awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Descent.Precoverage (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
J : GrothendieckTopology.{v, u} C
inst_1 : F.IsPrestack J
ι : Type t
S : C
X : ι → C
f : (i : ι) → X i ⟶ S
ι' : Type t'
X' : ι' → C
f' : (j : ι') → X' j ⟶ S
α : ι' → ι
p' : (j : ι') → X' j ⟶ X (α j)
w : ∀ (j : ι'), p' j ≫ f (α j) = f' j
hf' : Sieve.ofArrows X' f' ∈ (J : (a : C) → Set (Sieve a)) S
D₁ D₂ : F.DescentData f
φ :
  Quiver.Hom (V := F.DescentData f') (Functor.obj (pullFunctor F (p := 𝟙 S) (α := α) (p' := p') ⋯) D₁)
    (Functor.obj (pullFunctor F (p := 𝟙 S) (α := α) (p' := p') ⋯) D₂)
W : C
q : W ⟶ S
i₁ i₂ : ι
f₁ : W ⟶ X i₁
f₂ : W ⟶ X i₂
hf₁ : f₁ ≫ f i₁ = q
hf₂ : f₂ ≫ f i₂ = q
⊢ Functor.map (Cat.Hom.toFunctor (F.map f₁.op.toLoc)) (hom w hf' φ i₁) =
    D₁.hom q f₁ f₂ ⋯ ⋯ ≫ Functor.map (Cat.Hom.toFunctor (F.map f₂.op.toLoc)) (hom w hf' φ i₂) ≫ D₂.hom q f₂ f₁ ⋯ ⋯
have H : Membership.mem (α := Sieve (Over.mk f₁)) (γ := Set (Sieve (Over.mk f₁)))
  ((J.over (X i₁) : (a : Over (X i₁)) → Set (Sieve a)) (Over.mk f₁))
  (((Sieve.overEquiv (Over.mk f₁)).symm : (a : Sieve (Over.mk f₁).left) → Sieve (Over.mk f₁))
      (Sieve.pullback q (Sieve.ofArrows X' f')) :
    Sieve (Over.mk f₁))
Tactic: have H : (Sieve.overEquiv (Over.mk f₁)).symm
      (Sieve.pullback q (Sieve.ofArrows X' f')) ∈ J.over _ _ := by
    rw [J.mem_over_iff, Equiv.apply_symm_apply]
    exact J.pullback_stable _ hf'
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Descent.Precoverage (goal)
Goal:
C : Type u
inst : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
J : GrothendieckTopology.{v, u} C
inst_1 : F.IsPrestack J
ι : Type t
S : C
X : ι → C
f : (i : ι) → X i ⟶ S
ι' : Type t'
X' : ι' → C
f' : (j : ι') → X' j ⟶ S
α : ι' → ι
p' : (j : ι') → X' j ⟶ X (α j)
w : ∀ (j : ι'), p' j ≫ f (α j) = f' j
hf' : Sieve.ofArrows X' f' ∈ (J : (a : C) → Set (Sieve a)) S
D₁ D₂ : F.DescentData f
φ :
  Quiver.Hom (V := F.DescentData f') (Functor.obj (pullFunctor F (p := 𝟙 S) (α := α) (p' := p') ⋯) D₁)
    (Functor.obj (pullFunctor F (p := 𝟙 S) (α := α) (p' := p') ⋯) D₂)
W : C
q : W ⟶ S
i₁ i₂ : ι
f₁ : W ⟶ X i₁
f₂ : W ⟶ X i₂
hf₁ : f₁ ≫ f i₁ = q
hf₂ : f₂ ≫ f i₂ = q
H :
  Membership.mem (α := Sieve (Over.mk f₁)) (γ := Set (Sieve (Over.mk f₁)))
    ((J.over (X i₁) : (a : Over (X i₁)) → Set (Sieve a)) (Over.mk f₁))
    (((Sieve.overEquiv (Over.mk f₁)).symm : (a : Sieve (Over.mk f₁).left) → Sieve (Over.mk f₁))
        (Sieve.pullback q (Sieve.ofArrows X' f')) :
      Sieve (Over.mk f₁))
⊢ Functor.map (Cat.Hom.toFunctor (F.map f₁.op.toLoc)) (hom w hf' φ i₁) =
    D₁.hom q f₁ f₂ ⋯ ⋯ ≫ Functor.map (Cat.Hom.toFunctor (F.map f₂.op.toLoc)) (hom w hf' φ i₂) ≫ D₂.hom q f₂ f₁ ⋯ ⋯
have refine : ∀ ⦃Y : Over.{v, u} (X i₁)⦄ ⦃f_1 : Y ⟶ Over.mk f₁⦄,
  Sieve.arrows
      (((Sieve.overEquiv (Over.mk f₁)).symm : (a : Sieve (Over.mk f₁).left) → Sieve (Over.mk f₁))
        (Sieve.pullback q (Sieve.ofArrows X' f')))
      f_1 →
