diff --git "a/random/random_our/val.jsonl" "b/random/random_our/val.jsonl"
new file mode 100644--- /dev/null
+++ "b/random/random_our/val.jsonl"
@@ -0,0 +1,4231 @@
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+{"state": {"context": ["α : Type u", "β : Type u_1", "n : ℕ", "x : α", "inst✝ : StrictOrderedRing α", "hx : x < 0", "hx' : 0 < x + 1", "hn : 1 < n"], "goal": "0 < ∑ i ∈ range 2, x ^ i ∧ ∑ i ∈ range 2, x ^ i < 1"}, "premise": [145198], "module": ["Mathlib/Algebra/GeomSum.lean"]}
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+{"state": {"context": ["α : Type u_1", "inst✝ : PseudoEMetricSpace α", "δ : ℝ", "δ_pos : 0 < δ", "E : Set α", "x : α", "x_out : ENNReal.ofReal δ ≤ infEdist x E"], "goal": "1 - infEdist x E / ENNReal.ofReal δ ≤ ⊥"}, "premise": [14296, 18803, 143820, 103366], "module": ["Mathlib/Topology/MetricSpace/ThickenedIndicator.lean"]}
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+{"state": {"context": ["M : Type u_1", "inst✝¹ : CancelCommMonoidWithZero M", "N : Type u_2", "inst✝ : CancelCommMonoidWithZero N", "m : Associates M", "n : Associates N", "d : { l // l ≤ m } ≃o { l // l ≤ n }", "this : OrderBot { l // l ≤ m } := Subtype.orderBot ⋯"], "goal": "↑(d ⟨1, ⋯⟩) = 1"}, "premise": [18803], "module": ["Mathlib/RingTheory/ChainOfDivisors.lean"]}
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+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "σ : Type u_5", "inst✝⁴ : Primcodable α", "inst✝³ : Primcodable β", "inst✝² : Primcodable γ", "inst✝¹ : Primcodable δ", "inst✝ : Primcodable σ", "m n : ℕ"], "goal": "(m, n).1 - (m, n).2 * (fun x x_1 => x / x_1) (m, n).1 (m, n).2 = m % n"}, "premise": [3893, 4474, 119708], "module": ["Mathlib/Computability/Primrec.lean"]}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "σ : Type u_5", "inst✝⁴ : Primcodable α", "inst✝³ : Primcodable β", "inst✝² : Primcodable γ", "inst✝¹ : Primcodable δ", "inst✝ : Primcodable σ", "m n : ℕ"], "goal": "(m, n).1 = m % n + (m, n).2 * (fun x x_1 => x / x_1) (m, n).1 (m, n).2"}, "premise": [4474, 119708, 3893], "module": ["Mathlib/Computability/Primrec.lean"]}
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+{"state": {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "inst✝ : Abelian C", "S : ShortComplex C", "γ : kernel S.g ⟶ cokernel S.f := kernel.ι S.g ≫ cokernel.π S.f", "f' : S.X₁ ⟶ kernel S.g := kernel.lift S.g S.f ⋯", "hf' : f' = kernel.lift γ f' ⋯ ≫ kernel.ι γ", "wπ : f' ≫ cokernel.π (kernel.ι γ) = 0", "e : Abelian.image S.f ≅ kernel γ := S.abelianImageToKernelIsKernel.conePointUniqueUpToIso (limit.isLimit (parallelPair (kernel.ι S.g ≫ cokernel.π S.f) 0))"], "goal": "S.LeftHomologyData"}, "premise": [94253, 94851, 96173, 96191], "module": ["Mathlib/Algebra/Homology/ShortComplex/Abelian.lean"]}
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+{"state": {"context": ["R : Type u_1", "inst✝² : CommRing R", "S : Submonoid R", "P : Type u_2", "inst✝¹ : CommRing P", "inst✝ : Algebra R P", "loc : IsLocalization S P", "I : FractionalIdeal S P", "x : P", "hx : x ∈ 0"], "goal": "0 ∈ I"}, "premise": [1673, 75651, 75638], "module": ["Mathlib/RingTheory/FractionalIdeal/Basic.lean"]}
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+{"state": {"context": ["α : Type ua", "β : Type ub", "γ : Type uc", "δ : Type ud", "ι : Sort u_1", "inst✝¹ : UniformSpace α", "inst✝ : TopologicalSpace β", "f : β → α", "s : Set β"], "goal": "ContinuousOn f s ↔ ∀ b ∈ s, Tendsto (fun x => (f b, f x)) (𝓝[s] b) (𝓤 α)"}, "premise": [60680], "module": ["Mathlib/Topology/UniformSpace/Basic.lean"]}
+{"state": {"context": ["ι : Type u_1", "α : ι → Type u_2", "inst✝¹ : (i : ι) → Preorder (α i)", "x✝ y✝ : (i : ι) → α i", "inst✝ : DecidableEq ι", "x y : (i : ι) → α i", "i₀ : ι", "m : α i₀", "hm : m ≤ y i₀"], "goal": "(univ.pi fun i => Ioc (x i) (update y i₀ m i)) = {z | z i₀ ≤ m} ∩ univ.pi fun i => Ioc (x i) (y i)"}, "premise": [1674, 20207, 20259, 133445, 133455, 133444, 134029, 134035, 134038], "module": ["Mathlib/Order/Interval/Set/Pi.lean"]}
+{"state": {"context": ["ι : Type u_1", "α : ι → Type u_2", "inst✝¹ : (i : ι) → Preorder (α i)", "x✝ y✝ : (i : ι) → α i", "inst✝ : DecidableEq ι", "x y : (i : ι) → α i", "i₀ : ι", "m : α i₀", "hm : m ≤ y i₀", "this : Ioc (x i₀) m = Iic m ∩ Ioc (x i₀) (y i₀)"], "goal": "(univ.pi fun i => Ioc (x i) (update y i₀ m i)) = {z | z i₀ ≤ m} ∩ univ.pi fun i => Ioc (x i) (y i)"}, "premise": [20259, 133444, 133445, 1674, 134029, 133455, 20207, 134035, 134038], "module": ["Mathlib/Order/Interval/Set/Pi.lean"]}
+{"state": {"context": ["ι : Type u_1", "α : ι → Type u_2", "inst✝¹ : (i : ι) → Preorder (α i)", "x✝ y✝ : (i : ι) → α i", "inst✝ : DecidableEq ι", "x y : (i : ι) → α i", "i₀ : ι", "m : α i₀", "hm : m ≤ y i₀", "this : Ioc (x i₀) m = Iic m ∩ Ioc (x i₀) (y i₀)"], "goal": "({x_1 | x_1 i₀ ∈ Iic m ∩ Ioc (x i₀) (y i₀)} ∩ {i₀}ᶜ.pi fun j => Ioc (x j) (y j)) = {z | z i₀ ≤ m} ∩ {x_1 | x_1 i₀ ∈ Ioc (x i₀) (y i₀)} ∩ {i₀}ᶜ.pi fun j => Ioc (x j) (y j)"}, "premise": [133444, 134029, 134035, 134038], "module": ["Mathlib/Order/Interval/Set/Pi.lean"]}
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+{"state": {"context": ["R : Type u", "inst✝ : CommRing R", "X : CoalgebraCat R"], "goal": "ModuleCat.ofHom Coalgebra.comul ≫ ModuleCat.of R ↑X.toModuleCat ◁ ModuleCat.ofHom Coalgebra.counit = (ρ_ (ModuleCat.of R ↑X.toModuleCat)).inv"}, "premise": [79254, 112710], "module": ["Mathlib/Algebra/Category/CoalgebraCat/ComonEquivalence.lean"]}
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+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "J : GrothendieckTopology C", "D : Type w", "inst✝² : Category.{max v u, w} D", "inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)", "P : Cᵒᵖ ⥤ D", "inst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D", "X : Cᵒᵖ", "S : (J.Cover (unop X))ᵒᵖ", "e : unop S ⟶ ⊤ := homOfLE ⋯"], "goal": "(J.diagramNatTrans { app := fun X => ⊤.toMultiequalizer P ≫ colimit.ι (J.diagram P (unop X)) (op ⊤), naturality := ⋯ } (unop X)).app S = (colimit.ι (J.diagram P (unop X)) S ≫ ⊤.toMultiequalizer (J.plusObj P)) ≫ (J.diagram (J.plusObj P) (unop X)).map e.op"}, "premise": [95099, 95098], "module": ["Mathlib/CategoryTheory/Sites/Plus.lean"]}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "J : GrothendieckTopology C", "D : Type w", "inst✝² : Category.{max v u, w} D", "inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)", "P : Cᵒᵖ ⥤ D", "inst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D", "X : Cᵒᵖ", "S : (J.Cover (unop X))ᵒᵖ", "e : unop S ⟶ ⊤ := homOfLE ⋯", "I : ((unop S).index (J.plusObj P)).L"], "goal": "(J.diagramNatTrans { app := fun X => ⊤.toMultiequalizer P ≫ colimit.ι (J.diagram P (unop X)) (op ⊤), naturality := ⋯ } (unop X)).app S ≫ Multiequalizer.ι ((unop S).index (J.plusObj P)) I = ((colimit.ι (J.diagram P (unop X)) S ≫ ⊤.toMultiequalizer (J.plusObj P)) ≫ (J.diagram (J.plusObj P) (unop X)).map e.op) ≫ Multiequalizer.ι ((unop S).index (J.plusObj P)) I"}, "premise": [95098, 95099], "module": ["Mathlib/CategoryTheory/Sites/Plus.lean"]}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "J : GrothendieckTopology C", "D : Type w", "inst✝² : Category.{max v u, w} D", "inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)", "P : Cᵒᵖ ⥤ D", "inst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D", "X : Cᵒᵖ", "S : (J.Cover (unop X))ᵒᵖ", "e : unop S ⟶ ⊤ := homOfLE ⋯", "I : ((unop S).index (J.plusObj P)).L"], "goal": "Multiequalizer.ι ((unop S).index P) I ≫ { app := fun X => ⊤.toMultiequalizer P ≫ colimit.ι (J.diagram P (unop X)) (op ⊤), naturality := ⋯ }.app (op I.Y) = ((colimit.ι (J.diagram P (unop X)) S ≫ ⊤.toMultiequalizer (J.plusObj P)) ≫ (J.diagram (J.plusObj P) (unop X)).map e.op) ≫ Multiequalizer.ι ((unop S).index (J.plusObj P)) I"}, "premise": [130987, 95052, 96173, 130988, 91856, 95098, 93340], "module": ["Mathlib/CategoryTheory/Sites/Plus.lean"]}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "J : GrothendieckTopology C", "D : Type w", "inst✝² : Category.{max v u, w} D", "inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)", "P : Cᵒᵖ ⥤ D", "inst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D", "X : Cᵒᵖ", "S : (J.Cover (unop X))ᵒᵖ", "e : unop S ⟶ ⊤ := homOfLE ⋯", "I : ((unop S).index (J.plusObj P)).L"], "goal": "Multiequalizer.ι ((unop S).index P) I ≫ ⊤.toMultiequalizer P ≫ colimit.ι (J.diagram P I.Y) (op ⊤) = colimit.ι (J.diagram P (unop X)) S ≫ (J.plusObj P).map (Cover.Arrow.map I e.op.unop).f.op"}, "premise": [130987, 95052, 96173, 130988, 91856, 18777, 93340], "module": ["Mathlib/CategoryTheory/Sites/Plus.lean"]}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "J : GrothendieckTopology C", "D : Type w", "inst✝² : Category.{max v u, w} D", "inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)", "P : Cᵒᵖ ⥤ D", "inst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D", "X : Cᵒᵖ", "S : (J.Cover (unop X))ᵒᵖ", "e : unop S ⟶ ⊤ := homOfLE ⋯", "I : ((unop S).index (J.plusObj P)).L", "ee : (J.pullback (Cover.Arrow.map I e).f).obj (unop S) ⟶ ⊤ := homOfLE ⋯"], "goal": "Multiequalizer.ι ((unop S).index P) I ≫ ⊤.toMultiequalizer P ≫ colimit.ι (J.diagram P I.Y) (op ⊤) = colimit.ι (J.diagram P (unop X)) S ≫ (J.plusObj P).map (Cover.Arrow.map I e.op.unop).f.op"}, "premise": [93413, 96173, 93390, 91857, 18777], "module": ["Mathlib/CategoryTheory/Sites/Plus.lean"]}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "J : GrothendieckTopology C", "D : Type w", "inst✝² : Category.{max v u, w} D", "inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)", "P : Cᵒᵖ ⥤ D", "inst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D", "X : Cᵒᵖ", "S : (J.Cover (unop X))ᵒᵖ", "e : unop S ⟶ ⊤ := homOfLE ⋯", "I : ((unop S).index (J.plusObj P)).L", "ee : (J.pullback (Cover.Arrow.map I e).f).obj (unop S) ⟶ ⊤ := homOfLE ⋯"], "goal": "((Multiequalizer.ι ((unop S).index P) I ≫ ⊤.toMultiequalizer P) ≫ (J.diagram P I.Y).map ee.op) ≫ colimit.ι (J.diagram P I.Y) (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S))) = Multiequalizer.lift ((unop ((J.pullback (Cover.Arrow.map I e.op.unop).f.op.unop).op.obj S)).index P) ((J.diagram P (unop X)).obj S) (fun I_1 => Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I_1)) ⋯ ≫ colimit.ι (J.diagram P (unop (op (Cover.Arrow.map I e.op.unop).Y))) ((J.pullback (Cover.Arrow.map I e.op.unop).f.op.unop).op.obj S)"}, "premise": [91857, 93390, 93413, 96173], "module": ["Mathlib/CategoryTheory/Sites/Plus.lean"]}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "J : GrothendieckTopology C", "D : Type w", "inst✝² : Category.{max v u, w} D", "inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)", "P : Cᵒᵖ ⥤ D", "inst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D", "X : Cᵒᵖ", "S : (J.Cover (unop X))ᵒᵖ", "e : unop S ⟶ ⊤ := homOfLE ⋯", "I : ((unop S).index (J.plusObj P)).L", "ee : (J.pullback (Cover.Arrow.map I e).f).obj (unop S) ⟶ ⊤ := homOfLE ⋯"], "goal": "(Multiequalizer.ι ((unop S).index P) I ≫ ⊤.toMultiequalizer P) ≫ (J.diagram P I.Y).map ee.op = Multiequalizer.lift ((unop ((J.pullback (Cover.Arrow.map I e.op.unop).f.op.unop).op.obj S)).index P) ((J.diagram P (unop X)).obj S) (fun I_1 => Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I_1)) ⋯"}, "premise": [95099, 95097], "module": ["Mathlib/CategoryTheory/Sites/Plus.lean"]}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "J : GrothendieckTopology C", "D : Type w", "inst✝² : Category.{max v u, w} D", "inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)", "P : Cᵒᵖ ⥤ D", "inst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D", "X : Cᵒᵖ", "S : (J.Cover (unop X))ᵒᵖ", "e : unop S ⟶ ⊤ := homOfLE ⋯", "I : ((unop S).index (J.plusObj P)).L", "ee : (J.pullback (Cover.Arrow.map I e).f).obj (unop S) ⟶ ⊤ := homOfLE ⋯", "II : ((unop (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S)))).index P).L"], "goal": "((Multiequalizer.ι ((unop S).index P) I ≫ ⊤.toMultiequalizer P) ≫ (J.diagram P I.Y).map ee.op) ≫ Multiequalizer.ι ((unop (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S)))).index P) II = Multiequalizer.lift ((unop ((J.pullback (Cover.Arrow.map I e.op.unop).f.op.unop).op.obj S)).index P) ((J.diagram P (unop X)).obj S) (fun I_1 => Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I_1)) ⋯ ≫ Multiequalizer.ι ((unop (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S)))).index P) II"}, "premise": [95097, 95099], "module": ["Mathlib/CategoryTheory/Sites/Plus.lean"]}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "J : GrothendieckTopology C", "D : Type w", "inst✝² : Category.{max v u, w} D", "inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)", "P : Cᵒᵖ ⥤ D", "inst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D", "X : Cᵒᵖ", "S : (J.Cover (unop X))ᵒᵖ", "e : unop S ⟶ ⊤ := homOfLE ⋯", "I : ((unop S).index (J.plusObj P)).L", "ee : (J.pullback (Cover.Arrow.map I e).f).obj (unop S) ⟶ ⊤ := homOfLE ⋯", "II : ((unop (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S)))).index P).L", "e_1✝ : ((J.diagram P (unop X)).obj S ⟶ ((unop (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S)))).index P).left II) = (multiequalizer ((unop S).index P) ⟶ ((unop S).index P).right { fst := I, snd := Cover.Arrow.base II, r := { Z := II.Y, g₁ := II.f, g₂ := 𝟙 II.Y, w := ⋯ } })"], "goal": "((Multiequalizer.ι ((unop S).index P) I ≫ ⊤.toMultiequalizer P) ≫ (J.diagram P I.Y).map ee.op) ≫ Multiequalizer.ι ((unop (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S)))).index P) II = Multiequalizer.ι ((unop S).index P) (((unop S).index P).fstTo { fst := I, snd := Cover.Arrow.base II, r := { Z := II.Y, g₁ := II.f, g₂ := 𝟙 II.Y, w := ⋯ } }) ≫ ((unop S).index P).fst { fst := I, snd := Cover.Arrow.base II, r := { Z := II.Y, g₁ := II.f, g₂ := 𝟙 II.Y, w := ⋯ } }"}, "premise": [95097], "module": ["Mathlib/CategoryTheory/Sites/Plus.lean"]}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "J : GrothendieckTopology C", "D : Type w", "inst✝² : Category.{max v u, w} D", "inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)", "P : Cᵒᵖ ⥤ D", "inst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D", "X : Cᵒᵖ", "S : (J.Cover (unop X))ᵒᵖ", "e : unop S ⟶ ⊤ := homOfLE ⋯", "I : ((unop S).index (J.plusObj P)).L", "ee : (J.pullback (Cover.Arrow.map I e).f).obj (unop S) ⟶ ⊤ := homOfLE ⋯", "II : ((unop (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S)))).index P).L", "e_1✝ : ((J.diagram P (unop X)).obj S ⟶ ((unop (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S)))).index P).left II) = (multiequalizer ((unop S).index P) ⟶ ((unop S).index P).right { fst := I, snd := Cover.Arrow.base II, r := { Z := II.Y, g₁ := II.f, g₂ := 𝟙 II.Y, w := ⋯ } })"], "goal": "Multiequalizer.lift ((unop ((J.pullback (Cover.Arrow.map I e.op.unop).f.op.unop).op.obj S)).index P) ((J.diagram P (unop X)).obj S) (fun I_1 => Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I_1)) ⋯ ≫ Multiequalizer.ι ((unop (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S)))).index P) II = Multiequalizer.ι ((unop S).index P) (((unop S).index P).sndTo { fst := I, snd := Cover.Arrow.base II, r := { Z := II.Y, g₁ := II.f, g₂ := 𝟙 II.Y, w := ⋯ } }) ≫ ((unop S).index P).snd { fst := I, snd := Cover.Arrow.base II, r := { Z := II.Y, g₁ := II.f, g₂ := 𝟙 II.Y, w := ⋯ } }"}, "premise": [95097], "module": ["Mathlib/CategoryTheory/Sites/Plus.lean"]}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "l l₁ l₂ l₃ : List α", "a b : α", "m n : ℕ", "L : List (List (Option α))", "hm : m < L.length"], "goal": "take m L <+: take n L ↔ m ≤ n"}, "premise": [3059, 129670, 1793, 2134, 2581, 3503, 3679, 3731, 4576, 19701], "module": ["Mathlib/Data/List/Infix.lean"]}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "l l₁ l₂ l₃ : List α", "a b : α", "m n : ℕ", "L : List (List (Option α))", "hm : m < L.length"], "goal": "take m L = take (min m L.length) (take n L) ↔ m ≤ n"}, "premise": [4576, 1793, 129670, 3503, 3059, 3731, 2581, 2134, 19701, 3679], "module": ["Mathlib/Data/List/Infix.lean"]}
+{"state": {"context": ["𝓕 : Type u_1", "α : Type u_2", "E : Type u_3", "F : Type u_4", "inst✝¹ : SeminormedCommGroup E", "inst✝ : SeminormedCommGroup F", "a✝ a₁ a₂ b✝ b₁ b₂ : E", "r r₁ r₂ : ℝ", "a b : E"], "goal": "dist a (a * b) = ‖b‖"}, "premise": [119730, 61420, 42663], "module": ["Mathlib/Analysis/Normed/Group/Uniform.lean"]}
+{"state": {"context": ["x : SimplexCategory", "i : x ⟶ x", "inst✝ : Epi i"], "goal": "i = 𝟙 x"}, "premise": [47487, 47485], "module": ["Mathlib/AlgebraicTopology/SimplexCategory.lean"]}
+{"state": {"context": ["x : SimplexCategory", "i : x ⟶ x", "inst✝ : Epi i"], "goal": "IsIso i"}, "premise": [47485, 47487], "module": ["Mathlib/AlgebraicTopology/SimplexCategory.lean"]}
+{"state": {"context": ["x : SimplexCategory", "i : x ⟶ x", "inst✝ : Epi i"], "goal": "Function.Bijective (Hom.toOrderHom i).toFun"}, "premise": [47485], "module": ["Mathlib/AlgebraicTopology/SimplexCategory.lean"]}
