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)
h : E.IsStronglySheafFor F
refine_1 : ∀ (s t : (E.multifork F).pt) (hst : (E.multifork F).toSections s = (E.multifork F).toSections t), s = t
⊢ Function.Bijective (E.multifork F).toSections
have refine_2 : ∀ (s : (E.multicospanIndex F).sections), ∃ (a : (E.multifork F).pt), (E.multifork F).toSections a = s
Tactic: refine ⟨fun s t hst ↦ ?_, fun s ↦ ?_⟩
-- 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)
h : E.IsStronglySheafFor F
s : (E.multicospanIndex F).sections
⊢ (E.multifork F).toSections (h.amalgamate s.val ⋯) = s
have val : ∀ (x : E.multicospanShape.L), ((E.multifork F).toSections (h.amalgamate s.val sorry)).val x = s.val x
Tactic: ext
-- 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)
⊢ ∃! t_1 : F.obj (op X), t.IsAmalgamation (R := S.arrows) t_1
have choose : ∀ (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),
  ∃! t_1 : F.obj (op X), t.IsAmalgamation (R := S.arrows) t_1
Tactic: choose s hs huniq using fun i ↦ H i (t.pullback (E.f i)) (ht.pullback (E.f i))
-- 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
i j : E.I₀
Z : C
gi : Z ⟶ E.X i
gj : Z ⟶ E.X j
hgij : gi ≫ E.f i = gj ≫ E.f j
⊢ Eq (α := (fun (X : Type u_2) ↦ X) (F.obj (op Z)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map gi.op) :
          (a : (fun (X : Type u_2) ↦ X) (F.obj (op (E.X i)))) → (fun (X : Type u_2) ↦ X) (F.obj (op Z)))
        (s i) :
      (fun (X : Type u_2) ↦ X) (F.obj (op Z)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map gj.op) :
          (a : (fun (X : Type u_2) ↦ X) (F.obj (op (E.X j)))) → (fun (X : Type u_2) ↦ X) (F.obj (op Z)))
        (s j) :
      (fun (X : Type u_2) ↦ X) (F.obj (op Z)))
have refine : ∀ (Y : C) (f : Y ⟶ Z),
  (E.sieve₁ gi gj).arrows f →
    ∀ (k : E.I₁ i j) (h : Y ⟶ E.Y k) (hf₁ : f ≫ gi = h ≫ E.p₁ k) (hf₂ : f ≫ gj = h ≫ E.p₂ k),
      Eq (α := (fun (X : Type u_2) ↦ X) (F.obj (op Y)))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f.op) :
              (a : (fun (X : Type u_2) ↦ X) (F.obj (op Z))) → (fun (X : Type u_2) ↦ X) (F.obj (op Y)))
            ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map gi.op) :
                  (a : (fun (X : Type u_2) ↦ X) (F.obj (op (E.X i)))) → (fun (X : Type u_2) ↦ X) (F.obj (op Z)))
                (s i) :
              (fun (X : Type u_2) ↦ X) (F.obj (op Z))) :
          (fun (X : Type u_2) ↦ X) (F.obj (op Y)))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f.op) :
              (a : (fun (X : Type u_2) ↦ X) (F.obj (op Z))) → (fun (X : Type u_2) ↦ X) (F.obj (op Y)))
            ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map gj.op) :
                  (a : (fun (X : Type u_2) ↦ X) (F.obj (op (E.X j)))) → (fun (X : Type u_2) ↦ X) (F.obj (op Z)))
                (s j) :
              (fun (X : Type u_2) ↦ X) (F.obj (op Z))) :
          (fun (X : Type u_2) ↦ X) (F.obj (op Y)))
Tactic: refine (h₁.isSeparatedFor_sieve₁ gi gj hgij).ext fun Y f ⟨k, h, hf₁, hf₂⟩ ↦ ?_
-- 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
