awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Functor.KanExtension.Preserves (goal)
Goal:
A : Type u_1
B : Type u_2
C : Type u_3
D : Type u_4
inst : Category.{v_1, u_1} A
inst_1 : Category.{v_2, u_2} B
inst_2 : Category.{v_3, u_3} C
inst_3 : Category.{v_4, u_4} D
G : Functor.{v_2, v_4, u_2, u_4} B D
F_1 : Functor.{v_1, v_2, u_1, u_2} A B
L : Functor.{v_1, v_3, u_1, u_3} A C
inst_4 : G.PreservesLeftKanExtensions L
inst_5 : ∀ (F : Functor.{v_1, v_2, u_1, u_2} A B), L.HasLeftKanExtension F
inst_6 : ∀ (F : Functor.{v_1, v_4, u_1, u_4} A D), L.HasLeftKanExtension F
F F' : Functor.{v_1, v_2, u_1, u_2} A B
η : F ⟶ F'
⊢ (whiskerRight (L.leftKanExtensionUnit F) G ≫ (L.associator (L.leftKanExtension F) G).hom) ≫
      L.whiskerLeft
        ((L.lan ⋙ obj (whiskeringRight C B D) G).map η ≫
          Iso.hom
            ((fun (F : Functor.{v_1, v_2, u_1, u_2} A B) ↦
                (G.leftKanExtensionCompIsoOfPreserves F L : L.leftKanExtension F ⋙ G ≅ L.leftKanExtension (F ⋙ G)))
              F')) =
    (whiskerRight (L.leftKanExtensionUnit F) G ≫ (L.associator (L.leftKanExtension F) G).hom) ≫
      L.whiskerLeft
        (Iso.hom
            ((fun (F : Functor.{v_1, v_2, u_1, u_2} A B) ↦
                (G.leftKanExtensionCompIsoOfPreserves F L : L.leftKanExtension F ⋙ G ≅ L.leftKanExtension (F ⋙ G)))
              F) ≫
          (obj (whiskeringRight A B D) G ⋙ L.lan).map η)
have hγ : Eq (α := F ⋙ G ⟶ L ⋙ L.leftKanExtension (F' ⋙ G))
  ((whiskerRight (L.leftKanExtensionUnit F) G ≫ (L.associator (L.leftKanExtension F) G).hom) ≫
      CategoryStruct.comp (Y := L ⋙ L.leftKanExtension F' ⋙ G)
        (L.whiskerLeft
          (whiskerRight
            ((L.leftKanExtension F).descOfIsLeftKanExtension (L.leftKanExtensionUnit F) (L.leftKanExtension F')
              (η ≫ L.leftKanExtensionUnit F'))
            G))
        (L.whiskerLeft (Iso.hom (G.leftKanExtensionCompIsoOfPreserves F' L))) :
    F ⋙ G ⟶ L ⋙ L.leftKanExtension (F' ⋙ G))
  ((whiskerRight (L.leftKanExtensionUnit F) G ≫ (L.associator (L.leftKanExtension F) G).hom) ≫
      CategoryStruct.comp (Y := L ⋙ L.leftKanExtension (F ⋙ G))
        (L.whiskerLeft (Iso.hom (G.leftKanExtensionCompIsoOfPreserves F L)) :
          L ⋙ L.leftKanExtension F ⋙ G ⟶ L ⋙ L.leftKanExtension (F ⋙ G))
        (L.whiskerLeft
            ((L.leftKanExtension (F ⋙ G)).descOfIsLeftKanExtension (L.leftKanExtensionUnit (F ⋙ G))
              (L.leftKanExtension (F' ⋙ G)) (whiskerRight η G ≫ L.leftKanExtensionUnit (F' ⋙ G))) :
          L ⋙ L.leftKanExtension (F ⋙ G) ⟶ L ⋙ L.leftKanExtension (F' ⋙ G)) :
    F ⋙ G ⟶ L ⋙ L.leftKanExtension (F' ⋙ G))
Tactic: dsimp [lan]
-- Lean 4 proof step from Mathlib.CategoryTheory.Functor.KanExtension.Preserves (goal)
Goal:
A : Type u_1
B : Type u_2
C : Type u_3
D : Type u_4
inst : Category.{v_1, u_1} A
inst_1 : Category.{v_2, u_2} B
inst_2 : Category.{v_3, u_3} C
inst_3 : Category.{v_4, u_4} D
G : Functor.{v_2, v_4, u_2, u_4} B D
F : Functor.{v_1, v_2, u_1, u_2} A B
L : Functor.{v_1, v_3, u_1, u_3} A C
