mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 3
φ : R₁ ⟶ R₂
hR₁ :
  (mk₂ (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)
      (R₁.map' 2 3 mono_of_epi_of_mono_of_mono'._proof_4 mono_of_epi_of_mono_of_mono'._proof_7)).Exact
hR₂ :
  (mk₂ (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_11)
      (R₂.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)).Exact
hR₂' : R₂.map' 1 3 mono_of_epi_of_mono_of_mono'._proof_11 mono_of_epi_of_mono_of_mono'._proof_7 = 0
h₀ : Epi (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_13)
h₂✝ : Epi (app' φ 2 mono_of_epi_of_mono_of_mono'._proof_4)
h₃✝ : Mono (app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7)
A : C
g₁ : A ⟶ R₂.obj' 1 mono_of_epi_of_mono_of_mono'._proof_11
A₁ : C
π₁ : A₁ ⟶ A
w✝¹ : Epi π₁
f₂ : A₁ ⟶ R₁.obj' 2 mono_of_epi_of_mono_of_mono'._proof_4
h₁ : π₁ ≫ g₁ ≫ R₂.map' 1 2 ⋯ ⋯ = f₂ ≫ app' φ 2 mono_of_epi_of_mono_of_mono'._proof_4
h₂ : f₂ ≫ R₁.map' 2 3 ⋯ ⋯ = 0
A₂ : C
π₂ : A₂ ⟶ A₁
w✝ : Epi π₂
f₁ : A₂ ⟶ R₁.obj 1
h₃ : π₂ ≫ f₂ = f₁ ≫ R₁.map (homOfLE ⋯)
⊢ (π₂ ≫ π₁ ≫ g₁) ≫ R₂.map' 1 2 ⋯ ⋯ - (f₁ ≫ app' φ 1 mono_of_epi_of_mono_of_mono'._proof_11) ≫ R₂.map' 1 2 ⋯ ⋯ = 0

Tactic:
assoc,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 3
φ : R₁ ⟶ R₂
hR₁ :
  (mk₂ (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)
      (R₁.map' 2 3 mono_of_epi_of_mono_of_mono'._proof_4 mono_of_epi_of_mono_of_mono'._proof_7)).Exact
hR₂ :
  (mk₂ (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_11)
      (R₂.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)).Exact
hR₂' : R₂.map' 1 3 mono_of_epi_of_mono_of_mono'._proof_11 mono_of_epi_of_mono_of_mono'._proof_7 = 0
h₀ : Epi (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_13)
h₂✝ : Epi (app' φ 2 mono_of_epi_of_mono_of_mono'._proof_4)
h₃✝ : Mono (app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7)
A : C
g₁ : A ⟶ R₂.obj' 1 mono_of_epi_of_mono_of_mono'._proof_11
A₁ : C
π₁ : A₁ ⟶ A
w✝¹ : Epi π₁
f₂ : A₁ ⟶ R₁.obj' 2 mono_of_epi_of_mono_of_mono'._proof_4
h₁ : π₁ ≫ g₁ ≫ R₂.map' 1 2 ⋯ ⋯ = f₂ ≫ app' φ 2 mono_of_epi_of_mono_of_mono'._proof_4
h₂ : f₂ ≫ R₁.map' 2 3 ⋯ ⋯ = 0
A₂ : C
π₂ : A₂ ⟶ A₁
w✝ : Epi π₂
f₁ : A₂ ⟶ R₁.obj 1
h₃ : π₂ ≫ f₂ = f₁ ≫ R₁.map (homOfLE ⋯)
⊢ π₂ ≫ (π₁ ≫ g₁) ≫ R₂.map' 1 2 ⋯ ⋯ - (f₁ ≫ app' φ 1 mono_of_epi_of_mono_of_mono'._proof_11) ≫ R₂.map' 1 2 ⋯ ⋯ = 0
Module: Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 3
φ : R₁ ⟶ R₂
hR₁ :
  (mk₂ (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)
      (R₁.map' 2 3 mono_of_epi_of_mono_of_mono'._proof_4 mono_of_epi_of_mono_of_mono'._proof_7)).Exact
hR₂ :
  (mk₂ (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_11)
      (R₂.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)).Exact
hR₂' : R₂.map' 1 3 mono_of_epi_of_mono_of_mono'._proof_11 mono_of_epi_of_mono_of_mono'._proof_7 = 0
h₀ : Epi (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_13)
h₂✝ : Epi (app' φ 2 mono_of_epi_of_mono_of_mono'._proof_4)
h₃✝ : Mono (app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7)
A : C
g₁ : A ⟶ R₂.obj' 1 mono_of_epi_of_mono_of_mono'._proof_11
A₁ : C
π₁ : A₁ ⟶ A
w✝¹ : Epi π₁
f₂ : A₁ ⟶ R₁.obj' 2 mono_of_epi_of_mono_of_mono'._proof_4
h₁ : π₁ ≫ g₁ ≫ R₂.map' 1 2 ⋯ ⋯ = f₂ ≫ app' φ 2 mono_of_epi_of_mono_of_mono'._proof_4
h₂ : f₂ ≫ R₁.map' 2 3 ⋯ ⋯ = 0
A₂ : C
π₂ : A₂ ⟶ A₁
w✝ : Epi π₂
f₁ : A₂ ⟶ R₁.obj 1
h₃ : π₂ ≫ f₂ = f₁ ≫ R₁.map (homOfLE ⋯)
⊢ π₂ ≫ (π₁ ≫ g₁) ≫ R₂.map' 1 2 ⋯ ⋯ - (f₁ ≫ app' φ 1 mono_of_epi_of_mono_of_mono'._proof_11) ≫ R₂.map' 1 2 ⋯ ⋯ = 0

