awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.OfIsCofiltered (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : LocallySmall.{w, v, u} C
N : Type u'
inst_2 : Category.{v', u'} N
p : Functor.{v', v, u', u} N C
inst_3 : InitiallySmall.{w, v', u'} N
X : C
x : (fiber p).obj X
⊢ ∃ (U : N) (f : p.obj U ⟶ X), fiberMk f = x
have obtain : ∀ (U : Nᵒᵖ) (f : (p.op ⋙ Functor.obj shrinkYoneda.{w, v, u} X).obj U),
  ∃ (U_1 : N) (f_1 : p.obj U_1 ⟶ X),
    Eq.{w + 1} (α := (fiber p).obj X) (fiberMk f_1)
      ((hom (FC := TypeCat.Fun) ((colimit.cocone (p.op ⋙ Functor.obj shrinkYoneda.{w, v, u} X)).ι.app U) :
          (a : (fun (X : Type w) ↦ X) ((p.op ⋙ shrinkYoneda.{w, v, u}.obj X).obj U)) →
            (fun (X : Type w) ↦ X)
              (((Functor.const Nᵒᵖ).obj (colimit.cocone (p.op ⋙ shrinkYoneda.{w, v, u}.obj X)).pt).obj U))
        f)
Tactic: obtain ⟨U, f, rfl⟩ := Types.jointly_surjective_of_isColimit
    (colimit.isColimit (p.op ⋙ shrinkYoneda.{w}.obj X)) x
Cleared hypotheses: x
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.OfIsCofiltered (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : LocallySmall.{w, v, u} C
N : Type u'
inst_2 : Category.{v', u'} N
p : Functor.{v', v, u', u} N C
inst_3 : InitiallySmall.{w, v', u'} N
X : C
U : Nᵒᵖ
f : (p.op ⋙ Functor.obj shrinkYoneda.{w, v, u} X).obj U
⊢ ∃ (U_1 : N) (f_1 : p.obj U_1 ⟶ X),
    Eq.{w + 1} (α := (fiber p).obj X) (fiberMk f_1)
      ((hom (FC := TypeCat.Fun) ((colimit.cocone (p.op ⋙ Functor.obj shrinkYoneda.{w, v, u} X)).ι.app U) :
          (a : (fun (X : Type w) ↦ X) ((p.op ⋙ shrinkYoneda.{w, v, u}.obj X).obj U)) →
            (fun (X : Type w) ↦ X)
              (((Functor.const Nᵒᵖ).obj (colimit.cocone (p.op ⋙ shrinkYoneda.{w, v, u}.obj X)).pt).obj U))
        f)
have obtain : ∀ (f : unop (p.op.obj U) ⟶ X),
  ∃ (U_1 : N) (f_1 : p.obj U_1 ⟶ X),
    Eq.{w + 1} (α := (fiber p).obj X) (fiberMk f_1)
      ((hom (FC := TypeCat.Fun) ((colimit.cocone (p.op ⋙ Functor.obj shrinkYoneda.{w, v, u} X)).ι.app U) :
          (a : (fun (X : Type w) ↦ X) ((p.op ⋙ shrinkYoneda.{w, v, u}.obj X).obj U)) →
            (fun (X : Type w) ↦ X)
              (((Functor.const Nᵒᵖ).obj (colimit.cocone (p.op ⋙ shrinkYoneda.{w, v, u}.obj X)).pt).obj U))
        ((Equiv.symm shrinkYonedaObjObjEquiv :
              (a : unop (p.op.obj U) ⟶ X) → (shrinkYoneda.{w, v, u}.obj X).obj (p.op.obj U))
            f :
          Functor.obj.{v, w, u, w + 1} (Functor.obj shrinkYoneda.{w, v, u} X) (p.op.obj U)))
Tactic: obtain ⟨f, rfl⟩ := shrinkYonedaObjObjEquiv.symm.surjective f
Cleared hypotheses: f
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.OfIsCofiltered (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : LocallySmall.{w, v, u} C
N : Type u'
inst_2 : Category.{v', u'} N
p : Functor.{v', v, u', u} N C
inst_3 : InitiallySmall.{w, v', u'} N
inst_4 : IsCofiltered.{v', u'} N
U : N
X : C
f₁ f₂ : p.obj U ⟶ X
