ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/GroupTheory/GroupAction/Pointwise.lean
R : Type u_1
S : Type u_2
M : Type u_3
M₁ : Type u_4
M₂ : Type u_5
N : Type u_6
inst✝⁷ : Monoid R
inst✝⁶ : Monoid S
σ : R → S
inst✝⁵ : MulAction R M
inst✝⁴ : MulAction S N
inst✝³ : MulAction R M₁
inst✝² : MulAction R M₂
F : Type u_7
h : F
inst✝¹ : FunLike F M N
inst✝ : MulActionSemiHomClass F σ M N
c : R
s : Set M
t : Set N

⊢ ⇑h '' (c • s) = σ c • ⇑h '' s
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
θ : Angle

⊢ θ.toReal = π / 2 ↔ θ = ↑(π / 2)
-- Mathlib/Algebra/IsPrimePow.lean
R : Type u_1
inst✝¹ : CommMonoidWithZero R
n p : R
k : ℕ
inst✝ : NoZeroDivisors R

⊢ ¬IsPrimePow 0
-- Mathlib/Algebra/IsPrimePow.lean
R : Type u_1
inst✝¹ : CommMonoidWithZero R
n p : R
k : ℕ
inst✝ : NoZeroDivisors R

⊢ ∀ (x : R) (x_1 : ℕ), 0 < x_1 → x ^ x_1 = 0 → ¬Prime x
-- Mathlib/Algebra/IsPrimePow.lean
R : Type u_1
inst✝¹ : CommMonoidWithZero R
n✝ p : R
k : ℕ
inst✝ : NoZeroDivisors R
x : R
n : ℕ
_hn : 0 < n
hx : x ^ n = 0

⊢ ¬Prime x
-- Mathlib/Algebra/IsPrimePow.lean
R : Type u_1
inst✝¹ : CommMonoidWithZero R
n✝ p : R
k : ℕ
inst✝ : NoZeroDivisors R
x : R
n : ℕ
_hn : 0 < n
hx : x ^ n = 0

⊢ ¬Prime 0
-- Mathlib/Algebra/Lie/Engel.lean
R : Type u₁
L : Type u₂
L₂ : Type u₃
M : Type u₄
inst✝⁹ : CommRing R
inst✝⁸ : LieRing L
inst✝⁷ : LieAlgebra R L
inst✝⁶ : LieRing L₂
inst✝⁵ : LieAlgebra R L₂
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : LieRingModule L M
inst✝¹ : LieModule R L M
inst✝ : IsNoetherian R M

⊢ IsNilpotent R L M ↔ ∀ (x : L), _root_.IsNilpotent ((toEnd R L M) x)
-- Mathlib/Algebra/Lie/Engel.lean
R : Type u₁
L : Type u₂
L₂ : Type u₃
M : Type u₄
inst✝⁹ : CommRing R
inst✝⁸ : LieRing L
inst✝⁷ : LieAlgebra R L
inst✝⁶ : LieRing L₂
inst✝⁵ : LieAlgebra R L₂
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : LieRingModule L M
inst✝¹ : LieModule R L M
inst✝ : IsNoetherian R M

⊢ (∀ (x : ↥(toEnd R L M).range), _root_.IsNilpotent ((toEnd R (↥(toEnd R L M).range) M) x)) ↔ ∀ (x : L), _root_.IsNilpotent ((toEnd R L M) x)
-- Mathlib/GroupTheory/GroupAction/Basic.lean
G : Type u_1
α : Type u_2
β : Type u_3
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : MulAction G β
a₁ a₂ : α

⊢ a₁ ∈ orbit G a₂ ↔ a₂ ∈ orbit G a₁
-- Mathlib/CategoryTheory/Sites/LeftExact.lean
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁹ : Category.{max v u, w} D
inst✝⁸ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁶ : ConcreteCategory D
inst✝⁵ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
K : Type (max v u)
inst✝⁴ : SmallCategory K
inst✝³ : FinCategory K
inst✝² : HasLimitsOfShape K D
inst✝¹ : PreservesLimitsOfShape K (forget D)
inst✝ : ReflectsLimitsOfShape K (forget D)
F : K ⥤ Cᵒᵖ ⥤ D
X : C
S : Cone (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X))
k : K

