-- Mathlib/Algebra/Homology/DerivedCategory/Ext/ExactSequences.lean
/-- Alternative formulation of `covariant_sequence_exact₁` -/
lemma covariant_sequence_exact₁' :
    (ShortComplex.mk
      (AddCommGrp.ofHom (hS.extClass.postcomp X h))
      (AddCommGrp.ofHom ((mk₀ S.f).postcomp X (add_zero n₁))) (by
        ext x
        dsimp
        simp only [comp_assoc_of_third_deg_zero, ShortComplex.ShortExact.extClass_comp,
          comp_zero])).Exact

Proof state:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
X : C
S : ShortComplex C
hS : S.ShortExact
n₀ n₁ : ℕ
h : n₀ + 1 = n₁
this✝ : HasDerivedCategory C := HasDerivedCategory.standard C
this :
  Function.Exact
    ⇑(ConcreteCategory.hom
        (ShortComplex.mk
            ((preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).homologySequenceδ hS.singleTriangle ↑n₀ ↑n₁ ⋯)
            (((preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).shift ↑n₁).map hS.singleTriangle.mor₁) ⋯).f)
    ⇑(ConcreteCategory.hom
        (ShortComplex.mk
            ((preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).homologySequenceδ hS.singleTriangle ↑n₀ ↑n₁ ⋯)
            (((preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).shift ↑n₁).map hS.singleTriangle.mor₁) ⋯).g)
⊢ Function.Exact
    ⇑(ConcreteCategory.hom
        (ShortComplex.mk (AddCommGrp.ofHom (hS.extClass.postcomp X h)) (AddCommGrp.ofHom ((mk₀ S.f).postcomp X ⋯)) ⋯).f)
    ⇑(ConcreteCategory.hom
        (ShortComplex.mk (AddCommGrp.ofHom (hS.extClass.postcomp X h)) (AddCommGrp.ofHom ((mk₀ S.f).postcomp X ⋯)) ⋯).g)

Tactic:
apply Function.Exact.of_ladder_addEquiv_of_exact' (e₁ := Ext.homAddEquiv)
  (e₂ := Ext.homAddEquiv) (e₃ := Ext.homAddEquiv) (H := this)

Result:
case comm₁₂
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
X : C
S : ShortComplex C
hS : S.ShortExact
n₀ n₁ : ℕ
h : n₀ + 1 = n₁
this✝ : HasDerivedCategory C := HasDerivedCategory.standard C
this :
  Function.Exact
    ⇑(ConcreteCategory.hom
        (ShortComplex.mk
            ((preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).homologySequenceδ hS.singleTriangle ↑n₀ ↑n₁ ⋯)
            (((preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).shift ↑n₁).map hS.singleTriangle.mor₁) ⋯).f)
    ⇑(ConcreteCategory.hom
        (ShortComplex.mk
            ((preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).homologySequenceδ hS.singleTriangle ↑n₀ ↑n₁ ⋯)
            (((preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).shift ↑n₁).map hS.singleTriangle.mor₁) ⋯).g)
⊢ (ConcreteCategory.hom
          (ShortComplex.mk
              ((preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).homologySequenceδ hS.singleTriangle ↑n₀ ↑n₁ ⋯)
              (((preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).shift ↑n₁).map hS.singleTriangle.mor₁)
              ⋯).f).comp
      ↑homAddEquiv =
    (↑homAddEquiv).comp
      (ConcreteCategory.hom
        (ShortComplex.mk (AddCommGrp.ofHom (hS.extClass.postcomp X h)) (AddCommGrp.ofHom ((mk₀ S.f).postcomp X ⋯)) ⋯).f)

case comm₂₃
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
X : C
S : ShortComplex C
hS : S.ShortExact
n₀ n₁ : ℕ
h : n₀ + 1 = n₁
this✝ : HasDerivedCategory C := HasDerivedCategory.standard C
this :
  Function.Exact
    ⇑(ConcreteCategory.hom
        (ShortComplex.mk
            ((preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).homologySequenceδ hS.singleTriangle ↑n₀ ↑n₁ ⋯)
            (((preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).shift ↑n₁).map hS.singleTriangle.mor₁) ⋯).f)
    ⇑(ConcreteCategory.hom
        (ShortComplex.mk
            ((preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).homologySequenceδ hS.singleTriangle ↑n₀ ↑n₁ ⋯)
            (((preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).shift ↑n₁).map hS.singleTriangle.mor₁) ⋯).g)
⊢ (ConcreteCategory.hom
          (ShortComplex.mk
              ((preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).homologySequenceδ hS.singleTriangle ↑n₀ ↑n₁ ⋯)
              (((preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).shift ↑n₁).map hS.singleTriangle.mor₁)
              ⋯).g).comp
      ↑homAddEquiv =
    (↑homAddEquiv).comp
      (ConcreteCategory.hom
        (ShortComplex.mk (AddCommGrp.ofHom (hS.extClass.postcomp X h)) (AddCommGrp.ofHom ((mk₀ S.f).postcomp X ⋯)) ⋯).g)

