ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
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 → StieltjesFunction
inst✝ : IsFiniteKernel κ
hf : IsCondKernelCDF f κ ν
a : α
s : Set β
hs : MeasurableSet s
h_dir : Directed (fun x y => x ⊆ y) fun q => Iic ↑q
h_dir_prod : Directed (fun x y => x ⊆ y) fun q => s ×ˢ Iic ↑q

⊢ ∀ (b : ℚ), AEMeasurable (fun b_1 => ((IsCondKernelCDF.toKernel f hf) (a, b_1)) (Iic ↑b)) ((ν a).restrict s)
-- Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
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 → StieltjesFunction
inst✝ : IsFiniteKernel κ
hf : IsCondKernelCDF f κ ν
a : α
s : Set β
hs : MeasurableSet s
h_dir : Directed (fun x y => x ⊆ y) fun q => Iic ↑q
h_dir_prod : Directed (fun x y => x ⊆ y) fun q => s ×ˢ Iic ↑q

⊢ Directed (fun x x_1 => x ≤ x_1) fun i b => ((IsCondKernelCDF.toKernel f hf) (a, b)) (Iic ↑i)
-- Mathlib/Analysis/Calculus/FDeriv/Basic.lean
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
h : s ∈ 𝓝 x

⊢ HasFDerivWithinAt f f' s x ↔ HasFDerivAt f f' x
-- Mathlib/Order/UpperLower/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
κ : ι → Sort u_5
inst✝ : Preorder α
s : UpperSet α
t : Set α
a : α
hts : t ⊆ ↑s
hst : ∀ b ∈ s, ∀ c ∈ t, b ≤ c → b ∈ t

⊢ s.sdiff t ⊓ upperClosure t = s
-- Mathlib/Order/UpperLower/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
κ : ι → Sort u_5
inst✝ : Preorder α
s : UpperSet α
t : Set α
a✝ : α
hts : t ⊆ ↑s
hst : ∀ b ∈ s, ∀ c ∈ t, b ≤ c → b ∈ t
a : α
ha : a ∈ (⇑toDual ∘ SetLike.coe) s

⊢ a ∈ (⇑toDual ∘ SetLike.coe) (s.sdiff t ⊓ upperClosure t)
-- Mathlib/Order/Basic.lean
ι : Type u_1
α : Type u_2
β : Type u_3
π : ι → Type u_4
a b : α
inst✝² : LinearOrder β
inst✝¹ : DecidableEq α
f : α → β
inj : Injective f
inst✝ : Decidable (a < b)

⊢ compare (f a) (f b) = compareOfLessAndEq a b
-- Mathlib/Order/Basic.lean
ι : Type u_1
α : Type u_2
β : Type u_3
π : ι → Type u_4
a b : α
inst✝² : LinearOrder β
inst✝¹ : DecidableEq α
f : α → β
inj : Injective f
inst✝ : Decidable (a < b)
h : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)

⊢ compare (f a) (f b) = compareOfLessAndEq a b
-- Mathlib/Algebra/Polynomial/Coeff.lean
R : Type u
S : Type v
a b : R
n✝ m : ℕ
inst✝ : Semiring R
p✝ q r p : R[X]
n : ℕ

⊢ (X * p).coeff (n + 1) = p.coeff n
-- Mathlib/Data/Multiset/Basic.lean
α : Type u_1
β : Type v
γ : Type u_2
δ : Type u_3
r : α → β → Prop
p : γ → δ → Prop
as : Multiset α
b : β
bs : Multiset β

⊢ Rel r as (b ::ₘ bs) ↔ ∃ a as', r a b ∧ Rel r as' bs ∧ as = a ::ₘ as'
-- Mathlib/Data/Multiset/Basic.lean
α : Type u_1
β : Type v
γ : Type u_2
δ : Type u_3
r : α → β → Prop
p : γ → δ → Prop
as : Multiset α
b : β
bs : Multiset β

⊢ (∃ b_1 bs', flip r b b_1 ∧ Rel (flip r) bs bs' ∧ as = b_1 ::ₘ bs') ↔ ∃ a as', r a b ∧ Rel r as' bs ∧ as = a ::ₘ as'
-- Mathlib/Data/Multiset/Basic.lean
α : Type u_1
β : Type v
γ : Type u_2
δ : Type u_3
r : α → β → Prop
p : γ → δ → Prop
as : Multiset α
b : β
bs : Multiset β
a : α
as' : Multiset α

