awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Algebra.Homology.Augment (goal)
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)
⊢ 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
have h₁ : ∀ (n : ℕ),
  Hom.f
      (CategoryStruct.comp (Y := (Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) sorry)
        {
          f := fun (x : ℕ) ↦
            match (motive := (x : ℕ) → C.X x ⟶ ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) sorry).X x) x with
            | 0 => 𝟙 (C.X 0)
            | n.succ => 𝟙 (C.X (n + 1)),
          comm' := sorry }
        {
          f := fun (x : ℕ) ↦
            match (motive := (x : ℕ) → ((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) sorry).X x ⟶ C.X x) x with
            | 0 => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) sorry).X 0)
            | n.succ => 𝟙 (((Functor.obj truncate C).augment (X := C.X 0) (C.d 1 0) sorry).X (n + 1)),
          comm' := sorry })
      (n + 1) =
    Hom.f (𝟙 C) (n + 1)
Tactic: cases i
Cleared hypotheses: i
-- Lean 4 proof step from Mathlib.Algebra.Homology.Augment (goal)
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
⊢ Eq (α := C.X (i + 1) ⟶ C.X (j + 1)) (C.d (i + 1) (j + 1)) 0
have a : ¬(ComplexShape.up ℕ).Rel (i + 1) (j + 1)
Tactic: apply C.shape
-- Lean 4 proof step from Mathlib.Algebra.Homology.Augment (goal)
Goal:
V : Type u
inst : Category.{v, u} V
inst_1 : HasZeroMorphisms.{v, u} V
C : CochainComplex.{v, u, 0} V ℕ
X : V
f : X ⟶ C.X 0
w : Eq (α := X ⟶ C.X 1) (f ≫ C.d 0 1) 0
i j : ℕ
s : ¬(ComplexShape.up ℕ).Rel i j
⊢ Eq.{v + 1} (α :=
    Quiver.Hom.{v, u} (V := V)
      (match (motive := ℕ → V) i with
        | 0 => X
        | i.succ => C.X i :
        V)
      (match (motive := ℕ → V) j with
        | 0 => X
        | i.succ => C.X i :
        V))
    (match (motive :=
        (x x_1 : ℕ) →
          Quiver.Hom.{v, u} (V := V)
            (match (motive := ℕ → V) x with
              | 0 => X
              | i.succ => C.X i :
              V)
            (match (motive := ℕ → V) x_1 with
              | 0 => X
              | i.succ => C.X i :
              V))
        i, j with
      | 0, 1 => f
      | i.succ, j.succ => C.d i j
      | x, x_1 => 0 :
      Quiver.Hom.{v, u} (V := V)
        (match (motive := ℕ → V) i with
          | 0 => X
          | i.succ => C.X i :
          V)
        (match (motive := ℕ → V) j with
          | 0 => X
          | i.succ => C.X i :
          V))
    (0 :
      Quiver.Hom.{v, u} (V := V)
        (match (motive := ℕ → V) i with
          | 0 => X
          | i.succ => C.X i :
          V)
        (match (motive := ℕ → V) j with
          | 0 => X
          | i.succ => C.X i :
          V))
have zero : ∀ (s : ¬(ComplexShape.up ℕ).Rel i 0),
  Eq.{v + 1} (α :=
    Quiver.Hom.{v, u} (V := V)
