ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- 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
\u22a2 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
\u22a2 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
\u22a2 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
\u22a2 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
\u22a2 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
\u22a2 (((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
\u22a2 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)
\u22a2 (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
\u22a2 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
\u22a2 ((∃ 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
\u22a2 (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
\u22a2 (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
\u22a2 (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
\u22a2 (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
\u22a2 -((↑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))
-- Mathlib/Analysis/Convex/Strict.lean
𝕜 : Type u_1
𝕝 : Type u_2
E : Type u_3
F : Type u_4
β : Type u_5
inst✝⁵ : LinearOrderedField 𝕜
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup E
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
s : Set E
x✝ : E
h : StrictConvex 𝕜 s
x : E
hx : x ∈ s
y : E
hy : y ∈ s
hxy : x ≠ y
a b : 𝕜
ha : 0 < a
hb : 0 < b
\u22a2 (a / (a + b)) • x + (b / (a + b)) • y ∈ interior s
-- Mathlib/Analysis/Convex/Strict.lean
𝕜 : Type u_1
𝕝 : Type u_2
E : Type u_3
F : Type u_4
β : Type u_5
inst✝⁵ : LinearOrderedField 𝕜
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup E
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
s : Set E
x✝ : E
h : StrictConvex 𝕜 s
x : E
hx : x ∈ s
y : E
hy : y ∈ s
hxy : x ≠ y
a b : 𝕜
ha : 0 < a
hb : 0 < b
\u22a2 a / (a + b) + b / (a + b) = 1
-- Mathlib/Analysis/Convex/Strict.lean
𝕜 : Type u_1
𝕝 : Type u_2
E : Type u_3
F : Type u_4
β : Type u_5
inst✝⁵ : LinearOrderedField 𝕜
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup E
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
s : Set E
x✝ : E
h : StrictConvex 𝕜 s
x : E
hx : x ∈ s
y : E
hy : y ∈ s
hxy : x ≠ y
a b : 𝕜
ha : 0 < a
hb : 0 < b
\u22a2 (a + b) / (a + b) = 1
-- Mathlib/Analysis/Convex/Strict.lean
𝕜 : Type u_1
𝕝 : Type u_2
E : Type u_3
F : Type u_4
β : Type u_5
inst✝⁵ : LinearOrderedField 𝕜
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup E
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
s : Set E
x✝ : E
h : s.Pairwise fun x y => ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → (a / (a + b)) • x + (b / (a + b)) • y ∈ interior s
x : E
hx : x ∈ s
y : E
hy : y ∈ s
hxy : x ≠ y
a b : 𝕜
ha : 0 < a
hb : 0 < b
hab : a + b = 1
\u22a2 a • x + b • y ∈ interior s
-- Mathlib/AlgebraicGeometry/Properties.lean
X : Scheme
R : CommRingCat
h : IsReduced (Spec R)