hf : Eq.{w + 1} (α := (fiber p).obj X) (fiberMk f₁) (fiberMk f₂)
⊢ ∃ (V : N) (g : V ⟶ U), p.map g ≫ f₁ = p.map g ≫ f₂
have obtain : ∀ (V : Nᵒᵖ) (g : op U ⟶ V)
  (hg :
    Eq (α := (fun (X : Type w) ↦ X) ((p.op ⋙ Functor.obj shrinkYoneda.{w, v, u} X).obj V))
      ((hom (FC := TypeCat.Fun) ((p.op ⋙ Functor.obj shrinkYoneda.{w, v, u} X).map g) :
            (a : (fun (X : Type w) ↦ X) ((p.op ⋙ shrinkYoneda.{w, v, u}.obj X).obj (op U))) →
              (fun (X : Type w) ↦ X) ((p.op ⋙ shrinkYoneda.{w, v, u}.obj X).obj V))
          ((Equiv.symm shrinkYonedaObjObjEquiv :
                (a : unop (p.op.obj (op U)) ⟶ X) → (shrinkYoneda.{w, v, u}.obj X).obj (p.op.obj (op U)))
              f₁ :
            Functor.obj.{v, w, u, w + 1} (Functor.obj shrinkYoneda.{w, v, u} X) (p.op.obj (op U))) :
        (fun (X : Type w) ↦ X) ((p.op ⋙ Functor.obj shrinkYoneda.{w, v, u} X).obj V))
      ((hom (FC := TypeCat.Fun) ((p.op ⋙ Functor.obj shrinkYoneda.{w, v, u} X).map g) :
            (a : (fun (X : Type w) ↦ X) ((p.op ⋙ shrinkYoneda.{w, v, u}.obj X).obj (op U))) →
              (fun (X : Type w) ↦ X) ((p.op ⋙ shrinkYoneda.{w, v, u}.obj X).obj V))
          ((Equiv.symm shrinkYonedaObjObjEquiv :
                (a : unop (p.op.obj (op U)) ⟶ X) → (shrinkYoneda.{w, v, u}.obj X).obj (p.op.obj (op U)))
              f₂ :
            Functor.obj.{v, w, u, w + 1} (Functor.obj shrinkYoneda.{w, v, u} X) (p.op.obj (op U))) :
        (fun (X : Type w) ↦ X) ((p.op ⋙ Functor.obj shrinkYoneda.{w, v, u} X).obj V))),
  ∃ (V : N) (g : V ⟶ U), p.map g ≫ f₁ = p.map g ≫ f₂
Tactic: obtain ⟨V, g, hg⟩ :=
    (Types.FilteredColimit.isColimit_eq_iff'
      (colimit.isColimit (p.op ⋙ shrinkYoneda.{w}.obj X)) _ _).1 hf
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.OfIsCofiltered (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : LocallySmall.{w, v, u} C
N : Type u'
inst_2 : Category.{v', u'} N
p : Functor.{v', v, u', u} N C
inst_3 : InitiallySmall.{w, v', u'} N
inst_4 : IsCofiltered.{v', u'} N
U : N
X : C
f₁ f₂ : p.obj U ⟶ X
hf : Eq.{w + 1} (α := (fiber p).obj X) (fiberMk f₁) (fiberMk f₂)
V : Nᵒᵖ
g : op U ⟶ V
hg :
  Eq (α := (fun (X : Type w) ↦ X) ((p.op ⋙ Functor.obj shrinkYoneda.{w, v, u} X).obj V))
    ((hom (FC := TypeCat.Fun) ((p.op ⋙ Functor.obj shrinkYoneda.{w, v, u} X).map g) :
          (a : (fun (X : Type w) ↦ X) ((p.op ⋙ shrinkYoneda.{w, v, u}.obj X).obj (op U))) →
            (fun (X : Type w) ↦ X) ((p.op ⋙ shrinkYoneda.{w, v, u}.obj X).obj V))
        ((Equiv.symm shrinkYonedaObjObjEquiv :
              (a : unop (p.op.obj (op U)) ⟶ X) → (shrinkYoneda.{w, v, u}.obj X).obj (p.op.obj (op U)))
            f₁ :
          Functor.obj.{v, w, u, w + 1} (Functor.obj shrinkYoneda.{w, v, u} X) (p.op.obj (op U))) :
      (fun (X : Type w) ↦ X) ((p.op ⋙ Functor.obj shrinkYoneda.{w, v, u} X).obj V))
    ((hom (FC := TypeCat.Fun) ((p.op ⋙ Functor.obj shrinkYoneda.{w, v, u} X).map g) :
          (a : (fun (X : Type w) ↦ X) ((p.op ⋙ shrinkYoneda.{w, v, u}.obj X).obj (op U))) →
