awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Subcanonical (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
inst_1 : J.Subcanonical
ι : Type u_1
X : ι → C
c : Cofan.{u_1, v, u} X
H : Membership.mem (γ := Set (Sieve.{v, u} c.pt)) ((J : (a : C) → Set (Sieve a)) c.pt) (Sieve.ofArrows X c.inj)
inst_2 : ∀ (i : ι), Mono (c.inj i)
hempty : ∀ (Y : C) (a : IsInitial.{v, u} Y), ⊥ ∈ (J : (a : C) → Set (Sieve a)) Y
hdisj :
  ∀ {i j : ι}, i ≠ j → ∀ {Y : C} (a : Y ⟶ X i) (b : Y ⟶ X j), a ≫ c.inj i = b ≫ c.inj j → Nonempty (IsInitial.{v, u} Y)
s : Cofan (C := Sheaf J (Type v)) fun (i : ι) ↦ J.yoneda.obj (X i)
Y : C
i j : ι
a : Y ⟶ X i
b : Y ⟶ X j
hab : a ≫ c.inj i = b ≫ c.inj j
h_1 : ¬i = j
h : IsInitial.{v, u} Y
⊢ Eq (α := (fun (X : Type v) ↦ X) (s.pt.obj (P := Presheaf.IsSheaf J).obj (op Y)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((s.inj i).hom.app (op Y)) :
          (a : (fun (X : Type v) ↦ X) ((J.yoneda.obj (X i)).obj.obj (op Y))) →
            (fun (X : Type v) ↦ X) (s.pt.obj.obj (op Y)))
        a :
      (fun (X : Type v) ↦ X) (s.pt.obj (P := Presheaf.IsSheaf J).obj (op Y)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((s.inj j).hom.app (op Y)) :
          (a : (fun (X : Type v) ↦ X) ((J.yoneda.obj (X j)).obj.obj (op Y))) →
            (fun (X : Type v) ↦ X) (s.pt.obj.obj (op Y)))
        b :
      (fun (X : Type v) ↦ X) (s.pt.obj (P := Presheaf.IsSheaf J).obj (op Y)))
have neg : ∀ (this : Unique (s.pt.obj (P := Presheaf.IsSheaf J).obj (op Y))),
  Eq (α := (fun (X : Type v) ↦ X) (s.pt.obj (P := Presheaf.IsSheaf J).obj (op Y)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((s.inj i).hom.app (op Y)) :
          (a : (fun (X : Type v) ↦ X) ((J.yoneda.obj (X i)).obj.obj (op Y))) →
            (fun (X : Type v) ↦ X) (s.pt.obj.obj (op Y)))
        a :
      (fun (X : Type v) ↦ X) (s.pt.obj (P := Presheaf.IsSheaf J).obj (op Y)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((s.inj j).hom.app (op Y)) :
          (a : (fun (X : Type v) ↦ X) ((J.yoneda.obj (X j)).obj.obj (op Y))) →
            (fun (X : Type v) ↦ X) (s.pt.obj.obj (op Y)))
        b :
      (fun (X : Type v) ↦ X) (s.pt.obj (P := Presheaf.IsSheaf J).obj (op Y)))
Tactic: have := Types.isTerminalEquivUnique _ (Sheaf.isTerminalOfBotCover s.pt _ (hempty Y h))
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Subcanonical (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
inst_1 : J.Subcanonical
ι : Type u_1
X : ι → C
c : Cofan.{u_1, v, u} X
H : Membership.mem (γ := Set (Sieve.{v, u} c.pt)) ((J : (a : C) → Set (Sieve a)) c.pt) (Sieve.ofArrows X c.inj)
inst_2 : ∀ (i : ι), Mono (c.inj i)
hempty : ∀ (Y : C) (a : IsInitial.{v, u} Y), ⊥ ∈ (J : (a : C) → Set (Sieve a)) Y
hdisj :
  ∀ {i j : ι}, i ≠ j → ∀ {Y : C} (a : Y ⟶ X i) (b : Y ⟶ X j), a ≫ c.inj i = b ≫ c.inj j → Nonempty (IsInitial.{v, u} Y)
⊢ IsColimit (J := Discrete ι) (F := Discrete.functor fun (i : ι) ↦ J.yoneda.obj (X i))
    (Cofan.mk (J.yoneda.obj c.pt) fun (i : ι) ↦ J.yoneda.map (c.inj i))
