Theorem: HomologicalComplex₂.D₁_totalShift₁XIso_hom
File: Mathlib/Algebra/Homology/TotalComplexShift.lean

tactic: by_cases h : (up ℤ).Rel n₀ n₁
state_before: C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x y : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + x = n₀'
h₁ : n₁ + x = n₁'
⊢ ((shiftFunctor₁ C x).obj K).D₁ (up ℤ) n₀ n₁ ≫ (K.totalShift₁XIso x n₁ n₁' h₁).hom =
    x.negOnePow • (K.totalShift₁XIso x n₀ n₀' h₀).hom ≫ K.D₁ (up ℤ) n₀' n₁'
state_after: case pos
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x y : 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).D₁ (up ℤ) n₀ n₁ ≫ (K.totalShift₁XIso x n₁ n₁' h₁).hom =
    x.negOnePow • (K.totalShift₁XIso x n₀ n₀' h₀).hom ≫ K.D₁ (up ℤ) n₀' n₁'

case neg
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x y : 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).D₁ (up ℤ) n₀ n₁ ≫ (K.totalShift₁XIso x n₁ n₁' h₁).hom =
    x.negOnePow • (K.totalShift₁XIso x n₀ n₀' h₀).hom ≫ K.D₁ (up ℤ) n₀' n₁'

tactic: apply total.hom_ext
state_before: case pos
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x y : 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).D₁ (up ℤ) n₀ n₁ ≫ (K.totalShift₁XIso x n₁ n₁' h₁).hom =
    x.negOnePow • (K.totalShift₁XIso x n₀ n₀' h₀).hom ≫ K.D₁ (up ℤ) n₀' n₁'
state_after: case pos.h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x y : 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).D₁ (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.D₁ (up ℤ) n₀' n₁')

tactic: intro p q hpq
state_before: case pos.h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x y : 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).D₁ (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.D₁ (up ℤ) n₀' n₁')
state_after: case pos.h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x y : 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).D₁ (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.D₁ (up ℤ) n₀' n₁')

tactic: dsimp at h hpq
state_before: case pos.h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x y : 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).D₁ (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.D₁ (up ℤ) n₀' n₁')
state_after: case pos.h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x y : 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).D₁ (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.D₁ (up ℤ) n₀' n₁')

tactic: dsimp [totalShift₁XIso]
state_before: case pos.h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x y : 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).D₁ (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.D₁ (up ℤ) n₀' n₁')
state_after: case pos.h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x y : 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).D₁ (up ℤ) n₀ n₁ ≫
        ((shiftFunctor₁ C x).obj K).totalDesc fun p q hpq => K.ιTotal (up ℤ) (p + x) q n₁' ⋯) =
    ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) p q n₀ hpq ≫
      (x.negOnePow •
        (((shiftFunctor₁ C x).obj K).totalDesc fun p q hpq => K.ιTotal (up ℤ) (p + x) q n₀' ⋯) ≫ K.D₁ (up ℤ) n₀' n₁')

tactic: rw [ι_D₁_assoc, Linear.comp_units_smul, ι_totalDesc_assoc, ι_D₁,
  ((shiftFunctor₁ C x).obj K).d₁_eq _ rfl _ _ (by dsimp; omega),
  K.d₁_eq _ (show p + x + 1 = p + 1 + x by omega) _ _ (by dsimp; omega)]
state_before: case pos.h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x y : 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).D₁ (up ℤ) n₀ n₁ ≫
        ((shiftFunctor₁ C x).obj K).totalDesc fun p q hpq => K.ιTotal (up ℤ) (p + x) q n₁' ⋯) =
    ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) p q n₀ hpq ≫
      (x.negOnePow •
        (((shiftFunctor₁ C x).obj K).totalDesc fun p q hpq => K.ιTotal (up ℤ) (p + x) q n₀' ⋯) ≫ K.D₁ (up ℤ) n₀' n₁')
state_after: case pos.h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x y : 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).ε₁ (up ℤ) (up ℤ) (p, q) •
        (((shiftFunctor₁ C x).obj K).d p (p + 1)).f q ≫ ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) (p + 1) q n₁ ⋯) ≫
      ((shiftFunctor₁ C x).obj K).totalDesc fun p q hpq => K.ιTotal (up ℤ) (p + x) q n₁' ⋯) =
    x.negOnePow •
      (up ℤ).ε₁ (up ℤ) (up ℤ) (p + x, q) • (K.d (p + x) (p + 1 + x)).f q ≫ K.ιTotal (up ℤ) (p + 1 + x) q n₁' ⋯

