awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.Conservative (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
P : ObjectProperty (GrothendieckTopology.Point.{w, v, u} J)
inst_1 : LocallySmall.{w, v, u} C
inst_2 : HasSheafify J (Type w)
inst_3 : J.WEqualsLocallyBijective (Type w) (FA := TypeCat.Fun)
hP : P.IsConservativeFamilyOfPoints
X : C
ι : Type u_1
inst_4 : Small.{w, u_1} ι
U : ι → C
f : (i : ι) → U i ⟶ X
hf :
  ∀ (Φ : P.FullSubcategory) (x : Φ.obj (P := P).fiber.obj X),
    ∃ (i : ι) (y : Φ.obj (P := P).fiber.obj (U i)),
      (ConcreteCategory.hom (FC := TypeCat.Fun) (Φ.obj (P := P).fiber.map (f i)) :
            (a : (fun (X : Type w) ↦ X) (Φ.obj.fiber.obj (U i))) → (fun (X : Type w) ↦ X) (Φ.obj.fiber.obj X))
          y =
        x
Φ : P.FullSubcategory
x : Functor.obj Φ.obj (P := P).presheafFiber (Functor.obj shrinkYoneda.{w, v, u} X)
⊢ ∃ (a :
    Functor.obj (C := Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)) Φ.obj (P := P).presheafFiber
      (∐ fun (i : ι) ↦ Functor.obj shrinkYoneda.{w, v, u} (U i))),
    (ConcreteCategory.hom (FC := TypeCat.Fun)
            (Functor.map Φ.obj (P := P).presheafFiber
              (Sigma.desc fun (i : ι) ↦ Functor.map shrinkYoneda.{w, v, u} (f i))) :
          (a : Φ.obj.presheafFiber.obj (∐ fun (i : ι) ↦ shrinkYoneda.{w, v, u}.obj (U i))) →
            Φ.obj.presheafFiber.obj (shrinkYoneda.{w, v, u}.obj X))
        a =
      x
have refine_2 : ∀ (x : Φ.obj (P := P).fiber.obj X),
  ∃ (a :
    Functor.obj (C := Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)) Φ.obj (P := P).presheafFiber
      (∐ fun (i : ι) ↦ Functor.obj shrinkYoneda.{w, v, u} (U i))),
    Eq (α := Functor.obj Φ.obj (P := P).presheafFiber (Functor.obj shrinkYoneda.{w, v, u} X))
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
              (Functor.map Φ.obj (P := P).presheafFiber
                (Sigma.desc fun (i : ι) ↦ Functor.map shrinkYoneda.{w, v, u} (f i))) :
            (a : Φ.obj.presheafFiber.obj (∐ fun (i : ι) ↦ shrinkYoneda.{w, v, u}.obj (U i))) →
              Φ.obj.presheafFiber.obj (shrinkYoneda.{w, v, u}.obj X))
          a :
        Functor.obj Φ.obj (P := P).presheafFiber (Functor.obj shrinkYoneda.{w, v, u} X))
      ((Equiv.symm (Iso.toEquiv (Iso.app Φ.obj (P := P).shrinkYonedaCompPresheafFiberIso X)) :
            (a : Φ.obj.fiber.obj X) → (shrinkYoneda.{w, v, u} ⋙ Φ.obj.presheafFiber).obj X)
          x :
        (shrinkYoneda.{w, v, u} ⋙ Φ.obj (P := P).presheafFiber).obj X)
Tactic: obtain ⟨x, rfl⟩ := (Φ.obj.shrinkYonedaCompPresheafFiberIso.app X).toEquiv.symm.surjective x
Cleared hypotheses: x
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.Conservative (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
P : ObjectProperty (GrothendieckTopology.Point.{w, v, u} J)
inst_1 : LocallySmall.{w, v, u} C
inst_2 : HasSheafify J (Type w)
inst_3 : J.WEqualsLocallyBijective (Type w) (FA := TypeCat.Fun)
