awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Presentable.ColimitPresentation (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J_1 : Type u_1
I_1 : J_1 → Type u_2
inst_1 : Category.{v_1, u_1} J_1
inst_2 : (j : J_1) → Category.{?u.52389, u_2} (I_1 j)
D_1 : Functor.{v_1, v, u_1, u} J_1 C
P_1 : (j : J_1) → ColimitPresentation.{?u.52389, u_2, v, u} (I_1 j) (D_1.obj j)
J : Type w
I : J → Type w
inst_3 : SmallCategory J
inst_4 : (j : J) → SmallCategory (I j)
D : Functor.{w, v, w, u} J C
P_2 : (j : J) → ColimitPresentation.{w, w, v, u} (I j) (D.obj j)
X : C
P : ColimitPresentation.{w, w, v, u} J X
Q : (j : J) → ColimitPresentation.{w, w, v, u} (I j) (P.diag.obj j)
inst_5 : ∀ (j : J), IsFiltered.{w, w} (I j)
inst_6 : ∀ (j : J) (i : I j), IsFinitelyPresentable.{w, v, u} ((Q j).diag.obj i)
c :
  Cocone
    {
      obj := fun (k : Total (I := I) Q) ↦
        Functor.obj (diag (Q (k.fst (β := fun (j : J) ↦ I j)))) (k.snd (β := fun (j : J) ↦ I j)),
      map := fun {k l : Total (I := I) Q} (f : k ⟶ l) ↦ f.hom (I := I), map_id := ⋯, map_comp := ⋯ :
      Total (I := I) Q ⥤ C }
j j' : J
u : j ⟶ j'
i : I j
⊢ Eq (α := (Q j).diag.obj i ⟶ c.pt)
    ((Q j).ι.app i ≫
        P.diag.map u ≫
          (Q j').isColimit.desc
            { pt := c.pt,
              ι := { app := fun (i : I j') ↦ c.ι.app (Sigma.mk (β := fun (j : J) ↦ I j) j' i), naturality := ⋯ } } :
      (Q j).diag.obj i ⟶ c.pt)
    (c.ι.app (Sigma.mk (β := fun (j : J) ↦ I j) j i) :
      {
              obj := fun (k : Total (I := I) Q) ↦
                Functor.obj (diag (Q (k.fst (β := fun (j : J) ↦ I j)))) (k.snd (β := fun (j : J) ↦ I j)),
              map := fun {k l : Total (I := I) Q} (f : k ⟶ l) ↦ f.hom (I := I), map_id := ⋯, map_comp := ⋯ :
              Total (I := I) Q ⥤ C }.obj
          (Sigma.mk (β := fun (j : J) ↦ I j) j i) ⟶
        Functor.obj (Functor.obj (Functor.const (Total (I := I) Q)) c.pt) (Sigma.mk (β := fun (j : J) ↦ I j) j i))
have obtain : ∀ (i' : I j') (hom : Total.mk (I := I) Q j i ⟶ Total.mk (I := I) Q j' i'),
  Eq (α := (Q j).diag.obj i ⟶ c.pt)
    ((Q j).ι.app i ≫
        P.diag.map (hom.base (I := I)) ≫
          (Q j').isColimit.desc
            { pt := c.pt,
              ι := { app := fun (i : I j') ↦ c.ι.app (Sigma.mk (β := fun (j : J) ↦ I j) j' i), naturality := sorry } } :
      (Q j).diag.obj i ⟶ c.pt)
    (c.ι.app (Sigma.mk (β := fun (j : J) ↦ I j) j i) :
      {
              obj := fun (k : Total (I := I) Q) ↦
                Functor.obj (diag (Q (k.fst (β := fun (j : J) ↦ I j)))) (k.snd (β := fun (j : J) ↦ I j)),
              map := fun {k l : Total (I := I) Q} (f : k ⟶ l) ↦ f.hom (I := I), map_id := sorry, map_comp := sorry :
              Total (I := I) Q ⥤ C }.obj
          (Sigma.mk (β := fun (j : J) ↦ I j) j i) ⟶
        Functor.obj (Functor.obj (Functor.const (Total (I := I) Q)) c.pt) (Sigma.mk (β := fun (j : J) ↦ I j) j i))
Tactic: obtain ⟨i', hom, rfl⟩ := Total.exists_hom_of_hom (P := Q) i u
Cleared hypotheses: u
