-- Module: Mathlib.Algebra.Homology.Homotopy
Goal:
V : Type u
inst : Category.{v, u} V
inst_1 : Preadditive.{v, u} V
P Q : ChainComplex.{v, u, 0} 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 (β := fun (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)).fst
                (β := fun (f' : P.X (n + 1) ⟶ Q.X (n + 2)) ↦
                e.f (n + 1) =
                  P.d (n + 1) n ≫
                      p.fst (β := fun (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' ≫ Q.d (n + 2) (n + 1)) +
            f'' ≫ Q.d (n + 3) (n + 2)
i : ℕ
zero₁ :
  PSigma.fst (β := fun (f' : P.X 0 ⟶ xPrev Q 0) ↦
        e.f 0 =
          dFrom P 0 ≫
              PSigma.fst (β := fun (f : xNext P 0 ⟶ Q.X 0) ↦
                (f' : P.X 0 ⟶ xPrev Q 0) ×' e.f 0 = dFrom P 0 ≫ f + f' ≫ dTo Q 0)
                (mkInductiveAux₂ e zero comm_zero one comm_one succ 0) +
            f' ≫ dTo Q 0)
        (PSigma.snd (β := fun (f : xNext P 0 ⟶ Q.X 0) ↦
          (f' : P.X 0 ⟶ xPrev Q 0) ×' e.f 0 = dFrom P 0 ≫ f + f' ≫ dTo Q 0)
          (mkInductiveAux₂ e zero comm_zero one comm_one succ 0)) ≫
      (xPrevIso Q ⋯).hom =
    (xNextIso P ⋯).inv ≫
      PSigma.fst (β := fun (f : xNext P (HAdd.hAdd (α := ℕ) 0 1) ⟶ Q.X (HAdd.hAdd (α := ℕ) 0 1)) ↦
        (f' : P.X (HAdd.hAdd (α := ℕ) 0 1) ⟶ xPrev Q (HAdd.hAdd (α := ℕ) 0 1)) ×'
          e.f (HAdd.hAdd (α := ℕ) 0 1) = dFrom P (HAdd.hAdd (α := ℕ) 0 1) ≫ f + f' ≫ dTo Q (HAdd.hAdd (α := ℕ) 0 1))
        (mkInductiveAux₂ e zero comm_zero one comm_one succ (HAdd.hAdd (α := ℕ) 0 1))
succ₁ :
  PSigma.fst (β := fun (f' : P.X (HAdd.hAdd (α := ℕ) 0 1) ⟶ xPrev Q (HAdd.hAdd (α := ℕ) 0 1)) ↦
        e.f (HAdd.hAdd (α := ℕ) 0 1) =
          dFrom P (HAdd.hAdd (α := ℕ) 0 1) ≫
              PSigma.fst (β := fun (f : xNext P (HAdd.hAdd (α := ℕ) 0 1) ⟶ Q.X (HAdd.hAdd (α := ℕ) 0 1)) ↦
                (f' : P.X (HAdd.hAdd (α := ℕ) 0 1) ⟶ xPrev Q (HAdd.hAdd (α := ℕ) 0 1)) ×'
                  e.f (HAdd.hAdd (α := ℕ) 0 1) =
                    dFrom P (HAdd.hAdd (α := ℕ) 0 1) ≫ f + f' ≫ dTo Q (HAdd.hAdd (α := ℕ) 0 1))
                (mkInductiveAux₂ e zero comm_zero one comm_one succ (HAdd.hAdd (α := ℕ) 0 1)) +
            f' ≫ dTo Q (HAdd.hAdd (α := ℕ) 0 1))
        (PSigma.snd (β := fun (f : xNext P (HAdd.hAdd (α := ℕ) 0 1) ⟶ Q.X (HAdd.hAdd (α := ℕ) 0 1)) ↦
          (f' : P.X (HAdd.hAdd (α := ℕ) 0 1) ⟶ xPrev Q (HAdd.hAdd (α := ℕ) 0 1)) ×'
            e.f (HAdd.hAdd (α := ℕ) 0 1) = dFrom P (HAdd.hAdd (α := ℕ) 0 1) ≫ f + f' ≫ dTo Q (HAdd.hAdd (α := ℕ) 0 1))
