Module: Mathlib.Algebra.Homology.Augment

Goal state before:
pp: case zero.zero
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
⊢ (ComplexShape.up ℕ).Rel 0 0 →
    (match 0 with
        | 0 => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X 0)
        | n.succ => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X (n + 1))) ≫
        C.d 0 0 =
      ((truncate.obj C).augment (C.d 0 1) ⋯).d 0 0 ≫
        match 0 with
        | 0 => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X 0)
        | n.succ => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X (n + 1))

pp: case zero.succ
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
n✝ : ℕ
⊢ (ComplexShape.up ℕ).Rel (n✝ + 1) 0 →
    (match n✝ + 1 with
        | 0 => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X 0)
        | n.succ => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X (n + 1))) ≫
        C.d (n✝ + 1) 0 =
      ((truncate.obj C).augment (C.d 0 1) ⋯).d (n✝ + 1) 0 ≫
        match 0 with
        | 0 => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X 0)
        | n.succ => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X (n + 1))

pp: case succ.zero.zero
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
⊢ (ComplexShape.up ℕ).Rel 0 (0 + 1) →
    (match 0 with
        | 0 => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X 0)
        | n.succ => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X (n + 1))) ≫
        C.d 0 (0 + 1) =
      ((truncate.obj C).augment (C.d 0 1) ⋯).d 0 (0 + 1) ≫
        match 0 + 1 with
        | 0 => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X 0)
        | n.succ => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X (n + 1))

pp: case succ.zero.succ
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
n✝ : ℕ
⊢ (ComplexShape.up ℕ).Rel (n✝ + 1) (0 + 1) →
    (match n✝ + 1 with
        | 0 => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X 0)
        | n.succ => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X (n + 1))) ≫
        C.d (n✝ + 1) (0 + 1) =
      ((truncate.obj C).augment (C.d 0 1) ⋯).d (n✝ + 1) (0 + 1) ≫
        match 0 + 1 with
        | 0 => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X 0)
        | n.succ => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X (n + 1))

pp: case succ.succ.zero
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
j : ℕ
⊢ (ComplexShape.up ℕ).Rel 0 (j + 1 + 1) →
    (match 0 with
        | 0 => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X 0)
        | n.succ => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X (n + 1))) ≫
        C.d 0 (j + 1 + 1) =
      ((truncate.obj C).augment (C.d 0 1) ⋯).d 0 (j + 1 + 1) ≫
        match j + 1 + 1 with
        | 0 => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X 0)
        | n.succ => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X (n + 1))

pp: case succ.succ.succ
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
j n✝ : ℕ
⊢ (ComplexShape.up ℕ).Rel (n✝ + 1) (j + 1 + 1) →
    (match n✝ + 1 with
        | 0 => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X 0)
        | n.succ => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X (n + 1))) ≫
        C.d (n✝ + 1) (j + 1 + 1) =
      ((truncate.obj C).augment (C.d 0 1) ⋯).d (n✝ + 1) (j + 1 + 1) ≫
        match j + 1 + 1 with
        | 0 => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X 0)
        | n.succ => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X (n + 1))

Tactic:
<;>

Goal state after:
case zero.zero
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
⊢ (ComplexShape.up ℕ).Rel 0 0 →
    (match 0 with
        | 0 => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X 0)
        | n.succ => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X (n + 1))) ≫
        C.d 0 0 =
      ((truncate.obj C).augment (C.d 0 1) ⋯).d 0 0 ≫
        match 0 with
        | 0 => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X 0)
        | n.succ => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X (n + 1))

case zero.succ
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
n✝ : ℕ
⊢ (ComplexShape.up ℕ).Rel (n✝ + 1) 0 →
    (match n✝ + 1 with
        | 0 => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X 0)
        | n.succ => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X (n + 1))) ≫
        C.d (n✝ + 1) 0 =
      ((truncate.obj C).augment (C.d 0 1) ⋯).d (n✝ + 1) 0 ≫
        match 0 with
        | 0 => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X 0)
        | n.succ => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X (n + 1))

case succ.zero.zero
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
⊢ (ComplexShape.up ℕ).Rel 0 (0 + 1) →
    (match 0 with
        | 0 => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X 0)
        | n.succ => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X (n + 1))) ≫
        C.d 0 (0 + 1) =
      ((truncate.obj C).augment (C.d 0 1) ⋯).d 0 (0 + 1) ≫
        match 0 + 1 with
        | 0 => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X 0)
        | n.succ => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X (n + 1))

case succ.zero.succ
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
n✝ : ℕ
⊢ (ComplexShape.up ℕ).Rel (n✝ + 1) (0 + 1) →
    (match n✝ + 1 with
        | 0 => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X 0)
        | n.succ => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X (n + 1))) ≫
        C.d (n✝ + 1) (0 + 1) =
      ((truncate.obj C).augment (C.d 0 1) ⋯).d (n✝ + 1) (0 + 1) ≫
        match 0 + 1 with
        | 0 => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X 0)
        | n.succ => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X (n + 1))

case succ.succ.zero
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
j : ℕ
⊢ (ComplexShape.up ℕ).Rel 0 (j + 1 + 1) →
    (match 0 with
        | 0 => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X 0)
        | n.succ => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X (n + 1))) ≫
        C.d 0 (j + 1 + 1) =
      ((truncate.obj C).augment (C.d 0 1) ⋯).d 0 (j + 1 + 1) ≫
        match j + 1 + 1 with
        | 0 => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X 0)
        | n.succ => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X (n + 1))

case succ.succ.succ
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
j n✝ : ℕ
⊢ (ComplexShape.up ℕ).Rel (n✝ + 1) (j + 1 + 1) →
    (match n✝ + 1 with
        | 0 => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X 0)
        | n.succ => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X (n + 1))) ≫
        C.d (n✝ + 1) (j + 1 + 1) =
      ((truncate.obj C).augment (C.d 0 1) ⋯).d (n✝ + 1) (j + 1 + 1) ≫
        match j + 1 + 1 with
        | 0 => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X 0)
        | n.succ => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X (n + 1))