awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.WithTerminal.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
a b : Cat
f : a ⟶ b
X : (↑(prelaxfunctor.obj a) : Type u_1)
⊢ (prelaxfunctor.map₂ (Bicategory.leftUnitor f).hom).toNatTrans.app X =
    ((Cat.Hom.isoMk (mapComp (Cat.Hom.toFunctor (𝟙 a)) f.toFunctor)).hom ≫
            Bicategory.whiskerRight (Iso.hom (Cat.Hom.isoMk (mapId (↑a : Type u_1)))) (prelaxfunctor.map f) ≫
              (Bicategory.leftUnitor (prelaxfunctor.map f)).hom).toNatTrans.app
      X
have h : ∀ (a_1 : (↑a : Type u_1)),
  Eq (α := (prelaxfunctor.map (𝟙 a ≫ f)).toFunctor.obj (of a_1) ⟶ (prelaxfunctor.map f).toFunctor.obj (of a_1))
    ((prelaxfunctor.map₂ (Bicategory.leftUnitor f).hom).toNatTrans.app (of a_1))
    (((Cat.Hom.isoMk (mapComp (Cat.Hom.toFunctor (𝟙 a)) f.toFunctor)).hom ≫
            Bicategory.whiskerRight (Iso.hom (Cat.Hom.isoMk (mapId (↑a : Type u_1)))) (prelaxfunctor.map f) ≫
              (Bicategory.leftUnitor (prelaxfunctor.map f)).hom).toNatTrans.app
      (of a_1))
Tactic: cases X
Cleared hypotheses: X
-- Lean 4 proof step from Mathlib.CategoryTheory.WithTerminal.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
a b : Cat
f : a ⟶ b
X : (↑(prelaxfunctor.obj a) : Type u_1)
h :
  ∀ (a_1 : (↑a : Type u_1)),
    Eq (α := (prelaxfunctor.map (𝟙 a ≫ f)).toFunctor.obj (of a_1) ⟶ (prelaxfunctor.map f).toFunctor.obj (of a_1))
      ((prelaxfunctor.map₂ (Bicategory.leftUnitor f).hom).toNatTrans.app (of a_1))
      (((Cat.Hom.isoMk (mapComp (Cat.Hom.toFunctor (𝟙 a)) f.toFunctor)).hom ≫
              Bicategory.whiskerRight (Iso.hom (Cat.Hom.isoMk (mapId (↑a : Type u_1)))) (prelaxfunctor.map f) ≫
                (Bicategory.leftUnitor (prelaxfunctor.map f)).hom).toNatTrans.app
        (of a_1))
⊢ (prelaxfunctor.map₂ (Bicategory.leftUnitor f).hom).toNatTrans.app X =
    ((Cat.Hom.isoMk (mapComp (Cat.Hom.toFunctor (𝟙 a)) f.toFunctor)).hom ≫
            Bicategory.whiskerRight (Iso.hom (Cat.Hom.isoMk (mapId (↑a : Type u_1)))) (prelaxfunctor.map f) ≫
              (Bicategory.leftUnitor (prelaxfunctor.map f)).hom).toNatTrans.app
      X
have h₁ : Eq (α := (prelaxfunctor.map (𝟙 a ≫ f)).toFunctor.obj star ⟶ (prelaxfunctor.map f).toFunctor.obj star)
  ((prelaxfunctor.map₂ (Bicategory.leftUnitor f).hom).toNatTrans.app star)
  (((Cat.Hom.isoMk (mapComp (Cat.Hom.toFunctor (𝟙 a)) f.toFunctor)).hom ≫
          Bicategory.whiskerRight (Iso.hom (Cat.Hom.isoMk (mapId (↑a : Type u_1)))) (prelaxfunctor.map f) ≫
            (Bicategory.leftUnitor (prelaxfunctor.map f)).hom).toNatTrans.app
    star)
Tactic: cases X
Cleared hypotheses: X
-- Lean 4 proof step from Mathlib.CategoryTheory.WithTerminal.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
a b : Cat
f : a ⟶ b
⊢ prelaxfunctor.map₂ (Bicategory.rightUnitor f).hom =
    (Cat.Hom.isoMk (mapComp f.toFunctor (Cat.Hom.toFunctor (𝟙 b)))).hom ≫
      Bicategory.whiskerLeft (prelaxfunctor.map f) (Iso.hom (Cat.Hom.isoMk (mapId (↑b : Type u_1)))) ≫
