awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.SheafOfTypes (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : C
E : PreOneHypercover.{w, v_1, u_1} X
F : Functor.{v_1, u_2, u_1, u_2 + 1} Cᵒᵖ (Type u_2)
h₁ : E.IsStronglySheafFor F
h₂ :
  ∀ ⦃Y : C⦄ (f : Y ⟶ X),
    Presieve.IsSeparatedFor F (Sieve.pullback f (PreZeroHypercover.sieve₀.{w, v_1, u_1} E.toPreZeroHypercover)).arrows
S : Sieve.{v_1, u_1} X
H : ∀ (i : E.I₀), Presieve.IsSheafFor F (Sieve.pullback (E.f i) S).arrows
H' : ∀ ⦃i j : E.I₀⦄ (k : E.I₁ i j), Presieve.IsSeparatedFor F (Sieve.pullback (E.p₁ k ≫ E.f i) S).arrows
t : Presieve.FamilyOfElements F S.arrows
ht : t.Compatible (R := S.arrows)
s : (i : E.I₀) → F.obj (op (E.X i))
hs :
  ∀ (i : E.I₀),
    (fun (t_1 : F.obj (op (E.X i))) ↦
        (Presieve.FamilyOfElements.pullback (E.f i) t).IsAmalgamation (R := (Sieve.pullback (E.f i) S).arrows) t_1)
      (s i)
huniq :
  ∀ (i : E.I₀) (y : F.obj (op (E.X i))),
    (fun (t_1 : F.obj (op (E.X i))) ↦
          (Presieve.FamilyOfElements.pullback (E.f i) t).IsAmalgamation (R := (Sieve.pullback (E.f i) S).arrows) t_1)
        y →
      y = s i
hr : Presieve.Arrows.Compatible F E.f s
s' : Presieve.FamilyOfElements F (Presieve.ofArrows E.X E.f)
hs' : ∀ (i : E.I₀), s' (E.f i) ⋯ = s i
⊢ ∃! t_1 : F.obj (op X), t.IsAmalgamation (R := S.arrows) t_1
have obtain : ∀ (t' : F.obj (op X))
  (ht' :
    ∀ (i : E.I₀),
      (ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (E.f i).op) :
            (a : (fun (X : Type u_2) ↦ X) (F.obj (op X))) → (fun (X : Type u_2) ↦ X) (F.obj (op (E.X i))))
          t' =
        s i)
  (hunique :
    ∀ (y : (fun (X : Type u_2) ↦ X) (F.obj (op X))),
      (fun (t : (fun (X : Type u_2) ↦ X) (F.obj (op X))) ↦
            ∀ (i : E.I₀),
              (ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (E.f i).op) :
                    (a : (fun (X : Type u_2) ↦ X) (F.obj (op X))) → (fun (X : Type u_2) ↦ X) (F.obj (op (E.X i))))
                  t =
                s i)
          y →
        y = t'),
  ∃! t_1 : F.obj (op X), t.IsAmalgamation (R := S.arrows) t_1
Tactic: obtain ⟨t', ht', hunique⟩ := (Presieve.isSheafFor_arrows_iff _ _).mp h₁.isSheafFor_presieve₀ _ hr
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.SheafOfTypes (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : C
E : PreOneHypercover.{w, v_1, u_1} X
F : Functor.{v_1, u_2, u_1, u_2 + 1} Cᵒᵖ (Type u_2)
h₁ : E.IsStronglySheafFor F
h₂ :
  ∀ ⦃Y : C⦄ (f : Y ⟶ X),
    Presieve.IsSeparatedFor F (Sieve.pullback f (PreZeroHypercover.sieve₀.{w, v_1, u_1} E.toPreZeroHypercover)).arrows
S : Sieve.{v_1, u_1} X
H : ∀ (i : E.I₀), Presieve.IsSheafFor F (Sieve.pullback (E.f i) S).arrows
H' : ∀ ⦃i j : E.I₀⦄ (k : E.I₁ i j), Presieve.IsSeparatedFor F (Sieve.pullback (E.p₁ k ≫ E.f i) S).arrows
t : Presieve.FamilyOfElements F S.arrows
ht : t.Compatible (R := S.arrows)
s : (i : E.I₀) → F.obj (op (E.X i))
hs :
  ∀ (i : E.I₀),
    (fun (t_1 : F.obj (op (E.X i))) ↦