F' : Functor.{v_3, v_2, u_3, u_2} C B
α : L ⋙ F' ⟶ F
inst_4 : F'.IsRightKanExtension α
h_1 : (F' ⋙ G).IsRightKanExtension ((L.associator F' G).inv ≫ whiskerRight α G)
F'' : Functor.{v_3, v_2, u_3, u_2} C B
α' : L ⋙ F'' ⟶ F
h : F''.IsRightKanExtension α'
⊢ L.whiskerLeft (isoWhiskerRight (F'.rightKanExtensionUnique α F'' α') G).hom ≫
      (L.associator F'' G).inv ≫ whiskerRight α' G =
    (L.associator F' G).inv ≫ whiskerRight α G
have w : ∀ (x : A),
  (L.whiskerLeft (isoWhiskerRight (F'.rightKanExtensionUnique α F'' α') G).hom ≫
          (L.associator F'' G).inv ≫ whiskerRight α' G).app
      x =
    ((L.associator F' G).inv ≫ whiskerRight α G).app x
Tactic: ext x
-- Lean 4 proof step from Mathlib.CategoryTheory.Functor.KanExtension.Preserves (goal)
Goal:
A : Type u_1
B : Type u_2
C : Type u_3
D : Type u_4
inst : Category.{v_1, u_1} A
inst_1 : Category.{v_2, u_2} B
inst_2 : Category.{v_3, u_3} C
inst_3 : Category.{v_4, u_4} D
G : Functor.{v_2, v_4, u_2, u_4} B D
F_1 : Functor.{v_1, v_2, u_1, u_2} A B
L : Functor.{v_1, v_3, u_1, u_3} A C
inst_4 : G.PreservesRightKanExtensions L
inst_5 : ∀ (F : Functor.{v_1, v_2, u_1, u_2} A B), L.HasRightKanExtension F
inst_6 : ∀ (F : Functor.{v_1, v_4, u_1, u_4} A D), L.HasRightKanExtension F
F F' : Functor.{v_1, v_2, u_1, u_2} A B
η : F ⟶ F'
⊢ (L.ran ⋙ obj (whiskeringRight C B D) G).map η ≫
      Iso.hom
        ((fun (F : Functor.{v_1, v_2, u_1, u_2} A B) ↦
            (G.rightKanExtensionCompIsoOfPreserves F L : L.rightKanExtension F ⋙ G ≅ L.rightKanExtension (F ⋙ G)))
          F') =
    Iso.hom
        ((fun (F : Functor.{v_1, v_2, u_1, u_2} A B) ↦
            (G.rightKanExtensionCompIsoOfPreserves F L : L.rightKanExtension F ⋙ G ≅ L.rightKanExtension (F ⋙ G)))
          F) ≫
      (obj (whiskeringRight A B D) G ⋙ L.ran).map η
have hγ : CategoryStruct.comp (Y := L ⋙ L.rightKanExtension (F' ⋙ G))
    (L.whiskerLeft
      ((L.ran ⋙ obj (whiskeringRight C B D) G).map η ≫
        Iso.hom
          ((fun (F : Functor.{v_1, v_2, u_1, u_2} A B) ↦
              (G.rightKanExtensionCompIsoOfPreserves F L : L.rightKanExtension F ⋙ G ≅ L.rightKanExtension (F ⋙ G)))
            F')))
    (L.rightKanExtensionCounit (F' ⋙ G)) =
  L.whiskerLeft
      (Iso.hom
          ((fun (F : Functor.{v_1, v_2, u_1, u_2} A B) ↦
              (G.rightKanExtensionCompIsoOfPreserves F L : L.rightKanExtension F ⋙ G ≅ L.rightKanExtension (F ⋙ G)))
            F) ≫
        (obj (whiskeringRight A B D) G ⋙ L.ran).map η) ≫
    L.rightKanExtensionCounit (F' ⋙ G)
Tactic: apply hom_ext_of_isRightKanExtension
        (L.rightKanExtension <| F' ⋙ G)
        (L.rightKanExtensionCounit <| F' ⋙ G)
-- Lean 4 proof step from Mathlib.CategoryTheory.Functor.KanExtension.Preserves (goal)
Goal:
A : Type u_1
B : Type u_2
C : Type u_3
D : Type u_4
inst : Category.{v_1, u_1} A
inst_1 : Category.{v_2, u_2} B
inst_2 : Category.{v_3, u_3} C
inst_3 : Category.{v_4, u_4} D
G : Functor.{v_2, v_4, u_2, u_4} B D
F_1 : Functor.{v_1, v_2, u_1, u_2} A B
L : Functor.{v_1, v_3, u_1, u_3} A C
