l3lab/ntp-mathlib huggingface dataset:
/-- Augmenting a truncated complex with the original object and morphism is isomorphic
(with components the identity) to the original complex.
-/
def augmentTruncate (C : CochainComplex V ℕ) :
    augment (truncate.obj C) (C.d 0 1) (C.d_comp_d _ _ _) ≅ C where
  hom :=
    { f := fun i => by cases i <;> exact 𝟙 _
      comm' := fun i j => by
        rcases j with (_ | _ | j) <;> cases i <;>
          · dsimp
            -- Porting note: simp can't handle this now but aesop does
            

Proof state:
case succ.succ.zero
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
j : ℕ
⊢ 0 + 1 = Nat.succ (Nat.succ j) →
    𝟙 (X C 0) ≫ d C 0 (Nat.succ (Nat.succ j)) =
      d (augment (truncate.obj C) (d C 0 1) (_ : d C 0 1 ≫ d C 1 (1 + 1) = 0)) 0 (Nat.succ (Nat.succ j)) ≫
        𝟙 (X C (Nat.succ j + 1))

Next tactic:
aesop
/-- Augmenting a truncated complex with the original object and morphism is isomorphic
(with components the identity) to the original complex.
-/
def augmentTruncate (C : CochainComplex V ℕ) :
    augment (truncate.obj C) (C.d 0 1) (C.d_comp_d _ _ _) ≅ C where
  hom :=
    { f := fun i => by cases i <;> exact 𝟙 _
      comm' := fun i j => by
        rcases j with (_ | _ | j) <;> cases i <;>
          · 

Proof state:
case succ.succ.succ
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
j n✝ : ℕ
⊢ ComplexShape.Rel (ComplexShape.up ℕ) (Nat.succ n✝) (Nat.succ (Nat.succ j)) →
    (fun i =>
            Nat.casesOn (motive := fun t =>
              i = t → (X (augment (truncate.obj C) (d C 0 1) (_ : d C 0 1 ≫ d C 1 (1 + 1) = 0)) i ⟶ X C i)) i
              (fun h =>
                (_ : Nat.zero = i) ▸
                  𝟙 (X (augment (truncate.obj C) (d C 0 1) (_ : d C 0 1 ≫ d C 1 (1 + 1) = 0)) Nat.zero))
              (fun n h =>
                (_ : Nat.succ n = i) ▸
                  𝟙 (X (augment (truncate.obj C) (d C 0 1) (_ : d C 0 1 ≫ d C 1 (1 + 1) = 0)) (Nat.succ n)))
              (_ : i = i))
          (Nat.succ n✝) ≫
        d C (Nat.succ n✝) (Nat.succ (Nat.succ j)) =
      d (augment (truncate.obj C) (d C 0 1) (_ : d C 0 1 ≫ d C 1 (1 + 1) = 0)) (Nat.succ n✝) (Nat.succ (Nat.succ j)) ≫
        (fun i =>
            Nat.casesOn (motive := fun t =>
              i = t → (X (augment (truncate.obj C) (d C 0 1) (_ : d C 0 1 ≫ d C 1 (1 + 1) = 0)) i ⟶ X C i)) i
              (fun h =>
                (_ : Nat.zero = i) ▸
                  𝟙 (X (augment (truncate.obj C) (d C 0 1) (_ : d C 0 1 ≫ d C 1 (1 + 1) = 0)) Nat.zero))
              (fun n h =>
                (_ : Nat.succ n = i) ▸
                  𝟙 (X (augment (truncate.obj C) (d C 0 1) (_ : d C 0 1 ≫ d C 1 (1 + 1) = 0)) (Nat.succ n)))
              (_ : i = i))
          (Nat.succ (Nat.succ j))

Next tactic:
dsimp
/-- Augmenting a truncated complex with the original object and morphism is isomorphic
(with components the identity) to the original complex.
-/
def augmentTruncate (C : CochainComplex V ℕ) :
    augment (truncate.obj C) (C.d 0 1) (C.d_comp_d _ _ _) ≅ C where
  hom :=
    { f := fun i => by cases i <;> exact 𝟙 _
      comm' := fun i j => by
        rcases j with (_ | _ | j) <;> cases i <;>
          · dsimp
            -- Porting note: simp can't handle this now but aesop does
            

