awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Algebra.Homology.Augment
Goal:
V : Type u
inst : Category.{v, u} V
inst_1 : HasZeroMorphisms.{v, u} V
C : ChainComplex.{v, u, 0} V ℕ
i j n : ℕ
zero :
  (ComplexShape.down ℕ).Rel (n + 2) 0 →
    Eq.{v + 1} (α := C.X (n + 2) ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X 0)
      (CategoryStruct.comp (Y := ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X (n + 2))
          (match (motive := (x : ℕ) → C.X x ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X x)
            n + 2 with
          | 0 => 𝟙 (C.X 0)
          | n.succ => 𝟙 (C.X (n + 1)))
          (((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).d (n + 2) 0) :
        C.X (n + 2) ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X 0)
      (CategoryStruct.comp (Y := C.X 0) (C.d (n + 2) 0)
          (match (motive := (x : ℕ) → C.X x ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X x) 0 with
          | 0 => 𝟙 (C.X 0)
          | n.succ => 𝟙 (C.X (n + 1))) :
        C.X (n + 2) ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X 0)
succ :
  ∀ (j : ℕ),
    (ComplexShape.down ℕ).Rel (n + 2) (j + 1) →
      Eq.{v + 1} (α := C.X (n + 2) ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X (j + 1))
        (CategoryStruct.comp (Y := ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X (n + 2))
            (match (motive := (x : ℕ) → C.X x ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X x)
              n + 2 with
            | 0 => 𝟙 (C.X 0)
            | n.succ => 𝟙 (C.X (n + 1)))
            (((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).d (n + 2) (j + 1)) :
          C.X (n + 2) ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X (j + 1))
        (CategoryStruct.comp (Y := C.X (j + 1)) (C.d (n + 2) (j + 1))
            (match (motive := (x : ℕ) → C.X x ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X x)
              j + 1 with
            | 0 => 𝟙 (C.X 0)
            | n.succ => 𝟙 (C.X (n + 1))) :
          C.X (n + 2) ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X (j + 1))
⊢ (ComplexShape.down ℕ).Rel (n + 2) j →
    Eq.{v + 1} (α := C.X (n + 2) ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X j)
      (CategoryStruct.comp (Y := ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X (n + 2))
          (match (motive := (x : ℕ) → C.X x ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X x)
            n + 2 with
          | 0 => 𝟙 (C.X 0)
          | n.succ => 𝟙 (C.X (n + 1)))
          (((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).d (n + 2) j) :
        C.X (n + 2) ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X j)
      (C.d (n + 2) j ≫
          match (motive := (x : ℕ) → C.X x ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X x) j with
          | 0 => 𝟙 (C.X 0)
          | n.succ => 𝟙 (C.X (n + 1)) :
        C.X (n + 2) ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X j)

Tactic:
rcases j with - | j
-- Module: Mathlib.Algebra.Homology.Augment
Goal:
V : Type u
inst : Category.{v, u} V
inst_1 : HasZeroMorphisms.{v, u} V
C : ChainComplex.{v, u, 0} V ℕ
i n : ℕ
zero :
  1 = n + 2 →
    (𝟙 (C.X (n + 1 + 1)) ≫
        match (motive :=
          (x x_1 : ℕ) →
            Quiver.Hom.{v, u} (V := V)
              (match (motive := ℕ → V) x with
                | 0 => C.X 0
                | i.succ => C.X (i + 1) :
                V)
              (match (motive := ℕ → V) x_1 with
                | 0 => C.X 0
                | i.succ => C.X (i + 1) :
                V))
          n + 2, 0 with
        | 1, 0 => C.d 1 0
        | i.succ, j.succ => C.d (i + 1) (j + 1)
        | x, x_1 => 0) =
      C.d (n + 2) 0 ≫ 𝟙 (C.X 0)
⊢ (ComplexShape.down ℕ).Rel (n + 2) 0 →
    Eq.{v + 1} (α := C.X (n + 2) ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X 0)
      (CategoryStruct.comp (Y := ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X (n + 2))
          (match (motive := (x : ℕ) → C.X x ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X x)
            n + 2 with
          | 0 => 𝟙 (C.X 0)
          | n.succ => 𝟙 (C.X (n + 1)))
          (((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).d (n + 2) 0) :
        C.X (n + 2) ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X 0)
      (CategoryStruct.comp (Y := C.X 0) (C.d (n + 2) 0)
          (match (motive := (x : ℕ) → C.X x ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X x) 0 with
          | 0 => 𝟙 (C.X 0)
          | n.succ => 𝟙 (C.X (n + 1))) :
        C.X (n + 2) ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X 0)

