awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.RelCat (hypothesis)
Goal:
X Y : RelCat
r : X ⟶ Y
h : IsIso r
h1 :
  ∀ (a b : X),
    Membership.mem (γ := SetRel X X) (Hom.rel (Mathlib.Tactic.TermCongr.cHole (r ≫ inv r) (p := r ≫ inv r = 𝟙 X) ⋯))
        (a, b) =
      Membership.mem (γ := SetRel X X) (Hom.rel (Mathlib.Tactic.TermCongr.cHole (𝟙 X) (p := r ≫ inv r = 𝟙 X) ⋯)) (a, b)
h2 :
  ∀ (a b : Y),
    Membership.mem (γ := SetRel Y Y) (Hom.rel (Mathlib.Tactic.TermCongr.cHole (inv r ≫ r) (p := inv r ≫ r = 𝟙 Y) ⋯))
        (a, b) =
      Membership.mem (γ := SetRel Y Y) (Hom.rel (Mathlib.Tactic.TermCongr.cHole (𝟙 Y) (p := inv r ≫ r = 𝟙 Y) ⋯)) (a, b)
h1₁ : ∀ (a b : X), (∃ (y : Y), (a, y) ∈ r.rel ∧ (y, b) ∈ (inv r).rel) ↔ a = b
⊢ ∃ (f : X ≅ Y), graphFunctor.map f.hom = r
have h2₁ : ∀ (a b : Y), (∃ (y : X), (a, y) ∈ (inv r).rel ∧ (y, b) ∈ r.rel) ↔ a = b
Tactic: simp only [RelCat.Hom.rel_comp_apply₂, RelCat.Hom.rel_id_apply₂, eq_iff_iff] at h1 h2
Cleared hypotheses: h1, h2
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.RelCat (goal)
Goal:
X Y : RelCat
r : X ⟶ Y
h : IsIso r
h1 : ∀ (a b : X), (∃ (y : Y), (a, y) ∈ r.rel ∧ (y, b) ∈ (inv r).rel) ↔ a = b
h2 : ∀ (a b : Y), (∃ (y : X), (a, y) ∈ (inv r).rel ∧ (y, b) ∈ r.rel) ↔ a = b
⊢ ∃ (f : X ≅ Y), graphFunctor.map f.hom = r
have mp : ∀ (f : (x : X) → Y) (hf : ∀ (x : X), (x, f x) ∈ r.rel ∧ (f x, x) ∈ (inv r).rel),
  ∃ (f : X ≅ Y), graphFunctor.map f.hom = r
Tactic: obtain ⟨f, hf⟩ := Classical.axiomOfChoice (fun a => (h1 a a).mpr rfl)
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.RelCat (goal)
Goal:
X Y : RelCat
r : X ⟶ Y
h : IsIso r
h1 : ∀ (a b : X), (∃ (y : Y), (a, y) ∈ r.rel ∧ (y, b) ∈ (inv r).rel) ↔ a = b
h2 : ∀ (a b : Y), (∃ (y : X), (a, y) ∈ (inv r).rel ∧ (y, b) ∈ r.rel) ↔ a = b
f : (x : X) → Y
hf : ∀ (x : X), (x, f x) ∈ r.rel ∧ (f x, x) ∈ (inv r).rel
⊢ ∃ (f : X ≅ Y), graphFunctor.map f.hom = r
have mp : ∀ (g : (x : Y) → X) (hg : ∀ (x : Y), (x, g x) ∈ (inv r).rel ∧ (g x, x) ∈ r.rel),
  ∃ (f : X ≅ Y), graphFunctor.map f.hom = r
Tactic: obtain ⟨g, hg⟩ := Classical.axiomOfChoice (fun a => (h2 a a).mpr rfl)
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.RelCat (goal)
Goal:
X Y : RelCat
r : X ⟶ Y
h : IsIso r
h1 : ∀ (a b : X), (∃ (y : Y), (a, y) ∈ r.rel ∧ (y, b) ∈ (inv r).rel) ↔ a = b
h2 : ∀ (a b : Y), (∃ (y : X), (a, y) ∈ (inv r).rel ∧ (y, b) ∈ r.rel) ↔ a = b
f : (x : X) → Y
hf : ∀ (x : X), (x, f x) ∈ r.rel ∧ (f x, x) ∈ (inv r).rel
g : (x : Y) → X
hg : ∀ (x : Y), (x, g x) ∈ (inv r).rel ∧ (g x, x) ∈ r.rel
hif : IsIso (↾f)
⊢ ∃ (f : X ≅ Y), graphFunctor.map f.hom = r
have h₁ : graphFunctor.map (asIso (↾f)).hom = r
Tactic: use asIso (↾f)
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.RelCat (goal)
Goal:
X Y : RelCat
r : X ⟶ Y
h : IsIso r
h1 : ∀ (a b : X), (∃ (y : Y), (a, y) ∈ r.rel ∧ (y, b) ∈ (inv r).rel) ↔ a = b
h2 : ∀ (a b : Y), (∃ (y : X), (a, y) ∈ (inv r).rel ∧ (y, b) ∈ r.rel) ↔ a = b
f : (x : X) → Y
hf : ∀ (x : X), (x, f x) ∈ r.rel ∧ (f x, x) ∈ (inv r).rel
g : (x : Y) → X