        (Presieve.FamilyOfElements.pullback (E.f i) t).IsAmalgamation (R := (Sieve.pullback (E.f i) S).arrows) t_1)
      (s i)
huniq :
  ∀ (i : E.I₀) (y : F.obj (op (E.X i))),
    (fun (t_1 : F.obj (op (E.X i))) ↦
          (Presieve.FamilyOfElements.pullback (E.f i) t).IsAmalgamation (R := (Sieve.pullback (E.f i) S).arrows) t_1)
        y →
      y = s i
hr : Presieve.Arrows.Compatible F E.f s
s' : Presieve.FamilyOfElements F (Presieve.ofArrows E.X E.f)
hs' : ∀ (i : E.I₀), s' (E.f i) ⋯ = s i
t' : F.obj (op X)
ht' :
  ∀ (i : E.I₀),
    (ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (E.f i).op) :
          (a : (fun (X : Type u_2) ↦ X) (F.obj (op X))) → (fun (X : Type u_2) ↦ X) (F.obj (op (E.X i))))
        t' =
      s i
hunique :
  ∀ (y : (fun (X : Type u_2) ↦ X) (F.obj (op X))),
    (fun (t : (fun (X : Type u_2) ↦ X) (F.obj (op X))) ↦
          ∀ (i : E.I₀),
            (ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (E.f i).op) :
                  (a : (fun (X : Type u_2) ↦ X) (F.obj (op X))) → (fun (X : Type u_2) ↦ X) (F.obj (op (E.X i))))
                t =
              s i)
        y →
      y = t'
⊢ ∃! t_1 : F.obj (op X), t.IsAmalgamation (R := S.arrows) t_1
have refine_1 : ∀ (T : C) (f : T ⟶ X) (hf : S.arrows f) (Z : C) (g : Z ⟶ T)
  (hg : (Sieve.pullback f (PreZeroHypercover.sieve₀.{w, v_1, u_1} E.toPreZeroHypercover)).arrows g),
  Eq (α := (fun (X : Type u_2) ↦ X) (F.obj (op Z)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map g.op) :
          (a : (fun (X : Type u_2) ↦ X) (F.obj (op T))) → (fun (X : Type u_2) ↦ X) (F.obj (op Z)))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f.op) :
              (a : (fun (X : Type u_2) ↦ X) (F.obj (op X))) → (fun (X : Type u_2) ↦ X) (F.obj (op T)))
            t' :
          (fun (X : Type u_2) ↦ X) (F.obj (op T))) :
      (fun (X : Type u_2) ↦ X) (F.obj (op Z)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map g.op) :
          (a : (fun (X : Type u_2) ↦ X) (F.obj (op T))) → (fun (X : Type u_2) ↦ X) (F.obj (op Z)))
        (t f hf) :
      (fun (X : Type u_2) ↦ X) (F.obj (op Z)))
Tactic: refine ⟨t', fun T f hf ↦ (h₂ f).ext fun Z g hg ↦ ?_, fun y hy ↦ ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.SheafOfTypes (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : C
E : PreOneHypercover.{w, v_1, u_1} X
F : Functor.{v_1, u_2, u_1, u_2 + 1} Cᵒᵖ (Type u_2)
h₁ : E.IsStronglySheafFor F
h₂ :
  ∀ ⦃Y : C⦄ (f : Y ⟶ X),
    Presieve.IsSeparatedFor F (Sieve.pullback f (PreZeroHypercover.sieve₀.{w, v_1, u_1} E.toPreZeroHypercover)).arrows
S : Sieve.{v_1, u_1} X
H : ∀ (i : E.I₀), Presieve.IsSheafFor F (Sieve.pullback (E.f i) S).arrows
H' : ∀ ⦃i j : E.I₀⦄ (k : E.I₁ i j), Presieve.IsSeparatedFor F (Sieve.pullback (E.p₁ k ≫ E.f i) S).arrows
t : Presieve.FamilyOfElements F S.arrows
ht : t.Compatible (R := S.arrows)
s : (i : E.I₀) → F.obj (op (E.X i))
hs :
  ∀ (i : E.I₀),
    (fun (t_1 : F.obj (op (E.X i))) ↦
        (Presieve.FamilyOfElements.pullback (E.f i) t).IsAmalgamation (R := (Sieve.pullback (E.f i) S).arrows) t_1)
      (s i)