⊢ flip r b a ∧ Rel (flip r) bs as' ∧ as = a ::ₘ as' ↔ r a b ∧ Rel r as' bs ∧ as = a ::ₘ as'
-- .lake/packages/batteries/Batteries/Data/List/Lemmas.lean
α✝ : Type u_1
l : List α✝
a : α✝

⊢ l.Disjoint [a] ↔ ¬a ∈ l
-- Mathlib/CategoryTheory/Monoidal/Bimod.lean
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)

⊢ R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫ actLeft P Q = (α_ R.X P.X Q.X).inv ≫ P.actLeft ▷ Q.X ≫ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)
-- Mathlib/CategoryTheory/Monoidal/Bimod.lean
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)

⊢ ((α_ R.X P.X Q.X).inv ≫ P.actLeft ▷ Q.X) ≫ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) = (α_ R.X P.X Q.X).inv ≫ P.actLeft ▷ Q.X ≫ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)
-- Mathlib/Algebra/Homology/Single.lean
V : Type u
inst✝² : Category.{v, u} V
inst✝¹ : HasZeroMorphisms V
inst✝ : HasZeroObject V
A B : V
f : A ⟶ B

⊢ (HomologicalComplex.singleObjXSelf (ComplexShape.up ℕ) 0 A).hom ≫ f ≫ (HomologicalComplex.singleObjXSelf (ComplexShape.up ℕ) 0 B).inv = f
-- Mathlib/GroupTheory/Perm/Cycle/Concrete.lean
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
p : Perm α
x y : α

⊢ y ∈ p.toList x ↔ p.SameCycle x y ∧ x ∈ p.support
-- Mathlib/GroupTheory/Perm/Cycle/Concrete.lean
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
p : Perm α
x y : α

⊢ (∃ a < (p.cycleOf x).support.card, (p ^ a) x = y) ↔ p.SameCycle x y ∧ x ∈ p.support
-- Mathlib/Algebra/Polynomial/Reverse.lean
R : Type u_1
inst✝ : Semiring R
f p : R[X]

⊢ (X * p).reverse = p.reverse
-- Mathlib/MeasureTheory/Decomposition/Lebesgue.lean
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
f g : α → ℝ≥0∞
hf : f ∈ measurableLE μ ν
hg : g ∈ measurableLE μ ν

⊢ (fun a => f a ⊔ g a) ∈ measurableLE μ ν
-- Mathlib/MeasureTheory/Decomposition/Lebesgue.lean
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
f g : α → ℝ≥0∞
hf : f ∈ measurableLE μ ν
hg : g ∈ measurableLE μ ν

⊢ (fun a => max (f a) (g a)) ∈ measurableLE μ ν
-- Mathlib/MeasureTheory/Decomposition/Lebesgue.lean
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
f g : α → ℝ≥0∞
hf : f ∈ measurableLE μ ν
hg : g ∈ measurableLE μ ν
A : Set α
hA : MeasurableSet A

⊢ ∫⁻ (x : α) in A, (fun a => max (f a) (g a)) x ∂μ ≤ ν A
-- Mathlib/MeasureTheory/Decomposition/Lebesgue.lean
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
f g : α → ℝ≥0∞
hf : f ∈ measurableLE μ ν
hg : g ∈ measurableLE μ ν
A : Set α
hA : MeasurableSet A
h₁ : MeasurableSet (A ∩ {a | f a ≤ g a})

⊢ ∫⁻ (x : α) in A, (fun a => max (f a) (g a)) x ∂μ ≤ ν A
-- Mathlib/MeasureTheory/Decomposition/Lebesgue.lean
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
f g : α → ℝ≥0∞
hf : f ∈ measurableLE μ ν
hg : g ∈ measurableLE μ ν
A : Set α
hA : MeasurableSet A
h₁ : MeasurableSet (A ∩ {a | f a ≤ g a})
h₂ : MeasurableSet (A ∩ {a | g a < f a})

⊢ ∫⁻ (x : α) in A, (fun a => max (f a) (g a)) x ∂μ ≤ ν A
-- Mathlib/MeasureTheory/Decomposition/Lebesgue.lean
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
f g : α → ℝ≥0∞
hf : f ∈ measurableLE μ ν
hg : g ∈ measurableLE μ ν
A : Set α
hA : MeasurableSet A
h₁ : MeasurableSet (A ∩ {a | f a ≤ g a})
h₂ : MeasurableSet (A ∩ {a | g a < f a})