          (mkInductiveAux₂ e zero comm_zero one comm_one succ (HAdd.hAdd (α := ℕ) 0 1))) ≫
      (xPrevIso Q ⋯).hom =
    (xNextIso P ⋯).inv ≫
      PSigma.fst (β := fun (f : xNext P (HAdd.hAdd (α := ℕ) 0 1 + 1) ⟶ Q.X (HAdd.hAdd (α := ℕ) 0 1 + 1)) ↦
        (f' : P.X (HAdd.hAdd (α := ℕ) 0 1 + 1) ⟶ xPrev Q (HAdd.hAdd (α := ℕ) 0 1 + 1)) ×'
          e.f (HAdd.hAdd (α := ℕ) 0 1 + 1) =
            dFrom P (HAdd.hAdd (α := ℕ) 0 1 + 1) ≫ f + f' ≫ dTo Q (HAdd.hAdd (α := ℕ) 0 1 + 1))
        (mkInductiveAux₂ e zero comm_zero one comm_one succ (HAdd.hAdd (α := ℕ) 0 1 + 1))
succ₂ :
  ∀ (i : ℕ),
    PSigma.fst (β := fun (f' : P.X (i + 1 + 1) ⟶ xPrev Q (i + 1 + 1)) ↦
          e.f (i + 1 + 1) =
            dFrom P (i + 1 + 1) ≫
                PSigma.fst (β := fun (f : xNext P (i + 1 + 1) ⟶ Q.X (i + 1 + 1)) ↦
                  (f' : P.X (i + 1 + 1) ⟶ xPrev Q (i + 1 + 1)) ×'
                    e.f (i + 1 + 1) = dFrom P (i + 1 + 1) ≫ f + f' ≫ dTo Q (i + 1 + 1))
                  (mkInductiveAux₂ e zero comm_zero one comm_one succ (i + 1 + 1)) +
              f' ≫ dTo Q (i + 1 + 1))
          (PSigma.snd (β := fun (f : xNext P (i + 1 + 1) ⟶ Q.X (i + 1 + 1)) ↦
            (f' : P.X (i + 1 + 1) ⟶ xPrev Q (i + 1 + 1)) ×'
              e.f (i + 1 + 1) = dFrom P (i + 1 + 1) ≫ f + f' ≫ dTo Q (i + 1 + 1))
            (mkInductiveAux₂ e zero comm_zero one comm_one succ (i + 1 + 1))) ≫
        (xPrevIso Q ⋯).hom =
      (xNextIso P ⋯).inv ≫
        PSigma.fst (β := fun (f : xNext P (i + 1 + 1 + 1) ⟶ Q.X (i + 1 + 1 + 1)) ↦
          (f' : P.X (i + 1 + 1 + 1) ⟶ xPrev Q (i + 1 + 1 + 1)) ×'
            e.f (i + 1 + 1 + 1) = dFrom P (i + 1 + 1 + 1) ≫ f + f' ≫ dTo Q (i + 1 + 1 + 1))
          (mkInductiveAux₂ e zero comm_zero one comm_one succ (i + 1 + 1 + 1))
⊢ (mkInductiveAux₂ e zero comm_zero one comm_one succ i).snd (β := fun (f : xNext P i ⟶ Q.X i) ↦
            (f' : P.X i ⟶ xPrev Q i) ×' e.f i = dFrom P i ≫ f + f' ≫ dTo Q i).fst
        (β := fun (f' : P.X i ⟶ xPrev Q i) ↦
        e.f i =
          dFrom P i ≫
              (mkInductiveAux₂ e zero comm_zero one comm_one succ i).fst (β := fun (f : xNext P i ⟶ Q.X i) ↦
                (f' : P.X i ⟶ xPrev Q i) ×' e.f i = dFrom P i ≫ f + f' ≫ dTo Q i) +
            f' ≫ dTo Q i) ≫
      (xPrevIso Q ⋯).hom =
    (xNextIso P ⋯).inv ≫
      PSigma.fst (β := fun (f : xNext P (i + 1) ⟶ Q.X (i + 1)) ↦
        (f' : P.X (i + 1) ⟶ xPrev Q (i + 1)) ×' e.f (i + 1) = dFrom P (i + 1) ≫ f + f' ≫ dTo Q (i + 1))
        (mkInductiveAux₂ e zero comm_zero one comm_one succ (i + 1))

Tactic:
rcases i with (_ | _ | i)