ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Analysis/BoxIntegral/DivergenceTheorem.lean
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
n : ℕ
inst✝ : CompleteSpace E
I : Box (Fin (n + 1))
i✝ : Fin (n + 1)
f : (Fin (n + 1) → ℝ) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hs : ∀ x ∈ s, ContinuousWithinAt f (Box.Icc I) x
Hd : ∀ x ∈ Box.Icc I \ s, HasFDerivWithinAt f (f' x) (Box.Icc I) x
i : Fin (n + 1)
Hc : ContinuousOn f (Box.Icc I)
fI : ℝ → Box (Fin n) → E := fun y J => integral J GP (fun x => f (i.insertNth y x)) BoxAdditiveMap.volume
fb : ↑(Set.Icc (I.lower i) (I.upper i)) → Fin n →ᵇᵃ[↑(I.face i)] E := fun x => ⋯.toBoxAdditive
F : Fin (n + 1) →ᵇᵃ[↑I] E := BoxAdditiveMap.upperSubLower I i fI fb ⋯

⊢ Fin (n + 1) →ᵇᵃ[↑I] ℝ
-- Mathlib/Analysis/BoxIntegral/DivergenceTheorem.lean
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
n : ℕ
inst✝ : CompleteSpace E
I : Box (Fin (n + 1))
i✝ : Fin (n + 1)
f : (Fin (n + 1) → ℝ) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hs : ∀ x ∈ s, ContinuousWithinAt f (Box.Icc I) x
Hd : ∀ x ∈ Box.Icc I \ s, HasFDerivWithinAt f (f' x) (Box.Icc I) x
i : Fin (n + 1)
Hc : ContinuousOn f (Box.Icc I)
fI : ℝ → Box (Fin n) → E := fun y J => integral J GP (fun x => f (i.insertNth y x)) BoxAdditiveMap.volume
fb : ↑(Set.Icc (I.lower i) (I.upper i)) → Fin n →ᵇᵃ[↑(I.face i)] E := fun x => ⋯.toBoxAdditive
F : Fin (n + 1) →ᵇᵃ[↑I] E := BoxAdditiveMap.upperSubLower I i fI fb ⋯

⊢ ∀ (J : Box (Fin (n + 1))), 0 ≤ ?refine_1 J
-- Mathlib/Analysis/BoxIntegral/DivergenceTheorem.lean
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
n : ℕ
inst✝ : CompleteSpace E
I : Box (Fin (n + 1))
i✝ : Fin (n + 1)
f : (Fin (n + 1) → ℝ) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hs : ∀ x ∈ s, ContinuousWithinAt f (Box.Icc I) x
Hd : ∀ x ∈ Box.Icc I \ s, HasFDerivWithinAt f (f' x) (Box.Icc I) x
i : Fin (n + 1)
Hc : ContinuousOn f (Box.Icc I)
fI : ℝ → Box (Fin n) → E := fun y J => integral J GP (fun x => f (i.insertNth y x)) BoxAdditiveMap.volume
fb : ↑(Set.Icc (I.lower i) (I.upper i)) → Fin n →ᵇᵃ[↑(I.face i)] E := fun x => ⋯.toBoxAdditive
F : Fin (n + 1) →ᵇᵃ[↑I] E := BoxAdditiveMap.upperSubLower I i fI fb ⋯

⊢ ∀ (c : ℝ≥0), ∀ x ∈ Box.Icc I ∩ s, ∀ ε > 0, ∃ δ > 0, ∀ J ≤ I, Box.Icc J ⊆ closedBall x δ → x ∈ Box.Icc J → (GP.bDistortion = true → J.distortion ≤ c) → dist ((BoxAdditiveMap.volume J) ((f' x) (Pi.single i 1))) (F J) ≤ ε
-- Mathlib/Analysis/BoxIntegral/DivergenceTheorem.lean
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
n : ℕ
inst✝ : CompleteSpace E
I : Box (Fin (n + 1))
i✝ : Fin (n + 1)
f : (Fin (n + 1) → ℝ) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hs : ∀ x ∈ s, ContinuousWithinAt f (Box.Icc I) x
Hd : ∀ x ∈ Box.Icc I \ s, HasFDerivWithinAt f (f' x) (Box.Icc I) x
i : Fin (n + 1)
Hc : ContinuousOn f (Box.Icc I)
fI : ℝ → Box (Fin n) → E := fun y J => integral J GP (fun x => f (i.insertNth y x)) BoxAdditiveMap.volume
fb : ↑(Set.Icc (I.lower i) (I.upper i)) → Fin n →ᵇᵃ[↑(I.face i)] E := fun x => ⋯.toBoxAdditive
F : Fin (n + 1) →ᵇᵃ[↑I] E := BoxAdditiveMap.upperSubLower I i fI fb ⋯