    Eq (α := (fun (X : Type v') ↦ X) ((F.presheafHom (D₁.obj i₁) (D₂.obj i₁)).obj (op Y)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) ((F.presheafHom (D₁.obj i₁) (D₂.obj i₁)).map f_1.op) :
            (a : (fun (X : Type v') ↦ X) ((F.presheafHom (D₁.obj i₁) (D₂.obj i₁)).obj (op (Over.mk f₁)))) →
              (fun (X : Type v') ↦ X) ((F.presheafHom (D₁.obj i₁) (D₂.obj i₁)).obj (op Y)))
          (Functor.map (Cat.Hom.toFunctor (F.map f₁.op.toLoc)) (hom w hf' φ i₁)) :
        (fun (X : Type v') ↦ X) ((F.presheafHom (D₁.obj i₁) (D₂.obj i₁)).obj (op Y)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) ((F.presheafHom (D₁.obj i₁) (D₂.obj i₁)).map f_1.op) :
            (a : (fun (X : Type v') ↦ X) ((F.presheafHom (D₁.obj i₁) (D₂.obj i₁)).obj (op (Over.mk f₁)))) →
              (fun (X : Type v') ↦ X) ((F.presheafHom (D₁.obj i₁) (D₂.obj i₁)).obj (op Y)))
          (D₁.hom q f₁ f₂ sorry sorry ≫
            Functor.map (Cat.Hom.toFunctor (F.map f₂.op.toLoc)) (hom w hf' φ i₂) ≫ D₂.hom q f₂ f₁ sorry sorry) :
        (fun (X : Type v') ↦ X) ((F.presheafHom (D₁.obj i₁) (D₂.obj i₁)).obj (op Y)))
Tactic: refine ((isSheaf_iff_isSheaf_of_type _ _).1
    (IsPrestack.isSheaf J (D₁.obj i₁) (D₂.obj i₁)) _ H).isSeparatedFor.ext ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Descent.Precoverage (goal)
Goal:
C : Type u
inst : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
J : GrothendieckTopology.{v, u} C
inst_1 : F.IsPrestack J
ι : Type t
S : C
X : ι → C
f : (i : ι) → X i ⟶ S
ι' : Type t'
X' : ι' → C
f' : (j : ι') → X' j ⟶ S
α : ι' → ι
p' : (j : ι') → X' j ⟶ X (α j)
w : ∀ (j : ι'), p' j ≫ f (α j) = f' j
hf' : Sieve.ofArrows X' f' ∈ (J : (a : C) → Set (Sieve a)) S
D₁ D₂ : F.DescentData f
φ :
  Quiver.Hom (V := F.DescentData f') (Functor.obj (pullFunctor F (p := 𝟙 S) (α := α) (p' := p') ⋯) D₁)
    (Functor.obj (pullFunctor F (p := 𝟙 S) (α := α) (p' := p') ⋯) D₂)
W : C
q : W ⟶ S
i₁ i₂ : ι
f₁ : W ⟶ X i₁
f₂ : W ⟶ X i₂
hf₁ : f₁ ≫ f i₁ = q
hf₂ : f₂ ≫ f i₂ = q
H :
  Membership.mem (α := Sieve (Over.mk f₁)) (γ := Set (Sieve (Over.mk f₁)))
    ((J.over (X i₁) : (a : Over (X i₁)) → Set (Sieve a)) (Over.mk f₁))
    (((Sieve.overEquiv (Over.mk f₁)).symm : (a : Sieve (Over.mk f₁).left) → Sieve (Over.mk f₁))
        (Sieve.pullback q (Sieve.ofArrows X' f')) :
      Sieve (Over.mk f₁))
Y : C
j : ι'
T : C
g : T ⟶ X' j
p : T ⟶ W
fac : g ≫ f' j = p ≫ q
⊢ CategoryStruct.comp (Y :=
      Functor.obj (C := (↑(F.obj { as := op W : LocallyDiscrete Cᵒᵖ }) : Type u'))
        (Cat.Hom.toFunctor (F.map p.op.toLoc)) (Functor.obj (Cat.Hom.toFunctor (F.map f₁.op.toLoc)) (D₁.obj i₁)))
      (NatTrans.app (Cat.Hom₂.toNatTrans (Iso.hom (F.mapComp' f₁.op.toLoc p.op.toLoc (p ≫ f₁).op.toLoc ⋯))) (D₁.obj i₁))
      (Functor.map (Cat.Hom.toFunctor (F.map p.op.toLoc))
          (Functor.map (Cat.Hom.toFunctor (F.map f₁.op.toLoc)) (hom w hf' φ i₁)) ≫
        NatTrans.app (Cat.Hom₂.toNatTrans (Iso.inv (F.mapComp' f₁.op.toLoc p.op.toLoc (p ≫ f₁).op.toLoc ⋯)))
          (D₂.obj i₁)) =
    NatTrans.app (Cat.Hom₂.toNatTrans (Iso.hom (F.mapComp' f₁.op.toLoc p.op.toLoc (p ≫ f₁).op.toLoc ⋯))) (D₁.obj i₁) ≫