inst_4 : G.PreservesRightKanExtensions L
inst_5 : ∀ (F : Functor.{v_1, v_2, u_1, u_2} A B), L.HasRightKanExtension F
inst_6 : ∀ (F : Functor.{v_1, v_4, u_1, u_4} A D), L.HasRightKanExtension F
F F' : Functor.{v_1, v_2, u_1, u_2} A B
η : F ⟶ F'
⊢ CategoryStruct.comp (Y := L ⋙ L.rightKanExtension (F' ⋙ G))
      (L.whiskerLeft
        ((L.ran ⋙ obj (whiskeringRight C B D) G).map η ≫
          Iso.hom
            ((fun (F : Functor.{v_1, v_2, u_1, u_2} A B) ↦
                (G.rightKanExtensionCompIsoOfPreserves F L : L.rightKanExtension F ⋙ G ≅ L.rightKanExtension (F ⋙ G)))
              F')))
      (L.rightKanExtensionCounit (F' ⋙ G)) =
    L.whiskerLeft
        (Iso.hom
            ((fun (F : Functor.{v_1, v_2, u_1, u_2} A B) ↦
                (G.rightKanExtensionCompIsoOfPreserves F L : L.rightKanExtension F ⋙ G ≅ L.rightKanExtension (F ⋙ G)))
              F) ≫
          (obj (whiskeringRight A B D) G ⋙ L.ran).map η) ≫
      L.rightKanExtensionCounit (F' ⋙ G)
have hγ : Eq (α := L ⋙ L.rightKanExtension F ⋙ G ⟶ F' ⋙ G)
  (CategoryStruct.comp (Y := L ⋙ L.rightKanExtension (F' ⋙ G))
    (L.whiskerLeft
          (whiskerRight
            ((L.rightKanExtension F').liftOfIsRightKanExtension (L.rightKanExtensionCounit F') (L.rightKanExtension F)
              (L.rightKanExtensionCounit F ≫ η))
            G) ≫
        L.whiskerLeft (Iso.hom (G.rightKanExtensionCompIsoOfPreserves F' L)) :
      L ⋙ L.rightKanExtension F ⋙ G ⟶ L ⋙ L.rightKanExtension (F' ⋙ G))
    (L.rightKanExtensionCounit (F' ⋙ G) : L ⋙ L.rightKanExtension (F' ⋙ G) ⟶ F' ⋙ G))
  (CategoryStruct.comp (Y := L ⋙ L.rightKanExtension (F ⋙ G))
      (L.whiskerLeft (Iso.hom (G.rightKanExtensionCompIsoOfPreserves F L)) :
        L ⋙ L.rightKanExtension F ⋙ G ⟶ L ⋙ L.rightKanExtension (F ⋙ G))
      (L.whiskerLeft
          ((L.rightKanExtension (F' ⋙ G)).liftOfIsRightKanExtension (L.rightKanExtensionCounit (F' ⋙ G))
            (L.rightKanExtension (F ⋙ G)) (L.rightKanExtensionCounit (F ⋙ G) ≫ whiskerRight η G)) :
        L ⋙ L.rightKanExtension (F ⋙ G) ⟶ L ⋙ L.rightKanExtension (F' ⋙ G)) ≫
    L.rightKanExtensionCounit (F' ⋙ G))
Tactic: dsimp [ran]
-- Lean 4 proof step from Mathlib.CategoryTheory.Functor.KanExtension.Preserves (goal)
Goal:
A : Type u_1
B : Type u_2
C : Type u_3
D : Type u_4
inst : Category.{v_1, u_1} A
inst_1 : Category.{v_2, u_2} B
inst_2 : Category.{v_3, u_3} C
inst_3 : Category.{v_4, u_4} D
G : Functor.{v_2, v_4, u_2, u_4} B D
F_1 : Functor.{v_1, v_2, u_1, u_2} A B
L : Functor.{v_1, v_3, u_1, u_3} A C
inst_4 : G.PreservesRightKanExtensions L
inst_5 : ∀ (F : Functor.{v_1, v_2, u_1, u_2} A B), L.HasRightKanExtension F
inst_6 : ∀ (F : Functor.{v_1, v_4, u_1, u_4} A D), L.HasRightKanExtension F
F F' : Functor.{v_1, v_2, u_1, u_2} A B
η : F ⟶ F'
⊢ Eq (α := L ⋙ L.rightKanExtension F ⋙ G ⟶ F' ⋙ G)
    (CategoryStruct.comp (Y := L ⋙ L.rightKanExtension (F' ⋙ G))
      (L.whiskerLeft
            (whiskerRight
              ((L.rightKanExtension F').liftOfIsRightKanExtension (L.rightKanExtensionCounit F') (L.rightKanExtension F)