⊢ (limit.lift (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X)) S ≫ (HasLimit.isoOfNatIso (NatIso.ofComponents (fun k => colimitObjIsoColimitCompEvaluation (F ⋙ J.diagramFunctor D X).flip k) ⋯).symm).hom ≫ (colimitLimitIso (F ⋙ J.diagramFunctor D X)).inv ≫ (HasColimit.isoOfNatIso ((isLimitOfPreserves (J.diagramFunctor D X) (limit.isLimit F)).conePointUniqueUpToIso (limit.isLimit (F ⋙ J.diagramFunctor D X)))).inv) ≫ (J.plusMap (limit.π F k)).app (op X) = S.π.app k
-- Mathlib/CategoryTheory/Sites/LeftExact.lean
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁹ : Category.{max v u, w} D
inst✝⁸ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁶ : ConcreteCategory D
inst✝⁵ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
K : Type (max v u)
inst✝⁴ : SmallCategory K
inst✝³ : FinCategory K
inst✝² : HasLimitsOfShape K D
inst✝¹ : PreservesLimitsOfShape K (forget D)
inst✝ : ReflectsLimitsOfShape K (forget D)
F : K ⥤ Cᵒᵖ ⥤ D
X : C
S : Cone (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X))
k : K

⊢ limit.lift (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X)) S ≫ ((HasLimit.isoOfNatIso (NatIso.ofComponents (fun k => colimitObjIsoColimitCompEvaluation (F ⋙ J.diagramFunctor D X).flip k) ⋯).symm).hom ≫ (colimitLimitIso (F ⋙ J.diagramFunctor D X)).inv ≫ (HasColimit.isoOfNatIso ((isLimitOfPreserves (J.diagramFunctor D X) (limit.isLimit F)).conePointUniqueUpToIso (limit.isLimit (F ⋙ J.diagramFunctor D X)))).inv) ≫ (J.plusMap (limit.π F k)).app (op X) = (limit.isLimit (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X))).lift S ≫ (limit.cone (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X))).π.app k
-- Mathlib/CategoryTheory/Sites/LeftExact.lean
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁹ : Category.{max v u, w} D
inst✝⁸ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁶ : ConcreteCategory D
inst✝⁵ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
K : Type (max v u)
inst✝⁴ : SmallCategory K
inst✝³ : FinCategory K
inst✝² : HasLimitsOfShape K D
inst✝¹ : PreservesLimitsOfShape K (forget D)
inst✝ : ReflectsLimitsOfShape K (forget D)
F : K ⥤ Cᵒᵖ ⥤ D
X : C
S : Cone (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X))
k : K

⊢ ((HasLimit.isoOfNatIso (NatIso.ofComponents (fun k => colimitObjIsoColimitCompEvaluation (F ⋙ J.diagramFunctor D X).flip k) ⋯).symm).hom ≫ (colimitLimitIso (F ⋙ J.diagramFunctor D X)).inv ≫ (HasColimit.isoOfNatIso ((isLimitOfPreserves (J.diagramFunctor D X) (limit.isLimit F)).conePointUniqueUpToIso (limit.isLimit (F ⋙ J.diagramFunctor D X)))).inv) ≫ (J.plusMap (limit.π F k)).app (op X) = limit.π (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X)) k
-- Mathlib/CategoryTheory/Sites/LeftExact.lean
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁹ : Category.{max v u, w} D
inst✝⁸ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁶ : ConcreteCategory D
inst✝⁵ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
K : Type (max v u)
inst✝⁴ : SmallCategory K
inst✝³ : FinCategory K
inst✝² : HasLimitsOfShape K D
inst✝¹ : PreservesLimitsOfShape K (forget D)
inst✝ : ReflectsLimitsOfShape K (forget D)
F : K ⥤ Cᵒᵖ ⥤ D
X : C
S : Cone (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X))
k : K