            (fun (X : Type w) ↦ X) ((p.op ⋙ shrinkYoneda.{w, v, u}.obj X).obj V))
        ((Equiv.symm shrinkYonedaObjObjEquiv :
              (a : unop (p.op.obj (op U)) ⟶ X) → (shrinkYoneda.{w, v, u}.obj X).obj (p.op.obj (op U)))
            f₂ :
          Functor.obj.{v, w, u, w + 1} (Functor.obj shrinkYoneda.{w, v, u} X) (p.op.obj (op U))) :
      (fun (X : Type w) ↦ X) ((p.op ⋙ Functor.obj shrinkYoneda.{w, v, u} X).obj V))
⊢ ∃ (V : N) (g : V ⟶ U), p.map g ≫ f₁ = p.map g ≫ f₂
have refine : p.map g.unop ≫ f₁ = p.map g.unop ≫ f₂
Tactic: refine ⟨V.unop, g.unop, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.OfIsCofiltered (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : LocallySmall.{w, v, u} C
N : Type u'
inst_2 : Category.{v', u'} N
p : Functor.{v', v, u', u} N C
inst_3 : InitiallySmall.{w, v', u'} N
J : GrothendieckTopology.{v, u} C
inst_4 : IsCofiltered.{v', u'} N
⊢ Functor.Initial (D := (fiber p).Elements) (functor p)
have refine_1 : ∀ (d : (fiber p).Elements), ∃ (c : N), Nonempty.{v + 1} (Functor.obj (functor p) c ⟶ d)
Tactic: refine Functor.initial_of_exists_of_isCofiltered _ ?_ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.OfIsCofiltered (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : LocallySmall.{w, v, u} C
N : Type u'
inst_2 : Category.{v', u'} N
p : Functor.{v', v, u', u} N C
inst_3 : InitiallySmall.{w, v', u'} N
J : GrothendieckTopology.{v, u} C
inst_4 : IsCofiltered.{v', u'} N
refine_1 : ∀ (d : (fiber p).Elements), ∃ (c : N), Nonempty.{v + 1} (Functor.obj (functor p) c ⟶ d)
⊢ Functor.Initial (D := (fiber p).Elements) (functor p)
have refine_2 : ∀ {d : (fiber p).Elements} {c : N} (s s' : Functor.obj (functor p) c ⟶ d),
  ∃ (c' : N) (t : c' ⟶ c), Functor.map (functor p) t ≫ s = Functor.map (functor p) t ≫ s'
Tactic: refine Functor.initial_of_exists_of_isCofiltered _ ?_ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.OfIsCofiltered (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : LocallySmall.{w, v, u} C
N : Type u'
inst_2 : Category.{v', u'} N
p : Functor.{v', v, u', u} N C
inst_3 : InitiallySmall.{w, v', u'} N
J : GrothendieckTopology.{v, u} C
inst_4 : IsCofiltered.{v', u'} N
X : C
x : (fiber p).obj X
⊢ ∃ (c : N), Nonempty.{v + 1} (Functor.obj (functor p) c ⟶ Sigma.mk (β := fun (c : C) ↦ (fiber p).obj c) X x)
have refine_1 : ∀ (U : N) (f : p.obj U ⟶ X),
  ∃ (c : N),
    Nonempty.{v + 1}
      (Quiver.Hom (V := (fiber p).Elements) (Functor.obj (functor p) c)
        (Sigma.mk (β := fun (c : C) ↦ (fiber p).obj c) X (fiberMk f)))
Tactic: obtain ⟨U, f, rfl⟩ := fiberMk_jointly_surjective x
Cleared hypotheses: x
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.OfIsCofiltered (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : LocallySmall.{w, v, u} C
N : Type u'
inst_2 : Category.{v', u'} N
p : Functor.{v', v, u', u} N C
inst_3 : InitiallySmall.{w, v', u'} N
J : GrothendieckTopology.{v, u} C