                (L.rightKanExtensionCounit F ≫ η))
              G) ≫
          L.whiskerLeft (Iso.hom (G.rightKanExtensionCompIsoOfPreserves F' L)) :
        L ⋙ L.rightKanExtension F ⋙ G ⟶ L ⋙ L.rightKanExtension (F' ⋙ G))
      (L.rightKanExtensionCounit (F' ⋙ G) : L ⋙ L.rightKanExtension (F' ⋙ G) ⟶ F' ⋙ G))
    (CategoryStruct.comp (Y := L ⋙ L.rightKanExtension (F ⋙ G))
        (L.whiskerLeft (Iso.hom (G.rightKanExtensionCompIsoOfPreserves F L)) :
          L ⋙ L.rightKanExtension F ⋙ G ⟶ L ⋙ L.rightKanExtension (F ⋙ G))
        (L.whiskerLeft
            ((L.rightKanExtension (F' ⋙ G)).liftOfIsRightKanExtension (L.rightKanExtensionCounit (F' ⋙ G))
              (L.rightKanExtension (F ⋙ G)) (L.rightKanExtensionCounit (F ⋙ G) ≫ whiskerRight η G)) :
          L ⋙ L.rightKanExtension (F ⋙ G) ⟶ L ⋙ L.rightKanExtension (F' ⋙ G)) ≫
      L.rightKanExtensionCounit (F' ⋙ G))
have hγ : ∀ (x : A),
  Eq (α := (L ⋙ L.rightKanExtension F ⋙ G).obj x ⟶ (F' ⋙ G).obj x)
    (NatTrans.app
      (CategoryStruct.comp (Y := L ⋙ L.rightKanExtension (F' ⋙ G))
        (L.whiskerLeft
              (whiskerRight
                ((L.rightKanExtension F').liftOfIsRightKanExtension (L.rightKanExtensionCounit F')
                  (L.rightKanExtension F) (L.rightKanExtensionCounit F ≫ η))
                G) ≫
            L.whiskerLeft (Iso.hom (G.rightKanExtensionCompIsoOfPreserves F' L)) :
          L ⋙ L.rightKanExtension F ⋙ G ⟶ L ⋙ L.rightKanExtension (F' ⋙ G))
        (L.rightKanExtensionCounit (F' ⋙ G) : L ⋙ L.rightKanExtension (F' ⋙ G) ⟶ F' ⋙ G))
      x)
    (NatTrans.app
      (CategoryStruct.comp (Y := L ⋙ L.rightKanExtension (F ⋙ G))
          (L.whiskerLeft (Iso.hom (G.rightKanExtensionCompIsoOfPreserves F L)) :
            L ⋙ L.rightKanExtension F ⋙ G ⟶ L ⋙ L.rightKanExtension (F ⋙ G))
          (L.whiskerLeft
              ((L.rightKanExtension (F' ⋙ G)).liftOfIsRightKanExtension (L.rightKanExtensionCounit (F' ⋙ G))
                (L.rightKanExtension (F ⋙ G)) (L.rightKanExtensionCounit (F ⋙ G) ≫ whiskerRight η G)) :
            L ⋙ L.rightKanExtension (F ⋙ G) ⟶ L ⋙ L.rightKanExtension (F' ⋙ G)) ≫
        L.rightKanExtensionCounit (F' ⋙ G))
      x)
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Functor.KanExtension.Preserves (goal)
Goal:
A : Type u_1
B : Type u_2
C : Type u_3
D : Type u_4
inst : Category.{v_1, u_1} A
inst_1 : Category.{v_2, u_2} B
inst_2 : Category.{v_3, u_3} C
inst_3 : Category.{v_4, u_4} D
G : Functor.{v_2, v_4, u_2, u_4} B D
F_1 : Functor.{v_1, v_2, u_1, u_2} A B
L : Functor.{v_1, v_3, u_1, u_3} A C
inst_4 : G.PreservesRightKanExtensions L
inst_5 : ∀ (F : Functor.{v_1, v_2, u_1, u_2} A B), L.HasRightKanExtension F
inst_6 : ∀ (F : Functor.{v_1, v_4, u_1, u_4} A D), L.HasRightKanExtension F
F F' : Functor.{v_1, v_2, u_1, u_2} A B
η : F ⟶ F'
x : A
⊢ Eq (α := (L ⋙ L.rightKanExtension F ⋙ G).obj x ⟶ (F' ⋙ G).obj x)
    (NatTrans.app
      (CategoryStruct.comp (Y := L ⋙ L.rightKanExtension (F' ⋙ G))
        (L.whiskerLeft
