awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Filtered.FinallySmall (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : IsFiltered.{v, u} C
inst_2 : LocallySmall.{w, v, u} C
inst_3 : FinallySmall.{w, v, u} C
⊢ ∃ (D : Type w) (x : SmallCategory D) (_ : IsFiltered D) (F : Functor.{w, v, w, u} D C), F.Final
have this : ∀ (C₀ : Type u) [inst : Category.{w, u} C₀] [IsFiltered.{w, u} C₀] [FinallySmall.{w, w, u} C₀],
  ∃ (D : Type u) (_ : Small.{w, u} D) (x : Category.{w, u} D) (_ : IsFiltered.{w, u} D) (F : Functor.{w, w, u, u} D C₀),
    F.Final
Tactic: have (C₀ : Type u) [Category.{w} C₀] [IsFiltered C₀] [FinallySmall.{w} C₀] :
      ∃ (D : Type u) (_ : Small.{w} D) (_ : Category.{w} D) (_ : IsFiltered D) (F : D ⥤ C₀),
        F.Final := by
    /- For `D`, we can choose the full subcategory of `C₀` which is the strict image
    of the final functor `fromFinalModel.{w} C₀ : FinalModel.{w} C₀ ⥤ C₀`,
    where `FinalModel.{w} C₀` is a `w`-small category. -/
    let P : ObjectProperty C₀ := ObjectProperty.strictMap ⊤ (fromFinalModel.{w} C₀)
    have hP (X : C₀) : ∃ (Y : C₀) (hY : P Y), Nonempty (X ⟶ Y) := by
      let f : StructuredArrow X (fromFinalModel.{w} C₀) := Classical.arbitrary _
      exact ⟨_, ObjectProperty.strictMap_obj _ _ (X := f.right) (by tauto), ⟨f.hom⟩⟩
    have : IsFiltered P.FullSubcategory :=
      { nonempty := by
          obtain ⟨X, hX, _⟩ := hP (Classical.arbitrary C₀)
          exact ⟨X, hX⟩
        cocone_objs := by
          rintro ⟨X, hX⟩ ⟨Y, hY⟩
          obtain ⟨Z, hZ, ⟨f⟩⟩ := hP (max X Y)
          exact ⟨⟨Z, hZ⟩, ObjectProperty.homMk (leftToMax X Y ≫ f),
            ObjectProperty.homMk (rightToMax X Y ≫ f), by tauto⟩
        cocone_maps := by
          rintro ⟨X, hX⟩ ⟨Y, hY⟩ ⟨f₁ : X ⟶ Y⟩ ⟨f₂ : X ⟶ Y⟩
          obtain ⟨Z, hZ, ⟨g⟩⟩ := hP (coeq f₁ f₂)
          exact ⟨⟨Z, hZ⟩, ObjectProperty.homMk (coeqHom f₁ f₂ ≫ g),
            ObjectProperty.hom_ext _ (coeq_condition_assoc _ _ _) ⟩ }
    let G : FinalModel.{w} C₀ ⥤ P.FullSubcategory :=
      { obj X := ⟨(fromFinalModel.{w} C₀).obj X, by tauto⟩
        map f := ObjectProperty.homMk ((fromFinalModel.{w} C₀).map f) }
    have : (G ⋙ P.ι).Final := inferInstanceAs (fromFinalModel.{w} C₀).Final
    exact ⟨P.FullSubcategory, small_of_surjective (f := G.obj)
      (by rintro ⟨_, Y, _, rfl⟩; exact ⟨Y, rfl⟩), inferInstance, inferInstance, P.ι,
      Functor.final_of_comp_full_faithful' G P.ι ⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Filtered.FinallySmall (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : IsFiltered.{v, u} C
inst_2 : LocallySmall.{w, v, u} C
inst_3 : FinallySmall.{w, v, u} C
this :
  ∀ (C₀ : Type u) [inst : Category.{w, u} C₀] [IsFiltered.{w, u} C₀] [FinallySmall.{w, w, u} C₀],
    ∃ (D : Type u) (_ : Small.{w, u} D) (x : Category.{w, u} D) (_ : IsFiltered.{w, u} D) (F :
      Functor.{w, w, u, u} D C₀), F.Final
C₀ : Type u
inst_4 : Category.{w, u} C₀
h : IsFiltered.{w, u} C₀