have heq : ∀ (s : Cofan (C := Sheaf J (Type v)) fun (i : ι) ↦ J.yoneda.obj (X i)) {Y : C} {i j : ι} (a : Y ⟶ X i) (b : Y ⟶ X j)
  (hab : a ≫ c.inj i = b ≫ c.inj j),
  Eq (α := (fun (X : Type v) ↦ X) (s.pt.obj (P := Presheaf.IsSheaf J).obj (op Y)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((s.inj i).hom.app (op Y)) :
          (a : (fun (X : Type v) ↦ X) ((J.yoneda.obj (X i)).obj.obj (op Y))) →
            (fun (X : Type v) ↦ X) (s.pt.obj.obj (op Y)))
        a :
      (fun (X : Type v) ↦ X) (s.pt.obj (P := Presheaf.IsSheaf J).obj (op Y)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((s.inj j).hom.app (op Y)) :
          (a : (fun (X : Type v) ↦ X) ((J.yoneda.obj (X j)).obj.obj (op Y))) →
            (fun (X : Type v) ↦ X) (s.pt.obj.obj (op Y)))
        b :
      (fun (X : Type v) ↦ X) (s.pt.obj (P := Presheaf.IsSheaf J).obj (op Y)))
Tactic: have heq (s : Cofan fun i ↦ J.yoneda.obj (X i))
      {Y : C} {i j : ι} (a : Y ⟶ X i) (b : Y ⟶ X j) (hab : a ≫ c.inj i = b ≫ c.inj j) :
      (s.inj i).hom.app (op Y) a = (s.inj j).hom.app (op Y) b := by
    by_cases h : i = j
    · subst h
      rw [(cancel_mono _).mp hab]
    · obtain ⟨h⟩ := hdisj h a b hab
      have := Types.isTerminalEquivUnique _ (Sheaf.isTerminalOfBotCover s.pt _ (hempty Y h))
      exact Subsingleton.elim _ _
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Subcanonical (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
inst_1 : J.Subcanonical
ι : Type u_1
X : ι → C
c : Cofan.{u_1, v, u} X
H : Membership.mem (γ := Set (Sieve.{v, u} c.pt)) ((J : (a : C) → Set (Sieve a)) c.pt) (Sieve.ofArrows X c.inj)
inst_2 : ∀ (i : ι), Mono (c.inj i)
hempty : ∀ (Y : C) (a : IsInitial.{v, u} Y), ⊥ ∈ (J : (a : C) → Set (Sieve a)) Y
hdisj :
  ∀ {i j : ι}, i ≠ j → ∀ {Y : C} (a : Y ⟶ X i) (b : Y ⟶ X j), a ≫ c.inj i = b ≫ c.inj j → Nonempty (IsInitial.{v, u} Y)
heq :
  ∀ (s : Cofan (C := Sheaf J (Type v)) fun (i : ι) ↦ J.yoneda.obj (X i)) {Y : C} {i j : ι} (a : Y ⟶ X i) (b : Y ⟶ X j)
    (hab : a ≫ c.inj i = b ≫ c.inj j),
    Eq (α := (fun (X : Type v) ↦ X) (s.pt.obj (P := Presheaf.IsSheaf J).obj (op Y)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) ((s.inj i).hom.app (op Y)) :
            (a : (fun (X : Type v) ↦ X) ((J.yoneda.obj (X i)).obj.obj (op Y))) →
              (fun (X : Type v) ↦ X) (s.pt.obj.obj (op Y)))
          a :
        (fun (X : Type v) ↦ X) (s.pt.obj (P := Presheaf.IsSheaf J).obj (op Y)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) ((s.inj j).hom.app (op Y)) :
            (a : (fun (X : Type v) ↦ X) ((J.yoneda.obj (X j)).obj.obj (op Y))) →
              (fun (X : Type v) ↦ X) (s.pt.obj.obj (op Y)))
          b :
        (fun (X : Type v) ↦ X) (s.pt.obj (P := Presheaf.IsSheaf J).obj (op Y)))
⊢ IsColimit (J := Discrete ι) (F := Discrete.functor fun (i : ι) ↦ J.yoneda.obj (X i))
    (Cofan.mk (J.yoneda.obj c.pt) fun (i : ι) ↦ J.yoneda.map (c.inj i))
have refine_1 : (s : Cofan (C := Sheaf J (Type v)) fun (i : ι) ↦ J.yoneda.obj (X i)) →