      Functor.map (Cat.Hom.toFunctor (F.map p.op.toLoc))
          (D₁.hom q f₁ f₂ ⋯ ⋯ ≫
            Functor.map (Cat.Hom.toFunctor (F.map f₂.op.toLoc)) (hom w hf' φ i₂) ≫ D₂.hom q f₂ f₁ ⋯ ⋯) ≫
        NatTrans.app (Cat.Hom₂.toNatTrans (Iso.inv (F.mapComp' f₁.op.toLoc p.op.toLoc (p ≫ f₁).op.toLoc ⋯))) (D₂.obj i₁)
have simp : Functor.map (Cat.Hom.toFunctor (F.map (p ≫ f₁).op.toLoc)) (hom w hf' φ i₁) =
  NatTrans.app (Cat.Hom₂.toNatTrans (Iso.hom (F.mapComp' f₁.op.toLoc p.op.toLoc (p ≫ f₁).op.toLoc sorry))) (D₁.obj i₁) ≫
    Functor.map (Cat.Hom.toFunctor (F.map p.op.toLoc)) (D₁.hom q f₁ f₂ sorry sorry) ≫
      Functor.map (Cat.Hom.toFunctor (F.map p.op.toLoc))
          (Functor.map (Cat.Hom.toFunctor (F.map f₂.op.toLoc)) (hom w hf' φ i₂)) ≫
        Functor.map (Cat.Hom.toFunctor (F.map p.op.toLoc)) (D₂.hom q f₂ f₁ sorry sorry) ≫
          NatTrans.app (Cat.Hom₂.toNatTrans (Iso.inv (F.mapComp' f₁.op.toLoc p.op.toLoc (p ≫ f₁).op.toLoc sorry)))
            (D₂.obj i₁)
Tactic: simp only [mapComp'_inv_naturality, Cat.Hom.comp_toFunctor, Functor.comp_obj,
    Cat.Hom.hom_inv_id_toNatTrans_app_assoc, Functor.map_comp, Category.assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Descent.Precoverage (goal)
Goal:
C : Type u
inst : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
J : GrothendieckTopology.{v, u} C
inst_1 : F.IsPrestack J
ι : Type t
S : C
X : ι → C
f : (i : ι) → X i ⟶ S
ι' : Type t'
X' : ι' → C
f' : (j : ι') → X' j ⟶ S
α : ι' → ι
p' : (j : ι') → X' j ⟶ X (α j)
w : ∀ (j : ι'), p' j ≫ f (α j) = f' j
hf' : Sieve.ofArrows X' f' ∈ (J : (a : C) → Set (Sieve a)) S
D₁ D₂ : F.DescentData f
φ : (pullFunctor F ⋯).obj D₁ ⟶ (pullFunctor F ⋯).obj D₂
⊢ (pullFunctor F ⋯).map { hom := fun (i : ι) ↦ full_pullFunctor.hom w hf' φ i, comm := ⋯ : D₁.Hom D₂ } = φ
have h : ∀ (i : ι'),
  ((pullFunctor F sorry).map { hom := fun (i : ι) ↦ full_pullFunctor.hom w hf' φ i, comm := sorry : D₁.Hom D₂ }).hom i = φ.hom i
Tactic: ext i
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Descent.Precoverage (goal)
Goal:
C : Type u
inst : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
J : GrothendieckTopology.{v, u} C
inst_1 : F.IsPrestack J
ι : Type t
S : C
X : ι → C
f : (i : ι) → X i ⟶ S
ι' : Type t'
X' : ι' → C
f' : (j : ι') → X' j ⟶ S
α : ι' → ι
p' : (j : ι') → X' j ⟶ X (α j)
w : ∀ (j : ι'), p' j ≫ f (α j) = f' j
hf' : Sieve.ofArrows X' f' ∈ (J : (a : C) → Set (Sieve a)) S
D₁ D₂ : F.DescentData f
φ : (pullFunctor F ⋯).obj D₁ ⟶ (pullFunctor F ⋯).obj D₂
i : ι'
⊢ ((pullFunctor F ⋯).map { hom := fun (i : ι) ↦ full_pullFunctor.hom w hf' φ i, comm := ⋯ : D₁.Hom D₂ }).hom i = φ.hom i
have h : Functor.map (Cat.Hom.toFunctor (F.map (p' i).op.toLoc)) (full_pullFunctor.hom w hf' φ (α i)) = φ.hom i
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Descent.Precoverage (goal)
Goal:
C : Type u
inst : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
J : GrothendieckTopology.{v, u} C
inst_1 : F.IsPrestack J
ι : Type t
S : C
X : ι → C
f : (i : ι) → X i ⟶ S
ι' : Type t'
X' : ι' → C
