ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Algebra/Homology/Single.lean
V : Type u
inst✝² : Category.{v, u} V
inst✝¹ : HasZeroMorphisms V
inst✝ : HasZeroObject V
C : CochainComplex V ℕ
X : V
f : { f // f ≫ C.d 0 1 = 0 }
hi : (ComplexShape.up ℕ).Rel 0 1

⊢ ↑f ≫ C.d 0 1 = 0
-- Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
k : Type u_1
V : Type u_2
P : Type u_3
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
S : AffineSpace V P
p₁ p₂ : P
s : Set P
h₁ : s.Subsingleton
h₂ : affineSpan k s = ⊤

⊢ Subsingleton P
-- Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
k : Type u_1
V : Type u_2
P : Type u_3
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
S : AffineSpace V P
p₁ p₂ : P
s : Set P
h₁ : s.Subsingleton
h₂ : affineSpan k s = ⊤
p : P
hp : p ∈ s

⊢ Subsingleton P
-- Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
k : Type u_1
V : Type u_2
P : Type u_3
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
S : AffineSpace V P
p₁ p₂ : P
s : Set P
h₁ : s.Subsingleton
h₂ : affineSpan k s = ⊤
p : P
hp : p ∈ s
this : s = {p}

⊢ Subsingleton P
-- Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
k : Type u_1
V : Type u_2
P : Type u_3
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
S : AffineSpace V P
p₁ p₂ : P
s : Set P
h₁ : s.Subsingleton
p : P
h₂ : univ.Subsingleton
hp : p ∈ s
this : s = {p}

⊢ Subsingleton P
-- Mathlib/Data/Set/Card.lean
α : Type u_1
β : Type u_2
s✝ t✝ : Set α
f : α → β
s t : Set α
ht : autoParam t.Finite _auto✝

⊢ s.ncard ≤ (s \ t).ncard + t.ncard
-- Mathlib/Data/Set/Card.lean
α : Type u_1
β : Type u_2
s✝ t✝ : Set α
f : α → β
s t : Set α
ht : autoParam t.Finite _auto✝
hs : s.Finite

⊢ s.ncard ≤ (s \ t).ncard + t.ncard
-- Mathlib/Data/Set/Card.lean
α : Type u_1
β : Type u_2
s✝ t✝ : Set α
f : α → β
s t : Set α
ht : autoParam t.Finite _auto✝
hs : s.Infinite

⊢ s.ncard ≤ (s \ t).ncard + t.ncard
-- Mathlib/Data/Set/Card.lean
α : Type u_1
β : Type u_2
s✝ t✝ : Set α
f : α → β
s t : Set α
ht : autoParam t.Finite _auto✝
hs : s.Infinite

⊢ s.ncard = 0
-- Mathlib/RingTheory/MvPolynomial/Tower.lean
R : Type u_1
A : Type u_2
B : Type u_3
σ : Type u_4
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring A
inst✝⁴ : CommSemiring B
inst✝³ : Algebra R A
inst✝² : Algebra A B
inst✝¹ : Algebra R B
inst✝ : IsScalarTower R A B
x : σ → A
p : MvPolynomial σ R
h : Function.Injective ⇑(algebraMap A B)

⊢ (aeval (⇑(algebraMap A B) ∘ x)) p = 0 ↔ (aeval x) p = 0
-- Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
x y z : ℝ
n : ℕ
hx0 : 0 ≤ x
hx1 : x ≤ 1
hz : 0 ≤ z
hyz : z ≤ y

⊢ x ^ y ≤ x ^ z
-- Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
y z : ℝ
n : ℕ
hz : 0 ≤ z
hyz : z ≤ y
hx0 : 0 ≤ 0
hx1 : 0 ≤ 1

⊢ 0 ^ y ≤ 0 ^ z
-- Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
x y z : ℝ
n : ℕ
hx0 : 0 ≤ x
hx1 : x ≤ 1
hz : 0 ≤ z
hyz : z ≤ y
hx0' : 0 < x

⊢ x ^ y ≤ x ^ z
-- Mathlib/Logic/Equiv/Defs.lean
α : Sort u
β : Sort v
γ : Sort w
α₁ : Sort u_1
β₁ : Sort u_2
γ₁ : Sort u_3
α₂ : Sort u_4
β₂ : Sort u_5
γ₂ : Sort u_6
ea : α₁ ≃ α₂
eb : β₁ ≃ β₂
ec : γ₁ ≃ γ₂
f : α₁ → β₁
g : β₁ → γ₁
x✝ : α₂