        (Bicategory.rightUnitor (prelaxfunctor.map f)).hom
have h : ∀ (X : (↑(prelaxfunctor.obj a) : Type u_1)),
  (prelaxfunctor.map₂ (Bicategory.rightUnitor f).hom).toNatTrans.app X =
    ((Cat.Hom.isoMk (mapComp f.toFunctor (Cat.Hom.toFunctor (𝟙 b)))).hom ≫
            Bicategory.whiskerLeft (prelaxfunctor.map f) (Iso.hom (Cat.Hom.isoMk (mapId (↑b : Type u_1)))) ≫
              (Bicategory.rightUnitor (prelaxfunctor.map f)).hom).toNatTrans.app
      X
Tactic: ext X
-- Lean 4 proof step from Mathlib.CategoryTheory.WithTerminal.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
a b : Cat
f : a ⟶ b
X : (↑(prelaxfunctor.obj a) : Type u_1)
⊢ (prelaxfunctor.map₂ (Bicategory.rightUnitor f).hom).toNatTrans.app X =
    ((Cat.Hom.isoMk (mapComp f.toFunctor (Cat.Hom.toFunctor (𝟙 b)))).hom ≫
            Bicategory.whiskerLeft (prelaxfunctor.map f) (Iso.hom (Cat.Hom.isoMk (mapId (↑b : Type u_1)))) ≫
              (Bicategory.rightUnitor (prelaxfunctor.map f)).hom).toNatTrans.app
      X
have h : ∀ (a_1 : (↑a : Type u_1)),
  Eq (α := (prelaxfunctor.map (f ≫ 𝟙 b)).toFunctor.obj (of a_1) ⟶ (prelaxfunctor.map f).toFunctor.obj (of a_1))
    ((prelaxfunctor.map₂ (Bicategory.rightUnitor f).hom).toNatTrans.app (of a_1))
    (((Cat.Hom.isoMk (mapComp f.toFunctor (Cat.Hom.toFunctor (𝟙 b)))).hom ≫
            Bicategory.whiskerLeft (prelaxfunctor.map f) (Iso.hom (Cat.Hom.isoMk (mapId (↑b : Type u_1)))) ≫
              (Bicategory.rightUnitor (prelaxfunctor.map f)).hom).toNatTrans.app
      (of a_1))
Tactic: cases X
Cleared hypotheses: X
-- Lean 4 proof step from Mathlib.CategoryTheory.WithTerminal.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
a b : Cat
f : a ⟶ b
X : (↑(prelaxfunctor.obj a) : Type u_1)
h :
  ∀ (a_1 : (↑a : Type u_1)),
    Eq (α := (prelaxfunctor.map (f ≫ 𝟙 b)).toFunctor.obj (of a_1) ⟶ (prelaxfunctor.map f).toFunctor.obj (of a_1))
      ((prelaxfunctor.map₂ (Bicategory.rightUnitor f).hom).toNatTrans.app (of a_1))
      (((Cat.Hom.isoMk (mapComp f.toFunctor (Cat.Hom.toFunctor (𝟙 b)))).hom ≫
              Bicategory.whiskerLeft (prelaxfunctor.map f) (Iso.hom (Cat.Hom.isoMk (mapId (↑b : Type u_1)))) ≫
                (Bicategory.rightUnitor (prelaxfunctor.map f)).hom).toNatTrans.app
        (of a_1))
⊢ (prelaxfunctor.map₂ (Bicategory.rightUnitor f).hom).toNatTrans.app X =
    ((Cat.Hom.isoMk (mapComp f.toFunctor (Cat.Hom.toFunctor (𝟙 b)))).hom ≫
            Bicategory.whiskerLeft (prelaxfunctor.map f) (Iso.hom (Cat.Hom.isoMk (mapId (↑b : Type u_1)))) ≫
              (Bicategory.rightUnitor (prelaxfunctor.map f)).hom).toNatTrans.app
      X
have h₁ : Eq (α := (prelaxfunctor.map (f ≫ 𝟙 b)).toFunctor.obj star ⟶ (prelaxfunctor.map f).toFunctor.obj star)
  ((prelaxfunctor.map₂ (Bicategory.rightUnitor f).hom).toNatTrans.app star)
  (((Cat.Hom.isoMk (mapComp f.toFunctor (Cat.Hom.toFunctor (𝟙 b)))).hom ≫
          Bicategory.whiskerLeft (prelaxfunctor.map f) (Iso.hom (Cat.Hom.isoMk (mapId (↑b : Type u_1)))) ≫
            (Bicategory.rightUnitor (prelaxfunctor.map f)).hom).toNatTrans.app