Tactic:
assoc,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 3
φ : R₁ ⟶ R₂
hR₁ :
  (mk₂ (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)
      (R₁.map' 2 3 mono_of_epi_of_mono_of_mono'._proof_4 mono_of_epi_of_mono_of_mono'._proof_7)).Exact
hR₂ :
  (mk₂ (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_11)
      (R₂.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)).Exact
hR₂' : R₂.map' 1 3 mono_of_epi_of_mono_of_mono'._proof_11 mono_of_epi_of_mono_of_mono'._proof_7 = 0
h₀ : Epi (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_13)
h₂✝ : Epi (app' φ 2 mono_of_epi_of_mono_of_mono'._proof_4)
h₃✝ : Mono (app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7)
A : C
g₁ : A ⟶ R₂.obj' 1 mono_of_epi_of_mono_of_mono'._proof_11
A₁ : C
π₁ : A₁ ⟶ A
w✝¹ : Epi π₁
f₂ : A₁ ⟶ R₁.obj' 2 mono_of_epi_of_mono_of_mono'._proof_4
h₁ : π₁ ≫ g₁ ≫ R₂.map' 1 2 ⋯ ⋯ = f₂ ≫ app' φ 2 mono_of_epi_of_mono_of_mono'._proof_4
h₂ : f₂ ≫ R₁.map' 2 3 ⋯ ⋯ = 0
A₂ : C
π₂ : A₂ ⟶ A₁
w✝ : Epi π₂
f₁ : A₂ ⟶ R₁.obj 1
h₃ : π₂ ≫ f₂ = f₁ ≫ R₁.map (homOfLE ⋯)
⊢ π₂ ≫ π₁ ≫ g₁ ≫ R₂.map' 1 2 ⋯ ⋯ - (f₁ ≫ app' φ 1 mono_of_epi_of_mono_of_mono'._proof_11) ≫ R₂.map' 1 2 ⋯ ⋯ = 0
Module: Mathlib.Algebra.Homology.DerivedCategory.SingleTriangle

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
S : ShortComplex C
hS : S.ShortExact
S₁ S₂ : ShortComplex C
h₁ : S₁.ShortExact
h₂ : S₂.ShortExact
f : S₁ ⟶ S₂
⊢ (((singleFunctorsPostcompQIso C).hom.hom 0).app S₁.X₃ ≫
        triangleOfSESδ ⋯ ≫
          (shiftFunctor (DerivedCategory C) 1).map (((singleFunctorsPostcompQIso C).inv.hom 0).app S₁.X₁)) ≫
      (shiftFunctor (DerivedCategory C) 1).map ((singleFunctor C 0).map f.τ₁) =
    (singleFunctor C 0).map f.τ₃ ≫
      ((singleFunctorsPostcompQIso C).hom.hom 0).app S₂.X₃ ≫
        triangleOfSESδ ⋯ ≫
          (shiftFunctor (DerivedCategory C) 1).map (((singleFunctorsPostcompQIso C).inv.hom 0).app S₂.X₁)