⊢ (ea.arrowCongr ec) (g ∘ f) x✝ = ((eb.arrowCongr ec) g ∘ (ea.arrowCongr eb) f) x✝
-- Mathlib/LinearAlgebra/Prod.lean
R : Type u
K : Type u'
M : Type v
V : Type v'
M₂ : Type w
V₂ : Type w'
M₃ : Type y
V₃ : Type y'
M₄ : Type z
ι : Type x
M₅ : Type u_1
M₆ : Type u_2
S : Type u_3
inst✝¹³ : Semiring R
inst✝¹² : Semiring S
inst✝¹¹ : AddCommMonoid M
inst✝¹⁰ : AddCommMonoid M₂
inst✝⁹ : AddCommMonoid M₃
inst✝⁸ : AddCommMonoid M₄
inst✝⁷ : AddCommMonoid M₅
inst✝⁶ : AddCommMonoid M₆
inst✝⁵ : Module R M
inst✝⁴ : Module R M₂
inst✝³ : Module R M₃
inst✝² : Module R M₄
inst✝¹ : Module R M₅
inst✝ : Module R M₆
f✝ f : M →ₗ[R] M₂
g : M →ₗ[R] M₃
x y : M

⊢ Pi.prod (⇑f) (⇑g) (x + y) = Pi.prod (⇑f) (⇑g) x + Pi.prod (⇑f) (⇑g) y
-- Mathlib/LinearAlgebra/Prod.lean
R : Type u
K : Type u'
M : Type v
V : Type v'
M₂ : Type w
V₂ : Type w'
M₃ : Type y
V₃ : Type y'
M₄ : Type z
ι : Type x
M₅ : Type u_1
M₆ : Type u_2
S : Type u_3
inst✝¹³ : Semiring R
inst✝¹² : Semiring S
inst✝¹¹ : AddCommMonoid M
inst✝¹⁰ : AddCommMonoid M₂
inst✝⁹ : AddCommMonoid M₃
inst✝⁸ : AddCommMonoid M₄
inst✝⁷ : AddCommMonoid M₅
inst✝⁶ : AddCommMonoid M₆
inst✝⁵ : Module R M
inst✝⁴ : Module R M₂
inst✝³ : Module R M₃
inst✝² : Module R M₄
inst✝¹ : Module R M₅
inst✝ : Module R M₆
f✝ f : M →ₗ[R] M₂
g : M →ₗ[R] M₃
c : R
x : M

⊢ { toFun := Pi.prod ⇑f ⇑g, map_add' := ⋯ }.toFun (c • x) = (RingHom.id R) c • { toFun := Pi.prod ⇑f ⇑g, map_add' := ⋯ }.toFun x
-- Mathlib/Algebra/Order/Interval/Set/Instances.lean
α : Type u_1
inst✝ : OrderedSemiring α
x : ↑(Icc 0 1)

⊢ x = 1 ↔ ↑x = 1
-- Mathlib/Algebra/Homology/HomotopyCategory/ShortExact.lean
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n₀ n₁ : ℤ
h : n₀ + 1 = n₁

⊢ (homologyFunctor C (up ℤ) 0).homologySequenceδ (Triangle.mk ((quotient C (up ℤ)).map S.f) ((quotient C (up ℤ)).map (inr S.f)) ((quotient C (up ℤ)).map (triangle S.f).mor₃ ≫ ((quotient C (up ℤ)).commShiftIso 1).hom.app S.X₁)) n₀ n₁ h = (homologyFunctorFactors C (up ℤ) n₀).hom.app (mappingCone S.f) ≫ HomologicalComplex.homologyMap (descShortComplex S) n₀ ≫ hS.δ n₀ n₁ h ≫ (homologyFunctorFactors C (up ℤ) n₁).inv.app S.X₁
-- Mathlib/Algebra/Homology/HomotopyCategory/ShortExact.lean
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n₀ n₁ : ℤ
h : n₀ + 1 = n₁