Tactic:
dsimp [augment, truncate]

Premises used: ChainComplex.truncate, ChainComplex.augment
-- Module: Mathlib.Algebra.Homology.Augment
Goal:
V : Type u
inst : Category.{v, u} V
inst_1 : HasZeroMorphisms.{v, u} V
C : ChainComplex.{v, u, 0} V ℕ
i n j : ℕ
succ :
  j + 1 + 1 = n + 2 →
    (𝟙 (C.X (n + 1 + 1)) ≫
        match (motive :=
          (x x_1 : ℕ) →
            Quiver.Hom.{v, u} (V := V)
              (match (motive := ℕ → V) x with
                | 0 => C.X 0
                | i.succ => C.X (i + 1) :
                V)
              (match (motive := ℕ → V) x_1 with
                | 0 => C.X 0
                | i.succ => C.X (i + 1) :
                V))
          n + 2, j + 1 with
        | 1, 0 => C.d 1 0
        | i.succ, j.succ => C.d (i + 1) (j + 1)
        | x, x_1 => 0) =
      C.d (n + 2) (j + 1) ≫ 𝟙 (C.X (j + 1))
⊢ (ComplexShape.down ℕ).Rel (n + 2) (j + 1) →
    Eq.{v + 1} (α := C.X (n + 2) ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X (j + 1))
      (CategoryStruct.comp (Y := ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X (n + 2))
          (match (motive := (x : ℕ) → C.X x ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X x)
            n + 2 with
          | 0 => 𝟙 (C.X 0)
          | n.succ => 𝟙 (C.X (n + 1)))
          (((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).d (n + 2) (j + 1)) :
        C.X (n + 2) ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X (j + 1))
      (CategoryStruct.comp (Y := C.X (j + 1)) (C.d (n + 2) (j + 1))
          (match (motive := (x : ℕ) → C.X x ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X x)
            j + 1 with
          | 0 => 𝟙 (C.X 0)
          | n.succ => 𝟙 (C.X (n + 1))) :
        C.X (n + 2) ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X (j + 1))

Tactic:
dsimp [augment, truncate]

Premises used: ChainComplex.truncate, ChainComplex.augment
-- Module: Mathlib.Algebra.Homology.Augment
Goal:
V : Type u
inst : Category.{v, u} V
inst_1 : HasZeroMorphisms.{v, u} V
C : ChainComplex.{v, u, 0} V ℕ
i n : ℕ
⊢ 1 = n + 2 →
    (𝟙 (C.X (n + 1 + 1)) ≫
        match (motive :=
          (x x_1 : ℕ) →
            Quiver.Hom.{v, u} (V := V)
              (match (motive := ℕ → V) x with
                | 0 => C.X 0
                | i.succ => C.X (i + 1) :
                V)
              (match (motive := ℕ → V) x_1 with
                | 0 => C.X 0
                | i.succ => C.X (i + 1) :
                V))
          n + 2, 0 with
        | 1, 0 => C.d 1 0
        | i.succ, j.succ => C.d (i + 1) (j + 1)
        | x, x_1 => 0) =
      C.d (n + 2) 0 ≫ 𝟙 (C.X 0)