⊢ (J.plusMap (limit.π F k)).app (op X) = (HasColimit.isoOfNatIso ((isLimitOfPreserves (J.diagramFunctor D X) (limit.isLimit F)).conePointUniqueUpToIso (limit.isLimit (F ⋙ J.diagramFunctor D X)))).hom ≫ (colimitLimitIso (F ⋙ J.diagramFunctor D X)).hom ≫ (HasLimit.isoOfNatIso (NatIso.ofComponents (fun k => colimitObjIsoColimitCompEvaluation (F ⋙ J.diagramFunctor D X).flip k) ⋯).symm).inv ≫ limit.π (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X)) k
-- Mathlib/CategoryTheory/Sites/LeftExact.lean
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁹ : Category.{max v u, w} D
inst✝⁸ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁶ : ConcreteCategory D
inst✝⁵ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
K : Type (max v u)
inst✝⁴ : SmallCategory K
inst✝³ : FinCategory K
inst✝² : HasLimitsOfShape K D
inst✝¹ : PreservesLimitsOfShape K (forget D)
inst✝ : ReflectsLimitsOfShape K (forget D)
F : K ⥤ Cᵒᵖ ⥤ D
X : C
S : Cone (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X))
k : K
j : (J.Cover (unop (op X)))ᵒᵖ

⊢ colimit.ι (J.diagram (limit F) (unop (op X))) j ≫ (J.plusMap (limit.π F k)).app (op X) = colimit.ι (J.diagram (limit F) (unop (op X))) j ≫ (HasColimit.isoOfNatIso ((isLimitOfPreserves (J.diagramFunctor D X) (limit.isLimit F)).conePointUniqueUpToIso (limit.isLimit (F ⋙ J.diagramFunctor D X)))).hom ≫ (colimitLimitIso (F ⋙ J.diagramFunctor D X)).hom ≫ (HasLimit.isoOfNatIso (NatIso.ofComponents (fun k => colimitObjIsoColimitCompEvaluation (F ⋙ J.diagramFunctor D X).flip k) ⋯).symm).inv ≫ limit.π (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X)) k
-- Mathlib/CategoryTheory/Sites/LeftExact.lean
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁹ : Category.{max v u, w} D
inst✝⁸ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁶ : ConcreteCategory D
inst✝⁵ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
K : Type (max v u)
inst✝⁴ : SmallCategory K
inst✝³ : FinCategory K
inst✝² : HasLimitsOfShape K D
inst✝¹ : PreservesLimitsOfShape K (forget D)
inst✝ : ReflectsLimitsOfShape K (forget D)
F : K ⥤ Cᵒᵖ ⥤ D
X : C
S : Cone (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X))
k : K
j : (J.Cover (unop (op X)))ᵒᵖ

⊢ colimit.ι (J.diagram (limit F) X) j ≫ colimMap (J.diagramNatTrans (limit.π F k) X) = colimit.ι (J.diagram (limit F) X) j ≫ (HasColimit.isoOfNatIso ((isLimitOfPreserves (J.diagramFunctor D X) (limit.isLimit F)).conePointUniqueUpToIso (limit.isLimit (F ⋙ J.diagramFunctor D X)))).hom ≫ (colimitLimitIso (F ⋙ J.diagramFunctor D X)).hom ≫ (HasLimit.isoOfNatIso (NatIso.ofComponents (fun k => colimitObjIsoColimitCompEvaluation (F ⋙ J.diagramFunctor D X).flip k) ⋯).symm).inv ≫ limit.π (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X)) k
-- Mathlib/CategoryTheory/Sites/LeftExact.lean
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁹ : Category.{max v u, w} D
inst✝⁸ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁶ : ConcreteCategory D
inst✝⁵ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
K : Type (max v u)
inst✝⁴ : SmallCategory K
inst✝³ : FinCategory K
inst✝² : HasLimitsOfShape K D
inst✝¹ : PreservesLimitsOfShape K (forget D)
inst✝ : ReflectsLimitsOfShape K (forget D)
F : K ⥤ Cᵒᵖ ⥤ D
X : C
S : Cone (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X))
k : K
j : (J.Cover (unop (op X)))ᵒᵖ