hg : ∀ (x : Y), (x, g x) ∈ (inv r).rel ∧ (g x, x) ∈ r.rel
hif : IsIso (↾f)
⊢ graphFunctor.map (asIso (↾f)).hom = r
have h₁ : ∀ (x : graphFunctor.obj X) (y : graphFunctor.obj Y), (x, y) ∈ (graphFunctor.map (asIso (↾f)).hom).rel ↔ (x, y) ∈ r.rel
Tactic: ext ⟨x, y⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.RelCat (goal)
Goal:
X Y : RelCat
r : X ⟶ Y
h : IsIso r
h1 : ∀ (a b : X), (∃ (y : Y), (a, y) ∈ r.rel ∧ (y, b) ∈ (inv r).rel) ↔ a = b
h2 : ∀ (a b : Y), (∃ (y : X), (a, y) ∈ (inv r).rel ∧ (y, b) ∈ r.rel) ↔ a = b
f : (x : X) → Y
hf : ∀ (x : X), (x, f x) ∈ r.rel ∧ (f x, x) ∈ (inv r).rel
g : (x : Y) → X
hg : ∀ (x : Y), (x, g x) ∈ (inv r).rel ∧ (g x, x) ∈ r.rel
⊢ ∃ (f : X ≅ Y), graphFunctor.map f.hom = r
have mp : IsIso (↾f)
Tactic: suffices hif : IsIso (C := Type u) (↾f) by
      use asIso (↾f)
      ext ⟨x, y⟩
      exact ⟨by aesop, fun hxy ↦ (h2 (f x) y).1 ⟨x, (hf x).2, hxy⟩⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.RelCat (goal)
Goal:
X Y : RelCat
r : X ⟶ Y
h : IsIso r
h1 : ∀ (a b : X), (∃ (y : Y), (a, y) ∈ r.rel ∧ (y, b) ∈ (inv r).rel) ↔ a = b
h2 : ∀ (a b : Y), (∃ (y : X), (a, y) ∈ (inv r).rel ∧ (y, b) ∈ r.rel) ↔ a = b
f : (x : X) → Y
hf : ∀ (x : X), (x, f x) ∈ r.rel ∧ (f x, x) ∈ (inv r).rel
g : (x : Y) → X
hg : ∀ (x : Y), (x, g x) ∈ (inv r).rel ∧ (g x, x) ∈ r.rel
⊢ IsIso (↾f)
have h₁ : ↾f ≫ ↾g = 𝟙 X ∧ ↾g ≫ ↾f = 𝟙 Y
Tactic: use ↾g
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.RelCat (goal)
Goal:
X Y : RelCat
r : X ⟶ Y
h : IsIso r
h1 : ∀ (a b : X), (∃ (y : Y), (a, y) ∈ r.rel ∧ (y, b) ∈ (inv r).rel) ↔ a = b
h2 : ∀ (a b : Y), (∃ (y : X), (a, y) ∈ (inv r).rel ∧ (y, b) ∈ r.rel) ↔ a = b
f : (x : X) → Y
hf : ∀ (x : X), (x, f x) ∈ r.rel ∧ (f x, x) ∈ (inv r).rel
g : (x : Y) → X
hg : ∀ (x : Y), (x, g x) ∈ (inv r).rel ∧ (g x, x) ∈ r.rel
⊢ ↾f ≫ ↾g = 𝟙 X ∧ ↾g ≫ ↾f = 𝟙 Y
have h₁ : ↾f ≫ ↾g = 𝟙 X
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.RelCat (goal)
Goal:
X Y : RelCat
r : X ⟶ Y
h : IsIso r
h1 : ∀ (a b : X), (∃ (y : Y), (a, y) ∈ r.rel ∧ (y, b) ∈ (inv r).rel) ↔ a = b
h2 : ∀ (a b : Y), (∃ (y : X), (a, y) ∈ (inv r).rel ∧ (y, b) ∈ r.rel) ↔ a = b
f : (x : X) → Y
hf : ∀ (x : X), (x, f x) ∈ r.rel ∧ (f x, x) ∈ (inv r).rel
g : (x : Y) → X
hg : ∀ (x : Y), (x, g x) ∈ (inv r).rel ∧ (g x, x) ∈ r.rel
h₁ : ↾f ≫ ↾g = 𝟙 X
⊢ ↾f ≫ ↾g = 𝟙 X ∧ ↾g ≫ ↾f = 𝟙 Y
have h₂ : ↾g ≫ ↾f = 𝟙 Y
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.RelCat (goal)
Goal:
X Y : RelCat
r : X ⟶ Y
h : IsIso r
h1 : ∀ (a b : X), (∃ (y : Y), (a, y) ∈ r.rel ∧ (y, b) ∈ (inv r).rel) ↔ a = b
h2 : ∀ (a b : Y), (∃ (y : X), (a, y) ∈ (inv r).rel ∧ (y, b) ∈ r.rel) ↔ a = b
f : (x : X) → Y
hf : ∀ (x : X), (x, f x) ∈ r.rel ∧ (f x, x) ∈ (inv r).rel
g : (x : Y) → X
hg : ∀ (x : Y), (x, g x) ∈ (inv r).rel ∧ (g x, x) ∈ r.rel
⊢ ↾f ≫ ↾g = 𝟙 X
have h₁ : ∀ (x : X),
  (@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _ (↾f ≫ ↾g)).toFun x =
    (ConcreteCategory.hom (FC := TypeCat.Fun) (𝟙 X)).toFun x
Tactic: ext x
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.RelCat (goal)
