-- Mathlib/Algebra/Homology/TotalComplexShift.lean
@[reassoc]
lemma D₁_totalShift₂XIso_hom (n₀ n₁ n₀' n₁' : ℤ) (h₀ : n₀ + y = n₀') (h₁ : n₁ + y = n₁') :
    ((shiftFunctor₂ C y).obj K).D₁ (up ℤ) n₀ n₁ ≫ (K.totalShift₂XIso y n₁ n₁' h₁).hom =
      y.negOnePow • ((K.totalShift₂XIso y n₀ n₀' h₀).hom ≫ K.D₁ (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)
y : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + y = n₀'
h₁ : n₁ + y = n₁'
h : n₀ + 1 = n₁
p q : ℤ
hpq : p + q = n₀
⊢ ((p + 1) * y).negOnePow • (K.d p (p + 1)).f (q + y) ≫ K.ιTotal (up ℤ) (p + 1) (q + y) n₁' ⋯ =
    (y + p * y).negOnePow • (K.d p (p + 1)).f (q + y) ≫ K.ιTotal (up ℤ) (p + 1) (q + y) n₁' ⋯

Tactic:
congr 2

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

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)
y : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + y = n₀'
h₁ : n₁ + y = 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)
y : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + y = n₀'
h₁ : n₁ + y = n₁'
h : ¬(up ℤ).Rel n₀ n₁
h' : n₀' + 1 = n₁'
⊢ n₀ + 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)
y : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + y = n₀'
h₁ : n₁ + y = 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)
y : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + y = n₀'
h₁ : n₁ + y = n₁'
h : ¬(up ℤ).Rel n₀ n₁
h' : (up ℤ).Rel n₀' n₁'
⊢ (up ℤ).Rel 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)
y : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + y = n₀'
h₁ : n₁ + y = n₁'
h : n₀ + 1 = n₁
p q : ℤ
hpq : p + q = n₀
⊢ ((1 • (K.d p (p + 1)).f (q + y) ≫ ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) (p + 1) q n₁ ⋯) ≫
      ((shiftFunctor₂ C y).obj K).totalDesc fun p q hpq => (p * y).negOnePow • K.ιTotal (up ℤ) p (q + y) n₁' ⋯) =
    (y.negOnePow * (p * y).negOnePow) • 1 • (K.d p (p + 1)).f (q + y) ≫ K.ιTotal (up ℤ) (p + 1) (q + y) n₁' ⋯

Tactic:
rw [one_smul, one_smul, Category.assoc, ι_totalDesc, Linear.comp_units_smul,
  ← Int.negOnePow_add]

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)
y : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + y = n₀'
h₁ : n₁ + y = n₁'
h : n₀ + 1 = n₁
p q : ℤ
hpq : p + q = n₀
⊢ ((p + 1) * y).negOnePow • (K.d p (p + 1)).f (q + y) ≫ K.ιTotal (up ℤ) (p + 1) (q + y) n₁' ⋯ =
    (y + p * y).negOnePow • (K.d p (p + 1)).f (q + y) ≫ K.ιTotal (up ℤ) (p + 1) (q + y) 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)
y : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + y = n₀'
h₁ : n₁ + y = n₁'
h : n₀ + 1 = n₁
p q : ℤ
hpq : p + q = n₀
⊢ ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) p q n₀ hpq ≫
      ((shiftFunctor₂ C y).obj K).D₁ (up ℤ) n₀ n₁ ≫ (K.totalShift₂XIso y n₁ n₁' h₁).hom =
    ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) p q n₀ hpq ≫
      (y.negOnePow • (K.totalShift₂XIso y n₀ n₀' h₀).hom ≫ K.D₁ (up ℤ) n₀' n₁')

Tactic:
dsimp [totalShift₂XIso]