Proof state:
case succ.succ.succ
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
j n✝ : ℕ
⊢ Nat.succ n✝ + 1 = Nat.succ (Nat.succ j) →
    𝟙 (X C (n✝ + 1)) ≫ d C (Nat.succ n✝) (Nat.succ (Nat.succ j)) = d C (n✝ + 1) (j + 1 + 1) ≫ 𝟙 (X C (Nat.succ j + 1))

Next tactic:
aesop
/-- Augmenting a truncated complex with the original object and morphism is isomorphic
(with components the identity) to the original complex.
-/
def augmentTruncate (C : CochainComplex V ℕ) :
    augment (truncate.obj C) (C.d 0 1) (C.d_comp_d _ _ _) ≅ C where
  hom :=
    { f := fun i => by cases i <;> exact 𝟙 _
      comm' := fun i j => by
        rcases j with (_ | _ | j) <;> cases i <;>
          · dsimp
            -- Porting note: simp can't handle this now but aesop does
            aesop }
  inv :=
    { f := fun i => by 

Proof state:
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
i : ℕ
⊢ X C i ⟶ X (augment (truncate.obj C) (d C 0 1) (_ : d C 0 1 ≫ d C 1 (1 + 1) = 0)) i

Next tactic:
cases i
/-- Augmenting a truncated complex with the original object and morphism is isomorphic
(with components the identity) to the original complex.
-/
def augmentTruncate (C : CochainComplex V ℕ) :
    augment (truncate.obj C) (C.d 0 1) (C.d_comp_d _ _ _) ≅ C where
  hom :=
    { f := fun i => by cases i <;> exact 𝟙 _
      comm' := fun i j => by
        rcases j with (_ | _ | j) <;> cases i <;>
          · dsimp
            -- Porting note: simp can't handle this now but aesop does
            aesop }
  inv :=
    { f := fun i => by cases i <;> 

Proof state:
case zero
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
⊢ X C Nat.zero ⟶ X (augment (truncate.obj C) (d C 0 1) (_ : d C 0 1 ≫ d C 1 (1 + 1) = 0)) Nat.zero

Next tactic:
exact 𝟙 _
/-- Augmenting a truncated complex with the original object and morphism is isomorphic
(with components the identity) to the original complex.
-/
def augmentTruncate (C : CochainComplex V ℕ) :
    augment (truncate.obj C) (C.d 0 1) (C.d_comp_d _ _ _) ≅ C where
  hom :=
    { f := fun i => by cases i <;> exact 𝟙 _
      comm' := fun i j => by
        rcases j with (_ | _ | j) <;> cases i <;>
          · dsimp
            -- Porting note: simp can't handle this now but aesop does
            aesop }
  inv :=
    { f := fun i => by cases i <;> 

Proof state:
case succ
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
n✝ : ℕ
⊢ X C (Nat.succ n✝) ⟶ X (augment (truncate.obj C) (d C 0 1) (_ : d C 0 1 ≫ d C 1 (1 + 1) = 0)) (Nat.succ n✝)

Next tactic:
exact 𝟙 _
/-- Augmenting a truncated complex with the original object and morphism is isomorphic
(with components the identity) to the original complex.
-/
def augmentTruncate (C : CochainComplex V ℕ) :
    augment (truncate.obj C) (C.d 0 1) (C.d_comp_d _ _ _) ≅ C where
  hom :=
    { f := fun i => by cases i <;> exact 𝟙 _
      comm' := fun i j => by
        rcases j with (_ | _ | j) <;> cases i <;>
          · dsimp
            -- Porting note: simp can't handle this now but aesop does
            aesop }
  inv :=
    { f := fun i => by cases i <;> exact 𝟙 _
      comm' := fun i j => by
        