Goal:
X Y : RelCat
r : X ⟶ Y
h : IsIso r
h1 : ∀ (a b : X), (∃ (y : Y), (a, y) ∈ r.rel ∧ (y, b) ∈ (inv r).rel) ↔ a = b
h2 : ∀ (a b : Y), (∃ (y : X), (a, y) ∈ (inv r).rel ∧ (y, b) ∈ r.rel) ↔ a = b
f : (x : X) → Y
hf : ∀ (x : X), (x, f x) ∈ r.rel ∧ (f x, x) ∈ (inv r).rel
g : (x : Y) → X
hg : ∀ (x : Y), (x, g x) ∈ (inv r).rel ∧ (g x, x) ∈ r.rel
x : X
⊢ (@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _ (↾f ≫ ↾g)).toFun x =
    (ConcreteCategory.hom (FC := TypeCat.Fun) (𝟙 X)).toFun x
have h₁ : ∃ (y : Y),
  ((@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _ (↾f ≫ ↾g)).toFun x, y) ∈ r.rel ∧
    (y, (ConcreteCategory.hom (FC := TypeCat.Fun) (𝟙 X)).toFun x) ∈ (inv r).rel
Tactic: apply (h1 _ _).mp
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.RelCat (goal)
Goal:
X Y : RelCat
r : X ⟶ Y
h : IsIso r
h1 : ∀ (a b : X), (∃ (y : Y), (a, y) ∈ r.rel ∧ (y, b) ∈ (inv r).rel) ↔ a = b
h2 : ∀ (a b : Y), (∃ (y : X), (a, y) ∈ (inv r).rel ∧ (y, b) ∈ r.rel) ↔ a = b
f : (x : X) → Y
hf : ∀ (x : X), (x, f x) ∈ r.rel ∧ (f x, x) ∈ (inv r).rel
g : (x : Y) → X
hg : ∀ (x : Y), (x, g x) ∈ (inv r).rel ∧ (g x, x) ∈ r.rel
x : X
⊢ ∃ (y : Y),
    ((@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _ (↾f ≫ ↾g)).toFun x, y) ∈ r.rel ∧
      (y, (ConcreteCategory.hom (FC := TypeCat.Fun) (𝟙 X)).toFun x) ∈ (inv r).rel
have w : Y
Tactic: use f x, (hg _).2, (hf _).2
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.RelCat (goal)
Goal:
X Y : RelCat
r : X ⟶ Y
h : IsIso r
h1 : ∀ (a b : X), (∃ (y : Y), (a, y) ∈ r.rel ∧ (y, b) ∈ (inv r).rel) ↔ a = b
h2 : ∀ (a b : Y), (∃ (y : X), (a, y) ∈ (inv r).rel ∧ (y, b) ∈ r.rel) ↔ a = b
f : (x : X) → Y
hf : ∀ (x : X), (x, f x) ∈ r.rel ∧ (f x, x) ∈ (inv r).rel
g : (x : Y) → X
hg : ∀ (x : Y), (x, g x) ∈ (inv r).rel ∧ (g x, x) ∈ r.rel
x : X
w : Y
⊢ ∃ (y : Y),
    ((@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _ (↾f ≫ ↾g)).toFun x, y) ∈ r.rel ∧
      (y, (ConcreteCategory.hom (FC := TypeCat.Fun) (𝟙 X)).toFun x) ∈ (inv r).rel
have left : ((@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _ (↾f ≫ ↾g)).toFun x, f x) ∈ r.rel
Tactic: use f x, (hg _).2, (hf _).2
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.RelCat (goal)
Goal:
X Y : RelCat
r : X ⟶ Y
h : IsIso r
h1 : ∀ (a b : X), (∃ (y : Y), (a, y) ∈ r.rel ∧ (y, b) ∈ (inv r).rel) ↔ a = b
h2 : ∀ (a b : Y), (∃ (y : X), (a, y) ∈ (inv r).rel ∧ (y, b) ∈ r.rel) ↔ a = b
f : (x : X) → Y
hf : ∀ (x : X), (x, f x) ∈ r.rel ∧ (f x, x) ∈ (inv r).rel
g : (x : Y) → X
hg : ∀ (x : Y), (x, g x) ∈ (inv r).rel ∧ (g x, x) ∈ r.rel
x : X
w : Y
left : ((@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _ (↾f ≫ ↾g)).toFun x, f x) ∈ r.rel
⊢ ∃ (y : Y),
    ((@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _ (↾f ≫ ↾g)).toFun x, y) ∈ r.rel ∧
      (y, (ConcreteCategory.hom (FC := TypeCat.Fun) (𝟙 X)).toFun x) ∈ (inv r).rel
have right : (f x, (ConcreteCategory.hom (FC := TypeCat.Fun) (𝟙 X)).toFun x) ∈ (inv r).rel
Tactic: use f x, (hg _).2, (hf _).2
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.RelCat (goal)