-- Lean 4 proof step from Mathlib.CategoryTheory.Presentable.ColimitPresentation (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J_1 : Type u_1
I_1 : J_1 → Type u_2
inst_1 : Category.{v_1, u_1} J_1
inst_2 : (j : J_1) → Category.{?u.52389, u_2} (I_1 j)
D_1 : Functor.{v_1, v, u_1, u} J_1 C
P_1 : (j : J_1) → ColimitPresentation.{?u.52389, u_2, v, u} (I_1 j) (D_1.obj j)
J : Type w
I : J → Type w
inst_3 : SmallCategory J
inst_4 : (j : J) → SmallCategory (I j)
D : Functor.{w, v, w, u} J C
P_2 : (j : J) → ColimitPresentation.{w, w, v, u} (I j) (D.obj j)
X : C
P : ColimitPresentation.{w, w, v, u} J X
Q : (j : J) → ColimitPresentation.{w, w, v, u} (I j) (P.diag.obj j)
inst_5 : ∀ (j : J), IsFiltered.{w, w} (I j)
inst_6 : ∀ (j : J) (i : I j), IsFinitelyPresentable.{w, v, u} ((Q j).diag.obj i)
c :
  Cocone
    {
      obj := fun (k : Total (I := I) Q) ↦
        Functor.obj (diag (Q (k.fst (β := fun (j : J) ↦ I j)))) (k.snd (β := fun (j : J) ↦ I j)),
      map := fun {k l : Total (I := I) Q} (f : k ⟶ l) ↦ f.hom (I := I), map_id := ⋯, map_comp := ⋯ :
      Total (I := I) Q ⥤ C }
j j' : J
i : I j
i' : I j'
hom : Total.mk (I := I) Q j i ⟶ Total.mk (I := I) Q j' i'
⊢ Eq (α := (Q j).diag.obj i ⟶ c.pt)
    ((Q j).ι.app i ≫
        P.diag.map (hom.base (I := I)) ≫
          (Q j').isColimit.desc
            { pt := c.pt,
              ι := { app := fun (i : I j') ↦ c.ι.app (Sigma.mk (β := fun (j : J) ↦ I j) j' i), naturality := ⋯ } } :
      (Q j).diag.obj i ⟶ c.pt)
    (c.ι.app (Sigma.mk (β := fun (j : J) ↦ I j) j i) :
      {
              obj := fun (k : Total (I := I) Q) ↦
                Functor.obj (diag (Q (k.fst (β := fun (j : J) ↦ I j)))) (k.snd (β := fun (j : J) ↦ I j)),
              map := fun {k l : Total (I := I) Q} (f : k ⟶ l) ↦ f.hom (I := I), map_id := ⋯, map_comp := ⋯ :
              Total (I := I) Q ⥤ C }.obj
          (Sigma.mk (β := fun (j : J) ↦ I j) j i) ⟶
        Functor.obj (Functor.obj (Functor.const (Total (I := I) Q)) c.pt) (Sigma.mk (β := fun (j : J) ↦ I j) j i))
have rw : Eq (α := (Q j).diag.obj i ⟶ c.pt)
  (hom.hom (I := I) ≫
      NatTrans.app
        (Cocone.ι
          { pt := c.pt,
            ι := { app := fun (i : I j') ↦ c.ι.app (Sigma.mk (β := fun (j : J) ↦ I j) j' i), naturality := sorry } })
        i' :
    (Q j).diag.obj i ⟶ c.pt)
  (c.ι.app (Sigma.mk (β := fun (j : J) ↦ I j) j i) :
    {
            obj := fun (k : Total (I := I) Q) ↦
              Functor.obj (diag (Q (k.fst (β := fun (j : J) ↦ I j)))) (k.snd (β := fun (j : J) ↦ I j)),
            map := fun {k l : Total (I := I) Q} (f : k ⟶ l) ↦ f.hom (I := I), map_id := sorry, map_comp := sorry :
            Total (I := I) Q ⥤ C }.obj
        (Sigma.mk (β := fun (j : J) ↦ I j) j i) ⟶
      Functor.obj (Functor.obj (Functor.const (Total (I := I) Q)) c.pt) (Sigma.mk (β := fun (j : J) ↦ I j) j i))
Tactic: rw [reassoc_of% hom.w, (Q j').isColimit.fac]
-- Lean 4 proof step from Mathlib.CategoryTheory.Presentable.Dense (goal)
Goal:
C : Type u
inst : Category.{v, u} C
κ : Cardinal.{w}
inst_1 : Fact κ.IsRegular
inst_2 : IsCardinalAccessibleCategory.{w, v, u} C κ
⊢ ObjectProperty.IsCardinalFilteredGenerator.{w, v, u} (isCardinalPresentable C κ) κ
have obtain : ∀ (P : ObjectProperty.{v, u} C),
  ObjectProperty.EssentiallySmall.{w, v, u} P →
    ∀ (hP : P.IsCardinalFilteredGenerator κ),
      ObjectProperty.IsCardinalFilteredGenerator.{w, v, u} (isCardinalPresentable C κ) κ
Tactic: obtain ⟨P, _, hP⟩ := HasCardinalFilteredGenerator.exists_generator C κ
-- Lean 4 proof step from Mathlib.CategoryTheory.Presentable.Dense (goal)
Goal:
C : Type u
inst : Category.{v, u} C
κ : Cardinal.{w}
inst_1 : Fact κ.IsRegular
inst_2 : IsCardinalAccessibleCategory.{w, v, u} C κ
P : ObjectProperty.{v, u} C
w : ObjectProperty.EssentiallySmall.{w, v, u} P
hP : P.IsCardinalFilteredGenerator κ
⊢ ObjectProperty.IsCardinalFilteredGenerator.{w, v, u} (isCardinalPresentable C κ) κ
have refine : P ≤ ObjectProperty.isoClosure (isCardinalPresentable C κ)
Tactic: refine hP.of_le_isoClosure ?_ le_rfl
-- Lean 4 proof step from Mathlib.CategoryTheory.Presentable.Dense (goal)
Goal:
C : Type u
inst : Category.{v, u} C
κ : Cardinal.{w}
inst_1 : Fact κ.IsRegular
inst_2 : IsCardinalAccessibleCategory.{w, v, u} C κ
P : ObjectProperty.{v, u} C
w : ObjectProperty.EssentiallySmall.{w, v, u} P
hP : P.IsCardinalFilteredGenerator κ
⊢ P ≤ ObjectProperty.isoClosure (isCardinalPresentable C κ)
have rw : P ≤ isCardinalPresentable C κ
Tactic: rw [ObjectProperty.isoClosure_eq_self]
-- Lean 4 proof step from Mathlib.CategoryTheory.Presentable.Dense (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
κ : Cardinal.{w}
inst_1 : Fact κ.IsRegular
J : Type u'
inst_2 : Category.{v', u'} J
inst_3 : EssentiallySmall.{w, v', u'} J
inst_4 : IsCardinalFiltered.{w, v', u'} J κ
X : C
p : ObjectProperty.ColimitOfShape.{v', u', v, u} (isCardinalPresentable C κ) J X
⊢ p.toCostructuredArrow (P := isCardinalPresentable C κ).Final
have this : IsFiltered.{v', u'} J
Tactic: have := isFiltered_of_isCardinalFiltered J κ
-- Lean 4 proof step from Mathlib.CategoryTheory.Presentable.Dense (goal)
Goal:
C : Type u
inst : Category.{v, u} C
κ : Cardinal.{w}
inst_1 : Fact κ.IsRegular
J : Type u'
inst_2 : Category.{v', u'} J
inst_3 : EssentiallySmall.{w, v', u'} J
inst_4 : IsCardinalFiltered.{w, v', u'} J κ
X : C
p : ObjectProperty.ColimitOfShape.{v', u', v, u} (isCardinalPresentable C κ) J X
this : IsFiltered.{v', u'} J
⊢ p.toCostructuredArrow (P := isCardinalPresentable C κ).Final
have rw : (∀
    (d :
      CostructuredArrow.{v, v, u, u} (C := ObjectProperty.FullSubcategory.{v, u} (isCardinalPresentable C κ))
        (ObjectProperty.ι (isCardinalPresentable C κ)) X),
    ∃ (c : J), Nonempty.{v + 1} (d ⟶ p.toCostructuredArrow (P := isCardinalPresentable C κ).obj c)) ∧
  ∀
    {d :
      CostructuredArrow.{v, v, u, u} (C := ObjectProperty.FullSubcategory.{v, u} (isCardinalPresentable C κ))
        (ObjectProperty.ι (isCardinalPresentable C κ)) X}
