Theorem: CochainComplex.HomComplex.δ_δ
File: Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean

Proof steps:
state_before: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m n₀ n₁ n₂ : ℤ
z : Cochain F G n₀
⊢ δ n₁ n₂ (δ n₀ n₁ z) = 0
state_after: case pos
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m n₀ n₁ n₂ : ℤ
z : Cochain F G n₀
h₁₂ : n₁ + 1 = n₂
⊢ δ n₁ n₂ (δ n₀ n₁ z) = 0

case neg
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m n₀ n₁ n₂ : ℤ
z : Cochain F G n₀
h₁₂ : ¬n₁ + 1 = n₂
⊢ δ n₁ n₂ (δ n₀ n₁ z) = 0
tactic: by_cases h₁₂ : n₁ + 1 = n₂
state_before: case pos
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m n₀ n₁ n₂ : ℤ
z : Cochain F G n₀
h₁₂ : n₁ + 1 = n₂
⊢ δ n₁ n₂ (δ n₀ n₁ z) = 0

case neg
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m n₀ n₁ n₂ : ℤ
z : Cochain F G n₀
h₁₂ : ¬n₁ + 1 = n₂
⊢ δ n₁ n₂ (δ n₀ n₁ z) = 0
state_after: case neg
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m n₀ n₁ n₂ : ℤ
z : Cochain F G n₀
h₁₂ : ¬n₁ + 1 = n₂
⊢ δ n₁ n₂ (δ n₀ n₁ z) = 0

case pos
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m n₀ n₁ n₂ : ℤ
z : Cochain F G n₀
h₁₂ : n₁ + 1 = n₂
⊢ δ n₁ n₂ (δ n₀ n₁ z) = 0
tactic: swap
state_before: case pos
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m n₀ n₁ n₂ : ℤ
z : Cochain F G n₀
h₁₂ : n₁ + 1 = n₂
⊢ δ n₁ n₂ (δ n₀ n₁ z) = 0
state_after: case pos
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m n₀ n₁ n₂ : ℤ
z : Cochain F G n₀
h₁₂ : n₁ + 1 = n₂
h₀₁ : n₀ + 1 = n₁
⊢ δ n₁ n₂ (δ n₀ n₁ z) = 0

case neg
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m n₀ n₁ n₂ : ℤ
z : Cochain F G n₀
h₁₂ : n₁ + 1 = n₂
h₀₁ : ¬n₀ + 1 = n₁
⊢ δ n₁ n₂ (δ n₀ n₁ z) = 0
tactic: by_cases h₀₁ : n₀ + 1 = n₁
state_before: case pos
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m n₀ n₁ n₂ : ℤ
z : Cochain F G n₀
h₁₂ : n₁ + 1 = n₂
h₀₁ : n₀ + 1 = n₁
⊢ δ n₁ n₂ (δ n₀ n₁ z) = 0

case neg
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m n₀ n₁ n₂ : ℤ
z : Cochain F G n₀
h₁₂ : n₁ + 1 = n₂
h₀₁ : ¬n₀ + 1 = n₁
⊢ δ n₁ n₂ (δ n₀ n₁ z) = 0
state_after: case neg
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m n₀ n₁ n₂ : ℤ
z : Cochain F G n₀
h₁₂ : n₁ + 1 = n₂
h₀₁ : ¬n₀ + 1 = n₁
⊢ δ n₁ n₂ (δ n₀ n₁ z) = 0

case pos
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m n₀ n₁ n₂ : ℤ
z : Cochain F G n₀
h₁₂ : n₁ + 1 = n₂
h₀₁ : n₀ + 1 = n₁
⊢ δ n₁ n₂ (δ n₀ n₁ z) = 0
tactic: swap
state_before: case pos
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m n₀ n₁ n₂ : ℤ
z : Cochain F G n₀
h₁₂ : n₁ + 1 = n₂
h₀₁ : n₀ + 1 = n₁
⊢ δ n₁ n₂ (δ n₀ n₁ z) = 0
state_after: case pos.h
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m n₀ n₁ n₂ : ℤ
z : Cochain F G n₀
h₁₂ : n₁ + 1 = n₂
h₀₁ : n₀ + 1 = n₁
p q : ℤ
hpq : p + n₂ = q
⊢ (δ n₁ n₂ (δ n₀ n₁ z)).v p q hpq = Cochain.v 0 p q hpq
tactic: ext p q hpq
state_before: case pos.h
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m n₀ n₁ n₂ : ℤ
z : Cochain F G n₀
h₁₂ : n₁ + 1 = n₂
h₀₁ : n₀ + 1 = n₁
p q : ℤ
hpq : p + n₂ = q
⊢ (δ n₁ n₂ (δ n₀ n₁ z)).v p q hpq = Cochain.v 0 p q hpq
state_after: case pos.h
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m n₀ n₁ n₂ : ℤ
z : Cochain F G n₀
h₁₂ : n₁ + 1 = n₂
h₀₁ : n₀ + 1 = n₁
p q : ℤ
hpq : p + n₂ = q
⊢ (δ n₁ n₂ (δ n₀ n₁ z)).v p q hpq = 0
tactic: dsimp
state_before: case pos.h
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m n₀ n₁ n₂ : ℤ
z : Cochain F G n₀
h₁₂ : n₁ + 1 = n₂
h₀₁ : n₀ + 1 = n₁
p q : ℤ
hpq : p + n₂ = q
⊢ (δ n₁ n₂ (δ n₀ n₁ z)).v p q hpq = 0
state_after: no goals
tactic: simp only [δ_v n₁ n₂ h₁₂ _ p q hpq _ _ rfl rfl,
    δ_v n₀ n₁ h₀₁ z p (q-1) (by omega) (q-2) _ (by omega) rfl,
    δ_v n₀ n₁ h₀₁ z (p+1) q (by omega) _ (p+2) rfl (by omega),
    ← h₁₂, Int.negOnePow_succ, add_comp, assoc,
    HomologicalComplex.d_comp_d, comp_zero, zero_add, comp_add,
    HomologicalComplex.d_comp_d_assoc, zero_comp, smul_zero,
    add_zero, add_right_neg, Units.neg_smul,
    Linear.units_smul_comp, Linear.comp_units_smul]