⊢ (homologyFunctorFactors C (up ℤ) n₀).inv.app (mappingCone S.f) ≫ (homologyFunctor C (up ℤ) 0).homologySequenceδ (Triangle.mk ((quotient C (up ℤ)).map S.f) ((quotient C (up ℤ)).map (inr S.f)) ((quotient C (up ℤ)).map (triangle S.f).mor₃ ≫ ((quotient C (up ℤ)).commShiftIso 1).hom.app S.X₁)) n₀ n₁ h ≫ (homologyFunctorFactors C (up ℤ) n₁).hom.app S.X₁ = HomologicalComplex.homologyMap (descShortComplex S) n₀ ≫ hS.δ n₀ n₁ h ≫ 𝟙 ((HomologicalComplex.homologyFunctor C (up ℤ) n₁).obj S.X₁)
-- Mathlib/Algebra/Homology/HomotopyCategory/ShortExact.lean
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n₀ n₁ : ℤ
h : n₀ + 1 = n₁

⊢ yoneda.map ((homologyFunctorFactors C (up ℤ) n₀).inv.app (mappingCone S.f) ≫ (homologyFunctor C (up ℤ) 0).homologySequenceδ (Triangle.mk ((quotient C (up ℤ)).map S.f) ((quotient C (up ℤ)).map (inr S.f)) ((quotient C (up ℤ)).map (triangle S.f).mor₃ ≫ ((quotient C (up ℤ)).commShiftIso 1).hom.app S.X₁)) n₀ n₁ h ≫ (homologyFunctorFactors C (up ℤ) n₁).hom.app S.X₁) = yoneda.map (HomologicalComplex.homologyMap (descShortComplex S) n₀ ≫ hS.δ n₀ n₁ h ≫ 𝟙 ((HomologicalComplex.homologyFunctor C (up ℤ) n₁).obj S.X₁))
-- Mathlib/Algebra/Homology/HomotopyCategory/ShortExact.lean
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
A : C
x : A ⟶ (HomologicalComplex.homologyFunctor C (up ℤ) n₀).obj (mappingCone S.f)

⊢ (yoneda.map ((homologyFunctorFactors C (up ℤ) n₀).inv.app (mappingCone S.f) ≫ (homologyFunctor C (up ℤ) 0).homologySequenceδ (Triangle.mk ((quotient C (up ℤ)).map S.f) ((quotient C (up ℤ)).map (inr S.f)) ((quotient C (up ℤ)).map (triangle S.f).mor₃ ≫ ((quotient C (up ℤ)).commShiftIso 1).hom.app S.X₁)) n₀ n₁ h ≫ (homologyFunctorFactors C (up ℤ) n₁).hom.app S.X₁)).app (Opposite.op A) x = (yoneda.map (HomologicalComplex.homologyMap (descShortComplex S) n₀ ≫ hS.δ n₀ n₁ h ≫ 𝟙 ((HomologicalComplex.homologyFunctor C (up ℤ) n₁).obj S.X₁))).app (Opposite.op A) x
-- Mathlib/Algebra/Homology/HomotopyCategory/ShortExact.lean
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
A : C
x : A ⟶ (HomologicalComplex.homologyFunctor C (up ℤ) n₀).obj (mappingCone S.f)
A' : C
π : A' ⟶ A
w✝ : Epi π
x' : A' ⟶ (mappingCone S.f).X n₀
w : x' ≫ (mappingCone S.f).d n₀ n₁ = 0
hx' : π ≫ x = (mappingCone S.f).liftCycles x' n₁ ⋯ w ≫ (mappingCone S.f).homologyπ n₀

⊢ (yoneda.map ((homologyFunctorFactors C (up ℤ) n₀).inv.app (mappingCone S.f) ≫ (homologyFunctor C (up ℤ) 0).homologySequenceδ (Triangle.mk ((quotient C (up ℤ)).map S.f) ((quotient C (up ℤ)).map (inr S.f)) ((quotient C (up ℤ)).map (triangle S.f).mor₃ ≫ ((quotient C (up ℤ)).commShiftIso 1).hom.app S.X₁)) n₀ n₁ h ≫ (homologyFunctorFactors C (up ℤ) n₁).hom.app S.X₁)).app (Opposite.op A) x = (yoneda.map (HomologicalComplex.homologyMap (descShortComplex S) n₀ ≫ hS.δ n₀ n₁ h ≫ 𝟙 ((HomologicalComplex.homologyFunctor C (up ℤ) n₁).obj S.X₁))).app (Opposite.op A) x
-- Mathlib/Algebra/Homology/HomotopyCategory/ShortExact.lean
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
A : C
x : A ⟶ (HomologicalComplex.homologyFunctor C (up ℤ) n₀).obj (mappingCone S.f)
A' : C
π : A' ⟶ A
w✝ : Epi π
x' : A' ⟶ (mappingCone S.f).X n₀
w : x' ≫ (mappingCone S.f).d n₀ n₁ = 0
hx' : π ≫ x = (mappingCone S.f).liftCycles x' n₁ ⋯ w ≫ (mappingCone S.f).homologyπ n₀