Goal:
X Y : RelCat
r : X ⟶ Y
h : IsIso r
h1 : ∀ (a b : X), (∃ (y : Y), (a, y) ∈ r.rel ∧ (y, b) ∈ (inv r).rel) ↔ a = b
h2 : ∀ (a b : Y), (∃ (y : X), (a, y) ∈ (inv r).rel ∧ (y, b) ∈ r.rel) ↔ a = b
f : (x : X) → Y
hf : ∀ (x : X), (x, f x) ∈ r.rel ∧ (f x, x) ∈ (inv r).rel
g : (x : Y) → X
hg : ∀ (x : Y), (x, g x) ∈ (inv r).rel ∧ (g x, x) ∈ r.rel
⊢ ↾g ≫ ↾f = 𝟙 Y
have h₁ : ∀ (y : Y),
  (@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _ (↾g ≫ ↾f)).toFun y =
    (ConcreteCategory.hom (FC := TypeCat.Fun) (𝟙 Y)).toFun y
Tactic: ext y
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.RelCat (goal)
Goal:
X Y : RelCat
r : X ⟶ Y
h : IsIso r
h1 : ∀ (a b : X), (∃ (y : Y), (a, y) ∈ r.rel ∧ (y, b) ∈ (inv r).rel) ↔ a = b
h2 : ∀ (a b : Y), (∃ (y : X), (a, y) ∈ (inv r).rel ∧ (y, b) ∈ r.rel) ↔ a = b
f : (x : X) → Y
hf : ∀ (x : X), (x, f x) ∈ r.rel ∧ (f x, x) ∈ (inv r).rel
g : (x : Y) → X
hg : ∀ (x : Y), (x, g x) ∈ (inv r).rel ∧ (g x, x) ∈ r.rel
y : Y
⊢ (@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _ (↾g ≫ ↾f)).toFun y =
    (ConcreteCategory.hom (FC := TypeCat.Fun) (𝟙 Y)).toFun y
have h₁ : ∃ (y_1 : X),
  ((@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _ (↾g ≫ ↾f)).toFun y, y_1) ∈ (inv r).rel ∧
    (y_1, (ConcreteCategory.hom (FC := TypeCat.Fun) (𝟙 Y)).toFun y) ∈ r.rel
Tactic: apply (h2 _ _).mp
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.RelCat (goal)
Goal:
X Y : RelCat
r : X ⟶ Y
h : IsIso r
h1 : ∀ (a b : X), (∃ (y : Y), (a, y) ∈ r.rel ∧ (y, b) ∈ (inv r).rel) ↔ a = b
h2 : ∀ (a b : Y), (∃ (y : X), (a, y) ∈ (inv r).rel ∧ (y, b) ∈ r.rel) ↔ a = b
f : (x : X) → Y
hf : ∀ (x : X), (x, f x) ∈ r.rel ∧ (f x, x) ∈ (inv r).rel
g : (x : Y) → X
hg : ∀ (x : Y), (x, g x) ∈ (inv r).rel ∧ (g x, x) ∈ r.rel
y : Y
⊢ ∃ (y_1 : X),
    ((@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _ (↾g ≫ ↾f)).toFun y, y_1) ∈ (inv r).rel ∧
      (y_1, (ConcreteCategory.hom (FC := TypeCat.Fun) (𝟙 Y)).toFun y) ∈ r.rel
have w : X
Tactic: use g y, (hf (g y)).2, (hg y).2
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.RelCat (goal)
Goal:
X Y : RelCat
r : X ⟶ Y
h : IsIso r
h1 : ∀ (a b : X), (∃ (y : Y), (a, y) ∈ r.rel ∧ (y, b) ∈ (inv r).rel) ↔ a = b
h2 : ∀ (a b : Y), (∃ (y : X), (a, y) ∈ (inv r).rel ∧ (y, b) ∈ r.rel) ↔ a = b
f : (x : X) → Y
hf : ∀ (x : X), (x, f x) ∈ r.rel ∧ (f x, x) ∈ (inv r).rel
g : (x : Y) → X
hg : ∀ (x : Y), (x, g x) ∈ (inv r).rel ∧ (g x, x) ∈ r.rel
y : Y
w : X
⊢ ∃ (y_1 : X),
    ((@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _ (↾g ≫ ↾f)).toFun y, y_1) ∈ (inv r).rel ∧
      (y_1, (ConcreteCategory.hom (FC := TypeCat.Fun) (𝟙 Y)).toFun y) ∈ r.rel
have left : ((@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _ (↾g ≫ ↾f)).toFun y, g y) ∈ (inv r).rel
Tactic: use g y, (hf (g y)).2, (hg y).2
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.RelCat (goal)
Goal:
X Y : RelCat
r : X ⟶ Y
h : IsIso r
h1 : ∀ (a b : X), (∃ (y : Y), (a, y) ∈ r.rel ∧ (y, b) ∈ (inv r).rel) ↔ a = b
