-- Mathlib/Algebra/Homology/TotalComplexShift.lean
@[reassoc]
lemma D₂_totalShift₁XIso_hom (n₀ n₁ n₀' n₁' : ℤ) (h₀ : n₀ + x = n₀') (h₁ : n₁ + x = n₁') :
    ((shiftFunctor₁ C x).obj K).D₂ (up ℤ) n₀ n₁ ≫ (K.totalShift₁XIso x n₁ n₁' h₁).hom =
      x.negOnePow • ((K.totalShift₁XIso x n₀ n₀' h₀).hom ≫ K.D₂ (up ℤ) n₀' n₁')

Proof state:
case pos.h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
x : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + x = n₀'
h₁ : n₁ + x = n₁'
h : (up ℤ).Rel n₀ n₁
p q : ℤ
hpq : (up ℤ).π (up ℤ) (up ℤ) (p, q) = n₀
⊢ ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) p q n₀ hpq ≫
      ((shiftFunctor₁ C x).obj K).D₂ (up ℤ) n₀ n₁ ≫ (K.totalShift₁XIso x n₁ n₁' h₁).hom =
    ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) p q n₀ hpq ≫
      (x.negOnePow • (K.totalShift₁XIso x n₀ n₀' h₀).hom ≫ K.D₂ (up ℤ) n₀' n₁')

Tactic:
dsimp at h hpq

Result:
case pos.h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
x : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + x = n₀'
h₁ : n₁ + x = n₁'
h : n₀ + 1 = n₁
p q : ℤ
hpq : p + q = n₀
⊢ ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) p q n₀ hpq ≫
      ((shiftFunctor₁ C x).obj K).D₂ (up ℤ) n₀ n₁ ≫ (K.totalShift₁XIso x n₁ n₁' h₁).hom =
    ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) p q n₀ hpq ≫
      (x.negOnePow • (K.totalShift₁XIso x n₀ n₀' h₀).hom ≫ K.D₂ (up ℤ) n₀' n₁')

Proof state:
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
x : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + x = n₀'
h₁ : n₁ + x = n₁'
h : n₀ + 1 = n₁
p q : ℤ
hpq : p + q = n₀
⊢ (up ℤ).π (up ℤ) (up ℤ) (p, q + 1) = n₁

Tactic:
dsimp

Result:
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
x : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + x = n₀'
h₁ : n₁ + x = n₁'
h : n₀ + 1 = n₁
p q : ℤ
hpq : p + q = n₀
⊢ p + (q + 1) = n₁

Proof state:
case neg.h₁₂
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
x : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + x = n₀'
h₁ : n₁ + x = n₁'
h : ¬(up ℤ).Rel n₀ n₁
h' : n₀' + 1 = n₁'
⊢ n₀ + 1 = n₁

Tactic:
omega

Result:
no goals

Proof state:
case neg.h₁₂
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
x : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + x = n₀'
h₁ : n₁ + x = n₁'
h : ¬(up ℤ).Rel n₀ n₁
h' : (up ℤ).Rel n₀' n₁'
⊢ False

Tactic:
apply h

Result:
case neg.h₁₂
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
x : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + x = n₀'
h₁ : n₁ + x = n₁'
h : ¬(up ℤ).Rel n₀ n₁
h' : (up ℤ).Rel n₀' n₁'
⊢ (up ℤ).Rel n₀ n₁

Proof state:
case neg
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
x : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + x = n₀'
h₁ : n₁ + x = n₁'
h : ¬(up ℤ).Rel n₀ n₁
⊢ ((shiftFunctor₁ C x).obj K).D₂ (up ℤ) n₀ n₁ ≫ (K.totalShift₁XIso x n₁ n₁' h₁).hom =
    x.negOnePow • (K.totalShift₁XIso x n₀ n₀' h₀).hom ≫ K.D₂ (up ℤ) n₀' n₁'

Tactic:
rw [D₂_shape _ _ _ _ h, zero_comp, D₂_shape, comp_zero, smul_zero]

Result:
case neg.h₁₂
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
x : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + x = n₀'
h₁ : n₁ + x = n₁'
h : ¬(up ℤ).Rel n₀ n₁
⊢ ¬(up ℤ).Rel n₀' n₁'

Proof state:
case neg.h₁₂
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
x : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + x = n₀'
h₁ : n₁ + x = n₁'
h : ¬(up ℤ).Rel n₀ n₁
h' : (up ℤ).Rel n₀' n₁'
⊢ (up ℤ).Rel n₀ n₁

Tactic:
dsimp at h' ⊢

Result:
case neg.h₁₂
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
x : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + x = n₀'
h₁ : n₁ + x = n₁'
h : ¬(up ℤ).Rel n₀ n₁
h' : n₀' + 1 = n₁'
⊢ n₀ + 1 = n₁