hP : P.IsConservativeFamilyOfPoints
inst_4 : ObjectProperty.Small.{w, max u w, max (max u v) (w + 1)} P
X : C
ι : Type u_1
U : ι → C
f : (i : ι) → U i ⟶ X
⊢ Sieve.ofArrows U f ∈ (J : (a : C) → Set (Sieve a)) X ↔
    ∀ (Φ : P.FullSubcategory) (x : Φ.obj (P := P).fiber.obj X),
      ∃ (i : ι) (y : Φ.obj (P := P).fiber.obj (U i)),
        (ConcreteCategory.hom (FC := TypeCat.Fun) (Φ.obj (P := P).fiber.map (f i)) :
              (a : (fun (X : Type w) ↦ X) (Φ.obj.fiber.obj (U i))) → (fun (X : Type w) ↦ X) (Φ.obj.fiber.obj X))
            y =
          x
have refine_1 : ∀ (hf : Sieve.ofArrows U f ∈ (J : (a : C) → Set (Sieve a)) X) (Φ : P.FullSubcategory) (x : Φ.obj (P := P).fiber.obj X),
  ∃ (i : ι) (y : Φ.obj (P := P).fiber.obj (U i)),
    (ConcreteCategory.hom (FC := TypeCat.Fun) (Φ.obj (P := P).fiber.map (f i)) :
          (a : (fun (X : Type w) ↦ X) (Φ.obj.fiber.obj (U i))) → (fun (X : Type w) ↦ X) (Φ.obj.fiber.obj X))
        y =
      x
Tactic: refine ⟨fun hf Φ x ↦ ?_, fun hf ↦ ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.Conservative (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
P : ObjectProperty (GrothendieckTopology.Point.{w, v, u} J)
inst_1 : LocallySmall.{w, v, u} C
inst_2 : HasSheafify J (Type w)
inst_3 : J.WEqualsLocallyBijective (Type w) (FA := TypeCat.Fun)
hP : P.IsConservativeFamilyOfPoints
inst_4 : ObjectProperty.Small.{w, max u w, max (max u v) (w + 1)} P
X : C
ι : Type u_1
U : ι → C
f : (i : ι) → U i ⟶ X
refine_1 :
  ∀ (hf : Sieve.ofArrows U f ∈ (J : (a : C) → Set (Sieve a)) X) (Φ : P.FullSubcategory)
    (x : Φ.obj (P := P).fiber.obj X),
    ∃ (i : ι) (y : Φ.obj (P := P).fiber.obj (U i)),
      (ConcreteCategory.hom (FC := TypeCat.Fun) (Φ.obj (P := P).fiber.map (f i)) :
            (a : (fun (X : Type w) ↦ X) (Φ.obj.fiber.obj (U i))) → (fun (X : Type w) ↦ X) (Φ.obj.fiber.obj X))
          y =
        x
⊢ Sieve.ofArrows U f ∈ (J : (a : C) → Set (Sieve a)) X ↔
    ∀ (Φ : P.FullSubcategory) (x : Φ.obj (P := P).fiber.obj X),
      ∃ (i : ι) (y : Φ.obj (P := P).fiber.obj (U i)),
        (ConcreteCategory.hom (FC := TypeCat.Fun) (Φ.obj (P := P).fiber.map (f i)) :
              (a : (fun (X : Type w) ↦ X) (Φ.obj.fiber.obj (U i))) → (fun (X : Type w) ↦ X) (Φ.obj.fiber.obj X))
            y =
          x
have refine_2 : ∀
  (hf :
    ∀ (Φ : P.FullSubcategory) (x : Φ.obj (P := P).fiber.obj X),
      ∃ (i : ι) (y : Φ.obj (P := P).fiber.obj (U i)),
        (ConcreteCategory.hom (FC := TypeCat.Fun) (Φ.obj (P := P).fiber.map (f i)) :
              (a : (fun (X : Type w) ↦ X) (Φ.obj.fiber.obj (U i))) → (fun (X : Type w) ↦ X) (Φ.obj.fiber.obj X))
            y =
          x),
  Sieve.ofArrows U f ∈ (J : (a : C) → Set (Sieve a)) X
Tactic: refine ⟨fun hf Φ x ↦ ?_, fun hf ↦ ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.Conservative (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
P : ObjectProperty (GrothendieckTopology.Point.{w, v, u} J)
inst_1 : LocallySmall.{w, v, u} C