⊢ (J.diagramNatTrans (limit.π F k) X).app j ≫ colimit.ι (J.diagram (F.obj k) X) j = ((isLimitOfPreserves (J.diagramFunctor D X) (limit.isLimit F)).conePointUniqueUpToIso (limit.isLimit (F ⋙ J.diagramFunctor D X))).hom.app j ≫ colimit.ι (limit (F ⋙ J.diagramFunctor D X)) j ≫ (colimitLimitIso (F ⋙ J.diagramFunctor D X)).hom ≫ (HasLimit.isoOfNatIso (NatIso.ofComponents (fun k => colimitObjIsoColimitCompEvaluation (F ⋙ J.diagramFunctor D X).flip k) ⋯).symm).inv ≫ limit.π (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X)) k
-- Mathlib/CategoryTheory/Sites/LeftExact.lean
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁹ : Category.{max v u, w} D
inst✝⁸ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁶ : ConcreteCategory D
inst✝⁵ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
K : Type (max v u)
inst✝⁴ : SmallCategory K
inst✝³ : FinCategory K
inst✝² : HasLimitsOfShape K D
inst✝¹ : PreservesLimitsOfShape K (forget D)
inst✝ : ReflectsLimitsOfShape K (forget D)
F : K ⥤ Cᵒᵖ ⥤ D
X : C
S : Cone (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X))
k : K
j : (J.Cover (unop (op X)))ᵒᵖ

⊢ Multiequalizer.lift ((unop j).index (F.obj k)) (multiequalizer ((unop j).index (limit F))) (fun i => Multiequalizer.ι ((unop j).index (limit F)) i ≫ (limit.π F k).app (op i.Y)) ⋯ ≫ colimit.ι (J.diagram (F.obj k) X) j = (limit.lift (F ⋙ J.diagramFunctor D X) ((J.diagramFunctor D X).mapCone (limit.cone F))).app j ≫ colimit.ι (limit (F ⋙ J.diagramFunctor D X)) j ≫ (colimitLimitIso (F ⋙ J.diagramFunctor D X)).hom ≫ limit.lift (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X)) ((Cones.postcompose (NatIso.ofComponents (fun k => colimitObjIsoColimitCompEvaluation (F ⋙ J.diagramFunctor D X).flip k) ⋯).hom).obj (limit.cone (colimit (F ⋙ J.diagramFunctor D X).flip))) ≫ limit.π (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X)) k
-- Mathlib/CategoryTheory/Sites/LeftExact.lean
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁹ : Category.{max v u, w} D
inst✝⁸ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁶ : ConcreteCategory D
inst✝⁵ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
K : Type (max v u)
inst✝⁴ : SmallCategory K
inst✝³ : FinCategory K
inst✝² : HasLimitsOfShape K D
inst✝¹ : PreservesLimitsOfShape K (forget D)
inst✝ : ReflectsLimitsOfShape K (forget D)
F : K ⥤ Cᵒᵖ ⥤ D
X : C
S : Cone (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X))
k : K
j : (J.Cover (unop (op X)))ᵒᵖ

⊢ Multiequalizer.lift ((unop j).index (F.obj k)) (multiequalizer ((unop j).index (limit F))) (fun i => Multiequalizer.ι ((unop j).index (limit F)) i ≫ (limit.π F k).app (op i.Y)) ⋯ ≫ colimit.ι (J.diagram (F.obj k) X) j = (limit.lift (F ⋙ J.diagramFunctor D X) ((J.diagramFunctor D X).mapCone (limit.cone F))).app j ≫ colimit.ι (limit (F ⋙ J.diagramFunctor D X)) j ≫ (colimitLimitIso (F ⋙ J.diagramFunctor D X)).hom ≫ ((Cones.postcompose (NatIso.ofComponents (fun k => colimitObjIsoColimitCompEvaluation (F ⋙ J.diagramFunctor D X).flip k) ⋯).hom).obj (limit.cone (colimit (F ⋙ J.diagramFunctor D X).flip))).π.app k
-- Mathlib/CategoryTheory/Sites/LeftExact.lean
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁹ : Category.{max v u, w} D
inst✝⁸ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁶ : ConcreteCategory D
inst✝⁵ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
K : Type (max v u)
inst✝⁴ : SmallCategory K
inst✝³ : FinCategory K
inst✝² : HasLimitsOfShape K D
inst✝¹ : PreservesLimitsOfShape K (forget D)
inst✝ : ReflectsLimitsOfShape K (forget D)
F : K ⥤ Cᵒᵖ ⥤ D
X : C
S : Cone (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X))
k : K
j : (J.Cover (unop (op X)))ᵒᵖ