              (whiskerRight
                ((L.rightKanExtension F').liftOfIsRightKanExtension (L.rightKanExtensionCounit F')
                  (L.rightKanExtension F) (L.rightKanExtensionCounit F ≫ η))
                G) ≫
            L.whiskerLeft (Iso.hom (G.rightKanExtensionCompIsoOfPreserves F' L)) :
          L ⋙ L.rightKanExtension F ⋙ G ⟶ L ⋙ L.rightKanExtension (F' ⋙ G))
        (L.rightKanExtensionCounit (F' ⋙ G) : L ⋙ L.rightKanExtension (F' ⋙ G) ⟶ F' ⋙ G))
      x)
    (NatTrans.app
      (CategoryStruct.comp (Y := L ⋙ L.rightKanExtension (F ⋙ G))
          (L.whiskerLeft (Iso.hom (G.rightKanExtensionCompIsoOfPreserves F L)) :
            L ⋙ L.rightKanExtension F ⋙ G ⟶ L ⋙ L.rightKanExtension (F ⋙ G))
          (L.whiskerLeft
              ((L.rightKanExtension (F' ⋙ G)).liftOfIsRightKanExtension (L.rightKanExtensionCounit (F' ⋙ G))
                (L.rightKanExtension (F ⋙ G)) (L.rightKanExtensionCounit (F ⋙ G) ≫ whiskerRight η G)) :
            L ⋙ L.rightKanExtension (F ⋙ G) ⟶ L ⋙ L.rightKanExtension (F' ⋙ G)) ≫
        L.rightKanExtensionCounit (F' ⋙ G))
      x)
have hγ : Eq (α := G.obj ((L.rightKanExtension F).obj (L.obj x)) ⟶ G.obj (F'.obj x))
  (G.map
    (NatTrans.app
        ((L.rightKanExtension F').liftOfIsRightKanExtension (L.rightKanExtensionCounit F') (L.rightKanExtension F)
          (L.rightKanExtensionCounit F ≫ η))
        (L.obj x) ≫
      NatTrans.app (L.rightKanExtensionCounit F') x))
  (G.map (NatTrans.app (L.rightKanExtensionCounit F) x ≫ η.app x))
Tactic: simp only [comp_obj, Category.assoc, rightKanExtensionCompIsoOfPreserves_hom_fac,
        NatTrans.comp_app, whiskerLeft_app, whiskerRight_app, associator_inv_app, Category.id_comp,
        liftOfIsRightKanExtension_fac, rightKanExtensionCompIsoOfPreserves_hom_fac_assoc,
        ← G.map_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Functor.OfSequence (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : ℕ → C
f : (n : ℕ) → X n ⟶ X (n + 1)
i j : ℕ
h : i = j
⊢ X i = X j
have rw : X j = X j
Tactic: rw [h]
-- Lean 4 proof step from Mathlib.CategoryTheory.Functor.OfSequence (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : ℕ → C
f : (n : ℕ) → X n ⟶ X (n + 1)
i j : ℕ
h : i = j
⊢ X (j + 1) = X (i + 1)
have rw : X (j + 1) = X (j + 1)
Tactic: rw [h]
-- Lean 4 proof step from Mathlib.CategoryTheory.Functor.OfSequence (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : ℕ → C
f : (n : ℕ) → X n ⟶ X (n + 1)
i j : ℕ
h : i = j
⊢ f i = eqToHom ⋯ ≫ f j ≫ eqToHom ⋯
have subst : f i = eqToHom sorry ≫ f i ≫ eqToHom sorry
Tactic: subst h
Cleared hypotheses: j, h
-- Lean 4 proof step from Mathlib.CategoryTheory.Functor.OfSequence (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : ℕ → C
f : (n : ℕ) → X n ⟶ X (n + 1)
i : ℕ
⊢ map f i i ⋯ = 𝟙 (X i)
have revert : ∀ {X : ℕ → C} (f : (n : ℕ) → X n ⟶ X (n + 1)), map f i i sorry = 𝟙 (X i)
Tactic: revert X f
Cleared hypotheses: X, f
-- Lean 4 proof step from Mathlib.CategoryTheory.Functor.OfSequence (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