⊢ ∀ (c : ℝ≥0), ∀ x ∈ Box.Icc I \ s, ∀ ε > 0, ∃ δ > 0, ∀ J ≤ I, Box.Icc J ⊆ closedBall x δ → x ∈ Box.Icc J → (GP.bDistortion = true → J.distortion ≤ c) → dist ((BoxAdditiveMap.volume J) ((f' x) (Pi.single i 1))) (F J) ≤ ε * ?refine_1 J
-- Mathlib/Analysis/Convex/Function.lean
𝕜 : Type u_1
E : Type u_2
F : Type u_3
α : Type u_4
β : Type u_5
ι : Type u_6
inst✝⁵ : OrderedSemiring 𝕜
inst✝⁴ : AddCommMonoid E
inst✝³ : AddCommMonoid F
inst✝² : OrderedCancelAddCommMonoid β
inst✝¹ : SMul 𝕜 E
inst✝ : DistribMulAction 𝕜 β
s : Set E
f g : E → β
hf : StrictConvexOn 𝕜 s f
hg : StrictConvexOn 𝕜 s g
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

⊢ a • f x + b • f y + (a • g x + b • g y) = a • (f x + g x) + b • (f y + g y)
-- Mathlib/Order/KrullDimension.lean
α : Type u_1
β : Type u_2
inst✝² : Preorder α
inst✝¹ : Preorder β
inst✝ : InfiniteDimensionalOrder α
m : WithBot (WithTop ℕ)
hm✝ : ∀ (i : LTSeries α), ↑i.length ≤ m
hm : ∀ (i : LTSeries α), ↑i.length ≤ ⊥
this : Inhabited α

⊢ False
-- Mathlib/Order/KrullDimension.lean
α : Type u_1
β : Type u_2
inst✝² : Preorder α
inst✝¹ : Preorder β
inst✝ : InfiniteDimensionalOrder α
m✝ : WithBot (WithTop ℕ)
hm✝ : ∀ (i : LTSeries α), ↑i.length ≤ m✝
m : ℕ
hm : ∀ (i : LTSeries α), ↑i.length ≤ some (some m)

⊢ ⊤ ≤ some (some m)
-- Mathlib/Order/KrullDimension.lean
α : Type u_1
β : Type u_2
inst✝² : Preorder α
inst✝¹ : Preorder β
inst✝ : InfiniteDimensionalOrder α
m✝ : WithBot (WithTop ℕ)
hm✝ : ∀ (i : LTSeries α), ↑i.length ≤ m✝
m : ℕ
hm : ∀ (i : LTSeries α), ↑i.length ≤ some (some m)

⊢ some (some m) < ↑(LTSeries.withLength α (m + 1)).length
-- Mathlib/Order/KrullDimension.lean
α : Type u_1
β : Type u_2
inst✝² : Preorder α
inst✝¹ : Preorder β
inst✝ : InfiniteDimensionalOrder α
m✝ : WithBot (WithTop ℕ)
hm✝ : ∀ (i : LTSeries α), ↑i.length ≤ m✝
m : ℕ
hm : ∀ (i : LTSeries α), ↑i.length ≤ some (some m)

⊢ m < ↑(LTSeries.withLength α (m + 1)).length
-- Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
ι : Type u_4
p : ι → P
i0 : ι

⊢ vectorSpan k (range p) = Submodule.span k (range fun i => p i0 -ᵥ p i)
-- Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
ι : Type u_4
p : ι → P
i0 : ι

