-- 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
succ :
  (ComplexShape.up ℕ).Rel 0 (j + 1 + 1) →
    Eq.{v + 1} (α := ((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X 0 ⟶ C.X (j + 1 + 1))
      (CategoryStruct.comp (Y := 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) 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)))
          (C.d 0 (j + 1 + 1)) :
        ((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X 0 ⟶ 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 0 (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 0 ⟶ C.X (j + 1 + 1))
succ₁ :
  ∀ (n : ℕ),
    (ComplexShape.up ℕ).Rel (n + 1) (j + 1 + 1) →
      Eq.{v + 1} (α := ((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X (n + 1) ⟶ C.X (j + 1 + 1))
        (CategoryStruct.comp (Y := C.X (n + 1))
            (match (motive := (x : ℕ) → ((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X x ⟶ C.X x)
              n + 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)))
            (C.d (n + 1) (j + 1 + 1)) :
          ((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X (n + 1) ⟶ 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 (n + 1) (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 (n + 1) ⟶ C.X (j + 1 + 1))
⊢ (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))

Tactic:
cases i