i : ℕ
⊢ ∀ {X : ℕ → C} (f : (n : ℕ) → X n ⟶ X (n + 1)), map f i i ⋯ = 𝟙 (X i)
have zero : ∀ {X : ℕ → C} (f : (n : ℕ) → X n ⟶ X (n + 1)), map f 0 0 sorry = 𝟙 (X 0)
Tactic: induction i with
  | zero => intros; rfl
  | succ _ hi =>
      intro X f
      apply hi
Cleared hypotheses: i
-- Lean 4 proof step from Mathlib.CategoryTheory.Functor.OfSequence (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
i : ℕ
zero : ∀ {X : ℕ → C} (f : (n : ℕ) → X n ⟶ X (n + 1)), map f 0 0 ⋯ = 𝟙 (X 0)
⊢ ∀ {X : ℕ → C} (f : (n : ℕ) → X n ⟶ X (n + 1)), map f i i ⋯ = 𝟙 (X i)
have succ : ∀ (n : ℕ) (hi : ∀ {X : ℕ → C} (f : (n : ℕ) → X n ⟶ X (n + 1)), map f n n sorry = 𝟙 (X n)) {X : ℕ → C}
  (f : (n : ℕ) → X n ⟶ X (n + 1)), map f (n + 1) (n + 1) sorry = 𝟙 (X (n + 1))
Tactic: induction i with
  | zero => intros; rfl
  | succ _ hi =>
      intro X f
      apply hi
Cleared hypotheses: i
-- Lean 4 proof step from Mathlib.CategoryTheory.Functor.OfSequence (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : ℕ → C
f : (n : ℕ) → X n ⟶ X (n + 1)
i : ℕ
⊢ map f i (i + 1) ⋯ = f i
have revert : ∀ {X : ℕ → C} (f : (n : ℕ) → X n ⟶ X (n + 1)), map f i (i + 1) sorry = f i
Tactic: revert X f
Cleared hypotheses: X, f
-- Lean 4 proof step from Mathlib.CategoryTheory.Functor.OfSequence (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
i : ℕ
⊢ ∀ {X : ℕ → C} (f : (n : ℕ) → X n ⟶ X (n + 1)), map f i (i + 1) ⋯ = f i
have zero : ∀ {X : ℕ → C} (f : (n : ℕ) → X n ⟶ X (n + 1)),
  Eq (α := X 0 ⟶ X (HAdd.hAdd (α := ℕ) 0 1)) (map f 0 (HAdd.hAdd (α := ℕ) 0 1) sorry) (f 0)
Tactic: induction i with
  | zero => intros; rfl
  | succ _ hi =>
      intro X f
      apply hi
Cleared hypotheses: i
-- Lean 4 proof step from Mathlib.CategoryTheory.Functor.OfSequence (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
i : ℕ
zero :
  ∀ {X : ℕ → C} (f : (n : ℕ) → X n ⟶ X (n + 1)),
    Eq (α := X 0 ⟶ X (HAdd.hAdd (α := ℕ) 0 1)) (map f 0 (HAdd.hAdd (α := ℕ) 0 1) ⋯) (f 0)
⊢ ∀ {X : ℕ → C} (f : (n : ℕ) → X n ⟶ X (n + 1)), map f i (i + 1) ⋯ = f i
have succ : ∀ (n : ℕ) (hi : ∀ {X : ℕ → C} (f : (n : ℕ) → X n ⟶ X (n + 1)), map f n (n + 1) sorry = f n) {X : ℕ → C}
  (f : (n : ℕ) → X n ⟶ X (n + 1)), Eq (α := X (n + 1) ⟶ X (n + 1 + 1)) (map f (n + 1) (n + 1 + 1) sorry) (f (n + 1))
Tactic: induction i with
  | zero => intros; rfl
  | succ _ hi =>
      intro X f
      apply hi
Cleared hypotheses: i
-- Lean 4 proof step from Mathlib.CategoryTheory.Functor.OfSequence (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : ℕ → C
f : (n : ℕ) → X n ⟶ X (n + 1)
k : ℕ
hij : LE.le (α := ℕ) 0 0
hjk : 0 ≤ k
⊢ map f 0 k ⋯ = map f 0 0 hij ≫ map f 0 k hjk
have zero : map f 0 k sorry = 𝟙 (X 0) ≫ map f 0 k hjk
Tactic: rw [map_id, id_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Functor.OfSequence (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : ℕ → C
f : (n : ℕ) → X n ⟶ X (n + 1)
k : ℕ
hij : LE.le (α := ℕ) 0 0
hjk : 0 ≤ k
zero : map f 0 k ⋯ = 𝟙 (X 0) ≫ map f 0 k hjk
⊢ map f 0 k ⋯ = map f 0 0 hij ≫ map f 0 k hjk
