ = Termwise.mapMono K i ≫ Sigma.ι (summand K Δ') (Splitting.IndexSet.mk e)","state_after":"C : Type u_1\ninst✝⁴ : Category.{u_2, u_1} C\ninst✝³ : Preadditive C\nK K' : ChainComplex C ℕ\nf : K ⟶ K'\nΔ✝ Δ'✝ Δ''✝ : SimplexCategory\ninst✝² : HasFiniteCoproducts C\nΔ Δ' : SimplexCategoryᵒᵖ\nA : Splitting.IndexSet Δ\nθ : Δ ⟶ Δ'\nΔ'' : SimplexCategory\ne : unop Δ' ⟶ Δ''\ni : Δ'' ⟶ unop A.fst\ninst✝¹ : Epi e\ninst✝ : Mono i\nfac : e ≫ i = θ.unop ≫ A.e\n⊢ Termwise.mapMono K (image.ι (θ.unop ≫ A.e)) ≫ Sigma.ι (summand K Δ') (A.pull θ) =\n    Termwise.mapMono K i ≫ Sigma.ι (summand K Δ') (Splitting.IndexSet.mk e)","tactic":"simp only [map, colimit.ι_desc, Cofan.mk_ι_app]","premises":[{"full_name":"AlgebraicTopology.DoldKan.Γ₀.Obj.map","def_path":"Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean","def_pos":[170,4],"def_end_pos":[170,7]},{"full_name":"CategoryTheory.Limits.Cofan.mk_ι_app","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Products.lean","def_pos":[63,12],"def_end_pos":[63,17]},{"full_name":"CategoryTheory.Limits.colimit.ι_desc","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[691,8],"def_end_pos":[691,22]}]},{"state_before":"C : Type u_1\ninst✝⁴ : Category.{u_2, u_1} C\ninst✝³ : Preadditive C\nK K' : ChainComplex C ℕ\nf : K ⟶ K'\nΔ✝ Δ'✝ Δ''✝ : SimplexCategory\ninst✝² : HasFiniteCoproducts C\nΔ Δ' : SimplexCategoryᵒᵖ\nA : Splitting.IndexSet Δ\nθ : Δ ⟶ Δ'\nΔ'' : SimplexCategory\ne : unop Δ' ⟶ Δ''\ni : Δ'' ⟶ unop A.fst\ninst✝¹ : Epi e\ninst✝ : Mono i\nfac : e ≫ i = θ.unop ≫ A.e\n⊢ Termwise.mapMono K (image.ι (θ.unop ≫ A.e)) ≫ Sigma.ι (summand K Δ') (A.pull θ) =\n    Termwise.mapMono K i ≫ Sigma.ι (summand K Δ') (Splitting.IndexSet.mk e)","state_after":"C : Type u_1\ninst✝⁴ : Category.{u_2, u_1} C\ninst✝³ : Preadditive C\nK K' : ChainComplex C ℕ\nf : K ⟶ K'\nΔ✝ Δ'✝ Δ''✝ : SimplexCategory\ninst✝² : HasFiniteCoproducts C\nΔ Δ' : SimplexCategoryᵒᵖ\nA : Splitting.IndexSet Δ\nθ : Δ ⟶ Δ'\nΔ'' : SimplexCategory\ne : unop Δ' ⟶ Δ''\ni : Δ'' ⟶ unop A.fst\ninst✝¹ : Epi e\ninst✝ : Mono i\nfac : e ≫ i = θ.unop ≫ A.e\nh : image (θ.unop ≫ A.e) = Δ''\n⊢ Termwise.mapMono K (image.ι (θ.unop ≫ A.e)) ≫ Sigma.ι (summand K Δ') (A.pull θ) =\n    Termwise.mapMono K i ≫ Sigma.ι (summand K Δ') (Splitting.IndexSet.mk e)","tactic":"have h := SimplexCategory.image_eq fac","premises":[{"full_name":"SimplexCategory.image_eq","def_path":"Mathlib/AlgebraicTopology/SimplexCategory.lean","def_pos":[783,8],"def_end_pos":[783,16]}]},{"state_before":"C : Type u_1\ninst✝⁴ : Category.{u_2, u_1} C\ninst✝³ : Preadditive C\nK K' : ChainComplex C ℕ\nf : K ⟶ K'\nΔ✝ Δ'✝ Δ''✝ : SimplexCategory\ninst✝² : HasFiniteCoproducts C\nΔ Δ' : SimplexCategoryᵒᵖ\nA : Splitting.IndexSet Δ\nθ : Δ ⟶ Δ'\nΔ'' : SimplexCategory\ne : unop Δ' ⟶ Δ''\ni : Δ'' ⟶ unop A.fst\ninst✝¹ : Epi e\ninst✝ : Mono i\nfac : e ≫ i = θ.unop ≫ A.e\nh : image (θ.unop ≫ A.e) = Δ''\n⊢ Termwise.mapMono K (image.ι (θ.unop ≫ A.e)) ≫ Sigma.ι (summand K Δ') (A.pull θ) =\n    Termwise.mapMono K i ≫ Sigma.ι (summand K Δ') (Splitting.IndexSet.mk e)","state_after":"C : Type u_1\ninst✝⁴ : Category.{u_2, u_1} C\ninst✝³ : Preadditive C\nK K' : ChainComplex C ℕ\nf : K ⟶ K'\nΔ✝ Δ'✝ Δ'' : SimplexCategory\ninst✝² : HasFiniteCoproducts C\nΔ Δ' : SimplexCategoryᵒᵖ\nA : Splitting.IndexSet Δ\nθ : Δ ⟶ Δ'\ne : unop Δ' ⟶ image (θ.unop ≫ A.e)\ni : image (θ.unop ≫ A.e) ⟶ unop A.fst\ninst✝¹ : Epi e\ninst✝ : Mono i\nfac : e ≫ i = θ.unop ≫ A.e\n⊢ Termwise.mapMono K (image.ι (θ.unop ≫ A.e)) ≫ Sigma.ι (summand K Δ') (A.pull θ) =\n    Termwise.mapMono K i ≫ Sigma.ι (summand K Δ') (Splitting.IndexSet.mk e)","tactic":"subst h","premises":[]},{"state_before":"C : Type u_1\ninst✝⁴ : Category.{u_2, u_1} C\ninst✝³ : Preadditive C\nK K' : ChainComplex C ℕ\nf : K ⟶ K'\nΔ✝ Δ'✝ Δ'' : SimplexCategory\ninst✝² : HasFiniteCoproducts C\nΔ Δ' : SimplexCategoryᵒᵖ\nA : Splitting.IndexSet Δ\nθ : Δ ⟶ Δ'\ne : unop Δ' ⟶ image (θ.unop ≫ A.e)\ni : image (θ.unop ≫ A.e) ⟶ unop A.fst\ninst✝¹ : Epi e\ninst✝ : Mono i\nfac : e ≫ i = θ.unop ≫ A.e\n⊢ Termwise.mapMono K (image.ι (θ.unop ≫ A.e)) ≫ Sigma.ι (summand K Δ') (A.pull θ) =\n    Termwise.mapMono K i ≫ Sigma.ι (summand K Δ') (Splitting.IndexSet.mk e)","state_after":"case e_a.e_i\nC : Type u_1\ninst✝⁴ : Category.{u_2, u_1} C\ninst✝³ : Preadditive C\nK K' : ChainComplex C ℕ\nf : K ⟶ K'\nΔ✝ Δ'✝ Δ'' : SimplexCategory\ninst✝² : HasFiniteCoproducts C\nΔ Δ' : SimplexCategoryᵒᵖ\nA : Splitting.IndexSet Δ\nθ : Δ ⟶ Δ'\ne : unop Δ' ⟶ image (θ.unop ≫ A.e)\ni : image (θ.unop ≫ A.e) ⟶ unop A.fst\ninst✝¹ : Epi e\ninst✝ : Mono i\nfac : e ≫ i = θ.unop ≫ A.e\n⊢ image.ι (θ.unop ≫ A.e) = i\n\ncase e_a.h.e_6.h\nC : Type u_1\ninst✝⁴ : Category.{u_2, u_1} C\ninst✝³ : Preadditive C\nK K' : ChainComplex C ℕ\nf : K ⟶ K'\nΔ✝ Δ'✝ Δ'' : SimplexCategory\ninst✝² : HasFiniteCoproducts C\nΔ Δ' : SimplexCategoryᵒᵖ\nA : Splitting.IndexSet Δ\nθ : Δ ⟶ Δ'\ne : unop Δ' ⟶ image (θ.unop ≫ A.e)\ni : image (θ.unop ≫ A.e) ⟶ unop A.fst\ninst✝¹ : Epi e\ninst✝ : Mono i\nfac : e ≫ i = θ.unop ≫ A.e\n⊢ A.pull θ = Splitting.IndexSet.mk e","tactic":"congr","premises":[]}]}
+{"url":"Mathlib/GroupTheory/OrderOfElement.lean","commit":"","full_name":"AddSubgroup.nsmul_index_mem","start":[887,0],"end":[889,92],"file_path":"Mathlib/GroupTheory/OrderOfElement.lean","tactics":[{"state_before":"G✝ : Type u_1\nH✝ : Type u_2\nA : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝³ : Group G✝\ninst✝² : Fintype G✝\nx : G✝\nn : ℕ\nG : Type u_6\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : H.Normal\ng : G\n⊢ g ^ H.index ∈ H","state_after":"no goals","tactic":"rw [← eq_one_iff, QuotientGroup.mk_pow H, index, pow_card_eq_one']","premises":[{"full_name":"QuotientGroup.eq_one_iff","def_path":"Mathlib/GroupTheory/QuotientGroup.lean","def_pos":[110,8],"def_end_pos":[110,18]},{"full_name":"QuotientGroup.mk_pow","def_path":"Mathlib/GroupTheory/QuotientGroup.lean","def_pos":[157,8],"def_end_pos":[157,14]},{"full_name":"Subgroup.index","def_path":"Mathlib/GroupTheory/Index.lean","def_pos":[44,18],"def_end_pos":[44,23]},{"full_name":"pow_card_eq_one'","def_path":"Mathlib/GroupTheory/OrderOfElement.lean","def_pos":[880,8],"def_end_pos":[880,24]}]}]}
+{"url":"Mathlib/Algebra/MvPolynomial/Variables.lean","commit":"","full_name":"MvPolynomial.vars_map_of_injective","start":[199,0],"end":[200,44],"file_path":"Mathlib/Algebra/MvPolynomial/Variables.lean","tactics":[{"state_before":"R : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ninst✝ : CommSemiring S\nf : R →+* S\nhf : Injective ⇑f\n⊢ ((map f) p).vars = p.vars","state_after":"no goals","tactic":"simp [vars, degrees_map_of_injective _ hf]","premises":[{"full_name":"MvPolynomial.degrees_map_of_injective","def_path":"Mathlib/Algebra/MvPolynomial/Degrees.lean","def_pos":[189,8],"def_end_pos":[189,32]},{"full_name":"MvPolynomial.vars","def_path":"Mathlib/Algebra/MvPolynomial/Variables.lean","def_pos":[64,4],"def_end_pos":[64,8]}]}]}
+{"url":"Mathlib/RingTheory/Adjoin/PowerBasis.lean","commit":"","full_name":"PowerBasis.repr_gen_pow_isIntegral","start":[88,0],"end":[118,31],"file_path":"Mathlib/RingTheory/Adjoin/PowerBasis.lean","tactics":[{"state_before":"K : Type u_1\nS : Type u_2\ninst✝¹¹ : Field K\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra K S\nR : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsScalarTower R K S\nA : Type u_4\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst✝ : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : ℕ\n⊢ ∀ (i : Fin B.dim), IsIntegral R ((B.basis.repr (B.gen ^ n)) i)","state_after":"K : Type u_1\nS : Type u_2\ninst✝¹¹ : Field K\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra K S\nR : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsScalarTower R K S\nA : Type u_4\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst✝ : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : ℕ\ni : Fin B.dim\n⊢ IsIntegral R ((B.basis.repr (B.gen ^ n)) i)","tactic":"intro i","premises":[]},{"state_before":"K : Type u_1\nS : Type u_2\ninst✝¹¹ : Field K\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra K S\nR : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsScalarTower R K S\nA : Type u_4\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst✝ : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : ℕ\ni : Fin B.dim\n⊢ IsIntegral R ((B.basis.repr (B.gen ^ n)) i)","state_after":"K : Type u_1\nS : Type u_2\ninst✝¹¹ : Field K\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra K S\nR : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsScalarTower R K S\nA : Type u_4\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst✝ : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : ℕ\ni : Fin B.dim\nQ : R[X] := X ^ n %ₘ minpoly R B.gen\n⊢ IsIntegral R ((B.basis.repr (B.gen ^ n)) i)","tactic":"let Q := X ^ n %ₘ minpoly R B.gen","premises":[{"full_name":"Polynomial.X","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[474,4],"def_end_pos":[474,5]},{"full_name":"Polynomial.modByMonic","def_path":"Mathlib/Algebra/Polynomial/Div.lean","def_pos":[115,4],"def_end_pos":[115,14]},{"full_name":"PowerBasis.gen","def_path":"Mathlib/RingTheory/PowerBasis.lean","def_pos":[61,2],"def_end_pos":[61,5]},{"full_name":"minpoly","def_path":"Mathlib/FieldTheory/Minpoly/Basic.lean","def_pos":[36,18],"def_end_pos":[36,25]}]},{"state_before":"K : Type u_1\nS : Type u_2\ninst✝¹¹ : Field K\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra K S\nR : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsScalarTower R K S\nA : Type u_4\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst✝ : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : ℕ\ni : Fin B.dim\nQ : R[X] := X ^ n %ₘ minpoly R B.gen\n⊢ IsIntegral R ((B.basis.repr (B.gen ^ n)) i)","state_after":"K : Type u_1\nS : Type u_2\ninst✝¹¹ : Field K\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra K S\nR : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsScalarTower R K S\nA : Type u_4\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst✝ : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : ℕ\ni : Fin B.dim\nQ : R[X] := X ^ n %ₘ minpoly R B.gen\nthis : B.gen ^ n = (aeval B.gen) Q\n⊢ IsIntegral R ((B.basis.repr (B.gen ^ n)) i)","tactic":"have : B.gen ^ n = aeval B.gen Q := by\n    rw [← @aeval_X_pow R _ _ _ _ B.gen, ← modByMonic_add_div (X ^ n) (minpoly.monic hB)]\n    simp","premises":[{"full_name":"Polynomial.X","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[474,4],"def_end_pos":[474,5]},{"full_name":"Polynomial.aeval","def_path":"Mathlib/Algebra/Polynomial/AlgebraMap.lean","def_pos":[227,4],"def_end_pos":[227,9]},{"full_name":"Polynomial.aeval_X_pow","def_path":"Mathlib/Algebra/Polynomial/AlgebraMap.lean","def_pos":[262,8],"def_end_pos":[262,19]},{"full_name":"Polynomial.modByMonic_add_div","def_path":"Mathlib/Algebra/Polynomial/Div.lean","def_pos":[241,8],"def_end_pos":[241,26]},{"full_name":"PowerBasis.gen","def_path":"Mathlib/RingTheory/PowerBasis.lean","def_pos":[61,2],"def_end_pos":[61,5]},{"full_name":"minpoly.monic","def_path":"Mathlib/FieldTheory/Minpoly/Basic.lean","def_pos":[49,8],"def_end_pos":[49,13]}]},{"state_before":"K : Type u_1\nS : Type u_2\ninst✝¹¹ : Field K\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra K S\nR : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsScalarTower R K S\nA : Type u_4\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst✝ : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : ℕ\ni : Fin B.dim\nQ : R[X] := X ^ n %ₘ minpoly R B.gen\nthis : B.gen ^ n = (aeval B.gen) Q\n⊢ IsIntegral R ((B.basis.repr (B.gen ^ n)) i)","state_after":"case pos\nK : Type u_1\nS : Type u_2\ninst✝¹¹ : Field K\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra K S\nR : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsScalarTower R K S\nA : Type u_4\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst✝ : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : ℕ\ni : Fin B.dim\nQ : R[X] := X ^ n %ₘ minpoly R B.gen\nthis : B.gen ^ n = (aeval B.gen) Q\nhQ : Q = 0\n⊢ IsIntegral R ((B.basis.repr (B.gen ^ n)) i)\n\ncase neg\nK : Type u_1\nS : Type u_2\ninst✝¹¹ : Field K\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra K S\nR : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsScalarTower R K S\nA : Type u_4\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst✝ : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : ℕ\ni : Fin B.dim\nQ : R[X] := X ^ n %ₘ minpoly R B.gen\nthis : B.gen ^ n = (aeval B.gen) Q\nhQ : ¬Q = 0\n⊢ IsIntegral R ((B.basis.repr (B.gen ^ n)) i)","tactic":"by_cases hQ : Q = 0","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case neg\nK : Type u_1\nS : Type u_2\ninst✝¹¹ : Field K\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra K S\nR : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsScalarTower R K S\nA : Type u_4\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst✝ : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : ℕ\ni : Fin B.dim\nQ : R[X] := X ^ n %ₘ minpoly R B.gen\nthis : B.gen ^ n = (aeval B.gen) Q\nhQ : ¬Q = 0\n⊢ IsIntegral R ((B.basis.repr (B.gen ^ n)) i)","state_after":"case neg\nK : Type u_1\nS : Type u_2\ninst✝¹¹ : Field K\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra K S\nR : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsScalarTower R K S\nA : Type u_4\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst✝ : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : ℕ\ni : Fin B.dim\nQ : R[X] := X ^ n %ₘ minpoly R B.gen\nthis : B.gen ^ n = (aeval B.gen) Q\nhQ : ¬Q = 0\nhlt : Q.natDegree < B.dim\n⊢ IsIntegral R ((B.basis.repr (B.gen ^ n)) i)","tactic":"have hlt : Q.natDegree < B.dim := by\n    rw [← B.natDegree_minpoly, hmin, (minpoly.monic hB).natDegree_map,\n      natDegree_lt_natDegree_iff hQ]\n    letI : Nontrivial R := Nontrivial.of_polynomial_ne hQ\n    exact degree_modByMonic_lt _ (minpoly.monic hB)","premises":[{"full_name":"Nontrivial","def_path":"Mathlib/Logic/Nontrivial/Defs.lean","def_pos":[29,6],"def_end_pos":[29,16]},{"full_name":"Polynomial.Monic.natDegree_map","def_path":"Mathlib/Algebra/Polynomial/Monic.lean","def_pos":[275,8],"def_end_pos":[275,27]},{"full_name":"Polynomial.Nontrivial.of_polynomial_ne","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[681,8],"def_end_pos":[681,35]},{"full_name":"Polynomial.degree_modByMonic_lt","def_path":"Mathlib/Algebra/Polynomial/Div.lean","def_pos":[125,8],"def_end_pos":[125,28]},{"full_name":"Polynomial.natDegree","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[61,4],"def_end_pos":[61,13]},{"full_name":"Polynomial.natDegree_lt_natDegree_iff","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[532,8],"def_end_pos":[532,34]},{"full_name":"PowerBasis.dim","def_path":"Mathlib/RingTheory/PowerBasis.lean","def_pos":[62,2],"def_end_pos":[62,5]},{"full_name":"PowerBasis.natDegree_minpoly","def_path":"Mathlib/RingTheory/PowerBasis.lean","def_pos":[216,8],"def_end_pos":[216,25]},{"full_name":"minpoly.monic","def_path":"Mathlib/FieldTheory/Minpoly/Basic.lean","def_pos":[49,8],"def_end_pos":[49,13]}]},{"state_before":"case neg\nK : Type u_1\nS : Type u_2\ninst✝¹¹ : Field K\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra K S\nR : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsScalarTower R K S\nA : Type u_4\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst✝ : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : ℕ\ni : Fin B.dim\nQ : R[X] := X ^ n %ₘ minpoly R B.gen\nthis : B.gen ^ n = (aeval B.gen) Q\nhQ : ¬Q = 0\nhlt : Q.natDegree < B.dim\n⊢ IsIntegral R ((B.basis.repr (B.gen ^ n)) i)","state_after":"case neg\nK : Type u_1\nS : Type u_2\ninst✝¹¹ : Field K\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra K S\nR : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsScalarTower R K S\nA : Type u_4\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst✝ : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : ℕ\ni : Fin B.dim\nQ : R[X] := X ^ n %ₘ minpoly R B.gen\nthis : B.gen ^ n = (aeval B.gen) Q\nhQ : ¬Q = 0\nhlt : Q.natDegree < B.dim\n⊢ IsIntegral R ((B.basis.repr (∑ i ∈ Finset.range B.dim, Q.coeff i • B.gen ^ i)) i)","tactic":"rw [this, aeval_eq_sum_range' hlt]","premises":[{"full_name":"Polynomial.aeval_eq_sum_range'","def_path":"Mathlib/Algebra/Polynomial/AlgebraMap.lean","def_pos":[405,8],"def_end_pos":[405,27]}]},{"state_before":"case neg\nK : Type u_1\nS : Type u_2\ninst✝¹¹ : Field K\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra K S\nR : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsScalarTower R K S\nA : Type u_4\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst✝ : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : ℕ\ni : Fin B.dim\nQ : R[X] := X ^ n %ₘ minpoly R B.gen\nthis : B.gen ^ n = (aeval B.gen) Q\nhQ : ¬Q = 0\nhlt : Q.natDegree < B.dim\n⊢ IsIntegral R ((B.basis.repr (∑ i ∈ Finset.range B.dim, Q.coeff i • B.gen ^ i)) i)","state_after":"case neg\nK : Type u_1\nS : Type u_2\ninst✝¹¹ : Field K\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra K S\nR : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsScalarTower R K S\nA : Type u_4\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst✝ : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : ℕ\ni : Fin B.dim\nQ : R[X] := X ^ n %ₘ minpoly R B.gen\nthis : B.gen ^ n = (aeval B.gen) Q\nhQ : ¬Q = 0\nhlt : Q.natDegree < B.dim\n⊢ IsIntegral R (∑ k ∈ Finset.range B.dim, (B.basis.repr (Q.coeff k • B.gen ^ k)) i)","tactic":"simp only [map_sum, LinearEquiv.map_smulₛₗ, RingHom.id_apply, Finset.sum_apply']","premises":[{"full_name":"Finset.sum_apply'","def_path":"Mathlib/Algebra/BigOperators/Finsupp.lean","def_pos":[560,8],"def_end_pos":[560,25]},{"full_name":"LinearEquiv.map_smulₛₗ","def_path":"Mathlib/Algebra/Module/Equiv/Defs.lean","def_pos":[442,18],"def_end_pos":[442,28]},{"full_name":"RingHom.id_apply","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[549,8],"def_end_pos":[549,16]},{"full_name":"map_sum","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[286,2],"def_end_pos":[286,13]}]},{"state_before":"case neg\nK : Type u_1\nS : Type u_2\ninst✝¹¹ : Field K\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra K S\nR : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsScalarTower R K S\nA : Type u_4\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst✝ : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : ℕ\ni : Fin B.dim\nQ : R[X] := X ^ n %ₘ minpoly R B.gen\nthis : B.gen ^ n = (aeval B.gen) Q\nhQ : ¬Q = 0\nhlt : Q.natDegree < B.dim\n⊢ IsIntegral R (∑ k ∈ Finset.range B.dim, (B.basis.repr (Q.coeff k • B.gen ^ k)) i)","state_after":"case neg\nK : Type u_1\nS : Type u_2\ninst✝¹¹ : Field K\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra K S\nR : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsScalarTower R K S\nA : Type u_4\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst✝ : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : ℕ\ni : Fin B.dim\nQ : R[X] := X ^ n %ₘ minpoly R B.gen\nthis : B.gen ^ n = (aeval B.gen) Q\nhQ : ¬Q = 0\nhlt : Q.natDegree < B.dim\nj : ℕ\nhj : j ∈ Finset.range B.dim\n⊢ IsIntegral R ((B.basis.repr (Q.coeff j • B.gen ^ j)) i)","tactic":"refine IsIntegral.sum _ fun j hj => ?_","premises":[{"full_name":"IsIntegral.sum","def_path":"Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean","def_pos":[305,8],"def_end_pos":[305,22]}]},{"state_before":"case neg\nK : Type u_1\nS : Type u_2\ninst✝¹¹ : Field K\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra K S\nR : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsScalarTower R K S\nA : Type u_4\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst✝ : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : ℕ\ni : Fin B.dim\nQ : R[X] := X ^ n %ₘ minpoly R B.gen\nthis : B.gen ^ n = (aeval B.gen) Q\nhQ : ¬Q = 0\nhlt : Q.natDegree < B.dim\nj : ℕ\nhj : j ∈ Finset.range B.dim\n⊢ IsIntegral R ((B.basis.repr (Q.coeff j • B.gen ^ j)) i)","state_after":"case neg\nK : Type u_1\nS : Type u_2\ninst✝¹¹ : Field K\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra K S\nR : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsScalarTower R K S\nA : Type u_4\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst✝ : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : ℕ\ni : Fin B.dim\nQ : R[X] := X ^ n %ₘ minpoly R B.gen\nthis : B.gen ^ n = (aeval B.gen) Q\nhQ : ¬Q = 0\nhlt : Q.natDegree < B.dim\nj : ℕ\nhj : j < B.dim\n⊢ IsIntegral R ((B.basis.repr (Q.coeff j • B.gen ^ j)) i)","tactic":"replace hj := Finset.mem_range.1 hj","premises":[{"full_name":"Finset.mem_range","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2450,8],"def_end_pos":[2450,17]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]}]},{"state_before":"case neg\nK : Type u_1\nS : Type u_2\ninst✝¹¹ : Field K\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra K S\nR : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsScalarTower R K S\nA : Type u_4\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst✝ : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : ℕ\ni : Fin B.dim\nQ : R[X] := X ^ n %ₘ minpoly R B.gen\nthis : B.gen ^ n = (aeval B.gen) Q\nhQ : ¬Q = 0\nhlt : Q.natDegree < B.dim\nj : ℕ\nhj : j < B.dim\n⊢ IsIntegral R ((B.basis.repr (Q.coeff j • B.gen ^ j)) i)","state_after":"case neg\nK : Type u_1\nS : Type u_2\ninst✝¹¹ : Field K\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra K S\nR : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsScalarTower R K S\nA : Type u_4\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst✝ : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : ℕ\ni : Fin B.dim\nQ : R[X] := X ^ n %ₘ minpoly R B.gen\nthis : B.gen ^ n = (aeval B.gen) Q\nhQ : ¬Q = 0\nhlt : Q.natDegree < B.dim\nj : ℕ\nhj : j < B.dim\n⊢ IsIntegral R (((algebraMap R S) (Q.coeff ↑⟨j, hj⟩) • B.basis.repr (B.basis ⟨j, hj⟩)) i)","tactic":"rw [← Fin.val_mk hj, ← B.basis_eq_pow, Algebra.smul_def, IsScalarTower.algebraMap_apply R S A, ←\n    Algebra.smul_def, LinearEquiv.map_smul]","premises":[{"full_name":"Algebra.smul_def","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[270,8],"def_end_pos":[270,16]},{"full_name":"Fin.val_mk","def_path":".lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean","def_pos":[52,8],"def_end_pos":[52,14]},{"full_name":"IsScalarTower.algebraMap_apply","def_path":"Mathlib/Algebra/Algebra/Tower.lean","def_pos":[122,8],"def_end_pos":[122,24]},{"full_name":"LinearEquiv.map_smul","def_path":"Mathlib/Algebra/Module/Equiv/Defs.lean","def_pos":[445,8],"def_end_pos":[445,16]},{"full_name":"PowerBasis.basis_eq_pow","def_path":"Mathlib/RingTheory/PowerBasis.lean","def_pos":[64,2],"def_end_pos":[64,14]}]},{"state_before":"case neg\nK : Type u_1\nS : Type u_2\ninst✝¹¹ : Field K\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra K S\nR : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsScalarTower R K S\nA : Type u_4\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst✝ : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : ℕ\ni : Fin B.dim\nQ : R[X] := X ^ n %ₘ minpoly R B.gen\nthis : B.gen ^ n = (aeval B.gen) Q\nhQ : ¬Q = 0\nhlt : Q.natDegree < B.dim\nj : ℕ\nhj : j < B.dim\n⊢ IsIntegral R (((algebraMap R S) (Q.coeff ↑⟨j, hj⟩) • B.basis.repr (B.basis ⟨j, hj⟩)) i)","state_after":"case neg\nK : Type u_1\nS : Type u_2\ninst✝¹¹ : Field K\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra K S\nR : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsScalarTower R K S\nA : Type u_4\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst✝ : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : ℕ\ni : Fin B.dim\nQ : R[X] := X ^ n %ₘ minpoly R B.gen\nthis : B.gen ^ n = (aeval B.gen) Q\nhQ : ¬Q = 0\nhlt : Q.natDegree < B.dim\nj : ℕ\nhj : j < B.dim\n⊢ IsIntegral R (Q.coeff j • if ⟨j, hj⟩ = i then 1 else 0)","tactic":"simp only [algebraMap_smul, Finsupp.coe_smul, Pi.smul_apply, B.basis.repr_self_apply]","premises":[{"full_name":"Basis.repr_self_apply","def_path":"Mathlib/LinearAlgebra/Basis.lean","def_pos":[136,8],"def_end_pos":[136,23]},{"full_name":"Finsupp.coe_smul","def_path":"Mathlib/Data/Finsupp/Basic.lean","def_pos":[1298,8],"def_end_pos":[1298,16]},{"full_name":"Pi.smul_apply","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[104,59],"def_end_pos":[104,69]},{"full_name":"PowerBasis.basis","def_path":"Mathlib/RingTheory/PowerBasis.lean","def_pos":[63,2],"def_end_pos":[63,7]},{"full_name":"algebraMap_smul","def_path":"Mathlib/Algebra/Algebra/Basic.lean","def_pos":[330,8],"def_end_pos":[330,23]}]},{"state_before":"case neg\nK : Type u_1\nS : Type u_2\ninst✝¹¹ : Field K\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra K S\nR : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Algebra R 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: ℕ\ni : Fin B.dim\nQ : R[X] := X ^ n %ₘ minpoly R B.gen\nthis : B.gen ^ n = (aeval B.gen) Q\nhQ : ¬Q = 0\nhlt : Q.natDegree < B.dim\nj : ℕ\nhj : j < B.dim\nhij : ⟨j, hj⟩ = i\n⊢ IsIntegral R (Q.coeff j • if ⟨j, hj⟩ = i then 1 else 0)\n\ncase neg\nK : Type u_1\nS : Type u_2\ninst✝¹¹ : Field K\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra K S\nR : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsScalarTower R K S\nA : Type u_4\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst✝ : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : ℕ\ni : Fin B.dim\nQ : R[X] := X ^ n %ₘ minpoly R B.gen\nthis : B.gen ^ n = (aeval B.gen) Q\nhQ : ¬Q = 0\nhlt : Q.natDegree < B.dim\nj : ℕ\nhj : j < B.dim\nhij : ¬⟨j, hj⟩ = i\n⊢ IsIntegral R (Q.coeff j • if ⟨j, hj⟩ = i then 1 else 0)","tactic":"by_cases hij : (⟨j, hj⟩ : Fin _) = i","premises":[{"full_name":"Fin","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1831,10],"def_end_pos":[1831,13]},{"full_name":"Fin.mk","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1833,2],"def_end_pos":[1833,4]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
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+{"url":"Mathlib/CategoryTheory/Galois/GaloisObjects.lean","commit":"","full_name":"CategoryTheory.PreGaloisCategory.comp_autMap_apply","start":[148,0],"end":[152,71],"file_path":"Mathlib/CategoryTheory/Galois/GaloisObjects.lean","tactics":[{"state_before":"C : Type u₁\ninst✝⁴ : Category.{u₂, u₁} C\ninst✝³ : GaloisCategory C\nF : C ⥤ FintypeCat\ninst✝² : FiberFunctor F\nA B : C\ninst✝¹ : IsConnected A\ninst✝ : IsGalois B\nf : A ⟶ B\nσ : Aut A\na : ↑(F.obj A)\n⊢ F.map (autMap f σ).hom (F.map f a) = F.map f (F.map σ.hom a)","state_after":"no goals","tactic":"simpa [-comp_autMap] using congrFun (F.congr_map (comp_autMap f σ)) a","premises":[{"full_name":"CategoryTheory.PreGaloisCategory.comp_autMap","def_path":"Mathlib/CategoryTheory/Galois/GaloisObjects.lean","def_pos":[144,6],"def_end_pos":[144,17]},{"full_name":"Prefunctor.congr_map","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[123,8],"def_end_pos":[123,17]},{"full_name":"congrFun","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[376,8],"def_end_pos":[376,16]}]}]}
+{"url":"Mathlib/Topology/EMetricSpace/Basic.lean","commit":"","full_name":"EMetric.subset_countable_closure_of_compact","start":[743,0],"end":[749,87],"file_path":"Mathlib/Topology/EMetricSpace/Basic.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nX : Type u_1\ninst✝ : PseudoEMetricSpace α\nx y z : α\nε ε₁ ε₂ : ℝ≥0∞\ns✝ t s : Set α\nhs : IsCompact s\n⊢ ∃ t ⊆ s, t.Countable ∧ s ⊆ closure t","state_after":"α : Type u\nβ : Type v\nX : Type u_1\ninst✝ : PseudoEMetricSpace α\nx y z : α\nε✝ ε₁ ε₂ : ℝ≥0∞\ns✝ t s : Set α\nhs : IsCompact s\nε : ℝ≥0∞\nhε : ε > 0\n⊢ ∃ t, t.Countable ∧ s ⊆ ⋃ x ∈ t, closedBall x ε","tactic":"refine subset_countable_closure_of_almost_dense_set s fun ε hε => ?_","premises":[{"full_name":"EMetric.subset_countable_closure_of_almost_dense_set","def_path":"Mathlib/Topology/EMetricSpace/Basic.lean","def_pos":[690,8],"def_end_pos":[690,52]}]},{"state_before":"α : Type u\nβ : Type v\nX : Type u_1\ninst✝ : PseudoEMetricSpace α\nx y z : α\nε✝ ε₁ ε₂ : ℝ≥0∞\ns✝ t s : Set α\nhs : IsCompact s\nε : ℝ≥0∞\nhε : ε > 0\n⊢ ∃ t, t.Countable ∧ s ⊆ ⋃ x ∈ t, closedBall x ε","state_after":"case intro.intro.intro\nα : Type u\nβ : Type v\nX : Type u_1\ninst✝ : PseudoEMetricSpace α\nx y z : α\nε✝ ε₁ ε₂ : ℝ≥0∞\ns✝ t✝ s : Set α\nhs : IsCompact s\nε : ℝ≥0∞\nhε : ε > 0\nt : Set α\nhtf : t.Finite\nhst : s ⊆ ⋃ y ∈ t, ball y ε\n⊢ ∃ t, t.Countable ∧ s ⊆ ⋃ x ∈ t, closedBall x ε","tactic":"rcases totallyBounded_iff'.1 hs.totallyBounded ε hε with ⟨t, -, htf, hst⟩","premises":[{"full_name":"EMetric.totallyBounded_iff'","def_path":"Mathlib/Topology/EMetricSpace/Basic.lean","def_pos":[678,8],"def_end_pos":[678,27]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"IsCompact.totallyBounded","def_path":"Mathlib/Topology/UniformSpace/Cauchy.lean","def_pos":[625,18],"def_end_pos":[625,42]}]},{"state_before":"case intro.intro.intro\nα : Type u\nβ : Type v\nX : Type u_1\ninst✝ : PseudoEMetricSpace α\nx y z : α\nε✝ ε₁ ε₂ : ℝ≥0∞\ns✝ t✝ s : Set α\nhs : IsCompact s\nε : ℝ≥0∞\nhε : ε > 0\nt : Set α\nhtf : t.Finite\nhst : s ⊆ ⋃ y ∈ t, ball y ε\n⊢ ∃ t, t.Countable ∧ s ⊆ ⋃ x ∈ t, closedBall x ε","state_after":"no goals","tactic":"exact ⟨t, htf.countable, hst.trans <| iUnion₂_mono fun _ _ => ball_subset_closedBall⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"EMetric.ball_subset_closedBall","def_path":"Mathlib/Topology/EMetricSpace/Basic.lean","def_pos":[503,8],"def_end_pos":[503,30]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Set.Finite.countable","def_path":"Mathlib/Data/Set/Countable.lean","def_pos":[234,8],"def_end_pos":[234,24]},{"full_name":"Set.iUnion₂_mono","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[294,8],"def_end_pos":[294,20]}]}]}
+{"url":"Mathlib/CategoryTheory/Monoidal/Category.lean","commit":"","full_name":"CategoryTheory.MonoidalCategory.associator_inv_naturality","start":[631,0],"end":[634,22],"file_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","tactics":[{"state_before":"C✝ : Type u\n𝒞 : Category.{v, u} C✝\ninst✝² : MonoidalCategory C✝\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nU V W X✝ Y✝ Z✝ X Y Z X' Y' Z' : C\nf : X ⟶ X'\ng : Y ⟶ Y'\nh : Z ⟶ Z'\n⊢ (f ⊗ g ⊗ h) ≫ (α_ X' Y' Z').inv = (α_ X Y Z).inv ≫ ((f ⊗ g) ⊗ h)","state_after":"no goals","tactic":"simp [tensorHom_def]","premises":[{"full_name":"CategoryTheory.MonoidalCategory.tensorHom_def","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[157,2],"def_end_pos":[157,15]}]}]}
+{"url":"Mathlib/RingTheory/DiscreteValuationRing/Basic.lean","commit":"","full_name":"DiscreteValuationRing.aux_pid_of_ufd_of_unique_irreducible","start":[234,0],"end":[263,26],"file_path":"Mathlib/RingTheory/DiscreteValuationRing/Basic.lean","tactics":[{"state_before":"R✝ : Type u_1\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₁ : ∃ p, Irreducible p\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\n⊢ IsPrincipalIdealRing R","state_after":"case principal\nR✝ : Type u_1\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₁ : ∃ p, Irreducible p\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\n⊢ ∀ (S : Ideal R), Submodule.IsPrincipal S","tactic":"constructor","premises":[]},{"state_before":"case principal\nR✝ : Type u_1\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₁ : ∃ p, Irreducible p\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\n⊢ ∀ (S : Ideal R), Submodule.IsPrincipal S","state_after":"case principal\nR✝ : Type u_1\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₁ : ∃ p, Irreducible p\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\nI : Ideal R\n⊢ Submodule.IsPrincipal I","tactic":"intro I","premises":[]},{"state_before":"case principal\nR✝ : Type u_1\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₁ : ∃ p, Irreducible p\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\nI : Ideal R\n⊢ Submodule.IsPrincipal I","state_after":"case pos\nR✝ : Type u_1\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₁ : ∃ p, Irreducible p\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\nI : Ideal R\nI0 : I = ⊥\n⊢ Submodule.IsPrincipal I\n\ncase neg\nR✝ : Type u_1\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₁ : ∃ p, Irreducible p\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\nI : Ideal R\nI0 : ¬I = ⊥\n⊢ Submodule.IsPrincipal I","tactic":"by_cases I0 : I = ⊥","premises":[{"full_name":"Bot.bot","def_path":"Mathlib/Order/Notation.lean","def_pos":[100,2],"def_end_pos":[100,5]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case neg\nR✝ : Type u_1\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₁ : ∃ p, Irreducible p\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\nI : Ideal R\nI0 : ¬I = ⊥\n⊢ Submodule.IsPrincipal I","state_after":"case neg.intro.intro\nR✝ : Type u_1\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₁ : ∃ p, Irreducible p\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\nI : Ideal R\nI0 : ¬I = ⊥\nx : R\nhxI : x ∈ I\nhx0 : x ≠ 0\n⊢ Submodule.IsPrincipal I","tactic":"obtain ⟨x, hxI, hx0⟩ : ∃ x ∈ I, x ≠ (0 : R) := I.ne_bot_iff.mp I0","premises":[{"full_name":"And","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[516,10],"def_end_pos":[516,13]},{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Submodule.ne_bot_iff","def_path":"Mathlib/Algebra/Module/Submodule/Lattice.lean","def_pos":[88,18],"def_end_pos":[88,28]}]},{"state_before":"case neg.intro.intro\nR✝ : Type u_1\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₁ : ∃ p, Irreducible p\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\nI : Ideal R\nI0 : ¬I = ⊥\nx : R\nhxI : x ∈ I\nhx0 : x ≠ 0\n⊢ Submodule.IsPrincipal I","state_after":"case neg.intro.intro.intro.intro\nR✝ : Type u_1\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₁ : ∃ p, Irreducible p\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\nI : Ideal R\nI0 : ¬I = ⊥\nx : R\nhxI : x ∈ I\nhx0 : x ≠ 0\np : R\nleft✝ : Irreducible p\nH : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (p ^ n) x\n⊢ Submodule.IsPrincipal I","tactic":"obtain ⟨p, _, H⟩ := HasUnitMulPowIrreducibleFactorization.of_ufd_of_unique_irreducible h₁ h₂","premises":[{"full_name":"DiscreteValuationRing.HasUnitMulPowIrreducibleFactorization.of_ufd_of_unique_irreducible","def_path":"Mathlib/RingTheory/DiscreteValuationRing/Basic.lean","def_pos":[212,8],"def_end_pos":[212,36]}]},{"state_before":"case neg.intro.intro.intro.intro\nR✝ : Type u_1\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₁ : ∃ p, Irreducible p\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\nI : Ideal R\nI0 : ¬I = ⊥\nx : R\nhxI : x ∈ I\nhx0 : x ≠ 0\np : R\nleft✝ : Irreducible p\nH : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (p ^ n) x\n⊢ Submodule.IsPrincipal I","state_after":"case neg.intro.intro.intro.intro\nR✝ : Type u_1\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₁ : ∃ p, Irreducible p\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\nI : Ideal R\nI0 : ¬I = ⊥\nx : R\nhxI : x ∈ I\nhx0 : x ≠ 0\np : R\nleft✝ : Irreducible p\nH : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (p ^ n) x\nex : ∃ n, p ^ n ∈ I\n⊢ Submodule.IsPrincipal I","tactic":"have ex : ∃ n : ℕ, p ^ n ∈ I := by\n    obtain ⟨n, u, rfl⟩ := H hx0\n    refine ⟨n, ?_⟩\n    simpa only [Units.mul_inv_cancel_right] using I.mul_mem_right (↑u⁻¹) hxI","premises":[{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Ideal.mul_mem_right","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[486,8],"def_end_pos":[486,21]},{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]},{"full_name":"Units.mul_inv_cancel_right","def_path":"Mathlib/Algebra/Group/Units.lean","def_pos":[239,8],"def_end_pos":[239,28]}]},{"state_before":"case neg.intro.intro.intro.intro\nR✝ : Type u_1\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₁ : ∃ p, Irreducible p\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\nI : Ideal R\nI0 : ¬I = ⊥\nx : R\nhxI : x ∈ I\nhx0 : x ≠ 0\np : R\nleft✝ : Irreducible p\nH : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (p ^ n) x\nex : ∃ n, p ^ n ∈ I\n⊢ Submodule.IsPrincipal I","state_after":"case neg.intro.intro.intro.intro.principal'\nR✝ : Type u_1\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₁ : ∃ p, Irreducible p\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\nI : Ideal R\nI0 : ¬I = ⊥\nx : R\nhxI : x ∈ I\nhx0 : x ≠ 0\np : R\nleft✝ : Irreducible p\nH : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (p ^ n) x\nex : ∃ n, p ^ n ∈ I\n⊢ ∃ a, I = Submodule.span R {a}","tactic":"constructor","premises":[]},{"state_before":"case neg.intro.intro.intro.intro.principal'\nR✝ : Type u_1\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₁ : ∃ p, Irreducible p\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\nI : Ideal R\nI0 : ¬I = ⊥\nx : R\nhxI : x ∈ I\nhx0 : x ≠ 0\np : R\nleft✝ : Irreducible p\nH : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (p ^ n) x\nex : ∃ n, p ^ n ∈ I\n⊢ ∃ a, I = Submodule.span R {a}","state_after":"case h\nR✝ : Type u_1\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₁ : ∃ p, Irreducible p\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\nI : Ideal R\nI0 : ¬I = ⊥\nx : R\nhxI : x ∈ I\nhx0 : x ≠ 0\np : R\nleft✝ : Irreducible p\nH : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (p ^ n) x\nex : ∃ n, p ^ n ∈ I\n⊢ I = Submodule.span R {p ^ Nat.find ex}","tactic":"use p ^ Nat.find 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Irreducible q → Associated p q\nI : Ideal R\nI0 : ¬I = ⊥\nx : R\nhxI : x ∈ I\nhx0 : x ≠ 0\np : R\nleft✝ : Irreducible p\nH : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (p ^ n) x\nex : ∃ n, p ^ n ∈ I\n⊢ I = span {p ^ Nat.find ex}","tactic":"show I = Ideal.span _","premises":[{"full_name":"Ideal.span","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[101,4],"def_end_pos":[101,8]}]},{"state_before":"case h\nR✝ : Type u_1\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₁ : ∃ p, Irreducible p\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\nI : Ideal R\nI0 : ¬I = ⊥\nx : R\nhxI : x ∈ I\nhx0 : x ≠ 0\np : R\nleft✝ : Irreducible p\nH : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (p ^ n) x\nex : ∃ n, p ^ n ∈ I\n⊢ I = span {p ^ Nat.find ex}","state_after":"case h.a\nR✝ : Type u_1\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₁ : ∃ p, Irreducible p\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\nI : Ideal R\nI0 : ¬I = ⊥\nx : R\nhxI : x ∈ I\nhx0 : x ≠ 0\np : R\nleft✝ : Irreducible p\nH : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (p ^ n) x\nex : ∃ n, p ^ n ∈ I\n⊢ I ≤ span {p ^ Nat.find ex}\n\ncase h.a\nR✝ : Type u_1\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₁ : ∃ p, Irreducible p\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\nI : Ideal R\nI0 : ¬I = ⊥\nx : R\nhxI : x ∈ I\nhx0 : x ≠ 0\np : R\nleft✝ : Irreducible p\nH : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (p ^ n) x\nex : ∃ n, p ^ n ∈ I\n⊢ span {p ^ Nat.find ex} ≤ I","tactic":"apply le_antisymm","premises":[{"full_name":"le_antisymm","def_path":"Mathlib/Order/Defs.lean","def_pos":[156,8],"def_end_pos":[156,19]}]}]}
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+{"url":"Mathlib/Data/ZMod/Basic.lean","commit":"","full_name":"ZMod.cast_add_eq_ite","start":[279,0],"end":[291,7],"file_path":"Mathlib/Data/ZMod/Basic.lean","tactics":[{"state_before":"n✝ : ℕ\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na b : ZMod n\n⊢ (a + b).cast = if ↑n ≤ a.cast + b.cast then a.cast + b.cast - ↑n else a.cast + b.cast","state_after":"case zero\nn : ℕ\nR : Type u_1\ninst✝ : Ring R\na b : ZMod 0\n⊢ (a + b).cast = if ↑0 ≤ a.cast + b.cast then a.cast + b.cast - ↑0 else a.cast + b.cast\n\ncase succ\nn✝ : ℕ\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na b : ZMod (n + 1)\n⊢ (a + b).cast = if ↑(n + 1) ≤ a.cast + b.cast then a.cast + b.cast - ↑(n + 1) else a.cast + b.cast","tactic":"cases' n with n","premises":[]},{"state_before":"case succ\nn✝ : ℕ\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na b : ZMod (n + 1)\n⊢ (a + b).cast = if ↑(n + 1) ≤ a.cast + b.cast then a.cast + b.cast - ↑(n + 1) else a.cast + b.cast","state_after":"case succ\nn✝ : ℕ\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na b : Fin (n + 1)\n⊢ (a + b).cast = if ↑(n + 1) ≤ cast a + cast b then cast a + cast b - ↑(n + 1) else cast a + cast b","tactic":"change Fin (n + 1) at a b","premises":[{"full_name":"Fin","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1831,10],"def_end_pos":[1831,13]}]},{"state_before":"case succ\nn✝ : ℕ\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na b : Fin (n + 1)\n⊢ (a + b).cast = if ↑(n + 1) ≤ cast a + cast b then cast a + cast b - ↑(n + 1) else cast a + cast b","state_after":"case succ\nn✝ : ℕ\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na b : Fin (n + 1)\n⊢ ↑↑(a + b) = if ↑(n + 1) ≤ ↑↑a + ↑↑b then cast a + cast b - ↑(n + 1) else cast a + cast b","tactic":"change ((((a + b) : Fin (n + 1)) : ℕ) : ℤ) = if ((n + 1 : ℕ) : ℤ) ≤ (a : ℕ) + b then _ else _","premises":[{"full_name":"Fin","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1831,10],"def_end_pos":[1831,13]},{"full_name":"Int","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean","def_pos":[40,10],"def_end_pos":[40,13]},{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]},{"full_name":"ite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[970,20],"def_end_pos":[970,23]}]},{"state_before":"case succ\nn✝ : ℕ\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na b : Fin (n + 1)\n⊢ ↑↑(a + b) = if ↑(n + 1) ≤ ↑↑a + ↑↑b then cast a + cast b - ↑(n + 1) else cast a + cast b","state_after":"case succ\nn✝ : ℕ\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na b : Fin (n + 1)\n⊢ ↑(if n + 1 ≤ ↑a + ↑b then ↑a + ↑b - (n + 1) else ↑a + ↑b) =\n    if ↑n + 1 ≤ ↑↑a + ↑↑b then cast a + cast b - (↑n + 1) else cast a + cast b","tactic":"simp only [Fin.val_add_eq_ite, Int.ofNat_succ, Int.ofNat_le]","premises":[{"full_name":"Fin.val_add_eq_ite","def_path":"Mathlib/Data/Fin/Basic.lean","def_pos":[386,8],"def_end_pos":[386,22]},{"full_name":"Int.ofNat_le","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Order.lean","def_pos":[51,27],"def_end_pos":[51,35]},{"full_name":"Int.ofNat_succ","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean","def_pos":[27,8],"def_end_pos":[27,18]}]},{"state_before":"case succ\nn✝ : ℕ\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na b : Fin (n + 1)\n⊢ ↑(if n + 1 ≤ ↑a + ↑b then ↑a + ↑b - (n + 1) else ↑a + ↑b) =\n    if ↑n + 1 ≤ ↑↑a + ↑↑b then cast a + cast b - (↑n + 1) else cast a + cast b","state_after":"case succ\nn✝ : ℕ\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na b : Fin (n + 1)\n⊢ ↑(if n + 1 ≤ ↑a + ↑b then ↑a + ↑b - (n + 1) else ↑a + ↑b) =\n    if n + 1 ≤ ↑a + ↑b then cast a + cast b - ↑(n + 1) else cast a + cast b","tactic":"norm_cast","premises":[]},{"state_before":"case succ\nn✝ : ℕ\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na b : Fin (n + 1)\n⊢ ↑(if n + 1 ≤ ↑a + ↑b then ↑a + ↑b - (n + 1) else ↑a + ↑b) =\n    if n + 1 ≤ ↑a + ↑b then cast a + cast b - ↑(n + 1) else cast a + cast b","state_after":"case pos\nn✝ : ℕ\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na b : Fin (n + 1)\nh : n + 1 ≤ ↑a + ↑b\n⊢ ↑(↑a + ↑b - (n + 1)) = cast a + cast b - ↑(n + 1)\n\ncase neg\nn✝ : ℕ\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na b : Fin (n + 1)\nh : ¬n + 1 ≤ ↑a + ↑b\n⊢ ↑(↑a + ↑b) = cast a + cast b","tactic":"split_ifs with h","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
+{"url":"Mathlib/CategoryTheory/Preadditive/Yoneda/Basic.lean","commit":"","full_name":"CategoryTheory.preadditiveYoneda_map_app_apply","start":[47,0],"end":[61,37],"file_path":"Mathlib/CategoryTheory/Preadditive/Yoneda/Basic.lean","tactics":[{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nx✝ : C\n⊢ { obj := fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGrp,\n          map := fun {X Y} f =>\n            { app := fun X_1 => { toFun := fun g => g ≫ f, map_zero' := ⋯, map_add' := ⋯ }, naturality := ⋯ } }.map\n      (𝟙 x✝) =\n    𝟙\n      ({ obj := fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGrp,\n            map := fun {X Y} f =>\n              { app := fun X_1 => { toFun := fun g => g ≫ f, map_zero' := ⋯, map_add' := ⋯ }, naturality := ⋯ } }.obj\n        x✝)","state_after":"case w.h.w\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nx✝² : C\nx✝¹ : Cᵒᵖ\nx✝ :\n  ↑(({ obj := fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGrp,\n              map := fun {X Y} f =>\n                { app := fun X_1 => { toFun := fun g => g ≫ f, map_zero' := ⋯, map_add' := ⋯ }, naturality := ⋯ } }.obj\n          x✝²).obj\n      x✝¹)\n⊢ (({ obj := fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGrp,\n                map := fun {X Y} f =>\n                  { app := fun X_1 => { toFun := fun g => g ≫ f, map_zero' := ⋯, map_add' := ⋯ },\n                    naturality := ⋯ } }.map\n            (𝟙 x✝²)).app\n        x✝¹)\n      x✝ =\n    ((𝟙\n            ({ obj := fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGrp,\n                  map := fun {X Y} f =>\n                    { app := fun X_1 => { toFun := fun g => g ≫ f, map_zero' := ⋯, map_add' := ⋯ },\n                      naturality := ⋯ } }.obj\n              x✝²)).app\n        x✝¹)\n      x✝","tactic":"ext","premises":[]},{"state_before":"case w.h.w\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nx✝² : C\nx✝¹ : Cᵒᵖ\nx✝ :\n  ↑(({ obj := fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGrp,\n              map := fun {X Y} f =>\n                { app := fun X_1 => { toFun := fun g => g ≫ f, map_zero' := ⋯, map_add' := ⋯ }, naturality := ⋯ } }.obj\n          x✝²).obj\n      x✝¹)\n⊢ (({ obj := fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGrp,\n                map := fun {X Y} f =>\n                  { app := fun X_1 => { toFun := fun g => g ≫ f, map_zero' := ⋯, map_add' := ⋯ },\n                    naturality := ⋯ } }.map\n            (𝟙 x✝²)).app\n        x✝¹)\n      x✝ =\n    ((𝟙\n            ({ obj := fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGrp,\n                  map := fun {X Y} f =>\n                    { app := fun X_1 => { toFun := fun g => g ≫ f, map_zero' := ⋯, map_add' := ⋯ },\n                      naturality := ⋯ } }.obj\n              x✝²)).app\n        x✝¹)\n      x✝","state_after":"case w.h.w\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nx✝² : C\nx✝¹ : Cᵒᵖ\nx✝ :\n  ↑(({ obj := fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGrp,\n              map := fun {X Y} f =>\n                { app := fun X_1 => { toFun := fun g => g ≫ f, map_zero' := ⋯, map_add' := ⋯ }, naturality := ⋯ } }.obj\n          x✝²).obj\n      x✝¹)\n⊢ x✝ ≫ 𝟙 x✝² = x✝","tactic":"dsimp","premises":[]},{"state_before":"case w.h.w\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nx✝² : C\nx✝¹ : Cᵒᵖ\nx✝ :\n  ↑(({ obj := fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGrp,\n              map := fun {X Y} f =>\n                { app := fun X_1 => { toFun := fun g => g ≫ f, map_zero' := ⋯, map_add' := ⋯ }, naturality := ⋯ } }.obj\n          x✝²).obj\n      x✝¹)\n⊢ x✝ ≫ 𝟙 x✝² = x✝","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\n⊢ { obj := fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGrp,\n          map := fun {X Y} f =>\n            { app := fun X_1 => { toFun := fun g => g ≫ f, map_zero' := ⋯, map_add' := ⋯ }, naturality := ⋯ } }.map\n      (f ≫ g) =\n    { obj := fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGrp,\n            map := fun {X Y} f =>\n              { app := fun X_1 => { toFun := fun g => g ≫ f, map_zero' := ⋯, map_add' := ⋯ }, naturality := ⋯ } }.map\n        f ≫\n      { obj := fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGrp,\n            map := fun {X Y} f =>\n              { app := fun X_1 => { toFun := fun g => g ≫ f, map_zero' := ⋯, map_add' := ⋯ }, naturality := ⋯ } }.map\n        g","state_after":"case w.h.w\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\nx✝¹ : Cᵒᵖ\nx✝ :\n  ↑(({ obj := fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGrp,\n              map := fun {X Y} f =>\n                { app := fun X_1 => { toFun := fun g => g ≫ f, map_zero' := ⋯, map_add' := ⋯ }, naturality := ⋯ } }.obj\n          X✝).obj\n      x✝¹)\n⊢ (({ obj := fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGrp,\n                map := fun {X Y} f =>\n                  { app := fun X_1 => { toFun := fun g => g ≫ f, map_zero' := ⋯, map_add' := ⋯ },\n                    naturality := ⋯ } }.map\n            (f ≫ g)).app\n        x✝¹)\n      x✝ =\n    (({ obj := fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGrp,\n                  map := fun {X Y} f =>\n                    { app := fun X_1 => { toFun := fun g => g ≫ f, map_zero' := ⋯, map_add' := ⋯ },\n                      naturality := ⋯ } }.map\n              f ≫\n            { obj := fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGrp,\n                  map := fun {X Y} f =>\n                    { app := fun X_1 => { toFun := fun g => g ≫ f, map_zero' := ⋯, map_add' := ⋯ },\n                      naturality := ⋯ } }.map\n              g).app\n        x✝¹)\n      x✝","tactic":"ext","premises":[]},{"state_before":"case w.h.w\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\nx✝¹ : Cᵒᵖ\nx✝ :\n  ↑(({ obj := fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGrp,\n              map := fun {X Y} f =>\n                { app := fun X_1 => { toFun := fun g => g ≫ f, map_zero' := ⋯, map_add' := ⋯ }, naturality := ⋯ } }.obj\n          X✝).obj\n      x✝¹)\n⊢ (({ obj := fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGrp,\n                map := fun {X Y} f =>\n                  { app := fun X_1 => { toFun := fun g => g ≫ f, map_zero' := ⋯, map_add' := ⋯ },\n                    naturality := ⋯ } }.map\n            (f ≫ g)).app\n        x✝¹)\n      x✝ =\n    (({ obj := fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGrp,\n                  map := fun {X Y} f =>\n                    { app := fun X_1 => { toFun := fun g => g ≫ f, map_zero' := ⋯, map_add' := ⋯ },\n                      naturality := ⋯ } }.map\n              f ≫\n            { obj := fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGrp,\n                  map := fun {X Y} f =>\n                    { app := fun X_1 => { toFun := fun g => g ≫ f, map_zero' := ⋯, map_add' := ⋯ },\n                      naturality := ⋯ } }.map\n              g).app\n        x✝¹)\n      x✝","state_after":"case w.h.w\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\nx✝¹ : Cᵒᵖ\nx✝ :\n  ↑(({ obj := fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGrp,\n              map := fun {X Y} f =>\n                { app := fun X_1 => { toFun := fun g => g ≫ f, map_zero' := ⋯, map_add' := ⋯ }, naturality := ⋯ } }.obj\n          X✝).obj\n      x✝¹)\n⊢ x✝ ≫ f ≫ g = (x✝ ≫ f) ≫ g","tactic":"dsimp","premises":[]},{"state_before":"case w.h.w\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\nx✝¹ : Cᵒᵖ\nx✝ :\n  ↑(({ obj := fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGrp,\n              map := fun {X Y} f =>\n                { app := fun X_1 => { toFun := fun g => g ≫ f, map_zero' := ⋯, map_add' := ⋯ }, naturality := ⋯ } }.obj\n          X✝).obj\n      x✝¹)\n⊢ x✝ ≫ f ≫ g = (x✝ ≫ f) ≫ g","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/Data/Nat/Defs.lean","commit":"","full_name":"Nat.le_self_pow","start":[613,0],"end":[615,89],"file_path":"Mathlib/Data/Nat/Defs.lean","tactics":[{"state_before":"a✝ b c d m n k : ℕ\np q : ℕ → Prop\nhn : n ≠ 0\na : ℕ\n⊢ a + 1 ≤ (a + 1) ^ n","state_after":"no goals","tactic":"simpa using Nat.pow_le_pow_right a.succ_pos (Nat.one_le_iff_ne_zero.2 hn)","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Nat.one_le_iff_ne_zero","def_path":"Mathlib/Data/Nat/Defs.lean","def_pos":[132,6],"def_end_pos":[132,24]},{"full_name":"Nat.pow_le_pow_right","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean","def_pos":[629,17],"def_end_pos":[629,33]},{"full_name":"Nat.succ_pos","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1693,8],"def_end_pos":[1693,20]}]}]}
+{"url":"Mathlib/Combinatorics/SimpleGraph/Metric.lean","commit":"","full_name":"SimpleGraph.Reachable.dist_eq_zero_iff","start":[164,0],"end":[165,42],"file_path":"Mathlib/Combinatorics/SimpleGraph/Metric.lean","tactics":[{"state_before":"V : Type u_1\nG : SimpleGraph V\nu v w : V\nhr : G.Reachable u v\n⊢ G.dist u v = 0 ↔ u = v","state_after":"no goals","tactic":"simp [hr]","premises":[]}]}
+{"url":"Mathlib/MeasureTheory/Integral/Bochner.lean","commit":"","full_name":"MeasureTheory.integral_simpleFunc_larger_space","start":[1790,0],"end":[1796,9],"file_path":"Mathlib/MeasureTheory/Integral/Bochner.lean","tactics":[{"state_before":"α : Type u_1\nE : Type u_2\nF : Type u_3\n𝕜 : Type u_4\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\nhE : CompleteSpace E\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : SMulCommClass ℝ 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : CompleteSpace F\nG : Type u_5\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace ℝ G\nH : Type u_6\nβ : Type u_7\nγ : Type u_8\ninst✝ : NormedAddCommGroup H\nm m0 : MeasurableSpace β\nμ : Measure β\nhm : m ≤ m0\nf : β →ₛ F\nhf_int : Integrable (↑f) μ\n⊢ ∫ (x : β), ↑f x ∂μ = ∑ x ∈ f.range, (μ (↑f ⁻¹' {x})).toReal • x","state_after":"α : Type u_1\nE : Type u_2\nF : Type u_3\n𝕜 : Type u_4\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\nhE : CompleteSpace E\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : SMulCommClass ℝ 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : CompleteSpace F\nG : Type u_5\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace ℝ G\nH : Type u_6\nβ : Type u_7\nγ : Type u_8\ninst✝ : NormedAddCommGroup H\nm m0 : MeasurableSpace β\nμ : Measure β\nhm : m ≤ m0\nf : β →ₛ F\nhf_int : Integrable (↑f) μ\n⊢ ∫ (x : β), ↑(SimpleFunc.toLargerSpace hm f) x ∂μ =\n    ∑ x ∈ f.range, (μ (↑(SimpleFunc.toLargerSpace hm f) ⁻¹' {x})).toReal • x","tactic":"simp_rw [← f.coe_toLargerSpace_eq hm]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"MeasureTheory.SimpleFunc.coe_toLargerSpace_eq","def_path":"Mathlib/MeasureTheory/Integral/Bochner.lean","def_pos":[1787,8],"def_end_pos":[1787,39]}]},{"state_before":"α : Type u_1\nE : Type u_2\nF : Type u_3\n𝕜 : Type u_4\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\nhE : CompleteSpace E\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : SMulCommClass ℝ 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : CompleteSpace F\nG : Type u_5\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace ℝ G\nH : Type u_6\nβ : Type u_7\nγ : Type u_8\ninst✝ : NormedAddCommGroup H\nm m0 : MeasurableSpace β\nμ : Measure β\nhm : m ≤ m0\nf : β →ₛ F\nhf_int : Integrable (↑f) μ\n⊢ ∫ (x : β), ↑(SimpleFunc.toLargerSpace hm f) x ∂μ =\n    ∑ x ∈ f.range, (μ (↑(SimpleFunc.toLargerSpace hm f) ⁻¹' {x})).toReal • x","state_after":"α : Type u_1\nE : Type u_2\nF : Type u_3\n𝕜 : Type u_4\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\nhE : CompleteSpace E\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : SMulCommClass ℝ 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : CompleteSpace F\nG : Type u_5\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace ℝ G\nH : Type u_6\nβ : Type u_7\nγ : Type u_8\ninst✝ : NormedAddCommGroup H\nm m0 : MeasurableSpace β\nμ : Measure β\nhm : m ≤ m0\nf : β →ₛ F\nhf_int✝ : Integrable (↑f) μ\nhf_int : Integrable (↑(SimpleFunc.toLargerSpace hm f)) μ\n⊢ ∫ (x : β), ↑(SimpleFunc.toLargerSpace hm f) x ∂μ =\n    ∑ x ∈ f.range, (μ (↑(SimpleFunc.toLargerSpace hm f) ⁻¹' {x})).toReal • x","tactic":"have hf_int : Integrable (f.toLargerSpace hm) μ := by rwa [SimpleFunc.coe_toLargerSpace_eq]","premises":[{"full_name":"MeasureTheory.Integrable","def_path":"Mathlib/MeasureTheory/Function/L1Space.lean","def_pos":[389,4],"def_end_pos":[389,14]},{"full_name":"MeasureTheory.SimpleFunc.coe_toLargerSpace_eq","def_path":"Mathlib/MeasureTheory/Integral/Bochner.lean","def_pos":[1787,8],"def_end_pos":[1787,39]},{"full_name":"MeasureTheory.SimpleFunc.toLargerSpace","def_path":"Mathlib/MeasureTheory/Integral/Bochner.lean","def_pos":[1783,4],"def_end_pos":[1783,28]}]},{"state_before":"α : Type u_1\nE : Type u_2\nF : Type u_3\n𝕜 : Type u_4\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\nhE : CompleteSpace E\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : SMulCommClass ℝ 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : CompleteSpace F\nG : Type u_5\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace ℝ G\nH : Type u_6\nβ : Type u_7\nγ : Type u_8\ninst✝ : NormedAddCommGroup H\nm m0 : MeasurableSpace β\nμ : Measure β\nhm : m ≤ m0\nf : β →ₛ F\nhf_int✝ : Integrable (↑f) μ\nhf_int : Integrable (↑(SimpleFunc.toLargerSpace hm f)) μ\n⊢ ∫ (x : β), ↑(SimpleFunc.toLargerSpace hm f) x ∂μ =\n    ∑ x ∈ f.range, (μ (↑(SimpleFunc.toLargerSpace hm f) ⁻¹' {x})).toReal • x","state_after":"α : Type u_1\nE : Type u_2\nF : Type u_3\n𝕜 : Type u_4\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\nhE : CompleteSpace E\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : SMulCommClass ℝ 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : CompleteSpace F\nG : Type u_5\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace ℝ G\nH : Type u_6\nβ : Type u_7\nγ : Type u_8\ninst✝ : NormedAddCommGroup H\nm m0 : MeasurableSpace β\nμ : Measure β\nhm : m ≤ m0\nf : β →ₛ F\nhf_int✝ : Integrable (↑f) μ\nhf_int : Integrable (↑(SimpleFunc.toLargerSpace hm f)) μ\n⊢ ∑ x ∈ (SimpleFunc.toLargerSpace hm f).range, (μ (↑(SimpleFunc.toLargerSpace hm f) ⁻¹' {x})).toReal • x =\n    ∑ x ∈ f.range, (μ (↑(SimpleFunc.toLargerSpace hm f) ⁻¹' {x})).toReal • x","tactic":"rw [SimpleFunc.integral_eq_sum _ hf_int]","premises":[{"full_name":"MeasureTheory.SimpleFunc.integral_eq_sum","def_path":"Mathlib/MeasureTheory/Integral/Bochner.lean","def_pos":[1385,8],"def_end_pos":[1385,34]}]},{"state_before":"α : Type u_1\nE : Type u_2\nF : Type u_3\n𝕜 : Type u_4\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\nhE : CompleteSpace E\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : SMulCommClass ℝ 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : CompleteSpace F\nG : Type u_5\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace ℝ G\nH : Type u_6\nβ : Type u_7\nγ : Type u_8\ninst✝ : NormedAddCommGroup H\nm m0 : MeasurableSpace β\nμ : Measure β\nhm : m ≤ m0\nf : β →ₛ F\nhf_int✝ : Integrable (↑f) μ\nhf_int : Integrable (↑(SimpleFunc.toLargerSpace hm f)) μ\n⊢ ∑ x ∈ (SimpleFunc.toLargerSpace hm f).range, (μ (↑(SimpleFunc.toLargerSpace hm f) ⁻¹' {x})).toReal • x =\n    ∑ x ∈ f.range, (μ (↑(SimpleFunc.toLargerSpace hm f) ⁻¹' {x})).toReal • x","state_after":"no goals","tactic":"congr 1","premises":[]}]}
+{"url":"Mathlib/Algebra/Lie/TraceForm.lean","commit":"","full_name":"LieModule.traceForm_eq_sum_weightSpaceOf","start":[213,0],"end":[229,50],"file_path":"Mathlib/Algebra/Lie/TraceForm.lean","tactics":[{"state_before":"R : Type u_1\nK : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : LieRingModule L M\ninst✝⁶ : LieModule R L M\ninst✝⁵ : Module.Free R M\ninst✝⁴ : Module.Finite R M\ninst✝³ : LieAlgebra.IsNilpotent R L\ninst✝² : IsDomain R\ninst✝¹ : IsPrincipalIdealRing R\ninst✝ : IsTriangularizable R L M\nz : L\n⊢ traceForm R L M = ∑ χ ∈ ⋯.toFinset, traceForm R L ↥↑(weightSpaceOf M χ z)","state_after":"case H\nR : Type u_1\nK : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : LieRingModule L M\ninst✝⁶ : LieModule R L M\ninst✝⁵ : Module.Free R M\ninst✝⁴ : Module.Finite R M\ninst✝³ : LieAlgebra.IsNilpotent R L\ninst✝² : IsDomain R\ninst✝¹ : IsPrincipalIdealRing R\ninst✝ : IsTriangularizable R L M\nz x y : L\n⊢ ((traceForm R L M) x) y = ((∑ χ ∈ ⋯.toFinset, traceForm R L ↥↑(weightSpaceOf M χ z)) x) y","tactic":"ext x y","premises":[]},{"state_before":"case H\nR : Type u_1\nK : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : LieRingModule L M\ninst✝⁶ : LieModule R L M\ninst✝⁵ : Module.Free R M\ninst✝⁴ : Module.Finite R M\ninst✝³ : LieAlgebra.IsNilpotent R L\ninst✝² : IsDomain R\ninst✝¹ : IsPrincipalIdealRing R\ninst✝ : IsTriangularizable R L M\nz x y : L\n⊢ ((traceForm R L M) x) y = ((∑ χ ∈ ⋯.toFinset, traceForm R L ↥↑(weightSpaceOf M χ z)) x) y","state_after":"case H\nR : Type u_1\nK : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : LieRingModule L M\ninst✝⁶ : LieModule R L M\ninst✝⁵ : Module.Free R M\ninst✝⁴ : Module.Finite R M\ninst✝³ : LieAlgebra.IsNilpotent R L\ninst✝² : IsDomain R\ninst✝¹ : IsPrincipalIdealRing R\ninst✝ : IsTriangularizable R L M\nz x y : L\nhxy : ∀ (χ : R), MapsTo ⇑(φ x ∘ₗ φ y) ↑(weightSpaceOf M χ z) ↑(weightSpaceOf M χ z)\n⊢ ((traceForm R L M) x) y = ((∑ χ ∈ ⋯.toFinset, traceForm R L ↥↑(weightSpaceOf M χ z)) x) y","tactic":"have hxy : ∀ χ : R, MapsTo ((toEnd R L M x).comp (toEnd R L M y))\n      (weightSpaceOf M χ z) (weightSpaceOf M χ z) :=\n    fun χ m hm ↦ LieSubmodule.lie_mem _ <| LieSubmodule.lie_mem _ hm","premises":[{"full_name":"LieModule.toEnd","def_path":"Mathlib/Algebra/Lie/OfAssociative.lean","def_pos":[178,4],"def_end_pos":[178,19]},{"full_name":"LieModule.weightSpaceOf","def_path":"Mathlib/Algebra/Lie/Weights/Basic.lean","def_pos":[149,4],"def_end_pos":[149,17]},{"full_name":"LieSubmodule.lie_mem","def_path":"Mathlib/Algebra/Lie/Submodule.lean","def_pos":[45,2],"def_end_pos":[45,9]},{"full_name":"LinearMap.comp","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[489,4],"def_end_pos":[489,8]},{"full_name":"Set.MapsTo","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[232,4],"def_end_pos":[232,10]}]},{"state_before":"case H\nR : Type u_1\nK : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : LieRingModule L M\ninst✝⁶ : LieModule R L M\ninst✝⁵ : Module.Free R M\ninst✝⁴ : Module.Finite R M\ninst✝³ : LieAlgebra.IsNilpotent R L\ninst✝² : IsDomain R\ninst✝¹ : IsPrincipalIdealRing R\ninst✝ : IsTriangularizable R L M\nz x y : L\nhxy : ∀ (χ : R), MapsTo ⇑(φ x ∘ₗ φ y) ↑(weightSpaceOf M χ z) ↑(weightSpaceOf M χ z)\n⊢ ((traceForm R L M) x) y = ((∑ χ ∈ ⋯.toFinset, traceForm R L ↥↑(weightSpaceOf M χ z)) x) y","state_after":"case H\nR : Type u_1\nK : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : LieRingModule L M\ninst✝⁶ : LieModule R L M\ninst✝⁵ : Module.Free R M\ninst✝⁴ : Module.Finite R M\ninst✝³ : LieAlgebra.IsNilpotent R L\ninst✝² : IsDomain R\ninst✝¹ : IsPrincipalIdealRing R\ninst✝ : IsTriangularizable R L M\nz x y : L\nhxy : ∀ (χ : R), MapsTo ⇑(φ x ∘ₗ φ y) ↑(weightSpaceOf M χ z) ↑(weightSpaceOf M χ z)\nhfin : {χ | ↑(weightSpaceOf M χ z) ≠ ⊥}.Finite\n⊢ ((traceForm R L M) x) y = ((∑ χ ∈ ⋯.toFinset, traceForm R L ↥↑(weightSpaceOf M χ z)) x) y","tactic":"have hfin : {χ : R | (weightSpaceOf M χ z : Submodule R M) ≠ ⊥}.Finite := by\n    convert finite_weightSpaceOf_ne_bot R L M z\n    exact LieSubmodule.coeSubmodule_eq_bot_iff (weightSpaceOf M _ _)","premises":[{"full_name":"Bot.bot","def_path":"Mathlib/Order/Notation.lean","def_pos":[100,2],"def_end_pos":[100,5]},{"full_name":"LieModule.finite_weightSpaceOf_ne_bot","def_path":"Mathlib/Algebra/Lie/Weights/Basic.lean","def_pos":[692,6],"def_end_pos":[692,33]},{"full_name":"LieModule.weightSpaceOf","def_path":"Mathlib/Algebra/Lie/Weights/Basic.lean","def_pos":[149,4],"def_end_pos":[149,17]},{"full_name":"LieSubmodule.coeSubmodule_eq_bot_iff","def_path":"Mathlib/Algebra/Lie/Submodule.lean","def_pos":[331,8],"def_end_pos":[331,31]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Set.Finite","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[56,14],"def_end_pos":[56,20]},{"full_name":"Submodule","def_path":"Mathlib/Algebra/Module/Submodule/Basic.lean","def_pos":[36,10],"def_end_pos":[36,19]},{"full_name":"setOf","def_path":"Mathlib/Init/Set.lean","def_pos":[56,4],"def_end_pos":[56,9]}]},{"state_before":"case H\nR : Type u_1\nK : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : LieRingModule L M\ninst✝⁶ : LieModule R L M\ninst✝⁵ : Module.Free R M\ninst✝⁴ : Module.Finite R M\ninst✝³ : LieAlgebra.IsNilpotent R L\ninst✝² : IsDomain R\ninst✝¹ : IsPrincipalIdealRing R\ninst✝ : IsTriangularizable R L M\nz x y : L\nhxy : ∀ (χ : R), MapsTo ⇑(φ x ∘ₗ φ y) ↑(weightSpaceOf M χ z) ↑(weightSpaceOf M χ z)\nhfin : {χ | ↑(weightSpaceOf M χ z) ≠ ⊥}.Finite\n⊢ ((traceForm R L M) x) y = ((∑ χ ∈ ⋯.toFinset, traceForm R L ↥↑(weightSpaceOf M χ z)) x) y","state_after":"no goals","tactic":"classical\n  have hds := DirectSum.isInternal_submodule_of_independent_of_iSup_eq_top\n    (LieSubmodule.independent_iff_coe_toSubmodule.mp <| independent_weightSpaceOf R L M z)\n    (IsTriangularizable.iSup_eq_top z)\n  simp only [LinearMap.coeFn_sum, Finset.sum_apply, traceForm_apply_apply,\n    LinearMap.trace_eq_sum_trace_restrict' hds hfin hxy]\n  exact Finset.sum_congr (by simp) (fun χ _ ↦ rfl)","premises":[{"full_name":"DirectSum.isInternal_submodule_of_independent_of_iSup_eq_top","def_path":"Mathlib/Algebra/DirectSum/Module.lean","def_pos":[382,8],"def_end_pos":[382,58]},{"full_name":"Finset.sum_apply","def_path":"Mathlib/Algebra/BigOperators/Pi.lean","def_pos":[32,2],"def_end_pos":[32,13]},{"full_name":"Finset.sum_congr","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[380,2],"def_end_pos":[380,13]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"LieModule.IsTriangularizable.iSup_eq_top","def_path":"Mathlib/Algebra/Lie/Weights/Basic.lean","def_pos":[714,2],"def_end_pos":[714,13]},{"full_name":"LieModule.independent_weightSpaceOf","def_path":"Mathlib/Algebra/Lie/Weights/Basic.lean","def_pos":[687,6],"def_end_pos":[687,31]},{"full_name":"LieModule.traceForm_apply_apply","def_path":"Mathlib/Algebra/Lie/TraceForm.lean","def_pos":[51,6],"def_end_pos":[51,27]},{"full_name":"LieSubmodule.independent_iff_coe_toSubmodule","def_path":"Mathlib/Algebra/Lie/Submodule.lean","def_pos":[508,8],"def_end_pos":[508,39]},{"full_name":"LinearMap.coeFn_sum","def_path":"Mathlib/Algebra/Module/Submodule/LinearMap.lean","def_pos":[210,8],"def_end_pos":[210,17]},{"full_name":"LinearMap.trace_eq_sum_trace_restrict'","def_path":"Mathlib/Algebra/DirectSum/LinearMap.lean","def_pos":[73,6],"def_end_pos":[73,34]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
+{"url":"Mathlib/Geometry/Manifold/ContMDiff/Defs.lean","commit":"","full_name":"contMDiffWithinAt_iff_image","start":[353,0],"end":[363,99],"file_path":"Mathlib/Geometry/Manifold/ContMDiff/Defs.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝³⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³⁶ : NormedAddCommGroup E\ninst✝³⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝³⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝³³ : TopologicalSpace M\ninst✝³² : ChartedSpace H M\ninst✝³¹ : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst✝³⁰ : NormedAddCommGroup E'\ninst✝²⁹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝²⁸ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM' : Type u_7\ninst✝²⁷ : TopologicalSpace M'\ninst✝²⁶ : ChartedSpace H' M'\ninst✝²⁵ : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst✝²⁴ : NormedAddCommGroup E''\ninst✝²³ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝²² : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝²¹ : TopologicalSpace M''\ninst✝²⁰ : ChartedSpace H'' M''\nF : Type u_11\ninst✝¹⁹ : NormedAddCommGroup F\ninst✝¹⁸ : NormedSpace 𝕜 F\nG : Type u_12\ninst✝¹⁷ : TopologicalSpace G\nJ : ModelWithCorners 𝕜 F G\nN : Type u_13\ninst✝¹⁶ : TopologicalSpace N\ninst✝¹⁵ : ChartedSpace G N\ninst✝¹⁴ : SmoothManifoldWithCorners J N\nF' : Type u_14\ninst✝¹³ : NormedAddCommGroup F'\ninst✝¹² : NormedSpace 𝕜 F'\nG' : Type u_15\ninst✝¹¹ : TopologicalSpace G'\nJ' : ModelWithCorners 𝕜 F' G'\nN' : Type u_16\ninst✝¹⁰ : TopologicalSpace N'\ninst✝⁹ : ChartedSpace G' N'\ninst✝⁸ : SmoothManifoldWithCorners J' N'\nF₁ : Type u_17\ninst✝⁷ : NormedAddCommGroup F₁\ninst✝⁶ : NormedSpace 𝕜 F₁\nF₂ : Type u_18\ninst✝⁵ : NormedAddCommGroup F₂\ninst✝⁴ : NormedSpace 𝕜 F₂\nF₃ : Type u_19\ninst✝³ : NormedAddCommGroup F₃\ninst✝² : NormedSpace 𝕜 F₃\nF₄ : Type u_20\ninst✝¹ : NormedAddCommGroup F₄\ninst✝ : NormedSpace 𝕜 F₄\ne : PartialHomeomorph M H\ne' : PartialHomeomorph M' H'\nf f₁ : M → M'\ns s₁ t : Set M\nx✝ : M\nm n : ℕ∞\nx : M\nhe : e ∈ maximalAtlas I M\nhe' : e' ∈ maximalAtlas I' M'\nhs : s ⊆ e.source\nhx : x ∈ e.source\nhy : f x ∈ e'.source\n⊢ ContMDiffWithinAt I I' n f s x ↔\n    ContinuousWithinAt f s x ∧\n      ContDiffWithinAt 𝕜 n (↑(e'.extend I') ∘ f ∘ ↑(e.extend I).symm) (↑(e.extend I) '' s) (↑(e.extend I) x)","state_after":"𝕜 : Type u_1\ninst✝³⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³⁶ : NormedAddCommGroup E\ninst✝³⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝³⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝³³ : TopologicalSpace M\ninst✝³² : ChartedSpace H M\ninst✝³¹ : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst✝³⁰ : NormedAddCommGroup E'\ninst✝²⁹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝²⁸ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM' : Type u_7\ninst✝²⁷ : TopologicalSpace M'\ninst✝²⁶ : ChartedSpace H' M'\ninst✝²⁵ : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst✝²⁴ : NormedAddCommGroup E''\ninst✝²³ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝²² : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝²¹ : TopologicalSpace M''\ninst✝²⁰ : ChartedSpace H'' M''\nF : Type u_11\ninst✝¹⁹ : NormedAddCommGroup F\ninst✝¹⁸ : NormedSpace 𝕜 F\nG : Type u_12\ninst✝¹⁷ : TopologicalSpace G\nJ : ModelWithCorners 𝕜 F G\nN : Type u_13\ninst✝¹⁶ : TopologicalSpace N\ninst✝¹⁵ : ChartedSpace G N\ninst✝¹⁴ : SmoothManifoldWithCorners J N\nF' : Type u_14\ninst✝¹³ : NormedAddCommGroup F'\ninst✝¹² : NormedSpace 𝕜 F'\nG' : Type u_15\ninst✝¹¹ : TopologicalSpace G'\nJ' : ModelWithCorners 𝕜 F' G'\nN' : Type u_16\ninst✝¹⁰ : TopologicalSpace N'\ninst✝⁹ : ChartedSpace G' N'\ninst✝⁸ : SmoothManifoldWithCorners J' N'\nF₁ : Type u_17\ninst✝⁷ : NormedAddCommGroup F₁\ninst✝⁶ : NormedSpace 𝕜 F₁\nF₂ : Type u_18\ninst✝⁵ : NormedAddCommGroup F₂\ninst✝⁴ : NormedSpace 𝕜 F₂\nF₃ : Type u_19\ninst✝³ : NormedAddCommGroup F₃\ninst✝² : NormedSpace 𝕜 F₃\nF₄ : Type u_20\ninst✝¹ : NormedAddCommGroup F₄\ninst✝ : NormedSpace 𝕜 F₄\ne : PartialHomeomorph M H\ne' : PartialHomeomorph M' H'\nf f₁ : M → M'\ns s₁ t : Set M\nx✝ : M\nm n : ℕ∞\nx : M\nhe : e ∈ maximalAtlas I M\nhe' : e' ∈ maximalAtlas I' M'\nhs : s ⊆ e.source\nhx : x ∈ e.source\nhy : f x ∈ e'.source\n⊢ ContinuousWithinAt f s x →\n    (ContDiffWithinAt 𝕜 n (↑(e'.extend I') ∘ f ∘ ↑(e.extend I).symm) (↑(e.extend I).symm ⁻¹' s ∩ range ↑I)\n        (↑(e.extend I) x) ↔\n      ContDiffWithinAt 𝕜 n (↑(e'.extend I') ∘ f ∘ ↑(e.extend I).symm) (↑(e.extend I) '' s) (↑(e.extend I) x))","tactic":"rw [contMDiffWithinAt_iff_of_mem_maximalAtlas he he' hx hy, and_congr_right_iff]","premises":[{"full_name":"and_congr_right_iff","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[156,16],"def_end_pos":[156,35]},{"full_name":"contMDiffWithinAt_iff_of_mem_maximalAtlas","def_path":"Mathlib/Geometry/Manifold/ContMDiff/Defs.lean","def_pos":[345,8],"def_end_pos":[345,49]}]},{"state_before":"𝕜 : Type u_1\ninst✝³⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³⁶ : NormedAddCommGroup E\ninst✝³⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝³⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝³³ : TopologicalSpace M\ninst✝³² : ChartedSpace H M\ninst✝³¹ : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst✝³⁰ : NormedAddCommGroup E'\ninst✝²⁹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝²⁸ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM' : Type u_7\ninst✝²⁷ : TopologicalSpace M'\ninst✝²⁶ : ChartedSpace H' M'\ninst✝²⁵ : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst✝²⁴ : NormedAddCommGroup E''\ninst✝²³ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝²² : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝²¹ : TopologicalSpace M''\ninst✝²⁰ : ChartedSpace H'' M''\nF : Type u_11\ninst✝¹⁹ : NormedAddCommGroup F\ninst✝¹⁸ : NormedSpace 𝕜 F\nG : Type u_12\ninst✝¹⁷ : TopologicalSpace G\nJ : ModelWithCorners 𝕜 F G\nN : Type u_13\ninst✝¹⁶ : TopologicalSpace N\ninst✝¹⁵ : ChartedSpace G N\ninst✝¹⁴ : SmoothManifoldWithCorners J N\nF' : Type u_14\ninst✝¹³ : NormedAddCommGroup F'\ninst✝¹² : NormedSpace 𝕜 F'\nG' : Type u_15\ninst✝¹¹ : TopologicalSpace G'\nJ' : ModelWithCorners 𝕜 F' G'\nN' : Type u_16\ninst✝¹⁰ : TopologicalSpace N'\ninst✝⁹ : ChartedSpace G' N'\ninst✝⁸ : SmoothManifoldWithCorners J' N'\nF₁ : Type u_17\ninst✝⁷ : NormedAddCommGroup F₁\ninst✝⁶ : NormedSpace 𝕜 F₁\nF₂ : Type u_18\ninst✝⁵ : NormedAddCommGroup F₂\ninst✝⁴ : NormedSpace 𝕜 F₂\nF₃ : Type u_19\ninst✝³ : NormedAddCommGroup F₃\ninst✝² : NormedSpace 𝕜 F₃\nF₄ : Type u_20\ninst✝¹ : NormedAddCommGroup F₄\ninst✝ : NormedSpace 𝕜 F₄\ne : PartialHomeomorph M H\ne' : PartialHomeomorph M' H'\nf f₁ : M → M'\ns s₁ t : Set M\nx✝ : M\nm n : ℕ∞\nx : M\nhe : e ∈ maximalAtlas I M\nhe' : e' ∈ maximalAtlas I' M'\nhs : s ⊆ e.source\nhx : x ∈ e.source\nhy : f x ∈ e'.source\n⊢ ContinuousWithinAt f s x →\n    (ContDiffWithinAt 𝕜 n (↑(e'.extend I') ∘ f ∘ ↑(e.extend I).symm) (↑(e.extend I).symm ⁻¹' s ∩ range ↑I)\n        (↑(e.extend I) x) ↔\n      ContDiffWithinAt 𝕜 n (↑(e'.extend I') ∘ f ∘ ↑(e.extend I).symm) (↑(e.extend I) '' s) (↑(e.extend I) x))","state_after":"𝕜 : Type u_1\ninst✝³⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³⁶ : NormedAddCommGroup E\ninst✝³⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝³⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝³³ : TopologicalSpace M\ninst✝³² : ChartedSpace H M\ninst✝³¹ : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst✝³⁰ : NormedAddCommGroup E'\ninst✝²⁹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝²⁸ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM' : Type u_7\ninst✝²⁷ : TopologicalSpace M'\ninst✝²⁶ : ChartedSpace H' M'\ninst✝²⁵ : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst✝²⁴ : NormedAddCommGroup E''\ninst✝²³ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝²² : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝²¹ : TopologicalSpace M''\ninst✝²⁰ : ChartedSpace H'' M''\nF : Type u_11\ninst✝¹⁹ : NormedAddCommGroup F\ninst✝¹⁸ : NormedSpace 𝕜 F\nG : Type u_12\ninst✝¹⁷ : TopologicalSpace G\nJ : ModelWithCorners 𝕜 F G\nN : Type u_13\ninst✝¹⁶ : TopologicalSpace N\ninst✝¹⁵ : ChartedSpace G N\ninst✝¹⁴ : SmoothManifoldWithCorners J N\nF' : Type u_14\ninst✝¹³ : NormedAddCommGroup F'\ninst✝¹² : NormedSpace 𝕜 F'\nG' : Type u_15\ninst✝¹¹ : TopologicalSpace G'\nJ' : ModelWithCorners 𝕜 F' G'\nN' : Type u_16\ninst✝¹⁰ : TopologicalSpace N'\ninst✝⁹ : ChartedSpace G' N'\ninst✝⁸ : SmoothManifoldWithCorners J' N'\nF₁ : Type u_17\ninst✝⁷ : NormedAddCommGroup F₁\ninst✝⁶ : NormedSpace 𝕜 F₁\nF₂ : Type u_18\ninst✝⁵ : NormedAddCommGroup F₂\ninst✝⁴ : NormedSpace 𝕜 F₂\nF₃ : Type u_19\ninst✝³ : NormedAddCommGroup F₃\ninst✝² : NormedSpace 𝕜 F₃\nF₄ : Type u_20\ninst✝¹ : NormedAddCommGroup F₄\ninst✝ : NormedSpace 𝕜 F₄\ne : PartialHomeomorph M H\ne' : PartialHomeomorph M' H'\nf f₁ : M → M'\ns s₁ t : Set M\nx✝¹ : M\nm n : ℕ∞\nx : M\nhe : e ∈ maximalAtlas I M\nhe' : e' ∈ maximalAtlas I' M'\nhs : s ⊆ e.source\nhx : x ∈ e.source\nhy : f x ∈ e'.source\nx✝ : ContinuousWithinAt f s x\n⊢ 𝓝[↑(e.extend I).symm ⁻¹' s ∩ range ↑I] ↑(e.extend I) x = 𝓝[↑(e.extend I) '' s] ↑(e.extend I) x","tactic":"refine fun _ => contDiffWithinAt_congr_nhds ?_","premises":[{"full_name":"contDiffWithinAt_congr_nhds","def_path":"Mathlib/Analysis/Calculus/ContDiff/Defs.lean","def_pos":[474,8],"def_end_pos":[474,35]}]},{"state_before":"𝕜 : Type u_1\ninst✝³⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³⁶ : NormedAddCommGroup E\ninst✝³⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝³⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝³³ : TopologicalSpace M\ninst✝³² : ChartedSpace H M\ninst✝³¹ : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst✝³⁰ : NormedAddCommGroup E'\ninst✝²⁹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝²⁸ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM' : Type u_7\ninst✝²⁷ : TopologicalSpace M'\ninst✝²⁶ : ChartedSpace H' M'\ninst✝²⁵ : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst✝²⁴ : NormedAddCommGroup E''\ninst✝²³ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝²² : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝²¹ : TopologicalSpace M''\ninst✝²⁰ : ChartedSpace H'' M''\nF : Type u_11\ninst✝¹⁹ : NormedAddCommGroup F\ninst✝¹⁸ : NormedSpace 𝕜 F\nG : Type u_12\ninst✝¹⁷ : TopologicalSpace G\nJ : ModelWithCorners 𝕜 F G\nN : Type u_13\ninst✝¹⁶ : TopologicalSpace N\ninst✝¹⁵ : ChartedSpace G N\ninst✝¹⁴ : SmoothManifoldWithCorners J N\nF' : Type u_14\ninst✝¹³ : NormedAddCommGroup F'\ninst✝¹² : NormedSpace 𝕜 F'\nG' : Type u_15\ninst✝¹¹ : TopologicalSpace G'\nJ' : ModelWithCorners 𝕜 F' G'\nN' : Type u_16\ninst✝¹⁰ : TopologicalSpace N'\ninst✝⁹ : ChartedSpace G' N'\ninst✝⁸ : SmoothManifoldWithCorners J' N'\nF₁ : Type u_17\ninst✝⁷ : NormedAddCommGroup F₁\ninst✝⁶ : NormedSpace 𝕜 F₁\nF₂ : Type u_18\ninst✝⁵ : NormedAddCommGroup F₂\ninst✝⁴ : NormedSpace 𝕜 F₂\nF₃ : Type u_19\ninst✝³ : NormedAddCommGroup F₃\ninst✝² : NormedSpace 𝕜 F₃\nF₄ : Type u_20\ninst✝¹ : NormedAddCommGroup F₄\ninst✝ : NormedSpace 𝕜 F₄\ne : PartialHomeomorph M H\ne' : PartialHomeomorph M' H'\nf f₁ : M → M'\ns s₁ t : Set M\nx✝¹ : M\nm n : ℕ∞\nx : M\nhe : e ∈ maximalAtlas I M\nhe' : e' ∈ maximalAtlas I' M'\nhs : s ⊆ e.source\nhx : x ∈ e.source\nhy : f x ∈ e'.source\nx✝ : ContinuousWithinAt f s x\n⊢ 𝓝[↑(e.extend I).symm ⁻¹' s ∩ range ↑I] ↑(e.extend I) x = 𝓝[↑(e.extend I) '' s] ↑(e.extend I) x","state_after":"no goals","tactic":"simp_rw [nhdsWithin_eq_iff_eventuallyEq, e.extend_symm_preimage_inter_range_eventuallyEq I hs hx]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"PartialHomeomorph.extend_symm_preimage_inter_range_eventuallyEq","def_path":"Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean","def_pos":[965,8],"def_end_pos":[965,53]},{"full_name":"nhdsWithin_eq_iff_eventuallyEq","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[107,8],"def_end_pos":[107,38]}]}]}
+{"url":"Mathlib/Data/Finmap.lean","commit":"","full_name":"Finmap.lookup_union_left_of_not_in","start":[514,0],"end":[518,75],"file_path":"Mathlib/Data/Finmap.lean","tactics":[{"state_before":"α : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\na : α\ns₁ s₂ : Finmap β\nh : a ∉ s₂\n⊢ lookup a (s₁ ∪ s₂) = lookup a s₁","state_after":"case pos\nα : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\na : α\ns₁ s₂ : Finmap β\nh : a ∉ s₂\nh' : a ∈ s₁\n⊢ lookup a (s₁ ∪ s₂) = lookup a s₁\n\ncase neg\nα : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\na : α\ns₁ s₂ : Finmap β\nh : a ∉ s₂\nh' : a ∉ s₁\n⊢ lookup a (s₁ ∪ s₂) = lookup a s₁","tactic":"by_cases h' : a ∈ s₁","premises":[{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
+{"url":"Mathlib/Algebra/Homology/TotalComplexShift.lean","commit":"","full_name":"HomologicalComplex₂.totalShift₁Iso_hom_naturality","start":[218,0],"end":[228,53],"file_path":"Mathlib/Algebra/Homology/TotalComplexShift.lean","tactics":[{"state_before":"C : Type u_1\ninst✝³ : Category.{u_2, u_1} C\ninst✝² : Preadditive C\nK L : HomologicalComplex₂ C (up ℤ) (up ℤ)\nf : K ⟶ L\nx y : ℤ\ninst✝¹ : K.HasTotal (up ℤ)\ninst✝ : L.HasTotal (up ℤ)\n⊢ total.map ((shiftFunctor₁ C x).map f) (up ℤ) ≫ (L.totalShift₁Iso x).hom =\n    (K.totalShift₁Iso x).hom ≫ (shiftFunctor (HomologicalComplex C (up ℤ)) x).map (total.map f (up ℤ))","state_after":"case h.h\nC : Type u_1\ninst✝³ : Category.{u_2, u_1} C\ninst✝² : Preadditive C\nK L : HomologicalComplex₂ C (up ℤ) (up ℤ)\nf : K ⟶ L\nx y : ℤ\ninst✝¹ : K.HasTotal (up ℤ)\ninst✝ : L.HasTotal (up ℤ)\nn i₁ i₂ : ℤ\nh : (up ℤ).π (up ℤ) (up ℤ) (i₁, i₂) = n\n⊢ ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) i₁ i₂ n h ≫\n      (total.map ((shiftFunctor₁ C x).map f) (up ℤ) ≫ (L.totalShift₁Iso x).hom).f n =\n    ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) i₁ i₂ n h ≫\n      ((K.totalShift₁Iso x).hom ≫ (shiftFunctor (HomologicalComplex C (up ℤ)) x).map (total.map f (up ℤ))).f n","tactic":"ext n i₁ i₂ h","premises":[]},{"state_before":"case h.h\nC : Type u_1\ninst✝³ : Category.{u_2, u_1} C\ninst✝² : Preadditive C\nK L : HomologicalComplex₂ C (up ℤ) (up ℤ)\nf : K ⟶ L\nx y : ℤ\ninst✝¹ : K.HasTotal (up ℤ)\ninst✝ : L.HasTotal (up ℤ)\nn i₁ i₂ : ℤ\nh : (up ℤ).π (up ℤ) (up ℤ) (i₁, i₂) = n\n⊢ ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) i₁ i₂ n h ≫\n      (total.map ((shiftFunctor₁ C x).map f) (up ℤ) ≫ (L.totalShift₁Iso x).hom).f n =\n    ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) i₁ i₂ n h ≫\n      ((K.totalShift₁Iso x).hom ≫ (shiftFunctor (HomologicalComplex C (up ℤ)) x).map (total.map f (up ℤ))).f n","state_after":"case h.h\nC : Type u_1\ninst✝³ : Category.{u_2, u_1} C\ninst✝² : Preadditive C\nK L : HomologicalComplex₂ C (up ℤ) (up ℤ)\nf : K ⟶ L\nx y : ℤ\ninst✝¹ : K.HasTotal (up ℤ)\ninst✝ : L.HasTotal (up ℤ)\nn i₁ i₂ : ℤ\nh : i₁ + i₂ = n\n⊢ ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) i₁ i₂ n h ≫\n      (total.map ((shiftFunctor₁ C x).map f) (up ℤ) ≫ (L.totalShift₁Iso x).hom).f n =\n    ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) i₁ i₂ n h ≫\n      ((K.totalShift₁Iso x).hom ≫ (shiftFunctor (HomologicalComplex C (up ℤ)) x).map (total.map f (up ℤ))).f n","tactic":"dsimp at h","premises":[]},{"state_before":"case h.h\nC : Type u_1\ninst✝³ : Category.{u_2, u_1} C\ninst✝² : Preadditive C\nK L : HomologicalComplex₂ C (up ℤ) (up ℤ)\nf : K ⟶ L\nx y : ℤ\ninst✝¹ : K.HasTotal (up ℤ)\ninst✝ : L.HasTotal (up ℤ)\nn i₁ i₂ : ℤ\nh : i₁ + i₂ = n\n⊢ ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) i₁ i₂ n h ≫\n      (total.map ((shiftFunctor₁ C x).map f) (up ℤ) ≫ (L.totalShift₁Iso x).hom).f n =\n    ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) i₁ i₂ n h ≫\n      ((K.totalShift₁Iso x).hom ≫ (shiftFunctor (HomologicalComplex C (up ℤ)) x).map (total.map f (up ℤ))).f n","state_after":"case h.h\nC : Type u_1\ninst✝³ : Category.{u_2, u_1} C\ninst✝² : Preadditive C\nK L : HomologicalComplex₂ C (up ℤ) (up ℤ)\nf : K ⟶ L\nx y : ℤ\ninst✝¹ : K.HasTotal (up ℤ)\ninst✝ : L.HasTotal (up ℤ)\nn i₁ i₂ : ℤ\nh : i₁ + i₂ = n\n⊢ ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) i₁ i₂ n h ≫\n      (total.map ((shiftFunctor₁ C x).map f) (up ℤ)).f n ≫ (L.totalShift₁Iso x).hom.f n =\n    ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) i₁ i₂ n h ≫ (K.totalShift₁Iso x).hom.f n ≫ (total.map f (up ℤ)).f (n + x)","tactic":"dsimp","premises":[]},{"state_before":"case h.h\nC : Type u_1\ninst✝³ : Category.{u_2, u_1} C\ninst✝² : Preadditive C\nK L : HomologicalComplex₂ C (up ℤ) (up ℤ)\nf : K ⟶ L\nx y : ℤ\ninst✝¹ : K.HasTotal (up ℤ)\ninst✝ : L.HasTotal (up ℤ)\nn i₁ i₂ : ℤ\nh : i₁ + i₂ = n\n⊢ ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) i₁ i₂ n h ≫\n      (total.map ((shiftFunctor₁ C x).map f) (up ℤ)).f n ≫ (L.totalShift₁Iso x).hom.f n =\n    ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) i₁ i₂ n h ≫ (K.totalShift₁Iso x).hom.f n ≫ (total.map f (up ℤ)).f (n + x)","state_after":"case h.h\nC : Type u_1\ninst✝³ : Category.{u_2, u_1} C\ninst✝² : Preadditive C\nK L : HomologicalComplex₂ C (up ℤ) (up ℤ)\nf : K ⟶ L\nx y : ℤ\ninst✝¹ : K.HasTotal (up ℤ)\ninst✝ : L.HasTotal (up ℤ)\nn i₁ i₂ : ℤ\nh : i₁ + i₂ = n\n⊢ (((shiftFunctor₁ C x).map f).f i₁).f i₂ ≫\n      (L.shiftFunctor₁XXIso i₁ x (i₁ + x) ⋯ i₂).hom ≫\n        L.ιTotal (up ℤ) (i₁ + x) i₂ (n + x) ⋯ ≫\n          (CochainComplex.shiftFunctorObjXIso (L.total (up ℤ)) x n (n + x) ⋯).inv =\n    (K.shiftFunctor₁XXIso i₁ x (i₁ + x) ⋯ i₂).hom ≫\n      K.ιTotal (up ℤ) (i₁ + x) i₂ (n + x) ⋯ ≫\n        (CochainComplex.shiftFunctorObjXIso (K.total (up ℤ)) x n (n + x) ⋯).inv ≫ (total.map f (up ℤ)).f (n + x)","tactic":"rw [ιTotal_map_assoc, L.ι_totalShift₁Iso_hom_f x i₁ i₂ n h _ rfl _ rfl,\n    K.ι_totalShift₁Iso_hom_f_assoc x i₁ i₂ n h _ rfl _ rfl]","premises":[{"full_name":"HomologicalComplex₂.ι_totalShift₁Iso_hom_f","def_path":"Mathlib/Algebra/Homology/TotalComplexShift.lean","def_pos":[196,6],"def_end_pos":[196,28]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"case h.h\nC : Type u_1\ninst✝³ : Category.{u_2, u_1} C\ninst✝² : Preadditive C\nK L : HomologicalComplex₂ C (up ℤ) (up ℤ)\nf : K ⟶ L\nx y : ℤ\ninst✝¹ : K.HasTotal (up ℤ)\ninst✝ : L.HasTotal (up ℤ)\nn i₁ i₂ : ℤ\nh : i₁ + i₂ = n\n⊢ (((shiftFunctor₁ C x).map f).f i₁).f i₂ ≫\n      (L.shiftFunctor₁XXIso i₁ x (i₁ + x) ⋯ i₂).hom ≫\n        L.ιTotal (up ℤ) (i₁ + x) i₂ (n + x) ⋯ ≫\n          (CochainComplex.shiftFunctorObjXIso (L.total (up ℤ)) x n (n + x) ⋯).inv =\n    (K.shiftFunctor₁XXIso i₁ x (i₁ + x) ⋯ i₂).hom ≫\n      K.ιTotal (up ℤ) (i₁ + x) i₂ (n + x) ⋯ ≫\n        (CochainComplex.shiftFunctorObjXIso (K.total (up ℤ)) x n (n + x) ⋯).inv ≫ (total.map f (up ℤ)).f (n + x)","state_after":"case h.h\nC : Type u_1\ninst✝³ : Category.{u_2, u_1} C\ninst✝² : Preadditive C\nK L : HomologicalComplex₂ C (up ℤ) (up ℤ)\nf : K ⟶ L\nx y : ℤ\ninst✝¹ : K.HasTotal (up ℤ)\ninst✝ : L.HasTotal (up ℤ)\nn i₁ i₂ : ℤ\nh : i₁ + i₂ = n\n⊢ (f.f (i₁ + x)).f i₂ ≫\n      𝟙 ((L.X (i₁ + x)).X i₂) ≫ L.ιTotal (up ℤ) (i₁ + x) i₂ (n + x) ⋯ ≫ 𝟙 ((L.total (up ℤ)).X (n + x)) =\n    𝟙 ((K.X (i₁ + x)).X i₂) ≫\n      K.ιTotal (up ℤ) (i₁ + x) i₂ (n + x) ⋯ ≫ 𝟙 ((K.total (up ℤ)).X (n + x)) ≫ (total.map f (up ℤ)).f (n + x)","tactic":"dsimp","premises":[]},{"state_before":"case h.h\nC : Type u_1\ninst✝³ : Category.{u_2, u_1} C\ninst✝² : Preadditive C\nK L : HomologicalComplex₂ C (up ℤ) (up ℤ)\nf : K ⟶ L\nx y : ℤ\ninst✝¹ : K.HasTotal (up ℤ)\ninst✝ : L.HasTotal (up ℤ)\nn i₁ i₂ : ℤ\nh : i₁ + i₂ = n\n⊢ (f.f (i₁ + x)).f i₂ ≫\n      𝟙 ((L.X (i₁ + x)).X i₂) ≫ L.ιTotal (up ℤ) (i₁ + x) i₂ (n + x) ⋯ ≫ 𝟙 ((L.total (up ℤ)).X (n + x)) =\n    𝟙 ((K.X (i₁ + x)).X i₂) ≫\n      K.ιTotal (up ℤ) (i₁ + x) i₂ (n + x) ⋯ ≫ 𝟙 ((K.total (up ℤ)).X (n + x)) ≫ (total.map f (up ℤ)).f (n + x)","state_after":"no goals","tactic":"rw [id_comp, id_comp, id_comp, comp_id, ιTotal_map]","premises":[{"full_name":"CategoryTheory.Category.comp_id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[153,2],"def_end_pos":[153,9]},{"full_name":"CategoryTheory.Category.id_comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[151,2],"def_end_pos":[151,9]},{"full_name":"HomologicalComplex₂.ιTotal_map","def_path":"Mathlib/Algebra/Homology/TotalComplex.lean","def_pos":[429,6],"def_end_pos":[429,16]}]}]}
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+{"url":"Mathlib/MeasureTheory/Function/L1Space.lean","commit":"","full_name":"MeasureTheory.hasFiniteIntegral_smul_iff","start":[365,0],"end":[371,32],"file_path":"Mathlib/MeasureTheory/Function/L1Space.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁵ : MeasurableSpace δ\ninst✝⁴ : NormedAddCommGroup β\ninst✝³ : NormedAddCommGroup γ\n𝕜 : Type u_5\ninst✝² : NormedRing 𝕜\ninst✝¹ : MulActionWithZero 𝕜 β\ninst✝ : BoundedSMul 𝕜 β\nc : 𝕜\nhc : IsUnit c\nf : α → β\n⊢ HasFiniteIntegral (c • f) μ ↔ HasFiniteIntegral f μ","state_after":"case intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁵ : MeasurableSpace δ\ninst✝⁴ : NormedAddCommGroup β\ninst✝³ : NormedAddCommGroup γ\n𝕜 : Type u_5\ninst✝² : NormedRing 𝕜\ninst✝¹ : MulActionWithZero 𝕜 β\ninst✝ : BoundedSMul 𝕜 β\nf : α → β\nc : 𝕜ˣ\n⊢ HasFiniteIntegral (↑c • f) μ ↔ HasFiniteIntegral f μ","tactic":"obtain ⟨c, rfl⟩ := hc","premises":[]},{"state_before":"case intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁵ : MeasurableSpace δ\ninst✝⁴ : NormedAddCommGroup β\ninst✝³ : NormedAddCommGroup γ\n𝕜 : Type u_5\ninst✝² : NormedRing 𝕜\ninst✝¹ : MulActionWithZero 𝕜 β\ninst✝ : BoundedSMul 𝕜 β\nf : α → β\nc : 𝕜ˣ\n⊢ HasFiniteIntegral (↑c • f) μ ↔ HasFiniteIntegral f μ","state_after":"case intro.mp\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁵ : MeasurableSpace δ\ninst✝⁴ : NormedAddCommGroup β\ninst✝³ : NormedAddCommGroup γ\n𝕜 : Type u_5\ninst✝² : NormedRing 𝕜\ninst✝¹ : MulActionWithZero 𝕜 β\ninst✝ : BoundedSMul 𝕜 β\nf : α → β\nc : 𝕜ˣ\n⊢ HasFiniteIntegral (↑c • f) μ → HasFiniteIntegral f μ\n\ncase intro.mpr\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁵ : MeasurableSpace δ\ninst✝⁴ : NormedAddCommGroup β\ninst✝³ : NormedAddCommGroup γ\n𝕜 : Type u_5\ninst✝² : NormedRing 𝕜\ninst✝¹ : MulActionWithZero 𝕜 β\ninst✝ : BoundedSMul 𝕜 β\nf : α → β\nc : 𝕜ˣ\n⊢ HasFiniteIntegral f μ → HasFiniteIntegral (↑c • f) μ","tactic":"constructor","premises":[]},{"state_before":"case intro.mpr\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁵ : MeasurableSpace δ\ninst✝⁴ : NormedAddCommGroup β\ninst✝³ : NormedAddCommGroup γ\n𝕜 : Type u_5\ninst✝² : NormedRing 𝕜\ninst✝¹ : MulActionWithZero 𝕜 β\ninst✝ : BoundedSMul 𝕜 β\nf : α → β\nc : 𝕜ˣ\n⊢ HasFiniteIntegral f μ → HasFiniteIntegral (↑c • f) μ","state_after":"no goals","tactic":"exact HasFiniteIntegral.smul _","premises":[{"full_name":"MeasureTheory.HasFiniteIntegral.smul","def_path":"Mathlib/MeasureTheory/Function/L1Space.lean","def_pos":[351,8],"def_end_pos":[351,30]}]}]}
+{"url":"Mathlib/RingTheory/DedekindDomain/PID.lean","commit":"","full_name":"IsDedekindDomain.isPrincipalIdealRing_localization_over_prime","start":[238,0],"end":[254,100],"file_path":"Mathlib/RingTheory/DedekindDomain/PID.lean","tactics":[{"state_before":"R : Type u_1\ninst✝¹³ : CommRing R\ninst✝¹² : IsDedekindDomain R\nS : Type u_2\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R S\ninst✝⁹ : Module.Free R S\ninst✝⁸ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁷ : p.IsPrime\nSₚ : Type u_3\ninst✝⁶ : CommRing Sₚ\ninst✝⁵ : Algebra S Sₚ\ninst✝⁴ : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ\ninst✝³ : Algebra R Sₚ\ninst✝² : IsScalarTower R S Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : IsDomain S\n⊢ IsPrincipalIdealRing Sₚ","state_after":"R : Type u_1\ninst✝¹³ : CommRing R\ninst✝¹² : IsDedekindDomain R\nS : Type u_2\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R S\ninst✝⁹ : Module.Free R S\ninst✝⁸ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁷ : p.IsPrime\nSₚ : Type u_3\ninst✝⁶ : CommRing Sₚ\ninst✝⁵ : Algebra S Sₚ\ninst✝⁴ : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ\ninst✝³ : Algebra R Sₚ\ninst✝² : IsScalarTower R S Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : IsDomain S\nthis : DecidableEq (Ideal Sₚ) := Classical.decEq (Ideal Sₚ)\n⊢ IsPrincipalIdealRing Sₚ","tactic":"letI := Classical.decEq (Ideal Sₚ)","premises":[{"full_name":"Classical.decEq","def_path":"Mathlib/Logic/Basic.lean","def_pos":[737,18],"def_end_pos":[737,23]},{"full_name":"Ideal","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[39,7],"def_end_pos":[39,12]}]},{"state_before":"R : Type u_1\ninst✝¹³ : CommRing R\ninst✝¹² : IsDedekindDomain R\nS : Type u_2\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R S\ninst✝⁹ : Module.Free R S\ninst✝⁸ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁷ : p.IsPrime\nSₚ : Type u_3\ninst✝⁶ : CommRing Sₚ\ninst✝⁵ : Algebra S Sₚ\ninst✝⁴ : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ\ninst✝³ : Algebra R Sₚ\ninst✝² : IsScalarTower R S Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : IsDomain S\nthis : DecidableEq (Ideal Sₚ) := Classical.decEq (Ideal Sₚ)\n⊢ IsPrincipalIdealRing Sₚ","state_after":"R : Type u_1\ninst✝¹³ : CommRing R\ninst✝¹² : IsDedekindDomain R\nS : Type u_2\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R S\ninst✝⁹ : Module.Free R S\ninst✝⁸ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁷ : p.IsPrime\nSₚ : Type u_3\ninst✝⁶ : CommRing Sₚ\ninst✝⁵ : Algebra S Sₚ\ninst✝⁴ : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ\ninst✝³ : Algebra R Sₚ\ninst✝² : IsScalarTower R S Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : IsDomain S\nthis✝ : DecidableEq (Ideal Sₚ) := Classical.decEq (Ideal Sₚ)\nthis : DecidablePred fun P => P.IsPrime := Classical.decPred fun P => P.IsPrime\n⊢ IsPrincipalIdealRing Sₚ","tactic":"letI := Classical.decPred fun P : Ideal Sₚ => P.IsPrime","premises":[{"full_name":"Classical.decPred","def_path":"Mathlib/Logic/Basic.lean","def_pos":[731,18],"def_end_pos":[731,25]},{"full_name":"Ideal","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[39,7],"def_end_pos":[39,12]},{"full_name":"Ideal.IsPrime","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[203,6],"def_end_pos":[203,13]}]},{"state_before":"R : Type u_1\ninst✝¹³ : CommRing R\ninst✝¹² : IsDedekindDomain R\nS : Type u_2\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R S\ninst✝⁹ : Module.Free R S\ninst✝⁸ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁷ : p.IsPrime\nSₚ : Type u_3\ninst✝⁶ : CommRing Sₚ\ninst✝⁵ : Algebra S Sₚ\ninst✝⁴ : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ\ninst✝³ : Algebra R Sₚ\ninst✝² : IsScalarTower R S Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : IsDomain S\nthis✝ : DecidableEq (Ideal Sₚ) := Classical.decEq (Ideal Sₚ)\nthis : DecidablePred fun P => P.IsPrime := Classical.decPred fun P => P.IsPrime\n⊢ IsPrincipalIdealRing Sₚ","state_after":"R : Type u_1\ninst✝¹³ : CommRing R\ninst✝¹² : IsDedekindDomain R\nS : Type u_2\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R S\ninst✝⁹ : Module.Free R S\ninst✝⁸ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁷ : p.IsPrime\nSₚ : Type u_3\ninst✝⁶ : CommRing Sₚ\ninst✝⁵ : Algebra S Sₚ\ninst✝⁴ : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ\ninst✝³ : Algebra R Sₚ\ninst✝² : IsScalarTower R S Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : IsDomain S\nthis✝ : DecidableEq (Ideal Sₚ) := Classical.decEq (Ideal Sₚ)\nthis : DecidablePred fun P => P.IsPrime := Classical.decPred fun P => P.IsPrime\nP : Ideal Sₚ\n⊢ P ∈ Finset.filter (fun P => P.IsPrime) ({⊥} ∪ (normalizedFactors (map (algebraMap R Sₚ) p)).toFinset) ↔\n    P ∈ {I | I.IsPrime}","tactic":"refine\n    IsPrincipalIdealRing.of_finite_primes\n      (Set.Finite.ofFinset\n        (Finset.filter (fun P => P.IsPrime)\n          ({⊥} ∪ (normalizedFactors (Ideal.map (algebraMap R Sₚ) p)).toFinset))\n        fun P => ?_)","premises":[{"full_name":"Bot.bot","def_path":"Mathlib/Order/Notation.lean","def_pos":[100,2],"def_end_pos":[100,5]},{"full_name":"Finset.filter","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2144,4],"def_end_pos":[2144,10]},{"full_name":"Ideal.IsPrime","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[203,6],"def_end_pos":[203,13]},{"full_name":"Ideal.map","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[34,4],"def_end_pos":[34,7]},{"full_name":"IsPrincipalIdealRing.of_finite_primes","def_path":"Mathlib/RingTheory/DedekindDomain/PID.lean","def_pos":[176,8],"def_end_pos":[176,45]},{"full_name":"Multiset.toFinset","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2571,4],"def_end_pos":[2571,12]},{"full_name":"Set.Finite.ofFinset","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[74,18],"def_end_pos":[74,33]},{"full_name":"Singleton.singleton","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[467,2],"def_end_pos":[467,11]},{"full_name":"Union.union","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[402,2],"def_end_pos":[402,7]},{"full_name":"UniqueFactorizationMonoid.normalizedFactors","def_path":"Mathlib/RingTheory/UniqueFactorizationDomain.lean","def_pos":[550,18],"def_end_pos":[550,35]},{"full_name":"algebraMap","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[108,4],"def_end_pos":[108,14]}]},{"state_before":"R : Type u_1\ninst✝¹³ : CommRing R\ninst✝¹² : IsDedekindDomain R\nS : Type u_2\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R S\ninst✝⁹ : Module.Free R S\ninst✝⁸ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁷ : p.IsPrime\nSₚ : Type u_3\ninst✝⁶ : CommRing Sₚ\ninst✝⁵ : Algebra S Sₚ\ninst✝⁴ : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ\ninst✝³ : Algebra R Sₚ\ninst✝² : IsScalarTower R S Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : IsDomain S\nthis✝ : DecidableEq (Ideal Sₚ) := Classical.decEq (Ideal Sₚ)\nthis : DecidablePred fun P => P.IsPrime := Classical.decPred fun P => P.IsPrime\nP : Ideal Sₚ\n⊢ P ∈ Finset.filter (fun P => P.IsPrime) ({⊥} ∪ (normalizedFactors (map (algebraMap R Sₚ) p)).toFinset) ↔\n    P ∈ {I | I.IsPrime}","state_after":"R : Type u_1\ninst✝¹³ : CommRing R\ninst✝¹² : IsDedekindDomain R\nS : Type u_2\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R S\ninst✝⁹ : Module.Free R S\ninst✝⁸ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁷ : p.IsPrime\nSₚ : Type u_3\ninst✝⁶ : CommRing Sₚ\ninst✝⁵ : Algebra S Sₚ\ninst✝⁴ : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ\ninst✝³ : Algebra R Sₚ\ninst✝² : IsScalarTower R S Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : IsDomain S\nthis✝ : DecidableEq (Ideal Sₚ) := Classical.decEq (Ideal Sₚ)\nthis : DecidablePred fun P => P.IsPrime := Classical.decPred fun P => P.IsPrime\nP : Ideal Sₚ\n⊢ (P = ⊥ ∨ P ∈ normalizedFactors (map (algebraMap R Sₚ) p)) ∧ P.IsPrime ↔ P.IsPrime","tactic":"rw [Finset.mem_filter, Finset.mem_union, Finset.mem_singleton, Set.mem_setOf,\n    Multiset.mem_toFinset]","premises":[{"full_name":"Finset.mem_filter","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2158,8],"def_end_pos":[2158,18]},{"full_name":"Finset.mem_singleton","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[584,8],"def_end_pos":[584,21]},{"full_name":"Finset.mem_union","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[1191,8],"def_end_pos":[1191,17]},{"full_name":"Multiset.mem_toFinset","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2586,8],"def_end_pos":[2586,20]},{"full_name":"Set.mem_setOf","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[202,8],"def_end_pos":[202,17]}]},{"state_before":"R : Type u_1\ninst✝¹³ : CommRing R\ninst✝¹² : IsDedekindDomain R\nS : Type u_2\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R S\ninst✝⁹ : Module.Free R S\ninst✝⁸ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁷ : p.IsPrime\nSₚ : Type u_3\ninst✝⁶ : CommRing Sₚ\ninst✝⁵ : Algebra S Sₚ\ninst✝⁴ : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ\ninst✝³ : Algebra R Sₚ\ninst✝² : IsScalarTower R S Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : IsDomain S\nthis✝ : DecidableEq (Ideal Sₚ) := Classical.decEq (Ideal Sₚ)\nthis : DecidablePred fun P => P.IsPrime := Classical.decPred fun P => P.IsPrime\nP : Ideal Sₚ\n⊢ (P = ⊥ ∨ P ∈ normalizedFactors (map (algebraMap R Sₚ) p)) ∧ P.IsPrime ↔ P.IsPrime","state_after":"no goals","tactic":"exact\n    and_iff_right_of_imp fun hP =>\n      or_iff_not_imp_left.mpr (IsLocalization.OverPrime.mem_normalizedFactors_of_isPrime S p hp0 hP)","premises":[{"full_name":"Classical.or_iff_not_imp_left","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[147,8],"def_end_pos":[147,27]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"IsLocalization.OverPrime.mem_normalizedFactors_of_isPrime","def_path":"Mathlib/RingTheory/DedekindDomain/PID.lean","def_pos":[199,8],"def_end_pos":[199,65]},{"full_name":"and_iff_right_of_imp","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[162,8],"def_end_pos":[162,28]}]}]}
+{"url":"Mathlib/Algebra/ContinuedFractions/Computation/TerminatesIffRat.lean","commit":"","full_name":"GenContFract.IntFractPair.coe_of_rat_eq","start":[157,0],"end":[158,32],"file_path":"Mathlib/Algebra/ContinuedFractions/Computation/TerminatesIffRat.lean","tactics":[{"state_before":"K : Type u_1\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nv : K\nq : ℚ\nv_eq_q : v = ↑q\nn : ℕ\n⊢ mapFr Rat.cast (IntFractPair.of q) = IntFractPair.of v","state_after":"no goals","tactic":"simp [IntFractPair.of, v_eq_q]","premises":[{"full_name":"GenContFract.IntFractPair.of","def_path":"Mathlib/Algebra/ContinuedFractions/Computation/Basic.lean","def_pos":[122,14],"def_end_pos":[122,16]}]}]}
+{"url":"Mathlib/Algebra/Order/Antidiag/Pi.lean","commit":"","full_name":"Finset.map_nsmul_piAntidiag","start":[227,0],"end":[231,60],"file_path":"Mathlib/Algebra/Order/Antidiag/Pi.lean","tactics":[{"state_before":"ι : Type u_1\nμ : Type u_2\nμ' : Type u_3\ninst✝ : DecidableEq ι\ns : Finset ι\nm n : ℕ\nhn : n ≠ 0\n⊢ map { toFun := fun x => n • x, inj' := ⋯ } (s.piAntidiag m) =\n    filter (fun f => ∀ i ∈ s, n ∣ f i) (s.piAntidiag (n * m))","state_after":"no goals","tactic":"classical rw [map_eq_image]; exact nsmul_piAntidiag _ _ hn","premises":[{"full_name":"Finset.map_eq_image","def_path":"Mathlib/Data/Finset/Image.lean","def_pos":[312,8],"def_end_pos":[312,20]},{"full_name":"Finset.nsmul_piAntidiag","def_path":"Mathlib/Algebra/Order/Antidiag/Pi.lean","def_pos":[209,6],"def_end_pos":[209,22]}]}]}
+{"url":"Mathlib/CategoryTheory/Subobject/Lattice.lean","commit":"","full_name":"CategoryTheory.MonoOver.inf_map_app","start":[122,0],"end":[137,50],"file_path":"Mathlib/CategoryTheory/Subobject/Lattice.lean","tactics":[{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nX Y Z : C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : HasPullbacks C\nA : C\nX✝ Y✝ : MonoOver A\nk : X✝ ⟶ Y✝\ng : MonoOver A\n⊢ ((fun f => pullback f.arrow ⋙ map f.arrow) X✝).obj g ⟶ ((fun f => pullback f.arrow ⋙ map f.arrow) Y✝).obj g","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nX Y Z : C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : HasPullbacks C\nA : C\nX✝ Y✝ : MonoOver A\nk : X✝ ⟶ Y✝\ng : MonoOver A\n⊢ (((fun f => pullback f.arrow ⋙ map f.arrow) X✝).obj g).obj.left ⟶\n    (((fun f => pullback f.arrow ⋙ map f.arrow) Y✝).obj g).obj.left\n\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nX Y Z : C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : HasPullbacks C\nA : C\nX✝ Y✝ : MonoOver A\nk : X✝ ⟶ Y✝\ng : MonoOver A\n⊢ ?m.18614 ≫ (((fun f => pullback f.arrow ⋙ map f.arrow) Y✝).obj g).arrow =\n    (((fun f => pullback f.arrow ⋙ map f.arrow) X✝).obj g).arrow","tactic":"apply homMk _ _","premises":[{"full_name":"CategoryTheory.MonoOver.homMk","def_path":"Mathlib/CategoryTheory/Subobject/MonoOver.lean","def_pos":[118,7],"def_end_pos":[118,12]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nX Y Z : C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : HasPullbacks C\nA : C\nX✝ Y✝ : MonoOver A\nk : X✝ ⟶ Y✝\ng : MonoOver A\n⊢ pullback.lift (pullback.fst ((forget A).obj g).hom X✝.arrow) (pullback.snd ((forget A).obj g).hom X✝.arrow ≫ k.left)\n        ⋯ ≫\n      (((fun f => pullback f.arrow ⋙ map f.arrow) Y✝).obj g).arrow =\n    (((fun f => pullback f.arrow ⋙ map f.arrow) X✝).obj g).arrow","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nX Y Z : C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : HasPullbacks C\nA : C\nX✝ Y✝ : MonoOver A\nk : X✝ ⟶ Y✝\ng : MonoOver A\n⊢ pullback.lift (pullback.fst g.arrow X✝.arrow) (pullback.snd g.arrow X✝.arrow ≫ k.left) ⋯ ≫\n      pullback.snd g.arrow Y✝.arrow ≫ Y✝.arrow =\n    pullback.snd g.arrow X✝.arrow ≫ X✝.arrow","tactic":"dsimp","premises":[]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nX Y Z : C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : HasPullbacks C\nA : C\nX✝ Y✝ : MonoOver A\nk : X✝ ⟶ Y✝\ng : MonoOver A\n⊢ pullback.lift (pullback.fst g.arrow X✝.arrow) (pullback.snd g.arrow X✝.arrow ≫ k.left) ⋯ ≫\n      pullback.snd g.arrow Y✝.arrow ≫ Y✝.arrow =\n    pullback.snd g.arrow X✝.arrow ≫ X✝.arrow","state_after":"no goals","tactic":"rw [pullback.lift_snd_assoc, assoc, w k]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.MonoOver.w","def_path":"Mathlib/CategoryTheory/Subobject/MonoOver.lean","def_pos":[114,8],"def_end_pos":[114,9]}]}]}
+{"url":"Mathlib/Data/List/Forall2.lean","commit":"","full_name":"List.forall₂_drop_append","start":[204,0],"end":[208,23],"file_path":"Mathlib/Data/List/Forall2.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl : List α\nl₁ l₂ : List β\nh : Forall₂ R l (l₁ ++ l₂)\n⊢ Forall₂ R (drop l₁.length l) l₂","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl : List α\nl₁ l₂ : List β\nh : Forall₂ R l (l₁ ++ l₂)\nh' : Forall₂ R (drop l₁.length l) (drop l₁.length (l₁ ++ l₂))\n⊢ Forall₂ R (drop l₁.length l) l₂","tactic":"have h' : Forall₂ R (drop (length l₁) l) (drop (length l₁) (l₁ ++ l₂)) :=\n    forall₂_drop (length l₁) h","premises":[{"full_name":"List.Forall₂","def_path":".lake/packages/batteries/Batteries/Data/List/Basic.lean","def_pos":[559,10],"def_end_pos":[559,17]},{"full_name":"List.drop","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[732,4],"def_end_pos":[732,8]},{"full_name":"List.forall₂_drop","def_path":"Mathlib/Data/List/Forall2.lean","def_pos":[193,8],"def_end_pos":[193,20]},{"full_name":"List.length","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2316,4],"def_end_pos":[2316,15]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl : List α\nl₁ l₂ : List β\nh : Forall₂ R l (l₁ ++ l₂)\nh' : Forall₂ R (drop l₁.length l) (drop l₁.length (l₁ ++ l₂))\n⊢ Forall₂ R (drop l₁.length l) l₂","state_after":"no goals","tactic":"rwa [drop_left] at h'","premises":[{"full_name":"List.drop_left","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean","def_pos":[1709,8],"def_end_pos":[1709,17]}]}]}
+{"url":"Mathlib/NumberTheory/Padics/PadicNorm.lean","commit":"","full_name":"padicNorm.sub","start":[195,0],"end":[199,32],"file_path":"Mathlib/NumberTheory/Padics/PadicNorm.lean","tactics":[{"state_before":"p : ℕ\nhp : Fact (Nat.Prime p)\nq r : ℚ\n⊢ padicNorm p (q - r) ≤ max (padicNorm p q) (padicNorm p r)","state_after":"p : ℕ\nhp : Fact (Nat.Prime p)\nq r : ℚ\n⊢ padicNorm p (q + -r) ≤ max (padicNorm p q) (padicNorm p (-r))","tactic":"rw [sub_eq_add_neg, ← padicNorm.neg r]","premises":[{"full_name":"padicNorm.neg","def_path":"Mathlib/NumberTheory/Padics/PadicNorm.lean","def_pos":[110,18],"def_end_pos":[110,21]},{"full_name":"sub_eq_add_neg","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[905,2],"def_end_pos":[905,13]}]},{"state_before":"p : ℕ\nhp : Fact (Nat.Prime p)\nq r : ℚ\n⊢ padicNorm p (q + -r) ≤ max (padicNorm p q) (padicNorm p (-r))","state_after":"no goals","tactic":"exact padicNorm.nonarchimedean","premises":[{"full_name":"padicNorm.nonarchimedean","def_path":"Mathlib/NumberTheory/Padics/PadicNorm.lean","def_pos":[180,18],"def_end_pos":[180,32]}]}]}
+{"url":"Mathlib/NumberTheory/PythagoreanTriples.lean","commit":"","full_name":"PythagoreanTriple.isPrimitiveClassified_of_coprime_of_pos","start":[538,0],"end":[544,17],"file_path":"Mathlib/NumberTheory/PythagoreanTriples.lean","tactics":[{"state_before":"x y z : ℤ\nh : PythagoreanTriple x y z\nhc : x.gcd y = 1\nhzpos : 0 < z\n⊢ h.IsPrimitiveClassified","state_after":"case inl\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : x.gcd y = 1\nhzpos : 0 < z\nh1 : x % 2 = 0 ∧ y % 2 = 1\n⊢ h.IsPrimitiveClassified\n\ncase inr\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : x.gcd y = 1\nhzpos : 0 < z\nh2 : x % 2 = 1 ∧ y % 2 = 0\n⊢ h.IsPrimitiveClassified","tactic":"cases' h.even_odd_of_coprime hc with h1 h2","premises":[{"full_name":"PythagoreanTriple.even_odd_of_coprime","def_path":"Mathlib/NumberTheory/PythagoreanTriples.lean","def_pos":[118,8],"def_end_pos":[118,27]}]},{"state_before":"case inr\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : x.gcd y = 1\nhzpos : 0 < z\nh2 : x % 2 = 1 ∧ y % 2 = 0\n⊢ h.IsPrimitiveClassified","state_after":"case inr\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : y.gcd x = 1\nhzpos : 0 < z\nh2 : x % 2 = 1 ∧ y % 2 = 0\n⊢ h.IsPrimitiveClassified","tactic":"rw [Int.gcd_comm] at hc","premises":[{"full_name":"Int.gcd_comm","def_path":"Mathlib/Data/Int/GCD.lean","def_pos":[191,8],"def_end_pos":[191,16]}]},{"state_before":"case inr\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : y.gcd x = 1\nhzpos : 0 < z\nh2 : x % 2 = 1 ∧ y % 2 = 0\n⊢ h.IsPrimitiveClassified","state_after":"case inr.intro.intro\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : y.gcd x = 1\nhzpos : 0 < z\nh2 : x % 2 = 1 ∧ y % 2 = 0\nm n : ℤ\nH :\n  (y = m ^ 2 - n ^ 2 ∧ x = 2 * m * n ∨ y = 2 * m * n ∧ x = m ^ 2 - n ^ 2) ∧\n    m.gcd n = 1 ∧ (m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0)\n⊢ h.IsPrimitiveClassified","tactic":"obtain ⟨m, n, H⟩ := h.symm.isPrimitiveClassified_of_coprime_of_odd_of_pos hc h2.left hzpos","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"PythagoreanTriple.isPrimitiveClassified_of_coprime_of_odd_of_pos","def_path":"Mathlib/NumberTheory/PythagoreanTriples.lean","def_pos":[437,8],"def_end_pos":[437,54]},{"full_name":"PythagoreanTriple.symm","def_path":"Mathlib/NumberTheory/PythagoreanTriples.lean","def_pos":[65,8],"def_end_pos":[65,12]}]},{"state_before":"case inr.intro.intro\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : y.gcd x = 1\nhzpos : 0 < z\nh2 : x % 2 = 1 ∧ y % 2 = 0\nm n : ℤ\nH :\n  (y = m ^ 2 - n ^ 2 ∧ x = 2 * m * n ∨ y = 2 * m * n ∧ x = m ^ 2 - n ^ 2) ∧\n    m.gcd n = 1 ∧ (m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0)\n⊢ h.IsPrimitiveClassified","state_after":"case h\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : y.gcd x = 1\nhzpos : 0 < z\nh2 : x % 2 = 1 ∧ y % 2 = 0\nm n : ℤ\nH :\n  (y = m ^ 2 - n ^ 2 ∧ x = 2 * m * n ∨ y = 2 * m * n ∧ x = m ^ 2 - n ^ 2) ∧\n    m.gcd n = 1 ∧ (m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0)\n⊢ (x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n ∨ x = 2 * m * n ∧ y = m ^ 2 - n ^ 2) ∧\n    m.gcd n = 1 ∧ (m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0)","tactic":"use m, n","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]}]},{"state_before":"case h\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : y.gcd x = 1\nhzpos : 0 < z\nh2 : x % 2 = 1 ∧ y % 2 = 0\nm n : ℤ\nH :\n  (y = m ^ 2 - n ^ 2 ∧ x = 2 * m * n ∨ y = 2 * m * n ∧ x = m ^ 2 - n ^ 2) ∧\n    m.gcd n = 1 ∧ (m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0)\n⊢ (x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n ∨ x = 2 * m * n ∧ y = m ^ 2 - n ^ 2) ∧\n    m.gcd n = 1 ∧ (m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0)","state_after":"no goals","tactic":"tauto","premises":[{"full_name":"Classical.or_iff_not_imp_left","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[147,8],"def_end_pos":[147,27]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]}]}]}
+{"url":"Mathlib/Algebra/Homology/TotalComplex.lean","commit":"","full_name":"HomologicalComplex₂.total.mapIso_hom","start":[413,0],"end":[420,56],"file_path":"Mathlib/Algebra/Homology/TotalComplex.lean","tactics":[{"state_before":"C : Type u_1\ninst✝⁶ : Category.{?u.130934, u_1} C\ninst✝⁵ : Preadditive C\nI₁ : Type u_2\nI₂ : Type u_3\nI₁₂ : Type u_4\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nK L M : HomologicalComplex₂ C c₁ c₂\nφ : K ⟶ L\ne : K ≅ L\nψ : L ⟶ M\nc₁₂ : ComplexShape I₁₂\ninst✝⁴ : DecidableEq I₁₂\ninst✝³ : TotalComplexShape c₁ c₂ c₁₂\ninst✝² : K.HasTotal c₁₂\ninst✝¹ : L.HasTotal c₁₂\ninst✝ : M.HasTotal c₁₂\n⊢ map e.hom c₁₂ ≫ map e.inv c₁₂ = 𝟙 (K.total c₁₂)","state_after":"no goals","tactic":"rw [← map_comp, e.hom_inv_id, map_id]","premises":[{"full_name":"CategoryTheory.Iso.hom_inv_id","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[55,2],"def_end_pos":[55,12]},{"full_name":"HomologicalComplex₂.total.map_comp","def_path":"Mathlib/Algebra/Homology/TotalComplex.lean","def_pos":[409,6],"def_end_pos":[409,14]},{"full_name":"HomologicalComplex₂.total.map_id","def_path":"Mathlib/Algebra/Homology/TotalComplex.lean","def_pos":[402,6],"def_end_pos":[402,12]}]},{"state_before":"C : Type u_1\ninst✝⁶ : Category.{?u.130934, u_1} C\ninst✝⁵ : Preadditive C\nI₁ : Type u_2\nI₂ : Type u_3\nI₁₂ : Type u_4\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nK L M : HomologicalComplex₂ C c₁ c₂\nφ : K ⟶ L\ne : K ≅ L\nψ : L ⟶ M\nc₁₂ : ComplexShape I₁₂\ninst✝⁴ : DecidableEq I₁₂\ninst✝³ : TotalComplexShape c₁ c₂ c₁₂\ninst✝² : K.HasTotal c₁₂\ninst✝¹ : L.HasTotal c₁₂\ninst✝ : M.HasTotal c₁₂\n⊢ map e.inv c₁₂ ≫ map e.hom c₁₂ = 𝟙 (L.total c₁₂)","state_after":"no goals","tactic":"rw [← map_comp, e.inv_hom_id, map_id]","premises":[{"full_name":"CategoryTheory.Iso.inv_hom_id","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[58,2],"def_end_pos":[58,12]},{"full_name":"HomologicalComplex₂.total.map_comp","def_path":"Mathlib/Algebra/Homology/TotalComplex.lean","def_pos":[409,6],"def_end_pos":[409,14]},{"full_name":"HomologicalComplex₂.total.map_id","def_path":"Mathlib/Algebra/Homology/TotalComplex.lean","def_pos":[402,6],"def_end_pos":[402,12]}]}]}
+{"url":"Mathlib/Algebra/Module/LocalizedModule.lean","commit":"","full_name":"_private.Mathlib.Algebra.Module.LocalizedModule.0.LocalizedModule.add_smul_aux","start":[359,0],"end":[368,36],"file_path":"Mathlib/Algebra/Module/LocalizedModule.lean","tactics":[{"state_before":"R : Type u\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nT : Type u_1\ninst✝² : CommSemiring T\ninst✝¹ : Algebra R T\ninst✝ : IsLocalization S T\nx y : T\np : LocalizedModule S M\n⊢ (x + y) • p = x • p + y • p","state_after":"case h\nR : Type u\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nT : Type u_1\ninst✝² : CommSemiring T\ninst✝¹ : Algebra R T\ninst✝ : IsLocalization S T\nx y : T\nm : M\ns : ↥S\n⊢ (x + y) • mk m s = x • mk m s + y • mk m s","tactic":"induction' p with m s","premises":[]},{"state_before":"case h\nR : Type u\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nT : Type u_1\ninst✝² : CommSemiring T\ninst✝¹ : Algebra R T\ninst✝ : IsLocalization S T\nx y : T\nm : M\ns : ↥S\n⊢ (x + y) • mk m s = x • mk m s + y • mk m s","state_after":"case h\nR : Type u\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nT : Type u_1\ninst✝² : CommSemiring T\ninst✝¹ : Algebra R T\ninst✝ : IsLocalization S T\nx y : T\nm : M\ns : ↥S\n⊢ mk\n      ((((IsLocalization.sec S x).1 * ↑(IsLocalization.sec S y).2 +\n            (IsLocalization.sec S y).1 * ↑(IsLocalization.sec S x).2) *\n          ↑s) •\n        m)\n      ((IsLocalization.sec S x).2 * (IsLocalization.sec S y).2 * s * s) =\n    mk\n      ((((IsLocalization.sec S y).2 * s) • (IsLocalization.sec S x).1 +\n          ((IsLocalization.sec S x).2 * s) • (IsLocalization.sec S y).1) •\n        m)\n      ((IsLocalization.sec S x).2 * s * ((IsLocalization.sec S y).2 * s))","tactic":"rw [smul_def T x, smul_def T y, mk_add_mk, show (x + y) • _ =  IsLocalization.mk' T _ _ • _ by\n    rw [← IsLocalization.mk'_sec (M := S) T x, ← IsLocalization.mk'_sec (M := S) T y,\n      ← IsLocalization.mk'_add, IsLocalization.mk'_cancel _ _ s], mk'_smul_mk, ← smul_assoc,\n    ← smul_assoc, ← add_smul]","premises":[{"full_name":"IsLocalization.mk'","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[228,18],"def_end_pos":[228,21]},{"full_name":"IsLocalization.mk'_add","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[420,8],"def_end_pos":[420,15]},{"full_name":"IsLocalization.mk'_cancel","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[349,8],"def_end_pos":[349,18]},{"full_name":"IsLocalization.mk'_sec","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[232,8],"def_end_pos":[232,15]},{"full_name":"LocalizedModule.mk'_smul_mk","def_path":"Mathlib/Algebra/Module/LocalizedModule.lean","def_pos":[319,8],"def_end_pos":[319,19]},{"full_name":"LocalizedModule.mk_add_mk","def_path":"Mathlib/Algebra/Module/LocalizedModule.lean","def_pos":[151,8],"def_end_pos":[151,17]},{"full_name":"LocalizedModule.smul_def","def_path":"Mathlib/Algebra/Module/LocalizedModule.lean","def_pos":[316,8],"def_end_pos":[316,16]},{"full_name":"add_smul","def_path":"Mathlib/Algebra/Module/Defs.lean","def_pos":[87,8],"def_end_pos":[87,16]},{"full_name":"smul_assoc","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[219,6],"def_end_pos":[219,16]}]},{"state_before":"case h\nR : Type u\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nT : Type u_1\ninst✝² : CommSemiring T\ninst✝¹ : Algebra R T\ninst✝ : IsLocalization S T\nx y : T\nm : M\ns : ↥S\n⊢ mk\n      ((((IsLocalization.sec S x).1 * ↑(IsLocalization.sec S y).2 +\n            (IsLocalization.sec S y).1 * ↑(IsLocalization.sec S x).2) *\n          ↑s) •\n        m)\n      ((IsLocalization.sec S x).2 * (IsLocalization.sec S y).2 * s * s) =\n    mk\n      ((((IsLocalization.sec S y).2 * s) • (IsLocalization.sec S x).1 +\n          ((IsLocalization.sec S x).2 * s) • (IsLocalization.sec S y).1) •\n        m)\n      ((IsLocalization.sec S x).2 * s * ((IsLocalization.sec S y).2 * s))","state_after":"case h.e_m\nR : Type u\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nT : Type u_1\ninst✝² : CommSemiring T\ninst✝¹ : Algebra R T\ninst✝ : IsLocalization S T\nx y : T\nm : M\ns : ↥S\n⊢ (((IsLocalization.sec S x).1 * ↑(IsLocalization.sec S y).2 +\n          (IsLocalization.sec S y).1 * ↑(IsLocalization.sec S x).2) *\n        ↑s) •\n      m =\n    (((IsLocalization.sec S y).2 * s) • (IsLocalization.sec S x).1 +\n        ((IsLocalization.sec S x).2 * s) • (IsLocalization.sec S y).1) •\n      m\n\ncase h.e_s\nR : Type u\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nT : Type u_1\ninst✝² : CommSemiring T\ninst✝¹ : Algebra R T\ninst✝ : IsLocalization S T\nx y : T\nm : M\ns : ↥S\n⊢ (IsLocalization.sec S x).2 * (IsLocalization.sec S y).2 * s * s =\n    (IsLocalization.sec S x).2 * s * ((IsLocalization.sec S y).2 * s)","tactic":"congr 1","premises":[]}]}
+{"url":"Mathlib/Analysis/Normed/Group/Seminorm.lean","commit":"","full_name":"map_sub_le_max","start":[138,0],"end":[140,28],"file_path":"Mathlib/Analysis/Normed/Group/Seminorm.lean","tactics":[{"state_before":"ι : Type u_1\nR : Type u_2\nR' : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst✝² : AddGroup E\ninst✝¹ : FunLike F E ℝ\ninst✝ : NonarchAddGroupSeminormClass F E\nf : F\nx y : E\n⊢ f (x - y) ≤ max (f x) (f y)","state_after":"ι : Type u_1\nR : Type u_2\nR' : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst✝² : AddGroup E\ninst✝¹ : FunLike F E ℝ\ninst✝ : NonarchAddGroupSeminormClass F E\nf : F\nx y : E\n⊢ f (x + -y) ≤ max (f x) (f (-y))","tactic":"rw [sub_eq_add_neg, ← NonarchAddGroupSeminormClass.map_neg_eq_map' f y]","premises":[{"full_name":"NonarchAddGroupSeminormClass.map_neg_eq_map'","def_path":"Mathlib/Analysis/Normed/Group/Seminorm.lean","def_pos":[123,12],"def_end_pos":[123,27]},{"full_name":"sub_eq_add_neg","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[905,2],"def_end_pos":[905,13]}]},{"state_before":"ι : Type u_1\nR : Type u_2\nR' : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst✝² : AddGroup E\ninst✝¹ : FunLike F E ℝ\ninst✝ : NonarchAddGroupSeminormClass F E\nf : F\nx y : E\n⊢ f (x + -y) ≤ max (f x) (f (-y))","state_after":"no goals","tactic":"exact map_add_le_max _ _ _","premises":[{"full_name":"NonarchimedeanHomClass.map_add_le_max","def_path":"Mathlib/Algebra/Order/Hom/Basic.lean","def_pos":[100,2],"def_end_pos":[100,16]}]}]}
+{"url":"Mathlib/Order/Zorn.lean","commit":"","full_name":"zorn_preorder₀","start":[108,0],"end":[119,43],"file_path":"Mathlib/Order/Zorn.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nr : α → α → Prop\nc✝ : Set α\ninst✝ : Preorder α\ns : Set α\nih : ∀ c ⊆ s, IsChain (fun x x_1 => x ≤ x_1) c → ∃ ub ∈ s, ∀ z ∈ c, z ≤ ub\nc : Set ↑s\nhc : IsChain (fun x x_1 => x ≤ x_1) c\n⊢ IsChain (fun x x_1 => x ≤ x_1) (Subtype.val '' c)","state_after":"case intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nr : α → α → Prop\nc✝ : Set α\ninst✝ : Preorder α\ns : Set α\nih : ∀ c ⊆ s, IsChain (fun x x_1 => x ≤ x_1) c → ∃ ub ∈ s, ∀ z ∈ c, z ≤ ub\nc : Set ↑s\nhc : IsChain (fun x x_1 => x ≤ x_1) c\np : { x // x ∈ s }\nhpc : p ∈ c\nq : { x // x ∈ s }\nhqc : q ∈ c\nhpq : ↑p ≠ ↑q\n⊢ (fun x x_1 => x ≤ x_1) ↑p ↑q ∨ (fun x x_1 => x ≤ x_1) ↑q ↑p","tactic":"rintro _ ⟨p, hpc, rfl⟩ _ ⟨q, hqc, rfl⟩ hpq","premises":[]},{"state_before":"case intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nr : α → α → Prop\nc✝ : Set α\ninst✝ : Preorder α\ns : Set α\nih : ∀ c ⊆ s, IsChain (fun x x_1 => x ≤ x_1) c → ∃ ub ∈ s, ∀ z ∈ c, z ≤ ub\nc : Set ↑s\nhc : IsChain (fun x x_1 => x ≤ x_1) c\np : { x // x ∈ s }\nhpc : p ∈ c\nq : { x // x ∈ s }\nhqc : q ∈ c\nhpq : ↑p ≠ ↑q\n⊢ (fun x x_1 => x ≤ x_1) ↑p ↑q ∨ (fun x x_1 => x ≤ x_1) ↑q ↑p","state_after":"no goals","tactic":"exact hc hpc hqc fun t => hpq (Subtype.ext_iff.1 t)","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Subtype.ext_iff","def_path":"Mathlib/Data/Subtype.lean","def_pos":[62,18],"def_end_pos":[62,25]}]}]}
+{"url":"Mathlib/Analysis/InnerProductSpace/LinearPMap.lean","commit":"","full_name":"IsSelfAdjoint.dense_domain","start":[225,0],"end":[240,16],"file_path":"Mathlib/Analysis/InnerProductSpace/LinearPMap.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 F\ninst✝ : CompleteSpace E\nA : E →ₗ.[𝕜] E\nhA : IsSelfAdjoint A\n⊢ Dense ↑A.domain","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 F\ninst✝ : CompleteSpace E\nA : E →ₗ.[𝕜] E\nhA : IsSelfAdjoint A\nh : ¬Dense ↑A.domain\n⊢ False","tactic":"by_contra h","premises":[{"full_name":"Decidable.byContradiction","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[880,8],"def_end_pos":[880,23]},{"full_name":"Not","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[224,4],"def_end_pos":[224,7]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 F\ninst✝ : CompleteSpace E\nA : E →ₗ.[𝕜] E\nhA : IsSelfAdjoint A\nh : ¬Dense ↑A.domain\n⊢ False","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 F\ninst✝ : CompleteSpace E\nA : E →ₗ.[𝕜] E\nhA : A† = A\nh : ¬Dense ↑A.domain\n⊢ False","tactic":"rw [isSelfAdjoint_def] at hA","premises":[{"full_name":"LinearPMap.isSelfAdjoint_def","def_path":"Mathlib/Analysis/InnerProductSpace/LinearPMap.lean","def_pos":[223,8],"def_end_pos":[223,25]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 F\ninst✝ : CompleteSpace E\nA : E →ₗ.[𝕜] E\nhA : A† = A\nh : ¬Dense ↑A.domain\n⊢ False","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 F\ninst✝ : CompleteSpace E\nA : E →ₗ.[𝕜] E\nhA : A† = A\nh : ¬Dense ↑A.domain\nh' : A.domain = ⊤\n⊢ False","tactic":"have h' : A.domain = ⊤ := by\n    rw [← hA, Submodule.eq_top_iff']\n    intro x\n    rw [mem_adjoint_domain_iff, ← hA]\n    refine (innerSL 𝕜 x).cont.comp ?_\n    simp only [adjoint, h]\n    exact continuous_const","premises":[{"full_name":"Continuous.comp","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1389,8],"def_end_pos":[1389,23]},{"full_name":"ContinuousLinearMap.cont","def_path":"Mathlib/Topology/Algebra/Module/Basic.lean","def_pos":[227,2],"def_end_pos":[227,6]},{"full_name":"LinearPMap.adjoint","def_path":"Mathlib/Analysis/InnerProductSpace/LinearPMap.lean","def_pos":[146,4],"def_end_pos":[146,11]},{"full_name":"LinearPMap.domain","def_path":"Mathlib/LinearAlgebra/LinearPMap.lean","def_pos":[37,2],"def_end_pos":[37,8]},{"full_name":"LinearPMap.mem_adjoint_domain_iff","def_path":"Mathlib/Analysis/InnerProductSpace/LinearPMap.lean","def_pos":[152,8],"def_end_pos":[152,30]},{"full_name":"Submodule.eq_top_iff'","def_path":"Mathlib/Algebra/Module/Submodule/Lattice.lean","def_pos":[151,8],"def_end_pos":[151,19]},{"full_name":"Top.top","def_path":"Mathlib/Order/Notation.lean","def_pos":[94,2],"def_end_pos":[94,5]},{"full_name":"continuous_const","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1436,8],"def_end_pos":[1436,24]},{"full_name":"innerSL","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[1550,4],"def_end_pos":[1550,11]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 F\ninst✝ : CompleteSpace E\nA : E →ₗ.[𝕜] E\nhA : A† = A\nh : ¬Dense ↑A.domain\nh' : A.domain = ⊤\n⊢ False","state_after":"no goals","tactic":"simp [h'] at h","premises":[]}]}
+{"url":"Mathlib/LinearAlgebra/RootSystem/Defs.lean","commit":"","full_name":"RootPairing.ne_zero'","start":[126,0],"end":[127,48],"file_path":"Mathlib/LinearAlgebra/RootSystem/Defs.lean","tactics":[{"state_before":"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\ni j : ι\ninst✝ : CharZero R\nh : P.coroot i = 0\n⊢ False","state_after":"no goals","tactic":"simpa [h] using P.root_coroot_two i","premises":[{"full_name":"RootPairing.root_coroot_two","def_path":"Mathlib/LinearAlgebra/RootSystem/Defs.lean","def_pos":[96,2],"def_end_pos":[96,17]}]}]}
+{"url":"Mathlib/SetTheory/Ordinal/Basic.lean","commit":"","full_name":"Ordinal.typein_le_typein","start":[989,0],"end":[991,76],"file_path":"Mathlib/SetTheory/Ordinal/Basic.lean","tactics":[{"state_before":"α : Type u\nβ : Type u_1\nγ : Type u_2\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nr : α → α → Prop\ninst✝ : IsWellOrder α r\nx x' : α\n⊢ typein r x ≤ typein r x' ↔ ¬r x' x","state_after":"no goals","tactic":"rw [← not_lt, typein_lt_typein]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Ordinal.typein_lt_typein","def_path":"Mathlib/SetTheory/Ordinal/Basic.lean","def_pos":[414,8],"def_end_pos":[414,24]},{"full_name":"not_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[312,8],"def_end_pos":[312,14]}]}]}
+{"url":"Mathlib/RingTheory/Ideal/QuotientOperations.lean","commit":"","full_name":"Ideal.fst_comp_quotientMulEquivQuotientProd","start":[304,0],"end":[309,38],"file_path":"Mathlib/RingTheory/Ideal/QuotientOperations.lean","tactics":[{"state_before":"R : Type u\nS : Type v\nF : Type w\ninst✝¹ : CommRing R\ninst✝ : Semiring S\nι : Type u_1\nI J : Ideal R\ncoprime : IsCoprime I J\n⊢ (fst (R ⧸ I) (R ⧸ J)).comp ↑(I.quotientMulEquivQuotientProd J coprime) = factor (I * J) I ⋯","state_after":"case h\nR : Type u\nS : Type v\nF : Type w\ninst✝¹ : CommRing R\ninst✝ : Semiring S\nι : Type u_1\nI J : Ideal R\ncoprime : IsCoprime I J\n⊢ ((fst (R ⧸ I) (R ⧸ J)).comp ↑(I.quotientMulEquivQuotientProd J coprime)).comp (Quotient.mk (I * J)) =\n    (factor (I * J) I ⋯).comp (Quotient.mk (I * J))","tactic":"apply Quotient.ringHom_ext","premises":[{"full_name":"Ideal.Quotient.ringHom_ext","def_path":"Mathlib/RingTheory/Ideal/Quotient.lean","def_pos":[99,8],"def_end_pos":[99,19]}]},{"state_before":"case h\nR : Type u\nS : Type v\nF : Type w\ninst✝¹ : CommRing R\ninst✝ : Semiring S\nι : Type u_1\nI J : Ideal R\ncoprime : IsCoprime I J\n⊢ ((fst (R ⧸ I) (R ⧸ J)).comp ↑(I.quotientMulEquivQuotientProd J coprime)).comp (Quotient.mk (I * J)) =\n    (factor (I * J) I ⋯).comp (Quotient.mk (I * J))","state_after":"case h.a\nR : Type u\nS : Type v\nF : Type w\ninst✝¹ : CommRing R\ninst✝ : Semiring S\nι : Type u_1\nI J : Ideal R\ncoprime : IsCoprime I J\nx✝ : R\n⊢ (((fst (R ⧸ I) (R ⧸ J)).comp ↑(I.quotientMulEquivQuotientProd J coprime)).comp (Quotient.mk (I * J))) x✝ =\n    ((factor (I * J) I ⋯).comp (Quotient.mk (I * J))) x✝","tactic":"ext","premises":[]},{"state_before":"case h.a\nR : Type u\nS : Type v\nF : Type w\ninst✝¹ : CommRing R\ninst✝ : Semiring S\nι : Type u_1\nI J : Ideal R\ncoprime : IsCoprime I J\nx✝ : R\n⊢ (((fst (R ⧸ I) (R ⧸ J)).comp ↑(I.quotientMulEquivQuotientProd J coprime)).comp (Quotient.mk (I * J))) x✝ =\n    ((factor (I * J) I ⋯).comp (Quotient.mk (I * J))) x✝","state_after":"no goals","tactic":"rfl","premises":[]}]}
+{"url":"Mathlib/RingTheory/Polynomial/Vieta.lean","commit":"","full_name":"MvPolynomial.prod_X_add_C_coeff","start":[162,0],"end":[172,52],"file_path":"Mathlib/RingTheory/Polynomial/Vieta.lean","tactics":[{"state_before":"R : Type u_1\nσ : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : Fintype σ\nk : ℕ\nh : k ≤ Fintype.card σ\n⊢ (∏ i : σ, (Polynomial.X + Polynomial.C (X i))).coeff k = esymm σ R (Fintype.card σ - k)","state_after":"R : Type u_1\nσ : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : Fintype σ\nk : ℕ\nh : k ≤ Fintype.card σ\ns : Multiset (MvPolynomial σ R) := Multiset.map (fun i => X i) univ.val\n⊢ (∏ i : σ, (Polynomial.X + Polynomial.C (X i))).coeff k = esymm σ R (Fintype.card σ - k)","tactic":"let s := Finset.univ.val.map fun i => (MvPolynomial.X i : MvPolynomial σ R)","premises":[{"full_name":"Finset.univ","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[67,4],"def_end_pos":[67,8]},{"full_name":"Finset.val","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[135,2],"def_end_pos":[135,5]},{"full_name":"Multiset.map","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[1012,4],"def_end_pos":[1012,7]},{"full_name":"MvPolynomial","def_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","def_pos":[84,4],"def_end_pos":[84,16]},{"full_name":"MvPolynomial.X","def_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","def_pos":[178,4],"def_end_pos":[178,5]}]},{"state_before":"R : Type u_1\nσ : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : Fintype σ\nk : ℕ\nh : k ≤ Fintype.card σ\ns : Multiset (MvPolynomial σ R) := Multiset.map (fun i => X i) univ.val\n⊢ (∏ i : σ, (Polynomial.X + Polynomial.C (X i))).coeff k = esymm σ R (Fintype.card σ - k)","state_after":"R : Type u_1\nσ : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : Fintype σ\nk : ℕ\nh : k ≤ Fintype.card σ\ns : Multiset (MvPolynomial σ R) := Multiset.map (fun i => X i) univ.val\nthis : Fintype.card σ = Multiset.card s\n⊢ (∏ i : σ, (Polynomial.X + Polynomial.C (X i))).coeff k = esymm σ R (Fintype.card σ - k)","tactic":"have : Fintype.card σ = Multiset.card s := by\n    rw [Multiset.card_map, ← Finset.card_univ, Finset.card_def]","premises":[{"full_name":"Finset.card_def","def_path":"Mathlib/Data/Finset/Card.lean","def_pos":[43,8],"def_end_pos":[43,16]},{"full_name":"Finset.card_univ","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[228,8],"def_end_pos":[228,24]},{"full_name":"Fintype.card","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[62,4],"def_end_pos":[62,8]},{"full_name":"Multiset.card","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[659,4],"def_end_pos":[659,8]},{"full_name":"Multiset.card_map","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[1085,8],"def_end_pos":[1085,16]}]},{"state_before":"R : Type u_1\nσ : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : Fintype σ\nk : ℕ\nh : k ≤ Fintype.card σ\ns : Multiset (MvPolynomial σ R) := Multiset.map (fun i => X i) univ.val\nthis : Fintype.card σ = Multiset.card s\n⊢ (∏ i : σ, (Polynomial.X + Polynomial.C (X i))).coeff k = esymm σ R (Fintype.card σ - k)","state_after":"R : Type u_1\nσ : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : Fintype σ\nk : ℕ\ns : Multiset (MvPolynomial σ R) := Multiset.map (fun i => X i) univ.val\nh : k ≤ Multiset.card s\nthis : Fintype.card σ = Multiset.card s\n⊢ (∏ i : σ, (Polynomial.X + Polynomial.C (X i))).coeff k = esymm σ R (Multiset.card s - k)","tactic":"rw [this] at h ⊢","premises":[]},{"state_before":"R : Type u_1\nσ : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : Fintype σ\nk : ℕ\ns : Multiset (MvPolynomial σ R) := Multiset.map (fun i => X i) univ.val\nh : k ≤ Multiset.card s\nthis : Fintype.card σ = Multiset.card s\n⊢ (∏ i : σ, (Polynomial.X + Polynomial.C (X i))).coeff k = esymm σ R (Multiset.card s - k)","state_after":"R : Type u_1\nσ : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : Fintype σ\nk : ℕ\ns : Multiset (MvPolynomial σ R) := Multiset.map (fun i => X i) univ.val\nh : k ≤ Multiset.card s\nthis : Fintype.card σ = Multiset.card s\n⊢ (Multiset.map (fun i => Polynomial.X + Polynomial.C (X i)) univ.val).prod.coeff k =\n    (Multiset.map X univ.val).esymm (Multiset.card s - k)","tactic":"rw [MvPolynomial.esymm_eq_multiset_esymm σ R, Finset.prod_eq_multiset_prod]","premises":[{"full_name":"Finset.prod_eq_multiset_prod","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[267,8],"def_end_pos":[267,29]},{"full_name":"MvPolynomial.esymm_eq_multiset_esymm","def_path":"Mathlib/RingTheory/MvPolynomial/Symmetric.lean","def_pos":[172,8],"def_end_pos":[172,31]}]},{"state_before":"R : Type u_1\nσ : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : Fintype σ\nk : ℕ\ns : Multiset (MvPolynomial σ R) := Multiset.map (fun i => X i) univ.val\nh : k ≤ Multiset.card s\nthis : Fintype.card σ = Multiset.card s\n⊢ (Multiset.map (fun i => Polynomial.X + Polynomial.C (X i)) univ.val).prod.coeff k =\n    (Multiset.map X univ.val).esymm (Multiset.card s - k)","state_after":"case h.e'_2.h.e'_3.h.e'_3\nR : Type u_1\nσ : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : Fintype σ\nk : ℕ\ns : Multiset (MvPolynomial σ R) := Multiset.map (fun i => X i) univ.val\nh : k ≤ Multiset.card s\nthis : Fintype.card σ = Multiset.card s\n⊢ Multiset.map (fun i => Polynomial.X + Polynomial.C (X i)) univ.val =\n    Multiset.map (fun r => Polynomial.X + Polynomial.C r) s","tactic":"convert Multiset.prod_X_add_C_coeff s h","premises":[{"full_name":"Multiset.prod_X_add_C_coeff","def_path":"Mathlib/RingTheory/Polynomial/Vieta.lean","def_pos":[55,8],"def_end_pos":[55,26]}]},{"state_before":"case h.e'_2.h.e'_3.h.e'_3\nR : Type u_1\nσ : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : Fintype σ\nk : ℕ\ns : Multiset (MvPolynomial σ R) := Multiset.map (fun i => X i) univ.val\nh : k ≤ Multiset.card s\nthis : Fintype.card σ = Multiset.card s\n⊢ Multiset.map (fun i => Polynomial.X + Polynomial.C (X i)) univ.val =\n    Multiset.map (fun r => Polynomial.X + Polynomial.C r) s","state_after":"no goals","tactic":"simp_rw [s, Multiset.map_map, Function.comp_apply]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Function.comp_apply","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[35,16],"def_end_pos":[35,35]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Multiset.map_map","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[1129,8],"def_end_pos":[1129,15]}]}]}
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TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\ninst✝² : SmoothManifoldWithCorners I M\ninst✝¹ : T2Space M\nhi : Fintype ι\ns : Set M\nf✝ : SmoothBumpCovering ι I M s\ninst✝ : Finite ι\nf : SmoothBumpCovering ι I M\nval✝ : Fintype ι\n⊢ ∃ n e,\n    Smooth I 𝓘(ℝ, EuclideanSpace ℝ (Fin n)) e ∧\n      Injective e ∧ ∀ (x : M), Injective ⇑(mfderiv I 𝓘(ℝ, EuclideanSpace ℝ (Fin n)) e x)","tactic":"cases nonempty_fintype ι","premises":[{"full_name":"nonempty_fintype","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[390,8],"def_end_pos":[390,24]}]},{"state_before":"case intro\nι : Type uι\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : FiniteDimensional ℝ E\nH : Type uH\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\ninst✝² : SmoothManifoldWithCorners I M\ninst✝¹ : T2Space M\nhi : Fintype ι\ns : Set M\nf✝ : 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M\ninst✝³ : ChartedSpace H M\ninst✝² : SmoothManifoldWithCorners I M\ninst✝¹ : T2Space M\nhi : Fintype ι\ns : Set M\nf✝ : SmoothBumpCovering ι I M s\ninst✝ : Finite ι\nf : SmoothBumpCovering ι I M\nval✝ : Fintype ι\nF : Type := EuclideanSpace ℝ (Fin (finrank ℝ (ι → E × ℝ)))\nthis : IsNoetherian ℝ (E × ℝ) := IsNoetherian.iff_fg.mpr inferInstance\n⊢ ∃ n e,\n    Smooth I 𝓘(ℝ, EuclideanSpace ℝ (Fin n)) e ∧\n      Injective e ∧ ∀ (x : M), Injective ⇑(mfderiv I 𝓘(ℝ, EuclideanSpace ℝ (Fin n)) e x)","state_after":"case intro\nι : Type uι\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : FiniteDimensional ℝ E\nH : Type uH\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\ninst✝² : SmoothManifoldWithCorners I M\ninst✝¹ : T2Space M\nhi : Fintype ι\ns : Set M\nf✝ : SmoothBumpCovering ι I M s\ninst✝ : Finite ι\nf : SmoothBumpCovering ι I M\nval✝ : Fintype ι\nF : Type := EuclideanSpace ℝ (Fin (finrank ℝ (ι → E × ℝ)))\nthis✝ : IsNoetherian ℝ (E × ℝ) := IsNoetherian.iff_fg.mpr inferInstance\nthis : FiniteDimensional ℝ (ι → E × ℝ) := IsNoetherian.iff_fg.mp inferInstance\n⊢ ∃ n e,\n    Smooth I 𝓘(ℝ, EuclideanSpace ℝ (Fin n)) e ∧\n      Injective e ∧ ∀ (x : M), Injective ⇑(mfderiv I 𝓘(ℝ, EuclideanSpace ℝ (Fin n)) e x)","tactic":"letI : FiniteDimensional ℝ (ι → E × ℝ) := IsNoetherian.iff_fg.1 inferInstance","premises":[{"full_name":"FiniteDimensional","def_path":"Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean","def_pos":[77,7],"def_end_pos":[77,24]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"IsNoetherian.iff_fg","def_path":"Mathlib/FieldTheory/Finiteness.lean","def_pos":[91,8],"def_end_pos":[91,14]},{"full_name":"Prod","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[479,10],"def_end_pos":[479,14]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"inferInstance","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[99,7],"def_end_pos":[99,20]}]},{"state_before":"case intro\nι : Type uι\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : FiniteDimensional ℝ E\nH : Type uH\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\ninst✝² : SmoothManifoldWithCorners I M\ninst✝¹ : T2Space M\nhi : Fintype ι\ns : Set M\nf✝ : SmoothBumpCovering ι I M s\ninst✝ : Finite ι\nf : SmoothBumpCovering ι I M\nval✝ : Fintype ι\nF : Type := EuclideanSpace ℝ (Fin (finrank ℝ (ι → E × ℝ)))\nthis✝ : IsNoetherian ℝ (E × ℝ) := IsNoetherian.iff_fg.mpr inferInstance\nthis : FiniteDimensional ℝ (ι → E × ℝ) := IsNoetherian.iff_fg.mp inferInstance\n⊢ ∃ n e,\n    Smooth I 𝓘(ℝ, EuclideanSpace ℝ (Fin n)) e ∧\n      Injective e ∧ ∀ (x : M), Injective ⇑(mfderiv I 𝓘(ℝ, EuclideanSpace ℝ (Fin n)) e x)","state_after":"case intro\nι : Type uι\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : FiniteDimensional ℝ E\nH : Type uH\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\ninst✝² : SmoothManifoldWithCorners I M\ninst✝¹ : T2Space M\nhi : Fintype ι\ns : Set M\nf✝ : SmoothBumpCovering ι I M s\ninst✝ : Finite ι\nf : SmoothBumpCovering ι I M\nval✝ : Fintype ι\nF : Type := EuclideanSpace ℝ (Fin (finrank ℝ (ι → E × ℝ)))\nthis✝ : IsNoetherian ℝ (E × ℝ) := IsNoetherian.iff_fg.mpr inferInstance\nthis : FiniteDimensional ℝ (ι → E × ℝ) := IsNoetherian.iff_fg.mp inferInstance\neEF : (ι → E × ℝ) ≃L[ℝ] F := ContinuousLinearEquiv.ofFinrankEq ⋯\n⊢ ∃ n e,\n    Smooth I 𝓘(ℝ, EuclideanSpace ℝ (Fin n)) e ∧\n      Injective e ∧ ∀ (x : M), Injective ⇑(mfderiv I 𝓘(ℝ, EuclideanSpace ℝ (Fin n)) e x)","tactic":"set eEF : (ι → E × ℝ) ≃L[ℝ] F :=\n    ContinuousLinearEquiv.ofFinrankEq finrank_euclideanSpace_fin.symm","premises":[{"full_name":"ContinuousLinearEquiv","def_path":"Mathlib/Topology/Algebra/Module/Basic.lean","def_pos":[264,10],"def_end_pos":[264,31]},{"full_name":"ContinuousLinearEquiv.ofFinrankEq","def_path":"Mathlib/Topology/Algebra/Module/FiniteDimension.lean","def_pos":[406,4],"def_end_pos":[406,37]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Prod","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[479,10],"def_end_pos":[479,14]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"RingHom.id","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[538,4],"def_end_pos":[538,6]},{"full_name":"finrank_euclideanSpace_fin","def_path":"Mathlib/Analysis/InnerProductSpace/PiL2.lean","def_pos":[150,8],"def_end_pos":[150,34]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"case intro\nι : Type uι\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : FiniteDimensional ℝ E\nH : Type uH\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\ninst✝² : SmoothManifoldWithCorners I M\ninst✝¹ : T2Space M\nhi : Fintype ι\ns : Set M\nf✝ : SmoothBumpCovering ι I M s\ninst✝ : Finite ι\nf : SmoothBumpCovering ι I M\nval✝ : Fintype ι\nF : Type := EuclideanSpace ℝ (Fin (finrank ℝ (ι → E × ℝ)))\nthis✝ : IsNoetherian ℝ (E × ℝ) := IsNoetherian.iff_fg.mpr inferInstance\nthis : FiniteDimensional ℝ (ι → E × ℝ) := IsNoetherian.iff_fg.mp inferInstance\neEF : (ι → E × ℝ) ≃L[ℝ] F := ContinuousLinearEquiv.ofFinrankEq ⋯\n⊢ ∃ n e,\n    Smooth I 𝓘(ℝ, EuclideanSpace ℝ (Fin n)) e ∧\n      Injective e ∧ ∀ (x : M), Injective ⇑(mfderiv I 𝓘(ℝ, EuclideanSpace ℝ (Fin n)) e x)","state_after":"case intro\nι : Type uι\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : FiniteDimensional ℝ E\nH : Type uH\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\ninst✝² : SmoothManifoldWithCorners I M\ninst✝¹ : T2Space M\nhi : Fintype ι\ns : Set M\nf✝ : SmoothBumpCovering ι I M s\ninst✝ : Finite ι\nf : SmoothBumpCovering ι I M\nval✝ : Fintype ι\nF : Type := EuclideanSpace ℝ (Fin (finrank ℝ (ι → E × ℝ)))\nthis✝ : IsNoetherian ℝ (E × ℝ) := IsNoetherian.iff_fg.mpr inferInstance\nthis : FiniteDimensional ℝ (ι → E × ℝ) := IsNoetherian.iff_fg.mp inferInstance\neEF : (ι → E × ℝ) ≃L[ℝ] F := ContinuousLinearEquiv.ofFinrankEq ⋯\nx : M\n⊢ Injective ⇑(mfderiv I 𝓘(ℝ, EuclideanSpace ℝ (Fin (finrank ℝ (ι → E × ℝ)))) (⇑eEF ∘ ⇑f.embeddingPiTangent) x)","tactic":"refine ⟨_, eEF ∘ f.embeddingPiTangent,\n    eEF.toDiffeomorph.smooth.comp f.embeddingPiTangent.smooth,\n    eEF.injective.comp f.embeddingPiTangent_injective, fun x => ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"ContMDiffMap.smooth","def_path":"Mathlib/Geometry/Manifold/ContMDiffMap.lean","def_pos":[55,18],"def_end_pos":[55,24]},{"full_name":"ContinuousLinearEquiv.injective","def_path":"Mathlib/Topology/Algebra/Module/Basic.lean","def_pos":[1852,18],"def_end_pos":[1852,27]},{"full_name":"ContinuousLinearEquiv.toDiffeomorph","def_path":"Mathlib/Geometry/Manifold/Diffeomorph.lean","def_pos":[423,4],"def_end_pos":[423,17]},{"full_name":"Diffeomorph.smooth","def_path":"Mathlib/Geometry/Manifold/Diffeomorph.lean","def_pos":[128,18],"def_end_pos":[128,24]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Function.Injective.comp","def_path":"Mathlib/Logic/Function/Defs.lean","def_pos":[104,8],"def_end_pos":[104,22]},{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"Smooth.comp","def_path":"Mathlib/Geometry/Manifold/ContMDiff/Basic.lean","def_pos":[118,15],"def_end_pos":[118,26]},{"full_name":"SmoothBumpCovering.embeddingPiTangent","def_path":"Mathlib/Geometry/Manifold/WhitneyEmbedding.lean","def_pos":[53,4],"def_end_pos":[53,22]},{"full_name":"SmoothBumpCovering.embeddingPiTangent_injective","def_path":"Mathlib/Geometry/Manifold/WhitneyEmbedding.lean","def_pos":[73,8],"def_end_pos":[73,36]}]},{"state_before":"case intro\nι : Type uι\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : FiniteDimensional ℝ E\nH : Type uH\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\ninst✝² : SmoothManifoldWithCorners I M\ninst✝¹ : T2Space M\nhi : Fintype ι\ns : Set M\nf✝ : SmoothBumpCovering ι I M s\ninst✝ : Finite ι\nf : SmoothBumpCovering ι I M\nval✝ : Fintype ι\nF : Type := EuclideanSpace ℝ (Fin (finrank ℝ (ι → E × ℝ)))\nthis✝ : IsNoetherian ℝ (E × ℝ) := IsNoetherian.iff_fg.mpr inferInstance\nthis : FiniteDimensional ℝ (ι → E × ℝ) := IsNoetherian.iff_fg.mp inferInstance\neEF : (ι → E × ℝ) ≃L[ℝ] F := ContinuousLinearEquiv.ofFinrankEq ⋯\nx : M\n⊢ Injective ⇑(mfderiv I 𝓘(ℝ, EuclideanSpace ℝ (Fin (finrank ℝ (ι → E × ℝ)))) (⇑eEF ∘ ⇑f.embeddingPiTangent) x)","state_after":"case intro\nι : Type uι\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : FiniteDimensional ℝ E\nH : Type uH\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\ninst✝² : SmoothManifoldWithCorners I M\ninst✝¹ : T2Space M\nhi : Fintype ι\ns : Set M\nf✝ : SmoothBumpCovering ι I M s\ninst✝ : Finite ι\nf : SmoothBumpCovering ι I M\nval✝ : Fintype ι\nF : Type := EuclideanSpace ℝ (Fin (finrank ℝ (ι → E × ℝ)))\nthis✝ : IsNoetherian ℝ (E × ℝ) := IsNoetherian.iff_fg.mpr inferInstance\nthis : FiniteDimensional ℝ (ι → E × ℝ) := IsNoetherian.iff_fg.mp inferInstance\neEF : (ι → E × ℝ) ≃L[ℝ] F := ContinuousLinearEquiv.ofFinrankEq ⋯\nx : M\n⊢ Injective ⇑((↑eEF).comp (mfderiv I 𝓘(ℝ, ι → E × ℝ) (⇑f.embeddingPiTangent) x))","tactic":"rw [mfderiv_comp _ eEF.differentiableAt.mdifferentiableAt\n      f.embeddingPiTangent.smooth.mdifferentiableAt,\n    eEF.mfderiv_eq]","premises":[{"full_name":"ContMDiffMap.smooth","def_path":"Mathlib/Geometry/Manifold/ContMDiffMap.lean","def_pos":[55,18],"def_end_pos":[55,24]},{"full_name":"ContinuousLinearEquiv.differentiableAt","def_path":"Mathlib/Analysis/Calculus/FDeriv/Equiv.lean","def_pos":[64,18],"def_end_pos":[64,34]},{"full_name":"ContinuousLinearEquiv.mfderiv_eq","def_path":"Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean","def_pos":[91,8],"def_end_pos":[91,18]},{"full_name":"Smooth.mdifferentiableAt","def_path":"Mathlib/Geometry/Manifold/MFDeriv/Basic.lean","def_pos":[369,8],"def_end_pos":[369,32]},{"full_name":"SmoothBumpCovering.embeddingPiTangent","def_path":"Mathlib/Geometry/Manifold/WhitneyEmbedding.lean","def_pos":[53,4],"def_end_pos":[53,22]},{"full_name":"mfderiv_comp","def_path":"Mathlib/Geometry/Manifold/MFDeriv/Basic.lean","def_pos":[635,8],"def_end_pos":[635,20]}]},{"state_before":"case intro\nι : Type uι\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : FiniteDimensional ℝ E\nH : Type uH\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\ninst✝² : SmoothManifoldWithCorners I M\ninst✝¹ : T2Space M\nhi : Fintype ι\ns : Set M\nf✝ : SmoothBumpCovering ι I M s\ninst✝ : Finite ι\nf : SmoothBumpCovering ι I M\nval✝ : Fintype ι\nF : Type := EuclideanSpace ℝ (Fin (finrank ℝ (ι → E × ℝ)))\nthis✝ : IsNoetherian ℝ (E × ℝ) := IsNoetherian.iff_fg.mpr inferInstance\nthis : FiniteDimensional ℝ (ι → E × ℝ) := IsNoetherian.iff_fg.mp inferInstance\neEF : (ι → E × ℝ) ≃L[ℝ] F := ContinuousLinearEquiv.ofFinrankEq ⋯\nx : M\n⊢ Injective ⇑((↑eEF).comp (mfderiv I 𝓘(ℝ, ι → E × ℝ) (⇑f.embeddingPiTangent) x))","state_after":"no goals","tactic":"exact eEF.injective.comp (f.embeddingPiTangent_injective_mfderiv _ trivial)","premises":[{"full_name":"ContinuousLinearEquiv.injective","def_path":"Mathlib/Topology/Algebra/Module/Basic.lean","def_pos":[1852,18],"def_end_pos":[1852,27]},{"full_name":"Function.Injective.comp","def_path":"Mathlib/Logic/Function/Defs.lean","def_pos":[104,8],"def_end_pos":[104,22]},{"full_name":"SmoothBumpCovering.embeddingPiTangent_injective_mfderiv","def_path":"Mathlib/Geometry/Manifold/WhitneyEmbedding.lean","def_pos":[102,8],"def_end_pos":[102,44]},{"full_name":"trivial","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[645,34],"def_end_pos":[645,41]}]}]}
+{"url":"Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean","commit":"","full_name":"Polynomial.coprime_of_root_cyclotomic","start":[551,0],"end":[565,25],"file_path":"Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean","tactics":[{"state_before":"n : ℕ\nhpos : 0 < n\np : ℕ\nhprime : Fact (Nat.Prime p)\na : ℕ\nhroot : (cyclotomic n (ZMod p)).IsRoot ((Nat.castRingHom (ZMod p)) a)\n⊢ a.Coprime p","state_after":"case a\nn : ℕ\nhpos : 0 < n\np : ℕ\nhprime : Fact (Nat.Prime p)\na : ℕ\nhroot : (cyclotomic n (ZMod p)).IsRoot ((Nat.castRingHom (ZMod p)) a)\n⊢ p.Coprime a","tactic":"apply Nat.Coprime.symm","premises":[{"full_name":"Nat.Coprime.symm","def_path":".lake/packages/batteries/Batteries/Data/Nat/Gcd.lean","def_pos":[28,8],"def_end_pos":[28,20]}]},{"state_before":"case a\nn : ℕ\nhpos : 0 < n\np : ℕ\nhprime : Fact (Nat.Prime p)\na : ℕ\nhroot : (cyclotomic n (ZMod p)).IsRoot ((Nat.castRingHom (ZMod p)) a)\n⊢ p.Coprime a","state_after":"case a\nn : ℕ\nhpos : 0 < n\np : ℕ\nhprime : Fact (Nat.Prime p)\na : ℕ\nhroot : (cyclotomic n (ZMod p)).IsRoot ((Nat.castRingHom (ZMod p)) a)\n⊢ ¬p ∣ a","tactic":"rw [hprime.1.coprime_iff_not_dvd]","premises":[{"full_name":"Fact.out","def_path":"Mathlib/Logic/Basic.lean","def_pos":[92,2],"def_end_pos":[92,5]},{"full_name":"Nat.Prime.coprime_iff_not_dvd","def_path":"Mathlib/Data/Nat/Prime/Defs.lean","def_pos":[406,8],"def_end_pos":[406,33]}]},{"state_before":"case a\nn : ℕ\nhpos : 0 < n\np : ℕ\nhprime : Fact (Nat.Prime p)\na : ℕ\nhroot : (cyclotomic n (ZMod p)).IsRoot ((Nat.castRingHom (ZMod p)) a)\n⊢ ¬p ∣ a","state_after":"case a\nn : ℕ\nhpos : 0 < n\np : ℕ\nhprime : Fact (Nat.Prime p)\na : ℕ\nhroot : (cyclotomic n (ZMod p)).IsRoot ((Nat.castRingHom (ZMod p)) a)\nh : p ∣ a\n⊢ False","tactic":"intro h","premises":[]},{"state_before":"case a\nn : ℕ\nhpos : 0 < n\np : ℕ\nhprime : Fact (Nat.Prime p)\na : ℕ\nhroot : (cyclotomic n (ZMod p)).IsRoot ((Nat.castRingHom (ZMod p)) a)\nh : p ∣ a\n⊢ False","state_after":"case a\nn : ℕ\nhpos : 0 < n\np : ℕ\nhprime : Fact (Nat.Prime p)\na : ℕ\nhroot : (cyclotomic n (ZMod p)).IsRoot ((Nat.castRingHom (ZMod p)) a)\nh : ↑a = 0\n⊢ False","tactic":"replace h := (ZMod.natCast_zmod_eq_zero_iff_dvd a p).2 h","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"ZMod.natCast_zmod_eq_zero_iff_dvd","def_path":"Mathlib/Data/ZMod/Basic.lean","def_pos":[549,8],"def_end_pos":[549,36]}]},{"state_before":"case a\nn : ℕ\nhpos : 0 < n\np : ℕ\nhprime : Fact (Nat.Prime p)\na : ℕ\nhroot : (cyclotomic n (ZMod p)).IsRoot ((Nat.castRingHom (ZMod p)) a)\nh : ↑a = 0\n⊢ False","state_after":"case a\nn : ℕ\nhpos : 0 < n\np : ℕ\nhprime : Fact (Nat.Prime p)\na : ℕ\nhroot : (cyclotomic n (ZMod p)).coeff 0 = 0\nh : ↑a = 0\n⊢ False","tactic":"rw [IsRoot.def, eq_natCast, h, ← coeff_zero_eq_eval_zero] at hroot","premises":[{"full_name":"Polynomial.IsRoot.def","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[424,8],"def_end_pos":[424,18]},{"full_name":"Polynomial.coeff_zero_eq_eval_zero","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[430,8],"def_end_pos":[430,31]},{"full_name":"eq_natCast","def_path":"Mathlib/Data/Nat/Cast/Basic.lean","def_pos":[159,8],"def_end_pos":[159,18]}]},{"state_before":"case a\nn : ℕ\nhpos : 0 < n\np : ℕ\nhprime : Fact (Nat.Prime p)\na : ℕ\nhroot : (cyclotomic n (ZMod p)).coeff 0 = 0\nh : ↑a = 0\n⊢ False","state_after":"case pos\nn : ℕ\nhpos : 0 < n\np : ℕ\nhprime : Fact (Nat.Prime p)\na : ℕ\nhroot : (cyclotomic n (ZMod p)).coeff 0 = 0\nh : ↑a = 0\nhone : n = 1\n⊢ False\n\ncase neg\nn : ℕ\nhpos : 0 < n\np : ℕ\nhprime : Fact (Nat.Prime p)\na : ℕ\nhroot : (cyclotomic n (ZMod p)).coeff 0 = 0\nh : ↑a = 0\nhone : ¬n = 1\n⊢ False","tactic":"by_cases hone : n = 1","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case neg\nn : ℕ\nhpos : 0 < n\np : ℕ\nhprime : Fact (Nat.Prime p)\na : ℕ\nhroot : (cyclotomic n (ZMod p)).coeff 0 = 0\nh : ↑a = 0\nhone : ¬n = 1\n⊢ False","state_after":"case neg\nn : ℕ\nhpos : 0 < n\np : ℕ\nhprime : Fact (Nat.Prime p)\na : ℕ\nhroot : 1 = 0\nh : ↑a = 0\nhone : ¬n = 1\n⊢ False","tactic":"rw [cyclotomic_coeff_zero (ZMod p) (Nat.succ_le_of_lt\n    (lt_of_le_of_ne (Nat.succ_le_of_lt hpos) (Ne.symm hone)))] at hroot","premises":[{"full_name":"Nat.succ_le_of_lt","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[348,8],"def_end_pos":[348,21]},{"full_name":"Ne.symm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[704,8],"def_end_pos":[704,15]},{"full_name":"Polynomial.cyclotomic_coeff_zero","def_path":"Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean","def_pos":[522,8],"def_end_pos":[522,29]},{"full_name":"ZMod","def_path":"Mathlib/Data/ZMod/Defs.lean","def_pos":[89,4],"def_end_pos":[89,8]},{"full_name":"lt_of_le_of_ne","def_path":"Mathlib/Order/Defs.lean","def_pos":[164,8],"def_end_pos":[164,22]}]},{"state_before":"case neg\nn : ℕ\nhpos : 0 < n\np : ℕ\nhprime : Fact (Nat.Prime p)\na : ℕ\nhroot : 1 = 0\nh : ↑a = 0\nhone : ¬n = 1\n⊢ False","state_after":"no goals","tactic":"exact one_ne_zero hroot","premises":[{"full_name":"one_ne_zero","def_path":"Mathlib/Algebra/NeZero.lean","def_pos":[49,14],"def_end_pos":[49,25]}]}]}
+{"url":"Mathlib/RingTheory/OreLocalization/Basic.lean","commit":"","full_name":"AddOreLocalization.oreSub_zero_vadd","start":[543,0],"end":[546,69],"file_path":"Mathlib/RingTheory/OreLocalization/Basic.lean","tactics":[{"state_before":"R : Type u_1\nR' : Type u_2\nM : Type u_3\nX : Type u_4\ninst✝¹² : Monoid M\nS : Submonoid M\ninst✝¹¹ : OreSet S\ninst✝¹⁰ : MulAction M X\ninst✝⁹ : SMul R X\ninst✝⁸ : SMul R M\ninst✝⁷ : IsScalarTower R M M\ninst✝⁶ : IsScalarTower R M X\ninst✝⁵ : SMul R' X\ninst✝⁴ : SMul R' M\ninst✝³ : IsScalarTower R' M M\ninst✝² : IsScalarTower R' M X\ninst✝¹ : SMul R R'\ninst✝ : IsScalarTower R R' M\nr : M\nx : OreLocalization S X\n⊢ (r /ₒ 1) • x = r • x","state_after":"case c\nR : Type u_1\nR' : Type u_2\nM : Type u_3\nX : Type u_4\ninst✝¹² : Monoid M\nS : Submonoid M\ninst✝¹¹ : OreSet S\ninst✝¹⁰ : MulAction M X\ninst✝⁹ : SMul R X\ninst✝⁸ : SMul R M\ninst✝⁷ : IsScalarTower R M M\ninst✝⁶ : IsScalarTower R M X\ninst✝⁵ : SMul R' X\ninst✝⁴ : SMul R' M\ninst✝³ : IsScalarTower R' M M\ninst✝² : IsScalarTower R' M X\ninst✝¹ : SMul R R'\ninst✝ : IsScalarTower R R' M\nr : M\nr' : X\ns : ↥S\n⊢ (r /ₒ 1) • (r' /ₒ s) = r • (r' /ₒ s)","tactic":"induction' x using OreLocalization.ind with r' s","premises":[{"full_name":"OreLocalization.ind","def_path":"Mathlib/RingTheory/OreLocalization/Basic.lean","def_pos":[103,18],"def_end_pos":[103,21]}]},{"state_before":"case c\nR : Type u_1\nR' : Type u_2\nM : Type u_3\nX : Type u_4\ninst✝¹² : Monoid M\nS : Submonoid M\ninst✝¹¹ : OreSet S\ninst✝¹⁰ : MulAction M X\ninst✝⁹ : SMul R X\ninst✝⁸ : SMul R M\ninst✝⁷ : IsScalarTower R M M\ninst✝⁶ : IsScalarTower R M X\ninst✝⁵ : SMul R' X\ninst✝⁴ : SMul R' M\ninst✝³ : IsScalarTower R' M M\ninst✝² : IsScalarTower R' M X\ninst✝¹ : SMul R R'\ninst✝ : IsScalarTower R R' M\nr : M\nr' : X\ns : ↥S\n⊢ (r /ₒ 1) • (r' /ₒ s) = r • (r' /ₒ s)","state_after":"no goals","tactic":"rw [smul_oreDiv, oreDiv_smul_oreDiv, mul_one, smul_eq_mul, mul_one]","premises":[{"full_name":"OreLocalization.oreDiv_smul_oreDiv","def_path":"Mathlib/RingTheory/OreLocalization/Basic.lean","def_pos":[278,8],"def_end_pos":[278,26]},{"full_name":"OreLocalization.smul_oreDiv","def_path":"Mathlib/RingTheory/OreLocalization/Basic.lean","def_pos":[540,8],"def_end_pos":[540,19]},{"full_name":"mul_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[413,8],"def_end_pos":[413,15]},{"full_name":"smul_eq_mul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[79,6],"def_end_pos":[79,17]}]}]}
+{"url":"Mathlib/Analysis/SpecialFunctions/Integrals.lean","commit":"","full_name":"integral_cos_pow_aux","start":[653,0],"end":[674,57],"file_path":"Mathlib/Analysis/SpecialFunctions/Integrals.lean","tactics":[{"state_before":"a b : ℝ\nn : ℕ\n⊢ ∫ (x : ℝ) in a..b, cos x ^ (n + 2) =\n    (cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a + (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ n) -\n      (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ (n + 2)","state_after":"a b : ℝ\nn : ℕ\nC : ℝ := cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a\n⊢ ∫ (x : ℝ) in a..b, cos x ^ (n + 2) =\n    (cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a + (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ n) -\n      (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ (n + 2)","tactic":"let C := cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a","premises":[{"full_name":"Real.cos","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[112,11],"def_end_pos":[112,14]},{"full_name":"Real.sin","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[107,11],"def_end_pos":[107,14]}]},{"state_before":"a b : ℝ\nn : ℕ\nC : ℝ := cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a\n⊢ ∫ (x : ℝ) in a..b, cos x ^ (n + 2) =\n    (cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a + (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ n) -\n      (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ (n + 2)","state_after":"a b : ℝ\nn : ℕ\nC : ℝ := cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a\nh : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)\n⊢ ∫ (x : ℝ) in a..b, cos x ^ (n + 2) =\n    (cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a + (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ n) -\n      (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ (n + 2)","tactic":"have h : ∀ α β γ : ℝ, β * α * γ * α = β * (α * α * γ) := fun α β γ => by ring","premises":[{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]}]},{"state_before":"a b : ℝ\nn : ℕ\nC : ℝ := cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a\nh : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)\n⊢ ∫ (x : ℝ) in a..b, cos x ^ (n + 2) =\n    (cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a + (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ n) -\n      (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ (n + 2)","state_after":"a b : ℝ\nn : ℕ\nC : ℝ := cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a\nh : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)\nhu : ∀ x ∈ [[a, b]], HasDerivAt (fun y => cos y ^ (n + 1)) (-↑(n + 1) * sin x * cos x ^ n) x\n⊢ ∫ (x : ℝ) in a..b, cos x ^ (n + 2) =\n    (cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a + (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ n) -\n      (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ (n + 2)","tactic":"have hu : ∀ x ∈ [[a, b]],\n      HasDerivAt (fun y => cos y ^ (n + 1)) (-(n + 1 : ℕ) * sin x * cos x ^ n) x :=\n    fun x _ => by\n      simpa only [mul_right_comm, neg_mul, mul_neg] using (hasDerivAt_cos x).pow (n + 1)","premises":[{"full_name":"HasDerivAt","def_path":"Mathlib/Analysis/Calculus/Deriv/Basic.lean","def_pos":[121,4],"def_end_pos":[121,14]},{"full_name":"HasDerivAt.pow","def_path":"Mathlib/Analysis/Calculus/Deriv/Pow.lean","def_pos":[86,8],"def_end_pos":[86,22]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]},{"full_name":"Real.cos","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[112,11],"def_end_pos":[112,14]},{"full_name":"Real.hasDerivAt_cos","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean","def_pos":[499,8],"def_end_pos":[499,22]},{"full_name":"Real.sin","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[107,11],"def_end_pos":[107,14]},{"full_name":"Set.uIcc","def_path":"Mathlib/Order/Interval/Set/UnorderedInterval.lean","def_pos":[52,4],"def_end_pos":[52,8]},{"full_name":"mul_neg","def_path":"Mathlib/Algebra/Ring/Defs.lean","def_pos":[272,8],"def_end_pos":[272,15]},{"full_name":"mul_right_comm","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[156,8],"def_end_pos":[156,22]},{"full_name":"neg_mul","def_path":"Mathlib/Algebra/Ring/Defs.lean","def_pos":[268,8],"def_end_pos":[268,15]}]},{"state_before":"a b : ℝ\nn : ℕ\nC : ℝ := cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a\nh : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)\nhu : ∀ x ∈ [[a, b]], HasDerivAt (fun y => cos y ^ (n + 1)) (-↑(n + 1) * sin x * cos x ^ n) x\n⊢ ∫ (x : ℝ) in a..b, cos x ^ (n + 2) =\n    (cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a + (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ n) -\n      (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ (n + 2)","state_after":"a b : ℝ\nn : ℕ\nC : ℝ := cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a\nh : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)\nhu : ∀ x ∈ [[a, b]], HasDerivAt (fun y => cos y ^ (n + 1)) (-↑(n + 1) * sin x * cos x ^ n) x\nhv : ∀ x ∈ [[a, b]], HasDerivAt sin (cos x) x\n⊢ ∫ (x : ℝ) in a..b, cos x ^ (n + 2) =\n    (cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a + (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ n) -\n      (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ (n + 2)","tactic":"have hv : ∀ x ∈ [[a, b]], HasDerivAt sin (cos x) x := fun x _ => hasDerivAt_sin x","premises":[{"full_name":"HasDerivAt","def_path":"Mathlib/Analysis/Calculus/Deriv/Basic.lean","def_pos":[121,4],"def_end_pos":[121,14]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Real.cos","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[112,11],"def_end_pos":[112,14]},{"full_name":"Real.hasDerivAt_sin","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean","def_pos":[481,8],"def_end_pos":[481,22]},{"full_name":"Real.sin","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[107,11],"def_end_pos":[107,14]},{"full_name":"Set.uIcc","def_path":"Mathlib/Order/Interval/Set/UnorderedInterval.lean","def_pos":[52,4],"def_end_pos":[52,8]}]},{"state_before":"a b : ℝ\nn : ℕ\nC : ℝ := cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a\nh : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)\nhu : ∀ x ∈ [[a, b]], HasDerivAt (fun y => cos y ^ (n + 1)) (-↑(n + 1) * sin x * cos x ^ n) x\nhv : ∀ x ∈ [[a, b]], HasDerivAt sin (cos x) x\n⊢ ∫ (x : ℝ) in a..b, cos x ^ (n + 2) =\n    (cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a + (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ n) -\n      (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ (n + 2)","state_after":"case refine_3\na b : ℝ\nn : ℕ\nC : ℝ := cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a\nh : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)\nhu : ∀ x ∈ [[a, b]], HasDerivAt (fun y => cos y ^ (n + 1)) (-↑(n + 1) * sin x * cos x ^ n) x\nhv : ∀ x ∈ [[a, b]], HasDerivAt sin (cos x) x\nH :\n  ∫ (x : ℝ) in a..b, cos x ^ (n + 1) * cos x =\n    cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a - ∫ (x : ℝ) in a..b, -↑(n + 1) * sin x * cos x ^ n * sin x\n⊢ ∫ (x : ℝ) in a..b, cos x ^ (n + 2) =\n    (cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a + (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ n) -\n      (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ (n + 2)\n\ncase refine_1\na b : ℝ\nn : ℕ\nC : ℝ := cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a\nh : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)\nhu : ∀ x ∈ [[a, b]], HasDerivAt (fun y => cos y ^ (n + 1)) (-↑(n + 1) * sin x * cos x ^ n) x\nhv : ∀ x ∈ [[a, b]], HasDerivAt sin (cos x) x\n⊢ IntervalIntegrable (fun x => -↑(n + 1) * sin x * cos x ^ n) MeasureTheory.volume a b\n\ncase refine_2\na b : ℝ\nn : ℕ\nC : ℝ := cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a\nh : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)\nhu : ∀ x ∈ [[a, b]], HasDerivAt (fun y => cos y ^ (n + 1)) (-↑(n + 1) * sin x * cos x ^ n) x\nhv : ∀ x ∈ [[a, b]], HasDerivAt sin (cos x) x\n⊢ IntervalIntegrable cos MeasureTheory.volume a b","tactic":"have H := integral_mul_deriv_eq_deriv_mul hu hv ?_ ?_","premises":[{"full_name":"intervalIntegral.integral_mul_deriv_eq_deriv_mul","def_path":"Mathlib/MeasureTheory/Integral/FundThmCalculus.lean","def_pos":[1376,8],"def_end_pos":[1376,39]}]},{"state_before":"case refine_1\na b : ℝ\nn : ℕ\nC : ℝ := cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a\nh : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)\nhu : ∀ x ∈ [[a, b]], HasDerivAt (fun y => cos y ^ (n + 1)) (-↑(n + 1) * sin x * cos x ^ n) x\nhv : ∀ x ∈ [[a, b]], HasDerivAt sin (cos x) x\n⊢ IntervalIntegrable (fun x => -↑(n + 1) * sin x * cos x ^ n) MeasureTheory.volume a b\n\ncase refine_2\na b : ℝ\nn : ℕ\nC : ℝ := cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a\nh : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)\nhu : ∀ x ∈ [[a, b]], HasDerivAt (fun y => cos y ^ (n + 1)) (-↑(n + 1) * sin x * cos x ^ n) x\nhv : ∀ x ∈ [[a, b]], HasDerivAt sin (cos x) x\n⊢ IntervalIntegrable cos MeasureTheory.volume a b","state_after":"no goals","tactic":"all_goals apply Continuous.intervalIntegrable; fun_prop","premises":[{"full_name":"Continuous.intervalIntegrable","def_path":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","def_pos":[337,8],"def_end_pos":[337,37]}]}]}
+{"url":"Mathlib/Data/Int/Bitwise.lean","commit":"","full_name":"Int.bodd_subNatNat","start":[122,0],"end":[127,27],"file_path":"Mathlib/Data/Int/Bitwise.lean","tactics":[{"state_before":"m n : ℕ\n⊢ (subNatNat m n).bodd = xor m.bodd n.bodd","state_after":"no goals","tactic":"apply subNatNat_elim m n fun m n i => bodd i = xor m.bodd n.bodd <;>\n  intros i j <;>\n  simp only [Int.bodd, Int.bodd_coe, Nat.bodd_add] <;>\n  cases Nat.bodd i <;> simp","premises":[{"full_name":"Int.bodd","def_path":"Mathlib/Data/Int/Bitwise.lean","def_pos":[28,4],"def_end_pos":[28,8]},{"full_name":"Int.bodd_coe","def_path":"Mathlib/Data/Int/Bitwise.lean","def_pos":[119,8],"def_end_pos":[119,16]},{"full_name":"Int.subNatNat_elim","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean","def_pos":[94,8],"def_end_pos":[94,22]},{"full_name":"Nat.bodd","def_path":"Mathlib/Data/Nat/Bits.lean","def_pos":[47,4],"def_end_pos":[47,8]},{"full_name":"Nat.bodd_add","def_path":"Mathlib/Data/Nat/Bits.lean","def_pos":[62,6],"def_end_pos":[62,14]},{"full_name":"xor","def_path":".lake/packages/lean4/src/lean/Init/Data/Bool.lean","def_pos":[10,7],"def_end_pos":[10,10]}]}]}
+{"url":"Mathlib/Algebra/Order/Module/Defs.lean","commit":"","full_name":"smul_le_smul_of_nonpos_left","start":[808,0],"end":[810,63],"file_path":"Mathlib/Algebra/Order/Module/Defs.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\na a₁ a₂ : α\nb b₁ b₂ : β\ninst✝³ : OrderedRing α\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : Module α β\ninst✝ : PosSMulMono α β\nh : b₁ ≤ b₂\nha : a ≤ 0\n⊢ a • b₂ ≤ a • b₁","state_after":"α : Type u_1\nβ : Type u_2\na a₁ a₂ : α\nb b₁ b₂ : β\ninst✝³ : OrderedRing α\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : Module α β\ninst✝ : PosSMulMono α β\nh : b₁ ≤ b₂\nha : a ≤ 0\n⊢ -a • b₁ ≤ -a • b₂","tactic":"rw [← neg_neg a, neg_smul, neg_smul (-a), neg_le_neg_iff]","premises":[{"full_name":"neg_le_neg_iff","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Basic.lean","def_pos":[216,2],"def_end_pos":[216,13]},{"full_name":"neg_neg","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[733,2],"def_end_pos":[733,13]},{"full_name":"neg_smul","def_path":"Mathlib/Algebra/Module/Defs.lean","def_pos":[228,8],"def_end_pos":[228,16]}]},{"state_before":"α : Type u_1\nβ : Type u_2\na a₁ a₂ : α\nb b₁ b₂ : β\ninst✝³ : OrderedRing α\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : Module α β\ninst✝ : PosSMulMono α β\nh : b₁ ≤ b₂\nha : a ≤ 0\n⊢ -a • b₁ ≤ -a • b₂","state_after":"no goals","tactic":"exact smul_le_smul_of_nonneg_left h (neg_nonneg_of_nonpos ha)","premises":[{"full_name":"neg_nonneg_of_nonpos","def_path":"Mathlib/Algebra/Order/Group/Defs.lean","def_pos":[192,14],"def_end_pos":[192,34]},{"full_name":"smul_le_smul_of_nonneg_left","def_path":"Mathlib/Algebra/Order/Module/Defs.lean","def_pos":[272,16],"def_end_pos":[272,43]}]}]}
+{"url":"Mathlib/CategoryTheory/Subterminal.lean","commit":"","full_name":"CategoryTheory.subterminalsEquivMonoOverTerminal_functor_obj_obj","start":[136,0],"end":[159,41],"file_path":"Mathlib/CategoryTheory/Subterminal.lean","tactics":[{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nA : C\ninst✝ : HasTerminal C\nX✝ Y✝ : Subterminals C\nf : X✝ ⟶ Y✝\n⊢ f ≫ MonoOver.arrow ((fun X => { obj := Over.mk (terminal.from X.obj), property := ⋯ }) Y✝) =\n    MonoOver.arrow ((fun X => { obj := Over.mk (terminal.from X.obj), property := ⋯ }) X✝)","state_after":"no goals","tactic":"ext1 ⟨⟨⟩⟩","premises":[]},{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nA : C\ninst✝ : HasTerminal C\nX : MonoOver (⊤_ C)\nZ : C\nf g : Z ⟶ X.obj.left\n⊢ f = g","state_after":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nA : C\ninst✝ : HasTerminal C\nX : MonoOver (⊤_ C)\nZ : C\nf g : Z ⟶ X.obj.left\n⊢ f ≫ X.arrow = g ≫ X.arrow","tactic":"rw [← cancel_mono 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X.obj.left, property := ⋯ }, map := fun {X Y} f => f.left, map_id := ⋯,\n                map_comp := ⋯ }).map\n          f","state_after":"no goals","tactic":"subsingleton","premises":[]},{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nA : C\ninst✝ : HasTerminal C\n⊢ ∀ {X Y : MonoOver (⊤_ C)} (f : X ⟶ Y),\n    ({ obj := fun X => { obj := X.obj.left, property := ⋯ }, map := fun {X Y} f => f.left, map_id := ⋯,\n                map_comp := ⋯ } ⋙\n              { obj := fun X => { obj := Over.mk (terminal.from X.obj), property := ⋯ },\n                map := fun {X Y} f => MonoOver.homMk f ⋯, map_id := ⋯, map_comp := ⋯ }).map\n          f ≫\n        ((fun X =>\n              MonoOver.isoMk\n                (Iso.refl\n                  (({ obj := fun X => { obj := X.obj.left, property := ⋯ }, map := fun {X Y} f => f.left, map_id := ⋯,\n                              map_comp := ⋯ } ⋙\n                            { obj := fun X => { obj := Over.mk (terminal.from X.obj), property 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+{"url":"Mathlib/Algebra/Homology/ShortComplex/FunctorEquivalence.lean","commit":"","full_name":"CategoryTheory.ShortComplex.FunctorEquivalence.inverse_map_τ₃","start":[38,0],"end":[46,33],"file_path":"Mathlib/Algebra/Homology/ShortComplex/FunctorEquivalence.lean","tactics":[{"state_before":"J : Type u_1\nC : Type u_2\ninst✝² : Category.{?u.17393, u_1} J\ninst✝¹ : Category.{?u.17397, u_2} C\ninst✝ : HasZeroMorphisms C\nF : J ⥤ ShortComplex C\n⊢ whiskerLeft F π₁Toπ₂ ≫ whiskerLeft F π₂Toπ₃ = 0","state_after":"no goals","tactic":"aesop_cat","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]}]},{"state_before":"J : Type u_1\nC : Type u_2\ninst✝² : Category.{?u.17393, u_1} J\ninst✝¹ : Category.{?u.17397, u_2} C\ninst✝ : HasZeroMorphisms C\nX✝ Y✝ : J ⥤ ShortComplex C\nφ : X✝ ⟶ Y✝\n⊢ whiskerRight φ π₁ ≫ ((fun F => mk (whiskerLeft F π₁Toπ₂) (whiskerLeft F π₂Toπ₃) ⋯) Y✝).f =\n    ((fun F => mk (whiskerLeft F π₁Toπ₂) (whiskerLeft F π₂Toπ₃) ⋯) X✝).f ≫ whiskerRight φ π₂","state_after":"no goals","tactic":"aesop_cat","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]}]},{"state_before":"J : Type u_1\nC : Type u_2\ninst✝² : Category.{?u.17393, u_1} J\ninst✝¹ : Category.{?u.17397, u_2} C\ninst✝ : HasZeroMorphisms C\nX✝ Y✝ : J ⥤ ShortComplex C\nφ : X✝ ⟶ Y✝\n⊢ whiskerRight φ π₂ ≫ ((fun F => mk (whiskerLeft F π₁Toπ₂) (whiskerLeft F π₂Toπ₃) ⋯) Y✝).g =\n    ((fun F => mk (whiskerLeft F π₁Toπ₂) (whiskerLeft F π₂Toπ₃) ⋯) X✝).g ≫ whiskerRight φ π₃","state_after":"no goals","tactic":"aesop_cat","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]}]}]}
+{"url":"Mathlib/MeasureTheory/Measure/Haar/Unique.lean","commit":"","full_name":"MeasureTheory.Measure.measure_isHaarMeasure_eq_smul_of_isEverywherePos","start":[694,0],"end":[794,25],"file_path":"Mathlib/MeasureTheory/Measure/Haar/Unique.lean","tactics":[{"state_before":"G : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\n⊢ μ' s = μ'.haarScalarFactor μ • μ s","state_after":"G : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\nν : Measure G := μ'.haarScalarFactor μ • μ\n⊢ μ' s = μ'.haarScalarFactor μ • μ s","tactic":"let ν := haarScalarFactor μ' μ • μ","premises":[{"full_name":"MeasureTheory.Measure.haarScalarFactor","def_path":"Mathlib/MeasureTheory/Measure/Haar/Unique.lean","def_pos":[294,18],"def_end_pos":[294,34]}]},{"state_before":"G : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\nν : Measure G := μ'.haarScalarFactor μ • μ\n⊢ μ' s = μ'.haarScalarFactor μ • μ s","state_after":"G : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\nν : Measure G := μ'.haarScalarFactor μ • μ\n⊢ μ' s = ν s","tactic":"change μ' s = ν s","premises":[]},{"state_before":"G : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\nν : Measure G := μ'.haarScalarFactor μ • μ\n⊢ μ' s = ν s","state_after":"case intro.intro.intro\nG : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\nν : Measure G := μ'.haarScalarFactor μ • μ\nk : Set G\nk_comp : IsCompact k\nk_closed : IsClosed k\nk_mem : k ∈ 𝓝 1\n⊢ μ' s = ν s","tactic":"obtain ⟨k, k_comp, k_closed, k_mem⟩ : ∃ k, IsCompact k ∧ IsClosed k ∧ k ∈ 𝓝 (1 : G) := by\n    rcases exists_compact_mem_nhds (1 : G) with ⟨k, hk, hmem⟩\n    exact ⟨closure k, hk.closure, isClosed_closure, mem_of_superset hmem subset_closure⟩","premises":[{"full_name":"And","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[516,10],"def_end_pos":[516,13]},{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Filter.mem_of_superset","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[139,8],"def_end_pos":[139,23]},{"full_name":"IsClosed","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[93,6],"def_end_pos":[93,14]},{"full_name":"IsCompact","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[254,4],"def_end_pos":[254,13]},{"full_name":"IsCompact.closure","def_path":"Mathlib/Topology/Separation.lean","def_pos":[1089,18],"def_end_pos":[1089,35]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"WeaklyLocallyCompactSpace.exists_compact_mem_nhds","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[274,2],"def_end_pos":[274,25]},{"full_name":"closure","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[112,4],"def_end_pos":[112,11]},{"full_name":"isClosed_closure","def_path":"Mathlib/Topology/Basic.lean","def_pos":[344,8],"def_end_pos":[344,24]},{"full_name":"nhds","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[113,16],"def_end_pos":[113,20]},{"full_name":"subset_closure","def_path":"Mathlib/Topology/Basic.lean","def_pos":[347,8],"def_end_pos":[347,22]}]},{"state_before":"case intro.intro.intro\nG : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\nν : Measure G := μ'.haarScalarFactor μ • μ\nk : Set G\nk_comp : IsCompact k\nk_closed : IsClosed k\nk_mem : k ∈ 𝓝 1\n⊢ μ' s = ν s","state_after":"case intro.intro.intro\nG : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\nν : Measure G := μ'.haarScalarFactor μ • μ\nk : Set G\nk_comp : IsCompact k\nk_closed : IsClosed k\nk_mem : k ∈ 𝓝 1\none_k : 1 ∈ k\n⊢ μ' s = ν s","tactic":"have one_k : 1 ∈ k := mem_of_mem_nhds k_mem","premises":[{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"mem_of_mem_nhds","def_path":"Mathlib/Topology/Basic.lean","def_pos":[737,8],"def_end_pos":[737,23]}]},{"state_before":"case intro.intro.intro\nG : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\nν : Measure G := μ'.haarScalarFactor μ • μ\nk : Set G\nk_comp : IsCompact k\nk_closed : IsClosed k\nk_mem : k ∈ 𝓝 1\none_k : 1 ∈ k\n⊢ μ' s = ν s","state_after":"case intro.intro.intro\nG : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\nν : Measure G := μ'.haarScalarFactor μ • μ\nk : Set G\nk_comp : IsCompact k\nk_closed : IsClosed k\nk_mem : k ∈ 𝓝 1\none_k : 1 ∈ k\nA : Set (Set G) := {t | t ⊆ s ∧ t.PairwiseDisjoint fun x => x • k}\n⊢ μ' s = ν s","tactic":"let A : Set (Set G) := {t | t ⊆ s ∧ PairwiseDisjoint t (fun x ↦ x • k)}","premises":[{"full_name":"And","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[516,10],"def_end_pos":[516,13]},{"full_name":"HasSubset.Subset","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[384,2],"def_end_pos":[384,8]},{"full_name":"Set","def_path":"Mathlib/Init/Set.lean","def_pos":[53,4],"def_end_pos":[53,7]},{"full_name":"Set.PairwiseDisjoint","def_path":"Mathlib/Data/Set/Pairwise/Basic.lean","def_pos":[216,4],"def_end_pos":[216,20]},{"full_name":"setOf","def_path":"Mathlib/Init/Set.lean","def_pos":[56,4],"def_end_pos":[56,9]}]},{"state_before":"case intro.intro.intro.intro.intro\nG : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\nν : Measure G := μ'.haarScalarFactor μ • μ\nk : Set G\nk_comp : IsCompact k\nk_closed : IsClosed k\nk_mem : k ∈ 𝓝 1\none_k : 1 ∈ k\nA : Set (Set G) := {t | t ⊆ s ∧ t.PairwiseDisjoint fun x => x • k}\nm : Set G\nmA : m ∈ A\nm_max : ∀ a ∈ A, m ⊆ a → a = m\n⊢ μ' s = ν s","state_after":"case intro.intro.intro.intro.intro\nG : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\nν : Measure G := μ'.haarScalarFactor μ • μ\nk : Set G\nk_comp : IsCompact k\nk_closed : IsClosed k\nk_mem : k ∈ 𝓝 1\none_k : 1 ∈ k\nA : Set (Set G) := {t | t ⊆ s ∧ t.PairwiseDisjoint fun x => x • k}\nm : Set G\nm_max : ∀ a ∈ A, m ⊆ a → a = m\nmA : m ⊆ s ∧ m.PairwiseDisjoint fun x => x • k\n⊢ μ' s = ν s","tactic":"change m ⊆ s ∧ PairwiseDisjoint m (fun x ↦ x • k) at mA","premises":[{"full_name":"And","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[516,10],"def_end_pos":[516,13]},{"full_name":"HasSubset.Subset","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[384,2],"def_end_pos":[384,8]},{"full_name":"Set.PairwiseDisjoint","def_path":"Mathlib/Data/Set/Pairwise/Basic.lean","def_pos":[216,4],"def_end_pos":[216,20]}]},{"state_before":"case intro.intro.intro.intro.intro\nG : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\nν : Measure G := μ'.haarScalarFactor μ • μ\nk : Set G\nk_comp : IsCompact k\nk_closed : IsClosed k\nk_mem : k ∈ 𝓝 1\none_k : 1 ∈ k\nA : Set (Set G) := {t | t ⊆ s ∧ t.PairwiseDisjoint fun x => x • k}\nm : Set G\nm_max : ∀ a ∈ A, m ⊆ a → a = m\nmA : m ⊆ s ∧ m.PairwiseDisjoint fun x => x • k\nsm : s ⊆ ⋃ x ∈ m, x • (k * k⁻¹)\n⊢ μ' s = ν s","state_after":"case intro.intro.intro.intro.intro.inl\nG : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\nν : Measure G := μ'.haarScalarFactor μ • μ\nk : Set G\nk_comp : IsCompact k\nk_closed : IsClosed k\nk_mem : k ∈ 𝓝 1\none_k : 1 ∈ k\nA : Set (Set G) := {t | t ⊆ s ∧ t.PairwiseDisjoint fun x => x • k}\nm_max : ∀ a ∈ A, ∅ ⊆ a → a = ∅\nmA : ∅ ⊆ s ∧ ∅.PairwiseDisjoint fun x => x • k\nsm : s ⊆ ⋃ x ∈ ∅, x • (k * k⁻¹)\n⊢ μ' s = ν s\n\ncase intro.intro.intro.intro.intro.inr\nG : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\nν : Measure G := μ'.haarScalarFactor μ • μ\nk : Set G\nk_comp : IsCompact k\nk_closed : IsClosed k\nk_mem : k ∈ 𝓝 1\none_k : 1 ∈ k\nA : Set (Set G) := {t | t ⊆ s ∧ t.PairwiseDisjoint fun x => x • k}\nm : Set G\nm_max : ∀ a ∈ A, m ⊆ a → a = m\nmA : m ⊆ s ∧ m.PairwiseDisjoint fun x => x • k\nsm : s ⊆ ⋃ x ∈ m, x • (k * k⁻¹)\nhm : m.Nonempty\n⊢ μ' s = ν s","tactic":"rcases eq_empty_or_nonempty m with rfl|hm","premises":[{"full_name":"Set.eq_empty_or_nonempty","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[506,8],"def_end_pos":[506,28]}]},{"state_before":"case intro.intro.intro.intro.intro.inr\nG : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\nν : Measure G := μ'.haarScalarFactor μ • μ\nk : Set G\nk_comp : IsCompact k\nk_closed : IsClosed k\nk_mem : k ∈ 𝓝 1\none_k : 1 ∈ k\nA : Set (Set G) := {t | t ⊆ s ∧ t.PairwiseDisjoint fun x => x • k}\nm : Set G\nm_max : ∀ a ∈ A, m ⊆ a → a = m\nmA : m ⊆ s ∧ m.PairwiseDisjoint fun x => x • k\nsm : s ⊆ ⋃ x ∈ m, x • (k * k⁻¹)\nhm : m.Nonempty\n⊢ μ' s = ν s","state_after":"case pos\nG : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\nν : Measure G := μ'.haarScalarFactor μ • μ\nk : Set G\nk_comp : IsCompact k\nk_closed : IsClosed k\nk_mem : k ∈ 𝓝 1\none_k : 1 ∈ k\nA : Set (Set G) := {t | t ⊆ s ∧ t.PairwiseDisjoint fun x => x • k}\nm : Set G\nm_max : ∀ a ∈ A, m ⊆ a → a = m\nmA : m ⊆ s ∧ m.PairwiseDisjoint fun x => x • k\nsm : s ⊆ ⋃ x ∈ m, x • (k * k⁻¹)\nhm : m.Nonempty\nh'm : m.Countable\n⊢ μ' s = ν s\n\ncase neg\nG : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\nν : Measure G := μ'.haarScalarFactor μ • μ\nk : Set G\nk_comp : IsCompact k\nk_closed : IsClosed k\nk_mem : k ∈ 𝓝 1\none_k : 1 ∈ k\nA : Set (Set G) := {t | t ⊆ s ∧ t.PairwiseDisjoint fun x => x • k}\nm : Set G\nm_max : ∀ a ∈ A, m ⊆ a → a = m\nmA : m ⊆ s ∧ m.PairwiseDisjoint fun x => x • k\nsm : s ⊆ ⋃ x ∈ m, x • (k * k⁻¹)\nhm : m.Nonempty\nh'm : ¬m.Countable\n⊢ μ' s = ν s","tactic":"by_cases h'm : Set.Countable 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+{"url":"Mathlib/NumberTheory/LegendreSymbol/GaussEisensteinLemmas.lean","commit":"","full_name":"_private.Mathlib.NumberTheory.LegendreSymbol.GaussEisensteinLemmas.0.ZMod.eisenstein_lemma_aux₁","start":[113,0],"end":[139,11],"file_path":"Mathlib/NumberTheory/LegendreSymbol/GaussEisensteinLemmas.lean","tactics":[{"state_before":"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp2✝ : Fact (p % 2 = 1)\na : ℕ\nhap : ↑a ≠ 0\nhp2 : ↑p = ↑1\n⊢ ↑(∑ x ∈ Ico 1 (p / 2).succ, a * x) = ↑(∑ x ∈ Ico 1 (p / 2).succ, (a * x % p + p * (a * x / p)))","state_after":"no goals","tactic":"simp only [mod_add_div]","premises":[{"full_name":"Nat.mod_add_div","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Div.lean","def_pos":[201,8],"def_end_pos":[201,19]}]},{"state_before":"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp2✝ : Fact (p % 2 = 1)\na : ℕ\nhap : ↑a ≠ 0\nhp2 : ↑p = ↑1\n⊢ ↑(∑ x ∈ Ico 1 (p / 2).succ, (a * x % p + p * (a * x / p))) =\n    ↑(∑ x ∈ Ico 1 (p / 2).succ, (↑(a * x)).val) + ↑(∑ x ∈ Ico 1 (p / 2).succ, a * x / p)","state_after":"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp2✝ : Fact (p % 2 = 1)\na : ℕ\nhap : ↑a ≠ 0\nhp2 : ↑p = ↑1\n⊢ ↑(∑ x ∈ Ico 1 (p / 2).succ, (a * x % p + p * (a * x / p))) =\n    ↑(∑ x ∈ Ico 1 (p / 2).succ, a * x % p) + ↑(∑ x ∈ Ico 1 (p / 2).succ, a * x / p)","tactic":"simp only [val_natCast]","premises":[{"full_name":"ZMod.val_natCast","def_path":"Mathlib/Data/ZMod/Basic.lean","def_pos":[75,8],"def_end_pos":[75,19]}]},{"state_before":"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp2✝ : Fact (p % 2 = 1)\na : ℕ\nhap : ↑a ≠ 0\nhp2 : ↑p = ↑1\n⊢ ↑(∑ x ∈ Ico 1 (p / 2).succ, (a * x % p + p * (a * x / p))) =\n    ↑(∑ x ∈ Ico 1 (p / 2).succ, a * x % p) + ↑(∑ x ∈ Ico 1 (p / 2).succ, a * x / p)","state_after":"no goals","tactic":"simp [sum_add_distrib, ← mul_sum, Nat.cast_add, Nat.cast_mul, Nat.cast_sum, hp2]","premises":[{"full_name":"Finset.mul_sum","def_path":"Mathlib/Algebra/BigOperators/Ring.lean","def_pos":[44,6],"def_end_pos":[44,13]},{"full_name":"Finset.sum_add_distrib","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[768,2],"def_end_pos":[768,13]},{"full_name":"Nat.cast_add","def_path":"Mathlib/Data/Nat/Cast/Defs.lean","def_pos":[137,8],"def_end_pos":[137,16]},{"full_name":"Nat.cast_mul","def_path":"Mathlib/Data/Nat/Cast/Basic.lean","def_pos":[56,25],"def_end_pos":[56,33]},{"full_name":"Nat.cast_sum","def_path":"Mathlib/Algebra/BigOperators/Ring.lean","def_pos":[314,6],"def_end_pos":[314,14]}]},{"state_before":"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp2✝ : Fact (p % 2 = 1)\na : ℕ\nhap : ↑a ≠ 0\nhp2 : ↑p = ↑1\n⊢ ↑(∑ x ∈ Ico 1 (p / 2).succ, (↑(a * x)).val) =\n    ∑ x ∈ Ico 1 (p / 2).succ, ↑((↑a * ↑x).valMinAbs + ↑(if (↑a * ↑x).val ≤ p / 2 then 0 else p))","state_after":"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp2✝ : Fact (p % 2 = 1)\na : ℕ\nhap : ↑a ≠ 0\nhp2 : ↑p = ↑1\n⊢ ↑(∑ x ∈ Ico 1 (p / 2).succ, (↑(a * x)).val) = ∑ x ∈ Ico 1 (p / 2).succ, ↑↑(↑a * ↑x).val","tactic":"simp only [(val_eq_ite_valMinAbs _).symm]","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"ZMod.val_eq_ite_valMinAbs","def_path":"Mathlib/Data/ZMod/Basic.lean","def_pos":[1184,8],"def_end_pos":[1184,28]}]},{"state_before":"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp2✝ : Fact (p % 2 = 1)\na : ℕ\nhap : ↑a ≠ 0\nhp2 : ↑p = ↑1\n⊢ ↑(∑ x ∈ Ico 1 (p / 2).succ, (↑(a * x)).val) = ∑ x ∈ Ico 1 (p / 2).succ, ↑↑(↑a * ↑x).val","state_after":"no goals","tactic":"simp [Nat.cast_sum]","premises":[{"full_name":"Nat.cast_sum","def_path":"Mathlib/Algebra/BigOperators/Ring.lean","def_pos":[314,6],"def_end_pos":[314,14]}]},{"state_before":"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp2✝ : Fact (p % 2 = 1)\na : ℕ\nhap : ↑a ≠ 0\nhp2 : ↑p = ↑1\n⊢ ∑ x ∈ Ico 1 (p / 2).succ, ↑((↑a * ↑x).valMinAbs + ↑(if (↑a * ↑x).val ≤ p / 2 then 0 else p)) =\n    ↑(filter (fun x => p / 2 < (↑a * ↑x).val) (Ico 1 (p / 2).succ)).card +\n      ↑(∑ x ∈ Ico 1 (p / 2).succ, (↑a * ↑x).valMinAbs.natAbs)","state_after":"no goals","tactic":"simp [add_comm, sum_add_distrib, Finset.sum_ite, hp2, Nat.cast_sum]","premises":[{"full_name":"Finset.sum_add_distrib","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[768,2],"def_end_pos":[768,13]},{"full_name":"Finset.sum_ite","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[1051,2],"def_end_pos":[1051,13]},{"full_name":"Nat.cast_sum","def_path":"Mathlib/Algebra/BigOperators/Ring.lean","def_pos":[314,6],"def_end_pos":[314,14]},{"full_name":"add_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[293,2],"def_end_pos":[293,13]}]},{"state_before":"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp2✝ : Fact (p % 2 = 1)\na : ℕ\nhap : ↑a ≠ 0\nhp2 : ↑p = ↑1\n⊢ ↑(filter (fun x => p / 2 < (↑a * ↑x).val) (Ico 1 (p / 2).succ)).card +\n      ↑(∑ x ∈ Ico 1 (p / 2).succ, (↑a * ↑x).valMinAbs.natAbs) =\n    ↑(filter (fun x => p / 2 < (↑a * ↑x).val) (Ico 1 (p / 2).succ)).card + ↑(∑ x ∈ Ico 1 (p / 2).succ, x)","state_after":"no goals","tactic":"rw [Finset.sum_eq_multiset_sum, Ico_map_valMinAbs_natAbs_eq_Ico_map_id p a hap, ←\n              Finset.sum_eq_multiset_sum]","premises":[{"full_name":"Finset.sum_eq_multiset_sum","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[266,2],"def_end_pos":[266,13]},{"full_name":"ZMod.Ico_map_valMinAbs_natAbs_eq_Ico_map_id","def_path":"Mathlib/NumberTheory/LegendreSymbol/GaussEisensteinLemmas.lean","def_pos":[28,8],"def_end_pos":[28,46]}]}]}
+{"url":"Mathlib/Algebra/Order/Group/Defs.lean","commit":"","full_name":"inv_le_self_iff","start":[152,0],"end":[153,77],"file_path":"Mathlib/Algebra/Order/Group/Defs.lean","tactics":[{"state_before":"α : Type u\ninst✝ : LinearOrderedCommGroup α\na b c : α\n⊢ a⁻¹ ≤ a ↔ 1 ≤ a","state_after":"no goals","tactic":"simp [inv_le_iff_one_le_mul']","premises":[{"full_name":"inv_le_iff_one_le_mul'","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Basic.lean","def_pos":[60,8],"def_end_pos":[60,30]}]}]}
+{"url":"Mathlib/NumberTheory/NumberField/Units/Basic.lean","commit":"","full_name":"NumberField.Units.mem_torsion","start":[102,0],"end":[109,59],"file_path":"Mathlib/NumberTheory/NumberField/Units/Basic.lean","tactics":[{"state_before":"K : Type u_1\ninst✝¹ : Field K\nx : (𝓞 K)ˣ\ninst✝ : NumberField K\n⊢ x ∈ torsion K ↔ ∀ (w : InfinitePlace K), w ((algebraMap (𝓞 K) K) ↑x) = 1","state_after":"K : Type u_1\ninst✝¹ : Field K\nx : (𝓞 K)ˣ\ninst✝ : NumberField K\n⊢ IsOfFinOrder x ↔ ∀ (φ : K →+* ℂ), ‖φ ((algebraMap (𝓞 K) K) ↑x)‖ = 1","tactic":"rw [eq_iff_eq (x : K) 1, torsion, CommGroup.mem_torsion]","premises":[{"full_name":"CommGroup.mem_torsion","def_path":"Mathlib/GroupTheory/Torsion.lean","def_pos":[273,8],"def_end_pos":[273,19]},{"full_name":"NumberField.InfinitePlace.eq_iff_eq","def_path":"Mathlib/NumberTheory/NumberField/Embeddings.lean","def_pos":[291,8],"def_end_pos":[291,17]},{"full_name":"NumberField.Units.torsion","def_path":"Mathlib/NumberTheory/NumberField/Units/Basic.lean","def_pos":[100,4],"def_end_pos":[100,11]}]},{"state_before":"K : Type u_1\ninst✝¹ : Field K\nx : (𝓞 K)ˣ\ninst✝ : NumberField K\n⊢ IsOfFinOrder x ↔ ∀ (φ : K →+* ℂ), ‖φ ((algebraMap (𝓞 K) K) ↑x)‖ = 1","state_after":"K : Type u_1\ninst✝¹ : Field K\nx : (𝓞 K)ˣ\ninst✝ : NumberField K\nh : ∀ (φ : K →+* ℂ), ‖φ ((algebraMap (𝓞 K) K) ↑x)‖ = 1\n⊢ ∃ n, 0 < n ∧ x ^ n = 1","tactic":"refine ⟨fun hx φ ↦ (((φ.comp $ algebraMap (𝓞 K) K).toMonoidHom.comp $\n    Units.coeHom _).isOfFinOrder hx).norm_eq_one, fun h ↦ isOfFinOrder_iff_pow_eq_one.2 ?_⟩","premises":[{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"IsOfFinOrder.norm_eq_one","def_path":"Mathlib/Analysis/Normed/Field/Basic.lean","def_pos":[842,16],"def_end_pos":[842,40]},{"full_name":"MonoidHom.comp","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[691,4],"def_end_pos":[691,18]},{"full_name":"MonoidHom.isOfFinOrder","def_path":"Mathlib/GroupTheory/OrderOfElement.lean","def_pos":[96,8],"def_end_pos":[96,30]},{"full_name":"NumberField.RingOfIntegers","def_path":"Mathlib/NumberTheory/NumberField/Basic.lean","def_pos":[71,4],"def_end_pos":[71,18]},{"full_name":"RingHom.comp","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[563,4],"def_end_pos":[563,8]},{"full_name":"Units.coeHom","def_path":"Mathlib/Algebra/Group/Units/Hom.lean","def_pos":[84,4],"def_end_pos":[84,10]},{"full_name":"algebraMap","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[108,4],"def_end_pos":[108,14]},{"full_name":"isOfFinOrder_iff_pow_eq_one","def_path":"Mathlib/GroupTheory/OrderOfElement.lean","def_pos":[57,8],"def_end_pos":[57,35]}]},{"state_before":"K : Type u_1\ninst✝¹ : Field K\nx : (𝓞 K)ˣ\ninst✝ : NumberField K\nh : ∀ (φ : K →+* ℂ), ‖φ ((algebraMap (𝓞 K) K) ↑x)‖ = 1\n⊢ ∃ n, 0 < n ∧ x ^ n = 1","state_after":"case intro.intro\nK : Type u_1\ninst✝¹ : Field K\nx : (𝓞 K)ˣ\ninst✝ : NumberField K\nh : ∀ (φ : K →+* ℂ), ‖φ ((algebraMap (𝓞 K) K) ↑x)‖ = 1\nn : ℕ\nhn : 0 < n\nhx : (algebraMap (𝓞 K) K) ↑x ^ n = 1\n⊢ ∃ n, 0 < n ∧ x ^ n = 1","tactic":"obtain ⟨n, hn, hx⟩ := Embeddings.pow_eq_one_of_norm_eq_one K ℂ x.val.isIntegral_coe h","premises":[{"full_name":"Complex","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[27,10],"def_end_pos":[27,17]},{"full_name":"NumberField.Embeddings.pow_eq_one_of_norm_eq_one","def_path":"Mathlib/NumberTheory/NumberField/Embeddings.lean","def_pos":[114,8],"def_end_pos":[114,33]},{"full_name":"NumberField.RingOfIntegers.isIntegral_coe","def_path":"Mathlib/NumberTheory/NumberField/Basic.lean","def_pos":[181,8],"def_end_pos":[181,22]},{"full_name":"Units.val","def_path":"Mathlib/Algebra/Group/Units.lean","def_pos":[53,2],"def_end_pos":[53,5]}]},{"state_before":"case intro.intro\nK : Type u_1\ninst✝¹ : Field K\nx : (𝓞 K)ˣ\ninst✝ : NumberField K\nh : ∀ (φ : K →+* ℂ), ‖φ ((algebraMap (𝓞 K) K) ���x)‖ = 1\nn : ℕ\nhn : 0 < n\nhx : (algebraMap (𝓞 K) K) ↑x ^ n = 1\n⊢ ∃ n, 0 < n ∧ x ^ n = 1","state_after":"no goals","tactic":"exact ⟨n, hn, by ext; rw [NumberField.RingOfIntegers.coe_eq_algebraMap, coe_pow, hx,\n    NumberField.RingOfIntegers.coe_eq_algebraMap, coe_one]⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"NumberField.RingOfIntegers.coe_eq_algebraMap","def_path":"Mathlib/NumberTheory/NumberField/Basic.lean","def_pos":[104,6],"def_end_pos":[104,23]},{"full_name":"NumberField.Units.coe_one","def_path":"Mathlib/NumberTheory/NumberField/Units/Basic.lean","def_pos":[81,8],"def_end_pos":[81,15]},{"full_name":"NumberField.Units.coe_pow","def_path":"Mathlib/NumberTheory/NumberField/Units/Basic.lean","def_pos":[74,8],"def_end_pos":[74,15]}]}]}
+{"url":"Mathlib/Probability/Kernel/Composition.lean","commit":"","full_name":"ProbabilityTheory.Kernel.fst_map_id_prod","start":[883,0],"end":[886,55],"file_path":"Mathlib/Probability/Kernel/Composition.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nδ : Type u_4\nmδ : MeasurableSpace δ\nγ✝ : Type u_5\nmγ✝ : MeasurableSpace γ✝\nf✝ : β → γ✝\ng : γ✝ → α\nκ : Kernel α β\nγ : Type u_6\nmγ : MeasurableSpace γ\nf : β → γ\nhf : Measurable f\n⊢ (κ.map (fun a => (a, f a)) ⋯).fst = κ","state_after":"no goals","tactic":"rw [fst_map_prod _ measurable_id' hf, Kernel.map_id']","premises":[{"full_name":"ProbabilityTheory.Kernel.fst_map_prod","def_path":"Mathlib/Probability/Kernel/Composition.lean","def_pos":[875,6],"def_end_pos":[875,18]},{"full_name":"ProbabilityTheory.Kernel.map_id'","def_path":"Mathlib/Probability/Kernel/Composition.lean","def_pos":[608,6],"def_end_pos":[608,13]},{"full_name":"measurable_id'","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","def_pos":[496,8],"def_end_pos":[496,22]}]}]}
+{"url":"Mathlib/Algebra/Polynomial/Basic.lean","commit":"","full_name":"Polynomial.monomial_mul_X_pow","start":[538,0],"end":[543,47],"file_path":"Mathlib/Algebra/Polynomial/Basic.lean","tactics":[{"state_before":"R : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\nr : R\nk : ℕ\n⊢ (monomial n) r * X ^ k = (monomial (n + k)) r","state_after":"case zero\nR : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\nr : R\n⊢ (monomial n) r * X ^ 0 = (monomial (n + 0)) r\n\ncase succ\nR : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\nr : R\nk : ℕ\nih : (monomial n) r * X ^ k = (monomial (n + k)) r\n⊢ (monomial n) r * X ^ (k + 1) = (monomial (n + (k + 1))) r","tactic":"induction' k with k ih","premises":[]}]}
+{"url":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","commit":"","full_name":"MeasureTheory.liminf_ae_eq_of_forall_ae_eq","start":[641,0],"end":[650,12],"file_path":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t✝ : Set α\ns : ℕ → Set α\nt : Set α\nh : ∀ (n : ℕ), s n =ᶠ[ae μ] t\n⊢ liminf s atTop =ᶠ[ae μ] t","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t✝ : Set α\ns : ℕ → Set α\nt : Set α\nh : ∀ (n : ℕ), μ (s n \\ t) = 0 ∧ μ (t \\ s n) = 0\n⊢ μ (liminf s atTop \\ t) = 0 ∧ μ (t \\ liminf s atTop) = 0","tactic":"simp_rw [ae_eq_set] at h ⊢","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"MeasureTheory.ae_eq_set","def_path":"Mathlib/MeasureTheory/OuterMeasure/AE.lean","def_pos":[155,8],"def_end_pos":[155,17]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t✝ : Set α\ns : ℕ → Set α\nt : Set α\nh : ∀ (n : ℕ), μ (s n \\ t) = 0 ∧ μ (t \\ s n) = 0\n⊢ μ (liminf s atTop \\ t) = 0 ∧ μ (t \\ liminf s atTop) = 0","state_after":"case left\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t✝ : Set α\ns : ℕ → Set α\nt : Set α\nh : ∀ (n : ℕ), μ (s n \\ t) = 0 ∧ μ (t \\ s n) = 0\n⊢ μ (liminf s atTop \\ t) = 0\n\ncase right\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t✝ : Set α\ns : ℕ → Set α\nt : Set α\nh : ∀ (n : ℕ), μ (s n \\ t) = 0 ∧ μ (t \\ s n) = 0\n⊢ μ (t \\ liminf s atTop) = 0","tactic":"constructor","premises":[]}]}
+{"url":"Mathlib/RingTheory/UniqueFactorizationDomain.lean","commit":"","full_name":"Associates.dvd_of_mem_factors'","start":[1501,0],"end":[1506,34],"file_path":"Mathlib/RingTheory/UniqueFactorizationDomain.lean","tactics":[{"state_before":"α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\na : α\np : Associates α\nhp : Irreducible p\nhz : a ≠ 0\nh_mem : ⟨p, hp⟩ ∈ factors' a\n⊢ p ∣ Associates.mk a","state_after":"α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\na : α\np : Associates α\nhp : Irreducible p\nhz : a ≠ 0\nh_mem : ⟨p, hp⟩ ∈ factors' a\nthis : DecidableEq (Associates α)\n⊢ p ∣ Associates.mk a","tactic":"haveI := Classical.decEq (Associates α)","premises":[{"full_name":"Associates","def_path":"Mathlib/Algebra/Associated/Basic.lean","def_pos":[729,7],"def_end_pos":[729,17]},{"full_name":"Classical.decEq","def_path":"Mathlib/Logic/Basic.lean","def_pos":[737,18],"def_end_pos":[737,23]}]},{"state_before":"α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\na : α\np : Associates α\nhp : Irreducible p\nhz : a ≠ 0\nh_mem : ⟨p, hp⟩ ∈ factors' a\nthis : DecidableEq (Associates α)\n⊢ p ∣ Associates.mk a","state_after":"case hm\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\na : α\np : Associates α\nhp : Irreducible p\nhz : a ≠ 0\nh_mem : ⟨p, hp⟩ ∈ factors' a\nthis : DecidableEq (Associates α)\n⊢ p ∈ (Associates.mk a).factors","tactic":"apply dvd_of_mem_factors","premises":[{"full_name":"Associates.dvd_of_mem_factors","def_path":"Mathlib/RingTheory/UniqueFactorizationDomain.lean","def_pos":[1490,8],"def_end_pos":[1490,26]}]},{"state_before":"case hm\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\na : α\np : Associates α\nhp : Irreducible p\nhz : a ≠ 0\nh_mem : ⟨p, hp⟩ ∈ factors' a\nthis : DecidableEq (Associates α)\n⊢ p ∈ (Associates.mk a).factors","state_after":"case hm\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\na : α\np : Associates α\nhp : Irreducible p\nhz : a ≠ 0\nh_mem : ⟨p, hp⟩ ∈ factors' a\nthis : DecidableEq (Associates α)\n⊢ p ∈ ↑(factors' a)","tactic":"rw [factors_mk _ hz]","premises":[{"full_name":"Associates.factors_mk","def_path":"Mathlib/RingTheory/UniqueFactorizationDomain.lean","def_pos":[1359,8],"def_end_pos":[1359,18]}]},{"state_before":"case hm\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\na : α\np : Associates α\nhp : Irreducible p\nhz : a ≠ 0\nh_mem : ⟨p, hp⟩ ∈ factors' a\nthis : DecidableEq (Associates α)\n⊢ p ∈ ↑(factors' a)","state_after":"no goals","tactic":"apply mem_factorSet_some.2 h_mem","premises":[{"full_name":"Associates.mem_factorSet_some","def_path":"Mathlib/RingTheory/UniqueFactorizationDomain.lean","def_pos":[1261,8],"def_end_pos":[1261,26]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]}]}]}
+{"url":"Mathlib/Data/Set/Card.lean","commit":"","full_name":"Set.exists_subset_or_subset_of_two_mul_lt_ncard","start":[918,0],"end":[926,42],"file_path":"Mathlib/Data/Set/Card.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ns t : Set α\nf : α → β\nn : ℕ\nhst : 2 * n < (s ∪ t).ncard\n⊢ ∃ r, n < r.ncard ∧ (r ⊆ s ∨ r ⊆ t)","state_after":"no goals","tactic":"classical\n  have hu := finite_of_ncard_ne_zero ((Nat.zero_le _).trans_lt hst).ne.symm\n  rw [ncard_eq_toFinset_card _ hu,\n    Finite.toFinset_union (hu.subset subset_union_left)\n      (hu.subset subset_union_right)] at hst\n  obtain ⟨r', hnr', hr'⟩ := Finset.exists_subset_or_subset_of_two_mul_lt_card hst\n  exact ⟨r', by simpa, by simpa using hr'⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Finset.exists_subset_or_subset_of_two_mul_lt_card","def_path":"Mathlib/Data/Finset/Card.lean","def_pos":[583,8],"def_end_pos":[583,50]},{"full_name":"Nat.zero_le","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1663,8],"def_end_pos":[1663,19]},{"full_name":"Ne.symm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[704,8],"def_end_pos":[704,15]},{"full_name":"Set.Finite.subset","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[633,8],"def_end_pos":[633,21]},{"full_name":"Set.Finite.toFinset_union","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[222,18],"def_end_pos":[222,32]},{"full_name":"Set.finite_of_ncard_ne_zero","def_path":"Mathlib/Data/Set/Card.lean","def_pos":[535,8],"def_end_pos":[535,31]},{"full_name":"Set.ncard_eq_toFinset_card","def_path":"Mathlib/Data/Set/Card.lean","def_pos":[488,8],"def_end_pos":[488,30]},{"full_name":"Set.subset_union_left","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[657,8],"def_end_pos":[657,25]},{"full_name":"Set.subset_union_right","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[660,8],"def_end_pos":[660,26]}]}]}
+{"url":"Mathlib/Data/Finset/Pointwise.lean","commit":"","full_name":"Finset.preimage_inv","start":[270,0],"end":[272,66],"file_path":"Mathlib/Data/Finset/Pointwise.lean","tactics":[{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝¹ : DecidableEq α\ninst✝ : InvolutiveInv α\ns✝ : Finset α\na : α\ns : Finset α\n⊢ ↑(s.preimage (fun x => x⁻¹) ⋯) = ↑s⁻¹","state_after":"no goals","tactic":"rw [coe_preimage, Set.inv_preimage, coe_inv]","premises":[{"full_name":"Finset.coe_inv","def_path":"Mathlib/Data/Finset/Pointwise.lean","def_pos":[265,8],"def_end_pos":[265,15]},{"full_name":"Finset.coe_preimage","def_path":"Mathlib/Data/Finset/Preimage.lean","def_pos":[34,8],"def_end_pos":[34,20]},{"full_name":"Set.inv_preimage","def_path":"Mathlib/Data/Set/Pointwise/Basic.lean","def_pos":[159,8],"def_end_pos":[159,20]}]}]}
+{"url":"Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean","commit":"","full_name":"CategoryTheory.Limits.prodComparison_inv_natural","start":[1102,0],"end":[1108,83],"file_path":"Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean","tactics":[{"state_before":"C : Type u\ninst✝¹⁰ : Category.{v, u} C\nX Y : C\nD : Type u₂\ninst✝⁹ : Category.{w, u₂} D\nE : Type u₃\ninst✝⁸ : Category.{w', u₃} E\nF : C ⥤ D\nG : D ⥤ E\nA A' B B' : C\ninst✝⁷ : HasBinaryProduct A B\ninst✝⁶ : HasBinaryProduct A' B'\ninst✝⁵ : HasBinaryProduct (F.obj A) (F.obj B)\ninst✝⁴ : HasBinaryProduct (F.obj A') (F.obj B')\ninst✝³ : HasBinaryProduct (G.obj (F.obj A)) (G.obj (F.obj B))\ninst✝² : HasBinaryProduct ((F ⋙ G).obj A) ((F ⋙ G).obj B)\nf : A ⟶ A'\ng : B ⟶ B'\ninst✝¹ : IsIso (prodComparison F A B)\ninst✝ : IsIso (prodComparison F A' B')\n⊢ inv (prodComparison F A B) ≫ F.map (prod.map f g) = prod.map (F.map f) (F.map g) ≫ inv (prodComparison F A' B')","state_after":"no goals","tactic":"rw [IsIso.eq_comp_inv, Category.assoc, IsIso.inv_comp_eq, prodComparison_natural]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.IsIso.eq_comp_inv","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[385,8],"def_end_pos":[385,19]},{"full_name":"CategoryTheory.IsIso.inv_comp_eq","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[373,8],"def_end_pos":[373,19]},{"full_name":"CategoryTheory.Limits.prodComparison_natural","def_path":"Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean","def_pos":[1079,8],"def_end_pos":[1079,30]}]}]}
+{"url":"Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean","commit":"","full_name":"IsIntegral.tower_top","start":[50,0],"end":[55,96],"file_path":"Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean","tactics":[{"state_before":"R : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing A\ninst✝⁵ : Ring B\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\nf : R →+* S\ninst✝¹ : Algebra A B\ninst✝ : IsScalarTower R A B\nx : B\nhx : IsIntegral R x\np : R[X]\nhp : p.Monic\nhpx : eval₂ (algebraMap R B) x p = 0\n⊢ eval₂ (algebraMap A B) x (Polynomial.map (algebraMap R A) p) = 0","state_after":"no goals","tactic":"rw [← aeval_def, aeval_map_algebraMap, aeval_def, hpx]","premises":[{"full_name":"Polynomial.aeval_def","def_path":"Mathlib/Algebra/Polynomial/AlgebraMap.lean","def_pos":[242,8],"def_end_pos":[242,17]},{"full_name":"Polynomial.aeval_map_algebraMap","def_path":"Mathlib/RingTheory/Polynomial/Tower.lean","def_pos":[35,8],"def_end_pos":[35,28]}]}]}
+{"url":"Mathlib/Algebra/Group/Fin/Tuple.lean","commit":"","full_name":"Matrix.smul_cons","start":[48,0],"end":[49,88],"file_path":"Mathlib/Algebra/Group/Fin/Tuple.lean","tactics":[{"state_before":"α : Type u_1\nM : Type u_2\nn : ℕ\ninst✝ : SMul M α\nx : M\ny : α\nv : Fin n → α\n⊢ x • vecCons y v = vecCons (x • y) (x • v)","state_after":"case h\nα : Type u_1\nM : Type u_2\nn : ℕ\ninst✝ : SMul M α\nx : M\ny : α\nv : Fin n → α\ni : Fin n.succ\n⊢ (x • vecCons y v) i = vecCons (x • y) (x • v) i","tactic":"ext i","premises":[]},{"state_before":"case h\nα : Type u_1\nM : Type u_2\nn : ℕ\ninst✝ : SMul M α\nx : M\ny : α\nv : Fin n → α\ni : Fin n.succ\n⊢ (x • vecCons y v) i = vecCons (x • y) (x • v) i","state_after":"no goals","tactic":"refine i.cases ?_ ?_ <;> simp","premises":[{"full_name":"Fin.cases","def_path":".lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean","def_pos":[654,20],"def_end_pos":[654,25]}]}]}
+{"url":"Mathlib/CategoryTheory/Limits/IsLimit.lean","commit":"","full_name":"CategoryTheory.Limits.IsColimit.hom_desc","start":[648,0],"end":[660,18],"file_path":"Mathlib/CategoryTheory/Limits/IsLimit.lean","tactics":[{"state_before":"J : Type u₁\ninst✝² : Category.{v₁, u₁} J\nK : Type u₂\ninst✝¹ : Category.{v₂, u₂} K\nC : Type u₃\ninst✝ : Category.{v₃, u₃} C\nF : J ⥤ C\nt : Cocone F\nh : IsColimit t\nW : C\nm : t.pt ⟶ W\n⊢ ∀ ⦃X Y : J⦄ (f : X ⟶ Y), F.map f ≫ (fun b => t.ι.app b ≫ m) Y = (fun b => t.ι.app b ≫ m) X ≫ ((const J).obj W).map f","state_after":"J : Type u₁\ninst✝² : Category.{v₁, u₁} J\nK : Type u₂\ninst✝¹ : Category.{v₂, u₂} K\nC : Type u₃\ninst✝ : Category.{v₃, u₃} C\nF : J ⥤ C\nt : Cocone F\nh : IsColimit t\nW : C\nm : t.pt ⟶ W\nX✝ Y✝ : J\nf✝ : X✝ ⟶ Y✝\n⊢ F.map f✝ ≫ (fun b => t.ι.app b ≫ m) Y✝ = (fun b => t.ι.app b ≫ m) X✝ ≫ ((const J).obj W).map f✝","tactic":"intros","premises":[]},{"state_before":"J : Type u₁\ninst✝² : Category.{v₁, u₁} J\nK : Type u₂\ninst✝¹ : Category.{v₂, u₂} K\nC : Type u₃\ninst✝ : Category.{v₃, u₃} C\nF : J ⥤ C\nt : Cocone F\nh : IsColimit t\nW : C\nm : t.pt ⟶ W\nX✝ Y✝ : J\nf✝ : X✝ ⟶ Y✝\n⊢ F.map f✝ ≫ (fun b => t.ι.app b ≫ m) Y✝ = (fun b => t.ι.app b ≫ m) X✝ ≫ ((const J).obj W).map f✝","state_after":"no goals","tactic":"erw [← assoc, t.ι.naturality, comp_id, comp_id]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Category.comp_id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[153,2],"def_end_pos":[153,9]},{"full_name":"CategoryTheory.Limits.Cocone.ι","def_path":"Mathlib/CategoryTheory/Limits/Cones.lean","def_pos":[152,2],"def_end_pos":[152,3]},{"full_name":"CategoryTheory.NatTrans.naturality","def_path":"Mathlib/CategoryTheory/NatTrans.lean","def_pos":[50,2],"def_end_pos":[50,12]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]}]}]}
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+{"url":"Mathlib/Algebra/Order/AbsoluteValue.lean","commit":"","full_name":"IsAbsoluteValue.abv_sub_le","start":[352,0],"end":[353,69],"file_path":"Mathlib/Algebra/Order/AbsoluteValue.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nR✝ : Type u_3\nS✝ : Type u_4\nS : Type u_5\ninst✝² : OrderedRing S\nR : Type u_6\ninst✝¹ : Ring R\nabv : R → S\ninst✝ : IsAbsoluteValue abv\na b c : R\n⊢ abv (a - c) ≤ abv (a - b) + abv (b - c)","state_after":"no goals","tactic":"simpa [sub_eq_add_neg, add_assoc] using abv_add abv (a - b) (b - c)","premises":[{"full_name":"IsAbsoluteValue.abv_add","def_path":"Mathlib/Algebra/Order/AbsoluteValue.lean","def_pos":[289,6],"def_end_pos":[289,13]},{"full_name":"add_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[258,2],"def_end_pos":[258,13]},{"full_name":"sub_eq_add_neg","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[905,2],"def_end_pos":[905,13]}]}]}
+{"url":"Mathlib/Algebra/BigOperators/Group/Multiset.lean","commit":"","full_name":"Multiset.prod_induction_nonempty","start":[177,0],"end":[187,69],"file_path":"Mathlib/Algebra/BigOperators/Group/Multiset.lean","tactics":[{"state_before":"F : Type u_1\nι : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝¹ : CommMonoid α\ninst✝ : CommMonoid β\ns t : Multiset α\na : α\nm : Multiset ι\nf g : ι → α\np : α → Prop\np_mul : ∀ (a b : α), p a → p b → p (a * b)\nhs : s ≠ ∅\np_s : ∀ a ∈ s, p a\n⊢ p s.prod","state_after":"case empty\nF : Type u_1\nι : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝¹ : CommMonoid α\ninst✝ : CommMonoid β\ns t : Multiset α\na : α\nm : Multiset ι\nf g : ι → α\np : α → Prop\np_mul : ∀ (a b : α), p a → p b → p (a * b)\nhs : 0 ≠ ∅\np_s : ∀ a ∈ 0, p a\n⊢ p (prod 0)\n\ncase cons\nF : Type u_1\nι : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝¹ : CommMonoid α\ninst✝ : CommMonoid β\ns✝ t : Multiset α\na✝ : α\nm : Multiset ι\nf g : ι → α\np : α → Prop\np_mul : ∀ (a b : α), p a → p b → p (a * b)\na : α\ns : Multiset α\nhsa : s ≠ ∅ → (∀ a ∈ s, p a) → p s.prod\nhs : a ::ₘ s ≠ ∅\np_s : ∀ a_1 ∈ a ::ₘ s, p a_1\n⊢ p (a ::ₘ s).prod","tactic":"induction' s using Multiset.induction_on with a s hsa","premises":[{"full_name":"Multiset.induction_on","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[152,18],"def_end_pos":[152,30]}]},{"state_before":"case cons\nF : Type u_1\nι : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝¹ : CommMonoid α\ninst✝ : CommMonoid β\ns✝ t : Multiset α\na✝ : α\nm : Multiset ι\nf g : ι → α\np : α → Prop\np_mul : ∀ (a b : α), p a → p b → p (a * b)\na : α\ns : Multiset α\nhsa : s ≠ ∅ → (∀ a ∈ s, p a) → p s.prod\nhs : a ::ₘ s ≠ ∅\np_s : ∀ a_1 ∈ a ::ₘ s, p a_1\n⊢ p (a ::ₘ s).prod","state_after":"case cons\nF : Type u_1\nι : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝¹ : CommMonoid α\ninst✝ : CommMonoid β\ns✝ t : Multiset α\na✝ : α\nm : Multiset ι\nf g : ι → α\np : α → Prop\np_mul : ∀ (a b : α), p a → p b → p (a * b)\na : α\ns : Multiset α\nhsa : s ≠ ∅ → (∀ a ∈ s, p a) → p s.prod\nhs : a ::ₘ s ≠ ∅\np_s : ∀ a_1 ∈ a ::ₘ s, p a_1\n⊢ p (a * s.prod)","tactic":"rw [prod_cons]","premises":[{"full_name":"Multiset.prod_cons","def_path":"Mathlib/Algebra/BigOperators/Group/Multiset.lean","def_pos":[65,8],"def_end_pos":[65,17]}]},{"state_before":"case cons\nF : Type u_1\nι : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝¹ : CommMonoid α\ninst✝ : CommMonoid β\ns✝ t : Multiset α\na✝ : α\nm : Multiset ι\nf g : ι → α\np : α → Prop\np_mul : ∀ (a b : α), p a → p b → p (a * b)\na : α\ns : Multiset α\nhsa : s ≠ ∅ → (∀ a ∈ s, p a) → p s.prod\nhs : a ::ₘ s ≠ ∅\np_s : ∀ a_1 ∈ a ::ₘ s, p a_1\n⊢ p (a * s.prod)","state_after":"case pos\nF : Type u_1\nι : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝¹ : CommMonoid α\ninst✝ : CommMonoid β\ns✝ t : Multiset α\na✝ : α\nm : Multiset ι\nf g : ι → α\np : α → Prop\np_mul : ∀ (a b : α), p a → p b → p (a * b)\na : α\ns : Multiset α\nhsa : s ≠ ∅ → (∀ a ∈ s, p a) → p s.prod\nhs : a ::ₘ s ≠ ∅\np_s : ∀ a_1 ∈ a ::ₘ s, p a_1\nhs_empty : s = ∅\n⊢ p (a * s.prod)\n\ncase neg\nF : Type u_1\nι : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝¹ : CommMonoid α\ninst✝ : CommMonoid β\ns✝ t : Multiset α\na✝ : α\nm : Multiset ι\nf g : ι → α\np : α → Prop\np_mul : ∀ (a b : α), p a → p b → p (a * b)\na : α\ns : Multiset α\nhsa : s ≠ ∅ → (∀ a ∈ s, p a) → p s.prod\nhs : a ::ₘ s ≠ ∅\np_s : ∀ a_1 ∈ a ::ₘ s, p a_1\nhs_empty : ¬s = ∅\n⊢ p (a * s.prod)","tactic":"by_cases hs_empty : s = ∅","premises":[{"full_name":"EmptyCollection.emptyCollection","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[447,2],"def_end_pos":[447,17]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case neg\nF : Type u_1\nι : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝¹ : CommMonoid α\ninst✝ : CommMonoid β\ns✝ t : Multiset α\na✝ : α\nm : Multiset ι\nf g : ι → α\np : α → Prop\np_mul : ∀ (a b : α), p a → p b → p (a * b)\na : α\ns : Multiset α\nhsa : s ≠ ∅ → (∀ a ∈ s, p a) → p s.prod\nhs : a ::ₘ s ≠ ∅\np_s : ∀ a_1 ∈ a ::ₘ s, p a_1\nhs_empty : ¬s = ∅\n⊢ p (a * s.prod)","state_after":"case neg\nF : Type u_1\nι : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝¹ : CommMonoid α\ninst✝ : CommMonoid β\ns✝ t : Multiset α\na✝ : α\nm : Multiset ι\nf g : ι → α\np : α → Prop\np_mul : ∀ (a b : α), p a → p b → p (a * b)\na : α\ns : Multiset α\nhsa : s ≠ ∅ → (∀ a ∈ s, p a) → p s.prod\nhs : a ::ₘ s ≠ ∅\np_s : ∀ a_1 ∈ a ::ₘ s, p a_1\nhs_empty : ¬s = ∅\nhps : ∀ x ∈ s, p x\n⊢ p (a * s.prod)","tactic":"have hps : ∀ x, x ∈ s → p x := fun x hxs => p_s x (mem_cons_of_mem hxs)","premises":[{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Multiset.mem_cons_of_mem","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[221,8],"def_end_pos":[221,23]}]},{"state_before":"case neg\nF : Type u_1\nι : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝¹ : CommMonoid α\ninst✝ : CommMonoid β\ns✝ t : Multiset α\na✝ : α\nm : Multiset ι\nf g : ι → α\np : α → Prop\np_mul : ∀ (a b : α), p a → p b → p (a * b)\na : α\ns : Multiset α\nhsa : s ≠ ∅ → (∀ a ∈ s, p a) → p s.prod\nhs : a ::ₘ s ≠ ∅\np_s : ∀ a_1 ∈ a ::ₘ s, p a_1\nhs_empty : ¬s = ∅\nhps : ∀ x ∈ s, p x\n⊢ p (a * s.prod)","state_after":"no goals","tactic":"exact p_mul a s.prod (p_s a (mem_cons_self a s)) (hsa hs_empty hps)","premises":[{"full_name":"Multiset.mem_cons_self","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[225,8],"def_end_pos":[225,21]},{"full_name":"Multiset.prod","def_path":"Mathlib/Algebra/BigOperators/Group/Multiset.lean","def_pos":[38,4],"def_end_pos":[38,8]}]}]}
+{"url":"Mathlib/Analysis/Convex/Function.lean","commit":"","full_name":"ConvexOn.le_left_of_right_le","start":[703,0],"end":[706,54],"file_path":"Mathlib/Analysis/Convex/Function.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nα : Type u_4\nβ : Type u_5\nι : Type u_6\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : LinearOrderedCancelAddCommMonoid β\ninst✝² : SMul 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\ns : Set E\nf g : E → β\nhf : ConvexOn 𝕜 s f\nx y z : E\nhx : x ∈ s\nhy : y ∈ s\nhz : z ∈ openSegment 𝕜 x y\nhyz : f y ≤ f z\n⊢ f z ≤ f x","state_after":"case intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nα : Type u_4\nβ : Type u_5\nι : Type u_6\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : LinearOrderedCancelAddCommMonoid β\ninst✝² : SMul 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\ns : Set E\nf g : E → β\nhf : ConvexOn 𝕜 s f\nx y : E\nhx : x ∈ s\nhy : y ∈ s\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhyz : f y ≤ f (a • x + b • y)\n⊢ f (a • x + b • y) ≤ f x","tactic":"obtain ⟨a, b, ha, hb, hab, rfl⟩ := hz","premises":[]},{"state_before":"case intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nα : Type u_4\nβ : Type u_5\nι : Type u_6\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : LinearOrderedCancelAddCommMonoid β\ninst✝² : SMul 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\ns : Set E\nf g : E → β\nhf : ConvexOn 𝕜 s f\nx y : E\nhx : x ∈ s\nhy : y ∈ s\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhyz : f y ≤ f (a • x + b • y)\n⊢ f (a • x + b • y) ≤ f x","state_after":"no goals","tactic":"exact hf.le_left_of_right_le' hx hy ha hb.le hab hyz","premises":[{"full_name":"ConvexOn.le_left_of_right_le'","def_path":"Mathlib/Analysis/Convex/Function.lean","def_pos":[677,8],"def_end_pos":[677,37]}]}]}
+{"url":"Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean","commit":"","full_name":"Asymptotics.SuperpolynomialDecay.param_pow_mul","start":[104,0],"end":[108,56],"file_path":"Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nl : Filter α\nk f g g' : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : CommSemiring β\nhf : SuperpolynomialDecay l k f\nn : ℕ\n⊢ SuperpolynomialDecay l k (k ^ n * f)","state_after":"case zero\nα : Type u_1\nβ : Type u_2\nl : Filter α\nk f g g' : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : CommSemiring β\nhf : SuperpolynomialDecay l k f\n⊢ SuperpolynomialDecay l k (k ^ 0 * f)\n\ncase succ\nα : Type u_1\nβ : Type u_2\nl : Filter α\nk f g g' : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : CommSemiring β\nhf : SuperpolynomialDecay l k f\nn : ℕ\nhn : SuperpolynomialDecay l k (k ^ n * f)\n⊢ SuperpolynomialDecay l k (k ^ (n + 1) * f)","tactic":"induction' n with n hn","premises":[]}]}
+{"url":"Mathlib/Algebra/Lie/Submodule.lean","commit":"","full_name":"LieModuleHom.range_eq_top","start":[1281,0],"end":[1282,82],"file_path":"Mathlib/Algebra/Lie/Submodule.lean","tactics":[{"state_before":"R : Type u\nL : Type v\nM : Type w\nN : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : LieRingModule L N\ninst✝ : LieModule R L N\nf : M →ₗ⁅R,L⁆ N\n⊢ f.range = ⊤ ↔ Function.Surjective ⇑f","state_after":"no goals","tactic":"rw [SetLike.ext'_iff, coe_range, LieSubmodule.top_coe, Set.range_iff_surjective]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"LieModuleHom.coe_range","def_path":"Mathlib/Algebra/Lie/Submodule.lean","def_pos":[1267,8],"def_end_pos":[1267,17]},{"full_name":"LieSubmodule.top_coe","def_path":"Mathlib/Algebra/Lie/Submodule.lean","def_pos":[346,8],"def_end_pos":[346,15]},{"full_name":"Set.range_iff_surjective","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[586,8],"def_end_pos":[586,28]},{"full_name":"SetLike.ext'_iff","def_path":"Mathlib/Data/SetLike/Basic.lean","def_pos":[157,8],"def_end_pos":[157,16]}]}]}
+{"url":"Mathlib/RingTheory/NonUnitalSubring/Basic.lean","commit":"","full_name":"NonUnitalSubring.closure_induction₂","start":[704,0],"end":[717,67],"file_path":"Mathlib/RingTheory/NonUnitalSubring/Basic.lean","tactics":[{"state_before":"F : Type w\nR : Type u\nS : Type v\nT : Type u_1\ninst✝⁴ : NonUnitalNonAssocRing R\ninst✝³ : NonUnitalNonAssocRing S\ninst✝² : NonUnitalNonAssocRing T\ninst✝¹ : FunLike F R S\ninst✝ : NonUnitalRingHomClass F R S\ng : S →ₙ+* T\nf : R →ₙ+* S\ns : Set R\np : R → R → Prop\na b : R\nha : a ∈ closure s\nhb : b ∈ closure s\nHs : ∀ x ∈ s, ∀ y ∈ s, p x y\nH0_left : ∀ (x : R), p 0 x\nH0_right : ∀ (x : R), p x 0\nHneg_left : ∀ (x y : R), p x y → p (-x) y\nHneg_right : ∀ (x y : R), p x y → p x (-y)\nHadd_left : ∀ (x₁ x₂ y : R), p x₁ y → p x₂ y → p (x₁ + x₂) y\nHadd_right : ∀ (x y₁ y₂ : R), p x y₁ → p x y₂ → p x (y₁ + y₂)\nHmul_left : ∀ (x₁ x₂ y : R), p x₁ y → p x₂ y → p (x₁ * x₂) y\nHmul_right : ∀ (x y₁ y₂ : R), p x y₁ → p x y₂ → p x (y₁ * y₂)\n⊢ p a b","state_after":"F : Type w\nR : Type u\nS : Type v\nT : Type u_1\ninst✝⁴ : NonUnitalNonAssocRing R\ninst✝³ : NonUnitalNonAssocRing S\ninst✝² : NonUnitalNonAssocRing T\ninst✝¹ : FunLike F R S\ninst✝ : NonUnitalRingHomClass F R S\ng : S →ₙ+* T\nf : R →ₙ+* S\ns : Set R\np : R → R → Prop\na b : R\nha : a ∈ closure s\nhb : b ∈ closure s\nHs : ∀ x ∈ s, ∀ y ∈ s, p x y\nH0_left : ∀ (x : R), p 0 x\nH0_right : ∀ (x : R), p x 0\nHneg_left : ∀ (x y : R), p x y → p (-x) y\nHneg_right : ∀ (x y : R), p x y → p x (-y)\nHadd_left : ∀ (x₁ x₂ y : R), p x₁ y → p x₂ y → p (x₁ + x₂) y\nHadd_right : ∀ (x y₁ y₂ : R), p x y₁ → p x y₂ → p x (y₁ + y₂)\nHmul_left : ∀ (x₁ x₂ y : R), p x₁ y → p x₂ y → p (x₁ * x₂) y\nHmul_right : ∀ (x y₁ y₂ : R), p x y₁ → p x y₂ → p x (y₁ * y₂)\n⊢ ∀ x ∈ s, p a x","tactic":"refine closure_induction hb ?_ (H0_right _) (Hadd_right a) (Hneg_right a) (Hmul_right a)","premises":[{"full_name":"NonUnitalSubring.closure_induction","def_path":"Mathlib/RingTheory/NonUnitalSubring/Basic.lean","def_pos":[680,8],"def_end_pos":[680,25]}]},{"state_before":"F : Type w\nR : Type u\nS : Type v\nT : Type u_1\ninst✝⁴ : NonUnitalNonAssocRing R\ninst✝³ : NonUnitalNonAssocRing S\ninst✝² : NonUnitalNonAssocRing T\ninst✝¹ : FunLike F R S\ninst✝ : NonUnitalRingHomClass F R S\ng : S →ₙ+* T\nf : R →ₙ+* S\ns : Set R\np : R → R → Prop\na b : R\nha : a ∈ closure s\nhb : b ∈ closure s\nHs : ∀ x ∈ s, ∀ y ∈ s, p x y\nH0_left : ∀ (x : R), p 0 x\nH0_right : ∀ (x : R), p x 0\nHneg_left : ∀ (x y : R), p x y → p (-x) y\nHneg_right : ∀ (x y : R), p x y → p x (-y)\nHadd_left : ∀ (x₁ x₂ y : R), p x₁ y → p x₂ y → p (x₁ + x₂) y\nHadd_right : ∀ (x y₁ y₂ : R), p x y₁ → p x y₂ → p x (y₁ + y₂)\nHmul_left : ∀ (x₁ x₂ y : R), p x₁ y → p x₂ y → p (x₁ * x₂) y\nHmul_right : ∀ (x y₁ y₂ : R), p x y₁ → p x y₂ → p x (y₁ * y₂)\n⊢ ∀ x ∈ s, p a x","state_after":"case refine_1\nF : Type w\nR : Type u\nS : Type v\nT : Type u_1\ninst✝⁴ : NonUnitalNonAssocRing R\ninst✝³ : NonUnitalNonAssocRing S\ninst✝² : NonUnitalNonAssocRing T\ninst✝¹ : FunLike F R S\ninst✝ : NonUnitalRingHomClass F R S\ng : S →ₙ+* T\nf : R →ₙ+* S\ns : Set R\np : R → R → Prop\na b : R\nha : a ∈ closure s\nhb : b ∈ closure s\nHs : ∀ x ∈ s, ∀ y ∈ s, p x y\nH0_left : ∀ (x : R), p 0 x\nH0_right : ∀ (x : R), p x 0\nHneg_left : ∀ (x y : R), p x y → p (-x) y\nHneg_right : ∀ (x y : R), p x y → p x (-y)\nHadd_left : ∀ (x₁ x₂ y : R), p x₁ y → p x₂ y → p (x₁ + x₂) y\nHadd_right : ∀ (x y₁ y₂ : R), p x y₁ → p x y₂ → p x (y₁ + y₂)\nHmul_left : ∀ (x₁ x₂ y : R), p x₁ y → p x₂ y → p (x₁ * x₂) y\nHmul_right : ∀ (x y₁ y₂ : R), p x y₁ → p x y₂ → p x (y₁ * y₂)\n⊢ ∀ (x y : R), (∀ x_1 ∈ s, p x x_1) → (∀ x ∈ s, p y x) → ∀ x_1 ∈ s, p (x + y) x_1\n\ncase refine_2\nF : Type w\nR : Type u\nS : Type v\nT : Type u_1\ninst✝⁴ : NonUnitalNonAssocRing R\ninst✝³ : NonUnitalNonAssocRing S\ninst✝² : NonUnitalNonAssocRing T\ninst✝¹ : FunLike F R S\ninst✝ : NonUnitalRingHomClass F R S\ng : S →ₙ+* T\nf : R →ₙ+* S\ns : Set R\np : R → R → Prop\na b : R\nha : a ∈ closure s\nhb : b ∈ closure s\nHs : ∀ x ∈ s, ∀ y ∈ s, p x y\nH0_left : ∀ (x : R), p 0 x\nH0_right : ∀ (x : R), p x 0\nHneg_left : ∀ (x y : R), p x y → p (-x) y\nHneg_right : ∀ (x y : R), p x y → p x (-y)\nHadd_left : ∀ (x₁ x₂ y : R), p x₁ y → p x₂ y → p (x₁ + x₂) y\nHadd_right : ∀ (x y₁ y₂ : R), p x y₁ → p x y₂ → p x (y₁ + y₂)\nHmul_left : ∀ (x₁ x₂ y : R), p x₁ y → p x₂ y → p (x₁ * x₂) y\nHmul_right : ∀ (x y₁ y₂ : R), p x y₁ → p x y₂ → p x (y₁ * y₂)\n⊢ ∀ (x : R), (∀ x_1 ∈ s, p x x_1) → ∀ x_1 ∈ s, p (-x) x_1\n\ncase refine_3\nF : Type w\nR : Type u\nS : Type v\nT : Type u_1\ninst✝⁴ : NonUnitalNonAssocRing R\ninst✝³ : NonUnitalNonAssocRing S\ninst✝² : NonUnitalNonAssocRing T\ninst✝¹ : FunLike F R S\ninst✝ : NonUnitalRingHomClass F R S\ng : S →ₙ+* T\nf : R →ₙ+* S\ns : Set R\np : R → R → Prop\na b : R\nha : a ∈ closure s\nhb : b ∈ closure s\nHs : ∀ x ∈ s, ∀ y ∈ s, p x y\nH0_left : ∀ (x : R), p 0 x\nH0_right : ∀ (x : R), p x 0\nHneg_left : ∀ (x y : R), p x y → p (-x) y\nHneg_right : ∀ (x y : R), p x y → p x (-y)\nHadd_left : ∀ (x₁ x₂ y : R), p x₁ y → p x₂ y → p (x₁ + x₂) y\nHadd_right : ∀ (x y₁ y₂ : R), p x y₁ → p x y₂ → p x (y₁ + y₂)\nHmul_left : ∀ (x₁ x₂ y : R), p x₁ y → p x₂ y → p (x₁ * x₂) y\nHmul_right : ∀ (x y₁ y₂ : R), p x y₁ → p x y₂ → p x (y₁ * y₂)\n⊢ ∀ (x y : R), (∀ x_1 ∈ s, p x x_1) → (∀ x ∈ s, p y x) → ∀ x_1 ∈ s, p (x * y) x_1","tactic":"refine closure_induction ha Hs (fun x _ => H0_left x) ?_ ?_ ?_","premises":[{"full_name":"NonUnitalSubring.closure_induction","def_path":"Mathlib/RingTheory/NonUnitalSubring/Basic.lean","def_pos":[680,8],"def_end_pos":[680,25]}]}]}
+{"url":"Mathlib/Data/List/Sort.lean","commit":"","full_name":"List.orderedInsert_length","start":[207,0],"end":[211,48],"file_path":"Mathlib/Data/List/Sort.lean","tactics":[{"state_before":"α : Type u\nr : α → α → Prop\ninst✝ : DecidableRel r\nhd : α\ntl : List α\na : α\n⊢ (orderedInsert r a (hd :: tl)).length = (hd :: tl).length + 1","state_after":"α : Type u\nr : α → α → Prop\ninst✝ : DecidableRel r\nhd : α\ntl : List α\na : α\n⊢ (if r a hd then a :: hd :: tl else hd :: orderedInsert r a tl).length = tl.length + 1 + 1","tactic":"dsimp [orderedInsert]","premises":[{"full_name":"List.orderedInsert","def_path":"Mathlib/Data/List/Sort.lean","def_pos":[193,4],"def_end_pos":[193,17]}]},{"state_before":"α : Type u\nr : α → α → Prop\ninst✝ : DecidableRel r\nhd : α\ntl : List α\na : α\n⊢ (if r a hd then a :: hd :: tl else hd :: orderedInsert r a tl).length = tl.length + 1 + 1","state_after":"no goals","tactic":"split_ifs <;> simp [orderedInsert_length tl]","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
+{"url":"Mathlib/Algebra/Polynomial/FieldDivision.lean","commit":"","full_name":"Polynomial.eval₂_gcd_eq_zero","start":[394,0],"end":[398,63],"file_path":"Mathlib/Algebra/Polynomial/FieldDivision.lean","tactics":[{"state_before":"R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝² : Field R\np q : R[X]\ninst✝¹ : CommSemiring k\ninst✝ : DecidableEq R\nϕ : R →+* k\nf g : R[X]\nα : k\nhf : eval₂ ϕ α f = 0\nhg : eval₂ ϕ α g = 0\n⊢ eval₂ ϕ α (EuclideanDomain.gcd f g) = 0","state_after":"no goals","tactic":"rw [EuclideanDomain.gcd_eq_gcd_ab f g, Polynomial.eval₂_add, Polynomial.eval₂_mul,\n    Polynomial.eval₂_mul, hf, hg, zero_mul, zero_mul, zero_add]","premises":[{"full_name":"EuclideanDomain.gcd_eq_gcd_ab","def_path":"Mathlib/Algebra/EuclideanDomain/Basic.lean","def_pos":[192,8],"def_end_pos":[192,21]},{"full_name":"MulZeroClass.zero_mul","def_path":"Mathlib/Algebra/GroupWithZero/Defs.lean","def_pos":[35,2],"def_end_pos":[35,10]},{"full_name":"Polynomial.eval₂_add","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[77,8],"def_end_pos":[77,17]},{"full_name":"Polynomial.eval₂_mul","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[223,8],"def_end_pos":[223,17]},{"full_name":"zero_add","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[408,2],"def_end_pos":[408,13]}]}]}
+{"url":"Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Unital.lean","commit":"","full_name":"cfc_nonneg","start":[807,0],"end":[811,58],"file_path":"Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Unital.lean","tactics":[{"state_before":"R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁵ : OrderedCommSemiring R\ninst✝¹⁴ : StarRing R\ninst✝¹³ : StarOrderedRing R\ninst✝¹² : MetricSpace R\ninst✝¹¹ : TopologicalSemiring R\ninst✝¹⁰ : ContinuousStar R\ninst✝⁹ : ∀ (α : Type ?u.740550) [inst : TopologicalSpace α], StarOrderedRing C(α, R)\ninst✝⁸ : TopologicalSpace A\ninst✝⁷ : Ring A\ninst✝⁶ : StarRing A\ninst✝⁵ : PartialOrder A\ninst✝⁴ : StarOrderedRing A\ninst✝³ : Algebra R A\ninst✝² : StarModule R A\ninst✝¹ : ContinuousFunctionalCalculus R p\ninst✝ : NonnegSpectrumClass R A\nf : R → R\na : A\nh : ∀ x ∈ spectrum R a, 0 ≤ f x\n⊢ 0 ≤ cfc f a","state_after":"case pos\nR : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁵ : OrderedCommSemiring R\ninst✝¹⁴ : StarRing R\ninst✝¹³ : StarOrderedRing R\ninst✝¹² : MetricSpace R\ninst✝¹¹ : TopologicalSemiring R\ninst✝¹⁰ : ContinuousStar R\ninst✝⁹ : ∀ (α : Type ?u.740550) [inst : TopologicalSpace α], StarOrderedRing C(α, R)\ninst✝⁸ : TopologicalSpace A\ninst✝⁷ : Ring A\ninst✝⁶ : StarRing A\ninst✝⁵ : PartialOrder A\ninst✝⁴ : StarOrderedRing A\ninst✝³ : Algebra R A\ninst✝² : StarModule R A\ninst✝¹ : ContinuousFunctionalCalculus R p\ninst✝ : NonnegSpectrumClass R A\nf : R → R\na : A\nh : ∀ x ∈ spectrum R a, 0 ≤ f x\nhf : ContinuousOn f (spectrum R a)\n⊢ 0 ≤ cfc f a\n\ncase neg\nR : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁵ : OrderedCommSemiring R\ninst✝¹⁴ : StarRing R\ninst✝¹³ : StarOrderedRing R\ninst✝¹² : MetricSpace R\ninst✝¹¹ : TopologicalSemiring R\ninst✝¹⁰ : ContinuousStar R\ninst✝⁹ : ∀ (α : Type ?u.740550) [inst : TopologicalSpace α], StarOrderedRing C(α, R)\ninst✝⁸ : TopologicalSpace A\ninst✝⁷ : Ring A\ninst✝⁶ : StarRing A\ninst✝⁵ : PartialOrder A\ninst✝⁴ : StarOrderedRing A\ninst✝³ : Algebra R A\ninst✝² : StarModule R A\ninst✝¹ : ContinuousFunctionalCalculus R p\ninst✝ : NonnegSpectrumClass R A\nf : R → R\na : A\nh : ∀ x ∈ spectrum R a, 0 ≤ f x\nhf : ¬ContinuousOn f (spectrum R a)\n⊢ 0 ≤ cfc f a","tactic":"by_cases hf : ContinuousOn f (spectrum R a)","premises":[{"full_name":"ContinuousOn","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[164,4],"def_end_pos":[164,16]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]},{"full_name":"spectrum","def_path":"Mathlib/Algebra/Algebra/Spectrum.lean","def_pos":[65,4],"def_end_pos":[65,12]}]}]}
+{"url":"Mathlib/Analysis/Normed/Group/Basic.lean","commit":"","full_name":"mem_ball_iff_norm''","start":[507,0],"end":[508,94],"file_path":"Mathlib/Analysis/Normed/Group/Basic.lean","tactics":[{"state_before":"𝓕 : Type u_1\n𝕜 : Type u_2\nα : Type u_3\nι : Type u_4\nκ : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\n⊢ b ∈ ball a r ↔ ‖b / a‖ < r","state_after":"no goals","tactic":"rw [mem_ball, dist_eq_norm_div]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Metric.mem_ball","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[354,8],"def_end_pos":[354,16]},{"full_name":"dist_eq_norm_div","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[343,8],"def_end_pos":[343,24]}]}]}
+{"url":"Mathlib/AlgebraicGeometry/Cover/Open.lean","commit":"","full_name":"AlgebraicGeometry.Scheme.OpenCover.pullbackCover_J","start":[164,0],"end":[182,54],"file_path":"Mathlib/AlgebraicGeometry/Cover/Open.lean","tactics":[{"state_before":"X✝ Y Z : Scheme\n𝒰✝ : X✝.OpenCover\nf✝ : X✝ ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (x : 𝒰✝.J), HasPullback (𝒰✝.map x ≫ f✝) g\nX W : Scheme\n𝒰 : X.OpenCover\nf : W ⟶ X\nx : ↑↑W.toPresheafedSpace\n⊢ x ∈ Set.range ⇑((fun x => pullback.fst f (𝒰.map x)) ((fun x => 𝒰.f (f.val.base x)) x)).val.base","state_after":"X✝ Y Z : Scheme\n𝒰✝ : X✝.OpenCover\nf✝ : X✝ ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (x : 𝒰✝.J), HasPullback (𝒰✝.map x ≫ f✝) g\nX W : Scheme\n𝒰 : X.OpenCover\nf : W ⟶ X\nx : ↑↑W.toPresheafedSpace\n⊢ x ∈\n    Set.range\n      ⇑((PreservesPullback.iso forgetToTop f (𝒰.map (𝒰.f (f.val.base x)))).hom ≫\n          pullback.fst (forgetToTop.map f) (forgetToTop.map (𝒰.map (𝒰.f (f.val.base x)))))","tactic":"rw [←\n      show _ = (pullback.fst _ _ : pullback f (𝒰.map (𝒰.f (f.1.base x))) ⟶ _).1.base from\n        PreservesPullback.iso_hom_fst Scheme.forgetToTop f (𝒰.map (𝒰.f (f.1.base x)))]","premises":[{"full_name":"AlgebraicGeometry.LocallyRingedSpace.Hom.val","def_path":"Mathlib/Geometry/RingedSpace/LocallyRingedSpace.lean","def_pos":[75,2],"def_end_pos":[75,5]},{"full_name":"AlgebraicGeometry.PresheafedSpace.Hom.base","def_path":"Mathlib/Geometry/RingedSpace/PresheafedSpace.lean","def_pos":[90,2],"def_end_pos":[90,6]},{"full_name":"AlgebraicGeometry.Scheme.OpenCover.f","def_path":"Mathlib/AlgebraicGeometry/Cover/Open.lean","def_pos":[50,2],"def_end_pos":[50,3]},{"full_name":"AlgebraicGeometry.Scheme.OpenCover.map","def_path":"Mathlib/AlgebraicGeometry/Cover/Open.lean","def_pos":[48,2],"def_end_pos":[48,5]},{"full_name":"AlgebraicGeometry.Scheme.forgetToTop","def_path":"Mathlib/AlgebraicGeometry/Scheme.lean","def_pos":[194,4],"def_end_pos":[194,15]},{"full_name":"CategoryTheory.Limits.PreservesPullback.iso_hom_fst","def_path":"Mathlib/CategoryTheory/Limits/Preserves/Shapes/Pullbacks.lean","def_pos":[134,8],"def_end_pos":[134,37]},{"full_name":"CategoryTheory.Limits.pullback","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Pullback/HasPullback.lean","def_pos":[92,7],"def_end_pos":[92,15]},{"full_name":"CategoryTheory.Limits.pullback.fst","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Pullback/HasPullback.lean","def_pos":[108,7],"def_end_pos":[108,19]},{"full_name":"Quiver.Hom","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[43,2],"def_end_pos":[43,5]}]},{"state_before":"X✝ Y Z : Scheme\n𝒰✝ : X✝.OpenCover\nf✝ : X✝ ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (x : 𝒰✝.J), HasPullback (𝒰✝.map x ≫ f✝) g\nX W : Scheme\n𝒰 : X.OpenCover\nf : W ⟶ X\nx : ↑↑W.toPresheafedSpace\n⊢ x ∈\n    Set.range\n      ⇑((PreservesPullback.iso forgetToTop f (𝒰.map (𝒰.f (f.val.base x)))).hom ≫\n          pullback.fst (forgetToTop.map f) (forgetToTop.map (𝒰.map (𝒰.f (f.val.base x)))))","state_after":"X✝ Y Z : Scheme\n𝒰✝ : X✝.OpenCover\nf✝ : X✝ ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (x : 𝒰✝.J), HasPullback (𝒰✝.map x ≫ f✝) g\nX W : Scheme\n𝒰 : X.OpenCover\nf : W ⟶ X\nx : ↑↑W.toPresheafedSpace\n⊢ x ∈ {x_1 | ∃ y, (forgetToTop.map f) x_1 = (forgetToTop.map (𝒰.map (𝒰.f (f.val.base x)))) y}\n\nX✝ Y Z : Scheme\n𝒰✝ : X✝.OpenCover\nf✝ : X✝ ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (x : 𝒰✝.J), HasPullback (𝒰✝.map x ≫ f✝) g\nX W : Scheme\n𝒰 : X.OpenCover\nf : W ⟶ X\nx : ↑↑W.toPresheafedSpace\n⊢ Function.Surjective ⇑(PreservesPullback.iso forgetToTop f (𝒰.map (𝒰.f (f.val.base x)))).hom","tactic":"rw [TopCat.coe_comp, Set.range_comp, Set.range_iff_surjective.mpr, Set.image_univ,\n      TopCat.pullback_fst_range]","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Set.image_univ","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[596,8],"def_end_pos":[596,18]},{"full_name":"Set.range_comp","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[621,8],"def_end_pos":[621,18]},{"full_name":"Set.range_iff_surjective","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[586,8],"def_end_pos":[586,28]},{"full_name":"TopCat.coe_comp","def_path":"Mathlib/Topology/Category/TopCat/Basic.lean","def_pos":[71,16],"def_end_pos":[71,24]},{"full_name":"TopCat.pullback_fst_range","def_path":"Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean","def_pos":[229,8],"def_end_pos":[229,26]}]}]}
+{"url":"Mathlib/RingTheory/OreLocalization/Basic.lean","commit":"","full_name":"OreLocalization.mul_inv","start":[394,0],"end":[396,54],"file_path":"Mathlib/RingTheory/OreLocalization/Basic.lean","tactics":[{"state_before":"R : Type u_1\ninst✝² : Monoid R\nS : Submonoid R\ninst✝¹ : OreSet S\nX : Type ?u.97943\ninst✝ : MulAction R X\ns s' : ↥S\n⊢ ↑s /ₒ s' * (↑s' /ₒ s) = 1","state_after":"no goals","tactic":"simp [oreDiv_mul_char (s : R) s' s' s 1 1 (by simp)]","premises":[{"full_name":"OreLocalization.oreDiv_mul_char","def_path":"Mathlib/RingTheory/OreLocalization/Basic.lean","def_pos":[299,8],"def_end_pos":[299,23]}]}]}
+{"url":"Mathlib/Algebra/Ring/Hom/Defs.lean","commit":"","full_name":"RingHom.cancel_left","start":[612,0],"end":[615,93],"file_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","tactics":[{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nx✝² : NonAssocSemiring α\nx✝¹ : NonAssocSemiring β\nx✝ : NonAssocSemiring γ\ng : β →+* γ\nf₁ f₂ : α →+* β\nhg : Injective ⇑g\nh : g.comp f₁ = g.comp f₂\nx : α\n⊢ g (f₁ x) = g (f₂ x)","state_after":"no goals","tactic":"rw [← comp_apply, h, comp_apply]","premises":[{"full_name":"RingHom.comp_apply","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[575,8],"def_end_pos":[575,18]}]}]}
+{"url":"Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean","commit":"","full_name":"Finset.ruzsa_triangle_inequality_mul_div_div","start":[141,0],"end":[146,51],"file_path":"Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean","tactics":[{"state_before":"α : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B✝ C✝ A B C : Finset α\n⊢ (A * C).card * B.card ≤ (A / B).card * (B / C).card","state_after":"α : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B✝ C✝ A B C : Finset α\n⊢ (A * C).card * B⁻¹.card ≤ (A * B⁻¹).card * (B⁻¹ * C).card","tactic":"rw [div_eq_mul_inv, ← card_inv B, ← card_inv (B / C), inv_div', div_inv_eq_mul]","premises":[{"full_name":"Finset.card_inv","def_path":"Mathlib/Data/Finset/Pointwise.lean","def_pos":[268,8],"def_end_pos":[268,16]},{"full_name":"div_eq_mul_inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[907,8],"def_end_pos":[907,22]},{"full_name":"div_inv_eq_mul","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[501,8],"def_end_pos":[501,22]},{"full_name":"inv_div'","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[519,8],"def_end_pos":[519,16]}]},{"state_before":"α : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B✝ C✝ A B C : Finset α\n⊢ (A * C).card * B⁻¹.card ≤ (A * B⁻¹).card * (B⁻¹ * C).card","state_after":"no goals","tactic":"exact ruzsa_triangle_inequality_mul_mul_mul _ _ _","premises":[{"full_name":"Finset.ruzsa_triangle_inequality_mul_mul_mul","def_path":"Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean","def_pos":[121,8],"def_end_pos":[121,45]}]}]}
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fourierIntegral e μ L g = fourierIntegral e μ L (f + g)","state_after":"case h\n𝕜 : Type u_1\ninst✝¹⁷ : CommRing 𝕜\nV : Type u_2\ninst✝¹⁶ : AddCommGroup V\ninst✝¹⁵ : Module 𝕜 V\ninst✝¹⁴ : MeasurableSpace V\nW : Type u_3\ninst✝¹³ : AddCommGroup W\ninst✝¹² : Module 𝕜 W\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℂ E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace ℂ F\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace ℂ G\ninst✝⁵ : TopologicalSpace 𝕜\ninst✝⁴ : TopologicalRing 𝕜\ninst✝³ : TopologicalSpace V\ninst✝² : BorelSpace V\ninst✝¹ : TopologicalSpace W\ne : AddChar 𝕜 ↥𝕊\nμ : Measure V\nL : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜\ninst✝ : CompleteSpace E\nhe : Continuous ⇑e\nhL : Continuous fun p => (L p.1) p.2\nf g : V → E\nhf : Integrable f μ\nhg : Integrable g μ\nw : W\n⊢ (fourierIntegral e μ L f + fourierIntegral e μ L g) w = fourierIntegral e μ L (f + g) w","tactic":"ext1 w","premises":[]},{"state_before":"case h\n𝕜 : Type u_1\ninst✝¹⁷ : CommRing 𝕜\nV : Type u_2\ninst✝¹⁶ : AddCommGroup V\ninst✝¹⁵ : Module 𝕜 V\ninst✝¹⁴ : MeasurableSpace V\nW : Type u_3\ninst✝¹³ : AddCommGroup W\ninst✝¹² : Module 𝕜 W\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℂ E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace ℂ F\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace ℂ G\ninst✝⁵ : TopologicalSpace 𝕜\ninst✝⁴ : TopologicalRing 𝕜\ninst✝³ : TopologicalSpace V\ninst✝² : BorelSpace V\ninst✝¹ : TopologicalSpace W\ne : AddChar 𝕜 ↥𝕊\nμ : Measure V\nL : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜\ninst✝ : CompleteSpace E\nhe : Continuous ⇑e\nhL : Continuous fun p => (L p.1) p.2\nf g : V → E\nhf : Integrable f μ\nhg : Integrable g μ\nw : W\n⊢ (fourierIntegral e μ L f + fourierIntegral e μ L g) w = fourierIntegral e μ L (f + g) w","state_after":"case h\n𝕜 : Type u_1\ninst✝¹⁷ : CommRing 𝕜\nV : Type u_2\ninst✝¹⁶ : AddCommGroup V\ninst✝¹⁵ : Module 𝕜 V\ninst✝¹⁴ : MeasurableSpace V\nW : Type u_3\ninst✝¹³ : AddCommGroup W\ninst✝¹² : Module 𝕜 W\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℂ E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace ℂ F\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace ℂ G\ninst✝⁵ : TopologicalSpace 𝕜\ninst✝⁴ : TopologicalRing 𝕜\ninst✝³ : TopologicalSpace V\ninst✝² : BorelSpace V\ninst✝¹ : TopologicalSpace W\ne : AddChar 𝕜 ↥𝕊\nμ : Measure V\nL : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜\ninst✝ : CompleteSpace E\nhe : Continuous ⇑e\nhL : Continuous fun p => (L p.1) p.2\nf g : V → E\nhf : Integrable f μ\nhg : Integrable g μ\nw : W\n⊢ ∫ (v : V), e (-(L v) w) • f v ∂μ + ∫ (v : V), e (-(L v) w) • g v ∂μ = ∫ (v : V), e (-(L v) w) • (f v + g v) ∂μ","tactic":"dsimp only [Pi.add_apply, fourierIntegral]","premises":[{"full_name":"Pi.add_apply","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[81,2],"def_end_pos":[81,13]},{"full_name":"VectorFourier.fourierIntegral","def_path":"Mathlib/Analysis/Fourier/FourierTransform.lean","def_pos":[77,4],"def_end_pos":[77,19]}]},{"state_before":"case h\n𝕜 : Type u_1\ninst✝¹⁷ : CommRing 𝕜\nV : Type u_2\ninst✝¹⁶ : AddCommGroup V\ninst✝¹⁵ : Module 𝕜 V\ninst✝¹⁴ : MeasurableSpace V\nW : Type u_3\ninst✝¹³ : AddCommGroup W\ninst✝¹² : Module 𝕜 W\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℂ E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace ℂ F\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace ℂ G\ninst✝⁵ : TopologicalSpace 𝕜\ninst✝⁴ : TopologicalRing 𝕜\ninst✝³ : TopologicalSpace V\ninst✝² : BorelSpace V\ninst✝¹ : TopologicalSpace W\ne : AddChar 𝕜 ↥𝕊\nμ : Measure V\nL : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜\ninst✝ : CompleteSpace E\nhe : Continuous ⇑e\nhL : Continuous fun p => (L p.1) p.2\nf g : V → E\nhf : Integrable f μ\nhg : Integrable g μ\nw : W\n⊢ ∫ (v : V), e (-(L v) w) • f v ∂μ + ∫ (v : V), e (-(L v) w) • g v ∂μ = ∫ (v : V), e (-(L v) w) • (f v + g v) ∂μ","state_after":"case h\n𝕜 : Type u_1\ninst✝¹⁷ : CommRing 𝕜\nV : Type u_2\ninst✝¹⁶ : AddCommGroup V\ninst✝¹⁵ : Module 𝕜 V\ninst✝¹⁴ : MeasurableSpace V\nW : Type u_3\ninst✝¹³ : AddCommGroup W\ninst✝¹² : Module 𝕜 W\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℂ E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace ℂ F\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace ℂ G\ninst✝⁵ : TopologicalSpace 𝕜\ninst✝⁴ : TopologicalRing 𝕜\ninst✝³ : TopologicalSpace V\ninst✝² : BorelSpace V\ninst✝¹ : TopologicalSpace W\ne : AddChar 𝕜 ↥𝕊\nμ : Measure V\nL : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜\ninst✝ : CompleteSpace E\nhe : Continuous ⇑e\nhL : Continuous fun p => (L p.1) p.2\nf g : V → E\nhf : Integrable f μ\nhg : Integrable g μ\nw : W\n⊢ ∫ (v : V), e (-(L v) w) • f v ∂μ + ∫ (v : V), e (-(L v) w) • g v ∂μ =\n    ∫ (v : V), e (-(L v) w) • f v + e (-(L v) w) • g v ∂μ","tactic":"simp_rw [smul_add]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"smul_add","def_path":"Mathlib/Algebra/GroupWithZero/Action/Defs.lean","def_pos":[130,8],"def_end_pos":[130,16]}]},{"state_before":"case h\n𝕜 : Type u_1\ninst✝¹⁷ : CommRing 𝕜\nV : Type u_2\ninst✝¹⁶ : AddCommGroup V\ninst✝¹⁵ : Module 𝕜 V\ninst✝¹⁴ : MeasurableSpace V\nW : Type u_3\ninst✝¹³ : AddCommGroup W\ninst✝¹² : Module 𝕜 W\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℂ E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace ℂ F\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace ℂ G\ninst✝⁵ : TopologicalSpace 𝕜\ninst✝⁴ : TopologicalRing 𝕜\ninst✝³ : TopologicalSpace V\ninst✝² : BorelSpace V\ninst✝¹ : TopologicalSpace W\ne : AddChar 𝕜 ↥𝕊\nμ : Measure V\nL : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜\ninst✝ : CompleteSpace E\nhe : Continuous ⇑e\nhL : Continuous fun p => (L p.1) p.2\nf g : V → E\nhf : Integrable f μ\nhg : Integrable g μ\nw : W\n⊢ ∫ (v : V), e (-(L v) w) • f v ∂μ + ∫ (v : V), e (-(L v) w) • g v ∂μ =\n    ∫ (v : V), e (-(L v) w) • f v + e (-(L v) w) • g v ∂μ","state_after":"case h.hf\n𝕜 : Type u_1\ninst✝¹⁷ : CommRing 𝕜\nV : Type u_2\ninst✝¹⁶ : AddCommGroup V\ninst✝¹⁵ : Module 𝕜 V\ninst✝¹⁴ : MeasurableSpace V\nW : Type u_3\ninst✝¹³ : AddCommGroup W\ninst✝¹² : Module 𝕜 W\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℂ E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace ℂ F\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace ℂ G\ninst✝⁵ : TopologicalSpace 𝕜\ninst✝⁴ : TopologicalRing 𝕜\ninst✝³ : TopologicalSpace V\ninst✝² : BorelSpace V\ninst✝¹ : TopologicalSpace W\ne : AddChar 𝕜 ���𝕊\nμ : Measure V\nL : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜\ninst✝ : CompleteSpace E\nhe : Continuous ⇑e\nhL : Continuous fun p => (L p.1) p.2\nf g : V → E\nhf : Integrable f μ\nhg : Integrable g μ\nw : W\n⊢ Integrable (fun v => e (-(L v) w) • f v) μ\n\ncase h.hg\n𝕜 : Type u_1\ninst✝¹⁷ : CommRing 𝕜\nV : Type u_2\ninst✝¹⁶ : AddCommGroup V\ninst✝¹⁵ : Module 𝕜 V\ninst✝¹⁴ : MeasurableSpace V\nW : Type u_3\ninst✝¹³ : AddCommGroup W\ninst✝¹² : Module 𝕜 W\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℂ E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace ℂ F\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace ℂ G\ninst✝⁵ : TopologicalSpace 𝕜\ninst✝⁴ : TopologicalRing 𝕜\ninst✝³ : TopologicalSpace V\ninst✝² : BorelSpace V\ninst✝¹ : TopologicalSpace W\ne : AddChar 𝕜 ↥𝕊\nμ : Measure V\nL : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜\ninst✝ : CompleteSpace E\nhe : Continuous ⇑e\nhL : Continuous fun p => (L p.1) p.2\nf g : V → E\nhf : Integrable f μ\nhg : Integrable g μ\nw : W\n⊢ Integrable (fun v => e (-(L v) w) • g v) μ","tactic":"rw [integral_add]","premises":[{"full_name":"MeasureTheory.integral_add","def_path":"Mathlib/MeasureTheory/Integral/Bochner.lean","def_pos":[784,8],"def_end_pos":[784,20]}]}]}
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+{"url":"Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean","commit":"","full_name":"GaussianFourier.integrable_cexp_neg_mul_sum_add","start":[260,0],"end":[263,54],"file_path":"Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean","tactics":[{"state_before":"b : ℂ\nV : Type u_1\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : FiniteDimensional ℝ V\ninst✝² : MeasurableSpace V\ninst���¹ : BorelSpace V\nι : Type u_2\ninst✝ : Fintype ι\nhb : 0 < b.re\nc : ι → ℂ\n⊢ Integrable (fun v => cexp (-b * ∑ i : ι, ↑(v i) ^ 2 + ∑ i : ι, c i * ↑(v i))) volume","state_after":"b : ℂ\nV : Type u_1\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : FiniteDimensional ℝ V\ninst✝² : MeasurableSpace V\ninst✝¹ : BorelSpace V\nι : Type u_2\ninst✝ : Fintype ι\nhb : 0 < b.re\nc : ι → ℂ\n⊢ Integrable (fun v => cexp (-∑ i : ι, b * ↑(v i) ^ 2 + ∑ i : ι, c i * ↑(v i))) volume","tactic":"simp_rw [neg_mul, Finset.mul_sum]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Finset.mul_sum","def_path":"Mathlib/Algebra/BigOperators/Ring.lean","def_pos":[44,6],"def_end_pos":[44,13]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"neg_mul","def_path":"Mathlib/Algebra/Ring/Defs.lean","def_pos":[268,8],"def_end_pos":[268,15]}]},{"state_before":"b : ℂ\nV : Type u_1\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : FiniteDimensional ℝ V\ninst✝² : MeasurableSpace V\ninst✝¹ : BorelSpace V\nι : Type u_2\ninst✝ : Fintype ι\nhb : 0 < b.re\nc : ι → ℂ\n⊢ Integrable (fun v => cexp (-∑ i : ι, b * ↑(v i) ^ 2 + ∑ i : ι, c i * ↑(v i))) volume","state_after":"no goals","tactic":"exact integrable_cexp_neg_sum_mul_add (fun _ ↦ hb) c","premises":[{"full_name":"GaussianFourier.integrable_cexp_neg_sum_mul_add","def_path":"Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean","def_pos":[252,8],"def_end_pos":[252,39]}]}]}
+{"url":"Mathlib/Order/Filter/IndicatorFunction.lean","commit":"","full_name":"Monotone.mulIndicator_eventuallyEq_iUnion","start":[57,0],"end":[61,57],"file_path":"Mathlib/Order/Filter/IndicatorFunction.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nM : Type u_3\nE : Type u_4\nι : Type u_5\ninst✝¹ : Preorder ι\ninst✝ : One β\ns : ι → Set α\nhs : Monotone s\nf : α → β\na : α\n⊢ (fun i => (s i).mulIndicator f a) =ᶠ[atTop] fun x => (⋃ i, s i).mulIndicator f a","state_after":"no goals","tactic":"classical exact hs.piecewise_eventually_eq_iUnion f 1 a","premises":[{"full_name":"Monotone.piecewise_eventually_eq_iUnion","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[1778,8],"def_end_pos":[1778,47]}]}]}
+{"url":"Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean","commit":"","full_name":"AlgebraicGeometry.ProjectiveSpectrum.Proj.awayToΓ_ΓToStalk","start":[612,0],"end":[622,5],"file_path":"Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean","tactics":[{"state_before":"R : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : A\nx : ↑↑(Proj.restrict ⋯).toPresheafedSpace\n⊢ awayToΓ 𝒜 f ≫ (Proj.restrict ⋯).presheaf.Γgerm x =\n    HomogeneousLocalization.mapId 𝒜 ⋯ ≫ (Proj.stalkIso' 𝒜 ↑x).toCommRingCatIso.inv ≫ (Proj.restrictStalkIso ⋯ x).inv","state_after":"R : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : A\nx : ↑↑(Proj.restrict ⋯).toPresheafedSpace\n⊢ awayToSection 𝒜 f ≫\n      (structureSheaf 𝒜).val.map (homOfLE ⋯).op ≫\n        (Proj.restrict ⋯).presheaf.germ ⟨x, ⋯⟩ ≫ (Proj.restrictStalkIso ⋯ x).hom =\n    HomogeneousLocalization.mapId 𝒜 ⋯ ≫ (Proj.stalkIso' 𝒜 ↑x).toCommRingCatIso.inv","tactic":"rw [awayToΓ, Category.assoc, ← Category.assoc _ (Iso.inv _),\n    Iso.eq_comp_inv, Category.assoc, Category.assoc, Presheaf.Γgerm]","premises":[{"full_name":"AlgebraicGeometry.ProjectiveSpectrum.Proj.awayToΓ","def_path":"Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean","def_pos":[608,4],"def_end_pos":[608,11]},{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Iso.eq_comp_inv","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[187,8],"def_end_pos":[187,19]},{"full_name":"CategoryTheory.Iso.inv","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[53,2],"def_end_pos":[53,5]},{"full_name":"TopCat.Presheaf.Γgerm","def_path":"Mathlib/Topology/Sheaves/Stalks.lean","def_pos":[95,4],"def_end_pos":[95,9]}]},{"state_before":"R : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : A\nx : ↑↑(Proj.restrict ⋯).toPresheafedSpace\n⊢ awayToSection 𝒜 f ≫\n      (structureSheaf 𝒜).val.map (homOfLE ⋯).op ≫\n        (Proj.restrict ⋯).presheaf.germ ⟨x, ⋯⟩ ≫ (Proj.restrictStalkIso ⋯ x).hom =\n    HomogeneousLocalization.mapId 𝒜 ⋯ ≫ (Proj.stalkIso' 𝒜 ↑x).toCommRingCatIso.inv","state_after":"R : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : A\nx : ↑↑(Proj.restrict ⋯).toPresheafedSpace\n⊢ awayToSection 𝒜 f ≫ (structureSheaf 𝒜).val.map (homOfLE ⋯).op ≫ Proj.presheaf.germ ⟨(pbo f).inclusion x, ⋯⟩ =\n    HomogeneousLocalization.mapId 𝒜 ⋯ ≫ (Proj.stalkIso' 𝒜 ↑x).toCommRingCatIso.inv","tactic":"rw [LocallyRingedSpace.restrictStalkIso_hom_eq_germ]","premises":[{"full_name":"AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso_hom_eq_germ","def_path":"Mathlib/Geometry/RingedSpace/LocallyRingedSpace.lean","def_pos":[436,6],"def_end_pos":[436,34]}]},{"state_before":"R : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : A\nx : ↑↑(Proj.restrict ⋯).toPresheafedSpace\n⊢ awayToSection 𝒜 f ≫ (structureSheaf 𝒜).val.map (homOfLE ⋯).op ≫ Proj.presheaf.germ ⟨(pbo f).inclusion x, ⋯⟩ =\n    HomogeneousLocalization.mapId 𝒜 ⋯ ≫ (Proj.stalkIso' 𝒜 ↑x).toCommRingCatIso.inv","state_after":"R : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : A\nx : ↑↑(Proj.restrict ⋯).toPresheafedSpace\n⊢ awayToSection 𝒜 f ≫\n      (structureSheaf 𝒜).val.map (homOfLE ⋯).op ≫ Presheaf.germ (structureSheaf 𝒜).val ⟨(pbo f).inclusion x, ⋯⟩ =\n    HomogeneousLocalization.mapId 𝒜 ⋯ ≫ (Proj.stalkIso' 𝒜 ↑x).toCommRingCatIso.inv","tactic":"simp only [Proj.toLocallyRingedSpace, Proj.toSheafedSpace]","premises":[{"full_name":"AlgebraicGeometry.Proj.toLocallyRingedSpace","def_path":"Mathlib/AlgebraicGeometry/ProjectiveSpectrum/StructureSheaf.lean","def_pos":[369,4],"def_end_pos":[369,29]},{"full_name":"AlgebraicGeometry.Proj.toSheafedSpace","def_path":"Mathlib/AlgebraicGeometry/ProjectiveSpectrum/StructureSheaf.lean","def_pos":[235,4],"def_end_pos":[235,23]}]},{"state_before":"R : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : A\nx : ↑↑(Proj.restrict ⋯).toPresheafedSpace\n⊢ awayToSection 𝒜 f ≫\n      (structureSheaf 𝒜).val.map (homOfLE ⋯).op ≫ Presheaf.germ (structureSheaf 𝒜).val ⟨(pbo f).inclusion x, ⋯⟩ =\n    HomogeneousLocalization.mapId 𝒜 ⋯ ≫ (Proj.stalkIso' 𝒜 ↑x).toCommRingCatIso.inv","state_after":"R : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : A\nx : ↑↑(Proj.restrict ⋯).toPresheafedSpace\n⊢ HomogeneousLocalization.mapId 𝒜 ⋯ ≫\n      (Proj.stalkIso' 𝒜 ↑((fun x => ⟨↑x, ⋯⟩) ⟨(pbo f).inclusion x, ⋯⟩)).toCommRingCatIso.inv =\n    HomogeneousLocalization.mapId 𝒜 ⋯ ≫ (Proj.stalkIso' 𝒜 ↑x).toCommRingCatIso.inv","tactic":"rw [Presheaf.germ_res, awayToSection_germ]","premises":[{"full_name":"AlgebraicGeometry.ProjectiveSpectrum.Proj.awayToSection_germ","def_path":"Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean","def_pos":[594,6],"def_end_pos":[594,24]},{"full_name":"TopCat.Presheaf.germ_res","def_path":"Mathlib/Topology/Sheaves/Stalks.lean","def_pos":[99,8],"def_end_pos":[99,16]}]},{"state_before":"R : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : A\nx : ↑↑(Proj.restrict ⋯).toPresheafedSpace\n⊢ HomogeneousLocalization.mapId 𝒜 ⋯ ≫\n      (Proj.stalkIso' 𝒜 ↑((fun x => ⟨↑x, ⋯⟩) ⟨(pbo f).inclusion x, ⋯⟩)).toCommRingCatIso.inv =\n    HomogeneousLocalization.mapId 𝒜 ⋯ ≫ (Proj.stalkIso' 𝒜 ↑x).toCommRingCatIso.inv","state_after":"no goals","tactic":"rfl","premises":[]}]}
+{"url":"Mathlib/Algebra/Homology/TotalComplexShift.lean","commit":"","full_name":"HomologicalComplex₂.ι_totalShift₂Iso_inv_f","start":[317,0],"end":[328,34],"file_path":"Mathlib/Algebra/Homology/TotalComplexShift.lean","tactics":[{"state_before":"C : Type u_1\ninst✝² : Category.{?u.244750, u_1} C\ninst✝¹ : Preadditive C\nK L : HomologicalComplex₂ C (up ℤ) (up ℤ)\nf : K ⟶ L\nx y : ℤ\ninst✝ : K.HasTotal (up ℤ)\na b n : ℤ\nh : a + b = n\nb' n' : ℤ\nhb' : a + b' = n'\nhn' : n' = n + y\n⊢ b' = b + y","state_after":"no goals","tactic":"omega","premises":[]},{"state_before":"C : Type u_1\ninst✝² : Category.{u_2, u_1} C\ninst✝¹ : Preadditive C\nK L : HomologicalComplex₂ C (up ℤ) (up ℤ)\nf : K ⟶ L\nx y : ℤ\ninst✝ : K.HasTotal (up ℤ)\na b n : ℤ\nh : a + b = n\nb' n' : ℤ\nhb' : a + b' = n'\nhn' : n' = n + y\n⊢ K.ιTotal (up ℤ) a b' n' hb' ≫\n      (CochainComplex.shiftFunctorObjXIso (K.total (up ℤ)) y n n' hn').inv ≫ (K.totalShift₂Iso y).inv.f n =\n    (a * y).negOnePow • (K.shiftFunctor₂XXIso a b y b' ⋯).inv ≫ ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) a b n h","state_after":"C : Type u_1\ninst✝² : Category.{u_2, u_1} C\ninst✝¹ : Preadditive C\nK L : HomologicalComplex₂ C (up ℤ) (up ℤ)\nf : K ⟶ L\nx y : ℤ\ninst✝ : K.HasTotal (up ℤ)\na b n : ℤ\nh : a + b = n\nb' : ℤ\nhb' : a + b' = n + y\n⊢ K.ιTotal (up ℤ) a b' (n + y) hb' ≫\n      (CochainComplex.shiftFunctorObjXIso (K.total (up ℤ)) y n (n + y) ⋯).inv ≫ (K.totalShift₂Iso y).inv.f n =\n    (a * y).negOnePow • (K.shiftFunctor₂XXIso a b y b' ⋯).inv ≫ ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) a b n h","tactic":"subst hn'","premises":[]},{"state_before":"C : Type u_1\ninst✝² : Category.{u_2, u_1} C\ninst✝¹ : Preadditive C\nK L : HomologicalComplex₂ C (up ℤ) (up ℤ)\nf : K ⟶ L\nx y : ℤ\ninst✝ : K.HasTotal (up ℤ)\na b n : ℤ\nh : a + b = n\nb' : ℤ\nhb' : a + b' = n + y\n⊢ K.ιTotal (up ℤ) a b' (n + y) hb' ≫\n      (CochainComplex.shiftFunctorObjXIso (K.total (up ℤ)) y n (n + y) ⋯).inv ≫ (K.totalShift₂Iso y).inv.f n =\n    (a * y).negOnePow • (K.shiftFunctor₂XXIso a b y b' ⋯).inv ≫ ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) a b n h","state_after":"C : Type u_1\ninst✝² : Category.{u_2, u_1} C\ninst✝¹ : Preadditive C\nK L : HomologicalComplex₂ C (up ℤ) (up ℤ)\nf : K ⟶ L\nx y : ℤ\ninst✝ : K.HasTotal (up ℤ)\na n b' : ℤ\nhb' : a + b' = n + y\nh : a + (b' - y) = n\n⊢ K.ιTotal (up ℤ) a b' (n + y) hb' ≫\n      (CochainComplex.shiftFunctorObjXIso (K.total (up ℤ)) y n (n + y) ⋯).inv ≫ (K.totalShift₂Iso y).inv.f n =\n    (a * y).negOnePow •\n      (K.shiftFunctor₂XXIso a (b' - y) y b' ⋯).inv ≫ ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) a (b' - y) n h","tactic":"obtain rfl : b = b' - y := by omega","premises":[]},{"state_before":"C : Type u_1\ninst✝² : Category.{u_2, u_1} C\ninst✝¹ : Preadditive C\nK L : HomologicalComplex₂ C (up ℤ) (up ℤ)\nf : K ⟶ L\nx y : ℤ\ninst✝ : K.HasTotal (up ℤ)\na n b' : ℤ\nhb' : a + b' = n + y\nh : a + (b' - y) = n\n⊢ K.ιTotal (up ℤ) a b' (n + y) hb' ≫\n      (CochainComplex.shiftFunctorObjXIso (K.total (up ℤ)) y n (n + y) ⋯).inv ≫ (K.totalShift₂Iso y).inv.f n =\n    (a * y).negOnePow •\n      (K.shiftFunctor₂XXIso a (b' - y) y b' ⋯).inv ≫ ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) a (b' - y) n h","state_after":"C : Type u_1\ninst✝² : Category.{u_2, u_1} C\ninst✝¹ : Preadditive C\nK L : HomologicalComplex₂ C (up ℤ) (up ℤ)\nf : K ⟶ L\nx y : ℤ\ninst✝ : K.HasTotal (up ℤ)\na n b' : ℤ\nhb' : a + b' = n + y\nh : a + (b' - y) = n\n⊢ (K.ιTotal (up ℤ) a b' (n + y) hb' ≫\n      𝟙 ((K.total (up ℤ)).X (n + y)) ≫\n        K.totalDesc fun p q hpq =>\n          (p * y).negOnePow • eqToHom ⋯ ≫ ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) p (q - y) n ⋯) =\n    (a * y).negOnePow • eqToHom ⋯ ≫ ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) a (b' - y) n h","tactic":"dsimp [totalShift₂Iso, totalShift₂XIso, shiftFunctor₂XXIso, XXIsoOfEq]","premises":[{"full_name":"HomologicalComplex₂.XXIsoOfEq","def_path":"Mathlib/Algebra/Homology/HomologicalBicomplex.lean","def_pos":[202,4],"def_end_pos":[202,13]},{"full_name":"HomologicalComplex₂.shiftFunctor₂XXIso","def_path":"Mathlib/Algebra/Homology/TotalComplexShift.lean","def_pos":[62,4],"def_end_pos":[62,22]},{"full_name":"HomologicalComplex₂.totalShift₂Iso","def_path":"Mathlib/Algebra/Homology/TotalComplexShift.lean","def_pos":[298,18],"def_end_pos":[298,32]},{"full_name":"HomologicalComplex₂.totalShift₂XIso","def_path":"Mathlib/Algebra/Homology/TotalComplexShift.lean","def_pos":[233,18],"def_end_pos":[233,33]}]},{"state_before":"C : Type u_1\ninst✝² : Category.{u_2, u_1} C\ninst✝¹ : Preadditive C\nK L : HomologicalComplex₂ C (up ℤ) (up ℤ)\nf : K ⟶ L\nx y : ℤ\ninst✝ : K.HasTotal (up ℤ)\na n b' : ℤ\nhb' : a + b' = n + y\nh : a + (b' - y) = n\n⊢ (K.ιTotal (up ℤ) a b' (n + y) hb' ≫\n      𝟙 ((K.total (up ℤ)).X (n + y)) ≫\n        K.totalDesc fun p q hpq =>\n          (p * y).negOnePow • eqToHom ⋯ ≫ ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) p (q - y) n ⋯) =\n    (a * y).negOnePow • eqToHom ⋯ ≫ ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) a (b' - y) n h","state_after":"no goals","tactic":"simp only [id_comp, ι_totalDesc]","premises":[{"full_name":"CategoryTheory.Category.id_comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[151,2],"def_end_pos":[151,9]},{"full_name":"HomologicalComplex₂.ι_totalDesc","def_path":"Mathlib/Algebra/Homology/TotalComplex.lean","def_pos":[335,6],"def_end_pos":[335,17]}]}]}
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u_4\nX : Type u_5\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns t : Set M\nx✝ : M\nQ✝ : (H → H) → Set H → H → Prop\nhG✝ : G.LocalInvariantProp G' P\nQ : (H → H) → Set H → H → Prop\nhG : G.LocalInvariantProp G Q\nhQ : ∀ (y : H), Q id univ y\nU : Opens M\nx : ↥U\n⊢ LiftPropAt Q Subtype.val x","tactic":"intro x","premises":[]},{"state_before":"H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns t : Set M\nx✝ : M\nQ✝ : (H → H) → Set H → H → Prop\nhG✝ : G.LocalInvariantProp G' P\nQ : (H → H) → Set H → H → Prop\nhG : G.LocalInvariantProp G Q\nhQ : ∀ (y : H), Q id univ y\nU : Opens M\nx : ↥U\n⊢ LiftPropAt Q Subtype.val x","state_after":"H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns t : Set M\nx✝ : M\nQ✝ : (H → H) → Set H → H → Prop\nhG✝ : G.LocalInvariantProp G' P\nQ : (H → H) → Set H → H → Prop\nhG : G.LocalInvariantProp G Q\nhQ : ∀ (y : H), Q id univ y\nU : Opens M\nx : ↥U\n⊢ LiftPropAt Q (id ∘ Subtype.val) x","tactic":"show LiftPropAt Q (id ∘ Subtype.val) x","premises":[{"full_name":"ChartedSpace.LiftPropAt","def_path":"Mathlib/Geometry/Manifold/LocalInvariantProperties.lean","def_pos":[166,4],"def_end_pos":[166,14]},{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"Subtype.val","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[587,2],"def_end_pos":[587,5]},{"full_name":"id","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[33,14],"def_end_pos":[33,16]}]},{"state_before":"H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns t : Set M\nx✝ : M\nQ✝ : (H → H) → Set H → H → Prop\nhG✝ : G.LocalInvariantProp G' P\nQ : (H → H) → Set H → H → Prop\nhG : G.LocalInvariantProp G Q\nhQ : ∀ (y : H), Q id univ y\nU : Opens M\nx : ↥U\n⊢ LiftPropAt Q (id ∘ Subtype.val) x","state_after":"H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns t : Set M\nx✝ : M\nQ✝ : (H → H) → Set H → H → Prop\nhG✝ : G.LocalInvariantProp G' P\nQ : (H → H) → Set H → H → Prop\nhG : G.LocalInvariantProp G Q\nhQ : ∀ (y : H), Q id univ y\nU : Opens M\nx : ↥U\n⊢ LiftPropAt Q id ↑x","tactic":"rw [← hG.liftPropAt_iff_comp_subtype_val]","premises":[{"full_name":"StructureGroupoid.LocalInvariantProp.liftPropAt_iff_comp_subtype_val","def_path":"Mathlib/Geometry/Manifold/LocalInvariantProperties.lean","def_pos":[486,8],"def_end_pos":[486,39]}]},{"state_before":"H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns t : Set M\nx✝ : M\nQ✝ : (H → H) → Set H → H → Prop\nhG✝ : G.LocalInvariantProp G' P\nQ : (H → H) → Set H → H → Prop\nhG : G.LocalInvariantProp G Q\nhQ : ∀ (y : H), Q id univ y\nU : Opens M\nx : ↥U\n⊢ LiftPropAt Q id ↑x","state_after":"no goals","tactic":"apply hG.liftProp_id hQ","premises":[{"full_name":"StructureGroupoid.LocalInvariantProp.liftProp_id","def_path":"Mathlib/Geometry/Manifold/LocalInvariantProperties.lean","def_pos":[481,8],"def_end_pos":[481,19]}]}]}
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+{"url":"Mathlib/LinearAlgebra/Matrix/IsDiag.lean","commit":"","full_name":"Matrix.IsDiag.kronecker","start":[121,0],"end":[128,17],"file_path":"Mathlib/LinearAlgebra/Matrix/IsDiag.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst✝ : MulZeroClass α\nA : Matrix m m α\nB : Matrix n n α\nhA : A.IsDiag\nhB : B.IsDiag\n⊢ (kroneckerMap (fun x x_1 => x * x_1) A B).IsDiag","state_after":"case mk.mk\nα : Type u_1\nβ : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst✝ : MulZeroClass α\nA : Matrix m m α\nB : Matrix n n α\nhA : A.IsDiag\nhB : B.IsDiag\na : m\nb : n\nc : m\nd : n\nh : (a, b) ≠ (c, d)\n⊢ kroneckerMap (fun x x_1 => x * x_1) A B (a, b) (c, d) = 0","tactic":"rintro ⟨a, b⟩ ⟨c, d⟩ h","premises":[]},{"state_before":"case mk.mk\nα : Type u_1\nβ : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst✝ : MulZeroClass α\nA : Matrix m m α\nB : Matrix n n α\nhA : A.IsDiag\nhB : B.IsDiag\na : m\nb : n\nc : m\nd : n\nh : (a, b) ≠ (c, d)\n⊢ kroneckerMap (fun x x_1 => x * x_1) A B (a, b) (c, d) = 0","state_after":"case mk.mk\nα : Type u_1\nβ : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst✝ : MulZeroClass α\nA : Matrix m m α\nB : Matrix n n α\nhA : A.IsDiag\nhB : B.IsDiag\na : m\nb : n\nc : m\nd : n\nh : ¬a = c ∨ ¬b = d\n⊢ kroneckerMap (fun x x_1 => x * x_1) A B (a, b) (c, d) = 0","tactic":"simp only [Prod.mk.inj_iff, Ne, not_and_or] at h","premises":[{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Prod.mk.inj_iff","def_path":"Mathlib/Data/Prod/Basic.lean","def_pos":[85,8],"def_end_pos":[85,18]},{"full_name":"not_and_or","def_path":"Mathlib/Logic/Basic.lean","def_pos":[339,8],"def_end_pos":[339,18]}]},{"state_before":"case mk.mk\nα : Type u_1\nβ : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst✝ : MulZeroClass α\nA : Matrix m m α\nB : Matrix n n α\nhA : A.IsDiag\nhB : B.IsDiag\na : m\nb : n\nc : m\nd : n\nh : ¬a = c ∨ ¬b = d\n⊢ kroneckerMap (fun x x_1 => x * x_1) A B (a, b) (c, d) = 0","state_after":"case mk.mk.inl\nα : Type u_1\nβ : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst✝ : MulZeroClass α\nA : Matrix m m α\nB : Matrix n n α\nhA : A.IsDiag\nhB : B.IsDiag\na : m\nb : n\nc : m\nd : n\nhac : ¬a = c\n⊢ kroneckerMap (fun x x_1 => x * x_1) A B (a, b) (c, d) = 0\n\ncase mk.mk.inr\nα : Type u_1\nβ : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst✝ : MulZeroClass α\nA : Matrix m m α\nB : Matrix n n α\nhA : A.IsDiag\nhB : B.IsDiag\na : m\nb : n\nc : m\nd : n\nhbd : ¬b = d\n⊢ kroneckerMap (fun x x_1 => x * x_1) A B (a, b) (c, d) = 0","tactic":"cases' h with hac hbd","premises":[]}]}
+{"url":"Mathlib/Data/Nat/Bits.lean","commit":"","full_name":"Nat.testBit_bit_zero","start":[225,0],"end":[229,24],"file_path":"Mathlib/Data/Nat/Bits.lean","tactics":[{"state_before":"m n✝ : ℕ\nb : Bool\nn : ℕ\n⊢ (bit b n).testBit 0 = b","state_after":"m n✝ : ℕ\nb : Bool\nn : ℕ\n⊢ (1 &&& (bif b then fun x => 2 * x + 1 else fun x => 2 * x) n >>> 0 != 0) = b","tactic":"rw [testBit, bit]","premises":[{"full_name":"Nat.bit","def_path":"Mathlib/Data/Nat/Bits.lean","def_pos":[116,4],"def_end_pos":[116,7]},{"full_name":"Nat.testBit","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Bitwise/Basic.lean","def_pos":[81,4],"def_end_pos":[81,11]}]},{"state_before":"m n✝ : ℕ\nb : Bool\nn : ℕ\n⊢ (1 &&& (bif b then fun x => 2 * x + 1 else fun x => 2 * x) n >>> 0 != 0) = b","state_after":"case false\nm n✝ n : ℕ\n⊢ (1 &&& (bif false then fun x => 2 * x + 1 else fun x => 2 * x) n >>> 0 != 0) = false\n\ncase true\nm n✝ n : ℕ\n⊢ (1 &&& (bif true then fun x => 2 * x + 1 else fun x => 2 * x) n >>> 0 != 0) = true","tactic":"cases b","premises":[]}]}
+{"url":"Mathlib/Topology/MetricSpace/Holder.lean","commit":"","full_name":"HolderWith.dist_le_of_le","start":[214,0],"end":[219,29],"file_path":"Mathlib/Topology/MetricSpace/Holder.lean","tactics":[{"state_before":"X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝¹ : PseudoMetricSpace X\ninst✝ : PseudoMetricSpace Y\nC r : ℝ≥0\nf : X → Y\nhf : HolderWith C r f\nx y : X\nd : ℝ\nhd : dist x y ≤ d\n⊢ dist (f x) (f y) ≤ ↑C * d ^ ↑r","state_after":"case intro\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝¹ : PseudoMetricSpace X\ninst✝ : PseudoMetricSpace Y\nC r : ℝ≥0\nf : X → Y\nhf : HolderWith C r f\nx y : X\nd : ℝ≥0\nhd : dist x y ≤ ↑d\n⊢ dist (f x) (f y) ≤ ↑C * ↑d ^ ↑r","tactic":"lift d to ℝ≥0 using dist_nonneg.trans hd","premises":[{"full_name":"NNReal","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[60,4],"def_end_pos":[60,10]},{"full_name":"dist_nonneg","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[234,8],"def_end_pos":[234,19]}]},{"state_before":"case intro\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝¹ : PseudoMetricSpace X\ninst✝ : PseudoMetricSpace Y\nC r : ℝ≥0\nf : X → Y\nhf : HolderWith C r f\nx y : X\nd : ℝ≥0\nhd : dist x y ≤ ↑d\n⊢ dist (f x) (f y) ≤ ↑C * ↑d ^ ↑r","state_after":"case intro\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝¹ : PseudoMetricSpace X\ninst✝ : PseudoMetricSpace Y\nC r : ℝ≥0\nf : X → Y\nhf : HolderWith C r f\nx y : X\nd : ℝ≥0\nhd : ↑(nndist x y) ≤ ↑d\n⊢ ↑(nndist (f x) (f y)) ≤ ↑C * ↑d ^ ↑r","tactic":"rw [dist_nndist] at hd ⊢","premises":[{"full_name":"dist_nndist","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[269,8],"def_end_pos":[269,19]}]},{"state_before":"case intro\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝¹ : PseudoMetricSpace X\ninst✝ : PseudoMetricSpace Y\nC r : ℝ≥0\nf : X → Y\nhf : HolderWith C r f\nx y : X\nd : ℝ≥0\nhd : ↑(nndist x y) ≤ ↑d\n⊢ ↑(nndist (f x) (f y)) ≤ ↑C * ↑d ^ ↑r","state_after":"case intro\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝¹ : PseudoMetricSpace X\ninst✝ : PseudoMetricSpace Y\nC r : ℝ≥0\nf : X → Y\nhf : HolderWith C r f\nx y : X\nd : ℝ≥0\nhd : nndist x y ≤ d\n⊢ nndist (f x) (f y) ≤ C * d ^ ↑r","tactic":"norm_cast at hd ⊢","premises":[]},{"state_before":"case intro\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝¹ : PseudoMetricSpace X\ninst✝ : PseudoMetricSpace Y\nC r : ℝ≥0\nf : X → Y\nhf : HolderWith C r f\nx y : X\nd : ℝ≥0\nhd : nndist x y ≤ d\n⊢ nndist (f x) (f y) ≤ C * d ^ ↑r","state_after":"no goals","tactic":"exact hf.nndist_le_of_le hd","premises":[{"full_name":"HolderWith.nndist_le_of_le","def_path":"Mathlib/Topology/MetricSpace/Holder.lean","def_pos":[203,8],"def_end_pos":[203,23]}]}]}
+{"url":"Mathlib/Algebra/Polynomial/Basic.lean","commit":"","full_name":"Polynomial.C_0","start":[440,0],"end":[440,38],"file_path":"Mathlib/Algebra/Polynomial/Basic.lean","tactics":[{"state_before":"R : Type u\na b : R\nm n : ℕ\ninst✝ : Semiring R\np q : R[X]\n⊢ C 0 = 0","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/Algebra/Module/Submodule/LinearMap.lean","commit":"","full_name":"LinearMap.submodule_pow_eq_zero_of_pow_eq_zero","start":[218,0],"end":[224,16],"file_path":"Mathlib/Algebra/Module/Submodule/LinearMap.lean","tactics":[{"state_before":"R : Type u_1\nR₁ : Type u_2\nR₂ : Type u_3\nR₃ : Type u_4\nM : Type u_5\nM₁ : Type u_6\nM₂ : Type u_7\nM₃ : Type u_8\nι : Type u_9\ninst✝¹¹ : Semiring R\ninst✝¹⁰ : Semiring R₂\ninst✝⁹ : Semiring R₃\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : AddCommMonoid M₂\ninst✝⁵ : AddCommMonoid M₃\ninst✝⁴ : Module R M\ninst✝³ : Module R M₁\ninst✝² : Module R₂ M₂\ninst✝¹ : Module R₃ M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\ninst✝ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\nf : M →ₛₗ[σ₁₂] M₂\ng✝ : M₂ →ₛₗ[σ₂₃] M₃\nN : Submodule R M\ng : Module.End R ↥N\nG : Module.End R M\nh : G ∘ₗ N.subtype = N.subtype ∘ₗ g\nk : ℕ\nhG : G ^ k = 0\n⊢ g ^ k = 0","state_after":"case h.a\nR : Type u_1\nR₁ : Type u_2\nR₂ : Type u_3\nR₃ : Type u_4\nM : Type u_5\nM₁ : Type u_6\nM₂ : Type u_7\nM₃ : Type u_8\nι : Type u_9\ninst✝¹¹ : Semiring R\ninst✝¹⁰ : Semiring R₂\ninst✝⁹ : Semiring R₃\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : AddCommMonoid M₂\ninst✝⁵ : AddCommMonoid M₃\ninst✝⁴ : Module R M\ninst✝³ : Module R M₁\ninst✝² : Module R₂ M₂\ninst✝¹ : Module R₃ M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\ninst✝ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\nf : M →ₛₗ[σ₁₂] M₂\ng✝ : M₂ →ₛₗ[σ₂₃] M₃\nN : Submodule R M\ng : Module.End R ↥N\nG : Module.End R M\nh : G ∘ₗ N.subtype = N.subtype ∘ₗ g\nk : ℕ\nhG : G ^ k = 0\nm : ↥N\n⊢ ↑((g ^ k) m) = ↑(0 m)","tactic":"ext m","premises":[]},{"state_before":"case h.a\nR : Type u_1\nR₁ : Type u_2\nR₂ : Type u_3\nR₃ : Type u_4\nM : Type u_5\nM₁ : Type u_6\nM₂ : Type u_7\nM₃ : Type u_8\nι : Type u_9\ninst✝¹¹ : Semiring R\ninst✝¹⁰ : Semiring R₂\ninst✝⁹ : Semiring R₃\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : AddCommMonoid M₂\ninst✝⁵ : AddCommMonoid M₃\ninst✝⁴ : Module R M\ninst✝³ : Module R M₁\ninst✝² : Module R₂ M₂\ninst✝¹ : Module R₃ M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\ninst✝ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\nf : M →ₛₗ[σ₁₂] M₂\ng✝ : M₂ →ₛₗ[σ₂₃] M₃\nN : Submodule R M\ng : Module.End R ↥N\nG : Module.End R M\nh : G ∘ₗ N.subtype = N.subtype ∘ₗ g\nk : ℕ\nhG : G ^ k = 0\nm : ↥N\n⊢ ↑((g ^ k) m) = ↑(0 m)","state_after":"case h.a\nR : Type u_1\nR₁ : Type u_2\nR₂ : Type u_3\nR₃ : Type u_4\nM : Type u_5\nM₁ : Type u_6\nM₂ : Type u_7\nM₃ : Type u_8\nι : Type u_9\ninst✝¹¹ : Semiring R\ninst✝¹⁰ : Semiring R₂\ninst✝⁹ : Semiring R₃\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : AddCommMonoid M₂\ninst✝⁵ : AddCommMonoid M₃\ninst✝⁴ : Module R M\ninst✝³ : Module R M₁\ninst✝² : Module R₂ M₂\ninst✝¹ : Module R₃ M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\ninst✝ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\nf : M →ₛₗ[σ₁₂] M₂\ng✝ : M₂ →ₛₗ[σ₂₃] M₃\nN : Submodule R M\ng : Module.End R ↥N\nG : Module.End R M\nh : G ∘ₗ N.subtype = N.subtype ∘ₗ g\nk : ℕ\nhG : G ^ k = 0\nm : ↥N\nhg : (N.subtype ∘ₗ g ^ k) m = 0\n⊢ ↑((g ^ k) m) = ↑(0 m)","tactic":"have hg : N.subtype.comp (g ^ k) m = 0 := by\n    rw [← commute_pow_left_of_commute h, hG, zero_comp, zero_apply]","premises":[{"full_name":"LinearMap.commute_pow_left_of_commute","def_path":"Mathlib/Algebra/Module/LinearMap/End.lean","def_pos":[137,8],"def_end_pos":[137,35]},{"full_name":"LinearMap.comp","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[489,4],"def_end_pos":[489,8]},{"full_name":"LinearMap.zero_apply","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[786,8],"def_end_pos":[786,18]},{"full_name":"LinearMap.zero_comp","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[794,8],"def_end_pos":[794,17]},{"full_name":"Submodule.subtype","def_path":"Mathlib/Algebra/Module/Submodule/LinearMap.lean","def_pos":[69,14],"def_end_pos":[69,21]}]},{"state_before":"case h.a\nR : Type u_1\nR₁ : Type u_2\nR₂ : Type u_3\nR₃ : Type u_4\nM : Type u_5\nM₁ : Type u_6\nM₂ : Type u_7\nM₃ : Type u_8\nι : Type u_9\ninst✝¹¹ : Semiring R\ninst✝¹⁰ : Semiring R₂\ninst✝⁹ : Semiring R₃\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : AddCommMonoid M₂\ninst✝⁵ : AddCommMonoid M₃\ninst✝⁴ : Module R M\ninst✝³ : Module R M₁\ninst✝² : Module R₂ M₂\ninst✝¹ : Module R₃ M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\ninst✝ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\nf : M →ₛₗ[σ₁₂] M₂\ng✝ : M₂ →ₛₗ[σ₂₃] M₃\nN : Submodule R M\ng : Module.End R ↥N\nG : Module.End R M\nh : G ∘ₗ N.subtype = N.subtype ∘ₗ g\nk : ℕ\nhG : G ^ k = 0\nm : ↥N\nhg : (N.subtype ∘ₗ g ^ k) m = 0\n⊢ ↑((g ^ k) m) = ↑(0 m)","state_after":"no goals","tactic":"simpa using hg","premises":[]}]}
+{"url":".lake/packages/batteries/Batteries/Data/Rat/Basic.lean","commit":"","full_name":"Rat.add.aux","start":[184,0],"end":[210,69],"file_path":".lake/packages/batteries/Batteries/Data/Rat/Basic.lean","tactics":[{"state_before":"a b : Rat\ng ad bd : Nat\nhg : g = a.den.gcd b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\n⊢ let den := ad * b.den;\n  let num := a.num * ↑bd + b.num * ↑ad;\n  num.natAbs.gcd g = num.natAbs.gcd den","state_after":"a b : Rat\ng ad bd : Nat\nhg : g = a.den.gcd b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * ↑bd + b.num * ↑ad\n⊢ num.natAbs.gcd g = num.natAbs.gcd den","tactic":"intro den num","premises":[]},{"state_before":"a b : Rat\ng ad bd : Nat\nhg : g = a.den.gcd b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * ↑bd + b.num * ↑ad\n⊢ num.natAbs.gcd g = num.natAbs.gcd den","state_after":"a b : Rat\ng ad bd : Nat\nhg : g = a.den.gcd b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * ↑bd + b.num * ↑ad\nae : ad * g = a.den\n⊢ num.natAbs.gcd g = num.natAbs.gcd den","tactic":"have ae : ad * g = a.den := had ▸ Nat.div_mul_cancel (hg ▸ Nat.gcd_dvd_left ..)","premises":[{"full_name":"Nat.div_mul_cancel","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Dvd.lean","def_pos":[91,18],"def_end_pos":[91,32]},{"full_name":"Nat.gcd_dvd_left","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean","def_pos":[86,8],"def_end_pos":[86,20]},{"full_name":"Rat.den","def_path":".lake/packages/batteries/Batteries/Data/Rat/Basic.lean","def_pos":[24,2],"def_end_pos":[24,5]}]},{"state_before":"a b : Rat\ng ad bd : Nat\nhg : g = a.den.gcd b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * ↑bd + b.num * ↑ad\nae : ad * g = a.den\n⊢ num.natAbs.gcd g = num.natAbs.gcd den","state_after":"a b : Rat\ng ad bd : Nat\nhg : g = a.den.gcd b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * ↑bd + b.num * ↑ad\nae : ad * g = a.den\nbe : bd * g = b.den\n⊢ num.natAbs.gcd g = num.natAbs.gcd den","tactic":"have be : bd * g = b.den := hbd ▸ Nat.div_mul_cancel (hg ▸ Nat.gcd_dvd_right ..)","premises":[{"full_name":"Nat.div_mul_cancel","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Dvd.lean","def_pos":[91,18],"def_end_pos":[91,32]},{"full_name":"Nat.gcd_dvd_right","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean","def_pos":[88,8],"def_end_pos":[88,21]},{"full_name":"Rat.den","def_path":".lake/packages/batteries/Batteries/Data/Rat/Basic.lean","def_pos":[24,2],"def_end_pos":[24,5]}]},{"state_before":"a b : Rat\ng ad bd : Nat\nhg : g = a.den.gcd b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * ↑bd + b.num * ↑ad\nae : ad * g = a.den\nbe : bd * g = b.den\n⊢ num.natAbs.gcd g = num.natAbs.gcd den","state_after":"a b : Rat\ng ad bd : Nat\nhg : g = a.den.gcd b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * ↑bd + b.num * ↑ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\n⊢ num.natAbs.gcd g = num.natAbs.gcd den","tactic":"have hden : den = ad * bd * g := by rw [Nat.mul_assoc, be]","premises":[{"full_name":"Nat.mul_assoc","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[245,18],"def_end_pos":[245,27]}]},{"state_before":"a b : Rat\ng ad bd : Nat\nhg : g = a.den.gcd b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * ↑bd + b.num * ↑ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\n⊢ num.natAbs.gcd g = num.natAbs.gcd den","state_after":"case H\na b : Rat\ng ad bd : Nat\nhg : g = a.den.gcd b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * ↑bd + b.num * ↑ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\n⊢ (ad * bd).Coprime num.natAbs","tactic":"rw [hden, Nat.Coprime.gcd_mul_left_cancel_right]","premises":[{"full_name":"Nat.Coprime.gcd_mul_left_cancel_right","def_path":".lake/packages/batteries/Batteries/Data/Nat/Gcd.lean","def_pos":[49,8],"def_end_pos":[49,41]}]},{"state_before":"case H\na b : Rat\ng ad bd : Nat\nhg : g = a.den.gcd b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * ↑bd + b.num * ↑ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\n⊢ (ad * bd).Coprime num.natAbs","state_after":"case H\na b : Rat\ng ad bd : Nat\nhg : g = a.den.gcd b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * ↑bd + b.num * ↑ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\ncop : ad.Coprime bd\n⊢ (ad * bd).Coprime num.natAbs","tactic":"have cop : ad.Coprime bd := had ▸ hbd ▸ hg ▸\n    Nat.coprime_div_gcd_div_gcd (Nat.gcd_pos_of_pos_left _ a.den_pos)","premises":[{"full_name":"Nat.Coprime","def_path":".lake/packages/batteries/Batteries/Data/Nat/Gcd.lean","def_pos":[20,17],"def_end_pos":[20,24]},{"full_name":"Nat.coprime_div_gcd_div_gcd","def_path":".lake/packages/batteries/Batteries/Data/Nat/Gcd.lean","def_pos":[57,8],"def_end_pos":[57,31]},{"full_name":"Nat.gcd_pos_of_pos_left","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean","def_pos":[137,8],"def_end_pos":[137,27]},{"full_name":"Rat.den_pos","def_path":".lake/packages/batteries/Batteries/Data/Rat/Basic.lean","def_pos":[39,8],"def_end_pos":[39,19]}]},{"state_before":"case H\na b : Rat\ng ad bd : Nat\nhg : g = a.den.gcd b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * ↑bd + b.num * ↑ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\ncop : ad.Coprime bd\n⊢ (ad * bd).Coprime num.natAbs","state_after":"case H\na b : Rat\ng ad bd : Nat\nhg : g = a.den.gcd b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * ↑bd + b.num * ↑ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\ncop : ad.Coprime bd\nH1 : ∀ (d : Nat), d.gcd num.natAbs ∣ a.num.natAbs * bd ↔ d.gcd num.natAbs ∣ b.num.natAbs * ad\n⊢ (ad * bd).Coprime num.natAbs","tactic":"have H1 (d : Nat) :\n      d.gcd num.natAbs ∣ a.num.natAbs * bd ↔ d.gcd num.natAbs ∣ b.num.natAbs * ad := by\n    have := d.gcd_dvd_right num.natAbs\n    rw [← Int.ofNat_dvd, Int.dvd_natAbs] at this\n    have := Int.dvd_iff_dvd_of_dvd_add this\n    rwa [← Int.dvd_natAbs, Int.ofNat_dvd, Int.natAbs_mul,\n      ← Int.dvd_natAbs, Int.ofNat_dvd, Int.natAbs_mul] at this","premises":[{"full_name":"Dvd.dvd","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1344,2],"def_end_pos":[1344,5]},{"full_name":"Iff","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[114,10],"def_end_pos":[114,13]},{"full_name":"Int.dvd_iff_dvd_of_dvd_add","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean","def_pos":[94,18],"def_end_pos":[94,40]},{"full_name":"Int.dvd_natAbs","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean","def_pos":[107,8],"def_end_pos":[107,18]},{"full_name":"Int.natAbs","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean","def_pos":[262,4],"def_end_pos":[262,10]},{"full_name":"Int.natAbs_mul","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Order.lean","def_pos":[438,8],"def_end_pos":[438,18]},{"full_name":"Int.ofNat_dvd","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean","def_pos":[37,21],"def_end_pos":[37,30]},{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]},{"full_name":"Nat.gcd","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean","def_pos":[32,4],"def_end_pos":[32,7]},{"full_name":"Nat.gcd_dvd_right","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean","def_pos":[88,8],"def_end_pos":[88,21]},{"full_name":"Rat.num","def_path":".lake/packages/batteries/Batteries/Data/Rat/Basic.lean","def_pos":[22,2],"def_end_pos":[22,5]}]},{"state_before":"case H\na b : Rat\ng ad bd : Nat\nhg : g = a.den.gcd b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * ↑bd + b.num * ↑ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\ncop : ad.Coprime bd\nH1 : ∀ (d : Nat), d.gcd num.natAbs ∣ a.num.natAbs * bd ↔ d.gcd num.natAbs ∣ b.num.natAbs * ad\n⊢ (ad * bd).Coprime num.natAbs","state_after":"case H.H1\na b : Rat\ng ad bd : Nat\nhg : g = a.den.gcd b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * ↑bd + b.num * ↑ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\ncop : ad.Coprime bd\nH1 : ∀ (d : Nat), d.gcd num.natAbs ∣ a.num.natAbs * bd ↔ d.gcd num.natAbs ∣ b.num.natAbs * ad\n⊢ ad.Coprime num.natAbs\n\ncase H.H2\na b : Rat\ng ad bd : Nat\nhg : g = a.den.gcd b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * ↑bd + b.num * ↑ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\ncop : ad.Coprime bd\nH1 : ∀ (d : Nat), d.gcd num.natAbs ∣ a.num.natAbs * bd ↔ d.gcd num.natAbs ∣ b.num.natAbs * ad\n⊢ bd.Coprime num.natAbs","tactic":"apply Nat.Coprime.mul","premises":[{"full_name":"Nat.Coprime.mul","def_path":".lake/packages/batteries/Batteries/Data/Nat/Gcd.lean","def_pos":[81,8],"def_end_pos":[81,19]}]}]}
+{"url":"Mathlib/AlgebraicGeometry/Spec.lean","commit":"","full_name":"AlgebraicGeometry.stalkMap_toStalk","start":[195,0],"end":[202,5],"file_path":"Mathlib/AlgebraicGeometry/Spec.lean","tactics":[{"state_before":"R S : CommRingCat\nf : R ⟶ S\np : PrimeSpectrum ↑S\n⊢ toStalk (↑R) ((PrimeSpectrum.comap f) p) ≫ PresheafedSpace.Hom.stalkMap (Spec.sheafedSpaceMap f) p =\n    f ≫ toStalk (↑S) p","state_after":"R S : CommRingCat\nf : R ⟶ S\np : PrimeSpectrum ↑S\n⊢ CommRingCat.ofHom f ≫\n      toOpen (↑S) ((TopologicalSpace.Opens.comap (PrimeSpectrum.comap f)) ⊤) ≫\n        (Spec.sheafedSpaceObj S).presheaf.germ ⟨p, trivial⟩ =\n    f ≫ toOpen ↑S ⊤ ≫ (structureSheaf ↑S).presheaf.germ ⟨p, trivial⟩","tactic":"erw [← toOpen_germ S ⊤ ⟨p, trivial⟩, ← toOpen_germ R ⊤ ⟨PrimeSpectrum.comap f p, trivial⟩,\n    Category.assoc, PresheafedSpace.stalkMap_germ (Spec.sheafedSpaceMap f) ⊤ ⟨p, trivial⟩,\n    Spec.sheafedSpaceMap_c_app, toOpen_comp_comap_assoc]","premises":[{"full_name":"AlgebraicGeometry.PresheafedSpace.stalkMap_germ","def_path":"Mathlib/Geometry/RingedSpace/Stalks.lean","def_pos":[45,8],"def_end_pos":[45,21]},{"full_name":"AlgebraicGeometry.Spec.sheafedSpaceMap","def_path":"Mathlib/AlgebraicGeometry/Spec.lean","def_pos":[94,4],"def_end_pos":[94,24]},{"full_name":"AlgebraicGeometry.Spec.sheafedSpaceMap_c_app","def_path":"Mathlib/AlgebraicGeometry/Spec.lean","def_pos":[93,2],"def_end_pos":[93,7]},{"full_name":"AlgebraicGeometry.StructureSheaf.toOpen_germ","def_path":"Mathlib/AlgebraicGeometry/StructureSheaf.lean","def_pos":[408,8],"def_end_pos":[408,19]},{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"PrimeSpectrum.comap","def_path":"Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean","def_pos":[221,4],"def_end_pos":[221,9]},{"full_name":"Top.top","def_path":"Mathlib/Order/Notation.lean","def_pos":[94,2],"def_end_pos":[94,5]},{"full_name":"trivial","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[645,34],"def_end_pos":[645,41]}]},{"state_before":"R S : CommRingCat\nf : R ⟶ S\np : PrimeSpectrum ↑S\n⊢ CommRingCat.ofHom f ≫\n      toOpen (↑S) ((TopologicalSpace.Opens.comap (PrimeSpectrum.comap f)) ⊤) ≫\n        (Spec.sheafedSpaceObj S).presheaf.germ ⟨p, trivial⟩ =\n    f ≫ toOpen ↑S ⊤ ≫ (structureSheaf ↑S).presheaf.germ ⟨p, trivial⟩","state_after":"no goals","tactic":"rfl","premises":[]}]}
+{"url":"Mathlib/Algebra/Order/Group/PosPart.lean","commit":"","full_name":"negPart_zero","start":[75,0],"end":[75,88],"file_path":"Mathlib/Algebra/Order/Group/PosPart.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝¹ : Lattice α\ninst✝ : Group α\na : α\n⊢ 1⁻ᵐ = 1","state_after":"no goals","tactic":"simp [leOnePart]","premises":[{"full_name":"leOnePart","def_path":"Mathlib/Algebra/Order/Group/PosPart.lean","def_pos":[63,4],"def_end_pos":[63,13]}]}]}
+{"url":"Mathlib/GroupTheory/Index.lean","commit":"","full_name":"Subgroup.relindex_mul_relindex","start":[104,0],"end":[108,45],"file_path":"Mathlib/GroupTheory/Index.lean","tactics":[{"state_before":"G : Type u_1\nG' : Type u_2\ninst✝¹ : Group G\ninst✝ : Group G'\nH K L : Subgroup G\nhHK : H ≤ K\nhKL : K ≤ L\n⊢ H.relindex K * K.relindex L = H.relindex L","state_after":"G : Type u_1\nG' : Type u_2\ninst✝¹ : Group G\ninst✝ : Group G'\nH K L : Subgroup G\nhHK : H ≤ K\nhKL : K ≤ L\n⊢ (H.subgroupOf L).relindex (K.subgroupOf L) * K.relindex L = H.relindex L","tactic":"rw [← relindex_subgroupOf hKL]","premises":[{"full_name":"Subgroup.relindex_subgroupOf","def_path":"Mathlib/GroupTheory/Index.lean","def_pos":[98,8],"def_end_pos":[98,27]}]},{"state_before":"G : Type u_1\nG' : Type u_2\ninst✝¹ : Group G\ninst✝ : Group G'\nH K L : Subgroup G\nhHK : H ≤ K\nhKL : K ≤ L\n⊢ (H.subgroupOf L).relindex (K.subgroupOf L) * K.relindex L = H.relindex L","state_after":"no goals","tactic":"exact relindex_mul_index fun x hx => hHK hx","premises":[{"full_name":"Subgroup.relindex_mul_index","def_path":"Mathlib/GroupTheory/Index.lean","def_pos":[85,8],"def_end_pos":[85,26]}]}]}
+{"url":"Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean","commit":"","full_name":"InnerProductGeometry.tan_angle_add_of_inner_eq_zero","start":[139,0],"end":[143,65],"file_path":"Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean","tactics":[{"state_before":"V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : ⟪x, y⟫_ℝ = 0\n⊢ Real.tan (angle x (x + y)) = ‖y‖ / ‖x‖","state_after":"case pos\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : ⟪x, y⟫_ℝ = 0\nh0 : x = 0\n⊢ Real.tan (angle x (x + y)) = ‖y‖ / ‖x‖\n\ncase neg\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : ⟪x, y⟫_ℝ = 0\nh0 : ¬x = 0\n⊢ Real.tan (angle x (x + y)) = ‖y‖ / ‖x‖","tactic":"by_cases h0 : x = 0","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case neg\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : ⟪x, y⟫_ℝ = 0\nh0 : ¬x = 0\n⊢ Real.tan (angle x (x + y)) = ‖y‖ / ‖x‖","state_after":"no goals","tactic":"rw [angle_add_eq_arctan_of_inner_eq_zero h h0, Real.tan_arctan]","premises":[{"full_name":"InnerProductGeometry.angle_add_eq_arctan_of_inner_eq_zero","def_path":"Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean","def_pos":[87,8],"def_end_pos":[87,44]},{"full_name":"Real.tan_arctan","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean","def_pos":[103,8],"def_end_pos":[103,18]}]}]}
+{"url":"Mathlib/Algebra/Order/Group/Unbundled/Basic.lean","commit":"","full_name":"Left.neg_neg_iff","start":[87,0],"end":[90,53],"file_path":"Mathlib/Algebra/Order/Group/Unbundled/Basic.lean","tactics":[{"state_before":"α : Type u\ninst✝² : Group α\ninst✝¹ : LT α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b c : α\n⊢ a⁻¹ < 1 ↔ 1 < a","state_after":"no goals","tactic":"rw [← mul_lt_mul_iff_left a, mul_inv_self, mul_one]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"mul_inv_self","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[1055,8],"def_end_pos":[1055,20]},{"full_name":"mul_lt_mul_iff_left","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[105,8],"def_end_pos":[105,27]},{"full_name":"mul_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[413,8],"def_end_pos":[413,15]}]}]}
+{"url":"Mathlib/GroupTheory/PGroup.lean","commit":"","full_name":"IsPGroup.of_bot","start":[39,0],"end":[40,47],"file_path":"Mathlib/GroupTheory/PGroup.lean","tactics":[{"state_before":"p : ℕ\nG : Type u_1\ninst✝ : Group G\n⊢ Nat.card ↥⊥ = p ^ ?m.2519","state_after":"no goals","tactic":"rw [Subgroup.card_bot, pow_zero]","premises":[{"full_name":"Subgroup.card_bot","def_path":"Mathlib/Algebra/Group/Subgroup/Finite.lean","def_pos":[97,8],"def_end_pos":[97,16]},{"full_name":"pow_zero","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[563,8],"def_end_pos":[563,16]}]}]}
+{"url":"Mathlib/CategoryTheory/Limits/Preserves/Limits.lean","commit":"","full_name":"CategoryTheory.preserves_lift_mapCone","start":[39,0],"end":[42,84],"file_path":"Mathlib/CategoryTheory/Limits/Preserves/Limits.lean","tactics":[{"state_before":"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nG : C ⥤ D\nJ : Type w\ninst✝¹ : Category.{w', w} J\nF : J ⥤ C\ninst✝ : PreservesLimit F G\nc₁ c₂ : Cone F\nt : IsLimit c₁\n⊢ ∀ (j : J), G.map (t.lift c₂) ≫ (G.mapCone c₁).π.app j = (G.mapCone c₂).π.app j","state_after":"no goals","tactic":"simp [← G.map_comp]","premises":[{"full_name":"CategoryTheory.Functor.map_comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[41,2],"def_end_pos":[41,10]}]}]}