    star)
Tactic: cases X
Cleared hypotheses: X
-- Lean 4 proof step from Mathlib.CategoryTheory.WithTerminal.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u_1
inst_1 : Category.{v_1, u_1} D
Z : D
F : Functor.{v, v_1, u, u_1} C D
M : (x : C) → F.obj x ⟶ Z
hM : ∀ (x y : C) (f : x ⟶ y), F.map f ≫ M y = M x
x : C
⊢ (lift F M hM).map (Limits.IsTerminal.from starTerminal (Functor.obj.{v, v, u, u} incl x)) ≫ (liftStar F M hM).hom =
    (inclLift F M hM).hom.app x ≫ M x
have simp : (match (motive :=
    (X Y : WithTerminal C) →
      (X ⟶ Y) →
        Quiver.Hom.{v_1, u_1} (V := D)
          (match (motive := WithTerminal C → D) X with
            | of x => F.obj x
            | star => Z :
            D)
          (match (motive := WithTerminal C → D) Y with
            | of x => F.obj x
            | star => Z :
            D))
    Functor.obj.{v, v, u, u} incl x, star, Limits.IsTerminal.from starTerminal (Functor.obj.{v, v, u, u} incl x) with
  | of a, of a_1, f => F.map (down f)
  | of x, star, x_1 => M x
  | star, star, x => 𝟙 Z) =
  M x
Tactic: simp
-- Lean 4 proof step from Mathlib.CategoryTheory.WithTerminal.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u_1
inst_1 : Category.{v_1, u_1} D
Z : D
F : Functor.{v, v_1, u, u_1} C D
M : (x : C) → F.obj x ⟶ Z
hM : ∀ (x y : C) (f : x ⟶ y), F.map f ≫ M y = M x
G : Functor.{v, v_1, u, u_1} (WithTerminal C) D
h : incl ⋙ G ≅ F
hG : G.obj star ≅ Z
hh :
  ∀ (x : C), G.map (Limits.IsTerminal.from starTerminal (Functor.obj.{v, v, u, u} incl x)) ≫ hG.hom = h.hom.app x ≫ M x
X : C
f : of X ⟶ star
⊢ G.map f ≫
      Iso.hom
        ((fun (X : WithTerminal C) ↦
            (match (motive := (X : WithTerminal C) → G.obj X ≅ (lift F M hM).obj X) X with
              | of x => h.app x
              | star => hG :
              G.obj X ≅ (lift F M hM).obj X))
          star) =
    Iso.hom
        ((fun (X : WithTerminal C) ↦
            (match (motive := (X : WithTerminal C) → G.obj X ≅ (lift F M hM).obj X) X with
              | of x => h.app x
              | star => hG :
              G.obj X ≅ (lift F M hM).obj X))
          (of X)) ≫
      (lift F M hM).map f
have of : Eq (α := G.obj (of X) ⟶ (lift F M hM).obj star)
  (CategoryStruct.comp (Y := G.obj star) (G.map PUnit.unit)
    (Iso.hom
      ((fun (X : WithTerminal C) ↦
          (match (motive := (X : WithTerminal C) → G.obj X ≅ (lift F M hM).obj X) X with
            | of x => h.app x
            | star => hG :
            G.obj X ≅ (lift F M hM).obj X))
        star)))
  (CategoryStruct.comp (Y := (lift F M hM).obj (of X))
    (Iso.hom
      ((fun (X : WithTerminal C) ↦
          (match (motive := (X : WithTerminal C) → G.obj X ≅ (lift F M hM).obj X) X with
            | of x => h.app x
            | star => hG :
            G.obj X ≅ (lift F M hM).obj X))
        (of X)))
    ((lift F M hM).map PUnit.unit))
Tactic: cases f
Cleared hypotheses: f
-- Lean 4 proof step from Mathlib.CategoryTheory.WithTerminal.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u_1