⊢ Multiequalizer.lift ((unop j).index (F.obj k)) (multiequalizer ((unop j).index (limit F))) (fun i => Multiequalizer.ι ((unop j).index (limit F)) i ≫ (limit.π F k).app (op i.Y)) ⋯ ≫ colimit.ι (J.diagram (F.obj k) X) j = (limit.lift (F ⋙ J.diagramFunctor D X) ((J.diagramFunctor D X).mapCone (limit.cone F))).app j ≫ (limit.π (F ⋙ J.diagramFunctor D X) k).app j ≫ (colimit.ι (F ⋙ J.diagramFunctor D X).flip j).app k ≫ (colimitObjIsoColimitCompEvaluation (F ⋙ J.diagramFunctor D X).flip k).hom
-- Mathlib/CategoryTheory/Sites/LeftExact.lean
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁹ : Category.{max v u, w} D
inst✝⁸ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁶ : ConcreteCategory D
inst✝⁵ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
K : Type (max v u)
inst✝⁴ : SmallCategory K
inst✝³ : FinCategory K
inst✝² : HasLimitsOfShape K D
inst✝¹ : PreservesLimitsOfShape K (forget D)
inst✝ : ReflectsLimitsOfShape K (forget D)
F : K ⥤ Cᵒᵖ ⥤ D
X : C
S : Cone (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X))
k : K
j : (J.Cover (unop (op X)))ᵒᵖ

⊢ Multiequalizer.lift ((unop j).index (F.obj k)) (multiequalizer ((unop j).index (limit F))) (fun i => Multiequalizer.ι ((unop j).index (limit F)) i ≫ (limit.π F k).app (op i.Y)) ⋯ ≫ colimit.ι (J.diagram (F.obj k) X) j = ((limit.lift (F ⋙ J.diagramFunctor D X) ((J.diagramFunctor D X).mapCone (limit.cone F)) ≫ limit.π (F ⋙ J.diagramFunctor D X) k).app j ≫ (colimit.ι (F ⋙ J.diagramFunctor D X).flip j).app k) ≫ (colimitObjIsoColimitCompEvaluation (F ⋙ J.diagramFunctor D X).flip k).hom
-- Mathlib/CategoryTheory/Sites/LeftExact.lean
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁹ : Category.{max v u, w} D
inst✝⁸ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁶ : ConcreteCategory D
inst✝⁵ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
K : Type (max v u)
inst✝⁴ : SmallCategory K
inst✝³ : FinCategory K
inst✝² : HasLimitsOfShape K D
inst✝¹ : PreservesLimitsOfShape K (forget D)
inst✝ : ReflectsLimitsOfShape K (forget D)
F : K ⥤ Cᵒᵖ ⥤ D
X : C
S : Cone (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X))
k : K
j : (J.Cover (unop (op X)))ᵒᵖ

⊢ Multiequalizer.lift ((unop j).index (F.obj k)) (multiequalizer ((unop j).index (limit F))) (fun i => Multiequalizer.ι ((unop j).index (limit F)) i ≫ (limit.π F k).app (op i.Y)) ⋯ ≫ colimit.ι (J.diagram (F.obj k) X) j = (((J.diagramFunctor D X).mapCone (limit.cone F)).π.app k).app j ≫ (colimit.ι (F ⋙ J.diagramFunctor D X).flip j).app k ≫ (colimitObjIsoColimitCompEvaluation (F ⋙ J.diagramFunctor D X).flip k).hom
-- Mathlib/CategoryTheory/Sites/LeftExact.lean
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁹ : Category.{max v u, w} D
inst✝⁸ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁶ : ConcreteCategory D
inst✝⁵ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
K : Type (max v u)
inst✝⁴ : SmallCategory K
inst✝³ : FinCategory K
inst✝² : HasLimitsOfShape K D
inst✝¹ : PreservesLimitsOfShape K (forget D)
inst✝ : ReflectsLimitsOfShape K (forget D)
F : K ⥤ Cᵒᵖ ⥤ D
X : C
S : Cone (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X))
k : K
j : (J.Cover (unop (op X)))ᵒᵖ