  (Cocone.pt (J := Discrete ι) (F := Discrete.functor fun (i : ι) ↦ J.yoneda.obj (X i))
          (Cofan.mk (J.yoneda.obj c.pt) fun (i : ι) ↦ J.yoneda.map (c.inj i))).obj
      (P := Presheaf.IsSheaf J) ⟶
    s.pt.obj (P := Presheaf.IsSheaf J)
Tactic: refine Cofan.IsColimit.mk _ (fun s ↦ ⟨?_⟩) (fun s j ↦ ?_) fun s m hm ↦ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Subcanonical (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
inst_1 : J.Subcanonical
ι : Type u_1
X : ι → C
c : Cofan.{u_1, v, u} X
H : Membership.mem (γ := Set (Sieve.{v, u} c.pt)) ((J : (a : C) → Set (Sieve a)) c.pt) (Sieve.ofArrows X c.inj)
inst_2 : ∀ (i : ι), Mono (c.inj i)
hempty : ∀ (Y : C) (a : IsInitial.{v, u} Y), ⊥ ∈ (J : (a : C) → Set (Sieve a)) Y
hdisj :
  ∀ {i j : ι}, i ≠ j → ∀ {Y : C} (a : Y ⟶ X i) (b : Y ⟶ X j), a ≫ c.inj i = b ≫ c.inj j → Nonempty (IsInitial.{v, u} Y)
heq :
  ∀ (s : Cofan (C := Sheaf J (Type v)) fun (i : ι) ↦ J.yoneda.obj (X i)) {Y : C} {i j : ι} (a : Y ⟶ X i) (b : Y ⟶ X j)
    (hab : a ≫ c.inj i = b ≫ c.inj j),
    Eq (α := (fun (X : Type v) ↦ X) (s.pt.obj (P := Presheaf.IsSheaf J).obj (op Y)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) ((s.inj i).hom.app (op Y)) :
            (a : (fun (X : Type v) ↦ X) ((J.yoneda.obj (X i)).obj.obj (op Y))) →
              (fun (X : Type v) ↦ X) (s.pt.obj.obj (op Y)))
          a :
        (fun (X : Type v) ↦ X) (s.pt.obj (P := Presheaf.IsSheaf J).obj (op Y)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) ((s.inj j).hom.app (op Y)) :
            (a : (fun (X : Type v) ↦ X) ((J.yoneda.obj (X j)).obj.obj (op Y))) →
              (fun (X : Type v) ↦ X) (s.pt.obj.obj (op Y)))
          b :
        (fun (X : Type v) ↦ X) (s.pt.obj (P := Presheaf.IsSheaf J).obj (op Y)))
s : Cofan (C := Sheaf J (Type v)) fun (i : ι) ↦ J.yoneda.obj (X i)
⊢ (Cocone.pt (J := Discrete ι) (F := Discrete.functor fun (i : ι) ↦ J.yoneda.obj (X i))
          (Cofan.mk (J.yoneda.obj c.pt) fun (i : ι) ↦ J.yoneda.map (c.inj i))).obj
      (P := Presheaf.IsSheaf J) ⟶
    s.pt.obj (P := Presheaf.IsSheaf J)
have refine_1 : (Sieve.ofArrows X c.inj).functor ⟶ s.pt.obj (P := Presheaf.IsSheaf J)
Tactic: refine (s.pt.2.isSheafFor _ H).extend ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Subcanonical (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
inst_1 : J.Subcanonical
ι : Type u_1
X : ι → C
c : Cofan.{u_1, v, u} X
H : Membership.mem (γ := Set (Sieve.{v, u} c.pt)) ((J : (a : C) → Set (Sieve a)) c.pt) (Sieve.ofArrows X c.inj)
inst_2 : ∀ (i : ι), Mono (c.inj i)
hempty : ∀ (Y : C) (a : IsInitial.{v, u} Y), ⊥ ∈ (J : (a : C) → Set (Sieve a)) Y
hdisj :
  ∀ {i j : ι}, i ≠ j → ∀ {Y : C} (a : Y ⟶ X i) (b : Y ⟶ X j), a ≫ c.inj i = b ≫ c.inj j → Nonempty (IsInitial.{v, u} Y)
heq :
  ∀ (s : Cofan (C := Sheaf J (Type v)) fun (i : ι) ↦ J.yoneda.obj (X i)) {Y : C} {i j : ι} (a : Y ⟶ X i) (b : Y ⟶ X j)
    (hab : a ≫ c.inj i = b ≫ c.inj j),
    Eq (α := (fun (X : Type v) ↦ X) (s.pt.obj (P := Presheaf.IsSheaf J).obj (op Y)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) ((s.inj i).hom.app (op Y)) :