inst_1 : Category.{v_1, u_1} D
Z : D
F : Functor.{v, v_1, u, u_1} C D
M : (x : C) → F.obj x ⟶ Z
hM : ∀ (x y : C) (f : x ⟶ y), F.map f ≫ M y = M x
G : Functor.{v, v_1, u, u_1} (WithTerminal C) D
h : incl ⋙ G ≅ F
hG : G.obj star ≅ Z
hh :
  ∀ (x : C), G.map (Limits.IsTerminal.from starTerminal (Functor.obj.{v, v, u, u} incl x)) ≫ hG.hom = h.hom.app x ≫ M x
f : Quiver.Hom.{v, u} (V := WithTerminal C) star star
⊢ G.map f ≫
      Iso.hom
        ((fun (X : WithTerminal C) ↦
            (match (motive := (X : WithTerminal C) → G.obj X ≅ (lift F M hM).obj X) X with
              | of x => h.app x
              | star => hG :
              G.obj X ≅ (lift F M hM).obj X))
          star) =
    Iso.hom
        ((fun (X : WithTerminal C) ↦
            (match (motive := (X : WithTerminal C) → G.obj X ≅ (lift F M hM).obj X) X with
              | of x => h.app x
              | star => hG :
              G.obj X ≅ (lift F M hM).obj X))
          star) ≫
      (lift F M hM).map f
have star : Eq (α := G.obj star ⟶ (lift F M hM).obj star)
  (CategoryStruct.comp (Y := G.obj star) (G.map PUnit.unit)
    (Iso.hom
      ((fun (X : WithTerminal C) ↦
          (match (motive := (X : WithTerminal C) → G.obj X ≅ (lift F M hM).obj X) X with
            | of x => h.app x
            | star => hG :
            G.obj X ≅ (lift F M hM).obj X))
        star)))
  (CategoryStruct.comp (Y := (lift F M hM).obj star)
    (Iso.hom
      ((fun (X : WithTerminal C) ↦
          (match (motive := (X : WithTerminal C) → G.obj X ≅ (lift F M hM).obj X) X with
            | of x => h.app x
            | star => hG :
            G.obj X ≅ (lift F M hM).obj X))
        star))
    ((lift F M hM).map PUnit.unit))
Tactic: cases f
Cleared hypotheses: f
-- Lean 4 proof step from Mathlib.CategoryTheory.WithTerminal.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u_1
inst_1 : Category.{v_1, u_1} D
Z : D
F : Functor.{v, v_1, u, u_1} C D
M : (x : C) → F.obj x ⟶ Z
hM : ∀ (x y : C) (f : x ⟶ y), F.map f ≫ M y = M x
G : Functor.{v, v_1, u, u_1} (WithTerminal C) D
h : incl ⋙ G ≅ F
hG : G.obj star ≅ Z
hh :
  ∀ (x : C), G.map (Limits.IsTerminal.from starTerminal (Functor.obj.{v, v, u, u} incl x)) ≫ hG.hom = h.hom.app x ≫ M x
⊢ Eq (α := G.obj star ⟶ (lift F M hM).obj star)
    (CategoryStruct.comp (Y := G.obj star) (G.map PUnit.unit)
      (Iso.hom
        ((fun (X : WithTerminal C) ↦
            (match (motive := (X : WithTerminal C) → G.obj X ≅ (lift F M hM).obj X) X with
              | of x => h.app x
              | star => hG :
              G.obj X ≅ (lift F M hM).obj X))
          star)))
    (CategoryStruct.comp (Y := (lift F M hM).obj star)
      (Iso.hom
        ((fun (X : WithTerminal C) ↦
            (match (motive := (X : WithTerminal C) → G.obj X ≅ (lift F M hM).obj X) X with
              | of x => h.app x
              | star => hG :
              G.obj X ≅ (lift F M hM).obj X))
          star))
      ((lift F M hM).map PUnit.unit))
have star : G.map (𝟙 star) ≫ hG.hom = hG.hom ≫ 𝟙 Z