Tactic:
assoc,

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
S : ShortComplex C
hS : S.ShortExact
S₁ S₂ : ShortComplex C
h₁ : S₁.ShortExact
h₂ : S₂.ShortExact
f : S₁ ⟶ S₂
⊢ ((singleFunctorsPostcompQIso C).hom.hom 0).app S₁.X₃ ≫
      (triangleOfSESδ ⋯ ≫
          (shiftFunctor (DerivedCategory C) 1).map (((singleFunctorsPostcompQIso C).inv.hom 0).app S₁.X₁)) ≫
        (shiftFunctor (DerivedCategory C) 1).map ((singleFunctor C 0).map f.τ₁) =
    (singleFunctor C 0).map f.τ₃ ≫
      ((singleFunctorsPostcompQIso C).hom.hom 0).app S₂.X₃ ≫
        triangleOfSESδ ⋯ ≫
          (shiftFunctor (DerivedCategory C) 1).map (((singleFunctorsPostcompQIso C).inv.hom 0).app S₂.X₁)
Module: Mathlib.Algebra.Homology.DerivedCategory.SingleTriangle

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
S : ShortComplex C
hS : S.ShortExact
S₁ S₂ : ShortComplex C
h₁ : S₁.ShortExact
h₂ : S₂.ShortExact
f : S₁ ⟶ S₂
⊢ ((singleFunctorsPostcompQIso C).hom.hom 0).app S₁.X₃ ≫
      (triangleOfSESδ ⋯ ≫
          (shiftFunctor (DerivedCategory C) 1).map (((singleFunctorsPostcompQIso C).inv.hom 0).app S₁.X₁)) ≫
        (shiftFunctor (DerivedCategory C) 1).map ((singleFunctor C 0).map f.τ₁) =
    (singleFunctor C 0).map f.τ₃ ≫
      ((singleFunctorsPostcompQIso C).hom.hom 0).app S₂.X₃ ≫
        triangleOfSESδ ⋯ ≫
          (shiftFunctor (DerivedCategory C) 1).map (((singleFunctorsPostcompQIso C).inv.hom 0).app S₂.X₁)

Tactic:
assoc,

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
S : ShortComplex C
hS : S.ShortExact
S₁ S₂ : ShortComplex C
h₁ : S₁.ShortExact
h₂ : S₂.ShortExact
f : S₁ ⟶ S₂
⊢ ((singleFunctorsPostcompQIso C).hom.hom 0).app S₁.X₃ ≫
      triangleOfSESδ ⋯ ≫
        (shiftFunctor (DerivedCategory C) 1).map (((singleFunctorsPostcompQIso C).inv.hom 0).app S₁.X₁) ≫
          (shiftFunctor (DerivedCategory C) 1).map ((singleFunctor C 0).map f.τ₁) =
    (singleFunctor C 0).map f.τ₃ ≫
      ((singleFunctorsPostcompQIso C).hom.hom 0).app S₂.X₃ ≫
        triangleOfSESδ ⋯ ≫
          (shiftFunctor (DerivedCategory C) 1).map (((singleFunctorsPostcompQIso C).inv.hom 0).app S₂.X₁)
Module: Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 3
φ : R₁ ⟶ R₂
hR₁ :
  (mk₂ (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)
      (R₁.map' 2 3 mono_of_epi_of_mono_of_mono'._proof_4 mono_of_epi_of_mono_of_mono'._proof_7)).Exact
hR₂ :
  (mk₂ (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_11)
      (R₂.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)).Exact
hR₂' : R₂.map' 1 3 mono_of_epi_of_mono_of_mono'._proof_11 mono_of_epi_of_mono_of_mono'._proof_7 = 0
h₀ : Epi (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_13)
h₂✝ : Epi (app' φ 2 mono_of_epi_of_mono_of_mono'._proof_4)
h₃✝ : Mono (app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7)
A : C
g₁ : A ⟶ R₂.obj' 1 mono_of_epi_of_mono_of_mono'._proof_11
A₁ : C
π₁ : A₁ ⟶ A
w✝¹ : Epi π₁
f₂ : A₁ ⟶ R₁.obj' 2 mono_of_epi_of_mono_of_mono'._proof_4
h₁ : π₁ ≫ g₁ ≫ R₂.map' 1 2 ⋯ ⋯ = f₂ ≫ app' φ 2 mono_of_epi_of_mono_of_mono'._proof_4
h₂ : f₂ ≫ R₁.map' 2 3 ⋯ ⋯ = 0
A₂ : C
π₂ : A₂ ⟶ A₁
w✝ : Epi π₂
f₁ : A₂ ⟶ R₁.obj 1
h₃ : π₂ ≫ f₂ = f₁ ≫ R₁.map (homOfLE ⋯)
⊢ π₂ ≫ π₁ ≫ g₁ ≫ R₂.map' 1 2 ⋯ ⋯ - (f₁ ≫ app' φ 1 mono_of_epi_of_mono_of_mono'._proof_11) ≫ R₂.map' 1 2 ⋯ ⋯ = 0