            (a : (fun (X : Type v) ↦ X) ((J.yoneda.obj (X i)).obj.obj (op Y))) →
              (fun (X : Type v) ↦ X) (s.pt.obj.obj (op Y)))
          a :
        (fun (X : Type v) ↦ X) (s.pt.obj (P := Presheaf.IsSheaf J).obj (op Y)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) ((s.inj j).hom.app (op Y)) :
            (a : (fun (X : Type v) ↦ X) ((J.yoneda.obj (X j)).obj.obj (op Y))) →
              (fun (X : Type v) ↦ X) (s.pt.obj.obj (op Y)))
          b :
        (fun (X : Type v) ↦ X) (s.pt.obj (P := Presheaf.IsSheaf J).obj (op Y)))
s : Cofan (C := Sheaf J (Type v)) fun (i : ι) ↦ J.yoneda.obj (X i)
⊢ (Sieve.ofArrows X c.inj).functor ⟶ s.pt.obj (P := Presheaf.IsSheaf J)
have refine_1 : ∀ ⦃X_1 Y : Cᵒᵖ⦄ (f : X_1 ⟶ Y),
  (Sieve.ofArrows X c.inj).functor.map f ≫
      (fun (Y : Cᵒᵖ) ↦
          (↾fun (g : (Sieve.ofArrows X c.inj).functor.obj Y) ↦
              (ConcreteCategory.hom (FC := TypeCat.Fun)
                    (NatTrans.app (InducedCategory.Hom.hom (s.inj (Sieve.ofArrows.i sorry))) Y) :
                  (a : (fun (X : Type v) ↦ X) ((J.yoneda.obj (X (Sieve.ofArrows.i sorry))).obj.obj Y)) →
                    (fun (X : Type v) ↦ X) (s.pt.obj.obj Y))
                (Sieve.ofArrows.h sorry) :
            (Sieve.ofArrows X c.inj).functor.obj Y ⟶ s.pt.obj (P := Presheaf.IsSheaf J).obj Y))
        Y =
    (fun (Y : Cᵒᵖ) ↦
          (↾fun (g : (Sieve.ofArrows X c.inj).functor.obj Y) ↦
              (ConcreteCategory.hom (FC := TypeCat.Fun)
                    (NatTrans.app (InducedCategory.Hom.hom (s.inj (Sieve.ofArrows.i sorry))) Y) :
                  (a : (fun (X : Type v) ↦ X) ((J.yoneda.obj (X (Sieve.ofArrows.i sorry))).obj.obj Y)) →
                    (fun (X : Type v) ↦ X) (s.pt.obj.obj Y))
                (Sieve.ofArrows.h sorry) :
            (Sieve.ofArrows X c.inj).functor.obj Y ⟶ s.pt.obj (P := Presheaf.IsSheaf J).obj Y))
        X_1 ≫
      s.pt.obj (P := Presheaf.IsSheaf J).map f
Tactic: refine ⟨fun Y ↦ ↾fun g ↦ ((s.inj (Sieve.ofArrows.i g.2)).hom.app Y)
      (Sieve.ofArrows.h g.2), ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Subcanonical (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
inst_1 : J.Subcanonical
ι : Type u_1
X : ι → C
c : Cofan.{u_1, v, u} X
H : Membership.mem (γ := Set (Sieve.{v, u} c.pt)) ((J : (a : C) → Set (Sieve a)) c.pt) (Sieve.ofArrows X c.inj)
inst_2 : ∀ (i : ι), Mono (c.inj i)
hempty : ∀ (Y : C) (a : IsInitial.{v, u} Y), ⊥ ∈ (J : (a : C) → Set (Sieve a)) Y
hdisj :
  ∀ {i j : ι}, i ≠ j → ∀ {Y : C} (a : Y ⟶ X i) (b : Y ⟶ X j), a ≫ c.inj i = b ≫ c.inj j → Nonempty (IsInitial.{v, u} Y)
heq :
  ∀ (s : Cofan (C := Sheaf J (Type v)) fun (i : ι) ↦ J.yoneda.obj (X i)) {Y : C} {i j : ι} (a : Y ⟶ X i) (b : Y ⟶ X j)
    (hab : a ≫ c.inj i = b ≫ c.inj j),
    Eq (α := (fun (X : Type v) ↦ X) (s.pt.obj (P := Presheaf.IsSheaf J).obj (op Y)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) ((s.inj i).hom.app (op Y)) :
            (a : (fun (X : Type v) ↦ X) ((J.yoneda.obj (X i)).obj.obj (op Y))) →