Proof state:
case pos.h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
x : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + x = n₀'
h₁ : n₁ + x = n₁'
h : n₀ + 1 = n₁
p q : ℤ
hpq : p + q = n₀
⊢ (up ℤ).ε₂ (up ℤ) (up ℤ) (p, q) •
      (((shiftFunctor₁ C x).obj K).X p).d q (q + 1) ≫ K.ιTotal (up ℤ) (p + x) (q + 1) n₁' ⋯ =
    (x.negOnePow * (up ℤ).ε₂ (up ℤ) (up ℤ) (p + x, q)) •
      (K.X (p + x)).d q (q + 1) ≫ K.ιTotal (up ℤ) (p + x) (q + 1) n₁' ⋯

Tactic:
dsimp

Result:
case pos.h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
x : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + x = n₀'
h₁ : n₁ + x = n₁'
h : n₀ + 1 = n₁
p q : ℤ
hpq : p + q = n₀
⊢ p.negOnePow • (K.X (p + x)).d q (q + 1) ≫ K.ιTotal (up ℤ) (p + x) (q + 1) n₁' ⋯ =
    (x.negOnePow * (p + x).negOnePow) • (K.X (p + x)).d q (q + 1) ≫ K.ιTotal (up ℤ) (p + x) (q + 1) n₁' ⋯

Proof state:
case pos
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
x : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + x = n₀'
h₁ : n₁ + x = n₁'
h : (up ℤ).Rel n₀ n₁
⊢ ((shiftFunctor₁ C x).obj K).D₂ (up ℤ) n₀ n₁ ≫ (K.totalShift₁XIso x n₁ n₁' h₁).hom =
    x.negOnePow • (K.totalShift₁XIso x n₀ n₀' h₀).hom ≫ K.D₂ (up ℤ) n₀' n₁'

Tactic:
apply total.hom_ext

Result:
case pos.h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
x : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + x = n₀'
h₁ : n₁ + x = n₁'
h : (up ℤ).Rel n₀ n₁
⊢ ∀ (i₁ i₂ : ℤ) (hi : (up ℤ).π (up ℤ) (up ℤ) (i₁, i₂) = n₀),
    ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) i₁ i₂ n₀ hi ≫
        ((shiftFunctor₁ C x).obj K).D₂ (up ℤ) n₀ n₁ ≫ (K.totalShift₁XIso x n₁ n₁' h₁).hom =
      ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) i₁ i₂ n₀ hi ≫
        (x.negOnePow • (K.totalShift₁XIso x n₀ n₀' h₀).hom ≫ K.D₂ (up ℤ) n₀' n₁')

Proof state:
case pos.h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
x : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + x = n₀'
h₁ : n₁ + x = n₁'
h : (up ℤ).Rel n₀ n₁
⊢ ∀ (i₁ i₂ : ℤ) (hi : (up ℤ).π (up ℤ) (up ℤ) (i₁, i₂) = n₀),
    ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) i₁ i₂ n₀ hi ≫
        ((shiftFunctor₁ C x).obj K).D₂ (up ℤ) n₀ n₁ ≫ (K.totalShift₁XIso x n₁ n₁' h₁).hom =
      ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) i₁ i₂ n₀ hi ≫
        (x.negOnePow • (K.totalShift₁XIso x n₀ n₀' h₀).hom ≫ K.D₂ (up ℤ) n₀' n₁')

Tactic:
intro p q hpq

Result:
case pos.h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
x : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + x = n₀'
h₁ : n₁ + x = n₁'
h : (up ℤ).Rel n₀ n₁
p q : ℤ
hpq : (up ℤ).π (up ℤ) (up ℤ) (p, q) = n₀
⊢ ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) p q n₀ hpq ≫
      ((shiftFunctor₁ C x).obj K).D₂ (up ℤ) n₀ n₁ ≫ (K.totalShift₁XIso x n₁ n₁' h₁).hom =
    ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) p q n₀ hpq ≫
      (x.negOnePow • (K.totalShift₁XIso x n₀ n₀' h₀).hom ≫ K.D₂ (up ℤ) n₀' n₁')

Proof state:
case pos.h.e_a
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
x : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + x = n₀'
h₁ : n₁ + x = n₁'
h : n₀ + 1 = n₁
p q : ℤ
hpq : p + q = n₀
⊢ p.negOnePow = x.negOnePow * (p + x).negOnePow

Tactic:
rw [add_comm p, Int.negOnePow_add, ← mul_assoc, Int.units_mul_self, one_mul]

Result:
no goals