⊢ ∫⁻ (x : α) in A ∩ {x | f x ≤ g x}, g x ∂μ + ∫⁻ (x : α) in A ∩ {x | g x < f x}, f x ∂μ ≤ ν A
-- Mathlib/MeasureTheory/Decomposition/Lebesgue.lean
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
f g : α → ℝ≥0∞
hf : f ∈ measurableLE μ ν
hg : g ∈ measurableLE μ ν
A : Set α
hA : MeasurableSet A
h₁ : MeasurableSet (A ∩ {a | f a ≤ g a})
h₂ : MeasurableSet (A ∩ {a | g a < f a})

⊢ ν (A ∩ {a | f a ≤ g a}) + ν (A ∩ {a | g a < f a}) = ν A
-- Mathlib/MeasureTheory/Decomposition/Lebesgue.lean
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
f g : α → ℝ≥0∞
hf : f ∈ measurableLE μ ν
hg : g ∈ measurableLE μ ν
A : Set α
hA : MeasurableSet A
h₁ : MeasurableSet (A ∩ {a | f a ≤ g a})
h₂ : MeasurableSet (A ∩ {a | g a < f a})

⊢ ν (A ∩ {a | f a ≤ g a}) + ν (A \ {a | f a ≤ g a}) = ν A
-- Mathlib/RingTheory/Kaehler/Basic.lean
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module S M
inst✝ : IsScalarTower R S M
x : R
y : S

⊢ (y𝖣(algebraMap R S) x) = 0
-- Mathlib/RingTheory/Kaehler/Basic.lean
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module S M
inst✝ : IsScalarTower R S M
x : R
y : S

⊢ y • single ((algebraMap R S) x) 1 ∈ kerTotal R S
-- Mathlib/Topology/Order/T5.lean
X : Type u_1
inst✝² : LinearOrder X
inst✝¹ : TopologicalSpace X
inst✝ : OrderTopology X
a b c : X
s t : Set X
hd : Disjoint s (closure t)
ha : a ∈ s

⊢ (s.ordSeparatingSet t).ordConnectedSectionᶜ ∈ 𝓝 a
-- Mathlib/Topology/Order/T5.lean
X : Type u_1
inst✝² : LinearOrder X
inst✝¹ : TopologicalSpace X
inst✝ : OrderTopology X
a b c : X
s t : Set X
hd : Disjoint s (closure t)
ha : a ∈ s

⊢ (s.ordSeparatingSet t).ordConnectedSectionᶜ ∈ 𝓝[≤] a ∧ (s.ordSeparatingSet t).ordConnectedSectionᶜ ∈ 𝓝[≥] a
-- Mathlib/Topology/Algebra/InfiniteSum/Group.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝² : CommGroup α
inst✝¹ : UniformSpace α
inst✝ : UniformGroup α
f g : β → α
a a₁ a₂ : α

⊢ (CauchySeq fun s => ∏ b ∈ s, f b) ↔ ∀ e ∈ 𝓝 1, ∃ s, ∀ (t : Set β), Disjoint t ↑s → ∏' (b : ↑t), f ↑b ∈ e
-- Mathlib/Topology/Algebra/InfiniteSum/Group.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝² : CommGroup α
inst✝¹ : UniformSpace α
inst✝ : UniformGroup α
f g : β → α
a a₁ a₂ : α

⊢ (∀ e ∈ 𝓝 1, ∃ s, ∀ (t : Finset β), (∀ ⦃a : β⦄, a ∈ t → a ∉ s) → ∏ b ∈ t, f b ∈ e) ↔ ∀ e ∈ 𝓝 1, ∃ s, ∀ (t : Set β), (∀ ⦃a : β⦄, a ∈ t → a ∉ ↑s) → ∏' (b : ↑t), f ↑b ∈ e
-- Mathlib/CategoryTheory/Monad/Adjunction.lean
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
L : C ⥤ D
R : D ⥤ C
h : L ⊣ R
X : D

⊢ L.map (h.unit.app (R.obj X)) ≫ L.map (R.map (L.map (h.unit.app (R.obj X)))) = L.map (h.unit.app (R.obj X)) ≫ L.map (h.unit.app (R.obj (L.obj (R.obj X))))
-- Mathlib/CategoryTheory/Monad/Adjunction.lean
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
L : C ⥤ D
R : D ⥤ C
h : L ⊣ R
X : D