+{"url":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","commit":"","full_name":"List.get?_set_of_lt'","start":[449,0],"end":[451,79],"file_path":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","tactics":[{"state_before":"α : Type u_1\na : α\nm n : Nat\nl : List α\nh : m < l.length\n⊢ (l.set m a).get? n = if m = n then some a else l.get? n","state_after":"α : Type u_1\na : α\nm n : Nat\nl : List α\nh : m < l.length\n⊢ (if m = n then if m < l.length then some a else none else l[n]?) = if m = n then some a else l[n]?","tactic":"simp [getElem?_set]","premises":[{"full_name":"List.getElem?_set","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean","def_pos":[469,8],"def_end_pos":[469,20]}]},{"state_before":"α : Type u_1\na : α\nm n : Nat\nl : List α\nh : m < l.length\n⊢ (if m = n then if m < l.length then some a else none else l[n]?) = if m = n then some a else l[n]?","state_after":"no goals","tactic":"split <;> subst_vars <;> simp [*, getElem?_eq_getElem h]","premises":[{"full_name":"List.getElem?_eq_getElem","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean","def_pos":[178,16],"def_end_pos":[178,35]}]}]}
+{"url":"Mathlib/FieldTheory/KrullTopology.lean","commit":"","full_name":"IntermediateField.fixingSubgroup_bot","start":[266,0],"end":[269,40],"file_path":"Mathlib/FieldTheory/KrullTopology.lean","tactics":[{"state_before":"K : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\n⊢ ⊥.fixingSubgroup = ⊤","state_after":"case h\nK : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx✝ : L ≃ₐ[K] L\n⊢ x✝ ∈ ⊥.fixingSubgroup ↔ x✝ ∈ ⊤","tactic":"ext","premises":[]},{"state_before":"case h\nK : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx✝ : L ≃ₐ[K] L\n⊢ x✝ ∈ ⊥.fixingSubgroup ↔ x✝ ∈ ⊤","state_after":"no goals","tactic":"simp [mem_fixingSubgroup_iff, mem_bot]","premises":[{"full_name":"IntermediateField.mem_bot","def_path":"Mathlib/FieldTheory/Adjoin.lean","def_pos":[103,8],"def_end_pos":[103,15]},{"full_name":"IntermediateField.mem_fixingSubgroup_iff","def_path":"Mathlib/FieldTheory/KrullTopology.lean","def_pos":[108,8],"def_end_pos":[108,48]}]}]}
+{"url":"Mathlib/Analysis/CstarAlgebra/Exponential.lean","commit":"","full_name":"Commute.expUnitary","start":[47,0],"end":[52,64],"file_path":"Mathlib/Analysis/CstarAlgebra/Exponential.lean","tactics":[{"state_before":"A : Type u_1\ninst✝⁵ : NormedRing A\ninst✝⁴ : NormedAlgebra ℂ A\ninst✝³ : StarRing A\ninst✝² : ContinuousStar A\ninst✝¹ : CompleteSpace A\ninst✝ : StarModule ℂ A\na b : ↥(selfAdjoint A)\nh : Commute ↑a ↑b\n⊢ selfAdjoint.expUnitary a * selfAdjoint.expUnitary b = selfAdjoint.expUnitary b * selfAdjoint.expUnitary a","state_after":"no goals","tactic":"rw [← h.expUnitary_add, ← h.symm.expUnitary_add, add_comm]","premises":[{"full_name":"Commute.expUnitary_add","def_path":"Mathlib/Analysis/CstarAlgebra/Exponential.lean","def_pos":[39,8],"def_end_pos":[39,30]},{"full_name":"Commute.symm","def_path":"Mathlib/Algebra/Group/Commute/Defs.lean","def_pos":[62,18],"def_end_pos":[62,22]},{"full_name":"add_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[293,2],"def_end_pos":[293,13]}]}]}
+{"url":"Mathlib/Data/Prod/TProd.lean","commit":"","full_name":"Set.elim_preimage_pi","start":[156,0],"end":[163,7],"file_path":"Mathlib/Data/Prod/TProd.lean","tactics":[{"state_before":"ι : Type u\nα : ι → Type v\ni j : ι\nl✝ : List ι\nf : (i : ι) → α i\ninst✝ : DecidableEq ι\nl : List ι\nhnd : l.Nodup\nh : ∀ (i : ι), i ∈ l\nt : (i : ι) → Set (α i)\n⊢ TProd.elim' h ⁻¹' univ.pi t = Set.tprod l t","state_after":"ι : Type u\nα : ι → Type v\ni j : ι\nl✝ : List ι\nf : (i : ι) → α i\ninst✝ : DecidableEq ι\nl : List ι\nhnd : l.Nodup\nh : ∀ (i : ι), i ∈ l\nt : (i : ι) → Set (α i)\nh2 : {i | i ∈ l} = univ\n⊢ TProd.elim' h ⁻¹' univ.pi t = Set.tprod l t","tactic":"have h2 : { i | i ∈ l } = univ := by\n    ext i\n    simp [h]","premises":[{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Set.univ","def_path":"Mathlib/Init/Set.lean","def_pos":[157,4],"def_end_pos":[157,8]},{"full_name":"setOf","def_path":"Mathlib/Init/Set.lean","def_pos":[56,4],"def_end_pos":[56,9]}]},{"state_before":"ι : Type u\nα : ι → Type v\ni j : ι\nl✝ : List ι\nf : (i : ι) → α i\ninst✝ : DecidableEq ι\nl : List ι\nhnd : l.Nodup\nh : ∀ (i : ι), i ∈ l\nt : (i : ι) → Set (α i)\nh2 : {i | i ∈ l} = univ\n⊢ TProd.elim' h ⁻¹' univ.pi t = Set.tprod l t","state_after":"ι : Type u\nα : ι → Type v\ni j : ι\nl✝ : List ι\nf : (i : ι) → α i\ninst✝ : DecidableEq ι\nl : List ι\nhnd : l.Nodup\nh : ∀ (i : ι), i ∈ l\nt : (i : ι) → Set (α i)\nh2 : {i | i ∈ l} = univ\n⊢ (fun x => TProd.mk l (TProd.elim' h x)) ⁻¹' Set.tprod l t = Set.tprod l t","tactic":"rw [← h2, ← mk_preimage_tprod, preimage_preimage]","premises":[{"full_name":"Set.mk_preimage_tprod","def_path":"Mathlib/Data/Prod/TProd.lean","def_pos":[141,8],"def_end_pos":[141,25]},{"full_name":"Set.preimage_preimage","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[140,8],"def_end_pos":[140,25]}]},{"state_before":"ι : Type u\nα : ι → Type v\ni j : ι\nl✝ : List ι\nf : (i : ι) → α i\ninst✝ : DecidableEq ι\nl : List ι\nhnd : l.Nodup\nh : ∀ (i : ι), i ∈ l\nt : (i : ι) → Set (α i)\nh2 : {i | i ∈ l} = univ\n⊢ (fun x => TProd.mk l (TProd.elim' h x)) ⁻¹' Set.tprod l t = Set.tprod l t","state_after":"ι : Type u\nα : ι → Type v\ni j : ι\nl✝ : List ι\nf : (i : ι) → α i\ninst✝ : DecidableEq ι\nl : List ι\nhnd : l.Nodup\nh : ∀ (i : ι), i ∈ l\nt : (i : ι) → Set (α i)\nh2 : {i | i ∈ l} = univ\n⊢ (fun x => x) ⁻¹' Set.tprod l t = Set.tprod l t","tactic":"simp only [TProd.mk_elim hnd h]","premises":[{"full_name":"List.TProd.mk_elim","def_path":"Mathlib/Data/Prod/TProd.lean","def_pos":[120,8],"def_end_pos":[120,15]}]},{"state_before":"ι : Type u\nα : ι → Type v\ni j : ι\nl✝ : List ι\nf : (i : ι) → α i\ninst✝ : DecidableEq ι\nl : List ι\nhnd : l.Nodup\nh : ∀ (i : ι), i ∈ l\nt : (i : ι) → Set (α i)\nh2 : {i | i ∈ l} = univ\n⊢ (fun x => x) ⁻¹' Set.tprod l t = Set.tprod l t","state_after":"no goals","tactic":"dsimp","premises":[]}]}
+{"url":"Mathlib/NumberTheory/BernoulliPolynomials.lean","commit":"","full_name":"Polynomial.bernoulli_eval_one_add","start":[175,0],"end":[198,23],"file_path":"Mathlib/NumberTheory/BernoulliPolynomials.lean","tactics":[{"state_before":"n : ℕ\nx : ℚ\n⊢ eval (1 + x) (bernoulli n) = eval x (bernoulli n) + ↑n * x ^ (n - 1)","state_after":"n : ℕ\nx : ℚ\nd : ℕ\nhd : ∀ m < d, eval (1 + x) (bernoulli m) = eval x (bernoulli m) + ↑m * x ^ (m - 1)\n⊢ eval (1 + x) (bernoulli d) = eval x (bernoulli d) + ↑d * x ^ (d - 1)","tactic":"refine Nat.strong_induction_on n fun d hd => ?_","premises":[{"full_name":"Nat.strong_induction_on","def_path":"Mathlib/Init/Data/Nat/Lemmas.lean","def_pos":[83,18],"def_end_pos":[83,37]}]},{"state_before":"n : ℕ\nx : ℚ\nd : ℕ\nhd : ∀ m < d, eval (1 + x) (bernoulli m) = eval x (bernoulli m) + ↑m * x ^ (m - 1)\n⊢ eval (1 + x) (bernoulli d) = eval x (bernoulli d) + ↑d * x ^ (d - 1)","state_after":"n : ℕ\nx : ℚ\nd : ℕ\nhd : ∀ m < d, eval (1 + x) (bernoulli m) = eval x (bernoulli m) + ↑m * x ^ (m - 1)\nnz : ↑d.succ ≠ 0\n⊢ eval (1 + x) (bernoulli d) = eval x (bernoulli d) + ↑d * x ^ (d - 1)","tactic":"have nz : ((d.succ : ℕ) : ℚ) ≠ 0 := by\n    norm_cast","premises":[{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]},{"full_name":"Nat.succ","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1083,4],"def_end_pos":[1083,8]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Rat","def_path":".lake/packages/batteries/Batteries/Data/Rat/Basic.lean","def_pos":[17,10],"def_end_pos":[17,13]}]},{"state_before":"n : ℕ\nx : ℚ\nd : ℕ\nhd : ∀ m < d, eval (1 + x) (bernoulli m) = eval x (bernoulli m) + ↑m * x ^ (m - 1)\nnz : ↑d.succ ≠ 0\n⊢ eval (1 + x) (bernoulli d) = eval x (bernoulli d) + ↑d * x ^ (d - 1)","state_after":"n : ℕ\nx : ℚ\nd : ℕ\nhd : ∀ m < d, eval (1 + x) (bernoulli m) = eval x (bernoulli m) + ↑m * x ^ (m - 1)\nnz : ↑d.succ ≠ 0\n⊢ ↑d.succ * eval (1 + x) (bernoulli d) = ↑d.succ * (eval x (bernoulli d) + ↑d * x ^ (d - 1))","tactic":"apply (mul_right_inj' nz).1","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"mul_right_inj'","def_path":"Mathlib/Algebra/GroupWithZero/Defs.lean","def_pos":[113,8],"def_end_pos":[113,22]}]},{"state_before":"n : ℕ\nx : ℚ\nd : ℕ\nhd : ∀ m < d, eval (1 + x) (bernoulli m) = eval x (bernoulli m) + ↑m * x ^ (m - 1)\nnz : ↑d.succ ≠ 0\n⊢ ↑d.succ * eval (1 + x) (bernoulli d) = ↑d.succ * (eval x (bernoulli d) + ↑d * x ^ (d - 1))","state_after":"n : ℕ\nx : ℚ\nd : ℕ\nhd : ∀ m < d, eval (1 + x) (bernoulli m) = eval x (bernoulli m) + ↑m * x ^ (m - 1)\nnz : ↑d.succ ≠ 0\n⊢ eval (1 + x) ((monomial d) ↑d.succ) - ∑ i ∈ range d, eval (1 + x) (↑((d + 1).choose i) • bernoulli i) =\n    eval x ((monomial d) ↑d.succ - ∑ k ∈ range d, ↑((d + 1).choose k) • bernoulli k) + ↑d.succ * (↑d * x ^ (d - 1))","tactic":"rw [← smul_eq_mul, ← eval_smul, bernoulli_eq_sub_sum, mul_add, ← smul_eq_mul, ← eval_smul,\n    bernoulli_eq_sub_sum, eval_sub, eval_finset_sum]","premises":[{"full_name":"Polynomial.bernoulli_eq_sub_sum","def_path":"Mathlib/NumberTheory/BernoulliPolynomials.lean","def_pos":[146,8],"def_end_pos":[146,28]},{"full_name":"Polynomial.eval_finset_sum","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[412,8],"def_end_pos":[412,23]},{"full_name":"Polynomial.eval_smul","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[344,8],"def_end_pos":[344,17]},{"full_name":"Polynomial.eval_sub","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[1119,8],"def_end_pos":[1119,16]},{"full_name":"smul_eq_mul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[79,6],"def_end_pos":[79,17]}]},{"state_before":"n : ℕ\nx : ℚ\nd : ℕ\nhd : ∀ m < d, eval (1 + x) (bernoulli m) = eval x (bernoulli m) + ↑m * x ^ (m - 1)\nnz : ↑d.succ ≠ 0\n⊢ eval (1 + x) ((monomial d) ↑d.succ) -\n      ∑ x_1 ∈ range d, ↑((d + 1).choose x_1) • (eval x (bernoulli x_1) + ↑x_1 * x ^ (x_1 - 1)) =\n    eval x ((monomial d) ↑d.succ - ∑ k ∈ range d, ↑((d + 1).choose k) • bernoulli k) + ↑d.succ * (↑d * x ^ (d - 1))","state_after":"n : ℕ\nx : ℚ\nd : ℕ\nhd : ∀ m < d, eval (1 + x) (bernoulli m) = eval x (bernoulli m) + ↑m * x ^ (m - 1)\nnz : ↑d.succ ≠ 0\n⊢ eval (1 + x) ((monomial d) ↑d.succ) -\n      ∑ x_1 ∈ range d, ↑((d + 1).choose x_1) • (eval x (bernoulli x_1) + ↑x_1 * x ^ (x_1 - 1)) =\n    eval x ((monomial d) ↑d.succ) - ∑ i ∈ range d, eval x (↑((d + 1).choose i) • bernoulli i) +\n      ↑d.succ * (↑d * x ^ (d - 1))","tactic":"rw [eval_sub, eval_finset_sum]","premises":[{"full_name":"Polynomial.eval_finset_sum","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[412,8],"def_end_pos":[412,23]},{"full_name":"Polynomial.eval_sub","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[1119,8],"def_end_pos":[1119,16]}]},{"state_before":"n : ℕ\nx : ℚ\nd : ℕ\nhd : ∀ m < d, eval (1 + x) (bernoulli m) = eval x (bernoulli m) + ↑m * x ^ (m - 1)\nnz : ↑d.succ ≠ 0\n⊢ eval (1 + x) ((monomial d) ↑d.succ) -\n      ∑ x_1 ∈ range d, ↑((d + 1).choose x_1) • (eval x (bernoulli x_1) + ↑x_1 * x ^ (x_1 - 1)) =\n    eval x ((monomial d) ↑d.succ) - ∑ i ∈ range d, eval x (↑((d + 1).choose i) • bernoulli i) +\n      ↑d.succ * (↑d * x ^ (d - 1))","state_after":"n : ℕ\nx : ℚ\nd : ℕ\nhd : ∀ m < d, eval (1 + x) (bernoulli m) = eval x (bernoulli m) + ↑m * x ^ (m - 1)\nnz : ↑d.succ ≠ 0\n⊢ eval (1 + x) ((monomial d) ↑d.succ) -\n      ∑ x_1 ∈ range d,\n        (↑((d + 1).choose x_1) • eval x (bernoulli x_1) + ↑((d + 1).choose x_1) • (↑x_1 * x ^ (x_1 - 1))) =\n    eval x ((monomial d) ↑d.succ) - ∑ x_1 ∈ range d, ↑((d + 1).choose x_1) • eval x (bernoulli x_1) +\n      ↑d.succ * (↑d * x ^ (d - 1))","tactic":"simp_rw [eval_smul, smul_add]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Polynomial.eval_smul","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[344,8],"def_end_pos":[344,17]},{"full_name":"smul_add","def_path":"Mathlib/Algebra/GroupWithZero/Action/Defs.lean","def_pos":[130,8],"def_end_pos":[130,16]}]},{"state_before":"n : ℕ\nx : ℚ\nd : ℕ\nhd : ∀ m < d, eval (1 + x) (bernoulli m) = eval x (bernoulli m) + ↑m * x ^ (m - 1)\nnz : ↑d.succ ≠ 0\n⊢ eval (1 + x) ((monomial d) ↑d.succ) -\n      ∑ x_1 ∈ range d,\n        (↑((d + 1).choose x_1) • eval x (bernoulli x_1) + ↑((d + 1).choose x_1) • (↑x_1 * x ^ (x_1 - 1))) =\n    eval x ((monomial d) ↑d.succ) - ∑ x_1 ∈ range d, ↑((d + 1).choose x_1) • eval x (bernoulli x_1) +\n      ↑d.succ * (↑d * x ^ (d - 1))","state_after":"n : ℕ\nx : ℚ\nd : ℕ\nhd : ∀ m < d, eval (1 + x) (bernoulli m) = eval x (bernoulli m) + ↑m * x ^ (m - 1)\nnz : ↑d.succ ≠ 0\n⊢ eval (1 + x) ((monomial d) ↑d.succ) - eval x ((monomial d) ↑d.succ) =\n    ∑ x_1 ∈ range d, ↑((d + 1).choose x_1) • (↑x_1 * x ^ (x_1 - 1)) + ↑d.succ * (↑d * x ^ (d - 1))","tactic":"rw [sum_add_distrib, sub_add, sub_eq_sub_iff_sub_eq_sub, _root_.add_sub_sub_cancel]","premises":[{"full_name":"Finset.sum_add_distrib","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[768,2],"def_end_pos":[768,13]},{"full_name":"add_sub_sub_cancel","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[952,2],"def_end_pos":[952,13]},{"full_name":"sub_add","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[545,2],"def_end_pos":[545,13]},{"full_name":"sub_eq_sub_iff_sub_eq_sub","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[965,2],"def_end_pos":[965,13]}]},{"state_before":"n : ℕ\nx : ℚ\nd : ℕ\nhd : ∀ m < d, eval (1 + x) (bernoulli m) = eval x (bernoulli m) + ↑m * x ^ (m - 1)\nnz : ↑d.succ ≠ 0\n⊢ eval (1 + x) ((monomial d) ↑d.succ) - eval x ((monomial d) ↑d.succ) =\n    ∑ x_1 ∈ range d, ↑((d + 1).choose x_1) • (↑x_1 * x ^ (x_1 - 1)) + ↑((d + 1).choose d) * (↑d * x ^ (d - 1))","state_after":"n : ℕ\nx : ℚ\nd : ℕ\nhd : ∀ m < d, eval (1 + x) (bernoulli m) = eval x (bernoulli m) + ↑m * x ^ (m - 1)\nnz : ↑d.succ ≠ 0\n⊢ ∑ x_1 ∈ range (d + 1), ↑((d + 1).choose x_1) * (↑x_1 * x ^ (x_1 - 1)) =\n    ∑ x_1 ∈ range (d + 1), ↑((d + 1).choose x_1) • (↑x_1 * x ^ (x_1 - 1))","tactic":"rw [Nat.cast_succ, ← smul_eq_mul, ← sum_range_succ _ d, eval_monomial_one_add_sub]","premises":[{"full_name":"Finset.sum_range_succ","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[1296,2],"def_end_pos":[1296,13]},{"full_name":"Nat.cast_succ","def_path":"Mathlib/Data/Nat/Cast/Defs.lean","def_pos":[117,8],"def_end_pos":[117,17]},{"full_name":"Polynomial.eval_monomial_one_add_sub","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[359,8],"def_end_pos":[359,33]},{"full_name":"smul_eq_mul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[79,6],"def_end_pos":[79,17]}]},{"state_before":"n : ℕ\nx : ℚ\nd : ℕ\nhd : ∀ m < d, eval (1 + x) (bernoulli m) = eval x (bernoulli m) + ↑m * x ^ (m - 1)\nnz : ↑d.succ ≠ 0\n⊢ ∑ x_1 ∈ range (d + 1), ↑((d + 1).choose x_1) * (↑x_1 * x ^ (x_1 - 1)) =\n    ∑ x_1 ∈ range (d + 1), ↑((d + 1).choose x_1) • (↑x_1 * x ^ (x_1 - 1))","state_after":"no goals","tactic":"simp_rw [smul_eq_mul]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"smul_eq_mul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[79,6],"def_end_pos":[79,17]}]}]}
+{"url":"Mathlib/RingTheory/QuotientNilpotent.lean","commit":"","full_name":"Ideal.IsNilpotent.induction_on","start":[21,0],"end":[48,49],"file_path":"Mathlib/RingTheory/QuotientNilpotent.lean","tactics":[{"state_before":"R : Type u_1\nS : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nI : Ideal S\nhI : IsNilpotent I\nP : ⦃S : Type u_2⦄ → [inst : CommRing S] → Ideal S → Prop\nh₁ : ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I : Ideal S), I ^ 2 = ⊥ → P I\nh₂ : ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I J : Ideal S), I ≤ J → P I → P (map (Quotient.mk I) J) → P J\n⊢ P I","state_after":"case intro\nR : Type u_1\nS : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nI : Ideal S\nP : ⦃S : Type u_2⦄ → [inst : CommRing S] → Ideal S → Prop\nh₁ : ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I : Ideal S), I ^ 2 = ⊥ → P I\nh₂ : ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I J : Ideal S), I ≤ J → P I → P (map (Quotient.mk I) J) → P J\nn : ℕ\nhI : I ^ n = ⊥\n⊢ P I","tactic":"obtain ⟨n, hI : I ^ n = ⊥⟩ := hI","premises":[{"full_name":"Bot.bot","def_path":"Mathlib/Order/Notation.lean","def_pos":[100,2],"def_end_pos":[100,5]}]},{"state_before":"case intro\nR : Type u_1\nS : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nI : Ideal S\nP : ⦃S : Type u_2⦄ → [inst : CommRing S] → Ideal S → Prop\nh₁ : ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I : Ideal S), I ^ 2 = ⊥ → P I\nh₂ : ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I J : Ideal S), I ≤ J → P I → P (map (Quotient.mk I) J) → P J\nn : ℕ\nhI : I ^ n = ⊥\n⊢ P I","state_after":"case intro.h\nR : Type u_1\ninst✝² : CommSemiring R\nP : ⦃S : Type u_2⦄ → [inst : CommRing S] → Ideal S → Prop\nh₁ : ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I : Ideal S), I ^ 2 = ⊥ → P I\nh₂ : ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I J : Ideal S), I ≤ J → P I → P (map (Quotient.mk I) J) → P J\nn : ℕ\nH : ∀ m < n, ∀ {S : Type u_2} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = ⊥ → P I\nS : Type u_2\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nI : Ideal S\nhI : I ^ n = ⊥\n⊢ P I","tactic":"induction' n using Nat.strong_induction_on with n H generalizing S","premises":[{"full_name":"Nat.strong_induction_on","def_path":"Mathlib/Init/Data/Nat/Lemmas.lean","def_pos":[83,18],"def_end_pos":[83,37]}]},{"state_before":"case intro.h\nR : Type u_1\ninst✝² : CommSemiring R\nP : ⦃S : Type u_2⦄ → [inst : CommRing S] → Ideal S → Prop\nh₁ : ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I : Ideal S), I ^ 2 = ⊥ → P I\nh₂ : ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I J : Ideal S), I ≤ J → P I → P (map (Quotient.mk I) J) → P J\nn : ℕ\nH : ∀ m < n, ∀ {S : Type u_2} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = ⊥ → P I\nS : Type u_2\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nI : Ideal S\nhI : I ^ n = ⊥\n⊢ P I","state_after":"case pos\nR : Type u_1\ninst✝² : CommSemiring R\nP : ⦃S : Type u_2⦄ → [inst : CommRing S] → Ideal S → Prop\nh₁ : ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I : Ideal S), I ^ 2 = ⊥ → P I\nh₂ : ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I J : Ideal S), I ≤ J → P I → P (map (Quotient.mk I) J) → P J\nn : ℕ\nH : ∀ m < n, ∀ {S : Type u_2} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = ⊥ → P I\nS : Type u_2\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nI : Ideal S\nhI : I ^ n = ⊥\nhI' : I = ⊥\n⊢ P I\n\ncase neg\nR : Type u_1\ninst✝² : CommSemiring R\nP : ⦃S : Type u_2⦄ → [inst : CommRing S] → Ideal S → Prop\nh₁ : ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I : Ideal S), I ^ 2 = ⊥ → P I\nh₂ : ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I J : Ideal S), I ≤ J → P I → P (map (Quotient.mk I) J) → P J\nn : ℕ\nH : ∀ m < n, ∀ {S : Type u_2} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = ⊥ → P I\nS : Type u_2\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nI : Ideal S\nhI : I ^ n = ⊥\nhI' : ¬I = ⊥\n⊢ P I","tactic":"by_cases hI' : I = ⊥","premises":[{"full_name":"Bot.bot","def_path":"Mathlib/Order/Notation.lean","def_pos":[100,2],"def_end_pos":[100,5]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case neg\nR : Type u_1\ninst✝² : CommSemiring R\nP : ⦃S : Type u_2⦄ → [inst : CommRing S] → Ideal S → Prop\nh₁ : ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I : Ideal S), I ^ 2 = ⊥ → P I\nh₂ : ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I J : Ideal S), I ≤ J → P I → P (map (Quotient.mk I) J) → P J\nn : ℕ\nH : ∀ m < n, ∀ {S : Type u_2} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = ⊥ → P I\nS : Type u_2\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nI : Ideal S\nhI : I ^ n = ⊥\nhI' : ¬I = ⊥\n⊢ P I","state_after":"case neg.zero\nR : Type u_1\ninst✝² : CommSemiring R\nP : ⦃S : Type u_2⦄ → [inst : CommRing S] → Ideal S → Prop\nh₁ : ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I : Ideal S), I ^ 2 = ⊥ → P I\nh₂ : ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I J : Ideal S), I ≤ J → P I → P (map (Quotient.mk I) J) → P J\nS : Type u_2\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nI : Ideal S\nhI' : ¬I = ⊥\nH : ∀ m < 0, ∀ {S : Type u_2} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = ⊥ → P I\nhI : I ^ 0 = ⊥\n⊢ P I\n\ncase neg.succ\nR : Type u_1\ninst✝² : CommSemiring R\nP : ⦃S : Type u_2⦄ → [inst : CommRing S] → Ideal S → Prop\nh₁ : ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I : Ideal S), I ^ 2 = ⊥ → P I\nh₂ : ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I J : Ideal S), I ≤ J → P I → P (map (Quotient.mk I) J) → P J\nS : Type u_2\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nI : Ideal S\nhI' : ¬I = ⊥\nn : ℕ\nH : ∀ m < n + 1, ∀ {S : Type u_2} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = ⊥ → P I\nhI : I ^ (n + 1) = ⊥\n⊢ P I","tactic":"cases' n with n","premises":[]},{"state_before":"case neg.succ\nR : Type u_1\ninst✝² : CommSemiring R\nP : ⦃S : Type u_2⦄ → [inst : CommRing S] → Ideal S → Prop\nh₁ : ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I : Ideal S), I ^ 2 = ⊥ → P I\nh₂ : ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I J : Ideal S), I ≤ J → P I → P (map (Quotient.mk I) J) → P J\nS : Type u_2\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nI : Ideal S\nhI' : ¬I = ⊥\nn : ℕ\nH : ∀ m < n + 1, ∀ {S : Type u_2} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = ⊥ → P I\nhI : I ^ (n + 1) = ⊥\n⊢ P I","state_after":"case neg.succ.zero\nR : Type u_1\ninst✝² : CommSemiring R\nP : ⦃S : Type u_2⦄ → [inst : CommRing S] → Ideal S → Prop\nh₁ : ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I : Ideal S), I ^ 2 = ⊥ → P I\nh₂ : ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I J : Ideal S), I ≤ J → P I → P (map (Quotient.mk I) J) → P J\nS : Type u_2\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nI : Ideal S\nhI' : ¬I = ⊥\nH : ∀ m < 0 + 1, ∀ {S : Type u_2} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = ⊥ → P I\nhI : I ^ (0 + 1) = ⊥\n⊢ P I\n\ncase neg.succ.succ\nR : Type u_1\ninst✝² : CommSemiring R\nP : ⦃S : Type u_2⦄ → [inst : CommRing S] → Ideal S → Prop\nh₁ : ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I : Ideal S), I ^ 2 = ⊥ → P I\nh₂ : ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I J : Ideal S), I ≤ J → P I → P (map (Quotient.mk I) J) → P J\nS : Type u_2\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nI : Ideal S\nhI' : ¬I = ⊥\nn : ℕ\nH : ∀ m < n + 1 + 1, ∀ {S : Type u_2} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = ⊥ → P I\nhI : I ^ (n + 1 + 1) = ⊥\n⊢ P I","tactic":"cases' n with n","premises":[]},{"state_before":"case neg.succ.succ\nR : Type u_1\ninst✝² : CommSemiring R\nP : ⦃S : Type u_2⦄ → [inst : CommRing S] → Ideal S → Prop\nh₁ : ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I : Ideal S), I ^ 2 = ⊥ → P I\nh₂ : ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I J : Ideal S), I ≤ J → P I → P (map (Quotient.mk I) J) → P J\nS : Type u_2\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nI : Ideal S\nhI' : ¬I = ⊥\nn : ℕ\nH : ∀ m < n + 1 + 1, ∀ {S : Type u_2} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = ⊥ → P I\nhI : I ^ (n + 1 + 1) = ⊥\n⊢ P I","state_after":"case neg.succ.succ.a\nR : Type u_1\ninst✝² : CommSemiring R\nP : ⦃S : Type u_2⦄ → [inst : CommRing S] → Ideal S → Prop\nh₁ : ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I : Ideal S), I ^ 2 = ⊥ → P I\nh₂ : ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I J : Ideal S), I ≤ J → P I → P (map (Quotient.mk I) J) → P J\nS : Type u_2\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nI : Ideal S\nhI' : ¬I = ⊥\nn : ℕ\nH : ∀ m < n + 1 + 1, ∀ {S : Type u_2} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = ⊥ → P I\nhI : I ^ (n + 1 + 1) = ⊥\n⊢ P (I ^ 2)\n\ncase neg.succ.succ.a\nR : Type u_1\ninst✝² : CommSemiring R\nP : ⦃S : Type u_2⦄ → [inst : CommRing S] → Ideal S → Prop\nh₁ : ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I : Ideal S), I ^ 2 = ⊥ → P I\nh₂ : ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I J : Ideal S), I ≤ J → P I → P (map (Quotient.mk I) J) → P J\nS : Type u_2\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nI : Ideal S\nhI' : ¬I = ⊥\nn : ℕ\nH : ∀ m < n + 1 + 1, ∀ {S : Type u_2} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = ⊥ → P I\nhI : I ^ (n + 1 + 1) = ⊥\n⊢ P (map (Quotient.mk (I ^ 2)) I)","tactic":"apply h₂ (I ^ 2) _ (Ideal.pow_le_self two_ne_zero)","premises":[{"full_name":"Ideal.pow_le_self","def_path":"Mathlib/RingTheory/Ideal/Operations.lean","def_pos":[674,8],"def_end_pos":[674,19]},{"full_name":"two_ne_zero","def_path":"Mathlib/Algebra/NeZero.lean","def_pos":[54,6],"def_end_pos":[54,17]}]}]}
+{"url":"Mathlib/AlgebraicGeometry/Restrict.lean","commit":"","full_name":"AlgebraicGeometry.morphismRestrict_app","start":[405,0],"end":[418,56],"file_path":"Mathlib/AlgebraicGeometry/Restrict.lean","tactics":[{"state_before":"C : Type u₁\ninst✝ : Category.{v, u₁} C\nX Y : Scheme\nf : X ⟶ Y\nU : Y.Opens\nV : (↑U).Opens\n⊢ Hom.app (f ∣_ U) V = Hom.app f (U.ι ''ᵁ V) ≫ X.presheaf.map (eqToHom ⋯).op","state_after":"C : Type u₁\ninst✝ : Category.{v, u₁} C\nX Y : Scheme\nf : X ⟶ Y\nU : Y.Opens\nV : (↑U).Opens\nthis :\n  Hom.app (f ∣_ U ≫ U.ι) (U.ι ''ᵁ V) = Hom.app ((f ⁻¹ᵁ U).ι ≫ f) (U.ι ''ᵁ V) ≫ (↑(f ⁻¹ᵁ U)).presheaf.map (eqToHom ⋯).op\n⊢ Hom.app (f ∣_ U) V = Hom.app f (U.ι ''ᵁ V) ≫ X.presheaf.map (eqToHom ⋯).op","tactic":"have := Scheme.congr_app (morphismRestrict_ι f U) (U.ι ''ᵁ V)","premises":[{"full_name":"AlgebraicGeometry.Scheme.Hom.opensFunctor","def_path":"Mathlib/AlgebraicGeometry/OpenImmersion.lean","def_pos":[83,7],"def_end_pos":[83,19]},{"full_name":"AlgebraicGeometry.Scheme.Opens.ι","def_path":"Mathlib/AlgebraicGeometry/Restrict.lean","def_pos":[50,4],"def_end_pos":[50,5]},{"full_name":"AlgebraicGeometry.Scheme.congr_app","def_path":"Mathlib/AlgebraicGeometry/Scheme.lean","def_pos":[253,8],"def_end_pos":[253,17]},{"full_name":"AlgebraicGeometry.morphismRestrict_ι","def_path":"Mathlib/AlgebraicGeometry/Restrict.lean","def_pos":[337,8],"def_end_pos":[337,26]},{"full_name":"Prefunctor.obj","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[55,2],"def_end_pos":[55,5]}]},{"state_before":"C : Type u₁\ninst✝ : Category.{v, u₁} C\nX Y : Scheme\nf : X ⟶ Y\nU : Y.Opens\nV : (↑U).Opens\nthis :\n  Hom.app (f ∣_ U ≫ U.ι) (U.ι ''ᵁ V) = Hom.app ((f ⁻¹ᵁ U).ι ≫ f) (U.ι ''ᵁ V) ≫ (↑(f ⁻¹ᵁ U)).presheaf.map (eqToHom ⋯).op\n⊢ Hom.app (f ∣_ U) V = Hom.app f (U.ι ''ᵁ V) ≫ X.presheaf.map (eqToHom ⋯).op","state_after":"C : Type u₁\ninst✝ : Category.{v, u₁} C\nX Y : Scheme\nf : X ⟶ Y\nU : Y.Opens\nV : (↑U).Opens\nthis :\n  Y.presheaf.map (homOfLE ⋯).op ≫ Hom.appLE (f ∣_ U) (U.ι ⁻¹ᵁ U.ι ''ᵁ V) ((f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V) ⋯ =\n    (Hom.appLE f (U.ι ''ᵁ V) (f ⁻¹ᵁ U.ι ''ᵁ V) ⋯ ≫ X.presheaf.map (homOfLE ⋯).op) ≫\n      X.presheaf.map ((Hom.opensFunctor (f ⁻¹ᵁ U).ι).map (eqToHom ⋯)).op\n⊢ Hom.app (f ∣_ U) V = Hom.app f (U.ι ''ᵁ V) ≫ X.presheaf.map (eqToHom ⋯).op","tactic":"simp only [Scheme.preimage_comp, Opens.toScheme_presheaf_obj, Hom.app_eq_appLE, comp_appLE,\n    Opens.ι_appLE, eqToHom_op, Opens.toScheme_presheaf_map, eqToHom_unop] at this","premises":[{"full_name":"AlgebraicGeometry.Scheme.Hom.app_eq_appLE","def_path":"Mathlib/AlgebraicGeometry/Scheme.lean","def_pos":[133,6],"def_end_pos":[133,18]},{"full_name":"AlgebraicGeometry.Scheme.Opens.toScheme_presheaf_map","def_path":"Mathlib/AlgebraicGeometry/Restrict.lean","def_pos":[59,6],"def_end_pos":[59,27]},{"full_name":"AlgebraicGeometry.Scheme.Opens.toScheme_presheaf_obj","def_path":"Mathlib/AlgebraicGeometry/Restrict.lean","def_pos":[56,6],"def_end_pos":[56,27]},{"full_name":"AlgebraicGeometry.Scheme.Opens.ι_appLE","def_path":"Mathlib/AlgebraicGeometry/Restrict.lean","def_pos":[68,6],"def_end_pos":[68,13]},{"full_name":"AlgebraicGeometry.Scheme.comp_appLE","def_path":"Mathlib/AlgebraicGeometry/Scheme.lean","def_pos":[249,8],"def_end_pos":[249,18]},{"full_name":"AlgebraicGeometry.Scheme.preimage_comp","def_path":"Mathlib/AlgebraicGeometry/Scheme.lean","def_pos":[171,6],"def_end_pos":[171,19]},{"full_name":"CategoryTheory.eqToHom_op","def_path":"Mathlib/CategoryTheory/EqToHom.lean","def_pos":[156,8],"def_end_pos":[156,18]},{"full_name":"CategoryTheory.eqToHom_unop","def_path":"Mathlib/CategoryTheory/EqToHom.lean","def_pos":[161,8],"def_end_pos":[161,20]}]},{"state_before":"C : Type u₁\ninst✝ : Category.{v, u₁} C\nX Y : Scheme\nf : X ⟶ Y\nU : Y.Opens\nV : (↑U).Opens\nthis :\n  Y.presheaf.map (homOfLE ⋯).op ≫ Hom.appLE (f ∣_ U) (U.ι ⁻¹ᵁ U.ι ''ᵁ V) ((f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V) ⋯ =\n    (Hom.appLE f (U.ι ''ᵁ V) (f ⁻¹ᵁ U.ι ''ᵁ V) ⋯ ≫ X.presheaf.map (homOfLE ⋯).op) ≫\n      X.presheaf.map ((Hom.opensFunctor (f ⁻¹ᵁ U).ι).map (eqToHom ⋯)).op\n⊢ Hom.app (f ∣_ U) V = Hom.app f (U.ι ''ᵁ V) ≫ X.presheaf.map (eqToHom ⋯).op","state_after":"C : Type u₁\ninst✝ : Category.{v, u₁} C\nX Y : Scheme\nf : X ⟶ Y\nU : Y.Opens\nV : (↑U).Opens\nthis :\n  Y.presheaf.map (homOfLE ⋯).op ≫ Hom.appLE (f ∣_ U) (U.ι ⁻¹ᵁ U.ι ''ᵁ V) ((f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V) ⋯ =\n    (Hom.appLE f (U.ι ''ᵁ V) (f ⁻¹ᵁ U.ι ''ᵁ V) ⋯ ≫ X.presheaf.map (homOfLE ⋯).op) ≫\n      X.presheaf.map ((Hom.opensFunctor (f ⁻¹ᵁ U).ι).map (eqToHom ⋯)).op\ne : U.ι ⁻¹ᵁ U.ι ''ᵁ V = V\n⊢ Hom.app (f ∣_ U) V = Hom.app f (U.ι ''ᵁ V) ≫ X.presheaf.map (eqToHom ⋯).op","tactic":"have e : U.ι ⁻¹ᵁ (U.ι ''ᵁ V) = V :=\n    Opens.ext (Set.preimage_image_eq _ Subtype.coe_injective)","premises":[{"full_name":"AlgebraicGeometry.LocallyRingedSpace.Hom","def_path":"Mathlib/Geometry/RingedSpace/LocallyRingedSpace.lean","def_pos":[73,10],"def_end_pos":[73,13]},{"full_name":"AlgebraicGeometry.LocallyRingedSpace.Hom.val","def_path":"Mathlib/Geometry/RingedSpace/LocallyRingedSpace.lean","def_pos":[75,2],"def_end_pos":[75,5]},{"full_name":"AlgebraicGeometry.PresheafedSpace.Hom.base","def_path":"Mathlib/Geometry/RingedSpace/PresheafedSpace.lean","def_pos":[90,2],"def_end_pos":[90,6]},{"full_name":"AlgebraicGeometry.Scheme.Hom.opensFunctor","def_path":"Mathlib/AlgebraicGeometry/OpenImmersion.lean","def_pos":[83,7],"def_end_pos":[83,19]},{"full_name":"AlgebraicGeometry.Scheme.Opens","def_path":"Mathlib/AlgebraicGeometry/Scheme.lean","def_pos":[57,7],"def_end_pos":[57,12]},{"full_name":"AlgebraicGeometry.Scheme.Opens.ι","def_path":"Mathlib/AlgebraicGeometry/Restrict.lean","def_pos":[50,4],"def_end_pos":[50,5]},{"full_name":"Prefunctor.obj","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[55,2],"def_end_pos":[55,5]},{"full_name":"Set.preimage_image_eq","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[417,8],"def_end_pos":[417,25]},{"full_name":"Subtype.coe_injective","def_path":"Mathlib/Data/Subtype.lean","def_pos":[102,8],"def_end_pos":[102,21]},{"full_name":"TopologicalSpace.Opens.ext","def_path":"Mathlib/Topology/Sets/Opens.lean","def_pos":[100,8],"def_end_pos":[100,11]},{"full_name":"TopologicalSpace.Opens.map","def_path":"Mathlib/Topology/Category/TopCat/Opens.lean","def_pos":[133,4],"def_end_pos":[133,7]}]},{"state_before":"C : Type u₁\ninst✝ : Category.{v, u₁} C\nX Y : Scheme\nf : X ⟶ Y\nU : Y.Opens\nV : (↑U).Opens\nthis :\n  Y.presheaf.map (homOfLE ⋯).op ≫ Hom.appLE (f ∣_ U) (U.ι ⁻¹ᵁ U.ι ''ᵁ V) ((f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V) ⋯ =\n    (Hom.appLE f (U.ι ''ᵁ V) (f ⁻¹ᵁ U.ι ''ᵁ V) ⋯ ≫ X.presheaf.map (homOfLE ⋯).op) ≫\n      X.presheaf.map ((Hom.opensFunctor (f ⁻¹ᵁ U).ι).map (eqToHom ⋯)).op\ne : U.ι ⁻¹ᵁ U.ι ''ᵁ V = V\n⊢ Hom.app (f ∣_ U) V = Hom.app f (U.ι ''ᵁ V) ≫ X.presheaf.map (eqToHom ⋯).op","state_after":"C : Type u₁\ninst✝ : Category.{v, u₁} C\nX Y : Scheme\nf : X ⟶ Y\nU : Y.Opens\nV : (↑U).Opens\nthis :\n  Y.presheaf.map (homOfLE ⋯).op ≫ Hom.appLE (f ∣_ U) (U.ι ⁻¹ᵁ U.ι ''ᵁ V) ((f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V) ⋯ =\n    (Hom.appLE f (U.ι ''ᵁ V) (f ⁻¹ᵁ U.ι ''ᵁ V) ⋯ ≫ X.presheaf.map (homOfLE ⋯).op) ≫\n      X.presheaf.map ((Hom.opensFunctor (f ⁻¹ᵁ U).ι).map (eqToHom ⋯)).op\ne : U.ι ⁻¹ᵁ U.ι ''ᵁ V = V\ne' : (f ∣_ U) ⁻¹ᵁ V = (f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V\n⊢ Hom.app (f ∣_ U) V = Hom.app f (U.ι ''ᵁ V) ≫ X.presheaf.map (eqToHom ⋯).op","tactic":"have e' : (f ∣_ U) ⁻¹ᵁ V = (f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V := by rw [e]","premises":[{"full_name":"AlgebraicGeometry.LocallyRingedSpace.Hom","def_path":"Mathlib/Geometry/RingedSpace/LocallyRingedSpace.lean","def_pos":[73,10],"def_end_pos":[73,13]},{"full_name":"AlgebraicGeometry.LocallyRingedSpace.Hom.val","def_path":"Mathlib/Geometry/RingedSpace/LocallyRingedSpace.lean","def_pos":[75,2],"def_end_pos":[75,5]},{"full_name":"AlgebraicGeometry.PresheafedSpace.Hom.base","def_path":"Mathlib/Geometry/RingedSpace/PresheafedSpace.lean","def_pos":[90,2],"def_end_pos":[90,6]},{"full_name":"AlgebraicGeometry.Scheme.Hom.opensFunctor","def_path":"Mathlib/AlgebraicGeometry/OpenImmersion.lean","def_pos":[83,7],"def_end_pos":[83,19]},{"full_name":"AlgebraicGeometry.Scheme.Opens","def_path":"Mathlib/AlgebraicGeometry/Scheme.lean","def_pos":[57,7],"def_end_pos":[57,12]},{"full_name":"AlgebraicGeometry.Scheme.Opens.ι","def_path":"Mathlib/AlgebraicGeometry/Restrict.lean","def_pos":[50,4],"def_end_pos":[50,5]},{"full_name":"AlgebraicGeometry.morphismRestrict","def_path":"Mathlib/AlgebraicGeometry/Restrict.lean","def_pos":[325,4],"def_end_pos":[325,20]},{"full_name":"Prefunctor.obj","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[55,2],"def_end_pos":[55,5]},{"full_name":"TopologicalSpace.Opens.map","def_path":"Mathlib/Topology/Category/TopCat/Opens.lean","def_pos":[133,4],"def_end_pos":[133,7]}]},{"state_before":"C : Type u₁\ninst✝ : Category.{v, u₁} C\nX Y : Scheme\nf : X ⟶ Y\nU : Y.Opens\nV : (↑U).Opens\nthis :\n  Y.presheaf.map (homOfLE ⋯).op ≫ Hom.appLE (f ∣_ U) (U.ι ⁻¹ᵁ U.ι ''ᵁ V) ((f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V) ⋯ =\n    (Hom.appLE f (U.ι ''ᵁ V) (f ⁻¹ᵁ U.ι ''ᵁ V) ⋯ ≫ X.presheaf.map (homOfLE ⋯).op) ≫\n      X.presheaf.map ((Hom.opensFunctor (f ⁻¹ᵁ U).ι).map (eqToHom ⋯)).op\ne : U.ι ⁻¹ᵁ U.ι ''ᵁ V = V\ne' : (f ∣_ U) ⁻¹ᵁ V = (f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V\n⊢ Hom.app (f ∣_ U) V = Hom.app f (U.ι ''ᵁ V) ≫ X.presheaf.map (eqToHom ⋯).op","state_after":"C : Type u₁\ninst✝ : Category.{v, u₁} C\nX Y : Scheme\nf : X ⟶ Y\nU : Y.Opens\nV : (↑U).Opens\nthis :\n  Y.presheaf.map (homOfLE ⋯).op ≫ Hom.appLE (f ∣_ U) (U.ι ⁻¹ᵁ U.ι ''ᵁ V) ((f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V) ⋯ =\n    (Hom.appLE f (U.ι ''ᵁ V) (f ⁻¹ᵁ U.ι ''ᵁ V) ⋯ ≫ X.presheaf.map (homOfLE ⋯).op) ≫\n      X.presheaf.map ((Hom.opensFunctor (f ⁻¹ᵁ U).ι).map (eqToHom ⋯)).op\ne : U.ι ⁻¹ᵁ U.ι ''ᵁ V = V\ne' : (f ∣_ U) ⁻¹ᵁ V = (f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V\n⊢ Hom.appLE (f ∣_ U) V ((f ∣_ U) ⁻¹ᵁ V) ⋯ = Hom.appLE f (U.ι ''ᵁ V) ((f ⁻¹ᵁ U).ι ''ᵁ (f ∣_ U) ⁻¹ᵁ V) ⋯","tactic":"simp only [Opens.toScheme_presheaf_obj, Hom.app_eq_appLE, eqToHom_op, Hom.appLE_map]","premises":[{"full_name":"AlgebraicGeometry.Scheme.Hom.appLE_map","def_path":"Mathlib/AlgebraicGeometry/Scheme.lean","def_pos":[111,6],"def_end_pos":[111,15]},{"full_name":"AlgebraicGeometry.Scheme.Hom.app_eq_appLE","def_path":"Mathlib/AlgebraicGeometry/Scheme.lean","def_pos":[133,6],"def_end_pos":[133,18]},{"full_name":"AlgebraicGeometry.Scheme.Opens.toScheme_presheaf_obj","def_path":"Mathlib/AlgebraicGeometry/Restrict.lean","def_pos":[56,6],"def_end_pos":[56,27]},{"full_name":"CategoryTheory.eqToHom_op","def_path":"Mathlib/CategoryTheory/EqToHom.lean","def_pos":[156,8],"def_end_pos":[156,18]}]},{"state_before":"C : Type u₁\ninst✝ : Category.{v, u₁} C\nX Y : Scheme\nf : X ⟶ Y\nU : Y.Opens\nV : (↑U).Opens\nthis :\n  Y.presheaf.map (homOfLE ⋯).op ≫ Hom.appLE (f ∣_ U) (U.ι ⁻¹ᵁ U.ι ''ᵁ V) ((f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V) ⋯ =\n    (Hom.appLE f (U.ι ''ᵁ V) (f ⁻¹ᵁ U.ι ''ᵁ V) ⋯ ≫ X.presheaf.map (homOfLE ⋯).op) ≫\n      X.presheaf.map ((Hom.opensFunctor (f ⁻¹ᵁ U).ι).map (eqToHom ⋯)).op\ne : U.ι ⁻¹ᵁ U.ι ''ᵁ V = V\ne' : (f ∣_ U) ⁻¹ᵁ V = (f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V\n⊢ Hom.appLE (f ∣_ U) V ((f ∣_ U) ⁻¹ᵁ V) ⋯ = Hom.appLE f (U.ι ''ᵁ V) ((f ⁻¹ᵁ U).ι ''ᵁ (f ∣_ U) ⁻¹ᵁ V) ⋯","state_after":"C : Type u₁\ninst✝ : Category.{v, u₁} C\nX Y : Scheme\nf : X ⟶ Y\nU : Y.Opens\nV : (↑U).Opens\nthis :\n  Y.presheaf.map (homOfLE ⋯).op ≫ Hom.appLE (f ∣_ U) (U.ι ⁻¹ᵁ U.ι ''ᵁ V) ((f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V) ⋯ =\n    (Hom.appLE f (U.ι ''ᵁ V) (f ⁻¹ᵁ U.ι ''ᵁ V) ⋯ ≫ X.presheaf.map (homOfLE ⋯).op) ≫\n      X.presheaf.map ((Hom.opensFunctor (f ⁻¹ᵁ U).ι).map (eqToHom ⋯)).op\ne : U.ι ⁻¹ᵁ U.ι ''ᵁ V = V\ne' : (f ∣_ U) ⁻¹ᵁ V = (f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V\n⊢ ((↑U).presheaf.map (eqToHom e).op ≫ Hom.appLE (f ∣_ U) (U.ι ⁻¹ᵁ U.ι ''ᵁ V) ((f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V) ⋯) ≫\n      (↑(f ⁻¹ᵁ U)).presheaf.map (eqToHom e').op =\n    Hom.appLE f (U.ι ''ᵁ V) ((f ⁻¹ᵁ U).ι ''ᵁ (f ∣_ U) ⁻¹ᵁ V) ⋯","tactic":"rw [← (f ∣_ U).appLE_map' _ e', ← (f ∣_ U).map_appLE' _ e]","premises":[{"full_name":"AlgebraicGeometry.Scheme.Hom.appLE_map'","def_path":"Mathlib/AlgebraicGeometry/Scheme.lean","def_pos":[117,6],"def_end_pos":[117,16]},{"full_name":"AlgebraicGeometry.Scheme.Hom.map_appLE'","def_path":"Mathlib/AlgebraicGeometry/Scheme.lean","def_pos":[129,6],"def_end_pos":[129,16]},{"full_name":"AlgebraicGeometry.morphismRestrict","def_path":"Mathlib/AlgebraicGeometry/Restrict.lean","def_pos":[325,4],"def_end_pos":[325,20]}]},{"state_before":"C : Type u₁\ninst✝ : Category.{v, u₁} C\nX Y : Scheme\nf : X ⟶ Y\nU : Y.Opens\nV : (↑U).Opens\nthis :\n  Y.presheaf.map (homOfLE ⋯).op ≫ Hom.appLE (f ∣_ U) (U.ι ⁻¹ᵁ U.ι ''ᵁ V) ((f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V) ⋯ =\n    (Hom.appLE f (U.ι ''ᵁ V) (f ⁻¹ᵁ U.ι ''ᵁ V) ⋯ ≫ X.presheaf.map (homOfLE ⋯).op) ≫\n      X.presheaf.map ((Hom.opensFunctor (f ⁻¹ᵁ U).ι).map (eqToHom ⋯)).op\ne : U.ι ⁻¹ᵁ U.ι ''ᵁ V = V\ne' : (f ∣_ U) ⁻¹ᵁ V = (f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V\n⊢ ((↑U).presheaf.map (eqToHom e).op ≫ Hom.appLE (f ∣_ U) (U.ι ⁻¹ᵁ U.ι ''ᵁ V) ((f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V) ⋯) ≫\n      (↑(f ⁻¹ᵁ U)).presheaf.map (eqToHom e').op =\n    Hom.appLE f (U.ι ''ᵁ V) ((f ⁻¹ᵁ U).ι ''ᵁ (f ∣_ U) ⁻¹ᵁ V) ⋯","state_after":"C : Type u₁\ninst✝ : Category.{v, u₁} C\nX Y : Scheme\nf : X ⟶ Y\nU : Y.Opens\nV : (↑U).Opens\nthis :\n  Y.presheaf.map (homOfLE ⋯).op ≫ Hom.appLE (f ∣_ U) (U.ι ⁻¹ᵁ U.ι ''ᵁ V) ((f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V) ⋯ =\n    (Hom.appLE f (U.ι ''ᵁ V) (f ⁻¹ᵁ U.ι ''ᵁ V) ⋯ ≫ X.presheaf.map (homOfLE ⋯).op) ≫\n      X.presheaf.map ((Hom.opensFunctor (f ⁻¹ᵁ U).ι).map (eqToHom ⋯)).op\ne : U.ι ⁻¹ᵁ U.ι ''ᵁ V = V\ne' : (f ∣_ U) ⁻¹ᵁ V = (f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V\n⊢ (Y.presheaf.map (homOfLE ⋯).op ≫ Hom.appLE (f ∣_ U) (U.ι ⁻¹ᵁ U.ι ''ᵁ V) ((f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V) ⋯) ≫\n      X.presheaf.map (homOfLE ⋯).op =\n    Hom.appLE f (U.ι ''ᵁ V) ((f ⁻¹ᵁ U).ι ''ᵁ (f ∣_ U) ⁻¹ᵁ V) ⋯","tactic":"simp only [Opens.toScheme_presheaf_obj, eqToHom_eq_homOfLE, Opens.toScheme_presheaf_map,\n    Quiver.Hom.unop_op, Hom.opensFunctor_map_homOfLE]","premises":[{"full_name":"AlgebraicGeometry.Scheme.Hom.opensFunctor_map_homOfLE","def_path":"Mathlib/AlgebraicGeometry/OpenImmersion.lean","def_pos":[94,6],"def_end_pos":[94,30]},{"full_name":"AlgebraicGeometry.Scheme.Opens.toScheme_presheaf_map","def_path":"Mathlib/AlgebraicGeometry/Restrict.lean","def_pos":[59,6],"def_end_pos":[59,27]},{"full_name":"AlgebraicGeometry.Scheme.Opens.toScheme_presheaf_obj","def_path":"Mathlib/AlgebraicGeometry/Restrict.lean","def_pos":[56,6],"def_end_pos":[56,27]},{"full_name":"AlgebraicGeometry.eqToHom_eq_homOfLE","def_path":"Mathlib/AlgebraicGeometry/Restrict.lean","def_pos":[402,6],"def_end_pos":[402,24]},{"full_name":"Quiver.Hom.unop_op","def_path":"Mathlib/CategoryTheory/Opposites.lean","def_pos":[44,8],"def_end_pos":[44,26]}]},{"state_before":"C : Type u₁\ninst✝ : Category.{v, u₁} C\nX Y : Scheme\nf : X ⟶ Y\nU : Y.Opens\nV : (↑U).Opens\nthis :\n  Y.presheaf.map (homOfLE ⋯).op ≫ Hom.appLE (f ∣_ U) (U.ι ⁻¹ᵁ U.ι ''ᵁ V) ((f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V) ⋯ =\n    (Hom.appLE f (U.ι ''ᵁ V) (f ⁻¹ᵁ U.ι ''ᵁ V) ⋯ ≫ X.presheaf.map (homOfLE ⋯).op) ≫\n      X.presheaf.map ((Hom.opensFunctor (f ⁻¹ᵁ U).ι).map (eqToHom ⋯)).op\ne : U.ι ⁻¹ᵁ U.ι ''ᵁ V = V\ne' : (f ∣_ U) ⁻¹ᵁ V = (f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V\n⊢ (Y.presheaf.map (homOfLE ⋯).op ≫ Hom.appLE (f ∣_ U) (U.ι ⁻¹ᵁ U.ι ''ᵁ V) ((f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V) ⋯) ≫\n      X.presheaf.map (homOfLE ⋯).op =\n    Hom.appLE f (U.ι ''ᵁ V) ((f ⁻¹ᵁ U).ι ''ᵁ (f ∣_ U) ⁻¹ᵁ V) ⋯","state_after":"no goals","tactic":"rw [this, Hom.appLE_map, Hom.appLE_map, Hom.appLE_map]","premises":[{"full_name":"AlgebraicGeometry.Scheme.Hom.appLE_map","def_path":"Mathlib/AlgebraicGeometry/Scheme.lean","def_pos":[111,6],"def_end_pos":[111,15]}]}]}
+{"url":"Mathlib/Probability/ConditionalProbability.lean","commit":"","full_name":"ProbabilityTheory.cond_isProbabilityMeasure_of_finite","start":[85,0],"end":[93,40],"file_path":"Mathlib/Probability/ConditionalProbability.lean","tactics":[{"state_before":"Ω : Type u_1\nΩ' : Type u_2\nα : Type u_3\nm : MeasurableSpace Ω\nm' : MeasurableSpace Ω'\nμ : Measure Ω\ns t : Set Ω\nhcs : μ s ≠ 0\nhs : μ s ≠ ⊤\n⊢ μ[|s] univ = 1","state_after":"Ω : Type u_1\nΩ' : Type u_2\nα : Type u_3\nm : MeasurableSpace Ω\nm' : MeasurableSpace Ω'\nμ : Measure Ω\ns t : Set Ω\nhcs : μ s ≠ 0\nhs : μ s ≠ ⊤\n⊢ ((μ s)⁻¹ • μ.restrict s) univ = 1","tactic":"unfold ProbabilityTheory.cond","premises":[{"full_name":"ProbabilityTheory.cond","def_path":"Mathlib/Probability/ConditionalProbability.lean","def_pos":[71,4],"def_end_pos":[71,8]}]},{"state_before":"Ω : Type u_1\nΩ' : Type u_2\nα : Type u_3\nm : MeasurableSpace Ω\nm' : MeasurableSpace Ω'\nμ : Measure Ω\ns t : Set Ω\nhcs : μ s ≠ 0\nhs : μ s ≠ ⊤\n⊢ ((μ s)⁻¹ • μ.restrict s) univ = 1","state_after":"Ω : Type u_1\nΩ' : Type u_2\nα : Type u_3\nm : MeasurableSpace Ω\nm' : MeasurableSpace Ω'\nμ : Measure Ω\ns t : Set Ω\nhcs : μ s ≠ 0\nhs : μ s ≠ ⊤\n⊢ (μ s)⁻¹ * μ s = 1","tactic":"simp only [Measure.coe_smul, Pi.smul_apply, MeasurableSet.univ, Measure.restrict_apply,\n      Set.univ_inter, smul_eq_mul]","premises":[{"full_name":"MeasurableSet.univ","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","def_pos":[95,18],"def_end_pos":[95,36]},{"full_name":"MeasureTheory.Measure.coe_smul","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","def_pos":[807,8],"def_end_pos":[807,16]},{"full_name":"MeasureTheory.Measure.restrict_apply","def_path":"Mathlib/MeasureTheory/Measure/Restrict.lean","def_pos":[66,8],"def_end_pos":[66,22]},{"full_name":"Pi.smul_apply","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[104,59],"def_end_pos":[104,69]},{"full_name":"Set.univ_inter","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[803,8],"def_end_pos":[803,18]},{"full_name":"smul_eq_mul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[79,6],"def_end_pos":[79,17]}]},{"state_before":"Ω : Type u_1\nΩ' : Type u_2\nα : Type u_3\nm : MeasurableSpace Ω\nm' : MeasurableSpace Ω'\nμ : Measure Ω\ns t : Set Ω\nhcs : μ s ≠ 0\nhs : μ s ≠ ⊤\n⊢ (μ s)⁻¹ * μ s = 1","state_after":"no goals","tactic":"exact ENNReal.inv_mul_cancel hcs hs","premises":[{"full_name":"ENNReal.inv_mul_cancel","def_path":"Mathlib/Data/ENNReal/Inv.lean","def_pos":[91,18],"def_end_pos":[91,32]}]}]}
+{"url":"Mathlib/Analysis/SumOverResidueClass.lean","commit":"","full_name":"Finset.sum_indicator_mod","start":[21,0],"end":[25,32],"file_path":"Mathlib/Analysis/SumOverResidueClass.lean","tactics":[{"state_before":"R : Type u_1\ninst✝¹ : AddCommMonoid R\nm : ℕ\ninst✝ : NeZero m\nf : ℕ → R\n⊢ f = ∑ a : ZMod m, {n | ↑n = a}.indicator f","state_after":"case h\nR : Type u_1\ninst✝¹ : AddCommMonoid R\nm : ℕ\ninst✝ : NeZero m\nf : ℕ → R\nn : ℕ\n⊢ f n = (∑ a : ZMod m, {n | ↑n = a}.indicator f) n","tactic":"ext n","premises":[]},{"state_before":"case h\nR : Type u_1\ninst✝¹ : AddCommMonoid R\nm : ℕ\ninst✝ : NeZero m\nf : ℕ → R\nn : ℕ\n⊢ f n = (∑ a : ZMod m, {n | ↑n = a}.indicator f) n","state_after":"no goals","tactic":"simp only [Finset.sum_apply, Set.indicator_apply, Set.mem_setOf_eq, Finset.sum_ite_eq,\n    Finset.mem_univ, ↓reduceIte]","premises":[{"full_name":"Finset.mem_univ","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[71,8],"def_end_pos":[71,16]},{"full_name":"Finset.sum_apply","def_path":"Mathlib/Algebra/BigOperators/Pi.lean","def_pos":[32,2],"def_end_pos":[32,13]},{"full_name":"Finset.sum_ite_eq","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[1131,2],"def_end_pos":[1131,13]},{"full_name":"Set.indicator_apply","def_path":"Mathlib/Algebra/Group/Indicator.lean","def_pos":[55,2],"def_end_pos":[55,13]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]},{"full_name":"reduceIte","def_path":".lake/packages/lean4/src/lean/Lean/Meta/Tactic/Simp/BuiltinSimprocs/Core.lean","def_pos":[12,32],"def_end_pos":[12,41]}]}]}
+{"url":"Mathlib/Analysis/Normed/Lp/lpSpace.lean","commit":"","full_name":"lp.norm_apply_le_norm","start":[504,0],"end":[511,60],"file_path":"Mathlib/Analysis/Normed/Lp/lpSpace.lean","tactics":[{"state_before":"α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nhp : p ≠ 0\nf : ↥(lp E p)\ni : α\n⊢ ‖↑f i‖ ≤ ‖f‖","state_after":"case inl\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\ni : α\nhp : ⊤ ≠ 0\nf : ↥(lp E ⊤)\n⊢ ‖↑f i‖ ≤ ‖f‖\n\ncase inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nhp : p ≠ 0\nf : ↥(lp E p)\ni : α\nhp' : p ≠ ⊤\n⊢ ‖↑f i‖ ≤ ‖f‖","tactic":"rcases eq_or_ne p ∞ with (rfl | hp')","premises":[{"full_name":"ENNReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[96,4],"def_end_pos":[96,11]},{"full_name":"Top.top","def_path":"Mathlib/Order/Notation.lean","def_pos":[94,2],"def_end_pos":[94,5]},{"full_name":"eq_or_ne","def_path":"Mathlib/Logic/Basic.lean","def_pos":[167,8],"def_end_pos":[167,16]}]},{"state_before":"case inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nhp : p ≠ 0\nf : ↥(lp E p)\ni : α\nhp' : p ≠ ⊤\n⊢ ‖↑f i‖ ≤ ‖f‖","state_after":"case inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nhp : p ≠ 0\nf : ↥(lp E p)\ni : α\nhp' : p ≠ ⊤\nhp'' : 0 < p.toReal\n⊢ ‖↑f i‖ ≤ ‖f‖","tactic":"have hp'' : 0 < p.toReal := ENNReal.toReal_pos hp hp'","premises":[{"full_name":"ENNReal.toReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[184,14],"def_end_pos":[184,20]},{"full_name":"ENNReal.toReal_pos","def_path":"Mathlib/Data/ENNReal/Real.lean","def_pos":[155,8],"def_end_pos":[155,18]}]},{"state_before":"case inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nhp : p ≠ 0\nf : ↥(lp E p)\ni : α\nhp' : p ≠ ⊤\nhp'' : 0 < p.toReal\n⊢ ‖↑f i‖ ≤ ‖f‖","state_after":"case inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nhp : p ≠ 0\nf : ↥(lp E p)\ni : α\nhp' : p ≠ ⊤\nhp'' : 0 < p.toReal\nthis : ∀ (i : α), 0 ≤ ‖↑f i‖ ^ p.toReal\n⊢ ‖↑f i‖ ≤ ‖f‖","tactic":"have : ∀ i, 0 ≤ ‖f i‖ ^ p.toReal := fun i => Real.rpow_nonneg (norm_nonneg _) _","premises":[{"full_name":"ENNReal.toReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[184,14],"def_end_pos":[184,20]},{"full_name":"Norm.norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[59,2],"def_end_pos":[59,6]},{"full_name":"Real.rpow_nonneg","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","def_pos":[145,8],"def_end_pos":[145,19]},{"full_name":"norm_nonneg","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[401,29],"def_end_pos":[401,40]}]},{"state_before":"case inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nhp : p ≠ 0\nf : ↥(lp E p)\ni : α\nhp' : p ≠ ⊤\nhp'' : 0 < p.toReal\nthis : ∀ (i : α), 0 ≤ ‖↑f i‖ ^ p.toReal\n⊢ ‖↑f i‖ ≤ ‖f‖","state_after":"case inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nhp : p ≠ 0\nf : ↥(lp E p)\ni : α\nhp' : p ≠ ⊤\nhp'' : 0 < p.toReal\nthis : ∀ (i : α), 0 ≤ ‖↑f i‖ ^ p.toReal\n⊢ ‖↑f i‖ ^ p.toReal ≤ ‖f‖ ^ p.toReal","tactic":"rw [← Real.rpow_le_rpow_iff (norm_nonneg _) (norm_nonneg' _) hp'']","premises":[{"full_name":"Real.rpow_le_rpow_iff","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","def_pos":[516,8],"def_end_pos":[516,24]},{"full_name":"lp.norm_nonneg'","def_path":"Mathlib/Analysis/Normed/Lp/lpSpace.lean","def_pos":[388,8],"def_end_pos":[388,20]},{"full_name":"norm_nonneg","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[401,29],"def_end_pos":[401,40]}]},{"state_before":"case inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nhp : p ≠ 0\nf : ↥(lp E p)\ni : α\nhp' : p ≠ ⊤\nhp'' : 0 < p.toReal\nthis : ∀ (i : α), 0 ≤ ‖↑f i‖ ^ p.toReal\n⊢ ‖↑f i‖ ^ p.toReal ≤ ‖f‖ ^ p.toReal","state_after":"no goals","tactic":"convert le_hasSum (hasSum_norm hp'' f) i fun i _ => this i","premises":[{"full_name":"le_hasSum","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Order.lean","def_pos":[97,2],"def_end_pos":[97,13]},{"full_name":"lp.hasSum_norm","def_path":"Mathlib/Analysis/Normed/Lp/lpSpace.lean","def_pos":[383,8],"def_end_pos":[383,19]}]}]}
+{"url":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean","commit":"","full_name":"Real.Angle.cos_sub_pi_div_two","start":[393,0],"end":[395,33],"file_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean","tactics":[{"state_before":"θ : Angle\n⊢ (θ - ↑(π / 2)).cos = θ.sin","state_after":"case h\nx✝ : ℝ\n⊢ (↑x✝ - ↑(π / 2)).cos = (↑x✝).sin","tactic":"induction θ using Real.Angle.induction_on","premises":[{"full_name":"Real.Angle.induction_on","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean","def_pos":[68,18],"def_end_pos":[68,30]}]},{"state_before":"case h\nx✝ : ℝ\n⊢ (↑x✝ - ↑(π / 2)).cos = (↑x✝).sin","state_after":"no goals","tactic":"exact Real.cos_sub_pi_div_two _","premises":[{"full_name":"Real.cos_sub_pi_div_two","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","def_pos":[434,8],"def_end_pos":[434,26]}]}]}
+{"url":"Mathlib/LinearAlgebra/Quotient.lean","commit":"","full_name":"Submodule.Quotient.mk_eq_zero","start":[88,0],"end":[89,97],"file_path":"Mathlib/LinearAlgebra/Quotient.lean","tactics":[{"state_before":"R : Type u_1\nM : Type u_2\nr : R\nx y : M\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np p' : Submodule R M\n⊢ mk x = 0 ↔ x ∈ p","state_after":"no goals","tactic":"simpa using (Quotient.eq' p : mk x = 0 ↔ _)","premises":[{"full_name":"Iff","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[114,10],"def_end_pos":[114,13]},{"full_name":"Submodule.Quotient.eq'","def_path":"Mathlib/LinearAlgebra/Quotient.lean","def_pos":[69,18],"def_end_pos":[69,21]},{"full_name":"Submodule.Quotient.mk","def_path":"Mathlib/LinearAlgebra/Quotient.lean","def_pos":[50,4],"def_end_pos":[50,6]}]}]}
+{"url":"Mathlib/Algebra/DirectSum/Ring.lean","commit":"","full_name":"DirectSum.mul_eq_sum_support_ghas_mul","start":[305,0],"end":[310,67],"file_path":"Mathlib/Algebra/DirectSum/Ring.lean","tactics":[{"state_before":"ι : Type u_1\ninst✝⁴ : DecidableEq ι\nA : ι → Type u_2\ninst✝³ : (i : ι) → AddCommMonoid (A i)\ninst✝² : AddMonoid ι\ninst✝¹ : GSemiring A\ninst✝ : (i : ι) → (x : A i) → Decidable (x ≠ 0)\na a' : ⨁ (i : ι), A i\n⊢ a * a' =\n    ∑ ij ∈ DFinsupp.support a ×ˢ DFinsupp.support a', (of A (ij.1 + ij.2)) (GradedMonoid.GMul.mul (a ij.1) (a' ij.2))","state_after":"no goals","tactic":"simp only [mul_eq_dfinsupp_sum, DFinsupp.sum, Finset.sum_product]","premises":[{"full_name":"DFinsupp.sum","def_path":"Mathlib/Data/DFinsupp/Basic.lean","def_pos":[1497,2],"def_end_pos":[1497,13]},{"full_name":"DirectSum.mul_eq_dfinsupp_sum","def_path":"Mathlib/Algebra/DirectSum/Ring.lean","def_pos":[287,8],"def_end_pos":[287,27]},{"full_name":"Finset.sum_product","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[777,2],"def_end_pos":[777,13]}]}]}
+{"url":"Mathlib/Combinatorics/Additive/ETransform.lean","commit":"","full_name":"Finset.mulETransformLeft_inv","start":[158,0],"end":[160,72],"file_path":"Mathlib/Combinatorics/Additive/ETransform.lean","tactics":[{"state_before":"α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommGroup α\ne : α\nx : Finset α × Finset α\n⊢ mulETransformLeft e⁻¹ x = (mulETransformRight e x.swap).swap","state_after":"no goals","tactic":"simp [-op_inv, op_smul_eq_smul, mulETransformLeft, mulETransformRight]","premises":[{"full_name":"Finset.mulETransformLeft","def_path":"Mathlib/Combinatorics/Additive/ETransform.lean","def_pos":[105,4],"def_end_pos":[105,21]},{"full_name":"Finset.mulETransformRight","def_path":"Mathlib/Combinatorics/Additive/ETransform.lean","def_pos":[112,4],"def_end_pos":[112,22]},{"full_name":"IsCentralScalar.op_smul_eq_smul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[238,2],"def_end_pos":[238,17]},{"full_name":"MulOpposite.op_inv","def_path":"Mathlib/Algebra/Opposites.lean","def_pos":[196,36],"def_end_pos":[196,42]}]}]}
+{"url":".lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean","commit":"","full_name":"Rat.mul_one","start":[283,0],"end":[283,92],"file_path":".lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean","tactics":[{"state_before":"a : Rat\n⊢ a * 1 = a","state_after":"no goals","tactic":"simp [mul_def, normalize_self]","premises":[{"full_name":"Rat.mul_def","def_path":".lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean","def_pos":[264,8],"def_end_pos":[264,15]},{"full_name":"Rat.normalize_self","def_path":".lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean","def_pos":[50,8],"def_end_pos":[50,22]}]}]}
+{"url":"Mathlib/Data/List/Basic.lean","commit":"","full_name":"List.foldl_assoc","start":[1602,0],"end":[1607,65],"file_path":"Mathlib/Data/List/Basic.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\nop : α → α → α\nha : Std.Associative op\nhc : Std.Commutative op\na : α\nl : List α\na₁ a₂ : α\n⊢ ((a :: l) <*> op a₁ a₂) = l <*> op a₁ (op a₂ a)","state_after":"no goals","tactic":"simp only [foldl_cons, ha.assoc]","premises":[{"full_name":"List.foldl_cons","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[87,16],"def_end_pos":[87,26]},{"full_name":"Std.Associative.assoc","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1968,2],"def_end_pos":[1968,7]}]},{"state_before":"ι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\nop : α → α → α\nha : Std.Associative op\nhc : Std.Commutative op\na : α\nl : List α\na₁ a₂ : α\n⊢ (l <*> op a₁ (op a₂ a)) = op a₁ ((a :: l) <*> a₂)","state_after":"no goals","tactic":"rw [foldl_assoc, foldl_cons]","premises":[{"full_name":"List.foldl_cons","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[87,16],"def_end_pos":[87,26]}]}]}
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rfl⟩","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Submodule.mem_span_singleton","def_path":"Mathlib/LinearAlgebra/Span.lean","def_pos":[428,8],"def_end_pos":[428,26]}]},{"state_before":"case intro\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nn : ℕ\ns : Simplex ℝ P (n + 2)\ni₁ i₂ : Fin (n + 3)\nr : ℝ\nhv : r • (s.points i₁ -ᵥ s.points i₂) ∈ Submodule.span ℝ {s.points i₁ -ᵥ s.points i₂}\n⊢ ⟪s.mongePoint -ᵥ centroid ℝ {i₁, i₂}ᶜ s.points, r • (s.points i₁ -ᵥ s.points i₂)⟫_ℝ = 0","state_after":"no goals","tactic":"rw [inner_smul_right, s.inner_mongePoint_vsub_face_centroid_vsub, mul_zero]","premises":[{"full_name":"Affine.Simplex.inner_mongePoint_vsub_face_centroid_vsub","def_path":"Mathlib/Geometry/Euclidean/MongePoint.lean","def_pos":[197,8],"def_end_pos":[197,48]},{"full_name":"MulZeroClass.mul_zero","def_path":"Mathlib/Algebra/GroupWithZero/Defs.lean","def_pos":[37,2],"def_end_pos":[37,10]},{"full_name":"inner_smul_right","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[422,8],"def_end_pos":[422,24]}]}]}
+{"url":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","commit":"","full_name":"intervalIntegral.integral_Iic_sub_Iic","start":[860,0],"end":[866,64],"file_path":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","tactics":[{"state_before":"ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na b c d : ℝ\nf g : ℝ → E\nμ : Measure ℝ\nha : IntegrableOn f (Iic a) μ\nhb : IntegrableOn f (Iic b) μ\n⊢ ∫ (x : ℝ) in Iic b, f x ∂μ - ∫ (x : ℝ) in Iic a, f x ∂μ = ∫ (x : ℝ) in a..b, f x ∂μ","state_after":"case inr\nι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na b c d : ℝ\nf g : ℝ → E\nμ : Measure ℝ\nha : IntegrableOn f (Iic a) μ\nhb : IntegrableOn f (Iic b) μ\nthis :\n  ∀ {a b : ℝ},\n    IntegrableOn f (Iic a) μ →\n      IntegrableOn f (Iic b) μ →\n        a ≤ b → ∫ (x : ℝ) in Iic b, f x ∂μ - ∫ (x : ℝ) in Iic a, f x ∂μ = ∫ (x : ℝ) in a..b, f x ∂μ\nhab : ¬a ≤ b\n⊢ ∫ (x : ℝ) in Iic b, f x ∂μ - ∫ (x : ℝ) in Iic a, f x ∂μ = ∫ (x : ℝ) in a..b, f x ∂μ\n\nι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b✝ c d : ℝ\nf g : ℝ → E\nμ : Measure ℝ\na b : ℝ\nha : IntegrableOn f (Iic a) μ\nhb : IntegrableOn f (Iic b) μ\nhab : a ≤ b\n⊢ ∫ (x : ℝ) in Iic b, f x ∂μ - ∫ (x : ℝ) in Iic a, f x ∂μ = ∫ (x : ℝ) in a..b, f x ∂μ","tactic":"wlog hab : a ≤ b generalizing a b","premises":[]},{"state_before":"ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b✝ c d : ℝ\nf g : ℝ → E\nμ : Measure ℝ\na b : ℝ\nha : IntegrableOn f (Iic a) μ\nhb : IntegrableOn f (Iic b) μ\nhab : a ≤ b\n⊢ ∫ (x : ℝ) in Iic b, f x ∂μ - ∫ (x : ℝ) in Iic a, f x ∂μ = ∫ (x : ℝ) in a..b, f x ∂μ","state_after":"case ht\nι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b✝ c d : ℝ\nf g : ℝ → E\nμ : Measure ℝ\na b : ℝ\nha : IntegrableOn f (Iic a) μ\nhb : IntegrableOn f (Iic b) μ\nhab : a ≤ b\n⊢ MeasurableSet (Ioc a b)\n\ncase hfs\nι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b✝ c d : ℝ\nf g : ℝ → E\nμ : Measure ℝ\na b : ℝ\nha : IntegrableOn f (Iic a) μ\nhb : IntegrableOn f (Iic b) μ\nhab : a ≤ b\n⊢ IntegrableOn f (Iic a) μ\n\ncase hft\nι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b✝ c d : ℝ\nf g : ℝ → E\nμ : Measure ℝ\na b : ℝ\nha : IntegrableOn f (Iic a) μ\nhb : IntegrableOn f (Iic b) μ\nhab : a ≤ b\n⊢ IntegrableOn f (Ioc a b) μ","tactic":"rw [sub_eq_iff_eq_add', integral_of_le hab, ← integral_union (Iic_disjoint_Ioc le_rfl),\n    Iic_union_Ioc_eq_Iic hab]","premises":[{"full_name":"MeasureTheory.integral_union","def_path":"Mathlib/MeasureTheory/Integral/SetIntegral.lean","def_pos":[108,8],"def_end_pos":[108,22]},{"full_name":"Set.Iic_disjoint_Ioc","def_path":"Mathlib/Order/Interval/Set/Disjoint.lean","def_pos":[42,8],"def_end_pos":[42,24]},{"full_name":"Set.Iic_union_Ioc_eq_Iic","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[1167,8],"def_end_pos":[1167,28]},{"full_name":"intervalIntegral.integral_of_le","def_path":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","def_pos":[430,8],"def_end_pos":[430,22]},{"full_name":"le_rfl","def_path":"Mathlib/Order/Defs.lean","def_pos":[43,8],"def_end_pos":[43,14]},{"full_name":"sub_eq_iff_eq_add'","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[920,2],"def_end_pos":[920,13]}]},{"state_before":"case ht\nι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b✝ c d : ℝ\nf g : ℝ → E\nμ : Measure ℝ\na b : ℝ\nha : IntegrableOn f (Iic a) μ\nhb : IntegrableOn f (Iic b) μ\nhab : a ≤ b\n⊢ MeasurableSet (Ioc a b)\n\ncase hfs\nι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b✝ c d : ℝ\nf g : ℝ → E\nμ : Measure ℝ\na b : ℝ\nha : IntegrableOn f (Iic a) μ\nhb : IntegrableOn f (Iic b) μ\nhab : a ≤ b\n⊢ IntegrableOn f (Iic a) μ\n\ncase hft\nι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b✝ c d : ℝ\nf g : ℝ → E\nμ : Measure ℝ\na b : ℝ\nha : IntegrableOn f (Iic a) μ\nhb : IntegrableOn f (Iic b) μ\nhab : a ≤ b\n⊢ IntegrableOn f (Ioc a b) μ","state_after":"no goals","tactic":"exacts [measurableSet_Ioc, ha, hb.mono_set fun _ => And.right]","premises":[{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"MeasureTheory.IntegrableOn.mono_set","def_path":"Mathlib/MeasureTheory/Integral/IntegrableOn.lean","def_pos":[103,8],"def_end_pos":[103,29]},{"full_name":"measurableSet_Ioc","def_path":"Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean","def_pos":[178,8],"def_end_pos":[178,25]}]}]}
+{"url":"Mathlib/MeasureTheory/Function/Intersectivity.lean","commit":"","full_name":"bergelson'","start":[36,0],"end":[117,55],"file_path":"Mathlib/MeasureTheory/Function/Intersectivity.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\n⊢ ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)","state_after":"ι : Type u_1\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}\n⊢ ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)","tactic":"let M (f : α → ℝ) : Set α := {x | eLpNormEssSup f μ < ‖f x‖₊}","premises":[{"full_name":"MeasureTheory.eLpNormEssSup","def_path":"Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean","def_pos":[73,4],"def_end_pos":[73,17]},{"full_name":"NNNorm.nnnorm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[65,2],"def_end_pos":[65,8]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Set","def_path":"Mathlib/Init/Set.lean","def_pos":[53,4],"def_end_pos":[53,7]},{"full_name":"setOf","def_path":"Mathlib/Init/Set.lean","def_pos":[56,4],"def_end_pos":[56,9]}]},{"state_before":"ι : Type u_1\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}\n⊢ ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)","state_after":"ι : Type u_1\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)\n⊢ ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)","tactic":"let N : Set α := ⋃ u : Finset ℕ, M (Set.indicator (⋂ n ∈ u, s n) 1)","premises":[{"full_name":"Finset","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[133,10],"def_end_pos":[133,16]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]},{"full_name":"Set","def_path":"Mathlib/Init/Set.lean","def_pos":[53,4],"def_end_pos":[53,7]},{"full_name":"Set.iInter","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[182,4],"def_end_pos":[182,10]},{"full_name":"Set.iUnion","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[178,4],"def_end_pos":[178,10]},{"full_name":"Set.indicator","def_path":"Mathlib/Algebra/Group/Indicator.lean","def_pos":[45,2],"def_end_pos":[45,13]}]},{"state_before":"ι : Type u_1\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)\n⊢ ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)","state_after":"ι : Type u_1\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)\nhN₀ : μ N = 0\n⊢ ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)","tactic":"have hN₀ : μ N = 0 := measure_iUnion_null fun u ↦ meas_eLpNormEssSup_lt","premises":[{"full_name":"MeasureTheory.meas_eLpNormEssSup_lt","def_path":"Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean","def_pos":[966,8],"def_end_pos":[966,29]}]},{"state_before":"ι : Type u_1\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)\nhN₀ : μ N = 0\nhN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \\ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)\n⊢ ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)","state_after":"ι : Type u_1\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)\nhN₀ : μ N = 0\nhN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \\ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)\nf : ℕ → α → ℝ≥0∞ := fun n => (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1\n⊢ ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)","tactic":"let f (n : ℕ) : α → ℝ≥0∞ := (↑(n + 1) : ℝ≥0∞)⁻¹ • ∑ k in Finset.range (n + 1), (s k).indicator 1","premises":[{"full_name":"ENNReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[96,4],"def_end_pos":[96,11]},{"full_name":"Finset.range","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2442,4],"def_end_pos":[2442,9]},{"full_name":"Finset.sum","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[53,2],"def_end_pos":[53,13]},{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]},{"full_name":"Set.indicator","def_path":"Mathlib/Algebra/Group/Indicator.lean","def_pos":[45,2],"def_end_pos":[45,13]}]},{"state_before":"ι : Type u_1\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)\nhN₀ : μ N = 0\nhN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \\ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)\nf : ℕ → α → ℝ≥0∞ := fun n => (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1\n⊢ ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)","state_after":"ι : Type u_1\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)\nhN₀ : μ N = 0\nhN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \\ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)\nf : ℕ → α → ℝ≥0∞ := fun n => (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1\nhfapp : ∀ (n : ℕ) (a : α), f n a = (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a\n⊢ ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)","tactic":"have hfapp : ∀ n a, f n a = (↑(n + 1))⁻¹ * ∑ k in Finset.range (n + 1), (s k).indicator 1 a := by\n    simp only [f, Pi.natCast_def, Pi.smul_apply, Pi.inv_apply, Finset.sum_apply, eq_self_iff_true,\n    forall_const, imp_true_iff, smul_eq_mul]","premises":[{"full_name":"Finset.range","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2442,4],"def_end_pos":[2442,9]},{"full_name":"Finset.sum","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[53,2],"def_end_pos":[53,13]},{"full_name":"Finset.sum_apply","def_path":"Mathlib/Algebra/BigOperators/Pi.lean","def_pos":[32,2],"def_end_pos":[32,13]},{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"Pi.inv_apply","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[126,8],"def_end_pos":[126,17]},{"full_name":"Pi.natCast_def","def_path":"Mathlib/Data/Nat/Cast/Basic.lean","def_pos":[286,8],"def_end_pos":[286,19]},{"full_name":"Pi.smul_apply","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[104,59],"def_end_pos":[104,69]},{"full_name":"Set.indicator","def_path":"Mathlib/Algebra/Group/Indicator.lean","def_pos":[45,2],"def_end_pos":[45,13]},{"full_name":"eq_self_iff_true","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1380,8],"def_end_pos":[1380,24]},{"full_name":"forall_const","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[265,16],"def_end_pos":[265,28]},{"full_name":"imp_true_iff","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1412,8],"def_end_pos":[1412,20]},{"full_name":"smul_eq_mul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[79,6],"def_end_pos":[79,17]}]},{"state_before":"ι : Type u_1\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)\nhN₀ : μ N = 0\nhN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \\ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)\nf : ℕ → α → ℝ≥0∞ := fun n => (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1\nhfapp : ∀ (n : ℕ) (a : α), f n a = (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a\n⊢ ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)","state_after":"ι : Type u_1\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)\nhN₀ : μ N = 0\nhN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \\ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)\nf : ℕ → α → ℝ≥0∞ := fun n => (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1\nhfapp : ∀ (n : ℕ) (a : α), f n a = (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a\nhf : ∀ (n : ℕ), Measurable (f n)\n⊢ ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)","tactic":"have hf n : Measurable (f n) := Measurable.mul' (@measurable_const ℝ≥0∞ _ _ _ (↑(n + 1))⁻¹)\n      (Finset.measurable_sum' _ fun i _ ↦ measurable_one.indicator $ hs i)","premises":[{"full_name":"ENNReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[96,4],"def_end_pos":[96,11]},{"full_name":"Finset.measurable_sum'","def_path":"Mathlib/MeasureTheory/Group/Arithmetic.lean","def_pos":[787,2],"def_end_pos":[787,13]},{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"Measurable","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","def_pos":[479,4],"def_end_pos":[479,14]},{"full_name":"Measurable.indicator","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Basic.lean","def_pos":[288,8],"def_end_pos":[288,28]},{"full_name":"Measurable.mul'","def_path":"Mathlib/MeasureTheory/Group/Arithmetic.lean","def_pos":[114,8],"def_end_pos":[114,23]},{"full_name":"measurable_const","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","def_pos":[510,8],"def_end_pos":[510,24]},{"full_name":"measurable_one","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Basic.lean","def_pos":[220,8],"def_end_pos":[220,22]}]},{"state_before":"ι : Type u_1\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)\nhN₀ : μ N = 0\nhN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \\ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)\nf : ℕ → α → ℝ≥0∞ := fun n => (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1\nhfapp : ∀ (n : ℕ) (a : α), f n a = (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a\nhf : ∀ (n : ℕ), Measurable (f n)\n⊢ ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)","state_after":"ι : Type u_1\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)\nhN₀ : μ N = 0\nhN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \\ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)\nf : ℕ → α → ℝ≥0∞ := fun n => (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1\nhfapp : ∀ (n : ℕ) (a : α), f n a = (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a\nhf : ∀ (n : ℕ), Measurable (f n)\nhf₁ : ∀ (n : ℕ), f n ≤ 1\n⊢ ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)","tactic":"have hf₁ n : f n ≤ 1 := by\n    rintro a\n    rw [hfapp, ← ENNReal.div_eq_inv_mul]\n    refine (ENNReal.div_le_iff_le_mul (Or.inl $ Nat.cast_ne_zero.2 n.succ_ne_zero) $\n      Or.inr one_ne_zero).2 ?_\n    rw [mul_comm, ← nsmul_eq_mul, ← Finset.card_range n.succ]\n    exact Finset.sum_le_card_nsmul _ _ _ fun _ _ ↦ indicator_le (fun _ _ ↦ le_rfl) _","premises":[{"full_name":"ENNReal.div_eq_inv_mul","def_path":"Mathlib/Data/ENNReal/Inv.lean","def_pos":[41,18],"def_end_pos":[41,32]},{"full_name":"ENNReal.div_le_iff_le_mul","def_path":"Mathlib/Data/ENNReal/Inv.lean","def_pos":[270,18],"def_end_pos":[270,35]},{"full_name":"Finset.card_range","def_path":"Mathlib/Data/Finset/Card.lean","def_pos":[179,8],"def_end_pos":[179,18]},{"full_name":"Finset.sum_le_card_nsmul","def_path":"Mathlib/Algebra/Order/BigOperators/Group/Finset.lean","def_pos":[187,14],"def_end_pos":[187,31]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Nat.cast_ne_zero","def_path":"Mathlib/Algebra/CharZero/Defs.lean","def_pos":[76,8],"def_end_pos":[76,20]},{"full_name":"Nat.succ","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1083,4],"def_end_pos":[1083,8]},{"full_name":"Nat.succ_ne_zero","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[708,16],"def_end_pos":[708,28]},{"full_name":"Or.inl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[534,4],"def_end_pos":[534,7]},{"full_name":"Or.inr","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[536,4],"def_end_pos":[536,7]},{"full_name":"Set.indicator_le","def_path":"Mathlib/Algebra/Order/Group/Indicator.lean","def_pos":[187,2],"def_end_pos":[187,13]},{"full_name":"le_rfl","def_path":"Mathlib/Order/Defs.lean","def_pos":[43,8],"def_end_pos":[43,14]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]},{"full_name":"nsmul_eq_mul","def_path":"Mathlib/Data/Nat/Cast/Basic.lean","def_pos":[71,14],"def_end_pos":[71,33]},{"full_name":"one_ne_zero","def_path":"Mathlib/Algebra/NeZero.lean","def_pos":[49,14],"def_end_pos":[49,25]}]},{"state_before":"ι : Type u_1\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)\nhN₀ : μ N = 0\nhN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \\ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)\nf : ℕ → α → ℝ≥0∞ := fun n => (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1\nhfapp : ∀ (n : ℕ) (a : α), f n a = (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a\nhf : ∀ (n : ℕ), Measurable (f n)\nhf₁ : ∀ (n : ℕ), f n ≤ 1\n⊢ ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)","state_after":"ι : Type u_1\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)\nhN₀ : μ N = 0\nhN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \\ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)\nf : ℕ → α → ℝ≥0∞ := fun n => (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1\nhfapp : ∀ (n : ℕ) (a : α), f n a = (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a\nhf : ∀ (n : ℕ), Measurable (f n)\nhf₁ : ∀ (n : ℕ), f n ≤ 1\nhrf : ∀ (n : ℕ), r ≤ ∫⁻ (a : α), f n a ∂μ\n⊢ ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)","tactic":"have hrf n : r ≤ ∫⁻ a, f n a ∂μ := by\n    simp_rw [hfapp]\n    rw [lintegral_const_mul _ (Finset.measurable_sum _ fun _ _ ↦ measurable_one.indicator $ hs _),\n      lintegral_finset_sum _ fun _ _ ↦ measurable_one.indicator (hs _)]\n    simp only [lintegral_indicator_one (hs _)]\n    rw [← ENNReal.div_eq_inv_mul, ENNReal.le_div_iff_mul_le (by simp) (by simp), ← nsmul_eq_mul']\n    simpa using Finset.card_nsmul_le_sum (Finset.range (n + 1)) _ _ fun _ _ ↦ hr _","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"ENNReal.div_eq_inv_mul","def_path":"Mathlib/Data/ENNReal/Inv.lean","def_pos":[41,18],"def_end_pos":[41,32]},{"full_name":"ENNReal.le_div_iff_mul_le","def_path":"Mathlib/Data/ENNReal/Inv.lean","def_pos":[258,18],"def_end_pos":[258,35]},{"full_name":"Finset.card_nsmul_le_sum","def_path":"Mathlib/Algebra/Order/BigOperators/Group/Finset.lean","def_pos":[194,14],"def_end_pos":[194,31]},{"full_name":"Finset.measurable_sum","def_path":"Mathlib/MeasureTheory/Group/Arithmetic.lean","def_pos":[792,2],"def_end_pos":[792,13]},{"full_name":"Finset.range","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2442,4],"def_end_pos":[2442,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Measurable.indicator","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Basic.lean","def_pos":[288,8],"def_end_pos":[288,28]},{"full_name":"MeasureTheory.lintegral","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[59,16],"def_end_pos":[59,25]},{"full_name":"MeasureTheory.lintegral_const_mul","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[667,8],"def_end_pos":[667,27]},{"full_name":"MeasureTheory.lintegral_finset_sum","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[662,8],"def_end_pos":[662,28]},{"full_name":"MeasureTheory.lintegral_indicator_one","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[819,8],"def_end_pos":[819,31]},{"full_name":"measurable_one","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Basic.lean","def_pos":[220,8],"def_end_pos":[220,22]},{"full_name":"nsmul_eq_mul'","def_path":"Mathlib/Data/Nat/Cast/Basic.lean","def_pos":[66,6],"def_end_pos":[66,26]}]},{"state_before":"ι : Type u_1\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)\nhN₀ : μ N = 0\nhN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \\ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)\nf : ℕ → α → ℝ≥0∞ := fun n => (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1\nhfapp : ∀ (n : ℕ) (a : α), f n a = (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a\nhf : ∀ (n : ℕ), Measurable (f n)\nhf₁ : ∀ (n : ℕ), f n ≤ 1\nhrf : ∀ (n : ℕ), r ≤ ∫⁻ (a : α), f n a ∂μ\n⊢ ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)","state_after":"ι : Type u_1\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)\nhN₀ : μ N = 0\nhN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \\ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)\nf : ℕ → α → ℝ≥0∞ := fun n => (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1\nhfapp : ∀ (n : ℕ) (a : α), f n a = (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a\nhf : ∀ (n : ℕ), Measurable (f n)\nhf₁ : ∀ (n : ℕ), f n ≤ 1\nhrf : ∀ (n : ℕ), r ≤ ∫⁻ (a : α), f n a ∂μ\nhμ : μ ≠ 0\n⊢ ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)","tactic":"have hμ : μ ≠ 0 := by rintro rfl; exact hr₀ $ le_bot_iff.1 $ hr 0","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"le_bot_iff","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[276,8],"def_end_pos":[276,18]}]},{"state_before":"ι : Type u_1\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)\nhN₀ : μ N = 0\nhN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \\ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)\nf : ℕ → α → ℝ≥0∞ := fun n => (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1\nhfapp : ∀ (n : ℕ) (a : α), f n a = (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a\nhf : ∀ (n : ℕ), Measurable (f n)\nhf₁ : ∀ (n : ℕ), f n ≤ 1\nhrf : ∀ (n : ℕ), r ≤ ∫⁻ (a : α), f n a ∂μ\nhμ : μ ≠ 0\nthis : ∫⁻ (x : α), limsup (fun x_1 => f x_1 x) atTop ∂μ ≤ μ univ\n⊢ ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)","state_after":"case intro.intro\nι : Type u_1\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)\nhN₀ : μ N = 0\nhN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \\ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)\nf : ℕ → α → ℝ≥0∞ := fun n => (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1\nhfapp : ∀ (n : ℕ) (a : α), f n a = (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a\nhf : ∀ (n : ℕ), Measurable (f n)\nhf₁ : ∀ (n : ℕ), f n ≤ 1\nhrf : ∀ (n : ℕ), r ≤ ∫⁻ (a : α), f n a ∂μ\nhμ : μ ≠ 0\nthis : ∫⁻ (x : α), limsup (fun x_1 => f x_1 x) atTop ∂μ ≤ μ univ\nx : α\nhxN : x ∉ N\nhx : ⨍⁻ (a : α), limsup (fun x => f x a) atTop ∂μ ≤ limsup (fun x_1 => f x_1 x) atTop\n⊢ ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)","tactic":"obtain ⟨x, hxN, hx⟩ := exists_not_mem_null_laverage_le hμ\n    (ne_top_of_le_ne_top (measure_ne_top μ univ) this) hN₀","premises":[{"full_name":"MeasureTheory.exists_not_mem_null_laverage_le","def_path":"Mathlib/MeasureTheory/Integral/Average.lean","def_pos":[648,8],"def_end_pos":[648,39]},{"full_name":"MeasureTheory.measure_ne_top","def_path":"Mathlib/MeasureTheory/Measure/Typeclasses.lean","def_pos":[55,8],"def_end_pos":[55,22]},{"full_name":"Set.univ","def_path":"Mathlib/Init/Set.lean","def_pos":[157,4],"def_end_pos":[157,8]},{"full_name":"ne_top_of_le_ne_top","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[138,8],"def_end_pos":[138,27]}]}]}
+{"url":"Mathlib/GroupTheory/Perm/Cycle/Basic.lean","commit":"","full_name":"Equiv.Perm.sameCycle_pow_left","start":[132,0],"end":[134,42],"file_path":"Mathlib/GroupTheory/Perm/Cycle/Basic.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\nf g : Perm α\np : α → Prop\nx y z : α\nn : ℕ\n⊢ f.SameCycle ((f ^ n) x) y ↔ f.SameCycle x y","state_after":"no goals","tactic":"rw [← zpow_natCast, sameCycle_zpow_left]","premises":[{"full_name":"Equiv.Perm.sameCycle_zpow_left","def_path":"Mathlib/GroupTheory/Perm/Cycle/Basic.lean","def_pos":[125,8],"def_end_pos":[125,27]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"zpow_natCast","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[875,8],"def_end_pos":[875,20]}]}]}
+{"url":"Mathlib/LinearAlgebra/Span.lean","commit":"","full_name":"Submodule.span_int_eq","start":[259,0],"end":[261,99],"file_path":"Mathlib/LinearAlgebra/Span.lean","tactics":[{"state_before":"R : Type u_1\nR₂ : Type u_2\nK : Type u_3\nM✝ : Type u_4\nM₂ : Type u_5\nV : Type u_6\nS : Type u_7\ninst✝⁸ : Semiring R\ninst✝⁷ : AddCommMonoid M✝\ninst✝⁶ : Module R M✝\nx : M✝\np p' : Submodule R M✝\ninst✝⁵ : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : Module R₂ M₂\nF : Type u_8\ninst✝² : FunLike F M✝ M₂\ninst✝¹ : SemilinearMapClass F σ₁₂ M✝ M₂\ns✝ t : Set M✝\nM : Type u_9\ninst✝ : AddCommGroup M\ns : AddSubgroup M\n⊢ (span ℤ ↑s).toAddSubgroup = s","state_after":"no goals","tactic":"rw [span_int_eq_addSubgroup_closure, s.closure_eq]","premises":[{"full_name":"AddSubgroup.closure_eq","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[957,2],"def_end_pos":[957,13]},{"full_name":"Submodule.span_int_eq_addSubgroup_closure","def_path":"Mathlib/LinearAlgebra/Span.lean","def_pos":[252,8],"def_end_pos":[252,39]}]}]}
+{"url":"Mathlib/Order/Closure.lean","commit":"","full_name":"LowerAdjoint.mem_iff","start":[461,0],"end":[463,49],"file_path":"Mathlib/Order/Closure.lean","tactics":[{"state_before":"α : Type u_1\nι : Sort u_2\nκ : ι → Sort u_3\nβ : Type u_4\ninst✝ : SetLike α β\nl : LowerAdjoint SetLike.coe\ns : Set β\nx : β\n⊢ x ∈ l.toFun s ↔ ∀ (S : ��), s ⊆ ↑S → x ∈ S","state_after":"α : Type u_1\nι : Sort u_2\nκ : ι → Sort u_3\nβ : Type u_4\ninst✝ : SetLike α β\nl : LowerAdjoint SetLike.coe\ns : Set β\nx : β\n⊢ l.toFun {x} ≤ l.toFun s ↔ ∀ (S : α), l.toFun s ≤ S → l.toFun {x} ≤ S","tactic":"simp_rw [← SetLike.mem_coe, ← Set.singleton_subset_iff, ← l.le_iff_subset]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"LowerAdjoint.le_iff_subset","def_path":"Mathlib/Order/Closure.lean","def_pos":[458,8],"def_end_pos":[458,21]},{"full_name":"Set.singleton_subset_iff","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1047,8],"def_end_pos":[1047,28]},{"full_name":"SetLike.mem_coe","def_path":"Mathlib/Data/SetLike/Basic.lean","def_pos":[168,8],"def_end_pos":[168,15]}]},{"state_before":"α : Type u_1\nι : Sort u_2\nκ : ι → Sort u_3\nβ : Type u_4\ninst✝ : SetLike α β\nl : LowerAdjoint SetLike.coe\ns : Set β\nx : β\n⊢ l.toFun {x} ≤ l.toFun s ↔ ∀ (S : α), l.toFun s ≤ S → l.toFun {x} ≤ S","state_after":"no goals","tactic":"exact ⟨fun h S => h.trans, fun h => h _ le_rfl⟩","premises":[{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"le_rfl","def_path":"Mathlib/Order/Defs.lean","def_pos":[43,8],"def_end_pos":[43,14]}]}]}
+{"url":"Mathlib/CategoryTheory/Limits/Shapes/WidePullbacks.lean","commit":"","full_name":"CategoryTheory.Limits.WidePushout.arrow_ι","start":[364,0],"end":[366,81],"file_path":"Mathlib/CategoryTheory/Limits/Shapes/WidePullbacks.lean","tactics":[{"state_before":"J : Type w\nC✝ : Type u\ninst✝² : Category.{v, u} C✝\nC : Type u\ninst✝¹ : Category.{v, u} C\nB : C\nobjs : J → C\narrows : (j : J) → B ⟶ objs j\ninst✝ : HasWidePushout B objs arrows\nj : J\n⊢ arrows j ≫ ι arrows j = head arrows","state_after":"no goals","tactic":"apply colimit.w (WidePushoutShape.wideSpan _ _ _) (WidePushoutShape.Hom.init j)","premises":[{"full_name":"CategoryTheory.Limits.WidePushoutShape.Hom.init","def_path":"Mathlib/CategoryTheory/Limits/Shapes/WidePullbacks.lean","def_pos":[159,4],"def_end_pos":[159,8]},{"full_name":"CategoryTheory.Limits.WidePushoutShape.wideSpan","def_path":"Mathlib/CategoryTheory/Limits/Shapes/WidePullbacks.lean","def_pos":[211,4],"def_end_pos":[211,12]},{"full_name":"CategoryTheory.Limits.colimit.w","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[664,8],"def_end_pos":[664,17]}]}]}
+{"url":"Mathlib/SetTheory/Game/Basic.lean","commit":"","full_name":"SetTheory.PGame.leftMoves_mul_iff","start":[796,0],"end":[816,35],"file_path":"Mathlib/SetTheory/Game/Basic.lean","tactics":[{"state_before":"x y : PGame\nP : Game → Prop\n⊢ (∀ (k : (x * y).LeftMoves), P ⟦(x * y).moveLeft k⟧) ↔\n    (∀ (i : x.LeftMoves) (j : y.LeftMoves), P ⟦x.mulOption y i j⟧) ∧\n      ∀ (i : (-x).LeftMoves) (j : (-y).LeftMoves), P ⟦(-x).mulOption (-y) i j⟧","state_after":"case mk\ny : PGame\nP : Game → Prop\nα✝ β✝ : Type u_1\na✝¹ : α✝ → PGame\na✝ : β✝ → PGame\n⊢ (∀ (k : (mk α✝ β✝ a✝¹ a✝ * y).LeftMoves), P ⟦(mk α✝ β✝ a✝¹ a✝ * y).moveLeft k⟧) ↔\n    (∀ (i : (mk α✝ β✝ a✝¹ a✝).LeftMoves) (j : y.LeftMoves), P ⟦(mk α✝ β✝ a✝¹ a✝).mulOption y i j⟧) ∧\n      ∀ (i : (-mk α✝ β✝ a✝¹ a✝).LeftMoves) (j : (-y).LeftMoves), P ⟦(-mk α✝ β✝ a✝¹ a✝).mulOption (-y) i j⟧","tactic":"cases x","premises":[]},{"state_before":"case mk\ny : PGame\nP : Game → Prop\nα✝ β✝ : Type u_1\na✝¹ : α✝ → PGame\na✝ : β✝ → PGame\n⊢ (∀ (k : (mk α✝ β✝ a✝¹ a✝ * y).LeftMoves), P ⟦(mk α✝ β✝ a✝¹ a✝ * y).moveLeft k⟧) ↔\n    (∀ (i : (mk α✝ β✝ a✝¹ a✝).LeftMoves) (j : y.LeftMoves), P ⟦(mk α✝ β✝ a✝¹ a✝).mulOption y i j⟧) ∧\n      ∀ (i : (-mk α✝ β✝ a✝¹ a✝).LeftMoves) (j : (-y).LeftMoves), P ⟦(-mk α✝ β✝ a✝¹ a✝).mulOption (-y) i j⟧","state_after":"case mk.mk\nP : Game → Prop\nα✝¹ β✝¹ : Type u_1\na✝³ : α✝¹ → PGame\na✝² : β✝¹ → PGame\nα✝ β✝ : Type u_1\na✝¹ : α✝ → PGame\na✝ : β✝ → PGame\n⊢ (∀ (k : (mk α✝¹ β✝¹ a✝³ a✝² * mk α✝ β✝ a✝¹ a✝).LeftMoves), P ⟦(mk α✝¹ β✝¹ a✝³ a✝² * mk α✝ β✝ a✝¹ a✝).moveLeft k⟧) ↔\n    (∀ (i : (mk α✝¹ β✝¹ a✝³ a✝²).LeftMoves) (j : (mk α✝ β✝ a✝¹ a✝).LeftMoves),\n        P ⟦(mk α✝¹ β✝¹ a✝³ a✝²).mulOption (mk α✝ β✝ a✝¹ a✝) i j⟧) ∧\n      ∀ (i : (-mk α✝¹ β✝¹ a✝³ a✝²).LeftMoves) (j : (-mk α✝ β✝ a✝¹ a✝).LeftMoves),\n        P ⟦(-mk α✝¹ β✝¹ a✝³ a✝²).mulOption (-mk α✝ β✝ a✝¹ a✝) i j⟧","tactic":"cases y","premises":[]},{"state_before":"case mk.mk\nP : Game → Prop\nα✝¹ β✝¹ : Type u_1\na✝³ : α✝¹ → PGame\na✝² : β✝¹ → PGame\nα✝ β✝ : Type u_1\na✝¹ : α✝ → PGame\na✝ : β✝ → PGame\n⊢ (∀ (k : (mk α✝¹ β✝¹ a✝³ a✝² * mk α✝ β✝ a✝¹ a✝).LeftMoves), P ⟦(mk α✝¹ β✝¹ a✝³ a✝² * mk α✝ β✝ a✝¹ a✝).moveLeft k⟧) ↔\n    (∀ (i : (mk α✝¹ β✝¹ a✝³ a✝²).LeftMoves) (j : (mk α✝ β✝ a✝¹ a✝).LeftMoves),\n        P ⟦(mk α✝¹ β✝¹ a✝³ a✝²).mulOption (mk α✝ β✝ a✝¹ a✝) i j⟧) ∧\n      ∀ (i : (-mk α✝¹ β✝¹ a✝³ a✝²).LeftMoves) (j : (-mk α✝ β✝ a✝¹ a✝).LeftMoves),\n        P ⟦(-mk α✝¹ β✝¹ a✝³ a✝²).mulOption (-mk α✝ β✝ a✝¹ a✝) i j⟧","state_after":"case mk.mk.mp\nP : Game → Prop\nα✝¹ β✝¹ : Type u_1\na✝³ : α✝¹ → PGame\na✝² : β✝¹ → PGame\nα✝ β✝ : Type u_1\na✝¹ : α✝ → PGame\na✝ : β✝ → PGame\nh : ∀ (k : (mk α✝¹ β✝¹ a✝³ a✝² * mk α✝ β✝ a✝¹ a✝).LeftMoves), P ⟦(mk α✝¹ β✝¹ a✝³ a✝² * mk α✝ β✝ a✝¹ a✝).moveLeft k⟧\n⊢ (∀ (i : (mk α✝¹ β✝¹ a✝³ a✝²).LeftMoves) (j : (mk α✝ β✝ a✝¹ a✝).LeftMoves),\n      P ⟦(mk α✝¹ β✝¹ a✝³ a✝²).mulOption (mk α✝ β✝ a✝¹ a✝) i j⟧) ∧\n    ∀ (i : (-mk α✝¹ β✝¹ a✝³ a✝²).LeftMoves) (j : (-mk α✝ β✝ a✝¹ a✝).LeftMoves),\n      P ⟦(-mk α✝¹ β✝¹ a✝³ a✝²).mulOption (-mk α✝ β✝ a✝¹ a✝) i j⟧\n\ncase mk.mk.mpr\nP : Game → Prop\nα✝¹ β✝¹ : Type u_1\na✝³ : α✝¹ → PGame\na✝² : β✝¹ → PGame\nα✝ β✝ : Type u_1\na✝¹ : α✝ → PGame\na✝ : β✝ → PGame\nh :\n  (∀ (i : (mk α✝¹ β✝¹ a✝³ a✝²).LeftMoves) (j : (mk α✝ β✝ a✝¹ a✝).LeftMoves),\n      P ⟦(mk α✝¹ β✝¹ a✝³ a✝²).mulOption (mk α✝ β✝ a✝¹ a✝) i j⟧) ∧\n    ∀ (i : (-mk α✝¹ β✝¹ a✝³ a✝²).LeftMoves) (j : (-mk α✝ β✝ a✝¹ a✝).LeftMoves),\n      P ⟦(-mk α✝¹ β✝¹ a✝³ a✝²).mulOption (-mk α✝ β✝ a✝¹ a✝) i j⟧\n⊢ ∀ (k : (mk α✝¹ β✝¹ a✝³ a✝² * mk α✝ β✝ a✝¹ a✝).LeftMoves), P ⟦(mk α✝¹ β✝¹ a✝³ a✝² * mk α✝ β✝ a✝¹ a✝).moveLeft k⟧","tactic":"constructor <;> intro h","premises":[]},{"state_before":"case h.e'_1\nP : Game → Prop\nα✝¹ β✝¹ : Type u_1\na✝³ : α✝¹ → PGame\na✝² : β✝¹ → PGame\nα✝ β✝ : Type u_1\na✝¹ : α✝ → PGame\na✝ : β✝ → PGame\nh : ∀ (k : (mk α✝¹ β✝¹ a✝³ a✝² * mk α✝ β✝ a✝¹ a✝).LeftMoves), P ⟦(mk α✝¹ β✝¹ a✝³ a✝² * mk α✝ β✝ a✝¹ a✝).moveLeft k⟧\ni : (-mk α✝¹ β✝¹ a✝³ a✝²).LeftMoves\nj : (-mk α✝ β✝ a✝¹ a✝).LeftMoves\n⊢ ⟦(-mk α✝¹ β✝¹ a✝³ a✝²).mulOption (-mk α✝ β✝ a✝¹ a✝) i j⟧ =\n    ⟦(mk α✝¹ β✝¹ a✝³ a✝² * mk α✝ β✝ a✝¹ a✝).moveLeft (Sum.inr (i, j))⟧\n\ncase h.e'_1\nP : Game → Prop\nα✝¹ β✝¹ : Type u_1\na✝³ : α✝¹ → PGame\na✝² : β✝¹ → PGame\nα✝ β✝ : Type u_1\na✝¹ : α✝ → PGame\na✝ : β✝ → PGame\nh :\n  (∀ (i : (mk α✝¹ β✝¹ a✝³ a✝²).LeftMoves) (j : (mk α✝ β✝ a✝¹ a✝).LeftMoves),\n      P ⟦(mk α✝¹ β✝¹ a✝³ a✝²).mulOption (mk α✝ β✝ a✝¹ a✝) i j⟧) ∧\n    ∀ (i : (-mk α✝¹ β✝¹ a✝³ a✝²).LeftMoves) (j : (-mk α✝ β✝ a✝¹ a✝).LeftMoves),\n      P ⟦(-mk α✝¹ β✝¹ a✝³ a✝²).mulOption (-mk α✝ β✝ a✝¹ a✝) i j⟧\ni : β✝¹\nj : β✝\n⊢ ⟦(mk α✝¹ β✝¹ a✝³ a✝² * mk α✝ β✝ a✝¹ a✝).moveLeft (Sum.inr (i, j))⟧ =\n    ⟦(-mk α✝¹ β✝¹ a✝³ a✝²).mulOption (-mk α✝ β✝ a✝¹ a✝) i j⟧","state_after":"no goals","tactic":"all_goals\n    dsimp only [mk_mul_moveLeft_inr, quot_sub, quot_add, neg_def, mulOption, moveLeft_mk]\n    rw [← neg_def, ← neg_def]\n    congr 1\n    on_goal 1 => congr 1\n    all_goals rw [quot_neg_mul_neg]","premises":[{"full_name":"SetTheory.PGame.mk_mul_moveLeft_inr","def_path":"Mathlib/SetTheory/Game/Basic.lean","def_pos":[290,8],"def_end_pos":[290,27]},{"full_name":"SetTheory.PGame.moveLeft_mk","def_path":"Mathlib/SetTheory/Game/PGame.lean","def_pos":[140,8],"def_end_pos":[140,19]},{"full_name":"SetTheory.PGame.mulOption","def_path":"Mathlib/SetTheory/Game/Basic.lean","def_pos":[776,4],"def_end_pos":[776,13]},{"full_name":"SetTheory.PGame.neg_def","def_path":"Mathlib/SetTheory/Game/PGame.lean","def_pos":[1073,8],"def_end_pos":[1073,15]},{"full_name":"SetTheory.PGame.quot_add","def_path":"Mathlib/SetTheory/Game/Basic.lean","def_pos":[218,8],"def_end_pos":[218,16]},{"full_name":"SetTheory.PGame.quot_neg_mul_neg","def_path":"Mathlib/SetTheory/Game/Basic.lean","def_pos":[463,8],"def_end_pos":[463,24]},{"full_name":"SetTheory.PGame.quot_sub","def_path":"Mathlib/SetTheory/Game/Basic.lean","def_pos":[222,8],"def_end_pos":[222,16]}]}]}
+{"url":"Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean","commit":"","full_name":"Real.hasSum_pow_div_log_of_abs_lt_one","start":[243,0],"end":[266,94],"file_path":"Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean","tactics":[{"state_before":"x : ℝ\nh : |x| < 1\n⊢ HasSum (fun n => x ^ (n + 1) / (↑n + 1)) (-log (1 - x))","state_after":"x : ℝ\nh : |x| < 1\n⊢ Tendsto (fun n => ∑ i ∈ Finset.range n, x ^ (i + 1) / (↑i + 1)) atTop (𝓝 (-log (1 - x)))\n\nx : ℝ\nh : |x| < 1\n⊢ Summable fun n => x ^ (n + 1) / (↑n + 1)","tactic":"rw [Summable.hasSum_iff_tendsto_nat]","premises":[{"full_name":"Summable.hasSum_iff_tendsto_nat","def_path":"Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean","def_pos":[85,2],"def_end_pos":[85,13]}]},{"state_before":"x : ℝ\nh : |x| < 1\n⊢ Summable fun n => x ^ (n + 1) / (↑n + 1)","state_after":"x : ℝ\nh : |x| < 1\ni : ℕ\n⊢ ‖x ^ (i + 1) / (↑i + 1)‖ ≤ |x| ^ i","tactic":"refine .of_norm_bounded _ (summable_geometric_of_lt_one (abs_nonneg _) h) fun i => ?_","premises":[{"full_name":"Summable.of_norm_bounded","def_path":"Mathlib/Analysis/Normed/Group/InfiniteSum.lean","def_pos":[103,8],"def_end_pos":[103,32]},{"full_name":"abs_nonneg","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[95,29],"def_end_pos":[95,39]},{"full_name":"summable_geometric_of_lt_one","def_path":"Mathlib/Analysis/SpecificLimits/Basic.lean","def_pos":[260,8],"def_end_pos":[260,36]}]},{"state_before":"x : ℝ\nh : |x| < 1\ni : ℕ\n⊢ ‖x ^ (i + 1) / (↑i + 1)‖ ≤ |x| ^ i","state_after":"no goals","tactic":"calc\n    ‖x ^ (i + 1) / (i + 1)‖ = |x| ^ (i + 1) / (i + 1) := by\n      have : (0 : ℝ) ≤ i + 1 := le_of_lt (Nat.cast_add_one_pos i)\n      rw [norm_eq_abs, abs_div, ← pow_abs, abs_of_nonneg this]\n    _ ≤ |x| ^ (i + 1) / (0 + 1) := by\n      gcongr\n      exact i.cast_nonneg\n    _ ≤ |x| ^ i := by\n      simpa [pow_succ] using mul_le_of_le_one_right (pow_nonneg (abs_nonneg x) i) (le_of_lt h)","premises":[{"full_name":"Nat.cast_add_one_pos","def_path":"Mathlib/Data/Nat/Cast/Order/Basic.lean","def_pos":[56,8],"def_end_pos":[56,24]},{"full_name":"Nat.cast_nonneg","def_path":"Mathlib/Data/Nat/Cast/Order/Ring.lean","def_pos":[29,8],"def_end_pos":[29,19]},{"full_name":"Norm.norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[59,2],"def_end_pos":[59,6]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Real.norm_eq_abs","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[1132,8],"def_end_pos":[1132,19]},{"full_name":"abs","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[33,2],"def_end_pos":[33,13]},{"full_name":"abs_div","def_path":"Mathlib/Algebra/Order/Field/Basic.lean","def_pos":[809,8],"def_end_pos":[809,15]},{"full_name":"abs_nonneg","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[95,29],"def_end_pos":[95,39]},{"full_name":"abs_of_nonneg","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[76,2],"def_end_pos":[76,13]},{"full_name":"le_of_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[89,8],"def_end_pos":[89,16]},{"full_name":"mul_le_of_le_one_right","def_path":"Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean","def_pos":[673,8],"def_end_pos":[673,30]},{"full_name":"pow_abs","def_path":"Mathlib/Algebra/Order/Ring/Abs.lean","def_pos":[58,6],"def_end_pos":[58,13]},{"full_name":"pow_nonneg","def_path":"Mathlib/Algebra/Order/Ring/Unbundled/Basic.lean","def_pos":[141,8],"def_end_pos":[141,18]},{"full_name":"pow_succ","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[567,8],"def_end_pos":[567,16]}]}]}
+{"url":"Mathlib/RingTheory/Finiteness.lean","commit":"","full_name":"Submodule.fg_iff_add_subgroup_fg","start":[58,0],"end":[61,63],"file_path":"Mathlib/RingTheory/Finiteness.lean","tactics":[{"state_before":"R : Type u_1\nM : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nG : Type u_3\ninst✝ : AddCommGroup G\nP : Submodule ℤ G\nx✝ : P.FG\nS : Finset G\nhS : span ℤ ↑S = P\n⊢ AddSubgroup.closure ↑S = P.toAddSubgroup","state_after":"no goals","tactic":"simpa [← span_int_eq_addSubgroup_closure] using hS","premises":[{"full_name":"Submodule.span_int_eq_addSubgroup_closure","def_path":"Mathlib/LinearAlgebra/Span.lean","def_pos":[252,8],"def_end_pos":[252,39]}]},{"state_before":"R : Type u_1\nM : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nG : Type u_3\ninst✝ : AddCommGroup G\nP : Submodule ℤ G\nx✝ : P.toAddSubgroup.FG\nS : Finset G\nhS : AddSubgroup.closure ↑S = P.toAddSubgroup\n⊢ span ℤ ↑S = P","state_after":"no goals","tactic":"simpa [← span_int_eq_addSubgroup_closure] using hS","premises":[{"full_name":"Submodule.span_int_eq_addSubgroup_closure","def_path":"Mathlib/LinearAlgebra/Span.lean","def_pos":[252,8],"def_end_pos":[252,39]}]}]}
+{"url":"Mathlib/FieldTheory/IsAlgClosed/AlgebraicClosure.lean","commit":"","full_name":"AlgebraicClosure.toSplittingField_evalXSelf","start":[70,0],"end":[74,30],"file_path":"Mathlib/FieldTheory/IsAlgClosed/AlgebraicClosure.lean","tactics":[{"state_before":"k : Type u\ninst✝ : Field k\ns : Finset (MonicIrreducible k)\nf : MonicIrreducible k\nhf : f ∈ s\n⊢ (toSplittingField k s) (evalXSelf k f) = 0","state_after":"k : Type u\ninst✝ : Field k\ns : Finset (MonicIrreducible k)\nf : MonicIrreducible k\nhf : f ∈ s\n⊢ Polynomial.eval₂ (algebraMap k (∏ x ∈ s, ↑x).SplittingField)\n      (rootOfSplits (algebraMap k (∏ x ∈ s, ↑x).SplittingField) ⋯ ⋯) ↑f =\n    0","tactic":"rw [toSplittingField, evalXSelf, ← AlgHom.coe_toRingHom, hom_eval₂, AlgHom.coe_toRingHom,\n    MvPolynomial.aeval_X, dif_pos hf, ← MvPolynomial.algebraMap_eq, AlgHom.comp_algebraMap]","premises":[{"full_name":"AlgHom.coe_toRingHom","def_path":"Mathlib/Algebra/Algebra/Hom.lean","def_pos":[155,8],"def_end_pos":[155,21]},{"full_name":"AlgHom.comp_algebraMap","def_path":"Mathlib/Algebra/Algebra/Hom.lean","def_pos":[204,8],"def_end_pos":[204,23]},{"full_name":"AlgebraicClosure.evalXSelf","def_path":"Mathlib/FieldTheory/IsAlgClosed/AlgebraicClosure.lean","def_pos":[49,4],"def_end_pos":[49,13]},{"full_name":"AlgebraicClosure.toSplittingField","def_path":"Mathlib/FieldTheory/IsAlgClosed/AlgebraicClosure.lean","def_pos":[60,4],"def_end_pos":[60,20]},{"full_name":"MvPolynomial.aeval_X","def_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","def_pos":[1341,8],"def_end_pos":[1341,15]},{"full_name":"MvPolynomial.algebraMap_eq","def_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","def_pos":[172,8],"def_end_pos":[172,21]},{"full_name":"Polynomial.hom_eval₂","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[872,8],"def_end_pos":[872,17]},{"full_name":"dif_pos","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[949,8],"def_end_pos":[949,15]}]},{"state_before":"k : Type u\ninst✝ : Field k\ns : Finset (MonicIrreducible k)\nf : MonicIrreducible k\nhf : f ∈ s\n⊢ Polynomial.eval₂ (algebraMap k (∏ x ∈ s, ↑x).SplittingField)\n      (rootOfSplits (algebraMap k (∏ x ∈ s, ↑x).SplittingField) ⋯ ⋯) ↑f =\n    0","state_after":"no goals","tactic":"exact map_rootOfSplits _ _ _","premises":[{"full_name":"Polynomial.map_rootOfSplits","def_path":"Mathlib/Algebra/Polynomial/Splits.lean","def_pos":[263,8],"def_end_pos":[263,24]}]}]}
+{"url":"Mathlib/CategoryTheory/Triangulated/TriangleShift.lean","commit":"","full_name":"CategoryTheory.Pretriangulated.Triangle.shiftFunctorZero_inv_app_hom₂","start":[62,0],"end":[73,18],"file_path":"Mathlib/CategoryTheory/Triangulated/TriangleShift.lean","tactics":[{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\ninst✝¹ : HasShift C ℤ\ninst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive\nT : Triangle C\n⊢ ((shiftFunctor C 0).obj T).mor₁ ≫ ((CategoryTheory.shiftFunctorZero C ℤ).app T.obj₂).hom =\n    ((CategoryTheory.shiftFunctorZero C ℤ).app T.obj₁).hom ≫ ((𝟭 (Triangle C)).obj T).mor₁","state_after":"no goals","tactic":"aesop_cat","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]}]},{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\ninst✝¹ : HasShift C ℤ\ninst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive\nT : Triangle C\n⊢ ((shiftFunctor C 0).obj T).mor₂ ≫ ((CategoryTheory.shiftFunctorZero C ℤ).app T.obj₃).hom =\n    ((CategoryTheory.shiftFunctorZero C ℤ).app T.obj₂).hom ≫ ((𝟭 (Triangle C)).obj T).mor₂","state_after":"no goals","tactic":"aesop_cat","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]}]},{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\ninst✝¹ : HasShift C ℤ\ninst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive\nT : Triangle C\n⊢ ((shiftFunctor C 0).obj T).mor₃ ≫\n      (CategoryTheory.shiftFunctor C 1).map ((CategoryTheory.shiftFunctorZero C ℤ).app T.obj₁).hom =\n    ((CategoryTheory.shiftFunctorZero C ℤ).app T.obj₃).hom ≫ ((𝟭 (Triangle C)).obj T).mor₃","state_after":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\ninst✝¹ : HasShift C ℤ\ninst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive\nT : Triangle C\n⊢ (1 • (CategoryTheory.shiftFunctor C 0).map T.mor₃ ≫ (shiftFunctorComm C 1 0).hom.app T.obj₁) ≫\n      (CategoryTheory.shiftFunctor C 1).map ((CategoryTheory.shiftFunctorZero C ℤ).hom.app T.obj₁) =\n    (CategoryTheory.shiftFunctorZero C ℤ).hom.app T.obj₃ ≫ T.mor₃","tactic":"dsimp","premises":[]},{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\ninst✝¹ : HasShift C ℤ\ninst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive\nT : Triangle C\n⊢ (1 • (CategoryTheory.shiftFunctor C 0).map T.mor₃ ≫ (shiftFunctorComm C 1 0).hom.app T.obj₁) ≫\n      (CategoryTheory.shiftFunctor C 1).map ((CategoryTheory.shiftFunctorZero C ℤ).hom.app T.obj₁) =\n    (CategoryTheory.shiftFunctorZero C ℤ).hom.app T.obj₃ ≫ T.mor₃","state_after":"no goals","tactic":"simp only [one_smul, assoc, shiftFunctorComm_zero_hom_app,\n          ← Functor.map_comp, Iso.inv_hom_id_app, Functor.id_obj, Functor.map_id,\n          comp_id, NatTrans.naturality, Functor.id_map]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Category.comp_id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[153,2],"def_end_pos":[153,9]},{"full_name":"CategoryTheory.Functor.id_map","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[88,8],"def_end_pos":[88,14]},{"full_name":"CategoryTheory.Functor.id_obj","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[85,8],"def_end_pos":[85,14]},{"full_name":"CategoryTheory.Functor.map_comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[41,2],"def_end_pos":[41,10]},{"full_name":"CategoryTheory.Functor.map_id","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[39,2],"def_end_pos":[39,8]},{"full_name":"CategoryTheory.Iso.inv_hom_id_app","def_path":"Mathlib/CategoryTheory/NatIso.lean","def_pos":[64,8],"def_end_pos":[64,22]},{"full_name":"CategoryTheory.NatTrans.naturality","def_path":"Mathlib/CategoryTheory/NatTrans.lean","def_pos":[50,2],"def_end_pos":[50,12]},{"full_name":"CategoryTheory.shiftFunctorComm_zero_hom_app","def_path":"Mathlib/CategoryTheory/Shift/Basic.lean","def_pos":[569,6],"def_end_pos":[569,35]},{"full_name":"one_smul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[379,6],"def_end_pos":[379,14]}]},{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\ninst✝¹ : HasShift C ℤ\ninst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive\n⊢ ∀ {X Y : Triangle C} (f : X ⟶ Y),\n    (shiftFunctor C 0).map f ≫\n        ((fun T =>\n              ((shiftFunctor C 0).obj T).isoMk ((𝟭 (Triangle C)).obj T)\n                ((CategoryTheory.shiftFunctorZero C ℤ).app T.obj₁) ((CategoryTheory.shiftFunctorZero C ℤ).app T.obj₂)\n                ((CategoryTheory.shiftFunctorZero C ℤ).app T.obj₃) ⋯ ⋯ ⋯)\n            Y).hom =\n      ((fun T =>\n              ((shiftFunctor C 0).obj T).isoMk ((𝟭 (Triangle C)).obj T)\n                ((CategoryTheory.shiftFunctorZero C ℤ).app T.obj₁) ((CategoryTheory.shiftFunctorZero C ℤ).app T.obj₂)\n                ((CategoryTheory.shiftFunctorZero C ℤ).app T.obj₃) ⋯ ⋯ ⋯)\n            X).hom ≫\n        (𝟭 (Triangle C)).map f","state_after":"no goals","tactic":"aesop_cat","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]}]}]}
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R A L M) y) z) =\n    ((LieAlgebra.ExtendScalars.bracket' R A L M) (((LieAlgebra.ExtendScalars.bracket' R A L L) x) y)) z +\n      ((LieAlgebra.ExtendScalars.bracket' R A L M) y) (((LieAlgebra.ExtendScalars.bracket' R A L M) x) z)","tactic":"simp only [bracket_def]","premises":[{"full_name":"_private.Mathlib.Algebra.Lie.BaseChange.0.LieAlgebra.ExtendScalars.bracket_def","def_path":"Mathlib/Algebra/Lie/BaseChange.lean","def_pos":[54,16],"def_end_pos":[54,27]}]},{"state_before":"R : Type u_1\nA : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R A\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nx y : A ⊗[R] L\nz : A ⊗[R] M\n⊢ ((LieAlgebra.ExtendScalars.bracket' R A L M) x) (((LieAlgebra.ExtendScalars.bracket' R A L M) y) z) =\n    ((LieAlgebra.ExtendScalars.bracket' R A L M) (((LieAlgebra.ExtendScalars.bracket' R A L L) x) y)) z +\n      ((LieAlgebra.ExtendScalars.bracket' R A L M) y) (((LieAlgebra.ExtendScalars.bracket' R A L M) x) z)","state_after":"case refine_1\nR : Type u_1\nA : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R A\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nx y : A ⊗[R] L\nz : A ⊗[R] M\n⊢ ((LieAlgebra.ExtendScalars.bracket' R A L M) 0) (((LieAlgebra.ExtendScalars.bracket' R A L M) y) z) =\n    ((LieAlgebra.ExtendScalars.bracket' R A L M) (((LieAlgebra.ExtendScalars.bracket' R A L L) 0) y)) z +\n      ((LieAlgebra.ExtendScalars.bracket' R A L M) y) (((LieAlgebra.ExtendScalars.bracket' R A L M) 0) z)\n\ncase refine_2\nR : Type u_1\nA : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R A\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nx y : A ⊗[R] L\nz : A ⊗[R] M\n⊢ ∀ (x : A) (y_1 : L),\n    ((LieAlgebra.ExtendScalars.bracket' R A L M) (x ⊗ₜ[R] y_1)) (((LieAlgebra.ExtendScalars.bracket' R A L M) y) z) =\n      ((LieAlgebra.ExtendScalars.bracket' R A L M) (((LieAlgebra.ExtendScalars.bracket' R A L L) (x ⊗ₜ[R] y_1)) y)) z +\n        ((LieAlgebra.ExtendScalars.bracket' R A L M) y) (((LieAlgebra.ExtendScalars.bracket' R A L M) (x ⊗ₜ[R] y_1)) z)\n\ncase refine_3\nR : Type u_1\nA : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R A\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nx y : A ⊗[R] L\nz : A ⊗[R] M\n⊢ ∀ (x y_1 : A ⊗[R] L),\n    ((LieAlgebra.ExtendScalars.bracket' R A L M) x) (((LieAlgebra.ExtendScalars.bracket' R A L M) y) z) =\n        ((LieAlgebra.ExtendScalars.bracket' R A L M) (((LieAlgebra.ExtendScalars.bracket' R A L L) x) y)) z +\n          ((LieAlgebra.ExtendScalars.bracket' R A L M) y) (((LieAlgebra.ExtendScalars.bracket' R A L M) x) z) →\n      ((LieAlgebra.ExtendScalars.bracket' R A L M) y_1) (((LieAlgebra.ExtendScalars.bracket' R A L M) y) z) =\n          ((LieAlgebra.ExtendScalars.bracket' R A L M) (((LieAlgebra.ExtendScalars.bracket' R A L L) y_1) y)) z +\n            ((LieAlgebra.ExtendScalars.bracket' R A L M) y) (((LieAlgebra.ExtendScalars.bracket' R A L M) y_1) z) →\n        ((LieAlgebra.ExtendScalars.bracket' R A L M) (x + y_1)) (((LieAlgebra.ExtendScalars.bracket' R A L M) y) z) =\n          ((LieAlgebra.ExtendScalars.bracket' R A L M) (((LieAlgebra.ExtendScalars.bracket' R A L L) (x + y_1)) y)) z +\n            ((LieAlgebra.ExtendScalars.bracket' R A L M) y) (((LieAlgebra.ExtendScalars.bracket' R A L M) (x + y_1)) z)","tactic":"refine x.induction_on ?_ ?_ ?_","premises":[{"full_name":"TensorProduct.induction_on","def_path":"Mathlib/LinearAlgebra/TensorProduct/Basic.lean","def_pos":[135,18],"def_end_pos":[135,30]}]}]}
+{"url":"Mathlib/Analysis/BoxIntegral/Partition/Basic.lean","commit":"","full_name":"BoxIntegral.Prepartition.iUnion_filter_not","start":[543,0],"end":[552,27],"file_path":"Mathlib/Analysis/BoxIntegral/Partition/Basic.lean","tactics":[{"state_before":"ι : Type u_1\nI J J₁ J₂ : Box ι\nπ✝ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi πi₁ πi₂ : (J : Box ι) → Prepartition J\nπ : Prepartition I\np : Box ι → Prop\n⊢ (π.filter fun J => ¬p J).iUnion = π.iUnion \\ (π.filter p).iUnion","state_after":"ι : Type u_1\nI J J₁ J₂ : Box ι\nπ✝ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi πi₁ πi₂ : (J : Box ι) → Prepartition J\nπ : Prepartition I\np : Box ι → Prop\n⊢ ⋃ J ∈ π.filter fun J => ¬p J, ↑J = (⋃ J ∈ π, ↑J) \\ ⋃ J ∈ π.filter p, ↑J","tactic":"simp only [Prepartition.iUnion]","premises":[{"full_name":"BoxIntegral.Prepartition.iUnion","def_path":"Mathlib/Analysis/BoxIntegral/Partition/Basic.lean","def_pos":[191,14],"def_end_pos":[191,20]}]},{"state_before":"ι : Type u_1\nI J J₁ J₂ : Box ι\nπ✝ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi πi₁ πi₂ : (J : Box ι) → Prepartition J\nπ : Prepartition I\np : Box ι → Prop\n⊢ ⋃ J ∈ π.filter fun J => ¬p J, ↑J = (⋃ J ∈ π, ↑J) \\ ⋃ J ∈ π.filter p, ↑J","state_after":"case h.e'_2.h.e'_3.h.pq.a.a\nι : Type u_1\nI J J₁ J₂ : Box ι\nπ✝ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi πi₁ πi₂ : (J : Box ι) → Prepartition J\nπ : Prepartition I\np : Box ι → Prop\nx✝ : Box ι\n⊢ (x✝ ∈ π.filter fun J => ¬p J) ↔ x✝ ∈ ↑π.boxes \\ ↑(π.filter p).boxes\n\nι : Type u_1\nI J J₁ J₂ : Box ι\nπ✝ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi πi₁ πi₂ : (J : Box ι) → Prepartition J\nπ : Prepartition I\np : Box ι → Prop\n⊢ (↑π.boxes ∪ ↑(π.filter p).boxes).PairwiseDisjoint Box.toSet","tactic":"convert (@Set.biUnion_diff_biUnion_eq (ι → ℝ) (Box ι) π.boxes (π.filter p).boxes (↑) _).symm","premises":[{"full_name":"BoxIntegral.Box","def_path":"Mathlib/Analysis/BoxIntegral/Box/Basic.lean","def_pos":[73,10],"def_end_pos":[73,13]},{"full_name":"BoxIntegral.Prepartition.boxes","def_path":"Mathlib/Analysis/BoxIntegral/Partition/Basic.lean","def_pos":[55,2],"def_end_pos":[55,7]},{"full_name":"BoxIntegral.Prepartition.filter","def_path":"Mathlib/Analysis/BoxIntegral/Partition/Basic.lean","def_pos":[522,4],"def_end_pos":[522,10]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Set.biUnion_diff_biUnion_eq","def_path":"Mathlib/Data/Set/Pairwise/Lattice.lean","def_pos":[115,8],"def_end_pos":[115,31]}]}]}
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+{"url":"Mathlib/LinearAlgebra/TensorProduct/Subalgebra.lean","commit":"","full_name":"Subalgebra.finrank_sup_le_of_free","start":[195,0],"end":[212,97],"file_path":"Mathlib/LinearAlgebra/TensorProduct/Subalgebra.lean","tactics":[{"state_before":"R : Type u\nS : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nA B : Subalgebra R S\ninst✝¹ : Module.Free R ↥A\ninst✝ : Module.Free R ↥B\n⊢ finrank R ↥(A ⊔ B) ≤ finrank R ↥A * finrank R ↥B","state_after":"R : Type u\nS : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nA B : Subalgebra R S\ninst✝¹ : Module.Free R ↥A\ninst✝ : Module.Free R ↥B\na✝ : Nontrivial R\n⊢ finrank R ↥(A ⊔ B) ≤ finrank R ↥A * finrank R ↥B","tactic":"nontriviality R using finrank","premises":[{"full_name":"FiniteDimensional.finrank","def_path":"Mathlib/LinearAlgebra/Dimension/Finrank.lean","def_pos":[52,18],"def_end_pos":[52,25]}]},{"state_before":"R : Type u\nS : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nA B : Subalgebra R S\ninst✝¹ : Module.Free R ↥A\ninst✝ : Module.Free R ↥B\na✝ : Nontrivial R\n⊢ finrank R ↥(A ⊔ B) ≤ finrank R ↥A * finrank R ↥B","state_after":"case pos\nR : Type u\nS : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nA B : Subalgebra R S\ninst✝¹ : Module.Free R ↥A\ninst✝ : Module.Free R ↥B\na✝ : Nontrivial R\nh : Module.Finite R ↥A ∧ Module.Finite R ↥B\n⊢ finrank R ↥(A ⊔ B) ≤ finrank R ↥A * finrank R ↥B\n\ncase neg\nR : Type u\nS : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nA B : Subalgebra R S\ninst✝¹ : Module.Free R ↥A\ninst✝ : Module.Free R ↥B\na✝ : Nontrivial R\nh : ¬(Module.Finite R ↥A ∧ Module.Finite R ↥B)\n⊢ finrank R ↥(A ⊔ B) ≤ finrank R ↥A * finrank R ↥B","tactic":"by_cases h : Module.Finite R A ∧ Module.Finite R B","premises":[{"full_name":"And","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[516,10],"def_end_pos":[516,13]},{"full_name":"Module.Finite","def_path":"Mathlib/RingTheory/Finiteness.lean","def_pos":[500,6],"def_end_pos":[500,19]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case neg\nR : Type u\nS : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nA B : Subalgebra R S\ninst✝¹ : Module.Free R ↥A\ninst✝ : Module.Free R ↥B\na✝ : Nontrivial R\nh : ¬(Module.Finite R ↥A ∧ Module.Finite R ↥B)\n⊢ finrank R ↥(A ⊔ B) ≤ finrank R ↥A * finrank R ↥B","state_after":"case neg.inr\nR : Type u\nS : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nA B : Subalgebra R S\ninst✝¹ : Module.Free R ↥A\ninst✝ : Module.Free R ↥B\na✝ : Nontrivial R\nh : ¬(Module.Finite R ↥A ∧ Module.Finite R ↥B)\nthis :\n  ∀ (A B : Subalgebra R S) [inst : Module.Free R ↥A] [inst : Module.Free R ↥B],\n    ¬(Module.Finite R ↥A ∧ Module.Finite R ↥B) → ¬Module.Finite R ↥A → finrank R ↥(A ⊔ B) ≤ finrank R ↥A * finrank R ↥B\nhA : ¬¬Module.Finite R ↥A\n⊢ finrank R ↥(A ⊔ B) ≤ finrank R ↥A * finrank R ↥B\n\nR : Type u\nS : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nA✝ B✝ : Subalgebra R S\na✝ : Nontrivial R\nA B : Subalgebra R S\ninst✝¹ : Module.Free R ↥A\ninst✝ : Module.Free R ↥B\nh : ¬(Module.Finite R ↥A ∧ Module.Finite R ↥B)\nhA : ¬Module.Finite R ↥A\n⊢ finrank R ↥(A ⊔ B) ≤ finrank R ↥A * finrank R ↥B","tactic":"wlog hA : ¬ Module.Finite R A generalizing A B","premises":[{"full_name":"Module.Finite","def_path":"Mathlib/RingTheory/Finiteness.lean","def_pos":[500,6],"def_end_pos":[500,19]},{"full_name":"Not","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[224,4],"def_end_pos":[224,7]}]},{"state_before":"R : Type u\nS : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nA✝ B✝ : Subalgebra R S\na✝ : Nontrivial R\nA B : Subalgebra R S\ninst✝¹ : Module.Free R ↥A\ninst✝ : Module.Free R ↥B\nh : ¬(Module.Finite R ↥A ∧ Module.Finite R ↥B)\nhA : ¬Module.Finite R ↥A\n⊢ finrank R ↥(A ⊔ B) ≤ finrank R ↥A * finrank R ↥B","state_after":"R : Type u\nS : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nA✝ B✝ : Subalgebra R S\na✝ : Nontrivial R\nA B : Subalgebra R S\ninst✝¹ : Module.Free R ↥A\ninst✝ : Module.Free R ↥B\nh : ¬(Module.Finite R ↥A ∧ Module.Finite R ↥B)\nhA : Cardinal.aleph0 ≤ Module.rank R ↥A\n⊢ finrank R ↥(A ⊔ B) ≤ finrank R ↥A * finrank R ↥B","tactic":"rw [← Module.rank_lt_alpeh0_iff, not_lt] at hA","premises":[{"full_name":"Module.rank_lt_alpeh0_iff","def_path":"Mathlib/LinearAlgebra/Dimension/Free.lean","def_pos":[164,6],"def_end_pos":[164,31]},{"full_name":"not_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[312,8],"def_end_pos":[312,14]}]},{"state_before":"R : Type u\nS : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nA✝ B✝ : Subalgebra R S\na✝ : Nontrivial R\nA B : Subalgebra R S\ninst✝¹ : Module.Free R ↥A\ninst✝ : Module.Free R ↥B\nh : ¬(Module.Finite R ↥A ∧ Module.Finite R ↥B)\nhA : Cardinal.aleph0 ≤ Module.rank R ↥A\n⊢ finrank R ↥(A ⊔ B) ≤ finrank R ↥A * finrank R ↥B","state_after":"R : Type u\nS : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nA✝ B✝ : Subalgebra R S\na✝ : Nontrivial R\nA B : Subalgebra R S\ninst✝¹ : Module.Free R ↥A\ninst✝ : Module.Free R ↥B\nh : ¬(Module.Finite R ↥A ∧ Module.Finite R ↥B)\nhA : Cardinal.aleph0 ≤ Module.rank R ↥A\nthis : Module.rank R ↥(toSubmodule A) ≤ Module.rank R ↥(toSubmodule (A ⊔ B))\n⊢ finrank R ↥(A ⊔ B) ≤ finrank R ↥A * finrank R ↥B","tactic":"have := LinearMap.rank_le_of_injective _ <| Submodule.inclusion_injective <|\n    show toSubmodule A ≤ toSubmodule (A ⊔ B) by simp","premises":[{"full_name":"LinearMap.rank_le_of_injective","def_path":"Mathlib/LinearAlgebra/Dimension/Basic.lean","def_pos":[243,8],"def_end_pos":[243,38]},{"full_name":"Subalgebra.toSubmodule","def_path":"Mathlib/Algebra/Algebra/Subalgebra/Basic.lean","def_pos":[234,4],"def_end_pos":[234,15]},{"full_name":"Submodule.inclusion_injective","def_path":"Mathlib/Algebra/Module/Submodule/LinearMap.lean","def_pos":[288,8],"def_end_pos":[288,27]},{"full_name":"Sup.sup","def_path":"Mathlib/Order/Notation.lean","def_pos":[47,2],"def_end_pos":[47,5]}]},{"state_before":"R : Type u\nS : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nA✝ B✝ : Subalgebra R S\na✝ : Nontrivial R\nA B : Subalgebra R S\ninst✝¹ : Module.Free R ↥A\ninst✝ : Module.Free R ↥B\nh : ¬(Module.Finite R ↥A ∧ Module.Finite R ↥B)\nhA : Cardinal.aleph0 ≤ Module.rank R ↥A\nthis : Module.rank R ↥(toSubmodule A) ≤ Module.rank R ↥(toSubmodule (A ⊔ B))\n⊢ finrank R ↥(A ⊔ B) ≤ finrank R ↥A * finrank R ↥B","state_after":"no goals","tactic":"rw [show finrank R A = 0 from Cardinal.toNat_apply_of_aleph0_le hA,\n    show finrank R ↥(A ⊔ B) = 0 from Cardinal.toNat_apply_of_aleph0_le (hA.trans this), zero_mul]","premises":[{"full_name":"Cardinal.toNat_apply_of_aleph0_le","def_path":"Mathlib/SetTheory/Cardinal/ToNat.lean","def_pos":[52,8],"def_end_pos":[52,32]},{"full_name":"FiniteDimensional.finrank","def_path":"Mathlib/LinearAlgebra/Dimension/Finrank.lean","def_pos":[52,18],"def_end_pos":[52,25]},{"full_name":"MulZeroClass.zero_mul","def_path":"Mathlib/Algebra/GroupWithZero/Defs.lean","def_pos":[35,2],"def_end_pos":[35,10]},{"full_name":"Sup.sup","def_path":"Mathlib/Order/Notation.lean","def_pos":[47,2],"def_end_pos":[47,5]}]}]}
+{"url":"Mathlib/NumberTheory/FermatPsp.lean","commit":"","full_name":"Nat.coprime_of_fermatPsp","start":[107,0],"end":[114,49],"file_path":"Mathlib/NumberTheory/FermatPsp.lean","tactics":[{"state_before":"n b : ℕ\nh : n.FermatPsp b\nh₁ : 1 ≤ b\n⊢ n.Coprime b","state_after":"case intro.intro\nn b : ℕ\nh₁ : 1 ≤ b\nhp : n.ProbablePrime b\nleft✝ : ¬Prime n\nhn₂ : 1 < n\n⊢ n.Coprime b","tactic":"rcases h with ⟨hp, _, hn₂⟩","premises":[]},{"state_before":"case intro.intro\nn b : ℕ\nh₁ : 1 ≤ b\nhp : n.ProbablePrime b\nleft✝ : ¬Prime n\nhn₂ : 1 < n\n⊢ n.Coprime b","state_after":"no goals","tactic":"exact coprime_of_probablePrime hp (by omega) h₁","premises":[{"full_name":"Nat.coprime_of_probablePrime","def_path":"Mathlib/NumberTheory/FermatPsp.lean","def_pos":[69,8],"def_end_pos":[69,32]}]}]}
+{"url":"Mathlib/Data/Set/Image.lean","commit":"","full_name":"Set.exists_subtype_range_iff","start":[579,0],"end":[584,26],"file_path":"Mathlib/Data/Set/Image.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nι' : Sort u_5\nf : ι → α\ns t : Set α\np : ↑(range f) → Prop\nx✝ : ∃ a, p a\na : α\ni : ι\nhi : f i = a\nha : p ⟨a, ⋯⟩\n⊢ ∃ i, p ⟨f i, ⋯⟩","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nι' : Sort u_5\nf : ι → α\ns t : Set α\np : ↑(range f) → Prop\nx✝ : ∃ a, p a\ni : ι\nha : p ⟨f i, ⋯⟩\n⊢ ∃ i, p ⟨f i, ⋯⟩","tactic":"subst a","premises":[]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nι' : Sort u_5\nf : ι → α\ns t : Set α\np : ↑(range f) → Prop\nx✝ : ∃ a, p a\ni : ι\nha : p ⟨f i, ⋯⟩\n⊢ ∃ i, p ⟨f i, ⋯⟩","state_after":"no goals","tactic":"exact ⟨i, ha⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]}]}
+{"url":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","commit":"","full_name":"intervalIntegral.integral_smul","start":[541,0],"end":[545,55],"file_path":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","tactics":[{"state_before":"ι : Type u_1\n𝕜✝ : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedSpace ℝ E\na b : ℝ\nf✝ g : ℝ → E\nμ : Measure ℝ\n𝕜 : Type u_6\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : SMulCommClass ℝ 𝕜 E\nr : 𝕜\nf : ℝ → E\n⊢ ∫ (x : ℝ) in a..b, r • f x ∂μ = r • ∫ (x : ℝ) in a..b, f x ∂μ","state_after":"no goals","tactic":"simp only [intervalIntegral, integral_smul, smul_sub]","premises":[{"full_name":"MeasureTheory.integral_smul","def_path":"Mathlib/MeasureTheory/Integral/Bochner.lean","def_pos":[824,8],"def_end_pos":[824,21]},{"full_name":"intervalIntegral","def_path":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","def_pos":[414,4],"def_end_pos":[414,20]},{"full_name":"smul_sub","def_path":"Mathlib/Algebra/GroupWithZero/Action/Defs.lean","def_pos":[279,8],"def_end_pos":[279,16]}]}]}
+{"url":"Mathlib/Probability/CondCount.lean","commit":"","full_name":"ProbabilityTheory.condCount_singleton","start":[81,0],"end":[87,64],"file_path":"Mathlib/Probability/CondCount.lean","tactics":[{"state_before":"Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : MeasurableSingletonClass Ω\nω : Ω\nt : Set Ω\ninst✝ : Decidable (ω ∈ t)\n⊢ (condCount {ω}) t = if ω ∈ t then 1 else 0","state_after":"Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : MeasurableSingletonClass Ω\nω : Ω\nt : Set Ω\ninst✝ : Decidable (ω ∈ t)\n⊢ Measure.count ({ω} ∩ t) = if ω ∈ t then 1 else 0","tactic":"rw [condCount, cond_apply _ (measurableSet_singleton ω), Measure.count_singleton, inv_one,\n    one_mul]","premises":[{"full_name":"MeasurableSingletonClass.measurableSet_singleton","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","def_pos":[238,2],"def_end_pos":[238,25]},{"full_name":"MeasureTheory.Measure.count_singleton","def_path":"Mathlib/MeasureTheory/Measure/Count.lean","def_pos":[135,8],"def_end_pos":[135,23]},{"full_name":"ProbabilityTheory.condCount","def_path":"Mathlib/Probability/CondCount.lean","def_pos":[52,4],"def_end_pos":[52,13]},{"full_name":"ProbabilityTheory.cond_apply","def_path":"Mathlib/Probability/ConditionalProbability.lean","def_pos":[137,8],"def_end_pos":[137,18]},{"full_name":"inv_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[951,8],"def_end_pos":[951,15]},{"full_name":"one_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[409,8],"def_end_pos":[409,15]}]},{"state_before":"Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : MeasurableSingletonClass Ω\nω : Ω\nt : Set Ω\ninst✝ : Decidable (ω ∈ t)\n⊢ Measure.count ({ω} ∩ t) = if ω ∈ t then 1 else 0","state_after":"case pos\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : MeasurableSingletonClass Ω\nω : Ω\nt : Set Ω\ninst✝ : Decidable (ω ∈ t)\nh✝ : ω ∈ t\n⊢ Measure.count ({ω} ∩ t) = 1\n\ncase neg\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : MeasurableSingletonClass Ω\nω : Ω\nt : Set Ω\ninst✝ : Decidable (ω ∈ t)\nh✝ : ω ∉ t\n⊢ Measure.count ({ω} ∩ t) = 0","tactic":"split_ifs","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
+{"url":"Mathlib/Topology/Algebra/Order/Floor.lean","commit":"","full_name":"tendsto_ceil_left_pure","start":[72,0],"end":[73,69],"file_path":"Mathlib/Topology/Algebra/Order/Floor.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝³ : LinearOrderedRing α\ninst✝² : FloorRing α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nn : ℤ\n⊢ Tendsto ceil (𝓝[≤] ↑n) (pure n)","state_after":"no goals","tactic":"simpa only [ceil_intCast] using tendsto_ceil_left_pure_ceil (n : α)","premises":[{"full_name":"Int.ceil_intCast","def_path":"Mathlib/Algebra/Order/Floor.lean","def_pos":[1067,8],"def_end_pos":[1067,20]},{"full_name":"tendsto_ceil_left_pure_ceil","def_path":"Mathlib/Topology/Algebra/Order/Floor.lean","def_pos":[67,8],"def_end_pos":[67,35]}]}]}
+{"url":"Mathlib/LinearAlgebra/Multilinear/Basic.lean","commit":"","full_name":"MultilinearMap.map_sub","start":[1243,0],"end":[1246,70],"file_path":"Mathlib/LinearAlgebra/Multilinear/Basic.lean","tactics":[{"state_before":"R : Type uR\nS : Type uS\nι : Type uι\nn : ℕ\nM : Fin n.succ → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝⁵ : Semiring R\ninst✝⁴ : (i : ι) → AddCommGroup (M₁ i)\ninst✝³ : AddCommGroup M₂\ninst✝² : (i : ι) → Module R (M₁ i)\ninst✝¹ : Module R M₂\nf : MultilinearMap R M₁ M₂\ninst✝ : DecidableEq ι\nm : (i : ι) → M₁ i\ni : ι\nx y : M₁ i\n⊢ f (update m i (x - y)) = f (update m i x) - f (update m i y)","state_after":"no goals","tactic":"rw [sub_eq_add_neg, sub_eq_add_neg, MultilinearMap.map_add, map_neg]","premises":[{"full_name":"MultilinearMap.map_add","def_path":"Mathlib/LinearAlgebra/Multilinear/Basic.lean","def_pos":[149,18],"def_end_pos":[149,25]},{"full_name":"MultilinearMap.map_neg","def_path":"Mathlib/LinearAlgebra/Multilinear/Basic.lean","def_pos":[1238,8],"def_end_pos":[1238,15]},{"full_name":"sub_eq_add_neg","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[905,2],"def_end_pos":[905,13]}]}]}
+{"url":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","commit":"","full_name":"List.extractP_eq_find?_eraseP","start":[570,0],"end":[578,20],"file_path":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","tactics":[{"state_before":"α : Type u_1\np : α → Bool\nl : List α\nacc : Array α\nh : l = acc.data ++ []\n⊢ extractP.go p l [] acc = (find? p [], acc.data ++ eraseP p [])","state_after":"no goals","tactic":"simp [extractP.go, find?, eraseP, h]","premises":[{"full_name":"List.eraseP","def_path":".lake/packages/batteries/Batteries/Data/List/Basic.lean","def_pos":[663,4],"def_end_pos":[663,10]},{"full_name":"List.extractP.go","def_path":".lake/packages/batteries/Batteries/Data/List/Basic.lean","def_pos":[693,2],"def_end_pos":[693,4]},{"full_name":"List.find?","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[980,4],"def_end_pos":[980,9]}]},{"state_before":"α : Type u_1\np : α → Bool\nl : List α\nacc : Array α\nx : α\nxs : List α\n⊢ l = acc.data ++ x :: xs → extractP.go p l (x :: xs) acc = (find? p (x :: xs), acc.data ++ eraseP p (x :: xs))","state_after":"α : Type u_1\np : α → Bool\nl : List α\nacc : Array α\nx : α\nxs : List α\n⊢ l = acc.data ++ x :: xs →\n    (bif p x then (some x, acc.data ++ xs) else extractP.go p l xs (acc.push x)) =\n      (match p x with\n        | true => some x\n        | false => find? p xs,\n        acc.data ++ bif p x then xs else x :: eraseP p xs)","tactic":"simp [extractP.go, find?, eraseP]","premises":[{"full_name":"List.eraseP","def_path":".lake/packages/batteries/Batteries/Data/List/Basic.lean","def_pos":[663,4],"def_end_pos":[663,10]},{"full_name":"List.extractP.go","def_path":".lake/packages/batteries/Batteries/Data/List/Basic.lean","def_pos":[693,2],"def_end_pos":[693,4]},{"full_name":"List.find?","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[980,4],"def_end_pos":[980,9]}]},{"state_before":"α : Type u_1\np : α → Bool\nl : List α\nacc : Array α\nx : α\nxs : List α\n⊢ l = acc.data ++ x :: xs →\n    (bif p x then (some x, acc.data ++ xs) else extractP.go p l xs (acc.push x)) =\n      (match p x with\n        | true => some x\n        | false => find? p xs,\n        acc.data ++ bif p x then xs else x :: eraseP p xs)","state_after":"case false\nα : Type u_1\np : α → Bool\nl : List α\nacc : Array α\nx : α\nxs : List α\n⊢ l = acc.data ++ x :: xs → extractP.go p l xs (acc.push x) = (find? p xs, acc.data ++ x :: eraseP p xs)","tactic":"cases p x <;> simp","premises":[]},{"state_before":"α : Type u_1\np : α → Bool\nl : List α\n⊢ extractP p l = (find? p l, eraseP p l)","state_after":"no goals","tactic":"exact go #[] _ rfl","premises":[{"full_name":"List.nil","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2286,4],"def_end_pos":[2286,7]},{"full_name":"List.toArray","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2719,4],"def_end_pos":[2719,16]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
+{"url":"Mathlib/Data/Nat/PartENat.lean","commit":"","full_name":"PartENat.add_left_cancel_iff","start":[509,0],"end":[510,63],"file_path":"Mathlib/Data/Nat/PartENat.lean","tactics":[{"state_before":"a b c : PartENat\nha : a ≠ ⊤\n⊢ a + b = a + c ↔ b = c","state_after":"no goals","tactic":"rw [add_comm a, add_comm a, PartENat.add_right_cancel_iff ha]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"PartENat.add_right_cancel_iff","def_path":"Mathlib/Data/Nat/PartENat.lean","def_pos":[502,18],"def_end_pos":[502,38]},{"full_name":"add_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[293,2],"def_end_pos":[293,13]}]}]}
+{"url":"Mathlib/RingTheory/Nilpotent/Defs.lean","commit":"","full_name":"nilpotencyClass_eq_zero_of_subsingleton","start":[108,0],"end":[112,60],"file_path":"Mathlib/RingTheory/Nilpotent/Defs.lean","tactics":[{"state_before":"R : Type u_1\nS : Type u_2\nx y : R\ninst✝² : Zero R\ninst✝¹ : Pow R ℕ\ninst✝ : Subsingleton R\n⊢ nilpotencyClass x = 0","state_after":"R : Type u_1\nS : Type u_2\nx y : R\ninst✝² : Zero R\ninst✝¹ : Pow R ℕ\ninst✝ : Subsingleton R\ns : Set ℕ := {k | x ^ k = 0}\n⊢ nilpotencyClass x = 0","tactic":"let s : Set ℕ := {k | x ^ k = 0}","premises":[{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]},{"full_name":"Set","def_path":"Mathlib/Init/Set.lean","def_pos":[53,4],"def_end_pos":[53,7]},{"full_name":"setOf","def_path":"Mathlib/Init/Set.lean","def_pos":[56,4],"def_end_pos":[56,9]}]},{"state_before":"R : Type u_1\nS : Type u_2\nx y : R\ninst✝² : Zero R\ninst✝¹ : Pow R ℕ\ninst✝ : Subsingleton R\ns : Set ℕ := {k | x ^ k = 0}\n⊢ nilpotencyClass x = 0","state_after":"R : Type u_1\nS : Type u_2\nx y : R\ninst✝² : Zero R\ninst✝¹ : Pow R ℕ\ninst✝ : Subsingleton R\ns : Set ℕ := {k | x ^ k = 0}\n⊢ s = univ","tactic":"suffices s = univ by change sInf _ = 0; simp [s] at this; simp [this]","premises":[{"full_name":"InfSet.sInf","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[48,2],"def_end_pos":[48,6]},{"full_name":"Set.univ","def_path":"Mathlib/Init/Set.lean","def_pos":[157,4],"def_end_pos":[157,8]}]},{"state_before":"R : Type u_1\nS : Type u_2\nx y : R\ninst✝² : Zero R\ninst✝¹ : Pow R ℕ\ninst✝ : Subsingleton R\ns : Set ℕ := {k | x ^ k = 0}\n⊢ s = univ","state_after":"no goals","tactic":"exact eq_univ_iff_forall.mpr fun k ↦ Subsingleton.elim _ _","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Set.eq_univ_iff_forall","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[558,8],"def_end_pos":[558,26]},{"full_name":"Subsingleton.elim","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1015,18],"def_end_pos":[1015,35]}]}]}
+{"url":"Mathlib/MeasureTheory/Measure/WithDensity.lean","commit":"","full_name":"MeasureTheory.lintegral_withDensity_le_lintegral_mul","start":[426,0],"end":[432,91],"file_path":"Mathlib/MeasureTheory/Measure/WithDensity.lean","tactics":[{"state_before":"α : Type u_1\nm0 : MeasurableSpace α\nμ✝ μ : Measure α\nf : α → ℝ≥0∞\nf_meas : Measurable f\ng : α �� ℝ≥0∞\n⊢ ∫⁻ (a : α), g a ∂μ.withDensity f ≤ ∫⁻ (a : α), (f * g) a ∂μ","state_after":"α : Type u_1\nm0 : MeasurableSpace α\nμ✝ μ : Measure α\nf : α → ℝ≥0∞\nf_meas : Measurable f\ng : α → ℝ≥0∞\n⊢ ⨆ g_1, ⨆ (_ : Measurable g_1), ⨆ (_ : g_1 ≤ g), ∫⁻ (a : α), g_1 a ∂μ.withDensity f ≤\n    ⨆ g_1, ⨆ (_ : Measurable g_1), ⨆ (_ : g_1 ≤ f * g), ∫⁻ (a : α), g_1 a ∂μ","tactic":"rw [← iSup_lintegral_measurable_le_eq_lintegral, ← iSup_lintegral_measurable_le_eq_lintegral]","premises":[{"full_name":"MeasureTheory.iSup_lintegral_measurable_le_eq_lintegral","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[106,8],"def_end_pos":[106,49]}]},{"state_before":"α : Type u_1\nm0 : MeasurableSpace α\nμ✝ μ : Measure α\nf : α → ℝ≥0∞\nf_meas : Measurable f\ng : α → ℝ≥0∞\n⊢ ⨆ g_1, ⨆ (_ : Measurable g_1), ⨆ (_ : g_1 ≤ g), ∫⁻ (a : α), g_1 a ∂μ.withDensity f ≤\n    ⨆ g_1, ⨆ (_ : Measurable g_1), ⨆ (_ : g_1 ≤ f * g), ∫⁻ (a : α), g_1 a ∂μ","state_after":"α : Type u_1\nm0 : MeasurableSpace α\nμ✝ μ : Measure α\nf : α → ℝ≥0∞\nf_meas : Measurable f\ng i : α → ℝ≥0∞\ni_meas : Measurable i\nhi : i ≤ g\n⊢ ∫⁻ (a : α), i a ∂μ.withDensity f ≤ ⨆ g_1, ⨆ (_ : Measurable g_1), ⨆ (_ : g_1 ≤ f * g), ∫⁻ (a : α), g_1 a ∂μ","tactic":"refine iSup₂_le fun i i_meas => iSup_le fun hi => ?_","premises":[{"full_name":"iSup_le","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[661,8],"def_end_pos":[661,15]},{"full_name":"iSup₂_le","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[667,8],"def_end_pos":[667,16]}]},{"state_before":"α : Type u_1\nm0 : MeasurableSpace α\nμ✝ μ : Measure α\nf : α → ℝ≥0∞\nf_meas : Measurable f\ng i : α → ℝ≥0∞\ni_meas : Measurable i\nhi : i ≤ g\n⊢ ∫⁻ (a : α), i a ∂μ.withDensity f ≤ ⨆ g_1, ⨆ (_ : Measurable g_1), ⨆ (_ : g_1 ≤ f * g), ∫⁻ (a : α), g_1 a ∂μ","state_after":"α : Type u_1\nm0 : MeasurableSpace α\nμ✝ μ : Measure α\nf : α → ℝ≥0∞\nf_meas : Measurable f\ng i : α → ℝ≥0∞\ni_meas : Measurable i\nhi : i ≤ g\nA : f * i ≤ f * g\n⊢ ∫⁻ (a : α), i a ∂μ.withDensity f ≤ ⨆ g_1, ⨆ (_ : Measurable g_1), ⨆ (_ : g_1 ≤ f * g), ∫⁻ (a : α), g_1 a ∂μ","tactic":"have A : f * i ≤ f * g := fun x => mul_le_mul_left' (hi x) _","premises":[{"full_name":"mul_le_mul_left'","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[62,8],"def_end_pos":[62,24]}]},{"state_before":"α : Type u_1\nm0 : MeasurableSpace α\nμ✝ μ : Measure α\nf : α → ℝ≥0∞\nf_meas : Measurable f\ng i : α → ℝ≥0∞\ni_meas : Measurable i\nhi : i ≤ g\nA : f * i ≤ f * g\n⊢ ∫⁻ (a : α), i a ∂μ.withDensity f ≤ ⨆ g_1, ⨆ (_ : Measurable g_1), ⨆ (_ : g_1 ≤ f * g), ∫⁻ (a : α), g_1 a ∂μ","state_after":"α : Type u_1\nm0 : MeasurableSpace α\nμ✝ μ : Measure α\nf : α → ℝ≥0∞\nf_meas : Measurable f\ng i : α → ℝ≥0∞\ni_meas : Measurable i\nhi : i ≤ g\nA : f * i ≤ f * g\n⊢ ∫⁻ (a : α), i a ∂μ.withDensity f ≤ ⨆ (_ : f * i ≤ f * g), ∫⁻ (a : α), (f * i) a ∂μ","tactic":"refine le_iSup₂_of_le (f * i) (f_meas.mul i_meas) ?_","premises":[{"full_name":"Measurable.mul","def_path":"Mathlib/MeasureTheory/Group/Arithmetic.lean","def_pos":[119,8],"def_end_pos":[119,22]},{"full_name":"le_iSup₂_of_le","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[653,8],"def_end_pos":[653,22]}]},{"state_before":"α : Type u_1\nm0 : MeasurableSpace α\nμ✝ μ : Measure α\nf : α → ℝ≥0∞\nf_meas : Measurable f\ng i : α → ℝ≥0∞\ni_meas : Measurable i\nhi : i ≤ g\nA : f * i ≤ f * g\n⊢ ∫⁻ (a : α), i a ∂μ.withDensity f ≤ ⨆ (_ : f * i ≤ f * g), ∫⁻ (a : α), (f * i) a ∂μ","state_after":"no goals","tactic":"exact le_iSup_of_le A (le_of_eq (lintegral_withDensity_eq_lintegral_mul _ f_meas i_meas))","premises":[{"full_name":"MeasureTheory.lintegral_withDensity_eq_lintegral_mul","def_path":"Mathlib/MeasureTheory/Measure/WithDensity.lean","def_pos":[345,8],"def_end_pos":[345,46]},{"full_name":"le_iSup_of_le","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[641,8],"def_end_pos":[641,21]},{"full_name":"le_of_eq","def_path":"Mathlib/Order/Defs.lean","def_pos":[60,8],"def_end_pos":[60,16]}]}]}
+{"url":"Mathlib/Data/Real/EReal.lean","commit":"","full_name":"EReal.inv_inv","start":[1406,0],"end":[1407,91],"file_path":"Mathlib/Data/Real/EReal.lean","tactics":[{"state_before":"a : EReal\nh : a ≠ ⊥\nh' : a ≠ ⊤\n⊢ a⁻¹⁻¹ = a","state_after":"no goals","tactic":"rw [← coe_toReal h' h, ← coe_inv a.toReal, ← coe_inv a.toReal⁻¹, _root_.inv_inv a.toReal]","premises":[{"full_name":"EReal.coe_inv","def_path":"Mathlib/Data/Real/EReal.lean","def_pos":[1389,6],"def_end_pos":[1389,13]},{"full_name":"EReal.coe_toReal","def_path":"Mathlib/Data/Real/EReal.lean","def_pos":[358,8],"def_end_pos":[358,18]},{"full_name":"EReal.toReal","def_path":"Mathlib/Data/Real/EReal.lean","def_pos":[231,4],"def_end_pos":[231,10]},{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"inv_inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[734,8],"def_end_pos":[734,15]}]}]}
+{"url":"Mathlib/LinearAlgebra/AffineSpace/Combination.lean","commit":"","full_name":"Finset.map_affineCombination","start":[573,0],"end":[583,81],"file_path":"Mathlib/LinearAlgebra/AffineSpace/Combination.lean","tactics":[{"state_before":"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : Ring k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\nS : AffineSpace V P\nι : Type u_4\ns : Finset ι\nι₂ : Type u_5\ns₂ : Finset ι₂\nV₂ : Type u_6\nP₂ : Type u_7\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module k V₂\ninst✝ : AffineSpace V₂ P₂\np : ι → P\nw : ι → k\nhw : s.sum w = 1\nf : P →ᵃ[k] P₂\n⊢ f ((affineCombination k s p) w) = (affineCombination k s (⇑f ∘ p)) w","state_after":"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : Ring k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\nS : AffineSpace V P\nι : Type u_4\ns : Finset ι\nι₂ : Type u_5\ns₂ : Finset ι₂\nV₂ : Type u_6\nP₂ : Type u_7\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module k V₂\ninst✝ : AffineSpace V₂ P₂\np : ι → P\nw : ι → k\nhw : s.sum w = 1\nf : P →ᵃ[k] P₂\nb : P\n⊢ f ((affineCombination k s p) w) = (affineCombination k s (⇑f ∘ p)) w","tactic":"have b := Classical.choice (inferInstance : AffineSpace V P).nonempty","premises":[{"full_name":"AddTorsor","def_path":"Mathlib/Algebra/AddTorsor.lean","def_pos":[45,6],"def_end_pos":[45,15]},{"full_name":"AddTorsor.nonempty","def_path":"Mathlib/Algebra/AddTorsor.lean","def_pos":[47,3],"def_end_pos":[47,11]},{"full_name":"Classical.choice","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[735,6],"def_end_pos":[735,22]},{"full_name":"inferInstance","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[99,7],"def_end_pos":[99,20]}]},{"state_before":"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : Ring k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\nS : AffineSpace V P\nι : Type u_4\ns : Finset ι\nι₂ : Type u_5\ns₂ : Finset ι₂\nV₂ : Type u_6\nP₂ : Type u_7\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module k V₂\ninst✝ : AffineSpace V₂ P₂\np : ι → P\nw : ι → k\nhw : s.sum w = 1\nf : P →ᵃ[k] P₂\nb : P\n⊢ f ((affineCombination k s p) w) = (affineCombination k s (⇑f ∘ p)) w","state_after":"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : Ring k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\nS : AffineSpace V P\nι : Type u_4\ns : Finset ι\nι₂ : Type u_5\ns₂ : Finset ι₂\nV₂ : Type u_6\nP₂ : Type u_7\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module k V₂\ninst✝ : AffineSpace V₂ P₂\np : ι → P\nw : ι → k\nhw : s.sum w = 1\nf : P →ᵃ[k] P₂\nb : P\nb₂ : P₂\n⊢ f ((affineCombination k s p) w) = (affineCombination k s (⇑f ∘ p)) w","tactic":"have b₂ := Classical.choice (inferInstance : AffineSpace V₂ P₂).nonempty","premises":[{"full_name":"AddTorsor","def_path":"Mathlib/Algebra/AddTorsor.lean","def_pos":[45,6],"def_end_pos":[45,15]},{"full_name":"AddTorsor.nonempty","def_path":"Mathlib/Algebra/AddTorsor.lean","def_pos":[47,3],"def_end_pos":[47,11]},{"full_name":"Classical.choice","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[735,6],"def_end_pos":[735,22]},{"full_name":"inferInstance","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[99,7],"def_end_pos":[99,20]}]},{"state_before":"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : Ring k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\nS : AffineSpace V P\nι : Type u_4\ns : Finset ι\nι₂ : Type u_5\ns₂ : Finset ι₂\nV₂ : Type u_6\nP₂ : Type u_7\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module k V₂\ninst✝ : AffineSpace V₂ P₂\np : ι → P\nw : ι → k\nhw : s.sum w = 1\nf : P →ᵃ[k] P₂\nb : P\nb₂ : P₂\n⊢ f ((affineCombination k s p) w) = (affineCombination k s (⇑f ∘ p)) w","state_after":"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : Ring k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\nS : AffineSpace V P\nι : Type u_4\ns : Finset ι\nι₂ : Type u_5\ns₂ : Finset ι₂\nV₂ : Type u_6\nP₂ : Type u_7\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module k V₂\ninst✝ : AffineSpace V₂ P₂\np : ι → P\nw : ι → k\nhw : s.sum w = 1\nf : P →ᵃ[k] P₂\nb : P\nb₂ : P₂\n⊢ f ((s.weightedVSubOfPoint p b) w +ᵥ b) = (s.weightedVSubOfPoint (⇑f ∘ p) (f b)) w +ᵥ f b","tactic":"rw [s.affineCombination_eq_weightedVSubOfPoint_vadd_of_sum_eq_one w p hw b,\n    s.affineCombination_eq_weightedVSubOfPoint_vadd_of_sum_eq_one w (f ∘ p) hw b₂, ←\n    s.weightedVSubOfPoint_vadd_eq_of_sum_eq_one w (f ∘ p) hw (f b) b₂]","premises":[{"full_name":"Finset.affineCombination_eq_weightedVSubOfPoint_vadd_of_sum_eq_one","def_path":"Mathlib/LinearAlgebra/AffineSpace/Combination.lean","def_pos":[379,8],"def_end_pos":[379,67]},{"full_name":"Finset.weightedVSubOfPoint_vadd_eq_of_sum_eq_one","def_path":"Mathlib/LinearAlgebra/AffineSpace/Combination.lean","def_pos":[118,8],"def_end_pos":[118,49]},{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]}]},{"state_before":"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : Ring k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\nS : AffineSpace V P\nι : Type u_4\ns : Finset ι\nι₂ : Type u_5\ns₂ : Finset ι₂\nV₂ : Type u_6\nP₂ : Type u_7\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module k V₂\ninst✝ : AffineSpace V₂ P₂\np : ι → P\nw : ι → k\nhw : s.sum w = 1\nf : P →ᵃ[k] P₂\nb : P\nb₂ : P₂\n⊢ f ((s.weightedVSubOfPoint p b) w +ᵥ b) = (s.weightedVSubOfPoint (⇑f ∘ p) (f b)) w +ᵥ f b","state_after":"no goals","tactic":"simp only [weightedVSubOfPoint_apply, RingHom.id_apply, AffineMap.map_vadd,\n    LinearMap.map_smulₛₗ, AffineMap.linearMap_vsub, map_sum, Function.comp_apply]","premises":[{"full_name":"AffineMap.linearMap_vsub","def_path":"Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean","def_pos":[125,8],"def_end_pos":[125,22]},{"full_name":"AffineMap.map_vadd","def_path":"Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean","def_pos":[118,8],"def_end_pos":[118,16]},{"full_name":"Finset.weightedVSubOfPoint_apply","def_path":"Mathlib/LinearAlgebra/AffineSpace/Combination.lean","def_pos":[68,8],"def_end_pos":[68,33]},{"full_name":"Function.comp_apply","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[35,16],"def_end_pos":[35,35]},{"full_name":"LinearMap.map_smulₛₗ","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[345,18],"def_end_pos":[345,28]},{"full_name":"RingHom.id_apply","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[549,8],"def_end_pos":[549,16]},{"full_name":"map_sum","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[286,2],"def_end_pos":[286,13]}]}]}
+{"url":"Mathlib/Algebra/BigOperators/Ring/Multiset.lean","commit":"","full_name":"Commute.multiset_sum_right","start":[88,0],"end":[91,36],"file_path":"Mathlib/Algebra/BigOperators/Ring/Multiset.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : NonUnitalNonAssocSemiring α\ns : Multiset α\na : α\nh : ∀ b ∈ s, Commute a b\n⊢ Commute a s.sum","state_after":"case h\nι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : NonUnitalNonAssocSemiring α\ns : Multiset α\na : α\na✝ : List α\nh : ∀ b ∈ ⟦a✝⟧, Commute a b\n⊢ Commute a (sum ⟦a✝⟧)","tactic":"induction s using Quotient.inductionOn","premises":[{"full_name":"Quotient.inductionOn","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1629,18],"def_end_pos":[1629,29]}]},{"state_before":"case h\nι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : NonUnitalNonAssocSemiring α\ns : Multiset α\na : α\na✝ : List α\nh : ∀ b ∈ ⟦a✝⟧, Commute a b\n⊢ Commute a (sum ⟦a✝⟧)","state_after":"case h\nι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : NonUnitalNonAssocSemiring α\ns : Multiset α\na : α\na✝ : List α\nh : ∀ b ∈ ⟦a✝⟧, Commute a b\n⊢ Commute a a✝.sum","tactic":"rw [quot_mk_to_coe, sum_coe]","premises":[{"full_name":"Multiset.quot_mk_to_coe","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[43,8],"def_end_pos":[43,22]},{"full_name":"Multiset.sum_coe","def_path":"Mathlib/Algebra/BigOperators/Group/Multiset.lean","def_pos":[51,2],"def_end_pos":[51,13]}]},{"state_before":"case h\nι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : NonUnitalNonAssocSemiring α\ns : Multiset α\na : α\na✝ : List α\nh : ∀ b ∈ ⟦a✝⟧, Commute a b\n⊢ Commute a a✝.sum","state_after":"no goals","tactic":"exact Commute.list_sum_right _ _ h","premises":[{"full_name":"Commute.list_sum_right","def_path":"Mathlib/Algebra/BigOperators/Ring/List.lean","def_pos":[26,6],"def_end_pos":[26,20]}]}]}
+{"url":"Mathlib/Topology/Algebra/Group/Basic.lean","commit":"","full_name":"neg_mem_connectedComponent_zero","start":[662,0],"end":[669,45],"file_path":"Mathlib/Topology/Algebra/Group/Basic.lean","tactics":[{"state_before":"G✝ : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝⁶ : TopologicalSpace G✝\ninst✝⁵ : Group G✝\ninst✝⁴ : TopologicalGroup G✝\ninst✝³ : TopologicalSpace α\nf : α → G✝\ns : Set α\nx : α\nG : Type u_1\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : TopologicalGroup G\ng : G\nhg : g ∈ connectedComponent 1\n⊢ g⁻¹ ∈ connectedComponent 1","state_after":"G✝ : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝⁶ : TopologicalSpace G✝\ninst✝⁵ : Group G✝\ninst✝⁴ : TopologicalGroup G✝\ninst✝³ : TopologicalSpace α\nf : α → G✝\ns : Set α\nx : α\nG : Type u_1\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : TopologicalGroup G\ng : G\nhg : g ∈ connectedComponent 1\n⊢ g⁻¹ ∈ connectedComponent 1⁻¹","tactic":"rw [← inv_one]","premises":[{"full_name":"inv_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[951,8],"def_end_pos":[951,15]}]},{"state_before":"G✝ : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝⁶ : TopologicalSpace G✝\ninst✝⁵ : Group G✝\ninst✝⁴ : TopologicalGroup G✝\ninst✝³ : TopologicalSpace α\nf : α → G✝\ns : Set α\nx : α\nG : Type u_1\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : TopologicalGroup G\ng : G\nhg : g ∈ connectedComponent 1\n⊢ g⁻¹ ∈ connectedComponent 1⁻¹","state_after":"no goals","tactic":"exact\n    Continuous.image_connectedComponent_subset continuous_inv _\n      ((Set.mem_image _ _ _).mp ⟨g, hg, rfl⟩)","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Continuous.image_connectedComponent_subset","def_path":"Mathlib/Topology/Connected/Basic.lean","def_pos":[570,8],"def_end_pos":[570,50]},{"full_name":"ContinuousInv.continuous_inv","def_path":"Mathlib/Topology/Algebra/Group/Basic.lean","def_pos":[150,2],"def_end_pos":[150,16]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Set.mem_image","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[118,8],"def_end_pos":[118,17]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
+{"url":"Mathlib/LinearAlgebra/Matrix/Block.lean","commit":"","full_name":"Matrix.BlockTriangular.toBlock_inverse_mul_toBlock_eq_one","start":[266,0],"end":[280,28],"file_path":"Mathlib/LinearAlgebra/Matrix/Block.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nm : Type u_3\nn : Type u_4\no : Type u_5\nm' : α → Type u_6\nn' : α → Type u_7\nR : Type v\ninst✝⁶ : CommRing R\nM N : Matrix m m R\nb : m → α\ninst✝⁵ : DecidableEq m\ninst✝⁴ : Fintype m\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\ninst✝¹ : LinearOrder α\ninst✝ : Invertible M\nhM : M.BlockTriangular b\nk : α\n⊢ ((M⁻¹.toBlock (fun i => b i < k) fun i => b i < k) * M.toBlock (fun i => b i < k) fun i => b i < k) = 1","state_after":"α : Type u_1\nβ : Type u_2\nm : Type u_3\nn : Type u_4\no : Type u_5\nm' : α → Type u_6\nn' : α → Type u_7\nR : Type v\ninst✝⁶ : CommRing R\nM N : Matrix m m R\nb : m → α\ninst✝⁵ : DecidableEq m\ninst✝⁴ : Fintype m\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\ninst✝¹ : LinearOrder α\ninst✝ : Invertible M\nhM : M.BlockTriangular b\nk : α\np : m → Prop := fun i => b i < k\n⊢ ((M⁻¹.toBlock (fun i => b i < k) fun i => b i < k) * M.toBlock (fun i => b i < k) fun i => b i < k) = 1","tactic":"let p i := b i < k","premises":[]},{"state_before":"α : Type u_1\nβ : Type u_2\nm : Type u_3\nn : Type u_4\no : Type u_5\nm' : α → Type u_6\nn' : α → Type u_7\nR : Type v\ninst✝⁶ : CommRing R\nM N : Matrix m m R\nb : m → α\ninst✝⁵ : DecidableEq m\ninst✝⁴ : Fintype m\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\ninst✝¹ : LinearOrder α\ninst✝ : Invertible M\nhM : M.BlockTriangular b\nk : α\np : m → Prop := fun i => b i < k\n⊢ ((M⁻¹.toBlock (fun i => b i < k) fun i => b i < k) * M.toBlock (fun i => b i < k) fun i => b i < k) = 1","state_after":"α : Type u_1\nβ : Type u_2\nm : Type u_3\nn : Type u_4\no : Type u_5\nm' : α → Type u_6\nn' : α → Type u_7\nR : Type v\ninst✝⁶ : CommRing R\nM N : Matrix m m R\nb : m → α\ninst✝⁵ : DecidableEq m\ninst✝⁴ : Fintype m\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\ninst✝¹ : LinearOrder α\ninst✝ : Invertible M\nhM : M.BlockTriangular b\nk : α\np : m → Prop := fun i => b i < k\nh_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1\n⊢ ((M⁻¹.toBlock (fun i => b i < k) fun i => b i < k) * M.toBlock (fun i => b i < k) fun i => b i < k) = 1","tactic":"have h_sum :\n    M⁻¹.toBlock p p * M.toBlock p p +\n        (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p =\n      1 := by\n    rw [← toBlock_mul_eq_add, inv_mul_of_invertible M, toBlock_one_self]","premises":[{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"Matrix.inv_mul_of_invertible","def_path":"Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean","def_pos":[272,8],"def_end_pos":[272,29]},{"full_name":"Matrix.toBlock","def_path":"Mathlib/Data/Matrix/Block.lean","def_pos":[162,4],"def_end_pos":[162,11]},{"full_name":"Matrix.toBlock_mul_eq_add","def_path":"Mathlib/Data/Matrix/Block.lean","def_pos":[825,8],"def_end_pos":[825,26]},{"full_name":"Matrix.toBlock_one_self","def_path":"Mathlib/Data/Matrix/Block.lean","def_pos":[293,8],"def_end_pos":[293,24]},{"full_name":"Not","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[224,4],"def_end_pos":[224,7]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nm : Type u_3\nn : Type u_4\no : Type u_5\nm' : α → Type u_6\nn' : α → Type u_7\nR : Type v\ninst✝⁶ : CommRing R\nM N : Matrix m m R\nb : m → α\ninst✝⁵ : DecidableEq m\ninst✝⁴ : Fintype m\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\ninst✝¹ : LinearOrder α\ninst✝ : Invertible M\nhM : M.BlockTriangular b\nk : α\np : m → Prop := fun i => b i < k\nh_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1\n⊢ ((M⁻¹.toBlock (fun i => b i < k) fun i => b i < k) * M.toBlock (fun i => b i < k) fun i => b i < k) = 1","state_after":"α : Type u_1\nβ : Type u_2\nm : Type u_3\nn : Type u_4\no : Type u_5\nm' : α → Type u_6\nn' : α → Type u_7\nR : Type v\ninst✝⁶ : CommRing R\nM N : Matrix m m R\nb : m → α\ninst✝⁵ : DecidableEq m\ninst✝⁴ : Fintype m\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\ninst✝¹ : LinearOrder α\ninst✝ : Invertible M\nhM : M.BlockTriangular b\nk : α\np : m → Prop := fun i => b i < k\nh_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1\nh_zero : M.toBlock (fun i => ¬p i) p = 0\n⊢ ((M⁻¹.toBlock (fun i => b i < k) fun i => b i < k) * M.toBlock (fun i => b i < k) fun i => b i < k) = 1","tactic":"have h_zero : M.toBlock (fun i => ¬p i) p = 0 := by\n    ext i j\n    simpa using hM (lt_of_lt_of_le j.2 (le_of_not_lt i.2))","premises":[{"full_name":"Matrix.toBlock","def_path":"Mathlib/Data/Matrix/Block.lean","def_pos":[162,4],"def_end_pos":[162,11]},{"full_name":"Not","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[224,4],"def_end_pos":[224,7]},{"full_name":"Subtype.property","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[590,2],"def_end_pos":[590,10]},{"full_name":"le_of_not_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[275,8],"def_end_pos":[275,20]},{"full_name":"lt_of_lt_of_le","def_path":"Mathlib/Order/Defs.lean","def_pos":[93,8],"def_end_pos":[93,22]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nm : Type u_3\nn : Type u_4\no : Type u_5\nm' : α → Type u_6\nn' : α → Type u_7\nR : Type v\ninst✝⁶ : CommRing R\nM N : Matrix m m R\nb : m → α\ninst✝⁵ : DecidableEq m\ninst✝⁴ : Fintype m\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\ninst✝¹ : LinearOrder α\ninst✝ : Invertible M\nhM : M.BlockTriangular b\nk : α\np : m → Prop := fun i => b i < k\nh_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1\nh_zero : M.toBlock (fun i => ¬p i) p = 0\n⊢ ((M⁻¹.toBlock (fun i => b i < k) fun i => b i < k) * M.toBlock (fun i => b i < k) fun i => b i < k) = 1","state_after":"no goals","tactic":"simpa [h_zero] using h_sum","premises":[]}]}
+{"url":"Mathlib/Algebra/Category/Grp/EpiMono.lean","commit":"","full_name":"Grp.SurjectiveOfEpiAuxs.τ_symm_apply_infinity","start":[176,0],"end":[178,51],"file_path":"Mathlib/Algebra/Category/Grp/EpiMono.lean","tactics":[{"state_before":"A B : Grp\nf : A ⟶ B\n⊢ (Equiv.symm τ) ∞ = fromCoset ⟨↑(MonoidHom.range f), ⋯⟩","state_after":"no goals","tactic":"rw [tau, Equiv.symm_swap, Equiv.swap_apply_right]","premises":[{"full_name":"Equiv.swap_apply_right","def_path":"Mathlib/Logic/Equiv/Basic.lean","def_pos":[1402,8],"def_end_pos":[1402,24]},{"full_name":"Equiv.symm_swap","def_path":"Mathlib/Logic/Equiv/Basic.lean","def_pos":[1417,8],"def_end_pos":[1417,17]},{"full_name":"Grp.SurjectiveOfEpiAuxs.tau","def_path":"Mathlib/Algebra/Category/Grp/EpiMono.lean","def_pos":[158,18],"def_end_pos":[158,21]}]}]}
+{"url":"Mathlib/CategoryTheory/Abelian/DiagramLemmas/Four.lean","commit":"","full_name":"CategoryTheory.Abelian.mono_of_epi_of_mono_of_mono","start":[83,0],"end":[88,90],"file_path":"Mathlib/CategoryTheory/Abelian/DiagramLemmas/Four.lean","tactics":[{"state_before":"C : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : Abelian C\nR₁ R₂ : ComposableArrows C 3\nφ : R₁ ⟶ R₂\nhR₁ : R₁.Exact\nhR₂ : R₂.Exact\nh₀ : Epi (app' φ 0 ⋯)\nh₁ : Mono (app' φ 1 ⋯)\nh₃ : Mono (app' φ 3 ⋯)\n⊢ R₁.map' 0 2 ⋯ ⋯ = 0","state_after":"no goals","tactic":"simpa only [R₁.map'_comp 0 1 2] using hR₁.toIsComplex.zero 0","premises":[{"full_name":"CategoryTheory.ComposableArrows.IsComplex.zero","def_path":"Mathlib/Algebra/Homology/ExactSequence.lean","def_pos":[47,2],"def_end_pos":[47,6]},{"full_name":"CategoryTheory.ComposableArrows.map'_comp","def_path":"Mathlib/CategoryTheory/ComposableArrows.lean","def_pos":[87,6],"def_end_pos":[87,15]}]}]}
+{"url":"Mathlib/Data/Nat/Choose/Dvd.lean","commit":"","full_name":"Nat.Prime.dvd_choose_add","start":[22,0],"end":[27,62],"file_path":"Mathlib/Data/Nat/Choose/Dvd.lean","tactics":[{"state_before":"p a b k : ℕ\nhp : Prime p\nhap : a < p\nhbp : b < p\nh : p ≤ a + b\n⊢ p ∣ (a + b).choose a","state_after":"p a b k : ℕ\nhp : Prime p\nhap : a < p\nhbp : b < p\nh : p ≤ a + b\nh₁ : p ∣ (a + b)!\n⊢ p ∣ (a + b).choose a","tactic":"have h₁ : p ∣ (a + b)! := hp.dvd_factorial.2 h","premises":[{"full_name":"Dvd.dvd","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1344,2],"def_end_pos":[1344,5]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Nat.Prime.dvd_factorial","def_path":"Mathlib/Data/Nat/Prime/Basic.lean","def_pos":[199,8],"def_end_pos":[199,27]},{"full_name":"Nat.factorial","def_path":"Mathlib/Data/Nat/Factorial/Basic.lean","def_pos":[29,4],"def_end_pos":[29,13]}]},{"state_before":"p a b k : ℕ\nhp : Prime p\nhap : a < p\nhbp : b < p\nh : p ≤ a + b\nh₁ : p ∣ (a + b)!\n⊢ p ∣ (a + b).choose a","state_after":"p a b k : ℕ\nhp : Prime p\nhap : a < p\nhbp : b < p\nh : p ≤ a + b\nh₁ : (p ∣ (a + b).choose a ∨ p ≤ a) ∨ p ≤ b\n⊢ p ∣ (a + b).choose a","tactic":"rw [← add_choose_mul_factorial_mul_factorial, ← choose_symm_add, hp.dvd_mul, hp.dvd_mul,\n    hp.dvd_factorial, hp.dvd_factorial] at h₁","premises":[{"full_name":"Nat.Prime.dvd_factorial","def_path":"Mathlib/Data/Nat/Prime/Basic.lean","def_pos":[199,8],"def_end_pos":[199,27]},{"full_name":"Nat.Prime.dvd_mul","def_path":"Mathlib/Data/Nat/Prime/Defs.lean","def_pos":[410,8],"def_end_pos":[410,21]},{"full_name":"Nat.add_choose_mul_factorial_mul_factorial","def_path":"Mathlib/Data/Nat/Choose/Basic.lean","def_pos":[156,8],"def_end_pos":[156,46]},{"full_name":"Nat.choose_symm_add","def_path":"Mathlib/Data/Nat/Choose/Basic.lean","def_pos":[179,8],"def_end_pos":[179,23]}]},{"state_before":"p a b k : ℕ\nhp : Prime p\nhap : a < p\nhbp : b < p\nh : p ≤ a + b\nh₁ : (p ∣ (a + b).choose a ∨ p ≤ a) ∨ p ≤ b\n⊢ p ∣ (a + b).choose a","state_after":"no goals","tactic":"exact (h₁.resolve_right hbp.not_le).resolve_right hap.not_le","premises":[{"full_name":"Or.resolve_right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[556,8],"def_end_pos":[556,24]}]}]}
+{"url":"Mathlib/LinearAlgebra/Matrix/Block.lean","commit":"","full_name":"Matrix.twoBlockTriangular_det","start":[182,0],"end":[189,27],"file_path":"Mathlib/LinearAlgebra/Matrix/Block.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nm : Type u_3\nn : Type u_4\no : Type u_5\nm' : α → Type u_6\nn' : α → Type u_7\nR : Type v\ninst✝⁵ : CommRing R\nM✝ N : Matrix m m R\nb : m → α\ninst✝⁴ : DecidableEq m\ninst✝³ : Fintype m\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nM : Matrix m m R\np : m → Prop\ninst✝ : DecidablePred p\nh : ∀ (i : m), ¬p i → ∀ (j : m), p j → M i j = 0\n⊢ M.det = (M.toSquareBlockProp p).det * (M.toSquareBlockProp fun i => ¬p i).det","state_after":"α : Type u_1\nβ : Type u_2\nm : Type u_3\nn : Type u_4\no : Type u_5\nm' : α → Type u_6\nn' : α → Type u_7\nR : Type v\ninst✝⁵ : CommRing R\nM✝ N : Matrix m m R\nb : m → α\ninst✝⁴ : DecidableEq m\ninst✝³ : Fintype m\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nM : Matrix m m R\np : m → Prop\ninst✝ : DecidablePred p\nh : ∀ (i : m), ¬p i → ∀ (j : m), p j → M i j = 0\n⊢ (fromBlocks (M.toBlock p p) (M.toBlock p fun j => ¬p j) (M.toBlock (fun j => ¬p j) p)\n        (M.toBlock (fun j => ¬p j) fun j => ¬p j)).det =\n    (M.toSquareBlockProp p).det * (M.toSquareBlockProp fun i => ¬p i).det","tactic":"rw [det_toBlock M p]","premises":[{"full_name":"Matrix.det_toBlock","def_path":"Mathlib/LinearAlgebra/Matrix/Block.lean","def_pos":[169,8],"def_end_pos":[169,19]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nm : Type u_3\nn : Type u_4\no : Type u_5\nm' : α → Type u_6\nn' : α → Type u_7\nR : Type v\ninst✝⁵ : CommRing R\nM✝ N : Matrix m m R\nb : m → α\ninst✝⁴ : DecidableEq m\ninst✝³ : Fintype m\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nM : Matrix m m R\np : m → Prop\ninst✝ : DecidablePred p\nh : ∀ (i : m), ¬p i → ∀ (j : m), p j → M i j = 0\n⊢ (fromBlocks (M.toBlock p p) (M.toBlock p fun j => ¬p j) (M.toBlock (fun j => ¬p j) p)\n        (M.toBlock (fun j => ¬p j) fun j => ¬p j)).det =\n    (M.toSquareBlockProp p).det * (M.toSquareBlockProp fun i => ¬p i).det","state_after":"case h.e'_2.h.e'_6.h.e'_8\nα : Type u_1\nβ : Type u_2\nm : Type u_3\nn : Type u_4\no : Type u_5\nm' : α → Type u_6\nn' : α → Type u_7\nR : Type v\ninst✝⁵ : CommRing R\nM✝ N : Matrix m m R\nb : m → α\ninst✝⁴ : DecidableEq m\ninst✝³ : Fintype m\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nM : Matrix m m R\np : m → Prop\ninst✝ : DecidablePred p\nh : ∀ (i : m), ¬p i → ∀ (j : m), p j → M i j = 0\n⊢ M.toBlock (fun j => ¬p j) p = 0","tactic":"convert det_fromBlocks_zero₂₁ (toBlock M p p) (toBlock M p fun j => ¬p j)\n      (toBlock M (fun j => ¬p j) fun j => ¬p j)","premises":[{"full_name":"Matrix.det_fromBlocks_zero₂₁","def_path":"Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean","def_pos":[608,8],"def_end_pos":[608,29]},{"full_name":"Matrix.toBlock","def_path":"Mathlib/Data/Matrix/Block.lean","def_pos":[162,4],"def_end_pos":[162,11]},{"full_name":"Not","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[224,4],"def_end_pos":[224,7]}]},{"state_before":"case h.e'_2.h.e'_6.h.e'_8\nα : Type u_1\nβ : Type u_2\nm : Type u_3\nn : Type u_4\no : Type u_5\nm' : α → Type u_6\nn' : α → Type u_7\nR : Type v\ninst✝⁵ : CommRing R\nM✝ N : Matrix m m R\nb : m → α\ninst✝⁴ : DecidableEq m\ninst✝³ : Fintype m\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nM : Matrix m m R\np : m → Prop\ninst✝ : DecidablePred p\nh : ∀ (i : m), ¬p i → ∀ (j : m), p j → M i j = 0\n⊢ M.toBlock (fun j => ¬p j) p = 0","state_after":"case h.e'_2.h.e'_6.h.e'_8.a\nα : Type u_1\nβ : Type u_2\nm : Type u_3\nn : Type u_4\no : Type u_5\nm' : α → Type u_6\nn' : α → Type u_7\nR : Type v\ninst✝⁵ : CommRing R\nM✝ N : Matrix m m R\nb : m → α\ninst✝⁴ : DecidableEq m\ninst✝³ : Fintype m\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nM : Matrix m m R\np : m → Prop\ninst✝ : DecidablePred p\nh : ∀ (i : m), ¬p i → ∀ (j : m), p j → M i j = 0\ni : { a // ¬p a }\nj : { a // p a }\n⊢ M.toBlock (fun j => ¬p j) p i j = 0 i j","tactic":"ext i j","premises":[]},{"state_before":"case h.e'_2.h.e'_6.h.e'_8.a\nα : Type u_1\nβ : Type u_2\nm : Type u_3\nn : Type u_4\no : Type u_5\nm' : α → Type u_6\nn' : α → Type u_7\nR : Type v\ninst✝⁵ : CommRing R\nM✝ N : Matrix m m R\nb : m → α\ninst✝⁴ : DecidableEq m\ninst✝³ : Fintype m\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nM : Matrix m m R\np : m → Prop\ninst✝ : DecidablePred p\nh : ∀ (i : m), ¬p i → ∀ (j : m), p j → M i j = 0\ni : { a // ¬p a }\nj : { a // p a }\n⊢ M.toBlock (fun j => ¬p j) p i j = 0 i j","state_after":"no goals","tactic":"exact h (↑i) i.2 (↑j) j.2","premises":[{"full_name":"Subtype.property","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[590,2],"def_end_pos":[590,10]}]}]}
+{"url":"Mathlib/Order/ModularLattice.lean","commit":"","full_name":"eq_of_le_of_inf_le_of_le_sup","start":[204,0],"end":[208,58],"file_path":"Mathlib/Order/ModularLattice.lean","tactics":[{"state_before":"α : Type u_1\ninst✝¹ : Lattice α\ninst✝ : IsModularLattice α\nx y z : α\nhxy : x ≤ y\nhinf : y ⊓ z ≤ x\nhsup : y ≤ x ⊔ z\n⊢ x = y","state_after":"α : Type u_1\ninst✝¹ : Lattice α\ninst✝ : IsModularLattice α\nx y z : α\nhxy : x ≤ y\nhinf : y ⊓ z ≤ x\nhsup : y ≤ x ⊔ z\n⊢ y ≤ x","tactic":"refine hxy.antisymm ?_","premises":[]},{"state_before":"α : Type u_1\ninst✝¹ : Lattice α\ninst✝ : IsModularLattice α\nx y z : α\nhxy : x ≤ y\nhinf : y ⊓ z ≤ x\nhsup : y ≤ x ⊔ z\n⊢ y ≤ x","state_after":"α : Type u_1\ninst✝¹ : Lattice α\ninst✝ : IsModularLattice α\nx y z : α\nhxy : x ≤ y\nhinf : y ⊓ z ≤ x\nhsup : x ⊔ z ⊓ y = y\n⊢ y ≤ x","tactic":"rw [← inf_eq_right, sup_inf_assoc_of_le _ hxy] at hsup","premises":[{"full_name":"inf_eq_right","def_path":"Mathlib/Order/Lattice.lean","def_pos":[341,8],"def_end_pos":[341,20]},{"full_name":"sup_inf_assoc_of_le","def_path":"Mathlib/Order/ModularLattice.lean","def_pos":[182,8],"def_end_pos":[182,27]}]},{"state_before":"α : Type u_1\ninst✝¹ : Lattice α\ninst✝ : IsModularLattice α\nx y z : α\nhxy : x ≤ y\nhinf : y ⊓ z ≤ x\nhsup : x ⊔ z ⊓ y = y\n⊢ y ≤ x","state_after":"no goals","tactic":"rwa [← hsup, sup_le_iff, and_iff_right rfl.le, inf_comm]","premises":[{"full_name":"and_iff_right","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[70,8],"def_end_pos":[70,21]},{"full_name":"inf_comm","def_path":"Mathlib/Order/Lattice.lean","def_pos":[385,8],"def_end_pos":[385,16]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]},{"full_name":"sup_le_iff","def_path":"Mathlib/Order/Lattice.lean","def_pos":[133,8],"def_end_pos":[133,18]}]}]}
+{"url":"Mathlib/FieldTheory/AbelRuffini.lean","commit":"","full_name":"solvableByRad.induction","start":[222,0],"end":[254,75],"file_path":"Mathlib/FieldTheory/AbelRuffini.lean","tactics":[{"state_before":"F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nα✝ : E\nP : ↥(solvableByRad F E) → Prop\nbase : ∀ (α : F), P ((algebraMap F ↥(solvableByRad F E)) α)\nadd : ∀ (α β : ↥(solvableByRad F E)), P α → P β → P (α + β)\nneg : ∀ (α : ↥(solvableByRad F E)), P α → P (-α)\nmul : ∀ (α β : ↥(solvableByRad F E)), P α → P β → P (α * β)\ninv : ∀ (α : ↥(solvableByRad F E)), P α → P α⁻¹\nrad : ∀ (α : ↥(solvableByRad F E)) (n : ℕ), n ≠ 0 → P (α ^ n) → P α\nα : ↥(solvableByRad F E)\n⊢ P α","state_after":"F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nα : E\nP : ↥(solvableByRad F E) → Prop\nbase : ∀ (α : F), P ((algebraMap F ↥(solvableByRad F E)) α)\nadd : ∀ (α β : ↥(solvableByRad F E)), P α → P β → P (α + β)\nneg : ∀ (α : ↥(solvableByRad F E)), P α → P (-α)\nmul : ∀ (α β : ↥(solvableByRad F E)), P α → P β → P (α * β)\ninv : ∀ (α : ↥(solvableByRad F E)), P α → P α⁻¹\nrad : ∀ (α : ↥(solvableByRad F E)) (n : ℕ), n ≠ 0 → P (α ^ n) → P α\n⊢ ∀ (α : ↥(solvableByRad F E)), P α","tactic":"revert α","premises":[]},{"state_before":"F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nα : E\nP : ↥(solvableByRad F E) → Prop\nbase : ∀ (α : F), P ((algebraMap F ↥(solvableByRad F E)) α)\nadd : ∀ (α β : ↥(solvableByRad F E)), P α → P β → P (α + β)\nneg : ∀ (α : ↥(solvableByRad F E)), P α → P (-α)\nmul : ∀ (α β : ↥(solvableByRad F E)), P α → P β → P (α * β)\ninv : ∀ (α : ↥(solvableByRad F E)), P α → P α⁻¹\nrad : ∀ (α : ↥(solvableByRad F E)) (n : ℕ), n ≠ 0 → P (α ^ n) → P α\n⊢ ∀ (α : ↥(solvableByRad F E)), P α","state_after":"F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nα : E\nP : ↥(solvableByRad F E) → Prop\nbase : ∀ (α : F), P ((algebraMap F ↥(solvableByRad F E)) α)\nadd : ∀ (α β : ↥(solvableByRad F E)), P α → P β → P (α + β)\nneg : ∀ (α : ↥(solvableByRad F E)), P α → P (-α)\nmul : ∀ (α β : ↥(solvableByRad F E)), P α → P β → P (α * β)\ninv : ∀ (α : ↥(solvableByRad F E)), P α → P α⁻¹\nrad : ∀ (α : ↥(solvableByRad F E)) (n : ℕ), n ≠ 0 → P (α ^ n) → P α\n⊢ ∀ (α : E), IsSolvableByRad F α → ∃ β, ↑β = α ∧ P β","tactic":"suffices ∀ α : E, IsSolvableByRad F α → ∃ β : solvableByRad F E, ↑β = α ∧ P β by\n    intro α\n    obtain ⟨α₀, hα₀, Pα⟩ := this α (Subtype.mem α)\n    convert Pα\n    exact Subtype.ext hα₀.symm","premises":[{"full_name":"And","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[516,10],"def_end_pos":[516,13]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"IsSolvableByRad","def_path":"Mathlib/FieldTheory/AbelRuffini.lean","def_pos":[194,10],"def_end_pos":[194,25]},{"full_name":"Subtype.ext","def_path":"Mathlib/Data/Subtype.lean","def_pos":[59,18],"def_end_pos":[59,21]},{"full_name":"Subtype.mem","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[176,8],"def_end_pos":[176,19]},{"full_name":"solvableByRad","def_path":"Mathlib/FieldTheory/AbelRuffini.lean","def_pos":[205,4],"def_end_pos":[205,17]}]},{"state_before":"F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nα : E\nP : ↥(solvableByRad F E) → Prop\nbase : ∀ (α : F), P ((algebraMap F ↥(solvableByRad F E)) α)\nadd : ∀ (α β : ↥(solvableByRad F E)), P α → P β → P (α + β)\nneg : ∀ (α : ↥(solvableByRad F E)), P α → P (-α)\nmul : ∀ (α β : ↥(solvableByRad F E)), P α → P β → P (α * β)\ninv : ∀ (α : ↥(solvableByRad F E)), P α → P α⁻¹\nrad : ∀ (α : ↥(solvableByRad F E)) (n : ℕ), n ≠ 0 → P (α ^ n) → P α\n⊢ ∀ (α : E), IsSolvableByRad F α → ∃ β, ↑β = α ∧ P β","state_after":"case base\nF : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nα : E\nP : ↥(solvableByRad F E) → Prop\nbase : ∀ (α : F), P ((algebraMap F ↥(solvableByRad F E)) α)\nadd : ∀ (α β : ↥(solvableByRad F E)), P α → P β → P (α + β)\nneg : ∀ (α : ↥(solvableByRad F E)), P α → P (-α)\nmul : ∀ (α β : ↥(solvableByRad F E)), P α → P β → P (α * β)\ninv : ∀ (α : ↥(solvableByRad F E)), P α → P α⁻¹\nrad : ∀ (α : ↥(solvableByRad F E)) (n : ℕ), n ≠ 0 → P (α ^ n) → P α\n⊢ ∀ (α : F), ∃ β, ↑β = (algebraMap F E) α ∧ P β\n\ncase add\nF : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nα : E\nP : ↥(solvableByRad F E) → Prop\nbase : ∀ (α : F), P ((algebraMap F ↥(solvableByRad F E)) α)\nadd : ∀ (α β : ↥(solvableByRad F E)), P α → P β → P (α + β)\nneg : ∀ (α : ↥(solvableByRad F E)), P α → P (-α)\nmul : ∀ (α β : ↥(solvableByRad F E)), P α → P β → P (α * β)\ninv : ∀ (α : ↥(solvableByRad F E)), P α → P α⁻¹\nrad : ∀ (α : ↥(solvableByRad F E)) (n : ℕ), n ≠ 0 → P (α ^ n) → P α\n⊢ ∀ (α β : E),\n    IsSolvableByRad F α →\n      IsSolvableByRad F β → (∃ β, ↑β = α ∧ P β) → (∃ β_1, ↑β_1 = β ∧ P β_1) → ∃ β_1, ↑β_1 = α + β ∧ P β_1\n\ncase neg\nF : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nα : E\nP : ↥(solvableByRad F E) → Prop\nbase : ∀ (α : F), P ((algebraMap F ↥(solvableByRad F E)) α)\nadd : ∀ (α β : ↥(solvableByRad F E)), P α → P β → P (α + β)\nneg : ∀ (α : ↥(solvableByRad F E)), P α → P (-α)\nmul : ∀ (α β : ↥(solvableByRad F E)), P α → P β → P (α * β)\ninv : ∀ (α : ↥(solvableByRad F E)), P α → P α⁻¹\nrad : ∀ (α : ↥(solvableByRad F E)) (n : ℕ), n ≠ 0 → P (α ^ n) → P α\n⊢ ∀ (α : E), IsSolvableByRad F α → (∃ β, ↑β = α ∧ P β) → ∃ β, ↑β = -α ∧ P β\n\ncase mul\nF : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nα : E\nP : ↥(solvableByRad F E) → Prop\nbase : ∀ (α : F), P ((algebraMap F ↥(solvableByRad F E)) α)\nadd : ∀ (α β : ↥(solvableByRad F E)), P α → P β → P (α + β)\nneg : ∀ (α : ↥(solvableByRad F E)), P α → P (-α)\nmul : ∀ (α β : ↥(solvableByRad F E)), P α → P β → P (α * β)\ninv : ∀ (α : ↥(solvableByRad F E)), P α → P α⁻¹\nrad : ∀ (α : ↥(solvableByRad F E)) (n : ℕ), n ≠ 0 → P (α ^ n) → P α\n⊢ ∀ (α β : E),\n    IsSolvableByRad F α →\n      IsSolvableByRad F β → (∃ β, ↑β = α ∧ P β) → (∃ β_1, ↑β_1 = β ∧ P β_1) → ∃ β_1, ↑β_1 = α * β ∧ P β_1\n\ncase inv\nF : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nα : E\nP : ↥(solvableByRad F E) → Prop\nbase : ∀ (α : F), P ((algebraMap F ↥(solvableByRad F E)) α)\nadd : ∀ (α β : ↥(solvableByRad F E)), P α → P β → P (α + β)\nneg : ∀ (α : ↥(solvableByRad F E)), P α → P (-α)\nmul : ∀ (α β : ↥(solvableByRad F E)), P α → P β → P (α * β)\ninv : ∀ (α : ↥(solvableByRad F E)), P α → P α⁻¹\nrad : ∀ (α : ↥(solvableByRad F E)) (n : ℕ), n ≠ 0 → P (α ^ n) → P α\n⊢ ∀ (α : E), IsSolvableByRad F α → (∃ β, ↑β = α ∧ P β) → ∃ β, ↑β = α⁻¹ ∧ P β\n\ncase rad\nF : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nα : E\nP : ↥(solvableByRad F E) → Prop\nbase : ∀ (α : F), P ((algebraMap F ↥(solvableByRad F E)) α)\nadd : ∀ (α β : ↥(solvableByRad F E)), P α → P β → P (α + β)\nneg : ∀ (α : ↥(solvableByRad F E)), P α → P (-α)\nmul : ∀ (α β : ↥(solvableByRad F E)), P α → P β → P (α * β)\ninv : ∀ (α : ↥(solvableByRad F E)), P α → P α⁻¹\nrad : ∀ (α : ↥(solvableByRad F E)) (n : ℕ), n ≠ 0 → P (α ^ n) → P α\n⊢ ∀ (α : E) (n : ℕ), n ≠ 0 → IsSolvableByRad F (α ^ n) → (∃ β, ↑β = α ^ n ∧ P β) → ∃ β, ↑β = α ∧ P β","tactic":"apply IsSolvableByRad.rec","premises":[]}]}
+{"url":"Mathlib/Data/Nat/ModEq.lean","commit":"","full_name":"Nat.add_modEq_left","start":[198,0],"end":[198,79],"file_path":"Mathlib/Data/Nat/ModEq.lean","tactics":[{"state_before":"m n a b c d : ℕ\n⊢ n + a ≡ a [MOD n]","state_after":"no goals","tactic":"rw [ModEq, add_mod_left]","premises":[{"full_name":"Nat.ModEq","def_path":"Mathlib/Data/Nat/ModEq.lean","def_pos":[32,4],"def_end_pos":[32,9]},{"full_name":"Nat.add_mod_left","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Div.lean","def_pos":[271,16],"def_end_pos":[271,28]}]}]}
+{"url":"Mathlib/LinearAlgebra/AffineSpace/Slope.lean","commit":"","full_name":"lineMap_slope_lineMap_slope_lineMap","start":[112,0],"end":[120,19],"file_path":"Mathlib/LinearAlgebra/AffineSpace/Slope.lean","tactics":[{"state_before":"k : Type u_1\nE : Type u_2\nPE : Type u_3\ninst✝³ : Field k\ninst✝² : AddCommGroup E\ninst✝¹ : Module k E\ninst✝ : AddTorsor E PE\nf : k → PE\na b r : k\n⊢ (lineMap (slope f ((lineMap a b) r) b) (slope f a ((lineMap a b) r))) r = slope f a b","state_after":"case inl\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst✝³ : Field k\ninst✝² : AddCommGroup E\ninst✝¹ : Module k E\ninst✝ : AddTorsor E PE\nf : k → PE\na r : k\n⊢ (lineMap (slope f ((lineMap a a) r) a) (slope f a ((lineMap a a) r))) r = slope f a a\n\ncase inr\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst✝³ : Field k\ninst✝² : AddCommGroup E\ninst✝¹ : Module k E\ninst✝ : AddTorsor E PE\nf : k → PE\na b r : k\nhab : a ≠ b\n⊢ (lineMap (slope f ((lineMap a b) r) b) (slope f a ((lineMap a b) r))) r = slope f a b","tactic":"obtain rfl | hab : a = b ∨ a ≠ b := Classical.em _","premises":[{"full_name":"Classical.em","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[32,8],"def_end_pos":[32,10]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Or","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[532,10],"def_end_pos":[532,12]}]},{"state_before":"case inr\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst✝³ : Field k\ninst✝² : AddCommGroup E\ninst✝¹ : Module k E\ninst✝ : AddTorsor E PE\nf : k → PE\na b r : k\nhab : a ≠ b\n⊢ (lineMap (slope f ((lineMap a b) r) b) (slope f a ((lineMap a b) r))) r = slope f a b","state_after":"case inr\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst✝³ : Field k\ninst✝² : AddCommGroup E\ninst✝¹ : Module k E\ninst✝ : AddTorsor E PE\nf : k → PE\na b r : k\nhab : a ≠ b\n⊢ (lineMap (slope f b ((lineMap a b) r)) (slope f ((lineMap a b) r) a)) r = slope f b a","tactic":"rw [slope_comm _ a, slope_comm _ a, slope_comm _ _ b]","premises":[{"full_name":"slope_comm","def_path":"Mathlib/LinearAlgebra/AffineSpace/Slope.lean","def_pos":[77,8],"def_end_pos":[77,18]}]},{"state_before":"case inr\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst✝³ : Field k\ninst✝² : AddCommGroup E\ninst✝¹ : Module k E\ninst✝ : AddTorsor E PE\nf : k → PE\na b r : k\nhab : a ≠ b\n⊢ (lineMap (slope f b ((lineMap a b) r)) (slope f ((lineMap a b) r) a)) r = slope f b a","state_after":"case h.e'_2.h.e'_6\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst✝³ : Field k\ninst✝² : AddCommGroup E\ninst✝¹ : Module k E\ninst✝ : AddTorsor E PE\nf : k → PE\na b r : k\nhab : a ≠ b\n⊢ r = (a - (lineMap a b) r) / (a - b)","tactic":"convert lineMap_slope_slope_sub_div_sub f b (lineMap a b r) a hab.symm using 2","premises":[{"full_name":"AffineMap.lineMap","def_path":"Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean","def_pos":[451,4],"def_end_pos":[451,11]},{"full_name":"Ne.symm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[704,8],"def_end_pos":[704,15]},{"full_name":"lineMap_slope_slope_sub_div_sub","def_path":"Mathlib/LinearAlgebra/AffineSpace/Slope.lean","def_pos":[107,8],"def_end_pos":[107,39]}]},{"state_before":"case h.e'_2.h.e'_6\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst✝³ : Field k\ninst✝² : AddCommGroup E\ninst✝¹ : Module k E\ninst✝ : AddTorsor E PE\nf : k → PE\na b r : k\nhab : a ≠ b\n⊢ r = (a - (lineMap a b) r) / (a - b)","state_after":"no goals","tactic":"rw [lineMap_apply_ring, eq_div_iff (sub_ne_zero.2 hab), sub_mul, one_mul, mul_sub, ← sub_sub,\n    sub_sub_cancel]","premises":[{"full_name":"AffineMap.lineMap_apply_ring","def_path":"Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean","def_pos":[469,8],"def_end_pos":[469,26]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"eq_div_iff","def_path":"Mathlib/Algebra/GroupWithZero/Units/Basic.lean","def_pos":[299,21],"def_end_pos":[299,31]},{"full_name":"one_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[409,8],"def_end_pos":[409,15]},{"full_name":"sub_ne_zero","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[746,2],"def_end_pos":[746,13]},{"full_name":"sub_sub","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[542,2],"def_end_pos":[542,13]},{"full_name":"sub_sub_cancel","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[910,2],"def_end_pos":[910,13]}]}]}
+{"url":"Mathlib/Probability/IdentDistrib.lean","commit":"","full_name":"ProbabilityTheory.IdentDistrib.lintegral_eq","start":[166,0],"end":[170,64],"file_path":"Mathlib/Probability/IdentDistrib.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\ninst✝ : MeasurableSpace δ\nμ : Measure α\nν : Measure β\nf✝ : α → γ\ng✝ : β → γ\nf : α → ℝ≥0∞\ng : β → ℝ≥0∞\nh : IdentDistrib f g μ ν\n⊢ ∫⁻ (x : α), f x ∂μ = ∫⁻ (x : β), g x ∂ν","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\ninst✝ : MeasurableSpace δ\nμ : Measure α\nν : Measure β\nf✝ : α → γ\ng✝ : β → γ\nf : α → ℝ≥0∞\ng : β → ℝ≥0∞\nh : IdentDistrib f g μ ν\n⊢ ∫⁻ (x : α), id (f x) ∂μ = ∫⁻ (x : β), id (g x) ∂ν","tactic":"change ∫⁻ x, id (f x) ∂μ = ∫⁻ x, id (g x) ∂ν","premises":[{"full_name":"MeasureTheory.lintegral","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[59,16],"def_end_pos":[59,25]},{"full_name":"id","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[33,14],"def_end_pos":[33,16]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\ninst✝ : MeasurableSpace δ\nμ : Measure α\nν : Measure β\nf✝ : α → γ\ng✝ : β → γ\nf : α → ℝ≥0∞\ng : β → ℝ≥0∞\nh : IdentDistrib f g μ ν\n⊢ ∫⁻ (x : α), id (f x) ∂μ = ∫⁻ (x : β), id (g x) ∂ν","state_after":"no goals","tactic":"rw [← lintegral_map' aemeasurable_id h.aemeasurable_fst, ←\n    lintegral_map' aemeasurable_id h.aemeasurable_snd, h.map_eq]","premises":[{"full_name":"MeasureTheory.lintegral_map'","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[1364,8],"def_end_pos":[1364,22]},{"full_name":"ProbabilityTheory.IdentDistrib.aemeasurable_fst","def_path":"Mathlib/Probability/IdentDistrib.lean","def_pos":[69,2],"def_end_pos":[69,18]},{"full_name":"ProbabilityTheory.IdentDistrib.aemeasurable_snd","def_path":"Mathlib/Probability/IdentDistrib.lean","def_pos":[70,2],"def_end_pos":[70,18]},{"full_name":"ProbabilityTheory.IdentDistrib.map_eq","def_path":"Mathlib/Probability/IdentDistrib.lean","def_pos":[71,2],"def_end_pos":[71,8]},{"full_name":"aemeasurable_id","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[411,8],"def_end_pos":[411,23]}]}]}
+{"url":"Mathlib/Order/Filter/Bases.lean","commit":"","full_name":"Filter.antitone_seq_of_seq","start":[902,0],"end":[911,7],"file_path":"Mathlib/Order/Filter/Bases.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Sort u_5\ns : ℕ → Set α\n⊢ ∃ t, Antitone t ∧ ⨅ i, 𝓟 (s i) = ⨅ i, 𝓟 (t i)","state_after":"case h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Sort u_5\ns : ℕ → Set α\n⊢ (Antitone fun n => ⋂ m, ⋂ (_ : m ≤ n), s m) ∧ ⨅ i, 𝓟 (s i) = ⨅ i, 𝓟 (⋂ m, ⋂ (_ : m ≤ i), s m)","tactic":"use fun n => ⋂ m ≤ n, s m","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Set.iInter","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[182,4],"def_end_pos":[182,10]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]}]},{"state_before":"case h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Sort u_5\ns : ℕ → Set α\n⊢ (Antitone fun n => ⋂ m, ⋂ (_ : m ≤ n), s m) ∧ ⨅ i, 𝓟 (s i) = ⨅ i, 𝓟 (⋂ m, ⋂ (_ : m ≤ i), s m)","state_after":"case h.left\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Sort u_5\ns : ℕ → Set α\n⊢ Antitone fun n => ⋂ m, ⋂ (_ : m ≤ n), s m\n\ncase h.right\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Sort u_5\ns : ℕ → Set α\n⊢ ⨅ i, 𝓟 (s i) = ⨅ i, 𝓟 (⋂ m, ⋂ (_ : m ≤ i), s m)","tactic":"constructor","premises":[]},{"state_before":"case h.right\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Sort u_5\ns : ℕ → Set α\n⊢ ⨅ i, 𝓟 (s i) = ⨅ i, 𝓟 (⋂ m, ⋂ (_ : m ≤ i), s m)","state_after":"case h.right.a\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Sort u_5\ns : ℕ → Set α\ni : ℕ\n⊢ ⨅ i, 𝓟 (s i) ≤ 𝓟 (⋂ m, ⋂ (_ : m ≤ i), s m)\n\ncase h.right.a\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Sort u_5\ns : ℕ → Set α\ni : ℕ\n⊢ ⨅ i, 𝓟 (⋂ m, ⋂ (_ : m ≤ i), s m) ≤ 𝓟 (s i)","tactic":"apply le_antisymm <;> rw [le_iInf_iff] <;> intro i","premises":[{"full_name":"le_antisymm","def_path":"Mathlib/Order/Defs.lean","def_pos":[156,8],"def_end_pos":[156,19]},{"full_name":"le_iInf_iff","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[735,8],"def_end_pos":[735,19]}]}]}
+{"url":"Mathlib/Algebra/Order/Group/Abs.lean","commit":"","full_name":"abs_eq_neg_self","start":[199,0],"end":[201,56],"file_path":"Mathlib/Algebra/Order/Group/Abs.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\na b c d : α\n⊢ |a| = -a ↔ a ≤ 0","state_after":"no goals","tactic":"rw [abs_eq_max_neg, max_eq_right_iff, le_neg_self_iff]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"abs_eq_max_neg","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[188,2],"def_end_pos":[188,13]},{"full_name":"le_neg_self_iff","def_path":"Mathlib/Algebra/Order/Group/Defs.lean","def_pos":[158,2],"def_end_pos":[158,13]},{"full_name":"max_eq_right_iff","def_path":"Mathlib/Order/MinMax.lean","def_pos":[130,8],"def_end_pos":[130,24]}]}]}
+{"url":"Mathlib/Algebra/Order/UpperLower.lean","commit":"","full_name":"IsUpperSet.mul_left","start":[50,0],"end":[53,44],"file_path":"Mathlib/Algebra/Order/UpperLower.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : OrderedCommGroup α\ns t : Set α\na : α\nht : IsUpperSet t\n⊢ IsUpperSet (s * t)","state_after":"α : Type u_1\ninst✝ : OrderedCommGroup α\ns t : Set α\na : α\nht : IsUpperSet t\n⊢ IsUpperSet (⋃ a ∈ s, a • t)","tactic":"rw [← smul_eq_mul, ← Set.iUnion_smul_set]","premises":[{"full_name":"Set.iUnion_smul_set","def_path":"Mathlib/Data/Set/Pointwise/SMul.lean","def_pos":[222,8],"def_end_pos":[222,23]},{"full_name":"smul_eq_mul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[79,6],"def_end_pos":[79,17]}]},{"state_before":"α : Type u_1\ninst✝ : OrderedCommGroup α\ns t : Set α\na : α\nht : IsUpperSet t\n⊢ IsUpperSet (⋃ a ∈ s, a • t)","state_after":"no goals","tactic":"exact isUpperSet_iUnion₂ fun x _ ↦ ht.smul","premises":[{"full_name":"IsUpperSet.smul","def_path":"Mathlib/Algebra/Order/UpperLower.lean","def_pos":[40,8],"def_end_pos":[40,23]},{"full_name":"isUpperSet_iUnion₂","def_path":"Mathlib/Order/UpperLower/Basic.lean","def_pos":[116,8],"def_end_pos":[116,26]}]}]}
+{"url":"Mathlib/Data/Set/Functor.lean","commit":"","full_name":"Set.image2_def","start":[57,0],"end":[62,6],"file_path":"Mathlib/Data/Set/Functor.lean","tactics":[{"state_before":"α✝ β✝ : Type u\ns✝ : Set α✝\nf✝ : α✝ → Set β✝\ng : Set (α✝ → β✝)\nα β γ : Type u\nf : α → β → γ\ns : Set α\nt : Set β\n⊢ image2 f s t = Seq.seq (f <$> s) fun x => t","state_after":"case h\nα✝ β✝ : Type u\ns✝ : Set α✝\nf✝ : α✝ → Set β✝\ng : Set (α✝ → β✝)\nα β γ : Type u\nf : α → β → γ\ns : Set α\nt : Set β\nx✝ : γ\n⊢ x✝ ∈ image2 f s t ↔ x✝ ∈ Seq.seq (f <$> s) fun x => t","tactic":"ext","premises":[]},{"state_before":"case h\nα✝ β✝ : Type u\ns✝ : Set α✝\nf✝ : α✝ → Set β✝\ng : Set (α✝ → β✝)\nα β γ : Type u\nf : α → β → γ\ns : Set α\nt : Set β\nx✝ : γ\n⊢ x✝ ∈ image2 f s t ↔ x✝ ∈ Seq.seq (f <$> s) fun x => t","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/Analysis/SpecialFunctions/Log/Base.lean","commit":"","full_name":"Real.logb_rpow","start":[113,0],"end":[116,36],"file_path":"Mathlib/Analysis/SpecialFunctions/Log/Base.lean","tactics":[{"state_before":"b x y : ℝ\nb_pos : 0 < b\nb_ne_one : b ≠ 1\n⊢ logb b (b ^ x) = x","state_after":"b x y : ℝ\nb_pos : 0 < b\nb_ne_one : b ≠ 1\n⊢ log b ≠ 0","tactic":"rw [logb, div_eq_iff, log_rpow b_pos]","premises":[{"full_name":"Real.log_rpow","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","def_pos":[436,8],"def_end_pos":[436,16]},{"full_name":"Real.logb","def_path":"Mathlib/Analysis/SpecialFunctions/Log/Base.lean","def_pos":[38,18],"def_end_pos":[38,22]},{"full_name":"div_eq_iff","def_path":"Mathlib/Algebra/GroupWithZero/Units/Basic.lean","def_pos":[297,21],"def_end_pos":[297,31]}]},{"state_before":"b x y : ℝ\nb_pos : 0 < b\nb_ne_one : b ≠ 1\n⊢ log b ≠ 0","state_after":"no goals","tactic":"exact log_b_ne_zero b_pos b_ne_one","premises":[{"full_name":"_private.Mathlib.Analysis.SpecialFunctions.Log.Base.0.Real.log_b_ne_zero","def_path":"Mathlib/Analysis/SpecialFunctions/Log/Base.lean","def_pos":[108,16],"def_end_pos":[108,29]}]}]}
+{"url":"Mathlib/Analysis/MeanInequalities.lean","commit":"","full_name":"NNReal.geom_mean_le_arith_mean2_weighted","start":[205,0],"end":[211,61],"file_path":"Mathlib/Analysis/MeanInequalities.lean","tactics":[{"state_before":"ι : Type u\ns : Finset ι\nw₁ w₂ p₁ p₂ : ℝ≥0\n⊢ w₁ + w₂ = 1 → p₁ ^ ↑w₁ * p₂ ^ ↑w₂ ≤ w₁ * p₁ + w₂ * p₂","state_after":"no goals","tactic":"simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,\n    Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using\n    geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]","premises":[{"full_name":"Fin.cons_succ","def_path":"Mathlib/Data/Fin/Tuple/Basic.lean","def_pos":[115,8],"def_end_pos":[115,17]},{"full_name":"Fin.cons_zero","def_path":"Mathlib/Data/Fin/Tuple/Basic.lean","def_pos":[118,8],"def_end_pos":[118,17]},{"full_name":"Fin.prod_univ_succ","def_path":"Mathlib/Algebra/BigOperators/Fin.lean","def_pos":[68,8],"def_end_pos":[68,22]},{"full_name":"Fin.sum_univ_succ","def_path":"Mathlib/Algebra/BigOperators/Fin.lean","def_pos":[66,2],"def_end_pos":[66,13]},{"full_name":"Finset.prod_empty","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[314,8],"def_end_pos":[314,18]},{"full_name":"Finset.sum_empty","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[313,2],"def_end_pos":[313,13]},{"full_name":"Finset.univ","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[67,4],"def_end_pos":[67,8]},{"full_name":"Finset.univ_eq_empty","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[112,8],"def_end_pos":[112,21]},{"full_name":"Matrix.vecCons","def_path":"Mathlib/Data/Fin/VecNotation.lean","def_pos":[56,4],"def_end_pos":[56,11]},{"full_name":"Matrix.vecEmpty","def_path":"Mathlib/Data/Fin/VecNotation.lean","def_pos":[48,4],"def_end_pos":[48,12]},{"full_name":"NNReal.geom_mean_le_arith_mean_weighted","def_path":"Mathlib/Analysis/MeanInequalities.lean","def_pos":[199,8],"def_end_pos":[199,40]},{"full_name":"add_zero","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[412,2],"def_end_pos":[412,13]},{"full_name":"mul_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[413,8],"def_end_pos":[413,15]}]}]}
+{"url":"Mathlib/Data/Stream/Init.lean","commit":"","full_name":"Stream'.append_append_stream","start":[440,0],"end":[444,90],"file_path":"Mathlib/Data/Stream/Init.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nδ : Type w\na : α\nl₁ l₂ : List α\ns : Stream' α\n⊢ a :: l₁ ++ l₂ ++ₛ s = a :: l₁ ++ₛ (l₂ ++ₛ s)","state_after":"no goals","tactic":"rw [List.cons_append, cons_append_stream, cons_append_stream, append_append_stream l₁]","premises":[{"full_name":"List.cons_append","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[484,16],"def_end_pos":[484,27]},{"full_name":"Stream'.cons_append_stream","def_path":"Mathlib/Data/Stream/Init.lean","def_pos":[436,8],"def_end_pos":[436,26]}]}]}
+{"url":"Mathlib/Algebra/BigOperators/Finprod.lean","commit":"","full_name":"finsum_pos'","start":[487,0],"end":[493,68],"file_path":"Mathlib/Algebra/BigOperators/Finprod.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nG : Type u_4\nM✝ : Type u_5\nN : Type u_6\ninst✝² : CommMonoid M✝\ninst✝¹ : CommMonoid N\nf✝ g : α → M✝\na b : α\ns t : Set α\nM : Type u_7\ninst✝ : OrderedCancelCommMonoid M\nf : ι → M\nh : ∀ (i : ι), 1 ≤ f i\nh' : ∃ i, 1 < f i\nhf : (mulSupport f).Finite\n⊢ 1 < ∏ᶠ (i : ι), f i","state_after":"case intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nG : Type u_4\nM✝ : Type u_5\nN : Type u_6\ninst✝² : CommMonoid M✝\ninst✝¹ : CommMonoid N\nf✝ g : α → M✝\na b : α\ns t : Set α\nM : Type u_7\ninst✝ : OrderedCancelCommMonoid M\nf : ι → M\nh : ∀ (i : ι), 1 ≤ f i\nhf : (mulSupport f).Finite\ni : ι\nhi : 1 < f i\n⊢ 1 < ∏ᶠ (i : ι), f i","tactic":"rcases h' with ⟨i, hi⟩","premises":[]},{"state_before":"case intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nG : Type u_4\nM✝ : Type u_5\nN : Type u_6\ninst✝² : CommMonoid M✝\ninst✝¹ : CommMonoid N\nf✝ g : α → M✝\na b : α\ns t : Set α\nM : Type u_7\ninst✝ : OrderedCancelCommMonoid M\nf : ι → M\nh : ∀ (i : ι), 1 ≤ f i\nhf : (mulSupport f).Finite\ni : ι\nhi : 1 < f i\n⊢ 1 < ∏ᶠ (i : ι), f i","state_after":"case intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nG : Type u_4\nM✝ : Type u_5\nN : Type u_6\ninst✝² : CommMonoid M✝\ninst✝¹ : CommMonoid N\nf✝ g : α → M✝\na b : α\ns t : Set α\nM : Type u_7\ninst✝ : OrderedCancelCommMonoid M\nf : ι → M\nh : ∀ (i : ι), 1 ≤ f i\nhf : (mulSupport f).Finite\ni : ι\nhi : 1 < f i\n⊢ 1 < ∏ i ∈ hf.toFinset, f i","tactic":"rw [finprod_eq_prod _ hf]","premises":[{"full_name":"finprod_eq_prod","def_path":"Mathlib/Algebra/BigOperators/Finprod.lean","def_pos":[365,8],"def_end_pos":[365,23]}]},{"state_before":"case intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nG : Type u_4\nM✝ : Type u_5\nN : Type u_6\ninst✝² : CommMonoid M✝\ninst✝¹ : CommMonoid N\nf✝ g : α → M✝\na b : α\ns t : Set α\nM : Type u_7\ninst✝ : OrderedCancelCommMonoid M\nf : ι → M\nh : ∀ (i : ι), 1 ≤ f i\nhf : (mulSupport f).Finite\ni : ι\nhi : 1 < f i\n⊢ 1 < ∏ i ∈ hf.toFinset, f i","state_after":"case intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nG : Type u_4\nM✝ : Type u_5\nN : Type u_6\ninst✝² : CommMonoid M✝\ninst✝¹ : CommMonoid N\nf✝ g : α → M✝\na b : α\ns t : Set α\nM : Type u_7\ninst✝ : OrderedCancelCommMonoid M\nf : ι → M\nh : ∀ (i : ι), 1 ≤ f i\nhf : (mulSupport f).Finite\ni : ι\nhi : 1 < f i\n⊢ i ∈ hf.toFinset","tactic":"refine Finset.one_lt_prod' (fun i _ ↦ h i) ⟨i, ?_, hi⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Finset.one_lt_prod'","def_path":"Mathlib/Algebra/Order/BigOperators/Group/Finset.lean","def_pos":[445,8],"def_end_pos":[445,20]}]},{"state_before":"case intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nG : Type u_4\nM✝ : Type u_5\nN : Type u_6\ninst✝² : CommMonoid M✝\ninst✝¹ : CommMonoid N\nf✝ g : α → M✝\na b : α\ns t : Set α\nM : Type u_7\ninst✝ : OrderedCancelCommMonoid M\nf : ι → M\nh : ∀ (i : ι), 1 ≤ f i\nhf : (mulSupport f).Finite\ni : ι\nhi : 1 < f i\n⊢ i ∈ hf.toFinset","state_after":"no goals","tactic":"simpa only [Finite.mem_toFinset, mem_mulSupport] using ne_of_gt hi","premises":[{"full_name":"Function.mem_mulSupport","def_path":"Mathlib/Algebra/Group/Support.lean","def_pos":[44,8],"def_end_pos":[44,22]},{"full_name":"Set.Finite.mem_toFinset","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[146,18],"def_end_pos":[146,30]},{"full_name":"ne_of_gt","def_path":"Mathlib/Order/Defs.lean","def_pos":[85,8],"def_end_pos":[85,16]}]}]}
+{"url":"Mathlib/Algebra/GroupWithZero/Units/Basic.lean","commit":"","full_name":"Ring.mul_inverse_cancel_left","start":[109,0],"end":[110,51],"file_path":"Mathlib/Algebra/GroupWithZero/Units/Basic.lean","tactics":[{"state_before":"α : Type u_1\nM₀ : Type u_2\nG₀ : Type u_3\nM₀' : Type u_4\nG₀' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst✝ : MonoidWithZero M₀\nx y : M₀\nh : IsUnit x\n⊢ x * (inverse x * y) = y","state_after":"no goals","tactic":"rw [← mul_assoc, mul_inverse_cancel x h, one_mul]","premises":[{"full_name":"Ring.mul_inverse_cancel","def_path":"Mathlib/Algebra/GroupWithZero/Units/Basic.lean","def_pos":[95,8],"def_end_pos":[95,26]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]},{"full_name":"one_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[409,8],"def_end_pos":[409,15]}]}]}
+{"url":"Mathlib/Order/Heyting/Basic.lean","commit":"","full_name":"hnot_sdiff","start":[788,0],"end":[789,87],"file_path":"Mathlib/Order/Heyting/Basic.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : CoheytingAlgebra α\na✝ b c a : α\n⊢ ￢a \\ a = ￢a","state_after":"no goals","tactic":"rw [← top_sdiff', sdiff_sdiff, sup_idem]","premises":[{"full_name":"sdiff_sdiff","def_path":"Mathlib/Order/Heyting/Basic.lean","def_pos":[465,8],"def_end_pos":[465,19]},{"full_name":"sup_idem","def_path":"Mathlib/Order/Lattice.lean","def_pos":[189,8],"def_end_pos":[189,16]},{"full_name":"top_sdiff'","def_path":"Mathlib/Order/Heyting/Basic.lean","def_pos":[766,8],"def_end_pos":[766,18]}]}]}
+{"url":"Mathlib/Algebra/Group/Submonoid/Operations.lean","commit":"","full_name":"Nat.addSubmonoid_closure_one","start":[1179,0],"end":[1182,85],"file_path":"Mathlib/Algebra/Group/Submonoid/Operations.lean","tactics":[{"state_before":"M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\n⊢ closure {1} = ⊤","state_after":"M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\n⊢ ∀ n ∈ closure {1}, n.succ ∈ closure {1}","tactic":"refine (eq_top_iff' _).2 <| Nat.rec (zero_mem _) ?_","premises":[{"full_name":"AddSubmonoid.eq_top_iff'","def_path":"Mathlib/Algebra/Group/Submonoid/Operations.lean","def_pos":[992,2],"def_end_pos":[992,13]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"ZeroMemClass.zero_mem","def_path":"Mathlib/Algebra/Group/Submonoid/Basic.lean","def_pos":[77,2],"def_end_pos":[77,10]}]},{"state_before":"M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\n⊢ ∀ n ∈ closure {1}, n.succ ∈ closure {1}","state_after":"M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\n⊢ ∀ n ∈ closure {1}, n + 1 ∈ closure {1}","tactic":"simp_rw [Nat.succ_eq_add_one]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Nat.succ_eq_add_one","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[154,16],"def_end_pos":[154,31]}]},{"state_before":"M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\n⊢ ∀ n ∈ closure {1}, n + 1 ∈ closure {1}","state_after":"no goals","tactic":"exact fun n hn ↦ AddSubmonoid.add_mem _ hn <| subset_closure <| Set.mem_singleton _","premises":[{"full_name":"AddSubmonoid.add_mem","def_path":"Mathlib/Algebra/Group/Submonoid/Basic.lean","def_pos":[203,2],"def_end_pos":[203,13]},{"full_name":"AddSubmonoid.subset_closure","def_path":"Mathlib/Algebra/Group/Submonoid/Basic.lean","def_pos":[337,2],"def_end_pos":[337,13]},{"full_name":"Set.mem_singleton","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1015,8],"def_end_pos":[1015,21]}]}]}
+{"url":"Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean","commit":"","full_name":"Orientation.rotation_pi_apply","start":[137,0],"end":[138,66],"file_path":"Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean","tactics":[{"state_before":"V : Type u_1\nV' : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : NormedAddCommGroup V'\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : InnerProductSpace ℝ V'\ninst✝¹ : Fact (finrank ℝ V = 2)\ninst✝ : Fact (finrank ℝ V' = 2)\no : Orientation ℝ V (Fin 2)\nx : V\n⊢ (o.rotation ↑π) x = -x","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/Analysis/SpecialFunctions/NonIntegrable.lean","commit":"","full_name":"not_intervalIntegrable_of_sub_inv_isBigO_punctured","start":[167,0],"end":[181,74],"file_path":"Mathlib/Analysis/SpecialFunctions/NonIntegrable.lean","tactics":[{"state_before":"E : Type u_1\nF : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedAddCommGroup F\nf : ℝ → F\na b c : ℝ\nhf : (fun x => (x - c)⁻¹) =O[𝓝[≠] c] f\nhne : a ≠ b\nhc : c ∈ [[a, b]]\n⊢ ¬IntervalIntegrable f volume a b","state_after":"E : Type u_1\nF : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedAddCommGroup F\nf : ℝ → F\na b c : ℝ\nhf : (fun x => (x - c)⁻¹) =O[𝓝[≠] c] f\nhne : a ≠ b\nhc : c ∈ [[a, b]]\nA : ∀ᶠ (x : ℝ) in 𝓝[≠] c, HasDerivAt (fun x => Real.log (x - c)) (x - c)⁻¹ x\n⊢ ¬IntervalIntegrable f volume a b","tactic":"have A : ∀ᶠ x in ����[≠] c, HasDerivAt (fun x => Real.log (x - c)) (x - c)⁻¹ x := by\n    filter_upwards [self_mem_nhdsWithin] with x hx\n    simpa using ((hasDerivAt_id x).sub_const c).log (sub_ne_zero.2 hx)","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Filter.Eventually","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[948,14],"def_end_pos":[948,24]},{"full_name":"Filter.mp_mem","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[159,8],"def_end_pos":[159,14]},{"full_name":"HasCompl.compl","def_path":"Mathlib/Order/Notation.lean","def_pos":[34,2],"def_end_pos":[34,7]},{"full_name":"HasDerivAt","def_path":"Mathlib/Analysis/Calculus/Deriv/Basic.lean","def_pos":[121,4],"def_end_pos":[121,14]},{"full_name":"HasDerivAt.log","def_path":"Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean","def_pos":[93,8],"def_end_pos":[93,22]},{"full_name":"HasDerivAt.sub_const","def_path":"Mathlib/Analysis/Calculus/Deriv/Add.lean","def_pos":[287,15],"def_end_pos":[287,35]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"Real.log","def_path":"Mathlib/Analysis/SpecialFunctions/Log/Basic.lean","def_pos":[39,18],"def_end_pos":[39,21]},{"full_name":"Set","def_path":"Mathlib/Init/Set.lean","def_pos":[53,4],"def_end_pos":[53,7]},{"full_name":"Set.instSingletonSet","def_path":"Mathlib/Init/Set.lean","def_pos":[172,9],"def_end_pos":[172,25]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]},{"full_name":"Singleton.singleton","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[467,2],"def_end_pos":[467,11]},{"full_name":"hasDerivAt_id","def_path":"Mathlib/Analysis/Calculus/Deriv/Basic.lean","def_pos":[580,8],"def_end_pos":[580,21]},{"full_name":"nhdsWithin","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[121,4],"def_end_pos":[121,14]},{"full_name":"self_mem_nhdsWithin","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[124,8],"def_end_pos":[124,27]},{"full_name":"sub_ne_zero","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[746,2],"def_end_pos":[746,13]}]},{"state_before":"E : Type u_1\nF : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedAddCommGroup F\nf : ℝ → F\na b c : ℝ\nhf : (fun x => (x - c)⁻¹) =O[𝓝[≠] c] f\nhne : a ≠ b\nhc : c ∈ [[a, b]]\nA : ∀ᶠ (x : ℝ) in 𝓝[≠] c, HasDerivAt (fun x => Real.log (x - c)) (x - c)⁻¹ x\n⊢ ¬IntervalIntegrable f volume a b","state_after":"E : Type u_1\nF : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedAddCommGroup F\nf : ℝ → F\na b c : ℝ\nhf : (fun x => (x - c)⁻¹) =O[𝓝[≠] c] f\nhne : a ≠ b\nhc : c ∈ [[a, b]]\nA : ∀ᶠ (x : ℝ) in 𝓝[≠] c, HasDerivAt (fun x => Real.log (x - c)) (x - c)⁻¹ x\nB : Tendsto (fun x => ‖Real.log (x - c)‖) (𝓝[≠] c) atTop\n⊢ ¬IntervalIntegrable f volume a b","tactic":"have B : Tendsto (fun x => ‖Real.log (x - c)‖) (𝓝[≠] c) atTop := by\n    refine tendsto_abs_atBot_atTop.comp (Real.tendsto_log_nhdsWithin_zero.comp ?_)\n    rw [← sub_self c]\n    exact ((hasDerivAt_id c).sub_const c).tendsto_punctured_nhds one_ne_zero","premises":[{"full_name":"Filter.Tendsto","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2567,4],"def_end_pos":[2567,11]},{"full_name":"Filter.Tendsto.comp","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2652,8],"def_end_pos":[2652,20]},{"full_name":"Filter.atTop","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[40,4],"def_end_pos":[40,9]},{"full_name":"Filter.tendsto_abs_atBot_atTop","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[835,8],"def_end_pos":[835,31]},{"full_name":"HasCompl.compl","def_path":"Mathlib/Order/Notation.lean","def_pos":[34,2],"def_end_pos":[34,7]},{"full_name":"HasDerivAt.sub_const","def_path":"Mathlib/Analysis/Calculus/Deriv/Add.lean","def_pos":[287,15],"def_end_pos":[287,35]},{"full_name":"HasDerivAt.tendsto_punctured_nhds","def_path":"Mathlib/Analysis/Calculus/Deriv/Inverse.lean","def_pos":[97,8],"def_end_pos":[97,41]},{"full_name":"Norm.norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[59,2],"def_end_pos":[59,6]},{"full_name":"Real.log","def_path":"Mathlib/Analysis/SpecialFunctions/Log/Basic.lean","def_pos":[39,18],"def_end_pos":[39,21]},{"full_name":"Real.tendsto_log_nhdsWithin_zero","def_path":"Mathlib/Analysis/SpecialFunctions/Log/Basic.lean","def_pos":[294,8],"def_end_pos":[294,35]},{"full_name":"Set","def_path":"Mathlib/Init/Set.lean","def_pos":[53,4],"def_end_pos":[53,7]},{"full_name":"Set.instSingletonSet","def_path":"Mathlib/Init/Set.lean","def_pos":[172,9],"def_end_pos":[172,25]},{"full_name":"Singleton.singleton","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[467,2],"def_end_pos":[467,11]},{"full_name":"hasDerivAt_id","def_path":"Mathlib/Analysis/Calculus/Deriv/Basic.lean","def_pos":[580,8],"def_end_pos":[580,21]},{"full_name":"nhdsWithin","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[121,4],"def_end_pos":[121,14]},{"full_name":"one_ne_zero","def_path":"Mathlib/Algebra/NeZero.lean","def_pos":[49,14],"def_end_pos":[49,25]},{"full_name":"sub_self","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[695,29],"def_end_pos":[695,37]}]},{"state_before":"E : Type u_1\nF : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedAddCommGroup F\nf : ℝ → F\na b c : ℝ\nhf : (fun x => (x - c)⁻¹) =O[𝓝[≠] c] f\nhne : a ≠ b\nhc : c ∈ [[a, b]]\nA : ∀ᶠ (x : ℝ) in 𝓝[≠] c, HasDerivAt (fun x => Real.log (x - c)) (x - c)⁻¹ x\nB : Tendsto (fun x => ‖Real.log (x - c)‖) (𝓝[≠] c) atTop\n⊢ ¬IntervalIntegrable f volume a b","state_after":"no goals","tactic":"exact not_intervalIntegrable_of_tendsto_norm_atTop_of_deriv_isBigO_punctured\n    (A.mono fun x hx => hx.differentiableAt) B\n    (hf.congr' (A.mono fun x hx => hx.deriv.symm) EventuallyEq.rfl) hne hc","premises":[{"full_name":"Asymptotics.IsBigO.congr'","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[315,8],"def_end_pos":[315,21]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Filter.Eventually.mono","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1002,8],"def_end_pos":[1002,23]},{"full_name":"Filter.EventuallyEq.rfl","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1303,18],"def_end_pos":[1303,34]},{"full_name":"HasDerivAt.deriv","def_path":"Mathlib/Analysis/Calculus/Deriv/Basic.lean","def_pos":[382,8],"def_end_pos":[382,24]},{"full_name":"HasDerivAt.differentiableAt","def_path":"Mathlib/Analysis/Calculus/Deriv/Basic.lean","def_pos":[336,8],"def_end_pos":[336,35]},{"full_name":"not_intervalIntegrable_of_tendsto_norm_atTop_of_deriv_isBigO_punctured","def_path":"Mathlib/Analysis/SpecialFunctions/NonIntegrable.lean","def_pos":[159,8],"def_end_pos":[159,78]}]}]}
+{"url":"Mathlib/Algebra/Order/Rearrangement.lean","commit":"","full_name":"Monovary.sum_smul_comp_perm_eq_sum_smul_iff","start":[244,0],"end":[249,83],"file_path":"Mathlib/Algebra/Order/Rearrangement.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝⁴ : LinearOrderedRing α\ninst✝³ : LinearOrderedAddCommGroup β\ninst✝² : Module α β\ninst✝¹ : OrderedSMul α β\ns : Finset ι\nσ : Perm ι\nf : ι → α\ng : ι → β\ninst✝ : Fintype ι\nhfg : Monovary f g\n⊢ ∑ i : ι, f i • g (σ i) = ∑ i : ι, f i • g i ↔ Monovary f (g ∘ ⇑σ)","state_after":"no goals","tactic":"simp [(hfg.monovaryOn _).sum_smul_comp_perm_eq_sum_smul_iff fun _ _ ↦ mem_univ _]","premises":[{"full_name":"Finset.mem_univ","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[71,8],"def_end_pos":[71,16]},{"full_name":"Monovary.monovaryOn","def_path":"Mathlib/Order/Monotone/Monovary.lean","def_pos":[53,18],"def_end_pos":[53,37]},{"full_name":"MonovaryOn.sum_smul_comp_perm_eq_sum_smul_iff","def_path":"Mathlib/Algebra/Order/Rearrangement.lean","def_pos":[111,8],"def_end_pos":[111,53]}]}]}
+{"url":"Mathlib/Algebra/BigOperators/Group/Finset.lean","commit":"","full_name":"Multiset.disjoint_finset_sum_left","start":[2088,0],"end":[2091,41],"file_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","tactics":[{"state_before":"ι : Type u_1\nκ : Type u_2\nα : Type u_3\nβ✝ : Type u_4\nγ : Type u_5\ns s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β✝\nβ : Type u_6\ni : Finset β\nf : β → Multiset α\na : Multiset α\n⊢ (i.sum f).Disjoint a ↔ ∀ b ∈ i, (f b).Disjoint a","state_after":"case h.e'_2.a\nι : Type u_1\nκ : Type u_2\nα : Type u_3\nβ✝ : Type u_4\nγ : Type u_5\ns s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β✝\nβ : Type u_6\ni : Finset β\nf : β → Multiset α\na : Multiset α\n⊢ (∀ b ∈ i, (f b).Disjoint a) ↔ ∀ b ∈ map f i.val, b.Disjoint a","tactic":"convert @disjoint_sum_left _ a (map f i.val)","premises":[{"full_name":"Finset.val","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[135,2],"def_end_pos":[135,5]},{"full_name":"Multiset.disjoint_sum_left","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[2078,8],"def_end_pos":[2078,25]},{"full_name":"Multiset.map","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[1012,4],"def_end_pos":[1012,7]}]},{"state_before":"case h.e'_2.a\nι : Type u_1\nκ : Type u_2\nα : Type u_3\nβ✝ : Type u_4\nγ : Type u_5\ns s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β✝\nβ : Type u_6\ni : Finset β\nf : β → Multiset α\na : Multiset α\n⊢ (∀ b ∈ i, (f b).Disjoint a) ↔ ∀ b ∈ map f i.val, b.Disjoint a","state_after":"no goals","tactic":"simp [and_congr_left_iff, iff_self_iff]","premises":[{"full_name":"and_congr_left_iff","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[158,16],"def_end_pos":[158,34]},{"full_name":"iff_self_iff","def_path":"Mathlib/Init/Logic.lean","def_pos":[112,8],"def_end_pos":[112,20]}]}]}
+{"url":"Mathlib/Algebra/Divisibility/Basic.lean","commit":"","full_name":"dvd_mul","start":[192,0],"end":[196,25],"file_path":"Mathlib/Algebra/Divisibility/Basic.lean","tactics":[{"state_before":"α : Type u_1\ninst✝¹ : CommSemigroup α\na b c : α\ninst✝ : DecompositionMonoid α\nk m n : α\n⊢ k ∣ m * n ↔ ∃ d₁ d₂, d₁ ∣ m ∧ d₂ ∣ n ∧ k = d₁ * d₂","state_after":"α : Type u_1\ninst✝¹ : CommSemigroup α\na b c : α\ninst✝ : DecompositionMonoid α\nk m n : α\n⊢ (∃ d₁ d₂, d₁ ∣ m ∧ d₂ ∣ n ∧ k = d₁ * d₂) → k ∣ m * n","tactic":"refine ⟨exists_dvd_and_dvd_of_dvd_mul, ?_⟩","premises":[{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"exists_dvd_and_dvd_of_dvd_mul","def_path":"Mathlib/Algebra/Divisibility/Basic.lean","def_pos":[113,8],"def_end_pos":[113,37]}]},{"state_before":"α : Type u_1\ninst✝¹ : CommSemigroup α\na b c : α\ninst✝ : DecompositionMonoid α\nk m n : α\n⊢ (∃ d₁ d₂, d₁ ∣ m ∧ d₂ ∣ n ∧ k = d₁ * d₂) → k ∣ m * n","state_after":"case intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : CommSemigroup α\na b c : α\ninst✝ : DecompositionMonoid α\nm n d₁ d₂ : α\nhy : d₁ ∣ m\nhz : d₂ ∣ n\n⊢ d₁ * d₂ ∣ m * n","tactic":"rintro ⟨d₁, d₂, hy, hz, rfl⟩","premises":[]},{"state_before":"case intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : CommSemigroup α\na b c : α\ninst✝ : DecompositionMonoid α\nm n d₁ d₂ : α\nhy : d₁ ∣ m\nhz : d₂ ∣ n\n⊢ d₁ * d₂ ∣ m * n","state_after":"no goals","tactic":"exact mul_dvd_mul hy hz","premises":[{"full_name":"mul_dvd_mul","def_path":"Mathlib/Algebra/Divisibility/Basic.lean","def_pos":[186,8],"def_end_pos":[186,19]}]}]}
+{"url":"Mathlib/Algebra/BigOperators/Finprod.lean","commit":"","full_name":"finprod_mem_inter_mul_diff'","start":[847,0],"end":[853,83],"file_path":"Mathlib/Algebra/BigOperators/Finprod.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g : α → M\na b : α\ns t✝ t : Set α\nh : (s ∩ mulSupport f).Finite\n⊢ (∏ᶠ (i : α) (_ : i ∈ s ∩ t), f i) * ∏ᶠ (i : α) (_ : i ∈ s \\ t), f i = ∏ᶠ (i : α) (_ : i ∈ s), f i","state_after":"case hst\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g : α → M\na b : α\ns t✝ t : Set α\nh : (s ∩ mulSupport f).Finite\n⊢ Disjoint (s ∩ t) (s \\ t)\n\ncase hs\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g : α → M\na b : α\ns t✝ t : Set α\nh : (s ∩ mulSupport f).Finite\n⊢ (s ∩ t ∩ mulSupport f).Finite\n\ncase ht\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g : α → M\na b : α\ns t✝ t : Set α\nh : (s ∩ mulSupport f).Finite\n⊢ (s \\ t ∩ mulSupport f).Finite","tactic":"rw [← finprod_mem_union', inter_union_diff]","premises":[{"full_name":"Set.inter_union_diff","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1468,8],"def_end_pos":[1468,24]},{"full_name":"finprod_mem_union'","def_path":"Mathlib/Algebra/BigOperators/Finprod.lean","def_pos":[675,8],"def_end_pos":[675,26]}]},{"state_before":"case hs\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g : α → M\na b : α\ns t✝ t : Set α\nh : (s ∩ mulSupport f).Finite\n⊢ (s ∩ t ∩ mulSupport f).Finite\n\ncase ht\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g : α → M\na b : α\ns t✝ t : Set α\nh : (s ∩ mulSupport f).Finite\n⊢ (s \\ t ∩ mulSupport f).Finite","state_after":"no goals","tactic":"exacts [h.subset fun x hx => ⟨hx.1.1, hx.2⟩, h.subset fun x hx => ⟨hx.1.1, hx.2⟩]","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"Set.Finite.subset","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[633,8],"def_end_pos":[633,21]}]}]}
+{"url":"Mathlib/Analysis/InnerProductSpace/Basic.lean","commit":"","full_name":"InnerProductSpace.Core.inner_re_symm","start":[216,0],"end":[216,93],"file_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝² : _root_.RCLike 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : Core 𝕜 F\nx y : F\n⊢ re ⟪x, y⟫_𝕜 = re ⟪y, x⟫_𝕜","state_after":"no goals","tactic":"rw [← inner_conj_symm, conj_re]","premises":[{"full_name":"InnerProductSpace.Core.inner_conj_symm","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[196,8],"def_end_pos":[196,23]},{"full_name":"RCLike.conj_re","def_path":"Mathlib/Analysis/RCLike/Basic.lean","def_pos":[266,8],"def_end_pos":[266,15]}]}]}
+{"url":"Mathlib/Data/Finset/Density.lean","commit":"","full_name":"Finset.dens_eq_one","start":[110,0],"end":[111,53],"file_path":"Mathlib/Data/Finset/Density.lean","tactics":[{"state_before":"𝕜 : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : Fintype α\ns t : Finset α\na b : α\ninst✝ : Nonempty α\n⊢ s.dens = 1 ↔ s = univ","state_after":"no goals","tactic":"simp [dens, div_eq_one_iff_eq, card_eq_iff_eq_univ]","premises":[{"full_name":"Finset.card_eq_iff_eq_univ","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[235,8],"def_end_pos":[235,34]},{"full_name":"Finset.dens","def_path":"Mathlib/Data/Finset/Density.lean","def_pos":[59,4],"def_end_pos":[59,8]},{"full_name":"div_eq_one_iff_eq","def_path":"Mathlib/Algebra/GroupWithZero/Units/Basic.lean","def_pos":[310,6],"def_end_pos":[310,23]}]}]}
+{"url":"Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean","commit":"","full_name":"AffineMap.linear_eq_zero_iff_exists_const","start":[182,0],"end":[190,29],"file_path":"Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean","tactics":[{"state_before":"k : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V1\ninst✝¹⁰ : Module k V1\ninst✝⁹ : AffineSpace V1 P1\ninst✝⁸ : AddCommGroup V2\ninst✝⁷ : Module k V2\ninst✝⁶ : AffineSpace V2 P2\ninst✝⁵ : AddCommGroup V3\ninst✝⁴ : Module k V3\ninst✝³ : AffineSpace V3 P3\ninst✝² : AddCommGroup V4\ninst✝¹ : Module k V4\ninst✝ : AffineSpace V4 P4\nf : P1 →ᵃ[k] P2\n⊢ f.linear = 0 ↔ ∃ q, f = const k P1 q","state_after":"case refine_1\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V1\ninst✝¹⁰ : Module k V1\ninst✝⁹ : AffineSpace V1 P1\ninst✝⁸ : AddCommGroup V2\ninst✝⁷ : Module k V2\ninst✝⁶ : AffineSpace V2 P2\ninst✝⁵ : AddCommGroup V3\ninst✝⁴ : Module k V3\ninst✝³ : AffineSpace V3 P3\ninst✝² : AddCommGroup V4\ninst✝¹ : Module k V4\ninst✝ : AffineSpace V4 P4\nf : P1 →ᵃ[k] P2\nh : f.linear = 0\n⊢ ∃ q, f = const k P1 q\n\ncase refine_2\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V1\ninst✝¹⁰ : Module k V1\ninst✝⁹ : AffineSpace V1 P1\ninst✝⁸ : AddCommGroup V2\ninst✝⁷ : Module k V2\ninst✝⁶ : AffineSpace V2 P2\ninst✝⁵ : AddCommGroup V3\ninst✝⁴ : Module k V3\ninst✝³ : AffineSpace V3 P3\ninst✝² : AddCommGroup V4\ninst✝¹ : Module k V4\ninst✝ : AffineSpace V4 P4\nf : P1 →ᵃ[k] P2\nh : ∃ q, f = const k P1 q\n⊢ f.linear = 0","tactic":"refine ⟨fun h => ?_, fun h => ?_⟩","premises":[{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]}]}]}
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+{"url":"Mathlib/RingTheory/Polynomial/Basic.lean","commit":"","full_name":"Ideal.isPrime_map_C_iff_isPrime","start":[680,0],"end":[733,56],"file_path":"Mathlib/RingTheory/Polynomial/Basic.lean","tactics":[{"state_before":"R : Type u\nS : Type u_1\ninst✝ : CommRing R\nP : Ideal R\n⊢ (map C P).IsPrime ↔ P.IsPrime","state_after":"case mp\nR : Type u\nS : Type u_1\ninst✝ : CommRing R\nP : Ideal R\n⊢ (map C P).IsPrime → P.IsPrime\n\ncase mpr\nR : Type u\nS : Type u_1\ninst✝ : CommRing R\nP : Ideal R\n⊢ P.IsPrime → (map C P).IsPrime","tactic":"constructor","premises":[]}]}
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+{"url":"Mathlib/Algebra/Group/Submonoid/Operations.lean","commit":"","full_name":"Submonoid.prod_le_iff","start":[689,0],"end":[708,43],"file_path":"Mathlib/Algebra/Group/Submonoid/Operations.lean","tactics":[{"state_before":"M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\n⊢ s.prod t ≤ u ↔ map (inl M N) s ≤ u ∧ map (inr M N) t ≤ u","state_after":"case mp\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\n⊢ s.prod t ≤ u → map (inl M N) s ≤ u ∧ map (inr M N) t ≤ u\n\ncase mpr\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\n⊢ map (inl M N) s ≤ u ∧ map (inr M N) t ≤ u → s.prod t ≤ u","tactic":"constructor","premises":[]}]}
+{"url":"Mathlib/SetTheory/Cardinal/Ordinal.lean","commit":"","full_name":"Cardinal.add_le_add_iff_of_lt_aleph0","start":[835,0],"end":[840,74],"file_path":"Mathlib/SetTheory/Cardinal/Ordinal.lean","tactics":[{"state_before":"α β γ : Cardinal.{u_1}\nγ₀ : γ < ℵ₀\n⊢ α + γ ≤ β + γ ↔ α ≤ β","state_after":"α β γ : Cardinal.{u_1}\nγ₀ : γ < ℵ₀\nh : α + γ ≤ β + γ\n⊢ α ≤ β","tactic":"refine ⟨fun h => ?_, fun h => add_le_add_right h γ⟩","premises":[{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"add_le_add_right","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[74,31],"def_end_pos":[74,47]}]},{"state_before":"α β γ : Cardinal.{u_1}\nγ₀ : γ < ℵ₀\nh : α + γ ≤ β + γ\n⊢ α ≤ β","state_after":"α β γ : Cardinal.{u_1}\nγ₀ : γ < ℵ₀\nh : ¬α ≤ β\n⊢ ¬α + γ ≤ β + γ","tactic":"contrapose h","premises":[{"full_name":"Mathlib.Tactic.Contrapose.mtr","def_path":"Mathlib/Tactic/Contrapose.lean","def_pos":[24,6],"def_end_pos":[24,9]}]},{"state_before":"α β γ : Cardinal.{u_1}\nγ₀ : γ < ℵ₀\nh : ¬α ≤ β\n⊢ ¬α + γ ≤ β + γ","state_after":"α β γ : Cardinal.{u_1}\nγ₀ : γ < ℵ₀\nh : β ≤ α ∧ ¬β = α\n⊢ β + γ ≤ α + γ ∧ ¬β + γ = α + γ","tactic":"rw [not_le, lt_iff_le_and_ne, Ne] at h ⊢","premises":[{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"lt_iff_le_and_ne","def_path":"Mathlib/Order/Basic.lean","def_pos":[309,8],"def_end_pos":[309,24]},{"full_name":"not_le","def_path":"Mathlib/Order/Defs.lean","def_pos":[316,8],"def_end_pos":[316,14]}]},{"state_before":"α β γ : Cardinal.{u_1}\nγ₀ : γ < ℵ₀\nh : β ≤ α ∧ ¬β = α\n⊢ β + γ ≤ α + γ ∧ ¬β + γ = α + γ","state_after":"no goals","tactic":"exact ⟨add_le_add_right h.1 γ, mt (add_right_inj_of_lt_aleph0 γ₀).1 h.2⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"Cardinal.add_right_inj_of_lt_aleph0","def_path":"Mathlib/SetTheory/Cardinal/Ordinal.lean","def_pos":[824,8],"def_end_pos":[824,34]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"add_le_add_right","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[74,31],"def_end_pos":[74,47]},{"full_name":"mt","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[647,8],"def_end_pos":[647,10]}]}]}
+{"url":"Mathlib/NumberTheory/Padics/Hensel.lean","commit":"","full_name":"_private.Mathlib.NumberTheory.Padics.Hensel.0.deriv_norm_ne_zero","start":[110,0],"end":[111,47],"file_path":"Mathlib/NumberTheory/Padics/Hensel.lean","tactics":[{"state_before":"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nhnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2\nhnsol : Polynomial.eval a F ≠ 0\nh : ‖Polynomial.eval a (Polynomial.derivative F)‖ = 0\n⊢ ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2 = 0","state_after":"no goals","tactic":"simp [*, sq]","premises":[]}]}
+{"url":"Mathlib/LinearAlgebra/CliffordAlgebra/Grading.lean","commit":"","full_name":"CliffordAlgebra.evenOdd_induction","start":[143,0],"end":[186,29],"file_path":"Mathlib/LinearAlgebra/CliffordAlgebra/Grading.lean","tactics":[{"state_before":"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nn : ZMod 2\nmotive : (x : CliffordAlgebra Q) → x ∈ evenOdd Q n → Prop\nrange_ι_pow : ∀ (v : CliffordAlgebra Q) (h : v ∈ LinearMap.range (ι Q) ^ n.val), motive v ⋯\nadd :\n  ∀ (x y : CliffordAlgebra Q) (hx : x ∈ evenOdd Q n) (hy : y ∈ evenOdd Q n),\n    motive x hx → motive y hy → motive (x + y) ⋯\nι_mul_ι_mul :\n  ∀ (m₁ m₂ : M) (x : CliffordAlgebra Q) (hx : x ∈ evenOdd Q n), motive x hx → motive ((ι Q) m₁ * (ι Q) m₂ * x) ⋯\nx : CliffordAlgebra Q\nhx : x ∈ evenOdd Q n\n⊢ motive x hx","state_after":"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nn : ZMod 2\nmotive : (x : CliffordAlgebra Q) → x ∈ evenOdd Q n → Prop\nrange_ι_pow : ∀ (v : CliffordAlgebra Q) (h : v ∈ LinearMap.range (ι Q) ^ n.val), motive v ⋯\nadd :\n  ∀ (x y : CliffordAlgebra Q) (hx : x ∈ evenOdd Q n) (hy : y ∈ evenOdd Q n),\n    motive x hx → motive y hy → motive (x + y) ⋯\nι_mul_ι_mul :\n  ∀ (m₁ m₂ : M) (x : CliffordAlgebra Q) (hx : x ∈ evenOdd Q n), motive x hx → motive ((ι Q) m₁ * (ι Q) m₂ * x) ⋯\nx : CliffordAlgebra Q\nhx : x ∈ evenOdd Q n\n⊢ ∀ (i : { n_1 // ↑n_1 = n }) (x : CliffordAlgebra Q) (hx : x ∈ LinearMap.range (ι Q) ^ ↑i), motive x ⋯","tactic":"apply Submodule.iSup_induction' (C := motive) _ (range_ι_pow 0 (Submodule.zero_mem _)) add","premises":[{"full_name":"Submodule.iSup_induction'","def_path":"Mathlib/LinearAlgebra/Span.lean","def_pos":[629,8],"def_end_pos":[629,23]},{"full_name":"Submodule.zero_mem","def_path":"Mathlib/Algebra/Module/Submodule/Basic.lean","def_pos":[188,18],"def_end_pos":[188,26]}]},{"state_before":"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nn : ZMod 2\nmotive : (x : CliffordAlgebra Q) → x ∈ evenOdd Q n → Prop\nrange_ι_pow : ∀ (v : CliffordAlgebra Q) (h : v ∈ LinearMap.range (ι Q) ^ n.val), motive v ⋯\nadd :\n  ∀ (x y : CliffordAlgebra Q) (hx : x ∈ evenOdd Q n) (hy : y ∈ evenOdd Q n),\n    motive x hx → motive y hy → motive (x + y) ⋯\nι_mul_ι_mul :\n  ∀ (m₁ m₂ : M) (x : CliffordAlgebra Q) (hx : x ∈ evenOdd Q n), motive x hx → motive ((ι Q) m₁ * (ι Q) m₂ * x) ⋯\nx : CliffordAlgebra Q\nhx : x ∈ evenOdd Q n\n⊢ ∀ (i : { n_1 // ↑n_1 = n }) (x : CliffordAlgebra Q) (hx : x ∈ LinearMap.range (ι Q) ^ ↑i), motive x ⋯","state_after":"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nn : ZMod 2\nmotive : (x : CliffordAlgebra Q) → x ∈ evenOdd Q n → Prop\nrange_ι_pow : ∀ (v : CliffordAlgebra Q) (h : v ∈ LinearMap.range (ι Q) ^ n.val), motive v ⋯\nadd :\n  ∀ (x y : CliffordAlgebra Q) (hx : x ∈ evenOdd Q n) (hy : y ∈ evenOdd Q n),\n    motive x hx → motive y hy → motive (x + y) ⋯\nι_mul_ι_mul :\n  ∀ (m₁ m₂ : M) (x : CliffordAlgebra Q) (hx : x ∈ evenOdd Q n), motive x hx → motive ((ι Q) m₁ * (ι Q) m₂ * x) ⋯\nx : CliffordAlgebra Q\nhx : x ∈ evenOdd Q n\n⊢ ∀ (val : ℕ) (property : ↑val = n) (x : CliffordAlgebra Q) (hx : x ∈ LinearMap.range (ι Q) ^ ↑⟨val, property⟩),\n    motive x ⋯","tactic":"refine Subtype.rec ?_","premises":[]},{"state_before":"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nn : ZMod 2\nmotive : (x : CliffordAlgebra Q) → x ∈ evenOdd Q n → Prop\nrange_ι_pow : ∀ (v : CliffordAlgebra Q) (h : v ∈ LinearMap.range (ι Q) ^ n.val), motive v ⋯\nadd :\n  ∀ (x y : CliffordAlgebra Q) (hx : x ∈ evenOdd Q n) (hy : y ∈ evenOdd Q n),\n    motive x hx → motive y hy → motive (x + y) ⋯\nι_mul_ι_mul :\n  ∀ (m₁ m₂ : M) (x : CliffordAlgebra Q) (hx : x ∈ evenOdd Q n), motive x hx → motive ((ι Q) m₁ * (ι Q) m₂ * x) ⋯\nx : CliffordAlgebra Q\nhx : x ∈ evenOdd Q n\n⊢ ∀ (val : ℕ) (property : ↑val = n) (x : CliffordAlgebra Q) (hx : x ∈ LinearMap.range (ι Q) ^ ↑⟨val, property⟩),\n    motive x ⋯","state_after":"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nn : ZMod 2\nmotive : (x : CliffordAlgebra Q) → x ∈ evenOdd Q n → Prop\nrange_ι_pow : ∀ (v : CliffordAlgebra Q) (h : v ∈ LinearMap.range (ι Q) ^ n.val), motive v ⋯\nadd :\n  ∀ (x y : CliffordAlgebra Q) (hx : x ∈ evenOdd Q n) (hy : y ∈ evenOdd Q n),\n    motive x hx → motive y hy → motive (x + y) ⋯\nι_mul_ι_mul :\n  ∀ (m₁ m₂ : M) (x : CliffordAlgebra Q) (hx : x ∈ evenOdd Q n), motive x hx → motive ((ι Q) m₁ * (ι Q) m₂ * x) ⋯\nx : CliffordAlgebra Q\nhx : x ∈ evenOdd Q n\n⊢ ∀ (val : ℕ) (property : ∃ k, val = 2 * k + n.val) (x : CliffordAlgebra Q) (hx : x ∈ LinearMap.range (ι Q) ^ val),\n    motive x ⋯","tactic":"simp_rw [ZMod.natCast_eq_iff, add_comm n.val]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"ZMod.natCast_eq_iff","def_path":"Mathlib/Data/ZMod/Basic.lean","def_pos":[611,8],"def_end_pos":[611,22]},{"full_name":"ZMod.val","def_path":"Mathlib/Data/ZMod/Basic.lean","def_pos":[45,4],"def_end_pos":[45,7]},{"full_name":"add_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[293,2],"def_end_pos":[293,13]}]},{"state_before":"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nn : ZMod 2\nmotive : (x : CliffordAlgebra Q) → x ∈ evenOdd Q n → Prop\nrange_ι_pow : ∀ (v : CliffordAlgebra Q) (h : v ∈ LinearMap.range (ι Q) ^ n.val), motive v ⋯\nadd :\n  ∀ (x y : CliffordAlgebra Q) (hx : x ∈ evenOdd Q n) (hy : y ∈ evenOdd Q n),\n    motive x hx → motive y hy → motive (x + y) ⋯\nι_mul_ι_mul :\n  ∀ (m₁ m₂ : M) (x : CliffordAlgebra Q) (hx : x ∈ evenOdd Q n), motive x hx → motive ((ι Q) m₁ * (ι Q) m₂ * x) ⋯\nx : CliffordAlgebra Q\nhx : x ∈ evenOdd Q n\n⊢ ∀ (val : ℕ) (property : ∃ k, val = 2 * k + n.val) (x : CliffordAlgebra Q) (hx : x ∈ LinearMap.range (ι Q) ^ val),\n    motive x ⋯","state_after":"case intro\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nn : ZMod 2\nmotive : (x : CliffordAlgebra Q) → x ∈ evenOdd Q n → Prop\nrange_ι_pow : ∀ (v : CliffordAlgebra Q) (h : v ∈ LinearMap.range (ι Q) ^ n.val), motive v ⋯\nadd :\n  ∀ (x y : CliffordAlgebra Q) (hx : x ∈ evenOdd Q n) (hy : y ∈ evenOdd Q n),\n    motive x hx → motive y hy → motive (x + y) ⋯\nι_mul_ι_mul :\n  ∀ (m₁ m₂ : M) (x : CliffordAlgebra Q) (hx : x ∈ evenOdd Q n), motive x hx → motive ((ι Q) m₁ * (ι Q) m₂ * x) ⋯\nx : CliffordAlgebra Q\nhx : x ∈ evenOdd Q n\nk : ℕ\nxv : CliffordAlgebra Q\n⊢ ∀ (hx : xv ∈ LinearMap.range (ι Q) ^ (2 * k + n.val)), motive xv ⋯","tactic":"rintro n' ⟨k, rfl⟩ xv","premises":[]},{"state_before":"case intro\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nn : ZMod 2\nmotive : (x : CliffordAlgebra Q) → x ∈ evenOdd Q n → Prop\nrange_ι_pow : ∀ (v : CliffordAlgebra Q) (h : v ∈ LinearMap.range (ι Q) ^ n.val), motive v ⋯\nadd :\n  ∀ (x y : CliffordAlgebra Q) (hx : x ∈ evenOdd Q n) (hy : y ∈ evenOdd Q n),\n    motive x hx → motive y hy → motive (x + y) ⋯\nι_mul_ι_mul :\n  ∀ (m₁ m₂ : M) (x : CliffordAlgebra Q) (hx : x ∈ evenOdd Q n), motive x hx → motive ((ι Q) m₁ * (ι Q) m₂ * x) ⋯\nx : CliffordAlgebra Q\nhx : x ∈ evenOdd Q n\nk : ℕ\nxv : CliffordAlgebra Q\n⊢ ∀ (hx : xv ∈ LinearMap.range (ι Q) ^ (2 * k + n.val)), motive xv ⋯","state_after":"case intro\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nn : ZMod 2\nmotive : (x : CliffordAlgebra Q) → x ∈ evenOdd Q n → Prop\nrange_ι_pow : ∀ (v : CliffordAlgebra Q) (h : v ∈ LinearMap.range (ι Q) ^ n.val), motive v ⋯\nadd :\n  ∀ (x y : CliffordAlgebra Q) (hx : x ∈ evenOdd Q n) (hy : y ∈ evenOdd Q n),\n    motive x hx → motive y hy → motive (x + y) ⋯\nι_mul_ι_mul :\n  ∀ (m₁ m₂ : M) (x : CliffordAlgebra Q) (hx : x ∈ evenOdd Q n), motive x hx → motive ((ι Q) m₁ * (ι Q) m₂ * x) ⋯\nx : CliffordAlgebra Q\nhx : x ∈ evenOdd Q n\nk : ℕ\nxv : CliffordAlgebra Q\n⊢ ∀ (hx : xv ∈ (LinearMap.range (ι Q) ^ 2) ^ k * LinearMap.range (ι Q) ^ n.val), motive xv ⋯","tactic":"simp_rw [pow_add, pow_mul]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"pow_add","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[598,6],"def_end_pos":[598,13]},{"full_name":"pow_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[605,31],"def_end_pos":[605,38]}]},{"state_before":"case intro\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nn : ZMod 2\nmotive : (x : CliffordAlgebra Q) → x ∈ evenOdd Q n → Prop\nrange_ι_pow : ∀ (v : CliffordAlgebra Q) (h : v ∈ LinearMap.range (ι Q) ^ n.val), motive v ⋯\nadd :\n  ∀ (x y : CliffordAlgebra Q) (hx : x ∈ evenOdd Q n) (hy : y ∈ evenOdd Q n),\n    motive x hx → motive y hy → motive (x + y) ⋯\nι_mul_ι_mul :\n  ∀ (m₁ m₂ : M) (x : CliffordAlgebra Q) (hx : x ∈ evenOdd Q n), motive x hx → motive ((ι Q) m₁ * (ι Q) m₂ * x) ⋯\nx : CliffordAlgebra Q\nhx : x ∈ evenOdd Q n\nk : ℕ\nxv : CliffordAlgebra Q\n⊢ ∀ (hx : xv ∈ (LinearMap.range (ι Q) ^ 2) ^ k * LinearMap.range (ι Q) ^ n.val), motive xv ⋯","state_after":"case intro\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nn : ZMod 2\nmotive : (x : CliffordAlgebra Q) → x ∈ evenOdd Q n → Prop\nrange_ι_pow : ∀ (v : CliffordAlgebra Q) (h : v ∈ LinearMap.range (ι Q) ^ n.val), motive v ⋯\nadd :\n  ∀ (x y : CliffordAlgebra Q) (hx : x ∈ evenOdd Q n) (hy : y ∈ evenOdd Q n),\n    motive x hx → motive y hy → motive (x + y) ⋯\nι_mul_ι_mul :\n  ∀ (m₁ m₂ : M) (x : CliffordAlgebra Q) (hx : x ∈ evenOdd Q n), motive x hx → motive ((ι Q) m₁ * (ι Q) m₂ * x) ⋯\nx : CliffordAlgebra Q\nhx : x ∈ evenOdd Q n\nk : ℕ\nxv : CliffordAlgebra Q\nhxv : xv ∈ (LinearMap.range (ι Q) ^ 2) ^ k * LinearMap.range (ι Q) ^ n.val\n⊢ motive xv ⋯","tactic":"intro hxv","premises":[]},{"state_before":"case intro\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nn : ZMod 2\nmotive : (x : CliffordAlgebra Q) → x ∈ evenOdd Q n → Prop\nrange_ι_pow : ∀ (v : CliffordAlgebra Q) (h : v ∈ LinearMap.range (ι Q) ^ n.val), motive v ⋯\nadd :\n  ∀ (x y : CliffordAlgebra Q) (hx : x ∈ evenOdd Q n) (hy : y ∈ evenOdd Q n),\n    motive x hx → motive y hy → motive (x + y) ⋯\nι_mul_ι_mul :\n  ∀ (m₁ m₂ : M) (x : CliffordAlgebra Q) (hx : x ∈ evenOdd Q n), motive x hx → motive ((ι Q) m₁ * (ι Q) m₂ * x) ⋯\nx : CliffordAlgebra Q\nhx : x ∈ evenOdd Q n\nk : ℕ\nxv : CliffordAlgebra Q\nhxv : xv ∈ (LinearMap.range (ι Q) ^ 2) ^ k * LinearMap.range (ι Q) ^ n.val\n⊢ motive xv ⋯","state_after":"no goals","tactic":"induction hxv using Submodule.mul_induction_on' with\n  | mem_mul_mem a ha b hb =>\n    induction ha using Submodule.pow_induction_on_left' with\n    | algebraMap r =>\n      simp_rw [← Algebra.smul_def]\n      exact range_ι_pow _ (Submodule.smul_mem _ _ hb)\n    | add x y n hx hy ihx ihy =>\n      simp_rw [add_mul]\n      apply add _ _ _ _ ihx ihy\n    | mem_mul x hx n'' y hy ihy =>\n      revert hx\n      simp_rw [pow_two]\n      intro hx2\n      induction hx2 using Submodule.mul_induction_on' with\n      | mem_mul_mem m hm n hn =>\n        simp_rw [LinearMap.mem_range] at hm hn\n        obtain ⟨m₁, rfl⟩ := hm; obtain ⟨m₂, rfl⟩ := hn\n        simp_rw [mul_assoc _ y b]\n        exact ι_mul_ι_mul _ _ _ _ ihy\n      | add x hx y hy ihx ihy =>\n        simp_rw [add_mul]\n        apply add _ _ _ _ ihx ihy\n  | add x y hx hy ihx ihy =>\n    apply add _ _ _ _ ihx ihy","premises":[{"full_name":"Algebra.smul_def","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[270,8],"def_end_pos":[270,16]},{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"LinearMap.mem_range","def_path":"Mathlib/Algebra/Module/Submodule/Range.lean","def_pos":[67,8],"def_end_pos":[67,17]},{"full_name":"Submodule.add","def_path":"Mathlib/Algebra/Module/Submodule/Basic.lean","def_pos":[213,9],"def_end_pos":[213,12]},{"full_name":"Submodule.mul_induction_on'","def_path":"Mathlib/Algebra/Algebra/Operations.lean","def_pos":[161,18],"def_end_pos":[161,35]},{"full_name":"Submodule.pow_induction_on_left'","def_path":"Mathlib/Algebra/Algebra/Operations.lean","def_pos":[416,18],"def_end_pos":[416,40]},{"full_name":"Submodule.smul_mem","def_path":"Mathlib/Algebra/Module/Submodule/Basic.lean","def_pos":[194,8],"def_end_pos":[194,16]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]},{"full_name":"pow_two","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[581,31],"def_end_pos":[581,38]}]}]}
+{"url":"Mathlib/Algebra/Module/LocalizedModule.lean","commit":"","full_name":"IsLocalizedModule.fromLocalizedModule'_smul","start":[746,0],"end":[754,5],"file_path":"Mathlib/Algebra/Module/LocalizedModule.lean","tactics":[{"state_before":"R : Type u_1\ninst✝¹² : CommSemiring R\nS : Submonoid R\nM : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : AddCommMonoid M''\nA : Type u_5\ninst✝⁸ : CommSemiring A\ninst✝⁷ : Algebra R A\ninst✝⁶ : Module A M'\ninst✝⁵ : IsLocalization S A\ninst✝⁴ : Module R M\ninst✝³ : Module R M'\ninst✝² : Module R M''\ninst✝¹ : IsScalarTower R A M'\nf : M →ₗ[R] M'\ng : M →ₗ[R] M''\ninst✝ : IsLocalizedModule S f\nr : R\nx : LocalizedModule S M\n⊢ ∀ (m : M) (s : ↥S),\n    r • fromLocalizedModule' S f (LocalizedModule.mk m s) = fromLocalizedModule' S f (r • LocalizedModule.mk m s)","state_after":"R : Type u_1\ninst✝¹² : CommSemiring R\nS : Submonoid R\nM : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : AddCommMonoid M''\nA : Type u_5\ninst✝⁸ : CommSemiring A\ninst✝⁷ : Algebra R A\ninst✝⁶ : Module A M'\ninst✝⁵ : IsLocalization S A\ninst✝⁴ : Module R M\ninst✝³ : Module R M'\ninst✝² : Module R M''\ninst✝¹ : IsScalarTower R A M'\nf : M →ₗ[R] M'\ng : M →ₗ[R] M''\ninst✝ : IsLocalizedModule S f\nr : R\nx : LocalizedModule S M\na : M\nb : ↥S\n⊢ r • fromLocalizedModule' S f (LocalizedModule.mk a b) = fromLocalizedModule' S f (r • LocalizedModule.mk a b)","tactic":"intro a b","premises":[]},{"state_before":"R : Type u_1\ninst✝¹² : CommSemiring R\nS : Submonoid R\nM : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : AddCommMonoid M''\nA : Type u_5\ninst✝⁸ : CommSemiring A\ninst✝⁷ : Algebra R A\ninst✝⁶ : Module A M'\ninst✝⁵ : IsLocalization S A\ninst✝⁴ : Module R M\ninst✝³ : Module R M'\ninst✝² : Module R M''\ninst✝¹ : IsScalarTower R A M'\nf : M →ₗ[R] M'\ng : M →ₗ[R] M''\ninst✝ : IsLocalizedModule S f\nr : R\nx : LocalizedModule S M\na : M\nb : ↥S\n⊢ r • fromLocalizedModule' S f (LocalizedModule.mk a b) = fromLocalizedModule' S f (r • LocalizedModule.mk a b)","state_after":"R : Type u_1\ninst✝¹² : CommSemiring R\nS : Submonoid R\nM : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : AddCommMonoid M''\nA : Type u_5\ninst✝⁸ : CommSemiring A\ninst✝⁷ : Algebra R A\ninst✝⁶ : Module A M'\ninst✝⁵ : IsLocalization S A\ninst✝⁴ : Module R M\ninst✝³ : Module R M'\ninst✝² : Module R M''\ninst✝¹ : IsScalarTower R A M'\nf : M →ₗ[R] M'\ng : M →ₗ[R] M''\ninst✝ : IsLocalizedModule S f\nr : R\nx : LocalizedModule S M\na : M\nb : ↥S\n⊢ r • ↑⋯.unit⁻¹ (f a) = ↑⋯.unit⁻¹ (f (r • a))","tactic":"rw [fromLocalizedModule'_mk, LocalizedModule.smul'_mk, fromLocalizedModule'_mk]","premises":[{"full_name":"IsLocalizedModule.fromLocalizedModule'_mk","def_path":"Mathlib/Algebra/Module/LocalizedModule.lean","def_pos":[726,8],"def_end_pos":[726,31]},{"full_name":"LocalizedModule.smul'_mk","def_path":"Mathlib/Algebra/Module/LocalizedModule.lean","def_pos":[402,8],"def_end_pos":[402,16]}]},{"state_before":"R : Type u_1\ninst✝¹² : CommSemiring R\nS : Submonoid R\nM : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : AddCommMonoid M''\nA : Type u_5\ninst✝⁸ : CommSemiring A\ninst✝⁷ : Algebra R A\ninst✝⁶ : Module A M'\ninst✝⁵ : IsLocalization S A\ninst✝⁴ : Module R M\ninst✝³ : Module R M'\ninst✝² : Module R M''\ninst✝¹ : IsScalarTower R A M'\nf : M →ₗ[R] M'\ng : M →ₗ[R] M''\ninst✝ : IsLocalizedModule S f\nr : R\nx : LocalizedModule S M\na : M\nb : ↥S\n⊢ r • ↑⋯.unit⁻¹ (f a) = ↑⋯.unit⁻¹ (f (r • a))","state_after":"no goals","tactic":"rw [f.map_smul, map_smul]","premises":[{"full_name":"LinearMap.map_smul","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[348,18],"def_end_pos":[348,26]},{"full_name":"map_smul","def_path":"Mathlib/GroupTheory/GroupAction/Hom.lean","def_pos":[108,8],"def_end_pos":[108,16]}]}]}
+{"url":"Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean","commit":"","full_name":"cfcₙ_congr","start":[265,0],"end":[278,63],"file_path":"Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean","tactics":[{"state_before":"R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹² : CommSemiring R\ninst✝¹¹ : Nontrivial R\ninst✝¹⁰ : StarRing R\ninst✝⁹ : MetricSpace R\ninst✝⁸ : TopologicalSemiring R\ninst✝⁷ : ContinuousStar R\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : TopologicalSpace A\ninst✝³ : Module R A\ninst✝² : IsScalarTower R A A\ninst✝¹ : SMulCommClass R A A\ninst✝ : NonUnitalContinuousFunctionalCalculus R p\nf✝ g✝ : R → R\na✝ : A\nhf : autoParam (ContinuousOn f✝ (σₙ R a✝)) _auto✝\nhf0 : autoParam (f✝ 0 = 0) _auto✝\nhg : autoParam (ContinuousOn g✝ (σₙ R a✝)) _auto✝\nhg0 : autoParam (g✝ 0 = 0) _auto✝\nha : autoParam (p a✝) _auto✝\nf g : R → R\na : A\nhfg : Set.EqOn f g (σₙ R a)\n⊢ cfcₙ f a = cfcₙ g a","state_after":"case pos\nR : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹² : CommSemiring R\ninst✝¹¹ : Nontrivial R\ninst✝¹⁰ : StarRing R\ninst✝⁹ : MetricSpace R\ninst✝⁸ : TopologicalSemiring R\ninst✝⁷ : ContinuousStar R\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : TopologicalSpace A\ninst✝³ : Module R A\ninst✝² : IsScalarTower R A A\ninst✝¹ : SMulCommClass R A A\ninst✝ : NonUnitalContinuousFunctionalCalculus R p\nf✝ g✝ : R → R\na✝ : A\nhf : autoParam (ContinuousOn f✝ (σₙ R a✝)) _auto✝\nhf0 : autoParam (f✝ 0 = 0) _auto✝\nhg : autoParam (ContinuousOn g✝ (σₙ R a✝)) _auto✝\nhg0 : autoParam (g✝ 0 = 0) _auto✝\nha : autoParam (p a✝) _auto✝\nf g : R → R\na : A\nhfg : Set.EqOn f g (σₙ R a)\nh : p a ∧ ContinuousOn g (σₙ R a) ∧ g 0 = 0\n⊢ cfcₙ f a = cfcₙ g a\n\ncase neg\nR : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹² : CommSemiring R\ninst✝¹¹ : Nontrivial R\ninst✝¹⁰ : StarRing R\ninst✝⁹ : MetricSpace R\ninst✝⁸ : TopologicalSemiring R\ninst✝⁷ : ContinuousStar R\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : TopologicalSpace A\ninst✝³ : Module R A\ninst✝² : IsScalarTower R A A\ninst✝¹ : SMulCommClass R A A\ninst✝ : NonUnitalContinuousFunctionalCalculus R p\nf✝ g✝ : R → R\na✝ : A\nhf : autoParam (ContinuousOn f✝ (σₙ R a✝)) _auto✝\nhf0 : autoParam (f✝ 0 = 0) _auto✝\nhg : autoParam (ContinuousOn g✝ (σₙ R a✝)) _auto✝\nhg0 : autoParam (g✝ 0 = 0) _auto✝\nha : autoParam (p a✝) _auto✝\nf g : R → R\na : A\nhfg : Set.EqOn f g (σₙ R a)\nh : ¬(p a ∧ ContinuousOn g (σₙ R a) ∧ g 0 = 0)\n⊢ cfcₙ f a = cfcₙ g a","tactic":"by_cases h : p a ∧ ContinuousOn g (σₙ R a) ∧ g 0 = 0","premises":[{"full_name":"And","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[516,10],"def_end_pos":[516,13]},{"full_name":"ContinuousOn","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[164,4],"def_end_pos":[164,16]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]},{"full_name":"quasispectrum","def_path":"Mathlib/Algebra/Algebra/Quasispectrum.lean","def_pos":[246,4],"def_end_pos":[246,17]}]}]}
+{"url":"Mathlib/MeasureTheory/PiSystem.lean","commit":"","full_name":"IsPiSystem.insert_empty","start":[80,0],"end":[87,56],"file_path":"Mathlib/MeasureTheory/PiSystem.lean","tactics":[{"state_before":"α : Type u_1\nS : Set (Set α)\nh_pi : IsPiSystem S\n⊢ IsPiSystem (insert ∅ S)","state_after":"α : Type u_1\nS : Set (Set α)\nh_pi : IsPiSystem S\ns : Set α\nhs : s ∈ insert ∅ S\nt : Set α\nht : t ∈ insert ∅ S\nhst : (s ∩ t).Nonempty\n⊢ s ∩ t ∈ insert ∅ S","tactic":"intro s hs t ht hst","premises":[]},{"state_before":"α : Type u_1\nS : Set (Set α)\nh_pi : IsPiSystem S\ns : Set α\nhs : s ∈ insert ∅ S\nt : Set α\nht : t ∈ insert ∅ S\nhst : (s ∩ t).Nonempty\n⊢ s ∩ t ∈ insert ∅ S","state_after":"case inl\nα : Type u_1\nS : Set (Set α)\nh_pi : IsPiSystem S\ns t : Set α\nht : t ∈ insert ∅ S\nhst : (s ∩ t).Nonempty\nhs : s = ∅\n⊢ s ∩ t ∈ insert ∅ S\n\ncase inr\nα : Type u_1\nS : Set (Set α)\nh_pi : IsPiSystem S\ns t : Set α\nht : t ∈ insert ∅ S\nhst : (s ∩ t).Nonempty\nhs : s ∈ S\n⊢ s ∩ t ∈ insert ∅ S","tactic":"cases' hs with hs hs","premises":[]}]}
+{"url":"Mathlib/Algebra/Group/Equiv/TypeTags.lean","commit":"","full_name":"AddEquiv.toMultiplicative''_apply_apply","start":[78,0],"end":[95,28],"file_path":"Mathlib/Algebra/Group/Equiv/TypeTags.lean","tactics":[{"state_before":"G : Type u_1\nH : Type u_2\ninst✝¹ : AddZeroClass G\ninst✝ : MulOneClass H\nx : G ≃+ Additive H\n⊢ (fun f =>\n        { toFun := ⇑(AddMonoidHom.toMultiplicative''.symm f.toMonoidHom),\n          invFun := ⇑(AddMonoidHom.toMultiplicative'.symm f.symm.toMonoidHom), left_inv := ⋯, right_inv := ⋯,\n          map_add' := ⋯ })\n      ((fun f =>\n          { toFun := ⇑(AddMonoidHom.toMultiplicative'' f.toAddMonoidHom),\n            invFun := ⇑(AddMonoidHom.toMultiplicative' f.symm.toAddMonoidHom), left_inv := ⋯, right_inv := ⋯,\n            map_mul' := ⋯ })\n        x) =\n    x","state_after":"case h\nG : Type u_1\nH : Type u_2\ninst✝¹ : AddZeroClass G\ninst✝ : MulOneClass H\nx : G ≃+ Additive H\nx✝ : G\n⊢ ((fun f =>\n          { toFun := ⇑(AddMonoidHom.toMultiplicative''.symm f.toMonoidHom),\n            invFun := ⇑(AddMonoidHom.toMultiplicative'.symm f.symm.toMonoidHom), left_inv := ⋯, right_inv := ⋯,\n            map_add' := ⋯ })\n        ((fun f =>\n            { toFun := ⇑(AddMonoidHom.toMultiplicative'' f.toAddMonoidHom),\n              invFun := ⇑(AddMonoidHom.toMultiplicative' f.symm.toAddMonoidHom), left_inv := ⋯, right_inv := ⋯,\n              map_mul' := ⋯ })\n          x))\n      x✝ =\n    x x✝","tactic":"ext","premises":[]},{"state_before":"case h\nG : Type u_1\nH : Type u_2\ninst✝¹ : AddZeroClass G\ninst✝ : MulOneClass H\nx : G ≃+ Additive H\nx✝ : G\n⊢ ((fun f =>\n          { toFun := ⇑(AddMonoidHom.toMultiplicative''.symm f.toMonoidHom),\n            invFun := ⇑(AddMonoidHom.toMultiplicative'.symm f.symm.toMonoidHom), left_inv := ⋯, right_inv := ⋯,\n            map_add' := ⋯ })\n        ((fun f =>\n            { toFun := ⇑(AddMonoidHom.toMultiplicative'' f.toAddMonoidHom),\n              invFun := ⇑(AddMonoidHom.toMultiplicative' f.symm.toAddMonoidHom), left_inv := ⋯, right_inv := ⋯,\n              map_mul' := ⋯ })\n          x))\n      x✝ =\n    x x✝","state_after":"no goals","tactic":"rfl","premises":[]},{"state_before":"G : Type u_1\nH : Type u_2\ninst✝¹ : AddZeroClass G\ninst✝ : MulOneClass H\nx : Multiplicative G ≃* H\n⊢ (fun f =>\n        { toFun := ⇑(AddMonoidHom.toMultiplicative'' f.toAddMonoidHom),\n          invFun := ⇑(AddMonoidHom.toMultiplicative' f.symm.toAddMonoidHom), left_inv := ⋯, right_inv := ⋯,\n          map_mul' := ⋯ })\n      ((fun f =>\n          { toFun := ⇑(AddMonoidHom.toMultiplicative''.symm f.toMonoidHom),\n            invFun := ⇑(AddMonoidHom.toMultiplicative'.symm f.symm.toMonoidHom), left_inv := ⋯, right_inv := ⋯,\n            map_add' := ⋯ })\n        x) =\n    x","state_after":"case h\nG : Type u_1\nH : Type u_2\ninst✝¹ : AddZeroClass G\ninst✝ : MulOneClass H\nx : Multiplicative G ≃* H\nx✝ : Multiplicative G\n⊢ ((fun f =>\n          { toFun := ⇑(AddMonoidHom.toMultiplicative'' f.toAddMonoidHom),\n            invFun := ⇑(AddMonoidHom.toMultiplicative' f.symm.toAddMonoidHom), left_inv := ⋯, right_inv := ⋯,\n            map_mul' := ⋯ })\n        ((fun f =>\n            { toFun := ⇑(AddMonoidHom.toMultiplicative''.symm f.toMonoidHom),\n              invFun := ⇑(AddMonoidHom.toMultiplicative'.symm f.symm.toMonoidHom), left_inv := ⋯, right_inv := ⋯,\n              map_add' := ⋯ })\n          x))\n      x✝ =\n    x x✝","tactic":"ext","premises":[]},{"state_before":"case h\nG : Type u_1\nH : Type u_2\ninst✝¹ : AddZeroClass G\ninst✝ : MulOneClass H\nx : Multiplicative G ≃* H\nx✝ : Multiplicative G\n⊢ ((fun f =>\n          { toFun := ⇑(AddMonoidHom.toMultiplicative'' f.toAddMonoidHom),\n            invFun := ⇑(AddMonoidHom.toMultiplicative' f.symm.toAddMonoidHom), left_inv := ⋯, right_inv := ⋯,\n            map_mul' := ⋯ })\n        ((fun f =>\n            { toFun := ⇑(AddMonoidHom.toMultiplicative''.symm f.toMonoidHom),\n              invFun := ⇑(AddMonoidHom.toMultiplicative'.symm f.symm.toMonoidHom), left_inv := ⋯, right_inv := ⋯,\n              map_add' := ⋯ })\n          x))\n      x✝ =\n    x x✝","state_after":"no goals","tactic":"rfl","premises":[]}]}
+{"url":"Mathlib/NumberTheory/ArithmeticFunction.lean","commit":"","full_name":"ArithmeticFunction.sigma_one_apply_prime_pow","start":[825,0],"end":[827,33],"file_path":"Mathlib/NumberTheory/ArithmeticFunction.lean","tactics":[{"state_before":"R : Type u_1\np i : ℕ\nhp : Nat.Prime p\n⊢ (σ 1) (p ^ i) = ∑ k ∈ range (i + 1), p ^ k","state_after":"no goals","tactic":"simp [sigma_apply_prime_pow hp]","premises":[{"full_name":"ArithmeticFunction.sigma_apply_prime_pow","def_path":"Mathlib/NumberTheory/ArithmeticFunction.lean","def_pos":[819,8],"def_end_pos":[819,29]}]}]}
+{"url":"Mathlib/Data/Set/BoolIndicator.lean","commit":"","full_name":"Set.preimage_boolIndicator_eq_union","start":[40,0],"end":[44,24],"file_path":"Mathlib/Data/Set/BoolIndicator.lean","tactics":[{"state_before":"α : Type u_1\ns : Set α\nt : Set Bool\n⊢ s.boolIndicator ⁻¹' t = (if true ∈ t then s else ∅) ∪ if false ∈ t then sᶜ else ∅","state_after":"case h\nα : Type u_1\ns : Set α\nt : Set Bool\nx : α\n⊢ x ∈ s.boolIndicator ⁻¹' t ↔ x ∈ (if true ∈ t then s else ∅) ∪ if false ∈ t then sᶜ else ∅","tactic":"ext x","premises":[]},{"state_before":"case h\nα : Type u_1\ns : Set α\nt : Set Bool\nx : α\n⊢ x ∈ s.boolIndicator ⁻¹' t ↔ x ∈ (if true ∈ t then s else ∅) ∪ if false ∈ t then sᶜ else ∅","state_after":"case h\nα : Type u_1\ns : Set α\nt : Set Bool\nx : α\n⊢ (if x ∈ s then true else false) ∈ t ↔ x ∈ (if true ∈ t then s else ∅) ∪ if false ∈ t then sᶜ else ∅","tactic":"simp only [boolIndicator, mem_preimage]","premises":[{"full_name":"Set.boolIndicator","def_path":"Mathlib/Data/Set/BoolIndicator.lean","def_pos":[21,18],"def_end_pos":[21,31]},{"full_name":"Set.mem_preimage","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[112,8],"def_end_pos":[112,20]}]},{"state_before":"case h\nα : Type u_1\ns : Set α\nt : Set Bool\nx : α\n⊢ (if x ∈ s then true else false) ∈ t ↔ x ∈ (if true ∈ t then s else ∅) ∪ if false ∈ t then sᶜ else ∅","state_after":"no goals","tactic":"split_ifs <;> simp [*]","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
+{"url":"Mathlib/GroupTheory/OrderOfElement.lean","commit":"","full_name":"mod_natCard_nsmul","start":[901,0],"end":[903,98],"file_path":"Mathlib/GroupTheory/OrderOfElement.lean","tactics":[{"state_before":"G : Type u_1\nH : Type u_2\nA : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝¹ : Group G\ninst✝ : Fintype G\nx : G\nn✝ : ℕ\na : G\nn : ℕ\n⊢ a ^ (n % Nat.card G) = a ^ n","state_after":"no goals","tactic":"rw [eq_comm, ← pow_mod_orderOf, ← Nat.mod_mod_of_dvd n $ orderOf_dvd_natCard _, pow_mod_orderOf]","premises":[{"full_name":"Nat.mod_mod_of_dvd","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Dvd.lean","def_pos":[94,16],"def_end_pos":[94,30]},{"full_name":"eq_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[811,8],"def_end_pos":[811,15]},{"full_name":"orderOf_dvd_natCard","def_path":"Mathlib/GroupTheory/OrderOfElement.lean","def_pos":[859,8],"def_end_pos":[859,27]},{"full_name":"pow_mod_orderOf","def_path":"Mathlib/GroupTheory/OrderOfElement.lean","def_pos":[206,6],"def_end_pos":[206,21]}]}]}
+{"url":"Mathlib/Topology/EMetricSpace/Lipschitz.lean","commit":"","full_name":"lipschitzOnWith_iff_restrict","start":[92,0],"end":[93,88],"file_path":"Mathlib/Topology/EMetricSpace/Lipschitz.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nK : ℝ≥0\ns t : Set α\nf : α → β\n⊢ LipschitzOnWith K f s ↔ LipschitzWith K (s.restrict f)","state_after":"no goals","tactic":"simp only [LipschitzOnWith, LipschitzWith, SetCoe.forall', restrict, Subtype.edist_eq]","premises":[{"full_name":"LipschitzOnWith","def_path":"Mathlib/Topology/EMetricSpace/Lipschitz.lean","def_pos":[58,4],"def_end_pos":[58,19]},{"full_name":"LipschitzWith","def_path":"Mathlib/Topology/EMetricSpace/Lipschitz.lean","def_pos":[54,4],"def_end_pos":[54,17]},{"full_name":"Set.restrict","def_path":"Mathlib/Data/Set/Function.lean","def_pos":[45,4],"def_end_pos":[45,12]},{"full_name":"SetCoe.forall'","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[158,8],"def_end_pos":[158,22]},{"full_name":"Subtype.edist_eq","def_path":"Mathlib/Topology/EMetricSpace/Basic.lean","def_pos":[384,8],"def_end_pos":[384,24]}]}]}
+{"url":"Mathlib/AlgebraicGeometry/StructureSheaf.lean","commit":"","full_name":"AlgebraicGeometry.StructureSheaf.IsFraction.eq_mk'","start":[102,0],"end":[112,20],"file_path":"Mathlib/AlgebraicGeometry/StructureSheaf.lean","tactics":[{"state_before":"R : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\nf : (x : ↥U) → Localizations R ↑x\nhf : IsFraction f\n⊢ ∃ r s, ∀ (x : ↥U), ∃ (hs : s ∉ (↑x).asIdeal), f x = IsLocalization.mk' (Localization.AtPrime (↑x).asIdeal) r ⟨s, hs⟩","state_after":"case intro.intro\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\nf : (x : ↥U) → Localizations R ↑x\nr s : R\nh : ∀ (x : ↥U), s ∉ (↑x).asIdeal ∧ f x * (algebraMap R (Localizations R ↑x)) s = (algebraMap R (Localizations R ↑x)) r\n⊢ ∃ r s, ∀ (x : ↥U), ∃ (hs : s ∉ (↑x).asIdeal), f x = IsLocalization.mk' (Localization.AtPrime (↑x).asIdeal) r ⟨s, hs⟩","tactic":"rcases hf with ⟨r, s, h⟩","premises":[]},{"state_before":"case intro.intro\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\nf : (x : ↥U) → Localizations R ↑x\nr s : R\nh : ∀ (x : ↥U), s ∉ (↑x).asIdeal ∧ f x * (algebraMap R (Localizations R ↑x)) s = (algebraMap R (Localizations R ↑x)) r\n⊢ ∃ r s, ∀ (x : ↥U), ∃ (hs : s ∉ (↑x).asIdeal), f x = IsLocalization.mk' (Localization.AtPrime (↑x).asIdeal) r ⟨s, hs⟩","state_after":"case intro.intro\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\nf : (x : ↥U) → Localizations R ↑x\nr s : R\nh : ∀ (x : ↥U), s ∉ (↑x).asIdeal ∧ f x * (algebraMap R (Localizations R ↑x)) s = (algebraMap R (Localizations R ↑x)) r\nx : ↥U\n⊢ (algebraMap R (Localizations R ↑x)) r = f x * (algebraMap R (Localizations R ↑x)) ↑⟨s, ⋯⟩","tactic":"refine ⟨r, s, fun x => ⟨(h x).1, (IsLocalization.mk'_eq_iff_eq_mul.mpr ?_).symm⟩⟩","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"IsLocalization.mk'_eq_iff_eq_mul","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[265,8],"def_end_pos":[265,25]}]},{"state_before":"case intro.intro\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\nf : (x : ↥U) → Localizations R ↑x\nr s : R\nh : ∀ (x : ↥U), s ∉ (↑x).asIdeal ∧ f x * (algebraMap R (Localizations R ↑x)) s = (algebraMap R (Localizations R ↑x)) r\nx : ↥U\n⊢ (algebraMap R (Localizations R ↑x)) r = f x * (algebraMap R (Localizations R ↑x)) ↑⟨s, ⋯⟩","state_after":"no goals","tactic":"exact (h x).2.symm","premises":[{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]}]}]}
+{"url":"Mathlib/GroupTheory/Perm/Cycle/Concrete.lean","commit":"","full_name":"Equiv.Perm.nodup_toList","start":[251,0],"end":[281,38],"file_path":"Mathlib/GroupTheory/Perm/Cycle/Concrete.lean","tactics":[{"state_before":"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\n⊢ (p.toList x).Nodup","state_after":"case pos\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : p x = x\n⊢ (p.toList x).Nodup\n\ncase neg\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : ¬p x = x\n⊢ (p.toList x).Nodup","tactic":"by_cases hx : p x = x","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case neg\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : ¬p x = x\n⊢ (p.toList x).Nodup","state_after":"case neg\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : ¬p x = x\nhc : (p.cycleOf x).IsCycle\n⊢ (p.toList x).Nodup","tactic":"have hc : IsCycle (cycleOf p x) := isCycle_cycleOf p hx","premises":[{"full_name":"Equiv.Perm.IsCycle","def_path":"Mathlib/GroupTheory/Perm/Cycle/Basic.lean","def_pos":[230,4],"def_end_pos":[230,11]},{"full_name":"Equiv.Perm.cycleOf","def_path":"Mathlib/GroupTheory/Perm/Cycle/Factors.lean","def_pos":[42,4],"def_end_pos":[42,11]},{"full_name":"Equiv.Perm.isCycle_cycleOf","def_path":"Mathlib/GroupTheory/Perm/Cycle/Factors.lean","def_pos":[150,8],"def_end_pos":[150,23]}]},{"state_before":"case neg\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : ¬p x = x\nhc : (p.cycleOf x).IsCycle\n⊢ (p.toList x).Nodup","state_after":"case neg\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : ¬p x = x\nhc : (p.cycleOf x).IsCycle\n⊢ ∀ (i j : ℕ) (h₁ : i < (p.toList x).length) (h₂ : j < (p.toList x).length),\n    (p.toList x).nthLe i h₁ = (p.toList x).nthLe j h₂ → i = j","tactic":"rw [nodup_iff_nthLe_inj]","premises":[{"full_name":"List.nodup_iff_nthLe_inj","def_path":"Mathlib/Data/List/Nodup.lean","def_pos":[96,8],"def_end_pos":[96,27]}]},{"state_before":"case neg\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : ¬p x = x\nhc : (p.cycleOf x).IsCycle\n⊢ ∀ (i j : ℕ) (h₁ : i < (p.toList x).length) (h₂ : j < (p.toList x).length),\n    (p.toList x).nthLe i h₁ = (p.toList x).nthLe j h₂ → i = j","state_after":"case neg\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : ¬p x = x\nhc : (p.cycleOf x).IsCycle\nn m : ℕ\nhn : n < (p.toList x).length\nhm : m < (p.toList x).length\n⊢ (p.toList x).nthLe n hn = (p.toList x).nthLe m hm → n = m","tactic":"rintro n m hn hm","premises":[]},{"state_before":"case neg\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : ¬p x = x\nhc : (p.cycleOf x).IsCycle\nn m : ℕ\nhn : n < (p.toList x).length\nhm : m < (p.toList x).length\n⊢ (p.toList x).nthLe n hn = (p.toList x).nthLe m hm → n = m","state_after":"case neg\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : ¬p x = x\nhc : (p.cycleOf x).IsCycle\nn m : ℕ\nhn✝ : n < (p.toList x).length\nhn : n < orderOf (p.cycleOf x)\nhm✝ : m < (p.toList x).length\nhm : m < orderOf (p.cycleOf x)\n⊢ (p.toList x).nthLe n hn✝ = (p.toList x).nthLe m hm✝ → n = m","tactic":"rw [length_toList, ← hc.orderOf] at hm hn","premises":[{"full_name":"Equiv.Perm.IsCycle.orderOf","def_path":"Mathlib/GroupTheory/Perm/Cycle/Basic.lean","def_pos":[354,18],"def_end_pos":[354,33]},{"full_name":"Equiv.Perm.length_toList","def_path":"Mathlib/GroupTheory/Perm/Cycle/Concrete.lean","def_pos":[209,8],"def_end_pos":[209,21]}]},{"state_before":"case neg\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : ¬p x = x\nhc : (p.cycleOf x).IsCycle\nn m : ℕ\nhn✝ : n < (p.toList x).length\nhn : n < orderOf (p.cycleOf x)\nhm✝ : m < (p.toList x).length\nhm : m < orderOf (p.cycleOf x)\n⊢ (p.toList x).nthLe n hn✝ = (p.toList x).nthLe m hm✝ → n = m","state_after":"case neg\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : x ∈ (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn m : ℕ\nhn✝ : n < (p.toList x).length\nhn : n < orderOf (p.cycleOf x)\nhm✝ : m < (p.toList x).length\nhm : m < orderOf (p.cycleOf x)\n⊢ (p.toList x).nthLe n hn✝ = (p.toList x).nthLe m hm✝ → n = m","tactic":"rw [← cycleOf_apply_self, ← Ne, ← mem_support] at hx","premises":[{"full_name":"Equiv.Perm.cycleOf_apply_self","def_path":"Mathlib/GroupTheory/Perm/Cycle/Factors.lean","def_pos":[108,8],"def_end_pos":[108,26]},{"full_name":"Equiv.Perm.mem_support","def_path":"Mathlib/GroupTheory/Perm/Support.lean","def_pos":[271,8],"def_end_pos":[271,19]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]}]},{"state_before":"case neg\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : x ∈ (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn m : ℕ\nhn✝ : n < (p.toList x).length\nhn : n < orderOf (p.cycleOf x)\nhm✝ : m < (p.toList x).length\nhm : m < orderOf (p.cycleOf x)\n⊢ (p.toList x).nthLe n hn✝ = (p.toList x).nthLe m hm✝ → n = m","state_after":"case neg\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : x ∈ (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn m : ℕ\nhn✝ : n < (p.toList x).length\nhn : n < orderOf (p.cycleOf x)\nhm✝ : m < (p.toList x).length\nhm : m < orderOf (p.cycleOf x)\n⊢ (p.cycleOf x ^ n) x = (p.cycleOf x ^ m) x → n = m","tactic":"rw [nthLe_toList, nthLe_toList, ← cycleOf_pow_apply_self p x n, ←\n    cycleOf_pow_apply_self p x m]","premises":[{"full_name":"Equiv.Perm.cycleOf_pow_apply_self","def_path":"Mathlib/GroupTheory/Perm/Cycle/Factors.lean","def_pos":[61,8],"def_end_pos":[61,30]},{"full_name":"Equiv.Perm.nthLe_toList","def_path":"Mathlib/GroupTheory/Perm/Cycle/Concrete.lean","def_pos":[229,8],"def_end_pos":[229,20]}]},{"state_before":"case neg\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : x ∈ (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn m : ℕ\nhn✝ : n < (p.toList x).length\nhn : n < orderOf (p.cycleOf x)\nhm✝ : m < (p.toList x).length\nhm : m < orderOf (p.cycleOf x)\n⊢ (p.cycleOf x ^ n) x = (p.cycleOf x ^ m) x → n = m","state_after":"case neg.zero.zero\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : x ∈ (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nhn✝ : 0 < (p.toList x).length\nhn : 0 < orderOf (p.cycleOf x)\nhm✝ : 0 < (p.toList x).length\nhm : 0 < orderOf (p.cycleOf x)\n⊢ (p.cycleOf x ^ 0) x = (p.cycleOf x ^ 0) x → 0 = 0\n\ncase neg.zero.succ\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : x ∈ (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nhn✝ : 0 < (p.toList x).length\nhn : 0 < orderOf (p.cycleOf x)\nm : ℕ\nhm✝ : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\n⊢ (p.cycleOf x ^ 0) x = (p.cycleOf x ^ (m + 1)) x → 0 = m + 1\n\ncase neg.succ.zero\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : x ∈ (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : ℕ\nhn✝ : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nhm✝ : 0 < (p.toList x).length\nhm : 0 < orderOf (p.cycleOf x)\n⊢ (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ 0) x → n + 1 = 0\n\ncase neg.succ.succ\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : x ∈ (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : ℕ\nhn✝ : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : ℕ\nhm✝ : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\n⊢ (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x → n + 1 = m + 1","tactic":"cases' n with n <;> cases' m with m","premises":[]},{"state_before":"case neg.succ.succ\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : x ∈ (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : ℕ\nhn✝ : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : ℕ\nhm✝ : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\n⊢ (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x → n + 1 = m + 1","state_after":"case neg.succ.succ\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : x ∈ (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : ℕ\nhn✝ : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : ℕ\nhm✝ : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\nh : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x\n⊢ n + 1 = m + 1","tactic":"intro h","premises":[]},{"state_before":"case neg.succ.succ\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : x ∈ (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : ℕ\nhn✝ : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : ℕ\nhm✝ : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\nh : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x\n⊢ n + 1 = m + 1","state_after":"case neg.succ.succ\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : x ∈ (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : ℕ\nhn✝ : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : ℕ\nhm✝ : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\nh : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x\nhn' : ¬orderOf (p.cycleOf x) ∣ n.succ\n⊢ n + 1 = m + 1","tactic":"have hn' : ¬orderOf (p.cycleOf x) ∣ n.succ := Nat.not_dvd_of_pos_of_lt n.zero_lt_succ hn","premises":[{"full_name":"Dvd.dvd","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1344,2],"def_end_pos":[1344,5]},{"full_name":"Equiv.Perm.cycleOf","def_path":"Mathlib/GroupTheory/Perm/Cycle/Factors.lean","def_pos":[42,4],"def_end_pos":[42,11]},{"full_name":"Nat.not_dvd_of_pos_of_lt","def_path":"Mathlib/Data/Nat/Defs.lean","def_pos":[1031,6],"def_end_pos":[1031,26]},{"full_name":"Nat.succ","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1083,4],"def_end_pos":[1083,8]},{"full_name":"Nat.zero_lt_succ","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1671,8],"def_end_pos":[1671,24]},{"full_name":"Not","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[224,4],"def_end_pos":[224,7]},{"full_name":"orderOf","def_path":"Mathlib/GroupTheory/OrderOfElement.lean","def_pos":[129,18],"def_end_pos":[129,25]}]},{"state_before":"case neg.succ.succ\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : x ∈ (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : ℕ\nhn✝ : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : ℕ\nhm✝ : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\nh : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x\nhn' : ¬orderOf (p.cycleOf x) ∣ n.succ\n⊢ n + 1 = m + 1","state_after":"case neg.succ.succ\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : x ∈ (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : ℕ\nhn✝ : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : ℕ\nhm✝ : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\nh : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x\nhn' : ¬orderOf (p.cycleOf x) ∣ n.succ\nhm' : ¬orderOf (p.cycleOf x) ∣ m.succ\n⊢ n + 1 = m + 1","tactic":"have hm' : ¬orderOf (p.cycleOf x) ∣ m.succ := Nat.not_dvd_of_pos_of_lt m.zero_lt_succ hm","premises":[{"full_name":"Dvd.dvd","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1344,2],"def_end_pos":[1344,5]},{"full_name":"Equiv.Perm.cycleOf","def_path":"Mathlib/GroupTheory/Perm/Cycle/Factors.lean","def_pos":[42,4],"def_end_pos":[42,11]},{"full_name":"Nat.not_dvd_of_pos_of_lt","def_path":"Mathlib/Data/Nat/Defs.lean","def_pos":[1031,6],"def_end_pos":[1031,26]},{"full_name":"Nat.succ","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1083,4],"def_end_pos":[1083,8]},{"full_name":"Nat.zero_lt_succ","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1671,8],"def_end_pos":[1671,24]},{"full_name":"Not","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[224,4],"def_end_pos":[224,7]},{"full_name":"orderOf","def_path":"Mathlib/GroupTheory/OrderOfElement.lean","def_pos":[129,18],"def_end_pos":[129,25]}]},{"state_before":"case neg.succ.succ\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : x ∈ (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : ℕ\nhn✝ : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : ℕ\nhm✝ : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\nh : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x\nhn' : ¬orderOf (p.cycleOf x) ∣ n.succ\nhm' : ¬orderOf (p.cycleOf x) ∣ m.succ\n⊢ n + 1 = m + 1","state_after":"case neg.succ.succ\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : x ∈ (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : ℕ\nhn✝ : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : ℕ\nhm✝ : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\nh : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x\nhn' : (p.cycleOf x ^ n.succ).support = (p.cycleOf x).support\nhm' : (p.cycleOf x ^ m.succ).support = (p.cycleOf x).support\n⊢ n + 1 = m + 1","tactic":"rw [← hc.support_pow_eq_iff] at hn' hm'","premises":[{"full_name":"Equiv.Perm.IsCycle.support_pow_eq_iff","def_path":"Mathlib/GroupTheory/Perm/Cycle/Basic.lean","def_pos":[518,8],"def_end_pos":[518,34]}]},{"state_before":"case neg.succ.succ\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : x ∈ (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : ℕ\nhn✝ : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : ℕ\nhm✝ : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\nh : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x\nhn' : (p.cycleOf x ^ n.succ).support = (p.cycleOf x).support\nhm' : (p.cycleOf x ^ m.succ).support = (p.cycleOf x).support\n⊢ n + 1 = m + 1","state_after":"case neg.succ.succ\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : x ∈ (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : ℕ\nhn✝ : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : ℕ\nhm✝ : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\nh : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x\nhn' : (p.cycleOf x ^ n.succ).support = (p.cycleOf x).support\nhm' : (p.cycleOf x ^ m.succ).support = (p.cycleOf x).support\n⊢ p.cycleOf x ^ (n + 1) = p.cycleOf x ^ (m + 1)","tactic":"rw [← Nat.mod_eq_of_lt hn, ← Nat.mod_eq_of_lt hm, ← pow_inj_mod]","premises":[{"full_name":"Nat.mod_eq_of_lt","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Div.lean","def_pos":[131,8],"def_end_pos":[131,20]},{"full_name":"pow_inj_mod","def_path":"Mathlib/GroupTheory/OrderOfElement.lean","def_pos":[472,6],"def_end_pos":[472,17]}]},{"state_before":"case neg.succ.succ\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : x ∈ (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : ℕ\nhn✝ : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : ℕ\nhm✝ : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\nh : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x\nhn' : (p.cycleOf x ^ n.succ).support = (p.cycleOf x).support\nhm' : (p.cycleOf x ^ m.succ).support = (p.cycleOf x).support\n⊢ p.cycleOf x ^ (n + 1) = p.cycleOf x ^ (m + 1)","state_after":"case neg.succ.succ.refine_1\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : x ∈ (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : ℕ\nhn✝ : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : ℕ\nhm✝ : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\nh : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x\nhn' : (p.cycleOf x ^ n.succ).support = (p.cycleOf x).support\nhm' : (p.cycleOf x ^ m.succ).support = (p.cycleOf x).support\n⊢ (p.cycleOf x ^ (n + 1)).support ⊆ (p.cycleOf x ^ (m + 1)).support\n\ncase neg.succ.succ.refine_2\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : x ∈ (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : ℕ\nhn✝ : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : ℕ\nhm✝ : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\nh : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x\nhn' : (p.cycleOf x ^ n.succ).support = (p.cycleOf x).support\nhm' : (p.cycleOf x ^ m.succ).support = (p.cycleOf x).support\n⊢ ∀ x_1 ∈ (p.cycleOf x ^ (m + 1)).support, (p.cycleOf x ^ (n + 1)) x_1 = (p.cycleOf x ^ (m + 1)) x_1","tactic":"refine support_congr ?_ ?_","premises":[{"full_name":"Equiv.Perm.support_congr","def_path":"Mathlib/GroupTheory/Perm/Support.lean","def_pos":[292,8],"def_end_pos":[292,21]}]}]}
+{"url":"Mathlib/Tactic/Ring/Basic.lean","commit":"","full_name":"Mathlib.Tactic.Ring.sub_pf","start":[598,0],"end":[599,84],"file_path":"Mathlib/Tactic/Ring/Basic.lean","tactics":[{"state_before":"u✝ : Lean.Level\narg : Q(Type u✝)\nsα✝ : Q(CommSemiring «$arg»)\nu : Lean.Level\nα : Q(Type u)\nsα : Q(CommSemiring «$α»)\nR✝ : Type u_1\ninst✝¹ : CommSemiring R✝\na✝ a' a₁ a₂ a₃ b✝ b' b₁ b₂ b₃ c✝ c₁ c₂ : R✝\nR : Type u_2\ninst✝ : Ring R\na b c d : R\nx✝¹ : -b = c\nx✝ : a + c = d\n⊢ a - b = d","state_after":"u✝ : Lean.Level\narg : Q(Type u✝)\nsα✝ : Q(CommSemiring «$arg»)\nu : Lean.Level\nα : Q(Type u)\nsα : Q(CommSemiring «$α»)\nR✝ : Type u_1\ninst✝¹ : CommSemiring R✝\na✝ a' a₁ a₂ a₃ b✝ b' b₁ b₂ b₃ c c₁ c₂ : R✝\nR : Type u_2\ninst✝ : Ring R\na b : R\n⊢ a - b = a + -b","tactic":"subst_vars","premises":[]},{"state_before":"u✝ : Lean.Level\narg : Q(Type u✝)\nsα✝ : Q(CommSemiring «$arg»)\nu : Lean.Level\nα : Q(Type u)\nsα : Q(CommSemiring «$α»)\nR✝ : Type u_1\ninst✝¹ : CommSemiring R✝\na✝ a' a₁ a₂ a₃ b✝ b' b₁ b₂ b₃ c c₁ c₂ : R✝\nR : Type u_2\ninst✝ : Ring R\na b : R\n⊢ a - b = a + -b","state_after":"no goals","tactic":"simp [sub_eq_add_neg]","premises":[{"full_name":"sub_eq_add_neg","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[905,2],"def_end_pos":[905,13]}]}]}
+{"url":"Mathlib/CategoryTheory/Shift/Basic.lean","commit":"","full_name":"CategoryTheory.shiftFunctorComm_hom_app_comp_shift_shiftFunctorAdd_hom_app","start":[575,0],"end":[598,71],"file_path":"Mathlib/CategoryTheory/Shift/Basic.lean","tactics":[{"state_before":"C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddCommMonoid A\ninst✝ : HasShift C A\nX✝ Y : C\nf : X✝ ⟶ Y\nm₁ m₂ m₃ : A\nX : C\n⊢ (shiftFunctorComm C m₁ (m₂ + m₃)).hom.app X ≫ (shiftFunctor C m₁).map ((shiftFunctorAdd C m₂ m₃).hom.app X) =\n    (shiftFunctorAdd C m₂ m₃).hom.app ((shiftFunctor C m₁).obj X) ≫\n      (shiftFunctor C m₃).map ((shiftFunctorComm C m₁ m₂).hom.app X) ≫\n        (shiftFunctorComm C m₁ m₃).hom.app ((shiftFunctor C m₂).obj X)","state_after":"C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddCommMonoid A\ninst✝ : HasShift C A\nX✝ Y : C\nf : X✝ ⟶ Y\nm₁ m₂ m₃ : A\nX : C\n⊢ (((shiftFunctorComm C m₁ (m₂ + m₃)).hom.app X ≫ (shiftFunctor C m₁).map ((shiftFunctorAdd C m₂ m₃).hom.app X)) ≫\n        (shiftFunctorComm C m₁ m₃).inv.app ((shiftFunctor C m₂).obj X)) ≫\n      (shiftFunctor C m₃).map ((shiftFunctorComm C m₁ m₂).inv.app X) =\n    (((shiftFunctorAdd C m₂ m₃).hom.app ((shiftFunctor C m₁).obj X) ≫\n          (shiftFunctor C m₃).map ((shiftFunctorComm C m₁ m₂).hom.app X) ≫\n            (shiftFunctorComm C m₁ m₃).hom.app ((shiftFunctor C m₂).obj X)) ≫\n        (shiftFunctorComm C m₁ m₃).inv.app ((shiftFunctor C m₂).obj X)) ≫\n      (shiftFunctor C m₃).map ((shiftFunctorComm C m₁ m₂).inv.app X)","tactic":"rw [← cancel_mono ((shiftFunctorComm C m₁ m₃).inv.app (X⟦m₂⟧)),\n    ← cancel_mono (((shiftFunctorComm C m₁ m₂).inv.app X)⟦m₃⟧')]","premises":[{"full_name":"CategoryTheory.Iso.inv","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[53,2],"def_end_pos":[53,5]},{"full_name":"CategoryTheory.NatTrans.app","def_path":"Mathlib/CategoryTheory/NatTrans.lean","def_pos":[48,2],"def_end_pos":[48,5]},{"full_name":"CategoryTheory.cancel_mono","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[263,8],"def_end_pos":[263,19]},{"full_name":"CategoryTheory.shiftFunctor","def_path":"Mathlib/CategoryTheory/Shift/Basic.lean","def_pos":[159,4],"def_end_pos":[159,16]},{"full_name":"CategoryTheory.shiftFunctorComm","def_path":"Mathlib/CategoryTheory/Shift/Basic.lean","def_pos":[503,4],"def_end_pos":[503,20]},{"full_name":"Prefunctor.map","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[57,2],"def_end_pos":[57,5]},{"full_name":"Prefunctor.obj","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[55,2],"def_end_pos":[55,5]}]},{"state_before":"C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddCommMonoid A\ninst✝ : HasShift C A\nX✝ Y : C\nf : X✝ ⟶ Y\nm₁ m₂ m₃ : A\nX : C\n⊢ (((shiftFunctorComm C m₁ (m₂ + m₃)).hom.app X ≫ (shiftFunctor C m₁).map ((shiftFunctorAdd C m₂ m₃).hom.app X)) ≫\n        (shiftFunctorComm C m₁ m₃).inv.app ((shiftFunctor C m₂).obj X)) ≫\n      (shiftFunctor C m₃).map ((shiftFunctorComm C m₁ m₂).inv.app X) =\n    (((shiftFunctorAdd C m₂ m₃).hom.app ((shiftFunctor C m₁).obj X) ≫\n          (shiftFunctor C m₃).map ((shiftFunctorComm C m₁ m₂).hom.app X) ≫\n            (shiftFunctorComm C m₁ m₃).hom.app ((shiftFunctor C m₂).obj X)) ≫\n        (shiftFunctorComm C m₁ m₃).inv.app ((shiftFunctor C m₂).obj X)) ≫\n      (shiftFunctor C m₃).map ((shiftFunctorComm C m₁ m₂).inv.app X)","state_after":"C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddCommMonoid A\ninst✝ : HasShift C A\nX✝ Y : C\nf : X✝ ⟶ Y\nm₁ m₂ m₃ : A\nX : C\n⊢ (shiftFunctorComm C m₁ (m₂ + m₃)).hom.app X ≫\n      (shiftFunctor C m₁).map ((shiftFunctorAdd C m₂ m₃).hom.app X) ≫\n        (shiftFunctorComm C m₁ m₃).inv.app ((shiftFunctor C m₂).obj X) ≫\n          (shiftFunctor C m₃).map ((shiftFunctorComm C m₁ m₂).inv.app X) =\n    (shiftFunctorAdd C m₂ m₃).hom.app ((shiftFunctor C m₁).obj X) ≫\n      (shiftFunctor C m₃).map ((shiftFunctorComm C m₁ m₂).hom.app X) ≫\n        𝟙 ((shiftFunctor C m₁ ⋙ shiftFunctor C m₃).obj ((shiftFunctor C m₂).obj X)) ≫\n          (shiftFunctor C m₃).map ((shiftFunctorComm C m₁ m₂).inv.app X)","tactic":"simp only [Category.assoc, Iso.hom_inv_id_app]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Iso.hom_inv_id_app","def_path":"Mathlib/CategoryTheory/NatIso.lean","def_pos":[59,8],"def_end_pos":[59,22]}]},{"state_before":"C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddCommMonoid A\ninst✝ : HasShift C A\nX✝ Y : C\nf : X✝ ⟶ Y\nm₁ m₂ m₃ : A\nX : C\n⊢ (shiftFunctorComm C m₁ (m₂ + m₃)).hom.app X ≫\n      (shiftFunctor C m₁).map ((shiftFunctorAdd C m₂ m₃).hom.app X) ≫\n        (shiftFunctorComm C m₁ m₃).inv.app ((shiftFunctor C m₂).obj X) ≫\n          (shiftFunctor C m₃).map ((shiftFunctorComm C m₁ m₂).inv.app X) =\n    (shiftFunctorAdd C m₂ m₃).hom.app ((shiftFunctor C m₁).obj X) ≫\n      (shiftFunctor C m₃).map ((shiftFunctorComm C m₁ m₂).hom.app X) ≫\n        𝟙 ((shiftFunctor C m₁ ⋙ shiftFunctor C m₃).obj ((shiftFunctor C m₂).obj X)) ≫\n          (shiftFunctor C m₃).map ((shiftFunctorComm C m₁ m₂).inv.app X)","state_after":"C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddCommMonoid A\ninst✝ : HasShift C A\nX✝ Y : C\nf : X✝ ⟶ Y\nm₁ m₂ m₃ : A\nX : C\n⊢ (shiftFunctorComm C m₁ (m₂ + m₃)).hom.app X ≫\n      (shiftFunctor C m₁).map ((shiftFunctorAdd C m₂ m₃).hom.app X) ≫\n        (shiftFunctorComm C m₁ m₃).inv.app ((shiftFunctor C m₂).obj X) ≫\n          (shiftFunctor C m₃).map ((shiftFunctorComm C m₁ m₂).inv.app X) =\n    (shiftFunctorAdd C m₂ m₃).hom.app ((shiftFunctor C m₁).obj X) ≫\n      (shiftFunctor C m₃).map ((shiftFunctorComm C m₁ m₂).hom.app X) ≫\n        𝟙 ((shiftFunctor C m₃).obj ((shiftFunctor C m₁).obj ((shiftFunctor C m₂).obj X))) ≫\n          (shiftFunctor C m₃).map ((shiftFunctorComm C m₁ m₂).inv.app X)","tactic":"dsimp","premises":[]},{"state_before":"C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddCommMonoid A\ninst✝ : HasShift C A\nX✝ Y : C\nf : X✝ ⟶ Y\nm₁ m₂ m₃ : A\nX : C\n⊢ (shiftFunctorComm C m₁ (m₂ + m₃)).hom.app X ≫\n      (shiftFunctor C m₁).map ((shiftFunctorAdd C m₂ m₃).hom.app X) ≫\n        (shiftFunctorComm C m₁ m₃).inv.app ((shiftFunctor C m₂).obj X) ≫\n          (shiftFunctor C m₃).map ((shiftFunctorComm C m₁ m₂).inv.app X) =\n    (shiftFunctorAdd C m₂ m₃).hom.app ((shiftFunctor C m₁).obj X) ≫\n      (shiftFunctor C m₃).map ((shiftFunctorComm C m₁ m₂).hom.app X) ≫\n        𝟙 ((shiftFunctor C m₃).obj ((shiftFunctor C m₁).obj ((shiftFunctor C m₂).obj X))) ≫\n          (shiftFunctor C m₃).map ((shiftFunctorComm C m₁ m₂).inv.app X)","state_after":"C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddCommMonoid A\ninst✝ : HasShift C A\nX✝ Y : C\nf : X✝ ⟶ Y\nm₁ m₂ m₃ : A\nX : C\n⊢ (shiftFunctorComm C m₁ (m₂ + m₃)).hom.app X ≫\n      (shiftFunctor C m₁).map ((shiftFunctorAdd C m₂ m₃).hom.app X) ≫\n        (shiftFunctorComm C m₁ m₃).inv.app ((shiftFunctor C m₂).obj X) ≫\n          (shiftFunctor C m₃).map ((shiftFunctorComm C m₁ m₂).inv.app X) =\n    (shiftFunctorAdd C m₂ m₃).hom.app ((shiftFunctor C m₁).obj X) ≫\n      (shiftFunctor C m₃).map (𝟙 ((shiftFunctor C m₁ ⋙ shiftFunctor C m₂).obj X))","tactic":"simp only [Category.id_comp, ← Functor.map_comp, Iso.hom_inv_id_app]","premises":[{"full_name":"CategoryTheory.Category.id_comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[151,2],"def_end_pos":[151,9]},{"full_name":"CategoryTheory.Functor.map_comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[41,2],"def_end_pos":[41,10]},{"full_name":"CategoryTheory.Iso.hom_inv_id_app","def_path":"Mathlib/CategoryTheory/NatIso.lean","def_pos":[59,8],"def_end_pos":[59,22]}]},{"state_before":"C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddCommMonoid A\ninst✝ : HasShift C A\nX✝ Y : C\nf : X✝ ⟶ Y\nm₁ m₂ m₃ : A\nX : C\n⊢ (shiftFunctorComm C m₁ (m₂ + m₃)).hom.app X ≫\n      (shiftFunctor C m₁).map ((shiftFunctorAdd C m₂ m₃).hom.app X) ≫\n        (shiftFunctorComm C m₁ m₃).inv.app ((shiftFunctor C m₂).obj X) ≫\n          (shiftFunctor C m₃).map ((shiftFunctorComm C m₁ m₂).inv.app X) =\n    (shiftFunctorAdd C m₂ m₃).hom.app ((shiftFunctor C m₁).obj X) ≫\n      (shiftFunctor C m₃).map (𝟙 ((shiftFunctor C m₁ ⋙ shiftFunctor C m₂).obj X))","state_after":"C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddCommMonoid A\ninst✝ : HasShift C A\nX✝ Y : C\nf : X✝ ⟶ Y\nm₁ m₂ m₃ : A\nX : C\n⊢ (shiftFunctorComm C m₁ (m₂ + m₃)).hom.app X ≫\n      (shiftFunctor C m₁).map ((shiftFunctorAdd C m₂ m₃).hom.app X) ≫\n        (shiftFunctorComm C m₁ m₃).inv.app ((shiftFunctor C m₂).obj X) ≫\n          (shiftFunctor C m₃).map ((shiftFunctorComm C m₁ m₂).inv.app X) =\n    (shiftFunctorAdd C m₂ m₃).hom.app ((shiftFunctor C m₁).obj X) ≫\n      (shiftFunctor C m₃).map (𝟙 ((shiftFunctor C m₂).obj ((shiftFunctor C m₁).obj X)))","tactic":"dsimp","premises":[]},{"state_before":"C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddCommMonoid A\ninst✝ : HasShift C A\nX✝ Y : C\nf : X✝ ⟶ Y\nm₁ m₂ m₃ : A\nX : C\n⊢ (shiftFunctorComm C m₁ (m₂ + m₃)).hom.app X ≫\n      (shiftFunctor C m₁).map ((shiftFunctorAdd C m₂ m₃).hom.app X) ≫\n        (shiftFunctorComm C m₁ m₃).inv.app ((shiftFunctor C m₂).obj X) ≫\n          (shiftFunctor C m₃).map ((shiftFunctorComm C m₁ m₂).inv.app X) =\n    (shiftFunctorAdd C m₂ m₃).hom.app ((shiftFunctor C m₁).obj X) ≫\n      (shiftFunctor C m₃).map (𝟙 ((shiftFunctor C m₂).obj ((shiftFunctor C m₁).obj X)))","state_after":"C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddCommMonoid A\ninst✝ : HasShift C A\nX✝ Y : C\nf : X✝ ⟶ Y\nm₁ m₂ m₃ : A\nX : C\n⊢ ((shiftFunctorAdd' C m₁ (m₂ + m₃) (m₁ + (m₂ + m₃)) ⋯).symm ≪≫\n              shiftFunctorAdd' C (m₂ + m₃) m₁ (m₁ + (m₂ + m₃)) ⋯).hom.app\n        X ≫\n      (shiftFunctor C m₁).map ((shiftFunctorAdd' C m₂ m₃ (m₂ + m₃) ⋯).hom.app X) ≫\n        ((shiftFunctorAdd' C m₁ m₃ (m₁ + m₃) ⋯).symm ≪≫ shiftFunctorAdd' C m₃ m₁ (m₁ + m₃) ⋯).inv.app\n            ((shiftFunctor C m₂).obj X) ≫\n          (shiftFunctor C m₃).map\n            (((shiftFunctorAdd' C m₁ m₂ (m₁ + m₂) ⋯).symm ≪≫ shiftFunctorAdd' C m₂ m₁ (m₁ + m₂) ⋯).inv.app X) =\n    (shiftFunctorAdd' C m₂ m₃ (m₂ + m₃) ⋯).hom.app ((shiftFunctor C m₁).obj X)","tactic":"simp only [Functor.map_id, Category.comp_id,\n    shiftFunctorComm_eq C _ _ _ rfl, ← shiftFunctorAdd'_eq_shiftFunctorAdd]","premises":[{"full_name":"CategoryTheory.Category.comp_id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[153,2],"def_end_pos":[153,9]},{"full_name":"CategoryTheory.Functor.map_id","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[39,2],"def_end_pos":[39,8]},{"full_name":"CategoryTheory.shiftFunctorAdd'_eq_shiftFunctorAdd","def_path":"Mathlib/CategoryTheory/Shift/Basic.lean","def_pos":[171,6],"def_end_pos":[171,41]},{"full_name":"CategoryTheory.shiftFunctorComm_eq","def_path":"Mathlib/CategoryTheory/Shift/Basic.lean","def_pos":[508,6],"def_end_pos":[508,25]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddCommMonoid A\ninst✝ : HasShift C A\nX✝ Y : C\nf : X✝ ⟶ Y\nm₁ m₂ m��� : A\nX : C\n⊢ ((shiftFunctorAdd' C m₁ (m₂ + m₃) (m₁ + (m₂ + m₃)) ⋯).symm ≪≫\n              shiftFunctorAdd' C (m₂ + m₃) m₁ (m₁ + (m₂ + m₃)) ⋯).hom.app\n        X ≫\n      (shiftFunctor C m₁).map ((shiftFunctorAdd' C m₂ m₃ (m₂ + m₃) ⋯).hom.app X) ≫\n        ((shiftFunctorAdd' C m₁ m₃ (m₁ + m₃) ⋯).symm ≪≫ shiftFunctorAdd' C m₃ m₁ (m₁ + m₃) ⋯).inv.app\n            ((shiftFunctor C m₂).obj X) ≫\n          (shiftFunctor C m₃).map\n            (((shiftFunctorAdd' C m₁ m₂ (m₁ + m₂) ⋯).symm ≪≫ shiftFunctorAdd' C m₂ m₁ (m₁ + m₂) ⋯).inv.app X) =\n    (shiftFunctorAdd' C m₂ m₃ (m₂ + m₃) ⋯).hom.app ((shiftFunctor C m₁).obj X)","state_after":"C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddCommMonoid A\ninst✝ : HasShift C A\nX✝ Y : C\nf : X✝ ⟶ Y\nm₁ m₂ m₃ : A\nX : C\n⊢ ((shiftFunctorAdd' C m₁ (m₂ + m₃) (m₁ + (m₂ + m₃)) ⋯).inv.app X ≫\n        (shiftFunctorAdd' C (m₂ + m₃) m₁ (m₁ + (m₂ + m₃)) ⋯).hom.app X) ≫\n      (shiftFunctor C m₁).map ((shiftFunctorAdd' C m₂ m₃ (m₂ + m₃) ⋯).hom.app X) ≫\n        ((shiftFunctorAdd' C m₃ m₁ (m₁ + m₃) ⋯).inv.app ((shiftFunctor C m₂).obj X) ≫\n            (shiftFunctorAdd' C m₁ m₃ (m₁ + m₃) ⋯).hom.app ((shiftFunctor C m₂).obj X)) ≫\n          (shiftFunctor C m₃).map\n            ((shiftFunctorAdd' C m₂ m₁ (m₁ + m₂) ⋯).inv.app X ≫ (shiftFunctorAdd' C m₁ m₂ (m₁ + m₂) ⋯).hom.app X) =\n    (shiftFunctorAdd' C m₂ m₃ (m₂ + m₃) ⋯).hom.app ((shiftFunctor C m₁).obj X)","tactic":"dsimp","premises":[]},{"state_before":"C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddCommMonoid A\ninst✝ : HasShift C A\nX✝ Y : C\nf : X✝ ⟶ Y\nm₁ m₂ m₃ : A\nX : C\n⊢ ((shiftFunctorAdd' C m₁ (m₂ + m₃) (m₁ + (m₂ + m₃)) ⋯).inv.app X ≫\n        (shiftFunctorAdd' C (m₂ + m₃) m₁ (m₁ + (m₂ + m₃)) ⋯).hom.app X) ≫\n      (shiftFunctor C m₁).map ((shiftFunctorAdd' C m₂ m₃ (m₂ + m₃) ⋯).hom.app X) ≫\n        ((shiftFunctorAdd' C m₃ m₁ (m₁ + m₃) ⋯).inv.app ((shiftFunctor C m₂).obj X) ≫\n            (shiftFunctorAdd' C m₁ m₃ (m₁ + m₃) ⋯).hom.app ((shiftFunctor C m₂).obj X)) ≫\n          (shiftFunctor C m₃).map\n            ((shiftFunctorAdd' C m₂ m₁ (m₁ + m₂) ⋯).inv.app X ≫ (shiftFunctorAdd' C m₁ m₂ (m₁ + m₂) ⋯).hom.app X) =\n    (shiftFunctorAdd' C m₂ m₃ (m₂ + m₃) ⋯).hom.app ((shiftFunctor C m₁).obj X)","state_after":"no goals","tactic":"simp only [Category.assoc, Iso.hom_inv_id_app_assoc, Iso.inv_hom_id_app_assoc,\n    ← Functor.map_comp,\n    shiftFunctorAdd'_assoc_hom_app_assoc m₂ m₃ m₁ (m₂ + m₃) (m₁ + m₃) (m₁ + (m₂ + m₃)) rfl\n      (add_comm m₃ m₁) (add_comm _ m₁) X,\n    ← shiftFunctorAdd'_assoc_hom_app_assoc m₂ m₁ m₃ (m₁ + m₂) (m₁ + m₃)\n      (m₁ + (m₂ + m₃)) (add_comm _ _) rfl (by rw [add_comm m₂ m₁, add_assoc]) X,\n    shiftFunctorAdd'_assoc_hom_app m₁ m₂ m₃\n      (m₁ + m₂) (m₂ + m₃) (m₁ + (m₂ + m₃)) rfl rfl (add_assoc _ _ _) X]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Functor.map_comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[41,2],"def_end_pos":[41,10]},{"full_name":"CategoryTheory.shiftFunctorAdd'_assoc_hom_app","def_path":"Mathlib/CategoryTheory/Shift/Basic.lean","def_pos":[310,6],"def_end_pos":[310,36]},{"full_name":"add_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[258,2],"def_end_pos":[258,13]},{"full_name":"add_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[293,2],"def_end_pos":[293,13]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
+{"url":"Mathlib/GroupTheory/OrderOfElement.lean","commit":"","full_name":"addOrderOf_neg","start":[554,0],"end":[555,89],"file_path":"Mathlib/GroupTheory/OrderOfElement.lean","tactics":[{"state_before":"G : Type u_1\nH : Type u_2\nA : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝ : Group G\nx✝ y : G\ni : ℤ\nx : G\n⊢ orderOf x⁻¹ = orderOf x","state_after":"no goals","tactic":"simp [orderOf_eq_orderOf_iff]","premises":[{"full_name":"orderOf_eq_orderOf_iff","def_path":"Mathlib/GroupTheory/OrderOfElement.lean","def_pos":[289,8],"def_end_pos":[289,30]}]}]}
+{"url":"Mathlib/Order/Filter/FilterProduct.lean","commit":"","full_name":"Filter.Germ.lt_def","start":[78,0],"end":[80,14],"file_path":"Mathlib/Order/Filter/FilterProduct.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nφ : Ultrafilter α\ninst✝ : Preorder β\n⊢ (fun x x_1 => x < x_1) = LiftRel fun x x_1 => x < x_1","state_after":"case h.mk.h.mk.a\nα : Type u\nβ : Type v\nφ : Ultrafilter α\ninst✝ : Preorder β\nx✝¹ : (↑φ).Germ β\nf : α → β\nx✝ : (↑φ).Germ β\ng : α → β\n⊢ Quot.mk Setoid.r f < Quot.mk Setoid.r g ↔ LiftRel (fun x x_1 => x < x_1) (Quot.mk Setoid.r f) (Quot.mk Setoid.r g)","tactic":"ext ⟨f⟩ ⟨g⟩","premises":[]},{"state_before":"case h.mk.h.mk.a\nα : Type u\nβ : Type v\nφ : Ultrafilter α\ninst✝ : Preorder β\nx✝¹ : (↑φ).Germ β\nf : α → β\nx✝ : (↑φ).Germ β\ng : α → β\n⊢ Quot.mk Setoid.r f < Quot.mk Setoid.r g ↔ LiftRel (fun x x_1 => x < x_1) (Quot.mk Setoid.r f) (Quot.mk Setoid.r g)","state_after":"no goals","tactic":"exact coe_lt","premises":[{"full_name":"Filter.Germ.coe_lt","def_path":"Mathlib/Order/Filter/FilterProduct.lean","def_pos":[65,8],"def_end_pos":[65,14]}]}]}
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+{"url":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","commit":"","full_name":"Polynomial.leadingCoeff_C_mul_X","start":[739,0],"end":[740,57],"file_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","tactics":[{"state_before":"R : Type u\nS : Type v\na✝ b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ p q : R[X]\nι : Type u_1\na : R\n⊢ (C a * X).leadingCoeff = a","state_after":"no goals","tactic":"simpa only [pow_one] using leadingCoeff_C_mul_X_pow a 1","premises":[{"full_name":"Polynomial.leadingCoeff_C_mul_X_pow","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[736,8],"def_end_pos":[736,32]},{"full_name":"pow_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[571,6],"def_end_pos":[571,13]}]}]}
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+{"url":"Mathlib/Data/Nat/Totient.lean","commit":"","full_name":"Nat.totient_dvd_of_dvd","start":[328,0],"end":[336,87],"file_path":"Mathlib/Data/Nat/Totient.lean","tactics":[{"state_before":"a b : ℕ\nh : a ∣ b\n⊢ φ a ∣ φ b","state_after":"case inl\nb : ℕ\nh : 0 ∣ b\n⊢ φ 0 ∣ φ b\n\ncase inr\na b : ℕ\nh : a ∣ b\nha0 : a ≠ 0\n⊢ φ a ∣ φ b","tactic":"rcases eq_or_ne a 0 with (rfl | ha0)","premises":[{"full_name":"eq_or_ne","def_path":"Mathlib/Logic/Basic.lean","def_pos":[167,8],"def_end_pos":[167,16]}]},{"state_before":"case inr\na b : ℕ\nh : a ∣ b\nha0 : a ≠ 0\n⊢ φ a ∣ φ b","state_after":"case inr.inl\na : ℕ\nha0 : a ≠ 0\nh : a ∣ 0\n⊢ φ a ∣ φ 0\n\ncase inr.inr\na b : ℕ\nh : a ∣ b\nha0 : a ≠ 0\nhb0 : b ≠ 0\n⊢ φ a ∣ φ b","tactic":"rcases eq_or_ne b 0 with (rfl | hb0)","premises":[{"full_name":"eq_or_ne","def_path":"Mathlib/Logic/Basic.lean","def_pos":[167,8],"def_end_pos":[167,16]}]},{"state_before":"case inr.inr\na b : ℕ\nh : a ∣ b\nha0 : a ≠ 0\nhb0 : b ≠ 0\n⊢ φ a ∣ φ b","state_after":"case inr.inr\na b : ℕ\nh : a ∣ b\nha0 : a ≠ 0\nhb0 : b ≠ 0\nhab' : a.primeFactors ⊆ b.primeFactors\n⊢ φ a ∣ φ b","tactic":"have hab' := primeFactors_mono h hb0","premises":[{"full_name":"Nat.primeFactors_mono","def_path":"Mathlib/Data/Nat/PrimeFin.lean","def_pos":[41,6],"def_end_pos":[41,23]}]},{"state_before":"case inr.inr\na b : ℕ\nh : a ∣ b\nha0 : a ≠ 0\nhb0 : b ≠ 0\nhab' : a.primeFactors ⊆ b.primeFactors\n⊢ φ a ∣ φ b","state_after":"case inr.inr\na b : ℕ\nh : a ∣ b\nha0 : a ≠ 0\nhb0 : b ≠ 0\nhab' : a.primeFactors ⊆ b.primeFactors\n⊢ (a.factorization.prod fun p k => p ^ (k - 1) * (p - 1)) ∣ b.factorization.prod fun p k => p ^ (k - 1) * (p - 1)","tactic":"rw [totient_eq_prod_factorization ha0, totient_eq_prod_factorization hb0]","premises":[{"full_name":"Nat.totient_eq_prod_factorization","def_path":"Mathlib/Data/Nat/Totient.lean","def_pos":[259,8],"def_end_pos":[259,37]}]},{"state_before":"case inr.inr\na b : ℕ\nh : a ∣ b\nha0 : a ≠ 0\nhb0 : b ≠ 0\nhab' : a.primeFactors ⊆ b.primeFactors\n⊢ (a.factorization.prod fun p k => p ^ (k - 1) * (p - 1)) ∣ b.factorization.prod fun p k => p ^ (k - 1) * (p - 1)","state_after":"case inr.inr\na b : ℕ\nh : a ∣ b\nha0 : a ≠ 0\nhb0 : b ≠ 0\nhab' : a.primeFactors ⊆ b.primeFactors\np : ℕ\nx✝ : p ∈ a.factorization.support\n⊢ p ^ (a.factorization p - 1) ∣ p ^ (b.factorization p - 1)","tactic":"refine Finsupp.prod_dvd_prod_of_subset_of_dvd hab' fun p _ => mul_dvd_mul ?_ dvd_rfl","premises":[{"full_name":"Finsupp.prod_dvd_prod_of_subset_of_dvd","def_path":"Mathlib/Algebra/BigOperators/Finsupp.lean","def_pos":[501,8],"def_end_pos":[501,38]},{"full_name":"dvd_rfl","def_path":"Mathlib/Algebra/Divisibility/Basic.lean","def_pos":[125,8],"def_end_pos":[125,15]},{"full_name":"mul_dvd_mul","def_path":"Mathlib/Algebra/Divisibility/Basic.lean","def_pos":[186,8],"def_end_pos":[186,19]}]},{"state_before":"case inr.inr\na b : ℕ\nh : a ∣ b\nha0 : a ≠ 0\nhb0 : b ≠ 0\nhab' : a.primeFactors ⊆ b.primeFactors\np : ℕ\nx✝ : p ∈ a.factorization.support\n⊢ p ^ (a.factorization p - 1) ∣ p ^ (b.factorization p - 1)","state_after":"no goals","tactic":"exact pow_dvd_pow p (tsub_le_tsub_right ((factorization_le_iff_dvd ha0 hb0).2 h p) 1)","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Nat.factorization_le_iff_dvd","def_path":"Mathlib/Data/Nat/Factorization/Defs.lean","def_pos":[150,8],"def_end_pos":[150,32]},{"full_name":"pow_dvd_pow","def_path":"Mathlib/Algebra/Divisibility/Basic.lean","def_pos":[137,6],"def_end_pos":[137,17]},{"full_name":"tsub_le_tsub_right","def_path":"Mathlib/Algebra/Order/Sub/Defs.lean","def_pos":[97,18],"def_end_pos":[97,36]}]}]}
+{"url":"Mathlib/SetTheory/Cardinal/Basic.lean","commit":"","full_name":"Cardinal.sum_add_distrib","start":[761,0],"end":[765,12],"file_path":"Mathlib/SetTheory/Cardinal/Basic.lean","tactics":[{"state_before":"α β : Type u\nι : Type u_1\nf g : ι → Cardinal.{u_2}\n⊢ sum (f + g) = sum f + sum g","state_after":"α β : Type u\nι : Type u_1\nf g : ι → Cardinal.{u_2}\nthis :\n  #((i : ι) × ((Quotient.out ∘ f) i ⊕ (Quotient.out ∘ g) i)) =\n    #((i : ι) × (Quotient.out ∘ f) i ⊕ (i : ι) × (Quotient.out ∘ g) i)\n⊢ sum (f + g) = sum f + sum g","tactic":"have := mk_congr (Equiv.sigmaSumDistrib (Quotient.out ∘ f) (Quotient.out ∘ g))","premises":[{"full_name":"Cardinal.mk_congr","def_path":"Mathlib/SetTheory/Cardinal/Basic.lean","def_pos":[151,8],"def_end_pos":[151,16]},{"full_name":"Equiv.sigmaSumDistrib","def_path":"Mathlib/Logic/Equiv/Basic.lean","def_pos":[900,4],"def_end_pos":[900,19]},{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"Quotient.out","def_path":"Mathlib/Data/Quot.lean","def_pos":[339,18],"def_end_pos":[339,30]}]},{"state_before":"α β : Type u\nι : Type u_1\nf g : ι → Cardinal.{u_2}\nthis :\n  #((i : ι) × ((Quotient.out ∘ f) i ⊕ (Quotient.out ∘ g) i)) =\n    #((i : ι) × (Quotient.out ∘ f) i ⊕ (i : ι) × (Quotient.out ∘ g) i)\n⊢ sum (f + g) = sum f + sum g","state_after":"α β : Type u\nι : Type u_1\nf g : ι → Cardinal.{u_2}\nthis : (sum fun i => f i + g i) = (sum fun i => f i) + sum fun i => g i\n⊢ sum (f + g) = sum f + sum g","tactic":"simp only [comp_apply, mk_sigma, mk_sum, mk_out, lift_id] at this","premises":[{"full_name":"Cardinal.lift_id","def_path":"Mathlib/SetTheory/Cardinal/Basic.lean","def_pos":[203,8],"def_end_pos":[203,15]},{"full_name":"Cardinal.mk_out","def_path":"Mathlib/SetTheory/Cardinal/Basic.lean","def_pos":[144,8],"def_end_pos":[144,14]},{"full_name":"Cardinal.mk_sigma","def_path":"Mathlib/SetTheory/Cardinal/Basic.lean","def_pos":[747,8],"def_end_pos":[747,16]},{"full_name":"Cardinal.mk_sum","def_path":"Mathlib/SetTheory/Cardinal/Basic.lean","def_pos":[392,8],"def_end_pos":[392,14]},{"full_name":"Function.comp_apply","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[35,16],"def_end_pos":[35,35]}]},{"state_before":"α β : Type u\nι : Type u_1\nf g : ι → Cardinal.{u_2}\nthis : (sum fun i => f i + g i) = (sum fun i => f i) + sum fun i => g i\n⊢ sum (f + g) = sum f + sum g","state_after":"no goals","tactic":"exact this","premises":[]}]}
+{"url":"Mathlib/Topology/Filter.lean","commit":"","full_name":"Filter.isTopologicalBasis_Iic_principal","start":[55,0],"end":[62,28],"file_path":"Mathlib/Topology/Filter.lean","tactics":[{"state_before":"ι : Sort u_1\nα : Type u_2\nβ : Type u_3\nX : Type u_4\nY : Type u_5\n⊢ ∀ t₁ ∈ range (Iic ∘ 𝓟), ∀ t₂ ∈ range (Iic ∘ 𝓟), ∀ x ∈ t₁ ∩ t₂, ∃ t₃ ∈ range (Iic ∘ 𝓟), x ∈ t₃ ∧ t₃ ⊆ t₁ ∩ t₂","state_after":"case intro.intro\nι : Sort u_1\nα : Type u_2\nβ : Type u_3\nX : Type u_4\nY : Type u_5\ns t : Set α\nl : Filter α\nhl : l ∈ (Iic ∘ 𝓟) s ∩ (Iic ∘ 𝓟) t\n⊢ ∃ t₃ ∈ range (Iic ∘ 𝓟), l ∈ t₃ ∧ t₃ ⊆ (Iic ∘ 𝓟) s ∩ (Iic ∘ 𝓟) t","tactic":"rintro _ ⟨s, rfl⟩ _ ⟨t, rfl⟩ l hl","premises":[]},{"state_before":"case intro.intro\nι : Sort u_1\nα : Type u_2\nβ : Type u_3\nX : Type u_4\nY : Type u_5\ns t : Set α\nl : Filter α\nhl : l ∈ (Iic ∘ 𝓟) s ∩ (Iic ∘ 𝓟) t\n⊢ ∃ t₃ ∈ range (Iic ∘ 𝓟), l ∈ t₃ ∧ t₃ ⊆ (Iic ∘ 𝓟) s ∩ (Iic ∘ 𝓟) t","state_after":"no goals","tactic":"exact ⟨Iic (𝓟 s) ∩ Iic (𝓟 t), ⟨s ∩ t, by simp⟩, hl, Subset.rfl⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Filter.principal","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[269,4],"def_end_pos":[269,13]},{"full_name":"Inter.inter","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[407,2],"def_end_pos":[407,7]},{"full_name":"Set.Iic","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[58,4],"def_end_pos":[58,7]},{"full_name":"Set.Subset.rfl","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[292,8],"def_end_pos":[292,18]}]}]}
+{"url":"Mathlib/Data/Complex/Exponential.lean","commit":"","full_name":"Complex.tan_ofReal_im","start":[584,0],"end":[585,91],"file_path":"Mathlib/Data/Complex/Exponential.lean","tactics":[{"state_before":"x✝ y : ℂ\nx : ℝ\n⊢ (tan ↑x).im = 0","state_after":"no goals","tactic":"rw [← ofReal_tan_ofReal_re, ofReal_im]","premises":[{"full_name":"Complex.ofReal_im","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[89,8],"def_end_pos":[89,17]},{"full_name":"Complex.ofReal_tan_ofReal_re","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[569,8],"def_end_pos":[569,28]}]}]}