      (match (motive := ℕ → V) i with
        | 0 => X
        | i.succ => C.X i :
        V)
      (match (motive := ℕ → V) 0 with
        | 0 => X
        | i.succ => C.X i :
        V))
    (match (motive :=
        (x x_1 : ℕ) →
          Quiver.Hom.{v, u} (V := V)
            (match (motive := ℕ → V) x with
              | 0 => X
              | i.succ => C.X i :
              V)
            (match (motive := ℕ → V) x_1 with
              | 0 => X
              | i.succ => C.X i :
              V))
        i, 0 with
      | 0, 1 => f
      | i.succ, j.succ => C.d i j
      | x, x_1 => 0 :
      Quiver.Hom.{v, u} (V := V)
        (match (motive := ℕ → V) i with
          | 0 => X
          | i.succ => C.X i :
          V)
        (match (motive := ℕ → V) 0 with
          | 0 => X
          | i.succ => C.X i :
          V))
    (0 :
      Quiver.Hom.{v, u} (V := V)
        (match (motive := ℕ → V) i with
          | 0 => X
          | i.succ => C.X i :
          V)
        (match (motive := ℕ → V) 0 with
          | 0 => X
          | i.succ => C.X i :
          V))
Tactic: rcases j with (_ | _ | j)
Cleared hypotheses: j, s
-- Lean 4 proof step from Mathlib.Algebra.Homology.Augment (goal)
Goal:
V : Type u
inst : Category.{v, u} V
inst_1 : HasZeroMorphisms.{v, u} V
C : CochainComplex.{v, u, 0} V ℕ
X : V
f : X ⟶ C.X 0
w : Eq (α := X ⟶ C.X 1) (f ≫ C.d 0 1) 0
i : ℕ
s : ¬(ComplexShape.up ℕ).Rel i 0
⊢ Eq.{v + 1} (α :=
    Quiver.Hom.{v, u} (V := V)
      (match (motive := ℕ → V) i with
        | 0 => X
        | i.succ => C.X i :
        V)
      (match (motive := ℕ → V) 0 with
        | 0 => X
        | i.succ => C.X i :
        V))
    (match (motive :=
        (x x_1 : ℕ) →
          Quiver.Hom.{v, u} (V := V)
            (match (motive := ℕ → V) x with
              | 0 => X
              | i.succ => C.X i :
              V)
            (match (motive := ℕ → V) x_1 with
              | 0 => X
              | i.succ => C.X i :
              V))
        i, 0 with
      | 0, 1 => f
      | i.succ, j.succ => C.d i j
      | x, x_1 => 0 :
      Quiver.Hom.{v, u} (V := V)
        (match (motive := ℕ → V) i with
          | 0 => X
          | i.succ => C.X i :
          V)
        (match (motive := ℕ → V) 0 with
          | 0 => X
          | i.succ => C.X i :
          V))
    (0 :
      Quiver.Hom.{v, u} (V := V)
        (match (motive := ℕ → V) i with
          | 0 => X
          | i.succ => C.X i :
          V)
        (match (motive := ℕ → V) 0 with
          | 0 => X
          | i.succ => C.X i :
          V))
have zero : ∀ (s : ¬(ComplexShape.up ℕ).Rel 0 0),
  Eq.{v + 1} (α :=
    Quiver.Hom.{v, u} (V := V)
      (match (motive := ℕ → V) 0 with
        | 0 => X
        | i.succ => C.X i :
        V)
      (match (motive := ℕ → V) 0 with
        | 0 => X
        | i.succ => C.X i :