              (fun (X : Type v) ↦ X) (s.pt.obj.obj (op Y)))
          a :
        (fun (X : Type v) ↦ X) (s.pt.obj (P := Presheaf.IsSheaf J).obj (op Y)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) ((s.inj j).hom.app (op Y)) :
            (a : (fun (X : Type v) ↦ X) ((J.yoneda.obj (X j)).obj.obj (op Y))) →
              (fun (X : Type v) ↦ X) (s.pt.obj.obj (op Y)))
          b :
        (fun (X : Type v) ↦ X) (s.pt.obj (P := Presheaf.IsSheaf J).obj (op Y)))
s : Cofan (C := Sheaf J (Type v)) fun (i : ι) ↦ J.yoneda.obj (X i)
j : ι
u : (j : ι) → obj CategoryTheory.yoneda (X j) ⟶ (Sieve.ofArrows X c.inj).functor :=
  fun (j : ι) ↦ (Sieve.ofArrows X c.inj).toFunctor (c.inj j) ⋯
this : ∀ (j : ι), u j ≫ (Sieve.ofArrows X c.inj).functorInclusion = map CategoryTheory.yoneda (c.inj j)
⊢ map CategoryTheory.yoneda (c.inj j) ≫
      ⋯.extend
        {
          app := fun (Y : Cᵒᵖ) ↦
            ↾fun
                (g :
                  { g : unop Y ⟶ c.pt //
                    ∃ (Y_1 : C) (h : unop Y ⟶ Y_1) (g_1 : Y_1 ⟶ c.pt), Presieve.ofArrows X c.inj g_1 ∧ h ≫ g_1 = g }) ↦
              (ConcreteCategory.hom (FC := TypeCat.Fun)
                    (NatTrans.app (InducedCategory.Hom.hom (s.inj (Sieve.ofArrows.i ⋯))) Y) :
                  (a : unop Y ⟶ X (Sieve.ofArrows.i ⋯)) → s.pt.obj.obj Y)
                (Sieve.ofArrows.h ⋯),
          naturality := ⋯ : NatTrans (Sieve.ofArrows X c.inj).functor (s.pt.obj (P := Presheaf.IsSheaf J)) } =
    (s.inj j).hom
have refine_2 : u j ≫
    {
      app := fun (Y : Cᵒᵖ) ↦
        ↾fun
            (g :
              { g : unop Y ⟶ c.pt //
                ∃ (Y_1 : C) (h : unop Y ⟶ Y_1) (g_1 : Y_1 ⟶ c.pt), Presieve.ofArrows X c.inj g_1 ∧ h ≫ g_1 = g }) ↦
          (ConcreteCategory.hom (FC := TypeCat.Fun)
                (NatTrans.app (InducedCategory.Hom.hom (s.inj (Sieve.ofArrows.i sorry))) Y) :
              (a : unop Y ⟶ X (Sieve.ofArrows.i sorry)) → s.pt.obj.obj Y)
            (Sieve.ofArrows.h sorry),
      naturality := sorry } =
  (s.inj j).hom
Tactic: simp only [← this, Category.assoc, Presieve.IsSheafFor.functorInclusion_comp_extend]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Subcanonical (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
inst_1 : J.Subcanonical
ι : Type u_1
X : ι → C
c : Cofan.{u_1, v, u} X
H : Membership.mem (γ := Set (Sieve.{v, u} c.pt)) ((J : (a : C) → Set (Sieve a)) c.pt) (Sieve.ofArrows X c.inj)
inst_2 : ∀ (i : ι), Mono (c.inj i)
hempty : ∀ (Y : C) (a : IsInitial.{v, u} Y), ⊥ ∈ (J : (a : C) → Set (Sieve a)) Y
hdisj :
  ∀ {i j : ι}, i ≠ j → ∀ {Y : C} (a : Y ⟶ X i) (b : Y ⟶ X j), a ≫ c.inj i = b ≫ c.inj j → Nonempty (IsInitial.{v, u} Y)
heq :
  ∀ (s : Cofan (C := Sheaf J (Type v)) fun (i : ι) ↦ J.yoneda.obj (X i)) {Y : C} {i j : ι} (a : Y ⟶ X i) (b : Y ⟶ X j)
    (hab : a ≫ c.inj i = b ≫ c.inj j),
    Eq (α := (fun (X : Type v) ↦ X) (s.pt.obj (P := Presheaf.IsSheaf J).obj (op Y)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) ((s.inj i).hom.app (op Y)) :