⊢ colimit.ι (J.diagram (F.obj k) X) j = (colimit.ι (F ⋙ J.diagramFunctor D X).flip j).app k ≫ (colimitObjIsoColimitCompEvaluation (F ⋙ J.diagramFunctor D X).flip k).hom
-- Mathlib/CategoryTheory/Sites/LeftExact.lean
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁹ : Category.{max v u, w} D
inst✝⁸ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁶ : ConcreteCategory D
inst✝⁵ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
K : Type (max v u)
inst✝⁴ : SmallCategory K
inst✝³ : FinCategory K
inst✝² : HasLimitsOfShape K D
inst✝¹ : PreservesLimitsOfShape K (forget D)
inst✝ : ReflectsLimitsOfShape K (forget D)
F : K ⥤ Cᵒᵖ ⥤ D
X : C
S : Cone (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X))
k : K
j : (J.Cover (unop (op X)))ᵒᵖ

⊢ colimit.ι (J.diagram (F.obj k) X) j ≫ (colimitObjIsoColimitCompEvaluation (F ⋙ J.diagramFunctor D X).flip k).inv = (colimit.ι (F ⋙ J.diagramFunctor D X).flip j).app k
-- Mathlib/CategoryTheory/Sites/LeftExact.lean
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁹ : Category.{max v u, w} D
inst✝⁸ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁶ : ConcreteCategory D
inst✝⁵ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
K : Type (max v u)
inst✝⁴ : SmallCategory K
inst✝³ : FinCategory K
inst✝² : HasLimitsOfShape K D
inst✝¹ : PreservesLimitsOfShape K (forget D)
inst✝ : ReflectsLimitsOfShape K (forget D)
F : K ⥤ Cᵒᵖ ⥤ D
X : C
S : Cone (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X))
k : K
j : (J.Cover (unop (op X)))ᵒᵖ

⊢ (((evaluation K D).obj k).mapCocone (colimit.cocone (F ⋙ J.diagramFunctor D X).flip)).ι.app j = (colimit.ι (F ⋙ J.diagramFunctor D X).flip j).app k
-- Mathlib/SetTheory/Surreal/Basic.lean
x y : PGame
ox : x.Numeric
oy : y.Numeric

⊢ x ⧏ y → ¬y ⧏ x
-- Mathlib/SetTheory/Surreal/Basic.lean
x y : PGame
ox : x.Numeric
oy : y.Numeric
xl xr : Type u_1
xL : xl → PGame
xR : xr → PGame
hx : ∀ (i : xl) (j : xr), xL i < xR j
_oxl : ∀ (i : xl), (xL i).Numeric
_oxr : ∀ (i : xr), (xR i).Numeric
IHxl : ∀ (i : xl), (fun x => ∀ (z : PGame), z.Numeric → x ⧏ z → ¬z ⧏ x) (xL i)
IHxr : ∀ (i : xr), (fun x => ∀ (z : PGame), z.Numeric → x ⧏ z → ¬z ⧏ x) (xR i)

⊢ (fun x => ∀ (z : PGame), z.Numeric → x ⧏ z → ¬z ⧏ x) (mk xl xr xL xR)
-- Mathlib/SetTheory/Surreal/Basic.lean
x y : PGame
ox : x.Numeric
oy : y.Numeric
xl xr : Type u_1
xL : xl → PGame
xR : xr → PGame
hx : ∀ (i : xl) (j : xr), xL i < xR j
_oxl : ∀ (i : xl), (xL i).Numeric
_oxr : ∀ (i : xr), (xR i).Numeric
IHxl : ∀ (i : xl), (fun x => ∀ (z : PGame), z.Numeric → x ⧏ z → ¬z ⧏ x) (xL i)
IHxr : ∀ (i : xr), (fun x => ∀ (z : PGame), z.Numeric → x ⧏ z → ¬z ⧏ x) (xR i)
yl yr : Type u_1
yL : yl → PGame
yR : yr → PGame
hy : ∀ (i : yl) (j : yr), yL i < yR j
oyl : ∀ (i : yl), (yL i).Numeric
oyr : ∀ (i : yr), (yR i).Numeric
_IHyl : ∀ (i : yl), mk xl xr xL xR ⧏ yL i → ¬yL i ⧏ mk xl xr xL xR
_IHyr : ∀ (i : yr), mk xl xr xL xR ⧏ yR i → ¬yR i ⧏ mk xl xr xL xR