+{"url":"Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean","commit":"","full_name":"FiniteDimensional.exists_relation_sum_zero_pos_coefficient_of_finrank_succ_lt_card","start":[262,0],"end":[271,81],"file_path":"Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean","tactics":[{"state_before":"K : Type u\nV : Type v\ninst✝⁸ : DivisionRing K\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module K V\nV₂ : Type v'\ninst✝⁵ : AddCommGroup V₂\ninst✝⁴ : Module K V₂\nL : Type u_1\ninst✝³ : LinearOrderedField L\nW : Type v\ninst✝² : AddCommGroup W\ninst✝¹ : Module L W\ninst✝ : FiniteDimensional L W\nt : Finset W\nh : finrank L W + 1 < t.card\n⊢ ∃ f, ∑ e ∈ t, f e • e = 0 ∧ ∑ e ∈ t, f e = 0 ∧ ∃ x ∈ t, 0 < f x","state_after":"case intro.intro.intro\nK : Type u\nV : Type v\ninst✝⁸ : DivisionRing K\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module K V\nV₂ : Type v'\ninst✝⁵ : AddCommGroup V₂\ninst✝⁴ : Module K V₂\nL : Type u_1\ninst✝³ : LinearOrderedField L\nW : Type v\ninst✝² : AddCommGroup W\ninst✝¹ : Module L W\ninst✝ : FiniteDimensional L W\nt : Finset W\nh : finrank L W + 1 < t.card\nf : W → L\nsum : ∑ e ∈ t, f e • e = 0\ntotal : ∑ e ∈ t, f e = 0\nnonzero : ∃ x ∈ t, f x ≠ 0\n⊢ ∃ f, ∑ e ∈ t, f e • e = 0 ∧ ∑ e ∈ t, f e = 0 ∧ ∃ x ∈ t, 0 < f x","tactic":"obtain ⟨f, sum, total, nonzero⟩ :=\n    Module.exists_nontrivial_relation_sum_zero_of_finrank_succ_lt_card h","premises":[{"full_name":"Module.exists_nontrivial_relation_sum_zero_of_finrank_succ_lt_card","def_path":"Mathlib/LinearAlgebra/Dimension/Finite.lean","def_pos":[309,8],"def_end_pos":[309,74]}]},{"state_before":"case intro.intro.intro\nK : Type u\nV : Type v\ninst✝⁸ : DivisionRing K\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module K V\nV₂ : Type v'\ninst✝⁵ : AddCommGroup V₂\ninst✝⁴ : Module K V₂\nL : Type u_1\ninst✝³ : LinearOrderedField L\nW : Type v\ninst✝² : AddCommGroup W\ninst✝¹ : Module L W\ninst✝ : FiniteDimensional L W\nt : Finset W\nh : finrank L W + 1 < t.card\nf : W → L\nsum : ∑ e ∈ t, f e • e = 0\ntotal : ∑ e ∈ t, f e = 0\nnonzero : ∃ x ∈ t, f x ≠ 0\n⊢ ∃ f, ∑ e ∈ t, f e • e = 0 ∧ ∑ e ∈ t, f e = 0 ∧ ∃ x ∈ t, 0 < f x","state_after":"no goals","tactic":"exact ⟨f, sum, total, exists_pos_of_sum_zero_of_exists_nonzero f total nonzero⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Finset.exists_pos_of_sum_zero_of_exists_nonzero","def_path":"Mathlib/Algebra/Order/BigOperators/Group/Finset.lean","def_pos":[496,14],"def_end_pos":[496,54]}]}]}
+{"url":"Mathlib/Order/Interval/Finset/Box.lean","commit":"","full_name":"Finset.disjoint_box_succ_prod","start":[40,0],"end":[41,56],"file_path":"Mathlib/Order/Interval/Finset/Box.lean","tactics":[{"state_before":"α : Type u_1\ninst✝² : OrderedRing α\ninst✝¹ : LocallyFiniteOrder α\ninst✝ : DecidableEq α\nn✝ n : ℕ\n⊢ Disjoint (box (n + 1)) (Icc (-↑n) ↑n)","state_after":"α : Type u_1\ninst✝² : OrderedRing α\ninst✝¹ : LocallyFiniteOrder α\ninst✝ : DecidableEq α\nn✝ n : ℕ\n⊢ Disjoint (Icc (-↑n.succ) ↑n.succ \\ Icc (-↑n) ↑n) (Icc (-↑n) ↑n)","tactic":"rw [box_succ_eq_sdiff]","premises":[{"full_name":"Finset.box_succ_eq_sdiff","def_path":"Mathlib/Order/Interval/Finset/Box.lean","def_pos":[37,6],"def_end_pos":[37,23]}]},{"state_before":"α : Type u_1\ninst✝² : OrderedRing α\ninst✝¹ : LocallyFiniteOrder α\ninst✝ : DecidableEq α\nn✝ n : ℕ\n⊢ Disjoint (Icc (-↑n.succ) ↑n.succ \\ Icc (-↑n) ↑n) (Icc (-↑n) ↑n)","state_after":"no goals","tactic":"exact disjoint_sdiff_self_left","premises":[{"full_name":"disjoint_sdiff_self_left","def_path":"Mathlib/Order/BooleanAlgebra.lean","def_pos":[195,8],"def_end_pos":[195,32]}]}]}
+{"url":"Mathlib/LinearAlgebra/Matrix/ZPow.lean","commit":"","full_name":"Matrix.zero_zpow_eq","start":[85,0],"end":[88,22],"file_path":"Mathlib/LinearAlgebra/Matrix/ZPow.lean","tactics":[{"state_before":"n' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nn : ℤ\n⊢ 0 ^ n = if n = 0 then 1 else 0","state_after":"case pos\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nn : ℤ\nh : n = 0\n⊢ 0 ^ n = 1\n\ncase neg\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nn : ℤ\nh : ¬n = 0\n⊢ 0 ^ n = 0","tactic":"split_ifs with h","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
+{"url":"Mathlib/GroupTheory/PGroup.lean","commit":"","full_name":"IsPGroup.nonempty_fixed_point_of_prime_not_dvd_card","start":[191,0],"end":[201,45],"file_path":"Mathlib/GroupTheory/PGroup.lean","tactics":[{"state_before":"p : ℕ\nG : Type u_1\ninst✝² : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\nα : Type u_2\ninst✝¹ : MulAction G α\ninst✝ : Finite α\nhpα : ¬p ∣ Nat.card α\nthis : Finite α\n⊢ Nonempty ↑(fixedPoints G α)","state_after":"p : ℕ\nG : Type u_1\ninst✝² : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\nα : Type u_2\ninst✝¹ : MulAction G α\ninst✝ : Finite α\nhpα : ¬p ∣ Nat.card α\nthis : Finite α\n⊢ Nat.card ↑(fixedPoints G α) ≠ 0","tactic":"rw [← Finite.card_pos_iff, pos_iff_ne_zero]","premises":[{"full_name":"Finite.card_pos_iff","def_path":"Mathlib/Data/Finite/Card.lean","def_pos":[52,8],"def_end_pos":[52,27]},{"full_name":"pos_iff_ne_zero","def_path":"Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean","def_pos":[123,2],"def_end_pos":[123,13]}]},{"state_before":"p : ℕ\nG : Type u_1\ninst✝² : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\nα : Type u_2\ninst✝¹ : MulAction G α\ninst✝ : Finite α\nhpα : ¬p ∣ Nat.card α\nthis : Finite α\n⊢ Nat.card ↑(fixedPoints G α) ≠ 0","state_after":"p : ℕ\nG : Type u_1\ninst✝² : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\nα : Type u_2\ninst✝¹ : MulAction G α\ninst✝ this : Finite α\nhpα : Nat.card ↑(fixedPoints G α) = 0\n⊢ p ∣ Nat.card α","tactic":"contrapose! hpα","premises":[{"full_name":"Mathlib.Tactic.Contrapose.mtr","def_path":"Mathlib/Tactic/Contrapose.lean","def_pos":[24,6],"def_end_pos":[24,9]}]},{"state_before":"p : ℕ\nG : Type u_1\ninst✝² : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\nα : Type u_2\ninst✝¹ : MulAction G α\ninst✝ this : Finite α\nhpα : Nat.card ↑(fixedPoints G α) = 0\n⊢ p ∣ Nat.card α","state_after":"p : ℕ\nG : Type u_1\ninst✝² : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\nα : Type u_2\ninst✝¹ : MulAction G α\ninst✝ this : Finite α\nhpα : Nat.card ↑(fixedPoints G α) = 0\n⊢ Nat.card α ≡ Nat.card ↑(fixedPoints G α) [MOD p]","tactic":"rw [← Nat.modEq_zero_iff_dvd, ← hpα]","premises":[{"full_name":"Nat.modEq_zero_iff_dvd","def_path":"Mathlib/Data/Nat/ModEq.lean","def_pos":[70,8],"def_end_pos":[70,26]}]},{"state_before":"p : ℕ\nG : Type u_1\ninst✝² : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\nα : Type u_2\ninst✝¹ : MulAction G α\ninst✝ this : Finite α\nhpα : Nat.card ↑(fixedPoints G α) = 0\n⊢ Nat.card α ≡ Nat.card ↑(fixedPoints G α) [MOD p]","state_after":"no goals","tactic":"exact hG.card_modEq_card_fixedPoints α","premises":[{"full_name":"IsPGroup.card_modEq_card_fixedPoints","def_path":"Mathlib/GroupTheory/PGroup.lean","def_pos":[153,8],"def_end_pos":[153,35]}]}]}
+{"url":"Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean","commit":"","full_name":"MeasureTheory.OuterMeasure.isCaratheodory_compl","start":[60,0],"end":[61,42],"file_path":"Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean","tactics":[{"state_before":"α : Type u\nm : OuterMeasure α\ns s₁ s₂ : Set α\n⊢ m.IsCaratheodory s₁ → m.IsCaratheodory s₁ᶜ","state_after":"no goals","tactic":"simp [IsCaratheodory, diff_eq, add_comm]","premises":[{"full_name":"MeasureTheory.OuterMeasure.IsCaratheodory","def_path":"Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean","def_pos":[50,4],"def_end_pos":[50,18]},{"full_name":"Set.diff_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[86,8],"def_end_pos":[86,15]},{"full_name":"add_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[293,2],"def_end_pos":[293,13]}]}]}
+{"url":"Mathlib/LinearAlgebra/LinearIndependent.lean","commit":"","full_name":"LinearIndependent.map_of_surjective_injective","start":[287,0],"end":[300,23],"file_path":"Mathlib/LinearAlgebra/LinearIndependent.lean","tactics":[{"state_before":"ι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM'✝ : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : ι → M\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst���⁷ : AddCommMonoid M'✝\ninst✝⁶ : AddCommMonoid M''\ninst✝⁵ : Module R M\ninst✝⁴ : Module R M'✝\ninst✝³ : Module R M''\na b : R\nx y : M\nR' : Type u_8\nM' : Type u_9\ninst✝² : Semiring R'\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R' M'\nhv : LinearIndependent R v\ni : ZeroHom R R'\nj : M →+ M'\nhi : Surjective ⇑i\nhj : ∀ (m : M), j m = 0 → m = 0\nhc : ∀ (r : R) (m : M), j (r • m) = i r • j m\n⊢ LinearIndependent R' (⇑j ∘ v)","state_after":"case intro\nι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM'✝ : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : ι → M\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : AddCommMonoid M'✝\ninst✝⁶ : AddCommMonoid M''\ninst✝⁵ : Module R M\ninst✝⁴ : Module R M'✝\ninst✝³ : Module R M''\na b : R\nx y : M\nR' : Type u_8\nM' : Type u_9\ninst✝² : Semiring R'\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R' M'\nhv : LinearIndependent R v\ni : ZeroHom R R'\nj : M →+ M'\nhi : Surjective ⇑i\nhj : ∀ (m : M), j m = 0 → m = 0\nhc : ∀ (r : R) (m : M), j (r • m) = i r • j m\ni' : R' → R\nhi' : Function.RightInverse i' ⇑i\n⊢ LinearIndependent R' (⇑j ∘ v)","tactic":"obtain ⟨i', hi'⟩ := hi.hasRightInverse","premises":[{"full_name":"Function.Surjective.hasRightInverse","def_path":"Mathlib/Logic/Function/Basic.lean","def_pos":[426,8],"def_end_pos":[426,34]}]},{"state_before":"case intro\nι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM'✝ : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : ι → M\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : AddCommMonoid M'✝\ninst✝⁶ : AddCommMonoid M''\ninst✝⁵ : Module R M\ninst✝⁴ : Module R M'✝\ninst✝³ : Module R M''\na b : R\nx y : M\nR' : Type u_8\nM' : Type u_9\ninst✝² : Semiring R'\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R' M'\nhv : LinearIndependent R v\ni : ZeroHom R R'\nj : M →+ M'\nhi : Surjective ⇑i\nhj : ∀ (m : M), j m = 0 → m = 0\nhc : ∀ (r : R) (m : M), j (r • m) = i r • j m\ni' : R' → R\nhi' : Function.RightInverse i' ⇑i\n⊢ LinearIndependent R' (⇑j ∘ v)","state_after":"case intro.refine_1\nι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM'✝ : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : ι → M\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : AddCommMonoid M'✝\ninst✝⁶ : AddCommMonoid M''\ninst✝⁵ : Module R M\ninst✝⁴ : Module R M'✝\ninst✝³ : Module R M''\na b : R\nx y : M\nR' : Type u_8\nM' : Type u_9\ninst✝² : Semiring R'\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R' M'\nhv : LinearIndependent R v\ni : ZeroHom R R'\nj : M →+ M'\nhi : Surjective ⇑i\nhj : ∀ (m : M), j m = 0 → m = 0\nhc : ∀ (r : R) (m : M), j (r • m) = i r • j m\ni' : R' → R\nhi' : Function.RightInverse i' ⇑i\nx✝ : R'\nh : i' x✝ = 0\n⊢ x✝ = 0\n\ncase intro.refine_2\nι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM'✝ : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : ι → M\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : AddCommMonoid M'✝\ninst✝⁶ : AddCommMonoid M''\ninst✝⁵ : Module R M\ninst✝⁴ : Module R M'✝\ninst✝³ : Module R M''\na b : R\nx y : M\nR' : Type u_8\nM' : Type u_9\ninst✝² : Semiring R'\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R' M'\nhv : LinearIndependent R v\ni : ZeroHom R R'\nj : M →+ M'\nhi : Surjective ⇑i\nhj : ∀ (m : M), j m = 0 → m = 0\nhc : ∀ (r : R) (m : M), j (r • m) = i r • j m\ni' : R' → R\nhi' : Function.RightInverse i' ⇑i\nr : R'\nm : M\n⊢ j (i' r • m) = r • j m","tactic":"refine hv.map_of_injective_injective i' j (fun _ h ↦ ?_) hj fun r m ↦ ?_","premises":[{"full_name":"LinearIndependent.map_of_injective_injective","def_path":"Mathlib/LinearAlgebra/LinearIndependent.lean","def_pos":[278,8],"def_end_pos":[278,52]}]},{"state_before":"case intro.refine_2\nι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM'✝ : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : ι → M\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : AddCommMonoid M'✝\ninst✝⁶ : AddCommMonoid M''\ninst✝⁵ : Module R M\ninst✝⁴ : Module R M'✝\ninst✝³ : Module R M''\na b : R\nx y : M\nR' : Type u_8\nM' : Type u_9\ninst✝² : Semiring R'\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R' M'\nhv : LinearIndependent R v\ni : ZeroHom R R'\nj : M →+ M'\nhi : Surjective ⇑i\nhj : ∀ (m : M), j m = 0 → m = 0\nhc : ∀ (r : R) (m : M), j (r • m) = i r • j m\ni' : R' → R\nhi' : Function.RightInverse i' ⇑i\nr : R'\nm : M\n⊢ j (i' r • m) = r • j m","state_after":"no goals","tactic":"rw [hc (i' r) m, hi']","premises":[]}]}
+{"url":"Mathlib/CategoryTheory/Adjunction/Reflective.lean","commit":"","full_name":"CategoryTheory.equivEssImageOfReflective_unitIso","start":[181,0],"end":[209,13],"file_path":"Mathlib/CategoryTheory/Adjunction/Reflective.lean","tactics":[{"state_before":"C : Type u₁\nD : Type u₂\nE : Type u₃\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} E\ni : D ⥤ C\ninst✝ : Reflective i\n⊢ ∀ {X Y : D} (f : X ⟶ Y),\n    (𝟭 D).map f ≫ ((fun X => (asIso ((reflectorAdjunction i).counit.app X)).symm) Y).hom =\n      ((fun X => (asIso ((reflectorAdjunction i).counit.app X)).symm) X).hom ≫\n        (i.toEssImage ⋙ i.essImageInclusion ⋙ reflector i).map f","state_after":"C : Type u₁\nD : Type u₂\nE : Type u₃\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} E\ni : D ⥤ C\ninst✝ : Reflective i\nX Y : D\nf : X ⟶ Y\n⊢ (𝟭 D).map f ≫ ((fun X => (asIso ((reflectorAdjunction i).counit.app X)).symm) Y).hom =\n    ((fun X => (asIso ((reflectorAdjunction i).counit.app X)).symm) X).hom ≫\n      (i.toEssImage ⋙ i.essImageInclusion ⋙ reflector i).map f","tactic":"intro X Y f","premises":[]},{"state_before":"C : Type u₁\nD : Type u₂\nE : Type u₃\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} E\ni : D ⥤ C\ninst✝ : Reflective i\nX Y : D\nf : X ⟶ Y\n⊢ (𝟭 D).map f ≫ ((fun X => (asIso ((reflectorAdjunction i).counit.app X)).symm) Y).hom =\n    ((fun X => (asIso ((reflectorAdjunction i).counit.app X)).symm) X).hom ≫\n      (i.toEssImage ⋙ i.essImageInclusion ⋙ reflector i).map f","state_after":"C : Type u₁\nD : Type u₂\nE : Type u₃\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} E\ni : D ⥤ C\ninst✝ : Reflective i\nX Y : D\nf : X ⟶ Y\n⊢ f ≫ inv ((reflectorAdjunction i).counit.app Y) =\n    inv ((reflectorAdjunction i).counit.app X) ≫ (reflector i).map (i.map f)","tactic":"dsimp","premises":[]},{"state_before":"C : Type u₁\nD : Type u₂\nE : Type u₃\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} E\ni : D ⥤ C\ninst✝ : Reflective i\nX Y : D\nf : X ⟶ Y\n⊢ f ≫ inv ((reflectorAdjunction i).counit.app Y) =\n    inv ((reflectorAdjunction i).counit.app X) ≫ (reflector i).map (i.map f)","state_after":"C : Type u₁\nD : Type u₂\nE : Type u₃\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} E\ni : D ⥤ C\ninst✝ : Reflective i\nX Y : D\nf : X ⟶ Y\n⊢ (reflectorAdjunction i).counit.app X ≫ f = (reflector i).map (i.map f) ≫ (reflectorAdjunction i).counit.app Y","tactic":"rw [IsIso.comp_inv_eq, Category.assoc, IsIso.eq_inv_comp]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.IsIso.comp_inv_eq","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[381,8],"def_end_pos":[381,19]},{"full_name":"CategoryTheory.IsIso.eq_inv_comp","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[377,8],"def_end_pos":[377,19]}]},{"state_before":"C : Type u₁\nD : Type u₂\nE : Type u₃\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} E\ni : D ⥤ C\ninst✝ : Reflective i\nX Y : D\nf : X ⟶ Y\n⊢ (reflectorAdjunction i).counit.app X ≫ f = (reflector i).map (i.map f) ≫ (reflectorAdjunction i).counit.app Y","state_after":"no goals","tactic":"exact ((reflectorAdjunction i).counit.naturality f).symm","premises":[{"full_name":"CategoryTheory.Adjunction.counit","def_path":"Mathlib/CategoryTheory/Adjunction/Basic.lean","def_pos":[65,2],"def_end_pos":[65,8]},{"full_name":"CategoryTheory.NatTrans.naturality","def_path":"Mathlib/CategoryTheory/NatTrans.lean","def_pos":[50,2],"def_end_pos":[50,12]},{"full_name":"CategoryTheory.reflectorAdjunction","def_path":"Mathlib/CategoryTheory/Adjunction/Reflective.lean","def_pos":[46,4],"def_end_pos":[46,23]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]}]},{"state_before":"C : Type u₁\nD : Type u₂\nE : Type u₃\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} E\ni : D ⥤ C\ninst✝ : Reflective i\n⊢ ∀ {X Y : i.EssImageSubcategory} (f : X ⟶ Y),\n    ((i.essImageInclusion ⋙ reflector i) ⋙ i.toEssImage).map f ≫ (equivEssImageOfReflective_counitIso_app Y).hom =\n      (equivEssImageOfReflective_counitIso_app X).hom ≫ (𝟭 i.EssImageSubcategory).map f","state_after":"C : Type u₁\nD : Type u₂\nE : Type u₃\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} E\ni : D ⥤ C\ninst✝ : Reflective i\nX Y : i.EssImageSubcategory\nf : X ⟶ Y\n⊢ ((i.essImageInclusion ⋙ reflector i) ⋙ i.toEssImage).map f ≫ (equivEssImageOfReflective_counitIso_app Y).hom =\n    (equivEssImageOfReflective_counitIso_app X).hom ≫ (𝟭 i.EssImageSubcategory).map f","tactic":"intro X Y f","premises":[]},{"state_before":"C : Type u₁\nD : Type u₂\nE : Type u₃\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} E\ni : D ⥤ C\ninst✝ : Reflective i\nX Y : i.EssImageSubcategory\nf : X ⟶ Y\n⊢ ((i.essImageInclusion ⋙ reflector i) ⋙ i.toEssImage).map f ≫ (equivEssImageOfReflective_counitIso_app Y).hom =\n    (equivEssImageOfReflective_counitIso_app X).hom ≫ (𝟭 i.EssImageSubcategory).map f","state_after":"case a\nC : Type u₁\nD : Type u₂\nE : Type u₃\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} E\ni : D ⥤ C\ninst✝ : Reflective i\nX Y : i.EssImageSubcategory\nf : X ⟶ Y\n⊢ i.essImageInclusion.map\n      (((i.essImageInclusion ⋙ reflector i) ⋙ i.toEssImage).map f ≫ (equivEssImageOfReflective_counitIso_app Y).hom) =\n    i.essImageInclusion.map ((equivEssImageOfReflective_counitIso_app X).hom ≫ (𝟭 i.EssImageSubcategory).map f)","tactic":"apply (Functor.essImageInclusion i).map_injective","premises":[{"full_name":"CategoryTheory.Functor.essImageInclusion","def_path":"Mathlib/CategoryTheory/EssentialImage.lean","def_pos":[81,4],"def_end_pos":[81,21]},{"full_name":"CategoryTheory.Functor.map_injective","def_path":"Mathlib/CategoryTheory/Functor/FullyFaithful.lean","def_pos":[57,8],"def_end_pos":[57,21]}]},{"state_before":"case a\nC : Type u₁\nD : Type u₂\nE : Type u₃\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} E\ni : D ⥤ C\ninst✝ : Reflective i\nX Y : i.EssImageSubcategory\nf : X ⟶ Y\n⊢ i.essImageInclusion.map\n      (((i.essImageInclusion ⋙ reflector i) ⋙ i.toEssImage).map f ≫ (equivEssImageOfReflective_counitIso_app Y).hom) =\n    i.essImageInclusion.map ((equivEssImageOfReflective_counitIso_app X).hom ≫ (𝟭 i.EssImageSubcategory).map f)","state_after":"case a\nC : Type u₁\nD : Type u₂\nE : Type u₃\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} E\ni : D ⥤ C\ninst✝ : Reflective i\nX Y : i.EssImageSubcategory\nf : X ⟶ Y\nh :\n  (reflectorAdjunction i).unit.app X.obj ≫ (reflector i ⋙ i).map f =\n    (𝟭 C).map f ≫ (reflectorAdjunction i).unit.app Y.obj\n⊢ i.essImageInclusion.map\n      (((i.essImageInclusion ⋙ reflector i) ⋙ i.toEssImage).map f ≫ (equivEssImageOfReflective_counitIso_app Y).hom) =\n    i.essImageInclusion.map ((equivEssImageOfReflective_counitIso_app X).hom ≫ (𝟭 i.EssImageSubcategory).map f)","tactic":"have h := ((reflectorAdjunction i).unit.naturality f).symm","premises":[{"full_name":"CategoryTheory.Adjunction.unit","def_path":"Mathlib/CategoryTheory/Adjunction/Basic.lean","def_pos":[63,2],"def_end_pos":[63,6]},{"full_name":"CategoryTheory.NatTrans.naturality","def_path":"Mathlib/CategoryTheory/NatTrans.lean","def_pos":[50,2],"def_end_pos":[50,12]},{"full_name":"CategoryTheory.reflectorAdjunction","def_path":"Mathlib/CategoryTheory/Adjunction/Reflective.lean","def_pos":[46,4],"def_end_pos":[46,23]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]}]},{"state_before":"case a\nC : Type u₁\nD : Type u₂\nE : Type u₃\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} E\ni : D ⥤ C\ninst✝ : Reflective i\nX Y : i.EssImageSubcategory\nf : X ⟶ Y\nh :\n  (reflectorAdjunction i).unit.app X.obj ≫ (reflector i ⋙ i).map f =\n    (𝟭 C).map f ≫ (reflectorAdjunction i).unit.app Y.obj\n⊢ i.essImageInclusion.map\n      (((i.essImageInclusion ⋙ reflector i) ⋙ i.toEssImage).map f ≫ (equivEssImageOfReflective_counitIso_app Y).hom) =\n    i.essImageInclusion.map ((equivEssImageOfReflective_counitIso_app X).hom ≫ (𝟭 i.EssImageSubcategory).map f)","state_after":"case a\nC : Type u₁\nD : Type u₂\nE : Type u₃\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} E\ni : D ⥤ C\ninst✝ : Reflective i\nX Y : i.EssImageSubcategory\nf : X ⟶ Y\nh : (reflectorAdjunction i).unit.app X.obj ≫ (reflector i ⋙ i).map f = f ≫ (reflectorAdjunction i).unit.app Y.obj\n⊢ i.essImageInclusion.map\n      (((i.essImageInclusion ⋙ reflector i) ⋙ i.toEssImage).map f ≫ (equivEssImageOfReflective_counitIso_app Y).hom) =\n    i.essImageInclusion.map ((equivEssImageOfReflective_counitIso_app X).hom ≫ (𝟭 i.EssImageSubcategory).map f)","tactic":"rw [Functor.id_map] at h","premises":[{"full_name":"CategoryTheory.Functor.id_map","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[88,8],"def_end_pos":[88,14]}]},{"state_before":"case a\nC : Type u₁\nD : Type u₂\nE : Type u₃\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} E\ni : D ⥤ C\ninst✝ : Reflective i\nX Y : i.EssImageSubcategory\nf : X ⟶ Y\nh : (reflectorAdjunction i).unit.app X.obj ≫ (reflector i ⋙ i).map f = f ≫ (reflectorAdjunction i).unit.app Y.obj\n��� i.essImageInclusion.map\n      (((i.essImageInclusion ⋙ reflector i) ⋙ i.toEssImage).map f ≫ (equivEssImageOfReflective_counitIso_app Y).hom) =\n    i.essImageInclusion.map ((equivEssImageOfReflective_counitIso_app X).hom ≫ (𝟭 i.EssImageSubcategory).map f)","state_after":"no goals","tactic":"erw [Functor.map_comp, Functor.map_comp,\n          equivEssImageOfReflective_map_counitIso_app_hom,\n          equivEssImageOfReflective_map_counitIso_app_hom,\n          IsIso.comp_inv_eq, assoc, ← h, IsIso.inv_hom_id_assoc, Functor.comp_map]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Functor.comp_map","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[109,8],"def_end_pos":[109,16]},{"full_name":"CategoryTheory.Functor.map_comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[41,2],"def_end_pos":[41,10]},{"full_name":"CategoryTheory.IsIso.comp_inv_eq","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[381,8],"def_end_pos":[381,19]},{"full_name":"CategoryTheory.IsIso.inv_hom_id_assoc","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[263,8],"def_end_pos":[263,24]},{"full_name":"CategoryTheory.equivEssImageOfReflective_map_counitIso_app_hom","def_path":"Mathlib/CategoryTheory/Adjunction/Reflective.lean","def_pos":[167,6],"def_end_pos":[167,53]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]}]},{"state_before":"C : Type u₁\nD : Type u₂\nE : Type u₃\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} E\ni : D ⥤ C\ninst✝ : Reflective i\nX : D\n⊢ i.toEssImage.map ((NatIso.ofComponents (fun X => (asIso ((reflectorAdjunction i).counit.app X)).symm) ⋯).hom.app X) ≫\n      (NatIso.ofComponents equivEssImageOfReflective_counitIso_app ⋯).hom.app (i.toEssImage.obj X) =\n    𝟙 (i.toEssImage.obj X)","state_after":"case a\nC : Type u₁\nD : Type u₂\nE : Type u₃\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} E\ni : D ⥤ C\ninst✝ : Reflective i\nX : D\n⊢ i.essImageInclusion.map\n      (i.toEssImage.map\n          ((NatIso.ofComponents (fun X => (asIso ((reflectorAdjunction i).counit.app X)).symm) ⋯).hom.app X) ≫\n        (NatIso.ofComponents equivEssImageOfReflective_counitIso_app ⋯).hom.app (i.toEssImage.obj X)) =\n    i.essImageInclusion.map (𝟙 (i.toEssImage.obj X))","tactic":"apply (Functor.essImageInclusion i).map_injective","premises":[{"full_name":"CategoryTheory.Functor.essImageInclusion","def_path":"Mathlib/CategoryTheory/EssentialImage.lean","def_pos":[81,4],"def_end_pos":[81,21]},{"full_name":"CategoryTheory.Functor.map_injective","def_path":"Mathlib/CategoryTheory/Functor/FullyFaithful.lean","def_pos":[57,8],"def_end_pos":[57,21]}]},{"state_before":"case a\nC : Type u₁\nD : Type u₂\nE : Type u₃\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} E\ni : D ⥤ C\ninst✝ : Reflective i\nX : D\n⊢ i.essImageInclusion.map\n      (i.toEssImage.map\n          ((NatIso.ofComponents (fun X => (asIso ((reflectorAdjunction i).counit.app X)).symm) ⋯).hom.app X) ≫\n        (NatIso.ofComponents equivEssImageOfReflective_counitIso_app ⋯).hom.app (i.toEssImage.obj X)) =\n    i.essImageInclusion.map (𝟙 (i.toEssImage.obj X))","state_after":"case a\nC : Type u₁\nD : Type u₂\nE : Type u₃\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} E\ni : D ⥤ C\ninst✝ : Reflective i\nX : D\n⊢ i.essImageInclusion.map\n        (i.toEssImage.map\n          ((NatIso.ofComponents (fun X => (asIso ((reflectorAdjunction i).counit.app X)).symm) ⋯).hom.app X)) ≫\n      inv ((reflectorAdjunction i).unit.app ((𝟭 i.EssImageSubcategory).obj (i.toEssImage.obj X)).obj) =\n    i.essImageInclusion.map (𝟙 (i.toEssImage.obj X))","tactic":"erw [Functor.map_comp, equivEssImageOfReflective_map_counitIso_app_hom]","premises":[{"full_name":"CategoryTheory.Functor.map_comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[41,2],"def_end_pos":[41,10]},{"full_name":"CategoryTheory.equivEssImageOfReflective_map_counitIso_app_hom","def_path":"Mathlib/CategoryTheory/Adjunction/Reflective.lean","def_pos":[167,6],"def_end_pos":[167,53]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]}]},{"state_before":"case a\nC : Type u₁\nD : Type u₂\nE : Type u₃\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} E\ni : D ⥤ C\ninst✝ : Reflective i\nX : D\n⊢ i.essImageInclusion.map\n        (i.toEssImage.map\n          ((NatIso.ofComponents (fun X => (asIso ((reflectorAdjunction i).counit.app X)).symm) ⋯).hom.app X)) ≫\n      inv ((reflectorAdjunction i).unit.app ((𝟭 i.EssImageSubcategory).obj (i.toEssImage.obj X)).obj) =\n    i.essImageInclusion.map (𝟙 (i.toEssImage.obj X))","state_after":"no goals","tactic":"aesop_cat","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]}]}]}
+{"url":"Mathlib/CategoryTheory/Limits/Shapes/WidePullbacks.lean","commit":"","full_name":"CategoryTheory.Limits.WidePullback.hom_ext","start":[340,0],"end":[347,12],"file_path":"Mathlib/CategoryTheory/Limits/Shapes/WidePullbacks.lean","tactics":[{"state_before":"J : Type w\nC✝ : Type u\ninst✝² : Category.{v, u} C✝\nC : Type u\ninst✝¹ : Category.{v, u} C\nB : C\nobjs : J → C\narrows : (j : J) → objs j ⟶ B\ninst✝ : HasWidePullback B objs arrows\nX : C\nf : X ⟶ B\nfs : (j : J) → X ⟶ objs j\nw : ∀ (j : J), fs j ≫ arrows j = f\ng1 g2 : X ⟶ widePullback B objs arrows\n⊢ (∀ (j : J), g1 ≫ π arrows j = g2 ≫ π arrows j) → g1 ≫ base arrows = g2 ≫ base arrows → g1 = g2","state_after":"J : Type w\nC✝ : Type u\ninst✝² : Category.{v, u} C✝\nC : Type u\ninst✝¹ : Category.{v, u} C\nB : C\nobjs : J → C\narrows : (j : J) → objs j ⟶ B\ninst✝ : HasWidePullback B objs arrows\nX : C\nf : X ⟶ B\nfs : (j : J) → X ⟶ objs j\nw : ∀ (j : J), fs j ≫ arrows j = f\ng1 g2 : X ⟶ widePullback B objs arrows\nh1 : ∀ (j : J), g1 ≫ π arrows j = g2 ≫ π arrows j\nh2 : g1 ≫ base arrows = g2 ≫ base arrows\n⊢ g1 = g2","tactic":"intro h1 h2","premises":[]},{"state_before":"J : Type w\nC✝ : Type u\ninst✝² : Category.{v, u} C✝\nC : Type u\ninst✝¹ : Category.{v, u} C\nB : C\nobjs : J → C\narrows : (j : J) → objs j ⟶ B\ninst✝ : HasWidePullback B objs arrows\nX : C\nf : X ⟶ B\nfs : (j : J) → X ⟶ objs j\nw : ∀ (j : J), fs j ≫ arrows j = f\ng1 g2 : X ⟶ widePullback B objs arrows\nh1 : ∀ (j : J), g1 ≫ π arrows j = g2 ≫ π arrows j\nh2 : g1 ≫ base arrows = g2 ≫ base arrows\n⊢ g1 = g2","state_after":"case w\nJ : Type w\nC✝ : Type u\ninst✝² : Category.{v, u} C✝\nC : Type u\ninst✝¹ : Category.{v, u} C\nB : C\nobjs : J → C\narrows : (j : J) → objs j ⟶ B\ninst✝ : HasWidePullback B objs arrows\nX : C\nf : X ⟶ B\nfs : (j : J) → X ⟶ objs j\nw : ∀ (j : J), fs j ≫ arrows j = f\ng1 g2 : X ⟶ widePullback B objs arrows\nh1 : ∀ (j : J), g1 ≫ π arrows j = g2 ≫ π arrows j\nh2 : g1 ≫ base arrows = g2 ≫ base arrows\n⊢ ∀ (j : WidePullbackShape J),\n    g1 ≫ limit.π (WidePullbackShape.wideCospan B objs arrows) j =\n      g2 ≫ limit.π (WidePullbackShape.wideCospan B objs arrows) j","tactic":"apply limit.hom_ext","premises":[{"full_name":"CategoryTheory.Limits.limit.hom_ext","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[231,8],"def_end_pos":[231,21]}]},{"state_before":"case w\nJ : Type w\nC✝ : Type u\ninst✝² : Category.{v, u} C✝\nC : Type u\ninst✝¹ : Category.{v, u} C\nB : C\nobjs : J → C\narrows : (j : J) → objs j ⟶ B\ninst✝ : HasWidePullback B objs arrows\nX : C\nf : X ⟶ B\nfs : (j : J) → X ⟶ objs j\nw : ∀ (j : J), fs j ≫ arrows j = f\ng1 g2 : X ⟶ widePullback B objs arrows\nh1 : ∀ (j : J), g1 ≫ π arrows j = g2 ≫ π arrows j\nh2 : g1 ≫ base arrows = g2 ≫ base arrows\n⊢ ∀ (j : WidePullbackShape J),\n    g1 ≫ limit.π (WidePullbackShape.wideCospan B objs arrows) j =\n      g2 ≫ limit.π (WidePullbackShape.wideCospan B objs arrows) j","state_after":"case w.none\nJ : Type w\nC✝ : Type u\ninst✝² : Category.{v, u} C✝\nC : Type u\ninst✝¹ : Category.{v, u} C\nB : C\nobjs : J → C\narrows : (j : J) → objs j ⟶ B\ninst✝ : HasWidePullback B objs arrows\nX : C\nf : X ⟶ B\nfs : (j : J) → X ⟶ objs j\nw : ∀ (j : J), fs j ≫ arrows j = f\ng1 g2 : X ⟶ widePullback B objs arrows\nh1 : ∀ (j : J), g1 ≫ π arrows j = g2 ≫ π arrows j\nh2 : g1 ≫ base arrows = g2 ≫ base arrows\n⊢ g1 ≫ limit.π (WidePullbackShape.wideCospan B objs arrows) none =\n    g2 ≫ limit.π (WidePullbackShape.wideCospan B objs arrows) none\n\ncase w.some\nJ : Type w\nC✝ : Type u\ninst✝² : Category.{v, u} C✝\nC : Type u\ninst✝¹ : Category.{v, u} C\nB : C\nobjs : J → C\narrows : (j : J) → objs j ⟶ B\ninst✝ : HasWidePullback B objs arrows\nX : C\nf : X ⟶ B\nfs : (j : J) → X ⟶ objs j\nw : ∀ (j : J), fs j ≫ arrows j = f\ng1 g2 : X ⟶ widePullback B objs arrows\nh1 : ∀ (j : J), g1 ≫ π arrows j = g2 ≫ π arrows j\nh2 : g1 ≫ base arrows = g2 ≫ base arrows\nval✝ : J\n⊢ g1 ≫ limit.π (WidePullbackShape.wideCospan B objs arrows) (some val✝) =\n    g2 ≫ limit.π (WidePullbackShape.wideCospan B objs arrows) (some val✝)","tactic":"rintro (_ | _)","premises":[]}]}
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+{"url":"Mathlib/RingTheory/Smooth/Basic.lean","commit":"","full_name":"Algebra.FormallySmooth.exists_lift","start":[65,0],"end":[85,17],"file_path":"Mathlib/RingTheory/Smooth/Basic.lean","tactics":[{"state_before":"R : Type u\ninst✝⁶ : CommSemiring R\nA : Type u\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\nB✝ : Type u\ninst✝³ : CommRing B✝\ninst✝² : Algebra R B✝\nI✝ : Ideal B✝\nB : Type u\ninst✝¹ : CommRing B\n_RB : Algebra R B\ninst✝ : FormallySmooth R A\nI : Ideal B\nhI : IsNilpotent I\ng : A →ₐ[R] B ⧸ I\n⊢ ∃ f, (Ideal.Quotient.mkₐ R I).comp f = g","state_after":"R : Type u\ninst✝⁶ : CommSemiring R\nA : Type u\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\nB✝ : Type u\ninst✝³ : CommRing B✝\ninst✝² : Algebra R B✝\nI✝ : Ideal B✝\nB : Type u\ninst✝¹ : CommRing B\n_RB : Algebra R B\ninst✝ : FormallySmooth R A\nI : Ideal B\nhI : IsNilpotent I\n⊢ ∀ (g : A →ₐ[R] B ⧸ I), ∃ f, (Ideal.Quotient.mkₐ R I).comp f = g","tactic":"revert g","premises":[]},{"state_before":"R : Type u\ninst✝⁶ : CommSemiring R\nA : Type u\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\nB✝ : Type u\ninst✝³ : CommRing B✝\ninst✝² : Algebra R B✝\nI✝ : Ideal B✝\nB : Type u\ninst✝¹ : CommRing B\n_RB : Algebra R B\ninst✝ : FormallySmooth R A\nI : Ideal B\nhI : IsNilpotent I\n⊢ ∀ (g : A →ₐ[R] B ⧸ I), ∃ f, (Ideal.Quotient.mkₐ R I).comp f = g","state_after":"R : Type u\ninst✝⁶ : CommSemiring R\nA : Type u\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\nB✝ : Type u\ninst✝³ : CommRing B✝\ninst✝² : Algebra R B✝\nI✝ : Ideal B✝\nB : Type u\ninst✝¹ : CommRing B\n_RB : Algebra R B\ninst✝ : FormallySmooth R A\nI : Ideal B\nhI : IsNilpotent I\n⊢ Function.Surjective (Ideal.Quotient.mkₐ R I).comp","tactic":"change Function.Surjective (Ideal.Quotient.mkₐ R I).comp","premises":[{"full_name":"AlgHom.comp","def_path":"Mathlib/Algebra/Algebra/Hom.lean","def_pos":[274,4],"def_end_pos":[274,8]},{"full_name":"Function.Surjective","def_path":"Mathlib/Logic/Function/Defs.lean","def_pos":[109,4],"def_end_pos":[109,14]},{"full_name":"Ideal.Quotient.mkₐ","def_path":"Mathlib/RingTheory/Ideal/QuotientOperations.lean","def_pos":[342,4],"def_end_pos":[342,16]}]},{"state_before":"R : Type u\ninst✝⁶ : CommSemiring R\nA : Type u\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\nB✝ : Type u\ninst✝³ : CommRing B✝\ninst✝² : Algebra R B✝\nI✝ : Ideal B✝\nB : Type u\ninst✝¹ : CommRing B\n_RB : Algebra R B\ninst✝ : FormallySmooth R A\nI : Ideal B\nhI : IsNilpotent I\n⊢ Function.Surjective (Ideal.Quotient.mkₐ R I).comp","state_after":"R : Type u\ninst✝⁶ : CommSemiring R\nA : Type u\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\nB✝ : Type u\ninst✝³ : CommRing B✝\ninst✝² : Algebra R B✝\nI✝ : Ideal B✝\nB : Type u\ninst✝¹ : CommRing B\ninst✝ : FormallySmooth R A\nI : Ideal B\nhI : IsNilpotent I\n⊢ ∀ [_RB : Algebra R B], Function.Surjective (Ideal.Quotient.mkₐ R I).comp","tactic":"revert _RB","premises":[]},{"state_before":"R : Type u\ninst✝⁶ : CommSemiring R\nA : Type u\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\nB✝ : Type u\ninst✝³ : CommRing B✝\ninst✝² : Algebra R B✝\nI✝ : Ideal B✝\nB : Type u\ninst✝¹ : CommRing B\ninst✝ : FormallySmooth R A\nI : Ideal B\nhI : IsNilpotent I\n⊢ ∀ [_RB : Algebra R B], Function.Surjective (Ideal.Quotient.mkₐ R I).comp","state_after":"case h₁\nR : Type u\ninst✝⁶ : CommSemiring R\nA : Type u\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\nB✝ : Type u\ninst✝³ : CommRing B✝\ninst✝² : Algebra R B✝\nI✝ : Ideal B✝\nB : Type u\ninst✝¹ : CommRing B\ninst✝ : FormallySmooth R A\nI : Ideal B\nhI : IsNilpotent I\n⊢ ∀ ⦃S : Type u⦄ [inst : CommRing S] (I : Ideal S),\n    I ^ 2 = ⊥ → ∀ [_RB : Algebra R S], Function.Surjective (Ideal.Quotient.mkₐ R I).comp\n\ncase h₂\nR : Type u\ninst✝⁶ : CommSemiring R\nA : Type u\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\nB✝ : Type u\ninst✝³ : CommRing B✝\ninst✝² : Algebra R B✝\nI✝ : Ideal B✝\nB : Type u\ninst✝¹ : CommRing B\ninst✝ : FormallySmooth R A\nI : Ideal B\nhI : IsNilpotent I\n⊢ ∀ ⦃S : Type u⦄ [inst : CommRing S] (I J : Ideal S),\n    I ≤ J →\n      (∀ [_RB : Algebra R S], Function.Surjective (Ideal.Quotient.mkₐ R I).comp) →\n        (∀ [_RB : Algebra R (S ⧸ I)],\n            Function.Surjective (Ideal.Quotient.mkₐ R (Ideal.map (Ideal.Quotient.mk I) J)).comp) →\n          ∀ [_RB : Algebra R S], Function.Surjective (Ideal.Quotient.mkₐ R J).comp","tactic":"apply Ideal.IsNilpotent.induction_on (R := B) I hI","premises":[{"full_name":"Ideal.IsNilpotent.induction_on","def_path":"Mathlib/RingTheory/QuotientNilpotent.lean","def_pos":[23,8],"def_end_pos":[23,38]}]}]}
+{"url":"Mathlib/MeasureTheory/MeasurableSpace/Basic.lean","commit":"","full_name":"measurable_piEquivPiSubtypeProd_symm","start":[891,0],"end":[903,45],"file_path":"Mathlib/MeasureTheory/MeasurableSpace/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns t u : Set α\nπ : δ → Type u_6\ninst✝³ : MeasurableSpace α\ninst✝² : (a : δ) → MeasurableSpace (π a)\ninst✝¹ : MeasurableSpace γ\np : δ → Prop\ninst✝ : DecidablePred p\n⊢ Measurable ⇑(piEquivPiSubtypeProd p π).symm","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns t u : Set α\nπ : δ → Type u_6\ninst✝³ : MeasurableSpace α\ninst✝² : (a : δ) → MeasurableSpace (π a)\ninst✝¹ : MeasurableSpace γ\np : δ → Prop\ninst✝ : DecidablePred p\nj : δ\n⊢ Measurable fun x => (piEquivPiSubtypeProd p π).symm x j","tactic":"refine measurable_pi_iff.2 fun j => ?_","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"measurable_pi_iff","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Basic.lean","def_pos":[791,8],"def_end_pos":[791,25]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns t u : Set α\nπ : δ → Type u_6\ninst✝³ : MeasurableSpace α\ninst✝² : (a : δ) → MeasurableSpace (π a)\ninst✝¹ : MeasurableSpace γ\np : δ → Prop\ninst✝ : DecidablePred p\nj : δ\n⊢ Measurable fun x => (piEquivPiSubtypeProd p π).symm x j","state_after":"case pos\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns t u : Set α\nπ : δ → Type u_6\ninst✝³ : MeasurableSpace α\ninst✝² : (a : δ) → MeasurableSpace (π a)\ninst✝¹ : MeasurableSpace γ\np : δ → Prop\ninst✝ : DecidablePred p\nj : δ\nhj : p j\n⊢ Measurable fun x => (piEquivPiSubtypeProd p π).symm x j\n\ncase neg\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns t u : Set α\nπ : δ → Type u_6\ninst✝³ : MeasurableSpace α\ninst✝² : (a : δ) → MeasurableSpace (π a)\ninst✝¹ : MeasurableSpace γ\np : δ → Prop\ninst✝ : DecidablePred p\nj : δ\nhj : ¬p j\n⊢ Measurable fun x => (piEquivPiSubtypeProd p π).symm x j","tactic":"by_cases hj : p j","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
+{"url":"Mathlib/Probability/Cdf.lean","commit":"","full_name":"ProbabilityTheory.ofReal_cdf","start":[72,0],"end":[76,28],"file_path":"Mathlib/Probability/Cdf.lean","tactics":[{"state_before":"μ : Measure ℝ\ninst✝ : IsProbabilityMeasure μ\nx : ℝ\n⊢ ENNReal.ofReal (↑(cdf μ) x) = μ (Iic x)","state_after":"μ : Measure ℝ\ninst✝ : IsProbabilityMeasure μ\nx : ℝ\nthis : IsFiniteMeasure ((Measure.dirac ()).prod μ)\n⊢ ENNReal.ofReal (↑(cdf μ) x) = μ (Iic x)","tactic":"have := IsProbabilityMeasure.toIsFiniteMeasure (Measure.prod (Measure.dirac ()) μ)","premises":[{"full_name":"MeasureTheory.IsProbabilityMeasure.toIsFiniteMeasure","def_path":"Mathlib/MeasureTheory/Measure/Typeclasses.lean","def_pos":[217,27],"def_end_pos":[217,65]},{"full_name":"MeasureTheory.Measure.dirac","def_path":"Mathlib/MeasureTheory/Measure/Dirac.lean","def_pos":[29,4],"def_end_pos":[29,9]},{"full_name":"MeasureTheory.Measure.prod","def_path":"Mathlib/MeasureTheory/Constructions/Prod/Basic.lean","def_pos":[308,26],"def_end_pos":[308,30]},{"full_name":"Unit.unit","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[148,24],"def_end_pos":[148,33]}]},{"state_before":"μ : Measure ℝ\ninst✝ : IsProbabilityMeasure μ\nx : ℝ\nthis : IsFiniteMeasure ((Measure.dirac ()).prod μ)\n⊢ ENNReal.ofReal (↑(cdf μ) x) = μ (Iic x)","state_after":"μ : Measure ℝ\ninst✝ : IsProbabilityMeasure μ\nx : ℝ\nthis : IsFiniteMeasure ((Measure.dirac ()).prod μ)\nh :\n  ∫⁻ (a : Unit), ENNReal.ofReal (↑(condCDF ((Measure.dirac ()).prod μ) a) x) ∂((Measure.dirac ()).prod μ).fst =\n    ((Measure.dirac ()).prod μ) (univ ×ˢ Iic x)\n⊢ ENNReal.ofReal (↑(cdf μ) x) = μ (Iic x)","tactic":"have h := lintegral_condCDF ((Measure.dirac Unit.unit).prod μ) x","premises":[{"full_name":"MeasureTheory.Measure.dirac","def_path":"Mathlib/MeasureTheory/Measure/Dirac.lean","def_pos":[29,4],"def_end_pos":[29,9]},{"full_name":"MeasureTheory.Measure.prod","def_path":"Mathlib/MeasureTheory/Constructions/Prod/Basic.lean","def_pos":[308,26],"def_end_pos":[308,30]},{"full_name":"ProbabilityTheory.lintegral_condCDF","def_path":"Mathlib/Probability/Kernel/Disintegration/CondCdf.lean","def_pos":[310,8],"def_end_pos":[310,25]},{"full_name":"Unit.unit","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[148,24],"def_end_pos":[148,33]}]},{"state_before":"μ : Measure ℝ\ninst✝ : IsProbabilityMeasure μ\nx : ℝ\nthis : IsFiniteMeasure ((Measure.dirac ()).prod μ)\nh :\n  ∫⁻ (a : Unit), ENNReal.ofReal (↑(condCDF ((Measure.dirac ()).prod μ) a) x) ∂((Measure.dirac ()).prod μ).fst =\n    ((Measure.dirac ()).prod μ) (univ ×ˢ Iic x)\n⊢ ENNReal.ofReal (↑(cdf μ) x) = μ (Iic x)","state_after":"no goals","tactic":"simpa only [MeasureTheory.Measure.fst_prod, Measure.prod_prod, measure_univ, one_mul,\n    lintegral_dirac] using h","premises":[{"full_name":"MeasureTheory.IsProbabilityMeasure.measure_univ","def_path":"Mathlib/MeasureTheory/Measure/Typeclasses.lean","def_pos":[208,2],"def_end_pos":[208,14]},{"full_name":"MeasureTheory.Measure.fst_prod","def_path":"Mathlib/MeasureTheory/Constructions/Prod/Basic.lean","def_pos":[980,6],"def_end_pos":[980,14]},{"full_name":"MeasureTheory.Measure.prod_prod","def_path":"Mathlib/MeasureTheory/Constructions/Prod/Basic.lean","def_pos":[328,8],"def_end_pos":[328,17]},{"full_name":"MeasureTheory.lintegral_dirac","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[1526,8],"def_end_pos":[1526,23]},{"full_name":"one_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[409,8],"def_end_pos":[409,15]}]}]}
+{"url":"Mathlib/Data/Set/Lattice.lean","commit":"","full_name":"Set.image2_iUnion_left","start":[1565,0],"end":[1567,59],"file_path":"Mathlib/Data/Set/Lattice.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nι' : Sort u_5\nι₂ : Sort u_6\nκ : ι → Sort u_7\nκ₁ : ι → Sort u_8\nκ₂ : ι → Sort u_9\nκ' : ι' → Sort u_10\nf : α → β → γ\ns✝ : Set α\nt✝ : Set β\ns : ι → Set α\nt : Set β\n⊢ image2 f (⋃ i, s i) t = ⋃ i, image2 f (s i) t","state_after":"no goals","tactic":"simp only [← image_prod, iUnion_prod_const, image_iUnion]","premises":[{"full_name":"Set.iUnion_prod_const","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[1491,8],"def_end_pos":[1491,25]},{"full_name":"Set.image_iUnion","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[1384,8],"def_end_pos":[1384,20]},{"full_name":"Set.image_prod","def_path":"Mathlib/Data/Set/NAry.lean","def_pos":[67,6],"def_end_pos":[67,16]}]}]}
+{"url":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","commit":"","full_name":"lt_add_iff_pos_right","start":[463,0],"end":[467,53],"file_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝³ : MulOneClass α\ninst✝² : LT α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst✝ : ContravariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b : α\n⊢ a < a * b ↔ a * 1 < a * b","state_after":"no goals","tactic":"rw [mul_one]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"mul_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[413,8],"def_end_pos":[413,15]}]}]}
+{"url":"Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean","commit":"","full_name":"List.stronglyMeasurable_prod","start":[546,0],"end":[550,71],"file_path":"Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ninst✝³ : Countable ι\nf g : α → β\nM : Type u_5\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nm : MeasurableSpace α\nl : List (α → M)\nhl : ∀ f ∈ l, StronglyMeasurable f\n⊢ StronglyMeasurable fun x => (List.map (fun f => f x) l).prod","state_after":"no goals","tactic":"simpa only [← Pi.list_prod_apply] using l.stronglyMeasurable_prod' hl","premises":[{"full_name":"List.stronglyMeasurable_prod'","def_path":"Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean","def_pos":[539,8],"def_end_pos":[539,44]},{"full_name":"Pi.list_prod_apply","def_path":"Mathlib/Algebra/BigOperators/Pi.lean","def_pos":[21,8],"def_end_pos":[21,23]}]}]}
+{"url":"Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean","commit":"","full_name":"NumberField.mixedEmbedding.convexBodySum_isBounded","start":[376,0],"end":[380,50],"file_path":"Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean","tactics":[{"state_before":"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : ℝ\n⊢ Bornology.IsBounded (convexBodySum K B)","state_after":"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : ℝ\nx : ({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ)\nhx : x ∈ convexBodySum K B\ny : ({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ)\nhy : y ∈ convexBodySum K B\n⊢ dist x y ≤ B + B","tactic":"refine Metric.isBounded_iff.mpr ⟨B + B, fun x hx y hy => ?_⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Metric.isBounded_iff","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[588,8],"def_end_pos":[588,21]}]},{"state_before":"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : ℝ\nx : ({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ)\nhx : x ∈ convexBodySum K B\ny : ({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ)\nhy : y ∈ convexBodySum K B\n⊢ dist x y ≤ B + B","state_after":"case refine_1\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : ℝ\nx : ({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ)\nhx : x ∈ convexBodySum K B\ny : ({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ)\nhy : y ∈ convexBodySum K B\n⊢ ‖x‖ ≤ B\n\ncase refine_2\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : ℝ\nx : ({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ)\nhx : x ∈ convexBodySum K B\ny : ({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ)\nhy : y ∈ convexBodySum K B\n⊢ ‖y‖ ≤ B","tactic":"refine le_trans (norm_sub_le x y) (add_le_add ?_ ?_)","premises":[{"full_name":"add_le_add","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[182,31],"def_end_pos":[182,41]},{"full_name":"le_trans","def_path":"Mathlib/Order/Defs.lean","def_pos":[48,8],"def_end_pos":[48,16]},{"full_name":"norm_sub_le","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[453,2],"def_end_pos":[453,13]}]}]}
+{"url":"Mathlib/Algebra/Star/Subalgebra.lean","commit":"","full_name":"Subalgebra.star_mem_star_iff","start":[319,0],"end":[320,37],"file_path":"Mathlib/Algebra/Star/Subalgebra.lean","tactics":[{"state_before":"F : Type u_1\nR : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : StarRing A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : StarRing B\ninst✝ : StarModule R B\nS : Subalgebra R A\nx : A\n⊢ star x ∈ star S ↔ x ∈ S","state_after":"no goals","tactic":"simp only [mem_star_iff, star_star]","premises":[{"full_name":"Subalgebra.mem_star_iff","def_path":"Mathlib/Algebra/Star/Subalgebra.lean","def_pos":[315,8],"def_end_pos":[315,20]},{"full_name":"star_star","def_path":"Mathlib/Algebra/Star/Basic.lean","def_pos":[88,8],"def_end_pos":[88,17]}]}]}
+{"url":"Mathlib/Algebra/Homology/HomologicalComplex.lean","commit":"","full_name":"HomologicalComplex.d_comp_XIsoOfEq_inv","start":[138,0],"end":[140,66],"file_path":"Mathlib/Algebra/Homology/HomologicalComplex.lean","tactics":[{"state_before":"ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nK : HomologicalComplex V c\np₂ p₃ : ι\nh : p₃ = p₂\np₁ : ι\n⊢ K.d p₁ p₂ ≫ (K.XIsoOfEq h).inv = K.d p₁ p₃","state_after":"ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nK : HomologicalComplex V c\np₃ p₁ : ι\n⊢ K.d p₁ p₃ ≫ (K.XIsoOfEq ⋯).inv = K.d p₁ p₃","tactic":"subst h","premises":[]},{"state_before":"ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nK : HomologicalComplex V c\np₃ p₁ : ι\n⊢ K.d p₁ p₃ ≫ (K.XIsoOfEq ⋯).inv = K.d p₁ p₃","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/MeasureTheory/OuterMeasure/Basic.lean","commit":"","full_name":"MeasureTheory.OuterMeasure.mono'","start":[160,0],"end":[161,83],"file_path":"Mathlib/MeasureTheory/OuterMeasure/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nm✝ m : OuterMeasure α\ns₁ s₂ : Set α\nh : s₁ ⊆ s₂\n⊢ m s₁ ≤ m s₂","state_after":"no goals","tactic":"gcongr","premises":[]}]}
+{"url":"Mathlib/Algebra/BigOperators/Group/Finset.lean","commit":"","full_name":"Finset.sum_dite_of_true","start":[1068,0],"end":[1072,79],"file_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","tactics":[{"state_before":"ι : Type u_1\nκ : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ns s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\np : α → Prop\nx✝ : DecidablePred p\nh : ∀ i ∈ s, p i\nf : (i : α) → p i → β\ng : (i : α) → ¬p i → β\n⊢ (∏ i ∈ s, if hi : p i then f i hi else g i hi) = ∏ i : { x // x ∈ s }, f ↑i ⋯","state_after":"no goals","tactic":"refine prod_bij' (fun x hx => ⟨x, hx⟩) (fun x _ ↦ x) ?_ ?_ ?_ ?_ ?_ <;> aesop","premises":[{"full_name":"Finset.prod_bij'","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[591,8],"def_end_pos":[591,17]}]}]}
+{"url":"Mathlib/Algebra/Star/CHSH.lean","commit":"","full_name":"CHSH_inequality_of_comm","start":[111,0],"end":[133,53],"file_path":"Mathlib/Algebra/Star/CHSH.lean","tactics":[{"state_before":"R : Type u\ninst✝⁴ : OrderedCommRing R\ninst✝³ : StarRing R\ninst✝² : StarOrderedRing R\ninst✝¹ : Algebra ℝ R\ninst✝ : OrderedSMul ℝ R\nA₀ A₁ B₀ B₁ : R\nT : IsCHSHTuple A₀ A₁ B₀ B₁\n⊢ A₀ * B₀ + A₀ * B₁ + A₁ * B₀ - A₁ * B₁ ≤ 2","state_after":"R : Type u\ninst✝⁴ : OrderedCommRing R\ninst✝³ : StarRing R\ninst✝² : StarOrderedRing R\ninst✝¹ : Algebra ℝ R\ninst✝ : OrderedSMul ℝ R\nA₀ A₁ B₀ B₁ : R\nT : IsCHSHTuple A₀ A₁ B₀ B₁\nP : R := 2 - A₀ * B₀ - A₀ * B₁ - A₁ * B₀ + A₁ * B₁\n⊢ A₀ * B₀ + A₀ * B₁ + A₁ * B₀ - A₁ * B₁ ≤ 2","tactic":"let P := 2 - A₀ * B₀ - A₀ * B₁ - A₁ * B₀ + A₁ * B₁","premises":[]},{"state_before":"R : Type u\ninst✝⁴ : OrderedCommRing R\ninst✝³ : StarRing R\ninst✝² : StarOrderedRing R\ninst✝¹ : Algebra ℝ R\ninst✝ : OrderedSMul ℝ R\nA₀ A₁ B₀ B₁ : R\nT : IsCHSHTuple A₀ A₁ B₀ B₁\nP : R := 2 - A₀ * B₀ - A₀ * B₁ - A₁ * B₀ + A₁ * B₁\n⊢ A₀ * B₀ + A₀ * B₁ + A₁ * B₀ - A₁ * B₁ ≤ 2","state_after":"R : Type u\ninst✝⁴ : OrderedCommRing R\ninst✝³ : StarRing R\ninst✝² : StarOrderedRing R\ninst✝¹ : Algebra ℝ R\ninst✝ : OrderedSMul ℝ R\nA₀ A₁ B₀ B₁ : R\nT : IsCHSHTuple A₀ A₁ B₀ B₁\nP : R := 2 - A₀ * B₀ - A₀ * B₁ - A₁ * B₀ + A₁ * B₁\ni₁ : 0 ≤ P\n⊢ A₀ * B₀ + A₀ * B₁ + A₁ * B₀ - A₁ * B₁ ≤ 2","tactic":"have i₁ : 0 ≤ P := by\n    have idem : P * P = 4 * P := CHSH_id T.A₀_inv T.A₁_inv T.B₀_inv T.B₁_inv\n    have idem' : P = (1 / 4 : ℝ) • (P * P) := by\n      have h : 4 * P = (4 : ℝ) • P := by simp [Algebra.smul_def]\n      rw [idem, h, ← mul_smul]\n      norm_num\n    have sa : star P = P := by\n      dsimp [P]\n      simp only [star_add, star_sub, star_mul, star_ofNat, star_one, T.A₀_sa, T.A₁_sa, T.B₀_sa,\n        T.B₁_sa, mul_comm B₀, mul_comm B₁]\n    simpa only [← idem', sa]\n      using smul_nonneg (by norm_num : (0 : ℝ) ≤ 1 / 4) (star_mul_self_nonneg P)","premises":[{"full_name":"Algebra.smul_def","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[270,8],"def_end_pos":[270,16]},{"full_name":"CHSH_id","def_path":"Mathlib/Algebra/Star/CHSH.lean","def_pos":[101,8],"def_end_pos":[101,15]},{"full_name":"IsCHSHTuple.A₀_inv","def_path":"Mathlib/Algebra/Star/CHSH.lean","def_pos":[86,2],"def_end_pos":[86,8]},{"full_name":"IsCHSHTuple.A₀_sa","def_path":"Mathlib/Algebra/Star/CHSH.lean","def_pos":[90,2],"def_end_pos":[90,7]},{"full_name":"IsCHSHTuple.A₁_inv","def_path":"Mathlib/Algebra/Star/CHSH.lean","def_pos":[87,2],"def_end_pos":[87,8]},{"full_name":"IsCHSHTuple.A₁_sa","def_path":"Mathlib/Algebra/Star/CHSH.lean","def_pos":[91,2],"def_end_pos":[91,7]},{"full_name":"IsCHSHTuple.B₀_inv","def_path":"Mathlib/Algebra/Star/CHSH.lean","def_pos":[88,2],"def_end_pos":[88,8]},{"full_name":"IsCHSHTuple.B₀_sa","def_path":"Mathlib/Algebra/Star/CHSH.lean","def_pos":[92,2],"def_end_pos":[92,7]},{"full_name":"IsCHSHTuple.B₁_inv","def_path":"Mathlib/Algebra/Star/CHSH.lean","def_pos":[89,2],"def_end_pos":[89,8]},{"full_name":"IsCHSHTuple.B₁_sa","def_path":"Mathlib/Algebra/Star/CHSH.lean","def_pos":[93,2],"def_end_pos":[93,7]},{"full_name":"MulAction.mul_smul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[99,2],"def_end_pos":[99,10]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Star.star","def_path":"Mathlib/Algebra/Star/Basic.lean","def_pos":[46,2],"def_end_pos":[46,6]},{"full_name":"StarAddMonoid.star_add","def_path":"Mathlib/Algebra/Star/Basic.lean","def_pos":[227,2],"def_end_pos":[227,10]},{"full_name":"StarMul.star_mul","def_path":"Mathlib/Algebra/Star/Basic.lean","def_pos":[126,2],"def_end_pos":[126,10]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]},{"full_name":"smul_nonneg","def_path":"Mathlib/Algebra/Order/Module/Defs.lean","def_pos":[473,6],"def_end_pos":[473,17]},{"full_name":"star_mul_self_nonneg","def_path":"Mathlib/Algebra/Star/Order.lean","def_pos":[140,8],"def_end_pos":[140,28]},{"full_name":"star_ofNat","def_path":"Mathlib/Algebra/Star/Basic.lean","def_pos":[293,8],"def_end_pos":[293,18]},{"full_name":"star_one","def_path":"Mathlib/Algebra/Star/Basic.lean","def_pos":[180,8],"def_end_pos":[180,16]},{"full_name":"star_sub","def_path":"Mathlib/Algebra/Star/Basic.lean","def_pos":[260,8],"def_end_pos":[260,16]}]},{"state_before":"R : Type u\ninst✝⁴ : OrderedCommRing R\ninst✝³ : StarRing R\ninst✝² : StarOrderedRing R\ninst✝¹ : Algebra ℝ R\ninst✝ : OrderedSMul ℝ R\nA₀ A₁ B₀ B₁ : R\nT : IsCHSHTuple A₀ A₁ B₀ B₁\nP : R := 2 - A₀ * B₀ - A₀ * B₁ - A₁ * B₀ + A₁ * B₁\ni₁ : 0 ≤ P\n⊢ A₀ * B₀ + A₀ * B₁ + A₁ * B₀ - A₁ * B₁ ≤ 2","state_after":"case a\nR : Type u\ninst✝⁴ : OrderedCommRing R\ninst✝³ : StarRing R\ninst✝² : StarOrderedRing R\ninst✝¹ : Algebra ℝ R\ninst✝ : OrderedSMul ℝ R\nA₀ A₁ B₀ B₁ : R\nT : IsCHSHTuple A₀ A₁ B₀ B₁\nP : R := 2 - A₀ * B₀ - A₀ * B₁ - A₁ * B₀ + A₁ * B₁\ni₁ : 0 ≤ P\n⊢ 0 ≤ 2 - (A₀ * B₀ + A₀ * B₁ + A₁ * B₀ - A₁ * B₁)","tactic":"apply le_of_sub_nonneg","premises":[]},{"state_before":"case a\nR : Type u\ninst✝⁴ : OrderedCommRing R\ninst✝³ : StarRing R\ninst✝² : StarOrderedRing R\ninst✝¹ : Algebra ℝ R\ninst✝ : OrderedSMul ℝ R\nA₀ A₁ B₀ B₁ : R\nT : IsCHSHTuple A₀ A₁ B₀ B₁\nP : R := 2 - A₀ * B₀ - A₀ * B₁ - A₁ * B₀ + A₁ * B₁\ni₁ : 0 ≤ P\n⊢ 0 ≤ 2 - (A₀ * B₀ + A₀ * B₁ + A₁ * B₀ - A₁ * B₁)","state_after":"no goals","tactic":"simpa only [sub_add_eq_sub_sub, ← sub_add] using i₁","premises":[{"full_name":"sub_add","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[545,2],"def_end_pos":[545,13]},{"full_name":"sub_add_eq_sub_sub","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[554,2],"def_end_pos":[554,13]}]}]}
+{"url":"Mathlib/GroupTheory/OrderOfElement.lean","commit":"","full_name":"orderOf_one","start":[197,0],"end":[199,66],"file_path":"Mathlib/GroupTheory/OrderOfElement.lean","tactics":[{"state_before":"G : Type u_1\nH : Type u_2\nA : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝ : Monoid G\na b x y : G\nn m : ℕ\n⊢ orderOf 1 = 1","state_after":"no goals","tactic":"rw [orderOf, ← minimalPeriod_id (x := (1 : G)), ← one_mul_eq_id]","premises":[{"full_name":"Function.minimalPeriod_id","def_path":"Mathlib/Dynamics/PeriodicPts.lean","def_pos":[313,16],"def_end_pos":[313,32]},{"full_name":"one_mul_eq_id","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[138,8],"def_end_pos":[138,21]},{"full_name":"orderOf","def_path":"Mathlib/GroupTheory/OrderOfElement.lean","def_pos":[129,18],"def_end_pos":[129,25]}]}]}
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f).coconePointUniqueUpToIso hc).hom ≫ hc.desc c)\ne_3✝ : ∐ f = (mk (∐ f) (Sigma.ι f)).pt\n⊢ Sigma.desc c.inj = ((coproductIsCoproduct f).coconePointUniqueUpToIso hc).hom ≫ hc.desc c","state_after":"case h.e'_5.h.h\nβ : Type w\nα : Type w₂\nγ : Type w₃\nC : Type u\ninst✝¹ : Category.{v, u} C\nf : β → C\ninst✝ : HasCoproduct f\nc : Cofan f\nx✝ : Nonempty (IsColimit c)\nhc : IsColimit c\nthis : IsIso (((coproductIsCoproduct f).coconePointUniqueUpToIso hc).hom ≫ hc.desc c)\ne_3✝ : ∐ f = (mk (∐ f) (Sigma.ι f)).pt\nb✝ : β\n⊢ Sigma.ι f b✝ ≫ Sigma.desc c.inj =\n    Sigma.ι f b✝ ≫ ((coproductIsCoproduct f).coconePointUniqueUpToIso hc).hom ≫ hc.desc c","tactic":"ext","premises":[]},{"state_before":"case h.e'_5.h.h\nβ : Type w\nα : Type w₂\nγ : Type w₃\nC : Type u\ninst✝¹ : Category.{v, u} C\nf : β → C\ninst✝ : HasCoproduct f\nc : Cofan f\nx✝ : Nonempty (IsColimit c)\nhc : IsColimit c\nthis : IsIso (((coproductIsCoproduct f).coconePointUniqueUpToIso hc).hom ≫ hc.desc c)\ne_3✝ : ∐ f = (mk (∐ f) (Sigma.ι f)).pt\nb✝ : β\n⊢ Sigma.ι f b✝ ≫ Sigma.desc c.inj =\n    Sigma.ι f b✝ ≫ ((coproductIsCoproduct f).coconePointUniqueUpToIso hc).hom ≫ hc.desc c","state_after":"case h.e'_5.h.h\nβ : Type w\nα : Type w₂\nγ : Type w₃\nC : Type u\ninst✝¹ : Category.{v, u} C\nf : β → C\ninst✝ : HasCoproduct f\nc : Cofan f\nx✝ : Nonempty (IsColimit c)\nhc : IsColimit c\nthis : IsIso (((coproductIsCoproduct f).coconePointUniqueUpToIso hc).hom ≫ hc.desc c)\ne_3✝ : ∐ f = (mk (∐ f) (Sigma.ι f)).pt\nb✝ : β\n⊢ c.inj b✝ = c.ι.app { as := b✝ }","tactic":"simp only [colimit.ι_desc, mk_pt, mk_ι_app, IsColimit.coconePointUniqueUpToIso,\n    coproductIsCoproduct, colimit.cocone_x, Functor.mapIso_hom, IsColimit.uniqueUpToIso_hom,\n    Cocones.forget_map, IsColimit.descCoconeMorphism_hom, IsColimit.ofIsoColimit_desc,\n    Cocones.ext_inv_hom, Iso.refl_inv, colimit.isColimit_desc, Category.id_comp,\n    IsColimit.desc_self, Category.comp_id]","premises":[{"full_name":"CategoryTheory.Category.comp_id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[153,2],"def_end_pos":[153,9]},{"full_name":"CategoryTheory.Category.id_comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[151,2],"def_end_pos":[151,9]},{"full_name":"CategoryTheory.Functor.mapIso_hom","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[530,2],"def_end_pos":[530,7]},{"full_name":"CategoryTheory.Iso.refl_inv","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[113,8],"def_end_pos":[113,13]},{"full_name":"CategoryTheory.Limits.Cocones.ext_inv_hom","def_path":"Mathlib/CategoryTheory/Limits/Cones.lean","def_pos":[488,52],"def_end_pos":[488,57]},{"full_name":"CategoryTheory.Limits.Cocones.forget_map","def_path":"Mathlib/CategoryTheory/Limits/Cones.lean","def_pos":[597,2],"def_end_pos":[597,7]},{"full_name":"CategoryTheory.Limits.Cofan.mk_pt","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Products.lean","def_pos":[63,9],"def_end_pos":[63,11]},{"full_name":"CategoryTheory.Limits.Cofan.mk_ι_app","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Products.lean","def_pos":[63,12],"def_end_pos":[63,17]},{"full_name":"CategoryTheory.Limits.IsColimit.coconePointUniqueUpToIso","def_path":"Mathlib/CategoryTheory/Limits/IsLimit.lean","def_pos":[587,4],"def_end_pos":[587,28]},{"full_name":"CategoryTheory.Limits.IsColimit.descCoconeMorphism_hom","def_path":"Mathlib/CategoryTheory/Limits/IsLimit.lean","def_pos":[543,2],"def_end_pos":[543,7]},{"full_name":"CategoryTheory.Limits.IsColimit.desc_self","def_path":"Mathlib/CategoryTheory/Limits/IsLimit.lean","def_pos":[538,8],"def_end_pos":[538,17]},{"full_name":"CategoryTheory.Limits.IsColimit.ofIsoColimit_desc","def_path":"Mathlib/CategoryTheory/Limits/IsLimit.lean","def_pos":[616,8],"def_end_pos":[616,25]},{"full_name":"CategoryTheory.Limits.IsColimit.uniqueUpToIso_hom","def_path":"Mathlib/CategoryTheory/Limits/IsLimit.lean","def_pos":[575,2],"def_end_pos":[575,7]},{"full_name":"CategoryTheory.Limits.colimit.cocone_x","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[660,8],"def_end_pos":[660,24]},{"full_name":"CategoryTheory.Limits.colimit.isColimit_desc","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[677,8],"def_end_pos":[677,30]},{"full_name":"CategoryTheory.Limits.colimit.ι_desc","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[691,8],"def_end_pos":[691,22]},{"full_name":"CategoryTheory.Limits.coproductIsCoproduct","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Products.lean","def_pos":[219,4],"def_end_pos":[219,24]}]},{"state_before":"case h.e'_5.h.h\nβ : Type w\nα : Type w₂\nγ : Type w₃\nC : Type u\ninst✝¹ : Category.{v, u} C\nf : β → C\ninst✝ : HasCoproduct f\nc : Cofan f\nx✝ : Nonempty (IsColimit c)\nhc : IsColimit c\nthis : IsIso (((coproductIsCoproduct f).coconePointUniqueUpToIso hc).hom ≫ hc.desc c)\ne_3✝ : ∐ f = (mk (∐ f) (Sigma.ι f)).pt\nb✝ : β\n⊢ c.inj b✝ = c.ι.app { as := b✝ }","state_after":"no goals","tactic":"rfl","premises":[]}]}
+{"url":"Mathlib/LinearAlgebra/Lagrange.lean","commit":"","full_name":"Lagrange.sum_basis","start":[251,0],"end":[265,32],"file_path":"Mathlib/LinearAlgebra/Lagrange.lean","tactics":[{"state_before":"F : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns : Finset ι\nv : ι → F\ni j : ι\nhvs : Set.InjOn v ↑s\nhs : s.Nonempty\n⊢ ∑ j ∈ s, Lagrange.basis s v j = 1","state_after":"case refine_1\nF : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns : Finset ι\nv : ι → F\ni j : ι\nhvs : Set.InjOn v ↑s\nhs : s.Nonempty\n⊢ (s.sup fun b => (Lagrange.basis s v b).degree) < ↑s.card\n\ncase refine_2\nF : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns : Finset ι\nv : ι → F\ni j : ι\nhvs : Set.InjOn v ↑s\nhs : s.Nonempty\n⊢ degree 1 < ↑s.card\n\ncase refine_3\nF : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns : Finset ι\nv : ι → F\ni j : ι\nhvs : Set.InjOn v ↑s\nhs : s.Nonempty\n⊢ ∀ i ∈ s, eval (v i) (∑ j ∈ s, Lagrange.basis s v j) = eval (v i) 1","tactic":"refine eq_of_degrees_lt_of_eval_index_eq s hvs (lt_of_le_of_lt (degree_sum_le _ _) ?_) ?_ ?_","premises":[{"full_name":"Polynomial.degree_sum_le","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[695,8],"def_end_pos":[695,21]},{"full_name":"Polynomial.eq_of_degrees_lt_of_eval_index_eq","def_path":"Mathlib/LinearAlgebra/Lagrange.lean","def_pos":[105,8],"def_end_pos":[105,41]},{"full_name":"lt_of_le_of_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[99,8],"def_end_pos":[99,22]}]}]}
+{"url":"Mathlib/Data/Multiset/FinsetOps.lean","commit":"","full_name":"Multiset.coe_ndinter","start":[193,0],"end":[196,17],"file_path":"Mathlib/Data/Multiset/FinsetOps.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : DecidableEq α\ns : Multiset α\nl₁ l₂ : List α\n⊢ (↑l₁).ndinter ↑l₂ = ↑(l₁ ∩ l₂)","state_after":"α : Type u_1\ninst✝ : DecidableEq α\ns : Multiset α\nl₁ l₂ : List α\n⊢ List.filter (fun b => elem b l₂) l₁ ~ l₁ ∩ l₂","tactic":"simp only [ndinter, mem_coe, filter_coe, coe_eq_coe, ← elem_eq_mem]","premises":[{"full_name":"List.elem_eq_mem","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean","def_pos":[335,16],"def_end_pos":[335,27]},{"full_name":"Multiset.coe_eq_coe","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[60,8],"def_end_pos":[60,18]},{"full_name":"Multiset.filter_coe","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[1675,25],"def_end_pos":[1675,35]},{"full_name":"Multiset.mem_coe","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[211,8],"def_end_pos":[211,15]},{"full_name":"Multiset.ndinter","def_path":"Mathlib/Data/Multiset/FinsetOps.lean","def_pos":[190,4],"def_end_pos":[190,11]}]},{"state_before":"α : Type u_1\ninst✝ : DecidableEq α\ns : Multiset α\nl₁ l₂ : List α\n⊢ List.filter (fun b => elem b l₂) l₁ ~ l₁ ∩ l₂","state_after":"no goals","tactic":"apply Perm.refl","premises":[{"full_name":"List.Perm.refl","def_path":".lake/packages/batteries/Batteries/Data/List/Perm.lean","def_pos":[29,32],"def_end_pos":[29,41]}]}]}
+{"url":"Mathlib/Algebra/Free.lean","commit":"","full_name":"FreeMagma.toFreeSemigroup_comp_map","start":[653,0],"end":[655,93],"file_path":"Mathlib/Algebra/Free.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nf : α → β\n⊢ toFreeSemigroup.comp (map f) = (FreeSemigroup.map f).comp toFreeSemigroup","state_after":"case h\nα : Type u\nβ : Type v\nf : α → β\n⊢ ⇑(toFreeSemigroup.comp (map f)) ∘ of = ⇑((FreeSemigroup.map f).comp toFreeSemigroup) ∘ of","tactic":"ext1","premises":[]},{"state_before":"case h\nα : Type u\nβ : Type v\nf : α → β\n⊢ ⇑(toFreeSemigroup.comp (map f)) ∘ of = ⇑((FreeSemigroup.map f).comp toFreeSemigroup) ∘ of","state_after":"no goals","tactic":"rfl","premises":[]}]}
+{"url":"Mathlib/Combinatorics/SimpleGraph/Regularity/Chunk.lean","commit":"","full_name":"_private.Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk.0.SzemerediRegularity.le_sum_card_subset_chunk_parts","start":[197,0],"end":[203,63],"file_path":"Mathlib/Combinatorics/SimpleGraph/Regularity/Chunk.lean","tactics":[{"state_before":"α : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nU : Finset α\nhU : U ∈ P.parts\nV : Finset α\n𝒜 : Finset (Finset α)\ns : Finset α\nh𝒜 : 𝒜 ⊆ (chunk hP G ε hU).parts\nhs : s ∈ 𝒜\n⊢ ↑𝒜.card * ↑s.card * (↑m / (↑m + 1)) ≤ ↑(𝒜.sup id).card","state_after":"α : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nU : Finset α\nhU : U ∈ P.parts\nV : Finset α\n𝒜 : Finset (Finset α)\ns : Finset α\nh𝒜 : 𝒜 ⊆ (chunk hP G ε hU).parts\nhs : s ∈ 𝒜\n⊢ ↑𝒜.card * ↑m * ↑s.card ≤ ↑(𝒜.sup id).card * (↑m + 1)","tactic":"rw [mul_div_assoc', div_le_iff coe_m_add_one_pos, mul_right_comm]","premises":[{"full_name":"SzemerediRegularity.coe_m_add_one_pos","def_path":"Mathlib/Combinatorics/SimpleGraph/Regularity/Bound.lean","def_pos":[107,8],"def_end_pos":[107,25]},{"full_name":"div_le_iff","def_path":"Mathlib/Algebra/Order/Field/Basic.lean","def_pos":[51,8],"def_end_pos":[51,18]},{"full_name":"mul_div_assoc'","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[334,8],"def_end_pos":[334,22]},{"full_name":"mul_right_comm","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[156,8],"def_end_pos":[156,22]}]},{"state_before":"α : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nU : Finset α\nhU : U ∈ P.parts\nV : Finset α\n𝒜 : Finset (Finset α)\ns : Finset α\nh𝒜 : 𝒜 ⊆ (chunk hP G ε hU).parts\nhs : s ∈ 𝒜\n⊢ ↑𝒜.card * ↑m * ↑s.card ≤ ↑(𝒜.sup id).card * (↑m + 1)","state_after":"case refine_1\nα : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nU : Finset α\nhU : U ∈ P.parts\nV : Finset α\n𝒜 : Finset (Finset α)\ns : Finset α\nh𝒜 : 𝒜 ⊆ (chunk hP G ε hU).parts\nhs : s ∈ 𝒜\n⊢ ↑𝒜.card * ↑m ≤ ↑(𝒜.sup id).card\n\ncase refine_2\nα : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nU : Finset α\nhU : U ∈ P.parts\nV : Finset α\n𝒜 : Finset (Finset α)\ns : Finset α\nh𝒜 : 𝒜 ⊆ (chunk hP G ε hU).parts\nhs : s ∈ 𝒜\n⊢ ↑s.card ≤ ↑m + 1","tactic":"refine mul_le_mul ?_ ?_ (cast_nonneg _) (cast_nonneg _)","premises":[{"full_name":"Nat.cast_nonneg","def_path":"Mathlib/Data/Nat/Cast/Order/Ring.lean","def_pos":[29,8],"def_end_pos":[29,19]}]}]}
+{"url":"Mathlib/Algebra/Group/Submonoid/Operations.lean","commit":"","full_name":"Submonoid.closure_closure_coe_preimage","start":[576,0],"end":[583,33],"file_path":"Mathlib/Algebra/Group/Submonoid/Operations.lean","tactics":[{"state_before":"M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Set M\nx✝¹ : { x // x ∈ ↑(closure s) }\nx : M\nhx : x ∈ ↑(closure s)\nx✝ : ⟨x, hx⟩ ∈ ⊤\n⊢ ⟨x, hx⟩ ∈ closure (Subtype.val ⁻¹' s)","state_after":"case refine_1\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Set M\nx✝¹ : { x // x ∈ ↑(closure s) }\nx : M\nhx : x ∈ ↑(closure s)\nx✝ : ⟨x, hx⟩ ∈ ⊤\n⊢ (fun y hy => ⟨y, hy⟩ ∈ closure (Subtype.val ⁻¹' s)) 1 ⋯\n\ncase refine_2\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Set M\nx✝¹ : { x // x ∈ ↑(closure s) }\nx : M\nhx : x ∈ ↑(closure s)\nx✝ : ⟨x, hx⟩ ∈ ⊤\ng₁ : M\ng₂ : g₁ ∈ closure s\nhg₁ : M\nhg₂ : hg₁ ∈ closure s\n⊢ (fun y hy => ⟨y, hy⟩ ∈ closure (Subtype.val ⁻¹' s)) g₁ g₂ →\n    (fun y hy => ⟨y, hy⟩ ∈ closure (Subtype.val ⁻¹' s)) hg₁ hg₂ →\n      (fun y hy => ⟨y, hy⟩ ∈ closure (Subtype.val ⁻¹' s)) (g₁ * hg₁) ⋯","tactic":"refine closure_induction' (p := fun y hy ↦ ⟨y, hy⟩ ∈ closure (((↑) : closure s → M) ⁻¹' s))\n        (fun g hg => subset_closure hg) ?_ (fun g₁ g₂ hg₁ hg₂ => ?_) hx","premises":[{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Set.preimage","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[106,4],"def_end_pos":[106,12]},{"full_name":"Submonoid.closure","def_path":"Mathlib/Algebra/Group/Submonoid/Basic.lean","def_pos":[329,4],"def_end_pos":[329,11]},{"full_name":"Submonoid.closure_induction'","def_path":"Mathlib/Algebra/Group/Submonoid/Basic.lean","def_pos":[381,8],"def_end_pos":[381,26]},{"full_name":"Submonoid.subset_closure","def_path":"Mathlib/Algebra/Group/Submonoid/Basic.lean","def_pos":[339,8],"def_end_pos":[339,22]}]}]}
+{"url":"Mathlib/Combinatorics/SimpleGraph/Triangle/Counting.lean","commit":"","full_name":"_private.Mathlib.Combinatorics.SimpleGraph.Triangle.Counting.0.SimpleGraph.triple_eq_triple_of_mem","start":[134,0],"end":[147,17],"file_path":"Mathlib/Combinatorics/SimpleGraph/Triangle/Counting.lean","tactics":[{"state_before":"α : Type u_1\nG G' : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\ns t u : Finset α\ninst✝ : DecidableEq α\nhst : Disjoint s t\nhsu : Disjoint s u\nhtu : Disjoint t u\nx₁ x₂ y₁ y₂ z₁ z₂ : α\nh : {x₁, y₁, z₁} = {x₂, y₂, z₂}\nhx₁ : x₁ ∈ s\nhx₂ : x₂ ∈ s\nhy₁ : y₁ ∈ t\nhy₂ : y₂ ∈ t\nhz₁ : z₁ ∈ u\nhz₂ : z₂ ∈ u\n⊢ (x₁, y₁, z₁) = (x₂, y₂, z₂)","state_after":"α : Type u_1\nG G' : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\ns t u : Finset α\ninst✝ : DecidableEq α\nhst : Disjoint s t\nhsu : Disjoint s u\nhtu : Disjoint t u\nx₁ x₂ y₁ y₂ z₁ z₂ : α\nhx₁ : x₁ ∈ s\nhx₂ : x₂ ∈ s\nhy₁ : y₁ ∈ t\nhy₂ : y₂ ∈ t\nhz₁ : z₁ ∈ u\nhz₂ : z₂ ∈ u\nh :\n  ((x₁ = x₂ ∨ x₁ = y₂ ∨ x₁ = z₂) ∧ (y₁ = x₂ ∨ y₁ = y₂ ∨ y₁ = z₂) ∧ (z₁ = x₂ ∨ z₁ = y₂ ∨ z₁ = z₂)) ∧\n    (x₂ = x₁ ∨ x₂ = y₁ ∨ x₂ = z₁) ∧ (y₂ = x₁ ∨ y₂ = y₁ ∨ y₂ = z₁) ∧ (z₂ = x₁ ∨ z₂ = y₁ ∨ z₂ = z₁)\n⊢ (x₁, y₁, z₁) = (x₂, y₂, z₂)","tactic":"simp only [Finset.Subset.antisymm_iff, subset_iff, mem_insert, mem_singleton, forall_eq_or_imp,\n    forall_eq] at h","premises":[{"full_name":"Finset.Subset.antisymm_iff","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[343,8],"def_end_pos":[343,27]},{"full_name":"Finset.mem_insert","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[941,8],"def_end_pos":[941,18]},{"full_name":"Finset.mem_singleton","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[584,8],"def_end_pos":[584,21]},{"full_name":"Finset.subset_iff","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[332,8],"def_end_pos":[332,18]},{"full_name":"forall_eq","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[273,16],"def_end_pos":[273,25]},{"full_name":"forall_eq_or_imp","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[295,16],"def_end_pos":[295,32]}]},{"state_before":"α : Type u_1\nG G' : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\ns t u : Finset α\ninst✝ : DecidableEq α\nhst : Disjoint s t\nhsu : Disjoint s u\nhtu : Disjoint t u\nx₁ x₂ y₁ y₂ z₁ z₂ : α\nhx₁ : x₁ ∈ s\nhx₂ : x₂ ∈ s\nhy₁ : y₁ ∈ t\nhy₂ : y₂ ∈ t\nhz₁ : z₁ ∈ u\nhz₂ : z₂ ∈ u\nh :\n  ((x₁ = x₂ ∨ x₁ = y₂ ∨ x₁ = z₂) ∧ (y₁ = x₂ ∨ y₁ = y₂ ∨ y₁ = z₂) ∧ (z₁ = x₂ ∨ z₁ = y₂ ∨ z₁ = z₂)) ∧\n    (x₂ = x₁ ∨ x₂ = y₁ ∨ x₂ = z₁) ∧ (y₂ = x₁ ∨ y₂ = y₁ ∨ y₂ = z₁) ∧ (z₂ = x₁ ∨ z₂ = y₁ ∨ z₂ = z₁)\n⊢ (x₁, y₁, z₁) = (x₂, y₂, z₂)","state_after":"α : Type u_1\nG G' : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\ns t u : Finset α\ninst✝ : DecidableEq α\nhst : ∀ ⦃a : α⦄, a ∈ s → a ∉ t\nhsu : ∀ ⦃a : α⦄, a ∈ s → a ∉ u\nhtu : ∀ ⦃a : α⦄, a ∈ t → a ∉ u\nx₁ x₂ y₁ y₂ z₁ z₂ : α\nhx₁ : x₁ ∈ s\nhx₂ : x₂ ∈ s\nhy₁ : y₁ ∈ t\nhy₂ : y₂ ∈ t\nhz₁ : z₁ ∈ u\nhz₂ : z₂ ∈ u\nh :\n  ((x₁ = x₂ ∨ x₁ = y₂ ∨ x₁ = z₂) ∧ (y₁ = x₂ ∨ y₁ = y₂ ∨ y₁ = z₂) ∧ (z₁ = x₂ ∨ z₁ = y₂ ∨ z₁ = z₂)) ∧\n    (x₂ = x₁ ∨ x₂ = y₁ ∨ x₂ = z₁) ∧ (y₂ = x₁ ∨ y₂ = y₁ ∨ y₂ = z₁) ∧ (z₂ = x₁ ∨ z₂ = y₁ ∨ z₂ = z₁)\n⊢ (x₁, y₁, z₁) = (x₂, y₂, z₂)","tactic":"rw [disjoint_left] at hst hsu htu","premises":[{"full_name":"Finset.disjoint_left","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[814,8],"def_end_pos":[814,21]}]},{"state_before":"α : Type u_1\nG G' : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\ns t u : Finset α\ninst✝ : DecidableEq α\nhst : ∀ ⦃a : α⦄, a ∈ s → a ∉ t\nhsu : ∀ ⦃a : α⦄, a ∈ s → a ∉ u\nhtu : ∀ ⦃a : α⦄, a ∈ t → a ∉ u\nx₁ x₂ y₁ y₂ z₁ z₂ : α\nhx₁ : x₁ ∈ s\nhx₂ : x₂ ∈ s\nhy₁ : y₁ ∈ t\nhy₂ : y₂ ∈ t\nhz₁ : z₁ ∈ u\nhz₂ : z₂ ∈ u\nh :\n  ((x₁ = x₂ ∨ x₁ = y₂ ∨ x₁ = z₂) ∧ (y₁ = x₂ ∨ y₁ = y₂ ∨ y₁ = z₂) ∧ (z₁ = x₂ ∨ z₁ = y₂ ∨ z₁ = z₂)) ∧\n    (x₂ = x₁ ∨ x₂ = y₁ ∨ x₂ = z₁) ∧ (y₂ = x₁ ∨ y₂ = y₁ ∨ y₂ = z₁) ∧ (z₂ = x₁ ∨ z₂ = y₁ ∨ z₂ = z₁)\n⊢ (x₁, y₁, z₁) = (x₂, y₂, z₂)","state_after":"α : Type u_1\nG G' : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\ns t u : Finset α\ninst✝ : DecidableEq α\nhst : ∀ ⦃a : α⦄, a ∈ s → a ∉ t\nhsu : ∀ ⦃a : α⦄, a ∈ s → a ∉ u\nhtu : ∀ ⦃a : α⦄, a ∈ t → a ∉ u\nx₁ x₂ y₁ y₂ z₁ z₂ : α\nhx₁ : x₁ ∈ s\nhx₂ : x₂ ∈ s\nhy₁ : y₁ ∈ t\nhy₂ : y₂ ∈ t\nhz₁ : z₁ ∈ u\nhz₂ : z₂ ∈ u\nh :\n  ((x₁ = x₂ ∨ x₁ = y₂ ∨ x₁ = z₂) ∧ (y₁ = x₂ ∨ y₁ = y₂ ∨ y₁ = z₂) ∧ (z₁ = x₂ ∨ z₁ = y₂ ∨ z₁ = z₂)) ∧\n    (x₂ = x₁ ∨ x₂ = y₁ ∨ x₂ = z₁) ∧ (y₂ = x₁ ∨ y₂ = y₁ ∨ y₂ = z₁) ∧ (z₂ = x₁ ∨ z₂ = y₁ ∨ z₂ = z₁)\n⊢ x₁ = x₂ ∧ y₁ = y₂ ∧ z₁ = z₂","tactic":"rw [Prod.mk.inj_iff, Prod.mk.inj_iff]","premises":[{"full_name":"Prod.mk.inj_iff","def_path":"Mathlib/Data/Prod/Basic.lean","def_pos":[85,8],"def_end_pos":[85,18]}]},{"state_before":"α : Type u_1\nG G' : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\ns t u : Finset α\ninst✝ : DecidableEq α\nhst : ∀ ⦃a : α⦄, a ∈ s → a ∉ t\nhsu : ∀ ⦃a : α⦄, a ∈ s → a ∉ u\nhtu : ∀ ⦃a : α⦄, a ∈ t → a ∉ u\nx₁ x₂ y₁ y₂ z₁ z₂ : α\nhx₁ : x₁ ∈ s\nhx₂ : x₂ ∈ s\nhy₁ : y₁ ∈ t\nhy₂ : y₂ ∈ t\nhz₁ : z₁ ∈ u\nhz₂ : z₂ ∈ u\nh :\n  ((x₁ = x₂ ∨ x₁ = y₂ ∨ x₁ = z₂) ∧ (y₁ = x₂ ∨ y₁ = y₂ ∨ y₁ = z₂) ∧ (z₁ = x₂ ∨ z₁ = y₂ ∨ z₁ = z₂)) ∧\n    (x₂ = x₁ ∨ x₂ = y₁ ∨ x₂ = z₁) ∧ (y₂ = x₁ ∨ y₂ = y₁ ∨ y₂ = z₁) ∧ (z₂ = x₁ ∨ z₂ = y₁ ∨ z₂ = z₁)\n⊢ x₁ = x₂ ∧ y₁ = y₂ ∧ z₁ = z₂","state_after":"α : Type u_1\nG G' : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\ns t u : Finset α\ninst✝ : DecidableEq α\nhst : ∀ ⦃a : α⦄, a ∈ s → a ∉ t\nhsu : ∀ ⦃a : α⦄, a ∈ s → a ∉ u\nhtu : ∀ ⦃a : α⦄, a ∈ t → a ∉ u\nx₁ x₂ y₁ y₂ z₁ z₂ : α\nhx₁ : x₁ ∈ s\nhx₂ : x₂ ∈ s\nhy₁ : y₁ ∈ t\nhy₂ : y₂ ∈ t\nhz₁ : z₁ ∈ u\nhz₂ : z₂ ∈ u\nh :\n  (x₁ = x₂ ∨ x₁ = y₂ ∨ x₁ = z₂) ∧\n    (y₁ = y₂ ∨ y₁ = x₂ ∨ y₁ = z₂) ∧\n      (z₁ = z₂ ∨ z₁ = x₂ ∨ z₁ = y₂) ∧\n        (x₂ = x₁ ∨ x₂ = y₁ ∨ x₂ = z₁) ∧ (y₂ = x₁ ∨ y₂ = y₁ ∨ y₂ = z₁) ∧ (z₂ = x₁ ∨ z₂ = y₁ ∨ z₂ = z₁)\n⊢ x₁ = x₂ ∧ y₁ = y₂ ∧ z₁ = z₂","tactic":"simp only [and_assoc, @or_left_comm _ (y₁ = y₂), @or_comm _ (z₁ = z₂),\n    @or_left_comm _ (z₁ = z₂)] at h","premises":[{"full_name":"and_assoc","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[148,8],"def_end_pos":[148,17]},{"full_name":"or_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[823,8],"def_end_pos":[823,15]},{"full_name":"or_left_comm","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[81,8],"def_end_pos":[81,20]}]}]}
+{"url":"Mathlib/Data/Finset/Pointwise/Interval.lean","commit":"","full_name":"Finset.Icc_mul_Ico_subset'","start":[55,0],"end":[58,72],"file_path":"Mathlib/Data/Finset/Pointwise/Interval.lean","tactics":[{"state_before":"α : Type u_1\ninst✝⁵ : Mul α\ninst✝⁴ : PartialOrder α\ninst✝³ : DecidableEq α\ninst✝² : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst✝¹ : CovariantClass α α (Function.swap HMul.hMul) LT.lt\ninst✝ : LocallyFiniteOrder α\na b c d : α\n⊢ ↑(Icc a b * Ico c d) ⊆ ↑(Ico (a * c) (b * d))","state_after":"no goals","tactic":"simpa using Set.Icc_mul_Ico_subset' _ _ _ _","premises":[{"full_name":"Set.Icc_mul_Ico_subset'","def_path":"Mathlib/Data/Set/Pointwise/Interval.lean","def_pos":[66,8],"def_end_pos":[66,27]}]}]}
+{"url":"Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean","commit":"","full_name":"CategoryTheory.Subgroupoid.mem_im_objs_iff","start":[489,0],"end":[491,58],"file_path":"Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean","tactics":[{"state_before":"C : Type u\ninst✝¹ : Groupoid C\nS : Subgroupoid C\nD : Type u_1\ninst✝ : Groupoid D\nφ : C ⥤ D\nhφ : Function.Injective φ.obj\nd : D\n⊢ d ∈ (im φ hφ).objs ↔ ∃ c, φ.obj c = d","state_after":"no goals","tactic":"simp only [im, mem_map_objs_iff, mem_top_objs, true_and]","premises":[{"full_name":"CategoryTheory.Subgroupoid.im","def_path":"Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean","def_pos":[480,4],"def_end_pos":[480,6]},{"full_name":"CategoryTheory.Subgroupoid.mem_map_objs_iff","def_path":"Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean","def_pos":[464,8],"def_end_pos":[464,24]},{"full_name":"CategoryTheory.Subgroupoid.mem_top_objs","def_path":"Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean","def_pos":[185,8],"def_end_pos":[185,20]},{"full_name":"true_and","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[105,16],"def_end_pos":[105,24]}]}]}
+{"url":"Mathlib/Data/Nat/Squarefree.lean","commit":"","full_name":"Nat.minSqFacAux_has_prop","start":[157,0],"end":[193,29],"file_path":"Mathlib/Data/Nat/Squarefree.lean","tactics":[{"state_before":"s : Finset ℕ\nm n✝ p n k : ℕ\nn0 : 0 < n\ni : ℕ\ne : k = 2 * i + 3\nih : ∀ (m : ℕ), Prime m → m ∣ n → k ≤ m\n⊢ n.MinSqFacProp (n.minSqFacAux k)","state_after":"s : Finset ℕ\nm n✝ p n k : ℕ\nn0 : 0 < n\ni : ℕ\ne : k = 2 * i + 3\nih : ∀ (m : ℕ), Prime m → m ∣ n → k ≤ m\n⊢ n.MinSqFacProp\n    (if h : n < k * k then none\n    else\n      let_fun this := ⋯;\n      if k ∣ n then\n        let n' := n / k;\n        let_fun this := ⋯;\n        if k ∣ n' then some k else n'.minSqFacAux (k + 2)\n      else n.minSqFacAux (k + 2))","tactic":"rw [minSqFacAux]","premises":[{"full_name":"Nat.minSqFacAux","def_path":"Mathlib/Data/Nat/Squarefree.lean","def_pos":[113,4],"def_end_pos":[113,15]}]},{"state_before":"s : Finset ℕ\nm n✝ p n k : ℕ\nn0 : 0 < n\ni : ℕ\ne : k = 2 * i + 3\nih : ∀ (m : ℕ), Prime m → m ∣ n → k ≤ m\n⊢ n.MinSqFacProp\n    (if h : n < k * k then none\n    else\n      let_fun this := ⋯;\n      if k ∣ n then\n        let n' := n / k;\n        let_fun this := ⋯;\n        if k ∣ n' then some k else n'.minSqFacAux (k + 2)\n      else n.minSqFacAux (k + 2))","state_after":"case pos\ns : Finset ℕ\nm n✝ p n k : ℕ\nn0 : 0 < n\ni : ℕ\ne : k = 2 * i + 3\nih : ∀ (m : ℕ), Prime m → m ∣ n → k ≤ m\nh : n < k * k\n⊢ n.MinSqFacProp none\n\ncase neg\ns : Finset ℕ\nm n✝ p n k : ℕ\nn0 : 0 < n\ni : ℕ\ne : k = 2 * i + 3\nih : ∀ (m : ℕ), Prime m → m ∣ n → k ≤ m\nh : ¬n < k * k\n⊢ n.MinSqFacProp (if k ∣ n then if k ∣ n / k then some k else (n / k).minSqFacAux (k + 2) else n.minSqFacAux (k + 2))","tactic":"by_cases h : n < k * k <;> simp only [h, ↓reduceDIte]","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]},{"full_name":"reduceDIte","def_path":".lake/packages/lean4/src/lean/Lean/Meta/Tactic/Simp/BuiltinSimprocs/Core.lean","def_pos":[23,32],"def_end_pos":[23,42]}]},{"state_before":"case neg\ns : Finset ℕ\nm n✝ p n k : ℕ\nn0 : 0 < n\ni : ℕ\ne : k = 2 * i + 3\nih : ∀ (m : ℕ), Prime m → m ∣ n → k ≤ m\nh : ¬n < k * k\n⊢ n.MinSqFacProp (if k ∣ n then if k ∣ n / k then some k else (n / k).minSqFacAux (k + 2) else n.minSqFacAux (k + 2))","state_after":"case neg\ns : Finset ℕ\nm n✝ p n k : ℕ\nn0 : 0 < n\ni : ℕ\ne : k = 2 * i + 3\nih : ∀ (m : ℕ), Prime m → m ∣ n → k ≤ m\nh : ¬n < k * k\nk2 : 2 ≤ k\n⊢ n.MinSqFacProp (if k ∣ n then if k ∣ n / k then some k else (n / k).minSqFacAux (k + 2) else n.minSqFacAux (k + 2))","tactic":"have k2 : 2 ≤ k := by omega","premises":[]},{"state_before":"case neg\ns : Finset ℕ\nm n✝ p n k : ℕ\nn0 : 0 < n\ni : ℕ\ne : k = 2 * i + 3\nih : ∀ (m : ℕ), Prime m → m ∣ n → k ≤ m\nh : ¬n < k * k\nk2 : 2 ≤ k\n⊢ n.MinSqFacProp (if k ∣ n then if k ∣ n / k then some k else (n / k).minSqFacAux (k + 2) else n.minSqFacAux (k + 2))","state_after":"case neg\ns : Finset ℕ\nm n✝ p n k : ℕ\nn0 : 0 < n\ni : ℕ\ne : k = 2 * i + 3\nih : ∀ (m : ℕ), Prime m → m ∣ n → k ≤ m\nh : ¬n < k * k\nk2 : 2 ≤ k\nk0 : 0 < k\n⊢ n.MinSqFacProp (if k ∣ n then if k ∣ n / k then some k else (n / k).minSqFacAux (k + 2) else n.minSqFacAux (k + 2))","tactic":"have k0 : 0 < k := lt_of_lt_of_le (by decide) k2","premises":[{"full_name":"lt_of_lt_of_le","def_path":"Mathlib/Order/Defs.lean","def_pos":[93,8],"def_end_pos":[93,22]}]},{"state_before":"case neg\ns : Finset ℕ\nm n✝ p n k : ℕ\nn0 : 0 < n\ni : ℕ\ne : k = 2 * i + 3\nih : ∀ (m : ℕ), Prime m → m ∣ n → k ≤ m\nh : ¬n < k * k\nk2 : 2 ≤ k\nk0 : 0 < k\nIH : ∀ (n' : ℕ), n' ∣ n → ¬k ∣ n' → n'.MinSqFacProp (n'.minSqFacAux (k + 2))\n⊢ n.MinSqFacProp (if k ∣ n then if k ∣ n / k then some k else (n / k).minSqFacAux (k + 2) else n.minSqFacAux (k + 2))","state_after":"case neg\ns : Finset ℕ\nm n✝ p n k : ℕ\nn0 : 0 < n\ni : ℕ\ne : k = 2 * i + 3\nih : ∀ (m : ℕ), Prime m → m ∣ n → k ≤ m\nh : ¬n < k * k\nk2 : 2 ≤ k\nk0 : 0 < k\nIH : ∀ (n' : ℕ), n' ∣ n → ¬k ∣ n' → n'.MinSqFacProp (n'.minSqFacAux (k + 2))\npk : k ∣ n → Prime k\n⊢ n.MinSqFacProp (if k ∣ n then if k ∣ n / k then some k else (n / k).minSqFacAux (k + 2) else n.minSqFacAux (k + 2))","tactic":"have pk : k ∣ n → Prime k := by\n    refine fun dk => prime_def_minFac.2 ⟨k2, le_antisymm (minFac_le k0) ?_⟩\n    exact ih _ (minFac_prime (ne_of_gt k2)) (dvd_trans (minFac_dvd _) dk)","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Dvd.dvd","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1344,2],"def_end_pos":[1344,5]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Nat.Prime","def_path":"Mathlib/Data/Nat/Prime/Defs.lean","def_pos":[36,4],"def_end_pos":[36,9]},{"full_name":"Nat.minFac_dvd","def_path":"Mathlib/Data/Nat/Prime/Defs.lean","def_pos":[279,8],"def_end_pos":[279,18]},{"full_name":"Nat.minFac_le","def_path":"Mathlib/Data/Nat/Prime/Defs.lean","def_pos":[296,8],"def_end_pos":[296,17]},{"full_name":"Nat.minFac_prime","def_path":"Mathlib/Data/Nat/Prime/Defs.lean","def_pos":[282,8],"def_end_pos":[282,20]},{"full_name":"Nat.ne_of_gt","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[454,8],"def_end_pos":[454,16]},{"full_name":"Nat.prime_def_minFac","def_path":"Mathlib/Data/Nat/Prime/Defs.lean","def_pos":[311,8],"def_end_pos":[311,24]},{"full_name":"dvd_trans","def_path":"Mathlib/Algebra/Divisibility/Basic.lean","def_pos":[65,8],"def_end_pos":[65,17]},{"full_name":"le_antisymm","def_path":"Mathlib/Order/Defs.lean","def_pos":[156,8],"def_end_pos":[156,19]}]},{"state_before":"case neg\ns : Finset ℕ\nm n✝ p n k : ℕ\nn0 : 0 < n\ni : ℕ\ne : k = 2 * i + 3\nih : ∀ (m : ℕ), Prime m → m ∣ n → k ≤ m\nh : ¬n < k * k\nk2 : 2 ≤ k\nk0 : 0 < k\nIH : ∀ (n' : ℕ), n' ∣ n → ¬k ∣ n' → n'.MinSqFacProp (n'.minSqFacAux (k + 2))\npk : k ∣ n → Prime k\n⊢ n.MinSqFacProp (if k ∣ n then if k ∣ n / k then some k else (n / k).minSqFacAux (k + 2) else n.minSqFacAux (k + 2))","state_after":"case pos\ns : Finset ℕ\nm n✝ p n k : ℕ\nn0 : 0 < n\ni : ℕ\ne : k = 2 * i + 3\nih : ∀ (m : ℕ), Prime m → m ∣ n → k ≤ m\nh : ¬n < k * k\nk2 : 2 ≤ k\nk0 : 0 < k\nIH : ∀ (n' : ℕ), n' ∣ n → ¬k ∣ n' → n'.MinSqFacProp (n'.minSqFacAux (k + 2))\npk : k ∣ n → Prime k\ndk : k ∣ n\ndkk : k ∣ n / k\n⊢ n.MinSqFacProp (some k)\n\ncase neg\ns : Finset ℕ\nm n✝ p n k : ℕ\nn0 : 0 < n\ni : ℕ\ne : k = 2 * i + 3\nih : ∀ (m : ℕ), Prime m → m ∣ n → k ≤ m\nh : ¬n < k * k\nk2 : 2 ≤ k\nk0 : 0 < k\nIH : ∀ (n' : ℕ), n' ∣ n → ¬k ∣ n' → n'.MinSqFacProp (n'.minSqFacAux (k + 2))\npk : k ∣ n → Prime k\ndk : k ∣ n\ndkk : ¬k ∣ n / k\n⊢ n.MinSqFacProp ((n / k).minSqFacAux (k + 2))\n\ncase neg\ns : Finset ℕ\nm n✝ p n k : ℕ\nn0 : 0 < n\ni : ℕ\ne : k = 2 * i + 3\nih : ∀ (m : ℕ), Prime m → m ∣ n → k ≤ m\nh : ¬n < k * k\nk2 : 2 ≤ k\nk0 : 0 < k\nIH : ∀ (n' : ℕ), n' ∣ n → ¬k ∣ n' → n'.MinSqFacProp (n'.minSqFacAux (k + 2))\npk : k ∣ n → Prime k\ndk : ¬k ∣ n\n⊢ n.MinSqFacProp (n.minSqFacAux (k + 2))","tactic":"split_ifs with dk dkk","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
+{"url":"Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean","commit":"","full_name":"UniformOnFun.uniformContinuous_ofUniformFun","start":[775,0],"end":[779,51],"file_path":"Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\n⊢ UniformContinuous fun f => (ofFun 𝔖) (UniformFun.toFun f)","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\n⊢ ∀ i ∈ 𝔖,\n    ∀ i_2 ∈ 𝓤 β,\n      ∃ i_4 ∈ 𝓤 β,\n        ∀ ⦃x : (α →ᵤ β) × (α →ᵤ β)⦄,\n          x ∈ UniformFun.gen α β i_4 →\n            ((ofFun 𝔖) (UniformFun.toFun x.1), (ofFun 𝔖) (UniformFun.toFun x.2)) ∈ UniformOnFun.gen 𝔖 i i_2","tactic":"simp only [UniformContinuous, UniformOnFun.uniformity_eq, tendsto_iInf, tendsto_principal,\n    (UniformFun.hasBasis_uniformity _ _).eventually_iff]","premises":[{"full_name":"Filter.HasBasis.eventually_iff","def_path":"Mathlib/Order/Filter/Bases.lean","def_pos":[325,8],"def_end_pos":[325,31]},{"full_name":"Filter.tendsto_iInf","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2729,8],"def_end_pos":[2729,20]},{"full_name":"Filter.tendsto_principal","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2761,16],"def_end_pos":[2761,33]},{"full_name":"UniformContinuous","def_path":"Mathlib/Topology/UniformSpace/Basic.lean","def_pos":[944,4],"def_end_pos":[944,21]},{"full_name":"UniformFun.hasBasis_uniformity","def_path":"Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean","def_pos":[296,18],"def_end_pos":[296,37]},{"full_name":"UniformOnFun.uniformity_eq","def_path":"Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean","def_pos":[747,18],"def_end_pos":[747,31]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\n⊢ ∀ i ∈ 𝔖,\n    ∀ i_2 ∈ 𝓤 β,\n      ∃ i_4 ∈ 𝓤 β,\n        ∀ ⦃x : (α →ᵤ β) × (α →ᵤ β)⦄,\n          x ∈ UniformFun.gen α β i_4 →\n            ((ofFun 𝔖) (UniformFun.toFun x.1), (ofFun 𝔖) (UniformFun.toFun x.2)) ∈ UniformOnFun.gen 𝔖 i i_2","state_after":"no goals","tactic":"exact fun _ _ U hU ↦ ⟨U, hU, fun f hf x _ ↦ hf x⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]}]}
+{"url":"Mathlib/LinearAlgebra/Matrix/Basis.lean","commit":"","full_name":"basis_toMatrix_mul_linearMap_toMatrix_mul_basis_toMatrix","start":[204,0],"end":[208,83],"file_path":"Mathlib/LinearAlgebra/Matrix/Basis.lean","tactics":[{"state_before":"ι : Type u_1\nι' : Type u_2\nκ : Type u_3\nκ' : Type u_4\nR : Type u_5\nM : Type u_6\ninst✝¹³ : CommSemiring R\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : Module R M\nR₂ : Type u_7\nM₂ : Type u_8\ninst✝¹⁰ : CommRing R₂\ninst✝⁹ : AddCommGroup M₂\ninst✝⁸ : Module R₂ M₂\ne : Basis ι R M\nv : ι' → M\ni : ι\nj : ι'\nN : Type u_9\ninst✝⁷ : AddCommMonoid N\ninst✝⁶ : Module R N\nb : Basis ι R M\nb' : Basis ι' R M\nc : Basis κ R N\nc' : Basis κ' R N\nf : M →ₗ[R] N\ninst✝⁵ : Fintype ι'\ninst✝⁴ : Finite κ\ninst✝³ : Fintype ι\ninst✝² : Fintype κ'\ninst✝¹ : DecidableEq ι\ninst✝ : DecidableEq ι'\n⊢ c.toMatrix ⇑c' * (toMatrix b' c') f * b'.toMatrix ⇑b = (toMatrix b c) f","state_after":"case intro\nι : Type u_1\nι' : Type u_2\nκ : Type u_3\nκ' : Type u_4\nR : Type u_5\nM : Type u_6\ninst✝¹³ : CommSemiring R\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : Module R M\nR₂ : Type u_7\nM₂ : Type u_8\ninst✝¹⁰ : CommRing R₂\ninst✝⁹ : AddCommGroup M₂\ninst✝⁸ : Module R₂ M₂\ne : Basis ι R M\nv : ι' → M\ni : ι\nj : ι'\nN : Type u_9\ninst✝⁷ : AddCommMonoid N\ninst✝⁶ : Module R N\nb : Basis ι R M\nb' : Basis ι' R M\nc : Basis κ R N\nc' : Basis κ' R N\nf : M →ₗ[R] N\ninst✝⁵ : Fintype ι'\ninst✝⁴ : Finite κ\ninst✝³ : Fintype ι\ninst✝² : Fintype κ'\ninst✝¹ : DecidableEq ι\ninst✝ : DecidableEq ι'\nval✝ : Fintype κ\n⊢ c.toMatrix ⇑c' * (toMatrix b' c') f * b'.toMatrix ⇑b = (toMatrix b c) f","tactic":"cases nonempty_fintype κ","premises":[{"full_name":"nonempty_fintype","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[390,8],"def_end_pos":[390,24]}]},{"state_before":"case intro\nι : Type u_1\nι' : Type u_2\nκ : Type u_3\nκ' : Type u_4\nR : Type u_5\nM : Type u_6\ninst✝¹³ : CommSemiring R\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : Module R M\nR₂ : Type u_7\nM₂ : Type u_8\ninst✝¹⁰ : CommRing R₂\ninst✝⁹ : AddCommGroup M₂\ninst✝⁸ : Module R₂ M₂\ne : Basis ι R M\nv : ι' → M\ni : ι\nj : ι'\nN : Type u_9\ninst✝⁷ : AddCommMonoid N\ninst✝⁶ : Module R N\nb : Basis ι R M\nb' : Basis ι' R M\nc : Basis κ R N\nc' : Basis κ' R N\nf : M →ₗ[R] N\ninst✝⁵ : Fintype ι'\ninst✝⁴ : Finite κ\ninst✝³ : Fintype ι\ninst✝² : Fintype κ'\ninst✝¹ : DecidableEq ι\ninst✝ : DecidableEq ι'\nval✝ : Fintype κ\n⊢ c.toMatrix ⇑c' * (toMatrix b' c') f * b'.toMatrix ⇑b = (toMatrix b c) f","state_after":"no goals","tactic":"rw [basis_toMatrix_mul_linearMap_toMatrix, linearMap_toMatrix_mul_basis_toMatrix]","premises":[{"full_name":"basis_toMatrix_mul_linearMap_toMatrix","def_path":"Mathlib/LinearAlgebra/Matrix/Basis.lean","def_pos":[184,8],"def_end_pos":[184,45]},{"full_name":"linearMap_toMatrix_mul_basis_toMatrix","def_path":"Mathlib/LinearAlgebra/Matrix/Basis.lean","def_pos":[199,8],"def_end_pos":[199,45]}]}]}
+{"url":"Mathlib/LinearAlgebra/FreeProduct/Basic.lean","commit":"","full_name":"LinearAlgebra.FreeProduct.lift_apply","start":[207,0],"end":[226,28],"file_path":"Mathlib/LinearAlgebra/FreeProduct/Basic.lean","tactics":[{"state_before":"I : Type u\ninst✝⁵ : DecidableEq I\ni : I\nR : Type v\ninst✝⁴ : CommSemiring R\nA : I → Type w\ninst✝³ : (i : I) → Semiring (A i)\ninst✝² : (i : I) → Algebra R (A i)\nB : Type w'\ninst✝¹ : Semiring B\ninst✝ : Algebra R B\nmaps✝ maps : {i : I} → A i →ₐ[R] B\nx y : FreeTensorAlgebra R A\nr : rel R A x y\n⊢ ((TensorAlgebra.lift R) (toModule R I B fun x => maps.toLinearMap)) x =\n    ((TensorAlgebra.lift R) (toModule R I B fun x => maps.toLinearMap)) y","state_after":"no goals","tactic":"cases r with\n          | id => simp\n          | prod => simp","premises":[{"full_name":"LinearAlgebra.FreeProduct.rel.id","def_path":"Mathlib/LinearAlgebra/FreeProduct/Basic.lean","def_pos":[130,4],"def_end_pos":[130,6]},{"full_name":"LinearAlgebra.FreeProduct.rel.prod","def_path":"Mathlib/LinearAlgebra/FreeProduct/Basic.lean","def_pos":[131,4],"def_end_pos":[131,8]}]},{"state_before":"I : Type u\ninst✝⁵ : DecidableEq I\ni : I\nR : Type v\ninst✝⁴ : CommSemiring R\nA : I → Type w\ninst✝³ : (i : I) → Semiring (A i)\ninst✝² : (i : I) → Algebra R (A i)\nB : Type w'\ninst✝¹ : Semiring B\ninst✝ : Algebra R B\nmaps π : {i : I} → A i →ₐ[R] B\n⊢ (fun π i => π ∘ₐ ι R A i)\n      ((fun maps => (liftAlgHom R) ⟨(TensorAlgebra.lift R) (toModule R I B fun x => maps.toLinearMap), ⋯⟩) π) =\n    π","state_after":"case h.H\nI : Type u\ninst✝⁵ : DecidableEq I\ni✝ : I\nR : Type v\ninst✝⁴ : CommSemiring R\nA : I → Type w\ninst✝³ : (i : I) → Semiring (A i)\ninst✝² : (i : I) → Algebra R (A i)\nB : Type w'\ninst✝¹ : Semiring B\ninst✝ : Algebra R B\nmaps π : {i : I} → A i →ₐ[R] B\ni : I\naᵢ : A i\n⊢ ((fun π i => π ∘ₐ ι R A i)\n        ((fun maps => (liftAlgHom R) ⟨(TensorAlgebra.lift R) (toModule R I B fun x => maps.toLinearMap), ⋯⟩) π) i)\n      aᵢ =\n    π aᵢ","tactic":"ext i aᵢ","premises":[]},{"state_before":"case h.H\nI : Type u\ninst✝⁵ : DecidableEq I\ni✝ : I\nR : Type v\ninst✝⁴ : CommSemiring R\nA : I → Type w\ninst✝³ : (i : I) → Semiring (A i)\ninst✝² : (i : I) → Algebra R (A i)\nB : Type w'\ninst✝¹ : Semiring B\ninst✝ : Algebra R B\nmaps π : {i : I} → A i →ₐ[R] B\ni : I\naᵢ : A i\n⊢ ((fun π i => π ∘ₐ ι R A i)\n        ((fun maps => (liftAlgHom R) ⟨(TensorAlgebra.lift R) (toModule R I B fun x => maps.toLinearMap), ⋯⟩) π) i)\n      aᵢ =\n    π aᵢ","state_after":"no goals","tactic":"aesop (add simp [ι, ι'])","premises":[]},{"state_before":"I : Type u\ninst✝⁵ : DecidableEq I\ni : I\nR : Type v\ninst✝⁴ : CommSemiring R\nA : I → Type w\ninst✝³ : (i : I) → Semiring (A i)\ninst✝² : (i : I) → Algebra R (A i)\nB : Type w'\ninst✝¹ : Semiring B\ninst✝ : Algebra R B\nmaps✝ : {i : I} → A i →ₐ[R] B\nmaps : FreeProduct R A →ₐ[R] B\n⊢ (fun maps => (liftAlgHom R) ⟨(TensorAlgebra.lift R) (toModule R I B fun x => maps.toLinearMap), ⋯⟩)\n      ((fun π i => π ∘ₐ ι R A i) maps) =\n    maps","state_after":"case w.w.H.h\nI : Type u\ninst✝⁵ : DecidableEq I\ni✝ : I\nR : Type v\ninst✝⁴ : CommSemiring R\nA : I → Type w\ninst✝³ : (i : I) → Semiring (A i)\ninst✝² : (i : I) → Algebra R (A i)\nB : Type w'\ninst✝¹ : Semiring B\ninst✝ : Algebra R B\nmaps✝ : {i : I} → A i →ₐ[R] B\nmaps : FreeProduct R A →ₐ[R] B\ni : I\na : A i\n⊢ ((((fun maps => (liftAlgHom R) ⟨(TensorAlgebra.lift R) (toModule R I B fun x => maps.toLinearMap), ⋯⟩)\n                ((fun π i => π ∘ₐ ι R A i) maps) ∘ₐ\n              RingQuot.mkAlgHom R (rel R A)).toLinearMap ∘ₗ\n          TensorAlgebra.ι R) ∘ₗ\n        DirectSum.lof R I A i)\n      a =\n    (((maps ∘ₐ RingQuot.mkAlgHom R (rel R A)).toLinearMap ∘ₗ TensorAlgebra.ι R) ∘ₗ DirectSum.lof R I A i) a","tactic":"ext i a","premises":[]},{"state_before":"case w.w.H.h\nI : Type u\ninst✝⁵ : DecidableEq I\ni✝ : I\nR : Type v\ninst✝⁴ : CommSemiring R\nA : I → Type w\ninst✝³ : (i : I) → Semiring (A i)\ninst✝² : (i : I) → Algebra R (A i)\nB : Type w'\ninst✝¹ : Semiring B\ninst✝ : Algebra R B\nmaps✝ : {i : I} → A i →ₐ[R] B\nmaps : FreeProduct R A →ₐ[R] B\ni : I\na : A i\n⊢ ((((fun maps => (liftAlgHom R) ⟨(TensorAlgebra.lift R) (toModule R I B fun x => maps.toLinearMap), ⋯⟩)\n                ((fun π i => π ∘ₐ ι R A i) maps) ∘ₐ\n              RingQuot.mkAlgHom R (rel R A)).toLinearMap ∘ₗ\n          TensorAlgebra.ι R) ∘ₗ\n        DirectSum.lof R I A i)\n      a =\n    (((maps ∘ₐ RingQuot.mkAlgHom R (rel R A)).toLinearMap ∘ₗ TensorAlgebra.ι R) ∘ₗ DirectSum.lof R I A i) a","state_after":"no goals","tactic":"aesop (add simp [ι, ι'])","premises":[]}]}
+{"url":"Mathlib/MeasureTheory/Measure/WithDensity.lean","commit":"","full_name":"MeasureTheory.withDensity_tsum","start":[161,0],"end":[167,85],"file_path":"Mathlib/MeasureTheory/Measure/WithDensity.lean","tactics":[{"state_before":"α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nι : Type u_2\ninst✝ : Countable ι\nf : ι → α → ℝ≥0∞\nh : ∀ (i : ι), Measurable (f i)\n⊢ μ.withDensity (∑' (n : ι), f n) = sum fun n => μ.withDensity (f n)","state_after":"case h\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nι : Type u_2\ninst✝ : Countable ι\nf : ι → α → ℝ≥0∞\nh : ∀ (i : ι), Measurable (f i)\ns : Set α\nhs : MeasurableSet s\n⊢ (μ.withDensity (∑' (n : ι), f n)) s = (sum fun n => μ.withDensity (f n)) s","tactic":"ext1 s hs","premises":[]},{"state_before":"case h\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nι : Type u_2\ninst✝ : Countable ι\nf : ι → α → ℝ≥0∞\nh : ∀ (i : ι), Measurable (f i)\ns : Set α\nhs : MeasurableSet s\n⊢ (μ.withDensity (∑' (n : ι), f n)) s = (sum fun n => μ.withDensity (f n)) s","state_after":"case h\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nι : Type u_2\ninst✝ : Countable ι\nf : ι → α → ℝ≥0∞\nh : ∀ (i : ι), Measurable (f i)\ns : Set α\nhs : MeasurableSet s\n⊢ ∫⁻ (a : α) in s, tsum (fun n => f n) a ∂μ = ∑' (i : ι), ∫⁻ (a : α) in s, f i a ∂μ","tactic":"simp_rw [sum_apply _ hs, withDensity_apply _ hs]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"MeasureTheory.Measure.sum_apply","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","def_pos":[1334,8],"def_end_pos":[1334,17]},{"full_name":"MeasureTheory.withDensity_apply","def_path":"Mathlib/MeasureTheory/Measure/WithDensity.lean","def_pos":[38,8],"def_end_pos":[38,25]}]},{"state_before":"case h\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nι : Type u_2\ninst✝ : Countable ι\nf : ι → α → ℝ≥0∞\nh : ∀ (i : ι), Measurable (f i)\ns : Set α\nhs : MeasurableSet s\n⊢ ∫⁻ (a : α) in s, tsum (fun n => f n) a ∂μ = ∑' (i : ι), ∫⁻ (a : α) in s, f i a ∂μ","state_after":"case h\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nι : Type u_2\ninst✝ : Countable ι\nf : ι → α → ℝ≥0∞\nh : ∀ (i : ι), Measurable (f i)\ns : Set α\nhs : MeasurableSet s\n⊢ ∫⁻ (x : α) in s, tsum (fun n => f n) x ∂μ = ∑' (i : ι), ∫⁻ (x : α) in s, f i x ∂μ","tactic":"change ∫⁻ x in s, (∑' n, f n) x ∂μ = ∑' i, ∫⁻ x, f i x ∂μ.restrict s","premises":[{"full_name":"MeasureTheory.Measure.restrict","def_path":"Mathlib/MeasureTheory/Measure/Restrict.lean","def_pos":[43,18],"def_end_pos":[43,26]},{"full_name":"MeasureTheory.lintegral","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[59,16],"def_end_pos":[59,25]},{"full_name":"tsum","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Defs.lean","def_pos":[94,2],"def_end_pos":[94,13]}]},{"state_before":"case h\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nι : Type u_2\ninst✝ : Countable ι\nf : ι → α → ℝ≥0∞\nh : ∀ (i : ι), Measurable (f i)\ns : Set α\nhs : MeasurableSet s\n⊢ ∫⁻ (x : α) in s, tsum (fun n => f n) x ∂μ = ∑' (i : ι), ∫⁻ (x : α) in s, f i x ∂μ","state_after":"case h\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nι : Type u_2\ninst✝ : Countable ι\nf : ι → α → ℝ≥0∞\nh : ∀ (i : ι), Measurable (f i)\ns : Set α\nhs : MeasurableSet s\n⊢ ∫⁻ (x : α) in s, tsum (fun n => f n) x ∂μ = ∫⁻ (a : α) in s, ∑' (i : ι), f i a ∂μ","tactic":"rw [← lintegral_tsum fun i => (h i).aemeasurable]","premises":[{"full_name":"Measurable.aemeasurable","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[380,8],"def_end_pos":[380,31]},{"full_name":"MeasureTheory.lintegral_tsum","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[1259,8],"def_end_pos":[1259,22]}]},{"state_before":"case h\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nι : Type u_2\ninst✝ : Countable ι\nf : ι → α → ℝ≥0∞\nh : ∀ (i : ι), Measurable (f i)\ns : Set α\nhs : MeasurableSet s\n⊢ ∫⁻ (x : α) in s, tsum (fun n => f n) x ∂μ = ∫⁻ (a : α) in s, ∑' (i : ι), f i a ∂μ","state_after":"no goals","tactic":"exact lintegral_congr fun x => tsum_apply (Pi.summable.2 fun _ => ENNReal.summable)","premises":[{"full_name":"ENNReal.summable","def_path":"Mathlib/Topology/Instances/ENNReal.lean","def_pos":[705,18],"def_end_pos":[705,26]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"MeasureTheory.lintegral_congr","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[312,8],"def_end_pos":[312,23]},{"full_name":"Pi.summable","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean","def_pos":[205,2],"def_end_pos":[205,13]},{"full_name":"tsum_apply","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean","def_pos":[209,2],"def_end_pos":[209,13]}]}]}
+{"url":"Mathlib/Algebra/MvPolynomial/Degrees.lean","commit":"","full_name":"MvPolynomial.degrees_add_of_disjoint","start":[156,0],"end":[163,33],"file_path":"Mathlib/Algebra/MvPolynomial/Degrees.lean","tactics":[{"state_before":"R : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np✝ q✝ : MvPolynomial σ R\ninst✝ : DecidableEq σ\np q : MvPolynomial σ R\nh : p.degrees.Disjoint q.degrees\n⊢ (p + q).degrees = p.degrees ∪ q.degrees","state_after":"case a\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np✝ q✝ : MvPolynomial σ R\ninst✝ : DecidableEq σ\np q : MvPolynomial σ R\nh : p.degrees.Disjoint q.degrees\n⊢ (p + q).degrees ≤ p.degrees ∪ q.degrees\n\ncase a\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np✝ q✝ : MvPolynomial σ R\ninst✝ : DecidableEq σ\np q : MvPolynomial σ R\nh : p.degrees.Disjoint q.degrees\n⊢ p.degrees ∪ q.degrees ≤ (p + q).degrees","tactic":"apply le_antisymm","premises":[{"full_name":"le_antisymm","def_path":"Mathlib/Order/Defs.lean","def_pos":[156,8],"def_end_pos":[156,19]}]}]}
+{"url":"Mathlib/Topology/MetricSpace/Holder.lean","commit":"","full_name":"holderOnWith_one","start":[73,0],"end":[76,77],"file_path":"Mathlib/Topology/MetricSpace/Holder.lean","tactics":[{"state_before":"X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝² : PseudoEMetricSpace X\ninst✝¹ : PseudoEMetricSpace Y\ninst✝ : PseudoEMetricSpace Z\nC : ℝ≥0\nf : X → Y\ns : Set X\n⊢ HolderOnWith C 1 f s ↔ LipschitzOnWith C f s","state_after":"no goals","tactic":"simp only [HolderOnWith, LipschitzOnWith, NNReal.coe_one, ENNReal.rpow_one]","premises":[{"full_name":"ENNReal.rpow_one","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean","def_pos":[428,8],"def_end_pos":[428,16]},{"full_name":"HolderOnWith","def_path":"Mathlib/Topology/MetricSpace/Holder.lean","def_pos":[54,4],"def_end_pos":[54,16]},{"full_name":"LipschitzOnWith","def_path":"Mathlib/Topology/EMetricSpace/Lipschitz.lean","def_pos":[58,4],"def_end_pos":[58,19]},{"full_name":"NNReal.coe_one","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[157,25],"def_end_pos":[157,32]}]}]}
+{"url":"Mathlib/SetTheory/ZFC/Basic.lean","commit":"","full_name":"ZFSet.map_fval","start":[1498,0],"end":[1508,26],"file_path":"Mathlib/SetTheory/ZFC/Basic.lean","tactics":[{"state_before":"f : ZFSet → ZFSet\nH : PSet.Definable 1 f\nx y : ZFSet\nh : y ∈ x\nz : ZFSet\n⊢ Class.ToSet (fun x_1 => ↑(map f x) (x_1.pair z)) ↑y ↔ z = f y","state_after":"f : ZFSet → ZFSet\nH : PSet.Definable 1 f\nx y : ZFSet\nh : y ∈ x\nz : ZFSet\n⊢ (∃ z_1 ∈ x, z_1.pair (f z_1) = y.pair z) ↔ z = f y","tactic":"rw [Class.toSet_of_ZFSet, Class.coe_apply, mem_map]","premises":[{"full_name":"Class.coe_apply","def_path":"Mathlib/SetTheory/ZFC/Basic.lean","def_pos":[1364,8],"def_end_pos":[1364,17]},{"full_name":"Class.toSet_of_ZFSet","def_path":"Mathlib/SetTheory/ZFC/Basic.lean","def_pos":[1356,8],"def_end_pos":[1356,22]},{"full_name":"ZFSet.mem_map","def_path":"Mathlib/SetTheory/ZFC/Basic.lean","def_pos":[1155,8],"def_end_pos":[1155,15]}]},{"state_before":"f : ZFSet → ZFSet\nH : PSet.Definable 1 f\nx y : ZFSet\nh : y ∈ x\nz : ZFSet\n⊢ (∃ z_1 ∈ x, z_1.pair (f z_1) = y.pair z) ↔ z = f y","state_after":"no goals","tactic":"exact\n      ⟨fun ⟨w, _, pr⟩ => by\n        let ⟨wy, fw⟩ := ZFSet.pair_injective pr\n        rw [← fw, wy], fun e => by\n        subst e\n        exact ⟨_, h, rfl⟩⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"ZFSet.pair_injective","def_path":"Mathlib/SetTheory/ZFC/Basic.lean","def_pos":[1094,8],"def_end_pos":[1094,22]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
+{"url":"Mathlib/Order/PartialSups.lean","commit":"","full_name":"ciSup_partialSups_eq","start":[148,0],"end":[154,36],"file_path":"Mathlib/Order/PartialSups.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : ConditionallyCompleteLattice α\nf : ℕ → α\nh : BddAbove (Set.range f)\n⊢ ⨆ n, (partialSups f) n = ⨆ n, f n","state_after":"case refine_1\nα : Type u_1\ninst✝ : ConditionallyCompleteLattice α\nf : ℕ → α\nh : BddAbove (Set.range f)\nn : ℕ\n⊢ (partialSups f) n ≤ ⨆ n, f n\n\ncase refine_2\nα : Type u_1\ninst✝ : ConditionallyCompleteLattice α\nf : ℕ → α\nh : BddAbove (Set.range f)\n⊢ BddAbove (Set.range fun n => (partialSups f) n)","tactic":"refine (ciSup_le fun n => ?_).antisymm (ciSup_mono ?_ <| le_partialSups f)","premises":[{"full_name":"ciSup_le","def_path":"Mathlib/Order/ConditionallyCompleteLattice/Basic.lean","def_pos":[703,8],"def_end_pos":[703,16]},{"full_name":"ciSup_mono","def_path":"Mathlib/Order/ConditionallyCompleteLattice/Basic.lean","def_pos":[714,8],"def_end_pos":[714,18]},{"full_name":"le_partialSups","def_path":"Mathlib/Order/PartialSups.lean","def_pos":[70,8],"def_end_pos":[70,22]}]}]}
+{"url":"Mathlib/Data/Matroid/Map.lean","commit":"","full_name":"Matroid.map_id","start":[461,0],"end":[462,38],"file_path":"Mathlib/Data/Matroid/Map.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nf : α → β\nE I s : Set α\nM : Matroid α\nN : Matroid β\n⊢ M.map id ⋯ = M","state_after":"no goals","tactic":"simp [eq_iff_indep_iff_indep_forall]","premises":[{"full_name":"Matroid.eq_iff_indep_iff_indep_forall","def_path":"Mathlib/Data/Matroid/Basic.lean","def_pos":[669,8],"def_end_pos":[669,37]}]}]}
+{"url":"Mathlib/CategoryTheory/Localization/CalculusOfFractions/Fractions.lean","commit":"","full_name":"CategoryTheory.MorphismProperty.RightFraction₂.exists_leftFraction₂","start":[238,0],"end":[249,24],"file_path":"Mathlib/CategoryTheory/Localization/CalculusOfFractions/Fractions.lean","tactics":[{"state_before":"C : Type u_1\nD : Type u_2\ninst✝³ : Category.{u_3, u_1} C\ninst✝² : Category.{?u.15006, u_2} D\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\nX Y : C\nφ : W.RightFraction₂ X Y\ninst✝ : W.HasLeftCalculusOfFractions\n⊢ ∃ ψ, φ.f ≫ ψ.s = φ.s ≫ ψ.f ∧ φ.f' ≫ ψ.s = φ.s ≫ ψ.f'","state_after":"case intro\nC : Type u_1\nD : Type u_2\ninst✝³ : Category.{u_3, u_1} C\ninst✝² : Category.{?u.15006, u_2} D\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\nX Y : C\nφ : W.RightFraction₂ X Y\ninst✝ : W.HasLeftCalculusOfFractions\nψ₁ : W.LeftFraction X Y\nhψ₁ : φ.fst.f ≫ ψ₁.s = φ.fst.s ≫ ψ₁.f\n⊢ ∃ ψ, φ.f ≫ ψ.s = φ.s ≫ ψ.f ∧ φ.f' ≫ ψ.s = φ.s ≫ ψ.f'","tactic":"obtain ⟨ψ₁, hψ₁⟩ := φ.fst.exists_leftFraction","premises":[{"full_name":"CategoryTheory.MorphismProperty.RightFraction.exists_leftFraction","def_path":"Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean","def_pos":[197,6],"def_end_pos":[197,39]},{"full_name":"CategoryTheory.MorphismProperty.RightFraction₂.fst","def_path":"Mathlib/CategoryTheory/Localization/CalculusOfFractions/Fractions.lean","def_pos":[225,7],"def_end_pos":[225,10]}]},{"state_before":"case intro\nC : Type u_1\nD : Type u_2\ninst✝³ : Category.{u_3, u_1} C\ninst✝² : Category.{?u.15006, u_2} D\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\nX Y : C\nφ : W.RightFraction₂ X Y\ninst✝ : W.HasLeftCalculusOfFractions\nψ₁ : W.LeftFraction X Y\nhψ₁ : φ.fst.f ≫ ψ₁.s = φ.fst.s ≫ ψ₁.f\n⊢ ∃ ψ, φ.f ≫ ψ.s = φ.s ≫ ψ.f ∧ φ.f' ≫ ψ.s = φ.s ≫ ψ.f'","state_after":"case intro.intro\nC : Type u_1\nD : Type u_2\ninst✝³ : Category.{u_3, u_1} C\ninst✝² : Category.{?u.15006, u_2} D\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\nX Y : C\nφ : W.RightFraction₂ X Y\ninst✝ : W.HasLeftCalculusOfFractions\nψ₁ : W.LeftFraction X Y\nhψ₁ : φ.fst.f ≫ ψ₁.s = φ.fst.s ≫ ψ₁.f\nψ₂ : W.LeftFraction X Y\nhψ₂ : φ.snd.f ≫ ψ₂.s = φ.snd.s ≫ ψ₂.f\n⊢ ∃ ψ, φ.f ≫ ψ.s = φ.s ≫ ψ.f ∧ φ.f' ≫ ψ.s = φ.s ≫ ψ.f'","tactic":"obtain ⟨ψ₂, hψ₂⟩ := φ.snd.exists_leftFraction","premises":[{"full_name":"CategoryTheory.MorphismProperty.RightFraction.exists_leftFraction","def_path":"Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean","def_pos":[197,6],"def_end_pos":[197,39]},{"full_name":"CategoryTheory.MorphismProperty.RightFraction₂.snd","def_path":"Mathlib/CategoryTheory/Localization/CalculusOfFractions/Fractions.lean","def_pos":[232,7],"def_end_pos":[232,10]}]},{"state_before":"case intro.intro\nC : Type u_1\nD : Type u_2\ninst✝³ : Category.{u_3, u_1} C\ninst✝² : Category.{?u.15006, u_2} D\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\nX Y : C\nφ : W.RightFraction₂ X Y\ninst✝ : W.HasLeftCalculusOfFractions\nψ₁ : W.LeftFraction X Y\nhψ₁ : φ.fst.f ≫ ψ₁.s = φ.fst.s ≫ ψ₁.f\nψ₂ : W.LeftFraction X Y\nhψ₂ : φ.snd.f ≫ ψ₂.s = φ.snd.s ≫ ψ₂.f\n⊢ ∃ ψ, φ.f ≫ ψ.s = φ.s ≫ ψ.f ∧ φ.f' ≫ ψ.s = φ.s ≫ ψ.f'","state_after":"case intro.intro.intro\nC : Type u_1\nD : Type u_2\ninst✝³ : Category.{u_3, u_1} C\ninst✝² : Category.{?u.15006, u_2} D\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\nX Y : C\nφ : W.RightFraction₂ X Y\ninst✝ : W.HasLeftCalculusOfFractions\nψ₁ : W.LeftFraction X Y\nhψ₁ : φ.fst.f ≫ ψ₁.s = φ.fst.s ≫ ψ₁.f\nψ₂ : W.LeftFraction X Y\nhψ₂ : φ.snd.f ≫ ψ₂.s = φ.snd.s ≫ ψ₂.f\nα : W.LeftFraction ψ₁.Y' ψ₂.Y'\nhα : (RightFraction.mk ψ₁.s ⋯ ψ₂.s).f ≫ α.s = (RightFraction.mk ψ₁.s ⋯ ψ₂.s).s ≫ α.f\n⊢ ∃ ψ, φ.f ≫ ψ.s = φ.s ≫ ψ.f ∧ φ.f' ≫ ψ.s = φ.s ≫ ψ.f'","tactic":"obtain ⟨α, hα⟩ := (RightFraction.mk _ ψ₁.hs ψ₂.s).exists_leftFraction","premises":[{"full_name":"CategoryTheory.MorphismProperty.LeftFraction.hs","def_path":"Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean","def_pos":[47,2],"def_end_pos":[47,4]},{"full_name":"CategoryTheory.MorphismProperty.LeftFraction.s","def_path":"Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean","def_pos":[45,2],"def_end_pos":[45,3]},{"full_name":"CategoryTheory.MorphismProperty.RightFraction.exists_leftFraction","def_path":"Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean","def_pos":[197,6],"def_end_pos":[197,39]}]},{"state_before":"case intro.intro.intro\nC : Type u_1\nD : Type u_2\ninst✝³ : Category.{u_3, u_1} C\ninst✝² : Category.{?u.15006, u_2} D\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\nX Y : C\nφ : W.RightFraction₂ X Y\ninst✝ : W.HasLeftCalculusOfFractions\nψ₁ : W.LeftFraction X Y\nhψ₁ : φ.fst.f ≫ ψ₁.s = φ.fst.s ≫ ψ₁.f\nψ₂ : W.LeftFraction X Y\nhψ₂ : φ.snd.f ≫ ψ₂.s = φ.snd.s ≫ ψ₂.f\nα : W.LeftFraction ψ₁.Y' ψ₂.Y'\nhα : (RightFraction.mk ψ₁.s ⋯ ψ₂.s).f ≫ α.s = (RightFraction.mk ψ₁.s ⋯ ψ₂.s).s ≫ α.f\n⊢ ∃ ψ, φ.f ≫ ψ.s = φ.s ≫ ψ.f ∧ φ.f' ≫ ψ.s = φ.s ≫ ψ.f'","state_after":"case intro.intro.intro\nC : Type u_1\nD : Type u_2\ninst✝³ : Category.{u_3, u_1} C\ninst✝² : Category.{?u.15006, u_2} D\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\nX Y : C\nφ : W.RightFraction₂ X Y\ninst✝ : W.HasLeftCalculusOfFractions\nψ₁ : W.LeftFraction X Y\nhψ₁ : φ.f ≫ ψ₁.s = φ.s ≫ ψ₁.f\nψ₂ : W.LeftFraction X Y\nhψ₂ : φ.f' ≫ ψ₂.s = φ.s ≫ ψ₂.f\nα : W.LeftFraction ψ₁.Y' ψ₂.Y'\nhα : ψ₂.s ≫ α.s = ψ₁.s ≫ α.f\n⊢ ∃ ψ, φ.f ≫ ψ.s = φ.s ≫ ψ.f ∧ φ.f' ≫ ψ.s = φ.s ≫ ψ.f'","tactic":"dsimp at hψ₁ hψ₂ hα","premises":[]},{"state_before":"case intro.intro.intro\nC : Type u_1\nD : Type u_2\ninst✝³ : Category.{u_3, u_1} C\ninst✝² : Category.{?u.15006, u_2} D\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\nX Y : C\nφ : W.RightFraction₂ X Y\ninst✝ : W.HasLeftCalculusOfFractions\nψ₁ : W.LeftFraction X Y\nhψ₁ : φ.f ≫ ψ₁.s = φ.s ≫ ψ₁.f\nψ₂ : W.LeftFraction X Y\nhψ₂ : φ.f' ≫ ψ₂.s = φ.s ≫ ψ₂.f\nα : W.LeftFraction ψ₁.Y' ψ₂.Y'\nhα : ψ₂.s ≫ α.s = ψ₁.s ≫ α.f\n⊢ ∃ ψ, φ.f ≫ ψ.s = φ.s ≫ ψ.f ∧ φ.f' ≫ ψ.s = φ.s ≫ ψ.f'","state_after":"case intro.intro.intro.refine_1\nC : Type u_1\nD : Type u_2\ninst✝³ : Category.{u_3, u_1} C\ninst✝² : Category.{?u.15006, u_2} D\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\nX Y : C\nφ : W.RightFraction₂ X Y\ninst✝ : W.HasLeftCalculusOfFractions\nψ₁ : W.LeftFraction X Y\nhψ₁ : φ.f ≫ ψ₁.s = φ.s ≫ ψ₁.f\nψ₂ : W.LeftFraction X Y\nhψ₂ : φ.f' ≫ ψ₂.s = φ.s ≫ ψ₂.f\nα : W.LeftFraction ψ₁.Y' ψ₂.Y'\nhα : ψ₂.s ≫ α.s = ψ₁.s ≫ α.f\n⊢ φ.f ≫ (LeftFraction₂.mk (ψ₁.f ≫ α.f) (ψ₂.f ≫ α.s) (ψ₂.s ≫ α.s) ⋯).s =\n    φ.s ≫ (LeftFraction₂.mk (ψ₁.f ≫ α.f) (ψ₂.f ≫ α.s) (ψ₂.s ≫ α.s) ⋯).f\n\ncase intro.intro.intro.refine_2\nC : Type u_1\nD : Type u_2\ninst✝³ : Category.{u_3, u_1} C\ninst✝² : Category.{?u.15006, u_2} D\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\nX Y : C\nφ : W.RightFraction₂ X Y\ninst✝ : W.HasLeftCalculusOfFractions\nψ₁ : W.LeftFraction X Y\nhψ₁ : φ.f ≫ ψ₁.s = φ.s ≫ ψ₁.f\nψ₂ : W.LeftFraction X Y\nhψ₂ : φ.f' ≫ ψ₂.s = φ.s ≫ ψ₂.f\nα : W.LeftFraction ψ₁.Y' ψ₂.Y'\nhα : ψ₂.s ≫ α.s = ψ₁.s ≫ α.f\n⊢ φ.f' ≫ (LeftFraction₂.mk (ψ₁.f ≫ α.f) (ψ₂.f ≫ α.s) (ψ₂.s ≫ α.s) ⋯).s =\n    φ.s ≫ (LeftFraction₂.mk (ψ₁.f ≫ α.f) (ψ₂.f ≫ α.s) (ψ₂.s ≫ α.s) ⋯).f'","tactic":"refine ⟨LeftFraction₂.mk (ψ₁.f ≫ α.f) (ψ₂.f ≫ α.s) (ψ₂.s ≫ α.s)\n      (W.comp_mem _ _ ψ₂.hs α.hs), ?_, ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"CategoryTheory.CategoryStruct.comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]},{"full_name":"CategoryTheory.MorphismProperty.LeftFraction.f","def_path":"Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean","def_pos":[43,2],"def_end_pos":[43,3]},{"full_name":"CategoryTheory.MorphismProperty.LeftFraction.hs","def_path":"Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean","def_pos":[47,2],"def_end_pos":[47,4]},{"full_name":"CategoryTheory.MorphismProperty.LeftFraction.s","def_path":"Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean","def_pos":[45,2],"def_end_pos":[45,3]},{"full_name":"CategoryTheory.MorphismProperty.comp_mem","def_path":"Mathlib/CategoryTheory/MorphismProperty/Composition.lean","def_pos":[69,6],"def_end_pos":[69,14]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]}]}
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+{"url":"Mathlib/NumberTheory/FunctionField.lean","commit":"","full_name":"FunctionField.inftyValuation.C","start":[201,0],"end":[205,57],"file_path":"Mathlib/NumberTheory/FunctionField.lean","tactics":[{"state_before":"Fq F : Type\ninst✝² : Field Fq\ninst✝¹ : Field F\ninst✝ : DecidableEq (RatFunc Fq)\nk : Fq\nhk : k ≠ 0\n⊢ inftyValuationDef Fq (RatFunc.C k) = ↑(Multiplicative.ofAdd 0)","state_after":"Fq F : Type\ninst✝² : Field Fq\ninst✝¹ : Field F\ninst✝ : DecidableEq (RatFunc Fq)\nk : Fq\nhk : k ≠ 0\nhCk : RatFunc.C k ≠ 0\n⊢ inftyValuationDef Fq (RatFunc.C k) = ↑(Multiplicative.ofAdd 0)","tactic":"have hCk : RatFunc.C k ≠ 0 := (map_ne_zero _).mpr hk","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"RatFunc.C","def_path":"Mathlib/FieldTheory/RatFunc/AsPolynomial.lean","def_pos":[48,4],"def_end_pos":[48,5]},{"full_name":"map_ne_zero","def_path":"Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean","def_pos":[37,8],"def_end_pos":[37,19]}]},{"state_before":"Fq F : Type\ninst✝² : Field Fq\ninst✝¹ : Field F\ninst✝ : DecidableEq (RatFunc Fq)\nk : Fq\nhk : k ≠ 0\nhCk : RatFunc.C k ≠ 0\n⊢ inftyValuationDef Fq (RatFunc.C k) = ↑(Multiplicative.ofAdd 0)","state_after":"no goals","tactic":"rw [inftyValuationDef, if_neg hCk, RatFunc.intDegree_C]","premises":[{"full_name":"FunctionField.inftyValuationDef","def_path":"Mathlib/NumberTheory/FunctionField.lean","def_pos":[147,4],"def_end_pos":[147,21]},{"full_name":"RatFunc.intDegree_C","def_path":"Mathlib/FieldTheory/RatFunc/Degree.lean","def_pos":[51,8],"def_end_pos":[51,19]},{"full_name":"if_neg","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[937,8],"def_end_pos":[937,14]}]}]}
+{"url":"Mathlib/Analysis/Normed/Group/Hom.lean","commit":"","full_name":"NormedAddGroupHom.NormNoninc.zero","start":[730,0],"end":[730,75],"file_path":"Mathlib/Analysis/Normed/Group/Hom.lean","tactics":[{"state_before":"V : Type u_1\nW : Type u_2\nV₁ : Type u_3\nV₂ : Type u_4\nV₃ : Type u_5\ninst✝⁴ : SeminormedAddCommGroup V\ninst✝³ : SeminormedAddCommGroup W\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf : NormedAddGroupHom V W\nv : V₁\n⊢ ‖0 v‖ ≤ ‖v‖","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/CategoryTheory/Subobject/FactorThru.lean","commit":"","full_name":"CategoryTheory.Subobject.factorThru_zero","start":[142,0],"end":[144,65],"file_path":"Mathlib/CategoryTheory/Subobject/FactorThru.lean","tactics":[{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nX✝ Y✝ Z : C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : HasZeroMorphisms C\nX Y : C\nP : Subobject Y\nh : P.Factors 0\n⊢ P.factorThru 0 h = 0","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/RingTheory/Idempotents.lean","commit":"","full_name":"CompleteOrthogonalIdempotents.of_ker_isNilpotent_of_isMulCentral","start":[317,0],"end":[329,10],"file_path":"Mathlib/RingTheory/Idempotents.lean","tactics":[{"state_before":"R : Type u_1\nS : Type u_2\ninst✝² : Ring R\ninst✝¹ : Ring S\nf : R →+* S\nI : Type u_3\ne : I → R\nhe✝¹ : OrthogonalIdempotents e\ninst✝ : Fintype I\nhe✝ : CompleteOrthogonalIdempotents e\nh : ∀ x ∈ RingHom.ker f, IsNilpotent x\nhe : ∀ (i : I), IsIdempotentElem (e i)\nhe' : ∀ (i : I), IsMulCentral (e i)\nhe'' : CompleteOrthogonalIdempotents (⇑f ∘ e)\n⊢ CompleteOrthogonalIdempotents e","state_after":"case intro.intro\nR : Type u_1\nS : Type u_2\ninst✝² : Ring R\ninst✝¹ : Ring S\nf : R →+* S\nI : Type u_3\ne : I → R\nhe✝¹ : OrthogonalIdempotents e\ninst✝ : Fintype I\nhe✝ : CompleteOrthogonalIdempotents e\nh : ∀ x ∈ RingHom.ker f, IsNilpotent x\nhe : ∀ (i : I), IsIdempotentElem (e i)\nhe' : ∀ (i : I), IsMulCentral (e i)\nhe'' : CompleteOrthogonalIdempotents (⇑f ∘ e)\ne' : I → R\nh₁ : CompleteOrthogonalIdempotents e'\nh₂ : ⇑f ∘ e' = ⇑f ∘ e\n⊢ CompleteOrthogonalIdempotents e","tactic":"obtain ⟨e', h₁, h₂⟩ := lift_of_isNilpotent_ker f h he'' (fun _ ↦ ⟨_, rfl⟩)","premises":[{"full_name":"CompleteOrthogonalIdempotents.lift_of_isNilpotent_ker","def_path":"Mathlib/RingTheory/Idempotents.lean","def_pos":[294,6],"def_end_pos":[294,59]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"case intro.intro\nR : Type u_1\nS : Type u_2\ninst✝² : Ring R\ninst✝¹ : Ring S\nf : R →+* S\nI : Type u_3\ne : I → R\nhe✝¹ : OrthogonalIdempotents e\ninst✝ : Fintype I\nhe✝ : CompleteOrthogonalIdempotents e\nh : ∀ x ∈ RingHom.ker f, IsNilpotent x\nhe : ∀ (i : I), IsIdempotentElem (e i)\nhe' : ∀ (i : I), IsMulCentral (e i)\nhe'' : CompleteOrthogonalIdempotents (⇑f ∘ e)\ne' : I → R\nh₁ : CompleteOrthogonalIdempotents e'\nh₂ : ⇑f ∘ e' = ⇑f ∘ e\n⊢ CompleteOrthogonalIdempotents e","state_after":"case intro.intro\nR : Type u_1\nS : Type u_2\ninst✝² : Ring R\ninst✝¹ : Ring S\nf : R →+* S\nI : Type u_3\ne : I → R\nhe✝¹ : OrthogonalIdempotents e\ninst✝ : Fintype I\nhe✝ : CompleteOrthogonalIdempotents e\nh : ∀ x ∈ RingHom.ker f, IsNilpotent x\nhe : ∀ (i : I), IsIdempotentElem (e i)\nhe' : ∀ (i : I), IsMulCentral (e i)\nhe'' : CompleteOrthogonalIdempotents (⇑f ∘ e)\nh₁ : CompleteOrthogonalIdempotents e\nh₂ : ⇑f ∘ e = ⇑f ∘ e\n⊢ CompleteOrthogonalIdempotents e","tactic":"obtain rfl : e = e' := by\n    ext i\n    refine eq_of_isNilpotent_sub_of_isIdempotentElem_of_commute\n      (he _) (h₁.idem _) (h _ ?_) ((he' i).comm _)\n    simpa [RingHom.mem_ker, sub_eq_zero] using congr_fun h₂.symm i","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"IsMulCentral.comm","def_path":"Mathlib/Algebra/Group/Center.lean","def_pos":[60,2],"def_end_pos":[60,6]},{"full_name":"OrthogonalIdempotents.idem","def_path":"Mathlib/RingTheory/Idempotents.lean","def_pos":[111,2],"def_end_pos":[111,6]},{"full_name":"RingHom.mem_ker","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[524,8],"def_end_pos":[524,15]},{"full_name":"eq_of_isNilpotent_sub_of_isIdempotentElem_of_commute","def_path":"Mathlib/RingTheory/Idempotents.lean","def_pos":[303,8],"def_end_pos":[303,60]},{"full_name":"sub_eq_zero","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[738,2],"def_end_pos":[738,13]}]},{"state_before":"case intro.intro\nR : Type u_1\nS : Type u_2\ninst✝² : Ring R\ninst✝¹ : Ring S\nf : R →+* S\nI : Type u_3\ne : I → R\nhe✝¹ : OrthogonalIdempotents e\ninst✝ : Fintype I\nhe✝ : CompleteOrthogonalIdempotents e\nh : ∀ x ∈ RingHom.ker f, IsNilpotent x\nhe : ∀ (i : I), IsIdempotentElem (e i)\nhe' : ∀ (i : I), IsMulCentral (e i)\nhe'' : CompleteOrthogonalIdempotents (⇑f ∘ e)\nh₁ : CompleteOrthogonalIdempotents e\nh₂ : ⇑f ∘ e = ⇑f ∘ e\n⊢ CompleteOrthogonalIdempotents e","state_after":"no goals","tactic":"exact h₁","premises":[]}]}
+{"url":"Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean","commit":"","full_name":"MeasureTheory.ExistsSeqTendstoAe.exists_nat_measure_lt_two_inv","start":[133,0],"end":[137,100],"file_path":"Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean","tactics":[{"state_before":"α : Type u_1\nι : Type u_2\nE : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nn : ℕ\n⊢ ∃ N, ∀ m_1 ≥ N, μ {x | 2⁻¹ ^ n ≤ dist (f m_1 x) (g x)} ≤ 2⁻¹ ^ n","state_after":"α : Type u_1\nι : Type u_2\nE : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nn : ℕ\nhfg : Tendsto (fun i => μ {x | 2⁻¹ ^ n ≤ dist (f i x) (g x)}) atTop (𝓝 0)\n⊢ ∃ N, ∀ m_1 ≥ N, μ {x | 2⁻¹ ^ n ≤ dist (f m_1 x) (g x)} ≤ 2⁻¹ ^ n","tactic":"specialize hfg ((2⁻¹ : ℝ) ^ n) (by simp only [Real.rpow_natCast, inv_pos, zero_lt_two, pow_pos])","premises":[{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Real.rpow_natCast","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","def_pos":[63,8],"def_end_pos":[63,20]},{"full_name":"inv_pos","def_path":"Mathlib/Algebra/Order/Field/Unbundled/Basic.lean","def_pos":[23,14],"def_end_pos":[23,21]},{"full_name":"pow_pos","def_path":"Mathlib/Algebra/Order/Ring/Unbundled/Basic.lean","def_pos":[385,8],"def_end_pos":[385,15]},{"full_name":"zero_lt_two","def_path":"Mathlib/Algebra/Order/Monoid/NatCast.lean","def_pos":[62,14],"def_end_pos":[62,25]}]},{"state_before":"α : Type u_1\nι : Type u_2\nE : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nn : ℕ\nhfg : Tendsto (fun i => μ {x | 2⁻¹ ^ n ≤ dist (f i x) (g x)}) atTop (𝓝 0)\n⊢ ∃ N, ∀ m_1 ≥ N, μ {x | 2⁻¹ ^ n ≤ dist (f m_1 x) (g x)} ≤ 2⁻¹ ^ n","state_after":"α : Type u_1\nι : Type u_2\nE : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nn : ℕ\nhfg : ∀ ε > 0, ∃ N, ∀ n_1 ≥ N, μ {x | 2⁻¹ ^ n ≤ dist (f n_1 x) (g x)} ≤ ε\n⊢ ∃ N, ∀ m_1 ≥ N, μ {x | 2⁻¹ ^ n ≤ dist (f m_1 x) (g x)} ≤ 2⁻¹ ^ n","tactic":"rw [ENNReal.tendsto_atTop_zero] at hfg","premises":[{"full_name":"ENNReal.tendsto_atTop_zero","def_path":"Mathlib/Topology/Instances/ENNReal.lean","def_pos":[271,18],"def_end_pos":[271,36]}]},{"state_before":"α : Type u_1\nι : Type u_2\nE : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nn : ℕ\nhfg : ∀ ε > 0, ∃ N, ∀ n_1 ≥ N, μ {x | 2⁻¹ ^ n ≤ dist (f n_1 x) (g x)} ≤ ε\n⊢ ∃ N, ∀ m_1 ≥ N, μ {x | 2⁻¹ ^ n ≤ dist (f m_1 x) (g x)} ≤ 2⁻¹ ^ n","state_after":"no goals","tactic":"exact hfg ((2 : ℝ≥0∞)⁻¹ ^ n) (pos_iff_ne_zero.mpr fun h_zero => by simpa using pow_eq_zero h_zero)","premises":[{"full_name":"ENNReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[96,4],"def_end_pos":[96,11]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"pos_iff_ne_zero","def_path":"Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean","def_pos":[123,2],"def_end_pos":[123,13]},{"full_name":"pow_eq_zero","def_path":"Mathlib/Algebra/GroupWithZero/Basic.lean","def_pos":[162,6],"def_end_pos":[162,17]}]}]}
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+{"url":"Mathlib/Algebra/Group/Basic.lean","commit":"","full_name":"eq_iff_eq_of_div_eq_div","start":[759,0],"end":[761,34],"file_path":"Mathlib/Algebra/Group/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nG : Type u_3\nM : Type u_4\ninst✝ : Group G\na b c d : G\nn : ℤ\nH : a / b = c / d\n⊢ a = b ↔ c = d","state_after":"no goals","tactic":"rw [← div_eq_one, H, div_eq_one]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"div_eq_one","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[739,8],"def_end_pos":[739,18]}]}]}
+{"url":"Mathlib/Algebra/Order/Ring/Cast.lean","commit":"","full_name":"Int.cast_mono","start":[32,0],"end":[37,22],"file_path":"Mathlib/Algebra/Order/Ring/Cast.lean","tactics":[{"state_before":"R : Type u_1\ninst✝⁴ : AddCommGroupWithOne R\ninst✝³ : PartialOrder R\ninst✝² : CovariantClass R R (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : ZeroLEOneClass R\ninst✝ : NeZero 1\n⊢ Monotone Int.cast","state_after":"R : Type u_1\ninst✝⁴ : AddCommGroupWithOne R\ninst✝³ : PartialOrder R\ninst✝² : CovariantClass R R (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : ZeroLEOneClass R\ninst✝ : NeZero 1\nm n : ℤ\nh : m ≤ n\n⊢ ↑m ≤ ↑n","tactic":"intro m n h","premises":[]},{"state_before":"R : Type u_1\ninst✝⁴ : AddCommGroupWithOne R\ninst✝³ : PartialOrder R\ninst✝² : CovariantClass R R (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : ZeroLEOneClass R\ninst✝ : NeZero 1\nm n : ℤ\nh : m ≤ n\n⊢ ↑m ≤ ↑n","state_after":"R : Type u_1\ninst✝⁴ : AddCommGroupWithOne R\ninst✝³ : PartialOrder R\ninst✝² : CovariantClass R R (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : ZeroLEOneClass R\ninst✝ : NeZero 1\nm n : ℤ\nh : 0 ≤ n - m\n⊢ ↑m ≤ ↑n","tactic":"rw [← sub_nonneg] at h","premises":[{"full_name":"sub_nonneg","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Basic.lean","def_pos":[480,29],"def_end_pos":[480,39]}]},{"state_before":"R : Type u_1\ninst✝⁴ : AddCommGroupWithOne R\ninst✝³ : PartialOrder R\ninst✝² : CovariantClass R R (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : ZeroLEOneClass R\ninst✝ : NeZero 1\nm n : ℤ\nh : 0 ≤ n - m\n⊢ ↑m ≤ ↑n","state_after":"case intro\nR : Type u_1\ninst✝⁴ : AddCommGroupWithOne R\ninst✝³ : PartialOrder R\ninst✝² : CovariantClass R R (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : ZeroLEOneClass R\ninst✝ : NeZero 1\nm n : ℤ\nk : ℕ\nhk : ↑k = n - m\n⊢ ↑m ≤ ↑n","tactic":"lift n - m to ℕ using h with k hk","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]}]},{"state_before":"case intro\nR : Type u_1\ninst✝⁴ : AddCommGroupWithOne R\ninst✝³ : PartialOrder R\ninst✝² : CovariantClass R R (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : ZeroLEOneClass R\ninst✝ : NeZero 1\nm n : ℤ\nk : ℕ\nhk : ↑k = n - m\n⊢ ↑m ≤ ↑n","state_after":"case intro\nR : Type u_1\ninst✝⁴ : AddCommGroupWithOne R\ninst✝³ : PartialOrder R\ninst✝² : CovariantClass R R (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : ZeroLEOneClass R\ninst✝ : NeZero 1\nm n : ℤ\nk : ℕ\nhk : ↑k = n - m\n⊢ 0 ≤ ↑k","tactic":"rw [← sub_nonneg, ← cast_sub, ← hk, cast_natCast]","premises":[{"full_name":"Int.cast_natCast","def_path":"Mathlib/Data/Int/Cast/Basic.lean","def_pos":[59,8],"def_end_pos":[59,20]},{"full_name":"Int.cast_sub","def_path":"Mathlib/Data/Int/Cast/Basic.lean","def_pos":[107,8],"def_end_pos":[107,16]},{"full_name":"sub_nonneg","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Basic.lean","def_pos":[480,29],"def_end_pos":[480,39]}]},{"state_before":"case intro\nR : Type u_1\ninst✝⁴ : AddCommGroupWithOne R\ninst✝³ : PartialOrder R\ninst✝² : CovariantClass R R (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : ZeroLEOneClass R\ninst✝ : NeZero 1\nm n : ℤ\nk : ℕ\nhk : ↑k = n - m\n⊢ 0 ≤ ↑k","state_after":"no goals","tactic":"exact k.cast_nonneg'","premises":[{"full_name":"Nat.cast_nonneg'","def_path":"Mathlib/Data/Nat/Cast/Order/Basic.lean","def_pos":[44,8],"def_end_pos":[44,20]}]}]}
+{"url":"Mathlib/Algebra/Module/LocalizedModule.lean","commit":"","full_name":"LocalizedModule.divBy_apply","start":[468,0],"end":[486,10],"file_path":"Mathlib/Algebra/Module/LocalizedModule.lean","tactics":[{"state_before":"R : Type u\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nT : Type u_1\ninst✝² : CommSemiring T\ninst✝¹ : Algebra R T\ninst✝ : IsLocalization S T\ns : ↥S\np : LocalizedModule S M\nx✝² x✝¹ : M × ↥S\na : M\nb : ↥S\na' : M\nb' : ↥S\nx✝ : (a, b) ≈ (a', b')\nc : ↥S\neq1 : c • (a', b').2 • (a, b).1 = c • (a, b).2 • (a', b').1\n⊢ c • ((a', b').2 * s) • (a, b).1 = c • ((a, b).2 * s) • (a', b').1","state_after":"no goals","tactic":"rw [mul_smul, mul_smul, smul_comm _ s, smul_comm _ s, eq1, smul_comm _ s,\n        smul_comm _ s]","premises":[{"full_name":"MulAction.mul_smul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[99,2],"def_end_pos":[99,10]},{"full_name":"SMulCommClass.smul_comm","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[154,2],"def_end_pos":[154,11]}]},{"state_before":"R : Type u\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nT : Type u_1\ninst✝² : CommSemiring T\ninst✝¹ : Algebra R T\ninst✝ : IsLocalization S T\ns : ↥S\nx y : LocalizedModule S M\n⊢ (fun p => p.liftOn (fun p => mk p.1 (p.2 * s)) ⋯) (x + y) =\n    (fun p => p.liftOn (fun p => mk p.1 (p.2 * s)) ⋯) x + (fun p => p.liftOn (fun p => mk p.1 (p.2 * s)) ⋯) y","state_after":"R : Type u\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nT : Type u_1\ninst✝² : CommSemiring T\ninst✝¹ : Algebra R T\ninst✝ : IsLocalization S T\ns : ↥S\nx y : LocalizedModule S M\n⊢ ∀ (m m' : M) (s_1 s' : ↥S),\n    (fun p => p.liftOn (fun p => mk p.1 (p.2 * s)) ⋯) (mk m s_1 + mk m' s') =\n      (fun p => p.liftOn (fun p => mk p.1 (p.2 * s)) ⋯) (mk m s_1) +\n        (fun p => p.liftOn (fun p => mk p.1 (p.2 * s)) ⋯) (mk m' s')","tactic":"refine x.induction_on₂ ?_ y","premises":[{"full_name":"LocalizedModule.induction_on₂","def_path":"Mathlib/Algebra/Module/LocalizedModule.lean","def_pos":[98,8],"def_end_pos":[98,21]}]},{"state_before":"R : Type u\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nT : Type u_1\ninst✝² : CommSemiring T\ninst✝¹ : Algebra R T\ninst✝ : IsLocalization S T\ns : ↥S\nx y : LocalizedModule S M\n⊢ ∀ (m m' : M) (s_1 s' : ↥S),\n    (fun p => p.liftOn (fun p => mk p.1 (p.2 * s)) ⋯) (mk m s_1 + mk m' s') =\n      (fun p => p.liftOn (fun p => mk p.1 (p.2 * s)) ⋯) (mk m s_1) +\n        (fun p => p.liftOn (fun p => mk p.1 (p.2 * s)) ⋯) (mk m' s')","state_after":"R : Type u\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nT : Type u_1\ninst✝² : CommSemiring T\ninst✝¹ : Algebra R T\ninst✝ : IsLocalization S T\ns : ↥S\nx y : LocalizedModule S M\nm₁ m₂ : M\nt₁ t₂ : ↥S\n⊢ (fun p => p.liftOn (fun p => mk p.1 (p.2 * s)) ⋯) (mk m₁ t₁ + mk m₂ t₂) =\n    (fun p => p.liftOn (fun p => mk p.1 (p.2 * s)) ⋯) (mk m₁ t₁) +\n      (fun p => p.liftOn (fun p => mk p.1 (p.2 * s)) ⋯) (mk m₂ t₂)","tactic":"intro m₁ m₂ t₁ t₂","premises":[]},{"state_before":"R : Type u\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nT : Type u_1\ninst✝² : CommSemiring T\ninst✝¹ : Algebra R T\ninst✝ : IsLocalization S T\ns : ↥S\nx y : LocalizedModule S M\nm₁ m₂ : M\nt₁ t₂ : ↥S\n⊢ (fun p => p.liftOn (fun p => mk p.1 (p.2 * s)) ⋯) (mk m₁ t₁ + mk m₂ t₂) =\n    (fun p => p.liftOn (fun p => mk p.1 (p.2 * s)) ⋯) (mk m₁ t₁) +\n      (fun p => p.liftOn (fun p => mk p.1 (p.2 * s)) ⋯) (mk m₂ t₂)","state_after":"no goals","tactic":"simp_rw [mk_add_mk, LocalizedModule.liftOn_mk, mk_add_mk, mul_smul, mul_comm _ s, mul_assoc,\n      smul_comm _ s, ← smul_add, mul_left_comm s t₁ t₂, mk_cancel_common_left s]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"LocalizedModule.liftOn_mk","def_path":"Mathlib/Algebra/Module/LocalizedModule.lean","def_pos":[110,8],"def_end_pos":[110,17]},{"full_name":"LocalizedModule.mk_add_mk","def_path":"Mathlib/Algebra/Module/LocalizedModule.lean","def_pos":[151,8],"def_end_pos":[151,17]},{"full_name":"LocalizedModule.mk_cancel_common_left","def_path":"Mathlib/Algebra/Module/LocalizedModule.lean","def_pos":[385,8],"def_end_pos":[385,29]},{"full_name":"MulAction.mul_smul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[99,2],"def_end_pos":[99,10]},{"full_name":"SMulCommClass.smul_comm","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[154,2],"def_end_pos":[154,11]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]},{"full_name":"mul_left_comm","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[152,8],"def_end_pos":[152,21]},{"full_name":"smul_add","def_path":"Mathlib/Algebra/GroupWithZero/Action/Defs.lean","def_pos":[130,8],"def_end_pos":[130,16]}]},{"state_before":"R : Type u\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nT : Type u_1\ninst✝² : CommSemiring T\ninst✝¹ : Algebra R T\ninst✝ : IsLocalization S T\ns : ↥S\nr : R\nx : LocalizedModule S M\n⊢ { toFun := fun p => p.liftOn (fun p => mk p.1 (p.2 * s)) ⋯, map_add' := ⋯ }.toFun (r • x) =\n    (RingHom.id R) r • { toFun := fun p => p.liftOn (fun p => mk p.1 (p.2 * s)) ⋯, map_add' := ⋯ }.toFun x","state_after":"R : Type u\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nT : Type u_1\ninst✝² : CommSemiring T\ninst✝¹ : Algebra R T\ninst✝ : IsLocalization S T\ns : ↥S\nr : R\nx : LocalizedModule S M\nx✝¹ : M\nx✝ : ↥S\n⊢ { toFun := fun p => p.liftOn (fun p => mk p.1 (p.2 * s)) ⋯, map_add' := ⋯ }.toFun (r • mk x✝¹ x✝) =\n    (RingHom.id R) r • { toFun := fun p => p.liftOn (fun p => mk p.1 (p.2 * s)) ⋯, map_add' := ⋯ }.toFun (mk x✝¹ x✝)","tactic":"refine x.induction_on (fun _ _ ↦ ?_)","premises":[{"full_name":"LocalizedModule.induction_on","def_path":"Mathlib/Algebra/Module/LocalizedModule.lean","def_pos":[92,8],"def_end_pos":[92,20]}]},{"state_before":"R : Type u\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nT : Type u_1\ninst✝² : CommSemiring T\ninst✝¹ : Algebra R T\ninst✝ : IsLocalization S T\ns : ↥S\nr : R\nx : LocalizedModule S M\nx✝¹ : M\nx✝ : ↥S\n⊢ { toFun := fun p => p.liftOn (fun p => mk p.1 (p.2 * s)) ⋯, map_add' := ⋯ }.toFun (r • mk x✝¹ x✝) =\n    (RingHom.id R) r • { toFun := fun p => p.liftOn (fun p => mk p.1 (p.2 * s)) ⋯, map_add' := ⋯ }.toFun (mk x✝¹ x✝)","state_after":"R : Type u\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nT : Type u_1\ninst✝² : CommSemiring T\ninst✝¹ : Algebra R T\ninst✝ : IsLocalization S T\ns : ↥S\nr : R\nx : LocalizedModule S M\nx✝¹ : M\nx✝ : ↥S\n⊢ (r • mk x✝¹ x✝).liftOn (fun p => mk p.1 (p.2 * s)) ⋯ =\n    (RingHom.id R) r • (mk x✝¹ x✝).liftOn (fun p => mk p.1 (p.2 * s)) ⋯","tactic":"dsimp only","premises":[]},{"state_before":"R : Type u\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nT : Type u_1\ninst✝² : CommSemiring T\ninst✝¹ : Algebra R T\ninst✝ : IsLocalization S T\ns : ↥S\nr : R\nx : LocalizedModule S M\nx✝¹ : M\nx✝ : ↥S\n⊢ (r • mk x✝¹ x✝).liftOn (fun p => mk p.1 (p.2 * s)) ⋯ =\n    (RingHom.id R) r • (mk x✝¹ x✝).liftOn (fun p => mk p.1 (p.2 * s)) ⋯","state_after":"R : Type u\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nT : Type u_1\ninst✝² : CommSemiring T\ninst✝¹ : Algebra R T\ninst✝ : IsLocalization S T\ns : ↥S\nr : R\nx : LocalizedModule S M\nx✝¹ : M\nx✝ : ↥S\n⊢ (mk ((IsLocalization.sec S (↑(algebraMap R (Localization S)).toMonoidWithZeroHom r)).1 • (x✝¹, x✝).1)\n          ((IsLocalization.sec S (↑(algebraMap R (Localization S)).toMonoidWithZeroHom r)).2 * (x✝¹, x✝).2)).liftOn\n      (fun p => mk p.1 (p.2 * s)) ⋯ =\n    r • (mk x✝¹ x✝).liftOn (fun p => mk p.1 (p.2 * s)) ⋯","tactic":"change liftOn (mk _ _) _ _ = r • (liftOn (mk _ _) _ _)","premises":[{"full_name":"LocalizedModule.liftOn","def_path":"Mathlib/Algebra/Module/LocalizedModule.lean","def_pos":[106,4],"def_end_pos":[106,10]},{"full_name":"LocalizedModule.mk","def_path":"Mathlib/Algebra/Module/LocalizedModule.lean","def_pos":[85,4],"def_end_pos":[85,6]}]},{"state_before":"R : Type u\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nT : Type u_1\ninst✝² : CommSemiring T\ninst✝¹ : Algebra R T\ninst✝ : IsLocalization S T\ns : ↥S\nr : R\nx : LocalizedModule S M\nx✝¹ : M\nx✝ : ↥S\n⊢ (mk ((IsLocalization.sec S (↑(algebraMap R (Localization S)).toMonoidWithZeroHom r)).1 • (x✝¹, x✝).1)\n          ((IsLocalization.sec S (↑(algebraMap R (Localization S)).toMonoidWithZeroHom r)).2 * (x✝¹, x✝).2)).liftOn\n      (fun p => mk p.1 (p.2 * s)) ⋯ =\n    r • (mk x✝¹ x✝).liftOn (fun p => mk p.1 (p.2 * s)) ⋯","state_after":"R : Type u\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nT : Type u_1\ninst✝² : CommSemiring T\ninst✝¹ : Algebra R T\ninst✝ : IsLocalization S T\ns : ↥S\nr : R\nx : LocalizedModule S M\nx✝¹ : M\nx✝ : ↥S\n⊢ { toFun := (↑↑(algebraMap R (Localization S))).toFun, map_zero' := ⋯ } r • mk x✝¹ (x✝ * s) = r • mk x✝¹ (x✝ * s)","tactic":"simp_rw [liftOn_mk, mul_assoc, ← smul_def]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"LocalizedModule.liftOn_mk","def_path":"Mathlib/Algebra/Module/LocalizedModule.lean","def_pos":[110,8],"def_end_pos":[110,17]},{"full_name":"LocalizedModule.smul_def","def_path":"Mathlib/Algebra/Module/LocalizedModule.lean","def_pos":[316,8],"def_end_pos":[316,16]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]}]},{"state_before":"R : Type u\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nT : Type u_1\ninst✝² : CommSemiring T\ninst✝¹ : Algebra R T\ninst✝ : IsLocalization S T\ns : ↥S\nr : R\nx : LocalizedModule S M\nx✝¹ : M\nx✝ : ↥S\n⊢ { toFun := (↑↑(algebraMap R (Localization S))).toFun, map_zero' := ⋯ } r • mk x✝¹ (x✝ * s) = r • mk x✝�� (x✝ * s)","state_after":"no goals","tactic":"congr!","premises":[]}]}
+{"url":"Mathlib/AlgebraicGeometry/Pullbacks.lean","commit":"","full_name":"AlgebraicGeometry.Scheme.Pullback.gluing_t'","start":[193,0],"end":[212,38],"file_path":"Mathlib/AlgebraicGeometry/Pullbacks.lean","tactics":[{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\nX Y Z : Scheme\n𝒰 : X.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (i : 𝒰.J), HasPullback (𝒰.map i ≫ f) g\ni j k : 𝒰.J\n⊢ (fun i j k => t' 𝒰 f g i j k) i j k ≫\n      pullback.snd ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j k)\n        ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j i) =\n    pullback.fst ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i j)\n        ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i k) ≫\n      (fun i j => t 𝒰 f g i j) i j","state_after":"case h₀\nC : Type u\ninst✝¹ : Category.{v, u} C\nX Y Z : Scheme\n𝒰 : X.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (i : 𝒰.J), HasPullback (𝒰.map i ≫ f) g\ni j k : 𝒰.J\n⊢ ((fun i j k => t' 𝒰 f g i j k) i j k ≫\n        pullback.snd ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j k)\n          ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j i)) ≫\n      pullback.fst (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i) =\n    (pullback.fst ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i j)\n          ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i k) ≫\n        (fun i j => t 𝒰 f g i j) i j) ≫\n      pullback.fst (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)\n\ncase h₁\nC : Type u\ninst✝¹ : Category.{v, u} C\nX Y Z : Scheme\n𝒰 : X.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (i : 𝒰.J), HasPullback (𝒰.map i ≫ f) g\ni j k : 𝒰.J\n⊢ ((fun i j k => t' 𝒰 f g i j k) i j k ≫\n        pullback.snd ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j k)\n          ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j i)) ≫\n      pullback.snd (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i) =\n    (pullback.fst ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i j)\n          ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i k) ≫\n        (fun i j => t 𝒰 f g i j) i j) ≫\n      pullback.snd (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)","tactic":"apply pullback.hom_ext","premises":[{"full_name":"CategoryTheory.Limits.pullback.hom_ext","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Pullback/HasPullback.lean","def_pos":[212,8],"def_end_pos":[212,24]}]},{"state_before":"case h₀\nC : Type u\ninst✝¹ : Category.{v, u} C\nX Y Z : Scheme\n𝒰 : X.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (i : 𝒰.J), HasPullback (𝒰.map i ≫ f) g\ni j k : 𝒰.J\n⊢ ((fun i j k => t' 𝒰 f g i j k) i j k ≫\n        pullback.snd ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j k)\n          ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j i)) ≫\n      pullback.fst (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i) =\n    (pullback.fst ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i j)\n          ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i k) ≫\n        (fun i j => t 𝒰 f g i j) i j) ≫\n      pullback.fst (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)\n\ncase h₁\nC : Type u\ninst✝¹ : Category.{v, u} C\nX Y Z : Scheme\n𝒰 : X.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (i : 𝒰.J), HasPullback (𝒰.map i ≫ f) g\ni j k : 𝒰.J\n⊢ ((fun i j k => t' 𝒰 f g i j k) i j k ≫\n        pullback.snd ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j k)\n          ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j i)) ≫\n      pullback.snd (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i) =\n    (pullback.fst ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i j)\n          ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i k) ≫\n        (fun i j => t 𝒰 f g i j) i j) ≫\n      pullback.snd (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)","state_after":"case h₀.h₀\nC : Type u\ninst✝¹ : Category.{v, u} C\nX Y Z : Scheme\n𝒰 : X.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (i : 𝒰.J), HasPullback (���.map i ≫ f) g\ni j k : 𝒰.J\n⊢ (((fun i j k => t' 𝒰 f g i j k) i j k ≫\n          pullback.snd ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j k)\n            ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j i)) ≫\n        pullback.fst (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)) ≫\n      pullback.fst (𝒰.map j ≫ f) g =\n    ((pullback.fst ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i j)\n            ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i k) ≫\n          (fun i j => t 𝒰 f g i j) i j) ≫\n        pullback.fst (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)) ≫\n      pullback.fst (𝒰.map j ≫ f) g\n\ncase h₀.h₁\nC : Type u\ninst✝¹ : Category.{v, u} C\nX Y Z : Scheme\n𝒰 : X.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (i : 𝒰.J), HasPullback (𝒰.map i ≫ f) g\ni j k : 𝒰.J\n⊢ (((fun i j k => t' 𝒰 f g i j k) i j k ≫\n          pullback.snd ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j k)\n            ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j i)) ≫\n        pullback.fst (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)) ≫\n      pullback.snd (𝒰.map j ≫ f) g =\n    ((pullback.fst ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i j)\n            ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i k) ≫\n          (fun i j => t 𝒰 f g i j) i j) ≫\n        pullback.fst (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)) ≫\n      pullback.snd (𝒰.map j ≫ f) g\n\ncase h₁\nC : Type u\ninst✝¹ : Category.{v, u} C\nX Y Z : Scheme\n𝒰 : X.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (i : 𝒰.J), HasPullback (𝒰.map i ≫ f) g\ni j k : 𝒰.J\n⊢ ((fun i j k => t' 𝒰 f g i j k) i j k ≫\n        pullback.snd ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j k)\n          ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j i)) ≫\n      pullback.snd (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i) =\n    (pullback.fst ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i j)\n          ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i k) ≫\n        (fun i j => t 𝒰 f g i j) i j) ≫\n      pullback.snd (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)","tactic":"on_goal 1 => apply pullback.hom_ext","premises":[{"full_name":"CategoryTheory.Limits.pullback.hom_ext","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Pullback/HasPullback.lean","def_pos":[212,8],"def_end_pos":[212,24]}]},{"state_before":"case h₀.h₀\nC : Type u\ninst✝¹ : Category.{v, u} C\nX Y Z : Scheme\n𝒰 : X.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (i : 𝒰.J), HasPullback (𝒰.map i ≫ f) g\ni j k : 𝒰.J\n⊢ (((fun i j k => t' 𝒰 f g i j k) i j k ≫\n          pullback.snd ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j k)\n            ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j i)) ≫\n        pullback.fst (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)) ≫\n      pullback.fst (𝒰.map j ≫ f) g =\n    ((pullback.fst ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i j)\n            ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i k) ≫\n          (fun i j => t 𝒰 f g i j) i j) ≫\n        pullback.fst (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)) ≫\n      pullback.fst (𝒰.map j ≫ f) g\n\ncase h₀.h₁\nC : Type u\ninst✝¹ : Category.{v, u} C\nX Y Z : Scheme\n𝒰 : X.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (i : 𝒰.J), HasPullback (𝒰.map i ≫ f) g\ni j k : 𝒰.J\n⊢ (((fun i j k => t' 𝒰 f g i j k) i j k ≫\n          pullback.snd ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j k)\n            ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j i)) ≫\n        pullback.fst (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)) ≫\n      pullback.snd (𝒰.map j ≫ f) g =\n    ((pullback.fst ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i j)\n            ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i k) ≫\n          (fun i j => t 𝒰 f g i j) i j) ≫\n        pullback.fst (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)) ≫\n      pullback.snd (𝒰.map j ≫ f) g\n\ncase h���\nC : Type u\ninst✝¹ : Category.{v, u} C\nX Y Z : Scheme\n𝒰 : X.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (i : 𝒰.J), HasPullback (𝒰.map i ≫ f) g\ni j k : 𝒰.J\n⊢ ((fun i j k => t' 𝒰 f g i j k) i j k ≫\n        pullback.snd ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j k)\n          ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j i)) ≫\n      pullback.snd (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i) =\n    (pullback.fst ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i j)\n          ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i k) ≫\n        (fun i j => t 𝒰 f g i j) i j) ≫\n      pullback.snd (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)","state_after":"no goals","tactic":"all_goals\n      simp only [t'_snd_fst_fst, t'_snd_fst_snd, t'_snd_snd, t_fst_fst, t_fst_snd, t_snd,\n        Category.assoc]","premises":[{"full_name":"AlgebraicGeometry.Scheme.Pullback.t'_snd_fst_fst","def_path":"Mathlib/AlgebraicGeometry/Pullbacks.lean","def_pos":[125,8],"def_end_pos":[125,22]},{"full_name":"AlgebraicGeometry.Scheme.Pullback.t'_snd_fst_snd","def_path":"Mathlib/AlgebraicGeometry/Pullbacks.lean","def_pos":[133,8],"def_end_pos":[133,22]},{"full_name":"AlgebraicGeometry.Scheme.Pullback.t'_snd_snd","def_path":"Mathlib/AlgebraicGeometry/Pullbacks.lean","def_pos":[141,8],"def_end_pos":[141,18]},{"full_name":"AlgebraicGeometry.Scheme.Pullback.t_fst_fst","def_path":"Mathlib/AlgebraicGeometry/Pullbacks.lean","def_pos":[59,8],"def_end_pos":[59,17]},{"full_name":"AlgebraicGeometry.Scheme.Pullback.t_fst_snd","def_path":"Mathlib/AlgebraicGeometry/Pullbacks.lean","def_pos":[66,8],"def_end_pos":[66,17]},{"full_name":"AlgebraicGeometry.Scheme.Pullback.t_snd","def_path":"Mathlib/AlgebraicGeometry/Pullbacks.lean","def_pos":[72,8],"def_end_pos":[72,13]},{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]}]}]}
+{"url":"Mathlib/AlgebraicTopology/SimplicialSet.lean","commit":"","full_name":"SSet.horn.edge_coe","start":[207,0],"end":[224,23],"file_path":"Mathlib/AlgebraicTopology/SimplicialSet.lean","tactics":[{"state_before":"n : ℕ\ni a b : Fin (n + 1)\nhab : a ≤ b\nH : {i, a, b}.card ≤ n\n⊢ Λ[n, i] _[1]","state_after":"case range\nn : ℕ\ni a b : Fin (n + 1)\nhab : a ≤ b\nH : {i, a, b}.card ≤ n\n⊢ Set.range ⇑(asOrderHom (standardSimplex.edge n a b hab)) ∪ {i} ≠ Set.univ","tactic":"refine ⟨standardSimplex.edge n a b hab, ?range⟩","premises":[{"full_name":"SSet.standardSimplex.edge","def_path":"Mathlib/AlgebraicTopology/SimplicialSet.lean","def_pos":[120,4],"def_end_pos":[120,8]}]},{"state_before":"case range\nn : ℕ\ni a b : Fin (n + 1)\nhab : a ≤ b\nH : {i, a, b}.card ≤ n\n⊢ Set.range ⇑(asOrderHom (standardSimplex.edge n a b hab)) ∪ {i} ≠ Set.univ","state_after":"no goals","tactic":"case range =>\n    suffices ∃ x, ¬i = x ∧ ¬a = x ∧ ¬b = x by\n      simpa only [unop_op, SimplexCategory.len_mk, asOrderHom, SimplexCategory.Hom.toOrderHom_mk,\n        Set.union_singleton, ne_eq, ← Set.univ_subset_iff, Set.subset_def, Set.mem_univ,\n        Set.mem_insert_iff, @eq_comm _ _ i, Set.mem_range, forall_true_left, not_forall, not_or,\n        not_exists, Fin.forall_fin_two]\n    contrapose! H\n    replace H : univ ⊆ {i, a, b} :=\n      fun x _ ↦ by simpa [or_iff_not_imp_left, eq_comm] using H x\n    replace H := card_le_card H\n    rwa [card_fin] at H","premises":[{"full_name":"And","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[516,10],"def_end_pos":[516,13]},{"full_name":"Classical.not_forall","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[137,20],"def_end_pos":[137,30]},{"full_name":"Classical.or_iff_not_imp_left","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[147,8],"def_end_pos":[147,27]},{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"Fin.forall_fin_two","def_path":".lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean","def_pos":[681,8],"def_end_pos":[681,22]},{"full_name":"Finset.card_fin","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[291,8],"def_end_pos":[291,23]},{"full_name":"Finset.card_le_card","def_path":"Mathlib/Data/Finset/Card.lean","def_pos":[57,8],"def_end_pos":[57,20]},{"full_name":"Finset.univ","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[67,4],"def_end_pos":[67,8]},{"full_name":"HasSubset.Subset","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[384,2],"def_end_pos":[384,8]},{"full_name":"Insert.insert","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[458,2],"def_end_pos":[458,8]},{"full_name":"Mathlib.Tactic.Contrapose.mtr","def_path":"Mathlib/Tactic/Contrapose.lean","def_pos":[24,6],"def_end_pos":[24,9]},{"full_name":"Not","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[224,4],"def_end_pos":[224,7]},{"full_name":"Opposite.unop_op","def_path":"Mathlib/Data/Opposite.lean","def_pos":[64,8],"def_end_pos":[64,15]},{"full_name":"SSet.asOrderHom","def_path":"Mathlib/AlgebraicTopology/SimplicialSet.lean","def_pos":[148,4],"def_end_pos":[148,14]},{"full_name":"Set.mem_insert_iff","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[895,8],"def_end_pos":[895,22]},{"full_name":"Set.mem_range","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[146,16],"def_end_pos":[146,25]},{"full_name":"Set.mem_univ","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[80,28],"def_end_pos":[80,36]},{"full_name":"Set.subset_def","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[283,8],"def_end_pos":[283,18]},{"full_name":"Set.union_singleton","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1060,8],"def_end_pos":[1060,23]},{"full_name":"Set.univ_subset_iff","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[553,8],"def_end_pos":[553,23]},{"full_name":"SimplexCategory.Hom.toOrderHom_mk","def_path":"Mathlib/AlgebraicTopology/SimplexCategory.lean","def_pos":[111,8],"def_end_pos":[111,21]},{"full_name":"SimplexCategory.len_mk","def_path":"Mathlib/AlgebraicTopology/SimplexCategory.lean","def_pos":[73,8],"def_end_pos":[73,14]},{"full_name":"Singleton.singleton","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[467,2],"def_end_pos":[467,11]},{"full_name":"eq_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[811,8],"def_end_pos":[811,15]},{"full_name":"forall_true_left","def_path":"Mathlib/Logic/Basic.lean","def_pos":[692,16],"def_end_pos":[692,32]},{"full_name":"ne_eq","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[89,16],"def_end_pos":[89,21]},{"full_name":"not_exists","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[254,16],"def_end_pos":[254,26]},{"full_name":"not_or","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[134,16],"def_end_pos":[134,22]}]}]}
+{"url":"Mathlib/GroupTheory/NoncommPiCoprod.lean","commit":"","full_name":"MonoidHom.injective_noncommPiCoprod_of_independent","start":[192,0],"end":[206,37],"file_path":"Mathlib/GroupTheory/NoncommPiCoprod.lean","tactics":[{"state_before":"G : Type u_1\ninst✝¹ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝ : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf g : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\nhind : CompleteLattice.Independent fun i => (ϕ i).range\nhinj : ∀ (i : ι), Function.Injective ⇑(ϕ i)\n⊢ Function.Injective ⇑(noncommPiCoprod ϕ hcomm)","state_after":"no goals","tactic":"classical\n    apply (MonoidHom.ker_eq_bot_iff _).mp\n    rw [eq_bot_iff]\n    intro f heq1\n    have : ∀ i, i ∈ Finset.univ → ϕ i (f i) = 1 :=\n      Subgroup.eq_one_of_noncommProd_eq_one_of_independent _ _ (fun _ _ _ _ h => hcomm h _ _)\n        _ hind (by simp) heq1\n    ext i\n    apply hinj\n    simp [this i (Finset.mem_univ i)]","premises":[{"full_name":"Finset.mem_univ","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[71,8],"def_end_pos":[71,16]},{"full_name":"Finset.univ","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[67,4],"def_end_pos":[67,8]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"MonoidHom.ker_eq_bot_iff","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[2150,8],"def_end_pos":[2150,22]},{"full_name":"Subgroup.eq_one_of_noncommProd_eq_one_of_independent","def_path":"Mathlib/GroupTheory/NoncommPiCoprod.lean","def_pos":[51,8],"def_end_pos":[51,51]},{"full_name":"eq_bot_iff","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[282,8],"def_end_pos":[282,18]}]}]}
+{"url":"Mathlib/Analysis/Calculus/Deriv/Slope.lean","commit":"","full_name":"isSeparable_range_derivWithin","start":[138,0],"end":[144,60],"file_path":"Mathlib/Analysis/Calculus/Deriv/Slope.lean","tactics":[{"state_before":"𝕜 : Type u\ninst✝⁵ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\nE : Type w\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\nf✝ f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns✝ t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\ninst✝ : SeparableSpace 𝕜\nf : 𝕜 → F\ns : Set 𝕜\n⊢ IsSeparable (range (derivWithin f s))","state_after":"case intro.intro.intro\n𝕜 : Type u\ninst✝⁵ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\nE : Type w\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\nf✝ f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns✝ t✝ : Set 𝕜\nL L₁ L₂ : Filter 𝕜\ninst✝ : SeparableSpace 𝕜\nf : 𝕜 → F\ns t : Set 𝕜\nts : t ⊆ s\nt_count : t.Countable\nht : s ⊆ closure t\n⊢ IsSeparable (range (derivWithin f s))","tactic":"obtain ⟨t, ts, t_count, ht⟩ : ∃ t, t ⊆ s ∧ Set.Countable t ∧ s ⊆ closure t :=\n    (IsSeparable.of_separableSpace s).exists_countable_dense_subset","premises":[{"full_name":"And","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[516,10],"def_end_pos":[516,13]},{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"HasSubset.Subset","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[384,2],"def_end_pos":[384,8]},{"full_name":"Set.Countable","def_path":"Mathlib/Data/Set/Countable.lean","def_pos":[43,14],"def_end_pos":[43,23]},{"full_name":"TopologicalSpace.IsSeparable.exists_countable_dense_subset","def_path":"Mathlib/Topology/EMetricSpace/Basic.lean","def_pos":[722,8],"def_end_pos":[722,73]},{"full_name":"TopologicalSpace.IsSeparable.of_separableSpace","def_path":"Mathlib/Topology/Bases.lean","def_pos":[526,8],"def_end_pos":[526,37]},{"full_name":"closure","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[112,4],"def_end_pos":[112,11]}]},{"state_before":"case intro.intro.intro\n𝕜 : Type u\ninst✝⁵ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\nE : Type w\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\nf✝ f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns✝ t✝ : Set 𝕜\nL L₁ L₂ : Filter 𝕜\ninst✝ : SeparableSpace 𝕜\nf : 𝕜 → F\ns t : Set 𝕜\nts : t ⊆ s\nt_count : t.Countable\nht : s ⊆ closure t\n⊢ IsSeparable (range (derivWithin f s))","state_after":"case intro.intro.intro\n𝕜 : Type u\ninst✝⁵ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\nE : Type w\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\nf✝ f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns✝ t✝ : Set 𝕜\nL L₁ L₂ : Filter 𝕜\ninst✝ : SeparableSpace 𝕜\nf : 𝕜 → F\ns t : Set 𝕜\nts : t ⊆ s\nt_count : t.Countable\nht : s ⊆ closure t\nthis : s ⊆ closure (s ∩ t)\n⊢ IsSeparable (range (derivWithin f s))","tactic":"have : s ⊆ closure (s ∩ t) := by rwa [inter_eq_self_of_subset_right ts]","premises":[{"full_name":"HasSubset.Subset","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[384,2],"def_end_pos":[384,8]},{"full_name":"Inter.inter","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[407,2],"def_end_pos":[407,7]},{"full_name":"Set.inter_eq_self_of_subset_right","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[784,8],"def_end_pos":[784,37]},{"full_name":"closure","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[112,4],"def_end_pos":[112,11]}]},{"state_before":"case intro.intro.intro\n𝕜 : Type u\ninst✝⁵ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\nE : Type w\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\nf✝ f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns✝ t✝ : Set 𝕜\nL L₁ L₂ : Filter 𝕜\ninst✝ : SeparableSpace 𝕜\nf : 𝕜 → F\ns t : Set 𝕜\nts : t ⊆ s\nt_count : t.Countable\nht : s ⊆ closure t\nthis : s ⊆ closure (s ∩ t)\n⊢ IsSeparable (range (derivWithin f s))","state_after":"𝕜 : Type u\ninst✝⁵ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\nE : Type w\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\nf✝ f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns✝ t✝ : Set 𝕜\nL L₁ L₂ : Filter 𝕜\ninst✝ : SeparableSpace 𝕜\nf : 𝕜 → F\ns t : Set 𝕜\nts : t ⊆ s\nt_count : t.Countable\nht : s ⊆ closure t\nthis : s ⊆ closure (s ∩ t)\n⊢ IsSeparable (closure ↑(Submodule.span 𝕜 (f '' t)))","tactic":"apply IsSeparable.mono _ (range_derivWithin_subset_closure_span_image f this)","premises":[{"full_name":"TopologicalSpace.IsSeparable.mono","def_path":"Mathlib/Topology/Bases.lean","def_pos":[430,8],"def_end_pos":[430,24]},{"full_name":"range_derivWithin_subset_closure_span_image","def_path":"Mathlib/Analysis/Calculus/Deriv/Slope.lean","def_pos":[93,8],"def_end_pos":[93,51]}]},{"state_before":"𝕜 : Type u\ninst✝⁵ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\nE : Type w\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\nf✝ f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns✝ t✝ : Set 𝕜\nL L₁ L₂ : Filter 𝕜\ninst✝ : SeparableSpace 𝕜\nf : 𝕜 → F\ns t : Set 𝕜\nts : t ⊆ s\nt_count : t.Countable\nht : s ⊆ closure t\nthis : s ⊆ closure (s ∩ t)\n⊢ IsSeparable (closure ↑(Submodule.span 𝕜 (f '' t)))","state_after":"no goals","tactic":"exact (Countable.image t_count f).isSeparable.span.closure","premises":[{"full_name":"Set.Countable.image","def_path":"Mathlib/Data/Set/Countable.lean","def_pos":[149,8],"def_end_pos":[149,23]},{"full_name":"Set.Countable.isSeparable","def_path":"Mathlib/Topology/Bases.lean","def_pos":[459,8],"def_end_pos":[459,40]},{"full_name":"TopologicalSpace.IsSeparable.span","def_path":"Mathlib/Topology/Algebra/Module/Basic.lean","def_pos":[109,6],"def_end_pos":[109,39]}]}]}
+{"url":"Mathlib/AlgebraicGeometry/Gluing.lean","commit":"","full_name":"AlgebraicGeometry.Scheme.OpenCover.gluedCoverT'_snd_snd","start":[289,0],"end":[293,26],"file_path":"Mathlib/AlgebraicGeometry/Gluing.lean","tactics":[{"state_before":"X : Scheme\n𝒰 : X.OpenCover\nx y z : 𝒰.J\n⊢ 𝒰.gluedCoverT' x y z ≫\n      pullback.snd (pullback.fst (𝒰.map y) (𝒰.map z)) (pullback.fst (𝒰.map y) (𝒰.map x)) ≫\n        pullback.snd (𝒰.map y) (𝒰.map x) =\n    pullback.fst (pullback.fst (𝒰.map x) (𝒰.map y)) (pullback.fst (𝒰.map x) (𝒰.map z)) ≫\n      pullback.fst (𝒰.map x) (𝒰.map y)","state_after":"X : Scheme\n𝒰 : X.OpenCover\nx y z : 𝒰.J\n⊢ ((pullbackRightPullbackFstIso (𝒰.map x) (𝒰.map z) (pullback.fst (𝒰.map x) (𝒰.map y))).hom ≫\n        (pullback.map (pullback.fst (𝒰.map x) (𝒰.map y) ≫ 𝒰.map x) (𝒰.map z)\n              (pullback.fst (𝒰.map y) (𝒰.map x) ≫ 𝒰.map y) (𝒰.map z) (pullbackSymmetry (𝒰.map x) (𝒰.map y)).hom\n              (𝟙 (𝒰.obj z)) (𝟙 X) ⋯ ⋯ ≫\n            (pullbackRightPullbackFstIso (𝒰.map y) (𝒰.map z) (pullback.fst (𝒰.map y) (𝒰.map x))).inv) ≫\n          (pullbackSymmetry (pullback.fst (𝒰.map y) (𝒰.map x)) (pullback.fst (𝒰.map y) (𝒰.map z))).hom) ≫\n      pullback.snd (pullback.fst (𝒰.map y) (𝒰.map z)) (pullback.fst (𝒰.map y) (𝒰.map x)) ≫\n        pullback.snd (𝒰.map y) (𝒰.map x) =\n    pullback.fst (pullback.fst (𝒰.map x) (𝒰.map y)) (pullback.fst (𝒰.map x) (𝒰.map z)) ≫\n      pullback.fst (𝒰.map x) (𝒰.map y)","tactic":"delta gluedCoverT'","premises":[{"full_name":"AlgebraicGeometry.Scheme.OpenCover.gluedCoverT'","def_path":"Mathlib/AlgebraicGeometry/Gluing.lean","def_pos":[261,4],"def_end_pos":[261,16]}]},{"state_before":"X : Scheme\n𝒰 : X.OpenCover\nx y z : 𝒰.J\n⊢ ((pullbackRightPullbackFstIso (𝒰.map x) (𝒰.map z) (pullback.fst (𝒰.map x) (𝒰.map y))).hom ≫\n        (pullback.map (pullback.fst (𝒰.map x) (𝒰.map y) ≫ 𝒰.map x) (𝒰.map z)\n              (pullback.fst (𝒰.map y) (𝒰.map x) ≫ 𝒰.map y) (𝒰.map z) (pullbackSymmetry (𝒰.map x) (𝒰.map y)).hom\n              (𝟙 (𝒰.obj z)) (𝟙 X) ⋯ ⋯ ≫\n            (pullbackRightPullbackFstIso (𝒰.map y) (𝒰.map z) (pullback.fst (𝒰.map y) (𝒰.map x))).inv) ≫\n          (pullbackSymmetry (pullback.fst (𝒰.map y) (𝒰.map x)) (pullback.fst (𝒰.map y) (𝒰.map z))).hom) ≫\n      pullback.snd (pullback.fst (𝒰.map y) (𝒰.map z)) (pullback.fst (𝒰.map y) (𝒰.map x)) ≫\n        pullback.snd (𝒰.map y) (𝒰.map x) =\n    pullback.fst (pullback.fst (𝒰.map x) (𝒰.map y)) (pullback.fst (𝒰.map x) (𝒰.map z)) ≫\n      pullback.fst (𝒰.map x) (𝒰.map y)","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean","commit":"","full_name":"Real.Angle.two_zsmul_coe_pi","start":[144,0],"end":[145,88],"file_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean","tactics":[{"state_before":"⊢ 2 • ↑π = 0","state_after":"no goals","tactic":"simp [← intCast_mul_eq_zsmul]","premises":[{"full_name":"Real.Angle.intCast_mul_eq_zsmul","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean","def_pos":[98,8],"def_end_pos":[98,28]}]}]}
+{"url":"Mathlib/Probability/Kernel/Condexp.lean","commit":"","full_name":"ProbabilityTheory.condexpKernel_ae_eq_trim_condexp","start":[167,0],"end":[173,72],"file_path":"Mathlib/Probability/Kernel/Condexp.lean","tactics":[{"state_before":"Ω : Type u_1\nF : Type u_2\nm mΩ : MeasurableSpace Ω\ninst✝³ : StandardBorelSpace Ω\ninst✝² : Nonempty Ω\nμ : Measure Ω\ninst✝¹ inst✝ : IsFiniteMeasure μ\nhm : m ≤ mΩ\ns : Set Ω\nhs : MeasurableSet s\n⊢ (fun ω => (((condexpKernel μ m) ω) s).toReal) =ᶠ[ae (μ.trim hm)] μ[s.indicator fun ω => 1|m]","state_after":"Ω : Type u_1\nF : Type u_2\nm mΩ : MeasurableSpace Ω\ninst✝³ : StandardBorelSpace Ω\ninst✝² : Nonempty Ω\nμ : Measure Ω\ninst✝¹ inst✝ : IsFiniteMeasure μ\nhm : m ≤ mΩ\ns : Set Ω\nhs : MeasurableSet s\n⊢ (fun ω => (((condexpKernel μ m) ω) s).toReal) =ᶠ[ae μ] μ[s.indicator fun ω => 1|m]\n\nΩ : Type u_1\nF : Type u_2\nm mΩ : MeasurableSpace Ω\ninst✝³ : StandardBorelSpace Ω\ninst✝² : Nonempty Ω\nμ : Measure Ω\ninst✝¹ inst✝ : IsFiniteMeasure μ\nhm : m ≤ mΩ\ns : Set Ω\nhs : MeasurableSet s\n⊢ StronglyMeasurable fun ω => (((condexpKernel μ m) ω) s).toReal","tactic":"rw [ae_eq_trim_iff hm _ stronglyMeasurable_condexp]","premises":[{"full_name":"MeasureTheory.ae_eq_trim_iff","def_path":"Mathlib/MeasureTheory/Integral/Bochner.lean","def_pos":[1850,8],"def_end_pos":[1850,22]},{"full_name":"MeasureTheory.stronglyMeasurable_condexp","def_path":"Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean","def_pos":[165,8],"def_end_pos":[165,34]}]}]}
+{"url":"Mathlib/Data/Option/Basic.lean","commit":"","full_name":"Option.map_pbind","start":[147,0],"end":[149,42],"file_path":"Mathlib/Data/Option/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\np : α → Prop\nf✝ : (a : α) → p a → β\nx✝ : Option α\nf : β → γ\nx : Option α\ng : (a : α) → a ∈ x → Option β\n⊢ Option.map f (x.pbind g) = x.pbind fun a H => Option.map f (g a H)","state_after":"no goals","tactic":"cases x <;> simp only [pbind, map_none']","premises":[{"full_name":"Option.map_none'","def_path":".lake/packages/lean4/src/lean/Init/Data/Option/Basic.lean","def_pos":[114,16],"def_end_pos":[114,25]},{"full_name":"Option.pbind","def_path":".lake/packages/lean4/src/lean/Init/Data/Option/Instances.lean","def_pos":[59,4],"def_end_pos":[59,9]}]}]}
+{"url":"Mathlib/NumberTheory/Cyclotomic/Rat.lean","commit":"","full_name":"IsCyclotomicExtension.Rat.discr_prime_pow_eq_unit_mul_pow'","start":[57,0],"end":[64,83],"file_path":"Mathlib/NumberTheory/Cyclotomic/Rat.lean","tactics":[{"state_before":"p : ℕ+\nk : ℕ\nK : Type u\ninst✝² : Field K\ninst✝¹ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ k} ℚ K\nhζ : IsPrimitiveRoot ζ ↑(p ^ k)\n⊢ ∃ u n, Algebra.discr ℚ ⇑(IsPrimitiveRoot.subOnePowerBasis ℚ hζ).basis = ↑↑u * ↑↑p ^ n","state_after":"p : ℕ+\nk : ℕ\nK : Type u\ninst✝² : Field K\ninst✝¹ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ k} ℚ K\nhζ : IsPrimitiveRoot ζ ↑(p ^ k)\n⊢ ∃ u n, Algebra.discr ℚ ⇑(IsPrimitiveRoot.powerBasis ℚ hζ).basis = ↑↑u * ↑↑p ^ n","tactic":"rw [hζ.discr_zeta_eq_discr_zeta_sub_one.symm]","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"IsPrimitiveRoot.discr_zeta_eq_discr_zeta_sub_one","def_path":"Mathlib/NumberTheory/Cyclotomic/Discriminant.lean","def_pos":[35,8],"def_end_pos":[35,40]}]},{"state_before":"p : ℕ+\nk : ℕ\nK : Type u\ninst✝² : Field K\ninst✝¹ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ k} ℚ K\nhζ : IsPrimitiveRoot ζ ↑(p ^ k)\n⊢ ∃ u n, Algebra.discr ℚ ⇑(IsPrimitiveRoot.powerBasis ℚ hζ).basis = ↑↑u * ↑↑p ^ n","state_after":"no goals","tactic":"exact discr_prime_pow_eq_unit_mul_pow hζ (cyclotomic.irreducible_rat (p ^ k).pos)","premises":[{"full_name":"IsCyclotomicExtension.discr_prime_pow_eq_unit_mul_pow","def_path":"Mathlib/NumberTheory/Cyclotomic/Discriminant.lean","def_pos":[187,8],"def_end_pos":[187,39]},{"full_name":"PNat.pos","def_path":"Mathlib/Data/PNat/Defs.lean","def_pos":[129,8],"def_end_pos":[129,11]},{"full_name":"Polynomial.cyclotomic.irreducible_rat","def_path":"Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean","def_pos":[190,8],"def_end_pos":[190,34]}]}]}
+{"url":"Mathlib/MeasureTheory/OuterMeasure/Operations.lean","commit":"","full_name":"MeasureTheory.OuterMeasure.map_top","start":[351,0],"end":[354,25],"file_path":"Mathlib/MeasureTheory/OuterMeasure/Operations.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nm : OuterMeasure α\nf : α → β\ns : Set β\n⊢ ((map f) ⊤) s = ((restrict (range f)) ⊤) s","state_after":"no goals","tactic":"rw [map_apply, restrict_apply, ← image_preimage_eq_inter_range, top_apply', top_apply',\n      Set.image_eq_empty]","premises":[{"full_name":"MeasureTheory.OuterMeasure.map_apply","def_path":"Mathlib/MeasureTheory/OuterMeasure/Operations.lean","def_pos":[209,8],"def_end_pos":[209,17]},{"full_name":"MeasureTheory.OuterMeasure.restrict_apply","def_path":"Mathlib/MeasureTheory/OuterMeasure/Operations.lean","def_pos":[294,8],"def_end_pos":[294,22]},{"full_name":"MeasureTheory.OuterMeasure.top_apply'","def_path":"Mathlib/MeasureTheory/OuterMeasure/Operations.lean","def_pos":[344,8],"def_end_pos":[344,18]},{"full_name":"Set.image_eq_empty","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[297,8],"def_end_pos":[297,22]},{"full_name":"Set.image_preimage_eq_inter_range","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[666,8],"def_end_pos":[666,37]}]}]}
+{"url":"Mathlib/Algebra/Periodic.lean","commit":"","full_name":"Function.Antiperiodic.mul","start":[513,0],"end":[514,63],"file_path":"Mathlib/Algebra/Periodic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf g : α → β\nc c₁ c₂ x : α\ninst✝² : Add α\ninst✝¹ : Mul β\ninst✝ : HasDistribNeg β\nhf : Antiperiodic f c\nhg : Antiperiodic g c\n⊢ Periodic (f * g) c","state_after":"no goals","tactic":"simp_all","premises":[]}]}
+{"url":"Mathlib/MeasureTheory/Measure/LogLikelihoodRatio.lean","commit":"","full_name":"MeasureTheory.llr_smul_left","start":[83,0],"end":[98,6],"file_path":"Mathlib/MeasureTheory/Measure/LogLikelihoodRatio.lean","tactics":[{"state_before":"α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : μ.HaveLebesgueDecomposition ν\nhμν : μ ≪ ν\nc : ℝ≥0∞\nhc : c ≠ 0\nhc_ne_top : c ≠ ⊤\n⊢ llr (c • μ) ν =ᶠ[ae μ] fun x => llr μ ν x + log c.toReal","state_after":"α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : μ.HaveLebesgueDecomposition ν\nhμν : μ ≪ ν\nc : ℝ≥0∞\nhc : c ≠ 0\nhc_ne_top : c ≠ ⊤\n⊢ (fun x => log ((c • μ).rnDeriv ν x).toReal) =ᶠ[ae μ] fun x => log (μ.rnDeriv ν x).toReal + log c.toReal","tactic":"simp only [llr, llr_def]","premises":[{"full_name":"MeasureTheory.llr","def_path":"Mathlib/MeasureTheory/Measure/LogLikelihoodRatio.lean","def_pos":[33,18],"def_end_pos":[33,21]},{"full_name":"MeasureTheory.llr_def","def_path":"Mathlib/MeasureTheory/Measure/LogLikelihoodRatio.lean","def_pos":[35,6],"def_end_pos":[35,13]}]},{"state_before":"α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : μ.HaveLebesgueDecomposition ν\nhμν : μ ≪ ν\nc : ℝ≥0∞\nhc : c ≠ 0\nhc_ne_top : c ≠ ⊤\n⊢ (fun x => log ((c • μ).rnDeriv ν x).toReal) =ᶠ[ae μ] fun x => log (μ.rnDeriv ν x).toReal + log c.toReal","state_after":"α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : μ.HaveLebesgueDecomposition ν\nhμν : μ ≪ ν\nc : ℝ≥0∞\nhc : c ≠ 0\nhc_ne_top : c ≠ ⊤\nh : (c • μ).rnDeriv ν =ᶠ[ae ν] c • μ.rnDeriv ν\n⊢ (fun x => log ((c • μ).rnDeriv ν x).toReal) =ᶠ[ae μ] fun x => log (μ.rnDeriv ν x).toReal + log c.toReal","tactic":"have h := Measure.rnDeriv_smul_left_of_ne_top μ ν hc_ne_top","premises":[{"full_name":"MeasureTheory.Measure.rnDeriv_smul_left_of_ne_top","def_path":"Mathlib/MeasureTheory/Decomposition/Lebesgue.lean","def_pos":[581,8],"def_end_pos":[581,35]}]},{"state_before":"α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : μ.HaveLebesgueDecomposition ν\nhμν : μ ≪ ν\nc : ℝ≥0∞\nhc : c ≠ 0\nhc_ne_top : c ≠ ⊤\nh : (c • μ).rnDeriv ν =ᶠ[ae ν] c • μ.rnDeriv ν\n⊢ (fun x => log ((c • μ).rnDeriv ν x).toReal) =ᶠ[ae μ] fun x => log (μ.rnDeriv ν x).toReal + log c.toReal","state_after":"case h\nα : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : μ.HaveLebesgueDecomposition ν\nhμν : μ ≪ ν\nc : ℝ≥0∞\nhc : c ≠ 0\nhc_ne_top : c ≠ ⊤\nh : (c • μ).rnDeriv ν =ᶠ[ae ν] c • μ.rnDeriv ν\nx : α\nhx_eq : (c • μ).rnDeriv ν x = (c • μ.rnDeriv ν) x\nhx_pos : 0 < μ.rnDeriv ν x\nhx_ne_top : μ.rnDeriv ν x < ⊤\n⊢ log ((c • μ).rnDeriv ν x).toReal = log (μ.rnDeriv ν x).toReal + log c.toReal","tactic":"filter_upwards [hμν.ae_le h, Measure.rnDeriv_pos hμν, hμν.ae_le (Measure.rnDeriv_lt_top μ ν)]\n    with x hx_eq hx_pos hx_ne_top","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Filter.mp_mem","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[159,8],"def_end_pos":[159,14]},{"full_name":"MeasureTheory.Measure.rnDeriv_lt_top","def_path":"Mathlib/MeasureTheory/Decomposition/Lebesgue.lean","def_pos":[362,8],"def_end_pos":[362,22]},{"full_name":"MeasureTheory.Measure.rnDeriv_pos","def_path":"Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean","def_pos":[75,6],"def_end_pos":[75,17]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]}]},{"state_before":"case h\nα : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : μ.HaveLebesgueDecomposition ν\nhμν : μ ≪ ν\nc : ℝ≥0∞\nhc : c ≠ 0\nhc_ne_top : c ≠ ⊤\nh : (c • μ).rnDeriv ν =ᶠ[ae ν] c • μ.rnDeriv ν\nx : α\nhx_eq : (c • μ).rnDeriv ν x = (c • μ.rnDeriv ν) x\nhx_pos : 0 < μ.rnDeriv ν x\nhx_ne_top : μ.rnDeriv ν x < ⊤\n⊢ log ((c • μ).rnDeriv ν x).toReal = log (μ.rnDeriv ν x).toReal + log c.toReal","state_after":"case h\nα : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : μ.HaveLebesgueDecomposition ν\nhμν : μ ≪ ν\nc : ℝ≥0∞\nhc : c ≠ 0\nhc_ne_top : c ≠ ⊤\nh : (c • μ).rnDeriv ν =ᶠ[ae ν] c • μ.rnDeriv ν\nx : α\nhx_eq : (c • μ).rnDeriv ν x = (c • μ.rnDeriv ν) x\nhx_pos : 0 < μ.rnDeriv ν x\nhx_ne_top : μ.rnDeriv ν x < ⊤\n⊢ log ((c • μ.rnDeriv ν) x).toReal = log (μ.rnDeriv ν x).toReal + log c.toReal","tactic":"rw [hx_eq]","premises":[]},{"state_before":"case h\nα : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : μ.HaveLebesgueDecomposition ν\nhμν : μ ≪ ν\nc : ℝ≥0∞\nhc : c ≠ 0\nhc_ne_top : c ≠ ⊤\nh : (c • μ).rnDeriv ν =ᶠ[ae ν] c • μ.rnDeriv ν\nx : α\nhx_eq : (c • μ).rnDeriv ν x = (c • μ.rnDeriv ν) x\nhx_pos : 0 < μ.rnDeriv ν x\nhx_ne_top : μ.rnDeriv ν x < ⊤\n⊢ log ((c • μ.rnDeriv ν) x).toReal = log (μ.rnDeriv ν x).toReal + log c.toReal","state_after":"case h\nα : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : μ.HaveLebesgueDecomposition ν\nhμν : μ ≪ ν\nc : ℝ≥0∞\nhc : c ≠ 0\nhc_ne_top : c ≠ ⊤\nh : (c • μ).rnDeriv ν =ᶠ[ae ν] c • μ.rnDeriv ν\nx : α\nhx_eq : (c • μ).rnDeriv ν x = (c • μ.rnDeriv ν) x\nhx_pos : 0 < μ.rnDeriv ν x\nhx_ne_top : μ.rnDeriv ν x < ⊤\n⊢ log (c.toReal * (μ.rnDeriv ν x).toReal) = log (μ.rnDeriv ν x).toReal + log c.toReal","tactic":"simp only [Pi.smul_apply, smul_eq_mul, ENNReal.toReal_mul]","premises":[{"full_name":"ENNReal.toReal_mul","def_path":"Mathlib/Data/ENNReal/Real.lean","def_pos":[378,8],"def_end_pos":[378,18]},{"full_name":"Pi.smul_apply","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[104,59],"def_end_pos":[104,69]},{"full_name":"smul_eq_mul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[79,6],"def_end_pos":[79,17]}]},{"state_before":"case h\nα : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : μ.HaveLebesgueDecomposition ν\nhμν : μ ≪ ν\nc : ℝ≥0∞\nhc : c ≠ 0\nhc_ne_top : c ≠ ⊤\nh : (c • μ).rnDeriv ν =ᶠ[ae ν] c • μ.rnDeriv ν\nx : α\nhx_eq : (c • μ).rnDeriv ν x = (c • μ.rnDeriv ν) x\nhx_pos : 0 < μ.rnDeriv ν x\nhx_ne_top : μ.rnDeriv ν x < ⊤\n⊢ log (c.toReal * (μ.rnDeriv ν x).toReal) = log (μ.rnDeriv ν x).toReal + log c.toReal","state_after":"case h\nα : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : μ.HaveLebesgueDecomposition ν\nhμν : μ ≪ ν\nc : ℝ≥0∞\nhc : c ≠ 0\nhc_ne_top : c ≠ ⊤\nh : (c • μ).rnDeriv ν =ᶠ[ae ν] c • μ.rnDeriv ν\nx : α\nhx_eq : (c • μ).rnDeriv ν x = (c • μ.rnDeriv ν) x\nhx_pos : 0 < μ.rnDeriv ν x\nhx_ne_top : μ.rnDeriv ν x < ⊤\n⊢ log c.toReal + log (μ.rnDeriv ν x).toReal = log (μ.rnDeriv ν x).toReal + log c.toReal\n\ncase h.hx\nα : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : μ.HaveLebesgueDecomposition ν\nhμν : μ ≪ ν\nc : ℝ≥0∞\nhc : c ≠ 0\nhc_ne_top : c ≠ ⊤\nh : (c • μ).rnDeriv ν =ᶠ[ae ν] c • μ.rnDeriv ν\nx : α\nhx_eq : (c • μ).rnDeriv ν x = (c • μ.rnDeriv ν) x\nhx_pos : 0 < μ.rnDeriv ν x\nhx_ne_top : μ.rnDeriv ν x < ⊤\n⊢ c.toReal ≠ 0\n\ncase h.hy\nα : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : μ.HaveLebesgueDecomposition ν\nhμν : μ ≪ ν\nc : ℝ≥0∞\nhc : c ≠ 0\nhc_ne_top : c ≠ ⊤\nh : (c • μ).rnDeriv ν =ᶠ[ae ν] c • μ.rnDeriv ν\nx : α\nhx_eq : (c • μ).rnDeriv ν x = (c • μ.rnDeriv ν) x\nhx_pos : 0 < μ.rnDeriv ν x\nhx_ne_top : μ.rnDeriv ν x < ⊤\n⊢ (μ.rnDeriv ν x).toReal ≠ 0","tactic":"rw [log_mul]","premises":[{"full_name":"Real.log_mul","def_path":"Mathlib/Analysis/SpecialFunctions/Log/Basic.lean","def_pos":[106,8],"def_end_pos":[106,15]}]},{"state_before":"case h\nα : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : μ.HaveLebesgueDecomposition ν\nhμν : μ ≪ ν\nc : ℝ≥0∞\nhc : c ≠ 0\nhc_ne_top : c ≠ ⊤\nh : (c • μ).rnDeriv ν =ᶠ[ae ν] c • μ.rnDeriv ν\nx : α\nhx_eq : (c • μ).rnDeriv ν x = (c • μ.rnDeriv ν) x\nhx_pos : 0 < μ.rnDeriv ν x\nhx_ne_top : μ.rnDeriv ν x < ⊤\n⊢ log c.toReal + log (μ.rnDeriv ν x).toReal = log (μ.rnDeriv ν x).toReal + log c.toReal\n\ncase h.hx\nα : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : μ.HaveLebesgueDecomposition ν\nhμν : μ ≪ ν\nc : ℝ≥0∞\nhc : c ≠ 0\nhc_ne_top : c ≠ ⊤\nh : (c • μ).rnDeriv ν =ᶠ[ae ν] c • μ.rnDeriv ν\nx : α\nhx_eq : (c • μ).rnDeriv ν x = (c • μ.rnDeriv ν) x\nhx_pos : 0 < μ.rnDeriv ν x\nhx_ne_top : μ.rnDeriv ν x < ⊤\n⊢ c.toReal ≠ 0\n\ncase h.hy\nα : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : μ.HaveLebesgueDecomposition ν\nhμν : μ ≪ ν\nc : ℝ≥0∞\nhc : c ≠ 0\nhc_ne_top : c ≠ ⊤\nh : (c • μ).rnDeriv ν =ᶠ[ae ν] c • μ.rnDeriv ν\nx : α\nhx_eq : (c • μ).rnDeriv ν x = (c • μ.rnDeriv ν) x\nhx_pos : 0 < μ.rnDeriv ν x\nhx_ne_top : μ.rnDeriv ν x < ⊤\n⊢ (μ.rnDeriv ν x).toReal ≠ 0","state_after":"case h.hx\nα : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : μ.HaveLebesgueDecomposition ν\nhμν : μ ≪ ν\nc : ℝ≥0∞\nhc : c ≠ 0\nhc_ne_top : c ≠ ⊤\nh : (c • μ).rnDeriv ν =ᶠ[ae ν] c • μ.rnDeriv ν\nx : α\nhx_eq : (c • μ).rnDeriv ν x = (c • μ.rnDeriv ν) x\nhx_pos : 0 < μ.rnDeriv ν x\nhx_ne_top : μ.rnDeriv ν x < ⊤\n⊢ c.toReal ≠ 0\n\ncase h.hy\nα : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : μ.HaveLebesgueDecomposition ν\nhμν : μ ≪ ν\nc : ℝ≥0∞\nhc : c ≠ 0\nhc_ne_top : c ≠ ⊤\nh : (c • μ).rnDeriv ν =ᶠ[ae ν] c • μ.rnDeriv ν\nx : α\nhx_eq : (c • μ).rnDeriv ν x = (c • μ.rnDeriv ν) x\nhx_pos : 0 < μ.rnDeriv ν x\nhx_ne_top : μ.rnDeriv ν x < ⊤\n⊢ (μ.rnDeriv ν x).toReal ≠ 0\n\ncase h\nα : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : μ.HaveLebesgueDecomposition ν\nhμν : μ ≪ ν\nc : ℝ≥0∞\nhc : c ≠ 0\nhc_ne_top : c ≠ ⊤\nh : (c • μ).rnDeriv ν =ᶠ[ae ν] c • μ.rnDeriv ν\nx : α\nhx_eq : (c • μ).rnDeriv ν x = (c • μ.rnDeriv ν) x\nhx_pos : 0 < μ.rnDeriv ν x\nhx_ne_top : μ.rnDeriv ν x < ⊤\n⊢ log c.toReal + log (μ.rnDeriv ν x).toReal = log (μ.rnDeriv ν x).toReal + log c.toReal","tactic":"rotate_left","premises":[]},{"state_before":"case h\nα : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : μ.HaveLebesgueDecomposition ν\nhμν : μ ≪ ν\nc : ℝ≥0∞\nhc : c ≠ 0\nhc_ne_top : c ≠ ⊤\nh : (c • μ).rnDeriv ν =ᶠ[ae ν] c • μ.rnDeriv ν\nx : α\nhx_eq : (c • μ).rnDeriv ν x = (c • μ.rnDeriv ν) x\nhx_pos : 0 < μ.rnDeriv ν x\nhx_ne_top : μ.rnDeriv ν x < ⊤\n⊢ log c.toReal + log (μ.rnDeriv ν x).toReal = log (μ.rnDeriv ν x).toReal + log c.toReal","state_after":"no goals","tactic":"ring","premises":[]}]}
+{"url":"Mathlib/Algebra/Algebra/Subalgebra/Rank.lean","commit":"","full_name":"Subalgebra.finrank_sup_eq_finrank_right_mul_finrank_of_free","start":[51,0],"end":[53,64],"file_path":"Mathlib/Algebra/Algebra/Subalgebra/Rank.lean","tactics":[{"state_before":"R : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nA B : Subalgebra R S\ninst✝³ : Module.Free R ↥A\ninst✝² : Module.Free R ↥B\ninst✝¹ : Module.Free ↥A ↥(Algebra.adjoin ↥A ↑B)\ninst✝ : Module.Free ↥B ↥(Algebra.adjoin ↥B ↑A)\n⊢ finrank R ↥(A ⊔ B) = finrank R ↥B * finrank ↥B ↥(Algebra.adjoin ↥B ↑A)","state_after":"no goals","tactic":"rw [sup_comm, finrank_sup_eq_finrank_left_mul_finrank_of_free]","premises":[{"full_name":"Subalgebra.finrank_sup_eq_finrank_left_mul_finrank_of_free","def_path":"Mathlib/Algebra/Algebra/Subalgebra/Rank.lean","def_pos":[47,8],"def_end_pos":[47,55]},{"full_name":"sup_comm","def_path":"Mathlib/Order/Lattice.lean","def_pos":[193,8],"def_end_pos":[193,16]}]}]}
+{"url":"Mathlib/Data/Multiset/Basic.lean","commit":"","full_name":"Multiset.map_erase","start":[1170,0],"end":[1177,93],"file_path":"Mathlib/Data/Multiset/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type v\nγ : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nf : α → β\nhf : Injective f\nx : α\ns : Multiset α\n⊢ map f (s.erase x) = (map f s).erase (f x)","state_after":"case empty\nα : Type u_1\nβ : Type v\nγ : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nf : α → β\nhf : Injective f\nx : α\n⊢ map f (erase 0 x) = (map f 0).erase (f x)\n\ncase cons\nα : Type u_1\nβ : Type v\nγ : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nf : α → β\nhf : Injective f\nx y : α\ns : Multiset α\nih : map f (s.erase x) = (map f s).erase (f x)\n⊢ map f ((y ::ₘ s).erase x) = (map f (y ::ₘ s)).erase (f x)","tactic":"induction' s using Multiset.induction_on with y s ih","premises":[{"full_name":"Multiset.induction_on","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[152,18],"def_end_pos":[152,30]}]},{"state_before":"case cons\nα : Type u_1\nβ : Type v\nγ : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nf : α → β\nhf : Injective f\nx y : α\ns : Multiset α\nih : map f (s.erase x) = (map f s).erase (f x)\n⊢ map f ((y ::ₘ s).erase x) = (map f (y ::ₘ s)).erase (f x)","state_after":"case pos\nα : Type u_1\nβ : Type v\nγ : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nf : α → β\nhf : Injective f\nx y : α\ns : Multiset α\nih : map f (s.erase x) = (map f s).erase (f x)\nhxy : y = x\n⊢ map f ((y ::ₘ s).erase x) = (map f (y ::ₘ s)).erase (f x)\n\ncase neg\nα : Type u_1\nβ : Type v\nγ : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nf : α → β\nhf : Injective f\nx y : α\ns : Multiset α\nih : map f (s.erase x) = (map f s).erase (f x)\nhxy : ¬y = x\n⊢ map f ((y ::ₘ s).erase x) = (map f (y ::ₘ s)).erase (f x)","tactic":"by_cases hxy : y = x","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
+{"url":"Mathlib/Order/SupIndep.lean","commit":"","full_name":"CompleteLattice.setIndependent_iff","start":[318,0],"end":[322,31],"file_path":"Mathlib/Order/SupIndep.lean","tactics":[{"state_before":"α✝ : Type u_1\nβ : Type u_2\nι : Type u_3\nι' : Type u_4\ninst✝¹ : CompleteLattice α✝\ns✝ : Set α✝\nhs : SetIndependent s✝\nα : Type u_5\ninst✝ : CompleteLattice α\ns : Set α\n⊢ SetIndependent s ↔ Independent Subtype.val","state_after":"α✝ : Type u_1\nβ : Type u_2\nι : Type u_3\nι' : Type u_4\ninst✝¹ : CompleteLattice α✝\ns✝ : Set α✝\nhs : SetIndependent s✝\nα : Type u_5\ninst✝ : CompleteLattice α\ns : Set α\n⊢ (∀ ⦃a : α⦄, a ∈ s → Disjoint a (⨆ a_2 ∈ s \\ {a}, a_2)) ↔\n    ∀ (x : α) (h : x ∈ s), Disjoint x (⨆ j, ⨆ (_ : j ≠ ⟨x, h⟩), ↑j)","tactic":"simp_rw [Independent, SetIndependent, SetCoe.forall, sSup_eq_iSup]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"CompleteLattice.Independent","def_path":"Mathlib/Order/SupIndep.lean","def_pos":[315,4],"def_end_pos":[315,15]},{"full_name":"CompleteLattice.SetIndependent","def_path":"Mathlib/Order/SupIndep.lean","def_pos":[268,4],"def_end_pos":[268,18]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"SetCoe.forall","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[146,8],"def_end_pos":[146,21]},{"full_name":"sSup_eq_iSup","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[750,8],"def_end_pos":[750,20]}]},{"state_before":"α✝ : Type u_1\nβ : Type u_2\nι : Type u_3\nι' : Type u_4\ninst✝¹ : CompleteLattice α✝\ns✝ : Set α✝\nhs : SetIndependent s✝\nα : Type u_5\ninst✝ : CompleteLattice α\ns : Set α\n⊢ (∀ ⦃a : α⦄, a ∈ s → Disjoint a (⨆ a_2 ∈ s \\ {a}, a_2)) ↔\n    ∀ (x : α) (h : x ∈ s), Disjoint x (⨆ j, ⨆ (_ : j ≠ ⟨x, h⟩), ↑j)","state_after":"α✝ : Type u_1\nβ : Type u_2\nι : Type u_3\nι' : Type u_4\ninst✝¹ : CompleteLattice α✝\ns✝ : Set α✝\nhs : SetIndependent s✝\nα : Type u_5\ninst✝ : CompleteLattice α\ns : Set α\na : α\nha : a ∈ s\n⊢ Disjoint a (⨆ a_1 ∈ s \\ {a}, a_1) ↔ Disjoint a (⨆ j, ⨆ (_ : j ≠ ⟨a, ha⟩), ↑j)","tactic":"refine forall₂_congr fun a ha => ?_","premises":[{"full_name":"forall₂_congr","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[215,8],"def_end_pos":[215,21]}]},{"state_before":"α✝ : Type u_1\nβ : Type u_2\nι : Type u_3\nι' : Type u_4\ninst✝¹ : CompleteLattice α✝\ns✝ : Set α✝\nhs : SetIndependent s✝\nα : Type u_5\ninst✝ : CompleteLattice α\ns : Set α\na : α\nha : a ∈ s\n⊢ Disjoint a (⨆ a_1 ∈ s \\ {a}, a_1) ↔ Disjoint a (⨆ j, ⨆ (_ : j ≠ ⟨a, ha⟩), ↑j)","state_after":"no goals","tactic":"simp [iSup_subtype, iSup_and]","premises":[{"full_name":"iSup_and","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[1084,8],"def_end_pos":[1084,16]},{"full_name":"iSup_subtype","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[972,8],"def_end_pos":[972,20]}]}]}
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+{"url":"Mathlib/Algebra/Polynomial/Expand.lean","commit":"","full_name":"Polynomial.monic_expand_iff","start":[153,0],"end":[154,43],"file_path":"Mathlib/Algebra/Polynomial/Expand.lean","tactics":[{"state_before":"R : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\np✝ q p : ℕ\nf : R[X]\nhp : 0 < p\n⊢ ((expand R p) f).Monic ↔ f.Monic","state_after":"no goals","tactic":"simp only [Monic, leadingCoeff_expand hp]","premises":[{"full_name":"Polynomial.Monic","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[69,4],"def_end_pos":[69,9]},{"full_name":"Polynomial.leadingCoeff_expand","def_path":"Mathlib/Algebra/Polynomial/Expand.lean","def_pos":[149,8],"def_end_pos":[149,27]}]}]}
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+{"url":"Mathlib/GroupTheory/OrderOfElement.lean","commit":"","full_name":"orderOf_eq_prime_pow","start":[440,0],"end":[443,76],"file_path":"Mathlib/GroupTheory/OrderOfElement.lean","tactics":[{"state_before":"G : Type u_1\nH : Type u_2\nA : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝ : Monoid G\na b x y : G\nn m p : ℕ\nhp : Fact (Nat.Prime p)\nhnot : ¬x ^ p ^ n = 1\nhfin : x ^ p ^ (n + 1) = 1\n⊢ orderOf x = p ^ (n + 1)","state_after":"no goals","tactic":"apply minimalPeriod_eq_prime_pow <;> rwa [isPeriodicPt_mul_iff_pow_eq_one]","premises":[{"full_name":"Function.minimalPeriod_eq_prime_pow","def_path":"Mathlib/Dynamics/PeriodicPts.lean","def_pos":[358,8],"def_end_pos":[358,34]},{"full_name":"isPeriodicPt_mul_iff_pow_eq_one","def_path":"Mathlib/GroupTheory/OrderOfElement.lean","def_pos":[40,8],"def_end_pos":[40,39]}]}]}
+{"url":"Mathlib/Order/ConditionallyCompleteLattice/Basic.lean","commit":"","full_name":"WithTop.iInf_coe_eq_top","start":[1540,0],"end":[1540,87],"file_path":"Mathlib/Order/ConditionallyCompleteLattice/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\ninst✝ : ConditionallyCompleteLinearOrderBot α\nf : ι → α\n⊢ ⨅ x, ↑(f x) = ⊤ ↔ IsEmpty ι","state_after":"no goals","tactic":"simp [isEmpty_iff]","premises":[{"full_name":"isEmpty_iff","def_path":"Mathlib/Logic/IsEmpty.lean","def_pos":[94,8],"def_end_pos":[94,19]}]}]}
+{"url":"Mathlib/MeasureTheory/Measure/Portmanteau.lean","commit":"","full_name":"MeasureTheory.limsup_measure_compl_le_of_le_liminf_measure","start":[129,0],"end":[147,29],"file_path":"Mathlib/MeasureTheory/Measure/Portmanteau.lean","tactics":[{"state_before":"Ω : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nE : Set Ω\nE_mble : MeasurableSet E\nh : μ E ≤ liminf (fun i => (μs i) E) L\n⊢ limsup (fun i => (μs i) Eᶜ) L ≤ μ Eᶜ","state_after":"case inl\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nE : Set Ω\nE_mble : MeasurableSet E\nh : μ E ≤ liminf (fun i => (μs i) E) ⊥\n⊢ limsup (fun i => (μs i) Eᶜ) ⊥ ≤ μ Eᶜ\n\ncase inr\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nE : Set Ω\nE_mble : MeasurableSet E\nh : μ E ≤ liminf (fun i => (μs i) E) L\nhne : L.NeBot\n⊢ limsup (fun i => (μs i) Eᶜ) L ≤ μ Eᶜ","tactic":"rcases L.eq_or_neBot with rfl | hne","premises":[{"full_name":"Filter.eq_or_neBot","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[484,8],"def_end_pos":[484,19]}]},{"state_before":"case inr\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nE : Set Ω\nE_mble : MeasurableSet E\nh : μ E ≤ liminf (fun i => (μs i) E) L\nhne : L.NeBot\n⊢ limsup (fun i => (μs i) Eᶜ) L ≤ μ Eᶜ","state_after":"case inr\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nE : Set Ω\nE_mble : MeasurableSet E\nh : μ E ≤ liminf (fun i => (μs i) E) L\nhne : L.NeBot\nmeas_Ec : μ Eᶜ = 1 - μ E\n⊢ limsup (fun i => (μs i) Eᶜ) L ≤ μ Eᶜ","tactic":"have meas_Ec : μ Eᶜ = 1 - μ E := by\n    simpa only [measure_univ] using measure_compl E_mble (measure_lt_top μ E).ne","premises":[{"full_name":"HasCompl.compl","def_path":"Mathlib/Order/Notation.lean","def_pos":[34,2],"def_end_pos":[34,7]},{"full_name":"MeasureTheory.IsProbabilityMeasure.measure_univ","def_path":"Mathlib/MeasureTheory/Measure/Typeclasses.lean","def_pos":[208,2],"def_end_pos":[208,14]},{"full_name":"MeasureTheory.measure_compl","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","def_pos":[277,8],"def_end_pos":[277,21]},{"full_name":"MeasureTheory.measure_lt_top","def_path":"Mathlib/MeasureTheory/Measure/Typeclasses.lean","def_pos":[48,8],"def_end_pos":[48,22]}]},{"state_before":"case inr\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nE : Set Ω\nE_mble : MeasurableSet E\nh : μ E ≤ liminf (fun i => (μs i) E) L\nhne : L.NeBot\nmeas_Ec : μ Eᶜ = 1 - μ E\n⊢ limsup (fun i => (μs i) Eᶜ) L ≤ μ Eᶜ","state_after":"case inr\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nE : Set Ω\nE_mble : MeasurableSet E\nh : μ E ≤ liminf (fun i => (μs i) E) L\nhne : L.NeBot\nmeas_Ec : μ Eᶜ = 1 - μ E\nmeas_i_Ec : ∀ (i : ι), (μs i) Eᶜ = 1 - (μs i) E\n⊢ limsup (fun i => (μs i) Eᶜ) L ≤ μ Eᶜ","tactic":"have meas_i_Ec : ∀ i, μs i Eᶜ = 1 - μs i E := by\n    intro i\n    simpa only [measure_univ] using measure_compl E_mble (measure_lt_top (μs i) E).ne","premises":[{"full_name":"HasCompl.compl","def_path":"Mathlib/Order/Notation.lean","def_pos":[34,2],"def_end_pos":[34,7]},{"full_name":"MeasureTheory.IsProbabilityMeasure.measure_univ","def_path":"Mathlib/MeasureTheory/Measure/Typeclasses.lean","def_pos":[208,2],"def_end_pos":[208,14]},{"full_name":"MeasureTheory.measure_compl","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","def_pos":[277,8],"def_end_pos":[277,21]},{"full_name":"MeasureTheory.measure_lt_top","def_path":"Mathlib/MeasureTheory/Measure/Typeclasses.lean","def_pos":[48,8],"def_end_pos":[48,22]}]},{"state_before":"case inr\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nE : Set Ω\nE_mble : MeasurableSet E\nh : μ E ≤ liminf (fun i => (μs i) E) L\nhne : L.NeBot\nmeas_Ec : μ Eᶜ = 1 - μ E\nmeas_i_Ec : ∀ (i : ι), (μs i) Eᶜ = 1 - (μs i) E\n⊢ limsup (fun i => (μs i) Eᶜ) L ≤ μ Eᶜ","state_after":"case inr\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nE : Set Ω\nE_mble : MeasurableSet E\nh : μ E ≤ liminf (fun i => (μs i) E) L\nhne : L.NeBot\nmeas_Ec : μ Eᶜ = 1 - μ E\nmeas_i_Ec : ∀ (i : ι), (μs i) Eᶜ = 1 - (μs i) E\n⊢ limsup (fun i => 1 - (μs i) E) L ≤ 1 - μ E","tactic":"simp_rw [meas_Ec, meas_i_Ec]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]}]},{"state_before":"case inr\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nE : Set Ω\nE_mble : MeasurableSet E\nh : μ E ≤ liminf (fun i => (μs i) E) L\nhne : L.NeBot\nmeas_Ec : μ Eᶜ = 1 - μ E\nmeas_i_Ec : ∀ (i : ι), (μs i) Eᶜ = 1 - (μs i) E\n⊢ limsup (fun i => 1 - (μs i) E) L ≤ 1 - μ E","state_after":"case inr\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nE : Set Ω\nE_mble : MeasurableSet E\nh : μ E ≤ liminf (fun i => (μs i) E) L\nhne : L.NeBot\nmeas_Ec : μ Eᶜ = 1 - μ E\nmeas_i_Ec : ∀ (i : ι), (μs i) Eᶜ = 1 - (μs i) E\nobs : limsup (fun i => 1 - (μs i) E) L = limsup ((fun x => 1 - x) ∘ fun i => (μs i) E) L\n⊢ limsup (fun i => 1 - (μs i) E) L ≤ 1 - μ E","tactic":"have obs :\n    (L.limsup fun i : ι => 1 - μs i E) = L.limsup ((fun x => 1 - x) ∘ fun i : ι => μs i E) := rfl","premises":[{"full_name":"Filter.limsup","def_path":"Mathlib/Order/LiminfLimsup.lean","def_pos":[446,4],"def_end_pos":[446,10]},{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"case inr\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nE : Set Ω\nE_mble : MeasurableSet E\nh : μ E ≤ liminf (fun i => (μs i) E) L\nhne : L.NeBot\nmeas_Ec : μ Eᶜ = 1 - μ E\nmeas_i_Ec : ∀ (i : ι), (μs i) Eᶜ = 1 - (μs i) E\nobs : limsup (fun i => 1 - (μs i) E) L = limsup ((fun x => 1 - x) ∘ fun i => (μs i) E) L\n⊢ limsup (fun i => 1 - (μs i) E) L ≤ 1 - μ E","state_after":"case inr\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nE : Set Ω\nE_mble : MeasurableSet E\nh : μ E ≤ liminf (fun i => (μs i) E) L\nhne : L.NeBot\nmeas_Ec : μ Eᶜ = 1 - μ E\nmeas_i_Ec : ∀ (i : ι), (μs i) Eᶜ = 1 - (μs i) E\nobs : limsup (fun i => 1 - (μs i) E) L = limsup ((fun x => 1 - x) ∘ fun i => (μs i) E) L\n⊢ limsup ((fun x => 1 - x) ∘ fun i => (μs i) E) L ≤ 1 - μ E","tactic":"rw [obs]","premises":[]},{"state_before":"case inr\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nE : Set Ω\nE_mble : MeasurableSet E\nh : μ E ≤ liminf (fun i => (μs i) E) L\nhne : L.NeBot\nmeas_Ec : μ Eᶜ = 1 - μ E\nmeas_i_Ec : ∀ (i : ι), (μs i) Eᶜ = 1 - (μs i) E\nobs : limsup (fun i => 1 - (μs i) E) L = limsup ((fun x => 1 - x) ∘ fun i => (μs i) E) L\n⊢ limsup ((fun x => 1 - x) ∘ fun i => (μs i) E) L ≤ 1 - μ E","state_after":"case inr\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nE : Set Ω\nE_mble : MeasurableSet E\nh : μ E ≤ liminf (fun i => (μs i) E) L\nhne : L.NeBot\nmeas_Ec : μ Eᶜ = 1 - μ E\nmeas_i_Ec : ∀ (i : ι), (μs i) Eᶜ = 1 - (μs i) E\nobs : limsup (fun i => 1 - (μs i) E) L = limsup ((fun x => 1 - x) ∘ fun i => (μs i) E) L\nthis : 1 - liminf (fun i => (μs i) E) L = limsup ((fun x => 1 - x) ∘ fun i => (μs i) E) L\n⊢ limsup ((fun x => 1 - x) ∘ fun i => (μs i) E) L ≤ 1 - μ E","tactic":"have := antitone_const_tsub.map_liminf_of_continuousAt (F := L)\n    (fun i => μs i E) (ENNReal.continuous_sub_left ENNReal.one_ne_top).continuousAt","premises":[{"full_name":"Antitone.map_liminf_of_continuousAt","def_path":"Mathlib/Topology/Algebra/Order/LiminfLimsup.lean","def_pos":[370,8],"def_end_pos":[370,43]},{"full_name":"Continuous.continuousAt","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1424,8],"def_end_pos":[1424,31]},{"full_name":"ENNReal.continuous_sub_left","def_path":"Mathlib/Topology/Instances/ENNReal.lean","def_pos":[401,8],"def_end_pos":[401,27]},{"full_name":"ENNReal.one_ne_top","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[308,16],"def_end_pos":[308,26]},{"full_name":"antitone_const_tsub","def_path":"Mathlib/Algebra/Order/Sub/Defs.lean","def_pos":[116,8],"def_end_pos":[116,27]}]},{"state_before":"case inr\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nE : Set Ω\nE_mble : MeasurableSet E\nh : μ E ≤ liminf (fun i => (μs i) E) L\nhne : L.NeBot\nmeas_Ec : μ Eᶜ = 1 - μ E\nmeas_i_Ec : ∀ (i : ι), (μs i) Eᶜ = 1 - (μs i) E\nobs : limsup (fun i => 1 - (μs i) E) L = limsup ((fun x => 1 - x) ∘ fun i => (μs i) E) L\nthis : 1 - liminf (fun i => (μs i) E) L = limsup ((fun x => 1 - x) ∘ fun i => (μs i) E) L\n⊢ limsup ((fun x => 1 - x) ∘ fun i => (μs i) E) L ≤ 1 - μ E","state_after":"case inr\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nE : Set Ω\nE_mble : MeasurableSet E\nh : μ E ≤ liminf (fun i => (μs i) E) L\nhne : L.NeBot\nmeas_Ec : μ Eᶜ = 1 - μ E\nmeas_i_Ec : ∀ (i : ι), (μs i) Eᶜ = 1 - (μs i) E\nobs : limsup (fun i => 1 - (μs i) E) L = limsup ((fun x => 1 - x) ∘ fun i => (μs i) E) L\nthis : 1 - liminf (fun i => (μs i) E) L = limsup ((fun x => 1 - x) ∘ fun i => (μs i) E) L\n⊢ 1 - liminf (fun i => (μs i) E) L ≤ 1 - μ E","tactic":"simp_rw [← this]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]}]},{"state_before":"case inr\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nE : Set Ω\nE_mble : MeasurableSet E\nh : μ E ≤ liminf (fun i => (μs i) E) L\nhne : L.NeBot\nmeas_Ec : μ Eᶜ = 1 - μ E\nmeas_i_Ec : ∀ (i : ι), (μs i) Eᶜ = 1 - (μs i) E\nobs : limsup (fun i => 1 - (μs i) E) L = limsup ((fun x => 1 - x) ∘ fun i => (μs i) E) L\nthis : 1 - liminf (fun i => (μs i) E) L = limsup ((fun x => 1 - x) ∘ fun i => (μs i) E) L\n⊢ 1 - liminf (fun i => (μs i) E) L ≤ 1 - μ E","state_after":"no goals","tactic":"exact antitone_const_tsub h","premises":[{"full_name":"antitone_const_tsub","def_path":"Mathlib/Algebra/Order/Sub/Defs.lean","def_pos":[116,8],"def_end_pos":[116,27]}]}]}
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⊤","premises":[{"full_name":"Set.encard","def_path":"Mathlib/Data/Set/Card.lean","def_pos":[62,18],"def_end_pos":[62,24]},{"full_name":"Top.top","def_path":"Mathlib/Order/Notation.lean","def_pos":[94,2],"def_end_pos":[94,5]},{"full_name":"eq_or_ne","def_path":"Mathlib/Logic/Basic.lean","def_pos":[167,8],"def_end_pos":[167,16]}]},{"state_before":"case inr\nα : Type u_1\nβ : Type u_2\ns t : Set α\nk : ℕ∞\nhst : s ⊆ t\nhsk : s.encard ≤ k\nhkt : k ≤ t.encard\nhs : s.encard ≠ ⊤\n⊢ ∃ r, s ⊆ r ∧ r ⊆ t ∧ r.encard = k","state_after":"case inr.intro\nα : Type u_1\nβ : Type u_2\ns t : Set α\nhst : s ⊆ t\nhs : s.encard ≠ ⊤\nk : ℕ∞\nhsk : s.encard ≤ s.encard + k\nhkt : s.encard + k ≤ t.encard\n⊢ ∃ r, s ⊆ r ∧ r ⊆ t ∧ r.encard = s.encard + k","tactic":"obtain ⟨k, rfl⟩ := exists_add_of_le hsk","premises":[{"full_name":"ExistsAddOfLE.exists_add_of_le","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/ExistsOfLE.lean","def_pos":[30,2],"def_end_pos":[30,18]}]},{"state_before":"case inr.intro\nα : Type u_1\nβ : Type u_2\ns t : Set α\nhst : s ⊆ t\nhs : s.encard ≠ ⊤\nk : ℕ∞\nhsk : s.encard ≤ s.encard + k\nhkt : s.encard + k ≤ t.encard\n⊢ ∃ r, s ⊆ r ∧ r ⊆ t ∧ r.encard = s.encard + k","state_after":"case inr.intro.intro\nα : Type u_1\nβ : Type u_2\ns t : Set α\nhst : s ⊆ t\nhs : s.encard ≠ ⊤\nk : ℕ∞\nhsk : s.encard ≤ s.encard + k\nhkt : s.encard + k ≤ t.encard\nk' : ℕ∞\nhk' : t.encard = s.encard + k + k'\n⊢ ∃ r, s ⊆ r ∧ r ⊆ t ∧ r.encard = s.encard + k","tactic":"obtain ⟨k', hk'⟩ := exists_add_of_le hkt","premises":[{"full_name":"ExistsAddOfLE.exists_add_of_le","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/ExistsOfLE.lean","def_pos":[30,2],"def_end_pos":[30,18]}]},{"state_before":"case inr.intro.intro\nα : Type u_1\nβ : Type u_2\ns t : Set α\nhst : s ⊆ t\nhs : s.encard ≠ ⊤\nk : ℕ∞\nhsk : s.encard ≤ s.encard + k\nhkt : s.encard + k ≤ t.encard\nk' : ℕ∞\nhk' : t.encard = s.encard + k + k'\n⊢ ∃ r, s ⊆ r ∧ r ⊆ t ∧ r.encard = s.encard + k","state_after":"case inr.intro.intro\nα : Type u_1\nβ : Type u_2\ns t : Set α\nhst : s ⊆ t\nhs : s.encard ≠ ⊤\nk : ℕ∞\nhsk : s.encard ≤ s.encard + k\nhkt : s.encard + k ≤ t.encard\nk' : ℕ∞\nhk' : t.encard = s.encard + k + k'\nhk : k ≤ (t \\ s).encard\n⊢ ∃ r, s ⊆ r ∧ r ⊆ t ∧ r.encard = s.encard + k","tactic":"have hk : k ≤ encard (t \\ s) := by\n    rw [← encard_diff_add_encard_of_subset hst, add_comm] at hkt\n    exact WithTop.le_of_add_le_add_right hs hkt","premises":[{"full_name":"SDiff.sdiff","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[415,2],"def_end_pos":[415,7]},{"full_name":"Set.encard","def_path":"Mathlib/Data/Set/Card.lean","def_pos":[62,18],"def_end_pos":[62,24]},{"full_name":"Set.encard_diff_add_encard_of_subset","def_path":"Mathlib/Data/Set/Card.lean","def_pos":[157,8],"def_end_pos":[157,40]},{"full_name":"WithTop.le_of_add_le_add_right","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/WithTop.lean","def_pos":[187,18],"def_end_pos":[187,40]},{"full_name":"add_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[293,2],"def_end_pos":[293,13]}]},{"state_before":"case inr.intro.intro\nα : Type u_1\nβ : Type u_2\ns t : Set α\nhst : s ⊆ t\nhs : s.encard ≠ ⊤\nk : ℕ∞\nhsk : s.encard ≤ s.encard + k\nhkt : s.encard + k ≤ t.encard\nk' : ℕ∞\nhk' : t.encard = s.encard + k + k'\nhk : k ≤ (t \\ s).encard\n⊢ ∃ r, s ⊆ r ∧ r ⊆ t ∧ r.encard = s.encard + k","state_after":"case inr.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\ns t : Set α\nhst : s ⊆ t\nhs : s.encard ≠ ⊤\nk' : ℕ∞\nr' : Set α\nhr' : r' ⊆ t \\ s\nhsk : s.encard ≤ s.encard + r'.encard\nhkt : s.encard + r'.encard ≤ t.encard\nhk' : t.encard = s.encard + r'.encard + k'\nhk : r'.encard ≤ (t \\ s).encard\n⊢ ∃ r, s ⊆ r ∧ r ⊆ t ∧ r.encard = s.encard + r'.encard","tactic":"obtain ⟨r', hr', rfl⟩ := exists_subset_encard_eq hk","premises":[{"full_name":"Set.exists_subset_encard_eq","def_path":"Mathlib/Data/Set/Card.lean","def_pos":[350,8],"def_end_pos":[350,31]}]},{"state_before":"case inr.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\ns t : Set α\nhst : s ⊆ t\nhs : s.encard ≠ ⊤\nk' : ℕ∞\nr' : Set α\nhr' : r' ⊆ t \\ s\nhsk : s.encard ≤ s.encard + r'.encard\nhkt : s.encard + r'.encard ≤ t.encard\nhk' : t.encard = s.encard + r'.encard + k'\nhk : r'.encard ≤ (t \\ s).encard\n⊢ ∃ r, s ⊆ r ∧ r ⊆ t ∧ r.encard = s.encard + r'.encard","state_after":"case inr.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\ns t : Set α\nhst : s ⊆ t\nhs : s.encard ≠ ⊤\nk' : ℕ∞\nr' : Set α\nhr' : r' ⊆ t \\ s\nhsk : s.encard ≤ s.encard + r'.encard\nhkt : s.encard + r'.encard ≤ t.encard\nhk' : t.encard = s.encard + r'.encard + k'\nhk : r'.encard ≤ (t \\ s).encard\n⊢ (s ∪ r').encard = s.encard + r'.encard","tactic":"refine ⟨s ∪ r', subset_union_left, union_subset hst (hr'.trans diff_subset), ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Set.diff_subset","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1432,8],"def_end_pos":[1432,19]},{"full_name":"Set.subset_union_left","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[657,8],"def_end_pos":[657,25]},{"full_name":"Set.union_subset","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[662,8],"def_end_pos":[662,20]},{"full_name":"Union.union","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[402,2],"def_end_pos":[402,7]}]},{"state_before":"case inr.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\ns t : Set α\nhst : s ⊆ t\nhs : s.encard ≠ ⊤\nk' : ℕ∞\nr' : Set α\nhr' : r' ⊆ t \\ s\nhsk : s.encard ≤ s.encard + r'.encard\nhkt : s.encard + r'.encard ≤ t.encard\nhk' : t.encard = s.encard + r'.encard + k'\nhk : r'.encard ≤ (t \\ s).encard\n⊢ (s ∪ r').encard = s.encard + r'.encard","state_after":"no goals","tactic":"rw [encard_union_eq (disjoint_of_subset_right hr' disjoint_sdiff_right)]","premises":[{"full_name":"Set.disjoint_of_subset_right","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1237,6],"def_end_pos":[1237,30]},{"full_name":"Set.disjoint_sdiff_right","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1256,6],"def_end_pos":[1256,26]},{"full_name":"Set.encard_union_eq","def_path":"Mathlib/Data/Set/Card.lean","def_pos":[110,8],"def_end_pos":[110,23]}]}]}
+{"url":"Mathlib/Data/Finsupp/Basic.lean","commit":"","full_name":"Finsupp.support_smul_eq","start":[1355,0],"end":[1358,54],"file_path":"Mathlib/Data/Finsupp/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : NoZeroSMulDivisors R M\nb : R\nhb : b ≠ 0\ng : α →₀ M\na : α\n⊢ a ∈ (b • g).support ↔ a ∈ g.support","state_after":"no goals","tactic":"simp [Finsupp.smul_apply, hb]","premises":[{"full_name":"Finsupp.smul_apply","def_path":"Mathlib/Data/Finsupp/Basic.lean","def_pos":[1301,8],"def_end_pos":[1301,18]}]}]}
+{"url":"Mathlib/LinearAlgebra/Projection.lean","commit":"","full_name":"Submodule.prodEquivOfIsCompl_symm_apply_left","start":[106,0],"end":[109,55],"file_path":"Mathlib/LinearAlgebra/Projection.lean","tactics":[{"state_before":"R : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type u_4\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type u_5\ninst✝² : Semiring S\nM : Type u_6\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nh : IsCompl p q\nx : ↥p\n⊢ ↑x = (p.prodEquivOfIsCompl q h) (x, 0)","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/Data/Nat/PartENat.lean","commit":"","full_name":"PartENat.coe_lt_iff","start":[301,0],"end":[304,5],"file_path":"Mathlib/Data/Nat/PartENat.lean","tactics":[{"state_before":"n : ℕ\nx : PartENat\n⊢ ↑n < x ↔ ∀ (h : x.Dom), n < x.get h","state_after":"n : ℕ\nx : PartENat\n⊢ (∀ (hy : x.Dom), (↑n).get ⋯ < x.get hy) ↔ ∀ (h : x.Dom), n < x.get h","tactic":"simp only [lt_def, exists_prop_of_true, dom_some, forall_true_iff]","premises":[{"full_name":"PartENat.dom_some","def_path":"Mathlib/Data/Nat/PartENat.lean","def_pos":[84,8],"def_end_pos":[84,16]},{"full_name":"PartENat.lt_def","def_path":"Mathlib/Data/Nat/PartENat.lean","def_pos":[231,8],"def_end_pos":[231,14]},{"full_name":"exists_prop_of_true","def_path":"Mathlib/Logic/Basic.lean","def_pos":[641,8],"def_end_pos":[641,27]},{"full_name":"forall_true_iff","def_path":"Mathlib/Logic/Basic.lean","def_pos":[497,8],"def_end_pos":[497,23]}]},{"state_before":"n : ℕ\nx : PartENat\n⊢ (∀ (hy : x.Dom), (↑n).get ⋯ < x.get hy) ↔ ∀ (h : x.Dom), n < x.get h","state_after":"no goals","tactic":"rfl","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]}]}]}
+{"url":"Mathlib/Topology/Order/Lattice.lean","commit":"","full_name":"ContinuousWithinAt.finset_inf","start":[375,0],"end":[377,59],"file_path":"Mathlib/Topology/Order/Lattice.lean","tactics":[{"state_before":"L : Type u_1\nX : Type u_2\ninst✝⁴ : TopologicalSpace L\ninst✝³ : TopologicalSpace X\nι : Type u_3\ninst✝² : SemilatticeInf L\ninst✝¹ : OrderTop L\ninst✝ : ContinuousInf L\ns : Finset ι\nf : ι → X → L\nt : Set X\nx : X\nhs : ∀ i ∈ s, ContinuousWithinAt (f i) t x\n⊢ ContinuousWithinAt (s.inf f) t x","state_after":"no goals","tactic":"simpa only [← Finset.inf_apply] using finset_inf_apply hs","premises":[{"full_name":"ContinuousWithinAt.finset_inf_apply","def_path":"Mathlib/Topology/Order/Lattice.lean","def_pos":[370,6],"def_end_pos":[370,41]},{"full_name":"Finset.inf_apply","def_path":"Mathlib/Data/Finset/Lattice.lean","def_pos":[1050,18],"def_end_pos":[1050,27]}]}]}
+{"url":"Mathlib/MeasureTheory/Function/L1Space.lean","commit":"","full_name":"MeasureTheory.integrable_add_iff_integrable_left","start":[614,0],"end":[617,55],"file_path":"Mathlib/MeasureTheory/Function/L1Space.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf g : α → β\nhf : Integrable f μ\n⊢ Integrable (g + f) μ ↔ Integrable g μ","state_after":"no goals","tactic":"rw [add_comm, integrable_add_iff_integrable_right hf]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"MeasureTheory.integrable_add_iff_integrable_right","def_path":"Mathlib/MeasureTheory/Function/L1Space.lean","def_pos":[609,6],"def_end_pos":[609,41]},{"full_name":"add_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[293,2],"def_end_pos":[293,13]}]}]}
+{"url":"Mathlib/Combinatorics/Additive/AP/Three/Defs.lean","commit":"","full_name":"threeAPFree_iff_eq_right","start":[230,0],"end":[237,43],"file_path":"Mathlib/Combinatorics/Additive/AP/Three/Defs.lean","tactics":[{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\n𝕜 : Type u_4\nE : Type u_5\ns : Set ℕ\n⊢ ThreeAPFree s ↔ ∀ ⦃a : ℕ⦄, a ∈ s → ∀ ⦃b : ℕ⦄, b ∈ s → ∀ ⦃c : ℕ⦄, c ∈ s → a + c = b + b → a = c","state_after":"case refine_1\nF : Type u_1\nα : Type u_2\nβ : Type u_3\n𝕜 : Type u_4\nE : Type u_5\ns : Set ℕ\na : ℕ\n_ha : a ∈ s\nb : ℕ\nhb : b ∈ s\nc : ℕ\nhc : c ∈ s\nhabc : a + c = b + b\n⊢ a = b → a = c\n\ncase refine_2\nF : Type u_1\nα : Type u_2\nβ : Type u_3\n𝕜 : Type u_4\nE : Type u_5\ns : Set ℕ\na : ℕ\n_ha : a ∈ s\nb : ℕ\nhb : b ∈ s\nc : ℕ\nhc : c ∈ s\nhabc : a + c = b + b\n⊢ a = c → a = b","tactic":"refine forall₄_congr fun a _ha b hb => forall₃_congr fun c hc habc => ⟨?_, ?_⟩","premises":[{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"forall₃_congr","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[224,8],"def_end_pos":[224,21]},{"full_name":"forall₄_congr","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[233,8],"def_end_pos":[233,21]}]}]}
+{"url":"Mathlib/RingTheory/Int/Basic.lean","commit":"","full_name":"Int.gcd_eq_one_iff_coprime","start":[33,0],"end":[46,85],"file_path":"Mathlib/RingTheory/Int/Basic.lean","tactics":[{"state_before":"a b : ℤ\n⊢ a.gcd b = 1 ↔ IsCoprime a b","state_after":"case mp\na b : ℤ\n⊢ a.gcd b = 1 → IsCoprime a b\n\ncase mpr\na b : ℤ\n⊢ IsCoprime a b → a.gcd b = 1","tactic":"constructor","premises":[]}]}
+{"url":"Mathlib/Algebra/Polynomial/EraseLead.lean","commit":"","full_name":"Polynomial.map_natDegree_eq_sub","start":[307,0],"end":[314,17],"file_path":"Mathlib/Algebra/Polynomial/EraseLead.lean","tactics":[{"state_before":"R : Type u_1\ninst✝³ : Semiring R\nf : R[X]\nS : Type u_2\nF : Type u_3\ninst✝² : Semiring S\ninst✝¹ : FunLike F R[X] S[X]\ninst✝ : AddMonoidHomClass F R[X] S[X]\nφ : F\np : R[X]\nk : ℕ\nφ_k : ∀ (f : R[X]), f.natDegree < k → φ f = 0\nφ_mon : ∀ (n : ℕ) (c : R), c ≠ 0 → (φ ((monomial n) c)).natDegree = n - k\n⊢ ∀ {n : ℕ}, n ≤ k → (fun j => j - k) n = 0","state_after":"no goals","tactic":"simp_all","premises":[]}]}
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+{"url":"Mathlib/Algebra/Homology/ShortComplex/PreservesHomology.lean","commit":"","full_name":"CategoryTheory.ShortComplex.RightHomologyData.map_opcyclesMap'","start":[377,0],"end":[380,94],"file_path":"Mathlib/Algebra/Homology/ShortComplex/PreservesHomology.lean","tactics":[{"state_before":"C : Type u_1\nD : Type u_2\ninst✝⁷ : Category.{u_4, u_1} C\ninst✝⁶ : Category.{u_3, u_2} D\ninst✝⁵ : HasZeroMorphisms C\ninst✝⁴ : HasZeroMorphisms D\nS : ShortComplex C\nh₁✝ : S.LeftHomologyData\nh₂✝ : S.RightHomologyData\nF✝ : C ⥤ D\ninst✝³ : F✝.PreservesZeroMorphisms\nhl : S.LeftHomologyData\nhr : S.RightHomologyData\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nhl₁ : S₁.LeftHomologyData\nhr₁ : S₁.RightHomologyData\nhl₂ : S₂.LeftHomologyData\nhr₂ : S₂.RightHomologyData\nh₁ : S₁.HomologyData\nh₂ : S₂.HomologyData\nF : C ⥤ D\ninst✝² : F.PreservesZeroMorphisms\ninst✝¹ : hr₁.IsPreservedBy F\ninst✝ : hr₂.IsPreservedBy F\n⊢ F.map (opcyclesMap' φ hr₁ hr₂) = opcyclesMap' (F.mapShortComplex.map φ) (hr₁.map F) (hr₂.map F)","state_after":"C : Type u_1\nD : Type u_2\ninst✝⁷ : Category.{u_4, u_1} C\ninst✝⁶ : Category.{u_3, u_2} D\ninst✝⁵ : HasZeroMorphisms C\ninst✝⁴ : HasZeroMorphisms D\nS : ShortComplex C\nh₁✝ : S.LeftHomologyData\nh₂✝ : S.RightHomologyData\nF✝ : C ⥤ D\ninst✝³ : F✝.PreservesZeroMorphisms\nhl : S.LeftHomologyData\nhr : S.RightHomologyData\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nhl₁ : S₁.LeftHomologyData\nhr₁ : S₁.RightHomologyData\nhl₂ : S₂.LeftHomologyData\nhr₂ : S₂.RightHomologyData\nh₁ : S₁.HomologyData\nh₂ : S₂.HomologyData\nF : C ⥤ D\ninst✝² : F.PreservesZeroMorphisms\ninst✝¹ : hr₁.IsPreservedBy F\ninst✝ : hr₂.IsPreservedBy F\nγ : RightHomologyMapData φ hr₁ hr₂\n⊢ F.map (opcyclesMap' φ hr₁ hr₂) = opcyclesMap' (F.mapShortComplex.map φ) (hr₁.map F) (hr₂.map F)","tactic":"have γ : ShortComplex.RightHomologyMapData φ hr₁ hr₂ := default","premises":[{"full_name":"CategoryTheory.ShortComplex.RightHomologyMapData","def_path":"Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean","def_pos":[344,10],"def_end_pos":[344,30]},{"full_name":"Inhabited.default","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[697,2],"def_end_pos":[697,9]}]},{"state_before":"C : Type u_1\nD : Type u_2\ninst✝⁷ : Category.{u_4, u_1} C\ninst✝⁶ : Category.{u_3, u_2} D\ninst✝⁵ : HasZeroMorphisms C\ninst✝⁴ : HasZeroMorphisms D\nS : ShortComplex C\nh₁✝ : S.LeftHomologyData\nh₂✝ : S.RightHomologyData\nF✝ : C ⥤ D\ninst✝³ : F✝.PreservesZeroMorphisms\nhl : S.LeftHomologyData\nhr : S.RightHomologyData\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nhl₁ : S₁.LeftHomologyData\nhr₁ : S₁.RightHomologyData\nhl₂ : S₂.LeftHomologyData\nhr₂ : S₂.RightHomologyData\nh₁ : S₁.HomologyData\nh₂ : S₂.HomologyData\nF : C ⥤ D\ninst✝² : F.PreservesZeroMorphisms\ninst✝¹ : hr₁.IsPreservedBy F\ninst✝ : hr₂.IsPreservedBy F\nγ : RightHomologyMapData φ hr₁ hr₂\n⊢ F.map (opcyclesMap' φ hr₁ hr₂) = opcyclesMap' (F.mapShortComplex.map φ) (hr₁.map F) (hr₂.map F)","state_after":"no goals","tactic":"rw [γ.opcyclesMap'_eq, (γ.map F).opcyclesMap'_eq,  ShortComplex.RightHomologyMapData.map_φQ]","premises":[{"full_name":"CategoryTheory.ShortComplex.RightHomologyMapData.map","def_path":"Mathlib/Algebra/Homology/ShortComplex/PreservesHomology.lean","def_pos":[217,4],"def_end_pos":[217,28]},{"full_name":"CategoryTheory.ShortComplex.RightHomologyMapData.map_φQ","def_path":"Mathlib/Algebra/Homology/ShortComplex/PreservesHomology.lean","def_pos":[216,2],"def_end_pos":[216,7]},{"full_name":"CategoryTheory.ShortComplex.RightHomologyMapData.opcyclesMap'_eq","def_path":"Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean","def_pos":[640,6],"def_end_pos":[640,21]}]}]}
+{"url":"Mathlib/Combinatorics/Schnirelmann.lean","commit":"","full_name":"schnirelmannDensity_eq_one_iff","start":[113,0],"end":[126,32],"file_path":"Mathlib/Combinatorics/Schnirelmann.lean","tactics":[{"state_before":"A : Set ℕ\ninst✝ : DecidablePred fun x => x ∈ A\n⊢ schnirelmannDensity A = 1 ↔ {0}ᶜ ⊆ A","state_after":"A : Set ℕ\ninst✝ : DecidablePred fun x => x ∈ A\n⊢ 1 ≤ schnirelmannDensity A ↔ {0}ᶜ ⊆ A","tactic":"rw [le_antisymm_iff, and_iff_right schnirelmannDensity_le_one]","premises":[{"full_name":"and_iff_right","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[70,8],"def_end_pos":[70,21]},{"full_name":"le_antisymm_iff","def_path":"Mathlib/Order/Defs.lean","def_pos":[161,8],"def_end_pos":[161,23]},{"full_name":"schnirelmannDensity_le_one","def_path":"Mathlib/Combinatorics/Schnirelmann.lean","def_pos":[84,6],"def_end_pos":[84,32]}]},{"state_before":"A : Set ℕ\ninst✝ : DecidablePred fun x => x ∈ A\n⊢ 1 ≤ schnirelmannDensity A ↔ {0}ᶜ ⊆ A","state_after":"case mp\nA : Set ℕ\ninst✝ : DecidablePred fun x => x ∈ A\n⊢ 1 ≤ schnirelmannDensity A → {0}ᶜ ⊆ A\n\ncase mpr\nA : Set ℕ\ninst✝ : DecidablePred fun x => x ∈ A\n⊢ {0}ᶜ ⊆ A → 1 ≤ schnirelmannDensity A","tactic":"constructor","premises":[]}]}
+{"url":"Mathlib/RingTheory/Localization/NumDen.lean","commit":"","full_name":"IsFractionRing.isUnit_den_iff","start":[110,0],"end":[132,53],"file_path":"Mathlib/RingTheory/Localization/NumDen.lean","tactics":[{"state_before":"R : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommRing P\nA : Type u_4\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nh : IsUnit ↑(den A x)\n⊢ IsInteger A x","state_after":"case intro\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommRing P\nA : Type u_4\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nh : IsUnit ↑(IsFractionRing.den A x)\nden : A\nhd : ↑(IsFractionRing.den A x) * den = 1\n⊢ IsInteger A x","tactic":"obtain ⟨den, hd⟩ := IsUnit.exists_right_inv h","premises":[{"full_name":"IsUnit.exists_right_inv","def_path":"Mathlib/Algebra/Group/Units.lean","def_pos":[575,6],"def_end_pos":[575,29]}]},{"state_before":"case intro\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommRing P\nA : Type u_4\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nh : IsUnit ↑(IsFractionRing.den A x)\nden : A\nhd : ↑(IsFractionRing.den A x) * den = 1\n⊢ IsInteger A x","state_after":"case h\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommRing P\nA : Type u_4\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nh : IsUnit ↑(IsFractionRing.den A x)\nden : A\nhd : ↑(IsFractionRing.den A x) * den = 1\n⊢ (algebraMap A K) (num A x * den) = x","tactic":"use (num A x) * den","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"IsFractionRing.num","def_path":"Mathlib/RingTheory/Localization/NumDen.lean","def_pos":[48,18],"def_end_pos":[48,21]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]}]},{"state_before":"case h\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommRing P\nA : Type u_4\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nh : IsUnit ↑(IsFractionRing.den A x)\nden : A\nhd : ↑(IsFractionRing.den A x) * den = 1\n⊢ (algebraMap A K) (num A x * den) = x","state_after":"case h\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommRing P\nA : Type u_4\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nh : IsUnit ↑(IsFractionRing.den A x)\nden : A\nhd : ↑(IsFractionRing.den A x) * den = 1\n⊢ (algebraMap A K) (num A x * den) = (algebraMap A K) (num A x) / (algebraMap A K) ↑(IsFractionRing.den A x)","tactic":"conv => rhs; rw [← mk'_num_den' A x]","premises":[{"full_name":"IsFractionRing.mk'_num_den'","def_path":"Mathlib/RingTheory/Localization/NumDen.lean","def_pos":[63,8],"def_end_pos":[63,20]}]},{"state_before":"case h\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommRing P\nA : Type u_4\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nh : IsUnit ↑(IsFractionRing.den A x)\nden : A\nhd : ↑(IsFractionRing.den A x) * den = 1\n⊢ (algebraMap A K) (num A x * den) = (algebraMap A K) (num A x) / (algebraMap A K) ↑(IsFractionRing.den A x)","state_after":"case h\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommRing P\nA : Type u_4\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nh : IsUnit ↑(IsFractionRing.den A x)\nden : A\nhd : ↑(IsFractionRing.den A x) * den = 1\n⊢ (algebraMap A K) (num A x) * (algebraMap A K) den =\n    (algebraMap A K) (num A x) * ((algebraMap A K) ↑(IsFractionRing.den A x))⁻¹","tactic":"rw [map_mul, div_eq_mul_inv]","premises":[{"full_name":"div_eq_mul_inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[907,8],"def_end_pos":[907,22]},{"full_name":"map_mul","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[281,8],"def_end_pos":[281,15]}]},{"state_before":"case h\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommRing P\nA : Type u_4\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nh : IsUnit ↑(IsFractionRing.den A x)\nden : A\nhd : ↑(IsFractionRing.den A x) * den = 1\n⊢ (algebraMap A K) (num A x) * (algebraMap A K) den =\n    (algebraMap A K) (num A x) * ((algebraMap A K) ↑(IsFractionRing.den A x))⁻¹","state_after":"case h.e_a\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommRing P\nA : Type u_4\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nh : IsUnit ↑(IsFractionRing.den A x)\nden : A\nhd : ↑(IsFractionRing.den A x) * den = 1\n⊢ (algebraMap A K) den = ((algebraMap A K) ↑(IsFractionRing.den A x))⁻¹","tactic":"congr 1","premises":[]},{"state_before":"case h.e_a\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommRing P\nA : Type u_4\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nh : IsUnit ↑(IsFractionRing.den A x)\nden : A\nhd : ↑(IsFractionRing.den A x) * den = 1\n⊢ (algebraMap A K) den = ((algebraMap A K) ↑(IsFractionRing.den A x))⁻¹","state_after":"case h.e_a.h\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommRing P\nA : Type u_4\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nh : IsUnit ↑(IsFractionRing.den A x)\nden : A\nhd : ↑(IsFractionRing.den A x) * den = 1\n⊢ (algebraMap A K) ↑(IsFractionRing.den A x) * (algebraMap A K) den = 1","tactic":"apply eq_inv_of_mul_eq_one_right","premises":[{"full_name":"eq_inv_of_mul_eq_one_right","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[369,8],"def_end_pos":[369,34]}]},{"state_before":"case h.e_a.h\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommRing P\nA : Type u_4\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nh : IsUnit ↑(IsFractionRing.den A x)\nden : A\nhd : ↑(IsFractionRing.den A x) * den = 1\n⊢ (algebraMap A K) ↑(IsFractionRing.den A x) * (algebraMap A K) den = 1","state_after":"case h.e_a.h\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommRing P\nA : Type u_4\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nh : IsUnit ↑(IsFractionRing.den A x)\nden : A\nhd : ↑(IsFractionRing.den A x) * den = 1\n⊢ (algebraMap A K) (↑(IsFractionRing.den A x) * den) = 1","tactic":"rw [← map_mul]","premises":[{"full_name":"map_mul","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[281,8],"def_end_pos":[281,15]}]},{"state_before":"case h.e_a.h\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommRing P\nA : Type u_4\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nh : IsUnit ↑(IsFractionRing.den A x)\nden : A\nhd : ↑(IsFractionRing.den A x) * den = 1\n⊢ (algebraMap A K) (↑(IsFractionRing.den A x) * den) = 1","state_after":"no goals","tactic":"simp only [hd, OneMemClass.coe_one, map_one]","premises":[{"full_name":"OneMemClass.coe_one","def_path":"Mathlib/Algebra/Group/Submonoid/Operations.lean","def_pos":[408,8],"def_end_pos":[408,15]},{"full_name":"map_one","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[190,8],"def_end_pos":[190,15]}]},{"state_before":"R : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommRing P\nA : Type u_4\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : 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u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommRing P\nA : Type u_4\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nh✝ : IsInteger A x\nv : A\nh : (algebraMap A K) v = x\n⊢ Function.Injective ⇑(algebraMap A K)","state_after":"no goals","tactic":"exact NoZeroSMulDivisors.algebraMap_injective A K","premises":[{"full_name":"NoZeroSMulDivisors.algebraMap_injective","def_path":"Mathlib/Algebra/Algebra/Basic.lean","def_pos":[282,8],"def_end_pos":[282,28]}]}]}
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+{"url":"Mathlib/CategoryTheory/Triangulated/Functor.lean","commit":"","full_name":"CategoryTheory.Functor.mapTriangleInvRotateIso_inv_app_hom₃","start":[107,0],"end":[114,66],"file_path":"Mathlib/CategoryTheory/Triangulated/Functor.lean","tactics":[{"state_before":"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝¹¹ : Category.{?u.222118, u_1} C\ninst✝¹⁰ : Category.{?u.222122, u_2} D\ninst✝⁹ : Category.{?u.222126, u_3} E\ninst✝⁸ : HasShift C ℤ\ninst✝⁷ : HasShift D ℤ\ninst✝⁶ : HasShift E ℤ\nF : C ⥤ D\ninst✝⁵ : F.CommShift ℤ\nG : D ⥤ E\ninst✝⁴ : G.CommShift ℤ\ninst✝³ : Preadditive C\ninst✝² : Preadditive D\ninst✝¹ inst✝ : F.Additive\nT : Triangle C\n⊢ ((F.mapTriangle ⋙ invRotate D).obj T).mor₁ ≫ (Iso.refl ((F.mapTriangle ⋙ invRotate D).obj T).obj₂).hom =\n    ((F.commShiftIso (-1)).symm.app T.obj₃).hom ≫ ((invRotate C ⋙ F.mapTriangle).obj T).mor₁","state_after":"no goals","tactic":"aesop_cat","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]}]},{"state_before":"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝¹¹ : Category.{?u.222118, u_1} C\ninst✝¹⁰ : Category.{?u.222122, u_2} D\ninst✝⁹ : Category.{?u.222126, u_3} E\ninst✝⁸ : HasShift C ℤ\ninst✝⁷ : HasShift D ℤ\ninst✝⁶ : HasShift E ℤ\nF : C ⥤ D\ninst✝⁵ : F.CommShift ℤ\nG : D ⥤ E\ninst✝⁴ : G.CommShift ℤ\ninst✝³ : Preadditive C\ninst✝² : Preadditive D\ninst✝¹ inst✝ : F.Additive\nT : Triangle C\n⊢ ((F.mapTriangle ⋙ invRotate D).obj T).mor₂ ≫ (Iso.refl ((F.mapTriangle ⋙ invRotate D).obj T).obj₃).hom =\n    (Iso.refl ((F.mapTriangle ⋙ invRotate D).obj T).obj₂).hom ≫ ((invRotate C ⋙ F.mapTriangle).obj T).mor₂","state_after":"no goals","tactic":"aesop_cat","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]}]},{"state_before":"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝¹¹ : Category.{?u.222118, u_1} C\ninst✝¹⁰ : Category.{?u.222122, u_2} D\ninst✝⁹ : Category.{?u.222126, u_3} E\ninst✝⁸ : HasShift C ℤ\ninst✝⁷ : HasShift D ℤ\ninst✝⁶ : HasShift E ℤ\nF : C ⥤ D\ninst✝⁵ : F.CommShift ℤ\nG : D ⥤ E\ninst✝⁴ : G.CommShift ℤ\ninst✝³ : Preadditive C\ninst✝² : Preadditive D\ninst✝¹ inst✝ : F.Additive\nT : Triangle C\n⊢ ((F.mapTriangle ⋙ invRotate D).obj T).mor₃ ≫ (shiftFunctor D 1).map ((F.commShiftIso (-1)).symm.app T.obj₃).hom =\n    (Iso.refl ((F.mapTriangle ⋙ invRotate D).obj T).obj₃).hom ≫ ((invRotate C ⋙ F.mapTriangle).obj T).mor₃","state_after":"no goals","tactic":"aesop_cat","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]}]},{"state_before":"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝¹¹ : Category.{?u.222118, u_1} C\ninst✝¹⁰ : Category.{?u.222122, u_2} D\ninst✝⁹ : Category.{?u.222126, u_3} E\ninst✝⁸ : HasShift C ℤ\ninst✝⁷ : HasShift D ℤ\ninst✝⁶ : HasShift E ℤ\nF : C ⥤ D\ninst✝⁵ : F.CommShift ℤ\nG : D ⥤ E\ninst✝⁴ : G.CommShift ℤ\ninst✝³ : Preadditive C\ninst✝² : Preadditive D\ninst✝¹ inst✝ : F.Additive\n⊢ ∀ {X Y : Triangle C} (f : X ⟶ Y),\n    (F.mapTriangle ⋙ invRotate D).map f ≫\n        ((fun T =>\n              ((F.mapTriangle ⋙ invRotate D).obj T).isoMk ((invRotate C ⋙ F.mapTriangle).obj T)\n                ((F.commShiftIso (-1)).symm.app T.obj₃) (Iso.refl ((F.mapTriangle ⋙ invRotate D).obj T).obj₂)\n                (Iso.refl ((F.mapTriangle ⋙ invRotate D).obj T).obj₃) ⋯ ⋯ ⋯)\n            Y).hom =\n      ((fun T =>\n              ((F.mapTriangle ⋙ invRotate D).obj T).isoMk ((invRotate C ⋙ F.mapTriangle).obj T)\n                ((F.commShiftIso (-1)).symm.app T.obj₃) (Iso.refl ((F.mapTriangle ⋙ invRotate D).obj T).obj₂)\n                (Iso.refl ((F.mapTriangle ⋙ invRotate D).obj T).obj₃) ⋯ ⋯ ⋯)\n            X).hom ≫\n        (invRotate C ⋙ F.mapTriangle).map f","state_after":"no goals","tactic":"aesop_cat","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]}]}]}
+{"url":"Mathlib/Order/UpperLower/Basic.lean","commit":"","full_name":"UpperSet.coe_iSup","start":[585,0],"end":[586,90],"file_path":"Mathlib/Order/UpperLower/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nκ : ι → Sort u_5\ninst✝ : LE α\nS : Set (UpperSet α)\ns t : UpperSet α\na : α\nf : ι → UpperSet α\n⊢ ↑(⨆ i, f i) = ⋂ i, ↑(f i)","state_after":"no goals","tactic":"simp [iSup]","premises":[{"full_name":"iSup","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[56,4],"def_end_pos":[56,8]}]}]}
+{"url":"Mathlib/Algebra/Group/Subgroup/Basic.lean","commit":"","full_name":"AddSubgroup.prod_eq_bot_iff","start":[1368,0],"end":[1370,72],"file_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","tactics":[{"state_before":"G : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst✝⁵ : Group G\ninst✝⁴ : Group G'\ninst✝³ : Group G''\nA : Type u_4\ninst✝² : AddGroup A\nH✝ K✝ : Subgroup G\nk : Set G\nN : Type u_5\ninst✝¹ : Group N\nP : Type u_6\ninst✝ : Group P\nH : Subgroup G\nK : Subgroup N\n⊢ H.prod K = ⊥ ↔ H = ⊥ ∧ K = ⊥","state_after":"no goals","tactic":"simpa only [← Subgroup.toSubmonoid_eq] using Submonoid.prod_eq_bot_iff","premises":[{"full_name":"Subgroup.toSubmonoid_eq","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[370,8],"def_end_pos":[370,22]},{"full_name":"Submonoid.prod_eq_bot_iff","def_path":"Mathlib/Algebra/Group/Submonoid/Operations.lean","def_pos":[968,8],"def_end_pos":[968,23]}]}]}
+{"url":"Mathlib/AlgebraicGeometry/Morphisms/QuasiSeparated.lean","commit":"","full_name":"AlgebraicGeometry.is_localization_basicOpen_of_qcqs","start":[328,0],"end":[348,30],"file_path":"Mathlib/AlgebraicGeometry/Morphisms/QuasiSeparated.lean","tactics":[{"state_before":"X✝ Y : Scheme\nf✝ : X✝ ⟶ Y\nX : Scheme\nU : X.Opens\nhU : IsCompact U.carrier\nhU' : IsQuasiSeparated U.carrier\nf : ↑Γ(X, U)\n⊢ IsLocalization.Away f ↑Γ(X, X.basicOpen f)","state_after":"case map_units'\nX✝ Y : Scheme\nf✝ : X✝ ⟶ Y\nX : Scheme\nU : X.Opens\nhU : IsCompact U.carrier\nhU' : IsQuasiSeparated U.carrier\nf : ↑Γ(X, U)\n⊢ ∀ (y : ↥(Submonoid.powers f)), IsUnit ((algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) ↑y)\n\ncase surj'\nX✝ Y : Scheme\nf✝ : X✝ ⟶ Y\nX : Scheme\nU : X.Opens\nhU : IsCompact U.carrier\nhU' : IsQuasiSeparated U.carrier\nf : ↑Γ(X, U)\n⊢ ∀ (z : ↑Γ(X, X.basicOpen f)),\n    ∃ x, z * (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) ↑x.2 = (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) x.1\n\ncase exists_of_eq\nX✝ Y : Scheme\nf✝ : X✝ ⟶ Y\nX : Scheme\nU : X.Opens\nhU : IsCompact U.carrier\nhU' : IsQuasiSeparated U.carrier\nf : ↑Γ(X, U)\n⊢ ∀ {x y : ↑Γ(X, U)},\n    (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) x = (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) y → ∃ c, ↑c * x = ↑c * y","tactic":"constructor","premises":[]}]}
+{"url":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","commit":"","full_name":"Real.rpow_lt_rpow_left_iff_of_base_lt_one","start":[613,0],"end":[616,81],"file_path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","tactics":[{"state_before":"x y z : ℝ\nn : ℕ\nhx0 : 0 < x\nhx1 : x < 1\n⊢ x ^ y < x ^ z ↔ z < y","state_after":"no goals","tactic":"rw [lt_iff_not_le, rpow_le_rpow_left_iff_of_base_lt_one hx0 hx1, lt_iff_not_le]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Real.rpow_le_rpow_left_iff_of_base_lt_one","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","def_pos":[608,8],"def_end_pos":[608,44]},{"full_name":"lt_iff_not_le","def_path":"Mathlib/Order/Basic.lean","def_pos":[391,8],"def_end_pos":[391,21]}]}]}
+{"url":"Mathlib/Data/Finset/Basic.lean","commit":"","full_name":"Finset.ne_insert_of_not_mem","start":[1020,0],"end":[1022,10],"file_path":"Mathlib/Data/Finset/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : DecidableEq α\ns✝ t✝ u v : Finset α\na✝ b : α\ns t : Finset α\na : α\nh : a ∉ s\n⊢ s ≠ insert a t","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : DecidableEq α\ns✝ t✝ u v : Finset α\na✝ b : α\ns t : Finset α\na : α\nh : s = insert a t\n⊢ a ∈ s","tactic":"contrapose! h","premises":[{"full_name":"Mathlib.Tactic.Contrapose.mtr","def_path":"Mathlib/Tactic/Contrapose.lean","def_pos":[24,6],"def_end_pos":[24,9]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : DecidableEq α\ns✝ t✝ u v : Finset α\na✝ b : α\ns t : Finset α\na : α\nh : s = insert a t\n⊢ a ∈ s","state_after":"no goals","tactic":"simp [h]","premises":[]}]}
+{"url":"Mathlib/Data/Real/Hyperreal.lean","commit":"","full_name":"Hyperreal.infiniteNeg_mul_of_infiniteNeg_not_infinitesimal_pos","start":[695,0],"end":[698,60],"file_path":"Mathlib/Data/Real/Hyperreal.lean","tactics":[{"state_before":"x y : ℝ*\n⊢ x.InfiniteNeg → ¬y.Infinitesimal → 0 < y → (x * y).InfiniteNeg","state_after":"x y : ℝ*\n⊢ (-x).InfinitePos → ¬y.Infinitesimal → 0 < y → (-x * y).InfinitePos","tactic":"rw [← infinitePos_neg, ← infinitePos_neg, neg_mul_eq_neg_mul]","premises":[{"full_name":"Hyperreal.infinitePos_neg","def_path":"Mathlib/Data/Real/Hyperreal.lean","def_pos":[385,16],"def_end_pos":[385,31]},{"full_name":"neg_mul_eq_neg_mul","def_path":"Mathlib/Algebra/Ring/Defs.lean","def_pos":[277,8],"def_end_pos":[277,26]}]},{"state_before":"x y : ℝ*\n⊢ (-x).InfinitePos → ¬y.Infinitesimal → 0 < y → (-x * y).InfinitePos","state_after":"no goals","tactic":"exact infinitePos_mul_of_infinitePos_not_infinitesimal_pos","premises":[{"full_name":"Hyperreal.infinitePos_mul_of_infinitePos_not_infinitesimal_pos","def_path":"Mathlib/Data/Real/Hyperreal.lean","def_pos":[664,8],"def_end_pos":[664,60]}]}]}
+{"url":"Mathlib/Topology/Order/IsLUB.lean","commit":"","full_name":"exists_seq_tendsto_sSup","start":[189,0],"end":[193,25],"file_path":"Mathlib/Topology/Order/IsLUB.lean","tactics":[{"state_before":"α✝ : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁹ : TopologicalSpace α✝\ninst✝⁸ : TopologicalSpace β\ninst✝⁷ : LinearOrder α✝\ninst✝⁶ : LinearOrder β\ninst✝⁵ : OrderTopology α✝\ninst✝⁴ : OrderTopology β\nα : Type u_4\ninst✝³ : ConditionallyCompleteLinearOrder α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : FirstCountableTopology α\nS : Set α\nhS : S.Nonempty\nhS' : BddAbove S\n⊢ ∃ u, Monotone u ∧ Tendsto u atTop (𝓝 (sSup S)) ∧ ∀ (n : ℕ), u n ∈ S","state_after":"case intro\nα✝ : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁹ : TopologicalSpace α✝\ninst✝⁸ : TopologicalSpace β\ninst✝⁷ : LinearOrder α✝\ninst✝⁶ : LinearOrder β\ninst✝⁵ : OrderTopology α✝\ninst✝⁴ : OrderTopology β\nα : Type u_4\ninst✝³ : ConditionallyCompleteLinearOrder α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : FirstCountableTopology α\nS : Set α\nhS : S.Nonempty\nhS' : BddAbove S\nu : ℕ → α\nhu : Monotone u ∧ (∀ (n : ℕ), u n ≤ sSup S) ∧ Tendsto u atTop (𝓝 (sSup S)) ∧ ∀ (n : ℕ), u n ∈ S\n⊢ ∃ u, Monotone u ∧ Tendsto u atTop (𝓝 (sSup S)) ∧ ∀ (n : ℕ), u n ∈ S","tactic":"rcases (isLUB_csSup hS hS').exists_seq_monotone_tendsto hS with ⟨u, hu⟩","premises":[{"full_name":"IsLUB.exists_seq_monotone_tendsto","def_path":"Mathlib/Topology/Order/IsLUB.lean","def_pos":[161,8],"def_end_pos":[161,41]},{"full_name":"isLUB_csSup","def_path":"Mathlib/Order/ConditionallyCompleteLattice/Basic.lean","def_pos":[461,8],"def_end_pos":[461,19]}]},{"state_before":"case intro\nα✝ : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁹ : TopologicalSpace α✝\ninst✝⁸ : TopologicalSpace β\ninst✝⁷ : LinearOrder α✝\ninst✝⁶ : LinearOrder β\ninst✝⁵ : OrderTopology α✝\ninst✝⁴ : OrderTopology β\nα : Type u_4\ninst✝³ : ConditionallyCompleteLinearOrder α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : FirstCountableTopology α\nS : Set α\nhS : S.Nonempty\nhS' : BddAbove S\nu : ℕ → α\nhu : Monotone u ∧ (∀ (n : ℕ), u n ≤ sSup S) ∧ Tendsto u atTop (𝓝 (sSup S)) ∧ ∀ (n : ℕ), u n ∈ S\n⊢ ∃ u, Monotone u ∧ Tendsto u atTop (𝓝 (sSup S)) ∧ ∀ (n : ℕ), u n ∈ S","state_after":"no goals","tactic":"exact ⟨u, hu.1, hu.2.2⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]}]}
+{"url":"Mathlib/Data/Set/Pointwise/Interval.lean","commit":"","full_name":"Set.image_neg_Iic","start":[331,0],"end":[331,62],"file_path":"Mathlib/Data/Set/Pointwise/Interval.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\n⊢ Neg.neg '' Iic a = Ici (-a)","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/FieldTheory/Tower.lean","commit":"","full_name":"FiniteDimensional.right","start":[61,0],"end":[65,33],"file_path":"Mathlib/FieldTheory/Tower.lean","tactics":[{"state_before":"F : Type u\nK : Type v\nA : Type w\ninst✝⁶ : DivisionRing F\ninst✝⁵ : DivisionRing K\ninst✝⁴ : AddCommGroup A\ninst✝³ : Module F K\ninst✝² : Module K A\ninst✝¹ : Module F A\ninst✝ : IsScalarTower F K A\nhf : FiniteDimensional F A\nb : Finset A\nhb : span F ↑b = ⊤\n⊢ restrictScalars F (span K ↑b) = restrictScalars F ⊤","state_after":"F : Type u\nK : Type v\nA : Type w\ninst✝⁶ : DivisionRing F\ninst✝⁵ : DivisionRing K\ninst✝⁴ : AddCommGroup A\ninst✝³ : Module F K\ninst✝² : Module K A\ninst✝¹ : Module F A\ninst✝ : IsScalarTower F K A\nhf : FiniteDimensional F A\nb : Finset A\nhb : span F ↑b = ⊤\n⊢ ↑b ⊆ ↑(restrictScalars F (span K ↑b))","tactic":"rw [Submodule.restrictScalars_top, eq_top_iff, ← hb, Submodule.span_le]","premises":[{"full_name":"Submodule.restrictScalars_top","def_path":"Mathlib/Algebra/Module/Submodule/RestrictScalars.lean","def_pos":[97,8],"def_end_pos":[97,27]},{"full_name":"Submodule.span_le","def_path":"Mathlib/LinearAlgebra/Span.lean","def_pos":[74,8],"def_end_pos":[74,15]},{"full_name":"eq_top_iff","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[116,8],"def_end_pos":[116,18]}]},{"state_before":"F : Type u\nK : Type v\nA : Type w\ninst✝⁶ : DivisionRing F\ninst✝⁵ : DivisionRing K\ninst✝⁴ : AddCommGroup A\ninst✝³ : Module F K\ninst✝² : Module K A\ninst✝¹ : Module F A\ninst✝ : IsScalarTower F K A\nhf : FiniteDimensional F A\nb : Finset A\nhb : span F ↑b = ⊤\n⊢ ↑b ⊆ ↑(restrictScalars F (span K ↑b))","state_after":"no goals","tactic":"exact Submodule.subset_span","premises":[{"full_name":"Submodule.subset_span","def_path":"Mathlib/LinearAlgebra/Span.lean","def_pos":[72,8],"def_end_pos":[72,19]}]}]}
+{"url":"Mathlib/Algebra/GroupWithZero/Basic.lean","commit":"","full_name":"zpow_add_one₀","start":[388,0],"end":[394,14],"file_path":"Mathlib/Algebra/GroupWithZero/Basic.lean","tactics":[{"state_before":"α : Type u_1\nM₀ : Type u_2\nG₀ : Type u_3\nM₀' : Type u_4\nG₀' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst✝ : GroupWithZero G₀\na b c : G₀\nha : a ≠ 0\nn : ℕ\n⊢ a ^ (↑n + 1) = a ^ ↑n * a","state_after":"no goals","tactic":"simp only [← Int.ofNat_succ, zpow_natCast, pow_succ]","premises":[{"full_name":"Int.ofNat_succ","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean","def_pos":[27,8],"def_end_pos":[27,18]},{"full_name":"pow_succ","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[567,8],"def_end_pos":[567,16]},{"full_name":"zpow_natCast","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[875,8],"def_end_pos":[875,20]}]},{"state_before":"α : Type u_1\nM₀ : Type u_2\nG₀ : Type u_3\nM₀' : Type u_4\nG₀' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst✝ : GroupWithZero G₀\na b c : G₀\nha : a ≠ 0\n⊢ a ^ (Int.negSucc 0 + 1) = a ^ Int.negSucc 0 * a","state_after":"no goals","tactic":"erw [zpow_zero, zpow_negSucc, pow_one, inv_mul_cancel ha]","premises":[{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"inv_mul_cancel","def_path":"Mathlib/Algebra/GroupWithZero/NeZero.lean","def_pos":[50,8],"def_end_pos":[50,22]},{"full_name":"pow_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[571,6],"def_end_pos":[571,13]},{"full_name":"zpow_negSucc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[890,8],"def_end_pos":[890,20]},{"full_name":"zpow_zero","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[871,49],"def_end_pos":[871,58]}]},{"state_before":"α : Type u_1\nM₀ : Type u_2\nG₀ : Type u_3\nM₀' : Type u_4\nG₀' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst✝ : GroupWithZero G₀\na b c : G₀\nha : a ≠ 0\nn : ℕ\n⊢ a ^ (Int.negSucc (n + 1) + 1) = a ^ Int.negSucc (n + 1) * a","state_after":"no goals","tactic":"rw [Int.negSucc_eq, zpow_neg, Int.neg_add, Int.neg_add_cancel_right, zpow_neg, ← Int.ofNat_succ,\n      zpow_natCast, zpow_natCast, pow_succ' _ (n + 1), mul_inv_rev, mul_assoc, inv_mul_cancel ha,\n      mul_one]","premises":[{"full_name":"Int.negSucc_eq","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean","def_pos":[64,8],"def_end_pos":[64,18]},{"full_name":"Int.neg_add","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean","def_pos":[255,32],"def_end_pos":[255,39]},{"full_name":"Int.neg_add_cancel_right","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean","def_pos":[248,18],"def_end_pos":[248,38]},{"full_name":"Int.ofNat_succ","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean","def_pos":[27,8],"def_end_pos":[27,18]},{"full_name":"inv_mul_cancel","def_path":"Mathlib/Algebra/GroupWithZero/NeZero.lean","def_pos":[50,8],"def_end_pos":[50,22]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]},{"full_name":"mul_inv_rev","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[982,8],"def_end_pos":[982,19]},{"full_name":"mul_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[413,8],"def_end_pos":[413,15]},{"full_name":"pow_succ'","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[573,33],"def_end_pos":[573,42]},{"full_name":"zpow_natCast","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[875,8],"def_end_pos":[875,20]},{"full_name":"zpow_neg","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[429,6],"def_end_pos":[429,14]}]}]}
+{"url":"Mathlib/Algebra/Module/Submodule/Ker.lean","commit":"","full_name":"LinearMap.ker_le_comap","start":[92,0],"end":[95,36],"file_path":"Mathlib/Algebra/Module/Submodule/Ker.lean","tactics":[{"state_before":"R : Type u_1\nR₁ : Type u_2\nR₂ : Type u_3\nR₃ : Type u_4\nK : Type u_5\nM : Type u_6\nM₁ : Type u_7\nM₂ : Type u_8\nM₃ : Type u_9\nV : Type u_10\nV₂ : Type u_11\ninst✝¹² : Semiring R\ninst✝¹¹ : Semiring R₂\ninst✝¹⁰ : Semiring R₃\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : AddCommMonoid M₂\ninst✝⁷ : AddCommMonoid M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\ninst✝⁶ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝⁵ : Module R M\ninst✝⁴ : Module R₂ M₂\ninst✝³ : Module R₃ M₃\nσ₂₁ : R₂ →+* R\nτ₁₂ : R →+* R₂\nτ₂₃ : R₂ →+* R₃\nτ₁₃ : R →+* R₃\ninst✝² : RingHomCompTriple τ₁₂ τ₂₃ τ₁₃\nF : Type u_12\ninst✝¹ : FunLike F M M₂\ninst✝ : SemilinearMapClass F τ₁₂ M M₂\np : Submodule R₂ M₂\nf : M →ₛₗ[τ₁₂] M₂\nx : M\nhx : x ∈ ker f\n⊢ x ∈ comap f p","state_after":"no goals","tactic":"simp [mem_ker.mp hx]","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"LinearMap.mem_ker","def_path":"Mathlib/Algebra/Module/Submodule/Ker.lean","def_pos":[62,8],"def_end_pos":[62,15]}]}]}
+{"url":"Mathlib/Algebra/Tropical/Basic.lean","commit":"","full_name":"Tropical.succ_nsmul","start":[490,0],"end":[494,43],"file_path":"Mathlib/Algebra/Tropical/Basic.lean","tactics":[{"state_before":"R✝ : Type u\ninst✝² : LinearOrderedAddCommMonoidWithTop R✝\nR : Type u_1\ninst✝¹ : LinearOrder R\ninst✝ : OrderTop R\nx : Tropical R\nn : ℕ\n⊢ (n + 1) • x = x","state_after":"case zero\nR✝ : Type u\ninst✝² : LinearOrderedAddCommMonoidWithTop R✝\nR : Type u_1\ninst✝¹ : LinearOrder R\ninst✝ : OrderTop R\nx : Tropical R\n⊢ (0 + 1) • x = x\n\ncase succ\nR✝ : Type u\ninst✝² : LinearOrderedAddCommMonoidWithTop R✝\nR : Type u_1\ninst✝¹ : LinearOrder R\ninst✝ : OrderTop R\nx : Tropical R\nn : ℕ\nIH : (n + 1) • x = x\n⊢ (n + 1 + 1) • x = x","tactic":"induction' n with n IH","premises":[]}]}
+{"url":"Mathlib/Combinatorics/SimpleGraph/Triangle/Tripartite.lean","commit":"","full_name":"SimpleGraph.TripartiteFromTriangles.Graph.in₀₂_iff'","start":[93,0],"end":[96,68],"file_path":"Mathlib/Combinatorics/SimpleGraph/Triangle/Tripartite.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\n𝕜 : Type u_4\ninst✝ : LinearOrderedField 𝕜\nt : Finset (α × β × γ)\na a' : α\nb b' : β\nc c' : γ\nx : α × β × γ\nε : 𝕜\n⊢ (graph t).Adj (in₀ a) (in₂ c) → ∃ x ∈ t, x.1 = a ∧ x.2.2 = c","state_after":"case in₀₂\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\n𝕜 : Type u_4\ninst✝ : LinearOrderedField 𝕜\nt : Finset (α × β × γ)\na a' : α\nb b' : β\nc c' : γ\nx : α × β × γ\nε : 𝕜\nb✝ : β\na✝ : (a, b✝, c) ∈ t\n⊢ ∃ x ∈ t, x.1 = a ∧ x.2.2 = c","tactic":"rintro ⟨⟩","premises":[]},{"state_before":"case in₀₂\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\n𝕜 : Type u_4\ninst✝ : LinearOrderedField 𝕜\nt : Finset (α × β × γ)\na a' : α\nb b' : β\nc c' : γ\nx : α × β × γ\nε : 𝕜\nb✝ : β\na✝ : (a, b✝, c) ∈ t\n⊢ ∃ x ∈ t, x.1 = a ∧ x.2.2 = c","state_after":"no goals","tactic":"exact ⟨_, ‹_›, by simp⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\n𝕜 : Type u_4\ninst✝ : LinearOrderedField 𝕜\nt : Finset (α × β × γ)\na a' : α\nb b' : β\nc c' : γ\nx : α × β × γ\nε : 𝕜\n⊢ (∃ x ∈ t, x.1 = a ∧ x.2.2 = c) → (graph t).Adj (in₀ a) (in₂ c)","state_after":"case intro.mk.mk.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\n𝕜 : Type u_4\ninst✝ : LinearOrderedField 𝕜\nt : Finset (α × β × γ)\na' : α\nb✝ b' : β\nc' : γ\nx : α × β × γ\nε : 𝕜\na : α\nb : β\nc : γ\nh : (a, b, c) ∈ t\n⊢ (graph t).Adj (in₀ (a, b, c).1) (in₂ (a, b, c).2.2)","tactic":"rintro ⟨⟨a, b, c⟩, h, rfl, rfl⟩","premises":[]},{"state_before":"case intro.mk.mk.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\n𝕜 : Type u_4\ninst✝ : LinearOrderedField 𝕜\nt : Finset (α × β × γ)\na' : α\nb✝ b' : β\nc' : γ\nx : α × β × γ\nε : 𝕜\na : α\nb : β\nc : γ\nh : (a, b, c) ∈ t\n⊢ (graph t).Adj (in₀ (a, b, c).1) (in₂ (a, b, c).2.2)","state_after":"case intro.mk.mk.intro.intro.a\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\n𝕜 : Type u_4\ninst✝ : LinearOrderedField 𝕜\nt : Finset (α × β × γ)\na' : α\nb✝ b' : β\nc' : γ\nx : α × β × γ\nε : 𝕜\na : α\nb : β\nc : γ\nh : (a, b, c) ∈ t\n⊢ ((a, b, c).1, ?intro.mk.mk.intro.intro.b, (a, b, c).2.2) ∈ t\n\ncase intro.mk.mk.intro.intro.b\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\n𝕜 : Type u_4\ninst✝ : LinearOrderedField 𝕜\nt : Finset (α × β × γ)\na' : α\nb✝ b' : β\nc' : γ\nx : α × β × γ\nε : 𝕜\na : α\nb : β\nc : γ\nh : (a, b, c) ∈ t\n⊢ β","tactic":"constructor","premises":[]},{"state_before":"case intro.mk.mk.intro.intro.a\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\n𝕜 : Type u_4\ninst✝ : LinearOrderedField 𝕜\nt : Finset (α × β × γ)\na' : α\nb✝ b' : β\nc' : γ\nx : α × β × γ\nε : 𝕜\na : α\nb : β\nc : γ\nh : (a, b, c) ∈ t\n⊢ ((a, b, c).1, ?intro.mk.mk.intro.intro.b, (a, b, c).2.2) ∈ t\n\ncase intro.mk.mk.intro.intro.b\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\n𝕜 : Type u_4\ninst✝ : LinearOrderedField 𝕜\nt : Finset (α × β × γ)\na' : α\nb✝ b' : β\nc' : γ\nx : α × β × γ\nε : 𝕜\na : α\nb : β\nc : γ\nh : (a, b, c) ∈ t\n⊢ β","state_after":"no goals","tactic":"assumption","premises":[]}]}
+{"url":"Mathlib/Data/List/Enum.lean","commit":"","full_name":"List.enumFrom_cons'","start":[134,0],"end":[136,70],"file_path":"Mathlib/Data/List/Enum.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nn : ℕ\nx : α\nxs : List α\n⊢ enumFrom n (x :: xs) = (n, x) :: map (Prod.map Nat.succ id) (enumFrom n xs)","state_after":"no goals","tactic":"rw [enumFrom_cons, Nat.add_comm, ← map_fst_add_enumFrom_eq_enumFrom]","premises":[{"full_name":"List.enumFrom_cons","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[1188,16],"def_end_pos":[1188,29]},{"full_name":"List.map_fst_add_enumFrom_eq_enumFrom","def_path":"Mathlib/Data/List/Enum.lean","def_pos":[126,8],"def_end_pos":[126,40]},{"full_name":"Nat.add_comm","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[160,18],"def_end_pos":[160,26]}]}]}
+{"url":"Mathlib/GroupTheory/MonoidLocalization/Basic.lean","commit":"","full_name":"Submonoid.LocalizationMap.eq'","start":[582,0],"end":[585,38],"file_path":"Mathlib/GroupTheory/MonoidLocalization/Basic.lean","tactics":[{"state_before":"M : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type u_3\ninst✝ : CommMonoid P\nf : S.LocalizationMap N\na₁ b₁ : M\na₂ b₂ : ↥S\n⊢ f.mk' a₁ a₂ = f.mk' b₁ b₂ ↔ (Localization.r S) (a₁, a₂) (b₁, b₂)","state_after":"no goals","tactic":"rw [f.eq, Localization.r_iff_exists]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Localization.r_iff_exists","def_path":"Mathlib/GroupTheory/MonoidLocalization/Basic.lean","def_pos":[193,8],"def_end_pos":[193,20]},{"full_name":"Submonoid.LocalizationMap.eq","def_path":"Mathlib/GroupTheory/MonoidLocalization/Basic.lean","def_pos":[578,18],"def_end_pos":[578,20]}]}]}
+{"url":"Mathlib/Analysis/InnerProductSpace/Basic.lean","commit":"","full_name":"norm_inner_eq_norm_tfae","start":[1367,0],"end":[1390,13],"file_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : _root_.RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nx y : E\n⊢ [‖⟪x, y⟫_𝕜‖ = ‖x‖ * ‖y‖, x = 0 ∨ y = (⟪x, y⟫_𝕜 / ⟪x, x⟫_𝕜) • x, x = 0 ∨ ∃ r, y = r • x,\n      x = 0 ∨ y ∈ Submodule.span 𝕜 {x}].TFAE","state_after":"case tfae_1_to_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : _root_.RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nx y : E\n⊢ ‖⟪x, y⟫_𝕜‖ = ‖x‖ * ‖y‖ → x = 0 ∨ y = (⟪x, y⟫_𝕜 / ⟪x, x⟫_𝕜) • x\n\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : _root_.RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nx y : E\ntfae_1_to_2 : ‖⟪x, y⟫_𝕜‖ = ‖x‖ * ‖y‖ → x = 0 ∨ y = (⟪x, y⟫_𝕜 / ⟪x, x⟫_𝕜) • x\n⊢ [‖⟪x, y⟫_𝕜‖ = ‖x‖ * ‖y‖, x = 0 ∨ y = (⟪x, y⟫_𝕜 / ⟪x, x⟫_𝕜) • x, x = 0 ∨ ∃ r, y = r • x,\n      x = 0 ∨ y ∈ Submodule.span 𝕜 {x}].TFAE","tactic":"tfae_have 1 → 2","premises":[]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : _root_.RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nx y : E\ntfae_1_to_2 : ‖⟪x, y⟫_𝕜‖ = ‖x‖ * ‖y‖ → x = 0 ∨ y = (⟪x, y⟫_𝕜 / ⟪x, x⟫_𝕜) • x\n⊢ [‖⟪x, y⟫_𝕜‖ = ‖x‖ * ‖y‖, x = 0 ∨ y = (⟪x, y⟫_𝕜 / ⟪x, x⟫_𝕜) • x, x = 0 ∨ ∃ r, y = r • x,\n      x = 0 ∨ y ∈ Submodule.span 𝕜 {x}].TFAE","state_after":"case tfae_2_to_3\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : _root_.RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nx y : E\ntfae_1_to_2 : ‖⟪x, y⟫_𝕜‖ = ‖x‖ * ‖y‖ → x = 0 ∨ y = (⟪x, y⟫_𝕜 / ⟪x, x⟫_𝕜) • x\n⊢ x = 0 ∨ y = (⟪x, y⟫_𝕜 / ⟪x, x⟫_𝕜) • x → x = 0 ∨ ∃ r, y = r • x\n\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : _root_.RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nx y : E\ntfae_1_to_2 : ‖⟪x, y⟫_𝕜‖ = ‖x‖ * ‖y‖ → x = 0 ∨ y = (⟪x, y⟫_𝕜 / ⟪x, x⟫_𝕜) • x\ntfae_2_to_3 : x = 0 ∨ y = (⟪x, y⟫_𝕜 / ⟪x, x⟫_𝕜) • x → x = 0 ∨ ∃ r, y = r • x\n⊢ [‖⟪x, y⟫_𝕜‖ = ‖x‖ * ‖y‖, x = 0 ∨ y = (⟪x, y⟫_𝕜 / ⟪x, x⟫_𝕜) • x, x = 0 ∨ ∃ r, y = r • x,\n      x = 0 ∨ y ∈ Submodule.span 𝕜 {x}].TFAE","tactic":"tfae_have 2 → 3","premises":[]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : _root_.RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nx y : E\ntfae_1_to_2 : ‖⟪x, y⟫_𝕜‖ = ‖x‖ * ‖y‖ → x = 0 ∨ y = (⟪x, y⟫_𝕜 / ⟪x, x⟫_𝕜) • x\ntfae_2_to_3 : x = 0 ∨ y = (⟪x, y⟫_𝕜 / ⟪x, x⟫_𝕜) • x → x = 0 ∨ ∃ r, y = r • x\n⊢ [‖⟪x, y⟫_𝕜‖ = ‖x‖ * ‖y‖, x = 0 ∨ y = (⟪x, y⟫_𝕜 / ⟪x, x⟫_𝕜) • x, x = 0 ∨ ∃ r, y = r • x,\n      x = 0 ∨ y ∈ Submodule.span 𝕜 {x}].TFAE","state_after":"case tfae_3_to_1\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : _root_.RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nx y : E\ntfae_1_to_2 : ‖⟪x, y⟫_𝕜‖ = ‖x‖ * ‖y‖ → x = 0 ∨ y = (⟪x, y⟫_𝕜 / ⟪x, x⟫_𝕜) • x\ntfae_2_to_3 : x = 0 ∨ y = (⟪x, y⟫_𝕜 / ⟪x, x⟫_𝕜) • x → x = 0 ∨ ∃ r, y = r • x\n⊢ (x = 0 ∨ ∃ r, y = r • x) → ‖⟪x, y⟫_𝕜‖ = ‖x‖ * ‖y‖\n\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : _root_.RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nx y : E\ntfae_1_to_2 : ‖⟪x, y⟫_𝕜‖ = ‖x‖ * ‖y‖ → x = 0 ∨ y = (⟪x, y⟫_𝕜 / ⟪x, x⟫_𝕜) • x\ntfae_2_to_3 : x = 0 ∨ y = (⟪x, y⟫_𝕜 / ⟪x, x⟫_𝕜) • x → x = 0 ∨ ∃ r, y = r • x\ntfae_3_to_1 : (x = 0 ∨ ∃ r, y = r • x) → ‖⟪x, y⟫_𝕜‖ = ‖x‖ * ‖y‖\n⊢ [‖⟪x, y⟫_𝕜‖ = ‖x‖ * ‖y‖, x = 0 ∨ y = (⟪x, y⟫_𝕜 / ⟪x, x⟫_𝕜) • x, x = 0 ∨ ∃ r, y = r • x,\n      x = 0 ∨ y ∈ Submodule.span 𝕜 {x}].TFAE","tactic":"tfae_have 3 → 1","premises":[]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : _root_.RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nx y : E\ntfae_1_to_2 : ‖⟪x, y⟫_𝕜‖ = ‖x‖ * ‖y‖ → x = 0 ∨ y = (⟪x, y⟫_𝕜 / ⟪x, x⟫_𝕜) • x\ntfae_2_to_3 : x = 0 ∨ y = (⟪x, y⟫_𝕜 / ⟪x, x⟫_𝕜) • x → x = 0 ∨ ∃ r, y = r • x\ntfae_3_to_1 : (x = 0 ∨ ∃ r, y = r • x) → ‖⟪x, y⟫_𝕜‖ = ‖x‖ * ‖y‖\n⊢ [‖⟪x, y⟫_𝕜‖ = ‖x‖ * ‖y‖, x = 0 ∨ y = (⟪x, y⟫_𝕜 / ⟪x, x⟫_𝕜) • x, x = 0 ∨ ∃ r, y = r • x,\n      x = 0 ∨ y ∈ Submodule.span 𝕜 {x}].TFAE","state_after":"case tfae_3_iff_4\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : _root_.RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nx y : E\ntfae_1_to_2 : ‖⟪x, y⟫_𝕜‖ = ‖x‖ * ‖y‖ → x = 0 ∨ y = (⟪x, y⟫_𝕜 / ⟪x, x⟫_𝕜) • x\ntfae_2_to_3 : x = 0 ∨ y = (⟪x, y⟫_𝕜 / ⟪x, x⟫_𝕜) • x → x = 0 ∨ ∃ r, y = r • x\ntfae_3_to_1 : (x = 0 ∨ ∃ r, y = r • x) → ‖⟪x, y⟫_𝕜‖ = ‖x‖ * ‖y‖\n⊢ (x = 0 ∨ ∃ r, y = r • x) ↔ x = 0 ∨ y ∈ Submodule.span 𝕜 {x}\n\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : _root_.RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nx y : E\ntfae_1_to_2 : ‖⟪x, y⟫_𝕜‖ = ‖x‖ * ‖y‖ → x = 0 ∨ y = (⟪x, y⟫_𝕜 / ⟪x, x⟫_𝕜) • x\ntfae_2_to_3 : x = 0 ∨ y = (⟪x, y⟫_𝕜 / ⟪x, x⟫_𝕜) • x → x = 0 ∨ ∃ r, y = r • x\ntfae_3_to_1 : (x = 0 ∨ ∃ r, y = r • x) → ‖⟪x, y⟫_𝕜‖ = ‖x‖ * ‖y‖\ntfae_3_iff_4 : (x = 0 ∨ ∃ r, y = r • x) ↔ x = 0 ∨ y ∈ Submodule.span 𝕜 {x}\n⊢ [‖⟪x, y⟫_𝕜‖ = ‖x‖ * ‖y‖, x = 0 ∨ y = (⟪x, y⟫_𝕜 / ⟪x, x⟫_𝕜) • x, x = 0 ∨ ∃ r, y = r • x,\n      x = 0 ∨ y ∈ Submodule.span 𝕜 {x}].TFAE","tactic":"tfae_have 3 ↔ 4","premises":[]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : _root_.RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nx y : E\ntfae_1_to_2 : ‖⟪x, y⟫_𝕜‖ = ‖x‖ * ‖y‖ → x = 0 ∨ y = (⟪x, y⟫_𝕜 / ⟪x, x⟫_𝕜) • x\ntfae_2_to_3 : x = 0 ∨ y = (⟪x, y⟫_𝕜 / ⟪x, x⟫_𝕜) • x → x = 0 ∨ ∃ r, y = r • x\ntfae_3_to_1 : (x = 0 ∨ ∃ r, y = r • x) → ‖⟪x, y⟫_𝕜‖ = ‖x‖ * ��y‖\ntfae_3_iff_4 : (x = 0 ∨ ∃ r, y = r • x) ↔ x = 0 ∨ y ∈ Submodule.span 𝕜 {x}\n⊢ [‖⟪x, y⟫_𝕜‖ = ‖x‖ * ‖y‖, x = 0 ∨ y = (⟪x, y⟫_𝕜 / ⟪x, x⟫_𝕜) • x, x = 0 ∨ ∃ r, y = r • x,\n      x = 0 ∨ y ∈ Submodule.span 𝕜 {x}].TFAE","state_after":"no goals","tactic":"tfae_finish","premises":[]}]}
+{"url":"Mathlib/Topology/Algebra/Module/StrongTopology.lean","commit":"","full_name":"UniformConvergenceCLM.topologicalSpace_eq","start":[93,0],"end":[98,41],"file_path":"Mathlib/Topology/Algebra/Module/StrongTopology.lean","tactics":[{"state_before":"𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝⁸ : NormedField 𝕜₁\ninst✝⁷ : NormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nE : Type u_3\nF : Type u_4\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module 𝕜₁ E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : AddCommGroup F\ninst✝² : Module 𝕜₂ F\ninst✝¹ : UniformSpace F\ninst✝ : UniformAddGroup F\n𝔖 : Set (Set E)\n⊢ instTopologicalSpace σ F 𝔖 = TopologicalSpace.induced DFunLike.coe (UniformOnFun.topologicalSpace E F 𝔖)","state_after":"𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝⁸ : NormedField 𝕜₁\ninst✝⁷ : NormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nE : Type u_3\nF : Type u_4\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module 𝕜₁ E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : AddCommGroup F\ninst✝² : Module 𝕜₂ F\ninst✝¹ : UniformSpace F\ninst✝ : UniformAddGroup F\n𝔖 : Set (Set E)\n⊢ TopologicalSpace.induced DFunLike.coe (UniformOnFun.topologicalSpace E F 𝔖) =\n    TopologicalSpace.induced DFunLike.coe (UniformOnFun.topologicalSpace E F 𝔖)","tactic":"rw [instTopologicalSpace]","premises":[{"full_name":"UniformConvergenceCLM.instTopologicalSpace","def_path":"Mathlib/Topology/Algebra/Module/StrongTopology.lean","def_pos":[88,9],"def_end_pos":[88,29]}]},{"state_before":"𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝⁸ : NormedField 𝕜₁\ninst✝⁷ : NormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nE : Type u_3\nF : Type u_4\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module 𝕜₁ E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : AddCommGroup F\ninst✝² : Module 𝕜₂ F\ninst✝¹ : UniformSpace F\ninst✝ : UniformAddGroup F\n𝔖 : Set (Set E)\n⊢ TopologicalSpace.induced DFunLike.coe (UniformOnFun.topologicalSpace E F 𝔖) =\n    TopologicalSpace.induced DFunLike.coe (UniformOnFun.topologicalSpace E F 𝔖)","state_after":"case e_t.h.e_3.h\n𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝⁸ : NormedField 𝕜₁\ninst✝⁷ : NormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nE : Type u_3\nF : Type u_4\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module 𝕜₁ E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : AddCommGroup F\ninst✝² : Module 𝕜₂ F\ninst✝¹ : UniformSpace F\ninst✝ : UniformAddGroup F\n𝔖 : Set (Set E)\n⊢ TopologicalAddGroup.toUniformSpace F = inst✝¹","tactic":"congr","premises":[]},{"state_before":"case e_t.h.e_3.h\n𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝⁸ : NormedField 𝕜₁\ninst✝⁷ : NormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nE : Type u_3\nF : Type u_4\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module 𝕜₁ E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : AddCommGroup F\ninst✝² : Module 𝕜₂ F\ninst✝¹ : UniformSpace F\ninst✝ : UniformAddGroup F\n𝔖 : Set (Set E)\n⊢ TopologicalAddGroup.toUniformSpace F = inst✝¹","state_after":"no goals","tactic":"exact UniformAddGroup.toUniformSpace_eq","premises":[{"full_name":"UniformAddGroup.toUniformSpace_eq","def_path":"Mathlib/Topology/Algebra/UniformGroup.lean","def_pos":[582,2],"def_end_pos":[582,13]}]}]}
+{"url":"Mathlib/Analysis/Analytic/IsolatedZeros.lean","commit":"","full_name":"HasFPowerSeriesAt.locally_ne_zero","start":[100,0],"end":[105,67],"file_path":"Mathlib/Analysis/Analytic/IsolatedZeros.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ns : E\np q : FormalMultilinearSeries 𝕜 𝕜 E\nf g : 𝕜 → E\nn : ℕ\nz z₀ : 𝕜\nhp : HasFPowerSeriesAt f p z₀\nh : p ≠ 0\n⊢ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z ≠ 0","state_after":"𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ns : E\np q : FormalMultilinearSeries 𝕜 𝕜 E\nf g : 𝕜 → E\nn : ℕ\nz z₀ : 𝕜\nhp : HasFPowerSeriesAt f p z₀\nh : p ≠ 0\n⊢ ∀ᶠ (x : 𝕜) in 𝓝 z₀, x ∈ {z₀}ᶜ → f x ≠ 0","tactic":"rw [eventually_nhdsWithin_iff]","premises":[{"full_name":"eventually_nhdsWithin_iff","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[43,8],"def_end_pos":[43,33]}]},{"state_before":"𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ns : E\np q : FormalMultilinearSeries 𝕜 𝕜 E\nf g : 𝕜 → E\nn : ℕ\nz z₀ : 𝕜\nhp : HasFPowerSeriesAt f p z₀\nh : p ≠ 0\n⊢ ∀ᶠ (x : 𝕜) in 𝓝 z₀, x ∈ {z₀}ᶜ → f x ≠ 0","state_after":"𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ns : E\np q : FormalMultilinearSeries 𝕜 𝕜 E\nf g : 𝕜 → E\nn : ℕ\nz z₀ : 𝕜\nhp : HasFPowerSeriesAt f p z₀\nh : p ≠ 0\nh2 : ContinuousAt ((swap dslope z₀)^[p.order] f) z₀\n⊢ ∀ᶠ (x : 𝕜) in 𝓝 z₀, x ∈ {z₀}ᶜ → f x ≠ 0","tactic":"have h2 := (has_fpower_series_iterate_dslope_fslope p.order hp).continuousAt","premises":[{"full_name":"FormalMultilinearSeries.order","def_path":"Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean","def_pos":[225,18],"def_end_pos":[225,23]},{"full_name":"HasFPowerSeriesAt.continuousAt","def_path":"Mathlib/Analysis/Analytic/Basic.lean","def_pos":[834,18],"def_end_pos":[834,48]},{"full_name":"HasFPowerSeriesAt.has_fpower_series_iterate_dslope_fslope","def_path":"Mathlib/Analysis/Analytic/IsolatedZeros.lean","def_pos":[78,8],"def_end_pos":[78,47]}]},{"state_before":"𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ns : E\np q : FormalMultilinearSeries 𝕜 𝕜 E\nf g : 𝕜 → E\nn : ℕ\nz z₀ : 𝕜\nhp : HasFPowerSeriesAt f p z₀\nh : p ≠ 0\nh2 : ContinuousAt ((swap dslope z₀)^[p.order] f) z₀\n⊢ ∀ᶠ (x : 𝕜) in 𝓝 z₀, x ∈ {z₀}ᶜ → f x ≠ 0","state_after":"𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ns : E\np q : FormalMultilinearSeries 𝕜 𝕜 E\nf g : 𝕜 → E\nn : ℕ\nz z₀ : 𝕜\nhp : HasFPowerSeriesAt f p z₀\nh : p ≠ 0\nh2 : ContinuousAt ((swap dslope z₀)^[p.order] f) z₀\nh3 : ∀ᶠ (z : 𝕜) in 𝓝 z₀, (swap dslope z₀)^[p.order] f z ≠ 0\n⊢ ∀ᶠ (x : 𝕜) in 𝓝 z₀, x ∈ {z₀}ᶜ → f x ≠ 0","tactic":"have h3 := h2.eventually_ne (iterate_dslope_fslope_ne_zero hp h)","premises":[{"full_name":"ContinuousAt.eventually_ne","def_path":"Mathlib/Topology/Separation.lean","def_pos":[836,8],"def_end_pos":[836,34]},{"full_name":"HasFPowerSeriesAt.iterate_dslope_fslope_ne_zero","def_path":"Mathlib/Analysis/Analytic/IsolatedZeros.lean","def_pos":[84,8],"def_end_pos":[84,37]}]},{"state_before":"𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ns : E\np q : FormalMultilinearSeries 𝕜 𝕜 E\nf g : 𝕜 → E\nn : ℕ\nz z₀ : 𝕜\nhp : HasFPowerSeriesAt f p z₀\nh : p ≠ 0\nh2 : ContinuousAt ((swap dslope z₀)^[p.order] f) z₀\nh3 : ∀ᶠ (z : 𝕜) in 𝓝 z₀, (swap dslope z₀)^[p.order] f z ≠ 0\n⊢ ∀ᶠ (x : 𝕜) in 𝓝 z₀, x ∈ {z₀}ᶜ → f x ≠ 0","state_after":"case h\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ns : E\np q : FormalMultilinearSeries 𝕜 𝕜 E\nf g : 𝕜 → E\nn : ℕ\nz✝ z₀ : 𝕜\nhp : HasFPowerSeriesAt f p z₀\nh : p ≠ 0\nh2 : ContinuousAt ((swap dslope z₀)^[p.order] f) z₀\nh3 : ∀ᶠ (z : 𝕜) in 𝓝 z₀, (swap dslope z₀)^[p.order] f z ≠ 0\nz : 𝕜\ne1 : f z = (z - z₀) ^ p.order • (swap dslope z₀)^[p.order] f z\ne2 : (swap dslope z₀)^[p.order] f z ≠ 0\ne3 : z ∈ {z₀}ᶜ\n⊢ f z ≠ 0","tactic":"filter_upwards [eq_pow_order_mul_iterate_dslope hp, h3] with z e1 e2 e3","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Filter.mp_mem","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[159,8],"def_end_pos":[159,14]},{"full_name":"HasFPowerSeriesAt.eq_pow_order_mul_iterate_dslope","def_path":"Mathlib/Analysis/Analytic/IsolatedZeros.lean","def_pos":[89,8],"def_end_pos":[89,39]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]}]},{"state_before":"case h\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ns : E\np q : FormalMultilinearSeries 𝕜 𝕜 E\nf g : 𝕜 → E\nn : ℕ\nz✝ z₀ : 𝕜\nhp : HasFPowerSeriesAt f p z₀\nh : p ≠ 0\nh2 : ContinuousAt ((swap dslope z₀)^[p.order] f) z₀\nh3 : ∀ᶠ (z : 𝕜) in 𝓝 z₀, (swap dslope z₀)^[p.order] f z ≠ 0\nz : 𝕜\ne1 : f z = (z - z₀) ^ p.order • (swap dslope z₀)^[p.order] f z\ne2 : (swap dslope z₀)^[p.order] f z ≠ 0\ne3 : z ∈ {z₀}ᶜ\n⊢ f z ≠ 0","state_after":"no goals","tactic":"simpa [e1, e2, e3] using pow_ne_zero p.order (sub_ne_zero.mpr e3)","premises":[{"full_name":"FormalMultilinearSeries.order","def_path":"Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean","def_pos":[225,18],"def_end_pos":[225,23]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"pow_ne_zero","def_path":"Mathlib/Algebra/GroupWithZero/Basic.lean","def_pos":[172,6],"def_end_pos":[172,17]},{"full_name":"sub_ne_zero","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[746,2],"def_end_pos":[746,13]}]}]}
+{"url":"Mathlib/Analysis/Convex/Hull.lean","commit":"","full_name":"AffineMap.image_convexHull","start":[172,0],"end":[180,46],"file_path":"Mathlib/Analysis/Convex/Hull.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : OrderedRing 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nf : E →ᵃ[𝕜] F\ns : Set E\n⊢ ⇑f '' (convexHull 𝕜) s = (convexHull 𝕜) (⇑f '' s)","state_after":"case h₁\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : OrderedRing 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nf : E →ᵃ[𝕜] F\ns : Set E\n⊢ ⇑f '' (convexHull 𝕜) s ⊆ (convexHull 𝕜) (⇑f '' s)\n\ncase h₂\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : OrderedRing 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nf : E →ᵃ[𝕜] F\ns : Set E\n⊢ (convexHull 𝕜) (⇑f '' s) ⊆ ⇑f '' (convexHull 𝕜) s","tactic":"apply Set.Subset.antisymm","premises":[{"full_name":"Set.Subset.antisymm","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[302,8],"def_end_pos":[302,23]}]}]}
+{"url":"Mathlib/RingTheory/IsTensorProduct.lean","commit":"","full_name":"IsTensorProduct.inductionOn","start":[104,0],"end":[116,29],"file_path":"Mathlib/RingTheory/IsTensorProduct.lean","tactics":[{"state_before":"R : Type u_1\ninst✝¹⁴ : CommSemiring R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\nM' : Type u_5\ninst✝¹³ : AddCommMonoid M₁\ninst✝¹² : AddCommMonoid M₂\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nN₁ : Type u_6\nN₂ : Type u_7\nN : Type u_8\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N\ng : N₁ →ₗ[R] N₂ →ₗ[R] N\nh : IsTensorProduct f\nC : M → Prop\nm : M\nh0 : C 0\nhtmul : ∀ (x : M₁) (y : M₂), C ((f x) y)\nhadd : ∀ (x y : M), C x → C y → C (x + y)\n⊢ C m","state_after":"R : Type u_1\ninst✝¹⁴ : CommSemiring R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\nM' : Type u_5\ninst✝¹³ : AddCommMonoid M₁\ninst✝¹² : AddCommMonoid M₂\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nN₁ : Type u_6\nN₂ : Type u_7\nN : Type u_8\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N\ng : N₁ →ₗ[R] N₂ →ₗ[R] N\nh : IsTensorProduct f\nC : M → Prop\nm : M\nh0 : C 0\nhtmul : ∀ (x : M₁) (y : M₂), C ((f x) y)\nhadd : ∀ (x y : M), C x → C y → C (x + y)\n⊢ C ((↑h.equiv).toFun (h.equiv.invFun m))","tactic":"rw [← h.equiv.right_inv m]","premises":[{"full_name":"IsTensorProduct.equiv","def_path":"Mathlib/RingTheory/IsTensorProduct.lean","def_pos":[68,18],"def_end_pos":[68,39]}]},{"state_before":"R : Type u_1\ninst✝¹⁴ : CommSemiring R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\nM' : Type u_5\ninst✝¹³ : AddCommMonoid M₁\ninst✝¹² : AddCommMonoid M₂\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nN₁ : Type u_6\nN₂ : Type u_7\nN : Type u_8\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N\ng : N₁ →ₗ[R] N₂ →ₗ[R] N\nh : IsTensorProduct f\nC : M → Prop\nm : M\nh0 : C 0\nhtmul : ∀ (x : M₁) (y : M₂), C ((f x) y)\nhadd : ∀ (x y : M), C x → C y → C (x + y)\n⊢ C ((↑h.equiv).toFun (h.equiv.invFun m))","state_after":"R : Type u_1\ninst✝¹⁴ : CommSemiring R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\nM' : Type u_5\ninst✝¹³ : AddCommMonoid M₁\ninst✝¹² : AddCommMonoid M₂\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nN₁ : Type u_6\nN₂ : Type u_7\nN : Type u_8\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N\ng : N₁ →ₗ[R] N₂ →ₗ[R] N\nh : IsTensorProduct f\nC : M → Prop\nm : M\nh0 : C 0\nhtmul : ∀ (x : M₁) (y : M₂), C ((f x) y)\nhadd : ∀ (x y : M), C x → C y ��� C (x + y)\ny : M₁ ⊗[R] M₂\n⊢ C ((↑h.equiv).toFun y)","tactic":"generalize h.equiv.invFun m = y","premises":[{"full_name":"IsTensorProduct.equiv","def_path":"Mathlib/RingTheory/IsTensorProduct.lean","def_pos":[68,18],"def_end_pos":[68,39]}]},{"state_before":"R : Type u_1\ninst✝¹⁴ : CommSemiring R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\nM' : Type u_5\ninst✝¹³ : AddCommMonoid M₁\ninst✝¹² : AddCommMonoid M₂\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nN₁ : Type u_6\nN₂ : Type u_7\nN : Type u_8\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N\ng : N₁ →ₗ[R] N₂ →ₗ[R] N\nh : IsTensorProduct f\nC : M → Prop\nm : M\nh0 : C 0\nhtmul : ∀ (x : M₁) (y : M₂), C ((f x) y)\nhadd : ∀ (x y : M), C x → C y → C (x + y)\ny : M₁ ⊗[R] M₂\n⊢ C ((↑h.equiv).toFun y)","state_after":"R : Type u_1\ninst✝¹⁴ : CommSemiring R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\nM' : Type u_5\ninst✝¹³ : AddCommMonoid M₁\ninst✝¹² : AddCommMonoid M₂\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nN₁ : Type u_6\nN₂ : Type u_7\nN : Type u_8\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N\ng : N₁ →ₗ[R] N₂ →ₗ[R] N\nh : IsTensorProduct f\nC : M → Prop\nm : M\nh0 : C 0\nhtmul : ∀ (x : M₁) (y : M₂), C ((f x) y)\nhadd : ∀ (x y : M), C x → C y → C (x + y)\ny : M₁ ⊗[R] M₂\n⊢ C ((TensorProduct.lift f) y)","tactic":"change C (TensorProduct.lift f y)","premises":[{"full_name":"TensorProduct.lift","def_path":"Mathlib/LinearAlgebra/TensorProduct/Basic.lean","def_pos":[509,4],"def_end_pos":[509,8]}]},{"state_before":"R : Type u_1\ninst✝¹⁴ : CommSemiring R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\nM' : Type u_5\ninst✝¹³ : AddCommMonoid M₁\ninst✝¹² : AddCommMonoid M₂\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nN₁ : Type u_6\nN₂ : Type u_7\nN : Type u_8\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N\ng : N₁ →ₗ[R] N₂ →ₗ[R] N\nh : IsTensorProduct f\nC : M → Prop\nm : M\nh0 : C 0\nhtmul : ∀ (x : M₁) (y : M₂), C ((f x) y)\nhadd : ∀ (x y : M), C x → C y → C (x + y)\ny : M₁ ⊗[R] M₂\n⊢ C ((TensorProduct.lift f) y)","state_after":"no goals","tactic":"induction y with\n  | zero => rwa [map_zero]\n  | tmul _ _ =>\n    rw [TensorProduct.lift.tmul]\n    apply htmul\n  | add _ _ _ _ =>\n    rw [map_add]\n    apply hadd <;> assumption","premises":[{"full_name":"TensorProduct.lift.tmul","def_path":"Mathlib/LinearAlgebra/TensorProduct/Basic.lean","def_pos":[515,8],"def_end_pos":[515,17]},{"full_name":"map_add","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[280,2],"def_end_pos":[280,13]},{"full_name":"map_zero","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[189,2],"def_end_pos":[189,13]}]}]}
+{"url":"Mathlib/Topology/Algebra/Module/Basic.lean","commit":"","full_name":"ContinuousLinearMap.smulRight_one_pow","start":[1322,0],"end":[1327,71],"file_path":"Mathlib/Topology/Algebra/Module/Basic.lean","tactics":[{"state_before":"R : Type u_1\ninst✝¹⁵ : Ring R\nR₂ : Type u_2\ninst✝¹⁴ : Ring R₂\nR₃ : Type u_3\ninst✝¹³ : Ring R₃\nM : Type u_4\ninst✝¹² : TopologicalSpace M\ninst✝¹¹ : AddCommGroup M\nM₂ : Type u_5\ninst✝¹⁰ : TopologicalSpace M₂\ninst✝⁹ : AddCommGroup M₂\nM₃ : Type u_6\ninst✝⁸ : TopologicalSpace M₃\ninst✝⁷ : AddCommGroup M₃\nM₄ : Type u_7\ninst✝⁶ : TopologicalSpace M₄\ninst✝⁵ : AddCommGroup M₄\ninst✝⁴ : Module R M\ninst✝³ : Module R₂ M₂\ninst✝² : Module R₃ M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalRing R\nc : R\nn : ℕ\n⊢ smulRight 1 c ^ n = smulRight 1 (c ^ n)","state_after":"case zero\nR : Type u_1\ninst✝¹⁵ : Ring R\nR₂ : Type u_2\ninst✝¹⁴ : Ring R₂\nR₃ : Type u_3\ninst✝¹³ : Ring R₃\nM : Type u_4\ninst✝¹² : TopologicalSpace M\ninst✝¹¹ : AddCommGroup M\nM₂ : Type u_5\ninst✝¹⁰ : TopologicalSpace M₂\ninst✝⁹ : AddCommGroup M₂\nM₃ : Type u_6\ninst✝⁸ : TopologicalSpace M₃\ninst✝⁷ : AddCommGroup M₃\nM₄ : Type u_7\ninst✝⁶ : TopologicalSpace M₄\ninst✝⁵ : AddCommGroup M₄\ninst✝⁴ : Module R M\ninst✝³ : Module R₂ M₂\ninst✝² : Module R₃ M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalRing R\nc : R\n⊢ smulRight 1 c ^ 0 = smulRight 1 (c ^ 0)\n\ncase succ\nR : Type u_1\ninst✝¹⁵ : Ring R\nR₂ : Type u_2\ninst✝¹⁴ : Ring R₂\nR₃ : Type u_3\ninst✝¹³ : Ring R₃\nM : Type u_4\ninst✝¹² : TopologicalSpace M\ninst✝¹¹ : AddCommGroup M\nM₂ : Type u_5\ninst✝¹⁰ : TopologicalSpace M₂\ninst✝⁹ : AddCommGroup M₂\nM₃ : Type u_6\ninst✝⁸ : TopologicalSpace M₃\ninst✝⁷ : AddCommGroup M₃\nM₄ : Type u_7\ninst✝⁶ : TopologicalSpace M₄\ninst✝⁵ : AddCommGroup M₄\ninst✝⁴ : Module R M\ninst✝³ : Module R₂ M₂\ninst✝² : Module R₃ M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalRing R\nc : R\nn : ℕ\nihn : smulRight 1 c ^ n = smulRight 1 (c ^ n)\n⊢ smulRight 1 c ^ (n + 1) = smulRight 1 (c ^ (n + 1))","tactic":"induction' n with n ihn","premises":[]}]}
+{"url":"Mathlib/Data/Set/Basic.lean","commit":"","full_name":"Set.singleton_subset_singleton","start":[1050,0],"end":[1050,75],"file_path":"Mathlib/Data/Set/Basic.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\na b : α\ns s₁ s₂ t t₁ t₂ u : Set α\n⊢ {a} ⊆ {b} ↔ a = b","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/Data/Fin/Basic.lean","commit":"","full_name":"Fin.pred_succAbove_pred","start":[1168,0],"end":[1172,87],"file_path":"Mathlib/Data/Fin/Basic.lean","tactics":[{"state_before":"n m : ℕ\np : Fin (n + 1)\ni j : Fin n\na : Fin (n + 2)\nb : Fin (n + 1)\nha : a ≠ 0\nhb : b ≠ 0\nhk : optParam (a.succAbove b ≠ 0) ⋯\n⊢ (a.pred ha).succAbove (b.pred hb) = (a.succAbove b).pred hk","state_after":"no goals","tactic":"simp_rw [← succ_inj (b := pred (succAbove a b) hk), ← succ_succAbove_succ, succ_pred]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Fin.pred","def_path":".lake/packages/lean4/src/lean/Init/Data/Fin/Basic.lean","def_pos":[197,14],"def_end_pos":[197,18]},{"full_name":"Fin.succAbove","def_path":"Mathlib/Data/Fin/Basic.lean","def_pos":[944,4],"def_end_pos":[944,13]},{"full_name":"Fin.succ_inj","def_path":".lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean","def_pos":[246,16],"def_end_pos":[246,24]},{"full_name":"Fin.succ_pred","def_path":".lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean","def_pos":[521,16],"def_end_pos":[521,25]},{"full_name":"Fin.succ_succAbove_succ","def_path":"Mathlib/Data/Fin/Basic.lean","def_pos":[1154,14],"def_end_pos":[1154,33]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]}]}]}
+{"url":"Mathlib/Algebra/Order/Group/PosPart.lean","commit":"","full_name":"one_lt_oneLePart","start":[106,0],"end":[107,33],"file_path":"Mathlib/Algebra/Order/Group/PosPart.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝¹ : Lattice α\ninst✝ : Group α\na : α\nha : 1 < a\n⊢ 1 < a⁺ᵐ","state_after":"no goals","tactic":"rwa [oneLePart_eq_self.2 ha.le]","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"oneLePart_eq_self","def_path":"Mathlib/Algebra/Order/Group/PosPart.lean","def_pos":[86,36],"def_end_pos":[86,53]}]}]}
+{"url":"Mathlib/Data/Finset/Basic.lean","commit":"","full_name":"Finset.isEmpty_coe_sort","start":[539,0],"end":[540,39],"file_path":"Mathlib/Data/Finset/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ns✝ s : Finset α\n⊢ IsEmpty { x // x ∈ s } ↔ s = ∅","state_after":"no goals","tactic":"simpa using @Set.isEmpty_coe_sort α s","premises":[{"full_name":"Set.isEmpty_coe_sort","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[503,8],"def_end_pos":[503,24]}]}]}
+{"url":"Mathlib/Topology/Category/Compactum.lean","commit":"","full_name":"Compactum.le_nhds_of_str_eq","start":[336,0],"end":[337,60],"file_path":"Mathlib/Topology/Category/Compactum.lean","tactics":[{"state_before":"X : Compactum\nF : Ultrafilter X.A\nx : X.A\nh : X.str F = x\ns : Set X.A\nhx : x ∈ s\nhs : IsOpen s\n⊢ X.str F ∈ s","state_after":"no goals","tactic":"rwa [h]","premises":[]}]}
+{"url":"Mathlib/Data/Stream/Init.lean","commit":"","full_name":"Stream'.mem_interleave_left","start":[376,0],"end":[377,69],"file_path":"Mathlib/Data/Stream/Init.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nδ : Type w\na : α\ns₁ s₂ : Stream' α\nx✝ : a ∈ s₁\nn : ℕ\nh : (fun b => a = b) (s₁.get n)\n⊢ (fun b => a = b) ((s₁ ⋈ s₂).get (2 * n))","state_after":"no goals","tactic":"rw [h, get_interleave_left]","premises":[{"full_name":"Stream'.get_interleave_left","def_path":"Mathlib/Data/Stream/Init.lean","def_pos":[358,8],"def_end_pos":[358,27]}]}]}
+{"url":"Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean","commit":"","full_name":"NumberField.mixedEmbedding.exists_ne_zero_mem_ideal_lt","start":[496,0],"end":[511,69],"file_path":"Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean","tactics":[{"state_before":"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nf : InfinitePlace K → ℝ≥0\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nh : minkowskiBound K I < volume (convexBodyLT K f)\n⊢ ∃ a ∈ ↑I, a ≠ 0 ∧ ∀ (w : InfinitePlace K), w a < ↑(f w)","state_after":"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nf : InfinitePlace K → ℝ≥0\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nh : minkowskiBound K I < volume (convexBodyLT K f)\nh_fund :\n  IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup)\n    (fundamentalDomain (fractionalIdealLatticeBasis K I)) volume\n⊢ ∃ a ∈ ↑I, a ≠ 0 ∧ ∀ (w : InfinitePlace K), w a < ↑(f w)","tactic":"have h_fund := Zspan.isAddFundamentalDomain (fractionalIdealLatticeBasis K I) volume","premises":[{"full_name":"MeasureTheory.MeasureSpace.volume","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[326,2],"def_end_pos":[326,8]},{"full_name":"NumberField.mixedEmbedding.fractionalIdealLatticeBasis","def_path":"Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean","def_pos":[633,4],"def_end_pos":[633,31]},{"full_name":"Zspan.isAddFundamentalDomain","def_path":"Mathlib/Algebra/Module/Zlattice/Basic.lean","def_pos":[309,18],"def_end_pos":[309,40]}]},{"state_before":"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nf : InfinitePlace K → ℝ≥0\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nh : minkowskiBound K I < volume (convexBodyLT K f)\nh_fund :\n  IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup)\n    (fundamentalDomain (fractionalIdealLatticeBasis K I)) volume\n⊢ ∃ a ∈ ↑I, a ≠ 0 ∧ ∀ (w : InfinitePlace K), w a < ↑(f w)","state_after":"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nf : InfinitePlace K → ℝ≥0\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nh : minkowskiBound K I < volume (convexBodyLT K f)\nh_fund :\n  IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup)\n    (fundamentalDomain (fractionalIdealLatticeBasis K I)) volume\nthis : Countable ↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup\n⊢ ∃ a ∈ ↑I, a ≠ 0 ∧ ∀ (w : InfinitePlace K), w a < ↑(f w)","tactic":"have : Countable (span ℤ (Set.range (fractionalIdealLatticeBasis K I))).toAddSubgroup := by\n    change Countable (span ℤ (Set.range (fractionalIdealLatticeBasis K I)) : Set (E K))\n    infer_instance","premises":[{"full_name":"Complex","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[27,10],"def_end_pos":[27,17]},{"full_name":"Countable","def_path":"Mathlib/Data/Countable/Defs.lean","def_pos":[36,6],"def_end_pos":[36,15]},{"full_name":"Int","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean","def_pos":[40,10],"def_end_pos":[40,13]},{"full_name":"NumberField.InfinitePlace","def_path":"Mathlib/NumberTheory/NumberField/Embeddings.lean","def_pos":[246,4],"def_end_pos":[246,29]},{"full_name":"NumberField.InfinitePlace.IsComplex","def_path":"Mathlib/NumberTheory/NumberField/Embeddings.lean","def_pos":[337,4],"def_end_pos":[337,13]},{"full_name":"NumberField.InfinitePlace.IsReal","def_path":"Mathlib/NumberTheory/NumberField/Embeddings.lean","def_pos":[334,4],"def_end_pos":[334,10]},{"full_name":"NumberField.mixedEmbedding.fractionalIdealLatticeBasis","def_path":"Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean","def_pos":[633,4],"def_end_pos":[633,31]},{"full_name":"Prod","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[479,10],"def_end_pos":[479,14]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Set","def_path":"Mathlib/Init/Set.lean","def_pos":[53,4],"def_end_pos":[53,7]},{"full_name":"Set.range","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[144,4],"def_end_pos":[144,9]},{"full_name":"Submodule.span","def_path":"Mathlib/LinearAlgebra/Span.lean","def_pos":[49,4],"def_end_pos":[49,8]},{"full_name":"Submodule.toAddSubgroup","def_path":"Mathlib/Algebra/Module/Submodule/Basic.lean","def_pos":[338,4],"def_end_pos":[338,17]},{"full_name":"Subtype","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[583,10],"def_end_pos":[583,17]},{"full_name":"inferInstance","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[99,7],"def_end_pos":[99,20]}]},{"state_before":"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nf : InfinitePlace K → ℝ≥0\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nh : minkowskiBound K I < volume (convexBodyLT K f)\nh_fund :\n  IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup)\n    (fundamentalDomain (fractionalIdealLatticeBasis K I)) volume\nthis : Countable ↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup\n⊢ ∃ a ∈ ↑I, a ≠ 0 ∧ ∀ (w : InfinitePlace K), w a < ↑(f w)","state_after":"case intro.mk.intro\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nf : InfinitePlace K → ℝ≥0\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nh : minkowskiBound K I < volume (convexBodyLT K f)\nh_fund :\n  IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup)\n    (fundamentalDomain (fractionalIdealLatticeBasis K I)) volume\nthis : Countable ↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup\nx : ({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ)\nhx : x ∈ (span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup\nh_nz : ⟨x, hx⟩ ≠ 0\nh_mem : ↑⟨x, hx⟩ ∈ convexBodyLT K f\n⊢ ∃ a ∈ ↑I, a ≠ 0 ∧ ∀ (w : InfinitePlace K), w a < ↑(f w)","tactic":"obtain ⟨⟨x, hx⟩, h_nz, h_mem⟩ := exists_ne_zero_mem_lattice_of_measure_mul_two_pow_lt_measure\n    h_fund (convexBodyLT_neg_mem K f) (convexBodyLT_convex K f) h","premises":[{"full_name":"MeasureTheory.exists_ne_zero_mem_lattice_of_measure_mul_two_pow_lt_measure","def_path":"Mathlib/MeasureTheory/Group/GeometryOfNumbers.lean","def_pos":[61,8],"def_end_pos":[61,68]},{"full_name":"NumberField.mixedEmbedding.convexBodyLT_convex","def_path":"Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean","def_pos":[75,8],"def_end_pos":[75,27]},{"full_name":"NumberField.mixedEmbedding.convexBodyLT_neg_mem","def_path":"Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean","def_pos":[68,8],"def_end_pos":[68,28]}]},{"state_before":"case intro.mk.intro\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nf : InfinitePlace K → ℝ≥0\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nh : minkowskiBound K I < volume (convexBodyLT K f)\nh_fund :\n  IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup)\n    (fundamentalDomain (fractionalIdealLatticeBasis K I)) volume\nthis : Countable ↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup\nx : ({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ)\nhx : x ∈ (span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup\nh_nz : ⟨x, hx⟩ ≠ 0\nh_mem : ↑⟨x, hx⟩ ∈ convexBodyLT K f\n⊢ ∃ a ∈ ↑I, a ≠ 0 ∧ ∀ (w : InfinitePlace K), w a < ↑(f w)","state_after":"case intro.mk.intro\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nf : InfinitePlace K → ℝ≥0\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nh : minkowskiBound K I < volume (convexBodyLT K f)\nh_fund :\n  IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup)\n    (fundamentalDomain (fractionalIdealLatticeBasis K I)) volume\nthis : Countable ↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup\nx : ({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ)\nhx✝ : x ∈ (span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup\nhx : x ∈ ⇑(mixedEmbedding K) '' ↑↑I\nh_nz : ⟨x, hx✝⟩ ≠ 0\nh_mem : ↑⟨x, hx✝⟩ ∈ convexBodyLT K f\n⊢ ∃ a ∈ ↑I, a ≠ 0 ∧ ∀ (w : InfinitePlace K), w a < ↑(f w)","tactic":"rw [mem_toAddSubgroup, mem_span_fractionalIdealLatticeBasis] at hx","premises":[{"full_name":"NumberField.mixedEmbedding.mem_span_fractionalIdealLatticeBasis","def_path":"Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean","def_pos":[653,8],"def_end_pos":[653,44]},{"full_name":"Submodule.mem_toAddSubgroup","def_path":"Mathlib/Algebra/Module/Submodule/Basic.lean","def_pos":[346,8],"def_end_pos":[346,25]}]},{"state_before":"case intro.mk.intro\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nf : InfinitePlace K → ℝ≥0\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nh : minkowskiBound K I < volume (convexBodyLT K f)\nh_fund :\n  IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup)\n    (fundamentalDomain (fractionalIdealLatticeBasis K I)) volume\nthis : Countable ↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup\nx : ({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ)\nhx✝ : x ∈ (span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup\nhx : x ∈ ⇑(mixedEmbedding K) '' ↑↑I\nh_nz : ⟨x, hx✝⟩ ≠ 0\nh_mem : ↑⟨x, hx✝⟩ ∈ convexBodyLT K f\n⊢ ∃ a ∈ ↑I, a ≠ 0 ∧ ∀ (w : InfinitePlace K), w a < ↑(f w)","state_after":"case intro.mk.intro.intro.intro\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nf : InfinitePlace K → ℝ≥0\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nh : minkowskiBound K I < volume (convexBodyLT K f)\nh_fund :\n  IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup)\n    (fundamentalDomain (fractionalIdealLatticeBasis K I)) volume\nthis : Countable ↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup\na : K\nha : a ∈ ↑↑I\nhx : (mixedEmbedding K) a ∈ (span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup\nh_nz : ⟨(mixedEmbedding K) a, hx⟩ ≠ 0\nh_mem : ↑⟨(mixedEmbedding K) a, hx⟩ ∈ convexBodyLT K f\n⊢ ∃ a ∈ ↑I, a ≠ 0 ∧ ∀ (w : InfinitePlace K), w a < ↑(f w)","tactic":"obtain ⟨a, ha, rfl⟩ := hx","premises":[]},{"state_before":"case intro.mk.intro.intro.intro\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nf : InfinitePlace K → ℝ≥0\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nh : minkowskiBound K I < volume (convexBodyLT K f)\nh_fund :\n  IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup)\n    (fundamentalDomain (fractionalIdealLatticeBasis K I)) volume\nthis : Countable ↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup\na : K\nha : a ∈ ↑↑I\nhx : (mixedEmbedding K) a ∈ (span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup\nh_nz : ⟨(mixedEmbedding K) a, hx⟩ ≠ 0\nh_mem : ↑⟨(mixedEmbedding K) a, hx⟩ ∈ convexBodyLT K f\n⊢ ∃ a ∈ ↑I, a ≠ 0 ∧ ∀ (w : InfinitePlace K), w a < ↑(f w)","state_after":"no goals","tactic":"exact ⟨a, ha, by simpa using h_nz, (convexBodyLT_mem K f).mp h_mem⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"NumberField.mixedEmbedding.convexBodyLT_mem","def_path":"Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean","def_pos":[61,8],"def_end_pos":[61,24]}]}]}
+{"url":"Mathlib/RingTheory/Trace/Basic.lean","commit":"","full_name":"Algebra.traceMatrix_of_basis","start":[324,0],"end":[327,61],"file_path":"Mathlib/RingTheory/Trace/Basic.lean","tactics":[{"state_before":"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝²⁰ : CommRing R\ninst✝¹⁹ : CommRing S\ninst✝¹⁸ : CommRing T\ninst✝¹⁷ : Algebra R S\ninst✝¹⁶ : Algebra R T\nK : Type u_4\nL : Type u_5\ninst✝¹⁵ : Field K\ninst✝¹⁴ : Field L\ninst✝¹³ : Algebra K L\nι κ : Type w\ninst✝¹² : Fintype ι\nF : Type u_6\ninst✝¹¹ : Field F\ninst✝¹⁰ : Algebra R L\ninst✝⁹ : Algebra L F\ninst✝⁸ : Algebra R F\ninst✝⁷ : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst✝⁶ : CommRing A\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra A B\ninst✝³ : CommRing C\ninst✝² : Algebra A C\ninst✝¹ : Fintype κ\ninst✝ : DecidableEq κ\nb : Basis κ A B\n⊢ traceMatrix A ⇑b = (BilinForm.toMatrix b) (traceForm A B)","state_after":"case a\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝²⁰ : CommRing R\ninst✝¹⁹ : CommRing S\ninst✝¹⁸ : CommRing T\ninst✝¹⁷ : Algebra R S\ninst✝¹⁶ : Algebra R T\nK : Type u_4\nL : Type u_5\ninst✝¹⁵ : Field K\ninst✝¹⁴ : Field L\ninst✝¹³ : Algebra K L\nι κ : Type w\ninst✝¹² : Fintype ι\nF : Type u_6\ninst✝¹¹ : Field F\ninst✝¹⁰ : Algebra R L\ninst✝⁹ : Algebra L F\ninst✝⁸ : Algebra R F\ninst✝⁷ : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst✝⁶ : CommRing A\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra A B\ninst✝³ : CommRing C\ninst✝² : Algebra A C\ninst✝¹ : Fintype κ\ninst✝ : DecidableEq κ\nb : Basis κ A B\ni j : κ\n⊢ traceMatrix A (⇑b) i j = (BilinForm.toMatrix b) (traceForm A B) i j","tactic":"ext (i j)","premises":[]},{"state_before":"case a\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝²⁰ : CommRing R\ninst✝¹⁹ : CommRing S\ninst✝¹⁸ : CommRing T\ninst✝¹⁷ : Algebra R S\ninst✝¹⁶ : Algebra R T\nK : Type u_4\nL : Type u_5\ninst✝¹⁵ : Field K\ninst✝¹⁴ : Field L\ninst✝¹³ : Algebra K L\nι κ : Type w\ninst✝¹² : Fintype ι\nF : Type u_6\ninst✝¹¹ : Field F\ninst✝¹⁰ : Algebra R L\ninst✝⁹ : Algebra L F\ninst✝⁸ : Algebra R F\ninst✝⁷ : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst✝⁶ : CommRing A\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra A B\ninst✝³ : CommRing C\ninst✝² : Algebra A C\ninst✝¹ : Fintype κ\ninst✝ : DecidableEq κ\nb : Basis κ A B\ni j : κ\n⊢ traceMatrix A (⇑b) i j = (BilinForm.toMatrix b) (traceForm A B) i j","state_after":"no goals","tactic":"rw [traceMatrix_apply, traceForm_apply, traceForm_toMatrix]","premises":[{"full_name":"Algebra.traceForm_apply","def_path":"Mathlib/RingTheory/Trace/Defs.lean","def_pos":[163,8],"def_end_pos":[163,23]},{"full_name":"Algebra.traceForm_toMatrix","def_path":"Mathlib/RingTheory/Trace/Defs.lean","def_pos":[168,8],"def_end_pos":[168,26]},{"full_name":"Algebra.traceMatrix_apply","def_path":"Mathlib/RingTheory/Trace/Basic.lean","def_pos":[290,8],"def_end_pos":[290,25]}]}]}
+{"url":"Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean","commit":"","full_name":"rel_iSup_sum","start":[157,0],"end":[163,38],"file_path":"Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean","tactics":[{"state_before":"M : Type u_1\ninst✝⁵ : CommMonoid M\ninst✝⁴ : TopologicalSpace M\nm✝ m' : M\nG : Type u_2\ninst✝³ : CommGroup G\ng g' : G\ninst✝² : T2Space M\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝¹ : Countable β\ninst✝ : CompleteLattice α\nm : α → M\nm0 : m ⊥ = 1\nR : M → M → Prop\nm_iSup : ∀ (s : ℕ → α), R (m (⨆ i, s i)) (∏' (i : ℕ), m (s i))\ns : γ → α\nt : Finset γ\n⊢ R (m (⨆ d ∈ t, s d)) (∏ d ∈ t, m (s d))","state_after":"M : Type u_1\ninst✝⁵ : CommMonoid M\ninst✝⁴ : TopologicalSpace M\nm✝ m' : M\nG : Type u_2\ninst✝³ : CommGroup G\ng g' : G\ninst✝² : T2Space M\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝¹ : Countable β\ninst✝ : CompleteLattice α\nm : α → M\nm0 : m ⊥ = 1\nR : M → M → Prop\nm_iSup : ∀ (s : ℕ → α), R (m (⨆ i, s i)) (∏' (i : ℕ), m (s i))\ns : γ → α\nt : Finset γ\n⊢ R (m (⨆ x, s ↑x)) (∏' (x : { x // x ∈ t }), m (s ↑x))","tactic":"rw [iSup_subtype', ← Finset.tprod_subtype]","premises":[{"full_name":"Finset.tprod_subtype","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Basic.lean","def_pos":[421,8],"def_end_pos":[421,28]},{"full_name":"iSup_subtype'","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[979,8],"def_end_pos":[979,21]}]},{"state_before":"M : Type u_1\ninst✝⁵ : CommMonoid M\ninst✝⁴ : TopologicalSpace M\nm✝ m' : M\nG : Type u_2\ninst✝³ : CommGroup G\ng g' : G\ninst✝² : T2Space M\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝¹ : Countable β\ninst✝ : CompleteLattice α\nm : α → M\nm0 : m ⊥ = 1\nR : M → M → Prop\nm_iSup : ∀ (s : ℕ → α), R (m (⨆ i, s i)) (∏' (i : ℕ), m (s i))\ns : γ → α\nt : Finset γ\n⊢ R (m (⨆ x, s ↑x)) (∏' (x : { x // x ∈ t }), m (s ↑x))","state_after":"no goals","tactic":"exact rel_iSup_tprod m m0 R m_iSup _","premises":[{"full_name":"rel_iSup_tprod","def_path":"Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean","def_pos":[150,8],"def_end_pos":[150,22]}]}]}
+{"url":"Mathlib/Analysis/Normed/Group/AddTorsor.lean","commit":"","full_name":"nndist_vsub_vsub_le","start":[159,0],"end":[161,83],"file_path":"Mathlib/Analysis/Normed/Group/AddTorsor.lean","tactics":[{"state_before":"α : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst✝⁵ : SeminormedAddCommGroup V\ninst✝⁴ : PseudoMetricSpace P\ninst✝³ : NormedAddTorsor V P\ninst✝² : NormedAddCommGroup W\ninst✝¹ : MetricSpace Q\ninst✝ : NormedAddTorsor W Q\np₁ p₂ p₃ p₄ : P\n⊢ nndist (p₁ -ᵥ p₂) (p₃ -ᵥ p₄) ≤ nndist p₁ p₃ + nndist p₂ p₄","state_after":"no goals","tactic":"simp only [← NNReal.coe_le_coe, NNReal.coe_add, ← dist_nndist, dist_vsub_vsub_le]","premises":[{"full_name":"NNReal.coe_add","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[160,18],"def_end_pos":[160,25]},{"full_name":"NNReal.coe_le_coe","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[325,25],"def_end_pos":[325,35]},{"full_name":"dist_nndist","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[269,8],"def_end_pos":[269,19]},{"full_name":"dist_vsub_vsub_le","def_path":"Mathlib/Analysis/Normed/Group/AddTorsor.lean","def_pos":[154,8],"def_end_pos":[154,25]}]}]}
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+{"url":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","commit":"","full_name":"List.pairwise_reverse","start":[783,0],"end":[785,53],"file_path":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","tactics":[{"state_before":"α : Type u_1\nR : α → α → Prop\nl : List α\n⊢ Pairwise R l.reverse ↔ Pairwise (fun a b => R b a) l","state_after":"no goals","tactic":"induction l <;> simp [*, pairwise_append, and_comm]","premises":[{"full_name":"List.pairwise_append","def_path":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","def_pos":[779,8],"def_end_pos":[779,23]},{"full_name":"and_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[819,8],"def_end_pos":[819,16]}]}]}