h2 : ∀ (a b : Y), (∃ (y : X), (a, y) ∈ (inv r).rel ∧ (y, b) ∈ r.rel) ↔ a = b
f : (x : X) → Y
hf : ∀ (x : X), (x, f x) ∈ r.rel ∧ (f x, x) ∈ (inv r).rel
g : (x : Y) → X
hg : ∀ (x : Y), (x, g x) ∈ (inv r).rel ∧ (g x, x) ∈ r.rel
y : Y
w : X
left : ((@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _ (↾g ≫ ↾f)).toFun y, g y) ∈ (inv r).rel
⊢ ∃ (y_1 : X),
    ((@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _ (↾g ≫ ↾f)).toFun y, y_1) ∈ (inv r).rel ∧
      (y_1, (ConcreteCategory.hom (FC := TypeCat.Fun) (𝟙 Y)).toFun y) ∈ r.rel
have right : (g y, (ConcreteCategory.hom (FC := TypeCat.Fun) (𝟙 Y)).toFun y) ∈ r.rel
Tactic: use g y, (hf (g y)).2, (hg y).2
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.TwoP (goal)
Goal:
α : Type u_1
β : Type u_2
X : Pointed
Y : TwoP
f : pointedToTwoPFst.obj X ⟶ Y
⊢ (fun
          (f :
            X ⟶
              Functor.obj
                (forget₂ TwoP (FC := fun (X Y : TwoP.{u_3}) ↦ X.toBipointed.HomSubtype Y.toBipointed) (CC :=
                    fun (X : TwoP.{u_3}) ↦ X.toBipointed.X) Bipointed (FD := Bipointed.HomSubtype) (CD := Bipointed.X) ⋙
                  bipointedToPointedFst)
                Y) ↦
        ({
            hom :=
              {
                toFun := fun (o : (pointedToTwoPFst.obj X).toBipointed.X) ↦
                  Option.elim o Y.toTwoPointing.toProd.2 f.toFun,
                map_fst := ⋯, map_snd := ⋯ } } :
          InducedCategory.Hom (D := Bipointed) (F := TwoP.toBipointed) (pointedToTwoPFst.obj X) Y))
      ((fun (f : pointedToTwoPFst.obj X ⟶ Y) ↦
          { toFun := f.hom.toFun ∘ some, map_point := ⋯ :
            X.Hom
              (Functor.obj
                (forget₂ TwoP (FC := fun (X Y : TwoP.{u_3}) ↦ X.toBipointed.HomSubtype Y.toBipointed) (CC :=
                    fun (X : TwoP.{u_3}) ↦ X.toBipointed.X) Bipointed (FD := Bipointed.HomSubtype) (CD := Bipointed.X) ⋙
                  bipointedToPointedFst)
                Y) })
        f) =
    f
have h : (↑(ConcreteCategory.hom (FC := Bipointed.HomSubtype) (CC := Bipointed.X)
          (InducedCategory.Hom.hom
            ((fun
                  (f :
                    X ⟶
                      Functor.obj
                        (forget₂ TwoP (FC := fun (X Y : TwoP.{u_3}) ↦ X.toBipointed.HomSubtype Y.toBipointed) (CC :=
                            fun (X : TwoP.{u_3}) ↦ X.toBipointed.X) Bipointed (FD := Bipointed.HomSubtype) (CD :=
                            Bipointed.X) ⋙
                          bipointedToPointedFst)
                        Y) ↦
                ({
                    hom :=
                      {
                        toFun := fun (o : (pointedToTwoPFst.obj X).toBipointed.X) ↦
                          Option.elim o Y.toTwoPointing.toProd.2 f.toFun,
                        map_fst := sorry, map_snd := sorry : (pointedToTwoPFst.obj X).toBipointed.Hom Y.toBipointed } } :
                  InducedCategory.Hom (D := Bipointed) (F := TwoP.toBipointed) (pointedToTwoPFst.obj X) Y))
              ((fun (f : pointedToTwoPFst.obj X ⟶ Y) ↦