⊢ mk xl xr xL xR ⧏ mk yl yr yL yR → ¬mk yl yr yL yR ⧏ mk xl xr xL xR
-- Mathlib/SetTheory/Surreal/Basic.lean
x y : PGame
ox : x.Numeric
oy : y.Numeric
xl xr : Type u_1
xL : xl → PGame
xR : xr → PGame
hx : ∀ (i : xl) (j : xr), xL i < xR j
_oxl : ∀ (i : xl), (xL i).Numeric
_oxr : ∀ (i : xr), (xR i).Numeric
IHxl : ∀ (i : xl), (fun x => ∀ (z : PGame), z.Numeric → x ⧏ z → ¬z ⧏ x) (xL i)
IHxr : ∀ (i : xr), (fun x => ∀ (z : PGame), z.Numeric → x ⧏ z → ¬z ⧏ x) (xR i)
yl yr : Type u_1
yL : yl → PGame
yR : yr → PGame
hy : ∀ (i : yl) (j : yr), yL i < yR j
oyl : ∀ (i : yl), (yL i).Numeric
oyr : ∀ (i : yr), (yR i).Numeric
_IHyl : ∀ (i : yl), mk xl xr xL xR ⧏ yL i → ¬yL i ⧏ mk xl xr xL xR
_IHyr : ∀ (i : yr), mk xl xr xL xR ⧏ yR i → ¬yR i ⧏ mk xl xr xL xR

⊢ ((∃ i, mk xl xr xL xR ≤ yL i) ∨ ∃ j, xR j ≤ mk yl yr yL yR) → ¬((∃ i, mk yl yr yL yR ≤ xL i) ∨ ∃ j, yR j ≤ mk xl xr xL xR)
-- Mathlib/Analysis/SpecialFunctions/ExpDeriv.lean
n : ℕ
c : ℂ

⊢ (iteratedDeriv n fun s => cexp (c * s)) = fun s => c ^ n * cexp (c * s)
-- Mathlib/GroupTheory/DoubleCoset.lean
G : Type u_1
inst✝¹ : Group G
α : Type u_2
inst✝ : Mul α
J : Subgroup G
g : G
H : Subgroup G
a✝ b✝ : G

⊢ (setoid ↑H.toSubmonoid ↑⊥).Rel a✝ b✝ ↔ (QuotientGroup.rightRel H).Rel a✝ b✝
-- Mathlib/GroupTheory/DoubleCoset.lean
G : Type u_1
inst✝¹ : Group G
α : Type u_2
inst✝ : Mul α
J : Subgroup G
g : G
H : Subgroup G
a✝ b✝ : G

⊢ (setoid ↑H.toSubmonoid ↑⊥).Rel a✝ b✝ ↔ (setoid ↑H ↑⊥).Rel a✝ b✝
-- Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
C : VariableChange R

⊢ (W.variableChange C).Δ = ↑C.u⁻¹ ^ 12 * W.Δ
-- Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
C : VariableChange R