            (a : (fun (X : Type v) ↦ X) ((J.yoneda.obj (X i)).obj.obj (op Y))) →
              (fun (X : Type v) ↦ X) (s.pt.obj.obj (op Y)))
          a :
        (fun (X : Type v) ↦ X) (s.pt.obj (P := Presheaf.IsSheaf J).obj (op Y)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) ((s.inj j).hom.app (op Y)) :
            (a : (fun (X : Type v) ↦ X) ((J.yoneda.obj (X j)).obj.obj (op Y))) →
              (fun (X : Type v) ↦ X) (s.pt.obj.obj (op Y)))
          b :
        (fun (X : Type v) ↦ X) (s.pt.obj (P := Presheaf.IsSheaf J).obj (op Y)))
s : Cofan (C := Sheaf J (Type v)) fun (i : ι) ↦ J.yoneda.obj (X i)
j : ι
u : (j : ι) → obj CategoryTheory.yoneda (X j) ⟶ (Sieve.ofArrows X c.inj).functor :=
  fun (j : ι) ↦ (Sieve.ofArrows X c.inj).toFunctor (c.inj j) ⋯
this : ∀ (j : ι), u j ≫ (Sieve.ofArrows X c.inj).functorInclusion = map CategoryTheory.yoneda (c.inj j)
Z : Cᵒᵖ
g : unop Z ⟶ X j
⊢ TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (NatTrans.app
          (u j ≫
            {
              app := fun (Y : Cᵒᵖ) ↦
                ↾fun
                    (g :
                      { g : unop Y ⟶ c.pt //
                        ∃ (Y_1 : C) (h : unop Y ⟶ Y_1) (g_1 : Y_1 ⟶ c.pt),
                          Presieve.ofArrows X c.inj g_1 ∧ h ≫ g_1 = g }) ↦
                  (ConcreteCategory.hom (FC := TypeCat.Fun)
                        (NatTrans.app (InducedCategory.Hom.hom (s.inj (Sieve.ofArrows.i ⋯))) Y) :
                      (a : unop Y ⟶ X (Sieve.ofArrows.i ⋯)) → s.pt.obj.obj Y)
                    (Sieve.ofArrows.h ⋯),
              naturality := ⋯ : NatTrans (Sieve.ofArrows X c.inj).functor (s.pt.obj (P := Presheaf.IsSheaf J)) })
          Z))
      g =
    (ConcreteCategory.hom (FC := TypeCat.Fun) ((s.inj j).hom.app Z)).toFun g
have h : (Sieve.ofArrows X c.inj).arrows (g ≫ c.inj j)
Tactic: have h : Sieve.ofArrows X c.inj (g ≫ c.inj j) :=
      Sieve.downward_closed _ (Sieve.ofArrows_mk _ _ j) _
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Subcanonical (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
inst_1 : J.Subcanonical
ι : Type u_1
X : ι → C
c : Cofan.{u_1, v, u} X
H : Membership.mem (γ := Set (Sieve.{v, u} c.pt)) ((J : (a : C) → Set (Sieve a)) c.pt) (Sieve.ofArrows X c.inj)
inst_2 : ∀ (i : ι), Mono (c.inj i)
hempty : ∀ (Y : C) (a : IsInitial.{v, u} Y), ⊥ ∈ (J : (a : C) → Set (Sieve a)) Y
hdisj :
  ∀ {i j : ι}, i ≠ j → ∀ {Y : C} (a : Y ⟶ X i) (b : Y ⟶ X j), a ≫ c.inj i = b ≫ c.inj j → Nonempty (IsInitial.{v, u} Y)
heq :
  ∀ (s : Cofan (C := Sheaf J (Type v)) fun (i : ι) ↦ J.yoneda.obj (X i)) {Y : C} {i j : ι} (a : Y ⟶ X i) (b : Y ⟶ X j)
    (hab : a ≫ c.inj i = b ≫ c.inj j),
    Eq (α := (fun (X : Type v) ↦ X) (s.pt.obj (P := Presheaf.IsSheaf J).obj (op Y)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) ((s.inj i).hom.app (op Y)) :
            (a : (fun (X : Type v) ↦ X) ((J.yoneda.obj (X i)).obj.obj (op Y))) →
              (fun (X : Type v) ↦ X) (s.pt.obj.obj (op Y)))
          a :