f' : (j : ι') → X' j ⟶ S
α : ι' → ι
p' : (j : ι') → X' j ⟶ X (α j)
w : ∀ (j : ι'), p' j ≫ f (α j) = f' j
hf' : Sieve.ofArrows X' f' ∈ (J : (a : C) → Set (Sieve a)) S
D₁ D₂ : F.DescentData f
φ : (pullFunctor F ⋯).obj D₁ ⟶ (pullFunctor F ⋯).obj D₂
i : ι'
⊢ Functor.map (Cat.Hom.toFunctor (F.map (p' i).op.toLoc)) (full_pullFunctor.hom w hf' φ (α i)) = φ.hom i
have h : mor w φ (p' i) (𝟙 (X' i)) sorry = φ.hom i
Tactic: rw [map_hom _ _ _ _ (𝟙 _) (by cat_disch),
        mor_eq _ _ _ _ _ (f' i) (p' i), pullHom_id,
        D₁.hom_self _ _ (by cat_disch), D₂.hom_self _ _ (by cat_disch),
        Category.id_comp, Category.comp_id]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Descent.Precoverage (goal)
Goal:
C : Type u
inst : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
J : GrothendieckTopology.{v, u} C
inst_1 : F.IsPrestack J
ι : Type t
S : C
X : ι → C
f : (i : ι) → X i ⟶ S
ι' : Type t'
X' : ι' → C
f' : (j : ι') → X' j ⟶ S
α : ι' → ι
p' : (j : ι') → X' j ⟶ X (α j)
w : ∀ (j : ι'), p' j ≫ f (α j) = f' j
hf' : Sieve.ofArrows X' f' ∈ (J : (a : C) → Set (Sieve a)) S
D₁ D₂ : F.DescentData f
φ : (pullFunctor F ⋯).obj D₁ ⟶ (pullFunctor F ⋯).obj D₂
i : ι'
h : mor w φ (p' i) (𝟙 (X' i)) ⋯ = φ.hom i
⊢ Functor.map (Cat.Hom.toFunctor (F.map (p' i).op.toLoc)) (full_pullFunctor.hom w hf' φ (α i)) = φ.hom i
have h₁ : D₁.hom (f' i) (p' i) (p' i) sorry sorry ≫ pullHom (φ.hom i) (𝟙 (X' i)) (p' i) (p' i) sorry sorry ≫ D₂.hom (f' i) (p' i) (p' i) sorry sorry =
  φ.hom i
Tactic: rw [map_hom _ _ _ _ (𝟙 _) (by cat_disch),
        mor_eq _ _ _ _ _ (f' i) (p' i), pullHom_id,
        D₁.hom_self _ _ (by cat_disch), D₂.hom_self _ _ (by cat_disch),
        Category.id_comp, Category.comp_id]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Descent.Precoverage (goal)
Goal:
C : Type u
inst : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
J : GrothendieckTopology.{v, u} C
inst_1 : F.IsPrestack J
ι : Type t
S : C
X : ι → C
f : (i : ι) → X i ⟶ S
ι' : Type t'
X' : ι' → C
f' : (j : ι') → X' j ⟶ S
α : ι' → ι
p' : (j : ι') → X' j ⟶ X (α j)
w : ∀ (j : ι'), p' j ≫ f (α j) = f' j
hf' : Sieve.ofArrows X' f' ∈ (J : (a : C) → Set (Sieve a)) S
D₁ D₂ : F.DescentData f
φ : (pullFunctor F ⋯).obj D₁ ⟶ (pullFunctor F ⋯).obj D₂
i : ι'
h : mor w φ (p' i) (𝟙 (X' i)) ⋯ = φ.hom i
h₁ :
  D₁.hom (f' i) (p' i) (p' i) ⋯ ⋯ ≫ pullHom (φ.hom i) (𝟙 (X' i)) (p' i) (p' i) ⋯ ⋯ ≫ D₂.hom (f' i) (p' i) (p' i) ⋯ ⋯ =
    φ.hom i
⊢ Functor.map (Cat.Hom.toFunctor (F.map (p' i).op.toLoc)) (full_pullFunctor.hom w hf' φ (α i)) = φ.hom i
have h₂ : D₁.hom (f' i) (p' i) (p' i) sorry sorry ≫ φ.hom i ≫ D₂.hom (f' i) (p' i) (p' i) sorry sorry = φ.hom i
Tactic: rw [map_hom _ _ _ _ (𝟙 _) (by cat_disch),
        mor_eq _ _ _ _ _ (f' i) (p' i), pullHom_id,
        D₁.hom_self _ _ (by cat_disch), D₂.hom_self _ _ (by cat_disch),
        Category.id_comp, Category.comp_id]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Descent.Precoverage (goal)
Goal:
C : Type u
inst : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
