awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.SubcanonicalOver (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
J : GrothendieckTopology.{v_1, u_1} C
inst_1 : J.Subcanonical
X : C
E Z : Over.{v_1, u_1} X
ι : Type (max u_1 v_1)
T : ι → Over.{v_1, u_1} X
g : (i : ι) → T i ⟶ Z
hR :
  Membership.mem (γ := Set (Sieve.{v_1, u_1} Z.left)) ((J : (a : C) → Set (Sieve a)) Z.left)
    (Sieve.generate
      (Presieve.ofArrows (fun (i : ι) ↦ Functor.obj (Over.forget X) (T i)) fun (i : ι) ↦
        Functor.map (Over.forget X) (g i)))
t : Presieve.FamilyOfElements (Functor.obj CategoryTheory.yoneda E) (Sieve.ofArrows T g).arrows
ht : t.Compatible (R := (Sieve.ofArrows T g).arrows)
hg : Presieve.Arrows.Compatible (Functor.obj CategoryTheory.yoneda E.left) (B := Z.left) (X := fun (i : ι) ↦ (T i).left)
  (fun (i : ι) ↦ Over.Hom.left (g i)) fun (i : ι) ↦ Over.Hom.left (t (g i) ⋯) :=
  ⋯
a : Functor.obj (Functor.obj CategoryTheory.yoneda E.left) (Opposite.op Z.left)
ha :
  hg.familyOfElements.sieveExtend (R :=
        Presieve.ofArrows (fun (i : ι) ↦ (T i).left) fun (i : ι) ↦ Over.Hom.left (g i)).IsAmalgamation
    (R :=
    Sieve.arrows
      (Sieve.generate
        (Presieve.ofArrows (fun (i : ι) ↦ Functor.obj (Over.forget X) (T i)) fun (i : ι) ↦
          Functor.map (Over.forget X) (g i))))
    a
huniq :
  ∀ (y : Functor.obj (Functor.obj CategoryTheory.yoneda E.left) (Opposite.op Z.left)),
    (fun (t_1 : Functor.obj (Functor.obj CategoryTheory.yoneda E.left) (Opposite.op Z.left)) ↦
          hg.familyOfElements.sieveExtend (R :=
                Presieve.ofArrows (fun (i : ι) ↦ (T i).left) fun (i : ι) ↦ Over.Hom.left (g i)).IsAmalgamation
            (R :=
            Sieve.arrows
              (Sieve.generate
                (Presieve.ofArrows (fun (i : ι) ↦ Functor.obj (Over.forget X) (T i)) fun (i : ι) ↦
                  Functor.map (Over.forget X) (g i))))
            t_1)
        y →
      y = a
⊢ Functor.obj (Functor.obj CategoryTheory.yoneda E) (Opposite.op Z)
have refine_1 : a ≫ E.hom = (Opposite.unop (Opposite.op Z)).hom
Tactic: refine Over.homMk a ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.SubcanonicalOver (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
J : GrothendieckTopology.{v_1, u_1} C
inst_1 : J.Subcanonical
X : C
E Z : Over.{v_1, u_1} X
ι : Type (max u_1 v_1)
T : ι → Over.{v_1, u_1} X
g : (i : ι) → T i ⟶ Z
hR :
  Membership.mem (γ := Set (Sieve.{v_1, u_1} Z.left)) ((J : (a : C) → Set (Sieve a)) Z.left)
    (Sieve.generate
      (Presieve.ofArrows (fun (i : ι) ↦ Functor.obj (Over.forget X) (T i)) fun (i : ι) ↦
        Functor.map (Over.forget X) (g i)))
t : Presieve.FamilyOfElements (Functor.obj CategoryTheory.yoneda E) (Sieve.ofArrows T g).arrows
ht : t.Compatible (R := (Sieve.ofArrows T g).arrows)
hg : Presieve.Arrows.Compatible (Functor.obj CategoryTheory.yoneda E.left) (B := Z.left) (X := fun (i : ι) ↦ (T i).left)
  (fun (i : ι) ↦ Over.Hom.left (g i)) fun (i : ι) ↦ Over.Hom.left (t (g i) ⋯) :=
  ⋯
a : Functor.obj (Functor.obj CategoryTheory.yoneda E.left) (Opposite.op Z.left)
ha :
  hg.familyOfElements.sieveExtend (R :=
        Presieve.ofArrows (fun (i : ι) ↦ (T i).left) fun (i : ι) ↦ Over.Hom.left (g i)).IsAmalgamation
    (R :=
    Sieve.arrows
      (Sieve.generate
        (Presieve.ofArrows (fun (i : ι) ↦ Functor.obj (Over.forget X) (T i)) fun (i : ι) ↦
          Functor.map (Over.forget X) (g i))))
    a