⊢ L.map (h.unit.app (R.obj X) ≫ R.map (L.map (h.unit.app (R.obj X)))) = L.map (h.unit.app (R.obj X)) ≫ L.map (h.unit.app (R.obj (L.obj (R.obj X))))
-- Mathlib/CategoryTheory/Monad/Adjunction.lean
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
L : C ⥤ D
R : D ⥤ C
h : L ⊣ R
X : D

⊢ L.map (h.unit.app (R.obj X)) ≫ L.map (R.map (h.counit.app X)) = 𝟙 (L.obj (R.obj X))
-- Mathlib/CategoryTheory/Monad/Adjunction.lean
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
L : C ⥤ D
R : D ⥤ C
h : L ⊣ R
X : D

⊢ L.map (h.unit.app (R.obj X) ≫ R.map (h.counit.app X)) = 𝟙 (L.obj (R.obj X))
-- Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean
C : Type u_1
D : Type u_2
inst✝⁶ : Category.{u_3, u_1} C
inst✝⁵ : Category.{?u.24208, u_2} D
inst✝⁴ : ConcreteCategory D
inst✝³ : Preregular C
inst✝² : FinitaryPreExtensive C
F G : Cᵒᵖ ⥤ Type w
f : F ⟶ G
inst✝¹ : PreservesFiniteProducts F
inst✝ : PreservesFiniteProducts G
h : Presheaf.IsLocallySurjective (coherentTopology C) f
U✝ : C
y : (forget (Type w)).obj (G.obj (op U✝))

⊢ Presheaf.imageSieve f y ∈ (regularTopology C).sieves U✝
-- Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean
C : Type u_1
D : Type u_2
inst✝⁶ : Category.{u_3, u_1} C
inst✝⁵ : Category.{?u.24208, u_2} D
inst✝⁴ : ConcreteCategory D
inst✝³ : Preregular C
inst✝² : FinitaryPreExtensive C
F G : Cᵒᵖ ⥤ Type w
f : F ⟶ G
inst✝¹ : PreservesFiniteProducts F
inst✝ : PreservesFiniteProducts G
U✝ : C
y : (forget (Type w)).obj (G.obj (op U✝))
h : Presheaf.imageSieve f y ∈ (coherentTopology C).sieves U✝

⊢ Presheaf.imageSieve f y ∈ (regularTopology C).sieves U✝
-- Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean
C : Type u_1
D : Type u_2
inst✝⁶ : Category.{u_3, u_1} C
inst✝⁵ : Category.{?u.24208, u_2} D
inst✝⁴ : ConcreteCategory D
inst✝³ : Preregular C
inst✝² : FinitaryPreExtensive C
F G : Cᵒᵖ ⥤ Type w
f : F ⟶ G
inst✝¹ : PreservesFiniteProducts F
inst✝ : PreservesFiniteProducts G
U✝ : C
y : (forget (Type w)).obj (G.obj (op U✝))
h : ∃ α, ∃ (_ : Finite α), ∃ Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), (Presheaf.imageSieve f y).arrows (π a)

⊢ Presheaf.imageSieve f y ∈ (regularTopology C).sieves U✝
-- Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean
C : Type u_1
D : Type u_2
inst✝⁶ : Category.{u_3, u_1} C
inst✝⁵ : Category.{?u.24208, u_2} D
inst✝⁴ : ConcreteCategory D
inst✝³ : Preregular C
inst✝² : FinitaryPreExtensive C
F G : Cᵒᵖ ⥤ Type w
f : F ⟶ G
inst✝¹ : PreservesFiniteProducts F
inst✝ : PreservesFiniteProducts G
U✝ : C
y : (forget (Type w)).obj (G.obj (op U✝))
α : Type
w✝ : Finite α
Z : α → C
π : (a : α) → Z a ⟶ U✝
h : EffectiveEpiFamily Z π
h' : ∀ (a : α), (Presheaf.imageSieve f y).arrows (π a)