have zero₁ : map f 0 k sorry = map f 0 k hjk
Tactic: rw [map_id, id_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Functor.OfSequence (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
j : ℕ
hj :
  ∀ {X : ℕ → C} (f : (n : ℕ) → X n ⟶ X (n + 1)) (k : ℕ) (hij : 0 ≤ j) (hjk : j ≤ k),
    map f 0 k ⋯ = map f 0 j hij ≫ map f j k hjk
X : ℕ → C
f : (n : ℕ) → X n ⟶ X (n + 1)
k : ℕ
hij : 0 ≤ j + 1
hjk : j + 1 ≤ k
⊢ map f 0 k ⋯ = map f 0 (j + 1) hij ≫ map f (j + 1) k hjk
have zero : ∀ (hjk : j + 1 ≤ 0), map f 0 0 sorry = map f 0 (j + 1) hij ≫ map f (j + 1) 0 hjk
Tactic: obtain (_ | _ | k) := k
Cleared hypotheses: k, hjk
-- Lean 4 proof step from Mathlib.CategoryTheory.Functor.OfSequence (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
j : ℕ
hj :
  ∀ {X : ℕ → C} (f : (n : ℕ) → X n ⟶ X (n + 1)) (k : ℕ) (hij : 0 ≤ j) (hjk : j ≤ k),
    map f 0 k ⋯ = map f 0 j hij ≫ map f j k hjk
X : ℕ → C
f : (n : ℕ) → X n ⟶ X (n + 1)
k : ℕ
hij : 0 ≤ j + 1
hjk : j + 1 ≤ k
zero : ∀ (hjk : j + 1 ≤ 0), map f 0 0 ⋯ = map f 0 (j + 1) hij ≫ map f (j + 1) 0 hjk
⊢ map f 0 k ⋯ = map f 0 (j + 1) hij ≫ map f (j + 1) k hjk
have zero₁ : ∀ (hjk : j + 1 ≤ HAdd.hAdd (α := ℕ) 0 1),
  map f 0 (HAdd.hAdd (α := ℕ) 0 1) sorry = map f 0 (j + 1) hij ≫ map f (j + 1) (HAdd.hAdd (α := ℕ) 0 1) hjk
Tactic: obtain (_ | _ | k) := k
Cleared hypotheses: k, hjk
-- Lean 4 proof step from Mathlib.CategoryTheory.Functor.OfSequence (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
j : ℕ
hj :
  ∀ {X : ℕ → C} (f : (n : ℕ) → X n ⟶ X (n + 1)) (k : ℕ) (hij : 0 ≤ j) (hjk : j ≤ k),
    map f 0 k ⋯ = map f 0 j hij ≫ map f j k hjk
X : ℕ → C
f : (n : ℕ) → X n ⟶ X (n + 1)
k : ℕ
hij : 0 ≤ j + 1
hjk : j + 1 ≤ k
zero : ∀ (hjk : j + 1 ≤ 0), map f 0 0 ⋯ = map f 0 (j + 1) hij ≫ map f (j + 1) 0 hjk
zero₁ :
  ∀ (hjk : j + 1 ≤ HAdd.hAdd (α := ℕ) 0 1),
    map f 0 (HAdd.hAdd (α := ℕ) 0 1) ⋯ = map f 0 (j + 1) hij ≫ map f (j + 1) (HAdd.hAdd (α := ℕ) 0 1) hjk
⊢ map f 0 k ⋯ = map f 0 (j + 1) hij ≫ map f (j + 1) k hjk
have zero₂ : ∀ (k : ℕ) (hjk : j + 1 ≤ k + 1 + 1), map f 0 (k + 1 + 1) sorry = map f 0 (j + 1) hij ≫ map f (j + 1) (k + 1 + 1) hjk
Tactic: obtain (_ | _ | k) := k
Cleared hypotheses: k, hjk
-- Lean 4 proof step from Mathlib.CategoryTheory.Functor.OfSequence (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
j : ℕ
hj :
  ∀ {X : ℕ → C} (f : (n : ℕ) → X n ⟶ X (n + 1)) (k : ℕ) (hij : 0 ≤ j) (hjk : j ≤ k),
    map f 0 k ⋯ = map f 0 j hij ≫ map f j k hjk
X : ℕ → C
f : (n : ℕ) → X n ⟶ X (n + 1)
hij : 0 ≤ j + 1
hjk : j + 1 ≤ HAdd.hAdd (α := ℕ) 0 1
⊢ map f 0 (HAdd.hAdd (α := ℕ) 0 1) ⋯ = map f 0 (j + 1) hij ≫ map f (j + 1) (HAdd.hAdd (α := ℕ) 0 1) hjk
have zero : ∀
  (hj :
    ∀ {X : ℕ → C} (f : (n : ℕ) → X n ⟶ X (n + 1)) (k : ℕ) (hij : LE.le (α := ℕ) 0 0) (hjk : 0 ≤ k),
      map f 0 k sorry = map f 0 0 hij ≫ map f 0 k hjk)
  (hij : 0 ≤ HAdd.hAdd (α := ℕ) 0 1) (hjk : HAdd.hAdd (α := ℕ) 0 1 ≤ HAdd.hAdd (α := ℕ) 0 1),
  map f 0 (HAdd.hAdd (α := ℕ) 0 1) sorry =
    map f 0 (HAdd.hAdd (α := ℕ) 0 1) hij ≫ map f (HAdd.hAdd (α := ℕ) 0 1) (HAdd.hAdd (α := ℕ) 0 1) hjk