huniq :
  ∀ (y : Functor.obj (Functor.obj CategoryTheory.yoneda E.left) (Opposite.op Z.left)),
    (fun (t_1 : Functor.obj (Functor.obj CategoryTheory.yoneda E.left) (Opposite.op Z.left)) ↦
          hg.familyOfElements.sieveExtend (R :=
                Presieve.ofArrows (fun (i : ι) ↦ (T i).left) fun (i : ι) ↦ Over.Hom.left (g i)).IsAmalgamation
            (R :=
            Sieve.arrows
              (Sieve.generate
                (Presieve.ofArrows (fun (i : ι) ↦ Functor.obj (Over.forget X) (T i)) fun (i : ι) ↦
                  Functor.map (Over.forget X) (g i))))
            t_1)
        y →
      y = a
W Y : Over.{v_1, u_1} X
i : ι
v : W ⟶ T i
⊢ (ConcreteCategory.hom (FC := TypeCat.Fun) (Functor.map (Functor.obj CategoryTheory.yoneda E) (v ≫ g i).op) :
        (a : (fun (X : Type v_1) ↦ X) ((CategoryTheory.yoneda.obj E).obj (Opposite.op Z))) →
          (fun (X : Type v_1) ↦ X) ((CategoryTheory.yoneda.obj E).obj (Opposite.op W)))
      (Over.homMk a ⋯) =
    t (v ≫ g i) ⋯
have refine_2 : Over.Hom.left
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (Functor.map (Functor.obj CategoryTheory.yoneda E) (v ≫ g i).op) :
        (a : (fun (X : Type v_1) ↦ X) ((CategoryTheory.yoneda.obj E).obj (Opposite.op Z))) →
          (fun (X : Type v_1) ↦ X) ((CategoryTheory.yoneda.obj E).obj (Opposite.op W)))
      (Over.homMk a sorry)) =
  Over.Hom.left (t (v ≫ g i) sorry)
Tactic: refine Over.OverMorphism.ext ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.SubcanonicalOver (hypothesis)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
J : GrothendieckTopology.{v_1, u_1} C
inst_1 : J.Subcanonical
X : C
E Z : Over.{v_1, u_1} X
ι : Type (max u_1 v_1)
T : ι → Over.{v_1, u_1} X
g : (i : ι) → T i ⟶ Z
hR :
  Membership.mem (γ := Set (Sieve.{v_1, u_1} Z.left)) ((J : (a : C) → Set (Sieve a)) Z.left)
    (Sieve.generate
      (Presieve.ofArrows (fun (i : ι) ↦ Functor.obj (Over.forget X) (T i)) fun (i : ι) ↦
        Functor.map (Over.forget X) (g i)))
t : Presieve.FamilyOfElements (Functor.obj CategoryTheory.yoneda E) (Sieve.ofArrows T g).arrows
ht : t.Compatible (R := (Sieve.ofArrows T g).arrows)
hg : Presieve.Arrows.Compatible (Functor.obj CategoryTheory.yoneda E.left) (B := Z.left) (X := fun (i : ι) ↦ (T i).left)
  (fun (i : ι) ↦ Over.Hom.left (g i)) fun (i : ι) ↦ Over.Hom.left (t (g i) ⋯) :=
  ⋯
a : Functor.obj (Functor.obj CategoryTheory.yoneda E.left) (Opposite.op Z.left)
ha :
  hg.familyOfElements.sieveExtend (R :=
        Presieve.ofArrows (fun (i : ι) ↦ (T i).left) fun (i : ι) ↦ Over.Hom.left (g i)).IsAmalgamation
    (R :=
    Sieve.arrows
      (Sieve.generate
        (Presieve.ofArrows (fun (i : ι) ↦ Functor.obj (Over.forget X) (T i)) fun (i : ι) ↦
          Functor.map (Over.forget X) (g i))))
    a
huniq :
  ∀ (y : Functor.obj (Functor.obj CategoryTheory.yoneda E.left) (Opposite.op Z.left)),
    (fun (t_1 : Functor.obj (Functor.obj CategoryTheory.yoneda E.left) (Opposite.op Z.left)) ↦
          hg.familyOfElements.sieveExtend (R :=
                Presieve.ofArrows (fun (i : ι) ↦ (T i).left) fun (i : ι) ↦ Over.Hom.left (g i)).IsAmalgamation
            (R :=
            Sieve.arrows
              (Sieve.generate
                (Presieve.ofArrows (fun (i : ι) ↦ Functor.obj (Over.forget X) (T i)) fun (i : ι) ↦
                  Functor.map (Over.forget X) (g i))))
            t_1)
        y →
      y = a
W Y : Over.{v_1, u_1} X
i : ι
v : W ⟶ T i