⊢ Submodule.span k (range ((fun x => p i0 -ᵥ x) ∘ p)) = Submodule.span k (range fun i => p i0 -ᵥ p i)
-- Mathlib/NumberTheory/EllipticDivisibilitySequence.lean
R : Type u
S : Type v
inst✝¹ : CommRing R
inst✝ : CommRing S
W : ℤ → R
f : R →+* S
h : IsEllSequence W
x : R
m n r : ℤ

⊢ (x • W) (m + n) * (x • W) (m - n) * (x • W) r ^ 2 = (x • W) (m + r) * (x • W) (m - r) * (x • W) n ^ 2 - (x • W) (n + r) * (x • W) (n - r) * (x • W) m ^ 2
-- Mathlib/MeasureTheory/OuterMeasure/Basic.lean
α : Type u_1
ι : Type u_2
F : Type u_3
inst✝¹ : FunLike F (Set α) ℝ≥0∞
inst✝ : OuterMeasureClass F α
μ✝ : F
s✝ t : Set α
I : Set ι
μ : F
hI : I.Countable
s : ι → Set α
this : Countable ↑I

⊢ μ (⋃ i ∈ I, s i) ≤ ∑' (i : ↑I), μ (s ↑i)
-- Mathlib/MeasureTheory/OuterMeasure/Basic.lean
α : Type u_1
ι : Type u_2
F : Type u_3
inst✝¹ : FunLike F (Set α) ℝ≥0∞
inst✝ : OuterMeasureClass F α
μ✝ : F
s✝ t : Set α
I : Set ι
μ : F
hI : I.Countable
s : ι → Set α
this : Countable ↑I

⊢ μ (⋃ x, s ↑x) ≤ ∑' (i : ↑I), μ (s ↑i)
-- Mathlib/Algebra/Order/ToIntervalMod.lean
α : Type u_1
inst✝¹ : LinearOrderedRing α
inst✝ : Archimedean α
a : α

⊢ ⋃ n, Ico (a + ↑n) (a + ↑n + 1) = univ
-- Mathlib/Algebra/Lie/Submodule.lean
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
N N' : LieSubmodule R L M
I J : LieIdeal R L
s : Set M
m : M
hm : m ∈ s

⊢ m ∈ ↑(lieSpan R L s)
-- Mathlib/Algebra/Lie/Submodule.lean
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
N N' : LieSubmodule R L M
I J : LieIdeal R L
s : Set M
m : M
hm : m ∈ s

⊢ ∀ (N : LieSubmodule R L M), s ⊆ ↑N → m ∈ N
-- Mathlib/Topology/Instances/EReal.lean
α : Type u_1
inst✝ : TopologicalSpace α
s : Set EReal

⊢ (∃ i, True ∧ Iio ↑i ⊆ s) ↔ ∃ y, Iio ↑y ⊆ s
-- Mathlib/Topology/UniformSpace/Basic.lean
α✝ : Type ua
β : Type ub
γ : Type uc
δ : Type ud
ι : Sort u_1
α : Type u
u : Core α
x : α

⊢ 𝓝 x = comap (Prod.mk x) u.uniformity
-- Mathlib/Topology/UniformSpace/Basic.lean
α✝ : Type ua
β : Type ub
γ : Type uc
δ : Type ud
ι : Sort u_1
α : Type u
u : Core α
x : α

⊢ ∀ (a : α), ∀ i ∈ u.uniformity, a ∈ Prod.mk a ⁻¹' id i
-- Mathlib/Topology/UniformSpace/Basic.lean
α✝ : Type ua
β : Type ub
γ : Type uc
δ : Type ud
ι : Sort u_1
α : Type u
u : Core α
x : α

⊢ ∀ (a : α), ∀ i ∈ u.uniformity, ∀ᶠ (x : α) in comap (Prod.mk a) u.uniformity, Prod.mk a ⁻¹' id i ∈ comap (Prod.mk x) u.uniformity
-- Mathlib/Order/Minimal.lean
α : Type u_1
β : Type u_2
r r₁ r₂ : α → α → Prop
s t : Set α
a✝ b✝ x : α
ht : IsAntichain r t
h : maximals r s ⊆ t
hs : ∀ ⦃a : α⦄, a ∈ t → ∃ b ∈ maximals r s, r b a
a : α
ha : a ∈ t
b : α
hb : b ∈ maximals r s
hr : r b a