huniq :
  ∀ (i : E.I₀) (y : F.obj (op (E.X i))),
    (fun (t_1 : F.obj (op (E.X i))) ↦
          (Presieve.FamilyOfElements.pullback (E.f i) t).IsAmalgamation (R := (Sieve.pullback (E.f i) S).arrows) t_1)
        y →
      y = s i
hr : Presieve.Arrows.Compatible F E.f s
s' : Presieve.FamilyOfElements F (Presieve.ofArrows E.X E.f)
hs' : ∀ (i : E.I₀), s' (E.f i) ⋯ = s i
t' : F.obj (op X)
ht' :
  ∀ (i : E.I₀),
    (ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (E.f i).op) :
          (a : (fun (X : Type u_2) ↦ X) (F.obj (op X))) → (fun (X : Type u_2) ↦ X) (F.obj (op (E.X i))))
        t' =
      s i
hunique :
  ∀ (y : (fun (X : Type u_2) ↦ X) (F.obj (op X))),
    (fun (t : (fun (X : Type u_2) ↦ X) (F.obj (op X))) ↦
          ∀ (i : E.I₀),
            (ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (E.f i).op) :
                  (a : (fun (X : Type u_2) ↦ X) (F.obj (op X))) → (fun (X : Type u_2) ↦ X) (F.obj (op (E.X i))))
                t =
              s i)
        y →
      y = t'
refine_1 :
  ∀ (T : C) (f : T ⟶ X) (hf : S.arrows f) (Z : C) (g : Z ⟶ T)
    (hg : (Sieve.pullback f (PreZeroHypercover.sieve₀.{w, v_1, u_1} E.toPreZeroHypercover)).arrows g),
    Eq (α := (fun (X : Type u_2) ↦ X) (F.obj (op Z)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map g.op) :
            (a : (fun (X : Type u_2) ↦ X) (F.obj (op T))) → (fun (X : Type u_2) ↦ X) (F.obj (op Z)))
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f.op) :
                (a : (fun (X : Type u_2) ↦ X) (F.obj (op X))) → (fun (X : Type u_2) ↦ X) (F.obj (op T)))
              t' :
            (fun (X : Type u_2) ↦ X) (F.obj (op T))) :
        (fun (X : Type u_2) ↦ X) (F.obj (op Z)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map g.op) :
            (a : (fun (X : Type u_2) ↦ X) (F.obj (op T))) → (fun (X : Type u_2) ↦ X) (F.obj (op Z)))
          (t f hf) :
        (fun (X : Type u_2) ↦ X) (F.obj (op Z)))
⊢ ∃! t_1 : F.obj (op X), t.IsAmalgamation (R := S.arrows) t_1
have refine_2 : ∀ (y : F.obj (op X)) (hy : (fun (t_1 : F.obj (op X)) ↦ t.IsAmalgamation (R := S.arrows) t_1) y), y = t'
Tactic: refine ⟨t', fun T f hf ↦ (h₂ f).ext fun Z g hg ↦ ?_, fun y hy ↦ ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.SheafOfTypes (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : C
E : PreOneHypercover.{w, v_1, u_1} X
F : Functor.{v_1, u_2, u_1, u_2 + 1} Cᵒᵖ (Type u_2)
h₁ : E.IsStronglySheafFor F
h₂ :
  ∀ ⦃Y : C⦄ (f : Y ⟶ X),
    Presieve.IsSeparatedFor F (Sieve.pullback f (PreZeroHypercover.sieve₀.{w, v_1, u_1} E.toPreZeroHypercover)).arrows
S : Sieve.{v_1, u_1} X
H : ∀ (i : E.I₀), Presieve.IsSheafFor F (Sieve.pullback (E.f i) S).arrows
H' : ∀ ⦃i j : E.I₀⦄ (k : E.I₁ i j), Presieve.IsSeparatedFor F (Sieve.pullback (E.p₁ k ≫ E.f i) S).arrows
t : Presieve.FamilyOfElements F S.arrows
ht : t.Compatible (R := S.arrows)
s : (i : E.I₀) → F.obj (op (E.X i))
hs :
  ∀ (i : E.I₀),
    (fun (t_1 : F.obj (op (E.X i))) ↦