Tactic:
assoc,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 3
φ : R₁ ⟶ R₂
hR₁ :
  (mk₂ (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)
      (R₁.map' 2 3 mono_of_epi_of_mono_of_mono'._proof_4 mono_of_epi_of_mono_of_mono'._proof_7)).Exact
hR₂ :
  (mk₂ (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_11)
      (R₂.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)).Exact
hR₂' : R₂.map' 1 3 mono_of_epi_of_mono_of_mono'._proof_11 mono_of_epi_of_mono_of_mono'._proof_7 = 0
h₀ : Epi (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_13)
h₂✝ : Epi (app' φ 2 mono_of_epi_of_mono_of_mono'._proof_4)
h₃✝ : Mono (app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7)
A : C
g₁ : A ⟶ R₂.obj' 1 mono_of_epi_of_mono_of_mono'._proof_11
A₁ : C
π₁ : A₁ ⟶ A
w✝¹ : Epi π₁
f₂ : A₁ ⟶ R₁.obj' 2 mono_of_epi_of_mono_of_mono'._proof_4
h₁ : π₁ ≫ g₁ ≫ R₂.map' 1 2 ⋯ ⋯ = f₂ ≫ app' φ 2 mono_of_epi_of_mono_of_mono'._proof_4
h₂ : f₂ ≫ R₁.map' 2 3 ⋯ ⋯ = 0
A₂ : C
π₂ : A₂ ⟶ A₁
w✝ : Epi π₂
f₁ : A₂ ⟶ R₁.obj 1
h₃ : π₂ ≫ f₂ = f₁ ≫ R₁.map (homOfLE ⋯)
⊢ π₂ ≫ π₁ ≫ g₁ ≫ R₂.map' 1 2 ⋯ ⋯ - f₁ ≫ app' φ 1 mono_of_epi_of_mono_of_mono'._proof_11 ≫ R₂.map' 1 2 ⋯ ⋯ = 0
Module: Mathlib.Algebra.Homology.HomotopyCategory.ShortExact

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, 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 = HomologicalComplex.liftCycles (mappingCone S.f) x' n₁ ⋯ w✝ ≫ HomologicalComplex.homologyπ (mappingCone S.f) n₀
a : A' ⟶ S.X₁.X n₁
b : A' ⟶ S.X₂.X n₀
w :
  ((a ≫ (inl S.f).v n₁ n₀ ⋯) ≫ (mappingCone S.f).d n₀ n₁ + (b ≫ (inr S.f).f n₀) ≫ (mappingCone S.f).d n₀ n₁) ≫
        (↑(fst S.f)).v n₁ (n₁ + 1) ⋯ =
      0 ≫ (↑(fst S.f)).v n₁ (n₁ + 1) ⋯ ∧
    ((a ≫ (inl S.f).v n₁ n₀ ⋯) ≫ (mappingCone S.f).d n₀ n₁ + (b ≫ (inr S.f).f n₀) ≫ (mappingCone S.f).d n₀ n₁) ≫
        (snd S.f).v n₁ n₁ ⋯ =
      0 ≫ (snd S.f).v n₁ n₁ ⋯
hab : x' = a ≫ (inl S.f).v n₁ n₀ ⋯ + b ≫ (inr S.f).f n₀
⊢ HomologicalComplex.liftCycles (mappingCone S.f) x' n₁ ⋯ w✝ ≫
      HomologicalComplex.homologyπ (mappingCone S.f) 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

