-- 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 : ℕ
zero :
  (ComplexShape.up ℕ).Rel 0 0 →
    Eq.{v + 1} (α := ((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X 0 ⟶ C.X 0)
      (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 0) :
        ((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X 0 ⟶ 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 0 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 0 ⟶ C.X 0)
zero₁ :
  ∀ (n : ℕ),
    (ComplexShape.up ℕ).Rel (n + 1) 0 →
      Eq.{v + 1} (α := ((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X (n + 1) ⟶ C.X 0)
        (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) 0) :
          ((Functor.obj truncate C).augment (X := C.X 0) (C.d 0 1) ⋯).X (n + 1) ⟶ 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 (n + 1) 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 (n + 1) ⟶ C.X 0)
⊢ (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)

Tactic:
cases i