Proof state:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
X : C
S : ShortComplex C
hS : S.ShortExact
n₀ n₁ : ℕ
h : n₀ + 1 = n₁
this : HasDerivedCategory C := HasDerivedCategory.standard C
⊢ (ShortComplex.mk (AddCommGrp.ofHom (hS.extClass.postcomp X h)) (AddCommGrp.ofHom ((mk₀ S.f).postcomp X ⋯)) ⋯).Exact

Tactic:
have := (preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).homologySequence_exact₁ _
  (hS.singleTriangle_distinguished) n₀ n₁ (by omega)

Result:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
X : C
S : ShortComplex C
hS : S.ShortExact
n₀ n₁ : ℕ
h : n₀ + 1 = n₁
this✝ : HasDerivedCategory C := HasDerivedCategory.standard C
this :
  (ShortComplex.mk
      ((preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).homologySequenceδ hS.singleTriangle ↑n₀ ↑n₁ ⋯)
      (((preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).shift ↑n₁).map hS.singleTriangle.mor₁) ⋯).Exact
⊢ (ShortComplex.mk (AddCommGrp.ofHom (hS.extClass.postcomp X h)) (AddCommGrp.ofHom ((mk₀ S.f).postcomp X ⋯)) ⋯).Exact

Proof state:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
X : C
S : ShortComplex C
hS : S.ShortExact
n₀ n₁ : ℕ
h : n₀ + 1 = n₁
this✝ : HasDerivedCategory C := HasDerivedCategory.standard C
this :
  (ShortComplex.mk
      ((preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).homologySequenceδ hS.singleTriangle ↑n₀ ↑n₁ ⋯)
      (((preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).shift ↑n₁).map hS.singleTriangle.mor₁) ⋯).Exact
⊢ (ShortComplex.mk (AddCommGrp.ofHom (hS.extClass.postcomp X h)) (AddCommGrp.ofHom ((mk₀ S.f).postcomp X ⋯)) ⋯).Exact

Tactic:
rw [ShortComplex.ab_exact_iff_function_exact] at this ⊢

Result:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
X : C
S : ShortComplex C
hS : S.ShortExact
n₀ n₁ : ℕ
h : n₀ + 1 = n₁
this✝ : HasDerivedCategory C := HasDerivedCategory.standard C
this :
  Function.Exact
    ⇑(ConcreteCategory.hom
        (ShortComplex.mk
            ((preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).homologySequenceδ hS.singleTriangle ↑n₀ ↑n₁ ⋯)
            (((preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).shift ↑n₁).map hS.singleTriangle.mor₁) ⋯).f)
    ⇑(ConcreteCategory.hom
        (ShortComplex.mk
            ((preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).homologySequenceδ hS.singleTriangle ↑n₀ ↑n₁ ⋯)
            (((preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).shift ↑n₁).map hS.singleTriangle.mor₁) ⋯).g)
⊢ Function.Exact
    ⇑(ConcreteCategory.hom
        (ShortComplex.mk (AddCommGrp.ofHom (hS.extClass.postcomp X h)) (AddCommGrp.ofHom ((mk₀ S.f).postcomp X ⋯)) ⋯).f)
    ⇑(ConcreteCategory.hom
        (ShortComplex.mk (AddCommGrp.ofHom (hS.extClass.postcomp X h)) (AddCommGrp.ofHom ((mk₀ S.f).postcomp X ⋯)) ⋯).g)

Proof state:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
X : C
S : ShortComplex C
hS : S.ShortExact
n₀ n₁ : ℕ
h : n₀ + 1 = n₁
this : HasDerivedCategory C := HasDerivedCategory.standard C
⊢ ↑n₀ + 1 = ↑n₁

Tactic:
omega

Result:
no goals