Proof state:
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
i j : ℕ
⊢ ComplexShape.Rel (ComplexShape.up ℕ) i j →
    (fun i =>
            Nat.casesOn (motive := fun t =>
              i = t → (X C i ⟶ X (augment (truncate.obj C) (d C 0 1) (_ : d C 0 1 ≫ d C 1 (1 + 1) = 0)) i)) i
              (fun h => (_ : Nat.zero = i) ▸ 𝟙 (X C Nat.zero)) (fun n h => (_ : Nat.succ n = i) ▸ 𝟙 (X C (Nat.succ n)))
              (_ : i = i))
          i ≫
        d (augment (truncate.obj C) (d C 0 1) (_ : d C 0 1 ≫ d C 1 (1 + 1) = 0)) i j =
      d C i j ≫
        (fun i =>
            Nat.casesOn (motive := fun t =>
              i = t → (X C i ⟶ X (augment (truncate.obj C) (d C 0 1) (_ : d C 0 1 ≫ d C 1 (1 + 1) = 0)) i)) i
              (fun h => (_ : Nat.zero = i) ▸ 𝟙 (X C Nat.zero)) (fun n h => (_ : Nat.succ n = i) ▸ 𝟙 (X C (Nat.succ n)))
              (_ : i = i))
          j

Next tactic:
rcases j with (_ | _ | j)
/-- Augmenting a truncated complex with the original object and morphism is isomorphic
(with components the identity) to the original complex.
-/
def augmentTruncate (C : CochainComplex V ℕ) :
    augment (truncate.obj C) (C.d 0 1) (C.d_comp_d _ _ _) ≅ C where
  hom :=
    { f := fun i => by cases i <;> exact 𝟙 _
      comm' := fun i j => by
        rcases j with (_ | _ | j) <;> cases i <;>
          · dsimp
            -- Porting note: simp can't handle this now but aesop does
            aesop }
  inv :=
    { f := fun i => by cases i <;> exact 𝟙 _
      comm' := fun i j => by
        rcases j with (_ | _ | j) <;> 

Proof state:
case zero
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
i : ℕ
⊢ ComplexShape.Rel (ComplexShape.up ℕ) i Nat.zero →
    (fun i =>
            Nat.casesOn (motive := fun t =>
              i = t → (X C i ⟶ X (augment (truncate.obj C) (d C 0 1) (_ : d C 0 1 ≫ d C 1 (1 + 1) = 0)) i)) i
              (fun h => (_ : Nat.zero = i) ▸ 𝟙 (X C Nat.zero)) (fun n h => (_ : Nat.succ n = i) ▸ 𝟙 (X C (Nat.succ n)))
              (_ : i = i))
          i ≫
        d (augment (truncate.obj C) (d C 0 1) (_ : d C 0 1 ≫ d C 1 (1 + 1) = 0)) i Nat.zero =
      d C i Nat.zero ≫
        (fun i =>
            Nat.casesOn (motive := fun t =>
              i = t → (X C i ⟶ X (augment (truncate.obj C) (d C 0 1) (_ : d C 0 1 ≫ d C 1 (1 + 1) = 0)) i)) i
              (fun h => (_ : Nat.zero = i) ▸ 𝟙 (X C Nat.zero)) (fun n h => (_ : Nat.succ n = i) ▸ 𝟙 (X C (Nat.succ n)))
              (_ : i = i))
          Nat.zero

Next tactic:
cases' i with i
/-- Augmenting a truncated complex with the original object and morphism is isomorphic
(with components the identity) to the original complex.
-/
def augmentTruncate (C : CochainComplex V ℕ) :
    augment (truncate.obj C) (C.d 0 1) (C.d_comp_d _ _ _) ≅ C where
  hom :=
    { f := fun i => by cases i <;> exact 𝟙 _
      comm' := fun i j => by
        rcases j with (_ | _ | j) <;> cases i <;>
          · dsimp
            -- Porting note: simp can't handle this now but aesop does
            aesop }
  inv :=
    { f := fun i => by cases i <;> exact 𝟙 _
      comm' := fun i j => by
        rcases j with (_ | _ | j) <;> 