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EventuallyEq.rfl)","premises":[{"full_name":"Filter.EventuallyEq","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1260,4],"def_end_pos":[1260,16]},{"full_name":"Filter.EventuallyEq.mul","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1345,8],"def_end_pos":[1345,24]},{"full_name":"Filter.EventuallyEq.rfl","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1303,18],"def_end_pos":[1303,34]},{"full_name":"MeasureTheory.AEStronglyMeasurable'.ae_eq_mk","def_path":"Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean","def_pos":[111,8],"def_end_pos":[111,16]},{"full_name":"MeasureTheory.AEStronglyMeasurable'.mk","def_path":"Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean","def_pos":[104,18],"def_end_pos":[104,20]},{"full_name":"MeasureTheory.ae","def_path":"Mathlib/MeasureTheory/OuterMeasure/AE.lean","def_pos":[43,4],"def_end_pos":[43,6]},{"full_name":"MeasureTheory.condexp","def_path":"Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean","def_pos":[90,30],"def_end_pos":[90,37]},{"full_name":"MeasureTheory.condexp_congr_ae","def_path":"Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean","def_pos":[177,8],"def_end_pos":[177,24]}]},{"state_before":"α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\ninst✝ : IsFiniteMeasure μ\nf g : α → ℝ\nhf : AEStronglyMeasurable' m f μ\nhg : Integrable g μ\nc : ℝ\nhf_bound : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ c\nthis : μ[f * g|m] =ᶠ[ae μ] μ[AEStronglyMeasurable'.mk f hf * g|m]\n⊢ μ[f * g|m] =ᶠ[ae μ] f * μ[g|m]","state_after":"α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\ninst✝ : IsFiniteMeasure μ\nf g : α → ℝ\nhf : AEStronglyMeasurable' m f μ\nhg : Integrable g μ\nc : ℝ\nhf_bound : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ c\nthis : μ[f * g|m] =ᶠ[ae μ] μ[AEStronglyMeasurable'.mk f hf * g|m]\n⊢ μ[AEStronglyMeasurable'.mk f hf * g|m] =ᶠ[ae μ] f * μ[g|m]","tactic":"refine this.trans ?_","premises":[{"full_name":"Filter.EventuallyEq.trans","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1311,8],"def_end_pos":[1311,26]}]},{"state_before":"α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\ninst✝ : IsFiniteMeasure μ\nf g : α → ℝ\nhf : AEStronglyMeasurable' m f μ\nhg : Integrable g μ\nc : ℝ\nhf_bound : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ c\nthis : μ[f * g|m] =ᶠ[ae μ] μ[AEStronglyMeasurable'.mk f hf * g|m]\n⊢ μ[AEStronglyMeasurable'.mk f hf * g|m] =ᶠ[ae μ] f * μ[g|m]","state_after":"α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\ninst✝ : IsFiniteMeasure μ\nf g : α → ℝ\nhf : AEStronglyMeasurable' m f μ\nhg : Integrable g μ\nc : ℝ\nhf_bound : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ c\nthis✝ : μ[f * g|m] =ᶠ[ae μ] μ[AEStronglyMeasurable'.mk f hf * g|m]\nthis : f * μ[g|m] =ᶠ[ae μ] AEStronglyMeasurable'.mk f hf * μ[g|m]\n⊢ μ[AEStronglyMeasurable'.mk f hf * g|m] =ᶠ[ae μ] f * μ[g|m]","tactic":"have : f * μ[g|m] =ᵐ[μ] hf.mk f * μ[g|m] := EventuallyEq.mul hf.ae_eq_mk EventuallyEq.rfl","premises":[{"full_name":"Filter.EventuallyEq","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1260,4],"def_end_pos":[1260,16]},{"full_name":"Filter.EventuallyEq.mul","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1345,8],"def_end_pos":[1345,24]},{"full_name":"Filter.EventuallyEq.rfl","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1303,18],"def_end_pos":[1303,34]},{"full_name":"MeasureTheory.AEStronglyMeasurable'.ae_eq_mk","def_path":"Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean","def_pos":[111,8],"def_end_pos":[111,16]},{"full_name":"MeasureTheory.AEStronglyMeasurable'.mk","def_path":"Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean","def_pos":[104,18],"def_end_pos":[104,20]},{"full_name":"MeasureTheory.ae","def_path":"Mathlib/MeasureTheory/OuterMeasure/AE.lean","def_pos":[43,4],"def_end_pos":[43,6]},{"full_name":"MeasureTheory.condexp","def_path":"Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean","def_pos":[90,30],"def_end_pos":[90,37]}]},{"state_before":"α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\ninst✝ : IsFiniteMeasure μ\nf g : α → ℝ\nhf : AEStronglyMeasurable' m f μ\nhg : Integrable g μ\nc : ℝ\nhf_bound : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ c\nthis✝ : μ[f * g|m] =ᶠ[ae μ] μ[AEStronglyMeasurable'.mk f hf * g|m]\nthis : f * μ[g|m] =ᶠ[ae μ] AEStronglyMeasurable'.mk f hf * μ[g|m]\n⊢ μ[AEStronglyMeasurable'.mk f hf * g|m] =ᶠ[ae μ] f * μ[g|m]","state_after":"α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\ninst✝ : IsFiniteMeasure μ\nf g : α → ℝ\nhf : AEStronglyMeasurable' m f μ\nhg : Integrable g μ\nc : ℝ\nhf_bound : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ c\nthis✝ : μ[f * g|m] =ᶠ[ae μ] μ[AEStronglyMeasurable'.mk f hf * g|m]\nthis : f * μ[g|m] =ᶠ[ae μ] AEStronglyMeasurable'.mk f hf * μ[g|m]\n⊢ μ[AEStronglyMeasurable'.mk f hf * g|m] =ᶠ[ae μ] AEStronglyMeasurable'.mk f hf * μ[g|m]","tactic":"refine EventuallyEq.trans ?_ this.symm","premises":[{"full_name":"Filter.EventuallyEq.symm","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1307,8],"def_end_pos":[1307,25]},{"full_name":"Filter.EventuallyEq.trans","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1311,8],"def_end_pos":[1311,26]}]},{"state_before":"α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\ninst✝ : IsFiniteMeasure μ\nf g : α → ℝ\nhf : AEStronglyMeasurable' m f μ\nhg : Integrable g μ\nc : ℝ\nhf_bound : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ c\nthis✝ : μ[f * g|m] =ᶠ[ae μ] μ[AEStronglyMeasurable'.mk f hf * g|m]\nthis : f * μ[g|m] =ᶠ[ae μ] AEStronglyMeasurable'.mk f hf * μ[g|m]\n⊢ μ[AEStronglyMeasurable'.mk f hf * g|m] =ᶠ[ae μ] AEStronglyMeasurable'.mk f hf * μ[g|m]","state_after":"α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\ninst✝ : IsFiniteMeasure μ\nf g : α → ℝ\nhf : AEStronglyMeasurable' m f μ\nhg : Integrable g μ\nc : ℝ\nhf_bound : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ c\nthis✝ : μ[f * g|m] =ᶠ[ae μ] μ[AEStronglyMeasurable'.mk f hf * g|m]\nthis : f * μ[g|m] =ᶠ[ae μ] AEStronglyMeasurable'.mk f hf * μ[g|m]\n⊢ ∀ᵐ (x : α) ∂μ, ‖AEStronglyMeasurable'.mk f hf x‖ ≤ c","tactic":"refine condexp_stronglyMeasurable_mul_of_bound hm hf.stronglyMeasurable_mk hg c ?_","premises":[{"full_name":"MeasureTheory.AEStronglyMeasurable'.stronglyMeasurable_mk","def_path":"Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean","def_pos":[107,8],"def_end_pos":[107,29]},{"full_name":"MeasureTheory.condexp_stronglyMeasurable_mul_of_bound","def_path":"Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean","def_pos":[251,8],"def_end_pos":[251,47]}]},{"state_before":"α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\ninst✝ : IsFiniteMeasure μ\nf g : α → ℝ\nhf : AEStronglyMeasurable' m f μ\nhg : Integrable g μ\nc : ℝ\nhf_bound : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ c\nthis✝ : μ[f * g|m] =ᶠ[ae μ] μ[AEStronglyMeasurable'.mk f hf * g|m]\nthis : f * μ[g|m] =ᶠ[ae μ] AEStronglyMeasurable'.mk f hf * μ[g|m]\n⊢ ∀ᵐ (x : α) ∂μ, ‖AEStronglyMeasurable'.mk f hf x‖ ≤ c","state_after":"case h\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\ninst✝ : IsFiniteMeasure μ\nf g : α → ℝ\nhf : AEStronglyMeasurable' m f μ\nhg : Integrable g μ\nc : ℝ\nhf_bound : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ c\nthis✝ : μ[f * g|m] =ᶠ[ae μ] μ[AEStronglyMeasurable'.mk f hf * g|m]\nthis : f * μ[g|m] =ᶠ[ae μ] AEStronglyMeasurable'.mk f hf * μ[g|m]\nx : α\nhxc : ‖f x‖ ≤ c\nhx_eq : f x = AEStronglyMeasurable'.mk f hf x\n⊢ ‖AEStronglyMeasurable'.mk f hf x‖ ≤ c","tactic":"filter_upwards [hf_bound, hf.ae_eq_mk] with x hxc hx_eq","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Filter.mp_mem","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[159,8],"def_end_pos":[159,14]},{"full_name":"MeasureTheory.AEStronglyMeasurable'.ae_eq_mk","def_path":"Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean","def_pos":[111,8],"def_end_pos":[111,16]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]}]},{"state_before":"case h\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\ninst✝ : IsFiniteMeasure μ\nf g : α → ℝ\nhf : AEStronglyMeasurable' m f μ\nhg : Integrable g μ\nc : ℝ\nhf_bound : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ c\nthis✝ : μ[f * g|m] =ᶠ[ae μ] μ[AEStronglyMeasurable'.mk f hf * g|m]\nthis : f * μ[g|m] =ᶠ[ae μ] AEStronglyMeasurable'.mk f hf * μ[g|m]\nx : α\nhxc : ‖f x‖ ≤ c\nhx_eq : f x = AEStronglyMeasurable'.mk f hf x\n⊢ ‖AEStronglyMeasurable'.mk f hf x‖ ≤ c","state_after":"no goals","tactic":"rwa [← hx_eq]","premises":[]}]}
+{"url":"Mathlib/Algebra/Order/Group/PosPart.lean","commit":"","full_name":"posPart_add_negPart","start":[144,0],"end":[147,38],"file_path":"Mathlib/Algebra/Order/Group/PosPart.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝³ : Lattice α\ninst✝² : Group α\na✝ : α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na : α\n⊢ a⁺ᵐ * a⁻ᵐ = mabs a","state_after":"α : Type u_1\nβ : Type u_2\ninst✝³ : Lattice α\ninst✝² : Group α\na✝ : α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na : α\n⊢ a ⊔ a⁻¹ ⊔ 1 = mabs a","tactic":"rw [oneLePart, sup_mul, one_mul, leOnePart, mul_sup, mul_one, mul_inv_self, sup_assoc,\n    ← sup_assoc a, sup_eq_right.2 le_sup_right]","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"leOnePart","def_path":"Mathlib/Algebra/Order/Group/PosPart.lean","def_pos":[63,4],"def_end_pos":[63,13]},{"full_name":"le_sup_right","def_path":"Mathlib/Order/Lattice.lean","def_pos":[112,8],"def_end_pos":[112,20]},{"full_name":"mul_inv_self","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[1055,8],"def_end_pos":[1055,20]},{"full_name":"mul_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[413,8],"def_end_pos":[413,15]},{"full_name":"mul_sup","def_path":"Mathlib/Algebra/Order/Group/Lattice.lean","def_pos":[48,6],"def_end_pos":[48,13]},{"full_name":"oneLePart","def_path":"Mathlib/Algebra/Order/Group/PosPart.lean","def_pos":[57,4],"def_end_pos":[57,13]},{"full_name":"one_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[409,8],"def_end_pos":[409,15]},{"full_name":"sup_assoc","def_path":"Mathlib/Order/Lattice.lean","def_pos":[197,8],"def_end_pos":[197,17]},{"full_name":"sup_eq_right","def_path":"Mathlib/Order/Lattice.lean","def_pos":[142,8],"def_end_pos":[142,20]},{"full_name":"sup_mul","def_path":"Mathlib/Algebra/Order/Group/Lattice.lean","def_pos":[51,6],"def_end_pos":[51,13]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝³ : Lattice α\ninst✝² : Group α\na✝ : α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na : α\n⊢ a ⊔ a⁻¹ ⊔ 1 = mabs a","state_after":"no goals","tactic":"exact sup_eq_left.2 <| one_le_mabs a","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"one_le_mabs","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[95,47],"def_end_pos":[95,58]},{"full_name":"sup_eq_left","def_path":"Mathlib/Order/Lattice.lean","def_pos":[138,8],"def_end_pos":[138,19]}]}]}
+{"url":"Mathlib/Algebra/Module/LocalizedModuleIntegers.lean","commit":"","full_name":"IsLocalizedModule.exist_integer_multiples_of_finite","start":[72,0],"end":[77,46],"file_path":"Mathlib/Algebra/Module/LocalizedModuleIntegers.lean","tactics":[{"state_before":"R : Type u_1\ninst✝⁶ : CommSemiring R\nS : Submonoid R\nM : Type u_2\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nM' : Type u_3\ninst✝³ : AddCommMonoid M'\ninst✝² : Module R M'\nf : M →ₗ[R] M'\ninst✝¹ : IsLocalizedModule S f\nι : Type u_4\ninst✝ : Finite ι\ng : ι → M'\n⊢ ∃ b, ∀ (i : ι), IsInteger f (↑b • g i)","state_after":"case intro\nR : Type u_1\ninst✝⁶ : CommSemiring R\nS : Submonoid R\nM : Type u_2\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nM' : Type u_3\ninst✝³ : AddCommMonoid M'\ninst✝² : Module R M'\nf : M →ₗ[R] M'\ninst✝¹ : IsLocalizedModule S f\nι : Type u_4\ninst✝ : Finite ι\ng : ι → M'\nval✝ : Fintype ι\n⊢ ∃ b, ∀ (i : ι), IsInteger f (↑b • g i)","tactic":"cases nonempty_fintype ι","premises":[{"full_name":"nonempty_fintype","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[390,8],"def_end_pos":[390,24]}]},{"state_before":"case intro\nR : Type u_1\ninst✝⁶ : CommSemiring R\nS : Submonoid R\nM : Type u_2\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nM' : Type u_3\ninst✝³ : AddCommMonoid M'\ninst✝² : Module R M'\nf : M →ₗ[R] M'\ninst✝¹ : IsLocalizedModule S f\nι : Type u_4\ninst✝ : Finite ι\ng : ι → M'\nval✝ : Fintype ι\n⊢ ∃ b, ∀ (i : ι), IsInteger f (↑b • g i)","state_after":"case intro.intro\nR : Type u_1\ninst✝⁶ : CommSemiring R\nS : Submonoid R\nM : Type u_2\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nM' : Type u_3\ninst✝³ : AddCommMonoid M'\ninst✝² : Module R M'\nf : M →ₗ[R] M'\ninst✝¹ : IsLocalizedModule S f\nι : Type u_4\ninst✝ : Finite ι\ng : ι → M'\nval✝ : Fintype ι\nb : ↥S\nhb : ∀ i ∈ Finset.univ, IsInteger f (↑b • g i)\n⊢ ∃ b, ∀ (i : ι), IsInteger f (↑b • g i)","tactic":"obtain ⟨b, hb⟩ := exist_integer_multiples S f Finset.univ g","premises":[{"full_name":"Finset.univ","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[67,4],"def_end_pos":[67,8]},{"full_name":"IsLocalizedModule.exist_integer_multiples","def_path":"Mathlib/Algebra/Module/LocalizedModuleIntegers.lean","def_pos":[59,8],"def_end_pos":[59,31]}]},{"state_before":"case intro.intro\nR : Type u_1\ninst✝⁶ : CommSemiring R\nS : Submonoid R\nM : Type u_2\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nM' : Type u_3\ninst✝³ : AddCommMonoid M'\ninst✝² : Module R M'\nf : M →ₗ[R] M'\ninst✝¹ : IsLocalizedModule S f\nι : Type u_4\ninst✝ : Finite ι\ng : ι → M'\nval✝ : Fintype ι\nb : ↥S\nhb : ∀ i ∈ Finset.univ, IsInteger f (↑b • g i)\n⊢ ∃ b, ∀ (i : ι), IsInteger f (↑b • g i)","state_after":"no goals","tactic":"exact ⟨b, fun i => hb i (Finset.mem_univ _)⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Finset.mem_univ","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[71,8],"def_end_pos":[71,16]}]}]}
+{"url":"Mathlib/Data/Set/Card.lean","commit":"","full_name":"Set.ncard_image_le","start":[621,0],"end":[622,89],"file_path":"Mathlib/Data/Set/Card.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ns t : Set α\nf : α → β\nhs : autoParam s.Finite _auto✝\n⊢ (f '' s).ncard ≤ s.ncard","state_after":"α : Type u_1\nβ : Type u_2\ns t : Set α\nf : α → β\nhs : autoParam s.Finite _auto✝\n⊢ ↑(f '' s).ncard ≤ ↑s.ncard","tactic":"to_encard_tac","premises":[{"full_name":"ENat","def_path":"Mathlib/Data/ENat/Basic.lean","def_pos":[28,4],"def_end_pos":[28,8]},{"full_name":"Nat.cast_add","def_path":"Mathlib/Data/Nat/Cast/Defs.lean","def_pos":[137,8],"def_end_pos":[137,16]},{"full_name":"Nat.cast_inj","def_path":"Mathlib/Algebra/CharZero/Defs.lean","def_pos":[69,8],"def_end_pos":[69,16]},{"full_name":"Nat.cast_le","def_path":"Mathlib/Data/Nat/Cast/Order/Basic.lean","def_pos":[78,8],"def_end_pos":[78,15]},{"full_name":"Nat.cast_one","def_path":"Mathlib/Data/Nat/Cast/Defs.lean","def_pos":[133,8],"def_end_pos":[133,16]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ns t : Set α\nf : α → β\nhs : autoParam s.Finite _auto✝\n⊢ ↑(f '' s).ncard ≤ ↑s.ncard","state_after":"α : Type u_1\nβ : Type u_2\ns t : Set α\nf : α → β\nhs : autoParam s.Finite _auto✝\n⊢ (f '' s).encard ≤ s.encard","tactic":"rw [hs.cast_ncard_eq, (hs.image _).cast_ncard_eq]","premises":[{"full_name":"Set.Finite.cast_ncard_eq","def_path":"Mathlib/Data/Set/Card.lean","def_pos":[477,8],"def_end_pos":[477,28]},{"full_name":"Set.Finite.image","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[735,8],"def_end_pos":[735,20]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ns t : Set α\nf : α → β\nhs : autoParam s.Finite _auto✝\n⊢ (f '' s).encard ≤ s.encard","state_after":"no goals","tactic":"apply encard_image_le","premises":[{"full_name":"Set.encard_image_le","def_path":"Mathlib/Data/Set/Card.lean","def_pos":[394,8],"def_end_pos":[394,23]}]}]}
+{"url":"Mathlib/MeasureTheory/Function/L1Space.lean","commit":"","full_name":"MeasureTheory.HasFiniteIntegral.min_zero","start":[341,0],"end":[343,79],"file_path":"Mathlib/MeasureTheory/Function/L1Space.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → ℝ\nhf : HasFiniteIntegral f μ\nx : α\n⊢ ‖min (f x) 0‖ ≤ ‖f x‖","state_after":"no goals","tactic":"simpa [abs_le] using neg_abs_le _","premises":[{"full_name":"abs_le","def_path":"Mathlib/Algebra/Order/Group/Abs.lean","def_pos":[70,8],"def_end_pos":[70,14]},{"full_name":"neg_abs_le","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[220,2],"def_end_pos":[220,13]}]}]}
+{"url":"Mathlib/LinearAlgebra/Matrix/Reindex.lean","commit":"","full_name":"Matrix.reindexLinearEquiv_trans","start":[60,0],"end":[64,5],"file_path":"Mathlib/LinearAlgebra/Matrix/Reindex.lean","tactics":[{"state_before":"l : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nl' : Type u_5\nm' : Type u_6\nn' : Type u_7\no' : Type u_8\nm'' : Type u_9\nn'' : Type u_10\nR : Type u_11\nA : Type u_12\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid A\ninst✝ : Module R A\ne₁ : m ≃ m'\ne₂ : n ≃ n'\ne₁' : m' ≃ m''\ne₂' : n' ≃ n''\n⊢ reindexLinearEquiv R A e₁ e₂ ≪≫ₗ reindexLinearEquiv R A e₁' e₂' = reindexLinearEquiv R A (e₁.trans e₁') (e₂.trans e₂')","state_after":"case h.a\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nl' : Type u_5\nm' : Type u_6\nn' : Type u_7\no' : Type u_8\nm'' : Type u_9\nn'' : Type u_10\nR : Type u_11\nA : Type u_12\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid A\ninst✝ : Module R A\ne₁ : m ≃ m'\ne₂ : n ≃ n'\ne₁' : m' ≃ m''\ne₂' : n' ≃ n''\nx✝ : Matrix m n A\ni✝ : m''\nj✝ : n''\n⊢ (reindexLinearEquiv R A e₁ e₂ ≪≫ₗ reindexLinearEquiv R A e₁' e₂') x✝ i✝ j✝ =\n    (reindexLinearEquiv R A (e₁.trans e₁') (e₂.trans e₂')) x✝ i✝ j✝","tactic":"ext","premises":[]},{"state_before":"case h.a\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nl' : Type u_5\nm' : Type u_6\nn' : Type u_7\no' : Type u_8\nm'' : Type u_9\nn'' : Type u_10\nR : Type u_11\nA : Type u_12\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid A\ninst✝ : Module R A\ne₁ : m ≃ m'\ne₂ : n ≃ n'\ne₁' : m' ≃ m''\ne₂' : n' ≃ n''\nx✝ : Matrix m n A\ni✝ : m''\nj✝ : n''\n⊢ (reindexLinearEquiv R A e₁ e₂ ≪≫ₗ reindexLinearEquiv R A e₁' e₂') x✝ i✝ j✝ =\n    (reindexLinearEquiv R A (e₁.trans e₁') (e₂.trans e₂')) x✝ i✝ j✝","state_after":"no goals","tactic":"rfl","premises":[]}]}
+{"url":"Mathlib/LinearAlgebra/Basis.lean","commit":"","full_name":"Basis.constr_apply","start":[591,0],"end":[594,54],"file_path":"Mathlib/LinearAlgebra/Basis.lean","tactics":[{"state_before":"ι : Type u_1\nι' : Type u_2\nR : Type u_3\nR₂ : Type u_4\nK : Type u_5\nM : Type u_6\nM' : Type u_7\nM'' : Type u_8\nV : Type u\nV' : Type u_9\ninst✝⁷ : Semiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : Module R M'\nb b₁ : Basis ι R M\ni : ι\nc : R\nx✝ : M\nS : Type u_10\ninst✝² : Semiring S\ninst✝¹ : Module S M'\ninst✝ : SMulCommClass R S M'\nf : ι → M'\nx : M\n⊢ ((b.constr S) f) x = (b.repr x).sum fun b a => a • f b","state_after":"ι : Type u_1\nι' : Type u_2\nR : Type u_3\nR₂ : Type u_4\nK : Type u_5\nM : Type u_6\nM' : Type u_7\nM'' : Type u_8\nV : Type u\nV' : Type u_9\ninst✝⁷ : Semiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : Module R M'\nb b₁ : Basis ι R M\ni : ι\nc : R\nx✝ : M\nS : Type u_10\ninst✝² : Semiring S\ninst✝¹ : Module S M'\ninst✝ : SMulCommClass R S M'\nf : ι → M'\nx : M\n⊢ ((Finsupp.mapDomain f (↑b.repr x)).sum fun i a => a • id i) = (b.repr x).sum fun b a => a • f b","tactic":"simp only [constr_def, LinearMap.comp_apply, Finsupp.lmapDomain_apply, Finsupp.total_apply]","premises":[{"full_name":"Basis.constr_def","def_path":"Mathlib/LinearAlgebra/Basis.lean","def_pos":[587,8],"def_end_pos":[587,18]},{"full_name":"Finsupp.lmapDomain_apply","def_path":"Mathlib/LinearAlgebra/Finsupp.lean","def_pos":[512,8],"def_end_pos":[512,24]},{"full_name":"Finsupp.total_apply","def_path":"Mathlib/LinearAlgebra/Finsupp.lean","def_pos":[602,8],"def_end_pos":[602,19]},{"full_name":"LinearMap.comp_apply","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[504,8],"def_end_pos":[504,18]}]},{"state_before":"ι : Type u_1\nι' : Type u_2\nR : Type u_3\nR₂ : Type u_4\nK : Type u_5\nM : Type u_6\nM' : Type u_7\nM'' : Type u_8\nV : Type u\nV' : Type u_9\ninst✝⁷ : Semiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : Module R M'\nb b₁ : Basis ι R M\ni : ι\nc : R\nx✝ : M\nS : Type u_10\ninst✝² : Semiring S\ninst✝¹ : Module S M'\ninst✝ : SMulCommClass R S M'\nf : ι → M'\nx : M\n⊢ ((Finsupp.mapDomain f (↑b.repr x)).sum fun i a => a • id i) = (b.repr x).sum fun b a => a • f b","state_after":"no goals","tactic":"rw [Finsupp.sum_mapDomain_index] <;> simp [add_smul]","premises":[{"full_name":"Finsupp.sum_mapDomain_index","def_path":"Mathlib/Data/Finsupp/Basic.lean","def_pos":[506,2],"def_end_pos":[506,13]},{"full_name":"add_smul","def_path":"Mathlib/Algebra/Module/Defs.lean","def_pos":[87,8],"def_end_pos":[87,16]}]}]}
+{"url":"Mathlib/Topology/SeparatedMap.lean","commit":"","full_name":"isLocallyInjective_iff_nhds","start":[124,0],"end":[129,98],"file_path":"Mathlib/Topology/SeparatedMap.lean","tactics":[{"state_before":"X : Type u_1\nY : Type u_2\nA : Type ?u.14491\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace A\nf : X → Y\n⊢ IsLocallyInjective f ↔ ∀ (x : X), ∃ U ∈ 𝓝 x, Set.InjOn f U","state_after":"case mp\nX : Type u_1\nY : Type u_2\nA : Type ?u.14491\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace A\nf : X → Y\nh : IsLocallyInjective f\nx : X\n⊢ ∃ U ∈ 𝓝 x, Set.InjOn f U\n\ncase mpr\nX : Type u_1\nY : Type u_2\nA : Type ?u.14491\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace A\nf : X → Y\nh : ∀ (x : X), ∃ U ∈ 𝓝 x, Set.InjOn f U\nx : X\n⊢ ∃ U, IsOpen U ∧ x ∈ U ∧ Set.InjOn f U","tactic":"constructor <;> intro h x","premises":[]}]}
+{"url":"Mathlib/Topology/Inseparable.lean","commit":"","full_name":"ker_nhds_eq_specializes","start":[88,0],"end":[89,42],"file_path":"Mathlib/Topology/Inseparable.lean","tactics":[{"state_before":"X : Type u_1\nY : Type u_2\nZ : Type u_3\nα : Type u_4\nι : Type u_5\nπ : ι → Type u_6\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : (i : ι) → TopologicalSpace (π i)\nx y z : X\ns : Set X\nf g : X → Y\n⊢ (𝓝 x).ker = {y | y ⤳ x}","state_after":"case h\nX : Type u_1\nY : Type u_2\nZ : Type u_3\nα : Type u_4\nι : Type u_5\nπ : ι → Type u_6\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : (i : ι) → TopologicalSpace (π i)\nx y z : X\ns : Set X\nf g : X → Y\nx✝ : X\n⊢ x✝ ∈ (𝓝 x).ker ↔ x✝ ∈ {y | y ⤳ x}","tactic":"ext","premises":[]},{"state_before":"case h\nX : Type u_1\nY : Type u_2\nZ : Type u_3\nα : Type u_4\nι : Type u_5\nπ : ι → Type u_6\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : (i : ι) → TopologicalSpace (π i)\nx y z : X\ns : Set X\nf g : X → Y\nx✝ : X\n⊢ x✝ ∈ (𝓝 x).ker ↔ x✝ ∈ {y | y ⤳ x}","state_after":"no goals","tactic":"simp [specializes_iff_pure, le_def]","premises":[{"full_name":"Filter.le_def","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[311,8],"def_end_pos":[311,14]},{"full_name":"specializes_iff_pure","def_path":"Mathlib/Topology/Inseparable.lean","def_pos":[81,8],"def_end_pos":[81,28]}]}]}
+{"url":"Mathlib/RingTheory/DedekindDomain/Ideal.lean","commit":"","full_name":"FractionalIdeal.mul_inv_cancel_of_le_one","start":[431,0],"end":[444,66],"file_path":"Mathlib/RingTheory/DedekindDomain/Ideal.lean","tactics":[{"state_before":"R : Type u_1\nA : Type u_2\nK : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\n⊢ ↑I * (↑I)⁻¹ = 1","state_after":"case intro\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nJ : Ideal A\nhJ : ↑J = ↑I * (↑I)⁻¹\n⊢ ↑I * (↑I)⁻¹ = 1","tactic":"obtain ⟨J, hJ⟩ : ∃ J : Ideal A, (J : FractionalIdeal A⁰ K) = I * (I : FractionalIdeal A⁰ K)⁻¹ :=\n    le_one_iff_exists_coeIdeal.mp mul_one_div_le_one","premises":[{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"FractionalIdeal","def_path":"Mathlib/RingTheory/FractionalIdeal/Basic.lean","def_pos":[77,4],"def_end_pos":[77,19]},{"full_name":"FractionalIdeal.le_one_iff_exists_coeIdeal","def_path":"Mathlib/RingTheory/FractionalIdeal/Basic.lean","def_pos":[618,8],"def_end_pos":[618,34]},{"full_name":"FractionalIdeal.mul_one_div_le_one","def_path":"Mathlib/RingTheory/FractionalIdeal/Operations.lean","def_pos":[402,8],"def_end_pos":[402,26]},{"full_name":"Ideal","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[39,7],"def_end_pos":[39,12]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"nonZeroDivisors","def_path":"Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean","def_pos":[84,4],"def_end_pos":[84,19]}]},{"state_before":"case 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hs).tsum_eq.symm","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"HasSum.tsum_eq","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Defs.lean","def_pos":[165,2],"def_end_pos":[165,13]},{"full_name":"HurwitzZeta.hasSum_nat_completedCosZeta","def_path":"Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean","def_pos":[530,6],"def_end_pos":[530,33]}]},{"state_before":"s : ℂ\nhs : 1 < s.re\nthis : completedCosZeta (↑0) s = ∑' (b : ℕ), if b = 0 then 0 else s.Gammaℝ * ↑(Real.cos (2 * π * 0 * ↑b)) / ↑b ^ s\n⊢ completedRiemannZeta s = ↑π ^ (-s / 2) * Complex.Gamma (s / 2) * ∑' (n : ℕ), 1 / ↑n ^ s","state_after":"s : ℂ\nhs : 1 < s.re\nthis : completedRiemannZeta s = ∑' (b : ℕ), if b = 0 then 0 else s.Gammaℝ * ↑(Real.cos (2 * π * 0 * ↑b)) / ↑b ^ s\n⊢ completedRiemannZeta s = ↑π ^ (-s / 2) * Complex.Gamma (s / 2) * ∑' (n : ℕ), 1 / ↑n ^ s","tactic":"simp only [QuotientAddGroup.mk_zero, completedCosZeta_zero] at this","premises":[{"full_name":"HurwitzZeta.completedCosZeta_zero","def_path":"Mathlib/NumberTheory/LSeries/RiemannZeta.lean","def_pos":[72,6],"def_end_pos":[72,39]},{"full_name":"QuotientAddGroup.mk_zero","def_path":"Mathlib/GroupTheory/QuotientGroup.lean","def_pos":[140,2],"def_end_pos":[140,13]}]},{"state_before":"s : ℂ\nhs : 1 < s.re\nthis : completedRiemannZeta s = ∑' (b : ℕ), if b = 0 then 0 else s.Gammaℝ * ↑(Real.cos (2 * π * 0 * ↑b)) / ↑b ^ s\n⊢ completedRiemannZeta s = ↑π ^ (-s / 2) * Complex.Gamma (s / 2) * ∑' (n : ℕ), 1 / ↑n ^ s","state_after":"s : ℂ\nhs : 1 < s.re\nthis : completedRiemannZeta s = ∑' (b : ℕ), if b = 0 then 0 else s.Gammaℝ * ↑(Real.cos (2 * π * 0 * ↑b)) / ↑b ^ s\n⊢ (∑' (b : ℕ), if b = 0 then 0 else ↑π ^ (-s / 2) * Complex.Gamma (s / 2) / ↑b ^ s) =\n    ∑' (x : ℕ), ↑π ^ (-s / 2) * Complex.Gamma (s / 2) / ↑x ^ s","tactic":"simp only [this, Gammaℝ_def, mul_zero, zero_mul, Real.cos_zero, ofReal_one, mul_one, mul_one_div,\n    ← tsum_mul_left]","premises":[{"full_name":"Complex.Gammaℝ_def","def_path":"Mathlib/Analysis/SpecialFunctions/Gamma/Deligne.lean","def_pos":[43,6],"def_end_pos":[43,16]},{"full_name":"Complex.ofReal_one","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[154,8],"def_end_pos":[154,18]},{"full_name":"MulZeroClass.mul_zero","def_path":"Mathlib/Algebra/GroupWithZero/Defs.lean","def_pos":[37,2],"def_end_pos":[37,10]},{"full_name":"MulZeroClass.zero_mul","def_path":"Mathlib/Algebra/GroupWithZero/Defs.lean","def_pos":[35,2],"def_end_pos":[35,10]},{"full_name":"Real.cos_zero","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[732,8],"def_end_pos":[732,16]},{"full_name":"mul_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[413,8],"def_end_pos":[413,15]},{"full_name":"mul_one_div","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[326,8],"def_end_pos":[326,19]},{"full_name":"tsum_mul_left","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Ring.lean","def_pos":[95,8],"def_end_pos":[95,21]}]},{"state_before":"s : ℂ\nhs : 1 < s.re\nthis : completedRiemannZeta s = ∑' (b : ℕ), if b = 0 then 0 else s.Gammaℝ * ↑(Real.cos (2 * π * 0 * ↑b)) / ↑b ^ s\n⊢ (∑' (b : ℕ), if b = 0 then 0 else ↑π ^ (-s / 2) * Complex.Gamma (s / 2) / ↑b ^ s) =\n    ∑' (x : ℕ), ↑π ^ (-s / 2) * Complex.Gamma (s / 2) / ↑x ^ s","state_after":"case e_f.h\ns : ℂ\nhs : 1 < s.re\nthis : completedRiemannZeta s = ∑' (b : ℕ), if b = 0 then 0 else s.Gammaℝ * ↑(Real.cos (2 * π * 0 * ↑b)) / ↑b ^ s\nn : ℕ\n⊢ (if n = 0 then 0 else ↑π ^ (-s / 2) * Complex.Gamma (s / 2) / ↑n ^ s) = ↑π ^ (-s / 2) * Complex.Gamma (s / 2) / ↑n ^ s","tactic":"congr 1 with n","premises":[]},{"state_before":"case e_f.h\ns : ℂ\nhs : 1 < s.re\nthis : completedRiemannZeta s = ∑' (b : ℕ), if b = 0 then 0 else s.Gammaℝ * ↑(Real.cos (2 * π * 0 * ↑b)) / ↑b ^ s\nn : ℕ\n⊢ (if n = 0 then 0 else ↑π ^ (-s / 2) * Complex.Gamma (s / 2) / ↑n ^ s) = ↑π ^ (-s / 2) * Complex.Gamma (s / 2) / ↑n ^ s","state_after":"case pos\ns : ℂ\nhs : 1 < s.re\nthis : completedRiemannZeta s = ∑' (b : ℕ), if b = 0 then 0 else s.Gammaℝ * ↑(Real.cos (2 * π * 0 * ↑b)) / ↑b ^ s\nn : ℕ\nh : n = 0\n⊢ 0 = ↑π ^ (-s / 2) * Complex.Gamma (s / 2) / ↑n ^ s\n\ncase neg\ns : ℂ\nhs : 1 < s.re\nthis : completedRiemannZeta s = ∑' (b : ℕ), if b = 0 then 0 else s.Gammaℝ * ↑(Real.cos (2 * π * 0 * ↑b)) / ↑b ^ s\nn : ℕ\nh : ¬n = 0\n⊢ ↑π ^ (-s / 2) * Complex.Gamma (s / 2) / ↑n ^ s = ↑π ^ (-s / 2) * Complex.Gamma (s / 2) / ↑n ^ s","tactic":"split_ifs with h","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
+{"url":"Mathlib/Analysis/SpecialFunctions/Arsinh.lean","commit":"","full_name":"Real.hasStrictDerivAt_arsinh","start":[155,0],"end":[158,28],"file_path":"Mathlib/Analysis/SpecialFunctions/Arsinh.lean","tactics":[{"state_before":"x✝ y x : ℝ\n⊢ HasStrictDerivAt arsinh (√(1 + x ^ 2))⁻¹ x","state_after":"case h.e'_7.h.e'_3\nx✝ y x : ℝ\n⊢ √(1 + x ^ 2) = cosh (↑sinhHomeomorph.toPartialHomeomorph.symm x)","tactic":"convert sinhHomeomorph.toPartialHomeomorph.hasStrictDerivAt_symm (mem_univ x) (cosh_pos _).ne'\n    (hasStrictDerivAt_sinh _) using 2","premises":[{"full_name":"Homeomorph.toPartialHomeomorph","def_path":"Mathlib/Topology/PartialHomeomorph.lean","def_pos":[192,4],"def_end_pos":[192,41]},{"full_name":"LT.lt.ne'","def_path":"Mathlib/Order/Basic.lean","def_pos":[267,8],"def_end_pos":[267,11]},{"full_name":"PartialHomeomorph.hasStrictDerivAt_symm","def_path":"Mathlib/Analysis/Calculus/Deriv/Inverse.lean","def_pos":[68,8],"def_end_pos":[68,47]},{"full_name":"Real.cosh_pos","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[1046,8],"def_end_pos":[1046,16]},{"full_name":"Real.hasStrictDerivAt_sinh","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean","def_pos":[517,8],"def_end_pos":[517,29]},{"full_name":"Real.sinhHomeomorph","def_path":"Mathlib/Analysis/SpecialFunctions/Arsinh.lean","def_pos":[108,4],"def_end_pos":[108,18]},{"full_name":"Set.mem_univ","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[80,28],"def_end_pos":[80,36]}]},{"state_before":"case h.e'_7.h.e'_3\nx✝ y x : ℝ\n⊢ √(1 + x ^ 2) = cosh (↑sinhHomeomorph.toPartialHomeomorph.symm x)","state_after":"no goals","tactic":"exact (cosh_arsinh _).symm","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Real.cosh_arsinh","def_path":"Mathlib/Analysis/SpecialFunctions/Arsinh.lean","def_pos":[76,8],"def_end_pos":[76,19]}]}]}
+{"url":"Mathlib/Computability/AkraBazzi/AkraBazzi.lean","commit":"","full_name":"AkraBazziRecurrence.exists_eventually_const_mul_le_r","start":[199,0],"end":[202,96],"file_path":"Mathlib/Computability/AkraBazzi/AkraBazzi.lean","tactics":[{"state_before":"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\n⊢ ∃ c ∈ Set.Ioo 0 1, ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c * ↑n ≤ ↑(r i n)","state_after":"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\ngt_zero : 0 < b (min_bi b)\n⊢ ∃ c ∈ Set.Ioo 0 1, ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c * ↑n ≤ ↑(r i n)","tactic":"have gt_zero : 0 < b (min_bi b) := R.b_pos (min_bi b)","premises":[{"full_name":"AkraBazziRecurrence.b_pos","def_path":"Mathlib/Computability/AkraBazzi/AkraBazzi.lean","def_pos":[90,2],"def_end_pos":[90,7]},{"full_name":"AkraBazziRecurrence.min_bi","def_path":"Mathlib/Computability/AkraBazzi/AkraBazzi.lean","def_pos":[113,18],"def_end_pos":[113,24]}]},{"state_before":"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\ngt_zero : 0 < b (min_bi b)\n⊢ ∃ c ∈ Set.Ioo 0 1, ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c * ↑n ≤ ↑(r i n)","state_after":"no goals","tactic":"exact ⟨b (min_bi b) / 2, ⟨⟨by positivity, R.bi_min_div_two_lt_one⟩, R.eventually_bi_mul_le_r⟩⟩","premises":[{"full_name":"AkraBazziRecurrence.bi_min_div_two_lt_one","def_path":"Mathlib/Computability/AkraBazzi/AkraBazzi.lean","def_pos":[192,6],"def_end_pos":[192,27]},{"full_name":"AkraBazziRecurrence.eventually_bi_mul_le_r","def_path":"Mathlib/Computability/AkraBazzi/AkraBazzi.lean","def_pos":[176,6],"def_end_pos":[176,28]},{"full_name":"AkraBazziRecurrence.min_bi","def_path":"Mathlib/Computability/AkraBazzi/AkraBazzi.lean","def_pos":[113,18],"def_end_pos":[113,24]},{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]}]}
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+{"url":"Mathlib/Order/CompactlyGenerated/Basic.lean","commit":"","full_name":"CompleteLattice.Iic_coatomic_of_compact_element","start":[465,0],"end":[485,47],"file_path":"Mathlib/Order/CompactlyGenerated/Basic.lean","tactics":[{"state_before":"ι : Sort u_1\nα : Type u_2\ninst✝ : CompleteLattice α\nf : ι → α\nk : α\nh : IsCompactElement k\n⊢ IsCoatomic ↑(Iic k)","state_after":"case eq_top_or_exists_le_coatom\nι : Sort u_1\nα : Type u_2\ninst✝ : CompleteLattice α\nf : ι → α\nk : α\nh : IsCompactElement k\n⊢ ∀ (b : ↑(Iic k)), b = ⊤ ∨ ∃ a, IsCoatom a ∧ b ≤ a","tactic":"constructor","premises":[]},{"state_before":"case eq_top_or_exists_le_coatom\nι : Sort u_1\nα : Type u_2\ninst✝ : CompleteLattice α\nf : ι → α\nk : α\nh : IsCompactElement k\n⊢ ∀ (b : ↑(Iic k)), b = ⊤ ∨ ∃ a, IsCoatom a ∧ b ≤ a","state_after":"case eq_top_or_exists_le_coatom.mk\nι : Sort u_1\nα : Type u_2\ninst✝ : CompleteLattice α\nf : ι → α\nk : α\nh : IsCompactElement k\nb : α\nhbk : b ∈ Iic k\n⊢ ⟨b, hbk⟩ = ⊤ ∨ ∃ a, IsCoatom a ∧ ⟨b, hbk⟩ ≤ a","tactic":"rintro ⟨b, hbk⟩","premises":[]},{"state_before":"case eq_top_or_exists_le_coatom.mk\nι : Sort u_1\nα : Type u_2\ninst✝ : CompleteLattice α\nf : ι → α\nk : α\nh : IsCompactElement k\nb : α\nhbk : b ∈ Iic k\n⊢ ⟨b, hbk⟩ = ⊤ ∨ ∃ a, IsCoatom a ∧ ⟨b, hbk⟩ ≤ a","state_after":"case eq_top_or_exists_le_coatom.mk.inl\nι : Sort u_1\nα : Type u_2\ninst✝ : CompleteLattice α\nf : ι → α\nb : α\nh : IsCompactElement b\nhbk : b ∈ Iic b\n⊢ ⟨b, hbk⟩ = ⊤ ∨ ∃ a, IsCoatom a ∧ ⟨b, hbk⟩ ≤ a\n\ncase eq_top_or_exists_le_coatom.mk.inr\nι : Sort u_1\nα : Type u_2\ninst✝ : CompleteLattice α\nf : ι → α\nk : α\nh : IsCompactElement k\nb : α\nhbk : b ∈ Iic k\nH : b ≠ k\n⊢ ⟨b, hbk⟩ = ⊤ ∨ ∃ a, IsCoatom a ∧ ⟨b, hbk⟩ ≤ a","tactic":"obtain rfl | H := eq_or_ne b k","premises":[{"full_name":"eq_or_ne","def_path":"Mathlib/Logic/Basic.lean","def_pos":[167,8],"def_end_pos":[167,16]}]},{"state_before":"case eq_top_or_exists_le_coatom.mk.inr\nι : Sort u_1\nα : Type u_2\ninst✝ : CompleteLattice α\nf : ι → α\nk : α\nh : IsCompactElement k\nb : α\nhbk : b ∈ Iic k\nH : b ≠ k\n⊢ ⟨b, hbk⟩ = ⊤ ∨ ∃ a, IsCoatom a ∧ ⟨b, hbk⟩ ≤ a","state_after":"case eq_top_or_exists_le_coatom.mk.inr.h\nι : Sort u_1\nα : Type u_2\ninst✝ : CompleteLattice α\nf : ι → α\nk : α\nh : IsCompactElement k\nb : α\nhbk : b ∈ Iic k\nH : b ≠ k\n⊢ ∃ a, IsCoatom a ∧ ⟨b, hbk⟩ ≤ a","tactic":"right","premises":[]},{"state_before":"case eq_top_or_exists_le_coatom.mk.inr.h\nι : Sort u_1\nα : Type u_2\ninst✝ : CompleteLattice α\nf : ι → α\nk : α\nh : IsCompactElement k\nb : α\nhbk : b ∈ Iic k\nH : b ≠ k\n⊢ ∃ a, IsCoatom a ∧ ⟨b, hbk⟩ ≤ a","state_after":"case eq_top_or_exists_le_coatom.mk.inr.h.refine_2\nι : Sort u_1\nα : Type u_2\ninst✝ : CompleteLattice α\nf : ι → α\nk : α\nh✝ : IsCompactElement k\nb : α\nhbk : b ∈ Iic k\nH : b ≠ k\na : α\na₀ : a ∈ Iio k\nba : b ≤ a\nh : ∀ z ∈ Iio k, a ≤ z → z = a\n⊢ ∃ a, IsCoatom a ∧ ⟨b, hbk⟩ ≤ a\n\ncase eq_top_or_exists_le_coatom.mk.inr.h.refine_1\nι : Sort u_1\nα : Type u_2\ninst✝ : CompleteLattice α\nf : ι → α\nk : α\nh : IsCompactElement k\nb : α\nhbk : b ∈ Iic k\nH : b ≠ k\n⊢ ∀ c ⊆ Iio k, IsChain (fun x x_1 => x ≤ x_1) c → ∀ y ∈ c, ∃ ub ∈ Iio k, ∀ z ∈ c, z ≤ ub","tactic":"have ⟨a, a₀, ba, h⟩ := zorn_nonempty_partialOrder₀ (Set.Iio k) ?_ b (lt_of_le_of_ne hbk H)","premises":[{"full_name":"Set.Iio","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[50,4],"def_end_pos":[50,7]},{"full_name":"lt_of_le_of_ne","def_path":"Mathlib/Order/Defs.lean","def_pos":[164,8],"def_end_pos":[164,22]},{"full_name":"zorn_nonempty_partialOrder₀","def_path":"Mathlib/Order/Zorn.lean","def_pos":[165,8],"def_end_pos":[165,35]}]}]}
+{"url":"Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean","commit":"","full_name":"EllipticCurve.map_a₄","start":[601,0],"end":[604,100],"file_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean","tactics":[{"state_before":"R : Type u\ninst✝¹ : CommRing R\nE : EllipticCurve R\nA : Type v\ninst✝ : CommRing A\nφ : R →+* A\n⊢ ↑((Units.map ↑φ) E.Δ') = (E.map φ).Δ","state_after":"R : Type u\ninst✝¹ : CommRing R\nE : EllipticCurve R\nA : Type v\ninst✝ : CommRing A\nφ : R →+* A\n⊢ ↑φ E.Δ = φ E.Δ","tactic":"simp only [Units.coe_map, coe_Δ', E.map_Δ]","premises":[{"full_name":"EllipticCurve.coe_Δ'","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean","def_pos":[534,2],"def_end_pos":[534,8]},{"full_name":"Units.coe_map","def_path":"Mathlib/Algebra/Group/Units/Hom.lean","def_pos":[65,8],"def_end_pos":[65,15]},{"full_name":"WeierstrassCurve.map_Δ","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean","def_pos":[369,6],"def_end_pos":[369,11]}]},{"state_before":"R : Type u\ninst✝¹ : CommRing R\nE : EllipticCurve R\nA : Type v\ninst✝ : CommRing A\nφ : R →+* A\n⊢ ↑φ E.Δ = φ E.Δ","state_after":"no goals","tactic":"rfl","premises":[]}]}
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y","state_after":"case h\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type u_4\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type u_5\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\ny : F\nx✝ : E\nx : ↥f.domain\nh : ↑x = x✝ ∧ ↑f x = y\n⊢ ↑f x = y","tactic":"simp only at h","premises":[]},{"state_before":"case h\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type u_4\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type u_5\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\ny : F\nx✝ : E\nx : ↥f.domain\nh : ↑x = x✝ ∧ ↑f x = y\n⊢ ↑f x = y","state_after":"no goals","tactic":"rw [h.2]","premises":[{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]}]}]}
+{"url":"Mathlib/Algebra/Polynomial/Laurent.lean","commit":"","full_name":"LaurentPolynomial.reduce_to_polynomial_of_mul_T","start":[361,0],"end":[371,18],"file_path":"Mathlib/Algebra/Polynomial/Laurent.lean","tactics":[{"state_before":"R : Type u_1\ninst✝ : Semiring R\nf : R[T;T⁻¹]\nQ : R[T;T⁻¹] → Prop\nQf : ∀ (f : R[X]), Q (toLaurent f)\nQT : ∀ (f : R[T;T⁻¹]), Q (f * T 1) → Q f\n⊢ Q f","state_after":"case Qf\nR : Type u_1\ninst✝ : Semiring R\nQ : R[T;T⁻¹] → Prop\nQf : ∀ (f : R[X]), Q (toLaurent f)\nQT : ∀ (f : R[T;T⁻¹]), Q (f * T 1) → Q f\nf : R[X]\nn : ℕ\n⊢ Q (toLaurent f * T (-↑n))","tactic":"induction' f using LaurentPolynomial.induction_on_mul_T with f n","premises":[{"full_name":"LaurentPolynomial.induction_on_mul_T","def_path":"Mathlib/Algebra/Polynomial/Laurent.lean","def_pos":[354,8],"def_end_pos":[354,26]}]},{"state_before":"case Qf\nR : Type u_1\ninst✝ : Semiring R\nQ : R[T;T⁻¹] → Prop\nQf : ∀ (f : R[X]), Q (toLaurent f)\nQT : ∀ (f : R[T;T⁻¹]), Q (f * T 1) → Q f\nf : R[X]\nn : ℕ\n⊢ Q (toLaurent f * T (-↑n))","state_after":"case Qf.zero\nR : Type u_1\ninst✝ : Semiring R\nQ : R[T;T⁻¹] → Prop\nQf : ∀ (f : R[X]), Q (toLaurent f)\nQT : ∀ (f : R[T;T⁻¹]), Q (f * T 1) → Q f\nf : R[X]\n⊢ Q (toLaurent f * T (-↑0))\n\ncase Qf.succ\nR : Type u_1\ninst✝ : Semiring R\nQ : R[T;T⁻¹] → Prop\nQf : ∀ (f : R[X]), Q (toLaurent f)\nQT : ∀ (f : R[T;T⁻¹]), Q (f * T 1) → Q f\nf : R[X]\nn : ℕ\nhn : Q (toLaurent f * T (-↑n))\n⊢ Q (toLaurent f * T (-↑(n + 1)))","tactic":"induction' n with n hn","premises":[]}]}
+{"url":"Mathlib/Data/List/Basic.lean","commit":"","full_name":"List.foldl_eq_foldr'","start":[1575,0],"end":[1577,77],"file_path":"Mathlib/Data/List/Basic.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\nf : α → β → α\nhf : ∀ (a : α) (b c : β), f (f a b) c = f (f a c) b\na : α\nb : β\nl : List β\n⊢ foldl f a (b :: l) = foldr (flip f) a (b :: l)","state_after":"ι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\nf : α → β → α\nhf : ∀ (a : α) (b c : β), f (f a b) c = f (f a c) b\na : α\nb : β\nl : List β\n⊢ f (foldr (flip f) a l) b = flip f b (foldr (flip f) a l)","tactic":"rw [foldl_eq_of_comm' hf, foldr, foldl_eq_foldr' ..]","premises":[{"full_name":"List.foldl_eq_of_comm'","def_path":"Mathlib/Data/List/Basic.lean","def_pos":[1571,8],"def_end_pos":[1571,25]},{"full_name":"List.foldr","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[416,18],"def_end_pos":[416,23]}]},{"state_before":"ι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\nf : α → β → α\nhf : ∀ (a : α) (b c : β), f (f a b) c = f (f a c) b\na : α\nb : β\nl : List β\n⊢ f (foldr (flip f) a l) b = flip f b (foldr (flip f) a l)","state_after":"no goals","tactic":"rfl","premises":[]}]}
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f\n⊢ IsPretransitive E G","tactic":"let _ : MulAction E G := MulAction.compHom _ f","premises":[{"full_name":"MulAction","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[95,6],"def_end_pos":[95,15]},{"full_name":"MulAction.compHom","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[527,4],"def_end_pos":[527,11]}]},{"state_before":"M : Type u_1\nN : Type u_2\nG✝ : Type u_3\nH : Type u_4\nA : Type u_5\nB : Type u_6\nα : Type u_7\nβ : Type u_8\nγ : Type u_9\nδ : Type u_10\ninst✝⁵ : Monoid M\ninst✝⁴ : MulAction M α\nE : Type u_11\nF : Type u_12\nG : Type u_13\ninst✝³ : Monoid E\ninst✝² : Monoid F\ninst✝¹ : MulAction F G\ninst✝ : IsPretransitive F G\nf : E →* F\nhf : Surjective ⇑f\nx✝ : MulAction E G := compHom G f\n⊢ IsPretransitive E G","state_after":"M : Type u_1\nN : Type u_2\nG✝ : Type u_3\nH : Type u_4\nA : Type u_5\nB : Type u_6\nα : Type u_7\nβ : Type u_8\nγ : Type u_9\nδ : Type u_10\ninst✝⁵ : Monoid M\ninst✝⁴ : MulAction M α\nE : Type u_11\nF : Type u_12\nG : Type u_13\ninst✝³ : Monoid E\ninst✝² : Monoid F\ninst✝¹ : MulAction F G\ninst✝ : IsPretransitive F G\nf : E →* F\nhf : Surjective ⇑f\nx✝ : MulAction E G := compHom G f\nx y : G\n⊢ ∃ g, g • x = y","tactic":"refine ⟨fun x y ↦ ?_⟩","premises":[]},{"state_before":"M : Type u_1\nN : Type u_2\nG✝ : Type u_3\nH : Type u_4\nA : Type u_5\nB : Type u_6\nα : Type u_7\nβ : Type u_8\nγ : Type u_9\nδ : Type u_10\ninst✝⁵ : Monoid M\ninst✝⁴ : MulAction M α\nE : Type u_11\nF : Type u_12\nG : Type u_13\ninst✝³ : Monoid E\ninst✝² : Monoid F\ninst✝¹ : MulAction F G\ninst✝ : IsPretransitive F G\nf : E →* F\nhf : Surjective ⇑f\nx✝ : MulAction E G := compHom G f\nx y : G\n⊢ ∃ g, g • x = y","state_after":"case intro\nM : Type u_1\nN : Type u_2\nG✝ : Type u_3\nH : Type u_4\nA : Type u_5\nB : Type u_6\nα : Type u_7\nβ : Type u_8\nγ : Type u_9\nδ : Type u_10\ninst✝⁵ : Monoid M\ninst✝⁴ : MulAction M α\nE : Type u_11\nF : Type u_12\nG : Type u_13\ninst✝³ : Monoid E\ninst✝² : Monoid F\ninst✝¹ : MulAction F G\ninst✝ : IsPretransitive F G\nf : E →* F\nhf : Surjective ⇑f\nx✝ : MulAction E G := compHom G f\nx : G\nm : F\n⊢ ∃ g, g • x = m • x","tactic":"obtain ⟨m, rfl⟩ : ∃ m : F, m • x = y := exists_smul_eq F x y","premises":[{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"MulAction.exists_smul_eq","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[131,6],"def_end_pos":[131,20]}]},{"state_before":"case intro\nM : Type u_1\nN : Type u_2\nG✝ : Type u_3\nH : Type u_4\nA : Type u_5\nB : Type u_6\nα : Type u_7\nβ : Type u_8\nγ : Type u_9\nδ : Type u_10\ninst✝⁵ : Monoid M\ninst✝⁴ : MulAction M α\nE : Type u_11\nF : Type u_12\nG : Type u_13\ninst✝³ : Monoid E\ninst✝² : Monoid F\ninst✝¹ : MulAction F G\ninst✝ : IsPretransitive F G\nf : E →* F\nhf : Surjective ⇑f\nx✝ : MulAction E G := compHom G f\nx : G\nm : F\n⊢ ∃ g, g • x = m • x","state_after":"case intro.intro\nM : Type u_1\nN : Type u_2\nG✝ : Type u_3\nH : Type u_4\nA : Type u_5\nB : Type u_6\nα : Type u_7\nβ : Type u_8\nγ : Type u_9\nδ : Type u_10\ninst✝⁵ : Monoid M\ninst✝⁴ : MulAction M α\nE : Type u_11\nF : Type u_12\nG : Type u_13\ninst✝³ : Monoid E\ninst✝² : Monoid F\ninst✝¹ : MulAction F G\ninst✝ : IsPretransitive F G\nf : E →* F\nhf : Surjective ⇑f\nx✝ : MulAction E G := compHom G f\nx : G\ne : E\n⊢ ∃ g, g • x = f e • x","tactic":"obtain ⟨e, rfl⟩ : ∃ e, f e = m := hf m","premises":[{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]}]},{"state_before":"case intro.intro\nM : Type u_1\nN : Type u_2\nG✝ : Type u_3\nH : Type u_4\nA : Type u_5\nB : Type u_6\nα : Type u_7\nβ : Type u_8\nγ : Type u_9\nδ : Type u_10\ninst✝⁵ : Monoid M\ninst✝⁴ : MulAction M α\nE : Type u_11\nF : Type u_12\nG : Type u_13\ninst✝³ : Monoid E\ninst✝² : Monoid F\ninst✝¹ : MulAction F G\ninst✝ : IsPretransitive F G\nf : E →* F\nhf : Surjective ⇑f\nx✝ : MulAction E G := compHom G 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V\nX Y Z : V\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\n⊢ (imageSubobjectIso g.op).hom ≫ (imageUnopOp g.op).inv ≫ (factorThruImage g.op.unop).op =\n    (imageSubobjectIso g.op).hom ≫ (imageOpOp g).inv ≫ (factorThruImage g).op","tactic":"simp only [Iso.trans_hom, Iso.symm_hom, Iso.trans_inv, kernelOpOp_inv, Category.assoc,\n    imageToKernel_arrow, kernelSubobject_arrow', kernel.lift_ι, ← op_comp, cokernel.π_desc,\n    ← imageSubobject_arrow, ← imageUnopOp_inv_comp_op_factorThruImage 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q.natTrailingDegree), p.coeff x.1 * q.coeff x.2 =\n    p.trailingCoeff * q.trailingCoeff","state_after":"R : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np q r : R[X]\n⊢ ∀ b ∈ antidiagonal (p.natTrailingDegree + q.natTrailingDegree),\n    b ≠ (p.natTrailingDegree, q.natTrailingDegree) → p.coeff b.1 * q.coeff b.2 = 0","tactic":"refine\n    Finset.sum_eq_single (p.natTrailingDegree, q.natTrailingDegree) ?_ fun h =>\n      (h (mem_antidiagonal.mpr 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Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np q r : R[X]\n⊢ ∀ b ∈ antidiagonal (p.natTrailingDegree + q.natTrailingDegree),\n    b ≠ (p.natTrailingDegree, q.natTrailingDegree) → p.coeff b.1 * q.coeff b.2 = 0","state_after":"case mk\nR : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np q r : R[X]\ni j : ℕ\nh₁ : (i, j) ∈ antidiagonal (p.natTrailingDegree + q.natTrailingDegree)\nh₂ : (i, j) ≠ (p.natTrailingDegree, q.natTrailingDegree)\n⊢ p.coeff (i, j).1 * q.coeff (i, j).2 = 0","tactic":"rintro ⟨i, j⟩ h₁ h₂","premises":[]},{"state_before":"case mk\nR : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np q r : R[X]\ni j : ℕ\nh₁ : (i, j) ∈ antidiagonal (p.natTrailingDegree + q.natTrailingDegree)\nh₂ : (i, j) ≠ (p.natTrailingDegree, q.natTrailingDegree)\n⊢ p.coeff (i, j).1 * q.coeff (i, j).2 = 0","state_after":"case mk\nR : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np q r : R[X]\ni j : ℕ\nh₁ : (i, j).1 + (i, j).2 = p.natTrailingDegree + q.natTrailingDegree\nh₂ : (i, j) ≠ (p.natTrailingDegree, q.natTrailingDegree)\n⊢ p.coeff (i, j).1 * q.coeff (i, j).2 = 0","tactic":"rw [mem_antidiagonal] at h₁","premises":[{"full_name":"Finset.HasAntidiagonal.mem_antidiagonal","def_path":"Mathlib/Algebra/Order/Antidiag/Prod.lean","def_pos":[58,2],"def_end_pos":[58,18]}]},{"state_before":"case mk\nR : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np q r : R[X]\ni j : ℕ\nh₁ : (i, j).1 + (i, j).2 = p.natTrailingDegree + q.natTrailingDegree\nh₂ : (i, j) ≠ (p.natTrailingDegree, q.natTrailingDegree)\n⊢ p.coeff (i, j).1 * q.coeff (i, j).2 = 0","state_after":"case pos\nR : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np q r : R[X]\ni j : ℕ\nh₁ : (i, j).1 + (i, j).2 = p.natTrailingDegree + q.natTrailingDegree\nh₂ : (i, j) ≠ (p.natTrailingDegree, q.natTrailingDegree)\nhi : i < p.natTrailingDegree\n⊢ p.coeff (i, j).1 * q.coeff (i, j).2 = 0\n\ncase neg\nR : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np q r : R[X]\ni j : ℕ\nh₁ : (i, j).1 + (i, j).2 = p.natTrailingDegree + q.natTrailingDegree\nh₂ : (i, j) ≠ (p.natTrailingDegree, q.natTrailingDegree)\nhi : ¬i < p.natTrailingDegree\n⊢ p.coeff (i, j).1 * q.coeff (i, j).2 = 0","tactic":"by_cases hi : i < p.natTrailingDegree","premises":[{"full_name":"Polynomial.natTrailingDegree","def_path":"Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean","def_pos":[52,4],"def_end_pos":[52,21]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case neg\nR : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np q r : R[X]\ni j : ℕ\nh₁ : (i, j).1 + (i, j).2 = p.natTrailingDegree + q.natTrailingDegree\nh₂ : (i, j) ≠ (p.natTrailingDegree, q.natTrailingDegree)\nhi : ¬i < p.natTrailingDegree\n⊢ p.coeff (i, j).1 * q.coeff (i, j).2 = 0","state_after":"case pos\nR : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np q r : R[X]\ni j : ℕ\nh₁ : (i, j).1 + (i, j).2 = p.natTrailingDegree + q.natTrailingDegree\nh₂ : (i, j) ≠ (p.natTrailingDegree, q.natTrailingDegree)\nhi : ¬i < p.natTrailingDegree\nhj : j < q.natTrailingDegree\n⊢ p.coeff (i, j).1 * q.coeff (i, j).2 = 0\n\ncase neg\nR : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np q r : R[X]\ni j : ℕ\nh₁ : (i, j).1 + (i, j).2 = p.natTrailingDegree + q.natTrailingDegree\nh₂ : (i, j) ≠ (p.natTrailingDegree, q.natTrailingDegree)\nhi : ¬i < p.natTrailingDegree\nhj : ¬j < q.natTrailingDegree\n⊢ p.coeff (i, j).1 * q.coeff (i, j).2 = 0","tactic":"by_cases hj : j < q.natTrailingDegree","premises":[{"full_name":"Polynomial.natTrailingDegree","def_path":"Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean","def_pos":[52,4],"def_end_pos":[52,21]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case neg\nR : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np q r : R[X]\ni j : ℕ\nh₁ : (i, j).1 + (i, j).2 = p.natTrailingDegree + q.natTrailingDegree\nh₂ : (i, j) ≠ (p.natTrailingDegree, q.natTrailingDegree)\nhi : ¬i < p.natTrailingDegree\nhj : ¬j < q.natTrailingDegree\n⊢ p.coeff (i, j).1 * q.coeff (i, j).2 = 0","state_after":"case neg\nR : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np q r : R[X]\ni j : ℕ\nh₁ : (i, j).1 + (i, j).2 = p.natTrailingDegree + q.natTrailingDegree\nh₂ : (i, j) ≠ (p.natTrailingDegree, q.natTrailingDegree)\nhi : p.natTrailingDegree ≤ i\nhj : q.natTrailingDegree ≤ j\n⊢ p.coeff (i, j).1 * q.coeff (i, j).2 = 0","tactic":"rw [not_lt] at hi hj","premises":[{"full_name":"not_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[312,8],"def_end_pos":[312,14]}]},{"state_before":"case neg\nR : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np q r : R[X]\ni j : ℕ\nh₁ : (i, j).1 + (i, j).2 = p.natTrailingDegree + q.natTrailingDegree\nh₂ : (i, j) ≠ (p.natTrailingDegree, q.natTrailingDegree)\nhi : p.natTrailingDegree ≤ i\nhj : q.natTrailingDegree ≤ j\n⊢ p.coeff (i, j).1 * q.coeff (i, j).2 = 0","state_after":"case neg\nR : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np q r : R[X]\ni j : ℕ\nh₁ : (i, j).1 + (i, j).2 = p.natTrailingDegree + q.natTrailingDegree\nh₂ : (i, j) ≠ (p.natTrailingDegree, q.natTrailingDegree)\nhi : p.natTrailingDegree ≤ i\nhj : q.natTrailingDegree ≤ j\n⊢ (p.natTrailingDegree, q.natTrailingDegree).1 = (i, j).1 ∧ (p.natTrailingDegree, q.natTrailingDegree).2 = (i, j).2","tactic":"refine (h₂ (Prod.ext_iff.mpr 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(add_eq_add_iff_eq_and_eq hi hj).mp h₁.symm","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"add_eq_add_iff_eq_and_eq","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[270,2],"def_end_pos":[270,13]}]}]}
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+{"url":"Mathlib/Data/Set/Basic.lean","commit":"","full_name":"Set.sep_eq_self_iff_mem_true","start":[1124,0],"end":[1126,58],"file_path":"Mathlib/Data/Set/Basic.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\na b : α\ns s₁ s₂ t t₁ t₂ u : Set α\np q : α → Prop\nx : α\n⊢ {x | x ∈ s ∧ p x} = s ↔ ∀ x ∈ s, p x","state_after":"no goals","tactic":"simp_rw [Set.ext_iff, mem_sep_iff, and_iff_left_iff_imp]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Set.ext_iff","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[190,18],"def_end_pos":[190,25]},{"full_name":"Set.mem_sep_iff","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1112,8],"def_end_pos":[1112,19]},{"full_name":"and_iff_left_iff_imp","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[164,16],"def_end_pos":[164,36]}]}]}
+{"url":"Mathlib/Algebra/BigOperators/Fin.lean","commit":"","full_name":"Fin.sum_univ_seven","start":[128,0],"end":[132,5],"file_path":"Mathlib/Algebra/BigOperators/Fin.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid β\nf : Fin 7 → β\n⊢ ∏ i : Fin 7, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6","state_after":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid β\nf : Fin 7 → β\n⊢ f (castSucc 0) * f (castSucc 1) * f (castSucc 2) * f (castSucc 3) * f (castSucc 4) * f (castSucc 5) * f (last 6) =\n    f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6","tactic":"rw [prod_univ_castSucc, prod_univ_six]","premises":[{"full_name":"Fin.prod_univ_castSucc","def_path":"Mathlib/Algebra/BigOperators/Fin.lean","def_pos":[76,8],"def_end_pos":[76,26]},{"full_name":"Fin.prod_univ_six","def_path":"Mathlib/Algebra/BigOperators/Fin.lean","def_pos":[123,8],"def_end_pos":[123,21]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid β\nf : Fin 7 → β\n⊢ f (castSucc 0) * f (castSucc 1) * f (castSucc 2) * f (castSucc 3) * f (castSucc 4) * f (castSucc 5) * f (last 6) =\n    f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6","state_after":"no goals","tactic":"rfl","premises":[]}]}
+{"url":"Mathlib/Data/Option/Basic.lean","commit":"","full_name":"Option.bnot_isNone","start":[378,0],"end":[381,18],"file_path":"Mathlib/Data/Option/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\na : Option α\n⊢ (!a.isNone) = a.isSome","state_after":"no goals","tactic":"cases a <;> simp","premises":[]}]}
+{"url":"Mathlib/Data/Set/Pairwise/Lattice.lean","commit":"","full_name":"Set.PairwiseDisjoint.biUnion","start":[64,0],"end":[78,55],"file_path":"Mathlib/Data/Set/Pairwise/Lattice.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Type u_5\nκ : Sort u_6\nr p q : α → α → Prop\ninst✝ : CompleteLattice α\ns✝ : Set ι\nt s : Set ι'\ng : ι' → Set ι\nf : ι → α\nhs : s.PairwiseDisjoint fun i' => ⨆ i ∈ g i', f i\nhg : ∀ i ∈ s, (g i).PairwiseDisjoint f\n⊢ (⋃ i ∈ s, g i).PairwiseDisjoint f","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Type u_5\nκ : Sort u_6\nr p q : α → α → Prop\ninst✝ : CompleteLattice α\ns✝ : Set ι\nt s : Set ι'\ng : ι' → Set ι\nf : ι → α\nhs : s.PairwiseDisjoint fun i' => ⨆ i ∈ g i', f i\nhg : ∀ i ∈ s, (g i).PairwiseDisjoint f\na : ι\nha : a ∈ ⋃ i ∈ s, g i\nb : ι\nhb : b ∈ ⋃ i ∈ s, g i\nhab : a ≠ b\n⊢ (Disjoint on f) a b","tactic":"rintro a ha b hb hab","premises":[]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Type u_5\nκ : Sort u_6\nr p q : α → α → Prop\ninst✝ : CompleteLattice α\ns✝ : Set ι\nt s : Set ι'\ng : ι' → Set ι\nf : ι → α\nhs : s.PairwiseDisjoint fun i' => ⨆ i ∈ g i', f i\nhg : ∀ i ∈ s, (g i).PairwiseDisjoint f\na : ι\nha : a ∈ ⋃ i ∈ s, g i\nb : ι\nhb : b ∈ ⋃ i ∈ s, g i\nhab : a ≠ b\n⊢ (Disjoint on f) a b","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Type u_5\nκ : Sort u_6\nr p q : α → α → Prop\ninst✝ : CompleteLattice α\ns✝ : Set ι\nt s : Set ι'\ng : ι' → Set ι\nf : ι → α\nhs : s.PairwiseDisjoint fun i' => ⨆ i ∈ g i', f i\nhg : ∀ i ∈ s, (g i).PairwiseDisjoint f\na b : ι\nhab : a ≠ b\nha : ∃ i, ∃ (_ : i ∈ s), a ∈ g i\nhb : ∃ i, ∃ (_ : i ∈ s), b ∈ g i\n⊢ (Disjoint on f) a b","tactic":"simp_rw [Set.mem_iUnion] at ha hb","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Set.mem_iUnion","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[254,8],"def_end_pos":[254,18]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Type u_5\nκ : Sort u_6\nr p q : α → α → Prop\ninst✝ : CompleteLattice α\ns✝ : Set ι\nt s : Set ι'\ng : ι' → Set ι\nf : ι → α\nhs : s.PairwiseDisjoint fun i' => ⨆ i ∈ g i', f i\nhg : ∀ i ∈ s, (g i).PairwiseDisjoint f\na b : ι\nhab : a ≠ b\nha : ∃ i, ∃ (_ : i ∈ s), a ∈ g i\nhb : ∃ i, ∃ (_ : i ∈ s), b ∈ g i\n⊢ (Disjoint on f) a b","state_after":"case intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Type u_5\nκ : Sort u_6\nr p q : α → α → Prop\ninst✝ : CompleteLattice α\ns✝ : Set ι\nt s : Set ι'\ng : ι' → Set ι\nf : ι → α\nhs : s.PairwiseDisjoint fun i' => ⨆ i ∈ g i', f i\nhg : ∀ i ∈ s, (g i).PairwiseDisjoint f\na b : ι\nhab : a ≠ b\nhb : ∃ i, ∃ (_ : i ∈ s), b ∈ g i\nc : ι'\nhc : c ∈ s\nha : a ∈ g c\n⊢ (Disjoint on f) a b","tactic":"obtain ⟨c, hc, ha⟩ := ha","premises":[]},{"state_before":"case intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Type u_5\nκ : Sort u_6\nr p q : α → α → Prop\ninst✝ : CompleteLattice α\ns✝ : Set ι\nt s : Set ι'\ng : ι' → Set ι\nf : ι → α\nhs : s.PairwiseDisjoint fun i' => ⨆ i ∈ g i', f i\nhg : ∀ i ∈ s, (g i).PairwiseDisjoint f\na b : ι\nhab : a ≠ b\nhb : ∃ i, ∃ (_ : i ∈ s), b ∈ g i\nc : ι'\nhc : c ∈ s\nha : a ∈ g c\n⊢ (Disjoint on f) a b","state_after":"case intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Type u_5\nκ : Sort u_6\nr p q : α → α → Prop\ninst✝ : CompleteLattice α\ns✝ : Set ι\nt s : Set ι'\ng : ι' → Set ι\nf : ι → α\nhs : s.PairwiseDisjoint fun i' => ⨆ i ∈ g i', f i\nhg : ∀ i ∈ s, (g i).PairwiseDisjoint f\na b : ι\nhab : a ≠ b\nc : ι'\nhc : c ∈ s\nha : a ∈ g c\nd : ι'\nhd : d ∈ s\nhb : b ∈ g d\n⊢ (Disjoint on f) a b","tactic":"obtain ⟨d, hd, hb⟩ := hb","premises":[]},{"state_before":"case intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Type u_5\nκ : Sort u_6\nr p q : α → α → Prop\ninst✝ : CompleteLattice α\ns✝ : Set ι\nt s : Set ι'\ng : ι' → Set ι\nf : ι → α\nhs : s.PairwiseDisjoint fun i' => ⨆ i ∈ g i', f i\nhg : ∀ i ∈ s, (g i).PairwiseDisjoint f\na b : ι\nhab : a ≠ b\nc : ι'\nhc : c ∈ s\nha : a ∈ g c\nd : ι'\nhd : d ∈ s\nhb : b ∈ g d\n⊢ (Disjoint on f) a b","state_after":"case intro.intro.intro.intro.inl\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Type u_5\nκ : Sort u_6\nr p q : α → α → Prop\ninst✝ : CompleteLattice α\ns✝ : Set ι\nt s : Set ι'\ng : ι' → Set ι\nf : ι → α\nhs : s.PairwiseDisjoint fun i' => ⨆ i ∈ g i', f i\nhg : ∀ i ∈ s, (g i).PairwiseDisjoint f\na b : ι\nhab : a ≠ b\nc : ι'\nhc : c ∈ s\nha : a ∈ g c\nd : ι'\nhd : d ∈ s\nhb : b ∈ g d\nhcd : g c = g d\n⊢ (Disjoint on f) a b\n\ncase intro.intro.intro.intro.inr\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Type u_5\nκ : Sort u_6\nr p q : α → α → Prop\ninst✝ : CompleteLattice α\ns✝ : Set ι\nt s : Set ι'\ng : ι' → Set ι\nf : ι → α\nhs : s.PairwiseDisjoint fun i' => ⨆ i ∈ g i', f i\nhg : ∀ i ∈ s, (g i).PairwiseDisjoint f\na b : ι\nhab : a ≠ b\nc : ι'\nhc : c ∈ s\nha : a ∈ g c\nd : ι'\nhd : d ∈ s\nhb : b ∈ g d\nhcd : g c ≠ g d\n⊢ (Disjoint on f) a b","tactic":"obtain hcd | hcd := eq_or_ne (g c) (g d)","premises":[{"full_name":"eq_or_ne","def_path":"Mathlib/Logic/Basic.lean","def_pos":[167,8],"def_end_pos":[167,16]}]}]}
+{"url":"Mathlib/LinearAlgebra/Matrix/PosDef.lean","commit":"","full_name":"Matrix.PosSemidef.posSemidef_sqrt","start":[156,0],"end":[163,32],"file_path":"Mathlib/LinearAlgebra/Matrix/PosDef.lean","tactics":[{"state_before":"m : Type u_1\nn : Type u_2\nR : Type u_3\n𝕜 : Type u_4\ninst✝⁷ : Fintype m\ninst✝⁶ : Fintype n\ninst✝⁵ : CommRing R\ninst✝⁴ : PartialOrder R\ninst✝³ : StarRing R\ninst✝² : StarOrderedRing R\ninst✝¹ : RCLike 𝕜\ninst✝ : DecidableEq n\nA : Matrix n n 𝕜\nhA : A.PosSemidef\n⊢ hA.sqrt.PosSemidef","state_after":"case hA\nm : Type u_1\nn : Type u_2\nR : Type u_3\n𝕜 : Type u_4\ninst✝⁷ : Fintype m\ninst✝⁶ : Fintype n\ninst✝⁵ : CommRing R\ninst✝⁴ : PartialOrder R\ninst✝³ : StarRing R\ninst✝² : StarOrderedRing R\ninst✝¹ : RCLike 𝕜\ninst✝ : DecidableEq n\nA : Matrix n n 𝕜\nhA : A.PosSemidef\n⊢ (diagonal (RCLike.ofReal ∘ Real.sqrt ∘ ⋯.eigenvalues)).PosSemidef","tactic":"apply PosSemidef.mul_mul_conjTranspose_same","premises":[{"full_name":"Matrix.PosSemidef.mul_mul_conjTranspose_same","def_path":"Mathlib/LinearAlgebra/Matrix/PosDef.lean","def_pos":[73,6],"def_end_pos":[73,32]}]},{"state_before":"case hA\nm : Type u_1\nn : Type u_2\nR : Type u_3\n𝕜 : Type u_4\ninst✝⁷ : Fintype m\ninst✝⁶ : Fintype n\ninst✝⁵ : CommRing R\ninst✝⁴ : PartialOrder R\ninst✝³ : StarRing R\ninst✝² : StarOrderedRing R\ninst✝¹ : RCLike 𝕜\ninst✝ : DecidableEq n\nA : Matrix n n 𝕜\nhA : A.PosSemidef\n⊢ (diagonal (RCLike.ofReal ∘ Real.sqrt ∘ ⋯.eigenvalues)).PosSemidef","state_after":"case hA\nm : Type u_1\nn : Type u_2\nR : Type u_3\n𝕜 : Type u_4\ninst✝⁷ : Fintype m\ninst✝⁶ : Fintype n\ninst✝⁵ : CommRing R\ninst✝⁴ : PartialOrder R\ninst✝³ : StarRing R\ninst✝² : StarOrderedRing R\ninst✝¹ : RCLike 𝕜\ninst✝ : DecidableEq n\nA : Matrix n n 𝕜\nhA : A.PosSemidef\ni : n\n⊢ 0 ≤ (RCLike.ofReal ∘ Real.sqrt ∘ ⋯.eigenvalues) i","tactic":"refine posSemidef_diagonal_iff.mpr fun i ↦ ?_","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Matrix.posSemidef_diagonal_iff","def_path":"Mathlib/LinearAlgebra/Matrix/PosDef.lean","def_pos":[49,6],"def_end_pos":[49,29]}]},{"state_before":"case hA\nm : Type u_1\nn : Type u_2\nR : Type u_3\n𝕜 : Type u_4\ninst✝⁷ : Fintype m\ninst✝⁶ : Fintype n\ninst✝⁵ : CommRing R\ninst✝⁴ : PartialOrder R\ninst✝³ : StarRing R\ninst✝² : StarOrderedRing R\ninst✝¹ : RCLike 𝕜\ninst✝ : DecidableEq n\nA : Matrix n n 𝕜\nhA : A.PosSemidef\ni : n\n⊢ 0 ≤ (RCLike.ofReal ∘ Real.sqrt ∘ ⋯.eigenvalues) i","state_after":"case hA\nm : Type u_1\nn : Type u_2\nR : Type u_3\n𝕜 : Type u_4\ninst✝⁷ : Fintype m\ninst✝⁶ : Fintype n\ninst✝⁵ : CommRing R\ninst✝⁴ : PartialOrder R\ninst✝³ : StarRing R\ninst✝² : StarOrderedRing R\ninst✝¹ : RCLike 𝕜\ninst✝ : DecidableEq n\nA : Matrix n n 𝕜\nhA : A.PosSemidef\ni : n\n⊢ 0 ≤ RCLike.re ↑((Real.sqrt ∘ ⋯.eigenvalues) i) ∧ RCLike.im ↑((Real.sqrt ∘ ⋯.eigenvalues) i) = 0","tactic":"rw [Function.comp_apply, RCLike.nonneg_iff]","premises":[{"full_name":"Function.comp_apply","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[35,16],"def_end_pos":[35,35]},{"full_name":"RCLike.nonneg_iff","def_path":"Mathlib/Analysis/RCLike/Basic.lean","def_pos":[706,8],"def_end_pos":[706,18]}]},{"state_before":"case hA\nm : Type u_1\nn : Type u_2\nR : Type u_3\n𝕜 : Type u_4\ninst✝⁷ : Fintype m\ninst✝⁶ : Fintype n\ninst✝⁵ : CommRing R\ninst✝⁴ : PartialOrder R\ninst✝³ : StarRing R\ninst✝² : StarOrderedRing R\ninst✝¹ : RCLike 𝕜\ninst✝ : DecidableEq n\nA : Matrix n n 𝕜\nhA : A.PosSemidef\ni : n\n⊢ 0 ≤ RCLike.re ↑((Real.sqrt ∘ ⋯.eigenvalues) i) ∧ RCLike.im ↑((Real.sqrt ∘ ⋯.eigenvalues) i) = 0","state_after":"case hA.left\nm : Type u_1\nn : Type u_2\nR : Type u_3\n𝕜 : Type u_4\ninst✝⁷ : Fintype m\ninst✝⁶ : Fintype n\ninst✝⁵ : CommRing R\ninst✝⁴ : PartialOrder R\ninst✝³ : StarRing R\ninst✝² : StarOrderedRing R\ninst✝¹ : RCLike 𝕜\ninst✝ : DecidableEq n\nA : Matrix n n 𝕜\nhA : A.PosSemidef\ni : n\n⊢ 0 ≤ RCLike.re ↑((Real.sqrt ∘ ⋯.eigenvalues) i)\n\ncase hA.right\nm : Type u_1\nn : Type u_2\nR : Type u_3\n𝕜 : Type u_4\ninst✝⁷ : Fintype m\ninst✝⁶ : Fintype n\ninst✝⁵ : CommRing R\ninst✝⁴ : PartialOrder R\ninst✝³ : StarRing R\ninst✝² : StarOrderedRing R\ninst✝¹ : RCLike 𝕜\ninst✝ : DecidableEq n\nA : Matrix n n 𝕜\nhA : A.PosSemidef\ni : n\n⊢ RCLike.im ↑((Real.sqrt ∘ ⋯.eigenvalues) i) = 0","tactic":"constructor","premises":[]}]}
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+{"url":"Mathlib/Data/List/Sigma.lean","commit":"","full_name":"List.kerase_of_not_mem_keys","start":[351,0],"end":[353,78],"file_path":"Mathlib/Data/List/Sigma.lean","tactics":[{"state_before":"α : Type u\nβ : α → Type v\nl✝ l₁ l₂ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nl : List (Sigma β)\nh : a ∉ l.keys\n⊢ kerase a l = l","state_after":"no goals","tactic":"induction' l with _ _ ih <;> [rfl; (simp [not_or] at h; simp [h.1, ih h.2])]","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"not_or","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[134,16],"def_end_pos":[134,22]}]}]}
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+{"url":"Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean","commit":"","full_name":"MeasureTheory.Measure.ext_of_Ioc'","start":[367,0],"end":[374,24],"file_path":"Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean","tactics":[{"state_before":"α✝ : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Sort y\ns t u : Set α✝\ninst✝¹¹ : TopologicalSpace α✝\ninst✝¹⁰ : MeasurableSpace α✝\ninst✝⁹ : OpensMeasurableSpace α✝\ninst✝⁸ : MeasurableSpace δ\ninst✝⁷ : LinearOrder α✝\ninst✝⁶ : OrderClosedTopology α✝\na b x : α✝\nα : Type u_5\ninst✝⁵ : TopologicalSpace α\nm : MeasurableSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : BorelSpace α\ninst✝ : NoMinOrder α\nμ ν : Measure α\nhμ : ∀ ⦃a b : α⦄, a < b → μ (Ioc a b) ≠ ⊤\nh : ∀ ⦃a b : α⦄, a < b → μ (Ioc a b) = ν (Ioc a b)\n⊢ μ = ν","state_after":"case refine_1\nα✝ : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Sort y\ns t u : Set α✝\ninst✝¹¹ : TopologicalSpace α✝\ninst✝¹⁰ : MeasurableSpace α✝\ninst✝⁹ : OpensMeasurableSpace α✝\ninst✝⁸ : MeasurableSpace δ\ninst✝⁷ : LinearOrder α✝\ninst✝⁶ : OrderClosedTopology α✝\na✝ b✝ x : α✝\nα : Type u_5\ninst✝⁵ : TopologicalSpace α\nm : MeasurableSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : BorelSpace α\ninst✝ : NoMinOrder α\nμ ν : Measure α\nhμ : ∀ ⦃a b : α⦄, a < b → μ (Ioc a b) ≠ ⊤\nh : ∀ ⦃a b : α⦄, a < b → μ (Ioc a b) = ν (Ioc a b)\na b : αᵒᵈ\nhab : a < b\n⊢ μ (⇑OrderDual.ofDual ⁻¹' Ioc b a) ≠ ⊤\n\ncase refine_2\nα✝ : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Sort y\ns t u : Set α✝\ninst✝¹¹ : TopologicalSpace α✝\ninst✝¹⁰ : MeasurableSpace α✝\ninst✝⁹ : OpensMeasurableSpace α✝\ninst✝⁸ : MeasurableSpace δ\ninst✝⁷ : LinearOrder α✝\ninst✝⁶ : OrderClosedTopology α✝\na✝ b✝ x : α✝\nα : Type u_5\ninst✝⁵ : TopologicalSpace α\nm : MeasurableSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : BorelSpace α\ninst✝ : NoMinOrder α\nμ ν : Measure α\nhμ : ∀ ⦃a b : α⦄, a < b → μ (Ioc a b) ≠ ⊤\nh : ∀ ⦃a b : α⦄, a < b → μ (Ioc a b) = ν (Ioc a b)\na b : αᵒᵈ\nhab : a < b\n⊢ μ (⇑OrderDual.ofDual ⁻¹' Ioc b a) = ν (⇑OrderDual.ofDual ⁻¹' Ioc b a)","tactic":"refine @ext_of_Ico' αᵒᵈ _ _ _ _ _ ‹_› _ μ ν ?_ ?_ <;> intro a b hab <;> erw [dual_Ico (α := α)]","premises":[{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"MeasureTheory.Measure.ext_of_Ico'","def_path":"Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean","def_pos":[344,8],"def_end_pos":[344,19]},{"full_name":"OrderDual","def_path":"Mathlib/Order/Basic.lean","def_pos":[695,4],"def_end_pos":[695,13]},{"full_name":"Set.dual_Ico","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[206,8],"def_end_pos":[206,16]}]},{"state_before":"case refine_1\nα✝ : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Sort y\ns t u : Set α✝\ninst✝¹¹ : TopologicalSpace α✝\ninst✝¹⁰ : MeasurableSpace α✝\ninst✝⁹ : OpensMeasurableSpace α✝\ninst✝⁸ : MeasurableSpace δ\ninst✝⁷ : LinearOrder α✝\ninst✝⁶ : OrderClosedTopology α✝\na✝ b✝ x : α✝\nα : Type u_5\ninst✝⁵ : TopologicalSpace α\nm : MeasurableSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : BorelSpace α\ninst✝ : NoMinOrder α\nμ ν : Measure α\nhμ : ∀ ⦃a b : α⦄, a < b → μ (Ioc a b) ≠ ⊤\nh : ∀ ⦃a b : α⦄, a < b → μ (Ioc a b) = ν (Ioc a b)\na b : αᵒᵈ\nhab : a < b\n⊢ μ (⇑OrderDual.ofDual ⁻¹' Ioc b a) ≠ ⊤\n\ncase refine_2\nα✝ : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Sort y\ns t u : Set α✝\ninst✝¹¹ : TopologicalSpace α✝\ninst✝¹⁰ : MeasurableSpace α✝\ninst✝⁹ : OpensMeasurableSpace α✝\ninst✝⁸ : MeasurableSpace δ\ninst✝⁷ : LinearOrder α✝\ninst✝⁶ : OrderClosedTopology α✝\na✝ b✝ x : α✝\nα : Type u_5\ninst✝⁵ : TopologicalSpace α\nm : MeasurableSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : BorelSpace α\ninst✝ : NoMinOrder α\nμ ν : Measure α\nhμ : ∀ ⦃a b : α⦄, a < b → μ (Ioc a b) ≠ ⊤\nh : ∀ ⦃a b : α⦄, a < b → μ (Ioc a b) = ν (Ioc a b)\na b : αᵒᵈ\nhab : a < b\n⊢ μ (⇑OrderDual.ofDual ⁻¹' Ioc b a) = ν (⇑OrderDual.ofDual ⁻¹' Ioc b a)","state_after":"no goals","tactic":"exacts [hμ hab, h hab]","premises":[]}]}
+{"url":"Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean","commit":"","full_name":"NonUnitalSubalgebra.toSubalgebra_toNonUnitalSubalgebra","start":[73,0],"end":[75,14],"file_path":"Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean","tactics":[{"state_before":"R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : NonUnitalSubalgebra R A\nh1 : 1 ∈ S\n⊢ (S.toSubalgebra h1).toNonUnitalSubalgebra = S","state_after":"case mk\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\ntoNonUnitalSubsemiring✝ : NonUnitalSubsemiring A\nsmul_mem'✝ : ∀ (c : R) {x : A}, x ∈ toNonUnitalSubsemiring✝.carrier → c • x ∈ toNonUnitalSubsemiring✝.carrier\nh1 : 1 ∈ { toNonUnitalSubsemiring := toNonUnitalSubsemiring✝, smul_mem' := smul_mem'✝ }\n⊢ ({ toNonUnitalSubsemiring := toNonUnitalSubsemiring✝, smul_mem' := smul_mem'✝ }.toSubalgebra\n        h1).toNonUnitalSubalgebra =\n    { toNonUnitalSubsemiring := toNonUnitalSubsemiring✝, smul_mem' := smul_mem'✝ }","tactic":"cases S","premises":[]},{"state_before":"case mk\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\ntoNonUnitalSubsemiring✝ : NonUnitalSubsemiring A\nsmul_mem'✝ : ∀ (c : R) {x : A}, x ∈ toNonUnitalSubsemiring✝.carrier → c • x ∈ toNonUnitalSubsemiring✝.carrier\nh1 : 1 ∈ { toNonUnitalSubsemiring := toNonUnitalSubsemiring✝, smul_mem' := smul_mem'✝ }\n⊢ ({ toNonUnitalSubsemiring := toNonUnitalSubsemiring✝, smul_mem' := smul_mem'✝ }.toSubalgebra\n        h1).toNonUnitalSubalgebra =\n    { toNonUnitalSubsemiring := toNonUnitalSubsemiring✝, smul_mem' := smul_mem'✝ }","state_after":"no goals","tactic":"rfl","premises":[]}]}
+{"url":"Mathlib/Analysis/NormedSpace/AddTorsorBases.lean","commit":"","full_name":"AffineBasis.centroid_mem_interior_convexHull","start":[123,0],"end":[127,82],"file_path":"Mathlib/Analysis/NormedSpace/AddTorsorBases.lean","tactics":[{"state_before":"V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nι : Type u_3\ninst✝ : Fintype ι\nb : AffineBasis ι ℝ V\n⊢ Finset.centroid ℝ Finset.univ ⇑b ∈ interior ((convexHull ℝ) (range ⇑b))","state_after":"V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nι : Type u_3\ninst✝ : Fintype ι\nb : AffineBasis ι ℝ V\nthis : Nonempty ι\n⊢ Finset.centroid ℝ Finset.univ ⇑b ∈ interior ((convexHull ℝ) (range ⇑b))","tactic":"haveI := b.nonempty","premises":[{"full_name":"AffineBasis.nonempty","def_path":"Mathlib/LinearAlgebra/AffineSpace/Basis.lean","def_pos":[82,18],"def_end_pos":[82,26]}]},{"state_before":"V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nι : Type u_3\ninst✝ : Fintype ι\nb : AffineBasis ι ℝ V\nthis : Nonempty ι\n⊢ Finset.centroid ℝ Finset.univ ⇑b ∈ interior ((convexHull ℝ) (range ⇑b))","state_after":"no goals","tactic":"simp only [b.interior_convexHull, mem_setOf_eq, b.coord_apply_centroid (Finset.mem_univ _),\n    inv_pos, Nat.cast_pos, Finset.card_pos, Finset.univ_nonempty, forall_true_iff]","premises":[{"full_name":"AffineBasis.coord_apply_centroid","def_path":"Mathlib/LinearAlgebra/AffineSpace/Basis.lean","def_pos":[285,8],"def_end_pos":[285,28]},{"full_name":"AffineBasis.interior_convexHull","def_path":"Mathlib/Analysis/NormedSpace/AddTorsorBases.lean","def_pos":[53,8],"def_end_pos":[53,39]},{"full_name":"Finset.card_pos","def_path":"Mathlib/Data/Finset/Card.lean","def_pos":[65,14],"def_end_pos":[65,22]},{"full_name":"Finset.mem_univ","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[71,8],"def_end_pos":[71,16]},{"full_name":"Finset.univ_nonempty","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[97,8],"def_end_pos":[97,21]},{"full_name":"Nat.cast_pos","def_path":"Mathlib/Data/Nat/Cast/Order/Ring.lean","def_pos":[53,8],"def_end_pos":[53,16]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]},{"full_name":"forall_true_iff","def_path":"Mathlib/Logic/Basic.lean","def_pos":[497,8],"def_end_pos":[497,23]},{"full_name":"inv_pos","def_path":"Mathlib/Algebra/Order/Field/Unbundled/Basic.lean","def_pos":[23,14],"def_end_pos":[23,21]}]}]}
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+{"url":"Mathlib/RingTheory/Perfection.lean","commit":"","full_name":"PreTilt.valAux_eq","start":[475,0],"end":[490,5],"file_path":"Mathlib/RingTheory/Perfection.lean","tactics":[{"state_before":"K : Type u₁\ninst✝² : Field K\nv : Valuation K ℝ≥0\nO : Type u₂\ninst✝¹ : CommRing O\ninst✝ : Algebra O K\nhv : v.Integers O\np : ℕ\nhp : Fact (Nat.Prime p)\nhvp : Fact (v ↑p ≠ 1)\nf : PreTilt K v O hv p\nn : ℕ\nhfn : (coeff (ModP K v O hv p) p n) f ≠ 0\n⊢ valAux K v O hv p f = ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p n) f) ^ p ^ n","state_after":"K : Type u₁\ninst✝² : Field K\nv : Valuation K ℝ≥0\nO : Type u₂\ninst✝¹ : CommRing O\ninst✝ : Algebra O K\nhv : v.Integers O\np : ℕ\nhp : Fact (Nat.Prime p)\nhvp : Fact (v ↑p ≠ 1)\nf : PreTilt K v O hv p\nn : ℕ\nhfn : (coeff (ModP K v O hv p) p n) f ≠ 0\nh : ∃ n, (coeff (ModP K v O hv p) p n) f ≠ 0\n⊢ valAux K v O hv p f = ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p n) f) ^ p ^ n","tactic":"have h : ∃ n, coeff _ _ n f ≠ 0 := ⟨n, hfn⟩","premises":[{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Perfection.coeff","def_path":"Mathlib/RingTheory/Perfection.lean","def_pos":[83,4],"def_end_pos":[83,9]}]},{"state_before":"K : Type u₁\ninst✝² : Field K\nv : Valuation K ℝ≥0\nO : Type u₂\ninst✝¹ : CommRing O\ninst✝ : Algebra O K\nhv : v.Integers O\np : ℕ\nhp : Fact (Nat.Prime p)\nhvp : Fact (v ↑p ≠ 1)\nf : PreTilt K v O hv p\nn : ℕ\nhfn : (coeff (ModP K v O hv p) p n) f ≠ 0\nh : ∃ n, (coeff (ModP K v O hv p) p n) f ≠ 0\n⊢ valAux K v O hv p f = ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p n) f) ^ p ^ n","state_after":"K : Type u₁\ninst✝² : Field K\nv : Valuation K ℝ≥0\nO : Type u₂\ninst✝¹ : CommRing O\ninst✝ : Algebra O K\nhv : v.Integers O\np : ℕ\nhp : Fact (Nat.Prime p)\nhvp : Fact (v ↑p ≠ 1)\nf : PreTilt K v O hv p\nn : ℕ\nhfn : (coeff (ModP K v O hv p) p n) f ≠ 0\nh : ∃ n, (coeff (ModP K v O hv p) p n) f ≠ 0\n⊢ ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h =\n    ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p n) f) ^ p ^ n","tactic":"rw [valAux, dif_pos h]","premises":[{"full_name":"PreTilt.valAux","def_path":"Mathlib/RingTheory/Perfection.lean","def_pos":[464,18],"def_end_pos":[464,24]},{"full_name":"dif_pos","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[949,8],"def_end_pos":[949,15]}]},{"state_before":"K : Type u₁\ninst✝² : Field K\nv : Valuation K ℝ≥0\nO : Type u₂\ninst✝¹ : CommRing O\ninst✝ : Algebra O K\nhv : v.Integers O\np : ℕ\nhp : Fact (Nat.Prime p)\nhvp : Fact (v ↑p ≠ 1)\nf : PreTilt K v O hv p\nn : ℕ\nhfn : (coeff (ModP K v O hv p) p n) f ≠ 0\nh : ∃ n, (coeff (ModP K v O hv p) p n) f ≠ 0\n⊢ ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h =\n    ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p n) f) ^ p ^ n","state_after":"case intro\nK : Type u₁\ninst✝² : Field K\nv : Valuation K ℝ≥0\nO : Type u₂\ninst✝¹ : CommRing O\ninst✝ : Algebra O K\nhv : v.Integers O\np : ℕ\nhp : Fact (Nat.Prime p)\nhvp : Fact (v ↑p ≠ 1)\nf : PreTilt K v O hv p\nh : ∃ n, (coeff (ModP K v O hv p) p n) f ≠ 0\nk : ℕ\nhfn : (coeff (ModP K v O hv p) p (Nat.find h + k)) f ≠ 0\n��� ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h =\n    ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h + k)) f) ^ p ^ (Nat.find h + k)","tactic":"obtain ⟨k, rfl⟩ := Nat.exists_eq_add_of_le (Nat.find_min' h hfn)","premises":[{"full_name":"Nat.exists_eq_add_of_le","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean","def_pos":[192,18],"def_end_pos":[192,37]},{"full_name":"Nat.find_min'","def_path":"Mathlib/Data/Nat/Find.lean","def_pos":[71,18],"def_end_pos":[71,27]}]},{"state_before":"case intro\nK : Type u₁\ninst✝² : Field K\nv : Valuation K ℝ≥0\nO : Type u₂\ninst✝¹ : CommRing O\ninst✝ : Algebra O K\nhv : v.Integers O\np : ℕ\nhp : Fact (Nat.Prime p)\nhvp : Fact (v ↑p ≠ 1)\nf : PreTilt K v O hv p\nh : ∃ n, (coeff (ModP K v O hv p) p n) f ≠ 0\nk : ℕ\nhfn : (coeff (ModP K v O hv p) p (Nat.find h + k)) f ≠ 0\n⊢ ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h =\n    ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h + k)) f) ^ p ^ (Nat.find h + k)","state_after":"case intro.zero\nK : Type u₁\ninst✝² : Field K\nv : Valuation K ℝ≥0\nO : Type u₂\ninst✝¹ : CommRing O\ninst✝ : Algebra O K\nhv : v.Integers O\np : ℕ\nhp : Fact (Nat.Prime p)\nhvp : Fact (v ↑p ≠ 1)\nf : PreTilt K v O hv p\nh : ∃ n, (coeff (ModP K v O hv p) p n) f ≠ 0\nhfn : (coeff (ModP K v O hv p) p (Nat.find h + 0)) f ≠ 0\n⊢ ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h =\n    ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h + 0)) f) ^ p ^ (Nat.find h + 0)\n\ncase intro.succ\nK : Type u₁\ninst✝² : Field K\nv : Valuation K ℝ≥0\nO : Type u₂\ninst✝¹ : CommRing O\ninst✝ : Algebra O K\nhv : v.Integers O\np : ℕ\nhp : Fact (Nat.Prime p)\nhvp : Fact (v ↑p ≠ 1)\nf : PreTilt K v O hv p\nh : ∃ n, (coeff (ModP K v O hv p) p n) f ≠ 0\nk : ℕ\nih :\n  (coeff (ModP K v O hv p) p (Nat.find h + k)) f ≠ 0 →\n    ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h =\n      ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h + k)) f) ^ p ^ (Nat.find h + k)\nhfn : (coeff (ModP K v O hv p) p (Nat.find h + (k + 1))) f ≠ 0\n⊢ ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h =\n    ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h + (k + 1))) f) ^ p ^ (Nat.find h + (k + 1))","tactic":"induction' k with k ih","premises":[]},{"state_before":"case intro.succ\nK : Type u₁\ninst✝² : Field K\nv : Valuation K ℝ≥0\nO : Type u₂\ninst✝¹ : CommRing O\ninst✝ : Algebra O K\nhv : v.Integers O\np : ℕ\nhp : Fact (Nat.Prime p)\nhvp : Fact (v ↑p ≠ 1)\nf : PreTilt K v O hv p\nh : ∃ n, (coeff (ModP K v O hv p) p n) f ≠ 0\nk : ℕ\nih :\n  (coeff (ModP K v O hv p) p (Nat.find h + k)) f ≠ 0 →\n    ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h =\n      ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h + k)) f) ^ p ^ (Nat.find h + k)\nhfn : (coeff (ModP K v O hv p) p (Nat.find h + (k + 1))) f ≠ 0\n⊢ ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h =\n    ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h + (k + 1))) f) ^ p ^ (Nat.find h + (k + 1))","state_after":"case intro.succ.intro\nK : Type u₁\ninst✝² : Field K\nv : Valuation K ℝ≥0\nO : Type u₂\ninst✝¹ : CommRing O\ninst✝ : Algebra O K\nhv : v.Integers O\np : ℕ\nhp : Fact (Nat.Prime p)\nhvp : Fact (v ↑p ≠ 1)\nf : PreTilt K v O hv p\nh : ∃ n, (coeff (ModP K v O hv p) p n) f ≠ 0\nk : ℕ\nih :\n  (coeff (ModP K v O hv p) p (Nat.find h + k)) f ≠ 0 →\n    ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h =\n      ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h + k)) f) ^ p ^ (Nat.find h + k)\nhfn : (coeff (ModP K v O hv p) p (Nat.find h + (k + 1))) f ≠ 0\nx : O\nhx : (Ideal.Quotient.mk (Ideal.span {↑p})) x = (coeff (ModP K v O hv p) p (Nat.find h + k + 1)) f\n⊢ ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h =\n    ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h + (k + 1))) f) ^ p ^ (Nat.find h + (k + 1))","tactic":"obtain ⟨x, hx⟩ := Ideal.Quotient.mk_surjective (coeff (ModP K v O hv p) p (Nat.find h + k + 1) f)","premises":[{"full_name":"Ideal.Quotient.mk_surjective","def_path":"Mathlib/RingTheory/Ideal/Quotient.lean","def_pos":[141,8],"def_end_pos":[141,21]},{"full_name":"ModP","def_path":"Mathlib/RingTheory/Perfection.lean","def_pos":[332,4],"def_end_pos":[332,8]},{"full_name":"Nat.find","def_path":"Mathlib/Data/Nat/Find.lean","def_pos":[62,14],"def_end_pos":[62,18]},{"full_name":"Perfection.coeff","def_path":"Mathlib/RingTheory/Perfection.lean","def_pos":[83,4],"def_end_pos":[83,9]}]},{"state_before":"case intro.succ.intro\nK : Type u₁\ninst✝² : Field K\nv : Valuation K ℝ≥0\nO : Type u₂\ninst✝¹ : CommRing O\ninst✝ : Algebra O K\nhv : v.Integers O\np : ℕ\nhp : Fact (Nat.Prime p)\nhvp : Fact (v ↑p ≠ 1)\nf : PreTilt K v O hv p\nh : ∃ n, (coeff (ModP K v O hv p) p n) f ≠ 0\nk : ℕ\nih :\n  (coeff (ModP K v O hv p) p (Nat.find h + k)) f ≠ 0 →\n    ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h =\n      ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h + k)) f) ^ p ^ (Nat.find h + k)\nhfn : (coeff (ModP K v O hv p) p (Nat.find h + (k + 1))) f ≠ 0\nx : O\nhx : (Ideal.Quotient.mk (Ideal.span {↑p})) x = (coeff (ModP K v O hv p) p (Nat.find h + k + 1)) f\n⊢ ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h =\n    ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h + (k + 1))) f) ^ p ^ (Nat.find h + (k + 1))","state_after":"case intro.succ.intro\nK : Type u₁\ninst✝² : Field K\nv : Valuation K ℝ≥0\nO : Type u₂\ninst✝¹ : CommRing O\ninst✝ : Algebra O K\nhv : v.Integers O\np : ℕ\nhp : Fact (Nat.Prime p)\nhvp : Fact (v ↑p ≠ 1)\nf : PreTilt K v O hv p\nh : ∃ n, (coeff (ModP K v O hv p) p n) f ≠ 0\nk : ℕ\nih :\n  (coeff (ModP K v O hv p) p (Nat.find h + k)) f ≠ 0 →\n    ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h =\n      ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h + k)) f) ^ p ^ (Nat.find h + k)\nhfn : (coeff (ModP K v O hv p) p (Nat.find h + (k + 1))) f ≠ 0\nx : O\nhx : (Ideal.Quotient.mk (Ideal.span {↑p})) x = (coeff (ModP K v O hv p) p (Nat.find h + k + 1)) f\nh1 : (Ideal.Quotient.mk (Ideal.span {↑p})) x ≠ 0\n⊢ ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h =\n    ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h + (k + 1))) f) ^ p ^ (Nat.find h + (k + 1))","tactic":"have h1 : (Ideal.Quotient.mk _ x : ModP K v O hv p) ≠ 0 := hx.symm ▸ hfn","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Ideal.Quotient.mk","def_path":"Mathlib/RingTheory/Ideal/Quotient.lean","def_pos":[84,4],"def_end_pos":[84,6]},{"full_name":"ModP","def_path":"Mathlib/RingTheory/Perfection.lean","def_pos":[332,4],"def_end_pos":[332,8]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]}]},{"state_before":"case intro.succ.intro\nK : Type u₁\ninst✝² : Field K\nv : Valuation K ℝ≥0\nO : Type u₂\ninst✝¹ : CommRing O\ninst✝ : Algebra O K\nhv : v.Integers O\np : ℕ\nhp : Fact (Nat.Prime p)\nhvp : Fact (v ↑p ≠ 1)\nf : PreTilt K v O hv p\nh : ∃ n, (coeff (ModP K v O hv p) p n) f ≠ 0\nk : ℕ\nih :\n  (coeff (ModP K v O hv p) p (Nat.find h + k)) f ≠ 0 →\n    ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h =\n      ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h + k)) f) ^ p ^ (Nat.find h + k)\nhfn : (coeff (ModP K v O hv p) p (Nat.find h + (k + 1))) f ≠ 0\nx : O\nhx : (Ideal.Quotient.mk (Ideal.span {↑p})) x = (coeff (ModP K v O hv p) p (Nat.find h + k + 1)) f\nh1 : (Ideal.Quotient.mk (Ideal.span {↑p})) x ≠ 0\n⊢ ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h =\n    ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h + (k + 1))) f) ^ p ^ (Nat.find h + (k + 1))","state_after":"case intro.succ.intro\nK : Type u₁\ninst✝² : Field K\nv : Valuation K ℝ≥0\nO : Type u₂\ninst✝¹ : CommRing O\ninst✝ : Algebra O K\nhv : v.Integers O\np : ℕ\nhp : Fact (Nat.Prime p)\nhvp : Fact (v ↑p ≠ 1)\nf : PreTilt K v O hv p\nh : ∃ n, (coeff (ModP K v O hv p) p n) f ≠ 0\nk : ℕ\nih :\n  (coeff (ModP K v O hv p) p (Nat.find h + k)) f ≠ 0 →\n    ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h =\n      ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h + k)) f) ^ p ^ (Nat.find h + k)\nhfn : (coeff (ModP K v O hv p) p (Nat.find h + (k + 1))) f ≠ 0\nx : O\nhx : (Ideal.Quotient.mk (Ideal.span {↑p})) x = (coeff (ModP K v O hv p) p (Nat.find h + k + 1)) f\nh1 : (Ideal.Quotient.mk (Ideal.span {↑p})) x ≠ 0\nh2 : (Ideal.Quotient.mk (Ideal.span {↑p})) (x ^ p) ≠ 0\n⊢ ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h =\n    ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h + (k + 1))) f) ^ p ^ (Nat.find h + (k + 1))","tactic":"have h2 : (Ideal.Quotient.mk _ (x ^ p) : ModP K v O hv p) ≠ 0 := by\n    erw [RingHom.map_pow, hx, ← RingHom.map_pow, coeff_pow_p]\n    exact coeff_nat_find_add_ne_zero k","premises":[{"full_name":"Ideal.Quotient.mk","def_path":"Mathlib/RingTheory/Ideal/Quotient.lean","def_pos":[84,4],"def_end_pos":[84,6]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"ModP","def_path":"Mathlib/RingTheory/Perfection.lean","def_pos":[332,4],"def_end_pos":[332,8]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Perfection.coeff_pow_p","def_path":"Mathlib/RingTheory/Perfection.lean","def_pos":[114,8],"def_end_pos":[114,19]},{"full_name":"PreTilt.coeff_nat_find_add_ne_zero","def_path":"Mathlib/RingTheory/Perfection.lean","def_pos":[471,8],"def_end_pos":[471,34]},{"full_name":"RingHom.map_pow","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[622,16],"def_end_pos":[622,31]}]},{"state_before":"case intro.succ.intro\nK : Type u₁\ninst✝² : Field K\nv : Valuation K ℝ≥0\nO : Type u₂\ninst✝¹ : CommRing O\ninst✝ : Algebra O K\nhv : v.Integers O\np : ℕ\nhp : Fact (Nat.Prime p)\nhvp : Fact (v ↑p ≠ 1)\nf : PreTilt K v O hv p\nh : ∃ n, (coeff (ModP K v O hv p) p n) f ≠ 0\nk : ℕ\nih :\n  (coeff (ModP K v O hv p) p (Nat.find h + k)) f ≠ 0 →\n    ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h =\n      ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h + k)) f) ^ p ^ (Nat.find h + k)\nhfn : (coeff (ModP K v O hv p) p (Nat.find h + (k + 1))) f ≠ 0\nx : O\nhx : (Ideal.Quotient.mk (Ideal.span {↑p})) x = (coeff (ModP K v O hv p) p (Nat.find h + k + 1)) f\nh1 : (Ideal.Quotient.mk (Ideal.span {↑p})) x ≠ 0\nh2 : (Ideal.Quotient.mk (Ideal.span {↑p})) (x ^ p) ≠ 0\n⊢ ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h =\n    ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h + (k + 1))) f) ^ p ^ (Nat.find h + (k + 1))","state_after":"case intro.succ.intro\nK : Type u₁\ninst✝² : Field K\nv : Valuation K ℝ≥0\nO : Type u₂\ninst✝¹ : CommRing O\ninst✝ : Algebra O K\nhv : v.Integers O\np : ℕ\nhp : Fact (Nat.Prime p)\nhvp : Fact (v ↑p ≠ 1)\nf : PreTilt K v O hv p\nh : ∃ n, (coeff (ModP K v O hv p) p n) f ≠ 0\nk : ℕ\nih :\n  (coeff (ModP K v O hv p) p (Nat.find h + k)) f ≠ 0 →\n    ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h =\n      ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h + k)) f) ^ p ^ (Nat.find h + k)\nhfn : (coeff (ModP K v O hv p) p (Nat.find h + (k + 1))) f ≠ 0\nx : O\nhx : (Ideal.Quotient.mk (Ideal.span {↑p})) x = (coeff (ModP K v O hv p) p (Nat.find h + k + 1)) f\nh1 : (Ideal.Quotient.mk (Ideal.span {↑p})) x ≠ 0\nh2 : (Ideal.Quotient.mk (Ideal.span {↑p})) (x ^ p) ≠ 0\n⊢ v ((algebraMap O K) x) ^ (p * p ^ (Nat.find h + k)) = v ((algebraMap O K) x) ^ (p * p ^ (Nat.find h).add k)","tactic":"erw [ih (coeff_nat_find_add_ne_zero k), ← hx, ← coeff_pow_p, RingHom.map_pow, ← hx,\n    ← RingHom.map_pow, ModP.preVal_mk h1, ModP.preVal_mk h2, RingHom.map_pow, v.map_pow, ← pow_mul,\n    pow_succ']","premises":[{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"ModP.preVal_mk","def_path":"Mathlib/RingTheory/Perfection.lean","def_pos":[360,8],"def_end_pos":[360,17]},{"full_name":"Perfection.coeff_pow_p","def_path":"Mathlib/RingTheory/Perfection.lean","def_pos":[114,8],"def_end_pos":[114,19]},{"full_name":"PreTilt.coeff_nat_find_add_ne_zero","def_path":"Mathlib/RingTheory/Perfection.lean","def_pos":[471,8],"def_end_pos":[471,34]},{"full_name":"RingHom.map_pow","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[622,16],"def_end_pos":[622,31]},{"full_name":"Valuation.map_pow","def_path":"Mathlib/RingTheory/Valuation/Basic.lean","def_pos":[195,8],"def_end_pos":[195,15]},{"full_name":"pow_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[605,31],"def_end_pos":[605,38]},{"full_name":"pow_succ'","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[573,33],"def_end_pos":[573,42]}]},{"state_before":"case intro.succ.intro\nK : Type u₁\ninst✝² : Field K\nv : Valuation K ℝ≥0\nO : Type u₂\ninst✝¹ : CommRing O\ninst✝ : Algebra O K\nhv : v.Integers O\np : ℕ\nhp : Fact (Nat.Prime p)\nhvp : Fact (v ↑p ≠ 1)\nf : PreTilt K v O hv p\nh : ∃ n, (coeff (ModP K v O hv p) p n) f ≠ 0\nk : ℕ\nih :\n  (coeff (ModP K v O hv p) p (Nat.find h + k)) f ≠ 0 →\n    ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h =\n      ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h + k)) f) ^ p ^ (Nat.find h + k)\nhfn : (coeff (ModP K v O hv p) p (Nat.find h + (k + 1))) f ≠ 0\nx : O\nhx : (Ideal.Quotient.mk (Ideal.span {↑p})) x = (coeff (ModP K v O hv p) p (Nat.find h + k + 1)) f\nh1 : (Ideal.Quotient.mk (Ideal.span {↑p})) x ≠ 0\nh2 : (Ideal.Quotient.mk (Ideal.span {↑p})) (x ^ p) ≠ 0\n⊢ v ((algebraMap O K) x) ^ (p * p ^ (Nat.find h + k)) = v ((algebraMap O K) x) ^ (p * p ^ (Nat.find h).add k)","state_after":"no goals","tactic":"rfl","premises":[]}]}
+{"url":"Mathlib/Algebra/DirectSum/Module.lean","commit":"","full_name":"DirectSum.isInternal_biSup_submodule_of_independent","start":[411,0],"end":[423,69],"file_path":"Mathlib/Algebra/DirectSum/Module.lean","tactics":[{"state_before":"R : Type u\ninst✝² : Ring R\nι : Type v\ndec_ι : DecidableEq ι\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nA : ι → Submodule R M\ns : Set ι\nh : CompleteLattice.Independent fun i => A ↑i\n⊢ IsInternal fun i => Submodule.comap (⨆ i ∈ s, A i).subtype (A ↑i)","state_after":"R : Type u\ninst✝² : Ring R\nι : Type v\ndec_ι : DecidableEq ι\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nA : ι → Submodule R M\ns : Set ι\nh : CompleteLattice.Independent fun i => A ↑i\n⊢ CompleteLattice.Independent fun i => Submodule.comap (⨆ i ∈ s, A i).subtype (A ↑i)","tactic":"refine (isInternal_submodule_iff_independent_and_iSup_eq_top _).mpr ⟨?_, by simp [iSup_subtype]⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"DirectSum.isInternal_submodule_iff_independent_and_iSup_eq_top","def_path":"Mathlib/Algebra/DirectSum/Module.lean","def_pos":[391,8],"def_end_pos":[391,60]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"iSup_subtype","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[972,8],"def_end_pos":[972,20]}]},{"state_before":"R : Type u\ninst✝² : Ring R\nι : Type v\ndec_ι : DecidableEq ι\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nA : ι → Submodule R M\ns : Set ι\nh : CompleteLattice.Independent fun i => A ↑i\n⊢ CompleteLattice.Independent fun i => Submodule.comap (⨆ i ∈ s, A i).subtype (A ↑i)","state_after":"R : Type u\ninst✝² : Ring R\nι : Type v\ndec_ι : DecidableEq ι\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nA : ι → Submodule R M\ns : Set ι\nh : CompleteLattice.Independent fun i => A ↑i\np : Submodule R M := ⨆ i ∈ s, A i\n⊢ CompleteLattice.Independent fun i => Submodule.comap (⨆ i ∈ s, A i).subtype (A ↑i)","tactic":"let p := ⨆ i ∈ s, A i","premises":[{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"iSup","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[56,4],"def_end_pos":[56,8]}]},{"state_before":"R : Type u\ninst✝² : Ring R\nι : Type v\ndec_ι : DecidableEq ι\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nA : ι → Submodule R M\ns : Set ι\nh : CompleteLattice.Independent fun i => A ↑i\np : Submodule R M := ⨆ i ∈ s, A i\n⊢ CompleteLattice.Independent fun i => Submodule.comap (⨆ i ∈ s, A i).subtype (A ↑i)","state_after":"R : Type u\ninst✝² : Ring R\nι : Type v\ndec_ι : DecidableEq ι\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nA : ι → Submodule R M\ns : Set ι\nh : CompleteLattice.Independent fun i => A ↑i\np : Submodule R M := ⨆ i ∈ s, A i\nhp : ∀ i ∈ s, A i ≤ p\n⊢ CompleteLattice.Independent fun i => Submodule.comap (⨆ i ∈ s, A i).subtype (A ↑i)","tactic":"have hp : ∀ i ∈ s, A i ≤ p := fun i hi ↦ le_biSup A hi","premises":[{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"le_biSup","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[1019,6],"def_end_pos":[1019,14]}]},{"state_before":"R : Type u\ninst✝² : Ring R\nι : Type v\ndec_ι : DecidableEq ι\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nA : ι → Submodule R M\ns : Set ι\nh : CompleteLattice.Independent fun i => A ↑i\np : Submodule R M := ⨆ i ∈ s, A i\nhp : ∀ i ∈ s, A i ≤ p\n⊢ CompleteLattice.Independent fun i => Submodule.comap (⨆ i ∈ s, A i).subtype (A ↑i)","state_after":"R : Type u\ninst✝² : Ring R\nι : Type v\ndec_ι : DecidableEq ι\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nA : ι → Submodule R M\ns : Set ι\nh : CompleteLattice.Independent fun i => A ↑i\np : Submodule R M := ⨆ i ∈ s, A i\nhp : ∀ i ∈ s, A i ≤ p\ne : Submodule R ↥p ≃o ↑(Set.Iic p) := p.mapIic\n⊢ CompleteLattice.Independent fun i => Submodule.comap (⨆ i ∈ s, A i).subtype (A ↑i)","tactic":"let e : Submodule R p ≃o Set.Iic p := p.mapIic","premises":[{"full_name":"OrderIso","def_path":"Mathlib/Order/Hom/Basic.lean","def_pos":[97,7],"def_end_pos":[97,15]},{"full_name":"Set.Iic","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[58,4],"def_end_pos":[58,7]},{"full_name":"Submodule","def_path":"Mathlib/Algebra/Module/Submodule/Basic.lean","def_pos":[36,10],"def_end_pos":[36,19]},{"full_name":"Submodule.mapIic","def_path":"Mathlib/Algebra/Module/Submodule/Range.lean","def_pos":[313,4],"def_end_pos":[313,10]}]},{"state_before":"R : Type u\ninst✝² : Ring R\nι : Type v\ndec_ι : DecidableEq ι\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nA : ι → Submodule R M\ns : Set ι\nh : CompleteLattice.Independent fun i => A ↑i\np : Submodule R M := ⨆ i ∈ s, A i\nhp : ∀ i ∈ s, A i ≤ p\ne : Submodule R ↥p ≃o ↑(Set.Iic p) := p.mapIic\n⊢ CompleteLattice.Independent fun i => Submodule.comap (⨆ i ∈ s, A i).subtype (A ↑i)","state_after":"R : Type u\ninst✝² : Ring R\nι : Type v\ndec_ι : DecidableEq ι\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nA : ι → Submodule R M\ns : Set ι\nh : CompleteLattice.Independent fun i => A ↑i\np : Submodule R M := ⨆ i ∈ s, A i\nhp : ∀ i ∈ s, A i ≤ p\ne : Submodule R ↥p ≃o ↑(Set.Iic p) := p.mapIic\n⊢ (⇑e ∘ fun i => Submodule.comap p.subtype (A ↑i)) = fun i => ⟨A ↑i, ⋯⟩","tactic":"suffices (e ∘ fun i : s ↦ (A i).comap p.subtype) = fun i ↦ ⟨A i, hp i i.property⟩ by\n    rw [← CompleteLattice.independent_map_orderIso_iff e, this]\n    exact CompleteLattice.independent_of_independent_coe_Iic_comp h","premises":[{"full_name":"CompleteLattice.independent_map_orderIso_iff","def_path":"Mathlib/Order/SupIndep.lean","def_pos":[421,8],"def_end_pos":[421,36]},{"full_name":"CompleteLattice.independent_of_independent_coe_Iic_comp","def_path":"Mathlib/Order/SupIndep.lean","def_pos":[434,6],"def_end_pos":[434,45]},{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"Submodule.comap","def_path":"Mathlib/Algebra/Module/Submodule/Map.lean","def_pos":[161,4],"def_end_pos":[161,9]},{"full_name":"Submodule.subtype","def_path":"Mathlib/Algebra/Module/Submodule/LinearMap.lean","def_pos":[69,14],"def_end_pos":[69,21]},{"full_name":"Subtype.property","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[590,2],"def_end_pos":[590,10]}]},{"state_before":"R : Type u\ninst✝² : Ring R\nι : Type v\ndec_ι : DecidableEq ι\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nA : ι → Submodule R M\ns : Set ι\nh : CompleteLattice.Independent fun i => A ↑i\np : Submodule R M := ⨆ i ∈ s, A i\nhp : ∀ i ∈ s, A i ≤ p\ne : Submodule R ↥p ≃o ↑(Set.Iic p) := p.mapIic\n⊢ (⇑e ∘ fun i => Submodule.comap p.subtype (A ↑i)) = fun i => ⟨A ↑i, ⋯⟩","state_after":"case h.a.h\nR : Type u\ninst✝² : Ring R\nι : Type v\ndec_ι : DecidableEq ι\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nA : ι → Submodule R M\ns : Set ι\nh : CompleteLattice.Independent fun i => A ↑i\np : Submodule R M := ⨆ i ∈ s, A i\nhp : ∀ i ∈ s, A i ≤ p\ne : Submodule R ↥p ≃o ↑(Set.Iic p) := p.mapIic\ni : ↑s\nm : M\n⊢ m ∈ ↑((⇑e ∘ fun i => Submodule.comap p.subtype (A ↑i)) i) ↔ m ∈ ↑⟨A ↑i, ⋯⟩","tactic":"ext i m","premises":[]},{"state_before":"case h.a.h\nR : Type u\ninst✝² : Ring R\nι : Type v\ndec_ι : DecidableEq ι\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nA : ι → Submodule R M\ns : Set ι\nh : CompleteLattice.Independent fun i => A ↑i\np : Submodule R M := ⨆ i ∈ s, A i\nhp : ∀ i ∈ s, A i ≤ p\ne : Submodule R ↥p ≃o ↑(Set.Iic p) := p.mapIic\ni : ↑s\nm : M\n⊢ m ∈ ↑((⇑e ∘ fun i => Submodule.comap p.subtype (A ↑i)) i) ↔ m ∈ ↑⟨A ↑i, ⋯⟩","state_after":"case h.a.h\nR : Type u\ninst✝² : Ring R\nι : Type v\ndec_ι : DecidableEq ι\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nA : ι → Submodule R M\ns : Set ι\nh : CompleteLattice.Independent fun i => A ↑i\np : Submodule R M := ⨆ i ∈ s, A i\nhp : ∀ i ∈ s, A i ≤ p\ne : Submodule R ↥p ≃o ↑(Set.Iic p) := p.mapIic\ni : ↑s\nm : M\n⊢ m ∈ Submodule.map p.subtype (Submodule.comap p.subtype (A ↑i)) ↔ m ∈ ↑⟨A ↑i, ⋯⟩","tactic":"change m ∈ ((A i).comap p.subtype).map p.subtype ↔ _","premises":[{"full_name":"Iff","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[114,10],"def_end_pos":[114,13]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Submodule.comap","def_path":"Mathlib/Algebra/Module/Submodule/Map.lean","def_pos":[161,4],"def_end_pos":[161,9]},{"full_name":"Submodule.map","def_path":"Mathlib/Algebra/Module/Submodule/Map.lean","def_pos":[48,4],"def_end_pos":[48,7]},{"full_name":"Submodule.subtype","def_path":"Mathlib/Algebra/Module/Submodule/LinearMap.lean","def_pos":[69,14],"def_end_pos":[69,21]}]},{"state_before":"case h.a.h\nR : Type u\ninst✝² : Ring R\nι : Type v\ndec_ι : DecidableEq ι\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nA : ι → Submodule R M\ns : Set ι\nh : CompleteLattice.Independent fun i => A ↑i\np : Submodule R M := ⨆ i ∈ s, A i\nhp : ∀ i ∈ s, A i ≤ p\ne : Submodule R ↥p ≃o ↑(Set.Iic p) := p.mapIic\ni : ↑s\nm : M\n⊢ m ∈ Submodule.map p.subtype (Submodule.comap p.subtype (A ↑i)) ↔ m ∈ ↑⟨A ↑i, ⋯⟩","state_after":"no goals","tactic":"rw [Submodule.map_comap_subtype, inf_of_le_right (hp i i.property)]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Submodule.map_comap_subtype","def_path":"Mathlib/Algebra/Module/Submodule/Map.lean","def_pos":[371,8],"def_end_pos":[371,25]},{"full_name":"Subtype.property","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[590,2],"def_end_pos":[590,10]}]}]}
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+{"url":"Mathlib/FieldTheory/PurelyInseparable.lean","commit":"","full_name":"_private.Mathlib.FieldTheory.PurelyInseparable.0.LinearIndependent.map_pow_expChar_pow_of_fd_isSeparable","start":[727,0],"end":[744,59],"file_path":"Mathlib/FieldTheory/PurelyInseparable.lean","tactics":[{"state_before":"F : Type u\nE : Type v\ninst✝⁶ : Field F\ninst✝⁵ : Field E\ninst✝⁴ : Algebra F E\nK : Type w\ninst✝³ : Field K\ninst✝² : Algebra F K\nq n : ℕ\nhF : ExpChar F q\nι : Type u_1\nv : ι → E\ninst✝¹ : FiniteDimensional F E\ninst✝ : Algebra.IsSeparable F E\nh : LinearIndependent F v\n⊢ LinearIndependent F fun x => v x ^ q ^ n","state_after":"F : Type u\nE : Type v\ninst✝⁶ : Field F\ninst✝⁵ : Field E\ninst✝⁴ : Algebra F E\nK : Type w\ninst✝³ : Field K\ninst✝² : Algebra F K\nq n : ℕ\nhF : ExpChar F q\nι : Type u_1\nv : ι → E\ninst✝¹ : FiniteDimensional F E\ninst✝ : Algebra.IsSeparable F E\nh : LinearIndependent F v\nh' : LinearIndependent (ι := { x // x ∈ Set.range v }) F Subtype.val\n⊢ LinearIndependent F fun x => v x ^ q ^ n","tactic":"have h' := h.coe_range","premises":[{"full_name":"LinearIndependent.coe_range","def_path":"Mathlib/LinearAlgebra/LinearIndependent.lean","def_pos":[239,8],"def_end_pos":[239,35]}]},{"state_before":"F : Type u\nE : Type v\ninst✝⁶ : Field F\ninst✝⁵ : Field E\ninst✝⁴ : Algebra F E\nK : Type w\ninst✝³ : Field K\ninst✝² : Algebra F K\nq n : ℕ\nhF : ExpChar F q\nι : Type u_1\nv : ι → E\ninst✝¹ : FiniteDimensional F E\ninst✝ : Algebra.IsSeparable F E\nh : LinearIndependent F v\nh' : LinearIndependent (ι := { x // x ∈ Set.range v }) F Subtype.val\n⊢ LinearIndependent F fun x => v x ^ q ^ n","state_after":"F : Type u\nE : Type v\ninst✝⁶ : Field F\ninst✝⁵ : Field E\ninst✝⁴ : Algebra F E\nK : Type w\ninst✝³ : Field K\ninst✝² : Algebra F K\nq n : ℕ\nhF : ExpChar F q\nι : Type u_1\nv : ι → E\ninst✝¹ : FiniteDimensional F E\ninst✝ : Algebra.IsSeparable F E\nh : LinearIndependent F v\nh' : LinearIndependent (ι := { x // x ∈ Set.range v }) F Subtype.val\nι' : Set E := h'.extend ⋯\n⊢ LinearIndependent F fun x => v x ^ q ^ n","tactic":"let ι' := h'.extend (Set.range v).subset_univ","premises":[{"full_name":"LinearIndependent.extend","def_path":"Mathlib/LinearAlgebra/LinearIndependent.lean","def_pos":[1362,18],"def_end_pos":[1362,42]},{"full_name":"Set.range","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[144,4],"def_end_pos":[144,9]},{"full_name":"Set.subset_univ","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[550,8],"def_end_pos":[550,19]}]},{"state_before":"F : Type u\nE : Type v\ninst✝⁶ : Field F\ninst✝⁵ : Field E\ninst✝⁴ : Algebra F E\nK : Type w\ninst✝³ : Field K\ninst✝² : Algebra F K\nq n : ℕ\nhF : ExpChar F q\nι : Type u_1\nv : ι → E\ninst✝¹ : FiniteDimensional F E\ninst✝ : Algebra.IsSeparable F E\nh : LinearIndependent F v\nh' : LinearIndependent (ι := { x // x ∈ Set.range v }) F Subtype.val\nι' : Set E := h'.extend ⋯\n⊢ LinearIndependent F fun x => v x ^ q ^ n","state_after":"F : Type u\nE : Type v\ninst✝⁶ : Field F\ninst✝⁵ : Field E\ninst✝⁴ : Algebra F E\nK : Type w\ninst✝³ : Field K\ninst✝² : Algebra F K\nq n : ℕ\nhF : ExpChar F q\nι : Type u_1\nv : ι → E\ninst✝¹ : FiniteDimensional F E\ninst✝ : Algebra.IsSeparable F E\nh : LinearIndependent F v\nh' : LinearIndependent (ι := { x // x ∈ Set.range v }) F Subtype.val\nι' : Set E := h'.extend ⋯\nb : Basis (↑ι') F E := Basis.extend h'\n⊢ LinearIndependent F fun x => v x ^ q ^ n","tactic":"let b : Basis ι' F E := Basis.extend h'","premises":[{"full_name":"Basis","def_path":"Mathlib/LinearAlgebra/Basis.lean","def_pos":[86,10],"def_end_pos":[86,15]},{"full_name":"Basis.extend","def_path":"Mathlib/LinearAlgebra/Basis/VectorSpace.lean","def_pos":[47,18],"def_end_pos":[47,24]}]},{"state_before":"F : Type u\nE : Type v\ninst✝�� : Field F\ninst✝⁵ : Field E\ninst✝⁴ : Algebra F E\nK : Type w\ninst✝³ : Field K\ninst✝² : Algebra F K\nq n : ℕ\nhF : ExpChar F q\nι : Type u_1\nv : ι → E\ninst✝¹ : FiniteDimensional F E\ninst✝ : Algebra.IsSeparable F E\nh : LinearIndependent F v\nh' : LinearIndependent (ι := { x // x ∈ Set.range v }) F Subtype.val\nι' : Set E := h'.extend ⋯\nb : Basis (↑ι') F E := Basis.extend h'\n⊢ LinearIndependent F fun x => v x ^ q ^ n","state_after":"F : Type u\nE : Type v\ninst✝⁶ : Field F\ninst✝⁵ : Field E\ninst✝⁴ : Algebra F E\nK : Type w\ninst✝³ : Field K\ninst✝² : Algebra F K\nq n : ℕ\nhF : ExpChar F q\nι : Type u_1\nv : ι → E\ninst✝¹ : FiniteDimensional F E\ninst✝ : Algebra.IsSeparable F E\nh : LinearIndependent F v\nh' : LinearIndependent (ι := { x // x ∈ Set.range v }) F Subtype.val\nι' : Set E := h'.extend ⋯\nb : Basis (↑ι') F E := Basis.extend h'\nthis : Fintype ↑ι' := fintypeBasisIndex b\n⊢ LinearIndependent F fun x => v x ^ q ^ n","tactic":"letI : Fintype ι' := fintypeBasisIndex b","premises":[{"full_name":"FiniteDimensional.fintypeBasisIndex","def_path":"Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean","def_pos":[127,18],"def_end_pos":[127,35]},{"full_name":"Fintype","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[55,6],"def_end_pos":[55,13]}]},{"state_before":"F : Type u\nE : Type v\ninst✝⁶ : Field F\ninst✝⁵ : Field E\ninst✝⁴ : Algebra F E\nK : Type w\ninst✝³ : Field K\ninst✝² : Algebra F K\nq n : ℕ\nhF : ExpChar F q\nι : Type u_1\nv : ι → E\ninst✝¹ : FiniteDimensional F E\ninst✝ : Algebra.IsSeparable F E\nh : LinearIndependent F v\nh' : LinearIndependent (ι := { x // x ∈ Set.range v }) F Subtype.val\nι' : Set E := h'.extend ⋯\nb : Basis (↑ι') F E := Basis.extend h'\nthis : Fintype ↑ι' := fintypeBasisIndex b\n⊢ LinearIndependent F fun x => v x ^ q ^ n","state_after":"F : Type u\nE : Type v\ninst✝⁶ : Field F\ninst✝⁵ : Field E\ninst✝⁴ : Algebra F E\nK : Type w\ninst✝³ : Field K\ninst✝² : Algebra F K\nq n : ℕ\nhF : ExpChar F q\nι : Type u_1\nv : ι → E\ninst✝¹ : FiniteDimensional F E\ninst✝ : Algebra.IsSeparable F E\nh : LinearIndependent F v\nh' : LinearIndependent (ι := { x // x ∈ Set.range v }) F Subtype.val\nι' : Set E := h'.extend ⋯\nb : Basis (↑ι') F E := Basis.extend h'\nthis : Fintype ↑ι' := fintypeBasisIndex b\nH : LinearIndependent F fun (x : ↑ι') => b x ^ q ^ n\n⊢ LinearIndependent F fun x => v x ^ q ^ n","tactic":"have H := linearIndependent_of_top_le_span_of_card_eq_finrank\n    (span_map_pow_expChar_pow_eq_top_of_isSeparable q n b.span_eq).ge\n    (finrank_eq_card_basis b).symm","premises":[{"full_name":"Basis.span_eq","def_path":"Mathlib/LinearAlgebra/Basis.lean","def_pos":[540,18],"def_end_pos":[540,25]},{"full_name":"Eq.ge","def_path":"Mathlib/Order/Basic.lean","def_pos":[194,18],"def_end_pos":[194,20]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Field.span_map_pow_expChar_pow_eq_top_of_isSeparable","def_path":"Mathlib/FieldTheory/PurelyInseparable.lean","def_pos":[715,8],"def_end_pos":[715,60]},{"full_name":"FiniteDimensional.finrank_eq_card_basis","def_path":"Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean","def_pos":[388,8],"def_end_pos":[388,29]},{"full_name":"linearIndependent_of_top_le_span_of_card_eq_finrank","def_path":"Mathlib/LinearAlgebra/Dimension/DivisionRing.lean","def_pos":[119,8],"def_end_pos":[119,59]}]},{"state_before":"F : Type u\nE : Type v\ninst✝⁶ : Field F\ninst✝⁵ : Field E\ninst✝⁴ : Algebra F E\nK : Type w\ninst✝³ : Field K\ninst✝² : Algebra F K\nq n : ℕ\nhF : ExpChar F q\nι : Type u_1\nv : ι → E\ninst✝¹ : FiniteDimensional F E\ninst✝ : Algebra.IsSeparable F E\nh : LinearIndependent F v\nh' : LinearIndependent (ι := { x // x ∈ Set.range v }) F Subtype.val\nι' : Set E := h'.extend ⋯\nb : Basis (↑ι') F E := Basis.extend h'\nthis : Fintype ↑ι' := fintypeBasisIndex b\nH : LinearIndependent F fun (x : ↑ι') => b x ^ q ^ n\n⊢ LinearIndependent F fun x => v x ^ q ^ n","state_after":"F : Type u\nE : Type v\ninst✝⁶ : Field F\ninst✝⁵ : Field E\ninst✝⁴ : Algebra F E\nK : Type w\ninst✝³ : Field K\ninst✝² : Algebra F K\nq n : ℕ\nhF : ExpChar F q\nι : Type u_1\nv : ι → E\ninst✝¹ : FiniteDimensional F E\ninst✝ : Algebra.IsSeparable F E\nh : LinearIndependent F v\nh' : LinearIndependent (ι := { x // x ∈ Set.range v }) F Subtype.val\nι' : Set E := h'.extend ⋯\nb : Basis (↑ι') F E := Basis.extend h'\nthis : Fintype ↑ι' := fintypeBasisIndex b\nH : LinearIndependent F fun (x : ↑ι') => b x ^ q ^ n\nf : ι → ↑ι' := fun i => ⟨v i, ⋯⟩\n⊢ LinearIndependent F fun x => v x ^ q ^ n","tactic":"let f (i : ι) : ι' := ⟨v i, h'.subset_extend _ ⟨i, rfl⟩⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"LinearIndependent.subset_extend","def_path":"Mathlib/LinearAlgebra/LinearIndependent.lean","def_pos":[1371,8],"def_end_pos":[1371,39]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"F : Type u\nE : Type v\ninst✝⁶ : Field F\ninst✝⁵ : Field E\ninst✝⁴ : Algebra F E\nK : Type w\ninst✝³ : Field K\ninst✝² : Algebra F K\nq n : ℕ\nhF : ExpChar F q\nι : Type u_1\nv : ι → E\ninst✝¹ : FiniteDimensional F E\ninst✝ : Algebra.IsSeparable F E\nh : LinearIndependent F v\nh' : LinearIndependent (ι := { x // x ∈ Set.range v }) F Subtype.val\nι' : Set E := h'.extend ⋯\nb : Basis (↑ι') F E := Basis.extend h'\nthis : Fintype ↑ι' := fintypeBasisIndex b\nH : LinearIndependent F fun (x : ↑ι') => b x ^ q ^ n\nf : ι → ↑ι' := fun i => ⟨v i, ⋯⟩\n⊢ LinearIndependent F fun x => v x ^ q ^ n","state_after":"case h.e'_4.h\nF : Type u\nE : Type v\ninst✝⁶ : Field F\ninst✝⁵ : Field E\ninst✝⁴ : Algebra F E\nK : Type w\ninst✝³ : Field K\ninst✝² : Algebra F K\nq n : ℕ\nhF : ExpChar F q\nι : Type u_1\nv : ι → E\ninst✝¹ : FiniteDimensional F E\ninst✝ : Algebra.IsSeparable F E\nh : LinearIndependent F v\nh' : LinearIndependent (ι := { x // x ∈ Set.range v }) F Subtype.val\nι' : Set E := h'.extend ⋯\nb : Basis (↑ι') F E := Basis.extend h'\nthis : Fintype ↑ι' := fintypeBasisIndex b\nH : LinearIndependent F fun (x : ↑ι') => b x ^ q ^ n\nf : ι → ↑ι' := fun i => ⟨v i, ⋯⟩\nx✝ : ι\n⊢ v x✝ ^ q ^ n = ((fun x => b x ^ q ^ n) ∘ f) x✝","tactic":"convert H.comp f fun _ _ heq ↦ h.injective (by simpa only [f, Subtype.mk.injEq] using heq)","premises":[{"full_name":"LinearIndependent.comp","def_path":"Mathlib/LinearAlgebra/LinearIndependent.lean","def_pos":[221,8],"def_end_pos":[221,30]},{"full_name":"LinearIndependent.injective","def_path":"Mathlib/LinearAlgebra/LinearIndependent.lean","def_pos":[525,8],"def_end_pos":[525,35]}]},{"state_before":"case h.e'_4.h\nF : Type u\nE : Type v\ninst✝⁶ : Field F\ninst✝⁵ : Field E\ninst✝⁴ : Algebra F E\nK : Type w\ninst✝³ : Field K\ninst✝² : Algebra F K\nq n : ℕ\nhF : ExpChar F q\nι : Type u_1\nv : ι → E\ninst✝¹ : FiniteDimensional F E\ninst✝ : Algebra.IsSeparable F E\nh : LinearIndependent F v\nh' : LinearIndependent (ι := { x // x ∈ Set.range v }) F Subtype.val\nι' : Set E := h'.extend ⋯\nb : Basis (↑ι') F E := Basis.extend h'\nthis : Fintype ↑ι' := fintypeBasisIndex b\nH : LinearIndependent F fun (x : ↑ι') => b x ^ q ^ n\nf : ι → ↑ι' := fun i => ⟨v i, ⋯⟩\nx✝ : ι\n⊢ v x✝ ^ q ^ n = ((fun x => b x ^ q ^ n) ∘ f) x✝","state_after":"no goals","tactic":"simp_rw [Function.comp_apply, b, Basis.extend_apply_self]","premises":[{"full_name":"Basis.extend_apply_self","def_path":"Mathlib/LinearAlgebra/Basis/VectorSpace.lean","def_pos":[53,8],"def_end_pos":[53,25]},{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Function.comp_apply","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[35,16],"def_end_pos":[35,35]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]}]}]}
+{"url":"Mathlib/Data/Rat/Defs.lean","commit":"","full_name":"Rat.mk'_zero","start":[86,0],"end":[86,82],"file_path":"Mathlib/Data/Rat/Defs.lean","tactics":[{"state_before":"q : ℚ\nd : ℕ\nh : d ≠ 0\nw : (Int.natAbs 0).Coprime d\n⊢ { num := 0, den := d, den_nz := h, reduced := w } = 0","state_after":"case e_den\nq : ℚ\nd : ℕ\nh : d ≠ 0\nw : (Int.natAbs 0).Coprime d\n⊢ d = 1","tactic":"congr","premises":[]},{"state_before":"case e_den\nq : ℚ\nd : ℕ\nh : d ≠ 0\nw : (Int.natAbs 0).Coprime d\n⊢ d = 1","state_after":"no goals","tactic":"simp_all","premises":[]}]}
+{"url":"Mathlib/Geometry/RingedSpace/OpenImmersion.lean","commit":"","full_name":"AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.lift_uniq","start":[503,0],"end":[504,59],"file_path":"Mathlib/Geometry/RingedSpace/OpenImmersion.lean","tactics":[{"state_before":"C : Type u\ninst✝ : Category.{v, u} C\nX Y Z : PresheafedSpace C\nf : X ⟶ Z\nhf : IsOpenImmersion f\ng : Y ⟶ Z\ns : PullbackCone f g\nH : Set.range ⇑g.base ⊆ Set.range ⇑f.base\nl : Y ⟶ X\nhl : l ≫ f = g\n⊢ l = lift f g H","state_after":"no goals","tactic":"rw [← cancel_mono f, hl, lift_fac]","premises":[{"full_name":"AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.lift_fac","def_path":"Mathlib/Geometry/RingedSpace/OpenImmersion.lean","def_pos":[500,8],"def_end_pos":[500,16]},{"full_name":"CategoryTheory.cancel_mono","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[263,8],"def_end_pos":[263,19]}]}]}
+{"url":"Mathlib/NumberTheory/Padics/RingHoms.lean","commit":"","full_name":"PadicInt.lift_self","start":[620,0],"end":[624,29],"file_path":"Mathlib/NumberTheory/Padics/RingHoms.lean","tactics":[{"state_before":"p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), (ZMod.castHom ⋯ (ZMod (p ^ k1))).comp (f k2) = f k1\nz : ℤ_[p]\n⊢ (lift ⋯) z = z","state_after":"p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), (ZMod.castHom ⋯ (ZMod (p ^ k1))).comp (f k2) = f k1\nz : ℤ_[p]\n⊢ (lift ⋯) z = (RingHom.id ℤ_[p]) z","tactic":"show _ = RingHom.id _ z","premises":[{"full_name":"RingHom.id","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[538,4],"def_end_pos":[538,6]}]},{"state_before":"p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), (ZMod.castHom ⋯ (ZMod (p ^ k1))).comp (f k2) = f k1\nz : ℤ_[p]\n⊢ (lift ⋯) z = (RingHom.id ℤ_[p]) z","state_after":"p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), (ZMod.castHom ⋯ (ZMod (p ^ k1))).comp (f k2) = f k1\nz : ℤ_[p]\n⊢ ∀ (n : ℕ), (toZModPow n).comp (RingHom.id ℤ_[p]) = toZModPow n","tactic":"rw [@lift_unique p _ ℤ_[p] _ _ zmod_cast_comp_toZModPow (RingHom.id ℤ_[p])]","premises":[{"full_name":"PadicInt","def_path":"Mathlib/NumberTheory/Padics/PadicIntegers.lean","def_pos":[56,4],"def_end_pos":[56,12]},{"full_name":"PadicInt.lift_unique","def_path":"Mathlib/NumberTheory/Padics/RingHoms.lean","def_pos":[610,8],"def_end_pos":[610,19]},{"full_name":"PadicInt.zmod_cast_comp_toZModPow","def_path":"Mathlib/NumberTheory/Padics/RingHoms.lean","def_pos":[397,8],"def_end_pos":[397,32]},{"full_name":"RingHom.id","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[538,4],"def_end_pos":[538,6]}]},{"state_before":"p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), (ZMod.castHom ⋯ (ZMod (p ^ k1))).comp (f k2) = f k1\nz : ℤ_[p]\n⊢ ∀ (n : ℕ), (toZModPow n).comp (RingHom.id ℤ_[p]) = toZModPow n","state_after":"p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), (ZMod.castHom ⋯ (ZMod (p ^ k1))).comp (f k2) = f k1\nz : ℤ_[p]\nn✝ : ℕ\n⊢ (toZModPow n✝).comp (RingHom.id ℤ_[p]) = toZModPow n✝","tactic":"intro","premises":[]},{"state_before":"p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), (ZMod.castHom ⋯ (ZMod (p ^ k1))).comp (f k2) = f k1\nz : ℤ_[p]\nn✝ : ℕ\n⊢ (toZModPow n✝).comp (RingHom.id ℤ_[p]) = toZModPow n✝","state_after":"no goals","tactic":"rw [RingHom.comp_id]","premises":[{"full_name":"RingHom.comp_id","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[580,8],"def_end_pos":[580,15]}]}]}
+{"url":"Mathlib/Topology/Gluing.lean","commit":"","full_name":"TopCat.GlueData.isOpen_iff","start":[99,0],"end":[110,31],"file_path":"Mathlib/Topology/Gluing.lean","tactics":[{"state_before":"D : GlueData\nU : Set ↑D.glued\n⊢ IsOpen U ↔ ∀ (i : D.J), IsOpen (⇑(D.ι i) ⁻¹' U)","state_after":"D : GlueData\nU : Set ↑D.glued\n⊢ IsOpen U ↔ ∀ (i : D.J), IsOpen (⇑(Multicoequalizer.π D.diagram i) ⁻¹' U)","tactic":"delta CategoryTheory.GlueData.ι","premises":[{"full_name":"CategoryTheory.GlueData.ι","def_path":"Mathlib/CategoryTheory/GlueData.lean","def_pos":[178,4],"def_end_pos":[178,5]}]},{"state_before":"D : GlueData\nU : Set ↑D.glued\n⊢ IsOpen U ↔ ∀ (i : D.J), IsOpen (⇑(Multicoequalizer.π D.diagram i) ⁻¹' U)","state_after":"D : GlueData\nU : Set ↑D.glued\n⊢ IsOpen U ↔ ∀ (i : D.J), IsOpen (⇑(Sigma.ι D.diagram.right i ≫ Multicoequalizer.sigmaπ D.diagram) ⁻¹' U)","tactic":"simp_rw [← Multicoequalizer.ι_sigmaπ 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U)","tactic":"rw [← (homeoOfIso (Multicoequalizer.isoCoequalizer 𝖣.diagram).symm).isOpen_preimage]","premises":[{"full_name":"CategoryTheory.GlueData.diagram","def_path":"Mathlib/CategoryTheory/GlueData.lean","def_pos":[127,4],"def_end_pos":[127,11]},{"full_name":"CategoryTheory.Iso.symm","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[84,4],"def_end_pos":[84,8]},{"full_name":"CategoryTheory.Limits.Multicoequalizer.isoCoequalizer","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean","def_pos":[818,4],"def_end_pos":[818,18]},{"full_name":"Homeomorph.isOpen_preimage","def_path":"Mathlib/Topology/Homeomorph.lean","def_pos":[315,8],"def_end_pos":[315,23]},{"full_name":"TopCat.homeoOfIso","def_path":"Mathlib/Topology/Category/TopCat/Basic.lean","def_pos":[142,4],"def_end_pos":[142,14]}]},{"state_before":"D : GlueData\nU : Set ↑D.glued\n⊢ IsOpen (⇑(homeoOfIso (Multicoequalizer.isoCoequalizer D.diagram).symm) ⁻¹' U) ↔\n    ∀ (i : D.J), IsOpen (⇑(Sigma.ι D.diagram.right i ≫ Multicoequalizer.sigmaπ D.diagram) ⁻¹' U)","state_after":"D : GlueData\nU : Set ↑D.glued\n⊢ IsOpen\n      (⇑(colimit.ι (parallelPair D.diagram.fstSigmaMap D.diagram.sndSigmaMap) WalkingParallelPair.one) ⁻¹'\n        (⇑(homeoOfIso (Multicoequalizer.isoCoequalizer D.diagram).symm) ⁻¹' U)) ↔\n    ∀ (i : D.J), IsOpen (⇑(Sigma.ι D.diagram.right i ≫ Multicoequalizer.sigmaπ D.diagram) ⁻¹' U)","tactic":"rw [coequalizer_isOpen_iff]","premises":[{"full_name":"TopCat.coequalizer_isOpen_iff","def_path":"Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean","def_pos":[433,8],"def_end_pos":[433,30]}]},{"state_before":"D : GlueData\nU : Set ↑D.glued\n⊢ IsOpen\n      (⇑(colimit.ι (parallelPair D.diagram.fstSigmaMap D.diagram.sndSigmaMap) WalkingParallelPair.one) ⁻¹'\n        (⇑(homeoOfIso (Multicoequalizer.isoCoequalizer D.diagram).symm) ⁻¹' U)) ↔\n    ∀ (i : D.J), IsOpen (⇑(Sigma.ι D.diagram.right i ≫ Multicoequalizer.sigmaπ D.diagram) ⁻¹' U)","state_after":"D : GlueData\nU : Set ↑D.glued\n⊢ IsOpen\n      (⇑(colimit.ι (parallelPair D.diagram.fstSigmaMap D.diagram.sndSigmaMap) WalkingParallelPair.one) ⁻¹'\n        (⇑(homeoOfIso (Multicoequalizer.isoCoequalizer D.diagram).symm) ⁻¹' U)) ↔\n    ∀ (i : D.J), IsOpen (⇑(Sigma.ι D.U i ≫ Multicoequalizer.sigmaπ D.diagram) ⁻¹' U)","tactic":"dsimp only [GlueData.diagram_l, GlueData.diagram_left, GlueData.diagram_r, GlueData.diagram_right,\n    parallelPair_obj_one]","premises":[{"full_name":"CategoryTheory.GlueData.diagram_l","def_path":"Mathlib/CategoryTheory/GlueData.lean","def_pos":[138,8],"def_end_pos":[138,17]},{"full_name":"CategoryTheory.GlueData.diagram_left","def_path":"Mathlib/CategoryTheory/GlueData.lean","def_pos":[162,8],"def_end_pos":[162,20]},{"full_name":"CategoryTheory.GlueData.diagram_r","def_path":"Mathlib/CategoryTheory/GlueData.lean","def_pos":[142,8],"def_end_pos":[142,17]},{"full_name":"CategoryTheory.GlueData.diagram_right","def_path":"Mathlib/CategoryTheory/GlueData.lean","def_pos":[166,8],"def_end_pos":[166,21]},{"full_name":"CategoryTheory.Limits.parallelPair_obj_one","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean","def_pos":[210,8],"def_end_pos":[210,28]}]},{"state_before":"D : GlueData\nU : Set ↑D.glued\n⊢ IsOpen\n      (⇑(colimit.ι (parallelPair D.diagram.fstSigmaMap D.diagram.sndSigmaMap) WalkingParallelPair.one) ⁻¹'\n        (⇑(homeoOfIso (Multicoequalizer.isoCoequalizer D.diagram).symm) ⁻¹' U)) ↔\n    ∀ (i : D.J), IsOpen (⇑(Sigma.ι D.U i ≫ Multicoequalizer.sigmaπ D.diagram) ⁻¹' U)","state_after":"D : GlueData\nU : Set ↑D.glued\n⊢ (∀ (j : Discrete D.J),\n      IsOpen\n        (⇑(colimit.ι (Discrete.functor D.U) j) ⁻¹'\n          (⇑(colimit.ι (parallelPair D.diagram.fstSigmaMap D.diagram.sndSigmaMap) WalkingParallelPair.one) ⁻¹'\n            (⇑(homeoOfIso (Multicoequalizer.isoCoequalizer D.diagram).symm) ⁻¹' U)))) ↔\n    ∀ (i : D.J), IsOpen (⇑(Sigma.ι D.U i ≫ Multicoequalizer.sigmaπ D.diagram) ⁻¹' U)","tactic":"rw [colimit_isOpen_iff.{_,u}]","premises":[{"full_name":"TopCat.colimit_isOpen_iff","def_path":"Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean","def_pos":[427,8],"def_end_pos":[427,26]}]},{"state_before":"D : GlueData\nU : Set ↑D.glued\n⊢ (∀ (j : Discrete D.J),\n      IsOpen\n        (⇑(colimit.ι (Discrete.functor D.U) j) ⁻¹'\n          (⇑(colimit.ι (parallelPair D.diagram.fstSigmaMap D.diagram.sndSigmaMap) WalkingParallelPair.one) ⁻¹'\n            (⇑(homeoOfIso (Multicoequalizer.isoCoequalizer D.diagram).symm) ⁻¹' U)))) ↔\n    ∀ (i : D.J), IsOpen (⇑(Sigma.ι D.U i ≫ Multicoequalizer.sigmaπ D.diagram) ⁻¹' U)","state_after":"case mp\nD : GlueData\nU : Set ↑D.glued\n⊢ (∀ (j : Discrete D.J),\n      IsOpen\n        (⇑(colimit.ι (Discrete.functor D.U) j) ⁻¹'\n          (⇑(colimit.ι (parallelPair D.diagram.fstSigmaMap D.diagram.sndSigmaMap) WalkingParallelPair.one) ⁻¹'\n            (⇑(homeoOfIso (Multicoequalizer.isoCoequalizer D.diagram).symm) ⁻¹' U)))) →\n    ∀ (i : D.J), IsOpen (⇑(Sigma.ι D.U i ≫ Multicoequalizer.sigmaπ D.diagram) ⁻¹' U)\n\ncase mpr\nD : GlueData\nU : Set ↑D.glued\n⊢ (∀ (i : D.J), IsOpen (⇑(Sigma.ι D.U i ≫ Multicoequalizer.sigmaπ D.diagram) ⁻¹' U)) →\n    ∀ (j : Discrete D.J),\n      IsOpen\n        (⇑(colimit.ι (Discrete.functor D.U) j) ⁻¹'\n          (⇑(colimit.ι (parallelPair D.diagram.fstSigmaMap D.diagram.sndSigmaMap) WalkingParallelPair.one) ⁻¹'\n            (⇑(homeoOfIso (Multicoequalizer.isoCoequalizer D.diagram).symm) ⁻¹' U)))","tactic":"constructor","premises":[]}]}
+{"url":"Mathlib/CategoryTheory/Triangulated/Functor.lean","commit":"","full_name":"CategoryTheory.Functor.mapTriangle_map_hom₂","start":[32,0],"end":[47,40],"file_path":"Mathlib/CategoryTheory/Triangulated/Functor.lean","tactics":[{"state_before":"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁷ : Category.{?u.284, u_1} C\ninst✝⁶ : Category.{?u.288, u_2} D\ninst✝⁵ : Category.{?u.292, u_3} E\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasShift D ℤ\ninst✝² : HasShift E ℤ\nF : C ⥤ D\ninst✝¹ : F.CommShift ℤ\nG : D ⥤ E\ninst✝ : G.CommShift ℤ\nX✝ Y✝ : Triangle C\nf : X✝ ⟶ Y✝\n⊢ ((fun T => Triangle.mk (F.map T.mor₁) (F.map T.mor₂) (F.map T.mor₃ ≫ (F.commShiftIso 1).hom.app T.obj₁)) X✝).mor₁ ≫\n      F.map f.hom₂ =\n    F.map f.hom₁ ≫\n      ((fun T => Triangle.mk (F.map T.mor₁) (F.map T.mor₂) (F.map T.mor₃ ≫ (F.commShiftIso 1).hom.app T.obj₁)) Y✝).mor₁","state_after":"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁷ : Category.{?u.284, u_1} C\ninst✝⁶ : Category.{?u.288, u_2} D\ninst✝⁵ : Category.{?u.292, u_3} E\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasShift D ℤ\ninst✝² : HasShift E ℤ\nF : C ⥤ D\ninst✝¹ : F.CommShift ℤ\nG : D ⥤ E\ninst✝ : G.CommShift ℤ\nX✝ Y✝ : Triangle C\nf : X✝ ⟶ Y✝\n⊢ F.map X✝.mor₁ ≫ F.map f.hom₂ = F.map f.hom₁ ≫ F.map Y✝.mor₁","tactic":"dsimp","premises":[]},{"state_before":"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁷ : Category.{?u.284, u_1} C\ninst✝⁶ : Category.{?u.288, u_2} D\ninst✝⁵ : Category.{?u.292, u_3} E\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasShift D ℤ\ninst✝² : HasShift E ℤ\nF : C ⥤ D\ninst✝¹ : F.CommShift ℤ\nG : D ⥤ E\ninst✝ : G.CommShift ℤ\nX✝ Y✝ : Triangle C\nf : X✝ ⟶ Y✝\n⊢ F.map X✝.mor₁ ≫ F.map f.hom₂ = F.map f.hom₁ ≫ F.map Y✝.mor₁","state_after":"no goals","tactic":"simp only [← F.map_comp, f.comm₁]","premises":[{"full_name":"CategoryTheory.Functor.map_comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[41,2],"def_end_pos":[41,10]},{"full_name":"CategoryTheory.Pretriangulated.TriangleMorphism.comm₁","def_path":"Mathlib/CategoryTheory/Triangulated/Basic.lean","def_pos":[109,2],"def_end_pos":[109,7]}]},{"state_before":"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁷ : Category.{?u.284, u_1} C\ninst✝⁶ : Category.{?u.288, u_2} D\ninst✝⁵ : Category.{?u.292, u_3} E\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasShift D ℤ\ninst✝² : HasShift E ℤ\nF : C ⥤ D\ninst✝¹ : F.CommShift ℤ\nG : D ⥤ E\ninst✝ : G.CommShift ℤ\nX✝ Y✝ : Triangle C\nf : X✝ ⟶ Y✝\n⊢ ((fun T => Triangle.mk (F.map T.mor₁) (F.map T.mor₂) (F.map T.mor₃ ≫ (F.commShiftIso 1).hom.app T.obj₁)) X✝).mor₂ ≫\n      F.map f.hom₃ =\n    F.map f.hom₂ ≫\n      ((fun T => Triangle.mk (F.map T.mor₁) (F.map T.mor₂) (F.map T.mor₃ ≫ (F.commShiftIso 1).hom.app T.obj₁)) Y✝).mor₂","state_after":"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁷ : Category.{?u.284, u_1} C\ninst✝⁶ : Category.{?u.288, u_2} D\ninst✝⁵ : Category.{?u.292, u_3} E\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasShift D ℤ\ninst✝² : HasShift E ℤ\nF : C ⥤ D\ninst✝¹ : F.CommShift ℤ\nG : D ⥤ E\ninst✝ : G.CommShift ℤ\nX✝ Y✝ : Triangle C\nf : X✝ ⟶ Y✝\n⊢ F.map X✝.mor₂ ≫ F.map f.hom₃ = F.map f.hom₂ ≫ F.map Y✝.mor₂","tactic":"dsimp","premises":[]},{"state_before":"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁷ : Category.{?u.284, u_1} C\ninst✝⁶ : Category.{?u.288, u_2} D\ninst✝⁵ : Category.{?u.292, u_3} E\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasShift D ℤ\ninst✝² : HasShift E ℤ\nF : C ⥤ D\ninst✝¹ : F.CommShift ℤ\nG : D ⥤ E\ninst✝ : G.CommShift ℤ\nX✝ Y✝ : Triangle C\nf : X✝ ⟶ Y✝\n⊢ F.map X✝.mor₂ ≫ F.map f.hom₃ = F.map f.hom₂ ≫ F.map Y✝.mor₂","state_after":"no goals","tactic":"simp only [← F.map_comp, f.comm₂]","premises":[{"full_name":"CategoryTheory.Functor.map_comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[41,2],"def_end_pos":[41,10]},{"full_name":"CategoryTheory.Pretriangulated.TriangleMorphism.comm₂","def_path":"Mathlib/CategoryTheory/Triangulated/Basic.lean","def_pos":[111,2],"def_end_pos":[111,7]}]},{"state_before":"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁷ : Category.{?u.284, u_1} C\ninst✝⁶ : Category.{?u.288, u_2} D\ninst✝⁵ : Category.{?u.292, u_3} E\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasShift D ℤ\ninst✝² : HasShift E ℤ\nF : C ⥤ D\ninst✝¹ : F.CommShift ℤ\nG : D ⥤ E\ninst✝ : G.CommShift ℤ\nX✝ Y✝ : Triangle C\nf : X✝ ⟶ Y✝\n⊢ ((fun T => Triangle.mk (F.map T.mor₁) (F.map T.mor₂) (F.map T.mor₃ ≫ (F.commShiftIso 1).hom.app T.obj₁)) X✝).mor₃ ≫\n      (shiftFunctor D 1).map (F.map f.hom₁) =\n    F.map f.hom₃ ≫\n      ((fun T => Triangle.mk (F.map T.mor₁) (F.map T.mor₂) (F.map T.mor₃ ≫ (F.commShiftIso 1).hom.app T.obj₁)) Y✝).mor₃","state_after":"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁷ : Category.{?u.284, u_1} C\ninst✝⁶ : Category.{?u.288, u_2} D\ninst✝⁵ : Category.{?u.292, u_3} E\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasShift D ℤ\ninst✝² : HasShift E ℤ\nF : C ⥤ D\ninst✝¹ : F.CommShift ℤ\nG : D ⥤ E\ninst✝ : G.CommShift ℤ\nX✝ Y✝ : Triangle C\nf : X✝ ⟶ Y✝\n⊢ (F.map X✝.mor₃ ≫ (F.commShiftIso 1).hom.app X✝.obj₁) ≫ (shiftFunctor D 1).map (F.map f.hom₁) =\n    F.map f.hom₃ ≫ F.map Y✝.mor₃ ≫ (F.commShiftIso 1).hom.app Y✝.obj₁","tactic":"dsimp [Functor.comp]","premises":[{"full_name":"CategoryTheory.Functor.comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[100,4],"def_end_pos":[100,8]}]},{"state_before":"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁷ : Category.{?u.284, u_1} C\ninst✝⁶ : Category.{?u.288, u_2} D\ninst✝⁵ : Category.{?u.292, u_3} E\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasShift D ℤ\ninst✝² : HasShift E ℤ\nF : C ⥤ D\ninst✝¹ : F.CommShift ℤ\nG : D ⥤ E\ninst✝ : G.CommShift ℤ\nX✝ Y✝ : Triangle C\nf : X✝ ⟶ Y✝\n⊢ (F.map X✝.mor₃ ≫ (F.commShiftIso 1).hom.app X✝.obj₁) ≫ (shiftFunctor D 1).map (F.map f.hom₁) =\n    F.map f.hom₃ ≫ F.map Y✝.mor₃ ≫ (F.commShiftIso 1).hom.app Y✝.obj₁","state_after":"no goals","tactic":"simp only [Category.assoc, ← NatTrans.naturality,\n          ← F.map_comp_assoc, f.comm₃]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Functor.map_comp_assoc","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[57,6],"def_end_pos":[57,28]},{"full_name":"CategoryTheory.NatTrans.naturality","def_path":"Mathlib/CategoryTheory/NatTrans.lean","def_pos":[50,2],"def_end_pos":[50,12]},{"full_name":"CategoryTheory.Pretriangulated.TriangleMorphism.comm₃","def_path":"Mathlib/CategoryTheory/Triangulated/Basic.lean","def_pos":[113,2],"def_end_pos":[113,7]}]}]}
+{"url":"Mathlib/Probability/Kernel/Disintegration/Density.lean","commit":"","full_name":"ProbabilityTheory.Kernel.densityProcess_le_one","start":[176,0],"end":[180,59],"file_path":"Mathlib/Probability/Kernel/Disintegration/Density.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝ : CountablyGenerated γ\nκ : Kernel α (γ × β)\nν : Kernel α γ\nhκν : κ.fst ≤ ν\nn : ℕ\na : α\nx : γ\ns : Set β\n⊢ κ.densityProcess ν n a x s ≤ 1","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝ : CountablyGenerated γ\nκ : Kernel α (γ × β)\nν : Kernel α γ\nhκν : κ.fst ≤ ν\nn : ℕ\na : α\nx : γ\ns : Set β\n⊢ (κ a) (countablePartitionSet n x ×ˢ s) ≤ ENNReal.ofReal 1 * (ν a) (countablePartitionSet n x)","tactic":"refine ENNReal.toReal_le_of_le_ofReal zero_le_one (ENNReal.div_le_of_le_mul ?_)","premises":[{"full_name":"ENNReal.div_le_of_le_mul","def_path":"Mathlib/Data/ENNReal/Inv.lean","def_pos":[279,8],"def_end_pos":[279,24]},{"full_name":"ENNReal.toReal_le_of_le_ofReal","def_path":"Mathlib/Data/ENNReal/Real.lean","def_pos":[307,8],"def_end_pos":[307,30]},{"full_name":"zero_le_one","def_path":"Mathlib/Algebra/Order/ZeroLEOne.lean","def_pos":[23,14],"def_end_pos":[23,25]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝ : CountablyGenerated γ\nκ : Kernel α (γ × β)\nν : Kernel α γ\nhκν : κ.fst ≤ ν\nn : ℕ\na : α\nx : γ\ns : Set β\n⊢ (κ a) (countablePartitionSet n x ×ˢ s) ≤ ENNReal.ofReal 1 * (ν a) (countablePartitionSet n x)","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝ : CountablyGenerated γ\nκ : Kernel α (γ × β)\nν : Kernel α γ\nhκν : κ.fst ≤ ν\nn : ℕ\na : α\nx : γ\ns : Set β\n⊢ (κ a) (countablePartitionSet n x ×ˢ s) ≤ (ν a) (countablePartitionSet n x)","tactic":"rw [ENNReal.ofReal_one, one_mul]","premises":[{"full_name":"ENNReal.ofReal_one","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[242,16],"def_end_pos":[242,26]},{"full_name":"one_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[409,8],"def_end_pos":[409,15]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝ : CountablyGenerated γ\nκ : Kernel α (γ × β)\nν : Kernel α γ\nhκν : κ.fst ≤ ν\nn : ℕ\na : α\nx : γ\ns : Set β\n⊢ (κ a) (countablePartitionSet n x ×ˢ s) ≤ (ν a) (countablePartitionSet n x)","state_after":"no goals","tactic":"exact meas_countablePartitionSet_le_of_fst_le hκν n a x s","premises":[{"full_name":"ProbabilityTheory.Kernel.meas_countablePartitionSet_le_of_fst_le","def_path":"Mathlib/Probability/Kernel/Disintegration/Density.lean","def_pos":[165,6],"def_end_pos":[165,45]}]}]}
+{"url":"Mathlib/Topology/DiscreteQuotient.lean","commit":"","full_name":"DiscreteQuotient.map_ofLE","start":[300,0],"end":[304,5],"file_path":"Mathlib/Topology/DiscreteQuotient.lean","tactics":[{"state_before":"α : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nS : DiscreteQuotient X\nf : C(X, Y)\nA A' : DiscreteQuotient X\nB B' : DiscreteQuotient Y\ng : C(Y, Z)\nC : DiscreteQuotient Z\ncond : LEComap f A B\nh : A' ≤ A\nc : Quotient A'.toSetoid\n⊢ map f cond (ofLE h c) = map f ⋯ c","state_after":"case mk\nα : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nS : DiscreteQuotient X\nf : C(X, Y)\nA A' : DiscreteQuotient X\nB B' : DiscreteQuotient Y\ng : C(Y, Z)\nC : DiscreteQuotient Z\ncond : LEComap f A B\nh : A' ≤ A\nc : Quotient A'.toSetoid\na✝ : X\n⊢ map f cond (ofLE h (Quot.mk Setoid.r a✝)) = map f ⋯ (Quot.mk Setoid.r a✝)","tactic":"rcases c with ⟨⟩","premises":[]},{"state_before":"case mk\nα : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nS : DiscreteQuotient X\nf : C(X, Y)\nA A' : DiscreteQuotient X\nB B' : DiscreteQuotient Y\ng : C(Y, Z)\nC : DiscreteQuotient Z\ncond : LEComap f A B\nh : A' ≤ A\nc : Quotient A'.toSetoid\na✝ : X\n⊢ map f cond (ofLE h (Quot.mk Setoid.r a✝)) = map f ⋯ (Quot.mk Setoid.r a✝)","state_after":"no goals","tactic":"rfl","premises":[]}]}
+{"url":"Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean","commit":"","full_name":"Orientation.oangle_sign_smul_add_right","start":[792,0],"end":[824,59],"file_path":"Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean","tactics":[{"state_before":"V : Type u_1\nV' : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : NormedAddCommGroup V'\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : InnerProductSpace ℝ V'\ninst✝¹ : Fact (finrank ℝ V = 2)\ninst✝ : Fact (finrank ℝ V' = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nr : ℝ\n⊢ (o.oangle x (r • x + y)).sign = (o.oangle x y).sign","state_after":"case pos\nV : Type u_1\nV' : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : NormedAddCommGroup V'\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : InnerProductSpace ℝ V'\ninst✝¹ : Fact (finrank ℝ V = 2)\ninst✝ : Fact (finrank ℝ V' = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nr : ℝ\nh : o.oangle x y = 0 ∨ o.oangle x y = ↑π\n⊢ (o.oangle x (r • x + y)).sign = (o.oangle x y).sign\n\ncase neg\nV : Type u_1\nV' : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : NormedAddCommGroup V'\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : InnerProductSpace ℝ V'\ninst✝¹ : Fact (finrank ℝ V = 2)\ninst✝ : Fact (finrank ℝ V' = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nr : ℝ\nh : ¬(o.oangle x y = 0 ∨ o.oangle x y = ↑π)\n⊢ (o.oangle x (r • x + y)).sign = (o.oangle x y).sign","tactic":"by_cases h : o.oangle x y = 0 ∨ o.oangle x y = π","premises":[{"full_name":"Or","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[532,10],"def_end_pos":[532,12]},{"full_name":"Orientation.oangle","def_path":"Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean","def_pos":[51,4],"def_end_pos":[51,10]},{"full_name":"Real.pi","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","def_pos":[119,28],"def_end_pos":[119,30]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case neg\nV : Type u_1\nV' : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : NormedAddCommGroup V'\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : InnerProductSpace ℝ V'\ninst✝¹ : Fact (finrank ℝ V = 2)\ninst✝ : Fact (finrank ℝ V' = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nr : ℝ\nh : ¬(o.oangle x y = 0 ∨ o.oangle x y = ↑π)\n⊢ (o.oangle x (r • x + y)).sign = (o.oangle x y).sign","state_after":"case neg\nV : Type u_1\nV' : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : NormedAddCommGroup V'\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : InnerProductSpace ℝ V'\ninst✝¹ : Fact (finrank ℝ V = 2)\ninst✝ : Fact (finrank ℝ V' = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nr : ℝ\nh : ¬(o.oangle x y = 0 ∨ o.oangle x y = ↑π)\nh' : ∀ (r' : ℝ), o.oangle x (r' • x + y) ≠ 0 ∧ o.oangle x (r' • x + y) ≠ ↑π\n⊢ (o.oangle x (r • x + y)).sign = (o.oangle x y).sign","tactic":"have h' : ∀ r' : ℝ, o.oangle x (r' • x + y) ≠ 0 ∧ o.oangle x (r' • x + y) ≠ π := by\n    intro r'\n    rwa [← o.oangle_smul_add_right_eq_zero_or_eq_pi_iff r', not_or] at h","premises":[{"full_name":"And","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[516,10],"def_end_pos":[516,13]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Orientation.oangle","def_path":"Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean","def_pos":[51,4],"def_end_pos":[51,10]},{"full_name":"Orientation.oangle_smul_add_right_eq_zero_or_eq_pi_iff","def_path":"Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean","def_pos":[754,8],"def_end_pos":[754,50]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Real.pi","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","def_pos":[119,28],"def_end_pos":[119,30]},{"full_name":"not_or","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[134,16],"def_end_pos":[134,22]}]},{"state_before":"case neg\nV : Type u_1\nV' : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : NormedAddCommGroup V'\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : InnerProductSpace ℝ V'\ninst✝¹ : Fact (finrank ℝ V = 2)\ninst✝ : Fact (finrank ℝ V' = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nr : ℝ\nh : ¬(o.oangle x y = 0 ∨ o.oangle x y = ↑π)\nh' : ∀ (r' : ℝ), o.oangle x (r' • x + y) ≠ 0 ∧ o.oangle x (r' • x + y) ≠ ↑π\n⊢ (o.oangle x (r • x + y)).sign = (o.oangle x y).sign","state_after":"case neg\nV : Type u_1\nV' : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : NormedAddCommGroup V'\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : InnerProductSpace ℝ V'\ninst✝¹ : Fact (finrank ℝ V = 2)\ninst✝ : Fact (finrank ℝ V' = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nr : ℝ\nh : ¬(o.oangle x y = 0 ∨ o.oangle x y = ↑π)\nh' : ∀ (r' : ℝ), o.oangle x (r' • x + y) ≠ 0 ∧ o.oangle x (r' • x + y) ≠ ↑π\ns : Set (V × V) := (fun r' => (x, r' • x + y)) '' Set.univ\n⊢ (o.oangle x (r • x + y)).sign = (o.oangle x y).sign","tactic":"let s : Set (V × V) := (fun r' : ℝ => (x, r' • x + y)) '' Set.univ","premises":[{"full_name":"Prod","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[479,10],"def_end_pos":[479,14]},{"full_name":"Prod.mk","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[481,2],"def_end_pos":[481,4]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Set","def_path":"Mathlib/Init/Set.lean","def_pos":[53,4],"def_end_pos":[53,7]},{"full_name":"Set.image","def_path":"Mathlib/Init/Set.lean","def_pos":[208,4],"def_end_pos":[208,9]},{"full_name":"Set.univ","def_path":"Mathlib/Init/Set.lean","def_pos":[157,4],"def_end_pos":[157,8]}]},{"state_before":"case neg\nV : Type u_1\nV' : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : NormedAddCommGroup V'\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : InnerProductSpace ℝ V'\ninst✝¹ : Fact (finrank ℝ V = 2)\ninst✝ : Fact (finrank ℝ V' = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nr : ℝ\nh : ¬(o.oangle x y = 0 ∨ o.oangle x y = ↑π)\nh' : ∀ (r' : ℝ), o.oangle x (r' • x + y) ≠ 0 ∧ o.oangle x (r' • x + y) ≠ ↑π\ns : Set (V × V) := (fun r' => (x, r' • x + y)) '' Set.univ\n⊢ (o.oangle x (r • x + y)).sign = (o.oangle x y).sign","state_after":"case neg\nV : Type u_1\nV' : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : NormedAddCommGroup V'\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : InnerProductSpace ℝ V'\ninst✝¹ : Fact (finrank ℝ V = 2)\ninst✝ : Fact (finrank ℝ V' = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nr : ℝ\nh : ¬(o.oangle x y = 0 ∨ o.oangle x y = ↑π)\nh' : ∀ (r' : ℝ), o.oangle x (r' • x + y) ≠ 0 ∧ o.oangle x (r' • x + y) ≠ ↑π\ns : Set (V × V) := (fun r' => (x, r' • x + y)) '' Set.univ\nhc : IsConnected s\n⊢ (o.oangle x (r • x + y)).sign = (o.oangle x y).sign","tactic":"have hc : IsConnected s := isConnected_univ.image _ (continuous_const.prod_mk\n    ((continuous_id.smul continuous_const).add continuous_const)).continuousOn","premises":[{"full_name":"Continuous.add","def_path":"Mathlib/Topology/Algebra/Monoid.lean","def_pos":[93,2],"def_end_pos":[93,13]},{"full_name":"Continuous.continuousOn","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[810,8],"def_end_pos":[810,31]},{"full_name":"Continuous.prod_mk","def_path":"Mathlib/Topology/Constructions.lean","def_pos":[370,8],"def_end_pos":[370,26]},{"full_name":"Continuous.smul","def_path":"Mathlib/Topology/Algebra/MulAction.lean","def_pos":[119,8],"def_end_pos":[119,23]},{"full_name":"IsConnected","def_path":"Mathlib/Topology/Connected/Basic.lean","def_pos":[54,4],"def_end_pos":[54,15]},{"full_name":"IsConnected.image","def_path":"Mathlib/Topology/Connected/Basic.lean","def_pos":[299,18],"def_end_pos":[299,35]},{"full_name":"continuous_const","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1436,8],"def_end_pos":[1436,24]},{"full_name":"continuous_id","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1382,8],"def_end_pos":[1382,21]},{"full_name":"isConnected_univ","def_path":"Mathlib/Topology/Connected/Basic.lean","def_pos":[620,8],"def_end_pos":[620,24]}]},{"state_before":"case neg\nV : Type u_1\nV' : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : NormedAddCommGroup V'\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : InnerProductSpace ℝ V'\ninst✝¹ : Fact (finrank ℝ V = 2)\ninst✝ : Fact (finrank ℝ V' = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nr : ℝ\nh : ¬(o.oangle x y = 0 ∨ o.oangle x y = ↑π)\nh' : ∀ (r' : ℝ), o.oangle x (r' • x + y) ≠ 0 ∧ o.oangle x (r' • x + y) ≠ ↑π\ns : Set (V × V) := (fun r' => (x, r' • x + y)) '' Set.univ\nhc : IsConnected s\nhf : ContinuousOn (fun z => o.oangle z.1 z.2) s\n⊢ (o.oangle x (r • x + y)).sign = (o.oangle x y).sign","state_after":"case neg\nV : Type u_1\nV' : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : NormedAddCommGroup V'\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : InnerProductSpace ℝ V'\ninst✝¹ : Fact (finrank ℝ V = 2)\ninst✝ : Fact (finrank ℝ V' = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nr : ℝ\nh : ¬(o.oangle x y = 0 ∨ o.oangle x y = ↑π)\nh' : ∀ (r' : ℝ), o.oangle x (r' • x + y) ≠ 0 ∧ o.oangle x (r' • x + y) ≠ ↑π\ns : Set (V × V) := (fun r' => (x, r' • x + y)) '' Set.univ\nhc : IsConnected s\nhf : ContinuousOn (fun z => o.oangle z.1 z.2) s\nhs : ∀ z ∈ s, o.oangle z.1 z.2 ≠ 0 ∧ o.oangle z.1 z.2 ≠ ↑π\n⊢ (o.oangle x (r • x + y)).sign = (o.oangle x y).sign","tactic":"have hs : ∀ z : V × V, z ∈ s → o.oangle z.1 z.2 ≠ 0 ∧ o.oangle z.1 z.2 ≠ π := by\n    intro z hz\n    simp_rw [s, Set.mem_image] at hz\n    obtain ⟨r', -, rfl⟩ := hz\n    exact h' r'","premises":[{"full_name":"And","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[516,10],"def_end_pos":[516,13]},{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Orientation.oangle","def_path":"Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean","def_pos":[51,4],"def_end_pos":[51,10]},{"full_name":"Prod","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[479,10],"def_end_pos":[479,14]},{"full_name":"Prod.fst","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[483,2],"def_end_pos":[483,5]},{"full_name":"Prod.snd","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[485,2],"def_end_pos":[485,5]},{"full_name":"Real.pi","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","def_pos":[119,28],"def_end_pos":[119,30]},{"full_name":"Set.mem_image","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[118,8],"def_end_pos":[118,17]}]},{"state_before":"case neg\nV : Type u_1\nV' : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : NormedAddCommGroup V'\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : InnerProductSpace ℝ V'\ninst✝¹ : Fact (finrank ℝ V = 2)\ninst✝ : Fact (finrank ℝ V' = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nr : ℝ\nh : ¬(o.oangle x y = 0 ∨ o.oangle x y = ↑π)\nh' : ∀ (r' : ℝ), o.oangle x (r' • x + y) ≠ 0 ∧ o.oangle x (r' • x + y) ≠ ↑π\ns : Set (V × V) := (fun r' => (x, r' • x + y)) '' Set.univ\nhc : IsConnected s\nhf : ContinuousOn (fun z => o.oangle z.1 z.2) s\nhs : ∀ z ∈ s, o.oangle z.1 z.2 ≠ 0 ∧ o.oangle z.1 z.2 ≠ ↑π\n⊢ (o.oangle x (r • x + y)).sign = (o.oangle x y).sign","state_after":"case neg\nV : Type u_1\nV' : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : NormedAddCommGroup V'\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : InnerProductSpace ℝ V'\ninst✝¹ : Fact (finrank ℝ V = 2)\ninst✝ : Fact (finrank ℝ V' = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nr : ℝ\nh : ¬(o.oangle x y = 0 ∨ o.oangle x y = ↑π)\nh' : ∀ (r' : ℝ), o.oangle x (r' • x + y) ≠ 0 ∧ o.oangle x (r' • x + y) ≠ ↑π\ns : Set (V × V) := (fun r' => (x, r' • x + y)) '' Set.univ\nhc : IsConnected s\nhf : ContinuousOn (fun z => o.oangle z.1 z.2) s\nhs : ∀ z ∈ s, o.oangle z.1 z.2 ≠ 0 ∧ o.oangle z.1 z.2 ≠ ↑π\nhx : (x, y) ∈ s\n⊢ (o.oangle x (r • x + y)).sign = (o.oangle x y).sign","tactic":"have hx : (x, y) ∈ s := by\n    convert Set.mem_image_of_mem (fun r' : ℝ => (x, r' • x + y)) (Set.mem_univ 0)\n    simp","premises":[{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Prod.mk","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[481,2],"def_end_pos":[481,4]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Set.mem_image_of_mem","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[122,8],"def_end_pos":[122,24]},{"full_name":"Set.mem_univ","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[80,28],"def_end_pos":[80,36]}]},{"state_before":"case neg\nV : Type u_1\nV' : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : NormedAddCommGroup V'\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : InnerProductSpace ℝ V'\ninst✝¹ : Fact (finrank ℝ V = 2)\ninst✝ : Fact (finrank ℝ V' = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nr : ℝ\nh : ¬(o.oangle x y = 0 ∨ o.oangle x y = ↑π)\nh' : ∀ (r' : ℝ), o.oangle x (r' • x + y) ≠ 0 ∧ o.oangle x (r' • x + y) ≠ ↑π\ns : Set (V × V) := (fun r' => (x, r' • x + y)) '' Set.univ\nhc : IsConnected s\nhf : ContinuousOn (fun z => o.oangle z.1 z.2) s\nhs : ∀ z ∈ s, o.oangle z.1 z.2 ≠ 0 ∧ o.oangle z.1 z.2 ≠ ↑π\nhx : (x, y) ∈ s\n⊢ (o.oangle x (r • x + y)).sign = (o.oangle x y).sign","state_after":"case neg\nV : Type u_1\nV' : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : NormedAddCommGroup V'\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : InnerProductSpace ℝ V'\ninst✝¹ : Fact (finrank ℝ V = 2)\ninst✝ : Fact (finrank ℝ V' = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nr : ℝ\nh : ¬(o.oangle x y = 0 ∨ o.oangle x y = ↑π)\nh' : ∀ (r' : ℝ), o.oangle x (r' • x + y) ≠ 0 ∧ o.oangle x (r' • x + y) ≠ ↑π\ns : Set (V × V) := (fun r' => (x, r' • x + y)) '' Set.univ\nhc : IsConnected s\nhf : ContinuousOn (fun z => o.oangle z.1 z.2) s\nhs : ∀ z ∈ s, o.oangle z.1 z.2 ≠ 0 ∧ o.oangle z.1 z.2 ≠ ↑π\nhx : (x, y) ∈ s\nhy : (x, r • x + y) ∈ s\n⊢ (o.oangle x (r • x + y)).sign = (o.oangle x y).sign","tactic":"have hy : (x, r • x + y) ∈ s := Set.mem_image_of_mem _ (Set.mem_univ _)","premises":[{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Prod.mk","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[481,2],"def_end_pos":[481,4]},{"full_name":"Set.mem_image_of_mem","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[122,8],"def_end_pos":[122,24]},{"full_name":"Set.mem_univ","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[80,28],"def_end_pos":[80,36]}]},{"state_before":"case neg\nV : Type u_1\nV' : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : NormedAddCommGroup V'\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : InnerProductSpace ℝ V'\ninst✝¹ : Fact (finrank ℝ V = 2)\ninst✝ : Fact (finrank ℝ V' = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nr : ℝ\nh : ¬(o.oangle x y = 0 ∨ o.oangle x y = ↑π)\nh' : ∀ (r' : ℝ), o.oangle x (r' • x + y) ≠ 0 ∧ o.oangle x (r' • x + y) ≠ ↑π\ns : Set (V × V) := (fun r' => (x, r' • x + y)) '' Set.univ\nhc : IsConnected s\nhf : ContinuousOn (fun z => o.oangle z.1 z.2) s\nhs : ∀ z ∈ s, o.oangle z.1 z.2 ≠ 0 ∧ o.oangle z.1 z.2 ≠ ↑π\nhx : (x, y) ∈ s\nhy : (x, r • x + y) ∈ s\n⊢ (o.oangle x (r • x + y)).sign = (o.oangle x y).sign","state_after":"no goals","tactic":"convert Real.Angle.sign_eq_of_continuousOn hc hf hs hx hy","premises":[{"full_name":"Real.Angle.sign_eq_of_continuousOn","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean","def_pos":[886,8],"def_end_pos":[886,31]}]}]}
+{"url":"Mathlib/Data/List/Basic.lean","commit":"","full_name":"List.get_eq_get?","start":[575,0],"end":[577,23],"file_path":"Mathlib/Data/List/Basic.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ l : List α\ni : Fin l.length\n⊢ (l.get? ↑i).isSome = true","state_after":"no goals","tactic":"simp [getElem?_eq_getElem]","premises":[{"full_name":"List.getElem?_eq_getElem","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean","def_pos":[178,16],"def_end_pos":[178,35]}]},{"state_before":"ι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ l : List α\ni : Fin l.length\n⊢ l.get i = (l.get? ↑i).get ⋯","state_after":"no goals","tactic":"simp [getElem_eq_iff]","premises":[{"full_name":"List.getElem_eq_iff","def_path":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","def_pos":[246,8],"def_end_pos":[246,22]}]}]}
+{"url":"Mathlib/Order/UpperLower/Basic.lean","commit":"","full_name":"IsUpperSet.total","start":[420,0],"end":[426,26],"file_path":"Mathlib/Order/UpperLower/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nκ : ι → Sort u_5\ninst✝ : LinearOrder α\ns t : Set α\nhs : IsUpperSet s\nht : IsUpperSet t\n⊢ s ⊆ t ∨ t ⊆ s","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nκ : ι → Sort u_5\ninst✝ : LinearOrder α\ns t : Set α\nhs : IsUpperSet s\nht : IsUpperSet t\nh : ¬s ⊆ t ∧ ¬t ⊆ s\n⊢ False","tactic":"by_contra! h","premises":[{"full_name":"Classical.byContradiction","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[129,8],"def_end_pos":[129,23]},{"full_name":"Not","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[224,4],"def_end_pos":[224,7]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nκ : ι → Sort u_5\ninst✝ : LinearOrder α\ns t : Set α\nhs : IsUpperSet s\nht : IsUpperSet t\nh : ¬s ⊆ t ∧ ¬t ⊆ s\n⊢ False","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nκ : ι → Sort u_5\ninst✝ : LinearOrder α\ns t : Set α\nhs : IsUpperSet s\nht : IsUpperSet t\nh : (∃ a ∈ s, a ∉ t) ∧ ∃ a ∈ t, a ∉ s\n⊢ False","tactic":"simp_rw [Set.not_subset] at h","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Set.not_subset","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[318,8],"def_end_pos":[318,18]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nκ : ι → Sort u_5\ninst✝ : LinearOrder α\ns t : Set α\nhs : IsUpperSet s\nht : IsUpperSet t\nh : (∃ a ∈ s, a ∉ t) ∧ ∃ a ∈ t, a ∉ s\n⊢ False","state_after":"case intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nκ : ι → Sort u_5\ninst✝ : LinearOrder α\ns t : Set α\nhs : IsUpperSet s\nht : IsUpperSet t\na : α\nhas : a ∈ s\nhat : a ∉ t\nb : α\nhbt : b ∈ t\nhbs : b ∉ s\n⊢ False","tactic":"obtain ⟨⟨a, has, hat⟩, b, hbt, hbs⟩ := h","premises":[]},{"state_before":"case intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nκ : ι → Sort u_5\ninst✝ : LinearOrder α\ns t : Set α\nhs : IsUpperSet s\nht : IsUpperSet t\na : α\nhas : a ∈ s\nhat : a ∉ t\nb : α\nhbt : b ∈ t\nhbs : b ∉ s\n⊢ False","state_after":"case intro.intro.intro.intro.intro.inl\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nκ : ι → Sort u_5\ninst✝ : LinearOrder α\ns t : Set α\nhs : IsUpperSet s\nht : IsUpperSet t\na : α\nhas : a ∈ s\nhat : a ∉ t\nb : α\nhbt : b ∈ t\nhbs : b ∉ s\nhab : a ≤ b\n⊢ False\n\ncase intro.intro.intro.intro.intro.inr\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nκ : ι → Sort u_5\ninst✝ : LinearOrder α\ns t : Set α\nhs : IsUpperSet s\nht : IsUpperSet t\na : α\nhas : a ∈ s\nhat : a ∉ t\nb : α\nhbt : b ∈ t\nhbs : b ∉ s\nhba : b ≤ a\n⊢ False","tactic":"obtain hab | hba := le_total a b","premises":[{"full_name":"le_total","def_path":"Mathlib/Order/Defs.lean","def_pos":[254,8],"def_end_pos":[254,16]}]}]}
+{"url":"Mathlib/LinearAlgebra/Matrix/SchurComplement.lean","commit":"","full_name":"Matrix.inv_fromBlocks_zero₁₂_of_isUnit_iff","start":[203,0],"end":[218,79],"file_path":"Mathlib/LinearAlgebra/Matrix/SchurComplement.lean","tactics":[{"state_before":"l : Type u_1\nm : Type u_2\nn : Type u_3\nα : Type u_4\ninst✝⁶ : Fintype l\ninst✝⁵ : Fintype m\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq l\ninst✝² : DecidableEq m\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA : Matrix m m α\nC : Matrix n m α\nD : Matrix n n α\nhAD : IsUnit A ↔ IsUnit D\n⊢ (fromBlocks A 0 C D)⁻¹ = fromBlocks A⁻¹ 0 (-(D⁻¹ * C * A⁻¹)) D⁻¹","state_after":"case pos\nl : Type u_1\nm : Type u_2\nn : Type u_3\nα : Type u_4\ninst✝⁶ : Fintype l\ninst✝⁵ : Fintype m\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq l\ninst✝² : DecidableEq m\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA : Matrix m m α\nC : Matrix n m α\nD : Matrix n n α\nhAD : IsUnit A ↔ IsUnit D\nhA : IsUnit A\n⊢ (fromBlocks A 0 C D)⁻¹ = fromBlocks A⁻¹ 0 (-(D⁻¹ * C * A⁻¹)) D⁻¹\n\ncase neg\nl : Type u_1\nm : Type u_2\nn : Type u_3\nα : Type u_4\ninst✝⁶ : Fintype l\ninst✝⁵ : Fintype m\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq l\ninst✝² : DecidableEq m\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA : Matrix m m α\nC : Matrix n m α\nD : Matrix n n α\nhAD : IsUnit A ↔ IsUnit D\nhA : ¬IsUnit A\n⊢ (fromBlocks A 0 C D)⁻¹ = fromBlocks A⁻¹ 0 (-(D⁻¹ * C * A⁻¹)) D⁻¹","tactic":"by_cases hA : IsUnit A","premises":[{"full_name":"IsUnit","def_path":"Mathlib/Algebra/Group/Units.lean","def_pos":[523,4],"def_end_pos":[523,10]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
+{"url":"Mathlib/Data/Matrix/Basic.lean","commit":"","full_name":"Matrix.one_eq_pi_single","start":[528,0],"end":[529,49],"file_path":"Mathlib/Data/Matrix/Basic.lean","tactics":[{"state_before":"l : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nm' : o → Type u_5\nn' : o → Type u_6\nR : Type u_7\nS : Type u_8\nα : Type v\nβ : Type w\nγ : Type u_9\ninst✝² : DecidableEq n\ninst✝¹ : Zero α\ninst✝ : One α\ni j : n\n⊢ 1 i j = Pi.single i 1 j","state_after":"no goals","tactic":"simp only [one_apply, Pi.single_apply, eq_comm]","premises":[{"full_name":"Matrix.one_apply","def_path":"Mathlib/Data/Matrix/Basic.lean","def_pos":[506,8],"def_end_pos":[506,17]},{"full_name":"Pi.single_apply","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[290,2],"def_end_pos":[290,13]},{"full_name":"eq_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[811,8],"def_end_pos":[811,15]}]}]}
+{"url":"Mathlib/Data/Set/Lattice.lean","commit":"","full_name":"Set.surjOn_iInter","start":[1346,0],"end":[1350,23],"file_path":"Mathlib/Data/Set/Lattice.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nι' : Sort u_5\nι₂ : Sort u_6\nκ : ι → Sort u_7\nκ₁ : ι → Sort u_8\nκ₂ : ι → Sort u_9\nκ' : ι' → Sort u_10\ninst✝ : Nonempty ι\ns : ι → Set α\nt : Set β\nf : α → β\nH : ∀ (i : ι), SurjOn f (s i) t\nHinj : InjOn f (⋃ i, s i)\n⊢ SurjOn f (⋂ i, s i) t","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nι' : Sort u_5\nι₂ : Sort u_6\nκ : ι → Sort u_7\nκ₁ : ι → Sort u_8\nκ₂ : ι → Sort u_9\nκ' : ι' → Sort u_10\ninst✝ : Nonempty ι\ns : ι → Set α\nt : Set β\nf : α → β\nH : ∀ (i : ι), SurjOn f (s i) t\nHinj : InjOn f (⋃ i, s i)\ny : β\nhy : y ∈ t\n⊢ y ∈ f '' ⋂ i, s i","tactic":"intro y hy","premises":[]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nι' : Sort u_5\nι₂ : Sort u_6\nκ : ι → Sort u_7\nκ₁ : ι → Sort u_8\nκ₂ : ι → Sort u_9\nκ' : ι' → Sort u_10\ninst✝ : Nonempty ι\ns : ι → Set α\nt : Set β\nf : α → β\nH : ∀ (i : ι), SurjOn f (s i) t\nHinj : InjOn f (⋃ i, s i)\ny : β\nhy : y ∈ t\n⊢ y ∈ f '' ⋂ i, s i","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nι' : Sort u_5\nι₂ : Sort u_6\nκ : ι → Sort u_7\nκ₁ : ι → Sort u_8\nκ₂ : ι → Sort u_9\nκ' : ι' → Sort u_10\ninst✝ : Nonempty ι\ns : ι → Set α\nt : Set β\nf : α → β\nH : ∀ (i : ι), SurjOn f (s i) t\nHinj : InjOn f (⋃ i, s i)\ny : β\nhy : y ∈ t\n⊢ ∀ (i : ι), y ∈ f '' s i","tactic":"rw [Hinj.image_iInter_eq, mem_iInter]","premises":[{"full_name":"Set.InjOn.image_iInter_eq","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[1277,8],"def_end_pos":[1277,29]},{"full_name":"Set.mem_iInter","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[259,8],"def_end_pos":[259,18]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nι' : Sort u_5\nι₂ : Sort u_6\nκ : ι → Sort u_7\nκ₁ : ι → Sort u_8\nκ₂ : ι → Sort u_9\nκ' : ι' → Sort u_10\ninst✝ : Nonempty ι\ns : ι → Set α\nt : Set β\nf : α → β\nH : ∀ (i : ι), SurjOn f (s i) t\nHinj : InjOn f (⋃ i, s i)\ny : β\nhy : y ∈ t\n⊢ ∀ (i : ι), y ∈ f '' s i","state_after":"no goals","tactic":"exact fun i => H i hy","premises":[]}]}
+{"url":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","commit":"","full_name":"Real.rpow_lt_one_of_one_lt_of_neg","start":[626,0],"end":[628,26],"file_path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","tactics":[{"state_before":"x✝ y z✝ : ℝ\nn : ℕ\nx z : ℝ\nhx : 1 < x\nhz : z < 0\n⊢ x ^ z < 1","state_after":"case h.e'_4\nx✝ y z✝ : ℝ\nn : ℕ\nx z : ℝ\nhx : 1 < x\nhz : z < 0\n⊢ 1 = x ^ 0","tactic":"convert rpow_lt_rpow_of_exponent_lt hx hz","premises":[{"full_name":"Real.rpow_lt_rpow_of_exponent_lt","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","def_pos":[554,8],"def_end_pos":[554,35]}]},{"state_before":"case h.e'_4\nx✝ y z✝ : ℝ\nn : ℕ\nx z : ℝ\nhx : 1 < x\nhz : z < 0\n⊢ 1 = x ^ 0","state_after":"no goals","tactic":"exact (rpow_zero x).symm","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Real.rpow_zero","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","def_pos":[109,8],"def_end_pos":[109,17]}]}]}
+{"url":"Mathlib/CategoryTheory/Sites/Coverage.lean","commit":"","full_name":"CategoryTheory.Presieve.factorsThru_of_le","start":[86,0],"end":[88,44],"file_path":"Mathlib/CategoryTheory/Sites/Coverage.lean","tactics":[{"state_before":"C : Type u_2\nD : Type ?u.22848\ninst✝¹ : Category.{u_1, u_2} C\ninst✝ : Category.{?u.22856, ?u.22848} D\nX : C\nS T : Presieve X\nh : S ≤ T\nY : C\ng : Y ⟶ X\nhg : S g\n⊢ 𝟙 Y ≫ g = g","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/SetTheory/Game/Nim.lean","commit":"","full_name":"SetTheory.PGame.nim_fuzzy_zero_of_ne_zero","start":[192,0],"end":[195,51],"file_path":"Mathlib/SetTheory/Game/Nim.lean","tactics":[{"state_before":"o : Ordinal.{u_1}\nho : o ≠ 0\n⊢ nim o ‖ 0","state_after":"o : Ordinal.{u_1}\nho : o ≠ 0\n⊢ ∃ j,\n    (mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>\n            nim (typein (fun x x_1 => x < x_1) o₂)).moveRight\n        j ≤\n      0","tactic":"rw [Impartial.fuzzy_zero_iff_lf, nim_def, lf_zero_le]","premises":[{"full_name":"SetTheory.PGame.Impartial.fuzzy_zero_iff_lf","def_path":"Mathlib/SetTheory/Game/Impartial.lean","def_pos":[163,8],"def_end_pos":[163,25]},{"full_name":"SetTheory.PGame.lf_zero_le","def_path":"Mathlib/SetTheory/Game/PGame.lean","def_pos":[578,8],"def_end_pos":[578,18]},{"full_name":"SetTheory.PGame.nim_def","def_path":"Mathlib/SetTheory/Game/Nim.lean","def_pos":[56,8],"def_end_pos":[56,15]}]},{"state_before":"o : Ordinal.{u_1}\nho : o ≠ 0\n⊢ ∃ j,\n    (mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>\n            nim (typein (fun x x_1 => x < x_1) o₂)).moveRight\n        j ≤\n      0","state_after":"o : Ordinal.{u_1}\nho : 0 < o\n⊢ ∃ j,\n    (mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>\n            nim (typein (fun x x_1 => x < x_1) o₂)).moveRight\n        j ≤\n      0","tactic":"rw [← Ordinal.pos_iff_ne_zero] at ho","premises":[{"full_name":"Ordinal.pos_iff_ne_zero","def_path":"Mathlib/SetTheory/Ordinal/Basic.lean","def_pos":[353,18],"def_end_pos":[353,33]}]},{"state_before":"o : Ordinal.{u_1}\nho : 0 < o\n⊢ ∃ j,\n    (mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>\n            nim (typein (fun x x_1 => x < x_1) o₂)).moveRight\n        j ≤\n      0","state_after":"no goals","tactic":"exact ⟨(Ordinal.principalSegOut ho).top, by simp⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Ordinal.principalSegOut","def_path":"Mathlib/SetTheory/Ordinal/Basic.lean","def_pos":[378,4],"def_end_pos":[378,19]},{"full_name":"PrincipalSeg.top","def_path":"Mathlib/Order/InitialSeg.lean","def_pos":[211,2],"def_end_pos":[211,5]}]}]}
+{"url":"Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean","commit":"","full_name":"Complex.betaIntegral_scaled","start":[106,0],"end":[124,48],"file_path":"Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean","tactics":[{"state_before":"s t : ℂ\na : ℝ\nha : 0 < a\n⊢ ∫ (x : ℝ) in 0 ..a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) = ↑a ^ (s + t - 1) * s.betaIntegral t","state_after":"s t : ℂ\na : ℝ\nha : 0 < a\nha' : ↑a ≠ 0\n⊢ ∫ (x : ℝ) in 0 ..a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) = ↑a ^ (s + t - 1) * s.betaIntegral t","tactic":"have ha' : (a : ℂ) ≠ 0 := ofReal_ne_zero.mpr ha.ne'","premises":[{"full_name":"Complex","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[27,10],"def_end_pos":[27,17]},{"full_name":"Complex.ofReal_ne_zero","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[139,8],"def_end_pos":[139,22]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"LT.lt.ne'","def_path":"Mathlib/Order/Basic.lean","def_pos":[267,8],"def_end_pos":[267,11]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]}]},{"state_before":"s t : ℂ\na : ℝ\nha : 0 < a\nha' : ↑a ≠ 0\n⊢ ∫ (x : ℝ) in 0 ..a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) = ↑a ^ (s + t - 1) * s.betaIntegral t","state_after":"s t : ℂ\na : ℝ\nha : 0 < a\nha' : ↑a ≠ 0\n⊢ ∫ (x : ℝ) in 0 ..a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) =\n    ↑a ^ (s + t - 1) * ∫ (x : ℝ) in 0 ..1, ↑x ^ (s - 1) * (1 - ↑x) ^ (t - 1)","tactic":"rw [betaIntegral]","premises":[{"full_name":"Complex.betaIntegral","def_path":"Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean","def_pos":[55,18],"def_end_pos":[55,30]}]},{"state_before":"s t : ℂ\na : ℝ\nha : 0 < a\nha' : ↑a ≠ 0\n⊢ ∫ (x : ℝ) in 0 ..a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) =\n    ↑a ^ (s + t - 1) * ∫ (x : ℝ) in 0 ..1, ↑x ^ (s - 1) * (1 - ↑x) ^ (t - 1)","state_after":"s t : ℂ\na : ℝ\nha : 0 < a\nha' : ↑a ≠ 0\nA : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))\n⊢ ∫ (x : ℝ) in 0 ..a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) =\n    ↑a ^ (s + t - 1) * ∫ (x : ℝ) in 0 ..1, ↑x ^ (s - 1) * (1 - ↑x) ^ (t - 1)","tactic":"have A : (a : ℂ) ^ (s + t - 1) = a * ((a : ℂ) ^ (s - 1) * (a : ℂ) ^ (t - 1)) := by\n    rw [(by abel : s + t - 1 = 1 + (s - 1) + (t - 1)), cpow_add _ _ ha', cpow_add 1 _ ha', cpow_one,\n      mul_assoc]","premises":[{"full_name":"Complex","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[27,10],"def_end_pos":[27,17]},{"full_name":"Complex.cpow_add","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean","def_pos":[78,8],"def_end_pos":[78,16]},{"full_name":"Complex.cpow_one","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean","def_pos":[69,8],"def_end_pos":[69,16]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]}]},{"state_before":"s t : ℂ\na : ℝ\nha : 0 < a\nha' : ↑a ≠ 0\nA : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))\n⊢ ∫ (x : ℝ) in 0 ..a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) =\n    ↑a ^ (s + t - 1) * ∫ (x : ℝ) in 0 ..1, ↑x ^ (s - 1) * (1 - ↑x) ^ (t - 1)","state_after":"s t : ℂ\na : ℝ\nha : 0 < a\nha' : ↑a ≠ 0\nA : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))\n⊢ ∫ (x : ℝ) in 0 ..a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) =\n    ∫ (x : ℝ) in 0 ..a, ↑a ^ (s - 1) * ↑a ^ (t - 1) * (↑(x / a) ^ (s - 1) * (1 - ↑(x / a)) ^ (t - 1))","tactic":"rw [A, mul_assoc, ← intervalIntegral.integral_const_mul, ← real_smul, ← zero_div a, ←\n    div_self ha.ne', ← intervalIntegral.integral_comp_div _ ha.ne', zero_div]","premises":[{"full_name":"Complex.real_smul","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[313,8],"def_end_pos":[313,17]},{"full_name":"LT.lt.ne'","def_path":"Mathlib/Order/Basic.lean","def_pos":[267,8],"def_end_pos":[267,11]},{"full_name":"div_self","def_path":"Mathlib/Algebra/GroupWithZero/Units/Basic.lean","def_pos":[251,14],"def_end_pos":[251,22]},{"full_name":"intervalIntegral.integral_comp_div","def_path":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","def_pos":[656,8],"def_end_pos":[656,25]},{"full_name":"intervalIntegral.integral_const_mul","def_path":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","def_pos":[553,8],"def_end_pos":[553,26]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]},{"full_name":"zero_div","def_path":"Mathlib/Algebra/GroupWithZero/Basic.lean","def_pos":[298,8],"def_end_pos":[298,16]}]},{"state_before":"s t : ℂ\na : ℝ\nha : 0 < a\nha' : ↑a ≠ 0\nA : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))\n⊢ ∫ (x : ℝ) in 0 ..a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) =\n    ∫ (x : ℝ) in 0 ..a, ↑a ^ (s - 1) * ↑a ^ (t - 1) * (↑(x / a) ^ (s - 1) * (1 - ↑(x / a)) ^ (t - 1))","state_after":"s t : ℂ\na : ℝ\nha : 0 < a\nha' : ↑a ≠ 0\nA : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))\n⊢ ∫ (x : ℝ) in Ioc 0 a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) ∂volume =\n    ∫ (x : ℝ) in Ioc 0 a, ↑a ^ (s - 1) * ↑a ^ (t - 1) * (↑(x / a) ^ (s - 1) * (1 - ↑(x / a)) ^ (t - 1)) ∂volume","tactic":"simp_rw [intervalIntegral.integral_of_le ha.le]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"intervalIntegral.integral_of_le","def_path":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","def_pos":[430,8],"def_end_pos":[430,22]}]},{"state_before":"s t : ℂ\na : ℝ\nha : 0 < a\nha' : ↑a ≠ 0\nA : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))\n⊢ ∫ (x : ℝ) in Ioc 0 a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) ∂volume =\n    ∫ (x : ℝ) in Ioc 0 a, ↑a ^ (s - 1) * ↑a ^ (t - 1) * (↑(x / a) ^ (s - 1) * (1 - ↑(x / a)) ^ (t - 1)) ∂volume","state_after":"s t : ℂ\na : ℝ\nha : 0 < a\nha' : ↑a ≠ 0\nA : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))\nx : ℝ\nhx : x ∈ Ioc 0 a\n⊢ ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) = ↑a ^ (s - 1) * ↑a ^ (t - 1) * (↑(x / a) ^ (s - 1) * (1 - ↑(x / a)) ^ (t - 1))","tactic":"refine setIntegral_congr measurableSet_Ioc fun x hx => ?_","premises":[{"full_name":"MeasureTheory.setIntegral_congr","def_path":"Mathlib/MeasureTheory/Integral/SetIntegral.lean","def_pos":[90,8],"def_end_pos":[90,25]},{"full_name":"measurableSet_Ioc","def_path":"Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean","def_pos":[178,8],"def_end_pos":[178,25]}]},{"state_before":"s t : ℂ\na : ℝ\nha : 0 < a\nha' : ↑a ≠ 0\nA : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))\nx : ℝ\nhx : x ∈ Ioc 0 a\n⊢ ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) = ↑a ^ (s - 1) * ↑a ^ (t - 1) * (↑(x / a) ^ (s - 1) * (1 - ↑(x / a)) ^ (t - 1))","state_after":"s t : ℂ\na : ℝ\nha : 0 < a\nha' : ↑a ≠ 0\nA : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))\nx : ℝ\nhx : x ∈ Ioc 0 a\n⊢ ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) = ↑a ^ (s - 1) * ↑(x / a) ^ (s - 1) * (↑a ^ (t - 1) * (1 - ↑(x / a)) ^ (t - 1))","tactic":"rw [mul_mul_mul_comm]","premises":[{"full_name":"mul_mul_mul_comm","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[160,8],"def_end_pos":[160,24]}]},{"state_before":"s t : ℂ\na : ℝ\nha : 0 < a\nha' : ↑a ≠ 0\nA : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))\nx : ℝ\nhx : x ∈ Ioc 0 a\n⊢ ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) = ↑a ^ (s - 1) * ↑(x / a) ^ (s - 1) * (↑a ^ (t - 1) * (1 - ↑(x / a)) ^ (t - 1))","state_after":"case e_a\ns t : ℂ\na : ℝ\nha : 0 < a\nha' : ↑a ≠ 0\nA : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))\nx : ℝ\nhx : x ∈ Ioc 0 a\n⊢ ↑x ^ (s - 1) = ↑a ^ (s - 1) * ↑(x / a) ^ (s - 1)\n\ncase e_a\ns t : ℂ\na : ℝ\nha : 0 < a\nha' : ↑a ≠ 0\nA : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))\nx : ℝ\nhx : x ∈ Ioc 0 a\n⊢ (↑a - ↑x) ^ (t - 1) = ↑a ^ (t - 1) * (1 - ↑(x / a)) ^ (t - 1)","tactic":"congr 1","premises":[]}]}
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IsIntegrallyClosed R\nK : Type u_3\ninst✝¹ : CommRing K\ninst✝ : Algebra R K\nifr : IsFractionRing R K\nn : ℕ\nhn : n ≠ 0\na b : R\nx✝ : a ^ n ∣ b ^ n\nx : R\nhx : b ^ n = a ^ n * x\nha : ¬a = 0\n⊢ a ∣ b","tactic":"by_cases ha : a = 0","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case neg\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\nid : IsDomain R\niic : IsIntegrallyClosed R\nK : Type u_3\ninst✝¹ : CommRing K\ninst✝ : Algebra R K\nifr : IsFractionRing R K\nn : ℕ\nhn : n ≠ 0\na b : R\nx✝ : a ^ n ∣ b ^ n\nx : R\nhx : b ^ n = a ^ n * x\nha : ¬a = 0\n⊢ a ∣ b","state_after":"case neg\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\nid : IsDomain R\niic : IsIntegrallyClosed R\nK✝ : Type u_3\ninst✝¹ : CommRing K✝\ninst✝ : Algebra R K✝\nifr : IsFractionRing R K✝\nn : ℕ\nhn : n ≠ 0\na b : R\nx✝ : a ^ n ∣ b ^ n\nx : R\nhx : b ^ n = a ^ n * x\nha : ¬a = 0\nK : Type u_1 := FractionRing R\n⊢ a ∣ b","tactic":"let K := FractionRing R","premises":[{"full_name":"FractionRing","def_path":"Mathlib/RingTheory/Localization/FractionRing.lean","def_pos":[266,7],"def_end_pos":[266,19]}]},{"state_before":"case neg\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\nid : IsDomain R\niic : IsIntegrallyClosed R\nK✝ : Type u_3\ninst✝¹ : CommRing K✝\ninst✝ : Algebra R K✝\nifr : IsFractionRing R K✝\nn : ℕ\nhn : n ≠ 0\na b : R\nx✝ : a ^ n ∣ b ^ n\nx : R\nhx : b ^ n = a ^ n * x\nha : ¬a = 0\nK : Type u_1 := FractionRing R\n⊢ a ∣ b","state_after":"case neg\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\nid : IsDomain R\niic : IsIntegrallyClosed R\nK✝ : Type u_3\ninst✝¹ : CommRing K✝\ninst✝ : Algebra R K✝\nifr : IsFractionRing R K✝\nn : ℕ\nhn : n ≠ 0\na b : R\nx✝ : a ^ n ∣ b ^ n\nx : R\nhx : b ^ n = a ^ n * x\nK : Type u_1 := FractionRing R\nha : (algebraMap R K) a ≠ 0\n⊢ a ∣ b","tactic":"replace ha : algebraMap R K a ≠ 0 := fun h ↦\n    ha <| (injective_iff_map_eq_zero _).1 (IsFractionRing.injective R K) _ h","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"IsFractionRing.injective","def_path":"Mathlib/RingTheory/Localization/FractionRing.lean","def_pos":[77,18],"def_end_pos":[77,27]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"algebraMap","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[108,4],"def_end_pos":[108,14]},{"full_name":"injective_iff_map_eq_zero","def_path":"Mathlib/Algebra/Group/Hom/Basic.lean","def_pos":[105,2],"def_end_pos":[105,13]}]},{"state_before":"case neg\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\nid : IsDomain R\niic : IsIntegrallyClosed R\nK✝ : Type u_3\ninst✝¹ : CommRing K✝\ninst✝ : Algebra R K✝\nifr : IsFractionRing R K✝\nn : ℕ\nhn : n ≠ 0\na b : R\nx✝ : a ^ n ∣ b ^ n\nx : R\nhx : b ^ n = a ^ n * x\nK : Type u_1 := FractionRing R\nha : (algebraMap R K) a ≠ 0\n⊢ a ∣ b","state_after":"case neg\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\nid : IsDomain R\niic : IsIntegrallyClosed R\nK✝ : Type u_3\ninst✝¹ : CommRing K✝\ninst✝ : Algebra R K✝\nifr : IsFractionRing R K✝\nn : ℕ\nhn : n ≠ 0\na b : R\nx✝ : a ^ n ∣ b ^ n\nx : R\nhx : b ^ n = a ^ n * x\nK : Type u_1 := FractionRing R\nha : (algebraMap R K) a ≠ 0\ny : K := (algebraMap R K) b / (algebraMap R K) a\n⊢ a ∣ b","tactic":"let y := (algebraMap R K b) / (algebraMap R K a)","premises":[{"full_name":"algebraMap","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[108,4],"def_end_pos":[108,14]}]},{"state_before":"case neg\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\nid : IsDomain R\niic : IsIntegrallyClosed R\nK✝ : Type u_3\ninst✝¹ : CommRing K✝\ninst✝ : Algebra R K✝\nifr : IsFractionRing R K✝\nn : ℕ\nhn : n ≠ 0\na b : R\nx✝ : a ^ n ∣ b ^ n\nx : R\nhx : b ^ n = a ^ n * x\nK : Type u_1 := FractionRing R\nha : (algebraMap R K) a ≠ 0\ny : K := (algebraMap R K) b / (algebraMap R K) a\n⊢ a ∣ b","state_after":"case neg\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\nid : IsDomain R\niic : IsIntegrallyClosed R\nK✝ : Type u_3\ninst✝¹ : CommRing K✝\ninst✝ : Algebra R K✝\nifr : IsFractionRing R K✝\nn : ℕ\nhn : n ≠ 0\na b : R\nx✝ : a ^ n ∣ b ^ n\nx : R\nhx : b ^ n = a ^ n * x\nK : Type u_1 := FractionRing R\nha : (algebraMap R K) a ≠ 0\ny : K := (algebraMap R K) b / (algebraMap R K) a\nhy : IsIntegral R y\n⊢ a ∣ b","tactic":"have hy : IsIntegral R y := by\n    refine ⟨X ^ n - C x, monic_X_pow_sub_C _ hn, ?_⟩\n    simp only [y, map_pow, eval₂_sub, eval₂_X_pow, div_pow, eval₂_pow', eval₂_C]\n    replace hx := congr_arg (algebraMap R K) hx\n    rw [map_pow] at hx\n    field_simp [hx, ha]","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"IsIntegral","def_path":"Mathlib/RingTheory/IntegralClosure/IsIntegral/Defs.lean","def_pos":[46,4],"def_end_pos":[46,14]},{"full_name":"Polynomial.C","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[426,4],"def_end_pos":[426,5]},{"full_name":"Polynomial.X","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[474,4],"def_end_pos":[474,5]},{"full_name":"Polynomial.eval₂_C","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[61,8],"def_end_pos":[61,15]},{"full_name":"Polynomial.eval₂_X_pow","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[71,8],"def_end_pos":[71,19]},{"full_name":"Polynomial.eval₂_pow'","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[918,8],"def_end_pos":[918,18]},{"full_name":"Polynomial.eval₂_sub","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[1110,8],"def_end_pos":[1110,17]},{"full_name":"Polynomial.monic_X_pow_sub_C","def_path":"Mathlib/Algebra/Polynomial/Monic.lean","def_pos":[347,8],"def_end_pos":[347,25]},{"full_name":"algebraMap","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[108,4],"def_end_pos":[108,14]},{"full_name":"div_pow","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[589,6],"def_end_pos":[589,13]},{"full_name":"map_pow","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[423,8],"def_end_pos":[423,15]}]},{"state_before":"case neg\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\nid : IsDomain R\niic : IsIntegrallyClosed R\nK✝ : Type u_3\ninst✝¹ : CommRing K✝\ninst✝ : Algebra R K✝\nifr : IsFractionRing R K✝\nn : ℕ\nhn : n ≠ 0\na b : R\nx✝ : a ^ n ∣ b ^ n\nx : R\nhx : b ^ n = a ^ n * x\nK : Type u_1 := FractionRing R\nha : (algebraMap R K) a ≠ 0\ny : K := (algebraMap R K) b / (algebraMap R K) a\nhy : IsIntegral R y\n⊢ a ∣ b","state_after":"case neg.intro\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\nid : IsDomain R\niic : IsIntegrallyClosed R\nK✝ : Type u_3\ninst✝¹ : CommRing K✝\ninst✝ : Algebra R K✝\nifr : IsFractionRing R K✝\nn : ℕ\nhn : n ≠ 0\na b : R\nx✝ : a ^ n ∣ b ^ n\nx : R\nhx : b ^ n = a ^ n * x\nK : Type u_1 := FractionRing R\nha : (algebraMap R K) a ≠ 0\ny : K := (algebraMap R K) b / (algebraMap R K) a\nhy : IsIntegral R y\nk : R\nhk : (algebraMap R K) k = y\n⊢ a ∣ b","tactic":"obtain ⟨k, hk⟩ := algebraMap_eq_of_integral hy","premises":[{"full_name":"IsIntegrallyClosed.algebraMap_eq_of_integral","def_path":"Mathlib/RingTheory/IntegralClosure/IntegrallyClosed.lean","def_pos":[198,8],"def_end_pos":[198,33]}]},{"state_before":"case neg.intro\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\nid : IsDomain R\niic : IsIntegrallyClosed R\nK✝ : Type u_3\ninst✝¹ : CommRing K✝\ninst✝ : Algebra R K✝\nifr : IsFractionRing R K✝\nn : ℕ\nhn : n ≠ 0\na b : R\nx✝ : a ^ n ∣ b ^ n\nx : R\nhx : b ^ n = a ^ n * x\nK : Type u_1 := FractionRing R\nha : (algebraMap R K) a ≠ 0\ny : K := (algebraMap R K) b / (algebraMap R K) a\nhy : IsIntegral R y\nk : R\nhk : (algebraMap R K) k = y\n⊢ a ∣ b","state_after":"case neg.intro\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\nid : IsDomain R\niic : IsIntegrallyClosed R\nK✝ : Type u_3\ninst✝¹ : CommRing K✝\ninst✝ : Algebra R K✝\nifr : IsFractionRing R K✝\nn : ℕ\nhn : n ≠ 0\na b : R\nx✝ : a ^ n ∣ b ^ n\nx : R\nhx : b ^ n = a ^ n * x\nK : Type u_1 := FractionRing R\nha : (algebraMap R K) a ≠ 0\ny : K := (algebraMap R K) b / (algebraMap R K) a\nhy : IsIntegral R y\nk : R\nhk : (algebraMap R K) k = y\n⊢ (algebraMap R K) b = (algebraMap R K) (a * k)","tactic":"refine ⟨k, IsFractionRing.injective R K ?_⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"IsFractionRing.injective","def_path":"Mathlib/RingTheory/Localization/FractionRing.lean","def_pos":[77,18],"def_end_pos":[77,27]}]},{"state_before":"case neg.intro\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\nid : IsDomain R\niic : IsIntegrallyClosed R\nK✝ : Type u_3\ninst✝¹ : CommRing K✝\ninst✝ : Algebra R K✝\nifr : IsFractionRing R K✝\nn : ℕ\nhn : n ≠ 0\na b : R\nx✝ : a ^ n ∣ b ^ n\nx : R\nhx : b ^ n = a ^ n * x\nK : Type u_1 := FractionRing R\nha : (algebraMap R K) a ≠ 0\ny : K := (algebraMap R K) b / (algebraMap R K) a\nhy : IsIntegral R y\nk : R\nhk : (algebraMap R K) k = y\n⊢ (algebraMap R K) b = (algebraMap R K) (a * k)","state_after":"no goals","tactic":"rw [map_mul, hk, mul_div_cancel₀ _ ha]","premises":[{"full_name":"map_mul","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[281,8],"def_end_pos":[281,15]},{"full_name":"mul_div_cancel₀","def_path":"Mathlib/Algebra/GroupWithZero/Units/Basic.lean","def_pos":[424,6],"def_end_pos":[424,21]}]}]}
+{"url":"Mathlib/Order/Filter/Pointwise.lean","commit":"","full_name":"Filter.isAddUnit_iff","start":[612,0],"end":[621,19],"file_path":"Mathlib/Order/Filter/Pointwise.lean","tactics":[{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\nε : Type u_6\ninst✝ : DivisionMonoid α\nf g : Filter α\n⊢ IsUnit f ↔ ∃ a, f = pure a ∧ IsUnit a","state_after":"case mp\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\nε : Type u_6\ninst✝ : DivisionMonoid α\nf g : Filter α\n⊢ IsUnit f → ∃ a, f = pure a ∧ IsUnit a\n\ncase mpr\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\nε : Type u_6\ninst✝ : DivisionMonoid α\nf g : Filter α\n⊢ (∃ a, f = pure a ∧ IsUnit a) → IsUnit f","tactic":"constructor","premises":[]}]}
+{"url":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","commit":"","full_name":"mul_le_iff_le_one_right'","start":[393,0],"end":[397,53],"file_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝³ : MulOneClass α\ninst✝² : LE α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : ContravariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na b : α\n⊢ a * b ≤ a ↔ a * b ≤ a * 1","state_after":"no goals","tactic":"rw [mul_one]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"mul_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[413,8],"def_end_pos":[413,15]}]}]}
+{"url":"Mathlib/RingTheory/Localization/Integer.lean","commit":"","full_name":"IsLocalization.exist_integer_multiples_of_finite","start":[95,0],"end":[100,46],"file_path":"Mathlib/RingTheory/Localization/Integer.lean","tactics":[{"state_before":"R : Type u_1\ninst✝⁵ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝⁴ : CommSemiring S\ninst✝³ : Algebra R S\nP : Type u_3\ninst✝² : CommSemiring P\ninst✝¹ : IsLocalization M S\nι : Type u_4\ninst✝ : Finite ι\nf : ι → S\n⊢ ∃ b, ∀ (i : ι), IsInteger R (↑b • f i)","state_after":"case intro\nR : Type u_1\ninst✝⁵ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝⁴ : CommSemiring S\ninst✝³ : Algebra R S\nP : Type u_3\ninst✝² : CommSemiring P\ninst✝¹ : IsLocalization M S\nι : Type u_4\ninst✝ : Finite ι\nf : ι → S\nval✝ : Fintype ι\n⊢ ∃ b, ∀ (i : ι), IsInteger R (↑b • f i)","tactic":"cases nonempty_fintype ι","premises":[{"full_name":"nonempty_fintype","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[390,8],"def_end_pos":[390,24]}]},{"state_before":"case intro\nR : Type u_1\ninst✝⁵ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝⁴ : CommSemiring S\ninst✝³ : Algebra R S\nP : Type u_3\ninst✝² : CommSemiring P\ninst✝¹ : IsLocalization M S\nι : Type u_4\ninst✝ : Finite ι\nf : ι → S\nval✝ : Fintype ι\n⊢ ∃ b, ∀ (i : ι), IsInteger R (↑b • f i)","state_after":"case intro.intro\nR : Type u_1\ninst✝⁵ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝⁴ : CommSemiring S\ninst✝³ : Algebra R S\nP : Type u_3\ninst✝² : CommSemiring P\ninst✝¹ : IsLocalization M S\nι : Type u_4\ninst✝ : Finite ι\nf : ι → S\nval✝ : Fintype ι\nb : ↥M\nhb : ∀ i ∈ Finset.univ, IsInteger R (↑b • f i)\n⊢ ∃ b, ∀ (i : ι), IsInteger R (↑b • f i)","tactic":"obtain ⟨b, hb⟩ := exist_integer_multiples M Finset.univ f","premises":[{"full_name":"Finset.univ","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[67,4],"def_end_pos":[67,8]},{"full_name":"IsLocalization.exist_integer_multiples","def_path":"Mathlib/RingTheory/Localization/Integer.lean","def_pos":[81,8],"def_end_pos":[81,31]}]},{"state_before":"case intro.intro\nR : Type u_1\ninst✝⁵ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝⁴ : CommSemiring S\ninst✝³ : Algebra R S\nP : Type u_3\ninst✝² : CommSemiring P\ninst✝¹ : IsLocalization M S\nι : Type u_4\ninst✝ : Finite ι\nf : ι → S\nval✝ : Fintype ι\nb : ↥M\nhb : ∀ i ∈ Finset.univ, IsInteger R (↑b • f i)\n⊢ ∃ b, ∀ (i : ι), IsInteger R (↑b • f i)","state_after":"no goals","tactic":"exact ⟨b, fun i => hb i (Finset.mem_univ _)⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Finset.mem_univ","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[71,8],"def_end_pos":[71,16]}]}]}
+{"url":".lake/packages/batteries/Batteries/Data/RBMap/WF.lean","commit":"","full_name":"Batteries.RBNode.Ordered.balLeft","start":[294,0],"end":[305,26],"file_path":".lake/packages/batteries/Batteries/Data/RBMap/WF.lean","tactics":[{"state_before":"α : Type u_1\ncmp : α → α → Ordering\nl : RBNode α\nv : α\nr : RBNode α\nlv : All (fun x => cmpLT cmp x v) l\nvr : All (fun x => cmpLT cmp v x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\n⊢ Ordered cmp (l.balLeft v r)","state_after":"α : Type u_1\ncmp : α → α → Ordering\nl : RBNode α\nv : α\nr : RBNode α\nlv : All (fun x => cmpLT cmp x v) l\nvr : All (fun x => cmpLT cmp v x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\n⊢ Ordered cmp\n    (match l with\n    | node red a x b => node red (node black a x b) v r\n    | l =>\n      match r with\n      | node black a y b => l.balance2 v (node red a y b)\n      | node red (node black a y b) z c => node red (node black l v a) y (b.balance2 z c.setRed)\n      | r => node red l v r)","tactic":"unfold balLeft","premises":[{"full_name":"Batteries.RBNode.balLeft","def_path":".lake/packages/batteries/Batteries/Data/RBMap/Basic.lean","def_pos":[336,4],"def_end_pos":[336,11]}]},{"state_before":"α : Type u_1\ncmp : α → α → Ordering\nl : RBNode α\nv : α\nr : RBNode α\nlv : All (fun x => cmpLT cmp x v) l\nvr : All (fun x => cmpLT cmp v x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\n⊢ Ordered cmp\n    (match l with\n    | node red a x b => node red (node black a x b) v r\n    | l =>\n      match r with\n      | node black a y b => l.balance2 v (node red a y b)\n      | node red (node black a y b) z c => node red (node black l v a) y (b.balance2 z c.setRed)\n      | r => node red l v r)","state_after":"case h_1\nα : Type u_1\ncmp : α → α → Ordering\nv : α\nr : RBNode α\nvr : All (fun x => cmpLT cmp v x) r\nhr : Ordered cmp r\nl✝ a✝ : RBNode α\nx✝ : α\nb✝ : RBNode α\nlv : All (fun x => cmpLT cmp x v) (node red a✝ x✝ b✝)\nhl : Ordered cmp (node red a✝ x✝ b✝)\n⊢ Ordered cmp (node red (node black a✝ x✝ b✝) v r)\n\ncase h_2\nα : Type u_1\ncmp : α → α → Ordering\nl : RBNode α\nv : α\nr : RBNode α\nlv : All (fun x => cmpLT cmp x v) l\nvr : All (fun x => cmpLT cmp v x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nl✝ : RBNode α\nx✝ : ∀ (a : RBNode α) (x : α) (b : RBNode α), l = node red a x b → False\n⊢ Ordered cmp\n    (match r with\n    | node black a y b => l.balance2 v (node red a y b)\n    | node red (node black a y b) z c => node red (node black l v a) y (b.balance2 z c.setRed)\n    | r => node red l v r)","tactic":"split","premises":[]},{"state_before":"case h_2\nα : Type u_1\ncmp : α → α → Ordering\nl : RBNode α\nv : α\nr : RBNode α\nlv : All (fun x => cmpLT cmp x v) l\nvr : All (fun x => cmpLT cmp v x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nl✝ : RBNode α\nx✝ : ∀ (a : RBNode α) (x : α) (b : RBNode α), l = node red a x b → False\n⊢ Ordered cmp\n    (match r with\n    | node black a y b => l.balance2 v (node red a y b)\n    | node red (node black a y b) z c => node red (node black l v a) y (b.balance2 z c.setRed)\n    | r => node red l v r)","state_after":"case h_2.h_1\nα : Type u_1\ncmp : α → α → Ordering\nl : RBNode α\nv : α\nlv : All (fun x => cmpLT cmp x v) l\nhl : Ordered cmp l\nl✝ : RBNode α\nx✝ : ∀ (a : RBNode α) (x : α) (b : RBNode α), l = node red a x b → False\nr✝ a✝ : RBNode α\ny✝ : α\nb✝ : RBNode α\nvr : All (fun x => cmpLT cmp v x) (node black a✝ y✝ b✝)\nhr : Ordered cmp (node black a✝ y✝ b✝)\n⊢ Ordered cmp (l.balance2 v (node red a✝ y✝ b✝))\n\ncase h_2.h_2\nα : Type u_1\ncmp : α → α → Ordering\nl : RBNode α\nv : α\nlv : All (fun x => cmpLT cmp x v) l\nhl : Ordered cmp l\nl✝ : RBNode α\nx✝ : ∀ (a : RBNode α) (x : α) (b : RBNode α), l = node red a x b → False\nr✝ a✝ : RBNode α\ny✝ : α\nb✝ : RBNode α\nz✝ : α\nc✝ : RBNode α\nvr : All (fun x => cmpLT cmp v x) (node red (node black a✝ y✝ b✝) z✝ c✝)\nhr : Ordered cmp (node red (node black a✝ y✝ b✝) z✝ c✝)\n⊢ Ordered cmp (node red (node black l v a✝) y✝ (b✝.balance2 z✝ c✝.setRed))\n\ncase h_2.h_3\nα : Type u_1\ncmp : α → α → Ordering\nl : RBNode α\nv : α\nr : RBNode α\nlv : All (fun x => cmpLT cmp x v) l\nvr : All (fun x => cmpLT cmp v x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nl✝ : RBNode α\nx✝² : ∀ (a : RBNode α) (x : α) (b : RBNode α), l = node red a x b → False\nr✝ : RBNode α\nx✝¹ : ∀ (a : RBNode α) (y : α) (b : RBNode α), r = node black a y b → False\nx✝ : ∀ (a : RBNode α) (y : α) (b : RBNode α) (z : α) (c : RBNode α), r = node red (node black a y b) z c → False\n⊢ Ordered cmp (node red l v r)","tactic":"split","premises":[]}]}
+{"url":"Mathlib/Geometry/Manifold/MFDeriv/Basic.lean","commit":"","full_name":"mdifferentiableAt_iff_of_mem_source","start":[322,0],"end":[330,69],"file_path":"Mathlib/Geometry/Manifold/MFDeriv/Basic.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝¹⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁴ : NormedAddCommGroup E\ninst✝¹³ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹² : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹¹ : TopologicalSpace M\ninst✝¹⁰ : ChartedSpace H M\nE' : Type u_5\ninst✝⁹ : NormedAddCommGroup E'\ninst✝⁸ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝⁷ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM' : Type u_7\ninst✝⁶ : TopologicalSpace M'\ninst✝⁵ : ChartedSpace H' M'\nE'' : Type u_8\ninst✝⁴ : NormedAddCommGroup E''\ninst✝³ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝² : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝¹ : TopologicalSpace M''\ninst✝ : ChartedSpace H'' M''\nf f₀ f₁ : M → M'\nx : M\ns t : Set M\ng : M' → M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f₀' f₁' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) →L[𝕜] TangentSpace I'' (g (f x))\nx' : M\ny : M'\nhx : x' ∈ (chartAt H x).source\nhy : f x' ∈ (chartAt H' y).source\n⊢ ContinuousWithinAt f univ x' ∧\n      DifferentiableWithinAt 𝕜 (↑(extChartAt I' y) ∘ f ∘ ↑(extChartAt I x).symm)\n        (↑(extChartAt I x).symm ⁻¹' univ ∩ range ↑I) (↑(extChartAt I x) x') ↔\n    ContinuousAt f x' ∧\n      DifferentiableWithinAt 𝕜 (↑(extChartAt I' y) ∘ f ∘ ↑(extChartAt I x).symm) (range ↑I) (↑(extChartAt I x) x')","state_after":"no goals","tactic":"rw [continuousWithinAt_univ, Set.preimage_univ, Set.univ_inter]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Set.preimage_univ","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[59,8],"def_end_pos":[59,21]},{"full_name":"Set.univ_inter","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[803,8],"def_end_pos":[803,18]},{"full_name":"continuousWithinAt_univ","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[445,8],"def_end_pos":[445,31]}]}]}
+{"url":"Mathlib/MeasureTheory/Measure/WithDensity.lean","commit":"","full_name":"MeasureTheory.withDensity_apply_le","start":[42,0],"end":[50,49],"file_path":"Mathlib/MeasureTheory/Measure/WithDensity.lean","tactics":[{"state_before":"α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\ns : Set α\n⊢ ∫⁻ (a : α) in s, f a ∂μ ≤ (μ.withDensity f) s","state_after":"α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\ns : Set α\nt : Set α := toMeasurable (μ.withDensity f) s\n⊢ ∫⁻ (a : α) in s, f a ∂μ ≤ (μ.withDensity f) s","tactic":"let t := toMeasurable (μ.withDensity f) s","premises":[{"full_name":"MeasureTheory.Measure.withDensity","def_path":"Mathlib/MeasureTheory/Measure/WithDensity.lean","def_pos":[33,4],"def_end_pos":[33,23]},{"full_name":"MeasureTheory.toMeasurable","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[294,16],"def_end_pos":[294,28]}]},{"state_before":"α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\ns : Set α\nt : Set α := toMeasurable (μ.withDensity f) s\n⊢ ∫⁻ (a : α) in s, f a ∂μ ≤ (μ.withDensity f) s","state_after":"no goals","tactic":"calc\n  ∫⁻ a in s, f a ∂μ ≤ ∫⁻ a in t, f a ∂μ :=\n    lintegral_mono_set (subset_toMeasurable (withDensity μ f) s)\n  _ = μ.withDensity f t :=\n    (withDensity_apply f (measurableSet_toMeasurable (withDensity μ f) s)).symm\n  _ = μ.withDensity f s := measure_toMeasurable s","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"MeasureTheory.Measure.restrict","def_path":"Mathlib/MeasureTheory/Measure/Restrict.lean","def_pos":[43,18],"def_end_pos":[43,26]},{"full_name":"MeasureTheory.Measure.withDensity","def_path":"Mathlib/MeasureTheory/Measure/WithDensity.lean","def_pos":[33,4],"def_end_pos":[33,23]},{"full_name":"MeasureTheory.lintegral","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[59,16],"def_end_pos":[59,25]},{"full_name":"MeasureTheory.lintegral_mono_set","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[114,8],"def_end_pos":[114,26]},{"full_name":"MeasureTheory.measurableSet_toMeasurable","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[310,8],"def_end_pos":[310,34]},{"full_name":"MeasureTheory.measure_toMeasurable","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[317,8],"def_end_pos":[317,28]},{"full_name":"MeasureTheory.subset_toMeasurable","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[300,8],"def_end_pos":[300,27]},{"full_name":"MeasureTheory.withDensity_apply","def_path":"Mathlib/MeasureTheory/Measure/WithDensity.lean","def_pos":[38,8],"def_end_pos":[38,25]}]}]}
+{"url":"Mathlib/Data/ENNReal/Real.lean","commit":"","full_name":"ENNReal.iSup_zero_eq_zero","start":[581,0],"end":[582,74],"file_path":"Mathlib/Data/ENNReal/Real.lean","tactics":[{"state_before":"ι : Sort u_1\n⊢ ⨆ x, 0 = 0","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean","commit":"","full_name":"CategoryTheory.Limits.MultispanIndex.multicoforkEquivSigmaCofork_counitIso","start":[650,0],"end":[670,12],"file_path":"Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean","tactics":[{"state_before":"C : Type u\ninst✝² : Category.{v, u} C\nI : MultispanIndex C\ninst✝¹ : HasCoproduct I.left\ninst✝ : HasCoproduct I.right\nK : Multicofork I\n⊢ ∀ (j : WalkingMultispan I.fstFrom I.sndFrom),\n    ((𝟭 (Multicofork I)).obj K).ι.app j ≫ (Iso.refl ((𝟭 (Multicofork I)).obj K).pt).hom =\n      ((I.toSigmaCoforkFunctor ⋙ I.ofSigmaCoforkFunctor).obj K).ι.app j","state_after":"no goals","tactic":"rintro (_ | _) <;> simp","premises":[]},{"state_before":"C : Type u\ninst✝² : Category.{v, u} C\nI : MultispanIndex C\ninst✝¹ : HasCoproduct I.left\ninst✝ : HasCoproduct I.right\nK : Cofork I.fstSigmaMap I.sndSigmaMap\n⊢ ((I.ofSigmaCoforkFunctor ⋙ I.toSigmaCoforkFunctor).obj K).π ≫\n      (Iso.refl ((I.ofSigmaCoforkFunctor ⋙ I.toSigmaCoforkFunctor).obj K).pt).hom =\n    ((𝟭 (Cofork I.fstSigmaMap I.sndSigmaMap)).obj K).π","state_after":"case w\nC : Type u\ninst✝² : Category.{v, u} C\nI : MultispanIndex C\ninst✝¹ : HasCoproduct I.left\ninst✝ : HasCoproduct I.right\nK : Cofork I.fstSigmaMap I.sndSigmaMap\n⊢ ∀ (j : Discrete I.R),\n    colimit.ι (Discrete.functor I.right) j ≫\n        ((I.ofSigmaCoforkFunctor ⋙ I.toSigmaCoforkFunctor).obj K).π ≫\n          (Iso.refl ((I.ofSigmaCoforkFunctor ⋙ I.toSigmaCoforkFunctor).obj K).pt).hom =\n      colimit.ι (Discrete.functor I.right) j ≫ ((𝟭 (Cofork I.fstSigmaMap I.sndSigmaMap)).obj K).π","tactic":"apply Limits.colimit.hom_ext","premises":[{"full_name":"CategoryTheory.Limits.colimit.hom_ext","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[757,8],"def_end_pos":[757,23]}]},{"state_before":"case w\nC : Type u\ninst✝² : Category.{v, u} C\nI : MultispanIndex C\ninst✝¹ : HasCoproduct I.left\ninst✝ : HasCoproduct I.right\nK : Cofork I.fstSigmaMap I.sndSigmaMap\n⊢ ∀ (j : Discrete I.R),\n    colimit.ι (Discrete.functor I.right) j ≫\n        ((I.ofSigmaCoforkFunctor ⋙ I.toSigmaCoforkFunctor).obj K).π ≫\n          (Iso.refl ((I.ofSigmaCoforkFunctor ⋙ I.toSigmaCoforkFunctor).obj K).pt).hom =\n      colimit.ι (Discrete.functor I.right) j ≫ ((𝟭 (Cofork I.fstSigmaMap I.sndSigmaMap)).obj K).π","state_after":"case w.mk\nC : Type u\ninst✝² : Category.{v, u} C\nI : MultispanIndex C\ninst✝¹ : HasCoproduct I.left\ninst✝ : HasCoproduct I.right\nK : Cofork I.fstSigmaMap I.sndSigmaMap\nj : I.R\n⊢ colimit.ι (Discrete.functor I.right) { as := j } ≫\n      ((I.ofSigmaCoforkFunctor ⋙ I.toSigmaCoforkFunctor).obj K).π ≫\n        (Iso.refl ((I.ofSigmaCoforkFunctor ⋙ I.toSigmaCoforkFunctor).obj K).pt).hom =\n    colimit.ι (Discrete.functor I.right) { as := j } ≫ ((𝟭 (Cofork I.fstSigmaMap I.sndSigmaMap)).obj K).π","tactic":"rintro ⟨j⟩","premises":[]},{"state_before":"case w.mk\nC : Type u\ninst✝² : Category.{v, u} C\nI : MultispanIndex C\ninst✝¹ : HasCoproduct I.left\ninst✝ : HasCoproduct I.right\nK : Cofork I.fstSigmaMap I.sndSigmaMap\nj : I.R\n⊢ colimit.ι (Discrete.functor I.right) { as := j } ≫\n      ((I.ofSigmaCoforkFunctor ⋙ I.toSigmaCoforkFunctor).obj K).π ≫\n        (Iso.refl ((I.ofSigmaCoforkFunctor ⋙ I.toSigmaCoforkFunctor).obj K).pt).hom =\n    colimit.ι (Discrete.functor I.right) { as := j } ≫ ((𝟭 (Cofork I.fstSigmaMap I.sndSigmaMap)).obj K).π","state_after":"case w.mk\nC : Type u\ninst✝² : Category.{v, u} C\nI : MultispanIndex C\ninst✝¹ : HasCoproduct I.left\ninst✝ : HasCoproduct I.right\nK : Cofork I.fstSigmaMap I.sndSigmaMap\nj : I.R\n⊢ colimit.ι (Discrete.functor I.right) { as := j } ≫ Sigma.desc (Multicofork.ofSigmaCofork I K).π ≫ 𝟙 K.pt =\n    colimit.ι (Discrete.functor I.right) { as := j } ≫ K.π","tactic":"dsimp","premises":[]},{"state_before":"case w.mk\nC : Type u\ninst✝² : Category.{v, u} C\nI : MultispanIndex C\ninst✝¹ : HasCoproduct I.left\ninst✝ : HasCoproduct I.right\nK : 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+{"url":"Mathlib/Algebra/Order/CauSeq/Basic.lean","commit":"","full_name":"CauSeq.mul_limZero_right","start":[380,0],"end":[385,74],"file_path":"Mathlib/Algebra/Order/CauSeq/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝² : LinearOrderedField α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf g : CauSeq β abv\nhg : g.LimZero\nε : α\nε0 : ε > 0\nF : α\nF0 : F > 0\nhF : ∀ (i : ℕ), abv (↑f i) < F\ni : ℕ\nH : ∀ j ≥ i, abv (↑g j) < ε / F\nj : ℕ\nij : j ≥ i\n⊢ abv (↑(f * g) j) < ε","state_after":"α : Type u_1\nβ : Type u_2\ninst✝² : LinearOrderedField α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf g : CauSeq β abv\nhg : g.LimZero\nε : α\nε0 : ε > 0\nF : α\nF0 : F > 0\nhF : ∀ (i : ℕ), abv (↑f i) < F\ni : ℕ\nH : ∀ j ≥ i, abv (↑g j) < ε / F\nj : ℕ\nij : j ≥ i\nthis : abv (↑f j) * abv (↑g j) < F * (ε / F)\n⊢ abv (↑(f * g) j) < ε","tactic":"have := mul_lt_mul' (le_of_lt <| hF j) (H _ ij) (abv_nonneg abv _) F0","premises":[{"full_name":"IsAbsoluteValue.abv_nonneg","def_path":"Mathlib/Algebra/Order/AbsoluteValue.lean","def_pos":[277,6],"def_end_pos":[277,16]},{"full_name":"le_of_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[89,8],"def_end_pos":[89,16]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝² : LinearOrderedField α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf g : CauSeq β abv\nhg : g.LimZero\nε : α\nε0 : ε > 0\nF : α\nF0 : F > 0\nhF : ∀ (i : ℕ), abv (↑f i) < F\ni : ℕ\nH : ∀ j ≥ i, abv (↑g j) < ε / F\nj : ℕ\nij : j ≥ i\nthis : abv (↑f j) * abv (↑g j) < F * (ε / F)\n⊢ abv (↑(f * g) j) < ε","state_after":"no goals","tactic":"rwa [mul_comm F, div_mul_cancel₀ _ (ne_of_gt F0), ← abv_mul] at this","premises":[{"full_name":"IsAbsoluteValue.abv_mul","def_path":"Mathlib/Algebra/Order/AbsoluteValue.lean","def_pos":[291,6],"def_end_pos":[291,13]},{"full_name":"div_mul_cancel₀","def_path":"Mathlib/Algebra/GroupWithZero/Units/Basic.lean","def_pos":[289,14],"def_end_pos":[289,29]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]},{"full_name":"ne_of_gt","def_path":"Mathlib/Order/Defs.lean","def_pos":[85,8],"def_end_pos":[85,16]}]}]}
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TopologicalSpace M''\ninst✝¹ : ChartedSpace H'' M''\ninst✝ : SmoothManifoldWithCorners I'' M''\ns : Set M\nx✝ z : M\nf✝ g : M → E'\nf' g' : TangentSpace I z →L[𝕜] E'\nf : M → E'\nx : M\n⊢ mfderiv I 𝓘(𝕜, E') (-f) x = -mfderiv I 𝓘(𝕜, E') f x","state_after":"𝕜 : Type u_1\ninst✝¹⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁷ : NormedAddCommGroup E\ninst✝¹⁶ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁵ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹⁴ : TopologicalSpace M\ninst✝¹³ : ChartedSpace H M\ninst✝¹² : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst✝¹¹ : NormedAddCommGroup E'\ninst✝¹⁰ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝⁹ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM' : Type u_7\ninst✝⁸ : TopologicalSpace M'\ninst✝⁷ : ChartedSpace H' M'\ninst✝⁶ : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst✝⁵ : NormedAddCommGroup E''\ninst✝⁴ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝³ : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝² : TopologicalSpace M''\ninst✝¹ : ChartedSpace H'' M''\ninst✝ : SmoothManifoldWithCorners I'' M''\ns : Set M\nx✝ z : M\nf✝ g : M → E'\nf' g' : TangentSpace I z →L[𝕜] E'\nf : M → E'\nx : M\n⊢ (if MDifferentiableAt I 𝓘(𝕜, E') (-f) x then\n      fderivWithin 𝕜 (writtenInExtChartAt I 𝓘(𝕜, E') x (-f)) (range ↑I) (↑(extChartAt I x) x)\n    else 0) =\n    -if MDifferentiableAt I 𝓘(𝕜, E') f x then\n        fderivWithin 𝕜 (writtenInExtChartAt I 𝓘(𝕜, E') x f) (range ↑I) (↑(extChartAt I x) x)\n      else 0","tactic":"simp_rw [mfderiv]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"mfderiv","def_path":"Mathlib/Geometry/Manifold/MFDeriv/Defs.lean","def_pos":[312,4],"def_end_pos":[312,11]}]},{"state_before":"𝕜 : Type u_1\ninst✝¹⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁷ : NormedAddCommGroup E\ninst✝¹⁶ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁵ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹⁴ : TopologicalSpace M\ninst✝¹³ : ChartedSpace H M\ninst✝¹² : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst✝¹¹ : NormedAddCommGroup E'\ninst✝¹⁰ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝⁹ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM' : Type u_7\ninst✝⁸ : TopologicalSpace M'\ninst✝⁷ : ChartedSpace H' M'\ninst✝⁶ : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst✝⁵ : NormedAddCommGroup E''\ninst✝⁴ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝³ : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝² : TopologicalSpace M''\ninst✝¹ : ChartedSpace H'' M''\ninst✝ : SmoothManifoldWithCorners I'' M''\ns : Set M\nx✝ z : M\nf✝ g : M → E'\nf' g' : TangentSpace I z →L[𝕜] E'\nf : M → E'\nx : M\n⊢ (if MDifferentiableAt I 𝓘(𝕜, E') (-f) x then\n      fderivWithin 𝕜 (writtenInExtChartAt I 𝓘(𝕜, E') x (-f)) (range ↑I) (↑(extChartAt I x) x)\n    else 0) =\n    -if MDifferentiableAt I 𝓘(𝕜, E') f x then\n        fderivWithin 𝕜 (writtenInExtChartAt I 𝓘(𝕜, E') x f) (range ↑I) (↑(extChartAt I x) x)\n      else 0","state_after":"case pos\n𝕜 : Type u_1\ninst✝¹⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁷ : NormedAddCommGroup E\ninst✝¹⁶ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁵ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹⁴ : TopologicalSpace M\ninst✝¹³ : ChartedSpace H M\ninst✝¹² : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst✝¹¹ : NormedAddCommGroup E'\ninst✝¹⁰ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝⁹ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM' : Type u_7\ninst✝⁸ : TopologicalSpace M'\ninst✝⁷ : ChartedSpace H' M'\ninst✝⁶ : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst✝⁵ : NormedAddCommGroup E''\ninst✝⁴ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝³ : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝² : TopologicalSpace M''\ninst✝¹ : ChartedSpace H'' M''\ninst✝ : SmoothManifoldWithCorners I'' M''\ns : Set M\nx✝ z : M\nf✝ g : M → E'\nf' g' : TangentSpace I z →L[𝕜] E'\nf : M → E'\nx : M\nhf : MDifferentiableAt I 𝓘(𝕜, E') f x\n⊢ (if MDifferentiableAt I 𝓘(𝕜, E') (-f) x then\n      fderivWithin 𝕜 (writtenInExtChartAt I 𝓘(𝕜, E') x (-f)) (range ↑I) (↑(extChartAt I x) x)\n    else 0) =\n    -if MDifferentiableAt I 𝓘(𝕜, E') f x then\n        fderivWithin 𝕜 (writtenInExtChartAt I 𝓘(𝕜, E') x f) (range ↑I) (↑(extChartAt I x) x)\n      else 0\n\ncase neg\n𝕜 : Type u_1\ninst✝¹⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁷ : NormedAddCommGroup E\ninst✝¹⁶ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁵ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹⁴ : TopologicalSpace M\ninst✝¹³ : ChartedSpace H M\ninst✝¹² : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst✝¹¹ : NormedAddCommGroup E'\ninst✝¹⁰ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝⁹ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM' : Type u_7\ninst✝⁸ : TopologicalSpace M'\ninst✝⁷ : ChartedSpace H' M'\ninst✝⁶ : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst✝⁵ : NormedAddCommGroup E''\ninst✝⁴ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝³ : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝² : TopologicalSpace M''\ninst✝¹ : ChartedSpace H'' M''\ninst✝ : SmoothManifoldWithCorners I'' M''\ns : Set M\nx✝ z : M\nf✝ g : M → E'\nf' g' : TangentSpace I z →L[𝕜] E'\nf : M → E'\nx : M\nhf : ¬MDifferentiableAt I 𝓘(𝕜, E') f x\n⊢ (if MDifferentiableAt I 𝓘(𝕜, E') (-f) x then\n      fderivWithin 𝕜 (writtenInExtChartAt I 𝓘(𝕜, E') x (-f)) (range ↑I) (↑(extChartAt I x) x)\n    else 0) =\n    -if MDifferentiableAt I 𝓘(𝕜, E') f x then\n        fderivWithin 𝕜 (writtenInExtChartAt I 𝓘(𝕜, E') x f) (range ↑I) (↑(extChartAt I x) x)\n      else 0","tactic":"by_cases hf : MDifferentiableAt I 𝓘(𝕜, E') f x","premises":[{"full_name":"MDifferentiableAt","def_path":"Mathlib/Geometry/Manifold/MFDeriv/Defs.lean","def_pos":[226,4],"def_end_pos":[226,21]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]},{"full_name":"modelWithCornersSelf","def_path":"Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean","def_pos":[148,4],"def_end_pos":[148,24]}]}]}
+{"url":"Mathlib/NumberTheory/Divisors.lean","commit":"","full_name":"Nat.Prime.divisors","start":[331,0],"end":[333,99],"file_path":"Mathlib/NumberTheory/Divisors.lean","tactics":[{"state_before":"n p : ℕ\npp : Prime p\n⊢ p.divisors = {1, p}","state_after":"case a\nn p : ℕ\npp : Prime p\na✝ : ℕ\n⊢ a✝ ∈ p.divisors ↔ a✝ ∈ {1, p}","tactic":"ext","premises":[]},{"state_before":"case a\nn p : ℕ\npp : Prime p\na✝ : ℕ\n⊢ a✝ ∈ p.divisors ↔ a✝ ∈ {1, p}","state_after":"no goals","tactic":"rw [mem_divisors, dvd_prime pp, and_iff_left pp.ne_zero, Finset.mem_insert, Finset.mem_singleton]","premises":[{"full_name":"Finset.mem_insert","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[941,8],"def_end_pos":[941,18]},{"full_name":"Finset.mem_singleton","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[584,8],"def_end_pos":[584,21]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Nat.Prime.ne_zero","def_path":"Mathlib/Data/Nat/Prime/Defs.lean","def_pos":[48,8],"def_end_pos":[48,21]},{"full_name":"Nat.dvd_prime","def_path":"Mathlib/Data/Nat/Prime/Defs.lean","def_pos":[151,8],"def_end_pos":[151,17]},{"full_name":"Nat.mem_divisors","def_path":"Mathlib/NumberTheory/Divisors.lean","def_pos":[85,8],"def_end_pos":[85,20]},{"full_name":"and_iff_left","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[69,8],"def_end_pos":[69,20]}]}]}
+{"url":"Mathlib/LinearAlgebra/LinearDisjoint.lean","commit":"","full_name":"Submodule.LinearDisjoint.of_basis_right'","start":[199,0],"end":[207,11],"file_path":"Mathlib/LinearAlgebra/LinearDisjoint.lean","tactics":[{"state_before":"R : Type u\nS : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring S\ninst✝ : Algebra R S\nM N : Submodule R S\nι : Type u_1\nn : Basis ι R ↥N\nH : Function.Injective ⇑(M.mulRightMap ⇑n)\n⊢ M.LinearDisjoint N","state_after":"R : Type u\nS : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring S\ninst✝ : Algebra R S\nM N : Submodule R S\nι : Type u_1\nn : Basis ι R ↥N\nH : Function.Injective ⇑(M.mulMap N)\n⊢ M.LinearDisjoint N","tactic":"classical simp_rw [mulRightMap_eq_mulMap_comp, ← Basis.coe_repr_symm,\n    ← LinearEquiv.coe_lTensor, LinearEquiv.comp_coe, LinearMap.coe_comp,\n    LinearEquiv.coe_coe, EquivLike.injective_comp] at H","premises":[{"full_name":"Basis.coe_repr_symm","def_path":"Mathlib/LinearAlgebra/Basis.lean","def_pos":[147,8],"def_end_pos":[147,21]},{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"EquivLike.injective_comp","def_path":"Mathlib/Data/FunLike/Equiv.lean","def_pos":[178,8],"def_end_pos":[178,22]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"LinearEquiv.coe_coe","def_path":"Mathlib/Algebra/Module/Equiv/Defs.lean","def_pos":[208,8],"def_end_pos":[208,15]},{"full_name":"LinearEquiv.coe_lTensor","def_path":"Mathlib/LinearAlgebra/TensorProduct/Basic.lean","def_pos":[1279,16],"def_end_pos":[1279,27]},{"full_name":"LinearEquiv.comp_coe","def_path":"Mathlib/Algebra/Module/Equiv/Defs.lean","def_pos":[428,8],"def_end_pos":[428,16]},{"full_name":"LinearMap.coe_comp","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[508,8],"def_end_pos":[508,16]},{"full_name":"Submodule.mulRightMap_eq_mulMap_comp","def_path":"Mathlib/LinearAlgebra/TensorProduct/Submodule.lean","def_pos":[242,8],"def_end_pos":[242,34]}]},{"state_before":"R : Type u\nS : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring S\ninst✝ : Algebra R S\nM N : Submodule R S\nι : Type u_1\nn : Basis ι R ↥N\nH : Function.Injective ⇑(M.mulMap N)\n⊢ M.LinearDisjoint N","state_after":"no goals","tactic":"exact ⟨H⟩","premises":[]}]}
+{"url":"Mathlib/Algebra/Algebra/Spectrum.lean","commit":"","full_name":"spectrum.not_mem_iff","start":[100,0],"end":[102,33],"file_path":"Mathlib/Algebra/Algebra/Spectrum.lean","tactics":[{"state_before":"R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nr : R\na : A\n⊢ r ∉ σ a ↔ IsUnit (↑ₐ r - a)","state_after":"R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nr : R\na : A\n⊢ ¬r ∉ σ a ↔ ¬IsUnit (↑ₐ r - a)","tactic":"apply not_iff_not.mp","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"not_iff_not","def_path":"Mathlib/Logic/Basic.lean","def_pos":[319,8],"def_end_pos":[319,19]}]},{"state_before":"R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nr : R\na : A\n⊢ ¬r ∉ σ a ↔ ¬IsUnit (↑ₐ r - a)","state_after":"no goals","tactic":"simp [Set.not_not_mem, mem_iff]","premises":[{"full_name":"Set.not_not_mem","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[349,8],"def_end_pos":[349,19]},{"full_name":"spectrum.mem_iff","def_path":"Mathlib/Algebra/Algebra/Spectrum.lean","def_pos":[97,8],"def_end_pos":[97,15]}]}]}
+{"url":"Mathlib/GroupTheory/GroupAction/Basic.lean","commit":"","full_name":"MulAction.orbitRel.Quotient.mem_subgroup_orbit_iff'","start":[563,0],"end":[576,59],"file_path":"Mathlib/GroupTheory/GroupAction/Basic.lean","tactics":[{"state_before":"G : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝² : Group G\ninst✝¹ : MulAction G α\ninst✝ : MulAction G β\nH : Subgroup G\nx : Quotient G α\na b : ↑x.orbit\nc : α\nh : ⟦a⟧ = ⟦b⟧\n⊢ ↑a ∈ MulAction.orbit (↥H) c ↔ ↑b ∈ MulAction.orbit (↥H) c","state_after":"G : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝² : Group G\ninst✝¹ : MulAction G α\ninst✝ : MulAction G β\nH : Subgroup G\nx : Quotient G α\na b : ↑x.orbit\nc : α\nh : ⟦a⟧ = ⟦b⟧\n⊢ c ∈ MulAction.orbit ↥H ↑a ↔ c ∈ MulAction.orbit ↥H ↑b","tactic":"simp_rw [mem_orbit_symm (a₂ := c)]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"MulAction.mem_orbit_symm","def_path":"Mathlib/GroupTheory/GroupAction/Basic.lean","def_pos":[341,6],"def_end_pos":[341,20]}]},{"state_before":"G : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝² : Group G\ninst✝¹ : MulAction G α\ninst✝ : MulAction G β\nH : Subgroup G\nx : Quotient G α\na b : ↑x.orbit\nc : α\nh : ⟦a⟧ = ⟦b⟧\n⊢ c ∈ MulAction.orbit ↥H ↑a ↔ c ∈ MulAction.orbit ↥H ↑b","state_after":"case h.e'_2.h.e'_5\nG : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝² : Group G\ninst✝¹ : MulAction G α\ninst✝ : MulAction G β\nH : Subgroup G\nx : Quotient G α\na b : ↑x.orbit\nc : α\nh : ⟦a⟧ = ⟦b⟧\n⊢ MulAction.orbit ↥H ↑b = MulAction.orbit ↥H ↑a","tactic":"convert Iff.rfl using 2","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]}]},{"state_before":"case h.e'_2.h.e'_5\nG : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝² : Group G\ninst✝¹ : MulAction G α\ninst✝ : MulAction G β\nH : Subgroup G\nx : Quotient G α\na b : ↑x.orbit\nc : α\nh : ⟦a⟧ = ⟦b⟧\n⊢ MulAction.orbit ↥H ↑b = MulAction.orbit ↥H ↑a","state_after":"case h.e'_2.h.e'_5\nG : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝² : Group G\ninst✝¹ : MulAction G α\ninst✝ : MulAction G β\nH : Subgroup G\nx : Quotient G α\na b : ↑x.orbit\nc : α\nh : ⟦a⟧ = ⟦b⟧\n⊢ ↑b ∈ MulAction.orbit ↥H ↑a","tactic":"rw [orbit_eq_iff]","premises":[{"full_name":"MulAction.orbit_eq_iff","def_path":"Mathlib/GroupTheory/GroupAction/Basic.lean","def_pos":[320,8],"def_end_pos":[320,20]}]},{"state_before":"case h.e'_2.h.e'_5\nG : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝² : Group G\ninst✝¹ : MulAction G α\ninst✝ : MulAction G β\nH : Subgroup G\nx : Quotient G α\na b : ↑x.orbit\nc : α\nh : ⟦a⟧ = ⟦b⟧\n⊢ ↑b ∈ MulAction.orbit ↥H ↑a","state_after":"case h.e'_2.h.e'_5\nG : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝² : Group G\ninst✝¹ : MulAction G α\ninst✝ : MulAction G β\nH : Subgroup G\nx : Quotient G α\na b : ↑x.orbit\nc : α\nh : ⟦a⟧ = ⟦b⟧\n⊢ b ∈ orbit ⟦a⟧","tactic":"suffices hb : ↑b ∈ orbitRel.Quotient.orbit (⟦a⟧ : orbitRel.Quotient H x.orbit) by\n    rw [orbitRel.Quotient.orbit_eq_orbit_out (⟦a⟧ : orbitRel.Quotient H x.orbit) Quotient.out_eq']\n       at hb\n    rw [orbitRel.Quotient.mem_subgroup_orbit_iff]\n    convert hb using 1\n    rw [orbit_eq_iff, ← orbitRel_r_apply, ← Quotient.eq'', Quotient.out_eq', @Quotient.mk''_eq_mk]","premises":[{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"MulAction.orbitRel.Quotient","def_path":"Mathlib/GroupTheory/GroupAction/Basic.lean","def_pos":[433,4],"def_end_pos":[433,21]},{"full_name":"MulAction.orbitRel.Quotient.mem_subgroup_orbit_iff","def_path":"Mathlib/GroupTheory/GroupAction/Basic.lean","def_pos":[544,6],"def_end_pos":[544,46]},{"full_name":"MulAction.orbitRel.Quotient.orbit","def_path":"Mathlib/GroupTheory/GroupAction/Basic.lean","def_pos":[461,11],"def_end_pos":[461,34]},{"full_name":"MulAction.orbitRel.Quotient.orbit_eq_orbit_out","def_path":"Mathlib/GroupTheory/GroupAction/Basic.lean","def_pos":[478,8],"def_end_pos":[478,44]},{"full_name":"MulAction.orbitRel_r_apply","def_path":"Mathlib/GroupTheory/GroupAction/Basic.lean","def_pos":[374,6],"def_end_pos":[374,22]},{"full_name":"MulAction.orbit_eq_iff","def_path":"Mathlib/GroupTheory/GroupAction/Basic.lean","def_pos":[320,8],"def_end_pos":[320,20]},{"full_name":"Quotient.eq''","def_path":"Mathlib/Data/Quot.lean","def_pos":[695,18],"def_end_pos":[695,22]},{"full_name":"Quotient.mk","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1592,14],"def_end_pos":[1592,16]},{"full_name":"Quotient.mk''_eq_mk","def_path":"Mathlib/Data/Quot.lean","def_pos":[714,18],"def_end_pos":[714,28]},{"full_name":"Quotient.out_eq'","def_path":"Mathlib/Data/Quot.lean","def_pos":[704,8],"def_end_pos":[704,15]}]},{"state_before":"case h.e'_2.h.e'_5\nG : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝² : Group G\ninst✝¹ : MulAction G α\ninst✝ : MulAction G β\nH : Subgroup G\nx : Quotient G α\na b : ↑x.orbit\nc : α\nh : ⟦a⟧ = ⟦b⟧\n⊢ b ∈ orbit ⟦a⟧","state_after":"no goals","tactic":"rw [orbitRel.Quotient.mem_orbit, h, @Quotient.mk''_eq_mk]","premises":[{"full_name":"MulAction.orbitRel.Quotient.mem_orbit","def_path":"Mathlib/GroupTheory/GroupAction/Basic.lean","def_pos":[470,8],"def_end_pos":[470,35]},{"full_name":"Quotient.mk''_eq_mk","def_path":"Mathlib/Data/Quot.lean","def_pos":[714,18],"def_end_pos":[714,28]}]}]}
+{"url":"Mathlib/MeasureTheory/Measure/Tilted.lean","commit":"","full_name":"MeasureTheory.integral_tilted","start":[236,0],"end":[238,81],"file_path":"Mathlib/MeasureTheory/Measure/Tilted.lean","tactics":[{"state_before":"α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\nf✝ : α → ℝ\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ\ng : α → E\n⊢ ∫ (x : α), g x ∂μ.tilted f = ∫ (x : α), (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) • g x ∂μ","state_after":"no goals","tactic":"rw [← integral_univ, setIntegral_tilted' f g MeasurableSet.univ, integral_univ]","premises":[{"full_name":"MeasurableSet.univ","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","def_pos":[95,18],"def_end_pos":[95,36]},{"full_name":"MeasureTheory.integral_univ","def_path":"Mathlib/MeasureTheory/Integral/SetIntegral.lean","def_pos":[151,8],"def_end_pos":[151,21]},{"full_name":"MeasureTheory.setIntegral_tilted'","def_path":"Mathlib/MeasureTheory/Measure/Tilted.lean","def_pos":[196,6],"def_end_pos":[196,25]}]}]}
+{"url":"Mathlib/Analysis/LocallyConvex/WithSeminorms.lean","commit":"","full_name":"WithSeminorms.first_countable","start":[899,0],"end":[910,45],"file_path":"Mathlib/Analysis/LocallyConvex/WithSeminorms.lean","tactics":[{"state_before":"𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕝 : Type u_3\n𝕝₂ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nι : Type u_8\nι' : Type u_9\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Nonempty ι\ninst✝¹ : Countable ι\np : SeminormFamily 𝕜 E ι\ninst✝ : TopologicalSpace E\nhp : WithSeminorms p\n⊢ FirstCountableTopology E","state_after":"𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕝 : Type u_3\n𝕝₂ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nι : Type u_8\nι' : Type u_9\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Nonempty ι\ninst✝¹ : Countable ι\np : SeminormFamily 𝕜 E ι\ninst✝ : TopologicalSpace E\nhp : WithSeminorms p\nthis : TopologicalAddGroup E\n⊢ FirstCountableTopology E","tactic":"have := hp.topologicalAddGroup","premises":[{"full_name":"WithSeminorms.topologicalAddGroup","def_path":"Mathlib/Analysis/LocallyConvex/WithSeminorms.lean","def_pos":[272,8],"def_end_pos":[272,41]}]},{"state_before":"𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕝 : Type u_3\n𝕝₂ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nι : Type u_8\nι' : Type u_9\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Nonempty ι\ninst✝¹ : Countable ι\np : SeminormFamily 𝕜 E ι\ninst✝ : TopologicalSpace E\nhp : WithSeminorms p\nthis : TopologicalAddGroup E\n⊢ FirstCountableTopology E","state_after":"𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕝 : Type u_3\n𝕝₂ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nι : Type u_8\nι' : Type u_9\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Nonempty ι\ninst✝¹ : Countable ι\np : SeminormFamily 𝕜 E ι\ninst✝ : TopologicalSpace E\nhp : WithSeminorms p\nthis : TopologicalAddGroup E\nx✝ : UniformSpace E := TopologicalAddGroup.toUniformSpace E\n⊢ FirstCountableTopology E","tactic":"let _ : UniformSpace E := TopologicalAddGroup.toUniformSpace E","premises":[{"full_name":"TopologicalAddGroup.toUniformSpace","def_path":"Mathlib/Topology/Algebra/UniformGroup.lean","def_pos":[451,2],"def_end_pos":[451,13]},{"full_name":"UniformSpace","def_path":"Mathlib/Topology/UniformSpace/Basic.lean","def_pos":[280,6],"def_end_pos":[280,18]}]},{"state_before":"𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕝 : Type u_3\n𝕝₂ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nι : Type u_8\nι' : Type u_9\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Nonempty ι\ninst✝¹ : Countable ι\np : SeminormFamily 𝕜 E ι\ninst✝ : TopologicalSpace E\nhp : WithSeminorms p\nthis : TopologicalAddGroup E\nx✝ : UniformSpace E := TopologicalAddGroup.toUniformSpace E\n⊢ FirstCountableTopology E","state_after":"𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕝 : Type u_3\n𝕝₂ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nι : Type u_8\nι' : Type u_9\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Nonempty ι\ninst✝¹ : Countable ι\np : SeminormFamily 𝕜 E ι\ninst✝ : TopologicalSpace E\nhp : WithSeminorms p\nthis✝ : TopologicalAddGroup E\nx✝ : UniformSpace E := TopologicalAddGroup.toUniformSpace E\nthis : UniformAddGroup E\n⊢ FirstCountableTopology E","tactic":"have : UniformAddGroup E := comm_topologicalAddGroup_is_uniform","premises":[{"full_name":"UniformAddGroup","def_path":"Mathlib/Topology/Algebra/UniformGroup.lean","def_pos":[55,6],"def_end_pos":[55,21]},{"full_name":"comm_topologicalAddGroup_is_uniform","def_path":"Mathlib/Topology/Algebra/UniformGroup.lean","def_pos":[561,2],"def_end_pos":[561,13]}]},{"state_before":"𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕝 : Type u_3\n𝕝₂ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nι : Type u_8\nι' : Type u_9\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Nonempty ι\ninst✝¹ : Countable ι\np : SeminormFamily 𝕜 E ι\ninst✝ : TopologicalSpace E\nhp : WithSeminorms p\nthis✝ : TopologicalAddGroup E\nx✝ : UniformSpace E := TopologicalAddGroup.toUniformSpace E\nthis : UniformAddGroup E\n⊢ FirstCountableTopology E","state_after":"𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕝 : Type u_3\n𝕝₂ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nι : Type u_8\nι' : Type u_9\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Nonempty ι\ninst✝¹ : Countable ι\np : SeminormFamily 𝕜 E ι\ninst✝ : TopologicalSpace E\nhp : WithSeminorms p\nthis✝¹ : TopologicalAddGroup E\nx✝ : UniformSpace E := TopologicalAddGroup.toUniformSpace E\nthis✝ : UniformAddGroup E\nthis : (𝓝 0).IsCountablyGenerated\n⊢ FirstCountableTopology E","tactic":"have : (𝓝 (0 : E)).IsCountablyGenerated := by\n    rw [p.withSeminorms_iff_nhds_eq_iInf.mp hp]\n    exact Filter.iInf.isCountablyGenerated _","premises":[{"full_name":"Filter.IsCountablyGenerated","def_path":"Mathlib/Order/Filter/Bases.lean","def_pos":[866,6],"def_end_pos":[866,26]},{"full_name":"Filter.iInf.isCountablyGenerated","def_path":"Mathlib/Order/Filter/Bases.lean","def_pos":[1059,9],"def_end_pos":[1059,34]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"SeminormFamily.withSeminorms_iff_nhds_eq_iInf","def_path":"Mathlib/Analysis/LocallyConvex/WithSeminorms.lean","def_pos":[406,8],"def_end_pos":[406,53]},{"full_name":"nhds","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[113,16],"def_end_pos":[113,20]}]},{"state_before":"𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕝 : Type u_3\n𝕝₂ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nι : Type u_8\nι' : Type u_9\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Nonempty ι\ninst✝¹ : Countable ι\np : SeminormFamily 𝕜 E ι\ninst✝ : TopologicalSpace E\nhp : WithSeminorms p\nthis✝¹ : TopologicalAddGroup E\nx✝ : UniformSpace E := TopologicalAddGroup.toUniformSpace E\nthis✝ : UniformAddGroup E\nthis : (𝓝 0).IsCountablyGenerated\n⊢ FirstCountableTopology E","state_after":"𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕝 : Type u_3\n𝕝₂ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nι : Type u_8\nι' : Type u_9\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Nonempty ι\ninst✝¹ : Countable ι\np : SeminormFamily 𝕜 E ι\ninst✝ : TopologicalSpace E\nhp : WithSeminorms p\nthis✝² : TopologicalAddGroup E\nx✝ : UniformSpace E := TopologicalAddGroup.toUniformSpace E\nthis✝¹ : UniformAddGroup E\nthis✝ : (𝓝 0).IsCountablyGenerated\nthis : (𝓤 E).IsCountablyGenerated\n⊢ FirstCountableTopology E","tactic":"have : (uniformity E).IsCountablyGenerated := UniformAddGroup.uniformity_countably_generated","premises":[{"full_name":"Filter.IsCountablyGenerated","def_path":"Mathlib/Order/Filter/Bases.lean","def_pos":[866,6],"def_end_pos":[866,26]},{"full_name":"UniformAddGroup.uniformity_countably_generated","def_path":"Mathlib/Topology/Algebra/UniformGroup.lean","def_pos":[262,2],"def_end_pos":[262,13]},{"full_name":"uniformity","def_path":"Mathlib/Topology/UniformSpace/Basic.lean","def_pos":[293,4],"def_end_pos":[293,14]}]},{"state_before":"𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕝 : Type u_3\n𝕝₂ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nι : Type u_8\nι' : Type u_9\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Nonempty ι\ninst✝¹ : Countable ι\np : SeminormFamily 𝕜 E ι\ninst✝ : TopologicalSpace E\nhp : WithSeminorms p\nthis✝² : TopologicalAddGroup E\nx✝ : UniformSpace E := TopologicalAddGroup.toUniformSpace E\nthis✝¹ : UniformAddGroup E\nthis✝ : (𝓝 0).IsCountablyGenerated\nthis : (𝓤 E).IsCountablyGenerated\n⊢ FirstCountableTopology E","state_after":"no goals","tactic":"exact UniformSpace.firstCountableTopology E","premises":[{"full_name":"UniformSpace.firstCountableTopology","def_path":"Mathlib/Topology/UniformSpace/Cauchy.lean","def_pos":[763,27],"def_end_pos":[763,49]}]}]}
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+{"url":"Mathlib/CategoryTheory/Preadditive/Biproducts.lean","commit":"","full_name":"CategoryTheory.Limits.binaryBiconeOfIsSplitMonoOfCokernel_inl","start":[439,0],"end":[469,33],"file_path":"Mathlib/CategoryTheory/Preadditive/Biproducts.lean","tactics":[{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nX✝ Y✝ : C\ninst✝¹ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\nc : CokernelCofork f\ni : IsColimit c\n⊢ (𝟙 Y - (𝟙 Y - retraction f ≫ f)) ≫ Cofork.π c = 0","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nX✝ Y✝ : C\ninst✝¹ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\nc : CokernelCofork f\ni : IsColimit c\nc' : CokernelCofork (𝟙 Y - (𝟙 Y - retraction f ≫ f)) := CokernelCofork.ofπ (Cofork.π c) ⋯\n⊢ 𝟙 Y - (𝟙 Y - retraction f ≫ f) = retraction f ≫ f","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nX✝ Y✝ : C\ninst✝¹ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\nc : CokernelCofork f\ni : IsColimit c\nc' : CokernelCofork (𝟙 Y - (𝟙 Y - retraction f ≫ f)) := CokernelCofork.ofπ (Cofork.π c) ⋯\ni' : IsColimit c' := isCokernelEpiComp i (retraction f) ⋯\ni'' : IsColimit (coforkOfCokernelCofork c') := isColimitCoforkOfCokernelCofork i'\n⊢ (𝟙 Y - retraction f ≫ f) ≫ (𝟙 Y - retraction f ≫ f) = 𝟙 Y - retraction f ≫ f","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nX✝ Y✝ : C\ninst✝¹ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\nc : CokernelCofork f\ni : IsColimit c\n⊢ f ≫ retraction f = 𝟙 X","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nX✝ Y✝ : C\ninst✝¹ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\nc : CokernelCofork f\ni : IsColimit c\n⊢ f ≫ Cofork.π c = 0","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nX✝ Y✝ : C\ninst✝¹ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\nc : CokernelCofork f\ni : IsColimit c\n⊢ (let c' := CokernelCofork.ofπ (Cofork.π c) ⋯;\n      let i' := isCokernelEpiComp i (retraction f) ⋯;\n      let i'' := isColimitCoforkOfCokernelCofork i';\n      (splitEpiOfIdempotentOfIsColimitCofork C ⋯ i'').section_) ≫\n      retraction f =\n    0","state_after":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nX✝ Y✝ : C\ninst✝¹ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\nc : CokernelCofork f\ni : IsColimit c\n⊢ (splitEpiOfIdempotentOfIsColimitCofork C ⋯\n          (isColimitCoforkOfCokernelCofork (isCokernelEpiComp i (retraction f) ⋯))).section_ ≫\n      retraction f =\n    0","tactic":"dsimp only","premises":[]},{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nX✝ Y✝ : C\ninst✝¹ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\nc : CokernelCofork f\ni : IsColimit c\n⊢ (splitEpiOfIdempotentOfIsColimitCofork C ⋯\n          (isColimitCoforkOfCokernelCofork (isCokernelEpiComp i (retraction f) ⋯))).section_ ≫\n      retraction f =\n    0","state_after":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nX✝ Y✝ : C\ninst✝¹ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\nc : CokernelCofork f\ni : IsColimit c\n⊢ i.desc (Cofork.ofπ (Cofork.π (cokernelCoforkOfCofork (Cofork.ofπ (𝟙 Y - retraction f ≫ f) ⋯))) ⋯) ≫ retraction f = 0","tactic":"rw [splitEpiOfIdempotentOfIsColimitCofork_section_,\n      isColimitCoforkOfCokernelCofork_desc, isCokernelEpiComp_desc]","premises":[{"full_name":"CategoryTheory.Limits.isCokernelEpiComp_desc","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","def_pos":[596,8],"def_end_pos":[596,30]},{"full_name":"CategoryTheory.Limits.splitEpiOfIdempotentOfIsColimitCofork_section_","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean","def_pos":[1190,2],"def_end_pos":[1190,7]},{"full_name":"CategoryTheory.Preadditive.isColimitCoforkOfCokernelCofork_desc","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[368,8],"def_end_pos":[368,44]}]},{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nX✝ Y✝ : C\ninst✝¹ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\nc : CokernelCofork f\ni : IsColimit c\n⊢ i.desc (Cofork.ofπ (Cofork.π (cokernelCoforkOfCofork (Cofork.ofπ (𝟙 Y - retraction f ≫ f) ⋯))) ⋯) ≫ retraction f = 0","state_after":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nX✝ Y✝ : C\ninst✝¹ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\nc : CokernelCofork f\ni : IsColimit c\n⊢ i.desc (Cofork.ofπ (Cofork.π (CokernelCofork.ofπ (𝟙 Y - retraction f ≫ f) ⋯)) ⋯) ≫ retraction f = 0","tactic":"dsimp only [cokernelCoforkOfCofork_ofπ]","premises":[{"full_name":"CategoryTheory.Preadditive.cokernelCoforkOfCofork_ofπ","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[356,8],"def_end_pos":[356,34]}]},{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nX✝ Y✝ : C\ninst✝¹ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\nc : CokernelCofork f\ni : IsColimit c\n⊢ i.desc (Cofork.ofπ (Cofork.π (CokernelCofork.ofπ (𝟙 Y - retraction f ≫ f) ⋯)) ⋯) ≫ retraction f = 0","state_after":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nX✝ Y✝ : C\ninst✝¹ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\nc : CokernelCofork f\ni : IsColimit c\nthis : Epi (Cofork.π c) := epi_of_isColimit_cofork i\n⊢ i.desc (Cofork.ofπ (Cofork.π (CokernelCofork.ofπ (𝟙 Y - retraction f ≫ f) ⋯)) ⋯) ≫ retraction f = 0","tactic":"letI := epi_of_isColimit_cofork i","premises":[{"full_name":"CategoryTheory.Limits.epi_of_isColimit_cofork","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean","def_pos":[915,8],"def_end_pos":[915,31]}]},{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nX✝ Y✝ : C\ninst✝¹ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\nc : CokernelCofork f\ni : IsColimit c\nthis : Epi (Cofork.π c) := epi_of_isColimit_cofork i\n⊢ i.desc (Cofork.ofπ (Cofork.π (CokernelCofork.ofπ (𝟙 Y - retraction f ≫ f) ⋯)) ⋯) ≫ retraction f = 0","state_after":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nX✝ Y✝ : C\ninst✝¹ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\nc : CokernelCofork f\ni : IsColimit c\nthis : Epi (Cofork.π c) := epi_of_isColimit_cofork i\n⊢ Cofork.π c ≫ i.desc (Cofork.ofπ (Cofork.π (CokernelCofork.ofπ (𝟙 Y - retraction f ≫ f) ⋯)) ⋯) ≫ retraction f = 0","tactic":"apply zero_of_epi_comp c.π","premises":[{"full_name":"CategoryTheory.Limits.Cofork.π","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean","def_pos":[294,4],"def_end_pos":[294,12]},{"full_name":"CategoryTheory.Limits.zero_of_epi_comp","def_path":"Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean","def_pos":[130,8],"def_end_pos":[130,24]}]},{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nX✝ Y✝ : C\ninst✝¹ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\nc : CokernelCofork f\ni : IsColimit c\nthis : Epi (Cofork.π c) := epi_of_isColimit_cofork i\n⊢ Cofork.π c ≫ i.desc (Cofork.ofπ (Cofork.π (CokernelCofork.ofπ (𝟙 Y - retraction f ≫ f) ⋯)) ⋯) ≫ retraction f = 0","state_after":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nX✝ Y✝ : C\ninst✝¹ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\nc : CokernelCofork f\ni : IsColimit c\nthis : Epi (Cofork.π c) := epi_of_isColimit_cofork i\n⊢ 𝟙 Y ≫ retraction f - retraction f = 0","tactic":"simp only [sub_comp, comp_sub, Category.comp_id, Category.assoc, IsSplitMono.id, sub_self,\n      Cofork.IsColimit.π_desc_assoc, CokernelCofork.π_ofπ, IsSplitMono.id_assoc]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Category.comp_id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[153,2],"def_end_pos":[153,9]},{"full_name":"CategoryTheory.IsSplitMono.id","def_path":"Mathlib/CategoryTheory/EpiMono.lean","def_pos":[91,8],"def_end_pos":[91,22]},{"full_name":"CategoryTheory.Limits.CokernelCofork.π_ofπ","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","def_pos":[528,8],"def_end_pos":[528,28]},{"full_name":"CategoryTheory.Preadditive.comp_sub","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[133,8],"def_end_pos":[133,16]},{"full_name":"CategoryTheory.Preadditive.sub_comp","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[128,8],"def_end_pos":[128,16]},{"full_name":"sub_self","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[695,29],"def_end_pos":[695,37]}]},{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nX✝ Y✝ : C\ninst✝¹ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\nc : CokernelCofork f\ni : IsColimit c\nthis : Epi (Cofork.π c) := epi_of_isColimit_cofork i\n⊢ 𝟙 Y ≫ retraction f - retraction f = 0","state_after":"case a\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nX✝ Y✝ : C\ninst✝¹ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\nc : CokernelCofork f\ni : IsColimit c\nthis : Epi (Cofork.π c) := epi_of_isColimit_cofork i\n⊢ 𝟙 Y ≫ retraction f = retraction f","tactic":"apply sub_eq_zero_of_eq","premises":[]},{"state_before":"case a\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nX✝ Y✝ : C\ninst✝¹ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\nc : CokernelCofork f\ni : IsColimit c\nthis : Epi (Cofork.π c) := epi_of_isColimit_cofork i\n⊢ 𝟙 Y ≫ retraction f = retraction f","state_after":"no goals","tactic":"apply Category.id_comp","premises":[{"full_name":"CategoryTheory.Category.id_comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[151,2],"def_end_pos":[151,9]}]},{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nX✝ Y✝ : C\ninst✝¹ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\nc : CokernelCofork f\ni : IsColimit c\n⊢ (let c' := CokernelCofork.ofπ (Cofork.π c) ⋯;\n      let i' := isCokernelEpiComp i (retraction f) ⋯;\n      let i'' := isColimitCoforkOfCokernelCofork i';\n      (splitEpiOfIdempotentOfIsColimitCofork C ⋯ i'').section_) ≫\n      Cofork.π c =\n    𝟙 c.pt","state_after":"no goals","tactic":"apply SplitEpi.id","premises":[{"full_name":"CategoryTheory.SplitEpi.id","def_path":"Mathlib/CategoryTheory/EpiMono.lean","def_pos":[73,2],"def_end_pos":[73,4]}]}]}
+{"url":"Mathlib/Order/SymmDiff.lean","commit":"","full_name":"symmDiff_right_comm","start":[406,0],"end":[406,97],"file_path":"Mathlib/Order/SymmDiff.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\nπ : ι → Type u_4\ninst✝ : GeneralizedBooleanAlgebra α\na b c d : α\n⊢ a ∆ b ∆ c = a ∆ c ∆ b","state_after":"no goals","tactic":"simp_rw [symmDiff_assoc, symmDiff_comm]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"symmDiff_assoc","def_path":"Mathlib/Order/SymmDiff.lean","def_pos":[397,8],"def_end_pos":[397,22]},{"full_name":"symmDiff_comm","def_path":"Mathlib/Order/SymmDiff.lean","def_pos":[102,8],"def_end_pos":[102,21]}]}]}
+{"url":"Mathlib/Analysis/MellinTransform.lean","commit":"","full_name":"mellin_convergent_of_isBigO_scalar","start":[244,0],"end":[261,98],"file_path":"Mathlib/Analysis/MellinTransform.lean","tactics":[{"state_before":"E : Type u_1\ninst✝ : NormedAddCommGroup E\na b : ℝ\nf : ℝ → ℝ\ns : ℝ\nhfc : LocallyIntegrableOn f (Ioi 0) volume\nhf_top : f =O[atTop] fun x => x ^ (-a)\nhs_top : s < a\nhf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)\nhs_bot : b < s\n⊢ IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi 0) volume","state_after":"case intro.intro\nE : Type u_1\ninst✝ : NormedAddCommGroup E\na b : ℝ\nf : ℝ → ℝ\ns : ℝ\nhfc : LocallyIntegrableOn f (Ioi 0) volume\nhf_top : f =O[atTop] fun x => x ^ (-a)\nhs_top : s < a\nhf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)\nhs_bot : b < s\nc1 : ℝ\nhc1 : 0 < c1\nhc1' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume\n⊢ IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi 0) volume","tactic":"obtain ⟨c1, hc1, hc1'⟩ := mellin_convergent_top_of_isBigO hfc.aestronglyMeasurable hf_top hs_top","premises":[{"full_name":"MeasureTheory.LocallyIntegrableOn.aestronglyMeasurable","def_path":"Mathlib/MeasureTheory/Function/LocallyIntegrable.lean","def_pos":[123,8],"def_end_pos":[123,48]},{"full_name":"mellin_convergent_top_of_isBigO","def_path":"Mathlib/Analysis/MellinTransform.lean","def_pos":[190,8],"def_end_pos":[190,39]}]},{"state_before":"case intro.intro\nE : Type u_1\ninst✝ : NormedAddCommGroup E\na b : ℝ\nf : ℝ → ℝ\ns : ℝ\nhfc : LocallyIntegrableOn f (Ioi 0) volume\nhf_top : f =O[atTop] fun x => x ^ (-a)\nhs_top : s < a\nhf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)\nhs_bot : b < s\nc1 : ℝ\nhc1 : 0 < c1\nhc1' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume\n⊢ IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi 0) volume","state_after":"case intro.intro.intro.intro\nE : Type u_1\ninst✝ : NormedAddCommGroup E\na b : ℝ\nf : ℝ → ℝ\ns : ℝ\nhfc : LocallyIntegrableOn f (Ioi 0) volume\nhf_top : f =O[atTop] fun x => x ^ (-a)\nhs_top : s < a\nhf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)\nhs_bot : b < s\nc1 : ℝ\nhc1 : 0 < c1\nhc1' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume\nc2 : ℝ\nhc2 : 0 < c2\nhc2' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc 0 c2) volume\n⊢ IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi 0) volume","tactic":"obtain ⟨c2, hc2, hc2'⟩ :=\n    mellin_convergent_zero_of_isBigO hfc.aestronglyMeasurable hf_bot hs_bot","premises":[{"full_name":"MeasureTheory.LocallyIntegrableOn.aestronglyMeasurable","def_path":"Mathlib/MeasureTheory/Function/LocallyIntegrable.lean","def_pos":[123,8],"def_end_pos":[123,48]},{"full_name":"mellin_convergent_zero_of_isBigO","def_path":"Mathlib/Analysis/MellinTransform.lean","def_pos":[215,8],"def_end_pos":[215,40]}]},{"state_before":"case intro.intro.intro.intro\nE : Type u_1\ninst✝ : NormedAddCommGroup E\na b : ℝ\nf : ℝ → ℝ\ns : ℝ\nhfc : LocallyIntegrableOn f (Ioi 0) volume\nhf_top : f =O[atTop] fun x => x ^ (-a)\nhs_top : s < a\nhf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)\nhs_bot : b < s\nc1 : ℝ\nhc1 : 0 < c1\nhc1' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume\nc2 : ℝ\nhc2 : 0 < c2\nhc2' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc 0 c2) volume\n⊢ IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi 0) volume","state_after":"case intro.intro.intro.intro\nE : Type u_1\ninst✝ : NormedAddCommGroup E\na b : ℝ\nf : ℝ → ℝ\ns : ℝ\nhfc : LocallyIntegrableOn f (Ioi 0) volume\nhf_top : f =O[atTop] fun x => x ^ (-a)\nhs_top : s < a\nhf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)\nhs_bot : b < s\nc1 : ℝ\nhc1 : 0 < c1\nhc1' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume\nc2 : ℝ\nhc2 : 0 < c2\nhc2' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc 0 c2) volume\nthis : Ioi 0 = Ioc 0 c2 ∪ Ioc c2 c1 ∪ Ioi c1\n⊢ IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi 0) volume","tactic":"have : Ioi 0 = Ioc 0 c2 ∪ Ioc c2 c1 ∪ Ioi c1 := by\n    rw [union_assoc, Ioc_union_Ioi (le_max_right _ _),\n      Ioc_union_Ioi ((min_le_left _ _).trans (le_max_right _ _)), min_eq_left (lt_min hc2 hc1).le]","premises":[{"full_name":"Set.Ioc","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[62,4],"def_end_pos":[62,7]},{"full_name":"Set.Ioc_union_Ioi","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[1088,8],"def_end_pos":[1088,21]},{"full_name":"Set.Ioi","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[70,4],"def_end_pos":[70,7]},{"full_name":"Set.union_assoc","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[627,8],"def_end_pos":[627,19]},{"full_name":"Union.union","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[402,2],"def_end_pos":[402,7]},{"full_name":"le_max_right","def_path":"Mathlib/Init/Order/LinearOrder.lean","def_pos":[60,8],"def_end_pos":[60,20]},{"full_name":"lt_min","def_path":"Mathlib/Init/Order/LinearOrder.lean","def_pos":[138,8],"def_end_pos":[138,14]},{"full_name":"min_eq_left","def_path":"Mathlib/Init/Order/LinearOrder.lean","def_pos":[92,8],"def_end_pos":[92,19]},{"full_name":"min_le_left","def_path":"Mathlib/Init/Order/LinearOrder.lean","def_pos":[36,8],"def_end_pos":[36,19]}]},{"state_before":"case intro.intro.intro.intro\nE : Type u_1\ninst✝ : NormedAddCommGroup E\na b : ℝ\nf : ℝ → ℝ\ns : ℝ\nhfc : LocallyIntegrableOn f (Ioi 0) volume\nhf_top : f =O[atTop] fun x => x ^ (-a)\nhs_top : s < a\nhf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)\nhs_bot : b < s\nc1 : ℝ\nhc1 : 0 < c1\nhc1' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume\nc2 : ℝ\nhc2 : 0 < c2\nhc2' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc 0 c2) volume\nthis : Ioi 0 = Ioc 0 c2 ∪ Ioc c2 c1 ∪ Ioi c1\n⊢ IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi 0) volume","state_after":"case intro.intro.intro.intro\nE : Type u_1\ninst✝ : NormedAddCommGroup E\na b : ℝ\nf : ℝ → ℝ\ns : ℝ\nhfc : LocallyIntegrableOn f (Ioi 0) volume\nhf_top : f =O[atTop] fun x => x ^ (-a)\nhs_top : s < a\nhf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)\nhs_bot : b < s\nc1 : ℝ\nhc1 : 0 < c1\nhc1' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume\nc2 : ℝ\nhc2 : 0 < c2\nhc2' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc 0 c2) volume\nthis : Ioi 0 = Ioc 0 c2 ∪ Ioc c2 c1 ∪ Ioi c1\n⊢ (IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc 0 c2) volume ∧\n      IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc c2 c1) volume) ∧\n    IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume","tactic":"rw [this, integrableOn_union, integrableOn_union]","premises":[{"full_name":"MeasureTheory.integrableOn_union","def_path":"Mathlib/MeasureTheory/Integral/IntegrableOn.lean","def_pos":[161,8],"def_end_pos":[161,26]}]},{"state_before":"case intro.intro.intro.intro\nE : Type u_1\ninst✝ : NormedAddCommGroup E\na b : ℝ\nf : ℝ → ℝ\ns : ℝ\nhfc : LocallyIntegrableOn f (Ioi 0) volume\nhf_top : f =O[atTop] fun x => x ^ (-a)\nhs_top : s < a\nhf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)\nhs_bot : b < s\nc1 : ℝ\nhc1 : 0 < c1\nhc1' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume\nc2 : ℝ\nhc2 : 0 < c2\nhc2' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc 0 c2) volume\nthis : Ioi 0 = Ioc 0 c2 ∪ Ioc c2 c1 ∪ Ioi c1\n⊢ (IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc 0 c2) volume ∧\n      IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc c2 c1) volume) ∧\n    IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume","state_after":"case intro.intro.intro.intro\nE : Type u_1\ninst✝ : NormedAddCommGroup E\na b : ℝ\nf : ℝ → ℝ\ns : ℝ\nhfc : LocallyIntegrableOn f (Ioi 0) volume\nhf_top : f =O[atTop] fun x => x ^ (-a)\nhs_top : s < a\nhf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)\nhs_bot : b < s\nc1 : ℝ\nhc1 : 0 < c1\nhc1' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume\nc2 : ℝ\nhc2 : 0 < c2\nhc2' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc 0 c2) volume\nthis : Ioi 0 = Ioc 0 c2 ∪ Ioc c2 c1 ∪ Ioi c1\n⊢ IntegrableOn (fun t => t ^ (s - 1) * f t) (Icc c2 c1) volume","tactic":"refine ⟨⟨hc2', integrableOn_Icc_iff_integrableOn_Ioc.mp ?_⟩, hc1'⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"integrableOn_Icc_iff_integrableOn_Ioc","def_path":"Mathlib/MeasureTheory/Integral/IntegrableOn.lean","def_pos":[668,8],"def_end_pos":[668,45]}]},{"state_before":"case intro.intro.intro.intro\nE : Type u_1\ninst✝ : NormedAddCommGroup E\na b : ℝ\nf : ℝ → ℝ\ns : ℝ\nhfc : LocallyIntegrableOn f (Ioi 0) volume\nhf_top : f =O[atTop] fun x => x ^ (-a)\nhs_top : s < a\nhf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)\nhs_bot : b < s\nc1 : ℝ\nhc1 : 0 < c1\nhc1' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume\nc2 : ℝ\nhc2 : 0 < c2\nhc2' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc 0 c2) volume\nthis : Ioi 0 = Ioc 0 c2 ∪ Ioc c2 c1 ∪ Ioi c1\n⊢ IntegrableOn (fun t => t ^ (s - 1) * f t) (Icc c2 c1) volume","state_after":"case intro.intro.intro.intro\nE : Type u_1\ninst✝ : NormedAddCommGroup E\na b : ℝ\nf : ℝ → ℝ\ns : ℝ\nhfc : LocallyIntegrableOn f (Ioi 0) volume\nhf_top : f =O[atTop] fun x => x ^ (-a)\nhs_top : s < a\nhf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)\nhs_bot : b < s\nc1 : ℝ\nhc1 : 0 < c1\nhc1' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume\nc2 : ℝ\nhc2 : 0 < c2\nhc2' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc 0 c2) volume\nthis : Ioi 0 = Ioc 0 c2 ∪ Ioc c2 c1 ∪ Ioi c1\n⊢ ContinuousOn (fun t => t ^ (s - 1)) (Ioi 0)","tactic":"refine\n    (hfc.continuousOn_mul ?_ isOpen_Ioi).integrableOn_compact_subset\n      (fun t ht => (hc2.trans_le ht.1 : 0 < t)) isCompact_Icc","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"CompactIccSpace.isCompact_Icc","def_path":"Mathlib/Topology/Algebra/Order/Compact.lean","def_pos":[50,2],"def_end_pos":[50,15]},{"full_name":"MeasureTheory.LocallyIntegrableOn.continuousOn_mul","def_path":"Mathlib/MeasureTheory/Function/LocallyIntegrable.lean","def_pos":[620,8],"def_end_pos":[620,24]},{"full_name":"MeasureTheory.LocallyIntegrableOn.integrableOn_compact_subset","def_path":"Mathlib/MeasureTheory/Function/LocallyIntegrable.lean","def_pos":[68,8],"def_end_pos":[68,55]},{"full_name":"isOpen_Ioi","def_path":"Mathlib/Topology/Order/OrderClosed.lean","def_pos":[178,8],"def_end_pos":[178,18]}]},{"state_before":"case intro.intro.intro.intro\nE : Type u_1\ninst✝ : NormedAddCommGroup E\na b : ℝ\nf : ℝ → ℝ\ns : ℝ\nhfc : LocallyIntegrableOn f (Ioi 0) volume\nhf_top : f =O[atTop] fun x => x ^ (-a)\nhs_top : s < a\nhf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)\nhs_bot : b < s\nc1 : ℝ\nhc1 : 0 < c1\nhc1' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume\nc2 : ℝ\nhc2 : 0 < c2\nhc2' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc 0 c2) volume\nthis : Ioi 0 = Ioc 0 c2 ∪ Ioc c2 c1 ∪ Ioi c1\n⊢ ContinuousOn (fun t => t ^ (s - 1)) (Ioi 0)","state_after":"no goals","tactic":"exact ContinuousAt.continuousOn fun t ht => continuousAt_rpow_const _ _ <| Or.inl <| ne_of_gt ht","premises":[{"full_name":"ContinuousAt.continuousOn","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[772,8],"def_end_pos":[772,33]},{"full_name":"Or.inl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[534,4],"def_end_pos":[534,7]},{"full_name":"Real.continuousAt_rpow_const","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean","def_pos":[217,8],"def_end_pos":[217,31]},{"full_name":"ne_of_gt","def_path":"Mathlib/Order/Defs.lean","def_pos":[85,8],"def_end_pos":[85,16]}]}]}
+{"url":"Mathlib/RingTheory/OreLocalization/Basic.lean","commit":"","full_name":"AddOreLocalization.eq_of_num_factor_eq","start":[127,0],"end":[144,52],"file_path":"Mathlib/RingTheory/OreLocalization/Basic.lean","tactics":[{"state_before":"R : Type u_1\ninst✝² : Monoid R\nS : Submonoid R\ninst✝¹ : OreSet S\nX : Type ?u.24237\ninst✝ : MulAction R X\nr r' r₁ r₂ : R\ns t : ↥S\nh : ↑t * r = ↑t * r'\n⊢ r₁ * r * r₂ /ₒ s = r₁ * r' * r₂ /ₒ s","state_after":"case mk.mk\nR : Type u_1\ninst✝² : Monoid R\nS : Submonoid R\ninst✝¹ : OreSet S\nX : Type ?u.24237\ninst✝ : MulAction R X\nr r' r₁ r₂ : R\ns t : ↥S\nh : ↑t * r = ↑t * r'\nr₁' : R\nt' : ↥S\nhr₁ : ↑t' * r₁ = r₁' * ↑t\n⊢ r₁ * r * r₂ /ₒ s = r₁ * r' * r₂ /ₒ s","tactic":"rcases oreCondition r₁ t with ⟨r₁', t', hr₁⟩","premises":[{"full_name":"OreLocalization.oreCondition","def_path":"Mathlib/RingTheory/OreLocalization/OreSet.lean","def_pos":[91,4],"def_end_pos":[91,16]}]},{"state_before":"case mk.mk\nR : Type u_1\ninst✝² : Monoid R\nS : Submonoid R\ninst✝¹ : OreSet S\nX : Type ?u.24237\ninst✝ : MulAction R X\nr r' r₁ r₂ : R\ns t : ↥S\nh : ↑t * r = ↑t * r'\nr₁' : R\nt' : ↥S\nhr₁ : ↑t' * r₁ = r₁' * ↑t\n⊢ r₁ * r * r₂ /ₒ s = r₁ * r' * r₂ /ₒ s","state_after":"case mk.mk\nR : Type u_1\ninst✝² : Monoid R\nS : Submonoid R\ninst✝¹ : OreSet S\nX : Type ?u.24237\ninst✝ : MulAction R X\nr r' r₁ r₂ : R\ns t : ↥S\nh : ↑t * r = ↑t * r'\nr₁' : R\nt' : ↥S\nhr₁ : ↑t' * r₁ = r₁' * ↑t\n⊢ t' • (r₁ * r * r₂) /ₒ (t' * s) = t' • (r₁ * r' * r₂) /ₒ (t' * s)","tactic":"rw [OreLocalization.expand' _ s t', OreLocalization.expand' _ s t']","premises":[{"full_name":"OreLocalization.expand'","def_path":"Mathlib/RingTheory/OreLocalization/Basic.lean","def_pos":[124,18],"def_end_pos":[124,25]}]},{"state_before":"case mk.mk\nR : Type u_1\ninst✝² : Monoid R\nS : Submonoid R\ninst✝¹ : OreSet S\nX : Type ?u.24237\ninst✝ : MulAction R X\nr r' r₁ r₂ : R\ns t : ↥S\nh : ↑t * r = ↑t * r'\nr₁' : R\nt' : ↥S\nhr₁ : ↑t' * r₁ = r₁' * ↑t\n⊢ t' • (r₁ * r * r₂) /ₒ (t' * s) = t' • (r₁ * r' * r₂) /ₒ (t' * s)","state_after":"case mk.mk.e_r\nR : Type u_1\ninst✝² : Monoid R\nS : Submonoid R\ninst✝¹ : OreSet S\nX : Type ?u.24237\ninst✝ : MulAction R X\nr r' r₁ r₂ : R\ns t : ↥S\nh : ↑t * r = ↑t * r'\nr₁' : R\nt' : ↥S\nhr₁ : ↑t' * r₁ = r₁' * ↑t\n⊢ t' • (r₁ * r * r₂) = t' • (r₁ * r' * r₂)","tactic":"congr 1","premises":[]},{"state_before":"case mk.mk.e_r\nR : Type u_1\ninst✝² : Monoid R\nS : Submonoid R\ninst✝¹ : OreSet S\nX : Type ?u.24237\ninst✝ : MulAction R X\nr r' r₁ r₂ : R\ns t : ↥S\nh : ↑t * r = ↑t * r'\nr₁' : R\nt' : ↥S\nhr₁ : ↑t' * r₁ = r₁' * ↑t\n⊢ t' • (r₁ * r * r₂) = t' • (r₁ * r' * r₂)","state_after":"no goals","tactic":"calc (t' : R) * (r₁ * r * r₂)\n      = t' * r₁ * r * r₂ := by simp [← mul_assoc]\n    _ = r₁' * t * r * r₂ := by rw [hr₁]\n    _ = r₁' * (t * r) * r₂ := by simp [← mul_assoc]\n    _ = r₁' * (t * r') * r₂ := by rw [h]\n    _ = r₁' * t * r' * r₂ := by simp [← mul_assoc]\n    _ = t' * r₁ * r' * r₂ := by rw [hr₁]\n    _ = t' * (r₁ * r' * r₂) := by simp [← mul_assoc]","premises":[{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]}]}]}
+{"url":"Mathlib/CategoryTheory/FiberedCategory/HomLift.lean","commit":"","full_name":"CategoryTheory.IsHomLift.commSq","start":[90,0],"end":[91,75],"file_path":"Mathlib/CategoryTheory/FiberedCategory/HomLift.lean","tactics":[{"state_before":"𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₂} 𝒳\ninst✝¹ : Category.{v₂, u₁} 𝒮\np : 𝒳 ⥤ 𝒮\nR S : 𝒮\na b : 𝒳\nf : R ⟶ S\nφ : a ⟶ b\ninst✝ : p.IsHomLift f φ\n⊢ p.map φ ≫ eqToHom ⋯ = eqToHom ⋯ ≫ f","state_after":"no goals","tactic":"simp only [fac p f φ, eqToHom_trans_assoc, eqToHom_refl, id_comp]","premises":[{"full_name":"CategoryTheory.Category.id_comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[151,2],"def_end_pos":[151,9]},{"full_name":"CategoryTheory.IsHomLift.fac","def_path":"Mathlib/CategoryTheory/FiberedCategory/HomLift.lean","def_pos":[84,6],"def_end_pos":[84,9]},{"full_name":"CategoryTheory.eqToHom_refl","def_path":"Mathlib/CategoryTheory/EqToHom.lean","def_pos":[44,8],"def_end_pos":[44,20]}]}]}
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?u.5555\ninst✝⁷ : Ring R✝\ninst✝⁶ : AddCommGroup M✝\ninst✝⁵ : Module R✝ M✝\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R✝ N\nR M : Type u\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nι : Finset M\nhι₁ : Submodule.span R ↑ι = ⊤\nhι₂ : (LinearMap.ker (Finsupp.total { x // x ∈ ι } M R Subtype.val)).FG\n⊢ ∃ x,\n    Free R ({ x // x ∈ ι } →₀ R) ∧\n      Finite R ({ x // x ∈ ι } →₀ R) ∧ (LinearMap.ker (Finsupp.total { x // x ∈ ι } M R Subtype.val)).FG","state_after":"case h\nR✝ : Type ?u.4933\nM✝ : Type ?u.4936\nN : Type ?u.5555\ninst✝⁷ : Ring R✝\ninst✝⁶ : AddCommGroup M✝\ninst✝⁵ : Module R✝ M✝\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R✝ N\nR M : Type u\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nι : Finset M\nhι₁ : Submodule.span R ↑ι = ⊤\nhι₂ : (LinearMap.ker (Finsupp.total { x // x ∈ ι } M R Subtype.val)).FG\n⊢ Function.Surjective ⇑(Finsupp.total { x // x ∈ ι } M R Subtype.val)","tactic":"refine ⟨(LinearMap.quotKerEquivOfSurjective _ ?_).symm, 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Submodule.span R ↑ι = ⊤\nhι₂ : (LinearMap.ker (Finsupp.total { x // x ∈ ι } M R Subtype.val)).FG\n⊢ Function.Surjective ⇑(Finsupp.total { x // x ∈ ι } M R Subtype.val)","state_after":"case h\nR✝ : Type ?u.4933\nM✝ : Type ?u.4936\nN : Type ?u.5555\ninst✝⁷ : Ring R✝\ninst✝⁶ : AddCommGroup M✝\ninst✝⁵ : Module R✝ M✝\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R✝ N\nR M : Type u\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nι : Finset M\nhι₁ : Submodule.span R ↑ι = ⊤\nhι₂ : (LinearMap.ker (Finsupp.total { x // x ∈ ι } M R Subtype.val)).FG\n⊢ LinearMap.range (Finsupp.total { x // x ∈ ι } M R Subtype.val) = ⊤","tactic":"apply LinearMap.range_eq_top.mp","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"LinearMap.range_eq_top","def_path":"Mathlib/Algebra/Module/Submodule/Range.lean","def_pos":[89,8],"def_end_pos":[89,20]}]},{"state_before":"case h\nR✝ : Type ?u.4933\nM✝ : Type 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+{"url":"Mathlib/GroupTheory/CoprodI.lean","commit":"","full_name":"Monoid.CoprodI.Word.mem_of_mem_equivPair_tail","start":[459,0],"end":[464,70],"file_path":"Mathlib/GroupTheory/CoprodI.lean","tactics":[{"state_before":"ι : Type u_1\nM : ι → Type u_2\ninst✝³ : (i : ι) → Monoid (M i)\nN : Type u_3\ninst✝² : Monoid N\ninst✝¹ : (i : ι) → DecidableEq (M i)\ninst✝ : DecidableEq ι\ni j : ι\nw : Word M\nm : M i\n⊢ ⟨i, m⟩ ∈ ((equivPair j) w).tail.toList → ⟨i, m⟩ ∈ w.toList","state_after":"ι : Type u_1\nM : ι → Type u_2\ninst✝³ : (i : ι) → Monoid (M i)\nN : Type u_3\ninst✝² : Monoid N\ninst✝¹ : (i : ι) → DecidableEq (M i)\ninst✝ : DecidableEq ι\ni j : ι\nw : Word M\nm : M i\n⊢ (⟨i, m⟩ ∈ w.toList.tail ∨ i ≠ j ∧ ∃ (h : w.toList ≠ []), w.toList.head h = ⟨i, m⟩) → ⟨i, m⟩ ∈ w.toList","tactic":"rw [mem_equivPair_tail_iff]","premises":[{"full_name":"Monoid.CoprodI.Word.mem_equivPair_tail_iff","def_path":"Mathlib/GroupTheory/CoprodI.lean","def_pos":[444,8],"def_end_pos":[444,30]}]},{"state_before":"ι : Type u_1\nM : ι → Type u_2\ninst✝³ : (i : ι) → Monoid (M i)\nN : Type u_3\ninst✝² : Monoid N\ninst✝¹ : (i : ι) → DecidableEq (M i)\ninst✝ : DecidableEq ι\ni j : ι\nw : Word M\nm : M i\n⊢ (⟨i, m⟩ ∈ w.toList.tail ∨ i ≠ j ∧ ∃ (h : w.toList ≠ []), w.toList.head h = ⟨i, m⟩) → ⟨i, m⟩ ∈ w.toList","state_after":"case inl\nι : Type u_1\nM : ι → Type u_2\ninst✝³ : (i : ι) → Monoid (M i)\nN : Type u_3\ninst✝² : Monoid N\ninst✝¹ : (i : ι) → DecidableEq (M i)\ninst✝ : DecidableEq ι\ni j : ι\nw : Word M\nm : M i\nh : ⟨i, m⟩ ∈ w.toList.tail\n⊢ ⟨i, m⟩ ∈ w.toList\n\ncase inr\nι : Type u_1\nM : ι → Type u_2\ninst✝³ : (i : ι) → Monoid (M i)\nN : Type u_3\ninst✝² : Monoid N\ninst✝¹ : (i : ι) → DecidableEq (M i)\ninst✝ : DecidableEq ι\ni j : ι\nw : Word M\nm : M i\nh : i ≠ j ∧ ∃ (h : w.toList ≠ []), w.toList.head h = ⟨i, m⟩\n⊢ ⟨i, m⟩ ∈ w.toList","tactic":"rintro (h | h)","premises":[]}]}
+{"url":"Mathlib/Geometry/Manifold/Algebra/LieGroup.lean","commit":"","full_name":"ContMDiffAt.div₀","start":[310,0],"end":[312,50],"file_path":"Mathlib/Geometry/Manifold/Algebra/LieGroup.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝¹³ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝¹² : TopologicalSpace H\nE : Type u_3\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nG : Type u_4\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : ChartedSpace H G\ninst✝⁷ : GroupWithZero G\ninst✝⁶ : SmoothInv₀ I G\ninst✝⁵ : SmoothMul I G\nE' : Type u_5\ninst✝⁴ : NormedAddCommGroup E'\ninst✝³ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝² : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM : Type u_7\ninst✝¹ : TopologicalSpace M\ninst✝ : ChartedSpace H' M\nf g : M → G\ns : Set M\na : M\nn : ℕ∞\nhf : ContMDiffAt I' I n f a\nhg : ContMDiffAt I' I n g a\nh₀ : g a ≠ 0\n⊢ ContMDiffAt I' I n (f / g) a","state_after":"no goals","tactic":"simpa [div_eq_mul_inv] using hf.mul (hg.inv₀ h₀)","premises":[{"full_name":"ContMDiffAt.inv₀","def_path":"Mathlib/Geometry/Manifold/Algebra/LieGroup.lean","def_pos":[258,8],"def_end_pos":[258,24]},{"full_name":"ContMDiffAt.mul","def_path":"Mathlib/Geometry/Manifold/Algebra/Monoid.lean","def_pos":[95,15],"def_end_pos":[95,30]},{"full_name":"div_eq_mul_inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[907,8],"def_end_pos":[907,22]}]}]}
+{"url":"Mathlib/GroupTheory/Perm/Sign.lean","commit":"","full_name":"Equiv.Perm.sign_swap'","start":[399,0],"end":[401,72],"file_path":"Mathlib/GroupTheory/Perm/Sign.lean","tactics":[{"state_before":"α : Type u\ninst✝¹ : DecidableEq α\nβ : Type v\ninst✝ : Fintype α\nx y : α\nH : x = y\n⊢ sign (swap x y) = if x = y then 1 else -1","state_after":"no goals","tactic":"simp [H, swap_self]","premises":[{"full_name":"Equiv.swap_self","def_path":"Mathlib/Logic/Equiv/Basic.lean","def_pos":[1388,8],"def_end_pos":[1388,17]}]},{"state_before":"α : Type u\ninst✝¹ : DecidableEq α\nβ : Type v\ninst✝ : Fintype α\nx y : α\nH : ¬x = y\n⊢ sign (swap x y) = if x = y then 1 else -1","state_after":"no goals","tactic":"simp [sign_swap H, H]","premises":[{"full_name":"Equiv.Perm.sign_swap","def_path":"Mathlib/GroupTheory/Perm/Sign.lean","def_pos":[396,8],"def_end_pos":[396,17]}]}]}
+{"url":"Mathlib/LinearAlgebra/Prod.lean","commit":"","full_name":"LinearMap.coprod_inr","start":[202,0],"end":[204,74],"file_path":"Mathlib/LinearAlgebra/Prod.lean","tactics":[{"state_before":"R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type u_1\nM₆ : Type u_2\nS : Type u_3\ninst✝¹³ : Semiring R\ninst✝¹² : Semiring S\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M₂\ninst✝⁹ : AddCommMonoid M₃\ninst✝⁸ : AddCommMonoid M₄\ninst✝⁷ : AddCommMonoid M₅\ninst✝⁶ : AddCommMonoid M₆\ninst✝⁵ : Module R M\ninst✝⁴ : Module R M₂\ninst✝³ : Module R M₃\ninst✝² : Module R M₄\ninst✝¹ : Module R M₅\ninst✝ : Module R M₆\nf✝ : M →ₗ[R] M₂\nf : M →ₗ[R] M₃\ng : M₂ →ₗ[R] M₃\n⊢ f.coprod g ∘ₗ inr R M M₂ = g","state_after":"case h\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type u_1\nM₆ : Type u_2\nS : Type u_3\ninst✝¹³ : Semiring R\ninst✝¹² : Semiring S\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M₂\ninst✝⁹ : AddCommMonoid M₃\ninst✝⁸ : AddCommMonoid M₄\ninst✝⁷ : AddCommMonoid M₅\ninst✝⁶ : AddCommMonoid M₆\ninst✝⁵ : Module R M\ninst✝⁴ : Module R M₂\ninst✝³ : Module R M₃\ninst✝² : Module R M₄\ninst✝¹ : Module R M₅\ninst✝ : Module R M₆\nf✝ : M →ₗ[R] M₂\nf : M →ₗ[R] M₃\ng : M₂ →ₗ[R] M₃\nx✝ : M₂\n⊢ (f.coprod g ∘ₗ inr R M M₂) x✝ = g x✝","tactic":"ext","premises":[]},{"state_before":"case h\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type u_1\nM₆ : Type u_2\nS : Type u_3\ninst✝¹³ : Semiring R\ninst✝¹² : Semiring S\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M₂\ninst✝⁹ : AddCommMonoid M₃\ninst✝⁸ : AddCommMonoid M₄\ninst✝⁷ : AddCommMonoid M₅\ninst✝⁶ : AddCommMonoid M₆\ninst✝⁵ : Module R M\ninst✝⁴ : Module R M₂\ninst✝³ : Module R M₃\ninst✝² : Module R M₄\ninst✝¹ : Module R M₅\ninst✝ : Module R M₆\nf✝ : M →ₗ[R] M₂\nf : M →ₗ[R] M₃\ng : M₂ →ₗ[R] M₃\nx✝ : M₂\n⊢ (f.coprod g ∘ₗ inr R M M₂) x✝ = g x✝","state_after":"no goals","tactic":"simp only [map_zero, coprod_apply, inr_apply, zero_add, comp_apply]","premises":[{"full_name":"LinearMap.comp_apply","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[504,8],"def_end_pos":[504,18]},{"full_name":"LinearMap.coprod_apply","def_path":"Mathlib/LinearAlgebra/Prod.lean","def_pos":[194,8],"def_end_pos":[194,20]},{"full_name":"LinearMap.inr_apply","def_path":"Mathlib/LinearAlgebra/Prod.lean","def_pos":[176,8],"def_end_pos":[176,17]},{"full_name":"map_zero","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[189,2],"def_end_pos":[189,13]},{"full_name":"zero_add","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[408,2],"def_end_pos":[408,13]}]}]}
+{"url":"Mathlib/Topology/Category/Profinite/Nobeling.lean","commit":"","full_name":"Profinite.NobelingProof.GoodProducts.sum_equiv_comp_eval_eq_elim","start":[1409,0],"end":[1412,26],"file_path":"Mathlib/Topology/Category/Profinite/Nobeling.lean","tactics":[{"state_before":"I : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\no : Ordinal.{u}\nhC : IsClosed C\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x x_1 => x < x_1\n⊢ eval C ∘ (sum_equiv C hsC ho).toFun = Sum.elim (fun l => Products.eval C ↑l) fun l => Products.eval C ↑l","state_after":"no goals","tactic":"ext ⟨_,_⟩ <;> [rfl; rfl]","premises":[]}]}
+{"url":"Mathlib/Order/Part.lean","commit":"","full_name":"Antitone.partSeq","start":[52,0],"end":[53,88],"file_path":"Mathlib/Order/Part.lean","tactics":[{"state_before":"α : Type u_1\nβ✝ : Type u_2\nγ✝ : Type u_3\ninst✝ : Preorder α\nβ γ : Type u_4\nf : α → Part (β → γ)\ng : α → Part β\nhf : Antitone f\nhg : Antitone g\n⊢ Antitone fun x => Seq.seq (f x) fun x_1 => g x","state_after":"no goals","tactic":"simpa only [seq_eq_bind_map] using hf.partBind $ Antitone.of_apply₂ fun _ ↦ hg.partMap","premises":[{"full_name":"Antitone.partBind","def_path":"Mathlib/Order/Part.lean","def_pos":[27,6],"def_end_pos":[27,23]},{"full_name":"Antitone.partMap","def_path":"Mathlib/Order/Part.lean","def_pos":[41,6],"def_end_pos":[41,22]},{"full_name":"seq_eq_bind_map","def_path":".lake/packages/lean4/src/lean/Init/Control/Lawful/Basic.lean","def_pos":[95,8],"def_end_pos":[95,23]}]}]}
+{"url":"Mathlib/RingTheory/DedekindDomain/FiniteAdeleRing.lean","commit":"","full_name":"DedekindDomain.ProdAdicCompletions.IsFiniteAdele.one","start":[254,0],"end":[263,22],"file_path":"Mathlib/RingTheory/DedekindDomain/FiniteAdeleRing.lean","tactics":[{"state_before":"R : Type u_1\nK : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\n⊢ IsFiniteAdele 1","state_after":"R : Type u_1\nK : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\n⊢ {x | 1 x ∉ adicCompletionIntegers K x}.Finite","tactic":"rw [IsFiniteAdele, Filter.eventually_cofinite]","premises":[{"full_name":"DedekindDomain.ProdAdicCompletions.IsFiniteAdele","def_path":"Mathlib/RingTheory/DedekindDomain/FiniteAdeleRing.lean","def_pos":[184,4],"def_end_pos":[184,17]},{"full_name":"Filter.eventually_cofinite","def_path":"Mathlib/Order/Filter/Cofinite.lean","def_pos":[38,8],"def_end_pos":[38,27]}]},{"state_before":"R : Type u_1\nK : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\n⊢ {x | 1 x ∉ adicCompletionIntegers K x}.Finite","state_after":"R : Type u_1\nK : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nh_empty : {v | 1 ∉ adicCompletionIntegers K v} = ∅\n⊢ {x | 1 x ∉ adicCompletionIntegers K x}.Finite","tactic":"have h_empty :\n    {v : HeightOneSpectrum R | ¬(1 : v.adicCompletion K) ∈ v.adicCompletionIntegers K} = ∅ := by\n    ext v; rw [mem_empty_iff_false, iff_false_iff]; intro hv\n    rw [mem_setOf] at hv; apply hv; rw [mem_adicCompletionIntegers]\n    exact le_of_eq Valued.v.map_one'","premises":[{"full_name":"EmptyCollection.emptyCollection","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[447,2],"def_end_pos":[447,17]},{"full_name":"IsDedekindDomain.HeightOneSpectrum","def_path":"Mathlib/RingTheory/DedekindDomain/Ideal.lean","def_pos":[935,10],"def_end_pos":[935,27]},{"full_name":"IsDedekindDomain.HeightOneSpectrum.adicCompletion","def_path":"Mathlib/RingTheory/DedekindDomain/AdicValuation.lean","def_pos":[365,7],"def_end_pos":[365,21]},{"full_name":"IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegers","def_path":"Mathlib/RingTheory/DedekindDomain/AdicValuation.lean","def_pos":[393,4],"def_end_pos":[393,26]},{"full_name":"IsDedekindDomain.HeightOneSpectrum.mem_adicCompletionIntegers","def_path":"Mathlib/RingTheory/DedekindDomain/AdicValuation.lean","def_pos":[401,8],"def_end_pos":[401,34]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Not","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[224,4],"def_end_pos":[224,7]},{"full_name":"Set.mem_empty_iff_false","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[446,8],"def_end_pos":[446,27]},{"full_name":"Set.mem_setOf","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[202,8],"def_end_pos":[202,17]},{"full_name":"Valued.v","def_path":"Mathlib/Topology/Algebra/Valued/ValuationTopology.lean","def_pos":[92,2],"def_end_pos":[92,3]},{"full_name":"iff_false_iff","def_path":"Mathlib/Init/Logic.lean","def_pos":[108,8],"def_end_pos":[108,21]},{"full_name":"le_of_eq","def_path":"Mathlib/Order/Defs.lean","def_pos":[60,8],"def_end_pos":[60,16]},{"full_name":"setOf","def_path":"Mathlib/Init/Set.lean","def_pos":[56,4],"def_end_pos":[56,9]}]},{"state_before":"R : Type u_1\nK : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nh_empty : {v | 1 ∉ adicCompletionIntegers K v} = ∅\n⊢ {x | 1 x ∉ adicCompletionIntegers K x}.Finite","state_after":"no goals","tactic":"convert finite_empty","premises":[{"full_name":"Set.finite_empty","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[716,8],"def_end_pos":[716,20]}]}]}
+{"url":"Mathlib/Algebra/Order/Kleene.lean","commit":"","full_name":"nsmul_eq_self","start":[137,0],"end":[140,82],"file_path":"Mathlib/Algebra/Order/Kleene.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nπ : ι → Type u_4\ninst✝ : IdemSemiring α\na✝ b c : α\nn : ℕ\nx✝ : n + 2 ≠ 0\na : α\n⊢ (n + 2) • a = a","state_after":"no goals","tactic":"rw [succ_nsmul, nsmul_eq_self n.succ_ne_zero, add_idem]","premises":[{"full_name":"Nat.succ_ne_zero","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[708,16],"def_end_pos":[708,28]},{"full_name":"add_idem","def_path":"Mathlib/Algebra/Order/Kleene.lean","def_pos":[135,8],"def_end_pos":[135,16]},{"full_name":"succ_nsmul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[566,14],"def_end_pos":[566,24]}]}]}
+{"url":"Mathlib/Algebra/Module/Basic.lean","commit":"","full_name":"Function.support_smul_subset_left","start":[126,0],"end":[128,46],"file_path":"Mathlib/Algebra/Module/Basic.lean","tactics":[{"state_before":"α : Type u_1\nR : Type u_2\nM : Type u_3\nM₂ : Type u_4\ninst✝² : Zero R\ninst✝¹ : Zero M\ninst✝ : SMulWithZero R M\nf : α → R\ng : α → M\nx : α\nhfg : x ∈ support (f • g)\nhf : f x = 0\n⊢ (f • g) x = 0","state_after":"no goals","tactic":"rw [Pi.smul_apply', hf, zero_smul]","premises":[{"full_name":"Pi.smul_apply'","def_path":"Mathlib/Algebra/Group/Action/Pi.lean","def_pos":[32,6],"def_end_pos":[32,17]},{"full_name":"zero_smul","def_path":"Mathlib/Algebra/SMulWithZero.lean","def_pos":[67,8],"def_end_pos":[67,17]}]}]}
+{"url":"Mathlib/MeasureTheory/Function/L1Space.lean","commit":"","full_name":"MeasureTheory.Integrable.right_of_add_measure","start":[476,0],"end":[479,30],"file_path":"Mathlib/MeasureTheory/Function/L1Space.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → β\nh : Integrable f (μ + ν)\n⊢ Integrable f ν","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → β\nh : Memℒp f 1 (μ + ν)\n⊢ Memℒp f 1 ν","tactic":"rw [← memℒp_one_iff_integrable] at h ⊢","premises":[{"full_name":"MeasureTheory.memℒp_one_iff_integrable","def_path":"Mathlib/MeasureTheory/Function/L1Space.lean","def_pos":[392,8],"def_end_pos":[392,32]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → β\nh : Memℒp f 1 (μ + ν)\n⊢ Memℒp f 1 ν","state_after":"no goals","tactic":"exact h.right_of_add_measure","premises":[{"full_name":"MeasureTheory.Memℒp.right_of_add_measure","def_path":"Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean","def_pos":[884,8],"def_end_pos":[884,34]}]}]}
+{"url":"Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean","commit":"","full_name":"MeasureTheory.BoundedContinuousFunction.integral_eq_integral_meas_le_of_hasFiniteIntegral","start":[298,0],"end":[306,84],"file_path":"Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean","tactics":[{"state_before":"ι : Type u_1\nΩ : Type u_2\ninst✝⁵ : MeasurableSpace Ω\ninst✝⁴ : PseudoMetricSpace Ω\ninst✝³ : OpensMeasurableSpace Ω\nα : Type u_3\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\ninst✝ : OpensMeasurableSpace α\nf : α →ᵇ ℝ\nμ : Measure α\nf_nn : 0 ≤ᶠ[ae μ] ⇑f\nhf : HasFiniteIntegral (⇑f) μ\n⊢ ∫ (ω : α), f ω ∂μ = ∫ (t : ℝ) in Ioc 0 ‖f‖, (μ {a | t ≤ f a}).toReal","state_after":"ι : Type u_1\nΩ : Type u_2\ninst✝⁵ : MeasurableSpace Ω\ninst✝⁴ : PseudoMetricSpace Ω\ninst✝³ : OpensMeasurableSpace Ω\nα : Type u_3\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\ninst✝ : OpensMeasurableSpace α\nf : α →ᵇ ℝ\nμ : Measure α\nf_nn : 0 ≤ᶠ[ae μ] ⇑f\nhf : HasFiniteIntegral (⇑f) μ\n⊢ Integrable (⇑f) μ\n\nι : Type u_1\nΩ : Type u_2\ninst✝⁵ : MeasurableSpace Ω\ninst✝⁴ : PseudoMetricSpace Ω\ninst✝³ : OpensMeasurableSpace Ω\nα : Type u_3\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\ninst✝ : OpensMeasurableSpace α\nf : α →ᵇ ℝ\nμ : Measure α\nf_nn : 0 ≤ᶠ[ae μ] ⇑f\nhf : HasFiniteIntegral (⇑f) μ\n⊢ ⇑f ≤ᶠ[ae μ] fun x => ‖f‖","tactic":"rw [Integrable.integral_eq_integral_Ioc_meas_le (M := ‖f‖) ?_ f_nn ?_]","premises":[{"full_name":"MeasureTheory.Integrable.integral_eq_integral_Ioc_meas_le","def_path":"Mathlib/MeasureTheory/Integral/Layercake.lean","def_pos":[546,6],"def_end_pos":[546,49]},{"full_name":"Norm.norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[59,2],"def_end_pos":[59,6]}]}]}
+{"url":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","commit":"","full_name":"Polynomial.degree_eq_iff_natDegree_eq","start":[127,0],"end":[128,94],"file_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","tactics":[{"state_before":"R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nhp : p ≠ 0\n⊢ p.degree = ↑n ↔ p.natDegree = n","state_after":"R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nhp : p ≠ 0\n⊢ ↑p.natDegree = ↑n ↔ p.natDegree = n","tactic":"rw [degree_eq_natDegree hp]","premises":[{"full_name":"Polynomial.degree_eq_natDegree","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[114,8],"def_end_pos":[114,27]}]},{"state_before":"R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nhp : p ≠ 0\n⊢ ↑p.natDegree = ↑n ↔ p.natDegree = n","state_after":"no goals","tactic":"exact WithBot.coe_eq_coe","premises":[{"full_name":"WithBot.coe_eq_coe","def_path":"Mathlib/Order/WithBot.lean","def_pos":[115,8],"def_end_pos":[115,18]}]}]}
+{"url":"Mathlib/Order/PropInstances.lean","commit":"","full_name":"Pi.isCompl_iff","start":[76,0],"end":[78,72],"file_path":"Mathlib/Order/PropInstances.lean","tactics":[{"state_before":"ι : Type u_1\nα' : ι → Type u_2\ninst✝¹ : (i : ι) → PartialOrder (α' i)\ninst✝ : (i : ι) → BoundedOrder (α' i)\nf g : (i : ι) → α' i\n⊢ IsCompl f g ↔ ∀ (i : ι), IsCompl (f i) (g i)","state_after":"no goals","tactic":"simp_rw [_root_.isCompl_iff, disjoint_iff, codisjoint_iff, forall_and]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Pi.codisjoint_iff","def_path":"Mathlib/Order/PropInstances.lean","def_pos":[72,8],"def_end_pos":[72,22]},{"full_name":"Pi.disjoint_iff","def_path":"Mathlib/Order/PropInstances.lean","def_pos":[62,8],"def_end_pos":[62,20]},{"full_name":"forall_and","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[256,8],"def_end_pos":[256,18]},{"full_name":"isCompl_iff","def_path":"Mathlib/Order/Disjoint.lean","def_pos":[396,8],"def_end_pos":[396,19]}]}]}
+{"url":"Mathlib/Order/Interval/Set/ProjIcc.lean","commit":"","full_name":"Set.projIcc_of_right_le","start":[66,0],"end":[67,23],"file_path":"Mathlib/Order/Interval/Set/ProjIcc.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder α\na b : α\nh : a ≤ b\nx : α\nhx : b ≤ x\n⊢ projIcc a b h x = ⟨b, ⋯⟩","state_after":"no goals","tactic":"simp [projIcc, hx, h]","premises":[{"full_name":"Set.projIcc","def_path":"Mathlib/Order/Interval/Set/ProjIcc.lean","def_pos":[44,4],"def_end_pos":[44,11]}]}]}
+{"url":"Mathlib/Data/Finset/SMulAntidiagonal.lean","commit":"","full_name":"Set.IsPWO.vadd","start":[29,0],"end":[33,71],"file_path":"Mathlib/Data/Finset/SMulAntidiagonal.lean","tactics":[{"state_before":"G : Type u_1\nP : Type u_2\ninst✝³ : PartialOrder G\ninst✝² : PartialOrder P\ninst✝¹ : SMul G P\ninst✝ : IsOrderedCancelSMul G P\ns : Set G\nt : Set P\nhs : s.IsPWO\nht : t.IsPWO\n⊢ (s • t).IsPWO","state_after":"G : Type u_1\nP : Type u_2\ninst✝³ : PartialOrder G\ninst✝² : PartialOrder P\ninst✝¹ : SMul G P\ninst✝ : IsOrderedCancelSMul G P\ns : Set G\nt : Set P\nhs : s.IsPWO\nht : t.IsPWO\n⊢ ((fun x => x.1 • x.2) '' s ×ˢ t).IsPWO","tactic":"rw [← @image_smul_prod]","premises":[{"full_name":"Set.image_smul_prod","def_path":"Mathlib/Data/Set/Pointwise/SMul.lean","def_pos":[81,8],"def_end_pos":[81,23]}]},{"state_before":"G : Type u_1\nP : Type u_2\ninst✝³ : PartialOrder G\ninst✝² : PartialOrder P\ninst✝¹ : SMul G P\ninst✝ : IsOrderedCancelSMul G P\ns : Set G\nt : Set P\nhs : s.IsPWO\nht : t.IsPWO\n⊢ ((fun x => x.1 • x.2) '' s ×ˢ t).IsPWO","state_after":"no goals","tactic":"exact (hs.prod ht).image_of_monotone (monotone_fst.smul monotone_snd)","premises":[{"full_name":"Monotone.smul","def_path":"Mathlib/Algebra/Order/AddTorsor.lean","def_pos":[84,8],"def_end_pos":[84,21]},{"full_name":"Set.IsPWO.image_of_monotone","def_path":"Mathlib/Order/WellFoundedSet.lean","def_pos":[413,8],"def_end_pos":[413,31]},{"full_name":"Set.IsPWO.prod","def_path":"Mathlib/Order/WellFoundedSet.lean","def_pos":[406,15],"def_end_pos":[406,25]},{"full_name":"monotone_fst","def_path":"Mathlib/Order/Monotone/Basic.lean","def_pos":[1033,8],"def_end_pos":[1033,20]},{"full_name":"monotone_snd","def_path":"Mathlib/Order/Monotone/Basic.lean","def_pos":[1035,8],"def_end_pos":[1035,20]}]}]}
+{"url":"Mathlib/Order/Filter/AtTopBot.lean","commit":"","full_name":"OrderIso.comap_atTop","start":[394,0],"end":[396,40],"file_path":"Mathlib/Order/Filter/AtTopBot.lean","tactics":[{"state_before":"ι : Type u_1\nι' : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ne : α ≃o β\n⊢ comap (⇑e) atTop = atTop","state_after":"no goals","tactic":"simp [atTop, ← e.surjective.iInf_comp]","premises":[{"full_name":"Filter.atTop","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[40,4],"def_end_pos":[40,9]},{"full_name":"Function.Surjective.iInf_comp","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[579,8],"def_end_pos":[579,37]},{"full_name":"OrderIso.surjective","def_path":"Mathlib/Order/Hom/Basic.lean","def_pos":[729,18],"def_end_pos":[729,28]}]}]}
+{"url":"Mathlib/CategoryTheory/Sites/Over.lean","commit":"","full_name":"CategoryTheory.Sieve.overEquiv_pullback","start":[69,0],"end":[82,31],"file_path":"Mathlib/CategoryTheory/Sites/Over.lean","tactics":[{"state_before":"C : Type u\ninst✝ : Category.{v, u} C\nX : C\nY₁ Y₂ : Over X\nf : Y₁ ⟶ Y₂\nS : Sieve Y₂\n⊢ (overEquiv Y₁) (pullback f S) = pullback f.left ((overEquiv Y₂) S)","state_after":"case h\nC : Type u\ninst✝ : Category.{v, u} C\nX : C\nY₁ Y₂ : Over X\nf : Y₁ ⟶ Y₂\nS : Sieve Y₂\nZ : C\ng : Z ⟶ Y₁.left\n⊢ ((overEquiv Y₁) (pullback f S)).arrows g ↔ (pullback f.left ((overEquiv Y₂) S)).arrows g","tactic":"ext Z g","premises":[]},{"state_before":"case h\nC : Type u\ninst✝ : Category.{v, u} C\nX : C\nY₁ Y₂ : Over X\nf : Y₁ ⟶ Y₂\nS : Sieve Y₂\nZ : C\ng : Z ⟶ Y₁.left\n⊢ ((overEquiv Y₁) (pullback f S)).arrows g ↔ (pullback f.left ((overEquiv Y₂) S)).arrows g","state_after":"case h\nC : Type u\ninst✝ : Category.{v, u} C\nX : C\nY₁ Y₂ : Over X\nf : Y₁ ⟶ Y₂\nS : Sieve Y₂\nZ : C\ng : Z ⟶ Y₁.left\n⊢ (∃ Z_1 g_1 h, S.arrows (g_1 ≫ f) ∧ g = h ≫ g_1.left) ↔ ∃ Z_1 g_1 h, S.arrows g_1 ∧ g ≫ f.left = h ≫ g_1.left","tactic":"dsimp [overEquiv, Presieve.functorPushforward]","premises":[{"full_name":"CategoryTheory.Presieve.functorPushforward","def_path":"Mathlib/CategoryTheory/Sites/Sieves.lean","def_pos":[199,4],"def_end_pos":[199,22]},{"full_name":"CategoryTheory.Sieve.overEquiv","def_path":"Mathlib/CategoryTheory/Sites/Over.lean","def_pos":[32,4],"def_end_pos":[32,13]}]},{"state_before":"case h\nC : Type u\ninst✝ : Category.{v, u} C\nX : C\nY₁ Y₂ : Over X\nf : Y₁ ⟶ Y₂\nS : Sieve Y₂\nZ : C\ng : Z ⟶ Y₁.left\n⊢ (∃ Z_1 g_1 h, S.arrows (g_1 ≫ f) ∧ g = h ≫ g_1.left) ↔ ∃ Z_1 g_1 h, S.arrows g_1 ∧ g ≫ f.left = h ≫ g_1.left","state_after":"case h.mp\nC : Type u\ninst✝ : Category.{v, u} C\nX : C\nY₁ Y₂ : Over X\nf : Y₁ ⟶ Y₂\nS : Sieve Y₂\nZ : C\ng : Z ⟶ Y₁.left\n⊢ (∃ Z_1 g_1 h, S.arrows (g_1 ≫ f) ∧ g = h ≫ g_1.left) → ∃ Z_1 g_1 h, S.arrows g_1 ∧ g ≫ f.left = h ≫ g_1.left\n\ncase h.mpr\nC : Type u\ninst✝ : Category.{v, u} C\nX : C\nY₁ Y₂ : Over X\nf : Y₁ ⟶ Y₂\nS : Sieve Y₂\nZ : C\ng : Z ⟶ Y₁.left\n⊢ (∃ Z_1 g_1 h, S.arrows g_1 ∧ g ≫ f.left = h ≫ g_1.left) → ∃ Z_1 g_1 h, S.arrows (g_1 ≫ f) ∧ g = h ≫ g_1.left","tactic":"constructor","premises":[]}]}
+{"url":"Mathlib/Data/Matroid/Closure.lean","commit":"","full_name":"Matroid.Flat.iInter","start":[87,0],"end":[94,62],"file_path":"Mathlib/Data/Matroid/Closure.lean","tactics":[{"state_before":"ι✝ : Type u_1\nα : Type u_2\nM : Matroid α\nF X Y : Set α\ne : α\nι : Type u_3\ninst✝ : Nonempty ι\nFs : ι → Set α\nhFs : ∀ (i : ι), M.Flat (Fs i)\n⊢ M.Flat (⋂ i, Fs i)","state_after":"ι✝ : Type u_1\nα : Type u_2\nM : Matroid α\nF X✝ Y : Set α\ne : α\nι : Type u_3\ninst✝ : Nonempty ι\nFs : ι → Set α\nhFs : ∀ (i : ι), M.Flat (Fs i)\nI X : Set α\nhI : M.Basis I (⋂ i, Fs i)\nhIX : M.Basis I X\ni : ι\n⊢ X ⊆ Fs i","tactic":"refine ⟨fun I X hI hIX ↦ subset_iInter fun i ↦ ?_,\n    (iInter_subset _ (Classical.arbitrary _)).trans (hFs _).subset_ground⟩","premises":[{"full_name":"Classical.arbitrary","def_path":"Mathlib/Logic/Nonempty.lean","def_pos":[85,31],"def_end_pos":[85,50]},{"full_name":"Matroid.Flat.subset_ground","def_path":"Mathlib/Data/Matroid/Closure.lean","def_pos":[80,2],"def_end_pos":[80,15]},{"full_name":"Set.iInter_subset","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[249,8],"def_end_pos":[249,21]},{"full_name":"Set.subset_iInter","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[221,8],"def_end_pos":[221,21]}]},{"state_before":"ι✝ : Type u_1\nα : Type u_2\nM : Matroid α\nF X✝ Y : Set α\ne : α\nι : Type u_3\ninst✝ : Nonempty ι\nFs : ι → Set α\nhFs : ∀ (i : ι), M.Flat (Fs i)\nI X : Set α\nhI : M.Basis I (⋂ i, Fs i)\nhIX : M.Basis I X\ni : ι\n⊢ X ⊆ Fs i","state_after":"case intro.intro\nι✝ : Type u_1\nα : Type u_2\nM : Matroid α\nF X✝ Y : Set α\ne : α\nι : Type u_3\ninst✝ : Nonempty ι\nFs : ι → Set α\nhFs : ∀ (i : ι), M.Flat (Fs i)\nI X : Set α\nhI : M.Basis I (⋂ i, Fs i)\nhIX : M.Basis I X\ni : ι\nJ : Set α\nhIJ : M.Basis J (Fs i)\nhJ : I ⊆ J\n⊢ X ⊆ Fs i","tactic":"obtain ⟨J, hIJ, hJ⟩ := hI.indep.subset_basis_of_subset (hI.subset.trans (iInter_subset _ i))","premises":[{"full_name":"Matroid.Basis.indep","def_path":"Mathlib/Data/Matroid/Basic.lean","def_pos":[721,8],"def_end_pos":[721,19]},{"full_name":"Matroid.Basis.subset","def_path":"Mathlib/Data/Matroid/Basic.lean","def_pos":[724,8],"def_end_pos":[724,20]},{"full_name":"Matroid.Indep.subset_basis_of_subset","def_path":"Mathlib/Data/Matroid/Basic.lean","def_pos":[838,8],"def_end_pos":[838,36]},{"full_name":"Set.iInter_subset","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[249,8],"def_end_pos":[249,21]}]},{"state_before":"case intro.intro\nι✝ : Type u_1\nα : Type u_2\nM : Matroid α\nF X✝ Y : Set α\ne : α\nι : Type u_3\ninst✝ : Nonempty ι\nFs : ι → Set α\nhFs : ∀ (i : ι), M.Flat (Fs i)\nI X : Set α\nhI : M.Basis I (⋂ i, Fs i)\nhIX : M.Basis I X\ni : ι\nJ : Set α\nhIJ : M.Basis J (Fs i)\nhJ : I ⊆ J\n⊢ X ⊆ Fs i","state_after":"case intro.intro\nι✝ : Type u_1\nα : Type u_2\nM : Matroid α\nF X✝ Y : Set α\ne : α\nι : Type u_3\ninst✝ : Nonempty ι\nFs : ι → Set α\nhFs : ∀ (i : ι), M.Flat (Fs i)\nI X : Set α\nhI : M.Basis I (⋂ i, Fs i)\nhIX : M.Basis I X\ni : ι\nJ : Set α\nhIJ : M.Basis J (Fs i)\nhJ : I ⊆ J\n⊢ M.Basis J (Fs i ∪ X)","tactic":"refine subset_union_right.trans ((hFs i).1 (X := Fs i ∪ X) hIJ ?_)","premises":[{"full_name":"Matroid.Flat.subset_of_basis_of_basis","def_path":"Mathlib/Data/Matroid/Closure.lean","def_pos":[79,2],"def_end_pos":[79,26]},{"full_name":"Set.subset_union_right","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[660,8],"def_end_pos":[660,26]},{"full_name":"Union.union","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[402,2],"def_end_pos":[402,7]}]},{"state_before":"case intro.intro\nι✝ : Type u_1\nα : Type u_2\nM : Matroid α\nF X✝ Y : Set α\ne : α\nι : Type u_3\ninst✝ : Nonempty ι\nFs : ι → Set α\nhFs : ∀ (i : ι), M.Flat (Fs i)\nI X : Set α\nhI : M.Basis I (⋂ i, Fs i)\nhIX : M.Basis I X\ni : ι\nJ : Set α\nhIJ : M.Basis J (Fs i)\nhJ : I ⊆ J\n⊢ M.Basis J (Fs i ∪ X)","state_after":"case h.e'_4\nι✝ : Type u_1\nα : Type u_2\nM : Matroid α\nF X✝ Y : Set α\ne : α\nι : Type u_3\ninst✝ : Nonempty ι\nFs : ι → Set α\nhFs : ∀ (i : ι), M.Flat (Fs i)\nI X : Set α\nhI : M.Basis I (⋂ i, Fs i)\nhIX : M.Basis I X\ni : ι\nJ : Set α\nhIJ : M.Basis J (Fs i)\nhJ : I ⊆ J\n⊢ Fs i ∪ X = Fs i ∪ (J ∪ X)","tactic":"convert hIJ.basis_union (hIX.basis_union_of_subset hIJ.indep hJ) using 1","premises":[{"full_name":"Matroid.Basis.basis_union","def_path":"Mathlib/Data/Matroid/Basic.lean","def_pos":[951,8],"def_end_pos":[951,25]},{"full_name":"Matroid.Basis.basis_union_of_subset","def_path":"Mathlib/Data/Matroid/Basic.lean","def_pos":[954,8],"def_end_pos":[954,35]},{"full_name":"Matroid.Basis.indep","def_path":"Mathlib/Data/Matroid/Basic.lean","def_pos":[721,8],"def_end_pos":[721,19]}]},{"state_before":"case h.e'_4\nι✝ : Type u_1\nα : Type u_2\nM : Matroid α\nF X✝ Y : Set α\ne : α\nι : Type u_3\ninst✝ : Nonempty ι\nFs : ι → Set α\nhFs : ∀ (i : ι), M.Flat (Fs i)\nI X : Set α\nhI : M.Basis I (⋂ i, Fs i)\nhIX : M.Basis I X\ni : ι\nJ : Set α\nhIJ : M.Basis J (Fs i)\nhJ : I ⊆ J\n⊢ Fs i ∪ X = Fs i ∪ (J ∪ X)","state_after":"no goals","tactic":"rw [← union_assoc, union_eq_self_of_subset_right hIJ.subset]","premises":[{"full_name":"Matroid.Basis.subset","def_path":"Mathlib/Data/Matroid/Basic.lean","def_pos":[724,8],"def_end_pos":[724,20]},{"full_name":"Set.union_assoc","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[627,8],"def_end_pos":[627,19]},{"full_name":"Set.union_eq_self_of_subset_right","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[653,8],"def_end_pos":[653,37]}]}]}
+{"url":"Mathlib/Data/DFinsupp/Basic.lean","commit":"","full_name":"DFinsupp.update_self","start":[726,0],"end":[729,6],"file_path":"Mathlib/Data/DFinsupp/Basic.lean","tactics":[{"state_before":"ι : Type u\nγ : Type w\nβ : ι → Type v\nβ�� : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → Zero (β i)\ns : Finset ι\nx : (i : ↑↑s) → β ↑i\ni : ι\nf : Π₀ (i : ι), β i\nb : β i\nj : ι\n⊢ update i f (f i) = f","state_after":"case h\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → Zero (β i)\ns : Finset ι\nx : (i : ↑↑s) → β ↑i\ni : ι\nf : Π₀ (i : ι), β i\nb : β i\nj i✝ : ι\n⊢ (update i f (f i)) i✝ = f i✝","tactic":"ext","premises":[]},{"state_before":"case h\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → Zero (β i)\ns : Finset ι\nx : (i : ↑↑s) → β ↑i\ni : ι\nf : Π₀ (i : ι), β i\nb : β i\nj i✝ : ι\n⊢ (update i f (f i)) i✝ = f i✝","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean","commit":"","full_name":"rel_iSup_tsum","start":[147,0],"end":[155,16],"file_path":"Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean","tactics":[{"state_before":"M : Type u_1\ninst✝⁵ : CommMonoid M\ninst✝⁴ : TopologicalSpace M\nm✝ m' : M\nG : Type u_2\ninst✝³ : CommGroup G\ng g' : G\ninst✝² : T2Space M\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝¹ : Countable β\ninst✝ : CompleteLattice α\nm : α → M\nm0 : m ⊥ = 1\nR : M → M → Prop\nm_iSup : ∀ (s : ℕ → α), R (m (⨆ i, s i)) (∏' (i : ℕ), m (s i))\ns : β → α\n⊢ R (m (⨆ b, s b)) (∏' (b : β), m (s b))","state_after":"case intro\nM : Type u_1\ninst✝⁵ : CommMonoid M\ninst✝⁴ : TopologicalSpace M\nm✝ m' : M\nG : Type u_2\ninst✝³ : CommGroup G\ng g' : G\ninst✝² : T2Space M\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝¹ : Countable β\ninst✝ : CompleteLattice α\nm : α → M\nm0 : m ⊥ = 1\nR : M → M → Prop\nm_iSup : ∀ (s : ℕ → α), R (m (⨆ i, s i)) (∏' (i : ℕ), m (s i))\ns : β → α\nval✝ : Encodable β\n⊢ R (m (⨆ b, s b)) (∏' (b : β), m (s b))","tactic":"cases nonempty_encodable β","premises":[{"full_name":"nonempty_encodable","def_path":"Mathlib/Logic/Encodable/Basic.lean","def_pos":[399,8],"def_end_pos":[399,26]}]},{"state_before":"case intro\nM : Type u_1\ninst✝⁵ : CommMonoid M\ninst✝⁴ : TopologicalSpace M\nm✝ m' : M\nG : Type u_2\ninst✝³ : CommGroup G\ng g' : G\ninst✝² : T2Space M\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝¹ : Countable β\ninst✝ : CompleteLattice α\nm : α → M\nm0 : m ⊥ = 1\nR : M → M → Prop\nm_iSup : ∀ (s : ℕ → α), R (m (⨆ i, s i)) (∏' (i : ℕ), m (s i))\ns : β → α\nval✝ : Encodable β\n⊢ R (m (⨆ b, s b)) (∏' (b : β), m (s b))","state_after":"case intro\nM : Type u_1\ninst✝⁵ : CommMonoid M\ninst✝⁴ : TopologicalSpace M\nm✝ m' : M\nG : Type u_2\ninst✝³ : CommGroup G\ng g' : G\ninst✝² : T2Space M\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝¹ : Countable β\ninst✝ : CompleteLattice α\nm : α → M\nm0 : m ⊥ = 1\nR : M → M → Prop\nm_iSup : ∀ (s : ℕ → α), R (m (⨆ i, s i)) (∏' (i : ℕ), m (s i))\ns : β → α\nval✝ : Encodable β\n⊢ R (m (⨆ i, ⨆ b ∈ decode₂ β i, s b)) (∏' (i : ℕ), m (⨆ b ∈ decode₂ β i, s b))","tactic":"rw [← iSup_decode₂, ← tprod_iSup_decode₂ _ m0 s]","premises":[{"full_name":"Encodable.iSup_decode₂","def_path":"Mathlib/Logic/Encodable/Lattice.lean","def_pos":[28,8],"def_end_pos":[28,20]},{"full_name":"tprod_iSup_decode₂","def_path":"Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean","def_pos":[121,8],"def_end_pos":[121,26]}]},{"state_before":"case intro\nM : Type u_1\ninst✝⁵ : CommMonoid M\ninst✝⁴ : TopologicalSpace M\nm✝ m' : M\nG : Type u_2\ninst✝³ : CommGroup G\ng g' : G\ninst✝² : T2Space M\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝¹ : Countable β\ninst✝ : CompleteLattice α\nm : α → M\nm0 : m ⊥ = 1\nR : M → M → Prop\nm_iSup : ∀ (s : ℕ → α), R (m (⨆ i, s i)) (∏' (i : ℕ), m (s i))\ns : β → α\nval✝ : Encodable β\n⊢ R (m (⨆ i, ⨆ b ∈ decode₂ β i, s b)) (∏' (i : ℕ), m (⨆ b ∈ decode₂ β i, s b))","state_after":"no goals","tactic":"exact m_iSup _","premises":[]}]}
+{"url":"Mathlib/Algebra/Homology/ShortComplex/Exact.lean","commit":"","full_name":"CategoryTheory.ShortComplex.Splitting.op_s","start":[665,0],"end":[675,9],"file_path":"Mathlib/Algebra/Homology/ShortComplex/Exact.lean","tactics":[{"state_before":"C : Type u_1\nD : Type u_2\ninst✝³ : Category.{?u.121165, u_1} C\ninst✝² : Category.{?u.121169, u_2} D\ninst✝¹ : Preadditive C\ninst✝ : Preadditive D\nS : ShortComplex C\nh : S.Splitting\n⊢ (S.op.f ≫ h.s.op).unop = (𝟙 S.op.X₁).unop","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"C : Type u_1\nD : Type u_2\ninst✝³ : Category.{?u.121165, u_1} C\ninst✝² : Category.{?u.121169, u_2} D\ninst✝¹ : Preadditive C\ninst✝ : Preadditive D\nS : ShortComplex C\nh : S.Splitting\n⊢ (h.r.op ≫ S.op.g).unop = (𝟙 S.op.X₃).unop","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"C : Type u_1\nD : Type u_2\ninst✝³ : Category.{?u.121165, u_1} C\ninst✝² : Category.{?u.121169, u_2} D\ninst✝¹ : Preadditive C\ninst✝ : Preadditive D\nS : ShortComplex C\nh : S.Splitting\n⊢ (h.s.op ≫ S.op.f + S.op.g ≫ h.r.op).unop = (𝟙 S.op.X₂).unop","state_after":"C : Type u_1\nD : Type u_2\ninst✝³ : Category.{?u.121165, u_1} C\ninst✝² : Category.{?u.121169, u_2} D\ninst✝¹ : Preadditive C\ninst✝ : Preadditive D\nS : ShortComplex C\nh : S.Splitting\n⊢ S.g ≫ h.s + h.r ≫ S.f = h.r ≫ S.f + S.g ≫ h.s","tactic":"simp only [op_X₂, Opposite.unop_op, op_X₁, op_f, op_X₃, op_g, unop_add, unop_comp,\n      Quiver.Hom.unop_op, unop_id, ← h.id]","premises":[{"full_name":"CategoryTheory.ShortComplex.Splitting.id","def_path":"Mathlib/Algebra/Homology/ShortComplex/Exact.lean","def_pos":[480,2],"def_end_pos":[480,4]},{"full_name":"CategoryTheory.ShortComplex.op_X₁","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[220,2],"def_end_pos":[220,7]},{"full_name":"CategoryTheory.ShortComplex.op_X₂","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[220,2],"def_end_pos":[220,7]},{"full_name":"CategoryTheory.ShortComplex.op_X₃","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[220,2],"def_end_pos":[220,7]},{"full_name":"CategoryTheory.ShortComplex.op_f","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[220,2],"def_end_pos":[220,7]},{"full_name":"CategoryTheory.ShortComplex.op_g","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[220,2],"def_end_pos":[220,7]},{"full_name":"CategoryTheory.unop_add","def_path":"Mathlib/CategoryTheory/Preadditive/Opposite.lean","def_pos":[33,8],"def_end_pos":[33,16]},{"full_name":"CategoryTheory.unop_comp","def_path":"Mathlib/CategoryTheory/Opposites.lean","def_pos":[80,8],"def_end_pos":[80,17]},{"full_name":"CategoryTheory.unop_id","def_path":"Mathlib/CategoryTheory/Opposites.lean","def_pos":[84,8],"def_end_pos":[84,15]},{"full_name":"Opposite.unop_op","def_path":"Mathlib/Data/Opposite.lean","def_pos":[64,8],"def_end_pos":[64,15]},{"full_name":"Quiver.Hom.unop_op","def_path":"Mathlib/CategoryTheory/Opposites.lean","def_pos":[44,8],"def_end_pos":[44,26]}]},{"state_before":"C : Type u_1\nD : Type u_2\ninst✝³ : Category.{?u.121165, u_1} C\ninst✝² : Category.{?u.121169, u_2} D\ninst✝¹ : Preadditive C\ninst✝ : Preadditive D\nS : ShortComplex C\nh : S.Splitting\n⊢ S.g ≫ h.s + h.r ≫ S.f = h.r ≫ S.f + S.g ≫ h.s","state_after":"no goals","tactic":"abel","premises":[]}]}