inst_2 : HasSheafify J (Type w)
inst_3 : J.WEqualsLocallyBijective (Type w) (FA := TypeCat.Fun)
hP : P.IsConservativeFamilyOfPoints
inst_4 : ObjectProperty.Small.{w, max u w, max (max u v) (w + 1)} P
X : C
ι : Type u_1
U : ι → C
f : (i : ι) → U i ⟶ X
hf : Sieve.ofArrows U f ∈ (J : (a : C) → Set (Sieve a)) X
Φ : P.FullSubcategory
x : Φ.obj (P := P).fiber.obj X
⊢ ∃ (i : ι) (y : Φ.obj (P := P).fiber.obj (U i)),
    (ConcreteCategory.hom (FC := TypeCat.Fun) (Φ.obj (P := P).fiber.map (f i)) :
          (a : (fun (X : Type w) ↦ X) (Φ.obj.fiber.obj (U i))) → (fun (X : Type w) ↦ X) (Φ.obj.fiber.obj X))
        y =
      x
have refine_1 : ∀ (Z : C) ⦃Y : C⦄ (i : ι) (p : Z ⟶ U i) (z : Φ.obj (P := P).fiber.obj Z),
  ∃ (i_1 : ι) (y : Φ.obj (P := P).fiber.obj (U i_1)),
    Eq (α := (fun (X : Type w) ↦ X) (Φ.obj (P := P).fiber.obj X))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (Φ.obj (P := P).fiber.map (f i_1)) :
            (a : (fun (X : Type w) ↦ X) (Φ.obj.fiber.obj (U i_1))) → (fun (X : Type w) ↦ X) (Φ.obj.fiber.obj X))
          y :
        (fun (X : Type w) ↦ X) (Φ.obj (P := P).fiber.obj X))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (Φ.obj (P := P).fiber.map (p ≫ f i)) :
            (a : (fun (X : Type w) ↦ X) (Φ.obj.fiber.obj Z)) → (fun (X : Type w) ↦ X) (Φ.obj.fiber.obj X))
          z :
        (fun (X : Type w) ↦ X) (Φ.obj (P := P).fiber.obj X))
Tactic: obtain ⟨Z, _, ⟨_, p, _, ⟨i⟩, rfl⟩, z, rfl⟩ := Φ.obj.jointly_surjective _ hf x
Cleared hypotheses: x
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.Conservative (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
P : ObjectProperty (GrothendieckTopology.Point.{w, v, u} J)
inst_1 : LocallySmall.{w, v, u} C
inst_2 : HasSheafify J (Type w)
inst_3 : J.WEqualsLocallyBijective (Type w) (FA := TypeCat.Fun)
hP : P.IsConservativeFamilyOfPoints
inst_4 : ObjectProperty.Small.{w, max u w, max (max u v) (w + 1)} P
X : C
ι : Type u_1
U : ι → C
f : (i : ι) → U i ⟶ X
hf :
  ∀ (Φ : P.FullSubcategory) (x : Φ.obj (P := P).fiber.obj X),
    ∃ (i : ι) (y : Φ.obj (P := P).fiber.obj (U i)),
      (ConcreteCategory.hom (FC := TypeCat.Fun) (Φ.obj (P := P).fiber.map (f i)) :
            (a : (fun (X : Type w) ↦ X) (Φ.obj.fiber.obj (U i))) → (fun (X : Type w) ↦ X) (Φ.obj.fiber.obj X))
          y =
        x
ι' : Type (max w (max u v) (w + 1)) := (Φ : P.FullSubcategory) × Φ.obj (P := P).fiber.obj X
⊢ Sieve.ofArrows U f ∈ (J : (a : C) → Set (Sieve a)) X
have refine_2 : ∀ (i : (j : ι') → ι)
  (y :
    (j : ι') → j.fst (β := fun (Φ : P.FullSubcategory) ↦ Φ.obj (P := P).fiber.obj X).obj (P := P).fiber.obj (U (i j)))
  (hy :
    ∀ (j : ι'),
      (ConcreteCategory.hom (FC := TypeCat.Fun)
              (j.fst (β := fun (Φ : P.FullSubcategory) ↦ Φ.obj (P := P).fiber.obj X).obj (P := P).fiber.map (f (i j))) :
            (a : (fun (X : Type w) ↦ X) (j.fst.obj.fiber.obj (U (i j)))) →
              (fun (X : Type w) ↦ X) (j.fst.obj.fiber.obj X))
