Module: Mathlib.Algebra.Homology.Homotopy

Goal state before:
pp: case zero
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
P Q : ChainComplex V ℕ
e : P ⟶ Q
zero : P.X 0 ⟶ Q.X 1
comm_zero : e.f 0 = zero ≫ Q.d 1 0
one : P.X 1 ⟶ Q.X 2
comm_one : e.f 1 = P.d 1 0 ≫ zero + one ≫ Q.d 2 1
succ :
  (n : ℕ) →
    (p :
        (f : P.X n ⟶ Q.X (n + 1)) ×'
          (f' : P.X (n + 1) ⟶ Q.X (n + 2)) ×' e.f (n + 1) = P.d (n + 1) n ≫ f + f' ≫ Q.d (n + 2) (n + 1)) →
      (f'' : P.X (n + 2) ⟶ Q.X (n + 3)) ×' e.f (n + 2) = P.d (n + 2) (n + 1) ≫ p.snd.fst + f'' ≫ Q.d (n + 3) (n + 2)
⊢ (mkInductiveAux₂ e zero comm_zero one comm_one succ 0).snd.fst ≫ (xPrevIso Q ⋯).hom =
    (xNextIso P ⋯).inv ≫ (mkInductiveAux₂ e zero comm_zero one comm_one succ (0 + 1)).fst

pp: case succ.zero
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
P Q : ChainComplex V ℕ
e : P ⟶ Q
zero : P.X 0 ⟶ Q.X 1
comm_zero : e.f 0 = zero ≫ Q.d 1 0
one : P.X 1 ⟶ Q.X 2
comm_one : e.f 1 = P.d 1 0 ≫ zero + one ≫ Q.d 2 1
succ :
  (n : ℕ) →
    (p :
        (f : P.X n ⟶ Q.X (n + 1)) ×'
          (f' : P.X (n + 1) ⟶ Q.X (n + 2)) ×' e.f (n + 1) = P.d (n + 1) n ≫ f + f' ≫ Q.d (n + 2) (n + 1)) →
      (f'' : P.X (n + 2) ⟶ Q.X (n + 3)) ×' e.f (n + 2) = P.d (n + 2) (n + 1) ≫ p.snd.fst + f'' ≫ Q.d (n + 3) (n + 2)
⊢ (mkInductiveAux₂ e zero comm_zero one comm_one succ (0 + 1)).snd.fst ≫ (xPrevIso Q ⋯).hom =
    (xNextIso P ⋯).inv ≫ (mkInductiveAux₂ e zero comm_zero one comm_one succ (0 + 1 + 1)).fst

pp: case succ.succ
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
P Q : ChainComplex V ℕ
e : P ⟶ Q
zero : P.X 0 ⟶ Q.X 1
comm_zero : e.f 0 = zero ≫ Q.d 1 0
one : P.X 1 ⟶ Q.X 2
comm_one : e.f 1 = P.d 1 0 ≫ zero + one ≫ Q.d 2 1
succ :
  (n : ℕ) →
    (p :
        (f : P.X n ⟶ Q.X (n + 1)) ×'
          (f' : P.X (n + 1) ⟶ Q.X (n + 2)) ×' e.f (n + 1) = P.d (n + 1) n ≫ f + f' ≫ Q.d (n + 2) (n + 1)) →
      (f'' : P.X (n + 2) ⟶ Q.X (n + 3)) ×' e.f (n + 2) = P.d (n + 2) (n + 1) ≫ p.snd.fst + f'' ≫ Q.d (n + 3) (n + 2)
i : ℕ
⊢ (mkInductiveAux₂ e zero comm_zero one comm_one succ (i + 1 + 1)).snd.fst ≫ (xPrevIso Q ⋯).hom =
    (xNextIso P ⋯).inv ≫ (mkInductiveAux₂ e zero comm_zero one comm_one succ (i + 1 + 1 + 1)).fst

Tactic:
<;>

Goal state after:
case zero
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
P Q : ChainComplex V ℕ
e : P ⟶ Q
zero : P.X 0 ⟶ Q.X 1
comm_zero : e.f 0 = zero ≫ Q.d 1 0
one : P.X 1 ⟶ Q.X 2
comm_one : e.f 1 = P.d 1 0 ≫ zero + one ≫ Q.d 2 1
succ :
  (n : ℕ) →
    (p :
        (f : P.X n ⟶ Q.X (n + 1)) ×'
          (f' : P.X (n + 1) ⟶ Q.X (n + 2)) ×' e.f (n + 1) = P.d (n + 1) n ≫ f + f' ≫ Q.d (n + 2) (n + 1)) →
      (f'' : P.X (n + 2) ⟶ Q.X (n + 3)) ×' e.f (n + 2) = P.d (n + 2) (n + 1) ≫ p.snd.fst + f'' ≫ Q.d (n + 3) (n + 2)
⊢ (mkInductiveAux₂ e zero comm_zero one comm_one succ 0).snd.fst ≫ (xPrevIso Q ⋯).hom =
    (xNextIso P ⋯).inv ≫ (mkInductiveAux₂ e zero comm_zero one comm_one succ (0 + 1)).fst

case succ.zero
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
P Q : ChainComplex V ℕ
e : P ⟶ Q
zero : P.X 0 ⟶ Q.X 1
comm_zero : e.f 0 = zero ≫ Q.d 1 0
one : P.X 1 ⟶ Q.X 2
comm_one : e.f 1 = P.d 1 0 ≫ zero + one ≫ Q.d 2 1
succ :
  (n : ℕ) →
    (p :
        (f : P.X n ⟶ Q.X (n + 1)) ×'
          (f' : P.X (n + 1) ⟶ Q.X (n + 2)) ×' e.f (n + 1) = P.d (n + 1) n ≫ f + f' ≫ Q.d (n + 2) (n + 1)) →
      (f'' : P.X (n + 2) ⟶ Q.X (n + 3)) ×' e.f (n + 2) = P.d (n + 2) (n + 1) ≫ p.snd.fst + f'' ≫ Q.d (n + 3) (n + 2)
⊢ (mkInductiveAux₂ e zero comm_zero one comm_one succ (0 + 1)).snd.fst ≫ (xPrevIso Q ⋯).hom =
    (xNextIso P ⋯).inv ≫ (mkInductiveAux₂ e zero comm_zero one comm_one succ (0 + 1 + 1)).fst

case succ.succ
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
P Q : ChainComplex V ℕ
e : P ⟶ Q
zero : P.X 0 ⟶ Q.X 1
comm_zero : e.f 0 = zero ≫ Q.d 1 0
one : P.X 1 ⟶ Q.X 2
comm_one : e.f 1 = P.d 1 0 ≫ zero + one ≫ Q.d 2 1
succ :
  (n : ℕ) →
    (p :
        (f : P.X n ⟶ Q.X (n + 1)) ×'
          (f' : P.X (n + 1) ⟶ Q.X (n + 2)) ×' e.f (n + 1) = P.d (n + 1) n ≫ f + f' ≫ Q.d (n + 2) (n + 1)) →
      (f'' : P.X (n + 2) ⟶ Q.X (n + 3)) ×' e.f (n + 2) = P.d (n + 2) (n + 1) ≫ p.snd.fst + f'' ≫ Q.d (n + 3) (n + 2)
i : ℕ
⊢ (mkInductiveAux₂ e zero comm_zero one comm_one succ (i + 1 + 1)).snd.fst ≫ (xPrevIso Q ⋯).hom =
    (xNextIso P ⋯).inv ≫ (mkInductiveAux₂ e zero comm_zero one comm_one succ (i + 1 + 1 + 1)).fst