Tactic: change G.map (𝟙 _) ≫ hG.hom = hG.hom ≫ 𝟙 _
-- Lean 4 proof step from Mathlib.CategoryTheory.WithTerminal.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u_1
inst_1 : Category.{v_1, u_1} D
F : Functor.{v, v_1, u, u_1} (WithTerminal C) D
x y : C
f : x ⟶ y
⊢ (Functor.obj (𝟭 (Functor.{v, v_1, u, u_1} C D)) (incl ⋙ F)).map f ≫
      F.map (Limits.IsTerminal.from starTerminal (of y)) =
    F.map (Limits.IsTerminal.from starTerminal (of x)) ≫ Functor.map (Functor.obj (Functor.const C) (F.obj star)) f
have dsimp : F.map (Functor.map incl f) ≫ F.map (Limits.IsTerminal.from starTerminal (of y)) =
  F.map (Limits.IsTerminal.from starTerminal (of x)) ≫ 𝟙 (F.obj star)
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.WithTerminal.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u_1
inst_1 : Category.{v_1, u_1} D
F : Functor.{v, v_1, u, u_1} (WithTerminal C) D
x y : C
f : x ⟶ y
⊢ F.map (Functor.map incl f) ≫ F.map (Limits.IsTerminal.from starTerminal (of y)) =
    F.map (Limits.IsTerminal.from starTerminal (of x)) ≫ 𝟙 (F.obj star)
have rw : F.map (Functor.map incl f ≫ Limits.IsTerminal.from starTerminal (of y)) =
  F.map (Limits.IsTerminal.from starTerminal (of x))
Tactic: rw [Category.comp_id, ← F.map_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.WithTerminal.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u_1
inst_1 : Category.{v_1, u_1} D
c c' : Comma (A := Functor.{v, v_1, u, u_1} C D) (𝟭 (Functor.{v, v_1, u, u_1} C D)) (Functor.const C)
φ : c ⟶ c'
x y : WithTerminal C
f : x ⟶ y
a : C
x_1 : of a ⟶ star
⊢ Eq (α := (ofCommaObject c).obj (of a) ⟶ (ofCommaObject c').obj star)
    ((ofCommaObject c).map x_1 ≫
      match (motive := (x : WithTerminal C) → (ofCommaObject c).obj x ⟶ (ofCommaObject c').obj x) star with
      | of x => φ.left.app x
      | star => φ.right)
    ((match (motive := (x : WithTerminal C) → (ofCommaObject c).obj x ⟶ (ofCommaObject c').obj x) of a with
      | of x => φ.left.app x
      | star => φ.right) ≫
      (ofCommaObject c').map x_1)
have simp : c.hom.app a ≫ φ.right = φ.left.app a ≫ c'.hom.app a
Tactic: simp
-- Lean 4 proof step from Mathlib.CategoryTheory.WithTerminal.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u_1
inst_1 : Category.{v_1, u_1} D
F : Functor.{v, v_1, u, u_1} (WithTerminal C) D
x y : C
f : x ⟶ y
⊢ Eq (α := (incl ⋙ F).obj x ⟶ F.obj star)
    ((incl ⋙ F).map f ≫ (fun (x : C) ↦ F.map (Limits.IsTerminal.from starTerminal (of x))) y)
    ((fun (x : C) ↦ F.map (Limits.IsTerminal.from starTerminal (of x))) x)
have simp : F.map (Functor.map incl f) ≫ F.map (Limits.IsTerminal.from starTerminal (of y)) =
  F.map (Limits.IsTerminal.from starTerminal (of x))
Tactic: simp only [Functor.comp_obj, Functor.comp_map]
-- Lean 4 proof step from Mathlib.CategoryTheory.WithTerminal.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u_1
inst_1 : Category.{v_1, u_1} D
F : Functor.{v, v_1, u, u_1} (WithTerminal C) D
x y : C
f : x ⟶ y
⊢ F.map (Functor.map incl f) ≫ F.map (Limits.IsTerminal.from starTerminal (of y)) =