          (y j) =
        j.snd (β := fun (Φ : P.FullSubcategory) ↦ Φ.obj (P := P).fiber.obj X)),
  Sieve.ofArrows U f ∈ (J : (a : C) → Set (Sieve a)) X
Tactic: choose i y hy using fun (j : ι') ↦ hf j.1 j.2
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.Conservative (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
P : ObjectProperty (GrothendieckTopology.Point.{w, v, u} J)
inst_1 : LocallySmall.{w, v, u} C
inst_2 : HasSheafify J (Type w)
inst_3 : J.WEqualsLocallyBijective (Type w) (FA := TypeCat.Fun)
hP : P.IsConservativeFamilyOfPoints
inst_4 : ObjectProperty.Small.{w, max u w, max (max u v) (w + 1)} P
X : C
ι : Type u_1
U : ι → C
f : (i : ι) → U i ⟶ X
hf :
  ∀ (Φ : P.FullSubcategory) (x : Φ.obj (P := P).fiber.obj X),
    ∃ (i : ι) (y : Φ.obj (P := P).fiber.obj (U i)),
      (ConcreteCategory.hom (FC := TypeCat.Fun) (Φ.obj (P := P).fiber.map (f i)) :
            (a : (fun (X : Type w) ↦ X) (Φ.obj.fiber.obj (U i))) → (fun (X : Type w) ↦ X) (Φ.obj.fiber.obj X))
          y =
        x
ι' : Type (max w (max u v) (w + 1)) := (Φ : P.FullSubcategory) × Φ.obj (P := P).fiber.obj X
i : (j : ι') → ι
y : (j : ι') → j.fst (β := fun (Φ : P.FullSubcategory) ↦ Φ.obj (P := P).fiber.obj X).obj (P := P).fiber.obj (U (i j))
hy :
  ∀ (j : ι'),
    (ConcreteCategory.hom (FC := TypeCat.Fun)
            (j.fst (β := fun (Φ : P.FullSubcategory) ↦ Φ.obj (P := P).fiber.obj X).obj (P := P).fiber.map (f (i j))) :
          (a : (fun (X : Type w) ↦ X) (j.fst.obj.fiber.obj (U (i j)))) → (fun (X : Type w) ↦ X) (j.fst.obj.fiber.obj X))
        (y j) =
      j.snd (β := fun (Φ : P.FullSubcategory) ↦ Φ.obj (P := P).fiber.obj X)
⊢ Sieve.ofArrows U f ∈ (J : (a : C) → Set (Sieve a)) X
have refine_2 : (Sieve.ofArrows (fun (i' : ι') ↦ U (i i')) fun (i' : ι') ↦ f (i i')) ≤ Sieve.ofArrows U f
Tactic: refine J.superset_covering (S := Sieve.ofArrows _ (fun i' ↦ f (i i'))) ?_ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.Conservative (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
P : ObjectProperty (GrothendieckTopology.Point.{w, v, u} J)
inst_1 : LocallySmall.{w, v, u} C
inst_2 : HasSheafify J (Type w)
inst_3 : J.WEqualsLocallyBijective (Type w) (FA := TypeCat.Fun)
hP : P.IsConservativeFamilyOfPoints
inst_4 : ObjectProperty.Small.{w, max u w, max (max u v) (w + 1)} P
X : C
ι : Type u_1
U : ι → C
f : (i : ι) → U i ⟶ X
hf :
  ∀ (Φ : P.FullSubcategory) (x : Φ.obj (P := P).fiber.obj X),
    ∃ (i : ι) (y : Φ.obj (P := P).fiber.obj (U i)),
      (ConcreteCategory.hom (FC := TypeCat.Fun) (Φ.obj (P := P).fiber.map (f i)) :
            (a : (fun (X : Type w) ↦ X) (Φ.obj.fiber.obj (U i))) → (fun (X : Type w) ↦ X) (Φ.obj.fiber.obj X))
          y =
        x
ι' : Type (max w (max u v) (w + 1)) := (Φ : P.FullSubcategory) × Φ.obj (P := P).fiber.obj X
i : (j : ι') → ι
y : (j : ι') → j.fst (β := fun (Φ : P.FullSubcategory) ↦ Φ.obj (P := P).fiber.obj X).obj (P := P).fiber.obj (U (i j))
