ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/LinearAlgebra/QuadraticForm/Basic.lean
S : Type u_1
T : Type u_2
R : Type u_3
M : Type u_4
N : Type u_5
P : Type u_6
A : Type u_7
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
Q : QuadraticMap R M N
x : M

⊢ Q.IsOrtho x 0
-- Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean
V : Type u
V' : Type v
G : SimpleGraph V
G' : SimpleGraph V'
H H' : G.Subgraph
hle : H ≤ H'
hv : H.verts = H'.verts
h : H.Connected

⊢ H'.Connected
-- Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean
V : Type u
V' : Type v
G : SimpleGraph V
G' : SimpleGraph V'
H H' : G.Subgraph
hle : H ≤ H'
hv : H.verts = H'.verts
h : H.Connected

⊢ (H'.copy H.verts hv H'.Adj ⋯).Connected
-- Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean
V : Type u
V' : Type v
G : SimpleGraph V
G' : SimpleGraph V'
H H' : G.Subgraph
hle : H ≤ H'
hv : H.verts = H'.verts
h : H.Connected

⊢ H.coe ≤ (H'.copy H.verts hv H'.Adj ⋯).coe
-- Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean
V : Type u
V' : Type v
G : SimpleGraph V
G' : SimpleGraph V'
H H' : G.Subgraph
hle : H ≤ H'
hv✝ : H.verts = H'.verts
h : H.Connected
v : V
hv : v ∈ H.verts
w : V
hw : w ∈ H.verts
hvw : H.coe.Adj ⟨v, hv⟩ ⟨w, hw⟩

⊢ (H'.copy H.verts hv✝ H'.Adj ⋯).coe.Adj ⟨v, hv⟩ ⟨w, hw⟩
-- Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
n✝ : ℕ
f : 𝕜 → F
s : Set 𝕜
x : 𝕜
n : ℕ∞
h : ∀ (m : ℕ), ↑m ≤ n → DifferentiableOn 𝕜 (iteratedDerivWithin m f s) s

⊢ ContDiffOn 𝕜 n f s
-- Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
n✝ : ℕ
f : 𝕜 → F
s : Set 𝕜
x : 𝕜
n : ℕ∞
h : ∀ (m : ℕ), ↑m ≤ n → DifferentiableOn 𝕜 (iteratedDerivWithin m f s) s

⊢ ∀ (m : ℕ), ↑m ≤ n → DifferentiableOn 𝕜 (iteratedFDerivWithin 𝕜 m f s) s
-- Mathlib/Algebra/AddTorsor.lean
G : Type u_1
P : Type u_2
inst✝ : AddGroup G
T : AddTorsor G P
g₁ g₂ : G
p : P
h : g₁ +ᵥ p = g₂ +ᵥ p

⊢ g₁ = g₂
-- Mathlib/CategoryTheory/Comma/StructuredArrow.lean
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
S✝ S' S'' : D
Y Y' Y'' : C
T T' : C ⥤ D
A : Type u₃
inst✝¹ : Category.{v₃, u₃} A
B : Type u₄
inst✝ : Category.{v₄, u₄} B
S : C
F : B ⥤ C
G : C ⥤ D
X✝ Y✝ : StructuredArrow S F
f : X✝ ⟶ Y✝

⊢ ((fun X => mk (G.map X.hom)) X✝).hom ≫ (F ⋙ G).map f.right = ((fun X => mk (G.map X.hom)) Y✝).hom
-- Mathlib/NumberTheory/NumberField/Units/Basic.lean
x : 𝓞 ℚ

⊢ IsUnit x ↔ ↑x = 1 ∨ ↑x = -1
-- Mathlib/NumberTheory/NumberField/Units/Basic.lean
x : 𝓞 ℚ

⊢ ↑x = ↑1 ∨ ↑x = ↑(-1) ↔ ↑x = 1 ∨ ↑x = -1
-- Mathlib/GroupTheory/Perm/Cycle/Type.lean
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ : Perm α
h : σ.IsThreeCycle

⊢ σ.support.card = 3
-- Mathlib/Analysis/SpecialFunctions/Bernstein.lean
n ν : ℕ
x : ↑I

⊢ 0 ≤ (bernstein n ν) x
-- Mathlib/Analysis/SpecialFunctions/Bernstein.lean
n ν : ℕ
x : ↑I