                  { toFun := f.hom.toFun ∘ some, map_point := sorry :
                    X.Hom
                      (Functor.obj
                        (forget₂ TwoP (FC := fun (X Y : TwoP.{u_3}) ↦ X.toBipointed.HomSubtype Y.toBipointed) (CC :=
                            fun (X : TwoP.{u_3}) ↦ X.toBipointed.X) Bipointed (FD := Bipointed.HomSubtype) (CD :=
                            Bipointed.X) ⋙
                          bipointedToPointedFst)
                        Y) })
                f)))) :
      (pointedToTwoPFst.obj X).toBipointed.X → Y.toBipointed.X)
    none =
  (↑(ConcreteCategory.hom (FC := Bipointed.HomSubtype) (CC := Bipointed.X) f.hom) :
      (pointedToTwoPFst.obj X).toBipointed.X → Y.toBipointed.X)
    none
Tactic: ext (_ | _) : 4
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.TwoP (goal)
Goal:
α : Type u_1
β : Type u_2
X : Pointed
Y : TwoP
f : pointedToTwoPSnd.obj X ⟶ Y
⊢ (fun
          (f :
            X ⟶
              Functor.obj
                (forget₂ TwoP (FC := fun (X Y : TwoP.{u_3}) ↦ X.toBipointed.HomSubtype Y.toBipointed) (CC :=
                    fun (X : TwoP.{u_3}) ↦ X.toBipointed.X) Bipointed (FD := Bipointed.HomSubtype) (CD := Bipointed.X) ⋙
                  bipointedToPointedSnd)
                Y) ↦
        ({
            hom :=
              {
                toFun := fun (o : (pointedToTwoPSnd.obj X).toBipointed.X) ↦
                  Option.elim o Y.toTwoPointing.toProd.1 f.toFun,
                map_fst := ⋯, map_snd := ⋯ } } :
          InducedCategory.Hom (D := Bipointed) (F := TwoP.toBipointed) (pointedToTwoPSnd.obj X) Y))
      ((fun (f : pointedToTwoPSnd.obj X ⟶ Y) ↦
          { toFun := f.hom.toFun ∘ some, map_point := ⋯ :
            X.Hom
              (Functor.obj
                (forget₂ TwoP (FC := fun (X Y : TwoP.{u_3}) ↦ X.toBipointed.HomSubtype Y.toBipointed) (CC :=
                    fun (X : TwoP.{u_3}) ↦ X.toBipointed.X) Bipointed (FD := Bipointed.HomSubtype) (CD := Bipointed.X) ⋙
                  bipointedToPointedSnd)
                Y) })
        f) =
    f
have h : (↑(ConcreteCategory.hom (FC := Bipointed.HomSubtype) (CC := Bipointed.X)
          (InducedCategory.Hom.hom
            ((fun
                  (f :
                    X ⟶
                      Functor.obj
                        (forget₂ TwoP (FC := fun (X Y : TwoP.{u_3}) ↦ X.toBipointed.HomSubtype Y.toBipointed) (CC :=
                            fun (X : TwoP.{u_3}) ↦ X.toBipointed.X) Bipointed (FD := Bipointed.HomSubtype) (CD :=
                            Bipointed.X) ⋙
                          bipointedToPointedSnd)
                        Y) ↦
                ({
                    hom :=
                      {
                        toFun := fun (o : (pointedToTwoPSnd.obj X).toBipointed.X) ↦
                          Option.elim o Y.toTwoPointing.toProd.1 f.toFun,
                        map_fst := sorry, map_snd := sorry : (pointedToTwoPSnd.obj X).toBipointed.Hom Y.toBipointed } } :
