-- Module: Mathlib.Algebra.Homology.Augment
Goal:
V : Type u
inst : Category.{v, u} V
inst_1 : HasZeroMorphisms.{v, u} V
C : CochainComplex.{v, u, 0} V ℕ
i j : ℕ
zero :
  (ComplexShape.up ℕ).Rel i 0 →
    Eq.{v + 1} (α := ((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X i ⟶ C.X 0)
      ((match (motive := (x : ℕ) → ((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X x ⟶ C.X x) i with
          | 0 => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X 0)
          | n.succ => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X (n + 1))) ≫
          C.d i 0 :
        ((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X i ⟶ C.X 0)
      (CategoryStruct.comp (Y := ((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X 0)
          (((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).d i 0)
          (match (motive := (x : ℕ) → ((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X x ⟶ C.X x) 0 with
          | 0 => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X 0)
          | n.succ => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X (n + 1))) :
        ((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X i ⟶ C.X 0)
succ :
  (ComplexShape.up ℕ).Rel i (HAdd.hAdd (α := ℕ) 0 1) →
    Eq.{v + 1} (α := ((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X i ⟶ C.X (HAdd.hAdd (α := ℕ) 0 1))
      ((match (motive := (x : ℕ) → ((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X x ⟶ C.X x) i with
          | 0 => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X 0)
          | n.succ => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X (n + 1))) ≫
          C.d i (HAdd.hAdd (α := ℕ) 0 1) :
        ((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X i ⟶ C.X (HAdd.hAdd (α := ℕ) 0 1))
      (CategoryStruct.comp (Y := ((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X (HAdd.hAdd (α := ℕ) 0 1))
          (((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).d i (HAdd.hAdd (α := ℕ) 0 1))
          (match (motive := (x : ℕ) → ((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X x ⟶ C.X x)
            HAdd.hAdd (α := ℕ) 0 1 with
          | 0 => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X 0)
          | n.succ => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X (n + 1))) :
        ((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X i ⟶ C.X (HAdd.hAdd (α := ℕ) 0 1))
succ₁ :
  ∀ (j : ℕ),
    (ComplexShape.up ℕ).Rel i (j + 1 + 1) →
      Eq.{v + 1} (α := ((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X i ⟶ C.X (j + 1 + 1))
        ((match (motive := (x : ℕ) → ((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X x ⟶ C.X x) i with
            | 0 => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X 0)
            | n.succ => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X (n + 1))) ≫
            C.d i (j + 1 + 1) :
          ((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X i ⟶ C.X (j + 1 + 1))
        (CategoryStruct.comp (Y := ((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X (j + 1 + 1))
            (((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).d i (j + 1 + 1))
            (match (motive := (x : ℕ) → ((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X x ⟶ C.X x)
              j + 1 + 1 with
            | 0 => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X 0)
            | n.succ => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X (n + 1))) :
          ((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X i ⟶ C.X (j + 1 + 1))
⊢ (ComplexShape.up ℕ).Rel i j →
    Eq.{v + 1} (α := ((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X i ⟶ C.X j)
      ((match (motive := (x : ℕ) → ((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X x ⟶ C.X x) i with
          | 0 => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X 0)
          | n.succ => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X (n + 1))) ≫
          C.d i j :
        ((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X i ⟶ C.X j)
      (((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).d i j ≫
          match (motive := (x : ℕ) → ((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X x ⟶ C.X x) j with
          | 0 => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X 0)
          | n.succ => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X (n + 1)) :
        ((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X i ⟶ C.X j)

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