        V))
    (match (motive :=
        (x x_1 : ℕ) →
          Quiver.Hom.{v, u} (V := V)
            (match (motive := ℕ → V) x with
              | 0 => X
              | i.succ => C.X i :
              V)
            (match (motive := ℕ → V) x_1 with
              | 0 => X
              | i.succ => C.X i :
              V))
        0, 0 with
      | 0, 1 => f
      | i.succ, j.succ => C.d i j
      | x, x_1 => 0 :
      Quiver.Hom.{v, u} (V := V)
        (match (motive := ℕ → V) 0 with
          | 0 => X
          | i.succ => C.X i :
          V)
        (match (motive := ℕ → V) 0 with
          | 0 => X
          | i.succ => C.X i :
          V))
    (0 :
      Quiver.Hom.{v, u} (V := V)
        (match (motive := ℕ → V) 0 with
          | 0 => X
          | i.succ => C.X i :
          V)
        (match (motive := ℕ → V) 0 with
          | 0 => X
          | i.succ => C.X i :
          V))
Tactic: cases i
Cleared hypotheses: i, s
-- Lean 4 proof step from Mathlib.Algebra.Homology.Augment (goal)
Goal:
V : Type u
inst : Category.{v, u} V
inst_1 : HasZeroMorphisms.{v, u} V
C : CochainComplex.{v, u, 0} V ℕ
X : V
f : X ⟶ C.X 0
w : Eq (α := X ⟶ C.X 1) (f ≫ C.d 0 1) 0
i : ℕ
s : ¬(ComplexShape.up ℕ).Rel i (HAdd.hAdd (α := ℕ) 0 1)
⊢ Eq.{v + 1} (α :=
    Quiver.Hom.{v, u} (V := V)
      (match (motive := ℕ → V) i with
        | 0 => X
        | i.succ => C.X i :
        V)
      (match (motive := ℕ → V) HAdd.hAdd (α := ℕ) 0 1 with
        | 0 => X
        | i.succ => C.X i :
        V))
    (match (motive :=
        (x x_1 : ℕ) →
          Quiver.Hom.{v, u} (V := V)
            (match (motive := ℕ → V) x with
              | 0 => X
              | i.succ => C.X i :
              V)
            (match (motive := ℕ → V) x_1 with
              | 0 => X
              | i.succ => C.X i :
              V))
        i, HAdd.hAdd (α := ℕ) 0 1 with
      | 0, 1 => f
      | i.succ, j.succ => C.d i j
      | x, x_1 => 0 :
      Quiver.Hom.{v, u} (V := V)
        (match (motive := ℕ → V) i with
          | 0 => X
          | i.succ => C.X i :
          V)
        (match (motive := ℕ → V) HAdd.hAdd (α := ℕ) 0 1 with
          | 0 => X
          | i.succ => C.X i :
          V))
    (0 :
      Quiver.Hom.{v, u} (V := V)
        (match (motive := ℕ → V) i with
          | 0 => X
          | i.succ => C.X i :
          V)
        (match (motive := ℕ → V) HAdd.hAdd (α := ℕ) 0 1 with
          | 0 => X
          | i.succ => C.X i :
          V))
have succ : ∀ (s : ¬(ComplexShape.up ℕ).Rel 0 (HAdd.hAdd (α := ℕ) 0 1)),
  Eq.{v + 1} (α :=
    Quiver.Hom.{v, u} (V := V)
      (match (motive := ℕ → V) 0 with
        | 0 => X
        | i.succ => C.X i :
        V)
      (match (motive := ℕ → V) HAdd.hAdd (α := ℕ) 0 1 with
        | 0 => X
        | i.succ => C.X i :
        V))
    (match (motive :=