⊢ 0 ≤ ↑(n.choose ν) * ↑x ^ ν * (1 - ↑x) ^ (n - ν)
-- Mathlib/Analysis/SpecialFunctions/Bernstein.lean
n ν : ℕ
x : ↑I
h₁ : 0 ≤ ↑x

⊢ 0 ≤ ↑(n.choose ν) * ↑x ^ ν * (1 - ↑x) ^ (n - ν)
-- Mathlib/Analysis/SpecialFunctions/Bernstein.lean
n ν : ℕ
x : ↑I
h₁ : 0 ≤ ↑x
h₂ : 0 ≤ 1 - ↑x

⊢ 0 ≤ ↑(n.choose ν) * ↑x ^ ν * (1 - ↑x) ^ (n - ν)
-- Mathlib/CategoryTheory/Monad/Limits.lean
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
J : Type u
inst✝² : Category.{v, u} J
R : D ⥤ C
inst✝¹ : Coreflective R
inst✝ : HasLimitsOfShape J C
F : J ⥤ D
c : Cone ((F ⋙ R) ⋙ comonadicRightAdjoint R) := (comonadicRightAdjoint R).mapCone (limit.cone (F ⋙ R))
this : PreservesLimitsOfShape J (comonadicRightAdjoint R) := PreservesLimitsOfSize.preservesLimitsOfShape
t : IsLimit c := isLimitOfPreserves (comonadicRightAdjoint R) (limit.isLimit (F ⋙ R))

⊢ HasLimit F
-- Mathlib/CategoryTheory/Monad/Limits.lean
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
J : Type u
inst✝² : Category.{v, u} J
R : D ⥤ C
inst✝¹ : Coreflective R
inst✝ : HasLimitsOfShape J C
F : J ⥤ D
c : Cone ((F ⋙ R) ⋙ comonadicRightAdjoint R) := (comonadicRightAdjoint R).mapCone (limit.cone (F ⋙ R))
this : PreservesLimitsOfShape J (comonadicRightAdjoint R) := PreservesLimitsOfSize.preservesLimitsOfShape
t : IsLimit c := isLimitOfPreserves (comonadicRightAdjoint R) (limit.isLimit (F ⋙ R))

⊢ (F ⋙ R) ⋙ comonadicRightAdjoint R ≅ F
-- Mathlib/Analysis/MeanInequalitiesPow.lean
ι : Type u
s : Finset ι
f : ι → ℝ
n : ℕ
hf : ∀ a ∈ s, 0 ≤ f a

⊢ (∑ x ∈ s, f x) ^ (n + 1) / ↑s.card ^ n ≤ ∑ x ∈ s, f x ^ (n + 1)
-- Mathlib/Analysis/MeanInequalitiesPow.lean
ι : Type u
f : ι → ℝ
n : ℕ
hf : ∀ a ∈ ∅, 0 ≤ f a

⊢ (∑ x ∈ ∅, f x) ^ (n + 1) / ↑∅.card ^ n ≤ ∑ x ∈ ∅, f x ^ (n + 1)
-- Mathlib/Analysis/MeanInequalitiesPow.lean
ι : Type u
s : Finset ι
f : ι → ℝ
n : ℕ
hf : ∀ a ∈ s, 0 ≤ f a
hs : s.Nonempty

⊢ (∑ x ∈ s, f x) ^ (n + 1) / ↑s.card ^ n ≤ ∑ x ∈ s, f x ^ (n + 1)
-- Mathlib/Data/Real/Hyperreal.lean
x y : ℝ*

⊢ x.InfiniteNeg → ¬y.InfinitePos → (x + y).InfiniteNeg
-- Mathlib/Data/Real/Hyperreal.lean
x y : ℝ*

⊢ (-x).InfinitePos → ¬(-y).InfiniteNeg → (-x + -y).InfinitePos
-- Mathlib/Algebra/Order/UpperLower.lean
α : Type u_1
inst✝ : OrderedCommGroup α
s t : Set α
a : α

⊢ s * ↑(upperClosure t) = ↑(upperClosure (s * t))
-- Mathlib/Algebra/Order/UpperLower.lean
α : Type u_1
inst✝ : OrderedCommGroup α
s t : Set α
a : α