Tactic:
simp
-- Module: Mathlib.Algebra.Homology.Augment
Goal:
V : Type u
inst : Category.{v, u} V
inst_1 : HasZeroMorphisms.{v, u} V
C : ChainComplex.{v, u, 0} V ℕ
i n j : ℕ
⊢ j + 1 + 1 = n + 2 →
    (𝟙 (C.X (n + 1 + 1)) ≫
        match (motive :=
          (x x_1 : ℕ) →
            Quiver.Hom.{v, u} (V := V)
              (match (motive := ℕ → V) x with
                | 0 => C.X 0
                | i.succ => C.X (i + 1) :
                V)
              (match (motive := ℕ → V) x_1 with
                | 0 => C.X 0
                | i.succ => C.X (i + 1) :
                V))
          n + 2, j + 1 with
        | 1, 0 => C.d 1 0
        | i.succ, j.succ => C.d (i + 1) (j + 1)
        | x, x_1 => 0) =
      C.d (n + 2) (j + 1) ≫ 𝟙 (C.X (j + 1))

Tactic:
simp
-- Module: Mathlib.Algebra.Homology.Augment
Goal:
V : Type u
inst : Category.{v, u} V
inst_1 : HasZeroMorphisms.{v, u} V
C : ChainComplex.{v, u, 0} V ℕ
i j : ℕ
⊢ (ComplexShape.down ℕ).Rel i j →
    Eq.{v + 1} (α := C.X i ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X j)
      ((match (motive := (x : ℕ) → C.X x ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X x) i with
          | 0 => 𝟙 (C.X 0)
          | n.succ => 𝟙 (C.X (n + 1))) ≫
          ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).d i j :
        C.X i ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X j)
      (C.d i j ≫
          match (motive := (x : ℕ) → C.X x ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X x) j with
          | 0 => 𝟙 (C.X 0)
          | n.succ => 𝟙 (C.X (n + 1)) :
        C.X i ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X j)

Tactic:
match i with
          | 0 | 1 | n + 2 =>
          rcases j with - | j <;> dsimp [augment, truncate] <;> simp
-- Module: Mathlib.Algebra.Homology.Augment
Goal:
V : Type u
inst : Category.{v, u} V
inst_1 : HasZeroMorphisms.{v, u} V
C : ChainComplex.{v, u, 0} V ℕ
h :
  ∀ (i : ℕ),
    Hom.f
        (CategoryStruct.comp (Y := C)
          {
            f := fun (x : ℕ) ↦
              match (motive := (x : ℕ) → ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X x ⟶ C.X x) x with
              | 0 => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X 0)
              | n.succ => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X (n + 1)),
            comm' := ⋯ }
          {
            f := fun (x : ℕ) ↦
              match (motive := (x : ℕ) → C.X x ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X x) x with
              | 0 => 𝟙 (C.X 0)
              | n.succ => 𝟙 (C.X (n + 1)),
            comm' := ⋯ })
        i =
      Hom.f (𝟙 ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯)) i
⊢ CategoryStruct.comp (Y := C)
      {
        f := fun (x : ℕ) ↦
          match (motive := (x : ℕ) → ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X x ⟶ C.X x) x with
          | 0 => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X 0)
          | n.succ => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X (n + 1)),
        comm' := ⋯ }
      {
        f := fun (x : ℕ) ↦
          match (motive := (x : ℕ) → C.X x ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X x) x with
          | 0 => 𝟙 (C.X 0)
          | n.succ => 𝟙 (C.X (n + 1)),
        comm' := ⋯ } =
    𝟙 ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯)