        (x x_1 : ℕ) →
          Quiver.Hom.{v, u} (V := V)
            (match (motive := ℕ → V) x with
              | 0 => X
              | i.succ => C.X i :
              V)
            (match (motive := ℕ → V) x_1 with
              | 0 => X
              | i.succ => C.X i :
              V))
        0, HAdd.hAdd (α := ℕ) 0 1 with
      | 0, 1 => f
      | i.succ, j.succ => C.d i j
      | x, x_1 => 0 :
      Quiver.Hom.{v, u} (V := V)
        (match (motive := ℕ → V) 0 with
          | 0 => X
          | i.succ => C.X i :
          V)
        (match (motive := ℕ → V) HAdd.hAdd (α := ℕ) 0 1 with
          | 0 => X
          | i.succ => C.X i :
          V))
    (0 :
      Quiver.Hom.{v, u} (V := V)
        (match (motive := ℕ → V) 0 with
          | 0 => X
          | i.succ => C.X i :
          V)
        (match (motive := ℕ → V) HAdd.hAdd (α := ℕ) 0 1 with
          | 0 => X
          | i.succ => C.X i :
          V))
Tactic: cases i
Cleared hypotheses: i, s
-- Lean 4 proof step from Mathlib.Algebra.Homology.Augment (goal)
Goal:
V : Type u
inst : Category.{v, u} V
inst_1 : HasZeroMorphisms.{v, u} V
C : CochainComplex.{v, u, 0} V ℕ
X : V
f : X ⟶ C.X 0
w : Eq (α := X ⟶ C.X 1) (f ≫ C.d 0 1) 0
i j : ℕ
s : ¬(ComplexShape.up ℕ).Rel i (j + 1 + 1)
⊢ Eq.{v + 1} (α :=
    Quiver.Hom.{v, u} (V := V)
      (match (motive := ℕ → V) i with
        | 0 => X
        | i.succ => C.X i :
        V)
      (match (motive := ℕ → V) j + 1 + 1 with
        | 0 => X
        | i.succ => C.X i :
        V))
    (match (motive :=
        (x x_1 : ℕ) →
          Quiver.Hom.{v, u} (V := V)
            (match (motive := ℕ → V) x with
              | 0 => X
              | i.succ => C.X i :
              V)
            (match (motive := ℕ → V) x_1 with
              | 0 => X
              | i.succ => C.X i :
              V))
        i, j + 1 + 1 with
      | 0, 1 => f
      | i.succ, j.succ => C.d i j
      | x, x_1 => 0 :
      Quiver.Hom.{v, u} (V := V)
        (match (motive := ℕ → V) i with
          | 0 => X
          | i.succ => C.X i :
          V)
        (match (motive := ℕ → V) j + 1 + 1 with
          | 0 => X
          | i.succ => C.X i :
          V))
    (0 :
      Quiver.Hom.{v, u} (V := V)
        (match (motive := ℕ → V) i with
          | 0 => X
          | i.succ => C.X i :
          V)
        (match (motive := ℕ → V) j + 1 + 1 with
          | 0 => X
          | i.succ => C.X i :
          V))
have succ : ∀ (s : ¬(ComplexShape.up ℕ).Rel 0 (j + 1 + 1)),
  Eq.{v + 1} (α :=
    Quiver.Hom.{v, u} (V := V)
      (match (motive := ℕ → V) 0 with
        | 0 => X
        | i.succ => C.X i :
        V)
      (match (motive := ℕ → V) j + 1 + 1 with
        | 0 => X
        | i.succ => C.X i :
        V))
    (match (motive :=
        (x x_1 : ℕ) →
          Quiver.Hom.{v, u} (V := V)