⊢ a ∈ maximals r s
-- Mathlib/Data/Multiset/Basic.lean
α : Type u_1
β : Type v
γ : Type u_2
inst✝ : DecidableEq α
s✝¹ t✝ u : Multiset α
a✝ b : α
s✝ : Multiset α
a : α
s : Multiset α
IH : ∀ (t : Multiset α), s ∩ t ≤ t
t : Multiset α
h : a ∈ t

⊢ (a ::ₘ s) ∩ t ≤ t
-- Mathlib/CategoryTheory/Abelian/RightDerived.lean
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{u_2, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasInjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : InjectiveResolution X

⊢ (NatTrans.rightDerivedToHomotopyCategory α).app X = (P.isoRightDerivedToHomotopyCategoryObj F).hom ≫ (HomotopyCategory.quotient D (ComplexShape.up ℕ)).map ((NatTrans.mapHomologicalComplex α (ComplexShape.up ℕ)).app P.cocomplex) ≫ (P.isoRightDerivedToHomotopyCategoryObj G).inv
-- Mathlib/CategoryTheory/Abelian/RightDerived.lean
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{u_2, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasInjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : InjectiveResolution X

⊢ (NatTrans.rightDerivedToHomotopyCategory α).app X ≫ (P.isoRightDerivedToHomotopyCategoryObj G).hom = (P.isoRightDerivedToHomotopyCategoryObj F).hom ≫ (HomotopyCategory.quotient D (ComplexShape.up ℕ)).map ((NatTrans.mapHomologicalComplex α (ComplexShape.up ℕ)).app P.cocomplex)
-- Mathlib/CategoryTheory/Abelian/RightDerived.lean
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{u_2, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasInjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : InjectiveResolution X

⊢ (HomotopyCategory.quotient D (ComplexShape.up ℕ)).map ((NatTrans.mapHomologicalComplex α (ComplexShape.up ℕ)).app ((injectiveResolutions C).obj X).as) ≫ (G.mapHomotopyCategory (ComplexShape.up ℕ)).map P.iso.hom ≫ 𝟙 ((HomotopyCategory.quotient D (ComplexShape.up ℕ)).obj ((G.mapHomologicalComplex (ComplexShape.up ℕ)).obj P.cocomplex)) = ((F.mapHomotopyCategory (ComplexShape.up ℕ)).map P.iso.hom ≫ 𝟙 ((HomotopyCategory.quotient D (ComplexShape.up ℕ)).obj ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj P.cocomplex))) ≫ (HomotopyCategory.quotient D (ComplexShape.up ℕ)).map ((NatTrans.mapHomologicalComplex α (ComplexShape.up ℕ)).app P.cocomplex)
-- Mathlib/CategoryTheory/Abelian/RightDerived.lean
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{u_2, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasInjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : InjectiveResolution X

⊢ (HomotopyCategory.quotient D (ComplexShape.up ℕ)).map ((NatTrans.mapHomologicalComplex α (ComplexShape.up ℕ)).app ((injectiveResolutions C).obj X).as) ≫ (G.mapHomotopyCategory (ComplexShape.up ℕ)).map P.iso.hom ≫ 𝟙 ((HomotopyCategory.quotient D (ComplexShape.up ℕ)).obj ((G.mapHomologicalComplex (ComplexShape.up ℕ)).obj P.cocomplex)) = (F.mapHomotopyCategory (ComplexShape.up ℕ)).map P.iso.hom ≫ 𝟙 ((HomotopyCategory.quotient D (ComplexShape.up ℕ)).obj ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj P.cocomplex)) ≫ (HomotopyCategory.quotient D (ComplexShape.up ℕ)).map ((NatTrans.mapHomologicalComplex α (ComplexShape.up ℕ)).app P.cocomplex)
-- Mathlib/CategoryTheory/Abelian/RightDerived.lean
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{u_2, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasInjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : InjectiveResolution X