+{"url":"Mathlib/Deprecated/Subgroup.lean","commit":"","full_name":"IsGroupHom.injective_of_trivial_ker","start":[355,0],"end":[361,47],"file_path":"Mathlib/Deprecated/Subgroup.lean","tactics":[{"state_before":"G : Type u_1\nH : Type u_2\nA : Type u_3\na a₁ a₂ b c : G\ninst✝¹ : Group G\ninst✝ : Group H\nf : G → H\nhf : IsGroupHom f\nh : ker f = IsSubgroup.trivial G\n⊢ Injective f","state_after":"G : Type u_1\nH : Type u_2\nA : Type u_3\na a₁✝ a₂✝ b c : G\ninst✝¹ : Group G\ninst✝ : Group H\nf : G → H\nhf : IsGroupHom f\nh : ker f = IsSubgroup.trivial G\na₁ a₂ : G\nhfa : f a₁ = f a₂\n⊢ a₁ = a₂","tactic":"intro a₁ a₂ hfa","premises":[]},{"state_before":"G : Type u_1\nH : Type u_2\nA : Type u_3\na a₁✝ a₂✝ b c : G\ninst✝¹ : Group G\ninst✝ : Group H\nf : G → H\nhf : IsGroupHom f\nh : ker f = IsSubgroup.trivial G\na₁ a₂ : G\nhfa : f a₁ = f a₂\n⊢ a₁ = a₂","state_after":"G : Type u_1\nH : Type u_2\nA : Type u_3\na a₁✝ a₂✝ b c : G\ninst✝¹ : Group G\ninst✝ : Group H\nf : G → H\nhf : IsGroupHom f\na₁ a₂ : G\nhfa : f a₁ = f a₂\nh : ∀ (x : G), f x = 1 ↔ x = 1\n⊢ a₁ = a₂","tactic":"simp [Set.ext_iff, ker, IsSubgroup.trivial] at h","premises":[{"full_name":"IsGroupHom.ker","def_path":"Mathlib/Deprecated/Subgroup.lean","def_pos":[268,4],"def_end_pos":[268,7]},{"full_name":"IsSubgroup.trivial","def_path":"Mathlib/Deprecated/Subgroup.lean","def_pos":[194,4],"def_end_pos":[194,11]},{"full_name":"Set.ext_iff","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[190,18],"def_end_pos":[190,25]}]},{"state_before":"G : Type u_1\nH : Type u_2\nA : Type u_3\na a₁✝ a₂✝ b c : G\ninst✝¹ : Group G\ninst✝ : Group H\nf : G → H\nhf : IsGroupHom f\na₁ a₂ : G\nhfa : f a₁ = f a₂\nh : ∀ (x : G), f x = 1 ↔ x = 1\n⊢ a₁ = a₂","state_after":"G : Type u_1\nH : Type u_2\nA : Type u_3\na a₁✝ a₂✝ b c : G\ninst✝¹ : Group G\ninst✝ : Group H\nf : G → H\nhf : IsGroupHom f\na₁ a₂ : G\nhfa : f a₁ = f a₂\nh : ∀ (x : G), f x = 1 ↔ x = 1\nha : a₁ * a₂⁻¹ = 1\n⊢ a₁ = a₂","tactic":"have ha : a₁ * a₂⁻¹ = 1 := by rw [← h]; exact hf.inv_ker_one hfa","premises":[{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"IsGroupHom.inv_ker_one","def_path":"Mathlib/Deprecated/Subgroup.lean","def_pos":[291,8],"def_end_pos":[291,19]}]},{"state_before":"G : Type u_1\nH : Type u_2\nA : Type u_3\na a₁✝ a₂✝ b c : G\ninst✝¹ : Group G\ninst✝ : Group H\nf : G → H\nhf : IsGroupHom f\na₁ a₂ : G\nhfa : f a₁ = f a₂\nh : ∀ (x : G), f x = 1 ↔ x = 1\nha : a₁ * a₂⁻¹ = 1\n⊢ a₁ = a₂","state_after":"no goals","tactic":"rw [eq_inv_of_mul_eq_one_left ha, inv_inv a₂]","premises":[{"full_name":"eq_inv_of_mul_eq_one_left","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[994,8],"def_end_pos":[994,33]},{"full_name":"inv_inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[734,8],"def_end_pos":[734,15]}]}]}
+{"url":"Mathlib/Algebra/Order/Rearrangement.lean","commit":"","full_name":"Antivary.sum_smul_lt_sum_smul_comp_perm_iff","start":[294,0],"end":[299,83],"file_path":"Mathlib/Algebra/Order/Rearrangement.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝⁴ : LinearOrderedRing α\ninst✝³ : LinearOrderedAddCommGroup β\ninst✝² : Module α β\ninst✝¹ : OrderedSMul α β\ns : Finset ι\nσ : Perm ι\nf : ι → α\ng : ι → β\ninst✝ : Fintype ι\nhfg : Antivary f g\n⊢ ∑ i : ι, f i • g i < ∑ i : ι, f i • g (σ i) ↔ ¬Antivary f (g ∘ ⇑σ)","state_after":"no goals","tactic":"simp [(hfg.antivaryOn _).sum_smul_lt_sum_smul_comp_perm_iff fun _ _ ↦ mem_univ _]","premises":[{"full_name":"Antivary.antivaryOn","def_path":"Mathlib/Order/Monotone/Monovary.lean","def_pos":[56,18],"def_end_pos":[56,37]},{"full_name":"AntivaryOn.sum_smul_lt_sum_smul_comp_perm_iff","def_path":"Mathlib/Algebra/Order/Rearrangement.lean","def_pos":[202,8],"def_end_pos":[202,53]},{"full_name":"Finset.mem_univ","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[71,8],"def_end_pos":[71,16]}]}]}
+{"url":"Mathlib/Data/Fin/Basic.lean","commit":"","full_name":"Fin.cast_eq_cast","start":[624,0],"end":[629,5],"file_path":"Mathlib/Data/Fin/Basic.lean","tactics":[{"state_before":"n m : ℕ\nh : n = m\n⊢ cast h = _root_.cast ⋯","state_after":"n : ℕ\n⊢ cast ⋯ = _root_.cast ⋯","tactic":"subst h","premises":[]},{"state_before":"n : ℕ\n⊢ cast ⋯ = _root_.cast ⋯","state_after":"case h.h\nn : ℕ\nx✝ : Fin n\n⊢ ↑(cast ⋯ x✝) = ↑(_root_.cast ⋯ x✝)","tactic":"ext","premises":[]},{"state_before":"case h.h\nn : ℕ\nx✝ : Fin n\n⊢ ↑(cast ⋯ x✝) = ↑(_root_.cast ⋯ x✝)","state_after":"no goals","tactic":"rfl","premises":[]}]}
+{"url":"Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean","commit":"","full_name":"CochainComplex.HomComplex.Cochain.comp_neg","start":[356,0],"end":[360,64],"file_path":"Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean","tactics":[{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nF G K L : CochainComplex C ℤ\nn m n₁ n₂ n₁₂ : ℤ\nz₁ : Cochain F G n₁\nz₂ : Cochain G K n₂\nh : n₁ + n₂ = n₁₂\n⊢ z₁.comp (-z₂) h = -z₁.comp z₂ h","state_after":"case h\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nF G K L : CochainComplex C ℤ\nn m n₁ n₂ n₁₂ : ℤ\nz₁ : Cochain F G n₁\nz₂ : Cochain G K n₂\nh : n₁ + n₂ = n₁₂\np q : ℤ\nhpq : p + n₁₂ = q\n⊢ (z₁.comp (-z₂) h).v p q hpq = (-z₁.comp z₂ h).v p q hpq","tactic":"ext p q hpq","premises":[]},{"state_before":"case h\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nF G K L : CochainComplex C ℤ\nn m n₁ n₂ n₁₂ : ℤ\nz₁ : Cochain F G n₁\nz₂ : Cochain G K n₂\nh : n₁ + n₂ = n₁₂\np q : ℤ\nhpq : p + n₁₂ = q\n⊢ (z₁.comp (-z₂) h).v p q hpq = (-z₁.comp z₂ h).v p q hpq","state_after":"no goals","tactic":"simp only [comp_v _ _ h p _ q rfl (by omega), neg_v, comp_neg]","premises":[{"full_name":"CategoryTheory.Preadditive.comp_neg","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[143,8],"def_end_pos":[143,16]},{"full_name":"CochainComplex.HomComplex.Cochain.comp_v","def_path":"Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean","def_pos":[236,6],"def_end_pos":[236,12]},{"full_name":"CochainComplex.HomComplex.Cochain.neg_v","def_path":"Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean","def_pos":[117,6],"def_end_pos":[117,11]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
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+{"url":"Mathlib/Algebra/Group/Subgroup/Basic.lean","commit":"","full_name":"AddSubgroup.addSubgroupOf_map_subtype","start":[1268,0],"end":[1270,92],"file_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","tactics":[{"state_before":"G : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst✝⁵ : Group G\ninst✝⁴ : Group G'\ninst✝³ : Group G''\nA : Type u_4\ninst✝² : AddGroup A\nH✝ K✝ : Subgroup G\nk : Set G\nN : Type u_5\ninst✝¹ : Group N\nP : Type u_6\ninst✝ : Group P\nH K : Subgroup G\n⊢ ↑(map K.subtype (H.subgroupOf K)) = ↑(H ⊓ K)","state_after":"G : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst✝⁵ : Group G\ninst✝⁴ : Group G'\ninst✝³ : Group G''\nA : Type u_4\ninst✝² : AddGroup A\nH✝ K✝ : Subgroup G\nk : Set G\nN : Type u_5\ninst✝¹ : Group N\nP : Type u_6\ninst✝ : Group P\nH K : Subgroup G\n⊢ ↑K ∩ H.toSubsemigroup.1 = ↑(H ⊓ K)","tactic":"refine Subtype.image_preimage_coe _ _ |>.trans ?_","premises":[{"full_name":"Eq.trans","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[335,8],"def_end_pos":[335,16]},{"full_name":"Subtype.image_preimage_coe","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[1188,8],"def_end_pos":[1188,26]}]},{"state_before":"G : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst✝⁵ : Group G\ninst✝⁴ : Group G'\ninst✝³ : Group G''\nA : Type u_4\ninst✝² : AddGroup A\nH✝ K✝ : Subgroup G\nk : Set G\nN : Type u_5\ninst✝¹ : Group N\nP : Type u_6\ninst✝ : Group P\nH K : Subgroup G\n⊢ ↑K ∩ H.toSubsemigroup.1 = ↑(H ⊓ K)","state_after":"no goals","tactic":"apply Set.inter_comm","premises":[{"full_name":"Set.inter_comm","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[742,8],"def_end_pos":[742,18]}]}]}
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?_)","premises":[{"full_name":"ENNReal.inv_le_inv","def_path":"Mathlib/Data/ENNReal/Inv.lean","def_pos":[205,18],"def_end_pos":[205,28]},{"full_name":"ENNReal.le_of_forall_pos_nnreal_lt","def_path":"Mathlib/Data/ENNReal/Inv.lean","def_pos":[362,8],"def_end_pos":[362,34]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]}]},{"state_before":"𝕜 : Type u_1\nA : Type u_2\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra ℂ A\ninst✝ : CompleteSpace A\na : A\nr : ℝ≥0\nr_pos : 0 < r\nr_lt : ↑r < (spectralRadius ℂ a)⁻¹\n⊢ ↑r ≤ (limsup (fun n => ↑‖a ^ n‖₊ ^ (1 / ↑n)) atTop)⁻¹","state_after":"𝕜 : Type u_1\nA : Type u_2\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra ℂ A\ninst✝ : CompleteSpace A\na : A\nr : ℝ≥0\nr_pos : 0 < r\nr_lt : ↑r < (spectralRadius ℂ a)⁻¹\n⊢ ↑r ≤ liminf (fun i => 1 / ↑‖a ^ i‖₊ ^ (1 / ↑i)) atTop","tactic":"simp_rw [inv_limsup, ← one_div]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"ENNReal.inv_limsup","def_path":"Mathlib/Topology/Instances/ENNReal.lean","def_pos":[464,8],"def_end_pos":[464,18]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"one_div","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[338,8],"def_end_pos":[338,15]}]},{"state_before":"𝕜 : Type u_1\nA : Type u_2\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra ℂ A\ninst✝ : CompleteSpace A\na : A\nr : ℝ≥0\nr_pos : 0 < r\nr_lt : ↑r < (spectralRadius ℂ a)⁻¹\n⊢ ↑r ≤ liminf (fun i => 1 / ↑‖a ^ i‖₊ ^ (1 / ↑i)) atTop","state_after":"𝕜 : Type u_1\nA : Type u_2\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra ℂ A\ninst✝ : CompleteSpace A\na : A\nr : ℝ≥0\nr_pos : 0 < r\nr_lt : ↑r < (spectralRadius ℂ a)⁻¹\np : FormalMultilinearSeries ℂ ℂ A := fun n => ContinuousMultilinearMap.mkPiRing ℂ (Fin n) (a ^ n)\n⊢ ↑r ≤ liminf (fun i => 1 / ↑‖a ^ i‖₊ ^ (1 / ↑i)) atTop","tactic":"let p : FormalMultilinearSeries ℂ ℂ A := fun n =>\n    ContinuousMultilinearMap.mkPiRing ℂ (Fin n) (a ^ n)","premises":[{"full_name":"Complex","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[27,10],"def_end_pos":[27,17]},{"full_name":"ContinuousMultilinearMap.mkPiRing","def_path":"Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean","def_pos":[595,14],"def_end_pos":[595,22]},{"full_name":"Fin","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1831,10],"def_end_pos":[1831,13]},{"full_name":"FormalMultilinearSeries","def_path":"Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean","def_pos":[44,4],"def_end_pos":[44,27]}]},{"state_before":"𝕜 : Type u_1\nA : Type u_2\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra ℂ A\ninst✝ : CompleteSpace A\na : A\nr : ℝ≥0\nr_pos : 0 < r\nr_lt : ↑r < (spectralRadius ℂ a)⁻¹\np : FormalMultilinearSeries ℂ ℂ A := fun n => ContinuousMultilinearMap.mkPiRing ℂ (Fin n) (a ^ n)\n⊢ ↑r ≤ liminf (fun i => 1 / ↑‖a ^ i‖₊ ^ (1 / ↑i)) atTop","state_after":"𝕜 : Type u_1\nA : Type u_2\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra ℂ A\ninst✝ : CompleteSpace A\na : A\nr : ℝ≥0\nr_pos : 0 < r\nr_lt : ↑r < (spectralRadius ℂ a)⁻¹\np : FormalMultilinearSeries ℂ ℂ A := fun n => ContinuousMultilinearMap.mkPiRing ℂ (Fin n) (a ^ n)\n⊢ ↑r ≤ p.radius","tactic":"suffices h : (r : ℝ≥0∞) ≤ p.radius by\n    convert h\n    simp only [p, p.radius_eq_liminf, ← norm_toNNReal, norm_mkPiRing]\n    congr\n    ext n\n    rw [norm_toNNReal, ENNReal.coe_rpow_def ‖a ^ n‖₊ (1 / n : ℝ), if_neg]\n    exact fun ha => (lt_self_iff_false _).mp\n      (ha.2.trans_le (one_div_nonneg.mpr n.cast_nonneg : 0 ≤ (1 / n : ℝ)))","premises":[{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"ContinuousMultilinearMap.norm_mkPiRing","def_path":"Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean","def_pos":[810,8],"def_end_pos":[810,21]},{"full_name":"ENNReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[96,4],"def_end_pos":[96,11]},{"full_name":"ENNReal.coe_rpow_def","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean","def_pos":[423,8],"def_end_pos":[423,20]},{"full_name":"FormalMultilinearSeries.radius","def_path":"Mathlib/Analysis/Analytic/Basic.lean","def_pos":[125,4],"def_end_pos":[125,10]},{"full_name":"FormalMultilinearSeries.radius_eq_liminf","def_path":"Mathlib/Analysis/Analytic/RadiusLiminf.lean","def_pos":[33,8],"def_end_pos":[33,24]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"NNNorm.nnnorm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[65,2],"def_end_pos":[65,8]},{"full_name":"Nat.cast_nonneg","def_path":"Mathlib/Data/Nat/Cast/Order/Ring.lean","def_pos":[29,8],"def_end_pos":[29,19]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"if_neg","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[937,8],"def_end_pos":[937,14]},{"full_name":"lt_self_iff_false","def_path":"Mathlib/Order/Basic.lean","def_pos":[155,8],"def_end_pos":[155,25]},{"full_name":"norm_toNNReal","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[623,14],"def_end_pos":[623,27]},{"full_name":"one_div_nonneg","def_path":"Mathlib/Algebra/Order/Field/Unbundled/Basic.lean","def_pos":[41,6],"def_end_pos":[41,20]}]},{"state_before":"𝕜 : Type u_1\nA : Type u_2\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra ℂ A\ninst✝ : CompleteSpace A\na : A\nr : ℝ≥0\nr_pos : 0 < r\nr_lt : ↑r < (spectralRadius ℂ a)⁻¹\np : FormalMultilinearSeries ℂ ℂ A := fun n => ContinuousMultilinearMap.mkPiRing ℂ (Fin n) (a ^ n)\n⊢ ↑r ≤ p.radius","state_after":"𝕜 : Type u_1\nA : Type u_2\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra ℂ A\ninst✝ : CompleteSpace A\na : A\nr : ℝ≥0\nr_pos : 0 < r\nr_lt : ↑r < (spectralRadius ℂ a)⁻¹\np : FormalMultilinearSeries ℂ ℂ A := fun n => ContinuousMultilinearMap.mkPiRing ℂ (Fin n) (a ^ n)\nH₁ :\n  HasFPowerSeriesOnBall (fun z => Ring.inverse (1 - z • a)) (cauchyPowerSeries (fun z => Ring.inverse (1 - z • a)) 0 ↑r)\n    0 ↑r\n⊢ ↑r ≤ p.radius","tactic":"have H₁ := (differentiableOn_inverse_one_sub_smul r_lt).hasFPowerSeriesOnBall r_pos","premises":[{"full_name":"DifferentiableOn.hasFPowerSeriesOnBall","def_path":"Mathlib/Analysis/Complex/CauchyIntegral.lean","def_pos":[559,18],"def_end_pos":[559,63]},{"full_name":"spectrum.differentiableOn_inverse_one_sub_smul","def_path":"Mathlib/Analysis/Normed/Algebra/Spectrum.lean","def_pos":[319,8],"def_end_pos":[319,45]}]},{"state_before":"𝕜 : Type u_1\nA : Type u_2\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra ℂ A\ninst✝ : CompleteSpace A\na : A\nr : ℝ≥0\nr_pos : 0 < r\nr_lt : ↑r < (spectralRadius ℂ a)⁻¹\np : FormalMultilinearSeries ℂ ℂ A := fun n => ContinuousMultilinearMap.mkPiRing ℂ (Fin n) (a ^ n)\nH₁ :\n  HasFPowerSeriesOnBall (fun z => Ring.inverse (1 - z • a)) (cauchyPowerSeries (fun z => Ring.inverse (1 - z • a)) 0 ↑r)\n    0 ↑r\n⊢ ↑r ≤ p.radius","state_after":"no goals","tactic":"exact ((hasFPowerSeriesOnBall_inverse_one_sub_smul ℂ a).exchange_radius H₁).r_le","premises":[{"full_name":"Complex","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[27,10],"def_end_pos":[27,17]},{"full_name":"HasFPowerSeriesOnBall.exchange_radius","def_path":"Mathlib/Analysis/Analytic/Basic.lean","def_pos":[981,8],"def_end_pos":[981,45]},{"full_name":"HasFPowerSeriesOnBall.r_le","def_path":"Mathlib/Analysis/Analytic/Basic.lean","def_pos":[352,2],"def_end_pos":[352,6]},{"full_name":"spectrum.hasFPowerSeriesOnBall_inverse_one_sub_smul","def_path":"Mathlib/Analysis/Normed/Algebra/Spectrum.lean","def_pos":[275,8],"def_end_pos":[275,50]}]}]}
+{"url":"Mathlib/Algebra/Group/Equiv/Basic.lean","commit":"","full_name":"MulEquiv.monoidHomCongr_apply","start":[517,0],"end":[531,30],"file_path":"Mathlib/Algebra/Group/Equiv/Basic.lean","tactics":[{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nA : Type u_4\nB : Type u_5\nM✝ : Type u_6\nN✝ : Type u_7\nP✝ : Type u_8\nQ✝ : Type u_9\nG : Type u_10\nH : Type u_11\ninst✝⁴ : EquivLike F α β\nM : Type ?u.43696\nN : Type ?u.43699\nP : Type ?u.43702\nQ : Type ?u.43705\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : CommMonoid P\ninst✝ : CommMonoid Q\nf : M ≃* N\ng : P ≃* Q\nh : M →* P\n⊢ (fun k => g.symm.toMonoidHom.comp (k.comp f.toMonoidHom))\n      ((fun h => g.toMonoidHom.comp (h.comp f.symm.toMonoidHom)) h) =\n    h","state_after":"case h\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nA : Type u_4\nB : Type u_5\nM✝ : Type u_6\nN✝ : Type u_7\nP✝ : Type u_8\nQ✝ : Type u_9\nG : Type u_10\nH : Type u_11\ninst✝⁴ : EquivLike F α β\nM : Type ?u.43696\nN : Type ?u.43699\nP : Type ?u.43702\nQ : Type ?u.43705\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : CommMonoid P\ninst✝ : CommMonoid Q\nf : M ≃* N\ng : P ≃* Q\nh : M →* P\nx✝ : M\n⊢ ((fun k => g.symm.toMonoidHom.comp (k.comp f.toMonoidHom))\n        ((fun h => g.toMonoidHom.comp (h.comp f.symm.toMonoidHom)) h))\n      x✝ =\n    h x✝","tactic":"ext","premises":[]},{"state_before":"case h\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nA : Type u_4\nB : Type u_5\nM✝ : Type u_6\nN✝ : Type u_7\nP✝ : Type u_8\nQ✝ : Type u_9\nG : Type u_10\nH : Type u_11\ninst✝⁴ : EquivLike F α β\nM : Type ?u.43696\nN : Type ?u.43699\nP : Type ?u.43702\nQ : Type ?u.43705\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : CommMonoid P\ninst✝ : CommMonoid Q\nf : M ≃* N\ng : P ≃* Q\nh : M →* P\nx✝ : M\n⊢ ((fun k => g.symm.toMonoidHom.comp (k.comp f.toMonoidHom))\n        ((fun h => g.toMonoidHom.comp (h.comp f.symm.toMonoidHom)) h))\n      x✝ =\n    h x✝","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nA : Type u_4\nB : Type u_5\nM✝ : Type u_6\nN✝ : Type u_7\nP✝ : Type u_8\nQ✝ : Type u_9\nG : Type u_10\nH : Type u_11\ninst✝⁴ : EquivLike F α β\nM : Type ?u.43696\nN : Type ?u.43699\nP : Type ?u.43702\nQ : Type ?u.43705\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : CommMonoid P\ninst✝ : CommMonoid Q\nf : M ≃* N\ng : P ≃* Q\nk : N →* Q\n⊢ (fun h => g.toMonoidHom.comp (h.comp f.symm.toMonoidHom))\n      ((fun k => g.symm.toMonoidHom.comp (k.comp f.toMonoidHom)) k) =\n    k","state_after":"case h\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nA : Type u_4\nB : Type u_5\nM✝ : Type u_6\nN✝ : Type u_7\nP✝ : Type u_8\nQ✝ : Type u_9\nG : Type u_10\nH : Type u_11\ninst✝⁴ : EquivLike F α β\nM : Type ?u.43696\nN : Type ?u.43699\nP : Type ?u.43702\nQ : Type ?u.43705\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : CommMonoid P\ninst✝ : CommMonoid Q\nf : M ≃* N\ng : P ≃* Q\nk : N →* Q\nx✝ : N\n⊢ ((fun h => g.toMonoidHom.comp (h.comp f.symm.toMonoidHom))\n        ((fun k => g.symm.toMonoidHom.comp (k.comp f.toMonoidHom)) k))\n      x✝ =\n    k x✝","tactic":"ext","premises":[]},{"state_before":"case h\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nA : Type u_4\nB : Type u_5\nM✝ : Type u_6\nN✝ : Type u_7\nP✝ : Type u_8\nQ✝ : Type u_9\nG : Type u_10\nH : Type u_11\ninst✝⁴ : EquivLike F α β\nM : Type ?u.43696\nN : Type ?u.43699\nP : Type ?u.43702\nQ : Type ?u.43705\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : CommMonoid P\ninst✝ : CommMonoid Q\nf : M ≃* N\ng : P ≃* Q\nk : N →* Q\nx✝ : N\n⊢ ((fun h => g.toMonoidHom.comp (h.comp f.symm.toMonoidHom))\n        ((fun k => g.symm.toMonoidHom.comp (k.comp f.toMonoidHom)) k))\n      x✝ =\n    k x✝","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nA : Type u_4\nB : Type u_5\nM✝ : Type u_6\nN✝ : Type u_7\nP✝ : Type u_8\nQ✝ : Type u_9\nG : Type u_10\nH : Type u_11\ninst✝⁴ : EquivLike F α β\nM : Type ?u.43696\nN : Type ?u.43699\nP : Type ?u.43702\nQ : Type ?u.43705\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : CommMonoid P\ninst✝ : CommMonoid Q\nf : M ≃* N\ng : P ≃* Q\nh k : M →* P\n⊢ { toFun := fun h => g.toMonoidHom.comp (h.comp f.symm.toMonoidHom),\n          invFun := fun k => g.symm.toMonoidHom.comp (k.comp f.toMonoidHom), left_inv := ⋯, right_inv := ⋯ }.toFun\n      (h * k) =\n    { toFun := fun h => g.toMonoidHom.comp (h.comp f.symm.toMonoidHom),\n            invFun := fun k => g.symm.toMonoidHom.comp (k.comp f.toMonoidHom), left_inv := ⋯, right_inv := ⋯ }.toFun\n        h *\n      { toFun := fun h => g.toMonoidHom.comp (h.comp f.symm.toMonoidHom),\n            invFun := fun k => g.symm.toMonoidHom.comp (k.comp f.toMonoidHom), left_inv := ⋯, right_inv := ⋯ }.toFun\n        k","state_after":"case h\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nA : Type u_4\nB : Type u_5\nM✝ : Type u_6\nN✝ : Type u_7\nP✝ : Type u_8\nQ✝ : Type u_9\nG : Type u_10\nH : Type u_11\ninst✝⁴ : EquivLike F α β\nM : Type ?u.43696\nN : Type ?u.43699\nP : Type ?u.43702\nQ : Type ?u.43705\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : CommMonoid P\ninst✝ : CommMonoid Q\nf : M ≃* N\ng : P ≃* Q\nh k : M →* P\nx✝ : N\n⊢ ({ toFun := fun h => g.toMonoidHom.comp (h.comp f.symm.toMonoidHom),\n            invFun := fun k => g.symm.toMonoidHom.comp (k.comp f.toMonoidHom), left_inv := ⋯, right_inv := ⋯ }.toFun\n        (h * k))\n      x✝ =\n    ({ toFun := fun h => g.toMonoidHom.comp (h.comp f.symm.toMonoidHom),\n              invFun := fun k => g.symm.toMonoidHom.comp (k.comp f.toMonoidHom), left_inv := ⋯, right_inv := ⋯ }.toFun\n          h *\n        { toFun := fun h => g.toMonoidHom.comp (h.comp f.symm.toMonoidHom),\n              invFun := fun k => g.symm.toMonoidHom.comp (k.comp f.toMonoidHom), left_inv := ⋯, right_inv := ⋯ }.toFun\n          k)\n      x✝","tactic":"ext","premises":[]},{"state_before":"case h\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nA : Type u_4\nB : Type u_5\nM✝ : Type u_6\nN✝ : Type u_7\nP✝ : Type u_8\nQ✝ : Type u_9\nG : Type u_10\nH : Type u_11\ninst✝⁴ : EquivLike F α β\nM : Type ?u.43696\nN : Type ?u.43699\nP : Type ?u.43702\nQ : Type ?u.43705\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : CommMonoid P\ninst✝ : CommMonoid Q\nf : M ≃* N\ng : P ≃* Q\nh k : M →* P\nx✝ : N\n⊢ ({ toFun := fun h => g.toMonoidHom.comp (h.comp f.symm.toMonoidHom),\n            invFun := fun k => g.symm.toMonoidHom.comp (k.comp f.toMonoidHom), left_inv := ⋯, right_inv := ⋯ }.toFun\n        (h * k))\n      x✝ =\n    ({ toFun := fun h => g.toMonoidHom.comp (h.comp f.symm.toMonoidHom),\n              invFun := fun k => g.symm.toMonoidHom.comp (k.comp f.toMonoidHom), left_inv := ⋯, right_inv := ⋯ }.toFun\n          h *\n        { toFun := fun h => g.toMonoidHom.comp (h.comp f.symm.toMonoidHom),\n              invFun := fun k => g.symm.toMonoidHom.comp (k.comp f.toMonoidHom), left_inv := ⋯, right_inv := ⋯ }.toFun\n          k)\n      x✝","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/Combinatorics/SimpleGraph/Operations.lean","commit":"","full_name":"SimpleGraph.sup_edge_self","start":[153,0],"end":[155,46],"file_path":"Mathlib/Combinatorics/SimpleGraph/Operations.lean","tactics":[{"state_before":"V : Type u_1\ninst✝ : DecidableEq V\nG : SimpleGraph V\ns t : V\n⊢ G ⊔ edge s s = G","state_after":"no goals","tactic":"rw [edge_self_eq_bot, sup_of_le_left bot_le]","premises":[{"full_name":"SimpleGraph.edge_self_eq_bot","def_path":"Mathlib/Combinatorics/SimpleGraph/Operations.lean","def_pos":[150,6],"def_end_pos":[150,22]},{"full_name":"bot_le","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[192,8],"def_end_pos":[192,14]}]}]}
+{"url":"Mathlib/Analysis/Convex/Segment.lean","commit":"","full_name":"Prod.segment_subset","start":[554,0],"end":[556,94],"file_path":"Mathlib/Analysis/Convex/Segment.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\nπ : ι → Type u_6\ninst✝⁴ : OrderedSemiring 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nx y : E × F\n⊢ [x-[𝕜]y] ⊆ [x.1-[𝕜]y.1] ×ˢ [x.2-[𝕜]y.2]","state_after":"case intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\nπ : ι → Type u_6\ninst✝⁴ : OrderedSemiring 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nx y z : E × F\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nhz : a • x + b • y = z\n⊢ z ∈ [x.1-[𝕜]y.1] ×ˢ [x.2-[𝕜]y.2]","tactic":"rintro z ⟨a, b, ha, hb, hab, hz⟩","premises":[]},{"state_before":"case intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\nπ : ι → Type u_6\ninst✝⁴ : OrderedSemiring 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nx y z : E × F\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nhz : a • x + b • y = z\n⊢ z ∈ [x.1-[𝕜]y.1] ×ˢ [x.2-[𝕜]y.2]","state_after":"no goals","tactic":"exact ⟨⟨a, b, ha, hb, hab, congr_arg Prod.fst hz⟩, a, b, ha, hb, hab, congr_arg Prod.snd hz⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Prod.fst","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[483,2],"def_end_pos":[483,5]},{"full_name":"Prod.snd","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[485,2],"def_end_pos":[485,5]}]}]}
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+{"url":"Mathlib/AlgebraicGeometry/Morphisms/QuasiSeparated.lean","commit":"","full_name":"AlgebraicGeometry.quasiCompact_affineProperty_iff_quasiSeparatedSpace","start":[75,0],"end":[104,14],"file_path":"Mathlib/AlgebraicGeometry/Morphisms/QuasiSeparated.lean","tactics":[{"state_before":"X✝ Y✝ : Scheme\nf✝ : X✝ ⟶ Y✝\nX Y : Scheme\ninst✝ : IsAffine Y\nf : X ⟶ Y\n⊢ AffineTargetMorphismProperty.diagonal (fun X x x_1 x => CompactSpace ↑↑X.toPresheafedSpace) f ↔\n    QuasiSeparatedSpace ↑↑X.toPresheafedSpace","state_after":"X✝ Y✝ : Scheme\nf✝ : X✝ ⟶ Y✝\nX Y : Scheme\ninst✝ : IsAffine Y\nf : X ⟶ Y\n⊢ (∀ ⦃U₁ U₂ : Scheme⦄ (f₁ : U₁ ⟶ X) (f₂ : U₂ ⟶ X) [inst : IsAffine U₁] [inst_1 : IsAffine U₂]\n      [inst_2 : IsOpenImmersion f₁] [inst_3 : IsOpenImmersion f₂],\n      (fun X x x_1 x => CompactSpace ↑↑X.toPresheafedSpace) (pullback f₁ f₂) (pullback (f₁ ≫ f) (f₂ ≫ f))\n        (pullback.mapDesc f₁ f₂ f) ⋯) ↔\n    QuasiSeparatedSpace ↑↑X.toPresheafedSpace","tactic":"delta AffineTargetMorphismProperty.diagonal","premises":[{"full_name":"AlgebraicGeometry.AffineTargetMorphismProperty.diagonal","def_path":"Mathlib/AlgebraicGeometry/Morphisms/Constructors.lean","def_pos":[46,4],"def_end_pos":[46,41]}]},{"state_before":"X✝ Y✝ : Scheme\nf✝ : X✝ ⟶ Y✝\nX Y : Scheme\ninst��� : IsAffine Y\nf : X ⟶ Y\n⊢ (∀ ⦃U₁ U₂ : Scheme⦄ (f₁ : U₁ ⟶ X) (f₂ : U₂ ⟶ X) [inst : IsAffine U₁] [inst_1 : IsAffine U₂]\n      [inst_2 : IsOpenImmersion f₁] [inst_3 : IsOpenImmersion f₂],\n      (fun X x x_1 x => CompactSpace ↑↑X.toPresheafedSpace) (pullback f₁ f₂) (pullback (f₁ ≫ f) (f₂ ≫ f))\n        (pullback.mapDesc f₁ f₂ f) ⋯) ↔\n    QuasiSeparatedSpace ↑↑X.toPresheafedSpace","state_after":"X✝ Y✝ : Scheme\nf✝ : X✝ ⟶ Y✝\nX Y : Scheme\ninst✝ : IsAffine Y\nf : X ⟶ Y\n⊢ (∀ ⦃U₁ U₂ : Scheme⦄ (f₁ : U₁ ⟶ X) (f₂ : U₂ ⟶ X) [inst : IsAffine U₁] [inst_1 : IsAffine U₂]\n      [inst_2 : IsOpenImmersion f₁] [inst_3 : IsOpenImmersion f₂],\n      (fun X x x_1 x => CompactSpace ↑↑X.toPresheafedSpace) (pullback f₁ f₂) (pullback (f₁ ≫ f) (f₂ ≫ f))\n        (pullback.mapDesc f₁ f₂ f) ⋯) ↔\n    ∀ (U V : ↑X.affineOpens), IsCompact (↑↑U ∩ ↑↑V)","tactic":"rw [quasiSeparatedSpace_iff_affine]","premises":[{"full_name":"AlgebraicGeometry.quasiSeparatedSpace_iff_affine","def_path":"Mathlib/AlgebraicGeometry/Morphisms/QuasiSeparated.lean","def_pos":[49,8],"def_end_pos":[49,38]}]},{"state_before":"X✝ Y✝ : Scheme\nf✝ : X✝ ⟶ Y✝\nX Y : Scheme\ninst✝ : IsAffine Y\nf : X ⟶ Y\n⊢ (∀ ⦃U₁ U₂ : Scheme⦄ (f₁ : U₁ ⟶ X) (f₂ : U₂ ⟶ X) [inst : IsAffine U₁] [inst_1 : IsAffine U₂]\n      [inst_2 : IsOpenImmersion f₁] [inst_3 : IsOpenImmersion f₂],\n      (fun X x x_1 x => CompactSpace ↑↑X.toPresheafedSpace) (pullback f₁ f₂) (pullback (f₁ ≫ f) (f₂ ≫ f))\n        (pullback.mapDesc f₁ f₂ f) ⋯) ↔\n    ∀ (U V : ↑X.affineOpens), IsCompact (↑↑U ∩ ↑↑V)","state_after":"case mp\nX✝ Y✝ : Scheme\nf✝ : X✝ ⟶ Y✝\nX Y : Scheme\ninst✝ : IsAffine Y\nf : X ⟶ Y\n⊢ (∀ ⦃U₁ U₂ : Scheme⦄ (f₁ : U₁ ⟶ X) (f₂ : U₂ ⟶ X) [inst : IsAffine U₁] [inst_1 : IsAffine U₂]\n      [inst_2 : IsOpenImmersion f₁] [inst_3 : IsOpenImmersion f₂],\n      (fun X x x_1 x => CompactSpace ↑↑X.toPresheafedSpace) (pullback f₁ f₂) (pullback (f₁ ≫ f) (f₂ ≫ f))\n        (pullback.mapDesc f₁ f₂ f) ⋯) →\n    ∀ (U V : ↑X.affineOpens), IsCompact (↑↑U ∩ ↑↑V)\n\ncase mpr\nX✝ Y✝ : Scheme\nf✝ : X✝ ⟶ Y✝\nX Y : Scheme\ninst✝ : IsAffine Y\nf : X ⟶ Y\n⊢ (∀ (U V : ↑X.affineOpens), IsCompact (↑↑U ∩ ↑↑V)) →\n    ∀ ⦃U₁ U₂ : Scheme⦄ (f₁ : U₁ ⟶ X) (f₂ : U₂ ⟶ X) [inst : IsAffine U₁] [inst_1 : IsAffine U₂]\n      [inst_2 : IsOpenImmersion f₁] [inst_3 : IsOpenImmersion f₂],\n      (fun X x x_1 x => CompactSpace ↑↑X.toPresheafedSpace) (pullback f₁ f₂) (pullback (f₁ ≫ f) (f₂ ≫ f))\n        (pullback.mapDesc f₁ f₂ f) ⋯","tactic":"constructor","premises":[]}]}
+{"url":"Mathlib/CategoryTheory/Abelian/DiagramLemmas/Four.lean","commit":"","full_name":"CategoryTheory.Abelian.epi_of_epi_of_epi_of_mono","start":[119,0],"end":[124,78],"file_path":"Mathlib/CategoryTheory/Abelian/DiagramLemmas/Four.lean","tactics":[{"state_before":"C : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : Abelian C\nR₁ R₂ : ComposableArrows C 3\nφ : R₁ ⟶ R₂\nhR₁ : R₁.Exact\nhR₂ : R₂.Exact\nh₀ : Epi (app' φ 0 ⋯)\nh₂ : Epi (app' φ 2 ⋯)\nh₃ : Mono (app' φ 3 ⋯)\n⊢ R₂.map' 1 3 ⋯ ⋯ = 0","state_after":"no goals","tactic":"simpa only [R₂.map'_comp 1 2 3] using hR₂.toIsComplex.zero 1","premises":[{"full_name":"CategoryTheory.ComposableArrows.IsComplex.zero","def_path":"Mathlib/Algebra/Homology/ExactSequence.lean","def_pos":[47,2],"def_end_pos":[47,6]},{"full_name":"CategoryTheory.ComposableArrows.map'_comp","def_path":"Mathlib/CategoryTheory/ComposableArrows.lean","def_pos":[87,6],"def_end_pos":[87,15]}]}]}
+{"url":"Mathlib/RingTheory/DedekindDomain/AdicValuation.lean","commit":"","full_name":"IsDedekindDomain.HeightOneSpectrum.valuation_lt_one_iff_dvd","start":[327,0],"end":[330,69],"file_path":"Mathlib/RingTheory/DedekindDomain/AdicValuation.lean","tactics":[{"state_before":"R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nr : R\n⊢ v.valuation ((algebraMap R K) r) < 1 ↔ v.asIdeal ∣ Ideal.span {r}","state_after":"R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nr : R\n⊢ v.intValuation r < 1 ↔ v.asIdeal ∣ Ideal.span {r}","tactic":"rw [valuation_of_algebraMap]","premises":[{"full_name":"IsDedekindDomain.HeightOneSpectrum.valuation_of_algebraMap","def_path":"Mathlib/RingTheory/DedekindDomain/AdicValuation.lean","def_pos":[316,8],"def_end_pos":[316,31]}]},{"state_before":"R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nr : R\n⊢ v.intValuation r < 1 ↔ v.asIdeal ∣ Ideal.span {r}","state_after":"no goals","tactic":"exact v.intValuation_lt_one_iff_dvd r","premises":[{"full_name":"IsDedekindDomain.HeightOneSpectrum.intValuation_lt_one_iff_dvd","def_path":"Mathlib/RingTheory/DedekindDomain/AdicValuation.lean","def_pos":[132,8],"def_end_pos":[132,35]}]}]}
+{"url":"Mathlib/RingTheory/Valuation/Basic.lean","commit":"","full_name":"Valuation.map_sub_eq_of_lt_right","start":[302,0],"end":[304,15],"file_path":"Mathlib/RingTheory/Valuation/Basic.lean","tactics":[{"state_before":"K : Type u_1\nF : Type u_2\nR : Type u_3\ninst✝³ : DivisionRing K\nΓ₀ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx y z : R\nh : v x < v y\n⊢ v (x - y) = v y","state_after":"case h\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst✝³ : DivisionRing K\nΓ₀ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx y z : R\nh : v x < v y\n⊢ v x < v (-y)","tactic":"rw [sub_eq_add_neg, map_add_eq_of_lt_right, map_neg]","premises":[{"full_name":"Valuation.map_add_eq_of_lt_right","def_path":"Mathlib/RingTheory/Valuation/Basic.lean","def_pos":[296,8],"def_end_pos":[296,30]},{"full_name":"Valuation.map_neg","def_path":"Mathlib/RingTheory/Valuation/Basic.lean","def_pos":[260,8],"def_end_pos":[260,15]},{"full_name":"sub_eq_add_neg","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[905,2],"def_end_pos":[905,13]}]},{"state_before":"case h\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst✝³ : DivisionRing K\nΓ₀ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx y z : R\nh : v x < v y\n⊢ v x < v (-y)","state_after":"no goals","tactic":"rwa [map_neg]","premises":[{"full_name":"Valuation.map_neg","def_path":"Mathlib/RingTheory/Valuation/Basic.lean","def_pos":[260,8],"def_end_pos":[260,15]}]}]}
+{"url":"Mathlib/Algebra/ModEq.lean","commit":"","full_name":"AddCommGroup.ModEq.sub_iff_left","start":[163,0],"end":[166,99],"file_path":"Mathlib/Algebra/ModEq.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : AddCommGroup α\np a a₁ a₂ b b₁ b₂ c : α\nn : ℕ\nz : ℤ\nx✝ : a₁ ≡ b₁ [PMOD p]\nm : ℤ\nhm : b₁ - a₁ = m • p\n⊢ (∃ b, b₁ - b₂ - (a₁ - a₂) = (Equiv.subLeft m).symm.symm b • p) ↔ a₂ ≡ b₂ [PMOD p]","state_after":"no goals","tactic":"simp [sub_sub_sub_comm, hm, sub_smul, ModEq]","premises":[{"full_name":"AddCommGroup.ModEq","def_path":"Mathlib/Algebra/ModEq.lean","def_pos":[44,4],"def_end_pos":[44,9]},{"full_name":"sub_smul","def_path":"Mathlib/Algebra/Module/Defs.lean","def_pos":[245,8],"def_end_pos":[245,16]},{"full_name":"sub_sub_sub_comm","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[575,2],"def_end_pos":[575,13]}]}]}
+{"url":"Mathlib/Data/Nat/Defs.lean","commit":"","full_name":"Nat.pred_eq_succ_iff","start":[156,0],"end":[157,47],"file_path":"Mathlib/Data/Nat/Defs.lean","tactics":[{"state_before":"a b c d m n k : ℕ\np q : ℕ → Prop\n⊢ n - 1 = m + 1 ↔ n = m + 2","state_after":"no goals","tactic":"cases n <;> constructor <;> rintro ⟨⟩ <;> rfl","premises":[]}]}
+{"url":"Mathlib/Algebra/Group/Defs.lean","commit":"","full_name":"add_neg_cancel_left","start":[1062,0],"end":[1064,42],"file_path":"Mathlib/Algebra/Group/Defs.lean","tactics":[{"state_before":"G : Type u_1\ninst✝ : Group G\na✝ b✝ c a b : G\n⊢ a * (a⁻¹ * b) = b","state_after":"no goals","tactic":"rw [← mul_assoc, mul_right_inv, one_mul]","premises":[{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]},{"full_name":"mul_right_inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[1051,8],"def_end_pos":[1051,21]},{"full_name":"one_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[409,8],"def_end_pos":[409,15]}]}]}
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↥(Submonoid.closure s)), r ^ n₁ x • ↑x ∈ Submonoid.closure (r • s)\nn : ℕ\nhn : n ≥ l.support.sup n₁\na : ↑↑(Submonoid.closure s)\nha : a ∈ l.support\nthis : n ≥ n₁ a\n⊢ r ^ n₁ a • ↑a ∈ B'.toSubmonoid","tactic":"change _ ∈ B'.toSubmonoid","premises":[{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]}]},{"state_before":"R : Type uR\nS : Type uS\nA : Type uA\nB : Type uB\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring A\ninst✝⁴ : Algebra R A\ns✝ t : Set A\ninst✝³ : CommSemiring B\ninst✝² : Algebra R B\ninst✝¹ : Algebra A B\ninst✝ : IsScalarTower R A B\nr : A\ns : Set B\nB' : Subalgebra R B\nhs : r • s ⊆ ↑B'\nhr : (algebraMap A B) r ∈ B'\nl : ↑↑(Submonoid.closure s) →₀ R\nn₁ : ↥(Submonoid.closure s) → ℕ\nn₂ : ∀ (x : ↥(Submonoid.closure s)), r ^ n₁ x • ↑x ∈ Submonoid.closure (r • s)\nn : ℕ\nhn : n ≥ l.support.sup n₁\na : ↑↑(Submonoid.closure s)\nha : a ∈ l.support\nthis : n ≥ n₁ a\n⊢ r ^ n₁ a • ↑a ∈ B'.toSubmonoid","state_after":"R : Type uR\nS : Type uS\nA : Type uA\nB : Type uB\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring A\ninst✝⁴ : Algebra R A\ns✝ t : Set A\ninst✝³ : CommSemiring B\ninst✝² : Algebra R B\ninst✝¹ : Algebra A B\ninst✝ : IsScalarTower R A B\nr : A\ns : Set B\nB' : Subalgebra R B\nhs : r • s ⊆ ↑B'\nhr : (algebraMap A B) r ∈ B'\nl : ↑↑(Submonoid.closure s) →₀ R\nn₁ : ↥(Submonoid.closure s) → ℕ\nn₂ : ∀ (x : ↥(Submonoid.closure s)), r ^ n₁ x • ↑x ∈ Submonoid.closure (r • s)\nn : ℕ\nhn : n ≥ l.support.sup n₁\na : ↑↑(Submonoid.closure s)\nha : a ∈ l.support\nthis : n ≥ n₁ a\n⊢ r ^ n₁ a • ↑a ∈ Submonoid.closure ↑B'.toSubmonoid","tactic":"rw [← Submonoid.closure_eq B'.toSubmonoid]","premises":[{"full_name":"Submonoid.closure_eq","def_path":"Mathlib/Algebra/Group/Submonoid/Basic.lean","def_pos":[446,8],"def_end_pos":[446,18]}]},{"state_before":"R : Type uR\nS : Type uS\nA : Type uA\nB : Type uB\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring A\ninst✝⁴ : Algebra R A\ns✝ t : Set A\ninst✝³ : CommSemiring B\ninst✝² : Algebra R B\ninst✝¹ : Algebra A B\ninst✝ : IsScalarTower R A B\nr : A\ns : Set B\nB' : Subalgebra R B\nhs : r • s ⊆ ↑B'\nhr : (algebraMap A B) r ∈ B'\nl : ↑↑(Submonoid.closure s) →₀ R\nn₁ : ↥(Submonoid.closure s) → ℕ\nn₂ : ∀ (x : ↥(Submonoid.closure s)), r ^ n₁ x • ↑x ∈ Submonoid.closure (r • s)\nn : ℕ\nhn : n ≥ l.support.sup n₁\na : ↑↑(Submonoid.closure s)\nha : a ∈ l.support\nthis : n ≥ n₁ a\n⊢ r ^ n₁ a • ↑a ∈ Submonoid.closure ↑B'.toSubmonoid","state_after":"no goals","tactic":"apply Submonoid.closure_mono hs (n₂ a)","premises":[{"full_name":"Submonoid.closure_mono","def_path":"Mathlib/Algebra/Group/Submonoid/Basic.lean","def_pos":[360,8],"def_end_pos":[360,20]}]}]}
+{"url":"Mathlib/Algebra/Order/Group/PosPart.lean","commit":"","full_name":"negPart_nonpos","start":[104,0],"end":[104,80],"file_path":"Mathlib/Algebra/Order/Group/PosPart.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝¹ : Lattice α\ninst✝ : Group α\na : α\n⊢ a⁻ᵐ ≤ 1 ↔ a⁻¹ ≤ 1","state_after":"no goals","tactic":"simp [leOnePart]","premises":[{"full_name":"leOnePart","def_path":"Mathlib/Algebra/Order/Group/PosPart.lean","def_pos":[63,4],"def_end_pos":[63,13]}]}]}
+{"url":"Mathlib/FieldTheory/SeparableDegree.lean","commit":"","full_name":"Field.finSepDegree_eq_finrank_iff","start":[727,0],"end":[738,83],"file_path":"Mathlib/FieldTheory/SeparableDegree.lean","tactics":[{"state_before":"F : Type u\nE : Type v\ninst✝⁵ : Field F\ninst✝⁴ : Field E\ninst✝³ : Algebra F E\nK : Type w\ninst✝² : Field K\ninst✝¹ : Algebra F K\ninst✝ : FiniteDimensional F E\nheq : finSepDegree F E = finrank F E\nx : E\n⊢ IsSeparable F x","state_after":"F : Type u\nE : Type v\ninst✝⁵ : Field F\ninst✝⁴ : Field E\ninst✝³ : Algebra F E\nK : Type w\ninst✝² : Field K\ninst✝¹ : Algebra F K\ninst✝ : FiniteDimensional F E\nheq : finSepDegree F E = finrank F E\nx : E\nhalg : IsAlgebraic F x\n⊢ IsSeparable F x","tactic":"have halg := IsAlgebraic.of_finite F x","premises":[{"full_name":"IsAlgebraic.of_finite","def_path":"Mathlib/RingTheory/Algebraic.lean","def_pos":[276,8],"def_end_pos":[276,29]}]},{"state_before":"F : Type u\nE : Type v\ninst✝⁵ : Field F\ninst✝⁴ : Field E\ninst✝³ : Algebra F E\nK : Type w\ninst✝² : Field K\ninst✝¹ : Algebra F K\ninst✝ : FiniteDimensional F E\nheq : finSepDegree F E = finrank F E\nx : E\nhalg : IsAlgebraic F x\n⊢ IsSeparable F x","state_after":"F : Type u\nE : Type v\ninst✝⁵ : Field F\ninst✝⁴ : Field E\ninst✝³ : Algebra F E\nK : Type w\ninst✝² : Field K\ninst✝¹ : Algebra F K\ninst✝ : FiniteDimensional F E\nheq : finSepDegree F E = finrank F E\nx : E\nhalg : IsAlgebraic F x\nh : finSepDegree F ↥F⟮x⟯ < finrank F ↥F⟮x⟯\n⊢ False","tactic":"refine (finSepDegree_adjoin_simple_eq_finrank_iff F E x halg).1 <| le_antisymm\n      (finSepDegree_adjoin_simple_le_finrank F E x halg) <| le_of_not_lt fun h ↦ ?_","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"IntermediateField.finSepDegree_adjoin_simple_eq_finrank_iff","def_path":"Mathlib/FieldTheory/SeparableDegree.lean","def_pos":[669,8],"def_end_pos":[669,49]},{"full_name":"IntermediateField.finSepDegree_adjoin_simple_le_finrank","def_path":"Mathlib/FieldTheory/SeparableDegree.lean","def_pos":[662,8],"def_end_pos":[662,45]},{"full_name":"le_antisymm","def_path":"Mathlib/Order/Defs.lean","def_pos":[156,8],"def_end_pos":[156,19]},{"full_name":"le_of_not_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[275,8],"def_end_pos":[275,20]}]},{"state_before":"F : Type u\nE : Type v\ninst✝⁵ : Field F\ninst✝⁴ : Field E\ninst✝³ : Algebra F E\nK : Type w\ninst✝² : Field K\ninst✝¹ : Algebra F K\ninst✝ : FiniteDimensional F E\nheq : finSepDegree F E = finrank F E\nx : E\nhalg : IsAlgebraic F x\nh : finSepDegree F ↥F⟮x⟯ < finrank F ↥F⟮x⟯\n⊢ False","state_after":"F : Type u\nE : Type v\ninst✝⁵ : Field F\ninst✝⁴ : Field E\ninst✝³ : Algebra F E\nK : Type w\ninst✝² : Field K\ninst✝¹ : Algebra F K\ninst✝ : FiniteDimensional F E\nheq : finSepDegree F E = finrank F E\nx : E\nhalg : IsAlgebraic F x\nh : finSepDegree F ↥F⟮x⟯ < finrank F ↥F⟮x⟯\nthis : finSepDegree F ↥F⟮x⟯ * finSepDegree (↥F⟮x⟯) E < finrank F ↥F⟮x⟯ * finrank (↥F⟮x⟯) E\n⊢ False","tactic":"have := Nat.mul_lt_mul_of_lt_of_le' h (finSepDegree_le_finrank F⟮x⟯ E) Fin.size_pos'","premises":[{"full_name":"Field.finSepDegree_le_finrank","def_path":"Mathlib/FieldTheory/SeparableDegree.lean","def_pos":[694,8],"def_end_pos":[694,31]},{"full_name":"Fin.size_pos'","def_path":".lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean","def_pos":[29,8],"def_end_pos":[29,17]},{"full_name":"IntermediateField.adjoin","def_path":"Mathlib/FieldTheory/Adjoin.lean","def_pos":[42,4],"def_end_pos":[42,10]},{"full_name":"Nat.mul_lt_mul_of_lt_of_le'","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean","def_pos":[475,18],"def_end_pos":[475,41]},{"full_name":"Singleton.singleton","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[467,2],"def_end_pos":[467,11]}]},{"state_before":"F : Type u\nE : Type v\ninst✝⁵ : Field F\ninst✝⁴ : Field E\ninst✝³ : Algebra F E\nK : Type w\ninst✝² : Field K\ninst✝¹ : Algebra F K\ninst✝ : FiniteDimensional F E\nheq : finSepDegree F E = finrank F E\nx : E\nhalg : IsAlgebraic F x\nh : finSepDegree F ↥F⟮x⟯ < finrank F ↥F⟮x⟯\nthis : finSepDegree F ↥F⟮x⟯ * finSepDegree (↥F⟮x⟯) E < finrank F ↥F⟮x⟯ * finrank (↥F⟮x⟯) E\n⊢ False","state_after":"F : Type u\nE : Type v\ninst✝⁵ : Field F\ninst✝⁴ : Field E\ninst✝³ : Algebra F E\nK : Type w\ninst✝² : Field K\ninst✝¹ : Algebra F K\ninst✝ : FiniteDimensional F E\nheq : finSepDegree F E = finrank F E\nx : E\nhalg : IsAlgebraic F x\nh : finSepDegree F ↥F⟮x⟯ < finrank F ↥F⟮x⟯\nthis : finSepDegree F E < finrank F E\n⊢ False","tactic":"rw [finSepDegree_mul_finSepDegree_of_isAlgebraic F F⟮x⟯ E,\n      FiniteDimensional.finrank_mul_finrank F F⟮x⟯ E] at this","premises":[{"full_name":"Field.finSepDegree_mul_finSepDegree_of_isAlgebraic","def_path":"Mathlib/FieldTheory/SeparableDegree.lean","def_pos":[252,8],"def_end_pos":[252,52]},{"full_name":"FiniteDimensional.finrank_mul_finrank","def_path":"Mathlib/LinearAlgebra/Dimension/Free.lean","def_pos":[59,8],"def_end_pos":[59,45]},{"full_name":"IntermediateField.adjoin","def_path":"Mathlib/FieldTheory/Adjoin.lean","def_pos":[42,4],"def_end_pos":[42,10]},{"full_name":"Singleton.singleton","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[467,2],"def_end_pos":[467,11]}]},{"state_before":"F : Type u\nE : Type v\ninst✝⁵ : Field F\ninst✝⁴ : Field E\ninst✝³ : Algebra F E\nK : Type w\ninst✝² : Field K\ninst✝¹ : Algebra F K\ninst✝ : FiniteDimensional F E\nheq : finSepDegree F E = finrank F E\nx : E\nhalg : IsAlgebraic F x\nh : finSepDegree F ↥F⟮x⟯ < finrank F ↥F⟮x⟯\nthis : finSepDegree F E < finrank F E\n⊢ False","state_after":"no goals","tactic":"linarith only [heq, this]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]}]}]}
+{"url":"Mathlib/Tactic/Qify.lean","commit":"","full_name":"Mathlib.Tactic.Qify.intCast_eq","start":[67,0],"end":[67,99],"file_path":"Mathlib/Tactic/Qify.lean","tactics":[{"state_before":"a b : ℤ\n⊢ a = b ↔ ↑a = ↑b","state_after":"no goals","tactic":"simp only [Int.cast_inj]","premises":[{"full_name":"Int.cast_inj","def_path":"Mathlib/Data/Int/Cast/Lemmas.lean","def_pos":[67,6],"def_end_pos":[67,14]}]}]}
+{"url":"Mathlib/RingTheory/Artinian.lean","commit":"","full_name":"IsArtinianRing.localization_surjective","start":[450,0],"end":[463,72],"file_path":"Mathlib/RingTheory/Artinian.lean","tactics":[{"state_before":"R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\ninst✝ : IsLocalization S L\n⊢ Surjective ⇑(algebraMap R L)","state_after":"R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\ninst✝ : IsLocalization S L\nr' : L\n⊢ ∃ a, (algebraMap R L) a = r'","tactic":"intro r'","premises":[]},{"state_before":"R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\ninst✝ : IsLocalization S L\nr' : L\n⊢ ∃ a, (algebraMap R L) a = r'","state_after":"case intro.intro\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\ninst✝ : IsLocalization S L\nr₁ : R\ns : ↥S\n⊢ ∃ a, (algebraMap R L) a = IsLocalization.mk' L r₁ s","tactic":"obtain ⟨r₁, s, rfl⟩ := IsLocalization.mk'_surjective S r'","premises":[{"full_name":"IsLocalization.mk'_surjective","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[279,8],"def_end_pos":[279,22]}]},{"state_before":"case intro.intro\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\ninst✝ : IsLocalization S L\nr₁ : R\ns : ↥S\n⊢ ∃ a, (algebraMap R L) a = IsLocalization.mk' L r₁ s","state_after":"case intro.intro.intro\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\ninst✝ : IsLocalization S L\nr₁ : R\ns : ↥S\nr₂ : R\nh : IsLocalization.mk' L 1 s = (algebraMap R L) r₂\n⊢ ∃ a, (algebraMap R L) a = IsLocalization.mk' L r₁ s","tactic":"obtain ⟨r₂, h⟩ : ∃ r : R, IsLocalization.mk' L 1 s = algebraMap R L r := by\n    obtain ⟨n, r, hr⟩ := IsArtinian.exists_pow_succ_smul_dvd (s : R) (1 : R)\n    use r\n    rw [smul_eq_mul, smul_eq_mul, pow_succ, mul_assoc] at hr\n    apply_fun algebraMap R L at hr\n    simp only [map_mul] at hr\n    rw [← IsLocalization.mk'_one (M := S) L, IsLocalization.mk'_eq_iff_eq, mul_one,\n      Submonoid.coe_one, ← (IsLocalization.map_units L (s ^ n)).mul_left_cancel hr, map_mul]","premises":[{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"IsArtinian.exists_pow_succ_smul_dvd","def_path":"Mathlib/RingTheory/Artinian.lean","def_pos":[321,8],"def_end_pos":[321,32]},{"full_name":"IsLocalization.map_units","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[115,8],"def_end_pos":[115,17]},{"full_name":"IsLocalization.mk'","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[228,18],"def_end_pos":[228,21]},{"full_name":"IsLocalization.mk'_eq_iff_eq","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[297,8],"def_end_pos":[297,21]},{"full_name":"IsLocalization.mk'_one","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[238,8],"def_end_pos":[238,15]},{"full_name":"IsUnit.mul_left_cancel","def_path":"Mathlib/Algebra/Group/Units.lean","def_pos":[696,18],"def_end_pos":[696,33]},{"full_name":"Submonoid.coe_one","def_path":"Mathlib/Algebra/Group/Submonoid/Operations.lean","def_pos":[500,8],"def_end_pos":[500,15]},{"full_name":"algebraMap","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[108,4],"def_end_pos":[108,14]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]},{"full_name":"map_mul","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[281,8],"def_end_pos":[281,15]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]},{"full_name":"mul_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[413,8],"def_end_pos":[413,15]},{"full_name":"pow_succ","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[567,8],"def_end_pos":[567,16]},{"full_name":"smul_eq_mul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[79,6],"def_end_pos":[79,17]}]},{"state_before":"case intro.intro.intro\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\ninst✝ : IsLocalization S L\nr₁ : R\ns : ↥S\nr₂ : R\nh : IsLocalization.mk' L 1 s = (algebraMap R L) r₂\n⊢ ∃ a, (algebraMap R L) a = IsLocalization.mk' L r₁ s","state_after":"no goals","tactic":"exact ⟨r₁ * r₂, by rw [IsLocalization.mk'_eq_mul_mk'_one, map_mul, h]⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"IsLocalization.mk'_eq_mul_mk'_one","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[371,8],"def_end_pos":[371,26]},{"full_name":"map_mul","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[281,8],"def_end_pos":[281,15]}]}]}
+{"url":"Mathlib/Algebra/Module/LocalizedModule.lean","commit":"","full_name":"LocalizedModule.mkLinearMap_apply","start":[458,0],"end":[464,40],"file_path":"Mathlib/Algebra/Module/LocalizedModule.lean","tactics":[{"state_before":"R : Type u\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nT : Type u_1\ninst✝² : CommSemiring T\ninst✝¹ : Algebra R T\ninst✝ : IsLocalization S T\nx y : M\n⊢ (fun m => mk m 1) (x + y) = (fun m => mk m 1) x + (fun m => mk m 1) y","state_after":"no goals","tactic":"simp [mk_add_mk]","premises":[{"full_name":"LocalizedModule.mk_add_mk","def_path":"Mathlib/Algebra/Module/LocalizedModule.lean","def_pos":[151,8],"def_end_pos":[151,17]}]}]}
+{"url":"Mathlib/CategoryTheory/GradedObject.lean","commit":"","full_name":"CategoryTheory.GradedObject.eqToHom_apply","start":[179,0],"end":[182,5],"file_path":"Mathlib/CategoryTheory/GradedObject.lean","tactics":[{"state_before":"C : Type u\ninst✝ : Category.{v, u} C\nβ : Type w\nX Y : β → C\nh : X = Y\nb : β\n⊢ X b = Y b","state_after":"no goals","tactic":"rw [h]","premises":[]},{"state_before":"C : Type u\ninst✝ : Category.{v, u} C\nβ : Type w\nX Y : β → C\nh : X = Y\nb : β\n⊢ eqToHom h b = eqToHom ⋯","state_after":"C : Type u\ninst✝ : Category.{v, u} C\nβ : Type w\nX : β → C\nb : β\n⊢ eqToHom ⋯ b = eqToHom ⋯","tactic":"subst h","premises":[]},{"state_before":"C : Type u\ninst✝ : Category.{v, u} C\nβ : Type w\nX : β → C\nb : β\n⊢ eqToHom ⋯ b = eqToHom ⋯","state_after":"no goals","tactic":"rfl","premises":[]}]}
+{"url":"Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean","commit":"","full_name":"Path.Homotopy.transReflReparamAux_zero","start":[131,0],"end":[132,32],"file_path":"Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean","tactics":[{"state_before":"X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\n⊢ transReflReparamAux 0 = 0","state_after":"no goals","tactic":"norm_num [transReflReparamAux]","premises":[{"full_name":"Path.Homotopy.transReflReparamAux","def_path":"Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean","def_pos":[117,4],"def_end_pos":[117,23]}]}]}
+{"url":"Mathlib/Order/Basic.lean","commit":"","full_name":"Function.const_lt_const","start":[903,0],"end":[904,93],"file_path":"Mathlib/Order/Basic.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\nπ : ι → Type u_4\ninst✝¹ : Preorder α\ninst✝ : Nonempty β\na b : α\n⊢ const β a < const β b ↔ a < b","state_after":"no goals","tactic":"simpa [Pi.lt_def] using le_of_lt","premises":[{"full_name":"Pi.lt_def","def_path":"Mathlib/Order/Basic.lean","def_pos":[788,8],"def_end_pos":[788,17]},{"full_name":"le_of_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[89,8],"def_end_pos":[89,16]}]}]}
+{"url":"Mathlib/RingTheory/HahnSeries/Summable.lean","commit":"","full_name":"HahnSeries.isUnit_iff","start":[480,0],"end":[493,65],"file_path":"Mathlib/RingTheory/HahnSeries/Summable.lean","tactics":[{"state_before":"Γ : Type u_1\nR : Type u_2\ninst✝² : LinearOrderedAddCommGroup Γ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nx : HahnSeries Γ R\n⊢ IsUnit x ↔ IsUnit x.leadingCoeff","state_after":"case mp\nΓ : Type u_1\nR : Type u_2\ninst✝² : LinearOrderedAddCommGroup Γ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nx : HahnSeries Γ R\n⊢ IsUnit x → IsUnit x.leadingCoeff\n\ncase mpr\nΓ : Type u_1\nR : Type u_2\ninst✝² : LinearOrderedAddCommGroup Γ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nx : HahnSeries Γ R\n⊢ IsUnit x.leadingCoeff → IsUnit x","tactic":"constructor","premises":[]}]}
+{"url":"Mathlib/RingTheory/RootsOfUnity/Minpoly.lean","commit":"","full_name":"IsPrimitiveRoot.minpoly_dvd_pow_mod","start":[85,0],"end":[96,33],"file_path":"Mathlib/RingTheory/RootsOfUnity/Minpoly.lean","tactics":[{"state_before":"n : ℕ\nK : Type u_1\ninst✝² : CommRing K\nμ : K\nh : IsPrimitiveRoot μ n\ninst✝¹ : IsDomain K\ninst✝ : CharZero K\np : ℕ\nhprime : Fact (Nat.Prime p)\nhdiv : ¬p ∣ n\n⊢ map (Int.castRingHom (ZMod p)) (minpoly ℤ μ) ∣ map (Int.castRingHom (ZMod p)) (minpoly ℤ (μ ^ p)) ^ p","state_after":"n : ℕ\nK : Type u_1\ninst✝² : CommRing K\nμ : K\nh : IsPrimitiveRoot μ n\ninst✝¹ : IsDomain K\ninst✝ : CharZero K\np : ℕ\nhprime : Fact (Nat.Prime p)\nhdiv : ¬p ∣ n\nQ : ℤ[X] := minpoly ℤ (μ ^ p)\n⊢ map (Int.castRingHom (ZMod p)) (minpoly ℤ μ) ∣ map (Int.castRingHom (ZMod p)) Q ^ p","tactic":"set Q := minpoly ℤ (μ ^ p)","premises":[{"full_name":"Int","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean","def_pos":[40,10],"def_end_pos":[40,13]},{"full_name":"minpoly","def_path":"Mathlib/FieldTheory/Minpoly/Basic.lean","def_pos":[36,18],"def_end_pos":[36,25]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"n : ℕ\nK : Type u_1\ninst✝² : CommRing K\nμ : K\nh : IsPrimitiveRoot μ n\ninst✝¹ : IsDomain K\ninst✝ : CharZero K\np : ℕ\nhprime : Fact (Nat.Prime p)\nhdiv : ¬p ∣ n\nQ : ℤ[X] := minpoly ℤ (μ ^ p)\n⊢ map (Int.castRingHom (ZMod p)) (minpoly ℤ μ) ∣ map (Int.castRingHom (ZMod p)) Q ^ p","state_after":"n : ℕ\nK : Type u_1\ninst✝² : CommRing K\nμ : K\nh : IsPrimitiveRoot μ n\ninst✝¹ : IsDomain K\ninst✝ : CharZero K\np : ℕ\nhprime : Fact (Nat.Prime p)\nhdiv : ¬p ∣ n\nQ : ℤ[X] := minpoly ℤ (μ ^ p)\nhfrob : map (Int.castRingHom (ZMod p)) Q ^ p = map (Int.castRingHom (ZMod p)) ((expand ℤ p) Q)\n⊢ map (Int.castRingHom (ZMod p)) (minpoly ℤ μ) ∣ map (Int.castRingHom (ZMod p)) Q ^ p","tactic":"have hfrob :\n    map (Int.castRingHom (ZMod p)) Q ^ p = map (Int.castRingHom (ZMod p)) (expand ℤ p Q) := by\n    rw [← ZMod.expand_card, map_expand]","premises":[{"full_name":"Int","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean","def_pos":[40,10],"def_end_pos":[40,13]},{"full_name":"Int.castRingHom","def_path":"Mathlib/Data/Int/Cast/Lemmas.lean","def_pos":[84,4],"def_end_pos":[84,15]},{"full_name":"Polynomial.expand","def_path":"Mathlib/Algebra/Polynomial/Expand.lean","def_pos":[34,18],"def_end_pos":[34,24]},{"full_name":"Polynomial.map","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[597,4],"def_end_pos":[597,7]},{"full_name":"Polynomial.map_expand","def_path":"Mathlib/Algebra/Polynomial/Expand.lean","def_pos":[158,8],"def_end_pos":[158,18]},{"full_name":"ZMod","def_path":"Mathlib/Data/ZMod/Defs.lean","def_pos":[89,4],"def_end_pos":[89,8]},{"full_name":"ZMod.expand_card","def_path":"Mathlib/FieldTheory/Finite/Basic.lean","def_pos":[512,8],"def_end_pos":[512,19]}]},{"state_before":"n : ℕ\nK : Type u_1\ninst✝² : CommRing K\nμ : K\nh : IsPrimitiveRoot μ n\ninst✝¹ : IsDomain K\ninst✝ : CharZero K\np : ℕ\nhprime : Fact (Nat.Prime p)\nhdiv : ¬p ∣ n\nQ : ℤ[X] := minpoly ℤ (μ ^ p)\nhfrob : map (Int.castRingHom (ZMod p)) Q ^ p = map (Int.castRingHom (ZMod p)) ((expand ℤ p) Q)\n⊢ map (Int.castRingHom (ZMod p)) (minpoly ℤ μ) ∣ map (Int.castRingHom (ZMod p)) Q ^ p","state_after":"n : ℕ\nK : Type u_1\ninst✝² : CommRing K\nμ : K\nh : IsPrimitiveRoot μ n\ninst✝¹ : IsDomain K\ninst✝ : CharZero K\np : ℕ\nhprime : Fact (Nat.Prime p)\nhdiv : ¬p ∣ n\nQ : ℤ[X] := minpoly ℤ (μ ^ p)\nhfrob : map (Int.castRingHom (ZMod p)) Q ^ p = map (Int.castRingHom (ZMod p)) ((expand ℤ p) Q)\n⊢ map (Int.castRingHom (ZMod p)) (minpoly ℤ μ) ∣ map (Int.castRingHom (ZMod p)) ((expand ℤ p) Q)","tactic":"rw [hfrob]","premises":[]},{"state_before":"n : ℕ\nK : Type u_1\ninst✝² : CommRing K\nμ : K\nh : IsPrimitiveRoot μ n\ninst✝¹ : IsDomain K\ninst✝ : CharZero K\np : ℕ\nhprime : Fact (Nat.Prime p)\nhdiv : ¬p ∣ n\nQ : ℤ[X] := minpoly ℤ (μ ^ p)\nhfrob : map (Int.castRingHom (ZMod p)) Q ^ p = map (Int.castRingHom (ZMod p)) ((expand ℤ p) Q)\n⊢ map (Int.castRingHom (ZMod p)) (minpoly ℤ μ) ∣ map (Int.castRingHom (ZMod p)) ((expand ℤ p) Q)","state_after":"n : ℕ\nK : Type u_1\ninst✝² : CommRing K\nμ : K\nh : IsPrimitiveRoot μ n\ninst✝¹ : IsDomain K\ninst✝ : CharZero K\np : ℕ\nhprime : Fact (Nat.Prime p)\nhdiv : ¬p ∣ n\nQ : ℤ[X] := minpoly ℤ (μ ^ p)\nhfrob : map (Int.castRingHom (ZMod p)) Q ^ p = map (Int.castRingHom (ZMod p)) ((expand ℤ p) Q)\n⊢ minpoly ℤ μ ∣ (expand ℤ p) Q","tactic":"apply RingHom.map_dvd (mapRingHom (Int.castRingHom (ZMod p)))","premises":[{"full_name":"Int.castRingHom","def_path":"Mathlib/Data/Int/Cast/Lemmas.lean","def_pos":[84,4],"def_end_pos":[84,15]},{"full_name":"Polynomial.mapRingHom","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[641,4],"def_end_pos":[641,14]},{"full_name":"RingHom.map_dvd","def_path":"Mathlib/Algebra/Ring/Hom/Basic.lean","def_pos":[48,18],"def_end_pos":[48,25]},{"full_name":"ZMod","def_path":"Mathlib/Data/ZMod/Defs.lean","def_pos":[89,4],"def_end_pos":[89,8]}]},{"state_before":"n : ℕ\nK : Type u_1\ninst✝² : CommRing K\nμ : K\nh : IsPrimitiveRoot μ n\ninst✝¹ : IsDomain K\ninst✝ : CharZero K\np : ℕ\nhprime : Fact (Nat.Prime p)\nhdiv : ¬p ∣ n\nQ : ℤ[X] := minpoly ℤ (μ ^ p)\nhfrob : map (Int.castRingHom (ZMod p)) Q ^ p = map (Int.castRingHom (ZMod p)) ((expand ℤ p) Q)\n⊢ minpoly ℤ μ ∣ (expand ℤ p) Q","state_after":"no goals","tactic":"exact minpoly_dvd_expand h hdiv","premises":[{"full_name":"IsPrimitiveRoot.minpoly_dvd_expand","def_path":"Mathlib/RingTheory/RootsOfUnity/Minpoly.lean","def_pos":[75,8],"def_end_pos":[75,26]}]}]}
+{"url":"Mathlib/Data/Fin/Basic.lean","commit":"","full_name":"Fin.succ_zero_eq_one'","start":[487,0],"end":[491,7],"file_path":"Mathlib/Data/Fin/Basic.lean","tactics":[{"state_before":"n m : ℕ\ninst✝ : NeZero n\n⊢ succ 0 = 1","state_after":"case zero\nm : ℕ\ninst✝ : NeZero 0\n⊢ succ 0 = 1\n\ncase succ\nm n✝ : ℕ\ninst✝ : NeZero (n✝ + 1)\n⊢ succ 0 = 1","tactic":"cases n","premises":[]}]}
+{"url":"Mathlib/MeasureTheory/Measure/Tilted.lean","commit":"","full_name":"MeasureTheory.toReal_rnDeriv_tilted_right","start":[321,0],"end":[328,30],"file_path":"Mathlib/MeasureTheory/Measure/Tilted.lean","tactics":[{"state_before":"α : Type u_1\nmα : MeasurableSpace α\nμ✝ : Measure α\nf : α → ℝ\nμ ν : Measure α\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite ν\nhf : Integrable (fun x => rexp (f x)) ν\n⊢ (fun x => (μ.rnDeriv (ν.tilted f) x).toReal) =ᶠ[ae ν] fun x =>\n    (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * (μ.rnDeriv ν x).toReal","state_after":"case h\nα : Type u_1\nmα : MeasurableSpace α\nμ✝ : Measure α\nf : α → ℝ\nμ ν : Measure α\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite ν\nhf : Integrable (fun x => rexp (f x)) ν\nx : α\nhx : μ.rnDeriv (ν.tilted f) x = ENNReal.ofReal (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * μ.rnDeriv ν x\n⊢ (μ.rnDeriv (ν.tilted f) x).toReal = (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * (μ.rnDeriv ν x).toReal","tactic":"filter_upwards [rnDeriv_tilted_right μ ν hf] with x hx","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Filter.mp_mem","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[159,8],"def_end_pos":[159,14]},{"full_name":"MeasureTheory.rnDeriv_tilted_right","def_path":"Mathlib/MeasureTheory/Measure/Tilted.lean","def_pos":[303,6],"def_end_pos":[303,26]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]}]},{"state_before":"case h\nα : Type u_1\nmα : MeasurableSpace α\nμ✝ : Measure α\nf : α → ℝ\nμ ν : Measure α\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite ν\nhf : Integrable (fun x => rexp (f x)) ν\nx : α\nhx : μ.rnDeriv (ν.tilted f) x = ENNReal.ofReal (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * μ.rnDeriv ν x\n⊢ (μ.rnDeriv (ν.tilted f) x).toReal = (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * (μ.rnDeriv ν x).toReal","state_after":"case h\nα : Type u_1\nmα : MeasurableSpace α\nμ✝ : Measure α\nf : α → ℝ\nμ ν : Measure α\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite ν\nhf : Integrable (fun x => rexp (f x)) ν\nx : α\nhx : μ.rnDeriv (ν.tilted f) x = ENNReal.ofReal (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * μ.rnDeriv ν x\n⊢ (ENNReal.ofReal (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * μ.rnDeriv ν x).toReal =\n    (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * (μ.rnDeriv ν x).toReal","tactic":"rw [hx]","premises":[]},{"state_before":"case h\nα : Type u_1\nmα : MeasurableSpace α\nμ✝ : Measure α\nf : α → ℝ\nμ ν : Measure α\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite ν\nhf : Integrable (fun x => rexp (f x)) ν\nx : α\nhx : μ.rnDeriv (ν.tilted f) x = ENNReal.ofReal (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * μ.rnDeriv ν x\n⊢ (ENNReal.ofReal (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * μ.rnDeriv ν x).toReal =\n    (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * (μ.rnDeriv ν x).toReal","state_after":"case h\nα : Type u_1\nmα : MeasurableSpace α\nμ✝ : Measure α\nf : α → ℝ\nμ ν : Measure α\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite ν\nhf : Integrable (fun x => rexp (f x)) ν\nx : α\nhx : μ.rnDeriv (ν.tilted f) x = ENNReal.ofReal (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * μ.rnDeriv ν x\n⊢ 0 ≤ rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν ∨ (μ.rnDeriv ν x).toReal = 0","tactic":"simp only [ENNReal.toReal_mul, gt_iff_lt, mul_eq_mul_right_iff, ENNReal.toReal_ofReal_eq_iff]","premises":[{"full_name":"ENNReal.toReal_mul","def_path":"Mathlib/Data/ENNReal/Real.lean","def_pos":[378,8],"def_end_pos":[378,18]},{"full_name":"ENNReal.toReal_ofReal_eq_iff","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[299,8],"def_end_pos":[299,28]},{"full_name":"gt_iff_lt","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1949,16],"def_end_pos":[1949,25]},{"full_name":"mul_eq_mul_right_iff","def_path":"Mathlib/Algebra/GroupWithZero/Basic.lean","def_pos":[193,8],"def_end_pos":[193,28]}]},{"state_before":"case h\nα : Type u_1\nmα : MeasurableSpace α\nμ✝ : Measure α\nf : α → ℝ\nμ ν : Measure α\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite ν\nhf : Integrable (fun x => rexp (f x)) ν\nx : α\nhx : μ.rnDeriv (ν.tilted f) x = ENNReal.ofReal (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * μ.rnDeriv ν x\n⊢ 0 ≤ rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν ∨ (μ.rnDeriv ν x).toReal = 0","state_after":"no goals","tactic":"exact Or.inl (by positivity)","premises":[{"full_name":"Or.inl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[534,4],"def_end_pos":[534,7]}]}]}
+{"url":"Mathlib/Algebra/Divisibility/Basic.lean","commit":"","full_name":"mul_dvd_mul","start":[186,0],"end":[187,54],"file_path":"Mathlib/Algebra/Divisibility/Basic.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : CommSemigroup α\na✝ b c✝ a c e f : α\n⊢ a * e * (c * f) = a * c * (e * f)","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/SetTheory/ZFC/Basic.lean","commit":"","full_name":"ZFSet.omega_succ","start":[782,0],"end":[789,14],"file_path":"Mathlib/SetTheory/ZFC/Basic.lean","tactics":[{"state_before":"n✝ : ZFSet\nx : PSet\nx✝ : ⟦x⟧ ∈ omega\nn : ℕ\nh : x.Equiv (PSet.omega.Func { down := n })\n⊢ insert (mk x) (mk x) = insert (mk (ofNat n)) (mk (ofNat n))","state_after":"n✝ : ZFSet\nx : PSet\nx✝ : ⟦x⟧ ∈ omega\nn : ℕ\nh : x.Equiv (PSet.omega.Func { down := n })\n⊢ insert (mk (PSet.omega.Func { down := n })) (mk (PSet.omega.Func { down := n })) =\n    insert (mk (ofNat n)) (mk (ofNat n))","tactic":"rw [ZFSet.sound h]","premises":[{"full_name":"ZFSet.sound","def_path":"Mathlib/SetTheory/ZFC/Basic.lean","def_pos":[568,8],"def_end_pos":[568,13]}]},{"state_before":"n✝ : ZFSet\nx : PSet\nx✝ : ⟦x⟧ ∈ omega\nn : ℕ\nh : x.Equiv (PSet.omega.Func { down := n })\n⊢ insert (mk (PSet.omega.Func { down := n })) (mk (PSet.omega.Func { down := n })) =\n    insert (mk (ofNat n)) (mk (ofNat n))","state_after":"no goals","tactic":"rfl","premises":[]}]}
+{"url":"Mathlib/Order/Filter/Ultrafilter.lean","commit":"","full_name":"Filter.NeBot.eq_pure_iff","start":[351,0],"end":[353,36],"file_path":"Mathlib/Order/Filter/Ultrafilter.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nγ : Type u_1\nf : Filter α\ns : Set α\na : α\nhf : f.NeBot\nx : α\n⊢ f = pure x ↔ {x} ∈ f","state_after":"no goals","tactic":"rw [← hf.le_pure_iff, le_pure_iff]","premises":[{"full_name":"Filter.NeBot.le_pure_iff","def_path":"Mathlib/Order/Filter/Ultrafilter.lean","def_pos":[348,18],"def_end_pos":[348,35]},{"full_name":"Filter.le_pure_iff","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2415,8],"def_end_pos":[2415,19]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]}]}]}
+{"url":"Mathlib/Data/Matrix/Basic.lean","commit":"","full_name":"Matrix.add_vecMul","start":[1562,0],"end":[1565,22],"file_path":"Mathlib/Data/Matrix/Basic.lean","tactics":[{"state_before":"l : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nm' : o → Type u_5\nn' : o → Type u_6\nR : Type u_7\nS : Type u_8\nα : Type v\nβ : Type w\nγ : Type u_9\ninst✝¹ : NonUnitalNonAssocSemiring α\ninst✝ : Fintype m\nA : Matrix m n α\nx y : m → α\n⊢ (x + y) ᵥ* A = x ᵥ* A + y ᵥ* A","state_after":"case h\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nm' : o → Type u_5\nn' : o → Type u_6\nR : Type u_7\nS : Type u_8\nα : Type v\nβ : Type w\nγ : Type u_9\ninst✝¹ : NonUnitalNonAssocSemiring α\ninst✝ : Fintype m\nA : Matrix m n α\nx y : m → α\nx✝ : n\n⊢ ((x + y) ᵥ* A) x✝ = (x ᵥ* A + y ᵥ* A) x✝","tactic":"ext","premises":[]},{"state_before":"case h\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nm' : o → Type u_5\nn' : o → Type u_6\nR : Type u_7\nS : Type u_8\nα : Type v\nβ : Type w\nγ : Type u_9\ninst✝¹ : NonUnitalNonAssocSemiring α\ninst✝ : Fintype m\nA : Matrix m n α\nx y : m → α\nx✝ : n\n⊢ ((x + y) ᵥ* A) x✝ = (x ᵥ* A + y ᵥ* A) x✝","state_after":"no goals","tactic":"apply add_dotProduct","premises":[{"full_name":"Matrix.add_dotProduct","def_path":"Mathlib/Data/Matrix/Basic.lean","def_pos":[710,8],"def_end_pos":[710,22]}]}]}
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Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\nι : fsu → ssu := fun x => ↑x\nT0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}\nT1 : ssu := insert AA T0\n⊢ ↑F ≤ 𝓝 x → X.str F = x","tactic":"let T1 := insert AA T0","premises":[{"full_name":"Insert.insert","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[458,2],"def_end_pos":[458,8]}]},{"state_before":"X : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\nι : fsu → ssu := fun x => ↑x\nT0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}\nT1 : ssu := insert AA T0\n⊢ ↑F ≤ 𝓝 x → X.str F = x","state_after":"X : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\nι : fsu → ssu := fun x => ↑x\nT0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\n⊢ ↑F ≤ 𝓝 x → X.str F = x","tactic":"let T2 := finiteInterClosure T1","premises":[{"full_name":"FiniteInter.finiteInterClosure","def_path":"Mathlib/Data/Set/Constructions.lean","def_pos":[38,10],"def_end_pos":[38,28]}]},{"state_before":"X : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\nι : fsu → ssu := fun x => ↑x\nT0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\n⊢ ↑F ≤ 𝓝 x → X.str F = x","state_after":"X : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\nι : fsu → ssu := fun x => ↑x\nT0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : ↑F ≤ 𝓝 x\n⊢ X.str F = x","tactic":"intro cond","premises":[]},{"state_before":"X : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\nι : fsu → ssu := fun x => ↑x\nT0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : ↑F ≤ 𝓝 x\n⊢ X.str F = x","state_after":"X : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\nι : fsu → ssu := fun x => ↑x\nT0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : ↑F ≤ 𝓝 x\nclaim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A\n⊢ X.str F = x","tactic":"have claim1 : ∀ A : Set X, IsClosed A → A ∈ F → x ∈ A := by\n    intro A hA h\n    by_contra H\n    rw [le_nhds_iff] at cond\n    specialize cond Aᶜ H hA.isOpen_compl\n    rw [Ultrafilter.mem_coe, Ultrafilter.compl_mem_iff_not_mem] at cond\n    contradiction","premises":[{"full_name":"Decidable.byContradiction","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[880,8],"def_end_pos":[880,23]},{"full_name":"HasCompl.compl","def_path":"Mathlib/Order/Notation.lean","def_pos":[34,2],"def_end_pos":[34,7]},{"full_name":"IsClosed","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[93,6],"def_end_pos":[93,14]},{"full_name":"IsClosed.isOpen_compl","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[95,2],"def_end_pos":[95,14]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Not","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[224,4],"def_end_pos":[224,7]},{"full_name":"Set","def_path":"Mathlib/Init/Set.lean","def_pos":[53,4],"def_end_pos":[53,7]},{"full_name":"Ultrafilter.compl_mem_iff_not_mem","def_path":"Mathlib/Order/Filter/Ultrafilter.lean","def_pos":[116,8],"def_end_pos":[116,29]},{"full_name":"Ultrafilter.mem_coe","def_path":"Mathlib/Order/Filter/Ultrafilter.lean","def_pos":[70,8],"def_end_pos":[70,15]},{"full_name":"le_nhds_iff","def_path":"Mathlib/Topology/Basic.lean","def_pos":[709,8],"def_end_pos":[709,19]}]},{"state_before":"X : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\nι : fsu → ssu := fun x => ↑x\nT0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : ↑F ≤ 𝓝 x\nclaim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A\n⊢ X.str F = x","state_after":"X : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\nι : fsu → ssu := fun x => ↑x\nT0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : ↑F ≤ 𝓝 x\nclaim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A\nclaim2 : ∀ A ∈ F, x ∈ Compactum.cl A\n⊢ X.str F = x","tactic":"have claim2 : ∀ A : Set X, A ∈ F → x ∈ cl A := by\n    intro A hA\n    exact claim1 (cl A) (isClosed_cl A) (mem_of_superset hA (subset_cl A))","premises":[{"full_name":"Compactum.isClosed_cl","def_path":"Mathlib/Topology/Category/Compactum.lean","def_pos":[261,8],"def_end_pos":[261,19]},{"full_name":"Filter.mem_of_superset","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[139,8],"def_end_pos":[139,23]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Set","def_path":"Mathlib/Init/Set.lean","def_pos":[53,4],"def_end_pos":[53,7]},{"full_name":"_private.Mathlib.Topology.Category.Compactum.0.Compactum.cl","def_path":"Mathlib/Topology/Category/Compactum.lean","def_pos":[187,12],"def_end_pos":[187,14]},{"full_name":"_private.Mathlib.Topology.Category.Compactum.0.Compactum.subset_cl","def_path":"Mathlib/Topology/Category/Compactum.lean","def_pos":[199,16],"def_end_pos":[199,25]}]},{"state_before":"X : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\nι : fsu → ssu := fun x => ↑x\nT0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : ↑F ≤ 𝓝 x\nclaim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A\nclaim2 : ∀ A ∈ F, x ∈ Compactum.cl A\n⊢ X.str F = x","state_after":"X : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\nι : fsu → ssu := fun x => ↑x\nT0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : ↑F ≤ 𝓝 x\nclaim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A\nclaim2 : ∀ A ∈ F, x ∈ Compactum.cl A\nclaim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0\n⊢ X.str F = x","tactic":"have claim3 : ∀ (S1) (_ : S1 ∈ T0) (S2) (_ : S2 ∈ T0), S1 ∩ S2 ∈ T0 := by\n    rintro S1 ⟨S1, hS1, rfl⟩ S2 ⟨S2, hS2, rfl⟩\n    exact ⟨S1 ∩ S2, inter_mem hS1 hS2, by simp [basic_inter]⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Filter.inter_mem","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[145,8],"def_end_pos":[145,17]},{"full_name":"Inter.inter","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[407,2],"def_end_pos":[407,7]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"_private.Mathlib.Topology.Category.Compactum.0.Compactum.basic_inter","def_path":"Mathlib/Topology/Category/Compactum.lean","def_pos":[190,16],"def_end_pos":[190,27]}]},{"state_before":"X : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\nι : fsu → ssu := fun x => ↑x\nT0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : ↑F ≤ 𝓝 x\nclaim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A\nclaim2 : ∀ A ∈ F, x ∈ Compactum.cl A\nclaim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0\n⊢ X.str F = x","state_after":"X : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\nι : fsu → ssu := fun x => ↑x\nT0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : ↑F ≤ 𝓝 x\nclaim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A\nclaim2 : ∀ A ∈ F, x ∈ Compactum.cl A\nclaim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0\nclaim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty\n⊢ X.str F = x","tactic":"have claim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty := by\n    rintro S ⟨S, hS, rfl⟩\n    rcases claim2 _ hS with ⟨G, hG, hG2⟩\n    exact ⟨G, hG2, hG⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Inter.inter","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[407,2],"def_end_pos":[407,7]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Set.Nonempty","def_path":"Mathlib/Init/Set.lean","def_pos":[222,14],"def_end_pos":[222,22]}]},{"state_before":"X : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\nι : fsu → ssu := fun x => ↑x\nT0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : ↑F ≤ 𝓝 x\nclaim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A\nclaim2 : ∀ A ∈ F, x ∈ Compactum.cl A\nclaim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0\nclaim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty\n⊢ X.str F = x","state_after":"X : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\nι : fsu → ssu := fun x => ↑x\nT0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : ↑F ≤ 𝓝 x\nclaim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A\nclaim2 : ∀ A ∈ F, x ∈ Compactum.cl A\nclaim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0\nclaim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty\nclaim5 : ∀ S ∈ T0, S.Nonempty\n⊢ X.str F = x","tactic":"have claim5 : ∀ S ∈ T0, Set.Nonempty S := by\n    rintro S ⟨S, hS, rfl⟩\n    exact ⟨F, hS⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Set.Nonempty","def_path":"Mathlib/Init/Set.lean","def_pos":[222,14],"def_end_pos":[222,22]}]},{"state_before":"X : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\nι : fsu → ssu := fun x => ↑x\nT0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : ↑F ≤ 𝓝 x\nclaim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A\nclaim2 : ∀ A ∈ F, x ∈ Compactum.cl A\nclaim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0\nclaim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty\nclaim5 : ∀ S ∈ T0, S.Nonempty\nclaim6 : ∀ S ∈ T2, S.Nonempty\n⊢ X.str F = x","state_after":"X : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\nι : fsu → ssu := fun x => ↑x\nT0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : ↑F ≤ 𝓝 x\nclaim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A\nclaim2 : ∀ A ∈ F, x ∈ Compactum.cl A\nclaim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0\nclaim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty\nclaim5 : ∀ S ∈ T0, S.Nonempty\nclaim6 : ∀ S ∈ T2, S.Nonempty\n⊢ ∀ (F : fsu), ↑F ⊆ T1 → (⋂₀ ι F).Nonempty","tactic":"suffices ∀ F : fsu, ↑F ⊆ T1 → (⋂₀ ι F).Nonempty by\n    obtain ⟨G, h1⟩ := Ultrafilter.exists_ultrafilter_of_finite_inter_nonempty _ this\n    have c1 : X.join G = F := Ultrafilter.coe_le_coe.1 fun P hP => h1 (Or.inr ⟨P, hP, rfl⟩)\n    have c2 : G.map X.str = X.incl x := by\n      refine Ultrafilter.coe_le_coe.1 fun P hP => ?_\n      apply mem_of_superset (h1 (Or.inl rfl))\n      rintro x ⟨rfl⟩\n      exact hP\n    simp [← c1, c2]","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Compactum.incl","def_path":"Mathlib/Topology/Category/Compactum.lean","def_pos":[128,4],"def_end_pos":[128,8]},{"full_name":"Compactum.join","def_path":"Mathlib/Topology/Category/Compactum.lean","def_pos":[124,4],"def_end_pos":[124,8]},{"full_name":"Compactum.str","def_path":"Mathlib/Topology/Category/Compactum.lean","def_pos":[120,4],"def_end_pos":[120,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Filter.mem_of_superset","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[139,8],"def_end_pos":[139,23]},{"full_name":"HasSubset.Subset","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[384,2],"def_end_pos":[384,8]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Or.inl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[534,4],"def_end_pos":[534,7]},{"full_name":"Or.inr","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[536,4],"def_end_pos":[536,7]},{"full_name":"Set.Nonempty","def_path":"Mathlib/Init/Set.lean","def_pos":[222,14],"def_end_pos":[222,22]},{"full_name":"Set.sInter","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[156,4],"def_end_pos":[156,10]},{"full_name":"Ultrafilter.coe_le_coe","def_path":"Mathlib/Order/Filter/Ultrafilter.lean","def_pos":[80,8],"def_end_pos":[80,18]},{"full_name":"Ultrafilter.exists_ultrafilter_of_finite_inter_nonempty","def_path":"Mathlib/Order/Filter/Ultrafilter.lean","def_pos":[329,8],"def_end_pos":[329,51]},{"full_name":"Ultrafilter.map","def_path":"Mathlib/Order/Filter/Ultrafilter.lean","def_pos":[176,11],"def_end_pos":[176,14]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"X : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\nι : fsu → ssu := fun x => ↑x\nT0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : ↑F ≤ 𝓝 x\nclaim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A\nclaim2 : ∀ A ∈ F, x ∈ Compactum.cl A\nclaim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0\nclaim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty\nclaim5 : ∀ S ∈ T0, S.Nonempty\nclaim6 : ∀ S ∈ T2, S.Nonempty\n⊢ ∀ (F : fsu), ↑F ⊆ T1 → (⋂₀ ι F).Nonempty","state_after":"X : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\nι : fsu → ssu := fun x => ↑x\nT0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : ↑F ≤ 𝓝 x\nclaim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A\nclaim2 : ∀ A ∈ F, x ∈ Compactum.cl A\nclaim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0\nclaim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty\nclaim5 : ∀ S ∈ T0, S.Nonempty\nclaim6 : ∀ S ∈ T2, S.Nonempty\nT : fsu\nhT : ↑T ⊆ T1\n⊢ (⋂₀ ι T).Nonempty","tactic":"intro T hT","premises":[]},{"state_before":"X : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\nι : fsu → ssu := fun x => ↑x\nT0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : ↑F ≤ 𝓝 x\nclaim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A\nclaim2 : ∀ A ∈ F, x ∈ Compactum.cl A\nclaim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0\nclaim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty\nclaim5 : ∀ S ∈ T0, S.Nonempty\nclaim6 : ∀ S ∈ T2, S.Nonempty\nT : fsu\nhT : ↑T ⊆ T1\n⊢ (⋂₀ ι T).Nonempty","state_after":"X : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\nι : fsu → ssu := fun x => ↑x\nT0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : ↑F ≤ 𝓝 x\nclaim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A\nclaim2 : ∀ A ∈ F, x ∈ Compactum.cl A\nclaim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0\nclaim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty\nclaim5 : ∀ S ∈ T0, S.Nonempty\nclaim6 : ∀ S ∈ T2, S.Nonempty\nT : fsu\nhT : ↑T ⊆ T1\n⊢ ↑T ⊆ finiteInterClosure T1","tactic":"refine claim6 _ (finiteInter_mem (.finiteInterClosure_finiteInter _) _ ?_)","premises":[{"full_name":"FiniteInter.finiteInterClosure_finiteInter","def_path":"Mathlib/Data/Set/Constructions.lean","def_pos":[43,8],"def_end_pos":[43,38]},{"full_name":"FiniteInter.finiteInter_mem","def_path":"Mathlib/Data/Set/Constructions.lean","def_pos":[49,8],"def_end_pos":[49,23]}]},{"state_before":"X : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\nι : fsu → ssu := fun x => ↑x\nT0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : ↑F ≤ 𝓝 x\nclaim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A\nclaim2 : ∀ A ∈ F, x ∈ Compactum.cl A\nclaim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0\nclaim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty\nclaim5 : ∀ S ∈ T0, S.Nonempty\nclaim6 : ∀ S ∈ T2, S.Nonempty\nT : fsu\nhT : ↑T ⊆ T1\n⊢ ↑T ⊆ finiteInterClosure T1","state_after":"X : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\nι : fsu → ssu := fun x => ↑x\nT0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : ↑F ≤ 𝓝 x\nclaim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A\nclaim2 : ∀ A ∈ F, x ∈ Compactum.cl A\nclaim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0\nclaim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty\nclaim5 : ∀ S ∈ T0, S.Nonempty\nclaim6 : ∀ S ∈ T2, S.Nonempty\nT : fsu\nhT : ↑T ⊆ T1\nt : Set (Ultrafilter X.A)\nht : t ∈ ↑T\n⊢ t ∈ finiteInterClosure T1","tactic":"intro t ht","premises":[]},{"state_before":"X : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\nι : fsu → ssu := fun x => ↑x\nT0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : ↑F ≤ 𝓝 x\nclaim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A\nclaim2 : ∀ A ∈ F, x ∈ Compactum.cl A\nclaim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0\nclaim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty\nclaim5 : ∀ S ∈ T0, S.Nonempty\nclaim6 : ∀ S ∈ T2, S.Nonempty\nT : fsu\nhT : ↑T ⊆ T1\nt : Set (Ultrafilter X.A)\nht : t ∈ ↑T\n⊢ t ∈ finiteInterClosure T1","state_after":"no goals","tactic":"exact finiteInterClosure.basic (@hT t ht)","premises":[{"full_name":"FiniteInter.finiteInterClosure.basic","def_path":"Mathlib/Data/Set/Constructions.lean","def_pos":[39,4],"def_end_pos":[39,9]}]}]}
+{"url":"Mathlib/AlgebraicTopology/FundamentalGroupoid/SimplyConnected.lean","commit":"","full_name":"simply_connected_iff_unique_homotopic","start":[38,0],"end":[44,48],"file_path":"Mathlib/AlgebraicTopology/FundamentalGroupoid/SimplyConnected.lean","tactics":[{"state_before":"X : Type u_1\ninst✝ : TopologicalSpace X\n⊢ SimplyConnectedSpace X ↔ Nonempty X ∧ ∀ (x y : X), Nonempty (Unique (Path.Homotopic.Quotient x y))","state_after":"X : Type u_1\ninst✝ : TopologicalSpace X\n⊢ X →\n    ((∀ (x y : FundamentalGroupoid X), Nonempty (Unique (x ⟶ y))) ↔\n      ∀ (x y : X), Nonempty (Unique (Path.Homotopic.Quotient x y)))","tactic":"simp only [simply_connected_def, equiv_punit_iff_unique,\n    FundamentalGroupoid.nonempty_iff X, and_congr_right_iff, Nonempty.forall]","premises":[{"full_name":"CategoryTheory.equiv_punit_iff_unique","def_path":"Mathlib/CategoryTheory/PUnit.lean","def_pos":[66,8],"def_end_pos":[66,30]},{"full_name":"FundamentalGroupoid.nonempty_iff","def_path":"Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean","def_pos":[274,6],"def_end_pos":[274,18]},{"full_name":"Nonempty.forall","def_path":"Mathlib/Logic/Nonempty.lean","def_pos":[25,8],"def_end_pos":[25,23]},{"full_name":"and_congr_right_iff","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[156,16],"def_end_pos":[156,35]},{"full_name":"simply_connected_def","def_path":"Mathlib/AlgebraicTopology/FundamentalGroupoid/SimplyConnected.lean","def_pos":[34,9],"def_end_pos":[34,29]}]},{"state_before":"X : Type u_1\ninst✝ : TopologicalSpace X\n⊢ X →\n    ((∀ (x y : FundamentalGroupoid X), Nonempty (Unique (x ⟶ y))) ↔\n      ∀ (x y : X), Nonempty (Unique (Path.Homotopic.Quotient x y)))","state_after":"X : Type u_1\ninst✝ : TopologicalSpace X\na✝ : X\n⊢ (∀ (x y : FundamentalGroupoid X), Nonempty (Unique (x ⟶ y))) ↔\n    ∀ (x y : X), Nonempty (Unique (Path.Homotopic.Quotient x y))","tactic":"intros","premises":[]},{"state_before":"X : Type u_1\ninst✝ : TopologicalSpace X\na✝ : X\n⊢ (∀ (x y : FundamentalGroupoid X), Nonempty (Unique (x ⟶ y))) ↔\n    ∀ (x y : X), Nonempty (Unique (Path.Homotopic.Quotient x y))","state_after":"no goals","tactic":"exact ⟨fun h _ _ => h _ _, fun h _ _ => h _ _⟩","premises":[{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]}]}]}
+{"url":"Mathlib/Geometry/RingedSpace/SheafedSpace.lean","commit":"","full_name":"AlgebraicGeometry.SheafedSpace.mono_of_base_injective_of_stalk_epi","start":[233,0],"end":[242,9],"file_path":"Mathlib/Geometry/RingedSpace/SheafedSpace.lean","tactics":[{"state_before":"C : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : ConcreteCategory C\ninst✝⁴ : HasColimits C\ninst✝³ : HasLimits C\ninst✝² : PreservesLimits (CategoryTheory.forget C)\ninst✝¹ : PreservesFilteredColimits (CategoryTheory.forget C)\ninst✝ : (CategoryTheory.forget C).ReflectsIsomorphisms\nX Y : SheafedSpace C\nf : X ⟶ Y\nh₁ : Function.Injective ⇑f.base\nh₂ : ∀ (x : ↑↑X.toPresheafedSpace), Epi (PresheafedSpace.Hom.stalkMap f x)\n⊢ Mono f","state_after":"case right_cancellation\nC : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : ConcreteCategory C\ninst✝⁴ : HasColimits C\ninst✝³ : HasLimits C\ninst✝² : PreservesLimits (CategoryTheory.forget C)\ninst✝¹ : PreservesFilteredColimits (CategoryTheory.forget C)\ninst✝ : (CategoryTheory.forget C).ReflectsIsomorphisms\nX Y : SheafedSpace C\nf : X ⟶ Y\nh₁ : Function.Injective ⇑f.base\nh₂ : ∀ (x : ↑↑X.toPresheafedSpace), Epi (PresheafedSpace.Hom.stalkMap f x)\n⊢ ∀ {Z : SheafedSpace C} (g h : Z ⟶ X), g ≫ f = h ≫ f → g = h","tactic":"constructor","premises":[]},{"state_before":"case right_cancellation\nC : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : ConcreteCategory C\ninst✝⁴ : HasColimits C\ninst✝³ : HasLimits C\ninst✝² : PreservesLimits (CategoryTheory.forget C)\ninst✝¹ : PreservesFilteredColimits (CategoryTheory.forget C)\ninst✝ : (CategoryTheory.forget C).ReflectsIsomorphisms\nX Y : SheafedSpace C\nf : X ⟶ Y\nh₁ : Function.Injective ⇑f.base\nh₂ : ∀ (x : ↑↑X.toPresheafedSpace), Epi (PresheafedSpace.Hom.stalkMap f x)\n⊢ ∀ {Z : SheafedSpace C} (g h : Z ⟶ X), g ≫ f = h ≫ f → g = h","state_after":"case right_cancellation\nC : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : ConcreteCategory C\ninst✝⁴ : HasColimits C\ninst✝³ : HasLimits C\ninst✝² : PreservesLimits (CategoryTheory.forget C)\ninst✝¹ : PreservesFilteredColimits (CategoryTheory.forget C)\ninst✝ : (CategoryTheory.forget C).ReflectsIsomorphisms\nX Y : SheafedSpace C\nf : X ⟶ Y\nh₁ : Function.Injective ⇑f.base\nh₂ : ∀ (x : ↑↑X.toPresheafedSpace), Epi (PresheafedSpace.Hom.stalkMap f x)\nZ : SheafedSpace C\ng : ↑Z.toPresheafedSpace ⟶ ↑X.toPresheafedSpace\ngc : X.presheaf ⟶ (pushforward C g).obj Z.presheaf\nh : ↑Z.toPresheafedSpace ⟶ ↑X.toPresheafedSpace\nhc : X.presheaf ⟶ (pushforward C h).obj Z.presheaf\ne : { base := g, c := gc } ≫ f = { base := h, c := hc } ≫ f\n⊢ { base := g, c := gc } = { base := h, c := hc }","tactic":"intro Z ⟨g, gc⟩ ⟨h, hc⟩ e","premises":[]},{"state_before":"case right_cancellation\nC : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : ConcreteCategory C\ninst✝⁴ : HasColimits C\ninst✝³ : HasLimits C\ninst✝² : PreservesLimits (CategoryTheory.forget C)\ninst✝¹ : PreservesFilteredColimits (CategoryTheory.forget C)\ninst✝ : (CategoryTheory.forget C).ReflectsIsomorphisms\nX Y : SheafedSpace C\nf : X ⟶ Y\nh₁ : Function.Injective ⇑f.base\nh₂ : ∀ (x : ↑↑X.toPresheafedSpace), Epi (PresheafedSpace.Hom.stalkMap f x)\nZ : SheafedSpace C\ng : ↑Z.toPresheafedSpace ⟶ ↑X.toPresheafedSpace\ngc : X.presheaf ⟶ (pushforward C g).obj Z.presheaf\nh : ↑Z.toPresheafedSpace ⟶ ↑X.toPresheafedSpace\nhc : X.presheaf ⟶ (pushforward C h).obj Z.presheaf\ne : { base := g, c := gc } ≫ f = { base := h, c := hc } ≫ f\n⊢ { base := g, c := gc } = { base := h, c := hc }","state_after":"case right_cancellation\nC : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : ConcreteCategory C\ninst✝⁴ : HasColimits C\ninst✝³ : HasLimits C\ninst✝² : PreservesLimits (CategoryTheory.forget C)\ninst✝¹ : PreservesFilteredColimits (CategoryTheory.forget C)\ninst✝ : (CategoryTheory.forget C).ReflectsIsomorphisms\nX Y : SheafedSpace C\nf : X ⟶ Y\nh₁ : Function.Injective ⇑f.base\nh₂ : ∀ (x : ↑↑X.toPresheafedSpace), Epi (PresheafedSpace.Hom.stalkMap f x)\nZ : SheafedSpace C\ng : ↑Z.toPresheafedSpace ⟶ ↑X.toPresheafedSpace\ngc hc : X.presheaf ⟶ (pushforward C g).obj Z.presheaf\ne : { base := g, c := gc } ≫ f = { base := g, c := hc } ≫ f\n⊢ { base := g, c := gc } = { base := g, c := hc }","tactic":"obtain rfl : g = h := ConcreteCategory.hom_ext _ _ fun x ↦ h₁ congr(($e).base x)","premises":[{"full_name":"AlgebraicGeometry.PresheafedSpace.Hom.base","def_path":"Mathlib/Geometry/RingedSpace/PresheafedSpace.lean","def_pos":[90,2],"def_end_pos":[90,6]},{"full_name":"CategoryTheory.ConcreteCategory.hom_ext","def_path":"Mathlib/CategoryTheory/ConcreteCategory/Basic.lean","def_pos":[97,8],"def_end_pos":[97,32]}]},{"state_before":"case right_cancellation\nC : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : ConcreteCategory C\ninst✝⁴ : HasColimits C\ninst✝³ : HasLimits C\ninst✝² : PreservesLimits (CategoryTheory.forget C)\ninst✝¹ : PreservesFilteredColimits (CategoryTheory.forget C)\ninst✝ : (CategoryTheory.forget C).ReflectsIsomorphisms\nX Y : SheafedSpace C\nf : X ⟶ Y\nh₁ : Function.Injective ⇑f.base\nh₂ : ∀ (x : ↑↑X.toPresheafedSpace), Epi (PresheafedSpace.Hom.stalkMap f x)\nZ : SheafedSpace C\ng : ↑Z.toPresheafedSpace ⟶ ↑X.toPresheafedSpace\ngc hc : X.presheaf ⟶ (pushforward C g).obj Z.presheaf\ne : { base := g, c := gc } ≫ f = { base := g, c := hc } ≫ f\n⊢ { base := g, c := gc } = { base := g, c := hc }","state_after":"case right_cancellation\nC : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : ConcreteCategory C\ninst✝⁴ : HasColimits C\ninst✝³ : HasLimits C\ninst✝² : PreservesLimits (CategoryTheory.forget C)\ninst✝¹ : PreservesFilteredColimits (CategoryTheory.forget C)\ninst✝ : (CategoryTheory.forget C).ReflectsIsomorphisms\nX Y : SheafedSpace C\nf : X ⟶ Y\nh₁ : Function.Injective ⇑f.base\nh₂ : ∀ (x : ↑↑X.toPresheafedSpace), Epi (PresheafedSpace.Hom.stalkMap f x)\nZ : SheafedSpace C\ng : ↑Z.toPresheafedSpace ⟶ ↑X.toPresheafedSpace\ngc hc : X.presheaf ⟶ (pushforward C g).obj Z.presheaf\ne : { base := g, c := gc } ≫ f = { base := g, c := hc } ≫ f\nx : ↑↑Z.toPresheafedSpace\n⊢ { base := g, c := gc }.stalkMap x = (X.presheaf.stalkCongr ⋯).hom ≫ { base := g, c := hc }.stalkMap x","tactic":"refine SheafedSpace.hom_stalk_ext ⟨g, gc⟩ ⟨g, hc⟩ rfl fun x ↦ ?_","premises":[{"full_name":"AlgebraicGeometry.SheafedSpace.hom_stalk_ext","def_path":"Mathlib/Geometry/RingedSpace/SheafedSpace.lean","def_pos":[221,6],"def_end_pos":[221,19]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"case right_cancellation\nC : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : ConcreteCategory C\ninst✝⁴ : HasColimits C\ninst✝³ : HasLimits C\ninst✝² : PreservesLimits (CategoryTheory.forget C)\ninst✝¹ : PreservesFilteredColimits (CategoryTheory.forget C)\ninst✝ : (CategoryTheory.forget C).ReflectsIsomorphisms\nX Y : SheafedSpace C\nf : X ⟶ Y\nh₁ : Function.Injective ⇑f.base\nh₂ : ∀ (x : ↑↑X.toPresheafedSpace), Epi (PresheafedSpace.Hom.stalkMap f x)\nZ : SheafedSpace C\ng : ↑Z.toPresheafedSpace ⟶ ↑X.toPresheafedSpace\ngc hc : X.presheaf ⟶ (pushforward C g).obj Z.presheaf\ne : { base := g, c := gc } ≫ f = { base := g, c := hc } ≫ f\nx : ↑↑Z.toPresheafedSpace\n⊢ { base := g, c := gc }.stalkMap x = (X.presheaf.stalkCongr ⋯).hom ≫ { base := g, c := hc }.stalkMap x","state_after":"case right_cancellation\nC : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : ConcreteCategory C\ninst✝⁴ : HasColimits C\ninst✝³ : HasLimits C\ninst✝² : PreservesLimits (CategoryTheory.forget C)\ninst✝¹ : PreservesFilteredColimits (CategoryTheory.forget C)\ninst✝ : (CategoryTheory.forget C).ReflectsIsomorphisms\nX Y : SheafedSpace C\nf : X ⟶ Y\nh₁ : Function.Injective ⇑f.base\nh₂ : ∀ (x : ↑↑X.toPresheafedSpace), Epi (PresheafedSpace.Hom.stalkMap f x)\nZ : SheafedSpace C\ng : ↑Z.toPresheafedSpace ⟶ ↑X.toPresheafedSpace\ngc hc : X.presheaf ⟶ (pushforward C g).obj Z.presheaf\ne : { base := g, c := gc } ≫ f = { base := g, c := hc } ≫ f\nx : ↑↑Z.toPresheafedSpace\n⊢ 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↑(({ obj := fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGrp,\n              map := fun {X Y} f =>\n                { app := fun Y_1 => { toFun := fun g => f.unop ≫ g, map_zero' := ⋯, map_add' := ⋯ },\n                  naturality := ⋯ } }.obj\n          x✝²).obj\n      x✝¹)\n⊢ (({ obj := fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGrp,\n                map := fun {X Y} f =>\n                  { app := fun Y_1 => { toFun := fun g => f.unop ≫ g, map_zero' := ⋯, map_add' := ⋯ },\n                    naturality := ⋯ } }.map\n            (𝟙 x✝²)).app\n        x✝¹)\n      x✝ =\n    ((𝟙\n            ({ obj := fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGrp,\n                  map := fun {X Y} f =>\n                    { app := fun Y_1 => { toFun := fun g => f.unop ≫ g, map_zero' := ⋯, map_add' := ⋯ },\n                      naturality := ⋯ } }.obj\n              x✝²)).app\n        x✝¹)\n      x✝","tactic":"ext","premises":[]},{"state_before":"case w.h.w\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nx✝² : Cᵒᵖ\nx✝¹ : C\nx✝ :\n  ↑(({ obj := fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGrp,\n              map := fun {X Y} f =>\n                { app := fun Y_1 => { toFun := fun g => f.unop ≫ g, map_zero' := ⋯, map_add' := ⋯ },\n                  naturality := ⋯ } }.obj\n          x✝²).obj\n      x✝¹)\n⊢ (({ obj := fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGrp,\n                map := fun {X Y} f =>\n                  { app := fun Y_1 => { toFun := fun g => f.unop ≫ g, map_zero' := ⋯, map_add' := ⋯ },\n                    naturality := ⋯ } }.map\n            (𝟙 x✝²)).app\n        x✝¹)\n      x✝ =\n    ((𝟙\n            ({ obj := fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGrp,\n                  map := fun {X Y} f =>\n                    { app := fun Y_1 => { toFun := fun g => f.unop ≫ g, map_zero' := ⋯, map_add' := ⋯ },\n                      naturality := ⋯ } }.obj\n              x✝²)).app\n        x✝¹)\n      x✝","state_after":"case w.h.w\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nx✝² : Cᵒᵖ\nx✝¹ : C\nx✝ :\n  ↑(({ obj := fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGrp,\n              map := fun {X Y} f =>\n                { app := fun Y_1 => { toFun := fun g => f.unop ≫ g, map_zero' := ⋯, map_add' := ⋯ },\n                  naturality := ⋯ } }.obj\n          x✝²).obj\n      x✝¹)\n⊢ 𝟙 (unop x✝²) ≫ x✝ = x✝","tactic":"dsimp","premises":[]},{"state_before":"case w.h.w\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nx✝² : Cᵒᵖ\nx✝¹ : C\nx✝ :\n  ↑(({ obj := fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGrp,\n              map := fun {X Y} f =>\n                { app := fun Y_1 => { toFun := fun g => f.unop ≫ g, map_zero' := ⋯, map_add' := ⋯ },\n                  naturality := ⋯ } }.obj\n          x✝²).obj\n      x✝¹)\n⊢ 𝟙 (unop x✝²) ≫ x✝ = x✝","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nX✝ Y✝ Z✝ : Cᵒᵖ\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\n⊢ { obj := fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGrp,\n          map := fun {X Y} f =>\n            { app := fun Y_1 => { toFun := fun g => f.unop ≫ g, map_zero' := ⋯, map_add' := ⋯ }, naturality := ⋯ } }.map\n      (f ≫ g) =\n    { obj := fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGrp,\n            map := fun {X Y} f =>\n              { app := fun Y_1 => { toFun := fun g => f.unop ≫ g, map_zero' := ⋯, map_add' := ⋯ },\n                naturality := ⋯ } }.map\n        f ≫\n      { obj := fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGrp,\n            map := fun {X Y} f =>\n              { app := fun Y_1 => { toFun := fun g => f.unop ≫ g, map_zero' := ⋯, map_add' := ⋯ },\n                naturality := ⋯ } }.map\n        g","state_after":"case w.h.w\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nX✝ Y✝ Z✝ : Cᵒᵖ\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\nx✝¹ : C\nx✝ :\n  ↑(({ obj := fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGrp,\n              map := fun {X Y} f =>\n                { app := fun Y_1 => { toFun := fun g => f.unop ≫ g, map_zero' := ⋯, map_add' := ⋯ },\n                  naturality := ⋯ } }.obj\n          X✝).obj\n      x✝¹)\n⊢ (({ obj := fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGrp,\n                map := fun {X Y} f =>\n                  { app := fun Y_1 => { toFun := fun g => f.unop ≫ g, map_zero' := ⋯, map_add' := ⋯ },\n                    naturality := ⋯ } }.map\n            (f ≫ g)).app\n        x✝¹)\n      x✝ =\n    (({ obj := fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGrp,\n                  map := fun {X Y} f =>\n                    { app := fun Y_1 => { toFun := fun g => f.unop ≫ g, map_zero' := ⋯, map_add' := ⋯ },\n                      naturality := ⋯ } }.map\n              f ≫\n            { obj := fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGrp,\n                  map := fun {X Y} f =>\n                    { app := fun Y_1 => { toFun := fun g => f.unop ≫ g, map_zero' := ⋯, map_add' := ⋯ },\n                      naturality := ⋯ } }.map\n              g).app\n        x✝¹)\n      x✝","tactic":"ext","premises":[]},{"state_before":"case w.h.w\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nX✝ Y✝ Z✝ : Cᵒᵖ\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\nx✝¹ : C\nx✝ :\n  ↑(({ obj := fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGrp,\n              map := fun {X Y} f =>\n                { app := fun Y_1 => { toFun := fun g => f.unop ≫ g, map_zero' := ⋯, map_add' := ⋯ },\n                  naturality := ⋯ } }.obj\n          X✝).obj\n      x✝¹)\n⊢ (({ obj := fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGrp,\n                map := fun {X Y} f =>\n                  { app := fun Y_1 => { toFun := fun g => f.unop ≫ g, map_zero' := ⋯, map_add' := ⋯ },\n                    naturality := ⋯ } }.map\n            (f ≫ g)).app\n        x✝¹)\n      x✝ =\n    (({ obj := fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGrp,\n                  map := fun {X Y} f =>\n                    { app := fun Y_1 => { toFun := fun g => f.unop ≫ g, map_zero' := ⋯, map_add' := ⋯ },\n                      naturality := ⋯ } }.map\n              f ≫\n            { obj := fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGrp,\n                  map := fun {X Y} f =>\n                    { app := fun Y_1 => { toFun := fun g => f.unop ≫ g, map_zero' := ⋯, map_add' := ⋯ },\n                      naturality := ⋯ } }.map\n              g).app\n        x✝¹)\n      x✝","state_after":"case w.h.w\nC : Type u\ninst✝¹ : 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+{"url":"Mathlib/Analysis/Normed/Module/Dual.lean","commit":"","full_name":"NormedSpace.eq_iff_forall_dual_eq","start":[121,0],"end":[124,20],"file_path":"Mathlib/Analysis/Normed/Module/Dual.lean","tactics":[{"state_before":"𝕜 : Type v\ninst✝² : RCLike 𝕜\nE : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx y : E\n⊢ x = y ↔ ∀ (g : Dual 𝕜 E), g x = g y","state_after":"𝕜 : Type v\ninst✝² : RCLike 𝕜\nE : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx y : E\n⊢ (∀ (g : Dual 𝕜 E), g (x - y) = 0) ↔ ∀ (g : Dual 𝕜 E), g x = g y","tactic":"rw [← sub_eq_zero, eq_zero_iff_forall_dual_eq_zero 𝕜 (x - y)]","premises":[{"full_name":"NormedSpace.eq_zero_iff_forall_dual_eq_zero","def_path":"Mathlib/Analysis/Normed/Module/Dual.lean","def_pos":[118,8],"def_end_pos":[118,39]},{"full_name":"sub_eq_zero","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[738,2],"def_end_pos":[738,13]}]},{"state_before":"𝕜 : Type v\ninst✝² : RCLike 𝕜\nE : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx y : E\n⊢ (∀ (g : Dual 𝕜 E), g (x - y) = 0) ↔ ∀ (g : Dual 𝕜 E), g x = g y","state_after":"no goals","tactic":"simp [sub_eq_zero]","premises":[{"full_name":"sub_eq_zero","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[738,2],"def_end_pos":[738,13]}]}]}
+{"url":"Mathlib/Topology/Instances/ENNReal.lean","commit":"","full_name":"NNReal.not_summable_iff_tendsto_nat_atTop","start":[1022,0],"end":[1029,80],"file_path":"Mathlib/Topology/Instances/ENNReal.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : ℕ → ℝ≥0\n⊢ ¬Summable f ↔ Tendsto (fun n => ∑ i ∈ Finset.range n, f i) atTop atTop","state_after":"case mp\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : ℕ → ℝ≥0\n⊢ ¬Summable f → Tendsto (fun n => ∑ i ∈ Finset.range n, f i) atTop atTop\n\ncase mpr\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : ℕ → ℝ≥0\n⊢ Tendsto (fun n => ∑ i ∈ Finset.range n, f i) atTop atTop → ¬Summable f","tactic":"constructor","premises":[]}]}
+{"url":"Mathlib/MeasureTheory/Group/FundamentalDomain.lean","commit":"","full_name":"MeasureTheory.measure_map_restrict_apply","start":[640,0],"end":[645,97],"file_path":"Mathlib/MeasureTheory/Group/FundamentalDomain.lean","tactics":[{"state_before":"G : Type u_1\nH : Type u_2\nα : Type u_3\nβ : Type u_4\nE : Type u_5\ninst✝² : Group G\ninst✝¹ : MulAction G α\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\nU : Set (Quotient α_mod_G)\nmeas_U : MeasurableSet U\n⊢ (Measure.map (Quotient.mk α_mod_G) (μ.restrict s)) U = μ (Quotient.mk α_mod_G ⁻¹' U ∩ s)","state_after":"no goals","tactic":"rw [map_apply (f := π) (fun V hV ↦ measurableSet_quotient.mp hV) meas_U,\n    Measure.restrict_apply (t := (Quotient.mk α_mod_G ⁻¹' U)) (measurableSet_quotient.mp meas_U)]","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"MeasureTheory.Measure.map_apply","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","def_pos":[1159,8],"def_end_pos":[1159,17]},{"full_name":"MeasureTheory.Measure.restrict_apply","def_path":"Mathlib/MeasureTheory/Measure/Restrict.lean","def_pos":[66,8],"def_end_pos":[66,22]},{"full_name":"MulAction.orbitRel","def_path":"Mathlib/GroupTheory/GroupAction/Basic.lean","def_pos":[361,4],"def_end_pos":[361,12]},{"full_name":"Quotient.mk","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1592,14],"def_end_pos":[1592,16]},{"full_name":"Set.preimage","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[106,4],"def_end_pos":[106,12]},{"full_name":"measurableSet_quotient","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Basic.lean","def_pos":[421,8],"def_end_pos":[421,30]}]}]}
+{"url":"Mathlib/Algebra/Homology/ShortComplex/Exact.lean","commit":"","full_name":"CategoryTheory.ShortComplex.quasiIso_iff_of_zeros'","start":[856,0],"end":[875,7],"file_path":"Mathlib/Algebra/Homology/ShortComplex/Exact.lean","tactics":[{"state_before":"C : Type u_1\nD : Type u_2\ninst✝² : Category.{?u.209500, u_1} C\ninst✝¹ : Category.{?u.209504, u_2} D\ninst✝ : Abelian C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nhg₁ : S₁.g = 0\nhf₂ : S₂.f = 0\nhg₂ : S₂.g = 0\n⊢ S₁.f ≫ φ.τ₂ = 0","state_after":"no goals","tactic":"rw [← φ.comm₁₂, hf₂, comp_zero]","premises":[{"full_name":"CategoryTheory.Limits.comp_zero","def_path":"Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean","def_pos":[61,8],"def_end_pos":[61,17]},{"full_name":"CategoryTheory.ShortComplex.Hom.comm₁₂","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[60,2],"def_end_pos":[60,8]}]},{"state_before":"C : Type u_1\nD : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Category.{?u.209504, u_2} D\ninst✝ : Abelian C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nhg₁ : S₁.g = 0\nhf₂ : S₂.f = 0\nhg₂ : S₂.g = 0\n⊢ QuasiIso φ ↔ (mk S₁.f φ.τ₂ ⋯).Exact ∧ Epi φ.τ₂","state_after":"C : Type u_1\nD : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Category.{?u.209504, u_2} D\ninst✝ : Abelian C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nhg₁ : S₁.g = 0\nhf₂ : S₂.f = 0\nhg₂ : S₂.g = 0\n⊢ (mk (opMap φ).τ₂ S₁.op.g ⋯).Exact ∧ Mono (opMap φ).τ₂ ↔ (mk S₁.f φ.τ₂ ⋯).Exact ∧ Epi φ.τ₂\n\ncase hf₁\nC : Type u_1\nD : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Category.{?u.209504, u_2} D\ninst✝ : Abelian C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nhg₁ : S₁.g = 0\nhf₂ : S₂.f = 0\nhg₂ : S₂.g = 0\n⊢ S₂.op.f = 0\n\ncase hg₁\nC : Type u_1\nD : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Category.{?u.209504, u_2} D\ninst✝ : Abelian C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nhg₁ : S₁.g = 0\nhf₂ : S₂.f = 0\nhg₂ : S₂.g = 0\n⊢ S₂.op.g = 0\n\ncase hf₂\nC : Type u_1\nD : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Category.{?u.209504, u_2} D\ninst✝ : Abelian C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nhg₁ : S₁.g = 0\nhf₂ : S₂.f = 0\nhg₂ : S₂.g = 0\n⊢ S₁.op.f = 0","tactic":"rw [← quasiIso_opMap_iff, quasiIso_iff_of_zeros]","premises":[{"full_name":"CategoryTheory.ShortComplex.quasiIso_iff_of_zeros","def_path":"Mathlib/Algebra/Homology/ShortComplex/Exact.lean","def_pos":[836,6],"def_end_pos":[836,27]},{"full_name":"CategoryTheory.ShortComplex.quasiIso_opMap_iff","def_path":"Mathlib/Algebra/Homology/ShortComplex/QuasiIso.lean","def_pos":[144,6],"def_end_pos":[144,24]}]},{"state_before":"C : Type u_1\nD : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Category.{?u.209504, u_2} D\ninst✝ : Abelian C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nhg₁ : S₁.g = 0\nhf₂ : S₂.f = 0\nhg₂ : S₂.g = 0\n⊢ (mk (opMap φ).τ₂ S₁.op.g ⋯).Exact ∧ Mono (opMap φ).τ₂ ↔ (mk S₁.f φ.τ₂ ⋯).Exact ∧ Epi φ.τ₂\n\ncase hf₁\nC : Type u_1\nD : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Category.{?u.209504, u_2} D\ninst✝ : Abelian C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nhg₁ : S₁.g = 0\nhf₂ : S₂.f = 0\nhg₂ : S₂.g = 0\n⊢ S₂.op.f = 0\n\ncase hg₁\nC : Type u_1\nD : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Category.{?u.209504, u_2} D\ninst✝ : Abelian C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nhg₁ : S₁.g = 0\nhf₂ : S₂.f = 0\nhg₂ : S₂.g = 0\n⊢ S₂.op.g = 0\n\ncase hf₂\nC : Type u_1\nD : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Category.{?u.209504, u_2} D\ninst✝ : Abelian C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nhg₁ : S₁.g = 0\nhf₂ : S₂.f = 0\nhg₂ : S₂.g = 0\n⊢ S₁.op.f = 0","state_after":"case hf₁\nC : Type u_1\nD : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Category.{?u.209504, u_2} D\ninst✝ : Abelian C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nhg₁ : S₁.g = 0\nhf₂ : S₂.f = 0\nhg₂ : S₂.g = 0\n⊢ S₂.op.f = 0\n\ncase hg₁\nC : Type u_1\nD : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Category.{?u.209504, u_2} D\ninst✝ : Abelian C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nhg₁ : S₁.g = 0\nhf₂ : S₂.f = 0\nhg₂ : S₂.g = 0\n⊢ S₂.op.g = 0\n\ncase hf₂\nC : Type u_1\nD : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Category.{?u.209504, u_2} D\ninst✝ : Abelian C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nhg₁ : S₁.g = 0\nhf₂ : S₂.f = 0\nhg₂ : S₂.g = 0\n⊢ S₁.op.f = 0\n\nC : Type u_1\nD : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Category.{?u.209504, u_2} D\ninst✝ : Abelian C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nhg₁ : S₁.g = 0\nhf₂ : S₂.f = 0\nhg₂ : S₂.g = 0\n⊢ (mk (opMap φ).τ₂ S₁.op.g ⋯).Exact ∧ Mono (opMap φ).τ₂ ↔ (mk S₁.f φ.τ₂ ⋯).Exact ∧ Epi φ.τ₂","tactic":"rotate_left","premises":[]},{"state_before":"C : Type u_1\nD : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Category.{?u.209504, u_2} D\ninst✝ : Abelian C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nhg₁ : S₁.g = 0\nhf₂ : S₂.f = 0\nhg₂ : S₂.g = 0\n⊢ (mk (opMap φ).τ₂ S₁.op.g ⋯).Exact ∧ Mono (opMap φ).τ₂ ↔ (mk S₁.f φ.τ₂ ⋯).Exact ∧ Epi φ.τ₂","state_after":"C : Type u_1\nD : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Category.{?u.209504, u_2} D\ninst✝ : Abelian C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nhg₁ : S₁.g = 0\nhf₂ : S₂.f = 0\nhg₂ : S₂.g = 0\n⊢ (mk (opMap φ).τ₂ S₁.op.g ⋯).unop.Exact ∧ Mono (opMap φ).τ₂ ↔ (mk S₁.f φ.τ₂ ⋯).Exact ∧ Epi φ.τ₂","tactic":"rw [← exact_unop_iff]","premises":[{"full_name":"CategoryTheory.ShortComplex.exact_unop_iff","def_path":"Mathlib/Algebra/Homology/ShortComplex/Exact.lean","def_pos":[192,6],"def_end_pos":[192,20]}]},{"state_before":"C : Type u_1\nD : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Category.{?u.209504, u_2} D\ninst✝ : Abelian C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nhg₁ : S₁.g = 0\nhf₂ : S₂.f = 0\nhg₂ : S₂.g = 0\n⊢ (mk (opMap φ).τ₂ S₁.op.g ⋯).unop.Exact ∧ Mono (opMap φ).τ₂ ↔ (mk S₁.f φ.τ₂ ⋯).Exact ∧ Epi φ.τ₂","state_after":"C : Type u_1\nD : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Category.{?u.209504, u_2} D\ninst✝ : Abelian C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nhg₁ : S₁.g = 0\nhf₂ : S₂.f = 0\nhg₂ : S₂.g = 0\nthis : Mono φ.τ₂.op ↔ Epi φ.τ₂\n⊢ (mk (opMap φ).τ₂ S₁.op.g ⋯).unop.Exact ∧ Mono (opMap φ).τ₂ ↔ (mk S₁.f φ.τ₂ ⋯).Exact ∧ Epi φ.τ₂","tactic":"have : Mono φ.τ₂.op ↔ Epi φ.τ₂ :=\n    ⟨fun _ => unop_epi_of_mono φ.τ₂.op, fun _ => op_mono_of_epi 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+{"url":"Mathlib/MeasureTheory/Measure/Dirac.lean","commit":"","full_name":"MeasureTheory.Measure.map_const","start":[60,0],"end":[68,91],"file_path":"Mathlib/MeasureTheory/Measure/Dirac.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nδ : Type u_3\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\ns : Set α\na : α\nμ : Measure α\nc : β\n⊢ map (fun x => c) μ = μ univ • dirac c","state_after":"case h\nα : Type u_1\nβ : Type u_2\nδ : Type u_3\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\ns✝ : Set α\na : α\nμ : Measure α\nc : β\ns : Set β\nhs : MeasurableSet s\n⊢ (map (fun x => c) μ) s = (μ univ • dirac c) s","tactic":"ext s hs","premises":[]},{"state_before":"case h\nα : Type u_1\nβ : Type u_2\nδ : Type u_3\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\ns✝ : Set α\na : α\nμ : Measure α\nc : β\ns : Set β\nhs : MeasurableSet s\n⊢ (map (fun x => c) μ) s = (μ univ • dirac c) s","state_after":"case h\nα : Type u_1\nβ : Type u_2\nδ : Type u_3\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\ns✝ : Set α\na : α\nμ : Measure α\nc : β\ns : Set β\nhs : MeasurableSet s\n⊢ (map (fun x => c) μ) s = μ univ * s.indicator 1 c","tactic":"simp only [aemeasurable_const, measurable_const, Measure.coe_smul, Pi.smul_apply,\n    dirac_apply' _ hs, smul_eq_mul]","premises":[{"full_name":"MeasureTheory.Measure.coe_smul","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","def_pos":[807,8],"def_end_pos":[807,16]},{"full_name":"MeasureTheory.Measure.dirac_apply'","def_path":"Mathlib/MeasureTheory/Measure/Dirac.lean","def_pos":[38,8],"def_end_pos":[38,20]},{"full_name":"Pi.smul_apply","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[104,59],"def_end_pos":[104,69]},{"full_name":"aemeasurable_const","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[407,8],"def_end_pos":[407,26]},{"full_name":"measurable_const","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","def_pos":[510,8],"def_end_pos":[510,24]},{"full_name":"smul_eq_mul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[79,6],"def_end_pos":[79,17]}]}]}
+{"url":"Mathlib/CategoryTheory/Limits/Preserves/Ulift.lean","commit":"","full_name":"CategoryTheory.Limits.Types.descSet_spec","start":[73,0],"end":[82,59],"file_path":"Mathlib/CategoryTheory/Limits/Preserves/Ulift.lean","tactics":[{"state_before":"J : Type u_1\ninst✝ : Category.{u_2, u_1} J\nK : J ⥤ Type u\nc : Cocone K\nhc : IsColimit c\nlc : Cocone (K ⋙ uliftFunctor.{v, u})\ns : Set c.pt\nls : Set lc.pt\n⊢ descSet hc ls = s ↔ ∀ (j : J) (x : K.obj j), lc.ι.app j { down := x } ∈ ls ↔ c.ι.app j x ∈ s","state_after":"case refine_1\nJ : Type u_1\ninst✝ : Category.{u_2, u_1} J\nK : J ⥤ Type u\nc : Cocone K\nhc : IsColimit c\nlc : Cocone (K ⋙ uliftFunctor.{v, u})\ns : Set c.pt\nls : Set lc.pt\n⊢ descSet hc ls = s → ∀ (j : J) (x : K.obj j), lc.ι.app j { down := x } ∈ ls ↔ c.ι.app j x ∈ s\n\ncase refine_2\nJ : Type u_1\ninst✝ : Category.{u_2, u_1} J\nK : J ⥤ Type u\nc : Cocone K\nhc : IsColimit c\nlc : Cocone (K ⋙ uliftFunctor.{v, u})\ns : Set c.pt\nls : Set lc.pt\nhe : ∀ (j : J) (x : K.obj j), lc.ι.app j { down := x } ∈ ls ↔ c.ι.app j x ∈ s\nx : c.pt\n⊢ descSet hc ls x = s x","tactic":"refine ⟨?_, fun he ↦ funext fun x ↦ ?_⟩","premises":[{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"funext","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1817,8],"def_end_pos":[1817,14]}]}]}
+{"url":"Mathlib/Algebra/BigOperators/Group/Multiset.lean","commit":"","full_name":"Multiset.prod_hom","start":[126,0],"end":[130,96],"file_path":"Mathlib/Algebra/BigOperators/Group/Multiset.lean","tactics":[{"state_before":"F✝ : Type u_1\nι : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝⁴ : CommMonoid α\ninst✝³ : CommMonoid β\ns✝ t : Multiset α\na : α\nm : Multiset ι\nf✝ g : ι → α\ninst✝² : CommMonoid β\ns : Multiset α\nF : Type u_6\ninst✝¹ : FunLike F α β\ninst✝ : MonoidHomClass F α β\nf : F\nl : List α\n⊢ (map ⇑f ⟦l⟧).prod = f (prod ⟦l⟧)","state_after":"no goals","tactic":"simp only [l.prod_hom f, quot_mk_to_coe, map_coe, prod_coe]","premises":[{"full_name":"List.prod_hom","def_path":"Mathlib/Algebra/BigOperators/Group/List.lean","def_pos":[142,8],"def_end_pos":[142,16]},{"full_name":"Multiset.map_coe","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[1031,25],"def_end_pos":[1031,32]},{"full_name":"Multiset.prod_coe","def_path":"Mathlib/Algebra/BigOperators/Group/Multiset.lean","def_pos":[52,8],"def_end_pos":[52,16]},{"full_name":"Multiset.quot_mk_to_coe","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[43,8],"def_end_pos":[43,22]}]}]}
+{"url":"Mathlib/Algebra/Polynomial/Eval.lean","commit":"","full_name":"Polynomial.natCast_mul_comp","start":[533,0],"end":[535,33],"file_path":"Mathlib/Algebra/Polynomial/Eval.lean","tactics":[{"state_before":"R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np q r : R[X]\nn : ℕ\n⊢ (↑n * p).comp r = ↑n * p.comp r","state_after":"no goals","tactic":"rw [← C_eq_natCast, C_mul_comp]","premises":[{"full_name":"Polynomial.C_eq_natCast","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[459,8],"def_end_pos":[459,20]},{"full_name":"Polynomial.C_mul_comp","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[526,8],"def_end_pos":[526,18]}]}]}
+{"url":"Mathlib/Algebra/Ring/Subring/Basic.lean","commit":"","full_name":"Subring.closure_induction₂","start":[776,0],"end":[791,67],"file_path":"Mathlib/Algebra/Ring/Subring/Basic.lean","tactics":[{"state_before":"R : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ns : Set R\np : R → R → Prop\na b : R\nha : a ∈ closure s\nhb : b ∈ closure s\nHs : ∀ x ∈ s, ∀ y ∈ s, p x y\nH0_left : ∀ (x : R), p 0 x\nH0_right : ∀ (x : R), p x 0\nH1_left : ∀ (x : R), p 1 x\nH1_right : ∀ (x : R), p x 1\nHneg_left : ∀ (x y : R), p x y → p (-x) y\nHneg_right : ∀ (x y : R), p x y → p x (-y)\nHadd_left : ∀ (x₁ x₂ y : R), p x₁ y → p x₂ y → p (x₁ + x₂) y\nHadd_right : ∀ (x y₁ y₂ : R), p x y₁ → p x y₂ → p x (y₁ + y₂)\nHmul_left : ∀ (x₁ x₂ y : R), p x₁ y → p x₂ y → p (x₁ * x₂) y\nHmul_right : ∀ (x y₁ y₂ : R), p x y₁ → p x y₂ → p x (y₁ * y₂)\n⊢ p a b","state_after":"R : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ns : Set R\np : R → R → Prop\na b : R\nha : a ∈ closure s\nhb : b ∈ closure s\nHs : ∀ x ∈ s, ∀ y ∈ s, p x y\nH0_left : ∀ (x : R), p 0 x\nH0_right : ∀ (x : R), p x 0\nH1_left : ∀ (x : R), p 1 x\nH1_right : ∀ (x : R), p x 1\nHneg_left : ∀ (x y : R), p x y → p (-x) y\nHneg_right : ∀ (x y : R), p x y → p x (-y)\nHadd_left : ∀ (x₁ x₂ y : R), p x₁ y → p x₂ y → p (x₁ + x₂) y\nHadd_right : ∀ (x y₁ y₂ : R), p x y₁ → p x y₂ → p x (y₁ + y₂)\nHmul_left : ∀ (x₁ x₂ y : R), p x₁ y → p x₂ y → p (x₁ * x₂) y\nHmul_right : ∀ (x y₁ y₂ : R), p x y₁ → p x y₂ → p x (y₁ * y₂)\n⊢ ∀ x ∈ s, p a x","tactic":"refine\n    closure_induction hb ?_ (H0_right _) (H1_right _) (Hadd_right a) (Hneg_right a) (Hmul_right a)","premises":[{"full_name":"Subring.closure_induction","def_path":"Mathlib/Algebra/Ring/Subring/Basic.lean","def_pos":[754,8],"def_end_pos":[754,25]}]},{"state_before":"R : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ns : Set R\np : R → R → Prop\na b : R\nha : a ∈ closure s\nhb : b ∈ closure s\nHs : ∀ x ∈ s, ∀ y ∈ s, p x y\nH0_left : ∀ (x : R), p 0 x\nH0_right : ∀ (x : R), p x 0\nH1_left : ∀ (x : R), p 1 x\nH1_right : ∀ (x : R), p x 1\nHneg_left : ∀ (x y : R), p x y → p (-x) y\nHneg_right : ∀ (x y : R), p x y → p x (-y)\nHadd_left : ∀ (x₁ x₂ y : R), p x₁ y → p x₂ y → p (x₁ + x₂) y\nHadd_right : ∀ (x y₁ y₂ : R), p x y₁ → p x y₂ → p x (y₁ + y₂)\nHmul_left : ∀ (x₁ x₂ y : R), p x₁ y → p x₂ y → p (x₁ * x₂) y\nHmul_right : ∀ (x y₁ y₂ : R), p x y₁ → p x y₂ → p x (y₁ * y₂)\n⊢ ∀ x ∈ s, p a x","state_after":"case refine_1\nR : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ns : Set R\np : R → R → Prop\na b : R\nha : a ∈ closure s\nhb : b ∈ closure s\nHs : ∀ x ∈ s, ∀ y ∈ s, p x y\nH0_left : ∀ (x : R), p 0 x\nH0_right : ∀ (x : R), p x 0\nH1_left : ∀ (x : R), p 1 x\nH1_right : ∀ (x : R), p x 1\nHneg_left : ∀ (x y : R), p x y → p (-x) y\nHneg_right : ∀ (x y : R), p x y → p x (-y)\nHadd_left : ∀ (x₁ x₂ y : R), p x₁ y → p x₂ y → p (x₁ + x₂) y\nHadd_right : ∀ (x y₁ y₂ : R), p x y₁ → p x y₂ → p x (y₁ + y₂)\nHmul_left : ∀ (x₁ x₂ y : R), p x₁ y → p x₂ y → p (x₁ * x₂) y\nHmul_right : ∀ (x y₁ y₂ : R), p x y₁ → p x y₂ → p x (y₁ * y₂)\n⊢ ∀ (x y : R), (∀ x_1 ∈ s, p x x_1) → (∀ x ∈ s, p y x) → ∀ x_1 ∈ s, p (x + y) x_1\n\ncase refine_2\nR : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ns : Set R\np : R → R → Prop\na b : R\nha : a ∈ closure s\nhb : b ∈ closure s\nHs : ∀ x ∈ s, ∀ y ∈ s, p x y\nH0_left : ∀ (x : R), p 0 x\nH0_right : ∀ (x : R), p x 0\nH1_left : ∀ (x : R), p 1 x\nH1_right : ∀ (x : R), p x 1\nHneg_left : ∀ (x y : R), p x y → p (-x) y\nHneg_right : ∀ (x y : R), p x y → p x (-y)\nHadd_left : ∀ (x₁ x₂ y : R), p x₁ y → p x₂ y → p (x₁ + x₂) y\nHadd_right : ∀ (x y₁ y₂ : R), p x y₁ → p x y₂ → p x (y₁ + y₂)\nHmul_left : ∀ (x₁ x₂ y : R), p x₁ y → p x₂ y → p (x₁ * x₂) y\nHmul_right : ∀ (x y₁ y₂ : R), p x y₁ → p x y₂ → p x (y₁ * y₂)\n⊢ ∀ (x : R), (∀ x_1 ∈ s, p x x_1) → ∀ x_1 ∈ s, p (-x) x_1\n\ncase refine_3\nR : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ns : Set R\np : R → R → Prop\na b : R\nha : a ∈ closure s\nhb : b ∈ closure s\nHs : ∀ x ∈ s, ∀ y ∈ s, p x y\nH0_left : ∀ (x : R), p 0 x\nH0_right : ∀ (x : R), p x 0\nH1_left : ∀ (x : R), p 1 x\nH1_right : ∀ (x : R), p x 1\nHneg_left : ∀ (x y : R), p x y → p (-x) y\nHneg_right : ∀ (x y : R), p x y → p x (-y)\nHadd_left : ∀ (x₁ x₂ y : R), p x₁ y → p x₂ y → p (x₁ + x₂) y\nHadd_right : ∀ (x y₁ y₂ : R), p x y₁ → p x y₂ → p x (y₁ + y₂)\nHmul_left : ∀ (x₁ x₂ y : R), p x₁ y → p x₂ y → p (x₁ * x₂) y\nHmul_right : ∀ (x y₁ y₂ : R), p x y₁ → p x y₂ → p x (y₁ * y₂)\n⊢ ∀ (x y : R), (∀ x_1 ∈ s, p x x_1) → (∀ x ∈ s, p y x) → ∀ x_1 ∈ s, p (x * y) x_1","tactic":"refine closure_induction ha Hs (fun x _ => H0_left x) (fun x _ => H1_left x) ?_ ?_ ?_","premises":[{"full_name":"Subring.closure_induction","def_path":"Mathlib/Algebra/Ring/Subring/Basic.lean","def_pos":[754,8],"def_end_pos":[754,25]}]}]}
+{"url":"Mathlib/Order/IsWellOrderLimitElement.lean","commit":"","full_name":"IsWellOrderLimitElement.neq_succ","start":[86,0],"end":[88,57],"file_path":"Mathlib/Order/IsWellOrderLimitElement.lean","tactics":[{"state_before":"α : Type u_1\ninst✝² : LinearOrder α\na✝ : α\nha✝ : IsWellOrderLimitElement a✝\ninst✝¹ : IsWellOrder α fun x x_1 => x < x_1\na : α\nha : a < wellOrderSucc a\ninst✝ : IsWellOrderLimitElement (wellOrderSucc a)\n⊢ False","state_after":"no goals","tactic":"simpa using IsWellOrderLimitElement.wellOrderSucc_lt ha","premises":[{"full_name":"IsWellOrderLimitElement.wellOrderSucc_lt","def_path":"Mathlib/Order/IsWellOrderLimitElement.lean","def_pos":[73,6],"def_end_pos":[73,46]}]}]}
+{"url":"Mathlib/Algebra/Polynomial/Eval.lean","commit":"","full_name":"Polynomial.eval₂_eq_sum","start":[44,0],"end":[45,16],"file_path":"Mathlib/Algebra/Polynomial/Eval.lean","tactics":[{"state_before":"R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np q r : R[X]\ninst✝ : Semiring S\nf✝ : R →+* S\nx✝ : S\nf : R →+* S\nx : S\n⊢ eval₂ f x p = p.sum fun e a => f a * x ^ e","state_after":"no goals","tactic":"rw [eval₂_def]","premises":[{"full_name":"Polynomial.eval₂_def","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[41,16],"def_end_pos":[41,21]}]}]}
+{"url":"Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean","commit":"","full_name":"hasFDerivAt_of_tendstoLocallyUniformlyOn","start":[388,0],"end":[397,91],"file_path":"Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean","tactics":[{"state_before":"ι : Type u_1\nl : Filter ι\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedSpace 𝕜 E\nG : Type u_4\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → E → G\ng : E → G\nf' : ι → E → E →L[𝕜] G\ng' : E → E →L[𝕜] G\nx : E\ninst✝ : l.NeBot\ns : Set E\nhs : IsOpen s\nhf' : TendstoLocallyUniformlyOn f' g' l s\nhf : ∀ (n : ι), ∀ x ∈ s, HasFDerivAt (f n) (f' n x) x\nhfg : ∀ x ∈ s, Tendsto (fun n => f n x) l (𝓝 (g x))\nhx : x ∈ s\n⊢ HasFDerivAt g (g' x) x","state_after":"ι : Type u_1\nl : Filter ι\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedSpace 𝕜 E\nG : Type u_4\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → E → G\ng : E → G\nf' : ι → E → E →L[𝕜] G\ng' : E → E →L[𝕜] G\nx : E\ninst✝ : l.NeBot\ns : Set E\nhs : IsOpen s\nhf' : TendstoLocallyUniformlyOn f' g' l s\nhf : ∀ (n : ι), ∀ x ∈ s, HasFDerivAt (f n) (f' n x) x\nhfg : ∀ x ∈ s, Tendsto (fun n => f n x) l (𝓝 (g x))\nhx : x ∈ s\nh1 : s ∈ 𝓝 x\n⊢ HasFDerivAt g (g' x) x","tactic":"have h1 : s ∈ 𝓝 x := hs.mem_nhds hx","premises":[{"full_name":"IsOpen.mem_nhds","def_path":"Mathlib/Topology/Basic.lean","def_pos":[744,8],"def_end_pos":[744,23]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"nhds","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[113,16],"def_end_pos":[113,20]}]},{"state_before":"ι : Type u_1\nl : Filter ι\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedSpace 𝕜 E\nG : Type u_4\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → E → G\ng : E → G\nf' : ι → E → E →L[𝕜] G\ng' : E → E →L[𝕜] G\nx : E\ninst✝ : l.NeBot\ns : Set E\nhs : IsOpen s\nhf' : TendstoLocallyUniformlyOn f' g' l s\nhf : ∀ (n : ι), ∀ x ∈ s, HasFDerivAt (f n) (f' n x) x\nhfg : ∀ x ∈ s, Tendsto (fun n => f n x) l (𝓝 (g x))\nhx : x ∈ s\nh1 : s ∈ 𝓝 x\n⊢ HasFDerivAt g (g' x) x","state_after":"ι : Type u_1\nl : Filter ι\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedSpace 𝕜 E\nG : Type u_4\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → E → G\ng : E → G\nf' : ι → E → E →L[𝕜] G\ng' : E → E →L[𝕜] G\nx : E\ninst✝ : l.NeBot\ns : Set E\nhs : IsOpen s\nhf' : TendstoLocallyUniformlyOn f' g' l s\nhf : ∀ (n : ι), ∀ x ∈ s, HasFDerivAt (f n) (f' n x) x\nhfg : ∀ x ∈ s, Tendsto (fun n => f n x) l (𝓝 (g x))\nhx : x ∈ s\nh1 : s ∈ 𝓝 x\nh3 : Set.univ ×ˢ s ∈ l ×ˢ 𝓝 x\n⊢ HasFDerivAt g (g' x) x","tactic":"have h3 : Set.univ ×ˢ s ∈ l ×ˢ 𝓝 x := by simp only [h1, prod_mem_prod_iff, univ_mem, and_self_iff]","premises":[{"full_name":"Filter.prod_mem_prod_iff","def_path":"Mathlib/Order/Filter/Prod.lean","def_pos":[70,8],"def_end_pos":[70,25]},{"full_name":"Filter.univ_mem","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[136,8],"def_end_pos":[136,16]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Set.univ","def_path":"Mathlib/Init/Set.lean","def_pos":[157,4],"def_end_pos":[157,8]},{"full_name":"and_self_iff","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[35,8],"def_end_pos":[35,20]},{"full_name":"nhds","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[113,16],"def_end_pos":[113,20]}]},{"state_before":"ι : Type u_1\nl : Filter ι\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedSpace 𝕜 E\nG : Type u_4\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → E → G\ng : E → G\nf' : ι → E → E →L[𝕜] G\ng' : E → E →L[𝕜] G\nx : E\ninst✝ : l.NeBot\ns : Set E\nhs : IsOpen s\nhf' : TendstoLocallyUniformlyOn f' g' l s\nhf : ∀ (n : ι), ∀ x ∈ s, HasFDerivAt (f n) (f' n x) x\nhfg : ∀ x �� s, Tendsto (fun n => f n x) l (𝓝 (g x))\nhx : x ∈ s\nh1 : s ∈ 𝓝 x\nh3 : Set.univ ×ˢ s ∈ l ×ˢ 𝓝 x\n⊢ HasFDerivAt g (g' x) x","state_after":"ι : Type u_1\nl : Filter ι\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedSpace 𝕜 E\nG : Type u_4\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → E → G\ng : E → G\nf' : ι → E → E →L[𝕜] G\ng' : E → E →L[𝕜] G\nx : E\ninst✝ : l.NeBot\ns : Set E\nhs : IsOpen s\nhf' : TendstoLocallyUniformlyOn f' g' l s\nhf : ∀ (n : ι), ∀ x ∈ s, HasFDerivAt (f n) (f' n x) x\nhfg : ∀ x ∈ s, Tendsto (fun n => f n x) l (𝓝 (g x))\nhx : x ∈ s\nh1 : s ∈ 𝓝 x\nh3 : Set.univ ×ˢ s ∈ l ×ˢ 𝓝 x\nh4 : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2\n⊢ HasFDerivAt g (g' x) x","tactic":"have h4 : ∀ᶠ n : ι × E in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2 :=\n    eventually_of_mem h3 fun ⟨n, z⟩ ⟨_, hz⟩ => hf n z hz","premises":[{"full_name":"Filter.Eventually","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[948,14],"def_end_pos":[948,24]},{"full_name":"Filter.eventually_of_mem","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[969,8],"def_end_pos":[969,25]},{"full_name":"HasFDerivAt","def_path":"Mathlib/Analysis/Calculus/FDeriv/Basic.lean","def_pos":[149,4],"def_end_pos":[149,15]},{"full_name":"Prod","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[479,10],"def_end_pos":[479,14]},{"full_name":"Prod.fst","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[483,2],"def_end_pos":[483,5]},{"full_name":"Prod.snd","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[485,2],"def_end_pos":[485,5]},{"full_name":"nhds","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[113,16],"def_end_pos":[113,20]}]},{"state_before":"ι : Type u_1\nl : Filter ι\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedSpace 𝕜 E\nG : Type u_4\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → E → G\ng : E → G\nf' : ι → E → E →L[𝕜] G\ng' : E → E →L[𝕜] G\nx : E\ninst✝ : l.NeBot\ns : Set E\nhs : IsOpen s\nhf' : TendstoLocallyUniformlyOn f' g' l s\nhf : ∀ (n : ι), ∀ x ∈ s, HasFDerivAt (f n) (f' n x) x\nhfg : ∀ x ∈ s, Tendsto (fun n => f n x) l (𝓝 (g x))\nhx : x ∈ s\nh1 : s ∈ 𝓝 x\nh3 : Set.univ ×ˢ s ∈ l ×ˢ 𝓝 x\nh4 : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2\n⊢ HasFDerivAt g (g' x) x","state_after":"ι : Type u_1\nl : Filter ι\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedSpace 𝕜 E\nG : Type u_4\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → E → G\ng : E → G\nf' : ι → E → E →L[𝕜] G\ng' : E → E →L[𝕜] G\nx : E\ninst✝ : l.NeBot\ns : Set E\nhs : IsOpen s\nhf' : TendstoLocallyUniformlyOn f' g' l s\nhf : ∀ (n : ι), ∀ x ∈ s, HasFDerivAt (f n) (f' n x) x\nhfg : ∀ x ∈ s, Tendsto (fun n => f n x) l (𝓝 (g x))\nhx : x ∈ s\nh1 : s ∈ 𝓝 x\nh3 : Set.univ ×ˢ s ∈ l ×ˢ 𝓝 x\nh4 : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2\n⊢ TendstoUniformlyOnFilter f' g' l (𝓝 x)","tactic":"refine hasFDerivAt_of_tendstoUniformlyOnFilter ?_ h4 (eventually_of_mem h1 hfg)","premises":[{"full_name":"Filter.eventually_of_mem","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[969,8],"def_end_pos":[969,25]},{"full_name":"hasFDerivAt_of_tendstoUniformlyOnFilter","def_path":"Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean","def_pos":[301,8],"def_end_pos":[301,47]}]},{"state_before":"ι : Type u_1\nl : Filter ι\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedSpace 𝕜 E\nG : Type u_4\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → E → G\ng : E → G\nf' : ι → E → E →L[𝕜] G\ng' : E → E →L[𝕜] G\nx : E\ninst✝ : l.NeBot\ns : Set E\nhs : IsOpen s\nhf' : TendstoLocallyUniformlyOn f' g' l s\nhf : ∀ (n : ι), ∀ x ∈ s, HasFDerivAt (f n) (f' n x) x\nhfg : ∀ x ∈ s, Tendsto (fun n => f n x) l (𝓝 (g x))\nhx : x ∈ s\nh1 : s ∈ 𝓝 x\nh3 : Set.univ ×ˢ s ∈ l ×ˢ 𝓝 x\nh4 : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2\n⊢ TendstoUniformlyOnFilter f' g' l (𝓝 x)","state_after":"no goals","tactic":"simpa [IsOpen.nhdsWithin_eq hs hx] using tendstoLocallyUniformlyOn_iff_filter.mp hf' x hx","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"IsOpen.nhdsWithin_eq","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[177,8],"def_end_pos":[177,28]},{"full_name":"tendstoLocallyUniformlyOn_iff_filter","def_path":"Mathlib/Topology/UniformSpace/UniformConvergence.lean","def_pos":[677,8],"def_end_pos":[677,44]}]}]}
+{"url":"Mathlib/Topology/Order/LeftRightNhds.lean","commit":"","full_name":"mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset","start":[249,0],"end":[251,69],"file_path":"Mathlib/Topology/Order/LeftRightNhds.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na l' : α\ns : Set α\nhl' : l' < a\n⊢ [s ∈ 𝓝[≤] a, s ∈ 𝓝[Icc l' a] a, s ∈ 𝓝[Ioc l' a] a, ∃ l ∈ Ico l' a, Ioc l a ⊆ s, ∃ l ∈ Iio a, Ioc l a ⊆ s].get? 0 =\n    some (s ∈ 𝓝[≤] a)","state_after":"no goals","tactic":"norm_num","premises":[]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na l' : α\ns : Set α\nhl' : l' < a\n⊢ [s ∈ 𝓝[≤] a, s ∈ 𝓝[Icc l' a] a, s ∈ 𝓝[Ioc l' a] a, ∃ l ∈ Ico l' a, Ioc l a ⊆ s, ∃ l ∈ Iio a, Ioc l a ⊆ s].get? 3 =\n    some (∃ l ∈ Ico l' a, Ioc l a ⊆ s)","state_after":"no goals","tactic":"norm_num","premises":[]}]}
+{"url":"Mathlib/Analysis/BoxIntegral/Partition/Measure.lean","commit":"","full_name":"BoxIntegral.Prepartition.measure_iUnion_toReal","start":[76,0],"end":[80,81],"file_path":"Mathlib/Analysis/BoxIntegral/Partition/Measure.lean","tactics":[{"state_before":"ι : Type u_1\ninst✝¹ : Finite ι\nI : Box ι\nπ : Prepartition I\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\n⊢ (μ π.iUnion).toReal = ∑ J ∈ π.boxes, (μ ↑J).toReal","state_after":"ι : Type u_1\ninst✝¹ : Finite ι\nI : Box ι\nπ : Prepartition I\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\n⊢ ∀ b ∈ π.boxes, MeasurableSet ↑b\n\nι : Type u_1\ninst✝¹ : Finite ι\nI : Box ι\nπ : Prepartition I\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\n⊢ ∀ a ∈ π.boxes, μ ↑a ≠ ⊤","tactic":"erw [← ENNReal.toReal_sum, π.iUnion_def, measure_biUnion_finset π.pairwiseDisjoint]","premises":[{"full_name":"BoxIntegral.Prepartition.iUnion_def","def_path":"Mathlib/Analysis/BoxIntegral/Partition/Basic.lean","def_pos":[194,8],"def_end_pos":[194,18]},{"full_name":"BoxIntegral.Prepartition.pairwiseDisjoint","def_path":"Mathlib/Analysis/BoxIntegral/Partition/Basic.lean","def_pos":[59,2],"def_end_pos":[59,18]},{"full_name":"ENNReal.toReal_sum","def_path":"Mathlib/Data/ENNReal/Operations.lean","def_pos":[426,8],"def_end_pos":[426,18]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"MeasureTheory.measure_biUnion_finset","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","def_pos":[166,8],"def_end_pos":[166,30]}]},{"state_before":"ι : Type u_1\ninst✝¹ : Finite ι\nI : Box ι\nπ : Prepartition I\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\n⊢ ∀ b ∈ π.boxes, MeasurableSet ↑b\n\nι : Type u_1\ninst✝¹ : Finite ι\nI : Box ι\nπ : Prepartition I\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\n⊢ ∀ a ∈ π.boxes, μ ↑a ≠ ⊤","state_after":"no goals","tactic":"exacts [fun J _ => J.measurableSet_coe, fun J _ => (J.measure_coe_lt_top μ).ne]","premises":[{"full_name":"BoxIntegral.Box.measurableSet_coe","def_path":"Mathlib/Analysis/BoxIntegral/Partition/Measure.lean","def_pos":[53,8],"def_end_pos":[53,25]},{"full_name":"BoxIntegral.Box.measure_coe_lt_top","def_path":"Mathlib/Analysis/BoxIntegral/Partition/Measure.lean","def_pos":[46,8],"def_end_pos":[46,26]}]}]}
+{"url":"Mathlib/Analysis/Quaternion.lean","commit":"","full_name":"Quaternion.coe_real_complex_mul","start":[132,0],"end":[133,87],"file_path":"Mathlib/Analysis/Quaternion.lean","tactics":[{"state_before":"r : ℝ\nz : ℂ\n⊢ r • ↑z = ↑r * ↑z","state_after":"no goals","tactic":"ext <;> simp","premises":[]}]}
+{"url":"Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean","commit":"","full_name":"val_inv_unitsNonZeroDivisorsEquiv_symm_apply_coe","start":[273,0],"end":[280,20],"file_path":"Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean","tactics":[{"state_before":"M₀ : Type u_1\ninst✝¹ inst✝ : MonoidWithZero M₀\na b : ↥M₀⁰\nu : M₀ˣ\n⊢ ⟨↑u, ⋯⟩ * ⟨↑u⁻¹, ⋯⟩ = 1","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"M₀ : Type u_1\ninst✝¹ inst✝ : MonoidWithZero M₀\na b : ↥M₀⁰\nu : M₀ˣ\n⊢ ⟨↑u⁻¹, ⋯⟩ * ⟨↑u, ⋯⟩ = 1","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/MeasureTheory/Measure/AEDisjoint.lean","commit":"","full_name":"MeasureTheory.AEDisjoint.union_right_iff","start":[93,0],"end":[95,34],"file_path":"Mathlib/MeasureTheory/Measure/AEDisjoint.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ns t u v : Set α\n⊢ AEDisjoint μ s (t ∪ u) ↔ AEDisjoint μ s t ∧ AEDisjoint μ s u","state_after":"no goals","tactic":"simp [union_eq_iUnion, and_comm]","premises":[{"full_name":"Set.union_eq_iUnion","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[1110,8],"def_end_pos":[1110,23]},{"full_name":"and_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[819,8],"def_end_pos":[819,16]}]}]}
+{"url":"Mathlib/MeasureTheory/Order/UpperLower.lean","commit":"","full_name":"IsUpperSet.null_frontier","start":[121,0],"end":[130,73],"file_path":"Mathlib/MeasureTheory/Order/UpperLower.lean","tactics":[{"state_before":"ι : Type u_1\ninst✝ : Fintype ι\ns : Set (ι → ℝ)\nx y : ι → ℝ\nδ : ℝ\nhs : IsUpperSet s\n⊢ volume (frontier s) = 0","state_after":"ι : Type u_1\ninst✝ : Fintype ι\ns : Set (ι → ℝ)\nx✝ y : ι → ℝ\nδ : ℝ\nhs : IsUpperSet s\nx : ι → ℝ\nhx : x ∈ frontier s\n⊢ x ∈\n    {x |\n        (fun x =>\n            Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0)\n              (𝓝 ((closure s).indicator 1 x)))\n          x}ᶜ","tactic":"refine measure_mono_null (fun x hx ↦ ?_)\n    (Besicovitch.ae_tendsto_measure_inter_div_of_measurableSet _\n      (isClosed_closure (s := s)).measurableSet)","premises":[{"full_name":"Besicovitch.ae_tendsto_measure_inter_div_of_measurableSet","def_path":"Mathlib/MeasureTheory/Covering/Besicovitch.lean","def_pos":[1086,8],"def_end_pos":[1086,53]},{"full_name":"IsClosed.measurableSet","def_path":"Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean","def_pos":[261,8],"def_end_pos":[261,30]},{"full_name":"MeasureTheory.measure_mono_null","def_path":"Mathlib/MeasureTheory/OuterMeasure/Basic.lean","def_pos":[52,8],"def_end_pos":[52,25]},{"full_name":"isClosed_closure","def_path":"Mathlib/Topology/Basic.lean","def_pos":[344,8],"def_end_pos":[344,24]}]},{"state_before":"ι : Type u_1\ninst✝ : Fintype ι\ns : Set (ι → ℝ)\nx✝ y : ι → ℝ\nδ : ℝ\nhs : IsUpperSet s\nx : ι → ℝ\nhx : x ∈ frontier s\n⊢ x ∈\n    {x |\n        (fun x =>\n            Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0)\n              (𝓝 ((closure s).indicator 1 x)))\n          x}ᶜ","state_after":"case pos\nι : Type u_1\ninst✝ : Fintype ι\ns : Set (ι → ℝ)\nx✝ y : ι → ℝ\nδ : ℝ\nhs : IsUpperSet s\nx : ι → ℝ\nhx : x ∈ frontier s\nh : x ∈ closure s\n⊢ ¬Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 1)\n\ncase neg\nι : Type u_1\ninst✝ : Fintype ι\ns : Set (ι → ℝ)\nx✝ y : ι → ℝ\nδ : ℝ\nhs : IsUpperSet s\nx : ι → ℝ\nhx : x ∈ frontier s\nh : x ∉ closure s\n⊢ ¬Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 0)","tactic":"by_cases h : x ∈ closure s <;>\n    simp only [mem_compl_iff, mem_setOf, h, not_false_eq_true, indicator_of_not_mem,\n      indicator_of_mem, Pi.one_apply]","premises":[{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Pi.one_apply","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[60,8],"def_end_pos":[60,17]},{"full_name":"Set.indicator_of_mem","def_path":"Mathlib/Algebra/Group/Indicator.lean","def_pos":[61,2],"def_end_pos":[61,13]},{"full_name":"Set.indicator_of_not_mem","def_path":"Mathlib/Algebra/Group/Indicator.lean","def_pos":[65,2],"def_end_pos":[65,13]},{"full_name":"Set.mem_compl_iff","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[84,16],"def_end_pos":[84,29]},{"full_name":"Set.mem_setOf","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[202,8],"def_end_pos":[202,17]},{"full_name":"closure","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[112,4],"def_end_pos":[112,11]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]},{"full_name":"not_false_eq_true","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[134,16],"def_end_pos":[134,33]}]}]}
+{"url":"Mathlib/LinearAlgebra/Dimension/Basic.lean","commit":"","full_name":"lift_rank_map_le","start":[267,0],"end":[270,48],"file_path":"Mathlib/LinearAlgebra/Dimension/Basic.lean","tactics":[{"state_before":"R : Type u\nR' : Type u'\nM M₁ : Type v\nM' : Type v'\ninst✝⁹ : Ring R\ninst✝⁸ : Ring R'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup M'\ninst✝⁵ : AddCommGroup M₁\ninst✝⁴ : Module R M\ninst✝³ : Module R M'\ninst✝² : Module R M₁\ninst✝¹ : Module R' M'\ninst✝ : Module R' M₁\nf : M →ₗ[R] M'\np : Submodule R M\n⊢ lift.{v, v'} (Module.rank R ↥(Submodule.map f p)) ≤ lift.{v', v} (Module.rank R ↥p)","state_after":"R : Type u\nR' : Type u'\nM M₁ : Type v\nM' : Type v'\ninst✝⁹ : Ring R\ninst✝⁸ : Ring R'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup M'\ninst✝⁵ : AddCommGroup M₁\ninst✝⁴ : Module R M\ninst✝³ : Module R M'\ninst✝² : Module R M₁\ninst✝¹ : Module R' M'\ninst✝ : Module R' M₁\nf : M →ₗ[R] M'\np : Submodule R M\nh : lift.{v, v'} (Module.rank R ↥(LinearMap.range (f ∘ₗ p.subtype))) ≤ lift.{v', v} (Module.rank R ↥p)\n⊢ lift.{v, v'} (Module.rank R ↥(Submodule.map f p)) ≤ lift.{v', v} (Module.rank R ↥p)","tactic":"have h := lift_rank_range_le (f.comp (Submodule.subtype p))","premises":[{"full_name":"LinearMap.comp","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[489,4],"def_end_pos":[489,8]},{"full_name":"Submodule.subtype","def_path":"Mathlib/Algebra/Module/Submodule/LinearMap.lean","def_pos":[69,14],"def_end_pos":[69,21]},{"full_name":"lift_rank_range_le","def_path":"Mathlib/LinearAlgebra/Dimension/Basic.lean","def_pos":[250,8],"def_end_pos":[250,26]}]},{"state_before":"R : Type u\nR' : Type u'\nM M₁ : Type v\nM' : Type v'\ninst✝⁹ : Ring R\ninst✝⁸ : Ring R'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup M'\ninst✝⁵ : AddCommGroup M₁\ninst✝⁴ : Module R M\ninst✝³ : Module R M'\ninst✝² : Module R M₁\ninst✝¹ : Module R' M'\ninst✝ : Module R' M₁\nf : M →ₗ[R] M'\np : Submodule R M\nh : lift.{v, v'} (Module.rank R ↥(LinearMap.range (f ∘ₗ p.subtype))) ≤ lift.{v', v} (Module.rank R ↥p)\n⊢ lift.{v, v'} (Module.rank R ↥(Submodule.map f p)) ≤ lift.{v', v} (Module.rank R ↥p)","state_after":"no goals","tactic":"rwa [LinearMap.range_comp, range_subtype] at h","premises":[{"full_name":"LinearMap.range_comp","def_path":"Mathlib/Algebra/Module/Submodule/Range.lean","def_pos":[81,8],"def_end_pos":[81,18]},{"full_name":"Submodule.range_subtype","def_path":"Mathlib/Algebra/Module/Submodule/Range.lean","def_pos":[262,8],"def_end_pos":[262,21]}]}]}
+{"url":"Mathlib/LinearAlgebra/Orientation.lean","commit":"","full_name":"Basis.det_adjustToOrientation","start":[297,0],"end":[308,8],"file_path":"Mathlib/LinearAlgebra/Orientation.lean","tactics":[{"state_before":"R : Type u_1\ninst✝⁵ : LinearOrderedCommRing R\nM : Type u_2\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nι : Type u_3\ninst✝² : Fintype ι\ninst✝¹ : DecidableEq ι\ninst✝ : Nonempty ι\ne : Basis ι R M\nx : Orientation R M ι\n⊢ (e.adjustToOrientation x).det = e.det ∨ (e.adjustToOrientation x).det = -e.det","state_after":"R : Type u_1\ninst✝⁵ : LinearOrderedCommRing R\nM : Type u_2\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nι : Type u_3\ninst✝² : Fintype ι\ninst✝¹ : DecidableEq ι\ninst✝ : Nonempty ι\ne : Basis ι R M\nx : Orientation R M ι\n⊢ (if e.orientation = x then e else e.unitsSMul (Function.update 1 (Classical.arbitrary ι) (-1))).det = e.det ∨\n    (if e.orientation = x then e else e.unitsSMul (Function.update 1 (Classical.arbitrary ι) (-1))).det = -e.det","tactic":"dsimp [Basis.adjustToOrientation]","premises":[{"full_name":"Basis.adjustToOrientation","def_path":"Mathlib/LinearAlgebra/Orientation.lean","def_pos":[272,4],"def_end_pos":[272,23]}]},{"state_before":"R : Type u_1\ninst✝⁵ : LinearOrderedCommRing R\nM : Type u_2\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nι : Type u_3\ninst✝² : Fintype ι\ninst✝¹ : DecidableEq ι\ninst✝ : Nonempty ι\ne : Basis ι R M\nx : Orientation R M ι\n⊢ (if e.orientation = x then e else e.unitsSMul (Function.update 1 (Classical.arbitrary ι) (-1))).det = e.det ∨\n    (if e.orientation = x then e else e.unitsSMul (Function.update 1 (Classical.arbitrary ι) (-1))).det = -e.det","state_after":"case pos\nR : Type u_1\ninst✝⁵ : LinearOrderedCommRing R\nM : Type u_2\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nι : Type u_3\ninst✝² : Fintype ι\ninst✝¹ : DecidableEq ι\ninst✝ : Nonempty ι\ne : Basis ι R M\nx : Orientation R M ι\nh✝ : e.orientation = x\n⊢ e.det = e.det ∨ e.det = -e.det\n\ncase neg\nR : Type u_1\ninst✝⁵ : LinearOrderedCommRing R\nM : Type u_2\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nι : Type u_3\ninst✝² : Fintype ι\ninst✝¹ : DecidableEq ι\ninst✝ : Nonempty ι\ne : Basis ι R M\nx : Orientation R M ι\nh✝ : ¬e.orientation = x\n⊢ (e.unitsSMul (Function.update 1 (Classical.arbitrary ι) (-1))).det = e.det ∨\n    (e.unitsSMul (Function.update 1 (Classical.arbitrary ι) (-1))).det = -e.det","tactic":"split_ifs","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
+{"url":"Mathlib/Data/Real/Irrational.lean","commit":"","full_name":"Irrational.int_sub","start":[295,0],"end":[296,49],"file_path":"Mathlib/Data/Real/Irrational.lean","tactics":[{"state_before":"q : ℚ\nx y : ℝ\nh : Irrational x\nm : ℤ\n⊢ Irrational (↑m - x)","state_after":"no goals","tactic":"simpa only [Rat.cast_intCast] using h.rat_sub m","premises":[{"full_name":"Irrational.rat_sub","def_path":"Mathlib/Data/Real/Irrational.lean","def_pos":[283,8],"def_end_pos":[283,15]},{"full_name":"Rat.cast_intCast","def_path":"Mathlib/Data/Rat/Cast/Defs.lean","def_pos":[112,8],"def_end_pos":[112,20]}]}]}
+{"url":"Mathlib/Topology/Order/ScottTopology.lean","commit":"","full_name":"dirSupInacc_compl","start":[82,0],"end":[84,29],"file_path":"Mathlib/Topology/Order/ScottTopology.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝ : Preorder α\ns t : Set α\n⊢ DirSupInacc sᶜ ↔ DirSupClosed s","state_after":"no goals","tactic":"simp [DirSupInacc, DirSupClosed, ← not_disjoint_iff_nonempty_inter, not_imp_not,\n    disjoint_compl_right_iff]","premises":[{"full_name":"DirSupClosed","def_path":"Mathlib/Topology/Order/ScottTopology.lean","def_pos":[79,4],"def_end_pos":[79,16]},{"full_name":"DirSupInacc","def_path":"Mathlib/Topology/Order/ScottTopology.lean","def_pos":[72,4],"def_end_pos":[72,15]},{"full_name":"Set.not_disjoint_iff_nonempty_inter","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1224,6],"def_end_pos":[1224,37]},{"full_name":"disjoint_compl_right_iff","def_path":"Mathlib/Order/BooleanAlgebra.lean","def_pos":[669,8],"def_end_pos":[669,32]},{"full_name":"not_imp_not","def_path":"Mathlib/Logic/Basic.lean","def_pos":[290,8],"def_end_pos":[290,19]}]}]}
+{"url":"Mathlib/MeasureTheory/Measure/EverywherePos.lean","commit":"","full_name":"MeasureTheory.Measure.innerRegularWRT_preimage_one_hasCompactSupport_measure_ne_top_of_addGroup","start":[268,0],"end":[296,30],"file_path":"Mathlib/MeasureTheory/Measure/EverywherePos.lean","tactics":[{"state_before":"α : Type u_1\ninst✝¹⁰ : TopologicalSpace α\ninst✝⁹ : MeasurableSpace α\nμ✝ ν : Measure α\ns k : Set α\nG : Type u_2\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : TopologicalGroup G\ninst✝⁵ : LocallyCompactSpace G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\nμ : Measure G\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : IsFiniteMeasureOnCompacts μ\ninst✝ : μ.InnerRegularCompactLTTop\n⊢ μ.InnerRegularWRT (fun s => ∃ f, Continuous f ∧ HasCompactSupport f ∧ s = f ⁻¹' {1}) fun s =>\n    MeasurableSet s ∧ μ s ≠ ⊤","state_after":"α : Type u_1\ninst✝¹⁰ : TopologicalSpace α\ninst✝⁹ : MeasurableSpace α\nμ✝ ν : Measure α\ns k : Set α\nG : Type u_2\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : TopologicalGroup G\ninst✝⁵ : LocallyCompactSpace G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\nμ : Measure G\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : IsFiniteMeasureOnCompacts μ\ninst✝ : μ.InnerRegularCompactLTTop\n⊢ μ.InnerRegularWRT (fun s => ∃ f, Continuous f ∧ HasCompactSupport f ∧ s = f ⁻¹' {1}) fun s => IsCompact s ∧ IsClosed s","tactic":"apply InnerRegularWRT.trans _ innerRegularWRT_isCompact_isClosed_measure_ne_top_of_group","premises":[{"full_name":"MeasureTheory.Measure.InnerRegularWRT.trans","def_path":"Mathlib/MeasureTheory/Measure/Regular.lean","def_pos":[254,8],"def_end_pos":[254,13]},{"full_name":"MeasureTheory.innerRegularWRT_isCompact_isClosed_measure_ne_top_of_group","def_path":"Mathlib/MeasureTheory/Group/Measure.lean","def_pos":[686,6],"def_end_pos":[686,64]}]},{"state_before":"α : Type u_1\ninst✝¹⁰ : TopologicalSpace α\ninst✝⁹ : MeasurableSpace α\nμ✝ ν : Measure α\ns k : Set α\nG : Type u_2\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : TopologicalGroup G\ninst✝⁵ : LocallyCompactSpace G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\nμ : Measure G\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : IsFiniteMeasureOnCompacts μ\ninst✝ : μ.InnerRegularCompactLTTop\n⊢ μ.InnerRegularWRT (fun s => ∃ f, Continuous f ∧ HasCompactSupport f ∧ s = f ⁻¹' {1}) fun s => IsCompact s ∧ IsClosed s","state_after":"α : Type u_1\ninst✝¹⁰ : TopologicalSpace α\ninst✝⁹ : MeasurableSpace α\nμ✝ ν : Measure α\ns k : Set α\nG : Type u_2\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : TopologicalGroup G\ninst✝⁵ : LocallyCompactSpace G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\nμ : Measure G\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : IsFiniteMeasureOnCompacts μ\ninst✝ : μ.InnerRegularCompactLTTop\nK : Set G\nK_comp : IsCompact K\nK_closed : IsClosed K\nr : ℝ≥0∞\nhr : r < μ K\n⊢ ∃ K_1 ⊆ K, (fun s => ∃ f, Continuous f ∧ HasCompactSupport f ∧ s = f ⁻¹' {1}) K_1 ∧ r < μ K_1","tactic":"intro K ⟨K_comp, K_closed⟩ r hr","premises":[]},{"state_before":"α : Type u_1\ninst✝¹⁰ : TopologicalSpace α\ninst✝⁹ : MeasurableSpace α\nμ✝ ν : Measure α\ns k : Set α\nG : Type u_2\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : TopologicalGroup G\ninst✝⁵ : LocallyCompactSpace G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\nμ : Measure G\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : IsFiniteMeasureOnCompacts μ\ninst✝ : μ.InnerRegularCompactLTTop\nK : Set G\nK_comp : IsCompact K\nK_closed : IsClosed K\nr : ℝ≥0∞\nhr : r < μ K\n⊢ ∃ K_1 ⊆ K, (fun s => ∃ f, Continuous f ∧ HasCompactSupport f ∧ s = f ⁻¹' {1}) K_1 ∧ r < μ K_1","state_after":"α : Type u_1\ninst✝¹⁰ : TopologicalSpace α\ninst✝⁹ : MeasurableSpace α\nμ✝ ν : Measure α\ns k : Set α\nG : Type u_2\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : TopologicalGroup G\ninst✝⁵ : LocallyCompactSpace G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\nμ : Measure G\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : IsFiniteMeasureOnCompacts μ\ninst✝ : μ.InnerRegularCompactLTTop\nK : Set G\nK_comp : IsCompact K\nK_closed : IsClosed K\nr : ℝ≥0∞\nhr : r < μ K\nL : Set G := μ.everywherePosSubset K\n⊢ ∃ K_1 ⊆ K, (fun s => ∃ f, Continuous f ∧ HasCompactSupport f ∧ s = f ⁻¹' {1}) K_1 ∧ r < μ K_1","tactic":"let L := μ.everywherePosSubset K","premises":[{"full_name":"MeasureTheory.Measure.everywherePosSubset","def_path":"Mathlib/MeasureTheory/Measure/EverywherePos.lean","def_pos":[53,4],"def_end_pos":[53,23]}]},{"state_before":"α : Type u_1\ninst✝¹⁰ : TopologicalSpace α\ninst✝⁹ : MeasurableSpace α\nμ✝ ν : Measure α\ns k : Set α\nG : Type u_2\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : TopologicalGroup G\ninst✝⁵ : LocallyCompactSpace G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\nμ : Measure G\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : IsFiniteMeasureOnCompacts μ\ninst✝ : μ.InnerRegularCompactLTTop\nK : Set G\nK_comp : IsCompact K\nK_closed : IsClosed K\nr : ℝ≥0∞\nhr : r < μ K\nL : Set G := μ.everywherePosSubset K\n⊢ ∃ K_1 ⊆ K, (fun s => ∃ f, Continuous f ∧ HasCompactSupport f ∧ s = f ⁻¹' {1}) K_1 ∧ r < μ K_1","state_after":"α : Type u_1\ninst✝¹⁰ : TopologicalSpace α\ninst✝⁹ : MeasurableSpace α\nμ✝ ν : Measure α\ns k : Set α\nG : Type u_2\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : TopologicalGroup G\ninst✝⁵ : LocallyCompactSpace G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\nμ : Measure G\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : IsFiniteMeasureOnCompacts μ\ninst✝ : μ.InnerRegularCompactLTTop\nK : Set G\nK_comp : IsCompact K\nK_closed : IsClosed K\nr : ℝ≥0∞\nhr : r < μ K\nL : Set G := μ.everywherePosSubset K\nL_comp : IsCompact L\n⊢ ∃ K_1 ⊆ K, (fun s => ∃ f, Continuous f ∧ HasCompactSupport f ∧ s = f ⁻¹' {1}) K_1 ∧ r < μ K_1","tactic":"have L_comp : IsCompact L := K_comp.everywherePosSubset","premises":[{"full_name":"IsCompact","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[254,4],"def_end_pos":[254,13]},{"full_name":"IsCompact.everywherePosSubset","def_path":"Mathlib/MeasureTheory/Measure/EverywherePos.lean","def_pos":[90,16],"def_end_pos":[90,52]}]},{"state_before":"α : Type u_1\ninst✝¹⁰ : TopologicalSpace α\ninst✝⁹ : MeasurableSpace α\nμ✝ ν : Measure α\ns k : Set α\nG : Type u_2\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : TopologicalGroup G\ninst✝⁵ : LocallyCompactSpace G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\nμ : Measure G\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : IsFiniteMeasureOnCompacts μ\ninst✝ : μ.InnerRegularCompactLTTop\nK : Set G\nK_comp : IsCompact K\nK_closed : IsClosed K\nr : ℝ≥0∞\nhr : r < μ K\nL : Set G := μ.everywherePosSubset K\nL_comp : IsCompact L\n⊢ ∃ K_1 ⊆ K, (fun s => ∃ f, Continuous f ∧ HasCompactSupport f ∧ s = f ⁻¹' {1}) K_1 ∧ r < μ K_1","state_after":"α : Type u_1\ninst✝¹⁰ : TopologicalSpace α\ninst✝⁹ : MeasurableSpace α\nμ✝ ν : Measure α\ns k : Set α\nG : Type u_2\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : TopologicalGroup G\ninst✝⁵ : LocallyCompactSpace G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\nμ : Measure G\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : IsFiniteMeasureOnCompacts μ\ninst✝ : μ.InnerRegularCompactLTTop\nK : Set G\nK_comp : IsCompact K\nK_closed : IsClosed K\nr : ℝ≥0∞\nhr : r < μ K\nL : Set G := μ.everywherePosSubset K\nL_comp : IsCompact L\nL_closed : IsClosed L\n⊢ ∃ K_1 ⊆ K, (fun s => ∃ f, Continuous f ∧ HasCompactSupport f ∧ s = f ⁻¹' {1}) K_1 ∧ r < μ K_1","tactic":"have L_closed : IsClosed L := K_closed.everywherePosSubset","premises":[{"full_name":"IsClosed","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[93,6],"def_end_pos":[93,14]},{"full_name":"IsClosed.everywherePosSubset","def_path":"Mathlib/MeasureTheory/Measure/EverywherePos.lean","def_pos":[84,16],"def_end_pos":[84,51]}]},{"state_before":"α : Type u_1\ninst✝¹⁰ : TopologicalSpace α\ninst✝⁹ : MeasurableSpace α\nμ✝ ν : Measure α\ns k : Set α\nG : Type u_2\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : TopologicalGroup G\ninst✝⁵ : LocallyCompactSpace G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\nμ : Measure G\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : IsFiniteMeasureOnCompacts μ\ninst✝ : μ.InnerRegularCompactLTTop\nK : Set G\nK_comp : IsCompact K\nK_closed : IsClosed K\nr : ℝ≥0∞\nhr : r < μ K\nL : Set G := μ.everywherePosSubset K\nL_comp : IsCompact L\nL_closed : IsClosed L\n⊢ ∃ K_1 ⊆ K, (fun s => ∃ f, Continuous f ∧ HasCompactSupport f ∧ s = f ⁻¹' {1}) K_1 ∧ r < μ K_1","state_after":"case refine_1\nα : Type u_1\ninst✝¹⁰ : TopologicalSpace α\ninst✝⁹ : MeasurableSpace α\nμ✝ ν : Measure α\ns k : Set α\nG : Type u_2\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : TopologicalGroup G\ninst✝⁵ : LocallyCompactSpace G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\nμ : Measure G\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : IsFiniteMeasureOnCompacts μ\ninst✝ : μ.InnerRegularCompactLTTop\nK : Set G\nK_comp : IsCompact K\nK_closed : IsClosed K\nr : ℝ≥0∞\nhr : r < μ K\nL : Set G := μ.everywherePosSubset K\nL_comp : IsCompact L\nL_closed : IsClosed L\n⊢ (fun s => ∃ f, Continuous f ∧ HasCompactSupport f ∧ s = f ⁻¹' {1}) L\n\ncase refine_2\nα : Type u_1\ninst✝¹⁰ : TopologicalSpace α\ninst✝⁹ : MeasurableSpace α\nμ✝ ν : Measure α\ns k : Set α\nG : Type u_2\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : TopologicalGroup G\ninst✝⁵ : LocallyCompactSpace G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\nμ : Measure G\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : IsFiniteMeasureOnCompacts μ\ninst✝ : μ.InnerRegularCompactLTTop\nK : Set G\nK_comp : IsCompact K\nK_closed : IsClosed K\nr : ℝ≥0∞\nhr : r < μ K\nL : Set G := μ.everywherePosSubset K\nL_comp : IsCompact L\nL_closed : IsClosed L\n⊢ r < μ L","tactic":"refine ⟨L, everywherePosSubset_subset μ K, ?_, ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"MeasureTheory.Measure.everywherePosSubset_subset","def_path":"Mathlib/MeasureTheory/Measure/EverywherePos.lean","def_pos":[56,6],"def_end_pos":[56,32]}]}]}
+{"url":"Mathlib/Analysis/InnerProductSpace/Projection.lean","commit":"","full_name":"ker_orthogonalProjection","start":[547,0],"end":[549,45],"file_path":"Mathlib/Analysis/InnerProductSpace/Projection.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : _root_.RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : HasOrthogonalProjection K\n⊢ LinearMap.ker (orthogonalProjection K) = Kᗮ","state_after":"case h\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : _root_.RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : HasOrthogonalProjection K\nx✝ : E\n⊢ x✝ ∈ LinearMap.ker (orthogonalProjection K) ↔ x✝ ∈ Kᗮ","tactic":"ext","premises":[]},{"state_before":"case h\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : _root_.RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : HasOrthogonalProjection K\nx✝ : E\n⊢ x✝ ∈ LinearMap.ker (orthogonalProjection K) ↔ x✝ ∈ Kᗮ","state_after":"no goals","tactic":"exact orthogonalProjection_eq_zero_iff","premises":[{"full_name":"orthogonalProjection_eq_zero_iff","def_path":"Mathlib/Analysis/InnerProductSpace/Projection.lean","def_pos":[541,8],"def_end_pos":[541,40]}]}]}
+{"url":"Mathlib/Algebra/Module/Submodule/Bilinear.lean","commit":"","full_name":"Submodule.map₂_span_span","start":[54,0],"end":[68,60],"file_path":"Mathlib/Algebra/Module/Submodule/Bilinear.lean","tactics":[{"state_before":"ι : Sort uι\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : AddCommMonoid P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N →ₗ[R] P\ns : Set M\nt : Set N\n⊢ map₂ f (span R s) (span R t) = span R (image2 (fun m n => (f m) n) s t)","state_after":"case a\nι : Sort uι\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : AddCommMonoid P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N →ₗ[R] P\ns : Set M\nt : Set N\n⊢ map₂ f (span R s) (span R t) ≤ span R (image2 (fun m n => (f m) n) s t)\n\ncase a\nι : Sort uι\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : AddCommMonoid P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N →ₗ[R] P\ns : Set M\nt : Set N\n⊢ span R (image2 (fun m n => (f m) n) s t) ≤ map₂ f (span R s) (span R t)","tactic":"apply le_antisymm","premises":[{"full_name":"le_antisymm","def_path":"Mathlib/Order/Defs.lean","def_pos":[156,8],"def_end_pos":[156,19]}]}]}
+{"url":"Mathlib/Topology/PartialHomeomorph.lean","commit":"","full_name":"Homeomorph.toPartialHomeomorphOfImageEq_source","start":[178,0],"end":[188,55],"file_path":"Mathlib/Topology/PartialHomeomorph.lean","tactics":[{"state_before":"X : Type u_1\nX' : Type u_2\nY : Type u_3\nY' : Type u_4\nZ : Type u_5\nZ' : Type u_6\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace X'\ninst✝³ : TopologicalSpace Y\ninst✝² : TopologicalSpace Y'\ninst✝¹ : TopologicalSpace Z\ninst✝ : TopologicalSpace Z'\ne✝ : PartialHomeomorph X Y\ne : X ≃ₜ Y\ns : Set X\nhs : IsOpen s\nt : Set Y\nh : ⇑e '' s = t\n⊢ IsOpen (e.toPartialEquivOfImageEq s t h).target","state_after":"no goals","tactic":"simpa [← h]","premises":[]}]}
+{"url":"Mathlib/FieldTheory/SeparableDegree.lean","commit":"","full_name":"perfectField_iff_splits_of_natSepDegree_eq_one","start":[837,0],"end":[853,38],"file_path":"Mathlib/FieldTheory/SeparableDegree.lean","tactics":[{"state_before":"F✝ : Type u\nE : Type v\ninst✝⁵ : Field F✝\ninst✝⁴ : Field E\ninst✝³ : Algebra F✝ E\nK : Type w\ninst✝² : Field K\ninst✝¹ : Algebra F✝ K\nF : Type u_1\ninst✝ : Field F\n⊢ PerfectField F ↔ ∀ (f : F[X]), f.natSepDegree = 1 → Splits (RingHom.id F) f","state_after":"case refine_1\nF✝ : Type u\nE : Type v\ninst✝⁵ : Field F✝\ninst✝⁴ : Field E\ninst✝³ : Algebra F✝ E\nK : Type w\ninst✝² : Field K\ninst✝¹ : Algebra F✝ K\nF : Type u_1\ninst✝ : Field F\nx✝ : PerfectField F\nf : F[X]\nhf : f.natSepDegree = 1\nh : ∀ {f : F[X]}, Irreducible f → f.Separable\nhn : ¬Polynomial.map (RingHom.id F) f = 0\ng : F[X]\nhg : Irreducible g\nhd : g ∣ Polynomial.map (RingHom.id F) f\n⊢ g.degree = 1\n\ncase refine_2\nF✝ : Type u\nE : Type v\ninst✝⁵ : Field F✝\ninst✝⁴ : Field E\ninst✝³ : Algebra F✝ E\nK : Type w\ninst✝² : Field K\ninst✝¹ : Algebra F✝ K\nF : Type u_1\ninst✝ : Field F\nh : ∀ (f : F[X]), f.natSepDegree = 1 → Splits (RingHom.id F) f\n⊢ PerfectField F","tactic":"refine ⟨fun ⟨h⟩ f hf ↦ or_iff_not_imp_left.2 fun hn g hg hd ↦ ?_, fun h ↦ ?_⟩","premises":[{"full_name":"Classical.or_iff_not_imp_left","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[147,8],"def_end_pos":[147,27]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]}]},{"state_before":"case refine_2\nF✝ : Type u\nE : Type v\ninst✝⁵ : Field F✝\ninst✝⁴ : Field E\ninst✝³ : Algebra F✝ E\nK : Type w\ninst✝² : Field K\ninst✝¹ : Algebra F✝ K\nF : Type u_1\ninst✝ : Field F\nh : ∀ (f : F[X]), f.natSepDegree = 1 → Splits (RingHom.id F) f\n⊢ PerfectField F","state_after":"case refine_2.intro\nF✝ : Type u\nE : Type v\ninst✝⁵ : Field F✝\ninst✝⁴ : Field E\ninst✝³ : Algebra F✝ E\nK : Type w\ninst✝² : Field K\ninst✝¹ : Algebra F✝ K\nF : Type u_1\ninst✝ : Field F\nh : ∀ (f : F[X]), f.natSepDegree = 1 → Splits (RingHom.id F) f\np : ℕ\nh✝ : ExpChar F p\n⊢ PerfectField F","tactic":"obtain ⟨p, _⟩ := ExpChar.exists F","premises":[{"full_name":"ExpChar.exists","def_path":"Mathlib/Algebra/CharP/ExpChar.lean","def_pos":[151,8],"def_end_pos":[151,22]}]},{"state_before":"case refine_2.intro\nF✝ : Type u\nE : Type v\ninst✝⁵ : Field F✝\ninst✝⁴ : Field E\ninst✝³ : Algebra F✝ E\nK : Type w\ninst✝² : Field K\ninst✝¹ : Algebra F✝ K\nF : Type u_1\ninst✝ : Field F\nh : ∀ (f : F[X]), f.natSepDegree = 1 → Splits (RingHom.id F) f\np : ℕ\nh✝ : ExpChar F p\n⊢ PerfectField F","state_after":"case refine_2.intro\nF✝ : Type u\nE : Type v\ninst✝⁵ : Field F✝\ninst✝⁴ : Field E\ninst✝³ : Algebra F✝ E\nK : Type w\ninst✝² : Field K\ninst✝¹ : Algebra F✝ K\nF : Type u_1\ninst✝ : Field F\nh : ∀ (f : F[X]), f.natSepDegree = 1 → Splits (RingHom.id F) f\np : ℕ\nh✝ : ExpChar F p\nthis : PerfectRing F p\n⊢ PerfectField F","tactic":"haveI := PerfectRing.ofSurjective F p fun x ↦ by\n    obtain ⟨y, hy⟩ := exists_root_of_splits _\n      (h _ (pow_one p ▸ natSepDegree_X_pow_char_pow_sub_C p 1 x))\n      ((degree_X_pow_sub_C (expChar_pos F p) x).symm ▸ Nat.cast_pos.2 (expChar_pos F p)).ne'\n    exact ⟨y, by rwa [← eval, eval_sub, eval_pow, eval_X, eval_C, sub_eq_zero] at hy⟩","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"LT.lt.ne'","def_path":"Mathlib/Order/Basic.lean","def_pos":[267,8],"def_end_pos":[267,11]},{"full_name":"Nat.cast_pos","def_path":"Mathlib/Data/Nat/Cast/Order/Ring.lean","def_pos":[53,8],"def_end_pos":[53,16]},{"full_name":"PerfectRing.ofSurjective","def_path":"Mathlib/FieldTheory/Perfect.lean","def_pos":[51,6],"def_end_pos":[51,30]},{"full_name":"Polynomial.degree_X_pow_sub_C","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[1385,8],"def_end_pos":[1385,26]},{"full_name":"Polynomial.eval","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[266,4],"def_end_pos":[266,8]},{"full_name":"Polynomial.eval_C","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[308,8],"def_end_pos":[308,14]},{"full_name":"Polynomial.eval_X","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[324,8],"def_end_pos":[324,14]},{"full_name":"Polynomial.eval_pow","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[953,8],"def_end_pos":[953,16]},{"full_name":"Polynomial.eval_sub","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[1119,8],"def_end_pos":[1119,16]},{"full_name":"Polynomial.exists_root_of_splits","def_path":"Mathlib/Algebra/Polynomial/Splits.lean","def_pos":[245,8],"def_end_pos":[245,29]},{"full_name":"Polynomial.natSepDegree_X_pow_char_pow_sub_C","def_path":"Mathlib/FieldTheory/SeparableDegree.lean","def_pos":[443,8],"def_end_pos":[443,41]},{"full_name":"expChar_pos","def_path":"Mathlib/Algebra/CharP/ExpChar.lean","def_pos":[137,8],"def_end_pos":[137,19]},{"full_name":"pow_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[571,6],"def_end_pos":[571,13]},{"full_name":"sub_eq_zero","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[738,2],"def_end_pos":[738,13]}]},{"state_before":"case refine_2.intro\nF✝ : Type u\nE : Type v\ninst✝⁵ : Field F✝\ninst✝⁴ : Field E\ninst✝³ : Algebra F✝ E\nK : Type w\ninst✝² : Field K\ninst✝¹ : Algebra F✝ K\nF : Type u_1\ninst✝ : Field F\nh : ∀ (f : F[X]), f.natSepDegree = 1 → Splits (RingHom.id F) f\np : ℕ\nh✝ : ExpChar F p\nthis : PerfectRing F p\n⊢ PerfectField F","state_after":"no goals","tactic":"exact PerfectRing.toPerfectField F p","premises":[{"full_name":"PerfectRing.toPerfectField","def_path":"Mathlib/FieldTheory/Perfect.lean","def_pos":[182,6],"def_end_pos":[182,32]}]}]}
+{"url":"Mathlib/Analysis/Matrix.lean","commit":"","full_name":"Matrix.frobenius_nnnorm_mul","start":[607,0],"end":[618,31],"file_path":"Mathlib/Analysis/Matrix.lean","tactics":[{"state_before":"R : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\nα : Type u_5\nβ : Type u_6\nι : Type u_7\ninst✝⁴ : Fintype l\ninst✝³ : Fintype m\ninst✝² : Fintype n\ninst✝¹ : Unique ι\ninst✝ : RCLike α\nA : Matrix l m α\nB : Matrix m n α\n⊢ ‖A * B‖₊ ≤ ‖A‖₊ * ‖B‖₊","state_after":"R : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\nα : Type u_5\nβ : Type u_6\nι : Type u_7\ninst✝⁴ : Fintype l\ninst✝³ : Fintype m\ninst✝² : Fintype n\ninst✝¹ : Unique ι\ninst✝ : RCLike α\nA : Matrix l m α\nB : Matrix m n α\n⊢ (∑ x : l, ∑ x_1 : n, ‖∑ j : m, A x j * B j x_1‖₊ ^ 2) ^ (1 / 2) ≤\n    (∑ i : l, ∑ j : m, ‖A i j‖₊ ^ 2) ^ (1 / 2) * (∑ i : m, ∑ j : n, ‖B i j‖₊ ^ 2) ^ (1 / 2)","tactic":"simp_rw [frobenius_nnnorm_def, Matrix.mul_apply]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Matrix.frobenius_nnnorm_def","def_path":"Mathlib/Analysis/Matrix.lean","def_pos":[512,8],"def_end_pos":[512,28]},{"full_name":"Matrix.mul_apply","def_path":"Mathlib/Data/Matrix/Basic.lean","def_pos":[849,8],"def_end_pos":[849,17]}]},{"state_before":"R : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\nα : Type u_5\nβ : Type u_6\nι : Type u_7\ninst✝⁴ : Fintype l\ninst✝³ : Fintype m\ninst✝² : Fintype n\ninst✝¹ : Unique ι\ninst✝ : RCLike α\nA : Matrix l m α\nB : Matrix m n α\n⊢ (∑ x : l, ∑ x_1 : n, ‖∑ j : m, A x j * B j x_1‖₊ ^ 2) ^ (1 / 2) ≤\n    (∑ i : l, ∑ j : m, ‖A i j‖₊ ^ 2) ^ (1 / 2) * (∑ i : m, ∑ j : n, ‖B i j‖₊ ^ 2) ^ (1 / 2)","state_after":"R : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\nα : Type u_5\nβ : Type u_6\nι : Type u_7\ninst✝⁴ : Fintype l\ninst✝³ : Fintype m\ninst✝² : Fintype n\ninst✝¹ : Unique ι\ninst✝ : RCLike α\nA : Matrix l m α\nB : Matrix m n α\n⊢ (∑ x : l, ∑ x_1 : n, ‖∑ j : m, A x j * B j x_1‖₊ ^ 2) ^ (1 / 2) ≤\n    (∑ i : l, ∑ j : n, (∑ j : m, ‖A i j‖₊ ^ 2) * ∑ x : m, ‖B x j‖₊ ^ 2) ^ (1 / 2)","tactic":"rw [← NNReal.mul_rpow, @Finset.sum_comm _ _ m, Finset.sum_mul_sum]","premises":[{"full_name":"Finset.sum_comm","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[816,2],"def_end_pos":[816,13]},{"full_name":"Finset.sum_mul_sum","def_path":"Mathlib/Algebra/BigOperators/Ring.lean","def_pos":[47,6],"def_end_pos":[47,17]},{"full_name":"NNReal.mul_rpow","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean","def_pos":[126,8],"def_end_pos":[126,16]}]},{"state_before":"R : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\nα : Type u_5\nβ : Type u_6\nι : Type u_7\ninst✝⁴ : Fintype l\ninst✝³ : Fintype m\ninst✝² : Fintype n\ninst✝¹ : Unique ��\ninst✝ : RCLike α\nA : Matrix l m α\nB : Matrix m n α\n⊢ (∑ x : l, ∑ x_1 : n, ‖∑ j : m, A x j * B j x_1‖₊ ^ 2) ^ (1 / 2) ≤\n    (∑ i : l, ∑ j : n, (∑ j : m, ‖A i j‖₊ ^ 2) * ∑ x : m, ‖B x j‖₊ ^ 2) ^ (1 / 2)","state_after":"case h₁.a.h.h\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\nα : Type u_5\nβ : Type u_6\nι : Type u_7\ninst✝⁴ : Fintype l\ninst✝³ : Fintype m\ninst✝² : Fintype n\ninst✝¹ : Unique ι\ninst✝ : RCLike α\nA : Matrix l m α\nB : Matrix m n α\ni : l\na✝¹ : i ∈ Finset.univ\nj : n\na✝ : j ∈ Finset.univ\n⊢ ‖∑ j_1 : m, A i j_1 * B j_1 j‖₊ ^ 2 ≤ (∑ j : m, ‖A i j‖₊ ^ 2) * ∑ x : m, ‖B x j‖₊ ^ 2","tactic":"gcongr with i _ j","premises":[]},{"state_before":"case h₁.a.h.h\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\nα : Type u_5\nβ : Type u_6\nι : Type u_7\ninst✝⁴ : Fintype l\ninst✝³ : Fintype m\ninst✝² : Fintype n\ninst✝¹ : Unique ι\ninst✝ : RCLike α\nA : Matrix l m α\nB : Matrix m n α\ni : l\na✝¹ : i ∈ Finset.univ\nj : n\na✝ : j ∈ Finset.univ\n⊢ ‖∑ j_1 : m, A i j_1 * B j_1 j‖₊ ^ 2 ≤ (∑ j : m, ‖A i j‖₊ ^ 2) * ∑ x : m, ‖B x j‖₊ ^ 2","state_after":"case h₁.a.h.h\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\nα : Type u_5\nβ : Type u_6\nι : Type u_7\ninst✝⁴ : Fintype l\ninst✝³ : Fintype m\ninst✝² : Fintype n\ninst✝¹ : Unique ι\ninst✝ : RCLike α\nA : Matrix l m α\nB : Matrix m n α\ni : l\na✝¹ : i ∈ Finset.univ\nj : n\na✝ : j ∈ Finset.univ\n⊢ ‖∑ j_1 : m, A i j_1 * B j_1 j‖₊ ≤ (∑ j : m, ‖A i j‖₊ ^ 2) ^ (1 / 2) * (∑ x : m, ‖B x j‖₊ ^ 2) ^ (1 / 2)","tactic":"rw [← NNReal.rpow_le_rpow_iff one_half_pos, ← NNReal.rpow_mul,\n    mul_div_cancel₀ (1 : ℝ) two_ne_zero, NNReal.rpow_one, NNReal.mul_rpow]","premises":[{"full_name":"NNReal.mul_rpow","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean","def_pos":[126,8],"def_end_pos":[126,16]},{"full_name":"NNReal.rpow_le_rpow_iff","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean","def_pos":[199,8],"def_end_pos":[199,24]},{"full_name":"NNReal.rpow_mul","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean","def_pos":[81,8],"def_end_pos":[81,16]},{"full_name":"NNReal.rpow_one","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean","def_pos":[60,8],"def_end_pos":[60,16]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"mul_div_cancel₀","def_path":"Mathlib/Algebra/GroupWithZero/Units/Basic.lean","def_pos":[424,6],"def_end_pos":[424,21]},{"full_name":"one_half_pos","def_path":"Mathlib/Algebra/Order/Field/Basic.lean","def_pos":[354,8],"def_end_pos":[354,20]},{"full_name":"two_ne_zero","def_path":"Mathlib/Algebra/NeZero.lean","def_pos":[54,6],"def_end_pos":[54,17]}]},{"state_before":"case h₁.a.h.h\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\nα : Type u_5\nβ : Type u_6\nι : Type u_7\ninst✝⁴ : Fintype l\ninst✝³ : Fintype m\ninst✝² : Fintype n\ninst✝¹ : Unique ι\ninst✝ : RCLike α\nA : Matrix l m α\nB : Matrix m n α\ni : l\na✝¹ : i ∈ Finset.univ\nj : n\na✝ : j ∈ Finset.univ\n⊢ ‖∑ j_1 : m, A i j_1 * B j_1 j‖₊ ≤ (∑ j : m, ‖A i j‖₊ ^ 2) ^ (1 / 2) * (∑ x : m, ‖B x j‖₊ ^ 2) ^ (1 / 2)","state_after":"case h₁.a.h.h\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\nα : Type u_5\nβ : Type u_6\nι : Type u_7\ninst✝⁴ : Fintype l\ninst✝³ : Fintype m\ninst✝² : Fintype n\ninst✝¹ : Unique ι\ninst✝ : RCLike α\nA : Matrix l m α\nB : Matrix m n α\ni : l\na✝¹ : i ∈ Finset.univ\nj : n\na✝ : j ∈ Finset.univ\nthis :\n  ‖⟪(WithLp.equiv 2 (m → α)).symm fun j => star (A i j), (WithLp.equiv 2 (m → α)).symm fun k => B k j⟫_α‖₊ ≤\n    ‖(WithLp.equiv 2 (m → α)).symm fun j => star (A i j)‖₊ * ‖(WithLp.equiv 2 (m → α)).symm fun k => B k j‖₊\n⊢ ‖∑ j_1 : m, A i j_1 * B j_1 j‖₊ ≤ (∑ j : m, ‖A i j‖₊ ^ 2) ^ (1 / 2) * (∑ x : m, ‖B x j‖₊ ^ 2) ^ (1 / 2)","tactic":"have :=\n    @nnnorm_inner_le_nnnorm α _ _ _ _ ((WithLp.equiv 2 <| _ → α).symm fun j => star (A i j))\n      ((WithLp.equiv 2 <| _ → α).symm fun k => B k j)","premises":[{"full_name":"Equiv.symm","def_path":"Mathlib/Logic/Equiv/Defs.lean","def_pos":[146,14],"def_end_pos":[146,18]},{"full_name":"Star.star","def_path":"Mathlib/Algebra/Star/Basic.lean","def_pos":[46,2],"def_end_pos":[46,6]},{"full_name":"WithLp.equiv","def_path":"Mathlib/Analysis/Normed/Lp/WithLp.lean","def_pos":[55,14],"def_end_pos":[55,19]},{"full_name":"nnnorm_inner_le_nnnorm","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[936,8],"def_end_pos":[936,30]}]},{"state_before":"case h₁.a.h.h\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\nα : Type u_5\nβ : Type u_6\nι : Type u_7\ninst✝⁴ : Fintype l\ninst✝³ : Fintype m\ninst✝² : Fintype n\ninst✝¹ : Unique ι\ninst✝ : RCLike α\nA : Matrix l m α\nB : Matrix m n α\ni : l\na✝¹ : i ∈ Finset.univ\nj : n\na✝ : j ∈ Finset.univ\nthis :\n  ‖⟪(WithLp.equiv 2 (m → α)).symm fun j => star (A i j), (WithLp.equiv 2 (m → α)).symm fun k => B k j⟫_α‖₊ ≤\n    ‖(WithLp.equiv 2 (m → α)).symm fun j => star (A i j)‖₊ * ‖(WithLp.equiv 2 (m → α)).symm fun k => B k j‖₊\n⊢ ‖∑ j_1 : m, A i j_1 * B j_1 j‖₊ ≤ (∑ j : m, ‖A i j‖₊ ^ 2) ^ (1 / 2) * (∑ x : m, ‖B x j‖₊ ^ 2) ^ (1 / 2)","state_after":"no goals","tactic":"simpa only [WithLp.equiv_symm_pi_apply, PiLp.inner_apply, RCLike.inner_apply, starRingEnd_apply,\n    Pi.nnnorm_def, PiLp.nnnorm_eq_of_L2, star_star, nnnorm_star, NNReal.sqrt_eq_rpow,\n    NNReal.rpow_two] using this","premises":[{"full_name":"NNReal.rpow_two","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean","def_pos":[124,8],"def_end_pos":[124,16]},{"full_name":"NNReal.sqrt_eq_rpow","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean","def_pos":[107,8],"def_end_pos":[107,20]},{"full_name":"Pi.nnnorm_def","def_path":"Mathlib/Analysis/Normed/Group/Constructions.lean","def_pos":[298,14],"def_end_pos":[298,27]},{"full_name":"PiLp.inner_apply","def_path":"Mathlib/Analysis/InnerProductSpace/PiL2.lean","def_pos":[94,8],"def_end_pos":[94,24]},{"full_name":"PiLp.nnnorm_eq_of_L2","def_path":"Mathlib/Analysis/Normed/Lp/PiLp.lean","def_pos":[582,8],"def_end_pos":[582,23]},{"full_name":"RCLike.inner_apply","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[1696,8],"def_end_pos":[1696,26]},{"full_name":"WithLp.equiv_symm_pi_apply","def_path":"Mathlib/Analysis/Normed/Lp/PiLp.lean","def_pos":[134,9],"def_end_pos":[134,42]},{"full_name":"nnnorm_star","def_path":"Mathlib/Analysis/CstarAlgebra/Basic.lean","def_pos":[53,8],"def_end_pos":[53,19]},{"full_name":"starRingEnd_apply","def_path":"Mathlib/Algebra/Star/Basic.lean","def_pos":[331,8],"def_end_pos":[331,25]},{"full_name":"star_star","def_path":"Mathlib/Algebra/Star/Basic.lean","def_pos":[88,8],"def_end_pos":[88,17]}]}]}
+{"url":"Mathlib/LinearAlgebra/Determinant.lean","commit":"","full_name":"Basis.det_ne_zero","start":[484,0],"end":[485,94],"file_path":"Mathlib/LinearAlgebra/Determinant.lean","tactics":[{"state_before":"R : Type u_1\ninst✝⁷ : CommRing R\nM : Type u_2\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\nM' : Type u_3\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R M'\nι : Type u_4\ninst✝² : DecidableEq ι\ninst✝¹ : Fintype ι\ne : Basis ι R M\ninst✝ : Nontrivial R\nh : e.det = 0\n⊢ False","state_after":"no goals","tactic":"simpa [h] using e.det_self","premises":[{"full_name":"Basis.det_self","def_path":"Mathlib/LinearAlgebra/Determinant.lean","def_pos":[477,8],"def_end_pos":[477,22]}]}]}
+{"url":"Mathlib/AlgebraicGeometry/Limits.lean","commit":"","full_name":"AlgebraicGeometry.disjointGlueData_U","start":[144,0],"end":[151,28],"file_path":"Mathlib/AlgebraicGeometry/Limits.lean","tactics":[{"state_before":"ι : Type u\nf : ι → Scheme\ni j : __spread✝⁻⁰.J\n⊢ IsOpenImmersion (__spread✝⁻⁰.f i j)","state_after":"ι : Type u\nf : ι → Scheme\ni j : __spread✝⁻⁰.J\n⊢ IsOpenImmersion (if h : i = j then eqToHom ⋯ else eqToHom ⋯ ≫ (f i).emptyTo)","tactic":"dsimp only [GlueData.ofGlueData', GlueData'.f', disjointGlueData']","premises":[{"full_name":"AlgebraicGeometry.disjointGlueData'","def_path":"Mathlib/AlgebraicGeometry/Limits.lean","def_pos":[132,4],"def_end_pos":[132,21]},{"full_name":"CategoryTheory.GlueData'.f'","def_path":"Mathlib/CategoryTheory/GlueData.lean","def_pos":[393,7],"def_end_pos":[393,19]},{"full_name":"CategoryTheory.GlueData.ofGlueData'","def_path":"Mathlib/CategoryTheory/GlueData.lean","def_pos":[448,4],"def_end_pos":[448,24]}]},{"state_before":"ι : Type u\nf : ι → Scheme\ni j : __spread✝⁻⁰.J\n⊢ IsOpenImmersion (if h : i = j then eqToHom ⋯ else eqToHom ⋯ ≫ (f i).emptyTo)","state_after":"no goals","tactic":"split <;> infer_instance","premises":[{"full_name":"inferInstance","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[99,7],"def_end_pos":[99,20]}]}]}
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+{"url":"Mathlib/Analysis/Convex/Cone/InnerDual.lean","commit":"","full_name":"innerDualCone_iUnion","start":[98,0],"end":[104,17],"file_path":"Mathlib/Analysis/Convex/Cone/InnerDual.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℝ H\ns t : Set H\nι : Sort u_6\nf : ι → Set H\n⊢ (⋃ i, f i).innerDualCone = ⨅ i, (f i).innerDualCone","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℝ H\ns t : Set H\nι : Sort u_6\nf : ι → Set H\n⊢ ⨅ i, (f i).innerDualCone ≤ (⋃ i, f i).innerDualCone","tactic":"refine le_antisymm (le_iInf fun i x hx y hy => hx _ <| mem_iUnion_of_mem _ hy) ?_","premises":[{"full_name":"Set.mem_iUnion_of_mem","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[72,8],"def_end_pos":[72,25]},{"full_name":"le_antisymm","def_path":"Mathlib/Order/Defs.lean","def_pos":[156,8],"def_end_pos":[156,19]},{"full_name":"le_iInf","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[664,8],"def_end_pos":[664,15]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℝ H\ns t : Set H\nι : Sort u_6\nf : ι → Set H\n⊢ ⨅ i, (f i).innerDualCone ≤ (⋃ i, f i).innerDualCone","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℝ H\ns t : Set H\nι : Sort u_6\nf : ι → Set H\nx : H\nhx : x ∈ ⨅ i, (f i).innerDualCone\ny : H\nhy : y ∈ ⋃ i, f i\n⊢ 0 ≤ ⟪y, x⟫_ℝ","tactic":"intro x hx y hy","premises":[]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℝ H\ns t : Set H\nι : Sort u_6\nf : ι → Set H\nx : H\nhx : x ∈ ⨅ i, (f i).innerDualCone\ny : H\nhy : y ∈ ⋃ i, f i\n⊢ 0 ≤ ⟪y, x⟫_ℝ","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℝ H\ns t : Set H\nι : Sort u_6\nf : ι → Set H\nx : H\nhx : ∀ (i : ι), x ∈ (f i).innerDualCone\ny : H\nhy : y ∈ ⋃ i, f i\n⊢ 0 ≤ ⟪y, x⟫_ℝ","tactic":"rw [ConvexCone.mem_iInf] at hx","premises":[{"full_name":"ConvexCone.mem_iInf","def_path":"Mathlib/Analysis/Convex/Cone/Basic.lean","def_pos":[132,8],"def_end_pos":[132,16]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℝ H\ns t : Set H\nι : Sort u_6\nf : ι → Set H\nx : H\nhx : ∀ (i : ι), x ∈ (f i).innerDualCone\ny : H\nhy : y ∈ ⋃ i, f i\n⊢ 0 ≤ ⟪y, x⟫_ℝ","state_after":"case intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℝ H\ns t : Set H\nι : Sort u_6\nf : ι → Set H\nx : H\nhx : ∀ (i : ι), x ∈ (f i).innerDualCone\ny : H\nhy : y ∈ ⋃ i, f i\nj : ι\nhj : y ∈ f j\n⊢ 0 ≤ ⟪y, x⟫_ℝ","tactic":"obtain ⟨j, hj⟩ := mem_iUnion.mp hy","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Set.mem_iUnion","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[254,8],"def_end_pos":[254,18]}]},{"state_before":"case intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℝ H\ns t : Set H\nι : Sort u_6\nf : ι → Set H\nx : H\nhx : ∀ (i : ι), x ∈ (f i).innerDualCone\ny : H\nhy : y ∈ ⋃ i, f i\nj : ι\nhj : y ∈ f j\n⊢ 0 ≤ ⟪y, x⟫_ℝ","state_after":"no goals","tactic":"exact hx _ _ hj","premises":[]}]}
+{"url":"Mathlib/Analysis/Normed/Group/Basic.lean","commit":"","full_name":"NormedAddCommGroup.nhds_basis_norm_lt","start":[586,0],"end":[590,30],"file_path":"Mathlib/Analysis/Normed/Group/Basic.lean","tactics":[{"state_before":"𝓕 : Type u_1\n𝕜 : Type u_2\nα : Type u_3\nι : Type u_4\nκ : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nx : E\n⊢ (𝓝 x).HasBasis (fun ε => 0 < ε) fun ε => {y | ‖y / x‖ < ε}","state_after":"𝓕 : Type u_1\n𝕜 : Type u_2\nα : Type u_3\nι : Type u_4\nκ : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nx : E\n⊢ (𝓝 x).HasBasis (fun ε => 0 < ε) fun ε => ball x ε","tactic":"simp_rw [← ball_eq']","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"ball_eq'","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[500,8],"def_end_pos":[500,16]}]},{"state_before":"𝓕 : Type u_1\n𝕜 : Type u_2\nα : Type u_3\nι : Type u_4\nκ : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nx : E\n⊢ (𝓝 x).HasBasis (fun ε => 0 < ε) fun ε => ball x ε","state_after":"no goals","tactic":"exact Metric.nhds_basis_ball","premises":[{"full_name":"Metric.nhds_basis_ball","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[805,8],"def_end_pos":[805,23]}]}]}
+{"url":"Mathlib/Analysis/Fourier/Inversion.lean","commit":"","full_name":"Real.tendsto_integral_gaussian_smul","start":[71,0],"end":[102,14],"file_path":"Mathlib/Analysis/Fourier/Inversion.lean","tactics":[{"state_before":"V : Type u_1\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MeasurableSpace V\ninst✝⁴ : BorelSpace V\ninst✝³ : FiniteDimensional ℝ V\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nf : V → E\ninst✝ : CompleteSpace E\nhf : Integrable f volume\nh'f : Integrable (𝓕 f) volume\nv : V\nA : Tendsto (fun c => ∫ (w : V), cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ) • 𝓕 f w) atTop (𝓝 (𝓕⁻ (𝓕 f) v))\n⊢ Tendsto (fun c => ∫ (w : V), ((↑π * ↑c) ^ (↑(finrank ℝ V) / 2) * cexp (-↑π ^ 2 * ↑c * ↑‖v - w‖ ^ 2)) • f w) atTop\n    (𝓝 (𝓕⁻ (𝓕 f) v))","state_after":"V : Type u_1\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MeasurableSpace V\ninst✝⁴ : BorelSpace V\ninst✝³ : FiniteDimensional ℝ V\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nf : V → E\ninst✝ : CompleteSpace E\nhf : Integrable f volume\nh'f : Integrable (𝓕 f) volume\nv : V\nA : Tendsto (fun c => ∫ (w : V), cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ) • 𝓕 f w) atTop (𝓝 (𝓕⁻ (𝓕 f) v))\nB :\n  Tendsto (fun c => ∫ (w : V), 𝓕 (fun w => cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ)) w • f w) atTop\n    (𝓝 (𝓕⁻ (𝓕 f) v))\n⊢ Tendsto (fun c => ∫ (w : V), ((↑π * ↑c) ^ (↑(finrank ℝ V) / 2) * cexp (-↑π ^ 2 * ↑c * ↑‖v - w‖ ^ 2)) • f w) atTop\n    (𝓝 (𝓕⁻ (𝓕 f) v))","tactic":"have B : Tendsto (fun (c : ℝ) ↦ (∫ w : V,\n        𝓕 (fun w ↦ cexp (- c⁻¹ * ‖w‖^2 + 2 * π * I * ⟪v, w⟫)) w • f w)) atTop\n      (𝓝 (𝓕⁻ (𝓕 f) v)) := by\n    apply A.congr'\n    filter_upwards [Ioi_mem_atTop 0] with c (hc : 0 < c)\n    have J : Integrable (fun w ↦ cexp (- c⁻¹ * ‖w‖^2 + 2 * π * I * ⟪v, w⟫)) :=\n      GaussianFourier.integrable_cexp_neg_mul_sq_norm_add (by simpa) _ _\n    simpa using (VectorFourier.integral_fourierIntegral_smul_eq_flip (L := innerₗ V)\n      Real.continuous_fourierChar continuous_inner J hf).symm","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Complex.I","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[231,4],"def_end_pos":[231,5]},{"full_name":"Complex.exp","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[50,4],"def_end_pos":[50,7]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Filter.Ioi_mem_atTop","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[56,8],"def_end_pos":[56,21]},{"full_name":"Filter.Tendsto","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2567,4],"def_end_pos":[2567,11]},{"full_name":"Filter.Tendsto.congr'","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2634,8],"def_end_pos":[2634,22]},{"full_name":"Filter.atTop","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[40,4],"def_end_pos":[40,9]},{"full_name":"Filter.mp_mem","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[159,8],"def_end_pos":[159,14]},{"full_name":"GaussianFourier.integrable_cexp_neg_mul_sq_norm_add","def_path":"Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean","def_pos":[283,8],"def_end_pos":[283,43]},{"full_name":"Inner.inner","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[79,2],"def_end_pos":[79,7]},{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"MeasureTheory.Integrable","def_path":"Mathlib/MeasureTheory/Function/L1Space.lean","def_pos":[389,4],"def_end_pos":[389,14]},{"full_name":"MeasureTheory.MeasureSpace.volume","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[326,2],"def_end_pos":[326,8]},{"full_name":"MeasureTheory.integral","def_path":"Mathlib/MeasureTheory/Integral/Bochner.lean","def_pos":[714,16],"def_end_pos":[714,24]},{"full_name":"Norm.norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[59,2],"def_end_pos":[59,6]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Real.continuous_fourierChar","def_path":"Mathlib/Analysis/Fourier/FourierTransform.lean","def_pos":[326,8],"def_end_pos":[326,30]},{"full_name":"Real.fourierIntegral","def_path":"Mathlib/Analysis/Fourier/FourierTransform.lean","def_pos":[383,4],"def_end_pos":[383,19]},{"full_name":"Real.fourierIntegralInv","def_path":"Mathlib/Analysis/Fourier/FourierTransform.lean","def_pos":[388,4],"def_end_pos":[388,22]},{"full_name":"Real.pi","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","def_pos":[119,28],"def_end_pos":[119,30]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]},{"full_name":"VectorFourier.integral_fourierIntegral_smul_eq_flip","def_path":"Mathlib/Analysis/Fourier/FourierTransform.lean","def_pos":[224,8],"def_end_pos":[224,45]},{"full_name":"continuous_inner","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[1989,8],"def_end_pos":[1989,24]},{"full_name":"innerₗ","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[1537,4],"def_end_pos":[1537,10]},{"full_name":"nhds","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[113,16],"def_end_pos":[113,20]}]},{"state_before":"V : Type u_1\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MeasurableSpace V\ninst✝⁴ : BorelSpace V\ninst✝³ : FiniteDimensional ℝ V\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nf : V → E\ninst✝ : CompleteSpace E\nhf : Integrable f volume\nh'f : Integrable (𝓕 f) volume\nv : V\nA : Tendsto (fun c => ∫ (w : V), cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ) • 𝓕 f w) atTop (𝓝 (𝓕⁻ (𝓕 f) v))\nB :\n  Tendsto (fun c => ∫ (w : V), 𝓕 (fun w => cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ)) w • f w) atTop\n    (𝓝 (𝓕⁻ (𝓕 f) v))\n⊢ Tendsto (fun c => ∫ (w : V), ((↑π * ↑c) ^ (↑(finrank ℝ V) / 2) * cexp (-↑π ^ 2 * ↑c * ↑‖v - w‖ ^ 2)) • f w) atTop\n    (𝓝 (𝓕⁻ (𝓕 f) v))","state_after":"V : Type u_1\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MeasurableSpace V\ninst✝⁴ : BorelSpace V\ninst✝³ : FiniteDimensional ℝ V\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nf : V → E\ninst✝ : CompleteSpace E\nhf : Integrable f volume\nh'f : Integrable (𝓕 f) volume\nv : V\nA : Tendsto (fun c => ∫ (w : V), cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ) • 𝓕 f w) atTop (𝓝 (𝓕⁻ (𝓕 f) v))\nB :\n  Tendsto (fun c => ∫ (w : V), 𝓕 (fun w => cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ)) w • f w) atTop\n    (𝓝 (𝓕⁻ (𝓕 f) v))\n⊢ (fun c => ∫ (w : V), 𝓕 (fun w => cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ)) w • f w) =ᶠ[atTop] fun c =>\n    ∫ (w : V), ((↑π * ↑c) ^ (↑(finrank ℝ V) / 2) * cexp (-↑π ^ 2 * ↑c * ↑‖v - w‖ ^ 2)) • f w","tactic":"apply B.congr'","premises":[{"full_name":"Filter.Tendsto.congr'","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2634,8],"def_end_pos":[2634,22]}]},{"state_before":"V : Type u_1\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MeasurableSpace V\ninst✝⁴ : BorelSpace V\ninst✝³ : FiniteDimensional ℝ V\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nf : V → E\ninst✝ : CompleteSpace E\nhf : Integrable f volume\nh'f : Integrable (𝓕 f) volume\nv : V\nA : Tendsto (fun c => ∫ (w : V), cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ) • 𝓕 f w) atTop (𝓝 (𝓕⁻ (𝓕 f) v))\nB :\n  Tendsto (fun c => ∫ (w : V), 𝓕 (fun w => cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ)) w • f w) atTop\n    (𝓝 (𝓕⁻ (𝓕 f) v))\n⊢ (fun c => ∫ (w : V), 𝓕 (fun w => cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ)) w • f w) =ᶠ[atTop] fun c =>\n    ∫ (w : V), ((↑π * ↑c) ^ (↑(finrank ℝ V) / 2) * cexp (-↑π ^ 2 * ↑c * ↑‖v - w‖ ^ 2)) • f w","state_after":"case h\nV : Type u_1\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MeasurableSpace V\ninst✝⁴ : BorelSpace V\ninst✝³ : FiniteDimensional ℝ V\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nf : V → E\ninst✝ : CompleteSpace E\nhf : Integrable f volume\nh'f : Integrable (𝓕 f) volume\nv : V\nA : Tendsto (fun c => ∫ (w : V), cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ) • 𝓕 f w) atTop (𝓝 (𝓕⁻ (𝓕 f) v))\nB :\n  Tendsto (fun c => ∫ (w : V), 𝓕 (fun w => cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ)) w • f w) atTop\n    (𝓝 (𝓕⁻ (𝓕 f) v))\nc : ℝ\nhc : 0 < c\n⊢ ∫ (w : V), 𝓕 (fun w => cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ)) w • f w =\n    ∫ (w : V), ((↑π * ↑c) ^ (↑(finrank ℝ V) / 2) * cexp (-↑π ^ 2 * ↑c * ↑‖v - w‖ ^ 2)) • f w","tactic":"filter_upwards [Ioi_mem_atTop 0] with c (hc : 0 < c)","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Filter.Ioi_mem_atTop","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[56,8],"def_end_pos":[56,21]},{"full_name":"Filter.mp_mem","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[159,8],"def_end_pos":[159,14]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]}]},{"state_before":"case h\nV : Type u_1\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MeasurableSpace V\ninst✝⁴ : BorelSpace V\ninst✝³ : FiniteDimensional ℝ V\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nf : V → E\ninst✝ : CompleteSpace E\nhf : Integrable f volume\nh'f : Integrable (𝓕 f) volume\nv : V\nA : Tendsto (fun c => ∫ (w : V), cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ) • 𝓕 f w) atTop (𝓝 (𝓕⁻ (𝓕 f) v))\nB :\n  Tendsto (fun c => ∫ (w : V), 𝓕 (fun w => cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ)) w • f w) atTop\n    (𝓝 (𝓕⁻ (𝓕 f) v))\nc : ℝ\nhc : 0 < c\n⊢ ∫ (w : V), 𝓕 (fun w => cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ)) w • f w =\n    ∫ (w : V), ((↑π * ↑c) ^ (↑(finrank ℝ V) / 2) * cexp (-↑π ^ 2 * ↑c * ↑‖v - w‖ ^ 2)) • f w","state_after":"case h.e_f.h\nV : Type u_1\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MeasurableSpace V\ninst✝⁴ : BorelSpace V\ninst✝³ : FiniteDimensional ℝ V\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nf : V → E\ninst✝ : CompleteSpace E\nhf : Integrable f volume\nh'f : Integrable (𝓕 f) volume\nv : V\nA : Tendsto (fun c => ∫ (w : V), cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ) • 𝓕 f w) atTop (𝓝 (𝓕⁻ (𝓕 f) v))\nB :\n  Tendsto (fun c => ∫ (w : V), 𝓕 (fun w => cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ)) w • f w) atTop\n    (𝓝 (𝓕⁻ (𝓕 f) v))\nc : ℝ\nhc : 0 < c\nw : V\n⊢ 𝓕 (fun w => cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ)) w • f w =\n    ((↑π * ↑c) ^ (↑(finrank ℝ V) / 2) * cexp (-↑π ^ 2 * ↑c * ↑‖v - w‖ ^ 2)) • f w","tactic":"congr with w","premises":[]},{"state_before":"case h.e_f.h\nV : Type u_1\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MeasurableSpace V\ninst✝⁴ : BorelSpace V\ninst✝³ : FiniteDimensional ℝ V\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nf : V → E\ninst✝ : CompleteSpace E\nhf : Integrable f volume\nh'f : Integrable (𝓕 f) volume\nv : V\nA : Tendsto (fun c => ∫ (w : V), cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ) • 𝓕 f w) atTop (𝓝 (𝓕⁻ (𝓕 f) v))\nB :\n  Tendsto (fun c => ∫ (w : V), 𝓕 (fun w => cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ)) w • f w) atTop\n    (𝓝 (𝓕⁻ (𝓕 f) v))\nc : ℝ\nhc : 0 < c\nw : V\n⊢ 𝓕 (fun w => cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ)) w • f w =\n    ((↑π * ↑c) ^ (↑(finrank ℝ V) / 2) * cexp (-↑π ^ 2 * ↑c * ↑‖v - w‖ ^ 2)) • f w","state_after":"case h.e_f.h\nV : Type u_1\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MeasurableSpace V\ninst✝⁴ : BorelSpace V\ninst✝³ : FiniteDimensional ℝ V\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nf : V → E\ninst✝ : CompleteSpace E\nhf : Integrable f volume\nh'f : Integrable (𝓕 f) volume\nv : V\nA : Tendsto (fun c => ∫ (w : V), cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ) • 𝓕 f w) atTop (𝓝 (𝓕⁻ (𝓕 f) v))\nB :\n  Tendsto (fun c => ∫ (w : V), 𝓕 (fun w => cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ)) w • f w) atTop\n    (𝓝 (𝓕⁻ (𝓕 f) v))\nc : ℝ\nhc : 0 < c\nw : V\n⊢ ((↑π / ↑c⁻¹) ^ (↑(finrank ℝ V) / 2) * cexp (-↑π ^ 2 * ↑‖v - w‖ ^ 2 / ↑c⁻¹)) • f w =\n    ((↑π * ↑c) ^ (↑(finrank ℝ V) / 2) * cexp (-↑π ^ 2 * ↑c * ↑‖v - w‖ ^ 2)) • f w","tactic":"rw [fourierIntegral_gaussian_innerProductSpace' (by simpa)]","premises":[{"full_name":"fourierIntegral_gaussian_innerProductSpace'","def_path":"Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean","def_pos":[355,8],"def_end_pos":[355,58]}]},{"state_before":"case h.e_f.h\nV : Type u_1\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MeasurableSpace V\ninst✝⁴ : BorelSpace V\ninst✝³ : FiniteDimensional ℝ V\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nf : V → E\ninst✝ : CompleteSpace E\nhf : Integrable f volume\nh'f : Integrable (𝓕 f) volume\nv : V\nA : Tendsto (fun c => ∫ (w : V), cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ) • 𝓕 f w) atTop (𝓝 (𝓕⁻ (𝓕 f) v))\nB :\n  Tendsto (fun c => ∫ (w : V), 𝓕 (fun w => cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ)) w • f w) atTop\n    (𝓝 (𝓕⁻ (𝓕 f) v))\nc : ℝ\nhc : 0 < c\nw : V\n⊢ ((↑π / ↑c⁻¹) ^ (↑(finrank ℝ V) / 2) * cexp (-↑π ^ 2 * ↑‖v - w‖ ^ 2 / ↑c⁻¹)) • f w =\n    ((↑π * ↑c) ^ (↑(finrank ℝ V) / 2) * cexp (-↑π ^ 2 * ↑c * ↑‖v - w‖ ^ 2)) • f w","state_after":"case h.e_f.h.e_a.e_a.e_a\nV : Type u_1\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MeasurableSpace V\ninst✝⁴ : BorelSpace V\ninst✝³ : FiniteDimensional ℝ V\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nf : V → E\ninst✝ : CompleteSpace E\nhf : Integrable f volume\nh'f : Integrable (𝓕 f) volume\nv : V\nA : Tendsto (fun c => ∫ (w : V), cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ) • 𝓕 f w) atTop (𝓝 (𝓕⁻ (𝓕 f) v))\nB :\n  Tendsto (fun c => ∫ (w : V), 𝓕 (fun w => cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ)) w • f w) atTop\n    (𝓝 (𝓕⁻ (𝓕 f) v))\nc : ℝ\nhc : 0 < c\nw : V\n⊢ ↑π / ↑c⁻¹ = ↑π * ↑c\n\ncase h.e_f.h.e_a.e_a.e_z\nV : Type u_1\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MeasurableSpace V\ninst✝⁴ : BorelSpace V\ninst✝³ : FiniteDimensional ℝ V\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nf : V → E\ninst✝ : CompleteSpace E\nhf : Integrable f volume\nh'f : Integrable (𝓕 f) volume\nv : V\nA : Tendsto (fun c => ∫ (w : V), cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ) • 𝓕 f w) atTop (𝓝 (𝓕⁻ (𝓕 f) v))\nB :\n  Tendsto (fun c => ∫ (w : V), 𝓕 (fun w => cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ)) w • f w) atTop\n    (𝓝 (𝓕⁻ (𝓕 f) v))\nc : ℝ\nhc : 0 < c\nw : V\n⊢ -↑π ^ 2 * ↑‖v - w‖ ^ 2 / ↑c⁻¹ = -↑π ^ 2 * ↑c * ↑‖v - w‖ ^ 2","tactic":"congr","premises":[]}]}
+{"url":"Mathlib/Topology/Algebra/Group/Basic.lean","commit":"","full_name":"compact_covered_by_add_left_translates","start":[1479,0],"end":[1493,71],"file_path":"Mathlib/Topology/Algebra/Group/Basic.lean","tactics":[{"state_before":"case intro\nG : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : TopologicalGroup G\nK V : Set G\nhK : IsCompact K\nhV : (interior V).Nonempty\nt : Finset G\nht : K ⊆ ⋃ x ∈ t, interior ((fun x_1 => x * x_1) ⁻¹' V)\n⊢ ∃ t, K ⊆ ⋃ g ∈ t, (fun x => g * x) ⁻¹' V","state_after":"no goals","tactic":"exact ⟨t, Subset.trans ht <| iUnion₂_mono fun g _ => interior_subset⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Set.Subset.trans","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[296,8],"def_end_pos":[296,20]},{"full_name":"Set.iUnion₂_mono","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[294,8],"def_end_pos":[294,20]},{"full_name":"interior_subset","def_path":"Mathlib/Topology/Basic.lean","def_pos":[222,8],"def_end_pos":[222,23]}]}]}
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+{"url":"Mathlib/Data/Option/NAry.lean","commit":"","full_name":"Option.map₂_swap","start":[75,0],"end":[76,78],"file_path":"Mathlib/Data/Option/NAry.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nf✝ : α → β → γ\na✝ : Option α\nb✝ : Option β\nc : Option γ\nf : α → β → γ\na : Option α\nb : Option β\n⊢ map₂ f a b = map₂ (fun a b => f b a) b a","state_after":"no goals","tactic":"cases a <;> cases b <;> rfl","premises":[]}]}
+{"url":"Mathlib/MeasureTheory/Integral/SetToL1.lean","commit":"","full_name":"MeasureTheory.continuousOn_setToFun_of_dominated","start":[1618,0],"end":[1627,51],"file_path":"Mathlib/MeasureTheory/Integral/SetToL1.lean","tactics":[{"state_before":"α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : NormedAddCommGroup F'\ninst✝⁴ : NormedSpace ℝ F'\ninst✝³ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf g : α → E\nX : Type u_7\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nhT : DominatedFinMeasAdditive μ T C\nfs : X → α → E\nbound : α → ℝ\ns : Set X\nhfs_meas : ∀ x ∈ s, AEStronglyMeasurable (fs x) μ\nh_bound : ∀ x ∈ s, ∀ᵐ (a : α) ∂μ, ‖fs x a‖ ≤ bound a\nbound_integrable : Integrable bound μ\nh_cont : ∀ᵐ (a : α) ∂μ, ContinuousOn (fun x => fs x a) s\n⊢ ContinuousOn (fun x => setToFun μ T hT (fs x)) s","state_after":"α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : NormedAddCommGroup F'\ninst✝⁴ : NormedSpace ℝ F'\ninst✝³ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf g : α → E\nX : Type u_7\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nhT : DominatedFinMeasAdditive μ T C\nfs : X → α → E\nbound : α → ℝ\ns : Set X\nhfs_meas : ∀ x ∈ s, AEStronglyMeasurable (fs x) μ\nh_bound : ∀ x ∈ s, ∀ᵐ (a : α) ∂μ, ‖fs x a‖ ≤ bound a\nbound_integrable : Integrable bound μ\nh_cont : ∀ᵐ (a : α) ∂μ, ContinuousOn (fun x => fs x a) s\nx : X\nhx : x ∈ s\n⊢ ContinuousWithinAt (fun x => setToFun μ T hT (fs x)) s x","tactic":"intro x hx","premises":[]},{"state_before":"α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : NormedAddCommGroup F'\ninst✝⁴ : NormedSpace ℝ F'\ninst✝³ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf g : α → E\nX : Type u_7\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nhT : DominatedFinMeasAdditive μ T C\nfs : X → α → E\nbound : α → ℝ\ns : Set X\nhfs_meas : ∀ x ∈ s, AEStronglyMeasurable (fs x) μ\nh_bound : ∀ x ∈ s, ∀ᵐ (a : α) ∂μ, ‖fs x a‖ ≤ bound a\nbound_integrable : Integrable bound μ\nh_cont : ∀ᵐ (a : α) ∂μ, ContinuousOn (fun x => fs x a) s\nx : X\nhx : x ∈ s\n⊢ ContinuousWithinAt (fun x => setToFun μ T hT (fs x)) s x","state_after":"case refine_1\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : NormedAddCommGroup F'\ninst✝⁴ : NormedSpace ℝ F'\ninst✝³ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf g : α → E\nX : Type u_7\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nhT : DominatedFinMeasAdditive μ T C\nfs : X → α → E\nbound : α → ℝ\ns : Set X\nhfs_meas : ∀ x ∈ s, AEStronglyMeasurable (fs x) μ\nh_bound : ∀ x ∈ s, ∀ᵐ (a : α) ∂μ, ‖fs x a‖ ≤ bound a\nbound_integrable : Integrable bound μ\nh_cont : ∀ᵐ (a : α) ∂μ, ContinuousOn (fun x => fs x a) s\nx : X\nhx : x ∈ s\n⊢ ∀ᶠ (x : X) in 𝓝[s] x, AEStronglyMeasurable (fs x) μ\n\ncase refine_2\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : NormedAddCommGroup F'\ninst✝⁴ : NormedSpace ℝ F'\ninst✝³ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf g : α → E\nX : Type u_7\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nhT : DominatedFinMeasAdditive μ T C\nfs : X → α → E\nbound : α → ℝ\ns : Set X\nhfs_meas : ∀ x ∈ s, AEStronglyMeasurable (fs x) μ\nh_bound : ∀ x ∈ s, ∀ᵐ (a : α) ∂μ, ‖fs x a‖ ≤ bound a\nbound_integrable : Integrable bound μ\nh_cont : ∀ᵐ (a : α) ∂μ, ContinuousOn (fun x => fs x a) s\nx : X\nhx : x ∈ s\n⊢ ∀ᶠ (x : X) in 𝓝[s] x, ∀ᵐ (a : α) ∂μ, ‖fs x a‖ ≤ bound a\n\ncase refine_3\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : NormedAddCommGroup F'\ninst✝⁴ : NormedSpace ℝ F'\ninst✝³ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf g : α → E\nX : Type u_7\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nhT : DominatedFinMeasAdditive μ T C\nfs : X → α → E\nbound : α → ℝ\ns : Set X\nhfs_meas : ∀ x ∈ s, AEStronglyMeasurable (fs x) μ\nh_bound : ∀ x ∈ s, ∀ᵐ (a : α) ∂μ, ‖fs x a‖ ≤ bound a\nbound_integrable : Integrable bound μ\nh_cont : ∀ᵐ (a : α) ∂μ, ContinuousOn (fun x => fs x a) s\nx : X\nhx : x ∈ s\n⊢ ∀ᵐ (a : α) ∂μ, ContinuousWithinAt (fun x => fs x a) s x","tactic":"refine continuousWithinAt_setToFun_of_dominated hT ?_ ?_ bound_integrable ?_","premises":[{"full_name":"MeasureTheory.continuousWithinAt_setToFun_of_dominated","def_path":"Mathlib/MeasureTheory/Integral/SetToL1.lean","def_pos":[1603,8],"def_end_pos":[1603,48]}]}]}
+{"url":"Mathlib/Order/Filter/Basic.lean","commit":"","full_name":"Filter.mem_inf_principal","start":[896,0],"end":[897,83],"file_path":"Mathlib/Order/Filter/Basic.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nδ : Type u_1\nι : Sort x\nf✝ g : Filter α\ns✝ t✝ : Set α\nf : Filter α\ns t : Set α\n⊢ s ∈ f ⊓ 𝓟 t ↔ {x | x ∈ t → x ∈ s} ∈ f","state_after":"no goals","tactic":"simp only [mem_inf_principal', imp_iff_not_or, setOf_or, compl_def, setOf_mem_eq]","premises":[{"full_name":"Filter.mem_inf_principal'","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[892,8],"def_end_pos":[892,26]},{"full_name":"Set.compl_def","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1290,8],"def_end_pos":[1290,17]},{"full_name":"Set.setOf_mem_eq","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[215,8],"def_end_pos":[215,20]},{"full_name":"Set.setOf_or","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[243,8],"def_end_pos":[243,16]},{"full_name":"imp_iff_not_or","def_path":"Mathlib/Logic/Basic.lean","def_pos":[286,8],"def_end_pos":[286,22]}]}]}
+{"url":"Mathlib/GroupTheory/OrderOfElement.lean","commit":"","full_name":"infinite_not_isOfFinOrder","start":[478,0],"end":[494,69],"file_path":"Mathlib/GroupTheory/OrderOfElement.lean","tactics":[{"state_before":"G : Type u_1\nH : Type u_2\nA : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝ : LeftCancelMonoid G\nx✝ y a : G\nm n : ℕ\nx : G\nh : ¬IsOfFinOrder x\n⊢ {y | ¬IsOfFinOrder y}.Infinite","state_after":"G : Type u_1\nH : Type u_2\nA : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝ : LeftCancelMonoid G\nx✝ y a : G\nm n : ℕ\nx : G\nh : ¬IsOfFinOrder x\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\n⊢ {y | ¬IsOfFinOrder y}.Infinite","tactic":"let s := { n | 0 < n }.image fun n : ℕ => x ^ n","premises":[{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]},{"full_name":"Set.image","def_path":"Mathlib/Init/Set.lean","def_pos":[208,4],"def_end_pos":[208,9]},{"full_name":"setOf","def_path":"Mathlib/Init/Set.lean","def_pos":[56,4],"def_end_pos":[56,9]}]},{"state_before":"G : Type u_1\nH : Type u_2\nA : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝ : LeftCancelMonoid G\nx✝ y a : G\nm n : ℕ\nx : G\nh : ¬IsOfFinOrder x\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\n⊢ {y | ¬IsOfFinOrder y}.Infinite","state_after":"G : Type u_1\nH : Type u_2\nA : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝ : LeftCancelMonoid G\nx✝ y a : G\nm n : ℕ\nx : G\nh : ¬IsOfFinOrder x\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nhs : s ⊆ {y | ¬IsOfFinOrder y}\n⊢ {y | ¬IsOfFinOrder y}.Infinite","tactic":"have hs : s ⊆ { y : G | ¬IsOfFinOrder y } := by\n    rintro - ⟨n, hn : 0 < n, rfl⟩ (contra : IsOfFinOrder (x ^ n))\n    apply h\n    rw [isOfFinOrder_iff_pow_eq_one] at contra ⊢\n    obtain ⟨m, hm, hm'⟩ := contra\n    exact ⟨n * m, mul_pos hn hm, by rwa [pow_mul]⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"HasSubset.Subset","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[384,2],"def_end_pos":[384,8]},{"full_name":"IsOfFinOrder","def_path":"Mathlib/GroupTheory/OrderOfElement.lean","def_pos":[47,4],"def_end_pos":[47,16]},{"full_name":"Not","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[224,4],"def_end_pos":[224,7]},{"full_name":"isOfFinOrder_iff_pow_eq_one","def_path":"Mathlib/GroupTheory/OrderOfElement.lean","def_pos":[57,8],"def_end_pos":[57,35]},{"full_name":"pow_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[605,31],"def_end_pos":[605,38]},{"full_name":"setOf","def_path":"Mathlib/Init/Set.lean","def_pos":[56,4],"def_end_pos":[56,9]}]},{"state_before":"G : Type u_1\nH : Type u_2\nA : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝ : LeftCancelMonoid G\nx✝ y a : G\nm n : ℕ\nx : G\nh : ¬IsOfFinOrder x\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nhs : s ⊆ {y | ¬IsOfFinOrder y}\n⊢ {y | ¬IsOfFinOrder y}.Infinite","state_after":"G : Type u_1\nH : Type u_2\nA : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝ : LeftCancelMonoid G\nx✝ y a : G\nm n : ℕ\nx : G\nh : ¬IsOfFinOrder x\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nhs : s ⊆ {y | ¬IsOfFinOrder y}\n⊢ s.Infinite","tactic":"suffices s.Infinite by exact this.mono hs","premises":[{"full_name":"Set.Infinite","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[120,14],"def_end_pos":[120,22]},{"full_name":"Set.Infinite.mono","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[663,16],"def_end_pos":[663,29]}]},{"state_before":"G : Type u_1\nH : Type u_2\nA : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝ : LeftCancelMonoid G\nx✝ y a : G\nm n : ℕ\nx : G\nh : ¬IsOfFinOrder x\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nhs : s ⊆ {y | ¬IsOfFinOrder y}\n⊢ s.Infinite","state_after":"G : Type u_1\nH : Type u_2\nA : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝ : LeftCancelMonoid G\nx✝ y a : G\nm n : ℕ\nx : G\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nhs : s ⊆ {y | ¬IsOfFinOrder y}\nh : ¬s.Infinite\n⊢ IsOfFinOrder x","tactic":"contrapose! h","premises":[{"full_name":"Mathlib.Tactic.Contrapose.mtr","def_path":"Mathlib/Tactic/Contrapose.lean","def_pos":[24,6],"def_end_pos":[24,9]}]},{"state_before":"G : Type u_1\nH : Type u_2\nA : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝ : LeftCancelMonoid G\nx✝ y a : G\nm n : ℕ\nx : G\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nhs : s ⊆ {y | ¬IsOfFinOrder y}\nh : ¬s.Infinite\n⊢ IsOfFinOrder x","state_after":"G : Type u_1\nH : Type u_2\nA : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝ : LeftCancelMonoid G\nx✝ y a : G\nm n : ℕ\nx : G\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nhs : s ⊆ {y | ¬IsOfFinOrder y}\nh : ¬s.Infinite\nthis : ¬Injective fun n => x ^ n\n⊢ IsOfFinOrder x","tactic":"have : ¬Injective fun n : ℕ => x ^ n := by\n    have := Set.not_injOn_infinite_finite_image (Set.Ioi_infinite 0) (Set.not_infinite.mp h)\n    contrapose! this\n    exact Set.injOn_of_injective this","premises":[{"full_name":"Function.Injective","def_path":"Mathlib/Logic/Function/Defs.lean","def_pos":[101,4],"def_end_pos":[101,13]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Mathlib.Tactic.Contrapose.mtr","def_path":"Mathlib/Tactic/Contrapose.lean","def_pos":[24,6],"def_end_pos":[24,9]},{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]},{"full_name":"Not","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[224,4],"def_end_pos":[224,7]},{"full_name":"Set.Ioi_infinite","def_path":"Mathlib/Order/Interval/Set/Infinite.lean","def_pos":[77,8],"def_end_pos":[77,20]},{"full_name":"Set.injOn_of_injective","def_path":"Mathlib/Data/Set/Function.lean","def_pos":[571,8],"def_end_pos":[571,26]},{"full_name":"Set.not_infinite","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[124,8],"def_end_pos":[124,20]},{"full_name":"Set.not_injOn_infinite_finite_image","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[1265,8],"def_end_pos":[1265,39]}]},{"state_before":"G : Type u_1\nH : Type u_2\nA : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝ : LeftCancelMonoid G\nx✝ y a : G\nm n : ℕ\nx : G\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nhs : s ⊆ {y | ¬IsOfFinOrder y}\nh : ¬s.Infinite\nthis : ¬Injective fun n => x ^ n\n⊢ IsOfFinOrder x","state_after":"no goals","tactic":"rwa [injective_pow_iff_not_isOfFinOrder, Classical.not_not] at this","premises":[{"full_name":"Classical.not_not","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[135,16],"def_end_pos":[135,23]},{"full_name":"injective_pow_iff_not_isOfFinOrder","def_path":"Mathlib/GroupTheory/OrderOfElement.lean","def_pos":[467,6],"def_end_pos":[467,40]}]}]}
+{"url":"Mathlib/Algebra/Group/Subgroup/Pointwise.lean","commit":"","full_name":"AddSubgroup.inf_add_assoc","start":[223,0],"end":[235,36],"file_path":"Mathlib/Algebra/Group/Subgroup/Pointwise.lean","tactics":[{"state_before":"α : Type u_1\nG : Type u_2\nA✝ : Type u_3\nS : Type u_4\ninst✝¹ : Group G\ninst✝ : AddGroup A✝\ns : Set G\nA B C : Subgroup G\nh : C ≤ A\n⊢ ↑(A ⊓ B) * ↑C = ↑A ∩ (↑B * ↑C)","state_after":"case h\nα : Type u_1\nG : Type u_2\nA✝ : Type u_3\nS : Type u_4\ninst✝¹ : Group G\ninst✝ : AddGroup A✝\ns : Set G\nA B C : Subgroup G\nh : C ≤ A\nx✝ : G\n⊢ x✝ ∈ ↑(A ⊓ B) * ↑C ↔ x✝ ∈ ↑A ∩ (↑B * ↑C)","tactic":"ext","premises":[]},{"state_before":"case h\nα : Type u_1\nG : Type u_2\nA✝ : Type u_3\nS : Type u_4\ninst✝¹ : Group G\ninst✝ : AddGroup A✝\ns : Set G\nA B C : Subgroup G\nh : C ≤ A\nx✝ : G\n⊢ x✝ ∈ ↑(A ⊓ B) * ↑C ↔ x✝ ∈ ↑A ∩ (↑B * ↑C)","state_after":"case h\nα : Type u_1\nG : Type u_2\nA✝ : Type u_3\nS : Type u_4\ninst✝¹ : Group G\ninst✝ : AddGroup A✝\ns : Set G\nA B C : Subgroup G\nh : C ≤ A\nx✝ : G\n⊢ (∃ x, (x ∈ ↑A ∧ x ∈ ↑B) ∧ ∃ y ∈ ↑C, x * y = x✝) ↔ x✝ ∈ ↑A ∧ ∃ x ∈ ↑B, ∃ y ∈ ↑C, x * y = x✝","tactic":"simp only [coe_inf, Set.mem_mul, Set.mem_inter_iff]","premises":[{"full_name":"Set.mem_inter_iff","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[718,8],"def_end_pos":[718,21]},{"full_name":"Set.mem_mul","def_path":"Mathlib/Data/Set/Pointwise/Basic.lean","def_pos":[273,8],"def_end_pos":[273,15]},{"full_name":"Subgroup.coe_inf","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[746,8],"def_end_pos":[746,15]}]},{"state_before":"case h\nα : Type u_1\nG : Type u_2\nA✝ : Type u_3\nS : Type u_4\ninst✝¹ : Group G\ninst✝ : AddGroup A✝\ns : Set G\nA B C : Subgroup G\nh : C ≤ A\nx✝ : G\n⊢ (∃ x, (x ∈ ↑A ∧ x ∈ ↑B) ∧ ∃ y ∈ ↑C, x * y = x✝) ↔ x✝ ∈ ↑A ∧ ∃ x ∈ ↑B, ∃ y ∈ ↑C, x * y = x✝","state_after":"case h.mp\nα : Type u_1\nG : Type u_2\nA✝ : Type u_3\nS : Type u_4\ninst✝¹ : Group G\ninst✝ : AddGroup A✝\ns : Set G\nA B C : Subgroup G\nh : C ≤ A\nx✝ : G\n⊢ (∃ x, (x ∈ ↑A ∧ x ∈ ↑B) ∧ ∃ y ∈ ↑C, x * y = x✝) → x✝ ∈ ↑A ∧ ∃ x ∈ ↑B, ∃ y ∈ ↑C, x * y = x✝\n\ncase h.mpr\nα : Type u_1\nG : Type u_2\nA✝ : Type u_3\nS : Type u_4\ninst✝¹ : Group G\ninst✝ : AddGroup A✝\ns : Set G\nA B C : Subgroup G\nh : C ≤ A\nx✝ : G\n⊢ (x✝ ∈ ↑A ∧ ∃ x ∈ ↑B, ∃ y ∈ ↑C, x * y = x✝) → ∃ x, (x ∈ ↑A ∧ x ∈ ↑B) ∧ ∃ y ∈ ↑C, x * y = x✝","tactic":"constructor","premises":[]},{"state_before":"case h.mpr\nα : Type u_1\nG : Type u_2\nA✝ : Type u_3\nS : Type u_4\ninst✝¹ : Group G\ninst✝ : AddGroup A✝\ns : Set G\nA B C : Subgroup G\nh : C ≤ A\nx✝ : G\n⊢ (x✝ ∈ ↑A ∧ ∃ x ∈ ↑B, ∃ y ∈ ↑C, x * y = x✝) → ∃ x, (x ∈ ↑A ∧ x ∈ ↑B) ∧ ∃ y ∈ ↑C, x * y = x✝","state_after":"case h.mpr.intro.intro.intro.intro.intro\nα : Type u_1\nG : Type u_2\nA✝ : Type u_3\nS : Type u_4\ninst✝¹ : Group G\ninst✝ : AddGroup A✝\ns : Set G\nA B C : Subgroup G\nh : C ≤ A\ny : G\nhy : y ∈ ↑B\nz : G\nhz : z ∈ ↑C\nhyz : y * z ∈ ↑A\n⊢ ∃ x, (x ∈ ↑A ∧ x ∈ ↑B) ∧ ∃ y_1 ∈ ↑C, x * y_1 = y * z","tactic":"rintro ⟨hyz, y, hy, z, hz, rfl⟩","premises":[]},{"state_before":"case h.mpr.intro.intro.intro.intro.intro\nα : Type u_1\nG : Type u_2\nA✝ : Type u_3\nS : Type u_4\ninst✝¹ : Group G\ninst✝ : AddGroup A✝\ns : Set G\nA B C : Subgroup G\nh : C ≤ A\ny : G\nhy : y ∈ ↑B\nz : G\nhz : z ∈ ↑C\nhyz : y * z ∈ ↑A\n⊢ ∃ x, (x ∈ ↑A ∧ x ∈ ↑B) ∧ ∃ y_1 ∈ ↑C, x * y_1 = y * z","state_after":"case h.mpr.intro.intro.intro.intro.intro\nα : Type u_1\nG : Type u_2\nA✝ : Type u_3\nS : Type u_4\ninst✝¹ : Group G\ninst✝ : AddGroup A✝\ns : Set G\nA B C : Subgroup G\nh : C ≤ A\ny : G\nhy : y ∈ ↑B\nz : G\nhz : z ∈ ↑C\nhyz : y * z ∈ ↑A\n⊢ y ∈ ↑A","tactic":"refine ⟨y, ⟨?_, hy⟩, z, hz, rfl⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"case h.mpr.intro.intro.intro.intro.intro\nα : Type u_1\nG : Type u_2\nA✝ : Type u_3\nS : Type u_4\ninst✝¹ : Group G\ninst✝ : AddGroup A✝\ns : Set G\nA B C : Subgroup G\nh : C ≤ A\ny : G\nhy : y ∈ ↑B\nz : G\nhz : z ∈ ↑C\nhyz : y * z ∈ ↑A\n⊢ y ∈ ↑A","state_after":"case h.mpr.intro.intro.intro.intro.intro\nα : Type u_1\nG : Type u_2\nA✝ : Type u_3\nS : Type u_4\ninst✝¹ : Group G\ninst✝ : AddGroup A✝\ns : Set G\nA B C : Subgroup G\nh : C ≤ A\ny : G\nhy : y ∈ ↑B\nz : G\nhz : z ∈ ↑C\nhyz : y * z ∈ ↑A\n⊢ y * z * z⁻¹ ∈ A","tactic":"suffices y * z * z⁻¹ ∈ A by simpa","premises":[{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]}]},{"state_before":"case h.mpr.intro.intro.intro.intro.intro\nα : Type u_1\nG : Type u_2\nA✝ : Type u_3\nS : Type u_4\ninst✝¹ : Group G\ninst✝ : AddGroup A✝\ns : Set G\nA B C : Subgroup G\nh : C ≤ A\ny : G\nhy : y ∈ ↑B\nz : G\nhz : z ∈ ↑C\nhyz : y * z ∈ ↑A\n⊢ y * z * z⁻¹ ∈ A","state_after":"no goals","tactic":"exact mul_mem hyz (inv_mem (h hz))","premises":[{"full_name":"InvMemClass.inv_mem","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[97,2],"def_end_pos":[97,9]},{"full_name":"MulMemClass.mul_mem","def_path":"Mathlib/Algebra/Group/Subsemigroup/Basic.lean","def_pos":[62,2],"def_end_pos":[62,9]}]}]}
+{"url":"Mathlib/Algebra/Algebra/Equiv.lean","commit":"","full_name":"AlgEquiv.arrowCongr_refl","start":[410,0],"end":[413,5],"file_path":"Mathlib/Algebra/Algebra/Equiv.lean","tactics":[{"state_before":"R : Type uR\nA₁ : Type uA₁\nA₂ : Type uA₂\nA₃ : Type uA₃\nA₁' : Type uA₁'\nA₂' : Type uA₂'\nA₃' : Type uA₃'\ninst✝¹² : CommSemiring R\ninst✝¹¹ : Semiring A₁\ninst✝¹⁰ : Semiring A₂\ninst✝⁹ : Semiring A₃\ninst✝⁸ : Semiring A₁'\ninst✝⁷ : Semiring A₂'\ninst✝⁶ : Semiring A₃'\ninst✝⁵ : Algebra R A₁\ninst✝⁴ : Algebra R A₂\ninst✝³ : Algebra R A₃\ninst✝² : Algebra R A₁'\ninst✝¹ : Algebra R A₂'\ninst✝ : Algebra R A₃'\ne : A₁ ≃ₐ[R] A₂\n⊢ refl.arrowCongr refl = Equiv.refl (A₁ →ₐ[R] A₂)","state_after":"case H.H\nR : Type uR\nA₁ : Type uA₁\nA₂ : Type uA₂\nA₃ : Type uA₃\nA₁' : Type uA₁'\nA₂' : Type uA₂'\nA₃' : Type uA₃'\ninst✝¹² : CommSemiring R\ninst✝¹¹ : Semiring A₁\ninst✝¹⁰ : Semiring A₂\ninst✝⁹ : Semiring A₃\ninst✝⁸ : Semiring A₁'\ninst✝⁷ : Semiring A₂'\ninst✝⁶ : Semiring A₃'\ninst✝⁵ : Algebra R A₁\ninst✝⁴ : Algebra R A₂\ninst✝³ : Algebra R A₃\ninst✝² : Algebra R A₁'\ninst✝¹ : Algebra R A₂'\ninst✝ : Algebra R A₃'\ne : A₁ ≃ₐ[R] A₂\nx✝¹ : A₁ →ₐ[R] A₂\nx✝ : A₁\n⊢ ((refl.arrowCongr refl) x✝¹) x✝ = ((Equiv.refl (A₁ →ₐ[R] A₂)) x✝¹) x✝","tactic":"ext","premises":[]},{"state_before":"case H.H\nR : Type uR\nA₁ : Type uA₁\nA₂ : Type uA₂\nA₃ : Type uA₃\nA₁' : Type uA₁'\nA₂' : Type uA₂'\nA₃' : Type uA₃'\ninst✝¹² : CommSemiring R\ninst✝¹¹ : Semiring A₁\ninst✝¹⁰ : Semiring A₂\ninst✝⁹ : Semiring A₃\ninst✝⁸ : Semiring A₁'\ninst✝⁷ : Semiring A₂'\ninst✝⁶ : Semiring A₃'\ninst✝⁵ : Algebra R A₁\ninst✝⁴ : Algebra R A₂\ninst✝³ : Algebra R A₃\ninst✝² : Algebra R A₁'\ninst✝¹ : Algebra R A₂'\ninst✝ : Algebra R A₃'\ne : A₁ ≃ₐ[R] A₂\nx✝¹ : A₁ →ₐ[R] A₂\nx✝ : A₁\n⊢ ((refl.arrowCongr refl) x✝¹) x✝ = ((Equiv.refl (A₁ →ₐ[R] A₂)) x✝¹) x✝","state_after":"no goals","tactic":"rfl","premises":[]}]}
+{"url":"Mathlib/Data/Set/Sigma.lean","commit":"","full_name":"Set.singleton_sigma_singleton","start":[101,0],"end":[103,39],"file_path":"Mathlib/Data/Set/Sigma.lean","tactics":[{"state_before":"ι : Type u_1\nι' : Type u_2\nα : ι → Type u_3\nβ : ι → Type u_4\ns s₁ s₂ : Set ι\nt t₁ t₂ : (i : ι) → Set (α i)\nu : Set ((i : ι) × α i)\nx : (i : ι) × α i\ni j : ι\na✝ : α i\na : (i : ι) → α i\n⊢ ({i}.sigma fun i => {a i}) = {⟨i, a i⟩}","state_after":"no goals","tactic":"rw [sigma_singleton, image_singleton]","premises":[{"full_name":"Set.image_singleton","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[286,8],"def_end_pos":[286,23]},{"full_name":"Set.sigma_singleton","def_path":"Mathlib/Data/Set/Sigma.lean","def_pos":[96,8],"def_end_pos":[96,23]}]}]}
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1","tactic":"by_cases h : p = 0","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case neg\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np : R[X]\nt : R\nhpt : p.IsRoot t\nhnzd : ↑(rootMultiplicity t p) ∈ nonZeroDivisors R\nh : ¬p = 0\n⊢ rootMultiplicity t (derivative p) = rootMultiplicity t p - 1","state_after":"case neg.intro.intro\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np : R[X]\nt : R\nhpt : p.IsRoot t\nhnzd : ↑(rootMultiplicity t p) ∈ nonZeroDivisors R\nh : ¬p = 0\ng : R[X]\nhp : p = (X - C t) ^ rootMultiplicity t p * g\nhndvd : ¬X - C t ∣ g\n⊢ rootMultiplicity t (derivative p) = rootMultiplicity t p - 1","tactic":"obtain ⟨g, hp, hndvd⟩ := p.exists_eq_pow_rootMultiplicity_mul_and_not_dvd h t","premises":[{"full_name":"Polynomial.exists_eq_pow_rootMultiplicity_mul_and_not_dvd","def_path":"Mathlib/Algebra/Polynomial/Div.lean","def_pos":[539,8],"def_end_pos":[539,54]}]},{"state_before":"case neg.intro.intro\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np : R[X]\nt : R\nhpt : p.IsRoot t\nhnzd : ↑(rootMultiplicity t p) ∈ nonZeroDivisors R\nh : ¬p = 0\ng : R[X]\nhp : p = (X - C t) ^ rootMultiplicity t p * g\nhndvd : ¬X - C t ∣ g\n⊢ rootMultiplicity t (derivative p) = rootMultiplicity t p - 1","state_after":"case neg.intro.intro\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np : R[X]\nt : R\nhpt : p.IsRoot t\nh : ¬p = 0\ng : R[X]\nhndvd : ¬X - C t ∣ g\nm : ℕ := rootMultiplicity t p\nhnzd : ↑m ∈ nonZeroDivisors R\nhp : p = (X - C t) ^ m * g\n⊢ rootMultiplicity t (derivative p) = m - 1","tactic":"set m := p.rootMultiplicity t","premises":[{"full_name":"Polynomial.rootMultiplicity","def_path":"Mathlib/Algebra/Polynomial/Div.lean","def_pos":[483,4],"def_end_pos":[483,20]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"case neg.intro.intro\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np : R[X]\nt : R\nhpt : p.IsRoot t\nh : ¬p = 0\ng : R[X]\nhndvd : ¬X - C t ∣ g\nm : ℕ := rootMultiplicity t p\nhnzd : ↑m ∈ nonZeroDivisors R\nhp : p = (X - C t) ^ m * g\n⊢ rootMultiplicity t (derivative p) = m - 1","state_after":"case neg.intro.intro\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np : R[X]\nt : R\nhpt : p.IsRoot t\nh : ¬p = 0\ng : R[X]\nhndvd : ¬X - C t ∣ g\nm : ℕ := rootMultiplicity t p\nhnzd : ↑m ∈ nonZeroDivisors R\nhp : p = (X - C t) ^ m * g\nhm : m - 1 + 1 = m\n⊢ rootMultiplicity t (derivative p) = m - 1","tactic":"have hm : m - 1 + 1 = m := Nat.sub_add_cancel <| (rootMultiplicity_pos h).2 hpt","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Nat.sub_add_cancel","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[893,26],"def_end_pos":[893,40]},{"full_name":"Polynomial.rootMultiplicity_pos","def_path":"Mathlib/Algebra/Polynomial/Div.lean","def_pos":[629,8],"def_end_pos":[629,28]}]},{"state_before":"case neg.intro.intro\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np : R[X]\nt : R\nhpt : p.IsRoot t\nh : ¬p = 0\ng : R[X]\nhndvd : ¬X - C t ∣ g\nm : ℕ := rootMultiplicity t p\nhnzd : ↑m ∈ nonZeroDivisors R\nhp : p = (X - C t) ^ m * g\nhm : m - 1 + 1 = m\n⊢ rootMultiplicity t (derivative p) = m - 1","state_after":"case neg.intro.intro\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np : R[X]\nt : R\nhpt : p.IsRoot t\nh : ¬p = 0\ng : R[X]\nhndvd✝ : ¬X - C t ∣ g\nm : ℕ := rootMultiplicity t p\nhnzd : ↑m ∈ nonZeroDivisors R\nhp : p = (X - C t) ^ m * g\nhm : m - 1 + 1 = m\nhndvd : ¬(X - C t) ^ m ∣ derivative p\n⊢ rootMultiplicity t (derivative p) = m - 1","tactic":"have hndvd : ¬(X - C t) ^ m ∣ derivative p := by\n    rw [hp, derivative_mul, dvd_add_left (dvd_mul_right _ _),\n      derivative_X_sub_C_pow, ← hm, pow_succ, hm, mul_comm (C _), mul_assoc,\n      dvd_cancel_left_mem_nonZeroDivisors (monic_X_sub_C t |>.pow _ |>.mem_nonZeroDivisors)]\n    rw [dvd_iff_isRoot, IsRoot] at hndvd ⊢\n    rwa [eval_mul, eval_C, mul_left_mem_nonZeroDivisors_eq_zero_iff hnzd]","premises":[{"full_name":"Dvd.dvd","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1344,2],"def_end_pos":[1344,5]},{"full_name":"Not","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[224,4],"def_end_pos":[224,7]},{"full_name":"Polynomial.C","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[426,4],"def_end_pos":[426,5]},{"full_name":"Polynomial.IsRoot","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[417,4],"def_end_pos":[417,10]},{"full_name":"Polynomial.Monic.mem_nonZeroDivisors","def_path":"Mathlib/Algebra/Polynomial/RingDivision.lean","def_pos":[472,8],"def_end_pos":[472,33]},{"full_name":"Polynomial.Monic.pow","def_path":"Mathlib/Algebra/Polynomial/Monic.lean","def_pos":[110,8],"def_end_pos":[110,17]},{"full_name":"Polynomial.X","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[474,4],"def_end_pos":[474,5]},{"full_name":"Polynomial.derivative","def_path":"Mathlib/Algebra/Polynomial/Derivative.lean","def_pos":[38,4],"def_end_pos":[38,14]},{"full_name":"Polynomial.derivative_X_sub_C_pow","def_path":"Mathlib/Algebra/Polynomial/Derivative.lean","def_pos":[561,8],"def_end_pos":[561,30]},{"full_name":"Polynomial.derivative_mul","def_path":"Mathlib/Algebra/Polynomial/Derivative.lean","def_pos":[238,8],"def_end_pos":[238,22]},{"full_name":"Polynomial.dvd_iff_isRoot","def_path":"Mathlib/Algebra/Polynomial/Div.lean","def_pos":[578,8],"def_end_pos":[578,22]},{"full_name":"Polynomial.eval_C","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[308,8],"def_end_pos":[308,14]},{"full_name":"Polynomial.eval_mul","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[938,8],"def_end_pos":[938,16]},{"full_name":"Polynomial.monic_X_sub_C","def_path":"Mathlib/Algebra/Polynomial/Monic.lean","def_pos":[340,8],"def_end_pos":[340,21]},{"full_name":"dvd_add_left","def_path":"Mathlib/Algebra/Ring/Divisibility/Basic.lean","def_pos":[103,8],"def_end_pos":[103,20]},{"full_name":"dvd_cancel_left_mem_nonZeroDivisors","def_path":"Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean","def_pos":[152,8],"def_end_pos":[152,43]},{"full_name":"dvd_mul_right","def_path":"Mathlib/Algebra/Divisibility/Basic.lean","def_pos":[75,8],"def_end_pos":[75,21]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]},{"full_name":"mul_left_mem_nonZeroDivisors_eq_zero_iff","def_path":"Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean","def_pos":[123,8],"def_end_pos":[123,48]},{"full_name":"pow_succ","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[567,8],"def_end_pos":[567,16]}]},{"state_before":"case neg.intro.intro\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np : R[X]\nt : R\nhpt : p.IsRoot t\nh : ¬p = 0\ng : R[X]\nhndvd✝ : ¬X - C t ∣ g\nm : ℕ := rootMultiplicity t p\nhnzd : ↑m ∈ nonZeroDivisors R\nhp : p = (X - C t) ^ m * g\nhm : m - 1 + 1 = m\nhndvd : ¬(X - C t) ^ m ∣ derivative p\n⊢ rootMultiplicity t (derivative p) = m - 1","state_after":"case neg.intro.intro\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np : R[X]\nt : R\nhpt : p.IsRoot t\nh : ¬p = 0\ng : R[X]\nhndvd✝ : ¬X - C t ∣ g\nm : ℕ := rootMultiplicity t p\nhnzd : ↑m ∈ nonZeroDivisors R\nhp : p = (X - C t) ^ m * g\nhm : m - 1 + 1 = m\nhndvd : ¬(X - C t) ^ m ∣ derivative p\nhnezero : derivative p ≠ 0\n⊢ rootMultiplicity t (derivative p) = m - 1","tactic":"have hnezero : derivative p ≠ 0 := fun h ↦ hndvd (by rw [h]; exact dvd_zero _)","premises":[{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Polynomial.derivative","def_path":"Mathlib/Algebra/Polynomial/Derivative.lean","def_pos":[38,4],"def_end_pos":[38,14]},{"full_name":"dvd_zero","def_path":"Mathlib/Algebra/GroupWithZero/Divisibility.lean","def_pos":[37,8],"def_end_pos":[37,16]}]},{"state_before":"case neg.intro.intro\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np : R[X]\nt : R\nhpt : p.IsRoot t\nh : ¬p = 0\ng : R[X]\nhndvd✝ : ¬X - C t ∣ g\nm : ℕ := rootMultiplicity t p\nhnzd : ↑m ∈ nonZeroDivisors R\nhp : p = (X - C t) ^ m * g\nhm : m - 1 + 1 = m\nhndvd : ¬(X - C t) ^ m ∣ derivative p\nhnezero : derivative p ≠ 0\n⊢ rootMultiplicity t (derivative p) = m - 1","state_after":"no goals","tactic":"exact le_antisymm (by rwa [rootMultiplicity_le_iff hnezero, hm])\n    (rootMultiplicity_sub_one_le_derivative_rootMultiplicity_of_ne_zero _ t hnezero)","premises":[{"full_name":"Polynomial.rootMultiplicity_le_iff","def_path":"Mathlib/Algebra/Polynomial/RingDivision.lean","def_pos":[412,8],"def_end_pos":[412,31]},{"full_name":"Polynomial.rootMultiplicity_sub_one_le_derivative_rootMultiplicity_of_ne_zero","def_path":"Mathlib/Algebra/Polynomial/FieldDivision.lean","def_pos":[32,8],"def_end_pos":[32,74]},{"full_name":"le_antisymm","def_path":"Mathlib/Order/Defs.lean","def_pos":[156,8],"def_end_pos":[156,19]}]}]}
+{"url":"Mathlib/Combinatorics/SimpleGraph/Walk.lean","commit":"","full_name":"SimpleGraph.Walk.support_tail","start":[808,0],"end":[810,47],"file_path":"Mathlib/Combinatorics/SimpleGraph/Walk.lean","tactics":[{"state_before":"V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v w x y : V\np : G.Walk v v\nhp : ¬p.Nil\n⊢ (p.tail hp).support = p.support.tail","state_after":"no goals","tactic":"rw [← cons_support_tail p hp, List.tail_cons]","premises":[{"full_name":"List.tail_cons","def_path":".lake/packages/batteries/Batteries/Data/List/Basic.lean","def_pos":[47,16],"def_end_pos":[47,25]},{"full_name":"SimpleGraph.Walk.cons_support_tail","def_path":"Mathlib/Combinatorics/SimpleGraph/Walk.lean","def_pos":[796,14],"def_end_pos":[796,31]}]}]}
+{"url":"Mathlib/Probability/Kernel/Disintegration/CondCdf.lean","commit":"","full_name":"ProbabilityTheory.condCDF_eq_stieltjesOfMeasurableRat_unit_prod","start":[255,0],"end":[259,52],"file_path":"Mathlib/Probability/Kernel/Disintegration/CondCdf.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\na : α\n⊢ condCDF ρ a = stieltjesOfMeasurableRat (fun p r => (preCDF ρ r p.2).toReal) ⋯ ((), a)","state_after":"case h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\na : α\nx : ℝ\n⊢ ↑(condCDF ρ a) x = ↑(stieltjesOfMeasurableRat (fun p r => (preCDF ρ r p.2).toReal) ⋯ ((), a)) x","tactic":"ext x","premises":[]},{"state_before":"case h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\na : α\nx : ℝ\n⊢ ↑(condCDF ρ a) x = ↑(stieltjesOfMeasurableRat (fun p r => (preCDF ρ r p.2).toReal) ⋯ ((), a)) x","state_after":"no goals","tactic":"rw [condCDF, ← stieltjesOfMeasurableRat_unit_prod]","premises":[{"full_name":"ProbabilityTheory.condCDF","def_path":"Mathlib/Probability/Kernel/Disintegration/CondCdf.lean","def_pos":[252,18],"def_end_pos":[252,25]},{"full_name":"ProbabilityTheory.stieltjesOfMeasurableRat_unit_prod","def_path":"Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean","def_pos":[450,6],"def_end_pos":[450,40]}]}]}
+{"url":"Mathlib/RingTheory/DedekindDomain/Factorization.lean","commit":"","full_name":"Associates.finite_factors","start":[75,0],"end":[86,31],"file_path":"Mathlib/RingTheory/DedekindDomain/Factorization.lean","tactics":[{"state_before":"R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI : Ideal R\nhI : I ≠ 0\n⊢ ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite, ↑((Associates.mk v.asIdeal).count (Associates.mk I).factors) = 0","state_after":"R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI : Ideal R\nhI : I ≠ 0\nh_supp : {v | ¬↑((Associates.mk v.asIdeal).count (Associates.mk I).factors) = 0} = {v | v.asIdeal ∣ I}\n⊢ ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite, ↑((Associates.mk v.asIdeal).count (Associates.mk I).factors) = 0","tactic":"have h_supp : {v : HeightOneSpectrum R | ��((Associates.mk v.asIdeal).count\n      (Associates.mk I).factors : ℤ) = 0} = {v : HeightOneSpectrum R | v.asIdeal ∣ I} := by\n    ext v\n    simp_rw [Int.natCast_eq_zero]\n    exact Associates.count_ne_zero_iff_dvd hI v.irreducible","premises":[{"full_name":"Associates.count","def_path":"Mathlib/RingTheory/UniqueFactorizationDomain.lean","def_pos":[1220,4],"def_end_pos":[1220,9]},{"full_name":"Associates.count_ne_zero_iff_dvd","def_path":"Mathlib/RingTheory/UniqueFactorizationDomain.lean","def_pos":[1604,8],"def_end_pos":[1604,29]},{"full_name":"Associates.factors","def_path":"Mathlib/RingTheory/UniqueFactorizationDomain.lean","def_pos":[1343,18],"def_end_pos":[1343,25]},{"full_name":"Associates.mk","def_path":"Mathlib/Algebra/Associated/Basic.lean","def_pos":[737,17],"def_end_pos":[737,19]},{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Dvd.dvd","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1344,2],"def_end_pos":[1344,5]},{"full_name":"Int","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean","def_pos":[40,10],"def_end_pos":[40,13]},{"full_name":"Int.natCast_eq_zero","def_path":"Mathlib/Data/Int/Defs.lean","def_pos":[106,29],"def_end_pos":[106,44]},{"full_name":"IsDedekindDomain.HeightOneSpectrum","def_path":"Mathlib/RingTheory/DedekindDomain/Ideal.lean","def_pos":[935,10],"def_end_pos":[935,27]},{"full_name":"IsDedekindDomain.HeightOneSpectrum.asIdeal","def_path":"Mathlib/RingTheory/DedekindDomain/Ideal.lean","def_pos":[936,2],"def_end_pos":[936,9]},{"full_name":"IsDedekindDomain.HeightOneSpectrum.irreducible","def_path":"Mathlib/RingTheory/DedekindDomain/Ideal.lean","def_pos":[950,8],"def_end_pos":[950,19]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Not","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[224,4],"def_end_pos":[224,7]},{"full_name":"setOf","def_path":"Mathlib/Init/Set.lean","def_pos":[56,4],"def_end_pos":[56,9]}]},{"state_before":"R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI : Ideal R\nhI : I ≠ 0\nh_supp : {v | ¬↑((Associates.mk v.asIdeal).count (Associates.mk I).factors) = 0} = {v | v.asIdeal ∣ I}\n⊢ ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite, ↑((Associates.mk v.asIdeal).count (Associates.mk I).factors) = 0","state_after":"R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI : Ideal R\nhI : I ≠ 0\nh_supp : {v | ¬↑((Associates.mk v.asIdeal).count (Associates.mk I).factors) = 0} = {v | v.asIdeal ∣ I}\n⊢ {v | v.asIdeal ∣ I}.Finite","tactic":"rw [Filter.eventually_cofinite, h_supp]","premises":[{"full_name":"Filter.eventually_cofinite","def_path":"Mathlib/Order/Filter/Cofinite.lean","def_pos":[38,8],"def_end_pos":[38,27]}]},{"state_before":"R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI : Ideal R\nhI : I ≠ 0\nh_supp : {v | ¬↑((Associates.mk v.asIdeal).count (Associates.mk I).factors) = 0} = {v | v.asIdeal ∣ I}\n⊢ {v | v.asIdeal ∣ I}.Finite","state_after":"no goals","tactic":"exact Ideal.finite_factors hI","premises":[{"full_name":"Ideal.finite_factors","def_path":"Mathlib/RingTheory/DedekindDomain/Factorization.lean","def_pos":[65,8],"def_end_pos":[65,28]}]}]}
+{"url":"Mathlib/Topology/UniformSpace/Pi.lean","commit":"","full_name":"Pi.uniformSpace_eq","start":[27,0],"end":[29,14],"file_path":"Mathlib/Topology/UniformSpace/Pi.lean","tactics":[{"state_before":"ι : Type u_1\nι' : Type u_2\nβ : Type u_3\nα : ι → Type u\nU : (i : ι) → UniformSpace (α i)\ninst✝ : UniformSpace β\n⊢ uniformSpace α = ⨅ i, UniformSpace.comap (eval i) (U i)","state_after":"case h\nι : Type u_1\nι' : Type u_2\nβ : Type u_3\nα : ι → Type u\nU : (i : ι) → UniformSpace (α i)\ninst✝ : UniformSpace β\n⊢ 𝓤 ((i : ι) → α i) = 𝓤 ((i : ι) → α i)","tactic":"ext : 1","premises":[]},{"state_before":"case h\nι : Type u_1\nι' : Type u_2\nβ : Type u_3\nα : ι → Type u\nU : (i : ι) → UniformSpace (α i)\ninst✝ : UniformSpace β\n⊢ 𝓤 ((i : ι) → α i) = 𝓤 ((i : ι) → α i)","state_after":"no goals","tactic":"rfl","premises":[]}]}
+{"url":"Mathlib/RingTheory/FractionalIdeal/Basic.lean","commit":"","full_name":"FractionalIdeal.le_one_iff_exists_coeIdeal","start":[618,0],"end":[640,25],"file_path":"Mathlib/RingTheory/FractionalIdeal/Basic.lean","tactics":[{"state_before":"R : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\n⊢ J ≤ 1 ↔ ∃ I, ↑I = J","state_after":"case mp\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\n⊢ J ≤ 1 → ∃ I, ↑I = J\n\ncase mpr\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\n⊢ (∃ I, ↑I = J) → J ≤ 1","tactic":"constructor","premises":[]}]}
+{"url":"Mathlib/Data/Set/Image.lean","commit":"","full_name":"Set.range_inl","start":[751,0],"end":[751,86],"file_path":"Mathlib/Data/Set/Image.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nι' : Sort u_5\nf : ι → α\ns t : Set α\n⊢ range Sum.inl = {x | x.isLeft = true}","state_after":"no goals","tactic":"ext (_|_) <;> simp","premises":[]}]}
+{"url":"Mathlib/FieldTheory/RatFunc/Basic.lean","commit":"","full_name":"RatFunc.ofFractionRing_algebraMap","start":[562,0],"end":[564,24],"file_path":"Mathlib/FieldTheory/RatFunc/Basic.lean","tactics":[{"state_before":"K : Type u\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\nx : K[X]\n⊢ { toFractionRing := (algebraMap K[X] (FractionRing K[X])) x } = (algebraMap K[X] (RatFunc K)) x","state_after":"no goals","tactic":"rw [← mk_one, mk_one']","premises":[{"full_name":"RatFunc.mk_one","def_path":"Mathlib/FieldTheory/RatFunc/Basic.lean","def_pos":[559,8],"def_end_pos":[559,14]},{"full_name":"RatFunc.mk_one'","def_path":"Mathlib/FieldTheory/RatFunc/Defs.lean","def_pos":[172,8],"def_end_pos":[172,15]}]}]}
+{"url":"Mathlib/Algebra/RingQuot.lean","commit":"","full_name":"_private.Mathlib.Algebra.RingQuot.0.RingQuot.npow_def","start":[168,0],"end":[185,10],"file_path":"Mathlib/Algebra/RingQuot.lean","tactics":[{"state_before":"R : Type uR\ninst✝³ : Semiring R\nS : Type uS\ninst✝² : CommSemiring S\nT : Type uT\nA : Type uA\ninst✝¹ : Semiring A\ninst✝ : Algebra S A\nr : R → R → Prop\nn : ℕ\na✝ : Quot (Rel r)\na b : R\nh : Rel r a b\n⊢ (fun a => Quot.mk (Rel r) (a ^ n)) a = (fun a => Quot.mk (Rel r) (a ^ n)) b","state_after":"R : Type uR\ninst✝³ : Semiring R\nS : Type uS\ninst✝² : CommSemiring S\nT : Type uT\nA : Type uA\ninst✝¹ : Semiring A\ninst✝ : Algebra S A\nr : R → R → Prop\nn : ℕ\na✝ : Quot (Rel r)\na b : R\nh : Rel r a b\n⊢ Quot.mk (Rel r) (a ^ n) = Quot.mk (Rel r) (b ^ n)","tactic":"dsimp only","premises":[]},{"state_before":"R : Type uR\ninst✝³ : Semiring R\nS : Type uS\ninst✝² : CommSemiring S\nT : Type uT\nA : Type uA\ninst✝¹ : Semiring A\ninst✝ : Algebra S A\nr : R → R → Prop\nn : ℕ\na✝ : Quot (Rel r)\na b : R\nh : Rel r a b\n⊢ Quot.mk (Rel r) (a ^ n) = Quot.mk (Rel r) (b ^ n)","state_after":"no goals","tactic":"induction n with\n          | zero => rw [pow_zero, pow_zero]\n          | succ n ih =>\n            rw [pow_succ, pow_succ]\n            -- Porting note:\n            -- `simpa [mul_def] using congr_arg₂ (fun x y ↦ mul r ⟨x⟩ ⟨y⟩) (Quot.sound h) ih`\n            -- mysteriously doesn't work\n            have := congr_arg₂ (fun x y ↦ mul r ⟨x⟩ ⟨y⟩) ih (Quot.sound h)\n            dsimp only at this\n            simp? [mul_def] at this says simp only [mul_def, Quot.map₂_mk, mk.injEq] at this\n            exact this","premises":[{"full_name":"Quot.map₂_mk","def_path":"Mathlib/Data/Quot.lean","def_pos":[136,8],"def_end_pos":[136,15]},{"full_name":"Quot.sound","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1471,6],"def_end_pos":[1471,11]},{"full_name":"_private.Mathlib.Algebra.RingQuot.0.RingQuot.mul","def_path":"Mathlib/Algebra/RingQuot.lean","def_pos":[158,24],"def_end_pos":[158,27]},{"full_name":"_private.Mathlib.Algebra.RingQuot.0.RingQuot.mul_def","def_path":"Mathlib/Algebra/RingQuot.lean","def_pos":[158,24],"def_end_pos":[158,27]},{"full_name":"pow_succ","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[567,8],"def_end_pos":[567,16]},{"full_name":"pow_zero","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[563,8],"def_end_pos":[563,16]}]}]}
+{"url":"Mathlib/Data/Set/Pointwise/BigOperators.lean","commit":"","full_name":"Set.multiset_prod_subset_multiset_prod","start":[113,0],"end":[119,43],"file_path":"Mathlib/Data/Set/Pointwise/BigOperators.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\nF : Type u_4\ninst✝³ : FunLike F α β\ninst✝² : CommMonoid α\ninst✝¹ : CommMonoid β\ninst✝ : MonoidHomClass F α β\nt : Multiset ι\nf₁ f₂ : ι → Set α\nhf : ∀ i ∈ t, f₁ i ⊆ f₂ i\n⊢ (Multiset.map f₁ t).prod ⊆ (Multiset.map f₂ t).prod","state_after":"case h\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nF : Type u_4\ninst✝³ : FunLike F α β\ninst✝² : CommMonoid α\ninst✝¹ : CommMonoid β\ninst✝ : MonoidHomClass F α β\nf₁ f₂ : ι → Set α\na✝ : List ι\nhf : ∀ i ∈ ⟦a✝⟧, f₁ i ⊆ f₂ i\n⊢ (Multiset.map f₁ ⟦a✝⟧).prod ⊆ (Multiset.map f₂ ⟦a✝⟧).prod","tactic":"induction t using Quotient.inductionOn","premises":[{"full_name":"Quotient.inductionOn","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1629,18],"def_end_pos":[1629,29]}]},{"state_before":"case h\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nF : Type u_4\ninst✝³ : FunLike F α β\ninst✝² : CommMonoid α\ninst✝¹ : CommMonoid β\ninst✝ : MonoidHomClass F α β\nf₁ f₂ : ι → Set α\na✝ : List ι\nhf : ∀ i ∈ ⟦a✝⟧, f₁ i ⊆ f₂ i\n⊢ (Multiset.map f₁ ⟦a✝⟧).prod ⊆ (Multiset.map f₂ ⟦a✝⟧).prod","state_after":"case h\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nF : Type u_4\ninst✝³ : FunLike F α β\ninst✝² : CommMonoid α\ninst✝¹ : CommMonoid β\ninst✝ : MonoidHomClass F α β\nf₁ f₂ : ι → Set α\na✝ : List ι\nhf : ∀ i ∈ ⟦a✝⟧, f₁ i ⊆ f₂ i\n⊢ (List.map f₁ a✝).prod ⊆ (List.map f₂ a✝).prod","tactic":"simp_rw [Multiset.quot_mk_to_coe, Multiset.map_coe, Multiset.prod_coe]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Multiset.map_coe","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[1031,25],"def_end_pos":[1031,32]},{"full_name":"Multiset.prod_coe","def_path":"Mathlib/Algebra/BigOperators/Group/Multiset.lean","def_pos":[52,8],"def_end_pos":[52,16]},{"full_name":"Multiset.quot_mk_to_coe","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[43,8],"def_end_pos":[43,22]}]},{"state_before":"case h\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nF : Type u_4\ninst✝³ : FunLike F α β\ninst✝² : CommMonoid α\ninst✝¹ : CommMonoid β\ninst✝ : MonoidHomClass F α β\nf₁ f₂ : ι → Set α\na✝ : List ι\nhf : ∀ i ∈ ⟦a✝⟧, f₁ i ⊆ f₂ i\n⊢ (List.map f₁ a✝).prod ⊆ (List.map f₂ a✝).prod","state_after":"no goals","tactic":"exact list_prod_subset_list_prod _ _ _ hf","premises":[{"full_name":"Set.list_prod_subset_list_prod","def_path":"Mathlib/Data/Set/Pointwise/BigOperators.lean","def_pos":[92,8],"def_end_pos":[92,34]}]}]}
+{"url":"Mathlib/RingTheory/Polynomial/Pochhammer.lean","commit":"","full_name":"factorial_mul_ascPochhammer","start":[198,0],"end":[200,81],"file_path":"Mathlib/RingTheory/Polynomial/Pochhammer.lean","tactics":[{"state_before":"S✝ : Type u_1\ninst✝¹ : Semiring S✝\nr✝ n✝ : ℕ\nS : Type u_2\ninst✝ : Semiring S\nr n : ℕ\n⊢ ↑r ! * eval (↑r + 1) (ascPochhammer S n) = ↑(r + n)!","state_after":"no goals","tactic":"rw_mod_cast [ascPochhammer_nat_eq_ascFactorial, Nat.factorial_mul_ascFactorial]","premises":[{"full_name":"Nat.factorial_mul_ascFactorial","def_path":"Mathlib/Data/Nat/Factorial/Basic.lean","def_pos":[217,8],"def_end_pos":[217,34]},{"full_name":"ascPochhammer_nat_eq_ascFactorial","def_path":"Mathlib/RingTheory/Polynomial/Pochhammer.lean","def_pos":[148,8],"def_end_pos":[148,41]}]}]}
+{"url":"Mathlib/LinearAlgebra/Dual.lean","commit":"","full_name":"Subspace.comap_dualAnnihilator_dualAnnihilator","start":[963,0],"end":[965,74],"file_path":"Mathlib/LinearAlgebra/Dual.lean","tactics":[{"state_before":"K : Type u\nV : Type v\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nW : Subspace K V\n⊢ comap (Module.Dual.eval K V) (dualAnnihilator W).dualAnnihilator = W","state_after":"case h\nK : Type u\nV : Type v\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nW : Subspace K V\nx✝ : V\n⊢ x✝ ∈ comap (Module.Dual.eval K V) (dualAnnihilator W).dualAnnihilator ↔ x✝ ∈ W","tactic":"ext","premises":[]},{"state_before":"case h\nK : Type u\nV : Type v\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nW : Subspace K V\nx✝ : V\n⊢ x✝ ∈ comap (Module.Dual.eval K V) (dualAnnihilator W).dualAnnihilator ↔ x✝ ∈ W","state_after":"case h\nK : Type u\nV : Type v\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nW : Subspace K V\nx✝ : V\n⊢ (∀ φ ∈ dualAnnihilator W, φ x✝ = 0) ↔ x✝ ∈ comap (Module.Dual.eval K V) (dualAnnihilator W).dualAnnihilator","tactic":"rw [Iff.comm, ← forall_mem_dualAnnihilator_apply_eq_zero_iff]","premises":[{"full_name":"Iff.comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[814,8],"def_end_pos":[814,16]},{"full_name":"Subspace.forall_mem_dualAnnihilator_apply_eq_zero_iff","def_path":"Mathlib/LinearAlgebra/Dual.lean","def_pos":[959,8],"def_end_pos":[959,52]}]},{"state_before":"case h\nK : Type u\nV : Type v\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nW : Subspace K V\nx✝ : V\n⊢ (∀ φ ∈ dualAnnihilator W, φ x✝ = 0) ↔ x✝ ∈ comap (Module.Dual.eval K V) (dualAnnihilator W).dualAnnihilator","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/Algebra/Order/Rearrangement.lean","commit":"","full_name":"MonovaryOn.sum_smul_comp_perm_lt_sum_smul_iff","start":[136,0],"end":[143,42],"file_path":"Mathlib/Algebra/Order/Rearrangement.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : LinearOrderedRing α\ninst✝² : LinearOrderedAddCommGroup β\ninst✝¹ : Module α β\ninst✝ : OrderedSMul α β\ns : Finset ι\nσ : Perm ι\nf : ι → α\ng : ι → β\nhfg : MonovaryOn f g ↑s\nhσ : {x | σ x ≠ x} ⊆ ↑s\n⊢ ∑ i ∈ s, f i • g (σ i) < ∑ i ∈ s, f i • g i ↔ ¬MonovaryOn f (g ∘ ⇑σ) ↑s","state_after":"no goals","tactic":"simp [← hfg.sum_smul_comp_perm_eq_sum_smul_iff hσ, lt_iff_le_and_ne,\n    hfg.sum_smul_comp_perm_le_sum_smul hσ]","premises":[{"full_name":"MonovaryOn.sum_smul_comp_perm_eq_sum_smul_iff","def_path":"Mathlib/Algebra/Order/Rearrangement.lean","def_pos":[111,8],"def_end_pos":[111,53]},{"full_name":"MonovaryOn.sum_smul_comp_perm_le_sum_smul","def_path":"Mathlib/Algebra/Order/Rearrangement.lean","def_pos":[60,8],"def_end_pos":[60,49]},{"full_name":"lt_iff_le_and_ne","def_path":"Mathlib/Order/Basic.lean","def_pos":[309,8],"def_end_pos":[309,24]}]}]}
+{"url":"Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Unital.lean","commit":"","full_name":"cfc_eval_X","start":[497,0],"end":[498,24],"file_path":"Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Unital.lean","tactics":[{"state_before":"R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : MetricSpace R\ninst✝⁶ : TopologicalSemiring R\ninst✝⁵ : ContinuousStar R\ninst✝⁴ : TopologicalSpace A\ninst✝³ : Ring A\ninst✝² : StarRing A\ninst✝¹ : Algebra R A\ninst✝ : ContinuousFunctionalCalculus R p\nf g : R → R\na : A\nha✝ : autoParam (p a) _auto✝\nhf : autoParam (ContinuousOn f (spectrum R a)) _auto✝\nhg : autoParam (ContinuousOn g (spectrum R a)) _auto✝\nha : autoParam (p a) _auto✝\n⊢ cfc (fun a => eval a X) a = a","state_after":"no goals","tactic":"simpa using cfc_id R a","premises":[{"full_name":"cfc_id","def_path":"Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Unital.lean","def_pos":[325,6],"def_end_pos":[325,12]}]}]}
+{"url":"Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean","commit":"","full_name":"UpperHalfPlane.cosh_half_dist","start":[46,0],"end":[53,22],"file_path":"Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean","tactics":[{"state_before":"z✝ w✝ : ℍ\nr R : ℝ\nz w : ℍ\n⊢ cosh (dist z w / 2) = dist (↑z) ((starRingEnd ℂ) ↑w) / (2 * √(z.im * w.im))","state_after":"z✝ w✝ : ℍ\nr R : ℝ\nz w : ℍ\n⊢ (2 ^ 2 * (z.im * w.im) + dist ↑z ↑w ^ 2) / (2 ^ 2 * (z.im * w.im)) =\n    dist (↑z) ((starRingEnd ℂ) ↑w) ^ 2 / (2 ^ 2 * (z.im * w.im))\n\nz✝ w✝ : ℍ\nr R : ℝ\nz w : ℍ\n⊢ 0 ≤ z.im * w.im\n\nz✝ w✝ : ℍ\nr R : ℝ\nz w : ℍ\n⊢ (2 * √(z.im * w.im)) ^ 2 ≠ 0\n\ncase ha\nz✝ w✝ : ℍ\nr R : ℝ\nz w : ℍ\n⊢ 0 ≤ cosh (dist z w / 2)\n\ncase hb\nz✝ w✝ : ℍ\nr R : ℝ\nz w : ℍ\n⊢ 0 ≤ dist (↑z) ((starRingEnd ℂ) ↑w) / (2 * √(z.im * w.im))","tactic":"rw [← sq_eq_sq, cosh_sq', sinh_half_dist, div_pow, div_pow, one_add_div, mul_pow, sq_sqrt]","premises":[{"full_name":"Real.cosh_sq'","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[928,8],"def_end_pos":[928,16]},{"full_name":"Real.sq_sqrt","def_path":"Mathlib/Data/Real/Sqrt.lean","def_pos":[170,8],"def_end_pos":[170,15]},{"full_name":"UpperHalfPlane.sinh_half_dist","def_path":"Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean","def_pos":[42,8],"def_end_pos":[42,22]},{"full_name":"div_pow","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[589,6],"def_end_pos":[589,13]},{"full_name":"mul_pow","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[222,31],"def_end_pos":[222,38]},{"full_name":"one_add_div","def_path":"Mathlib/Algebra/Field/Basic.lean","def_pos":[37,8],"def_end_pos":[37,19]},{"full_name":"sq_eq_sq","def_path":"Mathlib/Algebra/Order/Ring/Basic.lean","def_pos":[219,8],"def_end_pos":[219,16]}]},{"state_before":"z✝ w✝ : ℍ\nr R : ℝ\nz w : ℍ\n⊢ 0 ≤ z.im * w.im\n\nz✝ w✝ : ℍ\nr R : ℝ\nz w : ℍ\n⊢ (2 * √(z.im * w.im)) ^ 2 ≠ 0\n\ncase ha\nz✝ w✝ : ℍ\nr R : ℝ\nz w : ℍ\n⊢ 0 ≤ cosh (dist z w / 2)\n\ncase hb\nz✝ w✝ : ℍ\nr R : ℝ\nz w : ℍ\n⊢ 0 ≤ dist (↑z) ((starRingEnd ℂ) ↑w) / (2 * √(z.im * w.im))","state_after":"no goals","tactic":"all_goals positivity","premises":[]}]}
+{"url":"Mathlib/Data/NNReal/Basic.lean","commit":"","full_name":"NNReal.mul_iSup","start":[941,0],"end":[943,55],"file_path":"Mathlib/Data/NNReal/Basic.lean","tactics":[{"state_before":"ι : Sort u_1\nf✝ f : ι → ℝ≥0\na : ℝ≥0\n⊢ a * ⨆ i, f i = ⨆ i, a * f i","state_after":"ι : Sort u_1\nf✝ f : ι → ℝ≥0\na : ℝ≥0\n⊢ ↑a * ⨆ i, ↑(f i) = ⨆ i, ↑(a * f i)","tactic":"rw [← coe_inj, NNReal.coe_mul, NNReal.coe_iSup, NNReal.coe_iSup]","premises":[{"full_name":"NNReal.coe_iSup","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[461,8],"def_end_pos":[461,16]},{"full_name":"NNReal.coe_inj","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[150,25],"def_end_pos":[150,32]},{"full_name":"NNReal.coe_mul","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[164,18],"def_end_pos":[164,25]}]},{"state_before":"ι : Sort u_1\nf✝ f : ι → ℝ≥0\na : ℝ≥0\n⊢ ↑a * ⨆ i, ↑(f i) = ⨆ i, ↑(a * f i)","state_after":"no goals","tactic":"exact Real.mul_iSup_of_nonneg (NNReal.coe_nonneg _) _","premises":[{"full_name":"NNReal.coe_nonneg","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[122,8],"def_end_pos":[122,18]},{"full_name":"Real.mul_iSup_of_nonneg","def_path":"Mathlib/Data/Real/Pointwise.lean","def_pos":[109,8],"def_end_pos":[109,31]}]}]}
+{"url":"Mathlib/Analysis/Fourier/FourierTransformDeriv.lean","commit":"","full_name":"VectorFourier.fourierIntegral_fderiv","start":[249,0],"end":[277,13],"file_path":"Mathlib/Analysis/Fourier/FourierTransformDeriv.lean","tactics":[{"state_before":"E : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : NormedSpace ℝ V\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\ninst✝³ : MeasurableSpace V\ninst✝² : BorelSpace V\ninst✝¹ : FiniteDimensional ℝ V\nμ : Measure V\ninst✝ : μ.IsAddHaarMeasure\nhf : Integrable f μ\nh'f : Differentiable ℝ f\nhf' : Integrable (fderiv ℝ f) μ\n⊢ fourierIntegral 𝐞 μ L.toLinearMap₂ (fderiv ℝ f) = fourierSMulRight (-L.flip) (fourierIntegral 𝐞 μ L.toLinearMap₂ f)","state_after":"case h.h\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : NormedSpace ℝ V\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\ninst✝³ : MeasurableSpace V\ninst✝² : BorelSpace V\ninst✝¹ : FiniteDimensional ℝ V\nμ : Measure V\ninst✝ : μ.IsAddHaarMeasure\nhf : Integrable f μ\nh'f : Differentiable ℝ f\nhf' : Integrable (fderiv ℝ f) μ\nw : W\ny : V\n⊢ (fourierIntegral 𝐞 μ L.toLinearMap₂ (fderiv ℝ f) w) y =\n    (fourierSMulRight (-L.flip) (fourierIntegral 𝐞 μ L.toLinearMap₂ f) w) y","tactic":"ext w y","premises":[]},{"state_before":"case h.h\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : NormedSpace ℝ V\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\ninst✝³ : MeasurableSpace V\ninst✝² : BorelSpace V\ninst✝¹ : FiniteDimensional ℝ V\nμ : Measure V\ninst✝ : μ.IsAddHaarMeasure\nhf : Integrable f μ\nh'f : Differentiable ℝ f\nhf' : Integrable (fderiv ℝ f) μ\nw : W\ny : V\n⊢ (fourierIntegral 𝐞 μ L.toLinearMap₂ (fderiv ℝ f) w) y =\n    (fourierSMulRight (-L.flip) (fourierIntegral 𝐞 μ L.toLinearMap₂ f) w) y","state_after":"case h.h\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : NormedSpace ℝ V\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\ninst✝³ : MeasurableSpace V\ninst✝² : BorelSpace V\ninst✝¹ : FiniteDimensional ℝ V\nμ : Measure V\ninst✝ : μ.IsAddHaarMeasure\nhf : Integrable f μ\nh'f : Differentiable ℝ f\nhf' : Integrable (fderiv ℝ f) μ\nw : W\ny : V\ng : V → ℂ := fun v => ↑(𝐞 (-(L v) w))\n⊢ (fourierIntegral 𝐞 μ L.toLinearMap₂ (fderiv ℝ f) w) y =\n    (fourierSMulRight (-L.flip) (fourierIntegral 𝐞 μ L.toLinearMap₂ f) w) y","tactic":"let g (v : V) : ℂ := 𝐞 (-L v w)","premises":[{"full_name":"Complex","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[27,10],"def_end_pos":[27,17]},{"full_name":"Real.fourierChar","def_path":"Mathlib/Analysis/Fourier/FourierTransform.lean","def_pos":[313,4],"def_end_pos":[313,15]}]},{"state_before":"case h.h\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : NormedSpace ℝ V\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\ninst✝³ : MeasurableSpace V\ninst✝² : BorelSpace V\ninst✝¹ : FiniteDimensional ℝ V\nμ : Measure V\ninst✝ : μ.IsAddHaarMeasure\nhf : Integrable f μ\nh'f : Differentiable ℝ f\nhf' : Integrable (fderiv ℝ f) μ\nw : W\ny : V\ng : V → ℂ := fun v => ↑(𝐞 (-(L v) w))\n⊢ (fourierIntegral 𝐞 μ L.toLinearMap₂ (fderiv ℝ f) w) y =\n    (fourierSMulRight (-L.flip) (fourierIntegral 𝐞 μ L.toLinearMap₂ f) w) y","state_after":"case h.h\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : NormedSpace ℝ V\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\ninst✝³ : MeasurableSpace V\ninst✝² : BorelSpace V\ninst✝¹ : FiniteDimensional ℝ V\nμ : Measure V\ninst✝ : μ.IsAddHaarMeasure\nhf : Integrable f μ\nh'f : Differentiable ℝ f\nhf' : Integrable (fderiv ℝ f) μ\nw : W\ny : V\ng : V → ℂ := fun v => ↑(𝐞 (-(L v) w))\n⊢ ∫ (x : V), g x • (fderiv ℝ f x) y ∂μ = ∫ (x : V), (2 * ↑π * I * ↑((L y) w) * g x) • f x ∂μ","tactic":"suffices ∫ x, g x • fderiv ℝ f x y ∂μ = ∫ x, (2 * ↑π * I * L y w * g x) • f x ∂μ by\n    rw [fourierIntegral_continuousLinearMap_apply' hf']\n    simpa only [fourierIntegral, ContinuousLinearMap.toLinearMap₂_apply, fourierSMulRight_apply,\n      ContinuousLinearMap.neg_apply, ContinuousLinearMap.flip_apply, ← integral_smul, neg_smul,\n      smul_neg, ← smul_smul, coe_smul, neg_neg]","premises":[{"full_name":"Complex.I","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[231,4],"def_end_pos":[231,5]},{"full_name":"Complex.coe_smul","def_path":"Mathlib/Data/Complex/Module.lean","def_pos":[183,8],"def_end_pos":[183,24]},{"full_name":"ContinuousLinearMap.flip_apply","def_path":"Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean","def_pos":[163,8],"def_end_pos":[163,18]},{"full_name":"ContinuousLinearMap.neg_apply","def_path":"Mathlib/Topology/Algebra/Module/Basic.lean","def_pos":[1248,8],"def_end_pos":[1248,17]},{"full_name":"ContinuousLinearMap.toLinearMap₂_apply","def_path":"Mathlib/Topology/Algebra/Module/StrongTopology.lean","def_pos":[343,14],"def_end_pos":[343,32]},{"full_name":"MeasureTheory.integral","def_path":"Mathlib/MeasureTheory/Integral/Bochner.lean","def_pos":[714,16],"def_end_pos":[714,24]},{"full_name":"MeasureTheory.integral_smul","def_path":"Mathlib/MeasureTheory/Integral/Bochner.lean","def_pos":[824,8],"def_end_pos":[824,21]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Real.fourierIntegral_continuousLinearMap_apply'","def_path":"Mathlib/Analysis/Fourier/FourierTransform.lean","def_pos":[360,8],"def_end_pos":[360,50]},{"full_name":"Real.pi","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","def_pos":[119,28],"def_end_pos":[119,30]},{"full_name":"VectorFourier.fourierIntegral","def_path":"Mathlib/Analysis/Fourier/FourierTransform.lean","def_pos":[77,4],"def_end_pos":[77,19]},{"full_name":"VectorFourier.fourierSMulRight_apply","def_path":"Mathlib/Analysis/Fourier/FourierTransformDeriv.lean","def_pos":[162,14],"def_end_pos":[162,36]},{"full_name":"fderiv","def_path":"Mathlib/Analysis/Calculus/FDeriv/Basic.lean","def_pos":[183,16],"def_end_pos":[183,22]},{"full_name":"neg_neg","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[733,2],"def_end_pos":[733,13]},{"full_name":"neg_smul","def_path":"Mathlib/Algebra/Module/Defs.lean","def_pos":[228,8],"def_end_pos":[228,16]},{"full_name":"smul_neg","def_path":"Mathlib/Algebra/GroupWithZero/Action/Defs.lean","def_pos":[276,8],"def_end_pos":[276,16]},{"full_name":"smul_smul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[374,6],"def_end_pos":[374,15]}]},{"state_before":"case h.h\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : NormedSpace ℝ V\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\ninst✝³ : MeasurableSpace V\ninst✝² : BorelSpace V\ninst✝¹ : FiniteDimensional ℝ V\nμ : Measure V\ninst✝ : μ.IsAddHaarMeasure\nhf : Integrable f μ\nh'f : Differentiable ℝ f\nhf' : Integrable (fderiv ℝ f) μ\nw : W\ny : V\ng : V → ℂ := fun v => ↑(𝐞 (-(L v) w))\n⊢ ∫ (x : V), g x • (fderiv ℝ f x) y ∂μ = ∫ (x : V), (2 * ↑π * I * ↑((L y) w) * g x) • f x ∂μ","state_after":"case h.h\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : NormedSpace ℝ V\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\ninst✝³ : MeasurableSpace V\ninst✝² : BorelSpace V\ninst✝¹ : FiniteDimensional ℝ V\nμ : Measure V\ninst✝ : μ.IsAddHaarMeasure\nhf : Integrable f μ\nh'f : Differentiable ℝ f\nhf' : Integrable (fderiv ℝ f) μ\nw : W\ny : V\ng : V → ℂ := fun v => ↑(𝐞 (-(L v) w))\nA : ∀ (x : V), (fderiv ℝ g x) y = -2 * ↑π * I * ↑((L y) w) * g x\n⊢ ∫ (x : V), g x • (fderiv ℝ f x) y ∂μ = ∫ (x : V), (2 * ↑π * I * ↑((L y) w) * g x) • f x ∂μ","tactic":"have A x : fderiv ℝ g x y = - 2 * ↑π * I * L y w * g x :=\n    fderiv_fourierChar_neg_bilinear_left_apply _ _ _ _","premises":[{"full_name":"Complex.I","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[231,4],"def_end_pos":[231,5]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Real.fderiv_fourierChar_neg_bilinear_left_apply","def_path":"Mathlib/Analysis/Fourier/FourierTransformDeriv.lean","def_pos":[137,6],"def_end_pos":[137,48]},{"full_name":"Real.pi","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","def_pos":[119,28],"def_end_pos":[119,30]},{"full_name":"fderiv","def_path":"Mathlib/Analysis/Calculus/FDeriv/Basic.lean","def_pos":[183,16],"def_end_pos":[183,22]}]},{"state_before":"case h.h\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : NormedSpace ℝ V\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\ninst✝³ : MeasurableSpace V\ninst✝² : BorelSpace V\ninst✝¹ : FiniteDimensional ℝ V\nμ : Measure V\ninst✝ : μ.IsAddHaarMeasure\nhf : Integrable f μ\nh'f : Differentiable ℝ f\nhf' : Integrable (fderiv ℝ f) μ\nw : W\ny : V\ng : V → ℂ := fun v => ↑(𝐞 (-(L v) w))\nA : ∀ (x : V), (fderiv ℝ g x) y = -2 * ↑π * I * ↑((L y) w) * g x\n⊢ ∫ (x : V), g x • (fderiv ℝ f x) y ∂μ = ∫ (x : V), (2 * ↑π * I * ↑((L y) w) * g x) • f x ∂μ","state_after":"case h.h\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : NormedSpace ℝ V\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\ninst✝³ : MeasurableSpace V\ninst✝² : BorelSpace V\ninst✝¹ : FiniteDimensional ℝ V\nμ : Measure V\ninst✝ : μ.IsAddHaarMeasure\nhf : Integrable f μ\nh'f : Differentiable ℝ f\nhf' : Integrable (fderiv ℝ f) μ\nw : W\ny : V\ng : V → ℂ := fun v => ↑(𝐞 (-(L v) w))\nA : ∀ (x : V), (fderiv ℝ g x) y = -2 * ↑π * I * ↑((L y) w) * g x\n⊢ ∫ (a : V), -((fderiv ℝ g a) y • f a) ∂μ = ∫ (x : V), (2 * ↑π * I * ↑((L y) w) * g x) • f x ∂μ\n\ncase h.h.hf'g\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : NormedSpace ℝ V\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\ninst✝³ : MeasurableSpace V\ninst✝² : BorelSpace V\ninst✝¹ : FiniteDimensional ℝ V\nμ : Measure V\ninst✝ : μ.IsAddHaarMeasure\nhf : Integrable f μ\nh'f : Differentiable ℝ f\nhf' : Integrable (fderiv ℝ f) μ\nw : W\ny : V\ng : V → ℂ := fun v => ↑(𝐞 (-(L v) w))\nA : ∀ (x : V), (fderiv ℝ g x) y = -2 * ↑π * I * ↑((L y) w) * g x\n⊢ Integrable (fun x => (fderiv ℝ g x) y • f x) μ\n\ncase h.h.hfg'\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : NormedSpace ℝ V\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\ninst✝³ : MeasurableSpace V\ninst✝² : BorelSpace V\ninst✝¹ : FiniteDimensional ℝ V\nμ : Measure V\ninst✝ : μ.IsAddHaarMeasure\nhf : Integrable f μ\nh'f : Differentiable ℝ f\nhf' : Integrable (fderiv ℝ f) μ\nw : W\ny : V\ng : V → ℂ := fun v => ↑(𝐞 (-(L v) w))\nA : ∀ (x : V), (fderiv ℝ g x) y = -2 * ↑π * I * ↑((L y) w) * g x\n⊢ Integrable (fun x => g x • (fderiv ℝ f x) y) μ\n\ncase h.h.hfg\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : NormedSpace ℝ V\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\ninst✝³ : MeasurableSpace V\ninst✝² : BorelSpace V\ninst✝¹ : FiniteDimensional ℝ V\nμ : Measure V\ninst✝ : μ.IsAddHaarMeasure\nhf : Integrable f μ\nh'f : Differentiable ℝ f\nhf' : Integrable (fderiv ℝ f) μ\nw : W\ny : V\ng : V → ℂ := fun v => ↑(𝐞 (-(L v) w))\nA : ∀ (x : V), (fderiv ℝ g x) y = -2 * ↑π * I * ↑((L y) w) * g x\n⊢ Integrable (fun x => g x • f x) μ\n\ncase h.h.hf\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : NormedSpace ℝ V\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\ninst✝³ : MeasurableSpace V\ninst✝² : BorelSpace V\ninst✝¹ : FiniteDimensional ℝ V\nμ : Measure V\ninst✝ : μ.IsAddHaarMeasure\nhf : Integrable f μ\nh'f : Differentiable ℝ f\nhf' : Integrable (fderiv ℝ f) μ\nw : W\ny : V\ng : V → ℂ := fun v => ↑(𝐞 (-(L v) w))\nA : ∀ (x : V), (fderiv ℝ g x) y = -2 * ↑π * I * ↑((L y) w) * g x\n⊢ Differentiable ℝ g\n\ncase h.h.hg\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : NormedSpace ℝ V\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\ninst✝³ : MeasurableSpace V\ninst✝² : BorelSpace V\ninst✝¹ : FiniteDimensional ℝ V\nμ : Measure V\ninst✝ : μ.IsAddHaarMeasure\nhf : Integrable f μ\nh'f : Differentiable ℝ f\nhf' : Integrable (fderiv ℝ f) μ\nw : W\ny : V\ng : V → ℂ := fun v => ↑(𝐞 (-(L v) w))\nA : ∀ (x : V), (fderiv ℝ g x) y = -2 * ↑π * I * ↑((L y) w) * g x\n⊢ Differentiable ℝ f","tactic":"rw [integral_smul_fderiv_eq_neg_fderiv_smul_of_integrable, ← integral_neg]","premises":[{"full_name":"MeasureTheory.integral_neg","def_path":"Mathlib/MeasureTheory/Integral/Bochner.lean","def_pos":[803,8],"def_end_pos":[803,20]},{"full_name":"integral_smul_fderiv_eq_neg_fderiv_smul_of_integrable","def_path":"Mathlib/Analysis/Calculus/LineDeriv/IntegrationByParts.lean","def_pos":[188,8],"def_end_pos":[188,61]}]}]}
+{"url":"Mathlib/Order/Interval/Set/Basic.lean","commit":"","full_name":"Set.Ioo_union_Ioi","start":[1040,0],"end":[1044,31],"file_path":"Mathlib/Order/Interval/Set/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nh : c < max a b\n⊢ Ioo a b ∪ Ioi c = Ioi (min a c)","state_after":"case inl\nα : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nhab : a ≤ b\nh : c < b\n⊢ Ioo a b ∪ Ioi c = Ioi (min a c)\n\ncase inr\nα : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nhab : b ≤ a\nh : c < a\n⊢ Ioo a b ∪ Ioi c = Ioi (min a c)","tactic":"rcases le_total a b with hab | hab <;> simp [hab] at h","premises":[{"full_name":"le_total","def_path":"Mathlib/Order/Defs.lean","def_pos":[254,8],"def_end_pos":[254,16]}]}]}
+{"url":"Mathlib/Topology/MetricSpace/HausdorffDistance.lean","commit":"","full_name":"Metric.hausdorffDist_image","start":[745,0],"end":[748,46],"file_path":"Mathlib/Topology/MetricSpace/HausdorffDistance.lean","tactics":[{"state_before":"ι : Sort u_1\nα : Type u\nβ : Type v\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\ns t u : Set α\nx y : α\nΦ : α → β\nh : Isometry Φ\n⊢ hausdorffDist (Φ '' s) (Φ '' t) = hausdorffDist s t","state_after":"no goals","tactic":"simp [hausdorffDist, hausdorffEdist_image h]","premises":[{"full_name":"EMetric.hausdorffEdist_image","def_path":"Mathlib/Topology/MetricSpace/HausdorffDistance.lean","def_pos":[317,8],"def_end_pos":[317,28]},{"full_name":"Metric.hausdorffDist","def_path":"Mathlib/Topology/MetricSpace/HausdorffDistance.lean","def_pos":[616,4],"def_end_pos":[616,17]}]}]}
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+{"url":"Mathlib/Algebra/TrivSqZeroExt.lean","commit":"","full_name":"TrivSqZeroExt.snd_pow_of_smul_comm","start":[571,0],"end":[590,92],"file_path":"Mathlib/Algebra/TrivSqZeroExt.lean","tactics":[{"state_before":"R : Type u\nM : Type v\ninst✝⁴ : Monoid R\ninst✝³ : AddMonoid M\ninst✝² : DistribMulAction R M\ninst✝¹ : DistribMulAction Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nx : tsze R M\nn : ℕ\nh : x.snd <• x.fst = x.fst •> x.snd\n⊢ (x ^ n).snd = n •> x.fst ^ n.pred •> x.snd","state_after":"R : Type u\nM : Type v\ninst✝⁴ : Monoid R\ninst✝³ : AddMonoid M\ninst✝² : DistribMulAction R M\ninst✝¹ : DistribMulAction Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nx : tsze R M\nn : ℕ\nh : x.snd <• x.fst = x.fst •> x.snd\n⊢ (List.map (fun i => x.fst ^ (n.pred - i + i) •> x.snd) (List.range n)).sum = n •> x.fst ^ n.pred •> x.snd","tactic":"simp_rw [snd_pow_eq_sum, ← smul_comm (_ : R) (_ : Rᵐᵒᵖ), aux, smul_smul, ← pow_add]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"MulOpposite","def_path":"Mathlib/Algebra/Opposites.lean","def_pos":[62,4],"def_end_pos":[62,15]},{"full_name":"SMulCommClass.smul_comm","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[154,2],"def_end_pos":[154,11]},{"full_name":"TrivSqZeroExt.snd_pow_eq_sum","def_path":"Mathlib/Algebra/TrivSqZeroExt.lean","def_pos":[566,8],"def_end_pos":[566,22]},{"full_name":"pow_add","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[598,6],"def_end_pos":[598,13]},{"full_name":"smul_smul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[374,6],"def_end_pos":[374,15]}]},{"state_before":"R : Type u\nM : Type v\ninst✝⁴ : Monoid R\ninst✝³ : AddMonoid M\ninst✝² : DistribMulAction R M\ninst✝¹ : DistribMulAction Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nx : tsze R M\nn : ℕ\nh : x.snd <• x.fst = x.fst •> x.snd\n⊢ ∀ (n : ℕ), x.snd <• x.fst ^ n = x.fst ^ n •> x.snd","state_after":"R : Type u\nM : Type v\ninst✝⁴ : Monoid R\ninst✝³ : AddMonoid M\ninst✝² : DistribMulAction R M\ninst✝¹ : DistribMulAction Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nx : tsze R M\nn✝ : ℕ\nh : x.snd <• x.fst = x.fst •> x.snd\nn : ℕ\n⊢ x.snd <• x.fst ^ n = x.fst ^ n •> x.snd","tactic":"intro n","premises":[]},{"state_before":"R : Type u\nM : Type v\ninst✝⁴ : Monoid R\ninst✝³ : AddMonoid M\ninst✝² : DistribMulAction R M\ninst✝¹ : DistribMulAction Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nx : tsze R M\nn✝ : ℕ\nh : x.snd <• x.fst = x.fst •> x.snd\nn : ℕ\n⊢ x.snd <• x.fst ^ n = x.fst ^ n •> x.snd","state_after":"case zero\nR : Type u\nM : Type v\ninst✝⁴ : Monoid R\ninst✝³ : AddMonoid M\ninst✝² : DistribMulAction R M\ninst✝¹ : DistribMulAction Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nx : tsze R M\nn : ℕ\nh : x.snd <• x.fst = x.fst •> x.snd\n⊢ x.snd <• x.fst ^ 0 = x.fst ^ 0 •> x.snd\n\ncase succ\nR : Type u\nM : Type v\ninst✝⁴ : Monoid R\ninst✝³ : AddMonoid M\ninst✝² : DistribMulAction R M\ninst✝¹ : DistribMulAction Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nx : tsze R M\nn✝ : ℕ\nh : x.snd <• x.fst = x.fst •> x.snd\nn : ℕ\nih : x.snd <• x.fst ^ n = x.fst ^ n •> x.snd\n⊢ x.snd <• x.fst ^ (n + 1) = x.fst ^ (n + 1) •> x.snd","tactic":"induction' n with n ih","premises":[]}]}
+{"url":"Mathlib/Geometry/Manifold/VectorBundle/Basic.lean","commit":"","full_name":"Bundle.contMDiffAt_totalSpace","start":[191,0],"end":[196,74],"file_path":"Mathlib/Geometry/Manifold/VectorBundle/Basic.lean","tactics":[{"state_before":"𝕜 : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B → Type u_6\ninst✝¹⁶ : NontriviallyNormedField 𝕜\ninst✝¹⁵ : NormedAddCommGroup F\ninst✝¹⁴ : NormedSpace 𝕜 F\ninst✝¹³ : TopologicalSpace (TotalSpace F E)\ninst✝¹² : (x : B) → TopologicalSpace (E x)\nEB : Type u_7\ninst✝¹¹ : NormedAddCommGroup EB\ninst✝¹⁰ : NormedSpace 𝕜 EB\nHB : Type u_8\ninst✝⁹ : TopologicalSpace HB\nIB : ModelWithCorners 𝕜 EB HB\nE' : B → Type u_9\ninst✝⁸ : (x : B) → Zero (E' x)\nEM : Type u_10\ninst✝⁷ : NormedAddCommGroup EM\ninst✝⁶ : NormedSpace 𝕜 EM\nHM : Type u_11\ninst✝⁵ : TopologicalSpace HM\nIM : ModelWithCorners 𝕜 EM HM\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : ℕ∞\ninst✝² : TopologicalSpace B\ninst✝¹ : ChartedSpace HB B\ninst✝ : FiberBundle F E\nf : M → TotalSpace F E\nx₀ : M\n⊢ ContMDiffAt IM (IB.prod 𝓘(𝕜, F)) n f x₀ ↔\n    ContMDiffAt IM IB n (fun x => (f x).proj) x₀ ∧\n      ContMDiffAt IM 𝓘(𝕜, F) n (fun x => (↑(trivializationAt F E (f x₀).proj) (f x)).2) x₀","state_after":"𝕜 : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B → Type u_6\ninst✝¹⁶ : NontriviallyNormedField 𝕜\ninst✝¹⁵ : NormedAddCommGroup F\ninst✝¹⁴ : NormedSpace 𝕜 F\ninst✝¹³ : TopologicalSpace (TotalSpace F E)\ninst✝¹² : (x : B) → TopologicalSpace (E x)\nEB : Type u_7\ninst✝¹¹ : NormedAddCommGroup EB\ninst✝¹⁰ : NormedSpace 𝕜 EB\nHB : Type u_8\ninst✝⁹ : TopologicalSpace HB\nIB : ModelWithCorners 𝕜 EB HB\nE' : B → Type u_9\ninst✝⁸ : (x : B) → Zero (E' x)\nEM : Type u_10\ninst✝⁷ : NormedAddCommGroup EM\ninst✝⁶ : NormedSpace 𝕜 EM\nHM : Type u_11\ninst✝⁵ : TopologicalSpace HM\nIM : ModelWithCorners 𝕜 EM HM\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : ℕ∞\ninst✝² : TopologicalSpace B\ninst✝¹ : ChartedSpace HB B\ninst✝ : FiberBundle F E\nf : M → TotalSpace F E\nx₀ : M\n⊢ ContMDiffWithinAt IM (IB.prod 𝓘(𝕜, F)) n f univ x₀ ↔\n    ContMDiffWithinAt IM IB n (fun x => (f x).proj) univ x₀ ∧\n      ContMDiffWithinAt IM 𝓘(𝕜, F) n (fun x => (↑(trivializationAt F E (f x₀).proj) (f x)).2) univ x₀","tactic":"simp_rw [← contMDiffWithinAt_univ]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"contMDiffWithinAt_univ","def_path":"Mathlib/Geometry/Manifold/ContMDiff/Defs.lean","def_pos":[269,8],"def_end_pos":[269,30]}]},{"state_before":"𝕜 : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B → Type u_6\ninst✝¹⁶ : NontriviallyNormedField 𝕜\ninst✝¹⁵ : NormedAddCommGroup F\ninst✝¹⁴ : NormedSpace 𝕜 F\ninst✝¹³ : TopologicalSpace (TotalSpace F E)\ninst✝¹² : (x : B) → TopologicalSpace (E x)\nEB : Type u_7\ninst✝¹¹ : NormedAddCommGroup EB\ninst✝¹⁰ : NormedSpace 𝕜 EB\nHB : Type u_8\ninst✝⁹ : TopologicalSpace HB\nIB : ModelWithCorners 𝕜 EB HB\nE' : B → Type u_9\ninst✝⁸ : (x : B) → Zero (E' x)\nEM : Type u_10\ninst✝⁷ : NormedAddCommGroup EM\ninst✝⁶ : NormedSpace 𝕜 EM\nHM : Type u_11\ninst✝⁵ : TopologicalSpace HM\nIM : ModelWithCorners 𝕜 EM HM\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : ℕ∞\ninst✝² : TopologicalSpace B\ninst✝¹ : ChartedSpace HB B\ninst✝ : FiberBundle F E\nf : M → TotalSpace F E\nx₀ : M\n⊢ ContMDiffWithinAt IM (IB.prod 𝓘(𝕜, F)) n f univ x₀ ↔\n    ContMDiffWithinAt IM IB n (fun x => (f x).proj) univ x₀ ∧\n      ContMDiffWithinAt IM 𝓘(𝕜, F) n (fun x => (↑(trivializationAt F E (f x₀).proj) (f x)).2) univ x₀","state_after":"no goals","tactic":"exact contMDiffWithinAt_totalSpace f","premises":[{"full_name":"Bundle.contMDiffWithinAt_totalSpace","def_path":"Mathlib/Geometry/Manifold/VectorBundle/Basic.lean","def_pos":[171,8],"def_end_pos":[171,36]}]}]}
+{"url":"Mathlib/Algebra/Order/Monovary.lean","commit":"","full_name":"antivaryOn_inv₀","start":[257,0],"end":[259,56],"file_path":"Mathlib/Algebra/Order/Monovary.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : LinearOrderedSemifield α\ninst✝ : LinearOrderedSemifield β\ns : Set ι\nf f₁ f₂ : ι → α\ng g₁ g₂ : ι → β\nhf : ∀ i ∈ s, 0 < f i\nhg : ∀ i ∈ s, 0 < g i\n⊢ AntivaryOn f⁻¹ g⁻¹ s ↔ AntivaryOn f g s","state_after":"no goals","tactic":"rw [antivaryOn_inv_left₀ hf, monovaryOn_inv_right₀ hg]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"antivaryOn_inv_left₀","def_path":"Mathlib/Algebra/Order/Monovary.lean","def_pos":[243,6],"def_end_pos":[243,26]},{"full_name":"monovaryOn_inv_right₀","def_path":"Mathlib/Algebra/Order/Monovary.lean","def_pos":[247,6],"def_end_pos":[247,27]}]}]}
+{"url":"Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean","commit":"","full_name":"CategoryTheory.ShortComplex.LeftHomologyData.ofEpiOfIsIsoOfMono'_i","start":[833,0],"end":[859,42],"file_path":"Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean","tactics":[{"state_before":"C : Type u_1\ninst✝⁴ : Category.{?u.164135, u_1} C\ninst✝³ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh : S₂.LeftHomologyData\ninst✝² : Epi φ.τ₁\ninst✝¹ : IsIso φ.τ₂\ninst✝ : Mono φ.τ₃\n⊢ S₁.LeftHomologyData","state_after":"C : Type u_1\ninst✝⁴ : Category.{?u.164135, u_1} C\ninst✝³ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh : S₂.LeftHomologyData\ninst✝² : Epi φ.τ₁\ninst✝¹ : IsIso φ.τ₂\ninst✝ : Mono φ.τ₃\ni : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂\n⊢ S₁.LeftHomologyData","tactic":"let i : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂","premises":[{"full_name":"CategoryTheory.CategoryStruct.comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]},{"full_name":"CategoryTheory.ShortComplex.Hom.τ₂","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[56,2],"def_end_pos":[56,4]},{"full_name":"CategoryTheory.ShortComplex.LeftHomologyData.K","def_path":"Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean","def_pos":[46,2],"def_end_pos":[46,3]},{"full_name":"CategoryTheory.ShortComplex.LeftHomologyData.i","def_path":"Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean","def_pos":[50,2],"def_end_pos":[50,3]},{"full_name":"CategoryTheory.ShortComplex.X₂","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[33,3],"def_end_pos":[33,5]},{"full_name":"CategoryTheory.inv","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[235,18],"def_end_pos":[235,21]},{"full_name":"Quiver.Hom","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[43,2],"def_end_pos":[43,5]}]},{"state_before":"C : Type u_1\ninst✝⁴ : Category.{?u.164135, u_1} C\ninst✝³ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh : S₂.LeftHomologyData\ninst✝² : Epi φ.τ₁\ninst✝¹ : IsIso φ.τ₂\ninst✝ : Mono φ.τ₃\ni : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂\n⊢ S₁.LeftHomologyData","state_after":"C : Type u_1\ninst✝⁴ : Category.{?u.164135, u_1} C\ninst✝³ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh : S₂.LeftHomologyData\ninst✝² : Epi φ.τ₁\ninst✝¹ : IsIso φ.τ₂\ninst✝ : Mono φ.τ₃\ni : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂\nwi : i ≫ S₁.g = 0\n⊢ S₁.LeftHomologyData","tactic":"have wi : i ≫ S₁.g = 0 := by\n    rw [assoc, ← cancel_mono φ.τ₃, zero_comp, assoc, assoc, ← φ.comm₂₃,\n      IsIso.inv_hom_id_assoc, h.wi]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.CategoryStruct.comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]},{"full_name":"CategoryTheory.IsIso.inv_hom_id_assoc","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[263,8],"def_end_pos":[263,24]},{"full_name":"CategoryTheory.Limits.zero_comp","def_path":"Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean","def_pos":[66,8],"def_end_pos":[66,17]},{"full_name":"CategoryTheory.ShortComplex.Hom.comm₂₃","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[62,2],"def_end_pos":[62,8]},{"full_name":"CategoryTheory.ShortComplex.Hom.τ₃","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[58,2],"def_end_pos":[58,4]},{"full_name":"CategoryTheory.ShortComplex.LeftHomologyData.wi","def_path":"Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean","def_pos":[54,2],"def_end_pos":[54,4]},{"full_name":"CategoryTheory.ShortComplex.g","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[39,2],"def_end_pos":[39,3]},{"full_name":"CategoryTheory.cancel_mono","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[263,8],"def_end_pos":[263,19]}]},{"state_before":"C : Type u_1\ninst✝⁴ : Category.{?u.164135, u_1} C\ninst✝³ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh : S₂.LeftHomologyData\ninst✝² : Epi φ.τ₁\ninst✝¹ : IsIso φ.τ₂\ninst✝ : Mono φ.τ₃\ni : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂\nwi : i ≫ S₁.g = 0\n⊢ S₁.LeftHomologyData","state_after":"C : Type u_1\ninst✝⁴ : Category.{?u.164135, u_1} C\ninst✝³ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh : S₂.LeftHomologyData\ninst✝² : Epi φ.τ₁\ninst✝¹ : IsIso φ.τ₂\ninst✝ : Mono φ.τ₃\ni : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂\nwi : i ≫ S₁.g = 0\nhi : IsLimit (KernelFork.ofι i wi)\n⊢ S₁.LeftHomologyData","tactic":"have hi : IsLimit (KernelFork.ofι i wi) := KernelFork.IsLimit.ofι _ _\n    (fun x hx => h.liftK (x ≫ φ.τ₂)\n      (by rw [assoc, φ.comm₂₃, reassoc_of% hx, zero_comp]))\n    (fun x hx => by simp [i])\n    (fun x hx b hb => by rw [← cancel_mono h.i, ← cancel_mono (inv φ.τ₂), assoc, assoc,\n      hb, liftK_i_assoc, assoc, IsIso.hom_inv_id, comp_id])","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Category.comp_id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[153,2],"def_end_pos":[153,9]},{"full_name":"CategoryTheory.CategoryStruct.comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]},{"full_name":"CategoryTheory.IsIso.hom_inv_id","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[241,8],"def_end_pos":[241,18]},{"full_name":"CategoryTheory.Limits.IsLimit","def_path":"Mathlib/CategoryTheory/Limits/IsLimit.lean","def_pos":[54,10],"def_end_pos":[54,17]},{"full_name":"CategoryTheory.Limits.KernelFork.IsLimit.ofι","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","def_pos":[143,4],"def_end_pos":[143,26]},{"full_name":"CategoryTheory.Limits.KernelFork.ofι","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","def_pos":[89,7],"def_end_pos":[89,21]},{"full_name":"CategoryTheory.Limits.zero_comp","def_path":"Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean","def_pos":[66,8],"def_end_pos":[66,17]},{"full_name":"CategoryTheory.ShortComplex.Hom.comm₂₃","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[62,2],"def_end_pos":[62,8]},{"full_name":"CategoryTheory.ShortComplex.Hom.τ₂","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[56,2],"def_end_pos":[56,4]},{"full_name":"CategoryTheory.ShortComplex.LeftHomologyData.i","def_path":"Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean","def_pos":[50,2],"def_end_pos":[50,3]},{"full_name":"CategoryTheory.ShortComplex.LeftHomologyData.liftK","def_path":"Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean","def_pos":[91,4],"def_end_pos":[91,9]},{"full_name":"CategoryTheory.cancel_mono","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[263,8],"def_end_pos":[263,19]},{"full_name":"CategoryTheory.inv","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[235,18],"def_end_pos":[235,21]}]},{"state_before":"C : Type u_1\ninst✝⁴ : Category.{?u.164135, u_1} C\ninst✝³ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh : S₂.LeftHomologyData\ninst✝² : Epi φ.τ₁\ninst✝¹ : IsIso φ.τ₂\ninst✝ : Mono φ.τ₃\ni : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂\nwi : i ≫ S₁.g = 0\nhi : IsLimit (KernelFork.ofι i wi)\n⊢ S₁.LeftHomologyData","state_after":"C : Type u_1\ninst✝⁴ : Category.{?u.164135, u_1} C\ninst✝³ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh : S₂.LeftHomologyData\ninst✝² : Epi φ.τ₁\ninst✝¹ : IsIso φ.τ₂\ninst✝ : Mono φ.τ₃\ni : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂\nwi : i ≫ S₁.g = 0\nhi : IsLimit (KernelFork.ofι i wi)\nf' : (KernelFork.ofι S₁.f ⋯).pt ⟶ (KernelFork.ofι i wi).pt := hi.lift (KernelFork.ofι S₁.f ⋯)\n⊢ S₁.LeftHomologyData","tactic":"let f' := hi.lift (KernelFork.ofι S₁.f S₁.zero)","premises":[{"full_name":"CategoryTheory.Limits.IsLimit.lift","def_path":"Mathlib/CategoryTheory/Limits/IsLimit.lean","def_pos":[56,2],"def_end_pos":[56,6]},{"full_name":"CategoryTheory.Limits.KernelFork.ofι","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","def_pos":[89,7],"def_end_pos":[89,21]},{"full_name":"CategoryTheory.ShortComplex.f","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[37,2],"def_end_pos":[37,3]},{"full_name":"CategoryTheory.ShortComplex.zero","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[41,2],"def_end_pos":[41,6]}]},{"state_before":"C : Type u_1\ninst✝⁴ : Category.{?u.164135, u_1} C\ninst✝³ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh : S₂.LeftHomologyData\ninst✝² : Epi φ.τ₁\ninst✝¹ : IsIso φ.τ₂\ninst✝ : Mono φ.τ₃\ni : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂\nwi : i ≫ S₁.g = 0\nhi : IsLimit (KernelFork.ofι i wi)\nf' : (KernelFork.ofι S₁.f ⋯).pt ⟶ (KernelFork.ofι i wi).pt := hi.lift (KernelFork.ofι S₁.f ⋯)\n⊢ S₁.LeftHomologyData","state_after":"C : Type u_1\ninst✝⁴ : Category.{?u.164135, u_1} C\ninst✝³ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh : S₂.LeftHomologyData\ninst✝² : Epi φ.τ₁\ninst✝¹ : IsIso φ.τ₂\ninst✝ : Mono φ.τ₃\ni : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂\nwi : i ≫ S₁.g = 0\nhi : IsLimit (KernelFork.ofι i wi)\nf' : (KernelFork.ofι S₁.f ⋯).pt ⟶ (KernelFork.ofι i wi).pt := hi.lift (KernelFork.ofι S₁.f ⋯)\nhf' : f' ≫ i = S₁.f\n⊢ S₁.LeftHomologyData","tactic":"have hf' : f' ≫ i = S₁.f := Fork.IsLimit.lift_ι _","premises":[{"full_name":"CategoryTheory.CategoryStruct.comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]},{"full_name":"CategoryTheory.Limits.Fork.IsLimit.lift_ι","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean","def_pos":[382,8],"def_end_pos":[382,27]},{"full_name":"CategoryTheory.ShortComplex.f","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[37,2],"def_end_pos":[37,3]}]},{"state_before":"C : Type u_1\ninst✝⁴ : Category.{?u.164135, u_1} C\ninst✝³ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh : S₂.LeftHomologyData\ninst✝² : Epi φ.τ₁\ninst✝¹ : IsIso φ.τ₂\ninst✝ : Mono φ.τ₃\ni : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂\nwi : i ≫ S₁.g = 0\nhi : IsLimit (KernelFork.ofι i wi)\nf' : (KernelFork.ofι S₁.f ⋯).pt ⟶ (KernelFork.ofι i wi).pt := hi.lift (KernelFork.ofι S₁.f ⋯)\nhf' : f' ≫ i = S₁.f\n⊢ S₁.LeftHomologyData","state_after":"C : Type u_1\ninst✝⁴ : Category.{?u.164135, u_1} C\ninst✝³ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh : S₂.LeftHomologyData\ninst✝² : Epi φ.τ₁\ninst✝¹ : IsIso φ.τ₂\ninst✝ : Mono φ.τ₃\ni : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂\nwi : i ≫ S₁.g = 0\nhi : IsLimit (KernelFork.ofι i wi)\nf' : (KernelFork.ofι S₁.f ⋯).pt ⟶ (KernelFork.ofι i wi).pt := hi.lift (KernelFork.ofι S₁.f ⋯)\nhf' : f' ≫ i = S₁.f\nhf'' : f' = φ.τ₁ ≫ h.f'\n⊢ S₁.LeftHomologyData","tactic":"have hf'' : f' = φ.τ₁ ≫ h.f' := by\n    rw [← cancel_mono h.i, ← cancel_mono (inv φ.τ₂), assoc, assoc, assoc, hf', f'_i_assoc,\n      φ.comm₁₂_assoc, IsIso.hom_inv_id, comp_id]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Category.comp_id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[153,2],"def_end_pos":[153,9]},{"full_name":"CategoryTheory.CategoryStruct.comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]},{"full_name":"CategoryTheory.IsIso.hom_inv_id","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[241,8],"def_end_pos":[241,18]},{"full_name":"CategoryTheory.ShortComplex.Hom.τ₁","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[54,2],"def_end_pos":[54,4]},{"full_name":"CategoryTheory.ShortComplex.Hom.τ₂","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[56,2],"def_end_pos":[56,4]},{"full_name":"CategoryTheory.ShortComplex.LeftHomologyData.f'","def_path":"Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean","def_pos":[103,4],"def_end_pos":[103,6]},{"full_name":"CategoryTheory.ShortComplex.LeftHomologyData.i","def_path":"Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean","def_pos":[50,2],"def_end_pos":[50,3]},{"full_name":"CategoryTheory.cancel_mono","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[263,8],"def_end_pos":[263,19]},{"full_name":"CategoryTheory.inv","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[235,18],"def_end_pos":[235,21]}]},{"state_before":"C : Type u_1\ninst✝⁴ : Category.{?u.164135, u_1} C\ninst✝³ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh : S₂.LeftHomologyData\ninst✝² : Epi φ.τ₁\ninst✝¹ : IsIso φ.τ₂\ninst✝ : Mono φ.τ₃\ni : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂\nwi : i ≫ S₁.g = 0\nhi : IsLimit (KernelFork.ofι i wi)\nf' : (KernelFork.ofι S₁.f ⋯).pt ⟶ (KernelFork.ofι i wi).pt := hi.lift (KernelFork.ofι S₁.f ⋯)\nhf' : f' ≫ i = S₁.f\nhf'' : f' = φ.τ₁ ≫ h.f'\n⊢ S₁.LeftHomologyData","state_after":"C : Type u_1\ninst✝⁴ : Category.{?u.164135, u_1} C\ninst✝³ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh : S₂.LeftHomologyData\ninst✝² : Epi φ.τ₁\ninst✝¹ : IsIso φ.τ₂\ninst✝ : Mono φ.τ₃\ni : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂\nwi : i ≫ S₁.g = 0\nhi : IsLimit (KernelFork.ofι i wi)\nf' : (KernelFork.ofι S₁.f ⋯).pt ⟶ (KernelFork.ofι i wi).pt := hi.lift (KernelFork.ofι S₁.f ⋯)\nhf' : f' ≫ i = S₁.f\nhf'' : f' = φ.τ₁ ≫ h.f'\nwπ : f' ≫ h.π = 0\n⊢ S₁.LeftHomologyData","tactic":"have wπ : f' ≫ h.π = 0 := by simp only [hf'', assoc, f'_π, comp_zero]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.CategoryStruct.comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]},{"full_name":"CategoryTheory.Limits.comp_zero","def_path":"Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean","def_pos":[61,8],"def_end_pos":[61,17]},{"full_name":"CategoryTheory.ShortComplex.LeftHomologyData.f'_π","def_path":"Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean","def_pos":[107,32],"def_end_pos":[107,36]},{"full_name":"CategoryTheory.ShortComplex.LeftHomologyData.π","def_path":"Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean","def_pos":[52,2],"def_end_pos":[52,3]}]},{"state_before":"C : Type u_1\ninst✝⁴ : Category.{?u.164135, u_1} C\ninst✝³ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh : S₂.LeftHomologyData\ninst✝² : Epi φ.τ₁\ninst✝¹ : IsIso φ.τ₂\ninst✝ : Mono φ.τ₃\ni : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂\nwi : i ≫ S₁.g = 0\nhi : IsLimit (KernelFork.ofι i wi)\nf' : (KernelFork.ofι S₁.f ⋯).pt ⟶ (KernelFork.ofι i wi).pt := hi.lift (KernelFork.ofι S₁.f ⋯)\nhf' : f' ≫ i = S₁.f\nhf'' : f' = φ.τ₁ ≫ h.f'\nwπ : f' ≫ h.π = 0\n⊢ S₁.LeftHomologyData","state_after":"C : Type u_1\ninst✝⁴ : Category.{?u.164135, u_1} C\ninst✝³ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh : S₂.LeftHomologyData\ninst✝² : Epi φ.τ₁\ninst✝¹ : IsIso φ.τ₂\ninst✝ : Mono φ.τ₃\ni : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂\nwi : i ≫ S₁.g = 0\nhi : IsLimit (KernelFork.ofι i wi)\nf' : (KernelFork.ofι S₁.f ⋯).pt ⟶ (KernelFork.ofι i wi).pt := hi.lift (KernelFork.ofι S₁.f ⋯)\nhf' : f' ≫ i = S₁.f\nhf'' : f' = φ.τ₁ ≫ h.f'\nwπ : f' ≫ h.π = 0\nhπ : IsColimit (CokernelCofork.ofπ h.π wπ)\n⊢ S₁.LeftHomologyData","tactic":"have hπ : IsColimit (CokernelCofork.ofπ h.π wπ) := CokernelCofork.IsColimit.ofπ _ _\n    (fun x hx => h.descH x (by rw [← cancel_epi φ.τ₁, ← reassoc_of% hf'', hx, comp_zero]))\n    (fun x hx => π_descH _ _ _)\n    (fun x hx b hx => by rw [← cancel_epi h.π, π_descH, hx])","premises":[{"full_name":"CategoryTheory.Limits.CokernelCofork.IsColimit.ofπ","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","def_pos":[563,4],"def_end_pos":[563,32]},{"full_name":"CategoryTheory.Limits.CokernelCofork.ofπ","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","def_pos":[524,7],"def_end_pos":[524,25]},{"full_name":"CategoryTheory.Limits.IsColimit","def_path":"Mathlib/CategoryTheory/Limits/IsLimit.lean","def_pos":[507,10],"def_end_pos":[507,19]},{"full_name":"CategoryTheory.Limits.comp_zero","def_path":"Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean","def_pos":[61,8],"def_end_pos":[61,17]},{"full_name":"CategoryTheory.ShortComplex.Hom.τ₁","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[54,2],"def_end_pos":[54,4]},{"full_name":"CategoryTheory.ShortComplex.LeftHomologyData.descH","def_path":"Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean","def_pos":[121,4],"def_end_pos":[121,9]},{"full_name":"CategoryTheory.ShortComplex.LeftHomologyData.π","def_path":"Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean","def_pos":[52,2],"def_end_pos":[52,3]},{"full_name":"CategoryTheory.ShortComplex.LeftHomologyData.π_descH","def_path":"Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean","def_pos":[125,6],"def_end_pos":[125,13]},{"full_name":"CategoryTheory.cancel_epi","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[260,8],"def_end_pos":[260,18]}]},{"state_before":"C : Type u_1\ninst✝⁴ : Category.{?u.164135, u_1} C\ninst✝³ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh : S₂.LeftHomologyData\ninst✝² : Epi φ.τ₁\ninst✝¹ : IsIso φ.τ₂\ninst✝ : Mono φ.τ₃\ni : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂\nwi : i ≫ S₁.g = 0\nhi : IsLimit (KernelFork.ofι i wi)\nf' : (KernelFork.ofι S₁.f ⋯).pt ⟶ (KernelFork.ofι i wi).pt := hi.lift (KernelFork.ofι S₁.f ⋯)\nhf' : f' ≫ i = S₁.f\nhf'' : f' = φ.τ₁ ≫ h.f'\nwπ : f' ≫ h.π = 0\nhπ : IsColimit (CokernelCofork.ofπ h.π wπ)\n⊢ S₁.LeftHomologyData","state_after":"no goals","tactic":"exact ⟨h.K, h.H, i, h.π, wi, hi, wπ, hπ⟩","premises":[{"full_name":"CategoryTheory.ShortComplex.LeftHomologyData.H","def_path":"Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean","def_pos":[48,2],"def_end_pos":[48,3]},{"full_name":"CategoryTheory.ShortComplex.LeftHomologyData.K","def_path":"Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean","def_pos":[46,2],"def_end_pos":[46,3]},{"full_name":"CategoryTheory.ShortComplex.LeftHomologyData.π","def_path":"Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean","def_pos":[52,2],"def_end_pos":[52,3]}]}]}
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+{"url":"Mathlib/Order/UpperLower/LocallyFinite.lean","commit":"","full_name":"Set.Finite.upperClosure","start":[20,0],"end":[23,42],"file_path":"Mathlib/Order/UpperLower/LocallyFinite.lean","tactics":[{"state_before":"α : Type u_1\ninst✝¹ : Preorder α\ns : Set α\ninst✝ : LocallyFiniteOrderTop α\nhs : s.Finite\n⊢ (↑(upperClosure s)).Finite","state_after":"α : Type u_1\ninst✝¹ : Preorder α\ns : Set α\ninst✝ : LocallyFiniteOrderTop α\nhs : s.Finite\n⊢ (⋃ a ∈ s, Ici a).Finite","tactic":"rw [coe_upperClosure]","premises":[{"full_name":"coe_upperClosure","def_path":"Mathlib/Order/UpperLower/Basic.lean","def_pos":[1256,8],"def_end_pos":[1256,24]}]},{"state_before":"α : Type u_1\ninst✝¹ : Preorder α\ns : Set α\ninst✝ : LocallyFiniteOrderTop α\nhs : s.Finite\n⊢ (⋃ a ∈ s, Ici a).Finite","state_after":"no goals","tactic":"exact hs.biUnion fun _ _ => finite_Ici _","premises":[{"full_name":"Set.Finite.biUnion","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[683,8],"def_end_pos":[683,22]},{"full_name":"Set.finite_Ici","def_path":"Mathlib/Order/Interval/Finset/Defs.lean","def_pos":[494,8],"def_end_pos":[494,18]}]}]}