⊢ Presheaf.imageSieve f y ∈ (regularTopology C).sieves U✝
-- Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean
C : Type u_1
D : Type u_2
inst✝⁶ : Category.{u_3, u_1} C
inst✝⁵ : Category.{?u.24208, u_2} D
inst✝⁴ : ConcreteCategory D
inst✝³ : Preregular C
inst✝² : FinitaryPreExtensive C
F G : Cᵒᵖ ⥤ Type w
f : F ⟶ G
inst✝¹ : PreservesFiniteProducts F
inst✝ : PreservesFiniteProducts G
U✝ : C
y : (forget (Type w)).obj (G.obj (op U✝))
α : Type
w✝ : Finite α
Z : α → C
π : (a : α) → Z a ⟶ U✝
h : EffectiveEpiFamily Z π
h' : ∀ (a : α), (Presheaf.imageSieve f y).arrows (π a)

⊢ ∃ Y π, EffectiveEpi π ∧ (Presheaf.imageSieve f y).arrows π
-- Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean
C : Type u_1
D : Type u_2
inst✝⁶ : Category.{u_3, u_1} C
inst✝⁵ : Category.{?u.24208, u_2} D
inst✝⁴ : ConcreteCategory D
inst✝³ : Preregular C
inst✝² : FinitaryPreExtensive C
F G : Cᵒᵖ ⥤ Type w
f : F ⟶ G
inst✝¹ : PreservesFiniteProducts F
inst✝ : PreservesFiniteProducts G
U✝ : C
y : (forget (Type w)).obj (G.obj (op U✝))
α : Type
w✝ : Finite α
Z : α → C
π : (a : α) → Z a ⟶ U✝
h : EffectiveEpiFamily Z π
h' : ∀ (a : α), (Presheaf.imageSieve f y).arrows (π a)
x : (a : α) → F.obj (op (Z a)) := fun a => Exists.choose ⋯
x✝ : Fintype α := Fintype.ofFinite α
i' : ((a : α) → F.obj (op (Z a))) ≅ F.obj (op (∐ Z)) := (Types.productIso fun a => F.obj (op (Z a))).symm ≪≫ (PreservesProduct.iso F fun a => op (Z a)).symm ≪≫ F.mapIso (opCoproductIsoProduct Z).symm
this : PreservesLimitsOfShape (Discrete α)ᵒᵖ G

⊢ (f.app (op (∐ Z))) (i'.hom x) = (G.map (Sigma.desc π).op) y
-- Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean
C : Type u_1
D : Type u_2
inst✝⁶ : Category.{u_3, u_1} C
inst✝⁵ : Category.{?u.24208, u_2} D
inst✝⁴ : ConcreteCategory D
inst✝³ : Preregular C
inst✝² : FinitaryPreExtensive C
F G : Cᵒᵖ ⥤ Type w
f : F ⟶ G
inst✝¹ : PreservesFiniteProducts F
inst✝ : PreservesFiniteProducts G
U✝ : C
y : (forget (Type w)).obj (G.obj (op U✝))
α : Type
w✝ : Finite α
Z : α → C
π : (a : α) → Z a ⟶ U✝
h : EffectiveEpiFamily Z π
h' : ∀ (a : α), (Presheaf.imageSieve f y).arrows (π a)
x : (a : α) → F.obj (op (Z a)) := fun a => Exists.choose ⋯
x✝ : Fintype α := Fintype.ofFinite α
i' : ((a : α) → F.obj (op (Z a))) ≅ F.obj (op (∐ Z)) := (Types.productIso fun a => F.obj (op (Z a))).symm ≪≫ (PreservesProduct.iso F fun a => op (Z a)).symm ≪≫ F.mapIso (opCoproductIsoProduct Z).symm
this : PreservesLimitsOfShape (Discrete α)ᵒᵖ G