    {c : J} (s s' : d ⟶ p.toCostructuredArrow (P := isCardinalPresentable C κ).obj c),
    ∃ (c' : J) (t : c ⟶ c'),
      s ≫ p.toCostructuredArrow (P := isCardinalPresentable C κ).map t =
        s' ≫ p.toCostructuredArrow (P := isCardinalPresentable C κ).map t
Tactic: rw [Functor.final_iff_of_isFiltered]
-- Lean 4 proof step from Mathlib.CategoryTheory.Presentable.Dense (goal)
Goal:
C : Type u
inst : Category.{v, u} C
κ : Cardinal.{w}
inst_1 : Fact κ.IsRegular
J : Type u'
inst_2 : Category.{v', u'} J
inst_3 : EssentiallySmall.{w, v', u'} J
inst_4 : IsCardinalFiltered.{w, v', u'} J κ
X : C
p : ObjectProperty.ColimitOfShape.{v', u', v, u} (isCardinalPresentable C κ) J X
this : IsFiltered.{v', u'} J
⊢ (∀
      (d :
        CostructuredArrow.{v, v, u, u} (C := ObjectProperty.FullSubcategory.{v, u} (isCardinalPresentable C κ))
          (ObjectProperty.ι (isCardinalPresentable C κ)) X),
      ∃ (c : J), Nonempty.{v + 1} (d ⟶ p.toCostructuredArrow (P := isCardinalPresentable C κ).obj c)) ∧
    ∀
      {d :
        CostructuredArrow.{v, v, u, u} (C := ObjectProperty.FullSubcategory.{v, u} (isCardinalPresentable C κ))
          (ObjectProperty.ι (isCardinalPresentable C κ)) X}
      {c : J} (s s' : d ⟶ p.toCostructuredArrow (P := isCardinalPresentable C κ).obj c),
      ∃ (c' : J) (t : c ⟶ c'),
        s ≫ p.toCostructuredArrow (P := isCardinalPresentable C κ).map t =
          s' ≫ p.toCostructuredArrow (P := isCardinalPresentable C κ).map t
have refine_1 : ∀
  (f :
    CostructuredArrow.{v, v, u, u} (C := ObjectProperty.FullSubcategory.{v, u} (isCardinalPresentable C κ))
      (ObjectProperty.ι (isCardinalPresentable C κ)) X),
  ∃ (c : J), Nonempty.{v + 1} (f ⟶ p.toCostructuredArrow (P := isCardinalPresentable C κ).obj c)
Tactic: refine ⟨fun f ↦ ?_, fun {f j} g₁ g₂ ↦ ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Presentable.Dense (goal)
Goal:
C : Type u
inst : Category.{v, u} C
κ : Cardinal.{w}
inst_1 : Fact κ.IsRegular
J : Type u'
inst_2 : Category.{v', u'} J
inst_3 : EssentiallySmall.{w, v', u'} J
inst_4 : IsCardinalFiltered.{w, v', u'} J κ
X : C
p : ObjectProperty.ColimitOfShape.{v', u', v, u} (isCardinalPresentable C κ) J X
this : IsFiltered.{v', u'} J
refine_1 :
  ∀
    (f :
      CostructuredArrow.{v, v, u, u} (C := ObjectProperty.FullSubcategory.{v, u} (isCardinalPresentable C κ))
        (ObjectProperty.ι (isCardinalPresentable C κ)) X),
    ∃ (c : J), Nonempty.{v + 1} (f ⟶ p.toCostructuredArrow (P := isCardinalPresentable C κ).obj c)
⊢ (∀
      (d :
        CostructuredArrow.{v, v, u, u} (C := ObjectProperty.FullSubcategory.{v, u} (isCardinalPresentable C κ))
          (ObjectProperty.ι (isCardinalPresentable C κ)) X),
      ∃ (c : J), Nonempty.{v + 1} (d ⟶ p.toCostructuredArrow (P := isCardinalPresentable C κ).obj c)) ∧
    ∀
      {d :
        CostructuredArrow.{v, v, u, u} (C := ObjectProperty.FullSubcategory.{v, u} (isCardinalPresentable C κ))
          (ObjectProperty.ι (isCardinalPresentable C κ)) X}