+{"state": {"context": ["x : SimplexCategory", "i : x ⟶ x", "inst✝ : Epi i"], "goal": "Function.Bijective ⇑(Hom.toOrderHom i)"}, "premise": [1793, 20002, 47480, 141421], "module": ["Mathlib/AlgebraicTopology/SimplexCategory.lean"]}
+{"state": {"context": ["x : SimplexCategory", "i : x ⟶ x", "inst✝ : Epi i"], "goal": "Epi i"}, "premise": [1793, 20002, 47480, 141421], "module": ["Mathlib/AlgebraicTopology/SimplexCategory.lean"]}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "E : Type u_4", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℝ E", "b : ℝ", "f : ℕ → ℝ", "z : ℕ → E", "hfa : Monotone f", "hf0 : Tendsto f atTop (𝓝 0)"], "goal": "CauchySeq fun n => ∑ i ∈ Finset.range n, (-1) ^ i * f i"}, "premise": [119707, 34528, 34530], "module": ["Mathlib/Analysis/SpecificLimits/Normed.lean"]}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "E : Type u_4", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℝ E", "b : ℝ", "f : ℕ → ℝ", "z : ℕ → E", "hfa : Monotone f", "hf0 : Tendsto f atTop (𝓝 0)"], "goal": "CauchySeq fun n => ∑ x ∈ Finset.range n, f x * (-1) ^ x"}, "premise": [34528, 34530, 119707], "module": ["Mathlib/Analysis/SpecificLimits/Normed.lean"]}
+{"state": {"context": ["it : Iterator"], "goal": "ValidFor it.s.data.reverse [] { s := it.s, i := it.s.endPos }"}, "premise": [2307, 5293], "module": [".lake/packages/batteries/Batteries/Data/String/Lemmas.lean"]}
+{"state": {"context": ["K : Type u_1", "inst✝¹ : Field K", "inst✝ : NumberField K", "i : Free.ChooseBasisIndex ℤ (𝓞 K)"], "goal": "(latticeBasis K) i = (canonicalEmbedding K) ((integralBasis K) i)"}, "premise": [1670, 22033, 70751, 81528], "module": ["Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean"]}
+{"state": {"context": ["ι : Sort u_1", "V : Type u", "W : Type v", "G : SimpleGraph V", "v w : V", "hvw : G.Adj v w", "v✝ w✝ : V", "h : (fun a b => s(v, w) = s(a, b)) v✝ w✝"], "goal": "G.Adj v✝ w✝"}, "premise": [50423], "module": ["Mathlib/Combinatorics/SimpleGraph/Subgraph.lean"]}
+{"state": {"context": ["ι : Sort u_1", "V : Type u", "W : Type v", "G : SimpleGraph V", "v w : V", "hvw : G.Adj v w", "v✝ w✝ : V", "h : (fun a b => s(v, w) = s(a, b)) v✝ w✝"], "goal": "s(v, w) ∈ G.edgeSet"}, "premise": [50423], "module": ["Mathlib/Combinatorics/SimpleGraph/Subgraph.lean"]}
+{"state": {"context": ["ι : Sort u_1", "V : Type u", "W : Type v", "G : SimpleGraph V", "v w : V", "hvw : G.Adj v w", "a b : V", "h : (a ∈ s(v, w)) = (a ∈ s(a, b))"], "goal": "a ∈ {v, w}"}, "premise": [1190, 1194, 1792, 128272], "module": ["Mathlib/Combinatorics/SimpleGraph/Subgraph.lean"]}
+{"state": {"context": ["ι : Sort u_1", "V : Type u", "W : Type v", "G : SimpleGraph V", "v w : V", "hvw : G.Adj v w", "a b : V", "h : a = v ∨ a = w"], "goal": "a ∈ {v, w}"}, "premise": [1190, 1194, 1792, 128272], "module": ["Mathlib/Combinatorics/SimpleGraph/Subgraph.lean"]}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "A : Type u_3", "K : Type u_4", "inst✝³ : Semiring S", "inst✝² : CommSemiring R", "inst✝¹ : Semiring A", "inst✝ : Field K", "p : R[X]", "r s : R", "n✝ : ℕ"], "goal": "(p.scaleRoots (r * s)).coeff n✝ = ((p.scaleRoots r).scaleRoots s).coeff n✝"}, "premise": [117764, 119703], "module": ["Mathlib/RingTheory/Polynomial/ScaleRoots.lean"]}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "inst✝³ : TopologicalSpace X", "inst✝² : TopologicalSpace Y", "f : X → Y", "inst✝¹ : T1Space X", "inst✝ : WeaklyLocallyCompactSpace Y", "hf' : Continuous f", "hf : Tendsto f cofinite (cocompact Y)"], "goal": "DiscreteTopology X"}, "premise": [1673, 57523, 54090], "module": ["Mathlib/Topology/DiscreteSubset.lean"]}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "inst✝³ : TopologicalSpace X", "inst✝² : TopologicalSpace Y", "f : X → Y", "inst✝¹ : T1Space X", "inst✝ : WeaklyLocallyCompactSpace Y", "hf' : Continuous f", "hf : Tendsto f cofinite (cocompact Y)", "x : X"], "goal": "IsOpen {x}"}, "premise": [1673, 54090, 57523], "module": ["Mathlib/Topology/DiscreteSubset.lean"]}
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+{"state": {"context": ["ho : 0 < 1"], "goal": "(sup fun n => 1 ^ ↑n) = 1 ^ ω"}, "premise": [1674, 14302, 49765], "module": ["Mathlib/SetTheory/Ordinal/Exponential.lean"]}
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+{"state": {"context": ["ι : Type u_1", "𝕜 : Type u_2", "E : Type u_3", "F : Type u_4", "A✝ : Type u_5", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace ℝ E", "f : ℝ → E", "a b : ℝ", "c ca cb : E", "l l' la la' lb lb' : Filter ℝ", "lt : Filter ι", "μ : Measure ℝ", "u v ua va ub vb : ι → ℝ", "inst✝³ : CompleteSpace E", "inst✝² : FTCFilter a la la'", "inst✝¹ : FTCFilter b lb lb'", "inst✝ : IsLocallyFiniteMeasure μ", "hab : IntervalIntegrable f μ a b", "hmeas_a : StronglyMeasurableAtFilter f la' μ", "hmeas_b : StronglyMeasurableAtFilter f lb' μ", "ha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)", "hb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)", "hua : Tendsto ua lt la", "hva : Tendsto va lt la", "hub : Tendsto ub lt lb", "hvb : Tendsto vb lt lb", "this✝ : la'.IsMeasurablyGenerated", "this : lb'.IsMeasurablyGenerated", "A : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ua t) (va t)", "A' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ a (ua t)", "B : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ub t) (vb t)", "B' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ b (ub t)"], "goal": "(fun x => -(∫ (x : ℝ) in ua x..va x, f x ∂μ - ∫ (x : ℝ) in ua x..va x, ca ∂μ) + (∫ (x : ℝ) in ub x..vb x, f x ∂μ - ∫ (x : ℝ) in ub x..vb x, cb ∂μ)) =ᶠ[lt] fun t => ∫ (x : ℝ) in va t..vb t, f x ∂μ - ∫ (x : ℝ) in ua t..ub t, f x ∂μ - (∫ (x : ℝ) in ub t..vb t, cb ∂μ - ∫ (x : ℝ) in ua t..va t, ca ∂μ)"}, "premise": [16384, 131585, 16358, 27280, 27281, 15889, 26331], "module": ["Mathlib/MeasureTheory/Integral/FundThmCalculus.lean"]}
+{"state": {"context": ["ι : Type u_1", "𝕜 : Type u_2", "E : Type u_3", "F : Type u_4", "A✝ : Type u_5", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace ℝ E", "f : ℝ → E", "a b : ℝ", "c ca cb : E", "l l' la la' lb lb' : Filter ℝ", "lt : Filter ι", "μ : Measure ℝ", "u v ua va ub vb : ι → ℝ", "inst✝³ : CompleteSpace E", "inst✝² : FTCFilter a la la'", "inst✝¹ : FTCFilter b lb lb'", "inst✝ : IsLocallyFiniteMeasure μ", "hab : IntervalIntegrable f μ a b", "hmeas_a : StronglyMeasurableAtFilter f la' μ", "hmeas_b : StronglyMeasurableAtFilter f lb' μ", "ha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)", "hb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)", "hua : Tendsto ua lt la", "hva : Tendsto va lt la", "hub : Tendsto ub lt lb", "hvb : Tendsto vb lt lb", "this✝ : la'.IsMeasurablyGenerated", "this : lb'.IsMeasurablyGenerated", "A : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ua t) (va t)", "A' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ a (ua t)", "B : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ub t) (vb t)", "B' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ b (ub t)", "a✝ : ι", "ua_va : IntervalIntegrable f μ (ua a✝) (va a✝)", "a_ua : IntervalIntegrable f μ a (ua a✝)", "ub_vb : IntervalIntegrable f μ (ub a✝) (vb a✝)", "b_ub : IntervalIntegrable f μ b (ub a✝)"], "goal": "-(∫ (x : ℝ) in ua a✝..va a✝, f x ∂μ - ∫ (x : ℝ) in ua a✝..va a✝, ca ∂μ) + (∫ (x : ℝ) in ub a✝..vb a✝, f x ∂μ - ∫ (x : ℝ) in ub a✝..vb a✝, cb ∂μ) = ∫ (x : ℝ) in va a✝..vb a✝, f x ∂μ - ∫ (x : ℝ) in ua a✝..ub a✝, f x ∂μ - (∫ (x : ℝ) in ub a✝..vb a✝, cb ∂μ - ∫ (x : ℝ) in ua a✝..va a✝, ca ∂μ)"}, "premise": [26406, 15889, 131585], "module": ["Mathlib/MeasureTheory/Integral/FundThmCalculus.lean"]}
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+{"state": {"context": ["E : Type u_1", "inst✝ : NormedAddCommGroup E", "f : C(ℝ, E)", "b : ℝ", "hb : 0 < b", "hf : ⇑f =O[atBot] fun x => |x| ^ (-b)", "R S : ℝ", "h1 : ⇑(f.comp { toFun := fun a => -a, continuous_toFun := ⋯ }) =O[atTop] fun x => |x| ^ (-b)", "h2 : ((fun x => ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖) ∘ Neg.neg) =O[atBot] fun x => |x| ^ (-b)", "this : (fun x => |x| ^ (-b)) ∘ Neg.neg = fun x => |x| ^ (-b)", "x✝ x : ℝ", "hx : x ∈ Icc (x✝ + R) (x✝ + S)"], "goal": "‖f x‖ ≤ ‖ContinuousMap.restrict (Icc (-x✝ + -S) (-x✝ + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖"}, "premise": [62144, 14277, 62349], "module": ["Mathlib/Analysis/Fourier/PoissonSummation.lean"]}
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+{"state": {"context": ["E : Type u_1", "inst✝ : NormedAddCommGroup E", "f : C(ℝ, E)", "b : ℝ", "hb : 0 < b", "hf : ⇑f =O[atBot] fun x => |x| ^ (-b)", "R S : ℝ", "h1 : ⇑(f.comp { toFun := fun a => -a, continuous_toFun := ⋯ }) =O[atTop] fun x => |x| ^ (-b)", "h2 : ((fun x => ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖) ∘ Neg.neg) =O[atBot] fun x => |x| ^ (-b)", "this : (fun x => |x| ^ (-b)) ∘ Neg.neg = fun x => |x| ^ (-b)", "x✝ x : ℝ", "hx : x ∈ Icc (x✝ + R) (x✝ + S)"], "goal": "-x ∈ Icc (-x✝ + -S) (-x✝ + -R)"}, "premise": [14277, 62349], "module": ["Mathlib/Analysis/Fourier/PoissonSummation.lean"]}
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+{"state": {"context": ["ι : Type u_1", "α : Type u", "β : Type v", "γ : Type w", "l₁ l₂ : List α", "a : α", "l : List α", "x : α", "n : ℕ", "hn : n ≤ l.length"], "goal": "l.length ≤ (insertNth n x l).length"}, "premise": [14317], "module": ["Mathlib/Data/List/InsertNth.lean"]}
+{"state": {"context": ["ι : Type u_1", "α : Type u", "β : Type v", "γ : Type w", "l₁ l₂ : List α", "a : α", "l : List α", "x : α", "n : ℕ", "hn : l.length < n"], "goal": "l.length ≤ (insertNth n x l).length"}, "premise": [14317], "module": ["Mathlib/Data/List/InsertNth.lean"]}
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+{"state": {"context": ["K : Type u_1", "inst✝⁴ : Field K", "inst✝³ : NumberField K", "inst✝² : IsCyclotomicExtension {3} ℚ K", "ζ : K", "hζ : IsPrimitiveRoot ζ ↑3", "S : Solution hζ", "S' : Solution' hζ", "inst✝¹ : DecidableRel fun a b => a ∣ b", "inst✝ : DecidableEq (𝓞 K)", "h : λ ^ (S.multiplicity - 1 + 1) ∣ S.Solution'_descent.c"], "goal": "λ ∣ FermatLastTheoremForThreeGen.Solution.X S"}, "premise": [2100, 24535, 79545], "module": ["Mathlib/NumberTheory/FLT/Three.lean"]}
+{"state": {"context": ["K : Type u_1", "inst✝⁴ : Field K", "inst✝³ : NumberField K", "inst✝² : IsCyclotomicExtension {3} ℚ K", "ζ : K", "hζ : IsPrimitiveRoot ζ ↑3", "S : Solution hζ", "S' : Solution' hζ", "inst✝¹ : DecidableRel fun a b => a ∣ b", "inst✝ : DecidableEq (𝓞 K)", "k : 𝓞 K", "hk : λ ^ (S.multiplicity - 1) * FermatLastTheoremForThreeGen.Solution.X S = λ ^ (S.multiplicity - 1 + 1) * k"], "goal": "λ ∣ FermatLastTheoremForThreeGen.Solution.X S"}, "premise": [119703, 119742, 1169, 1182, 1191, 24765, 108286, 108288, 125797], "module": ["Mathlib/NumberTheory/FLT/Three.lean"]}
+{"state": {"context": ["K : Type u_1", "inst✝⁴ : Field K", "inst✝³ : NumberField K", "inst✝² : IsCyclotomicExtension {3} ℚ K", "ζ : K", "hζ : IsPrimitiveRoot ζ ↑3", "S : Solution hζ", "S' : Solution' hζ", "inst✝¹ : DecidableRel fun a b => a ∣ b", "inst✝ : DecidableEq (𝓞 K)", "k : 𝓞 K", "hk : λ ^ (S.multiplicity - 1) * FermatLastTheoremForThreeGen.Solution.X S = λ ^ (S.multiplicity - 1) * (λ * k)"], "goal": "λ ∣ FermatLastTheoremForThreeGen.Solution.X S"}, "premise": [108288, 125797, 1191, 108286, 1169, 119742, 119703, 24765, 1182], "module": ["Mathlib/NumberTheory/FLT/Three.lean"]}
+{"state": {"context": ["K : Type u_1", "inst✝⁴ : Field K", "inst✝³ : NumberField K", "inst✝² : IsCyclotomicExtension {3} ℚ K", "ζ : K", "hζ : IsPrimitiveRoot ζ ↑3", "S : Solution hζ", "S' : Solution' hζ", "inst✝¹ : DecidableRel fun a b => a ∣ b", "inst✝ : DecidableEq (𝓞 K)", "k : 𝓞 K", "hk : FermatLastTheoremForThreeGen.Solution.X S = λ * k"], "goal": "λ ∣ FermatLastTheoremForThreeGen.Solution.X S"}, "premise": [1169, 1182, 1191, 24765, 108286, 108288, 125797], "module": ["Mathlib/NumberTheory/FLT/Three.lean"]}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "X✝ Y Z : C", "inst✝² : HasZeroMorphisms C", "f : X✝ ⟶ Y", "inst✝¹ : HasKernel f", "inst✝ : HasCokernels C", "X A B : C", "g : B ⟶ X", "hg : Mono g", "i : A ≅ B", "hf : Mono (i.hom ≫ g)"], "goal": "(fun A f x => Subobject.mk (cokernel.π f).op) A (i.hom ≫ g) hf = (fun A f x => Subobject.mk (cokernel.π f).op) B g hg"}, "premise": [89269, 89626], "module": ["Mathlib/CategoryTheory/Subobject/Limits.lean"]}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "X✝ Y Z : C", "inst✝² : HasZeroMorphisms C", "f : X✝ ⟶ Y", "inst✝¹ : HasKernel f", "inst✝ : HasCokernels C", "X A B : C", "g : B ⟶ X", "hg : Mono g", "i : A ≅ B", "hf : Mono (i.hom ≫ g)"], "goal": "cokernel g ≅ cokernel (i.hom ≫ g)"}, "premise": [89269, 89626], "module": ["Mathlib/CategoryTheory/Subobject/Limits.lean"]}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "X✝ Y Z : C", "inst✝² : HasZeroMorphisms C", "f : X✝ ⟶ Y", "inst✝¹ : HasKernel f", "inst✝ : HasCokernels C", "X A B : C", "g : B ⟶ X", "hg : Mono g", "i : A ≅ B", "hf : Mono (i.hom ≫ g)"], "goal": "(?refine_1.op.hom ≫ (cokernel.π g).op).unop = (cokernel.π (i.hom ≫ g)).op.unop"}, "premise": [89269, 89626], "module": ["Mathlib/CategoryTheory/Subobject/Limits.lean"]}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "X✝ Y Z : C", "inst✝² : HasZeroMorphisms C", "f✝ : X✝ ⟶ Y", "inst✝¹ : HasKernel f✝", "inst✝ : HasCokernels C", "X A B : C", "f : A ⟶ X", "g : B ⟶ X", "hf : Mono f", "hg : Mono g", "h : Subobject.mk f ≤ Subobject.mk g"], "goal": "Subobject.lift (fun A f x => Subobject.mk (cokernel.π f).op) ⋯ (Subobject.mk f) ≤ Subobject.lift (fun A f x => Subobject.mk (cokernel.π f).op) ⋯ (Subobject.mk g)"}, "premise": [89253, 89261], "module": ["Mathlib/CategoryTheory/Subobject/Limits.lean"]}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "X✝ Y Z : C", "inst✝² : HasZeroMorphisms C", "f✝ : X✝ ⟶ Y", "inst✝¹ : HasKernel f✝", "inst✝ : HasCokernels C", "X A B : C", "f : A ⟶ X", "g : B ⟶ X", "hf : Mono f", "hg : Mono g", "h : Subobject.mk f ≤ Subobject.mk g"], "goal": "Subobject.mk (cokernel.π f).op ≤ Subobject.mk (cokernel.π g).op"}, "premise": [89253, 89261], "module": ["Mathlib/CategoryTheory/Subobject/Limits.lean"]}
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+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "X✝ Y Z : C", "inst✝² : HasZeroMorphisms C", "f✝ : X✝ ⟶ Y", "inst✝¹ : HasKernel f✝", "inst✝ : HasCokernels C", "X A B : C", "f : A ⟶ X", "g : B ⟶ X", "hf : Mono f", "hg : Mono g", "h : Subobject.mk f ≤ Subobject.mk g"], "goal": "(cokernel.desc f (cokernel.π g) ?refine_1).op ≫ (cokernel.π f).op = (cokernel.π g).op"}, "premise": [89261], "module": ["Mathlib/CategoryTheory/Subobject/Limits.lean"]}
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+{"state": {"context": ["f : ℝ → ℝ", "x : ℝ", "n : ℕ", "hf_conv : ConvexOn ℝ (Ioi 0) f", "hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y", "hx : 0 < x", "hn : n ≠ 0"], "goal": "f 1 + (x * log ↑n + log ↑n !) ≤ f (↑(n + 1) + x)"}, "premise": [14273, 14277, 39598, 117807, 39600, 105713, 119708], "module": ["Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean"]}
+{"state": {"context": ["f : ℝ → ℝ", "x : ℝ", "n : ℕ", "hf_conv : ConvexOn ℝ (Ioi 0) f", "hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y", "hx : 0 < x", "hn : n ≠ 0"], "goal": "f 1 + (x * log ↑n + log ↑n !) = f ↑(n + 1) + x * log (↑(n + 1) - 1)"}, "premise": [14273, 14277, 39600], "module": ["Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean"]}
+{"state": {"context": ["f : ℝ → ℝ", "x : ℝ", "n : ℕ", "hf_conv : ConvexOn ℝ (Ioi 0) f", "hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y", "hx : 0 < x", "hn : n ≠ 0"], "goal": "2 ≤ n + 1"}, "premise": [14273, 14277, 39600], "module": ["Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean"]}
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+{"state": {"context": ["b x y : ℝ"], "goal": "(x = 0 ∨ x = 1 ∨ x = -1) ∨ b = 0 ∨ b = 1 ∨ b = -1 ↔ b = 0 ∨ b = 1 ∨ b = -1 ∨ x = 0 ∨ x = 1 ∨ x = -1"}, "premise": [108406, 1101, 37910, 1674], "module": ["Mathlib/Analysis/SpecialFunctions/Log/Base.lean"]}