J : GrothendieckTopology.{v, u} C
inst_1 : F.IsPrestack J
ι : Type t
S : C
X : ι → C
f : (i : ι) → X i ⟶ S
ι' : Type t'
X' : ι' → C
f' : (j : ι') → X' j ⟶ S
α : ι' → ι
p' : (j : ι') → X' j ⟶ X (α j)
w : ∀ (j : ι'), p' j ≫ f (α j) = f' j
hf' : Sieve.ofArrows X' f' ∈ (J : (a : C) → Set (Sieve a)) S
D₁ D₂ : F.DescentData f
φ : (pullFunctor F ⋯).obj D₁ ⟶ (pullFunctor F ⋯).obj D₂
i : ι'
h : mor w φ (p' i) (𝟙 (X' i)) ⋯ = φ.hom i
h₁ :
  D₁.hom (f' i) (p' i) (p' i) ⋯ ⋯ ≫ pullHom (φ.hom i) (𝟙 (X' i)) (p' i) (p' i) ⋯ ⋯ ≫ D₂.hom (f' i) (p' i) (p' i) ⋯ ⋯ =
    φ.hom i
h₂ : D₁.hom (f' i) (p' i) (p' i) ⋯ ⋯ ≫ φ.hom i ≫ D₂.hom (f' i) (p' i) (p' i) ⋯ ⋯ = φ.hom i
⊢ Functor.map (Cat.Hom.toFunctor (F.map (p' i).op.toLoc)) (full_pullFunctor.hom w hf' φ (α i)) = φ.hom i
have h₃ : 𝟙
      (Functor.obj (C := (↑(F.obj { as := op (X (α i)) : LocallyDiscrete Cᵒᵖ }) : Type u'))
        (Cat.Hom.toFunctor (F.map (p' i).op.toLoc)) (D₁.obj (α i))) ≫
    φ.hom i ≫ D₂.hom (f' i) (p' i) (p' i) sorry sorry =
  φ.hom i
Tactic: rw [map_hom _ _ _ _ (𝟙 _) (by cat_disch),
        mor_eq _ _ _ _ _ (f' i) (p' i), pullHom_id,
        D₁.hom_self _ _ (by cat_disch), D₂.hom_self _ _ (by cat_disch),
        Category.id_comp, Category.comp_id]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Descent.Precoverage (goal)
Goal:
C : Type u
inst : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
J : GrothendieckTopology.{v, u} C
inst_1 : F.IsPrestack J
ι : Type t
S : C
X : ι → C
f : (i : ι) → X i ⟶ S
ι' : Type t'
X' : ι' → C
f' : (j : ι') → X' j ⟶ S
α : ι' → ι
p' : (j : ι') → X' j ⟶ X (α j)
w : ∀ (j : ι'), p' j ≫ f (α j) = f' j
hf' : Sieve.ofArrows X' f' ∈ (J : (a : C) → Set (Sieve a)) S
D₁ D₂ : F.DescentData f
φ : (pullFunctor F ⋯).obj D₁ ⟶ (pullFunctor F ⋯).obj D₂
i : ι'
h : mor w φ (p' i) (𝟙 (X' i)) ⋯ = φ.hom i
h₁ :
  D₁.hom (f' i) (p' i) (p' i) ⋯ ⋯ ≫ pullHom (φ.hom i) (𝟙 (X' i)) (p' i) (p' i) ⋯ ⋯ ≫ D₂.hom (f' i) (p' i) (p' i) ⋯ ⋯ =
    φ.hom i
h₂ : D₁.hom (f' i) (p' i) (p' i) ⋯ ⋯ ≫ φ.hom i ≫ D₂.hom (f' i) (p' i) (p' i) ⋯ ⋯ = φ.hom i
h₃ :
  𝟙
        (Functor.obj (C := (↑(F.obj { as := op (X (α i)) : LocallyDiscrete Cᵒᵖ }) : Type u'))
          (Cat.Hom.toFunctor (F.map (p' i).op.toLoc)) (D₁.obj (α i))) ≫
      φ.hom i ≫ D₂.hom (f' i) (p' i) (p' i) ⋯ ⋯ =
    φ.hom i
⊢ Functor.map (Cat.Hom.toFunctor (F.map (p' i).op.toLoc)) (full_pullFunctor.hom w hf' φ (α i)) = φ.hom i
have h₄ : 𝟙
      (Functor.obj (C := (↑(F.obj { as := op (X (α i)) : LocallyDiscrete Cᵒᵖ }) : Type u'))
        (Cat.Hom.toFunctor (F.map (p' i).op.toLoc)) (D₁.obj (α i))) ≫
    φ.hom i ≫
      𝟙
        (Functor.obj (C := (↑(F.obj { as := op (X (α i)) : LocallyDiscrete Cᵒᵖ }) : Type u'))
          (Cat.Hom.toFunctor (F.map (p' i).op.toLoc)) (D₂.obj (α i))) =
  φ.hom i
Tactic: rw [map_hom _ _ _ _ (𝟙 _) (by cat_disch),
        mor_eq _ _ _ _ _ (f' i) (p' i), pullHom_id,
        D₁.hom_self _ _ (by cat_disch), D₂.hom_self _ _ (by cat_disch),
        Category.id_comp, Category.comp_id]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Descent.Precoverage (goal)