hy :
  ∀ (j : ι'),
    (ConcreteCategory.hom (FC := TypeCat.Fun)
            (j.fst (β := fun (Φ : P.FullSubcategory) ↦ Φ.obj (P := P).fiber.obj X).obj (P := P).fiber.map (f (i j))) :
          (a : (fun (X : Type w) ↦ X) (j.fst.obj.fiber.obj (U (i j)))) → (fun (X : Type w) ↦ X) (j.fst.obj.fiber.obj X))
        (y j) =
      j.snd (β := fun (Φ : P.FullSubcategory) ↦ Φ.obj (P := P).fiber.obj X)
refine_2 : (Sieve.ofArrows (fun (i' : ι') ↦ U (i i')) fun (i' : ι') ↦ f (i i')) ≤ Sieve.ofArrows U f
⊢ Sieve.ofArrows U f ∈ (J : (a : C) → Set (Sieve a)) X
have refine_2₁ : (Sieve.ofArrows (fun (i' : ι') ↦ U (i i')) fun (i' : ι') ↦ f (i i')) ∈ (J : (a : C) → Set (Sieve a)) X
Tactic: refine J.superset_covering (S := Sieve.ofArrows _ (fun i' ↦ f (i i'))) ?_ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.Conservative (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
P : ObjectProperty (GrothendieckTopology.Point.{w, v, u} J)
inst_1 : LocallySmall.{w, v, u} C
inst_2 : HasSheafify J (Type w)
inst_3 : J.WEqualsLocallyBijective (Type w) (FA := TypeCat.Fun)
hP : P.IsConservativeFamilyOfPoints
inst_4 : ObjectProperty.Small.{w, max u w, max (max u v) (w + 1)} P
X : C
ι : Type u_1
U : ι → C
f : (i : ι) → U i ⟶ X
hf :
  ∀ (Φ : P.FullSubcategory) (x : Φ.obj (P := P).fiber.obj X),
    ∃ (i : ι) (y : Φ.obj (P := P).fiber.obj (U i)),
      (ConcreteCategory.hom (FC := TypeCat.Fun) (Φ.obj (P := P).fiber.map (f i)) :
            (a : (fun (X : Type w) ↦ X) (Φ.obj.fiber.obj (U i))) → (fun (X : Type w) ↦ X) (Φ.obj.fiber.obj X))
          y =
        x
ι' : Type (max w (max u v) (w + 1)) := (Φ : P.FullSubcategory) × Φ.obj (P := P).fiber.obj X
i : (j : ι') → ι
y : (j : ι') → j.fst (β := fun (Φ : P.FullSubcategory) ↦ Φ.obj (P := P).fiber.obj X).obj (P := P).fiber.obj (U (i j))
hy :
  ∀ (j : ι'),
    (ConcreteCategory.hom (FC := TypeCat.Fun)
            (j.fst (β := fun (Φ : P.FullSubcategory) ↦ Φ.obj (P := P).fiber.obj X).obj (P := P).fiber.map (f (i j))) :
          (a : (fun (X : Type w) ↦ X) (j.fst.obj.fiber.obj (U (i j)))) → (fun (X : Type w) ↦ X) (j.fst.obj.fiber.obj X))
        (y j) =
      j.snd (β := fun (Φ : P.FullSubcategory) ↦ Φ.obj (P := P).fiber.obj X)
⊢ (Sieve.ofArrows (fun (i' : ι') ↦ U (i i')) fun (i' : ι') ↦ f (i i')) ≤ Sieve.ofArrows U f
have refine_2 : ∀ (i_1 : ι'), (Sieve.ofArrows U f).arrows (Y := U (i i_1)) (f (i i_1))
Tactic: rw [Sieve.generate_le_iff, Presieve.ofArrows_le_iff]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.Conservative (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
P : ObjectProperty (GrothendieckTopology.Point.{w, v, u} J)
inst_1 : LocallySmall.{w, v, u} C
inst_2 : HasSheafify J (Type w)
inst_3 : J.WEqualsLocallyBijective (Type w) (FA := TypeCat.Fun)
hP : P.IsConservativeFamilyOfPoints
inst_4 : ObjectProperty.Small.{w, max u w, max (max u v) (w + 1)} P
X : C
ι : Type u_1
U : ι → C
f : (i : ι) → U i ⟶ X
hf :