Proof state:
case succ.zero
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
i : ℕ
⊢ ComplexShape.Rel (ComplexShape.up ℕ) i (Nat.succ Nat.zero) →
    (fun i =>
            Nat.casesOn (motive := fun t =>
              i = t → (X C i ⟶ X (augment (truncate.obj C) (d C 0 1) (_ : d C 0 1 ≫ d C 1 (1 + 1) = 0)) i)) i
              (fun h => (_ : Nat.zero = i) ▸ 𝟙 (X C Nat.zero)) (fun n h => (_ : Nat.succ n = i) ▸ 𝟙 (X C (Nat.succ n)))
              (_ : i = i))
          i ≫
        d (augment (truncate.obj C) (d C 0 1) (_ : d C 0 1 ≫ d C 1 (1 + 1) = 0)) i (Nat.succ Nat.zero) =
      d C i (Nat.succ Nat.zero) ≫
        (fun i =>
            Nat.casesOn (motive := fun t =>
              i = t → (X C i ⟶ X (augment (truncate.obj C) (d C 0 1) (_ : d C 0 1 ≫ d C 1 (1 + 1) = 0)) i)) i
              (fun h => (_ : Nat.zero = i) ▸ 𝟙 (X C Nat.zero)) (fun n h => (_ : Nat.succ n = i) ▸ 𝟙 (X C (Nat.succ n)))
              (_ : i = i))
          (Nat.succ Nat.zero)

Next tactic:
cases' i with i
/-- Augmenting a truncated complex with the original object and morphism is isomorphic
(with components the identity) to the original complex.
-/
def augmentTruncate (C : CochainComplex V ℕ) :
    augment (truncate.obj C) (C.d 0 1) (C.d_comp_d _ _ _) ≅ C where
  hom :=
    { f := fun i => by cases i <;> exact 𝟙 _
      comm' := fun i j => by
        rcases j with (_ | _ | j) <;> cases i <;>
          · dsimp
            -- Porting note: simp can't handle this now but aesop does
            aesop }
  inv :=
    { f := fun i => by cases i <;> exact 𝟙 _
      comm' := fun i j => by
        rcases j with (_ | _ | j) <;> 

Proof state:
case succ.succ
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
i j : ℕ
⊢ ComplexShape.Rel (ComplexShape.up ℕ) i (Nat.succ (Nat.succ j)) →
    (fun i =>
            Nat.casesOn (motive := fun t =>
              i = t → (X C i ⟶ X (augment (truncate.obj C) (d C 0 1) (_ : d C 0 1 ≫ d C 1 (1 + 1) = 0)) i)) i
              (fun h => (_ : Nat.zero = i) ▸ 𝟙 (X C Nat.zero)) (fun n h => (_ : Nat.succ n = i) ▸ 𝟙 (X C (Nat.succ n)))
              (_ : i = i))
          i ≫
        d (augment (truncate.obj C) (d C 0 1) (_ : d C 0 1 ≫ d C 1 (1 + 1) = 0)) i (Nat.succ (Nat.succ j)) =
      d C i (Nat.succ (Nat.succ j)) ≫
        (fun i =>
            Nat.casesOn (motive := fun t =>
              i = t → (X C i ⟶ X (augment (truncate.obj C) (d C 0 1) (_ : d C 0 1 ≫ d C 1 (1 + 1) = 0)) i)) i
              (fun h => (_ : Nat.zero = i) ▸ 𝟙 (X C Nat.zero)) (fun n h => (_ : Nat.succ n = i) ▸ 𝟙 (X C (Nat.succ n)))
              (_ : i = i))
          (Nat.succ (Nat.succ j))

Next tactic:
cases' i with i
/-- Augmenting a truncated complex with the original object and morphism is isomorphic
(with components the identity) to the original complex.
-/
def augmentTruncate (C : CochainComplex V ℕ) :
    augment (truncate.obj C) (C.d 0 1) (C.d_comp_d _ _ _) ≅ C where
  hom :=
    { f := fun i => by cases i <;> exact 𝟙 _
      comm' := fun i j => by
        rcases j with (_ | _ | j) <;> cases i <;>
          · dsimp
            -- Porting note: simp can't handle this now but aesop does
            aesop }
  inv :=
    { f := fun i => by cases i <;> exact 𝟙 _
      comm' := fun i j => by
        rcases j with (_ | _ | j) <;> cases' i with i <;>
          · 