Tactic:
assoc,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, 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 = HomologicalComplex.liftCycles (mappingCone S.f) x' n₁ ⋯ w✝ ≫ HomologicalComplex.homologyπ (mappingCone S.f) n₀
a : A' ⟶ S.X₁.X n₁
b : A' ⟶ S.X₂.X n₀
w :
  (a ≫ (inl S.f).v n₁ n₀ ⋯ ≫ (mappingCone S.f).d n₀ n₁ + (b ≫ (inr S.f).f n₀) ≫ (mappingCone S.f).d n₀ n₁) ≫
        (↑(fst S.f)).v n₁ (n₁ + 1) ⋯ =
      0 ≫ (↑(fst S.f)).v n₁ (n₁ + 1) ⋯ ∧
    (a ≫ (inl S.f).v n₁ n₀ ⋯ ≫ (mappingCone S.f).d n₀ n₁ + (b ≫ (inr S.f).f n₀) ≫ (mappingCone S.f).d n₀ n₁) ≫
        (snd S.f).v n₁ n₁ ⋯ =
      0 ≫ (snd S.f).v n₁ n₁ ⋯
hab : x' = a ≫ (inl S.f).v n₁ n₀ ⋯ + b ≫ (inr S.f).f n₀
⊢ HomologicalComplex.liftCycles (mappingCone S.f) x' n₁ ⋯ w✝ ≫
      HomologicalComplex.homologyπ (mappingCone S.f) 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
Module: Mathlib.Algebra.Homology.HomotopyCategory.ShortExact

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, 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 = HomologicalComplex.liftCycles (mappingCone S.f) x' n₁ ⋯ w✝ ≫ HomologicalComplex.homologyπ (mappingCone S.f) n₀
a : A' ⟶ S.X₁.X n₁
b : A' ⟶ S.X₂.X n₀
w :
  (a ≫ (inl S.f).v n₁ n₀ ⋯ ≫ (mappingCone S.f).d n₀ n₁ + (b ≫ (inr S.f).f n₀) ≫ (mappingCone S.f).d n₀ n₁) ≫
        (↑(fst S.f)).v n₁ (n₁ + 1) ⋯ =
      0 ≫ (↑(fst S.f)).v n₁ (n₁ + 1) ⋯ ∧
    (a ≫ (inl S.f).v n₁ n₀ ⋯ ≫ (mappingCone S.f).d n₀ n₁ + (b ≫ (inr S.f).f n₀) ≫ (mappingCone S.f).d n₀ n₁) ≫
        (snd S.f).v n₁ n₁ ⋯ =
      0 ≫ (snd S.f).v n₁ n₁ ⋯
hab : x' = a ≫ (inl S.f).v n₁ n₀ ⋯ + b ≫ (inr S.f).f n₀
⊢ HomologicalComplex.liftCycles (mappingCone S.f) x' n₁ ⋯ w✝ ≫
      HomologicalComplex.homologyπ (mappingCone S.f) 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

Tactic:
assoc,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, 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 = HomologicalComplex.liftCycles (mappingCone S.f) x' n₁ ⋯ w✝ ≫ HomologicalComplex.homologyπ (mappingCone S.f) n₀
a : A' ⟶ S.X₁.X n₁
b : A' ⟶ S.X₂.X n₀
w :
  (a ≫ (inl S.f).v n₁ n₀ ⋯ ≫ (mappingCone S.f).d n₀ n₁ + b ≫ (inr S.f).f n₀ ≫ (mappingCone S.f).d n₀ n₁) ≫
        (↑(fst S.f)).v n₁ (n₁ + 1) ⋯ =
      0 ≫ (↑(fst S.f)).v n₁ (n₁ + 1) ⋯ ∧
    (a ≫ (inl S.f).v n₁ n₀ ⋯ ≫ (mappingCone S.f).d n₀ n₁ + b ≫ (inr S.f).f n₀ ≫ (mappingCone S.f).d n₀ n₁) ≫
        (snd S.f).v n₁ n₁ ⋯ =
      0 ≫ (snd S.f).v n₁ n₁ ⋯
hab : x' = a ≫ (inl S.f).v n₁ n₀ ⋯ + b ≫ (inr S.f).f n₀
⊢ HomologicalComplex.liftCycles (mappingCone S.f) x' n₁ ⋯ w✝ ≫
      HomologicalComplex.homologyπ (mappingCone S.f) 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
Module: Mathlib.Algebra.Homology.HomotopyCategory.ShortExact