⊢ ∀ (j : (Discrete α)ᵒᵖ), ((G.mapCone (Cocone.op (Cofan.mk (∐ Z) (Sigma.ι Z)))).π.app j) ((f.app (op (∐ Z))) (i'.hom x)) = ((G.mapCone (Cocone.op (Cofan.mk (∐ Z) (Sigma.ι Z)))).π.app j) ((G.map (Sigma.desc π).op) y)
-- Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean
C : Type u_1
D : Type u_2
inst✝⁶ : Category.{u_3, u_1} C
inst✝⁵ : Category.{?u.24208, u_2} D
inst✝⁴ : ConcreteCategory D
inst✝³ : Preregular C
inst✝² : FinitaryPreExtensive C
F G : Cᵒᵖ ⥤ Type w
f : F ⟶ G
inst✝¹ : PreservesFiniteProducts F
inst✝ : PreservesFiniteProducts G
U✝ : C
y : (forget (Type w)).obj (G.obj (op U✝))
α : Type
w✝ : Finite α
Z : α → C
π : (a : α) → Z a ⟶ U✝
h : EffectiveEpiFamily Z π
h' : ∀ (a : α), (Presheaf.imageSieve f y).arrows (π a)
x : (a : α) → F.obj (op (Z a)) := fun a => Exists.choose ⋯
x✝ : Fintype α := Fintype.ofFinite α
i' : ((a : α) → F.obj (op (Z a))) ≅ F.obj (op (∐ Z)) := (Types.productIso fun a => F.obj (op (Z a))).symm ≪≫ (PreservesProduct.iso F fun a => op (Z a)).symm ≪≫ F.mapIso (opCoproductIsoProduct Z).symm
this : PreservesLimitsOfShape (Discrete α)ᵒᵖ G
a : α

⊢ ((G.mapCone (Cocone.op (Cofan.mk (∐ Z) (Sigma.ι Z)))).π.app (op { as := a })) ((f.app (op (∐ Z))) (i'.hom x)) = ((G.mapCone (Cocone.op (Cofan.mk (∐ Z) (Sigma.ι Z)))).π.app (op { as := a })) ((G.map (Sigma.desc π).op) y)
-- Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean
C : Type u_1
D : Type u_2
inst✝⁶ : Category.{u_3, u_1} C
inst✝⁵ : Category.{?u.24208, u_2} D
inst✝⁴ : ConcreteCategory D
inst✝³ : Preregular C
inst✝² : FinitaryPreExtensive C
F G : Cᵒᵖ ⥤ Type w
f : F ⟶ G
inst✝¹ : PreservesFiniteProducts F
inst✝ : PreservesFiniteProducts G
U✝ : C
y : (forget (Type w)).obj (G.obj (op U✝))
α : Type
w✝ : Finite α
Z : α → C
π : (a : α) → Z a ⟶ U✝
h : EffectiveEpiFamily Z π
h' : ∀ (a : α), (Presheaf.imageSieve f y).arrows (π a)
x : (a : α) → F.obj (op (Z a)) := fun a => Exists.choose ⋯
x✝ : Fintype α := Fintype.ofFinite α
i' : ((a : α) → F.obj (op (Z a))) ≅ F.obj (op (∐ Z)) := (Types.productIso fun a => F.obj (op (Z a))).symm ≪≫ (PreservesProduct.iso F fun a => op (Z a)).symm ≪≫ F.mapIso (opCoproductIsoProduct Z).symm
this : PreservesLimitsOfShape (Discrete α)ᵒᵖ G
a : α

⊢ (f.app (op (Z a))) ((F.map (Sigma.ι Z a).op) (F.map (opCoproductIsoProduct Z).inv ((PreservesProduct.iso F fun a => op (Z a)).inv ((Types.productIso fun a => F.obj (op (Z a))).inv x)))) = (G.map (Sigma.ι Z a).op) ((G.map (Sigma.desc π).op) y)
-- Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean
C : Type u_1
D : Type u_2
inst✝⁶ : Category.{u_3, u_1} C
inst✝⁵ : Category.{?u.24208, u_2} D
inst✝⁴ : ConcreteCategory D
inst✝³ : Preregular C
inst✝² : FinitaryPreExtensive C
F G : Cᵒᵖ ⥤ Type w
f : F ⟶ G
inst✝¹ : PreservesFiniteProducts F
inst✝ : PreservesFiniteProducts G
U✝ : C
y : (forget (Type w)).obj (G.obj (op U✝))
α : Type
w✝ : Finite α
Z : α → C
π : (a : α) → Z a ⟶ U✝
h : EffectiveEpiFamily Z π
h' : ∀ (a : α), (Presheaf.imageSieve f y).arrows (π a)
x : (a : α) → F.obj (op (Z a)) := fun a => Exists.choose ⋯
x✝ : Fintype α := Fintype.ofFinite α
i' : ((a : α) → F.obj (op (Z a))) ≅ F.obj (op (∐ Z)) := (Types.productIso fun a => F.obj (op (Z a))).symm ≪≫ (PreservesProduct.iso F fun a => op (Z a)).symm ≪≫ F.mapIso (opCoproductIsoProduct Z).symm
this✝ : PreservesLimitsOfShape (Discrete α)ᵒᵖ G
a : α
this : f.app (op (Z a)) (x a) = G.map (π a).op y