Tactic: obtain rfl : j = 0 := by lia
Cleared hypotheses: j, hj, hij, hjk
-- Lean 4 proof step from Mathlib.CategoryTheory.Functor.OfSequence (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : ℕ → C
f : (n : ℕ) → X n ⟶ X (n + 1)
hj :
  ∀ {X : ℕ → C} (f : (n : ℕ) → X n ⟶ X (n + 1)) (k : ℕ) (hij : LE.le (α := ℕ) 0 0) (hjk : 0 ≤ k),
    map f 0 k ⋯ = map f 0 0 hij ≫ map f 0 k hjk
hij : 0 ≤ HAdd.hAdd (α := ℕ) 0 1
hjk : HAdd.hAdd (α := ℕ) 0 1 ≤ HAdd.hAdd (α := ℕ) 0 1
⊢ map f 0 (HAdd.hAdd (α := ℕ) 0 1) ⋯ =
    map f 0 (HAdd.hAdd (α := ℕ) 0 1) hij ≫ map f (HAdd.hAdd (α := ℕ) 0 1) (HAdd.hAdd (α := ℕ) 0 1) hjk
have zero : map f 0 (HAdd.hAdd (α := ℕ) 0 1) sorry = map f 0 (HAdd.hAdd (α := ℕ) 0 1) hij ≫ 𝟙 (X (HAdd.hAdd (α := ℕ) 0 1))
Tactic: rw [map_id, comp_id]
-- Lean 4 proof step from Mathlib.CategoryTheory.Functor.OfSequence (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : ℕ → C
f : (n : ℕ) → X n ⟶ X (n + 1)
hj :
  ∀ {X : ℕ → C} (f : (n : ℕ) → X n ⟶ X (n + 1)) (k : ℕ) (hij : LE.le (α := ℕ) 0 0) (hjk : 0 ≤ k),
    map f 0 k ⋯ = map f 0 0 hij ≫ map f 0 k hjk
hij : 0 ≤ HAdd.hAdd (α := ℕ) 0 1
hjk : HAdd.hAdd (α := ℕ) 0 1 ≤ HAdd.hAdd (α := ℕ) 0 1
zero : map f 0 (HAdd.hAdd (α := ℕ) 0 1) ⋯ = map f 0 (HAdd.hAdd (α := ℕ) 0 1) hij ≫ 𝟙 (X (HAdd.hAdd (α := ℕ) 0 1))
⊢ map f 0 (HAdd.hAdd (α := ℕ) 0 1) ⋯ =
    map f 0 (HAdd.hAdd (α := ℕ) 0 1) hij ≫ map f (HAdd.hAdd (α := ℕ) 0 1) (HAdd.hAdd (α := ℕ) 0 1) hjk
have zero₁ : map f 0 (HAdd.hAdd (α := ℕ) 0 1) sorry = map f 0 (HAdd.hAdd (α := ℕ) 0 1) hij
Tactic: rw [map_id, comp_id]
-- Lean 4 proof step from Mathlib.CategoryTheory.Functor.OfSequence (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
j : ℕ
hj :
  ∀ {X : ℕ → C} (f : (n : ℕ) → X n ⟶ X (n + 1)) (k : ℕ) (hij : 0 ≤ j) (hjk : j ≤ k),
    map f 0 k ⋯ = map f 0 j hij ≫ map f j k hjk
X : ℕ → C
f : (n : ℕ) → X n ⟶ X (n + 1)
hij : 0 ≤ j + 1
k : ℕ
hjk : j + 1 ≤ k + 1 + 1
⊢ map f 0 (k + 1 + 1) ⋯ = map f 0 (j + 1) hij ≫ map f (j + 1) (k + 1 + 1) hjk
have zero : Eq (α := X 0 ⟶ X (k + 1 + 1)) (f 0 ≫ map (fun (n : ℕ) ↦ f (n + 1)) 0 (k + 1) sorry)
  (map f 0 (j + 1) hij ≫ map (fun (n : ℕ) ↦ f (n + 1)) j (k + 1) sorry)
Tactic: simp only [map, Nat.reduceAdd]
-- Lean 4 proof step from Mathlib.CategoryTheory.Functor.OfSequence (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
j : ℕ
hj :
  ∀ {X : ℕ → C} (f : (n : ℕ) → X n ⟶ X (n + 1)) (k : ℕ) (hij : 0 ≤ j) (hjk : j ≤ k),
    map f 0 k ⋯ = map f 0 j hij ≫ map f j k hjk
X : ℕ → C
f : (n : ℕ) → X n ⟶ X (n + 1)
hij : 0 ≤ j + 1
k : ℕ
hjk : j + 1 ≤ k + 1 + 1
⊢ Eq (α := X 0 ⟶ X (k + 1 + 1)) (f 0 ≫ map (fun (n : ℕ) ↦ f (n + 1)) 0 (k + 1) ⋯)
    (map f 0 (j + 1) hij ≫ map (fun (n : ℕ) ↦ f (n + 1)) j (k + 1) ⋯)
have zero : Eq (α := X 0 ⟶ X (k + 1 + 1))
  (f 0 ≫
    CategoryStruct.comp (Y := X (j + 1)) (map (fun (n : ℕ) ↦ f (n + 1)) 0 j sorry)
      (map (fun (n : ℕ) ↦ f (n + 1)) j (k + 1) sorry))
  (map f 0 (j + 1) hij ≫ map (fun (n : ℕ) ↦ f (n + 1)) j (k + 1) sorry)
Tactic: rw [hj (fun n ↦ f (n + 1)) (k + 1) (by lia) (by lia)]