inst_4 : IsCofiltered.{v', u'} N
hp :
  ∀ ⦃X : C⦄,
    ∀ R ∈ (J : (a : C) → Set (Sieve a)) X,
      ∀ ⦃U : N⦄ (f : p.obj U ⟶ X),
        ∃ (Y : C) (g : Y ⟶ X) (_ : R.arrows g) (V : N) (q : V ⟶ U) (a : p.obj V ⟶ Y), a ≫ g = p.map q ≫ f
X : C
R : Sieve.{v, u} X
hR : R ∈ (J : (a : C) → Set (Sieve a)) X
x : (ofIsCofiltered.fiber p).obj X
⊢ ∃ (Y : C) (f : Y ⟶ X) (_ : R.arrows f) (y : (ofIsCofiltered.fiber p).obj Y),
    (hom (FC := TypeCat.Fun) ((ofIsCofiltered.fiber p).map f) :
          (a : (fun (X : Type w) ↦ X) ((ofIsCofiltered.fiber p).obj Y)) →
            (fun (X : Type w) ↦ X) ((ofIsCofiltered.fiber p).obj X))
        y =
      x
have obtain : ∀ (U : N) (f : p.obj U ⟶ X),
  ∃ (Y : C) (f_1 : Y ⟶ X) (_ : R.arrows f_1) (y : (ofIsCofiltered.fiber p).obj Y),
    Eq (α := (fun (X : Type w) ↦ X) ((ofIsCofiltered.fiber p).obj X))
      ((hom (FC := TypeCat.Fun) ((ofIsCofiltered.fiber p).map f_1) :
            (a : (fun (X : Type w) ↦ X) ((ofIsCofiltered.fiber p).obj Y)) →
              (fun (X : Type w) ↦ X) ((ofIsCofiltered.fiber p).obj X))
          y :
        (fun (X : Type w) ↦ X) ((ofIsCofiltered.fiber p).obj X))
      (fiberMk f : (ofIsCofiltered.fiber p).obj X)
Tactic: obtain ⟨U, f, rfl⟩ := fiberMk_jointly_surjective x
Cleared hypotheses: x
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.OfIsCofiltered (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : LocallySmall.{w, v, u} C
N : Type u'
inst_2 : Category.{v', u'} N
p : Functor.{v', v, u', u} N C
inst_3 : InitiallySmall.{w, v', u'} N
J : GrothendieckTopology.{v, u} C
inst_4 : IsCofiltered.{v', u'} N
hp :
  ∀ ⦃X : C⦄,
    ∀ R ∈ (J : (a : C) → Set (Sieve a)) X,
      ∀ ⦃U : N⦄ (f : p.obj U ⟶ X),
        ∃ (Y : C) (g : Y ⟶ X) (_ : R.arrows g) (V : N) (q : V ⟶ U) (a : p.obj V ⟶ Y), a ≫ g = p.map q ≫ f
X : C
R : Sieve.{v, u} X
hR : R ∈ (J : (a : C) → Set (Sieve a)) X
U : N
f : p.obj U ⟶ X
⊢ ∃ (Y : C) (f_1 : Y ⟶ X) (_ : R.arrows f_1) (y : (ofIsCofiltered.fiber p).obj Y),
    Eq (α := (fun (X : Type w) ↦ X) ((ofIsCofiltered.fiber p).obj X))
      ((hom (FC := TypeCat.Fun) ((ofIsCofiltered.fiber p).map f_1) :
            (a : (fun (X : Type w) ↦ X) ((ofIsCofiltered.fiber p).obj Y)) →
              (fun (X : Type w) ↦ X) ((ofIsCofiltered.fiber p).obj X))
          y :
        (fun (X : Type w) ↦ X) ((ofIsCofiltered.fiber p).obj X))
      (fiberMk f : (ofIsCofiltered.fiber p).obj X)
have obtain : ∀ (Y : C) (g : Y ⟶ X) (hg : R.arrows g) (V : N) (q : V ⟶ U) (a : p.obj V ⟶ Y) (ha : a ≫ g = p.map q ≫ f),
  ∃ (Y : C) (f_1 : Y ⟶ X) (_ : R.arrows f_1) (y : (ofIsCofiltered.fiber p).obj Y),
    Eq (α := (fun (X : Type w) ↦ X) ((ofIsCofiltered.fiber p).obj X))
      ((hom (FC := TypeCat.Fun) ((ofIsCofiltered.fiber p).map f_1) :
            (a : (fun (X : Type w) ↦ X) ((ofIsCofiltered.fiber p).obj Y)) →
              (fun (X : Type w) ↦ X) ((ofIsCofiltered.fiber p).obj X))
          y :
        (fun (X : Type w) ↦ X) ((ofIsCofiltered.fiber p).obj X))