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+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "inst✝² : Abelian C", "inst✝¹ : HasLimits C", "inst✝ : EnoughInjectives C", "G : C", "hG : IsSeparator G", "this✝ : WellPowered C", "this : HasProductsOfShape (Subobject (op G)) C", "T : C := Injective.under (∏ᶜ fun P => unop (Subobject.underlying.obj P))", "X Y : C", "f : X ⟶ Y", "hf : ∀ (h : Y ⟶ T), f ≫ h = 0", "h : G ⟶ X"], "goal": "h ≫ f = 0"}, "premise": [91592, 1673, 93605, 94368], "module": ["Mathlib/CategoryTheory/Abelian/Generator.lean"]}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "inst✝² : Abelian C", "inst✝¹ : HasLimits C", "inst✝ : EnoughInjectives C", "G : C", "hG : IsSeparator G", "this✝ : WellPowered C", "this : HasProductsOfShape (Subobject (op G)) C", "T : C := Injective.under (∏ᶜ fun P => unop (Subobject.underlying.obj P))", "X Y : C", "f : X ⟶ Y", "hf : ∀ (h : Y ⟶ T), f ≫ h = 0", "h : G ⟶ X"], "goal": "factorThruImage (h ≫ f) = 0"}, "premise": [93605, 94368], "module": ["Mathlib/CategoryTheory/Abelian/Generator.lean"]}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "inst✝² : Abelian C", "inst✝¹ : HasLimits C", "inst✝ : EnoughInjectives C", "G : C", "hG : IsSeparator G", "this✝ : WellPowered C", "this : HasProductsOfShape (Subobject (op G)) C", "T : C := Injective.under (∏ᶜ fun P => unop (Subobject.underlying.obj P))", "X Y : C", "f : X ⟶ Y", "hf : ∀ (h : Y ⟶ T), f ≫ h = 0", "h : G ⟶ X", "R : Subobject (op G) := Subobject.mk (factorThruImage (h ≫ f)).op", "q₁ : image (h ≫ f) ⟶ unop (Subobject.underlying.obj R) := (Subobject.underlyingIso (factorThruImage (h ≫ f)).op).unop.hom", "q₂ : unop (Subobject.underlying.obj R) ⟶ ∏ᶜ fun P => unop (Subobject.underlying.obj P) := section_ (Pi.π (fun P => unop (Subobject.underlying.obj P)) R)", "q : image (h ≫ f) ⟶ T := q₁ ≫ q₂ ≫ Injective.ι (∏ᶜ fun P => unop (Subobject.underlying.obj P))"], "goal": "factorThruImage (h ≫ f) = 0"}, "premise": [91650, 93604, 93610, 96173], "module": ["Mathlib/CategoryTheory/Abelian/Generator.lean"]}
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+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝² : UniformSpace α", "inst✝¹ : Group α", "inst✝ : UniformGroup α"], "goal": "comap ((fun x => x.1⁻¹ * x.2) ∘ Prod.swap) (𝓝 1) = comap (fun x => x.2⁻¹ * x.1) (𝓝 1)"}, "premise": [16208, 60501, 67160], "module": ["Mathlib/Topology/Algebra/UniformGroup.lean"]}
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+{"state": {"context": ["n : ℕ", "R : Type u", "S : Type v", "T : Type w", "inst✝⁵ : CommRing R", "inst✝⁴ : CommRing S", "inst✝³ : CommRing T", "inst✝² : IsJacobson S", "inst✝¹ : Algebra R S", "inst✝ : Algebra R T", "IH : ∀ (Q : Ideal S), Q.IsMaximal → (algebraMap R (S ⧸ Q)).IsIntegral", "v : S[X] ≃ₐ[R] T", "P : Ideal T", "hP : P.IsMaximal", "Q : Ideal S[X] := comap (↑v).toRingHom P", "hw : map v Q = P", "hQ : Q.IsMaximal", "w : (S[X] ⧸ Q) ≃ₐ[R] T ⧸ P := Q.quotientEquivAlg P v ⋯", "Q' : Ideal S := comap Polynomial.C Q", "w' : S ⧸ Q' →ₐ[R] S[X] ⧸ Q := quotientMapₐ Q (Ideal.MvPolynomial.Cₐ R S) ⋯", "h_eq : algebraMap R (T ⧸ P) = w.toRingEquiv.toRingHom.comp (w'.comp (algebraMap R (S ⧸ Q')))"], "goal": "w.toRingEquiv.toRingHom.IsIntegral"}, "premise": [81940], "module": ["Mathlib/RingTheory/Jacobson.lean"]}
+{"state": {"context": ["C : Type u", "inst✝² : Category.{v, u} C", "inst✝¹ : HasFiniteProducts C", "inst✝ : HasPullbacks C", "X Y : Dial C"], "goal": "(X.tensorObj Y).rel = (Subobject.pullback (prod.map (prod.braiding X.src Y.src).hom (prod.braiding X.tgt Y.tgt).hom)).obj (Y.tensorObj X).rel"}, "premise": [14579, 89300, 90541], "module": ["Mathlib/CategoryTheory/Dialectica/Monoidal.lean"]}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "R : Type x", "inst✝ : CommSemiring α", "a✝ b✝ c a b : α"], "goal": "(a + b) ^ 2 = a ^ 2 + b ^ 2 + 2 * a * b"}, "premise": [119704, 119708, 122236], "module": ["Mathlib/Algebra/Ring/Defs.lean"]}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "r : α → α → Prop", "s : β → β → Prop", "t : γ → γ → Prop", "o : Ordinal.{u_4}", "h : ∃ a, succ o = succ a"], "goal": "(succ o).pred = o"}, "premise": [1084, 1739, 2100, 17414], "module": ["Mathlib/SetTheory/Ordinal/Arithmetic.lean"]}
+{"state": {"context": ["α : Type u", "β : Type v", "ι : Sort w", "κ : ι → Sort w'", "minAx : MinimalAxioms α", "f : (a : ι) → κ a → α", "x✝ : CompleteLattice α := minAx.toCompleteLattice"], "goal": "let x := minAx.toCompleteLattice; ⨅ i, ⨆ j, f i j = ⨆ g, ⨅ i, f i (g i)"}, "premise": [14296, 14950], "module": ["Mathlib/Order/CompleteBooleanAlgebra.lean"]}
+{"state": {"context": ["α : Type u", "β : Type v", "ι : Sort w", "κ : ι → Sort w'", "minAx : MinimalAxioms α", "f : (a : ι) → κ a → α", "x✝ : CompleteLattice α := minAx.toCompleteLattice"], "goal": "⨅ i, ⨆ j, f i j ≤ ⨆ g, ⨅ i, f i (g i)"}, "premise": [14296, 14950], "module": ["Mathlib/Order/CompleteBooleanAlgebra.lean"]}
+{"state": {"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "inst✝¹ : MonoidalCategory C", "inst✝ : BraidedCategory C", "M N P : Mon_ C"], "goal": "(tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫ (tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫ (α_ M.X N.X P.X).hom = ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫ tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫ (M.mul ⊗ tensor_μ C (N.X, P.X) (N.X, P.X) ≫ (N.mul ⊗ P.mul))"}, "premise": [96173, 99312, 99340, 96173, 96175, 99215], "module": ["Mathlib/CategoryTheory/Monoidal/Mon_.lean"]}
+{"state": {"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "inst✝¹ : MonoidalCategory C", "inst✝ : BraidedCategory C", "M N P : Mon_ C"], "goal": "tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫ (tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul) ≫ (α_ M.X N.X P.X).hom = ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫ tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫ (M.mul ⊗ tensor_μ C (N.X, P.X) (N.X, P.X) ≫ (N.mul ⊗ P.mul))"}, "premise": [99340, 96173, 99215, 99312, 96175], "module": ["Mathlib/CategoryTheory/Monoidal/Mon_.lean"]}
+{"state": {"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "inst✝¹ : MonoidalCategory C", "inst✝ : BraidedCategory C", "M N P : Mon_ C"], "goal": "tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫ ((tensor_μ C (M.X, N.X) (M.X, N.X) ⊗ 𝟙 (P.X ⊗ P.X)) ≫ ((M.mul ⊗ N.mul) ⊗ P.mul)) ≫ (α_ M.X N.X P.X).hom = ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫ tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫ (M.mul ⊗ tensor_μ C (N.X, P.X) (N.X, P.X) ≫ (N.mul ⊗ P.mul))"}, "premise": [99215, 99210, 96173, 96175], "module": ["Mathlib/CategoryTheory/Monoidal/Mon_.lean"]}
+{"state": {"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "inst✝¹ : MonoidalCategory C", "inst✝ : BraidedCategory C", "M N P : Mon_ C"], "goal": "tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫ (tensor_μ C (M.X, N.X) (M.X, N.X) ⊗ 𝟙 (P.X ⊗ P.X)) ≫ (α_ (M.X ⊗ M.X) (N.X ⊗ N.X) (P.X ⊗ P.X)).hom ≫ (M.mul ⊗ N.mul ⊗ P.mul) = ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫ tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫ (M.mul ⊗ tensor_μ C (N.X, P.X) (N.X, P.X) ≫ (N.mul ⊗ P.mul))"}, "premise": [99215, 99210, 96173, 96175], "module": ["Mathlib/CategoryTheory/Monoidal/Mon_.lean"]}
+{"state": {"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "inst✝¹ : MonoidalCategory C", "inst✝ : BraidedCategory C", "M N P : Mon_ C"], "goal": "tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫ (tensor_μ C (M.X, N.X) (M.X, N.X) ⊗ 𝟙 (P.X ⊗ P.X)) ≫ (α_ (M.X ⊗ M.X) (N.X ⊗ N.X) (P.X ⊗ P.X)).hom ≫ (M.mul ⊗ N.mul ⊗ P.mul) = ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫ tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫ (𝟙 (M.X ⊗ M.X) ⊗ tensor_μ C (N.X, P.X) (N.X, P.X)) ≫ (M.mul ⊗ N.mul ⊗ P.mul)"}, "premise": [96173, 99215, 96175, 99217, 99218], "module": ["Mathlib/CategoryTheory/Monoidal/Mon_.lean"]}
+{"state": {"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "inst✝¹ : MonoidalCategory C", "inst✝ : BraidedCategory C", "M N P : Mon_ C"], "goal": "tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫ tensor_μ C (M.X, N.X) (M.X, N.X) ▷ (P.X ⊗ P.X) ≫ (α_ (M.X ⊗ M.X) (N.X ⊗ N.X) (P.X ⊗ P.X)).hom ≫ (M.mul ⊗ N.mul ⊗ P.mul) = ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫ tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫ (M.X ⊗ M.X) ◁ tensor_μ C (N.X, P.X) (N.X, P.X) ≫ (M.mul ⊗ N.mul ⊗ P.mul)"}, "premise": [99217, 99218, 96173, 107150], "module": ["Mathlib/CategoryTheory/Monoidal/Mon_.lean"]}
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+{"state": {"context": ["R : Type u", "inst✝ : CommRing R", "M N U : ModuleCat R", "X✝ Y✝ Z✝ : Type u", "f✝ : X✝ ⟶ Y✝", "g✝ : Y✝ ⟶ Z✝"], "goal": "FreeAlgebra.ι R ∘ (f✝ ≫ g✝) = ⇑((FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f✝) ≫ (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ g✝)) ∘ FreeAlgebra.ι R"}, "premise": [124473], "module": ["Mathlib/Algebra/Category/AlgebraCat/Basic.lean"]}
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+{"state": {"context": ["R : Type u", "inst✝ : CommRing R", "M N U : ModuleCat R", "X✝ Y✝ Z✝ : Type u", "f✝ : X✝ ⟶ Y✝", "g✝ : Y✝ ⟶ Z✝", "x✝ : X✝"], "goal": "(FreeAlgebra.ι R ∘ (f✝ ≫ g✝)) x✝ = ((⇑((FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ g✝)) ∘ ⇑((FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f✝))) ∘ FreeAlgebra.ι R) x✝"}, "premise": [91291, 1670, 99591], "module": ["Mathlib/Algebra/Category/AlgebraCat/Basic.lean"]}
+{"state": {"context": ["R : Type u", "inst✝ : CommRing R", "M N U : ModuleCat R", "X✝ Y✝ Z✝ : Type u", "f✝ : X✝ ⟶ Y✝", "g✝ : Y✝ ⟶ Z✝", "x✝ : X✝"], "goal": "FreeAlgebra.ι R (g✝ (f✝ x✝)) = ((FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ g✝)) (((FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f✝)) (FreeAlgebra.ι R x✝))"}, "premise": [124474, 91291, 1670, 99591], "module": ["Mathlib/Algebra/Category/AlgebraCat/Basic.lean"]}
+{"state": {"context": ["R : Type u", "inst✝ : CommRing R", "M N U : ModuleCat R", "X✝ Y✝ Z✝ : Type u", "f✝ : X✝ ⟶ Y✝", "g✝ : Y✝ ⟶ Z✝", "x✝ : X✝"], "goal": "FreeAlgebra.ι R (g✝ (f✝ x✝)) = (FreeAlgebra.ι R ∘ g✝) (f✝ x✝)"}, "premise": [124474], "module": ["Mathlib/Algebra/Category/AlgebraCat/Basic.lean"]}
+{"state": {"context": ["m n✝ n : ℤ"], "goal": "Even (n * (n + 1))"}, "premise": [117547, 121611], "module": ["Mathlib/Algebra/Ring/Int.lean"]}
+{"state": {"context": ["R : Type u", "X : Type v", "inst✝¹ : CommRing R", "inst✝ : Nontrivial R"], "goal": "Module.rank R (FreeAlgebra R X) = Cardinal.lift.{u, v} (Cardinal.mk (List X))"}, "premise": [2101, 48581, 48583, 85905], "module": ["Mathlib/LinearAlgebra/FreeAlgebra.lean"]}
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+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "ι : Type x", "inst✝² : PseudoEMetricSpace α", "inst✝¹ : PseudoEMetricSpace β", "inst✝ : PseudoEMetricSpace γ", "K : ℝ≥0", "f✝ : α → β", "x✝ y✝ : α", "r : ℝ≥0∞", "f : α → β", "Kf : ℝ≥0", "hf : LipschitzWith Kf f", "g : α → γ", "Kg : ℝ≥0", "hg : LipschitzWith Kg g", "x y : α"], "goal": "max (edist ((fun x => (f x, g x)) x).1 ((fun x => (f x, g x)) y).1) (edist ((fun x => (f x, g x)) x).2 ((fun x => (f x, g x)) y).2) ≤ max (↑Kf * edist x y) (↑Kg * edist x y)"}, "premise": [12960, 58465, 143206, 13001, 143480], "module": ["Mathlib/Topology/EMetricSpace/Lipschitz.lean"]}
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+{"state": {"context": ["α : Type u_1", "inst✝ : Group α", "s : Subgroup α"], "goal": "∀ (a b : α), Setoid.r a b ↔ b * a⁻¹ ∈ s"}, "premise": [1792, 6901], "module": ["Mathlib/GroupTheory/Coset.lean"]}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝ : PartialOrder α", "a b c x : α"], "goal": "x ∈ Icc a b \\ {b} ↔ x ∈ Ico a b"}, "premise": [1206, 11244], "module": ["Mathlib/Order/Interval/Set/Basic.lean"]}
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+{"state": {"context": ["ι : Type u_1", "G : ι → Type u_2", "H : Type u_3", "K : Type u_4", "inst✝² : Monoid K", "inst✝¹ : (i : ι) → Monoid (G i)", "inst✝ : Monoid H", "φ : (i : ι) → H →* G i", "x✝ : PushoutI φ →* K"], "goal": "∀ (i : ι), ((fun f => lift (↑f).1 (↑f).2 ⋯) ((fun f => ⟨(fun i => f.comp (of i), f.comp (base φ)), ⋯⟩) x✝)).comp (of i) = x✝.comp (of i)"}, "premise": [128588], "module": ["Mathlib/GroupTheory/PushoutI.lean"]}
+{"state": {"context": ["ι : Type u_1", "G : ι → Type u_2", "H : Type u_3", "K : Type u_4", "inst✝² : Monoid K", "inst✝¹ : (i : ι) → Monoid (G i)", "inst✝ : Monoid H", "φ : (i : ι) → H →* G i", "x✝ : PushoutI φ →* K"], "goal": "((fun f => lift (↑f).1 (↑f).2 ⋯) ((fun f => ⟨(fun i => f.comp (of i), f.comp (base φ)), ⋯⟩) x✝)).comp (base φ) = x✝.comp (base φ)"}, "premise": [128588], "module": ["Mathlib/GroupTheory/PushoutI.lean"]}
+{"state": {"context": ["ι : Type u_1", "G : ι → Type u_2", "H : Type u_3", "K : Type u_4", "inst✝² : Monoid K", "inst✝¹ : (i : ι) → Monoid (G i)", "inst✝ : Monoid H", "φ : (i : ι) → H →* G i", "x✝ : { f // ∀ (i : ι), (f.1 i).comp (φ i) = f.2 }", "fst✝ : (i : ι) → G i →* K", "snd✝ : H →* K", "property✝ : ∀ (i : ι), ((fst✝, snd✝).1 i).comp (φ i) = (fst✝, snd✝).2"], "goal": "(fun f => ⟨(fun i => f.comp (of i), f.comp (base φ)), ⋯⟩) ((fun f => lift (↑f).1 (↑f).2 ⋯) ⟨(fst✝, snd✝), property✝⟩) = ⟨(fst✝, snd✝), property✝⟩"}, "premise": [24, 128588], "module": ["Mathlib/GroupTheory/PushoutI.lean"]}
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+{"state": {"context": ["n m N : ℕ", "a : ZMod N", "hN : N ≠ 0", "this : NeZero N", "d : ℕ := a.val.gcd N", "hd : d ≠ 0"], "goal": "∃ d, d ∣ N ∧ ∃ u, IsUnit u ∧ a = u * ↑d"}, "premise": [3616, 3636], "module": ["Mathlib/Data/ZMod/Units.lean"]}
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+{"state": {"context": ["n m N : ℕ", "a : ZMod N", "hN : N ≠ 0", "this : NeZero N", "d : ℕ := a.val.gcd N", "hd : d ≠ 0", "a₀ : ℕ", "ha₀ : a.val = d * a₀", "N₀ : ℕ", "hN₀ : N = d * N₀"], "goal": "∃ u, IsUnit u ∧ a = u * ↑d"}, "premise": [1673, 1674, 1717, 2574, 74464, 111256, 119703, 119707, 128974, 138376, 142648, 119707, 136229], "module": ["Mathlib/Data/ZMod/Units.lean"]}
+{"state": {"context": ["n m N : ℕ", "a : ZMod N", "hN : N ≠ 0", "this : NeZero N", "d : ℕ := a.val.gcd N", "hd : d ≠ 0", "a₀ : ℕ", "ha₀ : a.val = d * a₀", "N₀ : ℕ", "hN₀ : N = d * N₀", "hu₀ : IsUnit ↑a₀"], "goal": "∃ u, IsUnit u ∧ a = u * ↑d"}, "premise": [74464, 142648, 136229, 138376, 1673, 1674, 2574, 128974, 1717, 119703, 111256, 119707], "module": ["Mathlib/Data/ZMod/Units.lean"]}
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+{"state": {"context": ["n m N : ℕ", "a : ZMod N", "hN : N ≠ 0", "this : NeZero N", "d : ℕ := a.val.gcd N", "hd : d ≠ 0", "a₀ : ℕ", "ha₀ : a.val = d * a₀", "N₀ : ℕ", "hN₀ : N = d * N₀", "hu₀ : IsUnit ↑a₀", "u : (ZMod N)ˣ", "hu : (castHom ⋯ (ZMod N₀)) ↑u = ↑a₀"], "goal": "∃ u, IsUnit u ∧ a = u * ↑d"}, "premise": [136225, 120487, 120521, 116758, 120376, 117086], "module": ["Mathlib/Data/ZMod/Units.lean"]}
+{"state": {"context": ["n m N : ℕ", "a : ZMod N", "hN : N ≠ 0", "this : NeZero N", "d : ℕ := a.val.gcd N", "hd : d ≠ 0", "a₀ : ℕ", "ha₀ : a.val = d * a₀", "N₀ : ℕ", "hN₀ : N = d * N₀", "hu₀ : IsUnit ↑a₀", "u : (ZMod N)ˣ", "hu : (castHom ⋯ (ZMod N₀)) ↑u = ↑a₀"], "goal": "a = ↑u * ↑d"}, "premise": [120487, 138292, 142648, 119707, 138333], "module": ["Mathlib/Data/ZMod/Units.lean"]}
+{"state": {"context": ["n m N : ℕ", "a : ZMod N", "hN : N ≠ 0", "this : NeZero N", "d : ℕ := a.val.gcd N", "hd : d ≠ 0", "a₀ : ℕ", "ha₀ : a.val = d * a₀", "N₀ : ℕ", "hN₀ : N = d * N₀", "hu₀ : IsUnit ↑a₀", "u : (ZMod N)ˣ", "hu : (castHom ⋯ (ZMod N₀)) ↑u = ↑a₀"], "goal": "d * a₀ % N = d * (↑u).val % N"}, "premise": [145102, 4497, 138292, 142648, 119707, 138333], "module": ["Mathlib/Data/ZMod/Units.lean"]}
+{"state": {"context": ["n m N : ℕ", "a : ZMod N", "hN : N ≠ 0", "this : NeZero N", "d : ℕ := a.val.gcd N", "hd : d ≠ 0", "a₀ : ℕ", "ha₀ : a.val = d * a₀", "N₀ : ℕ", "hN₀ : N = d * N₀", "hu₀ : IsUnit ↑a₀", "u : (ZMod N)ˣ", "hu : (castHom ⋯ (ZMod N₀)) ↑u = ↑a₀"], "goal": "a₀ % N₀ = (↑u).val % N₀"}, "premise": [138308, 145102, 4497, 138333, 138302], "module": ["Mathlib/Data/ZMod/Units.lean"]}
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+{"state": {"context": ["A : Type u_1", "B : Type u_2", "C : Type u_3", "inst✝² : CommRing A", "inst✝¹ : CommRing B", "inst✝ : CommRing C", "g : B →+* C", "f : A →+* B", "hg : g.FinitePresentation", "hf : f.FinitePresentation", "ins1 : Algebra A B := f.toAlgebra", "ins2 : Algebra B C := g.toAlgebra", "ins3 : Algebra A C := (g.comp f).toAlgebra", "a : A", "b : B", "c : C"], "goal": "(algebraMap B C) ((algebraMap A B) a) * ((algebraMap B C) b * c) = (algebraMap A C) a * ((algebraMap B C) b * c)"}, "premise": [119703, 121165], "module": ["Mathlib/RingTheory/FinitePresentation.lean"]}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "inst✝ : DivisionMonoid α", "s t : Set α"], "goal": "univ / univ = univ"}, "premise": [119790], "module": ["Mathlib/Data/Set/Pointwise/Basic.lean"]}
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+{"state": {"context": ["X✝ : LocallyRingedSpace", "r : ↑(Γ.obj (op X✝))", "X : LocallyRingedSpace", "R : CommRingCat", "f : R ⟶ Γ.obj (op X)", "β : X ⟶ Spec.locallyRingedSpaceObj R", "w : X.toΓSpec.val.base ≫ (Spec.locallyRingedSpaceMap f).val.base = β.val.base", "h : ∀ (r : ↑R), f ≫ X.presheaf.map (homOfLE ⋯).op = toOpen (↑R) (basicOpen r) ≫ β.val.c.app (op (basicOpen r))"], "goal": "(X.toΓSpec ≫ Spec.locallyRingedSpaceMap f).val = β.val"}, "premise": [128409], "module": ["Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean"]}