⊢ x ≫ (homologyFunctorFactors C (up ℤ) n₀).inv.app (mappingCone S.f) ≫ (homologyFunctorFactors C (up ℤ) n₀).hom.app (mappingCone S.f) ≫ (HomologicalComplex.homologyFunctor C (up ℤ) 0).shiftMap (triangle S.f).mor₃ n₀ n₁ ⋯ ≫ (homologyFunctorFactors C (up ℤ) n₁).inv.app S.X₁ ≫ (homologyFunctorFactors C (up ℤ) n₁).hom.app S.X₁ = x ≫ HomologicalComplex.homologyMap (descShortComplex S) n₀ ≫ hS.δ n₀ n₁ h ≫ 𝟙 (S.X₁.homology n₁)
-- Mathlib/Algebra/Homology/HomotopyCategory/ShortExact.lean
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
A : C
x : A ⟶ (HomologicalComplex.homologyFunctor C (up ℤ) n₀).obj (mappingCone S.f)
A' : C
π : A' ⟶ A
w✝ : Epi π
x' : A' ⟶ (mappingCone S.f).X n₀
w : x' ≫ (mappingCone S.f).d n₀ n₁ = 0
hx' : π ≫ x = (mappingCone S.f).liftCycles x' n₁ ⋯ w ≫ (mappingCone S.f).homologyπ n₀

⊢ x ≫ (HomologicalComplex.homologyFunctor C (up ℤ) 0).shiftMap (triangle S.f).mor₃ n₀ n₁ ⋯ ≫ 𝟙 ((HomologicalComplex.homologyFunctor C (up ℤ) n₁).obj S.X₁) = x ≫ HomologicalComplex.homologyMap (descShortComplex S) n₀ ≫ hS.δ n₀ n₁ h
-- Mathlib/Algebra/Homology/HomotopyCategory/ShortExact.lean
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
A : C
x : A ⟶ (HomologicalComplex.homologyFunctor C (up ℤ) n₀).obj (mappingCone S.f)
A' : C
π : A' ⟶ A
w✝ : Epi π
x' : A' ⟶ (mappingCone S.f).X n₀
w : x' ≫ (mappingCone S.f).d n₀ n₁ = 0
hx' : π ≫ x = (mappingCone S.f).liftCycles x' n₁ ⋯ w ≫ (mappingCone S.f).homologyπ n₀

⊢ x ≫ (HomologicalComplex.homologyFunctor C (up ℤ) 0).shiftMap (triangle S.f).mor₃ n₀ n₁ ⋯ = x ≫ HomologicalComplex.homologyMap (descShortComplex S) n₀ ≫ hS.δ n₀ n₁ h
-- Mathlib/Algebra/Homology/HomotopyCategory/ShortExact.lean
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
A : C
x : A ⟶ (HomologicalComplex.homologyFunctor C (up ℤ) n₀).obj (mappingCone S.f)
A' : C
π : A' ⟶ A
w✝ : Epi π
x' : A' ⟶ (mappingCone S.f).X n₀
w : x' ≫ (mappingCone S.f).d n₀ n₁ = 0
hx' : π ≫ x = (mappingCone S.f).liftCycles x' n₁ ⋯ w ≫ (mappingCone S.f).homologyπ n₀