Tactic:
ext i
-- Module: Mathlib.Algebra.Homology.Augment
Goal:
V : Type u
inst : Category.{v, u} V
inst_1 : HasZeroMorphisms.{v, u} V
C : ChainComplex.{v, u, 0} V ℕ
i : ℕ
h :
  Eq (α :=
    ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X 0 ⟶
      ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X 0)
    (Hom.f
      (CategoryStruct.comp (Y := C)
        {
          f := fun (x : ℕ) ↦
            match (motive := (x : ℕ) → ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X x ⟶ C.X x) x with
            | 0 => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X 0)
            | n.succ => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X (n + 1)),
          comm' := ⋯ }
        {
          f := fun (x : ℕ) ↦
            match (motive := (x : ℕ) → C.X x ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X x) x with
            | 0 => 𝟙 (C.X 0)
            | n.succ => 𝟙 (C.X (n + 1)),
          comm' := ⋯ })
      0)
    (Hom.f (𝟙 ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯)) 0)
h₁ :
  ∀ (n : ℕ),
    Hom.f
        (CategoryStruct.comp (Y := C)
          {
            f := fun (x : ℕ) ↦
              match (motive := (x : ℕ) → ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X x ⟶ C.X x) x with
              | 0 => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X 0)
              | n.succ => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X (n + 1)),
            comm' := ⋯ }
          {
            f := fun (x : ℕ) ↦
              match (motive := (x : ℕ) → C.X x ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X x) x with
              | 0 => 𝟙 (C.X 0)
              | n.succ => 𝟙 (C.X (n + 1)),
            comm' := ⋯ })
        (n + 1) =
      Hom.f (𝟙 ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯)) (n + 1)
⊢ Hom.f
      (CategoryStruct.comp (Y := C)
        {
          f := fun (x : ℕ) ↦
            match (motive := (x : ℕ) → ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X x ⟶ C.X x) x with
            | 0 => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X 0)
            | n.succ => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X (n + 1)),
          comm' := ⋯ }
        {
          f := fun (x : ℕ) ↦
            match (motive := (x : ℕ) → C.X x ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X x) x with
            | 0 => 𝟙 (C.X 0)
            | n.succ => 𝟙 (C.X (n + 1)),
          comm' := ⋯ })
      i =
    Hom.f (𝟙 ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯)) i

Tactic:
cases i
-- Module: Mathlib.Algebra.Homology.Augment
Goal:
V : Type u
inst : Category.{v, u} V
inst_1 : HasZeroMorphisms.{v, u} V
C : ChainComplex.{v, u, 0} V ℕ
⊢ Eq (α :=
    ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X 0 ⟶
      ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X 0)
    (Hom.f
      (CategoryStruct.comp (Y := C)
        {
          f := fun (x : ℕ) ↦
            match (motive := (x : ℕ) → ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X x ⟶ C.X x) x with
            | 0 => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X 0)
            | n.succ => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X (n + 1)),
          comm' := ⋯ }
        {
          f := fun (x : ℕ) ↦
            match (motive := (x : ℕ) → C.X x ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X x) x with
            | 0 => 𝟙 (C.X 0)
            | n.succ => 𝟙 (C.X (n + 1)),
          comm' := ⋯ })
      0)
    (Hom.f (𝟙 ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯)) 0)

Tactic:
simp
-- Module: Mathlib.Algebra.Homology.Augment
Goal:
V : Type u
inst : Category.{v, u} V
inst_1 : HasZeroMorphisms.{v, u} V
C : ChainComplex.{v, u, 0} V ℕ
n : ℕ
⊢ Hom.f
      (CategoryStruct.comp (Y := C)
        {
          f := fun (x : ℕ) ↦
            match (motive := (x : ℕ) → ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X x ⟶ C.X x) x with
            | 0 => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X 0)
            | n.succ => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X (n + 1)),
          comm' := ⋯ }
        {
          f := fun (x : ℕ) ↦
            match (motive := (x : ℕ) → C.X x ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X x) x with
            | 0 => 𝟙 (C.X 0)
            | n.succ => 𝟙 (C.X (n + 1)),
          comm' := ⋯ })
      (n + 1) =
    Hom.f (𝟙 ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯)) (n + 1)