\u22a2 _root_.IsReduced ↑R
-- Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean
α : Type u_1
β : Type u_2
l : Filter α
k f g g' : α → β
inst✝² : TopologicalSpace β
inst✝¹ : CommSemiring β
inst✝ : ContinuousAdd β
hf : SuperpolynomialDecay l k f
hg : SuperpolynomialDecay l k g
z : ℕ
\u22a2 Tendsto (fun a => k a ^ z * (f + g) a) l (𝓝 0)
-- Mathlib/Algebra/Group/Center.lean
M : Type u_1
S T : Set M
inst✝¹ : Monoid M
a : M
inst✝ : Invertible a
ha : a ∈ center M
\u22a2 ⅟a ∈ center M
-- Mathlib/Algebra/Group/Center.lean
M : Type u_1
S T : Set M
inst✝¹ : Monoid M
a : M
inst✝ : Invertible a
ha : ∀ (g : M), g * a = a * g
\u22a2 ∀ (g : M), g * ⅟a = ⅟a * g
-- Mathlib/NumberTheory/ModularForms/JacobiTheta/OneVariable.lean
τ : ℂ
hτ : 0 < τ.im
\u22a2 jacobiTheta τ = 1 + 2 * ∑' (n : ℕ), cexp (↑π * I * (↑n + 1) ^ 2 * τ)
-- Mathlib/MeasureTheory/Measure/Dirac.lean
α : Type u_1
β : Type u_2
δ : Type u_3
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
s✝ : Set α
a✝ : α
inst✝ : MeasurableSingletonClass α
a : α
s : Set α
\u22a2 s ∈ ae (dirac a) ↔ s ∈ pure a
-- Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean
K : Type u
V : Type v
inst✝⁵ : DivisionRing K
inst✝⁴ : AddCommGroup V
inst✝³ : Module K V
V₂ : Type v'
inst✝² : AddCommGroup V₂
inst✝¹ : Module K V₂
inst✝ : FiniteDimensional K V
f g : V →ₗ[K] V
hfg : f * g = 1
\u22a2 g * f = 1
-- Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean
K : Type u
V : Type v
inst✝⁵ : DivisionRing K
inst✝⁴ : AddCommGroup V
inst✝³ : Module K V
V₂ : Type v'
inst✝² : AddCommGroup V₂
inst✝¹ : Module K V₂
inst✝ : FiniteDimensional K V
f g : V →ₗ[K] V
hfg : f * g = 1
ginj : Injective ⇑g
\u22a2 g * f = 1
-- Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean
K : Type u
V : Type v
inst✝⁵ : DivisionRing K
inst✝⁴ : AddCommGroup V
inst✝³ : Module K V
V₂ : Type v'
inst✝² : AddCommGroup V₂
inst✝¹ : Module K V₂
inst✝ : FiniteDimensional K V
f g : V →ₗ[K] V
hfg : f * g = 1
ginj : Injective ⇑g
i : V →ₗ[K] V
hi : g ∘ₗ i = id
\u22a2 g * f = 1
-- Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean
K : Type u
V : Type v
inst✝⁵ : DivisionRing K
inst✝⁴ : AddCommGroup V
inst✝³ : Module K V
V₂ : Type v'
inst✝² : AddCommGroup V₂
inst✝¹ : Module K V₂
inst✝ : FiniteDimensional K V
f g : V →ₗ[K] V
hfg : f * g = 1
ginj : Injective ⇑g
i : V →ₗ[K] V
hi : g ∘ₗ i = id
this : f * (g * i) = f * 1
\u22a2 g * f = 1
-- Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean
K : Type u
V : Type v
inst✝⁵ : DivisionRing K
inst✝⁴ : AddCommGroup V
inst✝³ : Module K V
V₂ : Type v'
inst✝² : AddCommGroup V₂
inst✝¹ : Module K V₂
inst✝ : FiniteDimensional K V
f g : V →ₗ[K] V
hfg : f * g = 1
ginj : Injective ⇑g
i : V →ₗ[K] V
hi : g ∘ₗ i = id
this : i = f
\u22a2 g * f = 1
-- Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean
R✝ : Type u
inst✝¹¹ : CommRing R✝
W✝ : Affine R✝
R : Type r
inst✝¹⁰ : CommRing R
W : Affine R
S : Type s