⊢ Over.Hom.left
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (Functor.map (Functor.obj CategoryTheory.yoneda E) (v ≫ g i).op) :
          (a : (fun (X : Type v_1) ↦ X) ((CategoryTheory.yoneda.obj E).obj (Opposite.op Z))) →
            (fun (X : Type v_1) ↦ X) ((CategoryTheory.yoneda.obj E).obj (Opposite.op W)))
        (Over.homMk a ⋯)) =
    Over.Hom.left (t (v ≫ g i) ⋯)
have this : (ConcreteCategory.hom (FC := TypeCat.Fun)
        (Functor.map (Functor.obj CategoryTheory.yoneda E.left) (Over.Hom.left (g i)).op) :
      (a : (fun (X : Type v_1) ↦ X) ((CategoryTheory.yoneda.obj E.left).obj (Opposite.op Z.left))) →
        (fun (X : Type v_1) ↦ X) ((CategoryTheory.yoneda.obj E.left).obj (Opposite.op (T i).left)))
    a =
  hg.familyOfElements.sieveExtend (R := Presieve.ofArrows (fun (i : ι) ↦ (T i).left) fun (i : ι) ↦ Over.Hom.left (g i))
    (Over.Hom.left (g i)) sorry
Tactic: have := ha (g i).left (Sieve.ofArrows_mk _ _ i)
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.SubcanonicalOver (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
J : GrothendieckTopology.{v_1, u_1} C
inst_1 : J.Subcanonical
X : C
E Z : Over.{v_1, u_1} X
ι : Type (max u_1 v_1)
T : ι → Over.{v_1, u_1} X
g : (i : ι) → T i ⟶ Z
hR :
  Membership.mem (γ := Set (Sieve.{v_1, u_1} Z.left)) ((J : (a : C) → Set (Sieve a)) Z.left)
    (Sieve.generate
      (Presieve.ofArrows (fun (i : ι) ↦ Functor.obj (Over.forget X) (T i)) fun (i : ι) ↦
        Functor.map (Over.forget X) (g i)))
t : Presieve.FamilyOfElements (Functor.obj CategoryTheory.yoneda E) (Sieve.ofArrows T g).arrows
ht : t.Compatible (R := (Sieve.ofArrows T g).arrows)
hg : Presieve.Arrows.Compatible (Functor.obj CategoryTheory.yoneda E.left) (B := Z.left) (X := fun (i : ι) ↦ (T i).left)
  (fun (i : ι) ↦ Over.Hom.left (g i)) fun (i : ι) ↦ Over.Hom.left (t (g i) ⋯) :=
  ⋯
a : Functor.obj (Functor.obj CategoryTheory.yoneda E.left) (Opposite.op Z.left)
ha :
  hg.familyOfElements.sieveExtend (R :=
        Presieve.ofArrows (fun (i : ι) ↦ (T i).left) fun (i : ι) ↦ Over.Hom.left (g i)).IsAmalgamation
    (R :=
    Sieve.arrows
      (Sieve.generate
        (Presieve.ofArrows (fun (i : ι) ↦ Functor.obj (Over.forget X) (T i)) fun (i : ι) ↦
          Functor.map (Over.forget X) (g i))))
    a
huniq :
  ∀ (y : Functor.obj (Functor.obj CategoryTheory.yoneda E.left) (Opposite.op Z.left)),
    (fun (t_1 : Functor.obj (Functor.obj CategoryTheory.yoneda E.left) (Opposite.op Z.left)) ↦
          hg.familyOfElements.sieveExtend (R :=
                Presieve.ofArrows (fun (i : ι) ↦ (T i).left) fun (i : ι) ↦ Over.Hom.left (g i)).IsAmalgamation
            (R :=
            Sieve.arrows
              (Sieve.generate
                (Presieve.ofArrows (fun (i : ι) ↦ Functor.obj (Over.forget X) (T i)) fun (i : ι) ↦
                  Functor.map (Over.forget X) (g i))))
            t_1)
        y →
      y = a
y : Functor.obj (Functor.obj CategoryTheory.yoneda E) (Opposite.op Z)
hty :
  (fun (t_1 : Functor.obj (Functor.obj CategoryTheory.yoneda E) (Opposite.op Z)) ↦
      t.IsAmalgamation (R := (Sieve.ofArrows T g).arrows) t_1)
    y
⊢ y = Over.homMk a ⋯
have refine_3 : (fun (t_1 : Functor.obj (Functor.obj CategoryTheory.yoneda E.left) (Opposite.op Z.left)) ↦
    hg.familyOfElements.sieveExtend (R :=
          Presieve.ofArrows (fun (i : ι) ↦ (T i).left) fun (i : ι) ↦ Over.Hom.left (g i)).IsAmalgamation
      (R :=
      Sieve.arrows
