-- Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean
lemma δ_v (hnm : n + 1 = m) (z : Cochain F G n) (p q : ℤ) (hpq : p + m = q) (q₁ q₂ : ℤ)
    (hq₁ : q₁ = q - 1) (hq₂ : p + 1 = q₂) : (δ n m z).v p q hpq =
    z.v p q₁ (by rw [hq₁, ← hpq, ← hnm, ← add_assoc, add_sub_cancel_right]) ≫ G.d q₁ q
      + m.negOnePow • F.d p q₂ ≫ z.v q₂ q
          (by rw [← hq₂, add_assoc, add_comm 1, hnm, hpq])

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G : CochainComplex C ℤ
n m : ℤ
hnm : n + 1 = m
z : Cochain F G n
p q : ℤ
hpq : p + m = q
q₂ : ℤ
hq₂ : p + 1 = q₂
hq₁ : p + n = q - 1
⊢ q₂ = p + m - n

Tactic:
omega

Result:
no goals

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G : CochainComplex C ℤ
n m : ℤ
hnm : n + 1 = m
z : Cochain F G n
p q : ℤ
hpq : p + m = q
q₂ : ℤ
hq₂ : p + 1 = q₂
hq₁ : p + n = q - 1
⊢ (δ n m z).v p q hpq = z.v p (p + n) ⋯ ≫ G.d (p + n) q + m.negOnePow • F.d p q₂ ≫ z.v q₂ q ⋯

Tactic:
obtain rfl : q₂ = p + m - n := by omega

Result:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G : CochainComplex C ℤ
n m : ℤ
hnm : n + 1 = m
z : Cochain F G n
p q : ℤ
hpq : p + m = q
hq₁ : p + n = q - 1
hq₂ : p + 1 = p + m - n
⊢ (δ n m z).v p q hpq = z.v p (p + n) ⋯ ≫ G.d (p + n) q + m.negOnePow • F.d p (p + m - n) ≫ z.v (p + m - n) q ⋯

Proof state:
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 : ℤ
hnm : n + 1 = m
z : Cochain F G n
p q : ℤ
hpq : p + m = q
q₁ q₂ : ℤ
hq₁ : q₁ = q - 1
hq₂ : p + 1 = q₂
⊢ q₂ + n = q

Tactic:
rw [← hq₂, add_assoc, add_comm 1, hnm, hpq]

Result:
no goals

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G : CochainComplex C ℤ
n m : ℤ
hnm : n + 1 = m
z : Cochain F G n
p q : ℤ
hpq : p + m = q
hq₁ : p + n = q - 1
hq₂ : p + 1 = p + m - n
⊢ (δ n m z).v p q hpq = z.v p (p + n) ⋯ ≫ G.d (p + n) q + m.negOnePow • F.d p (p + m - n) ≫ z.v (p + m - n) q ⋯

Tactic:
rfl

Result:
no goals

Proof state:
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 : ℤ
hnm : n + 1 = m
z : Cochain F G n
p q : ℤ
hpq : p + m = q
q₁ q₂ : ℤ
hq₁ : q₁ = q - 1
hq₂ : p + 1 = q₂
⊢ p + n = q₁

Tactic:
rw [hq₁, ← hpq, ← hnm, ← add_assoc, add_sub_cancel_right]

Result:
no goals

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G : CochainComplex C ℤ
n m : ℤ
hnm : n + 1 = m
z : Cochain F G n
p q : ℤ
hpq : p + m = q
q₁ q₂ : ℤ
hq₁ : q₁ = q - 1
hq₂ : p + 1 = q₂
⊢ q₁ = p + n

Tactic:
omega

Result:
no goals

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G : CochainComplex C ℤ
n m : ℤ
hnm : n + 1 = m
z : Cochain F G n
p q : ℤ
hpq : p + m = q
q₁ q₂ : ℤ
hq₁ : q₁ = q - 1
hq₂ : p + 1 = q₂
⊢ (δ n m z).v p q hpq = z.v p q₁ ⋯ ≫ G.d q₁ q + m.negOnePow • F.d p q₂ ≫ z.v q₂ q ⋯

Tactic:
obtain rfl : q₁ = p + n := by omega

Result:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G : CochainComplex C ℤ
n m : ℤ
hnm : n + 1 = m
z : Cochain F G n
p q : ℤ
hpq : p + m = q
q₂ : ℤ
hq₂ : p + 1 = q₂
hq₁ : p + n = q - 1
⊢ (δ n m z).v p q hpq = z.v p (p + n) ⋯ ≫ G.d (p + n) q + m.negOnePow • F.d p q₂ ≫ z.v q₂ q ⋯