Goal:
C : Type u
inst : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
J : GrothendieckTopology.{v, u} C
inst_1 : F.IsPrestack J
ι : Type t
S : C
X : ι → C
f : (i : ι) → X i ⟶ S
ι' : Type t'
X' : ι' → C
f' : (j : ι') → X' j ⟶ S
α : ι' → ι
p' : (j : ι') → X' j ⟶ X (α j)
w : ∀ (j : ι'), p' j ≫ f (α j) = f' j
hf' : Sieve.ofArrows X' f' ∈ (J : (a : C) → Set (Sieve a)) S
D₁ D₂ : F.DescentData f
φ : (pullFunctor F ⋯).obj D₁ ⟶ (pullFunctor F ⋯).obj D₂
i : ι'
h : mor w φ (p' i) (𝟙 (X' i)) ⋯ = φ.hom i
h₁ :
  D₁.hom (f' i) (p' i) (p' i) ⋯ ⋯ ≫ pullHom (φ.hom i) (𝟙 (X' i)) (p' i) (p' i) ⋯ ⋯ ≫ D₂.hom (f' i) (p' i) (p' i) ⋯ ⋯ =
    φ.hom i
h₂ : D₁.hom (f' i) (p' i) (p' i) ⋯ ⋯ ≫ φ.hom i ≫ D₂.hom (f' i) (p' i) (p' i) ⋯ ⋯ = φ.hom i
h₃ :
  𝟙
        (Functor.obj (C := (↑(F.obj { as := op (X (α i)) : LocallyDiscrete Cᵒᵖ }) : Type u'))
          (Cat.Hom.toFunctor (F.map (p' i).op.toLoc)) (D₁.obj (α i))) ≫
      φ.hom i ≫ D₂.hom (f' i) (p' i) (p' i) ⋯ ⋯ =
    φ.hom i
h₄ :
  𝟙
        (Functor.obj (C := (↑(F.obj { as := op (X (α i)) : LocallyDiscrete Cᵒᵖ }) : Type u'))
          (Cat.Hom.toFunctor (F.map (p' i).op.toLoc)) (D₁.obj (α i))) ≫
      φ.hom i ≫
        𝟙
          (Functor.obj (C := (↑(F.obj { as := op (X (α i)) : LocallyDiscrete Cᵒᵖ }) : Type u'))
            (Cat.Hom.toFunctor (F.map (p' i).op.toLoc)) (D₂.obj (α i))) =
    φ.hom i
⊢ Functor.map (Cat.Hom.toFunctor (F.map (p' i).op.toLoc)) (full_pullFunctor.hom w hf' φ (α i)) = φ.hom i
have h₅ : φ.hom i ≫
    𝟙
      (Functor.obj (C := (↑(F.obj { as := op (X (α i)) : LocallyDiscrete Cᵒᵖ }) : Type u'))
        (Cat.Hom.toFunctor (F.map (p' i).op.toLoc)) (D₂.obj (α i))) =
  φ.hom i
Tactic: rw [map_hom _ _ _ _ (𝟙 _) (by cat_disch),
        mor_eq _ _ _ _ _ (f' i) (p' i), pullHom_id,
        D₁.hom_self _ _ (by cat_disch), D₂.hom_self _ _ (by cat_disch),
        Category.id_comp, Category.comp_id]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Descent.Precoverage (goal)
Goal:
C : Type u
inst : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
J : GrothendieckTopology.{v, u} C
inst_1 : F.IsPrestack J
ι : Type t
S : C
X : ι → C
f : (i : ι) → X i ⟶ S
ι' : Type t'
X' : ι' → C
f' : (j : ι') → X' j ⟶ S
α : ι' → ι
p' : (j : ι') → X' j ⟶ X (α j)
w : ∀ (j : ι'), p' j ≫ f (α j) = f' j
hf' : Sieve.ofArrows X' f' ∈ (J : (a : C) → Set (Sieve a)) S
D₁ D₂ : F.DescentData f
φ : (pullFunctor F ⋯).obj D₁ ⟶ (pullFunctor F ⋯).obj D₂
i : ι'
h : mor w φ (p' i) (𝟙 (X' i)) ⋯ = φ.hom i
h₁ :
  D₁.hom (f' i) (p' i) (p' i) ⋯ ⋯ ≫ pullHom (φ.hom i) (𝟙 (X' i)) (p' i) (p' i) ⋯ ⋯ ≫ D₂.hom (f' i) (p' i) (p' i) ⋯ ⋯ =
    φ.hom i
h₂ : D₁.hom (f' i) (p' i) (p' i) ⋯ ⋯ ≫ φ.hom i ≫ D₂.hom (f' i) (p' i) (p' i) ⋯ ⋯ = φ.hom i
h₃ :
  𝟙
        (Functor.obj (C := (↑(F.obj { as := op (X (α i)) : LocallyDiscrete Cᵒᵖ }) : Type u'))
          (Cat.Hom.toFunctor (F.map (p' i).op.toLoc)) (D₁.obj (α i))) ≫
      φ.hom i ≫ D₂.hom (f' i) (p' i) (p' i) ⋯ ⋯ =
    φ.hom i
h₄ :
  𝟙
        (Functor.obj (C := (↑(F.obj { as := op (X (α i)) : LocallyDiscrete Cᵒᵖ }) : Type u'))