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, 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 = HomologicalComplex.liftCycles (mappingCone S.f) x' n₁ ⋯ w✝ ≫ HomologicalComplex.homologyπ (mappingCone S.f) n₀
a : A' ⟶ S.X₁.X n₁
b : A' ⟶ S.X₂.X n₀
w :
  (a ≫ (inl S.f).v n₁ n₀ ⋯ ≫ (mappingCone S.f).d n₀ n₁) ≫ (↑(fst S.f)).v n₁ (n₁ + 1) ⋯ +
        (b ≫ S.X₂.d n₀ n₁ ≫ (inr S.f).f n₁) ≫ (↑(fst S.f)).v n₁ (n₁ + 1) ⋯ =
      0 ≫ (↑(fst S.f)).v n₁ (n₁ + 1) ⋯ ∧
    (a ≫ (inl S.f).v n₁ n₀ ⋯ ≫ (mappingCone S.f).d n₀ n₁ + b ≫ S.X₂.d n₀ n₁ ≫ (inr S.f).f n₁) ≫ (snd S.f).v n₁ n₁ ⋯ =
      0 ≫ (snd S.f).v n₁ n₁ ⋯
hab : x' = a ≫ (inl S.f).v n₁ n₀ ⋯ + b ≫ (inr S.f).f n₀
⊢ HomologicalComplex.liftCycles (mappingCone S.f) x' n₁ ⋯ w✝ ≫
      HomologicalComplex.homologyπ (mappingCone S.f) 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

Tactic:
assoc,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, 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 = HomologicalComplex.liftCycles (mappingCone S.f) x' n₁ ⋯ w✝ ≫ HomologicalComplex.homologyπ (mappingCone S.f) n₀
a : A' ⟶ S.X₁.X n₁
b : A' ⟶ S.X₂.X n₀
w :
  a ≫ ((inl S.f).v n₁ n₀ ⋯ ≫ (mappingCone S.f).d n₀ n₁) ≫ (↑(fst S.f)).v n₁ (n₁ + 1) ⋯ +
        (b ≫ S.X₂.d n₀ n₁ ≫ (inr S.f).f n₁) ≫ (↑(fst S.f)).v n₁ (n₁ + 1) ⋯ =
      0 ≫ (↑(fst S.f)).v n₁ (n₁ + 1) ⋯ ∧
    (a ≫ (inl S.f).v n₁ n₀ ⋯ ≫ (mappingCone S.f).d n₀ n₁ + b ≫ S.X₂.d n₀ n₁ ≫ (inr S.f).f n₁) ≫ (snd S.f).v n₁ n₁ ⋯ =
      0 ≫ (snd S.f).v n₁ n₁ ⋯
hab : x' = a ≫ (inl S.f).v n₁ n₀ ⋯ + b ≫ (inr S.f).f n₀
⊢ HomologicalComplex.liftCycles (mappingCone S.f) x' n₁ ⋯ w✝ ≫
      HomologicalComplex.homologyπ (mappingCone S.f) 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
Module: Mathlib.Algebra.Homology.HomotopyCategory.ShortExact

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, 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 = HomologicalComplex.liftCycles (mappingCone S.f) x' n₁ ⋯ w✝ ≫ HomologicalComplex.homologyπ (mappingCone S.f) n₀
a : A' ⟶ S.X₁.X n₁
b : A' ⟶ S.X₂.X n₀
w :
  a ≫ ((inl S.f).v n₁ n₀ ⋯ ≫ (mappingCone S.f).d n₀ n₁) ≫ (↑(fst S.f)).v n₁ (n₁ + 1) ⋯ +
        (b ≫ S.X₂.d n₀ n₁ ≫ (inr S.f).f n₁) ≫ (↑(fst S.f)).v n₁ (n₁ + 1) ⋯ =
      0 ≫ (↑(fst S.f)).v n₁ (n₁ + 1) ⋯ ∧
    (a ≫ (inl S.f).v n₁ n₀ ⋯ ≫ (mappingCone S.f).d n₀ n₁ + b ≫ S.X₂.d n₀ n₁ ≫ (inr S.f).f n₁) ≫ (snd S.f).v n₁ n₁ ⋯ =
      0 ≫ (snd S.f).v n₁ n₁ ⋯
hab : x' = a ≫ (inl S.f).v n₁ n₀ ⋯ + b ≫ (inr S.f).f n₀
⊢ HomologicalComplex.liftCycles (mappingCone S.f) x' n₁ ⋯ w✝ ≫
      HomologicalComplex.homologyπ (mappingCone S.f) 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