  ∀ (Φ : P.FullSubcategory) (x : Φ.obj (P := P).fiber.obj X),
    ∃ (i : ι) (y : Φ.obj (P := P).fiber.obj (U i)),
      (ConcreteCategory.hom (FC := TypeCat.Fun) (Φ.obj (P := P).fiber.map (f i)) :
            (a : (fun (X : Type w) ↦ X) (Φ.obj.fiber.obj (U i))) → (fun (X : Type w) ↦ X) (Φ.obj.fiber.obj X))
          y =
        x
ι' : Type (max w (max u v) (w + 1)) := (Φ : P.FullSubcategory) × Φ.obj (P := P).fiber.obj X
i : (j : ι') → ι
y : (j : ι') → j.fst (β := fun (Φ : P.FullSubcategory) ↦ Φ.obj (P := P).fiber.obj X).obj (P := P).fiber.obj (U (i j))
hy :
  ∀ (j : ι'),
    (ConcreteCategory.hom (FC := TypeCat.Fun)
            (j.fst (β := fun (Φ : P.FullSubcategory) ↦ Φ.obj (P := P).fiber.obj X).obj (P := P).fiber.map (f (i j))) :
          (a : (fun (X : Type w) ↦ X) (j.fst.obj.fiber.obj (U (i j)))) → (fun (X : Type w) ↦ X) (j.fst.obj.fiber.obj X))
        (y j) =
      j.snd (β := fun (Φ : P.FullSubcategory) ↦ Φ.obj (P := P).fiber.obj X)
⊢ (Sieve.ofArrows (fun (i' : ι') ↦ U (i i')) fun (i' : ι') ↦ f (i i')) ∈ (J : (a : C) → Set (Sieve a)) X
have refine_2 : ∀ (Φ : P.FullSubcategory) (x : Φ.obj (P := P).fiber.obj X),
  ∃ (i_1 : ι') (y : Φ.obj (P := P).fiber.obj (U (i i_1))),
    (ConcreteCategory.hom (FC := TypeCat.Fun) (Φ.obj (P := P).fiber.map (f (i i_1))) :
          (a : (fun (X : Type w) ↦ X) (Φ.obj.fiber.obj (U (i i_1)))) → (fun (X : Type w) ↦ X) (Φ.obj.fiber.obj X))
        y =
      x
Tactic: rw [hP.jointly_reflect_ofArrows_mem]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.Conservative (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
P : ObjectProperty (GrothendieckTopology.Point.{w, v, u} J)
inst_1 : LocallySmall.{w, v, u} C
hP :
  ∀ ⦃X : C⦄ (S : Sieve.{v, u} X),
    (∀ (Φ : P.FullSubcategory) (x : Φ.obj (P := P).fiber.obj X),
        ∃ (Y : C) (g : Y ⟶ X) (_ : S.arrows g) (y : Φ.obj (P := P).fiber.obj Y),
          (ConcreteCategory.hom (FC := TypeCat.Fun) (Φ.obj (P := P).fiber.map g) :
                (a : (fun (X : Type w) ↦ X) (Φ.obj.fiber.obj Y)) → (fun (X : Type w) ↦ X) (Φ.obj.fiber.obj X))
              y =
            x) →
      S ∈ (J : (a : C) → Set (Sieve a)) X
F₁ F₂ : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)
f : F₁ ⟶ F₂
inst_2 : Mono f
hf :
  ∀ (Φ : P.FullSubcategory),
    Function.Surjective (α := (fun (X : Type w) ↦ X) (Functor.obj Φ.obj (P := P).presheafFiber F₁)) (β :=
      (fun (X : Type w) ↦ X) (Functor.obj Φ.obj (P := P).presheafFiber F₂))
      (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (Functor.map Φ.obj (P := P).presheafFiber f)) :
        (a : (fun (X : Type w) ↦ X) (Φ.obj.presheafFiber.obj F₁)) → (fun (X : Type w) ↦ X) (Φ.obj.presheafFiber.obj F₂))
⊢ Presheaf.IsLocallySurjective J (FA := TypeCat.Fun) f
have inr : ∀
  (this :
    ∀ {F₁ F₂ : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)} (f : F₁ ⟶ F₂) [Mono f]
      (hf :
        ∀ (Φ : P.FullSubcategory),
          Function.Surjective (α := (fun (X : Type w) ↦ X) (Functor.obj Φ.obj (P := P).presheafFiber F₁)) (β :=