i j : E.I₀
Z : C
gi : Z ⟶ E.X i
gj : Z ⟶ E.X j
hgij : gi ≫ E.f i = gj ≫ E.f j
Y : C
f : Y ⟶ Z
h_1 : (E.sieve₁ gi gj).arrows f
k : E.I₁ i j
h : Y ⟶ E.Y k
hf₁ : f ≫ gi = h ≫ E.p₁ k
hf₂ : f ≫ gj = h ≫ E.p₂ k
⊢ Eq (α := (fun (X : Type u_2) ↦ X) (F.obj (op Y)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f.op) :
          (a : (fun (X : Type u_2) ↦ X) (F.obj (op Z))) → (fun (X : Type u_2) ↦ X) (F.obj (op Y)))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map gi.op) :
              (a : (fun (X : Type u_2) ↦ X) (F.obj (op (E.X i)))) → (fun (X : Type u_2) ↦ X) (F.obj (op Z)))
            (s i) :
          (fun (X : Type u_2) ↦ X) (F.obj (op Z))) :
      (fun (X : Type u_2) ↦ X) (F.obj (op Y)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f.op) :
          (a : (fun (X : Type u_2) ↦ X) (F.obj (op Z))) → (fun (X : Type u_2) ↦ X) (F.obj (op Y)))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map gj.op) :
              (a : (fun (X : Type u_2) ↦ X) (F.obj (op (E.X j)))) → (fun (X : Type u_2) ↦ X) (F.obj (op Z)))
            (s j) :
          (fun (X : Type u_2) ↦ X) (F.obj (op Z))) :
      (fun (X : Type u_2) ↦ X) (F.obj (op Y)))
have simp : Eq (α := F.obj (op Y))
  ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (h ≫ E.p₁ k).op) : (a : F.obj (op (E.X i))) → F.obj (op Y)) (s i) :
    F.obj (op Y))
  ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (h ≫ E.p₂ k).op) : (a : F.obj (op (E.X j))) → F.obj (op Y)) (s j) :
    F.obj (op Y))
Tactic: simp only [← comp_apply, ← Functor.map_comp, ← op_comp, hf₁, hf₂]
-- 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
i j : E.I₀
Z : C
gi : Z ⟶ E.X i
gj : Z ⟶ E.X j
hgij : gi ≫ E.f i = gj ≫ E.f j
Y : C
f : Y ⟶ Z
h_1 : (E.sieve₁ gi gj).arrows f
k : E.I₁ i j
h : Y ⟶ E.Y k
hf₁ : f ≫ gi = h ≫ E.p₁ k
hf₂ : f ≫ gj = h ≫ E.p₂ k
⊢ Eq (α := F.obj (op Y))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (h ≫ E.p₁ k).op) : (a : F.obj (op (E.X i))) → F.obj (op Y))
        (s i) :
      F.obj (op Y))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (h ≫ E.p₂ k).op) : (a : F.obj (op (E.X j))) → F.obj (op Y))
        (s j) :
      F.obj (op Y))
have simp : Eq (α := F.obj (op Y))
  ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map h.op) : (a : F.obj (op (E.Y k))) → F.obj (op Y))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (E.p₁ k).op) : (a : F.obj (op (E.X i))) → F.obj (op (E.Y k)))
          (s i) :
        F.obj (op (E.Y k))) :
    F.obj (op Y))
  ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map h.op) : (a : F.obj (op (E.Y k))) → F.obj (op Y))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (E.p₂ k).op) : (a : F.obj (op (E.X j))) → F.obj (op (E.Y k)))
          (s j) :
        F.obj (op (E.Y k))) :
    F.obj (op Y))
Tactic: simp only [op_comp, Functor.map_comp, comp_apply]
-- 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
i j : E.I₀
Z : C
gi : Z ⟶ E.X i
gj : Z ⟶ E.X j
hgij : gi ≫ E.f i = gj ≫ E.f j
Y : C
f : Y ⟶ Z
h_1 : (E.sieve₁ gi gj).arrows f
k : E.I₁ i j
h : Y ⟶ E.Y k
hf₁ : f ≫ gi = h ≫ E.p₁ k