Proof state:
case zero.zero
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
⊢ ComplexShape.Rel (ComplexShape.up ℕ) Nat.zero Nat.zero →
    (fun i =>
            Nat.casesOn (motive := fun t =>
              i = t → (X C i ⟶ X (augment (truncate.obj C) (d C 0 1) (_ : d C 0 1 ≫ d C 1 (1 + 1) = 0)) i)) i
              (fun h => (_ : Nat.zero = i) ▸ 𝟙 (X C Nat.zero)) (fun n h => (_ : Nat.succ n = i) ▸ 𝟙 (X C (Nat.succ n)))
              (_ : i = i))
          Nat.zero ≫
        d (augment (truncate.obj C) (d C 0 1) (_ : d C 0 1 ≫ d C 1 (1 + 1) = 0)) Nat.zero Nat.zero =
      d C Nat.zero Nat.zero ≫
        (fun i =>
            Nat.casesOn (motive := fun t =>
              i = t → (X C i ⟶ X (augment (truncate.obj C) (d C 0 1) (_ : d C 0 1 ≫ d C 1 (1 + 1) = 0)) i)) i
              (fun h => (_ : Nat.zero = i) ▸ 𝟙 (X C Nat.zero)) (fun n h => (_ : Nat.succ n = i) ▸ 𝟙 (X C (Nat.succ n)))
              (_ : i = i))
          Nat.zero

Next tactic:
dsimp
/-- Augmenting a truncated complex with the original object and morphism is isomorphic
(with components the identity) to the original complex.
-/
def augmentTruncate (C : CochainComplex V ℕ) :
    augment (truncate.obj C) (C.d 0 1) (C.d_comp_d _ _ _) ≅ C where
  hom :=
    { f := fun i => by cases i <;> exact 𝟙 _
      comm' := fun i j => by
        rcases j with (_ | _ | j) <;> cases i <;>
          · dsimp
            -- Porting note: simp can't handle this now but aesop does
            aesop }
  inv :=
    { f := fun i => by cases i <;> exact 𝟙 _
      comm' := fun i j => by
        rcases j with (_ | _ | j) <;> cases' i with i <;>
          · dsimp
            -- Porting note: simp can't handle this now but aesop does
            

Proof state:
case zero.zero
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
⊢ 0 + 1 = 0 →
    𝟙 (X C 0) ≫ d (augment (truncate.obj C) (d C 0 1) (_ : d C 0 1 ≫ d C 1 (1 + 1) = 0)) 0 0 = d C 0 0 ≫ 𝟙 (X C 0)

Next tactic:
aesop
/-- Augmenting a truncated complex with the original object and morphism is isomorphic
(with components the identity) to the original complex.
-/
def augmentTruncate (C : CochainComplex V ℕ) :
    augment (truncate.obj C) (C.d 0 1) (C.d_comp_d _ _ _) ≅ C where
  hom :=
    { f := fun i => by cases i <;> exact 𝟙 _
      comm' := fun i j => by
        rcases j with (_ | _ | j) <;> cases i <;>
          · dsimp
            -- Porting note: simp can't handle this now but aesop does
            aesop }
  inv :=
    { f := fun i => by cases i <;> exact 𝟙 _
      comm' := fun i j => by
        rcases j with (_ | _ | j) <;> cases' i with i <;>
          · 

Proof state:
case zero.succ
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
i : ℕ
⊢ ComplexShape.Rel (ComplexShape.up ℕ) (Nat.succ i) Nat.zero →
    (fun i =>
            Nat.casesOn (motive := fun t =>
              i = t → (X C i ⟶ X (augment (truncate.obj C) (d C 0 1) (_ : d C 0 1 ≫ d C 1 (1 + 1) = 0)) i)) i
              (fun h => (_ : Nat.zero = i) ▸ 𝟙 (X C Nat.zero)) (fun n h => (_ : Nat.succ n = i) ▸ 𝟙 (X C (Nat.succ n)))
              (_ : i = i))
          (Nat.succ i) ≫
        d (augment (truncate.obj C) (d C 0 1) (_ : d C 0 1 ≫ d C 1 (1 + 1) = 0)) (Nat.succ i) Nat.zero =
      d C (Nat.succ i) Nat.zero ≫
        (fun i =>
            Nat.casesOn (motive := fun t =>
              i = t → (X C i ⟶ X (augment (truncate.obj C) (d C 0 1) (_ : d C 0 1 ≫ d C 1 (1 + 1) = 0)) i)) i
              (fun h => (_ : Nat.zero = i) ▸ 𝟙 (X C Nat.zero)) (fun n h => (_ : Nat.succ n = i) ▸ 𝟙 (X C (Nat.succ n)))
              (_ : i = i))
          Nat.zero