Tactic:
simp
-- Module: Mathlib.Algebra.Homology.Augment
Goal:
V : Type u
inst : Category.{v, u} V
inst_1 : HasZeroMorphisms.{v, u} V
C : ChainComplex.{v, u, 0} V ℕ
h :
  ∀ (i : ℕ),
    Hom.f
        (CategoryStruct.comp (Y := (Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯)
          {
            f := fun (x : ℕ) ↦
              match (motive := (x : ℕ) → C.X x ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X x) x with
              | 0 => 𝟙 (C.X 0)
              | n.succ => 𝟙 (C.X (n + 1)),
            comm' := ⋯ }
          {
            f := fun (x : ℕ) ↦
              match (motive := (x : ℕ) → ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X x ⟶ C.X x) x with
              | 0 => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X 0)
              | n.succ => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X (n + 1)),
            comm' := ⋯ })
        i =
      Hom.f (𝟙 C) i
⊢ CategoryStruct.comp (Y := (Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯)
      {
        f := fun (x : ℕ) ↦
          match (motive := (x : ℕ) → C.X x ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X x) x with
          | 0 => 𝟙 (C.X 0)
          | n.succ => 𝟙 (C.X (n + 1)),
        comm' := ⋯ }
      {
        f := fun (x : ℕ) ↦
          match (motive := (x : ℕ) → ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X x ⟶ C.X x) x with
          | 0 => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X 0)
          | n.succ => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X (n + 1)),
        comm' := ⋯ } =
    𝟙 C

Tactic:
ext i
-- Module: Mathlib.Algebra.Homology.Augment
Goal:
V : Type u
inst : Category.{v, u} V
inst_1 : HasZeroMorphisms.{v, u} V
C : ChainComplex.{v, u, 0} V ℕ
i : ℕ
h :
  Eq (α := C.X 0 ⟶ C.X 0)
    (Hom.f
      (CategoryStruct.comp (Y := (Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯)
        {
          f := fun (x : ℕ) ↦
            match (motive := (x : ℕ) → C.X x ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X x) x with
            | 0 => 𝟙 (C.X 0)
            | n.succ => 𝟙 (C.X (n + 1)),
          comm' := ⋯ }
        {
          f := fun (x : ℕ) ↦
            match (motive := (x : ℕ) → ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X x ⟶ C.X x) x with
            | 0 => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X 0)
            | n.succ => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X (n + 1)),
          comm' := ⋯ })
      0)
    (Hom.f (𝟙 C) 0)
h₁ :
  ∀ (n : ℕ),
    Hom.f
        (CategoryStruct.comp (Y := (Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯)
          {
            f := fun (x : ℕ) ↦
              match (motive := (x : ℕ) → C.X x ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X x) x with
              | 0 => 𝟙 (C.X 0)
              | n.succ => 𝟙 (C.X (n + 1)),
            comm' := ⋯ }
          {
            f := fun (x : ℕ) ↦
              match (motive := (x : ℕ) → ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X x ⟶ C.X x) x with
              | 0 => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X 0)
              | n.succ => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X (n + 1)),
            comm' := ⋯ })
        (n + 1) =
      Hom.f (𝟙 C) (n + 1)
⊢ Hom.f
      (CategoryStruct.comp (Y := (Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯)
        {
          f := fun (x : ℕ) ↦
            match (motive := (x : ℕ) → C.X x ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X x) x with
            | 0 => 𝟙 (C.X 0)
            | n.succ => 𝟙 (C.X (n + 1)),
          comm' := ⋯ }
        {
          f := fun (x : ℕ) ↦
            match (motive := (x : ℕ) → ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X x ⟶ C.X x) x with
            | 0 => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X 0)
            | n.succ => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X (n + 1)),
          comm' := ⋯ })
      i =
    Hom.f (𝟙 C) i

Tactic:
cases i
-- Module: Mathlib.Algebra.Homology.Augment
Goal:
V : Type u
inst : Category.{v, u} V
inst_1 : HasZeroMorphisms.{v, u} V
C : ChainComplex.{v, u, 0} V ℕ
⊢ Eq (α := C.X 0 ⟶ C.X 0)
    (Hom.f
      (CategoryStruct.comp (Y := (Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯)
        {
          f := fun (x : ℕ) ↦
            match (motive := (x : ℕ) → C.X x ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X x) x with
            | 0 => 𝟙 (C.X 0)
            | n.succ => 𝟙 (C.X (n + 1)),
          comm' := ⋯ }
        {
          f := fun (x : ℕ) ↦
            match (motive := (x : ℕ) → ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X x ⟶ C.X x) x with
            | 0 => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X 0)
            | n.succ => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X (n + 1)),
          comm' := ⋯ })
      0)
    (Hom.f (𝟙 C) 0)