hf₂ : f ≫ gj = h ≫ E.p₂ k
⊢ Eq (α := F.obj (op Y))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map h.op) : (a : F.obj (op (E.Y k))) → F.obj (op Y))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (E.p₁ k).op) : (a : F.obj (op (E.X i))) → F.obj (op (E.Y k)))
            (s i) :
          F.obj (op (E.Y k))) :
      F.obj (op Y))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map h.op) : (a : F.obj (op (E.Y k))) → F.obj (op Y))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (E.p₂ k).op) : (a : F.obj (op (E.X j))) → F.obj (op (E.Y k)))
            (s j) :
          F.obj (op (E.Y k))) :
      F.obj (op Y))
have h₃ : Eq (α := F.obj (op (E.Y k)))
  ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (E.p₁ k).op) : (a : F.obj (op (E.X i))) → F.obj (op (E.Y k)))
      (s i) :
    F.obj (op (E.Y k)))
  ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (E.p₂ k).op) : (a : F.obj (op (E.X j))) → F.obj (op (E.Y k)))
      (s j) :
    F.obj (op (E.Y k)))
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
i j : E.I₀
Z : C
gi : Z ⟶ E.X i
gj : Z ⟶ E.X j
hgij : gi ≫ E.f i = gj ≫ E.f j
Y : C
f : Y ⟶ Z
h_1 : (E.sieve₁ gi gj).arrows f
k : E.I₁ i j
h : Y ⟶ E.Y k
hf₁ : f ≫ gi = h ≫ E.p₁ k
hf₂ : f ≫ gj = h ≫ E.p₂ k
⊢ Eq (α := F.obj (op (E.Y k)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (E.p₁ k).op) : (a : F.obj (op (E.X i))) → F.obj (op (E.Y k)))
        (s i) :
      F.obj (op (E.Y k)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (E.p₂ k).op) : (a : F.obj (op (E.X j))) → F.obj (op (E.Y k)))
        (s j) :
      F.obj (op (E.Y k)))
have h₃ : ∀ (W : C) (p : W ⟶ E.Y k) (hp : (Sieve.pullback (E.p₁ k ≫ E.f i) S).arrows p),
  Eq (α := (fun (X : Type u_2) ↦ X) (F.obj (op W)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map p.op) :
          (a : (fun (X : Type u_2) ↦ X) (F.obj (op (E.Y k)))) → (fun (X : Type u_2) ↦ X) (F.obj (op W)))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (E.p₁ k).op) : (a : F.obj (op (E.X i))) → F.obj (op (E.Y k)))
            (s i) :
          F.obj (op (E.Y k))) :
      (fun (X : Type u_2) ↦ X) (F.obj (op W)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map p.op) :
          (a : (fun (X : Type u_2) ↦ X) (F.obj (op (E.Y k)))) → (fun (X : Type u_2) ↦ X) (F.obj (op W)))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (E.p₂ k).op) : (a : F.obj (op (E.X j))) → F.obj (op (E.Y k)))
            (s j) :
          F.obj (op (E.Y k))) :
      (fun (X : Type u_2) ↦ X) (F.obj (op W)))
Tactic: refine (H' k).ext fun W p hp ↦ ?_
-- 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
i j : E.I₀
Z : C
gi : Z ⟶ E.X i
gj : Z ⟶ E.X j
hgij : gi ≫ E.f i = gj ≫ E.f j
Y : C
f : Y ⟶ Z
h_1 : (E.sieve₁ gi gj).arrows f
k : E.I₁ i j
h : Y ⟶ E.Y k
hf₁ : f ≫ gi = h ≫ E.p₁ k
hf₂ : f ≫ gj = h ≫ E.p₂ k
W : C
p : W ⟶ E.Y k
hp : (Sieve.pullback (E.p₁ k ≫ E.f i) S).arrows p
⊢ Eq (α := (fun (X : Type u_2) ↦ X) (F.obj (op W)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map p.op) :