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
A : Type u
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra S A
inst✝⁴ : IsScalarTower R S A
B : Type v
inst✝³ : CommRing B
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
x₁ x₂ L : A
\u22a2 (baseChange W B).toAffine.addX (f x₁) (f x₂) (f L) = f ((baseChange W A).toAffine.addX x₁ x₂ L)
-- Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean
R✝ : Type u
inst✝¹¹ : CommRing R✝
W✝ : Affine R✝
R : Type r
inst✝¹⁰ : CommRing R
W : Affine R
S : Type s
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
A : Type u
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra S A
inst✝⁴ : IsScalarTower R S A
B : Type v
inst✝³ : CommRing B
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
x₁ x₂ L : A
\u22a2 (baseChange W B).toAffine.addX (f x₁) (f x₂) (f L) = (baseChange W B).toAffine.addX (f.toRingHom x₁) (f.toRingHom x₂) (f.toRingHom L)
-- Mathlib/GroupTheory/Congruence/Basic.lean
M : Type u_1
N : Type u_2
P : Type u_3
inst✝² : Group M
inst✝¹ : Group N
inst✝ : Group P
c : Con M
x✝ y : M
h : c x✝ y
x : M
\u22a2 c (x⁻¹ * x) 1
-- Mathlib/LinearAlgebra/Eigenspace/Basic.lean
K R : Type v
V M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : Field K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : NoZeroSMulDivisors R M
f : End R M
μ₁ μ₂ : R
hμ : μ₁ ≠ μ₂
\u22a2 Disjoint (⨆ k, (f.genEigenspace μ₁) k) (⨆ k, (f.genEigenspace μ₂) k)
-- Mathlib/LinearAlgebra/Eigenspace/Basic.lean
K R : Type v
V M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : Field K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : NoZeroSMulDivisors R M
f : End R M
μ₁ μ₂ : R
hμ : μ₁ ≠ μ₂
\u22a2 ∀ (i i_1 : ℕ), Disjoint ((f.genEigenspace μ₁) i) ((f.genEigenspace μ₂) i_1)
-- Mathlib/CategoryTheory/Sites/Adjunction.lean
C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
J : GrothendieckTopology C
D : Type u₂
inst✝⁴ : Category.{v₂, u₂} D
E : Type u_1
inst✝³ : Category.{?u.49535, u_1} E
F : D ⥤ E
G✝ : E ⥤ D
inst✝² : HasWeakSheafify J D
inst✝¹ : ConcreteCategory D
inst✝ : J.HasSheafCompose (forget D)
G : Type (max v₁ u₁) ⥤ D
adj : G ⊣ forget D
X : Sheaf J D
\u22a2 ((adjunctionToTypes J adj).counit.app X).val = sheafifyLift J ((X.1.associator (forget D) G).hom ≫ (Adjunction.whiskerRight Cᵒᵖ adj).counit.app X.1) ⋯
-- Mathlib/CategoryTheory/Sites/Adjunction.lean
C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
J : GrothendieckTopology C
D : Type u₂
inst✝⁴ : Category.{v₂, u₂} D
E : Type u_1
inst✝³ : Category.{?u.49535, u_1} E
F : D ⥤ E
G✝ : E ⥤ D
inst✝² : HasWeakSheafify J D
inst✝¹ : ConcreteCategory D
inst✝ : J.HasSheafCompose (forget D)
G : Type (max v₁ u₁) ⥤ D
adj : G ⊣ forget D
X : Sheaf J D
\u22a2 toSheafify J (((whiskeringRight Cᵒᵖ (Type (max u₁ v₁)) D).obj G).obj ((sheafToPresheaf J (Type (max u₁ v₁))).obj ((sheafEquivSheafOfTypes J).inverse.obj ((sheafForget J).obj X)))) ≫ ((adjunctionToTypes J adj).counit.app X).val = (X.1.associator (forget D) G).hom ≫ (Adjunction.whiskerRight Cᵒᵖ adj).counit.app X.1