        (Presieve.FamilyOfElements.pullback (E.f i) t).IsAmalgamation (R := (Sieve.pullback (E.f i) S).arrows) t_1)
      (s i)
huniq :
  ∀ (i : E.I₀) (y : F.obj (op (E.X i))),
    (fun (t_1 : F.obj (op (E.X i))) ↦
          (Presieve.FamilyOfElements.pullback (E.f i) t).IsAmalgamation (R := (Sieve.pullback (E.f i) S).arrows) t_1)
        y →
      y = s i
hr : Presieve.Arrows.Compatible F E.f s
s' : Presieve.FamilyOfElements F (Presieve.ofArrows E.X E.f)
hs' : ∀ (i : E.I₀), s' (E.f i) ⋯ = s i
t' : F.obj (op X)
ht' :
  ∀ (i : E.I₀),
    (ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (E.f i).op) :
          (a : (fun (X : Type u_2) ↦ X) (F.obj (op X))) → (fun (X : Type u_2) ↦ X) (F.obj (op (E.X i))))
        t' =
      s i
hunique :
  ∀ (y : (fun (X : Type u_2) ↦ X) (F.obj (op X))),
    (fun (t : (fun (X : Type u_2) ↦ X) (F.obj (op X))) ↦
          ∀ (i : E.I₀),
            (ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (E.f i).op) :
                  (a : (fun (X : Type u_2) ↦ X) (F.obj (op X))) → (fun (X : Type u_2) ↦ X) (F.obj (op (E.X i))))
                t =
              s i)
        y →
      y = t'
T : C
f : T ⟶ X
hf : S.arrows f
Z : C
g : Z ⟶ T
Y : C
i : E.I₀
w : Z ⟶ E.X i
heq : w ≫ E.f i = g ≫ f
⊢ t (g ≫ f) ⋯ = t (w ≫ E.f i) ⋯
have e_f : g ≫ f = w ≫ E.f i
Tactic: congr 1
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.SheafOfTypes (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : C
E : PreOneHypercover.{w, v_1, u_1} X
F : Functor.{v_1, u_2, u_1, u_2 + 1} Cᵒᵖ (Type u_2)
h₁ : E.IsStronglySheafFor F
h₂ :
  ∀ ⦃Y : C⦄ (f : Y ⟶ X),
    Presieve.IsSeparatedFor F (Sieve.pullback f (PreZeroHypercover.sieve₀.{w, v_1, u_1} E.toPreZeroHypercover)).arrows
S : Sieve.{v_1, u_1} X
H : ∀ (i : E.I₀), Presieve.IsSheafFor F (Sieve.pullback (E.f i) S).arrows
H' : ∀ ⦃i j : E.I₀⦄ (k : E.I₁ i j), Presieve.IsSeparatedFor F (Sieve.pullback (E.p₁ k ≫ E.f i) S).arrows
t : Presieve.FamilyOfElements F S.arrows
ht : t.Compatible (R := S.arrows)
s : (i : E.I₀) → F.obj (op (E.X i))
hs :
  ∀ (i : E.I₀),
    (fun (t_1 : F.obj (op (E.X i))) ↦
        (Presieve.FamilyOfElements.pullback (E.f i) t).IsAmalgamation (R := (Sieve.pullback (E.f i) S).arrows) t_1)
      (s i)
huniq :
  ∀ (i : E.I₀) (y : F.obj (op (E.X i))),
    (fun (t_1 : F.obj (op (E.X i))) ↦
          (Presieve.FamilyOfElements.pullback (E.f i) t).IsAmalgamation (R := (Sieve.pullback (E.f i) S).arrows) t_1)
        y →
      y = s i
hr : Presieve.Arrows.Compatible F E.f s
s' : Presieve.FamilyOfElements F (Presieve.ofArrows E.X E.f)
hs' : ∀ (i : E.I₀), s' (E.f i) ⋯ = s i
t' : F.obj (op X)
ht' :
  ∀ (i : E.I₀),
    (ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (E.f i).op) :
          (a : (fun (X : Type u_2) ↦ X) (F.obj (op X))) → (fun (X : Type u_2) ↦ X) (F.obj (op (E.X i))))
        t' =
      s i
hunique :
  ∀ (y : (fun (X : Type u_2) ↦ X) (F.obj (op X))),
    (fun (t : (fun (X : Type u_2) ↦ X) (F.obj (op X))) ↦
          ∀ (i : E.I₀),
            (ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (E.f i).op) :