⊢ ⋃ a ∈ s, a • ↑(upperClosure t) = ⋃ i ∈ s, ↑(i • upperClosure t)
-- Mathlib/MeasureTheory/Decomposition/Jordan.lean
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
s : SignedMeasure α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
u v w : Set α
hu : MeasurableSet u
hv : MeasurableSet v
hsu : VectorMeasure.restrict s u ≤ VectorMeasure.restrict 0 u
hsv : VectorMeasure.restrict s v ≤ VectorMeasure.restrict 0 v
hs : ↑s (u ∆ v) = 0

⊢ ↑s (u \ v) = 0 ∧ ↑s (v \ u) = 0
-- Mathlib/MeasureTheory/Decomposition/Jordan.lean
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
s : SignedMeasure α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
u v w : Set α
hu : MeasurableSet u
hv : MeasurableSet v
hsu : VectorMeasure.restrict 0 u ≤ (-s).restrict u
hsv : VectorMeasure.restrict s v ≤ VectorMeasure.restrict 0 v
hs : ↑s (u ∆ v) = 0

⊢ ↑s (u \ v) = 0 ∧ ↑s (v \ u) = 0
-- Mathlib/MeasureTheory/Decomposition/Jordan.lean
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
s : SignedMeasure α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
u v w : Set α
hu : MeasurableSet u
hv : MeasurableSet v
hsu : VectorMeasure.restrict 0 u ≤ (-s).restrict u
hsv : VectorMeasure.restrict 0 v ≤ (-s).restrict v
hs : ↑s (u ∆ v) = 0

⊢ ↑s (u \ v) = 0 ∧ ↑s (v \ u) = 0
-- Mathlib/MeasureTheory/Decomposition/Jordan.lean
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
s : SignedMeasure α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
u v w : Set α
hu : MeasurableSet u
hv : MeasurableSet v
hsu : VectorMeasure.restrict 0 u ≤ (-s).restrict u
hsv : VectorMeasure.restrict 0 v ≤ (-s).restrict v
hs : ↑s (u ∆ v) = 0
this : ↑(-s) (u ∆ v) = 0 → ↑(-s) (u \ v) = 0 ∧ ↑(-s) (v \ u) = 0

⊢ ↑s (u \ v) = 0 ∧ ↑s (v \ u) = 0
-- Mathlib/MeasureTheory/Decomposition/Jordan.lean
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
s : SignedMeasure α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
u v w : Set α
hu : MeasurableSet u
hv : MeasurableSet v
hsu : VectorMeasure.restrict 0 u ≤ (-s).restrict u
hsv : VectorMeasure.restrict 0 v ≤ (-s).restrict v
hs : ↑s (u ∆ v) = 0
this : ↑s (u ∆ v) = 0 → ↑s (u \ v) = 0 ∧ ↑s (v \ u) = 0

⊢ ↑s (u \ v) = 0 ∧ ↑s (v \ u) = 0
-- Mathlib/Algebra/Polynomial/Degree/Lemmas.lean
R : Type u
S : Type v
ι : Type w
a b : R
m n : ℕ
K : Type u_1
inst✝ : DivisionRing K
p : K[X]

⊢ p.degree + (C p.leadingCoeff⁻¹).degree = p.degree
-- Mathlib/FieldTheory/RatFunc/Defs.lean
K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
p q : K[X]

⊢ RatFunc.mk p q = { toFractionRing := (algebraMap K[X] (FractionRing K[X])) p / (algebraMap K[X] (FractionRing K[X])) q }
-- Mathlib/Analysis/Normed/Group/HomCompletion.lean
G : Type u_1
inst✝³ : SeminormedAddCommGroup G
H : Type u_2
inst✝² : SeminormedAddCommGroup H
inst✝¹ : T0Space H
inst✝ : CompleteSpace H
f : NormedAddGroupHom G H
g : NormedAddGroupHom (Completion G) H
hg : ∀ (v : G), f v = g (↑G v)
v : Completion G

⊢ f.extension v = g v
-- Mathlib/Order/SuccPred/LinearLocallyFinite.lean
ι : Type u_1
inst✝ : LinearOrder ι
i : ι

⊢ i ≤ succFn i
-- Mathlib/Order/SuccPred/LinearLocallyFinite.lean
ι : Type u_1
inst✝ : LinearOrder ι
i : ι