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sin z / cos z * I)) = z","state_after":"z : ℂ\nh₀ : z ≠ ↑π / 2\nh₁ : -(π / 2) < z.re\nh₂ : z.re ≤ π / 2\nh : cos z ≠ 0\n⊢ -I / 2 * log ((cos z + sin z * I) / (cos z - sin z * I)) = z","tactic":"rw [← mul_div_mul_right (1 + _) _ h, add_mul, sub_mul, one_mul, ← mul_rotate, mul_div_cancel₀ _ h]","premises":[{"full_name":"mul_div_cancel₀","def_path":"Mathlib/Algebra/GroupWithZero/Units/Basic.lean","def_pos":[424,6],"def_end_pos":[424,21]},{"full_name":"mul_div_mul_right","def_path":"Mathlib/Algebra/GroupWithZero/Units/Basic.lean","def_pos":[319,6],"def_end_pos":[319,23]},{"full_name":"mul_rotate","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[164,8],"def_end_pos":[164,18]},{"full_name":"one_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[409,8],"def_end_pos":[409,15]}]},{"state_before":"z : ℂ\nh₀ : z ≠ ↑π / 2\nh₁ : -(π / 2) < z.re\nh₂ : z.re ≤ π / 2\nh : cos z ≠ 0\n⊢ -I / 2 * log ((cos z + sin z * I) / (cos z - sin z * I)) = z","state_after":"z : ℂ\nh₀ : z ≠ ↑π / 2\nh₁ : -(π / 2) < z.re\nh₂ : z.re ≤ π / 2\nh : cos z ≠ 0\n⊢ -I / 2 * log ((cos z + sin z * I) / (cos (-z) + sin (-z) * I)) = z","tactic":"conv_lhs =>\n    enter [2, 1, 2]\n    rw [sub_eq_add_neg, ← neg_mul, ← sin_neg, ← cos_neg]","premises":[{"full_name":"Complex.cos_neg","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[455,8],"def_end_pos":[455,15]},{"full_name":"Complex.sin_neg","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[415,8],"def_end_pos":[415,15]},{"full_name":"neg_mul","def_path":"Mathlib/Algebra/Ring/Defs.lean","def_pos":[268,8],"def_end_pos":[268,15]},{"full_name":"sub_eq_add_neg","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[905,2],"def_end_pos":[905,13]}]},{"state_before":"z : ℂ\nh₀ : z ≠ ↑π / 2\nh₁ : -(π / 2) < z.re\nh₂ : z.re ≤ π / 2\nh : cos z ≠ 0\n⊢ -I / 2 * log ((cos z + sin z * I) / (cos (-z) + sin (-z) * I)) = z","state_after":"case hx₁\nz : ℂ\nh₀ : z ≠ ↑π / 2\nh₁ : -(π / 2) < z.re\nh₂ : z.re ≤ π / 2\nh : cos z ≠ 0\n⊢ -π < (2 * (I * z)).im\n\ncase hx₂\nz : ℂ\nh₀ : z ≠ ↑π / 2\nh₁ : -(π / 2) < z.re\nh₂ : z.re ≤ π / 2\nh : cos z ≠ 0\n⊢ (2 * (I * z)).im ≤ π","tactic":"rw [← exp_mul_I, ← exp_mul_I, ← exp_sub, show z * I - 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+{"url":"Mathlib/LinearAlgebra/LinearPMap.lean","commit":"","full_name":"LinearPMap.inverse_range","start":[1006,0],"end":[1009,5],"file_path":"Mathlib/LinearAlgebra/LinearPMap.lean","tactics":[{"state_before":"R : Type u_1\ninst✝⁶ : Ring R\nE : Type u_2\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_3\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nG : Type u_4\ninst✝¹ : AddCommGroup G\ninst✝ : Module R G\nf : E →ₗ.[R] F\nhf : LinearMap.ker f.toFun = ⊥\n⊢ LinearMap.range f.inverse.toFun = f.domain","state_after":"R : Type u_1\ninst✝⁶ : Ring R\nE : Type u_2\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_3\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nG : Type u_4\ninst✝¹ : AddCommGroup G\ninst✝ : Module R G\nf : E →ₗ.[R] F\nhf : LinearMap.ker f.toFun = ⊥\n⊢ Submodule.map (LinearMap.snd R F E) (Submodule.map (LinearEquiv.prodComm R E F) f.graph) =\n    Submodule.map (LinearMap.snd R F E) (Submodule.map (↑(LinearEquiv.prodComm R E F)) f.graph)","tactic":"rw [inverse, Submodule.toLinearPMap_range _ (mem_inverse_graph_snd_eq_zero hf),\n    ← graph_map_fst_eq_domain, ← LinearEquiv.snd_comp_prodComm, Submodule.map_comp]","premises":[{"full_name":"LinearEquiv.snd_comp_prodComm","def_path":"Mathlib/LinearAlgebra/Prod.lean","def_pos":[656,8],"def_end_pos":[656,25]},{"full_name":"LinearPMap.graph_map_fst_eq_domain","def_path":"Mathlib/LinearAlgebra/LinearPMap.lean","def_pos":[702,8],"def_end_pos":[702,31]},{"full_name":"LinearPMap.inverse","def_path":"Mathlib/LinearAlgebra/LinearPMap.lean","def_pos":[977,18],"def_end_pos":[977,25]},{"full_name":"LinearPMap.mem_inverse_graph_snd_eq_zero","def_path":"Mathlib/LinearAlgebra/LinearPMap.lean","def_pos":[990,8],"def_end_pos":[990,37]},{"full_name":"Submodule.map_comp","def_path":"Mathlib/Algebra/Module/Submodule/Map.lean","def_pos":[98,8],"def_end_pos":[98,16]},{"full_name":"Submodule.toLinearPMap_range","def_path":"Mathlib/LinearAlgebra/LinearPMap.lean","def_pos":[963,8],"def_end_pos":[963,26]}]},{"state_before":"R : Type u_1\ninst✝⁶ : Ring R\nE : Type u_2\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_3\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nG : Type u_4\ninst✝¹ : AddCommGroup G\ninst✝ : Module R G\nf : E →ₗ.[R] F\nhf : LinearMap.ker f.toFun = ⊥\n⊢ Submodule.map (LinearMap.snd R F E) (Submodule.map (LinearEquiv.prodComm R E F) f.graph) =\n    Submodule.map (LinearMap.snd R F E) (Submodule.map (↑(LinearEquiv.prodComm R E F)) f.graph)","state_after":"no goals","tactic":"rfl","premises":[]}]}
+{"url":"Mathlib/Analysis/NormedSpace/Exponential.lean","commit":"","full_name":"NormedSpace.expSeries_radius_pos","start":[384,0],"end":[386,27],"file_path":"Mathlib/Analysis/NormedSpace/Exponential.lean","tactics":[{"state_before":"𝕂 : Type u_1\n𝔸 : Type u_2\n𝔹 : Type u_3\ninst✝⁴ : RCLike 𝕂\ninst✝³ : NormedRing 𝔸\ninst✝² : NormedAlgebra 𝕂 𝔸\ninst✝¹ : NormedRing 𝔹\ninst✝ : NormedAlgebra 𝕂 𝔹\n⊢ 0 < (expSeries 𝕂 𝔸).radius","state_after":"𝕂 : Type u_1\n𝔸 : Type u_2\n𝔹 : Type u_3\ninst✝⁴ : RCLike 𝕂\ninst✝³ : NormedRing 𝔸\ninst✝² : NormedAlgebra 𝕂 𝔸\ninst✝¹ : NormedRing 𝔹\ninst✝ : NormedAlgebra 𝕂 𝔹\n⊢ 0 < ⊤","tactic":"rw [expSeries_radius_eq_top]","premises":[{"full_name":"NormedSpace.expSeries_radius_eq_top","def_path":"Mathlib/Analysis/NormedSpace/Exponential.lean","def_pos":[372,8],"def_end_pos":[372,31]}]},{"state_before":"𝕂 : Type u_1\n𝔸 : Type u_2\n𝔹 : Type u_3\ninst✝⁴ : RCLike 𝕂\ninst✝³ : NormedRing 𝔸\ninst✝² : NormedAlgebra 𝕂 𝔸\ninst✝¹ : NormedRing 𝔹\ninst✝ : NormedAlgebra 𝕂 𝔹\n⊢ 0 < ⊤","state_after":"no goals","tactic":"exact WithTop.zero_lt_top","premises":[{"full_name":"WithTop.zero_lt_top","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/WithTop.lean","def_pos":[372,8],"def_end_pos":[372,19]}]}]}
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+{"url":"Mathlib/Algebra/BigOperators/Finprod.lean","commit":"","full_name":"finsum_induction","start":[237,0],"end":[246,66],"file_path":"Mathlib/Algebra/BigOperators/Finprod.lean","tactics":[{"state_before":"G : Type u_1\nM : Type u_2\nN : Type u_3\nα : Sort u_4\nβ : Sort u_5\nι : Sort u_6\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf : α → M\np : M → Prop\nhp₀ : p 1\nhp₁ : ∀ (x y : M), p x → p y → p (x * y)\nhp₂ : ∀ (i : α), p (f i)\n⊢ p (∏ᶠ (i : α), f i)","state_after":"G : Type u_1\nM : Type u_2\nN : Type u_3\nα : Sort u_4\nβ : Sort u_5\nι : Sort u_6\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf : α → M\np : M → Prop\nhp₀ : p 1\nhp₁ : ∀ (x y : M), p x → p y → p (x * y)\nhp₂ : ∀ (i : α), p (f i)\n⊢ p (if h : (mulSupport ((fun i => f i) ∘ PLift.down)).Finite then ∏ i ∈ h.toFinset, f i.down else 1)","tactic":"rw [finprod]","premises":[{"full_name":"finprod","def_path":"Mathlib/Algebra/BigOperators/Finprod.lean","def_pos":[99,30],"def_end_pos":[99,37]},{"full_name":"finprod_def'","def_path":"Mathlib/Algebra/BigOperators/Finprod.lean","def_pos":[99,48],"def_end_pos":[99,60]}]},{"state_before":"G : Type u_1\nM : Type u_2\nN : Type u_3\nα : Sort u_4\nβ : Sort u_5\nι : Sort u_6\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf : α → M\np : M → Prop\nhp₀ : p 1\nhp₁ : ∀ (x y : M), p x → p y → p (x * y)\nhp₂ : ∀ (i : α), p (f i)\n⊢ p (if h : (mulSupport ((fun i => f i) ∘ PLift.down)).Finite then ∏ i ∈ h.toFinset, f i.down else 1)","state_after":"case pos\nG : Type u_1\nM : Type u_2\nN : Type u_3\nα : Sort u_4\nβ : Sort u_5\nι : Sort u_6\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf : α → M\np : M → Prop\nhp₀ : p 1\nhp₁ : ∀ (x y : M), p x → p y → p (x * y)\nhp₂ : ∀ (i : α), p (f i)\nh✝ : (mulSupport ((fun i => f i) ∘ PLift.down)).Finite\n⊢ p (∏ i ∈ h✝.toFinset, f i.down)\n\ncase neg\nG : Type u_1\nM : Type u_2\nN : Type u_3\nα : Sort u_4\nβ : Sort u_5\nι : Sort u_6\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf : α → M\np : M → Prop\nhp₀ : p 1\nhp₁ : ∀ (x y : M), p x → p y → p (x * y)\nhp₂ : ∀ (i : α), p (f i)\nh✝ : ¬(mulSupport ((fun i => f i) ∘ PLift.down)).Finite\n⊢ p 1","tactic":"split_ifs","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case pos\nG : Type u_1\nM : Type u_2\nN : Type u_3\nα : Sort u_4\nβ : Sort u_5\nι : Sort u_6\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf : α → M\np : M → Prop\nhp₀ : p 1\nhp₁ : ∀ (x y : M), p x → p y → p (x * y)\nhp₂ : ∀ (i : α), p (f i)\nh✝ : (mulSupport ((fun i => f i) ∘ PLift.down)).Finite\n⊢ p (∏ i ∈ h✝.toFinset, f i.down)\n\ncase neg\nG : Type u_1\nM : Type u_2\nN : Type u_3\nα : Sort u_4\nβ : Sort u_5\nι : Sort u_6\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf : α → M\np : M → Prop\nhp₀ : p 1\nhp₁ : ∀ (x y : M), p x → p y → p (x * y)\nhp₂ : ∀ (i : α), p (f i)\nh✝ : ¬(mulSupport ((fun i => f i) ∘ PLift.down)).Finite\n⊢ p 1","state_after":"no goals","tactic":"exacts [Finset.prod_induction _ _ hp₁ hp₀ fun i _ => hp₂ _, hp₀]","premises":[{"full_name":"Finset.prod_induction","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[1403,8],"def_end_pos":[1403,22]}]}]}
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+{"url":"Mathlib/FieldTheory/RatFunc/Defs.lean","commit":"","full_name":"RatFunc.mk_zero","start":[146,0],"end":[147,45],"file_path":"Mathlib/FieldTheory/RatFunc/Defs.lean","tactics":[{"state_before":"K : Type u\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\np : K[X]\n⊢ RatFunc.mk p 0 = { toFractionRing := 0 }","state_after":"no goals","tactic":"rw [mk_eq_div', RingHom.map_zero, div_zero]","premises":[{"full_name":"RatFunc.mk_eq_div'","def_path":"Mathlib/FieldTheory/RatFunc/Defs.lean","def_pos":[143,8],"def_end_pos":[143,18]},{"full_name":"RingHom.map_zero","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[472,18],"def_end_pos":[472,26]},{"full_name":"div_zero","def_path":"Mathlib/Algebra/GroupWithZero/Basic.lean","def_pos":[301,8],"def_end_pos":[301,16]}]}]}
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F, V⟯\nk : ℕ\n⊢ ⇑(k • s) = k • ⇑s","state_after":"case zero\n𝕜 : Type u_1\ninst✝²⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝²⁴ : NormedAddCommGroup E\ninst✝²³ : NormedSpace 𝕜 E\nE' : Type u_3\ninst✝²² : NormedAddCommGroup E'\ninst✝²¹ : NormedSpace 𝕜 E'\nH : Type u_4\ninst✝²⁰ : TopologicalSpace H\nH' : Type u_5\ninst✝¹⁹ : TopologicalSpace H'\nI : ModelWithCorners 𝕜 E H\nI' : ModelWithCorners 𝕜 E' H'\nM : Type u_6\ninst✝¹⁸ : TopologicalSpace M\ninst✝¹⁷ : ChartedSpace H M\nM' : Type u_7\ninst✝¹⁶ : TopologicalSpace M'\ninst✝¹⁵ : ChartedSpace H' M'\nE'' : Type u_8\ninst✝¹⁴ : NormedAddCommGroup E''\ninst✝¹³ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝¹² : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝¹¹ : TopologicalSpace M''\ninst✝¹⁰ : ChartedSpace H'' M''\ninst✝⁹ : SmoothManifoldWithCorners I M\nF : Type u_11\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nn : ℕ∞\nV : M → Type u_12\ninst✝⁶ : TopologicalSpace (TotalSpace F V)\ninst✝⁵ : (x : M) → AddCommGroup (V x)\ninst✝⁴ : (x : M) → Module 𝕜 (V x)\ninst✝³ : (x : M) → TopologicalSpace (V x)\ninst✝² : FiberBundle F V\ninst✝¹ : VectorBundle 𝕜 F V\ninst✝ : SmoothVectorBundle F V I\ns✝ t s : Cₛ^n⟮I; F, V⟯\n⊢ ⇑(0 • s) = 0 • ⇑s\n\ncase succ\n𝕜 : Type u_1\ninst✝²⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝²⁴ : NormedAddCommGroup E\ninst✝²³ : NormedSpace 𝕜 E\nE' : Type u_3\ninst✝²² : NormedAddCommGroup E'\ninst✝²¹ : NormedSpace 𝕜 E'\nH : Type u_4\ninst✝²⁰ : TopologicalSpace H\nH' : Type u_5\ninst✝¹⁹ : TopologicalSpace H'\nI : ModelWithCorners 𝕜 E H\nI' : ModelWithCorners 𝕜 E' H'\nM : Type u_6\ninst✝¹⁸ : TopologicalSpace M\ninst✝¹⁷ : ChartedSpace H M\nM' : Type u_7\ninst✝¹⁶ : TopologicalSpace M'\ninst✝¹⁵ : ChartedSpace H' M'\nE'' : Type u_8\ninst✝¹⁴ : NormedAddCommGroup E''\ninst✝¹³ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝¹² : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝¹¹ : TopologicalSpace M''\ninst✝¹⁰ : ChartedSpace H'' M''\ninst✝⁹ : SmoothManifoldWithCorners I M\nF : Type u_11\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nn : ℕ∞\nV : M → Type u_12\ninst✝⁶ : TopologicalSpace (TotalSpace F V)\ninst✝⁵ : (x : M) → AddCommGroup (V x)\ninst✝⁴ : (x : M) → Module 𝕜 (V x)\ninst✝³ : (x : M) → TopologicalSpace (V x)\ninst✝² : FiberBundle F V\ninst✝¹ : VectorBundle 𝕜 F V\ninst✝ : SmoothVectorBundle F V I\ns✝ t s : Cₛ^n⟮I; F, V⟯\nk : ℕ\nih : ⇑(k • s) = k • ⇑s\n⊢ ⇑((k + 1) • s) = (k + 1) • ⇑s","tactic":"induction' k with k ih","premises":[]},{"state_before":"case succ\n𝕜 : Type u_1\ninst✝²⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝²⁴ : NormedAddCommGroup E\ninst✝²³ : NormedSpace 𝕜 E\nE' : Type u_3\ninst✝²² : NormedAddCommGroup E'\ninst✝²¹ : NormedSpace 𝕜 E'\nH : Type u_4\ninst✝²⁰ : TopologicalSpace H\nH' : Type u_5\ninst✝¹⁹ : TopologicalSpace H'\nI : ModelWithCorners 𝕜 E H\nI' : ModelWithCorners 𝕜 E' H'\nM : Type u_6\ninst✝¹⁸ : TopologicalSpace M\ninst✝¹⁷ : ChartedSpace H M\nM' : Type u_7\ninst✝¹⁶ : TopologicalSpace M'\ninst✝¹⁵ : ChartedSpace H' M'\nE'' : Type u_8\ninst✝¹⁴ : NormedAddCommGroup E''\ninst✝¹³ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝¹² : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝¹¹ : TopologicalSpace M''\ninst✝¹⁰ : ChartedSpace H'' M''\ninst✝⁹ : SmoothManifoldWithCorners I M\nF : Type u_11\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nn : ℕ∞\nV : M → Type u_12\ninst✝⁶ : TopologicalSpace (TotalSpace F V)\ninst✝⁵ : (x : M) → AddCommGroup (V x)\ninst✝⁴ : (x : M) → Module 𝕜 (V x)\ninst✝³ : (x : M) → TopologicalSpace (V x)\ninst✝² : FiberBundle F V\ninst✝¹ : VectorBundle 𝕜 F V\ninst✝ : SmoothVectorBundle F V I\ns✝ t s : Cₛ^n⟮I; F, V⟯\nk : ℕ\nih : ⇑(k • s) = k • ⇑s\n⊢ ⇑((k + 1) • s) = (k + 1) • ⇑s","state_after":"case succ\n𝕜 : Type u_1\ninst✝²⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝²⁴ : NormedAddCommGroup E\ninst✝²³ : NormedSpace 𝕜 E\nE' : Type u_3\ninst✝²² : NormedAddCommGroup E'\ninst✝²¹ : NormedSpace 𝕜 E'\nH : Type u_4\ninst✝²⁰ : TopologicalSpace H\nH' : Type u_5\ninst✝¹⁹ : TopologicalSpace H'\nI : ModelWithCorners 𝕜 E H\nI' : ModelWithCorners 𝕜 E' H'\nM : Type u_6\ninst✝¹⁸ : TopologicalSpace M\ninst✝¹⁷ : ChartedSpace H M\nM' : Type u_7\ninst✝¹⁶ : TopologicalSpace M'\ninst✝¹⁵ : ChartedSpace H' M'\nE'' : Type u_8\ninst✝¹⁴ : NormedAddCommGroup E''\ninst✝¹³ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝¹² : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝¹¹ : TopologicalSpace M''\ninst✝¹⁰ : ChartedSpace H'' M''\ninst✝⁹ : SmoothManifoldWithCorners I M\nF : Type u_11\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nn : ℕ∞\nV : M → Type u_12\ninst✝⁶ : TopologicalSpace (TotalSpace F V)\ninst✝⁵ : (x : M) → AddCommGroup (V x)\ninst✝⁴ : (x : M) → Module 𝕜 (V x)\ninst✝³ : (x : M) → TopologicalSpace (V x)\ninst✝² : FiberBundle F V\ninst✝¹ : VectorBundle 𝕜 F V\ninst✝ : SmoothVectorBundle F V I\ns✝ t s : Cₛ^n⟮I; F, V⟯\nk : ℕ\nih : ⇑(k • s) = k • ⇑s\n⊢ ⇑((k + 1) • s) = ⇑(k • s) + ⇑s","tactic":"simp_rw [succ_nsmul, ← ih]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"succ_nsmul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[566,14],"def_end_pos":[566,24]}]},{"state_before":"case succ\n𝕜 : Type u_1\ninst✝²⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝²⁴ : NormedAddCommGroup E\ninst✝²³ : NormedSpace 𝕜 E\nE' : Type u_3\ninst✝²² : NormedAddCommGroup E'\ninst✝²¹ : NormedSpace 𝕜 E'\nH : Type u_4\ninst✝²⁰ : TopologicalSpace H\nH' : Type u_5\ninst✝¹⁹ : TopologicalSpace H'\nI : ModelWithCorners 𝕜 E H\nI' : ModelWithCorners 𝕜 E' H'\nM : Type u_6\ninst✝¹⁸ : TopologicalSpace M\ninst✝¹⁷ : ChartedSpace H M\nM' : Type u_7\ninst✝¹⁶ : TopologicalSpace M'\ninst✝¹⁵ : ChartedSpace H' M'\nE'' : Type u_8\ninst✝¹⁴ : NormedAddCommGroup E''\ninst✝¹³ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝¹² : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝¹¹ : TopologicalSpace M''\ninst✝¹⁰ : ChartedSpace H'' M''\ninst✝⁹ : SmoothManifoldWithCorners I M\nF : Type u_11\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nn : ℕ∞\nV : M → Type u_12\ninst✝⁶ : TopologicalSpace (TotalSpace F V)\ninst✝⁵ : (x : M) → AddCommGroup (V x)\ninst✝⁴ : (x : M) → Module 𝕜 (V x)\ninst✝³ : (x : M) → TopologicalSpace (V x)\ninst✝² : FiberBundle F V\ninst✝¹ : VectorBundle 𝕜 F V\ninst✝ : SmoothVectorBundle F V I\ns✝ t s : Cₛ^n⟮I; F, V⟯\nk : ℕ\nih : ⇑(k • s) = k • ⇑s\n⊢ ⇑((k + 1) • s) = ⇑(k • s) + ⇑s","state_after":"no goals","tactic":"rfl","premises":[]}]}
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μ\nhf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ\nt : Set α := hf.sigmaFiniteSet\n⊢ 0 ≤ᶠ[ae μ] f","tactic":"let t := hf.sigmaFiniteSet","premises":[{"full_name":"MeasureTheory.AEFinStronglyMeasurable.sigmaFiniteSet","def_path":"Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean","def_pos":[1755,4],"def_end_pos":[1755,18]}]},{"state_before":"α : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\nμ : Measure α\ns t✝ : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf✝ f : α → ℝ\nhf : AEFinStronglyMeasurable f μ\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ\nhf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ\nt : Set α := hf.sigmaFiniteSet\n⊢ 0 ≤ᶠ[ae μ] f","state_after":"α : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\nμ : Measure α\ns t✝ : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf✝ f : α → ℝ\nhf : AEFinStronglyMeasurable f μ\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ\nhf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ\nt : Set α := hf.sigmaFiniteSet\n⊢ 0 ≤ᶠ[ae (μ.restrict t)] f","tactic":"suffices 0 ≤ᵐ[μ.restrict t] f from\n    ae_of_ae_restrict_of_ae_restrict_compl _ this 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: Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\nμ : Measure α\ns t✝ : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf✝ f : α → ℝ\nhf : AEFinStronglyMeasurable f μ\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ\nhf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ\nt : Set α := hf.sigmaFiniteSet\n⊢ 0 ≤ᶠ[ae (μ.restrict t)] f","state_after":"α : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\nμ : Measure α\ns t✝ : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf✝ f : α → ℝ\nhf : AEFinStronglyMeasurable f μ\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ\nhf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ\nt : Set α := hf.sigmaFiniteSet\nthis : SigmaFinite (μ.restrict t)\n⊢ 0 ≤ᶠ[ae (μ.restrict t)] f","tactic":"haveI : SigmaFinite (μ.restrict t) := hf.sigmaFinite_restrict","premises":[{"full_name":"MeasureTheory.AEFinStronglyMeasurable.sigmaFinite_restrict","def_path":"Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean","def_pos":[1766,9],"def_end_pos":[1766,29]},{"full_name":"MeasureTheory.Measure.restrict","def_path":"Mathlib/MeasureTheory/Measure/Restrict.lean","def_pos":[43,18],"def_end_pos":[43,26]},{"full_name":"MeasureTheory.SigmaFinite","def_path":"Mathlib/MeasureTheory/Measure/Typeclasses.lean","def_pos":[555,6],"def_end_pos":[555,17]}]},{"state_before":"α : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\nμ : Measure α\ns t✝ : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf✝ f : α → ℝ\nhf : AEFinStronglyMeasurable f μ\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ\nhf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ\nt : Set α := hf.sigmaFiniteSet\nthis : SigmaFinite (μ.restrict t)\n⊢ 0 ≤ᶠ[ae (μ.restrict t)] f","state_after":"case refine_1\nα : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t✝ : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf✝ f : α → ℝ\nhf : AEFinStronglyMeasurable f μ\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ\nhf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ\nt : Set α := hf.sigmaFiniteSet\nthis : SigmaFinite (μ.restrict t)\ns : Set α\nhs : MeasurableSet s\nhμts : (μ.restrict t) s < ⊤\n⊢ IntegrableOn f s (μ.restrict t)\n\ncase refine_2\nα : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t✝ : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf✝ f : α → ℝ\nhf : AEFinStronglyMeasurable f μ\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ\nhf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ\nt : Set α := hf.sigmaFiniteSet\nthis : SigmaFinite (μ.restrict t)\ns : Set α\nhs : MeasurableSet s\nhμts : (μ.restrict t) s < ⊤\n⊢ 0 ≤ ∫ (x : α) in s, f x ∂μ.restrict t","tactic":"refine\n    ae_nonneg_of_forall_setIntegral_nonneg_of_sigmaFinite (fun s hs hμts => ?_) fun s hs hμts => ?_","premises":[{"full_name":"MeasureTheory.ae_nonneg_of_forall_setIntegral_nonneg_of_sigmaFinite","def_path":"Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean","def_pos":[308,8],"def_end_pos":[308,61]}]}]}
+{"url":"Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean","commit":"","full_name":"affineIndependent_iff_le_finrank_vectorSpan","start":[178,0],"end":[187,73],"file_path":"Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean","tactics":[{"state_before":"k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ninst✝ : Fintype ι\np : ι → P\nn : ℕ\nhc : Fintype.card ι = n + 1\n⊢ AffineIndependent k p ↔ n ≤ finrank k ↥(vectorSpan k (Set.range p))","state_after":"k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ninst✝ : Fintype ι\np : ι → P\nn : ℕ\nhc : Fintype.card ι = n + 1\n⊢ finrank k ↥(vectorSpan k (Set.range p)) = n ↔ n ≤ finrank k ↥(vectorSpan k (Set.range p))","tactic":"rw [affineIndependent_iff_finrank_vectorSpan_eq k p hc]","premises":[{"full_name":"affineIndependent_iff_finrank_vectorSpan_eq","def_path":"Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean","def_pos":[165,8],"def_end_pos":[165,51]}]},{"state_before":"k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ninst✝ : Fintype ι\np : ι → P\nn : ℕ\nhc : Fintype.card ι = n + 1\n⊢ finrank k ↥(vectorSpan k (Set.range p)) = n ↔ n ≤ finrank k ↥(vectorSpan k (Set.range p))","state_after":"case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ninst✝ : Fintype ι\np : ι → P\nn : ℕ\nhc : Fintype.card ι = n + 1\n⊢ finrank k ↥(vectorSpan k (Set.range p)) = n → n ≤ finrank k ↥(vectorSpan k (Set.range p))\n\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ninst✝ : Fintype ι\np : ι → P\nn : ℕ\nhc : Fintype.card ι = n + 1\n⊢ n ≤ finrank k ↥(vectorSpan k (Set.range p)) → finrank k ↥(vectorSpan k (Set.range p)) = n","tactic":"constructor","premises":[]}]}
+{"url":"Mathlib/Topology/UniformSpace/UniformEmbedding.lean","commit":"","full_name":"isComplete_image_iff","start":[260,0],"end":[266,88],"file_path":"Mathlib/Topology/UniformSpace/UniformEmbedding.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nm : α → β\ns : Set α\nhm : UniformInducing m\n⊢ IsComplete (m '' s) ↔ IsComplete s","state_after":"α : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nm : α → β\ns : Set α\nhm : UniformInducing m\nfact1 : SurjOn (map m) (Iic (𝓟 s)) (Iic (𝓟 (m '' s)))\n⊢ IsComplete (m '' s) ↔ IsComplete s","tactic":"have fact1 : SurjOn (map m) (Iic <| 𝓟 s) (Iic <| 𝓟 <| m '' s) := surjOn_image .. |>.filter_map_Iic","premises":[{"full_name":"Filter.map","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1637,4],"def_end_pos":[1637,7]},{"full_name":"Filter.principal","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[269,4],"def_end_pos":[269,13]},{"full_name":"Set.Iic","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[58,4],"def_end_pos":[58,7]},{"full_name":"Set.SurjOn","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[256,4],"def_end_pos":[256,10]},{"full_name":"Set.image","def_path":"Mathlib/Init/Set.lean","def_pos":[208,4],"def_end_pos":[208,9]},{"full_name":"Set.surjOn_image","def_path":"Mathlib/Data/Set/Function.lean","def_pos":[753,8],"def_end_pos":[753,20]}]},{"state_before":"α : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nm : α → β\ns : Set α\nhm : UniformInducing m\nfact1 : SurjOn (map m) (Iic (𝓟 s)) (Iic (𝓟 (m '' s)))\n⊢ IsComplete (m '' s) ↔ IsComplete s","state_after":"α : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nm : α → β\ns : Set α\nhm : UniformInducing m\nfact1 : SurjOn (map m) (Iic (𝓟 s)) (Iic (𝓟 (m '' s)))\nfact2 : MapsTo (map m) (Iic (𝓟 s)) (Iic (𝓟 (m '' s)))\n⊢ IsComplete (m '' s) ↔ IsComplete s","tactic":"have fact2 : MapsTo (map m) (Iic <| 𝓟 s) (Iic <| 𝓟 <| m '' s) := mapsTo_image .. |>.filter_map_Iic","premises":[{"full_name":"Filter.map","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1637,4],"def_end_pos":[1637,7]},{"full_name":"Filter.principal","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[269,4],"def_end_pos":[269,13]},{"full_name":"Set.Iic","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[58,4],"def_end_pos":[58,7]},{"full_name":"Set.MapsTo","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[232,4],"def_end_pos":[232,10]},{"full_name":"Set.image","def_path":"Mathlib/Init/Set.lean","def_pos":[208,4],"def_end_pos":[208,9]},{"full_name":"Set.mapsTo_image","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[234,8],"def_end_pos":[234,20]}]},{"state_before":"α : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nm : α → β\ns : Set α\nhm : UniformInducing m\nfact1 : SurjOn (map m) (Iic (𝓟 s)) (Iic (𝓟 (m '' s)))\nfact2 : MapsTo (map m) (Iic (𝓟 s)) (Iic (𝓟 (m '' s)))\n⊢ IsComplete (m '' s) ↔ IsComplete s","state_after":"no goals","tactic":"simp_rw [IsComplete, imp.swap (a := Cauchy _), ← mem_Iic (b := 𝓟 _), fact1.forall fact2,\n    hm.cauchy_map_iff, exists_mem_image, map_le_iff_le_comap, hm.inducing.nhds_eq_comap]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Cauchy","def_path":"Mathlib/Topology/UniformSpace/Cauchy.lean","def_pos":[29,4],"def_end_pos":[29,10]},{"full_name":"Filter.map_le_iff_le_comap","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2023,8],"def_end_pos":[2023,27]},{"full_name":"Filter.principal","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[269,4],"def_end_pos":[269,13]},{"full_name":"Inducing.nhds_eq_comap","def_path":"Mathlib/Topology/Maps/Basic.lean","def_pos":[82,8],"def_end_pos":[82,21]},{"full_name":"IsComplete","def_path":"Mathlib/Topology/UniformSpace/Cauchy.lean","def_pos":[34,4],"def_end_pos":[34,14]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Set.SurjOn.forall","def_path":"Mathlib/Data/Set/Function.lean","def_pos":[865,8],"def_end_pos":[865,21]},{"full_name":"Set.exists_mem_image","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[198,8],"def_end_pos":[198,24]},{"full_name":"Set.mem_Iic","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[114,8],"def_end_pos":[114,15]},{"full_name":"UniformInducing.cauchy_map_iff","def_path":"Mathlib/Topology/UniformSpace/UniformEmbedding.lean","def_pos":[77,8],"def_end_pos":[77,38]},{"full_name":"UniformInducing.inducing","def_path":"Mathlib/Topology/UniformSpace/UniformEmbedding.lean","def_pos":[105,8],"def_end_pos":[105,32]},{"full_name":"imp.swap","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1422,8],"def_end_pos":[1422,16]}]}]}
+{"url":"Mathlib/Algebra/DirectSum/Internal.lean","commit":"","full_name":"DirectSum.coe_mul_of_apply","start":[256,0],"end":[259,81],"file_path":"Mathlib/Algebra/DirectSum/Internal.lean","tactics":[{"state_before":"ι : Type u_1\nσ : Type u_2\nS : Type u_3\nR : Type u_4\ninst✝⁹ : DecidableEq ι\ninst✝⁸ : Semiring R\ninst✝⁷ : SetLike σ R\ninst✝⁶ : AddSubmonoidClass σ R\nA : ι → σ\ninst✝⁵ : CanonicallyOrderedAddCommMonoid ι\ninst✝⁴ : SetLike.GradedMonoid A\ninst✝³ : Sub ι\ninst✝² : OrderedSub ι\ninst✝¹ : ContravariantClass ι ι (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nr : ⨁ (i : ι), ↥(A i)\ni : ι\nr' : ↥(A i)\nn : ι\ninst✝ : Decidable (i ≤ n)\n⊢ ↑((r * (of (fun i => ↥(A i)) i) r') n) = if i ≤ n then ↑(r (n - i)) * ↑r' else 0","state_after":"case pos\nι : Type u_1\nσ : Type u_2\nS : Type u_3\nR : Type u_4\ninst✝⁹ : DecidableEq ι\ninst✝⁸ : Semiring R\ninst✝⁷ : SetLike σ R\ninst✝⁶ : AddSubmonoidClass σ R\nA : ι → σ\ninst✝⁵ : CanonicallyOrderedAddCommMonoid ι\ninst✝⁴ : SetLike.GradedMonoid A\ninst✝³ : Sub ι\ninst✝² : OrderedSub ι\ninst✝¹ : ContravariantClass ι ι (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nr : ⨁ (i : ι), ↥(A i)\ni : ι\nr' : ↥(A i)\nn : ι\ninst✝ : Decidable (i ≤ n)\nh : i ≤ n\n⊢ ↑((r * (of (fun i => ↥(A i)) i) r') n) = ↑(r (n - i)) * ↑r'\n\ncase neg\nι : Type u_1\nσ : Type u_2\nS : Type u_3\nR : Type u_4\ninst✝⁹ : DecidableEq ι\ninst✝⁸ : Semiring R\ninst✝⁷ : SetLike σ R\ninst✝⁶ : AddSubmonoidClass σ R\nA : ι → σ\ninst✝⁵ : CanonicallyOrderedAddCommMonoid ι\ninst✝⁴ : SetLike.GradedMonoid A\ninst✝³ : Sub ι\ninst✝² : OrderedSub ι\ninst✝¹ : ContravariantClass ι ι (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nr : ⨁ (i : ι), ↥(A i)\ni : ι\nr' : ↥(A i)\nn : ι\ninst✝ : Decidable (i ≤ n)\nh : ¬i ≤ n\n⊢ ↑((r * (of (fun i => ↥(A i)) i) r') n) = 0","tactic":"split_ifs with h","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case pos\nι : Type u_1\nσ : Type u_2\nS : Type u_3\nR : Type u_4\ninst✝⁹ : DecidableEq ι\ninst✝⁸ : Semiring R\ninst✝⁷ : SetLike σ R\ninst✝⁶ : AddSubmonoidClass σ R\nA : ι → σ\ninst✝⁵ : CanonicallyOrderedAddCommMonoid ι\ninst✝⁴ : SetLike.GradedMonoid A\ninst✝³ : Sub ι\ninst✝² : OrderedSub ι\ninst✝¹ : ContravariantClass ι ι (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nr : ⨁ (i : ι), ↥(A i)\ni : ι\nr' : ↥(A i)\nn : ι\ninst✝ : Decidable (i ≤ n)\nh : i ≤ n\n⊢ ↑((r * (of (fun i => ↥(A i)) i) r') n) = ↑(r (n - i)) * ↑r'\n\ncase neg\nι : Type u_1\nσ : Type u_2\nS : Type u_3\nR : Type u_4\ninst✝⁹ : DecidableEq ι\ninst✝⁸ : Semiring R\ninst✝⁷ : SetLike σ R\ninst✝⁶ : AddSubmonoidClass σ R\nA : ι → σ\ninst✝⁵ : CanonicallyOrderedAddCommMonoid ι\ninst✝⁴ : SetLike.GradedMonoid A\ninst✝³ : Sub ι\ninst✝² : OrderedSub ι\ninst✝¹ : ContravariantClass ι ι (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nr : ⨁ (i : ι), ↥(A i)\ni : ι\nr' : ↥(A i)\nn : ι\ninst✝ : Decidable (i ≤ n)\nh : ¬i ≤ n\n⊢ ↑((r * (of (fun i => ↥(A i)) i) r') n) = 0","state_after":"no goals","tactic":"exacts [coe_mul_of_apply_of_le _ _ _ n h, coe_mul_of_apply_of_not_le _ _ _ n h]","premises":[{"full_name":"DirectSum.coe_mul_of_apply_of_le","def_path":"Mathlib/Algebra/DirectSum/Internal.lean","def_pos":[248,8],"def_end_pos":[248,30]},{"full_name":"DirectSum.coe_mul_of_apply_of_not_le","def_path":"Mathlib/Algebra/DirectSum/Internal.lean","def_pos":[233,8],"def_end_pos":[233,34]}]}]}
+{"url":"Mathlib/CategoryTheory/Sites/SheafOfTypes.lean","commit":"","full_name":"CategoryTheory.Sieve.forallYonedaIsSheaf_iff_colimit","start":[166,0],"end":[197,71],"file_path":"Mathlib/CategoryTheory/Sites/SheafOfTypes.lean","tactics":[{"state_before":"C : Type u\ninst✝ : Category.{v, u} C\nX : C\nS : Sieve X\n⊢ (∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows) ↔ Nonempty (IsColimit S.arrows.cocone)","state_after":"case mp\nC : Type u\ninst✝ : Category.{v, u} C\nX : C\nS : Sieve X\n⊢ (∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows) → Nonempty (IsColimit S.arrows.cocone)\n\ncase mpr\nC : Type u\ninst✝ : Category.{v, u} C\nX : C\nS : Sieve X\n⊢ Nonempty (IsColimit S.arrows.cocone) → ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows","tactic":"constructor","premises":[]}]}
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: Category.{u_3, u_2} A\ninst✝² : HasWeakSheafify J A\ninst✝¹ : (constantSheaf J A).Faithful\ninst✝ : (constantSheaf J A).Full\nt : C\nht : IsTerminal t\nL : A ⥤ Sheaf J A\nadj : L ⊣ (sheafSections J A).obj (op t)\nF : Sheaf J A\nthis : L.Faithful\n⊢ IsDiscrete J ht F ↔ IsIso (adj.counit.app F)","state_after":"C : Type u_1\ninst✝⁴ : Category.{u_4, u_1} C\nJ : GrothendieckTopology C\nA : Type u_2\ninst✝³ : Category.{u_3, u_2} A\ninst✝² : HasWeakSheafify J A\ninst✝¹ : (constantSheaf J A).Faithful\ninst✝ : (constantSheaf J A).Full\nt : C\nht : IsTerminal t\nL : A ⥤ Sheaf J A\nadj : L ⊣ (sheafSections J A).obj (op t)\nF : Sheaf J A\nthis✝ : L.Faithful\nthis : L.Full\n⊢ IsDiscrete J ht F ↔ IsIso (adj.counit.app F)","tactic":"have : L.Full := Functor.Full.of_iso (adj.leftAdjointUniq (constantSheafAdj _ _ ht)).symm","premises":[{"full_name":"CategoryTheory.Adjunction.leftAdjointUniq","def_path":"Mathlib/CategoryTheory/Adjunction/Unique.lean","def_pos":[112,4],"def_end_pos":[112,19]},{"full_name":"CategoryTheory.Functor.Full","def_path":"Mathlib/CategoryTheory/Functor/FullyFaithful.lean","def_pos":[43,6],"def_end_pos":[43,10]},{"full_name":"CategoryTheory.Functor.Full.of_iso","def_path":"Mathlib/CategoryTheory/Functor/FullyFaithful.lean","def_pos":[254,6],"def_end_pos":[254,17]},{"full_name":"CategoryTheory.Iso.symm","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[84,4],"def_end_pos":[84,8]},{"full_name":"CategoryTheory.constantSheafAdj","def_path":"Mathlib/CategoryTheory/Sites/ConstantSheaf.lean","def_pos":[48,18],"def_end_pos":[48,34]}]},{"state_before":"C : Type u_1\ninst✝⁴ : Category.{u_4, u_1} C\nJ : GrothendieckTopology C\nA : Type u_2\ninst✝³ : Category.{u_3, u_2} A\ninst✝² : HasWeakSheafify J A\ninst✝¹ : (constantSheaf J A).Faithful\ninst✝ : (constantSheaf J A).Full\nt : 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HasWeakSheafify J A\ninst✝¹ : (constantSheaf J A).Faithful\ninst✝ : (constantSheaf J A).Full\nt : C\nht : IsTerminal t\nL : A ⥤ Sheaf J A\nadj : L ⊣ (sheafSections J A).obj (op t)\nF : Sheaf J A\nthis✝ : L.Faithful\nthis : L.Full\n⊢ IsDiscrete J ht F ↔ F ∈ L.essImage","state_after":"no goals","tactic":"exact isDiscrete_iff_mem_essImage' _ _ adj _","premises":[{"full_name":"CategoryTheory.Sheaf.isDiscrete_iff_mem_essImage'","def_path":"Mathlib/CategoryTheory/Sites/Discrete.lean","def_pos":[59,6],"def_end_pos":[59,34]}]}]}
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: C(X₀, C(Y, Z)) := { toFun := fun p => g (f p.1, p.2), continuous_toFun := hg }.curry\nh : ∀ (x : X), Continuous fun y => g (x, y)\n⊢ Continuous g","tactic":"have h : ∀ x : X, Continuous fun y => g (x, y) := by\n    intro x\n    obtain ⟨x₀, rfl⟩ := hf.surjective x\n    exact (Gf x₀).continuous","premises":[{"full_name":"Continuous","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[141,10],"def_end_pos":[141,20]},{"full_name":"ContinuousMap.continuous","def_path":"Mathlib/Topology/ContinuousFunction/Basic.lean","def_pos":[123,18],"def_end_pos":[123,28]},{"full_name":"Prod.mk","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[481,2],"def_end_pos":[481,4]},{"full_name":"QuotientMap.surjective","def_path":"Mathlib/Topology/Maps/Basic.lean","def_pos":[264,18],"def_end_pos":[264,28]}]},{"state_before":"X₀ : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst✝⁴ : TopologicalSpace X₀\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : LocallyCompactSpace Y\nf : X₀ → X\nhf : QuotientMap f\ng : X × Y → Z\nhg : Continuous fun p => g (f p.1, p.2)\nGf : C(X₀, C(Y, Z)) := { toFun := fun p => g (f p.1, p.2), continuous_toFun := hg }.curry\nh : ∀ (x : X), Continuous fun y => g (x, y)\n⊢ Continuous g","state_after":"X₀ : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst✝⁴ : TopologicalSpace X₀\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : LocallyCompactSpace Y\nf : X₀ → X\nhf : QuotientMap f\ng : X × Y → Z\nhg : Continuous fun p => g (f p.1, p.2)\nGf : C(X₀, C(Y, Z)) := { toFun := fun p => g (f p.1, p.2), continuous_toFun := hg }.curry\nh : ∀ (x : X), Continuous fun y => g (x, y)\nG : X → C(Y, Z) := fun x => { toFun := fun y => g (x, y), continuous_toFun := ⋯ }\n⊢ Continuous g","tactic":"let G : X → C(Y, Z) := fun x => ⟨_, h 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p.2)\nGf : C(X₀, C(Y, Z)) := { toFun := fun p => g (f p.1, p.2), continuous_toFun := hg }.curry\nh : ∀ (x : X), Continuous fun y => g (x, y)\nG : X → C(Y, Z) := fun x => { toFun := fun y => g (x, y), continuous_toFun := ⋯ }\nthis : Continuous G\n⊢ Continuous g","tactic":"have : Continuous G := by\n    rw [hf.continuous_iff]\n    exact Gf.continuous","premises":[{"full_name":"Continuous","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[141,10],"def_end_pos":[141,20]},{"full_name":"ContinuousMap.continuous","def_path":"Mathlib/Topology/ContinuousFunction/Basic.lean","def_pos":[123,18],"def_end_pos":[123,28]},{"full_name":"QuotientMap.continuous_iff","def_path":"Mathlib/Topology/Maps/Basic.lean","def_pos":[258,18],"def_end_pos":[258,32]}]},{"state_before":"X₀ : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst✝⁴ : TopologicalSpace X₀\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : LocallyCompactSpace Y\nf : X₀ → X\nhf : QuotientMap f\ng : X × Y → Z\nhg : Continuous fun p => g (f p.1, p.2)\nGf : C(X₀, C(Y, Z)) := { toFun := fun p => g (f p.1, p.2), continuous_toFun := hg }.curry\nh : ∀ (x : X), Continuous fun y => g (x, y)\nG : X → C(Y, Z) := fun x => { toFun := fun y => g (x, y), continuous_toFun := ⋯ }\nthis : Continuous G\n⊢ Continuous g","state_after":"no goals","tactic":"exact ContinuousMap.continuous_uncurry_of_continuous ⟨G, this⟩","premises":[{"full_name":"ContinuousMap.continuous_uncurry_of_continuous","def_path":"Mathlib/Topology/CompactOpen.lean","def_pos":[376,8],"def_end_pos":[376,40]}]}]}
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IsFractionRing.to_map_ne_zero_of_mem_nonZeroDivisors (den A x).2","premises":[{"full_name":"IsFractionRing.den","def_path":"Mathlib/RingTheory/Localization/NumDen.lean","def_pos":[52,18],"def_end_pos":[52,21]},{"full_name":"IsFractionRing.to_map_ne_zero_of_mem_nonZeroDivisors","def_path":"Mathlib/RingTheory/Localization/FractionRing.lean","def_pos":[90,18],"def_end_pos":[90,55]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Subtype.property","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[590,2],"def_end_pos":[590,10]},{"full_name":"algebraMap","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[108,4],"def_end_pos":[108,14]}]},{"state_before":"case intro\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommRing P\nA : Type u_4\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nd : Aˣ\nhd : ↑d = ↑(den A x)\nd_ne_zero : (algebraMap A K) ↑(den A x) ≠ 0\n⊢ IsInteger A x","state_after":"case h\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommRing P\nA : Type u_4\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nd : Aˣ\nhd : ↑d = ↑(den A x)\nd_ne_zero : (algebraMap A K) ↑(den A x) ≠ 0\n⊢ (algebraMap A K) (↑d⁻¹ * num A x) = x","tactic":"use ↑d⁻¹ * num A x","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"IsFractionRing.num","def_path":"Mathlib/RingTheory/Localization/NumDen.lean","def_pos":[48,18],"def_end_pos":[48,21]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]}]},{"state_before":"case h\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommRing P\nA : Type u_4\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nd : Aˣ\nhd : ↑d = ↑(den A x)\nd_ne_zero : (algebraMap A K) ↑(den A x) ≠ 0\n⊢ (algebraMap A K) (↑d⁻¹ * num A x) = x","state_after":"case h\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommRing P\nA : Type u_4\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nd : Aˣ\nhd : ↑d = ↑(den A x)\nd_ne_zero : (algebraMap A K) ↑(den A x) ≠ 0\n⊢ (algebraMap A K) (↑d⁻¹ * num A x) = mk' K (num A x) (den A x)","tactic":"refine _root_.trans ?_ (mk'_num_den A x)","premises":[{"full_name":"IsFractionRing.mk'_num_den","def_path":"Mathlib/RingTheory/Localization/NumDen.lean","def_pos":[59,8],"def_end_pos":[59,19]},{"full_name":"trans","def_path":"Mathlib/Init/Algebra/Classes.lean","def_pos":[261,8],"def_end_pos":[261,13]}]},{"state_before":"case h\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommRing P\nA : Type u_4\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nd : Aˣ\nhd : ↑d = ↑(den A x)\nd_ne_zero : (algebraMap A K) ↑(den A x) ≠ 0\n⊢ (algebraMap A K) (↑d⁻¹ * num A x) = mk' K (num A x) (den A x)","state_after":"case h\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommRing P\nA : Type u_4\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nd : Aˣ\nhd : ↑d = ↑(den A x)\nd_ne_zero : (algebraMap A K) ↑(den A x) ≠ 0\n⊢ ((algebraMap A K) ↑(den A x))⁻¹ * (algebraMap A K) (num A x) = mk' K (num A x) (den A x)","tactic":"rw [map_mul, map_units_inv, 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A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nd : Aˣ\nhd : ↑d = ↑(den A x)\nd_ne_zero : (algebraMap A K) ↑(den A x) ≠ 0\n⊢ (algebraMap A K) ↑(den A x) * (((algebraMap A K) ↑(den A x))⁻¹ * (algebraMap A K) (num A x)) =\n    (algebraMap A K) ↑(den A x) * mk' K (num A x) (den A x)","tactic":"apply mul_left_cancel₀ d_ne_zero","premises":[{"full_name":"mul_left_cancel₀","def_path":"Mathlib/Algebra/GroupWithZero/Defs.lean","def_pos":[48,8],"def_end_pos":[48,24]}]},{"state_before":"case h\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommRing P\nA : Type u_4\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nd : Aˣ\nhd : ↑d = ↑(den A x)\nd_ne_zero : (algebraMap A K) ↑(den A x) ≠ 0\n⊢ (algebraMap A K) ↑(den A x) * (((algebraMap A K) ↑(den A x))⁻¹ * (algebraMap A K) (num A x)) =\n    (algebraMap A K) ↑(den A x) * mk' K (num A x) (den A x)","state_after":"no goals","tactic":"rw [← mul_assoc, mul_inv_cancel d_ne_zero, one_mul, mk'_spec']","premises":[{"full_name":"IsLocalization.mk'_spec'","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[246,8],"def_end_pos":[246,17]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]},{"full_name":"mul_inv_cancel","def_path":"Mathlib/Algebra/GroupWithZero/Defs.lean","def_pos":[195,14],"def_end_pos":[195,28]},{"full_name":"one_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[409,8],"def_end_pos":[409,15]}]}]}
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+{"url":"Mathlib/SetTheory/ZFC/Basic.lean","commit":"","full_name":"ZFSet.toSet_range","start":[1066,0],"end":[1070,6],"file_path":"Mathlib/SetTheory/ZFC/Basic.lean","tactics":[{"state_before":"α : Type u\nf : α → ZFSet\n⊢ (range f).toSet = Set.range f","state_after":"case h\nα : Type u\nf : α → ZFSet\nx✝ : ZFSet\n⊢ x✝ ∈ (range f).toSet ↔ x✝ ∈ Set.range f","tactic":"ext","premises":[]},{"state_before":"case h\nα : Type u\nf : α → ZFSet\nx✝ : ZFSet\n⊢ x✝ ∈ (range f).toSet ↔ x✝ ∈ Set.range f","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/CategoryTheory/Opposites.lean","commit":"","full_name":"CategoryTheory.Functor.leftOpRightOpEquiv_inverse_map","start":[566,0],"end":[584,62],"file_path":"Mathlib/CategoryTheory/Opposites.lean","tactics":[{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\n⊢ ∀ {X Y : (Cᵒᵖ ⥤ D)ᵒᵖ} (f : X ⟶ Y),\n    (𝟭 (Cᵒᵖ ⥤ D)ᵒᵖ).map f ≫ ((fun F => (unop F).rightOpLeftOpIso.op) Y).hom =\n      ((fun F => (unop F).rightOpLeftOpIso.op) X).hom ≫\n        ({ obj := fun F => (unop F).rightOp, map := fun {X Y} η => NatTrans.rightOp η.unop, map_id := ⋯,\n                map_comp := ⋯ } ⋙\n              { obj := fun F => op F.leftOp, map := fun {X Y} η => (NatTrans.leftOp η).op, map_id := ⋯,\n                map_comp := ⋯ }).map\n          f","state_after":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF G : (Cᵒᵖ ⥤ D)ᵒᵖ\nη : F ⟶ G\n⊢ (𝟭 (Cᵒᵖ ⥤ D)ᵒᵖ).map η ≫ ((fun F => (unop F).rightOpLeftOpIso.op) G).hom =\n    ((fun F => (unop F).rightOpLeftOpIso.op) F).hom ≫\n      ({ obj := fun F => (unop F).rightOp, map := fun {X Y} η => NatTrans.rightOp η.unop, map_id := ⋯, map_comp := ⋯ } ⋙\n            { obj := fun F => op F.leftOp, map := fun {X Y} η => (NatTrans.leftOp η).op, map_id := ⋯,\n              map_comp := ⋯ }).map\n        η","tactic":"intro F G η","premises":[]},{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF G : (Cᵒᵖ ⥤ D)ᵒᵖ\nη : F ⟶ G\n⊢ (𝟭 (Cᵒᵖ ⥤ D)ᵒᵖ).map η ≫ ((fun F => (unop F).rightOpLeftOpIso.op) G).hom =\n    ((fun F => (unop F).rightOpLeftOpIso.op) F).hom ≫\n      ({ obj := fun F => (unop F).rightOp, map := fun {X Y} η => NatTrans.rightOp η.unop, map_id := ⋯, map_comp := ⋯ } ⋙\n            { obj := fun F => op F.leftOp, map := fun {X Y} η => (NatTrans.leftOp η).op, map_id := ⋯,\n              map_comp := ⋯ }).map\n        η","state_after":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF G : (Cᵒᵖ ⥤ D)ᵒᵖ\nη : F ⟶ G\n⊢ η ≫ (unop G).rightOpLeftOpIso.hom.op =\n    (unop F).rightOpLeftOpIso.hom.op ≫ (NatTrans.leftOp (NatTrans.rightOp η.unop)).op","tactic":"dsimp","premises":[]},{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF G : (Cᵒᵖ ⥤ D)ᵒᵖ\nη : F ⟶ G\n⊢ η ≫ (unop G).rightOpLeftOpIso.hom.op =\n    (unop F).rightOpLeftOpIso.hom.op ≫ (NatTrans.leftOp (NatTrans.rightOp η.unop)).op","state_after":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF G : (Cᵒᵖ ⥤ D)ᵒᵖ\nη : F ⟶ G\n⊢ ((unop G).rightOpLeftOpIso.hom ≫ η.unop).op =\n    (NatTrans.leftOp (NatTrans.rightOp η.unop.op.unop) ≫ (unop F).rightOpLeftOpIso.hom).op","tactic":"rw [show η = η.unop.op by simp, ← op_comp, ← op_comp]","premises":[{"full_name":"CategoryTheory.op_comp","def_path":"Mathlib/CategoryTheory/Opposites.lean","def_pos":[72,8],"def_end_pos":[72,15]},{"full_name":"Quiver.Hom.op","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[136,4],"def_end_pos":[136,10]},{"full_name":"Quiver.Hom.unop","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[139,4],"def_end_pos":[139,12]}]},{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF G : (Cᵒᵖ ⥤ D)ᵒᵖ\nη : F ⟶ G\n⊢ ((unop G).rightOpLeftOpIso.hom ≫ η.unop).op =\n    (NatTrans.leftOp (NatTrans.rightOp η.unop.op.unop) ≫ (unop F).rightOpLeftOpIso.hom).op","state_after":"case e_f\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF G : (Cᵒᵖ ⥤ D)ᵒᵖ\nη : F ⟶ G\n⊢ (unop G).rightOpLeftOpIso.hom ≫ η.unop =\n    NatTrans.leftOp (NatTrans.rightOp η.unop.op.unop) ≫ (unop F).rightOpLeftOpIso.hom","tactic":"congr 1","premises":[]},{"state_before":"case e_f\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF G : (Cᵒᵖ ⥤ D)ᵒᵖ\nη : F ⟶ G\n⊢ (unop G).rightOpLeftOpIso.hom ≫ η.unop =\n    NatTrans.leftOp (NatTrans.rightOp η.unop.op.unop) ≫ (unop F).rightOpLeftOpIso.hom","state_after":"no goals","tactic":"aesop_cat","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]}]}]}
+{"url":"Mathlib/Data/Fin/Tuple/Sort.lean","commit":"","full_name":"Tuple.eq_sort_iff'","start":[148,0],"end":[156,87],"file_path":"Mathlib/Data/Fin/Tuple/Sort.lean","tactics":[{"state_before":"n : ℕ\nα : Type u_1\ninst✝ : LinearOrder α\nf : Fin n → α\nσ : Equiv.Perm (Fin n)\n⊢ σ = sort f ↔ StrictMono ⇑(Equiv.trans σ (graphEquiv₁ f))","state_after":"case mp\nn : ℕ\nα : Type u_1\ninst✝ : LinearOrder α\nf : Fin n → α\nσ : Equiv.Perm (Fin n)\nh : σ = sort f\n⊢ StrictMono ⇑(Equiv.trans σ (graphEquiv₁ f))\n\ncase mpr\nn : ℕ\nα : Type u_1\ninst✝ : LinearOrder α\nf : Fin n → α\nσ : Equiv.Perm (Fin n)\nh : StrictMono ⇑(Equiv.trans σ (graphEquiv₁ f))\n⊢ σ = sort f","tactic":"constructor <;> intro h","premises":[]}]}
+{"url":"Mathlib/GroupTheory/FreeGroup/Basic.lean","commit":"","full_name":"FreeGroup.red_invRev_iff","start":[513,0],"end":[515,76],"file_path":"Mathlib/GroupTheory/FreeGroup/Basic.lean","tactics":[{"state_before":"α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\nh : Red (invRev L₁) (invRev L₂)\n⊢ Red L₁ L₂","state_after":"no goals","tactic":"simpa only [invRev_invRev] using h.invRev","premises":[{"full_name":"FreeGroup.Red.invRev","def_path":"Mathlib/GroupTheory/FreeGroup/Basic.lean","def_pos":[505,8],"def_end_pos":[505,18]},{"full_name":"FreeGroup.invRev_invRev","def_path":"Mathlib/GroupTheory/FreeGroup/Basic.lean","def_pos":[464,8],"def_end_pos":[464,21]}]}]}
+{"url":"Mathlib/Data/Multiset/Basic.lean","commit":"","full_name":"Multiset.rel_of_forall","start":[2455,0],"end":[2467,20],"file_path":"Mathlib/Data/Multiset/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type v\nγ : Type u_2\nδ : Type u_3\nr✝ : α → β → Prop\np : γ → δ → Prop\nm1 m2 : Multiset α\nr : α → α → Prop\nh : ∀ (a b : α), a ∈ m1 → b ∈ m2 → r a b\nhc : card m1 = card m2\n⊢ Rel r m1 m2","state_after":"α : Type u_1\nβ : Type v\nγ : Type u_2\nδ : Type u_3\nr✝ : α → β → Prop\np : γ → δ → Prop\nm2 : Multiset α\nr : α → α → Prop\n⊢ ∀ {m1 : Multiset α}, (∀ (a b : α), a ∈ m1 → b ∈ m2 → r a b) → card m1 = card m2 → Rel r m1 m2","tactic":"revert m1","premises":[]},{"state_before":"α : Type u_1\nβ : Type v\nγ : Type u_2\nδ : Type u_3\nr✝ : α → β → Prop\np : γ → δ → Prop\nm2 : Multiset α\nr : α → α → Prop\n⊢ ∀ {m1 : Multiset α}, (∀ (a b : α), a ∈ m1 → b ∈ m2 → r a b) → card m1 = card m2 → Rel r m1 m2","state_after":"case refine_1\nα : Type u_1\nβ : Type v\nγ : Type u_2\nδ : Type u_3\nr✝ : α → β → Prop\np : γ → δ → Prop\nm2 : Multiset α\nr : α → α → Prop\n⊢ ∀ {m1 : Multiset α}, (∀ (a b : α), a ∈ m1 → b ∈ 0 → r a b) → card m1 = card 0 → Rel r m1 0\n\ncase refine_2\nα : Type u_1\nβ : Type v\nγ : Type u_2\nδ : Type u_3\nr✝ : α → β → Prop\np : γ → δ → Prop\nm2 : Multiset α\nr : α → α → Prop\n⊢ ∀ (a : α) (s : Multiset α),\n    (∀ {m1 : Multiset α}, (∀ (a b : α), a ∈ m1 → b ∈ s → r a b) → card m1 = card s → Rel r m1 s) →\n      ∀ {m1 : Multiset α},\n        (∀ (a_2 b : α), a_2 ∈ m1 → b ∈ a ::ₘ s → r a_2 b) → card m1 = card (a ::ₘ s) → Rel r m1 (a ::ₘ s)","tactic":"refine @(m2.induction_on ?_ ?_)","premises":[{"full_name":"Multiset.induction_on","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[152,18],"def_end_pos":[152,30]}]}]}
+{"url":"Mathlib/LinearAlgebra/Matrix/Hermitian.lean","commit":"","full_name":"Matrix.IsHermitian.fromBlocks","start":[95,0],"end":[102,49],"file_path":"Mathlib/LinearAlgebra/Matrix/Hermitian.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nm : Type u_3\nn : Type u_4\nA✝ : Matrix n n α\ninst✝ : InvolutiveStar α\nA : Matrix m m α\nB : Matrix m n α\nC : Matrix n m α\nD : Matrix n n α\nhA : A.IsHermitian\nhBC : Bᴴ = C\nhD : D.IsHermitian\n⊢ (Matrix.fromBlocks A B C D).IsHermitian","state_after":"α : Type u_1\nβ : Type u_2\nm : Type u_3\nn : Type u_4\nA✝ : Matrix n n α\ninst✝ : InvolutiveStar α\nA : Matrix m m α\nB : Matrix m n α\nC : Matrix n m α\nD : Matrix n n α\nhA : A.IsHermitian\nhBC : Bᴴ = C\nhD : D.IsHermitian\nhCB : Cᴴ = B\n⊢ (Matrix.fromBlocks A B C D).IsHermitian","tactic":"have hCB : Cᴴ = B := by rw [← hBC, conjTranspose_conjTranspose]","premises":[{"full_name":"Matrix.conjTranspose","def_path":"Mathlib/Data/Matrix/Basic.lean","def_pos":[152,4],"def_end_pos":[152,17]},{"full_name":"Matrix.conjTranspose_conjTranspose","def_path":"Mathlib/Data/Matrix/Basic.lean","def_pos":[1988,8],"def_end_pos":[1988,35]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nm : Type u_3\nn : Type u_4\nA✝ : Matrix n n α\ninst✝ : InvolutiveStar α\nA : Matrix m m α\nB : Matrix m n α\nC : Matrix n m α\nD : Matrix n n α\nhA : A.IsHermitian\nhBC : Bᴴ = C\nhD : D.IsHermitian\nhCB : Cᴴ = B\n⊢ (Matrix.fromBlocks A B C D).IsHermitian","state_after":"α : Type u_1\nβ : Type u_2\nm : Type u_3\nn : Type u_4\nA✝ : Matrix n n α\ninst✝ : InvolutiveStar α\nA : Matrix m m α\nB : Matrix m n α\nC : Matrix n m α\nD : Matrix n n α\nhA : A.IsHermitian\nhBC : Bᴴ = C\nhD : D.IsHermitian\nhCB : Cᴴ = B\n⊢ (Matrix.fromBlocks A B C D)ᴴ = Matrix.fromBlocks A B C D","tactic":"unfold Matrix.IsHermitian","premises":[{"full_name":"Matrix.IsHermitian","def_path":"Mathlib/LinearAlgebra/Matrix/Hermitian.lean","def_pos":[40,4],"def_end_pos":[40,15]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nm : Type u_3\nn : Type u_4\nA✝ : Matrix n n α\ninst✝ : InvolutiveStar α\nA : Matrix m m α\nB : Matrix m n α\nC : Matrix n m α\nD : Matrix n n α\nhA : A.IsHermitian\nhBC : Bᴴ = C\nhD : D.IsHermitian\nhCB : Cᴴ = B\n⊢ (Matrix.fromBlocks A B C D)ᴴ = Matrix.fromBlocks A B C D","state_after":"no goals","tactic":"rw [fromBlocks_conjTranspose, hBC, hCB, hA, hD]","premises":[{"full_name":"Matrix.fromBlocks_conjTranspose","def_path":"Mathlib/Data/Matrix/Block.lean","def_pos":[134,8],"def_end_pos":[134,32]}]}]}
+{"url":"Mathlib/Data/Matrix/Notation.lean","commit":"","full_name":"Matrix.one_fin_three","start":[404,0],"end":[406,37],"file_path":"Mathlib/Data/Matrix/Notation.lean","tactics":[{"state_before":"α : Type u\no n m : ℕ\nm' : Type uₘ\nn' : Type uₙ\no' : Type uₒ\na b : ℕ\ninst✝¹ : Zero α\ninst✝ : One α\n⊢ 1 = !![1, 0, 0; 0, 1, 0; 0, 0, 1]","state_after":"case a\nα : Type u\no n m : ℕ\nm' : Type uₘ\nn' : Type uₙ\no' : Type uₒ\na b : ℕ\ninst✝¹ : Zero α\ninst✝ : One α\ni j : Fin 3\n⊢ 1 i j = !![1, 0, 0; 0, 1, 0; 0, 0, 1] i j","tactic":"ext i j","premises":[]},{"state_before":"case a\nα : Type u\no n m : ℕ\nm' : Type uₘ\nn' : Type uₙ\no' : Type uₒ\na b : ℕ\ninst✝¹ : Zero α\ninst✝ : One α\ni j : Fin 3\n⊢ 1 i j = !![1, 0, 0; 0, 1, 0; 0, 0, 1] i j","state_after":"no goals","tactic":"fin_cases i <;> fin_cases j <;> rfl","premises":[]}]}
+{"url":"Mathlib/Analysis/SpecificLimits/Basic.lean","commit":"","full_name":"aux_hasSum_of_le_geometric","start":[425,0],"end":[429,48],"file_path":"Mathlib/Analysis/SpecificLimits/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝ : PseudoMetricSpace α\nr C : ℝ\nhr : r < 1\nf : ℕ → α\nhu : ∀ (n : ℕ), dist (f n) (f (n + 1)) ≤ C * r ^ n\n⊢ HasSum (fun n => C * r ^ n) (C / (1 - r))","state_after":"case inl\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝ : PseudoMetricSpace α\nr : ℝ\nhr : r < 1\nf : ℕ → α\nhu : ∀ (n : ℕ), dist (f n) (f (n + 1)) ≤ 0 * r ^ n\n⊢ HasSum (fun n => 0 * r ^ n) (0 / (1 - r))\n\ncase inr.intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝ : PseudoMetricSpace α\nr C : ℝ\nhr : r < 1\nf : ℕ → α\nhu : ∀ (n : ℕ), dist (f n) (f (n + 1)) ≤ C * r ^ n\nleft✝ : 0 < C\nr₀ : 0 ≤ r\n⊢ HasSum (fun n => C * r ^ n) (C / (1 - r))","tactic":"rcases sign_cases_of_C_mul_pow_nonneg fun n ↦ dist_nonneg.trans (hu n) with (rfl | ⟨_, r₀⟩)","premises":[{"full_name":"dist_nonneg","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[234,8],"def_end_pos":[234,19]},{"full_name":"sign_cases_of_C_mul_pow_nonneg","def_path":"Mathlib/Algebra/Order/Ring/Unbundled/Basic.lean","def_pos":[753,6],"def_end_pos":[753,36]}]}]}
+{"url":"Mathlib/RingTheory/Multiplicity.lean","commit":"","full_name":"multiplicity.pow","start":[545,0],"end":[547,80],"file_path":"Mathlib/RingTheory/Multiplicity.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np a : α\nhp : Prime p\n⊢ multiplicity p (a ^ 0) = 0 • multiplicity p a","state_after":"no goals","tactic":"simp [one_right hp.not_unit]","premises":[{"full_name":"Prime.not_unit","def_path":"Mathlib/Algebra/Associated/Basic.lean","def_pos":[50,8],"def_end_pos":[50,16]},{"full_name":"multiplicity.one_right","def_path":"Mathlib/RingTheory/Multiplicity.lean","def_pos":[285,8],"def_end_pos":[285,17]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np a : α\nhp : Prime p\nk : ℕ\n⊢ multiplicity p (a ^ k.succ) = k.succ • multiplicity p a","state_after":"no goals","tactic":"simp [_root_.pow_succ, succ_nsmul, pow hp, multiplicity.mul hp]","premises":[{"full_name":"multiplicity.mul","def_path":"Mathlib/RingTheory/Multiplicity.lean","def_pos":[515,18],"def_end_pos":[515,21]},{"full_name":"pow_succ","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[567,8],"def_end_pos":[567,16]},{"full_name":"succ_nsmul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[566,14],"def_end_pos":[566,24]}]}]}
+{"url":"Mathlib/Algebra/Order/Pointwise.lean","commit":"","full_name":"sInf_sub","start":[77,0],"end":[78,100],"file_path":"Mathlib/Algebra/Order/Pointwise.lean","tactics":[{"state_before":"α : Type u_1\ninst✝³ : CompleteLattice α\ninst✝² : Group α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ns t : Set α\n⊢ sInf (s / t) = sInf s / sSup t","state_after":"no goals","tactic":"simp_rw [div_eq_mul_inv, sInf_mul, sInf_inv]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"div_eq_mul_inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[907,8],"def_end_pos":[907,22]},{"full_name":"sInf_inv","def_path":"Mathlib/Algebra/Order/Pointwise.lean","def_pos":[60,8],"def_end_pos":[60,16]},{"full_name":"sInf_mul","def_path":"Mathlib/Algebra/Order/Pointwise.lean","def_pos":[70,8],"def_end_pos":[70,16]}]}]}
+{"url":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","commit":"","full_name":"_private.Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.0.WeierstrassCurve.Jacobian.nonsingular_add_of_Z_ne_zero","start":[1242,0],"end":[1250,100],"file_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","tactics":[{"state_before":"R : Type u\ninst✝¹ : CommRing R\nW' : Jacobian R\nF : Type v\ninst✝ : Field F\nW : Jacobian F\nP Q : Fin 3 → F\nhP : W.Nonsingular P\nhQ : W.Nonsingular Q\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhxy : P x * Q z ^ 2 = Q x * P z ^ 2 → P y * Q z ^ 3 ≠ W.negY Q * P z ^ 3\n⊢ P x / P z ^ 2 = Q x / Q z ^ 2 → P y / P z ^ 3 ≠ W.toAffine.negY (Q x / Q z ^ 2) (Q y / Q z ^ 3)","state_after":"no goals","tactic":"rwa [← X_eq_iff hPz hQz, ne_eq, ← Y_eq_iff' hPz hQz]","premises":[{"full_name":"WeierstrassCurve.Jacobian.X_eq_iff","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","def_pos":[217,6],"def_end_pos":[217,14]},{"full_name":"WeierstrassCurve.Jacobian.Y_eq_iff'","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","def_pos":[513,6],"def_end_pos":[513,15]},{"full_name":"ne_eq","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[89,16],"def_end_pos":[89,21]}]}]}
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+{"url":"Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean","commit":"","full_name":"CategoryTheory.Limits.coprod.map_comp_id","start":[771,0],"end":[774,81],"file_path":"Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean","tactics":[{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\nX✝ Y✝ X Y Z W : C\nf : X ⟶ Y\ng : Y ⟶ Z\ninst✝² : HasBinaryCoproduct Z W\ninst✝¹ : HasBinaryCoproduct Y W\ninst✝ : HasBinaryCoproduct X W\n⊢ map (f ≫ g) (𝟙 W) = map f (𝟙 W) ≫ map g (𝟙 W)","state_after":"no goals","tactic":"simp","premises":[]}]}
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+{"url":"Mathlib/Topology/Algebra/Monoid.lean","commit":"","full_name":"continuousAdd_of_comm_of_nhds_zero","start":[282,0],"end":[288,30],"file_path":"Mathlib/Topology/Algebra/Monoid.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nM✝ : Type u_3\nN : Type u_4\nX : Type u_5\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace M✝\ninst✝³ : Mul M✝\ninst✝² : ContinuousMul M✝\nM : Type u\ninst✝¹ : CommMonoid M\ninst✝ : TopologicalSpace M\nhmul : Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)\nhleft : ∀ (x₀ : M), 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1)\n⊢ ContinuousMul M","state_after":"ι : Type u_1\nα : Type u_2\nM✝ : Type u_3\nN : Type u_4\nX : Type u_5\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace M✝\ninst✝³ : Mul M✝\ninst✝² : ContinuousMul M✝\nM : Type u\ninst✝¹ : CommMonoid M\ninst✝ : TopologicalSpace M\nhmul : Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)\nhleft : ∀ (x₀ : M), 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1)\n⊢ ∀ (x₀ : M), 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)","tactic":"apply ContinuousMul.of_nhds_one hmul hleft","premises":[{"full_name":"ContinuousMul.of_nhds_one","def_path":"Mathlib/Topology/Algebra/Monoid.lean","def_pos":[252,8],"def_end_pos":[252,33]}]},{"state_before":"ι : Type u_1\nα : Type u_2\nM✝ : Type u_3\nN : Type u_4\nX : Type u_5\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace M✝\ninst✝³ : Mul M✝\ninst✝² : ContinuousMul M✝\nM : Type u\ninst✝¹ : CommMonoid M\ninst✝ : TopologicalSpace M\nhmul : Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)\nhleft : ∀ (x₀ : M), 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1)\n⊢ ∀ (x₀ : M), 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)","state_after":"ι : Type u_1\nα : Type u_2\nM✝ : Type u_3\nN : Type u_4\nX : Type u_5\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace M✝\ninst✝³ : Mul M✝\ninst✝² : ContinuousMul M✝\nM : Type u\ninst✝¹ : CommMonoid M\ninst✝ : TopologicalSpace M\nhmul : Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)\nhleft : ∀ (x₀ : M), 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1)\nx₀ : M\n⊢ 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)","tactic":"intro x₀","premises":[]},{"state_before":"ι : Type u_1\nα : Type u_2\nM✝ : Type u_3\nN : Type u_4\nX : Type u_5\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace M✝\ninst✝³ : Mul M✝\ninst✝² : ContinuousMul M✝\nM : Type u\ninst✝¹ : CommMonoid M\ninst✝ : TopologicalSpace M\nhmul : Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)\nhleft : ∀ (x₀ : M), 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1)\nx₀ : M\n⊢ 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)","state_after":"no goals","tactic":"simp_rw [mul_comm, hleft x₀]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]}]}]}
+{"url":"Mathlib/MeasureTheory/Constructions/Polish/Basic.lean","commit":"","full_name":"MeasureTheory.MeasurablySeparable.iUnion","start":[359,0],"end":[370,46],"file_path":"Mathlib/MeasureTheory/Constructions/Polish/Basic.lean","tactics":[{"state_before":"α✝ : Type u_1\nι : Type u_2\ninst✝² : TopologicalSpace α✝\ninst✝¹ : Countable ι\nα : Type u_3\ninst✝ : MeasurableSpace α\ns t : ι → Set α\nh : ∀ (m n : ι), MeasurablySeparable (s m) (t n)\n⊢ MeasurablySeparable (⋃ n, s n) (⋃ m, t m)","state_after":"α✝ : Type u_1\nι : Type u_2\ninst✝² : TopologicalSpace α✝\ninst✝¹ : Countable ι\nα : Type u_3\ninst✝ : MeasurableSpace α\ns t : ι → Set α\nu : ι → ι → Set α\nhsu : ∀ (m n : ι), s m ⊆ u m n\nhtu : ∀ (m n : ι), Disjoint (t n) (u m n)\nhu : ∀ (m n : ι), MeasurableSet (u m n)\n⊢ MeasurablySeparable (⋃ n, s n) (⋃ m, t m)","tactic":"choose u hsu htu hu using h","premises":[]},{"state_before":"α✝ : Type u_1\nι : Type u_2\ninst✝² : TopologicalSpace α✝\ninst✝¹ : Countable ι\nα : Type u_3\ninst✝ : MeasurableSpace α\ns t : ι → Set α\nu : ι → ι → Set α\nhsu : ∀ (m n : ι), s m ⊆ u m n\nhtu : ∀ (m n : ι), Disjoint (t n) (u m n)\nhu : ∀ (m n : ι), MeasurableSet (u m n)\n⊢ MeasurablySeparable (⋃ n, s n) (⋃ m, t m)","state_after":"case refine_1\nα✝ : Type u_1\nι : Type u_2\ninst✝² : TopologicalSpace α✝\ninst✝¹ : Countable ι\nα : Type u_3\ninst✝ : MeasurableSpace α\ns t : ι → Set α\nu : ι → ι → Set α\nhsu : ∀ (m n : ι), s m ⊆ u m n\nhtu : ∀ (m n : ι), Disjoint (t n) (u m n)\nhu : ∀ (m n : ι), MeasurableSet (u m n)\n⊢ ⋃ n, s n ⊆ ⋃ m, ⋂ n, u m n\n\ncase refine_2\nα✝ : Type u_1\nι : Type u_2\ninst✝² : TopologicalSpace α✝\ninst✝¹ : Countable ι\nα : Type u_3\ninst✝ : MeasurableSpace α\ns t : ι → Set α\nu : ι → ι → Set α\nhsu : ∀ (m n : ι), s m ⊆ u m n\nhtu : ∀ (m n : ι), Disjoint (t n) (u m n)\nhu : ∀ (m n : ι), MeasurableSet (u m n)\n⊢ Disjoint (⋃ m, t m) (⋃ m, ⋂ n, u m n)\n\ncase refine_3\nα✝ : Type u_1\nι : Type u_2\ninst✝² : TopologicalSpace α✝\ninst✝¹ : Countable ι\nα : Type u_3\ninst✝ : MeasurableSpace α\ns t : ι → Set α\nu : ι → ι → Set α\nhsu : ∀ (m n : ι), s m ⊆ u m n\nhtu : ∀ (m n : ι), Disjoint (t n) (u m n)\nhu : ∀ (m n : ι), MeasurableSet (u m n)\n⊢ MeasurableSet (⋃ m, ⋂ n, u m n)","tactic":"refine ⟨⋃ m, ⋂ n, u m n, ?_, ?_, ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Set.iInter","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[182,4],"def_end_pos":[182,10]},{"full_name":"Set.iUnion","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[178,4],"def_end_pos":[178,10]}]}]}
+{"url":"Mathlib/Algebra/Lie/Abelian.lean","commit":"","full_name":"LieModule.coe_maxTrivLinearMapEquivLieModuleHom","start":[202,0],"end":[204,68],"file_path":"Mathlib/Algebra/Lie/Abelian.lean","tactics":[{"state_before":"R : Type u\nL : Type v\nM : Type w\nN : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝��� : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : LieRingModule L N\ninst✝ : LieModule R L N\nf : ↥(maxTrivSubmodule R L (M →ₗ[R] N))\n⊢ ⇑(maxTrivLinearMapEquivLieModuleHom f) = ⇑↑f","state_after":"case h\nR : Type u\nL : Type v\nM : Type w\nN : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : LieRingModule L N\ninst✝ : LieModule R L N\nf : ↥(maxTrivSubmodule R L (M →ₗ[R] N))\nx✝ : M\n⊢ (maxTrivLinearMapEquivLieModuleHom f) x✝ = ↑f x✝","tactic":"ext","premises":[]},{"state_before":"case h\nR : Type u\nL : Type v\nM : Type w\nN : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : LieRingModule L N\ninst✝ : LieModule R L N\nf : ↥(maxTrivSubmodule R L (M →ₗ[R] N))\nx✝ : M\n⊢ (maxTrivLinearMapEquivLieModuleHom f) x✝ = ↑f x✝","state_after":"no goals","tactic":"rfl","premises":[]}]}
+{"url":"Mathlib/CategoryTheory/Comma/StructuredArrow.lean","commit":"","full_name":"CategoryTheory.CostructuredArrow.toStructuredArrow'_map","start":[837,0],"end":[851,31],"file_path":"Mathlib/CategoryTheory/Comma/StructuredArrow.lean","tactics":[{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nd : D\nX✝ Y✝ : (CostructuredArrow F.op (op d))ᵒᵖ\nf : X✝ ⟶ Y✝\n⊢ ((fun X => StructuredArrow.mk (unop X).hom.unop) X✝).hom ≫ F.map f.unop.left.unop =\n    ((fun X => StructuredArrow.mk (unop X).hom.unop) Y✝).hom","state_after":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nd : D\nX✝ Y✝ : (CostructuredArrow F.op (op d))ᵒᵖ\nf : X✝ ⟶ Y✝\n⊢ (unop X✝).hom.unop ≫ F.map f.unop.left.unop = (unop Y✝).hom.unop","tactic":"dsimp","premises":[]},{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nd : D\nX✝ Y✝ : (CostructuredArrow F.op (op d))ᵒᵖ\nf : X✝ ⟶ Y✝\n⊢ (unop X✝).hom.unop ≫ F.map f.unop.left.unop = (unop Y✝).hom.unop","state_after":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nd : D\nX✝ Y✝ : (CostructuredArrow F.op (op d))ᵒᵖ\nf : X✝ ⟶ Y✝\n⊢ ((unop Y✝).hom ≫ 𝟙 (op d)).unop = (unop Y✝).hom.unop","tactic":"rw [← Quiver.Hom.unop_op (F.map f.unop.left.unop), ← unop_comp, ← F.op_map, f.unop.w,\n          Functor.const_obj_map]","premises":[{"full_name":"CategoryTheory.CommaMorphism.left","def_path":"Mathlib/CategoryTheory/Comma/Basic.lean","def_pos":[80,2],"def_end_pos":[80,6]},{"full_name":"CategoryTheory.CommaMorphism.w","def_path":"Mathlib/CategoryTheory/Comma/Basic.lean","def_pos":[82,2],"def_end_pos":[82,3]},{"full_name":"CategoryTheory.Functor.const_obj_map","def_path":"Mathlib/CategoryTheory/Functor/Const.lean","def_pos":[32,2],"def_end_pos":[32,7]},{"full_name":"CategoryTheory.Functor.op_map","def_path":"Mathlib/CategoryTheory/Opposites.lean","def_pos":[158,2],"def_end_pos":[158,7]},{"full_name":"CategoryTheory.unop_comp","def_path":"Mathlib/CategoryTheory/Opposites.lean","def_pos":[80,8],"def_end_pos":[80,17]},{"full_name":"Prefunctor.map","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[57,2],"def_end_pos":[57,5]},{"full_name":"Quiver.Hom.unop","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[139,4],"def_end_pos":[139,12]},{"full_name":"Quiver.Hom.unop_op","def_path":"Mathlib/CategoryTheory/Opposites.lean","def_pos":[44,8],"def_end_pos":[44,26]}]},{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nd : D\nX✝ Y✝ : (CostructuredArrow F.op (op d))ᵒᵖ\nf : X✝ ⟶ Y✝\n⊢ ((unop Y✝).hom ≫ 𝟙 (op d)).unop = (unop Y✝).hom.unop","state_after":"no goals","tactic":"erw [Category.comp_id]","premises":[{"full_name":"CategoryTheory.Category.comp_id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[153,2],"def_end_pos":[153,9]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]}]}]}
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+{"url":"Mathlib/Data/Int/Bitwise.lean","commit":"","full_name":"Int.zero_shiftLeft","start":[437,0],"end":[440,47],"file_path":"Mathlib/Data/Int/Bitwise.lean","tactics":[{"state_before":"n : ℕ\n⊢ Nat.shiftLeft' false 0 n = 0","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"a✝ : ℕ\n⊢ 0 >>> a✝.succ = 0","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/NumberTheory/Padics/Hensel.lean","commit":"","full_name":"_private.Mathlib.NumberTheory.Padics.Hensel.0.newton_seq_succ_dist_weak","start":[274,0],"end":[292,42],"file_path":"Mathlib/NumberTheory/Padics/Hensel.lean","tactics":[{"state_before":"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nhnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2\nhnsol : Polynomial.eval a F ≠ 0\nn : ℕ\n⊢ 2 ≤ 2 ^ (n + 1)","state_after":"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nhnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2\nhnsol : Polynomial.eval a F ≠ 0\nn : ℕ\nthis : 2 ^ 1 ≤ 2 ^ (n + 1)\n⊢ 2 ≤ 2 ^ (n + 1)","tactic":"have := pow_le_pow_right (by norm_num : 1 ≤ 2) (Nat.le_add_left _ _ : 1 ≤ n + 1)","premises":[{"full_name":"Nat.le_add_left","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[399,8],"def_end_pos":[399,19]},{"full_name":"pow_le_pow_right","def_path":"Mathlib/Algebra/Order/Ring/Basic.lean","def_pos":[84,8],"def_end_pos":[84,24]}]},{"state_before":"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nhnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2\nhnsol : Polynomial.eval a F ≠ 0\nn : ℕ\nthis : 2 ^ 1 ≤ 2 ^ (n + 1)\n⊢ 2 ≤ 2 ^ (n + 1)","state_after":"no goals","tactic":"simpa using this","premises":[]},{"state_before":"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nhnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2\nhnsol : Polynomial.eval a F ≠ 0\nn : ℕ\nthis : 2 ≤ 2 ^ (n + 1)\n⊢ 1 < 2","state_after":"no goals","tactic":"norm_num","premises":[]},{"state_before":"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nhnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2\nhnsol : Polynomial.eval a F ≠ 0\nn : ℕ\nthis : 2 ≤ 2 ^ (n + 1)\n⊢ ‖Polynomial.eval a (Polynomial.derivative F)‖ * T_gen p F a ^ 1 =\n    ‖Polynomial.eval a F‖ / ‖Polynomial.eval a (Polynomial.derivative F)‖","state_after":"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nhnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2\nhnsol : Polynomial.eval a F ≠ 0\nn : ℕ\nthis : 2 ≤ 2 ^ (n + 1)\n⊢ ‖Polynomial.eval a (Polynomial.derivative F)‖ * ‖Polynomial.eval a F‖ /\n      (‖↑(Polynomial.eval a (Polynomial.derivative F))‖ * ‖↑(Polynomial.eval a (Polynomial.derivative F))‖) =\n    ‖Polynomial.eval a F‖ / ‖Polynomial.eval a (Polynomial.derivative F)‖","tactic":"rw [T_gen, sq, pow_one, norm_div, ← mul_div_assoc, PadicInt.padic_norm_e_of_padicInt,\n        PadicInt.coe_mul, padicNormE.mul]","premises":[{"full_name":"PadicInt.coe_mul","def_path":"Mathlib/NumberTheory/Padics/PadicIntegers.lean","def_pos":[118,8],"def_end_pos":[118,15]},{"full_name":"PadicInt.padic_norm_e_of_padicInt","def_path":"Mathlib/NumberTheory/Padics/PadicIntegers.lean","def_pos":[269,8],"def_end_pos":[269,32]},{"full_name":"_private.Mathlib.NumberTheory.Padics.Hensel.0.T_gen","def_path":"Mathlib/NumberTheory/Padics/Hensel.lean","def_pos":[98,12],"def_end_pos":[98,17]},{"full_name":"mul_div_assoc","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[330,8],"def_end_pos":[330,21]},{"full_name":"norm_div","def_path":"Mathlib/Analysis/Normed/Field/Basic.lean","def_pos":[708,8],"def_end_pos":[708,16]},{"full_name":"padicNormE.mul","def_path":"Mathlib/NumberTheory/Padics/PadicNumbers.lean","def_pos":[751,18],"def_end_pos":[751,21]},{"full_name":"pow_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[571,6],"def_end_pos":[571,13]}]},{"state_before":"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nhnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2\nhnsol : Polynomial.eval a F ≠ 0\nn : ℕ\nthis : 2 ≤ 2 ^ (n + 1)\n⊢ ‖Polynomial.eval a (Polynomial.derivative F)‖ * ‖Polynomial.eval a F‖ /\n      (‖↑(Polynomial.eval a (Polynomial.derivative F))‖ * ‖↑(Polynomial.eval a (Polynomial.derivative F))‖) =\n    ‖Polynomial.eval a F‖ / ‖Polynomial.eval a (Polynomial.derivative F)‖","state_after":"case hc\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nhnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2\nhnsol : Polynomial.eval a F ≠ 0\nn : ℕ\nthis : 2 ≤ 2 ^ (n + 1)\n⊢ ‖Polynomial.eval a (Polynomial.derivative F)‖ ≠ 0","tactic":"apply mul_div_mul_left","premises":[{"full_name":"mul_div_mul_left","def_path":"Mathlib/Algebra/GroupWithZero/Units/Basic.lean","def_pos":[427,6],"def_end_pos":[427,22]}]},{"state_before":"case hc\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nhnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2\nhnsol : Polynomial.eval a F ≠ 0\nn : ℕ\nthis : 2 ≤ 2 ^ (n + 1)\n⊢ ‖Polynomial.eval a (Polynomial.derivative F)‖ ≠ 0","state_after":"case hc.hnorm\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nhnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2\nhnsol : Polynomial.eval a F ≠ 0\nn : ℕ\nthis : 2 ≤ 2 ^ (n + 1)\n⊢ ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2","tactic":"apply deriv_norm_ne_zero","premises":[{"full_name":"_private.Mathlib.NumberTheory.Padics.Hensel.0.deriv_norm_ne_zero","def_path":"Mathlib/NumberTheory/Padics/Hensel.lean","def_pos":[110,16],"def_end_pos":[110,34]}]},{"state_before":"case hc.hnorm\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nhnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2\nhnsol : Polynomial.eval a F ≠ 0\nn : ℕ\nthis : 2 ≤ 2 ^ (n + 1)\n⊢ ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2","state_after":"no goals","tactic":"assumption","premises":[]}]}
+{"url":"Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean","commit":"","full_name":"MeasureTheory.SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge","start":[87,0],"end":[150,8],"file_path":"Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean","tactics":[{"state_before":"α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : μ.WeaklyRegular\nf : α →ₛ ℝ≥0\nε : ℝ≥0∞\nε0 : ε ≠ 0\n⊢ ∃ g, (∀ (x : α), ↑f x ≤ g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), ↑(g x) ∂μ ≤ ∫⁻ (x : α), ↑(↑f x) ∂μ + ε","state_after":"case h_ind\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : μ.WeaklyRegular\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nε : ℝ≥0∞\nε0 : ε ≠ 0\n⊢ ∃ g,\n    (∀ (x : α), ↑(piecewise s hs (const α c) (const α 0)) x ≤ g x) ∧\n      LowerSemicontinuous g ∧ ∫⁻ (x : α), ↑(g x) ∂μ ≤ ∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ + ε\n\ncase h_add\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : μ.WeaklyRegular\nf₁ f₂ : α →ₛ ℝ≥0\na✝ : Disjoint (Function.support ↑f₁) (Function.support ↑f₂)\nh₁ :\n  ∀ {ε : ℝ≥0∞},\n    ε ≠ 0 → ∃ g, (∀ (x : α), ↑f₁ x ≤ g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), ↑(g x) ∂μ ≤ ∫⁻ (x : α), ↑(↑f₁ x) ∂μ + ε\nh₂ :\n  ∀ {ε : ℝ≥0∞},\n    ε ≠ 0 → ∃ g, (∀ (x : α), ↑f₂ x ≤ g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), ↑(g x) ∂μ ≤ ∫⁻ (x : α), ↑(↑f₂ x) ∂μ + ε\nε : ℝ≥0∞\nε0 : ε ≠ 0\n⊢ ∃ g,\n    (∀ (x : α), ↑(f₁ + f₂) x ≤ g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), ↑(g x) ∂μ ≤ ∫⁻ (x : α), ↑(↑(f₁ + f₂) x) ∂μ + ε","tactic":"induction' f using MeasureTheory.SimpleFunc.induction with c s hs f₁ f₂ _ h₁ h₂ generalizing ε","premises":[{"full_name":"MeasureTheory.SimpleFunc.induction","def_path":"Mathlib/MeasureTheory/Function/SimpleFunc.lean","def_pos":[1077,18],"def_end_pos":[1077,27]}]}]}
+{"url":"Mathlib/NumberTheory/FLT/Three.lean","commit":"","full_name":"FermatLastTheoremForThreeGen.Solution.associated_of_dvd_a_add_b_of_dvd_a_add_eta_sq_mul_b","start":[420,0],"end":[431,6],"file_path":"Mathlib/NumberTheory/FLT/Three.lean","tactics":[{"state_before":"K : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\np : 𝓞 K\nhp : Prime p\nhpab : p ∣ S.a + S.b\nhpaηsqb : p ∣ S.a + ↑η ^ 2 * S.b\n⊢ Associated p λ","state_after":"K : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\np : 𝓞 K\nhp : Prime p\nhpab : p ∣ S.a + S.b\nhpaηsqb : p ∣ S.a + ↑η ^ 2 * S.b\n⊢ p ∣ λ","tactic":"suffices p_lam : p ∣ λ from hp.associated_of_dvd hζ.zeta_sub_one_prime' p_lam","premises":[{"full_name":"Dvd.dvd","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1344,2],"def_end_pos":[1344,5]},{"full_name":"IsPrimitiveRoot.toInteger","def_path":"Mathlib/NumberTheory/Cyclotomic/Rat.lean","def_pos":[170,7],"def_end_pos":[170,16]},{"full_name":"IsPrimitiveRoot.zeta_sub_one_prime'","def_path":"Mathlib/NumberTheory/Cyclotomic/Rat.lean","def_pos":[333,8],"def_end_pos":[333,27]},{"full_name":"Prime.associated_of_dvd","def_path":"Mathlib/Algebra/Associated/Basic.lean","def_pos":[614,8],"def_end_pos":[614,31]}]},{"state_before":"K : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\np : 𝓞 K\nhp : Prime p\nhpab : p ∣ S.a + S.b\nhpaηsqb : p ∣ S.a + ↑η ^ 2 * S.b\n⊢ p ∣ λ","state_after":"K : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\np : 𝓞 K\nhp : Prime p\nhpab : p ∣ 1 * S.a + 1 * S.b\nhpaηsqb : p ∣ S.a + ↑η ^ 2 * S.b\n⊢ p ∣ λ","tactic":"rw [← one_mul S.a, ← one_mul S.b] at hpab","premises":[{"full_name":"FermatLastTheoremForThreeGen.Solution'.a","def_path":"Mathlib/NumberTheory/FLT/Three.lean","def_pos":[184,2],"def_end_pos":[184,3]},{"full_name":"FermatLastTheoremForThreeGen.Solution'.b","def_path":"Mathlib/NumberTheory/FLT/Three.lean","def_pos":[185,2],"def_end_pos":[185,3]},{"full_name":"one_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[409,8],"def_end_pos":[409,15]}]},{"state_before":"K : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\np : 𝓞 K\nhp : Prime p\nhpab : p ∣ 1 * S.a + 1 * S.b\nhpaηsqb : p ∣ S.a + ↑η ^ 2 * S.b\n⊢ p ∣ λ","state_after":"K : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\np : 𝓞 K\nhp : Prime p\nhpab : p ∣ 1 * S.a + 1 * S.b\nhpaηsqb : p ∣ 1 * S.a + ↑η ^ 2 * S.b\n⊢ p ∣ λ","tactic":"rw [← one_mul S.a] at hpaηsqb","premises":[{"full_name":"FermatLastTheoremForThreeGen.Solution'.a","def_path":"Mathlib/NumberTheory/FLT/Three.lean","def_pos":[184,2],"def_end_pos":[184,3]},{"full_name":"one_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[409,8],"def_end_pos":[409,15]}]},{"state_before":"K : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\np : 𝓞 K\nhp : Prime p\nhpab : p ∣ 1 * S.a + 1 * S.b\nhpaηsqb : p ∣ 1 * S.a + ↑η ^ 2 * S.b\n⊢ p ∣ λ","state_after":"K : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\np : 𝓞 K\nhp : Prime p\nhpab : p ∣ 1 * S.a + 1 * S.b\nhpaηsqb : p ∣ 1 * S.a + ↑η ^ 2 * S.b\nthis : p ∣ (1 * ↑η ^ 2 - 1 * 1) * gcd S.a S.b\n⊢ p ∣ λ","tactic":"have := dvd_mul_sub_mul_mul_gcd_of_dvd hpab hpaηsqb","premises":[{"full_name":"dvd_mul_sub_mul_mul_gcd_of_dvd","def_path":"Mathlib/Algebra/Ring/Divisibility/Lemmas.lean","def_pos":[113,6],"def_end_pos":[113,36]}]},{"state_before":"K : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\np : 𝓞 K\nhp : Prime p\nhpab : p ∣ 1 * S.a + 1 * S.b\nhpaηsqb : p ∣ 1 * S.a + ↑η ^ 2 * S.b\nthis : p ∣ (1 * ↑η ^ 2 - 1 * 1) * gcd S.a S.b\n⊢ p ∣ λ","state_after":"K : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\np : 𝓞 K\nhp : Prime p\nhpab : p ∣ 1 * S.a + 1 * S.b\nhpaηsqb : p ∣ 1 * S.a + ↑η ^ 2 * S.b\nthis : p ∣ -(↑η ^ 2 - 1)\n⊢ p ∣ λ","tactic":"rw [one_mul, mul_one, IsUnit.dvd_mul_right <| (gcd_isUnit_iff _ _).2 S.coprime, ← dvd_neg] at this","premises":[{"full_name":"FermatLastTheoremForThreeGen.Solution'.coprime","def_path":"Mathlib/NumberTheory/FLT/Three.lean","def_pos":[191,2],"def_end_pos":[191,9]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"IsUnit.dvd_mul_right","def_path":"Mathlib/Algebra/Divisibility/Units.lean","def_pos":[79,8],"def_end_pos":[79,21]},{"full_name":"dvd_neg","def_path":"Mathlib/Algebra/Ring/Divisibility/Basic.lean","def_pos":[75,8],"def_end_pos":[75,15]},{"full_name":"gcd_isUnit_iff","def_path":"Mathlib/RingTheory/PrincipalIdealDomain.lean","def_pos":[402,8],"def_end_pos":[402,22]},{"full_name":"mul_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[413,8],"def_end_pos":[413,15]},{"full_name":"one_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[409,8],"def_end_pos":[409,15]}]},{"state_before":"K : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\np : 𝓞 K\nhp : Prime p\nhpab : p ∣ 1 * S.a + 1 * S.b\nhpaηsqb : p ∣ 1 * S.a + ↑η ^ 2 * S.b\nthis : p ∣ -(↑η ^ 2 - 1)\n⊢ p ∣ λ","state_after":"case h.e'_4\nK : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\np : 𝓞 K\nhp : Prime p\nhpab : p ∣ 1 * S.a + 1 * S.b\nhpaηsqb : p ∣ 1 * S.a + ↑η ^ 2 * S.b\nthis : p ∣ -(↑η ^ 2 - 1)\n⊢ λ = -(↑η ^ 2 - 1) * ↑η","tactic":"convert dvd_mul_of_dvd_left this η using 1","premises":[{"full_name":"IsPrimitiveRoot.isUnit","def_path":"Mathlib/RingTheory/RootsOfUnity/Basic.lean","def_pos":[319,8],"def_end_pos":[319,14]},{"full_name":"IsPrimitiveRoot.toInteger_isPrimitiveRoot","def_path":"Mathlib/NumberTheory/Cyclotomic/Rat.lean","def_pos":[193,6],"def_end_pos":[193,31]},{"full_name":"IsUnit.unit","def_path":"Mathlib/Algebra/Group/Units.lean","def_pos":[648,28],"def_end_pos":[648,32]},{"full_name":"dvd_mul_of_dvd_left","def_path":"Mathlib/Algebra/Divisibility/Basic.lean","def_pos":[78,8],"def_end_pos":[78,27]}]},{"state_before":"case h.e'_4\nK : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\np : 𝓞 K\nhp : Prime p\nhpab : p ∣ 1 * S.a + 1 * S.b\nhpaηsqb : p ∣ 1 * S.a + ↑η ^ 2 * S.b\nthis : p ∣ -(↑η ^ 2 - 1)\n⊢ λ = -(↑η ^ 2 - 1) * ↑η","state_after":"case h.e'_4\nK : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\np : 𝓞 K\nhp : Prime p\nhpab : p ∣ 1 * S.a + 1 * S.b\nhpaηsqb : p ∣ 1 * S.a + ↑η ^ 2 * S.b\nthis : p ∣ -(↑η ^ 2 - 1)\n⊢ λ = 1 * hζ.toInteger - (-(-hζ.toInteger - 1) - 1 * hζ.toInteger)","tactic":"rw [eta_sq, neg_sub, sub_mul, sub_mul, neg_mul, ← pow_two, eta_sq, coe_eta]","premises":[{"full_name":"IsCyclotomicExtension.Rat.Three.coe_eta","def_path":"Mathlib/NumberTheory/Cyclotomic/Three.lean","def_pos":[39,6],"def_end_pos":[39,13]},{"full_name":"IsCyclotomicExtension.Rat.Three.eta_sq","def_path":"Mathlib/NumberTheory/Cyclotomic/Three.lean","def_pos":[85,6],"def_end_pos":[85,12]},{"full_name":"neg_mul","def_path":"Mathlib/Algebra/Ring/Defs.lean","def_pos":[268,8],"def_end_pos":[268,15]},{"full_name":"neg_sub","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[399,2],"def_end_pos":[399,13]},{"full_name":"pow_two","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[581,31],"def_end_pos":[581,38]}]},{"state_before":"case h.e'_4\nK : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\np : 𝓞 K\nhp : Prime p\nhpab : p ∣ 1 * S.a + 1 * S.b\nhpaηsqb : p ∣ 1 * S.a + ↑η ^ 2 * S.b\nthis : p ∣ -(↑η ^ 2 - 1)\n⊢ λ = 1 * hζ.toInteger - (-(-hζ.toInteger - 1) - 1 * hζ.toInteger)","state_after":"no goals","tactic":"ring","premises":[]}]}
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+{"url":"Mathlib/SetTheory/Ordinal/Basic.lean","commit":"","full_name":"Ordinal.typein_lt_typein","start":[413,0],"end":[424,100],"file_path":"Mathlib/SetTheory/Ordinal/Basic.lean","tactics":[{"state_before":"α : Type u\nβ : Type u_1\nγ : Type u_2\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nr : α → α → Prop\ninst✝ : IsWellOrder α r\na b : α\nx✝ : typein r a < typein r b\nf : Subrel r {b | r b a} ≺i Subrel r {b_1 | r b_1 b}\n⊢ r a b","state_after":"α : Type u\nβ : Type u_1\nγ : Type u_2\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nr : α → α → Prop\ninst✝ : IsWellOrder α r\na b : α\nx✝ : typein r a < typein r b\nf : Subrel r {b | r b a} ≺i Subrel r {b_1 | r b_1 b}\nthis : ↑f.top = a\n⊢ r a b","tactic":"have : f.top.1 = a := by\n      let f' := PrincipalSeg.ofElement r a\n      let g' := f.trans (PrincipalSeg.ofElement r b)\n      have : g'.top = f'.top := by rw [Subsingleton.elim f' g']\n      exact this","premises":[{"full_name":"PrincipalSeg.ofElement","def_path":"Mathlib/Order/InitialSeg.lean","def_pos":[332,4],"def_end_pos":[332,13]},{"full_name":"PrincipalSeg.top","def_path":"Mathlib/Order/InitialSeg.lean","def_pos":[211,2],"def_end_pos":[211,5]},{"full_name":"PrincipalSeg.trans","def_path":"Mathlib/Order/InitialSeg.lean","def_pos":[276,14],"def_end_pos":[276,19]},{"full_name":"Subsingleton.elim","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1015,18],"def_end_pos":[1015,35]},{"full_name":"Subtype.val","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[587,2],"def_end_pos":[587,5]}]},{"state_before":"α : Type u\nβ : Type u_1\nγ : Type u_2\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nr : α → α → Prop\ninst✝ : IsWellOrder α r\na b : α\nx✝ : typein r a < typein r b\nf : Subrel r {b | r b a} ≺i Subrel r {b_1 | r b_1 b}\nthis : ↑f.top = a\n⊢ r a b","state_after":"α : Type u\nβ : Type u_1\nγ : Type u_2\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nr : α → α → Prop\ninst✝ : IsWellOrder α r\na b : α\nx✝ : typein r a < typein r b\nf : Subrel r {b | r b a} ≺i Subrel r {b_1 | r b_1 b}\nthis : ↑f.top = a\n⊢ r (↑f.top) b","tactic":"rw [← this]","premises":[]},{"state_before":"α : Type u\nβ : Type u_1\nγ : Type u_2\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nr : α → α → Prop\ninst✝ : IsWellOrder α r\na b : α\nx✝ : typein r a < typein r b\nf : Subrel r {b | r b a} ≺i Subrel r {b_1 | r b_1 b}\nthis : ↑f.top = a\n⊢ r (↑f.top) b","state_after":"no goals","tactic":"exact f.top.2","premises":[{"full_name":"PrincipalSeg.top","def_path":"Mathlib/Order/InitialSeg.lean","def_pos":[211,2],"def_end_pos":[211,5]},{"full_name":"Subtype.property","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[590,2],"def_end_pos":[590,10]}]}]}
+{"url":"Mathlib/RingTheory/Ideal/Maps.lean","commit":"","full_name":"RingHom.ker_eq_bot_iff_eq_zero","start":[562,0],"end":[563,62],"file_path":"Mathlib/RingTheory/Ideal/Maps.lean","tactics":[{"state_before":"R : Type u\nS : Type v\nT : Type w\nF : Type u_1\ninst✝² : Ring R\ninst✝¹ : Semiring S\ninst✝ : FunLike F R S\nrc : RingHomClass F R S\nf : F\n⊢ ker f = ⊥ ↔ ∀ (x : R), f x = 0 → x = 0","state_after":"no goals","tactic":"rw [← injective_iff_map_eq_zero f, injective_iff_ker_eq_bot]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"RingHom.injective_iff_ker_eq_bot","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[558,8],"def_end_pos":[558,32]},{"full_name":"injective_iff_map_eq_zero","def_path":"Mathlib/Algebra/Group/Hom/Basic.lean","def_pos":[105,2],"def_end_pos":[105,13]}]}]}
+{"url":"Mathlib/Data/Set/Sigma.lean","commit":"","full_name":"Set.sigma_univ_range_eq","start":[193,0],"end":[195,24],"file_path":"Mathlib/Data/Set/Sigma.lean","tactics":[{"state_before":"ι : Type u_1\nι' : Type u_2\nα : ι → Type u_3\nβ✝² : ι → Type u_4\ns s₁ s₂ : Set ι\nt t₁ t₂ : (i : ι) → Set (α i)\nu : Set ((i : ι) × α i)\nx : (i : ι) × α i\ni j : ι\na : α i\nβ✝¹ : Type u_5\ninst✝ : CompleteLattice β✝¹\nβ✝ : Type u_6\nβ : ι → Type u_7\nf : (i : ι) → α i → β i\n⊢ ∀ (x : (i : ι) × β i), (x ∈ univ.sigma fun i => range (f i)) ↔ x ∈ range fun x => ⟨x.fst, f x.fst x.snd⟩","state_after":"no goals","tactic":"simp [range]","premises":[{"full_name":"Set.range","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[144,4],"def_end_pos":[144,9]}]}]}
+{"url":"Mathlib/Topology/MetricSpace/Bounded.lean","commit":"","full_name":"Metric.diam_le_of_subset_closedBall","start":[454,0],"end":[460,46],"file_path":"Mathlib/Topology/MetricSpace/Bounded.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nX : Type u_1\nι : Type u_2\ninst✝ : PseudoMetricSpace α\ns : Set α\nx y z : α\nr : ℝ\nhr : 0 ≤ r\nh : s ⊆ closedBall x r\na : α\nha : a ∈ s\nb : α\nhb : b ∈ s\n⊢ r + r = 2 * r","state_after":"no goals","tactic":"simp [mul_two, mul_comm]","premises":[{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]},{"full_name":"mul_two","def_path":"Mathlib/Algebra/Ring/Defs.lean","def_pos":[159,8],"def_end_pos":[159,15]}]}]}
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+{"url":"Mathlib/Analysis/Normed/Ring/SeminormFromBounded.lean","commit":"","full_name":"seminormFromBounded_le","start":[115,0],"end":[124,15],"file_path":"Mathlib/Analysis/Normed/Ring/SeminormFromBounded.lean","tactics":[{"state_before":"R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx : R\n⊢ seminormFromBounded' f x ≤ c * f x","state_after":"R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx y : R\n⊢ f (x * y) / f y ≤ c * f x","tactic":"refine ciSup_le (fun y ↦ ?_)","premises":[{"full_name":"ciSup_le","def_path":"Mathlib/Order/ConditionallyCompleteLattice/Basic.lean","def_pos":[703,8],"def_end_pos":[703,16]}]},{"state_before":"R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx y : R\n⊢ f (x * y) / f y ≤ c * f x","state_after":"case inl\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx y : R\nhy : f y = 0 y\n⊢ f (x * y) / f y ≤ c * f x\n\ncase inr\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx y : R\nhy : 0 y < f y\n⊢ f (x * y) / f y ≤ c * f x","tactic":"rcases (f_nonneg y).eq_or_gt with hy | hy","premises":[]}]}
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+{"url":"Mathlib/Analysis/Convex/StrictConvexSpace.lean","commit":"","full_name":"StrictConvexSpace.of_norm_combo_ne_one","start":[103,0],"end":[114,73],"file_path":"Mathlib/Analysis/Convex/StrictConvexSpace.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : NormedLinearOrderedField 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace ℝ E\nh : ∀ (x y : E), ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ∃ a b, 0 ≤ a ∧ 0 ≤ b ∧ a + b = 1 ∧ ‖a • x + b • y‖ ≠ 1\n⊢ StrictConvexSpace ℝ E","state_after":"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : NormedLinearOrderedField 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace ℝ E\nh : ∀ (x y : E), ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ∃ a b, 0 ≤ a ∧ 0 ≤ b ∧ a + b = 1 ∧ ‖a • x + b • y‖ ≠ 1\n⊢ (closedBall 0 1 \\ interior (closedBall 0 1)).Pairwise fun x y => ([x-[ℝ]y] \\ frontier (closedBall 0 1)).Nonempty","tactic":"refine StrictConvexSpace.of_strictConvex_closed_unit_ball ℝ\n    ((convex_closedBall _ _).strictConvex ?_)","premises":[{"full_name":"Convex.strictConvex","def_path":"Mathlib/Analysis/Convex/Topology.lean","def_pos":[247,18],"def_end_pos":[247,37]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"StrictConvexSpace.of_strictConvex_closed_unit_ball","def_path":"Mathlib/Analysis/Convex/StrictConvexSpace.lean","def_pos":[83,8],"def_end_pos":[83,58]},{"full_name":"convex_closedBall","def_path":"Mathlib/Analysis/Convex/Normed.lean","def_pos":[59,8],"def_end_pos":[59,25]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : NormedLinearOrderedField 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace ℝ E\nh : ∀ (x y : E), ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ∃ a b, 0 ≤ a ∧ 0 ≤ b ∧ a + b = 1 ∧ ‖a • x + b • y‖ ≠ 1\n⊢ (closedBall 0 1 \\ interior (closedBall 0 1)).Pairwise fun x y => ([x-[ℝ]y] \\ frontier (closedBall 0 1)).Nonempty","state_after":"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : NormedLinearOrderedField 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace ℝ E\nh : ∀ (x y : E), ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ∃ a b, 0 ≤ a ∧ 0 ≤ b ∧ a + b = 1 ∧ ‖a • x + b • y‖ ≠ 1\n⊢ ∀ ⦃x : E⦄, ‖x‖ = 1 → ∀ ⦃y : E⦄, ‖y‖ = 1 → x ≠ y → ([x-[ℝ]y] \\ sphere 0 1).Nonempty","tactic":"simp only [interior_closedBall _ one_ne_zero, closedBall_diff_ball, Set.Pairwise,\n    frontier_closedBall _ one_ne_zero, mem_sphere_zero_iff_norm]","premises":[{"full_name":"Metric.closedBall_diff_ball","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[486,8],"def_end_pos":[486,28]},{"full_name":"Set.Pairwise","def_path":"Mathlib/Logic/Pairwise.lean","def_pos":[62,14],"def_end_pos":[62,22]},{"full_name":"frontier_closedBall","def_path":"Mathlib/Analysis/NormedSpace/Real.lean","def_pos":[93,8],"def_end_pos":[93,27]},{"full_name":"interior_closedBall","def_path":"Mathlib/Analysis/NormedSpace/Real.lean","def_pos":[74,8],"def_end_pos":[74,27]},{"full_name":"mem_sphere_zero_iff_norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[548,2],"def_end_pos":[548,13]},{"full_name":"one_ne_zero","def_path":"Mathlib/Algebra/NeZero.lean","def_pos":[49,14],"def_end_pos":[49,25]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : NormedLinearOrderedField 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace ℝ E\nh : ∀ (x y : E), ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ∃ a b, 0 ≤ a ∧ 0 ≤ b ∧ a + b = 1 ∧ ‖a • x + b • y‖ ≠ 1\n⊢ ∀ ⦃x : E⦄, ‖x‖ = 1 → ∀ ⦃y : E⦄, ‖y‖ = 1 → x ≠ y → ([x-[ℝ]y] \\ sphere 0 1).Nonempty","state_after":"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : NormedLinearOrderedField 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace ℝ E\nh : ∀ (x y : E), ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ∃ a b, 0 ≤ a ∧ 0 ≤ b ∧ a + b = 1 ∧ ‖a • x + b • y‖ ≠ 1\nx : E\nhx : ‖x‖ = 1\ny : E\nhy : ‖y‖ = 1\nhne : x ≠ y\n⊢ ([x-[ℝ]y] \\ sphere 0 1).Nonempty","tactic":"intro x hx y hy hne","premises":[]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : NormedLinearOrderedField 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace ℝ E\nh : ∀ (x y : E), ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ∃ a b, 0 ≤ a ∧ 0 ≤ b ∧ a + b = 1 ∧ ‖a • x + b • y‖ ≠ 1\nx : E\nhx : ‖x‖ = 1\ny : E\nhy : ‖y‖ = 1\nhne : x ≠ y\n⊢ ([x-[ℝ]y] \\ sphere 0 1).Nonempty","state_after":"case intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\ninst✝³ : NormedLinearOrderedField 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace ℝ E\nh : ∀ (x y : E), ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ∃ a b, 0 ≤ a ∧ 0 ≤ b ∧ a + b = 1 ∧ ‖a • x + b • y‖ ≠ 1\nx : E\nhx : ‖x‖ = 1\ny : E\nhy : ‖y‖ = 1\nhne : x ≠ y\na b : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nhne' : ‖a • x + b • y‖ ≠ 1\n⊢ ([x-[ℝ]y] \\ sphere 0 1).Nonempty","tactic":"rcases h x y hx hy hne with ⟨a, b, ha, hb, hab, hne'⟩","premises":[]},{"state_before":"case intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\ninst✝³ : NormedLinearOrderedField 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace ℝ E\nh : ∀ (x y : E), ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ∃ a b, 0 ≤ a ∧ 0 ≤ b ∧ a + b = 1 ∧ ‖a • x + b • y‖ ≠ 1\nx : E\nhx : ‖x‖ = 1\ny : E\nhy : ‖y‖ = 1\nhne : x ≠ y\na b : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nhne' : ‖a • x + b • y‖ ≠ 1\n⊢ ([x-[ℝ]y] \\ sphere 0 1).Nonempty","state_after":"no goals","tactic":"exact ⟨_, ⟨a, b, ha, hb, hab, rfl⟩, mt mem_sphere_zero_iff_norm.1 hne'⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"mem_sphere_zero_iff_norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[548,2],"def_end_pos":[548,13]},{"full_name":"mt","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[647,8],"def_end_pos":[647,10]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
+{"url":"Mathlib/FieldTheory/Finite/Polynomial.lean","commit":"","full_name":"MvPolynomial.frobenius_zmod","start":[30,0],"end":[35,40],"file_path":"Mathlib/FieldTheory/Finite/Polynomial.lean","tactics":[{"state_before":"σ : Type u_1\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : MvPolynomial σ (ZMod p)\n⊢ (frobenius (MvPolynomial σ (ZMod p)) p) f = (expand p) f","state_after":"case h_C\nσ : Type u_1\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : MvPolynomial σ (ZMod p)\n⊢ ∀ (a : ZMod p), (frobenius (MvPolynomial σ (ZMod p)) p) (C a) = (expand p) (C a)\n\ncase h_add\nσ : Type u_1\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : MvPolynomial σ (ZMod p)\n⊢ ∀ (p_1 q : MvPolynomial σ (ZMod p)),\n    (frobenius (MvPolynomial σ (ZMod p)) p) p_1 = (expand p) p_1 →\n      (frobenius (MvPolynomial σ (ZMod p)) p) q = (expand p) q →\n        (frobenius (MvPolynomial σ (ZMod p)) p) (p_1 + q) = (expand p) (p_1 + q)\n\ncase h_X\nσ : Type u_1\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : MvPolynomial σ (ZMod p)\n⊢ ∀ (p_1 : MvPolynomial σ (ZMod p)) (n : σ),\n    (frobenius (MvPolynomial σ (ZMod p)) p) p_1 = (expand p) p_1 →\n      (frobenius (MvPolynomial σ (ZMod p)) p) (p_1 * X n) = (expand p) (p_1 * X n)","tactic":"apply induction_on f","premises":[{"full_name":"MvPolynomial.induction_on","def_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","def_pos":[399,8],"def_end_pos":[399,20]}]}]}
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+{"url":"Mathlib/CategoryTheory/Adjunction/Mates.lean","commit":"","full_name":"CategoryTheory.mateEquiv_counit","start":[99,0],"end":[103,30],"file_path":"Mathlib/CategoryTheory/Adjunction/Mates.lean","tactics":[{"state_before":"C : Type u₁\nD : Type u₂\nE : Type u₃\nF : Type u₄\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} E\ninst✝ : Category.{v₄, u₄} F\nG : C ⥤ E\nH : D ⥤ F\nL₁ : C ⥤ D\nR₁ : D ⥤ C\nL₂ : E ⥤ F\nR₂ : F ⥤ E\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nα : G ⋙ L₂ ⟶ L₁ ⋙ H\nd : D\n⊢ L₂.map (((mateEquiv adj₁ adj₂) α).app d) ≫ adj₂.counit.app (H.obj d) = α.app (R₁.obj d) ≫ H.map (adj₁.counit.app d)","state_after":"C : Type u₁\nD : Type u₂\nE : Type u₃\nF : Type u₄\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} E\ninst✝ : Category.{v₄, u₄} F\nG : C ⥤ E\nH : D ⥤ F\nL₁ : C ⥤ D\nR₁ : D ⥤ C\nL₂ : E ⥤ F\nR₂ : F ⥤ E\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nα : G ⋙ L₂ ⟶ L₁ ⋙ H\nd : D\n⊢ (L₂.map ((whiskerLeft (R₁ ⋙ G) adj₂.unit).app d) ≫\n        L₂.map ((whiskerRight (whiskerLeft R₁ α) R₂ ≫ whiskerRight adj₁.counit (H ⋙ R₂)).app d)) ≫\n      adj₂.counit.app (H.obj d) =\n    α.app (R₁.obj d) ≫ H.map (adj₁.counit.app d)","tactic":"erw [Functor.map_comp]","premises":[{"full_name":"CategoryTheory.Functor.map_comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[41,2],"def_end_pos":[41,10]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]}]},{"state_before":"C : Type u₁\nD : Type u₂\nE : Type u₃\nF : Type u₄\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} E\ninst✝ : Category.{v₄, u₄} F\nG : C ⥤ E\nH : D ⥤ F\nL₁ : C ⥤ D\nR₁ : D ⥤ C\nL₂ : E ⥤ F\nR₂ : F ⥤ E\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nα : G ⋙ L₂ ⟶ L₁ ⋙ H\nd : D\n⊢ (L₂.map ((whiskerLeft (R₁ ⋙ G) adj₂.unit).app d) ≫\n        L₂.map ((whiskerRight (whiskerLeft R₁ α) R₂ ≫ whiskerRight adj₁.counit (H ⋙ R₂)).app d)) ≫\n      adj₂.counit.app (H.obj d) =\n    α.app (R₁.obj d) ≫ H.map (adj₁.counit.app d)","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/Algebra/Group/Submonoid/Membership.lean","commit":"","full_name":"AddSubmonoid.mem_sup_right","start":[205,0],"end":[208,20],"file_path":"Mathlib/Algebra/Group/Submonoid/Membership.lean","tactics":[{"state_before":"M : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝ : MulOneClass M\nS T : Submonoid M\n⊢ ∀ {x : M}, x ∈ T → x ∈ S ⊔ T","state_after":"M : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝ : MulOneClass M\nS T : Submonoid M\n⊢ T ≤ S ⊔ T","tactic":"rw [← SetLike.le_def]","premises":[{"full_name":"SetLike.le_def","def_path":"Mathlib/Data/SetLike/Basic.lean","def_pos":[191,8],"def_end_pos":[191,14]}]},{"state_before":"M : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝ : MulOneClass M\nS T : Submonoid M\n⊢ T ≤ S ⊔ T","state_after":"no goals","tactic":"exact le_sup_right","premises":[{"full_name":"le_sup_right","def_path":"Mathlib/Order/Lattice.lean","def_pos":[112,8],"def_end_pos":[112,20]}]}]}
+{"url":"Mathlib/Algebra/MvPolynomial/Derivation.lean","commit":"","full_name":"MvPolynomial.mkDerivationₗ_X","start":[47,0],"end":[48,49],"file_path":"Mathlib/Algebra/MvPolynomial/Derivation.lean","tactics":[{"state_before":"σ : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid A\ninst✝¹ : Module R A\ninst✝ : Module (MvPolynomial σ R) A\nf : σ → A\ni : σ\n⊢ (1 • (Finsupp.single i 1).sum fun i_1 k => (monomial (Finsupp.single i 1 - Finsupp.single i_1 1)) ↑k • f i_1) = f i","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/NumberTheory/ArithmeticFunction.lean","commit":"","full_name":"ArithmeticFunction.moebius_eq_or","start":[990,0],"end":[996,7],"file_path":"Mathlib/NumberTheory/ArithmeticFunction.lean","tactics":[{"state_before":"R : Type u_1\nn : ℕ\n⊢ μ n = 0 ∨ μ n = 1 ∨ μ n = -1","state_after":"R : Type u_1\nn : ℕ\n⊢ (if Squarefree n then (-1) ^ Ω n else 0) = 0 ∨\n    (if Squarefree n then (-1) ^ Ω n else 0) = 1 ∨ (if Squarefree n then (-1) ^ Ω n else 0) = -1","tactic":"simp only [moebius, coe_mk]","premises":[{"full_name":"ArithmeticFunction.coe_mk","def_path":"Mathlib/NumberTheory/ArithmeticFunction.lean","def_pos":[103,8],"def_end_pos":[103,14]},{"full_name":"ArithmeticFunction.moebius","def_path":"Mathlib/NumberTheory/ArithmeticFunction.lean","def_pos":[965,4],"def_end_pos":[965,11]}]},{"state_before":"R : Type u_1\nn : ℕ\n⊢ (if Squarefree n then (-1) ^ Ω n else 0) = 0 ∨\n    (if Squarefree n then (-1) ^ Ω n else 0) = 1 ∨ (if Squarefree n then (-1) ^ Ω n else 0) = -1","state_after":"case pos\nR : Type u_1\nn : ℕ\nh✝ : Squarefree n\n⊢ (-1) ^ Ω n = 0 ∨ (-1) ^ Ω n = 1 ∨ (-1) ^ Ω n = -1\n\ncase neg\nR : Type u_1\nn : ℕ\nh✝ : ¬Squarefree n\n⊢ 0 = 0 ∨ 0 = 1 ∨ False","tactic":"split_ifs","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
+{"url":"Mathlib/Geometry/Manifold/BumpFunction.lean","commit":"","full_name":"SmoothBumpFunction.isOpen_support","start":[108,0],"end":[110,50],"file_path":"Mathlib/Geometry/Manifold/BumpFunction.lean","tactics":[{"state_before":"E : Type uE\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : FiniteDimensional ℝ E\nH : Type uH\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\n⊢ IsOpen (support ↑f)","state_after":"E : Type uE\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : FiniteDimensional ℝ E\nH : Type uH\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\n⊢ IsOpen ((chartAt H c).source ∩ ↑(extChartAt I c) ⁻¹' ball (↑(extChartAt I c) c) f.rOut)","tactic":"rw [support_eq_inter_preimage]","premises":[{"full_name":"SmoothBumpFunction.support_eq_inter_preimage","def_path":"Mathlib/Geometry/Manifold/BumpFunction.lean","def_pos":[102,8],"def_end_pos":[102,33]}]},{"state_before":"E : Type uE\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : FiniteDimensional ℝ E\nH : Type uH\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\n⊢ IsOpen ((chartAt H c).source ∩ ↑(extChartAt I c) ⁻¹' ball (↑(extChartAt I c) c) f.rOut)","state_after":"no goals","tactic":"exact isOpen_extChartAt_preimage I c isOpen_ball","premises":[{"full_name":"Metric.isOpen_ball","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[860,8],"def_end_pos":[860,19]},{"full_name":"isOpen_extChartAt_preimage","def_path":"Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean","def_pos":[1170,8],"def_end_pos":[1170,34]}]}]}
+{"url":"Mathlib/CategoryTheory/Abelian/Basic.lean","commit":"","full_name":"CategoryTheory.Abelian.epi_of_cokernel_π_eq_zero","start":[315,0],"end":[318,80],"file_path":"Mathlib/CategoryTheory/Abelian/Basic.lean","tactics":[{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nP Q : C\nf : P ⟶ Q\nh : cokernel.π f = 0\n⊢ Epi f","state_after":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nP Q : C\nf : P ⟶ Q\nh : cokernel.π f = 0\n⊢ IsColimit (CokernelCofork.ofπ 0 ⋯)","tactic":"apply NormalMonoCategory.epi_of_zero_cokernel _ (cokernel f)","premises":[{"full_name":"CategoryTheory.Limits.cokernel","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","def_pos":[674,7],"def_end_pos":[674,15]},{"full_name":"CategoryTheory.NormalMonoCategory.epi_of_zero_cokernel","def_path":"Mathlib/CategoryTheory/Limits/Shapes/NormalMono/Equalizers.lean","def_pos":[147,8],"def_end_pos":[147,28]}]},{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nP Q : C\nf : P ⟶ Q\nh : cokernel.π f = 0\n⊢ IsColimit (CokernelCofork.ofπ 0 ⋯)","state_after":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nP Q : C\nf : P ⟶ Q\nh : cokernel.π f = 0\n⊢ IsColimit (CokernelCofork.ofπ (cokernel.π f) ⋯)","tactic":"simp_rw [← h]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]}]},{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nP Q : C\nf : P ⟶ Q\nh : cokernel.π f = 0\n⊢ IsColimit (CokernelCofork.ofπ (cokernel.π f) ⋯)","state_after":"no goals","tactic":"exact IsColimit.ofIsoColimit (colimit.isColimit (parallelPair f 0)) (isoOfπ _)","premises":[{"full_name":"CategoryTheory.Limits.IsColimit.ofIsoColimit","def_path":"Mathlib/CategoryTheory/Limits/IsLimit.lean","def_pos":[611,4],"def_end_pos":[611,16]},{"full_name":"CategoryTheory.Limits.colimit.isColimit","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[669,4],"def_end_pos":[669,21]},{"full_name":"CategoryTheory.Limits.isoOfπ","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","def_pos":[533,4],"def_end_pos":[533,10]},{"full_name":"CategoryTheory.Limits.parallelPair","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean","def_pos":[192,4],"def_end_pos":[192,16]}]}]}
+{"url":"Mathlib/Analysis/Convex/Slope.lean","commit":"","full_name":"StrictConcaveOn.slope_anti_adjacent","start":[78,0],"end":[85,12],"file_path":"Mathlib/Analysis/Convex/Slope.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : StrictConcaveOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\n⊢ (f z - f y) / (z - y) < (f y - f x) / (y - x)","state_after":"𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : StrictConcaveOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nthis : -(((-f) z - (-f) y) / (z - y)) < -(((-f) y - (-f) x) / (y - x))\n⊢ (f z - f y) / (z - y) < (f y - f x) / (y - x)","tactic":"have := neg_lt_neg (StrictConvexOn.slope_strict_mono_adjacent hf.neg hx hz hxy hyz)","premises":[{"full_name":"StrictConvexOn.slope_strict_mono_adjacent","def_path":"Mathlib/Analysis/Convex/Slope.lean","def_pos":[56,8],"def_end_pos":[56,49]},{"full_name":"neg_lt_neg","def_path":"Mathlib/Algebra/Order/Group/Defs.lean","def_pos":[177,31],"def_end_pos":[177,41]}]},{"state_before":"𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : StrictConcaveOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nthis : -(((-f) z - (-f) y) / (z - y)) < -(((-f) y - (-f) x) / (y - x))\n⊢ (f z - f y) / (z - y) < (f y - f x) / (y - x)","state_after":"𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : StrictConcaveOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nthis : (f z - f y) / (z - y) < (f y - f x) / (y - x)\n⊢ (f z - f y) / (z - y) < (f y - f x) / (y - x)","tactic":"simp only [Pi.neg_apply, ← neg_div, neg_sub', neg_neg] at this","premises":[{"full_name":"Pi.neg_apply","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[125,2],"def_end_pos":[125,13]},{"full_name":"neg_div","def_path":"Mathlib/Algebra/Field/Basic.lean","def_pos":[99,8],"def_end_pos":[99,15]},{"full_name":"neg_neg","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[733,2],"def_end_pos":[733,13]},{"full_name":"neg_sub'","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[518,2],"def_end_pos":[518,13]}]},{"state_before":"𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : StrictConcaveOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nthis : (f z - f y) / (z - y) < (f y - f x) / (y - x)\n⊢ (f z - f y) / (z - y) < (f y - f x) / (y - x)","state_after":"no goals","tactic":"exact this","premises":[]}]}
+{"url":"Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean","commit":"","full_name":"ModelWithCorners.locallyCompactSpace","start":[315,0],"end":[323,66],"file_path":"Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹ : TopologicalSpace H\nI✝ : ModelWithCorners 𝕜 E H\ninst✝ : LocallyCompactSpace E\nI : ModelWithCorners 𝕜 E H\n⊢ LocallyCompactSpace H","state_after":"𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹ : TopologicalSpace H\nI✝ : ModelWithCorners 𝕜 E H\ninst✝ : LocallyCompactSpace E\nI : ModelWithCorners 𝕜 E H\nthis : ∀ (x : H), (𝓝 x).HasBasis (fun s => s ∈ 𝓝 (↑I x) ∧ IsCompact s) fun s => ↑I.symm '' (s ∩ range ↑I)\n⊢ LocallyCompactSpace H","tactic":"have : ∀ x : H, (𝓝 x).HasBasis (fun s => s ∈ 𝓝 (I x) ∧ IsCompact s)\n      fun s => I.symm '' (s ∩ range I) := fun x ↦ by\n    rw [← I.symm_map_nhdsWithin_range]\n    exact ((compact_basis_nhds (I x)).inf_principal _).map _","premises":[{"full_name":"And","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[516,10],"def_end_pos":[516,13]},{"full_name":"Filter.HasBasis","def_path":"Mathlib/Order/Filter/Bases.lean","def_pos":[211,10],"def_end_pos":[211,18]},{"full_name":"Filter.HasBasis.inf_principal","def_path":"Mathlib/Order/Filter/Bases.lean","def_pos":[553,8],"def_end_pos":[553,30]},{"full_name":"Filter.HasBasis.map","def_path":"Mathlib/Order/Filter/Bases.lean","def_pos":[694,8],"def_end_pos":[694,20]},{"full_name":"Inter.inter","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[407,2],"def_end_pos":[407,7]},{"full_name":"IsCompact","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[254,4],"def_end_pos":[254,13]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"ModelWithCorners.symm","def_path":"Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean","def_pos":[176,14],"def_end_pos":[176,18]},{"full_name":"ModelWithCorners.symm_map_nhdsWithin_range","def_path":"Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean","def_pos":[290,8],"def_end_pos":[290,33]},{"full_name":"Set.image","def_path":"Mathlib/Init/Set.lean","def_pos":[208,4],"def_end_pos":[208,9]},{"full_name":"Set.range","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[144,4],"def_end_pos":[144,9]},{"full_name":"compact_basis_nhds","def_path":"Mathlib/Topology/Compactness/LocallyCompact.lean","def_pos":[54,8],"def_end_pos":[54,26]},{"full_name":"nhds","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[113,16],"def_end_pos":[113,20]}]},{"state_before":"𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹ : TopologicalSpace H\nI✝ : ModelWithCorners 𝕜 E H\ninst✝ : LocallyCompactSpace E\nI : ModelWithCorners 𝕜 E H\nthis : ∀ (x : H), (𝓝 x).HasBasis (fun s => s ∈ 𝓝 (↑I x) ∧ IsCompact s) fun s => ↑I.symm '' (s ∩ range ↑I)\n⊢ LocallyCompactSpace H","state_after":"𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹ : TopologicalSpace H\nI✝ : ModelWithCorners 𝕜 E H\ninst✝ : LocallyCompactSpace E\nI : ModelWithCorners 𝕜 E H\nthis : ∀ (x : H), (𝓝 x).HasBasis (fun s => s ∈ 𝓝 (↑I x) ∧ IsCompact s) fun s => ↑I.symm '' (s ∩ range ↑I)\n⊢ ∀ (x : H) (i : Set E), i ∈ 𝓝 (↑I x) ∧ IsCompact i → IsCompact (↑I.symm '' (i ∩ range ↑I))","tactic":"refine .of_hasBasis this ?_","premises":[{"full_name":"LocallyCompactSpace.of_hasBasis","def_path":"Mathlib/Topology/Compactness/LocallyCompact.lean","def_pos":[62,8],"def_end_pos":[62,39]}]},{"state_before":"𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹ : TopologicalSpace H\nI✝ : ModelWithCorners 𝕜 E H\ninst✝ : LocallyCompactSpace E\nI : ModelWithCorners 𝕜 E H\nthis : ∀ (x : H), (𝓝 x).HasBasis (fun s => s ∈ 𝓝 (↑I x) ∧ IsCompact s) fun s => ↑I.symm '' (s ∩ range ↑I)\n⊢ ∀ (x : H) (i : Set E), i ∈ 𝓝 (↑I x) ∧ IsCompact i → IsCompact (↑I.symm '' (i ∩ range ↑I))","state_after":"case intro\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹ : TopologicalSpace H\nI✝ : ModelWithCorners 𝕜 E H\ninst✝ : LocallyCompactSpace E\nI : ModelWithCorners 𝕜 E H\nthis : ∀ (x : H), (𝓝 x).HasBasis (fun s => s ∈ 𝓝 (↑I x) ∧ IsCompact s) fun s => ↑I.symm '' (s ∩ range ↑I)\nx : H\ns : Set E\nhsc : IsCompact s\n⊢ IsCompact (↑I.symm '' (s ∩ range ↑I))","tactic":"rintro x s ⟨-, hsc⟩","premises":[]},{"state_before":"case intro\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹ : TopologicalSpace H\nI✝ : ModelWithCorners 𝕜 E H\ninst✝ : LocallyCompactSpace E\nI : ModelWithCorners 𝕜 E H\nthis : ∀ (x : H), (𝓝 x).HasBasis (fun s => s ∈ 𝓝 (↑I x) ∧ IsCompact s) fun s => ↑I.symm '' (s ∩ range ↑I)\nx : H\ns : Set E\nhsc : IsCompact s\n⊢ IsCompact (↑I.symm '' (s ∩ range ↑I))","state_after":"no goals","tactic":"exact (hsc.inter_right I.isClosed_range).image I.continuous_symm","premises":[{"full_name":"IsCompact.image","def_path":"Mathlib/Topology/Compactness/Compact.lean","def_pos":[111,8],"def_end_pos":[111,23]},{"full_name":"IsCompact.inter_right","def_path":"Mathlib/Topology/Compactness/Compact.lean","def_pos":[76,8],"def_end_pos":[76,29]},{"full_name":"ModelWithCorners.continuous_symm","def_path":"Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean","def_pos":[222,8],"def_end_pos":[222,23]},{"full_name":"ModelWithCorners.isClosed_range","def_path":"Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean","def_pos":[273,8],"def_end_pos":[273,22]}]}]}
+{"url":"Mathlib/RingTheory/PowerSeries/Basic.lean","commit":"","full_name":"PowerSeries.ext","start":[150,0],"end":[156,7],"file_path":"Mathlib/RingTheory/PowerSeries/Basic.lean","tactics":[{"state_before":"R : Type u_1\ninst✝ : Semiring R\nφ ψ : R⟦X⟧\nh : ∀ (n : ℕ), (coeff R n) φ = (coeff R n) ψ\nn : Unit →₀ ℕ\n⊢ (MvPowerSeries.coeff R n) φ = (MvPowerSeries.coeff R n) ψ","state_after":"R : Type u_1\ninst✝ : Semiring R\nφ ψ : R⟦X⟧\nh : ∀ (n : ℕ), (coeff R n) φ = (coeff R n) ψ\nn : Unit →₀ ℕ\n⊢ (coeff R ?m.10640) φ = (coeff R ?m.10640) ψ\n\nR : Type u_1\ninst✝ : Semiring R\nφ ψ : R⟦X⟧\nh : ∀ (n : ℕ), (coeff R n) φ = (coeff R n) ψ\nn : Unit →₀ ℕ\n⊢ n () = ?m.10640\n\nR : Type u_1\ninst✝ : Semiring R\nφ ψ : R⟦X⟧\nh : ∀ (n : ℕ), (coeff R n) φ = (coeff R n) ψ\nn : Unit →₀ ℕ\n⊢ ℕ","tactic":"rw [← coeff_def]","premises":[{"full_name":"PowerSeries.coeff_def","def_path":"Mathlib/RingTheory/PowerSeries/Basic.lean","def_pos":[147,8],"def_end_pos":[147,17]}]},{"state_before":"R : Type u_1\ninst✝ : Semiring R\nφ ψ : R⟦X⟧\nh : ∀ (n : ℕ), (coeff R n) φ = (coeff R n) ψ\nn : Unit →₀ ℕ\n⊢ n () = ?m.10640\n\nR : Type u_1\ninst✝ : Semiring R\nφ ψ : R⟦X⟧\nh : ∀ (n : ℕ), (coeff R n) φ = (coeff R n) ψ\nn : Unit →₀ ℕ\n⊢ ℕ","state_after":"no goals","tactic":"rfl","premises":[]}]}
+{"url":"Mathlib/Logic/Basic.lean","commit":"","full_name":"forall_apply_eq_imp_iff'","start":[588,0],"end":[589,52],"file_path":"Mathlib/Logic/Basic.lean","tactics":[{"state_before":"α : Sort u_1\nβ : Sort u_2\np✝ q : α → Prop\nf : α → β\np : β → Prop\n⊢ (∀ (a : α) (b : β), f a = b → p b) ↔ ∀ (a : α), p (f a)","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/Algebra/Group/Submonoid/Operations.lean","commit":"","full_name":"MonoidHom.map_mrange","start":[770,0],"end":[772,64],"file_path":"Mathlib/Algebra/Group/Submonoid/Operations.lean","tactics":[{"state_before":"M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝⁴ : MulOneClass M\ninst✝³ : MulOneClass N\ninst✝² : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst✝¹ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\nF : Type u_5\ninst✝ : FunLike F M N\nmc : MonoidHomClass F M N\ng : N →* P\nf : M →* N\n⊢ map g (mrange f) = mrange (g.comp f)","state_after":"no goals","tactic":"simpa only [mrange_eq_map] using (⊤ : Submonoid M).map_map g f","premises":[{"full_name":"MonoidHom.mrange_eq_map","def_path":"Mathlib/Algebra/Group/Submonoid/Operations.lean","def_pos":[763,8],"def_end_pos":[763,21]},{"full_name":"Submonoid","def_path":"Mathlib/Algebra/Group/Submonoid/Basic.lean","def_pos":[88,10],"def_end_pos":[88,19]},{"full_name":"Submonoid.map_map","def_path":"Mathlib/Algebra/Group/Submonoid/Operations.lean","def_pos":[218,8],"def_end_pos":[218,15]},{"full_name":"Top.top","def_path":"Mathlib/Order/Notation.lean","def_pos":[94,2],"def_end_pos":[94,5]}]}]}
+{"url":"Mathlib/CategoryTheory/Monoidal/Bimod.lean","commit":"","full_name":"Bimod.whisker_assoc_bimod","start":[849,0],"end":[875,41],"file_path":"Mathlib/CategoryTheory/Monoidal/Bimod.lean","tactics":[{"state_before":"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ whiskerRight (M.whiskerLeft f) P =\n    (M.associatorBimod N P).hom ≫ M.whiskerLeft (whiskerRight f P) ≫ (M.associatorBimod N' P).inv","state_after":"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ whiskerRight (M.whiskerLeft f) P =\n    (isoOfIso { hom := AssociatorBimod.hom M N P, inv := AssociatorBimod.inv M N P, hom_inv_id := ⋯, inv_hom_id := ⋯ } ⋯\n          ⋯).hom ≫\n      M.whiskerLeft (whiskerRight f P) ≫\n        (isoOfIso\n            { hom := AssociatorBimod.hom M N' P, inv := AssociatorBimod.inv M N' P, hom_inv_id := ⋯, inv_hom_id := ⋯ } ⋯\n            ⋯).inv","tactic":"dsimp [tensorHom, tensorBimod, associatorBimod]","premises":[{"full_name":"Bimod.associatorBimod","def_path":"Mathlib/CategoryTheory/Monoidal/Bimod.lean","def_pos":[698,18],"def_end_pos":[698,33]},{"full_name":"Bimod.tensorBimod","def_path":"Mathlib/CategoryTheory/Monoidal/Bimod.lean","def_pos":[340,18],"def_end_pos":[340,29]},{"full_name":"CategoryTheory.MonoidalCategoryStruct.tensorHom","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[89,2],"def_end_pos":[89,11]}]},{"state_before":"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ whiskerRight (M.whiskerLeft f) P =\n    (isoOfIso { hom := AssociatorBimod.hom M N P, inv := AssociatorBimod.inv M N P, hom_inv_id := ⋯, inv_hom_id := ⋯ } ⋯\n          ⋯).hom ≫\n      M.whiskerLeft (whiskerRight f P) ≫\n        (isoOfIso\n            { hom := AssociatorBimod.hom M N' P, inv := AssociatorBimod.inv M N' P, hom_inv_id := ⋯, inv_hom_id := ⋯ } ⋯\n            ⋯).inv","state_after":"case h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ (whiskerRight (M.whiskerLeft f) P).hom =\n    ((isoOfIso { hom := AssociatorBimod.hom M N P, inv := AssociatorBimod.inv M N P, hom_inv_id := ⋯, inv_hom_id := ⋯ }\n            ⋯ ⋯).hom ≫\n        M.whiskerLeft (whiskerRight f P) ≫\n          (isoOfIso\n              { hom := AssociatorBimod.hom M N' P, inv := AssociatorBimod.inv M N' P, hom_inv_id := ⋯, inv_hom_id := ⋯ }\n              ⋯ ⋯).inv).hom","tactic":"ext","premises":[]},{"state_before":"case h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ (whiskerRight (M.whiskerLeft f) P).hom =\n    ((isoOfIso { hom := AssociatorBimod.hom M N P, inv := AssociatorBimod.inv M N P, hom_inv_id := ⋯, inv_hom_id := ⋯ }\n            ⋯ ⋯).hom ≫\n        M.whiskerLeft (whiskerRight f P) ≫\n          (isoOfIso\n              { hom := AssociatorBimod.hom M N' P, inv := AssociatorBimod.inv M N' P, hom_inv_id := ⋯, inv_hom_id := ⋯ }\n              ⋯ ⋯).inv).hom","state_after":"case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ coequalizer.π\n        ({ X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := ⋯, left_assoc := ⋯,\n              actRight := TensorBimod.actRight M N, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ }.actRight ▷\n          P.X)\n        ((α_\n              { X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := ⋯, left_assoc := ⋯,\n                  actRight := TensorBimod.actRight M N, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ }.X\n              Y.X P.X).hom ≫\n          { X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := ⋯, left_assoc := ⋯,\n                actRight := TensorBimod.actRight M N, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ }.X ◁\n            P.actLeft) ≫\n      (whiskerRight (M.whiskerLeft f) P).hom =\n    coequalizer.π\n        ({ X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := ⋯, left_assoc := ⋯,\n              actRight := TensorBimod.actRight M N, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ }.actRight ▷\n          P.X)\n        ((α_\n              { X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := ⋯, left_assoc := ⋯,\n                  actRight := TensorBimod.actRight M N, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ }.X\n              Y.X P.X).hom ≫\n          { X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := ⋯, left_assoc := ⋯,\n                actRight := TensorBimod.actRight M N, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ }.X ◁\n            P.actLeft) ≫\n      ((isoOfIso\n              { hom := AssociatorBimod.hom M N P, inv := AssociatorBimod.inv M N P, hom_inv_id := ⋯, inv_hom_id := ⋯ } ⋯\n              ⋯).hom ≫\n          M.whiskerLeft (whiskerRight f P) ≫\n            (isoOfIso\n                { hom := AssociatorBimod.hom M N' P, inv := AssociatorBimod.inv M N' P, hom_inv_id := ⋯,\n                  inv_hom_id := ⋯ }\n                ⋯ ⋯).inv).hom","tactic":"apply coequalizer.hom_ext","premises":[{"full_name":"CategoryTheory.Limits.coequalizer.hom_ext","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean","def_pos":[896,8],"def_end_pos":[896,27]}]},{"state_before":"case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ coequalizer.π\n        ({ X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := ⋯, left_assoc := ⋯,\n              actRight := TensorBimod.actRight M N, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ }.actRight ▷\n          P.X)\n        ((α_\n              { X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := ⋯, left_assoc := ⋯,\n                  actRight := TensorBimod.actRight M N, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ }.X\n              Y.X P.X).hom ≫\n          { X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := ⋯, left_assoc := ⋯,\n                actRight := TensorBimod.actRight M N, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ }.X ◁\n            P.actLeft) ≫\n      (whiskerRight (M.whiskerLeft f) P).hom =\n    coequalizer.π\n        ({ X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := ⋯, left_assoc := ⋯,\n              actRight := TensorBimod.actRight M N, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ }.actRight ▷\n          P.X)\n        ((α_\n              { X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := ⋯, left_assoc := ⋯,\n                  actRight := TensorBimod.actRight M N, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ }.X\n              Y.X P.X).hom ≫\n          { X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := ⋯, left_assoc := ⋯,\n                actRight := TensorBimod.actRight M N, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ }.X ◁\n            P.actLeft) ≫\n      ((isoOfIso\n              { hom := AssociatorBimod.hom M N P, inv := AssociatorBimod.inv M N P, hom_inv_id := ⋯, inv_hom_id := ⋯ } ⋯\n              ⋯).hom ≫\n          M.whiskerLeft (whiskerRight f P) ≫\n            (isoOfIso\n                { hom := AssociatorBimod.hom M N' P, inv := AssociatorBimod.inv M N' P, hom_inv_id := ⋯,\n                  inv_hom_id := ⋯ }\n                ⋯ ⋯).inv).hom","state_after":"case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ coequalizer.π (TensorBimod.actRight M N ▷ P.X)\n        ((α_ (TensorBimod.X M N) Y.X P.X).hom ≫ TensorBimod.X M N ◁ P.actLeft) ≫\n      colimMap\n        (parallelPairHom (TensorBimod.actRight M N ▷ P.X)\n          ((α_ (TensorBimod.X M N) Y.X P.X).hom ≫ TensorBimod.X M N ◁ P.actLeft) (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft)\n          (colimMap\n                (parallelPairHom (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) (M.actRight ▷ N'.X)\n                  ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ((M.X ⊗ X.X) ◁ f.hom) (M.X ◁ f.hom) ⋯ ⋯) ▷\n              Y.X ▷\n            P.X)\n          (colimMap\n              (parallelPairHom (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) (M.actRight ▷ N'.X)\n                ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ((M.X ⊗ X.X) ◁ f.hom) (M.X ◁ f.hom) ⋯ ⋯) ▷\n            P.X)\n          ⋯ ⋯) =\n    coequalizer.π (TensorBimod.actRight M N ▷ P.X)\n        ((α_ (TensorBimod.X M N) Y.X P.X).hom ≫ TensorBimod.X M N ◁ P.actLeft) ≫\n      AssociatorBimod.hom M N P ≫\n        colimMap\n            (parallelPairHom (M.actRight ▷ TensorBimod.X N P)\n              ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P) (M.actRight ▷ TensorBimod.X N' P)\n              ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P)\n              ((M.X ⊗ X.X) ◁\n                colimMap\n                  (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                    ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n              (M.X ◁\n                colimMap\n                  (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                    ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n              ⋯ ⋯) ≫\n          AssociatorBimod.inv M N' P","tactic":"dsimp","premises":[]},{"state_before":"case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ coequalizer.π (TensorBimod.actRight M N ▷ P.X)\n        ((α_ (TensorBimod.X M N) Y.X P.X).hom ≫ TensorBimod.X M N ◁ P.actLeft) ≫\n      colimMap\n        (parallelPairHom (TensorBimod.actRight M N ▷ P.X)\n          ((α_ (TensorBimod.X M N) Y.X P.X).hom ≫ TensorBimod.X M N ◁ P.actLeft) (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft)\n          (colimMap\n                (parallelPairHom (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) (M.actRight ▷ N'.X)\n                  ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ((M.X ⊗ X.X) ◁ f.hom) (M.X ◁ f.hom) ⋯ ⋯) ▷\n              Y.X ▷\n            P.X)\n          (colimMap\n              (parallelPairHom (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) (M.actRight ▷ N'.X)\n                ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ((M.X ⊗ X.X) ◁ f.hom) (M.X ◁ f.hom) ⋯ ⋯) ▷\n            P.X)\n          ⋯ ⋯) =\n    coequalizer.π (TensorBimod.actRight M N ▷ P.X)\n        ((α_ (TensorBimod.X M N) Y.X P.X).hom ≫ TensorBimod.X M N ◁ P.actLeft) ≫\n      AssociatorBimod.hom M N P ≫\n        colimMap\n            (parallelPairHom (M.actRight ▷ TensorBimod.X N P)\n              ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P) (M.actRight ▷ TensorBimod.X N' P)\n              ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P)\n              ((M.X ⊗ X.X) ◁\n                colimMap\n                  (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                    ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n              (M.X ◁\n                colimMap\n                  (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                    ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n              ⋯ ⋯) ≫\n          AssociatorBimod.inv M N' P","state_after":"case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ colimMap\n          (parallelPairHom (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) (M.actRight ▷ N'.X)\n            ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ((M.X ⊗ X.X) ◁ f.hom) (M.X ◁ f.hom) ⋯ ⋯) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    coequalizer.π (TensorBimod.actRight M N ▷ P.X)\n        ((α_ (TensorBimod.X M N) Y.X P.X).hom ≫ TensorBimod.X M N ◁ P.actLeft) ≫\n      AssociatorBimod.hom M N P ≫\n        colimMap\n            (parallelPairHom (M.actRight ▷ TensorBimod.X N P)\n              ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P) (M.actRight ▷ TensorBimod.X N' P)\n              ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P)\n              ((M.X ⊗ X.X) ◁\n                colimMap\n                  (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                    ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n              (M.X ◁\n                colimMap\n                  (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                    ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n              ⋯ ⋯) ≫\n          AssociatorBimod.inv M N' P","tactic":"slice_lhs 1 2 => rw [ι_colimMap, parallelPairHom_app_one]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Limits.parallelPairHom_app_one","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean","def_pos":[246,8],"def_end_pos":[246,31]},{"full_name":"CategoryTheory.Limits.ι_colimMap","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[705,8],"def_end_pos":[705,18]}]},{"state_before":"case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ colimMap\n          (parallelPairHom (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) (M.actRight ▷ N'.X)\n            ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ((M.X ⊗ X.X) ◁ f.hom) (M.X ◁ f.hom) ⋯ ⋯) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    coequalizer.π (TensorBimod.actRight M N ▷ P.X)\n        ((α_ (TensorBimod.X M N) Y.X P.X).hom ≫ TensorBimod.X M N ◁ P.actLeft) ≫\n      AssociatorBimod.hom M N P ≫\n        colimMap\n            (parallelPairHom (M.actRight ▷ TensorBimod.X N P)\n              ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P) (M.actRight ▷ TensorBimod.X N' P)\n              ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P)\n              ((M.X ⊗ X.X) ◁\n                colimMap\n                  (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                    ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n              (M.X ◁\n                colimMap\n                  (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                    ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n              ⋯ ⋯) ≫\n          AssociatorBimod.inv M N' P","state_after":"case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ colimMap\n          (parallelPairHom (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) (M.actRight ▷ N'.X)\n            ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ((M.X ⊗ X.X) ◁ f.hom) (M.X ◁ f.hom) ⋯ ⋯) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    coequalizer.π (TensorBimod.actRight M N ▷ P.X)\n        ((α_ (TensorBimod.X M N) Y.X P.X).hom ≫ TensorBimod.X M N ◁ P.actLeft) ≫\n      coequalizer.desc (AssociatorBimod.homAux M N P) ⋯ ≫\n        colimMap\n            (parallelPairHom (M.actRight ▷ TensorBimod.X N P)\n              ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P) (M.actRight ▷ TensorBimod.X N' P)\n              ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P)\n              ((M.X ⊗ X.X) ◁\n                colimMap\n                  (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                    ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n              (M.X ◁\n                colimMap\n                  (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                    ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n              ⋯ ⋯) ≫\n          AssociatorBimod.inv M N' P","tactic":"dsimp [AssociatorBimod.hom]","premises":[{"full_name":"Bimod.AssociatorBimod.hom","def_path":"Mathlib/CategoryTheory/Monoidal/Bimod.lean","def_pos":[443,18],"def_end_pos":[443,21]}]},{"state_before":"case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ colimMap\n          (parallelPairHom (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) (M.actRight ▷ N'.X)\n            ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ((M.X ⊗ X.X) ◁ f.hom) (M.X ◁ f.hom) ⋯ ⋯) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    coequalizer.π (TensorBimod.actRight M N ▷ P.X)\n        ((α_ (TensorBimod.X M N) Y.X P.X).hom ≫ TensorBimod.X M N ◁ P.actLeft) ≫\n      coequalizer.desc (AssociatorBimod.homAux M N P) ⋯ ≫\n        colimMap\n            (parallelPairHom (M.actRight ▷ TensorBimod.X N P)\n              ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P) (M.actRight ▷ TensorBimod.X N' P)\n              ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P)\n              ((M.X ⊗ X.X) ◁\n                colimMap\n                  (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                    ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n              (M.X ◁\n                colimMap\n                  (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                    ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n              ⋯ ⋯) ≫\n          AssociatorBimod.inv M N' P","state_after":"case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ colimMap\n          (parallelPairHom (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) (M.actRight ▷ N'.X)\n            ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ((M.X ⊗ X.X) ◁ f.hom) (M.X ◁ f.hom) ⋯ ⋯) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    (AssociatorBimod.homAux M N P ≫\n        colimMap\n          (parallelPairHom (M.actRight ▷ TensorBimod.X N P)\n            ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P) (M.actRight ▷ TensorBimod.X N' P)\n            ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P)\n            ((M.X ⊗ X.X) ◁\n              colimMap\n                (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                  ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n            (M.X ◁\n              colimMap\n                (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                  ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n            ⋯ ⋯)) ≫\n      AssociatorBimod.inv M N' P","tactic":"slice_rhs 1 2 => rw [coequalizer.π_desc]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Limits.coequalizer.π_desc","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean","def_pos":[876,8],"def_end_pos":[876,26]}]},{"state_before":"case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ colimMap\n          (parallelPairHom (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) (M.actRight ▷ N'.X)\n            ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ((M.X ⊗ X.X) ◁ f.hom) (M.X ◁ f.hom) ⋯ ⋯) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    (AssociatorBimod.homAux M N P ≫\n        colimMap\n          (parallelPairHom (M.actRight ▷ TensorBimod.X N P)\n            ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P) (M.actRight ▷ TensorBimod.X N' P)\n            ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P)\n            ((M.X ⊗ X.X) ◁\n              colimMap\n                (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                  ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n            (M.X ◁\n              colimMap\n                (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                  ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n            ⋯ ⋯)) ≫\n      AssociatorBimod.inv M N' P","state_after":"case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ colimMap\n          (parallelPairHom (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) (M.actRight ▷ N'.X)\n            ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ((M.X ⊗ X.X) ◁ f.hom) (M.X ◁ f.hom) ⋯ ⋯) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    (((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫\n          coequalizer.desc\n            ((α_ M.X N.X P.X).hom ≫\n              M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) ≫\n                coequalizer.π (M.actRight ▷ TensorBimod.X N P)\n                  ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P))\n            ⋯) ≫\n        colimMap\n          (parallelPairHom (M.actRight ▷ TensorBimod.X N P)\n            ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P) (M.actRight ▷ TensorBimod.X N' P)\n            ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P)\n            ((M.X ⊗ X.X) ◁\n              colimMap\n                (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                  ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n            (M.X ◁\n              colimMap\n                (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                  ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n            ⋯ ⋯)) ≫\n      AssociatorBimod.inv M N' P","tactic":"dsimp [AssociatorBimod.homAux]","premises":[{"full_name":"Bimod.AssociatorBimod.homAux","def_path":"Mathlib/CategoryTheory/Monoidal/Bimod.lean","def_pos":[428,18],"def_end_pos":[428,24]}]},{"state_before":"case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ colimMap\n          (parallelPairHom (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) (M.actRight ▷ N'.X)\n            ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ((M.X ⊗ X.X) ◁ f.hom) (M.X ◁ f.hom) ⋯ ⋯) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    (((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫\n          coequalizer.desc\n            ((α_ M.X N.X P.X).hom ≫\n              M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) ≫\n                coequalizer.π (M.actRight ▷ TensorBimod.X N P)\n                  ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P))\n            ⋯) ≫\n        colimMap\n          (parallelPairHom (M.actRight ▷ TensorBimod.X N P)\n            ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P) (M.actRight ▷ TensorBimod.X N' P)\n            ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P)\n            ((M.X ⊗ X.X) ◁\n              colimMap\n                (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                  ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n            (M.X ◁\n              colimMap\n                (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                  ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n            ⋯ ⋯)) ≫\n      AssociatorBimod.inv M N' P","state_after":"case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ (tensorRight P.X).map (coequalizer.π (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft)) ≫\n      colimMap\n            (parallelPairHom (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) (M.actRight ▷ N'.X)\n              ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ((M.X ⊗ X.X) ◁ f.hom) (M.X ◁ f.hom) ⋯ ⋯) ▷\n          P.X ≫\n        colimit.ι\n          (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n            ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n          WalkingParallelPair.one =\n    (tensorRight P.X).map (coequalizer.π (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft)) ≫\n      (((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫\n            coequalizer.desc\n              ((α_ M.X N.X P.X).hom ≫\n                M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) ≫\n                  coequalizer.π (M.actRight ▷ TensorBimod.X N P)\n                    ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P))\n              ⋯) ≫\n          colimMap\n            (parallelPairHom (M.actRight ▷ TensorBimod.X N P)\n              ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P) (M.actRight ▷ TensorBimod.X N' P)\n              ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P)\n              ((M.X ⊗ X.X) ◁\n                colimMap\n                  (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                    ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n              (M.X ◁\n                colimMap\n                  (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                    ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n              ⋯ ⋯)) ≫\n        AssociatorBimod.inv M N' P","tactic":"refine (cancel_epi ((tensorRight _).map (coequalizer.π _ _))).1 ?_","premises":[{"full_name":"CategoryTheory.Limits.coequalizer.π","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean","def_pos":[841,21],"def_end_pos":[841,34]},{"full_name":"CategoryTheory.MonoidalCategory.tensorRight","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[823,4],"def_end_pos":[823,15]},{"full_name":"CategoryTheory.cancel_epi","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[260,8],"def_end_pos":[260,18]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Prefunctor.map","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[57,2],"def_end_pos":[57,5]}]},{"state_before":"case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ (tensorRight P.X).map (coequalizer.π (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft)) ≫\n      colimMap\n            (parallelPairHom (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) (M.actRight ▷ N'.X)\n              ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ((M.X ⊗ X.X) ◁ f.hom) (M.X ◁ f.hom) ⋯ ⋯) ▷\n          P.X ≫\n        colimit.ι\n          (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n            ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n          WalkingParallelPair.one =\n    (tensorRight P.X).map (coequalizer.π (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft)) ≫\n      (((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫\n            coequalizer.desc\n              ((α_ M.X N.X P.X).hom ≫\n                M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) ≫\n                  coequalizer.π (M.actRight ▷ TensorBimod.X N P)\n                    ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P))\n              ⋯) ≫\n          colimMap\n            (parallelPairHom (M.actRight ▷ TensorBimod.X N P)\n              ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P) (M.actRight ▷ TensorBimod.X N' P)\n              ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P)\n              ((M.X ⊗ X.X) ◁\n                colimMap\n                  (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                    ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n              (M.X ◁\n                colimMap\n                  (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                    ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n              ⋯ ⋯)) ≫\n        AssociatorBimod.inv M N' P","state_after":"case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ coequalizer.π (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) ▷ P.X ≫\n      colimMap\n            (parallelPairHom (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) (M.actRight ▷ N'.X)\n              ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ((M.X ⊗ X.X) ◁ f.hom) (M.X ◁ f.hom) ⋯ ⋯) ▷\n          P.X ≫\n        colimit.ι\n          (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n            ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n          WalkingParallelPair.one =\n    coequalizer.π (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) ▷ P.X ≫\n      (((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫\n            coequalizer.desc\n              ((α_ M.X N.X P.X).hom ≫\n                M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) ≫\n                  coequalizer.π (M.actRight ▷ TensorBimod.X N P)\n                    ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P))\n              ⋯) ≫\n          colimMap\n            (parallelPairHom (M.actRight ▷ TensorBimod.X N P)\n              ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P) (M.actRight ▷ TensorBimod.X N' P)\n              ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P)\n              ((M.X ⊗ X.X) ◁\n                colimMap\n                  (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                    ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n              (M.X ◁\n                colimMap\n                  (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                    ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n              ⋯ ⋯)) ≫\n        AssociatorBimod.inv M N' P","tactic":"rw [tensorRight_map]","premises":[{"full_name":"CategoryTheory.MonoidalCategory.tensorRight_map","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[822,2],"def_end_pos":[822,8]}]},{"state_before":"case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ coequalizer.π (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) ▷ P.X ≫\n      colimMap\n            (parallelPairHom (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) (M.actRight ▷ N'.X)\n              ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ((M.X ⊗ X.X) ◁ f.hom) (M.X ◁ f.hom) ⋯ ⋯) ▷\n          P.X ≫\n        colimit.ι\n          (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n            ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n          WalkingParallelPair.one =\n    coequalizer.π (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) ▷ P.X ≫\n      (((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫\n            coequalizer.desc\n              ((α_ M.X N.X P.X).hom ≫\n                M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) ≫\n                  coequalizer.π (M.actRight ▷ TensorBimod.X N P)\n                    ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P))\n              ⋯) ≫\n          colimMap\n            (parallelPairHom (M.actRight ▷ TensorBimod.X N P)\n              ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P) (M.actRight ▷ TensorBimod.X N' P)\n              ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P)\n              ((M.X ⊗ X.X) ◁\n                colimMap\n                  (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                    ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n              (M.X ◁\n                colimMap\n                  (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                    ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n              ⋯ ⋯)) ≫\n        AssociatorBimod.inv M N' P","state_after":"case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ (M.X ◁ f.hom ≫\n          colimit.ι (parallelPair (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft))\n            WalkingParallelPair.one) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    coequalizer.π (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) ▷ P.X ≫\n      (((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫\n            coequalizer.desc\n              ((α_ M.X N.X P.X).hom ≫\n                M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) ≫\n                  coequalizer.π (M.actRight ▷ TensorBimod.X N P)\n                    ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P))\n              ⋯) ≫\n          colimMap\n            (parallelPairHom (M.actRight ▷ TensorBimod.X N P)\n              ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P) (M.actRight ▷ TensorBimod.X N' P)\n              ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P)\n              ((M.X ⊗ X.X) ◁\n                colimMap\n                  (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                    ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n              (M.X ◁\n                colimMap\n                  (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                    ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n              ⋯ ⋯)) ≫\n        AssociatorBimod.inv M N' P","tactic":"slice_lhs 1 2 => rw [← comp_whiskerRight, ι_colimMap, parallelPairHom_app_one]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Limits.parallelPairHom_app_one","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean","def_pos":[246,8],"def_end_pos":[246,31]},{"full_name":"CategoryTheory.Limits.ι_colimMap","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[705,8],"def_end_pos":[705,18]},{"full_name":"CategoryTheory.MonoidalCategory.comp_whiskerRight","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[249,8],"def_end_pos":[249,25]}]},{"state_before":"case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ (M.X ◁ f.hom ≫\n          colimit.ι (parallelPair (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft))\n            WalkingParallelPair.one) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    coequalizer.π (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) ▷ P.X ≫\n      (((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫\n            coequalizer.desc\n              ((α_ M.X N.X P.X).hom ≫\n                M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) ≫\n                  coequalizer.π (M.actRight ▷ TensorBimod.X N P)\n                    ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P))\n              ⋯) ≫\n          colimMap\n            (parallelPairHom (M.actRight ▷ TensorBimod.X N P)\n              ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P) (M.actRight ▷ TensorBimod.X N' P)\n              ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P)\n              ((M.X ⊗ X.X) ◁\n                colimMap\n                  (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                    ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n              (M.X ◁\n                colimMap\n                  (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                    ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n              ⋯ ⋯)) ≫\n        AssociatorBimod.inv M N' P","state_after":"case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ (M.X ◁ f.hom ≫\n          colimit.ι (parallelPair (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft))\n            WalkingParallelPair.one) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    (((α_ M.X N.X P.X).hom ≫\n          M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) ≫\n            coequalizer.π (M.actRight ▷ TensorBimod.X N P)\n              ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P)) ≫\n        colimMap\n          (parallelPairHom (M.actRight ▷ TensorBimod.X N P)\n            ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P) (M.actRight ▷ TensorBimod.X N' P)\n            ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P)\n            ((M.X ⊗ X.X) ◁\n              colimMap\n                (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                  ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n            (M.X ◁\n              colimMap\n                (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                  ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n            ⋯ ⋯)) ≫\n      AssociatorBimod.inv M N' P","tactic":"slice_rhs 1 3 => rw [π_tensor_id_preserves_coequalizer_inv_desc]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"π_tensor_id_preserves_coequalizer_inv_desc","def_path":"Mathlib/CategoryTheory/Monoidal/Bimod.lean","def_pos":[55,8],"def_end_pos":[55,50]}]},{"state_before":"case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ (M.X ◁ f.hom ≫\n          colimit.ι (parallelPair (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft))\n            WalkingParallelPair.one) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    (((α_ M.X N.X P.X).hom ≫\n          M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) ≫\n            coequalizer.π (M.actRight �� TensorBimod.X N P)\n              ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P)) ≫\n        colimMap\n          (parallelPairHom (M.actRight ▷ TensorBimod.X N P)\n            ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P) (M.actRight ▷ TensorBimod.X N' P)\n            ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P)\n            ((M.X ⊗ X.X) ◁\n              colimMap\n                (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                  ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n            (M.X ◁\n              colimMap\n                (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                  ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n            ⋯ ⋯)) ≫\n      AssociatorBimod.inv M N' P","state_after":"case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ (M.X ◁ f.hom ≫\n          colimit.ι (parallelPair (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft))\n            WalkingParallelPair.one) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    (α_ M.X N.X P.X).hom ≫\n      M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) ≫\n        (M.X ◁\n              colimMap\n                (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                  ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯) ≫\n            colimit.ι\n              (parallelPair (M.actRight ▷ TensorBimod.X N' P)\n                ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P))\n              WalkingParallelPair.one) ≫\n          AssociatorBimod.inv M N' P","tactic":"slice_rhs 3 4 => rw [ι_colimMap, parallelPairHom_app_one]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Limits.parallelPairHom_app_one","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean","def_pos":[246,8],"def_end_pos":[246,31]},{"full_name":"CategoryTheory.Limits.ι_colimMap","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[705,8],"def_end_pos":[705,18]}]},{"state_before":"case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ (M.X ◁ f.hom ≫\n          colimit.ι (parallelPair (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft))\n            WalkingParallelPair.one) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    (α_ M.X N.X P.X).hom ≫\n      M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) ≫\n        (M.X ◁\n              colimMap\n                (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                  ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯) ≫\n            colimit.ι\n              (parallelPair (M.actRight ▷ TensorBimod.X N' P)\n                ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P))\n              WalkingParallelPair.one) ≫\n          AssociatorBimod.inv M N' P","state_after":"case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ (M.X ◁ f.hom ≫\n          colimit.ι (parallelPair (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft))\n            WalkingParallelPair.one) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    (α_ M.X N.X P.X).hom ≫\n      (M.X ◁\n            (f.hom ▷ P.X ≫\n              colimit.ι (parallelPair (N'.actRight ▷ P.X) ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft))\n                WalkingParallelPair.one) ≫\n          colimit.ι\n            (parallelPair (M.actRight ▷ TensorBimod.X N' P)\n              ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P))\n            WalkingParallelPair.one) ≫\n        AssociatorBimod.inv M N' P","tactic":"slice_rhs 2 3 => rw [← MonoidalCategory.whiskerLeft_comp, ι_colimMap, parallelPairHom_app_one]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Limits.parallelPairHom_app_one","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean","def_pos":[246,8],"def_end_pos":[246,31]},{"full_name":"CategoryTheory.Limits.ι_colimMap","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[705,8],"def_end_pos":[705,18]},{"full_name":"CategoryTheory.MonoidalCategory.whiskerLeft_comp","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[232,8],"def_end_pos":[232,24]}]},{"state_before":"case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ (M.X ◁ f.hom ≫\n          colimit.ι (parallelPair (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft))\n            WalkingParallelPair.one) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    (α_ M.X N.X P.X).hom ≫\n      (M.X ◁\n            (f.hom ▷ P.X ≫\n              colimit.ι (parallelPair (N'.actRight ▷ P.X) ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft))\n                WalkingParallelPair.one) ≫\n          colimit.ι\n            (parallelPair (M.actRight ▷ TensorBimod.X N' P)\n              ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P))\n            WalkingParallelPair.one) ≫\n        AssociatorBimod.inv M N' P","state_after":"case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ (M.X ◁ f.hom ≫\n          colimit.ι (parallelPair (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft))\n            WalkingParallelPair.one) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    (α_ M.X N.X P.X).hom ≫\n      (M.X ◁\n            (f.hom ▷ P.X ≫\n              colimit.ι (parallelPair (N'.actRight ▷ P.X) ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft))\n                WalkingParallelPair.one) ≫\n          colimit.ι\n            (parallelPair (M.actRight ▷ TensorBimod.X N' P)\n              ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P))\n            WalkingParallelPair.one) ≫\n        coequalizer.desc (AssociatorBimod.invAux M N' P) ⋯","tactic":"dsimp [AssociatorBimod.inv]","premises":[{"full_name":"Bimod.AssociatorBimod.inv","def_path":"Mathlib/CategoryTheory/Monoidal/Bimod.lean","def_pos":[535,18],"def_end_pos":[535,21]}]},{"state_before":"case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ (M.X ◁ f.hom ≫\n          colimit.ι (parallelPair (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft))\n            WalkingParallelPair.one) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    (α_ M.X N.X P.X).hom ≫\n      (M.X ◁\n            (f.hom ▷ P.X ≫\n              colimit.ι (parallelPair (N'.actRight ▷ P.X) ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft))\n                WalkingParallelPair.one) ≫\n          colimit.ι\n            (parallelPair (M.actRight ▷ TensorBimod.X N' P)\n              ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P))\n            WalkingParallelPair.one) ≫\n        coequalizer.desc (AssociatorBimod.invAux M N' P) ⋯","state_after":"case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ (M.X ◁ f.hom ≫\n          colimit.ι (parallelPair (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft))\n            WalkingParallelPair.one) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    (α_ M.X N.X P.X).hom ≫\n      M.X ◁\n          (f.hom ▷ P.X ≫\n            colimit.ι (parallelPair (N'.actRight ▷ P.X) ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft))\n              WalkingParallelPair.one) ≫\n        AssociatorBimod.invAux M N' P","tactic":"slice_rhs 3 4 => rw [coequalizer.π_desc]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Limits.coequalizer.π_desc","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean","def_pos":[876,8],"def_end_pos":[876,26]}]},{"state_before":"case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ (M.X ◁ f.hom ≫\n          colimit.ι (parallelPair (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft))\n            WalkingParallelPair.one) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    (α_ M.X N.X P.X).hom ≫\n      M.X ◁\n          (f.hom ▷ P.X ≫\n            colimit.ι (parallelPair (N'.actRight ▷ P.X) ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft))\n              WalkingParallelPair.one) ≫\n        AssociatorBimod.invAux M N' P","state_after":"case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ (M.X ◁ f.hom ≫\n          colimit.ι (parallelPair (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft))\n            WalkingParallelPair.one) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    (α_ M.X N.X P.X).hom ≫\n      M.X ◁\n          (f.hom ▷ P.X ≫\n            colimit.ι (parallelPair (N'.actRight ▷ P.X) ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft))\n              WalkingParallelPair.one) ≫\n        (PreservesCoequalizer.iso (tensorLeft M.X) (N'.actRight ▷ P.X) ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft)).inv ≫\n          coequalizer.desc\n            ((α_ M.X N'.X P.X).inv ≫\n              coequalizer.π (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ▷ P.X ≫\n                coequalizer.π (TensorBimod.actRight M N' ▷ P.X)\n                  ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n            ⋯","tactic":"dsimp [AssociatorBimod.invAux]","premises":[{"full_name":"Bimod.AssociatorBimod.invAux","def_path":"Mathlib/CategoryTheory/Monoidal/Bimod.lean","def_pos":[517,18],"def_end_pos":[517,24]}]},{"state_before":"case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ (M.X ◁ f.hom ≫\n          colimit.ι (parallelPair (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft))\n            WalkingParallelPair.one) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    (α_ M.X N.X P.X).hom ≫\n      M.X ◁\n          (f.hom ▷ P.X ≫\n            colimit.ι (parallelPair (N'.actRight ▷ P.X) ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft))\n              WalkingParallelPair.one) ≫\n        (PreservesCoequalizer.iso (tensorLeft M.X) (N'.actRight ▷ P.X) ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft)).inv ≫\n          coequalizer.desc\n            ((α_ M.X N'.X P.X).inv ≫\n              coequalizer.π (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ▷ P.X ≫\n                coequalizer.π (TensorBimod.actRight M N' ▷ P.X)\n                  ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n            ⋯","state_after":"case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ (M.X ◁ f.hom ≫\n          colimit.ι (parallelPair (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft))\n            WalkingParallelPair.one) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    (α_ M.X N.X P.X).hom ≫\n      ((M.X ◁ f.hom ▷ P.X ≫\n            M.X ◁\n              colimit.ι (parallelPair (N'.actRight ▷ P.X) ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft))\n                WalkingParallelPair.one) ≫\n          (PreservesCoequalizer.iso (tensorLeft M.X) (N'.actRight ▷ P.X)\n              ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft)).inv) ≫\n        coequalizer.desc\n          ((α_ M.X N'.X P.X).inv ≫\n            coequalizer.π (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ▷ P.X ≫\n              coequalizer.π (TensorBimod.actRight M N' ▷ P.X)\n                ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n          ⋯","tactic":"slice_rhs 2 2 => rw [MonoidalCategory.whiskerLeft_comp]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.MonoidalCategory.whiskerLeft_comp","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[232,8],"def_end_pos":[232,24]}]},{"state_before":"case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ (M.X ◁ f.hom ≫\n          colimit.ι (parallelPair (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft))\n            WalkingParallelPair.one) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    (α_ M.X N.X P.X).hom ≫\n      ((M.X ◁ f.hom ▷ P.X ≫\n            M.X ◁\n              colimit.ι (parallelPair (N'.actRight ▷ P.X) ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft))\n                WalkingParallelPair.one) ≫\n          (PreservesCoequalizer.iso (tensorLeft M.X) (N'.actRight ▷ P.X)\n              ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft)).inv) ≫\n        coequalizer.desc\n          ((α_ M.X N'.X P.X).inv ≫\n            coequalizer.π (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ▷ P.X ≫\n              coequalizer.π (TensorBimod.actRight M N' ▷ P.X)\n                ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n          ⋯","state_after":"case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ (M.X ◁ f.hom ≫\n          colimit.ι (parallelPair (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft))\n            WalkingParallelPair.one) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    (α_ M.X N.X P.X).hom ≫\n      M.X ◁ f.hom ▷ P.X ≫\n        (α_ M.X N'.X P.X).inv ≫\n          coequalizer.π (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ▷ P.X ≫\n            coequalizer.π (TensorBimod.actRight M N' ▷ P.X)\n              ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft)","tactic":"slice_rhs 3 5 => rw [id_tensor_π_preserves_coequalizer_inv_desc]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"id_tensor_π_preserves_coequalizer_inv_desc","def_path":"Mathlib/CategoryTheory/Monoidal/Bimod.lean","def_pos":[33,8],"def_end_pos":[33,50]}]},{"state_before":"case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ (M.X ◁ f.hom ≫\n          colimit.ι (parallelPair (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft))\n            WalkingParallelPair.one) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    (α_ M.X N.X P.X).hom ≫\n      M.X ◁ f.hom ▷ P.X ≫\n        (α_ M.X N'.X P.X).inv ≫\n          coequalizer.π (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ▷ P.X ≫\n            coequalizer.π (TensorBimod.actRight M N' ▷ P.X)\n              ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft)","state_after":"case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ (M.X ◁ f.hom ≫\n          colimit.ι (parallelPair (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft))\n            WalkingParallelPair.one) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    (α_ M.X N.X P.X).hom ≫\n      (((α_ M.X N.X P.X).inv ≫ (M.X ◁ f.hom) ▷ P.X) ≫\n          coequalizer.π (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ▷ P.X) ≫\n        coequalizer.π (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft)","tactic":"slice_rhs 2 3 => rw [associator_inv_naturality_middle]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.MonoidalCategory.associator_inv_naturality_middle","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[468,8],"def_end_pos":[468,40]}]},{"state_before":"case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ (M.X ◁ f.hom ≫\n          colimit.ι (parallelPair (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft))\n            WalkingParallelPair.one) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    (α_ M.X N.X P.X).hom ≫\n      (((α_ M.X N.X P.X).inv ≫ (M.X ◁ f.hom) ▷ P.X) ≫\n          coequalizer.π (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ▷ P.X) ≫\n        coequalizer.π (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft)","state_after":"case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ (M.X ◁ f.hom ≫\n          colimit.ι (parallelPair (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft))\n            WalkingParallelPair.one) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    ((M.X ◁ f.hom) ▷ P.X ≫ coequalizer.π (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ▷ P.X) ≫\n      coequalizer.π (TensorBimod.actRight M N' ▷ P.X)\n        ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft)","tactic":"slice_rhs 1 3 => rw [Iso.hom_inv_id_assoc]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]}]},{"state_before":"case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ (M.X ◁ f.hom ≫\n          colimit.ι (parallelPair (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft))\n            WalkingParallelPair.one) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    ((M.X ◁ f.hom) ▷ P.X ≫ coequalizer.π (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ▷ P.X) ≫\n      coequalizer.π (TensorBimod.actRight M N' ▷ P.X)\n        ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft)","state_after":"no goals","tactic":"slice_lhs 1 1 => rw [comp_whiskerRight]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.MonoidalCategory.comp_whiskerRight","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[249,8],"def_end_pos":[249,25]}]}]}
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↑(Submodule.map (↑↑(f.coprodSubtypeLEquivOfIsCompl h hker)) (⊤.prod ⊥))","state_after":"𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹⁴ : NontriviallyNormedField 𝕜\ninst✝¹³ : NontriviallyNormedField 𝕜'\nσ : 𝕜 →+* 𝕜'\nσ' : 𝕜' →+* 𝕜\ninst✝¹² : RingHomInvPair σ σ'\ninst✝¹¹ : RingHomInvPair σ' σ\ninst✝¹⁰ : RingHomIsometric σ\ninst✝⁹ : RingHomIsometric σ'\nE : Type u_3\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF✝ : Type u_4\ninst✝⁶ : NormedAddCommGroup F✝\ninst✝⁵ : NormedSpace 𝕜' F✝\nf✝ : E →SL[σ] F✝\ninst✝⁴ : CompleteSpace F✝\ninst✝³ : CompleteSpace E\nF : Type u_5\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\nf : E →L[𝕜] F\nG : Submodule 𝕜 F\nh : IsCompl (LinearMap.range f) G\nhG : IsClosed ↑G\nhker : LinearMap.ker f = ⊥\nthis : CompleteSpace ↥G\ng : (E × ↥G) ≃L[𝕜] F := f.coprodSubtypeLEquivOfIsCompl h hker\n⊢ IsClosed ↑(⊤.prod ⊥)","tactic":"apply 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NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\nf : E →L[𝕜] F\nG : Submodule 𝕜 F\nh : IsCompl (LinearMap.range f) G\nhG : IsClosed ↑G\nhker : LinearMap.ker f = ⊥\nthis : CompleteSpace ↥G\ng : (E × ↥G) ≃L[𝕜] F := f.coprodSubtypeLEquivOfIsCompl h hker\n⊢ IsClosed ↑(⊤.prod ⊥)","state_after":"no goals","tactic":"exact isClosed_univ.prod isClosed_singleton","premises":[{"full_name":"IsClosed.prod","def_path":"Mathlib/Topology/Constructions.lean","def_pos":[746,8],"def_end_pos":[746,21]},{"full_name":"isClosed_singleton","def_path":"Mathlib/Topology/Separation.lean","def_pos":[524,8],"def_end_pos":[524,26]},{"full_name":"isClosed_univ","def_path":"Mathlib/Topology/Basic.lean","def_pos":[157,16],"def_end_pos":[157,29]}]}]}
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+{"url":"Mathlib/Data/Multiset/Pi.lean","commit":"","full_name":"Multiset.Pi.cons_injective","start":[87,0],"end":[96,62],"file_path":"Mathlib/Data/Multiset/Pi.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : DecidableEq α\nδ : α → Sort u_2\nm : Multiset α\na✝ a : α\nb : δ a\ns : Multiset α\nhs : a ∉ s\nf₁ f₂ : (a : α) → a ∈ s → δ a\neq : cons s a b f₁ = cons s a b f₂\na' : α\nh' : a' ∈ s\nne : a ≠ a'\nthis : a' ∈ a ::ₘ s\n⊢ f₁ a' h' = cons s a b f₁ a' this","state_after":"no goals","tactic":"rw [Pi.cons_ne this ne.symm]","premises":[{"full_name":"Multiset.Pi.cons_ne","def_path":"Mathlib/Data/Multiset/Pi.lean","def_pos":[45,8],"def_end_pos":[45,15]},{"full_name":"Ne.symm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[704,8],"def_end_pos":[704,15]}]},{"state_before":"α : Type u_1\ninst✝ : DecidableEq α\nδ : α → Sort u_2\nm : Multiset α\na✝ a : α\nb : δ a\ns : Multiset α\nhs : a ∉ s\nf₁ f₂ : (a : α) → a ∈ s → δ a\neq : cons s a b f₁ = cons s a b f₂\na' : α\nh' : a' ∈ s\nne : a ≠ a'\nthis : a' ∈ a ::ₘ s\n⊢ cons s a b f₁ a' this = cons s a b f₂ a' this","state_after":"no goals","tactic":"rw [eq]","premises":[]},{"state_before":"α : Type u_1\ninst✝ : DecidableEq α\nδ : α → Sort u_2\nm : Multiset α\na✝ a : α\nb : δ a\ns : Multiset α\nhs : a ∉ s\nf₁ f₂ : (a : α) → a ∈ s → δ a\neq : cons s a b f₁ = cons s a b f₂\na' : α\nh' : a' ∈ s\nne : a ≠ a'\nthis : a' ∈ a ::ₘ s\n⊢ cons s a b f₂ a' this = f₂ a' h'","state_after":"no goals","tactic":"rw [Pi.cons_ne this ne.symm]","premises":[{"full_name":"Multiset.Pi.cons_ne","def_path":"Mathlib/Data/Multiset/Pi.lean","def_pos":[45,8],"def_end_pos":[45,15]},{"full_name":"Ne.symm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[704,8],"def_end_pos":[704,15]}]}]}
+{"url":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","commit":"","full_name":"MeasureTheory.subset_toMeasurable","start":[300,0],"end":[302,94],"file_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Sort u_5\ninst✝ : MeasurableSpace α\nμ✝ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t : Set α\nμ : Measure α\ns : Set α\n⊢ s ⊆ toMeasurable μ s","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Sort u_5\ninst✝ : MeasurableSpace α\nμ✝ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t : Set α\nμ : Measure α\ns : Set α\n⊢ s ⊆\n    if h : ∃ t ⊇ s, MeasurableSet t ∧ t =ᶠ[ae μ] s then h.choose\n    else\n      if h' : ∃ t ⊇ s, MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → μ (t ∩ u) = μ (s ∩ u) then h'.choose\n      else ⋯.choose","tactic":"rw [toMeasurable_def]","premises":[{"full_name":"MeasureTheory.toMeasurable_def","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[294,16],"def_end_pos":[294,28]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Sort u_5\ninst✝ : MeasurableSpace α\nμ✝ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t : Set α\nμ : Measure α\ns : Set α\n⊢ s ⊆\n    if h : ∃ t ⊇ s, MeasurableSet t ∧ t =ᶠ[ae μ] s then h.choose\n    else\n      if h' : ∃ t ⊇ s, MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → μ (t ∩ u) = μ (s ∩ u) then h'.choose\n      else ⋯.choose","state_after":"case pos\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Sort u_5\ninst✝ : MeasurableSpace α\nμ✝ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t : Set α\nμ : Measure α\ns : Set α\nhs : ∃ t ⊇ s, MeasurableSet t ∧ t =ᶠ[ae μ] s\n⊢ s ⊆ hs.choose\n\ncase pos\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Sort u_5\ninst✝ : MeasurableSpace α\nμ✝ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t : Set α\nμ : Measure α\ns : Set α\nhs : ¬∃ t ⊇ s, MeasurableSet t ∧ t =ᶠ[ae μ] s\nh's : ∃ t ⊇ s, MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → μ (t ∩ u) = μ (s ∩ u)\n⊢ s ⊆ h's.choose\n\ncase neg\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Sort u_5\ninst✝ : MeasurableSpace α\nμ✝ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t : Set α\nμ : Measure α\ns : Set α\nhs : ¬∃ t ⊇ s, MeasurableSet t ∧ t =ᶠ[ae μ] s\nh's : ¬∃ t ⊇ s, MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → μ (t ∩ u) = μ (s ∩ u)\n⊢ s ⊆ ⋯.choose","tactic":"split_ifs with hs h's","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case pos\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Sort u_5\ninst✝ : MeasurableSpace α\nμ✝ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t : Set α\nμ : Measure α\ns : Set α\nhs : ∃ t ⊇ s, MeasurableSet t ∧ t =ᶠ[ae μ] s\n⊢ s ⊆ hs.choose\n\ncase pos\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Sort u_5\ninst✝ : MeasurableSpace α\nμ✝ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t : Set α\nμ : Measure α\ns : Set α\nhs : ¬∃ t ⊇ s, MeasurableSet t ∧ t =ᶠ[ae μ] s\nh's : ∃ t ⊇ s, MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → μ (t ∩ u) = μ (s ∩ u)\n⊢ s ⊆ h's.choose\n\ncase neg\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Sort u_5\ninst✝ : MeasurableSpace α\nμ✝ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t : Set α\nμ : Measure α\ns : Set α\nhs : ¬∃ t ⊇ s, MeasurableSet t ∧ t =ᶠ[ae μ] s\nh's : ¬∃ t ⊇ s, MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → μ (t ∩ u) = μ (s ∩ u)\n⊢ s ⊆ ⋯.choose","state_after":"no goals","tactic":"exacts [hs.choose_spec.1, h's.choose_spec.1, (exists_measurable_superset μ s).choose_spec.1]","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"Exists.choose_spec","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[177,8],"def_end_pos":[177,26]},{"full_name":"MeasureTheory.exists_measurable_superset","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[188,8],"def_end_pos":[188,34]}]}]}
+{"url":"Mathlib/CategoryTheory/Abelian/Refinements.lean","commit":"","full_name":"CategoryTheory.ShortComplex.exact_iff_exact_up_to_refinements","start":[93,0],"end":[103,89],"file_path":"Mathlib/CategoryTheory/Abelian/Refinements.lean","tactics":[{"state_before":"C : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : Abelian C\nX Y : C\nS S₁ S₂ : ShortComplex C\n⊢ S.Exact ↔ ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂), x₂ ≫ S.g = 0 → ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ S.f","state_after":"C : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : Abelian C\nX Y : C\nS S₁ S₂ : ShortComplex C\n⊢ (∀ ⦃A : C⦄ (y : A ⟶ S.cycles), ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ y = x ≫ S.toCycles) ↔\n    ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂), x₂ ≫ S.g = 0 → ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ S.f","tactic":"rw [S.exact_iff_epi_toCycles, epi_iff_surjective_up_to_refinements]","premises":[{"full_name":"CategoryTheory.ShortComplex.exact_iff_epi_toCycles","def_path":"Mathlib/Algebra/Homology/ShortComplex/Exact.lean","def_pos":[382,6],"def_end_pos":[382,28]},{"full_name":"CategoryTheory.epi_iff_surjective_up_to_refinements","def_path":"Mathlib/CategoryTheory/Abelian/Refinements.lean","def_pos":[78,6],"def_end_pos":[78,42]}]},{"state_before":"C : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : Abelian C\nX Y : C\nS S₁ S₂ : ShortComplex C\n⊢ (∀ ⦃A : C⦄ (y : A ⟶ S.cycles), ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ y = x ≫ S.toCycles) ↔\n    ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂), x₂ ≫ S.g = 0 → ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ S.f","state_after":"case mp\nC : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : Abelian C\nX Y : C\nS S₁ S₂ : ShortComplex C\n⊢ (∀ ⦃A : C⦄ (y : A ⟶ S.cycles), ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ y = x ≫ S.toCycles) →\n    ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂), x₂ ≫ S.g = 0 → ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ S.f\n\ncase mpr\nC : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : Abelian C\nX Y : C\nS S₁ S₂ : ShortComplex C\n⊢ (∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂), x₂ ≫ S.g = 0 → ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ S.f) →\n    ∀ ⦃A : C⦄ (y : A ⟶ S.cycles), ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ y = x ≫ S.toCycles","tactic":"constructor","premises":[]}]}
+{"url":"Mathlib/Analysis/Calculus/FDeriv/Extend.lean","commit":"","full_name":"hasDerivAt_of_hasDerivAt_of_ne","start":[181,0],"end":[212,23],"file_path":"Mathlib/Analysis/Calculus/FDeriv/Extend.lean","tactics":[{"state_before":"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf g : ℝ → E\nx : ℝ\nf_diff : ∀ (y : ℝ), y ≠ x → HasDerivAt f (g y) y\nhf : ContinuousAt f x\nhg : ContinuousAt g x\n⊢ HasDerivAt f (g x) x","state_after":"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf g : ℝ → E\nx : ℝ\nf_diff : ∀ (y : ℝ), y ≠ x → HasDerivAt f (g y) y\nhf : ContinuousAt f x\nhg : ContinuousAt g x\nA : HasDerivWithinAt f (g x) (Ici x) x\n⊢ HasDerivAt f (g x) x","tactic":"have A : HasDerivWithinAt f (g x) (Ici x) x := by\n    have diff : DifferentiableOn ℝ f (Ioi x) := fun y hy =>\n      (f_diff y (ne_of_gt hy)).differentiableAt.differentiableWithinAt\n    -- next line is the nontrivial bit of this proof, appealing to differentiability\n    -- extension results.\n    apply\n      hasDerivWithinAt_Ici_of_tendsto_deriv diff hf.continuousWithinAt\n        self_mem_nhdsWithin\n    have : Tendsto g (𝓝[>] x) (𝓝 (g x)) := tendsto_inf_left hg\n    apply this.congr' _\n    apply mem_of_superset self_mem_nhdsWithin fun y hy => _\n    intros y hy\n    exact (f_diff y (ne_of_gt hy)).deriv.symm","premises":[{"full_name":"ContinuousAt.continuousWithinAt","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[752,8],"def_end_pos":[752,39]},{"full_name":"DifferentiableAt.differentiableWithinAt","def_path":"Mathlib/Analysis/Calculus/FDeriv/Basic.lean","def_pos":[576,8],"def_end_pos":[576,47]},{"full_name":"DifferentiableOn","def_path":"Mathlib/Analysis/Calculus/FDeriv/Basic.lean","def_pos":[188,4],"def_end_pos":[188,20]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Filter.Tendsto","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2567,4],"def_end_pos":[2567,11]},{"full_name":"Filter.Tendsto.congr'","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2634,8],"def_end_pos":[2634,22]},{"full_name":"Filter.mem_of_superset","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[139,8],"def_end_pos":[139,23]},{"full_name":"Filter.tendsto_inf_left","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2716,8],"def_end_pos":[2716,24]},{"full_name":"HasDerivAt.deriv","def_path":"Mathlib/Analysis/Calculus/Deriv/Basic.lean","def_pos":[382,8],"def_end_pos":[382,24]},{"full_name":"HasDerivAt.differentiableAt","def_path":"Mathlib/Analysis/Calculus/Deriv/Basic.lean","def_pos":[336,8],"def_end_pos":[336,35]},{"full_name":"HasDerivWithinAt","def_path":"Mathlib/Analysis/Calculus/Deriv/Basic.lean","def_pos":[114,4],"def_end_pos":[114,20]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Set.Ici","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[66,4],"def_end_pos":[66,7]},{"full_name":"Set.Ioi","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[70,4],"def_end_pos":[70,7]},{"full_name":"hasDerivWithinAt_Ici_of_tendsto_deriv","def_path":"Mathlib/Analysis/Calculus/FDeriv/Extend.lean","def_pos":[108,8],"def_end_pos":[108,45]},{"full_name":"ne_of_gt","def_path":"Mathlib/Order/Defs.lean","def_pos":[85,8],"def_end_pos":[85,16]},{"full_name":"nhds","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[113,16],"def_end_pos":[113,20]},{"full_name":"nhdsWithin","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[121,4],"def_end_pos":[121,14]},{"full_name":"self_mem_nhdsWithin","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[124,8],"def_end_pos":[124,27]}]},{"state_before":"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf g : ℝ → E\nx : ℝ\nf_diff : ∀ (y : ℝ), y ≠ x → HasDerivAt f (g y) y\nhf : ContinuousAt f x\nhg : ContinuousAt g x\nA : HasDerivWithinAt f (g x) (Ici x) x\n⊢ HasDerivAt f (g x) x","state_after":"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf g : ℝ → E\nx : ℝ\nf_diff : ∀ (y : ℝ), y ≠ x → HasDerivAt f (g y) y\nhf : ContinuousAt f x\nhg : ContinuousAt g x\nA : HasDerivWithinAt f (g x) (Ici x) x\nB : HasDerivWithinAt f (g x) (Iic x) x\n⊢ HasDerivAt f (g x) x","tactic":"have B : HasDerivWithinAt f (g x) (Iic x) x := by\n    have diff : DifferentiableOn ℝ f (Iio x) := fun y hy =>\n      (f_diff y (ne_of_lt hy)).differentiableAt.differentiableWithinAt\n    -- next line is the nontrivial bit of this proof, appealing to differentiability\n    -- extension results.\n    apply\n      hasDerivWithinAt_Iic_of_tendsto_deriv diff hf.continuousWithinAt\n        self_mem_nhdsWithin\n    have : Tendsto g (𝓝[<] x) (𝓝 (g x)) := tendsto_inf_left hg\n    apply this.congr' _\n    apply mem_of_superset self_mem_nhdsWithin fun y hy => _\n    intros y hy\n    exact (f_diff y (ne_of_lt hy)).deriv.symm","premises":[{"full_name":"ContinuousAt.continuousWithinAt","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[752,8],"def_end_pos":[752,39]},{"full_name":"DifferentiableAt.differentiableWithinAt","def_path":"Mathlib/Analysis/Calculus/FDeriv/Basic.lean","def_pos":[576,8],"def_end_pos":[576,47]},{"full_name":"DifferentiableOn","def_path":"Mathlib/Analysis/Calculus/FDeriv/Basic.lean","def_pos":[188,4],"def_end_pos":[188,20]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Filter.Tendsto","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2567,4],"def_end_pos":[2567,11]},{"full_name":"Filter.Tendsto.congr'","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2634,8],"def_end_pos":[2634,22]},{"full_name":"Filter.mem_of_superset","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[139,8],"def_end_pos":[139,23]},{"full_name":"Filter.tendsto_inf_left","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2716,8],"def_end_pos":[2716,24]},{"full_name":"HasDerivAt.deriv","def_path":"Mathlib/Analysis/Calculus/Deriv/Basic.lean","def_pos":[382,8],"def_end_pos":[382,24]},{"full_name":"HasDerivAt.differentiableAt","def_path":"Mathlib/Analysis/Calculus/Deriv/Basic.lean","def_pos":[336,8],"def_end_pos":[336,35]},{"full_name":"HasDerivWithinAt","def_path":"Mathlib/Analysis/Calculus/Deriv/Basic.lean","def_pos":[114,4],"def_end_pos":[114,20]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Set.Iic","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[58,4],"def_end_pos":[58,7]},{"full_name":"Set.Iio","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[50,4],"def_end_pos":[50,7]},{"full_name":"hasDerivWithinAt_Iic_of_tendsto_deriv","def_path":"Mathlib/Analysis/Calculus/FDeriv/Extend.lean","def_pos":[145,8],"def_end_pos":[145,45]},{"full_name":"ne_of_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[83,8],"def_end_pos":[83,16]},{"full_name":"nhds","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[113,16],"def_end_pos":[113,20]},{"full_name":"nhdsWithin","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[121,4],"def_end_pos":[121,14]},{"full_name":"self_mem_nhdsWithin","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[124,8],"def_end_pos":[124,27]}]},{"state_before":"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf g : ℝ → E\nx : ℝ\nf_diff : ∀ (y : ℝ), y ≠ x → HasDerivAt f (g y) y\nhf : ContinuousAt f x\nhg : ContinuousAt g x\nA : HasDerivWithinAt f (g x) (Ici x) x\nB : HasDerivWithinAt f (g x) (Iic x) x\n⊢ HasDerivAt f (g x) x","state_after":"no goals","tactic":"simpa using B.union A","premises":[{"full_name":"HasDerivWithinAt.union","def_path":"Mathlib/Analysis/Calculus/Deriv/Basic.lean","def_pos":[354,8],"def_end_pos":[354,30]}]}]}
+{"url":"Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean","commit":"","full_name":"Matrix.charpoly_map","start":[98,0],"end":[101,28],"file_path":"Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean","tactics":[{"state_before":"R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\nm : Type u_3\nn : Type u_4\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : Fintype m\ninst✝ : Fintype n\nM₁₁ : Matrix m m R\nM₁₂ : Matrix m n R\nM₂₁ : Matrix n m R\nM₂₂ M✝ : Matrix n n R\ni j : n\nM : Matrix n n R\nf : R →+* S\n⊢ (M.map ⇑f).charpoly = Polynomial.map f M.charpoly","state_after":"no goals","tactic":"rw [charpoly, charmatrix_map, ← Polynomial.coe_mapRingHom, charpoly, RingHom.map_det,\n    RingHom.mapMatrix_apply]","premises":[{"full_name":"Matrix.charmatrix_map","def_path":"Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean","def_pos":[77,6],"def_end_pos":[77,20]},{"full_name":"Matrix.charpoly","def_path":"Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean","def_pos":[90,4],"def_end_pos":[90,12]},{"full_name":"Polynomial.coe_mapRingHom","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[649,8],"def_end_pos":[649,22]},{"full_name":"RingHom.mapMatrix_apply","def_path":"Mathlib/Data/Matrix/Basic.lean","def_pos":[1344,2],"def_end_pos":[1344,7]},{"full_name":"RingHom.map_det","def_path":"Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean","def_pos":[284,8],"def_end_pos":[284,30]}]}]}
+{"url":"Mathlib/Control/Functor.lean","commit":"","full_name":"Functor.ext","start":[44,0],"end":[54,26],"file_path":"Mathlib/Control/Functor.lean","tactics":[{"state_before":"F✝ : Type u → Type v\nα β γ : Type u\ninst✝¹ : Functor F✝\ninst✝ : LawfulFunctor F✝\nF : Type u_1 → Type u_2\nm : {α β : Type u_1} → (α → β) → F α → F β\nmc : {α β : Type u_1} → α → F β → F α\nm' : {α β : Type u_1} → (α → β) → F α → F β\nmc' : {α β : Type u_1} → α → F β → F α\nH1 : LawfulFunctor F\nH2 : LawfulFunctor F\nH : ∀ (α β : Type u_1) (f : α → β) (x : F α), f <$> x = f <$> x\n⊢ { map := m, mapConst := mc } = { map := m', mapConst := mc' }","state_after":"case refl\nF✝ : Type u → Type v\nα β γ : Type u\ninst✝¹ : Functor F✝\ninst✝ : LawfulFunctor F✝\nF : Type u_1 → Type u_2\nm : {α β : Type u_1} → (α → β) → F α → F β\nmc mc' : {α β : Type u_1} → α → F β → F α\nH1 : LawfulFunctor F\nH2 : LawfulFunctor F\nH : ∀ (α β : Type u_1) (f : α → β) (x : F α), f <$> x = f <$> x\n⊢ { map := m, mapConst := mc } = { map := m, mapConst := mc' }","tactic":"cases show @m = @m' by funext α β f x; apply H","premises":[{"full_name":"funext","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1817,8],"def_end_pos":[1817,14]}]},{"state_before":"case refl\nF✝ : Type u → Type v\nα β γ : Type u\ninst✝¹ : Functor F✝\ninst✝ : LawfulFunctor F✝\nF : Type u_1 → Type u_2\nm : {α β : Type u_1} → (α → β) → F α → F β\nmc mc' : {α β : Type u_1} → α → F β → F α\nH1 : LawfulFunctor F\nH2 : LawfulFunctor F\nH : ∀ (α β : Type u_1) (f : α → β) (x : F α), f <$> x = f <$> x\n⊢ { map := m, mapConst := mc } = { map := m, mapConst := mc' }","state_after":"case refl.e_mapConst\nF✝ : Type u → Type v\nα β γ : Type u\ninst✝¹ : Functor F✝\ninst✝ : LawfulFunctor F✝\nF : Type u_1 → Type u_2\nm : {α β : Type u_1} → (α → β) → F α → F β\nmc mc' : {α β : Type u_1} → α → F β → F α\nH1 : LawfulFunctor F\nH2 : LawfulFunctor F\nH : ∀ (α β : Type u_1) (f : α → β) (x : F α), f <$> x = f <$> x\n⊢ mc = mc'","tactic":"congr","premises":[]},{"state_before":"case refl.e_mapConst\nF✝ : Type u → Type v\nα β γ : Type u\ninst✝¹ : Functor F✝\ninst✝ : LawfulFunctor F✝\nF : Type u_1 → Type u_2\nm : {α β : Type u_1} → (α → β) → F α → F β\nmc mc' : {α β : Type u_1} → α → F β → F α\nH1 : LawfulFunctor F\nH2 : LawfulFunctor F\nH : ∀ (α β : Type u_1) (f : α → β) (x : F α), f <$> x = f <$> x\n⊢ mc = mc'","state_after":"case refl.e_mapConst.h.h\nF✝ : Type u → Type v\nα✝ β✝ γ : Type u\ninst✝¹ : Functor F✝\ninst✝ : LawfulFunctor F✝\nF : Type u_1 → Type u_2\nm : {α β : Type u_1} → (α → β) → F α → F β\nmc mc' : {α β : Type u_1} → α → F β → F α\nH1 : LawfulFunctor F\nH2 : LawfulFunctor F\nH : ∀ (α β : Type u_1) (f : α → β) (x : F α), f <$> x = f <$> x\nα β : Type u_1\n⊢ mc = mc'","tactic":"funext α β","premises":[{"full_name":"funext","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1817,8],"def_end_pos":[1817,14]}]},{"state_before":"case refl.e_mapConst.h.h\nF✝ : Type u → Type v\nα✝ β✝ γ : Type u\ninst✝¹ : Functor F✝\ninst✝ : LawfulFunctor F✝\nF : Type u_1 → Type u_2\nm : {α β : Type u_1} → (α → β) → F α → F β\nmc mc' : {α β : Type u_1} → α → F β → F α\nH1 : LawfulFunctor F\nH2 : LawfulFunctor F\nH : ∀ (α β : Type u_1) (f : α → β) (x : F α), f <$> x = f <$> x\nα β : Type u_1\n⊢ mc = mc'","state_after":"case refl.e_mapConst.h.h\nF✝ : Type u → Type v\nα✝ β✝ γ : Type u\ninst✝¹ : Functor F✝\ninst✝ : LawfulFunctor F✝\nF : Type u_1 → Type u_2\nm : {α β : Type u_1} → (α → β) → F α → F β\nmc mc' : {α β : Type u_1} → α → F β → F α\nH1 : LawfulFunctor F\nH2 : LawfulFunctor F\nH : ∀ (α β : Type u_1) (f : α → β) (x : F α), f <$> x = f <$> x\nα β : Type u_1\nE1 : ∀ {α β : Type u_1}, mapConst = map ∘ Function.const β\n⊢ mc = mc'","tactic":"have E1 := @map_const _ ⟨@m, @mc⟩ H1","premises":[{"full_name":"LawfulFunctor.map_const","def_path":".lake/packages/lean4/src/lean/Init/Control/Lawful/Basic.lean","def_pos":[25,2],"def_end_pos":[25,11]}]},{"state_before":"case refl.e_mapConst.h.h\nF✝ : Type u → Type v\nα✝ β✝ γ : Type u\ninst✝¹ : Functor F✝\ninst✝ : LawfulFunctor F✝\nF : Type u_1 → Type u_2\nm : {α β : Type u_1} → (α → β) → F α → F β\nmc mc' : {α β : Type u_1} → α → F β → F α\nH1 : LawfulFunctor F\nH2 : LawfulFunctor F\nH : ∀ (α β : Type u_1) (f : α → β) (x : F α), f <$> x = f <$> x\nα β : Type u_1\nE1 : ∀ {α β : Type u_1}, mapConst = map ∘ Function.const β\n⊢ mc = mc'","state_after":"case refl.e_mapConst.h.h\nF✝ : Type u → Type v\nα✝ β✝ γ : Type u\ninst✝¹ : Functor F✝\ninst✝ : LawfulFunctor F✝\nF : Type u_1 → Type u_2\nm : {α β : Type u_1} → (α → β) → F α → F β\nmc mc' : {α β : Type u_1} → α → F β → F α\nH1 : LawfulFunctor F\nH2 : LawfulFunctor F\nH : ∀ (α β : Type u_1) (f : α → β) (x : F α), f <$> x = f <$> x\nα β : Type u_1\nE1 : ∀ {α β : Type u_1}, mapConst = map ∘ Function.const β\nE2 : ∀ {α β : Type u_1}, mapConst = map ∘ Function.const β\n⊢ mc = mc'","tactic":"have E2 := @map_const _ ⟨@m, @mc'⟩ H2","premises":[{"full_name":"LawfulFunctor.map_const","def_path":".lake/packages/lean4/src/lean/Init/Control/Lawful/Basic.lean","def_pos":[25,2],"def_end_pos":[25,11]}]},{"state_before":"case refl.e_mapConst.h.h\nF✝ : Type u → Type v\nα✝ β✝ γ : Type u\ninst✝¹ : Functor F✝\ninst✝ : LawfulFunctor F✝\nF : Type u_1 → Type u_2\nm : {α β : Type u_1} → (α → β) → F α → F β\nmc mc' : {α β : Type u_1} → α → F β → F α\nH1 : LawfulFunctor F\nH2 : LawfulFunctor F\nH : ∀ (α β : Type u_1) (f : α → β) (x : F α), f <$> x = f <$> x\nα β : Type u_1\nE1 : ∀ {α β : Type u_1}, mapConst = map ∘ Function.const β\nE2 : ∀ {α β : Type u_1}, mapConst = map ∘ Function.const β\n⊢ mc = mc'","state_after":"no goals","tactic":"exact E1.trans E2.symm","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Eq.trans","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[335,8],"def_end_pos":[335,16]}]}]}
+{"url":"Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean","commit":"","full_name":"UniformOnFun.hasBasis_uniformity_of_covering_of_basis","start":[662,0],"end":[680,59],"file_path":"Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι✝ : Type u_4\ns s' : Set α\nx : α\np✝ : Filter ι✝\ng : ι✝ → α\ninst✝¹ : UniformSpace β\n𝔖 : Set (Set α)\nι : Type u_5\nι' : Type u_6\ninst✝ : Nonempty ι\nt : ι → Set α\np : ι' → Prop\nV : ι' → Set (β × β)\nht : ∀ (i : ι), t i ∈ 𝔖\nhdir : Directed (fun x x_1 => x ⊆ x_1) t\nhex : ∀ s ∈ 𝔖, ∃ i, s ⊆ t i\nhb : (𝓤 β).HasBasis p V\n⊢ (𝓤 (α →ᵤ[𝔖] β)).HasBasis (fun i => p i.2) fun i => UniformOnFun.gen 𝔖 (t i.1) (V i.2)","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι✝ : Type u_4\ns s' : Set α\nx : α\np✝ : Filter ι✝\ng : ι✝ → α\ninst✝¹ : UniformSpace β\n𝔖 : Set (Set α)\nι : Type u_5\nι' : Type u_6\ninst✝ : Nonempty ι\nt : ι → Set α\np : ι' → Prop\nV : ι' → Set (β × β)\nht : ∀ (i : ι), t i ∈ 𝔖\nhdir : Directed (fun x x_1 => x ⊆ x_1) t\nhex : ∀ s ∈ 𝔖, ∃ i, s ⊆ t i\nhb : (𝓤 β).HasBasis p V\nhne : 𝔖.Nonempty\n⊢ (𝓤 (α →ᵤ[𝔖] β)).HasBasis (fun i => p i.2) fun i => UniformOnFun.gen 𝔖 (t i.1) (V i.2)","tactic":"have hne : 𝔖.Nonempty := (range_nonempty t).mono (range_subset_iff.2 ht)","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Set.Nonempty","def_path":"Mathlib/Init/Set.lean","def_pos":[222,14],"def_end_pos":[222,22]},{"full_name":"Set.Nonempty.mono","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[380,8],"def_end_pos":[380,21]},{"full_name":"Set.range_nonempty","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[641,8],"def_end_pos":[641,22]},{"full_name":"Set.range_subset_iff","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[623,8],"def_end_pos":[623,24]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι✝ : Type u_4\ns s' : Set α\nx : α\np✝ : Filter ι✝\ng : ι✝ → α\ninst✝¹ : UniformSpace β\n𝔖 : Set (Set α)\nι : Type u_5\nι' : Type u_6\ninst✝ : Nonempty ι\nt : ι → Set α\np : ι' → Prop\nV : ι' → Set (β × β)\nht : ∀ (i : ι), t i ∈ 𝔖\nhdir : Directed (fun x x_1 => x ⊆ x_1) t\nhex : ∀ s ∈ 𝔖, ∃ i, s ⊆ t i\nhb : (𝓤 β).HasBasis p V\nhne : 𝔖.Nonempty\nhd : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖\n⊢ (𝓤 (α →ᵤ[𝔖] β)).HasBasis (fun i => p i.2) fun i => UniformOnFun.gen 𝔖 (t i.1) (V i.2)","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι✝ : Type u_4\ns✝ s' : Set α\nx : α\np✝ : Filter ι✝\ng : ι✝ → α\ninst✝¹ : UniformSpace β\n𝔖 : Set (Set α)\nι : Type u_5\nι' : Type u_6\ninst✝ : Nonempty ι\nt : ι → Set α\np : ι' → Prop\nV : ι' → Set (β × β)\nht : ∀ (i : ι), t i ∈ 𝔖\nhdir : Directed (fun x x_1 => x ⊆ x_1) t\nhex : ∀ s ∈ 𝔖, ∃ i, s ⊆ t i\nhb : (𝓤 β).HasBasis p V\nhne : 𝔖.Nonempty\nhd : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖\nx✝¹ : Set α × ι'\ns : Set α\ni' : ι'\nx✝ : (s, i').1 ∈ 𝔖 ∧ p (s, i').2\nhs : (s, i').1 ∈ 𝔖\nhi' : p (s, i').2\n⊢ ∃ i'_1, p i'_1.2 ∧ UniformOnFun.gen 𝔖 (t i'_1.1) (V i'_1.2) ⊆ UniformOnFun.gen 𝔖 (s, i').1 (V (s, i').2)","tactic":"refine (UniformOnFun.hasBasis_uniformity_of_basis α β 𝔖 hne hd hb).to_hasBasis\n    (fun ⟨s, i'⟩ ⟨hs, hi'⟩ ↦ ?_) fun ⟨i, i'⟩ hi' ↦ ⟨(t i, i'), ⟨ht i, hi'⟩, Subset.rfl⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Filter.HasBasis.to_hasBasis","def_path":"Mathlib/Order/Filter/Bases.lean","def_pos":[310,8],"def_end_pos":[310,28]},{"full_name":"Prod.mk","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[481,2],"def_end_pos":[481,4]},{"full_name":"Set.Subset.rfl","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[292,8],"def_end_pos":[292,18]},{"full_name":"UniformOnFun.hasBasis_uniformity_of_basis","def_path":"Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean","def_pos":[643,18],"def_end_pos":[643,46]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι✝ : Type u_4\ns✝ s' : Set α\nx : α\np✝ : Filter ι✝\ng : ι✝ → α\ninst✝¹ : UniformSpace β\n𝔖 : Set (Set α)\nι : Type u_5\nι' : Type u_6\ninst✝ : Nonempty ι\nt : ι → Set α\np : ι' → Prop\nV : ι' → Set (β × β)\nht : ∀ (i : ι), t i ∈ 𝔖\nhdir : Directed (fun x x_1 => x ⊆ x_1) t\nhex : ∀ s ∈ 𝔖, ∃ i, s ⊆ t i\nhb : (𝓤 β).HasBasis p V\nhne : 𝔖.Nonempty\nhd : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖\nx✝¹ : Set α × ι'\ns : Set α\ni' : ι'\nx✝ : (s, i').1 ∈ 𝔖 ∧ p (s, i').2\nhs : (s, i').1 ∈ 𝔖\nhi' : p (s, i').2\n⊢ ∃ i'_1, p i'_1.2 ∧ UniformOnFun.gen 𝔖 (t i'_1.1) (V i'_1.2) ⊆ UniformOnFun.gen 𝔖 (s, i').1 (V (s, i').2)","state_after":"case intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι✝ : Type u_4\ns✝ s' : Set α\nx : α\np✝ : Filter ι✝\ng : ι✝ → α\ninst✝¹ : UniformSpace β\n𝔖 : Set (Set α)\nι : Type u_5\nι' : Type u_6\ninst✝ : Nonempty ι\nt : ι → Set α\np : ι' → Prop\nV : ι' → Set (β × β)\nht : ∀ (i : ι), t i ∈ 𝔖\nhdir : Directed (fun x x_1 => x ⊆ x_1) t\nhex : ∀ s ∈ 𝔖, ∃ i, s ⊆ t i\nhb : (𝓤 β).HasBasis p V\nhne : 𝔖.Nonempty\nhd : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖\nx✝¹ : Set α × ι'\ns : Set α\ni' : ι'\nx✝ : (s, i').1 ∈ 𝔖 ∧ p (s, i').2\nhs : (s, i').1 ∈ 𝔖\nhi' : p (s, i').2\ni : ι\nhi : s ⊆ t i\n⊢ ∃ i'_1, p i'_1.2 ∧ UniformOnFun.gen 𝔖 (t i'_1.1) (V i'_1.2) ⊆ UniformOnFun.gen 𝔖 (s, i').1 (V (s, i').2)","tactic":"rcases hex s hs with ⟨i, hi⟩","premises":[]},{"state_before":"case intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι✝ : Type u_4\ns✝ s' : Set α\nx : α\np✝ : Filter ι✝\ng : ι✝ → α\ninst✝¹ : UniformSpace β\n𝔖 : Set (Set α)\nι : Type u_5\nι' : Type u_6\ninst✝ : Nonempty ι\nt : ι → Set α\np : ι' → Prop\nV : ι' → Set (β × β)\nht : ∀ (i : ι), t i ∈ 𝔖\nhdir : Directed (fun x x_1 => x ⊆ x_1) t\nhex : ∀ s ∈ 𝔖, ∃ i, s ⊆ t i\nhb : (𝓤 β).HasBasis p V\nhne : 𝔖.Nonempty\nhd : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖\nx✝¹ : Set α × ι'\ns : Set α\ni' : ι'\nx✝ : (s, i').1 ∈ 𝔖 ∧ p (s, i').2\nhs : (s, i').1 ∈ 𝔖\nhi' : p (s, i').2\ni : ι\nhi : s ⊆ t i\n⊢ ∃ i'_1, p i'_1.2 ∧ UniformOnFun.gen 𝔖 (t i'_1.1) (V i'_1.2) ⊆ UniformOnFun.gen 𝔖 (s, i').1 (V (s, i').2)","state_after":"no goals","tactic":"exact ⟨(i, i'), hi', UniformOnFun.gen_mono hi Subset.rfl⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Prod.mk","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[481,2],"def_end_pos":[481,4]},{"full_name":"Set.Subset.rfl","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[292,8],"def_end_pos":[292,18]},{"full_name":"UniformOnFun.gen_mono","def_path":"Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean","def_pos":[576,18],"def_end_pos":[576,26]}]}]}
+{"url":"Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean","commit":"","full_name":"MeasureTheory.aefinStronglyMeasurable_iff_aemeasurable","start":[1791,0],"end":[1795,87],"file_path":"Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ninst✝⁵ : Countable ι\nG : Type u_5\np : ℝ≥0∞\nm m0 : MeasurableSpace α\nμ✝ : Measure α\ninst✝⁴ : SeminormedAddCommGroup G\ninst✝³ : MeasurableSpace G\ninst✝² : BorelSpace G\ninst✝¹ : SecondCountableTopology G\nf : α → G\n_m0 : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\n⊢ AEFinStronglyMeasurable f μ ↔ AEMeasurable f μ","state_after":"no goals","tactic":"simp_rw [AEFinStronglyMeasurable, AEMeasurable, finStronglyMeasurable_iff_measurable]","premises":[{"full_name":"AEMeasurable","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[376,4],"def_end_pos":[376,16]},{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"MeasureTheory.AEFinStronglyMeasurable","def_path":"Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean","def_pos":[94,4],"def_end_pos":[94,27]},{"full_name":"MeasureTheory.finStronglyMeasurable_iff_measurable","def_path":"Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean","def_pos":[1780,8],"def_end_pos":[1780,44]}]}]}
+{"url":"Mathlib/Topology/Connected/TotallyDisconnected.lean","commit":"","full_name":"exists_isClopen_of_totally_separated","start":[213,0],"end":[221,28],"file_path":"Mathlib/Topology/Connected/TotallyDisconnected.lean","tactics":[{"state_before":"α✝ : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝² : TopologicalSpace α✝\ns t u v : Set α✝\nα : Type u_3\ninst✝¹ : TopologicalSpace α\ninst✝ : TotallySeparatedSpace α\nx y : α\nhxy : x ≠ y\n⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∈ Uᶜ","state_after":"case intro.intro.intro.intro.intro.intro.intro\nα✝ : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝² : TopologicalSpace α✝\ns t u v : Set α✝\nα : Type u_3\ninst✝¹ : TopologicalSpace α\ninst✝ : TotallySeparatedSpace α\nx y : α\nhxy : x ≠ y\nU V : Set α\nhU : IsOpen U\nhV : IsOpen V\nUx : x ∈ U\nVy : y ∈ V\nf : univ ⊆ U ∪ V\ndisj : Disjoint U V\n⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∈ Uᶜ","tactic":"obtain ⟨U, V, hU, hV, Ux, Vy, f, disj⟩ :=\n    TotallySeparatedSpace.isTotallySeparated_univ (Set.mem_univ x) (Set.mem_univ y) hxy","premises":[{"full_name":"Set.mem_univ","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[80,28],"def_end_pos":[80,36]},{"full_name":"TotallySeparatedSpace.isTotallySeparated_univ","def_path":"Mathlib/Topology/Connected/TotallyDisconnected.lean","def_pos":[200,2],"def_end_pos":[200,25]}]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro\nα✝ : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝² : TopologicalSpace α✝\ns t u v : Set α✝\nα : Type u_3\ninst✝¹ : TopologicalSpace α\ninst✝ : TotallySeparatedSpace α\nx y : α\nhxy : x ≠ y\nU V : Set α\nhU : IsOpen U\nhV : IsOpen V\nUx : x ∈ U\nVy : y ∈ V\nf : univ ⊆ U ∪ V\ndisj : Disjoint U V\n⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∈ Uᶜ","state_after":"case intro.intro.intro.intro.intro.intro.intro\nα✝ : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝² : TopologicalSpace α✝\ns t u v : Set α✝\nα : Type u_3\ninst✝¹ : TopologicalSpace α\ninst✝ : TotallySeparatedSpace α\nx y : α\nhxy : x ≠ y\nU V : Set α\nhU✝ : IsOpen U\nhV : IsOpen V\nUx : x ∈ U\nVy : y ∈ V\nf : univ ⊆ U ∪ V\ndisj : Disjoint U V\nhU : IsClopen (univ ∩ U)\n⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∈ Uᶜ","tactic":"have hU := isClopen_inter_of_disjoint_cover_clopen isClopen_univ f hU hV disj","premises":[{"full_name":"isClopen_inter_of_disjoint_cover_clopen","def_path":"Mathlib/Topology/Clopen.lean","def_pos":[96,8],"def_end_pos":[96,47]},{"full_name":"isClopen_univ","def_path":"Mathlib/Topology/Clopen.lean","def_pos":[44,8],"def_end_pos":[44,21]}]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro\nα✝ : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝² : TopologicalSpace α✝\ns t u v : Set α✝\nα : Type u_3\ninst✝¹ : TopologicalSpace α\ninst✝ : TotallySeparatedSpace α\nx y : α\nhxy : x ≠ y\nU V : Set α\nhU✝ : IsOpen U\nhV : IsOpen V\nUx : x ∈ U\nVy : y ∈ V\nf : univ ⊆ U ∪ V\ndisj : Disjoint U V\nhU : IsClopen (univ ∩ U)\n⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∈ Uᶜ","state_after":"case intro.intro.intro.intro.intro.intro.intro\nα✝ : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝² : TopologicalSpace α✝\ns t u v : Set α✝\nα : Type u_3\ninst✝¹ : TopologicalSpace α\ninst✝ : TotallySeparatedSpace α\nx y : α\nhxy : x ≠ y\nU V : Set α\nhU✝ : IsOpen U\nhV : IsOpen V\nUx : x ∈ U\nVy : y ∈ V\nf : univ ⊆ U ∪ V\ndisj : Disjoint U V\nhU : IsClopen U\n⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∈ Uᶜ","tactic":"rw [univ_inter _] at hU","premises":[{"full_name":"Set.univ_inter","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[803,8],"def_end_pos":[803,18]}]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro\nα✝ : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝² : TopologicalSpace α✝\ns t u v : Set α✝\nα : Type u_3\ninst✝¹ : TopologicalSpace α\ninst✝ : TotallySeparatedSpace α\nx y : α\nhxy : x ≠ y\nU V : Set α\nhU✝ : IsOpen U\nhV : IsOpen V\nUx : x ∈ U\nVy : y ∈ V\nf : univ ⊆ U ∪ V\ndisj : Disjoint U V\nhU : IsClopen U\n⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∈ Uᶜ","state_after":"case intro.intro.intro.intro.intro.intro.intro\nα✝ : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝² : TopologicalSpace α✝\ns t u v : Set α✝\nα : Type u_3\ninst✝¹ : TopologicalSpace α\ninst✝ : TotallySeparatedSpace α\nx y : α\nhxy : x ≠ y\nU V : Set α\nhU✝ : IsOpen U\nhV : IsOpen V\nUx : x ∈ U\nVy : y ∈ V\nf : univ ⊆ U ∪ V\ndisj : V ⊆ Uᶜ\nhU : IsClopen U\n⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∈ Uᶜ","tactic":"rw [← Set.subset_compl_iff_disjoint_right, subset_compl_comm] at disj","premises":[{"full_name":"Set.subset_compl_comm","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1372,8],"def_end_pos":[1372,25]},{"full_name":"Set.subset_compl_iff_disjoint_right","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1384,8],"def_end_pos":[1384,39]}]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro\nα✝ : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝² : TopologicalSpace α✝\ns t u v : Set α✝\nα : Type u_3\ninst✝¹ : TopologicalSpace α\ninst✝ : TotallySeparatedSpace α\nx y : α\nhxy : x ≠ y\nU V : Set α\nhU✝ : IsOpen U\nhV : IsOpen V\nUx : x ∈ U\nVy : y ∈ V\nf : univ ⊆ U ∪ V\ndisj : V ⊆ Uᶜ\nhU : IsClopen U\n⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∈ Uᶜ","state_after":"no goals","tactic":"exact ⟨U, hU, Ux, disj Vy⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]}]}
+{"url":"Mathlib/GroupTheory/PGroup.lean","commit":"","full_name":"IsPGroup.center_nontrivial","start":[216,0],"end":[224,54],"file_path":"Mathlib/GroupTheory/PGroup.lean","tactics":[{"state_before":"p : ℕ\nG : Type u_1\ninst✝⁴ : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\nα : Type u_2\ninst✝³ : MulAction G α\ninst✝² : Finite α\ninst✝¹ : Nontrivial G\ninst✝ : Finite G\n⊢ Nontrivial ↥(Subgroup.center G)","state_after":"no goals","tactic":"classical\n    have := (hG.of_equiv ConjAct.toConjAct).exists_fixed_point_of_prime_dvd_card_of_fixed_point G\n    rw [ConjAct.fixedPoints_eq_center] at this\n    have dvd : p ∣ Nat.card G := by\n      obtain ⟨n, hn0, hn⟩ := hG.nontrivial_iff_card.mp inferInstance\n      exact hn.symm ▸ dvd_pow_self _ (ne_of_gt hn0)\n    obtain ⟨g, hg⟩ := this dvd (Subgroup.center G).one_mem\n    exact ⟨⟨1, ⟨g, hg.1⟩, mt Subtype.ext_iff.mp hg.2⟩⟩","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"ConjAct.fixedPoints_eq_center","def_path":"Mathlib/GroupTheory/GroupAction/ConjAct.lean","def_pos":[271,8],"def_end_pos":[271,29]},{"full_name":"ConjAct.toConjAct","def_path":"Mathlib/GroupTheory/GroupAction/ConjAct.lean","def_pos":[76,4],"def_end_pos":[76,13]},{"full_name":"Dvd.dvd","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1344,2],"def_end_pos":[1344,5]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"IsPGroup.exists_fixed_point_of_prime_dvd_card_of_fixed_point","def_path":"Mathlib/GroupTheory/PGroup.lean","def_pos":[205,8],"def_end_pos":[205,59]},{"full_name":"IsPGroup.nontrivial_iff_card","def_path":"Mathlib/GroupTheory/PGroup.lean","def_pos":[129,8],"def_end_pos":[129,27]},{"full_name":"IsPGroup.of_equiv","def_path":"Mathlib/GroupTheory/PGroup.lean","def_pos":[77,8],"def_end_pos":[77,16]},{"full_name":"Nat.card","def_path":"Mathlib/SetTheory/Cardinal/Finite.lean","def_pos":[33,14],"def_end_pos":[33,18]},{"full_name":"Subgroup.center","def_path":"Mathlib/GroupTheory/Subgroup/Center.lean","def_pos":[29,4],"def_end_pos":[29,10]},{"full_name":"Subgroup.one_mem","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[457,18],"def_end_pos":[457,25]},{"full_name":"Subtype.ext_iff","def_path":"Mathlib/Data/Subtype.lean","def_pos":[62,18],"def_end_pos":[62,25]},{"full_name":"dvd_pow_self","def_path":"Mathlib/Algebra/Divisibility/Basic.lean","def_pos":[146,6],"def_end_pos":[146,18]},{"full_name":"inferInstance","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[99,7],"def_end_pos":[99,20]},{"full_name":"mt","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[647,8],"def_end_pos":[647,10]},{"full_name":"ne_of_gt","def_path":"Mathlib/Order/Defs.lean","def_pos":[85,8],"def_end_pos":[85,16]}]}]}
+{"url":"Mathlib/Topology/Constructions.lean","commit":"","full_name":"QuotientMap.restrictPreimage_isOpen","start":[1071,0],"end":[1078,40],"file_path":"Mathlib/Topology/Constructions.lean","tactics":[{"state_before":"X : Type u\nY : Type v\nZ : Type u_1\nW : Type u_2\nε : Type u_3\nζ : Type u_4\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\np : X → Prop\nf : X → Y\nhf : QuotientMap f\ns : Set Y\nhs : IsOpen s\n⊢ QuotientMap (s.restrictPreimage f)","state_after":"X : Type u\nY : Type v\nZ : Type u_1\nW : Type u_2\nε : Type u_3\nζ : Type u_4\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\np : X → Prop\nf : X → Y\nhf : QuotientMap f\ns : Set Y\nhs : IsOpen s\nU : Set ↑s\n⊢ IsOpen U ↔ IsOpen (s.restrictPreimage f ⁻¹' U)","tactic":"refine quotientMap_iff.2 ⟨hf.surjective.restrictPreimage _, fun U ↦ ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"QuotientMap.surjective","def_path":"Mathlib/Topology/Maps/Basic.lean","def_pos":[264,18],"def_end_pos":[264,28]},{"full_name":"quotientMap_iff","def_path":"Mathlib/Topology/Maps/Basic.lean","def_pos":[231,8],"def_end_pos":[231,23]}]},{"state_before":"X : Type u\nY : Type v\nZ : Type u_1\nW : Type u_2\nε : Type u_3\nζ : Type u_4\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\np : X → Prop\nf : X → Y\nhf : QuotientMap f\ns : Set Y\nhs : IsOpen s\nU : Set ↑s\n⊢ IsOpen U ↔ IsOpen (s.restrictPreimage f ⁻¹' U)","state_after":"no goals","tactic":"rw [hs.openEmbedding_subtype_val.open_iff_image_open, ← hf.isOpen_preimage,\n    (hs.preimage hf.continuous).openEmbedding_subtype_val.open_iff_image_open,\n    image_val_preimage_restrictPreimage]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"IsOpen.openEmbedding_subtype_val","def_path":"Mathlib/Topology/Constructions.lean","def_pos":[959,8],"def_end_pos":[959,40]},{"full_name":"IsOpen.preimage","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1327,8],"def_end_pos":[1327,23]},{"full_name":"OpenEmbedding.open_iff_image_open","def_path":"Mathlib/Topology/Maps/Basic.lean","def_pos":[478,8],"def_end_pos":[478,41]},{"full_name":"QuotientMap.continuous","def_path":"Mathlib/Topology/Maps/Basic.lean","def_pos":[261,18],"def_end_pos":[261,28]},{"full_name":"QuotientMap.isOpen_preimage","def_path":"Mathlib/Topology/Maps/Basic.lean","def_pos":[267,18],"def_end_pos":[267,33]},{"full_name":"Set.image_val_preimage_restrictPreimage","def_path":"Mathlib/Data/Set/Function.lean","def_pos":[494,8],"def_end_pos":[494,43]}]}]}
+{"url":"Mathlib/Order/Heyting/Basic.lean","commit":"","full_name":"le_compl_self","start":[684,0],"end":[685,48],"file_path":"Mathlib/Order/Heyting/Basic.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : HeytingAlgebra α\na b c : α\n⊢ a ≤ aᶜ ↔ a = ⊥","state_after":"no goals","tactic":"rw [le_compl_iff_disjoint_left, disjoint_self]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"disjoint_self","def_path":"Mathlib/Order/Disjoint.lean","def_pos":[73,8],"def_end_pos":[73,21]},{"full_name":"le_compl_iff_disjoint_left","def_path":"Mathlib/Order/Heyting/Basic.lean","def_pos":[628,8],"def_end_pos":[628,34]}]}]}
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+{"url":"Mathlib/AlgebraicGeometry/Morphisms/Constructors.lean","commit":"","full_name":"AlgebraicGeometry.HasAffineProperty.diagonal_of_openCover","start":[63,0],"end":[84,48],"file_path":"Mathlib/AlgebraicGeometry/Morphisms/Constructors.lean","tactics":[{"state_before":"P : MorphismProperty Scheme\nQ : AffineTargetMorphismProperty\ninst✝² : HasAffineProperty P Q\nX Y : Scheme\nf : X ⟶ Y\n𝒰 : Y.OpenCover\ninst✝¹ : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)\n𝒰' : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover\ninst✝ : ∀ (i : 𝒰.J) (j : (𝒰' i).J), IsAffine ((𝒰' i).obj j)\nh𝒰' : ∀ (i : 𝒰.J) (j k : (𝒰' i).J), Q (pullback.mapDesc ((𝒰' i).map j) ((𝒰' i).map k) (𝒰.pullbackHom f i))\n⊢ P.diagonal f","state_after":"P : MorphismProperty Scheme\nQ : AffineTargetMorphismProperty\ninst✝² : HasAffineProperty P Q\nX Y : Scheme\nf : X ⟶ Y\n𝒰 : Y.OpenCover\ninst✝¹ : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)\n𝒰' : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover\ninst✝ : ∀ (i : 𝒰.J) (j : (𝒰' i).J), IsAffine ((𝒰' i).obj j)\nh𝒰' : ∀ (i : 𝒰.J) (j k : (𝒰' i).J), Q (pullback.mapDesc ((𝒰' i).map j) ((𝒰' i).map k) (𝒰.pullbackHom f i))\nthis : Q.IsLocal := isLocal_affineProperty P\n⊢ P.diagonal f","tactic":"letI := isLocal_affineProperty P","premises":[{"full_name":"AlgebraicGeometry.HasAffineProperty.isLocal_affineProperty","def_path":"Mathlib/AlgebraicGeometry/Morphisms/Basic.lean","def_pos":[421,2],"def_end_pos":[421,24]}]},{"state_before":"P : MorphismProperty Scheme\nQ : AffineTargetMorphismProperty\ninst✝² : HasAffineProperty P Q\nX Y : Scheme\nf : X ⟶ Y\n𝒰 : Y.OpenCover\ninst✝¹ : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)\n𝒰' : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover\ninst✝ : ∀ (i : 𝒰.J) (j : (𝒰' i).J), IsAffine ((𝒰' i).obj j)\nh𝒰' : ∀ (i : 𝒰.J) (j k : (𝒰' i).J), Q (pullback.mapDesc ((𝒰' i).map j) ((𝒰' i).map k) (𝒰.pullbackHom f i))\nthis : Q.IsLocal := isLocal_affineProperty P\n⊢ P.diagonal f","state_after":"P : MorphismProperty Scheme\nQ : AffineTargetMorphismProperty\ninst✝² : HasAffineProperty P Q\nX Y : Scheme\nf : X ⟶ Y\n𝒰 : Y.OpenCover\ninst✝¹ : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)\n𝒰' : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover\ninst✝ : ∀ (i : 𝒰.J) (j : (𝒰' i).J), IsAffine ((𝒰' i).obj j)\nh𝒰' : ∀ (i : 𝒰.J) (j k : (𝒰' i).J), Q (pullback.mapDesc ((𝒰' i).map j) ((𝒰' i).map k) (𝒰.pullbackHom f i))\nthis : Q.IsLocal := isLocal_affineProperty P\n𝒱 : (pullback f f).OpenCover :=\n  (Scheme.Pullback.openCoverOfBase 𝒰 f f).bind fun i =>\n    Scheme.Pullback.openCoverOfLeftRight (𝒰' i) (𝒰' i) (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i))\n⊢ P.diagonal f","tactic":"let 𝒱 := (Scheme.Pullback.openCoverOfBase 𝒰 f f).bind fun i =>\n    Scheme.Pullback.openCoverOfLeftRight.{u} (𝒰' i) (𝒰' i) (pullback.snd _ _) (pullback.snd _ _)","premises":[{"full_name":"AlgebraicGeometry.Scheme.OpenCover.bind","def_path":"Mathlib/AlgebraicGeometry/Cover/Open.lean","def_pos":[97,4],"def_end_pos":[97,18]},{"full_name":"AlgebraicGeometry.Scheme.Pullback.openCoverOfBase","def_path":"Mathlib/AlgebraicGeometry/Pullbacks.lean","def_pos":[535,4],"def_end_pos":[535,19]},{"full_name":"AlgebraicGeometry.Scheme.Pullback.openCoverOfLeftRight","def_path":"Mathlib/AlgebraicGeometry/Pullbacks.lean","def_pos":[498,4],"def_end_pos":[498,24]},{"full_name":"CategoryTheory.Limits.pullback.snd","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Pullback/HasPullback.lean","def_pos":[112,7],"def_end_pos":[112,19]}]},{"state_before":"P : MorphismProperty Scheme\nQ : AffineTargetMorphismProperty\ninst✝² : HasAffineProperty P Q\nX Y : Scheme\nf : X ⟶ Y\n𝒰 : Y.OpenCover\ninst✝¹ : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)\n𝒰' : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover\ninst✝ : ∀ (i : 𝒰.J) (j : (𝒰' i).J), IsAffine ((𝒰' i).obj j)\nh𝒰' : ∀ (i : 𝒰.J) (j k : (𝒰' i).J), Q (pullback.mapDesc ((𝒰' i).map j) ((𝒰' i).map k) (𝒰.pullbackHom f i))\nthis : Q.IsLocal := isLocal_affineProperty P\n𝒱 : (pullback f f).OpenCover :=\n  (Scheme.Pullback.openCoverOfBase 𝒰 f f).bind fun i =>\n    Scheme.Pullback.openCoverOfLeftRight (𝒰' i) (𝒰' i) (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i))\n⊢ P.diagonal f","state_after":"P : MorphismProperty Scheme\nQ : AffineTargetMorphismProperty\ninst✝² : HasAffineProperty P Q\nX Y : Scheme\nf : X ⟶ Y\n𝒰 : Y.OpenCover\ninst✝¹ : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)\n𝒰' : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover\ninst✝ : ∀ (i : 𝒰.J) (j : (𝒰' i).J), IsAffine ((𝒰' i).obj j)\nh𝒰' : ∀ (i : 𝒰.J) (j k : (𝒰' i).J), Q (pullback.mapDesc ((𝒰' i).map j) ((𝒰' i).map k) (𝒰.pullbackHom f i))\nthis : Q.IsLocal := isLocal_affineProperty P\n𝒱 : (pullback f f).OpenCover :=\n  (Scheme.Pullback.openCoverOfBase 𝒰 f f).bind fun i =>\n    Scheme.Pullback.openCoverOfLeftRight (𝒰' i) (𝒰' i) (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i))\ni1 : ∀ (i : 𝒱.J), IsAffine (𝒱.obj i)\n⊢ P.diagonal f","tactic":"have i1 : ∀ i, IsAffine (𝒱.obj i) := fun i => by dsimp [𝒱]; infer_instance","premises":[{"full_name":"AlgebraicGeometry.IsAffine","def_path":"Mathlib/AlgebraicGeometry/AffineScheme.lean","def_pos":[54,6],"def_end_pos":[54,14]},{"full_name":"AlgebraicGeometry.Scheme.OpenCover.obj","def_path":"Mathlib/AlgebraicGeometry/Cover/Open.lean","def_pos":[46,2],"def_end_pos":[46,5]},{"full_name":"inferInstance","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[99,7],"def_end_pos":[99,20]}]},{"state_before":"P : MorphismProperty Scheme\nQ : AffineTargetMorphismProperty\ninst✝² : HasAffineProperty P Q\nX Y : Scheme\nf : X ⟶ Y\n𝒰 : Y.OpenCover\ninst✝¹ : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)\n𝒰' : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover\ninst✝ : ∀ (i : 𝒰.J) (j : (𝒰' i).J), IsAffine ((𝒰' i).obj j)\nh𝒰' : ∀ (i : 𝒰.J) (j k : (𝒰' i).J), Q (pullback.mapDesc ((𝒰' i).map j) ((𝒰' i).map k) (𝒰.pullbackHom f i))\nthis : Q.IsLocal := isLocal_affineProperty P\n𝒱 : (pullback f f).OpenCover :=\n  (Scheme.Pullback.openCoverOfBase 𝒰 f f).bind fun i =>\n    Scheme.Pullback.openCoverOfLeftRight (𝒰' i) (𝒰' i) (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i))\ni1 : ∀ (i : 𝒱.J), IsAffine (𝒱.obj i)\n⊢ P.diagonal f","state_after":"P : MorphismProperty Scheme\nQ : AffineTargetMorphismProperty\ninst✝² : HasAffineProperty P Q\nX Y : Scheme\nf : X ⟶ Y\n𝒰 : Y.OpenCover\ninst✝¹ : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)\n𝒰' : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover\ninst✝ : ∀ (i : 𝒰.J) (j : (𝒰' i).J), IsAffine ((𝒰' i).obj j)\nh𝒰' : ∀ (i : 𝒰.J) (j k : (𝒰' i).J), Q (pullback.mapDesc ((𝒰' i).map j) ((𝒰' i).map k) (𝒰.pullbackHom f i))\nthis : Q.IsLocal := isLocal_affineProperty P\n𝒱 : (pullback f f).OpenCover :=\n  (Scheme.Pullback.openCoverOfBase 𝒰 f f).bind fun i =>\n    Scheme.Pullback.openCoverOfLeftRight (𝒰' i) (𝒰' i) (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i))\ni1 : ∀ (i : 𝒱.J), IsAffine (𝒱.obj i)\n⊢ ∀ (i : (𝒱.pullbackCover (pullback.diagonal f)).J), Q (𝒱.pullbackHom (pullback.diagonal f) i)","tactic":"apply of_openCover 𝒱","premises":[{"full_name":"AlgebraicGeometry.HasAffineProperty.of_openCover","def_path":"Mathlib/AlgebraicGeometry/Morphisms/Basic.lean","def_pos":[499,8],"def_end_pos":[499,20]}]},{"state_before":"P : MorphismProperty Scheme\nQ : AffineTargetMorphismProperty\ninst✝² : HasAffineProperty P Q\nX Y : Scheme\nf : X ⟶ Y\n𝒰 : Y.OpenCover\ninst✝¹ : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)\n𝒰' : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover\ninst✝ : ∀ (i : 𝒰.J) (j : (𝒰' i).J), IsAffine ((𝒰' i).obj j)\nh𝒰' : ∀ (i : 𝒰.J) (j k : (𝒰' i).J), Q (pullback.mapDesc ((𝒰' i).map j) ((𝒰' i).map k) (𝒰.pullbackHom f i))\nthis : Q.IsLocal := isLocal_affineProperty P\n𝒱 : (pullback f f).OpenCover :=\n  (Scheme.Pullback.openCoverOfBase 𝒰 f f).bind fun i =>\n    Scheme.Pullback.openCoverOfLeftRight (𝒰' i) (𝒰' i) (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i))\ni1 : ∀ (i : 𝒱.J), IsAffine (𝒱.obj i)\n⊢ ∀ (i : (𝒱.pullbackCover (pullback.diagonal f)).J), Q (𝒱.pullbackHom (pullback.diagonal f) i)","state_after":"case mk.mk\nP : MorphismProperty Scheme\nQ : AffineTargetMorphismProperty\ninst✝² : HasAffineProperty P Q\nX Y : Scheme\nf : X ⟶ Y\n𝒰 : Y.OpenCover\ninst✝¹ : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)\n𝒰' : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover\ninst✝ : ∀ (i : 𝒰.J) (j : (𝒰' i).J), IsAffine ((𝒰' i).obj j)\nh𝒰' : ∀ (i : 𝒰.J) (j k : (𝒰' i).J), Q (pullback.mapDesc ((𝒰' i).map j) ((𝒰' i).map k) (𝒰.pullbackHom f i))\nthis : Q.IsLocal := isLocal_affineProperty P\n𝒱 : (pullback f f).OpenCover :=\n  (Scheme.Pullback.openCoverOfBase 𝒰 f f).bind fun i =>\n    Scheme.Pullback.openCoverOfLeftRight (𝒰' i) (𝒰' i) (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i))\ni1 : ∀ (i : 𝒱.J), IsAffine (𝒱.obj i)\ni : (Scheme.Pullback.openCoverOfBase 𝒰 f f).J\nj k : (𝒰' i).J\n⊢ Q (𝒱.pullbackHom (pullback.diagonal f) ⟨i, (j, k)⟩)","tactic":"rintro ⟨i, j, k⟩","premises":[]},{"state_before":"case mk.mk\nP : MorphismProperty Scheme\nQ : AffineTargetMorphismProperty\ninst✝² : HasAffineProperty P Q\nX Y : Scheme\nf : X ⟶ Y\n𝒰 : Y.OpenCover\ninst✝¹ : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)\n𝒰' : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover\ninst✝ : ∀ (i : 𝒰.J) (j : (𝒰' i).J), IsAffine ((𝒰' i).obj j)\nh𝒰' : ∀ (i : 𝒰.J) (j k : (𝒰' i).J), Q (pullback.mapDesc ((𝒰' i).map j) ((𝒰' i).map k) (𝒰.pullbackHom f i))\nthis : Q.IsLocal := isLocal_affineProperty P\n𝒱 : (pullback f f).OpenCover :=\n  (Scheme.Pullback.openCoverOfBase 𝒰 f f).bind fun i =>\n    Scheme.Pullback.openCoverOfLeftRight (𝒰' i) (𝒰' i) (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i))\ni1 : ∀ (i : 𝒱.J), IsAffine (𝒱.obj i)\ni : (Scheme.Pullback.openCoverOfBase 𝒰 f f).J\nj k : (𝒰' i).J\n⊢ Q (𝒱.pullbackHom (pullback.diagonal f) ⟨i, (j, k)⟩)","state_after":"case mk.mk\nP : MorphismProperty Scheme\nQ : AffineTargetMorphismProperty\ninst✝² : HasAffineProperty P Q\nX Y : Scheme\nf : X ⟶ Y\n𝒰 : Y.OpenCover\ninst✝¹ : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)\n𝒰' : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover\ninst✝ : ∀ (i : 𝒰.J) (j : (𝒰' i).J), IsAffine ((𝒰' i).obj j)\nh𝒰' : ∀ (i : 𝒰.J) (j k : (𝒰' i).J), Q (pullback.mapDesc ((𝒰' i).map j) ((𝒰' i).map k) (𝒰.pullbackHom f i))\nthis : Q.IsLocal := isLocal_affineProperty P\n𝒱 : (pullback f f).OpenCover :=\n  (Scheme.Pullback.openCoverOfBase 𝒰 f f).bind fun i =>\n    Scheme.Pullback.openCoverOfLeftRight (𝒰' i) (𝒰' i) (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i))\ni1 : ∀ (i : 𝒱.J), IsAffine (𝒱.obj i)\ni : (Scheme.Pullback.openCoverOfBase 𝒰 f f).J\nj k : (𝒰' i).J\n⊢ Q\n    (((Scheme.Pullback.openCoverOfBase 𝒰 f f).bind fun i =>\n          Scheme.Pullback.openCoverOfLeftRight (𝒰' i) (𝒰' i) (pullback.snd f (𝒰.map i))\n            (pullback.snd f (𝒰.map i))).pullbackHom\n      (pullback.diagonal f) ⟨i, (j, k)⟩)","tactic":"dsimp [𝒱]","premises":[]},{"state_before":"case mk.mk\nP : MorphismProperty Scheme\nQ : AffineTargetMorphismProperty\ninst✝² : HasAffineProperty P Q\nX Y : Scheme\nf : X ⟶ Y\n𝒰 : Y.OpenCover\ninst✝¹ : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)\n𝒰' : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover\ninst✝ : ∀ (i : 𝒰.J) (j : (𝒰' i).J), IsAffine ((𝒰' i).obj j)\nh𝒰' : ∀ (i : 𝒰.J) (j k : (𝒰' i).J), Q (pullback.mapDesc ((𝒰' i).map j) ((𝒰' i).map k) (𝒰.pullbackHom f i))\nthis : Q.IsLocal := isLocal_affineProperty P\n𝒱 : (pullback f f).OpenCover :=\n  (Scheme.Pullback.openCoverOfBase 𝒰 f f).bind fun i =>\n    Scheme.Pullback.openCoverOfLeftRight (𝒰' i) (𝒰' i) (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i))\ni1 : ∀ (i : 𝒱.J), IsAffine (𝒱.obj i)\ni : (Scheme.Pullback.openCoverOfBase 𝒰 f f).J\nj k : (𝒰' i).J\n⊢ Q\n    (((Scheme.Pullback.openCoverOfBase 𝒰 f f).bind fun i =>\n          Scheme.Pullback.openCoverOfLeftRight (𝒰' i) (𝒰' i) (pullback.snd f (𝒰.map i))\n            (pullback.snd f (𝒰.map i))).pullbackHom\n      (pullback.diagonal f) ⟨i, (j, k)⟩)","state_after":"case mk.mk.convert_1\nP : MorphismProperty Scheme\nQ : AffineTargetMorphismProperty\ninst✝² : HasAffineProperty P Q\nX Y : Scheme\nf : X ⟶ Y\n𝒰 : Y.OpenCover\ninst✝¹ : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)\n𝒰' : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover\ninst✝ : ∀ (i : 𝒰.J) (j : (𝒰' i).J), IsAffine ((𝒰' i).obj j)\nh𝒰' : ∀ (i : 𝒰.J) (j k : (𝒰' i).J), Q (pullback.mapDesc ((𝒰' i).map j) ((𝒰' i).map k) (𝒰.pullbackHom f i))\nthis : Q.IsLocal := isLocal_affineProperty P\n𝒱 : (pullback f f).OpenCover :=\n  (Scheme.Pullback.openCoverOfBase 𝒰 f f).bind fun i =>\n    Scheme.Pullback.openCoverOfLeftRight (𝒰' i) (𝒰' i) (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i))\ni1 : ∀ (i : 𝒱.J), IsAffine (𝒱.obj i)\ni : (Scheme.Pullback.openCoverOfBase 𝒰 f f).J\nj k : (𝒰' i).J\n⊢ pullback.diagonal f ≫ 𝟙 (pullback.diagonalObj f) = 𝟙 X ≫ pullback.diagonal f\n\ncase mk.mk.convert_2\nP : MorphismProperty Scheme\nQ : AffineTargetMorphismProperty\ninst✝² : HasAffineProperty P Q\nX Y : Scheme\nf : X ⟶ Y\n𝒰 : Y.OpenCover\ninst✝¹ : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)\n𝒰' : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover\ninst✝ : ∀ (i : 𝒰.J) (j : (𝒰' i).J), IsAffine ((𝒰' i).obj j)\nh𝒰' : ∀ (i : 𝒰.J) (j k : (𝒰' i).J), Q (pullback.mapDesc ((𝒰' i).map j) ((𝒰' i).map k) (𝒰.pullbackHom f i))\nthis : Q.IsLocal := isLocal_affineProperty P\n𝒱 : (pullback f f).OpenCover :=\n  (Scheme.Pullback.openCoverOfBase 𝒰 f f).bind fun i =>\n    Scheme.Pullback.openCoverOfLeftRight (𝒰' i) (𝒰' i) (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i))\ni1 : ∀ (i : 𝒱.J), IsAffine (𝒱.obj i)\ni : (Scheme.Pullback.openCoverOfBase 𝒰 f f).J\nj k : (𝒰' i).J\n⊢ pullback.map ((𝒰' i).map j ≫ pullback.snd f (𝒰.map i)) ((𝒰' i).map k ≫ pullback.snd f (𝒰.map i)) f f\n        ((𝒰' i).map j ≫ pullback.fst f (𝒰.map i)) ((𝒰' i).map k ≫ pullback.fst f (𝒰.map i)) (𝒰.map i) ⋯ ⋯ ≫\n      𝟙 (pullback.diagonalObj f) =\n    𝟙 (pullback ((𝒰' i).map j ≫ pullback.snd f (𝒰.map i)) ((𝒰' i).map k ≫ pullback.snd f (𝒰.map i))) ≫\n      pullback.map ((𝒰' i).map j ≫ pullback.snd f (𝒰.map i)) ((𝒰' i).map k ≫ pullback.snd f (𝒰.map i))\n          (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i)) ((𝒰' i).map j) ((𝒰' i).map k) (𝟙 (𝒰.obj i)) ⋯ ⋯ ≫\n        pullback.map (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i)) f f (pullback.fst f (𝒰.map i))\n          (pullback.fst f (𝒰.map i)) (𝒰.map i) ⋯ ⋯\n\ncase mk.mk.convert_6\nP : MorphismProperty Scheme\nQ : AffineTargetMorphismProperty\ninst✝² : HasAffineProperty P Q\nX Y : Scheme\nf : X ⟶ Y\n𝒰 : Y.OpenCover\ninst✝¹ : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)\n𝒰' : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover\ninst✝ : ∀ (i : 𝒰.J) (j : (𝒰' i).J), IsAffine ((𝒰' i).obj j)\nh𝒰' : ∀ (i : 𝒰.J) (j k : (𝒰' i).J), Q (pullback.mapDesc ((𝒰' i).map j) ((𝒰' i).map k) (𝒰.pullbackHom f i))\nthis : Q.IsLocal := isLocal_affineProperty P\n𝒱 : (pullback f f).OpenCover :=\n  (Scheme.Pullback.openCoverOfBase 𝒰 f f).bind fun i =>\n    Scheme.Pullback.openCoverOfLeftRight (𝒰' i) (𝒰' i) (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i))\ni1 : ∀ (i : 𝒱.J), IsAffine (𝒱.obj i)\ni : (Scheme.Pullback.openCoverOfBase 𝒰 f f).J\nj k : (𝒰' i).J\n⊢ Q\n    (((pullbackDiagonalMapIso f (𝒰.map i) ((𝒰' i).map j) ((𝒰' i).map k)).inv ≫\n        pullback.map (pullback.diagonal f)\n          (pullback.map ((𝒰' i).map j ≫ pullback.snd f (𝒰.map i)) ((𝒰' i).map k ≫ pullback.snd f (𝒰.map i)) f f\n            ((𝒰' i).map j ≫ pullback.fst f (𝒰.map i)) ((𝒰' i).map k ≫ pullback.fst f (𝒰.map i)) (𝒰.map i) ⋯ ⋯)\n          (pullback.diagonal f)\n          (pullback.map ((𝒰' i).map j ≫ pullback.snd f (𝒰.map i)) ((𝒰' i).map k ≫ pullback.snd f (𝒰.map i))\n              (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i)) ((𝒰' i).map j) ((𝒰' i).map k) (𝟙 (𝒰.obj i)) ⋯ ⋯ ≫\n            pullback.map (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i)) f f (pullback.fst f (𝒰.map i))\n              (pullback.fst f (𝒰.map i)) (𝒰.map i) ⋯ ⋯)\n          (𝟙 X) (𝟙 (pullback ((𝒰' i).map j ≫ pullback.snd f (𝒰.map i)) ((𝒰' i).map k ≫ pullback.snd f (𝒰.map i))))\n          (𝟙 (pullback.diagonalObj f)) ?mk.mk.convert_1 ?mk.mk.convert_2) ≫\n      pullback.snd (pullback.diagonal f)\n        (pullback.map ((𝒰' i).map j ≫ pullback.snd f (𝒰.map i)) ((𝒰' i).map k ≫ pullback.snd f (𝒰.map i))\n            (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i)) ((𝒰' i).map j) ((𝒰' i).map k) (𝟙 (𝒰.obj i)) ⋯ ⋯ ≫\n          pullback.map (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i)) f f (pullback.fst f (𝒰.map i))\n            (pullback.fst f (𝒰.map i)) (𝒰.map i) ⋯ ⋯))","tactic":"convert (Q.cancel_left_of_respectsIso\n    ((pullbackDiagonalMapIso _ _ ((𝒰' i).map j) ((𝒰' i).map k)).inv ≫\n      pullback.map _ _ _ _ (𝟙 _) (𝟙 _) (𝟙 _) _ _) (pullback.snd _ _)).mp _ using 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+{"url":"Mathlib/GroupTheory/Archimedean.lean","commit":"","full_name":"AddSubgroup.exists_isLeast_pos","start":[53,0],"end":[83,40],"file_path":"Mathlib/GroupTheory/Archimedean.lean","tactics":[{"state_before":"G : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup G\ninst✝ : Archimedean G\nH : AddSubgroup G\nhbot : H ≠ ⊥\na : G\nh₀ : 0 < a\nhd : Disjoint (↑H) (Ioo 0 a)\nhex : ∀ g > 0, ∃ n, g ∈ Ioc (n • a) ((n + 1) • a)\n⊢ ∃ b, IsLeast {g | g ∈ H ∧ 0 < g} b","state_after":"G : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup G\ninst✝ : Archimedean G\nH : AddSubgroup G\nhbot : H ≠ ⊥\na : G\nh₀ : 0 < a\nhd : Disjoint (↑H) (Ioo 0 a)\nhex : ∀ g > 0, ∃ n, g ∈ Ioc (n • a) ((n + 1) • a)\nthis : ∃ n, (↑H ∩ Ioc (n • a) ((n + 1) • a)).Nonempty\n⊢ ∃ b, IsLeast {g | g ∈ H ∧ 0 < g} b","tactic":"have : ∃ n : ℕ, Set.Nonempty (H ∩ Ioc (n • a) ((n + 1) • a)) := by\n    rcases (bot_or_exists_ne_zero H).resolve_left hbot with ⟨g, hgH, hg₀⟩\n    rcases hex |g| (abs_pos.2 hg₀) with ⟨n, hn⟩\n    exact ⟨n, _, (@abs_mem_iff (AddSubgroup G) G _ _).2 hgH, hn⟩","premises":[{"full_name":"AddSubgroup","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[299,10],"def_end_pos":[299,21]},{"full_name":"AddSubgroup.bot_or_exists_ne_zero","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[728,2],"def_end_pos":[728,13]},{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Inter.inter","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[407,2],"def_end_pos":[407,7]},{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]},{"full_name":"Or.resolve_left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[555,8],"def_end_pos":[555,23]},{"full_name":"Set.Ioc","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[62,4],"def_end_pos":[62,7]},{"full_name":"Set.Nonempty","def_path":"Mathlib/Init/Set.lean","def_pos":[222,14],"def_end_pos":[222,22]},{"full_name":"abs","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[33,2],"def_end_pos":[33,13]},{"full_name":"abs_mem_iff","def_path":"Mathlib/Algebra/Group/Subgroup/Order.lean","def_pos":[17,16],"def_end_pos":[17,27]},{"full_name":"abs_pos","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[208,29],"def_end_pos":[208,36]}]},{"state_before":"G : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup G\ninst✝ : Archimedean G\nH : AddSubgroup G\nhbot : H ≠ ⊥\na : G\nh₀ : 0 < a\nhd : Disjoint (↑H) (Ioo 0 a)\nhex : ∀ g > 0, ∃ n, g ∈ Ioc (n • a) ((n + 1) • a)\nthis : ∃ n, (↑H ∩ Ioc (n • a) ((n + 1) • a)).Nonempty\n⊢ ∃ b, IsLeast {g | g ∈ H ∧ 0 < g} b","state_after":"case mk.intro.intro.intro.intro\nG : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup G\ninst✝ : Archimedean G\nH : AddSubgroup G\nhbot : H ≠ ⊥\na : G\nh₀ : 0 < a\nhd : Disjoint (↑H) (Ioo 0 a)\nhex : ∀ g > 0, ∃ n, g ∈ Ioc (n • a) ((n + 1) • a)\nthis : ∃ n, (↑H ∩ Ioc (n • a) ((n + 1) • a)).Nonempty\nn : ℕ\nhmin : ∀ m < n, ¬(↑H ∩ Ioc (m • a) ((m + 1) • a)).Nonempty\nx : G\nhxH : x ∈ ↑H\nhnx : n • a < x\nhxn : x ≤ (n + 1) • a\n⊢ ∃ b, IsLeast {g | g ∈ H ∧ 0 < g} b","tactic":"classical rcases Nat.findX this with ⟨n, ⟨x, hxH, hnx, hxn⟩, hmin⟩","premises":[{"full_name":"Nat.findX","def_path":"Mathlib/Data/Nat/Find.lean","def_pos":[41,14],"def_end_pos":[41,19]}]},{"state_before":"case mk.intro.intro.intro.intro\nG : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup G\ninst✝ : Archimedean G\nH : AddSubgroup G\nhbot : H ≠ ⊥\na : G\nh₀ : 0 < a\nhd : Disjoint (↑H) (Ioo 0 a)\nhex : ∀ g > 0, ∃ n, g ∈ Ioc (n • a) ((n + 1) • a)\nthis : ∃ n, (↑H ∩ Ioc (n • a) ((n + 1) • a)).Nonempty\nn : ℕ\nhmin : ∀ m < n, ¬(↑H ∩ Ioc (m • a) ((m + 1) • a)).Nonempty\nx : G\nhxH : x ∈ ↑H\nhnx : n • a < x\nhxn : x ≤ (n + 1) • a\n⊢ ∃ b, IsLeast {g | g ∈ H ∧ 0 < g} b","state_after":"case mk.intro.intro.intro.intro\nG : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup G\ninst✝ : Archimedean G\nH : AddSubgroup G\nhbot : H ≠ ⊥\na : G\nh₀ : 0 < a\nhd : Disjoint (↑H) (Ioo 0 a)\nhex : ∀ g > 0, ∃ n, g ∈ Ioc (n • a) ((n + 1) • a)\nthis : ∃ n, (↑H ∩ Ioc (n • a) ((n + 1) • a)).Nonempty\nn : ℕ\nhmin : ∀ m < n, ¬(↑H ∩ Ioc (m • a) ((m + 1) • a)).Nonempty\nx : G\nhxH : x ∈ ↑H\nhnx : n • a < x\nhxn : x ≤ (n + 1) • a\nhxmin : ¬∃ b, IsLeast {g | g ∈ H ∧ 0 < g} b\n⊢ False","tactic":"by_contra hxmin","premises":[{"full_name":"Classical.byContradiction","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[129,8],"def_end_pos":[129,23]},{"full_name":"Not","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[224,4],"def_end_pos":[224,7]}]},{"state_before":"case mk.intro.intro.intro.intro\nG : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup G\ninst✝ : Archimedean G\nH : AddSubgroup G\nhbot : H ≠ ⊥\na : G\nh₀ : 0 < a\nhd : Disjoint (↑H) (Ioo 0 a)\nhex : ∀ g > 0, ∃ n, g ∈ Ioc (n • a) ((n + 1) • a)\nthis : ∃ n, (↑H ∩ Ioc (n • a) ((n + 1) • a)).Nonempty\nn : ℕ\nhmin : ∀ m < n, ¬(↑H ∩ Ioc (m • a) ((m + 1) • a)).Nonempty\nx : G\nhxH : x ∈ ↑H\nhnx : n • a < x\nhxn : x ≤ (n + 1) • a\nhxmin : ¬∃ b, IsLeast {g | g ∈ H ∧ 0 < g} b\n⊢ False","state_after":"case mk.intro.intro.intro.intro\nG : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup G\ninst✝ : Archimedean G\nH : AddSubgroup G\nhbot : H ≠ ⊥\na : G\nh₀ : 0 < a\nhd : Disjoint (↑H) (Ioo 0 a)\nhex : ∀ g > 0, ∃ n, g ∈ Ioc (n • a) ((n + 1) • a)\nthis : ∃ n, (↑H ∩ Ioc (n • a) ((n + 1) • a)).Nonempty\nn : ℕ\nhmin : ∀ m < n, ¬(↑H ∩ Ioc (m • a) ((m + 1) • a)).Nonempty\nx : G\nhxH : x ∈ ↑H\nhnx : n • a < x\nhxn : x ≤ (n + 1) • a\nhxmin : ∀ (x : G), x ∈ H ∧ 0 < x → ∃ x_1, ∃ (_ : x_1 ∈ H ∧ 0 < x_1), x_1 < x\n⊢ False","tactic":"simp only [IsLeast, not_and, mem_setOf_eq, mem_lowerBounds, not_exists, not_forall,\n    not_le] at hxmin","premises":[{"full_name":"Classical.not_forall","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[137,20],"def_end_pos":[137,30]},{"full_name":"IsLeast","def_path":"Mathlib/Order/Bounds/Basic.lean","def_pos":[60,4],"def_end_pos":[60,11]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]},{"full_name":"mem_lowerBounds","def_path":"Mathlib/Order/Bounds/Basic.lean","def_pos":[78,8],"def_end_pos":[78,23]},{"full_name":"not_and","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[116,16],"def_end_pos":[116,23]},{"full_name":"not_exists","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[254,16],"def_end_pos":[254,26]},{"full_name":"not_le","def_path":"Mathlib/Order/Defs.lean","def_pos":[316,8],"def_end_pos":[316,14]}]},{"state_before":"case mk.intro.intro.intro.intro\nG : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup G\ninst✝ : Archimedean G\nH : AddSubgroup G\nhbot : H ≠ ⊥\na : G\nh₀ : 0 < a\nhd : Disjoint (↑H) (Ioo 0 a)\nhex : ∀ g > 0, ∃ n, g ∈ Ioc (n • a) ((n + 1) • a)\nthis : ∃ n, (↑H ∩ Ioc (n • a) ((n + 1) • a)).Nonempty\nn : ℕ\nhmin : ∀ m < n, ¬(↑H ∩ Ioc (m • a) ((m + 1) • a)).Nonempty\nx : G\nhxH : x ∈ ↑H\nhnx : n • a < x\nhxn : x ≤ (n + 1) • a\nhxmin : ∀ (x : G), x ∈ H ��� 0 < x → ∃ x_1, ∃ (_ : x_1 ∈ H ∧ 0 < x_1), x_1 < x\n⊢ False","state_after":"case mk.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup G\ninst✝ : Archimedean G\nH : AddSubgroup G\nhbot : H ≠ ⊥\na : G\nh₀ : 0 < a\nhd : Disjoint (↑H) (Ioo 0 a)\nhex : ∀ g > 0, ∃ n, g ∈ Ioc (n • a) ((n + 1) • a)\nthis : ∃ n, (↑H ∩ Ioc (n • a) ((n + 1) • a)).Nonempty\nn : ℕ\nhmin : ∀ m < n, ¬(↑H ∩ Ioc (m • a) ((m + 1) • a)).Nonempty\nx : G\nhxH : x ∈ ↑H\nhnx : n • a < x\nhxn : x ≤ (n + 1) • a\nhxmin : ∀ (x : G), x ∈ H ∧ 0 < x → ∃ x_1, ∃ (_ : x_1 ∈ H ∧ 0 < x_1), x_1 < x\ny : G\nhxy : y < x\nhyH : y ∈ H\nhy₀ : 0 < y\n⊢ False","tactic":"rcases hxmin x ⟨hxH, (nsmul_nonneg h₀.le _).trans_lt hnx⟩ with ⟨y, ⟨hyH, hy₀⟩, hxy⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"nsmul_nonneg","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Pow.lean","def_pos":[39,14],"def_end_pos":[39,26]}]},{"state_before":"case mk.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup G\ninst✝ : Archimedean G\nH : AddSubgroup G\nhbot : H ≠ ⊥\na : G\nh₀ : 0 < a\nhd : Disjoint (↑H) (Ioo 0 a)\nhex : ∀ g > 0, ∃ n, g ∈ Ioc (n • a) ((n + 1) • a)\nthis : ∃ n, (↑H ∩ Ioc (n • a) ((n + 1) • a)).Nonempty\nn : ℕ\nhmin : ∀ m < n, ¬(↑H ∩ Ioc (m • a) ((m + 1) • a)).Nonempty\nx : G\nhxH : x ∈ ↑H\nhnx : n • a < x\nhxn : x ≤ (n + 1) • a\nhxmin : ∀ (x : G), x ∈ H ∧ 0 < x → ∃ x_1, ∃ (_ : x_1 ∈ H ∧ 0 < x_1), x_1 < x\ny : G\nhxy : y < x\nhyH : y ∈ H\nhy₀ : 0 < y\n⊢ False","state_after":"case mk.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup G\ninst✝ : Archimedean G\nH : AddSubgroup G\nhbot : H ≠ ⊥\na : G\nh₀ : 0 < a\nhd : Disjoint (↑H) (Ioo 0 a)\nhex : ∀ g > 0, ∃ n, g ∈ Ioc (n • a) ((n + 1) • a)\nthis : ∃ n, (↑H ∩ Ioc (n • a) ((n + 1) • a)).Nonempty\nn : ℕ\nhmin : ∀ m < n, ¬(↑H ∩ Ioc (m • a) ((m + 1) • a)).Nonempty\nx : G\nhxH : x ∈ ↑H\nhnx : n • a < x\nhxn : x ≤ (n + 1) • a\nhxmin : ∀ (x : G), x ∈ H ∧ 0 < x → ∃ x_1, ∃ (_ : x_1 ∈ H ∧ 0 < x_1), x_1 < x\ny : G\nhxy : y < x\nhyH : y ∈ H\nhy₀ : 0 < y\nm : ℕ\nhm : y ∈ Ioc (m • a) ((m + 1) • a)\n⊢ False","tactic":"rcases hex y hy₀ with ⟨m, hm⟩","premises":[]},{"state_before":"case mk.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup G\ninst✝ : Archimedean G\nH : AddSubgroup G\nhbot : H ≠ ⊥\na : G\nh₀ : 0 < a\nhd : Disjoint (↑H) (Ioo 0 a)\nhex : ∀ g > 0, ∃ n, g ∈ Ioc (n • a) ((n + 1) • a)\nthis : ∃ n, (↑H ∩ Ioc (n • a) ((n + 1) • a)).Nonempty\nn : ℕ\nhmin : ∀ m < n, ¬(↑H ∩ Ioc (m • a) ((m + 1) • a)).Nonempty\nx : G\nhxH : x ∈ ↑H\nhnx : n • a < x\nhxn : x ≤ (n + 1) • a\nhxmin : ∀ (x : G), x ∈ H ∧ 0 < x → ∃ x_1, ∃ (_ : x_1 ∈ H ∧ 0 < x_1), x_1 < x\ny : G\nhxy : y < x\nhyH : y ∈ H\nhy₀ : 0 < y\nm : ℕ\nhm : y ∈ Ioc (m • a) ((m + 1) • a)\n⊢ False","state_after":"case mk.intro.intro.intro.intro.intro.intro.intro.intro.inl\nG : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup G\ninst✝ : Archimedean G\nH : AddSubgroup G\nhbot : H ≠ ⊥\na : G\nh₀ : 0 < a\nhd : Disjoint (↑H) (Ioo 0 a)\nhex : ∀ g > 0, ∃ n, g ∈ Ioc (n • a) ((n + 1) • a)\nthis : ∃ n, (↑H ∩ Ioc (n • a) ((n + 1) • a)).Nonempty\nn : ℕ\nhmin : ∀ m < n, ¬(↑H ∩ Ioc (m • a) ((m + 1) • a)).Nonempty\nx : G\nhxH : x ∈ ↑H\nhnx : n • a < x\nhxn : x ≤ (n + 1) • a\nhxmin : ∀ (x : G), x ∈ H ∧ 0 < x → ∃ x_1, ∃ (_ : x_1 ∈ H ∧ 0 < x_1), x_1 < x\ny : G\nhxy : y < x\nhyH : y ∈ H\nhy₀ : 0 < y\nm : ℕ\nhm : y ∈ Ioc (m • a) ((m + 1) • a)\nhmn : m < n\n⊢ False\n\ncase mk.intro.intro.intro.intro.intro.intro.intro.intro.inr\nG : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup G\ninst✝ : Archimedean G\nH : AddSubgroup G\nhbot : H ≠ ⊥\na : G\nh₀ : 0 < a\nhd : Disjoint (↑H) (Ioo 0 a)\nhex : ∀ g > 0, ∃ n, g ∈ Ioc (n • a) ((n + 1) • a)\nthis : ∃ n, (↑H ∩ Ioc (n • a) ((n + 1) • a)).Nonempty\nn : ℕ\nhmin : ∀ m < n, ¬(↑H ∩ Ioc (m • a) ((m + 1) • a)).Nonempty\nx : G\nhxH : x ∈ ↑H\nhnx : n • a < x\nhxn : x ≤ (n + 1) • a\nhxmin : ∀ (x : G), x ∈ H ∧ 0 < x → ∃ x_1, ∃ (_ : x_1 ∈ H ∧ 0 < x_1), x_1 < x\ny : G\nhxy : y < x\nhyH : y ∈ H\nhy₀ : 0 < y\nm : ℕ\nhm : y ∈ Ioc (m • a) ((m + 1) • a)\nhnm : n ≤ m\n⊢ False","tactic":"cases' lt_or_le m n with hmn hnm","premises":[{"full_name":"lt_or_le","def_path":"Mathlib/Order/Defs.lean","def_pos":[287,8],"def_end_pos":[287,16]}]}]}
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e (π C fun x => x ∈ s) a_1) = a) ∧\n          ((Pi.evalMonoidHom (fun a => ℤ) x).comp LocallyConstant.coeFnMonoidHom) a = x_1) →\n      x_1 = 1","tactic":"simp only [h, List.mem_map, LocallyConstant.evalMonoidHom, factors]","premises":[{"full_name":"List.mem_map","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean","def_pos":[750,16],"def_end_pos":[750,23]},{"full_name":"LocallyConstant.evalMonoidHom","def_path":"Mathlib/Topology/LocallyConstant/Algebra.lean","def_pos":[291,4],"def_end_pos":[291,17]},{"full_name":"Profinite.NobelingProof.factors","def_path":"Mathlib/Topology/Category/Profinite/Nobeling.lean","def_pos":[521,4],"def_end_pos":[521,11]}]},{"state_before":"case hl\nI : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\ns : Finset I\nx y : ↑(π C fun x => x ∈ s)\nh : y = x\n⊢ ∀ (x_1 : ℤ),\n    (∃ a,\n        (∃ a_1 ∈ Finset.sort (fun x x_2 => x ≥ x_2) s,\n            (if ↑x a_1 = true then e (π C fun x => x ∈ s) a_1 else 1 - e (π C fun x => x ∈ s) a_1) = a) ∧\n          ((Pi.evalMonoidHom (fun a => ℤ) x).comp LocallyConstant.coeFnMonoidHom) a = x_1) →\n      x_1 = 1","state_after":"case hl.intro.intro.intro.intro\nI : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\ns : Finset I\nx y : ↑(π C fun x => x ∈ s)\nh : y = x\nb : I\nleft✝ : b ∈ Finset.sort (fun x x_1 => x ≥ x_1) s\n⊢ ((Pi.evalMonoidHom (fun a => ℤ) x).comp LocallyConstant.coeFnMonoidHom)\n      (if ↑x b = true then e (π C fun x => x ∈ s) b else 1 - e (π C fun x => x ∈ s) b) =\n    1","tactic":"rintro _ ⟨a, ⟨b, _, rfl⟩, rfl⟩","premises":[]},{"state_before":"case hl.intro.intro.intro.intro\nI : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\ns : Finset I\nx y : ↑(π C fun x => x ∈ s)\nh : y = x\nb : I\nleft✝ : b ∈ Finset.sort (fun x x_1 => x ≥ x_1) s\n⊢ ((Pi.evalMonoidHom (fun a => ℤ) x).comp LocallyConstant.coeFnMonoidHom)\n      (if ↑x b = true then e (π C fun x => x ∈ s) b else 1 - e (π C fun x => x ∈ s) b) =\n    1","state_after":"case hl.intro.intro.intro.intro\nI : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\ns : Finset I\nx y : ↑(π C fun x => x ∈ s)\nh : y = x\nb : I\nleft✝ : b ∈ Finset.sort (fun x x_1 => x ≥ x_1) s\n⊢ (if ↑x b = true then e (π C fun x => x ∈ s) b else 1 - e (π C fun x => x ∈ s) b) x = 1","tactic":"dsimp","premises":[]},{"state_before":"case hl.intro.intro.intro.intro\nI : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\ns : Finset I\nx y : ↑(π C fun x => x ∈ s)\nh : y = x\nb : I\nleft✝ : b ∈ Finset.sort (fun x x_1 => x ≥ x_1) s\n⊢ (if ↑x b = true then e (π C fun x => x ∈ s) b else 1 - e (π C fun x => x ∈ s) b) x = 1","state_after":"case pos\nI : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\ns : Finset I\nx y : ↑(π C fun x => x ∈ s)\nh : y = x\nb : I\nleft✝ : b ∈ Finset.sort (fun x x_1 => x ≥ x_1) s\nhh : ↑x b = true\n⊢ (e (π C fun x => x ∈ s) b) x = 1\n\ncase neg\nI : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\ns : Finset I\nx y : ↑(π C fun x => x ∈ s)\nh : y = x\nb : I\nleft✝ : b ∈ Finset.sort (fun x x_1 => x ≥ x_1) s\nhh : ¬↑x b = true\n⊢ (1 - e (π C fun x => x ∈ s) b) x = 1","tactic":"split_ifs with hh","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
+{"url":"Mathlib/Testing/SlimCheck/Gen.lean","commit":"","full_name":"SlimCheck.Gen.chooseNatLt_aux","start":[51,0],"end":[57,52],"file_path":"Mathlib/Testing/SlimCheck/Gen.lean","tactics":[{"state_before":"lo hi a : ℕ\nh : lo.succ ≤ a ∧ a ≤ hi\n⊢ a.pred.succ ≤ hi","state_after":"lo hi a : ℕ\nh : lo.succ ≤ a ∧ a ≤ hi\n⊢ a ≤ hi\n\nlo hi a : ℕ\nh : lo.succ ≤ a ∧ a ≤ hi\n⊢ 0 < a","tactic":"rw [Nat.succ_pred_eq_of_pos]","premises":[{"full_name":"Nat.succ_pred_eq_of_pos","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[830,8],"def_end_pos":[830,27]}]}]}
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+{"url":"Mathlib/Order/CompleteLattice.lean","commit":"","full_name":"iSup₂_comm","start":[928,0],"end":[931,50],"file_path":"Mathlib/Order/CompleteLattice.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nβ₂ : Type u_3\nγ : Type u_4\nι : Sort u_5\nι' : Sort u_6\nκ : ι → Sort u_7\nκ' : ι' → Sort u_8\ninst✝ : CompleteLattice α\nf✝ g s t : ι → α\na b : α\nι₁ : Sort u_9\nι₂ : Sort u_10\nκ₁ : ι₁ → Sort u_11\nκ₂ : ι₂ → Sort u_12\nf : (i₁ : ι₁) → κ₁ i₁ → (i₂ : ι₂) → κ₂ i₂ → α\n⊢ ⨆ i₁, ⨆ j₁, ⨆ i₂, ⨆ j₂, f i₁ j₁ i₂ j₂ = ⨆ i₂, ⨆ j₂, ⨆ i₁, ⨆ j₁, f i₁ j₁ i₂ j₂","state_after":"no goals","tactic":"simp only [@iSup_comm _ (κ₁ _), @iSup_comm _ ι₁]","premises":[{"full_name":"iSup_comm","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[921,8],"def_end_pos":[921,17]}]}]}
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+{"url":"Mathlib/Algebra/Polynomial/Monic.lean","commit":"","full_name":"Polynomial.monic_of_degree_le","start":[80,0],"end":[84,87],"file_path":"Mathlib/Algebra/Polynomial/Monic.lean","tactics":[{"state_before":"R : Type u\nS : Type v\na b : R\nm n✝ : ℕ\nι : Type y\ninst✝ : Semiring R\np q r : R[X]\nn : ℕ\nH1 : p.degree ≤ ↑n\nH2 : p.coeff n = 1\nH : ¬p.degree < ↑n\n⊢ p.Monic","state_after":"no goals","tactic":"rwa [Monic, Polynomial.leadingCoeff, natDegree, (lt_or_eq_of_le H1).resolve_left H]","premises":[{"full_name":"Or.resolve_left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[555,8],"def_end_pos":[555,23]},{"full_name":"Polynomial.Monic","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[69,4],"def_end_pos":[69,9]},{"full_name":"Polynomial.leadingCoeff","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[65,4],"def_end_pos":[65,16]},{"full_name":"Polynomial.natDegree","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[61,4],"def_end_pos":[61,13]},{"full_name":"lt_or_eq_of_le","def_path":"Mathlib/Order/Defs.lean","def_pos":[191,8],"def_end_pos":[191,22]}]}]}
+{"url":"Mathlib/CategoryTheory/Sites/Continuous.lean","commit":"","full_name":"CategoryTheory.Functor.op_comp_isSheaf_of_preservesOneHypercovers","start":[154,0],"end":[161,43],"file_path":"Mathlib/CategoryTheory/Sites/Continuous.lean","tactics":[{"state_before":"C : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D\nE : Type u₃\ninst✝³ : Category.{v₃, u₃} E\nF : C ⥤ D\nA : Type u\ninst✝² : Category.{t, u} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst✝¹ : F.PreservesOneHypercovers J K\ninst✝ : J.IsGeneratedByOneHypercovers\nP : Dᵒᵖ ⥤ A\nhP : Presheaf.IsSheaf K P\n⊢ Presheaf.IsSheaf J (F.op ⋙ P)","state_after":"C : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D\nE : Type u₃\ninst✝³ : Category.{v₃, u₃} E\nF : C ⥤ D\nA : Type u\ninst✝² : Category.{t, u} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst✝¹ : F.PreservesOneHypercovers J K\ninst✝ : J.IsGeneratedByOneHypercovers\nP : Dᵒᵖ ⥤ A\nhP : Presheaf.IsSheaf K P\n⊢ ∀ ⦃X : C⦄ (E : J.OneHypercover X), Nonempty (IsLimit (E.multifork (F.op ⋙ P)))","tactic":"rw [Presheaf.isSheaf_iff_of_isGeneratedByOneHypercovers.{w}]","premises":[{"full_name":"CategoryTheory.Presheaf.isSheaf_iff_of_isGeneratedByOneHypercovers","def_path":"Mathlib/CategoryTheory/Sites/IsSheafOneHypercover.lean","def_pos":[156,6],"def_end_pos":[156,48]}]},{"state_before":"C : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D\nE : Type u₃\ninst✝³ : Category.{v₃, u₃} E\nF : C ⥤ D\nA : Type u\ninst✝² : Category.{t, u} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst✝¹ : F.PreservesOneHypercovers J K\ninst✝ : J.IsGeneratedByOneHypercovers\nP : Dᵒᵖ ⥤ A\nhP : Presheaf.IsSheaf K P\n⊢ ∀ ⦃X : C⦄ (E : J.OneHypercover X), Nonempty (IsLimit (E.multifork (F.op ⋙ P)))","state_after":"C : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D\nE✝ : Type u₃\ninst✝³ : Category.{v₃, u₃} E✝\nF : C ⥤ D\nA : Type u\ninst✝² : Category.{t, u} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst✝¹ : F.PreservesOneHypercovers J K\ninst✝ : J.IsGeneratedByOneHypercovers\nP : Dᵒᵖ ⥤ A\nhP : Presheaf.IsSheaf K P\nX : C\nE : J.OneHypercover X\n⊢ Nonempty (IsLimit (E.multifork (F.op ⋙ P)))","tactic":"intro X E","premises":[]},{"state_before":"C : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D\nE✝ : Type u₃\ninst✝³ : Category.{v₃, u₃} E✝\nF : C ⥤ D\nA : Type u\ninst✝² : Category.{t, u} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst✝¹ : F.PreservesOneHypercovers J K\ninst✝ : J.IsGeneratedByOneHypercovers\nP : Dᵒᵖ ⥤ A\nhP : Presheaf.IsSheaf K P\nX : C\nE : J.OneHypercover X\n⊢ Nonempty (IsLimit (E.multifork (F.op ⋙ P)))","state_after":"no goals","tactic":"exact ⟨(E.toPreOneHypercover.isLimitMapMultiforkEquiv F P)\n    ((E.map F K).isLimitMultifork ⟨P, hP⟩)⟩","premises":[{"full_name":"CategoryTheory.GrothendieckTopology.OneHypercover.isLimitMultifork","def_path":"Mathlib/CategoryTheory/Sites/OneHypercover.lean","def_pos":[202,18],"def_end_pos":[202,34]},{"full_name":"CategoryTheory.GrothendieckTopology.OneHypercover.map","def_path":"Mathlib/CategoryTheory/Sites/Continuous.lean","def_pos":[91,4],"def_end_pos":[91,7]},{"full_name":"CategoryTheory.PreOneHypercover.isLimitMapMultiforkEquiv","def_path":"Mathlib/CategoryTheory/Sites/Continuous.lean","def_pos":[68,4],"def_end_pos":[68,28]},{"full_name":"Nonempty.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[711,4],"def_end_pos":[711,9]}]}]}
+{"url":"Mathlib/Order/Ideal.lean","commit":"","full_name":"IsCoatom.isMaximal","start":[214,0],"end":[215,80],"file_path":"Mathlib/Order/Ideal.lean","tactics":[{"state_before":"P : Type u_1\ninst✝² : LE P\ninst✝¹ : IsDirected P fun x x_1 => x ≤ x_1\ninst✝ : Nonempty P\nI : Ideal P\nhI : IsCoatom I\nx✝ : Ideal P\nhJ : I < x✝\n⊢ ↑x✝ = univ","state_after":"no goals","tactic":"simp [hI.2 _ hJ]","premises":[{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]}]}]}
+{"url":"Mathlib/Data/List/Sym.lean","commit":"","full_name":"List.sym2_map","start":[40,0],"end":[44,55],"file_path":"Mathlib/Data/List/Sym.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nf : α → β\nxs : List α\n⊢ (map f xs).sym2 = map (Sym2.map f) xs.sym2","state_after":"no goals","tactic":"induction xs with\n  | nil => simp [List.sym2]\n  | cons x xs ih => simp [List.sym2, ih, Function.comp]","premises":[{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"List.cons","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2289,4],"def_end_pos":[2289,8]},{"full_name":"List.nil","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2286,4],"def_end_pos":[2286,7]},{"full_name":"List.sym2","def_path":"Mathlib/Data/List/Sym.lean","def_pos":[36,14],"def_end_pos":[36,18]}]}]}
+{"url":"Mathlib/MeasureTheory/Measure/Tilted.lean","commit":"","full_name":"MeasureTheory.tilted_neg_same'","start":[280,0],"end":[283,29],"file_path":"Mathlib/MeasureTheory/Measure/Tilted.lean","tactics":[{"state_before":"α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : Integrable (fun x => rexp (f x)) μ\n⊢ (μ.tilted f).tilted (-f) = (μ Set.univ)⁻¹ • μ","state_after":"α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : Integrable (fun x => rexp (f x)) μ\n⊢ μ.tilted (f + -f) = (μ Set.univ)⁻¹ • μ","tactic":"rw [tilted_tilted hf]","premises":[{"full_name":"MeasureTheory.tilted_tilted","def_path":"Mathlib/MeasureTheory/Measure/Tilted.lean","def_pos":[257,6],"def_end_pos":[257,19]}]},{"state_before":"α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : Integrable (fun x => rexp (f x)) μ\n⊢ μ.tilted (f + -f) = (μ Set.univ)⁻¹ • μ","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/Analysis/Complex/AbsMax.lean","commit":"","full_name":"Complex.norm_eqOn_of_isPreconnected_of_isMaxOn","start":[220,0],"end":[239,61],"file_path":"Mathlib/Analysis/Complex/AbsMax.lean","tactics":[{"state_before":"E : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : E → F\nU : Set E\nc : E\nhc : IsPreconnected U\nho : IsOpen U\nhd : DifferentiableOn ℂ f U\nhcU : c ∈ U\nhm : IsMaxOn (norm ∘ f) U c\n⊢ EqOn (norm ∘ f) (Function.const E ‖f c‖) U","state_after":"E : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : E → F\nU : Set E\nc : E\nhc : IsPreconnected U\nho : IsOpen U\nhd : DifferentiableOn ℂ f U\nhcU : c ∈ U\nhm : IsMaxOn (norm ∘ f) U c\nV : Set E := U ∩ {z | IsMaxOn (norm ∘ f) U z}\n⊢ EqOn (norm ∘ f) (Function.const E ‖f c‖) U","tactic":"set V := U ∩ {z | IsMaxOn (norm ∘ f) U z}","premises":[{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"Inter.inter","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[407,2],"def_end_pos":[407,7]},{"full_name":"IsMaxOn","def_path":"Mathlib/Order/Filter/Extr.lean","def_pos":[110,4],"def_end_pos":[110,11]},{"full_name":"Norm.norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[59,2],"def_end_pos":[59,6]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]},{"full_name":"setOf","def_path":"Mathlib/Init/Set.lean","def_pos":[56,4],"def_end_pos":[56,9]}]},{"state_before":"E : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : E → F\nU : Set E\nc : E\nhc : IsPreconnected U\nho : IsOpen U\nhd : DifferentiableOn ℂ f U\nhcU : c ∈ U\nhm : IsMaxOn (norm ∘ f) U c\nV : Set E := U ∩ {z | IsMaxOn (norm ∘ f) U z}\n⊢ EqOn (norm ∘ f) (Function.const E ‖f c‖) U","state_after":"E : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : E → F\nU : Set E\nc : E\nhc : IsPreconnected U\nho : IsOpen U\nhd : DifferentiableOn ℂ f U\nhcU : c ∈ U\nhm : IsMaxOn (norm ∘ f) U c\nV : Set E := U ∩ {z | IsMaxOn (norm ∘ f) U z}\nhV : ∀ x ∈ V, ‖f x‖ = ‖f c‖\n⊢ EqOn (norm ∘ f) (Function.const E ‖f c‖) U","tactic":"have hV : ∀ x ∈ V, ‖f x‖ = ‖f c‖ := fun x hx => le_antisymm (hm hx.1) (hx.2 hcU)","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Norm.norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[59,2],"def_end_pos":[59,6]},{"full_name":"le_antisymm","def_path":"Mathlib/Order/Defs.lean","def_pos":[156,8],"def_end_pos":[156,19]}]},{"state_before":"E : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : E → F\nU : Set E\nc : E\nhc : IsPreconnected U\nho : IsOpen U\nhd : DifferentiableOn ℂ f U\nhcU : c ∈ U\nhm : IsMaxOn (norm ∘ f) U c\nV : Set E := U ∩ {z | IsMaxOn (norm ∘ f) U z}\nhV : ∀ x ∈ V, ‖f x‖ = ‖f c‖\n⊢ EqOn (norm ∘ f) (Function.const E ‖f c‖) U","state_after":"E : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : E → F\nU : Set E\nc : E\nhc : IsPreconnected U\nho : IsOpen U\nhd : DifferentiableOn ℂ f U\nhcU : c ∈ U\nhm : IsMaxOn (norm ∘ f) U c\nV : Set E := U ∩ {z | IsMaxOn (norm ∘ f) U z}\nhV : ∀ x ∈ V, ‖f x‖ = ‖f c‖\n⊢ U ⊆ V","tactic":"suffices U ⊆ V from fun x hx => hV x (this hx)","premises":[{"full_name":"HasSubset.Subset","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[384,2],"def_end_pos":[384,8]}]},{"state_before":"E : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : E → F\nU : Set E\nc : E\nhc : IsPreconnected U\nho : IsOpen U\nhd : DifferentiableOn ℂ f U\nhcU : c ∈ U\nhm : IsMaxOn (norm ∘ f) U c\nV : Set E := U ∩ {z | IsMaxOn (norm ∘ f) U z}\nhV : ∀ x ∈ V, ‖f x‖ = ‖f c‖\n⊢ U ⊆ V","state_after":"E : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : E → F\nU : Set E\nc : E\nhc : IsPreconnected U\nho : IsOpen U\nhd : DifferentiableOn ℂ f U\nhcU : c ∈ U\nhm : IsMaxOn (norm ∘ f) U c\nV : Set E := U ∩ {z | IsMaxOn (norm ∘ f) U z}\nhV : ∀ x ∈ V, ‖f x‖ = ‖f c‖\nhVo : IsOpen V\n⊢ U ⊆ V","tactic":"have hVo : IsOpen V := by\n    simpa only [ho.mem_nhds_iff, setOf_and, setOf_mem_eq]\n      using isOpen_setOf_mem_nhds_and_isMaxOn_norm hd","premises":[{"full_name":"Complex.isOpen_setOf_mem_nhds_and_isMaxOn_norm","def_path":"Mathlib/Analysis/Complex/AbsMax.lean","def_pos":[213,8],"def_end_pos":[213,46]},{"full_name":"IsOpen","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[82,4],"def_end_pos":[82,10]},{"full_name":"IsOpen.mem_nhds_iff","def_path":"Mathlib/Topology/Basic.lean","def_pos":[747,18],"def_end_pos":[747,37]},{"full_name":"Set.setOf_and","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[240,8],"def_end_pos":[240,17]},{"full_name":"Set.setOf_mem_eq","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[215,8],"def_end_pos":[215,20]}]},{"state_before":"E : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : E → F\nU : Set E\nc : E\nhc : IsPreconnected U\nho : IsOpen U\nhd : DifferentiableOn ℂ f U\nhcU : c ∈ U\nhm : IsMaxOn (norm ∘ f) U c\nV : Set E := U ∩ {z | IsMaxOn (norm ∘ f) U z}\nhV : ∀ x ∈ V, ‖f x‖ = ‖f c‖\nhVo : IsOpen V\n⊢ U ⊆ V","state_after":"E : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : E → F\nU : Set E\nc : E\nhc : IsPreconnected U\nho : IsOpen U\nhd : DifferentiableOn ℂ f U\nhcU : c ∈ U\nhm : IsMaxOn (norm ∘ f) U c\nV : Set E := U ∩ {z | IsMaxOn (norm ∘ f) U z}\nhV : ∀ x ∈ V, ‖f x‖ = ‖f c‖\nhVo : IsOpen V\nhVne : (U ∩ V).Nonempty\n⊢ U ⊆ V","tactic":"have hVne : (U ∩ V).Nonempty := ⟨c, hcU, hcU, hm⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Inter.inter","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[407,2],"def_end_pos":[407,7]},{"full_name":"Set.Nonempty","def_path":"Mathlib/Init/Set.lean","def_pos":[222,14],"def_end_pos":[222,22]}]},{"state_before":"E : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : E → F\nU : Set E\nc : E\nhc : IsPreconnected U\nho : IsOpen U\nhd : DifferentiableOn ℂ f U\nhcU : c ∈ U\nhm : IsMaxOn (norm ∘ f) U c\nV : Set E := U ∩ {z | IsMaxOn (norm ∘ f) U z}\nhV : ∀ x ∈ V, ‖f x‖ = ‖f c‖\nhVo : IsOpen V\nhVne : (U ∩ V).Nonempty\n⊢ U ⊆ V","state_after":"E : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : E → F\nU : Set E\nc : E\nhc : IsPreconnected U\nho : IsOpen U\nhd : DifferentiableOn ℂ f U\nhcU : c ∈ U\nhm : IsMaxOn (norm ∘ f) U c\nV : Set E := U ∩ {z | IsMaxOn (norm ∘ f) U z}\nhV : ∀ x ∈ V, ‖f x‖ = ‖f c‖\nhVo : IsOpen V\nhVne : (U ∩ V).Nonempty\nW : Set E := U ∩ {z | ‖f z‖ ≠ ‖f c‖}\n⊢ U ⊆ V","tactic":"set W := U ∩ {z | ‖f z‖ ≠ ‖f c‖}","premises":[{"full_name":"Inter.inter","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[407,2],"def_end_pos":[407,7]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Norm.norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[59,2],"def_end_pos":[59,6]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]},{"full_name":"setOf","def_path":"Mathlib/Init/Set.lean","def_pos":[56,4],"def_end_pos":[56,9]}]},{"state_before":"E : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : E → F\nU : Set E\nc : E\nhc : IsPreconnected U\nho : IsOpen U\nhd : DifferentiableOn ℂ f U\nhcU : c ∈ U\nhm : IsMaxOn (norm ∘ f) U c\nV : Set E := U ∩ {z | IsMaxOn (norm ∘ f) U z}\nhV : ∀ x ∈ V, ‖f x‖ = ‖f c‖\nhVo : IsOpen V\nhVne : (U ∩ V).Nonempty\nW : Set E := U ∩ {z | ‖f z‖ ≠ ‖f c‖}\n⊢ U ⊆ V","state_after":"E : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : E → F\nU : Set E\nc : E\nhc : IsPreconnected U\nho : IsOpen U\nhd : DifferentiableOn ℂ f U\nhcU : c ∈ U\nhm : IsMaxOn (norm ∘ f) U c\nV : Set E := U ∩ {z | IsMaxOn (norm ∘ f) U z}\nhV : ∀ x ∈ V, ‖f x‖ = ‖f c‖\nhVo : IsOpen V\nhVne : (U ∩ V).Nonempty\nW : Set E := U ∩ {z | ‖f z‖ ≠ ‖f c‖}\nhWo : IsOpen W\n⊢ U ⊆ V","tactic":"have hWo : IsOpen W := hd.continuousOn.norm.isOpen_inter_preimage ho isOpen_ne","premises":[{"full_name":"ContinuousOn.isOpen_inter_preimage","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[918,8],"def_end_pos":[918,42]},{"full_name":"ContinuousOn.norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[820,33],"def_end_pos":[820,50]},{"full_name":"DifferentiableOn.continuousOn","def_path":"Mathlib/Analysis/Calculus/FDeriv/Basic.lean","def_pos":[715,8],"def_end_pos":[715,37]},{"full_name":"IsOpen","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[82,4],"def_end_pos":[82,10]},{"full_name":"isOpen_ne","def_path":"Mathlib/Topology/Separation.lean","def_pos":[530,8],"def_end_pos":[530,17]}]},{"state_before":"E : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : E → F\nU : Set E\nc : E\nhc : IsPreconnected U\nho : IsOpen U\nhd : DifferentiableOn ℂ f U\nhcU : c ∈ U\nhm : IsMaxOn (norm ∘ f) U c\nV : Set E := U ∩ {z | IsMaxOn (norm ∘ f) U z}\nhV : ∀ x ∈ V, ‖f x‖ = ‖f c‖\nhVo : IsOpen V\nhVne : (U ∩ V).Nonempty\nW : Set E := U ∩ {z | ‖f z‖ ≠ ‖f c‖}\nhWo : IsOpen W\n⊢ U ⊆ V","state_after":"E : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : E → F\nU : Set E\nc : E\nhc : IsPreconnected U\nho : IsOpen U\nhd : DifferentiableOn ℂ f U\nhcU : c ∈ U\nhm : IsMaxOn (norm ∘ f) U c\nV : Set E := U ∩ {z | IsMaxOn (norm ∘ f) U z}\nhV : ∀ x ∈ V, ‖f x‖ = ‖f c‖\nhVo : IsOpen V\nhVne : (U ∩ V).Nonempty\nW : Set E := U ∩ {z | ‖f z‖ ≠ ‖f c‖}\nhWo : IsOpen W\nhdVW : Disjoint V W\n⊢ U ⊆ V","tactic":"have hdVW : Disjoint V W := disjoint_left.mpr fun x hxV hxW => hxW.2 (hV x hxV)","premises":[{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"Disjoint","def_path":"Mathlib/Order/Disjoint.lean","def_pos":[40,4],"def_end_pos":[40,12]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Set.disjoint_left","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1216,8],"def_end_pos":[1216,21]}]},{"state_before":"E : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : E → F\nU : Set E\nc : E\nhc : IsPreconnected U\nho : IsOpen U\nhd : DifferentiableOn ℂ f U\nhcU : c ∈ U\nhm : IsMaxOn (norm ∘ f) U c\nV : Set E := U ∩ {z | IsMaxOn (norm ∘ f) U z}\nhV : ∀ x ∈ V, ‖f x‖ = ‖f c‖\nhVo : IsOpen V\nhVne : (U ∩ V).Nonempty\nW : Set E := U ∩ {z | ‖f z‖ ≠ ‖f c‖}\nhWo : IsOpen W\nhdVW : Disjoint V W\n⊢ U ⊆ V","state_after":"E : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : E → F\nU : Set E\nc : E\nhc : IsPreconnected U\nho : IsOpen U\nhd : DifferentiableOn ℂ f U\nhcU : c ∈ U\nhm : IsMaxOn (norm ∘ f) U c\nV : Set E := U ∩ {z | IsMaxOn (norm ∘ f) U z}\nhV : ∀ x ∈ V, ‖f x‖ = ‖f c‖\nhVo : IsOpen V\nhVne : (U ∩ V).Nonempty\nW : Set E := U ∩ {z | ‖f z‖ ≠ ‖f c‖}\nhWo : IsOpen W\nhdVW : Disjoint V W\nhUVW : U ⊆ V ∪ W\n⊢ U ⊆ V","tactic":"have hUVW : U ⊆ V ∪ W := fun x hx =>\n    (eq_or_ne ‖f x‖ ‖f c‖).imp (fun h => ⟨hx, fun y hy => (hm hy).out.trans_eq h.symm⟩)\n      (And.intro hx)","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"HasSubset.Subset","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[384,2],"def_end_pos":[384,8]},{"full_name":"Membership.mem.out","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[208,8],"def_end_pos":[208,33]},{"full_name":"Norm.norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[59,2],"def_end_pos":[59,6]},{"full_name":"Or.imp","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[172,8],"def_end_pos":[172,14]},{"full_name":"Union.union","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[402,2],"def_end_pos":[402,7]},{"full_name":"eq_or_ne","def_path":"Mathlib/Logic/Basic.lean","def_pos":[167,8],"def_end_pos":[167,16]}]},{"state_before":"E : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : E → F\nU : Set E\nc : E\nhc : IsPreconnected U\nho : IsOpen U\nhd : DifferentiableOn ℂ f U\nhcU : c ∈ U\nhm : IsMaxOn (norm ∘ f) U c\nV : Set E := U ∩ {z | IsMaxOn (norm ∘ f) U z}\nhV : ∀ x ∈ V, ‖f x‖ = ‖f c‖\nhVo : IsOpen V\nhVne : (U ∩ V).Nonempty\nW : Set E := U ∩ {z | ‖f z‖ ≠ ‖f c‖}\nhWo : IsOpen W\nhdVW : Disjoint V W\nhUVW : U ⊆ V ∪ W\n⊢ U ⊆ V","state_after":"no goals","tactic":"exact hc.subset_left_of_subset_union hVo hWo hdVW hUVW hVne","premises":[{"full_name":"IsPreconnected.subset_left_of_subset_union","def_path":"Mathlib/Topology/Connected/Basic.lean","def_pos":[382,8],"def_end_pos":[382,50]}]}]}
+{"url":".lake/packages/batteries/Batteries/Data/RBMap/WF.lean","commit":"","full_name":"Batteries.RBNode.Ordered.append","start":[354,0],"end":[385,30],"file_path":".lake/packages/batteries/Batteries/Data/RBMap/WF.lean","tactics":[{"state_before":"α : Type u_1\ncmp : α → α → Ordering\nl : RBNode α\nv : α\nr : RBNode α\nlv : All (fun x => cmpLT cmp x v) l\nvr : All (fun x => cmpLT cmp v x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\n⊢ Ordered cmp (l.append r)","state_after":"α : Type u_1\ncmp : α → α → Ordering\nl : RBNode α\nv : α\nr : RBNode α\nlv : All (fun x => cmpLT cmp x v) l\nvr : All (fun x => cmpLT cmp v x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\n⊢ Ordered cmp\n    (match l, r with\n    | nil, x => x\n    | x, nil => x\n    | node red a x b, node red c y d =>\n      match b.append c with\n      | node red b' z c' => node red (node red a x b') z (node red c' y d)\n      | bc => node red a x (node red bc y d)\n    | node black a x b, node black c y d =>\n      match b.append c with\n      | node red b' z c' => node red (node black a x b') z (node black c' y d)\n      | bc => a.balLeft x (node black bc y d)\n    | a@h:(node black l v r), node red b x c => node red (a.append b) x c\n    | node red a x b, c@h:(node black l v r) => node red a x (b.append c))","tactic":"unfold append","premises":[{"full_name":"Batteries.RBNode.append","def_path":".lake/packages/batteries/Batteries/Data/RBMap/Basic.lean","def_pos":[359,4],"def_end_pos":[359,10]}]},{"state_before":"α : Type u_1\ncmp : α → α → Ordering\nl : RBNode α\nv : α\nr : RBNode α\nlv : All (fun x => cmpLT cmp x v) l\nvr : All (fun x => cmpLT cmp v x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\n⊢ Ordered cmp\n    (match l, r with\n    | nil, x => x\n    | x, nil => x\n    | node red a x b, node red c y d =>\n      match b.append c with\n      | node red b' z c' => node red (node red a x b') z (node red c' y d)\n      | bc => node red a x (node red bc y d)\n    | node black a x b, node black c y d =>\n      match b.append c with\n      | node red b' z c' => node red (node black a x b') z (node black c' y d)\n      | bc => a.balLeft x (node black bc y d)\n    | a@h:(node black l v r), node red b x c => node red (a.append b) x c\n    | node red a x b, c@h:(node black l v r) => node red a x (b.append c))","state_after":"case h_1\nα : Type u_1\ncmp : α → α → Ordering\nv : α\nr : RBNode α\nvr : All (fun x => cmpLT cmp v x) r\nhr : Ordered cmp r\nx✝¹ x✝ : RBNode α\nlv : All (fun x => cmpLT cmp x v) nil\nhl : Ordered cmp nil\n⊢ Ordered cmp r\n\ncase h_2\nα : Type u_1\ncmp : α → α → Ordering\nl : RBNode α\nv : α\nlv : All (fun x => cmpLT cmp x v) l\nhl : Ordered cmp l\nx✝² x✝¹ : RBNode α\nx✝ : l = nil → False\nvr : All (fun x => cmpLT cmp v x) nil\nhr : Ordered cmp nil\n⊢ Ordered cmp l\n\ncase h_3\nα : Type u_1\ncmp : α → α → Ordering\nv : α\nx✝² x✝¹ a✝ : RBNode α\nx✝ : α\nb✝ c✝ : RBNode α\ny✝ : α\nd✝ : RBNode α\nlv : All (fun x => cmpLT cmp x v) (node red a✝ x✝ b✝)\nhl : Ordered cmp (node red a✝ x✝ b✝)\nvr : All (fun x => cmpLT cmp v x) (node red c✝ y✝ d✝)\nhr : Ordered cmp (node red c✝ y✝ d✝)\n⊢ Ordered cmp\n    (match b✝.append c✝ with\n    | node red b' z c' => node red (node red a✝ x✝ b') z (node red c' y✝ d✝)\n    | bc => node red a✝ x✝ (node red bc y✝ d✝))\n\ncase h_4\nα : Type u_1\ncmp : α → α → Ordering\nv : α\nx✝² x✝¹ a✝ : RBNode α\nx✝ : α\nb✝ c✝ : RBNode α\ny✝ : α\nd✝ : RBNode α\nlv : All (fun x => cmpLT cmp x v) (node black a✝ x✝ b✝)\nhl : Ordered cmp (node black a✝ x✝ b✝)\nvr : All (fun x => cmpLT cmp v x) (node black c✝ y✝ d✝)\nhr : Ordered cmp (node black c✝ y✝ d✝)\n⊢ Ordered cmp\n    (match b✝.append c✝ with\n    | node red b' z c' => node red (node black a✝ x✝ b') z (node black c' y✝ d✝)\n    | bc => a✝.balLeft x✝ (node black bc y✝ d✝))\n\ncase h_5\nα : Type u_1\ncmp : α → α → Ordering\nv : α\nx✝² x✝¹ l✝ : RBNode α\nv✝ : α\nr✝ b✝ : RBNode α\nx✝ : α\nc✝ : RBNode α\nlv : All (fun x => cmpLT cmp x v) (node black l✝ v✝ r✝)\nhl : Ordered cmp (node black l✝ v✝ r✝)\nvr : All (fun x => cmpLT cmp v x) (node red b✝ x✝ c✝)\nhr : Ordered cmp (node red b✝ x✝ c✝)\n⊢ Ordered cmp (node red ((node black l✝ v✝ r✝).append b✝) x✝ c✝)\n\ncase h_6\nα : Type u_1\ncmp : α → α → Ordering\nv : α\nx✝² x✝¹ a✝ : RBNode α\nx✝ : α\nb✝ l✝ : RBNode α\nv✝ : α\nr✝ : RBNode α\nlv : All (fun x => cmpLT cmp x v) (node red a✝ x✝ b✝)\nhl : Ordered cmp (node red a✝ x✝ b✝)\nvr : All (fun x => cmpLT cmp v x) (node black l✝ v✝ r✝)\nhr : Ordered cmp (node black l✝ v✝ r✝)\n⊢ Ordered cmp (node red a✝ x✝ (b✝.append (node black l✝ v✝ r✝)))","tactic":"split","premises":[]}]}
+{"url":"Mathlib/Analysis/Analytic/Inverse.lean","commit":"","full_name":"FormalMultilinearSeries.leftInv_comp","start":[89,0],"end":[142,55],"file_path":"Mathlib/Analysis/Analytic/Inverse.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\ni : E ≃L[𝕜] F\nh : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i\n⊢ (p.leftInv i).comp p = id 𝕜 E","state_after":"case h.H\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\ni : E ≃L[𝕜] F\nh : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i\nn : ℕ\nv : Fin n → E\n⊢ ((p.leftInv i).comp p n) v = (id 𝕜 E n) v","tactic":"ext (n v)","premises":[]}]}
+{"url":"Mathlib/Algebra/Pointwise/Stabilizer.lean","commit":"","full_name":"MulAction.map_stabilizer_le","start":[44,0],"end":[50,31],"file_path":"Mathlib/Algebra/Pointwise/Stabilizer.lean","tactics":[{"state_before":"G : Type u_1\nH : Type u_2\nα : Type u_3\ninst✝² : Group G\ninst✝¹ : Group H\ninst✝ : MulAction G α\na : G\nf : G →* H\ns : Set G\n⊢ Subgroup.map f (stabilizer G s) ≤ stabilizer H (⇑f '' s)","state_after":"G : Type u_1\nH : Type u_2\nα : Type u_3\ninst✝² : Group G\ninst✝¹ : Group H\ninst✝ : MulAction G α\na✝ : G\nf : G →* H\ns : Set G\na : H\n⊢ a ∈ Subgroup.map f (stabilizer G s) → a ∈ stabilizer H (⇑f '' s)","tactic":"rintro a","premises":[]},{"state_before":"G : Type u_1\nH : Type u_2\nα : Type u_3\ninst✝² : Group G\ninst✝¹ : Group H\ninst✝ : MulAction G α\na✝ : G\nf : G →* H\ns : Set G\na : H\n⊢ a ∈ Subgroup.map f (stabilizer G s) → a ∈ stabilizer H (⇑f '' s)","state_after":"G : Type u_1\nH : Type u_2\nα : Type u_3\ninst✝² : Group G\ninst✝¹ : Group H\ninst✝ : MulAction G α\na✝ : G\nf : G →* H\ns : Set G\na : H\n⊢ ∀ (x : G), x • s = s → f x = a → a • ⇑f '' s = ⇑f '' s","tactic":"simp only [Subgroup.mem_map, mem_stabilizer_iff, exists_prop, forall_exists_index, and_imp]","premises":[{"full_name":"MulAction.mem_stabilizer_iff","def_path":"Mathlib/GroupTheory/GroupAction/Basic.lean","def_pos":[641,8],"def_end_pos":[641,26]},{"full_name":"Subgroup.mem_map","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[1103,8],"def_end_pos":[1103,15]},{"full_name":"and_imp","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[115,16],"def_end_pos":[115,23]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]},{"full_name":"forall_exists_index","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[189,16],"def_end_pos":[189,35]}]},{"state_before":"G : Type u_1\nH : Type u_2\nα : Type u_3\ninst✝² : Group G\ninst✝¹ : Group H\ninst✝ : MulAction G α\na✝ : G\nf : G →* H\ns : Set G\na : H\n⊢ ∀ (x : G), x • s = s → f x = a → a • ⇑f '' s = ⇑f '' s","state_after":"G : Type u_1\nH : Type u_2\nα : Type u_3\ninst✝² : Group G\ninst✝¹ : Group H\ninst✝ : MulAction G α\na✝ : G\nf : G →* H\ns : Set G\na : G\nha : a • s = s\n⊢ f a • ⇑f '' s = ⇑f '' s","tactic":"rintro a ha rfl","premises":[]},{"state_before":"G : Type u_1\nH : Type u_2\nα : Type u_3\ninst✝² : Group G\ninst✝¹ : Group H\ninst✝ : MulAction G α\na✝ : G\nf : G →* H\ns : Set G\na : G\nha : a • s = s\n⊢ f a • ⇑f '' s = ⇑f '' s","state_after":"no goals","tactic":"rw [← image_smul_distrib, ha]","premises":[{"full_name":"Set.image_smul_distrib","def_path":"Mathlib/Data/Set/Pointwise/SMul.lean","def_pos":[597,8],"def_end_pos":[597,26]}]}]}
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LinearMap.det f ≠ 0\nι : Type := Fin (finrank ℝ E)\n⊢ map (⇑f) μ = ENNReal.ofReal |(LinearMap.det f)⁻¹| • μ","tactic":"let ι := Fin (finrank ℝ E)","premises":[{"full_name":"Fin","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1831,10],"def_end_pos":[1831,13]},{"full_name":"FiniteDimensional.finrank","def_path":"Mathlib/LinearAlgebra/Dimension/Finrank.lean","def_pos":[52,18],"def_end_pos":[52,25]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]}]},{"state_before":"E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nhf : LinearMap.det f ≠ 0\nι : Type := Fin (finrank ℝ E)\n⊢ map (⇑f) μ = ENNReal.ofReal |(LinearMap.det f)⁻¹| • μ","state_after":"E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF : Type u_2\ninst��² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nhf : LinearMap.det f ≠ 0\nι : Type := Fin (finrank ℝ E)\nthis : FiniteDimensional ℝ (ι → ℝ)\n⊢ map (⇑f) μ = ENNReal.ofReal |(LinearMap.det f)⁻¹| • μ","tactic":"haveI : FiniteDimensional ℝ (ι → ℝ) := by infer_instance","premises":[{"full_name":"FiniteDimensional","def_path":"Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean","def_pos":[77,7],"def_end_pos":[77,24]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"inferInstance","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[99,7],"def_end_pos":[99,20]}]},{"state_before":"E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nhf : LinearMap.det f ≠ 0\nι : Type := Fin (finrank ℝ E)\nthis : FiniteDimensional ℝ (ι → ℝ)\n⊢ map (⇑f) μ = ENNReal.ofReal |(LinearMap.det f)⁻¹| • μ","state_after":"E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nhf : LinearMap.det f ≠ 0\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\n⊢ map (⇑f) μ = ENNReal.ofReal |(LinearMap.det f)⁻¹| • μ","tactic":"have : finrank ℝ E = finrank ℝ (ι → ℝ) := by simp [ι]","premises":[{"full_name":"FiniteDimensional.finrank","def_path":"Mathlib/LinearAlgebra/Dimension/Finrank.lean","def_pos":[52,18],"def_end_pos":[52,25]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]}]},{"state_before":"E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nhf : LinearMap.det f ≠ 0\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\n⊢ map (⇑f) μ = ENNReal.ofReal |(LinearMap.det f)⁻¹| • μ","state_after":"E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nhf : LinearMap.det f ≠ 0\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\n⊢ map (⇑f) μ = ENNReal.ofReal |(LinearMap.det f)⁻¹| • μ","tactic":"have e : E ≃ₗ[ℝ] ι → ℝ := LinearEquiv.ofFinrankEq E (ι → ℝ) this","premises":[{"full_name":"LinearEquiv","def_path":"Mathlib/Algebra/Module/Equiv/Defs.lean","def_pos":[51,10],"def_end_pos":[51,21]},{"full_name":"LinearEquiv.ofFinrankEq","def_path":"Mathlib/LinearAlgebra/Dimension/Free.lean","def_pos":[157,18],"def_end_pos":[157,41]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"RingHom.id","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[538,4],"def_end_pos":[538,6]}]},{"state_before":"E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nhf : LinearMap.det f ≠ 0\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\n⊢ map (⇑f) μ = ENNReal.ofReal |(LinearMap.det f)⁻¹| • μ","state_after":"case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nhf : LinearMap.det f ≠ 0\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ��)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhg : g = ↑e ∘ₗ f ∘ₗ ↑e.symm\n⊢ map (⇑f) μ = ENNReal.ofReal |(LinearMap.det f)⁻¹| • μ","tactic":"obtain ⟨g, hg⟩ : ∃ g, g = (e : E →ₗ[ℝ] ι → ℝ).comp (f.comp (e.symm : (ι → ℝ) →ₗ[ℝ] E)) := ⟨_, rfl⟩","premises":[{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"LinearEquiv.symm","def_path":"Mathlib/Algebra/Module/Equiv/Defs.lean","def_pos":[258,4],"def_end_pos":[258,8]},{"full_name":"LinearMap","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[83,10],"def_end_pos":[83,19]},{"full_name":"LinearMap.comp","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[489,4],"def_end_pos":[489,8]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"RingHom.id","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[538,4],"def_end_pos":[538,6]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nhf : LinearMap.det f ≠ 0\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhg : g = ↑e ∘ₗ f ∘ₗ ↑e.symm\n⊢ map (⇑f) μ = ENNReal.ofReal |(LinearMap.det f)⁻¹| • μ","state_after":"case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nhf : LinearMap.det f ≠ 0\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhg : g = ↑e ∘ₗ f ∘ₗ ↑e.symm\ngdet : LinearMap.det g = LinearMap.det f\n⊢ map (⇑f) μ = ENNReal.ofReal |(LinearMap.det f)⁻¹| • μ","tactic":"have gdet : LinearMap.det g = LinearMap.det f := by rw [hg]; exact LinearMap.det_conj f e","premises":[{"full_name":"LinearMap.det","def_path":"Mathlib/LinearAlgebra/Determinant.lean","def_pos":[161,26],"def_end_pos":[161,29]},{"full_name":"LinearMap.det_conj","def_path":"Mathlib/LinearAlgebra/Determinant.lean","def_pos":[278,8],"def_end_pos":[278,16]}]},{"state_before":"case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nhf : LinearMap.det f ≠ 0\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhg : g = ↑e ∘ₗ f ∘ₗ ↑e.symm\ngdet : LinearMap.det g = LinearMap.det f\n⊢ map (⇑f) μ = ENNReal.ofReal |(LinearMap.det f)⁻¹| • μ","state_after":"case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : LinearMap.det g ≠ 0\nhg : g = ↑e ∘ₗ f ∘ₗ ↑e.symm\ngdet : LinearMap.det g = LinearMap.det f\n⊢ map (⇑f) μ = ENNReal.ofReal |(LinearMap.det g)⁻¹| • μ","tactic":"rw [← gdet] at hf ⊢","premises":[]},{"state_before":"case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : LinearMap.det g ≠ 0\nhg : g = ↑e ∘ₗ f ∘ₗ ↑e.symm\ngdet : LinearMap.det g = LinearMap.det f\n⊢ map (⇑f) μ = ENNReal.ofReal |(LinearMap.det g)⁻¹| • μ","state_after":"case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : LinearMap.det g ≠ 0\nhg : g = ↑e ∘ₗ f ∘ₗ ↑e.symm\ngdet : LinearMap.det g = LinearMap.det f\nfg : f = ↑e.symm ∘ₗ g ∘ₗ ↑e\n⊢ map (⇑f) μ = ENNReal.ofReal |(LinearMap.det g)⁻¹| • μ","tactic":"have fg : f = (e.symm : (ι → ℝ) →ₗ[ℝ] E).comp (g.comp (e : E →ₗ[ℝ] ι → ℝ)) := by\n    ext x\n    simp only [LinearEquiv.coe_coe, Function.comp_apply, LinearMap.coe_comp,\n      LinearEquiv.symm_apply_apply, hg]","premises":[{"full_name":"Function.comp_apply","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[35,16],"def_end_pos":[35,35]},{"full_name":"LinearEquiv.coe_coe","def_path":"Mathlib/Algebra/Module/Equiv/Defs.lean","def_pos":[208,8],"def_end_pos":[208,15]},{"full_name":"LinearEquiv.symm","def_path":"Mathlib/Algebra/Module/Equiv/Defs.lean","def_pos":[258,4],"def_end_pos":[258,8]},{"full_name":"LinearEquiv.symm_apply_apply","def_path":"Mathlib/Algebra/Module/Equiv/Defs.lean","def_pos":[346,8],"def_end_pos":[346,24]},{"full_name":"LinearMap","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[83,10],"def_end_pos":[83,19]},{"full_name":"LinearMap.coe_comp","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[508,8],"def_end_pos":[508,16]},{"full_name":"LinearMap.comp","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[489,4],"def_end_pos":[489,8]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"RingHom.id","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[538,4],"def_end_pos":[538,6]}]},{"state_before":"case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : LinearMap.det g ≠ 0\nhg : g = ↑e ∘ₗ f ∘ₗ ↑e.symm\ngdet : LinearMap.det g = LinearMap.det f\nfg : f = ↑e.symm ∘ₗ g ∘ₗ ↑e\n⊢ map (⇑f) μ = ENNReal.ofReal |(LinearMap.det g)⁻¹| • μ","state_after":"case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : LinearMap.det g ≠ 0\nhg : g = ↑e ∘ₗ f ∘ₗ ↑e.symm\ngdet : LinearMap.det g = LinearMap.det f\nfg : f = ↑e.symm ∘ₗ g ∘ₗ ↑e\n⊢ map (⇑e.symm ∘ ⇑g ∘ ⇑e) μ = ENNReal.ofReal |(LinearMap.det g)⁻¹| • μ","tactic":"simp only [fg, LinearEquiv.coe_coe, LinearMap.coe_comp]","premises":[{"full_name":"LinearEquiv.coe_coe","def_path":"Mathlib/Algebra/Module/Equiv/Defs.lean","def_pos":[208,8],"def_end_pos":[208,15]},{"full_name":"LinearMap.coe_comp","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[508,8],"def_end_pos":[508,16]}]},{"state_before":"case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι �� ℝ) →ₗ[ℝ] ι → ℝ\nhf : LinearMap.det g ≠ 0\nhg : g = ↑e ∘ₗ f ∘ₗ ↑e.symm\ngdet : LinearMap.det g = LinearMap.det f\nfg : f = ↑e.symm ∘ₗ g ∘ₗ ↑e\n⊢ map (⇑e.symm ∘ ⇑g ∘ ⇑e) μ = ENNReal.ofReal |(LinearMap.det g)⁻¹| • μ","state_after":"case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : LinearMap.det g ≠ 0\nhg : g = ↑e ∘ₗ f ∘ₗ ↑e.symm\ngdet : LinearMap.det g = LinearMap.det f\nfg : f = ↑e.symm ∘ₗ g ∘ₗ ↑e\nCe : Continuous ⇑e\n⊢ map (⇑e.symm ∘ ⇑g ∘ ⇑e) μ = ENNReal.ofReal |(LinearMap.det g)⁻¹| • μ","tactic":"have Ce : Continuous e := (e : E →ₗ[ℝ] ι → ℝ).continuous_of_finiteDimensional","premises":[{"full_name":"Continuous","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[141,10],"def_end_pos":[141,20]},{"full_name":"LinearMap","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[83,10],"def_end_pos":[83,19]},{"full_name":"LinearMap.continuous_of_finiteDimensional","def_path":"Mathlib/Topology/Algebra/Module/FiniteDimension.lean","def_pos":[239,8],"def_end_pos":[239,49]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"RingHom.id","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[538,4],"def_end_pos":[538,6]}]},{"state_before":"case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : LinearMap.det g ≠ 0\nhg : g = ↑e ∘ₗ f ∘ₗ ↑e.symm\ngdet : LinearMap.det g = LinearMap.det f\nfg : f = ↑e.symm ∘ₗ g ∘ₗ ↑e\nCe : Continuous ⇑e\n⊢ map (⇑e.symm ∘ ⇑g ∘ ⇑e) μ = ENNReal.ofReal |(LinearMap.det g)⁻¹| • μ","state_after":"case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : LinearMap.det g ≠ 0\nhg : g = ↑e ∘ₗ f ∘ₗ ↑e.symm\ngdet : LinearMap.det g = LinearMap.det f\nfg : f = ↑e.symm ∘ₗ g ∘ₗ ↑e\nCe : Continuous ⇑e\nCg : Continuous ⇑g\n⊢ map (⇑e.symm ∘ ⇑g ∘ ⇑e) μ = ENNReal.ofReal |(LinearMap.det g)⁻¹| • μ","tactic":"have Cg : Continuous g := LinearMap.continuous_of_finiteDimensional g","premises":[{"full_name":"Continuous","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[141,10],"def_end_pos":[141,20]},{"full_name":"LinearMap.continuous_of_finiteDimensional","def_path":"Mathlib/Topology/Algebra/Module/FiniteDimension.lean","def_pos":[239,8],"def_end_pos":[239,49]}]},{"state_before":"case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : LinearMap.det g ≠ 0\nhg : g = ↑e ∘ₗ f ∘ₗ ↑e.symm\ngdet : LinearMap.det g = LinearMap.det f\nfg : f = ↑e.symm ∘ₗ g ∘ₗ ↑e\nCe : Continuous ⇑e\nCg : Continuous ⇑g\n⊢ map (⇑e.symm ∘ ⇑g ∘ ⇑e) μ = ENNReal.ofReal |(LinearMap.det g)⁻¹| • μ","state_after":"case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : LinearMap.det g ≠ 0\nhg : g = ↑e ∘ₗ f ∘ₗ ↑e.symm\ngdet : LinearMap.det g = LinearMap.det f\nfg : f = ↑e.symm ∘ₗ g ∘ₗ ↑e\nCe : Continuous ⇑e\nCg : Continuous ⇑g\nCesymm : Continuous ⇑e.symm\n⊢ map (⇑e.symm ∘ ⇑g ∘ ⇑e) μ = ENNReal.ofReal |(LinearMap.det g)⁻¹| • μ","tactic":"have Cesymm : Continuous e.symm := (e.symm : (ι → ℝ) →ₗ[ℝ] E).continuous_of_finiteDimensional","premises":[{"full_name":"Continuous","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[141,10],"def_end_pos":[141,20]},{"full_name":"LinearEquiv.symm","def_path":"Mathlib/Algebra/Module/Equiv/Defs.lean","def_pos":[258,4],"def_end_pos":[258,8]},{"full_name":"LinearMap","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[83,10],"def_end_pos":[83,19]},{"full_name":"LinearMap.continuous_of_finiteDimensional","def_path":"Mathlib/Topology/Algebra/Module/FiniteDimension.lean","def_pos":[239,8],"def_end_pos":[239,49]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"RingHom.id","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[538,4],"def_end_pos":[538,6]}]},{"state_before":"case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : LinearMap.det g ≠ 0\nhg : g = ↑e ∘ₗ f ∘ₗ ↑e.symm\ngdet : LinearMap.det g = LinearMap.det f\nfg : f = ↑e.symm ∘ₗ g ∘ₗ ↑e\nCe : Continuous ⇑e\nCg : Continuous ⇑g\nCesymm : Continuous ⇑e.symm\n⊢ map (⇑e.symm ∘ ⇑g ∘ ⇑e) μ = ENNReal.ofReal |(LinearMap.det g)⁻¹| • μ","state_after":"case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : LinearMap.det g ≠ 0\nhg : g = ↑e ∘ₗ f ∘ₗ ↑e.symm\ngdet : LinearMap.det g = LinearMap.det f\nfg : f = ↑e.symm ∘ₗ g ∘ₗ ↑e\nCe : Continuous ⇑e\nCg : Continuous ⇑g\nCesymm : Continuous ⇑e.symm\n⊢ map (⇑e.symm) (map (⇑g) (map (⇑e) μ)) = ENNReal.ofReal |(LinearMap.det g)⁻¹| • μ","tactic":"rw [← map_map Cesymm.measurable (Cg.comp Ce).measurable, ← map_map Cg.measurable Ce.measurable]","premises":[{"full_name":"Continuous.comp","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1389,8],"def_end_pos":[1389,23]},{"full_name":"Continuous.measurable","def_path":"Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean","def_pos":[465,8],"def_end_pos":[465,29]},{"full_name":"MeasureTheory.Measure.map_map","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","def_pos":[1199,8],"def_end_pos":[1199,15]}]},{"state_before":"case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : LinearMap.det g ≠ 0\nhg : g = ↑e ∘ₗ f ∘ₗ ↑e.symm\ngdet : LinearMap.det g = LinearMap.det f\nfg : f = ↑e.symm ∘ₗ g ∘ₗ ↑e\nCe : Continuous ⇑e\nCg : Continuous ⇑g\nCesymm : Continuous ⇑e.symm\n⊢ map (⇑e.symm) (map (⇑g) (map (⇑e) μ)) = ENNReal.ofReal |(LinearMap.det g)⁻¹| • μ","state_after":"case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝¹ : FiniteDimensional ℝ (ι → ℝ)\nthis✝ : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : LinearMap.det g ≠ 0\nhg : g = ↑e ∘ₗ f ∘ₗ ↑e.symm\ngdet : LinearMap.det g = LinearMap.det f\nfg : f = ↑e.symm ∘ₗ g ∘ₗ ↑e\nCe : Continuous ⇑e\nCg : Continuous ⇑g\nCesymm : Continuous ⇑e.symm\nthis : (map (⇑e) μ).IsAddHaarMeasure\n⊢ map (⇑e.symm) (map (⇑g) (map (⇑e) μ)) = ENNReal.ofReal |(LinearMap.det g)⁻¹| • μ","tactic":"haveI : IsAddHaarMeasure (map e μ) := (e : E ≃+ (ι → ℝ)).isAddHaarMeasure_map μ Ce Cesymm","premises":[{"full_name":"AddEquiv","def_path":"Mathlib/Algebra/Group/Equiv/Basic.lean","def_pos":[65,10],"def_end_pos":[65,18]},{"full_name":"AddEquiv.isAddHaarMeasure_map","def_path":"Mathlib/MeasureTheory/Group/Measure.lean","def_pos":[819,2],"def_end_pos":[819,13]},{"full_name":"MeasureTheory.Measure.IsAddHaarMeasure","def_path":"Mathlib/MeasureTheory/Group/Measure.lean","def_pos":[725,6],"def_end_pos":[725,22]},{"full_name":"MeasureTheory.Measure.map","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","def_pos":[1090,16],"def_end_pos":[1090,19]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]}]},{"state_before":"case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝¹ : FiniteDimensional ℝ (ι → ℝ)\nthis✝ : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : LinearMap.det g ≠ 0\nhg : g = ↑e ∘ₗ f ∘ₗ ↑e.symm\ngdet : LinearMap.det g = LinearMap.det f\nfg : f = ↑e.symm ∘ₗ g ∘ₗ ↑e\nCe : Continuous ⇑e\nCg : Continuous ⇑g\nCesymm : Continuous ⇑e.symm\nthis : (map (⇑e) μ).IsAddHaarMeasure\n⊢ map (⇑e.symm) (map (⇑g) (map (⇑e) μ)) = ENNReal.ofReal |(LinearMap.det g)⁻¹| • μ","state_after":"case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝¹ : FiniteDimensional ℝ (ι → ℝ)\nthis✝ : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : LinearMap.det g ≠ 0\nhg : g = ↑e ∘ₗ f ∘ₗ ↑e.symm\ngdet : LinearMap.det g = LinearMap.det f\nfg : f = ↑e.symm ∘ₗ g ∘ₗ ↑e\nCe : Continuous ⇑e\nCg : Continuous ⇑g\nCesymm : Continuous ⇑e.symm\nthis : (map (⇑e) μ).IsAddHaarMeasure\necomp : ⇑e.symm ∘ ⇑e = id\n⊢ map (⇑e.symm) (map (⇑g) (map (⇑e) μ)) = ENNReal.ofReal |(LinearMap.det g)⁻¹| • μ","tactic":"have ecomp : e.symm ∘ e = id := by\n    ext x; simp only [id, Function.comp_apply, LinearEquiv.symm_apply_apply]","premises":[{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"Function.comp_apply","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[35,16],"def_end_pos":[35,35]},{"full_name":"LinearEquiv.symm","def_path":"Mathlib/Algebra/Module/Equiv/Defs.lean","def_pos":[258,4],"def_end_pos":[258,8]},{"full_name":"LinearEquiv.symm_apply_apply","def_path":"Mathlib/Algebra/Module/Equiv/Defs.lean","def_pos":[346,8],"def_end_pos":[346,24]},{"full_name":"id","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[33,14],"def_end_pos":[33,16]}]},{"state_before":"case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝¹ : FiniteDimensional ℝ (ι → ℝ)\nthis✝ : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : LinearMap.det g ≠ 0\nhg : g = ↑e ∘ₗ f ∘ₗ ↑e.symm\ngdet : LinearMap.det g = LinearMap.det f\nfg : f = ↑e.symm ∘ₗ g ∘ₗ ↑e\nCe : Continuous ⇑e\nCg : Continuous ⇑g\nCesymm : Continuous ⇑e.symm\nthis : (map (⇑e) μ).IsAddHaarMeasure\necomp : ⇑e.symm ∘ ⇑e = id\n⊢ map (⇑e.symm) (map (⇑g) (map (⇑e) μ)) = ENNReal.ofReal |(LinearMap.det g)⁻¹| • μ","state_after":"no goals","tactic":"rw [map_linearMap_addHaar_pi_eq_smul_addHaar hf (map e μ), Measure.map_smul,\n    map_map Cesymm.measurable Ce.measurable, ecomp, Measure.map_id]","premises":[{"full_name":"Continuous.measurable","def_path":"Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean","def_pos":[465,8],"def_end_pos":[465,29]},{"full_name":"MeasureTheory.Measure.map","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","def_pos":[1090,16],"def_end_pos":[1090,19]},{"full_name":"MeasureTheory.Measure.map_id","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","def_pos":[1192,8],"def_end_pos":[1192,14]},{"full_name":"MeasureTheory.Measure.map_linearMap_addHaar_pi_eq_smul_addHaar","def_path":"Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean","def_pos":[215,8],"def_end_pos":[215,48]},{"full_name":"MeasureTheory.Measure.map_map","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","def_pos":[1199,8],"def_end_pos":[1199,15]},{"full_name":"MeasureTheory.Measure.map_smul","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","def_pos":[1123,18],"def_end_pos":[1123,26]}]}]}