                  (a : (fun (X : Type u_2) ↦ X) (F.obj (op X))) → (fun (X : Type u_2) ↦ X) (F.obj (op (E.X i))))
                t =
              s i)
        y →
      y = t'
T : C
f : T ⟶ X
hf : S.arrows f
Z : C
g : Z ⟶ T
Y : C
i : E.I₀
w : Z ⟶ E.X i
heq : w ≫ E.f i = g ≫ f
⊢ g ≫ f = w ≫ E.f i
have e_f : g ≫ f = g ≫ f
Tactic: rw [heq]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.SheafOfTypes (hypothesis)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : C
E : PreOneHypercover.{w, v_1, u_1} X
F : Functor.{v_1, u_2, u_1, u_2 + 1} Cᵒᵖ (Type u_2)
h₁ : E.IsStronglySheafFor F
h₂ :
  ∀ ⦃Y : C⦄ (f : Y ⟶ X),
    Presieve.IsSeparatedFor F (Sieve.pullback f (PreZeroHypercover.sieve₀.{w, v_1, u_1} E.toPreZeroHypercover)).arrows
S : Sieve.{v_1, u_1} X
H : ∀ (i : E.I₀), Presieve.IsSheafFor F (Sieve.pullback (E.f i) S).arrows
H' : ∀ ⦃i j : E.I₀⦄ (k : E.I₁ i j), Presieve.IsSeparatedFor F (Sieve.pullback (E.p₁ k ≫ E.f i) S).arrows
t : Presieve.FamilyOfElements F S.arrows
ht : t.Compatible (R := S.arrows)
s : (i : E.I₀) → F.obj (op (E.X i))
hs :
  ∀ (i : E.I₀),
    (fun (t_1 : F.obj (op (E.X i))) ↦
        (Presieve.FamilyOfElements.pullback (E.f i) t).IsAmalgamation (R := (Sieve.pullback (E.f i) S).arrows) t_1)
      (s i)
huniq :
  ∀ (i : E.I₀) (y : F.obj (op (E.X i))),
    (fun (t_1 : F.obj (op (E.X i))) ↦
          (Presieve.FamilyOfElements.pullback (E.f i) t).IsAmalgamation (R := (Sieve.pullback (E.f i) S).arrows) t_1)
        y →
      y = s i
hr : Presieve.Arrows.Compatible F E.f s
s' : Presieve.FamilyOfElements F (Presieve.ofArrows E.X E.f)
hs' : ∀ (i : E.I₀), s' (E.f i) ⋯ = s i
t' : F.obj (op X)
ht' :
  ∀ (i : E.I₀),
    (ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (E.f i).op) :
          (a : (fun (X : Type u_2) ↦ X) (F.obj (op X))) → (fun (X : Type u_2) ↦ X) (F.obj (op (E.X i))))
        t' =
      s i
hunique :
  ∀ (y : (fun (X : Type u_2) ↦ X) (F.obj (op X))),
    (fun (t : (fun (X : Type u_2) ↦ X) (F.obj (op X))) ↦
          ∀ (i : E.I₀),
            (ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (E.f i).op) :
                  (a : (fun (X : Type u_2) ↦ X) (F.obj (op X))) → (fun (X : Type u_2) ↦ X) (F.obj (op (E.X i))))
                t =
              s i)
        y →
      y = t'
T : C
f : T ⟶ X
hf : S.arrows f
Z : C
g : Z ⟶ T
Y : C
i : E.I₀
w : Z ⟶ E.X i
heq : w ≫ E.f i = g ≫ f
⊢ Eq (α := (fun (X : Type u_2) ↦ X) (F.obj (op Z)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (g ≫ f).op) :
          (a : (fun (X : Type u_2) ↦ X) (F.obj (op X))) → F.obj (op Z))
        t' :
      F.obj (op Z))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map g.op) :
          (a : (fun (X : Type u_2) ↦ X) (F.obj (op T))) → (fun (X : Type u_2) ↦ X) (F.obj (op Z)))
        (t f hf) :
      (fun (X : Type u_2) ↦ X) (F.obj (op Z)))
have this : t (g ≫ f) sorry = t (w ≫ E.f i) sorry
Tactic: have : t (g ≫ f) (by simp [hf]) = t (w ≫ E.f i) (by simp [heq, hf]) := by
      congr 1
      rw [heq]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.SheafOfTypes (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : C