+{"state": {"context": ["X✝ : LocallyRingedSpace", "r : ↑(Γ.obj (op X✝))", "X : LocallyRingedSpace", "R : CommRingCat", "f : R ⟶ Γ.obj (op X)", "β : X ⟶ Spec.locallyRingedSpaceObj R", "w : X.toΓSpec.val.base ≫ (Spec.locallyRingedSpaceMap f).val.base = β.val.base", "h : ∀ (r : ↑R), f ≫ X.presheaf.map (homOfLE ⋯).op = toOpen (↑R) (basicOpen r) ≫ β.val.c.app (op (basicOpen r))"], "goal": "∀ (r : ↑R), let U := basicOpen r; (toOpen (↑R) U ≫ (X.toΓSpec ≫ Spec.locallyRingedSpaceMap f).val.c.app (op U)) ≫ X.presheaf.map (eqToHom ⋯) = toOpen (↑R) U ≫ β.val.c.app (op U)"}, "premise": [128409], "module": ["Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean"]}
+{"state": {"context": ["X✝ : LocallyRingedSpace", "r✝ : ↑(Γ.obj (op X✝))", "X : LocallyRingedSpace", "R : CommRingCat", "f : R ⟶ Γ.obj (op X)", "β : X ⟶ Spec.locallyRingedSpaceObj R", "w : X.toΓSpec.val.base ≫ (Spec.locallyRingedSpaceMap f).val.base = β.val.base", "h : ∀ (r : ↑R), f ≫ X.presheaf.map (homOfLE ⋯).op = toOpen (↑R) (basicOpen r) ≫ β.val.c.app (op (basicOpen r))", "r : ↑R", "U : Opens (PrimeSpectrum ↑R) := basicOpen r"], "goal": "(toOpen (↑R) U ≫ (X.toΓSpec ≫ Spec.locallyRingedSpaceMap f).val.c.app (op U)) ≫ X.presheaf.map (eqToHom ⋯) = toOpen (↑R) U ≫ β.val.c.app (op U)"}, "premise": [68444], "module": ["Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean"]}
+{"state": {"context": ["X✝ : LocallyRingedSpace", "r✝ : ↑(Γ.obj (op X✝))", "X : LocallyRingedSpace", "R : CommRingCat", "f : R ⟶ Γ.obj (op X)", "β : X ⟶ Spec.locallyRingedSpaceObj R", "w : X.toΓSpec.val.base ≫ (Spec.locallyRingedSpaceMap f).val.base = β.val.base", "h : ∀ (r : ↑R), f ≫ X.presheaf.map (homOfLE ⋯).op = toOpen (↑R) (basicOpen r) ≫ β.val.c.app (op (basicOpen r))", "r : ↑R", "U : Opens (PrimeSpectrum ↑R) := basicOpen r"], "goal": "(toOpen (↑R) U ≫ (Spec.locallyRingedSpaceMap f).val.c.app (op U) ≫ X.toΓSpec.val.c.app (op ((Opens.map (Spec.locallyRingedSpaceMap f).val.base).obj (unop (op U))))) ≫ X.presheaf.map (eqToHom ⋯) = toOpen (↑R) U ≫ β.val.c.app (op U)"}, "premise": [68444], "module": ["Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean"]}
+{"state": {"context": ["X✝ : LocallyRingedSpace", "r✝ : ↑(Γ.obj (op X✝))", "X : LocallyRingedSpace", "R : CommRingCat", "f : R ⟶ Γ.obj (op X)", "β : X ⟶ Spec.locallyRingedSpaceObj R", "w : X.toΓSpec.val.base ≫ (Spec.locallyRingedSpaceMap f).val.base = β.val.base", "h : ∀ (r : ↑R), f ≫ X.presheaf.map (homOfLE ⋯).op = toOpen (↑R) (basicOpen r) ≫ β.val.c.app (op (basicOpen r))", "r : ↑R", "U : Opens (PrimeSpectrum ↑R) := basicOpen r"], "goal": "(CommRingCat.ofHom f ≫ toOpen (↑(Γ.obj (op X))) ((Opens.comap (PrimeSpectrum.comap f)) U) ≫ X.toΓSpec.val.c.app (op ((Opens.map (Spec.locallyRingedSpaceMap f).val.base).obj (unop (op U))))) ≫ X.presheaf.map (eqToHom ⋯) = toOpen (↑R) U ≫ β.val.c.app (op U)"}, "premise": [96173, 99919, 129578], "module": ["Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean"]}
+{"state": {"context": ["X✝ : LocallyRingedSpace", "r✝ : ↑(Γ.obj (op X✝))", "X : LocallyRingedSpace", "R : CommRingCat", "f : R ⟶ Γ.obj (op X)", "β : X ⟶ Spec.locallyRingedSpaceObj R", "w : X.toΓSpec.val.base ≫ (Spec.locallyRingedSpaceMap f).val.base = β.val.base", "h : ∀ (r : ↑R), f ≫ X.presheaf.map (homOfLE ⋯).op = toOpen (↑R) (basicOpen r) ≫ β.val.c.app (op (basicOpen r))", "r : ↑R", "U : Opens (PrimeSpectrum ↑R) := basicOpen r"], "goal": "CommRingCat.ofHom f ≫ (toOpen (↑(Γ.obj (op X))) ((Opens.comap (PrimeSpectrum.comap f)) U) ≫ X.toΓSpec.val.c.app (op ((Opens.map (Spec.locallyRingedSpaceMap f).val.base).obj (unop (op U))))) ≫ X.presheaf.map (eqToHom ⋯) = toOpen (↑R) U ≫ β.val.c.app (op U)"}, "premise": [129578, 96173, 99919], "module": ["Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean"]}
+{"state": {"context": ["X✝ : LocallyRingedSpace", "r✝ : ↑(Γ.obj (op X✝))", "X : LocallyRingedSpace", "R : CommRingCat", "f : R ⟶ Γ.obj (op X)", "β : X ⟶ Spec.locallyRingedSpaceObj R", "w : X.toΓSpec.val.base ≫ (Spec.locallyRingedSpaceMap f).val.base = β.val.base", "h : ∀ (r : ↑R), f ≫ X.presheaf.map (homOfLE ⋯).op = toOpen (↑R) (basicOpen r) ≫ β.val.c.app (op (basicOpen r))", "r : ↑R", "U : Opens (PrimeSpectrum ↑R) := basicOpen r"], "goal": "CommRingCat.ofHom f ≫ X.presheaf.map ((X.toΓSpecMapBasicOpen (f r)).leTop.op ≫ eqToHom ⋯) = toOpen (↑R) U ≫ β.val.c.app (op U)"}, "premise": [99919, 129578], "module": ["Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean"]}
+{"state": {"context": ["X : Scheme"], "goal": "Scheme.Hom.app (adjunction.unit.app X) ⊤ = (Scheme.ΓSpecIso Γ(X, ⊤)).hom"}, "premise": [95803], "module": ["Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean"]}
+{"state": {"context": ["X : Scheme", "this : Scheme.Γ.rightOp.map (adjunction.unit.app X) ≫ adjunction.counit.app (Scheme.Γ.rightOp.obj X) = 𝟙 (Scheme.Γ.rightOp.obj X)"], "goal": "Scheme.Hom.app (adjunction.unit.app X) ⊤ = (Scheme.ΓSpecIso Γ(X, ⊤)).hom"}, "premise": [95803], "module": ["Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean"]}
+{"state": {"context": ["X : Scheme", "this : (Scheme.Hom.app (adjunction.unit.app X) ⊤).op ≫ (Scheme.ΓSpecIso Γ(X, ⊤)).inv.op = 𝟙 (op Γ(X, ⊤))"], "goal": "Scheme.Hom.app (adjunction.unit.app X) ⊤ = (Scheme.ΓSpecIso Γ(X, ⊤)).hom"}, "premise": [88802, 89664, 96175, 99922, 99924, 126596, 129597], "module": ["Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean"]}
+{"state": {"context": ["X : Scheme", "this : (Scheme.Hom.app (adjunction.unit.app X) ⊤).op = 𝟙 (op Γ(X, ⊤)) ≫ inv (Scheme.ΓSpecIso Γ(X, ⊤)).inv.op"], "goal": "Scheme.Hom.app (adjunction.unit.app X) ⊤ = (Scheme.ΓSpecIso Γ(X, ⊤)).hom"}, "premise": [89664, 88802, 126596, 96175, 99922, 99924, 129597], "module": ["Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean"]}
+{"state": {"context": ["X : Scheme", "this : (Scheme.Hom.app (adjunction.unit.app X) ⊤).op = inv (Scheme.ΓSpecIso Γ(X, ⊤)).inv.op"], "goal": "Scheme.Hom.app (adjunction.unit.app X) ⊤ = (Scheme.ΓSpecIso Γ(X, ⊤)).hom"}, "premise": [89664, 126596, 88797, 96175, 89648, 99922, 99924, 89625, 71387, 129597], "module": ["Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean"]}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "κ : Kernel α β", "f : β → ℝ≥0∞", "μ : Measure β", "a : α", "s : Set β"], "goal": "∫⁻ (x : β) in s, f x ∂(const α μ) a = ∫⁻ (x : β) in s, f x ∂μ"}, "premise": [72658], "module": ["Mathlib/Probability/Kernel/Basic.lean"]}
+{"state": {"context": ["R : Type u_1", "K : Type u_2", "L : Type u_3", "inst✝¹⁰ : CommRing R", "inst✝⁹ : Field K", "inst✝⁸ : Field L", "inst✝⁷ : DecidableEq L", "inst✝⁶ : Algebra R K", "inst✝⁵ : IsFractionRing R K", "inst✝⁴ : Algebra K L", "inst✝³ : FiniteDimensional K L", "inst✝² : Algebra R L", "inst✝¹ : IsScalarTower R K L", "inst✝ : IsDomain R", "I : (FractionalIdeal R⁰ K)ˣ"], "goal": "mk I = 1 ↔ (↑↑I).IsPrincipal"}, "premise": [71387, 119870, 10329, 10339, 76828, 80874, 80882, 116661, 117064, 119903, 120378, 122978, 123694], "module": ["Mathlib/RingTheory/ClassGroup.lean"]}
+{"state": {"context": ["R : Type u_1", "K : Type u_2", "L : Type u_3", "inst✝¹⁰ : CommRing R", "inst✝⁹ : Field K", "inst✝⁸ : Field L", "inst✝⁷ : DecidableEq L", "inst✝⁶ : Algebra R K", "inst✝⁵ : IsFractionRing R K", "inst✝⁴ : Algebra K L", "inst✝³ : FiniteDimensional K L", "inst✝² : Algebra R L", "inst✝¹ : IsScalarTower R K L", "inst✝ : IsDomain R", "I : (FractionalIdeal R⁰ K)ˣ"], "goal": "(equiv K) (mk I) = (equiv K) 1 ↔ (↑↑I).IsPrincipal"}, "premise": [122978, 10339, 117064, 80874, 123694, 80882, 116661, 10329, 120378, 71387, 76828, 119870, 119903], "module": ["Mathlib/RingTheory/ClassGroup.lean"]}
+{"state": {"context": ["R : Type u_1", "K : Type u_2", "L : Type u_3", "inst✝¹⁰ : CommRing R", "inst✝⁹ : Field K", "inst✝⁸ : Field L", "inst✝⁷ : DecidableEq L", "inst✝⁶ : Algebra R K", "inst✝⁵ : IsFractionRing R K", "inst✝⁴ : Algebra K L", "inst✝³ : FiniteDimensional K L", "inst✝² : Algebra R L", "inst✝¹ : IsScalarTower R K L", "inst✝ : IsDomain R", "I : (FractionalIdeal R⁰ K)ˣ"], "goal": "(∃ x, spanSingleton R⁰ ↑x = ↑I) ↔ (↑↑I).IsPrincipal"}, "premise": [122978, 10339, 117064, 80874, 123694, 80882, 116661, 10329, 120378, 76828, 76862, 119903], "module": ["Mathlib/RingTheory/ClassGroup.lean"]}
+{"state": {"context": ["R : Type u_1", "K : Type u_2", "L : Type u_3", "inst✝¹⁰ : CommRing R", "inst✝⁹ : Field K", "inst✝⁸ : Field L", "inst✝⁷ : DecidableEq L", "inst✝⁶ : Algebra R K", "inst✝⁵ : IsFractionRing R K", "inst✝⁴ : Algebra K L", "inst✝³ : FiniteDimensional K L", "inst✝² : Algebra R L", "inst✝¹ : IsScalarTower R K L", "inst✝ : IsDomain R", "I : (FractionalIdeal R⁰ K)ˣ"], "goal": "(↑↑I).IsPrincipal → ∃ x, spanSingleton R⁰ ↑x = ↑I"}, "premise": [76862], "module": ["Mathlib/RingTheory/ClassGroup.lean"]}
+{"state": {"context": ["R : Type u_1", "K : Type u_2", "L : Type u_3", "inst✝¹⁰ : CommRing R", "inst✝⁹ : Field K", "inst✝⁸ : Field L", "inst✝⁷ : DecidableEq L", "inst✝⁶ : Algebra R K", "inst✝⁵ : IsFractionRing R K", "inst✝⁴ : Algebra K L", "inst✝³ : FiniteDimensional K L", "inst✝² : Algebra R L", "inst✝¹ : IsScalarTower R K L", "inst✝ : IsDomain R", "I : (FractionalIdeal R⁰ K)ˣ", "hI : (↑↑I).IsPrincipal"], "goal": "∃ x, spanSingleton R⁰ ↑x = ↑I"}, "premise": [86681, 75639, 76862, 137138], "module": ["Mathlib/RingTheory/ClassGroup.lean"]}
+{"state": {"context": ["R : Type u_1", "K : Type u_2", "L : Type u_3", "inst✝¹⁰ : CommRing R", "inst✝⁹ : Field K", "inst✝⁸ : Field L", "inst✝⁷ : DecidableEq L", "inst✝⁶ : Algebra R K", "inst✝⁵ : IsFractionRing R K", "inst✝⁴ : Algebra K L", "inst✝³ : FiniteDimensional K L", "inst✝² : Algebra R L", "inst✝¹ : IsScalarTower R K L", "inst✝ : IsDomain R", "I : (FractionalIdeal R⁰ K)ˣ", "hI : (↑↑I).IsPrincipal", "x : K", "hx : ↑↑I = Submodule.span R {x}"], "goal": "∃ x, spanSingleton R⁰ ↑x = ↑I"}, "premise": [86681, 137138, 76862, 75639], "module": ["Mathlib/RingTheory/ClassGroup.lean"]}
+{"state": {"context": ["R : Type u_1", "K : Type u_2", "L : Type u_3", "inst✝¹⁰ : CommRing R", "inst✝⁹ : Field K", "inst✝⁸ : Field L", "inst✝⁷ : DecidableEq L", "inst✝⁶ : Algebra R K", "inst✝⁵ : IsFractionRing R K", "inst✝⁴ : Algebra K L", "inst✝³ : FiniteDimensional K L", "inst✝² : Algebra R L", "inst✝¹ : IsScalarTower R K L", "inst✝ : IsDomain R", "I : (FractionalIdeal R⁰ K)ˣ", "hI : (↑↑I).IsPrincipal", "x : K", "hx : ↑↑I = Submodule.span R {x}", "hx' : ↑I = spanSingleton R⁰ x"], "goal": "∃ x, spanSingleton R⁰ ↑x = ↑I"}, "premise": [75639, 76862, 137138], "module": ["Mathlib/RingTheory/ClassGroup.lean"]}
+{"state": {"context": ["α : Type u", "β : Type v", "inst✝¹ : Group α", "inst✝ : Group β", "f : α → β", "hf✝ hf : IsGroupHom f", "a : α"], "goal": "f a⁻¹ * f a = 1"}, "premise": [119820, 125433, 125473], "module": ["Mathlib/Deprecated/Group.lean"]}
+{"state": {"context": ["ι : Type u_1", "I J : Box ι", "x y : ι → ℝ", "a✝¹ a✝ : Box ι"], "goal": "↑(mk' (a✝¹.lower ⊔ a✝.lower) (a✝¹.upper ⊓ a✝.upper)) = ↑↑a✝¹ ∩ ↑↑a✝"}, "premise": [14641, 14643, 20463, 34341, 34365, 34372, 134012], "module": ["Mathlib/Analysis/BoxIntegral/Box/Basic.lean"]}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "inst✝³ : TopologicalSpace α", "inst✝² : TopologicalSpace β", "i✝ : α → β", "di : DenseInducing i✝", "inst✝¹ : TopologicalSpace δ", "f : γ → α", "g : γ → δ", "h : δ → β", "inst✝ : TopologicalSpace γ", "i : α → β", "c : Continuous i", "dense : ∀ (x : β), x ∈ closure (range i)", "H : ∀ (a : α), ∀ s ∈ 𝓝 a, ∃ t ∈ 𝓝 (i a), ∀ (b : α), i b ∈ t → b ∈ s", "a : α"], "goal": "comap i (𝓝 (i a)) ≤ 𝓝 a"}, "premise": [15904], "module": ["Mathlib/Topology/DenseEmbedding.lean"]}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "E : Type u_3", "J : Type u_4", "inst✝² : Category.{u_5, u_1} C", "inst✝¹ : Category.{?u.5761, u_2} D", "inst✝ : Category.{?u.5765, u_3} E", "X Y : GradedObject J C", "e : X ≅ Y", "j : J"], "goal": "e.hom j ≫ e.inv j = 𝟙 (X j)"}, "premise": [88743, 97567, 97568], "module": ["Mathlib/CategoryTheory/GradedObject.lean"]}
+{"state": {"context": ["ι : Type u_1", "α : Type u", "β : Type v", "γ : Type w", "l₁ l₂ : List α", "inst✝ : DecidableEq α", "a b : α", "l : List α", "x✝ : b ≠ a", "h : b ≠ a := x✝"], "goal": "indexOf a (b :: l) = (indexOf a l).succ"}, "premise": [1171, 1397, 1769, 2854], "module": ["Mathlib/Data/List/Basic.lean"]}
+{"state": {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "inst✝ : HasColimits C", "X Y : PresheafedSpace C", "α : X ⟶ Y", "U : Opens ↑↑Y", "x : ↥((Opens.map α.base).obj U)"], "goal": "Y.presheaf.germ ⟨α.base ↑x, ⋯⟩ ≫ Hom.stalkMap α ↑x = α.c.app (op U) ≫ X.presheaf.germ x"}, "premise": [64381], "module": ["Mathlib/Geometry/RingedSpace/Stalks.lean"]}
+{"state": {"context": ["a : UnitAddCircle", "r : ℝ"], "goal": "(fun x => (ofReal' ∘ oddKernel a) x - 0) =O[atTop] fun x => x ^ r"}, "premise": [22986, 43455], "module": ["Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean"]}
+{"state": {"context": ["a : UnitAddCircle", "r v : ℝ", "hv : 0 < v", "hv' : oddKernel a =O[atTop] fun x => rexp (-v * x)"], "goal": "(fun x => (ofReal' ∘ oddKernel a) x - 0) =O[atTop] fun x => x ^ r"}, "premise": [22986, 43455], "module": ["Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean"]}
+{"state": {"context": ["a : UnitAddCircle", "r v : ℝ", "hv : 0 < v", "hv' : (fun x => ‖oddKernel a x‖) =O[atTop] fun x => rexp (-v * x)"], "goal": "(fun x => ‖(ofReal' ∘ oddKernel a) x - 0‖) =O[atTop] fun x => x ^ r"}, "premise": [1671, 117816, 46181, 43455], "module": ["Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean"]}
+{"state": {"context": ["a : UnitAddCircle", "r v : ℝ", "hv : 0 < v", "hv' : (fun x => ‖oddKernel a x‖) =O[atTop] fun x => rexp (-v * x)"], "goal": "(fun x => ‖oddKernel a x‖) =O[atTop] fun x => x ^ r"}, "premise": [43365, 46181, 1671, 39208, 43411, 117816], "module": ["Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean"]}
+{"state": {"context": ["a : UnitAddCircle", "r : ℝ"], "goal": "(fun x => (ofReal' ∘ sinKernel a) x - 0) =O[atTop] fun x => x ^ r"}, "premise": [22987, 43455], "module": ["Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean"]}
+{"state": {"context": ["a : UnitAddCircle", "r v : ℝ", "hv : 0 < v", "hv' : sinKernel a =O[atTop] fun x => rexp (-v * x)"], "goal": "(fun x => (ofReal' ∘ sinKernel a) x - 0) =O[atTop] fun x => x ^ r"}, "premise": [22987, 43455], "module": ["Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean"]}
+{"state": {"context": ["a : UnitAddCircle", "r v : ℝ", "hv : 0 < v", "hv' : (fun x => ‖sinKernel a x‖) =O[atTop] fun x => rexp (-v * x)"], "goal": "(fun x => ‖(ofReal' ∘ sinKernel a) x - 0‖) =O[atTop] fun x => x ^ r"}, "premise": [1671, 117816, 46181, 43455], "module": ["Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean"]}
+{"state": {"context": ["a : UnitAddCircle", "r v : ℝ", "hv : 0 < v", "hv' : (fun x => ‖sinKernel a x‖) =O[atTop] fun x => rexp (-v * x)"], "goal": "(fun x => ‖sinKernel a x‖) =O[atTop] fun x => x ^ r"}, "premise": [43365, 46181, 1671, 39208, 43411, 117816], "module": ["Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean"]}
+{"state": {"context": ["a : UnitAddCircle", "x : ℝ", "hx : x ∈ Ioi 0"], "goal": "(ofReal' ∘ oddKernel a) (1 / x) = (1 * ↑(x ^ (3 / 2))) • (ofReal' ∘ sinKernel a) x"}, "premise": [1670, 14286, 22982, 40068, 117810, 118863, 119728, 119770, 148304], "module": ["Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean"]}
+{"state": {"context": ["m n : ℤ", "hn : n ≠ 0"], "goal": "(↑m / ↑n).den = 1 ↔ n ∣ m"}, "premise": [1673, 147952], "module": ["Mathlib/Data/Rat/Lemmas.lean"]}
+{"state": {"context": ["m n : ℤ", "hn : ↑n ≠ 0"], "goal": "(↑m / ↑n).den = 1 ↔ n ∣ m"}, "premise": [1673, 147952], "module": ["Mathlib/Data/Rat/Lemmas.lean"]}
+{"state": {"context": ["d m n : Nat", "dgt1 : 1 < d", "Hm : d ∣ m", "Hn : d ∣ n", "co : m.Coprime n"], "goal": "d ∣ 1"}, "premise": [394, 3620], "module": [".lake/packages/batteries/Batteries/Data/Nat/Gcd.lean"]}
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+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝² : Preorder α", "inst✝¹ : TopologicalSpace α", "inst✝ : IsLawson α", "lawsonBasis_image2 : lawsonBasis α = image2 (fun x x_1 => ⇑WithLower.toLower ⁻¹' x ∩ ⇑WithScott.toScott ⁻¹' x_1) (IsLower.lowerBasis (WithLower α)) {U | IsOpen U}"], "goal": "lower α ⊓ scott α = induced (⇑WithLower.toLower) WithLower.instTopologicalSpace ⊓ induced (⇑WithScott.toScott) WithScott.instTopologicalSpace"}, "premise": [19], "module": ["Mathlib/Topology/Order/LawsonTopology.lean"]}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝² : Preorder α", "inst✝¹ : TopologicalSpace α", "inst✝ : IsLawson α", "lawsonBasis_image2 : lawsonBasis α = image2 (fun x x_1 => ⇑WithLower.toLower ⁻¹' x ∩ ⇑WithScott.toScott ⁻¹' x_1) (IsLower.lowerBasis (WithLower α)) {U | IsOpen U}"], "goal": "lower α = induced (⇑WithLower.toLower) WithLower.instTopologicalSpace"}, "premise": [19], "module": ["Mathlib/Topology/Order/LawsonTopology.lean"]}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝² : Preorder α", "inst✝¹ : TopologicalSpace α", "inst✝ : IsLawson α", "lawsonBasis_image2 : lawsonBasis α = image2 (fun x x_1 => ⇑WithLower.toLower ⁻¹' x ∩ ⇑WithScott.toScott ⁻¹' x_1) (IsLower.lowerBasis (WithLower α)) {U | IsOpen U}"], "goal": "scott α = induced (⇑WithScott.toScott) WithScott.instTopologicalSpace"}, "premise": [19], "module": ["Mathlib/Topology/Order/LawsonTopology.lean"]}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝² : Preorder α", "inst✝¹ : TopologicalSpace α", "inst✝ : IsLawson α", "lawsonBasis_image2 : lawsonBasis α = image2 (fun x x_1 => ⇑WithLower.toLower ⁻¹' x ∩ ⇑WithScott.toScott ⁻¹' x_1) (IsLower.lowerBasis (WithLower α)) {U | IsOpen U}", "x✝ : TopologicalSpace α := scott α"], "goal": "scott α = induced (⇑WithScott.toScott) WithScott.instTopologicalSpace"}, "premise": [54002, 65595], "module": ["Mathlib/Topology/Order/LawsonTopology.lean"]}