⊢ (mappingCone S.f).liftCycles x' n₁ ⋯ w ≫ (mappingCone S.f).homologyπ n₀ ≫ (HomologicalComplex.homologyFunctor C (up ℤ) 0).shiftMap (triangle S.f).mor₃ n₀ n₁ ⋯ = HomologicalComplex.liftCycles S.X₃ (x' ≫ (descShortComplex S).f n₀) n₁ ⋯ ⋯ ≫ HomologicalComplex.homologyπ S.X₃ n₀ ≫ hS.δ n₀ n₁ h
-- Mathlib/Algebra/Homology/HomotopyCategory/ShortExact.lean
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
A : C
x : A ⟶ (HomologicalComplex.homologyFunctor C (up ℤ) n₀).obj (mappingCone S.f)
A' : C
π : A' ⟶ A
w✝ : Epi π
x' : A' ⟶ (mappingCone S.f).X n₀
w : x' ≫ (mappingCone S.f).d n₀ n₁ = 0
hx' : π ≫ x = (mappingCone S.f).liftCycles x' n₁ ⋯ w ≫ (mappingCone S.f).homologyπ n₀
a : A' ⟶ S.X₁.X n₁
b : A' ⟶ S.X₂.X n₀
hab : x' = a ≫ (inl S.f).v n₁ n₀ ⋯ + b ≫ (inr S.f).f n₀

⊢ (mappingCone S.f).liftCycles x' n₁ ⋯ w ≫ (mappingCone S.f).homologyπ n₀ ≫ (HomologicalComplex.homologyFunctor C (up ℤ) 0).shiftMap (triangle S.f).mor₃ n₀ n₁ ⋯ = HomologicalComplex.liftCycles S.X₃ (x' ≫ (descShortComplex S).f n₀) n₁ ⋯ ⋯ ≫ HomologicalComplex.homologyπ S.X₃ n₀ ≫ hS.δ n₀ n₁ h
-- Mathlib/Algebra/Homology/HomotopyCategory/ShortExact.lean
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
A : C
x : A ⟶ (HomologicalComplex.homologyFunctor C (up ℤ) n₀).obj (mappingCone S.f)
A' : C
π : A' ⟶ A
w✝¹ : Epi π
x' : A' ⟶ (mappingCone S.f).X n₀
w✝ : x' ≫ (mappingCone S.f).d n₀ n₁ = 0
hx' : π ≫ x = (mappingCone S.f).liftCycles x' n₁ ⋯ w✝ ≫ (mappingCone S.f).homologyπ n₀
a : A' ⟶ S.X₁.X n₁
b : A' ⟶ S.X₂.X n₀
w : a ≫ S.X₁.d n₁ (n₁ + 1) = 0 ∧ a ≫ S.f.f n₁ + b ≫ S.X₂.d n₀ n₁ = 0
hab : x' = a ≫ (inl S.f).v n₁ n₀ ⋯ + b ≫ (inr S.f).f n₀

⊢ (mappingCone S.f).liftCycles x' n₁ ⋯ w✝ ≫ (mappingCone S.f).homologyπ n₀ ≫ (HomologicalComplex.homologyFunctor C (up ℤ) 0).shiftMap (triangle S.f).mor₃ n₀ n₁ ⋯ = HomologicalComplex.liftCycles S.X₃ (x' ≫ (descShortComplex S).f n₀) n₁ ⋯ ⋯ ≫ HomologicalComplex.homologyπ S.X₃ n₀ ≫ hS.δ n₀ n₁ h
-- Mathlib/Algebra/Homology/HomotopyCategory/ShortExact.lean
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
A : C
x : A ⟶ (HomologicalComplex.homologyFunctor C (up ℤ) n₀).obj (mappingCone S.f)
A' : C
π : A' ⟶ A
w✝¹ : Epi π
x' : A' ⟶ (mappingCone S.f).X n₀
w✝ : x' ≫ (mappingCone S.f).d n₀ n₁ = 0
hx' : π ≫ x = (mappingCone S.f).liftCycles x' n₁ ⋯ w✝ ≫ (mappingCone S.f).homologyπ n₀
a : A' ⟶ S.X₁.X n₁
b : A' ⟶ S.X₂.X n₀
w : a ≫ S.X₁.d n₁ (n₁ + 1) = 0 ∧ a ≫ S.f.f n₁ + b ≫ S.X₂.d n₀ n₁ = 0
hab : x' = a ≫ (inl S.f).v n₁ n₀ ⋯ + b ≫ (inr S.f).f n₀