            (fun (X : Type w) ↦ X) (Functor.obj Φ.obj (P := P).presheafFiber F₂))
            (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (Functor.map Φ.obj (P := P).presheafFiber f)) :
              (a : (fun (X : Type w) ↦ X) (Φ.obj.presheafFiber.obj F₁)) →
                (fun (X : Type w) ↦ X) (Φ.obj.presheafFiber.obj F₂)))
      (hF₂ : ∃ (U : C), F₂ = Functor.obj shrinkYoneda.{w, v, u} U),
      Presheaf.IsLocallySurjective J (FA := TypeCat.Fun) f)
  (hF₂ : ¬∃ (U : C), F₂ = Functor.obj shrinkYoneda.{w, v, u} U), Presheaf.IsLocallySurjective J (FA := TypeCat.Fun) f
Tactic: wlog hF₂ : ∃ (U : C), F₂ = shrinkYoneda.obj U generalizing F₁ F₂
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.Conservative (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
P : ObjectProperty (GrothendieckTopology.Point.{w, v, u} J)
inst_1 : LocallySmall.{w, v, u} C
hP :
  ∀ ⦃X : C⦄ (S : Sieve.{v, u} X),
    (∀ (Φ : P.FullSubcategory) (x : Φ.obj (P := P).fiber.obj X),
        ∃ (Y : C) (g : Y ⟶ X) (_ : S.arrows g) (y : Φ.obj (P := P).fiber.obj Y),
          (ConcreteCategory.hom (FC := TypeCat.Fun) (Φ.obj (P := P).fiber.map g) :
                (a : (fun (X : Type w) ↦ X) (Φ.obj.fiber.obj Y)) → (fun (X : Type w) ↦ X) (Φ.obj.fiber.obj X))
              y =
            x) →
      S ∈ (J : (a : C) → Set (Sieve a)) X
F₁ F₂ : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)
f : F₁ ⟶ F₂
inst_2 : Mono f
hf :
  ∀ (Φ : P.FullSubcategory),
    Function.Surjective (α := (fun (X : Type w) ↦ X) (Functor.obj Φ.obj (P := P).presheafFiber F₁)) (β :=
      (fun (X : Type w) ↦ X) (Functor.obj Φ.obj (P := P).presheafFiber F₂))
      (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (Functor.map Φ.obj (P := P).presheafFiber f)) :
        (a : (fun (X : Type w) ↦ X) (Φ.obj.presheafFiber.obj F₁)) → (fun (X : Type w) ↦ X) (Φ.obj.presheafFiber.obj F₂))
inr :
  ∀
    (this :
      ∀ {F₁ F₂ : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)} (f : F₁ ⟶ F₂) [Mono f]
        (hf :
          ∀ (Φ : P.FullSubcategory),
            Function.Surjective (α := (fun (X : Type w) ↦ X) (Functor.obj Φ.obj (P := P).presheafFiber F₁)) (β :=
              (fun (X : Type w) ↦ X) (Functor.obj Φ.obj (P := P).presheafFiber F₂))
              (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (Functor.map Φ.obj (P := P).presheafFiber f)) :
                (a : (fun (X : Type w) ↦ X) (Φ.obj.presheafFiber.obj F₁)) →
                  (fun (X : Type w) ↦ X) (Φ.obj.presheafFiber.obj F₂)))
        (hF₂ : ∃ (U : C), F₂ = Functor.obj shrinkYoneda.{w, v, u} U),
        Presheaf.IsLocallySurjective J (FA := TypeCat.Fun) f)
    (hF₂ : ¬∃ (U : C), F₂ = Functor.obj shrinkYoneda.{w, v, u} U), Presheaf.IsLocallySurjective J (FA := TypeCat.Fun) f
⊢ Presheaf.IsLocallySurjective J (FA := TypeCat.Fun) f
have wlog : ∀ {F₁ F₂ : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)} (f : F₁ ⟶ F₂) [Mono f]