        (Sieve.generate
          (Presieve.ofArrows (fun (i : ι) ↦ Functor.obj (Over.forget X) (T i)) fun (i : ι) ↦
            Functor.map (Over.forget X) (g i))))
      t_1)
  (Over.Hom.left y)
Tactic: refine Over.OverMorphism.ext (huniq _ ?_)
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Subsheaf (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
F F' F'' : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)
G G' : Subfunctor F
U V : Cᵒᵖ
i : U ⟶ V
s : F.obj U
hs :
  Membership.mem (γ := Set (F.obj U))
    {s : F.obj U |
      Membership.mem (γ := Set (Sieve.{v, u} (unop U))) ((J : (a : C) → Set (Sieve a)) (unop U)) (G.sieveOfSection s)}
    s
⊢ Membership.mem (γ := Set (F.obj U))
    (Set.preimage (β := (fun (X : Type w) ↦ X) (F.obj V))
      (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (F.map i)) :
        (a : (fun (X : Type w) ↦ X) (F.obj U)) → (fun (X : Type w) ↦ X) (F.obj V))
      {s : F.obj V |
        Membership.mem (γ := Set (Sieve.{v, u} (unop V))) ((J : (a : C) → Set (Sieve a)) (unop V))
          (G.sieveOfSection s)})
    s
have refine : Sieve.pullback i.unop (G.sieveOfSection s) ≤
  G.sieveOfSection
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map i) :
        (a : (fun (X : Type w) ↦ X) (F.obj U)) → (fun (X : Type w) ↦ X) (F.obj V))
      s)
Tactic: refine J.superset_covering ?_ (J.pullback_stable i.unop hs)
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Subsheaf (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
F F' F'' : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)
G G' : Subfunctor F
U V : Cᵒᵖ
i : U ⟶ V
s : F.obj U
hs :
  Membership.mem (γ := Set (F.obj U))
    {s : F.obj U |
      Membership.mem (γ := Set (Sieve.{v, u} (unop U))) ((J : (a : C) → Set (Sieve a)) (unop U)) (G.sieveOfSection s)}
    s
Y : C
f : Y ⟶ unop V
h : (Sieve.pullback i.unop (G.sieveOfSection s)).arrows f
⊢ Sieve.arrows
    (G.sieveOfSection
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map i) :
          (a : (fun (X : Type w) ↦ X) (F.obj U)) → (fun (X : Type w) ↦ X) (F.obj V))
        s))
    f
have dsimp : Membership.mem (α := F.obj (op Y)) (γ := Set (F.obj (op Y))) (G.obj (op Y))
  ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f.op) : (a : F.obj V) → F.obj (op Y))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map i) : (a : F.obj U) → F.obj V) s : F.obj V) :
    F.obj (op Y))
Tactic: dsimp at h ⊢
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Subsheaf (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
F F' F'' : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)
G G' : Subfunctor F
U V : Cᵒᵖ
i : U ⟶ V
s : F.obj U
hs :
  Membership.mem (γ := Set (F.obj U))
    {s : F.obj U |
      Membership.mem (γ := Set (Sieve.{v, u} (unop U))) ((J : (a : C) → Set (Sieve a)) (unop U)) (G.sieveOfSection s)}
    s
Y : C
f : Y ⟶ unop V
h :
  Membership.mem (α := F.obj (op Y)) (γ := Set (F.obj (op Y))) (G.obj (op Y))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (i ≫ f.op)) : (a : F.obj U) → F.obj (op Y)) s : F.obj (op Y))
⊢ Membership.mem (α := F.obj (op Y)) (γ := Set (F.obj (op Y))) (G.obj (op Y))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f.op) : (a : F.obj V) → F.obj (op Y))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map i) : (a : F.obj U) → F.obj V) s : F.obj V) :
      F.obj (op Y))
have rwa : Membership.mem (α := F.obj (op Y)) (γ := Set (F.obj (op Y))) (G.obj (op Y))
  ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map i ≫ F.map f.op) : (a : F.obj U) → F.obj (op Y)) s : F.obj (op Y))
Tactic: rwa [← comp_apply, ← Functor.map_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Subsheaf (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
F : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)
G : Subfunctor F
U : Cᵒᵖ
s : F.obj U
hs : s ∈ G.obj U
⊢ Membership.mem (γ := Set (Sieve.{v, u} (unop U))) ((J : (a : C) → Set (Sieve a)) (unop U)) (G.sieveOfSection s)
have h : G.sieveOfSection s = ⊤
Tactic: convert! J.top_mem U.unop
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Subsheaf (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
F : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)
G : Subfunctor F
U : Cᵒᵖ
s : F.obj U
hs : s ∈ G.obj U
⊢ G.sieveOfSection s = ⊤
have h : ⊤ ≤ G.sieveOfSection s
Tactic: rw [eq_top_iff]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Subsheaf (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
F : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)
G : Subfunctor F
h : Presieve.IsSheaf J F
hG : Presieve.IsSheaf.{w, v, u} J G.toFunctor
⊢ G = sheafify J G
have apply : sheafify J G ≤ G
Tactic: apply (G.le_sheafify J).antisymm
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Subsheaf (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
F : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)
G : Subfunctor F
h : Presieve.IsSheaf J F
hG : Presieve.IsSheaf.{w, v, u} J G.toFunctor
U : Cᵒᵖ
s : F.obj U
hs : s ∈ (sheafify J G).obj U
this : (↑(⋯.amalgamate (R := (G.sieveOfSection s).arrows) (G.familyOfElementsOfSection s) ⋯) : F.obj (op (unop U))) = s
⊢ s ∈ G.obj U
have rw : (↑(sorry.amalgamate (R := (G.sieveOfSection s).arrows) (G.familyOfElementsOfSection s) sorry) : F.obj (op (unop U))) ∈ G.obj U
Tactic: rw [← this]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Subsheaf (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
F : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)
G : Subfunctor F
h : Presieve.IsSheaf J F
hG : Presieve.IsSheaf.{w, v, u} J G.toFunctor
U : Cᵒᵖ
s : F.obj U
hs : s ∈ (sheafify J G).obj U
⊢ s ∈ G.obj U
have suffices : (↑(sorry.amalgamate (R := (G.sieveOfSection s).arrows) (G.familyOfElementsOfSection s) sorry) : F.obj (op (unop U))) = s
Tactic: suffices ((hG _ hs).amalgamate _ (G.family_of_elements_compatible s)).1 = s by
    rw [← this]
    exact ((hG _ hs).amalgamate _ (G.family_of_elements_compatible s)).2
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Subsheaf (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
F : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)
G : Subfunctor F
h : Presieve.IsSheaf J F
hG : Presieve.IsSheaf.{w, v, u} J G.toFunctor
U : Cᵒᵖ
s : F.obj U
hs : s ∈ (sheafify J G).obj U
⊢ (↑(⋯.amalgamate (R := (G.sieveOfSection s).arrows) (G.familyOfElementsOfSection s) ⋯) : F.obj (op (unop U))) = s
have apply : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ unop U⦄,
  (G.sieveOfSection s).arrows f →
    Eq (α := (fun (X : Type w) ↦ X) (F.obj (op Y)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f.op) :
            (a : (fun (X : Type w) ↦ X) (F.obj (op (unop U)))) → (fun (X : Type w) ↦ X) (F.obj (op Y)))
          (↑(sorry.amalgamate (R := (G.sieveOfSection s).arrows) (G.familyOfElementsOfSection s) sorry) : F.obj (op (unop U))) :
        (fun (X : Type w) ↦ X) (F.obj (op Y)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f.op) :