+{"state": {"context": ["C : Type u_1", "inst✝¹ : Category.{?u.133780, u_1} C", "inst✝ : Preadditive C", "S₁ S₂ S₃ : ShortComplex C", "φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂", "h : Homotopy φ₁ φ₂", "h' : Homotopy φ₃ φ₄"], "goal": "(φ₁ - φ₃).τ₁ = S₁.f ≫ (h.h₁ - h'.h₁) + (h.h₀ - h'.h₀) + (φ₂ - φ₄).τ₁"}, "premise": [91601, 114861, 114910], "module": ["Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean"]}
+{"state": {"context": ["C : Type u_1", "inst✝¹ : Category.{?u.133780, u_1} C", "inst✝ : Preadditive C", "S₁ S₂ S₃ : ShortComplex C", "φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂", "h : Homotopy φ₁ φ₂", "h' : Homotopy φ₃ φ₄"], "goal": "S₁.f ≫ h.h₁ + h.h₀ + φ₂.τ₁ - (S₁.f ≫ h'.h₁ + h'.h₀ + φ₄.τ₁) = S₁.f ≫ h.h₁ - S₁.f ≫ h'.h₁ + (h.h₀ - h'.h₀) + (φ₂.τ₁ - φ₄.τ₁)"}, "premise": [91601, 114861, 114910], "module": ["Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean"]}
+{"state": {"context": ["C : Type u_1", "inst✝¹ : Category.{?u.133780, u_1} C", "inst✝ : Preadditive C", "S₁ S₂ S₃ : ShortComplex C", "φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂", "h : Homotopy φ₁ φ₂", "h' : Homotopy φ₃ φ₄"], "goal": "(φ₁ - φ₃).τ₂ = S₁.g ≫ (h.h₂ - h'.h₂) + (h.h₁ - h'.h₁) ≫ S₂.f + (φ₂ - φ₄).τ₂"}, "premise": [91600, 91601, 114862, 114912], "module": ["Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean"]}
+{"state": {"context": ["C : Type u_1", "inst✝¹ : Category.{?u.133780, u_1} C", "inst✝ : Preadditive C", "S₁ S₂ S₃ : ShortComplex C", "φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂", "h : Homotopy φ₁ φ₂", "h' : Homotopy φ₃ φ₄"], "goal": "S₁.g ≫ h.h₂ + h.h₁ ≫ S₂.f + φ₂.τ₂ - (S₁.g ≫ h'.h₂ + h'.h₁ ≫ S₂.f + φ₄.τ₂) = S₁.g ≫ h.h₂ - S₁.g ≫ h'.h₂ + (h.h₁ ≫ S₂.f - h'.h₁ ≫ S₂.f) + (φ₂.τ₂ - φ₄.τ₂)"}, "premise": [91600, 91601, 114862, 114912], "module": ["Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean"]}
+{"state": {"context": ["C : Type u_1", "inst✝¹ : Category.{?u.133780, u_1} C", "inst✝ : Preadditive C", "S₁ S₂ S₃ : ShortComplex C", "φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂", "h : Homotopy φ₁ φ₂", "h' : Homotopy φ₃ φ₄"], "goal": "(φ₁ - φ₃).τ₃ = h.h₃ - h'.h₃ + (h.h₂ - h'.h₂) ≫ S₂.g + (φ₂ - φ₄).τ₃"}, "premise": [91600, 114863, 114911], "module": ["Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean"]}
+{"state": {"context": ["C : Type u_1", "inst✝¹ : Category.{?u.133780, u_1} C", "inst✝ : Preadditive C", "S₁ S₂ S₃ : ShortComplex C", "φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂", "h : Homotopy φ₁ φ₂", "h' : Homotopy φ₃ φ₄"], "goal": "h.h₃ + h.h₂ ≫ S₂.g + φ₂.τ₃ - (h'.h₃ + h'.h₂ ≫ S₂.g + φ₄.τ₃) = h.h₃ - h'.h₃ + (h.h₂ ≫ S₂.g - h'.h₂ ≫ S₂.g) + (φ₂.τ₃ - φ₄.τ₃)"}, "premise": [91600, 114863, 114911], "module": ["Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean"]}
+{"state": {"context": ["R : Type u_1", "inst✝ : CommRing R", "f : R → ℝ", "c : ℝ", "f_ne_zero : f ≠ 0", "f_nonneg : 0 ≤ f", "h1 : f 1 = 0", "f_mul : ∀ (y : R), f (1 * y) ≤ c * f 1 * f y"], "goal": "False"}, "premise": [108557, 108558, 119728, 1673, 71383], "module": ["Mathlib/Analysis/Normed/Ring/SeminormFromBounded.lean"]}
+{"state": {"context": ["R : Type u_1", "inst✝ : CommRing R", "f : R → ℝ", "c : ℝ", "f_ne_zero : f ≠ 0", "f_nonneg : 0 ≤ f", "h1 : f 1 = 0", "f_mul : ∀ (y : R), f y ≤ 0"], "goal": "False"}, "premise": [1673, 108557, 108558, 119728, 71383], "module": ["Mathlib/Analysis/Normed/Ring/SeminormFromBounded.lean"]}
+{"state": {"context": ["R : Type u_1", "inst✝ : CommRing R", "f : R → ℝ", "c : ℝ", "f_ne_zero : f ≠ 0", "f_nonneg : 0 ≤ f", "h1 : f 1 = 0", "f_mul : ∀ (y : R), f y ≤ 0", "z : R", "hz : f z ≠ 0 z"], "goal": "False"}, "premise": [1673, 71383], "module": ["Mathlib/Analysis/Normed/Ring/SeminormFromBounded.lean"]}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "M : Type u_3", "H : Type u_4", "E' : Type u_5", "M' : Type u_6", "H' : Type u_7", "inst✝¹⁰ : NontriviallyNormedField 𝕜", "inst✝⁹ : NormedAddCommGroup E", "inst✝⁸ : NormedSpace 𝕜 E", "inst✝⁷ : TopologicalSpace H", "inst✝⁶ : TopologicalSpace M", "f f' : PartialHomeomorph M H", "I : ModelWithCorners 𝕜 E H", "inst✝⁵ : NormedAddCommGroup E'", "inst✝⁴ : NormedSpace 𝕜 E'", "inst✝³ : TopologicalSpace H'", "inst✝² : TopologicalSpace M'", "I' : ModelWithCorners 𝕜 E' H'", "s t : Set M", "inst✝¹ : ChartedSpace H M", "inst✝ : ChartedSpace H' M'", "x : M"], "goal": "x ∈ (extChartAt I x).source"}, "premise": [67287, 67848], "module": ["Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean"]}
+{"state": {"context": ["R : Type u_1", "inst✝ : CommMonoidWithZero R", "n✝ p : R", "k n a b : ℕ", "hab : a.Coprime b", "hn : IsPrimePow n"], "goal": "n ∣ a * b ↔ n ∣ a ∨ n ∣ b"}, "premise": [70039, 70039], "module": ["Mathlib/Data/Nat/Factorization/PrimePow.lean"]}
+{"state": {"context": ["R : Type u_1", "inst✝ : CommMonoidWithZero R", "n✝ p : R", "k n b : ℕ", "hn : IsPrimePow n", "hab : Coprime 0 b"], "goal": "n ∣ 0 * b ↔ n ∣ 0 ∨ n ∣ b"}, "premise": [70039, 70039], "module": ["Mathlib/Data/Nat/Factorization/PrimePow.lean"]}
+{"state": {"context": ["R : Type u_1", "inst✝ : CommMonoidWithZero R", "n✝ p : R", "k n a b : ℕ", "hab : a.Coprime b", "hn : IsPrimePow n", "ha : a ≠ 0"], "goal": "n ∣ a * b ↔ n ∣ a ∨ n ∣ b"}, "premise": [70039], "module": ["Mathlib/Data/Nat/Factorization/PrimePow.lean"]}
+{"state": {"context": ["R : Type u_1", "inst✝ : CommMonoidWithZero R", "n✝ p : R", "k n a : ℕ", "hn : IsPrimePow n", "ha : a ≠ 0", "hab : a.Coprime 0"], "goal": "n ∣ a * 0 ↔ n ∣ a ∨ n ∣ 0"}, "premise": [70039, 395, 1674, 2110, 108885, 143611, 143612], "module": ["Mathlib/Data/Nat/Factorization/PrimePow.lean"]}
+{"state": {"context": ["R : Type u_1", "inst✝ : CommMonoidWithZero R", "n✝ p : R", "k n a b : ℕ", "hab : a.Coprime b", "hn : IsPrimePow n", "ha : a ≠ 0", "hb : b ≠ 0"], "goal": "n ∣ a * b ↔ n ∣ a ∨ n ∣ b"}, "premise": [1674, 395, 108885, 70039, 143611, 143612, 2110], "module": ["Mathlib/Data/Nat/Factorization/PrimePow.lean"]}
+{"state": {"context": ["R : Type u_1", "inst✝ : CommMonoidWithZero R", "n✝ p : R", "k n a b : ℕ", "hab : a.Coprime b", "hn : IsPrimePow n", "ha : a ≠ 0", "hb : b ≠ 0"], "goal": "n ∣ a * b → n ∣ a ∨ n ∣ b"}, "premise": [1673, 1674, 395, 111243, 108885, 143611, 143612, 2110], "module": ["Mathlib/Data/Nat/Factorization/PrimePow.lean"]}
+{"state": {"context": ["R : Type u_1", "inst✝ : CommMonoidWithZero R", "n p✝ : R", "k✝ a b : ℕ", "hab : a.Coprime b", "ha : a ≠ 0", "hb : b ≠ 0", "p k : ℕ", "hp : Prime p", "left✝ : 0 < k", "hn : IsPrimePow (p ^ k)"], "goal": "p ^ k ∣ a * b → p ^ k ∣ a ∨ p ^ k ∣ b"}, "premise": [148168, 1673, 120650, 111243, 144202, 108269, 144694], "module": ["Mathlib/Data/Nat/Factorization/PrimePow.lean"]}
+{"state": {"context": ["R : Type u_1", "inst✝ : CommMonoidWithZero R", "n p✝ : R", "k✝ a b : ℕ", "hab : a.Coprime b", "ha : a ≠ 0", "hb : b ≠ 0", "p k : ℕ", "hp : Prime p", "left✝ : 0 < k", "hn : IsPrimePow (p ^ k)"], "goal": "k ≤ a.factorization p + b.factorization p → k ≤ a.factorization p ∨ k ≤ b.factorization p"}, "premise": [148065, 148168, 70089, 120650, 144202, 13484, 108269, 138900, 144694, 144182], "module": ["Mathlib/Data/Nat/Factorization/PrimePow.lean"]}
+{"state": {"context": ["R : Type u_1", "inst✝ : CommMonoidWithZero R", "n p✝ : R", "k✝ a b : ℕ", "hab : a.Coprime b", "ha : a ≠ 0", "hb : b ≠ 0", "p k : ℕ", "hp : Prime p", "left✝ : 0 < k", "hn : IsPrimePow (p ^ k)", "this : a.factorization p = 0 ∨ b.factorization p = 0"], "goal": "k ≤ a.factorization p + b.factorization p → k ≤ a.factorization p ∨ k ≤ b.factorization p"}, "premise": [148065, 70089, 13484, 138900, 144182, 70078], "module": ["Mathlib/Data/Nat/Factorization/PrimePow.lean"]}
+{"state": {"context": ["s : ℂ", "hs : 0 < s.re"], "goal": "Gamma s ≠ 0"}, "premise": [40597, 53688], "module": ["Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean"]}
+{"state": {"context": ["s : ℂ", "hs : 0 < s.re", "m : ℕ"], "goal": "s ≠ -↑m"}, "premise": [53688, 40597], "module": ["Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean"]}
+{"state": {"context": ["s : ℂ", "m : ℕ", "hs : s = -↑m"], "goal": "s.re ≤ 0"}, "premise": [148299, 148332, 142636, 148280, 53688], "module": ["Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean"]}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "M : Type u_3", "H : Type u_4", "E' : Type u_5", "M' : Type u_6", "H' : Type u_7", "inst✝⁹ : NontriviallyNormedField 𝕜", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : NormedSpace 𝕜 E", "inst✝⁶ : TopologicalSpace H", "inst✝⁵ : TopologicalSpace M", "f f' : PartialHomeomorph M H", "I : ModelWithCorners 𝕜 E H", "inst✝⁴ : NormedAddCommGroup E'", "inst✝³ : NormedSpace 𝕜 E'", "inst✝² : TopologicalSpace H'", "inst✝¹ : TopologicalSpace M'", "I' : ModelWithCorners 𝕜 E' H'", "s t : Set M", "inst✝ : ChartedSpace H M", "hf : f ∈ maximalAtlas I M", "hf' : f' ∈ maximalAtlas I M", "x : M", "hxf : x ∈ f.source", "hxf' : x ∈ f'.source"], "goal": "ContDiffWithinAt 𝕜 ⊤ (↑(f.extend I) ∘ ↑(f'.extend I).symm) (range ↑I) (↑(f'.extend I) x)"}, "premise": [67844, 67840], "module": ["Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean"]}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "M : Type u_3", "H : Type u_4", "E' : Type u_5", "M' : Type u_6", "H' : Type u_7", "inst✝⁹ : NontriviallyNormedField 𝕜", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : NormedSpace 𝕜 E", "inst✝⁶ : TopologicalSpace H", "inst✝⁵ : TopologicalSpace M", "f f' : PartialHomeomorph M H", "I : ModelWithCorners 𝕜 E H", "inst✝⁴ : NormedAddCommGroup E'", "inst✝³ : NormedSpace 𝕜 E'", "inst✝² : TopologicalSpace H'", "inst✝¹ : TopologicalSpace M'", "I' : ModelWithCorners 𝕜 E' H'", "s t : Set M", "inst✝ : ChartedSpace H M", "hf : f ∈ maximalAtlas I M", "hf' : f' ∈ maximalAtlas I M", "x : M", "hxf : x ∈ f.source", "hxf' : x ∈ f'.source"], "goal": "↑(f'.extend I) x ∈ ((f'.extend I).symm ≫ f.extend I).source"}, "premise": [67840, 67844], "module": ["Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean"]}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "M : Type u_3", "H : Type u_4", "E' : Type u_5", "M' : Type u_6", "H' : Type u_7", "inst✝⁹ : NontriviallyNormedField 𝕜", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : NormedSpace 𝕜 E", "inst✝⁶ : TopologicalSpace H", "inst✝⁵ : TopologicalSpace M", "f f' : PartialHomeomorph M H", "I : ModelWithCorners 𝕜 E H", "inst✝⁴ : NormedAddCommGroup E'", "inst✝³ : NormedSpace 𝕜 E'", "inst✝² : TopologicalSpace H'", "inst✝¹ : TopologicalSpace M'", "I' : ModelWithCorners 𝕜 E' H'", "s t : Set M", "inst✝ : ChartedSpace H M", "hf : f ∈ maximalAtlas I M", "hf' : f' ∈ maximalAtlas I M", "x : M", "hxf : x ∈ f.source", "hxf' : x ∈ f'.source"], "goal": "↑(f'.extend I) x ∈ ↑(f'.extend I) '' (f'.source ∩ f.source)"}, "premise": [67840, 131593], "module": ["Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean"]}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝¹ : LinearOrderedRing α", "inst✝ : FloorRing α", "z : ℤ", "a x : α"], "goal": "fract (-x) = 0 ↔ fract x = 0"}, "premise": [14271, 20003, 101702, 103416, 105158, 1674, 2045], "module": ["Mathlib/Algebra/Order/Floor.lean"]}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝¹ : LinearOrderedRing α", "inst✝ : FloorRing α", "z : ℤ", "a x : α"], "goal": "(∃ z, -x = ↑z) ↔ ∃ z, x = ↑z"}, "premise": [20003, 105158, 101702, 1674, 103416, 2045, 14271], "module": ["Mathlib/Algebra/Order/Floor.lean"]}
+{"state": {"context": ["R : Type u", "A : Type v", "B : Type w", "inst✝⁹ : CommSemiring R", "inst✝⁸ : NonUnitalNonAssocSemiring A", "inst✝⁷ : Module R A", "inst✝⁶ : IsScalarTower R A A", "inst✝⁵ : SMulCommClass R A A", "inst✝⁴ : NonUnitalNonAssocSemiring B", "inst✝³ : Module R B", "inst✝² : IsScalarTower R B B", "inst✝¹ : SMulCommClass R B B", "S : NonUnitalSubalgebra R A", "ι : Type u_1", "inst✝ : Nonempty ι", "K : ι → NonUnitalSubalgebra R A", "dir : Directed (fun x x_1 => x ≤ x_1) K", "f : (i : ι) → ↥(K i) →ₙₐ[R] B", "hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)", "i : ι", "x : ↥(K i)", "hx : ↑x ∈ iSup K"], "goal": "Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x) ⋯ ↑(iSup K) ⋯ ⟨↑x, hx⟩ = (f i) x"}, "premise": [133070], "module": ["Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean"]}
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+{"state": {"context": ["k : Type u_1", "inst✝³ : Field k", "K : Type u_2", "inst✝² : Field K", "F : Type u_3", "inst✝¹ : Field F", "inst✝ : NumberField K"], "goal": "card { φ // ComplexEmbedding.IsReal φ } ≤ card (K →+* ℂ)"}, "premise": [141443, 141446, 3884, 23345, 23347, 23289], "module": ["Mathlib/NumberTheory/NumberField/Embeddings.lean"]}
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+{"state": {"context": ["C : Type u_1", "inst✝¹ : Category.{?u.111346, u_1} C", "inst✝ : Preadditive C", "S₁ S₂ S₃ : ShortComplex C", "φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂", "h₁₂ : Homotopy φ₁ φ₂", "h₂₃ : Homotopy φ₂ φ₃"], "goal": "S₁.f ≫ h₁₂.h₁ + h₁₂.h₀ + (S₁.f ≫ h₂₃.h₁ + h₂₃.h₀ + φ₃.τ₁) = S₁.f ≫ h₁₂.h₁ + S₁.f ≫ h₂₃.h₁ + (h₁₂.h₀ + h₂₃.h₀) + φ₃.τ₁"}, "premise": [91599, 114910], "module": ["Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean"]}
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+{"state": {"context": ["C : Type u_1", "inst✝¹ : Category.{?u.111346, u_1} C", "inst✝ : Preadditive C", "S₁ S₂ S₃ : ShortComplex C", "φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂", "h₁₂ : Homotopy φ₁ φ₂", "h₂₃ : Homotopy φ₂ φ₃"], "goal": "S₁.g ≫ h₁₂.h₂ + h₁₂.h₁ ≫ S₂.f + (S₁.g ≫ h₂₃.h₂ + h₂₃.h₁ ≫ S₂.f + φ₃.τ₂) = S₁.g ≫ h₁₂.h₂ + S₁.g ≫ h₂₃.h₂ + (h₁₂.h₁ ≫ S₂.f + h₂₃.h₁ ≫ S₂.f) + φ₃.τ₂"}, "premise": [91598, 91599, 114912], "module": ["Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean"]}
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+{"state": {"context": ["C : Type u_1", "inst✝¹ : Category.{?u.111346, u_1} C", "inst✝ : Preadditive C", "S₁ S₂ S₃ : ShortComplex C", "φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂", "h₁₂ : Homotopy φ₁ φ₂", "h₂₃ : Homotopy φ₂ φ₃"], "goal": "h₁₂.h₃ + h₁₂.h₂ ≫ S₂.g + (h₂₃.h₃ + h₂₃.h₂ ≫ S₂.g + φ₃.τ₃) = h₁₂.h₃ + h₂₃.h₃ + (h₁₂.h₂ ≫ S₂.g + h₂₃.h₂ ≫ S₂.g) + φ₃.τ₃"}, "premise": [91598, 114911], "module": ["Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean"]}
+{"state": {"context": ["R✝ : Type u_1", "inst✝³¹ : CommSemiring R✝", "S✝ : Submonoid R✝", "M✝ : Type u_2", "M' : Type u_3", "M'' : Type u_4", "inst✝³⁰ : AddCommMonoid M✝", "inst✝²⁹ : AddCommMonoid M'", "inst✝²⁸ : AddCommMonoid M''", "A : Type u_5", "inst✝²⁷ : CommSemiring A", "inst✝²⁶ : Algebra R✝ A", "inst✝²⁵ : Module A M'", "inst✝²⁴ : IsLocalization S✝ A", "inst✝²³ : Module R✝ M✝", "inst✝²² : Module R✝ M'", "inst✝²¹ : Module R✝ M''", "inst✝²⁰ : IsScalarTower R✝ A M'", "f✝ : M✝ →ₗ[R✝] M'", "g : M✝ →ₗ[R✝] M''", "M₀ : Type ?u.1244734", "M₀' : Type ?u.1244737", "inst✝¹⁹ : AddCommGroup M₀", "inst✝¹⁸ : AddCommGroup M₀'", "inst✝¹⁷ : Module R✝ M₀", "inst✝¹⁶ : Module R✝ M₀'", "f₀ : M₀ →ₗ[R✝] M₀'", "inst✝¹⁵ : IsLocalizedModule S✝ f₀", "M₁ : Type ?u.1246805", "M₁' : Type ?u.1246808", "inst✝¹⁴ : AddCommGroup M₁", "inst✝¹³ : AddCommGroup M₁'", "inst✝¹² : Module R✝ M₁", "inst✝¹¹ : Module R✝ M₁'", "f₁ : M₁ →ₗ[R✝] M₁'", "inst✝¹⁰ : IsLocalizedModule S✝ f₁", "M₂ : Type ?u.1248876", "M₂' : Type ?u.1248879", "inst✝⁹ : AddCommGroup M₂", "inst✝⁸ : AddCommGroup M₂'", "inst✝⁷ : Module R✝ M₂", "inst✝⁶ : Module R✝ M₂'", "f₂ : M₂ →ₗ[R✝] M₂'", "inst✝⁵ : IsLocalizedModule S✝ f₂", "R : Type u_6", "S : Type u_7", "S' : Type u_8", "inst✝⁴ : CommRing R", "inst✝³ : CommRing S", "inst✝² : CommRing S'", "inst✝¹ : Algebra R S", "inst✝ : Algebra R S'", "M : Submonoid R", "f : S →ₐ[R] S'", "h₁ : ∀ x ∈ M, IsUnit ((algebraMap R S') x)", "h₂ : ∀ (y : S'), ∃ x, x.2 • y = f x.1", "h₃ : ∀ (x : S), f x = 0 → ∃ m, m • x = 0"], "goal": "IsLocalizedModule M f.toLinearMap"}, "premise": [119625, 120533, 121056, 121165], "module": ["Mathlib/Algebra/Module/LocalizedModule.lean"]}