h_1 : FinallySmall.{w, w, u} C₀
⊢ ∃ (D : Type w) (x : SmallCategory D) (_ : IsFiltered D) (F : D ⥤ C₀), F.Final
have obtain : ∀ (D : Type u),
  Small.{w, u} D →
    ∀ (w : Category.{w, u} D),
      IsFiltered.{w, u} D →
        ∀ (F : Functor.{w, w, u, u} D C₀),
          F.Final → ∃ (D : Type w) (x : SmallCategory D) (_ : IsFiltered D) (F : D ⥤ C₀), F.Final
Tactic: obtain ⟨D, _, _, _, F, _⟩ := this C₀
-- Lean 4 proof step from Mathlib.CategoryTheory.Filtered.FinallySmall (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : IsFiltered.{v, u} C
inst_2 : LocallySmall.{w, v, u} C
inst_3 : FinallySmall.{w, v, u} C
this :
  ∀ (C₀ : Type u) [inst : Category.{w, u} C₀] [IsFiltered.{w, u} C₀] [FinallySmall.{w, w, u} C₀],
    ∃ (D : Type u) (_ : Small.{w, u} D) (x : Category.{w, u} D) (_ : IsFiltered.{w, u} D) (F :
      Functor.{w, w, u, u} D C₀), F.Final
⊢ ∃ (D : Type w) (x : SmallCategory D) (_ : IsFiltered D) (F : Functor.{w, v, w, u} D C), F.Final
have this₁ : ∀ (C₀ : Type u) [inst : Category.{w, u} C₀],
  IsFiltered.{w, u} C₀ →
    FinallySmall.{w, w, u} C₀ → ∃ (D : Type w) (x : SmallCategory D) (_ : IsFiltered D) (F : D ⥤ C₀), F.Final
Tactic: have (C₀ : Type u) [Category.{w} C₀] (_ : IsFiltered C₀) (_ : FinallySmall.{w} C₀) :
      ∃ (D : Type w) (_ : SmallCategory D) (_ : IsFiltered D) (F : D ⥤ C₀), F.Final := by
    obtain ⟨D, _, _, _, F, _⟩ := this C₀
    let e := equivSmallModel.{w} D
    exact ⟨_, _, of_equivalence e, e.inverse ⋙ F, inferInstance⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Filtered.FinallySmall (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : IsFiltered.{v, u} C
inst_2 : LocallySmall.{w, v, u} C
inst_3 : FinallySmall.{w, v, u} C
this_1 :
  ∀ (C₀ : Type u) [inst : Category.{w, u} C₀] [IsFiltered.{w, u} C₀] [FinallySmall.{w, w, u} C₀],
    ∃ (D : Type u) (_ : Small.{w, u} D) (x : Category.{w, u} D) (_ : IsFiltered.{w, u} D) (F :
      Functor.{w, w, u, u} D C₀), F.Final
this :
  ∀ (C₀ : Type u) [inst : Category.{w, u} C₀],
    IsFiltered.{w, u} C₀ →
      FinallySmall.{w, w, u} C₀ → ∃ (D : Type w) (x : SmallCategory D) (_ : IsFiltered D) (F : D ⥤ C₀), F.Final
e : Equivalence.{v, w, u, u} C (ShrinkHoms.{u} C) := ShrinkHoms.equivalence C
⊢ ∃ (D : Type w) (x : SmallCategory D) (_ : IsFiltered D) (F : Functor.{w, v, w, u} D C), F.Final
have obtain : ∀ (D : Type w) (w : SmallCategory D),
  IsFiltered D →
    ∀ (F : Functor.{w, w, w, u} D (ShrinkHoms.{u} C)),
      F.Final → ∃ (D : Type w) (x : SmallCategory D) (_ : IsFiltered D) (F : Functor.{w, v, w, u} D C), F.Final
Tactic: obtain ⟨D, _, _, F, _⟩ := this (ShrinkHoms C)
    (of_equivalence e) (finallySmall_of_final_of_finallySmall e.functor)
-- Lean 4 proof step from Mathlib.CategoryTheory.Filtered.FinallySmall (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : IsCofiltered.{v, u} C
inst_2 : LocallySmall.{w, v, u} C
inst_3 : InitiallySmall.{w, v, u} C