⊢ ∀ x ∈ Set.Ioi i, i ≤ x
-- Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι
inst✝¹¹ : Fintype ι'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁵ : (i : ι') → SeminormedAddCommGroup (E' i)
inst✝⁴ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : SeminormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f : G →ₗ[𝕜] MultilinearMap 𝕜 E G'
C : ℝ
H : ∀ (x : G) (m : (i : ι) → E i), ‖(f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖

⊢ ‖{ toFun := fun x => (f x).mkContinuous (C * ‖x‖) ⋯, map_add' := ⋯, map_smul' := ⋯ }.mkContinuous (max C 0) ⋯‖ ≤ max C 0
-- Mathlib/RingTheory/RootsOfUnity/Basic.lean
M : Type u_1
N : Type u_2
G : Type u_3
R : Type u_4
S : Type u_5
F : Type u_6
inst✝² : CommMonoid M
inst✝¹ : CommMonoid N
inst✝ : DivisionCommMonoid G
k l : ℕ
ζ✝ : M
f : F
h : IsPrimitiveRoot ζ✝ k
ζ : Mˣ

⊢ IsPrimitiveRoot (↑ζ) k ↔ IsPrimitiveRoot ζ k
-- Mathlib/Geometry/RingedSpace/Basic.lean
X : RingedSpace
U : Opens ↑↑X.toPresheafedSpace
f : ↑(X.presheaf.obj (op U))
x : ↥U
left✝ : x ∈ {x | IsUnit ((X.presheaf.germ x) f)}

⊢ ↑x ∈ ↑U
-- Mathlib/Algebra/Homology/HomologicalComplex.lean
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
c : ComplexShape ι
C : HomologicalComplex V c
i j j' : ι
rij : c.Rel i j
rij' : c.Rel i j'

⊢ C.d i j' ≫ eqToHom ⋯ = C.d i j
-- Mathlib/Algebra/Homology/HomologicalComplex.lean
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
c : ComplexShape ι
C : HomologicalComplex V c
i j : ι
rij rij' : c.Rel i j

⊢ C.d i j ≫ eqToHom ⋯ = C.d i j
-- Mathlib/RingTheory/DedekindDomain/Different.lean
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁹ : CommRing A
inst✝¹⁸ : Field K
inst✝¹⁷ : CommRing B
inst✝¹⁶ : Field L
inst✝¹⁵ : Algebra A K
inst✝¹⁴ : Algebra B L
inst✝¹³ : Algebra A B
inst✝¹² : Algebra K L
inst✝¹¹ : Algebra A L
inst✝¹⁰ : IsScalarTower A K L
inst✝⁹ : IsScalarTower A B L
inst✝⁸ : IsDomain A
inst✝⁷ : IsFractionRing A K
inst✝⁶ : IsIntegralClosure B A L
inst✝⁵ : IsFractionRing B L
inst✝⁴ : FiniteDimensional K L
inst✝³ : Algebra.IsSeparable K L
inst✝² : IsIntegrallyClosed A
inst✝¹ : IsDedekindDomain B
I : FractionalIdeal B⁰ L
hI : I ≠ 0
inst✝ : NoZeroSMulDivisors A B

⊢ ↑(differentIdeal A B) ≤ I ↔ Submodule.map (↑A (Algebra.trace K L)) (restrictScalars A ↑I⁻¹) ≤ 1
-- Mathlib/RingTheory/DedekindDomain/Different.lean
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁹ : CommRing A
inst✝¹⁸ : Field K
inst✝¹⁷ : CommRing B
inst✝¹⁶ : Field L
inst✝¹⁵ : Algebra A K
inst✝¹⁴ : Algebra B L
inst✝¹³ : Algebra A B
inst✝¹² : Algebra K L
inst✝¹¹ : Algebra A L
inst✝¹⁰ : IsScalarTower A K L
inst✝⁹ : IsScalarTower A B L
inst✝⁸ : IsDomain A
inst✝⁷ : IsFractionRing A K
inst✝⁶ : IsIntegralClosure B A L
inst✝⁵ : IsFractionRing B L
inst✝⁴ : FiniteDimensional K L
inst✝³ : Algebra.IsSeparable K L
inst✝² : IsIntegrallyClosed A
inst✝¹ : IsDedekindDomain B
I : FractionalIdeal B⁰ L
hI : I ≠ 0
inst✝ : NoZeroSMulDivisors A B