⊢ (mappingCone S.f).liftCycles x' n₁ ⋯ w✝ ≫ (mappingCone S.f).homologyπ n₀ ≫ (HomologicalComplex.homologyFunctor C (up ℤ) 0).shiftMap (triangle S.f).mor₃ n₀ n₁ ⋯ = HomologicalComplex.liftCycles S.X₃ (b ≫ S.g.f n₀) n₁ ⋯ ⋯ ≫ HomologicalComplex.homologyπ S.X₃ n₀ ≫ hS.δ n₀ n₁ h
-- Mathlib/Algebra/Homology/HomotopyCategory/ShortExact.lean
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
A : C
x : A ⟶ (HomologicalComplex.homologyFunctor C (up ℤ) n₀).obj (mappingCone S.f)
A' : C
π : A' ⟶ A
w✝¹ : Epi π
x' : A' ⟶ (mappingCone S.f).X n₀
w✝ : x' ≫ (mappingCone S.f).d n₀ n₁ = 0
hx' : π ≫ x = (mappingCone S.f).liftCycles x' n₁ ⋯ w✝ ≫ (mappingCone S.f).homologyπ n₀
a : A' ⟶ S.X₁.X n₁
b : A' ⟶ S.X₂.X n₀
w : a ≫ S.X₁.d n₁ (n₁ + 1) = 0 ∧ a ≫ S.f.f n₁ + b ≫ S.X₂.d n₀ n₁ = 0
hab : x' = a ≫ (inl S.f).v n₁ n₀ ⋯ + b ≫ (inr S.f).f n₀

⊢ (mappingCone S.f).liftCycles x' n₁ ⋯ w✝ ≫ (mappingCone S.f).homologyπ n₀ ≫ (HomologicalComplex.homologyFunctor C (up ℤ) 0).shiftMap (triangle S.f).mor₃ n₀ n₁ ⋯ = S.X₁.liftCycles (-a) (n₁ + 1) ⋯ ⋯ ≫ S.X₁.homologyπ n₁
-- Mathlib/Algebra/Homology/HomotopyCategory/ShortExact.lean
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
A : C
x : A ⟶ (HomologicalComplex.homologyFunctor C (up ℤ) n₀).obj (mappingCone S.f)
A' : C
π : A' ⟶ A
w✝¹ : Epi π
x' : A' ⟶ (mappingCone S.f).X n₀
w✝ : x' ≫ (mappingCone S.f).d n₀ n₁ = 0
hx' : π ≫ x = (mappingCone S.f).liftCycles x' n₁ ⋯ w✝ ≫ (mappingCone S.f).homologyπ n₀
a : A' ⟶ S.X₁.X n₁
b : A' ⟶ S.X₂.X n₀
w : a ≫ S.X₁.d n₁ (n₁ + 1) = 0 ∧ a ≫ S.f.f n₁ + b ≫ S.X₂.d n₀ n₁ = 0
hab : x' = a ≫ (inl S.f).v n₁ n₀ ⋯ + b ≫ (inr S.f).f n₀

⊢ (mappingCone S.f).liftCycles x' n₁ ⋯ w✝ ≫ (mappingCone S.f).homologyπ n₀ ≫ HomologicalComplex.homologyMap (triangle S.f).mor₃ n₀ ≫ ((HomologicalComplex.homologyFunctor C (up ℤ) 0).shiftIso 1 n₀ n₁ ⋯).hom.app S.X₁ = S.X₁.liftCycles (-a) (n₁ + 1) ⋯ ⋯ ≫ S.X₁.homologyπ n₁
-- Mathlib/Algebra/Homology/HomotopyCategory/ShortExact.lean
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
A : C
x : A ⟶ (HomologicalComplex.homologyFunctor C (up ℤ) n₀).obj (mappingCone S.f)
A' : C
π : A' ⟶ A
w✝¹ : Epi π
x' : A' ⟶ (mappingCone S.f).X n₀
w✝ : x' ≫ (mappingCone S.f).d n₀ n₁ = 0
hx' : π ≫ x = (mappingCone S.f).liftCycles x' n₁ ⋯ w✝ ≫ (mappingCone S.f).homologyπ n₀
a : A' ⟶ S.X₁.X n₁
b : A' ⟶ S.X₂.X n₀
w : a ≫ S.X₁.d n₁ (n₁ + 1) = 0 ∧ a ≫ S.f.f n₁ + b ≫ S.X₂.d n₀ n₁ = 0
hab : x' = a ≫ (inl S.f).v n₁ n₀ ⋯ + b ≫ (inr S.f).f n₀