⊢ ∃ (D : Type w) (x : SmallCategory D) (_ : IsCofiltered D) (F : Functor.{w, v, w, u} D C), F.Initial
have obtain : ∀ (D : Type w) (w : SmallCategory D),
  IsFiltered D →
    ∀ (F : Functor.{w, v, w, u} D Cᵒᵖ),
      F.Final → ∃ (D : Type w) (x : SmallCategory D) (_ : IsCofiltered D) (F : Functor.{w, v, w, u} D C), F.Initial
Tactic: obtain ⟨D, _, _, F, _⟩ := FinallySmall.exists_of_isFiltered.{w} Cᵒᵖ
-- Lean 4 proof step from Mathlib.CategoryTheory.Filtered.Flat (hypothesis)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
F : Functor.{v₁, v₂, u₁, u₂} C D
inst_2 : IsCofiltered.{v₂, u₂} D
inst_3 : RepresentablyFlat F
⊢ IsCofiltered.{v₁, u₁} C
have this : IsFiltered.{v₁, u₁} Cᵒᵖ
Tactic: have := isFiltered_of_representablyCoflat F.op
-- Lean 4 proof step from Mathlib.CategoryTheory.Filtered.Grothendieck (goal)
Goal:
C : Type u
inst : Category.{v, u} C
F : C ⥤ Cat
inst_1 : IsFilteredOrEmpty.{v, u} C
inst_2 : ∀ (c : C), IsFilteredOrEmpty.{u_1, u_2} (↑(F.obj c) : Type u_2)
⊢ IsFilteredOrEmpty (Grothendieck F)
have refine_1 : ∀ (X Y : Grothendieck F), ∃ (Z : Grothendieck F) (x : X ⟶ Z) (x : Y ⟶ Z), True
Tactic: refine ⟨?_, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Filtered.Grothendieck (goal)
Goal:
C : Type u
inst : Category.{v, u} C
F : C ⥤ Cat
inst_1 : IsFilteredOrEmpty.{v, u} C
inst_2 : ∀ (c : C), IsFilteredOrEmpty.{u_1, u_2} (↑(F.obj c) : Type u_2)
refine_1 : ∀ (X Y : Grothendieck F), ∃ (Z : Grothendieck F) (x : X ⟶ Z) (x : Y ⟶ Z), True
⊢ IsFilteredOrEmpty (Grothendieck F)
have refine_2 : ∀ ⦃X Y : Grothendieck F⦄ (f g : X ⟶ Y), ∃ (Z : Grothendieck F) (h : Y ⟶ Z), f ≫ h = g ≫ h
Tactic: refine ⟨?_, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Filtered.Grothendieck (goal)
Goal:
C : Type u
inst : Category.{v, u} C
F : C ⥤ Cat
inst_1 : IsFilteredOrEmpty.{v, u} C
inst_2 : ∀ (c : C), IsFilteredOrEmpty.{u_1, u_2} (↑(F.obj c) : Type u_2)
c : C
f : (↑(F.obj c) : Type u_2)
d : C
g : (↑(F.obj d) : Type u_2)
u : { base := c, fiber := f : Grothendieck F }.base ⟶ { base := d, fiber := g : Grothendieck F }.base
x : (F.map u).toFunctor.obj { base := c, fiber := f : Grothendieck F }.fiber ⟶ { base := d, fiber := g }.fiber
v : { base := c, fiber := f : Grothendieck F }.base ⟶ { base := d, fiber := g : Grothendieck F }.base
y : (F.map v).toFunctor.obj { base := c, fiber := f : Grothendieck F }.fiber ⟶ { base := d, fiber := g }.fiber
⊢ ∃ (Z : Grothendieck F) (h : { base := d, fiber := g : Grothendieck F } ⟶ Z),
    Eq (α := { base := c, fiber := f : Grothendieck F } ⟶ Z)
      ({ base := u, fiber := x :
          { base := c, fiber := f : Grothendieck F }.Hom { base := d, fiber := g : Grothendieck F } } ≫
        h)
      ({ base := v, fiber := y :
          { base := c, fiber := f : Grothendieck F }.Hom { base := d, fiber := g : Grothendieck F } } ≫
        h)
have refine_2 : Eq (α := (↑(F.obj (coeq u v)) : Type u_2))
  (Functor.obj (Cat.Hom.toFunctor (F.map (coeqHom u v)))