          (Cat.Hom.toFunctor (F.map (p' i).op.toLoc)) (D₁.obj (α i))) ≫
      φ.hom i ≫
        𝟙
          (Functor.obj (C := (↑(F.obj { as := op (X (α i)) : LocallyDiscrete Cᵒᵖ }) : Type u'))
            (Cat.Hom.toFunctor (F.map (p' i).op.toLoc)) (D₂.obj (α i))) =
    φ.hom i
h₅ :
  φ.hom i ≫
      𝟙
        (Functor.obj (C := (↑(F.obj { as := op (X (α i)) : LocallyDiscrete Cᵒᵖ }) : Type u'))
          (Cat.Hom.toFunctor (F.map (p' i).op.toLoc)) (D₂.obj (α i))) =
    φ.hom i
⊢ Functor.map (Cat.Hom.toFunctor (F.map (p' i).op.toLoc)) (full_pullFunctor.hom w hf' φ (α i)) = φ.hom i
have h₆ : φ.hom i = φ.hom i
Tactic: rw [map_hom _ _ _ _ (𝟙 _) (by cat_disch),
        mor_eq _ _ _ _ _ (f' i) (p' i), pullHom_id,
        D₁.hom_self _ _ (by cat_disch), D₂.hom_self _ _ (by cat_disch),
        Category.id_comp, Category.comp_id]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Descent.Precoverage (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
J : GrothendieckTopology.{v, u} C
inst_1 : F.IsPrestack J
ι : Type t
S : C
X : ι → C
f : (i : ι) → X i ⟶ S
ι' : Type t'
X' : ι' → C
f' : (j : ι') → X' j ⟶ S
α : ι' → ι
p' : (j : ι') → X' j ⟶ X (α j)
w : ∀ (j : ι'), p' j ≫ f (α j) = f' j
hf' : Sieve.ofArrows X' f' ∈ (J : (a : C) → Set (Sieve a)) S
⊢ (pullFunctor F ⋯).FullyFaithful
have this : (pullFunctor F sorry).Faithful
Tactic: have := faithful_pullFunctor F w hf'
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Descent.Precoverage (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
J : GrothendieckTopology.{v, u} C
inst_1 : F.IsPrestack J
ι : Type t
S : C
X : ι → C
f : (i : ι) → X i ⟶ S
ι' : Type t'
X' : ι' → C
f' : (j : ι') → X' j ⟶ S
α : ι' → ι
p' : (j : ι') → X' j ⟶ X (α j)
w : ∀ (j : ι'), p' j ≫ f (α j) = f' j
hf' : Sieve.ofArrows X' f' ∈ (J : (a : C) → Set (Sieve a)) S
this : (pullFunctor F ⋯).Faithful
⊢ (pullFunctor F ⋯).FullyFaithful
have this₁ : (pullFunctor F sorry).Full
Tactic: have := full_pullFunctor F w hf'
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Descent.Precoverage (goal)
Goal:
C : Type u
inst : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
J : GrothendieckTopology.{v, u} C
inst_1 : F.IsPrestack J
ι : Type t
S : C
X : ι → C
f : (i : ι) → X i ⟶ S
ι' : Type t'
X' : ι' → C
f' : (j : ι') → X' j ⟶ S
h₁ : Sieve.ofArrows X' f' ∈ (J : (a : C) → Set (Sieve a)) S
h₂ : Sieve.ofArrows X' f' ≤ Sieve.ofArrows X f
inst_2 : (F.toDescentData f').IsEquivalence
i' : ι'
⊢ ∃ (i : ι) (p : X' i' ⟶ X i), p ≫ f i = f' i'
have obtain : ∀ ⦃Y : C⦄ (i : ι) (p : X' i' ⟶ X i) (fac : p ≫ f i = f' i'), ∃ (i : ι) (p : X' i' ⟶ X i), p ≫ f i = f' i'
Tactic: obtain ⟨_, p, _, ⟨i⟩, fac⟩ := h₂ _ (Sieve.ofArrows_mk _ f' i')
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Descent.Precoverage (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
J : GrothendieckTopology.{v, u} C
inst_1 : F.IsPrestack J
ι : Type t
S : C
X : ι → C
f : (i : ι) → X i ⟶ S
ι' : Type t'
X' : ι' → C
f' : (j : ι') → X' j ⟶ S
h₁ : Sieve.ofArrows X' f' ∈ (J : (a : C) → Set (Sieve a)) S
