-- 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
⊢ AddCommGrp.ofHom (hS.extClass.postcomp X h) ≫ AddCommGrp.ofHom ((mk₀ S.f).postcomp X ⋯) = 0

Tactic:
ext x

Result:
case hf.h
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
x : ↑(AddCommGrp.of (Ext X S.X₃ n₀))
⊢ (AddCommGrp.Hom.hom (AddCommGrp.ofHom (hS.extClass.postcomp X h) ≫ AddCommGrp.ofHom ((mk₀ S.f).postcomp X ⋯))) x =
    (AddCommGrp.Hom.hom 0) x

Proof state:
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)

Tactic:
ext x

Result:
case comm₁₂.h
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)
x : Ext X S.X₃ n₀
⊢ ((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)
      x =
    ((↑homAddEquiv).comp
        (ConcreteCategory.hom
          (ShortComplex.mk (AddCommGrp.ofHom (hS.extClass.postcomp X h)) (AddCommGrp.ofHom ((mk₀ S.f).postcomp X ⋯))
              ⋯).f))
      x

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
⊢ (ShortComplex.mk (AddCommGrp.ofHom (hS.extClass.postcomp X h)) (AddCommGrp.ofHom ((mk₀ S.f).postcomp X ⋯)) ⋯).Exact

Tactic:
letI := HasDerivedCategory.standard C

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
⊢ (ShortComplex.mk (AddCommGrp.ofHom (hS.extClass.postcomp X h)) (AddCommGrp.ofHom ((mk₀ S.f).postcomp X ⋯)) ⋯).Exact

Proof state:
case comm₂₃.h
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)
x : Ext X S.X₁ n₁
⊢ ((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)
      x =
    ((↑homAddEquiv).comp
        (ConcreteCategory.hom
          (ShortComplex.mk (AddCommGrp.ofHom (hS.extClass.postcomp X h)) (AddCommGrp.ofHom ((mk₀ S.f).postcomp X ⋯))
              ⋯).g))
      x

Tactic:
apply hom_comp_singleFunctor_map_shift (C := C)

Result:
no goals