    ((F.map v).toFunctor.obj { base := c, fiber := f : Grothendieck F }.fiber))
  (Functor.obj (Cat.Hom.toFunctor (F.map (coeqHom u v)))
    ((F.map u).toFunctor.obj { base := c, fiber := f : Grothendieck F }.fiber))
Tactic: refine ⟨⟨coeq u v, coeq (eqToHom ?_ ≫
        (F.map (coeqHom u v)).toFunctor.map x) ((F.map (coeqHom u v)).toFunctor.map y)⟩,
          ⟨coeqHom u v, coeqHom _ _⟩, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Filtered.Grothendieck (goal)
Goal:
C : Type u
inst : Category.{v, u} C
F : C ⥤ Cat
inst_1 : IsFilteredOrEmpty.{v, u} C
inst_2 : ∀ (c : C), IsFilteredOrEmpty.{u_1, u_2} (↑(F.obj c) : Type u_2)
c : C
f : (↑(F.obj c) : Type u_2)
d : C
g : (↑(F.obj d) : Type u_2)
u : { base := c, fiber := f : Grothendieck F }.base ⟶ { base := d, fiber := g : Grothendieck F }.base
x : (F.map u).toFunctor.obj { base := c, fiber := f : Grothendieck F }.fiber ⟶ { base := d, fiber := g }.fiber
v : { base := c, fiber := f : Grothendieck F }.base ⟶ { base := d, fiber := g : Grothendieck F }.base
y : (F.map v).toFunctor.obj { base := c, fiber := f : Grothendieck F }.fiber ⟶ { base := d, fiber := g }.fiber
refine_2 :
  Eq (α := (↑(F.obj (coeq u v)) : Type u_2))
    (Functor.obj (Cat.Hom.toFunctor (F.map (coeqHom u v)))
      ((F.map v).toFunctor.obj { base := c, fiber := f : Grothendieck F }.fiber))
    (Functor.obj (Cat.Hom.toFunctor (F.map (coeqHom u v)))
      ((F.map u).toFunctor.obj { base := c, fiber := f : Grothendieck F }.fiber))
⊢ ∃ (Z : Grothendieck F) (h : { base := d, fiber := g : Grothendieck F } ⟶ Z),
    Eq (α := { base := c, fiber := f : Grothendieck F } ⟶ Z)
      ({ base := u, fiber := x :
          { base := c, fiber := f : Grothendieck F }.Hom { base := d, fiber := g : Grothendieck F } } ≫
        h)
      ({ base := v, fiber := y :
          { base := c, fiber := f : Grothendieck F }.Hom { base := d, fiber := g : Grothendieck F } } ≫
        h)
have refine_2₁ : Eq (α :=
  { base := c, fiber := f : Grothendieck F } ⟶
    { base := coeq u v,
      fiber :=
        coeq (j :=
          Functor.obj (Cat.Hom.toFunctor (F.map (coeqHom u v)))
            ((F.map v).toFunctor.obj { base := c, fiber := f : Grothendieck F }.fiber))
          (eqToHom sorry ≫ Functor.map (Cat.Hom.toFunctor (F.map (coeqHom u v))) x)
          (Functor.map (Cat.Hom.toFunctor (F.map (coeqHom u v))) y) })
  ({ base := u, fiber := x :
      { base := c, fiber := f : Grothendieck F }.Hom { base := d, fiber := g : Grothendieck F } } ≫
    { base := coeqHom u v,
      fiber :=
        coeqHom (eqToHom sorry ≫ Functor.map (Cat.Hom.toFunctor (F.map (coeqHom u v))) x)
          (Functor.map (Cat.Hom.toFunctor (F.map (coeqHom u v))) y) })
  ({ base := v, fiber := y :
      { base := c, fiber := f : Grothendieck F }.Hom { base := d, fiber := g : Grothendieck F } } ≫
    { base := coeqHom u v,
      fiber :=
        coeqHom (eqToHom sorry ≫ Functor.map (Cat.Hom.toFunctor (F.map (coeqHom u v))) x)