            (match (motive := ℕ → V) x with
              | 0 => X
              | i.succ => C.X i :
              V)
            (match (motive := ℕ → V) x_1 with
              | 0 => X
              | i.succ => C.X i :
              V))
        0, j + 1 + 1 with
      | 0, 1 => f
      | i.succ, j.succ => C.d i j
      | x, x_1 => 0 :
      Quiver.Hom.{v, u} (V := V)
        (match (motive := ℕ → V) 0 with
          | 0 => X
          | i.succ => C.X i :
          V)
        (match (motive := ℕ → V) j + 1 + 1 with
          | 0 => X
          | i.succ => C.X i :
          V))
    (0 :
      Quiver.Hom.{v, u} (V := V)
        (match (motive := ℕ → V) 0 with
          | 0 => X
          | i.succ => C.X i :
          V)
        (match (motive := ℕ → V) j + 1 + 1 with
          | 0 => X
          | i.succ => C.X i :
          V))
Tactic: cases i
Cleared hypotheses: i, s
-- Lean 4 proof step from Mathlib.Algebra.Homology.Augment (goal)
Goal:
V : Type u
inst : Category.{v, u} V
inst_1 : HasZeroMorphisms.{v, u} V
C : CochainComplex.{v, u, 0} V ℕ
X : V
f : X ⟶ C.X 0
w : Eq (α := X ⟶ C.X 1) (f ≫ C.d 0 1) 0
i j k_1 : ℕ
hij : (ComplexShape.up ℕ).Rel i j
hjk : (ComplexShape.up ℕ).Rel j k_1
k : ℕ
zero : Eq (α := X ⟶ C.X (HAdd.hAdd (α := ℕ) 0 1)) (f ≫ C.d 0 (HAdd.hAdd (α := ℕ) 0 1)) 0
⊢ Eq (α := X ⟶ C.X (k + 1)) (f ≫ C.d 0 (k + 1)) 0
have succ : ∀ (n : ℕ), Eq (α := X ⟶ C.X (n + 1 + 1)) (f ≫ C.d 0 (n + 1 + 1)) 0
Tactic: cases k
Cleared hypotheses: k
-- Lean 4 proof step from Mathlib.Algebra.Homology.Augment (goal)
Goal:
V : Type u
inst : Category.{v, u} V
inst_1 : HasZeroMorphisms.{v, u} V
C : CochainComplex.{v, u, 0} V ℕ
X : V
f : X ⟶ C.X 0
w : Eq (α := X ⟶ C.X 1) (f ≫ C.d 0 1) 0
i j k : ℕ
hij : (ComplexShape.up ℕ).Rel i j
hjk : (ComplexShape.up ℕ).Rel j k
n : ℕ
⊢ Eq (α := X ⟶ C.X (n + 1 + 1)) (f ≫ C.d 0 (n + 1 + 1)) 0
have succ : ¬(ComplexShape.up ℕ).Rel 0 (n + 1 + 1)
Tactic: rw [C.shape, comp_zero]
-- Lean 4 proof step from Mathlib.Algebra.Homology.Augment (hypothesis)
Goal:
V : Type u
inst : Category.{v, u} V
inst_1 : HasZeroMorphisms.{v, u} V
C : CochainComplex.{v, u, 0} V ℕ
X : V
f : X ⟶ C.X 0
w : Eq (α := X ⟶ C.X 1) (f ≫ C.d 0 1) 0
i j k : ℕ
hij : (ComplexShape.up ℕ).Rel i j
hjk : (ComplexShape.up ℕ).Rel j k
⊢ Eq.{v + 1} (α :=
    Quiver.Hom.{v, u} (V := V)
      (match (motive := ℕ → V) i with
        | 0 => X
        | i.succ => C.X i :
        V)
      (match (motive := ℕ → V) k with
        | 0 => X
        | i.succ => C.X i :
        V))
    (CategoryStruct.comp (Y :=
        match (motive := ℕ → V) j with
        | 0 => X
        | i.succ => C.X i)
        (match (motive :=
          (x x_1 : ℕ) →
            Quiver.Hom.{v, u} (V := V)
              (match (motive := ℕ → V) x with
                | 0 => X
                | i.succ => C.X i :
                V)