      (fiberMk f : (ofIsCofiltered.fiber p).obj X)
Tactic: obtain ⟨Y, g, hg, V, q, a, ha⟩ := hp R hR f
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.Over (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
inst_1 : LocallySmall.{w, v, u} C
Φ : Point.{w, v, u} J
X : C
x : Φ.fiber.obj X
U : Over.{v, u} X
R : Sieve U
hR : R ∈ (J.over X : (a : Over X) → Set (Sieve a)) U
u : (Over.forget X ⋙ Φ.fiber).obj U
hu : u ∈ (FunctorToTypes.fromOverSubfunctor Φ.fiber x).obj U
⊢ ∃ (Y : Over.{v, u} X) (f : Y ⟶ U) (_ : R.arrows f) (y : (FunctorToTypes.fromOverFunctor Φ.fiber x).obj Y),
    (ConcreteCategory.hom (FC := TypeCat.Fun) ((FunctorToTypes.fromOverFunctor Φ.fiber x).map f) :
          (a : (fun (X : Type w) ↦ X) ((FunctorToTypes.fromOverFunctor Φ.fiber x).obj Y)) →
            (fun (X : Type w) ↦ X) ((FunctorToTypes.fromOverFunctor Φ.fiber x).obj U))
        y =
      Subtype.mk (p := fun (x_2 : (Over.forget X ⋙ Φ.fiber).obj U) ↦
        x_2 ∈ (FunctorToTypes.fromOverSubfunctor Φ.fiber x).obj U) u hu
have obtain : ∀ (R : Sieve.{v, u} U.left)
  (hR :
    Membership.mem (α := Sieve U) ((J.over X : (a : Over X) → Set (Sieve a)) U)
      (((Sieve.overEquiv U).symm : (a : Sieve U.left) → Sieve U) R : Sieve U)),
  ∃ (Y : Over.{v, u} X) (f : Y ⟶ U) (_ : Sieve.arrows (((Sieve.overEquiv U).symm : (a : Sieve U.left) → Sieve U) R) f)
    (y : (FunctorToTypes.fromOverFunctor Φ.fiber x).obj Y),
    (ConcreteCategory.hom (FC := TypeCat.Fun) ((FunctorToTypes.fromOverFunctor Φ.fiber x).map f) :
          (a : (fun (X : Type w) ↦ X) ((FunctorToTypes.fromOverFunctor Φ.fiber x).obj Y)) →
            (fun (X : Type w) ↦ X) ((FunctorToTypes.fromOverFunctor Φ.fiber x).obj U))
        y =
      Subtype.mk (p := fun (x_2 : (Over.forget X ⋙ Φ.fiber).obj U) ↦
        x_2 ∈ (FunctorToTypes.fromOverSubfunctor Φ.fiber x).obj U) u hu
Tactic: obtain ⟨R, rfl⟩ := (Sieve.overEquiv _).symm.surjective R
Cleared hypotheses: R, hR
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.Over (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
inst_1 : LocallySmall.{w, v, u} C
Φ : Point.{w, v, u} J
X : C
x : Φ.fiber.obj X
U : Over.{v, u} X
u : (Over.forget X ⋙ Φ.fiber).obj U
hu : u ∈ (FunctorToTypes.fromOverSubfunctor Φ.fiber x).obj U
R : Sieve.{v, u} U.left
hR :
  Membership.mem (α := Sieve U) ((J.over X : (a : Over X) → Set (Sieve a)) U)
    (((Sieve.overEquiv U).symm : (a : Sieve U.left) → Sieve U) R : Sieve U)
⊢ ∃ (Y : Over.{v, u} X) (f : Y ⟶ U) (_ : Sieve.arrows (((Sieve.overEquiv U).symm : (a : Sieve U.left) → Sieve U) R) f)
    (y : (FunctorToTypes.fromOverFunctor Φ.fiber x).obj Y),
    (ConcreteCategory.hom (FC := TypeCat.Fun) ((FunctorToTypes.fromOverFunctor Φ.fiber x).map f) :
          (a : (fun (X : Type w) ↦ X) ((FunctorToTypes.fromOverFunctor Φ.fiber x).obj Y)) →
            (fun (X : Type w) ↦ X) ((FunctorToTypes.fromOverFunctor Φ.fiber x).obj U))
        y =
