-- Mathlib/Algebra/Homology/HomotopyCategory/ShortExact.lean
lemma quasiIso_descShortComplex : QuasiIso (descShortComplex S) where
  quasiIsoAt n

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
n : ℤ
φ : ((homologyFunctor C (up ℤ) 0).homologySequenceComposableArrows₅ (triangleh S.f) n (n + 1) ⋯).δlast ⟶
  (composableArrows₅ hS n (n + 1) ⋯).δlast :=
  homMk₄ ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₂)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₃ ≫
      HomologicalComplex.homologyMap (descShortComplex S) n)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₂) ⋯ ⋯ ⋯ ⋯
⊢ IsIso
    ((homologyFunctorFactors C (up ℤ) n).hom.app (mappingCone S.f) ≫
      HomologicalComplex.homologyMap (descShortComplex S) n)

Tactic:
apply Abelian.isIso_of_epi_of_isIso_of_isIso_of_mono
  ((homologyFunctor C (up ℤ) 0).homologySequenceComposableArrows₅_exact _
    (mappingCone_triangleh_distinguished S.f) n _ rfl).δlast
  (composableArrows₅_exact hS n _ rfl).δlast φ

Result:
case h₀
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n : ℤ
φ : ((homologyFunctor C (up ℤ) 0).homologySequenceComposableArrows₅ (triangleh S.f) n (n + 1) ⋯).δlast ⟶
  (composableArrows₅ hS n (n + 1) ⋯).δlast :=
  homMk₄ ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₂)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₃ ≫
      HomologicalComplex.homologyMap (descShortComplex S) n)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₂) ⋯ ⋯ ⋯ ⋯
⊢ Epi (app' φ 0 ⋯)

case h₁
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n : ℤ
φ : ((homologyFunctor C (up ℤ) 0).homologySequenceComposableArrows₅ (triangleh S.f) n (n + 1) ⋯).δlast ⟶
  (composableArrows₅ hS n (n + 1) ⋯).δlast :=
  homMk₄ ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₂)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₃ ≫
      HomologicalComplex.homologyMap (descShortComplex S) n)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₂) ⋯ ⋯ ⋯ ⋯
⊢ IsIso (app' φ 1 ⋯)

case h₂
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n : ℤ
φ : ((homologyFunctor C (up ℤ) 0).homologySequenceComposableArrows₅ (triangleh S.f) n (n + 1) ⋯).δlast ⟶
  (composableArrows₅ hS n (n + 1) ⋯).δlast :=
  homMk₄ ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₂)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₃ ≫
      HomologicalComplex.homologyMap (descShortComplex S) n)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₂) ⋯ ⋯ ⋯ ⋯
⊢ IsIso (app' φ 3 ⋯)

case h₃
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n : ℤ
φ : ((homologyFunctor C (up ℤ) 0).homologySequenceComposableArrows₅ (triangleh S.f) n (n + 1) ⋯).δlast ⟶
  (composableArrows₅ hS n (n + 1) ⋯).δlast :=
  homMk₄ ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₂)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₃ ≫
      HomologicalComplex.homologyMap (descShortComplex S) n)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₂) ⋯ ⋯ ⋯ ⋯
⊢ Mono (app' φ 4 ⋯)

Proof state:
case h₃
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n : ℤ
φ : ((homologyFunctor C (up ℤ) 0).homologySequenceComposableArrows₅ (triangleh S.f) n (n + 1) ⋯).δlast ⟶
  (composableArrows₅ hS n (n + 1) ⋯).δlast :=
  homMk₄ ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₂)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₃ ≫
      HomologicalComplex.homologyMap (descShortComplex S) n)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₂) ⋯ ⋯ ⋯ ⋯
⊢ Mono (app' φ 4 ⋯)

Tactic:
dsimp [φ]

Result:
case h₃
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n : ℤ
φ : ((homologyFunctor C (up ℤ) 0).homologySequenceComposableArrows₅ (triangleh S.f) n (n + 1) ⋯).δlast ⟶
  (composableArrows₅ hS n (n + 1) ⋯).δlast :=
  homMk₄ ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₂)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₃ ≫
      HomologicalComplex.homologyMap (descShortComplex S) n)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₂) ⋯ ⋯ ⋯ ⋯
⊢ Mono ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app S.X₂)

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
n : ℤ
⊢ IsIso (HomologicalComplex.homologyMap (descShortComplex S) n)

Tactic:
let φ : ((homologyFunctor C (up ℤ) 0).homologySequenceComposableArrows₅
    (triangleh S.f) n _ rfl).δlast ⟶ (composableArrows₅ hS n _ rfl).δlast :=
  homMk₄ ((homologyFunctorFactors C (up ℤ) _).hom.app _)
    ((homologyFunctorFactors C (up ℤ) _).hom.app _)
    ((homologyFunctorFactors C (up ℤ) _).hom.app _ ≫
      HomologicalComplex.homologyMap (descShortComplex S) n)
    ((homologyFunctorFactors C (up ℤ) _).hom.app _)
    ((homologyFunctorFactors C (up ℤ) _).hom.app _)
    ((homologyFunctorFactors C (up ℤ) _).hom.naturality S.f)
    (by
      erw [(homologyFunctorFactors C (up ℤ) n).hom.naturality_assoc]
      dsimp
      rw [← HomologicalComplex.homologyMap_comp, inr_descShortComplex])
    (by
      dsimp
      erw [homologySequenceδ_triangleh hS]
      simp only [Functor.comp_obj, HomologicalComplex.homologyFunctor_obj, assoc,
        Iso.inv_hom_id_app, comp_id])
    ((homologyFunctorFactors C (up ℤ) _).hom.naturality S.f)

Result:
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n : ℤ
φ : ((homologyFunctor C (up ℤ) 0).homologySequenceComposableArrows₅ (triangleh S.f) n (n + 1) ⋯).δlast ⟶
  (composableArrows₅ hS n (n + 1) ⋯).δlast :=
  homMk₄ ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₂)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₃ ≫
      HomologicalComplex.homologyMap (descShortComplex S) n)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₂) ⋯ ⋯ ⋯ ⋯
⊢ IsIso (HomologicalComplex.homologyMap (descShortComplex S) n)