-- Mathlib/CategoryTheory/Sites/Adjunction.lean
C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
J : GrothendieckTopology C
D : Type u₂
inst✝⁴ : Category.{v₂, u₂} D
E : Type u_1
inst✝³ : Category.{?u.49535, u_1} E
F : D ⥤ E
G✝ : E ⥤ D
inst✝² : HasWeakSheafify J D
inst✝¹ : ConcreteCategory D
inst✝ : J.HasSheafCompose (forget D)
G : Type (max v₁ u₁) ⥤ D
adj : G ⊣ forget D
X : Sheaf J D
\u22a2 toSheafify J (((whiskeringRight Cᵒᵖ (Type (max u₁ v₁)) D).obj G).obj ((sheafToPresheaf J (Type (max u₁ v₁))).obj ((sheafEquivSheafOfTypes J).inverse.obj ((sheafForget J).obj X)))) ≫ (((sheafCompose J (forget D)).associator (sheafEquivSheafOfTypes J).symm.inverse ((sheafEquivSheafOfTypes J).symm.functor ⋙ composeAndSheafify J G)).hom.app X).val ≫ ((whiskerLeft (sheafCompose J (forget D)) (whiskerRight (sheafEquivSheafOfTypes J).symm.toAdjunction.counit (composeAndSheafify J G))).app X).val ≫ ((adjunction J adj).counit.app X).val = (X.1.associator (forget D) G).hom ≫ (Adjunction.whiskerRight Cᵒᵖ adj).counit.app X.1
-- Mathlib/CategoryTheory/Sites/Adjunction.lean
C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
J : GrothendieckTopology C
D : Type u₂
inst✝⁴ : Category.{v₂, u₂} D
E : Type u_1
inst✝³ : Category.{?u.49535, u_1} E
F : D ⥤ E
G✝ : E ⥤ D
inst✝² : HasWeakSheafify J D
inst✝¹ : ConcreteCategory D
inst✝ : J.HasSheafCompose (forget D)
G : Type (max v₁ u₁) ⥤ D
adj : G ⊣ forget D
X : Sheaf J D
\u22a2 toSheafify J (((whiskeringRight Cᵒᵖ (Type (max u₁ v₁)) D).obj G).obj ((sheafToPresheaf J (Type (max u₁ v₁))).obj ((sheafEquivSheafOfTypes J).inverse.obj ((sheafForget J).obj X)))) ≫ (((sheafCompose J (forget D)).associator (sheafEquivSheafOfTypes J).symm.inverse ((sheafEquivSheafOfTypes J).symm.functor ⋙ composeAndSheafify J G)).hom.app X).val ≫ ((whiskerLeft (sheafCompose J (forget D)) (whiskerRight (sheafEquivSheafOfTypes J).symm.toAdjunction.counit (composeAndSheafify J G))).app X).val ≫ sheafifyLift J (((Adjunction.whiskerRight Cᵒᵖ adj).homEquiv (X.val ⋙ forget D) X.val).symm (𝟙 (X.val ⋙ forget D))) ⋯ = (X.1.associator (forget D) G).hom ≫ (Adjunction.whiskerRight Cᵒᵖ adj).counit.app X.1
-- Mathlib/CategoryTheory/Sites/Adjunction.lean
C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
J : GrothendieckTopology C
D : Type u₂
inst✝⁴ : Category.{v₂, u₂} D
E : Type u_1
inst✝³ : Category.{?u.49535, u_1} E
F : D ⥤ E
G✝ : E ⥤ D
inst✝² : HasWeakSheafify J D
inst✝¹ : ConcreteCategory D
inst✝ : J.HasSheafCompose (forget D)
G : Type (max v₁ u₁) ⥤ D
adj : G ⊣ forget D
X : Sheaf J D
\u22a2 ((whiskeringRight Cᵒᵖ (Type (max u₁ v₁)) D).obj G).map ((sheafToPresheaf J (Type (max u₁ v₁))).map ((sheafEquivSheafOfTypes J).symm.toAdjunction.counit.app ((sheafCompose J (forget D)).obj X))) ≫ ((Adjunction.whiskerRight Cᵒᵖ adj).homEquiv (X.val ⋙ forget D) X.val).symm (𝟙 (X.val ⋙ forget D)) = (X.1.associator (forget D) G).hom ≫ (Adjunction.whiskerRight Cᵒᵖ adj).counit.app X.1