Next tactic:
dsimp
/-- Augmenting a truncated complex with the original object and morphism is isomorphic
(with components the identity) to the original complex.
-/
def augmentTruncate (C : CochainComplex V ℕ) :
    augment (truncate.obj C) (C.d 0 1) (C.d_comp_d _ _ _) ≅ C where
  hom :=
    { f := fun i => by cases i <;> exact 𝟙 _
      comm' := fun i j => by
        rcases j with (_ | _ | j) <;> cases i <;>
          · dsimp
            -- Porting note: simp can't handle this now but aesop does
            aesop }
  inv :=
    { f := fun i => by cases i <;> exact 𝟙 _
      comm' := fun i j => by
        rcases j with (_ | _ | j) <;> cases' i with i <;>
          · dsimp
            -- Porting note: simp can't handle this now but aesop does
            

Proof state:
case zero.succ
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
i : ℕ
⊢ Nat.succ i + 1 = 0 →
    𝟙 (X C (Nat.succ i)) ≫ d (augment (truncate.obj C) (d C 0 1) (_ : d C 0 1 ≫ d C 1 (1 + 1) = 0)) (Nat.succ i) 0 =
      d C (Nat.succ i) 0 ≫ 𝟙 (X C 0)

Next tactic:
aesop
/-- Augmenting a truncated complex with the original object and morphism is isomorphic
(with components the identity) to the original complex.
-/
def augmentTruncate (C : CochainComplex V ℕ) :
    augment (truncate.obj C) (C.d 0 1) (C.d_comp_d _ _ _) ≅ C where
  hom :=
    { f := fun i => by cases i <;> exact 𝟙 _
      comm' := fun i j => by
        rcases j with (_ | _ | j) <;> cases i <;>
          · dsimp
            -- Porting note: simp can't handle this now but aesop does
            aesop }
  inv :=
    { f := fun i => by cases i <;> exact 𝟙 _
      comm' := fun i j => by
        rcases j with (_ | _ | j) <;> cases' i with i <;>
          · 

Proof state:
case succ.zero.zero
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
⊢ ComplexShape.Rel (ComplexShape.up ℕ) Nat.zero (Nat.succ Nat.zero) →
    (fun i =>
            Nat.casesOn (motive := fun t =>
              i = t → (X C i ⟶ X (augment (truncate.obj C) (d C 0 1) (_ : d C 0 1 ≫ d C 1 (1 + 1) = 0)) i)) i
              (fun h => (_ : Nat.zero = i) ▸ 𝟙 (X C Nat.zero)) (fun n h => (_ : Nat.succ n = i) ▸ 𝟙 (X C (Nat.succ n)))
              (_ : i = i))
          Nat.zero ≫
        d (augment (truncate.obj C) (d C 0 1) (_ : d C 0 1 ≫ d C 1 (1 + 1) = 0)) Nat.zero (Nat.succ Nat.zero) =
      d C Nat.zero (Nat.succ Nat.zero) ≫
        (fun i =>
            Nat.casesOn (motive := fun t =>
              i = t → (X C i ⟶ X (augment (truncate.obj C) (d C 0 1) (_ : d C 0 1 ≫ d C 1 (1 + 1) = 0)) i)) i
              (fun h => (_ : Nat.zero = i) ▸ 𝟙 (X C Nat.zero)) (fun n h => (_ : Nat.succ n = i) ▸ 𝟙 (X C (Nat.succ n)))
              (_ : i = i))
          (Nat.succ Nat.zero)