          (Functor.map (Cat.Hom.toFunctor (F.map (coeqHom u v))) y) })
Tactic: refine ⟨⟨coeq u v, coeq (eqToHom ?_ ≫
        (F.map (coeqHom u v)).toFunctor.map x) ((F.map (coeqHom u v)).toFunctor.map y)⟩,
          ⟨coeqHom u v, coeqHom _ _⟩, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Filtered.Grothendieck (goal)
Goal:
C : Type u
inst : Category.{v, u} C
F : C ⥤ Cat
inst_1 : IsFilteredOrEmpty.{v, u} C
inst_2 : ∀ (c : C), IsFilteredOrEmpty.{u_1, u_2} (↑(F.obj c) : Type u_2)
c : C
f : (↑(F.obj c) : Type u_2)
d : C
g : (↑(F.obj d) : Type u_2)
u : { base := c, fiber := f : Grothendieck F }.base ⟶ { base := d, fiber := g : Grothendieck F }.base
x : (F.map u).toFunctor.obj { base := c, fiber := f : Grothendieck F }.fiber ⟶ { base := d, fiber := g }.fiber
v : { base := c, fiber := f : Grothendieck F }.base ⟶ { base := d, fiber := g : Grothendieck F }.base
y : (F.map v).toFunctor.obj { base := c, fiber := f : Grothendieck F }.fiber ⟶ { base := d, fiber := g }.fiber
| Functor.obj (Cat.Hom.toFunctor (F.map (coeqHom u v)))
    ((F.map u).toFunctor.obj { base := c, fiber := f : Grothendieck F }.fiber)
have rw : Functor.obj (Cat.Hom.toFunctor (F.map (coeqHom u v)))
  ((F.map v).toFunctor.obj { base := c, fiber := f : Grothendieck F }.fiber)
Tactic: rw [← Cat.Hom.comp_obj, ← F.map_comp, coeq_condition, F.map_comp,
        Cat.Hom.comp_obj]
-- Lean 4 proof step from Mathlib.CategoryTheory.Filtered.Grothendieck (goal)
Goal:
C : Type u
inst : Category.{v, u} C
F : C ⥤ Cat
inst_1 : IsFilteredOrEmpty.{v, u} C
inst_2 : ∀ (c : C), IsFilteredOrEmpty.{u_1, u_2} (↑(F.obj c) : Type u_2)
c : C
f : (↑(F.obj c) : Type u_2)
d : C
g : (↑(F.obj d) : Type u_2)
u : { base := c, fiber := f : Grothendieck F }.base ⟶ { base := d, fiber := g : Grothendieck F }.base
x : (F.map u).toFunctor.obj { base := c, fiber := f : Grothendieck F }.fiber ⟶ { base := d, fiber := g }.fiber
v : { base := c, fiber := f : Grothendieck F }.base ⟶ { base := d, fiber := g : Grothendieck F }.base
y : (F.map v).toFunctor.obj { base := c, fiber := f : Grothendieck F }.fiber ⟶ { base := d, fiber := g }.fiber
⊢ Eq (α := (↑(F.obj (coeq u v)) : Type u_2))
    (Functor.obj (Cat.Hom.toFunctor (F.map (coeqHom u v)))
      ((F.map v).toFunctor.obj { base := c, fiber := f : Grothendieck F }.fiber))
    (Functor.obj (Cat.Hom.toFunctor (F.map (coeqHom u v)))
      ((F.map u).toFunctor.obj { base := c, fiber := f : Grothendieck F }.fiber))
have conv_rhs : Eq (α := (↑(F.obj (coeq u v)) : Type u_2))
  (Functor.obj (Cat.Hom.toFunctor (F.map (coeqHom u v)))
    ((F.map v).toFunctor.obj { base := c, fiber := f : Grothendieck F }.fiber))
  (Functor.obj (Cat.Hom.toFunctor (F.map (coeqHom u v)))
    ((F.map u).toFunctor.obj { base := c, fiber := f : Grothendieck F }.fiber))
Tactic: conv_rhs => rw [← Cat.Hom.comp_obj, ← F.map_comp, coeq_condition, F.map_comp,
        Cat.Hom.comp_obj]
-- Lean 4 proof step from Mathlib.CategoryTheory.Filtered.Grothendieck (goal)
Goal:
C : Type u