+{"state": {"context": ["R✝ : Type u_1", "inst✝³¹ : CommSemiring R✝", "S✝ : Submonoid R✝", "M✝ : Type u_2", "M' : Type u_3", "M'' : Type u_4", "inst✝³⁰ : AddCommMonoid M✝", "inst✝²⁹ : AddCommMonoid M'", "inst✝²⁸ : AddCommMonoid M''", "A : Type u_5", "inst✝²⁷ : CommSemiring A", "inst✝²⁶ : Algebra R✝ A", "inst✝²⁵ : Module A M'", "inst✝²⁴ : IsLocalization S✝ A", "inst✝²³ : Module R✝ M✝", "inst✝²² : Module R✝ M'", "inst✝²¹ : Module R✝ M''", "inst✝²⁰ : IsScalarTower R✝ A M'", "f✝ : M✝ →ₗ[R✝] M'", "g : M✝ →ₗ[R✝] M''", "M₀ : Type ?u.1244734", "M₀' : Type ?u.1244737", "inst✝¹⁹ : AddCommGroup M₀", "inst✝¹⁸ : AddCommGroup M₀'", "inst✝¹⁷ : Module R✝ M₀", "inst✝¹⁶ : Module R✝ M₀'", "f₀ : M₀ →ₗ[R✝] M₀'", "inst✝¹⁵ : IsLocalizedModule S✝ f₀", "M₁ : Type ?u.1246805", "M₁' : Type ?u.1246808", "inst✝¹⁴ : AddCommGroup M₁", "inst✝¹³ : AddCommGroup M₁'", "inst✝¹² : Module R✝ M₁", "inst✝¹¹ : Module R✝ M₁'", "f₁ : M₁ →ₗ[R✝] M₁'", "inst✝¹⁰ : IsLocalizedModule S✝ f₁", "M₂ : Type ?u.1248876", "M₂' : Type ?u.1248879", "inst✝⁹ : AddCommGroup M₂", "inst✝⁸ : AddCommGroup M₂'", "inst✝⁷ : Module R✝ M₂", "inst✝⁶ : Module R✝ M₂'", "f₂ : M₂ →ₗ[R✝] M₂'", "inst✝⁵ : IsLocalizedModule S✝ f₂", "R : Type u_6", "S : Type u_7", "S' : Type u_8", "inst✝⁴ : CommRing R", "inst✝³ : CommRing S", "inst✝² : CommRing S'", "inst✝¹ : Algebra R S", "inst✝ : Algebra R S'", "M : Submonoid R", "f : S →ₐ[R] S'", "h₁ : ∀ x ∈ M, IsUnit ((algebraMap R S') x)", "h₂ : ∀ (y : S'), ∃ x, x.2 • y = f x.1", "h₃ : ∀ (x : S), f x = 0 ↔ ∃ m, m • x = 0"], "goal": "IsLocalizedModule M f.toLinearMap"}, "premise": [119625, 120533, 121056, 121165], "module": ["Mathlib/Algebra/Module/LocalizedModule.lean"]}
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+{"state": {"context": ["C : Type u_1", "ι✝ : Type u_2", "inst✝² : Category.{u_3, u_1} C", "inst✝¹ : Preadditive C", "c : ComplexShape ι✝", "K : HomologicalComplex C c", "i j : ι✝", "hij : c.Rel i j", "inst✝ : CategoryWithHomology C", "S : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.opcyclesToCycles i j) ⋯", "S' : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.fromOpcycles i j) ⋯", "ι : S ⟶ S' := { τ₁ := 𝟙 S.X₁, τ₂ := 𝟙 S.X₂, τ₃ := K.iCycles j, comm₁₂ := ⋯, comm₂₃ := ⋯ }", "hS : S.Exact"], "goal": "(composableArrows₃ K i j).Exact"}, "premise": [113350, 114550, 114588], "module": ["Mathlib/Algebra/Homology/HomologySequence.lean"]}
+{"state": {"context": ["C : Type u_1", "ι✝ : Type u_2", "inst✝² : Category.{u_3, u_1} C", "inst✝¹ : Preadditive C", "c : ComplexShape ι✝", "K : HomologicalComplex C c", "i j : ι✝", "hij : c.Rel i j", "inst✝ : CategoryWithHomology C", "S : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.opcyclesToCycles i j) ⋯", "S' : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.fromOpcycles i j) ⋯", "ι : S ⟶ S' := { τ₁ := 𝟙 S.X₁, τ₂ := 𝟙 S.X₂, τ₃ := K.iCycles j, comm₁₂ := ⋯, comm₂₃ := ⋯ }", "hS : S.Exact", "T : ShortComplex C := ShortComplex.mk (K.opcyclesToCycles i j) (K.homologyπ j) ⋯", "T' : ShortComplex C := ShortComplex.mk (K.toCycles i j) (K.homologyπ j) ⋯", "π : T' ⟶ T := { τ₁ := K.pOpcycles i, τ₂ := 𝟙 T'.X₂, τ₃ := 𝟙 T'.X₃, comm₁₂ := ⋯, comm₂₃ := ⋯ }"], "goal": "(composableArrows₃ K i j).Exact"}, "premise": [113351, 114550, 114589, 115189], "module": ["Mathlib/Algebra/Homology/HomologySequence.lean"]}
+{"state": {"context": ["C : Type u_1", "ι✝ : Type u_2", "inst✝² : Category.{u_3, u_1} C", "inst✝¹ : Preadditive C", "c : ComplexShape ι✝", "K : HomologicalComplex C c", "i j : ι✝", "hij : c.Rel i j", "inst✝ : CategoryWithHomology C", "S : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.opcyclesToCycles i j) ⋯", "S' : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.fromOpcycles i j) ⋯", "ι : S ⟶ S' := { τ₁ := 𝟙 S.X₁, τ₂ := 𝟙 S.X₂, τ₃ := K.iCycles j, comm₁₂ := ⋯, comm₂₃ := ⋯ }", "hS : S.Exact", "T : ShortComplex C := ShortComplex.mk (K.opcyclesToCycles i j) (K.homologyπ j) ⋯", "T' : ShortComplex C := ShortComplex.mk (K.toCycles i j) (K.homologyπ j) ⋯", "π : T' ⟶ T := { τ₁ := K.pOpcycles i, τ₂ := 𝟙 T'.X₂, τ₃ := 𝟙 T'.X₃, comm₁₂ := ⋯, comm₂₃ := ⋯ }", "hT : T.Exact"], "goal": "(composableArrows₃ K i j).Exact"}, "premise": [115189, 114589, 114550, 113351], "module": ["Mathlib/Algebra/Homology/HomologySequence.lean"]}
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+{"state": {"context": ["C : Type u_1", "ι✝ : Type u_2", "inst✝² : Category.{u_3, u_1} C", "inst✝¹ : Preadditive C", "c : ComplexShape ι✝", "K : HomologicalComplex C c", "i j : ι✝", "hij : c.Rel i j", "inst✝ : CategoryWithHomology C", "S : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.opcyclesToCycles i j) ⋯", "S' : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.fromOpcycles i j) ⋯", "ι : S ⟶ S' := { τ₁ := 𝟙 S.X₁, τ₂ := 𝟙 S.X₂, τ₃ := K.iCycles j, comm₁₂ := ⋯, comm₂₃ := ⋯ }", "hS : S.Exact", "T : ShortComplex C := ShortComplex.mk (K.opcyclesToCycles i j) (K.homologyπ j) ⋯", "T' : ShortComplex C := ShortComplex.mk (K.toCycles i j) (K.homologyπ j) ⋯", "π : T' ⟶ T := { τ₁ := K.pOpcycles i, τ₂ := 𝟙 T'.X₂, τ₃ := 𝟙 T'.X₃, comm₁₂ := ⋯, comm₂₃ := ⋯ }", "hT : T.Exact"], "goal": "(composableArrows₃ K i j).δ₀.Exact"}, "premise": [115189], "module": ["Mathlib/Algebra/Homology/HomologySequence.lean"]}
+{"state": {"context": ["V : Type u", "V' : Type v", "G : SimpleGraph V", "G' : SimpleGraph V'", "u v w : V", "p : G.Walk u v"], "goal": "(p.toSubgraph.neighborSet w).Finite"}, "premise": [52795, 52845, 52852, 134992, 135040, 135041], "module": ["Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean"]}
+{"state": {"context": ["α : Type", "d : ℕ", "ds ds₁ ds₂ ds₃ : List ℕ", "inst✝ : AddCommMonoid α", "β : Type", "i : ℕ", "hid : i < d", "s : Finset β", "f : β → Holor α (d :: ds)", "this : DecidableEq β := Classical.decEq β"], "goal": "∑ x ∈ s, (f x).slice i hid = (∑ x ∈ s, f x).slice i hid"}, "premise": [138847], "module": ["Mathlib/Data/Holor.lean"]}
+{"state": {"context": ["α : Type", "d : ℕ", "ds ds₁ ds₂ ds₃ : List ℕ", "inst✝ : AddCommMonoid α", "β : Type", "i : ℕ", "hid : i < d", "s : Finset β", "f : β → Holor α (d :: ds)", "this : DecidableEq β := Classical.decEq β"], "goal": "∑ x ∈ ∅, (f x).slice i hid = (∑ x ∈ ∅, f x).slice i hid"}, "premise": [138847], "module": ["Mathlib/Data/Holor.lean"]}
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+{"state": {"context": ["C : Type u_1", "D : Type u_2", "E : Type u_3", "J : Type u_4", "inst✝² : Category.{u_5, u_1} C", "inst✝¹ : Category.{u_7, u_2} D", "inst✝ : Category.{u_6, u_3} E", "X Y✝ : GradedObject J C", "e : X ≅ Y✝", "F : C ⥤ D ⥤ E", "j : J", "Y : D"], "goal": "(F.map (e.hom j)).app Y ≫ (F.map (e.inv j)).app Y = 𝟙 ((F.obj (X j)).obj Y)"}, "premise": [97575, 99919, 99920, 100030, 100031], "module": ["Mathlib/CategoryTheory/GradedObject.lean"]}
+{"state": {"context": ["R : Type uR", "S : Type uS", "ι : Type uι", "n✝ : ℕ", "M : Fin n✝.succ → Type v", "M₁ : ι → Type v₁", "M₂ : Type v₂", "M₃ : Type v₃", "M' : Type v'", "inst✝⁶ : CommSemiring R", "inst✝⁵ : (i : Fin n✝.succ) → AddCommMonoid (M i)", "inst✝⁴ : AddCommMonoid M'", "inst✝³ : AddCommMonoid M₂", "inst✝² : (i : Fin n✝.succ) → Module R (M i)", "inst✝¹ : Module R M'", "inst✝ : Module R M₂", "ι' : Type u_1", "k l n : ℕ", "s : Finset (Fin n)", "hk : s.card = k", "hl : sᶜ.card = l", "f : MultilinearMap R (fun x => M') M₂", "x y : M'"], "goal": "((((curryFinFinset R M₂ M' hk hl) f) fun x_1 => x) fun x => y) = f (s.piecewise (fun x_1 => x) fun x => y)"}, "premise": [2100, 2101, 86202, 110524], "module": ["Mathlib/LinearAlgebra/Multilinear/Basic.lean"]}
+{"state": {"context": ["R : Type uR", "S : Type uS", "ι : Type uι", "n✝ : ℕ", "M : Fin n✝.succ → Type v", "M₁ : ι → Type v₁", "M₂ : Type v₂", "M₃ : Type v₃", "M' : Type v'", "inst✝⁶ : CommSemiring R", "inst✝⁵ : (i : Fin n✝.succ) → AddCommMonoid (M i)", "inst✝⁴ : AddCommMonoid M'", "inst✝³ : AddCommMonoid M₂", "inst✝² : (i : Fin n✝.succ) → Module R (M i)", "inst✝¹ : Module R M'", "inst✝ : Module R M₂", "ι' : Type u_1", "k l n : ℕ", "s : Finset (Fin n)", "hk : s.card = k", "hl : sᶜ.card = l", "f : MultilinearMap R (fun x => M') M₂", "x y : M'"], "goal": "((curryFinFinset R M₂ M' hk hl).symm ((curryFinFinset R M₂ M' hk hl) f)) (s.piecewise (fun x_1 => x) fun x => y) = f (s.piecewise (fun x_1 => x) fun x => y)"}, "premise": [86202, 2100, 2101, 110524], "module": ["Mathlib/LinearAlgebra/Multilinear/Basic.lean"]}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "inst✝¹ : Semigroup R", "inst✝ : StarMul R", "x : R", "hx : IsSelfAdjoint x", "z : R"], "goal": "IsSelfAdjoint (z * x * star z)"}, "premise": [110975, 110982, 111581, 111582, 119703], "module": ["Mathlib/Algebra/Star/SelfAdjoint.lean"]}
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+{"state": {"context": ["α : Type u_1", "ι : Type u_2", "π : ι → Set (Set α)", "hpi : ∀ (x : ι), IsPiSystem (π x)", "S : Set ι", "t1 : Set α", "p1 : Finset ι", "hp1S : ↑p1 ⊆ S", "f1 : ι → Set α", "hf1m : ∀ x ∈ p1, f1 x ∈ π x", "ht1_eq : t1 = ⋂ x ∈ p1, f1 x", "t2 : Set α", "p2 : Finset ι", "hp2S : ↑p2 ⊆ S", "f2 : ι → Set α", "hf2m : ∀ x ∈ p2, f2 x ∈ π x", "ht2_eq : t2 = ⋂ x ∈ p2, f2 x", "h_nonempty : (t1 ∩ t2).Nonempty"], "goal": "∃ t, ∃ (_ : ↑t ⊆ S), ∃ f, ∃ (_ : ∀ x ∈ t, f x ∈ π x), t1 ∩ t2 = ⋂ x ∈ t, f x"}, "premise": [131585], "module": ["Mathlib/MeasureTheory/PiSystem.lean"]}
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+{"state": {"context": ["K : Type u", "V : Type v", "inst✝³ : Field K", "inst✝² : AddCommGroup V", "inst✝¹ : Module K V", "W✝ : Subspace K V", "inst✝ : FiniteDimensional K V", "W : Subspace K V"], "goal": "(dualAnnihilator W).dualAnnihilator = (Module.mapEvalEquiv K V) W"}, "premise": [88113, 88071], "module": ["Mathlib/LinearAlgebra/Dual.lean"]}
+{"state": {"context": ["K : Type u", "V : Type v", "inst✝³ : Field K", "inst✝² : AddCommGroup V", "inst✝¹ : Module K V", "W✝ : Subspace K V", "inst✝ : FiniteDimensional K V", "W : Subspace K V", "this : (dualAnnihilator W).dualCoannihilator = W"], "goal": "(dualAnnihilator W).dualAnnihilator = (Module.mapEvalEquiv K V) W"}, "premise": [88113, 88071], "module": ["Mathlib/LinearAlgebra/Dual.lean"]}
+{"state": {"context": ["K : Type u", "V : Type v", "inst✝³ : Field K", "inst��² : AddCommGroup V", "inst✝¹ : Module K V", "W✝ : Subspace K V", "inst✝ : FiniteDimensional K V", "W : Subspace K V", "this : (Module.mapEvalEquiv K V).symm (dualAnnihilator W).dualAnnihilator = W"], "goal": "(dualAnnihilator W).dualAnnihilator = (Module.mapEvalEquiv K V) W"}, "premise": [11079, 88071], "module": ["Mathlib/LinearAlgebra/Dual.lean"]}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "a b : R", "n : ℕ", "inst✝¹ : Semiring R", "inst✝ : NoZeroDivisors R", "p✝ q✝ p q : R[X]", "h₁ : p ∣ q", "h₂ : q.degree < p.degree"], "goal": "q = 0"}, "premise": [1094, 1673, 14322, 103441], "module": ["Mathlib/Algebra/Polynomial/RingDivision.lean"]}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "a b : R", "n : ℕ", "inst✝¹ : Semiring R", "inst✝ : NoZeroDivisors R", "p✝ q✝ p q : R[X]", "h₁ : p ∣ q", "h₂ : q.degree < p.degree", "hc : ¬q = 0"], "goal": "False"}, "premise": [1673, 14322, 103441, 1094], "module": ["Mathlib/Algebra/Polynomial/RingDivision.lean"]}
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+{"state": {"context": ["K : Type u_1", "inst✝¹ : LinearOrderedField K", "inst✝ : FloorRing K", "v : K", "q : ℚ", "v_eq_q : v = ↑q", "n : ℕ"], "goal": "↑(match (IntFractPair.of q, Stream'.Seq.tail ⟨IntFractPair.stream q, ⋯⟩) with | (h, s) => { h := ↑h.b, s := Stream'.Seq.map (fun p => { a := 1, b := ↑p.b }) s }).h = (match (IntFractPair.mapFr Rat.cast (IntFractPair.of q), Stream'.Seq.tail ⟨IntFractPair.stream v, ⋯⟩) with | (h, s) => { h := ↑h.b, s := Stream'.Seq.map (fun p => { a := 1, b := ↑p.b }) s }).h"}, "premise": [116193], "module": ["Mathlib/Algebra/ContinuedFractions/Computation/TerminatesIffRat.lean"]}
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+{"state": {"context": ["C : Type u", "inst✝⁴ : Category.{v, u} C", "J : GrothendieckTopology C", "A : Type u'", "inst✝³ : Category.{v', u'} A", "inst✝² : ConcreteCategory A", "F₁ F₂ F₃ : Cᵒᵖ ⥤ A", "f₁ : F₁ ⟶ F₂", "f₂ : F₂ ⟶ F₃", "inst✝¹ : IsLocallyInjective J (f₁ ≫ f₂)", "inst✝ : IsLocallySurjective J f₁", "X : Cᵒᵖ", "x₁ x₂ : (forget A).obj (F₂.obj X)", "h : (f₂.app X) x₁ = (f₂.app X) x₂", "S : Sieve (unop X) := imageSieve f₁ x₁ ⊓ imageSieve f₁ x₂", "hS : S ∈ J.sieves (unop X)"], "goal": "equalizerSieve x₁ x₂ ∈ J.sieves (unop X)"}, "premise": [90673, 2106, 2107, 89783], "module": ["Mathlib/CategoryTheory/Sites/LocallySurjective.lean"]}
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+{"state": {"context": ["C : Type u", "inst✝⁴ : Category.{v, u} C", "J : GrothendieckTopology C", "A : Type u'", "inst✝³ : Category.{v', u'} A", "inst✝² : ConcreteCategory A", "F₁ F₂ F₃ : Cᵒᵖ ⥤ A", "f₁ : F₁ ⟶ F₂", "f₂ : F₂ ⟶ F₃", "inst✝¹ : IsLocallyInjective J (f₁ ≫ f₂)", "inst✝ : IsLocallySurjective J f₁", "X : Cᵒᵖ", "x₁ x₂ : (forget A).obj (F₂.obj X)", "h : (f₂.app X) x₁ = (f₂.app X) x₂", "S : Sieve (unop X) := imageSieve f₁ x₁ ⊓ imageSieve f₁ x₂", "hS : S ∈ J.sieves (unop X)", "T : ⦃Y : C⦄ → (f : Y ⟶ unop X) → S.arrows f → Sieve Y := fun Y f hf => equalizerSieve (localPreimage f₁ x₁ f ⋯) (localPreimage f₁ x₂ f ⋯)"], "goal": "∀ ⦃Y : C⦄ ⦃f : Y ⟶ unop X⦄, S.arrows f → Sieve.pullback f (Sieve.bind S.arrows T) ∈ J.sieves Y"}, "premise": [89780, 89782], "module": ["Mathlib/CategoryTheory/Sites/LocallySurjective.lean"]}
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+{"state": {"context": [], "goal": "Convex ℝ (log ∘ Gamma '' Ioi 0)"}, "premise": [38145, 19753, 37865, 39595, 35981, 133391, 39891, 149301, 34998, 38137, 38141], "module": ["Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean"]}
+{"state": {"context": [], "goal": "IsPreconnected (log ∘ Gamma '' Ioi 0)"}, "premise": [65874, 35981, 57310, 56501], "module": ["Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean"]}
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+{"state": {"context": ["x : ℝ", "hx : x ∈ Ioi 0", "m : ℕ"], "goal": "x ≠ -↑m"}, "premise": [11234, 20164, 46377, 1673, 1674, 105642, 142636, 37934, 39891, 39896], "module": ["Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean"]}
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+{"state": {"context": ["C : Type u_1", "inst✝¹ : Category.{u_2, u_1} C", "X : SimplicialObject C", "s : Splitting X", "inst✝ : Preadditive C", "Δ : SimplexCategoryᵒᵖ", "A : IndexSet Δ"], "goal": "∀ b ∈ Finset.univ, b ≠ A → (s.cofan Δ).inj A ≫ s.πSummand b ≫ (s.cofan Δ).inj b = 0"}, "premise": [91609, 96174, 127034], "module": ["Mathlib/AlgebraicTopology/DoldKan/SplitSimplicialObject.lean"]}
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+{"state": {"context": ["α : Type u_1", "E : α → Type u_2", "p q : ℝ≥0∞", "inst✝⁴ : (i : α) → NormedAddCommGroup (E i)", "𝕜 : Type u_3", "inst✝³ : NormedRing 𝕜", "inst✝² : (i : α) → Module 𝕜 (E i)", "inst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)", "inst✝ : DecidableEq α", "hp : 0 < p.toReal", "f : (i : α) → E i", "s : Finset α"], "goal": "HasSum (fun i => ‖↑(∑ i ∈ s, lp.single p i (f i)) i‖ ^ p.toReal) (∑ i ∈ s, ‖f i‖ ^ p.toReal)"}, "premise": [45188, 45195, 45230, 123837, 127093, 63037], "module": ["Mathlib/Analysis/Normed/Lp/lpSpace.lean"]}
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+{"state": {"context": ["α : Type u_1", "E : α → Type u_2", "p q : ℝ≥0∞", "inst✝⁴ : (i : α) → NormedAddCommGroup (E i)", "𝕜 : Type u_3", "inst✝³ : NormedRing 𝕜", "inst✝² : (i : α) → Module 𝕜 (E i)", "inst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)", "inst✝ : DecidableEq α", "hp : 0 < p.toReal", "f : (i : α) → E i", "s : Finset α", "h : ∀ i ∉ s, ‖if i ∈ s then f i else 0‖ ^ p.toReal = 0"], "goal": "HasSum (fun i => ‖if i ∈ s then f i else 0‖ ^ p.toReal) (∑ i ∈ s, ‖f i‖ ^ p.toReal)"}, "premise": [1737, 1738, 40011, 11234], "module": ["Mathlib/Analysis/Normed/Lp/lpSpace.lean"]}
+{"state": {"context": ["α : Type u_1", "E : α → Type u_2", "p q : ℝ≥0∞", "inst✝⁴ : (i : α) → NormedAddCommGroup (E i)", "𝕜 : Type u_3", "inst✝³ : NormedRing 𝕜", "inst✝² : (i : α) → Module 𝕜 (E i)", "inst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)", "inst✝ : DecidableEq α", "hp : 0 < p.toReal", "f : (i : α) → E i", "s : Finset α", "h : ∀ i ∉ s, ‖if i ∈ s then f i else 0‖ ^ p.toReal = 0", "h' : ∀ i ∈ s, ‖f i‖ ^ p.toReal = ‖if i ∈ s then f i else 0‖ ^ p.toReal"], "goal": "HasSum (fun i => ‖if i ∈ s then f i else 0‖ ^ p.toReal) (∑ i ∈ s, ‖f i‖ ^ p.toReal)"}, "premise": [126920, 1737, 63030], "module": ["Mathlib/Analysis/Normed/Lp/lpSpace.lean"]}
+{"state": {"context": ["α : Type u_1", "inst✝² : Zero α", "inst✝¹ : Preorder α", "inst✝ : DecidableRel fun x x_1 => x < x_1", "a : α", "ha : 0 < a"], "goal": "sign a = 1"}, "premise": [1737, 146315], "module": ["Mathlib/Data/Sign.lean"]}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : CommGroup α", "inst✝ : DecidableEq α", "A B✝ C : Finset α", "hA : A.Nonempty", "B : Finset α", "m n : ℕ"], "goal": "↑(B ^ m / B ^ n).card ≤ (↑(A / B).card / ↑A.card) ^ (m + n) * ↑A.card"}, "premise": [117845, 117881, 119790, 141598, 53240], "module": ["Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean"]}