⊢ (HomotopyCategory.quotient D (ComplexShape.up ℕ)).map ((NatTrans.mapHomologicalComplex α (ComplexShape.up ℕ)).app ((injectiveResolutions C).obj X).as) ≫ (G.mapHomotopyCategory (ComplexShape.up ℕ)).map P.iso.hom = (F.mapHomotopyCategory (ComplexShape.up ℕ)).map P.iso.hom ≫ (HomotopyCategory.quotient D (ComplexShape.up ℕ)).map ((NatTrans.mapHomologicalComplex α (ComplexShape.up ℕ)).app P.cocomplex)
-- Mathlib/CategoryTheory/Abelian/RightDerived.lean
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{u_2, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasInjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : InjectiveResolution X
β : (injectiveResolution X).cocomplex ⟶ P.cocomplex
hβ : (HomotopyCategory.quotient C (ComplexShape.up ℕ)).map β = P.iso.hom

⊢ (HomotopyCategory.quotient D (ComplexShape.up ℕ)).map ((NatTrans.mapHomologicalComplex α (ComplexShape.up ℕ)).app ((injectiveResolutions C).obj X).as) ≫ (G.mapHomotopyCategory (ComplexShape.up ℕ)).map P.iso.hom = (F.mapHomotopyCategory (ComplexShape.up ℕ)).map P.iso.hom ≫ (HomotopyCategory.quotient D (ComplexShape.up ℕ)).map ((NatTrans.mapHomologicalComplex α (ComplexShape.up ℕ)).app P.cocomplex)
-- Mathlib/CategoryTheory/Abelian/RightDerived.lean
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{u_2, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasInjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : InjectiveResolution X
β : (injectiveResolution X).cocomplex ⟶ P.cocomplex
hβ : (HomotopyCategory.quotient C (ComplexShape.up ℕ)).map β = P.iso.hom

⊢ (HomotopyCategory.quotient D (ComplexShape.up ℕ)).map ((NatTrans.mapHomologicalComplex α (ComplexShape.up ℕ)).app ((injectiveResolutions C).obj X).as) ≫ (HomotopyCategory.quotient D (ComplexShape.up ℕ)).map ((G.mapHomologicalComplex (ComplexShape.up ℕ)).map β) = (HomotopyCategory.quotient D (ComplexShape.up ℕ)).map ((F.mapHomologicalComplex (ComplexShape.up ℕ)).map β) ≫ (HomotopyCategory.quotient D (ComplexShape.up ℕ)).map ((NatTrans.mapHomologicalComplex α (ComplexShape.up ℕ)).app P.cocomplex)
-- Mathlib/CategoryTheory/Abelian/RightDerived.lean
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{u_2, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasInjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : InjectiveResolution X
β : (injectiveResolution X).cocomplex ⟶ P.cocomplex
hβ : (HomotopyCategory.quotient C (ComplexShape.up ℕ)).map β = P.iso.hom

⊢ (HomotopyCategory.quotient D (ComplexShape.up ℕ)).map ((NatTrans.mapHomologicalComplex α (ComplexShape.up ℕ)).app ((injectiveResolutions C).obj X).as ≫ (G.mapHomologicalComplex (ComplexShape.up ℕ)).map β) = (HomotopyCategory.quotient D (ComplexShape.up ℕ)).map ((NatTrans.mapHomologicalComplex α (ComplexShape.up ℕ)).app (injectiveResolution X).cocomplex ≫ (G.mapHomologicalComplex (ComplexShape.up ℕ)).map β)
-- Mathlib/MeasureTheory/OuterMeasure/Basic.lean
α : Type u_1
ι✝ : Type u_2
F : Type u_3
inst✝² : FunLike F (Set α) ℝ≥0∞
inst✝¹ : OuterMeasureClass F α
μ : F
s✝ t : Set α
ι : Sort u_4
inst✝ : Countable ι
s : ι → Set α

⊢ μ (⋃ i, s i) = 0 ↔ ∀ (i : ι), μ (s i) = 0
-- Mathlib/Order/Filter/Prod.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Sort u_5
f : Filter α
g : Filter β
α₁ : Type u
α₂ : Type v
β₁ : Type w
β₂ : Type x
f₁ : Filter α₁
f₂ : Filter α₂
m₁ : α₁ → β₁
m₂ : α₂ → β₂
s : Set (β₁ × β₂)