            (a : (fun (X : Type w) ↦ X) (F.obj (op (unop U)))) → (fun (X : Type w) ↦ X) (F.obj (op Y)))
          s :
        (fun (X : Type w) ↦ X) (F.obj (op Y)))
Tactic: apply (h _ hs).isSeparatedFor.ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Subsheaf (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
F : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)
G : Subfunctor F
h : Presieve.IsSheaf J F
⊢ Presieve.IsSheaf.{w, v, u} J G.toFunctor ↔
    ∀ (U : Cᵒᵖ) (s : F.obj U),
      Membership.mem (γ := Set (Sieve.{v, u} (unop U))) ((J : (a : C) → Set (Sieve a)) (unop U)) (G.sieveOfSection s) →
        s ∈ G.obj U
have rw : G = sheafify J G ↔
  ∀ (U : Cᵒᵖ) (s : F.obj U),
    Membership.mem (γ := Set (Sieve.{v, u} (unop U))) ((J : (a : C) → Set (Sieve a)) (unop U)) (G.sieveOfSection s) →
      s ∈ G.obj U
Tactic: rw [← G.eq_sheafify_iff h]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Subsheaf (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
F F' F'' : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)
G G' : Subfunctor F
f : G.toFunctor ⟶ F'
h : Presieve.IsSheaf J F'
U V : Cᵒᵖ
i : U ⟶ V
s : (sheafify J G).toFunctor.obj U
⊢ TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        ((sheafify J G).toFunctor.map i ≫
          ↾fun (s : (sheafify J G).toFunctor.obj V) ↦
            ⋯.amalgamate (R := (G.sieveOfSection (↑s : F.obj V)).arrows)
              ((G.familyOfElementsOfSection (↑s : F.obj V)).map (R := (G.sieveOfSection (↑s : F.obj V)).arrows) f) ⋯))
      s =
    TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        ((↾fun (s : (sheafify J G).toFunctor.obj U) ↦
            ⋯.amalgamate (R := (G.sieveOfSection (↑s : F.obj U)).arrows)
              ((G.familyOfElementsOfSection (↑s : F.obj U)).map (R := (G.sieveOfSection (↑s : F.obj U)).arrows) f) ⋯) ≫
          F'.map i))
      s
have h₁ : ∀ ⦃Y : C⦄ ⦃f_1 : Y ⟶ unop V⦄,
  Sieve.arrows
      (G.sieveOfSection
        (↑((ConcreteCategory.hom (FC := TypeCat.Fun) ((sheafify J G).toFunctor.map i) :
                (a : (fun (X : Type w) ↦ X) ((sheafify J G).toFunctor.obj U)) →
                  (fun (X : Type w) ↦ X) ((sheafify J G).toFunctor.obj V))
              s) :
          F.obj V))
      f_1 →
    Eq (α := (fun (X : Type w) ↦ X) (F'.obj (op Y)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (F'.map f_1.op) :
            (a : (fun (X : Type w) ↦ X) (F'.obj (op (unop V)))) → (fun (X : Type w) ↦ X) (F'.obj (op Y)))
          (TypeCat.Fun.toFun
            (ConcreteCategory.hom (FC := TypeCat.Fun)
              ((sheafify J G).toFunctor.map i ≫
                ↾fun (s : (sheafify J G).toFunctor.obj V) ↦
                  sorry.amalgamate (R := (G.sieveOfSection (↑s : F.obj V)).arrows)
                    ((G.familyOfElementsOfSection (↑s : F.obj V)).map (R := (G.sieveOfSection (↑s : F.obj V)).arrows) f)
                    sorry))
            s) :
        (fun (X : Type w) ↦ X) (F'.obj (op Y)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (F'.map f_1.op) :
            (a : (fun (X : Type w) ↦ X) (F'.obj (op (unop V)))) → (fun (X : Type w) ↦ X) (F'.obj (op Y)))
          (TypeCat.Fun.toFun
            (ConcreteCategory.hom (FC := TypeCat.Fun)
              ((↾fun (s : (sheafify J G).toFunctor.obj U) ↦
                  sorry.amalgamate (R := (G.sieveOfSection (↑s : F.obj U)).arrows)