      {c : J} (s s' : d ⟶ p.toCostructuredArrow (P := isCardinalPresentable C κ).obj c),
      ∃ (c' : J) (t : c ⟶ c'),
        s ≫ p.toCostructuredArrow (P := isCardinalPresentable C κ).map t =
          s' ≫ p.toCostructuredArrow (P := isCardinalPresentable C κ).map t
have refine_2 : ∀
  {f :
    CostructuredArrow.{v, v, u, u} (C := ObjectProperty.FullSubcategory.{v, u} (isCardinalPresentable C κ))
      (ObjectProperty.ι (isCardinalPresentable C κ)) X}
  {j : J} (g₁ g₂ : f ⟶ p.toCostructuredArrow (P := isCardinalPresentable C κ).obj j),
  ∃ (c' : J) (t : j ⟶ c'),
    g₁ ≫ p.toCostructuredArrow (P := isCardinalPresentable C κ).map t =
      g₂ ≫ p.toCostructuredArrow (P := isCardinalPresentable C κ).map t
Tactic: refine ⟨fun f ↦ ?_, fun {f j} g₁ g₂ ↦ ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Presentable.Dense (goal)
Goal:
C : Type u
inst : Category.{v, u} C
κ : Cardinal.{w}
inst_1 : Fact κ.IsRegular
J : Type u'
inst_2 : Category.{v', u'} J
inst_3 : EssentiallySmall.{w, v', u'} J
inst_4 : IsCardinalFiltered.{w, v', u'} J κ
X : C
p : ObjectProperty.ColimitOfShape.{v', u', v, u} (isCardinalPresentable C κ) J X
this : IsFiltered.{v', u'} J
f :
  CostructuredArrow.{v, v, u, u} (C := ObjectProperty.FullSubcategory.{v, u} (isCardinalPresentable C κ))
    (ObjectProperty.ι (isCardinalPresentable C κ)) X
⊢ ∃ (c : J), Nonempty.{v + 1} (f ⟶ p.toCostructuredArrow (P := isCardinalPresentable C κ).obj c)
have refine_1 : ∀ (j : J)
  (g :
    Functor.obj (ObjectProperty.ι (isCardinalPresentable C κ)) f.left ⟶
      p.toColimitPresentation (P := isCardinalPresentable C κ).diag.obj j)
  (hg :
    g ≫
        { pt := X, ι := p.toColimitPresentation (P := isCardinalPresentable C κ).ι :
                Cocone p.toColimitPresentation (P := isCardinalPresentable C κ).diag }.ι.app
          j =
      f.hom),
  ∃ (c : J), Nonempty.{v + 1} (f ⟶ p.toCostructuredArrow (P := isCardinalPresentable C κ).obj c)
Tactic: obtain ⟨j, g, hg⟩ := IsCardinalPresentable.exists_hom_of_isColimit κ p.isColimit f.hom
-- Lean 4 proof step from Mathlib.CategoryTheory.Presentable.Dense (goal)
Goal:
C : Type u
inst : Category.{v, u} C
κ : Cardinal.{w}
inst_1 : Fact κ.IsRegular
J : Type u'
inst_2 : Category.{v', u'} J
inst_3 : EssentiallySmall.{w, v', u'} J
inst_4 : IsCardinalFiltered.{w, v', u'} J κ
X : C
p : ObjectProperty.ColimitOfShape.{v', u', v, u} (isCardinalPresentable C κ) J X
this : IsFiltered.{v', u'} J
f :
  CostructuredArrow.{v, v, u, u} (C := ObjectProperty.FullSubcategory.{v, u} (isCardinalPresentable C κ))
    (ObjectProperty.ι (isCardinalPresentable C κ)) X
j : J
g₁ g₂ : f ⟶ p.toCostructuredArrow (P := isCardinalPresentable C κ).obj j
⊢ ∃ (c' : J) (t : j ⟶ c'),
    g₁ ≫ p.toCostructuredArrow (P := isCardinalPresentable C κ).map t =
      g₂ ≫ p.toCostructuredArrow (P := isCardinalPresentable C κ).map t
have refine_2 : ∀ (k : J) (a : j ⟶ k)
  (h :
    g₁.left.hom ≫ p.toColimitPresentation (P := isCardinalPresentable C κ).diag.map a =