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+{"state": {"context": ["K : Type u_1", "v : K", "n : ℕ", "inst✝¹ : LinearOrderedField K", "inst✝ : FloorRing K", "b : K", "nth_partDenom_eq : (of v).partDens.get? n = some b"], "goal": "b * (of v).dens n ≤ (of v).dens (n + 1)"}, "premise": [116137, 116139, 118585], "module": ["Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean"]}
+{"state": {"context": ["K : Type u_1", "v : K", "n : ℕ", "inst✝¹ : LinearOrderedField K", "inst✝ : FloorRing K", "b : K", "nth_partDenom_eq : (of v).partDens.get? n = some b"], "goal": "b * ((of v).contsAux (n + 1)).b ≤ (of v).dens (n + 1)"}, "premise": [116137, 116139, 118585], "module": ["Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean"]}
+{"state": {"context": ["R : Type u_1", "inst✝² : CommMonoid R", "R' : Type u_2", "inst✝¹ : CommRing R'", "R'' : Type u_3", "inst✝ : CommRing R''", "f : R' →+* R''", "hf : Function.Injective ⇑f"], "goal": "Function.Injective fun x => x.ringHomComp f"}, "premise": [23020, 23024, 117118, 117125], "module": ["Mathlib/NumberTheory/MulChar/Basic.lean"]}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝¹ : LinearOrderedRing α", "inst✝ : FloorRing α", "z : ℤ", "a : α", "ha : fract a ≠ 0"], "goal": "↑⌈a⌉ - a = 1 - fract a"}, "premise": [1673, 1990, 105210], "module": ["Mathlib/Algebra/Order/Floor.lean"]}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝¹ : LinearOrderedRing α", "inst✝ : FloorRing α", "z : ℤ", "a : α", "ha : fract a ≠ 0"], "goal": "↑⌈a⌉ - a = 1 - (a + 1 - ↑⌈a⌉)"}, "premise": [1673, 1990, 105210], "module": ["Mathlib/Algebra/Order/Floor.lean"]}
+{"state": {"context": ["K : Type u_1", "L : Type u_2", "L' : Type u_3", "inst✝⁴ : Field K", "inst✝³ : Field L", "inst✝² : Field L'", "inst✝¹ : Algebra K L", "inst✝ : Algebra K L'", "S✝ S S' : IntermediateField K L", "h : S.toSubalgebra = S'.toSubalgebra", "x✝ : L"], "goal": "x✝ ∈ S ↔ x✝ ∈ S'"}, "premise": [1713, 88419], "module": ["Mathlib/FieldTheory/IntermediateField.lean"]}
+{"state": {"context": ["ι : Type u_1", "κ : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "s✝ s₁ s₂ : Finset α", "a : α", "f✝ g : α → β", "f : α → Multiset β", "s : Finset α", "b : β"], "goal": "b ∈ ∑ x ∈ s, f x ↔ ∃ a ∈ s, b ∈ f a"}, "premise": [1995, 2027, 126901, 138847], "module": ["Mathlib/Algebra/BigOperators/Group/Finset.lean"]}
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+{"state": {"context": ["m n : ℕ+"], "goal": "n.gcd m = 1 → n.gcd m = 1"}, "premise": [141185], "module": ["Mathlib/Data/PNat/Prime.lean"]}
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+{"state": {"context": ["𝕜 : Type u_1", "inst✝ : LinearOrderedField 𝕜", "s : Set 𝕜", "f : 𝕜 → 𝕜", "hf : ConvexOn 𝕜 s f", "a x y : 𝕜", "ha : a ∈ s", "hx : x ∈ s", "hy : y ∈ s", "hxa : x ≠ a", "hya : y ≠ a", "hxy✝ : x ≤ y", "hxy : x < y"], "goal": "(f x - f a) / (x - a) ≤ (f y - f a) / (y - a)"}, "premise": [11248, 14320], "module": ["Mathlib/Analysis/Convex/Slope.lean"]}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝ : LinearOrderedField 𝕜", "s : Set 𝕜", "f : 𝕜 → 𝕜", "hf : ConvexOn 𝕜 s f", "a x y : 𝕜", "ha : a ∈ s", "hx : x ∈ s", "hy : y ∈ s", "hxa✝ : x ≠ a", "hya : y ≠ a", "hxy✝ : x ≤ y", "hxy : x < y", "hxa : x < a"], "goal": "(f x - f a) / (x - a) ≤ (f y - f a) / (y - a)"}, "premise": [14320], "module": ["Mathlib/Analysis/Convex/Slope.lean"]}
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+{"state": {"context": ["M : Type u_1", "inst✝⁵ : CancelCommMonoidWithZero M", "N : Type u_2", "inst✝⁴ : CancelCommMonoidWithZero N", "inst✝³ : UniqueFactorizationMonoid N", "inst✝² : UniqueFactorizationMonoid M", "inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1", "inst✝ : DecidableRel fun x x_1 => x ∣ x_1", "m p : Associates M", "n : Associates N", "hn : n ≠ 0", "hp : p ∈ normalizedFactors m", "d : ↑(Set.Iic m) ≃o ↑(Set.Iic n)"], "goal": "multiplicity (↑(d ⟨p, ⋯⟩)) n ≤ multiplicity (↑(d.symm (d ⟨p, ⋯⟩))) m"}, "premise": [11076, 76126, 137134], "module": ["Mathlib/RingTheory/ChainOfDivisors.lean"]}
+{"state": {"context": ["M : Type u_1", "inst✝⁵ : CancelCommMonoidWithZero M", "N : Type u_2", "inst✝⁴ : CancelCommMonoidWithZero N", "inst✝³ : UniqueFactorizationMonoid N", "inst✝² : UniqueFactorizationMonoid M", "inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1", "inst✝ : DecidableRel fun x x_1 => x ∣ x_1", "m p : Associates M", "n : Associates N", "hn : n ≠ 0", "hp : p ∈ normalizedFactors m", "d : ↑(Set.Iic m) ≃o ↑(Set.Iic n)", "this : DecidableEq (Associates N) := Classical.decEq (Associates N)"], "goal": "multiplicity (↑(d ⟨p, ⋯⟩)) n ≤ multiplicity (↑(d.symm (d ⟨p, ⋯⟩))) m"}, "premise": [77031, 77032, 137133], "module": ["Mathlib/RingTheory/ChainOfDivisors.lean"]}
+{"state": {"context": ["S : Set ℝ", "f : ℝ → ℝ", "x y f' : ℝ", "hfc : ConcaveOn ℝ S f", "hx : x ∈ S", "hy : y ∈ S", "hxy : x < y", "hf' : HasDerivWithinAt f f' (Iio y) y"], "goal": "f' ≤ slope f x y"}, "premise": [35087, 44476, 82753, 104741, 119769, 120673], "module": ["Mathlib/Analysis/Convex/Deriv.lean"]}
+{"state": {"context": ["R : Type u", "S✝ : Type v", "T : Type w", "ι : Type y", "a b : R", "m n : ℕ", "inst✝¹ : Semiring R", "p q r✝ : R[X]", "x : R", "S : Type u_1", "inst✝ : Semiring S", "f : R →+* S", "r : R"], "goal": "eval₂ f (f r) p = f (eval r p)"}, "premise": [102824, 102860, 126889, 121567, 121592], "module": ["Mathlib/Algebra/Polynomial/Eval.lean"]}
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+{"state": {"context": ["α : Type u_1", "inst✝¹ : LinearOrder α", "E : Type u_2", "inst✝ : PseudoEMetricSpace E", "f : ℝ → E", "s : Set ℝ", "l : ℝ≥0"], "goal": "(∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → eVariationOn f (s ∩ Icc x y) = 0) ↔ eVariationOn f s = 0"}, "premise": [42108, 108557, 143177], "module": ["Mathlib/Analysis/ConstantSpeed.lean"]}
+{"state": {"context": ["α β : Type u", "c : Cardinal.{u_1}", "h : ∀ (n : ℕ), ↑n ≤ c", "hn : c < ℵ₀"], "goal": "False"}, "premise": [1673, 48770, 1673, 3738, 48752], "module": ["Mathlib/SetTheory/Cardinal/Basic.lean"]}
+{"state": {"context": ["α β : Type u", "n : ℕ", "h : ∀ (n_1 : ℕ), ↑n_1 ≤ ↑n", "hn : ↑n < ℵ₀"], "goal": "False"}, "premise": [48752, 1673, 48770, 3738], "module": ["Mathlib/SetTheory/Cardinal/Basic.lean"]}
+{"state": {"context": ["α : Type u_1", "M : Matroid α", "E B I X R J : Set α", "hIE : I ⊆ E"], "goal": "(uniqueBaseOn I E).Base B ↔ B = I"}, "premise": [1713, 133456, 137592, 139363], "module": ["Mathlib/Data/Matroid/Constructions.lean"]}
+{"state": {"context": ["M : Type u_1", "inst✝² : CommMonoid M", "S : Submonoid M", "N : Type u_2", "inst✝¹ : CommMonoid N", "P : Type u_3", "inst✝ : CommMonoid P", "f : S.LocalizationMap N", "g : M →* P", "hg : ∀ (y : ↥S), IsUnit (g ↑y)", "x✝ : M"], "goal": "((f.lift hg).comp f.toMap) x✝ = g x✝"}, "premise": [9220], "module": ["Mathlib/GroupTheory/MonoidLocalization/Basic.lean"]}
+{"state": {"context": [], "goal": "3.141592 < π"}, "premise": [146588, 146589], "module": ["Mathlib/Data/Real/Pi/Bounds.lean"]}
+{"state": {"context": ["R : Type u_1", "inst✝¹² : CommRing R", "M✝ : Type u_2", "inst✝¹¹ : AddCommGroup M✝", "inst✝¹⁰ : Module R M✝", "M' : Type u_3", "inst✝⁹ : AddCommGroup M'", "inst✝⁸ : Module R M'", "ι : Type u_4", "inst✝⁷ : DecidableEq ι", "inst✝⁶ : Fintype ι", "e : Basis ι R M✝", "A : Type u_5", "inst✝⁵ : CommRing A", "inst✝⁴ : Module A M✝", "κ : Type u_6", "inst✝³ : Fintype κ", "𝕜 : Type u_7", "inst✝² : Field 𝕜", "M : Type u_8", "inst✝¹ : AddCommGroup M", "inst✝ : Module 𝕜 M"], "goal": "LinearMap.det 0 = 0 ^ FiniteDimensional.finrank 𝕜 M"}, "premise": [83451, 115738, 117161, 119730], "module": ["Mathlib/LinearAlgebra/Determinant.lean"]}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝⁴ : RCLike 𝕜", "inst✝³ : NormedAddCommGroup E", "inst✝² : InnerProductSpace 𝕜 E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : InnerProductSpace 𝕜 F", "K : Submodule 𝕜 E"], "goal": "Kᗮ = ⨅ v, LinearMap.ker ((innerSL 𝕜) ↑v)"}, "premise": [14296], "module": ["Mathlib/Analysis/InnerProductSpace/Orthogonal.lean"]}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝⁴ : RCLike 𝕜", "inst✝³ : NormedAddCommGroup E", "inst✝² : InnerProductSpace 𝕜 E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : InnerProductSpace 𝕜 F", "K : Submodule 𝕜 E"], "goal": "Kᗮ ≤ ⨅ v, LinearMap.ker ((innerSL 𝕜) ↑v)"}, "premise": [14296], "module": ["Mathlib/Analysis/InnerProductSpace/Orthogonal.lean"]}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝⁴ : RCLike 𝕜", "inst✝³ : NormedAddCommGroup E", "inst✝² : InnerProductSpace 𝕜 E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : InnerProductSpace 𝕜 F", "K : Submodule 𝕜 E"], "goal": "⨅ v, LinearMap.ker ((innerSL 𝕜) ↑v) ≤ Kᗮ"}, "premise": [14296], "module": ["Mathlib/Analysis/InnerProductSpace/Orthogonal.lean"]}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "f g : α → β", "c c₁ c₂ x✝ : α", "inst✝¹ : AddSemigroup α", "inst✝ : Neg β", "h : Antiperiodic f c", "a x : α"], "goal": "(fun x => f (a + x)) (x + c) = -(fun x => f (a + x)) x"}, "premise": [119704], "module": ["Mathlib/Algebra/Periodic.lean"]}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "o : Part α", "inst✝ : Decidable o.Dom", "a : α"], "goal": "o.toOption = Option.some a ↔ a ∈ o"}, "premise": [1713, 3120, 131082], "module": ["Mathlib/Data/Part.lean"]}
+{"state": {"context": ["n : ℕ", "v : Vector ℕ n.succ"], "goal": "v.get 0 = v.head"}, "premise": [134773], "module": ["Mathlib/Computability/Primrec.lean"]}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "κ : Kernel α (β × ℝ)", "ν : Kernel α β", "f : α × β → ℚ → ℝ", "inst✝ : IsFiniteKernel κ", "hf : IsRatCondKernelCDF f κ ν", "a : α", "x : ℝ"], "goal": "∫⁻ (b : β), ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a = (κ a) (univ ×ˢ Iic x)"}, "premise": [27956, 30288, 73864], "module": ["Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean"]}
+{"state": {"context": ["R : Type u_1", "A✝ : Type u_2", "B : Type u_3", "A : Type u_4", "inst✝⁴ : CommSemiring R", "inst✝³ : Semiring A", "inst✝² : Semiring B", "inst✝¹ : Algebra R A", "inst✝ : Algebra R B", "f g : R[ε] →ₐ[R] A", "hε : f ε = g ε"], "goal": "f (1 • ε) = g (1 • ε)"}, "premise": [118910], "module": ["Mathlib/Algebra/DualNumber.lean"]}
+{"state": {"context": ["α : Type u_1", "inst✝ : LinearOrderedAddCommGroup α", "hα : Archimedean α", "p : α", "hp : 0 < p", "a✝ b✝ c : α", "n : ℤ", "a b : α"], "goal": "toIcoDiv hp a (-b) = -(toIocDiv hp (-a) b + 1)"}, "premise": [105865, 105878, 117878, 119789, 1717, 117795], "module": ["Mathlib/Algebra/Order/ToIntervalMod.lean"]}
+{"state": {"context": ["α : Type u_1", "inst✝ : LinearOrderedAddCommGroup α", "hα : Archimedean α", "p : α", "hp : 0 < p", "a✝ b✝ c : α", "n : ℤ", "a b : α"], "goal": "toIcoDiv hp a (-b) = -toIocDiv hp (-(a + p)) b"}, "premise": [117795, 105865, 119789, 1717, 117878, 105878], "module": ["Mathlib/Algebra/Order/ToIntervalMod.lean"]}
+{"state": {"context": ["α : Type u_1", "inst✝ : LinearOrderedAddCommGroup α", "hα : Archimedean α", "p : α", "hp : 0 < p", "a✝ b✝ c : α", "n : ℤ", "a b : α"], "goal": "toIocDiv hp (-(a + p)) b = -toIcoDiv hp a (-b)"}, "premise": [117795, 1717, 105823], "module": ["Mathlib/Algebra/Order/ToIntervalMod.lean"]}
+{"state": {"context": ["α : Type u_1", "inst✝ : LinearOrderedAddCommGroup α", "hα : Archimedean α", "p : α", "hp : 0 < p", "a✝ b✝ c : α", "n : ℤ", "a b : α"], "goal": "b - -toIcoDiv hp a (-b) • p ∈ Set.Ioc (-(a + p)) (-(a + p) + p)"}, "premise": [105820, 105823], "module": ["Mathlib/Algebra/Order/ToIntervalMod.lean"]}
+{"state": {"context": ["α : Type u_1", "inst✝ : LinearOrderedAddCommGroup α", "hα : Archimedean α", "p : α", "hp : 0 < p", "a✝ b✝ c : α", "n : ℤ", "a b : α", "hc : a ≤ -b - toIcoDiv hp a (-b) • p", "ho : -b - toIcoDiv hp a (-b) • p < a + p"], "goal": "b - -toIcoDiv hp a (-b) • p ∈ Set.Ioc (-(a + p)) (-(a + p) + p)"}, "premise": [105664, 110032, 117880, 119769, 105820], "module": ["Mathlib/Algebra/Order/ToIntervalMod.lean"]}
+{"state": {"context": ["α : Type u_1", "inst✝ : LinearOrderedAddCommGroup α", "hα : Archimedean α", "p : α", "hp : 0 < p", "a✝ b✝ c : α", "n : ℤ", "a b : α", "hc : a ≤ -b - toIcoDiv hp a (-b) • p", "ho : -(a + p) < b - -toIcoDiv hp a (-b) • p"], "goal": "b - -toIcoDiv hp a (-b) • p ∈ Set.Ioc (-(a + p)) (-(a + p) + p)"}, "premise": [105664, 105657, 110032, 117880, 119769], "module": ["Mathlib/Algebra/Order/ToIntervalMod.lean"]}
+{"state": {"context": ["α : Type u_1", "inst✝ : LinearOrderedAddCommGroup α", "hα : Archimedean α", "p : α", "hp : 0 < p", "a✝ b✝ c : α", "n : ℤ", "a b : α", "hc : b - -toIcoDiv hp a (-b) • p ≤ -a", "ho : -(a + p) < b - -toIcoDiv hp a (-b) • p"], "goal": "b - -toIcoDiv hp a (-b) • p ∈ Set.Ioc (-(a + p)) (-(a + p) + p)"}, "premise": [105657, 110032, 117880, 119769], "module": ["Mathlib/Algebra/Order/ToIntervalMod.lean"]}
+{"state": {"context": ["α : Type u_1", "inst✝ : LinearOrderedAddCommGroup α", "hα : Archimedean α", "p : α", "hp : 0 < p", "a✝ b✝ c : α", "n : ℤ", "a b : α", "hc : b - -toIcoDiv hp a (-b) • p ≤ -a", "ho : -(a + p) < b - -toIcoDiv hp a (-b) • p"], "goal": "-a = -(a + p) + p"}, "premise": [117878, 119834], "module": ["Mathlib/Algebra/Order/ToIntervalMod.lean"]}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝ : GeneralizedCoheytingAlgebra α", "a b✝ c d b : α"], "goal": "a \\ ⊥ ≤ b ↔ a ≤ b"}, "premise": [1713, 15060, 18834], "module": ["Mathlib/Order/Heyting/Basic.lean"]}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "inst✝² : PartialOrder α", "inst✝¹ : LocallyFiniteOrder α", "a✝ b✝ c : α", "inst✝ : DecidableEq α", "a b : α"], "goal": "Icc a b \\ {a, b} = Ioo a b"}, "premise": [138674], "module": ["Mathlib/Order/Interval/Finset/Basic.lean"]}
+{"state": {"context": ["ι : Type u_1", "κ : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "s✝ s₁ s₂ : Finset α", "a : α", "f✝ g✝ : α → β", "inst✝¹ : CommMonoid β", "inst✝ : DecidableEq α", "s t : Finset α", "f g : α → β"], "goal": "∏ x ∈ s, t.piecewise f g x = (∏ x ∈ s ∩ t, f x) * ∏ x ∈ s \\ t, g x"}, "premise": [127075, 139122, 139132], "module": ["Mathlib/Algebra/BigOperators/Group/Finset.lean"]}
+{"state": {"context": ["C : Type u", "inst✝ : Category.{v, u} C", "A : Cᵒᵖ ⥤ Type v", "F : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v", "X : C", "G : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v", "η : F ⟶ G", "p : YonedaCollection F X"], "goal": "(map₁ η p).yonedaEquivFst = p.yonedaEquivFst"}, "premise": [96845, 96849], "module": ["Mathlib/CategoryTheory/Comma/Presheaf.lean"]}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "R : Type u_4", "m n✝ n : ℕ"], "goal": "∃ k, n = 2 * k ∨ n = 2 * k + 1"}, "premise": [2027, 121817, 122222], "module": ["Mathlib/Algebra/Ring/Parity.lean"]}
+{"state": {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "inst✝ : HasColimits C", "X Y : PresheafedSpace C", "f : X ⟶ Y", "x y : ↑↑X", "h : x ⤳ y"], "goal": "colimit.desc ((OpenNhds.inclusion (f.base y)).op ⋙ Y.presheaf) { pt := colimit ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf), ι := { app := fun U => colimit.ι ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf) (op { obj := (unop U).obj, property := ⋯ }), naturality := ⋯ } } ≫ colimMap (whiskerLeft (OpenNhds.inclusion (f.base x)).op f.c) ≫ colimMap (whiskerRight (𝟙 (OpenNhds.map f.base x ⋙ OpenNhds.inclusion x).op) X.presheaf) ≫ colimit.pre ((OpenNhds.inclusion x).op ⋙ X.presheaf) (OpenNhds.map f.base x).op = (colimMap (whiskerLeft (OpenNhds.inclusion (f.base y)).op f.c) ≫ colimMap (whiskerRight (𝟙 (OpenNhds.map f.base y ⋙ OpenNhds.inclusion y).op) X.presheaf) ≫ colimit.pre ((OpenNhds.inclusion y).op ⋙ X.presheaf) (OpenNhds.map f.base y).op) ≫ colimit.desc ((OpenNhds.inclusion y).op ⋙ X.presheaf) { pt := colimit ((OpenNhds.inclusion x).op ⋙ X.presheaf), ι := { app := fun U => colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf) (op { obj := (unop U).obj, property := ⋯ }), naturality := ⋯ } }"}, "premise": [93401], "module": ["Mathlib/Geometry/RingedSpace/Stalks.lean"]}
+{"state": {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "inst✝ : HasColimits C", "X Y : PresheafedSpace C", "f : X ⟶ Y", "x y : ↑↑X", "h : x ⤳ y", "j : (OpenNhds (f.base y))ᵒᵖ"], "goal": "colimit.ι (((whiskeringLeft (OpenNhds (f.base y))ᵒᵖ (Opens ↑↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (f.base y)).op).obj Y.presheaf) j ≫ colimit.desc ((OpenNhds.inclusion (f.base y)).op ⋙ Y.presheaf) { pt := colimit ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf), ι := { app := fun U => colimit.ι ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf) (op { obj := (unop U).obj, property := ⋯ }), naturality := ⋯ } } ≫ colimMap (whiskerLeft (OpenNhds.inclusion (f.base x)).op f.c) ≫ colimMap (whiskerRight (𝟙 (OpenNhds.map f.base x ⋙ OpenNhds.inclusion x).op) X.presheaf) ≫ colimit.pre ((OpenNhds.inclusion x).op ⋙ X.presheaf) (OpenNhds.map f.base x).op = colimit.ι (((whiskeringLeft (OpenNhds (f.base y))ᵒᵖ (Opens ↑↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (f.base y)).op).obj Y.presheaf) j ≫ (colimMap (whiskerLeft (OpenNhds.inclusion (f.base y)).op f.c) ≫ colimMap (whiskerRight (𝟙 (OpenNhds.map f.base y ⋙ OpenNhds.inclusion y).op) X.presheaf) ≫ colimit.pre ((OpenNhds.inclusion y).op ⋙ X.presheaf) (OpenNhds.map f.base y).op) ≫ colimit.desc ((OpenNhds.inclusion y).op ⋙ X.presheaf) { pt := colimit ((OpenNhds.inclusion x).op ⋙ X.presheaf), ι := { app := fun U => colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf) (op { obj := (unop U).obj, property := ⋯ }), naturality := ⋯ } }"}, "premise": [93401], "module": ["Mathlib/Geometry/RingedSpace/Stalks.lean"]}