tactic: dsimp
state_before: case pos.h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x y : 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).ε₁ (up ℤ) (up ℤ) (p, q) •
        (((shiftFunctor₁ C x).obj K).d p (p + 1)).f q ≫ ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) (p + 1) q n₁ ⋯) ≫
      ((shiftFunctor₁ C x).obj K).totalDesc fun p q hpq => K.ιTotal (up ℤ) (p + x) q n₁' ⋯) =
    x.negOnePow •
      (up ℤ).ε₁ (up ℤ) (up ℤ) (p + x, q) • (K.d (p + x) (p + 1 + x)).f q ≫ K.ιTotal (up ℤ) (p + 1 + x) q n₁' ⋯
state_after: case pos.h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x y : 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
⊢ ((1 • (x.negOnePow • (K.d (p + x) (p + 1 + x)).f q) ≫ ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) (p + 1) q n₁ ⋯) ≫
      ((shiftFunctor₁ C x).obj K).totalDesc fun p q hpq => K.ιTotal (up ℤ) (p + x) q n₁' ⋯) =
    x.negOnePow • 1 • (K.d (p + x) (p + 1 + x)).f q ≫ K.ιTotal (up ℤ) (p + 1 + x) q n₁' ⋯

tactic: rw [one_smul, assoc, ι_totalDesc, one_smul, Linear.units_smul_comp]
state_before: case pos.h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x y : 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
⊢ ((1 • (x.negOnePow • (K.d (p + x) (p + 1 + x)).f q) ≫ ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) (p + 1) q n₁ ⋯) ≫
      ((shiftFunctor₁ C x).obj K).totalDesc fun p q hpq => K.ιTotal (up ℤ) (p + x) q n₁' ⋯) =
    x.negOnePow • 1 • (K.d (p + x) (p + 1 + x)).f q ≫ K.ιTotal (up ℤ) (p + 1 + x) q n₁' ⋯
state_after: no goals

tactic: dsimp
state_before: C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x y : 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 + 1, q) = n₁
state_after: C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x y : 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 + 1 + q = n₁

tactic: omega
state_before: C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x y : 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 + 1 + q = n₁
state_after: no goals

tactic: omega
state_before: C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x y : 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 + x + 1 = p + 1 + x
state_after: no goals

tactic: dsimp
state_before: C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x y : 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 + 1 + x, q) = n₁'
state_after: C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x y : 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 + 1 + x + q = n₁'

tactic: omega
state_before: C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x y : 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 + 1 + x + q = n₁'
state_after: no goals

tactic: rw [D₁_shape _ _ _ _ h, zero_comp, D₁_shape, comp_zero, smul_zero]
state_before: case neg
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x y : 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).D₁ (up ℤ) n₀ n₁ ≫ (K.totalShift₁XIso x n₁ n₁' h₁).hom =
    x.negOnePow • (K.totalShift₁XIso x n₀ n₀' h₀).hom ≫ K.D₁ (up ℤ) n₀' n₁'
state_after: case neg.h₁₂
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x y : 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'

tactic: intro h'
state_before: case neg.h₁₂
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x y : 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'
state_after: case neg.h₁₂
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x y : 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
state_before: case neg.h₁₂
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x y : 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
state_after: case neg.h₁₂
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x y : 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' ⊢
state_before: case neg.h₁₂
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x y : 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
state_after: case neg.h₁₂
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x y : 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
state_before: case neg.h₁₂
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x y : 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
state_after: no goals