⊢ s ∈ (map m₁ f₁).coprod (map m₂ f₂) → s ∈ map (Prod.map m₁ m₂) (f₁.coprod f₂)
-- Mathlib/Order/Filter/Prod.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Sort u_5
f : Filter α
g : Filter β
α₁ : Type u
α₂ : Type v
β₁ : Type w
β₂ : Type x
f₁ : Filter α₁
f₂ : Filter α₂
m₁ : α₁ → β₁
m₂ : α₂ → β₂
s : Set (β₁ × β₂)

⊢ ((∃ t₁, m₁ ⁻¹' t₁ ∈ f₁ ∧ Prod.fst ⁻¹' t₁ ⊆ s) ∧ ∃ t₂, m₂ ⁻¹' t₂ ∈ f₂ ∧ Prod.snd ⁻¹' t₂ ⊆ s) → (∃ t₁ ∈ f₁, Prod.fst ⁻¹' t₁ ⊆ Prod.map m₁ m₂ ⁻¹' s) ∧ ∃ t₂ ∈ f₂, Prod.snd ⁻¹' t₂ ⊆ Prod.map m₁ m₂ ⁻¹' s
-- 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

⊢ S₁.rightHomologyIso.hom ≫ homologyMap φ = rightHomologyMap φ ≫ S₂.rightHomologyIso.hom
-- Mathlib/Computability/Reduce.lean
α : Type u
inst✝⁵ : Primcodable α
inst✝⁴ : Inhabited α
β : Type v
inst✝³ : Primcodable β
inst✝² : Inhabited β
γ : Type w
inst✝¹ : Primcodable γ
inst✝ : Inhabited γ
d₁ d₂ d₃ : ManyOneDegree

⊢ d₁ + d₂ ≤ d₃ ↔ d₁ ≤ d₃ ∧ d₂ ≤ d₃
-- Mathlib/Computability/Reduce.lean
α : Type u
inst✝⁵ : Primcodable α
inst✝⁴ : Inhabited α
β : Type v
inst✝³ : Primcodable β
inst✝² : Inhabited β
γ : Type w
inst✝¹ : Primcodable γ
inst✝ : Inhabited γ
d₂ d₃ : ManyOneDegree
p✝ : Set ℕ

⊢ of p✝ + d₂ ≤ d₃ ↔ of p✝ ≤ d₃ ∧ d₂ ≤ d₃
-- Mathlib/Computability/Reduce.lean
α : Type u
inst✝⁵ : Primcodable α
inst✝⁴ : Inhabited α
β : Type v
inst✝³ : Primcodable β
inst✝² : Inhabited β
γ : Type w
inst✝¹ : Primcodable γ
inst✝ : Inhabited γ
d₃ : ManyOneDegree
p✝¹ p✝ : Set ℕ

⊢ of p✝¹ + of p✝ ≤ d₃ ↔ of p✝¹ ≤ d₃ ∧ of p✝ ≤ d₃
-- Mathlib/Computability/Reduce.lean
α : Type u
inst✝⁵ : Primcodable α
inst✝⁴ : Inhabited α
β : Type v
inst✝³ : Primcodable β
inst✝² : Inhabited β
γ : Type w
inst✝¹ : Primcodable γ
inst✝ : Inhabited γ
p✝² p✝¹ p✝ : Set ℕ

⊢ of p✝² + of p✝¹ ≤ of p✝ ↔ of p✝² ≤ of p✝ ∧ of p✝¹ ≤ of p✝
-- Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLattice α
s t : Set α
a b : α
inst✝ : Nonempty ι
f : ι → α
h₁ : ∀ (i : ι), b ≤ f i
h₂ : ∀ (w : α), b < w → ∃ i, f i < w

⊢ ⨅ i, f i = b
-- Mathlib/Algebra/Order/Group/Unbundled/Basic.lean
α : Type u
inst✝² : Group α
inst✝¹ : LE α
inst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
a b c d : α