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)
y : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + y = n₀'
h₁ : n₁ + y = n₁'
h : n₀ + 1 = n₁
p q : ℤ
hpq : p + q = n₀
⊢ (((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) p q n₀ hpq ≫
      ((shiftFunctor₂ C y).obj K).D₁ (up ℤ) n₀ n₁ ≫
        ((shiftFunctor₂ C y).obj K).totalDesc fun p q hpq => (p * y).negOnePow • K.ιTotal (up ℤ) p (q + y) n₁' ⋯) =
    ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) p q n₀ hpq ≫
      (y.negOnePow •
        (((shiftFunctor₂ C y).obj K).totalDesc fun p q hpq => (p * y).negOnePow • K.ιTotal (up ℤ) p (q + y) n₀' ⋯) ≫
          K.D₁ (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)
y : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + y = n₀'
h₁ : n₁ + y = n₁'
h : (up ℤ).Rel n₀ n₁
⊢ ∀ (i₁ i₂ : ℤ) (hi : (up ℤ).π (up ℤ) (up ℤ) (i₁, i₂) = n₀),
    ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) i₁ i₂ n₀ hi ≫
        ((shiftFunctor₂ C y).obj K).D₁ (up ℤ) n₀ n₁ ≫ (K.totalShift₂XIso y n₁ n₁' h₁).hom =
      ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) i₁ i₂ n₀ hi ≫
        (y.negOnePow • (K.totalShift₂XIso y n₀ n₀' h₀).hom ≫ K.D₁ (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)
y : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + y = n₀'
h₁ : n₁ + y = n₁'
h : (up ℤ).Rel n₀ n₁
p q : ℤ
hpq : (up ℤ).π (up ℤ) (up ℤ) (p, q) = n₀
⊢ ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) p q n₀ hpq ≫
      ((shiftFunctor₂ C y).obj K).D₁ (up ℤ) n₀ n₁ ≫ (K.totalShift₂XIso y n₁ n₁' h₁).hom =
    ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) p q n₀ hpq ≫
      (y.negOnePow • (K.totalShift₂XIso y n₀ n₀' h₀).hom ≫ K.D₁ (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)
y : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + y = n₀'
h₁ : n₁ + y = n₁'
h : n₀ + 1 = n₁
p q : ℤ
hpq : p + q = n₀
⊢ p + 1 + q = n₁

Tactic:
omega

Result:
no goals

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)
y : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + y = n₀'
h₁ : n₁ + y = n₁'
h : n₀ + 1 = n₁
p q : ℤ
hpq : p + q = n₀
⊢ (((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) p q n₀ hpq ≫
      ((shiftFunctor₂ C y).obj K).D₁ (up ℤ) n₀ n₁ ≫
        ((shiftFunctor₂ C y).obj K).totalDesc fun p q hpq => (p * y).negOnePow • K.ιTotal (up ℤ) p (q + y) n₁' ⋯) =
    ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) p q n₀ hpq ≫
      (y.negOnePow •
        (((shiftFunctor₂ C y).obj K).totalDesc fun p q hpq => (p * y).negOnePow • K.ιTotal (up ℤ) p (q + y) n₀' ⋯) ≫
          K.D₁ (up ℤ) n₀' n₁')

Tactic:
rw [ι_D₁_assoc, Linear.comp_units_smul, ι_totalDesc_assoc, Linear.units_smul_comp,
  ι_D₁, smul_smul, ((shiftFunctor₂ C y).obj K).d₁_eq _ rfl _ _ (by dsimp; omega),
  K.d₁_eq _ rfl _ _ (by dsimp; omega)]

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)
y : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + y = n₀'
h₁ : n₁ + y = n₁'
h : n₀ + 1 = n₁
p q : ℤ
hpq : p + q = n₀
⊢ (((up ℤ).ε₁ (up ℤ) (up ℤ) (p, q) •
        (((shiftFunctor₂ C y).obj K).d p (p + 1)).f q ≫ ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) (p + 1) q n₁ ⋯) ≫
      ((shiftFunctor₂ C y).obj K).totalDesc fun p q hpq => (p * y).negOnePow • K.ιTotal (up ℤ) p (q + y) n₁' ⋯) =
    (y.negOnePow * (p * y).negOnePow) •
      (up ℤ).ε₁ (up ℤ) (up ℤ) (p, q + y) • (K.d p (p + 1)).f (q + y) ≫ K.ιTotal (up ℤ) (p + 1) (q + y) n₁' ⋯