⊢ ↑I⁻¹ ≤ traceDual A K 1 ↔ Submodule.map (↑A (Algebra.trace K L)) (restrictScalars A ↑I⁻¹) ≤ 1
-- Mathlib/RingTheory/DedekindDomain/Different.lean
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁹ : CommRing A
inst✝¹⁸ : Field K
inst✝¹⁷ : CommRing B
inst✝¹⁶ : Field L
inst✝¹⁵ : Algebra A K
inst✝¹⁴ : Algebra B L
inst✝¹³ : Algebra A B
inst✝¹² : Algebra K L
inst✝¹¹ : Algebra A L
inst✝¹⁰ : IsScalarTower A K L
inst✝⁹ : IsScalarTower A B L
inst✝⁸ : IsDomain A
inst✝⁷ : IsFractionRing A K
inst✝⁶ : IsIntegralClosure B A L
inst✝⁵ : IsFractionRing B L
inst✝⁴ : FiniteDimensional K L
inst✝³ : Algebra.IsSeparable K L
inst✝² : IsIntegrallyClosed A
inst✝¹ : IsDedekindDomain B
I : FractionalIdeal B⁰ L
hI : I ≠ 0
inst✝ : NoZeroSMulDivisors A B

⊢ Submodule.map (↑A (Algebra.trace K L)) (restrictScalars A (↑I⁻¹ * 1)) ≤ 1 ↔ Submodule.map (↑A (Algebra.trace K L)) (restrictScalars A ↑I⁻¹) ≤ 1
-- Mathlib/Topology/Separation.lean
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : FirstCountableTopology X
s : Set X
inst✝ : DiscreteTopology ↑s
x : X
hx : x ∈ s

⊢ {x} ∈ 𝓝[s] x
-- Mathlib/Topology/Separation.lean
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : FirstCountableTopology X
s : Set X
inst✝ : DiscreteTopology ↑s
x : X
hx : x ∈ s
this : {⟨x, hx⟩} ∈ 𝓝 ⟨x, hx⟩

⊢ {x} ∈ 𝓝[s] x
-- Mathlib/CategoryTheory/Bicategory/Functor/Pseudofunctor.lean
B : Type u₁
inst✝⁴ : Bicategory B
C : Type u₂
inst✝³ : Bicategory C
D : Type u₃
inst✝² : Bicategory D
F✝ : Pseudofunctor B C
F : OplaxFunctor B C
inst✝¹ : ∀ (a : B), IsIso (F.mapId a)
inst✝ : ∀ {a b c : B} (f : a ⟶ b) (g : b ⟶ c), IsIso (F.mapComp f g)
a✝ b✝ c✝ : B
f : a✝ ⟶ b✝
g h : b✝ ⟶ c✝
η : g ⟶ h

⊢ F.map₂ (f ◁ η) = F.mapComp f g ≫ F.map f ◁ F.map₂ η ≫ inv (F.mapComp f h)
-- Mathlib/CategoryTheory/Bicategory/Functor/Pseudofunctor.lean
B : Type u₁
inst✝⁴ : Bicategory B
C : Type u₂
inst✝³ : Bicategory C
D : Type u₃
inst✝² : Bicategory D
F✝ : Pseudofunctor B C
F : OplaxFunctor B C
inst✝¹ : ∀ (a : B), IsIso (F.mapId a)
inst✝ : ∀ {a b c : B} (f : a ⟶ b) (g : b ⟶ c), IsIso (F.mapComp f g)
a✝ b✝ c✝ : B
f✝ g✝ : a✝ ⟶ b✝
η : f✝ ⟶ g✝
h : b✝ ⟶ c✝

⊢ F.map₂ (η ▷ h) = F.mapComp f✝ h ≫ F.map₂ η ▷ F.map h ≫ inv (F.mapComp g✝ h)
-- Mathlib/CategoryTheory/Bicategory/Functor/Pseudofunctor.lean
B : Type u₁
inst✝⁴ : Bicategory B
C : Type u₂
inst✝³ : Bicategory C
D : Type u₃
inst✝² : Bicategory D
F✝ : Pseudofunctor B C
F : OplaxFunctor B C
inst✝¹ : ∀ (a : B), IsIso (F.mapId a)
inst✝ : ∀ {a b c : B} (f : a ⟶ b) (g : b ⟶ c), IsIso (F.mapComp f g)
a✝ b✝ c✝ d✝ : B
f : a✝ ⟶ b✝
g : b✝ ⟶ c✝
h : c✝ ⟶ d✝