⊢ (f.app (op (Z a))) ((F.map (Sigma.ι Z a).op) (F.map (opCoproductIsoProduct Z).inv ((PreservesProduct.iso F fun a => op (Z a)).inv ((Types.productIso fun a => F.obj (op (Z a))).inv x)))) = (G.map (Sigma.ι Z a).op) ((G.map (Sigma.desc π).op) y)
-- Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean
p : ℕ
x✝ : Fact (Nat.Prime p)
a : ℤ

⊢ legendreSym p a = fastLegendreSym p a
-- Mathlib/Analysis/SpecialFunctions/Log/Basic.lean
x✝ y x : ℝ
hx : 0 < x

⊢ cosh (log x) = (x + x⁻¹) / 2
-- Mathlib/FieldTheory/Cardinality.lean
α : Type u_1
inst✝¹ : Fintype α
inst✝ : Field α

⊢ IsPrimePow ‖α‖
-- Mathlib/FieldTheory/Cardinality.lean
α : Type u_1
inst✝¹ : Fintype α
inst✝ : Field α
p : ℕ
h✝ : CharP α p

⊢ IsPrimePow ‖α‖
-- Mathlib/FieldTheory/Cardinality.lean
α : Type u_1
inst✝¹ : Fintype α
inst✝ : Field α
p : ℕ
h✝ : CharP α p
hp : Fact (Nat.Prime p)

⊢ IsPrimePow ‖α‖
-- Mathlib/FieldTheory/Cardinality.lean
α : Type u_1
inst✝¹ : Fintype α
inst✝ : Field α
p : ℕ
h✝ : CharP α p
hp : Fact (Nat.Prime p)
this : Algebra (ZMod p) α := ZMod.algebra α p
b : Basis { x // x ∈ IsNoetherian.finsetBasisIndex (ZMod p) α } (ZMod p) α := IsNoetherian.finsetBasis (ZMod p) α

⊢ IsPrimePow ‖α‖
-- Mathlib/FieldTheory/Cardinality.lean
α : Type u_1
inst✝¹ : Fintype α
inst✝ : Field α
p : ℕ
h✝ : CharP α p
hp : Fact (Nat.Prime p)
this : Algebra (ZMod p) α := ZMod.algebra α p
b : Basis { x // x ∈ IsNoetherian.finsetBasisIndex (ZMod p) α } (ZMod p) α := IsNoetherian.finsetBasis (ZMod p) α

⊢ IsPrimePow p
-- Mathlib/FieldTheory/Cardinality.lean
α : Type u_1
inst✝¹ : Fintype α
inst✝ : Field α
p : ℕ
h✝ : CharP α p
hp : Fact (Nat.Prime p)
this : Algebra (ZMod p) α := ZMod.algebra α p
b : Basis { x // x ∈ IsNoetherian.finsetBasisIndex (ZMod p) α } (ZMod p) α := IsNoetherian.finsetBasis (ZMod p) α

⊢ ‖{ x // x ∈ IsNoetherian.finsetBasisIndex (ZMod p) α }‖ ≠ 0
-- Mathlib/FieldTheory/Cardinality.lean
α : Type u_1
inst✝¹ : Fintype α
inst✝ : Field α
p : ℕ
h✝ : CharP α p
hp : Fact (Nat.Prime p)
this : Algebra (ZMod p) α := ZMod.algebra α p
b : Basis { x // x ∈ IsNoetherian.finsetBasisIndex (ZMod p) α } (ZMod p) α := IsNoetherian.finsetBasis (ZMod p) α

⊢ FiniteDimensional.finrank (ZMod p) α ≠ 0
-- Mathlib/MeasureTheory/Integral/Lebesgue.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
ι : Type u_5
inst✝ : Countable ι
f : α → ℝ≥0∞
s : ι → Set α
hd : Directed (fun x x_1 => x ⊆ x_1) s

⊢ ∫⁻ (x : α) in ⋃ i, s i, f x ∂μ = ⨆ i, ∫⁻ (x : α) in s i, f x ∂μ
-- .lake/packages/batteries/Batteries/Data/String/Lemmas.lean
x✝¹ : Substring
h✝ : x✝¹.Valid
x✝ : Nat
w✝² w✝¹ w✝ : List Char
h : ValidFor w✝² w✝¹ w✝ x✝¹