    F.map (Limits.IsTerminal.from starTerminal (of x))
have rw : F.map (Functor.map incl f ≫ Limits.IsTerminal.from starTerminal (of y)) =
  F.map (Limits.IsTerminal.from starTerminal (of x))
Tactic: rw [← F.map_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.WithTerminal.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u_1
inst_1 : Category.{v_1, u_1} D
F : Functor.{v, v_1, u, u_1} (WithTerminal C) D
x : C
⊢ F.map (Limits.IsTerminal.from starTerminal (Functor.obj.{v, v, u, u} incl x)) ≫ Iso.hom (Iso.refl (F.obj star)) =
    NatTrans.app (Iso.hom (Iso.refl (incl ⋙ F))) x ≫
      (fun (x : C) ↦ F.map (Limits.IsTerminal.from starTerminal (of x))) x
have simp : F.map (Limits.IsTerminal.from starTerminal (Functor.obj.{v, v, u, u} incl x)) =
  F.map (Limits.IsTerminal.from starTerminal (of x))
Tactic: simp only [Iso.refl_hom, Category.id_comp, Functor.comp_obj,
            NatTrans.id_app, Category.comp_id]
-- Lean 4 proof step from Mathlib.CategoryTheory.WithTerminal.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u_1
inst_1 : Category.{v_1, u_1} D
x : Functor.{v, v_1, u, u_1} (WithTerminal C) D
⊢ mkCommaMorphism
      (Iso.hom
        (NatIso.ofComponents
          (fun (X : WithTerminal C) ↦
            match (motive :=
              (X : WithTerminal C) →
                x.obj X ≅
                  match (motive := WithTerminal C → D) X with
                  | of x_1 => x.obj (Functor.obj.{v, v, u, u} incl x_1)
                  | star => x.obj star)
              X with
            | of x_1 => Iso.app (Iso.refl (incl ⋙ x)) x_1
            | star => Iso.refl (x.obj star))
          ⋯)) =
    𝟙 (mkCommaObject x)
have h₁ : ∀ (x_1 : C),
  NatTrans.app
      (CommaMorphism.left
        (mkCommaMorphism
          (Iso.hom
            (NatIso.ofComponents
              (fun (X : WithTerminal C) ↦
                match (motive :=
                  (X : WithTerminal C) →
                    x.obj X ≅
                      match (motive := WithTerminal C → D) X with
                      | of x_2 => x.obj (Functor.obj.{v, v, u, u} incl x_2)
                      | star => x.obj star)
                  X with
                | of x_2 => Iso.app (Iso.refl (incl ⋙ x)) x_2
                | star => Iso.refl (x.obj star))
              sorry))))
      x_1 =
    (CommaMorphism.left (𝟙 (mkCommaObject x))).app x_1
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.WithTerminal.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u_1
inst_1 : Category.{v_1, u_1} D
x : Functor.{v, v_1, u, u_1} (WithTerminal C) D
h₁ :
  ∀ (x_1 : C),
    NatTrans.app
        (CommaMorphism.left
          (mkCommaMorphism
            (Iso.hom
              (NatIso.ofComponents
                (fun (X : WithTerminal C) ↦
                  match (motive :=
                    (X : WithTerminal C) →
                      x.obj X ≅
                        match (motive := WithTerminal C → D) X with