⊢ a / c ≤ b ↔ a ≤ b * c
-- Mathlib/Analysis/Calculus/FDeriv/Prod.lean
𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
G : Type u_4
inst✝⁶ : NormedAddCommGroup G
inst✝⁵ : NormedSpace 𝕜 G
G' : Type u_5
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : 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
ι : Type u_6
inst✝² : Fintype ι
F' : ι → Type u_7
inst✝¹ : (i : ι) → NormedAddCommGroup (F' i)
inst✝ : (i : ι) → NormedSpace 𝕜 (F' i)
φ : (i : ι) → E → F' i
φ' : (i : ι) → E →L[𝕜] F' i
Φ : E → (i : ι) → F' i
Φ' : E →L[𝕜] (i : ι) → F' i

⊢ HasFDerivAtFilter Φ Φ' x L ↔ ∀ (i : ι), HasFDerivAtFilter (fun x => Φ x i) ((proj i).comp Φ') x L
-- Mathlib/Analysis/Calculus/FDeriv/Prod.lean
𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
G : Type u_4
inst✝⁶ : NormedAddCommGroup G
inst✝⁵ : NormedSpace 𝕜 G
G' : Type u_5
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : 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
ι : Type u_6
inst✝² : Fintype ι
F' : ι → Type u_7
inst✝¹ : (i : ι) → NormedAddCommGroup (F' i)
inst✝ : (i : ι) → NormedSpace 𝕜 (F' i)
φ : (i : ι) → E → F' i
φ' : (i : ι) → E →L[𝕜] F' i
Φ : E → (i : ι) → F' i
Φ' : E →L[𝕜] (i : ι) → F' i

⊢ ((fun x' => Φ x' - Φ x - Φ' (x' - x)) =o[L] fun x' => x' - x) ↔ ∀ (i : ι), (fun x' => Φ x' i - Φ x i - ((proj i).comp Φ') (x' - x)) =o[L] fun x' => x' - x
-- 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✝¹ : E
s t : Set E
L L₁ L₂ : Filter E
c : F
x : E
x✝ : E × E

⊢ 0 = (fun x => c) x✝.1 - (fun x => c) x✝.2 - 0 (x✝.1 - x✝.2)
-- Mathlib/Data/Set/Pointwise/Interval.lean
α : Type u_1
inst✝ : LinearOrderedField α
a✝ a b c : α
h : c < 0

⊢ (fun x => c * x) ⁻¹' Ioo a b = Ioo (b / c) (a / c)
-- Mathlib/RingTheory/AdjoinRoot.lean
R : Type u
S : Type v
K : Type w
inst✝ : CommRing R
f✝ f : R[X]

⊢ (Polynomial.map (of f) f).IsRoot (root f)
-- Mathlib/LinearAlgebra/Span.lean
R : Type u_1
R₂ : Type u_2
K : Type u_3
M : Type u_4
M₂ : Type u_5
V : Type u_6
S : Type u_7
inst✝⁷ : Semiring R
inst✝⁶ : AddCommMonoid M
inst✝⁵ : Module R M
x : M
p p' : Submodule R M
inst✝⁴ : Semiring R₂
σ₁₂ : R →+* R₂
inst✝³ : AddCommMonoid M₂
inst✝² : Module R₂ M₂
F : Type u_8
inst✝¹ : FunLike F M M₂
inst✝ : SemilinearMapClass F σ₁₂ M M₂
s t : Set M

⊢ span R 0 = ⊥
-- Mathlib/Analysis/Calculus/FDeriv/Measurable.lean
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
ε : ℝ
hε : 0 < ε
x : ℝ
hx : DifferentiableWithinAt ℝ f (Ici x) x

⊢ ∃ R > 0, ∀ r ∈ Ioo 0 R, x ∈ A f (derivWithin f (Ici x) x) r ε
-- Mathlib/Analysis/Calculus/FDeriv/Measurable.lean
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
ε : ℝ
hε : 0 < ε
x : ℝ
hx : DifferentiableWithinAt ℝ f (Ici x) x
this : HasDerivWithinAt f (derivWithin f (Ici x) x) (Ici x) x

⊢ ∃ R > 0, ∀ r ∈ Ioo 0 R, x ∈ A f (derivWithin f (Ici x) x) r ε
-- Mathlib/Analysis/Calculus/FDeriv/Measurable.lean
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
ε : ℝ
hε : 0 < ε
x : ℝ
hx : DifferentiableWithinAt ℝ f (Ici x) x
this : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : ℝ) in 𝓝[≥] x, ‖f x_1 - f x - (x_1 - x) • derivWithin f (Ici x) x‖ ≤ c * ‖x_1 - x‖