  (hf :
    ∀ (Φ : P.FullSubcategory),
      Function.Surjective (α := (fun (X : Type w) ↦ X) (Functor.obj Φ.obj (P := P).presheafFiber F₁)) (β :=
        (fun (X : Type w) ↦ X) (Functor.obj Φ.obj (P := P).presheafFiber F₂))
        (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (Functor.map Φ.obj (P := P).presheafFiber f)) :
          (a : (fun (X : Type w) ↦ X) (Φ.obj.presheafFiber.obj F₁)) →
            (fun (X : Type w) ↦ X) (Φ.obj.presheafFiber.obj F₂)))
  (hF₂ : ∃ (U : C), F₂ = Functor.obj shrinkYoneda.{w, v, u} U), Presheaf.IsLocallySurjective J (FA := TypeCat.Fun) f
Tactic: wlog hF₂ : ∃ (U : C), F₂ = shrinkYoneda.obj U generalizing F₁ F₂
Cleared hypotheses: F₁, F₂, f, inst_2, hf
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.Conservative (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
P : ObjectProperty (GrothendieckTopology.Point.{w, v, u} J)
inst_1 : LocallySmall.{w, v, u} C
hP :
  ∀ ⦃X : C⦄ (S : Sieve.{v, u} X),
    (∀ (Φ : P.FullSubcategory) (x : Φ.obj (P := P).fiber.obj X),
        ∃ (Y : C) (g : Y ⟶ X) (_ : S.arrows g) (y : Φ.obj (P := P).fiber.obj Y),
          (ConcreteCategory.hom (FC := TypeCat.Fun) (Φ.obj (P := P).fiber.map g) :
                (a : (fun (X : Type w) ↦ X) (Φ.obj.fiber.obj Y)) → (fun (X : Type w) ↦ X) (Φ.obj.fiber.obj X))
              y =
            x) →
      S ∈ (J : (a : C) → Set (Sieve a)) X
F₁ F₂ : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)
f : F₁ ⟶ F₂
inst_2 : Mono f
hf :
  ∀ (Φ : P.FullSubcategory),
    Function.Surjective (α := (fun (X : Type w) ↦ X) (Functor.obj Φ.obj (P := P).presheafFiber F₁)) (β :=
      (fun (X : Type w) ↦ X) (Functor.obj Φ.obj (P := P).presheafFiber F₂))
      (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (Functor.map Φ.obj (P := P).presheafFiber f)) :
        (a : (fun (X : Type w) ↦ X) (Φ.obj.presheafFiber.obj F₁)) → (fun (X : Type w) ↦ X) (Φ.obj.presheafFiber.obj F₂))
this :
  ∀ {F₁ F₂ : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)} (f : F₁ ⟶ F₂) [Mono f]
    (hf :
      ∀ (Φ : P.FullSubcategory),
        Function.Surjective (α := (fun (X : Type w) ↦ X) (Functor.obj Φ.obj (P := P).presheafFiber F₁)) (β :=
          (fun (X : Type w) ↦ X) (Functor.obj Φ.obj (P := P).presheafFiber F₂))
          (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (Functor.map Φ.obj (P := P).presheafFiber f)) :
            (a : (fun (X : Type w) ↦ X) (Φ.obj.presheafFiber.obj F₁)) →
              (fun (X : Type w) ↦ X) (Φ.obj.presheafFiber.obj F₂)))
    (hF₂ : ∃ (U : C), F₂ = Functor.obj shrinkYoneda.{w, v, u} U), Presheaf.IsLocallySurjective J (FA := TypeCat.Fun) f
hF₂ : ¬∃ (U : C), F₂ = Functor.obj shrinkYoneda.{w, v, u} U
⊢ Presheaf.IsLocallySurjective J (FA := TypeCat.Fun) f
have inr : ∀ {U : C} (s : ToType (FC := TypeCat.Fun) (F₂.obj (op U))),
  Presheaf.imageSieve (FA := TypeCat.Fun) f s ∈ (J : (a : C) → Set (Sieve a)) U
Tactic: refine ⟨fun {U} s ↦ ?_⟩