              (match (motive := ℕ → V) x_1 with
                | 0 => X
                | i.succ => C.X i :
                V))
          i, j with
        | 0, 1 => f
        | i.succ, j.succ => C.d i j
        | x, x_1 => 0)
        (match (motive :=
          (x x_1 : ℕ) →
            Quiver.Hom.{v, u} (V := V)
              (match (motive := ℕ → V) x with
                | 0 => X
                | i.succ => C.X i :
                V)
              (match (motive := ℕ → V) x_1 with
                | 0 => X
                | i.succ => C.X i :
                V))
          j, k with
        | 0, 1 => f
        | i.succ, j.succ => C.d i j
        | x, x_1 => 0) :
      Quiver.Hom.{v, u} (V := V)
        (match (motive := ℕ → V) i with
          | 0 => X
          | i.succ => C.X i :
          V)
        (match (motive := ℕ → V) k with
          | 0 => X
          | i.succ => C.X i :
          V))
    (0 :
      Quiver.Hom.{v, u} (V := V)
        (match (motive := ℕ → V) i with
          | 0 => X
          | i.succ => C.X i :
          V)
        (match (motive := ℕ → V) k with
          | 0 => X
          | i.succ => C.X i :
          V))
have this : ∀ (k : ℕ), Eq (α := X ⟶ C.X (k + 1)) (f ≫ C.d 0 (k + 1)) 0
Tactic: have (k : ℕ) : f ≫ C.d 0 (k + 1) = 0 := by
      cases k
      · exact w
      · rw [C.shape, comp_zero]
        simp only [ComplexShape.up_Rel, zero_add]
        exact (Nat.one_lt_succ_succ _).ne
-- Lean 4 proof step from Mathlib.Algebra.Homology.Augment (goal)
Goal:
V : Type u
inst : Category.{v, u} V
inst_1 : HasZeroMorphisms.{v, u} V
C : CochainComplex.{v, u, 0} V ℕ
X : V
f : X ⟶ C.X 0
w : Eq (α := X ⟶ C.X 1) (f ≫ C.d 0 1) 0
i j : ℕ
⊢ (ComplexShape.up ℕ).Rel i j → 𝟙 (C.X i) ≫ (Functor.obj truncate (C.augment f w)).d i j = C.d i j ≫ 𝟙 (C.X j)
have zero : (ComplexShape.up ℕ).Rel i 0 → 𝟙 (C.X i) ≫ (Functor.obj truncate (C.augment f w)).d i 0 = C.d i 0 ≫ 𝟙 (C.X 0)
Tactic: cases j
Cleared hypotheses: j
-- Lean 4 proof step from Mathlib.Algebra.Homology.Augment (goal)
Goal:
V : Type u
inst : Category.{v, u} V
inst_1 : HasZeroMorphisms.{v, u} V
C : CochainComplex.{v, u, 0} V ℕ
X : V
f : X ⟶ C.X 0
w : Eq (α := X ⟶ C.X 1) (f ≫ C.d 0 1) 0
i j : ℕ
zero : (ComplexShape.up ℕ).Rel i 0 → 𝟙 (C.X i) ≫ (Functor.obj truncate (C.augment f w)).d i 0 = C.d i 0 ≫ 𝟙 (C.X 0)
⊢ (ComplexShape.up ℕ).Rel i j → 𝟙 (C.X i) ≫ (Functor.obj truncate (C.augment f w)).d i j = C.d i j ≫ 𝟙 (C.X j)
have succ : ∀ (n : ℕ),
  (ComplexShape.up ℕ).Rel i (n + 1) →
    𝟙 (C.X i) ≫ (Functor.obj truncate (C.augment f w)).d i (n + 1) = C.d i (n + 1) ≫ 𝟙 (C.X (n + 1))
Tactic: cases j
Cleared hypotheses: j
-- Lean 4 proof step from Mathlib.Algebra.Homology.Augment (goal)
Goal:
V : Type u
inst : Category.{v, u} V
inst_1 : HasZeroMorphisms.{v, u} V
C : CochainComplex.{v, u, 0} V ℕ
X : V
f : X ⟶ C.X 0