      Subtype.mk (p := fun (x_2 : (Over.forget X ⋙ Φ.fiber).obj U) ↦
        x_2 ∈ (FunctorToTypes.fromOverSubfunctor Φ.fiber x).obj U) u hu
have hR₁ : Membership.mem (γ := Set (Sieve.{v, u} U.left)) ((J : (a : C) → Set (Sieve a)) U.left) R
Tactic: simp only [mem_over_iff, Equiv.apply_symm_apply] at hR
Cleared hypotheses: hR
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.Over (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
inst_1 : LocallySmall.{w, v, u} C
Φ : Point.{w, v, u} J
X : C
x : Φ.fiber.obj X
U : Over.{v, u} X
u : (Over.forget X ⋙ Φ.fiber).obj U
hu : u ∈ (FunctorToTypes.fromOverSubfunctor Φ.fiber x).obj U
R : Sieve.{v, u} U.left
hR : Membership.mem (γ := Set (Sieve.{v, u} U.left)) ((J : (a : C) → Set (Sieve a)) U.left) R
⊢ ∃ (Y : Over.{v, u} X) (f : Y ⟶ U) (_ : Sieve.arrows (((Sieve.overEquiv U).symm : (a : Sieve U.left) → Sieve U) R) f)
    (y : (FunctorToTypes.fromOverFunctor Φ.fiber x).obj Y),
    (ConcreteCategory.hom (FC := TypeCat.Fun) ((FunctorToTypes.fromOverFunctor Φ.fiber x).map f) :
          (a : (fun (X : Type w) ↦ X) ((FunctorToTypes.fromOverFunctor Φ.fiber x).obj Y)) →
            (fun (X : Type w) ↦ X) ((FunctorToTypes.fromOverFunctor Φ.fiber x).obj U))
        y =
      Subtype.mk (p := fun (x_2 : (Over.forget X ⋙ Φ.fiber).obj U) ↦
        x_2 ∈ (FunctorToTypes.fromOverSubfunctor Φ.fiber x).obj U) u hu
have obtain : ∀ (Y : C) (f : Y ⟶ U.left) (hf : R.arrows f) (v : Φ.fiber.obj Y)
  (hu :
    Membership.mem (α := (Over.forget X ⋙ Φ.fiber).obj U) ((FunctorToTypes.fromOverSubfunctor Φ.fiber x).obj U)
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (Φ.fiber.map f) :
            (a : (fun (X : Type w) ↦ X) (Φ.fiber.obj Y)) → (fun (X : Type w) ↦ X) (Φ.fiber.obj U.left))
          v :
        (fun (X : Type w) ↦ X) (Φ.fiber.obj U.left))),
  ∃ (Y_1 : Over.{v, u} X) (f_1 : Y_1 ⟶ U) (_ :
    Sieve.arrows (((Sieve.overEquiv U).symm : (a : Sieve U.left) → Sieve U) R) f_1) (y :
    (FunctorToTypes.fromOverFunctor Φ.fiber x).obj Y_1),
    Eq (α := (fun (X : Type w) ↦ X) ((FunctorToTypes.fromOverFunctor Φ.fiber x).obj U))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) ((FunctorToTypes.fromOverFunctor Φ.fiber x).map f_1) :
            (a : (fun (X : Type w) ↦ X) ((FunctorToTypes.fromOverFunctor Φ.fiber x).obj Y_1)) →
              (fun (X : Type w) ↦ X) ((FunctorToTypes.fromOverFunctor Φ.fiber x).obj U))
          y :
        (fun (X : Type w) ↦ X) ((FunctorToTypes.fromOverFunctor Φ.fiber x).obj U))
      (Subtype.mk (p := fun (x_2 : (Over.forget X ⋙ Φ.fiber).obj U) ↦
          x_2 ∈ (FunctorToTypes.fromOverSubfunctor Φ.fiber x).obj U)
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (Φ.fiber.map f) :
              (a : (fun (X : Type w) ↦ X) (Φ.fiber.obj Y)) → (fun (X : Type w) ↦ X) (Φ.fiber.obj U.left))
            v)
          hu :
        { x_2 : (Over.forget X ⋙ Φ.fiber).obj U // x_2 ∈ (FunctorToTypes.fromOverSubfunctor Φ.fiber x).obj U })