                        | of x_2 => x.obj (Functor.obj.{v, v, u, u} incl x_2)
                        | star => x.obj star)
                    X with
                  | of x_2 => Iso.app (Iso.refl (incl ⋙ x)) x_2
                  | star => Iso.refl (x.obj star))
                ⋯))))
        x_1 =
      (CommaMorphism.left (𝟙 (mkCommaObject x))).app x_1
⊢ mkCommaMorphism
      (Iso.hom
        (NatIso.ofComponents
          (fun (X : WithTerminal C) ↦
            match (motive :=
              (X : WithTerminal C) →
                x.obj X ≅
                  match (motive := WithTerminal C → D) X with
                  | of x_1 => x.obj (Functor.obj.{v, v, u, u} incl x_1)
                  | star => x.obj star)
              X with
            | of x_1 => Iso.app (Iso.refl (incl ⋙ x)) x_1
            | star => Iso.refl (x.obj star))
          ⋯)) =
    𝟙 (mkCommaObject x)
have h₂ : CommaMorphism.right
    (mkCommaMorphism
      (Iso.hom
        (NatIso.ofComponents
          (fun (X : WithTerminal C) ↦
            match (motive :=
              (X : WithTerminal C) →
                x.obj X ≅
                  match (motive := WithTerminal C → D) X with
                  | of x_1 => x.obj (Functor.obj.{v, v, u, u} incl x_1)
                  | star => x.obj star)
              X with
            | of x_1 => Iso.app (Iso.refl (incl ⋙ x)) x_1
            | star => Iso.refl (x.obj star))
          sorry))) =
  CommaMorphism.right (𝟙 (mkCommaObject x))
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.WithTerminal.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : Type u_1
x x_1 : WithTerminal (Discrete J)
⊢ Subsingleton.{u_1 + 1} (x ⟶ x_1)
have allEq : ∀ (a b : x ⟶ x_1), a = b
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.WithTerminal.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : Type u_1
x x_1 : WithTerminal (Discrete J)
a b : x ⟶ x_1
⊢ a = b
have allEq : ∀ (x x_2 : Discrete J) (a b : PLift (x.as = x_2.as)),
  { down := a : ULift.{u_1, 0} (PLift (x.as = x_2.as)) } = { down := b }
Tactic: casesm* WithTerminal _, (_ : WithTerminal _) ⟶ (_ : WithTerminal _)
Cleared hypotheses: x, x_1, a, b
-- Lean 4 proof step from Mathlib.CategoryTheory.WithTerminal.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : Type u_1
x x_1 : WithTerminal (Discrete J)
a b : x ⟶ x_1
allEq :
  ∀ (x x_2 : Discrete J) (a b : PLift (x.as = x_2.as)),
    { down := a : ULift.{u_1, 0} (PLift (x.as = x_2.as)) } = { down := b }
⊢ a = b
have allEq₁ : ∀ (x : Discrete J), PUnit.unit = PUnit.unit
Tactic: casesm* WithTerminal _, (_ : WithTerminal _) ⟶ (_ : WithTerminal _)
Cleared hypotheses: x, x_1, a, b
-- Lean 4 proof step from Mathlib.CategoryTheory.WithTerminal.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : Type u_1
x x_1 : WithTerminal (Discrete J)
a b : x ⟶ x_1
allEq :
  ∀ (x x_2 : Discrete J) (a b : PLift (x.as = x_2.as)),
    { down := a : ULift.{u_1, 0} (PLift (x.as = x_2.as)) } = { down := b }
allEq₁ : ∀ (x : Discrete J), PUnit.unit = PUnit.unit
⊢ a = b
have allEq₂ : PUnit.unit = PUnit.unit