      g₂.left.hom ≫ p.toColimitPresentation (P := isCardinalPresentable C κ).diag.map a),
  ∃ (c' : J) (t : j ⟶ c'),
    g₁ ≫ p.toCostructuredArrow (P := isCardinalPresentable C κ).map t =
      g₂ ≫ p.toCostructuredArrow (P := isCardinalPresentable C κ).map t
Tactic: obtain ⟨k, a, h⟩ := IsCardinalPresentable.exists_eq_of_isColimit' κ p.isColimit
      g₁.left.hom g₂.left.hom ((CostructuredArrow.w g₁).trans (CostructuredArrow.w g₂).symm)
-- Lean 4 proof step from Mathlib.CategoryTheory.Presentable.Dense (goal)
Goal:
C : Type u
inst : Category.{v, u} C
κ : Cardinal.{w}
inst_1 : Fact κ.IsRegular
inst_2 : IsCardinalAccessibleCategory.{w, v, u} C κ
X : C
⊢ Functor.isDenseAt.{v, v, u, u} (C := ObjectProperty.FullSubcategory.{v, u} (isCardinalPresentable C κ))
    (ObjectProperty.ι (isCardinalPresentable C κ)) X
have obtain : ∀ (J : Type w) (w : SmallCategory J),
  IsCardinalFiltered J κ →
    ∀ (p : ObjectProperty.ColimitOfShape.{w, w, v, u} (isCardinalPresentable C κ) J X),
      Functor.isDenseAt.{v, v, u, u} (C := ObjectProperty.FullSubcategory.{v, u} (isCardinalPresentable C κ))
        (ObjectProperty.ι (isCardinalPresentable C κ)) X
Tactic: obtain ⟨J, _, _, ⟨p⟩⟩ :=
      (isCardinalFilteredGenerator_isCardinalPresentable C κ).exists_colimitsOfShape X
-- Lean 4 proof step from Mathlib.CategoryTheory.Presentable.Dense (goal)
Goal:
C : Type u
inst : Category.{v, u} C
κ : Cardinal.{w}
inst_1 : Fact κ.IsRegular
inst_2 : IsCardinalAccessibleCategory.{w, v, u} C κ
X : C
⊢ IsCardinalFiltered
    (CostructuredArrow.{v, v, u, u} (C := ObjectProperty.FullSubcategory.{v, u} (isCardinalPresentable C κ))
      (ObjectProperty.ι (isCardinalPresentable C κ)) X)
    κ
have obtain : ∀ (J : Type w) (w : SmallCategory J),
  IsCardinalFiltered J κ →
    ∀ (p : ObjectProperty.ColimitOfShape.{w, w, v, u} (isCardinalPresentable C κ) J X),
      IsCardinalFiltered
        (CostructuredArrow.{v, v, u, u} (C := ObjectProperty.FullSubcategory.{v, u} (isCardinalPresentable C κ))
          (ObjectProperty.ι (isCardinalPresentable C κ)) X)
        κ
Tactic: obtain ⟨J, _, _, ⟨p⟩⟩ :=
    (isCardinalFilteredGenerator_isCardinalPresentable C κ).exists_colimitsOfShape X
-- Lean 4 proof step from Mathlib.CategoryTheory.Presentable.Dense (goal)
Goal:
C : Type u
inst : Category.{v, u} C
κ : Cardinal.{w}
inst_1 : Fact κ.IsRegular
inst_2 : IsFinitelyAccessibleCategory.{w, v, u} C
X : C
⊢ IsFiltered
    (CostructuredArrow.{v, v, u, u} (C :=
      ObjectProperty.FullSubcategory.{v, u} (ObjectProperty.isFinitelyPresentable C))
      (ObjectProperty.ι (ObjectProperty.isFinitelyPresentable C)) X)
have rw : IsCardinalFiltered
  (CostructuredArrow.{v, v, u, u} (C := ObjectProperty.FullSubcategory.{v, u} (isCardinalPresentable C Cardinal.aleph0))
    (ObjectProperty.ι (isCardinalPresentable C Cardinal.aleph0)) X)
  Cardinal.aleph0
Tactic: rw [← CategoryTheory.isCardinalFiltered_aleph0_iff.{w},
    ObjectProperty.isFinitelyPresentable_eq_isCardinalPresentable]