        (fun (X : Type v) ↦ X) (s.pt.obj (P := Presheaf.IsSheaf J).obj (op Y)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) ((s.inj j).hom.app (op Y)) :
            (a : (fun (X : Type v) ↦ X) ((J.yoneda.obj (X j)).obj.obj (op Y))) →
              (fun (X : Type v) ↦ X) (s.pt.obj.obj (op Y)))
          b :
        (fun (X : Type v) ↦ X) (s.pt.obj (P := Presheaf.IsSheaf J).obj (op Y)))
s : Cofan (C := Sheaf J (Type v)) fun (i : ι) ↦ J.yoneda.obj (X i)
m :
  Cocone.pt (J := Discrete ι) (F := Discrete.functor fun (i : ι) ↦ J.yoneda.obj (X i))
      (Cofan.mk (J.yoneda.obj c.pt) fun (i : ι) ↦ J.yoneda.map (c.inj i)) ⟶
    s.pt
hm :
  ∀ (j : ι),
    Cofan.inj (f := fun (i : ι) ↦ J.yoneda.obj (X i))
          (Cofan.mk (J.yoneda.obj c.pt) fun (i : ι) ↦ J.yoneda.map (c.inj i)) j ≫
        m =
      s.inj j
⊢ m.hom =
    ⋯.extend
      {
        app := fun (Y : Cᵒᵖ) ↦
          ↾fun
              (g :
                { g : unop Y ⟶ c.pt //
                  ∃ (Y_1 : C) (h : unop Y ⟶ Y_1) (g_1 : Y_1 ⟶ c.pt), Presieve.ofArrows X c.inj g_1 ∧ h ≫ g_1 = g }) ↦
            (ConcreteCategory.hom (FC := TypeCat.Fun)
                  (NatTrans.app (InducedCategory.Hom.hom (s.inj (Sieve.ofArrows.i ⋯))) Y) :
                (a : unop Y ⟶ X (Sieve.ofArrows.i ⋯)) → s.pt.obj.obj Y)
              (Sieve.ofArrows.h ⋯),
        naturality := ⋯ }
have refine_3 : (Sieve.ofArrows X c.inj).functorInclusion ≫ m.hom =
  {
    app := fun (Y : Cᵒᵖ) ↦
      ↾fun
          (g :
            { g : unop Y ⟶ c.pt //
              ∃ (Y_1 : C) (h : unop Y ⟶ Y_1) (g_1 : Y_1 ⟶ c.pt), Presieve.ofArrows X c.inj g_1 ∧ h ≫ g_1 = g }) ↦
        (ConcreteCategory.hom (FC := TypeCat.Fun)
              (NatTrans.app (InducedCategory.Hom.hom (s.inj (Sieve.ofArrows.i sorry))) Y) :
            (a : unop Y ⟶ X (Sieve.ofArrows.i sorry)) → s.pt.obj.obj Y)
          (Sieve.ofArrows.h sorry),
    naturality := sorry }
Tactic: apply Presieve.IsSheafFor.unique_extend
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Subcanonical (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
inst_1 : J.Subcanonical
ι : Type u_1
inst_2 : CoproductsOfShapeDisjoint.{v, u, u_1} C ι
inst_3 : HasPullbacks.{v, u} C
inst_4 : HasStrictInitialObjects.{v, u} C
hcov :
  ∀ {X : ι → C} {c : Cofan.{u_1, v, u} X} (x : IsColimit c),
    Membership.mem (γ := Set (Sieve.{v, u} c.pt)) ((J : (a : C) → Set (Sieve a)) c.pt) (Sieve.ofArrows X c.inj)
htriv : ∀ (Y : C) (a : IsInitial.{v, u} Y), ⊥ ∈ (J : (a : C) → Set (Sieve a)) Y
⊢ PreservesColimitsOfShape (Discrete ι) J.yoneda
have inst : ∀ (f : ι → C), PreservesColimit (Discrete.functor f) J.yoneda
Tactic: apply (config := { allowSynthFailures := true }) preservesColimitsOfShape_of_discrete
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Subcanonical (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
inst_1 : J.Subcanonical
ι : Type u_1
inst_2 : CoproductsOfShapeDisjoint.{v, u, u_1} C ι
inst_3 : HasPullbacks.{v, u} C
inst_4 : HasStrictInitialObjects.{v, u} C
hcov :