⊢ F.map₂ (α_ f g h).hom = F.mapComp (f ≫ g) h ≫ F.mapComp f g ▷ F.map h ≫ (α_ (F.map f) (F.map g) (F.map h)).hom ≫ F.map f ◁ inv (F.mapComp g h) ≫ inv (F.mapComp f (g ≫ h))
-- Mathlib/CategoryTheory/Bicategory/Functor/Pseudofunctor.lean
B : Type u₁
inst✝⁴ : Bicategory B
C : Type u₂
inst✝³ : Bicategory C
D : Type u₃
inst✝² : Bicategory D
F✝ : Pseudofunctor B C
F : OplaxFunctor B C
inst✝¹ : ∀ (a : B), IsIso (F.mapId a)
inst✝ : ∀ {a b c : B} (f : a ⟶ b) (g : b ⟶ c), IsIso (F.mapComp f g)
a✝ b✝ c✝ d✝ : B
f : a✝ ⟶ b✝
g : b✝ ⟶ c✝
h : c✝ ⟶ d✝

⊢ F.map₂ (α_ f g h).hom = (((F.mapComp (f ≫ g) h ≫ F.mapComp f g ▷ F.map h) ≫ (α_ (F.map f) (F.map g) (F.map h)).hom) ≫ F.map f ◁ inv (F.mapComp g h)) ≫ inv (F.mapComp f (g ≫ h))
-- Mathlib/CategoryTheory/Bicategory/Functor/Pseudofunctor.lean
B : Type u₁
inst✝⁴ : Bicategory B
C : Type u₂
inst✝³ : Bicategory C
D : Type u₃
inst✝² : Bicategory D
F✝ : Pseudofunctor B C
F : OplaxFunctor B C
inst✝¹ : ∀ (a : B), IsIso (F.mapId a)
inst✝ : ∀ {a b c : B} (f : a ⟶ b) (g : b ⟶ c), IsIso (F.mapComp f g)
a✝ b✝ c✝ d✝ : B
f : a✝ ⟶ b✝
g : b✝ ⟶ c✝
h : c✝ ⟶ d✝

⊢ (F.map₂ (α_ f g h).hom ≫ F.mapComp f (g ≫ h)) ≫ F.map f ◁ F.mapComp g h = (F.mapComp (f ≫ g) h ≫ F.mapComp f g ▷ F.map h) ≫ (α_ (F.map f) (F.map g) (F.map h)).hom
-- Mathlib/Topology/EMetricSpace/Basic.lean
α : Type u
β : Type v
X : Type u_1
inst✝ : PseudoEMetricSpace α
x y z : α
ε ε₁ ε₂ : ℝ≥0∞
s t : Set α

⊢ Inseparable x y ↔ edist x y = 0
-- Mathlib/Topology/VectorBundle/Hom.lean
𝕜₁ : Type u_1
inst✝²⁰ : NontriviallyNormedField 𝕜₁
𝕜₂ : Type u_2
inst✝¹⁹ : NontriviallyNormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
iσ : RingHomIsometric σ
B : Type u_3
F₁ : Type u_4
inst✝¹⁸ : NormedAddCommGroup F₁
inst✝¹⁷ : NormedSpace 𝕜₁ F₁
E₁ : B → Type u_5
inst✝¹⁶ : (x : B) → AddCommGroup (E₁ x)
inst✝¹⁵ : (x : B) → Module 𝕜₁ (E₁ x)
inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst✝¹³ : NormedAddCommGroup F₂
inst✝¹² : NormedSpace 𝕜₂ F₂
E₂ : B → Type u_7
inst✝¹¹ : (x : B) → AddCommGroup (E₂ x)
inst✝¹⁰ : (x : B) → Module 𝕜₂ (E₂ x)
inst✝⁹ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁸ : TopologicalSpace B
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
inst✝⁷ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁶ : FiberBundle F₁ E₁
inst✝⁵ : (x : B) → TopologicalSpace (E₂ x)
ita : ∀ (x : B), TopologicalAddGroup (E₂ x)
inst✝⁴ : FiberBundle F₂ E₂
inst✝³ : Trivialization.IsLinear 𝕜₁ e₁
inst✝² : Trivialization.IsLinear 𝕜₁ e₁'
inst✝¹ : Trivialization.IsLinear 𝕜₂ e₂
inst✝ : Trivialization.IsLinear 𝕜₂ e₂'
b : B
hb : b ∈ e₁.baseSet ∩ e₂.baseSet
L : F₁ →SL[σ] F₂