inst : Category.{v, u} C
F : C ⥤ Cat
inst_1 : IsFilteredOrEmpty.{v, u} C
inst_2 : ∀ (c : C), IsFilteredOrEmpty.{u_1, u_2} (↑(F.obj c) : Type u_2)
c : C
f : (↑(F.obj c) : Type u_2)
d : C
g : (↑(F.obj d) : Type u_2)
u : { base := c, fiber := f : Grothendieck F }.base ⟶ { base := d, fiber := g : Grothendieck F }.base
x : (F.map u).toFunctor.obj { base := c, fiber := f : Grothendieck F }.fiber ⟶ { base := d, fiber := g }.fiber
v : { base := c, fiber := f : Grothendieck F }.base ⟶ { base := d, fiber := g : Grothendieck F }.base
y : (F.map v).toFunctor.obj { base := c, fiber := f : Grothendieck F }.fiber ⟶ { base := d, fiber := g }.fiber
conv_rhs :
  Eq (α := (↑(F.obj (coeq u v)) : Type u_2))
    (Functor.obj (Cat.Hom.toFunctor (F.map (coeqHom u v)))
      ((F.map v).toFunctor.obj { base := c, fiber := f : Grothendieck F }.fiber))
    (Functor.obj (Cat.Hom.toFunctor (F.map (coeqHom u v)))
      ((F.map u).toFunctor.obj { base := c, fiber := f : Grothendieck F }.fiber))
⊢ Eq (α := (↑(F.obj (coeq u v)) : Type u_2))
    (Functor.obj (Cat.Hom.toFunctor (F.map (coeqHom u v)))
      ((F.map v).toFunctor.obj { base := c, fiber := f : Grothendieck F }.fiber))
    (Functor.obj (Cat.Hom.toFunctor (F.map (coeqHom u v)))
      ((F.map u).toFunctor.obj { base := c, fiber := f : Grothendieck F }.fiber))
have refine_2 : Eq (α := (↑(F.obj (coeq u v)) : Type u_2))
  (Functor.obj (Cat.Hom.toFunctor (F.map (coeqHom u v)))
    ((F.map v).toFunctor.obj { base := c, fiber := f : Grothendieck F }.fiber))
  (Functor.obj (Cat.Hom.toFunctor (F.map (coeqHom u v)))
    ((F.map v).toFunctor.obj { base := c, fiber := f : Grothendieck F }.fiber))
Tactic: conv_rhs => rw [← Cat.Hom.comp_obj, ← F.map_comp, coeq_condition, F.map_comp,
        Cat.Hom.comp_obj]
-- Lean 4 proof step from Mathlib.CategoryTheory.Filtered.Grothendieck (goal)
Goal:
C : Type u
inst : Category.{v, u} C
F : C ⥤ Cat
inst_1 : IsFiltered.{v, u} C
inst_2 : ∀ (c : C), IsFiltered.{u_1, u_2} (↑(F.obj c) : Type u_2)
⊢ Nonempty (Grothendieck F)
have obtain : ∀ (c : C), Nonempty (Grothendieck F)
Tactic: obtain ⟨c⟩ : Nonempty C := IsFiltered.nonempty
-- Lean 4 proof step from Mathlib.CategoryTheory.Filtered.Grothendieck (goal)
Goal:
C : Type u
inst : Category.{v, u} C
F : C ⥤ Cat
inst_1 : IsFiltered.{v, u} C
inst_2 : ∀ (c : C), IsFiltered.{u_1, u_2} (↑(F.obj c) : Type u_2)
c : C
⊢ Nonempty (Grothendieck F)
have obtain : ∀ (f : (↑(F.obj c) : Type u_2)), Nonempty (Grothendieck F)
Tactic: obtain ⟨f⟩ : Nonempty (F.obj c) := IsFiltered.nonempty
-- Lean 4 proof step from Mathlib.CategoryTheory.Filtered.Grothendieck (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
F : C ⥤ Cat
inst_1 : IsFiltered.{v, u} C
inst_2 : ∀ (c : C), IsFiltered.{u_1, u_2} (↑(F.obj c) : Type u_2)
⊢ IsFiltered (Grothendieck F)
have this : Nonempty (Grothendieck F)
Tactic: have : Nonempty (Grothendieck F) := by
    obtain ⟨c⟩ : Nonempty C := IsFiltered.nonempty
    obtain ⟨f⟩ : Nonempty (F.obj c) := IsFiltered.nonempty