Tactic: obtain ⟨Y, f, hf, v, rfl⟩ := Φ.jointly_surjective R hR u
Cleared hypotheses: u, hu
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.Over (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
inst_1 : LocallySmall.{w, v, u} C
Φ : Point.{w, v, u} J
X : C
x : Φ.fiber.obj X
U : Over.{v, u} X
R : Sieve.{v, u} U.left
hR : Membership.mem (γ := Set (Sieve.{v, u} U.left)) ((J : (a : C) → Set (Sieve a)) U.left) R
Y : C
f : Y ⟶ U.left
hf : R.arrows f
v : Φ.fiber.obj Y
hu :
  Membership.mem (α := (Over.forget X ⋙ Φ.fiber).obj U) ((FunctorToTypes.fromOverSubfunctor Φ.fiber x).obj U)
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (Φ.fiber.map f) :
          (a : (fun (X : Type w) ↦ X) (Φ.fiber.obj Y)) → (fun (X : Type w) ↦ X) (Φ.fiber.obj U.left))
        v :
      (fun (X : Type w) ↦ X) (Φ.fiber.obj U.left))
⊢ ∃ (Y_1 : Over.{v, u} X) (f_1 : Y_1 ⟶ U) (_ :
    Sieve.arrows (((Sieve.overEquiv U).symm : (a : Sieve U.left) → Sieve U) R) f_1) (y :
    (FunctorToTypes.fromOverFunctor Φ.fiber x).obj Y_1),
    Eq (α := (fun (X : Type w) ↦ X) ((FunctorToTypes.fromOverFunctor Φ.fiber x).obj U))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) ((FunctorToTypes.fromOverFunctor Φ.fiber x).map f_1) :
            (a : (fun (X : Type w) ↦ X) ((FunctorToTypes.fromOverFunctor Φ.fiber x).obj Y_1)) →
              (fun (X : Type w) ↦ X) ((FunctorToTypes.fromOverFunctor Φ.fiber x).obj U))
          y :
        (fun (X : Type w) ↦ X) ((FunctorToTypes.fromOverFunctor Φ.fiber x).obj U))
      (Subtype.mk (p := fun (x_2 : (Over.forget X ⋙ Φ.fiber).obj U) ↦
          x_2 ∈ (FunctorToTypes.fromOverSubfunctor Φ.fiber x).obj U)
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (Φ.fiber.map f) :
              (a : (fun (X : Type w) ↦ X) (Φ.fiber.obj Y)) → (fun (X : Type w) ↦ X) (Φ.fiber.obj U.left))
            v)
          hu :
        { x_2 : (Over.forget X ⋙ Φ.fiber).obj U // x_2 ∈ (FunctorToTypes.fromOverSubfunctor Φ.fiber x).obj U })
have refine : Membership.mem (γ := Set ((Over.forget X ⋙ Φ.fiber).obj (Over.mk (f ≫ U.hom))))
  ((FunctorToTypes.fromOverSubfunctor Φ.fiber x).obj (Over.mk (f ≫ U.hom))) v
Tactic: refine ⟨Over.mk (f ≫ U.hom), Over.homMk f, hf, ⟨v, ?_⟩, rfl⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.Over (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
inst_1 : LocallySmall.{w, v, u} C
Φ : Point.{w, v, u} J
X : C
x : Φ.fiber.obj X
U : Over.{v, u} X
R : Sieve.{v, u} U.left
hR : Membership.mem (γ := Set (Sieve.{v, u} U.left)) ((J : (a : C) → Set (Sieve a)) U.left) R
Y : C
f : Y ⟶ U.left
hf : R.arrows f
v : Φ.fiber.obj Y
hu :
  Membership.mem (α := (Over.forget X ⋙ Φ.fiber).obj U) ((FunctorToTypes.fromOverSubfunctor Φ.fiber x).obj U)
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (Φ.fiber.map f) :
          (a : (fun (X : Type w) ↦ X) (Φ.fiber.obj Y)) → (fun (X : Type w) ↦ X) (Φ.fiber.obj U.left))
        v :
      (fun (X : Type w) ↦ X) (Φ.fiber.obj U.left))