  ∀ {X : ι → C} {c : Cofan.{u_1, v, u} X} (x : IsColimit c),
    Membership.mem (γ := Set (Sieve.{v, u} c.pt)) ((J : (a : C) → Set (Sieve a)) c.pt) (Sieve.ofArrows X c.inj)
htriv : ∀ (Y : C) (a : IsInitial.{v, u} Y), ⊥ ∈ (J : (a : C) → Set (Sieve a)) Y
⊢ ∀ (f : ι → C), PreservesColimit (Discrete.functor f) J.yoneda
have inst : (X : ι → C) →
  {c : Cofan.{u_1, v, u} X} →
    (hc : IsColimit c) →
      IsColimit (J := Discrete ι) (F := Discrete.functor fun (i : ι) ↦ J.yoneda.obj (X i))
        (Cofan.mk (J.yoneda.obj c.pt) fun (i : ι) ↦ J.yoneda.map (c.inj i))
Tactic: refine fun X ↦ ⟨fun {c : Cofan X} hc ↦ ⟨(Limits.Cofan.isColimitMapCoconeEquiv _ _ _).symm ?_⟩⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Subcanonical (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
inst_1 : J.Subcanonical
ι : Type u_1
inst_2 : CoproductsOfShapeDisjoint.{v, u, u_1} C ι
inst_3 : HasPullbacks.{v, u} C
inst_4 : HasStrictInitialObjects.{v, u} C
hcov :
  ∀ {X : ι → C} {c : Cofan.{u_1, v, u} X} (x : IsColimit c),
    Membership.mem (γ := Set (Sieve.{v, u} c.pt)) ((J : (a : C) → Set (Sieve a)) c.pt) (Sieve.ofArrows X c.inj)
htriv : ∀ (Y : C) (a : IsInitial.{v, u} Y), ⊥ ∈ (J : (a : C) → Set (Sieve a)) Y
X : ι → C
c : Cofan.{u_1, v, u} X
hc : IsColimit c
⊢ IsColimit (J := Discrete ι) (F := Discrete.functor fun (i : ι) ↦ J.yoneda.obj (X i))
    (Cofan.mk (J.yoneda.obj c.pt) fun (i : ι) ↦ J.yoneda.map (c.inj i))
have this : ∀ (i : ι), Mono (c.inj i)
Tactic: have (i : ι) : Mono (c.inj i) := .of_coproductDisjoint hc _
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Subcanonical (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
inst_1 : J.Subcanonical
ι : Type u_1
inst_2 : CoproductsOfShapeDisjoint.{v, u, u_1} C ι
inst_3 : HasPullbacks.{v, u} C
inst_4 : HasStrictInitialObjects.{v, u} C
hcov :
  ∀ {X : ι → C} {c : Cofan.{u_1, v, u} X} (x : IsColimit c),
    Membership.mem (γ := Set (Sieve.{v, u} c.pt)) ((J : (a : C) → Set (Sieve a)) c.pt) (Sieve.ofArrows X c.inj)
htriv : ∀ (Y : C) (a : IsInitial.{v, u} Y), ⊥ ∈ (J : (a : C) → Set (Sieve a)) Y
X : ι → C
c : Cofan.{u_1, v, u} X
hc : IsColimit c
this : ∀ (i : ι), Mono (c.inj i)
⊢ IsColimit (J := Discrete ι) (F := Discrete.functor fun (i : ι) ↦ J.yoneda.obj (X i))
    (Cofan.mk (J.yoneda.obj c.pt) fun (i : ι) ↦ J.yoneda.map (c.inj i))
have inst : {i j : ι} →
  (hij : i ≠ j) → (Y : C) → (a : Y ⟶ X i) → (b : Y ⟶ X j) → (hab : a ≫ c.inj i = b ≫ c.inj j) → IsInitial.{v, u} Y
Tactic: refine isColimitCofanMkYoneda _ _ (hcov hc) htriv fun hij Y a b hab ↦ ⟨?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Subcanonical (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u_1
inst_1 : Category.{v', u_1} D
F : Functor.{v, v', u, u_1} C D
J : GrothendieckTopology.{v, u} C
K : GrothendieckTopology.{v', u_1} D
inst_2 : F.Full
inst_3 : F.Faithful
inst_4 : F.IsContinuous J K
inst_5 : K.Subcanonical
⊢ J.Subcanonical
have refine : ∀ (Y : C), Presieve.IsSheaf J (obj CategoryTheory.yoneda Y)
Tactic: refine .of_isSheaf_yoneda_obj _ fun Y ↦ ?_