-- Lean 4 proof step from Mathlib.CategoryTheory.Presentable.Directed (goal)
Goal:
J : Type w
inst : SmallCategory J
κ : Cardinal.{w}
D : Diagram J κ
e : J
h₁ h₂ : D.IsTerminal e
⊢ @lift J inst κ D e h₁ = @lift J inst κ D e h₂
have h : ∀ (j : J) (hj : D.P j), h₁.lift hj = h₂.lift hj
Tactic: ext j hj
-- Lean 4 proof step from Mathlib.CategoryTheory.Presentable.Directed (hypothesis)
Goal:
J : Type w
inst : SmallCategory J
κ : Cardinal.{w}
D : Diagram J κ
e : J
h₁ h₂ : D.IsTerminal e
⊢ h₁ = h₂
have this : @lift J inst κ D e h₁ = @lift J inst κ D e h₂
Tactic: have : @h₁.lift = @h₂.lift := by
      ext j hj
      exact h₁.uniq hj (h₂.hlift hj)
-- Lean 4 proof step from Mathlib.CategoryTheory.Presentable.Directed (goal)
Goal:
J : Type w
inst : SmallCategory J
κ : Cardinal.{w}
D : Diagram J κ
e : J
h₁ h₂ : D.IsTerminal e
this : @lift J inst κ D e h₁ = @lift J inst κ D e h₂
⊢ h₁ = h₂
have mk : ∀ (prop_id : D.W (𝟙 e)) (lift : {j : J} → (hj : D.P j) → j ⟶ e) (hlift : ∀ {j : J} (hj : D.P j), D.W (lift hj))
  (uniq : ∀ {j : J} (hj : D.P j) {φ : j ⟶ e} (hφ : D.W φ), lift hj = φ)
  (comm : ∀ {i j : J} (f : i ⟶ j) (hf : D.W f), f ≫ lift sorry = lift sorry)
  (this :
    @IsTerminal.lift J inst κ D e
        { prop_id := prop_id, lift := @lift, hlift := @hlift, uniq := @uniq, comm := @comm : D.IsTerminal e } =
      @IsTerminal.lift J inst κ D e h₂),
  { prop_id := prop_id, lift := @lift, hlift := @hlift, uniq := @uniq, comm := @comm : D.IsTerminal e } = h₂
Tactic: cases h₁
Cleared hypotheses: h₁, this
-- Lean 4 proof step from Mathlib.CategoryTheory.Presentable.Directed (goal)
Goal:
J : Type w
inst : SmallCategory J
κ : Cardinal.{w}
D : Diagram J κ
e : J
h₂ : D.IsTerminal e
prop_id : D.W (𝟙 e)
lift : {j : J} → (hj : D.P j) → j ⟶ e
hlift : ∀ {j : J} (hj : D.P j), D.W (lift hj)
uniq : ∀ {j : J} (hj : D.P j) {φ : j ⟶ e} (hφ : D.W φ), lift hj = φ
comm : ∀ {i j : J} (f : i ⟶ j) (hf : D.W f), f ≫ lift ⋯ = lift ⋯
this :
  @IsTerminal.lift J inst κ D e
      { prop_id := prop_id, lift := @lift, hlift := @hlift, uniq := @uniq, comm := @comm : D.IsTerminal e } =
    @IsTerminal.lift J inst κ D e h₂
⊢ { prop_id := prop_id, lift := @lift, hlift := @hlift, uniq := @uniq, comm := @comm : D.IsTerminal e } = h₂
have mk : ∀ (prop_id_1 : D.W (𝟙 e)) (lift_1 : {j : J} → (hj : D.P j) → j ⟶ e) (hlift_1 : ∀ {j : J} (hj : D.P j), D.W (lift_1 hj))
  (uniq_1 : ∀ {j : J} (hj : D.P j) {φ : j ⟶ e} (hφ : D.W φ), lift_1 hj = φ)
  (comm_1 : ∀ {i j : J} (f : i ⟶ j) (hf : D.W f), f ≫ lift_1 sorry = lift_1 sorry)
  (this :
    @IsTerminal.lift J inst κ D e
        { prop_id := prop_id, lift := @lift, hlift := @hlift, uniq := @uniq, comm := @comm : D.IsTerminal e } =
      @IsTerminal.lift J inst κ D e
        { prop_id := prop_id_1, lift := @lift_1, hlift := @hlift_1, uniq := @uniq_1, comm := @comm_1 }),
  { prop_id := prop_id, lift := @lift, hlift := @hlift, uniq := @uniq, comm := @comm : D.IsTerminal e } =
    { prop_id := prop_id_1, lift := @lift_1, hlift := @hlift_1, uniq := @uniq_1, comm := @comm_1 }