E : PreOneHypercover.{w, v_1, u_1} X
F : Functor.{v_1, u_2, u_1, u_2 + 1} Cᵒᵖ (Type u_2)
h₁ : E.IsStronglySheafFor F
h₂ :
  ∀ ⦃Y : C⦄ (f : Y ⟶ X),
    Presieve.IsSeparatedFor F (Sieve.pullback f (PreZeroHypercover.sieve₀.{w, v_1, u_1} E.toPreZeroHypercover)).arrows
S : Sieve.{v_1, u_1} X
H : ∀ (i : E.I₀), Presieve.IsSheafFor F (Sieve.pullback (E.f i) S).arrows
H' : ∀ ⦃i j : E.I₀⦄ (k : E.I₁ i j), Presieve.IsSeparatedFor F (Sieve.pullback (E.p₁ k ≫ E.f i) S).arrows
t : Presieve.FamilyOfElements F S.arrows
ht : t.Compatible (R := S.arrows)
s : (i : E.I₀) → F.obj (op (E.X i))
hs :
  ∀ (i : E.I₀),
    (fun (t_1 : F.obj (op (E.X i))) ↦
        (Presieve.FamilyOfElements.pullback (E.f i) t).IsAmalgamation (R := (Sieve.pullback (E.f i) S).arrows) t_1)
      (s i)
huniq :
  ∀ (i : E.I₀) (y : F.obj (op (E.X i))),
    (fun (t_1 : F.obj (op (E.X i))) ↦
          (Presieve.FamilyOfElements.pullback (E.f i) t).IsAmalgamation (R := (Sieve.pullback (E.f i) S).arrows) t_1)
        y →
      y = s i
hr : Presieve.Arrows.Compatible F E.f s
s' : Presieve.FamilyOfElements F (Presieve.ofArrows E.X E.f)
hs' : ∀ (i : E.I₀), s' (E.f i) ⋯ = s i
t' : F.obj (op X)
ht' :
  ∀ (i : E.I₀),
    (ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (E.f i).op) :
          (a : (fun (X : Type u_2) ↦ X) (F.obj (op X))) → (fun (X : Type u_2) ↦ X) (F.obj (op (E.X i))))
        t' =
      s i
hunique :
  ∀ (y : (fun (X : Type u_2) ↦ X) (F.obj (op X))),
    (fun (t : (fun (X : Type u_2) ↦ X) (F.obj (op X))) ↦
          ∀ (i : E.I₀),
            (ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (E.f i).op) :
                  (a : (fun (X : Type u_2) ↦ X) (F.obj (op X))) → (fun (X : Type u_2) ↦ X) (F.obj (op (E.X i))))
                t =
              s i)
        y →
      y = t'
y : F.obj (op X)
hy : (fun (t_1 : F.obj (op X)) ↦ t.IsAmalgamation (R := S.arrows) t_1) y
⊢ y = t'
have refine_2 : ∀ (i : E.I₀) (Z : C) (g : Z ⟶ E.X i) (hg : (Sieve.pullback (E.f i) S).arrows g),
  (ConcreteCategory.hom (FC := TypeCat.Fun) (F.map g.op) :
        (a : (fun (X : Type u_2) ↦ X) (F.obj (op (E.X i)))) → (fun (X : Type u_2) ↦ X) (F.obj (op Z)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (E.f i).op) :
            (a : (fun (X : Type u_2) ↦ X) (F.obj (op X))) → (fun (X : Type u_2) ↦ X) (F.obj (op (E.X i))))
          y :
        (fun (X : Type u_2) ↦ X) (F.obj (op (E.X i)))) =
    Presieve.FamilyOfElements.pullback (E.f i) t g hg
Tactic: refine hunique _ fun i ↦ huniq _ _ fun Z g hg ↦ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.SheafOfTypes (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : C
E : PreOneHypercover.{w, v_1, u_1} X
F : Functor.{v_1, u_2, u_1, u_2 + 1} Cᵒᵖ (Type u_2)
h₁ : E.IsStronglySheafFor F
h₂ :
  ∀ ⦃Y : C⦄ (f : Y ⟶ X),
    Presieve.IsSeparatedFor F (Sieve.pullback f (PreZeroHypercover.sieve₀.{w, v_1, u_1} E.toPreZeroHypercover)).arrows
R : Presieve.{v_1, u_1} X