          (a : (fun (X : Type u_2) ↦ X) (F.obj (op (E.Y k)))) → (fun (X : Type u_2) ↦ X) (F.obj (op W)))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (E.p₁ k).op) : (a : F.obj (op (E.X i))) → F.obj (op (E.Y k)))
            (s i) :
          F.obj (op (E.Y k))) :
      (fun (X : Type u_2) ↦ X) (F.obj (op W)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map p.op) :
          (a : (fun (X : Type u_2) ↦ X) (F.obj (op (E.Y k)))) → (fun (X : Type u_2) ↦ X) (F.obj (op W)))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (E.p₂ k).op) : (a : F.obj (op (E.X j))) → F.obj (op (E.Y k)))
            (s j) :
          F.obj (op (E.Y k))) :
      (fun (X : Type u_2) ↦ X) (F.obj (op W)))
have h₃ : Presieve.FamilyOfElements.pullback (E.f i) t (p ≫ E.p₁ k) sorry =
  Presieve.FamilyOfElements.pullback (E.f j) t (p ≫ E.p₂ k) sorry
Tactic: simp only [← comp_apply, ← Functor.map_comp, ← op_comp, hs i (p ≫ E.p₁ k) (by simpa),
      hs j (p ≫ E.p₂ k) (by simpa [← E.w])]
-- 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
i j : E.I₀
Z : C
gi : Z ⟶ E.X i
gj : Z ⟶ E.X j
hgij : gi ≫ E.f i = gj ≫ E.f j
Y : C
f : Y ⟶ Z
h_1 : (E.sieve₁ gi gj).arrows f
k : E.I₁ i j
h : Y ⟶ E.Y k
hf₁ : f ≫ gi = h ≫ E.p₁ k
hf₂ : f ≫ gj = h ≫ E.p₂ k
W : C
p : W ⟶ E.Y k
hp : (Sieve.pullback (E.p₁ k ≫ E.f i) S).arrows p
⊢ Presieve.FamilyOfElements.pullback (E.f i) t (p ≫ E.p₁ k) ⋯ =
    Presieve.FamilyOfElements.pullback (E.f j) t (p ≫ E.p₂ k) ⋯
have h₃ : t ((p ≫ E.p₁ k) ≫ E.f i) sorry = t ((p ≫ E.p₂ k) ≫ E.f j) sorry
Tactic: dsimp only [Presieve.FamilyOfElements.pullback]
-- 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
i j : E.I₀
Z : C
gi : Z ⟶ E.X i
gj : Z ⟶ E.X j
hgij : gi ≫ E.f i = gj ≫ E.f j
Y : C
f : Y ⟶ Z
h_1 : (E.sieve₁ gi gj).arrows f
k : E.I₁ i j
h : Y ⟶ E.Y k
hf₁ : f ≫ gi = h ≫ E.p₁ k
hf₂ : f ≫ gj = h ≫ E.p₂ k
W : C
p : W ⟶ E.Y k
hp : (Sieve.pullback (E.p₁ k ≫ E.f i) S).arrows p
⊢ t ((p ≫ E.p₁ k) ≫ E.f i) ⋯ = t ((p ≫ E.p₂ k) ≫ E.f j) ⋯
have h₃ : (p ≫ E.p₁ k) ≫ E.f i = (p ≫ E.p₂ k) ≫ E.f j
Tactic: congr 1
-- 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
⊢ ∃! t_1 : F.obj (op X), t.IsAmalgamation (R := S.arrows) t_1
have hr : Presieve.Arrows.Compatible F E.f s
Tactic: have hr : Presieve.Arrows.Compatible _ E.f s := by
    intro i j Z gi gj hgij
    refine (h₁.isSeparatedFor_sieve₁ gi gj hgij).ext fun Y f ⟨k, h, hf₁, hf₂⟩ ↦ ?_
    simp only [← comp_apply, ← Functor.map_comp, ← op_comp, hf₁, hf₂]
    simp only [op_comp, Functor.map_comp, comp_apply]
    congr! 1
    refine (H' k).ext fun W p hp ↦ ?_
    simp only [← comp_apply, ← Functor.map_comp, ← op_comp, hs i (p ≫ E.p₁ k) (by simpa),
      hs j (p ≫ E.p₂ k) (by simpa [← E.w])]
    dsimp only [Presieve.FamilyOfElements.pullback]
    congr 1
    simp [E.w]
-- 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
⊢ ∃! t_1 : F.obj (op X), t.IsAmalgamation (R := S.arrows) t_1
have obtain : ∀ (s' : Presieve.FamilyOfElements F (Presieve.ofArrows E.X E.f)) (hs' : ∀ (i : E.I₀), s' (E.f i) sorry = s i),
  ∃! t_1 : F.obj (op X), t.IsAmalgamation (R := S.arrows) t_1
Tactic: obtain ⟨s', hs'⟩ := hr.exists_familyOfElements