    exact ⟨⟨c, f⟩⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Filtered.Small (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : IsCofilteredOrEmpty.{v, u} C
α : Type w
f : α → C
⊢ Small.{max v w, u} (ObjectProperty.FullSubcategory.{v, u} (cofilteredClosure f))
have refine : ∀ (b : ObjectProperty.FullSubcategory.{v, u} (cofilteredClosure f)),
  ∃ (a : CofilteredClosureSmall.AbstractCofilteredClosure.{w, v, u} f),
    CofilteredClosureSmall.abstractCofilteredClosureRealization f a = b.obj (P := cofilteredClosure f)
Tactic: refine small_of_injective_of_exists
    (CofilteredClosureSmall.abstractCofilteredClosureRealization f)
    (fun _ _ => ObjectProperty.FullSubcategory.ext) ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Filtered.Small (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : IsCofilteredOrEmpty.{v, u} C
α : Type w
f : α → C
j : C
x : α
⊢ ∃ (a : CofilteredClosureSmall.AbstractCofilteredClosure.{w, v, u} f),
    CofilteredClosureSmall.abstractCofilteredClosureRealization f a =
      ObjectProperty.FullSubcategory.obj (P := cofilteredClosure f)
        (ObjectProperty.FullSubcategory.mk (P := cofilteredClosure f) (f x) ⋯)
have base : (CofilteredClosureSmall.bundledAbstractCofilteredClosure.{w, v, u} f 0).fst (β := fun (t : Type (max v w)) ↦ t → C)
Tactic: refine ⟨⟨0, ?_⟩,?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Filtered.Small (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : IsCofilteredOrEmpty.{v, u} C
α : Type w
f : α → C
j : C
x : α
base :
  (CofilteredClosureSmall.bundledAbstractCofilteredClosure.{w, v, u} f 0).fst (β := fun (t : Type (max v w)) ↦ t → C)
⊢ ∃ (a : CofilteredClosureSmall.AbstractCofilteredClosure.{w, v, u} f),
    CofilteredClosureSmall.abstractCofilteredClosureRealization f a =
      ObjectProperty.FullSubcategory.obj (P := cofilteredClosure f)
        (ObjectProperty.FullSubcategory.mk (P := cofilteredClosure f) (f x) ⋯)
have base₁ : CofilteredClosureSmall.abstractCofilteredClosureRealization.{w, v, u} f
    (Sigma.mk (β := fun (n : ℕ) ↦
      (CofilteredClosureSmall.bundledAbstractCofilteredClosure.{w, v, u} f n).fst (β := fun (t : Type (max v w)) ↦
        t → C))
      0 (id { down := x : ULift.{v, w} α })) =
  ObjectProperty.FullSubcategory.obj (P := cofilteredClosure f)
    (ObjectProperty.FullSubcategory.mk (P := cofilteredClosure f) (f x) sorry)
Tactic: refine ⟨⟨0, ?_⟩,?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Filtered.Small (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : IsCofilteredOrEmpty.{v, u} C
α : Type w
f : α → C
j : C
x : α
⊢ (CofilteredClosureSmall.bundledAbstractCofilteredClosure.{w, v, u} f 0).fst (β := fun (t : Type (max v w)) ↦ t → C)
have base : ULift.{v, w} α
Tactic: simp only [CofilteredClosureSmall.bundledAbstractCofilteredClosure]
-- Lean 4 proof step from Mathlib.CategoryTheory.Filtered.Small (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : IsCofilteredOrEmpty.{v, u} C