H : ∀ (i : E.I₀), Presieve.IsSheafFor F (Sieve.pullback (E.f i) (Sieve.generate R)).arrows
H' :
  ∀ ⦃i j : E.I₀⦄ (k : E.I₁ i j), Presieve.IsSeparatedFor F (Sieve.pullback (E.p₁ k ≫ E.f i) (Sieve.generate R)).arrows
⊢ Presieve.IsSheafFor F R
have rw : Presieve.IsSheafFor F (Sieve.generate R).arrows
Tactic: rw [Presieve.isSheafFor_iff_generate]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.SheafOfTypes (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
J : GrothendieckTopology.{v_1, u_1} C
X : C
E : OneHypercover.{w, v_1, u_1} J X
F : Functor.{v_1, u_2, u_1, u_2 + 1} Cᵒᵖ (Type u_2)
hf : Presieve.IsSeparated J F
⊢ Presieve.IsSeparatedFor F (PreZeroHypercover.presieve₀.{w, v_1, u_1} E.toPreZeroHypercover)
have rw : Presieve.IsSeparatedFor F (Sieve.generate (PreZeroHypercover.presieve₀.{w, v_1, u_1} E.toPreZeroHypercover)).arrows
Tactic: rw [Presieve.isSeparatedFor_iff_generate]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.SheafOfTypes (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
J : GrothendieckTopology.{v_1, u_1} C
X : C
E : OneHypercover.{w, v_1, u_1} J X
F : Functor.{v_1, u_2, u_1, u_2 + 1} Cᵒᵖ (Type u_2)
hf : Presieve.IsSheaf J F
⊢ Presieve.IsSheafFor F (PreZeroHypercover.presieve₀.{w, v_1, u_1} E.toPreZeroHypercover)
have rw : Presieve.IsSheafFor F (Sieve.generate (PreZeroHypercover.presieve₀.{w, v_1, u_1} E.toPreZeroHypercover)).arrows
Tactic: rw [Presieve.isSheafFor_iff_generate]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.SheafOfTypes (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
J : GrothendieckTopology.{v_1, u_1} C
X : C
E : OneHypercover.{w, v_1, u_1} J X
F : Functor.{v_1, u_2, u_1, u_2 + 1} Cᵒᵖ (Type u_2)
hF : Presieve.IsSheaf J F
S : Sieve.{v_1, u_1} X
h₁ : ∀ (i : E.I₀), Presieve.IsSheafFor F (Sieve.pullback (E.f i) S).arrows
h₂ : ∀ ⦃i j : E.I₀⦄ (k : E.I₁ i j), Presieve.IsSeparatedFor F (Sieve.pullback (E.p₁ k ≫ E.f i) S).arrows
⊢ Presieve.IsSheafFor F S.arrows
have refine : ∀ ⦃Y : C⦄ (f : Y ⟶ X),
  Presieve.IsSeparatedFor F (Sieve.pullback f (PreZeroHypercover.sieve₀.{w, v_1, u_1} E.toPreZeroHypercover)).arrows
Tactic: refine (E.isStronglySheafFor hF).isSheafFor_sieve_of_pullback ?_ h₁ h₂
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.SheafOfTypes (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
J : GrothendieckTopology.{v_1, u_1} C
X : C
E : OneHypercover.{w, v_1, u_1} J X
F : Functor.{v_1, u_2, u_1, u_2 + 1} Cᵒᵖ (Type u_2)
hF : Presieve.IsSheaf J F
R : Presieve.{v_1, u_1} X
h₁ : ∀ (i : E.I₀), Presieve.IsSheafFor F (Sieve.pullback (E.f i) (Sieve.generate R)).arrows
h₂ :
  ∀ ⦃i j : E.I₀⦄ (k : E.I₁ i j), Presieve.IsSeparatedFor F (Sieve.pullback (E.p₁ k ≫ E.f i) (Sieve.generate R)).arrows
⊢ Presieve.IsSheafFor F R
have rw : Presieve.IsSheafFor F (Sieve.generate R).arrows
Tactic: rw [Presieve.isSheafFor_iff_generate]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Subcanonical (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
J : Precoverage.{v, u} C
inst_1 : J.toGrothendieck.Subcanonical
X Y : C