⊢ (continuousLinearMap σ e₁ e₂).symm b L = (Trivialization.symmL 𝕜₂ e₂ b).comp (L.comp (Trivialization.continuousLinearMapAt 𝕜₁ e₁ b))
-- Mathlib/Topology/VectorBundle/Hom.lean
𝕜₁ : Type u_1
inst✝²⁰ : NontriviallyNormedField 𝕜₁
𝕜₂ : Type u_2
inst✝¹⁹ : NontriviallyNormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
iσ : RingHomIsometric σ
B : Type u_3
F₁ : Type u_4
inst✝¹⁸ : NormedAddCommGroup F₁
inst✝¹⁷ : NormedSpace 𝕜₁ F₁
E₁ : B → Type u_5
inst✝¹⁶ : (x : B) → AddCommGroup (E₁ x)
inst✝¹⁵ : (x : B) → Module 𝕜₁ (E₁ x)
inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst✝¹³ : NormedAddCommGroup F₂
inst✝¹² : NormedSpace 𝕜₂ F₂
E₂ : B → Type u_7
inst✝¹¹ : (x : B) → AddCommGroup (E₂ x)
inst✝¹⁰ : (x : B) → Module 𝕜₂ (E₂ x)
inst✝⁹ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁸ : TopologicalSpace B
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
inst✝⁷ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁶ : FiberBundle F₁ E₁
inst✝⁵ : (x : B) → TopologicalSpace (E₂ x)
ita : ∀ (x : B), TopologicalAddGroup (E₂ x)
inst✝⁴ : FiberBundle F₂ E₂
inst✝³ : Trivialization.IsLinear 𝕜₁ e₁
inst✝² : Trivialization.IsLinear 𝕜₁ e₁'
inst✝¹ : Trivialization.IsLinear 𝕜₂ e₂
inst✝ : Trivialization.IsLinear 𝕜₂ e₂'
b : B
hb : b ∈ e₁.baseSet ∩ e₂.baseSet
L : F₁ →SL[σ] F₂

⊢ cast ⋯ (↑(continuousLinearMap σ e₁ e₂).symm (b, L)).snd = (Trivialization.symmL 𝕜₂ e₂ b).comp (L.comp (Trivialization.continuousLinearMapAt 𝕜₁ e₁ b))
-- Mathlib/Topology/VectorBundle/Hom.lean
𝕜₁ : Type u_1
inst✝²⁰ : NontriviallyNormedField 𝕜₁
𝕜₂ : Type u_2
inst✝¹⁹ : NontriviallyNormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
iσ : RingHomIsometric σ
B : Type u_3
F₁ : Type u_4
inst✝¹⁸ : NormedAddCommGroup F₁
inst✝¹⁷ : NormedSpace 𝕜₁ F₁
E₁ : B → Type u_5
inst✝¹⁶ : (x : B) → AddCommGroup (E₁ x)
inst✝¹⁵ : (x : B) → Module 𝕜₁ (E₁ x)
inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst✝¹³ : NormedAddCommGroup F₂
inst✝¹² : NormedSpace 𝕜₂ F₂
E₂ : B → Type u_7
inst✝¹¹ : (x : B) → AddCommGroup (E₂ x)
inst✝¹⁰ : (x : B) → Module 𝕜₂ (E₂ x)
inst✝⁹ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁸ : TopologicalSpace B
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
inst✝⁷ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁶ : FiberBundle F₁ E₁
inst✝⁵ : (x : B) → TopologicalSpace (E₂ x)
ita : ∀ (x : B), TopologicalAddGroup (E₂ x)
inst✝⁴ : FiberBundle F₂ E₂
inst✝³ : Trivialization.IsLinear 𝕜₁ e₁
inst✝² : Trivialization.IsLinear 𝕜₁ e₁'
inst✝¹ : Trivialization.IsLinear 𝕜₂ e₂
inst✝ : Trivialization.IsLinear 𝕜₂ e₂'
b : B
hb : b ∈ e₁.baseSet ∩ e₂.baseSet
L : F₁ →SL[σ] F₂

⊢ b ∈ (continuousLinearMap σ e₁ e₂).baseSet
-- Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
α : Type u
a : α
inst✝² : Group α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
ha : ∀ (x : α), x ∈ zpowers a