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+{"state": {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "inst✝ : HasColimits C", "X Y : PresheafedSpace C", "f : X ⟶ Y", "x y : ↑↑X", "h : x ⤳ y", "j : OpenNhds (f.base y)"], "goal": "f.c.app ((OpenNhds.inclusion (f.base x)).op.obj (op { obj := j.obj, property := ⋯ })) ≫ X.presheaf.map (𝟙 ((OpenNhds.map f.base x ⋙ OpenNhds.inclusion x).op.obj (op { obj := j.obj, property := ⋯ }))) ≫ colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf) ((OpenNhds.map f.base x).op.obj (op { obj := j.obj, property := ⋯ })) = f.c.app ((OpenNhds.inclusion (f.base y)).op.obj (op j)) ≫ X.presheaf.map (𝟙 ((OpenNhds.map f.base y ⋙ OpenNhds.inclusion y).op.obj (op j))) ≫ colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf) (op { obj := (unop ((OpenNhds.map f.base y).op.obj (op j))).obj, property := ⋯ })"}, "premise": [95617, 93413, 93414, 97705, 97706, 96173, 96175, 93392, 93425, 99920, 100031, 100030, 95551], "module": ["Mathlib/Geometry/RingedSpace/Stalks.lean"]}
+{"state": {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "inst✝ : HasColimits C", "X Y : PresheafedSpace C", "f : X ⟶ Y", "x y : ↑↑X", "h : x ⤳ y", "j : OpenNhds (f.base y)"], "goal": "f.c.app ((OpenNhds.inclusion (f.base x)).op.obj (op { obj := j.obj, property := ⋯ })) ≫ colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf) ((OpenNhds.map f.base x).op.obj (op { obj := j.obj, property := ⋯ })) = f.c.app ((OpenNhds.inclusion (f.base y)).op.obj (op j)) ≫ colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf) ((OpenNhds.map f.base x).op.obj (op { obj := j.obj, property := ⋯ }))"}, "premise": [96175, 99920], "module": ["Mathlib/Geometry/RingedSpace/Stalks.lean"]}
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+{"state": {"context": ["l : Type u_1", "m : Type u_2", "n : Type u_3", "R : Type u_4", "α : Type u_5", "inst✝⁵ : DecidableEq l", "inst✝⁴ : DecidableEq m", "inst✝³ : DecidableEq n", "inst✝² : Semiring α", "inst✝¹ : Finite m", "inst✝ : Finite n", "P : Matrix m n α → Prop", "M : Matrix m n α", "h_zero : P 0", "h_add : ∀ (p q : Matrix m n α), P p → P q → P (p + q)", "h_std_basis : ∀ (i : m) (j : n) (x : α), P (stdBasisMatrix i j x)", "val✝¹ : Fintype m", "val✝ : Fintype n"], "goal": "∀ x ∈ Finset.univ ×ˢ Finset.univ, P (stdBasisMatrix x.1 x.2 (M x.1 x.2))"}, "premise": [127164], "module": ["Mathlib/Data/Matrix/Basis.lean"]}
+{"state": {"context": ["m n : ℕ"], "goal": "(m + n)# = m# * ∏ p ∈ filter Nat.Prime (Ico (m + 1) (m + n + 1)), p"}, "premise": [19112, 20654, 126931, 139120, 2134, 3747, 20613, 103886, 139114, 2134, 3747, 20613, 103886, 139114, 2134, 3747, 20613, 103886, 139114], "module": ["Mathlib/NumberTheory/Primorial.lean"]}
+{"state": {"context": ["m n : ℕ"], "goal": "0 ≤ m + 1"}, "premise": [3747, 20613, 19112, 139114, 20654, 103886, 139120, 126931, 2134], "module": ["Mathlib/NumberTheory/Primorial.lean"]}
+{"state": {"context": ["m n : ℕ"], "goal": "m + 1 ≤ m + n + 1"}, "premise": [3747, 20613, 19112, 139114, 20654, 103886, 139120, 126931, 2134], "module": ["Mathlib/NumberTheory/Primorial.lean"]}
+{"state": {"context": ["m n : ℕ"], "goal": "Disjoint (filter Nat.Prime (Ico 0 (m + 1))) (filter Nat.Prime (Ico (m + 1) (m + n + 1)))"}, "premise": [3747, 20613, 19112, 139114, 20654, 103886, 139120, 126931, 2134], "module": ["Mathlib/NumberTheory/Primorial.lean"]}
+{"state": {"context": ["n : Type u", "inst✝⁴ : DecidableEq n", "inst✝³ : Fintype n", "R✝ : Type v", "inst✝² : CommRing R✝", "S : Type u_1", "inst✝¹ : CommRing S", "R : Type u_2", "inst✝ : Field R", "g : SL(2, R)", "hg : ↑g 1 0 = 0"], "goal": "∃ a b, ∃ (h : a ≠ 0), g = ⟨!![a, b; 0, a⁻¹], ⋯⟩"}, "premise": [85121], "module": ["Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean"]}
+{"state": {"context": ["n : Type u", "inst✝⁴ : DecidableEq n", "inst✝³ : Fintype n", "R✝ : Type v", "inst✝² : CommRing R✝", "S : Type u_1", "inst✝¹ : CommRing S", "R : Type u_2", "inst✝ : Field R", "a b c d : R", "h_det : a * d - b * c = 1", "hg : ↑⟨!![a, b; c, d], ⋯⟩ 1 0 = 0"], "goal": "∃ a_1 b_1, ∃ (h : a_1 ≠ 0), ⟨!![a, b; c, d], ⋯⟩ = ⟨!![a_1, b_1; 0, a_1⁻¹], ⋯⟩"}, "premise": [85121], "module": ["Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean"]}
+{"state": {"context": ["n : Type u", "inst✝⁴ : DecidableEq n", "inst✝³ : Fintype n", "R✝ : Type v", "inst✝² : CommRing R✝", "S : Type u_1", "inst✝¹ : CommRing S", "R : Type u_2", "inst✝ : Field R", "a b c d : R", "h_det : a * d - b * c = 1", "hg : c = 0"], "goal": "∃ a_1 b_1, ∃ (h : a_1 ≠ 0), ⟨!![a, b; c, d], ⋯⟩ = ⟨!![a_1, b_1; 0, a_1⁻¹], ⋯⟩"}, "premise": [108558, 117816, 108274], "module": ["Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean"]}
+{"state": {"context": ["n : Type u", "inst✝⁴ : DecidableEq n", "inst✝³ : Fintype n", "R✝ : Type v", "inst✝² : CommRing R✝", "S : Type u_1", "inst✝¹ : CommRing S", "R : Type u_2", "inst✝ : Field R", "a b c d : R", "h_det : a * d - b * c = 1", "hg : c = 0", "had : a * d = 1"], "goal": "∃ a_1 b_1, ∃ (h : a_1 ≠ 0), ⟨!![a, b; c, d], ⋯⟩ = ⟨!![a_1, b_1; 0, a_1⁻¹], ⋯⟩"}, "premise": [117816, 108274, 108558], "module": ["Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean"]}
+{"state": {"context": ["n : Type u", "inst✝⁴ : DecidableEq n", "inst✝³ : Fintype n", "R✝ : Type v", "inst✝² : CommRing R✝", "S : Type u_1", "inst✝¹ : CommRing S", "R : Type u_2", "inst✝ : Field R", "a b c d : R", "h_det : a * d - b * c = 1", "hg : c = 0", "had : a * d = 1"], "goal": "⟨!![a, b; c, d], ⋯⟩ = ⟨!![a, b; 0, a⁻¹], ⋯⟩"}, "premise": [108274, 117820], "module": ["Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean"]}
+{"state": {"context": ["α β : Type", "n : ℕ", "f g : (α → ℕ) → ℕ", "df : DiophFn f", "dg : DiophFn g"], "goal": "DiophFn fun v => f v ^ g v"}, "premise": [22658, 22660, 22661, 22672, 22678, 22679, 22682, 22683, 22684, 22685, 22686, 22688, 22691, 22693], "module": ["Mathlib/NumberTheory/Dioph.lean"]}
+{"state": {"context": ["α β : Type", "n : ℕ", "f g : (α → ℕ) → ℕ", "df : DiophFn f", "dg : DiophFn g", "proof : Dioph (((fun v => v &2 = const (Fin2 3 → ℕ) 0 v) ∩ fun v => v &0 = const (Fin2 3 → ℕ) 1 v) ∪ {v | const (Fin2 3 → ℕ) 0 v < v &2} ∩ (((fun v => v &1 = const (Fin2 3 → ℕ) 0 v) ∩ fun v => v &0 = const (Fin2 3 → ℕ) 0 v) ∪ {v | const (Fin2 3 → ℕ) 0 v < v &1} ∩ {v | ∃ x, (x :: v) ∈ {v | ∃ x, (x :: v) ∈ {v | ∃ x, (x :: v) ∈ {v | ∃ x, (x :: v) ∈ {v | ∃ x, (x :: v) ∈ {v | ∃ x, (x :: v) ∈ {v | (v ∘ &4 :: &8 :: &1 :: &0 :: []) ∈ fun v => ∃ (h : 1 < v &0), xn h (v &1) = v &2 ∧ yn h (v &1) = v &3} ∩ ((fun v => v &1 ≡ v &0 * (v &4 - v &7) + v &6 [MOD v &3]) ∩ ((fun v => const (Fin2 (succ 8) → ℕ) 2 v * v &4 * v &7 = v &3 + (v &7 * v &7 + const (Fin2 (succ 8) → ℕ) 1 v)) ∩ ({v | v &6 < v &3} ∩ ({v | v &7 ≤ v &5} ∩ ({v | v &8 ≤ v &5} ∩ fun v => v &4 * v &4 - ((v &5 + const (Fin2 (succ 8) → ℕ) 1 v) * (v &5 + const (Fin2 (succ 8) → ℕ) 1 v) - const (Fin2 (succ 8) → ℕ) 1 v) * (v &5 * v &2) * (v &5 * v &2) = const (Fin2 (succ 8) → ℕ) 1 v)))))}}}}}}))"], "goal": "DiophFn fun v => f v ^ g v"}, "premise": [22658, 22660, 22661, 22672, 22678, 22679, 22682, 22683, 22684, 22685, 22686, 22688, 22691, 22693], "module": ["Mathlib/NumberTheory/Dioph.lean"]}
+{"state": {"context": ["α β : Type", "n : ℕ", "f g : (α → ℕ) → ℕ", "df : DiophFn f", "dg : DiophFn g", "proof : Dioph (((fun v => v &2 = const (Fin2 3 → ℕ) 0 v) ∩ fun v => v &0 = const (Fin2 3 → ℕ) 1 v) ∪ {v | const (Fin2 3 → ℕ) 0 v < v &2} ∩ (((fun v => v &1 = const (Fin2 3 → ℕ) 0 v) ∩ fun v => v &0 = const (Fin2 3 → ℕ) 0 v) ∪ {v | const (Fin2 3 → ℕ) 0 v < v &1} ∩ {v | ∃ x, (x :: v) ∈ {v | ∃ x, (x :: v) ∈ {v | ∃ x, (x :: v) ∈ {v | ∃ x, (x :: v) ∈ {v | ∃ x, (x :: v) ∈ {v | ∃ x, (x :: v) ∈ {v | (v ∘ &4 :: &8 :: &1 :: &0 :: []) ∈ fun v => ∃ (h : 1 < v &0), xn h (v &1) = v &2 ∧ yn h (v &1) = v &3} ∩ ((fun v => v &1 ≡ v &0 * (v &4 - v &7) + v &6 [MOD v &3]) ∩ ((fun v => const (Fin2 (succ 8) → ℕ) 2 v * v &4 * v &7 = v &3 + (v &7 * v &7 + const (Fin2 (succ 8) → ℕ) 1 v)) ∩ ({v | v &6 < v &3} ∩ ({v | v &7 ≤ v &5} ∩ ({v | v &8 ≤ v &5} ∩ fun v => v &4 * v &4 - ((v &5 + const (Fin2 (succ 8) → ℕ) 1 v) * (v &5 + const (Fin2 (succ 8) → ℕ) 1 v) - const (Fin2 (succ 8) → ℕ) 1 v) * (v &5 * v &2) * (v &5 * v &2) = const (Fin2 (succ 8) → ℕ) 1 v)))))}}}}}}))", "this : Dioph {v | v &2 = 0 ∧ v &0 = 1 ∨ 0 < v &2 ∧ (v &1 = 0 ∧ v &0 = 0 ∨ 0 < v &1 ∧ ∃ w a t z x y, (∃ (a1 : 1 < a), xn a1 (v &2) = x ∧ yn a1 (v &2) = y) ∧ x ≡ y * (a - v &1) + v &0 [MOD t] ∧ 2 * a * v &1 = t + (v &1 * v &1 + 1) ∧ v &0 < t ∧ v &1 ≤ w ∧ v &2 ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1)}"], "goal": "DiophFn fun v => f v ^ g v"}, "premise": [1674, 1713, 1715, 1718, 1963, 1980, 22546, 22654, 22673, 22681], "module": ["Mathlib/NumberTheory/Dioph.lean"]}
+{"state": {"context": ["α : Type u_1", "l : Filter α", "f : α → ℝ"], "goal": "Tendsto (fun x => (f x).toNNReal) l atTop ↔ Tendsto f l atTop"}, "premise": [1671, 1713, 16363, 56432], "module": ["Mathlib/Topology/Instances/NNReal.lean"]}
+{"state": {"context": ["z : ↥circle"], "goal": "normSq ↑z = 1"}, "premise": [148258], "module": ["Mathlib/Analysis/Complex/Circle.lean"]}
+{"state": {"context": ["X Y Z : Scheme", "𝒰✝ : X.OpenCover", "f✝ : X ⟶ Z", "g : Y ⟶ Z", "inst✝ : ∀ (x : 𝒰✝.J), HasPullback (𝒰✝.map x ≫ f✝) g", "f : X ⟶ Y", "𝒰 : Y.OpenCover", "i : (𝒰.affineRefinement.openCover.pullbackCover f).J"], "goal": "(pullbackCoverAffineRefinementObjIso f 𝒰 i).inv ≫ 𝒰.affineRefinement.openCover.pullbackHom f i = (𝒰.obj i.fst).affineCover.pullbackHom (𝒰.pullbackHom f i.fst) i.snd"}, "premise": [2098, 88746, 88755, 88789, 88799, 93340, 93753, 93754, 93898, 96173, 96174, 128130, 128141, 128142, 93897], "module": ["Mathlib/AlgebraicGeometry/Cover/Open.lean"]}
+{"state": {"context": ["X Y Z : Scheme", "𝒰✝ : X.OpenCover", "f✝ : X ⟶ Z", "g : Y ⟶ Z", "inst✝ : ∀ (x : 𝒰✝.J), HasPullback (𝒰✝.map x ≫ f✝) g", "f : X ⟶ Y", "𝒰 : Y.OpenCover", "i : (𝒰.affineRefinement.openCover.pullbackCover f).J"], "goal": "(pullbackSymmetry ((𝒰.obj i.fst).affineCover.map i.snd) (pullback.fst (𝒰.map i.fst) f)).inv ≫ (pullbackRightPullbackFstIso (𝒰.map i.fst) f ((𝒰.obj i.fst).affineCover.map i.snd)).hom ≫ pullback.fst (𝒰.affineRefinement.map i) f = pullback.snd (pullback.fst (𝒰.map i.fst) f) ((𝒰.obj i.fst).affineCover.map i.snd)"}, "premise": [128130, 93897, 88746, 93898, 96173, 96174, 128141, 128142, 2098, 88755, 88789, 93753, 93754, 93340, 88799], "module": ["Mathlib/AlgebraicGeometry/Cover/Open.lean"]}
+{"state": {"context": ["X Y Z : Scheme", "𝒰✝ : X.OpenCover", "f✝ : X ⟶ Z", "g : Y ⟶ Z", "inst✝ : ∀ (x : 𝒰✝.J), HasPullback (𝒰✝.map x ≫ f✝) g", "f : X ⟶ Y", "𝒰 : Y.OpenCover", "i : (𝒰.affineRefinement.openCover.pullbackCover f).J", "e_1✝ : (pullback (pullback.fst (𝒰.map i.fst) f) ((𝒰.obj i.fst).affineCover.map i.snd) ⟶ Spec (𝒰.affineRefinement.obj i)) = (pullback (pullback.fst (𝒰.map i.fst) f) ((𝒰.obj i.fst).affineCover.map i.snd) ⟶ (𝒰.obj i.fst).affineCover.obj i.snd)", "e_5✝ : Spec (𝒰.affineRefinement.obj i) = (𝒰.obj i.fst).affineCover.obj i.snd"], "goal": "(pullbackRightPullbackFstIso (𝒰.map i.fst) f ((𝒰.obj i.fst).affineCover.map i.snd)).hom ≫ pullback.fst (𝒰.affineRefinement.map i) f = pullback.fst ((𝒰.obj i.fst).affineCover.map i.snd) (pullback.fst (𝒰.map i.fst) f)"}, "premise": [94200, 93897], "module": ["Mathlib/AlgebraicGeometry/Cover/Open.lean"]}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝³ : Fintype α", "inst✝² : Fintype β", "inst✝¹ : DecidableEq α", "inst✝ : DecidableEq β", "G : SimpleGraph α", "a b : α", "p : G.Walk a b", "hp : p.IsHamiltonian", "c : α"], "goal": "c ∈ p.support"}, "premise": [375, 3786], "module": ["Mathlib/Combinatorics/SimpleGraph/Hamiltonian.lean"]}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "inst✝² : Preadditive C", "R : Type u_1", "inst✝¹ : Ring R", "inst✝ : Linear R C", "K L M : CochainComplex C ℤ", "n : ℤ", "γ γ₁ γ₂ : Cochain K L n", "a n' : ℤ", "hn' : n + a = n'", "m t t' : ℤ", "γ' : Cochain L M m", "h : n + m = t", "ht' : t + a = t'"], "goal": "n' + m = t'"}, "premise": [119704, 119708], "module": ["Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean"]}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "inst✝² : Preadditive C", "R : Type u_1", "inst✝¹ : Ring R", "inst✝ : Linear R C", "K L M : CochainComplex C ℤ", "n : ℤ", "γ γ₁ γ₂ : Cochain K L n", "a n' : ℤ", "hn' : n + a = n'", "m t t' : ℤ", "γ' : Cochain L M m", "h : n + m = t", "ht' : t + a = t'", "p q : ℤ", "hpq : p + t' = q", "h' : n' + m = t'"], "goal": "((γ.comp γ' h).leftShift a t' ht').v p q hpq = (a * m).negOnePow • ((γ.leftShift a n' hn').comp γ' ⋯).v p q hpq"}, "premise": [96173, 97903, 114452, 115392, 118909, 119703, 122189], "module": ["Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean"]}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "inst✝² : Preadditive C", "R : Type u_1", "inst✝¹ : Ring R", "inst✝ : Linear R C", "K L M : CochainComplex C ℤ", "n : ℤ", "γ γ₁ γ₂ : Cochain K L n", "a n' : ℤ", "hn' : n + a = n'", "m t t' : ℤ", "γ' : Cochain L M m", "h : n + m = t", "ht' : t + a = t'", "p q : ℤ", "hpq : p + t' = q", "h' : n' + m = t'"], "goal": "((a * (n' + m)).negOnePow * (a * (a - 1) / 2).negOnePow) • (K.shiftFunctorObjXIso a p (p + a) ⋯).hom ≫ γ.v (p + a) (p + n') ⋯ ≫ γ'.v (p + n') q ⋯ = ((a * m).negOnePow * (a * n').negOnePow * (a * (a - 1) / 2).negOnePow) • (K.shiftFunctorObjXIso a p (p + a) ⋯).hom ≫ γ.v (p + a) (p + n') ⋯ ≫ γ'.v (p + n') q ⋯"}, "premise": [96173, 97903, 114452, 115392, 118909, 119703, 122189], "module": ["Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean"]}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "inst✝² : Preadditive C", "R : Type u_1", "inst✝¹ : Ring R", "inst✝ : Linear R C", "K L M : CochainComplex C ℤ", "n : ℤ", "γ γ₁ γ₂ : Cochain K L n", "a n' : ℤ", "hn' : n + a = n'", "m t t' : ℤ", "γ' : Cochain L M m", "h : n + m = t", "ht' : t + a = t'", "p q : ℤ", "hpq : p + t' = q", "h' : n' + m = t'"], "goal": "(a * (n' + m)).negOnePow = (a * m).negOnePow * (a * n').negOnePow"}, "premise": [119708, 122189], "module": ["Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean"]}
+{"state": {"context": ["R : Type u_1", "inst✝⁸ : CommRing R", "M : Type u_2", "inst✝⁷ : AddCommGroup M", "inst✝⁶ : Module R M", "M' : Type u_3", "inst✝⁵ : AddCommGroup M'", "inst✝⁴ : Module R M'", "ι : Type u_4", "inst✝³ : DecidableEq ι", "inst✝² : Fintype ι", "e : Basis ι R M", "𝕜 : Type u_5", "inst✝¹ : Field 𝕜", "inst✝ : Module 𝕜 M", "f : M ≃ₗ[𝕜] M"], "goal": "LinearMap.det ↑f.symm = (LinearMap.det ↑f)⁻¹"}, "premise": [83470, 108373], "module": ["Mathlib/LinearAlgebra/Determinant.lean"]}
+{"state": {"context": ["G : Type u_1", "M : Type u_2", "inst✝ : Monoid M", "a✝ b c : M", "m n✝ : ℕ", "a : M", "n : ℕ"], "goal": "a ^ n * a = a * a ^ n"}, "premise": [119742, 119745], "module": ["Mathlib/Algebra/Group/Defs.lean"]}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "M : Type u_3", "inst✝⁶ : CommSemiring R", "inst✝⁵ : CommSemiring A", "inst✝⁴ : Algebra R A", "inst✝³ : AddCommMonoid M", "inst✝² : Module A M", "inst✝¹ : Module R M", "inst✝ : IsScalarTower R A M", "d : Derivation R A M", "a : A"], "goal": "∀ (a_1 b : R[X]), { toFun := fun f => (AEval.of R M a) (d ((aeval a) f)), map_add' := ⋯, map_smul' := ⋯ } (a_1 * b) = a_1 • { toFun := fun f => (AEval.of R M a) (d ((aeval a) f)), map_add' := ⋯, map_smul' := ⋯ } b + b • { toFun := fun f => (AEval.of R M a) (d ((aeval a) f)), map_add' := ⋯, map_smul' := ⋯ } a_1"}, "premise": [79729, 101165], "module": ["Mathlib/Algebra/Polynomial/Derivation.lean"]}
+{"state": {"context": ["R : Type u_1", "inst✝ : CommRing R", "n : ℕ", "P : Ideal R", "q : R[X]", "c : (R ⧸ P)[X]", "hq : map (mk P) q = c * X ^ n", "hn0 : n ≠ 0"], "goal": "eval 0 q ∈ P"}, "premise": [80142, 102870, 102886, 102928, 102966, 102969, 108276, 108558], "module": ["Mathlib/RingTheory/EisensteinCriterion.lean"]}
+{"state": {"context": ["G : Type u", "M : Type v", "α : Type w", "s✝ : Set α", "m : MeasurableSpace α", "inst✝³ : MeasurableSpace G", "inst✝² : Group G", "inst✝¹ : MulAction G α", "μ : Measure α", "inst✝ : SMulInvariantMeasure G α μ", "c : G", "s : Set α"], "goal": "EventuallyConst (c • s) (ae μ) ↔ EventuallyConst s (ae μ)"}, "premise": [1713, 11602, 13833, 30524, 132973], "module": ["Mathlib/MeasureTheory/Group/Action.lean"]}
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