⊢ HomologicalComplex.liftCycles S.X₁ ((x' ≫ (triangle S.f).mor₃.f n₀) ≫ (S.X₁.shiftFunctorObjXIso 1 n₀ n₁ ⋯).hom) (n₁ + 1) ⋯ ⋯ ≫ HomologicalComplex.homologyπ S.X₁ n₁ ≫ 𝟙 (((HomologicalComplex.homologyFunctor C (up ℤ) 0).shift n₁).obj S.X₁) = S.X₁.liftCycles (-a) (n₁ + 1) ⋯ ⋯ ≫ S.X₁.homologyπ n₁
-- Mathlib/Algebra/Homology/HomotopyCategory/ShortExact.lean
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
A : C
x : A ⟶ (HomologicalComplex.homologyFunctor C (up ℤ) n₀).obj (mappingCone S.f)
A' : C
π : A' ⟶ A
w✝¹ : Epi π
x' : A' ⟶ (mappingCone S.f).X n₀
w✝ : x' ≫ (mappingCone S.f).d n₀ n₁ = 0
hx' : π ≫ x = (mappingCone S.f).liftCycles x' n₁ ⋯ w✝ ≫ (mappingCone S.f).homologyπ n₀
a : A' ⟶ S.X₁.X n₁
b : A' ⟶ S.X₂.X n₀
w : a ≫ S.X₁.d n₁ (n₁ + 1) = 0 ∧ a ≫ S.f.f n₁ + b ≫ S.X₂.d n₀ n₁ = 0
hab : x' = a ≫ (inl S.f).v n₁ n₀ ⋯ + b ≫ (inr S.f).f n₀

⊢ HomologicalComplex.liftCycles S.X₁ ((x' ≫ (triangle S.f).mor₃.f n₀) ≫ (HomologicalComplex.XIsoOfEq S.X₁ ⋯).hom) (n₁ + 1) ⋯ ⋯ ≫ HomologicalComplex.homologyπ S.X₁ n₁ ≫ 𝟙 (HomologicalComplex.homology S.X₁ n₁) = S.X₁.liftCycles (-a) (n₁ + 1) ⋯ ⋯ ≫ S.X₁.homologyπ n₁
-- Mathlib/SetTheory/Cardinal/Cofinality.lean
α : Type u_1
r : α → α → Prop
a o : Ordinal.{u}
f : (b : Ordinal.{u}) → b < o → Ordinal.{u}
hf : a.IsFundamentalSequence o f
i j : Ordinal.{u}
hi : i < o
hj : j < o
hij : i ≤ j

⊢ f i hi ≤ f j hj
-- Mathlib/SetTheory/Cardinal/Cofinality.lean
α : Type u_1
r : α → α → Prop
a o : Ordinal.{u}
f : (b : Ordinal.{u}) → b < o → Ordinal.{u}
hf : a.IsFundamentalSequence o f
i j : Ordinal.{u}
hi : i < o
hj : j < o
hij✝ : i ≤ j
hij : i < j

⊢ f i hi ≤ f j hj
-- Mathlib/SetTheory/Cardinal/Cofinality.lean
α : Type u_1
r : α → α → Prop
a o : Ordinal.{u}
f : (b : Ordinal.{u}) → b < o → Ordinal.{u}
hf : a.IsFundamentalSequence o f
i : Ordinal.{u}
hi hj : i < o
hij : i ≤ i

⊢ f i hi ≤ f i hj
-- Mathlib/Data/Seq/WSeq.lean
α : Type u
β : Type v
γ : Type w
R : α → α → Prop
H : Symmetric R
s1 s2 : WSeq α
h : Function.swap (LiftRel R) s2 s1

⊢ LiftRel R s2 s1
-- Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean
C : Type u
inst✝² : Category.{v, u} C
I : MultispanIndex C
inst✝¹ : HasCoproduct I.left
inst✝ : HasCoproduct I.right
K₁ K₂ : Multicofork I
f : K₁ ⟶ K₂