α : Type w
f : α → C
j : C
x : α
⊢ CofilteredClosureSmall.abstractCofilteredClosureRealization.{w, v, u} f
      (Sigma.mk (β := fun (n : ℕ) ↦
        (CofilteredClosureSmall.bundledAbstractCofilteredClosure.{w, v, u} f n).fst (β := fun (t : Type (max v w)) ↦
          t → C))
        0 (id { down := x : ULift.{v, w} α })) =
    ObjectProperty.FullSubcategory.obj (P := cofilteredClosure f)
      (ObjectProperty.FullSubcategory.mk (P := cofilteredClosure f) (f x) ⋯)
have base : (f ∘ ULift.down) (id { down := x : ULift.{v, w} α }) = f x
Tactic: simp only [CofilteredClosureSmall.abstractCofilteredClosureRealization,
        CofilteredClosureSmall.bundledAbstractCofilteredClosure]
-- Lean 4 proof step from Mathlib.CategoryTheory.Filtered.Small (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : IsCofilteredOrEmpty.{v, u} C
α : Type w
f : α → C
j j_1 j' : C
hj₁ : cofilteredClosure.{w, v, u} f j_1
hj₂ : cofilteredClosure.{w, v, u} f j'
ih :
  ∃ (a : CofilteredClosureSmall.AbstractCofilteredClosure.{w, v, u} f),
    CofilteredClosureSmall.abstractCofilteredClosureRealization f a =
      (ObjectProperty.FullSubcategory.mk (P := cofilteredClosure f) j_1 hj₁).obj (P := cofilteredClosure f)
ih' :
  ∃ (a : CofilteredClosureSmall.AbstractCofilteredClosure.{w, v, u} f),
    CofilteredClosureSmall.abstractCofilteredClosureRealization f a =
      (ObjectProperty.FullSubcategory.mk (P := cofilteredClosure f) j' hj₂).obj (P := cofilteredClosure f)
⊢ ∃ (a : CofilteredClosureSmall.AbstractCofilteredClosure.{w, v, u} f),
    CofilteredClosureSmall.abstractCofilteredClosureRealization f a =
      (ObjectProperty.FullSubcategory.mk (P := cofilteredClosure f) (min.{v, u} j_1 j') ⋯).obj (P :=
        cofilteredClosure f)
have min : ∀ (n : ℕ)
  (x :
    (CofilteredClosureSmall.bundledAbstractCofilteredClosure.{w, v, u} f n).fst (β := fun (t : Type (max v w)) ↦ t → C))
  (hj₁ :
    cofilteredClosure.{w, v, u} f
      (CofilteredClosureSmall.abstractCofilteredClosureRealization.{w, v, u} f
        (Sigma.mk (β := fun (n : ℕ) ↦
          (CofilteredClosureSmall.bundledAbstractCofilteredClosure.{w, v, u} f n).fst (β := fun (t : Type (max v w)) ↦
            t → C))
          n x))),
  ∃ (a : CofilteredClosureSmall.AbstractCofilteredClosure.{w, v, u} f),
    CofilteredClosureSmall.abstractCofilteredClosureRealization f a =
      (ObjectProperty.FullSubcategory.mk (P := cofilteredClosure f)
            (min.{v, u}
              (CofilteredClosureSmall.abstractCofilteredClosureRealization.{w, v, u} f
                (Sigma.mk (β := fun (n : ℕ) ↦
                  (CofilteredClosureSmall.bundledAbstractCofilteredClosure.{w, v, u} f n).fst (β :=
                    fun (t : Type (max v w)) ↦ t → C))
                  n x))
              j')
            sorry).obj
        (P := cofilteredClosure f)
Tactic: rcases ih with ⟨⟨n, x⟩, rfl⟩
Cleared hypotheses: j_1, hj₁, ih