+{"url":"Mathlib/NumberTheory/ArithmeticFunction.lean","commit":"","full_name":"ArithmeticFunction.zeta_mul_pow_eq_sigma","start":[834,0],"end":[841,11],"file_path":"Mathlib/NumberTheory/ArithmeticFunction.lean","tactics":[{"state_before":"R : Type u_1\nk : ℕ\n⊢ ζ * pow k = σ k","state_after":"case h\nR : Type u_1\nk x✝ : ℕ\n⊢ (ζ * pow k) x✝ = (σ k) x✝","tactic":"ext","premises":[]},{"state_before":"case h\nR : Type u_1\nk x✝ : ℕ\n⊢ (ζ * pow k) x✝ = (σ k) x✝","state_after":"case h\nR : Type u_1\nk x✝ : ℕ\n⊢ ∑ i ∈ x✝.divisors, (pow k) i = { toFun := fun n => ∑ d ∈ n.divisors, d ^ k, map_zero' := ⋯ } x✝","tactic":"rw [sigma, zeta_mul_apply]","premises":[{"full_name":"ArithmeticFunction.sigma","def_path":"Mathlib/NumberTheory/ArithmeticFunction.lean","def_pos":[807,4],"def_end_pos":[807,9]},{"full_name":"ArithmeticFunction.zeta_mul_apply","def_path":"Mathlib/NumberTheory/ArithmeticFunction.lean","def_pos":[439,8],"def_end_pos":[439,22]}]},{"state_before":"case h\nR : Type u_1\nk x✝ : ℕ\n⊢ ∑ i ∈ x✝.divisors, (pow k) i = { toFun := fun n => ∑ d ∈ n.divisors, d ^ k, map_zero' := ⋯ } x✝","state_after":"case h\nR : Type u_1\nk x✝ : ℕ\n⊢ ∀ x ∈ x✝.divisors, (pow k) x = x ^ k","tactic":"apply sum_congr rfl","premises":[{"full_name":"Finset.sum_congr","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[380,2],"def_end_pos":[380,13]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"case h\nR : Type u_1\nk x✝ : ℕ\n⊢ ∀ x ∈ x✝.divisors, (pow k) x = x ^ k","state_after":"case h\nR : Type u_1\nk x✝ x : ℕ\nhx : x ∈ x✝.divisors\n⊢ (pow k) x = x ^ k","tactic":"intro x hx","premises":[]},{"state_before":"case h\nR : Type u_1\nk x✝ x : ℕ\nhx : x ∈ x✝.divisors\n⊢ (pow k) x = x ^ k","state_after":"R : Type u_1\nk x✝ x : ℕ\nhx : x ∈ x✝.divisors\n⊢ ¬x = 0","tactic":"rw [pow_apply, if_neg (not_and_of_not_right _ _)]","premises":[{"full_name":"ArithmeticFunction.pow_apply","def_path":"Mathlib/NumberTheory/ArithmeticFunction.lean","def_pos":[796,8],"def_end_pos":[796,17]},{"full_name":"if_neg","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[937,8],"def_end_pos":[937,14]},{"full_name":"not_and_of_not_right","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[49,8],"def_end_pos":[49,28]}]},{"state_before":"R : Type u_1\nk x✝ x : ℕ\nhx : x ∈ x✝.divisors\n⊢ ¬x = 0","state_after":"R : Type u_1\nk x✝ x : ℕ\nhx : x = 0\n⊢ x ∉ x✝.divisors","tactic":"contrapose! hx","premises":[{"full_name":"Mathlib.Tactic.Contrapose.mtr","def_path":"Mathlib/Tactic/Contrapose.lean","def_pos":[24,6],"def_end_pos":[24,9]}]},{"state_before":"R : Type u_1\nk x✝ x : ℕ\nhx : x = 0\n⊢ x ∉ x✝.divisors","state_after":"no goals","tactic":"simp [hx]","premises":[]}]}
+{"url":"Mathlib/Combinatorics/SimpleGraph/Walk.lean","commit":"","full_name":"SimpleGraph.Walk.append_nil","start":[199,0],"end":[203,39],"file_path":"Mathlib/Combinatorics/SimpleGraph/Walk.lean","tactics":[{"state_before":"V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\np : G.Walk u v\n⊢ p.append nil = p","state_after":"no goals","tactic":"induction p with\n  | nil => rfl\n  | cons _ _ ih => rw [cons_append, ih]","premises":[{"full_name":"SimpleGraph.Walk.cons","def_path":"Mathlib/Combinatorics/SimpleGraph/Walk.lean","def_pos":[55,4],"def_end_pos":[55,8]},{"full_name":"SimpleGraph.Walk.cons_append","def_path":"Mathlib/Combinatorics/SimpleGraph/Walk.lean","def_pos":[192,8],"def_end_pos":[192,19]},{"full_name":"SimpleGraph.Walk.nil","def_path":"Mathlib/Combinatorics/SimpleGraph/Walk.lean","def_pos":[54,4],"def_end_pos":[54,7]}]}]}
+{"url":"Mathlib/Order/Hom/Order.lean","commit":"","full_name":"OrderHom.coe_iInf","start":[92,0],"end":[95,29],"file_path":"Mathlib/Order/Hom/Order.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\nι : Sort u_3\ninst✝ : CompleteLattice β\nf : ι → α →o β\n⊢ ⇑(⨅ i, f i) = ⨅ i, ⇑(f i)","state_after":"case h\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\nι : Sort u_3\ninst✝ : CompleteLattice β\nf : ι → α →o β\nx : α\n⊢ (⨅ i, f i) x = (⨅ i, ⇑(f i)) x","tactic":"funext x","premises":[{"full_name":"funext","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1817,8],"def_end_pos":[1817,14]}]},{"state_before":"case h\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\nι : Sort u_3\ninst✝ : CompleteLattice β\nf : ι → α →o β\nx : α\n⊢ (⨅ i, f i) x = (⨅ i, ⇑(f i)) x","state_after":"no goals","tactic":"simp [iInf_apply]","premises":[{"full_name":"OrderHom.iInf_apply","def_path":"Mathlib/Order/Hom/Order.lean","def_pos":[88,8],"def_end_pos":[88,18]}]}]}
+{"url":"Mathlib/Order/Category/NonemptyFinLinOrd.lean","commit":"","full_name":"NonemptyFinLinOrd.epi_iff_surjective","start":[147,0],"end":[185,48],"file_path":"Mathlib/Order/Category/NonemptyFinLinOrd.lean","tactics":[{"state_before":"A B : NonemptyFinLinOrd\nf : A ⟶ B\n⊢ Epi f ↔ Function.Surjective ⇑f","state_after":"case mp\nA B : NonemptyFinLinOrd\nf : A ⟶ B\n⊢ Epi f → Function.Surjective ⇑f\n\ncase mpr\nA B : NonemptyFinLinOrd\nf : A ⟶ B\n⊢ Function.Surjective ⇑f → Epi f","tactic":"constructor","premises":[]}]}
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Category.{v, u} C\nK : Type u_1\ninst✝² : Category.{?u.254173, u_1} K\nD : Type u_2\ninst✝¹ : Category.{?u.254180, u_2} D\ninst✝ : HasInitial C\nH : IsUniversalColimit (BinaryCofan.mk (𝟙 (⊥_ C)) (𝟙 (⊥_ C)))\nA : C\nf : A ⟶ ⊥_ C\n⊢ IsIso f","state_after":"J : Type v'\ninst✝⁴ : Category.{u', v'} J\nC : Type u\ninst✝³ : Category.{v, u} C\nK : Type u_1\ninst✝² : Category.{?u.254173, u_1} K\nD : Type u_2\ninst✝¹ : Category.{?u.254180, u_2} D\ninst✝ : HasInitial C\nH : IsUniversalColimit (BinaryCofan.mk (𝟙 (⊥_ C)) (𝟙 (⊥_ C)))\nA : C\nf : A ⟶ ⊥_ C\n⊢ IsColimit (BinaryCofan.mk (𝟙 A) (𝟙 A))","tactic":"suffices IsColimit (BinaryCofan.mk (𝟙 A) (𝟙 A)) by\n        obtain ⟨l, h₁, h₂⟩ := Limits.BinaryCofan.IsColimit.desc' this (f ≫ initial.to A) (𝟙 A)\n        rcases(Category.id_comp _).symm.trans h₂ with rfl\n        exact ⟨⟨_, ((Category.id_comp _).symm.trans h₁).symm, initialIsInitial.hom_ext _ _⟩⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"CategoryTheory.Category.id_comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[151,2],"def_end_pos":[151,9]},{"full_name":"CategoryTheory.CategoryStruct.comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]},{"full_name":"CategoryTheory.CategoryStruct.id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[87,2],"def_end_pos":[87,4]},{"full_name":"CategoryTheory.Limits.BinaryCofan.IsColimit.desc'","def_path":"Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean","def_pos":[340,4],"def_end_pos":[340,31]},{"full_name":"CategoryTheory.Limits.BinaryCofan.mk","def_path":"Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean","def_pos":[265,4],"def_end_pos":[265,18]},{"full_name":"CategoryTheory.Limits.IsColimit","def_path":"Mathlib/CategoryTheory/Limits/IsLimit.lean","def_pos":[507,10],"def_end_pos":[507,19]},{"full_name":"CategoryTheory.Limits.IsInitial.hom_ext","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Terminal.lean","def_pos":[164,8],"def_end_pos":[164,25]},{"full_name":"CategoryTheory.Limits.initial.to","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Terminal.lean","def_pos":[330,7],"def_end_pos":[330,17]},{"full_name":"CategoryTheory.Limits.initialIsInitial","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Terminal.lean","def_pos":[338,4],"def_end_pos":[338,20]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Eq.trans","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[335,8],"def_end_pos":[335,16]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]},{"state_before":"J : Type v'\ninst✝⁴ : Category.{u', v'} J\nC : Type u\ninst✝³ : Category.{v, u} C\nK : Type u_1\ninst✝² : Category.{?u.254173, u_1} K\nD : Type u_2\ninst✝¹ : Category.{?u.254180, u_2} D\ninst✝ : HasInitial C\nH : IsUniversalColimit (BinaryCofan.mk (𝟙 (⊥_ C)) (𝟙 (⊥_ C)))\nA : C\nf : A ⟶ ⊥_ C\n⊢ IsColimit (BinaryCofan.mk (𝟙 A) (𝟙 A))","state_after":"J : Type v'\ninst✝⁴ : Category.{u', v'} J\nC : Type u\ninst✝³ : Category.{v, u} C\nK : Type u_1\ninst✝² : Category.{?u.254173, u_1} K\nD : Type u_2\ninst✝¹ : Category.{?u.254180, u_2} D\ninst✝ : HasInitial C\nH : IsUniversalColimit (BinaryCofan.mk (𝟙 (⊥_ C)) (𝟙 (⊥_ C)))\nA : C\nf : A ⟶ ⊥_ C\n⊢ ∀ (j : Discrete WalkingPair),\n    IsPullback ((BinaryCofan.mk (𝟙 A) (𝟙 A)).ι.app j) ((mapPair f f).app j) f\n      ((BinaryCofan.mk (𝟙 (⊥_ C)) (𝟙 (⊥_ C))).ι.app j)","tactic":"refine (H (BinaryCofan.mk (𝟙 _) (𝟙 _)) (mapPair f f) f (by ext ⟨⟨⟩⟩ <;> dsimp <;> simp)\n        (mapPair_equifibered _) ?_).some","premises":[{"full_name":"CategoryTheory.CategoryStruct.id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[87,2],"def_end_pos":[87,4]},{"full_name":"CategoryTheory.Limits.BinaryCofan.mk","def_path":"Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean","def_pos":[265,4],"def_end_pos":[265,18]},{"full_name":"CategoryTheory.Limits.mapPair","def_path":"Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean","def_pos":[134,4],"def_end_pos":[134,11]},{"full_name":"CategoryTheory.mapPair_equifibered","def_path":"Mathlib/CategoryTheory/Limits/VanKampen.lean","def_pos":[69,8],"def_end_pos":[69,27]},{"full_name":"Nonempty.some","def_path":"Mathlib/Logic/Nonempty.lean","def_pos":[81,31],"def_end_pos":[81,44]}]},{"state_before":"J : Type v'\ninst✝⁴ : Category.{u', v'} J\nC : Type u\ninst✝³ : Category.{v, u} C\nK : Type u_1\ninst✝² : Category.{?u.254173, u_1} K\nD : Type u_2\ninst✝¹ : Category.{?u.254180, u_2} D\ninst✝ : HasInitial C\nH : IsUniversalColimit (BinaryCofan.mk (𝟙 (⊥_ C)) (𝟙 (⊥_ C)))\nA : C\nf : A ⟶ ⊥_ C\n⊢ ∀ (j : Discrete WalkingPair),\n    IsPullback ((BinaryCofan.mk (𝟙 A) (𝟙 A)).ι.app j) ((mapPair f f).app j) f\n      ((BinaryCofan.mk (𝟙 (⊥_ C)) (𝟙 (⊥_ C))).ι.app j)","state_after":"no goals","tactic":"rintro ⟨⟨⟩⟩ <;> dsimp <;>\n        exact IsPullback.of_horiz_isIso ⟨(Category.id_comp _).trans (Category.comp_id 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+{"url":"Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean","commit":"","full_name":"Finpartition.IsEquipartition.card_biUnion_offDiag_le'","start":[309,0],"end":[325,63],"file_path":"Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean","tactics":[{"state_before":"α : Type u_1\n𝕜 : Type u_2\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\nA : Finset α\nP : Finpartition A\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε δ : 𝕜\nu v s t : Finset α\nhP : P.IsEquipartition\n⊢ ↑(P.parts.biUnion offDiag).card ≤ ↑A.card * (↑A.card + ↑P.parts.card) / ↑P.parts.card","state_after":"case inl\nα : Type u_1\n𝕜 : Type u_2\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\nA : Finset α\nP : Finpartition A\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε δ : 𝕜\nu v s t : Finset α\nhP : P.IsEquipartition\nh : P.parts = ∅\n⊢ ↑(P.parts.biUnion offDiag).card ≤ ↑A.card * (↑A.card + ↑P.parts.card) / ↑P.parts.card\n\ncase inr\nα : Type u_1\n𝕜 : Type u_2\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\nA : Finset α\nP : Finpartition A\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε δ : 𝕜\nu v s t : Finset α\nhP : P.IsEquipartition\nh : P.parts.Nonempty\n⊢ ↑(P.parts.biUnion offDiag).card ≤ ↑A.card * (↑A.card + ↑P.parts.card) / ↑P.parts.card","tactic":"obtain h | h := P.parts.eq_empty_or_nonempty","premises":[{"full_name":"Finpartition.parts","def_path":"Mathlib/Order/Partition/Finpartition.lean","def_pos":[65,2],"def_end_pos":[65,7]},{"full_name":"Finset.eq_empty_or_nonempty","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[528,8],"def_end_pos":[528,28]}]},{"state_before":"case inr\nα : Type u_1\n𝕜 : Type u_2\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\nA : Finset α\nP : Finpartition A\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε δ : 𝕜\nu v s t : Finset α\nhP : P.IsEquipartition\nh : P.parts.Nonempty\n⊢ ↑(P.parts.biUnion offDiag).card ≤ ↑A.card * (↑A.card + ↑P.parts.card) / ↑P.parts.card","state_after":"case inr.calc_1\nα : Type u_1\n𝕜 : Type u_2\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\nA : Finset α\nP : Finpartition A\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε δ : 𝕜\nu v s t : Finset α\nhP : P.IsEquipartition\nh : P.parts.Nonempty\n⊢ ↑(A.card / P.parts.card + 1) ≤ ↑A.card / ↑P.parts.card + 1\n\ncase inr.calc_2\nα : Type u_1\n𝕜 : Type u_2\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\nA : Finset α\nP : Finpartition A\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε δ : 𝕜\nu v s t : Finset α\nhP : P.IsEquipartition\nh : P.parts.Nonempty\nU : Finset α\nhU : U ∈ P.parts\n⊢ U.offDiag.card ≤ A.card / P.parts.card * (A.card / P.parts.card + 1)","tactic":"calc\n    _ ≤ (P.parts.card : 𝕜) * (↑(A.card / P.parts.card) * ↑(A.card / P.parts.card + 1)) :=\n        mod_cast card_biUnion_le_card_mul _ _ _ fun U hU ↦ ?_\n    _ = P.parts.card * ↑(A.card / P.parts.card) * ↑(A.card / P.parts.card + 1) := by rw [mul_assoc]\n    _ ≤ A.card * (A.card / P.parts.card + 1) :=\n        mul_le_mul (mod_cast Nat.mul_div_le _ _) ?_ (by positivity) (by positivity)\n    _ = _ := by rw [← div_add_same (mod_cast h.card_pos.ne'), mul_div_assoc]","premises":[{"full_name":"Finpartition.parts","def_path":"Mathlib/Order/Partition/Finpartition.lean","def_pos":[65,2],"def_end_pos":[65,7]},{"full_name":"Finset.card","def_path":"Mathlib/Data/Finset/Card.lean","def_pos":[40,4],"def_end_pos":[40,8]},{"full_name":"Finset.card_biUnion_le_card_mul","def_path":"Mathlib/Algebra/Order/BigOperators/Group/Finset.lean","def_pos":[198,8],"def_end_pos":[198,32]},{"full_name":"LT.lt.ne'","def_path":"Mathlib/Order/Basic.lean","def_pos":[267,8],"def_end_pos":[267,11]},{"full_name":"Nat.mul_div_le","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Div.lean","def_pos":[389,8],"def_end_pos":[389,18]},{"full_name":"div_add_same","def_path":"Mathlib/Algebra/Field/Basic.lean","def_pos":[35,8],"def_end_pos":[35,20]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]},{"full_name":"mul_div_assoc","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[330,8],"def_end_pos":[330,21]}]},{"state_before":"case inr.calc_2\nα : Type u_1\n𝕜 : Type u_2\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\nA : Finset α\nP : Finpartition A\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε δ : 𝕜\nu v s t : Finset α\nhP : P.IsEquipartition\nh : P.parts.Nonempty\nU : Finset α\nhU : U ∈ P.parts\n⊢ U.offDiag.card ≤ A.card / P.parts.card * (A.card / P.parts.card + 1)","state_after":"case inr.calc_2\nα : Type u_1\n𝕜 : Type u_2\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\nA : Finset α\nP : Finpartition A\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε δ : 𝕜\nu v s t : Finset α\nhP : P.IsEquipartition\nh : P.parts.Nonempty\nU : Finset α\nhU : U ∈ P.parts\n⊢ (U.card - 1) * U.card ≤ A.card / P.parts.card * (A.card / P.parts.card + 1)","tactic":"suffices (U.card - 1) * U.card ≤ A.card / P.parts.card * (A.card / P.parts.card + 1) by\n    rwa [Nat.mul_sub_right_distrib, one_mul, ← offDiag_card] at this","premises":[{"full_name":"Finpartition.parts","def_path":"Mathlib/Order/Partition/Finpartition.lean","def_pos":[65,2],"def_end_pos":[65,7]},{"full_name":"Finset.card","def_path":"Mathlib/Data/Finset/Card.lean","def_pos":[40,4],"def_end_pos":[40,8]},{"full_name":"Finset.offDiag_card","def_path":"Mathlib/Data/Finset/Prod.lean","def_pos":[293,8],"def_end_pos":[293,20]},{"full_name":"Nat.mul_sub_right_distrib","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[1103,18],"def_end_pos":[1103,39]},{"full_name":"one_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[409,8],"def_end_pos":[409,15]}]},{"state_before":"case inr.calc_2\nα : Type u_1\n𝕜 : Type u_2\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\nA : Finset α\nP : Finpartition A\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε δ : 𝕜\nu v s t : Finset α\nhP : P.IsEquipartition\nh : P.parts.Nonempty\nU : Finset α\nhU : U ∈ P.parts\n⊢ (U.card - 1) * U.card ≤ A.card / P.parts.card * (A.card / P.parts.card + 1)","state_after":"case inr.calc_2\nα : Type u_1\n𝕜 : Type u_2\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\nA : Finset α\nP : Finpartition A\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε δ : 𝕜\nu v s t : Finset α\nhP : P.IsEquipartition\nh : P.parts.Nonempty\nU : Finset α\nhU : U ∈ P.parts\nthis : U.card ≤ A.card / P.parts.card + 1\n⊢ (U.card - 1) * U.card ≤ A.card / P.parts.card * (A.card / P.parts.card + 1)","tactic":"have := hP.card_part_le_average_add_one hU","premises":[{"full_name":"Finpartition.IsEquipartition.card_part_le_average_add_one","def_path":"Mathlib/Order/Partition/Equipartition.lean","def_pos":[67,8],"def_end_pos":[67,52]}]},{"state_before":"case inr.calc_2\nα : Type u_1\n𝕜 : Type u_2\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\nA : Finset α\nP : Finpartition A\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε δ : 𝕜\nu v s t : Finset α\nhP : P.IsEquipartition\nh : P.parts.Nonempty\nU : Finset α\nhU : U ∈ P.parts\nthis : U.card ≤ A.card / P.parts.card + 1\n⊢ (U.card - 1) * U.card ≤ A.card / P.parts.card * (A.card / P.parts.card + 1)","state_after":"case inr.calc_2\nα : Type u_1\n𝕜 : Type u_2\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\nA : Finset α\nP : Finpartition A\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε δ : 𝕜\nu v s t : Finset α\nhP : P.IsEquipartition\nh : P.parts.Nonempty\nU : Finset α\nhU : U ∈ P.parts\nthis : U.card ≤ A.card / P.parts.card + 1\n⊢ A.card / P.parts.card + 1 - 1 ≤ A.card / P.parts.card","tactic":"refine Nat.mul_le_mul ((Nat.sub_le_sub_right this 1).trans ?_) this","premises":[{"full_name":"Nat.mul_le_mul","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[722,18],"def_end_pos":[722,28]},{"full_name":"Nat.sub_le_sub_right","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[1029,18],"def_end_pos":[1029,34]}]},{"state_before":"case inr.calc_2\nα : Type u_1\n𝕜 : Type u_2\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\nA : Finset α\nP : Finpartition A\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε δ : 𝕜\nu v s t : Finset α\nhP : P.IsEquipartition\nh : P.parts.Nonempty\nU : Finset α\nhU : U ∈ P.parts\nthis : U.card ≤ A.card / P.parts.card + 1\n⊢ A.card / P.parts.card + 1 - 1 ≤ A.card / P.parts.card","state_after":"no goals","tactic":"simp only [Nat.add_succ_sub_one, add_zero, card_univ, le_rfl]","premises":[{"full_name":"Finset.card_univ","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[228,8],"def_end_pos":[228,24]},{"full_name":"Nat.add_succ_sub_one","def_path":"Mathlib/Data/Nat/Defs.lean","def_pos":[186,14],"def_end_pos":[186,30]},{"full_name":"add_zero","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[412,2],"def_end_pos":[412,13]},{"full_name":"le_rfl","def_path":"Mathlib/Order/Defs.lean","def_pos":[43,8],"def_end_pos":[43,14]}]}]}
+{"url":"Mathlib/Data/Vector/MapLemmas.lean","commit":"","full_name":"Mathlib.Vector.mapAccumr₂_mapAccumr₂_left_left","start":[108,0],"end":[118,79],"file_path":"Mathlib/Data/Vector/MapLemmas.lean","tactics":[{"state_before":"α : Type\nn : ℕ\nβ : Type\nxs : Vector α n\nys : Vector β n\nγ σ₁ φ σ₂ : Type\ns₂ : σ₂\ns₁ : σ₁\nf₁ : γ → α → σ₁ → σ₁ × φ\nf₂ : α → β → σ₂ → σ₂ × γ\n⊢ mapAccumr₂ f₁ (mapAccumr₂ f₂ xs ys s₂).2 xs s₁ =\n    let m :=\n      mapAccumr₂\n        (fun x y x_1 =>\n          match x_1 with\n          | (s₁, s₂) =>\n            let r₂ := f₂ x y s₂;\n            let r₁ := f₁ r₂.2 x s₁;\n            ((r₁.1, r₂.1), r₁.2))\n        xs ys (s₁, s₂);\n    (m.1.1, m.2)","state_after":"no goals","tactic":"induction xs, ys using Vector.revInductionOn₂ generalizing s₁ s₂ <;> simp_all","premises":[{"full_name":"Mathlib.Vector.revInductionOn₂","def_path":"Mathlib/Data/Vector/Snoc.lean","def_pos":[94,4],"def_end_pos":[94,19]}]}]}
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+{"url":"Mathlib/Computability/Primrec.lean","commit":"","full_name":"Primrec.vector_get","start":[1205,0],"end":[1209,9],"file_path":"Mathlib/Computability/Primrec.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nσ : Type u_4\ninst✝³ : Primcodable α\ninst✝² : Primcodable β\ninst✝¹ : Primcodable γ\ninst✝ : Primcodable σ\nn : ℕ\na : Vector α n × Fin n\n⊢ a.1.toList.get? ↑a.2 = some (a.1.get a.2)","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nσ : Type u_4\ninst✝³ : Primcodable α\ninst✝² : Primcodable β\ninst✝¹ : Primcodable γ\ninst✝ : Primcodable σ\nn : ℕ\na : Vector α n × Fin n\n⊢ a.1.toList.get? ↑a.2 = a.1.toList.get? ↑(Fin.cast ⋯ a.2)","tactic":"rw [Vector.get_eq_get, ← List.get?_eq_get]","premises":[{"full_name":"List.get?_eq_get","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean","def_pos":[143,8],"def_end_pos":[143,19]},{"full_name":"Mathlib.Vector.get_eq_get","def_path":"Mathlib/Data/Vector/Basic.lean","def_pos":[101,8],"def_end_pos":[101,18]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nσ : Type u_4\ninst✝³ : Primcodable α\ninst✝² : Primcodable β\ninst✝¹ : Primcodable γ\ninst✝ : Primcodable σ\nn : ℕ\na : Vector α n × Fin n\n⊢ a.1.toList.get? ↑a.2 = a.1.toList.get? ↑(Fin.cast ⋯ a.2)","state_after":"no goals","tactic":"rfl","premises":[]}]}
+{"url":"Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean","commit":"","full_name":"iteratedDeriv_eq_equiv_comp","start":[197,0],"end":[201,12],"file_path":"Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nn : ℕ\nf : 𝕜 → F\ns : Set 𝕜\nx : 𝕜\n⊢ iteratedDeriv n f = ⇑(ContinuousMultilinearMap.piFieldEquiv 𝕜 (Fin n) F).symm ∘ iteratedFDeriv 𝕜 n f","state_after":"case h\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nn : ℕ\nf : 𝕜 → F\ns : Set 𝕜\nx✝ x : 𝕜\n⊢ iteratedDeriv n f x = (⇑(ContinuousMultilinearMap.piFieldEquiv 𝕜 (Fin n) F).symm ∘ iteratedFDeriv 𝕜 n f) x","tactic":"ext x","premises":[]},{"state_before":"case h\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nn : ℕ\nf : 𝕜 → F\ns : Set 𝕜\nx✝ x : 𝕜\n⊢ iteratedDeriv n f x = (⇑(ContinuousMultilinearMap.piFieldEquiv 𝕜 (Fin n) F).symm ∘ iteratedFDeriv 𝕜 n f) x","state_after":"no goals","tactic":"rfl","premises":[]}]}
+{"url":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","commit":"","full_name":"Polynomial.coeff_mul_add_eq_of_natDegree_le","start":[979,0],"end":[991,22],"file_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","tactics":[{"state_before":"R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ p q : R[X]\nι : Type u_1\ndf dg : ℕ\nf g : R[X]\nhdf : f.natDegree ≤ df\nhdg : g.natDegree ≤ dg\n⊢ (f * g).coeff (df + dg) = f.coeff df * g.coeff dg","state_after":"case h\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ p q : R[X]\nι : Type u_1\ndf dg : ℕ\nf g : R[X]\nhdf : f.natDegree ≤ df\nhdg : g.natDegree ≤ dg\n⊢ (df, dg) ∈ antidiagonal (df + dg)\n\ncase h₀\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ p q : R[X]\nι : Type u_1\ndf dg : ℕ\nf g : R[X]\nhdf : f.natDegree ≤ df\nhdg : g.natDegree ≤ dg\n⊢ ∀ b ∈ antidiagonal (df + dg), b ≠ (df, dg) → f.coeff b.1 * g.coeff b.2 = 0","tactic":"rw [coeff_mul, Finset.sum_eq_single_of_mem (df, dg)]","premises":[{"full_name":"Finset.sum_eq_single_of_mem","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[854,2],"def_end_pos":[854,13]},{"full_name":"Polynomial.coeff_mul","def_path":"Mathlib/Algebra/Polynomial/Coeff.lean","def_pos":[113,8],"def_end_pos":[113,17]},{"full_name":"Prod.mk","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[481,2],"def_end_pos":[481,4]}]},{"state_before":"case h₀\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ p q : R[X]\nι : Type u_1\ndf dg : ℕ\nf g : R[X]\nhdf : f.natDegree ≤ df\nhdg : g.natDegree ≤ dg\n⊢ ∀ b ∈ antidiagonal (df + dg), b ≠ (df, dg) → f.coeff b.1 * g.coeff b.2 = 0","state_after":"case h₀.mk\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ p q : R[X]\nι : Type u_1\ndf dg : ℕ\nf g : R[X]\nhdf : f.natDegree ≤ df\nhdg : g.natDegree ≤ dg\ndf' dg' : ℕ\nhmem : (df', dg') ∈ antidiagonal (df + dg)\nhne : (df', dg') ≠ (df, dg)\n⊢ f.coeff (df', dg').1 * g.coeff (df', dg').2 = 0","tactic":"rintro ⟨df', dg'⟩ hmem hne","premises":[]},{"state_before":"case h₀.mk\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ p q : R[X]\nι : Type u_1\ndf dg : ℕ\nf g : R[X]\nhdf : f.natDegree ≤ df\nhdg : g.natDegree ≤ dg\ndf' dg' : ℕ\nhmem : (df', dg') ∈ antidiagonal (df + dg)\nhne : (df', dg') ≠ (df, dg)\n⊢ f.coeff (df', dg').1 * g.coeff (df', dg').2 = 0","state_after":"case h₀.mk.inl\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ p q : R[X]\nι : Type u_1\ndf dg : ℕ\nf g : R[X]\nhdf : f.natDegree ≤ df\nhdg : g.natDegree ≤ dg\ndf' dg' : ℕ\nhmem : (df', dg') ∈ antidiagonal (df + dg)\nhne : (df', dg') ≠ (df, dg)\nh : df < df'\n⊢ f.coeff (df', dg').1 * g.coeff (df', dg').2 = 0\n\ncase h₀.mk.inr\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ p q : R[X]\nι : Type u_1\ndf dg : ℕ\nf g : R[X]\nhdf : f.natDegree ≤ df\nhdg : g.natDegree ≤ dg\ndf' dg' : ℕ\nhmem : (df', dg') ∈ antidiagonal (df + dg)\nhne : (df', dg') ≠ (df, dg)\nhdf' : df' ≤ df\n⊢ f.coeff (df', dg').1 * g.coeff (df', dg').2 = 0","tactic":"obtain h | hdf' := lt_or_le df df'","premises":[{"full_name":"lt_or_le","def_path":"Mathlib/Order/Defs.lean","def_pos":[287,8],"def_end_pos":[287,16]}]},{"state_before":"case h₀.mk.inr\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ p q : R[X]\nι : Type u_1\ndf dg : ℕ\nf g : R[X]\nhdf : f.natDegree ≤ df\nhdg : g.natDegree ≤ dg\ndf' dg' : ℕ\nhmem : (df', dg') ∈ antidiagonal (df + dg)\nhne : (df', dg') ≠ (df, dg)\nhdf' : df' ≤ df\n⊢ f.coeff (df', dg').1 * g.coeff (df', dg').2 = 0","state_after":"case h₀.mk.inr.inl\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ p q : R[X]\nι : Type u_1\ndf dg : ℕ\nf g : R[X]\nhdf : f.natDegree ≤ df\nhdg : g.natDegree ≤ dg\ndf' dg' : ℕ\nhmem : (df', dg') ∈ antidiagonal (df + dg)\nhne : (df', dg') ≠ (df, dg)\nhdf' : df' ≤ df\nh : dg < dg'\n⊢ f.coeff (df', dg').1 * g.coeff (df', dg').2 = 0\n\ncase h₀.mk.inr.inr\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ p q : R[X]\nι : Type u_1\ndf dg : ℕ\nf g : R[X]\nhdf : f.natDegree ≤ df\nhdg : g.natDegree ≤ dg\ndf' dg' : ℕ\nhmem : (df', dg') ∈ antidiagonal (df + dg)\nhne : (df', dg') ≠ (df, dg)\nhdf' : df' ≤ df\nhdg' : dg' ≤ dg\n⊢ f.coeff (df', dg').1 * g.coeff (df', dg').2 = 0","tactic":"obtain h | hdg' := lt_or_le dg dg'","premises":[{"full_name":"lt_or_le","def_path":"Mathlib/Order/Defs.lean","def_pos":[287,8],"def_end_pos":[287,16]}]},{"state_before":"case h₀.mk.inr.inr\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ p q : R[X]\nι : Type u_1\ndf dg : ℕ\nf g : R[X]\nhdf : f.natDegree ≤ df\nhdg : g.natDegree ≤ dg\ndf' dg' : ℕ\nhmem : (df', dg') ∈ antidiagonal (df + dg)\nhne : (df', dg') ≠ (df, dg)\nhdf' : df' ≤ df\nhdg' : dg' ≤ dg\n⊢ f.coeff (df', dg').1 * g.coeff (df', dg').2 = 0","state_after":"case h₀.mk.inr.inr.intro\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ p q : R[X]\nι : Type u_1\nf g : R[X]\ndf' dg' : ℕ\nhdf : f.natDegree ≤ df'\nhdf' : df' ≤ df'\nhdg : g.natDegree ≤ dg'\nhdg' : dg' ≤ dg'\nhmem : (df', dg') ∈ antidiagonal (df' + dg')\nhne : (df', dg') ≠ (df', dg')\n⊢ f.coeff (df', dg').1 * g.coeff (df', dg').2 = 0","tactic":"obtain ⟨rfl, rfl⟩ :=\n    (add_eq_add_iff_eq_and_eq hdf' hdg').mp (mem_antidiagonal.1 hmem)","premises":[{"full_name":"Finset.HasAntidiagonal.mem_antidiagonal","def_path":"Mathlib/Algebra/Order/Antidiag/Prod.lean","def_pos":[58,2],"def_end_pos":[58,18]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"add_eq_add_iff_eq_and_eq","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[270,2],"def_end_pos":[270,13]}]},{"state_before":"case h₀.mk.inr.inr.intro\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ p q : R[X]\nι : Type u_1\nf g : R[X]\ndf' dg' : ℕ\nhdf : f.natDegree ≤ df'\nhdf' : df' ≤ df'\nhdg : g.natDegree ≤ dg'\nhdg' : dg' ≤ dg'\nhmem : (df', dg') ∈ antidiagonal (df' + dg')\nhne : (df', dg') ≠ (df', dg')\n⊢ f.coeff (df', dg').1 * g.coeff (df', dg').2 = 0","state_after":"no goals","tactic":"exact (hne rfl).elim","premises":[{"full_name":"False.elim","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[236,20],"def_end_pos":[236,30]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
+{"url":"Mathlib/Data/List/Enum.lean","commit":"","full_name":"List.get_enum","start":[63,0],"end":[65,6],"file_path":"Mathlib/Data/List/Enum.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nl : List α\ni : Fin l.enum.length\n⊢ l.enum.get i = (↑i, l.get (Fin.cast ⋯ i))","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/Order/Filter/Basic.lean","commit":"","full_name":"Filter.mem_comap_prod_mk","start":[1732,0],"end":[1735,89],"file_path":"Mathlib/Order/Filter/Basic.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nδ : Type u_1\nι : Sort x\nl✝ : Filter α\nf : α → β\nl : Filter β\np : α → Prop\ns✝ : Set α\nx : α\ns : Set β\nF : Filter (α × β)\n⊢ s ∈ comap (Prod.mk x) F ↔ {p | p.1 = x → p.2 ∈ s} ∈ F","state_after":"no goals","tactic":"simp_rw [mem_comap', Prod.ext_iff, and_imp, @forall_swap β (_ = _), forall_eq, eq_comm]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Filter.mem_comap'","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1723,8],"def_end_pos":[1723,18]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Prod.ext_iff","def_path":"Mathlib/Data/Prod/Basic.lean","def_pos":[101,18],"def_end_pos":[101,25]},{"full_name":"and_imp","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[115,16],"def_end_pos":[115,23]},{"full_name":"eq_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[811,8],"def_end_pos":[811,15]},{"full_name":"forall_eq","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[273,16],"def_end_pos":[273,25]},{"full_name":"forall_swap","def_path":"Mathlib/Logic/Basic.lean","def_pos":[463,8],"def_end_pos":[463,19]}]}]}
+{"url":"Mathlib/CategoryTheory/Adjunction/Over.lean","commit":"","full_name":"CategoryTheory.Under.pushout_map","start":[128,0],"end":[135,38],"file_path":"Mathlib/CategoryTheory/Adjunction/Over.lean","tactics":[{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\nX✝ : C\ninst✝ : HasPushouts C\nX Y : C\nf : X ⟶ Y\nx x' : Under X\nu : x ⟶ x'\n⊢ x.hom ≫ u.right ≫ pushout.inl x'.hom f = f ≫ pushout.inr x'.hom f","state_after":"no goals","tactic":"simp [← pushout.condition]","premises":[{"full_name":"CategoryTheory.Limits.pushout.condition","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Pullback/HasPullback.lean","def_pos":[205,8],"def_end_pos":[205,25]}]}]}
+{"url":"Mathlib/ModelTheory/Satisfiability.lean","commit":"","full_name":"FirstOrder.Language.Theory.ModelsBoundedFormula.realize_sentence","start":[316,0],"end":[325,29],"file_path":"Mathlib/ModelTheory/Satisfiability.lean","tactics":[{"state_before":"L : Language\nT : L.Theory\nα : Type w\nn : ℕ\nφ : L.Sentence\nh : T ⊨ᵇ φ\nM : Type u_1\ninst✝² : L.Structure M\ninst✝¹ : M ⊨ T\ninst✝ : Nonempty M\n⊢ M ⊨ φ","state_after":"L : Language\nT : L.Theory\nα : Type w\nn : ℕ\nφ : L.Sentence\nh : ¬(T ∪ {Formula.not φ}).IsSatisfiable\nM : Type u_1\ninst✝² : L.Structure M\ninst✝¹ : M ⊨ T\ninst✝ : Nonempty M\n⊢ M ⊨ φ","tactic":"rw [models_iff_not_satisfiable] at h","premises":[{"full_name":"FirstOrder.Language.Theory.models_iff_not_satisfiable","def_path":"Mathlib/ModelTheory/Satisfiability.lean","def_pos":[299,8],"def_end_pos":[299,34]}]},{"state_before":"L : Language\nT : L.Theory\nα : Type w\nn : ℕ\nφ : L.Sentence\nh : ¬(T ∪ {Formula.not φ}).IsSatisfiable\nM : Type u_1\ninst✝² : L.Structure M\ninst✝¹ : M ⊨ T\ninst✝ : Nonempty M\n⊢ M ⊨ φ","state_after":"L : Language\nT : L.Theory\nα : Type w\nn : ℕ\nφ : L.Sentence\nM : Type u_1\ninst✝² : L.Structure M\ninst✝¹ : M ⊨ T\ninst✝ : Nonempty M\nh : ¬M ⊨ φ\n⊢ (T ∪ {Formula.not φ}).IsSatisfiable","tactic":"contrapose! h","premises":[{"full_name":"Mathlib.Tactic.Contrapose.mtr","def_path":"Mathlib/Tactic/Contrapose.lean","def_pos":[24,6],"def_end_pos":[24,9]}]},{"state_before":"L : Language\nT : L.Theory\nα : Type w\nn : ℕ\nφ : L.Sentence\nM : Type u_1\ninst✝² : L.Structure M\ninst✝¹ : M ⊨ T\ninst✝ : Nonempty M\nh : ¬M ⊨ φ\n⊢ (T ∪ {Formula.not φ}).IsSatisfiable","state_after":"L : Language\nT : L.Theory\nα : Type w\nn : ℕ\nφ : L.Sentence\nM : Type u_1\ninst✝² : L.Structure M\ninst✝¹ : M ⊨ T\ninst✝ : Nonempty M\nh : ¬M ⊨ φ\nthis : M ⊨ T ∪ {Formula.not φ}\n⊢ (T ∪ {Formula.not φ}).IsSatisfiable","tactic":"have : M ⊨ T ∪ {Formula.not φ} := by\n    simp only [Set.union_singleton, model_iff, Set.mem_insert_iff, forall_eq_or_imp,\n      Sentence.realize_not]\n    rw [← model_iff]\n    exact ⟨h, inferInstance⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"FirstOrder.Language.Formula.not","def_path":"Mathlib/ModelTheory/Syntax.lean","def_pos":[723,24],"def_end_pos":[723,27]},{"full_name":"FirstOrder.Language.Sentence.realize_not","def_path":"Mathlib/ModelTheory/Semantics.lean","def_pos":[594,8],"def_end_pos":[594,28]},{"full_name":"FirstOrder.Language.Theory.Model","def_path":"Mathlib/ModelTheory/Semantics.lean","def_pos":[658,6],"def_end_pos":[658,18]},{"full_name":"FirstOrder.Language.Theory.model_iff","def_path":"Mathlib/ModelTheory/Semantics.lean","def_pos":[668,8],"def_end_pos":[668,24]},{"full_name":"Set.mem_insert_iff","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[895,8],"def_end_pos":[895,22]},{"full_name":"Set.union_singleton","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1060,8],"def_end_pos":[1060,23]},{"full_name":"Singleton.singleton","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[467,2],"def_end_pos":[467,11]},{"full_name":"Union.union","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[402,2],"def_end_pos":[402,7]},{"full_name":"forall_eq_or_imp","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[295,16],"def_end_pos":[295,32]},{"full_name":"inferInstance","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[99,7],"def_end_pos":[99,20]}]},{"state_before":"L : Language\nT : L.Theory\nα : Type w\nn : ℕ\nφ : L.Sentence\nM : Type u_1\ninst✝² : L.Structure M\ninst✝¹ : M ⊨ T\ninst✝ : Nonempty M\nh : ¬M ⊨ φ\nthis : M ⊨ T ∪ {Formula.not φ}\n⊢ (T ∪ {Formula.not φ}).IsSatisfiable","state_after":"no goals","tactic":"exact Model.isSatisfiable M","premises":[{"full_name":"FirstOrder.Language.Theory.Model.isSatisfiable","def_path":"Mathlib/ModelTheory/Satisfiability.lean","def_pos":[70,8],"def_end_pos":[70,27]}]}]}
+{"url":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","commit":"","full_name":"Polynomial.leadingCoeff_monic_mul","start":[911,0],"end":[916,60],"file_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","tactics":[{"state_before":"R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝¹ p✝ q✝ : R[X]\nι : Type u_1\np q : R[X]\nhp : p.Monic\n⊢ (p * q).leadingCoeff = q.leadingCoeff","state_after":"case inl\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝¹ p✝ q : R[X]\nι : Type u_1\np : R[X]\nhp : p.Monic\n⊢ (p * 0).leadingCoeff = leadingCoeff 0\n\ncase inr\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝¹ p✝ q✝ : R[X]\nι : Type u_1\np q : R[X]\nhp : p.Monic\nH : q ≠ 0\n⊢ (p * q).leadingCoeff = q.leadingCoeff","tactic":"rcases eq_or_ne q 0 with (rfl | H)","premises":[{"full_name":"eq_or_ne","def_path":"Mathlib/Logic/Basic.lean","def_pos":[167,8],"def_end_pos":[167,16]}]}]}
+{"url":"Mathlib/Data/Nat/Bitwise.lean","commit":"","full_name":"Nat.bit_mod_two_eq_one_iff","start":[101,0],"end":[104,42],"file_path":"Mathlib/Data/Nat/Bitwise.lean","tactics":[{"state_before":"f : Bool → Bool → Bool\na : Bool\nx : ℕ\n⊢ bit a x % 2 = 1 ↔ a = true","state_after":"f : Bool → Bool → Bool\na : Bool\nx : ℕ\n⊢ (if a = true then 1 else 0) = 1 ↔ a = true","tactic":"rw [bit_mod_two]","premises":[{"full_name":"Nat.bit_mod_two","def_path":"Mathlib/Data/Nat/Bitwise.lean","def_pos":[88,6],"def_end_pos":[88,17]}]},{"state_before":"f : Bool → Bool → Bool\na : Bool\nx : ℕ\n⊢ (if a = true then 1 else 0) = 1 ↔ a = true","state_after":"no goals","tactic":"split_ifs <;> simp_all","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
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+{"url":"Mathlib/Data/Set/Finite.lean","commit":"","full_name":"Set.empty_card'","start":[1091,0],"end":[1092,6],"file_path":"Mathlib/Data/Set/Finite.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\nh : Fintype ↑∅\n⊢ Fintype.card ↑∅ = 0","state_after":"no goals","tactic":"simp","premises":[]}]}
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+{"url":"Mathlib/CategoryTheory/Sites/Sieves.lean","commit":"","full_name":"CategoryTheory.Sieve.functorPushforward_functor","start":[729,0],"end":[736,96],"file_path":"Mathlib/CategoryTheory/Sites/Sieves.lean","tactics":[{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y Z : C\nf : Y ⟶ X\nS✝ R : Sieve X\nE : Type u₃\ninst✝ : Category.{v₃, u₃} E\nG : D ⥤ E\nS : Sieve X\ne : C ≌ D\n⊢ functorPushforward e.functor S = functorPullback e.inverse (pullback (e.unitInv.app X) S)","state_after":"case h\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y✝ Z : C\nf : Y✝ ⟶ X\nS✝ R : Sieve X\nE : Type u₃\ninst✝ : Category.{v₃, u₃} E\nG : D ⥤ E\nS : Sieve X\ne : C ≌ D\nY : D\niYX : Y ⟶ e.functor.obj X\n⊢ (functorPushforward e.functor S).arrows iYX ↔ (functorPullback e.inverse (pullback (e.unitInv.app X) S)).arrows iYX","tactic":"ext Y iYX","premises":[]},{"state_before":"case h\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y✝ Z : C\nf : Y✝ ⟶ X\nS✝ R : Sieve X\nE : Type u₃\ninst✝ : Category.{v₃, u₃} E\nG : D ⥤ E\nS : Sieve X\ne : C ≌ D\nY : D\niYX : Y ⟶ e.functor.obj X\n⊢ (functorPushforward e.functor S).arrows iYX ↔ (functorPullback e.inverse (pullback (e.unitInv.app X) S)).arrows iYX","state_after":"case h.mp\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y✝ Z : C\nf : Y✝ ⟶ X\nS✝ R : Sieve X\nE : Type u₃\ninst✝ : Category.{v₃, u₃} E\nG : D ⥤ E\nS : Sieve X\ne : C ≌ D\nY : D\niYX : Y ⟶ e.functor.obj X\n⊢ (functorPushforward e.functor S).arrows iYX → (functorPullback e.inverse (pullback (e.unitInv.app X) S)).arrows iYX\n\ncase h.mpr\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y✝ Z : C\nf : Y✝ ⟶ X\nS✝ R : Sieve X\nE : Type u₃\ninst✝ : Category.{v₃, u₃} E\nG : D ⥤ E\nS : Sieve X\ne : C ≌ D\nY : D\niYX : Y ⟶ e.functor.obj X\n⊢ (functorPullback e.inverse (pullback (e.unitInv.app X) S)).arrows iYX → (functorPushforward e.functor S).arrows iYX","tactic":"constructor","premises":[]}]}
+{"url":"Mathlib/RingTheory/Perfection.lean","commit":"","full_name":"Perfection.pthRoot_frobenius","start":[132,0],"end":[134,93],"file_path":"Mathlib/RingTheory/Perfection.lean","tactics":[{"state_before":"R : Type u₁\ninst✝¹ : CommSemiring R\np : ℕ\nhp : Fact (Nat.Prime p)\ninst✝ : CharP R p\nx : Ring.Perfection R p\nn : ℕ\n⊢ (coeff R p n) (((pthRoot R p).comp (frobenius (Ring.Perfection R p) p)) x) =\n    (coeff R p n) ((RingHom.id (Ring.Perfection R p)) x)","state_after":"no goals","tactic":"rw [RingHom.comp_apply, RingHom.id_apply, coeff_pthRoot, coeff_frobenius]","premises":[{"full_name":"Perfection.coeff_frobenius","def_path":"Mathlib/RingTheory/Perfection.lean","def_pos":[120,8],"def_end_pos":[120,23]},{"full_name":"Perfection.coeff_pthRoot","def_path":"Mathlib/RingTheory/Perfection.lean","def_pos":[111,8],"def_end_pos":[111,21]},{"full_name":"RingHom.comp_apply","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[575,8],"def_end_pos":[575,18]},{"full_name":"RingHom.id_apply","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[549,8],"def_end_pos":[549,16]}]}]}
+{"url":"Mathlib/AlgebraicTopology/DoldKan/SplitSimplicialObject.lean","commit":"","full_name":"SimplicialObject.Splitting.σ_comp_πSummand_id_eq_zero","start":[66,0],"end":[79,7],"file_path":"Mathlib/AlgebraicTopology/DoldKan/SplitSimplicialObject.lean","tactics":[{"state_before":"C : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\nX : SimplicialObject C\ns : Splitting X\ninst✝ : Preadditive C\nn : ℕ\ni : Fin (n + 1)\n⊢ X.σ i ≫ s.πSummand (IndexSet.id (op [n + 1])) = 0","state_after":"case h\nC : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\nX : SimplicialObject C\ns : Splitting X\ninst✝ : Preadditive C\nn : ℕ\ni : Fin (n + 1)\n⊢ ∀ (A : IndexSet (op [n])),\n    (s.cofan (op [n])).inj A ≫ X.σ i ≫ s.πSummand (IndexSet.id (op [n + 1])) = (s.cofan (op [n])).inj A ≫ 0","tactic":"apply s.hom_ext'","premises":[{"full_name":"SimplicialObject.Splitting.hom_ext'","def_path":"Mathlib/AlgebraicTopology/SplitSimplicialObject.lean","def_pos":[246,8],"def_end_pos":[246,16]}]},{"state_before":"case h\nC : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\nX : SimplicialObject C\ns : Splitting X\ninst✝ : Preadditive C\nn : ℕ\ni : Fin (n + 1)\n⊢ ∀ (A : IndexSet (op [n])),\n    (s.cofan (op [n])).inj A ≫ X.σ i ≫ s.πSummand (IndexSet.id (op [n + 1])) = (s.cofan (op [n])).inj A ≫ 0","state_after":"case h\nC : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\nX : SimplicialObject C\ns : Splitting X\ninst✝ : Preadditive C\nn : ℕ\ni : Fin (n + 1)\nA : IndexSet (op [n])\n⊢ (s.cofan (op [n])).inj A ≫ X.σ i ≫ s.πSummand (IndexSet.id (op [n + 1])) = (s.cofan (op [n])).inj A ≫ 0","tactic":"intro A","premises":[]},{"state_before":"case h\nC : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\nX : SimplicialObject C\ns : Splitting X\ninst✝ : Preadditive C\nn : ℕ\ni : Fin (n + 1)\nA : IndexSet (op [n])\n⊢ (s.cofan (op [n])).inj A ≫ X.σ i ≫ s.πSummand (IndexSet.id (op [n + 1])) = (s.cofan (op [n])).inj A ≫ 0","state_after":"case h\nC : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\nX : SimplicialObject C\ns : Splitting X\ninst✝ : Preadditive C\nn : ℕ\ni : Fin (n + 1)\nA : IndexSet (op [n])\n⊢ (s.cofan (op [n])).inj A ≫ X.map (SimplexCategory.σ i).op ≫ s.πSummand (IndexSet.id (op [n + 1])) =\n    (s.cofan (op [n])).inj A ≫ 0","tactic":"dsimp only [SimplicialObject.σ]","premises":[{"full_name":"CategoryTheory.SimplicialObject.σ","def_path":"Mathlib/AlgebraicTopology/SimplicialObject.lean","def_pos":[86,4],"def_end_pos":[86,5]}]},{"state_before":"case h\nC : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\nX : SimplicialObject C\ns : Splitting X\ninst✝ : Preadditive C\nn : ℕ\ni : Fin (n + 1)\nA : IndexSet (op [n])\n⊢ (s.cofan (op [n])).inj A ≫ X.map (SimplexCategory.σ i).op ≫ s.πSummand (IndexSet.id (op [n + 1])) =\n    (s.cofan (op [n])).inj A ≫ 0","state_after":"case h.h\nC : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\nX : SimplicialObject C\ns : Splitting X\ninst✝ : Preadditive C\nn : ℕ\ni : Fin (n + 1)\nA : IndexSet (op [n])\n⊢ IndexSet.id (op [n + 1]) ≠ A.epiComp (SimplexCategory.σ i).op","tactic":"rw [comp_zero, s.cofan_inj_epi_naturality_assoc A (SimplexCategory.σ i).op,\n    cofan_inj_πSummand_eq_zero]","premises":[{"full_name":"CategoryTheory.Limits.comp_zero","def_path":"Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean","def_pos":[61,8],"def_end_pos":[61,17]},{"full_name":"Quiver.Hom.op","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[136,4],"def_end_pos":[136,10]},{"full_name":"SimplexCategory.σ","def_path":"Mathlib/AlgebraicTopology/SimplexCategory.lean","def_pos":[199,4],"def_end_pos":[199,5]},{"full_name":"SimplicialObject.Splitting.cofan_inj_πSummand_eq_zero","def_path":"Mathlib/AlgebraicTopology/DoldKan/SplitSimplicialObject.lean","def_pos":[49,8],"def_end_pos":[49,34]}]},{"state_before":"case h.h\nC : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\nX : SimplicialObject C\ns : Splitting X\ninst✝ : Preadditive C\nn : ℕ\ni : Fin (n + 1)\nA : IndexSet (op [n])\n⊢ IndexSet.id (op [n + 1]) ≠ A.epiComp (SimplexCategory.σ i).op","state_after":"case h.h\nC : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\nX : SimplicialObject C\ns : Splitting X\ninst✝ : Preadditive C\nn : ℕ\ni : Fin (n + 1)\nA : IndexSet (op [n])\n⊢ A.epiComp (SimplexCategory.σ i).op ≠ IndexSet.id (op [n + 1])","tactic":"rw [ne_comm]","premises":[{"full_name":"ne_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[706,8],"def_end_pos":[706,15]}]},{"state_before":"case h.h\nC : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\nX : SimplicialObject C\ns : Splitting X\ninst✝ : Preadditive C\nn : ℕ\ni : Fin (n + 1)\nA : IndexSet (op [n])\n⊢ A.epiComp (SimplexCategory.σ i).op ≠ IndexSet.id (op [n + 1])","state_after":"case h.h\nC : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\nX : SimplicialObject C\ns : Splitting X\ninst✝ : Preadditive C\nn : ℕ\ni : Fin (n + 1)\nA : IndexSet (op [n])\n⊢ ¬(A.epiComp (SimplexCategory.σ i).op).EqId","tactic":"change ¬(A.epiComp (SimplexCategory.σ i).op).EqId","premises":[{"full_name":"Not","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[224,4],"def_end_pos":[224,7]},{"full_name":"Quiver.Hom.op","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[136,4],"def_end_pos":[136,10]},{"full_name":"SimplexCategory.σ","def_path":"Mathlib/AlgebraicTopology/SimplexCategory.lean","def_pos":[199,4],"def_end_pos":[199,5]},{"full_name":"SimplicialObject.Splitting.IndexSet.EqId","def_path":"Mathlib/AlgebraicTopology/SplitSimplicialObject.lean","def_pos":[115,4],"def_end_pos":[115,8]},{"full_name":"SimplicialObject.Splitting.IndexSet.epiComp","def_path":"Mathlib/AlgebraicTopology/SplitSimplicialObject.lean","def_pos":[165,4],"def_end_pos":[165,11]}]},{"state_before":"case h.h\nC : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\nX : SimplicialObject C\ns : Splitting X\ninst✝ : Preadditive C\nn : ℕ\ni : Fin (n + 1)\nA : IndexSet (op [n])\n⊢ ¬(A.epiComp (SimplexCategory.σ i).op).EqId","state_after":"case h.h\nC : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\nX : SimplicialObject C\ns : Splitting X\ninst✝ : Preadditive C\nn : ℕ\ni : Fin (n + 1)\nA : IndexSet (op [n])\n⊢ ¬(unop (A.epiComp (SimplexCategory.σ i).op).fst).len = (unop (op [n + 1])).len","tactic":"rw [IndexSet.eqId_iff_len_eq]","premises":[{"full_name":"SimplicialObject.Splitting.IndexSet.eqId_iff_len_eq","def_path":"Mathlib/AlgebraicTopology/SplitSimplicialObject.lean","def_pos":[133,8],"def_end_pos":[133,23]}]},{"state_before":"case h.h\nC : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\nX : SimplicialObject C\ns : Splitting X\ninst✝ : Preadditive C\nn : ℕ\ni : Fin (n + 1)\nA : IndexSet (op [n])\n⊢ ¬(unop (A.epiComp (SimplexCategory.σ i).op).fst).len = (unop (op [n + 1])).len","state_after":"case h.h\nC : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\nX : SimplicialObject C\ns : Splitting X\ninst✝ : Preadditive C\nn : ℕ\ni : Fin (n + 1)\nA : IndexSet (op [n])\nh : (unop A.fst).len ≤ (unop (op [n])).len\n⊢ ¬(unop (A.epiComp (SimplexCategory.σ i).op).fst).len = (unop (op [n + 1])).len","tactic":"have h := SimplexCategory.len_le_of_epi (inferInstance : Epi A.e)","premises":[{"full_name":"CategoryTheory.Epi","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[241,6],"def_end_pos":[241,9]},{"full_name":"SimplexCategory.len_le_of_epi","def_path":"Mathlib/AlgebraicTopology/SimplexCategory.lean","def_pos":[533,8],"def_end_pos":[533,21]},{"full_name":"SimplicialObject.Splitting.IndexSet.e","def_path":"Mathlib/AlgebraicTopology/SplitSimplicialObject.lean","def_pos":[63,4],"def_end_pos":[63,5]},{"full_name":"inferInstance","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[99,7],"def_end_pos":[99,20]}]},{"state_before":"case h.h\nC : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\nX : SimplicialObject C\ns : Splitting X\ninst✝ : Preadditive C\nn : ℕ\ni : Fin (n + 1)\nA : IndexSet (op [n])\nh : (unop A.fst).len ≤ (unop (op [n])).len\n⊢ ¬(unop (A.epiComp (SimplexCategory.σ i).op).fst).len = (unop (op [n + 1])).len","state_after":"case h.h\nC : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\nX : SimplicialObject C\ns : Splitting X\ninst✝ : Preadditive C\nn : ℕ\ni : Fin (n + 1)\nA : IndexSet (op [n])\nh : (unop A.fst).len ≤ n\n⊢ ¬(unop A.fst).len = n + 1","tactic":"dsimp at h ⊢","premises":[]},{"state_before":"case h.h\nC : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\nX : SimplicialObject C\ns : Splitting X\ninst✝ : Preadditive C\nn : ℕ\ni : Fin (n + 1)\nA : IndexSet (op [n])\nh : (unop A.fst).len ≤ n\n⊢ ¬(unop A.fst).len = n + 1","state_after":"no goals","tactic":"omega","premises":[]}]}
+{"url":"Mathlib/RingTheory/AdicCompletion/Algebra.lean","commit":"","full_name":"AdicCompletion.val_smul_eq_evalₐ_smul","start":[214,0],"end":[217,45],"file_path":"Mathlib/RingTheory/AdicCompletion/Algebra.lean","tactics":[{"state_before":"R : Type u_1\ninst✝² : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nn : ℕ\nr : AdicCompletion I R\nx : M ⧸ I ^ n • ⊤\n⊢ ↑r n • x = (evalₐ I n) r • x","state_after":"R : Type u_1\ninst✝² : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nn : ℕ\nr✝ : AdicCompletion I R\nx : M ⧸ I ^ n • ⊤\nr : AdicCauchySequence I R\n⊢ ↑((mk I R) r) n • x = (evalₐ I n) ((mk I R) r) • x","tactic":"apply induction_on I R r (fun r ↦ ?_)","premises":[{"full_name":"AdicCompletion.induction_on","def_path":"Mathlib/RingTheory/AdicCompletion/Basic.lean","def_pos":[490,8],"def_end_pos":[490,20]}]},{"state_before":"R : Type u_1\ninst✝² : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nn : ℕ\nr✝ : AdicCompletion I R\nx : M ⧸ I ^ n • ⊤\nr : AdicCauchySequence I R\n⊢ ↑((mk I R) r) n • x = (evalₐ I n) ((mk I R) r) • x","state_after":"no goals","tactic":"exact Quotient.inductionOn' x (fun x ↦ rfl)","premises":[{"full_name":"Quotient.inductionOn'","def_path":"Mathlib/Data/Quot.lean","def_pos":[597,18],"def_end_pos":[597,30]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
+{"url":"Mathlib/Data/List/MinMax.lean","commit":"","full_name":"List.argmax_eq_none","start":[146,0],"end":[147,71],"file_path":"Mathlib/Data/List/MinMax.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder β\ninst✝ : DecidableRel fun x x_1 => x < x_1\nf : α → β\nl : List α\no : Option α\na m : α\n⊢ argmax f l = none ↔ l = []","state_after":"no goals","tactic":"simp [argmax]","premises":[{"full_name":"List.argmax","def_path":"Mathlib/Data/List/MinMax.lean","def_pos":[96,4],"def_end_pos":[96,10]}]}]}
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+{"url":"Mathlib/Analysis/InnerProductSpace/l2Space.lean","commit":"","full_name":"HilbertBasis.repr_apply_apply","start":[396,0],"end":[399,6],"file_path":"Mathlib/Analysis/InnerProductSpace/l2Space.lean","tactics":[{"state_before":"ι : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : RCLike 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type u_4\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nb : HilbertBasis ι 𝕜 E\nv : E\ni : ι\n⊢ ↑(b.repr v) i = ⟪(fun i => b.repr.symm (lp.single 2 i 1)) i, v⟫_𝕜","state_after":"ι : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : RCLike 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type u_4\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nb : HilbertBasis ι 𝕜 E\nv : E\ni : ι\n⊢ ↑(b.repr v) i = ⟪1, ↑(b.repr v) i⟫_𝕜","tactic":"rw [← b.repr.inner_map_map (b i) v, b.repr_self, lp.inner_single_left]","premises":[{"full_name":"HilbertBasis.repr","def_path":"Mathlib/Analysis/InnerProductSpace/l2Space.lean","def_pos":[374,2],"def_end_pos":[374,6]},{"full_name":"HilbertBasis.repr_self","def_path":"Mathlib/Analysis/InnerProductSpace/l2Space.lean","def_pos":[392,18],"def_end_pos":[392,27]},{"full_name":"LinearIsometryEquiv.inner_map_map","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[1093,8],"def_end_pos":[1093,41]},{"full_name":"lp.inner_single_left","def_path":"Mathlib/Analysis/InnerProductSpace/l2Space.lean","def_pos":[159,8],"def_end_pos":[159,25]}]},{"state_before":"ι : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : RCLike 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type u_4\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nb : HilbertBasis ι 𝕜 E\nv : E\ni : ι\n⊢ ↑(b.repr v) i = ⟪1, ↑(b.repr v) i⟫_𝕜","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/Topology/Sequences.lean","commit":"","full_name":"UniformSpace.compactSpace_iff_seqCompactSpace","start":[375,0],"end":[376,96],"file_path":"Mathlib/Topology/Sequences.lean","tactics":[{"state_before":"X : Type u_1\nY : Type u_2\ninst✝¹ : UniformSpace X\ns : Set X\ninst✝ : (𝓤 X).IsCountablyGenerated\n⊢ CompactSpace X ↔ SeqCompactSpace X","state_after":"no goals","tactic":"simp only [← isCompact_univ_iff, seqCompactSpace_iff, UniformSpace.isCompact_iff_isSeqCompact]","premises":[{"full_name":"UniformSpace.isCompact_iff_isSeqCompact","def_path":"Mathlib/Topology/Sequences.lean","def_pos":[372,18],"def_end_pos":[372,57]},{"full_name":"isCompact_univ_iff","def_path":"Mathlib/Topology/Compactness/Compact.lean","def_pos":[724,8],"def_end_pos":[724,26]},{"full_name":"seqCompactSpace_iff","def_path":"Mathlib/Topology/Defs/Sequences.lean","def_pos":[73,2],"def_end_pos":[73,8]}]}]}
+{"url":"Mathlib/SetTheory/Ordinal/Notation.lean","commit":"","full_name":"ONote.fastGrowingε₀_zero","start":[1134,0],"end":[1134,75],"file_path":"Mathlib/SetTheory/Ordinal/Notation.lean","tactics":[{"state_before":"⊢ fastGrowingε₀ 0 = 1","state_after":"no goals","tactic":"simp [fastGrowingε₀]","premises":[{"full_name":"ONote.fastGrowingε₀","def_path":"Mathlib/SetTheory/Ordinal/Notation.lean","def_pos":[1131,4],"def_end_pos":[1131,17]}]}]}
+{"url":"Mathlib/FieldTheory/PerfectClosure.lean","commit":"","full_name":"PerfectClosure.natCast","start":[346,0],"end":[356,15],"file_path":"Mathlib/FieldTheory/PerfectClosure.lean","tactics":[{"state_before":"K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nn x : ℕ\n⊢ ↑x = mk K p (n, ↑x)","state_after":"case zero\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx : ℕ\n⊢ ↑x = mk K p (0, ↑x)\n\ncase succ\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx n : ℕ\nih : ↑x = mk K p (n, ↑x)\n⊢ ↑x = mk K p (n + 1, ↑x)","tactic":"induction' n with n ih","premises":[]},{"state_before":"case succ\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx n : ℕ\nih : ↑x = mk K p (n, ↑x)\n⊢ ↑x = mk K p (n + 1, ↑x)","state_after":"case succ\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx n : ℕ\nih : ↑x = mk K p (n, ↑x)\n⊢ mk K p (n, ↑x) = mk K p (n + 1, ↑x)","tactic":"rw [ih]","premises":[]},{"state_before":"case succ\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx n : ℕ\nih : ↑x = mk K p (n, ↑x)\n⊢ mk K p (n, ↑x) = mk K p (n + 1, ↑x)","state_after":"case succ.a\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx n : ℕ\nih : ↑x = mk K p (n, ↑x)\n⊢ R K p (n, ↑x) (n + 1, ↑x)","tactic":"apply Quot.sound","premises":[{"full_name":"Quot.sound","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1471,6],"def_end_pos":[1471,11]}]},{"state_before":"case succ.a\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx n : ℕ\nih : ↑x = mk K p (n, ↑x)\n⊢ R K p (n, ↑x) (n + 1, ↑x)","state_after":"case succ.a\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx n : ℕ\nih : ↑x = mk K p (n, ↑x)\n⊢ R K p (n, ↑x) (n.succ, (frobenius K p) ↑x)","tactic":"suffices R K p (n, (x : K)) (Nat.succ n, frobenius K p (x : K)) by\n    rwa [frobenius_natCast K p x] at this","premises":[{"full_name":"Nat.succ","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1083,4],"def_end_pos":[1083,8]},{"full_name":"PerfectClosure.R","def_path":"Mathlib/FieldTheory/PerfectClosure.lean","def_pos":[60,10],"def_end_pos":[60,26]},{"full_name":"Prod.mk","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[481,2],"def_end_pos":[481,4]},{"full_name":"frobenius","def_path":"Mathlib/Algebra/CharP/ExpChar.lean","def_pos":[268,4],"def_end_pos":[268,13]},{"full_name":"frobenius_natCast","def_path":"Mathlib/Algebra/CharP/ExpChar.lean","def_pos":[376,8],"def_end_pos":[376,25]}]},{"state_before":"case succ.a\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx n : ℕ\nih : ↑x = mk K p (n, ↑x)\n⊢ R K p (n, ↑x) (n.succ, (frobenius K p) ↑x)","state_after":"no goals","tactic":"apply R.intro","premises":[{"full_name":"PerfectClosure.R.intro","def_path":"Mathlib/FieldTheory/PerfectClosure.lean","def_pos":[61,4],"def_end_pos":[61,9]}]}]}
+{"url":"Mathlib/Algebra/BigOperators/Group/List.lean","commit":"","full_name":"List.sum_add_sum_eq_sum_zipWith_add_sum_drop","start":[345,0],"end":[357,20],"file_path":"Mathlib/Algebra/BigOperators/Group/List.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst✝ : CommMonoid M\na : M\nl l₁ l₂ ys : List M\n⊢ [].prod * ys.prod = (zipWith (fun x x_1 => x * x_1) [] ys).prod * (drop ys.length []).prod * (drop [].length ys).prod","state_after":"no goals","tactic":"simp [Nat.zero_le]","premises":[{"full_name":"Nat.zero_le","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1663,8],"def_end_pos":[1663,19]}]},{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst✝ : CommMonoid M\na : M\nl l₁ l₂ xs : List M\n⊢ xs.prod * [].prod = (zipWith (fun x x_1 => x * x_1) xs []).prod * (drop [].length xs).prod * (drop xs.length []).prod","state_after":"no goals","tactic":"simp [Nat.zero_le]","premises":[{"full_name":"Nat.zero_le","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1663,8],"def_end_pos":[1663,19]}]},{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst✝ : CommMonoid M\na : M\nl l₁ l₂ : List M\nx : M\nxs : List M\ny : M\nys : List M\n⊢ (x :: xs).prod * (y :: ys).prod =\n    (zipWith (fun x x_1 => x * x_1) (x :: xs) (y :: ys)).prod * (drop (y :: ys).length (x :: xs)).prod *\n      (drop (x :: xs).length (y :: ys)).prod","state_after":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst✝ : CommMonoid M\na : M\nl l₁ l₂ : List M\nx : M\nxs : List M\ny : M\nys : List M\n⊢ x * xs.prod * (y * ys.prod) =\n    x * y * (zipWith (fun x x_1 => x * x_1) xs ys).prod * (drop ys.length xs).prod * (drop xs.length ys).prod","tactic":"simp only [drop, length, zipWith_cons_cons, prod_cons]","premises":[{"full_name":"List.drop","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[732,4],"def_end_pos":[732,8]},{"full_name":"List.length","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2316,4],"def_end_pos":[2316,15]},{"full_name":"List.prod_cons","def_path":"Mathlib/Algebra/BigOperators/Group/List.lean","def_pos":[85,8],"def_end_pos":[85,17]},{"full_name":"List.zipWith_cons_cons","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[1096,16],"def_end_pos":[1096,33]}]},{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst✝ : CommMonoid M\na : M\nl l₁ l₂ : List M\nx : M\nxs : List M\ny : M\nys : List M\n⊢ x * xs.prod * (y * ys.prod) =\n    x * y * (zipWith (fun x x_1 => x * x_1) xs ys).prod * (drop ys.length xs).prod * (drop xs.length ys).prod","state_after":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst✝ : CommMonoid M\na : M\nl l₁ l₂ : List M\nx : M\nxs : List M\ny : M\nys : List M\n⊢ x * (y * ((zipWith (fun x x_1 => x * x_1) xs ys).prod * (drop ys.length xs).prod * (drop xs.length ys).prod)) =\n    x * y * (zipWith (fun x x_1 => x * x_1) xs ys).prod * (drop ys.length xs).prod * (drop xs.length ys).prod","tactic":"conv =>\n      lhs; rw [mul_assoc]; right; rw [mul_comm, mul_assoc]; right\n      rw [mul_comm, prod_mul_prod_eq_prod_zipWith_mul_prod_drop xs ys]","premises":[{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]}]},{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst✝ : CommMonoid M\na : M\nl l₁ l₂ : List M\nx : M\nxs : List M\ny : M\nys : List M\n⊢ x * (y * ((zipWith (fun x x_1 => x * x_1) xs ys).prod * (drop ys.length xs).prod * (drop xs.length ys).prod)) =\n    x * y * (zipWith (fun x x_1 => x * x_1) xs ys).prod * (drop ys.length xs).prod * (drop xs.length ys).prod","state_after":"no goals","tactic":"simp [mul_assoc]","premises":[{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]}]}]}
+{"url":"Mathlib/Algebra/Order/Interval/Multiset.lean","commit":"","full_name":"Multiset.map_add_left_Ioo","start":[32,0],"end":[34,60],"file_path":"Mathlib/Algebra/Order/Interval/Multiset.lean","tactics":[{"state_before":"α : Type u_1\ninst✝² : OrderedCancelAddCommMonoid α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : LocallyFiniteOrder α\na b c : α\n⊢ map (fun x => c + x) (Ioo a b) = Ioo (c + a) (c + b)","state_after":"no goals","tactic":"classical rw [Ioo, Ioo, ← Finset.image_add_left_Ioo, Finset.image_val,\n      ((Finset.nodup _).map <| add_right_injective c).dedup]","premises":[{"full_name":"Finset.image_add_left_Ioo","def_path":"Mathlib/Algebra/Order/Interval/Finset.lean","def_pos":[73,14],"def_end_pos":[73,32]},{"full_name":"Finset.image_val","def_path":"Mathlib/Data/Finset/Image.lean","def_pos":[293,8],"def_end_pos":[293,17]},{"full_name":"Finset.nodup","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[137,2],"def_end_pos":[137,7]},{"full_name":"Multiset.Ioo","def_path":"Mathlib/Order/Interval/Multiset.lean","def_pos":[56,4],"def_end_pos":[56,7]},{"full_name":"Multiset.Nodup.map","def_path":"Mathlib/Data/Multiset/Nodup.lean","def_pos":[113,8],"def_end_pos":[113,17]},{"full_name":"add_right_injective","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[60,2],"def_end_pos":[60,13]}]}]}
+{"url":"Mathlib/Order/CompactlyGenerated/Basic.lean","commit":"","full_name":"CompleteLattice.isCompactElement_finsetSup","start":[177,0],"end":[190,28],"file_path":"Mathlib/Order/CompactlyGenerated/Basic.lean","tactics":[{"state_before":"ι : Sort u_1\nα✝ : Type u_2\ninst✝¹ : CompleteLattice α✝\nf✝ : ι → α✝\nα : Type u_3\nβ : Type u_4\ninst✝ : CompleteLattice α\nf : β → α\ns : Finset β\nh : ∀ x ∈ s, IsCompactElement (f x)\n⊢ IsCompactElement (s.sup f)","state_after":"no goals","tactic":"classical\n    rw [isCompactElement_iff_le_of_directed_sSup_le]\n    intro d hemp hdir hsup\n    rw [← Function.id_comp f]\n    rw [← Finset.sup_image]\n    apply Finset.sup_le_of_le_directed d hemp hdir\n    rintro x hx\n    obtain ⟨p, ⟨hps, rfl⟩⟩ := Finset.mem_image.mp hx\n    specialize h p hps\n    rw [isCompactElement_iff_le_of_directed_sSup_le] at h\n    specialize h d hemp hdir (le_trans (Finset.le_sup hps) hsup)\n    simpa only [exists_prop]","premises":[{"full_name":"CompleteLattice.isCompactElement_iff_le_of_directed_sSup_le","def_path":"Mathlib/Order/CompactlyGenerated/Basic.lean","def_pos":[104,8],"def_end_pos":[104,51]},{"full_name":"Finset.le_sup","def_path":"Mathlib/Data/Finset/Lattice.lean","def_pos":[103,8],"def_end_pos":[103,14]},{"full_name":"Finset.mem_image","def_path":"Mathlib/Data/Finset/Image.lean","def_pos":[303,8],"def_end_pos":[303,17]},{"full_name":"Finset.sup_image","def_path":"Mathlib/Data/Finset/Lattice.lean","def_pos":[61,8],"def_end_pos":[61,17]},{"full_name":"Finset.sup_le_of_le_directed","def_path":"Mathlib/Data/Finset/Lattice.lean","def_pos":[219,8],"def_end_pos":[219,29]},{"full_name":"Function.id_comp","def_path":"Mathlib/Logic/Function/Defs.lean","def_pos":[80,8],"def_end_pos":[80,15]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]},{"full_name":"le_trans","def_path":"Mathlib/Order/Defs.lean","def_pos":[48,8],"def_end_pos":[48,16]}]}]}
+{"url":"Mathlib/Data/List/DropRight.lean","commit":"","full_name":"List.rtakeWhile_eq_self_iff","start":[190,0],"end":[192,35],"file_path":"Mathlib/Data/List/DropRight.lean","tactics":[{"state_before":"α : Type u_1\np : α → Bool\nl : List α\nn : ℕ\n⊢ rtakeWhile p l = l ↔ ∀ (x : α), x ∈ l → p x = true","state_after":"no goals","tactic":"simp [rtakeWhile, reverse_eq_iff]","premises":[{"full_name":"List.reverse_eq_iff","def_path":"Mathlib/Data/List/Basic.lean","def_pos":[330,8],"def_end_pos":[330,22]},{"full_name":"List.rtakeWhile","def_path":"Mathlib/Data/List/DropRight.lean","def_pos":[165,4],"def_end_pos":[165,14]}]}]}
+{"url":"Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean","commit":"","full_name":"WeierstrassCurve.natDegree_preΨ₄_pos","start":[136,0],"end":[137,37],"file_path":"Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean","tactics":[{"state_before":"R : Type u\ninst✝ : CommRing R\nW : WeierstrassCurve R\nh : 2 ≠ 0\n⊢ 0 < W.preΨ₄.natDegree","state_after":"no goals","tactic":"linarith only [W.natDegree_preΨ₄ h]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"WeierstrassCurve.natDegree_preΨ₄","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean","def_pos":[133,6],"def_end_pos":[133,21]}]}]}
+{"url":"Mathlib/Data/Set/Finite.lean","commit":"","full_name":"Set.Finite.fin_embedding","start":[971,0],"end":[974,96],"file_path":"Mathlib/Data/Set/Finite.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ns : Set α\nh : s.Finite\n⊢ range ⇑(Fintype.equivFin ↑↑h.toFinset).symm.asEmbedding = s","state_after":"no goals","tactic":"simp only [Finset.coe_sort_coe, Equiv.asEmbedding_range, Finite.coe_toFinset, setOf_mem_eq]","premises":[{"full_name":"Equiv.asEmbedding_range","def_path":"Mathlib/Logic/Embedding/Set.lean","def_pos":[26,8],"def_end_pos":[26,31]},{"full_name":"Finset.coe_sort_coe","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[258,8],"def_end_pos":[258,20]},{"full_name":"Set.Finite.coe_toFinset","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[150,18],"def_end_pos":[150,30]},{"full_name":"Set.setOf_mem_eq","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[215,8],"def_end_pos":[215,20]}]}]}
+{"url":"Mathlib/Analysis/Calculus/ParametricIntegral.lean","commit":"","full_name":"hasFDerivAt_integral_of_dominated_loc_of_lip'","start":[68,0],"end":[153,45],"file_path":"Mathlib/Analysis/Calculus/ParametricIntegral.lean","tactics":[{"state_before":"α : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\n⊢ Integrable F' μ ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀","state_after":"α : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\n⊢ Integrable F' μ ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀","tactic":"have x₀_in : x₀ ∈ ball x₀ ε := mem_ball_self ε_pos","premises":[{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Metric.ball","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[350,4],"def_end_pos":[350,8]},{"full_name":"Metric.mem_ball_self","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[362,8],"def_end_pos":[362,21]}]},{"state_before":"α : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\n⊢ Integrable F' μ ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀","state_after":"α : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\n⊢ Integrable F' μ ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀","tactic":"have nneg : ∀ x, 0 ≤ ‖x - x₀‖⁻¹ := fun x ↦ inv_nonneg.mpr (norm_nonneg _)","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"Norm.norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[59,2],"def_end_pos":[59,6]},{"full_name":"inv_nonneg","def_path":"Mathlib/Algebra/Order/Field/Unbundled/Basic.lean","def_pos":[29,14],"def_end_pos":[29,24]},{"full_name":"norm_nonneg","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[401,29],"def_end_pos":[401,40]}]},{"state_before":"α : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\n⊢ Integrable F' μ ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀","state_after":"α : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\n⊢ Integrable F' μ ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀","tactic":"set b : α → ℝ := fun a ↦ |bound a|","premises":[{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"abs","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[33,2],"def_end_pos":[33,13]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"α : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\n⊢ Integrable F' μ ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀","state_after":"α : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\nb_int : Integrable b μ\n⊢ Integrable F' μ ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀","tactic":"have b_int : Integrable b μ := bound_integrable.norm","premises":[{"full_name":"MeasureTheory.Integrable","def_path":"Mathlib/MeasureTheory/Function/L1Space.lean","def_pos":[389,4],"def_end_pos":[389,14]},{"full_name":"MeasureTheory.Integrable.norm","def_path":"Mathlib/MeasureTheory/Function/L1Space.lean","def_pos":[659,8],"def_end_pos":[659,23]}]},{"state_before":"α : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\nb_int : Integrable b μ\n⊢ Integrable F' μ ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀","state_after":"α : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\nb_int : Integrable b μ\nb_nonneg : ∀ (a : α), 0 ≤ b a\n⊢ Integrable F' μ ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀","tactic":"have b_nonneg : ∀ a, 0 ≤ b a := fun a ↦ abs_nonneg _","premises":[{"full_name":"abs_nonneg","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[95,29],"def_end_pos":[95,39]}]},{"state_before":"α : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\nb_int : Integrable b μ\nb_nonneg : ∀ (a : α), 0 ≤ b a\n⊢ Integrable F' μ ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀","state_after":"α : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\nb_int : Integrable b μ\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\n⊢ Integrable F' μ ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀","tactic":"replace h_lipsch : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ :=\n    h_lipsch.mono fun a ha x hx ↦\n      (ha x hx).trans <| mul_le_mul_of_nonneg_right (le_abs_self _) (norm_nonneg _)","premises":[{"full_name":"Filter.Eventually","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[948,14],"def_end_pos":[948,24]},{"full_name":"Filter.Eventually.mono","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1002,8],"def_end_pos":[1002,23]},{"full_name":"MeasureTheory.ae","def_path":"Mathlib/MeasureTheory/OuterMeasure/AE.lean","def_pos":[43,4],"def_end_pos":[43,6]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Metric.ball","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[350,4],"def_end_pos":[350,8]},{"full_name":"Norm.norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[59,2],"def_end_pos":[59,6]},{"full_name":"le_abs_self","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[63,2],"def_end_pos":[63,13]},{"full_name":"mul_le_mul_of_nonneg_right","def_path":"Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean","def_pos":[194,8],"def_end_pos":[194,34]},{"full_name":"norm_nonneg","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[401,29],"def_end_pos":[401,40]}]},{"state_before":"α : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\nb_int : Integrable b μ\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\n⊢ Integrable F' μ ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀","state_after":"α : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\nb_int : Integrable b μ\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ\n⊢ Integrable F' μ ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀","tactic":"have hF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ := fun x x_in ↦ by\n    have : ∀ᵐ a ∂μ, ‖F x₀ a - F x a‖ ≤ ε * b a := by\n      simp only [norm_sub_rev (F x₀ _)]\n      refine h_lipsch.mono fun a ha ↦ (ha x x_in).trans ?_\n      rw [mul_comm ε]\n      rw [mem_ball, dist_eq_norm] at x_in\n      exact mul_le_mul_of_nonneg_left x_in.le (b_nonneg _)\n    exact integrable_of_norm_sub_le (hF_meas x x_in) hF_int\n      (bound_integrable.norm.const_mul ε) this","premises":[{"full_name":"Filter.Eventually","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[948,14],"def_end_pos":[948,24]},{"full_name":"Filter.Eventually.mono","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1002,8],"def_end_pos":[1002,23]},{"full_name":"MeasureTheory.Integrable","def_path":"Mathlib/MeasureTheory/Function/L1Space.lean","def_pos":[389,4],"def_end_pos":[389,14]},{"full_name":"MeasureTheory.Integrable.const_mul","def_path":"Mathlib/MeasureTheory/Function/L1Space.lean","def_pos":[1080,8],"def_end_pos":[1080,28]},{"full_name":"MeasureTheory.Integrable.norm","def_path":"Mathlib/MeasureTheory/Function/L1Space.lean","def_pos":[659,8],"def_end_pos":[659,23]},{"full_name":"MeasureTheory.ae","def_path":"Mathlib/MeasureTheory/OuterMeasure/AE.lean","def_pos":[43,4],"def_end_pos":[43,6]},{"full_name":"MeasureTheory.integrable_of_norm_sub_le","def_path":"Mathlib/MeasureTheory/Function/L1Space.lean","def_pos":[730,8],"def_end_pos":[730,33]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Metric.ball","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[350,4],"def_end_pos":[350,8]},{"full_name":"Metric.mem_ball","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[354,8],"def_end_pos":[354,16]},{"full_name":"Norm.norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[59,2],"def_end_pos":[59,6]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]},{"full_name":"mul_le_mul_of_nonneg_left","def_path":"Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean","def_pos":[190,8],"def_end_pos":[190,33]},{"full_name":"norm_sub_rev","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[371,2],"def_end_pos":[371,13]}]},{"state_before":"α : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\nb_int : Integrable b μ\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ\n⊢ Integrable F' μ ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀","state_after":"α : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\nb_int : Integrable b μ\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ\nhF'_int : Integrable F' μ\n⊢ Integrable F' μ ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀","tactic":"have hF'_int : Integrable F' μ :=\n    have : ∀ᵐ a ∂μ, ‖F' a‖ ≤ b a := by\n      apply (h_diff.and h_lipsch).mono\n      rintro a ⟨ha_diff, ha_lip⟩\n      exact ha_diff.le_of_lip' (b_nonneg a) (mem_of_superset (ball_mem_nhds _ ε_pos) <| ha_lip)\n    b_int.mono' hF'_meas this","premises":[{"full_name":"Filter.Eventually","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[948,14],"def_end_pos":[948,24]},{"full_name":"Filter.Eventually.and","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[973,18],"def_end_pos":[973,32]},{"full_name":"Filter.Eventually.mono","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1002,8],"def_end_pos":[1002,23]},{"full_name":"Filter.mem_of_superset","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[139,8],"def_end_pos":[139,23]},{"full_name":"HasFDerivAt.le_of_lip'","def_path":"Mathlib/Analysis/Calculus/FDeriv/Basic.lean","def_pos":[314,8],"def_end_pos":[314,30]},{"full_name":"MeasureTheory.Integrable","def_path":"Mathlib/MeasureTheory/Function/L1Space.lean","def_pos":[389,4],"def_end_pos":[389,14]},{"full_name":"MeasureTheory.Integrable.mono'","def_path":"Mathlib/MeasureTheory/Function/L1Space.lean","def_pos":[410,8],"def_end_pos":[410,24]},{"full_name":"MeasureTheory.ae","def_path":"Mathlib/MeasureTheory/OuterMeasure/AE.lean","def_pos":[43,4],"def_end_pos":[43,6]},{"full_name":"Metric.ball_mem_nhds","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[863,8],"def_end_pos":[863,21]},{"full_name":"Norm.norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[59,2],"def_end_pos":[59,6]}]},{"state_before":"α : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\nb_int : Integrable b μ\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ\nhF'_int : Integrable F' μ\n⊢ Integrable F' μ ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀","state_after":"α : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\nb_int : Integrable b μ\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ\nhF'_int : Integrable F' μ\n⊢ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀","tactic":"refine ⟨hF'_int, ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]}]},{"state_before":"α : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\nb_int : Integrable b μ\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ\nhF'_int : Integrable F' μ\n⊢ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀","state_after":"case pos\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\nb_int : Integrable b μ\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ\nhF'_int : Integrable F' μ\nhE : CompleteSpace E\n⊢ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀\n\ncase neg\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\nb_int : Integrable b μ\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ\nhF'_int : Integrable F' μ\nhE : ¬CompleteSpace E\n⊢ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀","tactic":"by_cases hE : CompleteSpace E","premises":[{"full_name":"CompleteSpace","def_path":"Mathlib/Topology/UniformSpace/Cauchy.lean","def_pos":[360,6],"def_end_pos":[360,19]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case pos\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\nb_int : Integrable b μ\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ\nhF'_int : Integrable F' μ\nhE : CompleteSpace E\n⊢ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀\n\ncase neg\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\nb_int : Integrable b μ\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ\nhF'_int : Integrable F' μ\nhE : ¬CompleteSpace E\n⊢ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀","state_after":"case neg\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\nb_int : Integrable b μ\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ\nhF'_int : Integrable F' μ\nhE : ¬CompleteSpace E\n⊢ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀\n\ncase pos\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\nb_int : Integrable b μ\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ\nhF'_int : Integrable F' μ\nhE : CompleteSpace E\n⊢ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀","tactic":"swap","premises":[]},{"state_before":"case pos\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\nb_int : Integrable b μ\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ\nhF'_int : Integrable F' μ\nhE : CompleteSpace E\n⊢ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀","state_after":"case pos\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝��� : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\nb_int : Integrable b μ\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ\nhF'_int : Integrable F' μ\nhE : CompleteSpace E\nh_ball : ball x₀ ε ∈ 𝓝 x₀\n⊢ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀","tactic":"have h_ball : ball x₀ ε ∈ 𝓝 x₀ := ball_mem_nhds x₀ ε_pos","premises":[{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Metric.ball","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[350,4],"def_end_pos":[350,8]},{"full_name":"Metric.ball_mem_nhds","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[863,8],"def_end_pos":[863,21]},{"full_name":"nhds","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[113,16],"def_end_pos":[113,20]}]},{"state_before":"case pos\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\nb_int : Integrable b μ\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ\nhF'_int : Integrable F' μ\nhE : CompleteSpace E\nh_ball : ball x₀ ε ∈ 𝓝 x₀\n⊢ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀","state_after":"case pos\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\nb_int : Integrable b μ\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ\nhF'_int : Integrable F' μ\nhE : CompleteSpace E\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n  ∀ᶠ (x : H) in 𝓝 x₀,\n    ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - (∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - (F' a) (x - x₀)) ∂μ‖\n⊢ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀","tactic":"have : ∀ᶠ x in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖((∫ a, F x a ∂μ) - ∫ a, F x₀ a ∂μ) - (∫ a, F' a ∂μ) (x - x₀)‖ =\n      ‖∫ a, ‖x - x₀‖⁻¹ • (F x a - F x₀ a - F' a (x - x₀)) ∂μ‖ := by\n    apply mem_of_superset (ball_mem_nhds _ ε_pos)\n    intro x x_in; simp only\n    rw [Set.mem_setOf_eq, ← norm_smul_of_nonneg (nneg _), integral_smul, integral_sub, integral_sub,\n      ← ContinuousLinearMap.integral_apply hF'_int]\n    exacts [hF_int' x x_in, hF_int, (hF_int' x x_in).sub hF_int,\n      hF'_int.apply_continuousLinearMap _]","premises":[{"full_name":"ContinuousLinearMap.integral_apply","def_path":"Mathlib/MeasureTheory/Integral/SetIntegral.lean","def_pos":[1196,8],"def_end_pos":[1196,22]},{"full_name":"Filter.Eventually","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[948,14],"def_end_pos":[948,24]},{"full_name":"Filter.mem_of_superset","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[139,8],"def_end_pos":[139,23]},{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"MeasureTheory.Integrable.apply_continuousLinearMap","def_path":"Mathlib/MeasureTheory/Function/L1Space.lean","def_pos":[1432,8],"def_end_pos":[1432,58]},{"full_name":"MeasureTheory.Integrable.sub","def_path":"Mathlib/MeasureTheory/Function/L1Space.lean","def_pos":[656,8],"def_end_pos":[656,22]},{"full_name":"MeasureTheory.integral","def_path":"Mathlib/MeasureTheory/Integral/Bochner.lean","def_pos":[714,16],"def_end_pos":[714,24]},{"full_name":"MeasureTheory.integral_smul","def_path":"Mathlib/MeasureTheory/Integral/Bochner.lean","def_pos":[824,8],"def_end_pos":[824,21]},{"full_name":"MeasureTheory.integral_sub","def_path":"Mathlib/MeasureTheory/Integral/Bochner.lean","def_pos":[812,8],"def_end_pos":[812,20]},{"full_name":"Metric.ball_mem_nhds","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[863,8],"def_end_pos":[863,21]},{"full_name":"Norm.norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[59,2],"def_end_pos":[59,6]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]},{"full_name":"nhds","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[113,16],"def_end_pos":[113,20]},{"full_name":"norm_smul_of_nonneg","def_path":"Mathlib/Analysis/NormedSpace/Real.lean","def_pos":[42,8],"def_end_pos":[42,27]}]},{"state_before":"case pos\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\nb_int : Integrable b μ\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ\nhF'_int : Integrable F' μ\nhE : CompleteSpace E\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n  ∀ᶠ (x : H) in 𝓝 x₀,\n    ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - (∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - (F' a) (x - x₀)) ∂μ‖\n⊢ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀","state_after":"case pos\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\nb_int : Integrable b μ\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ\nhF'_int : Integrable F' μ\nhE : CompleteSpace E\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n  ∀ᶠ (x : H) in 𝓝 x₀,\n    ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - (∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - (F' a) (x - x₀)) ∂μ‖\n⊢ Tendsto (fun x => ∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - (F' a) (x - x₀)) ∂μ) (𝓝 x₀)\n    (𝓝 (∫ (a : α), ‖x₀ - x₀‖⁻¹ • (F x₀ a - F x₀ a - (F' a) (x₀ - x₀)) ∂μ))","tactic":"rw [hasFDerivAt_iff_tendsto, tendsto_congr' this, ← tendsto_zero_iff_norm_tendsto_zero, ←\n    show (∫ a : α, ‖x₀ - x₀‖⁻¹ • (F x₀ a - F x₀ a - (F' a) (x₀ - x₀)) ∂μ) = 0 by simp]","premises":[{"full_name":"Filter.tendsto_congr'","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2631,8],"def_end_pos":[2631,22]},{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"MeasureTheory.integral","def_path":"Mathlib/MeasureTheory/Integral/Bochner.lean","def_pos":[714,16],"def_end_pos":[714,24]},{"full_name":"Norm.norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[59,2],"def_end_pos":[59,6]},{"full_name":"hasFDerivAt_iff_tendsto","def_path":"Mathlib/Analysis/Calculus/FDeriv/Basic.lean","def_pos":[302,8],"def_end_pos":[302,31]},{"full_name":"tendsto_zero_iff_norm_tendsto_zero","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[715,2],"def_end_pos":[715,13]}]},{"state_before":"case pos\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\nb_int : Integrable b μ\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ\nhF'_int : Integrable F' μ\nhE : CompleteSpace E\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n  ∀ᶠ (x : H) in 𝓝 x₀,\n    ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - (∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - (F' a) (x - x₀)) ∂μ‖\n⊢ Tendsto (fun x => ∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - (F' a) (x - x₀)) ∂μ) (𝓝 x₀)\n    (𝓝 (∫ (a : α), ‖x₀ - x₀‖⁻¹ • (F x₀ a - F x₀ a - (F' a) (x₀ - x₀)) ∂μ))","state_after":"case pos.hF_meas\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\nb_int : Integrable b μ\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ\nhF'_int : Integrable F' μ\nhE : CompleteSpace E\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n  ∀ᶠ (x : H) in 𝓝 x₀,\n    ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - (∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - (F' a) (x - x₀)) ∂μ‖\n⊢ ∀ᶠ (n : H) in 𝓝 x₀, AEStronglyMeasurable (fun a => ‖n - x₀‖⁻¹ • (F n a - F x₀ a - (F' a) (n - x₀))) μ\n\ncase pos.h_bound\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\nb_int : Integrable b μ\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ\nhF'_int : Integrable F' μ\nhE : CompleteSpace E\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n  ∀ᶠ (x : H) in 𝓝 x₀,\n    ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - (∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - (F' a) (x - x₀)) ∂μ‖\n⊢ ∀ᶠ (n : H) in 𝓝 x₀, ∀ᵐ (a : α) ∂μ, ‖‖n - x₀‖⁻¹ • (F n a - F x₀ a - (F' a) (n - x₀))‖ ≤ ?pos.bound✝ a\n\ncase pos.bound_integrable\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\nb_int : Integrable b μ\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ\nhF'_int : Integrable F' μ\nhE : CompleteSpace E\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n  ∀ᶠ (x : H) in 𝓝 x₀,\n    ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - (∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - (F' a) (x - x₀)) ∂μ‖\n⊢ Integrable ?pos.bound✝ μ\n\ncase pos.h_lim\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\nb_int : Integrable b μ\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ\nhF'_int : Integrable F' μ\nhE : CompleteSpace E\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n  ∀ᶠ (x : H) in 𝓝 x₀,\n    ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - (∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - (F' a) (x - x₀)) ∂μ‖\n⊢ ∀ᵐ (a : α) ∂μ,\n    Tendsto (fun n => ‖n - x₀‖⁻¹ • (F n a - F x₀ a - (F' a) (n - x₀))) (𝓝 x₀)\n      (𝓝 (‖x₀ - x₀‖⁻¹ • (F x₀ a - F x₀ a - (F' a) (x₀ - x₀))))\n\ncase pos.bound\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\nb_int : Integrable b μ\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ\nhF'_int : Integrable F' μ\nhE : CompleteSpace E\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n  ∀ᶠ (x : H) in 𝓝 x₀,\n    ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - (∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - (F' a) (x - x₀)) ∂μ‖\n⊢ α → ℝ","tactic":"apply tendsto_integral_filter_of_dominated_convergence","premises":[{"full_name":"MeasureTheory.tendsto_integral_filter_of_dominated_convergence","def_path":"Mathlib/MeasureTheory/Integral/DominatedConvergence.lean","def_pos":[65,8],"def_end_pos":[65,56]}]}]}
+{"url":"Mathlib/Topology/Basic.lean","commit":"","full_name":"frontier_empty","start":[617,0],"end":[618,71],"file_path":"Mathlib/Topology/Basic.lean","tactics":[{"state_before":"X : Type u\nY : Type v\nι : Sort w\nα : Type u_1\nβ : Type u_2\nx : X\ns s₁ s₂ t : Set X\np p₁ p₂ : X → Prop\ninst✝ : TopologicalSpace X\n�� frontier ∅ = ∅","state_after":"no goals","tactic":"simp [frontier]","premises":[{"full_name":"frontier","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[116,4],"def_end_pos":[116,12]}]}]}
+{"url":"Mathlib/CategoryTheory/Subobject/Lattice.lean","commit":"","full_name":"CategoryTheory.Subobject.le_sSup","start":[606,0],"end":[611,35],"file_path":"Mathlib/CategoryTheory/Subobject/Lattice.lean","tactics":[{"state_before":"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nX Y Z : C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\ninst✝² : WellPowered C\ninst✝¹ : HasCoproducts C\ninst✝ : HasImages C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nhf : f ∈ s\n⊢ f ≤ sSup s","state_after":"case f\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nX Y Z : C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\ninst✝² : WellPowered C\ninst✝¹ : HasCoproducts C\ninst✝ : HasImages C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nhf : f ∈ s\n⊢ underlying.obj f ⟶ underlying.obj (sSup s)\n\ncase w\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nX Y Z : C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\ninst✝² : WellPowered C\ninst✝¹ : HasCoproducts C\ninst✝ : HasImages C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nhf : f ∈ s\n⊢ ?f ≫ (sSup s).arrow = f.arrow","tactic":"fapply le_of_comm","premises":[{"full_name":"CategoryTheory.Subobject.le_of_comm","def_path":"Mathlib/CategoryTheory/Subobject/Basic.lean","def_pos":[239,8],"def_end_pos":[239,18]}]}]}
+{"url":"Mathlib/Algebra/Group/AddChar.lean","commit":"","full_name":"AddChar.map_neg_eq_inv","start":[292,0],"end":[295,61],"file_path":"Mathlib/Algebra/Group/AddChar.lean","tactics":[{"state_before":"A : Type u_1\nM : Type u_2\ninst✝¹ : AddGroup A\ninst✝ : DivisionMonoid M\nψ : AddChar A M\na : A\n⊢ ψ (-a) = (ψ a)⁻¹","state_after":"case h\nA : Type u_1\nM : Type u_2\ninst✝¹ : AddGroup A\ninst✝ : DivisionMonoid M\nψ : AddChar A M\na : A\n⊢ ψ (-a) * ψ a = 1","tactic":"apply eq_inv_of_mul_eq_one_left","premises":[{"full_name":"eq_inv_of_mul_eq_one_left","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[994,8],"def_end_pos":[994,33]}]},{"state_before":"case h\nA : Type u_1\nM : Type u_2\ninst✝¹ : AddGroup A\ninst✝ : DivisionMonoid M\nψ : AddChar A M\na : A\n⊢ ψ (-a) * ψ a = 1","state_after":"no goals","tactic":"simp only [← map_add_eq_mul, add_left_neg, map_zero_eq_one]","premises":[{"full_name":"AddChar.map_add_eq_mul","def_path":"Mathlib/Algebra/Group/AddChar.lean","def_pos":[93,6],"def_end_pos":[93,20]},{"full_name":"AddChar.map_zero_eq_one","def_path":"Mathlib/Algebra/Group/AddChar.lean","def_pos":[90,14],"def_end_pos":[90,29]},{"full_name":"add_left_neg","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[1038,2],"def_end_pos":[1038,13]}]}]}
+{"url":"Mathlib/Order/SuccPred/Limit.lean","commit":"","full_name":"Order.isSuccLimitRecOn_limit","start":[150,0],"end":[152,28],"file_path":"Mathlib/Order/SuccPred/Limit.lean","tactics":[{"state_before":"α : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : SuccOrder α\na b : α\nC : α → Sort u_2\nhs : (a : α) → ¬IsMax a → C (succ a)\nhl : (a : α) → IsSuccLimit a → C a\nhb : IsSuccLimit b\n⊢ isSuccLimitRecOn b hs hl = hl b hb","state_after":"no goals","tactic":"classical exact dif_pos hb","premises":[{"full_name":"dif_pos","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[949,8],"def_end_pos":[949,15]}]}]}
+{"url":"Mathlib/Data/Set/Prod.lean","commit":"","full_name":"Set.singleton_pi","start":[697,0],"end":[700,11],"file_path":"Mathlib/Data/Set/Prod.lean","tactics":[{"state_before":"ι : Type u_1\nα : ι → Type u_2\nβ : ι → Type u_3\ns s₁ s₂ : Set ι\nt✝ t₁ t₂ : (i : ι) → Set (α i)\ni✝ i : ι\nt : (i : ι) → Set (α i)\n⊢ {i}.pi t = eval i ⁻¹' t i","state_after":"case h\nι : Type u_1\nα : ι → Type u_2\nβ : ι → Type u_3\ns s₁ s₂ : Set ι\nt✝ t₁ t₂ : (i : ι) → Set (α i)\ni✝ i : ι\nt : (i : ι) → Set (α i)\nx✝ : (i : ι) → α i\n⊢ x✝ ∈ {i}.pi t ↔ x✝ ∈ eval i ⁻¹' t i","tactic":"ext","premises":[]},{"state_before":"case h\nι : Type u_1\nα : ι → Type u_2\nβ : ι → Type u_3\ns s₁ s₂ : Set ι\nt✝ t₁ t₂ : (i : ι) → Set (α i)\ni✝ i : ι\nt : (i : ι) → Set (α i)\nx✝ : (i : ι) → α i\n⊢ x✝ ∈ {i}.pi t ↔ x✝ ∈ eval i ⁻¹' t i","state_after":"no goals","tactic":"simp [pi]","premises":[{"full_name":"Set.pi","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[218,4],"def_end_pos":[218,6]}]}]}
+{"url":"Mathlib/GroupTheory/MonoidLocalization/Basic.lean","commit":"","full_name":"AddSubmonoid.LocalizationMap.map_left_cancel","start":[507,0],"end":[510,70],"file_path":"Mathlib/GroupTheory/MonoidLocalization/Basic.lean","tactics":[{"state_before":"M : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type u_3\ninst✝ : CommMonoid P\nf : S.LocalizationMap N\nx y : M\nc : ↥S\nh : f.toMap (x * ↑c) = f.toMap (y * ↑c)\n⊢ f.toMap (↑c * x) = f.toMap (↑c * y)","state_after":"no goals","tactic":"rw [mul_comm _ x, mul_comm _ y, h]","premises":[{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]}]}]}
+{"url":"Mathlib/Algebra/MonoidAlgebra/Basic.lean","commit":"","full_name":"MonoidAlgebra.single_algebraMap_eq_algebraMap_mul_of","start":[768,0],"end":[770,90],"file_path":"Mathlib/Algebra/MonoidAlgebra/Basic.lean","tactics":[{"state_before":"k : Type u₁\nG : Type u₂\nH : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝³ : CommSemiring k\ninst✝² : Semiring A\ninst✝¹ : Algebra k A\ninst✝ : Monoid G\na : G\nb : k\n⊢ single a ((algebraMap k A) b) = (algebraMap k (MonoidAlgebra A G)) b * (of A G) a","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/MeasureTheory/Function/L1Space.lean","commit":"","full_name":"MeasureTheory.integrable_smul_measure","start":[505,0],"end":[510,31],"file_path":"Mathlib/MeasureTheory/Function/L1Space.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → β\nc : ℝ≥0∞\nh₁ : c ≠ 0\nh₂ : c ≠ ⊤\nh : Integrable f (c • μ)\n⊢ Integrable f μ","state_after":"no goals","tactic":"simpa only [smul_smul, ENNReal.inv_mul_cancel h₁ h₂, one_smul] using\n      h.smul_measure (ENNReal.inv_ne_top.2 h₁)","premises":[{"full_name":"ENNReal.inv_mul_cancel","def_path":"Mathlib/Data/ENNReal/Inv.lean","def_pos":[91,18],"def_end_pos":[91,32]},{"full_name":"ENNReal.inv_ne_top","def_path":"Mathlib/Data/ENNReal/Inv.lean","def_pos":[115,8],"def_end_pos":[115,18]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"MeasureTheory.Integrable.smul_measure","def_path":"Mathlib/MeasureTheory/Function/L1Space.lean","def_pos":[495,8],"def_end_pos":[495,31]},{"full_name":"one_smul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[379,6],"def_end_pos":[379,14]},{"full_name":"smul_smul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[374,6],"def_end_pos":[374,15]}]}]}
+{"url":"Mathlib/RingTheory/Polynomial/Content.lean","commit":"","full_name":"Polynomial.content_eq_zero_iff","start":[134,0],"end":[144,12],"file_path":"Mathlib/RingTheory/Polynomial/Content.lean","tactics":[{"state_before":"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np : R[X]\n⊢ p.content = 0 ↔ p = 0","state_after":"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np : R[X]\n⊢ (∀ x ∈ p.support, p.coeff x = 0) ↔ p = 0","tactic":"rw [content, Finset.gcd_eq_zero_iff]","premises":[{"full_name":"Finset.gcd_eq_zero_iff","def_path":"Mathlib/Algebra/GCDMonoid/Finset.lean","def_pos":[178,8],"def_end_pos":[178,23]},{"full_name":"Polynomial.content","def_path":"Mathlib/RingTheory/Polynomial/Content.lean","def_pos":[70,4],"def_end_pos":[70,11]}]},{"state_before":"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np : R[X]\n⊢ (∀ x ∈ p.support, p.coeff x = 0) ↔ p = 0","state_after":"case mp\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np : R[X]\nh : ∀ x ∈ p.support, p.coeff x = 0\n⊢ p = 0\n\ncase mpr\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np : R[X]\nh : p = 0\n⊢ ∀ x ∈ p.support, p.coeff x = 0","tactic":"constructor <;> intro h","premises":[]}]}
+{"url":"Mathlib/RingTheory/Coprime/Ideal.lean","commit":"","full_name":"Ideal.iSup_iInf_eq_top_iff_pairwise","start":[21,0],"end":[110,18],"file_path":"Mathlib/RingTheory/Coprime/Ideal.lean","tactics":[{"state_before":"ι : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nt : Finset ι\nh : t.Nonempty\nI : ι → Ideal R\n⊢ ⨆ i ∈ t, ⨅ j ∈ t, ⨅ (_ : j ≠ i), I j = ⊤ ↔ (↑t).Pairwise fun i j => I i ⊔ I j = ⊤","state_after":"ι : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nt : Finset ι\nh : t.Nonempty\nI : ι → Ideal R\nthis : DecidableEq ι\n⊢ ⨆ i ∈ t, ⨅ j ∈ t, ⨅ (_ : j ≠ i), I j = ⊤ ↔ (↑t).Pairwise fun i j => I i ⊔ I j = ⊤","tactic":"haveI : DecidableEq ι := Classical.decEq ι","premises":[{"full_name":"Classical.decEq","def_path":"Mathlib/Logic/Basic.lean","def_pos":[737,18],"def_end_pos":[737,23]},{"full_name":"DecidableEq","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[868,7],"def_end_pos":[868,18]}]},{"state_before":"ι : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nt : Finset ι\nh : t.Nonempty\nI : ι → Ideal R\nthis : DecidableEq ι\n⊢ ⨆ i ∈ t, ⨅ j ∈ t, ⨅ (_ : j ≠ i), I j = ⊤ ↔ (↑t).Pairwise fun i j => I i ⊔ I j = ⊤","state_after":"ι : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nt : Finset ι\nh : t.Nonempty\nI : ι → Ideal R\nthis : DecidableEq ι\n⊢ (∃ μ, ∑ i ∈ t, ���(μ i) = 1) ↔ (↑t).Pairwise fun i j => I i ⊔ I j = ⊤","tactic":"rw [eq_top_iff_one, Submodule.mem_iSup_finset_iff_exists_sum]","premises":[{"full_name":"Ideal.eq_top_iff_one","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[87,8],"def_end_pos":[87,22]},{"full_name":"Submodule.mem_iSup_finset_iff_exists_sum","def_path":"Mathlib/LinearAlgebra/DFinsupp.lean","def_pos":[347,8],"def_end_pos":[347,38]}]},{"state_before":"ι : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nt : Finset ι\nh : t.Nonempty\nI : ι → Ideal R\nthis : DecidableEq ι\n⊢ (∃ μ, ∑ i ∈ t, ↑(μ i) = 1) ↔ (↑t).Pairwise fun i j => I i ⊔ I j = ⊤","state_after":"case refine_1\nι : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nI : ι → Ideal R\nthis : DecidableEq ι\n⊢ ∀ (a : ι), (∃ μ, ∑ i ∈ {a}, ↑(μ i) = 1) ↔ (↑{a}).Pairwise fun i j => I i ⊔ I j = ⊤\n\ncase refine_2\nι : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nI : ι → Ideal R\nthis : DecidableEq ι\n⊢ ∀ (a : ι) (s : Finset ι) (h : a ∉ s),\n    s.Nonempty →\n      ((∃ μ, ∑ i ∈ s, ↑(μ i) = 1) ↔ (↑s).Pairwise fun i j => I i ⊔ I j = ⊤) →\n        ((∃ μ, ∑ i ∈ Finset.cons a s h, ↑(μ i) = 1) ↔ (↑(Finset.cons a s h)).Pairwise fun i j => I i ⊔ I j = ⊤)","tactic":"refine h.cons_induction ?_ ?_ <;> clear t h","premises":[{"full_name":"Finset.Nonempty.cons_induction","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[1098,8],"def_end_pos":[1098,31]}]},{"state_before":"case refine_2\nι : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nI : ι → Ideal R\nthis : DecidableEq ι\n⊢ ∀ (a : ι) (s : Finset ι) (h : a ∉ s),\n    s.Nonempty →\n      ((∃ μ, ∑ i ∈ s, ↑(μ i) = 1) ↔ (↑s).Pairwise fun i j => I i ⊔ I j = ⊤) →\n        ((∃ μ, ∑ i ∈ Finset.cons a s h, ↑(μ i) = 1) ↔ (↑(Finset.cons a s h)).Pairwise fun i j => I i ⊔ I j = ⊤)","state_after":"case refine_2\nι : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nI : ι → Ideal R\nthis : DecidableEq ι\na : ι\nt : Finset ι\nhat : a ∉ t\nh : t.Nonempty\nih : (∃ μ, ∑ i ∈ t, ↑(μ i) = 1) ↔ (↑t).Pairwise fun i j => I i ⊔ I j = ⊤\n⊢ (∃ μ, ∑ i ∈ Finset.cons a t hat, ↑(μ i) = 1) ↔ (↑(Finset.cons a t hat)).Pairwise fun i j => I i ⊔ I j = ⊤","tactic":"intro a t hat h ih","premises":[]},{"state_before":"case refine_2\nι : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nI : ι → Ideal R\nthis : DecidableEq ι\na : ι\nt : Finset ι\nhat : a ∉ t\nh : t.Nonempty\nih : (∃ μ, ∑ i ∈ t, ↑(μ i) = 1) ↔ (↑t).Pairwise fun i j => I i ⊔ I j = ⊤\n⊢ (∃ μ, ∑ i ∈ Finset.cons a t hat, ↑(μ i) = 1) ↔ (↑(Finset.cons a t hat)).Pairwise fun i j => I i ⊔ I j = ⊤","state_after":"case refine_2\nι : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nI : ι → Ideal R\nthis : DecidableEq ι\na : ι\nt : Finset ι\nhat : a ∉ t\nh : t.Nonempty\nih : (∃ μ, ∑ i ∈ t, ↑(μ i) = 1) ↔ (↑t).Pairwise fun i j => I i ⊔ I j = ⊤\n⊢ (∃ μ, ∑ i ∈ Finset.cons a t hat, ↑(μ i) = 1) ↔\n    ((↑t).Pairwise fun i j => I i ⊔ I j = ⊤) ∧ ∀ b ∈ ↑t, a ≠ b → I a ⊔ I b = ⊤","tactic":"rw [Finset.coe_cons,\n    Set.pairwise_insert_of_symmetric fun i j (h : I i ⊔ I j = ⊤) ↦ (sup_comm _ _).trans h]","premises":[{"full_name":"Eq.trans","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[335,8],"def_end_pos":[335,16]},{"full_name":"Finset.coe_cons","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[783,8],"def_end_pos":[783,16]},{"full_name":"Set.pairwise_insert_of_symmetric","def_path":"Mathlib/Data/Set/Pairwise/Basic.lean","def_pos":[147,8],"def_end_pos":[147,36]},{"full_name":"Sup.sup","def_path":"Mathlib/Order/Notation.lean","def_pos":[47,2],"def_end_pos":[47,5]},{"full_name":"Top.top","def_path":"Mathlib/Order/Notation.lean","def_pos":[94,2],"def_end_pos":[94,5]},{"full_name":"sup_comm","def_path":"Mathlib/Order/Lattice.lean","def_pos":[193,8],"def_end_pos":[193,16]}]},{"state_before":"case refine_2\nι : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nI : ι → Ideal R\nthis : DecidableEq ι\na : ι\nt : Finset ι\nhat : a ∉ t\nh : t.Nonempty\nih : (∃ μ, ∑ i ∈ t, ↑(μ i) = 1) ↔ (↑t).Pairwise fun i j => I i ⊔ I j = ⊤\n⊢ (∃ μ, ∑ i ∈ Finset.cons a t hat, ↑(μ i) = 1) ↔\n    ((↑t).Pairwise fun i j => I i ⊔ I j = ⊤) ∧ ∀ b ∈ ↑t, a ≠ b → I a ⊔ I b = ⊤","state_after":"case refine_2.mp\nι : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nI : ι → Ideal R\nthis : DecidableEq ι\na : ι\nt : Finset ι\nhat : a ∉ t\nh : t.Nonempty\nih : (∃ μ, ∑ i ∈ t, ↑(μ i) = 1) ↔ (↑t).Pairwise fun i j => I i ⊔ I j = ⊤\n⊢ (∃ μ, ∑ i ∈ Finset.cons a t hat, ↑(μ i) = 1) →\n    ((↑t).Pairwise fun i j => I i ⊔ I j = ⊤) ∧ ∀ b ∈ ↑t, a ≠ b → I a ⊔ I b = ⊤\n\ncase refine_2.mpr\nι : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nI : ι → Ideal R\nthis : DecidableEq ι\na : ι\nt : Finset ι\nhat : a ∉ t\nh : t.Nonempty\nih : (∃ μ, ∑ i ∈ t, ↑(μ i) = 1) ↔ (↑t).Pairwise fun i j => I i ⊔ I j = ⊤\n⊢ (((↑t).Pairwise fun i j => I i ⊔ I j = ⊤) ∧ ∀ b ∈ ↑t, a ≠ b → I a ⊔ I b = ⊤) →\n    ∃ μ, ∑ i ∈ Finset.cons a t hat, ↑(μ i) = 1","tactic":"constructor","premises":[]}]}
+{"url":"Mathlib/NumberTheory/NumberField/Embeddings.lean","commit":"","full_name":"NumberField.InfinitePlace.card_complex_embeddings","start":[564,0],"end":[579,62],"file_path":"Mathlib/NumberTheory/NumberField/Embeddings.lean","tactics":[{"state_before":"k : Type u_1\ninst✝³ : Field k\nK : Type u_2\ninst✝² : Field K\nF : Type u_3\ninst✝¹ : Field F\ninst✝ : NumberField K\n⊢ card { φ // ¬ComplexEmbedding.IsReal φ } = 2 * NrComplexPlaces K","state_after":"k : Type u_1\ninst✝³ : Field k\nK : Type u_2\ninst✝² : Field K\nF : Type u_3\ninst✝¹ : Field F\ninst✝ : NumberField K\n⊢ ∀ (w : { w // w.IsComplex }), (Finset.filter (fun φ => mkComplex φ = w) Finset.univ).card = 2","tactic":"suffices ∀ w : { w : InfinitePlace K // IsComplex w }, (Finset.univ.filter\n      fun φ : { φ // ¬ ComplexEmbedding.IsReal φ } => mkComplex φ = w).card = 2 by\n    rw [Fintype.card, Finset.card_eq_sum_ones, ← Finset.sum_fiberwise _ (fun φ => mkComplex φ)]\n    simp_rw [Finset.sum_const, this, smul_eq_mul, mul_one, Fintype.card, Finset.card_eq_sum_ones,\n      Finset.mul_sum, Finset.sum_const, smul_eq_mul, mul_one]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Finset.card","def_path":"Mathlib/Data/Finset/Card.lean","def_pos":[40,4],"def_end_pos":[40,8]},{"full_name":"Finset.card_eq_sum_ones","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[1768,8],"def_end_pos":[1768,24]},{"full_name":"Finset.filter","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2144,4],"def_end_pos":[2144,10]},{"full_name":"Finset.mul_sum","def_path":"Mathlib/Algebra/BigOperators/Ring.lean","def_pos":[44,6],"def_end_pos":[44,13]},{"full_name":"Finset.sum_const","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[1469,2],"def_end_pos":[1469,13]},{"full_name":"Finset.sum_fiberwise","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[701,2],"def_end_pos":[701,13]},{"full_name":"Finset.univ","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[67,4],"def_end_pos":[67,8]},{"full_name":"Fintype.card","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[62,4],"def_end_pos":[62,8]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Not","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[224,4],"def_end_pos":[224,7]},{"full_name":"NumberField.ComplexEmbedding.IsReal","def_path":"Mathlib/NumberTheory/NumberField/Embeddings.lean","def_pos":[162,7],"def_end_pos":[162,13]},{"full_name":"NumberField.InfinitePlace","def_path":"Mathlib/NumberTheory/NumberField/Embeddings.lean","def_pos":[246,4],"def_end_pos":[246,29]},{"full_name":"NumberField.InfinitePlace.IsComplex","def_path":"Mathlib/NumberTheory/NumberField/Embeddings.lean","def_pos":[337,4],"def_end_pos":[337,13]},{"full_name":"NumberField.InfinitePlace.mkComplex","def_path":"Mathlib/NumberTheory/NumberField/Embeddings.lean","def_pos":[469,18],"def_end_pos":[469,27]},{"full_name":"Subtype","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[583,10],"def_end_pos":[583,17]},{"full_name":"mul_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[413,8],"def_end_pos":[413,15]},{"full_name":"smul_eq_mul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[79,6],"def_end_pos":[79,17]}]},{"state_before":"k : Type u_1\ninst✝³ : Field k\nK : Type u_2\ninst✝² : Field K\nF : Type u_3\ninst✝¹ : Field F\ninst✝ : NumberField K\n⊢ ∀ (w : { w // w.IsComplex }), (Finset.filter (fun φ => mkComplex φ = w) Finset.univ).card = 2","state_after":"case mk\nk : Type u_1\ninst✝³ : Field k\nK : Type u_2\ninst✝² : Field K\nF : Type u_3\ninst✝¹ : Field F\ninst✝ : NumberField K\nw : InfinitePlace K\nhw : w.IsComplex\n⊢ (Finset.filter (fun φ => mkComplex φ = ⟨w, hw⟩) Finset.univ).card = 2","tactic":"rintro ⟨w, hw⟩","premises":[]},{"state_before":"case mk\nk : Type u_1\ninst✝³ : Field k\nK : Type u_2\ninst✝² : Field K\nF : Type u_3\ninst✝¹ : Field F\ninst✝ : NumberField K\nw : InfinitePlace K\nhw : w.IsComplex\n⊢ (Finset.filter (fun φ => mkComplex φ = ⟨w, hw⟩) Finset.univ).card = 2","state_after":"case h.e'_2\nk : Type u_1\ninst✝³ : Field k\nK : Type u_2\ninst✝² : Field K\nF : Type u_3\ninst✝¹ : Field F\ninst✝ : NumberField K\nw : InfinitePlace K\nhw : w.IsComplex\n⊢ (Finset.filter (fun φ => mkComplex φ = ⟨w, hw⟩) Finset.univ).card =\n    (Finset.filter (fun φ => mk φ = w) Finset.univ).card\n\ncase h.e'_3\nk : Type u_1\ninst✝³ : Field k\nK : Type u_2\ninst✝² : Field K\nF : Type u_3\ninst✝¹ : Field F\ninst✝ : NumberField K\nw : InfinitePlace K\nhw : w.IsComplex\n⊢ 2 = w.mult","tactic":"convert card_filter_mk_eq w","premises":[{"full_name":"NumberField.InfinitePlace.card_filter_mk_eq","def_path":"Mathlib/NumberTheory/NumberField/Embeddings.lean","def_pos":[435,8],"def_end_pos":[435,25]}]}]}
+{"url":"Mathlib/FieldTheory/Finite/GaloisField.lean","commit":"","full_name":"GaloisField.card","start":[140,0],"end":[142,97],"file_path":"Mathlib/FieldTheory/Finite/GaloisField.lean","tactics":[{"state_before":"p✝ : ℕ\ninst✝ : Fact (Nat.Prime p✝)\nn✝ p : ℕ\nh_prime : Fact (Nat.Prime p)\nn : ℕ\nh : n ≠ 0\n⊢ Fintype.card (GaloisField p n) = p ^ n","state_after":"p✝ : ℕ\ninst✝ : Fact (Nat.Prime p✝)\nn✝ p : ℕ\nh_prime : Fact (Nat.Prime p)\nn : ℕ\nh : n ≠ 0\nb : Basis { x // x ∈ IsNoetherian.finsetBasisIndex (ZMod p) (GaloisField p n) } (ZMod p) (GaloisField p n) :=\n  IsNoetherian.finsetBasis (ZMod p) (GaloisField p n)\n⊢ Fintype.card (GaloisField p n) = p ^ n","tactic":"let b := IsNoetherian.finsetBasis (ZMod p) (GaloisField p n)","premises":[{"full_name":"GaloisField","def_path":"Mathlib/FieldTheory/Finite/GaloisField.lean","def_pos":[64,4],"def_end_pos":[64,15]},{"full_name":"IsNoetherian.finsetBasis","def_path":"Mathlib/FieldTheory/Finiteness.lean","def_pos":[80,18],"def_end_pos":[80,29]},{"full_name":"ZMod","def_path":"Mathlib/Data/ZMod/Defs.lean","def_pos":[89,4],"def_end_pos":[89,8]}]},{"state_before":"p✝ : ℕ\ninst✝ : Fact (Nat.Prime p✝)\nn✝ p : ℕ\nh_prime : Fact (Nat.Prime p)\nn : ℕ\nh : n ≠ 0\nb : Basis { x // x ∈ IsNoetherian.finsetBasisIndex (ZMod p) (GaloisField p n) } (ZMod p) (GaloisField p n) :=\n  IsNoetherian.finsetBasis (ZMod p) (GaloisField p n)\n⊢ Fintype.card (GaloisField p n) = p ^ n","state_after":"no goals","tactic":"rw [Module.card_fintype b, ← FiniteDimensional.finrank_eq_card_basis b, ZMod.card, finrank p h]","premises":[{"full_name":"FiniteDimensional.finrank_eq_card_basis","def_path":"Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean","def_pos":[388,8],"def_end_pos":[388,29]},{"full_name":"GaloisField.finrank","def_path":"Mathlib/FieldTheory/Finite/GaloisField.lean","def_pos":[91,8],"def_end_pos":[91,15]},{"full_name":"Module.card_fintype","def_path":"Mathlib/LinearAlgebra/Basis.lean","def_pos":[830,8],"def_end_pos":[830,27]},{"full_name":"ZMod.card","def_path":"Mathlib/Data/ZMod/Defs.lean","def_pos":[113,8],"def_end_pos":[113,12]}]}]}
+{"url":"Mathlib/Logic/Equiv/Basic.lean","commit":"","full_name":"Function.piCongrLeft'_symm_update","start":[1801,0],"end":[1804,62],"file_path":"Mathlib/Logic/Equiv/Basic.lean","tactics":[{"state_before":"α : Sort u_2\nβ : Sort u_3\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nP : α → Sort u_1\ne : α ≃ β\nf : (b : β) → P (e.symm b)\nb : β\nx : P (e.symm b)\n⊢ (Equiv.piCongrLeft' P e).symm (update f b x) = update ((Equiv.piCongrLeft' P e).symm f) (e.symm b) x","state_after":"no goals","tactic":"simp [(e.piCongrLeft' P).symm_apply_eq, piCongrLeft'_update]","premises":[{"full_name":"Equiv.piCongrLeft'","def_path":"Mathlib/Logic/Equiv/Basic.lean","def_pos":[1559,4],"def_end_pos":[1559,16]},{"full_name":"Equiv.symm_apply_eq","def_path":"Mathlib/Logic/Equiv/Defs.lean","def_pos":[289,8],"def_end_pos":[289,21]},{"full_name":"Function.piCongrLeft'_update","def_path":"Mathlib/Logic/Equiv/Basic.lean","def_pos":[1786,8],"def_end_pos":[1786,27]}]}]}
+{"url":"Mathlib/SetTheory/Game/PGame.lean","commit":"","full_name":"SetTheory.PGame.insertRight_le","start":[1680,0],"end":[1683,25],"file_path":"Mathlib/SetTheory/Game/PGame.lean","tactics":[{"state_before":"xl xr : Type u\nx x' : PGame\n⊢ x.insertRight x' ≤ x","state_after":"xl xr : Type u\nx x' : PGame\n⊢ -x ≤ (-x).insertLeft (-x')","tactic":"rw [← neg_le_neg_iff, ← neg_insertLeft_neg]","premises":[{"full_name":"SetTheory.PGame.neg_insertLeft_neg","def_path":"Mathlib/SetTheory/Game/PGame.lean","def_pos":[1677,8],"def_end_pos":[1677,26]},{"full_name":"SetTheory.PGame.neg_le_neg_iff","def_path":"Mathlib/SetTheory/Game/PGame.lean","def_pos":[1188,8],"def_end_pos":[1188,22]}]},{"state_before":"xl xr : Type u\nx x' : PGame\n⊢ -x ≤ (-x).insertLeft (-x')","state_after":"no goals","tactic":"exact le_insertLeft _ _","premises":[{"full_name":"SetTheory.PGame.le_insertLeft","def_path":"Mathlib/SetTheory/Game/PGame.lean","def_pos":[1625,6],"def_end_pos":[1625,19]}]}]}
+{"url":"Mathlib/RingTheory/Localization/AtPrime.lean","commit":"","full_name":"IsLocalization.AtPrime.localRing","start":[73,0],"end":[97,85],"file_path":"Mathlib/RingTheory/Localization/AtPrime.lean","tactics":[{"state_before":"R : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP✝ : Type u_3\ninst✝¹ : CommSemiring P✝\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nthis : _root_.Nontrivial S := Nontrivial S P\n⊢ ∀ (a b : S), a ∈ nonunits S → b ∈ nonunits S → a + b ∈ nonunits S","state_after":"R : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP✝ : Type u_3\ninst✝¹ : CommSemiring P✝\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nthis : _root_.Nontrivial S := Nontrivial S P\nx y : S\nhx : x ∈ nonunits S\nhy : y ∈ nonunits S\nhu : IsUnit (x + y)\n⊢ False","tactic":"intro x y hx hy hu","premises":[]},{"state_before":"R : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP✝ : Type u_3\ninst✝¹ : CommSemiring P✝\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nthis : _root_.Nontrivial S := Nontrivial S P\nx y : S\nhx : x ∈ nonunits S\nhy : y ∈ nonunits S\nhu : IsUnit (x + y)\n⊢ False","state_after":"case intro\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP✝ : Type u_3\ninst✝¹ : CommSemiring P✝\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nthis : _root_.Nontrivial S := Nontrivial S P\nx y : S\nhx : x ∈ nonunits S\nhy : y ∈ nonunits S\nhu : IsUnit (x + y)\nz : S\nhxyz : (x + y) * z = 1\n⊢ False","tactic":"cases' isUnit_iff_exists_inv.1 hu with z hxyz","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"isUnit_iff_exists_inv","def_path":"Mathlib/Algebra/Group/Units.lean","def_pos":[598,8],"def_end_pos":[598,29]}]},{"state_before":"case intro\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP✝ : Type u_3\ninst✝¹ : CommSemiring P✝\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nthis : _root_.Nontrivial S := Nontrivial S P\nx y : S\nhx : x ∈ nonunits S\nhy : y ∈ nonunits S\nhu : IsUnit (x + y)\nz : S\nhxyz : (x + y) * z = 1\n⊢ False","state_after":"case intro\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP✝ : Type u_3\ninst✝¹ : CommSemiring P✝\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nthis✝ : _root_.Nontrivial S := Nontrivial S P\nx y : S\nhx : x ∈ nonunits S\nhy : y ∈ nonunits S\nhu : IsUnit (x + y)\nz : S\nhxyz : (x + y) * z = 1\nthis : ∀ {r : R} {s : ↥P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P\n⊢ False","tactic":"have : ∀ {r : R} {s : P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P := fun {r s} =>\n        not_imp_comm.1 fun nr => isUnit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : P.primeCompl),\n          mk'_mul_mk'_eq_one' _ _ <| show r ∈ P.primeCompl from nr⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Ideal.primeCompl","def_path":"Mathlib/RingTheory/Localization/AtPrime.lean","def_pos":[44,4],"def_end_pos":[44,14]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"IsLocalization.mk'","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[228,18],"def_end_pos":[228,21]},{"full_name":"IsLocalization.mk'_mul_mk'_eq_one'","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[385,8],"def_end_pos":[385,27]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"isUnit_iff_exists_inv","def_path":"Mathlib/Algebra/Group/Units.lean","def_pos":[598,8],"def_end_pos":[598,29]},{"full_name":"nonunits","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[783,4],"def_end_pos":[783,12]},{"full_name":"not_imp_comm","def_path":"Mathlib/Logic/Basic.lean","def_pos":[213,8],"def_end_pos":[213,20]}]},{"state_before":"case intro\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP✝ : Type u_3\ninst✝¹ : CommSemiring P✝\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nthis✝ : _root_.Nontrivial S := Nontrivial S P\nx y : S\nhx : x ∈ nonunits S\nhy : y ∈ nonunits S\nhu : IsUnit (x + y)\nz : S\nhxyz : (x + y) * z = 1\nthis : ∀ {r : R} {s : ↥P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P\n⊢ False","state_after":"case intro.intro.intro\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP✝ : Type u_3\ninst✝¹ : CommSemiring P✝\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nthis✝ : _root_.Nontrivial S := Nontrivial S P\nx y : S\nhx : x ∈ nonunits S\nhy : y ∈ nonunits S\nhu : IsUnit (x + y)\nz : S\nhxyz : (x + y) * z = 1\nthis : ∀ {r : R} {s : ↥P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P\nrx : R\nsx : ↥P.primeCompl\nhrx : mk' S rx sx = x\n⊢ False","tactic":"rcases mk'_surjective P.primeCompl x with ⟨rx, sx, hrx⟩","premises":[{"full_name":"Ideal.primeCompl","def_path":"Mathlib/RingTheory/Localization/AtPrime.lean","def_pos":[44,4],"def_end_pos":[44,14]},{"full_name":"IsLocalization.mk'_surjective","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[279,8],"def_end_pos":[279,22]}]},{"state_before":"case intro.intro.intro\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP✝ : Type u_3\ninst✝¹ : CommSemiring P✝\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nthis✝ : _root_.Nontrivial S := Nontrivial S P\nx y : S\nhx : x ∈ nonunits S\nhy : y ∈ nonunits S\nhu : IsUnit (x + y)\nz : S\nhxyz : (x + y) * z = 1\nthis : ∀ {r : R} {s : ↥P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P\nrx : R\nsx : ↥P.primeCompl\nhrx : mk' S rx sx = x\n⊢ False","state_after":"case intro.intro.intro.intro.intro\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP✝ : Type u_3\ninst✝¹ : CommSemiring P✝\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nthis✝ : _root_.Nontrivial S := Nontrivial S P\nx y : S\nhx : x ∈ nonunits S\nhy : y ∈ nonunits S\nhu : IsUnit (x + y)\nz : S\nhxyz : (x + y) * z = 1\nthis : ∀ {r : R} {s : ↥P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P\nrx : R\nsx : ↥P.primeCompl\nhrx : mk' S rx sx = x\nry : R\nsy : ↥P.primeCompl\nhry : mk' S ry sy = y\n⊢ False","tactic":"rcases mk'_surjective P.primeCompl y with ⟨ry, sy, hry⟩","premises":[{"full_name":"Ideal.primeCompl","def_path":"Mathlib/RingTheory/Localization/AtPrime.lean","def_pos":[44,4],"def_end_pos":[44,14]},{"full_name":"IsLocalization.mk'_surjective","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[279,8],"def_end_pos":[279,22]}]},{"state_before":"case intro.intro.intro.intro.intro\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP✝ : Type u_3\ninst✝¹ : CommSemiring P✝\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nthis✝ : _root_.Nontrivial S := Nontrivial S P\nx y : S\nhx : x ∈ nonunits S\nhy : y ∈ nonunits S\nhu : IsUnit (x + y)\nz : S\nhxyz : (x + y) * z = 1\nthis : ∀ {r : R} {s : ↥P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P\nrx : R\nsx : ↥P.primeCompl\nhrx : mk' S rx sx = x\nry : R\nsy : ↥P.primeCompl\nhry : mk' S ry sy = y\n⊢ False","state_after":"case intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP✝ : Type u_3\ninst✝¹ : CommSemiring P✝\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nthis✝ : _root_.Nontrivial S := Nontrivial S P\nx y : S\nhx : x ∈ nonunits S\nhy : y ∈ nonunits S\nhu : IsUnit (x + y)\nz : S\nhxyz : (x + y) * z = 1\nthis : ∀ {r : R} {s : ↥P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P\nrx : R\nsx : ↥P.primeCompl\nhrx : mk' S rx sx = x\nry : R\nsy : ↥P.primeCompl\nhry : mk' S ry sy = y\nrz : R\nsz : ↥P.primeCompl\nhrz : mk' S rz sz = z\n⊢ False","tactic":"rcases mk'_surjective P.primeCompl z with ⟨rz, sz, hrz⟩","premises":[{"full_name":"Ideal.primeCompl","def_path":"Mathlib/RingTheory/Localization/AtPrime.lean","def_pos":[44,4],"def_end_pos":[44,14]},{"full_name":"IsLocalization.mk'_surjective","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[279,8],"def_end_pos":[279,22]}]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP✝ : Type u_3\ninst✝¹ : CommSemiring P✝\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nthis✝ : _root_.Nontrivial S := Nontrivial S P\nx y : S\nhx : x ∈ nonunits S\nhy : y ∈ nonunits S\nhu : IsUnit (x + y)\nz : S\nhxyz : (x + y) * z = 1\nthis : ∀ {r : R} {s : ↥P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P\nrx : R\nsx : ↥P.primeCompl\nhrx : mk' S rx sx = x\nry : R\nsy : ↥P.primeCompl\nhry : mk' S ry sy = y\nrz : R\nsz : ↥P.primeCompl\nhrz : mk' S rz sz = z\n⊢ False","state_after":"case intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP✝ : Type u_3\ninst✝¹ : CommSemiring P✝\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nthis✝ : _root_.Nontrivial S := Nontrivial S P\nx y : S\nhx : x ∈ nonunits S\nhy : y ∈ nonunits S\nhu : IsUnit (x + y)\nz : S\nthis : ∀ {r : R} {s : ↥P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P\nrx : R\nsx : ↥P.primeCompl\nhrx : mk' S rx sx = x\nry : R\nsy : ↥P.primeCompl\nhry : mk' S ry sy = y\nrz : R\nsz : ↥P.primeCompl\nhxyz : mk' S ((rx * ↑sy + ry * ↑sx) * rz) (sx * sy * sz) = mk' S 1 ⟨1, ⋯⟩\nhrz : mk' S rz sz = z\n⊢ False","tactic":"rw [← hrx, ← hry, ← hrz, ← mk'_add, ← mk'_mul, ← mk'_self S P.primeCompl.one_mem] at hxyz","premises":[{"full_name":"Ideal.primeCompl","def_path":"Mathlib/RingTheory/Localization/AtPrime.lean","def_pos":[44,4],"def_end_pos":[44,14]},{"full_name":"IsLocalization.mk'_add","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[420,8],"def_end_pos":[420,15]},{"full_name":"IsLocalization.mk'_mul","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[235,8],"def_end_pos":[235,15]},{"full_name":"IsLocalization.mk'_self","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[355,8],"def_end_pos":[355,16]},{"full_name":"Submonoid.one_mem","def_path":"Mathlib/Algebra/Group/Submonoid/Basic.lean","def_pos":[199,18],"def_end_pos":[199,25]}]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP✝ : Type u_3\ninst✝¹ : CommSemiring P✝\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nthis✝ : _root_.Nontrivial S := Nontrivial S P\nx y : S\nhx : x ∈ nonunits S\nhy : y ∈ nonunits S\nhu : IsUnit (x + y)\nz : S\nthis : ∀ {r : R} {s : ↥P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P\nrx : R\nsx : ↥P.primeCompl\nhrx : mk' S rx sx = x\nry : R\nsy : ↥P.primeCompl\nhry : mk' S ry sy = y\nrz : R\nsz : ↥P.primeCompl\nhxyz : mk' S ((rx * ↑sy + ry * ↑sx) * rz) (sx * sy * sz) = mk' S 1 ⟨1, ⋯⟩\nhrz : mk' S rz sz = z\n⊢ False","state_after":"case intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP✝ : Type u_3\ninst✝¹ : CommSemiring P✝\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nthis✝ : _root_.Nontrivial S := Nontrivial S P\nx y : S\nhy : y ∈ nonunits S\nhu : IsUnit (x + y)\nz : S\nthis : ∀ {r : R} {s : ↥P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P\nrx : R\nsx : ↥P.primeCompl\nhx : mk' S rx sx ∈ nonunits S\nhrx : mk' S rx sx = x\nry : R\nsy : ↥P.primeCompl\nhry : mk' S ry sy = y\nrz : R\nsz : ↥P.primeCompl\nhxyz : mk' S ((rx * ↑sy + ry * ↑sx) * rz) (sx * sy * sz) = mk' S 1 ⟨1, ⋯⟩\nhrz : mk' S rz sz = z\n⊢ False","tactic":"rw [← hrx] at hx","premises":[]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP✝ : Type u_3\ninst✝¹ : CommSemiring P✝\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nthis✝ : _root_.Nontrivial S := Nontrivial S P\nx y : S\nhy : y ∈ nonunits S\nhu : IsUnit (x + y)\nz : S\nthis : ∀ {r : R} {s : ↥P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P\nrx : R\nsx : ↥P.primeCompl\nhx : mk' S rx sx ∈ nonunits S\nhrx : mk' S rx sx = x\nry : R\nsy : ↥P.primeCompl\nhry : mk' S ry sy = y\nrz : R\nsz : ↥P.primeCompl\nhxyz : mk' S ((rx * ↑sy + ry * ↑sx) * rz) (sx * sy * sz) = mk' S 1 ⟨1, ⋯⟩\nhrz : mk' S rz sz = z\n⊢ False","state_after":"case intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP✝ : Type u_3\ninst✝¹ : CommSemiring P✝\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nthis✝ : _root_.Nontrivial S := Nontrivial S P\nx y : S\nhu : IsUnit (x + y)\nz : S\nthis : ∀ {r : R} {s : ↥P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P\nrx : R\nsx : ↥P.primeCompl\nhx : mk' S rx sx ∈ nonunits S\nhrx : mk' S rx sx = x\nry : R\nsy : ↥P.primeCompl\nhy : mk' S ry sy ∈ nonunits S\nhry : mk' S ry sy = y\nrz : R\nsz : ↥P.primeCompl\nhxyz : mk' S ((rx * ↑sy + ry * ↑sx) * rz) (sx * sy * sz) = mk' S 1 ⟨1, ⋯⟩\nhrz : mk' S rz sz = z\n⊢ False","tactic":"rw [← hry] at hy","premises":[]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP✝ : Type u_3\ninst✝¹ : CommSemiring P✝\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nthis✝ : _root_.Nontrivial S := Nontrivial S P\nx y : S\nhu : IsUnit (x + y)\nz : S\nthis : ∀ {r : R} {s : ↥P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P\nrx : R\nsx : ↥P.primeCompl\nhx : mk' S rx sx ∈ nonunits S\nhrx : mk' S rx sx = x\nry : R\nsy : ↥P.primeCompl\nhy : mk' S ry sy ∈ nonunits S\nhry : mk' S ry sy = y\nrz : R\nsz : ↥P.primeCompl\nhxyz : mk' S ((rx * ↑sy + ry * ↑sx) * rz) (sx * sy * sz) = mk' S 1 ⟨1, ⋯⟩\nhrz : mk' S rz sz = z\n⊢ False","state_after":"case intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP✝ : Type u_3\ninst✝¹ : CommSemiring P✝\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nthis✝ : _root_.Nontrivial S := Nontrivial S P\nx y : S\nhu : IsUnit (x + y)\nz : S\nthis : ∀ {r : R} {s : ↥P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P\nrx : R\nsx : ↥P.primeCompl\nhx : mk' S rx sx ∈ nonunits S\nhrx : mk' S rx sx = x\nry : R\nsy : ↥P.primeCompl\nhy : mk' S ry sy ∈ nonunits S\nhry : mk' S ry sy = y\nrz : R\nsz : ↥P.primeCompl\nhxyz : mk' S ((rx * ↑sy + ry * ↑sx) * rz) (sx * sy * sz) = mk' S 1 ⟨1, ⋯⟩\nhrz : mk' S rz sz = z\nt : ↥P.primeCompl\nht : ↑t * (↑⟨1, ⋯⟩ * ((rx * ↑sy + ry * ↑sx) * rz)) = ↑t * (↑(sx * sy * sz) * 1)\n⊢ False","tactic":"obtain ⟨t, ht⟩ := IsLocalization.eq.1 hxyz","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"IsLocalization.eq","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[314,18],"def_end_pos":[314,20]}]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP✝ : Type u_3\ninst✝¹ : CommSemiring P✝\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nthis✝ : _root_.Nontrivial S := Nontrivial S P\nx y : S\nhu : IsUnit (x + y)\nz : S\nthis : ∀ {r : R} {s : ↥P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P\nrx : R\nsx : ↥P.primeCompl\nhx : mk' S rx sx ∈ nonunits S\nhrx : mk' S rx sx = x\nry : R\nsy : ↥P.primeCompl\nhy : mk' S ry sy ∈ nonunits S\nhry : mk' S ry sy = y\nrz : R\nsz : ↥P.primeCompl\nhxyz : mk' S ((rx * ↑sy + ry * ↑sx) * rz) (sx * sy * sz) = mk' S 1 ⟨1, ⋯⟩\nhrz : mk' S rz sz = z\nt : ↥P.primeCompl\nht : ↑t * (↑⟨1, ⋯⟩ * ((rx * ↑sy + ry * ↑sx) * rz)) = ↑t * (↑(sx * sy * sz) * 1)\n⊢ False","state_after":"case intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP✝ : Type u_3\ninst✝¹ : CommSemiring P✝\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nthis✝ : _root_.Nontrivial S := Nontrivial S P\nx y : S\nhu : IsUnit (x + y)\nz : S\nthis : ∀ {r : R} {s : ↥P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P\nrx : R\nsx : ↥P.primeCompl\nhx : mk' S rx sx ∈ nonunits S\nhrx : mk' S rx sx = x\nry : R\nsy : ↥P.primeCompl\nhy : mk' S ry sy ∈ nonunits S\nhry : mk' S ry sy = y\nrz : R\nsz : ↥P.primeCompl\nhxyz : mk' S ((rx * ↑sy + ry * ↑sx) * rz) (sx * sy * sz) = mk' S 1 ⟨1, ⋯⟩\nhrz : mk' S rz sz = z\nt : ↥P.primeCompl\nht : ↑t * ((rx * ↑sy + ry * ↑sx) * rz) = ↑t * (↑sx * ↑sy * ↑sz)\n⊢ False","tactic":"simp only [mul_one, one_mul, Submonoid.coe_mul, Subtype.coe_mk] at ht","premises":[{"full_name":"Submonoid.coe_mul","def_path":"Mathlib/Algebra/Group/Submonoid/Operations.lean","def_pos":[496,8],"def_end_pos":[496,15]},{"full_name":"Subtype.coe_mk","def_path":"Mathlib/Data/Subtype.lean","def_pos":[86,8],"def_end_pos":[86,14]},{"full_name":"mul_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[413,8],"def_end_pos":[413,15]},{"full_name":"one_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[409,8],"def_end_pos":[409,15]}]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP✝ : Type u_3\ninst✝¹ : CommSemiring P✝\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nthis✝ : _root_.Nontrivial S := Nontrivial S P\nx y : S\nhu : IsUnit (x + y)\nz : S\nthis : ∀ {r : R} {s : ↥P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P\nrx : R\nsx : ↥P.primeCompl\nhx : mk' S rx sx ∈ nonunits S\nhrx : mk' S rx sx = x\nry : R\nsy : ↥P.primeCompl\nhy : mk' S ry sy ∈ nonunits S\nhry : mk' S ry sy = y\nrz : R\nsz : ↥P.primeCompl\nhxyz : mk' S ((rx * ↑sy + ry * ↑sx) * rz) (sx * sy * sz) = mk' S 1 ⟨1, ⋯⟩\nhrz : mk' S rz sz = z\nt : ↥P.primeCompl\nht : ↑t * ((rx * ↑sy + ry * ↑sx) * rz) = ↑t * (↑sx * ↑sy * ↑sz)\n⊢ False","state_after":"case intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP✝ : Type u_3\ninst✝¹ : CommSemiring P✝\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nthis✝ : _root_.Nontrivial S := Nontrivial S P\nx y : S\nhu : IsUnit (x + y)\nz : S\nthis : ∀ {r : R} {s : ↥P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P\nrx : R\nsx : ↥P.primeCompl\nhx : mk' S rx sx ∈ nonunits S\nhrx : mk' S rx sx = x\nry : R\nsy : ↥P.primeCompl\nhy : mk' S ry sy ∈ nonunits S\nhry : mk' S ry sy = y\nrz : R\nsz : ↥P.primeCompl\nhxyz : mk' S ((rx * ↑sy + ry * ↑sx) * rz) (sx * sy * sz) = mk' S 1 ⟨1, ⋯⟩\nhrz : mk' S rz sz = z\nt : ↥P.primeCompl\nht : ↑t * ((rx * ↑sy + ry * ↑sx) * rz) = ↑t * (↑sx * ↑sy * ↑sz)\n⊢ ↑t * (��sx * ↑sy * ↑sz) ∈ P","tactic":"suffices (t : R) * (sx * sy * sz) ∈ P from\n        not_or_of_not (mt hp.mem_or_mem <| not_or_of_not sx.2 sy.2) sz.2\n          (hp.mem_or_mem <| (hp.mem_or_mem this).resolve_left t.2)","premises":[{"full_name":"Ideal.IsPrime.mem_or_mem","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[215,8],"def_end_pos":[215,26]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Or.resolve_left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[555,8],"def_end_pos":[555,23]},{"full_name":"Subtype.property","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[590,2],"def_end_pos":[590,10]},{"full_name":"mt","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[647,8],"def_end_pos":[647,10]},{"full_name":"not_or_of_not","def_path":"Mathlib/Init/Logic.lean","def_pos":[103,8],"def_end_pos":[103,21]}]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP✝ : Type u_3\ninst✝¹ : CommSemiring P✝\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nthis✝ : _root_.Nontrivial S := Nontrivial S P\nx y : S\nhu : IsUnit (x + y)\nz : S\nthis : ∀ {r : R} {s : ↥P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P\nrx : R\nsx : ↥P.primeCompl\nhx : mk' S rx sx ∈ nonunits S\nhrx : mk' S rx sx = x\nry : R\nsy : ↥P.primeCompl\nhy : mk' S ry sy ∈ nonunits S\nhry : mk' S ry sy = y\nrz : R\nsz : ↥P.primeCompl\nhxyz : mk' S ((rx * ↑sy + ry * ↑sx) * rz) (sx * sy * sz) = mk' S 1 ⟨1, ⋯⟩\nhrz : mk' S rz sz = z\nt : ↥P.primeCompl\nht : ↑t * ((rx * ↑sy + ry * ↑sx) * rz) = ↑t * (↑sx * ↑sy * ↑sz)\n⊢ ↑t * (↑sx * ↑sy * ↑sz) ∈ P","state_after":"case intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP✝ : Type u_3\ninst✝¹ : CommSemiring P✝\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nthis✝ : _root_.Nontrivial S := Nontrivial S P\nx y : S\nhu : IsUnit (x + y)\nz : S\nthis : ∀ {r : R} {s : ↥P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P\nrx : R\nsx : ↥P.primeCompl\nhx : mk' S rx sx ∈ nonunits S\nhrx : mk' S rx sx = x\nry : R\nsy : ↥P.primeCompl\nhy : mk' S ry sy ∈ nonunits S\nhry : mk' S ry sy = y\nrz : R\nsz : ↥P.primeCompl\nhxyz : mk' S ((rx * ↑sy + ry * ↑sx) * rz) (sx * sy * sz) = mk' S 1 ⟨1, ⋯⟩\nhrz : mk' S rz sz = z\nt : ↥P.primeCompl\nht : ↑t * ((rx * ↑sy + ry * ↑sx) * rz) = ↑t * (↑sx * ↑sy * ↑sz)\n⊢ ↑t * ((rx * ↑sy + ry * ↑sx) * rz) ∈ P","tactic":"rw [← ht]","premises":[]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP✝ : Type u_3\ninst✝¹ : CommSemiring P✝\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nthis✝ : _root_.Nontrivial S := Nontrivial S P\nx y : S\nhu : IsUnit (x + y)\nz : S\nthis : ∀ {r : R} {s : ↥P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P\nrx : R\nsx : ↥P.primeCompl\nhx : mk' S rx sx ∈ nonunits S\nhrx : mk' S rx sx = x\nry : R\nsy : ↥P.primeCompl\nhy : mk' S ry sy ∈ nonunits S\nhry : mk' S ry sy = y\nrz : R\nsz : ↥P.primeCompl\nhxyz : mk' S ((rx * ↑sy + ry * ↑sx) * rz) (sx * sy * sz) = mk' S 1 ⟨1, ⋯⟩\nhrz : mk' S rz sz = z\nt : ↥P.primeCompl\nht : ↑t * ((rx * ↑sy + ry * ↑sx) * rz) = ↑t * (↑sx * ↑sy * ↑sz)\n⊢ ↑t * ((rx * ↑sy + ry * ↑sx) * rz) ∈ P","state_after":"no goals","tactic":"exact\n        P.mul_mem_left _ <| P.mul_mem_right _ <|\n            P.add_mem (P.mul_mem_right _ <| this hx) <| P.mul_mem_right _ <| this hy","premises":[{"full_name":"Ideal.add_mem","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[58,18],"def_end_pos":[58,25]},{"full_name":"Ideal.mul_mem_left","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[63,8],"def_end_pos":[63,20]},{"full_name":"Ideal.mul_mem_right","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[486,8],"def_end_pos":[486,21]}]}]}
+{"url":"Mathlib/CategoryTheory/Elements.lean","commit":"","full_name":"CategoryTheory.CategoryOfElements.map_obj_snd","start":[128,0],"end":[133,98],"file_path":"Mathlib/CategoryTheory/Elements.lean","tactics":[{"state_before":"C : Type u\ninst✝ : Category.{v, u} C\nF F₁ F₂ : C ⥤ Type w\nα : F₁ ⟶ F₂\nt₁ t₂ : F₁.Elements\nk : t₁ ⟶ t₂\n⊢ F₂.map (↑k) ((fun t => ⟨t.fst, α.app t.fst t.snd⟩) t₁).snd = ((fun t => ⟨t.fst, α.app t.fst t.snd⟩) t₂).snd","state_after":"no goals","tactic":"simpa [map_snd] using (FunctorToTypes.naturality _ _ α k.1 t₁.2).symm","premises":[{"full_name":"CategoryTheory.CategoryOfElements.map_snd","def_path":"Mathlib/CategoryTheory/Elements.lean","def_pos":[89,8],"def_end_pos":[89,15]},{"full_name":"CategoryTheory.FunctorToTypes.naturality","def_path":"Mathlib/CategoryTheory/Types.lean","def_pos":[145,8],"def_end_pos":[145,18]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Sigma.snd","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[177,2],"def_end_pos":[177,5]},{"full_name":"Subtype.val","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[587,2],"def_end_pos":[587,5]}]}]}
+{"url":"Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean","commit":"","full_name":"_private.Mathlib.MeasureTheory.Function.LpSeminorm.Basic.0.MeasureTheory.le_mul_iff_eq_zero_of_nonneg_of_neg_of_nonneg","start":[1121,0],"end":[1129,17],"file_path":"Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean","tactics":[{"state_before":"α✝ : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace α✝\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α✝\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\nα : Type u_5\ninst✝ : LinearOrderedSemiring α\na b c : α\nha : 0 ≤ a\nhb : b < 0\nhc : 0 ≤ c\n⊢ a ≤ b * c ↔ a = 0 ∧ c = 0","state_after":"case mp\nα✝ : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace α✝\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α✝\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\nα : Type u_5\ninst✝ : LinearOrderedSemiring α\na b c : α\nha : 0 ≤ a\nhb : b < 0\nhc : 0 ≤ c\n⊢ a ≤ b * c → a = 0 ∧ c = 0\n\ncase mpr\nα✝ : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace α✝\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α✝\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\nα : Type u_5\ninst✝ : LinearOrderedSemiring α\na b c : α\nha : 0 ≤ a\nhb : b < 0\nhc : 0 ≤ c\n⊢ a = 0 ∧ c = 0 → a ≤ b * c","tactic":"constructor","premises":[]}]}
+{"url":"Mathlib/GroupTheory/OrderOfElement.lean","commit":"","full_name":"Function.Injective.isOfFinOrder_iff","start":[305,0],"end":[308,69],"file_path":"Mathlib/GroupTheory/OrderOfElement.lean","tactics":[{"state_before":"G : Type u_1\nH : Type u_2\nA : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝¹ : Monoid G\na b x y : G\nn m : ℕ\ninst✝ : Monoid H\nf : G →* H\nhf : Injective ⇑f\n⊢ IsOfFinOrder (f x) ↔ IsOfFinOrder x","state_after":"no goals","tactic":"rw [← orderOf_pos_iff, orderOf_injective f hf x, ← orderOf_pos_iff]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"orderOf_injective","def_path":"Mathlib/GroupTheory/OrderOfElement.lean","def_pos":[295,8],"def_end_pos":[295,25]},{"full_name":"orderOf_pos_iff","def_path":"Mathlib/GroupTheory/OrderOfElement.lean","def_pos":[179,8],"def_end_pos":[179,23]}]}]}
+{"url":"Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean","commit":"","full_name":"MeasurableEmbedding.measurable_comp_iff","start":[113,0],"end":[117,54],"file_path":"Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns t u : Set α\nmα : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf : α → β\ng : β → γ\nhg : MeasurableEmbedding g\n⊢ Measurable (g ∘ f) ↔ Measurable f","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns t u : Set α\nmα : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf : α → β\ng : β → γ\nhg : MeasurableEmbedding g\nH : Measurable (g ∘ f)\n⊢ Measurable f","tactic":"refine ⟨fun H => ?_, hg.measurable.comp⟩","premises":[{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"Measurable.comp","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","def_pos":[498,18],"def_end_pos":[498,33]},{"full_name":"MeasurableEmbedding.measurable","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean","def_pos":[59,12],"def_end_pos":[59,22]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns t u : Set α\nmα : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf : α → β\ng : β → γ\nhg : MeasurableEmbedding g\nH : Measurable (g ∘ f)\n⊢ Measurable f","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns t u : Set α\nmα : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf : α → β\ng : β → γ\nhg : MeasurableEmbedding g\nH : Measurable (g ∘ f)\n⊢ Measurable ((rangeSplitting g ∘ rangeFactorization g) ∘ f)","tactic":"suffices Measurable ((rangeSplitting g ∘ rangeFactorization g) ∘ f) by\n    rwa [(rightInverse_rangeSplitting hg.injective).comp_eq_id] at this","premises":[{"full_name":"Function.RightInverse.comp_eq_id","def_path":"Mathlib/Logic/Function/Basic.lean","def_pos":[284,8],"def_end_pos":[284,31]},{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"Measurable","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","def_pos":[479,4],"def_end_pos":[479,14]},{"full_name":"MeasurableEmbedding.injective","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean","def_pos":[57,12],"def_end_pos":[57,21]},{"full_name":"Set.rangeFactorization","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[151,4],"def_end_pos":[151,22]},{"full_name":"Set.rangeSplitting","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[156,18],"def_end_pos":[156,32]},{"full_name":"Set.rightInverse_rangeSplitting","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[964,8],"def_end_pos":[964,35]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns t u : Set α\nmα : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf : α → β\ng : β → γ\nhg : MeasurableEmbedding g\nH : Measurable (g ∘ f)\n⊢ Measurable ((rangeSplitting g ∘ rangeFactorization g) ∘ f)","state_after":"no goals","tactic":"exact hg.measurable_rangeSplitting.comp H.subtype_mk","premises":[{"full_name":"Measurable.comp","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","def_pos":[498,18],"def_end_pos":[498,33]},{"full_name":"Measurable.subtype_mk","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Basic.lean","def_pos":[494,8],"def_end_pos":[494,29]},{"full_name":"MeasurableEmbedding.measurable_rangeSplitting","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean","def_pos":[93,8],"def_end_pos":[93,33]}]}]}
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MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp : 1 ≤ p\nhn : 0 < finrank ℝ E\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑(finrank ℝ E))⁻¹\nhp'0 : ¬p' = 0\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","tactic":"by_cases hp'0 : p' = 0","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp : 1 ≤ p\nhn : 0 < finrank ℝ E\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑(finrank ℝ E))⁻¹\nhp'0 : ¬p' = 0\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","state_after":"case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","tactic":"set n := finrank ℝ E","premises":[{"full_name":"FiniteDimensional.finrank","def_path":"Mathlib/LinearAlgebra/Dimension/Finrank.lean","def_pos":[52,18],"def_end_pos":[52,25]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","state_after":"case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","tactic":"let n' := NNReal.conjExponent n","premises":[{"full_name":"NNReal.conjExponent","def_path":"Mathlib/Data/Real/ConjExponents.lean","def_pos":[135,18],"def_end_pos":[135,30]}]},{"state_before":"case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","state_after":"case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","tactic":"have h2p : (p : ℝ) < n := by\n    have : 0 < p⁻¹ - (n : ℝ)⁻¹ :=\n      NNReal.coe_lt_coe.mpr (pos_iff_ne_zero.mpr (inv_ne_zero hp'0)) |>.trans_eq hp'\n    rwa [NNReal.coe_inv, sub_pos,\n      inv_lt_inv _ (zero_lt_one.trans_le (NNReal.coe_le_coe.mpr hp))] at this\n    exact_mod_cast hn","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"NNReal.coe_inv","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[168,18],"def_end_pos":[168,25]},{"full_name":"NNReal.coe_le_coe","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[325,25],"def_end_pos":[325,35]},{"full_name":"NNReal.coe_lt_coe","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[327,25],"def_end_pos":[327,35]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"inv_lt_inv","def_path":"Mathlib/Algebra/Order/Field/Basic.lean","def_pos":[167,8],"def_end_pos":[167,18]},{"full_name":"inv_ne_zero","def_path":"Mathlib/Algebra/GroupWithZero/NeZero.lean","def_pos":[45,8],"def_end_pos":[45,19]},{"full_name":"pos_iff_ne_zero","def_path":"Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean","def_pos":[123,2],"def_end_pos":[123,13]},{"full_name":"sub_pos","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Basic.lean","def_pos":[603,29],"def_end_pos":[603,36]},{"full_name":"zero_lt_one","def_path":"Mathlib/Algebra/Order/ZeroLEOne.lean","def_pos":[34,14],"def_end_pos":[34,25]}]},{"state_before":"case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","state_after":"case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","tactic":"have h0n : 2 ≤ n := Nat.succ_le_of_lt <| Nat.one_lt_cast.mp <| hp.trans_lt h2p","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Nat.one_lt_cast","def_path":"Mathlib/Data/Nat/Cast/Order/Basic.lean","def_pos":[86,8],"def_end_pos":[86,19]},{"full_name":"Nat.succ_le_of_lt","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[348,8],"def_end_pos":[348,21]}]},{"state_before":"case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","state_after":"case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","tactic":"have hn : NNReal.IsConjExponent n n' := .conjExponent (by norm_cast)","premises":[{"full_name":"NNReal.IsConjExponent","def_path":"Mathlib/Data/Real/ConjExponents.lean","def_pos":[130,10],"def_end_pos":[130,24]},{"full_name":"NNReal.IsConjExponent.conjExponent","def_path":"Mathlib/Data/Real/ConjExponents.lean","def_pos":[201,16],"def_end_pos":[201,43]}]},{"state_before":"case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","state_after":"case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","tactic":"have h1n : 1 ≤ (n : ℝ≥0) := hn.one_le","premises":[{"full_name":"NNReal","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[60,4],"def_end_pos":[60,10]},{"full_name":"NNReal.IsConjExponent.one_le","def_path":"Mathlib/Data/Real/ConjExponents.lean","def_pos":[150,6],"def_end_pos":[150,12]}]},{"state_before":"case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","state_after":"case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","tactic":"have h2n : (0 : ℝ) < n - 1 := by simp_rw [sub_pos]; exact hn.coe.one_lt","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Real.IsConjExponent.one_lt","def_path":"Mathlib/Data/Real/ConjExponents.lean","def_pos":[39,2],"def_end_pos":[39,8]},{"full_name":"sub_pos","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Basic.lean","def_pos":[603,29],"def_end_pos":[603,36]}]},{"state_before":"case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","state_after":"case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","tactic":"have hnp : (0 : ℝ) < n - p := by simp_rw [sub_pos]; exact h2p","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"sub_pos","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Basic.lean","def_pos":[603,29],"def_end_pos":[603,36]}]},{"state_before":"case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","state_after":"case neg.inl\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np' : ℝ≥0\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nn' : ℝ≥0 := (↑n).conjExponent\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhp : 1 ≤ 1\nhp' : (↑p')⁻¹ = ↑1⁻¹ - (↑n)⁻¹\nh2p : ↑1 < ↑n\nhnp : 0 < ↑n - ↑1\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑1) * eLpNorm (fderiv ℝ u) (↑1) μ\n\ncase neg.inr\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","tactic":"rcases hp.eq_or_lt with rfl|hp","premises":[]},{"state_before":"case neg.inr\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","state_after":"case neg.inr\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","tactic":"let q := Real.conjExponent p","premises":[{"full_name":"Real.conjExponent","def_path":"Mathlib/Data/Real/ConjExponents.lean","def_pos":[43,4],"def_end_pos":[43,16]}]},{"state_before":"case neg.inr\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (���n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","state_after":"case neg.inr\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","tactic":"have hq : Real.IsConjExponent p q := .conjExponent hp","premises":[{"full_name":"Real.IsConjExponent","def_path":"Mathlib/Data/Real/ConjExponents.lean","def_pos":[38,10],"def_end_pos":[38,24]},{"full_name":"Real.IsConjExponent.conjExponent","def_path":"Mathlib/Data/Real/ConjExponents.lean","def_pos":[116,6],"def_end_pos":[116,33]}]},{"state_before":"case neg.inr\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","state_after":"case neg.inr\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","tactic":"have h0p : p ≠ 0 := zero_lt_one.trans hp |>.ne'","premises":[{"full_name":"LT.lt.ne'","def_path":"Mathlib/Order/Basic.lean","def_pos":[267,8],"def_end_pos":[267,11]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"zero_lt_one","def_path":"Mathlib/Algebra/Order/ZeroLEOne.lean","def_pos":[34,14],"def_end_pos":[34,25]}]},{"state_before":"case neg.inr\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","state_after":"case neg.inr\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","tactic":"have h1p : (p : ℝ) ≠ 1 := hq.one_lt.ne'","premises":[{"full_name":"LT.lt.ne'","def_path":"Mathlib/Order/Basic.lean","def_pos":[267,8],"def_end_pos":[267,11]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Real.IsConjExponent.one_lt","def_path":"Mathlib/Data/Real/ConjExponents.lean","def_pos":[39,2],"def_end_pos":[39,8]}]},{"state_before":"case neg.inr\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","state_after":"case neg.inr\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","tactic":"have h3p : (p : ��) - 1 ≠ 0 := sub_ne_zero_of_ne h1p","premises":[{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"sub_ne_zero_of_ne","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[387,2],"def_end_pos":[387,13]}]},{"state_before":"case neg.inr\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","state_after":"case neg.inr\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","tactic":"have h0p' : p' ≠ 0 := by\n    suffices 0 < (p' : ℝ) from (show 0 < p' from this) |>.ne'\n    rw [← inv_pos, hp', sub_pos]\n    exact inv_lt_inv_of_lt hq.pos h2p","premises":[{"full_name":"LT.lt.ne'","def_path":"Mathlib/Order/Basic.lean","def_pos":[267,8],"def_end_pos":[267,11]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Real.IsConjExponent.pos","def_path":"Mathlib/Data/Real/ConjExponents.lean","def_pos":[53,8],"def_end_pos":[53,11]},{"full_name":"inv_lt_inv_of_lt","def_path":"Mathlib/Algebra/Order/Field/Basic.lean","def_pos":[171,8],"def_end_pos":[171,24]},{"full_name":"inv_pos","def_path":"Mathlib/Algebra/Order/Field/Unbundled/Basic.lean","def_pos":[23,14],"def_end_pos":[23,21]},{"full_name":"sub_pos","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Basic.lean","def_pos":[603,29],"def_end_pos":[603,36]}]},{"state_before":"case neg.inr\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","state_after":"case neg.inr\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","tactic":"have h2q : 1 / n' - 1 / q = 1 / p' := by\n    simp_rw (config := {zeta := false}) [one_div, hp']\n    rw [← hq.one_sub_inv, ← hn.coe.one_sub_inv, sub_sub_sub_cancel_left]\n    simp only [NNReal.coe_natCast, NNReal.coe_inv]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Lean.Meta.Simp.Config.arith","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[182,2],"def_end_pos":[182,7]},{"full_name":"Lean.Meta.Simp.Config.autoUnfold","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[187,2],"def_end_pos":[187,12]},{"full_name":"Lean.Meta.Simp.Config.beta","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[157,2],"def_end_pos":[157,6]},{"full_name":"Lean.Meta.Simp.Config.contextual","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[136,2],"def_end_pos":[136,12]},{"full_name":"Lean.Meta.Simp.Config.decide","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[180,2],"def_end_pos":[180,8]},{"full_name":"Lean.Meta.Simp.Config.dsimp","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[193,2],"def_end_pos":[193,7]},{"full_name":"Lean.Meta.Simp.Config.eta","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[162,2],"def_end_pos":[162,5]},{"full_name":"Lean.Meta.Simp.Config.etaStruct","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[167,2],"def_end_pos":[167,11]},{"full_name":"Lean.Meta.Simp.Config.ground","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[205,2],"def_end_pos":[205,8]},{"full_name":"Lean.Meta.Simp.Config.implicitDefEqProofs","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[228,2],"def_end_pos":[228,21]},{"full_name":"Lean.Meta.Simp.Config.index","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[220,2],"def_end_pos":[220,7]},{"full_name":"Lean.Meta.Simp.Config.iota","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[171,2],"def_end_pos":[171,6]},{"full_name":"Lean.Meta.Simp.Config.maxDischargeDepth","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[131,2],"def_end_pos":[131,19]},{"full_name":"Lean.Meta.Simp.Config.maxSteps","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[126,2],"def_end_pos":[126,10]},{"full_name":"Lean.Meta.Simp.Config.memoize","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[140,2],"def_end_pos":[140,9]},{"full_name":"Lean.Meta.Simp.Config.proj","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[175,2],"def_end_pos":[175,6]},{"full_name":"Lean.Meta.Simp.Config.singlePass","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[146,2],"def_end_pos":[146,12]},{"full_name":"Lean.Meta.Simp.Config.unfoldPartialApp","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[210,2],"def_end_pos":[210,18]},{"full_name":"Lean.Meta.Simp.Config.zeta","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[152,2],"def_end_pos":[152,6]},{"full_name":"Lean.Meta.Simp.Config.zetaDelta","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[215,2],"def_end_pos":[215,11]},{"full_name":"NNReal.coe_inv","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[168,18],"def_end_pos":[168,25]},{"full_name":"NNReal.coe_natCast","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[303,18],"def_end_pos":[303,29]},{"full_name":"Real.IsConjExponent.one_sub_inv","def_path":"Mathlib/Data/Real/ConjExponents.lean","def_pos":[80,6],"def_end_pos":[80,17]},{"full_name":"one_div","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[338,8],"def_end_pos":[338,15]},{"full_name":"sub_sub_sub_cancel_left","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[956,2],"def_end_pos":[956,13]}]},{"state_before":"case neg.inr\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","state_after":"case neg.inr\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\nγ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","tactic":"let γ : ℝ≥0 := ⟨p * (n - 1) / (n - p), by positivity⟩","premises":[{"full_name":"NNReal","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[60,4],"def_end_pos":[60,10]}]},{"state_before":"case neg.inr\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\nγ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","state_after":"case neg.inr\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\nγ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩\nh0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","tactic":"have h0γ : (γ : ℝ) = p * (n - 1) / (n - p) := rfl","premises":[{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"case neg.inr\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\nγ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩\nh0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","state_after":"case neg.inr\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\nγ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩\nh0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)\nh1γ : 1 < ↑γ\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","tactic":"have h1γ : 1 < (γ : ℝ) := by\n    rwa [h0γ, one_lt_div hnp, mul_sub, mul_one, sub_lt_sub_iff_right, lt_mul_iff_one_lt_left]\n    exact hn.coe.pos","premises":[{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Real.IsConjExponent.pos","def_path":"Mathlib/Data/Real/ConjExponents.lean","def_pos":[53,8],"def_end_pos":[53,11]},{"full_name":"lt_mul_iff_one_lt_left","def_path":"Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean","def_pos":[648,8],"def_end_pos":[648,30]},{"full_name":"mul_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[413,8],"def_end_pos":[413,15]},{"full_name":"one_lt_div","def_path":"Mathlib/Algebra/Order/Field/Basic.lean","def_pos":[301,8],"def_end_pos":[301,18]},{"full_name":"sub_lt_sub_iff_right","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Basic.lean","def_pos":[595,2],"def_end_pos":[595,13]}]},{"state_before":"case neg.inr\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\nγ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩\nh0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)\nh1γ : 1 < ↑γ\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","state_after":"case neg.inr\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\nγ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩\nh0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)\nh1γ : 1 < ↑γ\nh2γ : γ * n' = p'\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","tactic":"have h2γ : γ * n' = p' := by\n    rw [← NNReal.coe_inj, ← inv_inj, hp', NNReal.coe_mul, h0γ, hn.coe.conj_eq]\n    field_simp; ring","premises":[{"full_name":"NNReal.coe_inj","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[150,25],"def_end_pos":[150,32]},{"full_name":"NNReal.coe_mul","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[164,18],"def_end_pos":[164,25]},{"full_name":"Real.IsConjExponent.conj_eq","def_path":"Mathlib/Data/Real/ConjExponents.lean","def_pos":[73,8],"def_end_pos":[73,15]},{"full_name":"inv_inj","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[295,8],"def_end_pos":[295,15]}]},{"state_before":"case neg.inr\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\nγ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩\nh0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)\nh1γ : 1 < ↑γ\nh2γ : γ * n' = p'\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","state_after":"case neg.inr\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\nγ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩\nh0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)\nh1γ : 1 < ↑γ\nh2γ : γ * n' = p'\nh3γ : (↑γ - 1) * q = ↑p'\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","tactic":"have h3γ : (γ - 1) * q = p' := by\n    rw [← inv_inj, hp', h0γ, hq.conj_eq]\n    have : (p : ℝ) * (n - 1) - (n - p) = n * (p - 1) := by ring\n    field_simp [this]; ring","premises":[{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Real.IsConjExponent.conj_eq","def_path":"Mathlib/Data/Real/ConjExponents.lean","def_pos":[73,8],"def_end_pos":[73,15]},{"full_name":"inv_inj","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[295,8],"def_end_pos":[295,15]}]},{"state_before":"case neg.inr\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\nγ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩\nh0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)\nh1γ : 1 < ↑γ\nh2γ : γ * n' = p'\nh3γ : (↑γ - 1) * q = ↑p'\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","state_after":"case neg.inr\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\nγ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩\nh0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)\nh1γ : 1 < ↑γ\nh2γ : γ * n' = p'\nh3γ : (↑γ - 1) * q = ↑p'\nh4γ : ↑γ ≠ 0\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","tactic":"have h4γ : (γ : ℝ) ≠ 0 := (zero_lt_one.trans h1γ).ne'","premises":[{"full_name":"LT.lt.ne'","def_path":"Mathlib/Order/Basic.lean","def_pos":[267,8],"def_end_pos":[267,11]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"zero_lt_one","def_path":"Mathlib/Algebra/Order/ZeroLEOne.lean","def_pos":[34,14],"def_end_pos":[34,25]}]},{"state_before":"case neg.inr\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\nγ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩\nh0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)\nh1γ : 1 < ↑γ\nh2γ : γ * n' = p'\nh3γ : (↑γ - 1) * q = ↑p'\nh4γ : ↑γ ≠ 0\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","state_after":"case pos\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\nγ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩\nh0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)\nh1γ : 1 < ↑γ\nh2γ : γ * n' = p'\nh3γ : (↑γ - 1) * q = ↑p'\nh4γ : ↑γ ≠ 0\nh3u : ∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ = 0\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ\n\ncase neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\nγ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩\nh0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)\nh1γ : 1 < ↑γ\nh2γ : γ * n' = p'\nh3γ : (↑γ - 1) * q = ↑p'\nh4γ : ↑γ ≠ 0\nh3u : ¬∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ = 0\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","tactic":"by_cases h3u : ∫⁻ x, ‖u x‖₊ ^ (p' : ℝ) ∂μ = 0","premises":[{"full_name":"MeasureTheory.lintegral","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[59,16],"def_end_pos":[59,25]},{"full_name":"NNNorm.nnnorm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[65,2],"def_end_pos":[65,8]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\nγ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩\nh0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)\nh1γ : 1 < ↑γ\nh2γ : γ * n' = p'\nh3γ : (↑γ - 1) * q = ↑p'\nh4γ : ↑γ ≠ 0\nh3u : ¬∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ = 0\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","state_after":"case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\nγ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩\nh0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)\nh1γ : 1 < ↑γ\nh2γ : γ * n' = p'\nh3γ : (↑γ - 1) * q = ↑p'\nh4γ : ↑γ ≠ 0\nh3u : ¬∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ = 0\nh4u : ∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ ≠ ⊤\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","tactic":"have h4u : ∫⁻ x, ‖u x‖₊ ^ (p' : ℝ) ∂μ ≠ ∞ := by\n    refine lintegral_rpow_nnnorm_lt_top_of_eLpNorm'_lt_top (pos_iff_ne_zero.mpr h0p') ?_ |>.ne\n    dsimp only\n    rw [NNReal.val_eq_coe, ← eLpNorm_nnreal_eq_eLpNorm' h0p']\n    exact hu.continuous.memℒp_of_hasCompactSupport (μ := μ) h2u |>.eLpNorm_lt_top","premises":[{"full_name":"ContDiff.continuous","def_path":"Mathlib/Analysis/Calculus/ContDiff/Defs.lean","def_pos":[1374,8],"def_end_pos":[1374,27]},{"full_name":"Continuous.memℒp_of_hasCompactSupport","def_path":"Mathlib/MeasureTheory/Function/LpSpace.lean","def_pos":[732,8],"def_end_pos":[732,52]},{"full_name":"ENNReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[96,4],"def_end_pos":[96,11]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"MeasureTheory.Memℒp.eLpNorm_lt_top","def_path":"Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean","def_pos":[150,8],"def_end_pos":[150,28]},{"full_name":"MeasureTheory.eLpNorm_nnreal_eq_eLpNorm'","def_path":"Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean","def_pos":[88,6],"def_end_pos":[88,32]},{"full_name":"MeasureTheory.lintegral","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[59,16],"def_end_pos":[59,25]},{"full_name":"MeasureTheory.lintegral_rpow_nnnorm_lt_top_of_eLpNorm'_lt_top","def_path":"Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean","def_pos":[162,8],"def_end_pos":[162,55]},{"full_name":"NNNorm.nnnorm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[65,2],"def_end_pos":[65,8]},{"full_name":"NNReal.val_eq_coe","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[86,8],"def_end_pos":[86,18]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Top.top","def_path":"Mathlib/Order/Notation.lean","def_pos":[94,2],"def_end_pos":[94,5]},{"full_name":"pos_iff_ne_zero","def_path":"Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean","def_pos":[123,2],"def_end_pos":[123,13]}]},{"state_before":"case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\nγ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩\nh0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)\nh1γ : 1 < ↑γ\nh2γ : γ * n' = p'\nh3γ : (↑γ - 1) * q = ↑p'\nh4γ : ↑γ ≠ 0\nh3u : ¬∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ = 0\nh4u : ∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ ≠ ⊤\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","state_after":"case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\nγ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩\nh0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)\nh1γ : 1 < ↑γ\nh2γ : γ * n' = p'\nh3γ : (↑γ - 1) * q = ↑p'\nh4γ : ↑γ ≠ 0\nh3u : ¬∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ = 0\nh4u : ∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ ≠ ⊤\nh5u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ 0\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","tactic":"have h5u : (∫⁻ x, ‖u x‖₊ ^ (p' : ℝ) ∂μ) ^ (1 / q) ≠ 0 :=\n    ENNReal.rpow_pos (pos_iff_ne_zero.mpr h3u) h4u |>.ne'","premises":[{"full_name":"ENNReal.rpow_pos","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean","def_pos":[744,8],"def_end_pos":[744,16]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"LT.lt.ne'","def_path":"Mathlib/Order/Basic.lean","def_pos":[267,8],"def_end_pos":[267,11]},{"full_name":"MeasureTheory.lintegral","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[59,16],"def_end_pos":[59,25]},{"full_name":"NNNorm.nnnorm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[65,2],"def_end_pos":[65,8]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"pos_iff_ne_zero","def_path":"Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean","def_pos":[123,2],"def_end_pos":[123,13]}]},{"state_before":"case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\nγ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩\nh0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)\nh1γ : 1 < ↑γ\nh2γ : γ * n' = p'\nh3γ : (↑γ - 1) * q = ↑p'\nh4γ : ↑γ ≠ 0\nh3u : ¬∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ = 0\nh4u : ∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ ≠ ⊤\nh5u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ 0\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","state_after":"case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\nγ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩\nh0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)\nh1γ : 1 < ↑γ\nh2γ : γ * n' = p'\nh3γ : (↑γ - 1) * q = ↑p'\nh4γ : ↑γ ≠ 0\nh3u : ¬∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ = 0\nh4u : ∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ ≠ ⊤\nh5u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ 0\nh6u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ ⊤\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","tactic":"have h6u : (∫⁻ x, ‖u x‖₊ ^ (p' : ℝ) ∂μ) ^ (1 / q) ≠ ∞ :=\n    ENNReal.rpow_ne_top_of_nonneg (div_nonneg zero_le_one hq.symm.nonneg) h4u","premises":[{"full_name":"ENNReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[96,4],"def_end_pos":[96,11]},{"full_name":"ENNReal.rpow_ne_top_of_nonneg","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean","def_pos":[477,8],"def_end_pos":[477,29]},{"full_name":"MeasureTheory.lintegral","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[59,16],"def_end_pos":[59,25]},{"full_name":"NNNorm.nnnorm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[65,2],"def_end_pos":[65,8]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Real.IsConjExponent.nonneg","def_path":"Mathlib/Data/Real/ConjExponents.lean","def_pos":[55,8],"def_end_pos":[55,14]},{"full_name":"Real.IsConjExponent.symm","def_path":"Mathlib/Data/Real/ConjExponents.lean","def_pos":[89,24],"def_end_pos":[89,28]},{"full_name":"Top.top","def_path":"Mathlib/Order/Notation.lean","def_pos":[94,2],"def_end_pos":[94,5]},{"full_name":"div_nonneg","def_path":"Mathlib/Algebra/Order/Field/Unbundled/Basic.lean","def_pos":[48,6],"def_end_pos":[48,16]},{"full_name":"zero_le_one","def_path":"Mathlib/Algebra/Order/ZeroLEOne.lean","def_pos":[23,14],"def_end_pos":[23,25]}]},{"state_before":"case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\nγ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩\nh0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)\nh1γ : 1 < ↑γ\nh2γ : γ * n' = p'\nh3γ : (↑γ - 1) * q = ↑p'\nh4γ : ↑γ ≠ 0\nh3u : ¬∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ = 0\nh4u : ∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ ≠ ⊤\nh5u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ 0\nh6u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ ⊤\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","state_after":"case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\nγ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩\nh0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)\nh1γ : 1 < ↑γ\nh2γ : γ * n' = p'\nh3γ : (↑γ - 1) * q = ↑p'\nh4γ : ↑γ ≠ 0\nh3u : ¬∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ = 0\nh4u : ∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ ≠ ⊤\nh5u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ 0\nh6u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ ⊤\nh7u : Continuous u\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","tactic":"have h7u := hu.continuous","premises":[{"full_name":"ContDiff.continuous","def_path":"Mathlib/Analysis/Calculus/ContDiff/Defs.lean","def_pos":[1374,8],"def_end_pos":[1374,27]}]},{"state_before":"case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\nγ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩\nh0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)\nh1γ : 1 < ↑γ\nh2γ : γ * n' = p'\nh3γ : (↑γ - 1) * q = ↑p'\nh4γ : ↑γ ≠ 0\nh3u : ¬∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ = 0\nh4u : ∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ ≠ ⊤\nh5u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ 0\nh6u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ ⊤\nh7u : Continuous u\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","state_after":"case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\nγ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩\nh0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)\nh1γ : 1 < ↑γ\nh2γ : γ * n' = p'\nh3γ : (↑γ - 1) * q = ↑p'\nh4γ : ↑γ ≠ 0\nh3u : ¬∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ = 0\nh4u : ∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ ≠ ⊤\nh5u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ 0\nh6u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ ⊤\nh7u : Continuous u\nh8u : Continuous (fderiv ℝ u)\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","tactic":"have h8u := (hu.fderiv_right (m := 0) le_rfl).continuous","premises":[{"full_name":"ContDiff.continuous","def_path":"Mathlib/Analysis/Calculus/ContDiff/Defs.lean","def_pos":[1374,8],"def_end_pos":[1374,27]},{"full_name":"ContDiff.fderiv_right","def_path":"Mathlib/Analysis/Calculus/ContDiff/Basic.lean","def_pos":[1014,8],"def_end_pos":[1014,29]},{"full_name":"le_rfl","def_path":"Mathlib/Order/Defs.lean","def_pos":[43,8],"def_end_pos":[43,14]}]},{"state_before":"case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\nγ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩\nh0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)\nh1γ : 1 < ↑γ\nh2γ : γ * n' = p'\nh3γ : (↑γ - 1) * q = ↑p'\nh4γ : ↑γ ≠ 0\nh3u : ¬∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ = 0\nh4u : ∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ ≠ ⊤\nh5u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ 0\nh6u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ ⊤\nh7u : Continuous u\nh8u : Continuous (fderiv ℝ u)\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","state_after":"case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\nγ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩\nh0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)\nh1γ : 1 < ↑γ\nh2γ : γ * n' = p'\nh3γ : (↑γ - 1) * q = ↑p'\nh4γ : ↑γ ≠ 0\nh3u : ¬∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ = 0\nh4u : ∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ ≠ ⊤\nh5u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ 0\nh6u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ ⊤\nh7u : Continuous u\nh8u : Continuous (fderiv ℝ u)\nv : E → ℝ := fun x => ‖u x‖ ^ ↑γ\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","tactic":"let v : E → ℝ := fun x ↦ ‖u x‖ ^ (γ : ℝ)","premises":[{"full_name":"Norm.norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[59,2],"def_end_pos":[59,6]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]}]},{"state_before":"case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\nγ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩\nh0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)\nh1γ : 1 < ↑γ\nh2γ : γ * n' = p'\nh3γ : (↑γ - 1) * q = ↑p'\nh4γ : ↑γ ≠ 0\nh3u : ¬∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ = 0\nh4u : ∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ ≠ ⊤\nh5u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ 0\nh6u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ ⊤\nh7u : Continuous u\nh8u : Continuous (fderiv ℝ u)\nv : E → ℝ := fun x => ‖u x‖ ^ ↑γ\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","state_after":"case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\nγ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩\nh0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)\nh1γ : 1 < ↑γ\nh2γ : γ * n' = p'\nh3γ : (↑γ - 1) * q = ↑p'\nh4γ : ↑γ ≠ 0\nh3u : ¬∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ = 0\nh4u : ∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ ≠ ⊤\nh5u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ 0\nh6u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ ⊤\nh7u : Continuous u\nh8u : Continuous (fderiv ℝ u)\nv : E → ℝ := fun x => ‖u x‖ ^ ↑γ\nhv : ContDiff ℝ 1 v\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","tactic":"have hv : ContDiff ℝ 1 v := hu.norm_rpow h1γ","premises":[{"full_name":"ContDiff","def_path":"Mathlib/Analysis/Calculus/ContDiff/Defs.lean","def_pos":[1313,4],"def_end_pos":[1313,12]},{"full_name":"ContDiff.norm_rpow","def_path":"Mathlib/Analysis/InnerProductSpace/NormPow.lean","def_pos":[117,8],"def_end_pos":[117,26]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]}]},{"state_before":"case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\nγ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩\nh0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)\nh1γ : 1 < ↑γ\nh2γ : γ * n' = p'\nh3γ : (↑γ - 1) * q = ↑p'\nh4γ : ↑γ ≠ 0\nh3u : ¬∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ = 0\nh4u : ∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ ≠ ⊤\nh5u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ 0\nh6u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ ⊤\nh7u : Continuous u\nh8u : Continuous (fderiv ℝ u)\nv : E → ℝ := fun x => ‖u x‖ ^ ↑γ\nhv : ContDiff ℝ 1 v\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","state_after":"case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\nγ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩\nh0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)\nh1γ : 1 < ↑γ\nh2γ : γ * n' = p'\nh3γ : (↑γ - 1) * q = ↑p'\nh4γ : ↑γ ≠ 0\nh3u : ¬∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ = 0\nh4u : ∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ ≠ ⊤\nh5u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ 0\nh6u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ ⊤\nh7u : Continuous u\nh8u : Continuous (fderiv ℝ u)\nv : E → ℝ := fun x => ‖u x‖ ^ ↑γ\nhv : ContDiff ℝ 1 v\nh2v : HasCompactSupport v\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","tactic":"have h2v : HasCompactSupport v := h2u.norm.rpow_const h4γ","premises":[{"full_name":"HasCompactSupport","def_path":"Mathlib/Topology/Support.lean","def_pos":[126,2],"def_end_pos":[126,13]},{"full_name":"HasCompactSupport.rpow_const","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","def_pos":[228,18],"def_end_pos":[228,53]}]},{"state_before":"case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\nγ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩\nh0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)\nh1γ : 1 < ↑γ\nh2γ : γ * n' = p'\nh3γ : (↑γ - 1) * q = ↑p'\nh4γ : ↑γ ≠ 0\nh3u : ¬∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ = 0\nh4u : ∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ ≠ ⊤\nh5u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ 0\nh6u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ ⊤\nh7u : Continuous u\nh8u : Continuous (fderiv ℝ u)\nv : E → ℝ := fun x => ‖u x‖ ^ ↑γ\nhv : ContDiff ℝ 1 v\nh2v : HasCompactSupport v\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","state_after":"case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\nγ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩\nh0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)\nh1γ : 1 < ↑γ\nh2γ : γ * n' = p'\nh3γ : (↑γ - 1) * q = ↑p'\nh4γ : ↑γ ≠ 0\nh3u : ¬∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ = 0\nh4u : ∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ ≠ ⊤\nh5u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ 0\nh6u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ ⊤\nh7u : Continuous u\nh8u : Continuous (fderiv ℝ u)\nv : E → ℝ := fun x => ‖u x‖ ^ ↑γ\nhv : ContDiff ℝ 1 v\nh2v : HasCompactSupport v\nC : ℝ≥0 := eLpNormLESNormFDerivOneConst μ ↑n'\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","tactic":"set C := eLpNormLESNormFDerivOneConst μ n'","premises":[{"full_name":"MeasureTheory.eLpNormLESNormFDerivOneConst","def_path":"Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean","def_pos":[437,16],"def_end_pos":[437,44]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\nγ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩\nh0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)\nh1γ : 1 < ↑γ\nh2γ : γ * n' = p'\nh3γ : (↑γ - 1) * q = ↑p'\nh4γ : ↑γ ≠ 0\nh3u : ¬∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ = 0\nh4u : ∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ ≠ ⊤\nh5u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ 0\nh6u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ ⊤\nh7u : Continuous u\nh8u : Continuous (fderiv ℝ u)\nv : E → ℝ := fun x => ‖u x‖ ^ ↑γ\nhv : ContDiff ℝ 1 v\nh2v : HasCompactSupport v\nC : ℝ≥0 := eLpNormLESNormFDerivOneConst μ ↑n'\nthis :\n  (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / ↑n') ≤\n    ↑C * ↑γ * (∫⁻ (x : E), ↑‖fderiv ℝ u x‖₊ ^ ↑p ∂μ) ^ (1 / ↑p) * (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q)\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ","state_after":"no goals","tactic":"calc\n    eLpNorm u p' μ\n      = (∫⁻ x, ‖u x‖₊ ^ (p' : ℝ) ∂μ) ^ (1 / (p' : ℝ)) := eLpNorm_nnreal_eq_lintegral h0p'\n    _ ≤ C * γ * (∫⁻ x, ‖fderiv ℝ u x‖₊ ^ (p : ℝ) ∂μ) ^ (1 / (p : ℝ)) := by\n      rwa [← h2q, ENNReal.rpow_sub _ _ h3u h4u, ENNReal.div_le_iff h5u h6u]\n    _ = eLpNormLESNormFDerivOfEqInnerConst μ p *  eLpNorm (fderiv ℝ u) (↑p) μ := by\n      suffices (C : ℝ) * γ = eLpNormLESNormFDerivOfEqInnerConst μ p by\n        rw [eLpNorm_nnreal_eq_lintegral h0p]\n        congr\n        norm_cast at this ⊢\n      simp_rw [eLpNormLESNormFDerivOfEqInnerConst, γ]\n      refold_let n n' C\n      rw [NNReal.coe_mul, NNReal.coe_mk, Real.coe_toNNReal', mul_eq_mul_left_iff, eq_comm,\n        max_eq_left_iff]\n      left\n      positivity","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"ENNReal.div_le_iff","def_path":"Mathlib/Data/ENNReal/Inv.lean","def_pos":[141,18],"def_end_pos":[141,28]},{"full_name":"ENNReal.rpow_sub","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean","def_pos":[509,8],"def_end_pos":[509,16]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"MeasureTheory.eLpNorm","def_path":"Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean","def_pos":[78,4],"def_end_pos":[78,11]},{"full_name":"MeasureTheory.eLpNormLESNormFDerivOfEqInnerConst","def_path":"Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean","def_pos":[460,4],"def_end_pos":[460,38]},{"full_name":"MeasureTheory.eLpNorm_nnreal_eq_lintegral","def_path":"Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean","def_pos":[101,6],"def_end_pos":[101,33]},{"full_name":"MeasureTheory.lintegral","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[59,16],"def_end_pos":[59,25]},{"full_name":"NNNorm.nnnorm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[65,2],"def_end_pos":[65,8]},{"full_name":"NNReal.coe_mk","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[124,27],"def_end_pos":[124,33]},{"full_name":"NNReal.coe_mul","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[164,18],"def_end_pos":[164,25]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Real.coe_toNNReal'","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[557,8],"def_end_pos":[557,21]},{"full_name":"eq_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[811,8],"def_end_pos":[811,15]},{"full_name":"fderiv","def_path":"Mathlib/Analysis/Calculus/FDeriv/Basic.lean","def_pos":[183,16],"def_end_pos":[183,22]},{"full_name":"max_eq_left_iff","def_path":"Mathlib/Order/MinMax.lean","def_pos":[126,8],"def_end_pos":[126,23]},{"full_name":"mul_eq_mul_left_iff","def_path":"Mathlib/Algebra/GroupWithZero/Basic.lean","def_pos":[197,8],"def_end_pos":[197,27]}]}]}
+{"url":"Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean","commit":"","full_name":"Orientation.neg_rotation","start":[173,0],"end":[175,47],"file_path":"Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean","tactics":[{"state_before":"V : Type u_1\nV' : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : NormedAddCommGroup V'\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : InnerProductSpace ℝ V'\ninst✝¹ : Fact (finrank ℝ V = 2)\ninst✝ : Fact (finrank ℝ V' = 2)\no : Orientation ℝ V (Fin 2)\nθ : Real.Angle\nx : V\n⊢ -(o.rotation θ) x = (o.rotation (↑π + θ)) x","state_after":"no goals","tactic":"rw [← o.rotation_pi_apply, rotation_rotation]","premises":[{"full_name":"Orientation.rotation_pi_apply","def_path":"Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean","def_pos":[138,8],"def_end_pos":[138,25]},{"full_name":"Orientation.rotation_rotation","def_path":"Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean","def_pos":[147,8],"def_end_pos":[147,25]}]}]}
+{"url":"Mathlib/Data/Set/Pointwise/SMul.lean","commit":"","full_name":"Set.op_vadd_set_vadd_eq_vadd_vadd_set","start":[607,0],"end":[611,34],"file_path":"Mathlib/Data/Set/Pointwise/SMul.lean","tactics":[{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : SMul αᵐᵒᵖ β\ninst✝¹ : SMul β γ\ninst✝ : SMul α γ\na : α\ns : Set β\nt : Set γ\nh : ∀ (a : α) (b : β) (c : γ), (op a • b) • c = b • a • c\n⊢ (op a • s) • t = s • a • t","state_after":"case h\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : SMul αᵐᵒᵖ β\ninst✝¹ : SMul β γ\ninst✝ : SMul α γ\na : α\ns : Set β\nt : Set γ\nh : ∀ (a : α) (b : β) (c : γ), (op a • b) • c = b • a • c\nx✝ : γ\n⊢ x✝ ∈ (op a • s) • t ↔ x✝ ∈ s • a • t","tactic":"ext","premises":[]},{"state_before":"case h\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : SMul αᵐᵒᵖ β\ninst✝¹ : SMul β γ\ninst✝ : SMul α γ\na : α\ns : Set β\nt : Set γ\nh : ∀ (a : α) (b : β) (c : γ), (op a • b) • c = b • a • c\nx✝ : γ\n⊢ x✝ ∈ (op a • s) • t ↔ x✝ ∈ s • a • t","state_after":"no goals","tactic":"simp [mem_smul, mem_smul_set, h]","premises":[{"full_name":"Set.mem_smul","def_path":"Mathlib/Data/Set/Pointwise/SMul.lean","def_pos":[85,8],"def_end_pos":[85,16]},{"full_name":"Set.mem_smul_set","def_path":"Mathlib/Data/Set/Pointwise/SMul.lean","def_pos":[238,8],"def_end_pos":[238,20]}]}]}
+{"url":"Mathlib/Data/Option/Basic.lean","commit":"","full_name":"Option.exists_mem_map","start":[56,0],"end":[57,54],"file_path":"Mathlib/Data/Option/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nf : α → β\no : Option α\np : β → Prop\n⊢ (∃ y, y ∈ Option.map f o ∧ p y) ↔ ∃ x, x ∈ o ∧ p (f x)","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/LinearAlgebra/Matrix/LDL.lean","commit":"","full_name":"LDL.lowerInv_eq_gramSchmidtBasis","start":[53,0],"end":[62,5],"file_path":"Mathlib/LinearAlgebra/Matrix/LDL.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝⁴ : RCLike 𝕜\nn : Type u_2\ninst✝³ : LinearOrder n\ninst✝² : IsWellOrder n fun x x_1 => x < x_1\ninst✝¹ : LocallyFiniteOrderBot n\nS : Matrix n n 𝕜\ninst✝ : Fintype n\nhS : S.PosDef\n⊢ lowerInv hS = ((Pi.basisFun 𝕜 n).toMatrix ⇑(gramSchmidtBasis (Pi.basisFun 𝕜 n)))ᵀ","state_after":"𝕜 : Type u_1\ninst✝⁴ : RCLike 𝕜\nn : Type u_2\ninst✝³ : LinearOrder n\ninst✝² : IsWellOrder n fun x x_1 => x < x_1\ninst✝¹ : LocallyFiniteOrderBot n\nS : Matrix n n 𝕜\ninst✝ : Fintype n\nhS : S.PosDef\nthis : NormedAddCommGroup (n → 𝕜) := NormedAddCommGroup.ofMatrix ⋯\n⊢ lowerInv hS = ((Pi.basisFun 𝕜 n).toMatrix ⇑(gramSchmidtBasis (Pi.basisFun 𝕜 n)))ᵀ","tactic":"letI := NormedAddCommGroup.ofMatrix hS.transpose","premises":[{"full_name":"Matrix.NormedAddCommGroup.ofMatrix","def_path":"Mathlib/LinearAlgebra/Matrix/PosDef.lean","def_pos":[386,21],"def_end_pos":[386,48]},{"full_name":"Matrix.PosDef.transpose","def_path":"Mathlib/LinearAlgebra/Matrix/PosDef.lean","def_pos":[310,8],"def_end_pos":[310,17]}]},{"state_before":"𝕜 : Type u_1\ninst✝⁴ : RCLike 𝕜\nn : Type u_2\ninst✝³ : LinearOrder n\ninst✝² : IsWellOrder n fun x x_1 => x < x_1\ninst✝¹ : LocallyFiniteOrderBot n\nS : Matrix n n 𝕜\ninst✝ : Fintype n\nhS : S.PosDef\nthis : NormedAddCommGroup (n → 𝕜) := NormedAddCommGroup.ofMatrix ⋯\n⊢ lowerInv hS = ((Pi.basisFun 𝕜 n).toMatrix ⇑(gramSchmidtBasis (Pi.basisFun 𝕜 n)))ᵀ","state_after":"𝕜 : Type u_1\ninst✝⁴ : RCLike 𝕜\nn : Type u_2\ninst✝³ : LinearOrder n\ninst✝² : IsWellOrder n fun x x_1 => x < x_1\ninst✝¹ : LocallyFiniteOrderBot n\nS : Matrix n n 𝕜\ninst✝ : Fintype n\nhS : S.PosDef\nthis✝ : NormedAddCommGroup (n → 𝕜) := NormedAddCommGroup.ofMatrix ⋯\nthis : InnerProductSpace 𝕜 (n → 𝕜) := InnerProductSpace.ofMatrix ⋯\n⊢ lowerInv hS = ((Pi.basisFun 𝕜 n).toMatrix ⇑(gramSchmidtBasis (Pi.basisFun 𝕜 n)))ᵀ","tactic":"letI := InnerProductSpace.ofMatrix hS.transpose","premises":[{"full_name":"Matrix.InnerProductSpace.ofMatrix","def_path":"Mathlib/LinearAlgebra/Matrix/PosDef.lean","def_pos":[407,4],"def_end_pos":[407,30]},{"full_name":"Matrix.PosDef.transpose","def_path":"Mathlib/LinearAlgebra/Matrix/PosDef.lean","def_pos":[310,8],"def_end_pos":[310,17]}]},{"state_before":"𝕜 : Type u_1\ninst✝⁴ : RCLike 𝕜\nn : Type u_2\ninst✝³ : LinearOrder n\ninst✝² : IsWellOrder n fun x x_1 => x < x_1\ninst✝¹ : LocallyFiniteOrderBot n\nS : Matrix n n 𝕜\ninst✝ : Fintype n\nhS : S.PosDef\nthis✝ : NormedAddCommGroup (n → 𝕜) := NormedAddCommGroup.ofMatrix ⋯\nthis : InnerProductSpace 𝕜 (n → 𝕜) := InnerProductSpace.ofMatrix ⋯\n⊢ lowerInv hS = ((Pi.basisFun 𝕜 n).toMatrix ⇑(gramSchmidtBasis (Pi.basisFun 𝕜 n)))ᵀ","state_after":"case a\n𝕜 : Type u_1\ninst✝⁴ : RCLike 𝕜\nn : Type u_2\ninst✝³ : LinearOrder n\ninst✝² : IsWellOrder n fun x x_1 => x < x_1\ninst✝¹ : LocallyFiniteOrderBot n\nS : Matrix n n 𝕜\ninst✝ : Fintype n\nhS : S.PosDef\nthis✝ : NormedAddCommGroup (n → 𝕜) := NormedAddCommGroup.ofMatrix ⋯\nthis : InnerProductSpace 𝕜 (n → 𝕜) := InnerProductSpace.ofMatrix ⋯\ni j : n\n⊢ lowerInv hS i j = ((Pi.basisFun 𝕜 n).toMatrix ⇑(gramSchmidtBasis (Pi.basisFun 𝕜 n)))ᵀ i j","tactic":"ext i j","premises":[]},{"state_before":"case a\n𝕜 : Type u_1\ninst✝⁴ : RCLike 𝕜\nn : Type u_2\ninst✝³ : LinearOrder n\ninst✝² : IsWellOrder n fun x x_1 => x < x_1\ninst✝¹ : LocallyFiniteOrderBot n\nS : Matrix n n 𝕜\ninst✝ : Fintype n\nhS : S.PosDef\nthis✝ : NormedAddCommGroup (n → 𝕜) := NormedAddCommGroup.ofMatrix ⋯\nthis : InnerProductSpace 𝕜 (n → 𝕜) := InnerProductSpace.ofMatrix ⋯\ni j : n\n⊢ lowerInv hS i j = ((Pi.basisFun 𝕜 n).toMatrix ⇑(gramSchmidtBasis (Pi.basisFun 𝕜 n)))ᵀ i j","state_after":"case a\n𝕜 : Type u_1\ninst✝⁴ : RCLike 𝕜\nn : Type u_2\ninst✝³ : LinearOrder n\ninst✝² : IsWellOrder n fun x x_1 => x < x_1\ninst✝¹ : LocallyFiniteOrderBot n\nS : Matrix n n 𝕜\ninst✝ : Fintype n\nhS : S.PosDef\nthis✝ : NormedAddCommGroup (n → 𝕜) := NormedAddCommGroup.ofMatrix ⋯\nthis : InnerProductSpace 𝕜 (n → 𝕜) := InnerProductSpace.ofMatrix ⋯\ni j : n\n⊢ gramSchmidt 𝕜 (⇑(Pi.basisFun 𝕜 n)) i j = (gramSchmidt 𝕜 ⇑(Pi.basisFun 𝕜 n))ᵀᵀ i j","tactic":"rw [LDL.lowerInv, Basis.coePiBasisFun.toMatrix_eq_transpose, coe_gramSchmidtBasis]","premises":[{"full_name":"Basis.coePiBasisFun.toMatrix_eq_transpose","def_path":"Mathlib/LinearAlgebra/Matrix/Basis.lean","def_pos":[67,8],"def_end_pos":[67,43]},{"full_name":"LDL.lowerInv","def_path":"Mathlib/LinearAlgebra/Matrix/LDL.lean","def_pos":[49,18],"def_end_pos":[49,30]},{"full_name":"coe_gramSchmidtBasis","def_path":"Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean","def_pos":[229,8],"def_end_pos":[229,28]}]},{"state_before":"case a\n𝕜 : Type u_1\ninst✝⁴ : RCLike 𝕜\nn : Type u_2\ninst✝³ : LinearOrder n\ninst✝² : IsWellOrder n fun x x_1 => x < x_1\ninst✝¹ : LocallyFiniteOrderBot n\nS : Matrix n n 𝕜\ninst✝ : Fintype n\nhS : S.PosDef\nthis✝ : NormedAddCommGroup (n → 𝕜) := NormedAddCommGroup.ofMatrix ⋯\nthis : InnerProductSpace 𝕜 (n → 𝕜) := InnerProductSpace.ofMatrix ⋯\ni j : n\n⊢ gramSchmidt 𝕜 (⇑(Pi.basisFun 𝕜 n)) i j = (gramSchmidt 𝕜 ⇑(Pi.basisFun 𝕜 n))ᵀᵀ i j","state_after":"no goals","tactic":"rfl","premises":[]}]}
+{"url":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","commit":"","full_name":"WeierstrassCurve.Jacobian.add_of_Z_eq_zero","start":[1199,0],"end":[1201,87],"file_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","tactics":[{"state_before":"R : Type u\ninst✝¹ : CommRing R\nW' : Jacobian R\nF : Type v\ninst✝ : Field F\nW : Jacobian F\nP Q : Fin 3 → F\nhP : W.Nonsingular P\nhQ : W.Nonsingular Q\nhPz : P z = 0\nhQz : Q z = 0\n⊢ W.add P Q = P x ^ 2 • ![1, 1, 0]","state_after":"no goals","tactic":"rw [add, if_pos <| equiv_of_Z_eq_zero hP hQ hPz hQz, dblXYZ_of_Z_eq_zero hP.left hPz]","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"WeierstrassCurve.Jacobian.add","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","def_pos":[1159,18],"def_end_pos":[1159,21]},{"full_name":"WeierstrassCurve.Jacobian.dblXYZ_of_Z_eq_zero","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","def_pos":[738,6],"def_end_pos":[738,25]},{"full_name":"WeierstrassCurve.Jacobian.equiv_of_Z_eq_zero","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","def_pos":[434,6],"def_end_pos":[434,24]},{"full_name":"if_pos","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[932,8],"def_end_pos":[932,14]}]}]}
+{"url":"Mathlib/Topology/Algebra/ConstMulAction.lean","commit":"","full_name":"set_smul_mem_nhds_smul_iff","start":[485,0],"end":[489,49],"file_path":"Mathlib/Topology/Algebra/ConstMulAction.lean","tactics":[{"state_before":"M : Type u_1\nα : Type u_2\nβ : Type u_3\nΓ : Type u_4\ninst✝⁶ : Group Γ\nT : Type u_5\ninst✝⁵ : TopologicalSpace T\ninst✝⁴ : MulAction Γ T\nG₀ : Type u_6\ninst✝³ : GroupWithZero G₀\ninst✝² : MulAction G₀ α\ninst✝¹ : TopologicalSpace α\ninst✝ : ContinuousConstSMul G₀ α\nc : G₀\ns : Set α\nx : α\nhc : c ≠ 0\n⊢ c • s ∈ 𝓝 (c • x) ↔ s ∈ 𝓝 x","state_after":"M : Type u_1\nα : Type u_2\nβ : Type u_3\nΓ : Type u_4\ninst✝⁶ : Group Γ\nT : Type u_5\ninst✝⁵ : TopologicalSpace T\ninst✝⁴ : MulAction Γ T\nG₀ : Type u_6\ninst✝³ : GroupWithZero G₀\ninst✝² : MulAction G₀ α\ninst✝¹ : TopologicalSpace α\ninst✝ : ContinuousConstSMul G₀ α\nc : G₀\ns : Set α\nx : α\nhc : c ≠ 0\nh : c • s ∈ 𝓝 (c • x)\n⊢ s ∈ 𝓝 x","tactic":"refine ⟨fun h => ?_, fun h => set_smul_mem_nhds_smul h hc⟩","premises":[{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"set_smul_mem_nhds_smul","def_path":"Mathlib/Topology/Algebra/ConstMulAction.lean","def_pos":[479,8],"def_end_pos":[479,30]}]},{"state_before":"M : Type u_1\nα : Type u_2\nβ : Type u_3\nΓ : Type u_4\ninst✝⁶ : Group Γ\nT : Type u_5\ninst✝⁵ : TopologicalSpace T\ninst✝⁴ : MulAction Γ T\nG₀ : Type u_6\ninst✝³ : GroupWithZero G₀\ninst✝² : MulAction G₀ α\ninst✝¹ : TopologicalSpace α\ninst✝ : ContinuousConstSMul G₀ α\nc : G₀\ns : Set α\nx : α\nhc : c ≠ 0\nh : c • s ∈ 𝓝 (c • x)\n⊢ s ∈ 𝓝 x","state_after":"M : Type u_1\nα : Type u_2\nβ : Type u_3\nΓ : Type u_4\ninst✝⁶ : Group Γ\nT : Type u_5\ninst✝⁵ : TopologicalSpace T\ninst✝⁴ : MulAction Γ T\nG₀ : Type u_6\ninst✝³ : GroupWithZero G₀\ninst✝² : MulAction G₀ α\ninst✝¹ : TopologicalSpace α\ninst✝ : ContinuousConstSMul G₀ α\nc : G₀\ns : Set α\nx : α\nhc : c ≠ 0\nh : c • s ∈ 𝓝 (c • x)\n⊢ c⁻¹ • c • s ∈ 𝓝 (c⁻¹ • c • x)","tactic":"rw [← inv_smul_smul₀ hc x, ← inv_smul_smul₀ hc s]","premises":[{"full_name":"inv_smul_smul₀","def_path":"Mathlib/GroupTheory/GroupAction/Group.lean","def_pos":[35,8],"def_end_pos":[35,22]}]},{"state_before":"M : Type u_1\nα : Type u_2\nβ : Type u_3\nΓ : Type u_4\ninst✝⁶ : Group Γ\nT : Type u_5\ninst✝⁵ : TopologicalSpace T\ninst✝⁴ : MulAction Γ T\nG₀ : Type u_6\ninst✝³ : GroupWithZero G₀\ninst✝² : MulAction G₀ α\ninst✝¹ : TopologicalSpace α\ninst✝ : ContinuousConstSMul G₀ α\nc : G₀\ns : Set α\nx : α\nhc : c ≠ 0\nh : c • s ∈ 𝓝 (c • x)\n⊢ c⁻¹ • c • s ∈ 𝓝 (c⁻¹ • c • x)","state_after":"no goals","tactic":"exact set_smul_mem_nhds_smul h (inv_ne_zero hc)","premises":[{"full_name":"inv_ne_zero","def_path":"Mathlib/Algebra/GroupWithZero/NeZero.lean","def_pos":[45,8],"def_end_pos":[45,19]},{"full_name":"set_smul_mem_nhds_smul","def_path":"Mathlib/Topology/Algebra/ConstMulAction.lean","def_pos":[479,8],"def_end_pos":[479,30]}]}]}
+{"url":".lake/packages/batteries/Batteries/Classes/SatisfiesM.lean","commit":"","full_name":"SatisfiesM_ExceptT_eq","start":[176,0],"end":[182,96],"file_path":".lake/packages/batteries/Batteries/Classes/SatisfiesM.lean","tactics":[{"state_before":"m : Type u_1 → Type u_2\nα ρ : Type u_1\np : α → Prop\nx : ExceptT ρ m α\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\n⊢ SatisfiesM p x ↔ SatisfiesM (fun x => ∀ (a : α), x = Except.ok a → p a) x","state_after":"case refine_1\nm : Type u_1 → Type u_2\nα ρ : Type u_1\np : α → Prop\nx : ExceptT ρ m α\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nx✝ : SatisfiesM p x\nf : ExceptT ρ m { a // p a }\neq : Subtype.val <$> f = x\n⊢ SatisfiesM (fun x => ∀ (a : α), x = Except.ok a → p a) (Subtype.val <$> f)\n\ncase refine_2\nm : Type u_1 → Type u_2\nα ρ : Type u_1\np : α → Prop\nx : ExceptT ρ m α\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nx✝ : SatisfiesM (fun x => ∀ (a : α), x = Except.ok a → p a) x\nf : m { a // (fun x => ∀ (a : α), x = Except.ok a → p a) a }\neq : Subtype.val <$> f = x\n⊢ SatisfiesM p (Subtype.val <$> f)","tactic":"refine ⟨fun ⟨f, eq⟩ => eq ▸ ?_, fun ⟨f, eq⟩ => eq ▸ ?_⟩","premises":[{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]}]}]}
+{"url":"Mathlib/NumberTheory/FLT/Three.lean","commit":"","full_name":"_private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.lambda_pow_dvd_a_add_b","start":[468,0],"end":[479,77],"file_path":"Mathlib/NumberTheory/FLT/Three.lean","tactics":[{"state_before":"K : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\n⊢ λ ^ (3 * S.multiplicity - 2) ∣ S.a + S.b","state_after":"K : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\nh : λ ^ S.multiplicity ∣ S.c\n⊢ λ ^ (3 * S.multiplicity - 2) ∣ S.a + S.b","tactic":"have h : λ ^ S.multiplicity ∣ S.c := multiplicity.pow_multiplicity_dvd _","premises":[{"full_name":"Dvd.dvd","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1344,2],"def_end_pos":[1344,5]},{"full_name":"FermatLastTheoremForThreeGen.Solution'.c","def_path":"Mathlib/NumberTheory/FLT/Three.lean","def_pos":[186,2],"def_end_pos":[186,3]},{"full_name":"FermatLastTheoremForThreeGen.Solution.multiplicity","def_path":"Mathlib/NumberTheory/FLT/Three.lean","def_pos":[218,4],"def_end_pos":[218,25]},{"full_name":"IsPrimitiveRoot.toInteger","def_path":"Mathlib/NumberTheory/Cyclotomic/Rat.lean","def_pos":[170,7],"def_end_pos":[170,16]},{"full_name":"multiplicity.pow_multiplicity_dvd","def_path":"Mathlib/RingTheory/Multiplicity.lean","def_pos":[98,8],"def_end_pos":[98,28]}]},{"state_before":"K : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\nh : λ ^ S.multiplicity ∣ S.c\n⊢ λ ^ (3 * S.multiplicity - 2) ∣ S.a + S.b","state_after":"K : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\nh : (λ ^ S.multiplicity) ^ 3 ∣ ↑S.u * S.c ^ 3\n⊢ λ ^ (3 * S.multiplicity - 2) ∣ S.a + S.b","tactic":"replace h : (λ ^ multiplicity S) ^ 3 ∣ S.u * S.c ^ 3 := by simp [h]","premises":[{"full_name":"Dvd.dvd","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1344,2],"def_end_pos":[1344,5]},{"full_name":"FermatLastTheoremForThreeGen.Solution'.c","def_path":"Mathlib/NumberTheory/FLT/Three.lean","def_pos":[186,2],"def_end_pos":[186,3]},{"full_name":"FermatLastTheoremForThreeGen.Solution'.u","def_path":"Mathlib/NumberTheory/FLT/Three.lean","def_pos":[187,2],"def_end_pos":[187,3]},{"full_name":"FermatLastTheoremForThreeGen.Solution.multiplicity","def_path":"Mathlib/NumberTheory/FLT/Three.lean","def_pos":[218,4],"def_end_pos":[218,25]},{"full_name":"IsPrimitiveRoot.toInteger","def_path":"Mathlib/NumberTheory/Cyclotomic/Rat.lean","def_pos":[170,7],"def_end_pos":[170,16]}]},{"state_before":"K : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\nh : (λ ^ S.multiplicity) ^ 3 ∣ ↑S.u * S.c ^ 3\n⊢ λ ^ (3 * S.multiplicity - 2) ∣ S.a + S.b","state_after":"K : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\nh :\n  λ ^ (3 * S.multiplicity) ∣\n    (S.a + S.b) * (λ * FermatLastTheoremForThreeGen.Solution.y S) * (λ * FermatLastTheoremForThreeGen.Solution.z S)\n⊢ λ ^ (3 * S.multiplicity - 2) ∣ S.a + S.b","tactic":"rw [← S.H, a_cube_add_b_cube_eq_mul, ← pow_mul, mul_comm, y_spec, z_spec] at h","premises":[{"full_name":"FermatLastTheoremForThreeGen.Solution'.H","def_path":"Mathlib/NumberTheory/FLT/Three.lean","def_pos":[193,2],"def_end_pos":[193,3]},{"full_name":"FermatLastTheoremForThreeGen.a_cube_add_b_cube_eq_mul","def_path":"Mathlib/NumberTheory/FLT/Three.lean","def_pos":[297,6],"def_end_pos":[297,30]},{"full_name":"_private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.y_spec","def_path":"Mathlib/NumberTheory/FLT/Three.lean","def_pos":[450,14],"def_end_pos":[450,20]},{"full_name":"_private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.z_spec","def_path":"Mathlib/NumberTheory/FLT/Three.lean","def_pos":[455,14],"def_end_pos":[455,20]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]},{"full_name":"pow_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[605,31],"def_end_pos":[605,38]}]},{"state_before":"K : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\nh :\n  λ ^ (3 * S.multiplicity) ∣\n    (S.a + S.b) * (λ * FermatLastTheoremForThreeGen.Solution.y S) * (λ * FermatLastTheoremForThreeGen.Solution.z S)\n⊢ λ ^ (3 * S.multiplicity - 2) ∣ S.a + S.b","state_after":"K : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\nh :\n  λ ^ (3 * S.multiplicity) ∣\n    (S.a + S.b) * (λ * FermatLastTheoremForThreeGen.Solution.y S) * (λ * FermatLastTheoremForThreeGen.Solution.z S)\n⊢ λ ^ (3 * S.multiplicity - 2) ∣ FermatLastTheoremForThreeGen.Solution.z S * (S.a + S.b)","tactic":"apply hζ.zeta_sub_one_prime'.pow_dvd_of_dvd_mul_left _ S.lambda_not_dvd_z","premises":[{"full_name":"IsPrimitiveRoot.zeta_sub_one_prime'","def_path":"Mathlib/NumberTheory/Cyclotomic/Rat.lean","def_pos":[333,8],"def_end_pos":[333,27]},{"full_name":"Prime.pow_dvd_of_dvd_mul_left","def_path":"Mathlib/Algebra/Associated/Basic.lean","def_pos":[124,8],"def_end_pos":[124,37]},{"full_name":"_private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.lambda_not_dvd_z","def_path":"Mathlib/NumberTheory/FLT/Three.lean","def_pos":[463,14],"def_end_pos":[463,30]}]},{"state_before":"K : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\nh :\n  λ ^ (3 * S.multiplicity) ∣\n    (S.a + S.b) * (λ * FermatLastTheoremForThreeGen.Solution.y S) * (λ * FermatLastTheoremForThreeGen.Solution.z S)\n⊢ λ ^ (3 * S.multiplicity - 2) ∣ FermatLastTheoremForThreeGen.Solution.z S * (S.a + S.b)","state_after":"K : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\nh :\n  λ ^ (3 * S.multiplicity) ∣\n    (S.a + S.b) * (λ * FermatLastTheoremForThreeGen.Solution.y S) * (λ * FermatLastTheoremForThreeGen.Solution.z S)\n⊢ λ ^ (3 * S.multiplicity - 2) ∣\n    FermatLastTheoremForThreeGen.Solution.y S * (FermatLastTheoremForThreeGen.Solution.z S * (S.a + S.b))","tactic":"apply hζ.zeta_sub_one_prime'.pow_dvd_of_dvd_mul_left _ S.lambda_not_dvd_y","premises":[{"full_name":"IsPrimitiveRoot.zeta_sub_one_prime'","def_path":"Mathlib/NumberTheory/Cyclotomic/Rat.lean","def_pos":[333,8],"def_end_pos":[333,27]},{"full_name":"Prime.pow_dvd_of_dvd_mul_left","def_path":"Mathlib/Algebra/Associated/Basic.lean","def_pos":[124,8],"def_end_pos":[124,37]},{"full_name":"_private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.lambda_not_dvd_y","def_path":"Mathlib/NumberTheory/FLT/Three.lean","def_pos":[458,14],"def_end_pos":[458,30]}]},{"state_before":"K : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\nh :\n  λ ^ (3 * S.multiplicity) ∣\n    (S.a + S.b) * (λ * FermatLastTheoremForThreeGen.Solution.y S) * (λ * FermatLastTheoremForThreeGen.Solution.z S)\n⊢ λ ^ (3 * S.multiplicity - 2) ∣\n    FermatLastTheoremForThreeGen.Solution.y S * (FermatLastTheoremForThreeGen.Solution.z S * (S.a + S.b))","state_after":"K : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\nh :\n  λ ^ (3 * S.multiplicity) ∣\n    (S.a + S.b) * (λ * FermatLastTheoremForThreeGen.Solution.y S) * (λ * FermatLastTheoremForThreeGen.Solution.z S)\nthis : 2 ≤ S.multiplicity\n⊢ λ ^ (3 * S.multiplicity - 2) ∣\n    FermatLastTheoremForThreeGen.Solution.y S * (FermatLastTheoremForThreeGen.Solution.z S * (S.a + S.b))","tactic":"have := S.two_le_multiplicity","premises":[{"full_name":"FermatLastTheoremForThreeGen.Solution.two_le_multiplicity","def_path":"Mathlib/NumberTheory/FLT/Three.lean","def_pos":[293,6],"def_end_pos":[293,34]}]},{"state_before":"K : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\nh :\n  λ ^ (3 * S.multiplicity) ∣\n    (S.a + S.b) * (λ * FermatLastTheoremForThreeGen.Solution.y S) * (λ * FermatLastTheoremForThreeGen.Solution.z S)\nthis : 2 ≤ S.multiplicity\n⊢ λ ^ (3 * S.multiplicity - 2) ∣\n    FermatLastTheoremForThreeGen.Solution.y S * (FermatLastTheoremForThreeGen.Solution.z S * (S.a + S.b))","state_after":"K : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\nh :\n  λ ^ (3 * S.multiplicity - 2) * λ * λ ∣\n    (S.a + S.b) * FermatLastTheoremForThreeGen.Solution.y S * FermatLastTheoremForThreeGen.Solution.z S * λ * λ\nthis : 2 ≤ S.multiplicity\n⊢ λ ^ (3 * S.multiplicity - 2) ∣\n    FermatLastTheoremForThreeGen.Solution.y S * (FermatLastTheoremForThreeGen.Solution.z S * (S.a + S.b))","tactic":"rw [show 3 * multiplicity S = 3 * multiplicity S - 2 + 1 + 1 by omega, pow_succ, pow_succ,\n    show (S.a + S.b) * (λ * y S) * (λ * z S) = (S.a + S.b) * y S * z S * λ * λ by ring] at h","premises":[{"full_name":"FermatLastTheoremForThreeGen.Solution'.a","def_path":"Mathlib/NumberTheory/FLT/Three.lean","def_pos":[184,2],"def_end_pos":[184,3]},{"full_name":"FermatLastTheoremForThreeGen.Solution'.b","def_path":"Mathlib/NumberTheory/FLT/Three.lean","def_pos":[185,2],"def_end_pos":[185,3]},{"full_name":"FermatLastTheoremForThreeGen.Solution.multiplicity","def_path":"Mathlib/NumberTheory/FLT/Three.lean","def_pos":[218,4],"def_end_pos":[218,25]},{"full_name":"IsPrimitiveRoot.toInteger","def_path":"Mathlib/NumberTheory/Cyclotomic/Rat.lean","def_pos":[170,7],"def_end_pos":[170,16]},{"full_name":"_private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.y","def_path":"Mathlib/NumberTheory/FLT/Three.lean","def_pos":[449,26],"def_end_pos":[449,27]},{"full_name":"_private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.z","def_path":"Mathlib/NumberTheory/FLT/Three.lean","def_pos":[454,26],"def_end_pos":[454,27]},{"full_name":"pow_succ","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[567,8],"def_end_pos":[567,16]}]},{"state_before":"K : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\nh :\n  λ ^ (3 * S.multiplicity - 2) * λ * λ ∣\n    (S.a + S.b) * FermatLastTheoremForThreeGen.Solution.y S * FermatLastTheoremForThreeGen.Solution.z S * λ * λ\nthis : 2 ≤ S.multiplicity\n⊢ λ ^ (3 * S.multiplicity - 2) ∣\n    FermatLastTheoremForThreeGen.Solution.y S * (FermatLastTheoremForThreeGen.Solution.z S * (S.a + S.b))","state_after":"K : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\nthis : 2 ≤ S.multiplicity\nh :\n  λ ^ (3 * S.multiplicity - 2) ∣\n    (S.a + S.b) * FermatLastTheoremForThreeGen.Solution.y S * FermatLastTheoremForThreeGen.Solution.z S\n⊢ λ ^ (3 * S.multiplicity - 2) ∣\n    FermatLastTheoremForThreeGen.Solution.y S * (FermatLastTheoremForThreeGen.Solution.z S * (S.a + S.b))","tactic":"simp only [mul_dvd_mul_iff_right hζ.zeta_sub_one_prime'.ne_zero] at h","premises":[{"full_name":"IsPrimitiveRoot.zeta_sub_one_prime'","def_path":"Mathlib/NumberTheory/Cyclotomic/Rat.lean","def_pos":[333,8],"def_end_pos":[333,27]},{"full_name":"Prime.ne_zero","def_path":"Mathlib/Algebra/Associated/Basic.lean","def_pos":[47,8],"def_end_pos":[47,15]},{"full_name":"mul_dvd_mul_iff_right","def_path":"Mathlib/Algebra/GroupWithZero/Divisibility.lean","def_pos":[50,8],"def_end_pos":[50,29]}]},{"state_before":"K : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\nthis : 2 ≤ S.multiplicity\nh :\n  λ ^ (3 * S.multiplicity - 2) ∣\n    (S.a + S.b) * FermatLastTheoremForThreeGen.Solution.y S * FermatLastTheoremForThreeGen.Solution.z S\n⊢ λ ^ (3 * S.multiplicity - 2) ∣\n    FermatLastTheoremForThreeGen.Solution.y S * (FermatLastTheoremForThreeGen.Solution.z S * (S.a + S.b))","state_after":"no goals","tactic":"rwa [show (S.a + S.b) * y S * z S = y S * (z S * (S.a + S.b)) by ring] at h","premises":[{"full_name":"FermatLastTheoremForThreeGen.Solution'.a","def_path":"Mathlib/NumberTheory/FLT/Three.lean","def_pos":[184,2],"def_end_pos":[184,3]},{"full_name":"FermatLastTheoremForThreeGen.Solution'.b","def_path":"Mathlib/NumberTheory/FLT/Three.lean","def_pos":[185,2],"def_end_pos":[185,3]},{"full_name":"_private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.y","def_path":"Mathlib/NumberTheory/FLT/Three.lean","def_pos":[449,26],"def_end_pos":[449,27]},{"full_name":"_private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.z","def_path":"Mathlib/NumberTheory/FLT/Three.lean","def_pos":[454,26],"def_end_pos":[454,27]}]}]}
+{"url":"Mathlib/Computability/AkraBazzi/AkraBazzi.lean","commit":"","full_name":"AkraBazziRecurrence.smoothingFn_mul_asympBound_isBigO_T","start":[1294,0],"end":[1439,59],"file_path":"Mathlib/Computability/AkraBazzi/AkraBazzi.lean","tactics":[{"state_before":"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\n⊢ (fun n => (1 + ε ↑n) * asympBound g a b n) =O[atTop] T","state_after":"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := b (min_bi b) / 2\n⊢ (fun n => (1 + ε ↑n) * asympBound g a b n) =O[atTop] T","tactic":"let b' := b (min_bi b) / 2","premises":[{"full_name":"AkraBazziRecurrence.min_bi","def_path":"Mathlib/Computability/AkraBazzi/AkraBazzi.lean","def_pos":[113,18],"def_end_pos":[113,24]}]},{"state_before":"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := b (min_bi b) / 2\n⊢ (fun n => (1 + ε ↑n) * asympBound g a b n) =O[atTop] T","state_after":"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := b (min_bi b) / 2\nhb_pos : 0 < b'\n⊢ (fun n => (1 + ε ↑n) * asympBound g a b n) =O[atTop] T","tactic":"have hb_pos : 0 < b' := R.bi_min_div_two_pos","premises":[{"full_name":"AkraBazziRecurrence.bi_min_div_two_pos","def_path":"Mathlib/Computability/AkraBazzi/AkraBazzi.lean","def_pos":[197,6],"def_end_pos":[197,24]}]},{"state_before":"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := b (min_bi b) / 2\nhb_pos : 0 < b'\n⊢ (fun n => (1 + ε ↑n) * asympBound g a b n) =O[atTop] T","state_after":"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := b (min_bi b) / 2\nhb_pos : 0 < b'\n⊢ ∀ᶠ (n₀ : ℕ) in atTop, ∃ c > 0, ∀ n ≥ n₀, c * ‖(1 + ε ↑n) * asympBound g a b n‖ ≤ ‖T n‖","tactic":"rw [isBigO_atTop_iff_eventually_exists_pos]","premises":[{"full_name":"Asymptotics.isBigO_atTop_iff_eventually_exists_pos","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[1876,8],"def_end_pos":[1876,46]}]},{"state_before":"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := b (min_bi b) / 2\nhb_pos : 0 < b'\n⊢ ∀ᶠ (n₀ : ℕ) in atTop, ∃ c > 0, ∀ n ≥ n₀, c * ‖(1 + ε ↑n) * asympBound g a b n‖ ≤ ‖T n‖","state_after":"case intro.intro\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := b (min_bi b) / 2\nhb_pos : 0 < b'\nc₁ : ℝ\nhc₁ : c₁ > 0\nh_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n\n⊢ ∀ᶠ (n₀ : ℕ) in atTop, ∃ c > 0, ∀ n ≥ n₀, c * ‖(1 + ε ↑n) * asympBound g a b n‖ ≤ ‖T n‖","tactic":"obtain ⟨c₁, hc₁, h_sumTransform_aux⟩ := R.eventually_atTop_sumTransform_le","premises":[{"full_name":"AkraBazziRecurrence.eventually_atTop_sumTransform_le","def_path":"Mathlib/Computability/AkraBazzi/AkraBazzi.lean","def_pos":[605,6],"def_end_pos":[605,38]}]},{"state_before":"case intro.intro\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := b (min_bi b) / 2\nhb_pos : 0 < b'\nc₁ : ℝ\nhc₁ : c₁ > 0\nh_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n\n⊢ ∀ᶠ (n₀ : ℕ) in atTop, ∃ c > 0, ∀ n ≥ n₀, c * ‖(1 + ε ↑n) * asympBound g a b n‖ ≤ ‖T n‖","state_after":"case h\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := b (min_bi b) / 2\nhb_pos : 0 < b'\nc₁ : ℝ\nhc₁ : c₁ > 0\nh_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n\nn₀ : ℕ\nn₀_ge_Rn₀ : R.n₀ ≤ n₀\nh_b_floor : 0 < ⌊b' * ↑n₀⌋₊\nh_smoothing_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < 1 + ε ↑y\nh_smoothing_pos' : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_asympBound_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < asympBound g a b y\nh_asympBound_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < asympBound g a b (r i y)\nh_asympBound_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < asympBound g a b y\nn₀_pos : 0 < n₀\nh_smoothing_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < 1 + ε ↑(r i y)\nbound2 : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b i ^ p a b * ↑y ^ p a b * (1 + ε ↑y) ≤ ↑(r i y) ^ p a b * (1 + ε ↑(r i y))\nh_smoothingFn_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_sumTransform : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), sumTransform (p a b) g (r i y) y ≤ c₁ * g ↑y\nh_bi_le_r : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b (min_bi b) / 2 * ↑y ≤ ↑(r i y)\nh_exp : ∀ (y : ℕ), n₀ ≤ y → ⌈rexp 1⌉₊ ≤ y\n⊢ ∃ c > 0, ∀ n ≥ n₀, c * ‖(1 + ε ↑n) * asympBound g a b n‖ ≤ ‖T n‖","tactic":"filter_upwards [eventually_ge_atTop R.n₀,                                 -- n₀_ge_Rn₀\n      (tendsto_nat_floor_mul_atTop b' hb_pos).eventually_gt_atTop 0,        -- h_b_floor\n      eventually_forall_ge_atTop.mpr eventually_one_add_smoothingFn_pos,    -- h_smoothing_pos\n      (tendsto_nat_floor_mul_atTop b' hb_pos).eventually_forall_ge_atTop\n        eventually_one_add_smoothingFn_pos,                                 -- h_smoothing_pos'\n      eventually_forall_ge_atTop.mpr R.eventually_asympBound_pos,            -- h_asympBound_pos\n      eventually_forall_ge_atTop.mpr R.eventually_asympBound_r_pos,          -- h_asympBound_r_pos\n      (tendsto_nat_floor_mul_atTop b' hb_pos).eventually_forall_ge_atTop\n        R.eventually_asympBound_pos,                                         -- h_asympBound_floor\n      eventually_gt_atTop 0,                                                -- n₀_pos\n      eventually_forall_ge_atTop.mpr R.eventually_one_add_smoothingFn_r_pos,  -- h_smoothing_r_pos\n      eventually_forall_ge_atTop.mpr R.rpow_p_mul_one_add_smoothingFn_ge,   -- bound2\n      (tendsto_nat_floor_mul_atTop b' hb_pos).eventually_forall_ge_atTop\n        eventually_one_add_smoothingFn_pos,                                 -- h_smoothingFn_floor\n      eventually_forall_ge_atTop.mpr h_sumTransform_aux,                    -- h_sumTransform\n      eventually_forall_ge_atTop.mpr R.eventually_bi_mul_le_r,              -- h_bi_le_r\n      eventually_forall_ge_atTop.mpr (eventually_ge_atTop ⌈exp 1⌉₊)]        -- h_exp\n    with n₀ n₀_ge_Rn₀ h_b_floor h_smoothing_pos h_smoothing_pos' h_asympBound_pos h_asympBound_r_pos\n      h_asympBound_floor n₀_pos h_smoothing_r_pos bound2 h_smoothingFn_floor h_sumTransform\n      h_bi_le_r h_exp","premises":[{"full_name":"AkraBazziRecurrence.eventually_asympBound_pos","def_path":"Mathlib/Computability/AkraBazzi/AkraBazzi.lean","def_pos":[597,6],"def_end_pos":[597,31]},{"full_name":"AkraBazziRecurrence.eventually_asympBound_r_pos","def_path":"Mathlib/Computability/AkraBazzi/AkraBazzi.lean","def_pos":[601,6],"def_end_pos":[601,33]},{"full_name":"AkraBazziRecurrence.eventually_bi_mul_le_r","def_path":"Mathlib/Computability/AkraBazzi/AkraBazzi.lean","def_pos":[176,6],"def_end_pos":[176,28]},{"full_name":"AkraBazziRecurrence.eventually_one_add_smoothingFn_pos","def_path":"Mathlib/Computability/AkraBazzi/AkraBazzi.lean","def_pos":[428,6],"def_end_pos":[428,40]},{"full_name":"AkraBazziRecurrence.eventually_one_add_smoothingFn_r_pos","def_path":"Mathlib/Computability/AkraBazzi/AkraBazzi.lean","def_pos":[437,6],"def_end_pos":[437,42]},{"full_name":"AkraBazziRecurrence.n₀","def_path":"Mathlib/Computability/AkraBazzi/AkraBazzi.lean","def_pos":[84,2],"def_end_pos":[84,4]},{"full_name":"AkraBazziRecurrence.rpow_p_mul_one_add_smoothingFn_ge","def_path":"Mathlib/Computability/AkraBazzi/AkraBazzi.lean","def_pos":[1050,6],"def_end_pos":[1050,39]},{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Filter.Tendsto.eventually_forall_ge_atTop","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[222,8],"def_end_pos":[222,42]},{"full_name":"Filter.Tendsto.eventually_gt_atTop","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[176,18],"def_end_pos":[176,45]},{"full_name":"Filter.eventually_forall_ge_atTop","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[210,8],"def_end_pos":[210,34]},{"full_name":"Filter.eventually_ge_atTop","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[164,8],"def_end_pos":[164,27]},{"full_name":"Filter.eventually_gt_atTop","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[170,8],"def_end_pos":[170,27]},{"full_name":"Filter.mp_mem","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[159,8],"def_end_pos":[159,14]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Nat.ceil","def_path":"Mathlib/Algebra/Order/Floor.lean","def_pos":[102,4],"def_end_pos":[102,8]},{"full_name":"Real.exp","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[102,11],"def_end_pos":[102,14]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]},{"full_name":"tendsto_nat_floor_mul_atTop","def_path":"Mathlib/Analysis/SpecificLimits/Basic.lean","def_pos":[615,6],"def_end_pos":[615,33]}]},{"state_before":"case h\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := b (min_bi b) / 2\nhb_pos : 0 < b'\nc₁ : ℝ\nhc₁ : c₁ > 0\nh_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n\nn₀ : ℕ\nn₀_ge_Rn₀ : R.n₀ ≤ n₀\nh_b_floor : 0 < ⌊b' * ↑n₀⌋₊\nh_smoothing_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < 1 + ε ↑y\nh_smoothing_pos' : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_asympBound_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < asympBound g a b y\nh_asympBound_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < asympBound g a b (r i y)\nh_asympBound_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < asympBound g a b y\nn₀_pos : 0 < n₀\nh_smoothing_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < 1 + ε ↑(r i y)\nbound2 : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b i ^ p a b * ↑y ^ p a b * (1 + ε ↑y) ≤ ↑(r i y) ^ p a b * (1 + ε ↑(r i y))\nh_smoothingFn_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_sumTransform : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), sumTransform (p a b) g (r i y) y ≤ c₁ * g ↑y\nh_bi_le_r : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b (min_bi b) / 2 * ↑y ≤ ↑(r i y)\nh_exp : ∀ (y : ℕ), n₀ ≤ y → ⌈rexp 1⌉₊ ≤ y\n⊢ ∃ c > 0, ∀ n ≥ n₀, c * ‖(1 + ε ↑n) * asympBound g a b n‖ ≤ ‖T n‖","state_after":"case h\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ �� ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := b (min_bi b) / 2\nhb_pos : 0 < b'\nc₁ : ℝ\nhc₁ : c₁ > 0\nh_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n\nn₀ : ℕ\nn₀_ge_Rn₀ : R.n₀ ≤ n₀\nh_b_floor : 0 < ⌊b' * ↑n₀⌋₊\nh_smoothing_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < 1 + ε ↑y\nh_smoothing_pos' : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_asympBound_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < asympBound g a b y\nh_asympBound_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < asympBound g a b (r i y)\nh_asympBound_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < asympBound g a b y\nn₀_pos : 0 < n₀\nh_smoothing_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < 1 + ε ↑(r i y)\nbound2 : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b i ^ p a b * ↑y ^ p a b * (1 + ε ↑y) ≤ ↑(r i y) ^ p a b * (1 + ε ↑(r i y))\nh_smoothingFn_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_sumTransform : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), sumTransform (p a b) g (r i y) y ≤ c₁ * g ↑y\nh_bi_le_r : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b (min_bi b) / 2 * ↑y ≤ ↑(r i y)\nh_exp : ∀ (y : ℕ), n₀ ≤ y → ⌈rexp 1⌉₊ ≤ y\nh_base_nonempty : (Ico ⌊b (min_bi b) / 2 * ↑n₀⌋₊ n₀).Nonempty\n⊢ ∃ c > 0, ∀ n ≥ n₀, c * ‖(1 + ε ↑n) * asympBound g a b n‖ ≤ ‖T n‖","tactic":"have h_base_nonempty := R.base_nonempty n₀_pos","premises":[{"full_name":"AkraBazziRecurrence.base_nonempty","def_path":"Mathlib/Computability/AkraBazzi/AkraBazzi.lean","def_pos":[1152,6],"def_end_pos":[1152,19]}]},{"state_before":"case h\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := b (min_bi b) / 2\nhb_pos : 0 < b'\nc₁ : ℝ\nhc₁ : c₁ > 0\nh_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n\nn₀ : ℕ\nn₀_ge_Rn₀ : R.n₀ ≤ n₀\nh_b_floor : 0 < ⌊b' * ↑n₀⌋₊\nh_smoothing_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < 1 + ε ↑y\nh_smoothing_pos' : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_asympBound_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < asympBound g a b y\nh_asympBound_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < asympBound g a b (r i y)\nh_asympBound_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < asympBound g a b y\nn₀_pos : 0 < n₀\nh_smoothing_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < 1 + ε ↑(r i y)\nbound2 : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b i ^ p a b * ↑y ^ p a b * (1 + ε ↑y) ≤ ↑(r i y) ^ p a b * (1 + ε ↑(r i y))\nh_smoothingFn_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_sumTransform : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), sumTransform (p a b) g (r i y) y ≤ c₁ * g ↑y\nh_bi_le_r : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b (min_bi b) / 2 * ↑y ≤ ↑(r i y)\nh_exp : ∀ (y : ℕ), n₀ ≤ y → ⌈rexp 1⌉₊ ≤ y\nh_base_nonempty : (Ico ⌊b (min_bi b) / 2 * ↑n₀⌋₊ n₀).Nonempty\n⊢ ∃ c > 0, ∀ n ≥ n₀, c * ‖(1 + ε ↑n) * asympBound g a b n‖ ≤ ‖T n‖","state_after":"case h\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := b (min_bi b) / 2\nhb_pos : 0 < b'\nc₁ : ℝ\nhc₁ : c₁ > 0\nh_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n\nn₀ : ℕ\nn₀_ge_Rn₀ : R.n₀ ≤ n₀\nh_b_floor : 0 < ⌊b' * ↑n₀⌋₊\nh_smoothing_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < 1 + ε ↑y\nh_smoothing_pos' : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_asympBound_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < asympBound g a b y\nh_asympBound_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < asympBound g a b (r i y)\nh_asympBound_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < asympBound g a b y\nn₀_pos : 0 < n₀\nh_smoothing_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < 1 + ε ↑(r i y)\nbound2 : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b i ^ p a b * ↑y ^ p a b * (1 + ε ↑y) ≤ ↑(r i y) ^ p a b * (1 + ε ↑(r i y))\nh_smoothingFn_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_sumTransform : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), sumTransform (p a b) g (r i y) y ≤ c₁ * g ↑y\nh_bi_le_r : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b (min_bi b) / 2 * ↑y ≤ ↑(r i y)\nh_exp : ∀ (y : ℕ), n₀ ≤ y → ⌈rexp 1⌉₊ ≤ y\nh_base_nonempty : (Ico ⌊b (min_bi b) / 2 * ↑n₀⌋₊ n₀).Nonempty\nbase_min : ℝ := (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)\nbase_min_def : base_min = (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)\n⊢ ∃ c > 0, ∀ n ≥ n₀, c * ‖(1 + ε ↑n) * asympBound g a b n‖ ≤ ‖T n‖","tactic":"set base_min : ℝ :=\n    (Finset.Ico (⌊b' * n₀⌋₊) n₀).inf' h_base_nonempty\n      (fun n => T n / ((1 + ε n) * asympBound g a b n)) with base_min_def","premises":[{"full_name":"AkraBazziRecurrence.asympBound","def_path":"Mathlib/Computability/AkraBazzi/AkraBazzi.lean","def_pos":[577,18],"def_end_pos":[577,28]},{"full_name":"AkraBazziRecurrence.smoothingFn","def_path":"Mathlib/Computability/AkraBazzi/AkraBazzi.lean","def_pos":[297,18],"def_end_pos":[297,29]},{"full_name":"Finset.Ico","def_path":"Mathlib/Order/Interval/Finset/Defs.lean","def_pos":[281,4],"def_end_pos":[281,7]},{"full_name":"Finset.inf'","def_path":"Mathlib/Data/Finset/Lattice.lean","def_pos":[872,4],"def_end_pos":[872,8]},{"full_name":"Nat.floor","def_path":"Mathlib/Algebra/Order/Floor.lean","def_pos":[98,4],"def_end_pos":[98,9]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"case h\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := b (min_bi b) / 2\nhb_pos : 0 < b'\nc₁ : ℝ\nhc₁ : c₁ > 0\nh_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n\nn₀ : ℕ\nn₀_ge_Rn₀ : R.n₀ ≤ n₀\nh_b_floor : 0 < ⌊b' * ↑n₀⌋₊\nh_smoothing_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < 1 + ε ↑y\nh_smoothing_pos' : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_asympBound_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < asympBound g a b y\nh_asympBound_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < asympBound g a b (r i y)\nh_asympBound_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < asympBound g a b y\nn₀_pos : 0 < n₀\nh_smoothing_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < 1 + ε ↑(r i y)\nbound2 : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b i ^ p a b * ↑y ^ p a b * (1 + ε ↑y) ≤ ↑(r i y) ^ p a b * (1 + ε ↑(r i y))\nh_smoothingFn_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_sumTransform : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), sumTransform (p a b) g (r i y) y ≤ c₁ * g ↑y\nh_bi_le_r : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b (min_bi b) / 2 * ↑y ≤ ↑(r i y)\nh_exp : ∀ (y : ℕ), n₀ ≤ y → ⌈rexp 1⌉₊ ≤ y\nh_base_nonempty : (Ico ⌊b (min_bi b) / 2 * ↑n₀⌋₊ n₀).Nonempty\nbase_min : ℝ := (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)\nbase_min_def : base_min = (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)\n⊢ ∃ c > 0, ∀ n ≥ n₀, c * ‖(1 + ε ↑n) * asympBound g a b n‖ ≤ ‖T n‖","state_after":"case h\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := b (min_bi b) / 2\nhb_pos : 0 < b'\nc₁ : ℝ\nhc₁ : c₁ > 0\nh_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n\nn₀ : ℕ\nn₀_ge_Rn₀ : R.n₀ ≤ n₀\nh_b_floor : 0 < ⌊b' * ↑n₀⌋₊\nh_smoothing_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < 1 + ε ↑y\nh_smoothing_pos' : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_asympBound_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < asympBound g a b y\nh_asympBound_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < asympBound g a b (r i y)\nh_asympBound_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < asympBound g a b y\nn₀_pos : 0 < n₀\nh_smoothing_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < 1 + ε ↑(r i y)\nbound2 : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b i ^ p a b * ↑y ^ p a b * (1 + ε ↑y) ≤ ↑(r i y) ^ p a b * (1 + ε ↑(r i y))\nh_smoothingFn_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_sumTransform : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), sumTransform (p a b) g (r i y) y ≤ c₁ * g ↑y\nh_bi_le_r : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b (min_bi b) / 2 * ↑y ≤ ↑(r i y)\nh_exp : ∀ (y : ℕ), n₀ ≤ y → ⌈rexp 1⌉₊ ≤ y\nh_base_nonempty : (Ico ⌊b (min_bi b) / 2 * ↑n₀⌋₊ n₀).Nonempty\nbase_min : ℝ := (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)\nbase_min_def : base_min = (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)\nC : ℝ := min (2 * c₁)⁻¹ base_min\n⊢ ∃ c > 0, ∀ n ≥ n₀, c * ‖(1 + ε ↑n) * asympBound g a b n‖ ≤ ‖T n‖","tactic":"let C := min (2 * c₁)⁻¹ base_min","premises":[{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"Min.min","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1142,2],"def_end_pos":[1142,5]}]},{"state_before":"case h\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := b (min_bi b) / 2\nhb_pos : 0 < b'\nc₁ : ℝ\nhc₁ : c₁ > 0\nh_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n\nn₀ : ℕ\nn₀_ge_Rn₀ : R.n₀ ≤ n₀\nh_b_floor : 0 < ⌊b' * ↑n₀⌋₊\nh_smoothing_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < 1 + ε ↑y\nh_smoothing_pos' : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_asympBound_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < asympBound g a b y\nh_asympBound_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < asympBound g a b (r i y)\nh_asympBound_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < asympBound g a b y\nn₀_pos : 0 < n₀\nh_smoothing_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < 1 + ε ↑(r i y)\nbound2 : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b i ^ p a b * ↑y ^ p a b * (1 + ε ↑y) ≤ ↑(r i y) ^ p a b * (1 + ε ↑(r i y))\nh_smoothingFn_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_sumTransform : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), sumTransform (p a b) g (r i y) y ≤ c₁ * g ↑y\nh_bi_le_r : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b (min_bi b) / 2 * ↑y ≤ ↑(r i y)\nh_exp : ∀ (y : ℕ), n₀ ≤ y → ⌈rexp 1⌉₊ ≤ y\nh_base_nonempty : (Ico ⌊b (min_bi b) / 2 * ↑n₀⌋₊ n₀).Nonempty\nbase_min : ℝ := (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)\nbase_min_def : base_min = (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)\nC : ℝ := min (2 * c₁)⁻¹ base_min\n⊢ ∃ c > 0, ∀ n ≥ n₀, c * ‖(1 + ε ↑n) * asympBound g a b n‖ ≤ ‖T n‖","state_after":"case h\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := b (min_bi b) / 2\nhb_pos : 0 < b'\nc₁ : ℝ\nhc₁ : c₁ > 0\nh_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n\nn₀ : ℕ\nn₀_ge_Rn₀ : R.n₀ ≤ n₀\nh_b_floor : 0 < ⌊b' * ↑n₀⌋₊\nh_smoothing_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < 1 + ε ↑y\nh_smoothing_pos' : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_asympBound_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < asympBound g a b y\nh_asympBound_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < asympBound g a b (r i y)\nh_asympBound_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < asympBound g a b y\nn₀_pos : 0 < n₀\nh_smoothing_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < 1 + ε ↑(r i y)\nbound2 : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b i ^ p a b * ↑y ^ p a b * (1 + ε ↑y) ≤ ↑(r i y) ^ p a b * (1 + ε ↑(r i y))\nh_smoothingFn_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_sumTransform : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), sumTransform (p a b) g (r i y) y ≤ c₁ * g ↑y\nh_bi_le_r : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b (min_bi b) / 2 * ↑y ≤ ↑(r i y)\nh_exp : ∀ (y : ℕ), n₀ ≤ y → ⌈rexp 1⌉₊ ≤ y\nh_base_nonempty : (Ico ⌊b (min_bi b) / 2 * ↑n₀⌋₊ n₀).Nonempty\nbase_min : ℝ := (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)\nbase_min_def : base_min = (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)\nC : ℝ := min (2 * c₁)⁻¹ base_min\nhC_pos : 0 < C\n⊢ ∃ c > 0, ∀ n ≥ n₀, c * ‖(1 + ε ↑n) * asympBound g a b n‖ ≤ ‖T n‖","tactic":"have hC_pos : 0 < C := by\n    refine lt_min (by positivity) ?_\n    obtain ⟨m, hm_mem, hm⟩ :=\n      Finset.exists_mem_eq_inf' h_base_nonempty (fun n => T n / ((1 + ε n) * asympBound g a b n))\n    calc 0 < T m / ((1 + ε m) * asympBound g a b m) := by\n              have H₁ : 0 < T m := by exact R.T_pos _\n              have H₂ : 0 < 1 + ε m := by rw [Finset.mem_Ico] at hm_mem\n                                          exact h_smoothing_pos' m hm_mem.1\n              have H₃ : 0 < asympBound g a b m := by\n                refine R.asympBound_pos m ?_\n                calc 0 < ⌊b' * n₀⌋₊ := by exact h_b_floor\n                     _ ≤ m := by rw [Finset.mem_Ico] at hm_mem; exact hm_mem.1\n              positivity\n         _ = base_min := by rw [base_min_def, hm]","premises":[{"full_name":"AkraBazziRecurrence.T_pos","def_path":"Mathlib/Computability/AkraBazzi/AkraBazzi.lean","def_pos":[266,26],"def_end_pos":[266,31]},{"full_name":"AkraBazziRecurrence.asympBound","def_path":"Mathlib/Computability/AkraBazzi/AkraBazzi.lean","def_pos":[577,18],"def_end_pos":[577,28]},{"full_name":"AkraBazziRecurrence.asympBound_pos","def_path":"Mathlib/Computability/AkraBazzi/AkraBazzi.lean","def_pos":[587,6],"def_end_pos":[587,20]},{"full_name":"AkraBazziRecurrence.smoothingFn","def_path":"Mathlib/Computability/AkraBazzi/AkraBazzi.lean","def_pos":[297,18],"def_end_pos":[297,29]},{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"Finset.exists_mem_eq_inf'","def_path":"Mathlib/Data/Finset/Lattice.lean","def_pos":[1158,8],"def_end_pos":[1158,26]},{"full_name":"Finset.mem_Ico","def_path":"Mathlib/Order/Interval/Finset/Defs.lean","def_pos":[299,8],"def_end_pos":[299,15]},{"full_name":"Nat.floor","def_path":"Mathlib/Algebra/Order/Floor.lean","def_pos":[98,4],"def_end_pos":[98,9]},{"full_name":"lt_min","def_path":"Mathlib/Init/Order/LinearOrder.lean","def_pos":[138,8],"def_end_pos":[138,14]}]},{"state_before":"case h\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := b (min_bi b) / 2\nhb_pos : 0 < b'\nc₁ : ℝ\nhc₁ : c₁ > 0\nh_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n\nn₀ : ℕ\nn₀_ge_Rn₀ : R.n₀ ≤ n₀\nh_b_floor : 0 < ⌊b' * ↑n₀⌋₊\nh_smoothing_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < 1 + ε ↑y\nh_smoothing_pos' : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_asympBound_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < asympBound g a b y\nh_asympBound_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < asympBound g a b (r i y)\nh_asympBound_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < asympBound g a b y\nn₀_pos : 0 < n₀\nh_smoothing_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < 1 + ε ↑(r i y)\nbound2 : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b i ^ p a b * ↑y ^ p a b * (1 + ε ↑y) ≤ ↑(r i y) ^ p a b * (1 + ε ↑(r i y))\nh_smoothingFn_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_sumTransform : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), sumTransform (p a b) g (r i y) y ≤ c₁ * g ↑y\nh_bi_le_r : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b (min_bi b) / 2 * ↑y ≤ ↑(r i y)\nh_exp : ∀ (y : ℕ), n₀ ≤ y → ⌈rexp 1⌉₊ ≤ y\nh_base_nonempty : (Ico ⌊b (min_bi b) / 2 * ↑n₀⌋₊ n₀).Nonempty\nbase_min : ℝ := (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)\nbase_min_def : base_min = (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)\nC : ℝ := min (2 * c₁)⁻¹ base_min\nhC_pos : 0 < C\n⊢ ∃ c > 0, ∀ n ≥ n₀, c * ‖(1 + ε ↑n) * asympBound g a b n‖ ≤ ‖T n‖","state_after":"case h\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := b (min_bi b) / 2\nhb_pos : 0 < b'\nc₁ : ℝ\nhc₁ : c₁ > 0\nh_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n\nn₀ : ℕ\nn₀_ge_Rn₀ : R.n₀ ≤ n₀\nh_b_floor : 0 < ⌊b' * ↑n₀⌋₊\nh_smoothing_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < 1 + ε ↑y\nh_smoothing_pos' : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_asympBound_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < asympBound g a b y\nh_asympBound_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < asympBound g a b (r i y)\nh_asympBound_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < asympBound g a b y\nn₀_pos : 0 < n₀\nh_smoothing_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < 1 + ε ↑(r i y)\nbound2 : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b i ^ p a b * ↑y ^ p a b * (1 + ε ↑y) ≤ ↑(r i y) ^ p a b * (1 + ε ↑(r i y))\nh_smoothingFn_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_sumTransform : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), sumTransform (p a b) g (r i y) y ≤ c₁ * g ↑y\nh_bi_le_r : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b (min_bi b) / 2 * ↑y ≤ ↑(r i y)\nh_exp : ∀ (y : ℕ), n₀ ≤ y → ⌈rexp 1⌉₊ ≤ y\nh_base_nonempty : (Ico ⌊b (min_bi b) / 2 * ↑n₀⌋₊ n₀).Nonempty\nbase_min : ℝ := (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)\nbase_min_def : base_min = (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)\nC : ℝ := min (2 * c₁)⁻¹ base_min\nhC_pos : 0 < C\nn : ℕ\nhn : n ≥ n₀\n⊢ C * ‖(1 + ε ↑n) * asympBound g a b n‖ ≤ ‖T n‖","tactic":"refine ⟨C, hC_pos, fun n hn => ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]},{"state_before":"case h\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := b (min_bi b) / 2\nhb_pos : 0 < b'\nc₁ : ℝ\nhc₁ : c₁ > 0\nh_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n\nn₀ : ℕ\nn₀_ge_Rn₀ : R.n₀ ≤ n₀\nh_b_floor : 0 < ⌊b' * ↑n₀⌋₊\nh_smoothing_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < 1 + ε ↑y\nh_smoothing_pos' : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_asympBound_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < asympBound g a b y\nh_asympBound_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < asympBound g a b (r i y)\nh_asympBound_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < asympBound g a b y\nn₀_pos : 0 < n₀\nh_smoothing_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < 1 + ε ↑(r i y)\nbound2 : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b i ^ p a b * ↑y ^ p a b * (1 + ε ↑y) ≤ ↑(r i y) ^ p a b * (1 + ε ↑(r i y))\nh_smoothingFn_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_sumTransform : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), sumTransform (p a b) g (r i y) y ≤ c₁ * g ↑y\nh_bi_le_r : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b (min_bi b) / 2 * ↑y ≤ ↑(r i y)\nh_exp : ∀ (y : ℕ), n₀ ≤ y → ⌈rexp 1⌉₊ ≤ y\nh_base_nonempty : (Ico ⌊b (min_bi b) / 2 * ↑n₀⌋₊ n₀).Nonempty\nbase_min : ℝ := (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)\nbase_min_def : base_min = (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)\nC : ℝ := min (2 * c₁)⁻¹ base_min\nhC_pos : 0 < C\nn : ℕ\nhn : n ≥ n₀\n⊢ C * ‖(1 + ε ↑n) * asympBound g a b n‖ ≤ ‖T n‖","state_after":"case h\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := b (min_bi b) / 2\nhb_pos : 0 < b'\nc₁ : ℝ\nhc₁ : c₁ > 0\nh_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n\nn₀ : ℕ\nn₀_ge_Rn₀ : R.n₀ ≤ n₀\nh_b_floor : 0 < ⌊b' * ↑n₀⌋₊\nh_smoothing_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < 1 + ε ↑y\nh_smoothing_pos' : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_asympBound_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < asympBound g a b y\nh_asympBound_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < asympBound g a b (r i y)\nh_asympBound_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < asympBound g a b y\nn₀_pos : 0 < n₀\nh_smoothing_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < 1 + ε ↑(r i y)\nbound2 : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b i ^ p a b * ↑y ^ p a b * (1 + ε ↑y) ≤ ↑(r i y) ^ p a b * (1 + ε ↑(r i y))\nh_smoothingFn_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_sumTransform : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), sumTransform (p a b) g (r i y) y ≤ c₁ * g ↑y\nh_bi_le_r : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b (min_bi b) / 2 * ↑y ≤ ↑(r i y)\nh_exp : ∀ (y : ℕ), n₀ ≤ y → ⌈rexp 1⌉₊ ≤ y\nh_base_nonempty : (Ico ⌊b (min_bi b) / 2 * ↑n₀⌋₊ n₀).Nonempty\nbase_min : ℝ := (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)\nbase_min_def : base_min = (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)\nC : ℝ := min (2 * c₁)⁻¹ base_min\nhC_pos : 0 < C\nn : ℕ\nhn : n ≥ n₀\nh_base : ∀ n ∈ Ico ⌊b' * ↑n₀⌋₊ n₀, C * ((1 + ε ↑n) * asympBound g a b n) ≤ T n\n⊢ C * ‖(1 + ε ↑n) * asympBound g a b n‖ ≤ ‖T n‖","tactic":"have h_base : ∀ n ∈ Finset.Ico (⌊b' * n₀⌋₊) n₀, C * ((1 + ε n) * asympBound g a b n) ≤ T n := by\n    intro n hn\n    rw [Finset.mem_Ico] at hn\n    have htmp1 : 0 < 1 + ε n := h_smoothingFn_floor n hn.1\n    have htmp2 : 0 < asympBound g a b n := h_asympBound_floor n hn.1\n    rw [← _root_.le_div_iff (by positivity)]\n    rw [← Finset.mem_Ico] at hn\n    calc T n / ((1 + ε ↑n) * asympBound g a b n)\n           ≥ (Finset.Ico (⌊b' * n₀⌋₊) n₀).inf' h_base_nonempty\n                  fun z => T z / ((1 + ε z) * asympBound g a b z) :=\n                    Finset.inf'_le_of_le _ (b := n) hn <| le_refl _\n         _ ≥ C := min_le_right _ _","premises":[{"full_name":"AkraBazziRecurrence.asympBound","def_path":"Mathlib/Computability/AkraBazzi/AkraBazzi.lean","def_pos":[577,18],"def_end_pos":[577,28]},{"full_name":"AkraBazziRecurrence.smoothingFn","def_path":"Mathlib/Computability/AkraBazzi/AkraBazzi.lean","def_pos":[297,18],"def_end_pos":[297,29]},{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"Finset.Ico","def_path":"Mathlib/Order/Interval/Finset/Defs.lean","def_pos":[281,4],"def_end_pos":[281,7]},{"full_name":"Finset.inf'","def_path":"Mathlib/Data/Finset/Lattice.lean","def_pos":[872,4],"def_end_pos":[872,8]},{"full_name":"Finset.inf'_le_of_le","def_path":"Mathlib/Data/Finset/Lattice.lean","def_pos":[905,8],"def_end_pos":[905,21]},{"full_name":"Finset.mem_Ico","def_path":"Mathlib/Order/Interval/Finset/Defs.lean","def_pos":[299,8],"def_end_pos":[299,15]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Nat.floor","def_path":"Mathlib/Algebra/Order/Floor.lean","def_pos":[98,4],"def_end_pos":[98,9]},{"full_name":"le_div_iff","def_path":"Mathlib/Algebra/Order/Field/Basic.lean","def_pos":[41,8],"def_end_pos":[41,18]},{"full_name":"le_refl","def_path":"Mathlib/Order/Defs.lean","def_pos":[39,8],"def_end_pos":[39,15]},{"full_name":"min_le_right","def_path":"Mathlib/Init/Order/LinearOrder.lean","def_pos":[42,8],"def_end_pos":[42,20]}]},{"state_before":"case h\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := b (min_bi b) / 2\nhb_pos : 0 < b'\nc₁ : ℝ\nhc₁ : c₁ > 0\nh_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n\nn₀ : ℕ\nn₀_ge_Rn₀ : R.n₀ ≤ n₀\nh_b_floor : 0 < ⌊b' * ↑n₀⌋₊\nh_smoothing_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < 1 + ε ↑y\nh_smoothing_pos' : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_asympBound_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < asympBound g a b y\nh_asympBound_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < asympBound g a b (r i y)\nh_asympBound_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < asympBound g a b y\nn₀_pos : 0 < n₀\nh_smoothing_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < 1 + ε ↑(r i y)\nbound2 : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b i ^ p a b * ↑y ^ p a b * (1 + ε ↑y) ≤ ↑(r i y) ^ p a b * (1 + ε ↑(r i y))\nh_smoothingFn_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_sumTransform : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), sumTransform (p a b) g (r i y) y ≤ c₁ * g ↑y\nh_bi_le_r : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b (min_bi b) / 2 * ↑y ≤ ↑(r i y)\nh_exp : ∀ (y : ℕ), n₀ ≤ y → ⌈rexp 1⌉₊ ≤ y\nh_base_nonempty : (Ico ⌊b (min_bi b) / 2 * ↑n₀⌋₊ n₀).Nonempty\nbase_min : ℝ := (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)\nbase_min_def : base_min = (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)\nC : ℝ := min (2 * c₁)⁻¹ base_min\nhC_pos : 0 < C\nn : ℕ\nhn : n ≥ n₀\nh_base : ∀ n ∈ Ico ⌊b' * ↑n₀⌋₊ n₀, C * ((1 + ε ↑n) * asympBound g a b n) ≤ T n\n⊢ C * ‖(1 + ε ↑n) * asympBound g a b n‖ ≤ ‖T n‖","state_after":"case h\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := b (min_bi b) / 2\nhb_pos : 0 < b'\nc₁ : ℝ\nhc₁ : c₁ > 0\nh_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n\nn₀ : ℕ\nn₀_ge_Rn₀ : R.n₀ ≤ n₀\nh_b_floor : 0 < ⌊b' * ↑n₀⌋₊\nh_smoothing_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < 1 + ε ↑y\nh_smoothing_pos' : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_asympBound_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < asympBound g a b y\nh_asympBound_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < asympBound g a b (r i y)\nh_asympBound_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < asympBound g a b y\nn₀_pos : 0 < n₀\nh_smoothing_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < 1 + ε ↑(r i y)\nbound2 : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b i ^ p a b * ↑y ^ p a b * (1 + ε ↑y) ≤ ↑(r i y) ^ p a b * (1 + ε ↑(r i y))\nh_smoothingFn_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_sumTransform : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), sumTransform (p a b) g (r i y) y ≤ c₁ * g ↑y\nh_bi_le_r : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b (min_bi b) / 2 * ↑y ≤ ↑(r i y)\nh_exp : ∀ (y : ℕ), n₀ ≤ y → ⌈rexp 1⌉₊ ≤ y\nh_base_nonempty : (Ico ⌊b (min_bi b) / 2 * ↑n₀⌋₊ n₀).Nonempty\nbase_min : ℝ := (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)\nbase_min_def : base_min = (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)\nC : ℝ := min (2 * c₁)⁻¹ base_min\nhC_pos : 0 < C\nn : ℕ\nhn : n ≥ n₀\nh_base : ∀ n ∈ Ico ⌊b' * ↑n₀⌋₊ n₀, C * ((1 + ε ↑n) * asympBound g a b n) ≤ T n\nh_asympBound_pos' : 0 < asympBound g a b n\n⊢ C * ‖(1 + ε ↑n) * asympBound g a b n‖ ≤ ‖T n‖","tactic":"have h_asympBound_pos' : 0 < asympBound g a b n := h_asympBound_pos n hn","premises":[{"full_name":"AkraBazziRecurrence.asympBound","def_path":"Mathlib/Computability/AkraBazzi/AkraBazzi.lean","def_pos":[577,18],"def_end_pos":[577,28]}]},{"state_before":"case h\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := b (min_bi b) / 2\nhb_pos : 0 < b'\nc₁ : ℝ\nhc₁ : c₁ > 0\nh_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n\nn₀ : ℕ\nn₀_ge_Rn₀ : R.n₀ ≤ n₀\nh_b_floor : 0 < ⌊b' * ↑n₀⌋₊\nh_smoothing_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < 1 + ε ↑y\nh_smoothing_pos' : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_asympBound_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < asympBound g a b y\nh_asympBound_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < asympBound g a b (r i y)\nh_asympBound_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < asympBound g a b y\nn₀_pos : 0 < n₀\nh_smoothing_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < 1 + ε ↑(r i y)\nbound2 : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b i ^ p a b * ↑y ^ p a b * (1 + ε ↑y) ≤ ↑(r i y) ^ p a b * (1 + ε ↑(r i y))\nh_smoothingFn_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_sumTransform : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), sumTransform (p a b) g (r i y) y ≤ c₁ * g ↑y\nh_bi_le_r : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b (min_bi b) / 2 * ↑y ≤ ↑(r i y)\nh_exp : ∀ (y : ℕ), n₀ ≤ y → ⌈rexp 1⌉₊ ≤ y\nh_base_nonempty : (Ico ⌊b (min_bi b) / 2 * ↑n₀⌋₊ n₀).Nonempty\nbase_min : ℝ := (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)\nbase_min_def : base_min = (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)\nC : ℝ := min (2 * c₁)⁻¹ base_min\nhC_pos : 0 < C\nn : ℕ\nhn : n ≥ n₀\nh_base : ∀ n ∈ Ico ⌊b' * ↑n₀⌋₊ n₀, C * ((1 + ε ↑n) * asympBound g a b n) ≤ T n\nh_asympBound_pos' : 0 < asympBound g a b n\n⊢ C * ‖(1 + ε ↑n) * asympBound g a b n‖ ≤ ‖T n‖","state_after":"case h\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := b (min_bi b) / 2\nhb_pos : 0 < b'\nc₁ : ℝ\nhc₁ : c₁ > 0\nh_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n\nn₀ : ℕ\nn₀_ge_Rn₀ : R.n₀ ≤ n₀\nh_b_floor : 0 < ⌊b' * ↑n₀⌋₊\nh_smoothing_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < 1 + ε ↑y\nh_smoothing_pos' : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_asympBound_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < asympBound g a b y\nh_asympBound_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < asympBound g a b (r i y)\nh_asympBound_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < asympBound g a b y\nn₀_pos : 0 < n₀\nh_smoothing_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < 1 + ε ↑(r i y)\nbound2 : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b i ^ p a b * ↑y ^ p a b * (1 + ε ↑y) ≤ ↑(r i y) ^ p a b * (1 + ε ↑(r i y))\nh_smoothingFn_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_sumTransform : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), sumTransform (p a b) g (r i y) y ≤ c₁ * g ↑y\nh_bi_le_r : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b (min_bi b) / 2 * ↑y ≤ ↑(r i y)\nh_exp : ∀ (y : ℕ), n₀ ≤ y → ⌈rexp 1⌉₊ ≤ y\nh_base_nonempty : (Ico ⌊b (min_bi b) / 2 * ↑n₀⌋₊ n₀).Nonempty\nbase_min : ℝ := (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)\nbase_min_def : base_min = (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)\nC : ℝ := min (2 * c₁)⁻¹ base_min\nhC_pos : 0 < C\nn : ℕ\nhn : n ≥ n₀\nh_base : ∀ n ∈ Ico ⌊b' * ↑n₀⌋₊ n₀, C * ((1 + ε ↑n) * asympBound g a b n) ≤ T n\nh_asympBound_pos' : 0 < asympBound g a b n\nh_one_sub_smoothingFn_pos' : 0 < 1 + ε ↑n\n⊢ C * ‖(1 + ε ↑n) * asympBound g a b n‖ ≤ ‖T n‖","tactic":"have h_one_sub_smoothingFn_pos' : 0 < 1 + ε n := h_smoothing_pos n hn","premises":[{"full_name":"AkraBazziRecurrence.smoothingFn","def_path":"Mathlib/Computability/AkraBazzi/AkraBazzi.lean","def_pos":[297,18],"def_end_pos":[297,29]}]},{"state_before":"case h\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := b (min_bi b) / 2\nhb_pos : 0 < b'\nc₁ : ℝ\nhc₁ : c₁ > 0\nh_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n\nn₀ : ℕ\nn₀_ge_Rn₀ : R.n₀ ≤ n₀\nh_b_floor : 0 < ⌊b' * ↑n₀⌋₊\nh_smoothing_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < 1 + ε ↑y\nh_smoothing_pos' : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_asympBound_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < asympBound g a b y\nh_asympBound_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < asympBound g a b (r i y)\nh_asympBound_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < asympBound g a b y\nn₀_pos : 0 < n₀\nh_smoothing_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < 1 + ε ↑(r i y)\nbound2 : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b i ^ p a b * ↑y ^ p a b * (1 + ε ↑y) ≤ ↑(r i y) ^ p a b * (1 + ε ↑(r i y))\nh_smoothingFn_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_sumTransform : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), sumTransform (p a b) g (r i y) y ≤ c₁ * g ↑y\nh_bi_le_r : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b (min_bi b) / 2 * ↑y ≤ ↑(r i y)\nh_exp : ∀ (y : ℕ), n₀ ≤ y → ⌈rexp 1⌉₊ ≤ y\nh_base_nonempty : (Ico ⌊b (min_bi b) / 2 * ↑n₀⌋₊ n₀).Nonempty\nbase_min : ℝ := (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)\nbase_min_def : base_min = (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)\nC : ℝ := min (2 * c₁)⁻¹ base_min\nhC_pos : 0 < C\nn : ℕ\nhn : n ≥ n₀\nh_base : ∀ n ∈ Ico ⌊b' * ↑n₀⌋₊ n₀, C * ((1 + ε ↑n) * asympBound g a b n) ≤ T n\nh_asympBound_pos' : 0 < asympBound g a b n\nh_one_sub_smoothingFn_pos' : 0 < 1 + ε ↑n\n⊢ C * ‖(1 + ε ↑n) * asympBound g a b n‖ ≤ ‖T n‖","state_after":"case h\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := b (min_bi b) / 2\nhb_pos : 0 < b'\nc₁ : ℝ\nhc₁ : c₁ > 0\nh_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n\nn₀ : ℕ\nn₀_ge_Rn₀ : R.n₀ ≤ n₀\nh_b_floor : 0 < ⌊b' * ↑n₀⌋₊\nh_smoothing_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < 1 + ε ↑y\nh_smoothing_pos' : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_asympBound_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < asympBound g a b y\nh_asympBound_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < asympBound g a b (r i y)\nh_asympBound_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < asympBound g a b y\nn₀_pos : 0 < n₀\nh_smoothing_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < 1 + ε ↑(r i y)\nbound2 : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b i ^ p a b * ↑y ^ p a b * (1 + ε ↑y) ≤ ↑(r i y) ^ p a b * (1 + ε ↑(r i y))\nh_smoothingFn_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_sumTransform : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), sumTransform (p a b) g (r i y) y ≤ c₁ * g ↑y\nh_bi_le_r : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b (min_bi b) / 2 * ↑y ≤ ↑(r i y)\nh_exp : ∀ (y : ℕ), n₀ ≤ y → ⌈rexp 1⌉₊ ≤ y\nh_base_nonempty : (Ico ⌊b (min_bi b) / 2 * ↑n₀⌋₊ n₀).Nonempty\nbase_min : ℝ := (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)\nbase_min_def : base_min = (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)\nC : ℝ := min (2 * c₁)⁻¹ base_min\nhC_pos : 0 < C\nn : ℕ\nhn : n ≥ n₀\nh_base : ∀ n ∈ Ico ⌊b' * ↑n₀⌋₊ n₀, C * ((1 + ε ↑n) * asympBound g a b n) ≤ T n\nh_asympBound_pos' : 0 < asympBound g a b n\nh_one_sub_smoothingFn_pos' : 0 < 1 + ε ↑n\n⊢ C * ((1 + ε ↑n) * asympBound g a b n) ≤ T n","tactic":"rw [Real.norm_of_nonneg (R.T_nonneg n), Real.norm_of_nonneg (by positivity)]","premises":[{"full_name":"AkraBazziRecurrence.T_nonneg","def_path":"Mathlib/Computability/AkraBazzi/AkraBazzi.lean","def_pos":[278,6],"def_end_pos":[278,14]},{"full_name":"Real.norm_of_nonneg","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[1138,8],"def_end_pos":[1138,22]}]}]}
+{"url":"Mathlib/LinearAlgebra/Lagrange.lean","commit":"","full_name":"Lagrange.interpolate_singleton","start":[304,0],"end":[305,65],"file_path":"Mathlib/LinearAlgebra/Lagrange.lean","tactics":[{"state_before":"F : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns t : Finset ι\ni j : ι\nv r r' : ι → F\n⊢ (interpolate {i} v) r = C (r i)","state_after":"no goals","tactic":"rw [interpolate_apply, sum_singleton, basis_singleton, mul_one]","premises":[{"full_name":"Finset.sum_singleton","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[353,2],"def_end_pos":[353,13]},{"full_name":"Lagrange.basis_singleton","def_path":"Mathlib/LinearAlgebra/Lagrange.lean","def_pos":[201,8],"def_end_pos":[201,23]},{"full_name":"Lagrange.interpolate_apply","def_path":"Mathlib/LinearAlgebra/Lagrange.lean","def_pos":[286,2],"def_end_pos":[286,7]},{"full_name":"mul_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[413,8],"def_end_pos":[413,15]}]}]}
+{"url":"Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean","commit":"","full_name":"quadraticCharFun_eq_zero_iff","start":[62,0],"end":[68,69],"file_path":"Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean","tactics":[{"state_before":"F : Type u_1\ninst✝² : Field F\ninst✝¹ : Fintype F\ninst✝ : DecidableEq F\na : F\n⊢ quadraticCharFun F a = 0 ↔ a = 0","state_after":"F : Type u_1\ninst✝² : Field F\ninst✝¹ : Fintype F\ninst✝ : DecidableEq F\na : F\n⊢ (if a = 0 then 0 else if IsSquare a then 1 else -1) = 0 ↔ a = 0","tactic":"simp only [quadraticCharFun]","premises":[{"full_name":"quadraticCharFun","def_path":"Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean","def_pos":[41,4],"def_end_pos":[41,20]}]},{"state_before":"F : Type u_1\ninst✝² : Field F\ninst✝¹ : Fintype F\ninst✝ : DecidableEq F\na : F\n⊢ (if a = 0 then 0 else if IsSquare a then 1 else -1) = 0 ↔ a = 0","state_after":"case pos\nF : Type u_1\ninst✝² : Field F\ninst✝¹ : Fintype F\ninst✝ : DecidableEq F\na : F\nha : a = 0\n⊢ (if a = 0 then 0 else if IsSquare a then 1 else -1) = 0 ↔ a = 0\n\ncase neg\nF : Type u_1\ninst✝² : Field F\ninst✝¹ : Fintype F\ninst✝ : DecidableEq F\na : F\nha : ¬a = 0\n⊢ (if a = 0 then 0 else if IsSquare a then 1 else -1) = 0 ↔ a = 0","tactic":"by_cases ha : a = 0","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
+{"url":"Mathlib/Algebra/Order/Floor.lean","commit":"","full_name":"Nat.floor_lt_one","start":[134,0],"end":[135,45],"file_path":"Mathlib/Algebra/Order/Floor.lean","tactics":[{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : LinearOrderedSemiring α\ninst✝ : FloorSemiring α\na : α\nn : ℕ\nha : 0 ≤ a\n⊢ a < ↑1 ↔ a < 1","state_after":"no goals","tactic":"rw [Nat.cast_one]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Nat.cast_one","def_path":"Mathlib/Data/Nat/Cast/Defs.lean","def_pos":[133,8],"def_end_pos":[133,16]}]}]}
+{"url":"Mathlib/RingTheory/FractionalIdeal/Norm.lean","commit":"","full_name":"FractionalIdeal.absNorm_eq_zero_iff","start":[90,0],"end":[95,33],"file_path":"Mathlib/RingTheory/FractionalIdeal/Norm.lean","tactics":[{"state_before":"R : Type u_1\ninst✝⁷ : CommRing R\ninst✝⁶ : IsDedekindDomain R\ninst✝⁵ : Module.Free ℤ R\ninst✝⁴ : Module.Finite ℤ R\nK : Type u_2\ninst✝³ : CommRing K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : NoZeroDivisors K\nI : FractionalIdeal R⁰ K\n⊢ absNorm I = 0 ↔ I = 0","state_after":"R : Type u_1\ninst✝⁷ : CommRing R\ninst✝⁶ : IsDedekindDomain R\ninst✝⁵ : Module.Free ℤ R\ninst✝⁴ : Module.Finite ℤ R\nK : Type u_2\ninst✝³ : CommRing K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : NoZeroDivisors K\nI : FractionalIdeal R⁰ K\nh : absNorm I = 0\n⊢ I.num = ⊥","tactic":"refine ⟨fun h ↦ zero_of_num_eq_bot zero_not_mem_nonZeroDivisors ?_, fun h ↦ h ▸ absNorm_bot⟩","premises":[{"full_name":"FractionalIdeal.absNorm_bot","def_path":"Mathlib/RingTheory/FractionalIdeal/Norm.lean","def_pos":[86,8],"def_end_pos":[86,19]},{"full_name":"FractionalIdeal.zero_of_num_eq_bot","def_path":"Mathlib/RingTheory/FractionalIdeal/Basic.lean","def_pos":[331,8],"def_end_pos":[331,26]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"zero_not_mem_nonZeroDivisors","def_path":"Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean","def_pos":[159,8],"def_end_pos":[159,36]}]},{"state_before":"R : Type u_1\ninst✝⁷ : CommRing R\ninst✝⁶ : IsDedekindDomain R\ninst✝⁵ : Module.Free ℤ R\ninst✝⁴ : Module.Finite ℤ R\nK : Type u_2\ninst✝³ : CommRing K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : NoZeroDivisors K\nI : FractionalIdeal R⁰ K\nh : absNorm I = 0\n⊢ I.num = ⊥","state_after":"R : Type u_1\ninst✝⁷ : CommRing R\ninst✝⁶ : IsDedekindDomain R\ninst✝⁵ : Module.Free ℤ R\ninst✝⁴ : Module.Finite ℤ R\nK : Type u_2\ninst✝³ : CommRing K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : NoZeroDivisors K\nI : FractionalIdeal R⁰ K\nh : ↑(Ideal.absNorm I.num) = 0 ∨ ↑|(Algebra.norm ℤ) ↑I.den| = 0\n⊢ I.num = ⊥","tactic":"rw [absNorm_eq, div_eq_zero_iff] at h","premises":[{"full_name":"FractionalIdeal.absNorm_eq","def_path":"Mathlib/RingTheory/FractionalIdeal/Norm.lean","def_pos":[75,8],"def_end_pos":[75,18]},{"full_name":"div_eq_zero_iff","def_path":"Mathlib/Algebra/GroupWithZero/Units/Basic.lean","def_pos":[246,8],"def_end_pos":[246,23]}]},{"state_before":"R : Type u_1\ninst✝⁷ : CommRing R\ninst✝⁶ : IsDedekindDomain R\ninst✝⁵ : Module.Free ℤ R\ninst✝⁴ : Module.Finite ℤ R\nK : Type u_2\ninst✝³ : CommRing K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : NoZeroDivisors K\nI : FractionalIdeal R⁰ K\nh : ↑(Ideal.absNorm I.num) = 0 ∨ ↑|(Algebra.norm ℤ) ↑I.den| = 0\n⊢ I.num = ⊥","state_after":"R : Type u_1\ninst✝⁷ : CommRing R\ninst✝⁶ : IsDedekindDomain R\ninst✝⁵ : Module.Free ℤ R\ninst✝⁴ : Module.Finite ℤ R\nK : Type u_2\ninst✝³ : CommRing K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : NoZeroDivisors K\nI : FractionalIdeal R⁰ K\nh : ↑(Ideal.absNorm I.num) = 0 ∨ ↑|(Algebra.norm ℤ) ↑I.den| = 0\n⊢ ¬↑|(Algebra.norm ℤ) ↑I.den| = 0","tactic":"refine Ideal.absNorm_eq_zero_iff.mp <| Nat.cast_eq_zero.mp <| h.resolve_right ?_","premises":[{"full_name":"Ideal.absNorm_eq_zero_iff","def_path":"Mathlib/RingTheory/Ideal/Norm.lean","def_pos":[340,8],"def_end_pos":[340,27]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Nat.cast_eq_zero","def_path":"Mathlib/Algebra/CharZero/Defs.lean","def_pos":[73,8],"def_end_pos":[73,20]},{"full_name":"Or.resolve_right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[556,8],"def_end_pos":[556,24]}]},{"state_before":"R : Type u_1\ninst✝⁷ : CommRing R\ninst✝⁶ : IsDedekindDomain R\ninst✝⁵ : Module.Free ℤ R\ninst✝⁴ : Module.Finite ℤ R\nK : Type u_2\ninst✝³ : CommRing K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : NoZeroDivisors K\nI : FractionalIdeal R⁰ K\nh : ↑(Ideal.absNorm I.num) = 0 ∨ ↑|(Algebra.norm ℤ) ↑I.den| = 0\n⊢ ¬↑|(Algebra.norm ℤ) ↑I.den| = 0","state_after":"no goals","tactic":"simp [Algebra.norm_eq_zero_iff]","premises":[{"full_name":"Algebra.norm_eq_zero_iff","def_path":"Mathlib/RingTheory/Norm/Basic.lean","def_pos":[93,8],"def_end_pos":[93,24]}]}]}
+{"url":"Mathlib/GroupTheory/Perm/Cycle/Factors.lean","commit":"","full_name":"Equiv.Perm.isCycleOn_support_cycleOf","start":[251,0],"end":[262,35],"file_path":"Mathlib/GroupTheory/Perm/Cycle/Factors.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\nf✝ g : Perm α\nx✝ y : α\ninst✝³ : DecidableRel f✝.SameCycle\ninst✝² : DecidableRel g.SameCycle\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf : Perm α\nx : α\n⊢ ∀ (a : α), f a ∈ ↑(f.cycleOf x).support ↔ a ∈ ↑(f.cycleOf x).support","state_after":"case refine_1\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nf✝ g : Perm α\nx✝¹ y : α\ninst✝³ : DecidableRel f✝.SameCycle\ninst✝² : DecidableRel g.SameCycle\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf : Perm α\nx x✝ : α\nh : f x✝ ∈ ↑(f.cycleOf x).support\n⊢ f.SameCycle x x✝ ∧ x ∈ f.support\n\ncase refine_2\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nf✝ g : Perm α\nx✝¹ y : α\ninst✝³ : DecidableRel f✝.SameCycle\ninst✝² : DecidableRel g.SameCycle\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf : Perm α\nx x✝ : α\nh : x✝ ∈ ↑(f.cycleOf x).support\n⊢ f.SameCycle x (f x✝) ∧ x ∈ f.support","tactic":"refine fun _ ↦ ⟨fun h ↦ mem_support_cycleOf_iff.2 ?_, fun h ↦ mem_support_cycleOf_iff.2 ?_⟩","premises":[{"full_name":"Equiv.Perm.mem_support_cycleOf_iff","def_path":"Mathlib/GroupTheory/Perm/Cycle/Factors.lean","def_pos":[215,8],"def_end_pos":[215,31]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]}]},{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\nf✝ g : Perm α\nx✝ y : α\ninst✝³ : DecidableRel f✝.SameCycle\ninst✝² : DecidableRel g.SameCycle\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf : Perm α\nx a : α\nha : a ∈ ↑(f.cycleOf x).support\nb : α\nhb : b ∈ ↑(f.cycleOf x).support\n⊢ f.SameCycle a b","state_after":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\nf✝ g : Perm α\nx✝ y : α\ninst✝³ : DecidableRel f✝.SameCycle\ninst✝² : DecidableRel g.SameCycle\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf : Perm α\nx a : α\nha : f.SameCycle x a ∧ x ∈ f.support\nb : α\nhb : f.SameCycle x b ∧ x ∈ f.support\n⊢ f.SameCycle a b","tactic":"rw [mem_coe, mem_support_cycleOf_iff] at ha hb","premises":[{"full_name":"Equiv.Perm.mem_support_cycleOf_iff","def_path":"Mathlib/GroupTheory/Perm/Cycle/Factors.lean","def_pos":[215,8],"def_end_pos":[215,31]},{"full_name":"Finset.mem_coe","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[195,8],"def_end_pos":[195,15]}]},{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\nf✝ g : Perm α\nx✝ y : α\ninst✝³ : DecidableRel f✝.SameCycle\ninst✝² : DecidableRel g.SameCycle\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf : Perm α\nx a : α\nha : f.SameCycle x a ∧ x ∈ f.support\nb : α\nhb : f.SameCycle x b ∧ x ∈ f.support\n⊢ f.SameCycle a b","state_after":"no goals","tactic":"exact ha.1.symm.trans hb.1","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"Equiv.Perm.SameCycle.symm","def_path":"Mathlib/GroupTheory/Perm/Cycle/Basic.lean","def_pos":[62,8],"def_end_pos":[62,22]},{"full_name":"Equiv.Perm.SameCycle.trans","def_path":"Mathlib/GroupTheory/Perm/Cycle/Basic.lean","def_pos":[69,8],"def_end_pos":[69,23]}]}]}
+{"url":"Mathlib/GroupTheory/Torsion.lean","commit":"","full_name":"CommMonoid.primaryComponent_coe","start":[182,0],"end":[196,21],"file_path":"Mathlib/GroupTheory/Torsion.lean","tactics":[{"state_before":"G : Type u_1\nH : Type u_2\ninst✝ : CommMonoid G\np : ℕ\nhp : Fact (Nat.Prime p)\n⊢ orderOf 1 = p ^ 0","state_after":"no goals","tactic":"rw [pow_zero, orderOf_one]","premises":[{"full_name":"orderOf_one","def_path":"Mathlib/GroupTheory/OrderOfElement.lean","def_pos":[198,8],"def_end_pos":[198,19]},{"full_name":"pow_zero","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[563,8],"def_end_pos":[563,16]}]},{"state_before":"G : Type u_1\nH : Type u_2\ninst✝ : CommMonoid G\np : ℕ\nhp : Fact (Nat.Prime p)\na✝ b✝ : G\nhg₁ : a✝ ∈ {g | ∃ n, orderOf g = p ^ n}\nhg₂ : b✝ ∈ {g | ∃ n, orderOf g = p ^ n}\n⊢ ∃ m, (a✝ * b✝) ^ p ^ m = 1","state_after":"case intro\nG : Type u_1\nH : Type u_2\ninst✝ : CommMonoid G\np : ℕ\nhp : Fact (Nat.Prime p)\na✝ b✝ : G\nhg₁ : a✝ ∈ {g | ∃ n, orderOf g = p ^ n}\nhg₂ : b✝ ∈ {g | ∃ n, orderOf g = p ^ n}\nm : ℕ\nhm : a✝ ^ p ^ m = 1\n⊢ ∃ m, (a✝ * b✝) ^ p ^ m = 1","tactic":"obtain ⟨m, hm⟩ := exists_orderOf_eq_prime_pow_iff.mp hg₁","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"exists_orderOf_eq_prime_pow_iff","def_path":"Mathlib/GroupTheory/OrderOfElement.lean","def_pos":[446,8],"def_end_pos":[446,39]}]},{"state_before":"case intro\nG : Type u_1\nH : Type u_2\ninst✝ : CommMonoid G\np : ℕ\nhp : Fact (Nat.Prime p)\na✝ b✝ : G\nhg₁ : a✝ ∈ {g | ∃ n, orderOf g = p ^ n}\nhg₂ : b✝ ∈ {g | ∃ n, orderOf g = p ^ n}\nm : ℕ\nhm : a✝ ^ p ^ m = 1\n⊢ ∃ m, (a✝ * b✝) ^ p ^ m = 1","state_after":"case intro.intro\nG : Type u_1\nH : Type u_2\ninst✝ : CommMonoid G\np : ℕ\nhp : Fact (Nat.Prime p)\na✝ b✝ : G\nhg₁ : a✝ ∈ {g | ∃ n, orderOf g = p ^ n}\nhg₂ : b✝ ∈ {g | ∃ n, orderOf g = p ^ n}\nm : ℕ\nhm : a✝ ^ p ^ m = 1\nn : ℕ\nhn : b✝ ^ p ^ n = 1\n⊢ ∃ m, (a✝ * b✝) ^ p ^ m = 1","tactic":"obtain ⟨n, hn⟩ := exists_orderOf_eq_prime_pow_iff.mp hg₂","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"exists_orderOf_eq_prime_pow_iff","def_path":"Mathlib/GroupTheory/OrderOfElement.lean","def_pos":[446,8],"def_end_pos":[446,39]}]},{"state_before":"case intro.intro\nG : Type u_1\nH : Type u_2\ninst✝ : CommMonoid G\np : ℕ\nhp : Fact (Nat.Prime p)\na✝ b✝ : G\nhg₁ : a✝ ∈ {g | ∃ n, orderOf g = p ^ n}\nhg₂ : b✝ ∈ {g | ∃ n, orderOf g = p ^ n}\nm : ℕ\nhm : a✝ ^ p ^ m = 1\nn : ℕ\nhn : b✝ ^ p ^ n = 1\n⊢ ∃ m, (a✝ * b✝) ^ p ^ m = 1","state_after":"no goals","tactic":"exact\n        ⟨m + n, by\n          rw [mul_pow, pow_add, pow_mul, hm, one_pow, Monoid.one_mul, mul_comm, pow_mul, hn,\n            one_pow]⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]},{"full_name":"mul_pow","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[222,31],"def_end_pos":[222,38]},{"full_name":"one_pow","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[593,38],"def_end_pos":[593,45]},{"full_name":"pow_add","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[598,6],"def_end_pos":[598,13]},{"full_name":"pow_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[605,31],"def_end_pos":[605,38]}]}]}
+{"url":"Mathlib/Order/Filter/AtTopBot.lean","commit":"","full_name":"Filter.inf_map_atTop_neBot_iff","start":[434,0],"end":[436,80],"file_path":"Mathlib/Order/Filter/AtTopBot.lean","tactics":[{"state_before":"ι : Type u_1\nι' : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝¹ : SemilatticeSup α\ninst✝ : Nonempty α\nF : Filter β\nu : α → β\n⊢ (F ⊓ map u atTop).NeBot ↔ ∀ U ∈ F, ∀ (N : α), ∃ n ≥ N, u n ∈ U","state_after":"ι : Type u_1\nι' : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝¹ : SemilatticeSup α\ninst✝ : Nonempty α\nF : Filter β\nu : α → β\n⊢ (∀ {p : β → Prop}, (∀ᶠ (x : β) in F, p x) → ∀ (a : α), ∃ b ≥ a, p (u b)) ↔ ∀ U ∈ F, ∀ (N : α), ∃ n ≥ N, u n ∈ U","tactic":"simp_rw [inf_neBot_iff_frequently_left, frequently_map, frequently_atTop]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Filter.frequently_atTop","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[310,8],"def_end_pos":[310,24]},{"full_name":"Filter.frequently_map","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1654,8],"def_end_pos":[1654,22]},{"full_name":"Filter.inf_neBot_iff_frequently_left","def_path":"Mathlib/Order/Filter/Bases.lean","def_pos":[653,8],"def_end_pos":[653,37]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]}]},{"state_before":"ι : Type u_1\nι' : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝¹ : SemilatticeSup α\ninst✝ : Nonempty α\nF : Filter β\nu : α → β\n⊢ (∀ {p : β → Prop}, (∀ᶠ (x : β) in F, p x) → ∀ (a : α), ∃ b ≥ a, p (u b)) ↔ ∀ U ∈ F, ∀ (N : α), ∃ n ≥ N, u n ∈ U","state_after":"no goals","tactic":"rfl","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]}]}]}
+{"url":"Mathlib/GroupTheory/Perm/Cycle/Basic.lean","commit":"","full_name":"Equiv.Perm.sameCycle_apply_left","start":[107,0],"end":[110,62],"file_path":"Mathlib/GroupTheory/Perm/Cycle/Basic.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\nf g : Perm α\np : α → Prop\nx y z : α\n⊢ (∃ b, (f ^ (Equiv.symm (Equiv.addRight 1)) b) (f x) = y) ↔ f.SameCycle x y","state_after":"no goals","tactic":"simp [zpow_sub, SameCycle, Int.add_neg_one, Function.comp]","premises":[{"full_name":"Equiv.Perm.SameCycle","def_path":"Mathlib/GroupTheory/Perm/Cycle/Basic.lean","def_pos":[49,4],"def_end_pos":[49,13]},{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"Int.add_neg_one","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean","def_pos":[89,8],"def_end_pos":[89,19]},{"full_name":"zpow_sub","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[832,31],"def_end_pos":[832,39]}]}]}
+{"url":"Mathlib/Algebra/Ring/Subring/Basic.lean","commit":"","full_name":"Subring.comap_map_eq_self","start":[1317,0],"end":[1320,31],"file_path":"Mathlib/Algebra/Ring/Subring/Basic.lean","tactics":[{"state_before":"R : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\nf : R →+* S\ns : Subring R\nh : ⇑f ⁻¹' {0} ⊆ ↑s\n⊢ comap f (map f s) = s","state_after":"case h.e'_3\nR : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\nf : R →+* S\ns : Subring R\nh : ⇑f ⁻¹' {0} ⊆ ↑s\n⊢ s = s ⊔ closure (⇑f ⁻¹' {0})","tactic":"convert comap_map_eq f s","premises":[{"full_name":"Subring.comap_map_eq","def_path":"Mathlib/Algebra/Ring/Subring/Basic.lean","def_pos":[1303,8],"def_end_pos":[1303,20]}]},{"state_before":"case h.e'_3\nR : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\nf : R →+* S\ns : Subring R\nh : ⇑f ⁻¹' {0} ⊆ ↑s\n⊢ s = s ⊔ closure (⇑f ⁻¹' {0})","state_after":"no goals","tactic":"rwa [left_eq_sup, closure_le]","premises":[{"full_name":"Subring.closure_le","def_path":"Mathlib/Algebra/Ring/Subring/Basic.lean","def_pos":[738,8],"def_end_pos":[738,18]},{"full_name":"left_eq_sup","def_path":"Mathlib/Order/Lattice.lean","def_pos":[146,8],"def_end_pos":[146,19]}]}]}
+{"url":"Mathlib/MeasureTheory/Constructions/BorelSpace/Metrizable.lean","commit":"","full_name":"measurableSet_of_tendsto_indicator","start":[137,0],"end":[144,76],"file_path":"Mathlib/MeasureTheory/Constructions/BorelSpace/Metrizable.lean","tactics":[{"state_before":"α✝ : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α✝\nα : Type u_3\ninst✝² : MeasurableSpace α\nA : Set α\nι : Type u_4\nL : Filter ι\ninst✝¹ : L.IsCountablyGenerated\nAs : ι → Set α\ninst✝ : L.NeBot\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nh_lim : ∀ (x : α), ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A\n⊢ MeasurableSet A","state_after":"α✝ : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α✝\nα : Type u_3\ninst✝² : MeasurableSpace α\nA : Set α\nι : Type u_4\nL : Filter ι\ninst✝¹ : L.IsCountablyGenerated\nAs : ι → Set α\ninst✝ : L.NeBot\nh_lim : ∀ (x : α), ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A\nAs_mble : ∀ (i : ι), Measurable ((As i).indicator fun x => 1)\n⊢ Measurable (A.indicator fun x => 1)","tactic":"simp_rw [← measurable_indicator_const_iff (1 : ℝ≥0∞)] at As_mble ⊢","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"ENNReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[96,4],"def_end_pos":[96,11]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"measurable_indicator_const_iff","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Basic.lean","def_pos":[294,6],"def_end_pos":[294,36]}]},{"state_before":"α✝ : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α✝\nα : Type u_3\ninst✝² : MeasurableSpace α\nA : Set α\nι : Type u_4\nL : Filter ι\ninst✝¹ : L.IsCountablyGenerated\nAs : ι → Set α\ninst✝ : L.NeBot\nh_lim : ∀ (x : α), ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A\nAs_mble : ∀ (i : ι), Measurable ((As i).indicator fun x => 1)\n⊢ Measurable (A.indicator fun x => 1)","state_after":"no goals","tactic":"exact ENNReal.measurable_of_tendsto' L As_mble\n    ((tendsto_indicator_const_iff_forall_eventually L (1 : ℝ≥0∞)).mpr h_lim)","premises":[{"full_name":"ENNReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[96,4],"def_end_pos":[96,11]},{"full_name":"ENNReal.measurable_of_tendsto'","def_path":"Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean","def_pos":[272,8],"def_end_pos":[272,30]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"tendsto_indicator_const_iff_forall_eventually","def_path":"Mathlib/Topology/IndicatorConstPointwise.lean","def_pos":[102,14],"def_end_pos":[102,59]}]}]}
+{"url":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","commit":"","full_name":"lt_add_iff_pos_left","start":[469,0],"end":[472,54],"file_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝³ : MulOneClass α\ninst✝² : LT α\ninst✝¹ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst✝ : ContravariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b : α\n⊢ a < b * a ↔ 1 * a < b * a","state_after":"no goals","tactic":"rw [one_mul]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"one_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[409,8],"def_end_pos":[409,15]}]}]}
+{"url":"Mathlib/Order/Height.lean","commit":"","full_name":"Set.singleton_mem_subchain_iff","start":[71,0],"end":[72,96],"file_path":"Mathlib/Order/Height.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl : List α\na : α\n⊢ [a] ∈ s.subchain ↔ a ∈ s","state_after":"no goals","tactic":"simp [cons_mem_subchain_iff]","premises":[{"full_name":"Set.cons_mem_subchain_iff","def_path":"Mathlib/Order/Height.lean","def_pos":[66,8],"def_end_pos":[66,29]}]}]}
+{"url":"Mathlib/Data/List/Sublists.lean","commit":"","full_name":"List.sublists_map","start":[344,0],"end":[350,37],"file_path":"Mathlib/Data/List/Sublists.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nf : α → β\n⊢ (map f []).sublists = map (map f) [].sublists","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nf : α → β\na : α\nl : List α\n⊢ (map f (a :: l)).sublists = map (map f) (a :: l).sublists","state_after":"α : Type u\nβ : Type v\nγ : Type w\nf : α → β\na : α\nl : List α\n⊢ ((l.sublists.bind fun x => [map f x]).bind fun x => [x, f a :: x]) =\n    (l.sublists.bind fun x => [x, a :: x]).bind fun x => [map f x]","tactic":"rw [map_cons, sublists_cons, bind_eq_bind, sublists_map f l, sublists_cons,\n      bind_eq_bind, map_eq_bind, map_eq_bind]","premises":[{"full_name":"List.bind_eq_bind","def_path":"Mathlib/Data/List/Basic.lean","def_pos":[291,8],"def_end_pos":[291,20]},{"full_name":"List.map_cons","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[366,16],"def_end_pos":[366,24]},{"full_name":"List.map_eq_bind","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean","def_pos":[1281,8],"def_end_pos":[1281,19]},{"full_name":"List.sublists_cons","def_path":"Mathlib/Data/List/Sublists.lean","def_pos":[143,8],"def_end_pos":[143,21]}]},{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nf : α → β\na : α\nl : List α\n⊢ ((l.sublists.bind fun x => [map f x]).bind fun x => [x, f a :: x]) =\n    (l.sublists.bind fun x => [x, a :: x]).bind fun x => [map f x]","state_after":"no goals","tactic":"induction sublists l <;> simp [*]","premises":[{"full_name":"List.sublists","def_path":".lake/packages/batteries/Batteries/Data/List/Basic.lean","def_pos":[535,4],"def_end_pos":[535,12]}]}]}
+{"url":"Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean","commit":"","full_name":"FormalMultilinearSeries.coeff_iterate_fslope","start":[303,0],"end":[307,37],"file_path":"Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean","tactics":[{"state_before":"𝕜 : Type u\n𝕜' : Type u'\nE : Type v\nF : Type w\nG : Type x\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\np : FormalMultilinearSeries 𝕜 𝕜 E\nn✝ k n : ℕ\n⊢ (fslope^[k] p).coeff n = p.coeff (n + k)","state_after":"no goals","tactic":"induction k generalizing p with\n  | zero => rfl\n  | succ k ih => simp [ih, add_assoc]","premises":[{"full_name":"add_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[258,2],"def_end_pos":[258,13]}]}]}
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s\nhp' : dist p' s.center ≤ s.radius\nh : ¬p' = p\n⊢ Wbtw ℝ p p' (s.secondInter p (p' -ᵥ p))","tactic":"by_cases h : p' = p","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case neg\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\np p' : P\nhp : p ∈ s\nhp' : dist p' s.center ≤ s.radius\nh : ¬p' = p\n⊢ Wbtw ℝ p p' (s.secondInter p (p' -ᵥ p))","state_after":"case neg\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\np p' : P\nhp : p ∈ s\nhp' : dist p' s.center ≤ s.radius\nh : ¬p' = p\n⊢ p ≠ s.secondInter p (p' -ᵥ p)","tactic":"refine\n    wbtw_of_collinear_of_dist_center_le_radius (s.secondInter_collinear p p') hp hp'\n      ((Sphere.secondInter_mem _).2 hp) ?_","premises":[{"full_name":"EuclideanGeometry.Sphere.secondInter_collinear","def_path":"Mathlib/Geometry/Euclidean/Sphere/SecondInter.lean","def_pos":[136,8],"def_end_pos":[136,36]},{"full_name":"EuclideanGeometry.Sphere.secondInter_mem","def_path":"Mathlib/Geometry/Euclidean/Sphere/SecondInter.lean","def_pos":[50,8],"def_end_pos":[50,30]},{"full_name":"EuclideanGeometry.wbtw_of_collinear_of_dist_center_le_radius","def_path":"Mathlib/Geometry/Euclidean/Sphere/Basic.lean","def_pos":[339,8],"def_end_pos":[339,50]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]}]},{"state_before":"case neg\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\np p' : P\nhp : p ∈ s\nhp' : dist p' s.center ≤ s.radius\nh : ¬p' = p\n⊢ p ≠ s.secondInter p (p' -ᵥ p)","state_after":"case neg\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\np p' : P\nhp : p ∈ s\nhp' : dist p' s.center ≤ s.radius\nh : ¬p' = p\nhe : p = s.secondInter p (p' -ᵥ p)\n⊢ False","tactic":"intro he","premises":[]},{"state_before":"case neg\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\np p' : P\nhp : p ∈ s\nhp' : dist p' s.center ≤ s.radius\nh : ¬p' = p\nhe : p = s.secondInter p (p' -ᵥ p)\n⊢ False","state_after":"case neg\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\np p' : P\nhp : p ∈ s\nhp' : dist p' s.center ≤ s.radius\nh : ¬p' = p\nhe : 0 = ⟪p -ᵥ p', p -ᵥ s.center⟫_ℝ\n⊢ False","tactic":"rw [eq_comm, Sphere.secondInter_eq_self_iff, ← neg_neg (p' -ᵥ p), inner_neg_left,\n    neg_vsub_eq_vsub_rev, 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NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\np p' : P\nhp : p ∈ s\nhp' : dist p' s.center ≤ s.radius\nh : ¬p' = p\nhe : 0 = ⟪p -ᵥ p', p -ᵥ s.center⟫_ℝ\n⊢ False","state_after":"no goals","tactic":"exact ((inner_pos_or_eq_of_dist_le_radius hp hp').resolve_right (Ne.symm h)).ne he","premises":[{"full_name":"EuclideanGeometry.inner_pos_or_eq_of_dist_le_radius","def_path":"Mathlib/Geometry/Euclidean/Sphere/Basic.lean","def_pos":[293,8],"def_end_pos":[293,41]},{"full_name":"Ne.symm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[704,8],"def_end_pos":[704,15]},{"full_name":"Or.resolve_right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[556,8],"def_end_pos":[556,24]}]}]}
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+{"url":"Mathlib/Algebra/Field/Subfield.lean","commit":"","full_name":"Subfield.coe_sSup_of_directedOn","start":[699,0],"end":[701,58],"file_path":"Mathlib/Algebra/Field/Subfield.lean","tactics":[{"state_before":"K : Type u\nL : Type v\nM : Type w\ninst✝² : DivisionRing K\ninst✝¹ : DivisionRing L\ninst✝ : DivisionRing M\nS : Set (Subfield K)\nSne : S.Nonempty\nhS : DirectedOn (fun x x_1 => x ≤ x_1) S\nx : K\n⊢ x ∈ ↑(sSup S) ↔ x ∈ ⋃ s ∈ S, ↑s","state_after":"no goals","tactic":"simp [mem_sSup_of_directedOn Sne hS]","premises":[{"full_name":"Subfield.mem_sSup_of_directedOn","def_path":"Mathlib/Algebra/Field/Subfield.lean","def_pos":[694,8],"def_end_pos":[694,30]}]}]}
+{"url":"Mathlib/RingTheory/Polynomial/Pochhammer.lean","commit":"","full_name":"monic_descPochhammer","start":[239,0],"end":[246,37],"file_path":"Mathlib/RingTheory/Polynomial/Pochhammer.lean","tactics":[{"state_before":"R : Type u\ninst✝² : Ring R\nn : ℕ\ninst✝¹ : Nontrivial R\ninst✝ : NoZeroDivisors R\n⊢ (descPochhammer R n).Monic","state_after":"case zero\nR : Type u\ninst✝² : Ring R\ninst✝¹ : Nontrivial R\ninst✝ : NoZeroDivisors R\n⊢ (descPochhammer R 0).Monic\n\ncase succ\nR : Type u\ninst✝² : Ring R\ninst✝¹ : Nontrivial R\ninst✝ : NoZeroDivisors R\nn : ℕ\nhn : (descPochhammer R n).Monic\n⊢ (descPochhammer R (n + 1)).Monic","tactic":"induction' n with n hn","premises":[]}]}
+{"url":"Mathlib/Data/TypeVec.lean","commit":"","full_name":"TypeVec.eq_of_drop_last_eq","start":[146,0],"end":[152,22],"file_path":"Mathlib/Data/TypeVec.lean","tactics":[{"state_before":"n : ℕ\nα : TypeVec.{u_1} (n + 1)\nβ : TypeVec.{u_2} (n + 1)\nf g : α ⟹ β\nh₀ : dropFun f = dropFun g\nh₁ : lastFun f = lastFun g\n⊢ f = g","state_after":"n : ℕ\nα : TypeVec.{u_1} (n + 1)\nβ : TypeVec.{u_2} (n + 1)\nf g : α ⟹ β\nh₀ : dropFun f = dropFun g\nh₁ : lastFun f = lastFun g\nx : Fin2 (n + 1)\n⊢ f x = g x","tactic":"refine funext (fun x => ?_)","premises":[{"full_name":"funext","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1817,8],"def_end_pos":[1817,14]}]},{"state_before":"n : ℕ\nα : TypeVec.{u_1} (n + 1)\nβ : TypeVec.{u_2} (n + 1)\nf g : α ⟹ β\nh₀ : dropFun f = dropFun g\nh₁ : lastFun f = lastFun g\nx : Fin2 (n + 1)\n⊢ f x = g x","state_after":"case fz\nn : ℕ\nα : TypeVec.{u_1} (n + 1)\nβ : TypeVec.{u_2} (n + 1)\nf g : α ⟹ β\nh₀ : dropFun f = dropFun g\nh₁ : lastFun f = lastFun g\n⊢ f Fin2.fz = g Fin2.fz\n\ncase fs\nn : ℕ\nα : TypeVec.{u_1} (n + 1)\nβ : TypeVec.{u_2} (n + 1)\nf g : α ⟹ β\nh₀ : dropFun f = dropFun g\nh₁ : lastFun f = lastFun g\na✝ : Fin2 n\n⊢ f a✝.fs = g a✝.fs","tactic":"cases x","premises":[]}]}