                  InducedCategory.Hom (D := Bipointed) (F := TwoP.toBipointed) (pointedToTwoPSnd.obj X) Y))
              ((fun (f : pointedToTwoPSnd.obj X ⟶ Y) ↦
                  { toFun := f.hom.toFun ∘ some, map_point := sorry :
                    X.Hom
                      (Functor.obj
                        (forget₂ TwoP (FC := fun (X Y : TwoP.{u_3}) ↦ X.toBipointed.HomSubtype Y.toBipointed) (CC :=
                            fun (X : TwoP.{u_3}) ↦ X.toBipointed.X) Bipointed (FD := Bipointed.HomSubtype) (CD :=
                            Bipointed.X) ⋙
                          bipointedToPointedSnd)
                        Y) })
                f)))) :
      (pointedToTwoPSnd.obj X).toBipointed.X → Y.toBipointed.X)
    none =
  (↑(ConcreteCategory.hom (FC := Bipointed.HomSubtype) (CC := Bipointed.X) f.hom) :
      (pointedToTwoPSnd.obj X).toBipointed.X → Y.toBipointed.X)
    none
Tactic: ext (_ | _) : 4
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.ULift (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
X Y : AsSmall.{u_1, v₁, u₁} C
h : X = Y
⊢ ULift.down (eqToHom h) = eqToHom ⋯
have subst : ULift.down (eqToHom sorry) = eqToHom sorry
Tactic: subst h
Cleared hypotheses: Y, h
-- Lean 4 proof step from Mathlib.CategoryTheory.Center.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
x y : CatCenter.{v, u} C
X : C
⊢ (x * y).app X = x.app X ≫ y.app X
have rw : y.app X ≫ x.app X = x.app X ≫ y.app X
Tactic: rw [mul_app']
-- Lean 4 proof step from Mathlib.CategoryTheory.Center.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
x y : CatCenter.{v, u} C
⊢ x * y = y * x
have h : ∀ (X : C), (x * y).app X = (y * x).app X
Tactic: ext X
-- Lean 4 proof step from Mathlib.CategoryTheory.Center.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
x y : CatCenter.{v, u} C
X : C
⊢ (x * y).app X = (y * x).app X
have h : y.app X ≫ x.app X = (y * x).app X
Tactic: rw [mul_app' x y, mul_app y x]
-- Lean 4 proof step from Mathlib.CategoryTheory.Center.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
x y : CatCenter.{v, u} C
X : C
h : y.app X ≫ x.app X = (y * x).app X
⊢ (x * y).app X = (y * x).app X
have h₁ : y.app X ≫ x.app X = y.app X ≫ x.app X
Tactic: rw [mul_app' x y, mul_app y x]
-- Lean 4 proof step from Mathlib.CategoryTheory.Center.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y : C
z : (CatCenter.{v, u} C)ˣ
e : X ≅ Y
⊢ ((↑z : CatCenter C) • e.hom) ≫ ((↑z⁻¹ : CatCenter C) • e.inv) = 𝟙 X
have rw : e.hom ≫ ((↑z : CatCenter C) * (↑z⁻¹ : CatCenter C)).app Y ≫ e.inv = 𝟙 X
Tactic: rw [smul_eq, smul_eq', Category.assoc, ← mul_app_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Center.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y : C
z : (CatCenter.{v, u} C)ˣ
e : X ≅ Y
⊢ ((↑z⁻¹ : CatCenter C) • e.inv) ≫ ((↑z : CatCenter C) • e.hom) = 𝟙 Y
have rw : e.inv ≫ ((↑z⁻¹ : CatCenter C) * (↑z : CatCenter C)).app X ≫ e.hom = 𝟙 Y
Tactic: rw [smul_eq, smul_eq', Category.assoc, ← mul_app_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Center.Linear (goal)