                    ((G.familyOfElementsOfSection (↑s : F.obj U)).map (R := (G.sieveOfSection (↑s : F.obj U)).arrows) f)
                    sorry) ≫
                F'.map i))
            s) :
        (fun (X : Type w) ↦ X) (F'.obj (op Y)))
Tactic: apply (h _ ((Subfunctor.sheafify J G).toFunctor.map i s).prop).isSeparatedFor.ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Subsheaf (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
F F' F'' : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)
G G' : Subfunctor F
f : G.toFunctor ⟶ F'
h : Presieve.IsSheaf J F'
U V : Cᵒᵖ
i : U ⟶ V
s : (sheafify J G).toFunctor.obj U
W : C
j : W ⟶ unop V
hj :
  Sieve.arrows
    (G.sieveOfSection
      (↑((ConcreteCategory.hom (FC := TypeCat.Fun) ((sheafify J G).toFunctor.map i) :
              (a : (fun (X : Type w) ↦ X) ((sheafify J G).toFunctor.obj U)) →
                (fun (X : Type w) ↦ X) ((sheafify J G).toFunctor.obj V))
            s) :
        F.obj V))
    j
⊢ Presieve.FamilyOfElements.map (R :=
      Sieve.arrows
        (G.sieveOfSection
          (↑((ConcreteCategory.hom (FC := TypeCat.Fun) ((sheafify J G).toFunctor.map i) :
                  (a : (fun (X : Type w) ↦ X) ((sheafify J G).toFunctor.obj U)) →
                    (fun (X : Type w) ↦ X) ((sheafify J G).toFunctor.obj V))
                s) :
            F.obj V)))
      (G.familyOfElementsOfSection
        (↑((ConcreteCategory.hom (FC := TypeCat.Fun) ((sheafify J G).toFunctor.map i) :
                (a : (fun (X : Type w) ↦ X) ((sheafify J G).toFunctor.obj U)) →
                  (fun (X : Type w) ↦ X) ((sheafify J G).toFunctor.obj V))
              s) :
          F.obj V))
      f j hj =
    (ConcreteCategory.hom (FC := TypeCat.Fun) (F'.map j.op) :
        (a : (fun (X : Type w) ↦ X) (F'.obj (op (unop V)))) → (fun (X : Type w) ↦ X) (F'.obj (op W)))
      (TypeCat.Fun.toFun
        (ConcreteCategory.hom (FC := TypeCat.Fun)
          ((↾fun (s : (sheafify J G).toFunctor.obj U) ↦
              ⋯.amalgamate (R := (G.sieveOfSection (↑s : F.obj U)).arrows)
                ((G.familyOfElementsOfSection (↑s : F.obj U)).map (R := (G.sieveOfSection (↑s : F.obj U)).arrows) f)
                ⋯) ≫
            F'.map i))
        s)
have h₁ : Eq (α := F'.obj (op W))
  ((ConcreteCategory.hom (FC := TypeCat.Fun) (f.app (op W)) : (a : (↑(G.obj (op W)) : Type w)) → F'.obj (op W))
      (G.familyOfElementsOfSection
        (↑(({
                  toFun := fun (x : (↑((sheafify J G).obj U) : Type w)) ↦
                    Subtype.mk (p := fun (x : F.obj V) ↦ x ∈ (sheafify J G).obj V)
                      ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map i) : (a : F.obj U) → F.obj V) (↑x : F.obj U))
                      sorry :
                  TypeCat.Fun (↑((sheafify J G).obj U) : Type w) (↑((sheafify J G).obj V) : Type w) } :
                (a : (↑((sheafify J G).obj U) : Type w)) → (↑((sheafify J G).obj V) : Type w))
              s) :
          F.obj V)
        j hj) :
    F'.obj (op W))
  ((ConcreteCategory.hom (FC := TypeCat.Fun) (F'.map j.op) : (a : F'.obj V) → F'.obj (op W))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (F'.map i) : (a : F'.obj U) → F'.obj V)
          (sorry.amalgamate (R := (G.sieveOfSection (↑s : F.obj U)).arrows)
            ((G.familyOfElementsOfSection (↑s : F.obj U)).map (R := (G.sieveOfSection (↑s : F.obj U)).arrows) f) sorry) :
        F'.obj V) :
    F'.obj (op W))
Tactic: dsimp