⊢ ∃ R > 0, ∀ r ∈ Ioo 0 R, x ∈ A f (derivWithin f (Ici x) x) r ε
-- Mathlib/Analysis/Calculus/FDeriv/Measurable.lean
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
ε : ℝ
hε : 0 < ε
x : ℝ
hx : DifferentiableWithinAt ℝ f (Ici x) x
this : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : ℝ) in 𝓝[≥] x, ‖f x_1 - f x - (x_1 - x) • derivWithin f (Ici x) x‖ ≤ c * ‖x_1 - x‖
m : ℝ
xm : m ∈ Ioi x
hm : Ico x m ⊆ {x_1 | (fun x_2 => ‖f x_2 - f x - (x_2 - x) • derivWithin f (Ici x) x‖ ≤ ε / 2 * ‖x_2 - x‖) x_1}

⊢ ∃ R > 0, ∀ r ∈ Ioo 0 R, x ∈ A f (derivWithin f (Ici x) x) r ε
-- Mathlib/Analysis/Calculus/FDeriv/Measurable.lean
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
ε : ℝ
hε : 0 < ε
x : ℝ
hx : DifferentiableWithinAt ℝ f (Ici x) x
this : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : ℝ) in 𝓝[≥] x, ‖f x_1 - f x - (x_1 - x) • derivWithin f (Ici x) x‖ ≤ c * ‖x_1 - x‖
m : ℝ
xm : m ∈ Ioi x
hm : Ico x m ⊆ {x_1 | (fun x_2 => ‖f x_2 - f x - (x_2 - x) • derivWithin f (Ici x) x‖ ≤ ε / 2 * ‖x_2 - x‖) x_1}
r : ℝ
hr : r ∈ Ioo 0 (m - x)

⊢ x ∈ A f (derivWithin f (Ici x) x) r ε
-- Mathlib/Analysis/Calculus/FDeriv/Measurable.lean
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
ε : ℝ
hε : 0 < ε
x : ℝ
hx : DifferentiableWithinAt ℝ f (Ici x) x
this✝ : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : ℝ) in 𝓝[≥] x, ‖f x_1 - f x - (x_1 - x) • derivWithin f (Ici x) x‖ ≤ c * ‖x_1 - x‖
m : ℝ
xm : m ∈ Ioi x
hm : Ico x m ⊆ {x_1 | (fun x_2 => ‖f x_2 - f x - (x_2 - x) • derivWithin f (Ici x) x‖ ≤ ε / 2 * ‖x_2 - x‖) x_1}
r : ℝ
hr : r ∈ Ioo 0 (m - x)
this : r ∈ Ioc (r / 2) r

⊢ x ∈ A f (derivWithin f (Ici x) x) r ε
-- Mathlib/Data/Rat/Floor.lean
n d : ℕ
n_coprime_d : n.Coprime d

⊢ (↑n - ↑d * ⌊↑n / ↑d⌋).natAbs.Coprime d
-- Mathlib/Data/Rat/Floor.lean
n d : ℕ
n_coprime_d : n.Coprime d
this : ↑n % ↑d = ↑n - ↑d * ⌊↑n / ↑d⌋

⊢ (↑n - ↑d * ⌊↑n / ↑d⌋).natAbs.Coprime d
-- Mathlib/Data/Rat/Floor.lean
n d : ℕ
n_coprime_d : n.Coprime d
this : ↑n % ↑d = ↑n - ↑d * ⌊↑n / ↑d⌋

⊢ (↑n % ↑d).natAbs.Coprime d
-- Mathlib/Data/Rat/Floor.lean
n d : ℕ
n_coprime_d : n.Coprime d
this✝ : ↑n % ↑d = ↑n - ↑d * ⌊↑n / ↑d⌋
this : d.Coprime n

⊢ (↑n % ↑d).natAbs.Coprime d
-- 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 K : Subgroup G
a b : Quotient ↑H.toSubmonoid ↑K

⊢ a ≠ b → Disjoint (doset (Quotient.out' a) ↑H ↑K) (doset (Quotient.out' b) ↑H ↑K)