-- Mathlib/Algebra/Homology/ShortComplex/Homology.lean
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
S S₁ S₂ S₃ S₄ : ShortComplex C
φ✝ : S₁ ⟶ S₂
h₁ : S₁.HomologyData
h₂ : S₂.HomologyData
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
\u22a2 S₁.rightHomologyIso.inv ≫ rightHomologyMap φ = homologyMap φ ≫ S₂.rightHomologyIso.inv
-- Mathlib/Combinatorics/SimpleGraph/ConcreteColorings.lean
n : ℕ
h : 2 ≤ n
\u22a2 (pathGraph n).chromaticNumber = 2
-- Mathlib/Combinatorics/SimpleGraph/ConcreteColorings.lean
n : ℕ
h : 2 ≤ n
hc : (pathGraph n).Colorable (Fintype.card Bool)
\u22a2 (pathGraph n).chromaticNumber = 2
-- Mathlib/Combinatorics/SimpleGraph/ConcreteColorings.lean
n : ℕ
h : 2 ≤ n
hc : (pathGraph n).Colorable (Fintype.card Bool)
\u22a2 (pathGraph n).chromaticNumber ≤ 2
-- Mathlib/Combinatorics/SimpleGraph/ConcreteColorings.lean
n : ℕ
h : 2 ≤ n
hc : (pathGraph n).Colorable (Fintype.card Bool)
\u22a2 2 ≤ (pathGraph n).chromaticNumber
-- Mathlib/Probability/Kernel/Composition.lean
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
δ : Type u_5
mγ : MeasurableSpace γ
mδ : MeasurableSpace δ
f : β → γ
g : γ → α
κ : Kernel α β
a : α
s✝ : Set β
a✝ : MeasurableSet s✝
\u22a2 ((κ.map id ⋯) a) s✝ = (κ a) s✝
-- Mathlib/Probability/Kernel/Composition.lean
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
δ : Type u_5
mγ : MeasurableSpace γ
mδ : MeasurableSpace δ
f : β → γ
g : γ → α
κ : Kernel α β
a : α
s✝ : Set β
a✝ : MeasurableSet s✝
\u22a2 (Measure.map id (κ a)) s✝ = (κ a) s✝
-- Mathlib/Data/Real/Cardinality.lean
c : ℝ
f g : ℕ → Bool
n : ℕ
a : ℝ
\u22a2 #↑(Iio a) = 𝔠
-- Mathlib/Data/Real/Cardinality.lean
c : ℝ
f g : ℕ → Bool
n : ℕ
a : ℝ
\u22a2 𝔠 ≤ #↑(Iio a)
-- Mathlib/Data/Real/Cardinality.lean
c : ℝ
f g : ℕ → Bool
n : ℕ
a : ℝ
h2 : (fun x => a + a - x) '' Iio a = Ioi a
\u22a2 𝔠 ≤ #↑(Iio a)
-- Mathlib/Topology/Order.lean
α : Type u_1
t t₁ t₂ : TopologicalSpace α
s✝ : Set α
h : ∀ (x : α), 𝓝 x ≤ 𝓝 x
s : Set α
\u22a2 IsOpen s → IsOpen s
-- Mathlib/Topology/Order.lean
α : Type u_1
t t₁ t₂ : TopologicalSpace α
s✝ : Set α
h : ∀ (x : α), 𝓝 x ≤ 𝓝 x
s : Set α
\u22a2 (∀ x ∈ s, s ∈ 𝓝 x) → ∀ x ∈ s, s ∈ 𝓝 x
-- Mathlib/LinearAlgebra/Matrix/ToLin.lean
R : Type u_1
inst✝¹⁰ : CommSemiring R
l : Type u_2
m : Type u_3
n : Type u_4
inst✝⁹ : Fintype n
inst✝⁸ : Finite m
inst✝⁷ : DecidableEq n
M₁ : Type u_5
M₂ : Type u_6
inst✝⁶ : AddCommMonoid M₁
inst✝⁵ : AddCommMonoid M₂
inst✝⁴ : Module R M₁
inst✝³ : Module R M₂
v₁ : Basis n R M₁
v₂ : Basis m R M₂
G : Type u_7
inst✝² : Group G
inst✝¹ : DistribMulAction G M₁
inst✝ : SMulCommClass G R M₁
g : G
f : M₁ →ₗ[R] M₂
i✝ : m
j✝ : n
\u22a2 (toMatrix (g • v₁) v₂) f i✝ j✝ = (toMatrix v₁ v₂) (f ∘ₗ DistribMulAction.toLinearMap R M₁ g) i✝ j✝