⊢ (x✝¹.take x✝).toString.data = List.take x✝ x✝¹.toString.data
-- Mathlib/Analysis/SpecialFunctions/Log/Base.lean
b✝ x y : ℝ
b : ℕ
r : ℝ
hb : 1 < b
hr✝ : 0 ≤ r
hr : 0 < r

⊢ ⌊logb (↑b) r⌋ = Int.log b r
-- Mathlib/Analysis/SpecialFunctions/Log/Base.lean
b✝ x y : ℝ
b : ℕ
r : ℝ
hb : 1 < b
hr✝ : 0 ≤ r
hr : 0 < r
hb1' : 1 < ↑b

⊢ ⌊logb (↑b) r⌋ = Int.log b r
-- Mathlib/Analysis/SpecialFunctions/Log/Base.lean
b✝ x y : ℝ
b : ℕ
r : ℝ
hb : 1 < b
hr✝ : 0 ≤ r
hr : 0 < r
hb1' : 1 < ↑b

⊢ ⌊logb (↑b) r⌋ ≤ Int.log b r
-- Mathlib/Analysis/SpecialFunctions/Log/Base.lean
b✝ x y : ℝ
b : ℕ
r : ℝ
hb : 1 < b
hr✝ : 0 ≤ r
hr : 0 < r
hb1' : 1 < ↑b

⊢ Int.log b r ≤ ⌊logb (↑b) r⌋
-- Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean
F : Type u_1
inst✝² : Field F
inst✝¹ : Fintype F
inst✝ : DecidableEq F
hF : ringChar F ≠ 2

⊢ quadraticChar F ≠ 1
-- Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean
F : Type u_1
inst✝² : Field F
inst✝¹ : Fintype F
inst✝ : DecidableEq F
hF : ringChar F ≠ 2
a : Fˣ
ha : (quadraticChar F) ↑a = -1

⊢ quadraticChar F ≠ 1
-- Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean
F : Type u_1
inst✝² : Field F
inst✝¹ : Fintype F
inst✝ : DecidableEq F
hF : ringChar F ≠ 2
a : Fˣ
ha : (quadraticChar F) ↑a = -1
hχ : quadraticChar F = 1

⊢ False
-- Mathlib/RingTheory/Multiplicity.lean
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
k : ℕ

⊢ multiplicity a b < ↑k ↔ ¬a ^ k ∣ b
-- Mathlib/Analysis/NormedSpace/Connected.lean
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
h : 1 < Module.rank ℝ E
x : E
r : ℝ
hr : 0 ≤ r
rpos : 0 < r
f : E → E := fun y => x + (r * ‖y‖⁻¹) • y

⊢ IsPathConnected (sphere x r)
-- Mathlib/Analysis/NormedSpace/Connected.lean
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
h : 1 < Module.rank ℝ E
x : E
r : ℝ
hr : 0 ≤ r
rpos : 0 < r
f : E → E := fun y => x + (r * ‖y‖⁻¹) • y
A : ContinuousOn f {0}ᶜ

⊢ IsPathConnected (sphere x r)
-- Mathlib/Analysis/NormedSpace/Connected.lean
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
h : 1 < Module.rank ℝ E
x : E
r : ℝ
hr : 0 ≤ r
rpos : 0 < r
f : E → E := fun y => x + (r * ‖y‖⁻¹) • y
A : ContinuousOn f {0}ᶜ
B : IsPathConnected {0}ᶜ

⊢ IsPathConnected (sphere x r)
-- Mathlib/Analysis/NormedSpace/Connected.lean
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
h : 1 < Module.rank ℝ E
x : E
r : ℝ
hr : 0 ≤ r
rpos : 0 < r
f : E → E := fun y => x + (r * ‖y‖⁻¹) • y
A : ContinuousOn f {0}ᶜ
B : IsPathConnected {0}ᶜ
C : IsPathConnected (f '' {0}ᶜ)

⊢ IsPathConnected (sphere x r)
-- Mathlib/MeasureTheory/Measure/MeasureSpace.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t✝ s t u : Set α
hs : MeasurableSet s
h : μ t = μ u
htu : t ⊆ u
ht_ne_top : μ u ≠ ⊤

⊢ μ (t ∩ s) = μ (u ∩ s)