w : Eq (α := X ⟶ C.X 1) (f ≫ C.d 0 1) 0
⊢ { f := fun (x : ℕ) ↦ 𝟙 ((Functor.obj truncate (C.augment f w)).X x), comm' := ⋯ :
        Hom (Functor.obj truncate (C.augment f w)) C } ≫
      { f := fun (x : ℕ) ↦ 𝟙 (C.X x), comm' := ⋯ } =
    𝟙 (Functor.obj truncate (C.augment f w))
have h : ∀ (i : ℕ),
  Hom.f
      ({ f := fun (x : ℕ) ↦ 𝟙 ((Functor.obj truncate (C.augment f w)).X x), comm' := sorry :
          Hom (Functor.obj truncate (C.augment f w)) C } ≫
        { f := fun (x : ℕ) ↦ 𝟙 (C.X x), comm' := sorry })
      i =
    Hom.f (𝟙 (Functor.obj truncate (C.augment f w))) i
Tactic: ext i
-- Lean 4 proof step from Mathlib.Algebra.Homology.Augment (goal)
Goal:
V : Type u
inst : Category.{v, u} V
inst_1 : HasZeroMorphisms.{v, u} V
C : CochainComplex.{v, u, 0} V ℕ
X : V
f : X ⟶ C.X 0
w : Eq (α := X ⟶ C.X 1) (f ≫ C.d 0 1) 0
i : ℕ
⊢ Hom.f
      ({ f := fun (x : ℕ) ↦ 𝟙 ((Functor.obj truncate (C.augment f w)).X x), comm' := ⋯ :
          Hom (Functor.obj truncate (C.augment f w)) C } ≫
        { f := fun (x : ℕ) ↦ 𝟙 (C.X x), comm' := ⋯ })
      i =
    Hom.f (𝟙 (Functor.obj truncate (C.augment f w))) i
have h : Eq (α := (Functor.obj truncate (C.augment f w)).X 0 ⟶ (Functor.obj truncate (C.augment f w)).X 0)
  (Hom.f
    ({ f := fun (x : ℕ) ↦ 𝟙 ((Functor.obj truncate (C.augment f w)).X x), comm' := sorry :
        Hom (Functor.obj truncate (C.augment f w)) C } ≫
      { f := fun (x : ℕ) ↦ 𝟙 (C.X x), comm' := sorry })
    0)
  (Hom.f (𝟙 (Functor.obj truncate (C.augment f w))) 0)
Tactic: cases i
Cleared hypotheses: i
-- Lean 4 proof step from Mathlib.Algebra.Homology.Augment (goal)
Goal:
V : Type u
inst : Category.{v, u} V
inst_1 : HasZeroMorphisms.{v, u} V
C : CochainComplex.{v, u, 0} V ℕ
X : V
f : X ⟶ C.X 0
w : Eq (α := X ⟶ C.X 1) (f ≫ C.d 0 1) 0
i : ℕ
h :
  Eq (α := (Functor.obj truncate (C.augment f w)).X 0 ⟶ (Functor.obj truncate (C.augment f w)).X 0)
    (Hom.f
      ({ f := fun (x : ℕ) ↦ 𝟙 ((Functor.obj truncate (C.augment f w)).X x), comm' := ⋯ :
          Hom (Functor.obj truncate (C.augment f w)) C } ≫
        { f := fun (x : ℕ) ↦ 𝟙 (C.X x), comm' := ⋯ })
      0)
    (Hom.f (𝟙 (Functor.obj truncate (C.augment f w))) 0)
⊢ Hom.f
      ({ f := fun (x : ℕ) ↦ 𝟙 ((Functor.obj truncate (C.augment f w)).X x), comm' := ⋯ :
          Hom (Functor.obj truncate (C.augment f w)) C } ≫
        { f := fun (x : ℕ) ↦ 𝟙 (C.X x), comm' := ⋯ })
      i =
    Hom.f (𝟙 (Functor.obj truncate (C.augment f w))) i
have h₁ : ∀ (n : ℕ),
  Hom.f
      ({ f := fun (x : ℕ) ↦ 𝟙 ((Functor.obj truncate (C.augment f w)).X x), comm' := sorry :
          Hom (Functor.obj truncate (C.augment f w)) C } ≫
        { f := fun (x : ℕ) ↦ 𝟙 (C.X x), comm' := sorry })
      (n + 1) =
    Hom.f (𝟙 (Functor.obj truncate (C.augment f w))) (n + 1)
Tactic: cases i
Cleared hypotheses: i
-- Lean 4 proof step from Mathlib.Algebra.Homology.Augment (goal)