Tactic: casesm* WithTerminal _, (_ : WithTerminal _) ⟶ (_ : WithTerminal _)
Cleared hypotheses: x, x_1, a, b
-- Lean 4 proof step from Mathlib.CategoryTheory.WithTerminal.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : Type u_1
x : WidePullbackShape J
⊢ (fun (x : WithTerminal (Discrete J)) ↦
        (match (motive := WithTerminal (Discrete J) → WidePullbackShape J) x with
          | of x => some x.as
          | star => none :
          WidePullbackShape J))
      ((fun (x : WidePullbackShape J) ↦
          (match (motive := WidePullbackShape J → WithTerminal (Discrete J)) x with
            | some x => of { as := x }
            | none => star :
            WithTerminal (Discrete J)))
        x) =
    x
have none : Eq.{u_1 + 1} (α := WidePullbackShape J)
  ((fun (x : WithTerminal (Discrete J)) ↦
      (match (motive := WithTerminal (Discrete J) → WidePullbackShape J) x with
        | of x => some x.as
        | star => none :
        WidePullbackShape J))
    ((fun (x : WidePullbackShape J) ↦
        (match (motive := WidePullbackShape J → WithTerminal (Discrete J)) x with
          | some x => of { as := x }
          | none => star :
          WithTerminal (Discrete J)))
      none))
  none
Tactic: cases x
Cleared hypotheses: x
-- Lean 4 proof step from Mathlib.CategoryTheory.WithTerminal.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : Type u_1
x : WidePullbackShape J
none :
  Eq.{u_1 + 1} (α := WidePullbackShape J)
    ((fun (x : WithTerminal (Discrete J)) ↦
        (match (motive := WithTerminal (Discrete J) → WidePullbackShape J) x with
          | of x => some x.as
          | star => Option.none :
          WidePullbackShape J))
      ((fun (x : WidePullbackShape J) ↦
          (match (motive := WidePullbackShape J → WithTerminal (Discrete J)) x with
            | some x => of { as := x }
            | Option.none => star :
            WithTerminal (Discrete J)))
        Option.none))
    Option.none
⊢ (fun (x : WithTerminal (Discrete J)) ↦
        (match (motive := WithTerminal (Discrete J) → WidePullbackShape J) x with
          | of x => some x.as
          | star => Option.none :
          WidePullbackShape J))
      ((fun (x : WidePullbackShape J) ↦
          (match (motive := WidePullbackShape J → WithTerminal (Discrete J)) x with
            | some x => of { as := x }
            | Option.none => star :
            WithTerminal (Discrete J)))
        x) =
    x
have some : ∀ (val : J),
  (fun (x : WithTerminal (Discrete J)) ↦
        (match (motive := WithTerminal (Discrete J) → WidePullbackShape J) x with
          | of x => some x.as
          | star => Option.none :
          WidePullbackShape J))
      ((fun (x : WidePullbackShape J) ↦
          (match (motive := WidePullbackShape J → WithTerminal (Discrete J)) x with
            | some x => of { as := x }
            | Option.none => star :
            WithTerminal (Discrete J)))
        (some val)) =
    some val
Tactic: cases x
Cleared hypotheses: x