⊢ K₁.toSigmaCofork.ι.app WalkingParallelPair.zero ≫ f.hom = K₂.toSigmaCofork.ι.app WalkingParallelPair.zero
-- Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean
C : Type u
inst✝² : Category.{v, u} C
I : MultispanIndex C
inst✝¹ : HasCoproduct I.left
inst✝ : HasCoproduct I.right
K₁ K₂ : Multicofork I
f : K₁ ⟶ K₂

⊢ K₁.toSigmaCofork.ι.app WalkingParallelPair.one ≫ f.hom = K₂.toSigmaCofork.ι.app WalkingParallelPair.one
-- Mathlib/MeasureTheory/Covering/Differentiation.lean
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
f : α → E
hf : Integrable f μ
h'f : StronglyMeasurable f
A : μ.FiniteSpanningSetsIn {K | IsOpen K} := μ.finiteSpanningSetsInOpen'

⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / μ a) (v.filterAt x) (𝓝 0)
-- Mathlib/MeasureTheory/Covering/Differentiation.lean
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
f : α → E
hf : Integrable f μ
h'f : StronglyMeasurable f
A : μ.FiniteSpanningSetsIn {K | IsOpen K} := μ.finiteSpanningSetsInOpen'
t : Set E
t_count : t.Countable
ht : range f ⊆ closure t

⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / μ a) (v.filterAt x) (𝓝 0)
-- Mathlib/MeasureTheory/Covering/Differentiation.lean
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
f : α → E
hf : Integrable f μ
h'f : StronglyMeasurable f
A : μ.FiniteSpanningSetsIn {K | IsOpen K} := μ.finiteSpanningSetsInOpen'
t : Set E
t_count : t.Countable
ht : range f ⊆ closure t
main : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), ∀ c ∈ t, Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - (A.set n).indicator (fun x => c) y‖₊ ∂μ) / μ a) (v.filterAt x) (𝓝 ↑‖f x - (A.set n).indicator (fun x => c) x‖₊)

⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / μ a) (v.filterAt x) (𝓝 0)
-- Mathlib/MeasureTheory/Covering/Differentiation.lean
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
f : α → E
hf : Integrable f μ
h'f : StronglyMeasurable f
A : μ.FiniteSpanningSetsIn {K | IsOpen K} := μ.finiteSpanningSetsInOpen'
t : Set E
t_count : t.Countable
ht : range f ⊆ closure t
main : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), ∀ c ∈ t, Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - (A.set n).indicator (fun x => c) y‖₊ ∂μ) / μ a) (v.filterAt x) (𝓝 ↑‖f x - (A.set n).indicator (fun x => c) x‖₊)
x : α
hx : ∀ (n : ℕ), ∀ c ∈ t, Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - (A.set n).indicator (fun x => c) y‖₊ ∂μ) / μ a) (v.filterAt x) (𝓝 ↑‖f x - (A.set n).indicator (fun x => c) x‖₊)
h'x : ∀ᶠ (a : Set α) in v.filterAt x, 0 < μ a

⊢ Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / μ a) (v.filterAt x) (𝓝 0)
-- Mathlib/MeasureTheory/Covering/Differentiation.lean
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
f : α → E
hf : Integrable f μ
h'f : StronglyMeasurable f
A : μ.FiniteSpanningSetsIn {K | IsOpen K} := μ.finiteSpanningSetsInOpen'
t : Set E
t_count : t.Countable
ht : range f ⊆ closure t
main : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), ∀ c ∈ t, Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - (A.set n).indicator (fun x => c) y‖₊ ∂μ) / μ a) (v.filterAt x) (𝓝 ↑‖f x - (A.set n).indicator (fun x => c) x‖₊)
x : α
hx : ∀ (n : ℕ), ∀ c ∈ t, Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - (A.set n).indicator (fun x => c) y‖₊ ∂μ) / μ a) (v.filterAt x) (𝓝 ↑‖f x - (A.set n).indicator (fun x => c) x‖₊)
h'x : ∀ᶠ (a : Set α) in v.filterAt x, 0 < μ a
M : ∀ c ∈ t, Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - c‖₊ ∂μ) / μ a) (v.filterAt x) (𝓝 ↑‖f x - c‖₊)

⊢ Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / μ a) (v.filterAt x) (𝓝 0)