Goal:
V : Type u
inst : Category.{v, u} V
inst_1 : HasZeroMorphisms.{v, u} V
C : CochainComplex.{v, u, 0} V ℕ
X : V
f : X ⟶ C.X 0
w : Eq (α := X ⟶ C.X 1) (f ≫ C.d 0 1) 0
⊢ { f := fun (x : ℕ) ↦ 𝟙 (C.X x), comm' := ⋯ } ≫
      { f := fun (x : ℕ) ↦ 𝟙 ((Functor.obj truncate (C.augment f w)).X x), comm' := ⋯ :
        Hom (Functor.obj truncate (C.augment f w)) C } =
    𝟙 C
have h : ∀ (i : ℕ),
  Hom.f
      ({ f := fun (x : ℕ) ↦ 𝟙 (C.X x), comm' := sorry } ≫
        { f := fun (x : ℕ) ↦ 𝟙 ((Functor.obj truncate (C.augment f w)).X x), comm' := sorry :
          Hom (Functor.obj truncate (C.augment f w)) C })
      i =
    Hom.f (𝟙 C) i
Tactic: ext i
-- Lean 4 proof step from Mathlib.Algebra.Homology.Augment (goal)
Goal:
V : Type u
inst : Category.{v, u} V
inst_1 : HasZeroMorphisms.{v, u} V
C : CochainComplex.{v, u, 0} V ℕ
X : V
f : X ⟶ C.X 0
w : Eq (α := X ⟶ C.X 1) (f ≫ C.d 0 1) 0
i : ℕ
⊢ Hom.f
      ({ f := fun (x : ℕ) ↦ 𝟙 (C.X x), comm' := ⋯ } ≫
        { f := fun (x : ℕ) ↦ 𝟙 ((Functor.obj truncate (C.augment f w)).X x), comm' := ⋯ :
          Hom (Functor.obj truncate (C.augment f w)) C })
      i =
    Hom.f (𝟙 C) i
have h : Eq (α := C.X 0 ⟶ C.X 0)
  (Hom.f
    ({ f := fun (x : ℕ) ↦ 𝟙 (C.X x), comm' := sorry } ≫
      { f := fun (x : ℕ) ↦ 𝟙 ((Functor.obj truncate (C.augment f w)).X x), comm' := sorry :
        Hom (Functor.obj truncate (C.augment f w)) C })
    0)
  (Hom.f (𝟙 C) 0)
Tactic: cases i
Cleared hypotheses: i
-- Lean 4 proof step from Mathlib.Algebra.Homology.Augment (goal)
Goal:
V : Type u
inst : Category.{v, u} V
inst_1 : HasZeroMorphisms.{v, u} V
C : CochainComplex.{v, u, 0} V ℕ
X : V
f : X ⟶ C.X 0
w : Eq (α := X ⟶ C.X 1) (f ≫ C.d 0 1) 0
i : ℕ
h :
  Eq (α := C.X 0 ⟶ C.X 0)
    (Hom.f
      ({ f := fun (x : ℕ) ↦ 𝟙 (C.X x), comm' := ⋯ } ≫
        { f := fun (x : ℕ) ↦ 𝟙 ((Functor.obj truncate (C.augment f w)).X x), comm' := ⋯ :
          Hom (Functor.obj truncate (C.augment f w)) C })
      0)
    (Hom.f (𝟙 C) 0)
⊢ Hom.f
      ({ f := fun (x : ℕ) ↦ 𝟙 (C.X x), comm' := ⋯ } ≫
        { f := fun (x : ℕ) ↦ 𝟙 ((Functor.obj truncate (C.augment f w)).X x), comm' := ⋯ :
          Hom (Functor.obj truncate (C.augment f w)) C })
      i =
    Hom.f (𝟙 C) i
have h₁ : ∀ (n : ℕ),
  Hom.f
      ({ f := fun (x : ℕ) ↦ 𝟙 (C.X x), comm' := sorry } ≫
        { f := fun (x : ℕ) ↦ 𝟙 ((Functor.obj truncate (C.augment f w)).X x), comm' := sorry :
          Hom (Functor.obj truncate (C.augment f w)) C })
      (n + 1) =
    Hom.f (𝟙 C) (n + 1)
Tactic: cases i
Cleared hypotheses: i
-- Lean 4 proof step from Mathlib.Algebra.Homology.Augment (goal)
Goal:
V : Type u
inst : Category.{v, u} V
inst_1 : HasZeroMorphisms.{v, u} V
C : CochainComplex.{v, u, 0} V ℕ
i : ℕ
⊢ Eq (α := C.X 0 ⟶ C.X (i + 2)) (C.d 0 (i + 2)) 0
have a : ¬(ComplexShape.up ℕ).Rel 0 (i + 2)
Tactic: rw [C.shape]