+{"url":"Mathlib/NumberTheory/Cyclotomic/Basic.lean","commit":"","full_name":"IsCyclotomicExtension.singleton_one_of_bot_eq_top","start":[249,0],"end":[253,6],"file_path":"Mathlib/NumberTheory/Cyclotomic/Basic.lean","tactics":[{"state_before":"n : ℕ+\nS T : Set ℕ+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra A B\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nh : ⊥ = ⊤\n⊢ IsCyclotomicExtension {1} A B","state_after":"case h.e'_1\nn : ℕ+\nS T : Set ℕ+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra A B\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nh : ⊥ = ⊤\n⊢ {1} = ∅ ∪ {1}","tactic":"convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"IsCyclotomicExtension.iff_union_singleton_one","def_path":"Mathlib/NumberTheory/Cyclotomic/Basic.lean","def_pos":[236,8],"def_end_pos":[236,31]},{"full_name":"IsCyclotomicExtension.singleton_zero_of_bot_eq_top","def_path":"Mathlib/NumberTheory/Cyclotomic/Basic.lean","def_pos":[113,8],"def_end_pos":[113,36]}]},{"state_before":"case h.e'_1\nn : ℕ+\nS T : Set ℕ+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra A B\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nh : ⊥ = ⊤\n⊢ {1} = ∅ ∪ {1}","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/Computability/TuringMachine.lean","commit":"","full_name":"Turing.TM2to1.tr_respects_aux","start":[2427,0],"end":[2445,10],"file_path":"Mathlib/Computability/TuringMachine.lean","tactics":[{"state_before":"K : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type u_3\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nq : Stmt₂\nv : σ\nT : ListBlank ((i : K) → Option (Γ i))\nk : K\nS : (k : K) → List (Γ k)\nhT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse\no : StAct k\nIH :\n  ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))},\n    (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) →\n      ∃ b,\n        TrCfg (TM2.stepAux q v S) b ∧\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b\n⊢ ∃ b,\n    TrCfg (TM2.stepAux (stRun o q) v S) b ∧\n      Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal (stRun o q)) v (Tape.mk' ∅ (addBottom T))) b","state_after":"K : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type u_3\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nq : Stmt₂\nv : σ\nT : ListBlank ((i : K) → Option (Γ i))\nk : K\nS : (k : K) → List (Γ k)\nhT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse\no : StAct k\nIH :\n  ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))},\n    (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) →\n      ∃ b,\n        TrCfg (TM2.stepAux q v S) b ∧\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b\n⊢ ∃ b,\n    TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b ∧\n      Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' ∅ (addBottom T))) b","tactic":"simp only [trNormal_run, step_run]","premises":[{"full_name":"Turing.TM2to1.step_run","def_path":"Mathlib/Computability/TuringMachine.lean","def_pos":[2263,8],"def_end_pos":[2263,16]},{"full_name":"Turing.TM2to1.trNormal_run","def_path":"Mathlib/Computability/TuringMachine.lean","def_pos":[2281,8],"def_end_pos":[2281,20]}]},{"state_before":"K : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type u_3\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nq : Stmt₂\nv : σ\nT : ListBlank ((i : K) → Option (Γ i))\nk : K\nS : (k : K) → List (Γ k)\nhT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse\no : StAct k\nIH :\n  ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))},\n    (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) →\n      ∃ b,\n        TrCfg (TM2.stepAux q v S) b ∧\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b\n⊢ ∃ b,\n    TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b ∧\n      Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' ∅ (addBottom T))) b","state_after":"K : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type u_3\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nq : Stmt₂\nv : σ\nT : ListBlank ((i : K) → Option (Γ i))\nk : K\nS : (k : K) → List (Γ k)\nhT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse\no : StAct k\nIH :\n  ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))},\n    (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) →\n      ∃ b,\n        TrCfg (TM2.stepAux q v S) b ∧\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b\nhgo :\n  Reaches₀ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) }\n    { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)) }\n⊢ ∃ b,\n    TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b ∧\n      Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' ∅ (addBottom T))) b","tactic":"have hgo := tr_respects_aux₁ M o q v (hT k) _ le_rfl","premises":[{"full_name":"Turing.TM2to1.tr_respects_aux₁","def_path":"Mathlib/Computability/TuringMachine.lean","def_pos":[2404,8],"def_end_pos":[2404,24]},{"full_name":"le_rfl","def_path":"Mathlib/Order/Defs.lean","def_pos":[43,8],"def_end_pos":[43,14]}]},{"state_before":"K : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type u_3\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nq : Stmt₂\nv : σ\nT : ListBlank ((i : K) → Option (Γ i))\nk : K\nS : (k : K) → List (Γ k)\nhT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse\no : StAct k\nIH :\n  ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))},\n    (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) →\n      ∃ b,\n        TrCfg (TM2.stepAux q v S) b ∧\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b\nhgo :\n  Reaches₀ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) }\n    { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)) }\n⊢ ∃ b,\n    TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b ∧\n      Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' ∅ (addBottom T))) b","state_after":"case intro.intro\nK : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type u_3\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nq : Stmt₂\nv : σ\nT : ListBlank ((i : K) → Option (Γ i))\nk : K\nS : (k : K) → List (Γ k)\nhT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse\no : StAct k\nIH :\n  ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))},\n    (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) →\n      ∃ b,\n        TrCfg (TM2.stepAux q v S) b ∧\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b\nhgo :\n  Reaches₀ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) }\n    { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)) }\nT' : ListBlank ((k : K) → Option (Γ k))\nhT' :\n  ∀ (k_1 : K),\n    ListBlank.map (proj k_1) T' = ListBlank.mk (List.map some (update S k (stWrite ?m.379536 (S k) o) k_1)).reverse\nhrun :\n  TM1.stepAux (trStAct ?m.379535 o) ?m.379536 ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T))) =\n    TM1.stepAux ?m.379535 (stVar ?m.379536 (S k) o)\n      ((Tape.move Dir.right)^[(update S k (stWrite ?m.379536 (S k) o) k).length] (Tape.mk' ∅ (addBottom T')))\n⊢ ∃ b,\n    TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b ∧\n      Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' ∅ (addBottom T))) b","tactic":"obtain ⟨T', hT', hrun⟩ := tr_respects_aux₂ hT o","premises":[{"full_name":"Turing.TM2to1.tr_respects_aux₂","def_path":"Mathlib/Computability/TuringMachine.lean","def_pos":[2300,8],"def_end_pos":[2300,24]}]},{"state_before":"case intro.intro\nK : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type u_3\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nq : Stmt₂\nv : σ\nT : ListBlank ((i : K) → Option (Γ i))\nk : K\nS : (k : K) → List (Γ k)\nhT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse\no : StAct k\nIH :\n  ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))},\n    (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) →\n      ∃ b,\n        TrCfg (TM2.stepAux q v S) b ∧\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b\nhgo :\n  Reaches₀ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) }\n    { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)) }\nT' : ListBlank ((k : K) → Option (Γ k))\nhT' :\n  ∀ (k_1 : K),\n    ListBlank.map (proj k_1) T' = ListBlank.mk (List.map some (update S k (stWrite ?m.379536 (S k) o) k_1)).reverse\nhrun :\n  TM1.stepAux (trStAct ?m.379535 o) ?m.379536 ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T))) =\n    TM1.stepAux ?m.379535 (stVar ?m.379536 (S k) o)\n      ((Tape.move Dir.right)^[(update S k (stWrite ?m.379536 (S k) o) k).length] (Tape.mk' ∅ (addBottom T')))\n⊢ ∃ b,\n    TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b ∧\n      Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' ∅ (addBottom T))) b","state_after":"case intro.intro\nK : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type u_3\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nq : Stmt₂\nv : σ\nT : ListBlank ((i : K) → Option (Γ i))\nk : K\nS : (k : K) → List (Γ k)\nhT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse\no : StAct k\nIH :\n  ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))},\n    (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) →\n      ∃ b,\n        TrCfg (TM2.stepAux q v S) b ∧\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b\nhgo :\n  Reaches₀ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) }\n    { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)) }\nT' : ListBlank ((k : K) → Option (Γ k))\nhT' :\n  ∀ (k_1 : K),\n    ListBlank.map (proj k_1) T' = ListBlank.mk (List.map some (update S k (stWrite ?m.379536 (S k) o) k_1)).reverse\nhrun :\n  TM1.stepAux (trStAct ?m.379535 o) ?m.379536 ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T))) =\n    TM1.stepAux ?m.379535 (stVar ?m.379536 (S k) o)\n      ((Tape.move Dir.right)^[(update S k (stWrite ?m.379536 (S k) o) k).length] (Tape.mk' ∅ (addBottom T')))\nthis :\n  Reaches₁ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) }\n    (TM1.stepAux (tr M (go k o q)) v ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T))))\n⊢ ∃ b,\n    TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b ∧\n      Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' ∅ (addBottom T))) b","tactic":"have := hgo.tail' rfl","premises":[{"full_name":"Turing.Reaches₀.tail'","def_path":"Mathlib/Computability/TuringMachine.lean","def_pos":[718,8],"def_end_pos":[718,22]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"case intro.intro\nK : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type u_3\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nq : Stmt₂\nv : σ\nT : ListBlank ((i : K) → Option (Γ i))\nk : K\nS : (k : K) → List (Γ k)\nhT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse\no : StAct k\nIH :\n  ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))},\n    (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) →\n      ∃ b,\n        TrCfg (TM2.stepAux q v S) b ∧\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b\nhgo :\n  Reaches₀ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) }\n    { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)) }\nT' : ListBlank ((k : K) → Option (Γ k))\nhT' :\n  ∀ (k_1 : K),\n    ListBlank.map (proj k_1) T' = ListBlank.mk (List.map some (update S k (stWrite ?m.379536 (S k) o) k_1)).reverse\nhrun :\n  TM1.stepAux (trStAct ?m.379535 o) ?m.379536 ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T))) =\n    TM1.stepAux ?m.379535 (stVar ?m.379536 (S k) o)\n      ((Tape.move Dir.right)^[(update S k (stWrite ?m.379536 (S k) o) k).length] (Tape.mk' ∅ (addBottom T')))\nthis :\n  Reaches₁ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) }\n    (TM1.stepAux (tr M (go k o q)) v ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T))))\n⊢ ∃ b,\n    TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b ∧\n      Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' ∅ (addBottom T))) b","state_after":"case intro.intro\nK : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type u_3\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nq : Stmt₂\nv : σ\nT : ListBlank ((i : K) → Option (Γ i))\nk : K\nS : (k : K) → List (Γ k)\nhT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse\no : StAct k\nIH :\n  ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))},\n    (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) →\n      ∃ b,\n        TrCfg (TM2.stepAux q v S) b ∧\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b\nhgo :\n  Reaches₀ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) }\n    { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)) }\nT' : ListBlank ((k : K) → Option (Γ k))\nhT' :\n  ∀ (k_1 : K), ListBlank.map (proj k_1) T' = ListBlank.mk (List.map some (update S k (stWrite v (S k) o) k_1)).reverse\nhrun :\n  TM1.stepAux (trStAct (goto fun x x => ret q) o) v ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T))) =\n    TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n      ((Tape.move Dir.right)^[(update S k (stWrite v (S k) o) k).length] (Tape.mk' ∅ (addBottom T')))\nthis :\n  Reaches₁ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) }\n    (match\n      match none with\n      | some val => false\n      | none => true with\n    | true =>\n      TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n        ((Tape.move Dir.right)^[(update S k (stWrite v (S k) o) k).length] (Tape.mk' ∅ (addBottom T')))\n    | false =>\n      TM1.stepAux (goto fun x x => go k o q) v\n        (Tape.move Dir.right ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)))))\n⊢ ∃ b,\n    TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b ∧\n      Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' ∅ (addBottom T))) b","tactic":"rw [tr, TM1.stepAux, Tape.move_right_n_head, Tape.mk'_nth_nat, addBottom_nth_snd,\n    stk_nth_val _ (hT k), List.get?_len_le (le_of_eq (List.length_reverse _)), Option.isNone, cond,\n    hrun, TM1.stepAux] at this","premises":[{"full_name":"List.get?_len_le","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean","def_pos":[139,8],"def_end_pos":[139,19]},{"full_name":"List.length_reverse","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean","def_pos":[1435,16],"def_end_pos":[1435,30]},{"full_name":"Option.isNone","def_path":".lake/packages/lean4/src/lean/Init/Data/Option/Basic.lean","def_pos":[32,14],"def_end_pos":[32,20]},{"full_name":"Turing.TM1.stepAux","def_path":"Mathlib/Computability/TuringMachine.lean","def_pos":[1143,4],"def_end_pos":[1143,11]},{"full_name":"Turing.TM2to1.addBottom_nth_snd","def_path":"Mathlib/Computability/TuringMachine.lean","def_pos":[2157,8],"def_end_pos":[2157,25]},{"full_name":"Turing.TM2to1.stk_nth_val","def_path":"Mathlib/Computability/TuringMachine.lean","def_pos":[2106,8],"def_end_pos":[2106,19]},{"full_name":"Turing.TM2to1.tr","def_path":"Mathlib/Computability/TuringMachine.lean","def_pos":[2388,4],"def_end_pos":[2388,6]},{"full_name":"Turing.Tape.mk'_nth_nat","def_path":"Mathlib/Computability/TuringMachine.lean","def_pos":[559,8],"def_end_pos":[559,24]},{"full_name":"Turing.Tape.move_right_n_head","def_path":"Mathlib/Computability/TuringMachine.lean","def_pos":[581,8],"def_end_pos":[581,30]},{"full_name":"cond","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1006,20],"def_end_pos":[1006,24]},{"full_name":"le_of_eq","def_path":"Mathlib/Order/Defs.lean","def_pos":[60,8],"def_end_pos":[60,16]}]},{"state_before":"case intro.intro\nK : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type u_3\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nq : Stmt₂\nv : σ\nT : ListBlank ((i : K) → Option (Γ i))\nk : K\nS : (k : K) → List (Γ k)\nhT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse\no : StAct k\nIH :\n  ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))},\n    (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) →\n      ∃ b,\n        TrCfg (TM2.stepAux q v S) b ∧\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b\nhgo :\n  Reaches₀ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) }\n    { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)) }\nT' : ListBlank ((k : K) → Option (Γ k))\nhT' :\n  ∀ (k_1 : K), ListBlank.map (proj k_1) T' = ListBlank.mk (List.map some (update S k (stWrite v (S k) o) k_1)).reverse\nhrun :\n  TM1.stepAux (trStAct (goto fun x x => ret q) o) v ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T))) =\n    TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n      ((Tape.move Dir.right)^[(update S k (stWrite v (S k) o) k).length] (Tape.mk' ∅ (addBottom T')))\nthis :\n  Reaches₁ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) }\n    (match\n      match none with\n      | some val => false\n      | none => true with\n    | true =>\n      TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n        ((Tape.move Dir.right)^[(update S k (stWrite v (S k) o) k).length] (Tape.mk' ∅ (addBottom T')))\n    | false =>\n      TM1.stepAux (goto fun x x => go k o q) v\n        (Tape.move Dir.right ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)))))\n⊢ ∃ b,\n    TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b ∧\n      Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' ∅ (addBottom T))) b","state_after":"case intro.intro.intro.intro\nK : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type u_3\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nq : Stmt₂\nv : σ\nT : ListBlank ((i : K) → Option (Γ i))\nk : K\nS : (k : K) → List (Γ k)\nhT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse\no : StAct k\nIH :\n  ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))},\n    (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) →\n      ∃ b,\n        TrCfg (TM2.stepAux q v S) b ∧\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b\nhgo :\n  Reaches₀ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) }\n    { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)) }\nT' : ListBlank ((k : K) → Option (Γ k))\nhT' :\n  ∀ (k_1 : K), ListBlank.map (proj k_1) T' = ListBlank.mk (List.map some (update S k (stWrite v (S k) o) k_1)).reverse\nhrun :\n  TM1.stepAux (trStAct (goto fun x x => ret q) o) v ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T))) =\n    TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n      ((Tape.move Dir.right)^[(update S k (stWrite v (S k) o) k).length] (Tape.mk' ∅ (addBottom T')))\nthis :\n  Reaches₁ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) }\n    (match\n      match none with\n      | some val => false\n      | none => true with\n    | true =>\n      TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n        ((Tape.move Dir.right)^[(update S k (stWrite v (S k) o) k).length] (Tape.mk' ∅ (addBottom T')))\n    | false =>\n      TM1.stepAux (goto fun x x => go k o q) v\n        (Tape.move Dir.right ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)))))\nc : TM1.Cfg Γ' Λ' σ\ngc : TrCfg (TM2.stepAux q ?m.383145 (update S k (stWrite v (S k) o))) c\nrc : Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) ?m.383145 (Tape.mk' ∅ (addBottom T'))) c\n⊢ ∃ b,\n    TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b ∧\n      Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' ∅ (addBottom T))) b","tactic":"obtain ⟨c, gc, rc⟩ := IH hT'","premises":[]},{"state_before":"case intro.intro.intro.intro\nK : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type u_3\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nq : Stmt₂\nv : σ\nT : ListBlank ((i : K) → Option (Γ i))\nk : K\nS : (k : K) → List (Γ k)\nhT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse\no : StAct k\nIH :\n  ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))},\n    (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) →\n      ∃ b,\n        TrCfg (TM2.stepAux q v S) b ∧\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b\nhgo :\n  Reaches₀ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) }\n    { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)) }\nT' : ListBlank ((k : K) → Option (Γ k))\nhT' :\n  ∀ (k_1 : K), ListBlank.map (proj k_1) T' = ListBlank.mk (List.map some (update S k (stWrite v (S k) o) k_1)).reverse\nhrun :\n  TM1.stepAux (trStAct (goto fun x x => ret q) o) v ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T))) =\n    TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n      ((Tape.move Dir.right)^[(update S k (stWrite v (S k) o) k).length] (Tape.mk' ∅ (addBottom T')))\nthis :\n  Reaches₁ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) }\n    (match\n      match none with\n      | some val => false\n      | none => true with\n    | true =>\n      TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n        ((Tape.move Dir.right)^[(update S k (stWrite v (S k) o) k).length] (Tape.mk' ∅ (addBottom T')))\n    | false =>\n      TM1.stepAux (goto fun x x => go k o q) v\n        (Tape.move Dir.right ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)))))\nc : TM1.Cfg Γ' Λ' σ\ngc : TrCfg (TM2.stepAux q ?m.383145 (update S k (stWrite v (S k) o))) c\nrc : Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) ?m.383145 (Tape.mk' ∅ (addBottom T'))) c\n⊢ ∃ b,\n    TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b ∧\n      Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' ∅ (addBottom T))) b","state_after":"case intro.intro.intro.intro\nK : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type u_3\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nq : Stmt₂\nv : σ\nT : ListBlank ((i : K) → Option (Γ i))\nk : K\nS : (k : K) → List (Γ k)\nhT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse\no : StAct k\nIH :\n  ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))},\n    (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) →\n      ∃ b,\n        TrCfg (TM2.stepAux q v S) b ∧\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b\nhgo :\n  Reaches₀ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) }\n    { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)) }\nT' : ListBlank ((k : K) → Option (Γ k))\nhT' :\n  ∀ (k_1 : K), ListBlank.map (proj k_1) T' = ListBlank.mk (List.map some (update S k (stWrite v (S k) o) k_1)).reverse\nhrun :\n  TM1.stepAux (trStAct (goto fun x x => ret q) o) v ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T))) =\n    TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n      ((Tape.move Dir.right)^[(update S k (stWrite v (S k) o) k).length] (Tape.mk' ∅ (addBottom T')))\nthis :\n  Reaches₁ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) }\n    (match\n      match none with\n      | some val => false\n      | none => true with\n    | true =>\n      TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n        ((Tape.move Dir.right)^[(update S k (stWrite v (S k) o) k).length] (Tape.mk' ∅ (addBottom T')))\n    | false =>\n      TM1.stepAux (goto fun x x => go k o q) v\n        (Tape.move Dir.right ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)))))\nc : TM1.Cfg Γ' Λ' σ\ngc : TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) c\nrc : Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) (stVar v (S k) o) (Tape.mk' ∅ (addBottom T'))) c\n⊢ ReflTransGen (fun a b => b ∈ TM1.step (tr M) a)\n    (TM1.stepAux (tr M (ret q)) (stVar v (S k) o) (Tape.mk' ∅ (addBottom T'))) c","tactic":"refine ⟨c, gc, (this.to₀.trans (tr_respects_aux₃ M _) c (TransGen.head' rfl ?_)).to_reflTransGen⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Relation.TransGen.head'","def_path":"Mathlib/Logic/Relation.lean","def_pos":[349,8],"def_end_pos":[349,13]},{"full_name":"Relation.TransGen.to_reflTransGen","def_path":"Mathlib/Logic/Relation.lean","def_pos":[328,8],"def_end_pos":[328,23]},{"full_name":"Turing.Reaches₀.trans","def_path":"Mathlib/Computability/TuringMachine.lean","def_pos":[690,8],"def_end_pos":[690,22]},{"full_name":"Turing.Reaches₁.to₀","def_path":"Mathlib/Computability/TuringMachine.lean","def_pos":[712,8],"def_end_pos":[712,20]},{"full_name":"Turing.TM2to1.tr_respects_aux₃","def_path":"Mathlib/Computability/TuringMachine.lean","def_pos":[2417,8],"def_end_pos":[2417,24]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"case intro.intro.intro.intro\nK : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type u_3\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nq : Stmt₂\nv : σ\nT : ListBlank ((i : K) → Option (Γ i))\nk : K\nS : (k : K) → List (Γ k)\nhT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse\no : StAct k\nIH :\n  ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))},\n    (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) →\n      ∃ b,\n        TrCfg (TM2.stepAux q v S) b ∧\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b\nhgo :\n  Reaches₀ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) }\n    { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)) }\nT' : ListBlank ((k : K) → Option (Γ k))\nhT' :\n  ∀ (k_1 : K), ListBlank.map (proj k_1) T' = ListBlank.mk (List.map some (update S k (stWrite v (S k) o) k_1)).reverse\nhrun :\n  TM1.stepAux (trStAct (goto fun x x => ret q) o) v ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T))) =\n    TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n      ((Tape.move Dir.right)^[(update S k (stWrite v (S k) o) k).length] (Tape.mk' ∅ (addBottom T')))\nthis :\n  Reaches₁ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) }\n    (match\n      match none with\n      | some val => false\n      | none => true with\n    | true =>\n      TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n        ((Tape.move Dir.right)^[(update S k (stWrite v (S k) o) k).length] (Tape.mk' ∅ (addBottom T')))\n    | false =>\n      TM1.stepAux (goto fun x x => go k o q) v\n        (Tape.move Dir.right ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)))))\nc : TM1.Cfg Γ' Λ' σ\ngc : TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) c\nrc : Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) (stVar v (S k) o) (Tape.mk' ∅ (addBottom T'))) c\n⊢ ReflTransGen (fun a b => b ∈ TM1.step (tr M) a)\n    (TM1.stepAux (tr M (ret q)) (stVar v (S k) o) (Tape.mk' ∅ (addBottom T'))) c","state_after":"case intro.intro.intro.intro\nK : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type u_3\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nq : Stmt₂\nv : σ\nT : ListBlank ((i : K) → Option (Γ i))\nk : K\nS : (k : K) → List (Γ k)\nhT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse\no : StAct k\nIH :\n  ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))},\n    (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) →\n      ∃ b,\n        TrCfg (TM2.stepAux q v S) b ∧\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b\nhgo :\n  Reaches₀ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) }\n    { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)) }\nT' : ListBlank ((k : K) → Option (Γ k))\nhT' :\n  ∀ (k_1 : K), ListBlank.map (proj k_1) T' = ListBlank.mk (List.map some (update S k (stWrite v (S k) o) k_1)).reverse\nhrun :\n  TM1.stepAux (trStAct (goto fun x x => ret q) o) v ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T))) =\n    TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n      ((Tape.move Dir.right)^[(update S k (stWrite v (S k) o) k).length] (Tape.mk' ∅ (addBottom T')))\nthis :\n  Reaches₁ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) }\n    (match\n      match none with\n      | some val => false\n      | none => true with\n    | true =>\n      TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n        ((Tape.move Dir.right)^[(update S k (stWrite v (S k) o) k).length] (Tape.mk' ∅ (addBottom T')))\n    | false =>\n      TM1.stepAux (goto fun x x => go k o q) v\n        (Tape.move Dir.right ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)))))\nc : TM1.Cfg Γ' Λ' σ\ngc : TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) c\nrc : Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) (stVar v (S k) o) (Tape.mk' ∅ (addBottom T'))) c\n⊢ ReflTransGen (fun a b => b ∈ TM1.step (tr M) a)\n    (bif true then TM1.stepAux (trNormal q) (stVar v (S k) o) (Tape.mk' ∅ (addBottom T'))\n    else TM1.stepAux (move Dir.left (goto fun x x => ret q)) (stVar v (S k) o) (Tape.mk' ∅ (addBottom T')))\n    c","tactic":"rw [tr, TM1.stepAux, Tape.mk'_head, addBottom_head_fst]","premises":[{"full_name":"Turing.TM1.stepAux","def_path":"Mathlib/Computability/TuringMachine.lean","def_pos":[1143,4],"def_end_pos":[1143,11]},{"full_name":"Turing.TM2to1.addBottom_head_fst","def_path":"Mathlib/Computability/TuringMachine.lean","def_pos":[2165,8],"def_end_pos":[2165,26]},{"full_name":"Turing.TM2to1.tr","def_path":"Mathlib/Computability/TuringMachine.lean","def_pos":[2388,4],"def_end_pos":[2388,6]},{"full_name":"Turing.Tape.mk'_head","def_path":"Mathlib/Computability/TuringMachine.lean","def_pos":[502,8],"def_end_pos":[502,21]}]},{"state_before":"case intro.intro.intro.intro\nK : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type u_3\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nq : Stmt₂\nv : σ\nT : ListBlank ((i : K) → Option (Γ i))\nk : K\nS : (k : K) → List (Γ k)\nhT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse\no : StAct k\nIH :\n  ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))},\n    (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) →\n      ∃ b,\n        TrCfg (TM2.stepAux q v S) b ∧\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b\nhgo :\n  Reaches₀ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) }\n    { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)) }\nT' : ListBlank ((k : K) → Option (Γ k))\nhT' :\n  ∀ (k_1 : K), ListBlank.map (proj k_1) T' = ListBlank.mk (List.map some (update S k (stWrite v (S k) o) k_1)).reverse\nhrun :\n  TM1.stepAux (trStAct (goto fun x x => ret q) o) v ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T))) =\n    TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n      ((Tape.move Dir.right)^[(update S k (stWrite v (S k) o) k).length] (Tape.mk' ∅ (addBottom T')))\nthis :\n  Reaches₁ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) }\n    (match\n      match none with\n      | some val => false\n      | none => true with\n    | true =>\n      TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n        ((Tape.move Dir.right)^[(update S k (stWrite v (S k) o) k).length] (Tape.mk' ∅ (addBottom T')))\n    | false =>\n      TM1.stepAux (goto fun x x => go k o q) v\n        (Tape.move Dir.right ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)))))\nc : TM1.Cfg Γ' Λ' σ\ngc : TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) c\nrc : Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) (stVar v (S k) o) (Tape.mk' ∅ (addBottom T'))) c\n⊢ ReflTransGen (fun a b => b ∈ TM1.step (tr M) a)\n    (bif true then TM1.stepAux (trNormal q) (stVar v (S k) o) (Tape.mk' ∅ (addBottom T'))\n    else TM1.stepAux (move Dir.left (goto fun x x => ret q)) (stVar v (S k) o) (Tape.mk' ∅ (addBottom T')))\n    c","state_after":"no goals","tactic":"exact rc","premises":[]}]}
+{"url":"Mathlib/InformationTheory/Hamming.lean","commit":"","full_name":"hammingDist_triangle_right","start":[70,0],"end":[74,34],"file_path":"Mathlib/InformationTheory/Hamming.lean","tactics":[{"state_before":"α : Type u_1\nι : Type u_2\nβ : ι → Type u_3\ninst✝² : Fintype ι\ninst✝¹ : (i : ι) → DecidableEq (β i)\nγ : ι → Type u_4\ninst✝ : (i : ι) → DecidableEq (γ i)\nx y z : (i : ι) → β i\n⊢ hammingDist x y ≤ hammingDist x z + hammingDist y z","state_after":"α : Type u_1\nι : Type u_2\nβ : ι → Type u_3\ninst✝² : Fintype ι\ninst✝¹ : (i : ι) → DecidableEq (β i)\nγ : ι → Type u_4\ninst✝ : (i : ι) → DecidableEq (γ i)\nx y z : (i : ι) → β i\n⊢ hammingDist x y ≤ hammingDist x z + hammingDist z y","tactic":"rw [hammingDist_comm y]","premises":[{"full_name":"hammingDist_comm","def_path":"Mathlib/InformationTheory/Hamming.lean","def_pos":[52,8],"def_end_pos":[52,24]}]},{"state_before":"α : Type u_1\nι : Type u_2\nβ : ι → Type u_3\ninst✝² : Fintype ι\ninst✝¹ : (i : ι) → DecidableEq (β i)\nγ : ι → Type u_4\ninst✝ : (i : ι) → DecidableEq (γ i)\nx y z : (i : ι) → β i\n⊢ hammingDist x y ≤ hammingDist x z + hammingDist z y","state_after":"no goals","tactic":"exact hammingDist_triangle _ _ _","premises":[{"full_name":"hammingDist_triangle","def_path":"Mathlib/InformationTheory/Hamming.lean","def_pos":[56,8],"def_end_pos":[56,28]}]}]}
+{"url":"Mathlib/LinearAlgebra/Eigenspace/Basic.lean","commit":"","full_name":"Module.End.HasEigenvector.pow_apply","start":[104,0],"end":[106,80],"file_path":"Mathlib/LinearAlgebra/Eigenspace/Basic.lean","tactics":[{"state_before":"K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End R M\nμ : R\nv : M\nhv : f.HasEigenvector μ v\nn : ℕ\n⊢ (f ^ n) v = μ ^ n • v","state_after":"no goals","tactic":"induction n <;> simp [*, pow_succ f, hv.apply_eq_smul, smul_smul, pow_succ' μ]","premises":[{"full_name":"Module.End.HasEigenvector.apply_eq_smul","def_path":"Mathlib/LinearAlgebra/Eigenspace/Basic.lean","def_pos":[100,8],"def_end_pos":[100,36]},{"full_name":"pow_succ","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[567,8],"def_end_pos":[567,16]},{"full_name":"pow_succ'","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[573,33],"def_end_pos":[573,42]},{"full_name":"smul_smul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[374,6],"def_end_pos":[374,15]}]}]}
+{"url":"Mathlib/Data/Finset/Pointwise.lean","commit":"","full_name":"Finset.image_mul_right'","start":[1009,0],"end":[1011,82],"file_path":"Mathlib/Data/Finset/Pointwise.lean","tactics":[{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝⁵ : DecidableEq α\ninst✝⁴ : DecidableEq β\ninst✝³ : Group α\ninst✝² : DivisionMonoid β\ninst✝¹ : FunLike F α β\ninst✝ : MonoidHomClass F α β\nf : F\ns t : Finset α\na b : α\n⊢ image (fun x => x * b⁻¹) t = t.preimage (fun x => x * b) ⋯","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/Data/List/ProdSigma.lean","commit":"","full_name":"List.length_sigma'","start":[78,0],"end":[83,76],"file_path":"Mathlib/Data/List/ProdSigma.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nσ : α → Type u_3\nl₁ : List α\nl₂ : (a : α) → List (σ a)\n⊢ (l₁.sigma l₂).length = Nat.sum (map (fun a => (l₂ a).length) l₁)","state_after":"case nil\nα : Type u_1\nβ : Type u_2\nσ : α → Type u_3\nl₂ : (a : α) → List (σ a)\n⊢ ([].sigma l₂).length = Nat.sum (map (fun a => (l₂ a).length) [])\n\ncase cons\nα : Type u_1\nβ : Type u_2\nσ : α → Type u_3\nl₂ : (a : α) → List (σ a)\nx : α\nl₁ : List α\nIH : (l₁.sigma l₂).length = Nat.sum (map (fun a => (l₂ a).length) l₁)\n⊢ ((x :: l₁).sigma l₂).length = Nat.sum (map (fun a => (l₂ a).length) (x :: l₁))","tactic":"induction' l₁ with x l₁ IH","premises":[]}]}
+{"url":"Mathlib/Algebra/Group/Units.lean","commit":"","full_name":"IsUnit.isUnit_iff_mulRight_bijective","start":[718,0],"end":[723,33],"file_path":"Mathlib/Algebra/Group/Units.lean","tactics":[{"state_before":"α : Type u\nM : Type u_1\nN : Type u_2\ninst✝ : Monoid M\na✝ b c a : M\nh : IsUnit a\ny : M\n⊢ (fun x => x * a) (y * ↑h.unit⁻¹) = y","state_after":"no goals","tactic":"simp [mul_assoc]","premises":[{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]}]}]}
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+{"url":"Mathlib/LinearAlgebra/TensorProduct/Subalgebra.lean","commit":"","full_name":"Subalgebra.mulMap_comm","start":[151,0],"end":[152,14],"file_path":"Mathlib/LinearAlgebra/TensorProduct/Subalgebra.lean","tactics":[{"state_before":"R : Type u\nS : Type v\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : Algebra R S\nA B : Subalgebra R S\n⊢ B.mulMap A = (A.mulMap B).comp ↑(Algebra.TensorProduct.comm R ↥B ↥A)","state_after":"no goals","tactic":"ext <;> simp","premises":[]}]}
+{"url":"Mathlib/LinearAlgebra/AffineSpace/Combination.lean","commit":"","full_name":"Finset.affineCombination_eq_linear_combination","start":[421,0],"end":[426,79],"file_path":"Mathlib/LinearAlgebra/AffineSpace/Combination.lean","tactics":[{"state_before":"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\nι : Type u_4\ns✝ : Finset ι\nι₂ : Type u_5\ns₂ : Finset ι₂\ns : Finset ι\np : ι → V\nw : ι → k\nhw : ∑ i ∈ s, w i = 1\n⊢ (affineCombination k s p) w = ∑ i ∈ s, w i • p i","state_after":"no goals","tactic":"simp [s.affineCombination_eq_weightedVSubOfPoint_vadd_of_sum_eq_one w p hw 0]","premises":[{"full_name":"Finset.affineCombination_eq_weightedVSubOfPoint_vadd_of_sum_eq_one","def_path":"Mathlib/LinearAlgebra/AffineSpace/Combination.lean","def_pos":[379,8],"def_end_pos":[379,67]}]}]}
+{"url":"Mathlib/SetTheory/Cardinal/Basic.lean","commit":"","full_name":"Cardinal.mk_biUnion_le_lift","start":[1757,0],"end":[1760,25],"file_path":"Mathlib/SetTheory/Cardinal/Basic.lean","tactics":[{"state_before":"α✝ β : Type u\nc : Cardinal.{?u.203972}\nα : Type u\nι : Type v\nA : ι → Set α\ns : Set ι\n⊢ lift.{v, u} #↑(⋃ x ∈ s, A x) ≤ lift.{u, v} #↑s * ⨆ x, lift.{v, u} #↑(A ↑x)","state_after":"α✝ β : Type u\nc : Cardinal.{?u.203972}\nα : Type u\nι : Type v\nA : ι → Set α\ns : Set ι\n⊢ lift.{v, u} #↑(⋃ x, A ↑x) ≤ lift.{u, v} #↑s * ⨆ x, lift.{v, u} #↑(A ↑x)","tactic":"rw [biUnion_eq_iUnion]","premises":[{"full_name":"Set.biUnion_eq_iUnion","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[729,8],"def_end_pos":[729,25]}]},{"state_before":"α✝ β : Type u\nc : Cardinal.{?u.203972}\nα : Type u\nι : Type v\nA : ι → Set α\ns : Set ι\n⊢ lift.{v, u} #↑(⋃ x, A ↑x) ≤ lift.{u, v} #↑s * ⨆ x, lift.{v, u} #↑(A ↑x)","state_after":"no goals","tactic":"apply mk_iUnion_le_lift","premises":[{"full_name":"Cardinal.mk_iUnion_le_lift","def_path":"Mathlib/SetTheory/Cardinal/Basic.lean","def_pos":[1743,8],"def_end_pos":[1743,25]}]}]}
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+{"url":"Mathlib/RingTheory/EssentialFiniteness.lean","commit":"","full_name":"Algebra.EssFiniteType.of_comp","start":[187,0],"end":[195,70],"file_path":"Mathlib/RingTheory/EssentialFiniteness.lean","tactics":[{"state_before":"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : CommRing T\ninst✝³ : Algebra R S\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nh : EssFiniteType R T\n⊢ EssFiniteType S T","state_after":"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : CommRing T\ninst✝³ : Algebra R S\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nh : ∃ σ, ∀ (s : T), ∃ t ∈ adjoin R ↑σ, IsUnit t ∧ s * t ∈ adjoin R ↑σ\n⊢ ∃ σ, ∀ (s : T), ∃ t ∈ adjoin S ↑σ, IsUnit t ∧ s * t ∈ adjoin S ↑σ","tactic":"rw [essFiniteType_iff] at h ⊢","premises":[{"full_name":"Algebra.essFiniteType_iff","def_path":"Mathlib/RingTheory/EssentialFiniteness.lean","def_pos":[80,6],"def_end_pos":[80,23]}]},{"state_before":"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : CommRing T\ninst✝³ : Algebra R S\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nh : ∃ σ, ∀ (s : T), ∃ t ∈ adjoin R ↑σ, IsUnit t ∧ s * t ∈ adjoin R ↑σ\n⊢ ∃ σ, ∀ (s : T), ∃ t ∈ adjoin S ↑σ, IsUnit t ∧ s * t ∈ adjoin S ↑σ","state_after":"no goals","tactic":"classical\n  obtain ⟨σ, hσ⟩ := h\n  use σ\n  intro x\n  obtain ⟨y, hy₁, hy₂, hy₃⟩ := hσ x\n  simp_rw [← Algebra.adjoin_adjoin_of_tower R (S := S) (σ : Set T)]\n  exact ⟨y, Algebra.subset_adjoin hy₁, hy₂, Algebra.subset_adjoin hy₃⟩","premises":[{"full_name":"Algebra.adjoin_adjoin_of_tower","def_path":"Mathlib/RingTheory/Adjoin/Basic.lean","def_pos":[311,8],"def_end_pos":[311,30]},{"full_name":"Algebra.subset_adjoin","def_path":"Mathlib/RingTheory/Adjoin/Basic.lean","def_pos":[44,8],"def_end_pos":[44,21]},{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Set","def_path":"Mathlib/Init/Set.lean","def_pos":[53,4],"def_end_pos":[53,7]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]}]}]}
+{"url":"Mathlib/Analysis/LocallyConvex/WithSeminorms.lean","commit":"","full_name":"SeminormFamily.basisSets_zero","start":[108,0],"end":[111,10],"file_path":"Mathlib/Analysis/LocallyConvex/WithSeminorms.lean","tactics":[{"state_before":"𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕝 : Type u_3\n𝕝₂ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nι : Type u_8\nι' : Type u_9\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nU : Set E\nhU : U ∈ p.basisSets\n⊢ 0 ∈ U","state_after":"case intro.intro.intro\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕝 : Type u_3\n𝕝₂ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nι : Type u_8\nι'✝ : Type u_9\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nU : Set E\nhU✝ : U ∈ p.basisSets\nι' : Finset ι\nr : ℝ\nhr : 0 < r\nhU : U = (ι'.sup p).ball 0 r\n⊢ 0 ∈ U","tactic":"rcases p.basisSets_iff.mp hU with ⟨ι', r, hr, hU⟩","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"SeminormFamily.basisSets_iff","def_path":"Mathlib/Analysis/LocallyConvex/WithSeminorms.lean","def_pos":[77,8],"def_end_pos":[77,21]}]},{"state_before":"case intro.intro.intro\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕝 : Type u_3\n𝕝₂ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nι : Type u_8\nι'✝ : Type u_9\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nU : Set E\nhU✝ : U ∈ p.basisSets\nι' : Finset ι\nr : ℝ\nhr : 0 < r\nhU : U = (ι'.sup p).ball 0 r\n⊢ 0 ∈ U","state_after":"case intro.intro.intro\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕝 : Type u_3\n𝕝₂ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nι : Type u_8\nι'✝ : Type u_9\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nU : Set E\nhU✝ : U ∈ p.basisSets\nι' : Finset ι\nr : ℝ\nhr : 0 < r\nhU : U = (ι'.sup p).ball 0 r\n⊢ 0 < r","tactic":"rw [hU, mem_ball_zero, map_zero]","premises":[{"full_name":"Seminorm.mem_ball_zero","def_path":"Mathlib/Analysis/Seminorm.lean","def_pos":[631,8],"def_end_pos":[631,21]},{"full_name":"map_zero","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[189,2],"def_end_pos":[189,13]}]},{"state_before":"case intro.intro.intro\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕝 : Type u_3\n𝕝₂ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nι : Type u_8\nι'✝ : Type u_9\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nU : Set E\nhU✝ : U ∈ p.basisSets\nι' : Finset ι\nr : ℝ\nhr : 0 < r\nhU : U = (ι'.sup p).ball 0 r\n⊢ 0 < r","state_after":"no goals","tactic":"exact hr","premises":[]}]}
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+{"url":"Mathlib/LinearAlgebra/CliffordAlgebra/Contraction.lean","commit":"","full_name":"CliffordAlgebra.changeForm_self_apply","start":[299,0],"end":[305,15],"file_path":"Mathlib/LinearAlgebra/CliffordAlgebra/Contraction.lean","tactics":[{"state_before":"R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ Q' Q'' : QuadraticForm R M\nB B' : BilinForm R M\nh : BilinMap.toQuadraticMap B = Q' - Q\nh' : BilinMap.toQuadraticMap B' = Q'' - Q'\nx : CliffordAlgebra Q\n⊢ (changeForm ⋯) x = x","state_after":"case algebraMap\nR : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ Q' Q'' : QuadraticForm R M\nB B' : BilinForm R M\nh : BilinMap.toQuadraticMap B = Q' - Q\nh' : BilinMap.toQuadraticMap B' = Q'' - Q'\nr : R\n⊢ (changeForm ⋯) ((algebraMap R (CliffordAlgebra Q)) r) = (algebraMap R (CliffordAlgebra Q)) r\n\ncase add\nR : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ Q' Q'' : QuadraticForm R M\nB B' : BilinForm R M\nh : BilinMap.toQuadraticMap B = Q' - Q\nh' : BilinMap.toQuadraticMap B' = Q'' - Q'\nx y : CliffordAlgebra Q\nhx : (changeForm ⋯) x = x\nhy : (changeForm ⋯) y = y\n⊢ (changeForm ⋯) (x + y) = x + y\n\ncase ι_mul\nR : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ Q' Q'' : QuadraticForm R M\nB B' : BilinForm R M\nh : BilinMap.toQuadraticMap B = Q' - Q\nh' : BilinMap.toQuadraticMap B' = Q'' - Q'\nm : CliffordAlgebra Q\nx : M\nhx : (changeForm ⋯) m = m\n⊢ (changeForm ⋯) ((ι Q) x * m) = (ι Q) x * m","tactic":"induction' x using CliffordAlgebra.left_induction with r x y hx hy m x hx","premises":[{"full_name":"CliffordAlgebra.left_induction","def_path":"Mathlib/LinearAlgebra/CliffordAlgebra/Fold.lean","def_pos":[144,8],"def_end_pos":[144,22]}]}]}
+{"url":"Mathlib/Data/Nat/Cast/Order/Basic.lean","commit":"","full_name":"Nat.mono_cast","start":[31,0],"end":[34,64],"file_path":"Mathlib/Data/Nat/Cast/Order/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝³ : AddMonoidWithOne α\ninst✝² : PartialOrder α\ninst✝¹ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝ : ZeroLEOneClass α\nn : ℕ\n⊢ ↑n ≤ ↑(n + 1)","state_after":"α : Type u_1\nβ : Type u_2\ninst✝³ : AddMonoidWithOne α\ninst✝² : PartialOrder α\ninst✝¹ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝ : ZeroLEOneClass α\nn : ℕ\n⊢ ↑n ≤ ↑n + 1","tactic":"rw [Nat.cast_succ]","premises":[{"full_name":"Nat.cast_succ","def_path":"Mathlib/Data/Nat/Cast/Defs.lean","def_pos":[117,8],"def_end_pos":[117,17]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝³ : AddMonoidWithOne α\ninst✝² : PartialOrder α\ninst✝¹ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝ : ZeroLEOneClass α\nn : ℕ\n⊢ ↑n ≤ ↑n + 1","state_after":"no goals","tactic":"exact le_add_of_nonneg_right zero_le_one","premises":[{"full_name":"le_add_of_nonneg_right","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[331,14],"def_end_pos":[331,36]},{"full_name":"zero_le_one","def_path":"Mathlib/Algebra/Order/ZeroLEOne.lean","def_pos":[23,14],"def_end_pos":[23,25]}]}]}
+{"url":"Mathlib/Algebra/Group/Subgroup/Basic.lean","commit":"","full_name":"MonoidHom.eq_liftOfRightInverse","start":[2586,0],"end":[2591,65],"file_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","tactics":[{"state_before":"G : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst✝⁶ : Group G\ninst✝⁵ : Group G'\ninst✝⁴ : Group G''\nA : Type u_4\ninst✝³ : AddGroup A\nG₁ : Type u_5\nG₂ : Type u_6\nG₃ : Type u_7\ninst✝² : Group G₁\ninst✝¹ : Group G₂\ninst✝ : Group G₃\nf : G₁ →* G₂\nf_inv : G₂ → G₁\nhf : Function.RightInverse f_inv ⇑f\ng : G₁ →* G₃\nhg : f.ker ≤ g.ker\nh : G₂ →* G₃\nhh : h.comp f = g\n⊢ h = (f.liftOfRightInverse f_inv hf) ⟨g, hg⟩","state_after":"G : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst✝⁶ : Group G\ninst✝⁵ : Group G'\ninst✝⁴ : Group G''\nA : Type u_4\ninst✝³ : AddGroup A\nG₁ : Type u_5\nG₂ : Type u_6\nG₃ : Type u_7\ninst✝² : Group G₁\ninst✝¹ : Group G₂\ninst✝ : Group G₃\nf : G₁ →* G₂\nf_inv : G₂ → G₁\nhf : Function.RightInverse f_inv ⇑f\ng : G₁ →* G₃\nhg : f.ker ≤ g.ker\nh : G₂ →* G₃\nhh : h.comp f = g\n⊢ h = (f.liftOfRightInverse f_inv hf) ⟨h.comp f, ⋯⟩","tactic":"simp_rw [← hh]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]}]},{"state_before":"G : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst✝⁶ : Group G\ninst✝⁵ : Group G'\ninst✝⁴ : Group G''\nA : Type u_4\ninst✝³ : AddGroup A\nG₁ : Type u_5\nG₂ : Type u_6\nG₃ : Type u_7\ninst✝² : Group G₁\ninst✝¹ : Group G₂\ninst✝ : Group G₃\nf : G₁ →* G₂\nf_inv : G₂ → G₁\nhf : Function.RightInverse f_inv ⇑f\ng : G₁ →* G₃\nhg : f.ker ≤ g.ker\nh : G₂ →* G₃\nhh : h.comp f = g\n⊢ h = (f.liftOfRightInverse f_inv hf) ⟨h.comp f, ⋯⟩","state_after":"no goals","tactic":"exact ((f.liftOfRightInverse f_inv hf).apply_symm_apply _).symm","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Equiv.apply_symm_apply","def_path":"Mathlib/Logic/Equiv/Defs.lean","def_pos":[239,16],"def_end_pos":[239,32]},{"full_name":"MonoidHom.liftOfRightInverse","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[2555,4],"def_end_pos":[2555,22]}]}]}
+{"url":"Mathlib/Analysis/Convex/Function.lean","commit":"","full_name":"StrictConvexOn.add_convexOn","start":[484,0],"end":[490,92],"file_path":"Mathlib/Analysis/Convex/Function.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nα : Type u_4\nβ : Type u_5\nι : Type u_6\ninst✝⁵ : OrderedSemiring 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid F\ninst✝² : OrderedCancelAddCommMonoid β\ninst✝¹ : SMul 𝕜 E\ninst✝ : DistribMulAction 𝕜 β\ns : Set E\nf g : E → β\nhf : StrictConvexOn 𝕜 s f\nhg : ConvexOn 𝕜 s g\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\nhxy : x ≠ y\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ a • f x + b • f y + (a • g x + b • g y) = a • (f x + g x) + b • (f y + g y)","state_after":"no goals","tactic":"rw [smul_add, smul_add, add_add_add_comm]","premises":[{"full_name":"add_add_add_comm","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[159,2],"def_end_pos":[159,13]},{"full_name":"smul_add","def_path":"Mathlib/Algebra/GroupWithZero/Action/Defs.lean","def_pos":[130,8],"def_end_pos":[130,16]}]}]}
+{"url":".lake/packages/batteries/Batteries/Data/Nat/Lemmas.lean","commit":"","full_name":"Nat.recDiagAux_zero_right","start":[57,0],"end":[61,96],"file_path":".lake/packages/batteries/Batteries/Data/Nat/Lemmas.lean","tactics":[{"state_before":"motive : Nat → Nat → Sort u_1\nzero_left : (n : Nat) → motive 0 n\nzero_right : (m : Nat) → motive m 0\nsucc_succ : (m n : Nat) → motive m n → motive (m + 1) (n + 1)\nm : Nat\nh : autoParam (zero_left 0 = zero_right 0) _auto✝\n⊢ Nat.recDiagAux zero_left zero_right succ_succ m 0 = zero_right m","state_after":"case zero\nmotive : Nat → Nat → Sort u_1\nzero_left : (n : Nat) → motive 0 n\nzero_right : (m : Nat) → motive m 0\nsucc_succ : (m n : Nat) → motive m n → motive (m + 1) (n + 1)\nh : autoParam (zero_left 0 = zero_right 0) _auto✝\n⊢ Nat.recDiagAux zero_left zero_right succ_succ 0 0 = zero_right 0\n\ncase succ\nmotive : Nat → Nat → Sort u_1\nzero_left : (n : Nat) → motive 0 n\nzero_right : (m : Nat) → motive m 0\nsucc_succ : (m n : Nat) → motive m n → motive (m + 1) (n + 1)\nh : autoParam (zero_left 0 = zero_right 0) _auto✝\nn✝ : Nat\n⊢ Nat.recDiagAux zero_left zero_right succ_succ (n✝ + 1) 0 = zero_right (n✝ + 1)","tactic":"cases m","premises":[]},{"state_before":"case zero\nmotive : Nat → Nat → Sort u_1\nzero_left : (n : Nat) → motive 0 n\nzero_right : (m : Nat) → motive m 0\nsucc_succ : (m n : Nat) → motive m n → motive (m + 1) (n + 1)\nh : autoParam (zero_left 0 = zero_right 0) _auto✝\n⊢ Nat.recDiagAux zero_left zero_right succ_succ 0 0 = zero_right 0\n\ncase succ\nmotive : Nat → Nat → Sort u_1\nzero_left : (n : Nat) → motive 0 n\nzero_right : (m : Nat) → motive m 0\nsucc_succ : (m n : Nat) → motive m n → motive (m + 1) (n + 1)\nh : autoParam (zero_left 0 = zero_right 0) _auto✝\nn✝ : Nat\n⊢ Nat.recDiagAux zero_left zero_right succ_succ (n✝ + 1) 0 = zero_right (n✝ + 1)","state_after":"case succ\nmotive : Nat → Nat → Sort u_1\nzero_left : (n : Nat) → motive 0 n\nzero_right : (m : Nat) → motive m 0\nsucc_succ : (m n : Nat) → motive m n → motive (m + 1) (n + 1)\nh : autoParam (zero_left 0 = zero_right 0) _auto✝\nn✝ : Nat\n⊢ Nat.recDiagAux zero_left zero_right succ_succ (n✝ + 1) 0 = zero_right (n✝ + 1)","tactic":"exact h","premises":[]},{"state_before":"case succ\nmotive : Nat → Nat → Sort u_1\nzero_left : (n : Nat) → motive 0 n\nzero_right : (m : Nat) → motive m 0\nsucc_succ : (m n : Nat) → motive m n → motive (m + 1) (n + 1)\nh : autoParam (zero_left 0 = zero_right 0) _auto✝\nn✝ : Nat\n⊢ Nat.recDiagAux zero_left zero_right succ_succ (n✝ + 1) 0 = zero_right (n✝ + 1)","state_after":"no goals","tactic":"rfl","premises":[]}]}
+{"url":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","commit":"","full_name":"WeierstrassCurve.Jacobian.map_negDblY","start":[1605,0],"end":[1605,94],"file_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","tactics":[{"state_before":"R : Type u\ninst✝² : CommRing R\nW' : Jacobian R\nF : Type v\ninst✝¹ : Field F\nW : Jacobian F\nS : Type u_1\ninst✝ : CommRing S\nf : R →+* S\nP Q : Fin 3 → R\n⊢ negDblY (map W' f) (⇑f ∘ P) = f (W'.negDblY P)","state_after":"no goals","tactic":"simp [negDblY]","premises":[{"full_name":"WeierstrassCurve.Jacobian.negDblY","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","def_pos":[668,18],"def_end_pos":[668,25]}]}]}
+{"url":"Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean","commit":"","full_name":"coplanar_iff_finrank_le_two","start":[626,0],"end":[632,18],"file_path":"Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean","tactics":[{"state_before":"k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ns : Set P\ninst✝ : FiniteDimensional k ↥(vectorSpan k s)\n⊢ Coplanar k s ↔ finrank k ↥(vectorSpan k s) ≤ 2","state_after":"k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ns : Set P\ninst✝ : FiniteDimensional k ↥(vectorSpan k s)\nh : Coplanar k s ↔ Module.rank k ↥(vectorSpan k s) ≤ 2\n⊢ Coplanar k s ↔ finrank k ↥(vectorSpan k s) ≤ 2","tactic":"have h : Coplanar k s ↔ Module.rank k (vectorSpan k s) ≤ 2 := Iff.rfl","premises":[{"full_name":"Coplanar","def_path":"Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean","def_pos":[610,4],"def_end_pos":[610,12]},{"full_name":"Iff","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[114,10],"def_end_pos":[114,13]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Module.rank","def_path":"Mathlib/LinearAlgebra/Dimension/Basic.lean","def_pos":[47,0],"def_end_pos":[59,66]},{"full_name":"vectorSpan","def_path":"Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean","def_pos":[58,4],"def_end_pos":[58,14]}]},{"state_before":"k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ns : Set P\ninst✝ : FiniteDimensional k ↥(vectorSpan k s)\nh : Coplanar k s ↔ Module.rank k ↥(vectorSpan k s) ≤ 2\n⊢ Coplanar k s ↔ finrank k ↥(vectorSpan k s) ≤ 2","state_after":"k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ns : Set P\ninst✝ : FiniteDimensional k ↥(vectorSpan k s)\nh : Coplanar k s ↔ ↑(finrank k ↥(vectorSpan k s)) ≤ 2\n⊢ Coplanar k s ↔ finrank k ↥(vectorSpan k s) ≤ 2","tactic":"rw [← finrank_eq_rank] at h","premises":[{"full_name":"finrank_eq_rank","def_path":"Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean","def_pos":[443,8],"def_end_pos":[443,23]}]},{"state_before":"k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ns : Set P\ninst✝ : FiniteDimensional k ↥(vectorSpan k s)\nh : Coplanar k s ↔ ↑(finrank k ↥(vectorSpan k s)) ≤ 2\n⊢ Coplanar k s ↔ finrank k ↥(vectorSpan k s) ≤ 2","state_after":"no goals","tactic":"exact mod_cast h","premises":[]}]}
+{"url":"Mathlib/RingTheory/HahnSeries/Multiplication.lean","commit":"","full_name":"HahnSeries.order_pow","start":[584,0],"end":[591,64],"file_path":"Mathlib/RingTheory/HahnSeries/Multiplication.lean","tactics":[{"state_before":"Γ✝ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\ninst✝³ : OrderedCancelAddCommMonoid Γ✝\nΓ : Type u_5\ninst✝² : LinearOrderedCancelAddCommMonoid Γ\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\nx : HahnSeries Γ R\nn : ℕ\n⊢ (x ^ n).order = n • x.order","state_after":"case zero\nΓ✝ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\ninst✝³ : OrderedCancelAddCommMonoid Γ✝\nΓ : Type u_5\ninst✝² : LinearOrderedCancelAddCommMonoid Γ\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\nx : HahnSeries Γ R\n⊢ (x ^ 0).order = 0 • x.order\n\ncase succ\nΓ✝ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\ninst✝³ : OrderedCancelAddCommMonoid Γ✝\nΓ : Type u_5\ninst✝² : LinearOrderedCancelAddCommMonoid Γ\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\nx : HahnSeries Γ R\nh : ℕ\nIH : (x ^ h).order = h • x.order\n⊢ (x ^ (h + 1)).order = (h + 1) • x.order","tactic":"induction' n with h IH","premises":[]},{"state_before":"case succ\nΓ✝ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\ninst✝³ : OrderedCancelAddCommMonoid Γ✝\nΓ : Type u_5\ninst✝² : LinearOrderedCancelAddCommMonoid Γ\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\nx : HahnSeries Γ R\nh : ℕ\nIH : (x ^ h).order = h • x.order\n⊢ (x ^ (h + 1)).order = (h + 1) • x.order","state_after":"case succ.inl\nΓ✝ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\ninst✝³ : OrderedCancelAddCommMonoid Γ✝\nΓ : Type u_5\ninst✝² : LinearOrderedCancelAddCommMonoid Γ\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\nh : ℕ\nIH : (0 ^ h).order = h • order 0\n⊢ (0 ^ (h + 1)).order = (h + 1) • order 0\n\ncase succ.inr\nΓ✝ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\ninst✝³ : OrderedCancelAddCommMonoid Γ✝\nΓ : Type u_5\ninst✝² : LinearOrderedCancelAddCommMonoid Γ\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\nx : HahnSeries Γ R\nh : ℕ\nIH : (x ^ h).order = h • x.order\nhx : x ≠ 0\n⊢ (x ^ (h + 1)).order = (h + 1) • x.order","tactic":"rcases eq_or_ne x 0 with (rfl | hx)","premises":[{"full_name":"eq_or_ne","def_path":"Mathlib/Logic/Basic.lean","def_pos":[167,8],"def_end_pos":[167,16]}]},{"state_before":"case succ.inr\nΓ✝ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\ninst✝³ : OrderedCancelAddCommMonoid Γ✝\nΓ : Type u_5\ninst✝² : LinearOrderedCancelAddCommMonoid Γ\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\nx : HahnSeries Γ R\nh : ℕ\nIH : (x ^ h).order = h • x.order\nhx : x ≠ 0\n⊢ (x ^ (h + 1)).order = (h + 1) • x.order","state_after":"no goals","tactic":"rw [pow_succ, order_mul (pow_ne_zero _ hx) hx, succ_nsmul, IH]","premises":[{"full_name":"HahnSeries.order_mul","def_path":"Mathlib/RingTheory/HahnSeries/Multiplication.lean","def_pos":[574,8],"def_end_pos":[574,17]},{"full_name":"pow_ne_zero","def_path":"Mathlib/Algebra/GroupWithZero/Basic.lean","def_pos":[172,6],"def_end_pos":[172,17]},{"full_name":"pow_succ","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[567,8],"def_end_pos":[567,16]},{"full_name":"succ_nsmul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[566,14],"def_end_pos":[566,24]}]}]}
+{"url":"Mathlib/RingTheory/DedekindDomain/Ideal.lean","commit":"","full_name":"multiplicity_normalizedFactorsEquivSpanNormalizedFactors_eq_multiplicity","start":[1407,0],"end":[1416,44],"file_path":"Mathlib/RingTheory/DedekindDomain/Ideal.lean","tactics":[{"state_before":"R : Type u_1\nA : Type u_2\nK : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsPrincipalIdealRing R\ninst✝⁴ : DecidableEq R\ninst✝³ : DecidableEq (Ideal R)\ninst✝² : NormalizationMonoid R\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\nr d : R\nhr : r ≠ 0\nhd : d ∈ normalizedFactors r\n⊢ multiplicity d r = multiplicity (↑((normalizedFactorsEquivSpanNormalizedFactors hr) ⟨d, hd⟩)) (span {r})","state_after":"no goals","tactic":"simp only [normalizedFactorsEquivSpanNormalizedFactors, multiplicity_eq_multiplicity_span,\n    Subtype.coe_mk, Equiv.ofBijective_apply]","premises":[{"full_name":"Equiv.ofBijective_apply","def_path":"Mathlib/Logic/Equiv/Defs.lean","def_pos":[783,8],"def_end_pos":[783,13]},{"full_name":"Subtype.coe_mk","def_path":"Mathlib/Data/Subtype.lean","def_pos":[86,8],"def_end_pos":[86,14]},{"full_name":"multiplicity_eq_multiplicity_span","def_path":"Mathlib/RingTheory/DedekindDomain/Ideal.lean","def_pos":[1358,8],"def_end_pos":[1358,41]},{"full_name":"normalizedFactorsEquivSpanNormalizedFactors","def_path":"Mathlib/RingTheory/DedekindDomain/Ideal.lean","def_pos":[1381,18],"def_end_pos":[1381,61]}]}]}
+{"url":"Mathlib/Probability/Cdf.lean","commit":"","full_name":"ProbabilityTheory.measure_cdf","start":[86,0],"end":[89,82],"file_path":"Mathlib/Probability/Cdf.lean","tactics":[{"state_before":"μ : Measure ℝ\ninst✝ : IsProbabilityMeasure μ\n⊢ (cdf μ).measure = μ","state_after":"μ : Measure ℝ\ninst✝ : IsProbabilityMeasure μ\na : ℝ\n⊢ (cdf μ).measure (Iic a) = μ (Iic a)","tactic":"refine Measure.ext_of_Iic (cdf μ).measure μ (fun a ↦ ?_)","premises":[{"full_name":"MeasureTheory.Measure.ext_of_Iic","def_path":"Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean","def_pos":[394,8],"def_end_pos":[394,18]},{"full_name":"ProbabilityTheory.cdf","def_path":"Mathlib/Probability/Cdf.lean","def_pos":[51,4],"def_end_pos":[51,7]},{"full_name":"StieltjesFunction.measure","def_path":"Mathlib/MeasureTheory/Measure/Stieltjes.lean","def_pos":[355,26],"def_end_pos":[355,33]}]},{"state_before":"μ : Measure ℝ\ninst✝ : IsProbabilityMeasure μ\na : ℝ\n⊢ (cdf μ).measure (Iic a) = μ (Iic a)","state_after":"no goals","tactic":"rw [StieltjesFunction.measure_Iic _ (tendsto_cdf_atBot μ), sub_zero, ofReal_cdf]","premises":[{"full_name":"ProbabilityTheory.ofReal_cdf","def_path":"Mathlib/Probability/Cdf.lean","def_pos":[72,6],"def_end_pos":[72,16]},{"full_name":"ProbabilityTheory.tendsto_cdf_atBot","def_path":"Mathlib/Probability/Cdf.lean","def_pos":[67,6],"def_end_pos":[67,23]},{"full_name":"StieltjesFunction.measure_Iic","def_path":"Mathlib/MeasureTheory/Measure/Stieltjes.lean","def_pos":[427,8],"def_end_pos":[427,19]},{"full_name":"sub_zero","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[353,2],"def_end_pos":[353,13]}]}]}
+{"url":"Mathlib/Topology/Maps/Proper/UniversallyClosed.lean","commit":"","full_name":"isProperMap_iff_universally_closed","start":[42,0],"end":[56,22],"file_path":"Mathlib/Topology/Maps/Proper/UniversallyClosed.lean","tactics":[{"state_before":"X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\n⊢ IsProperMap f ↔ Continuous f ∧ ∀ (Z : Type u) [inst : TopologicalSpace Z], IsClosedMap (Prod.map f id)","state_after":"case mp\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\nH : IsProperMap f\n⊢ Continuous f ∧ ∀ (Z : Type u) [inst : TopologicalSpace Z], IsClosedMap (Prod.map f id)\n\ncase mpr\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\nH : Continuous f ∧ ∀ (Z : Type u) [inst : TopologicalSpace Z], IsClosedMap (Prod.map f id)\n�� IsProperMap f","tactic":"constructor <;> intro H","premises":[]}]}
+{"url":"Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean","commit":"","full_name":"_private.Mathlib.RingTheory.MvPolynomial.NewtonIdentities.0.MvPolynomial.NewtonIdentities.weight_add_weight_pairMap","start":[111,0],"end":[133,8],"file_path":"Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean","tactics":[{"state_before":"σ : Type u_1\ninst✝² : Fintype σ\ninst✝¹ : DecidableEq σ\nR : Type u_2\ninst✝ : CommRing R\nk : ℕ\nt : Finset σ × σ\nh : t ∈ MvPolynomial.NewtonIdentities.pairs σ k\n⊢ MvPolynomial.NewtonIdentities.weight σ R k t +\n      MvPolynomial.NewtonIdentities.weight σ R k (MvPolynomial.NewtonIdentities.pairMap σ t) =\n    0","state_after":"σ : Type u_1\ninst✝² : Fintype σ\ninst✝¹ : DecidableEq σ\nR : Type u_2\ninst✝ : CommRing R\nk : ℕ\nt : Finset σ × σ\nh : t ∈ MvPolynomial.NewtonIdentities.pairs σ k\n⊢ (-1) ^ t.1.card * ((∏ a ∈ t.1, X a) * X t.2 ^ (k - t.1.card)) +\n      (-1) ^ (MvPolynomial.NewtonIdentities.pairMap σ t).1.card *\n        ((∏ a ∈ (MvPolynomial.NewtonIdentities.pairMap σ t).1, X a) *\n          X (MvPolynomial.NewtonIdentities.pairMap σ t).2 ^ (k - (MvPolynomial.NewtonIdentities.pairMap σ t).1.card)) =\n    0","tactic":"rw [weight, weight]","premises":[{"full_name":"_private.Mathlib.RingTheory.MvPolynomial.NewtonIdentities.0.MvPolynomial.NewtonIdentities.weight","def_path":"Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean","def_pos":[65,12],"def_end_pos":[65,18]}]},{"state_before":"σ : Type u_1\ninst✝² : Fintype σ\ninst✝¹ : DecidableEq σ\nR : Type u_2\ninst✝ : CommRing R\nk : ℕ\nt : Finset σ × σ\nh : t ∈ MvPolynomial.NewtonIdentities.pairs σ k\n⊢ (-1) ^ t.1.card * ((∏ a ∈ t.1, X a) * X t.2 ^ (k - t.1.card)) +\n      (-1) ^ (MvPolynomial.NewtonIdentities.pairMap σ t).1.card *\n        ((∏ a ∈ (MvPolynomial.NewtonIdentities.pairMap σ t).1, X a) *\n          X (MvPolynomial.NewtonIdentities.pairMap σ t).2 ^ (k - (MvPolynomial.NewtonIdentities.pairMap σ t).1.card)) =\n    0","state_after":"σ : Type u_1\ninst✝² : Fintype σ\ninst✝¹ : DecidableEq σ\nR : Type u_2\ninst✝ : CommRing R\nk : ℕ\nt : Finset σ × σ\nh : t.1.card ≤ k ∧ (t.1.card = k → t.2 ∈ t.1)\n⊢ (-1) ^ t.1.card * ((∏ a ∈ t.1, X a) * X t.2 ^ (k - t.1.card)) +\n      (-1) ^ (MvPolynomial.NewtonIdentities.pairMap σ t).1.card *\n        ((∏ a ∈ (MvPolynomial.NewtonIdentities.pairMap σ t).1, X a) *\n          X (MvPolynomial.NewtonIdentities.pairMap σ t).2 ^ (k - (MvPolynomial.NewtonIdentities.pairMap σ t).1.card)) =\n    0","tactic":"rw [mem_pairs] at h","premises":[{"full_name":"_private.Mathlib.RingTheory.MvPolynomial.NewtonIdentities.0.MvPolynomial.NewtonIdentities.mem_pairs","def_path":"Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean","def_pos":[61,14],"def_end_pos":[61,23]}]},{"state_before":"σ : Type u_1\ninst✝² : Fintype σ\ninst✝¹ : DecidableEq σ\nR : Type u_2\ninst✝ : CommRing R\nk : ℕ\nt : Finset σ × σ\nh : t.1.card ≤ k ∧ (t.1.card = k → t.2 ∈ t.1)\n⊢ (-1) ^ t.1.card * ((∏ a ∈ t.1, X a) * X t.2 ^ (k - t.1.card)) +\n      (-1) ^ (MvPolynomial.NewtonIdentities.pairMap σ t).1.card *\n        ((∏ a ∈ (MvPolynomial.NewtonIdentities.pairMap σ t).1, X a) *\n          X (MvPolynomial.NewtonIdentities.pairMap σ t).2 ^ (k - (MvPolynomial.NewtonIdentities.pairMap σ t).1.card)) =\n    0","state_after":"σ : Type u_1\ninst✝² : Fintype σ\ninst✝¹ : DecidableEq σ\nR : Type u_2\ninst✝ : CommRing R\nk : ℕ\nt : Finset σ × σ\nh : t.1.card ≤ k ∧ (t.1.card = k → t.2 ∈ t.1)\nh2 : ∀ (n : ℕ), -(-1) ^ n = (-1) ^ (n + 1)\n⊢ (-1) ^ t.1.card * ((∏ a ∈ t.1, X a) * X t.2 ^ (k - t.1.card)) +\n      (-1) ^ (MvPolynomial.NewtonIdentities.pairMap σ t).1.card *\n        ((∏ a ∈ (MvPolynomial.NewtonIdentities.pairMap σ t).1, X a) *\n          X (MvPolynomial.NewtonIdentities.pairMap σ t).2 ^ (k - (MvPolynomial.NewtonIdentities.pairMap σ t).1.card)) =\n    0","tactic":"have h2 (n : ℕ) : -(-1 : MvPolynomial σ R) ^ n = (-1) ^ (n + 1) := by\n    rw [← neg_one_mul ((-1 : MvPolynomial σ R) ^ n), pow_add, pow_one, mul_comm]","premises":[{"full_name":"MvPolynomial","def_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","def_pos":[84,4],"def_end_pos":[84,16]},{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]},{"full_name":"neg_one_mul","def_path":"Mathlib/Algebra/Ring/Defs.lean","def_pos":[299,8],"def_end_pos":[299,19]},{"full_name":"pow_add","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[598,6],"def_end_pos":[598,13]},{"full_name":"pow_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[571,6],"def_end_pos":[571,13]}]},{"state_before":"σ : Type u_1\ninst✝² : Fintype σ\ninst✝¹ : DecidableEq σ\nR : Type u_2\ninst✝ : CommRing R\nk : ℕ\nt : Finset σ × σ\nh : t.1.card ≤ k ∧ (t.1.card = k → t.2 ∈ t.1)\nh2 : ∀ (n : ℕ), -(-1) ^ n = (-1) ^ (n + 1)\n⊢ (-1) ^ t.1.card * ((∏ a ∈ t.1, X a) * X t.2 ^ (k - t.1.card)) +\n      (-1) ^ (MvPolynomial.NewtonIdentities.pairMap σ t).1.card *\n        ((∏ a ∈ (MvPolynomial.NewtonIdentities.pairMap σ t).1, X a) *\n          X (MvPolynomial.NewtonIdentities.pairMap σ t).2 ^ (k - (MvPolynomial.NewtonIdentities.pairMap σ t).1.card)) =\n    0","state_after":"case inl\nσ : Type u_1\ninst✝² : Fintype σ\ninst✝¹ : DecidableEq σ\nR : Type u_2\ninst✝ : CommRing R\nk : ℕ\nt : Finset σ × σ\nh : t.1.card ≤ k ∧ (t.1.card = k → t.2 ∈ t.1)\nh2 : ∀ (n : ℕ), -(-1) ^ n = (-1) ^ (n + 1)\nh1 : t.2 ∈ t.1\n⊢ (-1) ^ t.1.card * ((∏ a ∈ t.1, X a) * X t.2 ^ (k - t.1.card)) +\n      (-1) ^ (MvPolynomial.NewtonIdentities.pairMap σ t).1.card *\n        ((∏ a ∈ (MvPolynomial.NewtonIdentities.pairMap σ t).1, X a) *\n          X (MvPolynomial.NewtonIdentities.pairMap σ t).2 ^ (k - (MvPolynomial.NewtonIdentities.pairMap σ t).1.card)) =\n    0\n\ncase inr\nσ : Type u_1\ninst✝² : Fintype σ\ninst✝¹ : DecidableEq σ\nR : Type u_2\ninst✝ : CommRing R\nk : ℕ\nt : Finset σ × σ\nh : t.1.card ≤ k ∧ (t.1.card = k → t.2 ∈ t.1)\nh2 : ∀ (n : ℕ), -(-1) ^ n = (-1) ^ (n + 1)\nh1 : t.2 ∉ t.1\n⊢ (-1) ^ t.1.card * ((∏ a ∈ t.1, X a) * X t.2 ^ (k - t.1.card)) +\n      (-1) ^ (MvPolynomial.NewtonIdentities.pairMap σ t).1.card *\n        ((∏ a ∈ (MvPolynomial.NewtonIdentities.pairMap σ t).1, X a) *\n          X (MvPolynomial.NewtonIdentities.pairMap σ t).2 ^ (k - (MvPolynomial.NewtonIdentities.pairMap σ t).1.card)) =\n    0","tactic":"rcases (em (t.snd ∈ t.fst)) with h1 | h1","premises":[{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Prod.fst","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[483,2],"def_end_pos":[483,5]},{"full_name":"Prod.snd","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[485,2],"def_end_pos":[485,5]}]}]}
+{"url":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","commit":"","full_name":"MeasureTheory.lintegral_const_mul","start":[666,0],"end":[684,94],"file_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nr : ℝ≥0∞\nf : α → ℝ≥0∞\nhf : Measurable f\n⊢ ∫⁻ (a : α), r * f a ∂μ = ∫⁻ (a : α), ⨆ n, ↑(const α r * eapprox f n) a ∂μ","state_after":"case e_f\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nr : ℝ≥0∞\nf : α → ℝ≥0∞\nhf : Measurable f\n⊢ (fun a => r * f a) = fun a => ⨆ n, ↑(const α r * eapprox f n) a","tactic":"congr","premises":[]},{"state_before":"case e_f\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nr : ℝ≥0∞\nf : α → ℝ≥0∞\nhf : Measurable f\n⊢ (fun a => r * f a) = fun a => ⨆ n, ↑(const α r * eapprox f n) a","state_after":"case e_f.h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nr : ℝ≥0∞\nf : α → ℝ≥0∞\nhf : Measurable f\na : α\n⊢ r * f a = ⨆ n, ↑(const α r * eapprox f n) a","tactic":"funext a","premises":[{"full_name":"funext","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1817,8],"def_end_pos":[1817,14]}]},{"state_before":"case e_f.h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nr : ℝ≥0∞\nf : α → ℝ≥0∞\nhf : Measurable f\na : α\n⊢ r * f a = ⨆ n, ↑(const α r * eapprox f n) a","state_after":"case e_f.h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nr : ℝ≥0∞\nf : α → ℝ≥0∞\nhf : Measurable f\na : α\n⊢ ⨆ i, r * ↑(eapprox f i) a = ⨆ n, ↑(const α r * eapprox f n) a","tactic":"rw [← iSup_eapprox_apply f hf, ENNReal.mul_iSup]","premises":[{"full_name":"ENNReal.mul_iSup","def_path":"Mathlib/Topology/Instances/ENNReal.lean","def_pos":[571,8],"def_end_pos":[571,16]},{"full_name":"MeasureTheory.SimpleFunc.iSup_eapprox_apply","def_path":"Mathlib/MeasureTheory/Function/SimpleFunc.lean","def_pos":[746,8],"def_end_pos":[746,26]}]},{"state_before":"case e_f.h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nr : ℝ≥0∞\nf : α → ℝ≥0∞\nhf : Measurable f\na : α\n⊢ ⨆ i, r * ↑(eapprox f i) a = ⨆ n, ↑(const α r * eapprox f n) a","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nr : ℝ≥0∞\nf : α → ℝ≥0∞\nhf : Measurable f\n⊢ ∫⁻ (a : α), ⨆ n, ↑(const α r * eapprox f n) a ∂μ = ⨆ n, r * (eapprox f n).lintegral μ","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nr : ℝ≥0∞\nf : α → ℝ≥0∞\nhf : Measurable f\n⊢ ⨆ n, ∫⁻ (a : α), ↑(const α r * eapprox f n) a ∂μ = ⨆ n, r * (eapprox f n).lintegral μ\n\ncase hf\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nr : ℝ≥0∞\nf : α → ℝ≥0∞\nhf : Measurable f\n⊢ ∀ (n : ℕ), Measurable ↑(const α r * eapprox f n)\n\ncase h_mono\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nr : ℝ≥0∞\nf : α → ℝ≥0∞\nhf : Measurable f\n⊢ Monotone fun n => ↑(const α r * eapprox f n)","tactic":"rw [lintegral_iSup]","premises":[{"full_name":"MeasureTheory.lintegral_iSup","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[349,8],"def_end_pos":[349,22]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nr : ℝ≥0∞\nf : α → ℝ≥0∞\nhf : Measurable f\n⊢ ⨆ n, r * (eapprox f n).lintegral μ = r * ∫⁻ (a : α), f a ∂μ","state_after":"no goals","tactic":"rw [← ENNReal.mul_iSup, lintegral_eq_iSup_eapprox_lintegral hf]","premises":[{"full_name":"ENNReal.mul_iSup","def_path":"Mathlib/Topology/Instances/ENNReal.lean","def_pos":[571,8],"def_end_pos":[571,16]},{"full_name":"MeasureTheory.lintegral_eq_iSup_eapprox_lintegral","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[444,8],"def_end_pos":[444,43]}]}]}
+{"url":"Mathlib/Data/Set/Lattice.lean","commit":"","full_name":"Set.biInter_and'","start":[671,0],"end":[675,42],"file_path":"Mathlib/Data/Set/Lattice.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nι' : Sort u_5\nι₂ : Sort u_6\nκ : ι → Sort u_7\nκ₁ : ι → Sort u_8\nκ₂ : ι → Sort u_9\nκ' : ι' → Sort u_10\np : ι' → Prop\nq : ι → ι' → Prop\ns : (x : ι) → (y : ι') → p y ∧ q x y → Set α\n⊢ ⋂ x, ⋂ y, ⋂ (h : p y ∧ q x y), s x y h = ⋂ y, ⋂ (hy : p y), ⋂ x, ⋂ (hx : q x y), s x y ⋯","state_after":"no goals","tactic":"simp only [iInter_and, @iInter_comm _ ι]","premises":[{"full_name":"Set.iInter_and","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[614,8],"def_end_pos":[614,18]},{"full_name":"Set.iInter_comm","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[624,8],"def_end_pos":[624,19]}]}]}
+{"url":"Mathlib/Algebra/Group/Indicator.lean","commit":"","full_name":"Set.mulIndicator_image","start":[195,0],"end":[198,56],"file_path":"Mathlib/Algebra/Group/Indicator.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝¹ : One M\ninst✝ : One N\ns✝ t : Set α\nf✝ g✝ : α → M\na : α\ns : Set α\nf : β → M\ng : α → β\nhg : Injective g\nx : α\n⊢ (g '' s).mulIndicator f (g x) = s.mulIndicator (f ∘ g) x","state_after":"no goals","tactic":"rw [← mulIndicator_comp_right, preimage_image_eq _ hg]","premises":[{"full_name":"Set.mulIndicator_comp_right","def_path":"Mathlib/Algebra/Group/Indicator.lean","def_pos":[190,8],"def_end_pos":[190,31]},{"full_name":"Set.preimage_image_eq","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[417,8],"def_end_pos":[417,25]}]}]}
+{"url":"Mathlib/Combinatorics/SetFamily/AhlswedeZhang.lean","commit":"","full_name":"Finset.truncatedInf_empty","start":[197,0],"end":[197,100],"file_path":"Mathlib/Combinatorics/SetFamily/AhlswedeZhang.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝⁵ : SemilatticeInf α\ninst✝⁴ : BoundedOrder α\ninst✝³ : DecidableRel fun x x_1 => x ≤ x_1\ninst✝² : SemilatticeInf β\ninst✝¹ : BoundedOrder β\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns t : Finset α\na✝ a : α\n⊢ a ∉ upperClosure ↑∅","state_after":"no goals","tactic":"simp","premises":[]}]}
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CompleteSpace ↥(U t)\nhU : Monotone U\nx : E\nhU' : (⨆ t, U t).topologicalClosure = ⊤\n⊢ x = ↑((orthogonalProjection (⨆ i, U i).topologicalClosure) x)","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁷ : _root_.RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_4\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (t : ι), CompleteSpace ↥(U t)\nhU : Monotone U\nx : E\nhU' : (⨆ t, U t).topologicalClosure = ⊤\n⊢ x ∈ (⨆ i, U i).topologicalClosure","tactic":"rw [orthogonalProjection_eq_self_iff.mpr _]","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"orthogonalProjection_eq_self_iff","def_path":"Mathlib/Analysis/InnerProductSpace/Projection.lean","def_pos":[534,8],"def_end_pos":[534,40]}]},{"state_before":"𝕜 : Type u_1\nE : Type 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+{"url":"Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean","commit":"","full_name":"_private.Mathlib.Topology.MetricSpace.GromovHausdorffRealized.0.GromovHausdorff.HD_bound_aux1","start":[289,0],"end":[295,73],"file_path":"Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean","tactics":[{"state_before":"X : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf✝ : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\nf : GromovHausdorff.Cb X Y\nC : ℝ\n⊢ BddAbove (range fun x => ⨅ y, f (inl x, inr y) + C)","state_after":"case intro\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf✝ : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\nf : GromovHausdorff.Cb X Y\nC Cf : ℝ\nhCf : Cf ∈ upperBounds (range ⇑f)\n⊢ BddAbove (range fun x => ⨅ y, f (inl x, inr y) + C)","tactic":"obtain ⟨Cf, hCf⟩ := f.isBounded_range.bddAbove","premises":[{"full_name":"Bornology.IsBounded.bddAbove","def_path":"Mathlib/Topology/Order/Bornology.lean","def_pos":[62,16],"def_end_pos":[62,44]},{"full_name":"BoundedContinuousFunction.isBounded_range","def_path":"Mathlib/Topology/ContinuousFunction/Bounded.lean","def_pos":[104,8],"def_end_pos":[104,23]}]},{"state_before":"case intro\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf✝ : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\nf : GromovHausdorff.Cb X Y\nC Cf : ℝ\nhCf : Cf ∈ upperBounds (range ⇑f)\n⊢ BddAbove (range fun x => ⨅ y, f (inl x, inr y) + C)","state_after":"case intro\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf✝ : GromovHausdorff.ProdSpaceFun X Y\nx✝ y z t : X ⊕ Y\nf : GromovHausdorff.Cb X Y\nC Cf : ℝ\nhCf : Cf ∈ upperBounds (range ⇑f)\nx : X\n⊢ ⨅ y, f (inl x, inr y) + C ≤ Cf + C","tactic":"refine ⟨Cf + C, forall_mem_range.2 fun x => ?_⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Set.forall_mem_range","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[564,8],"def_end_pos":[564,24]}]},{"state_before":"case intro\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf✝ : GromovHausdorff.ProdSpaceFun X Y\nx✝ y z t : X ⊕ Y\nf : GromovHausdorff.Cb X Y\nC Cf : ℝ\nhCf : Cf ∈ upperBounds (range ⇑f)\nx : X\n⊢ ⨅ y, f (inl x, inr y) + C ≤ Cf + C","state_after":"no goals","tactic":"calc\n    ⨅ y, f (inl x, inr y) + C ≤ f (inl x, inr default) + C := ciInf_le (HD_below_aux1 C) default\n    _ ≤ Cf + C := add_le_add ((fun x => hCf (mem_range_self x)) _) le_rfl","premises":[{"full_name":"GromovHausdorff.HD_below_aux1","def_path":"Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean","def_pos":[284,8],"def_end_pos":[284,21]},{"full_name":"Inhabited.default","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[697,2],"def_end_pos":[697,9]},{"full_name":"Prod.mk","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[481,2],"def_end_pos":[481,4]},{"full_name":"Set.mem_range_self","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[148,22],"def_end_pos":[148,36]},{"full_name":"Sum.inl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[132,4],"def_end_pos":[132,7]},{"full_name":"Sum.inr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[134,4],"def_end_pos":[134,7]},{"full_name":"add_le_add","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[182,31],"def_end_pos":[182,41]},{"full_name":"ciInf_le","def_path":"Mathlib/Order/ConditionallyCompleteLattice/Basic.lean","def_pos":[733,8],"def_end_pos":[733,16]},{"full_name":"iInf","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[60,4],"def_end_pos":[60,8]},{"full_name":"le_rfl","def_path":"Mathlib/Order/Defs.lean","def_pos":[43,8],"def_end_pos":[43,14]}]}]}
+{"url":"Mathlib/Init/Order/LinearOrder.lean","commit":"","full_name":"le_min","start":[48,0],"end":[52,41],"file_path":"Mathlib/Init/Order/LinearOrder.lean","tactics":[{"state_before":"α : Type u\ninst✝ : LinearOrder α\na b c : α\nh₁ : c ≤ a\nh₂ : c ≤ b\n⊢ c ≤ min a b","state_after":"no goals","tactic":"if h : a ≤ b\n  then simp [min_def, if_pos h]; exact h₁\n  else simp [min_def, if_neg h]; exact h₂","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]},{"full_name":"if_neg","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[937,8],"def_end_pos":[937,14]},{"full_name":"if_pos","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[932,8],"def_end_pos":[932,14]},{"full_name":"min_def","def_path":"Mathlib/Init/Order/LinearOrder.lean","def_pos":[30,8],"def_end_pos":[30,15]}]}]}
+{"url":"Mathlib/LinearAlgebra/FreeModule/PID.lean","commit":"","full_name":"Submodule.basisOfPid_bot","start":[303,0],"end":[308,58],"file_path":"Mathlib/LinearAlgebra/FreeModule/PID.lean","tactics":[{"state_before":"ι✝ : Type u_1\nR : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsPrincipalIdealRing R\nM : Type u_3\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nb✝ : ι✝ → M\nι : Type u_4\ninst✝ : Finite ι\nb : Basis ι R M\n⊢ basisOfPid b ⊥ = ⟨0, Basis.empty ↥⊥⟩","state_after":"case mk\nι✝ : Type u_1\nR : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsPrincipalIdealRing R\nM : Type u_3\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nb✝ : ι✝ → M\nι : Type u_4\ninst✝ : Finite ι\nb : Basis ι R M\nn : ℕ\nb' : Basis (Fin n) R ↥⊥\n⊢ ⟨n, b'⟩ = ⟨0, Basis.empty ↥⊥⟩","tactic":"obtain ⟨n, b'⟩ := Submodule.basisOfPid b ⊥","premises":[{"full_name":"Bot.bot","def_path":"Mathlib/Order/Notation.lean","def_pos":[100,2],"def_end_pos":[100,5]},{"full_name":"Submodule.basisOfPid","def_path":"Mathlib/LinearAlgebra/FreeModule/PID.lean","def_pos":[299,18],"def_end_pos":[299,38]}]},{"state_before":"case mk\nι✝ : Type u_1\nR : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsPrincipalIdealRing R\nM : Type u_3\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nb✝ : ι✝ → M\nι : Type u_4\ninst✝ : Finite ι\nb : Basis ι R M\nn : ℕ\nb' : Basis (Fin n) R ↥⊥\n⊢ ⟨n, b'⟩ = ⟨0, Basis.empty ↥⊥⟩","state_after":"case mk\nι✝ : Type u_1\nR : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsPrincipalIdealRing R\nM : Type u_3\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nb✝ : ι✝ → M\nι : Type u_4\ninst✝ : Finite ι\nb : Basis ι R M\nn : ℕ\nb' : Basis (Fin n) R ↥⊥\ne : Fin n ≃ Fin 0 := b'.indexEquiv (Basis.empty ↥⊥)\n⊢ ⟨n, b'⟩ = ⟨0, Basis.empty ↥⊥⟩","tactic":"let e : Fin n ≃ Fin 0 := b'.indexEquiv (Basis.empty _ : Basis (Fin 0) R (⊥ : Submodule R M))","premises":[{"full_name":"Basis","def_path":"Mathlib/LinearAlgebra/Basis.lean","def_pos":[86,10],"def_end_pos":[86,15]},{"full_name":"Basis.empty","def_path":"Mathlib/LinearAlgebra/Basis.lean","def_pos":[798,14],"def_end_pos":[798,19]},{"full_name":"Basis.indexEquiv","def_path":"Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean","def_pos":[85,4],"def_end_pos":[85,20]},{"full_name":"Bot.bot","def_path":"Mathlib/Order/Notation.lean","def_pos":[100,2],"def_end_pos":[100,5]},{"full_name":"Equiv","def_path":"Mathlib/Logic/Equiv/Defs.lean","def_pos":[61,10],"def_end_pos":[61,15]},{"full_name":"Fin","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1831,10],"def_end_pos":[1831,13]},{"full_name":"Submodule","def_path":"Mathlib/Algebra/Module/Submodule/Basic.lean","def_pos":[36,10],"def_end_pos":[36,19]}]},{"state_before":"case mk\nι✝ : Type u_1\nR : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsPrincipalIdealRing R\nM : Type u_3\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nb✝ : ι✝ → M\nι : Type u_4\ninst✝ : Finite ι\nb : Basis ι R M\nn : ℕ\nb' : Basis (Fin n) R ↥⊥\ne : Fin n ≃ Fin 0 := b'.indexEquiv (Basis.empty ↥⊥)\n⊢ ⟨n, b'⟩ = ⟨0, Basis.empty ↥⊥⟩","state_after":"case mk\nι✝ : Type u_1\nR : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsPrincipalIdealRing R\nM : Type u_3\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nb✝ : ι✝ → M\nι : Type u_4\ninst✝ : Finite ι\nb : Basis ι R M\nb' : Basis (Fin 0) R ↥⊥\ne : Fin 0 ≃ Fin 0 := b'.indexEquiv (Basis.empty ↥⊥)\n⊢ ⟨0, b'⟩ = ⟨0, Basis.empty ↥⊥⟩","tactic":"obtain rfl : n = 0 := by simpa using Fintype.card_eq.mpr ⟨e⟩","premises":[{"full_name":"Fintype.card_eq","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[190,8],"def_end_pos":[190,15]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Nonempty.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[711,4],"def_end_pos":[711,9]}]},{"state_before":"case mk\nι✝ : Type u_1\nR : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsPrincipalIdealRing R\nM : Type u_3\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nb✝ : ι✝ → M\nι : Type u_4\ninst✝ : Finite ι\nb : Basis ι R M\nb' : Basis (Fin 0) R ↥⊥\ne : Fin 0 ≃ Fin 0 := b'.indexEquiv (Basis.empty ↥⊥)\n⊢ ⟨0, b'⟩ = ⟨0, Basis.empty ↥⊥⟩","state_after":"no goals","tactic":"exact Sigma.eq rfl (Basis.eq_of_apply_eq <| finZeroElim)","premises":[{"full_name":"Basis.eq_of_apply_eq","def_path":"Mathlib/LinearAlgebra/Basis.lean","def_pos":[296,8],"def_end_pos":[296,22]},{"full_name":"Sigma.eq","def_path":"Mathlib/Init/Data/Sigma/Basic.lean","def_pos":[23,18],"def_end_pos":[23,26]},{"full_name":"finZeroElim","def_path":"Mathlib/Data/Fin/Basic.lean","def_pos":[77,4],"def_end_pos":[77,15]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
+{"url":"Mathlib/Algebra/Homology/HomotopyCategory/ShiftSequence.lean","commit":"","full_name":"CochainComplex.shiftShortComplexFunctor'_hom_app_τ₃","start":[33,0],"end":[44,86],"file_path":"Mathlib/Algebra/Homology/HomotopyCategory/ShiftSequence.lean","tactics":[{"state_before":"C : Type u_1\ninst✝¹ : Category.{?u.87, u_1} C\ninst✝ : Preadditive C\nn i j k i' j' k' : ℤ\nhi : n + i = i'\nhj : n + j = j'\nhk : n + k = k'\nK : CochainComplex C ℤ\n⊢ (CategoryTheory.shiftFunctor (CochainComplex C ℤ) n ⋙ shortComplexFunctor' C (up ℤ) i j k).obj K ≅\n    (shortComplexFunctor' C (up ℤ) i' j' k').obj K","state_after":"C : Type u_1\ninst✝¹ : Category.{?u.87, u_1} C\ninst✝ : Preadditive C\nn i j k i' j' k' : ℤ\nhi : n + i = i'\nhj : n + j = j'\nhk : n + k = k'\nK : CochainComplex C ℤ\n⊢ ShortComplex.mk (n.negOnePow • K.d (i + n) (j + n)) (n.negOnePow • K.d (j + n) (k + n)) ⋯ ≅\n    ShortComplex.mk (K.d i' j') (K.d j' k') ⋯","tactic":"dsimp [shortComplexFunctor']","premises":[{"full_name":"HomologicalComplex.shortComplexFunctor'","def_path":"Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean","def_pos":[34,4],"def_end_pos":[34,24]}]},{"state_before":"C : Type u_1\ninst✝¹ : Category.{?u.87, u_1} C\ninst✝ : Preadditive C\nn i j k i' j' k' : ℤ\nhi : n + i = i'\nhj : n + j = j'\nhk : n + k = k'\nK : CochainComplex C ℤ\n⊢ ShortComplex.mk (n.negOnePow • K.d (i + n) (j + n)) (n.negOnePow • K.d (j + n) (k + n)) ⋯ ≅\n    ShortComplex.mk (K.d i' j') (K.d j' k') ⋯","state_after":"no goals","tactic":"exact ShortComplex.isoMk\n      (n.negOnePow • ((shiftEval C n i i' hi).app K))\n      ((shiftEval C n j j' hj).app K) (n.negOnePow • ((shiftEval C n k k' hk).app K))","premises":[{"full_name":"CategoryTheory.Iso.app","def_path":"Mathlib/CategoryTheory/NatIso.lean","def_pos":[51,4],"def_end_pos":[51,7]},{"full_name":"CategoryTheory.ShortComplex.isoMk","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[205,4],"def_end_pos":[205,9]},{"full_name":"CochainComplex.shiftEval","def_path":"Mathlib/Algebra/Homology/HomotopyCategory/Shift.lean","def_pos":[205,4],"def_end_pos":[205,13]},{"full_name":"Int.negOnePow","def_path":"Mathlib/Algebra/Ring/NegOnePow.lean","def_pos":[22,4],"def_end_pos":[22,13]}]}]}
+{"url":"Mathlib/Order/Filter/CountableSeparatingOn.lean","commit":"","full_name":"HasCountableSeparatingOn.of_subtype","start":[116,0],"end":[124,35],"file_path":"Mathlib/Order/Filter/CountableSeparatingOn.lean","tactics":[{"state_before":"α : Type u_1\np : Set α → Prop\nt : Set α\nq : Set ↑t → Prop\nh : HasCountableSeparatingOn (↑t) q univ\nhpq : ∀ (U : Set ↑t), q U → ∃ V, p V ∧ Subtype.val ⁻¹' V = U\n⊢ HasCountableSeparatingOn α p t","state_after":"case intro.intro.intro\nα : Type u_1\np : Set α → Prop\nt : Set α\nq : Set ↑t → Prop\nh : HasCountableSeparatingOn (↑t) q univ\nhpq : ∀ (U : Set ↑t), q U → ∃ V, p V ∧ Subtype.val ⁻¹' V = U\nS : Set (Set ↑t)\nhSc : S.Countable\nhSq : ∀ s ∈ S, q s\nhS : ∀ x ∈ univ, ∀ y ∈ univ, (∀ s ∈ S, x ∈ s ↔ y ∈ s) → x = y\n⊢ HasCountableSeparatingOn α p t","tactic":"rcases h.1 with ⟨S, hSc, hSq, hS⟩","premises":[{"full_name":"HasCountableSeparatingOn.exists_countable_separating","def_path":"Mathlib/Order/Filter/CountableSeparatingOn.lean","def_pos":[84,2],"def_end_pos":[84,29]}]},{"state_before":"case intro.intro.intro\nα : Type u_1\np : Set α → Prop\nt : Set α\nq : Set ↑t → Prop\nh : HasCountableSeparatingOn (↑t) q univ\nhpq : ∀ (U : Set ↑t), q U → ∃ V, p V ∧ Subtype.val ⁻¹' V = U\nS : Set (Set ↑t)\nhSc : S.Countable\nhSq : ∀ s ∈ S, q s\nhS : ∀ x ∈ univ, ∀ y ∈ univ, (∀ s ∈ S, x ∈ s ↔ y ∈ s) → x = y\n⊢ HasCountableSeparatingOn α p t","state_after":"case intro.intro.intro\nα : Type u_1\np : Set α → Prop\nt : Set α\nq : Set ↑t → Prop\nh : HasCountableSeparatingOn (↑t) q univ\nhpq : ∀ (U : Set ↑t), q U → ∃ V, p V ∧ Subtype.val ⁻¹' V = U\nS : Set (Set ↑t)\nhSc : S.Countable\nhSq : ∀ s ∈ S, q s\nhS : ∀ x ∈ univ, ∀ y ∈ univ, (∀ s ∈ S, x ∈ s ↔ y ∈ s) → x = y\nV : Set ↑t → Set α\nhpV : ∀ s ∈ S, p (V s)\nhV : ∀ s ∈ S, Subtype.val ⁻¹' V s = s\n⊢ HasCountableSeparatingOn α p t","tactic":"choose! V hpV hV using fun s hs ↦ hpq s (hSq s hs)","premises":[]},{"state_before":"case intro.intro.intro\nα : Type u_1\np : Set α → Prop\nt : Set α\nq : Set ↑t → Prop\nh : HasCountableSeparatingOn (↑t) q univ\nhpq : ∀ (U : Set ↑t), q U → ∃ V, p V ∧ Subtype.val ⁻¹' V = U\nS : Set (Set ↑t)\nhSc : S.Countable\nhSq : ∀ s ∈ S, q s\nhS : ∀ x ∈ univ, ∀ y ∈ univ, (∀ s ∈ S, x ∈ s ↔ y ∈ s) → x = y\nV : Set ↑t → Set α\nhpV : ∀ s ∈ S, p (V s)\nhV : ∀ s ∈ S, Subtype.val ⁻¹' V s = s\n⊢ HasCountableSeparatingOn α p t","state_after":"case intro.intro.intro\nα : Type u_1\np : Set α → Prop\nt : Set α\nq : Set ↑t → Prop\nh✝ : HasCountableSeparatingOn (↑t) q univ\nhpq : ∀ (U : Set ↑t), q U → ∃ V, p V ∧ Subtype.val ⁻¹' V = U\nS : Set (Set ↑t)\nhSc : S.Countable\nhSq : ∀ s ∈ S, q s\nhS : ∀ x ∈ univ, ∀ y ∈ univ, (∀ s ∈ S, x ∈ s ↔ y ∈ s) → x = y\nV : Set ↑t → Set α\nhpV : ∀ s ∈ S, p (V s)\nhV : ∀ s ∈ S, Subtype.val ⁻¹' V s = s\nx : α\nhx : x ∈ t\ny : α\nhy : y ∈ t\nh : ∀ s ∈ V '' S, x ∈ s ↔ y ∈ s\n⊢ x = y","tactic":"refine ⟨⟨V '' S, hSc.image _, forall_mem_image.2 hpV, fun x hx y hy h ↦ ?_⟩⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Set.Countable.image","def_path":"Mathlib/Data/Set/Countable.lean","def_pos":[149,8],"def_end_pos":[149,23]},{"full_name":"Set.forall_mem_image","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[195,8],"def_end_pos":[195,24]},{"full_name":"Set.image","def_path":"Mathlib/Init/Set.lean","def_pos":[208,4],"def_end_pos":[208,9]}]},{"state_before":"case intro.intro.intro\nα : Type u_1\np : Set α → Prop\nt : Set α\nq : Set ↑t → Prop\nh✝ : HasCountableSeparatingOn (↑t) q univ\nhpq : ∀ (U : Set ↑t), q U → ∃ V, p V ∧ Subtype.val ⁻¹' V = U\nS : Set (Set ↑t)\nhSc : S.Countable\nhSq : ∀ s ∈ S, q s\nhS : ∀ x ∈ univ, ∀ y ∈ univ, (∀ s ∈ S, x ∈ s ↔ y ∈ s) → x = y\nV : Set ↑t → Set α\nhpV : ∀ s ∈ S, p (V s)\nhV : ∀ s ∈ S, Subtype.val ⁻¹' V s = s\nx : α\nhx : x ∈ t\ny : α\nhy : y ∈ t\nh : ∀ s ∈ V '' S, x ∈ s ↔ y ∈ s\n⊢ x = y","state_after":"case intro.intro.intro\nα : Type u_1\np : Set α → Prop\nt : Set α\nq : Set ↑t → Prop\nh✝ : HasCountableSeparatingOn (↑t) q univ\nhpq : ∀ (U : Set ↑t), q U → ∃ V, p V ∧ Subtype.val ⁻¹' V = U\nS : Set (Set ↑t)\nhSc : S.Countable\nhSq : ∀ s ∈ S, q s\nhS : ∀ x ∈ univ, ∀ y ∈ univ, (∀ s ∈ S, x ∈ s ↔ y ∈ s) → x = y\nV : Set ↑t → Set α\nhpV : ∀ s ∈ S, p (V s)\nhV : ∀ s ∈ S, Subtype.val ⁻¹' V s = s\nx : α\nhx : x ∈ t\ny : α\nhy : y ∈ t\nh : ∀ s ∈ V '' S, x ∈ s ↔ y ∈ s\nU : Set ↑t\nhU : U ∈ S\n⊢ ⟨x, hx⟩ ∈ U ↔ ⟨y, hy⟩ ∈ U","tactic":"refine congr_arg Subtype.val (hS ⟨x, hx⟩ trivial ⟨y, hy⟩ trivial fun U hU ↦ 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Subtype.val ⁻¹' V = U\nS : Set (Set ↑t)\nhSc : S.Countable\nhSq : ∀ s ∈ S, q s\nhS : ∀ x ∈ univ, ∀ y ∈ univ, (∀ s ∈ S, x ∈ s ↔ y ∈ s) → x = y\nV : Set ↑t → Set α\nhpV : ∀ s ∈ S, p (V s)\nhV : ∀ s ∈ S, Subtype.val ⁻¹' V s = s\nx : α\nhx : x ∈ t\ny : α\nhy : y ∈ t\nh : ∀ s ∈ V '' S, x ∈ s ↔ y ∈ s\nU : Set ↑t\nhU : U ∈ S\n⊢ ⟨x, hx⟩ ∈ Subtype.val ⁻¹' V U ↔ ⟨y, hy⟩ ∈ Subtype.val ⁻¹' V U","tactic":"rw [← hV U hU]","premises":[]},{"state_before":"case intro.intro.intro\nα : Type u_1\np : Set α → Prop\nt : Set α\nq : Set ↑t → Prop\nh✝ : HasCountableSeparatingOn (↑t) q univ\nhpq : ∀ (U : Set ↑t), q U → ∃ V, p V ∧ Subtype.val ⁻¹' V = U\nS : Set (Set ↑t)\nhSc : S.Countable\nhSq : ∀ s ∈ S, q s\nhS : ∀ x ∈ univ, ∀ y ∈ univ, (∀ s ∈ S, x ∈ s ↔ y ∈ s) → x = y\nV : Set ↑t → Set α\nhpV : ∀ s ∈ S, p (V s)\nhV : ∀ s ∈ S, Subtype.val ⁻¹' V s = s\nx : α\nhx : x ∈ t\ny : α\nhy : y ∈ t\nh : ∀ s ∈ V '' S, x ∈ s ↔ y ∈ s\nU : Set ↑t\nhU : U ∈ S\n⊢ ⟨x, hx⟩ ∈ Subtype.val ⁻¹' V U ↔ ⟨y, hy⟩ ∈ Subtype.val ⁻¹' V U","state_after":"no goals","tactic":"exact h _ (mem_image_of_mem _ hU)","premises":[{"full_name":"Set.mem_image_of_mem","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[122,8],"def_end_pos":[122,24]}]}]}
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hom_inv_id_assoc]","premises":[{"full_name":"CategoryTheory.LaxFunctor.PseudoCore.mapCompIso_inv","def_path":"Mathlib/CategoryTheory/Bicategory/Functor/Lax.lean","def_pos":[172,2],"def_end_pos":[172,16]},{"full_name":"CategoryTheory.LaxFunctor.mapComp_naturality_right","def_path":"Mathlib/CategoryTheory/Bicategory/Functor/Lax.lean","def_pos":[66,2],"def_end_pos":[66,26]}]},{"state_before":"B : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nD : Type u₃\ninst✝ : Bicategory D\nF✝ : Pseudofunctor B C\nF : LaxFunctor B C\nF' : F.PseudoCore\na✝ b✝ c✝ : B\nf✝ g✝ : a✝ ⟶ b✝\nη : f✝ ⟶ g✝\nh : b✝ ⟶ c✝\n⊢ F.map₂ (η ▷ h) =\n    ((fun {a b c} => F'.mapCompIso) f✝ h).hom ≫ F.map₂ η ▷ F.map h ≫ ((fun {a b c} => F'.mapCompIso) g✝ h).inv","state_after":"no goals","tactic":"rw [F'.mapCompIso_inv, ← LaxFunctor.mapComp_naturality_left, ← F'.mapCompIso_inv,\n      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Type u₃\ninst✝ : Bicategory D\nF✝ : Pseudofunctor B C\nF : LaxFunctor B C\nF' : F.PseudoCore\na b c d : B\nf : a ⟶ b\ng : b ⟶ c\nh : c ⟶ d\n⊢ inv (((fun {a b c} => F'.mapCompIso) f g).hom ▷ F.map h) ≫\n      ((fun {a b c} => F'.mapCompIso) (f ≫ g) h).inv ≫ F.map₂ (α_ f g h).hom =\n    (α_ (F.map f) (F.map g) (F.map h)).hom ≫ F.map f ◁ F.mapComp g h ≫ F.mapComp f (g ≫ h)","tactic":"rw [F'.mapCompIso_inv, F'.mapCompIso_inv, ← inv_comp_eq, ← IsIso.inv_comp_eq]","premises":[{"full_name":"CategoryTheory.IsIso.inv_comp_eq","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[373,8],"def_end_pos":[373,19]},{"full_name":"CategoryTheory.Iso.inv_comp_eq","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[178,8],"def_end_pos":[178,19]},{"full_name":"CategoryTheory.LaxFunctor.PseudoCore.mapCompIso_inv","def_path":"Mathlib/CategoryTheory/Bicategory/Functor/Lax.lean","def_pos":[172,2],"def_end_pos":[172,16]}]},{"state_before":"B : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nD : Type u₃\ninst✝ : Bicategory D\nF✝ : Pseudofunctor B C\nF : LaxFunctor B C\nF' : F.PseudoCore\na b c d : B\nf : a ⟶ b\ng : b ⟶ c\nh : c ⟶ d\n⊢ inv (((fun {a b c} => F'.mapCompIso) f g).hom ▷ F.map h) ≫\n      ((fun {a b c} => F'.mapCompIso) (f ≫ g) h).inv ≫ F.map₂ (α_ f g h).hom =\n    (α_ (F.map f) (F.map g) (F.map h)).hom ≫ F.map f ◁ F.mapComp g h ≫ F.mapComp f (g ≫ h)","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"B : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nD : Type u₃\ninst✝ : Bicategory D\nF✝ : Pseudofunctor B C\nF : LaxFunctor B C\nF' : F.PseudoCore\na b : B\nf : a ⟶ b\n⊢ F.map₂ (λ_ f).hom = ((fun {a b c} => F'.mapCompIso) (𝟙 a) f).hom ≫ (F'.mapIdIso a).hom ▷ F.map f ≫ (λ_ (F.map f)).hom","state_after":"B : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nD : Type u₃\ninst✝ : Bicategory D\nF✝ : Pseudofunctor B C\nF : LaxFunctor B C\nF' : F.PseudoCore\na b : B\nf : a ⟶ b\n⊢ F.map₂ (inv (λ_ f).hom) =\n    inv (((fun {a b c} => F'.mapCompIso) (𝟙 a) f).hom ≫ (F'.mapIdIso a).hom ▷ F.map f ≫ (λ_ (F.map f)).hom)","tactic":"rw [← IsIso.inv_eq_inv, ← F.map₂_inv]","premises":[{"full_name":"CategoryTheory.IsIso.inv_eq_inv","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[418,8],"def_end_pos":[418,24]},{"full_name":"CategoryTheory.PrelaxFunctor.map₂_inv","def_path":"Mathlib/CategoryTheory/Bicategory/Functor/Prelax.lean","def_pos":[177,6],"def_end_pos":[177,14]}]},{"state_before":"B : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nD : Type u₃\ninst✝ : Bicategory D\nF✝ : Pseudofunctor B C\nF : LaxFunctor B C\nF' : F.PseudoCore\na b : B\nf : a ⟶ b\n⊢ F.map₂ (inv (λ_ f).hom) =\n    inv (((fun {a b c} => F'.mapCompIso) (𝟙 a) f).hom ≫ (F'.mapIdIso a).hom ▷ F.map f ≫ (λ_ (F.map f)).hom)","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"B : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nD : Type u₃\ninst✝ : Bicategory 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: Type u₃\ninst✝ : Bicategory D\nF✝ : Pseudofunctor B C\nF : LaxFunctor B C\nF' : F.PseudoCore\na b : B\nf : a ⟶ b\n⊢ F.map₂ (inv (ρ_ f).hom) =\n    inv (((fun {a b c} => F'.mapCompIso) f (𝟙 b)).hom ≫ F.map f ◁ (F'.mapIdIso b).hom ≫ (ρ_ (F.map f)).hom)","state_after":"no goals","tactic":"simp","premises":[]}]}
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+{"url":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","commit":"","full_name":"MeasureTheory.lintegral_count","start":[1555,0],"end":[1559,43],"file_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\nf : α → ℝ≥0∞\n⊢ ∫⁻ (a : α), f a ∂count = ∑' (a : α), f a","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\nf : α → ℝ≥0∞\n⊢ ∑' (i : α), ∫⁻ (a : α), f a ∂dirac i = ∑' (a : α), f a","tactic":"rw [count, lintegral_sum_measure]","premises":[{"full_name":"MeasureTheory.Measure.count","def_path":"Mathlib/MeasureTheory/Measure/Count.lean","def_pos":[26,4],"def_end_pos":[26,9]},{"full_name":"MeasureTheory.lintegral_sum_measure","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[592,8],"def_end_pos":[592,29]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\nf : α → ℝ≥0∞\n⊢ ∑' (i : α), ∫⁻ (a : α), f a ∂dirac i = ∑' (a : α), f a","state_after":"case e_f\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\nf : α → ℝ≥0∞\n⊢ (fun i => ∫⁻ (a : α), f a ∂dirac i) = fun a => f a","tactic":"congr","premises":[]},{"state_before":"case e_f\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\nf : α → ℝ≥0∞\n⊢ (fun i => ∫⁻ (a : α), f a ∂dirac i) = fun a => f a","state_after":"no goals","tactic":"exact funext fun a => lintegral_dirac a f","premises":[{"full_name":"MeasureTheory.lintegral_dirac","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[1526,8],"def_end_pos":[1526,23]},{"full_name":"funext","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1817,8],"def_end_pos":[1817,14]}]}]}
+{"url":"Mathlib/Data/Finset/NAry.lean","commit":"","full_name":"Finset.image₂_union_left","start":[136,0],"end":[139,27],"file_path":"Mathlib/Data/Finset/NAry.lean","tactics":[{"state_before":"α : Type u_1\nα' : Type u_2\nβ : Type u_3\nβ' : Type u_4\nγ : Type u_5\nγ' : Type u_6\nδ : Type u_7\nδ' : Type u_8\nε : Type u_9\nε' : Type u_10\nζ : Type u_11\nζ' : Type u_12\nν : Type u_13\ninst✝⁸ : DecidableEq α'\ninst✝⁷ : DecidableEq β'\ninst✝⁶ : DecidableEq γ\ninst✝⁵ : DecidableEq γ'\ninst✝⁴ : DecidableEq δ\ninst✝³ : DecidableEq δ'\ninst✝² : DecidableEq ε\ninst✝¹ : DecidableEq ε'\nf f' : α → β → γ\ng g' : α → β → γ → δ\ns s' : Finset α\nt t' : Finset β\nu u' : Finset γ\na a' : α\nb b' : β\nc : γ\ninst✝ : DecidableEq α\n⊢ ↑(image₂ f (s ∪ s') t) = ↑(image₂ f s t ∪ image₂ f s' t)","state_after":"α : Type u_1\nα' : Type u_2\nβ : Type u_3\nβ' : Type u_4\nγ : Type u_5\nγ' : Type u_6\nδ : Type u_7\nδ' : Type u_8\nε : Type u_9\nε' : Type u_10\nζ : Type u_11\nζ' : Type u_12\nν : Type u_13\ninst✝⁸ : DecidableEq α'\ninst✝⁷ : DecidableEq β'\ninst✝⁶ : DecidableEq γ\ninst✝⁵ : DecidableEq γ'\ninst✝⁴ : DecidableEq δ\ninst✝³ : DecidableEq δ'\ninst✝² : DecidableEq ε\ninst✝¹ : DecidableEq ε'\nf f' : α → β → γ\ng g' : α → β → γ → δ\ns s' : Finset α\nt t' : Finset β\nu u' : Finset γ\na a' : α\nb b' : β\nc : γ\ninst✝ : DecidableEq α\n⊢ image2 f (↑s ∪ ↑s') ↑t = image2 f ↑s ↑t ∪ image2 f ↑s' ↑t","tactic":"push_cast","premises":[]},{"state_before":"α : Type u_1\nα' : Type u_2\nβ : Type u_3\nβ' : Type u_4\nγ : Type u_5\nγ' : Type u_6\nδ : Type u_7\nδ' : Type u_8\nε : Type u_9\nε' : Type u_10\nζ : Type u_11\nζ' : Type u_12\nν : Type u_13\ninst✝⁸ : DecidableEq α'\ninst✝⁷ : DecidableEq β'\ninst✝⁶ : DecidableEq γ\ninst✝⁵ : DecidableEq γ'\ninst✝⁴ : DecidableEq δ\ninst✝³ : DecidableEq δ'\ninst✝² : DecidableEq ε\ninst✝¹ : DecidableEq ε'\nf f' : α → β → γ\ng g' : α → β → γ → δ\ns s' : Finset α\nt t' : Finset β\nu u' : Finset γ\na a' : α\nb b' : β\nc : γ\ninst✝ : DecidableEq α\n⊢ image2 f (↑s ∪ ↑s') ↑t = image2 f ↑s ↑t ∪ image2 f ↑s' ↑t","state_after":"no goals","tactic":"exact image2_union_left","premises":[{"full_name":"Set.image2_union_left","def_path":"Mathlib/Data/Set/NAry.lean","def_pos":[86,8],"def_end_pos":[86,25]}]}]}
+{"url":"Mathlib/GroupTheory/Perm/Cycle/Type.lean","commit":"","full_name":"exists_prime_orderOf_dvd_card","start":[422,0],"end":[454,36],"file_path":"Mathlib/GroupTheory/Perm/Cycle/Type.lean","tactics":[{"state_before":"α : Type u_1\ninst✝³ : Fintype α\nG✝ : Type u_2\ninst✝² : Group G✝\nn : ℕ\nG : Type u_3\ninst✝¹ : Group G\ninst✝ : Fintype G\np : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ∣ Fintype.card G\n⊢ ∃ x, orderOf x = p","state_after":"α : Type u_1\ninst✝³ : Fintype α\nG✝ : Type u_2\ninst✝² : Group G✝\nn : ℕ\nG : Type u_3\ninst✝¹ : Group G\ninst✝ : Fintype G\np : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ∣ Fintype.card G\nhp' : p - 1 ≠ 0\n⊢ ∃ x, orderOf x = p","tactic":"have hp' : p - 1 ≠ 0 := mt tsub_eq_zero_iff_le.mp (not_le_of_lt hp.out.one_lt)","premises":[{"full_name":"Fact.out","def_path":"Mathlib/Logic/Basic.lean","def_pos":[92,2],"def_end_pos":[92,5]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Nat.Prime.one_lt","def_path":"Mathlib/Data/Nat/Prime/Defs.lean","def_pos":[59,8],"def_end_pos":[59,20]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"mt","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[647,8],"def_end_pos":[647,10]},{"full_name":"not_le_of_lt","def_path":"Mathlib/Order/Basic.lean","def_pos":[290,8],"def_end_pos":[290,20]},{"full_name":"tsub_eq_zero_iff_le","def_path":"Mathlib/Algebra/Order/Sub/Canonical.lean","def_pos":[275,8],"def_end_pos":[275,27]}]},{"state_before":"α : Type u_1\ninst✝³ : Fintype α\nG✝ : Type u_2\ninst✝² : Group G✝\nn : ℕ\nG : Type u_3\ninst✝¹ : Group G\ninst✝ : Fintype G\np : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ∣ Fintype.card G\nhp' : p - 1 ≠ 0\n⊢ ∃ x, orderOf x = p","state_after":"α : Type u_1\ninst✝³ : Fintype α\nG✝ : Type u_2\ninst✝² : Group G✝\nn : ℕ\nG : Type u_3\ninst✝¹ : Group G\ninst✝ : Fintype G\np : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ∣ Fintype.card G\nhp' : p - 1 ≠ 0\nScard : p ∣ Fintype.card ↑(vectorsProdEqOne G p)\n⊢ ∃ x, orderOf x = p","tactic":"have Scard :=\n    calc\n      p ∣ Fintype.card G ^ (p - 1) := hdvd.trans (dvd_pow (dvd_refl _) hp')\n      _ = Fintype.card (vectorsProdEqOne G p) := (VectorsProdEqOne.card G p).symm","premises":[{"full_name":"Dvd.dvd","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1344,2],"def_end_pos":[1344,5]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Equiv.Perm.VectorsProdEqOne.card","def_path":"Mathlib/GroupTheory/Perm/Cycle/Type.lean","def_pos":[400,8],"def_end_pos":[400,12]},{"full_name":"Equiv.Perm.vectorsProdEqOne","def_path":"Mathlib/GroupTheory/Perm/Cycle/Type.lean","def_pos":[351,4],"def_end_pos":[351,20]},{"full_name":"Fintype.card","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[62,4],"def_end_pos":[62,8]},{"full_name":"dvd_pow","def_path":"Mathlib/Algebra/Divisibility/Basic.lean","def_pos":[140,6],"def_end_pos":[140,13]},{"full_name":"dvd_refl","def_path":"Mathlib/Algebra/Divisibility/Basic.lean","def_pos":[122,8],"def_end_pos":[122,16]}]},{"state_before":"α : Type u_1\ninst✝³ : Fintype α\nG✝ : Type u_2\ninst✝² : Group G✝\nn : ℕ\nG : Type u_3\ninst✝¹ : Group G\ninst✝ : Fintype G\np : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ∣ Fintype.card G\nhp' : p - 1 ≠ 0\nScard : p ∣ Fintype.card ↑(vectorsProdEqOne G p)\n⊢ ∃ x, orderOf x = p","state_after":"α : Type u_1\ninst✝³ : Fintype α\nG✝ : Type u_2\ninst✝² : Group G✝\nn : ℕ\nG : Type u_3\ninst✝¹ : Group G\ninst✝ : Fintype G\np : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ∣ Fintype.card G\nhp' : p - 1 ≠ 0\nScard : p ∣ Fintype.card ↑(vectorsProdEqOne G p)\nf : ℕ → ↑(vectorsProdEqOne G p) → ↑(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k\n⊢ ∃ x, orderOf x = p","tactic":"let f : ℕ → vectorsProdEqOne G p → vectorsProdEqOne G p := fun k v =>\n    VectorsProdEqOne.rotate v k","premises":[{"full_name":"Equiv.Perm.VectorsProdEqOne.rotate","def_path":"Mathlib/GroupTheory/Perm/Cycle/Type.lean","def_pos":[407,4],"def_end_pos":[407,10]},{"full_name":"Equiv.Perm.vectorsProdEqOne","def_path":"Mathlib/GroupTheory/Perm/Cycle/Type.lean","def_pos":[351,4],"def_end_pos":[351,20]},{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]}]},{"state_before":"α : Type u_1\ninst✝³ : Fintype α\nG✝ : Type u_2\ninst✝² : Group G✝\nn : ℕ\nG : Type u_3\ninst✝¹ : Group G\ninst✝ : Fintype G\np : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ∣ Fintype.card G\nhp' : p - 1 ≠ 0\nScard : p ∣ Fintype.card ↑(vectorsProdEqOne G p)\nf : ℕ → ↑(vectorsProdEqOne G p) → ↑(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k\n⊢ ∃ x, orderOf x = p","state_after":"α : Type u_1\ninst✝³ : Fintype α\nG✝ : Type u_2\ninst✝² : Group G✝\nn : ℕ\nG : Type u_3\ninst✝¹ : Group G\ninst✝ : Fintype G\np : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ∣ Fintype.card G\nhp' : p - 1 ≠ 0\nScard : p ∣ Fintype.card ↑(vectorsProdEqOne G p)\nf : ℕ → ↑(vectorsProdEqOne G p) → ↑(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k\nhf1 : ∀ (v : ↑(vectorsProdEqOne G p)), f 0 v = v\n⊢ ∃ x, orderOf x = p","tactic":"have hf1 : ∀ v, f 0 v = v := VectorsProdEqOne.rotate_zero","premises":[{"full_name":"Equiv.Perm.VectorsProdEqOne.rotate_zero","def_path":"Mathlib/GroupTheory/Perm/Cycle/Type.lean","def_pos":[410,8],"def_end_pos":[410,19]}]},{"state_before":"α : Type u_1\ninst✝³ : Fintype α\nG✝ : Type u_2\ninst✝² : Group G✝\nn : ℕ\nG : Type u_3\ninst✝¹ : Group G\ninst�� : Fintype G\np : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ∣ Fintype.card G\nhp' : p - 1 ≠ 0\nScard : p ∣ Fintype.card ↑(vectorsProdEqOne G p)\nf : ℕ → ↑(vectorsProdEqOne G p) → ↑(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k\nhf1 : ∀ (v : ↑(vectorsProdEqOne G p)), f 0 v = v\n⊢ ∃ x, orderOf x = p","state_after":"α : Type u_1\ninst✝³ : Fintype α\nG✝ : Type u_2\ninst✝² : Group G✝\nn : ℕ\nG : Type u_3\ninst✝¹ : Group G\ninst✝ : Fintype G\np : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ∣ Fintype.card G\nhp' : p - 1 ≠ 0\nScard : p ∣ Fintype.card ↑(vectorsProdEqOne G p)\nf : ℕ → ↑(vectorsProdEqOne G p) → ↑(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k\nhf1 : ∀ (v : ↑(vectorsProdEqOne G p)), f 0 v = v\nhf2 : ∀ (j k : ℕ) (v : ↑(vectorsProdEqOne G p)), f k (f j v) = f (j + k) v\n⊢ ∃ x, orderOf x = p","tactic":"have hf2 : ∀ j k v, f k (f j v) = f (j + k) v := fun j k v =>\n    VectorsProdEqOne.rotate_rotate v j k","premises":[{"full_name":"Equiv.Perm.VectorsProdEqOne.rotate_rotate","def_path":"Mathlib/GroupTheory/Perm/Cycle/Type.lean","def_pos":[413,8],"def_end_pos":[413,21]}]},{"state_before":"α : Type u_1\ninst✝³ : Fintype α\nG✝ : Type u_2\ninst✝² : Group G✝\nn : ℕ\nG : Type u_3\ninst✝¹ : Group G\ninst✝ : Fintype G\np : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ∣ Fintype.card G\nhp' : p - 1 ≠ 0\nScard : p ∣ Fintype.card ↑(vectorsProdEqOne G p)\nf : ℕ → ↑(vectorsProdEqOne G p) → ↑(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k\nhf1 : ∀ (v : ↑(vectorsProdEqOne G p)), f 0 v = v\nhf2 : ∀ (j k : ℕ) (v : ↑(vectorsProdEqOne G p)), f k (f j v) = f (j + k) v\n⊢ ∃ x, orderOf x = p","state_after":"α : Type u_1\ninst✝³ : Fintype α\nG✝ : Type u_2\ninst✝² : Group G✝\nn : ℕ\nG : Type u_3\ninst✝¹ : Group G\ninst✝ : Fintype G\np : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ∣ Fintype.card G\nhp' : p - 1 ≠ 0\nScard : p ∣ Fintype.card ↑(vectorsProdEqOne G p)\nf : ℕ → ↑(vectorsProdEqOne G p) → ↑(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k\nhf1 : ∀ (v : ↑(vectorsProdEqOne G p)), f 0 v = v\nhf2 : ∀ (j k : ℕ) (v : ↑(vectorsProdEqOne G p)), f k (f j v) = f (j + k) v\nhf3 : ∀ (v : ↑(vectorsProdEqOne G p)), f p v = v\n⊢ ∃ x, orderOf x = p","tactic":"have hf3 : ∀ v, f p v = v := VectorsProdEqOne.rotate_length","premises":[{"full_name":"Equiv.Perm.VectorsProdEqOne.rotate_length","def_path":"Mathlib/GroupTheory/Perm/Cycle/Type.lean","def_pos":[416,8],"def_end_pos":[416,21]}]},{"state_before":"α : Type u_1\ninst✝³ : Fintype α\nG✝ : Type u_2\ninst✝² : Group G✝\nn : ℕ\nG : Type u_3\ninst✝¹ : Group G\ninst✝ : Fintype G\np : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ∣ Fintype.card G\nhp' : p - 1 ≠ 0\nScard : p ∣ Fintype.card ↑(vectorsProdEqOne G p)\nf : ℕ → ↑(vectorsProdEqOne G p) → ↑(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k\nhf1 : ∀ (v : ↑(vectorsProdEqOne G p)), f 0 v = v\nhf2 : ∀ (j k : ℕ) (v : ↑(vectorsProdEqOne G p)), f k (f j v) = f (j + k) v\nhf3 : ∀ (v : ↑(vectorsProdEqOne G p)), f p v = v\n⊢ ∃ x, orderOf x = p","state_after":"α : Type u_1\ninst✝³ : Fintype α\nG✝ : Type u_2\ninst✝² : Group G✝\nn : ℕ\nG : Type u_3\ninst✝¹ : Group G\ninst✝ : Fintype G\np : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ∣ Fintype.card G\nhp' : p - 1 ≠ 0\nScard : p ∣ Fintype.card ↑(vectorsProdEqOne G p)\nf : ℕ → ↑(vectorsProdEqOne G p) → ↑(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k\nhf1 : ∀ (v : ↑(vectorsProdEqOne G p)), f 0 v = v\nhf2 : ∀ (j k : ℕ) (v : ↑(vectorsProdEqOne G p)), f k (f j v) = f (j + k) v\nhf3 : ∀ (v : ↑(vectorsProdEqOne G p)), f p v = v\nσ : ↑(vectorsProdEqOne G p) ≃ ↑(vectorsProdEqOne G p) :=\n  { toFun := f 1, invFun := f (p - 1), left_inv := ⋯, right_inv := ⋯ }\n⊢ ∃ x, orderOf x = p","tactic":"let σ :=\n    Equiv.mk (f 1) (f (p - 1)) (fun s => by rw [hf2, add_tsub_cancel_of_le hp.out.one_lt.le, hf3])\n      fun s => by rw [hf2, tsub_add_cancel_of_le hp.out.one_lt.le, hf3]","premises":[{"full_name":"Fact.out","def_path":"Mathlib/Logic/Basic.lean","def_pos":[92,2],"def_end_pos":[92,5]},{"full_name":"Nat.Prime.one_lt","def_path":"Mathlib/Data/Nat/Prime/Defs.lean","def_pos":[59,8],"def_end_pos":[59,20]},{"full_name":"add_tsub_cancel_of_le","def_path":"Mathlib/Algebra/Order/Sub/Canonical.lean","def_pos":[23,8],"def_end_pos":[23,29]},{"full_name":"tsub_add_cancel_of_le","def_path":"Mathlib/Algebra/Order/Sub/Canonical.lean","def_pos":[28,8],"def_end_pos":[28,29]}]},{"state_before":"α : Type u_1\ninst✝³ : Fintype α\nG✝ : Type u_2\ninst✝² : Group G✝\nn : ℕ\nG : Type u_3\ninst✝¹ : Group G\ninst✝ : Fintype G\np : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ∣ Fintype.card G\nhp' : p - 1 ≠ 0\nScard : p ∣ Fintype.card ↑(vectorsProdEqOne G p)\nf : ℕ → ↑(vectorsProdEqOne G p) → ↑(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k\nhf1 : ∀ (v : ↑(vectorsProdEqOne G p)), f 0 v = v\nhf2 : ∀ (j k : ℕ) (v : ↑(vectorsProdEqOne G p)), f k (f j v) = f (j + k) v\nhf3 : ∀ (v : ↑(vectorsProdEqOne G p)), f p v = v\nσ : ↑(vectorsProdEqOne G p) ≃ ↑(vectorsProdEqOne G p) :=\n  { toFun := f 1, invFun := f (p - 1), left_inv := ⋯, right_inv := ⋯ }\n⊢ ∃ x, orderOf x = p","state_after":"α : Type u_1\ninst✝³ : Fintype α\nG✝ : Type u_2\ninst✝² : Group G✝\nn : ℕ\nG : Type u_3\ninst✝¹ : Group G\ninst✝ : Fintype G\np : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ∣ Fintype.card G\nhp' : p - 1 ≠ 0\nScard : p ∣ Fintype.card ↑(vectorsProdEqOne G p)\nf : ℕ → ↑(vectorsProdEqOne G p) → ↑(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k\nhf1 : ∀ (v : ↑(vectorsProdEqOne G p)), f 0 v = v\nhf2 : ∀ (j k : ℕ) (v : ↑(vectorsProdEqOne G p)), f k (f j v) = f (j + k) v\nhf3 : ∀ (v : ↑(vectorsProdEqOne G p)), f p v = v\nσ : ↑(vectorsProdEqOne G p) ≃ ↑(vectorsProdEqOne G p) :=\n  { toFun := f 1, invFun := f (p - 1), left_inv := ⋯, right_inv := ⋯ }\nhσ : ∀ (k : ℕ) (v : ↑(vectorsProdEqOne G p)), (σ ^ k) v = f k v\n⊢ ∃ x, orderOf x = p","tactic":"have hσ : ∀ k v, (σ ^ k) v = f k v := fun k =>\n    Nat.rec (fun v => (hf1 v).symm) (fun k hk v => by\n      rw [pow_succ, Perm.mul_apply, hk (σ v), Nat.succ_eq_one_add, ← hf2 1 k]\n      simp only [σ, coe_fn_mk]) k","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Equiv.Perm.mul_apply","def_path":"Mathlib/GroupTheory/Perm/Basic.lean","def_pos":[65,8],"def_end_pos":[65,17]},{"full_name":"Equiv.coe_fn_mk","def_path":"Mathlib/Logic/Equiv/Defs.lean","def_pos":[109,16],"def_end_pos":[109,25]},{"full_name":"Nat.succ_eq_one_add","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean","def_pos":[29,8],"def_end_pos":[29,23]},{"full_name":"pow_succ","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[567,8],"def_end_pos":[567,16]}]},{"state_before":"α : Type u_1\ninst✝³ : Fintype α\nG✝ : Type u_2\ninst✝² : Group G✝\nn : ℕ\nG : Type u_3\ninst✝¹ : Group G\ninst✝ : Fintype G\np : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ∣ Fintype.card G\nhp' : p - 1 ≠ 0\nScard : p ∣ Fintype.card ↑(vectorsProdEqOne G p)\nf : ℕ → ↑(vectorsProdEqOne G p) → ↑(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k\nhf1 : ∀ (v : ↑(vectorsProdEqOne G p)), f 0 v = v\nhf2 : ∀ (j k : ℕ) (v : ↑(vectorsProdEqOne G p)), f k (f j v) = f (j + k) v\nhf3 : ∀ (v : ↑(vectorsProdEqOne G p)), f p v = v\nσ : ↑(vectorsProdEqOne G p) ≃ ↑(vectorsProdEqOne G p) :=\n  { toFun := f 1, invFun := f (p - 1), left_inv := ⋯, right_inv := ⋯ }\nhσ : ∀ (k : ℕ) (v : ↑(vectorsProdEqOne G p)), (σ ^ k) v = f k v\n⊢ ∃ x, orderOf x = p","state_after":"α : Type u_1\ninst✝³ : Fintype α\nG✝ : Type u_2\ninst✝² : Group G✝\nn : ℕ\nG : Type u_3\ninst✝¹ : Group G\ninst✝ : Fintype G\np : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ∣ Fintype.card G\nhp' : p - 1 ≠ 0\nScard : p ∣ Fintype.card ↑(vectorsProdEqOne G p)\nf : ℕ → ↑(vectorsProdEqOne G p) → ↑(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k\nhf1 : ∀ (v : ↑(vectorsProdEqOne G p)), f 0 v = v\nhf2 : ∀ (j k : ℕ) (v : ↑(vectorsProdEqOne G p)), f k (f j v) = f (j + k) v\nhf3 : ∀ (v : ↑(vectorsProdEqOne G p)), f p v = v\nσ : ↑(vectorsProdEqOne G p) ≃ ↑(vectorsProdEqOne G p) :=\n  { toFun := f 1, invFun := f (p - 1), left_inv := ⋯, right_inv := ⋯ }\nhσ : σ ^ p ^ 1 = 1\n⊢ ∃ x, orderOf x = p","tactic":"replace hσ : σ ^ p ^ 1 = 1 := Perm.ext fun v => by rw [pow_one, hσ, hf3, one_apply]","premises":[{"full_name":"Equiv.Perm.ext","def_path":"Mathlib/Logic/Equiv/Defs.lean","def_pos":[129,15],"def_end_pos":[129,23]},{"full_name":"Equiv.Perm.one_apply","def_path":"Mathlib/GroupTheory/Perm/Basic.lean","def_pos":[68,8],"def_end_pos":[68,17]},{"full_name":"pow_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[571,6],"def_end_pos":[571,13]}]},{"state_before":"α : Type u_1\ninst✝³ : Fintype α\nG✝ : Type u_2\ninst✝² : Group G✝\nn : ℕ\nG : Type u_3\ninst✝¹ : Group G\ninst✝ : Fintype G\np : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ∣ Fintype.card G\nhp' : p - 1 ≠ 0\nScard : p ∣ Fintype.card ↑(vectorsProdEqOne G p)\nf : ℕ → ↑(vectorsProdEqOne G p) → ↑(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k\nhf1 : ∀ (v : ↑(vectorsProdEqOne G p)), f 0 v = v\nhf2 : ∀ (j k : ℕ) (v : ↑(vectorsProdEqOne G p)), f k (f j v) = f (j + k) v\nhf3 : ∀ (v : ↑(vectorsProdEqOne G p)), f p v = v\nσ : ↑(vectorsProdEqOne G p) ≃ ↑(vectorsProdEqOne G p) :=\n  { toFun := f 1, invFun := f (p - 1), left_inv := ⋯, right_inv := ⋯ }\nhσ : σ ^ p ^ 1 = 1\n⊢ ∃ x, orderOf x = p","state_after":"α : Type u_1\ninst✝³ : Fintype α\nG✝ : Type u_2\ninst✝² : Group G✝\nn : ℕ\nG : Type u_3\ninst✝¹ : Group G\ninst✝ : Fintype G\np : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ∣ Fintype.card G\nhp' : p - 1 ≠ 0\nScard : p ∣ Fintype.card ↑(vectorsProdEqOne G p)\nf : ℕ → ↑(vectorsProdEqOne G p) → ↑(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k\nhf1 : ∀ (v : ↑(vectorsProdEqOne G p)), f 0 v = v\nhf2 : ∀ (j k : ℕ) (v : ↑(vectorsProdEqOne G p)), f k (f j v) = f (j + k) v\nhf3 : ∀ (v : ↑(vectorsProdEqOne G p)), f p v = v\nσ : ↑(vectorsProdEqOne G p) ≃ ↑(vectorsProdEqOne G p) :=\n  { toFun := f 1, invFun := f (p - 1), left_inv := ⋯, right_inv := ⋯ }\nhσ : σ ^ p ^ 1 = 1\nv₀ : ↑(vectorsProdEqOne G p) := ⟨Vector.replicate p 1, ⋯⟩\n⊢ ∃ x, orderOf x = p","tactic":"let v₀ : vectorsProdEqOne G p :=\n    ⟨Vector.replicate p 1, (List.prod_replicate p 1).trans (one_pow p)⟩","premises":[{"full_name":"Eq.trans","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[335,8],"def_end_pos":[335,16]},{"full_name":"Equiv.Perm.vectorsProdEqOne","def_path":"Mathlib/GroupTheory/Perm/Cycle/Type.lean","def_pos":[351,4],"def_end_pos":[351,20]},{"full_name":"List.prod_replicate","def_path":"Mathlib/Algebra/BigOperators/Group/List.lean","def_pos":[121,8],"def_end_pos":[121,22]},{"full_name":"Mathlib.Vector.replicate","def_path":"Mathlib/Data/Vector/Defs.lean","def_pos":[110,4],"def_end_pos":[110,13]},{"full_name":"one_pow","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[593,38],"def_end_pos":[593,45]}]},{"state_before":"α : Type u_1\ninst✝³ : Fintype α\nG✝ : Type u_2\ninst✝² : Group G✝\nn : ℕ\nG : Type u_3\ninst✝¹ : Group G\ninst✝ : Fintype G\np : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ∣ Fintype.card G\nhp' : p - 1 ≠ 0\nScard : p ∣ Fintype.card ↑(vectorsProdEqOne G p)\nf : ℕ → ↑(vectorsProdEqOne G p) → ↑(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k\nhf1 : ∀ (v : ↑(vectorsProdEqOne G p)), f 0 v = v\nhf2 : ∀ (j k : ℕ) (v : ↑(vectorsProdEqOne G p)), f k (f j v) = f (j + k) v\nhf3 : ∀ (v : ↑(vectorsProdEqOne G p)), f p v = v\nσ : ↑(vectorsProdEqOne G p) ≃ ↑(vectorsProdEqOne G p) :=\n  { toFun := f 1, invFun := f (p - 1), left_inv := ⋯, right_inv := ⋯ }\nhσ : σ ^ p ^ 1 = 1\nv₀ : ↑(vectorsProdEqOne G p) := ⟨Vector.replicate p 1, ⋯⟩\n⊢ ∃ x, orderOf x = p","state_after":"α : Type u_1\ninst✝³ : Fintype α\nG✝ : Type u_2\ninst✝² : Group G✝\nn : ℕ\nG : Type u_3\ninst✝¹ : Group G\ninst✝ : Fintype G\np : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ∣ Fintype.card G\nhp' : p - 1 ≠ 0\nScard : p ∣ Fintype.card ↑(vectorsProdEqOne G p)\nf : ℕ → ↑(vectorsProdEqOne G p) → ↑(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k\nhf1 : ∀ (v : ↑(vectorsProdEqOne G p)), f 0 v = v\nhf2 : ∀ (j k : ℕ) (v : ↑(vectorsProdEqOne G p)), f k (f j v) = f (j + k) v\nhf3 : ∀ (v : ↑(vectorsProdEqOne G p)), f p v = v\nσ : ↑(vectorsProdEqOne G p) ≃ ↑(vectorsProdEqOne G p) :=\n  { toFun := f 1, invFun := f (p - 1), left_inv := ⋯, right_inv := ⋯ }\nhσ : σ ^ p ^ 1 = 1\nv₀ : ↑(vectorsProdEqOne G p) := ⟨Vector.replicate p 1, ⋯⟩\nhv₀ : σ v₀ = v₀\n⊢ ∃ x, orderOf x = p","tactic":"have hv₀ : σ v₀ = v₀ := Subtype.ext (Subtype.ext (List.rotate_replicate (1 : G) p 1))","premises":[{"full_name":"List.rotate_replicate","def_path":"Mathlib/Data/List/Rotate.lean","def_pos":[119,8],"def_end_pos":[119,24]},{"full_name":"Subtype.ext","def_path":"Mathlib/Data/Subtype.lean","def_pos":[59,18],"def_end_pos":[59,21]}]},{"state_before":"α : Type u_1\ninst✝³ : Fintype α\nG✝ : Type u_2\ninst✝² : Group G✝\nn : ℕ\nG : Type u_3\ninst✝¹ : Group G\ninst✝ : Fintype G\np : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ∣ Fintype.card G\nhp' : p - 1 ≠ 0\nScard : p ∣ Fintype.card ↑(vectorsProdEqOne G p)\nf : ℕ → ↑(vectorsProdEqOne G p) → ↑(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k\nhf1 : ∀ (v : ↑(vectorsProdEqOne G p)), f 0 v = v\nhf2 : ∀ (j k : ℕ) (v : ↑(vectorsProdEqOne G p)), f k (f j v) = f (j + k) v\nhf3 : ∀ (v : ↑(vectorsProdEqOne G p)), f p v = v\nσ : ↑(vectorsProdEqOne G p) ≃ ↑(vectorsProdEqOne G p) :=\n  { toFun := f 1, invFun := f (p - 1), left_inv := ⋯, right_inv := ⋯ }\nhσ : σ ^ p ^ 1 = 1\nv₀ : ↑(vectorsProdEqOne G p) := ⟨Vector.replicate p 1, ⋯⟩\nhv₀ : σ v₀ = v₀\n⊢ ∃ x, orderOf x = p","state_after":"case intro.intro\nα : Type u_1\ninst✝³ : Fintype α\nG✝ : Type u_2\ninst✝² : Group G✝\nn : ℕ\nG : Type u_3\ninst✝¹ : Group G\ninst✝ : Fintype G\np : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ∣ Fintype.card G\nhp' : p - 1 ≠ 0\nScard : p ∣ Fintype.card ↑(vectorsProdEqOne G p)\nf : ℕ → ↑(vectorsProdEqOne G p) → ↑(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k\nhf1 : ∀ (v : ↑(vectorsProdEqOne G p)), f 0 v = v\nhf2 : ∀ (j k : ℕ) (v : ↑(vectorsProdEqOne G p)), f k (f j v) = f (j + k) v\nhf3 : ∀ (v : ↑(vectorsProdEqOne G p)), f p v = v\nσ : ↑(vectorsProdEqOne G p) ≃ ↑(vectorsProdEqOne G p) :=\n  { toFun := f 1, invFun := f (p - 1), left_inv := ⋯, right_inv := ⋯ }\nhσ : σ ^ p ^ 1 = 1\nv₀ : ↑(vectorsProdEqOne G p) := ⟨Vector.replicate p 1, ⋯⟩\nhv₀ : σ v₀ = v₀\nv : ↑(vectorsProdEqOne G p)\nhv1 : σ v = v\nhv2 : v ≠ v₀\n⊢ ∃ x, orderOf x = p","tactic":"obtain ⟨v, hv1, hv2⟩ := exists_fixed_point_of_prime' Scard hσ hv₀","premises":[{"full_name":"Equiv.Perm.exists_fixed_point_of_prime'","def_path":"Mathlib/GroupTheory/Perm/Cycle/Type.lean","def_pos":[326,8],"def_end_pos":[326,36]}]},{"state_before":"case intro.intro\nα : Type u_1\ninst✝³ : Fintype α\nG✝ : Type u_2\ninst✝² : Group G✝\nn : ℕ\nG : Type u_3\ninst✝¹ : Group G\ninst✝ : Fintype G\np : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ∣ Fintype.card G\nhp' : p - 1 ≠ 0\nScard : p ∣ Fintype.card ↑(vectorsProdEqOne G p)\nf : ℕ → ↑(vectorsProdEqOne G p) → ↑(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k\nhf1 : ∀ (v : ↑(vectorsProdEqOne G p)), f 0 v = v\nhf2 : ∀ (j k : ℕ) (v : ↑(vectorsProdEqOne G p)), f k (f j v) = f (j + k) v\nhf3 : ∀ (v : ↑(vectorsProdEqOne G p)), f p v = v\nσ : ↑(vectorsProdEqOne G p) ≃ ↑(vectorsProdEqOne G p) :=\n  { toFun := f 1, invFun := f (p - 1), left_inv := ⋯, right_inv := ⋯ }\nhσ : σ ^ p ^ 1 = 1\nv₀ : ↑(vectorsProdEqOne G p) := ⟨Vector.replicate p 1, ⋯⟩\nhv₀ : σ v₀ = v₀\nv : ↑(vectorsProdEqOne G p)\nhv1 : σ v = v\nhv2 : v ≠ v₀\n⊢ ∃ x, orderOf x = p","state_after":"case intro.intro.refine_1\nα : Type u_1\ninst✝³ : Fintype α\nG✝ : Type u_2\ninst✝² : Group G✝\nn : ℕ\nG : Type u_3\ninst✝¹ : Group G\ninst✝ : Fintype G\np : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ∣ Fintype.card G\nhp' : p - 1 ≠ 0\nScard : p ∣ Fintype.card ↑(vectorsProdEqOne G p)\nf : ℕ → ↑(vectorsProdEqOne G p) → ↑(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k\nhf1 : ∀ (v : ↑(vectorsProdEqOne G p)), f 0 v = v\nhf2 : ∀ (j k : ℕ) (v : ↑(vectorsProdEqOne G p)), f k (f j v) = f (j + k) v\nhf3 : ∀ (v : ↑(vectorsProdEqOne G p)), f p v = v\nσ : ↑(vectorsProdEqOne G p) ≃ ↑(vectorsProdEqOne G p) :=\n  { toFun := f 1, invFun := f (p - 1), left_inv := ⋯, right_inv := ⋯ }\nhσ : σ ^ p ^ 1 = 1\nv₀ : ↑(vectorsProdEqOne G p) := ⟨Vector.replicate p 1, ⋯⟩\nhv₀ : σ v₀ = v₀\nv : ↑(vectorsProdEqOne G p)\nhv1 : σ v = v\nhv2 : v ≠ v₀\ng : G\nhg : ↑↑v = List.replicate (↑↑v).length g\n⊢ g ^ p = 1\n\ncase intro.intro.refine_2\nα : Type u_1\ninst✝³ : Fintype α\nG✝ : Type u_2\ninst✝² : Group G✝\nn : ℕ\nG : Type u_3\ninst✝¹ : Group G\ninst✝ : Fintype G\np : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ∣ Fintype.card G\nhp' : p - 1 ≠ 0\nScard : p ∣ Fintype.card ↑(vectorsProdEqOne G p)\nf : ℕ → ↑(vectorsProdEqOne G p) → ↑(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k\nhf1 : ∀ (v : ↑(vectorsProdEqOne G p)), f 0 v = v\nhf2 : ∀ (j k : ℕ) (v : ↑(vectorsProdEqOne G p)), f k (f j v) = f (j + k) v\nhf3 : ∀ (v : ↑(vectorsProdEqOne G p)), f p v = v\nσ : ↑(vectorsProdEqOne G p) ≃ ↑(vectorsProdEqOne G p) :=\n  { toFun := f 1, invFun := f (p - 1), left_inv := ⋯, right_inv := ⋯ }\nhσ : σ ^ p ^ 1 = 1\nv₀ : ↑(vectorsProdEqOne G p) := ⟨Vector.replicate p 1, ⋯⟩\nhv₀ : σ v₀ = v₀\nv : ↑(vectorsProdEqOne G p)\nhv1 : σ v = v\nhv2 : v ≠ v₀\ng : G\nhg : ↑↑v = List.replicate (↑↑v).length g\nhg' : g = 1\n⊢ v = v₀","tactic":"refine\n    Exists.imp (fun g hg => orderOf_eq_prime ?_ fun hg' => hv2 ?_)\n      (List.rotate_one_eq_self_iff_eq_replicate.mp (Subtype.ext_iff.mp (Subtype.ext_iff.mp hv1)))","premises":[{"full_name":"Exists.imp","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[193,8],"def_end_pos":[193,18]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"List.rotate_one_eq_self_iff_eq_replicate","def_path":"Mathlib/Data/List/Rotate.lean","def_pos":[281,8],"def_end_pos":[281,43]},{"full_name":"Subtype.ext_iff","def_path":"Mathlib/Data/Subtype.lean","def_pos":[62,18],"def_end_pos":[62,25]},{"full_name":"orderOf_eq_prime","def_path":"Mathlib/GroupTheory/OrderOfElement.lean","def_pos":[436,8],"def_end_pos":[436,24]}]}]}
+{"url":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean","commit":"","full_name":"Real.Angle.two_nsmul_toReal_eq_two_mul_sub_two_pi","start":[582,0],"end":[588,81],"file_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean","tactics":[{"state_before":"θ : Angle\n⊢ (2 • θ).toReal = 2 * θ.toReal - 2 * π ↔ π / 2 < θ.toReal","state_after":"θ : Angle\n⊢ (2 • ↑θ.toReal).toReal = 2 * θ.toReal - 2 * π ↔ π / 2 < θ.toReal","tactic":"nth_rw 1 [← coe_toReal θ]","premises":[{"full_name":"Real.Angle.coe_toReal","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean","def_pos":[464,8],"def_end_pos":[464,18]}]},{"state_before":"θ : Angle\n⊢ (2 • ↑θ.toReal).toReal = 2 * θ.toReal - 2 * π ↔ π / 2 < θ.toReal","state_after":"θ : Angle\n⊢ π < 2 * θ.toReal ∧ 2 * θ.toReal ≤ 3 * π ↔ π / 2 < θ.toReal","tactic":"rw [← coe_nsmul, two_nsmul, ← two_mul, toReal_coe_eq_self_sub_two_pi_iff, Set.mem_Ioc]","premises":[{"full_name":"Real.Angle.coe_nsmul","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean","def_pos":[87,8],"def_end_pos":[87,17]},{"full_name":"Real.Angle.toReal_coe_eq_self_sub_two_pi_iff","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean","def_pos":[574,8],"def_end_pos":[574,41]},{"full_name":"Set.mem_Ioc","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[118,8],"def_end_pos":[118,15]},{"full_name":"two_mul","def_path":"Mathlib/Algebra/Ring/Defs.lean","def_pos":[155,8],"def_end_pos":[155,15]},{"full_name":"two_nsmul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[581,14],"def_end_pos":[581,23]}]},{"state_before":"θ : Angle\n⊢ π < 2 * θ.toReal ∧ 2 * θ.toReal ≤ 3 * π ↔ π / 2 < θ.toReal","state_after":"no goals","tactic":"exact\n    ⟨fun h => by linarith, fun h =>\n      ⟨(div_lt_iff' (zero_lt_two' ℝ)).1 h, by linarith [pi_pos, toReal_le_pi θ]⟩⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Real.Angle.toReal_le_pi","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean","def_pos":[472,8],"def_end_pos":[472,20]},{"full_name":"Real.pi_pos","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","def_pos":[148,8],"def_end_pos":[148,14]},{"full_name":"div_lt_iff'","def_path":"Mathlib/Algebra/Order/Field/Basic.lean","def_pos":[78,8],"def_end_pos":[78,19]},{"full_name":"zero_lt_two'","def_path":"Mathlib/Algebra/Order/Monoid/NatCast.lean","def_pos":[77,6],"def_end_pos":[77,18]}]}]}
+{"url":"Mathlib/Data/Nat/Prime/Defs.lean","commit":"","full_name":"Nat.prime_of_coprime","start":[122,0],"end":[128,44],"file_path":"Mathlib/Data/Nat/Prime/Defs.lean","tactics":[{"state_before":"n✝ n : ℕ\nh1 : 1 < n\nh : ∀ (m : ℕ), m < n → m ≠ 0 → n.Coprime m\n⊢ Prime n","state_after":"n✝ n : ℕ\nh1 : 1 < n\nh : ∀ (m : ℕ), m < n → m ≠ 0 → n.Coprime m\nm : ℕ\nmlt : m < n\nmdvd : m ∣ n\n⊢ m = 1","tactic":"refine prime_def_lt.mpr ⟨h1, fun m mlt mdvd => ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Nat.prime_def_lt","def_path":"Mathlib/Data/Nat/Prime/Defs.lean","def_pos":[90,8],"def_end_pos":[90,20]}]},{"state_before":"n✝ n : ℕ\nh1 : 1 < n\nh : ∀ (m : ℕ), m < n → m ≠ 0 → n.Coprime m\nm : ℕ\nmlt : m < n\nmdvd : m ∣ n\n⊢ m = 1","state_after":"n✝ n : ℕ\nh1 : 1 < n\nh : ∀ (m : ℕ), m < n → m ≠ 0 → n.Coprime m\nm : ℕ\nmlt : m < n\nmdvd : m ∣ n\nhm : m ≠ 0\n⊢ m = 1","tactic":"have hm : m ≠ 0 := by\n    rintro rfl\n    rw [zero_dvd_iff] at mdvd\n    exact mlt.ne' mdvd","premises":[{"full_name":"LT.lt.ne'","def_path":"Mathlib/Order/Basic.lean","def_pos":[267,8],"def_end_pos":[267,11]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"zero_dvd_iff","def_path":"Mathlib/Algebra/GroupWithZero/Divisibility.lean","def_pos":[31,8],"def_end_pos":[31,20]}]},{"state_before":"n✝ n : ℕ\nh1 : 1 < n\nh : ∀ (m : ℕ), m < n → m ≠ 0 → n.Coprime m\nm : ℕ\nmlt : m < n\nmdvd : m ∣ n\nhm : m ≠ 0\n⊢ m = 1","state_after":"no goals","tactic":"exact (h m mlt hm).symm.eq_one_of_dvd mdvd","premises":[{"full_name":"Nat.Coprime.eq_one_of_dvd","def_path":".lake/packages/batteries/Batteries/Data/Nat/Gcd.lean","def_pos":[168,8],"def_end_pos":[168,29]},{"full_name":"Nat.Coprime.symm","def_path":".lake/packages/batteries/Batteries/Data/Nat/Gcd.lean","def_pos":[28,8],"def_end_pos":[28,20]}]}]}
+{"url":"Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean","commit":"","full_name":"IsPrimitiveRoot.norm_pow_sub_one_two","start":[476,0],"end":[494,6],"file_path":"Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean","tactics":[{"state_before":"p n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk : ℕ\nhζ : IsPrimitiveRoot ζ (2 ^ (k + 1))\ninst✝ : IsCyclotomicExtension {2 ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (2 ^ (k + 1)) K)\n⊢ (Algebra.norm K) (ζ ^ 2 ^ k - 1) = (-2) ^ 2 ^ k","state_after":"p n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk : ℕ\nhζ : IsPrimitiveRoot ζ (2 ^ (k + 1))\ninst✝ : IsCyclotomicExtension {2 ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (2 ^ (k + 1)) K)\nthis : IsPrimitiveRoot (ζ ^ 2 ^ k) (2 ^ (k + 1) / 2 ^ k)\n⊢ (Algebra.norm K) (ζ ^ 2 ^ k - 1) = (-2) ^ 2 ^ k","tactic":"have := hζ.pow_of_dvd (fun h => two_ne_zero (pow_eq_zero h)) (pow_dvd_pow 2 (le_succ k))","premises":[{"full_name":"IsPrimitiveRoot.pow_of_dvd","def_path":"Mathlib/RingTheory/RootsOfUnity/Basic.lean","def_pos":[432,8],"def_end_pos":[432,18]},{"full_name":"Nat.le_succ","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1684,8],"def_end_pos":[1684,19]},{"full_name":"pow_dvd_pow","def_path":"Mathlib/Algebra/Divisibility/Basic.lean","def_pos":[137,6],"def_end_pos":[137,17]},{"full_name":"pow_eq_zero","def_path":"Mathlib/Algebra/GroupWithZero/Basic.lean","def_pos":[162,6],"def_end_pos":[162,17]},{"full_name":"two_ne_zero","def_path":"Mathlib/Algebra/NeZero.lean","def_pos":[54,6],"def_end_pos":[54,17]}]},{"state_before":"p n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk : ℕ\nhζ : IsPrimitiveRoot ζ (2 ^ (k + 1))\ninst✝ : IsCyclotomicExtension {2 ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (2 ^ (k + 1)) K)\nthis : IsPrimitiveRoot (ζ ^ 2 ^ k) (2 ^ (k + 1) / 2 ^ k)\n⊢ (Algebra.norm K) (ζ ^ 2 ^ k - 1) = (-2) ^ 2 ^ k","state_after":"p n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk : ℕ\nhζ : IsPrimitiveRoot ζ (2 ^ (k + 1))\ninst✝ : IsCyclotomicExtension {2 ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (2 ^ (k + 1)) K)\nthis : IsPrimitiveRoot (ζ ^ 2 ^ k) 2\n⊢ (Algebra.norm K) (ζ ^ 2 ^ k - 1) = (-2) ^ 2 ^ k","tactic":"rw [Nat.pow_div (le_succ k) zero_lt_two, Nat.succ_sub (le_refl k), Nat.sub_self, pow_one] at this","premises":[{"full_name":"Nat.le_succ","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1684,8],"def_end_pos":[1684,19]},{"full_name":"Nat.pow_div","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean","def_pos":[779,18],"def_end_pos":[779,25]},{"full_name":"Nat.sub_self","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[309,26],"def_end_pos":[309,34]},{"full_name":"Nat.succ_sub","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[1013,8],"def_end_pos":[1013,16]},{"full_name":"le_refl","def_path":"Mathlib/Order/Defs.lean","def_pos":[39,8],"def_end_pos":[39,15]},{"full_name":"pow_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[571,6],"def_end_pos":[571,13]},{"full_name":"zero_lt_two","def_path":"Mathlib/Algebra/Order/Monoid/NatCast.lean","def_pos":[62,14],"def_end_pos":[62,25]}]},{"state_before":"p n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk : ℕ\nhζ : IsPrimitiveRoot ζ (2 ^ (k + 1))\ninst✝ : IsCyclotomicExtension {2 ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (2 ^ (k + 1)) K)\nthis : IsPrimitiveRoot (ζ ^ 2 ^ k) 2\n⊢ (Algebra.norm K) (ζ ^ 2 ^ k - 1) = (-2) ^ 2 ^ k","state_after":"p n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk : ℕ\nhζ : IsPrimitiveRoot ζ (2 ^ (k + 1))\ninst✝ : IsCyclotomicExtension {2 ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (2 ^ (k + 1)) K)\nthis : IsPrimitiveRoot (ζ ^ 2 ^ k) 2\nH : -1 - 1 = (algebraMap K L) (-2)\n⊢ (Algebra.norm K) (ζ ^ 2 ^ k - 1) = (-2) ^ 2 ^ k","tactic":"have H : (-1 : L) - (1 : L) = algebraMap K L (-2) := by\n    simp only [map_neg, map_ofNat]\n    ring","premises":[{"full_name":"algebraMap","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[108,4],"def_end_pos":[108,14]},{"full_name":"map_neg","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[395,2],"def_end_pos":[395,13]},{"full_name":"map_ofNat","def_path":"Mathlib/Data/Nat/Cast/Basic.lean","def_pos":[168,8],"def_end_pos":[168,17]}]},{"state_before":"p n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk : ℕ\nhζ : IsPrimitiveRoot ζ (2 ^ (k + 1))\ninst✝ : IsCyclotomicExtension {2 ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (2 ^ (k + 1)) K)\nthis : IsPrimitiveRoot (ζ ^ 2 ^ k) 2\nH : -1 - 1 = (algebraMap K L) (-2)\n⊢ (Algebra.norm K) (ζ ^ 2 ^ k - 1) = (-2) ^ 2 ^ k","state_after":"p n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk : ℕ\nhζ : IsPrimitiveRoot ζ (2 ^ (k + 1))\ninst✝ : IsCyclotomicExtension {2 ^ (k + 1)} K L\nthis : IsPrimitiveRoot (ζ ^ 2 ^ k) 2\nH : -1 - 1 = (algebraMap K L) (-2)\nhirr : Irreducible (cyclotomic (↑(2 ^ (k + 1))) K)\n⊢ (Algebra.norm K) (ζ ^ 2 ^ k - 1) = (-2) ^ 2 ^ k","tactic":"replace hirr : Irreducible (cyclotomic ((2 : ℕ+) ^ (k + 1) : ℕ+) K) := by simpa using hirr","premises":[{"full_name":"Irreducible","def_path":"Mathlib/Algebra/Associated/Basic.lean","def_pos":[174,10],"def_end_pos":[174,21]},{"full_name":"PNat","def_path":"Mathlib/Data/PNat/Defs.lean","def_pos":[24,4],"def_end_pos":[24,8]},{"full_name":"Polynomial.cyclotomic","def_path":"Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean","def_pos":[231,4],"def_end_pos":[231,14]}]},{"state_before":"p n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk : ℕ\nhζ : IsPrimitiveRoot ζ (2 ^ (k + 1))\ninst✝ : IsCyclotomicExtension {2 ^ (k + 1)} K L\nthis : IsPrimitiveRoot (ζ ^ 2 ^ k) 2\nH : -1 - 1 = (algebraMap K L) (-2)\nhirr : Irreducible (cyclotomic (↑(2 ^ (k + 1))) K)\n⊢ (Algebra.norm K) (ζ ^ 2 ^ k - 1) = (-2) ^ 2 ^ k","state_after":"p n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk : ℕ\nhζ : IsPrimitiveRoot ζ (2 ^ (k + 1))\ninst✝ : IsCyclotomicExtension {2 ^ (k + 1)} K L\nthis : IsPrimitiveRoot (ζ ^ 2 ^ k) 2\nH : -1 - 1 = (algebraMap K L) (-2)\nhirr : Irreducible (cyclotomic (↑(2 ^ (k + 1))) K)\n⊢ (-2) ^ (2 ^ k * (2 - 1)) = (-2) ^ 2 ^ k","tactic":"rw [this.eq_neg_one_of_two_right, H, Algebra.norm_algebraMap,\n    IsCyclotomicExtension.finrank L hirr, pow_coe, show ((2 : ℕ+) : ℕ) = 2 from rfl,\n      totient_prime_pow Nat.prime_two (zero_lt_succ k), succ_sub_succ_eq_sub, tsub_zero]","premises":[{"full_name":"Algebra.norm_algebraMap","def_path":"Mathlib/RingTheory/Norm/Defs.lean","def_pos":[91,18],"def_end_pos":[91,33]},{"full_name":"IsCyclotomicExtension.finrank","def_path":"Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean","def_pos":[173,8],"def_end_pos":[173,15]},{"full_name":"IsPrimitiveRoot.eq_neg_one_of_two_right","def_path":"Mathlib/RingTheory/RootsOfUnity/Basic.lean","def_pos":[616,8],"def_end_pos":[616,31]},{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]},{"full_name":"Nat.prime_two","def_path":"Mathlib/Data/Nat/Prime/Defs.lean","def_pos":[143,8],"def_end_pos":[143,17]},{"full_name":"Nat.succ_sub_succ_eq_sub","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[276,8],"def_end_pos":[276,28]},{"full_name":"Nat.totient_prime_pow","def_path":"Mathlib/Data/Nat/Totient.lean","def_pos":[199,8],"def_end_pos":[199,25]},{"full_name":"Nat.zero_lt_succ","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1671,8],"def_end_pos":[1671,24]},{"full_name":"PNat","def_path":"Mathlib/Data/PNat/Defs.lean","def_pos":[24,4],"def_end_pos":[24,8]},{"full_name":"PNat.pow_coe","def_path":"Mathlib/Data/PNat/Basic.lean","def_pos":[226,8],"def_end_pos":[226,15]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]},{"full_name":"tsub_zero","def_path":"Mathlib/Algebra/Order/Sub/Defs.lean","def_pos":[384,8],"def_end_pos":[384,17]}]},{"state_before":"p n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk : ℕ\nhζ : IsPrimitiveRoot ζ (2 ^ (k + 1))\ninst✝ : IsCyclotomicExtension {2 ^ (k + 1)} K L\nthis : IsPrimitiveRoot (ζ ^ 2 ^ k) 2\nH : -1 - 1 = (algebraMap K L) (-2)\nhirr : Irreducible (cyclotomic (↑(2 ^ (k + 1))) K)\n⊢ (-2) ^ (2 ^ k * (2 - 1)) = (-2) ^ 2 ^ k","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/Algebra/BigOperators/Finprod.lean","commit":"","full_name":"finsum_mem_eq_toFinset_sum","start":[425,0],"end":[428,77],"file_path":"Mathlib/Algebra/BigOperators/Finprod.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\nf : α → M\ns : Set α\ninst✝ : Fintype ↑s\n⊢ s ∩ mulSupport f = ↑s.toFinset ∩ mulSupport f","state_after":"no goals","tactic":"simp_rw [coe_toFinset s]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Set.coe_toFinset","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[557,8],"def_end_pos":[557,20]}]}]}
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NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\nF' : Type u_7\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : Fintype ι\nb : OrthonormalBasis ι 𝕜 E\nx y : E\nthis : ((innerSL 𝕜) x) (∑ i : ι, b.repr y i • b i) = ((innerSL 𝕜) x) y\n⊢ ∑ i : ι, ⟪x, b i⟫_𝕜 * ⟪b i, y⟫_𝕜 = ⟪x, y⟫_𝕜","tactic":"have := congr_arg (innerSL 𝕜 x) (b.sum_repr y)","premises":[{"full_name":"OrthonormalBasis.sum_repr","def_path":"Mathlib/Analysis/InnerProductSpace/PiL2.lean","def_pos":[385,18],"def_end_pos":[385,26]},{"full_name":"innerSL","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[1550,4],"def_end_pos":[1550,11]}]},{"state_before":"ι : Type u_1\nι' : Type u_2\n𝕜 : Type u_3\ninst✝⁹ : _root_.RCLike 𝕜\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\nE' : Type u_5\ninst✝⁶ : NormedAddCommGroup E'\ninst✝⁵ : InnerProductSpace 𝕜 E'\nF : Type u_6\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\nF' : Type u_7\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : Fintype ι\nb : OrthonormalBasis ι 𝕜 E\nx y : E\nthis : ((innerSL 𝕜) x) (∑ i : ι, b.repr y i • b i) = ((innerSL 𝕜) x) y\n⊢ ∑ i : ι, ⟪x, b i⟫_𝕜 * ⟪b i, y⟫_𝕜 = ⟪x, y⟫_𝕜","state_after":"ι : Type u_1\nι' : Type u_2\n𝕜 : Type u_3\ninst✝⁹ : _root_.RCLike 𝕜\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\nE' : Type u_5\ninst✝⁶ : NormedAddCommGroup E'\ninst✝⁵ : InnerProductSpace 𝕜 E'\nF : Type u_6\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\nF' : Type u_7\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : Fintype ι\nb : OrthonormalBasis ι 𝕜 E\nx y : E\nthis : ∑ x_1 : ι, ((innerSL 𝕜) x) (b.repr y x_1 • b x_1) = ((innerSL 𝕜) x) y\n⊢ ∑ i : ι, ⟪x, b i⟫_𝕜 * ⟪b i, y⟫_𝕜 = ⟪x, y⟫_𝕜","tactic":"rw [map_sum] at this","premises":[{"full_name":"map_sum","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[286,2],"def_end_pos":[286,13]}]},{"state_before":"ι : Type u_1\nι' : Type u_2\n𝕜 : Type u_3\ninst✝⁹ : _root_.RCLike 𝕜\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\nE' : Type u_5\ninst✝⁶ : NormedAddCommGroup E'\ninst✝⁵ : InnerProductSpace 𝕜 E'\nF : Type u_6\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\nF' : Type u_7\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : Fintype ι\nb : OrthonormalBasis ι 𝕜 E\nx y : E\nthis : ∑ x_1 : ι, ((innerSL 𝕜) x) (b.repr y x_1 • b x_1) = ((innerSL 𝕜) x) y\n⊢ ∑ i : ι, ⟪x, b i⟫_𝕜 * ⟪b i, y⟫_𝕜 = ⟪x, y⟫_𝕜","state_after":"case h.e'_2.a\nι : Type u_1\nι' : Type u_2\n𝕜 : Type u_3\ninst✝⁹ : _root_.RCLike 𝕜\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\nE' : Type u_5\ninst✝⁶ : NormedAddCommGroup E'\ninst✝⁵ : InnerProductSpace 𝕜 E'\nF : Type u_6\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\nF' : Type u_7\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : Fintype ι\nb : OrthonormalBasis ι 𝕜 E\nx y : E\nthis : ∑ x_1 : ι, ((innerSL 𝕜) x) (b.repr y x_1 • b x_1) = ((innerSL 𝕜) x) y\nx✝ : ι\na✝ : x✝ ∈ Finset.univ\n⊢ ⟪x, b x✝⟫_𝕜 * ⟪b x✝, y⟫_𝕜 = ((innerSL 𝕜) x) (b.repr y x✝ • b x✝)","tactic":"convert this","premises":[]},{"state_before":"case h.e'_2.a\nι : Type u_1\nι' : Type u_2\n𝕜 : Type u_3\ninst✝⁹ : _root_.RCLike 𝕜\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\nE' : Type u_5\ninst✝⁶ : NormedAddCommGroup E'\ninst✝⁵ : InnerProductSpace 𝕜 E'\nF : Type u_6\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\nF' : Type u_7\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : Fintype ι\nb : OrthonormalBasis ι 𝕜 E\nx y : E\nthis : ∑ x_1 : ι, ((innerSL 𝕜) x) (b.repr y x_1 • b x_1) = ((innerSL 𝕜) x) y\nx✝ : ι\na✝ : x✝ ∈ Finset.univ\n⊢ ⟪x, b x✝⟫_𝕜 * ⟪b x✝, y⟫_𝕜 = ((innerSL 𝕜) x) (b.repr y x✝ • b x✝)","state_after":"case h.e'_2.a\nι : Type u_1\nι' : Type u_2\n𝕜 : Type u_3\ninst✝⁹ : _root_.RCLike 𝕜\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\nE' : Type u_5\ninst✝⁶ : NormedAddCommGroup E'\ninst✝⁵ : InnerProductSpace 𝕜 E'\nF : Type u_6\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\nF' : Type u_7\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : Fintype ι\nb : OrthonormalBasis ι 𝕜 E\nx y : E\nthis : ∑ x_1 : ι, ((innerSL 𝕜) x) (b.repr y x_1 • b x_1) = ((innerSL 𝕜) x) y\nx✝ : ι\na✝ : x✝ ∈ Finset.univ\n⊢ ⟪b x✝, y⟫_𝕜 * ⟪x, b x✝⟫_𝕜 = ⟪b x✝, y⟫_𝕜 • ((innerSL 𝕜) x) (b x✝)","tactic":"rw [map_smul, b.repr_apply_apply, mul_comm]","premises":[{"full_name":"OrthonormalBasis.repr_apply_apply","def_path":"Mathlib/Analysis/InnerProductSpace/PiL2.lean","def_pos":[347,18],"def_end_pos":[347,34]},{"full_name":"map_smul","def_path":"Mathlib/GroupTheory/GroupAction/Hom.lean","def_pos":[108,8],"def_end_pos":[108,16]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]}]},{"state_before":"case h.e'_2.a\nι : Type u_1\nι' : Type u_2\n𝕜 : Type u_3\ninst✝⁹ : _root_.RCLike 𝕜\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\nE' : Type u_5\ninst✝⁶ : NormedAddCommGroup E'\ninst✝⁵ : InnerProductSpace 𝕜 E'\nF : Type u_6\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\nF' : Type u_7\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : Fintype ι\nb : OrthonormalBasis ι 𝕜 E\nx y : E\nthis : ∑ x_1 : ι, ((innerSL 𝕜) x) (b.repr y x_1 • b x_1) = ((innerSL 𝕜) x) y\nx✝ : ι\na✝ : x✝ ∈ Finset.univ\n⊢ ⟪b x✝, y⟫_𝕜 * ⟪x, b x✝⟫_𝕜 = ⟪b x✝, y⟫_𝕜 • ((innerSL 𝕜) x) (b x✝)","state_after":"no goals","tactic":"simp only [innerSL_apply, smul_eq_mul]","premises":[{"full_name":"innerSL_apply","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[1559,8],"def_end_pos":[1559,21]},{"full_name":"smul_eq_mul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[79,6],"def_end_pos":[79,17]}]}]}
+{"url":"Mathlib/MeasureTheory/Measure/Complex.lean","commit":"","full_name":"MeasureTheory.ComplexMeasure.equivSignedMeasureₗ_apply","start":[91,0],"end":[101,32],"file_path":"Mathlib/MeasureTheory/Measure/Complex.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nR : Type u_3\ninst✝³ : Semiring R\ninst✝² : Module R ℝ\ninst✝¹ : ContinuousConstSMul R ℝ\ninst✝ : ContinuousConstSMul R ℂ\nc d : ComplexMeasure α\n⊢ __src✝.toFun (c + d) = __src✝.toFun c + __src✝.toFun d","state_after":"no goals","tactic":"rfl","premises":[]},{"state_before":"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nR : Type u_3\ninst✝³ : Semiring R\ninst✝² : Module R ℝ\ninst✝¹ : ContinuousConstSMul R ℝ\ninst✝ : ContinuousConstSMul R ℂ\n⊢ ∀ (m_1 : R) (x : ComplexMeasure α),\n    { toFun := __src✝.toFun, map_add' := ⋯ }.toFun (m_1 • x) =\n      (RingHom.id R) m_1 • { toFun := __src✝.toFun, map_add' := ⋯ }.toFun x","state_after":"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nR : Type u_3\ninst✝³ : Semiring R\ninst✝² : Module R ℝ\ninst✝¹ : ContinuousConstSMul R ℝ\ninst✝ : ContinuousConstSMul R ℂ\nr : R\nc : ComplexMeasure α\n⊢ { toFun := __src✝.toFun, map_add' := ⋯ }.toFun (r • c) =\n    (RingHom.id R) r • { toFun := __src✝.toFun, map_add' := ⋯ }.toFun c","tactic":"intro r c","premises":[]},{"state_before":"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nR : Type u_3\ninst✝³ : Semiring R\ninst✝² : Module R ℝ\ninst✝¹ : ContinuousConstSMul R ℝ\ninst✝ : ContinuousConstSMul R ℂ\nr : R\nc : ComplexMeasure α\n⊢ { toFun := __src✝.toFun, map_add' := ⋯ }.toFun (r • c) =\n    (RingHom.id R) r • { toFun := __src✝.toFun, map_add' := ⋯ }.toFun c","state_after":"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nR : Type u_3\ninst✝³ : Semiring R\ninst✝² : Module R ℝ\ninst✝¹ : ContinuousConstSMul R ℝ\ninst✝ : ContinuousConstSMul R ℂ\nr : R\nc : ComplexMeasure α\n⊢ (mapRange (r • c) Complex.reLm.toAddMonoidHom Complex.continuous_re,\n      mapRange (r • c) Complex.imLm.toAddMonoidHom Complex.continuous_im) =\n    (r • mapRange c Complex.reLm.toAddMonoidHom Complex.continuous_re,\n      r • mapRange c Complex.imLm.toAddMonoidHom Complex.continuous_im)","tactic":"dsimp","premises":[]},{"state_before":"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nR : Type u_3\ninst✝³ : Semiring R\ninst✝² : Module R ℝ\ninst✝¹ : ContinuousConstSMul R ℝ\ninst✝ : ContinuousConstSMul R ℂ\nr : R\nc : ComplexMeasure α\n⊢ (mapRange (r • c) Complex.reLm.toAddMonoidHom Complex.continuous_re,\n      mapRange (r • c) Complex.imLm.toAddMonoidHom Complex.continuous_im) =\n    (r • mapRange c Complex.reLm.toAddMonoidHom Complex.continuous_re,\n      r • mapRange c Complex.imLm.toAddMonoidHom Complex.continuous_im)","state_after":"case a.h\nα : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nR : Type u_3\ninst✝³ : Semiring R\ninst✝² : Module R ℝ\ninst✝¹ : ContinuousConstSMul R ℝ\ninst✝ : ContinuousConstSMul R ℂ\nr : R\nc : ComplexMeasure α\ni✝ : Set α\na✝ : MeasurableSet i✝\n⊢ ↑(mapRange (r • c) Complex.reLm.toAddMonoidHom Complex.continuous_re,\n            mapRange (r • c) Complex.imLm.toAddMonoidHom Complex.continuous_im).1\n      i✝ =\n    ↑(r • mapRange c Complex.reLm.toAddMonoidHom Complex.continuous_re,\n            r • mapRange c Complex.imLm.toAddMonoidHom Complex.continuous_im).1\n      i✝\n\ncase a.h\nα : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nR : Type u_3\ninst✝³ : Semiring R\ninst✝² : Module R ℝ\ninst✝¹ : ContinuousConstSMul R ℝ\ninst✝ : ContinuousConstSMul R ℂ\nr : R\nc : ComplexMeasure α\ni✝ : Set α\na✝ : MeasurableSet i✝\n⊢ ↑(mapRange (r • c) Complex.reLm.toAddMonoidHom Complex.continuous_re,\n            mapRange (r • c) Complex.imLm.toAddMonoidHom Complex.continuous_im).2\n      i✝ =\n    ↑(r • mapRange c Complex.reLm.toAddMonoidHom Complex.continuous_re,\n            r • mapRange c Complex.imLm.toAddMonoidHom Complex.continuous_im).2\n      i✝","tactic":"ext","premises":[]}]}
+{"url":"Mathlib/Algebra/Homology/ExactSequence.lean","commit":"","full_name":"CategoryTheory.ComposableArrows.exact_iff_δ₀","start":[216,0],"end":[234,24],"file_path":"Mathlib/Algebra/Homology/ExactSequence.lean","tactics":[{"state_before":"C : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : HasZeroMorphisms C\nn : ℕ\nS✝ : ComposableArrows C n\nS : ComposableArrows C (n + 2)\n⊢ S.Exact ↔ (mk₂ (S.map' 0 1 ⋯ ⋯) (S.map' 1 2 ⋯ ⋯)).Exact ∧ S.δ₀.Exact","state_after":"case mp\nC : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : HasZeroMorphisms C\nn : ℕ\nS✝ : ComposableArrows C n\nS : ComposableArrows C (n + 2)\n⊢ S.Exact → (mk₂ (S.map' 0 1 ⋯ ⋯) (S.map' 1 2 ⋯ ⋯)).Exact ∧ S.δ₀.Exact\n\ncase mpr\nC : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : HasZeroMorphisms C\nn : ℕ\nS✝ : ComposableArrows C n\nS : ComposableArrows C (n + 2)\n⊢ (mk₂ (S.map' 0 1 ⋯ ⋯) (S.map' 1 2 ⋯ ⋯)).Exact ∧ S.δ₀.Exact → S.Exact","tactic":"constructor","premises":[]}]}
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+{"url":"Mathlib/Algebra/Module/LocalizedModule.lean","commit":"","full_name":"LocalizedModule.mk_cancel_common_right","start":[395,0],"end":[397,35],"file_path":"Mathlib/Algebra/Module/LocalizedModule.lean","tactics":[{"state_before":"R : Type u\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nT : Type u_1\ninst✝² : CommSemiring T\ninst✝¹ : Algebra R T\ninst✝ : IsLocalization S T\ns s' : ↥S\nm : M\n⊢ 1 • s • s' • m = 1 • (s * s') • m","state_after":"no goals","tactic":"simp [mul_smul]","premises":[{"full_name":"MulAction.mul_smul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[99,2],"def_end_pos":[99,10]}]}]}
+{"url":"Mathlib/Topology/Algebra/OpenSubgroup.lean","commit":"","full_name":"OpenAddSubgroup.isClosed","start":[135,0],"end":[144,18],"file_path":"Mathlib/Topology/Algebra/OpenSubgroup.lean","tactics":[{"state_before":"G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\nU✝ V : OpenSubgroup G\ng : G\ninst✝ : ContinuousMul G\nU : OpenSubgroup G\n⊢ IsClosed ↑U","state_after":"G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\nU✝ V : OpenSubgroup G\ng : G\ninst✝ : ContinuousMul G\nU : OpenSubgroup G\n⊢ IsOpen (↑U)ᶜ","tactic":"apply isOpen_compl_iff.1","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"isOpen_compl_iff","def_path":"Mathlib/Topology/Basic.lean","def_pos":[143,16],"def_end_pos":[143,32]}]},{"state_before":"G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\nU✝ V : OpenSubgroup G\ng : G\ninst✝ : ContinuousMul G\nU : OpenSubgroup G\n⊢ IsOpen (↑U)ᶜ","state_after":"case refine_1\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\nU✝ V : OpenSubgroup G\ng : G\ninst✝ : ContinuousMul G\nU : OpenSubgroup G\nx : G\nhx : x ∈ (↑U)ᶜ\n⊢ (fun y => y * x⁻¹) ⁻¹' ↑U ⊆ (↑U)ᶜ\n\ncase refine_2\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\nU✝ V : OpenSubgroup G\ng : G\ninst✝ : ContinuousMul G\nU : OpenSubgroup G\nx : G\nhx : x ∈ (↑U)ᶜ\n⊢ IsOpen ((fun y => y * x⁻¹) ⁻¹' ↑U)\n\ncase refine_3\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\nU✝ V : OpenSubgroup G\ng : G\ninst✝ : ContinuousMul G\nU : OpenSubgroup G\nx : G\nhx : x ∈ (↑U)ᶜ\n⊢ x ∈ (fun y => y * x⁻¹) ⁻¹' ↑U","tactic":"refine isOpen_iff_forall_mem_open.2 fun x hx => ⟨(fun y => y * x⁻¹) ⁻¹' U, ?_, ?_, ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"Set.preimage","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[106,4],"def_end_pos":[106,12]},{"full_name":"isOpen_iff_forall_mem_open","def_path":"Mathlib/Topology/Basic.lean","def_pos":[311,8],"def_end_pos":[311,34]}]}]}
+{"url":"Mathlib/Analysis/RCLike/Basic.lean","commit":"","full_name":"RCLike.im_eq_zero","start":[993,0],"end":[995,6],"file_path":"Mathlib/Analysis/RCLike/Basic.lean","tactics":[{"state_before":"K : Type u_1\nE : Type u_2\ninst✝ : RCLike K\nh : I = 0\nz : K\n⊢ im z = 0","state_after":"K : Type u_1\nE : Type u_2\ninst✝ : RCLike K\nh : I = 0\nz : K\n⊢ im (↑(re z) + ↑(im z) * 0) = 0","tactic":"rw [← re_add_im z, h]","premises":[{"full_name":"RCLike.re_add_im","def_path":"Mathlib/Analysis/RCLike/Basic.lean","def_pos":[108,8],"def_end_pos":[108,17]}]},{"state_before":"K : Type u_1\nE : Type u_2\ninst✝ : RCLike K\nh : I = 0\nz : K\n⊢ im (↑(re z) + ↑(im z) * 0) = 0","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/Analysis/Seminorm.lean","commit":"","full_name":"Seminorm.uniformity_eq_of_hasBasis","start":[1159,0],"end":[1164,50],"file_path":"Mathlib/Analysis/Seminorm.lean","tactics":[{"state_before":"R : Type u_1\nR' : Type u_2\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type u_5\n𝕝 : Type u_6\nE : Type u_7\nE₂ : Type u_8\nE₃ : Type u_9\nF : Type u_10\nG : Type u_11\nι✝ : Type u_12\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : SeminormedRing 𝕝\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Module 𝕝 E\nι : Sort u_13\ninst✝² : UniformSpace E\ninst✝¹ : UniformAddGroup E\ninst✝ : ContinuousConstSMul 𝕜 E\np' : ι → Prop\ns : ι → Set E\np : Seminorm 𝕜 E\nhb : (𝓝 0).HasBasis p' s\nh₁ : ∃ r, p.closedBall 0 r ∈ 𝓝 0\nh₂ : ∀ (i : ι), p' i → ∃ r > 0, p.ball 0 r ⊆ s i\n⊢ 𝓤 E = ⨅ r, ⨅ (_ : r > 0), 𝓟 {x | p (x.1 - x.2) < r}","state_after":"no goals","tactic":"rfl","premises":[]}]}
+{"url":"Mathlib/GroupTheory/Perm/Cycle/Basic.lean","commit":"","full_name":"Set.prod_self_eq_iUnion_perm","start":[915,0],"end":[923,44],"file_path":"Mathlib/GroupTheory/Perm/Cycle/Basic.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\nf : Perm α\ns : Set α\nhf : f.IsCycleOn s\n⊢ s ×ˢ s = ⋃ n, (fun a => (a, (f ^ n) a)) '' s","state_after":"case h.mk\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nf : Perm α\ns : Set α\nhf : f.IsCycleOn s\na b : α\n⊢ (a, b) ∈ s ×ˢ s ↔ (a, b) ∈ ⋃ n, (fun a => (a, (f ^ n) a)) '' s","tactic":"ext ⟨a, b⟩","premises":[]},{"state_before":"case h.mk\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nf : Perm α\ns : Set α\nhf : f.IsCycleOn s\na b : α\n⊢ (a, b) ∈ s ×ˢ s ↔ (a, b) ∈ ⋃ n, (fun a => (a, (f ^ n) a)) '' s","state_after":"case h.mk\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nf : Perm α\ns : Set α\nhf : f.IsCycleOn s\na b : α\n⊢ a ∈ s ∧ b ∈ s ↔ ∃ i, ∃ x ∈ s, (x, (f ^ i) x) = (a, b)","tactic":"simp only [Set.mem_prod, Set.mem_iUnion, Set.mem_image]","premises":[{"full_name":"Set.mem_iUnion","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[254,8],"def_end_pos":[254,18]},{"full_name":"Set.mem_image","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[118,8],"def_end_pos":[118,17]},{"full_name":"Set.mem_prod","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[184,8],"def_end_pos":[184,16]}]},{"state_before":"case h.mk\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nf : Perm α\ns : Set α\nhf : f.IsCycleOn s\na b : α\n⊢ a ∈ s ∧ b ∈ s ↔ ∃ i, ∃ x ∈ s, (x, (f ^ i) x) = (a, b)","state_after":"case h.mk.refine_1\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nf : Perm α\ns : Set α\nhf : f.IsCycleOn s\na b : α\nhx : a ∈ s ∧ b ∈ s\n⊢ ∃ i, ∃ x ∈ s, (x, (f ^ i) x) = (a, b)\n\ncase h.mk.refine_2\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nf : Perm α\ns : Set α\nhf : f.IsCycleOn s\na b : α\n⊢ (∃ i, ∃ x ∈ s, (x, (f ^ i) x) = (a, b)) → a ∈ s ∧ b ∈ s","tactic":"refine ⟨fun hx => ?_, ?_⟩","premises":[{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]}]}]}
+{"url":"Mathlib/Analysis/Complex/UpperHalfPlane/Topology.lean","commit":"","full_name":"UpperHalfPlane.ModularGroup_T_zpow_mem_verticalStrip","start":[102,0],"end":[117,52],"file_path":"Mathlib/Analysis/Complex/UpperHalfPlane/Topology.lean","tactics":[{"state_before":"z : ℍ\nN : ℕ\nhn : 0 < N\n⊢ ∃ n, ModularGroup.T ^ (↑N * n) • z ∈ verticalStrip (↑N) z.im","state_after":"z : ℍ\nN : ℕ\nhn : 0 < N\nn : ℤ := ⌊z.re / ↑N⌋\n⊢ ∃ n, ModularGroup.T ^ (↑N * n) • z ∈ verticalStrip (↑N) z.im","tactic":"let n := Int.floor (z.re/N)","premises":[{"full_name":"Int.floor","def_path":"Mathlib/Algebra/Order/Floor.lean","def_pos":[563,4],"def_end_pos":[563,9]},{"full_name":"UpperHalfPlane.re","def_path":"Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean","def_pos":[76,4],"def_end_pos":[76,6]}]},{"state_before":"z : ℍ\nN : ℕ\nhn : 0 < N\nn : ℤ := ⌊z.re / ↑N⌋\n⊢ ∃ n, ModularGroup.T ^ (↑N * n) • z ∈ verticalStrip (↑N) z.im","state_after":"case h\nz : ℍ\nN : ℕ\nhn : 0 < N\nn : ℤ := ⌊z.re / ↑N⌋\n⊢ ModularGroup.T ^ (↑N * -n) • z ∈ verticalStrip (↑N) z.im","tactic":"use -n","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]}]},{"state_before":"case h\nz : ℍ\nN : ℕ\nhn : 0 < N\nn : ℤ := ⌊z.re / ↑N⌋\n⊢ ModularGroup.T ^ (↑N * -n) • z ∈ verticalStrip (↑N) z.im","state_after":"case h\nz : ℍ\nN : ℕ\nhn : 0 < N\nn : ℤ := ⌊z.re / ↑N⌋\n⊢ ↑(↑N * -n) +ᵥ z ∈ verticalStrip (↑N) z.im","tactic":"rw [modular_T_zpow_smul z (N * -n)]","premises":[{"full_name":"UpperHalfPlane.modular_T_zpow_smul","def_path":"Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean","def_pos":[457,8],"def_end_pos":[457,27]}]},{"state_before":"case h\nz : ℍ\nN : ℕ\nhn : 0 < N\nn : ℤ := ⌊z.re / ↑N⌋\n⊢ ↑(↑N * -n) +ᵥ z ∈ verticalStrip (↑N) z.im","state_after":"case h\nz : ℍ\nN : ℕ\nhn : 0 < N\nn : ℤ := ⌊z.re / ↑N⌋\n⊢ |(↑(↑N * -n) +ᵥ z).re| ≤ ↑N","tactic":"refine ⟨?_, (by simp only [mul_neg, Int.cast_neg, Int.cast_mul, Int.cast_natCast, vadd_im,\n    le_refl])⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Int.cast_mul","def_path":"Mathlib/Algebra/Ring/Int.lean","def_pos":[55,6],"def_end_pos":[55,14]},{"full_name":"Int.cast_natCast","def_path":"Mathlib/Data/Int/Cast/Basic.lean","def_pos":[59,8],"def_end_pos":[59,20]},{"full_name":"Int.cast_neg","def_path":"Mathlib/Data/Int/Cast/Basic.lean","def_pos":[75,8],"def_end_pos":[75,16]},{"full_name":"UpperHalfPlane.vadd_im","def_path":"Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean","def_pos":[447,8],"def_end_pos":[447,15]},{"full_name":"le_refl","def_path":"Mathlib/Order/Defs.lean","def_pos":[39,8],"def_end_pos":[39,15]},{"full_name":"mul_neg","def_path":"Mathlib/Algebra/Ring/Defs.lean","def_pos":[272,8],"def_end_pos":[272,15]}]},{"state_before":"case h\nz : ℍ\nN : ℕ\nhn : 0 < N\nn : ℤ := ⌊z.re / ↑N⌋\n⊢ |(↑(↑N * -n) +ᵥ z).re| ≤ ↑N","state_after":"case h\nz : ℍ\nN : ℕ\nhn : 0 < N\nn : ℤ := ⌊z.re / ↑N⌋\nh : (↑N * -↑n +ᵥ z).re = -↑N * ↑⌊z.re / ↑N⌋ + z.re\n⊢ |(↑(↑N * -n) +ᵥ z).re| ≤ ↑N","tactic":"have h : (N * (-n : ℝ) +ᵥ z).re = -N * Int.floor (z.re / N) + z.re := by\n    simp only [Int.cast_natCast, mul_neg, vadd_re, neg_mul]","premises":[{"full_name":"HVAdd.hVAdd","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[61,2],"def_end_pos":[61,7]},{"full_name":"Int.cast_natCast","def_path":"Mathlib/Data/Int/Cast/Basic.lean","def_pos":[59,8],"def_end_pos":[59,20]},{"full_name":"Int.floor","def_path":"Mathlib/Algebra/Order/Floor.lean","def_pos":[563,4],"def_end_pos":[563,9]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"UpperHalfPlane.re","def_path":"Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean","def_pos":[76,4],"def_end_pos":[76,6]},{"full_name":"UpperHalfPlane.vadd_re","def_path":"Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean","def_pos":[443,8],"def_end_pos":[443,15]},{"full_name":"mul_neg","def_path":"Mathlib/Algebra/Ring/Defs.lean","def_pos":[272,8],"def_end_pos":[272,15]},{"full_name":"neg_mul","def_path":"Mathlib/Algebra/Ring/Defs.lean","def_pos":[268,8],"def_end_pos":[268,15]}]},{"state_before":"case h\nz : ℍ\nN : ℕ\nhn : 0 < N\nn : ℤ := ⌊z.re / ↑N⌋\nh : (↑N * -↑n +ᵥ z).re = -↑N * ↑⌊z.re / ↑N⌋ + z.re\n⊢ |(↑(↑N * -n) +ᵥ z).re| ≤ ↑N","state_after":"case h\nz : ℍ\nN : ℕ\nhn : 0 < N\nn : ℤ := ⌊z.re / ↑N⌋\nh : (↑(↑N * -n) +ᵥ z).re = ↑(-↑N * ⌊z.re / ↑N⌋) + z.re\n⊢ |(↑(↑N * -n) +ᵥ z).re| ≤ ↑N","tactic":"norm_cast at *","premises":[]},{"state_before":"case h\nz : ℍ\nN : ℕ\nhn : 0 < N\nn : ℤ := ⌊z.re / ↑N⌋\nh : (↑(↑N * -n) +ᵥ z).re = ↑(-↑N * ⌊z.re / ↑N⌋) + z.re\n⊢ |(↑(↑N * -n) +ᵥ z).re| ≤ ↑N","state_after":"case h\nz : ℍ\nN : ℕ\nhn : 0 < N\nn : ℤ := ⌊z.re / ↑N⌋\nh : (↑(↑N * -n) +ᵥ z).re = ↑(-↑N * ⌊z.re / ↑N⌋) + z.re\n⊢ |z.re + ↑(-↑N * ⌊z.re / ↑N⌋)| ≤ ↑N","tactic":"rw [h, add_comm]","premises":[{"full_name":"add_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[293,2],"def_end_pos":[293,13]}]},{"state_before":"case h\nz : ℍ\nN : ℕ\nhn : 0 < N\nn : ℤ := ⌊z.re / ↑N⌋\nh : (↑(↑N * -n) +ᵥ z).re = ↑(-↑N * ⌊z.re / ↑N⌋) + z.re\n⊢ |z.re + ↑(-↑N * ⌊z.re / ↑N⌋)| ≤ ↑N","state_after":"case h\nz : ℍ\nN : ℕ\nhn : 0 < N\nn : ℤ := ⌊z.re / ↑N⌋\nh : (↑(↑N * -n) +ᵥ z).re = ↑(-↑N * ⌊z.re / ↑N⌋) + z.re\n⊢ |z.re + -(↑N * ↑⌊z.re / ↑N⌋)| ≤ ↑N","tactic":"simp only [neg_mul, Int.cast_neg, Int.cast_mul, Int.cast_natCast]","premises":[{"full_name":"Int.cast_mul","def_path":"Mathlib/Algebra/Ring/Int.lean","def_pos":[55,6],"def_end_pos":[55,14]},{"full_name":"Int.cast_natCast","def_path":"Mathlib/Data/Int/Cast/Basic.lean","def_pos":[59,8],"def_end_pos":[59,20]},{"full_name":"Int.cast_neg","def_path":"Mathlib/Data/Int/Cast/Basic.lean","def_pos":[75,8],"def_end_pos":[75,16]},{"full_name":"neg_mul","def_path":"Mathlib/Algebra/Ring/Defs.lean","def_pos":[268,8],"def_end_pos":[268,15]}]},{"state_before":"case h\nz : ℍ\nN : ℕ\nhn : 0 < N\nn : ℤ := ⌊z.re / ↑N⌋\nh : (↑(↑N * -n) +ᵥ z).re = ↑(-↑N * ⌊z.re / ↑N⌋) + z.re\n⊢ |z.re + -(↑N * ↑⌊z.re / ↑N⌋)| ≤ ↑N","state_after":"case h\nz : ℍ\nN : ℕ\nhn : 0 < N\nn : ℤ := ⌊z.re / ↑N⌋\nh : (↑(↑N * -n) +ᵥ z).re = ↑(-↑N * ⌊z.re / ↑N⌋) + z.re\nhnn : 0 < ↑N\n⊢ |z.re + -(↑N * ↑⌊z.re / ↑N⌋)| ≤ ↑N","tactic":"have hnn : (0 : ℝ) < (N : ℝ) := by norm_cast at *","premises":[{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]}]},{"state_before":"case h\nz : ℍ\nN : ℕ\nhn : 0 < N\nn : ℤ := ⌊z.re / ↑N⌋\nh : (↑(↑N * -n) +ᵥ z).re = ↑(-↑N * ⌊z.re / ↑N⌋) + z.re\nhnn : 0 < ↑N\n⊢ |z.re + -(↑N * ↑⌊z.re / ↑N⌋)| ≤ ↑N","state_after":"case h\nz : ℍ\nN : ℕ\nhn : 0 < N\nn : ℤ := ⌊z.re / ↑N⌋\nh : (↑(↑N * -n) +ᵥ z).re = ↑(-↑N * ⌊z.re / ↑N⌋) + z.re\nhnn : 0 < ↑N\nh2 : z.re + -(↑N * ↑n) = z.re - ↑n * ↑N\n⊢ |z.re + -(↑N * ↑⌊z.re / ↑N⌋)| ≤ ↑N","tactic":"have h2 : z.re + -(N * n) =  z.re - n * N := by ring","premises":[{"full_name":"UpperHalfPlane.re","def_path":"Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean","def_pos":[76,4],"def_end_pos":[76,6]}]},{"state_before":"case h\nz : ℍ\nN : ℕ\nhn : 0 < N\nn : ℤ := ⌊z.re / ↑N⌋\nh : (↑(↑N * -n) +ᵥ z).re = ↑(-↑N * ⌊z.re / ↑N⌋) + z.re\nhnn : 0 < ↑N\nh2 : z.re + -(↑N * ↑n) = z.re - ↑n * ↑N\n⊢ |z.re + -(↑N * ↑⌊z.re / ↑N⌋)| ≤ ↑N","state_after":"case h\nz : ℍ\nN : ℕ\nhn : 0 < N\nn : ℤ := ⌊z.re / ↑N⌋\nh : (↑(↑N * -n) +ᵥ z).re = ↑(-↑N * ⌊z.re / ↑N⌋) + z.re\nhnn : 0 < ↑N\nh2 : z.re + -(↑N * ↑n) = z.re - ↑n * ↑N\n⊢ z.re - ↑⌊z.re / ↑N⌋ * ↑N ≤ ↑N","tactic":"rw [h2, abs_eq_self.2 (Int.sub_floor_div_mul_nonneg (z.re : ℝ) hnn)]","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Int.sub_floor_div_mul_nonneg","def_path":"Mathlib/Algebra/Order/Floor.lean","def_pos":[978,8],"def_end_pos":[978,32]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"UpperHalfPlane.re","def_path":"Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean","def_pos":[76,4],"def_end_pos":[76,6]},{"full_name":"abs_eq_self","def_path":"Mathlib/Algebra/Order/Group/Abs.lean","def_pos":[196,8],"def_end_pos":[196,19]}]},{"state_before":"case h\nz : ℍ\nN : ℕ\nhn : 0 < N\nn : ℤ := ⌊z.re / ↑N⌋\nh : (↑(↑N * -n) +ᵥ z).re = ↑(-↑N * ⌊z.re / ↑N⌋) + z.re\nhnn : 0 < ↑N\nh2 : z.re + -(↑N * ↑n) = z.re - ↑n * ↑N\n⊢ z.re - ↑⌊z.re / ↑N⌋ * ↑N ≤ ↑N","state_after":"no goals","tactic":"apply (Int.sub_floor_div_mul_lt (z.re : ℝ) hnn).le","premises":[{"full_name":"Int.sub_floor_div_mul_lt","def_path":"Mathlib/Algebra/Order/Floor.lean","def_pos":[981,8],"def_end_pos":[981,28]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"UpperHalfPlane.re","def_path":"Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean","def_pos":[76,4],"def_end_pos":[76,6]}]}]}
+{"url":"Mathlib/CategoryTheory/Dialectica/Basic.lean","commit":"","full_name":"CategoryTheory.Dial.isoMk_hom_F","start":[110,0],"end":[125,3],"file_path":"Mathlib/CategoryTheory/Dialectica/Basic.lean","tactics":[{"state_before":"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasFiniteProducts C\ninst✝ : HasPullbacks C\nX Y : Dial C\ne₁ : X.src ≅ Y.src\ne₂ : X.tgt ≅ Y.tgt\neq : X.rel = (Subobject.pullback (prod.map e₁.hom e₂.hom)).obj Y.rel\n⊢ (Subobject.pullback π(π₁, π₂ ≫ e₂.inv)).obj X.rel ≤ (Subobject.pullback (prod.map e₁.hom (𝟙 Y.tgt))).obj Y.rel","state_after":"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasFiniteProducts C\ninst✝ : HasPullbacks C\nX Y : Dial C\ne₁ : X.src ≅ Y.src\ne₂ : X.tgt ≅ Y.tgt\neq : X.rel = (Subobject.pullback (prod.map e₁.hom e₂.hom)).obj Y.rel\n⊢ (Subobject.pullback (π(π₁, π₂ ≫ e₂.inv) ≫ prod.map e₁.hom e₂.hom)).obj Y.rel ≤\n    (Subobject.pullback (prod.map e₁.hom (𝟙 Y.tgt))).obj Y.rel","tactic":"rw [eq, ← Subobject.pullback_comp]","premises":[{"full_name":"CategoryTheory.Subobject.pullback_comp","def_path":"Mathlib/CategoryTheory/Subobject/Basic.lean","def_pos":[498,8],"def_end_pos":[498,21]}]},{"state_before":"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasFiniteProducts C\ninst✝ : HasPullbacks C\nX Y : Dial C\ne₁ : X.src ≅ Y.src\ne₂ : X.tgt ≅ Y.tgt\neq : X.rel = (Subobject.pullback (prod.map e₁.hom e₂.hom)).obj Y.rel\n⊢ (Subobject.pullback (π(π₁, π₂ ≫ e₂.inv) ≫ prod.map e₁.hom e₂.hom)).obj Y.rel ≤\n    (Subobject.pullback (prod.map e₁.hom (𝟙 Y.tgt))).obj Y.rel","state_after":"case a\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasFiniteProducts C\ninst✝ : HasPullbacks C\nX Y : Dial C\ne₁ : X.src ≅ Y.src\ne₂ : X.tgt ≅ Y.tgt\neq : X.rel = (Subobject.pullback (prod.map e₁.hom e₂.hom)).obj Y.rel\n⊢ (Subobject.pullback (π(π₁, π₂ ≫ e₂.inv) ≫ prod.map e₁.hom e₂.hom)).obj Y.rel =\n    (Subobject.pullback (prod.map e₁.hom (𝟙 Y.tgt))).obj Y.rel","tactic":"apply le_of_eq","premises":[{"full_name":"le_of_eq","def_path":"Mathlib/Order/Defs.lean","def_pos":[60,8],"def_end_pos":[60,16]}]},{"state_before":"case a\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasFiniteProducts C\ninst✝ : HasPullbacks C\nX Y : Dial C\ne₁ : X.src ≅ Y.src\ne₂ : X.tgt ≅ Y.tgt\neq : X.rel = (Subobject.pullback (prod.map e₁.hom e₂.hom)).obj Y.rel\n⊢ (Subobject.pullback (π(π₁, π₂ ≫ e₂.inv) ≫ prod.map e₁.hom e₂.hom)).obj Y.rel =\n    (Subobject.pullback (prod.map e₁.hom (𝟙 Y.tgt))).obj Y.rel","state_after":"case a.e_self.e_self.e_f\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasFiniteProducts C\ninst✝ : HasPullbacks C\nX Y : Dial C\ne₁ : X.src ≅ Y.src\ne₂ : X.tgt ≅ Y.tgt\neq : X.rel = (Subobject.pullback (prod.map e₁.hom e₂.hom)).obj Y.rel\n⊢ π(π₁, π₂ ≫ e₂.inv) ≫ prod.map e₁.hom e₂.hom = prod.map e₁.hom (𝟙 Y.tgt)","tactic":"congr","premises":[]},{"state_before":"case a.e_self.e_self.e_f\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasFiniteProducts C\ninst✝ : HasPullbacks C\nX Y : Dial C\ne₁ : X.src ≅ Y.src\ne₂ : X.tgt ≅ Y.tgt\neq : X.rel = (Subobject.pullback (prod.map e₁.hom e₂.hom)).obj Y.rel\n⊢ π(π₁, π₂ ≫ e₂.inv) ≫ prod.map e₁.hom e₂.hom = prod.map e₁.hom (𝟙 Y.tgt)","state_after":"no goals","tactic":"ext <;> simp","premises":[]},{"state_before":"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasFiniteProducts C\ninst✝ : HasPullbacks C\nX Y : Dial C\ne₁ : X.src ≅ Y.src\ne₂ : X.tgt ≅ Y.tgt\neq : X.rel = (Subobject.pullback (prod.map e₁.hom e₂.hom)).obj Y.rel\n⊢ (Subobject.pullback π(π₁, π₂ ≫ e₂.hom)).obj Y.rel ≤ (Subobject.pullback (prod.map e₁.inv (𝟙 X.tgt))).obj X.rel","state_after":"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasFiniteProducts C\ninst✝ : HasPullbacks C\nX Y : Dial C\ne₁ : X.src ≅ Y.src\ne₂ : X.tgt ≅ Y.tgt\neq : X.rel = (Subobject.pullback (prod.map e₁.hom e₂.hom)).obj Y.rel\n⊢ (Subobject.pullback π(π₁, π₂ ≫ e₂.hom)).obj Y.rel ≤\n    (Subobject.pullback (prod.map e₁.inv (𝟙 X.tgt) ≫ prod.map e₁.hom e₂.hom)).obj Y.rel","tactic":"rw [eq, ← Subobject.pullback_comp]","premises":[{"full_name":"CategoryTheory.Subobject.pullback_comp","def_path":"Mathlib/CategoryTheory/Subobject/Basic.lean","def_pos":[498,8],"def_end_pos":[498,21]}]},{"state_before":"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasFiniteProducts C\ninst✝ : HasPullbacks C\nX Y : Dial C\ne₁ : X.src ≅ Y.src\ne₂ : X.tgt ≅ Y.tgt\neq : X.rel = (Subobject.pullback (prod.map e₁.hom e₂.hom)).obj Y.rel\n⊢ (Subobject.pullback π(π₁, π₂ ≫ e₂.hom)).obj Y.rel ≤\n    (Subobject.pullback (prod.map e₁.inv (𝟙 X.tgt) ≫ prod.map e₁.hom e₂.hom)).obj Y.rel","state_after":"case a\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasFiniteProducts C\ninst✝ : HasPullbacks C\nX Y : Dial C\ne₁ : X.src ≅ Y.src\ne₂ : X.tgt ≅ Y.tgt\neq : X.rel = (Subobject.pullback (prod.map e₁.hom e₂.hom)).obj Y.rel\n⊢ (Subobject.pullback π(π₁, π₂ ≫ e₂.hom)).obj Y.rel =\n    (Subobject.pullback (prod.map e₁.inv (𝟙 X.tgt) ≫ prod.map e₁.hom e₂.hom)).obj Y.rel","tactic":"apply le_of_eq","premises":[{"full_name":"le_of_eq","def_path":"Mathlib/Order/Defs.lean","def_pos":[60,8],"def_end_pos":[60,16]}]},{"state_before":"case a\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasFiniteProducts C\ninst✝ : HasPullbacks C\nX Y : Dial C\ne₁ : X.src ≅ Y.src\ne₂ : X.tgt ≅ Y.tgt\neq : X.rel = (Subobject.pullback (prod.map e₁.hom e₂.hom)).obj Y.rel\n⊢ (Subobject.pullback π(π₁, π₂ ≫ e₂.hom)).obj Y.rel =\n    (Subobject.pullback (prod.map e₁.inv (𝟙 X.tgt) ≫ prod.map e₁.hom e₂.hom)).obj Y.rel","state_after":"case a.e_self.e_self.e_f\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasFiniteProducts C\ninst✝ : HasPullbacks C\nX Y : Dial C\ne₁ : X.src ≅ Y.src\ne₂ : X.tgt ≅ Y.tgt\neq : X.rel = (Subobject.pullback (prod.map e₁.hom e₂.hom)).obj Y.rel\n⊢ π(π₁, π₂ ≫ e₂.hom) = prod.map e₁.inv (𝟙 X.tgt) ≫ prod.map e₁.hom e₂.hom","tactic":"congr","premises":[]},{"state_before":"case a.e_self.e_self.e_f\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasFiniteProducts C\ninst✝ : HasPullbacks C\nX Y : Dial C\ne₁ : X.src ≅ Y.src\ne₂ : X.tgt ≅ Y.tgt\neq : X.rel = (Subobject.pullback (prod.map e₁.hom e₂.hom)).obj Y.rel\n⊢ π(π₁, π₂ ≫ e₂.hom) = prod.map e₁.inv (𝟙 X.tgt) ≫ prod.map e₁.hom e₂.hom","state_after":"no goals","tactic":"ext <;> simp","premises":[]}]}
+{"url":"Mathlib/MeasureTheory/Measure/Stieltjes.lean","commit":"","full_name":"StieltjesFunction.measurableSet_Ioi","start":[301,0],"end":[318,75],"file_path":"Mathlib/MeasureTheory/Measure/Stieltjes.lean","tactics":[{"state_before":"f : StieltjesFunction\nc : ℝ\n⊢ MeasurableSet (Ioi c)","state_after":"f : StieltjesFunction\nc : ℝ\nt : Set ℝ\n⊢ f.length (t ∩ Ioi c) + f.length (t \\ Ioi c) ≤ f.length t","tactic":"refine OuterMeasure.ofFunction_caratheodory fun t => ?_","premises":[{"full_name":"MeasureTheory.OuterMeasure.ofFunction_caratheodory","def_path":"Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean","def_pos":[158,8],"def_end_pos":[158,31]}]},{"state_before":"f : StieltjesFunction\nc : ℝ\nt : Set ℝ\n⊢ f.length (t ∩ Ioi c) + f.length (t \\ Ioi c) ≤ f.length t","state_after":"f : StieltjesFunction\nc : ℝ\nt : Set ℝ\na b : ℝ\nh : t ⊆ Ioc a b\n⊢ f.length (t ∩ Ioi c) + f.length (t \\ Ioi c) ≤ ofReal (↑f b - ↑f a)","tactic":"refine le_iInf fun a => le_iInf fun b => le_iInf fun h => ?_","premises":[{"full_name":"le_iInf","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[664,8],"def_end_pos":[664,15]}]},{"state_before":"f : StieltjesFunction\nc : ℝ\nt : Set ℝ\na b : ℝ\nh : t ⊆ Ioc a b\n⊢ f.length (t ∩ Ioi c) + f.length (t \\ Ioi c) ≤ ofReal (↑f b - ↑f a)","state_after":"f : StieltjesFunction\nc : ℝ\nt : Set ℝ\na b : ℝ\nh : t ⊆ Ioc a b\n⊢ f.length (Ioc a b ∩ Ioi c) + f.length (Ioc a b \\ Ioi c) ≤ ofReal (↑f b - ↑f a)","tactic":"refine\n    le_trans\n      (add_le_add (f.length_mono <| inter_subset_inter_left _ h)\n        (f.length_mono <| diff_subset_diff_left h)) ?_","premises":[{"full_name":"Set.diff_subset_diff_left","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1485,8],"def_end_pos":[1485,29]},{"full_name":"Set.inter_subset_inter_left","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[810,8],"def_end_pos":[810,31]},{"full_name":"StieltjesFunction.length_mono","def_path":"Mathlib/MeasureTheory/Measure/Stieltjes.lean","def_pos":[189,8],"def_end_pos":[189,19]},{"full_name":"add_le_add","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[182,31],"def_end_pos":[182,41]},{"full_name":"le_trans","def_path":"Mathlib/Order/Defs.lean","def_pos":[48,8],"def_end_pos":[48,16]}]},{"state_before":"f : StieltjesFunction\nc : ℝ\nt : Set ℝ\na b : ℝ\nh : t ⊆ Ioc a b\n⊢ f.length (Ioc a b ∩ Ioi c) + f.length (Ioc a b \\ Ioi c) ≤ ofReal (↑f b - ↑f a)","state_after":"case inl.inl\nf : StieltjesFunction\nc : ℝ\nt : Set ℝ\na b : ℝ\nh : t ⊆ Ioc a b\nhac : a ≤ c\nhbc : b ≤ c\n⊢ f.length (Ioc a b ∩ Ioi c) + f.length (Ioc a b \\ Ioi c) ≤ ofReal (↑f b - ↑f a)\n\ncase inl.inr\nf : StieltjesFunction\nc : ℝ\nt : Set ℝ\na b : ℝ\nh : t ⊆ Ioc a b\nhac : a ≤ c\nhbc : c ≤ b\n⊢ f.length (Ioc a b ∩ Ioi c) + f.length (Ioc a b \\ Ioi c) ≤ ofReal (↑f b - ↑f a)\n\ncase inr.inl\nf : StieltjesFunction\nc : ℝ\nt : Set ℝ\na b : ℝ\nh : t ⊆ Ioc a b\nhac : c ≤ a\nhbc : b ≤ c\n⊢ f.length (Ioc a b ∩ Ioi c) + f.length (Ioc a b \\ Ioi c) ≤ ofReal (↑f b - ↑f a)\n\ncase inr.inr\nf : StieltjesFunction\nc : ℝ\nt : Set ℝ\na b : ℝ\nh : t ⊆ Ioc a b\nhac : c ≤ a\nhbc : c ≤ b\n⊢ f.length (Ioc a b ∩ Ioi c) + f.length (Ioc a b \\ Ioi c) ≤ ofReal (↑f b - ↑f a)","tactic":"rcases le_total a c with hac | hac <;> rcases le_total b c with hbc | hbc","premises":[{"full_name":"le_total","def_path":"Mathlib/Order/Defs.lean","def_pos":[254,8],"def_end_pos":[254,16]}]}]}
+{"url":"Mathlib/Algebra/Homology/HomologicalComplex.lean","commit":"","full_name":"HomologicalComplex.XIsoOfEq_hom_naturality","start":[355,0],"end":[357,80],"file_path":"Mathlib/Algebra/Homology/HomologicalComplex.lean","tactics":[{"state_before":"ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC K L : HomologicalComplex V c\nφ : K ⟶ L\nn n' : ι\nh : n = n'\n⊢ φ.f n ≫ (L.XIsoOfEq h).hom = (K.XIsoOfEq h).hom ≫ φ.f n'","state_after":"ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC K L : HomologicalComplex V c\nφ : K ⟶ L\nn : ι\n⊢ φ.f n ≫ (L.XIsoOfEq ⋯).hom = (K.XIsoOfEq ⋯).hom ≫ φ.f n","tactic":"subst h","premises":[]},{"state_before":"ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC K L : HomologicalComplex V c\nφ : K ⟶ L\nn : ι\n⊢ φ.f n ≫ (L.XIsoOfEq ⋯).hom = (K.XIsoOfEq ⋯).hom ≫ φ.f n","state_after":"no goals","tactic":"simp","premises":[]}]}
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: Type u₁\ninst✝² : Category.{v₁, u₁} J\nK : Type u₂\ninst✝¹ : Category.{v₂, u₂} K\nC : Type u₃\ninst✝ : Category.{v₃, u₃} C\nF✝ : J ⥤ C\nt✝ : Cone F✝\nF : J ⥤ C\ns : Cone F\nG : K ⥤ C\nt : Cone G\nP : IsLimit s\nQ : IsLimit t\ne : J ≌ K\nw : e.functor ⋙ G ≅ F\nw' : e.inverse ⋙ F ≅ G := (isoWhiskerLeft e.inverse w).symm ≪≫ e.invFunIdAssoc G\nj : J\n⊢ s.π.app (e.inverse.obj (e.functor.obj j)) ≫\n      w.inv.app (e.inverse.obj (e.functor.obj j)) ≫ G.map (e.counit.app (e.functor.obj j)) ≫ w.hom.app j =\n    s.π.app j","state_after":"J : Type u₁\ninst✝² : Category.{v₁, u₁} J\nK : Type u₂\ninst✝¹ : Category.{v₂, u₂} K\nC : Type u₃\ninst✝ : Category.{v₃, u₃} C\nF✝ : J ⥤ C\nt✝ : Cone F✝\nF : J ⥤ C\ns : Cone F\nG : K ⥤ C\nt : Cone G\nP : IsLimit s\nQ : IsLimit t\ne : J ≌ K\nw : e.functor ⋙ G ≅ F\nw' : e.inverse ⋙ F ≅ G := (isoWhiskerLeft e.inverse w).symm ≪≫ e.invFunIdAssoc G\nj : J\n⊢ s.π.app (e.inverse.obj (e.functor.obj j)) ≫\n      w.inv.app (e.inverse.obj (e.functor.obj j)) ≫ (e.functor ⋙ G).map (e.unitInv.app j) ≫ w.hom.app j =\n    s.π.app j","tactic":"rw [counit_app_functor, ← Functor.comp_map]","premises":[{"full_name":"CategoryTheory.Equivalence.counit_app_functor","def_path":"Mathlib/CategoryTheory/Equivalence.lean","def_pos":[156,8],"def_end_pos":[156,26]},{"full_name":"CategoryTheory.Functor.comp_map","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[109,8],"def_end_pos":[109,16]}]},{"state_before":"J : Type u₁\ninst✝² : Category.{v₁, u₁} J\nK : Type u₂\ninst✝¹ : Category.{v₂, u₂} K\nC : Type u₃\ninst✝ : Category.{v₃, u₃} C\nF✝ : J ⥤ C\nt✝ : Cone F✝\nF : J ⥤ C\ns : Cone F\nG : K ⥤ C\nt : Cone G\nP : IsLimit s\nQ : IsLimit t\ne : J ≌ K\nw : e.functor ⋙ G ≅ F\nw' : e.inverse ⋙ F ≅ G := (isoWhiskerLeft e.inverse w).symm ≪≫ e.invFunIdAssoc G\nj : J\n⊢ s.π.app (e.inverse.obj (e.functor.obj j)) ≫\n      w.inv.app (e.inverse.obj (e.functor.obj j)) ≫ (e.functor ⋙ G).map (e.unitInv.app j) ≫ w.hom.app j =\n    s.π.app j","state_after":"J : Type u₁\ninst✝² : Category.{v₁, u₁} J\nK : Type u₂\ninst✝¹ : Category.{v₂, u₂} K\nC : Type u₃\ninst✝ : Category.{v₃, u₃} C\nF✝ : J ⥤ C\nt✝ : Cone F✝\nF : J ⥤ C\ns : Cone F\nG : K ⥤ C\nt : Cone G\nP : IsLimit s\nQ : IsLimit t\ne : J ≌ K\nw : e.functor ⋙ G ≅ F\nw' : e.inverse ⋙ F ≅ G := (isoWhiskerLeft e.inverse w).symm ≪≫ e.invFunIdAssoc G\nj : J\nl : w.hom.app j = w.hom.app ((𝟭 J).obj j)\n⊢ s.π.app (e.inverse.obj (e.functor.obj j)) ≫\n      w.inv.app (e.inverse.obj (e.functor.obj j)) ≫ (e.functor ⋙ G).map (e.unitInv.app j) ≫ w.hom.app j =\n    s.π.app j","tactic":"have l :\n        NatTrans.app w.hom j = NatTrans.app w.hom (Prefunctor.obj (𝟭 J).toPrefunctor j) := by dsimp","premises":[{"full_name":"CategoryTheory.Functor.id","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[72,14],"def_end_pos":[72,16]},{"full_name":"CategoryTheory.Iso.hom","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[51,2],"def_end_pos":[51,5]},{"full_name":"CategoryTheory.NatTrans.app","def_path":"Mathlib/CategoryTheory/NatTrans.lean","def_pos":[48,2],"def_end_pos":[48,5]},{"full_name":"Prefunctor.obj","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[55,2],"def_end_pos":[55,5]}]},{"state_before":"J : Type u₁\ninst✝² : Category.{v₁, u₁} J\nK : Type u₂\ninst✝¹ : Category.{v₂, u₂} K\nC : Type u₃\ninst✝ : Category.{v₃, u₃} C\nF✝ : J ⥤ C\nt✝ : Cone F✝\nF : J ⥤ C\ns : Cone F\nG : K ⥤ C\nt : Cone G\nP : IsLimit s\nQ : IsLimit t\ne : J ≌ K\nw : e.functor ⋙ G ≅ F\nw' : e.inverse ⋙ F ≅ G := (isoWhiskerLeft e.inverse w).symm ≪≫ e.invFunIdAssoc G\nj : J\nl : w.hom.app j = w.hom.app ((𝟭 J).obj j)\n⊢ s.π.app (e.inverse.obj (e.functor.obj j)) ≫\n      w.inv.app (e.inverse.obj (e.functor.obj j)) ≫ (e.functor ⋙ G).map (e.unitInv.app j) ≫ w.hom.app j =\n    s.π.app j","state_after":"J : Type u₁\ninst✝² : Category.{v₁, u₁} J\nK : Type u₂\ninst✝¹ : Category.{v₂, u₂} K\nC : Type u₃\ninst✝ : Category.{v₃, u₃} C\nF✝ : J ⥤ C\nt✝ : Cone F✝\nF : J ⥤ C\ns : Cone F\nG : K ⥤ C\nt : Cone G\nP : IsLimit s\nQ : IsLimit t\ne : J ≌ K\nw : e.functor ⋙ G ≅ F\nw' : e.inverse ⋙ F ≅ G := (isoWhiskerLeft e.inverse w).symm ≪≫ e.invFunIdAssoc G\nj : J\nl : w.hom.app j = w.hom.app ((𝟭 J).obj j)\n⊢ s.π.app (e.inverse.obj (e.functor.obj j)) ≫\n      w.inv.app (e.inverse.obj (e.functor.obj j)) ≫\n        w.hom.app ((e.functor ⋙ e.inverse).obj j) ≫ F.map (e.unitInv.app j) =\n    s.π.app j","tactic":"rw [l,w.hom.naturality]","premises":[{"full_name":"CategoryTheory.Iso.hom","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[51,2],"def_end_pos":[51,5]},{"full_name":"CategoryTheory.NatTrans.naturality","def_path":"Mathlib/CategoryTheory/NatTrans.lean","def_pos":[50,2],"def_end_pos":[50,12]}]},{"state_before":"J : Type u₁\ninst✝² : Category.{v₁, u₁} J\nK : Type u₂\ninst✝¹ : Category.{v₂, u₂} K\nC : Type u₃\ninst✝ : Category.{v₃, u₃} C\nF✝ : J ⥤ C\nt✝ : Cone F✝\nF : J ⥤ C\ns : Cone F\nG : K ⥤ C\nt : Cone G\nP : IsLimit s\nQ : IsLimit t\ne : J ≌ K\nw : e.functor ⋙ G ≅ F\nw' : e.inverse ⋙ F ≅ G := (isoWhiskerLeft e.inverse w).symm ≪≫ e.invFunIdAssoc G\nj : J\nl : w.hom.app j = w.hom.app ((𝟭 J).obj j)\n⊢ s.π.app (e.inverse.obj (e.functor.obj j)) ≫\n      w.inv.app (e.inverse.obj (e.functor.obj j)) ≫\n        w.hom.app ((e.functor ⋙ e.inverse).obj j) ≫ F.map (e.unitInv.app j) =\n    s.π.app j","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"J : Type u₁\ninst✝² : Category.{v₁, u₁} J\nK : Type u₂\ninst✝¹ : Category.{v₂, u₂} K\nC : Type u₃\ninst✝ : Category.{v₃, u₃} C\nF✝ : J ⥤ C\nt✝ : Cone F✝\nF : J ⥤ C\ns : Cone F\nG : K ⥤ C\nt : Cone G\nP : IsLimit s\nQ : IsLimit t\ne : J ≌ K\nw : e.functor ⋙ G ≅ F\nw' : e.inverse ⋙ F ≅ G := (isoWhiskerLeft e.inverse w).symm ≪≫ e.invFunIdAssoc G\n⊢ P.lift ((Cones.equivalenceOfReindexing e w).functor.obj t) ≫\n      Q.lift ((Cones.equivalenceOfReindexing e.symm w').functor.obj s) =\n    𝟙 t.pt","state_after":"J : Type u₁\ninst✝² : Category.{v₁, u₁} J\nK : Type u₂\ninst✝¹ : Category.{v₂, u₂} K\nC : Type u₃\ninst✝ : Category.{v₃, u₃} C\nF✝ : J ⥤ C\nt✝ : Cone F✝\nF : J ⥤ C\ns : Cone F\nG : K ⥤ C\nt : Cone G\nP : IsLimit s\nQ : IsLimit t\ne : J ≌ K\nw : e.functor ⋙ G ≅ F\nw' : e.inverse ⋙ F ≅ G := (isoWhiskerLeft e.inverse w).symm ≪≫ e.invFunIdAssoc G\n⊢ ∀ (j : K),\n    (P.lift ((Cones.equivalenceOfReindexing e w).functor.obj t) ≫\n          Q.lift ((Cones.equivalenceOfReindexing e.symm w').functor.obj s)) ≫\n        t.π.app j =\n      𝟙 t.pt ≫ t.π.app j","tactic":"apply hom_ext Q","premises":[{"full_name":"CategoryTheory.Limits.IsLimit.hom_ext","def_path":"Mathlib/CategoryTheory/Limits/IsLimit.lean","def_pos":[201,8],"def_end_pos":[201,15]}]},{"state_before":"J : Type u₁\ninst✝² : Category.{v₁, u₁} J\nK : Type u₂\ninst✝¹ : Category.{v₂, u₂} K\nC : Type u₃\ninst✝ : Category.{v₃, u₃} C\nF✝ : J ⥤ C\nt✝ : Cone F✝\nF : J ⥤ C\ns : Cone F\nG : K ⥤ C\nt : Cone G\nP : IsLimit s\nQ : IsLimit t\ne : J ≌ K\nw : e.functor ⋙ G ≅ F\nw' : e.inverse ⋙ F ≅ G := (isoWhiskerLeft e.inverse w).symm ≪≫ e.invFunIdAssoc G\n⊢ ∀ (j : K),\n    (P.lift ((Cones.equivalenceOfReindexing e w).functor.obj t) ≫\n          Q.lift ((Cones.equivalenceOfReindexing e.symm w').functor.obj s)) ≫\n        t.π.app j =\n      𝟙 t.pt ≫ t.π.app j","state_after":"no goals","tactic":"aesop_cat","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]}]}]}
+{"url":"Mathlib/Data/Set/Finite.lean","commit":"","full_name":"Set.finite_of_forall_not_lt_lt","start":[1521,0],"end":[1524,91],"file_path":"Mathlib/Data/Set/Finite.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ninst✝ : LinearOrder α\ns : Set α\nh : ∀ x ∈ s, ∀ y ∈ s, ∀ z ∈ s, x < y → y < z → False\n⊢ ∀ ⦃x y z : ↑s⦄, x < y → y < z → False","state_after":"no goals","tactic":"simpa only [SetCoe.forall'] using h","premises":[{"full_name":"SetCoe.forall'","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[158,8],"def_end_pos":[158,22]}]}]}
+{"url":"Mathlib/CategoryTheory/Limits/Shapes/Images.lean","commit":"","full_name":"CategoryTheory.Limits.IsImage.isoExt_hom_m","start":[202,0],"end":[202,66],"file_path":"Mathlib/CategoryTheory/Limits/Shapes/Images.lean","tactics":[{"state_before":"C : Type u\ninst✝ : Category.{v, u} C\nX Y : C\nf : X ⟶ Y\nF F' : MonoFactorisation f\nhF : IsImage F\nhF' : IsImage F'\n⊢ (hF.isoExt hF').hom ≫ F'.m = F.m","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/Algebra/ModEq.lean","commit":"","full_name":"AddCommGroup.ModEq.nsmul","start":[132,0],"end":[133,58],"file_path":"Mathlib/Algebra/ModEq.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : AddCommGroup α\np a a₁ a₂ b b₁ b₂ c : α\nn : ℕ\nz m : ℤ\nhm : b - a = m • p\n⊢ n • b - n • a = m • n • p","state_after":"no goals","tactic":"rw [← smul_sub, hm, smul_comm]","premises":[{"full_name":"SMulCommClass.smul_comm","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[154,2],"def_end_pos":[154,11]},{"full_name":"smul_sub","def_path":"Mathlib/Algebra/GroupWithZero/Action/Defs.lean","def_pos":[279,8],"def_end_pos":[279,16]}]}]}
+{"url":"Mathlib/RingTheory/AdicCompletion/Basic.lean","commit":"","full_name":"AdicCompletion.transitionMap_mk","start":[310,0],"end":[315,5],"file_path":"Mathlib/RingTheory/AdicCompletion/Basic.lean","tactics":[{"state_before":"R : Type u_1\ninst✝⁴ : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u_3\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nm n : ℕ\nhmn : m ≤ n\nx : M\n⊢ (transitionMap I M hmn) (Submodule.Quotient.mk x) = Submodule.Quotient.mk x","state_after":"no goals","tactic":"rfl","premises":[]}]}
+{"url":"Mathlib/MeasureTheory/Integral/SetToL1.lean","commit":"","full_name":"MeasureTheory.setToFun_smul_left","start":[1187,0],"end":[1191,44],"file_path":"Mathlib/MeasureTheory/Integral/SetToL1.lean","tactics":[{"state_before":"α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nhT : DominatedFinMeasAdditive μ T C\nc : ℝ\nf : α → E\n⊢ setToFun μ (fun s => c • T s) ⋯ f = c • setToFun μ T hT f","state_after":"case pos\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nhT : DominatedFinMeasAdditive μ T C\nc : ℝ\nf : α → E\nhf : Integrable f μ\n⊢ setToFun μ (fun s => c • T s) ⋯ f = c • setToFun μ T hT f\n\ncase neg\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nhT : DominatedFinMeasAdditive μ T C\nc : ℝ\nf : α → E\nhf : ¬Integrable f μ\n⊢ setToFun μ (fun s => c • T s) ⋯ f = c • setToFun μ T hT f","tactic":"by_cases hf : Integrable f μ","premises":[{"full_name":"MeasureTheory.Integrable","def_path":"Mathlib/MeasureTheory/Function/L1Space.lean","def_pos":[389,4],"def_end_pos":[389,14]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
+{"url":"Mathlib/Topology/MetricSpace/Basic.lean","commit":"","full_name":"MetricSpace.ext","start":[40,0],"end":[44,44],"file_path":"Mathlib/Topology/MetricSpace/Basic.lean","tactics":[{"state_before":"α✝ : Type u\nβ : Type v\nX : Type u_1\nι : Type u_2\ninst✝ : PseudoMetricSpace α✝\nα : Type u_3\nm m' : MetricSpace α\nh : PseudoMetricSpace.toDist = PseudoMetricSpace.toDist\n⊢ m = m'","state_after":"case mk\nα✝ : Type u\nβ : Type v\nX : Type u_1\nι : Type u_2\ninst✝ : PseudoMetricSpace α✝\nα : Type u_3\nm' : MetricSpace α\ntoPseudoMetricSpace✝ : PseudoMetricSpace α\neq_of_dist_eq_zero✝ : ∀ {x y : α}, dist x y = 0 → x = y\nh : PseudoMetricSpace.toDist = PseudoMetricSpace.toDist\n⊢ mk eq_of_dist_eq_zero✝ = m'","tactic":"cases m","premises":[]},{"state_before":"case mk\nα✝ : Type u\nβ : Type v\nX : Type u_1\nι : Type u_2\ninst✝ : PseudoMetricSpace α✝\nα : Type u_3\nm' : MetricSpace α\ntoPseudoMetricSpace✝ : PseudoMetricSpace α\neq_of_dist_eq_zero✝ : ∀ {x y : α}, dist x y = 0 → x = y\nh : PseudoMetricSpace.toDist = PseudoMetricSpace.toDist\n⊢ mk eq_of_dist_eq_zero✝ = m'","state_after":"case mk.mk\nα✝ : Type u\nβ : Type v\nX : Type u_1\nι : Type u_2\ninst✝ : PseudoMetricSpace α✝\nα : Type u_3\ntoPseudoMetricSpace✝¹ : PseudoMetricSpace α\neq_of_dist_eq_zero✝¹ : ∀ {x y : α}, dist x y = 0 → x = y\ntoPseudoMetricSpace✝ : PseudoMetricSpace α\neq_of_dist_eq_zero✝ : ∀ {x y : α}, dist x y = 0 → x = y\nh : PseudoMetricSpace.toDist = PseudoMetricSpace.toDist\n⊢ mk eq_of_dist_eq_zero✝¹ = mk eq_of_dist_eq_zero✝","tactic":"cases m'","premises":[]},{"state_before":"case mk.mk\nα✝ : Type u\nβ : Type v\nX : Type u_1\nι : Type u_2\ninst✝ : PseudoMetricSpace α✝\nα : Type u_3\ntoPseudoMetricSpace✝¹ : PseudoMetricSpace α\neq_of_dist_eq_zero✝¹ : ∀ {x y : α}, dist x y = 0 → x = y\ntoPseudoMetricSpace✝ : PseudoMetricSpace α\neq_of_dist_eq_zero✝ : ∀ {x y : α}, dist x y = 0 → x = y\nh : PseudoMetricSpace.toDist = PseudoMetricSpace.toDist\n⊢ mk eq_of_dist_eq_zero✝¹ = mk eq_of_dist_eq_zero✝","state_after":"case mk.mk.e_toPseudoMetricSpace\nα✝ : Type u\nβ : Type v\nX : Type u_1\nι : Type u_2\ninst✝ : PseudoMetricSpace α✝\nα : Type u_3\ntoPseudoMetricSpace✝¹ : PseudoMetricSpace α\neq_of_dist_eq_zero✝¹ : ∀ {x y : α}, dist x y = 0 → x = y\ntoPseudoMetricSpace✝ : PseudoMetricSpace α\neq_of_dist_eq_zero✝ : ∀ {x y : α}, dist x y = 0 → x = y\nh : PseudoMetricSpace.toDist = PseudoMetricSpace.toDist\n⊢ toPseudoMetricSpace✝¹ = toPseudoMetricSpace✝","tactic":"congr","premises":[]},{"state_before":"case mk.mk.e_toPseudoMetricSpace\nα✝ : Type u\nβ : Type v\nX : Type u_1\nι : Type u_2\ninst✝ : PseudoMetricSpace α✝\nα : Type u_3\ntoPseudoMetricSpace✝¹ : PseudoMetricSpace α\neq_of_dist_eq_zero✝¹ : ∀ {x y : α}, dist x y = 0 → x = y\ntoPseudoMetricSpace✝ : PseudoMetricSpace α\neq_of_dist_eq_zero✝ : ∀ {x y : α}, dist x y = 0 → x = y\nh : PseudoMetricSpace.toDist = PseudoMetricSpace.toDist\n⊢ toPseudoMetricSpace✝¹ = toPseudoMetricSpace✝","state_after":"case mk.mk.e_toPseudoMetricSpace.h\nα✝ : Type u\nβ : Type v\nX : Type u_1\nι : Type u_2\ninst✝ : PseudoMetricSpace α✝\nα : Type u_3\ntoPseudoMetricSpace✝¹ : PseudoMetricSpace α\neq_of_dist_eq_zero✝¹ : ∀ {x y : α}, dist x y = 0 → x = y\ntoPseudoMetricSpace✝ : PseudoMetricSpace α\neq_of_dist_eq_zero✝ : ∀ {x y : α}, dist x y = 0 → x = y\nh : PseudoMetricSpace.toDist = PseudoMetricSpace.toDist\n⊢ PseudoMetricSpace.toDist = PseudoMetricSpace.toDist","tactic":"ext1","premises":[]},{"state_before":"case mk.mk.e_toPseudoMetricSpace.h\nα✝ : Type u\nβ : Type v\nX : Type u_1\nι : Type u_2\ninst✝ : PseudoMetricSpace α✝\nα : Type u_3\ntoPseudoMetricSpace✝¹ : PseudoMetricSpace α\neq_of_dist_eq_zero✝¹ : ∀ {x y : α}, dist x y = 0 → x = y\ntoPseudoMetricSpace✝ : PseudoMetricSpace α\neq_of_dist_eq_zero✝ : ∀ {x y : α}, dist x y = 0 → x = y\nh : PseudoMetricSpace.toDist = PseudoMetricSpace.toDist\n⊢ PseudoMetricSpace.toDist = PseudoMetricSpace.toDist","state_after":"no goals","tactic":"assumption","premises":[]}]}
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(fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nthis : a ≤ 0 ∨ 0 ≤ a\n⊢ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0 ↔ (a ≤ 0 ∨ 0 ≤ b) ∧ (b ≤ 0 ∨ 0 ≤ a)","state_after":"α : Type u\nβ : Type u_1\ninst✝⁶ : Semiring α\ninst✝⁵ : LinearOrder α\na b c d : α\ninst✝⁴ : ExistsAddOfLE α\ninst✝³ : PosMulStrictMono α\ninst✝² : MulPosStrictMono α\ninst✝¹ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝ : ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nthis✝ : a ≤ 0 ∨ 0 ≤ a\nthis : b ≤ 0 ∨ 0 ≤ b\n⊢ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0 ↔ (a ≤ 0 ∨ 0 ≤ b) ∧ (b ≤ 0 ∨ 0 ≤ a)","tactic":"have := le_total b 0","premises":[{"full_name":"le_total","def_path":"Mathlib/Order/Defs.lean","def_pos":[254,8],"def_end_pos":[254,16]}]},{"state_before":"α : Type u\nβ : Type u_1\ninst✝⁶ : Semiring α\ninst✝⁵ : LinearOrder α\na b c d : α\ninst✝⁴ : ExistsAddOfLE α\ninst✝³ : PosMulStrictMono α\ninst✝² : MulPosStrictMono α\ninst✝¹ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝ : 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+{"url":"Mathlib/GroupTheory/Coset.lean","commit":"","full_name":"rightAddCoset_zero","start":[107,0],"end":[109,35],"file_path":"Mathlib/GroupTheory/Coset.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : Monoid α\ns : Set α\n⊢ ∀ (x : α), x ∈ op 1 • s ↔ x ∈ s","state_after":"no goals","tactic":"simp [← image_smul]","premises":[{"full_name":"Set.image_smul","def_path":"Mathlib/Data/Set/Pointwise/SMul.lean","def_pos":[232,8],"def_end_pos":[232,18]}]}]}
+{"url":"Mathlib/GroupTheory/PushoutI.lean","commit":"","full_name":"Monoid.PushoutI.NormalWord.summand_smul_def","start":[458,0],"end":[464,5],"file_path":"Mathlib/GroupTheory/PushoutI.lean","tactics":[{"state_before":"ι : Type u_1\nG : ι → Type u_2\nH : Type u_3\nK : Type u_4\ninst✝⁴ : Monoid K\ninst✝³ : (i : ι) → Group (G i)\ninst✝² : Group H\nφ : (i : ι) → H →* G i\nd : Transversal φ\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → DecidableEq (G i)\ni : ι\ng : G i\nw : NormalWord d\n⊢ (of i) g • w =\n    (equivPair i).symm\n      (let __src := (equivPair i) w;\n      { head := g * ((equivPair i) w).head, tail := __src.tail, fstIdx_ne := ⋯, normalized := ⋯ })","state_after":"ι : Type u_1\nG : ι → Type u_2\nH : Type u_3\nK : Type u_4\ninst✝⁴ : Monoid K\ninst✝³ : (i : ι) → Group (G i)\ninst✝² : Group H\nφ : (i : ι) → H →* G i\nd : Transversal φ\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → DecidableEq (G i)\ni : ι\ng : G i\nw : NormalWord d\n⊢ (lift (fun i => MulAction.toEndHom) MulAction.toEndHom ⋯) ((of i) g) w =\n    (equivPair i).symm\n      { head := g * ((equivPair i) w).head, tail := ((equivPair i) w).tail, fstIdx_ne := ⋯, normalized := ⋯ }","tactic":"dsimp [NormalWord.mulAction, instHSMul, SMul.smul]","premises":[{"full_name":"Monoid.PushoutI.NormalWord.mulAction","def_path":"Mathlib/GroupTheory/PushoutI.lean","def_pos":[436,23],"def_end_pos":[436,32]},{"full_name":"SMul.smul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[101,2],"def_end_pos":[101,6]},{"full_name":"instHSMul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[149,9],"def_end_pos":[149,18]}]},{"state_before":"ι : Type u_1\nG : ι → Type u_2\nH : Type u_3\nK : Type u_4\ninst✝⁴ : Monoid K\ninst✝³ : (i : ι) → Group (G i)\ninst✝² : Group H\nφ : (i : ι) → H →* G i\nd : Transversal φ\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → DecidableEq (G i)\ni : ι\ng : G i\nw : NormalWord d\n⊢ (lift (fun i => MulAction.toEndHom) MulAction.toEndHom ⋯) ((of i) g) w =\n    (equivPair i).symm\n      { head := g * ((equivPair i) w).head, tail := ((equivPair i) w).tail, fstIdx_ne := ⋯, normalized := ⋯ }","state_after":"ι : Type u_1\nG : ι → Type u_2\nH : Type u_3\nK : Type u_4\ninst✝⁴ : Monoid K\ninst✝³ : (i : ι) → Group (G i)\ninst✝² : Group H\nφ : (i : ι) → H →* G i\nd : Transversal φ\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → DecidableEq (G i)\ni : ι\ng : G i\nw : NormalWord d\n⊢ MulAction.toEndHom g w =\n    (equivPair i).symm\n      { head := g * ((equivPair i) w).head, tail := ((equivPair i) w).tail, fstIdx_ne := ⋯, normalized := ⋯ }","tactic":"rw [lift_of]","premises":[{"full_name":"Monoid.PushoutI.lift_of","def_path":"Mathlib/GroupTheory/PushoutI.lean","def_pos":[110,8],"def_end_pos":[110,15]}]},{"state_before":"ι : Type u_1\nG : ι → Type u_2\nH : Type u_3\nK : Type u_4\ninst✝⁴ : Monoid K\ninst✝³ : (i : ι) → Group (G i)\ninst✝² : Group H\nφ : (i : ι) → H →* G i\nd : Transversal φ\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → DecidableEq (G i)\ni : ι\ng : G i\nw : NormalWord d\n⊢ MulAction.toEndHom g w =\n    (equivPair i).symm\n      { head := g * ((equivPair i) w).head, tail := ((equivPair i) w).tail, fstIdx_ne := ⋯, normalized := ⋯ }","state_after":"no goals","tactic":"rfl","premises":[]}]}
+{"url":"Mathlib/Data/Int/Interval.lean","commit":"","full_name":"Int.card_uIcc","start":[113,0],"end":[121,97],"file_path":"Mathlib/Data/Int/Interval.lean","tactics":[{"state_before":"a b : ℤ\n⊢ ofNat (range (a ⊔ b + 1 - a ⊓ b).toNat).card = ofNat ((b - a).natAbs + 1)","state_after":"a b : ℤ\n⊢ ↑(range (a ⊔ b + 1 - a ⊓ b).toNat).card = ↑((b - a).natAbs + 1)","tactic":"change ((↑) : ℕ → ℤ) _ = ((↑) : ℕ → ℤ) _","premises":[{"full_name":"Int","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean","def_pos":[40,10],"def_end_pos":[40,13]},{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]}]},{"state_before":"a b : ℤ\n⊢ ↑(range (a ⊔ b + 1 - a ⊓ b).toNat).card = ↑((b - a).natAbs + 1)","state_after":"no goals","tactic":"rw [card_range, sup_eq_max, inf_eq_min,\n        Int.toNat_of_nonneg (sub_nonneg_of_le <| le_add_one min_le_max), Int.ofNat_add,\n        Int.natCast_natAbs, add_comm, add_sub_assoc, max_sub_min_eq_abs, add_comm, Int.ofNat_one]","premises":[{"full_name":"Finset.card_range","def_path":"Mathlib/Data/Finset/Card.lean","def_pos":[179,8],"def_end_pos":[179,18]},{"full_name":"Int.le_add_one","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Order.lean","def_pos":[596,8],"def_end_pos":[596,18]},{"full_name":"Int.natCast_natAbs","def_path":"Mathlib/Algebra/Order/Group/Int.lean","def_pos":[50,25],"def_end_pos":[50,39]},{"full_name":"Int.ofNat_add","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean","def_pos":[25,21],"def_end_pos":[25,30]},{"full_name":"Int.ofNat_one","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean","def_pos":[75,16],"def_end_pos":[75,25]},{"full_name":"Int.toNat_of_nonneg","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Order.lean","def_pos":[468,16],"def_end_pos":[468,31]},{"full_name":"add_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[293,2],"def_end_pos":[293,13]},{"full_name":"add_sub_assoc","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[329,2],"def_end_pos":[329,13]},{"full_name":"inf_eq_min","def_path":"Mathlib/Order/Lattice.lean","def_pos":[668,8],"def_end_pos":[668,18]},{"full_name":"max_sub_min_eq_abs","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[257,2],"def_end_pos":[257,13]},{"full_name":"min_le_max","def_path":"Mathlib/Order/MinMax.lean","def_pos":[114,8],"def_end_pos":[114,18]},{"full_name":"sup_eq_max","def_path":"Mathlib/Order/Lattice.lean","def_pos":[665,8],"def_end_pos":[665,18]}]}]}
+{"url":"Mathlib/Data/DFinsupp/Basic.lean","commit":"","full_name":"DFinsupp.induction","start":[824,0],"end":[856,17],"file_path":"Mathlib/Data/DFinsupp/Basic.lean","tactics":[{"state_before":"ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → AddZeroClass (β i)\np : (Π₀ (i : ι), β i) → Prop\nf : Π₀ (i : ι), β i\nh0 : p 0\nha : ∀ (i : ι) (b : β i) (f : Π₀ (i : ι), β i), f i = 0 → b ≠ 0 → p f → p (single i b + f)\n⊢ p f","state_after":"case mk'\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → AddZeroClass (β i)\np : (Π₀ (i : ι), β i) → Prop\nh0 : p 0\nha : ∀ (i : ι) (b : β i) (f : Π₀ (i : ι), β i), f i = 0 → b ≠ 0 → p f → p (single i b + f)\nf : (i : ι) → β i\ns : Trunc { s // ∀ (i : ι), i ∈ s ∨ f i = 0 }\n⊢ p { toFun := f, support' := s }","tactic":"cases' f with f s","premises":[]},{"state_before":"case mk'\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → AddZeroClass (β i)\np : (Π₀ (i : ι), β i) → Prop\nh0 : p 0\nha : ∀ (i : ι) (b : β i) (f : Π₀ (i : ι), β i), f i = 0 → b ≠ 0 → p f → p (single i b + f)\nf : (i : ι) → β i\ns : Trunc { s // ∀ (i : ι), i ∈ s ∨ f i = 0 }\n⊢ p { toFun := f, support' := s }","state_after":"case mk'.h\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → AddZeroClass (β i)\np : (Π₀ (i : ι), β i) → Prop\nh0 : p 0\nha : ∀ (i : ι) (b : β i) (f : Π₀ (i : ι), β i), f i = 0 → b ≠ 0 → p f → p (single i b + f)\nf : (i : ι) → β i\ns : { s // ∀ (i : ι), i ∈ s ∨ f i = 0 }\n⊢ p { toFun := f, support' := Trunc.mk s }","tactic":"induction' s using Trunc.induction_on with s","premises":[{"full_name":"Trunc.induction_on","def_path":"Mathlib/Data/Quot.lean","def_pos":[448,18],"def_end_pos":[448,30]}]},{"state_before":"case mk'.h\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → AddZeroClass (β i)\np : (Π₀ (i : ι), β i) → Prop\nh0 : p 0\nha : ∀ (i : ι) (b : β i) (f : Π₀ (i : ι), β i), f i = 0 → b ≠ 0 → p f → p (single i b + f)\nf : (i : ι) → β i\ns : { s // ∀ (i : ι), i ∈ s ∨ f i = 0 }\n⊢ p { toFun := f, support' := Trunc.mk s }","state_after":"case mk'.h.mk\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → AddZeroClass (β i)\np : (Π₀ (i : ι), β i) → Prop\nh0 : p 0\nha : ∀ (i : ι) (b : β i) (f : Π₀ (i : ι), β i), f i = 0 → b ≠ 0 → p f → p (single i b + f)\nf : (i : ι) → β i\ns : Multiset ι\nH : ∀ (i : ι), i ∈ s ∨ f i = 0\n⊢ p { toFun := f, support' := Trunc.mk ⟨s, H⟩ }","tactic":"cases' s with s H","premises":[]},{"state_before":"case mk'.h.mk\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → AddZeroClass (β i)\np : (Π₀ (i : ι), β i) → Prop\nh0 : p 0\nha : ∀ (i : ι) (b : β i) (f : Π₀ (i : ι), β i), f i = 0 → b ≠ 0 → p f → p (single i b + f)\nf : (i : ι) → β i\ns : Multiset ι\nH : ∀ (i : ι), i ∈ s ∨ f i = 0\n⊢ p { toFun := f, support' := Trunc.mk ⟨s, H⟩ }","state_after":"case mk'.h.mk.empty\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → AddZeroClass (β i)\np : (Π₀ (i : ι), β i) → Prop\nh0 : p 0\nha : ∀ (i : ι) (b : β i) (f : Π₀ (i : ι), β i), f i = 0 → b ≠ 0 → p f → p (single i b + f)\nf : (i : ι) → β i\nH : ∀ (i : ι), i ∈ 0 ∨ f i = 0\n⊢ p { toFun := f, support' := Trunc.mk ⟨0, H⟩ }\n\ncase mk'.h.mk.cons\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → AddZeroClass (β i)\np : (Π₀ (i : ι), β i) → Prop\nh0 : p 0\nha : ∀ (i : ι) (b : β i) (f : Π₀ (i : ι), β i), f i = 0 → b ≠ 0 → p f → p (single i b + f)\ni : ι\ns : Multiset ι\nih : ∀ (f : (i : ι) → β i) (H : ∀ (i : ι), i ∈ s ∨ f i = 0), p { toFun := f, support' := Trunc.mk ⟨s, H⟩ }\nf : (i : ι) → β i\nH : ∀ (i_1 : ι), i_1 ∈ i ::ₘ s ∨ f i_1 = 0\n⊢ p { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ }","tactic":"induction' s using Multiset.induction_on with i s ih generalizing f","premises":[{"full_name":"Multiset.induction_on","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[152,18],"def_end_pos":[152,30]}]},{"state_before":"case mk'.h.mk.cons\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → AddZeroClass (β i)\np : (Π₀ (i : ι), β i) → Prop\nh0 : p 0\nha : ∀ (i : ι) (b : β i) (f : Π₀ (i : ι), β i), f i = 0 → b ≠ 0 → p f → p (single i b + f)\ni : ι\ns : Multiset ι\nih : ∀ (f : (i : ι) → β i) (H : ∀ (i : ι), i ∈ s ∨ f i = 0), p { toFun := f, support' := Trunc.mk ⟨s, H⟩ }\nf : (i : ι) → β i\nH : ∀ (i_1 : ι), i_1 ∈ i ::ₘ s ∨ f i_1 = 0\nH2 : p (erase i { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ })\n⊢ p { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ }","state_after":"case mk'.h.mk.cons\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → AddZeroClass (β i)\np : (Π₀ (i : ι), β i) → Prop\nh0 : p 0\nha : ∀ (i : ι) (b : β i) (f : Π₀ (i : ι), β i), f i = 0 → b ≠ 0 → p f → p (single i b + f)\ni : ι\ns : Multiset ι\nih : ∀ (f : (i : ι) → β i) (H : ∀ (i : ι), i ∈ s ∨ f i = 0), p { toFun := f, support' := Trunc.mk ⟨s, H⟩ }\nf : (i : ι) → β i\nH : ∀ (i_1 : ι), i_1 ∈ i ::ₘ s ∨ f i_1 = 0\nH2 : p (erase i { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ })\nH3 :\n  single i ({ toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ } i) +\n      erase i { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ } =\n    { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ }\n⊢ p { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ }","tactic":"have H3 : single i _ + _ = (⟨f, Trunc.mk ⟨i ::ₘ s, H⟩⟩ : Π₀ i, β i) := single_add_erase _ _","premises":[{"full_name":"DFinsupp","def_path":"Mathlib/Data/DFinsupp/Basic.lean","def_pos":[60,10],"def_end_pos":[60,18]},{"full_name":"DFinsupp.mk'","def_path":"Mathlib/Data/DFinsupp/Basic.lean","def_pos":[60,58],"def_end_pos":[60,61]},{"full_name":"DFinsupp.single","def_path":"Mathlib/Data/DFinsupp/Basic.lean","def_pos":[539,4],"def_end_pos":[539,10]},{"full_name":"DFinsupp.single_add_erase","def_path":"Mathlib/Data/DFinsupp/Basic.lean","def_pos":[810,8],"def_end_pos":[810,24]},{"full_name":"Multiset.cons","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[118,4],"def_end_pos":[118,8]},{"full_name":"Trunc.mk","def_path":"Mathlib/Data/Quot.lean","def_pos":[425,4],"def_end_pos":[425,6]}]},{"state_before":"case mk'.h.mk.cons\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → AddZeroClass (β i)\np : (Π₀ (i : ι), β i) → Prop\nh0 : p 0\nha : ∀ (i : ι) (b : β i) (f : Π₀ (i : ι), β i), f i = 0 → b ≠ 0 → p f → p (single i b + f)\ni : ι\ns : Multiset ι\nih : ∀ (f : (i : ι) → β i) (H : ∀ (i : ι), i ∈ s ∨ f i = 0), p { toFun := f, support' := Trunc.mk ⟨s, H⟩ }\nf : (i : ι) → β i\nH : ∀ (i_1 : ι), i_1 ∈ i ::ₘ s ∨ f i_1 = 0\nH2 : p (erase i { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ })\nH3 :\n  single i ({ toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ } i) +\n      erase i { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ } =\n    { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ }\n⊢ p { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ }","state_after":"case mk'.h.mk.cons\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → AddZeroClass (β i)\np : (Π₀ (i : ι), β i) → Prop\nh0 : p 0\nha : ∀ (i : ι) (b : β i) (f : Π₀ (i : ι), β i), f i = 0 → b ≠ 0 → p f → p (single i b + f)\ni : ι\ns : Multiset ι\nih : ∀ (f : (i : ι) → β i) (H : ∀ (i : ι), i ∈ s ∨ f i = 0), p { toFun := f, support' := Trunc.mk ⟨s, H⟩ }\nf : (i : ι) → β i\nH : ∀ (i_1 : ι), i_1 ∈ i ::ₘ s ∨ f i_1 = 0\nH2 : p (erase i { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ })\nH3 :\n  single i ({ toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ } i) +\n      erase i { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ } =\n    { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ }\n⊢ p\n    (single i ({ toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ } i) +\n      erase i { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ })","tactic":"rw [← H3]","premises":[]},{"state_before":"case mk'.h.mk.cons\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → AddZeroClass (β i)\np : (Π₀ (i : ι), β i) → Prop\nh0 : p 0\nha : ∀ (i : ι) (b : β i) (f : Π₀ (i : ι), β i), f i = 0 → b ≠ 0 → p f → p (single i b + f)\ni : ι\ns : Multiset ι\nih : ∀ (f : (i : ι) → β i) (H : ∀ (i : ι), i ∈ s ∨ f i = 0), p { toFun := f, support' := Trunc.mk ⟨s, H⟩ }\nf : (i : ι) → β i\nH : ∀ (i_1 : ι), i_1 ∈ i ::ₘ s ∨ f i_1 = 0\nH2 : p (erase i { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ })\nH3 :\n  single i ({ toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ } i) +\n      erase i { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ } =\n    { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ }\n⊢ p\n    (single i ({ toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ } i) +\n      erase i { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ })","state_after":"case mk'.h.mk.cons\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → AddZeroClass (β i)\np : (Π₀ (i : ι), β i) → Prop\nh0 : p 0\nha : ∀ (i : ι) (b : β i) (f : Π₀ (i : ι), β i), f i = 0 → b ≠ 0 → p f → p (single i b + f)\ni : ι\ns : Multiset ι\nih : ∀ (f : (i : ι) → β i) (H : ∀ (i : ι), i ∈ s ∨ f i = 0), p { toFun := f, support' := Trunc.mk ⟨s, H⟩ }\nf : (i : ι) → β i\nH : ∀ (i_1 : ι), i_1 ∈ i ::ₘ s ∨ f i_1 = 0\nH2 : p (erase i { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ })\nH3 :\n  single i ({ toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ } i) +\n      erase i { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ } =\n    { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ }\n⊢ p (single i (f i) + erase i { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ })","tactic":"change p (single i (f i) + _)","premises":[{"full_name":"DFinsupp.single","def_path":"Mathlib/Data/DFinsupp/Basic.lean","def_pos":[539,4],"def_end_pos":[539,10]}]},{"state_before":"case mk'.h.mk.cons\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → AddZeroClass (β i)\np : (Π₀ (i : ι), β i) → Prop\nh0 : p 0\nha : ∀ (i : ι) (b : β i) (f : Π₀ (i : ι), β i), f i = 0 → b ≠ 0 → p f → p (single i b + f)\ni : ι\ns : Multiset ι\nih : ∀ (f : (i : ι) → β i) (H : ∀ (i : ι), i ∈ s ∨ f i = 0), p { toFun := f, support' := Trunc.mk ⟨s, H⟩ }\nf : (i : ι) → β i\nH : ∀ (i_1 : ι), i_1 ∈ i ::ₘ s ∨ f i_1 = 0\nH2 : p (erase i { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ })\nH3 :\n  single i ({ toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ } i) +\n      erase i { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ } =\n    { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ }\n⊢ p (single i (f i) + erase i { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ })","state_after":"case mk'.h.mk.cons.inl\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → AddZeroClass (β i)\np : (Π₀ (i : ι), β i) → Prop\nh0 : p 0\nha : ∀ (i : ι) (b : β i) (f : Π₀ (i : ι), β i), f i = 0 → b ≠ 0 → p f → p (single i b + f)\ni : ι\ns : Multiset ι\nih : ∀ (f : (i : ι) → β i) (H : ∀ (i : ι), i ∈ s ∨ f i = 0), p { toFun := f, support' := Trunc.mk ⟨s, H⟩ }\nf : (i : ι) → β i\nH : ∀ (i_1 : ι), i_1 ∈ i ::ₘ s ∨ f i_1 = 0\nH2 : p (erase i { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ })\nH3 :\n  single i ({ toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ } i) +\n      erase i { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ } =\n    { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ }\nh : f i = 0\n⊢ p (single i (f i) + erase i { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ })\n\ncase mk'.h.mk.cons.inr\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → AddZeroClass (β i)\np : (Π₀ (i : ι), β i) → Prop\nh0 : p 0\nha : ∀ (i : ι) (b : β i) (f : Π₀ (i : ι), β i), f i = 0 → b ≠ 0 → p f → p (single i b + f)\ni : ι\ns : Multiset ι\nih : ∀ (f : (i : ι) → β i) (H : ∀ (i : ι), i ∈ s ∨ f i = 0), p { toFun := f, support' := Trunc.mk ⟨s, H⟩ }\nf : (i : ι) → β i\nH : ∀ (i_1 : ι), i_1 ∈ i ::ₘ s ∨ f i_1 = 0\nH2 : p (erase i { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ })\nH3 :\n  single i ({ toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ } i) +\n      erase i { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ } =\n    { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ }\nh : ¬f i = 0\n⊢ p (single i (f i) + erase i { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ })","tactic":"cases' Classical.em (f i = 0) with h h","premises":[{"full_name":"Classical.em","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[32,8],"def_end_pos":[32,10]}]},{"state_before":"case mk'.h.mk.cons.inr\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → AddZeroClass (β i)\np : (Π₀ (i : ι), β i) → Prop\nh0 : p 0\nha : ∀ (i : ι) (b : β i) (f : Π₀ (i : ι), β i), f i = 0 → b ≠ 0 → p f → p (single i b + f)\ni : ι\ns : Multiset ι\nih : ∀ (f : (i : ι) → β i) (H : ∀ (i : ι), i ∈ s ∨ f i = 0), p { toFun := f, support' := Trunc.mk ⟨s, H⟩ }\nf : (i : ι) → β i\nH : ∀ (i_1 : ι), i_1 ∈ i ::ₘ s ∨ f i_1 = 0\nH2 : p (erase i { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ })\nH3 :\n  single i ({ toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ } i) +\n      erase i { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ } =\n    { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ }\nh : ¬f i = 0\n⊢ p (single i (f i) + erase i { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ })","state_after":"case mk'.h.mk.cons.inr\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → AddZeroClass (β i)\np : (Π₀ (i : ι), β i) → Prop\nh0 : p 0\nha : ∀ (i : ι) (b : β i) (f : Π₀ (i : ι), β i), f i = 0 → b ≠ 0 → p f → p (single i b + f)\ni : ι\ns : Multiset ι\nih : ∀ (f : (i : ι) → β i) (H : ∀ (i : ι), i ∈ s ∨ f i = 0), p { toFun := f, support' := Trunc.mk ⟨s, H⟩ }\nf : (i : ι) → β i\nH : ∀ (i_1 : ι), i_1 ∈ i ::ₘ s ∨ f i_1 = 0\nH2 : p (erase i { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ })\nH3 :\n  single i ({ toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ } i) +\n      erase i { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ } =\n    { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ }\nh : ¬f i = 0\n⊢ (erase i { toFun := f, support' := Trunc.mk ⟨i ::ₘ s, H⟩ }) i = 0","tactic":"refine ha _ _ _ ?_ h H2","premises":[]},{"state_before":"case mk'.h.mk.cons.inr\nι : Type u\nγ : Type w\nβ : ι → 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+{"url":"Mathlib/Analysis/Normed/Lp/lpSpace.lean","commit":"","full_name":"zero_memℓp","start":[119,0],"end":[127,47],"file_path":"Mathlib/Analysis/Normed/Lp/lpSpace.lean","tactics":[{"state_before":"α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\n⊢ Memℓp 0 p","state_after":"case inl\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\n⊢ Memℓp 0 0\n\ncase inr.inl\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\n⊢ Memℓp 0 ⊤\n\ncase inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nhp : 0 < p.toReal\n⊢ Memℓp 0 p","tactic":"rcases p.trichotomy with (rfl | rfl | hp)","premises":[{"full_name":"ENNReal.trichotomy","def_path":"Mathlib/Data/ENNReal/Real.lean","def_pos":[412,18],"def_end_pos":[412,28]}]}]}
+{"url":"Mathlib/Algebra/Quaternion.lean","commit":"","full_name":"QuaternionAlgebra.star_add_self'","start":[609,0],"end":[609,85],"file_path":"Mathlib/Algebra/Quaternion.lean","tactics":[{"state_before":"S : Type u_1\nT : Type u_2\nR : Type u_3\ninst✝ : CommRing R\nc₁ c₂ r x y z : R\na b c : ℍ[R,c₁,c₂]\n⊢ star a + a = ↑(2 * a.re)","state_after":"no goals","tactic":"rw [add_comm, self_add_star']","premises":[{"full_name":"QuaternionAlgebra.self_add_star'","def_path":"Mathlib/Algebra/Quaternion.lean","def_pos":[605,8],"def_end_pos":[605,22]},{"full_name":"add_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[293,2],"def_end_pos":[293,13]}]}]}
+{"url":"Mathlib/Order/Lattice.lean","commit":"","full_name":"Monotone.map_sup","start":[915,0],"end":[918,39],"file_path":"Mathlib/Order/Lattice.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\ninst✝¹ : LinearOrder α\ninst✝ : SemilatticeSup β\nf : α → β\nhf : Monotone f\nx y : α\nh : x ≤ y\n⊢ f (x ⊔ y) = f x ⊔ f y","state_after":"no goals","tactic":"simp only [h, hf h, sup_of_le_right]","premises":[]},{"state_before":"α : Type u\nβ : Type v\ninst✝¹ : LinearOrder α\ninst✝ : SemilatticeSup β\nf : α → β\nhf : Monotone f\nx y : α\nh : y ≤ x\n⊢ f (x ⊔ y) = f x ⊔ f y","state_after":"no goals","tactic":"simp only [h, hf h, sup_of_le_left]","premises":[]}]}
+{"url":"Mathlib/NumberTheory/EulerProduct/Basic.lean","commit":"","full_name":"EulerProduct.eulerProduct_hasProd_mulIndicator","start":[176,0],"end":[185,46],"file_path":"Mathlib/NumberTheory/EulerProduct/Basic.lean","tactics":[{"state_before":"R : Type u_1\ninst✝¹ : NormedCommRing R\ninst✝ : CompleteSpace R\nf : ℕ → R\nhf₁ : f 1 = 1\nhmul : ∀ {m n : ℕ}, m.Coprime n → f (m * n) = f m * f n\nhsum : Summable fun x => ‖f x‖\nhf₀ : f 0 = 0\n⊢ HasProd ({p | Nat.Prime p}.mulIndicator fun p => ∑' (e : ℕ), f (p ^ e)) (∑' (n : ℕ), f n)","state_after":"R : Type u_1\ninst✝¹ : NormedCommRing R\ninst✝ : CompleteSpace R\nf : ℕ → R\nhf₁ : f 1 = 1\nhmul : ∀ {m n : ℕ}, m.Coprime n → f (m * n) = f m * f n\nhsum : Summable fun x => ‖f x‖\nhf₀ : f 0 = 0\n⊢ HasProd ((fun p => ∑' (e : ℕ), f (p ^ e)) ∘ Subtype.val) (∑' (n : ℕ), f n)","tactic":"rw [← hasProd_subtype_iff_mulIndicator]","premises":[{"full_name":"hasProd_subtype_iff_mulIndicator","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Basic.lean","def_pos":[92,8],"def_end_pos":[92,40]}]},{"state_before":"R : Type u_1\ninst✝¹ : NormedCommRing R\ninst✝ : CompleteSpace R\nf : ℕ → R\nhf₁ : f 1 = 1\nhmul : ∀ {m n : ℕ}, m.Coprime n → f (m * n) = f m * f n\nhsum : Summable fun x => ‖f x‖\nhf₀ : f 0 = 0\n⊢ HasProd ((fun p => ∑' (e : ℕ), f (p ^ e)) ∘ Subtype.val) (∑' (n : ℕ), f n)","state_after":"no goals","tactic":"exact eulerProduct_hasProd hf₁ hmul hsum hf₀","premises":[{"full_name":"EulerProduct.eulerProduct_hasProd","def_path":"Mathlib/NumberTheory/EulerProduct/Basic.lean","def_pos":[160,8],"def_end_pos":[160,28]}]}]}
+{"url":"Mathlib/Algebra/Homology/HomotopyCategory/ShiftSequence.lean","commit":"","full_name":"CochainComplex.shiftShortComplexFunctorIso_inv_app_τ₁","start":[46,0],"end":[51,64],"file_path":"Mathlib/Algebra/Homology/HomotopyCategory/ShiftSequence.lean","tactics":[{"state_before":"C : Type u_1\ninst✝¹ : Category.{?u.71408, u_1} C\ninst✝ : Preadditive C\nn i i' : ℤ\nhi : n + i = i'\n⊢ n + (up ℤ).prev i = (up ℤ).prev i'","state_after":"C : Type u_1\ninst✝¹ : Category.{?u.71408, u_1} C\ninst✝ : Preadditive C\nn i i' : ℤ\nhi : n + i = i'\n⊢ n + (i - 1) = i' - 1","tactic":"simp only [prev]","premises":[{"full_name":"CochainComplex.prev","def_path":"Mathlib/Algebra/Homology/HomologicalComplex.lean","def_pos":[184,8],"def_end_pos":[184,12]}]},{"state_before":"C : Type u_1\ninst✝¹ : Category.{?u.71408, u_1} C\ninst✝ : Preadditive C\nn i i' : ℤ\nhi : n + i = i'\n⊢ n + (i - 1) = i' - 1","state_after":"no goals","tactic":"omega","premises":[]},{"state_before":"C : Type u_1\ninst✝¹ : Category.{?u.71408, u_1} C\ninst✝ : Preadditive C\nn i i' : ℤ\nhi : n + i = i'\n⊢ n + (up ℤ).next i = (up ℤ).next i'","state_after":"C : Type u_1\ninst✝¹ : Category.{?u.71408, u_1} C\ninst✝ : Preadditive C\nn i i' : ℤ\nhi : n + i = i'\n⊢ n + (i + 1) = i' + 1","tactic":"simp only [next]","premises":[{"full_name":"CochainComplex.next","def_path":"Mathlib/Algebra/Homology/HomologicalComplex.lean","def_pos":[188,8],"def_end_pos":[188,12]}]},{"state_before":"C : Type u_1\ninst✝¹ : Category.{?u.71408, u_1} C\ninst✝ : Preadditive C\nn i i' : ℤ\nhi : n + i = i'\n⊢ n + (i + 1) = i' + 1","state_after":"no goals","tactic":"omega","premises":[]}]}
+{"url":"Mathlib/Order/Heyting/Basic.lean","commit":"","full_name":"sdiff_sup_self","start":[420,0],"end":[421,98],"file_path":"Mathlib/Order/Heyting/Basic.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : GeneralizedCoheytingAlgebra α\na✝ b✝ c d a b : α\n⊢ b \\ a ⊔ a = b ⊔ a","state_after":"no goals","tactic":"rw [sup_comm, sup_sdiff_self, sup_comm]","premises":[{"full_name":"sup_comm","def_path":"Mathlib/Order/Lattice.lean","def_pos":[193,8],"def_end_pos":[193,16]},{"full_name":"sup_sdiff_self","def_path":"Mathlib/Order/Heyting/Basic.lean","def_pos":[417,8],"def_end_pos":[417,22]}]}]}
+{"url":"Mathlib/Algebra/Polynomial/RingDivision.lean","commit":"","full_name":"Polynomial.rootMultiplicity_X_sub_C_pow","start":[493,0],"end":[497,62],"file_path":"Mathlib/Algebra/Polynomial/RingDivision.lean","tactics":[{"state_before":"R : Type u\nS : Type v\nT : Type w\na✝ b : R\nn✝ : ℕ\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\na : R\nn : ℕ\n⊢ rootMultiplicity a ((X - C a) ^ n) = n","state_after":"R : Type u\nS : Type v\nT : Type w\na✝ b : R\nn✝ : ℕ\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\na : R\nn : ℕ\nthis : rootMultiplicity a (C 1 * (X - C a) ^ n) = rootMultiplicity a (C 1) + n\n⊢ rootMultiplicity a ((X - C a) ^ n) = n","tactic":"have := rootMultiplicity_mul_X_sub_C_pow (a := a) (n := n) C.map_one_ne_zero","premises":[{"full_name":"Polynomial.C","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[426,4],"def_end_pos":[426,5]},{"full_name":"Polynomial.rootMultiplicity_mul_X_sub_C_pow","def_path":"Mathlib/Algebra/Polynomial/RingDivision.lean","def_pos":[483,8],"def_end_pos":[483,40]},{"full_name":"RingHom.map_one_ne_zero","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[508,8],"def_end_pos":[508,23]}]},{"state_before":"R : Type u\nS : Type v\nT : Type w\na✝ b : R\nn✝ : ℕ\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\na : R\nn : ℕ\nthis : rootMultiplicity a (C 1 * (X - C a) ^ n) = rootMultiplicity a (C 1) + n\n⊢ rootMultiplicity a ((X - C a) ^ n) = n","state_after":"no goals","tactic":"rwa [rootMultiplicity_C, map_one, one_mul, zero_add] at this","premises":[{"full_name":"Polynomial.rootMultiplicity_C","def_path":"Mathlib/Algebra/Polynomial/Div.lean","def_pos":[513,8],"def_end_pos":[513,26]},{"full_name":"map_one","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[190,8],"def_end_pos":[190,15]},{"full_name":"one_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[409,8],"def_end_pos":[409,15]},{"full_name":"zero_add","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[408,2],"def_end_pos":[408,13]}]}]}
+{"url":"Mathlib/MeasureTheory/Constructions/Pi.lean","commit":"","full_name":"MeasureTheory.Measure.pi_eq_generateFrom","start":[333,0],"end":[348,58],"file_path":"Mathlib/MeasureTheory/Constructions/Pi.lean","tactics":[{"state_before":"ι : Type u_1\nι' : Type u_2\nα : ι → Type u_3\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\nC : (i : ι) → Set (Set (α i))\ni : ι\n⊢ MeasurableSpace (α i)","state_after":"no goals","tactic":"apply_assumption","premises":[{"full_name":"congrArg","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[362,8],"def_end_pos":[362,16]},{"full_name":"congrFun","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[376,8],"def_end_pos":[376,16]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]},{"full_name":"trivial","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[645,34],"def_end_pos":[645,41]}]},{"state_before":"ι : Type u_1\nι' : Type u_2\nα : ι → Type u_3\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\nC : (i : ι) → Set (Set (α i))\nhC : ∀ (i : ι), generateFrom (C i) = inst✝ i\nh2C : ∀ (i : ι), IsPiSystem (C i)\nh3C : (i : ι) → (μ i).FiniteSpanningSetsIn (C i)\nμν : Measure ((i : ι) → α i)\nh₁ : ∀ (s : (i : ι) → Set (α i)), (∀ (i : ι), s i ∈ C i) → μν (univ.pi s) = ∏ i : ι, (μ i) (s i)\n⊢ Measure.pi μ = μν","state_after":"ι : Type u_1\nι' : Type u_2\nα : ι → Type u_3\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\nC : (i : ι) → Set (Set (α i))\nhC : ∀ (i : ι), generateFrom (C i) = inst✝ i\nh2C : ∀ (i : ι), IsPiSystem (C i)\nh3C : (i : ι) → (μ i).FiniteSpanningSetsIn (C i)\nμν : Measure ((i : ι) → α i)\nh₁ : ∀ (s : (i : ι) → Set (α i)), (∀ (i : ι), s i ∈ C i) → μν (univ.pi s) = ∏ i : ι, (μ i) (s i)\nh4C : ∀ (i : ι), ∀ s ∈ C i, MeasurableSet s\n⊢ Measure.pi μ = μν","tactic":"have h4C : ∀ (i) (s : Set (α i)), s ∈ C i → MeasurableSet s := by\n    intro i s hs; rw [← hC]; exact measurableSet_generateFrom hs","premises":[{"full_name":"MeasurableSet","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","def_pos":[61,4],"def_end_pos":[61,17]},{"full_name":"MeasurableSpace.measurableSet_generateFrom","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","def_pos":[328,8],"def_end_pos":[328,34]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Set","def_path":"Mathlib/Init/Set.lean","def_pos":[53,4],"def_end_pos":[53,7]}]},{"state_before":"ι : Type u_1\nι' : Type u_2\nα : ι → Type u_3\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\nC : (i : ι) → Set (Set (α i))\nhC : ∀ (i : ι), generateFrom (C i) = inst✝ i\nh2C : ∀ (i : ι), IsPiSystem (C i)\nh3C : (i : ι) → (μ i).FiniteSpanningSetsIn (C i)\nμν : Measure ((i : ι) → α i)\nh₁ : ∀ (s : (i : ι) → Set (α i)), (∀ (i : ι), s i ∈ C i) → μν (univ.pi s) = ∏ i : ι, (μ i) (s i)\nh4C : ∀ (i : ι), ∀ s ∈ C i, MeasurableSet s\n⊢ Measure.pi μ = μν","state_after":"ι : Type u_1\nι' : Type u_2\nα : ι → Type u_3\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\nC : (i : ι) → Set (Set (α i))\nhC : ∀ (i : ι), generateFrom (C i) = inst✝ i\nh2C : ∀ (i : ι), IsPiSystem (C i)\nh3C : (i : ι) → (μ i).FiniteSpanningSetsIn (C i)\nμν : Measure ((i : ι) → α i)\nh₁ : ∀ (s : (i : ι) → Set (α i)), (∀ (i : ι), s i ∈ C i) → μν (univ.pi s) = ∏ i : ι, (μ i) (s i)\nh4C : ∀ (i : ι), ∀ s ∈ C i, MeasurableSet s\n⊢ ∀ s ∈ univ.pi '' univ.pi C, (Measure.pi μ) s = μν s","tactic":"refine\n    (FiniteSpanningSetsIn.pi h3C).ext\n      (generateFrom_eq_pi hC fun i => (h3C i).isCountablySpanning).symm (IsPiSystem.pi h2C) ?_","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"IsPiSystem.pi","def_path":"Mathlib/MeasureTheory/Constructions/Pi.lean","def_pos":[67,8],"def_end_pos":[67,21]},{"full_name":"MeasureTheory.Measure.FiniteSpanningSetsIn.ext","def_path":"Mathlib/MeasureTheory/Measure/Typeclasses.lean","def_pos":[935,18],"def_end_pos":[935,21]},{"full_name":"MeasureTheory.Measure.FiniteSpanningSetsIn.isCountablySpanning","def_path":"Mathlib/MeasureTheory/Measure/Typeclasses.lean","def_pos":[939,18],"def_end_pos":[939,37]},{"full_name":"MeasureTheory.Measure.FiniteSpanningSetsIn.pi","def_path":"Mathlib/MeasureTheory/Constructions/Pi.lean","def_pos":[308,4],"def_end_pos":[308,27]},{"full_name":"generateFrom_eq_pi","def_path":"Mathlib/MeasureTheory/Constructions/Pi.lean","def_pos":[126,8],"def_end_pos":[126,26]}]},{"state_before":"ι : Type u_1\nι' : Type u_2\nα : ι → Type u_3\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\nC : (i : ι) → Set (Set (α i))\nhC : ∀ (i : ι), generateFrom (C i) = inst✝ i\nh2C : ∀ (i : ι), IsPiSystem (C i)\nh3C : (i : ι) → (μ i).FiniteSpanningSetsIn (C i)\nμν : Measure ((i : ι) → α i)\nh₁ : ∀ (s : (i : ι) → Set (α i)), (∀ (i : ι), s i ∈ C i) → μν (univ.pi s) = ∏ i : ι, (μ i) (s i)\nh4C : ∀ (i : ι), ∀ s ∈ C i, MeasurableSet s\n⊢ ∀ s ∈ univ.pi '' univ.pi C, (Measure.pi μ) s = μν s","state_after":"case intro.intro\nι : Type u_1\nι' : Type u_2\nα : ι → Type u_3\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\nC : (i : ι) → Set (Set (α i))\nhC : ∀ (i : ι), generateFrom (C i) = inst✝ i\nh2C : ∀ (i : ι), IsPiSystem (C i)\nh3C : (i : ι) → (μ i).FiniteSpanningSetsIn (C i)\nμν : Measure ((i : ι) → α i)\nh₁ : ∀ (s : (i : ι) → Set (α i)), (∀ (i : ι), s i ∈ C i) → μν (univ.pi s) = ∏ i : ι, (μ i) (s i)\nh4C : ∀ (i : ι), ∀ s ∈ C i, MeasurableSet s\ns : (i : ι) → Set (α i)\nhs : s ∈ univ.pi C\n⊢ (Measure.pi μ) (univ.pi s) = μν (univ.pi s)","tactic":"rintro _ ⟨s, hs, rfl⟩","premises":[]},{"state_before":"case intro.intro\nι : Type u_1\nι' : Type u_2\nα : ι → Type u_3\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\nC : (i : ι) → Set (Set (α i))\nhC : ∀ (i : ι), generateFrom (C i) = inst✝ i\nh2C : ∀ (i : ι), IsPiSystem (C i)\nh3C : (i : ι) → (μ i).FiniteSpanningSetsIn (C i)\nμν : Measure ((i : ι) → α i)\nh₁ : ∀ (s : (i : ι) → Set (α i)), (∀ (i : ι), s i ∈ C i) → μν (univ.pi s) = ∏ i : ι, (μ i) (s i)\nh4C : ∀ (i : ι), ∀ s ∈ C i, MeasurableSet s\ns : (i : ι) → Set (α i)\nhs : s ∈ univ.pi C\n⊢ (Measure.pi μ) (univ.pi s) = μν (univ.pi s)","state_after":"case intro.intro\nι : Type u_1\nι' : Type u_2\nα : ι → Type u_3\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\nC : (i : ι) → Set (Set (α i))\nhC : ∀ (i : ι), generateFrom (C i) = inst✝ i\nh2C : ∀ (i : ι), IsPiSystem (C i)\nh3C : (i : ι) → (μ i).FiniteSpanningSetsIn (C i)\nμν : Measure ((i : ι) → α i)\nh₁ : ∀ (s : (i : ι) → Set (α i)), (∀ (i : ι), s i ∈ C i) → μν (univ.pi s) = ∏ i : ι, (μ i) (s i)\nh4C : ∀ (i : ι), ∀ s ∈ C i, MeasurableSet s\ns : (i : ι) → Set (α i)\nhs : ∀ (i : ι), s i ∈ C i\n⊢ (Measure.pi μ) (univ.pi s) = μν (univ.pi s)","tactic":"rw [mem_univ_pi] at hs","premises":[{"full_name":"Set.mem_univ_pi","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[224,8],"def_end_pos":[224,19]}]},{"state_before":"case intro.intro\nι : Type u_1\nι' : Type u_2\nα : ι → Type u_3\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\nC : (i : ι) → Set (Set (α i))\nhC : ∀ (i : ι), generateFrom (C i) = inst✝ i\nh2C : ∀ (i : ι), IsPiSystem (C i)\nh3C : (i : ι) → (μ i).FiniteSpanningSetsIn (C i)\nμν : Measure ((i : ι) → α i)\nh₁ : ∀ (s : (i : ι) → Set (α i)), (∀ (i : ι), s i ∈ C i) → μν (univ.pi s) = ∏ i : ι, (μ i) (s i)\nh4C : ∀ (i : ι), ∀ s ∈ C i, MeasurableSet s\ns : (i : ι) → Set (α i)\nhs : ∀ (i : ι), s i ∈ C i\n⊢ (Measure.pi μ) (univ.pi s) = μν (univ.pi s)","state_after":"case intro.intro\nι : Type u_1\nι' : Type u_2\nα : ι → Type u_3\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\nC : (i : ι) → Set (Set (α i))\nhC : ∀ (i : ι), generateFrom (C i) = inst✝ i\nh2C : ∀ (i : ι), IsPiSystem (C i)\nh3C : (i : ι) → (μ i).FiniteSpanningSetsIn (C i)\nμν : Measure ((i : ι) → α i)\nh₁ : ∀ (s : (i : ι) → Set (α i)), (∀ (i : ι), s i ∈ C i) → μν (univ.pi s) = ∏ i : ι, (μ i) (s i)\nh4C : ∀ (i : ι), ∀ s ∈ C i, MeasurableSet s\ns : (i : ι) → Set (α i)\nhs : ∀ (i : ι), s i ∈ C i\nthis : ∀ (i : ι), SigmaFinite (μ i)\n⊢ (Measure.pi μ) (univ.pi s) = μν (univ.pi s)","tactic":"haveI := fun i => (h3C i).sigmaFinite","premises":[{"full_name":"MeasureTheory.Measure.FiniteSpanningSetsIn.sigmaFinite","def_path":"Mathlib/MeasureTheory/Measure/Typeclasses.lean","def_pos":[930,18],"def_end_pos":[930,29]}]},{"state_before":"case intro.intro\nι : Type u_1\nι' : Type u_2\nα : ι → Type u_3\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\nC : (i : ι) → Set (Set (α i))\nhC : ∀ (i : ι), generateFrom (C i) = inst✝ i\nh2C : ∀ (i : ι), IsPiSystem (C i)\nh3C : (i : ι) → (μ i).FiniteSpanningSetsIn (C i)\nμν : Measure ((i : ι) → α i)\nh₁ : ∀ (s : (i : ι) → Set (α i)), (∀ (i : ι), s i ∈ C i) → μν (univ.pi s) = ∏ i : ι, (μ i) (s i)\nh4C : ∀ (i : ι), ∀ s ∈ C i, MeasurableSet s\ns : (i : ι) → Set (α i)\nhs : ∀ (i : ι), s i ∈ C i\nthis : ∀ (i : ι), SigmaFinite (μ i)\n⊢ (Measure.pi μ) (univ.pi s) = μν (univ.pi s)","state_after":"no goals","tactic":"simp_rw [h₁ s hs, pi_pi_aux μ s fun i => h4C i _ (hs i)]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"MeasureTheory.Measure.pi_pi_aux","def_path":"Mathlib/MeasureTheory/Constructions/Pi.lean","def_pos":[291,8],"def_end_pos":[291,17]}]}]}
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Type ?u.43702\nQ : Type ?u.43705\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : CommMonoid P\ninst✝ : CommMonoid Q\nf : M ≃* N\ng : P ≃* Q\nh : M →* P\nx✝ : M\n⊢ ((fun k => g.symm.toMonoidHom.comp (k.comp f.toMonoidHom))\n        ((fun h => g.toMonoidHom.comp (h.comp f.symm.toMonoidHom)) h))\n      x✝ =\n    h x✝","tactic":"ext","premises":[]},{"state_before":"case h\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nA : Type u_4\nB : Type u_5\nM✝ : Type u_6\nN✝ : Type u_7\nP✝ : Type u_8\nQ✝ : Type u_9\nG : Type u_10\nH : Type u_11\ninst✝⁴ : EquivLike F α β\nM : Type ?u.43696\nN : Type ?u.43699\nP : Type ?u.43702\nQ : Type ?u.43705\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : CommMonoid P\ninst✝ : CommMonoid Q\nf : M ≃* N\ng : P ≃* Q\nh : M →* P\nx✝ : M\n⊢ ((fun k => g.symm.toMonoidHom.comp (k.comp f.toMonoidHom))\n        ((fun h => g.toMonoidHom.comp (h.comp f.symm.toMonoidHom)) h))\n      x✝ =\n    h x✝","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nA : Type u_4\nB : Type u_5\nM✝ : Type u_6\nN✝ : Type u_7\nP✝ : Type u_8\nQ✝ : Type u_9\nG : Type u_10\nH : Type u_11\ninst✝⁴ : EquivLike F α β\nM : Type ?u.43696\nN : Type ?u.43699\nP : Type ?u.43702\nQ : Type ?u.43705\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : CommMonoid P\ninst✝ : CommMonoid Q\nf : M ≃* N\ng : P ≃* Q\nk : N →* Q\n⊢ (fun h => g.toMonoidHom.comp (h.comp f.symm.toMonoidHom))\n      ((fun k => g.symm.toMonoidHom.comp (k.comp f.toMonoidHom)) k) =\n    k","state_after":"case h\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nA : Type u_4\nB : Type u_5\nM✝ : Type u_6\nN✝ : Type u_7\nP✝ : Type u_8\nQ✝ : Type u_9\nG : Type u_10\nH : Type u_11\ninst✝⁴ : EquivLike F α β\nM : Type ?u.43696\nN : Type ?u.43699\nP : Type ?u.43702\nQ : Type ?u.43705\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : CommMonoid P\ninst✝ : CommMonoid Q\nf : M ≃* N\ng : P ≃* Q\nk : N →* Q\nx✝ : N\n⊢ ((fun h => g.toMonoidHom.comp (h.comp f.symm.toMonoidHom))\n        ((fun k => g.symm.toMonoidHom.comp (k.comp f.toMonoidHom)) k))\n      x✝ =\n    k x✝","tactic":"ext","premises":[]},{"state_before":"case h\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nA : Type u_4\nB : Type u_5\nM✝ : Type u_6\nN✝ : Type u_7\nP✝ : Type u_8\nQ✝ : Type u_9\nG : Type u_10\nH : Type u_11\ninst✝⁴ : EquivLike F α β\nM : Type ?u.43696\nN : Type ?u.43699\nP : Type ?u.43702\nQ : Type ?u.43705\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : CommMonoid P\ninst✝ : CommMonoid Q\nf : M ≃* N\ng : P ≃* Q\nk : N →* Q\nx✝ : N\n⊢ ((fun h => g.toMonoidHom.comp (h.comp f.symm.toMonoidHom))\n        ((fun k => g.symm.toMonoidHom.comp (k.comp f.toMonoidHom)) k))\n      x✝ =\n    k x✝","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nA : Type u_4\nB : Type u_5\nM✝ : Type u_6\nN✝ : Type u_7\nP✝ : Type u_8\nQ✝ : Type u_9\nG : Type u_10\nH : Type u_11\ninst✝⁴ : EquivLike F α β\nM : Type ?u.43696\nN : Type ?u.43699\nP : Type ?u.43702\nQ : Type ?u.43705\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : CommMonoid P\ninst✝ : CommMonoid Q\nf : M ≃* N\ng : P ≃* Q\nh k : M →* P\n⊢ { toFun := fun h => g.toMonoidHom.comp (h.comp f.symm.toMonoidHom),\n          invFun := fun k => g.symm.toMonoidHom.comp (k.comp f.toMonoidHom), left_inv := ⋯, right_inv := ⋯ }.toFun\n      (h * k) =\n    { toFun := fun h => g.toMonoidHom.comp (h.comp f.symm.toMonoidHom),\n            invFun := fun k => g.symm.toMonoidHom.comp (k.comp f.toMonoidHom), left_inv := ⋯, right_inv := ⋯ }.toFun\n        h *\n      { toFun := fun h => g.toMonoidHom.comp (h.comp f.symm.toMonoidHom),\n            invFun := fun k => g.symm.toMonoidHom.comp (k.comp f.toMonoidHom), left_inv := ⋯, right_inv := ⋯ }.toFun\n        k","state_after":"case h\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nA : Type u_4\nB : Type u_5\nM✝ : Type u_6\nN✝ : Type u_7\nP✝ : Type u_8\nQ✝ : Type u_9\nG : Type u_10\nH : Type u_11\ninst✝⁴ : EquivLike F α β\nM : Type ?u.43696\nN : Type ?u.43699\nP : Type ?u.43702\nQ : Type ?u.43705\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : CommMonoid P\ninst✝ : CommMonoid Q\nf : M ≃* N\ng : P ≃* Q\nh k : M →* P\nx✝ : N\n⊢ ({ toFun := fun h => g.toMonoidHom.comp (h.comp f.symm.toMonoidHom),\n            invFun := fun k => g.symm.toMonoidHom.comp (k.comp f.toMonoidHom), left_inv := ⋯, right_inv := ⋯ }.toFun\n        (h * k))\n      x✝ =\n    ({ toFun := fun h => g.toMonoidHom.comp (h.comp f.symm.toMonoidHom),\n              invFun := fun k => g.symm.toMonoidHom.comp (k.comp f.toMonoidHom), left_inv := ⋯, right_inv := ⋯ }.toFun\n          h *\n        { toFun := fun h => g.toMonoidHom.comp (h.comp f.symm.toMonoidHom),\n              invFun := fun k => g.symm.toMonoidHom.comp (k.comp f.toMonoidHom), left_inv := ⋯, right_inv := ⋯ }.toFun\n          k)\n      x✝","tactic":"ext","premises":[]},{"state_before":"case h\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nA : Type u_4\nB : Type u_5\nM✝ : Type u_6\nN✝ : Type u_7\nP✝ : Type u_8\nQ✝ : Type u_9\nG : Type u_10\nH : Type u_11\ninst✝⁴ : EquivLike F α β\nM : Type ?u.43696\nN : Type ?u.43699\nP : Type ?u.43702\nQ : Type ?u.43705\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : CommMonoid P\ninst✝ : CommMonoid Q\nf : M ≃* N\ng : P ≃* Q\nh k : M →* P\nx✝ : N\n⊢ ({ toFun := fun h => g.toMonoidHom.comp (h.comp f.symm.toMonoidHom),\n            invFun := fun k => g.symm.toMonoidHom.comp (k.comp f.toMonoidHom), left_inv := ⋯, right_inv := ⋯ }.toFun\n        (h * k))\n      x✝ =\n    ({ toFun := fun h => g.toMonoidHom.comp (h.comp f.symm.toMonoidHom),\n              invFun := fun k => g.symm.toMonoidHom.comp (k.comp f.toMonoidHom), left_inv := ⋯, right_inv := ⋯ }.toFun\n          h *\n        { toFun := fun h => g.toMonoidHom.comp (h.comp f.symm.toMonoidHom),\n              invFun := fun k => g.symm.toMonoidHom.comp (k.comp f.toMonoidHom), left_inv := ⋯, right_inv := ⋯ }.toFun\n          k)\n      x✝","state_after":"no goals","tactic":"simp","premises":[]}]}
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Y => AddCommGrp.of (Ext X Y n),\n          map := fun {X_1 Y} f => AddCommGrp.ofHom ((Ext.mk₀ f).postcomp X ⋯) }.obj\n      X✝)\n⊢ ({ obj := fun Y => AddCommGrp.of (Ext X Y n),\n            map := fun {X_1 Y} f => AddCommGrp.ofHom ((Ext.mk₀ f).postcomp X ⋯) }.map\n        (f ≫ f'))\n      α =\n    ({ obj := fun Y => AddCommGrp.of (Ext X Y n),\n              map := fun {X_1 Y} f => AddCommGrp.ofHom ((Ext.mk₀ f).postcomp X ⋯) }.map\n          f ≫\n        { obj := fun Y => AddCommGrp.of (Ext X Y n),\n              map := fun {X_1 Y} f => AddCommGrp.ofHom ((Ext.mk₀ f).postcomp X ⋯) }.map\n          f')\n      α","tactic":"ext α","premises":[]},{"state_before":"case w\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX : C\nn : ℕ\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\nf' : Y✝ ⟶ Z✝\nα :\n  ↑({ obj := fun Y => AddCommGrp.of (Ext X Y n),\n          map := fun {X_1 Y} f => AddCommGrp.ofHom ((Ext.mk₀ f).postcomp X ⋯) }.obj\n      X✝)\n⊢ ({ obj := fun Y => AddCommGrp.of (Ext X Y n),\n            map := fun {X_1 Y} f => AddCommGrp.ofHom ((Ext.mk₀ f).postcomp X ⋯) }.map\n        (f ≫ f'))\n      α =\n    ({ obj := fun Y => AddCommGrp.of (Ext X Y n),\n              map := fun {X_1 Y} f => AddCommGrp.ofHom ((Ext.mk₀ f).postcomp X ⋯) }.map\n          f ≫\n        { obj := fun Y => AddCommGrp.of (Ext X Y n),\n              map := fun {X_1 Y} f => AddCommGrp.ofHom ((Ext.mk₀ f).postcomp X ⋯) }.map\n          f')\n      α","state_after":"case w\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX : C\nn : ℕ\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\nf' : Y✝ ⟶ Z✝\nα :\n  ↑({ obj := fun Y => AddCommGrp.of (Ext X Y n),\n          map := fun {X_1 Y} f => AddCommGrp.ofHom ((Ext.mk₀ f).postcomp X ⋯) }.obj\n      X✝)\n⊢ Ext.comp α (Ext.mk₀ (f ≫ f')) ⋯ = (Ext.comp α (Ext.mk₀ f) ⋯).comp (Ext.mk₀ f') ⋯","tactic":"dsimp [AddCommGrp.ofHom]","premises":[{"full_name":"AddCommGrp.ofHom","def_path":"Mathlib/Algebra/Category/Grp/Basic.lean","def_pos":[269,2],"def_end_pos":[269,13]}]},{"state_before":"case w\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX : C\nn : ℕ\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\nf' : Y✝ ⟶ Z✝\nα :\n  ↑({ obj := fun Y => AddCommGrp.of (Ext X Y n),\n          map := fun {X_1 Y} f => AddCommGrp.ofHom ((Ext.mk₀ f).postcomp X ⋯) }.obj\n      X✝)\n⊢ Ext.comp α (Ext.mk₀ (f ≫ f')) ⋯ = (Ext.comp α (Ext.mk₀ f) ⋯).comp (Ext.mk₀ f') ⋯","state_after":"case w\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX : C\nn : ℕ\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\nf' : Y✝ ⟶ Z✝\nα :\n  ↑({ obj := fun Y => AddCommGrp.of (Ext X Y n),\n          map := fun {X_1 Y} f => AddCommGrp.ofHom ((Ext.mk₀ f).postcomp X ⋯) }.obj\n      X✝)\n⊢ Ext.comp α ((Ext.mk₀ f).comp (Ext.mk₀ f') ⋯) ⋯ = (Ext.comp α (Ext.mk₀ f) ⋯).comp (Ext.mk₀ f') ⋯","tactic":"rw [← Ext.mk₀_comp_mk₀]","premises":[{"full_name":"CategoryTheory.Abelian.Ext.mk₀_comp_mk₀","def_path":"Mathlib/Algebra/Homology/DerivedCategory/Ext.lean","def_pos":[147,6],"def_end_pos":[147,18]}]},{"state_before":"case w\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX : C\nn : ℕ\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\nf' : Y✝ ⟶ Z✝\nα :\n  ↑({ obj := fun Y => AddCommGrp.of (Ext X Y n),\n          map := fun {X_1 Y} f => AddCommGrp.ofHom ((Ext.mk₀ f).postcomp X ⋯) }.obj\n      X✝)\n⊢ Ext.comp α ((Ext.mk₀ f).comp (Ext.mk₀ f') ⋯) ⋯ = (Ext.comp α (Ext.mk₀ f) ⋯).comp (Ext.mk₀ f') ⋯","state_after":"case w\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX : C\nn : ℕ\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\nf' : Y✝ ⟶ Z✝\nα :\n  ↑({ obj := fun Y => AddCommGrp.of (Ext X Y n),\n          map := fun {X_1 Y} f => AddCommGrp.ofHom ((Ext.mk₀ f).postcomp X ⋯) }.obj\n      X✝)\n⊢ (Ext.comp α (Ext.mk₀ f) ⋯).comp (Ext.mk₀ f') ⋯ = Ext.comp α ((Ext.mk₀ f).comp (Ext.mk₀ f') ⋯) ⋯","tactic":"symm","premises":[]},{"state_before":"case w\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX : C\nn : ℕ\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\nf' : Y✝ ⟶ Z✝\nα :\n  ↑({ obj := fun Y => AddCommGrp.of (Ext X Y n),\n          map := fun {X_1 Y} f => AddCommGrp.ofHom ((Ext.mk₀ f).postcomp X ⋯) }.obj\n      X✝)\n⊢ (Ext.comp α (Ext.mk₀ f) ⋯).comp (Ext.mk₀ f') ⋯ = Ext.comp α ((Ext.mk₀ f).comp (Ext.mk₀ f') ⋯) ⋯","state_after":"case w.h\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX : C\nn : ℕ\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\nf' : Y✝ ⟶ Z✝\nα :\n  ↑({ obj := fun Y => AddCommGrp.of (Ext X Y n),\n          map := fun {X_1 Y} f => AddCommGrp.ofHom ((Ext.mk₀ f).postcomp X ⋯) }.obj\n      X✝)\n⊢ n + 0 + 0 = n","tactic":"apply Ext.comp_assoc","premises":[{"full_name":"CategoryTheory.Abelian.Ext.comp_assoc","def_path":"Mathlib/Algebra/Homology/DerivedCategory/Ext.lean","def_pos":[100,6],"def_end_pos":[100,16]}]},{"state_before":"case w.h\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX : C\nn : ℕ\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\nf' : Y✝ ⟶ Z✝\nα :\n  ↑({ obj := fun Y => AddCommGrp.of (Ext X Y n),\n          map := fun {X_1 Y} f => AddCommGrp.ofHom ((Ext.mk₀ f).postcomp X ⋯) }.obj\n      X✝)\n⊢ n + 0 + 0 = n","state_after":"no goals","tactic":"omega","premises":[]}]}
+{"url":"Mathlib/Data/Nat/Factorization/Basic.lean","commit":"","full_name":"Nat.ord_compl_dvd_ord_compl_of_dvd","start":[276,0],"end":[291,50],"file_path":"Mathlib/Data/Nat/Factorization/Basic.lean","tactics":[{"state_before":"a✝ b✝ m n p✝ a b : ℕ\nhab : a ∣ b\np : ℕ\n⊢ a / p ^ a.factorization p ∣ b / p ^ b.factorization p","state_after":"case inl\na✝ b✝ m n p✝ a b : ℕ\nhab : a ∣ b\np : ℕ\npp : ¬Prime p\n⊢ a / p ^ a.factorization p ∣ b / p ^ b.factorization p\n\ncase inr\na✝ b✝ m n p✝ a b : ℕ\nhab : a ∣ b\np : ℕ\npp : Prime p\n⊢ a / p ^ a.factorization p ∣ b / p ^ b.factorization p","tactic":"rcases em' p.Prime with (pp | pp)","premises":[{"full_name":"Nat.Prime","def_path":"Mathlib/Data/Nat/Prime/Defs.lean","def_pos":[36,4],"def_end_pos":[36,9]},{"full_name":"em'","def_path":"Mathlib/Logic/Basic.lean","def_pos":[157,8],"def_end_pos":[157,11]}]},{"state_before":"case inr\na✝ b✝ m n p✝ a b : ℕ\nhab : a ∣ b\np : ℕ\npp : Prime p\n⊢ a / p ^ a.factorization p ∣ b / p ^ b.factorization p","state_after":"case inr.inl\na✝ b m n p✝ a p : ℕ\npp : Prime p\nhab : a ∣ 0\n⊢ a / p ^ a.factorization p ∣ 0 / p ^ (factorization 0) p\n\ncase inr.inr\na✝ b✝ m n p✝ a b : ℕ\nhab : a ∣ b\np : ℕ\npp : Prime p\nhb0 : b ≠ 0\n⊢ a / p ^ a.factorization p ∣ b / p ^ b.factorization p","tactic":"rcases eq_or_ne b 0 with (rfl | hb0)","premises":[{"full_name":"eq_or_ne","def_path":"Mathlib/Logic/Basic.lean","def_pos":[167,8],"def_end_pos":[167,16]}]},{"state_before":"case inr.inr\na✝ b✝ m n p✝ a b : ℕ\nhab : a ∣ b\np : ℕ\npp : Prime p\nhb0 : b ≠ 0\n⊢ a / p ^ a.factorization p ∣ b / p ^ b.factorization p","state_after":"case inr.inr.inl\na b✝ m n p✝ b p : ℕ\npp : Prime p\nhb0 : b ≠ 0\nhab : 0 ∣ b\n⊢ 0 / p ^ (factorization 0) p ∣ b / p ^ b.factorization p\n\ncase inr.inr.inr\na✝ b✝ m n p✝ a b : ℕ\nhab : a ∣ b\np : ℕ\npp : Prime p\nhb0 : b ≠ 0\nha0 : a ≠ 0\n⊢ a / p ^ a.factorization p ∣ b / p ^ b.factorization p","tactic":"rcases eq_or_ne a 0 with (rfl | ha0)","premises":[{"full_name":"eq_or_ne","def_path":"Mathlib/Logic/Basic.lean","def_pos":[167,8],"def_end_pos":[167,16]}]},{"state_before":"case inr.inr.inr\na✝ b✝ m n p✝ a b : ℕ\nhab : a ∣ b\np : ℕ\npp : Prime p\nhb0 : b ≠ 0\nha0 : a ≠ 0\n⊢ a / p ^ a.factorization p ∣ b / p ^ b.factorization p","state_after":"case inr.inr.inr\na✝ b✝ m n p✝ a b : ℕ\nhab : a ∣ b\np : ℕ\npp : Prime p\nhb0 : b ≠ 0\nha0 : a ≠ 0\nha : a / p ^ a.factorization p ≠ 0\n⊢ a / p ^ a.factorization p ∣ b / p ^ b.factorization p","tactic":"have ha := (Nat.div_pos (ord_proj_le p ha0) (ord_proj_pos a p)).ne'","premises":[{"full_name":"LT.lt.ne'","def_path":"Mathlib/Order/Basic.lean","def_pos":[267,8],"def_end_pos":[267,11]},{"full_name":"Nat.div_pos","def_path":"Mathlib/Data/Nat/Defs.lean","def_pos":[435,16],"def_end_pos":[435,23]},{"full_name":"Nat.ord_proj_le","def_path":"Mathlib/Data/Nat/Factorization/Basic.lean","def_pos":[105,8],"def_end_pos":[105,19]},{"full_name":"Nat.ord_proj_pos","def_path":"Mathlib/Data/Nat/Factorization/Basic.lean","def_pos":[102,8],"def_end_pos":[102,20]}]},{"state_before":"case inr.inr.inr\na✝ b✝ m n p✝ a b : ℕ\nhab : a ∣ b\np : ℕ\npp : Prime p\nhb0 : b ≠ 0\nha0 : a ≠ 0\nha : a / p ^ a.factorization p ≠ 0\n⊢ a / p ^ a.factorization p ∣ b / p ^ b.factorization p","state_after":"case inr.inr.inr\na✝ b✝ m n p✝ a b : ℕ\nhab : a ∣ b\np : ℕ\npp : Prime p\nhb0 : b ≠ 0\nha0 : a ≠ 0\nha : a / p ^ a.factorization p ≠ 0\nhb : b / p ^ b.factorization p ≠ 0\n⊢ a / p ^ a.factorization p ∣ b / p ^ b.factorization p","tactic":"have hb := (Nat.div_pos (ord_proj_le p hb0) (ord_proj_pos b p)).ne'","premises":[{"full_name":"LT.lt.ne'","def_path":"Mathlib/Order/Basic.lean","def_pos":[267,8],"def_end_pos":[267,11]},{"full_name":"Nat.div_pos","def_path":"Mathlib/Data/Nat/Defs.lean","def_pos":[435,16],"def_end_pos":[435,23]},{"full_name":"Nat.ord_proj_le","def_path":"Mathlib/Data/Nat/Factorization/Basic.lean","def_pos":[105,8],"def_end_pos":[105,19]},{"full_name":"Nat.ord_proj_pos","def_path":"Mathlib/Data/Nat/Factorization/Basic.lean","def_pos":[102,8],"def_end_pos":[102,20]}]},{"state_before":"case inr.inr.inr\na✝ b✝ m n p✝ a b : ℕ\nhab : a ∣ b\np : ℕ\npp : Prime p\nhb0 : b ≠ 0\nha0 : a ≠ 0\nha : a / p ^ a.factorization p ≠ 0\nhb : b / p ^ b.factorization p ≠ 0\n⊢ a / p ^ a.factorization p ∣ b / p ^ b.factorization p","state_after":"case inr.inr.inr\na✝ b✝ m n p✝ a b : ℕ\nhab : a ∣ b\np : ℕ\npp : Prime p\nhb0 : b ≠ 0\nha0 : a ≠ 0\nha : a / p ^ a.factorization p ≠ 0\nhb : b / p ^ b.factorization p ≠ 0\n⊢ Finsupp.erase p a.factorization ≤ Finsupp.erase p b.factorization","tactic":"rw [← factorization_le_iff_dvd ha hb, factorization_ord_compl a p, factorization_ord_compl b p]","premises":[{"full_name":"Nat.factorization_le_iff_dvd","def_path":"Mathlib/Data/Nat/Factorization/Defs.lean","def_pos":[150,8],"def_end_pos":[150,32]},{"full_name":"Nat.factorization_ord_compl","def_path":"Mathlib/Data/Nat/Factorization/Basic.lean","def_pos":[207,8],"def_end_pos":[207,31]}]},{"state_before":"case inr.inr.inr\na✝ b✝ m n p✝ a b : ℕ\nhab : a ∣ b\np : ℕ\npp : Prime p\nhb0 : b ≠ 0\nha0 : a ≠ 0\nha : a / p ^ a.factorization p ≠ 0\nhb : b / p ^ b.factorization p ≠ 0\n⊢ Finsupp.erase p a.factorization ≤ Finsupp.erase p b.factorization","state_after":"case inr.inr.inr\na✝ b✝ m n p✝ a b : ℕ\nhab : a ∣ b\np : ℕ\npp : Prime p\nhb0 : b ≠ 0\nha0 : a ≠ 0\nha : a / p ^ a.factorization p ≠ 0\nhb : b / p ^ b.factorization p ≠ 0\nq : ℕ\n⊢ (Finsupp.erase p a.factorization) q ≤ (Finsupp.erase p b.factorization) q","tactic":"intro q","premises":[]},{"state_before":"case inr.inr.inr\na✝ b✝ m n p✝ a b : ℕ\nhab : a ∣ b\np : ℕ\npp : Prime p\nhb0 : b ≠ 0\nha0 : a ≠ 0\nha : a / p ^ a.factorization p ≠ 0\nhb : b / p ^ b.factorization p ≠ 0\nq : ℕ\n⊢ (Finsupp.erase p a.factorization) q ≤ (Finsupp.erase p b.factorization) q","state_after":"case inr.inr.inr.inl\na✝ b✝ m n p a b : ℕ\nhab : a ∣ b\nhb0 : b ≠ 0\nha0 : a ≠ 0\nq : ℕ\npp : Prime q\nha : a / q ^ a.factorization q ≠ 0\nhb : b / q ^ b.factorization q ≠ 0\n⊢ (Finsupp.erase q a.factorization) q ≤ (Finsupp.erase q b.factorization) q\n\ncase inr.inr.inr.inr\na✝ b✝ m n p✝ a b : ℕ\nhab : a ∣ b\np : ℕ\npp : Prime p\nhb0 : b ≠ 0\nha0 : a ≠ 0\nha : a / p ^ a.factorization p ≠ 0\nhb : b / p ^ b.factorization p ≠ 0\nq : ℕ\nhqp : q ≠ p\n⊢ (Finsupp.erase p a.factorization) q ≤ (Finsupp.erase p b.factorization) q","tactic":"rcases eq_or_ne q p with (rfl | hqp)","premises":[{"full_name":"eq_or_ne","def_path":"Mathlib/Logic/Basic.lean","def_pos":[167,8],"def_end_pos":[167,16]}]},{"state_before":"case inr.inr.inr.inr\na✝ b✝ m n p✝ a b : ℕ\nhab : a ∣ b\np : ℕ\npp : Prime p\nhb0 : b ≠ 0\nha0 : a ≠ 0\nha : a / p ^ a.factorization p ≠ 0\nhb : b / p ^ b.factorization p ≠ 0\nq : ℕ\nhqp : q ≠ p\n⊢ (Finsupp.erase p a.factorization) q ≤ (Finsupp.erase p b.factorization) q","state_after":"case inr.inr.inr.inr\na✝ b✝ m n p✝ a b : ℕ\nhab : a ∣ b\np : ℕ\npp : Prime p\nhb0 : b ≠ 0\nha0 : a ≠ 0\nha : a / p ^ a.factorization p ≠ 0\nhb : b / p ^ b.factorization p ≠ 0\nq : ℕ\nhqp : q ≠ p\n⊢ a.factorization q ≤ b.factorization q","tactic":"simp_rw [erase_ne hqp]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Finsupp.erase_ne","def_path":"Mathlib/Data/Finsupp/Defs.lean","def_pos":[561,8],"def_end_pos":[561,16]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]}]},{"state_before":"case inr.inr.inr.inr\na✝ b✝ m n p✝ a b : ℕ\nhab : a ∣ b\np : ℕ\npp : Prime p\nhb0 : b ≠ 0\nha0 : a ≠ 0\nha : a / p ^ a.factorization p ≠ 0\nhb : b / p ^ b.factorization p ≠ 0\nq : ℕ\nhqp : q ≠ p\n⊢ a.factorization q ≤ b.factorization q","state_after":"no goals","tactic":"exact (factorization_le_iff_dvd ha0 hb0).2 hab q","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Nat.factorization_le_iff_dvd","def_path":"Mathlib/Data/Nat/Factorization/Defs.lean","def_pos":[150,8],"def_end_pos":[150,32]}]}]}
+{"url":"Mathlib/Analysis/Complex/PhragmenLindelof.lean","commit":"","full_name":"PhragmenLindelof.eqOn_right_half_plane_of_superexponential_decay","start":[800,0],"end":[825,63],"file_path":"Mathlib/Analysis/Complex/PhragmenLindelof.lean","tactics":[{"state_before":"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b C : ℝ\nf g✝ : ℂ → E\nz : ℂ\ng : ℂ → E\nhfd : DiffContOnCl ℂ f {z | 0 < z.re}\nhgd : DiffContOnCl ℂ g {z | 0 < z.re}\nhfexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * Complex.abs z ^ c)\nhgexp : ∃ c < 2, ∃ B, g =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * Complex.abs z ^ c)\nhre : SuperpolynomialDecay atTop expR fun x => ‖f ↑x - g ↑x‖\nhfim : ∃ C, ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C\nhgim : ∃ C, ∀ (x : ℝ), ‖g (↑x * I)‖ ≤ C\n⊢ EqOn f g {z | 0 ≤ z.re}","state_after":"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b C : ℝ\nf g✝ : ℂ → E\nz : ℂ\ng : ℂ → E\nhfd : DiffContOnCl ℂ f {z | 0 < z.re}\nhgd : DiffContOnCl ℂ g {z | 0 < z.re}\nhfexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * Complex.abs z ^ c)\nhgexp : ∃ c < 2, ∃ B, g =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * Complex.abs z ^ c)\nhre : SuperpolynomialDecay atTop expR fun x => ‖f ↑x - g ↑x‖\nhfim : ∃ C, ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C\nhgim : ∃ C, ∀ (x : ℝ), ‖g (↑x * I)‖ ≤ C\n⊢ EqOn (f - g) 0 {z | 0 ≤ z.re}","tactic":"suffices EqOn (f - g) 0 {z : ℂ | 0 ≤ z.re} by\n    simpa only [EqOn, Pi.sub_apply, Pi.zero_apply, sub_eq_zero] using this","premises":[{"full_name":"Complex","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[27,10],"def_end_pos":[27,17]},{"full_name":"Complex.re","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[29,2],"def_end_pos":[29,4]},{"full_name":"Pi.sub_apply","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[144,2],"def_end_pos":[144,13]},{"full_name":"Pi.zero_apply","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[59,2],"def_end_pos":[59,13]},{"full_name":"Set.EqOn","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[229,4],"def_end_pos":[229,8]},{"full_name":"setOf","def_path":"Mathlib/Init/Set.lean","def_pos":[56,4],"def_end_pos":[56,9]},{"full_name":"sub_eq_zero","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[738,2],"def_end_pos":[738,13]}]},{"state_before":"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b C : ℝ\nf g✝ : ℂ → E\nz : ℂ\ng : ℂ → E\nhfd : DiffContOnCl ℂ f {z | 0 < z.re}\nhgd : DiffContOnCl ℂ g {z | 0 < z.re}\nhfexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * Complex.abs z ^ c)\nhgexp : ∃ c < 2, ∃ B, g =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * Complex.abs z ^ c)\nhre : SuperpolynomialDecay atTop expR fun x => ‖f ↑x - g ↑x‖\nhfim : ∃ C, ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C\nhgim : ∃ C, ∀ (x : ℝ), ‖g (↑x * I)‖ ≤ C\n⊢ EqOn (f - g) 0 {z | 0 ≤ z.re}","state_after":"case refine_1\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b C : ℝ\nf g✝ : ℂ → E\nz : ℂ\ng : ℂ → E\nhfd : DiffContOnCl ℂ f {z | 0 < z.re}\nhgd : DiffContOnCl ℂ g {z | 0 < z.re}\nhfexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * Complex.abs z ^ c)\nhgexp : ∃ c < 2, ∃ B, g =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * Complex.abs z ^ c)\nhre : SuperpolynomialDecay atTop expR fun x => ‖f ↑x - g ↑x‖\nhfim : ∃ C, ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C\nhgim : ∃ C, ∀ (x : ℝ), ‖g (↑x * I)‖ ≤ C\n⊢ ∃ c < 2, ∃ B, (f - g) =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * Complex.abs z ^ c)\n\ncase refine_2\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b C : ℝ\nf g✝ : ℂ → E\nz : ℂ\ng : ℂ → E\nhfd : DiffContOnCl ℂ f {z | 0 < z.re}\nhgd : DiffContOnCl ℂ g {z | 0 < z.re}\nhfexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * Complex.abs z ^ c)\nhgexp : ∃ c < 2, ∃ B, g =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * Complex.abs z ^ c)\nhre : SuperpolynomialDecay atTop expR fun x => ‖f ↑x - g ↑x‖\nhfim : ∃ C, ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C\nhgim : ∃ C, ∀ (x : ℝ), ‖g (↑x * I)‖ ≤ C\n⊢ ∃ C, ∀ (x : ℝ), ‖(f - g) (↑x * I)‖ ≤ C","tactic":"refine eq_zero_on_right_half_plane_of_superexponential_decay (hfd.sub hgd) ?_ hre ?_","premises":[{"full_name":"DiffContOnCl.sub","def_path":"Mathlib/Analysis/Calculus/DiffContOnCl.lean","def_pos":[89,8],"def_end_pos":[89,11]},{"full_name":"PhragmenLindelof.eq_zero_on_right_half_plane_of_superexponential_decay","def_path":"Mathlib/Analysis/Complex/PhragmenLindelof.lean","def_pos":[758,8],"def_end_pos":[758,61]}]}]}
+{"url":"Mathlib/Logic/Equiv/Set.lean","commit":"","full_name":"Equiv.ofLeftInverse'_eq_ofInjective","start":[561,0],"end":[564,6],"file_path":"Mathlib/Logic/Equiv/Set.lean","tactics":[{"state_before":"α✝ : Sort u\nβ✝ : Sort v\nγ : Sort w\nα : Type u_1\nβ : Type u_2\nf : α → β\nf_inv : β → α\nhf : LeftInverse f_inv f\n⊢ ofLeftInverse' f f_inv hf = ofInjective f ⋯","state_after":"case H.a\nα✝ : Sort u\nβ✝ : Sort v\nγ : Sort w\nα : Type u_1\nβ : Type u_2\nf : α → β\nf_inv : β → α\nhf : LeftInverse f_inv f\nx✝ : α\n⊢ ↑((ofLeftInverse' f f_inv hf) x✝) = ↑((ofInjective f ⋯) x✝)","tactic":"ext","premises":[]},{"state_before":"case H.a\nα✝ : Sort u\nβ✝ : Sort v\nγ : Sort w\nα : Type u_1\nβ : Type u_2\nf : α → β\nf_inv : β → α\nhf : LeftInverse f_inv f\nx✝ : α\n⊢ ↑((ofLeftInverse' f f_inv hf) x✝) = ↑((ofInjective f ⋯) x✝)","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/Data/List/Perm.lean","commit":"","full_name":"List.bind_append_perm","start":[292,0],"end":[299,39],"file_path":"Mathlib/Data/List/Perm.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nl✝ l₁ l₂ : List α\na : α\nl : List α\nf g : α → List β\n⊢ l.bind f ++ l.bind g ~ l.bind fun x => f x ++ g x","state_after":"case nil\nα : Type u_1\nβ : Type u_2\nl l₁ l₂ : List α\na : α\nf g : α → List β\n⊢ [].bind f ++ [].bind g ~ [].bind fun x => f x ++ g x\n\ncase cons\nα : Type u_1\nβ : Type u_2\nl✝ l₁ l₂ : List α\na✝ : α\nf g : α → List β\na : α\nl : List α\nIH : l.bind f ++ l.bind g ~ l.bind fun x => f x ++ g x\n⊢ (a :: l).bind f ++ (a :: l).bind g ~ (a :: l).bind fun x => f x ++ g x","tactic":"induction' l with a l IH","premises":[]},{"state_before":"case cons\nα : Type u_1\nβ : Type u_2\nl✝ l₁ l₂ : List α\na✝ : α\nf g : α → List β\na : α\nl : List α\nIH : l.bind f ++ l.bind g ~ l.bind fun x => f x ++ g x\n⊢ (a :: l).bind f ++ (a :: l).bind g ~ (a :: l).bind fun x => f x ++ g x","state_after":"case cons\nα : Type u_1\nβ : Type u_2\nl✝ l₁ l₂ : List α\na✝ : α\nf g : α → List β\na : α\nl : List α\nIH : l.bind f ++ l.bind g ~ l.bind fun x => f x ++ g x\n⊢ f a ++ (l.bind f ++ (g a ++ l.bind g)) ~ f a ++ (g a ++ l.bind fun x => f x ++ g x)","tactic":"simp only [bind_cons, append_assoc]","premises":[{"full_name":"List.append_assoc","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[501,16],"def_end_pos":[501,28]},{"full_name":"List.bind_cons","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[554,16],"def_end_pos":[554,25]}]},{"state_before":"case cons\nα : Type u_1\nβ : Type u_2\nl✝ l₁ l₂ : List α\na✝ : α\nf g : α → List β\na : α\nl : List α\nIH : l.bind f ++ l.bind g ~ l.bind fun x => f x ++ g x\n⊢ f a ++ (l.bind f ++ (g a ++ l.bind g)) ~ f a ++ (g a ++ l.bind fun x => f x ++ g x)","state_after":"case cons\nα : Type u_1\nβ : Type u_2\nl✝ l₁ l₂ : List α\na✝ : α\nf g : α → List β\na : α\nl : List α\nIH : l.bind f ++ l.bind g ~ l.bind fun x => f x ++ g x\n⊢ l.bind f ++ (g a ++ l.bind g) ~ g a ++ (l.bind f ++ l.bind g)","tactic":"refine (Perm.trans ?_ (IH.append_left _)).append_left _","premises":[{"full_name":"List.Perm.append_left","def_path":".lake/packages/batteries/Batteries/Data/List/Perm.lean","def_pos":[85,8],"def_end_pos":[85,24]},{"full_name":"List.Perm.trans","def_path":".lake/packages/batteries/Batteries/Data/List/Basic.lean","def_pos":[1358,4],"def_end_pos":[1358,9]}]},{"state_before":"case cons\nα : Type u_1\nβ : Type u_2\nl✝ l₁ l₂ : List α\na✝ : α\nf g : α → List β\na : α\nl : List α\nIH : l.bind f ++ l.bind g ~ l.bind fun x => f x ++ g x\n⊢ l.bind f ++ (g a ++ l.bind g) ~ g a ++ (l.bind f ++ l.bind g)","state_after":"case cons\nα : Type u_1\nβ : Type u_2\nl✝ l₁ l₂ : List α\na✝ : α\nf g : α → List β\na : α\nl : List α\nIH : l.bind f ++ l.bind g ~ l.bind fun x => f x ++ g x\n⊢ l.bind f ++ g a ++ l.bind g ~ g a ++ l.bind f ++ l.bind g","tactic":"rw [← append_assoc, ← append_assoc]","premises":[{"full_name":"List.append_assoc","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[501,16],"def_end_pos":[501,28]}]},{"state_before":"case cons\nα : Type u_1\nβ : Type u_2\nl✝ l₁ l₂ : List α\na✝ : α\nf g : α → List β\na : α\nl : List α\nIH : l.bind f ++ l.bind g ~ l.bind fun x => f x ++ g x\n⊢ l.bind f ++ g a ++ l.bind g ~ g a ++ l.bind f ++ l.bind g","state_after":"no goals","tactic":"exact perm_append_comm.append_right _","premises":[{"full_name":"List.Perm.append_right","def_path":".lake/packages/batteries/Batteries/Data/List/Perm.lean","def_pos":[78,8],"def_end_pos":[78,25]},{"full_name":"List.perm_append_comm","def_path":".lake/packages/batteries/Batteries/Data/List/Perm.lean","def_pos":[102,8],"def_end_pos":[102,24]}]}]}
+{"url":"Mathlib/Data/Seq/WSeq.lean","commit":"","full_name":"Stream'.WSeq.mem_of_mem_tail","start":[839,0],"end":[849,7],"file_path":"Mathlib/Data/Seq/WSeq.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\na : α\n⊢ a ∈ s.tail → a ∈ s","state_after":"α : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\na : α\nh : a ∈ s.tail\n⊢ a ∈ s","tactic":"intro h","premises":[]},{"state_before":"α : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\na : α\nh : a ∈ s.tail\n⊢ a ∈ s","state_after":"α : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\na : α\nh this : a ∈ s.tail\n⊢ a ∈ s","tactic":"have := h","premises":[]},{"state_before":"α : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\na : α\nh this : a ∈ s.tail\n⊢ a ∈ s","state_after":"case intro\nα : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\na : α\nthis : a ∈ s.tail\nn : ℕ\ne : (fun b => some (some a) = b) ((↑s.tail).get n)\n⊢ a ∈ s","tactic":"cases' h with n e","premises":[]},{"state_before":"case intro\nα : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\na : α\nthis : a ∈ s.tail\nn : ℕ\ne : (fun b => some (some a) = b) ((↑s.tail).get n)\n⊢ a ∈ s","state_after":"case intro\nα : Type u\nβ : Type v\nγ : Type w\na : α\nn : ℕ\n⊢ ∀ {s : WSeq α}, a ∈ s.tail → (fun b => some (some a) = b) ((↑s.tail).get n) → a ∈ s","tactic":"revert s","premises":[]},{"state_before":"case intro\nα : Type u\nβ : Type v\nγ : Type w\na : α\nn : ℕ\n⊢ ∀ {s : WSeq α}, a ∈ s.tail → (fun b => some (some a) = b) ((↑s.tail).get n) → a ∈ s","state_after":"case intro\nα : Type u\nβ : Type v\nγ : Type w\na : α\nn : ℕ\n⊢ ∀ {s : WSeq α}, a ∈ s.tail → some (some a) = ↑s.tail n → a ∈ s","tactic":"simp only [Stream'.get]","premises":[{"full_name":"Stream'.get","def_path":"Mathlib/Data/Stream/Defs.lean","def_pos":[32,4],"def_end_pos":[32,7]}]},{"state_before":"case intro\nα : Type u\nβ : Type v\nγ : Type w\na : α\nn : ℕ\n⊢ ∀ {s : WSeq α}, a ∈ s.tail → some (some a) = ↑s.tail n → a ∈ s","state_after":"case intro.zero.h2\nα : Type u\nβ : Type v\nγ : Type w\na x : α\ns : WSeq α\nm : a ∈ s\ne : some (some a) = ↑s 0\n⊢ a = x ∨ a ∈ s\n\ncase intro.succ.h2\nα : Type u\nβ : Type v\nγ : Type w\na : α\nn : ℕ\nIH : ∀ {s : WSeq α}, a ∈ s.tail → some (some a) = ↑s.tail n → a ∈ s\nx : α\ns : WSeq α\nm : a ∈ s\ne : some (some a) = ↑s (n + 1)\n⊢ a = x ∨ a ∈ s\n\ncase intro.succ.h3\nα : Type u\nβ : Type v\nγ : Type w\na : α\nn : ℕ\nIH : ∀ {s : WSeq α}, a ∈ s.tail → some (some a) = ↑s.tail n → a ∈ s\ns : WSeq α\nm : a ∈ s.tail\ne : some (some a) = ↑s.tail.think (n + 1)\n⊢ a ∈ s","tactic":"induction' n with n IH <;> intro s <;> induction' s using WSeq.recOn with x s s <;>\n    simp <;> intro m e <;>\n    injections","premises":[{"full_name":"Stream'.WSeq.recOn","def_path":"Mathlib/Data/Seq/WSeq.lean","def_pos":[104,4],"def_end_pos":[104,9]}]}]}
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+{"url":"Mathlib/Analysis/Calculus/Implicit.lean","commit":"","full_name":"ImplicitFunctionData.map_nhds_eq","start":[180,0],"end":[182,74],"file_path":"Mathlib/Analysis/Calculus/Implicit.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : CompleteSpace E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : CompleteSpace F\nG : Type u_4\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\ninst✝ : CompleteSpace G\nφ : ImplicitFunctionData 𝕜 E F G\n⊢ map (Prod.fst ∘ φ.prodFun) (𝓝 φ.pt) = 𝓝 (φ.prodFun φ.pt).1","state_after":"no goals","tactic":"rw [← map_map, φ.hasStrictFDerivAt.map_nhds_eq_of_equiv, map_fst_nhds]","premises":[{"full_name":"Filter.map_map","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1695,8],"def_end_pos":[1695,15]},{"full_name":"HasStrictFDerivAt.map_nhds_eq_of_equiv","def_path":"Mathlib/Analysis/Calculus/InverseFunctionTheorem/FDeriv.lean","def_pos":[134,8],"def_end_pos":[134,28]},{"full_name":"ImplicitFunctionData.hasStrictFDerivAt","def_path":"Mathlib/Analysis/Calculus/Implicit.lean","def_pos":[128,18],"def_end_pos":[128,35]},{"full_name":"map_fst_nhds","def_path":"Mathlib/Topology/Constructions.lean","def_pos":[673,8],"def_end_pos":[673,20]}]}]}
+{"url":"Mathlib/CategoryTheory/Category/Preorder.lean","commit":"","full_name":"OrderIso.equivalence_inverse","start":[174,0],"end":[180,62],"file_path":"Mathlib/CategoryTheory/Category/Preorder.lean","tactics":[{"state_before":"X : Type u\nY : Type v\ninst✝¹ : Preorder X\ninst✝ : Preorder Y\ne : X ≃o Y\nx✝ : X\n⊢ (𝟭 X).obj x✝ = (⋯.functor ⋙ ⋯.functor).obj x✝","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"X : Type u\nY : Type v\ninst✝¹ : Preorder X\ninst✝ : Preorder Y\ne : X ≃o Y\nx✝ : Y\n⊢ (⋯.functor ⋙ ⋯.functor).obj x✝ = (𝟭 Y).obj x✝","state_after":"no goals","tactic":"simp","premises":[]}]}
+{"url":"Mathlib/Data/Prod/Basic.lean","commit":"","full_name":"Prod.mk.inj_right","start":[92,0],"end":[95,61],"file_path":"Mathlib/Data/Prod/Basic.lean","tactics":[{"state_before":"α✝ : Type u_1\nβ✝ : Type u_2\nγ : Type u_3\nδ : Type u_4\nα : Type u_5\nβ : Type u_6\nb : β\n⊢ Function.Injective fun a => (a, b)","state_after":"α✝ : Type u_1\nβ✝ : Type u_2\nγ : Type u_3\nδ : Type u_4\nα : Type u_5\nβ : Type u_6\nb : β\nb₁ b₂ : α\nh : (fun a => (a, b)) b₁ = (fun a => (a, b)) b₂\n⊢ b₁ = b₂","tactic":"intro b₁ b₂ h","premises":[]},{"state_before":"α✝ : Type u_1\nβ✝ : Type u_2\nγ : Type u_3\nδ : Type u_4\nα : Type u_5\nβ : Type u_6\nb : β\nb₁ b₂ : α\nh : (fun a => (a, b)) b₁ = (fun a => (a, b)) b₂\n⊢ b₁ = b₂","state_after":"no goals","tactic":"simpa only [and_true, eq_self_iff_true, mk.inj_iff] using h","premises":[{"full_name":"Prod.mk.inj_iff","def_path":"Mathlib/Data/Prod/Basic.lean","def_pos":[85,8],"def_end_pos":[85,18]},{"full_name":"and_true","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[104,16],"def_end_pos":[104,24]},{"full_name":"eq_self_iff_true","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1380,8],"def_end_pos":[1380,24]}]}]}
+{"url":"Mathlib/Data/Nat/Count.lean","commit":"","full_name":"Nat.count_le_cardinal","start":[101,0],"end":[103,48],"file_path":"Mathlib/Data/Nat/Count.lean","tactics":[{"state_before":"p : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\n⊢ ↑(count p n) ≤ Cardinal.mk ↑{k | p k}","state_after":"p : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\n⊢ Cardinal.mk { k // k < n ∧ p k } ≤ Cardinal.mk ↑{k | p k}","tactic":"rw [count_eq_card_fintype, ← Cardinal.mk_fintype]","premises":[{"full_name":"Cardinal.mk_fintype","def_path":"Mathlib/SetTheory/Cardinal/Basic.lean","def_pos":[404,8],"def_end_pos":[404,18]},{"full_name":"Nat.count_eq_card_fintype","def_path":"Mathlib/Data/Nat/Count.lean","def_pos":[53,8],"def_end_pos":[53,29]}]},{"state_before":"p : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\n⊢ Cardinal.mk { k // k < n ∧ p k } ≤ Cardinal.mk ↑{k | p k}","state_after":"no goals","tactic":"exact Cardinal.mk_subtype_mono fun x hx ↦ hx.2","premises":[{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"Cardinal.mk_subtype_mono","def_path":"Mathlib/SetTheory/Cardinal/Basic.lean","def_pos":[1825,8],"def_end_pos":[1825,23]}]}]}
+{"url":"Mathlib/Algebra/Module/LocalizedModule.lean","commit":"","full_name":"LocalizedModule.induction_on₂","start":[97,0],"end":[101,19],"file_path":"Mathlib/Algebra/Module/LocalizedModule.lean","tactics":[{"state_before":"R : Type u\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nT : Type u_1\ninst✝² : CommSemiring T\ninst✝¹ : Algebra R T\ninst✝ : IsLocalization S T\nβ : LocalizedModule S M → LocalizedModule S M → Prop\nh : ∀ (m m' : M) (s s' : ↥S), β (mk m s) (mk m' s')\n⊢ ∀ (x y : LocalizedModule S M), β x y","state_after":"case mk.mk.mk.mk\nR : Type u\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nT : Type u_1\ninst✝² : CommSemiring T\ninst✝¹ : Algebra R T\ninst✝ : IsLocalization S T\nβ : LocalizedModule S M → LocalizedModule S M → Prop\nh : ∀ (m m' : M) (s s' : ↥S), β (mk m s) (mk m' s')\nx✝ : LocalizedModule S M\nm : M\ns : ↥S\ny✝ : LocalizedModule S M\nm' : M\ns' : ↥S\n⊢ β (Quot.mk Setoid.r (m, s)) (Quot.mk Setoid.r (m', s'))","tactic":"rintro ⟨⟨m, s⟩⟩ ⟨⟨m', s'⟩⟩","premises":[]},{"state_before":"case mk.mk.mk.mk\nR : Type u\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nT : Type u_1\ninst✝² : CommSemiring T\ninst✝¹ : Algebra R T\ninst✝ : IsLocalization S T\nβ : LocalizedModule S M → LocalizedModule S M → Prop\nh : ∀ (m m' : M) (s s' : ↥S), β (mk m s) (mk m' s')\nx✝ : LocalizedModule S M\nm : M\ns : ↥S\ny✝ : LocalizedModule S M\nm' : M\ns' : ↥S\n⊢ β (Quot.mk Setoid.r (m, s)) (Quot.mk Setoid.r (m', s'))","state_after":"no goals","tactic":"exact h m m' s s'","premises":[]}]}
+{"url":"Mathlib/RingTheory/Localization/Away/Basic.lean","commit":"","full_name":"exists_reduced_fraction'","start":[371,0],"end":[393,56],"file_path":"Mathlib/RingTheory/Localization/Away/Basic.lean","tactics":[{"state_before":"R : Type u_1\ninst✝⁵ : CommRing R\nx : R\nB : Type u_2\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R B\ninst✝² : IsLocalization.Away x B\ninst✝¹ : IsDomain R\ninst✝ : WfDvdMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\n⊢ ∃ a n, ¬x ∣ a ∧ selfZPow x B n * (algebraMap R B) a = b","state_after":"case intro.mk\nR : Type u_1\ninst✝⁵ : CommRing R\nx : R\nB : Type u_2\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R B\ninst✝² : IsLocalization.Away x B\ninst✝¹ : IsDomain R\ninst✝ : WfDvdMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : ↥(Submonoid.powers x)\nH : b * (algebraMap R B) ↑(a₀, y).2 = (algebraMap R B) (a₀, y).1\n⊢ ∃ a n, ¬x ∣ a ∧ selfZPow x B n * (algebraMap R B) a = b","tactic":"obtain ⟨⟨a₀, y⟩, H⟩ := surj (Submonoid.powers x) b","premises":[{"full_name":"IsLocalization.surj","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[120,8],"def_end_pos":[120,12]},{"full_name":"Submonoid.powers","def_path":"Mathlib/Algebra/Group/Submonoid/Membership.lean","def_pos":[393,4],"def_end_pos":[393,10]}]},{"state_before":"case intro.mk\nR : Type u_1\ninst✝⁵ : CommRing R\nx : R\nB : Type u_2\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R B\ninst✝² : IsLocalization.Away x B\ninst✝¹ : IsDomain R\ninst✝ : WfDvdMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : ↥(Submonoid.powers x)\nH : b * (algebraMap R B) ↑(a₀, y).2 = (algebraMap R B) (a₀, y).1\n⊢ ∃ a n, ¬x ∣ a ∧ selfZPow x B n * (algebraMap R B) a = b","state_after":"case intro.mk.intro\nR : Type u_1\ninst✝⁵ : CommRing R\nx : R\nB : Type u_2\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R B\ninst✝² : IsLocalization.Away x B\ninst✝¹ : IsDomain R\ninst✝ : WfDvdMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : ↥(Submonoid.powers x)\nH : b * (algebraMap R B) ↑(a₀, y).2 = (algebraMap R B) (a₀, y).1\nd : ℕ\nhy : x ^ d = ↑y\n⊢ ∃ a n, ¬x ∣ a ∧ selfZPow x B n * (algebraMap R B) a = b","tactic":"obtain ⟨d, hy⟩ := (Submonoid.mem_powers_iff y.1 x).mp y.2","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Submonoid.mem_powers_iff","def_path":"Mathlib/Algebra/Group/Submonoid/Membership.lean","def_pos":[403,8],"def_end_pos":[403,22]},{"full_name":"Subtype.property","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[590,2],"def_end_pos":[590,10]},{"full_name":"Subtype.val","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[587,2],"def_end_pos":[587,5]}]},{"state_before":"case intro.mk.intro\nR : Type u_1\ninst✝⁵ : CommRing R\nx : R\nB : Type u_2\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R B\ninst✝² : IsLocalization.Away x B\ninst✝¹ : IsDomain R\ninst✝ : WfDvdMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : ↥(Submonoid.powers x)\nH : b * (algebraMap R B) ↑(a₀, y).2 = (algebraMap R B) (a₀, y).1\nd : ℕ\nhy : x ^ d = ↑y\nha₀ : a₀ ≠ 0\n⊢ ∃ a n, ¬x ∣ a ∧ selfZPow x B n * (algebraMap R B) a = b","state_after":"case intro.mk.intro\nR : Type u_1\ninst✝⁵ : CommRing R\nx : R\nB : Type u_2\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R B\ninst✝² : IsLocalization.Away x B\ninst✝¹ : IsDomain R\ninst✝ : WfDvdMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : ↥(Submonoid.powers x)\nd : ℕ\nhy : x ^ d = ↑y\nha₀ : a₀ ≠ 0\nH : b * (algebraMap R B) (x ^ d) = (algebraMap R B) a₀\n⊢ ∃ a n, ¬x ∣ a ∧ selfZPow x B n * (algebraMap R B) a = b","tactic":"simp only [← hy] at H","premises":[]},{"state_before":"case intro.mk.intro\nR : Type u_1\ninst✝⁵ : CommRing R\nx : R\nB : Type u_2\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R B\ninst✝² : IsLocalization.Away x B\ninst✝¹ : IsDomain R\ninst✝ : WfDvdMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : ↥(Submonoid.powers x)\nd : ℕ\nhy : x ^ d = ↑y\nha₀ : a₀ ≠ 0\nH : b * (algebraMap R B) (x ^ d) = (algebraMap R B) a₀\n⊢ ∃ a n, ¬x ∣ a ∧ selfZPow x B n * (algebraMap R B) a = b","state_after":"case intro.mk.intro.intro.intro.intro\nR : Type u_1\ninst✝⁵ : CommRing R\nx : R\nB : Type u_2\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R B\ninst✝² : IsLocalization.Away x B\ninst✝¹ : IsDomain R\ninst✝ : WfDvdMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : ↥(Submonoid.powers x)\nd : ℕ\nhy : x ^ d = ↑y\nha₀ : a₀ ≠ 0\nH : b * (algebraMap R B) (x ^ d) = (algebraMap R B) a₀\nm : ℕ\na : R\nhyp1 : ¬x ∣ a\nhyp2 : a₀ = x ^ m * a\n⊢ ∃ a n, ¬x ∣ a ∧ selfZPow x B n * (algebraMap R B) a = b","tactic":"obtain ⟨m, a, hyp1, hyp2⟩ := WfDvdMonoid.max_power_factor ha₀ hx","premises":[{"full_name":"WfDvdMonoid.max_power_factor","def_path":"Mathlib/RingTheory/UniqueFactorizationDomain.lean","def_pos":[147,8],"def_end_pos":[147,36]}]},{"state_before":"case intro.mk.intro.intro.intro.intro\nR : Type u_1\ninst✝⁵ : CommRing R\nx : R\nB : Type u_2\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R B\ninst✝² : IsLocalization.Away x B\ninst✝¹ : IsDomain R\ninst✝ : WfDvdMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : ↥(Submonoid.powers x)\nd : ℕ\nhy : x ^ d = ↑y\nha₀ : a₀ ≠ 0\nH : b * (algebraMap R B) (x ^ d) = (algebraMap R B) a₀\nm : ℕ\na : R\nhyp1 : ¬x ∣ a\nhyp2 : a₀ = x ^ m * a\n⊢ ∃ a n, ¬x ∣ a ∧ selfZPow x B n * (algebraMap R B) a = b","state_after":"case intro.mk.intro.intro.intro.intro\nR : Type u_1\ninst✝⁵ : CommRing R\nx : R\nB : Type u_2\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R B\ninst✝² : IsLocalization.Away x B\ninst✝¹ : IsDomain R\ninst✝ : WfDvdMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : ↥(Submonoid.powers x)\nd : ℕ\nhy : x ^ d = ↑y\nha₀ : a₀ ≠ 0\nH : b * (algebraMap R B) (x ^ d) = (algebraMap R B) a₀\nm : ℕ\na : R\nhyp1 : ¬x ∣ a\nhyp2 : a₀ = x ^ m * a\n⊢ ¬x ∣ a ∧ selfZPow x B (↑m - ↑d) * (algebraMap R B) a = b","tactic":"refine ⟨a, m - d, ?_⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]},{"state_before":"case intro.mk.intro.intro.intro.intro\nR : Type u_1\ninst✝⁵ : CommRing R\nx : R\nB : Type u_2\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R B\ninst✝² : IsLocalization.Away x B\ninst✝¹ : IsDomain R\ninst✝ : WfDvdMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : ↥(Submonoid.powers x)\nd : ℕ\nhy : x ^ d = ↑y\nha₀ : a₀ ≠ 0\nH : b * (algebraMap R B) (x ^ d) = (algebraMap R B) a₀\nm : ℕ\na : R\nhyp1 : ¬x ∣ a\nhyp2 : a₀ = x ^ m * a\n⊢ ¬x ∣ a ∧ selfZPow x B (↑m - ↑d) * (algebraMap R B) a = b","state_after":"case intro.mk.intro.intro.intro.intro\nR : Type u_1\ninst✝⁵ : CommRing R\nx : R\nB : Type u_2\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R B\ninst✝² : IsLocalization.Away x B\ninst✝¹ : IsDomain R\ninst✝ : WfDvdMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : ↥(Submonoid.powers x)\nd : ℕ\nhy : x ^ d = ↑y\nha₀ : a₀ ≠ 0\nH : b * (algebraMap R B) (x ^ d) = (algebraMap R B) a₀\nm : ℕ\na : R\nhyp1 : ¬x ∣ a\nhyp2 : a₀ = x ^ m * a\n⊢ ¬x ∣ a ∧ (algebraMap R B) x ^ m * mk' B a 1 = (algebraMap R B) x ^ m * (algebraMap R B) a","tactic":"rw [← mk'_one (M := Submonoid.powers x) B, selfZPow_pow_sub, selfZPow_natCast, selfZPow_natCast,\n    ← map_pow _ _ d, mul_comm _ b, H, hyp2, map_mul, map_pow _ _ m]","premises":[{"full_name":"IsLocalization.mk'_one","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[238,8],"def_end_pos":[238,15]},{"full_name":"Submonoid.powers","def_path":"Mathlib/Algebra/Group/Submonoid/Membership.lean","def_pos":[393,4],"def_end_pos":[393,10]},{"full_name":"map_mul","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[281,8],"def_end_pos":[281,15]},{"full_name":"map_pow","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[423,8],"def_end_pos":[423,15]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]},{"full_name":"selfZPow_natCast","def_path":"Mathlib/RingTheory/Localization/Away/Basic.lean","def_pos":[286,8],"def_end_pos":[286,24]},{"full_name":"selfZPow_pow_sub","def_path":"Mathlib/RingTheory/Localization/Away/Basic.lean","def_pos":[355,8],"def_end_pos":[355,24]}]},{"state_before":"case intro.mk.intro.intro.intro.intro\nR : Type u_1\ninst✝⁵ : CommRing R\nx : R\nB : Type u_2\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R B\ninst✝² : IsLocalization.Away x B\ninst✝¹ : IsDomain R\ninst✝ : WfDvdMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : ↥(Submonoid.powers x)\nd : ℕ\nhy : x ^ d = ↑y\nha₀ : a₀ ≠ 0\nH : b * (algebraMap R B) (x ^ d) = (algebraMap R B) a₀\nm : ℕ\na : R\nhyp1 : ¬x ∣ a\nhyp2 : a₀ = x ^ m * a\n⊢ ¬x ∣ a ∧ (algebraMap R B) x ^ m * mk' B a 1 = (algebraMap R B) x ^ m * (algebraMap R B) a","state_after":"no goals","tactic":"exact ⟨hyp1, congr_arg _ (IsLocalization.mk'_one _ _)⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"IsLocalization.mk'_one","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[238,8],"def_end_pos":[238,15]}]}]}
+{"url":"Mathlib/LinearAlgebra/PiTensorProduct.lean","commit":"","full_name":"PiTensorProduct.reindex_symm","start":[743,0],"end":[750,28],"file_path":"Mathlib/LinearAlgebra/PiTensorProduct.lean","tactics":[{"state_before":"ι : Type u_1\nι₂ : Type u_2\nι₃ : Type u_3\nR : Type u_4\ninst✝⁷ : CommSemiring R\nR₁ : Type u_5\nR₂ : Type u_6\ns : ι → Type u_7\ninst✝⁶ : (i : ι) → AddCommMonoid (s i)\ninst✝⁵ : (i : ι) → Module R (s i)\nM : Type u_8\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nE : Type u_9\ninst✝² : AddCommMonoid E\ninst✝¹ : Module R E\nF : Type u_10\ninst✝ : AddCommMonoid F\ne : ι ≃ ι₂\n⊢ (reindex R (fun x => M) e).symm = reindex R (fun x => M) e.symm","state_after":"case h\nι : Type u_1\nι₂ : Type u_2\nι₃ : Type u_3\nR : Type u_4\ninst✝⁷ : CommSemiring R\nR₁ : Type u_5\nR₂ : Type u_6\ns : ι → Type u_7\ninst✝⁶ : (i : ι) → AddCommMonoid (s i)\ninst✝⁵ : (i : ι) → Module R (s i)\nM : Type u_8\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nE : Type u_9\ninst✝² : AddCommMonoid E\ninst✝¹ : Module R E\nF : Type u_10\ninst✝ : AddCommMonoid F\ne : ι ≃ ι₂\nx : ⨂[R] (i : ι₂), M\n⊢ (reindex R (fun x => M) e).symm x = (reindex R (fun x => M) e.symm) x","tactic":"ext x","premises":[]},{"state_before":"case h\nι : Type u_1\nι₂ : Type u_2\nι₃ : Type u_3\nR : Type u_4\ninst✝⁷ : CommSemiring R\nR₁ : Type u_5\nR₂ : Type u_6\ns : ι → Type u_7\ninst✝⁶ : (i : ι) → AddCommMonoid (s i)\ninst✝⁵ : (i : ι) → Module R (s i)\nM : Type u_8\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nE : Type u_9\ninst✝² : AddCommMonoid E\ninst✝¹ : Module R E\nF : Type u_10\ninst✝ : AddCommMonoid F\ne : ι ≃ ι₂\nx : ⨂[R] (i : ι₂), M\n⊢ (reindex R (fun x => M) e).symm x = (reindex R (fun x => M) e.symm) x","state_after":"no goals","tactic":"simp only [reindex, domDomCongrLinearEquiv', LinearEquiv.coe_symm_mk, LinearEquiv.coe_mk,\n    LinearEquiv.ofLinear_symm_apply, Equiv.symm_symm_apply, LinearEquiv.ofLinear_apply,\n    Equiv.piCongrLeft'_symm]","premises":[{"full_name":"Equiv.piCongrLeft'_symm","def_path":"Mathlib/Logic/Equiv/Basic.lean","def_pos":[1575,8],"def_end_pos":[1575,25]},{"full_name":"Equiv.symm_symm_apply","def_path":"Mathlib/Logic/Equiv/Defs.lean","def_pos":[268,31],"def_end_pos":[268,46]},{"full_name":"LinearEquiv.coe_mk","def_path":"Mathlib/Algebra/Module/Equiv/Defs.lean","def_pos":[189,8],"def_end_pos":[189,14]},{"full_name":"LinearEquiv.coe_symm_mk","def_path":"Mathlib/Algebra/Module/Equiv/Defs.lean","def_pos":[480,8],"def_end_pos":[480,19]},{"full_name":"LinearEquiv.ofLinear_apply","def_path":"Mathlib/Algebra/Module/Equiv/Basic.lean","def_pos":[443,8],"def_end_pos":[443,22]},{"full_name":"LinearEquiv.ofLinear_symm_apply","def_path":"Mathlib/Algebra/Module/Equiv/Basic.lean","def_pos":[447,8],"def_end_pos":[447,27]},{"full_name":"MultilinearMap.domDomCongrLinearEquiv'","def_path":"Mathlib/LinearAlgebra/Multilinear/Basic.lean","def_pos":[866,4],"def_end_pos":[866,27]},{"full_name":"PiTensorProduct.reindex","def_path":"Mathlib/LinearAlgebra/PiTensorProduct.lean","def_pos":[685,4],"def_end_pos":[685,11]}]}]}
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𝕜 F\ninst✝⁷ : TopologicalSpace (TotalSpace F E)\ninst✝⁶ : (x : B) → TopologicalSpace (E x)\ninst✝⁵ : FiberBundle F E\ninst✝⁴ : VectorBundle 𝕜 F E\ninst✝³ : SmoothVectorBundle F E IB\ne✝ e' : Trivialization F TotalSpace.proj\ninst✝² : MemTrivializationAtlas e✝\ninst✝¹ : MemTrivializationAtlas e'\ne : Trivialization F TotalSpace.proj\ninst✝ : MemTrivializationAtlas e\n⊢ SmoothOn (IB.prod 𝓘(𝕜, F)) (IB.prod 𝓘(𝕜, F)) (↑e) e.source","state_after":"𝕜 : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B → Type u_6\ninst✝²³ : NontriviallyNormedField 𝕜\nEB : Type u_7\ninst✝²² : NormedAddCommGroup EB\ninst✝²¹ : NormedSpace 𝕜 EB\nHB : Type u_8\ninst✝²⁰ : TopologicalSpace HB\nIB : ModelWithCorners 𝕜 EB HB\ninst✝¹⁹ : TopologicalSpace B\ninst✝¹⁸ : ChartedSpace HB B\ninst✝¹⁷ : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst✝¹⁶ : NormedAddCommGroup EM\ninst✝¹⁵ : NormedSpace 𝕜 EM\nHM : Type u_10\ninst✝¹⁴ : TopologicalSpace HM\nIM : ModelWithCorners 𝕜 EM HM\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : ℕ∞\ninst✝¹¹ : (x : B) → AddCommMonoid (E x)\ninst✝¹⁰ : (x : B) → Module 𝕜 (E x)\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\ninst✝⁷ : TopologicalSpace (TotalSpace F E)\ninst✝⁶ : (x : B) → TopologicalSpace (E x)\ninst✝⁵ : FiberBundle F E\ninst✝⁴ : VectorBundle 𝕜 F E\ninst✝³ : SmoothVectorBundle F E IB\ne✝ e' : Trivialization F TotalSpace.proj\ninst✝�� : MemTrivializationAtlas e✝\ninst✝¹ : MemTrivializationAtlas e'\ne : Trivialization F TotalSpace.proj\ninst✝ : MemTrivializationAtlas e\nthis : SmoothOn (IB.prod 𝓘(𝕜, F)) (IB.prod 𝓘(𝕜, F)) id e.source\n⊢ SmoothOn (IB.prod 𝓘(𝕜, F)) (IB.prod 𝓘(𝕜, F)) (↑e) e.source","tactic":"have : SmoothOn (IB.prod 𝓘(𝕜, F)) (IB.prod 𝓘(𝕜, F)) id e.source := smoothOn_id","premises":[{"full_name":"ModelWithCorners.prod","def_path":"Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean","def_pos":[362,4],"def_end_pos":[362,25]},{"full_name":"PartialEquiv.source","def_path":"Mathlib/Logic/Equiv/PartialEquiv.lean","def_pos":[124,2],"def_end_pos":[124,8]},{"full_name":"SmoothOn","def_path":"Mathlib/Geometry/Manifold/ContMDiff/Defs.lean","def_pos":[198,7],"def_end_pos":[198,15]},{"full_name":"id","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[33,14],"def_end_pos":[33,16]},{"full_name":"modelWithCornersSelf","def_path":"Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean","def_pos":[148,4],"def_end_pos":[148,24]},{"full_name":"smoothOn_id","def_path":"Mathlib/Geometry/Manifold/ContMDiff/Basic.lean","def_pos":[194,8],"def_end_pos":[194,19]}]},{"state_before":"𝕜 : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B → Type u_6\ninst✝²³ : NontriviallyNormedField 𝕜\nEB : Type u_7\ninst✝²² : NormedAddCommGroup EB\ninst✝²¹ : NormedSpace 𝕜 EB\nHB : Type u_8\ninst✝²⁰ : TopologicalSpace HB\nIB : ModelWithCorners 𝕜 EB HB\ninst✝¹⁹ : TopologicalSpace B\ninst✝¹⁸ : ChartedSpace HB B\ninst✝¹⁷ : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst✝¹⁶ : NormedAddCommGroup EM\ninst✝¹⁵ : NormedSpace 𝕜 EM\nHM : Type u_10\ninst✝¹⁴ : TopologicalSpace HM\nIM : ModelWithCorners 𝕜 EM HM\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : ℕ∞\ninst✝¹¹ : (x : B) → AddCommMonoid (E x)\ninst✝¹⁰ : (x : B) → Module 𝕜 (E x)\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\ninst✝⁷ : TopologicalSpace (TotalSpace F E)\ninst✝⁶ : (x : B) → TopologicalSpace (E x)\ninst✝⁵ : FiberBundle F E\ninst✝⁴ : VectorBundle 𝕜 F E\ninst✝³ : SmoothVectorBundle F E IB\ne✝ e' : Trivialization F TotalSpace.proj\ninst✝² : MemTrivializationAtlas e✝\ninst✝¹ : MemTrivializationAtlas e'\ne : Trivialization F TotalSpace.proj\ninst✝ : MemTrivializationAtlas e\nthis : SmoothOn (IB.prod 𝓘(𝕜, F)) (IB.prod 𝓘(𝕜, F)) id e.source\n⊢ SmoothOn (IB.prod 𝓘(𝕜, F)) (IB.prod 𝓘(𝕜, F)) (↑e) e.source","state_after":"𝕜 : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B → Type u_6\ninst✝²³ : NontriviallyNormedField 𝕜\nEB : Type u_7\ninst✝²² : NormedAddCommGroup EB\ninst✝²¹ : NormedSpace 𝕜 EB\nHB : Type u_8\ninst✝²⁰ : TopologicalSpace HB\nIB : ModelWithCorners 𝕜 EB HB\ninst✝¹⁹ : TopologicalSpace B\ninst✝¹⁸ : ChartedSpace HB B\ninst✝¹⁷ : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst✝¹⁶ : NormedAddCommGroup EM\ninst✝¹⁵ : NormedSpace 𝕜 EM\nHM : Type u_10\ninst✝¹⁴ : TopologicalSpace HM\nIM : ModelWithCorners 𝕜 EM HM\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : ℕ∞\ninst✝¹¹ : (x : B) → AddCommMonoid (E x)\ninst✝¹⁰ : (x : B) → Module 𝕜 (E x)\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\ninst✝⁷ : TopologicalSpace (TotalSpace F E)\ninst✝⁶ : (x : B) → TopologicalSpace (E x)\ninst✝⁵ : FiberBundle F E\ninst✝⁴ : VectorBundle 𝕜 F E\ninst✝³ : SmoothVectorBundle F E IB\ne✝ e' : Trivialization F TotalSpace.proj\ninst✝² : MemTrivializationAtlas e✝\ninst✝¹ : MemTrivializationAtlas e'\ne : Trivialization F TotalSpace.proj\ninst✝ : MemTrivializationAtlas e\nthis :\n  SmoothOn (IB.prod 𝓘(𝕜, F)) IB (fun x => (id x).proj) e.source ∧\n    SmoothOn (IB.prod 𝓘(𝕜, F)) 𝓘(𝕜, F) (fun x => (↑e (id x)).2) e.source\n⊢ SmoothOn (IB.prod 𝓘(𝕜, F)) (IB.prod 𝓘(𝕜, F)) (↑e) e.source","tactic":"rw [e.smoothOn_iff IB (mapsTo_id _)] at this","premises":[{"full_name":"Set.mapsTo_id","def_path":"Mathlib/Data/Set/Function.lean","def_pos":[372,8],"def_end_pos":[372,17]},{"full_name":"Trivialization.smoothOn_iff","def_path":"Mathlib/Geometry/Manifold/VectorBundle/Basic.lean","def_pos":[525,8],"def_end_pos":[525,35]}]},{"state_before":"𝕜 : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B → Type u_6\ninst✝²³ : NontriviallyNormedField 𝕜\nEB : Type u_7\ninst✝²² : NormedAddCommGroup EB\ninst✝²¹ : NormedSpace 𝕜 EB\nHB : Type u_8\ninst✝²⁰ : TopologicalSpace HB\nIB : ModelWithCorners 𝕜 EB HB\ninst✝¹⁹ : TopologicalSpace B\ninst✝¹⁸ : ChartedSpace HB B\ninst✝¹⁷ : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst✝¹⁶ : NormedAddCommGroup EM\ninst✝¹⁵ : NormedSpace 𝕜 EM\nHM : Type u_10\ninst✝¹⁴ : TopologicalSpace HM\nIM : ModelWithCorners 𝕜 EM HM\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : ℕ∞\ninst✝¹¹ : (x : B) → AddCommMonoid (E x)\ninst✝¹⁰ : (x : B) → Module 𝕜 (E x)\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\ninst✝⁷ : TopologicalSpace (TotalSpace F E)\ninst✝⁶ : (x : B) → TopologicalSpace (E x)\ninst✝⁵ : FiberBundle F E\ninst✝⁴ : VectorBundle 𝕜 F E\ninst✝³ : SmoothVectorBundle F E IB\ne✝ e' : Trivialization F TotalSpace.proj\ninst✝² : MemTrivializationAtlas e✝\ninst✝¹ : MemTrivializationAtlas e'\ne : Trivialization F TotalSpace.proj\ninst✝ : MemTrivializationAtlas e\nthis :\n  SmoothOn (IB.prod 𝓘(𝕜, F)) IB (fun x => (id x).proj) e.source ∧\n    SmoothOn (IB.prod 𝓘(𝕜, F)) 𝓘(𝕜, F) (fun x => (↑e (id x)).2) e.source\n⊢ SmoothOn (IB.prod 𝓘(𝕜, F)) (IB.prod 𝓘(𝕜, F)) (↑e) e.source","state_after":"no goals","tactic":"exact (this.1.prod_mk this.2).congr fun x hx ↦ (e.mk_proj_snd hx).symm","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"ContMDiffOn.congr","def_path":"Mathlib/Geometry/Manifold/ContMDiff/Defs.lean","def_pos":[790,8],"def_end_pos":[790,25]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"SmoothOn.prod_mk","def_path":"Mathlib/Geometry/Manifold/ContMDiff/Product.lean","def_pos":[113,15],"def_end_pos":[113,31]},{"full_name":"Trivialization.mk_proj_snd","def_path":"Mathlib/Topology/FiberBundle/Trivialization.lean","def_pos":[318,8],"def_end_pos":[318,19]}]}]}
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Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\n⊢ Odd (Fintype.card V) ↔ Odd (Nat.card ↑{c | Odd (Nat.card ↑c.supp)})","state_after":"V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\n⊢ Odd (Finset.univ.biUnion fun x => x.supp.toFinset).card ↔ Odd (Nat.card ↑{x | Odd (Nat.card ↑x.supp)})","tactic":"simp only [← (set_fintype_card_eq_univ_iff _).mpr G.iUnion_connectedComponentSupp,\n    ConnectedComponent.mem_supp_iff, Fintype.card_subtype_compl,\n    ← Set.toFinset_card, Set.toFinset_iUnion ConnectedComponent.supp]","premises":[{"full_name":"Fintype.card_subtype_compl","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[768,8],"def_end_pos":[768,34]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Set.toFinset_card","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[222,8],"def_end_pos":[222,21]},{"full_name":"Set.toFinset_iUnion","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[343,6],"def_end_pos":[343,21]},{"full_name":"SimpleGraph.ConnectedComponent.mem_supp_iff","def_path":"Mathlib/Combinatorics/SimpleGraph/Path.lean","def_pos":[939,8],"def_end_pos":[939,20]},{"full_name":"SimpleGraph.ConnectedComponent.supp","def_path":"Mathlib/Combinatorics/SimpleGraph/Path.lean","def_pos":[918,4],"def_end_pos":[918,8]},{"full_name":"SimpleGraph.iUnion_connectedComponentSupp","def_path":"Mathlib/Combinatorics/SimpleGraph/Path.lean","def_pos":[976,6],"def_end_pos":[976,35]},{"full_name":"set_fintype_card_eq_univ_iff","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[690,8],"def_end_pos":[690,36]}]},{"state_before":"V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\n⊢ Odd (Finset.univ.biUnion fun x => x.supp.toFinset).card ↔ Odd (Nat.card ↑{x | Odd (Nat.card ↑x.supp)})","state_after":"V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\n⊢ Odd (∑ u : G.ConnectedComponent, u.supp.toFinset.card) ↔ Odd (Nat.card ↑{x | Odd (Nat.card ↑x.supp)})","tactic":"rw [Finset.card_biUnion\n    (fun x _ y _ hxy ↦ Set.disjoint_toFinset.mpr (pairwise_disjoint_supp_connectedComponent _ hxy))]","premises":[{"full_name":"Finset.card_biUnion","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[1848,8],"def_end_pos":[1848,20]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Set.disjoint_toFinset","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[597,8],"def_end_pos":[597,25]},{"full_name":"SimpleGraph.pairwise_disjoint_supp_connectedComponent","def_path":"Mathlib/Combinatorics/SimpleGraph/Path.lean","def_pos":[967,6],"def_end_pos":[967,47]}]},{"state_before":"V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\n⊢ Odd (∑ u : G.ConnectedComponent, u.supp.toFinset.card) ↔ Odd (Nat.card ↑{x | Odd (Nat.card ↑x.supp)})","state_after":"V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\n⊢ Odd (∑ x : G.ConnectedComponent, Nat.card ↑x.supp) ↔ Odd (Nat.card ↑{x | Odd (Nat.card ↑x.supp)})","tactic":"simp_rw [Set.toFinset_card, ← Nat.card_eq_fintype_card]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Nat.card_eq_fintype_card","def_path":"Mathlib/SetTheory/Cardinal/Finite.lean","def_pos":[37,8],"def_end_pos":[37,28]},{"full_name":"Set.toFinset_card","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[222,8],"def_end_pos":[222,21]}]},{"state_before":"V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\n⊢ Odd (∑ x : G.ConnectedComponent, Nat.card ↑x.supp) ↔ Odd (Nat.card ↑{x | Odd (Nat.card ↑x.supp)})","state_after":"V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\n⊢ Odd (∑ x : G.ConnectedComponent, Nat.card ↑x.supp) ↔\n    Odd (Finset.filter (fun x => x ∈ {x | Odd (Nat.card ↑x.supp)}) Finset.univ).card","tactic":"rw [Nat.card_eq_fintype_card, Fintype.card_ofFinset]","premises":[{"full_name":"Fintype.card_ofFinset","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[126,8],"def_end_pos":[126,21]},{"full_name":"Nat.card_eq_fintype_card","def_path":"Mathlib/SetTheory/Cardinal/Finite.lean","def_pos":[37,8],"def_end_pos":[37,28]}]},{"state_before":"V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\n⊢ Odd (∑ x : G.ConnectedComponent, Nat.card ↑x.supp) ↔\n    Odd (Finset.filter (fun x => x ∈ {x | Odd (Nat.card ↑x.supp)}) Finset.univ).card","state_after":"no goals","tactic":"exact (Finset.odd_sum_iff_odd_card_odd (fun x : G.ConnectedComponent ↦ Nat.card x.supp))","premises":[{"full_name":"Finset.odd_sum_iff_odd_card_odd","def_path":"Mathlib/Algebra/BigOperators/Ring/Nat.lean","def_pos":[28,6],"def_end_pos":[28,30]},{"full_name":"Nat.card","def_path":"Mathlib/SetTheory/Cardinal/Finite.lean","def_pos":[33,14],"def_end_pos":[33,18]},{"full_name":"SimpleGraph.ConnectedComponent","def_path":"Mathlib/Combinatorics/SimpleGraph/Path.lean","def_pos":[765,4],"def_end_pos":[765,22]},{"full_name":"SimpleGraph.ConnectedComponent.supp","def_path":"Mathlib/Combinatorics/SimpleGraph/Path.lean","def_pos":[918,4],"def_end_pos":[918,8]}]}]}
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+{"url":"Mathlib/Analysis/PSeries.lean","commit":"","full_name":"ENNReal.tsum_schlomilch_le","start":[154,0],"end":[164,17],"file_path":"Mathlib/Analysis/PSeries.lean","tactics":[{"state_before":"u : ℕ → ℕ\nf : ℕ → ℝ≥0∞\nC : ℕ\nhf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m\nh_pos : ∀ (n : ℕ), 0 < u n\nh_nonneg : ∀ (n : ℕ), 0 ≤ f n\nhu : Monotone u\nh_succ_diff : SuccDiffBounded C u\n⊢ ∑' (k : ℕ), (↑(u (k + 1)) - ↑(u k)) * f (u k) ≤ (↑(u 1) - ↑(u 0)) * f (u 0) + ↑C * ∑' (k : ℕ), f k","state_after":"u : ℕ → ℕ\nf : ℕ → ℝ≥0∞\nC : ℕ\nhf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m\nh_pos : ∀ (n : ℕ), 0 < u n\nh_nonneg : ∀ (n : ℕ), 0 ≤ f n\nhu : Monotone u\nh_succ_diff : SuccDiffBounded C u\n⊢ ⨆ i, ∑ a ∈ range i.succ, (↑(u (a + 1)) - ↑(u a)) * f (u a) ≤ (↑(u 1) - ↑(u 0)) * f (u 0) + ↑C * ∑' (k : ℕ), f k","tactic":"rw [ENNReal.tsum_eq_iSup_nat' (tendsto_atTop_mono Nat.le_succ tendsto_id)]","premises":[{"full_name":"ENNReal.tsum_eq_iSup_nat'","def_path":"Mathlib/Topology/Instances/ENNReal.lean","def_pos":[755,18],"def_end_pos":[755,35]},{"full_name":"Filter.tendsto_atTop_mono","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[358,8],"def_end_pos":[358,26]},{"full_name":"Filter.tendsto_id","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2649,8],"def_end_pos":[2649,18]},{"full_name":"Nat.le_succ","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1684,8],"def_end_pos":[1684,19]}]},{"state_before":"u : ℕ → ℕ\nf : ℕ → ℝ≥0∞\nC : ℕ\nhf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m\nh_pos : ∀ (n : ℕ), 0 < u n\nh_nonneg : ∀ (n : ℕ), 0 ≤ f n\nhu : Monotone u\nh_succ_diff : SuccDiffBounded C u\n⊢ ⨆ i, ∑ a ∈ range i.succ, (↑(u (a + 1)) - ↑(u a)) * f (u a) ≤ (↑(u 1) - ↑(u 0)) * f (u 0) + ↑C * ∑' (k : ℕ), f k","state_after":"case refine_1\nu : ℕ → ℕ\nf : ℕ → ℝ≥0∞\nC : ℕ\nhf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m\nh_pos : ∀ (n : ℕ), 0 < u n\nh_nonneg : ∀ (n : ℕ), 0 ≤ f n\nhu : Monotone u\nh_succ_diff : SuccDiffBounded C u\nn : ℕ\n⊢ ∑ a ∈ range n.succ, (↑(u (a + 1)) - ↑(u a)) * f (u a) ≤\n    (↑(u 1) - ↑(u 0)) * f (u 0) + ↑C * ∑ x ∈ Ico (u 0 + 1) (u n + 1), f x\n\ncase refine_2\nu : ℕ → ℕ\nf : ℕ → ℝ≥0∞\nC : ℕ\nhf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m\nh_pos : ∀ (n : ℕ), 0 < u n\nh_nonneg : ∀ (n : ℕ), 0 ≤ f n\nhu : Monotone u\nh_succ_diff : SuccDiffBounded C u\nn : ℕ\n⊢ 0 ≤ ↑C","tactic":"refine\n    iSup_le fun n =>\n      le_trans ?_\n        (add_le_add_left\n          (mul_le_mul_of_nonneg_left (ENNReal.sum_le_tsum <| Finset.Ico (u 0 + 1) (u n + 1)) ?_) _)","premises":[{"full_name":"ENNReal.sum_le_tsum","def_path":"Mathlib/Topology/Instances/ENNReal.lean","def_pos":[752,18],"def_end_pos":[752,29]},{"full_name":"Finset.Ico","def_path":"Mathlib/Order/Interval/Finset/Defs.lean","def_pos":[281,4],"def_end_pos":[281,7]},{"full_name":"add_le_add_left","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[61,31],"def_end_pos":[61,46]},{"full_name":"iSup_le","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[661,8],"def_end_pos":[661,15]},{"full_name":"le_trans","def_path":"Mathlib/Order/Defs.lean","def_pos":[48,8],"def_end_pos":[48,16]},{"full_name":"mul_le_mul_of_nonneg_left","def_path":"Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean","def_pos":[190,8],"def_end_pos":[190,33]}]},{"state_before":"case refine_1\nu : ℕ → ℕ\nf : ℕ → ℝ≥0∞\nC : ℕ\nhf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m\nh_pos : ∀ (n : ℕ), 0 < u n\nh_nonneg : ∀ (n : ℕ), 0 ≤ f n\nhu : Monotone u\nh_succ_diff : SuccDiffBounded C u\nn : ℕ\n⊢ ∑ a ∈ range n.succ, (↑(u (a + 1)) - ↑(u a)) * f (u a) ≤\n    (↑(u 1) - ↑(u 0)) * f (u 0) + ↑C * ∑ x ∈ Ico (u 0 + 1) (u n + 1), f x\n\ncase refine_2\nu : ℕ → ℕ\nf : ℕ → ℝ≥0∞\nC : ℕ\nhf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m\nh_pos : ∀ (n : ℕ), 0 < u n\nh_nonneg : ∀ (n : ℕ), 0 ≤ f n\nhu : Monotone u\nh_succ_diff : SuccDiffBounded C u\nn : ℕ\n⊢ 0 ≤ ↑C","state_after":"case refine_2\nu : ℕ → ℕ\nf : ℕ → ℝ≥0∞\nC : ℕ\nhf : �� ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m\nh_pos : ∀ (n : ℕ), 0 < u n\nh_nonneg : ∀ (n : ℕ), 0 ≤ f n\nhu : Monotone u\nh_succ_diff : SuccDiffBounded C u\nn : ℕ\n⊢ 0 ≤ ↑C","tactic":"simpa using Finset.sum_schlomilch_le hf h_pos h_nonneg hu h_succ_diff n","premises":[{"full_name":"Finset.sum_schlomilch_le","def_path":"Mathlib/Analysis/PSeries.lean","def_pos":[102,8],"def_end_pos":[102,25]}]},{"state_before":"case refine_2\nu : ℕ → ℕ\nf : ℕ → ℝ≥0∞\nC : ℕ\nhf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m\nh_pos : ∀ (n : ℕ), 0 < u n\nh_nonneg : ∀ (n : ℕ), 0 ≤ f n\nhu : Monotone u\nh_succ_diff : SuccDiffBounded C u\nn : ℕ\n⊢ 0 ≤ ↑C","state_after":"no goals","tactic":"exact zero_le _","premises":[{"full_name":"zero_le","def_path":"Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean","def_pos":[105,29],"def_end_pos":[105,36]}]}]}
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