⊢ Membership.mem (γ := Set ((Over.forget X ⋙ Φ.fiber).obj (Over.mk (f ≫ U.hom))))
    ((FunctorToTypes.fromOverSubfunctor Φ.fiber x).obj (Over.mk (f ≫ U.hom))) v
have rw : ∀
  (hu :
    (ConcreteCategory.hom (FC := TypeCat.Fun) (Φ.fiber.map U.hom) :
          (a : (fun (X : Type w) ↦ X) (Φ.fiber.obj U.left)) → (fun (X : Type w) ↦ X) (Φ.fiber.obj X))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (Φ.fiber.map f) :
              (a : (fun (X : Type w) ↦ X) (Φ.fiber.obj Y)) → (fun (X : Type w) ↦ X) (Φ.fiber.obj U.left))
            v :
          (fun (X : Type w) ↦ X) (Φ.fiber.obj U.left)) =
      x),
  (ConcreteCategory.hom (FC := TypeCat.Fun) (Φ.fiber.map (Over.mk (f ≫ U.hom)).hom) :
        (a : (fun (X : Type w) ↦ X) (Φ.fiber.obj (Over.mk (f ≫ U.hom)).left)) → (fun (X : Type w) ↦ X) (Φ.fiber.obj X))
      v =
    x
Tactic: rw [FunctorToTypes.mem_fromOverSubfunctor_iff] at hu ⊢
Cleared hypotheses: hu
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.Over (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
inst_1 : LocallySmall.{w, v, u} C
Φ : Point.{w, v, u} J
X_1 : C
x : Φ.fiber.obj X_1
inst_2 : J.HasEnoughPoints
inst_3 : J.WEqualsLocallyBijective (Type w) (FA := TypeCat.Fun)
inst_4 : HasSheafify J (Type w)
X : C
inst_5 : HasSheafify (J.over X) (Type w)
⊢ (J.over X).HasEnoughPoints
have obtain : ∀ (P : ObjectProperty (Point.{w, v, u} J)),
  ObjectProperty.Small.{w, max u w, max (max u v) (w + 1)} P →
    ∀ (hP : P.IsConservativeFamilyOfPoints), (J.over X).HasEnoughPoints
Tactic: obtain ⟨P, _, hP⟩ := HasEnoughPoints.exists_objectProperty J
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.Presheaf (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : LocallySmall.{w, v, u} C
X U : C
R : Sieve.{v, u} U
hR : R ∈ (⊥ : (a : C) → Set (Sieve a)) U
x : Functor.obj.{v, w, u, w + 1} (Functor.obj (Functor.flip shrinkYoneda.{w, v, u}) (op X)) U
⊢ ∃ (Y : C) (f : Y ⟶ U) (_ : R.arrows f) (y :
    Functor.obj.{v, w, u, w + 1} (Functor.obj (Functor.flip shrinkYoneda.{w, v, u}) (op X)) Y),
    (ConcreteCategory.hom (FC := TypeCat.Fun)
            (Functor.map (Functor.obj (Functor.flip shrinkYoneda.{w, v, u}) (op X)) f) :
          (a : (fun (X : Type w) ↦ X) ((shrinkYoneda.{w, v, u}.flip.obj (op X)).obj Y)) →
            (fun (X : Type w) ↦ X) ((shrinkYoneda.{w, v, u}.flip.obj (op X)).obj U))
        y =
      x
have obtain : ∀ (hR : ⊤ ∈ (⊥ : (a : C) → Set (Sieve a)) U),
  ∃ (Y : C) (f : Y ⟶ U) (_ : Sieve.arrows ⊤ f) (y :
    Functor.obj.{v, w, u, w + 1} (Functor.obj (Functor.flip shrinkYoneda.{w, v, u}) (op X)) Y),
    (ConcreteCategory.hom (FC := TypeCat.Fun)
            (Functor.map (Functor.obj (Functor.flip shrinkYoneda.{w, v, u}) (op X)) f) :
          (a : (fun (X : Type w) ↦ X) ((shrinkYoneda.{w, v, u}.flip.obj (op X)).obj Y)) →
            (fun (X : Type w) ↦ X) ((shrinkYoneda.{w, v, u}.flip.obj (op X)).obj U))
        y =
      x
Tactic: obtain rfl : R = ⊤ := by simpa using hR
Cleared hypotheses: R, hR