Tactic:
simp
-- Module: Mathlib.Algebra.Homology.Augment
Goal:
V : Type u
inst : Category.{v, u} V
inst_1 : HasZeroMorphisms.{v, u} V
C : ChainComplex.{v, u, 0} V ℕ
n : ℕ
⊢ Hom.f
      (CategoryStruct.comp (Y := (Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯)
        {
          f := fun (x : ℕ) ↦
            match (motive := (x : ℕ) → C.X x ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X x) x with
            | 0 => 𝟙 (C.X 0)
            | n.succ => 𝟙 (C.X (n + 1)),
          comm' := ⋯ }
        {
          f := fun (x : ℕ) ↦
            match (motive := (x : ℕ) → ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X x ⟶ C.X x) x with
            | 0 => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X 0)
            | n.succ => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X (n + 1)),
          comm' := ⋯ })
      (n + 1) =
    Hom.f (𝟙 C) (n + 1)

Tactic:
simp
-- Module: Mathlib.Algebra.Homology.Augment
Goal:
V : Type u
inst : Category.{v, u} V
inst_1 : HasZeroMorphisms.{v, u} V
C : ChainComplex.{v, u, 0} V ℕ
⊢ Eq (α := ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X 0 ⟶ C.X 0) (C.augmentTruncate.hom.f 0)
    (𝟙 (C.X 0))

Tactic:
rfl

Premises used: rfl
-- Module: Mathlib.Algebra.Homology.Augment
Goal:
V : Type u
inst : Category.{v, u} V
inst_1 : HasZeroMorphisms.{v, u} V
C : ChainComplex.{v, u, 0} V ℕ
i : ℕ
⊢ Eq (α := ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) ⋯).X (i + 1) ⟶ C.X (i + 1))
    (C.augmentTruncate.hom.f (i + 1)) (𝟙 (C.X (i + 1)))

Tactic:
rfl

Premises used: rfl
-- Module: Mathlib.Algebra.Homology.Augment
Goal:
V : Type u
inst : Category.{v, u} V
inst_1 : HasZeroMorphisms.{v, u} V
C : ChainComplex.{v, u, 0} V ℕ
⊢ C.augmentTruncate.inv.f 0 = 𝟙 (C.X 0)

Tactic:
rfl

Premises used: rfl
-- Module: Mathlib.Algebra.Homology.Augment
Goal:
V : Type u
inst : Category.{v, u} V
inst_1 : HasZeroMorphisms.{v, u} V
C : ChainComplex.{v, u, 0} V ℕ
i : ℕ
⊢ C.augmentTruncate.inv.f (i + 1) = 𝟙 (C.X (i + 1))

Tactic:
rfl

Premises used: rfl
-- 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
w : ¬(ComplexShape.up ℕ).Rel i j
a : ¬(ComplexShape.up ℕ).Rel (i + 1) (j + 1)
⊢ Eq (α := C.X (i + 1) ⟶ C.X (j + 1)) (C.d (i + 1) (j + 1)) 0

Tactic:
apply C.shape

Premises used: HomologicalComplex.shape
-- 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
w : ¬(ComplexShape.up ℕ).Rel i j
⊢ ¬(ComplexShape.up ℕ).Rel (i + 1) (j + 1)

Tactic:
simpa
-- Module: Mathlib.Algebra.Homology.Augment
Goal:
V : Type u
inst : Category.{v, u} V
inst_1 : HasZeroObject.{v, u} V
inst_2 : HasZeroMorphisms.{v, u} V
C : CochainComplex.{v, u, 0} V ℕ
⊢ Eq (α := C.X 0 ⟶ (Functor.obj truncate C).X 1)
    (CategoryStruct.comp (Y := (Functor.obj truncate C).X 0) (C.d 0 1) ((Functor.obj truncate C).d 0 1)) 0

Tactic:
simp