Next tactic:
dsimp
/-- Augmenting a truncated complex with the original object and morphism is isomorphic
(with components the identity) to the original complex.
-/
def augmentTruncate (C : CochainComplex V ℕ) :
    augment (truncate.obj C) (C.d 0 1) (C.d_comp_d _ _ _) ≅ C where
  hom :=
    { f := fun i => by cases i <;> exact 𝟙 _
      comm' := fun i j => by
        rcases j with (_ | _ | j) <;> cases i <;>
          · dsimp
            -- Porting note: simp can't handle this now but aesop does
            aesop }
  inv :=
    { f := fun i => by cases i <;> exact 𝟙 _
      comm' := fun i j => by
        rcases j with (_ | _ | j) <;> cases' i with i <;>
          · dsimp
            -- Porting note: simp can't handle this now but aesop does
            

Proof state:
case succ.zero.zero
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
⊢ 0 + 1 = Nat.succ 0 → 𝟙 (X C 0) ≫ d C 0 1 = d C 0 (Nat.succ 0) ≫ 𝟙 (X C (Nat.succ 0))

Next tactic:
aesop
/-- Augmenting a truncated complex with the original object and morphism is isomorphic
(with components the identity) to the original complex.
-/
def augmentTruncate (C : CochainComplex V ℕ) :
    augment (truncate.obj C) (C.d 0 1) (C.d_comp_d _ _ _) ≅ C where
  hom :=
    { f := fun i => by cases i <;> exact 𝟙 _
      comm' := fun i j => by
        rcases j with (_ | _ | j) <;> cases i <;>
          · dsimp
            -- Porting note: simp can't handle this now but aesop does
            aesop }
  inv :=
    { f := fun i => by cases i <;> exact 𝟙 _
      comm' := fun i j => by
        rcases j with (_ | _ | j) <;> cases' i with i <;>
          · 

Proof state:
case succ.zero.succ
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
i : ℕ
⊢ ComplexShape.Rel (ComplexShape.up ℕ) (Nat.succ i) (Nat.succ Nat.zero) →
    (fun i =>
            Nat.casesOn (motive := fun t =>
              i = t → (X C i ⟶ X (augment (truncate.obj C) (d C 0 1) (_ : d C 0 1 ≫ d C 1 (1 + 1) = 0)) i)) i
              (fun h => (_ : Nat.zero = i) ▸ 𝟙 (X C Nat.zero)) (fun n h => (_ : Nat.succ n = i) ▸ 𝟙 (X C (Nat.succ n)))
              (_ : i = i))
          (Nat.succ i) ≫
        d (augment (truncate.obj C) (d C 0 1) (_ : d C 0 1 ≫ d C 1 (1 + 1) = 0)) (Nat.succ i) (Nat.succ Nat.zero) =
      d C (Nat.succ i) (Nat.succ Nat.zero) ≫
        (fun i =>
            Nat.casesOn (motive := fun t =>
              i = t → (X C i ⟶ X (augment (truncate.obj C) (d C 0 1) (_ : d C 0 1 ≫ d C 1 (1 + 1) = 0)) i)) i
              (fun h => (_ : Nat.zero = i) ▸ 𝟙 (X C Nat.zero)) (fun n h => (_ : Nat.succ n = i) ▸ 𝟙 (X C (Nat.succ n)))
              (_ : i = i))
          (Nat.succ Nat.zero)

Next tactic:
dsimp
/-- Augmenting a truncated complex with the original object and morphism is isomorphic
(with components the identity) to the original complex.
-/
def augmentTruncate (C : CochainComplex V ℕ) :
    augment (truncate.obj C) (C.d 0 1) (C.d_comp_d _ _ _) ≅ C where
  hom :=
    { f := fun i => by cases i <;> exact 𝟙 _
      comm' := fun i j => by
        rcases j with (_ | _ | j) <;> cases i <;>
          · dsimp
            -- Porting note: simp can't handle this now but aesop does
            aesop }
  inv :=
    { f := fun i => by cases i <;> exact 𝟙 _
      comm' := fun i j => by
        rcases j with (_ | _ | j) <;> cases' i with i <;>
          · dsimp
            -- Porting note: simp can't handle this now but aesop does
            

Proof state:
case succ.zero.succ
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
i : ℕ
⊢ Nat.succ i + 1 = Nat.succ 0 →
    𝟙 (X C (Nat.succ i)) ≫ d C (i + 1) (0 + 1) = d C (Nat.succ i) (Nat.succ 0) ≫ 𝟙 (X C (Nat.succ 0))

Next tactic:
aesop