-- Mathlib/Algebra/Homology/HomotopyCategory/ShortExact.lean
lemma homologySequenceδ_triangleh (n₀ : ℤ) (n₁ : ℤ) (h : n₀ + 1 = n₁) :
    (homologyFunctor C (up ℤ) 0).homologySequenceδ (triangleh S.f) n₀ n₁ h =
      (homologyFunctorFactors C (up ℤ) n₀).hom.app _ ≫
        HomologicalComplex.homologyMap (descShortComplex S) n₀ ≫ hS.δ n₀ n₁ h ≫
          (homologyFunctorFactors C (up ℤ) n₁).inv.app _

Proof state:
case a.w.h.op.h.intro.intro.intro.intro.intro.intro.intro
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

Tactic:
rw [HomologicalComplex.homologyπ_naturality_assoc,
  HomologicalComplex.liftCycles_comp_cyclesMap_assoc,
  S.X₁.liftCycles_shift_homologyπ_assoc _ _ _ _ n₁ (by omega) (n₁ + 1) (by simp),
  Iso.inv_hom_id_app]

Result:
case a.w.h.op.h.intro.intro.intro.intro.intro.intro.intro
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

Proof state:
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)

Tactic:
apply yoneda.map_injective

Result:
case a
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))

Proof state:
case a.w.h.op.h.intro.intro.intro.intro.intro.intro.intro
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

Tactic:
dsimp [Functor.shiftMap, homologyFunctor_shift]

Result:
case a.w.h.op.h.intro.intro.intro.intro.intro.intro.intro
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

Proof state:
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)
⊢ (up ℤ).next n₀ = n₁

Tactic:
simpa using h

Result:
no goals

Proof state:
case a.w.h.op.h.intro.intro.intro.intro.intro
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

Tactic:
rw [← cancel_epi π, reassoc_of% hx', reassoc_of% hx',
  HomologicalComplex.homologyπ_naturality_assoc,
  HomologicalComplex.liftCycles_comp_cyclesMap_assoc]

Result:
case a.w.h.op.h.intro.intro.intro.intro.intro
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