Tactic: cases h₂
Cleared hypotheses: h₂, this
-- Lean 4 proof step from Mathlib.CategoryTheory.Presentable.Directed (goal)
Goal:
J : Type w
inst : SmallCategory J
κ : Cardinal.{w}
D : Diagram J κ
e : J
prop_id : D.W (𝟙 e)
h₁ : ∀ ⦃j : J⦄, D.P j → ∃ (lift : j ⟶ e), D.W lift
h₂ : ∀ ⦃j : J⦄, D.P j → ∀ (l₁ l₂ : j ⟶ e), D.W l₁ → D.W l₂ → l₁ = l₂
h₃ : ∀ ⦃i j : J⦄ (f : i ⟶ j), D.W f → ∃ (li : i ⟶ e) (lj : j ⟶ e), D.W li ∧ D.W lj ∧ f ≫ lj = li
i j : J
x : i ⟶ j
hf : D.W x
⊢ x ≫ ⋯.choose (p := D.W) = ⋯.choose (p := D.W)
have obtain : ∀ (li : i ⟶ e) (lj : j ⟶ e) (hli : D.W li) (hlj : D.W lj) (fac : x ≫ lj = li),
  x ≫ sorry.choose (p := D.W) = sorry.choose (p := D.W)
Tactic: obtain ⟨li, lj, hli, hlj, fac⟩ := h₃ _ hf
-- Lean 4 proof step from Mathlib.CategoryTheory.Presentable.Directed (goal)
Goal:
J : Type w
inst : SmallCategory J
κ : Cardinal.{w}
D : Diagram J κ
e : J
prop_id : D.W (𝟙 e)
h₁ : ∀ ⦃j : J⦄, D.P j → ∃ (lift : j ⟶ e), D.W lift
h₂ : ∀ ⦃j : J⦄, D.P j → ∀ (l₁ l₂ : j ⟶ e), D.W l₁ → D.W l₂ → l₁ = l₂
h₃ : ∀ ⦃i j : J⦄ (f : i ⟶ j), D.W f → ∃ (li : i ⟶ e) (lj : j ⟶ e), D.W li ∧ D.W lj ∧ f ≫ lj = li
i j : J
x : i ⟶ j
hf : D.W x
li : i ⟶ e
lj : j ⟶ e
hli : D.W li
hlj : D.W lj
fac : x ≫ lj = li
⊢ x ≫ ⋯.choose (p := D.W) = ⋯.choose (p := D.W)
have rw : x ≫ sorry.choose (p := D.W) = li
Tactic: rw [h₂ (D.src hf) _ li (h₁ (D.src hf)).choose_spec hli,
      h₂ (D.tgt hf) _ lj (h₁ (D.tgt hf)).choose_spec hlj, fac]
-- Lean 4 proof step from Mathlib.CategoryTheory.Presentable.Directed (goal)
Goal:
J : Type w
inst : SmallCategory J
κ : Cardinal.{w}
D : Diagram J κ
e : J
prop_id : D.W (𝟙 e)
h₁ : ∀ ⦃j : J⦄, D.P j → ∃ (lift : j ⟶ e), D.W lift
h₂ : ∀ ⦃j : J⦄, D.P j → ∀ (l₁ l₂ : j ⟶ e), D.W l₁ → D.W l₂ → l₁ = l₂
h₃ : ∀ ⦃i j : J⦄ (f : i ⟶ j), D.W f → ∃ (li : i ⟶ e) (lj : j ⟶ e), D.W li ∧ D.W lj ∧ f ≫ lj = li
i j : J
x : i ⟶ j
hf : D.W x
li : i ⟶ e
lj : j ⟶ e
hli : D.W li
hlj : D.W lj
fac : x ≫ lj = li
rw : x ≫ ⋯.choose (p := D.W) = li
rw₁ : x ≫ lj = li
⊢ x ≫ ⋯.choose (p := D.W) = ⋯.choose (p := D.W)
have rw₂ : li = li
Tactic: rw [h₂ (D.src hf) _ li (h₁ (D.src hf)).choose_spec hli,
      h₂ (D.tgt hf) _ lj (h₁ (D.tgt hf)).choose_spec hlj, fac]
-- Lean 4 proof step from Mathlib.CategoryTheory.Presentable.Directed (goal)
Goal:
J : Type w
inst : SmallCategory J
κ : Cardinal.{w}
D₁ D₂ : DiagramWithUniqueTerminal J κ
hW : D₁.W = D₂.W
hP : D₁.P = D₂.P
⊢ D₁ = D₂
have obtain : ∀ (D₁ : Diagram J κ) (top : J) (h₁ : D₁.IsTerminal top) (h₁' : ∀ (j : J) (hj : D₁.IsTerminal j), j = top)
  (hW :
    { toDiagram := D₁, top := top, isTerminal := h₁, uniq_terminal := h₁' : DiagramWithUniqueTerminal J κ }.W = D₂.W)
  (hP :
    { toDiagram := D₁, top := top, isTerminal := h₁, uniq_terminal := h₁' : DiagramWithUniqueTerminal J κ }.P = D₂.P),
  { toDiagram := D₁, top := top, isTerminal := h₁, uniq_terminal := h₁' : DiagramWithUniqueTerminal J κ } = D₂
Tactic: obtain ⟨D₁, top, h₁, h₁'⟩ := D₁
Cleared hypotheses: D₁, hW, hP