Tactic:
assoc,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, 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 = HomologicalComplex.liftCycles (mappingCone S.f) x' n₁ ⋯ w✝ ≫ HomologicalComplex.homologyπ (mappingCone S.f) n₀
a : A' ⟶ S.X₁.X n₁
b : A' ⟶ S.X₂.X n₀
w :
  a ≫ (inl S.f).v n₁ n₀ ⋯ ≫ (mappingCone S.f).d n₀ n₁ ≫ (↑(fst S.f)).v n₁ (n₁ + 1) ⋯ +
        (b ≫ S.X₂.d n₀ n₁ ≫ (inr S.f).f n₁) ≫ (↑(fst S.f)).v n₁ (n₁ + 1) ⋯ =
      0 ≫ (↑(fst S.f)).v n₁ (n₁ + 1) ⋯ ∧
    (a ≫ (inl S.f).v n₁ n₀ ⋯ ≫ (mappingCone S.f).d n₀ n₁ + b ≫ S.X₂.d n₀ n₁ ≫ (inr S.f).f n₁) ≫ (snd S.f).v n₁ n₁ ⋯ =
      0 ≫ (snd S.f).v n₁ n₁ ⋯
hab : x' = a ≫ (inl S.f).v n₁ n₀ ⋯ + b ≫ (inr S.f).f n₀
⊢ HomologicalComplex.liftCycles (mappingCone S.f) x' n₁ ⋯ w✝ ≫
      HomologicalComplex.homologyπ (mappingCone S.f) 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
Module: Mathlib.Algebra.Homology.HomotopyCategory.ShortExact

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, 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 = HomologicalComplex.liftCycles (mappingCone S.f) x' n₁ ⋯ w✝ ≫ HomologicalComplex.homologyπ (mappingCone S.f) n₀
a : A' ⟶ S.X₁.X n₁
b : A' ⟶ S.X₂.X n₀
w :
  a ≫ (inl S.f).v n₁ n₀ ⋯ ≫ (mappingCone S.f).d n₀ n₁ ≫ (↑(fst S.f)).v n₁ (n₁ + 1) ⋯ +
        (b ≫ S.X₂.d n₀ n₁ ≫ (inr S.f).f n₁) ≫ (↑(fst S.f)).v n₁ (n₁ + 1) ⋯ =
      0 ≫ (↑(fst S.f)).v n₁ (n₁ + 1) ⋯ ∧
    (a ≫ (inl S.f).v n₁ n₀ ⋯ ≫ (mappingCone S.f).d n₀ n₁ + b ≫ S.X₂.d n₀ n₁ ≫ (inr S.f).f n₁) ≫ (snd S.f).v n₁ n₁ ⋯ =
      0 ≫ (snd S.f).v n₁ n₁ ⋯
hab : x' = a ≫ (inl S.f).v n₁ n₀ ⋯ + b ≫ (inr S.f).f n₀
⊢ HomologicalComplex.liftCycles (mappingCone S.f) x' n₁ ⋯ w✝ ≫
      HomologicalComplex.homologyπ (mappingCone S.f) 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

Tactic:
assoc,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, 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 = HomologicalComplex.liftCycles (mappingCone S.f) x' n₁ ⋯ w✝ ≫ HomologicalComplex.homologyπ (mappingCone S.f) n₀
a : A' ⟶ S.X₁.X n₁
b : A' ⟶ S.X₂.X n₀
w :
  a ≫ (inl S.f).v n₁ n₀ ⋯ ≫ (mappingCone S.f).d n₀ n₁ ≫ (↑(fst S.f)).v n₁ (n₁ + 1) ⋯ +
        b ≫ (S.X₂.d n₀ n₁ ≫ (inr S.f).f n₁) ≫ (↑(fst S.f)).v n₁ (n₁ + 1) ⋯ =
      0 ≫ (↑(fst S.f)).v n₁ (n₁ + 1) ⋯ ∧
    (a ≫ (inl S.f).v n₁ n₀ ⋯ ≫ (mappingCone S.f).d n₀ n₁ + b ≫ S.X₂.d n₀ n₁ ≫ (inr S.f).f n₁) ≫ (snd S.f).v n₁ n₁ ⋯ =
      0 ≫ (snd S.f).v n₁ n₁ ⋯
hab : x' = a ≫ (inl S.f).v n₁ n₀ ⋯ + b ≫ (inr S.f).f n₀
⊢ HomologicalComplex.liftCycles (mappingCone S.f) x' n₁ ⋯ w✝ ≫
      HomologicalComplex.homologyπ (mappingCone S.f) 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