𝒰 : ZeroHypercover.{u_1, v, u} J X
f g : X ⟶ Y
h : ∀ (i : 𝒰.I₀), 𝒰.f i ≫ f = 𝒰.f i ≫ g
⊢ f = g
have hs : Presieve.IsSheafFor (Functor.obj yoneda Y) (PreZeroHypercover.presieve₀.{u_1, v, u} 𝒰.toPreZeroHypercover)
Tactic: have hs : 𝒰.presieve₀.IsSheafFor (yoneda.obj Y) :=
    (GrothendieckTopology.Subcanonical.isSheaf_of_isRepresentable _).isSheafFor_of_mem_precoverage
      𝒰.mem₀
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Subcanonical (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : Precoverage.{v, u} C
inst_1 : J.toGrothendieck.Subcanonical
inst_2 : Limits.HasPullbacks.{v, u} C
inst_3 : J.IsStableUnderBaseChange
⊢ IsLocalAtTarget (isomorphisms C) J
have refine_1 : {X Y : C} →
  (f : X ⟶ Y) →
    (𝒰 : J.ZeroHypercover Y) → (H : ∀ (i : 𝒰.I₀), IsIso (X := pullback f (𝒰.f i)) (pullback.snd f (𝒰.f i))) → Y ⟶ X
Tactic: refine .mk_of_isStableUnderBaseChange fun {X Y} f 𝒰 (H : ∀ i, IsIso _) ↦ ⟨?_, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Subcanonical (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : Precoverage.{v, u} C
inst_1 : J.toGrothendieck.Subcanonical
inst_2 : Limits.HasPullbacks.{v, u} C
inst_3 : J.IsStableUnderBaseChange
refine_1 :
  {X Y : C} →
    (f : X ⟶ Y) →
      (𝒰 : J.ZeroHypercover Y) → (H : ∀ (i : 𝒰.I₀), IsIso (X := pullback f (𝒰.f i)) (pullback.snd f (𝒰.f i))) → Y ⟶ X
⊢ IsLocalAtTarget (isomorphisms C) J
have refine_2 : ∀ {X Y : C} (f : X ⟶ Y) (𝒰 : J.ZeroHypercover Y)
  (H : ∀ (i : 𝒰.I₀), IsIso (X := pullback f (𝒰.f i)) (pullback.snd f (𝒰.f i))),
  f ≫
      𝒰.glueMorphisms
        (fun (i : 𝒰.I₀) ↦
          CategoryStruct.comp (Y := pullback f (𝒰.f i)) (inv (pullback.snd f (𝒰.f i))) (pullback.fst f (𝒰.f i)))
        sorry =
    𝟙 X
Tactic: refine .mk_of_isStableUnderBaseChange fun {X Y} f 𝒰 (H : ∀ i, IsIso _) ↦ ⟨?_, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Subcanonical (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : Precoverage.{v, u} C
inst_1 : J.toGrothendieck.Subcanonical
inst_2 : Limits.HasPullbacks.{v, u} C
inst_3 : J.IsStableUnderBaseChange
refine_1 :
  {X Y : C} →
    (f : X ⟶ Y) →
      (𝒰 : J.ZeroHypercover Y) → (H : ∀ (i : 𝒰.I₀), IsIso (X := pullback f (𝒰.f i)) (pullback.snd f (𝒰.f i))) → Y ⟶ X
refine_2 :
  ∀ {X Y : C} (f : X ⟶ Y) (𝒰 : J.ZeroHypercover Y)
    (H : ∀ (i : 𝒰.I₀), IsIso (X := pullback f (𝒰.f i)) (pullback.snd f (𝒰.f i))),
    f ≫
        𝒰.glueMorphisms
          (fun (i : 𝒰.I₀) ↦
            CategoryStruct.comp (Y := pullback f (𝒰.f i)) (inv (pullback.snd f (𝒰.f i))) (pullback.fst f (𝒰.f i)))
          ⋯ =
      𝟙 X
⊢ IsLocalAtTarget (isomorphisms C) J
have refine_3 : ∀ {X Y : C} (f : X ⟶ Y) (𝒰 : J.ZeroHypercover Y)
  (H : ∀ (i : 𝒰.I₀), IsIso (X := pullback f (𝒰.f i)) (pullback.snd f (𝒰.f i))),
  𝒰.glueMorphisms
        (fun (i : 𝒰.I₀) ↦
          CategoryStruct.comp (Y := pullback f (𝒰.f i)) (inv (pullback.snd f (𝒰.f i))) (pullback.fst f (𝒰.f i)))
        sorry ≫
      f =
    𝟙 Y
Tactic: refine .mk_of_isStableUnderBaseChange fun {X Y} f 𝒰 (H : ∀ i, IsIso _) ↦ ⟨?_, ?_, ?_⟩