⊢ -((↑C.u⁻¹ * (W.a₁ + 2 * C.s)) ^ 2 + 4 * (↑C.u⁻¹ ^ 2 * (W.a₂ - C.s * W.a₁ + 3 * C.r - C.s ^ 2))) ^ 2 * ((↑C.u⁻¹ * (W.a₁ + 2 * C.s)) ^ 2 * (↑C.u⁻¹ ^ 6 * (W.a₆ + C.r * W.a₄ + C.r ^ 2 * W.a₂ + C.r ^ 3 - C.t * W.a₃ - C.t ^ 2 - C.r * C.t * W.a₁)) + 4 * (↑C.u⁻¹ ^ 2 * (W.a₂ - C.s * W.a₁ + 3 * C.r - C.s ^ 2)) * (↑C.u⁻¹ ^ 6 * (W.a₆ + C.r * W.a₄ + C.r ^ 2 * W.a₂ + C.r ^ 3 - C.t * W.a₃ - C.t ^ 2 - C.r * C.t * W.a₁)) - ↑C.u⁻¹ * (W.a₁ + 2 * C.s) * (↑C.u⁻¹ ^ 3 * (W.a₃ + C.r * W.a₁ + 2 * C.t)) * (↑C.u⁻¹ ^ 4 * (W.a₄ - C.s * W.a₃ + 2 * C.r * W.a₂ - (C.t + C.r * C.s) * W.a₁ + 3 * C.r ^ 2 - 2 * C.s * C.t)) + ↑C.u⁻¹ ^ 2 * (W.a₂ - C.s * W.a₁ + 3 * C.r - C.s ^ 2) * (↑C.u⁻¹ ^ 3 * (W.a₃ + C.r * W.a₁ + 2 * C.t)) ^ 2 - (↑C.u⁻¹ ^ 4 * (W.a₄ - C.s * W.a₃ + 2 * C.r * W.a₂ - (C.t + C.r * C.s) * W.a₁ + 3 * C.r ^ 2 - 2 * C.s * C.t)) ^ 2) - 8 * (2 * (↑C.u⁻¹ ^ 4 * (W.a₄ - C.s * W.a₃ + 2 * C.r * W.a₂ - (C.t + C.r * C.s) * W.a₁ + 3 * C.r ^ 2 - 2 * C.s * C.t)) + ↑C.u⁻¹ * (W.a₁ + 2 * C.s) * (↑C.u⁻¹ ^ 3 * (W.a₃ + C.r * W.a₁ + 2 * C.t))) ^ 3 - 27 * ((↑C.u⁻¹ ^ 3 * (W.a₃ + C.r * W.a₁ + 2 * C.t)) ^ 2 + 4 * (↑C.u⁻¹ ^ 6 * (W.a₆ + C.r * W.a₄ + C.r ^ 2 * W.a₂ + C.r ^ 3 - C.t * W.a₃ - C.t ^ 2 - C.r * C.t * W.a₁))) ^ 2 + 9 * ((↑C.u⁻¹ * (W.a₁ + 2 * C.s)) ^ 2 + 4 * (↑C.u⁻¹ ^ 2 * (W.a₂ - C.s * W.a₁ + 3 * C.r - C.s ^ 2))) * (2 * (↑C.u⁻¹ ^ 4 * (W.a₄ - C.s * W.a₃ + 2 * C.r * W.a₂ - (C.t + C.r * C.s) * W.a₁ + 3 * C.r ^ 2 - 2 * C.s * C.t)) + ↑C.u⁻¹ * (W.a₁ + 2 * C.s) * (↑C.u⁻¹ ^ 3 * (W.a₃ + C.r * W.a₁ + 2 * C.t))) * ((↑C.u⁻¹ ^ 3 * (W.a₃ + C.r * W.a₁ + 2 * C.t)) ^ 2 + 4 * (↑C.u⁻¹ ^ 6 * (W.a₆ + C.r * W.a₄ + C.r ^ 2 * W.a₂ + C.r ^ 3 - C.t * W.a₃ - C.t ^ 2 - C.r * C.t * W.a₁))) = ↑C.u⁻¹ ^ 12 * (-(W.a₁ ^ 2 + 4 * W.a₂) ^ 2 * (W.a₁ ^ 2 * W.a₆ + 4 * W.a₂ * W.a₆ - W.a₁ * W.a₃ * W.a₄ + W.a₂ * W.a₃ ^ 2 - W.a₄ ^ 2) - 8 * (2 * W.a₄ + W.a₁ * W.a₃) ^ 3 - 27 * (W.a₃ ^ 2 + 4 * W.a₆) ^ 2 + 9 * (W.a₁ ^ 2 + 4 * W.a₂) * (2 * W.a₄ + W.a₁ * W.a₃) * (W.a₃ ^ 2 + 4 * W.a₆))