h₂ : Sieve.ofArrows X' f' ≤ Sieve.ofArrows X f
inst_2 : (F.toDescentData f').IsEquivalence
⊢ (F.toDescentData f).IsEquivalence
have H : ∀ (i' : ι'), ∃ (i : ι) (p : X' i' ⟶ X i), p ≫ f i = f' i'
Tactic: have H (i' : ι') : ∃ (i : ι) (p : X' i' ⟶ X i), p ≫ f i = f' i' := by
    obtain ⟨_, p, _, ⟨i⟩, fac⟩ := h₂ _ (Sieve.ofArrows_mk _ f' i')
    exact ⟨_, _, fac⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Descent.Precoverage (goal)
Goal:
C : Type u
inst : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
J : GrothendieckTopology.{v, u} C
inst_1 : F.IsPrestack J
ι : Type t
S : C
X : ι → C
f : (i : ι) → X i ⟶ S
ι' : Type t'
X' : ι' → C
f' : (j : ι') → X' j ⟶ S
h₁ : Sieve.ofArrows X' f' ∈ (J : (a : C) → Set (Sieve a)) S
h₂ : Sieve.ofArrows X' f' ≤ Sieve.ofArrows X f
inst_2 : (F.toDescentData f').IsEquivalence
H : ∀ (i' : ι'), ∃ (i : ι) (p : X' i' ⟶ X i), p ≫ f i = f' i'
⊢ (F.toDescentData f).IsEquivalence
have choose : ∀ (α : (i' : ι') → ι) (p : (i' : ι') → X' i' ⟶ X (α i')) (hp : ∀ (i' : ι'), p i' ≫ f (α i') = f' i'),
  (F.toDescentData f).IsEquivalence
Tactic: choose α p hp using H
Cleared hypotheses: H
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Descent.Precoverage (goal)
Goal:
C : Type u
inst : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
J : GrothendieckTopology.{v, u} C
inst_1 : F.IsPrestack J
ι : Type t
S : C
X : ι → C
f : (i : ι) → X i ⟶ S
ι' : Type t'
X' : ι' → C
f' : (j : ι') → X' j ⟶ S
h₁ : Sieve.ofArrows X' f' ∈ (J : (a : C) → Set (Sieve a)) S
h₂ : Sieve.ofArrows X' f' ≤ Sieve.ofArrows X f
inst_2 : (F.toDescentData f').IsEquivalence
α : (i' : ι') → ι
p : (i' : ι') → X' i' ⟶ X (α i')
hp : ∀ (i' : ι'), p i' ≫ f (α i') = f' i'
⊢ ∃ (H : F.DescentData f ⥤ F.DescentData f') (x : H.FullyFaithful),
    Nonempty (F.toDescentData f ⋙ H ≅ F.toDescentData f')
have refine_1 : Functor.FullyFaithful (C := F.DescentData f) (D := F.DescentData f') (pullFunctor F (p := 𝟙 S) (α := α) (p' := p) sorry)
Tactic: refine ⟨pullFunctor F (p := 𝟙 _) (p' := p) (by cat_disch), ?_, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Descent.Precoverage (goal)
Goal:
C : Type u
inst : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
J : GrothendieckTopology.{v, u} C
inst_1 : F.IsPrestack J
ι : Type t
S : C
X : ι → C
f : (i : ι) → X i ⟶ S
ι' : Type t'
X' : ι' → C
f' : (j : ι') → X' j ⟶ S
h₁ : Sieve.ofArrows X' f' ∈ (J : (a : C) → Set (Sieve a)) S
h₂ : Sieve.ofArrows X' f' ≤ Sieve.ofArrows X f
inst_2 : (F.toDescentData f').IsEquivalence
α : (i' : ι') → ι
p : (i' : ι') → X' i' ⟶ X (α i')
hp : ∀ (i' : ι'), p i' ≫ f (α i') = f' i'
refine_1 :
  Functor.FullyFaithful (C := F.DescentData f) (D := F.DescentData f') (pullFunctor F (p := 𝟙 S) (α := α) (p' := p) ⋯)
⊢ ∃ (H : F.DescentData f ⥤ F.DescentData f') (x : H.FullyFaithful),
    Nonempty (F.toDescentData f ⋙ H ≅ F.toDescentData f')
have refine_2 : Nonempty (F.toDescentData f ⋙ pullFunctor F (p := 𝟙 S) (α := α) (p' := p) sorry ≅ F.toDescentData f')
Tactic: refine ⟨pullFunctor F (p := 𝟙 _) (p' := p) (by cat_disch), ?_, ?_⟩