Goal:
R : Type w
inst : Ring R
C : Type u
inst_1 : Category.{v, u} C
inst_2 : Preadditive.{v, u} C
inst_3 : Linear.{w, v, u} R C
a b : R
⊢ { app := fun (X : C) ↦ HSMul.hSMul (β := X ⟶ X) (a * b) (𝟙 X), naturality := ⋯ } =
    HMul.hMul (α := CatCenter.{v, u} C)
      { app := fun (X : C) ↦ HSMul.hSMul (β := X ⟶ X) a (𝟙 X), naturality := ⋯ : NatTrans.{v, v, u, u} (𝟭 C) (𝟭 C) }
      { app := fun (X : C) ↦ HSMul.hSMul (β := X ⟶ X) b (𝟙 X), naturality := ⋯ : NatTrans.{v, v, u, u} (𝟭 C) (𝟭 C) }
have rw : { app := fun (X : C) ↦ HSMul.hSMul (β := X ⟶ X) (a * b) (𝟙 X), naturality := sorry } =
  HMul.hMul (α := CatCenter.{v, u} C)
    { app := fun (X : C) ↦ HSMul.hSMul (β := X ⟶ X) b (𝟙 X), naturality := sorry : NatTrans.{v, v, u, u} (𝟭 C) (𝟭 C) }
    { app := fun (X : C) ↦ HSMul.hSMul (β := X ⟶ X) a (𝟙 X), naturality := sorry : NatTrans.{v, v, u, u} (𝟭 C) (𝟭 C) }
Tactic: rw [mul_comm]
-- Lean 4 proof step from Mathlib.CategoryTheory.Center.Linear (goal)
Goal:
R : Type w
inst : Ring R
C : Type u
inst_1 : Category.{v, u} C
inst_2 : Preadditive.{v, u} C
inst_3 : Linear.{w, v, u} R C
a b : R
⊢ { app := fun (X : C) ↦ HSMul.hSMul (β := X ⟶ X) (a * b) (𝟙 X), naturality := ⋯ } =
    HMul.hMul (α := CatCenter.{v, u} C)
      { app := fun (X : C) ↦ HSMul.hSMul (β := X ⟶ X) b (𝟙 X), naturality := ⋯ : NatTrans.{v, v, u, u} (𝟭 C) (𝟭 C) }
      { app := fun (X : C) ↦ HSMul.hSMul (β := X ⟶ X) a (𝟙 X), naturality := ⋯ : NatTrans.{v, v, u, u} (𝟭 C) (𝟭 C) }
have h : ∀ (X : C),
  CatCenter.app { app := fun (X : C) ↦ HSMul.hSMul (β := X ⟶ X) (a * b) (𝟙 X), naturality := sorry } X =
    (HMul.hMul (α := CatCenter.{v, u} C)
          { app := fun (X : C) ↦ HSMul.hSMul (β := X ⟶ X) b (𝟙 X), naturality := sorry : NatTrans.{v, v, u, u} (𝟭 C) (𝟭 C) }
          { app := fun (X : C) ↦ HSMul.hSMul (β := X ⟶ X) a (𝟙 X), naturality := sorry :
            NatTrans.{v, v, u, u} (𝟭 C) (𝟭 C) }).app
      X
Tactic: ext X
-- Lean 4 proof step from Mathlib.CategoryTheory.Center.Linear (goal)
Goal:
R : Type w
inst : Ring R
C : Type u
inst_1 : Category.{v, u} C
inst_2 : Preadditive.{v, u} C
inst_3 : Linear.{w, v, u} R C
a b : R
X : C
⊢ CatCenter.app { app := fun (X : C) ↦ HSMul.hSMul (β := X ⟶ X) (a * b) (𝟙 X), naturality := ⋯ } X =
    (HMul.hMul (α := CatCenter.{v, u} C)
          { app := fun (X : C) ↦ HSMul.hSMul (β := X ⟶ X) b (𝟙 X), naturality := ⋯ : NatTrans.{v, v, u, u} (𝟭 C) (𝟭 C) }
          { app := fun (X : C) ↦ HSMul.hSMul (β := X ⟶ X) a (𝟙 X), naturality := ⋯ :
            NatTrans.{v, v, u, u} (𝟭 C) (𝟭 C) }).app
      X
have h : Eq.{v + 1} (α := X ⟶ X)
  (CatCenter.app { app := fun (X : C) ↦ HSMul.hSMul (β := X ⟶ X) (a * b) (𝟙 X), naturality := sorry } X)
  (CatCenter.app
      { app := fun (X : C) ↦ HSMul.hSMul (β := X ⟶ X) a (𝟙 X), naturality := sorry : NatTrans.{v, v, u, u} (𝟭 C) (𝟭 C) } X ≫
    CatCenter.app { app := fun (X : C) ↦ HSMul.hSMul (β := X ⟶ X) b (𝟙 X), naturality := sorry } X)
Tactic: dsimp only [CatCenter.mul_app']