⊢ image (fun i => a ^ i) (range (orderOf a)) = univ
-- Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
α : Type u
a : α
inst✝² : Group α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
ha : ∀ (x : α), x ∈ ↑(zpowers a)

⊢ image (fun i => a ^ i) (range (orderOf a)) = univ
-- Mathlib/Topology/Algebra/Group/Basic.lean
G : Type w
H : Type x
α : Type u
β : Type v
ι : Sort u_1
inst✝ : Group G
t₁ t₂ : TopologicalSpace G
h₁ : TopologicalGroup G
h₂ : TopologicalGroup G

⊢ TopologicalGroup G
-- Mathlib/Topology/Algebra/Group/Basic.lean
G : Type w
H : Type x
α : Type u
β : Type v
ι : Sort u_1
inst✝ : Group G
t₁ t₂ : TopologicalSpace G
h₁ : TopologicalGroup G
h₂ : TopologicalGroup G
b : Bool

⊢ TopologicalGroup G
-- Mathlib/CategoryTheory/Monoidal/Bimod.lean
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
W X Y Z : Mon_ C
M M' : Bimod W X
f : M ⟶ M'
N : Bimod X Y
P : Bimod Y Z

⊢ (whiskerRight f { X := TensorBimod.X N P, actLeft := TensorBimod.actLeft N P, one_actLeft := ⋯, left_assoc := ⋯, actRight := TensorBimod.actRight N P, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ }).hom = ((isoOfIso { hom := AssociatorBimod.hom M N P, inv := AssociatorBimod.inv M N P, hom_inv_id := ⋯, inv_hom_id := ⋯ } ⋯ ⋯).inv ≫ whiskerRight (whiskerRight f N) P ≫ (isoOfIso { hom := AssociatorBimod.hom M' N P, inv := AssociatorBimod.inv M' N P, hom_inv_id := ⋯, inv_hom_id := ⋯ } ⋯ ⋯).hom).hom
-- Mathlib/CategoryTheory/Monoidal/Bimod.lean
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
W X Y Z : Mon_ C
M M' : Bimod W X
f : M ⟶ M'
N : Bimod X Y
P : Bimod Y Z

⊢ coequalizer.π (M.actRight ▷ { X := TensorBimod.X N P, actLeft := TensorBimod.actLeft N P, one_actLeft := ⋯, left_assoc := ⋯, actRight := TensorBimod.actRight N P, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ }.X) ((α_ M.X X.X { X := TensorBimod.X N P, actLeft := TensorBimod.actLeft N P, one_actLeft := ⋯, left_assoc := ⋯, actRight := TensorBimod.actRight N P, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ }.X).hom ≫ M.X ◁ { X := TensorBimod.X N P, actLeft := TensorBimod.actLeft N P, one_actLeft := ⋯, left_assoc := ⋯, actRight := TensorBimod.actRight N P, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ }.actLeft) ≫ (whiskerRight f { X := TensorBimod.X N P, actLeft := TensorBimod.actLeft N P, one_actLeft := ⋯, left_assoc := ⋯, actRight := TensorBimod.actRight N P, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ }).hom = coequalizer.π (M.actRight ▷ { X := TensorBimod.X N P, actLeft := TensorBimod.actLeft N P, one_actLeft := ⋯, left_assoc := ⋯, actRight := TensorBimod.actRight N P, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ }.X) ((α_ M.X X.X { X := TensorBimod.X N P, actLeft := TensorBimod.actLeft N P, one_actLeft := ⋯, left_assoc := ⋯, actRight := TensorBimod.actRight N P, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ }.X).hom ≫ M.X ◁ { X := TensorBimod.X N P, actLeft := TensorBimod.actLeft N P, one_actLeft := ⋯, left_assoc := ⋯, actRight := TensorBimod.actRight N P, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ }.actLeft) ≫ ((isoOfIso { hom := AssociatorBimod.hom M N P, inv := AssociatorBimod.inv M N P, hom_inv_id := ⋯, inv_hom_id := ⋯ } ⋯ ⋯).inv ≫ whiskerRight (whiskerRight f N) P ≫ (isoOfIso { hom := AssociatorBimod.hom M' N P, inv := AssociatorBimod.inv M' N P, hom_inv_id := ⋯, inv_hom_id := ⋯ } ⋯ ⋯).hom).hom