awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Closed (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
X : C
S : Sieve.{v, u} X
H : Presieve.IsSheafFor (Functor.closedSieves J).toFunctor S.arrows
this_1 : J.IsClosed (X := X) ⊤
this : Subtype.mk (p := J.IsClosed) (J.close S) ⋯ = Subtype.mk (p := J.IsClosed) ⊤ this_1
⊢ J.close S = ⊤
have this₁ : (↑(Subtype.mk (p := J.IsClosed) (J.close S) sorry) : Sieve X) = (↑(Subtype.mk (p := J.IsClosed) ⊤ this_1) : Sieve X)
Tactic: rw [Subtype.ext_iff] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Closed (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
X : C
S : Sieve.{v, u} X
H : Presieve.IsSheafFor (Functor.closedSieves J).toFunctor S.arrows
this : J.IsClosed (X := X) ⊤
⊢ J.close S = ⊤
have suffices : Subtype.mk (p := J.IsClosed) (J.close S) sorry = Subtype.mk (p := J.IsClosed) ⊤ this
Tactic: suffices (⟨J.close S, J.close_isClosed S⟩ : Subtype _) = ⟨⊤, this⟩ by
    rw [Subtype.ext_iff] at this
    exact this
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Closed (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
X : C
S : Sieve.{v, u} X
H : Presieve.IsSheafFor (Functor.closedSieves J).toFunctor S.arrows
this : J.IsClosed (X := X) ⊤
⊢ Subtype.mk (p := J.IsClosed) (J.close S) ⋯ = Subtype.mk (p := J.IsClosed) ⊤ this
have refine : ∀ (Y : C) (f : Y ⟶ X) (hf : S.arrows f),
  Eq (α := (fun (X : Type (max u v)) ↦ X) ((Functor.closedSieves J).toFunctor.obj (Opposite.op Y)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((Functor.closedSieves J).toFunctor.map f.op) :
          (a : (fun (X : Type (max u v)) ↦ X) ((Functor.closedSieves J).toFunctor.obj (Opposite.op X))) →
            (fun (X : Type (max u v)) ↦ X) ((Functor.closedSieves J).toFunctor.obj (Opposite.op Y)))
        (Subtype.mk (p := J.IsClosed) (J.close S) sorry) :
      (fun (X : Type (max u v)) ↦ X) ((Functor.closedSieves J).toFunctor.obj (Opposite.op Y)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((Functor.closedSieves J).toFunctor.map f.op) :
          (a : (fun (X : Type (max u v)) ↦ X) ((Functor.closedSieves J).toFunctor.obj (Opposite.op X))) →
            (fun (X : Type (max u v)) ↦ X) ((Functor.closedSieves J).toFunctor.obj (Opposite.op Y)))
        (Subtype.mk (p := J.IsClosed) ⊤ this) :
      (fun (X : Type (max u v)) ↦ X) ((Functor.closedSieves J).toFunctor.obj (Opposite.op Y)))
Tactic: refine H.isSeparatedFor.ext fun Y f hf ↦ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Closed (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
X : C
S : Sieve.{v, u} X
H : Presieve.IsSheafFor (Functor.closedSieves J).toFunctor S.arrows
this : J.IsClosed (X := X) ⊤
Y : C
f : Y ⟶ X
hf : S.arrows f
⊢ Eq (α := (fun (X : Type (max u v)) ↦ X) ((Functor.closedSieves J).toFunctor.obj (Opposite.op Y)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((Functor.closedSieves J).toFunctor.map f.op) :
          (a : (fun (X : Type (max u v)) ↦ X) ((Functor.closedSieves J).toFunctor.obj (Opposite.op X))) →
            (fun (X : Type (max u v)) ↦ X) ((Functor.closedSieves J).toFunctor.obj (Opposite.op Y)))
        (Subtype.mk (p := J.IsClosed) (J.close S) ⋯) :
      (fun (X : Type (max u v)) ↦ X) ((Functor.closedSieves J).toFunctor.obj (Opposite.op Y)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((Functor.closedSieves J).toFunctor.map f.op) :
          (a : (fun (X : Type (max u v)) ↦ X) ((Functor.closedSieves J).toFunctor.obj (Opposite.op X))) →
            (fun (X : Type (max u v)) ↦ X) ((Functor.closedSieves J).toFunctor.obj (Opposite.op Y)))
        (Subtype.mk (p := J.IsClosed) ⊤ this) :
      (fun (X : Type (max u v)) ↦ X) ((Functor.closedSieves J).toFunctor.obj (Opposite.op Y)))
have simp : Eq (α := { x : Sieve.{v, u} Y // J.IsClosed x })
  ((ConcreteCategory.hom (FC := TypeCat.Fun) ((Functor.closedSieves J).toFunctor.map f.op) :
        (a : { x : Sieve X // J.IsClosed x }) → { x : Sieve Y // J.IsClosed x })
      (Subtype.mk (p := J.IsClosed) (J.close S) sorry) :
    { x : Sieve.{v, u} Y // J.IsClosed x })
  ((ConcreteCategory.hom (FC := TypeCat.Fun) ((Functor.closedSieves J).toFunctor.map f.op) :
        (a : { x : Sieve X // J.IsClosed x }) → { x : Sieve Y // J.IsClosed x })
      (Subtype.mk (p := J.IsClosed) ⊤ this) :
    { x : Sieve.{v, u} Y // J.IsClosed x })
Tactic: simp only [Subfunctor.toFunctor_obj, Functor.sieves_obj, Functor.closedSieves_obj, Set.coe_setOf]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Closed (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
X : C
S : Sieve.{v, u} X
H : Presieve.IsSheafFor (Functor.closedSieves J).toFunctor S.arrows
this : J.IsClosed (X := X) ⊤
Y : C
f : Y ⟶ X
hf : S.arrows f
⊢ Eq (α := { x : Sieve.{v, u} Y // J.IsClosed x })
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((Functor.closedSieves J).toFunctor.map f.op) :
          (a : { x : Sieve X // J.IsClosed x }) → { x : Sieve Y // J.IsClosed x })
        (Subtype.mk (p := J.IsClosed) (J.close S) ⋯) :
      { x : Sieve.{v, u} Y // J.IsClosed x })
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((Functor.closedSieves J).toFunctor.map f.op) :
          (a : { x : Sieve X // J.IsClosed x }) → { x : Sieve Y // J.IsClosed x })
        (Subtype.mk (p := J.IsClosed) ⊤ this) :
      { x : Sieve.{v, u} Y // J.IsClosed x })
have a : Eq (α := Sieve.{v, u} Y)
  (↑((ConcreteCategory.hom (FC := TypeCat.Fun) ((Functor.closedSieves J).toFunctor.map f.op) :
          (a : { x : Sieve X // J.IsClosed x }) → { x : Sieve Y // J.IsClosed x })
        (Subtype.mk (p := J.IsClosed) (J.close S) sorry)) :
    Sieve Y)
  (↑((ConcreteCategory.hom (FC := TypeCat.Fun) ((Functor.closedSieves J).toFunctor.map f.op) :
          (a : { x : Sieve X // J.IsClosed x }) → { x : Sieve Y // J.IsClosed x })
        (Subtype.mk (p := J.IsClosed) ⊤ this)) :
    Sieve Y)
Tactic: ext1
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Closed (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
X : C
S : Sieve.{v, u} X
H : Presieve.IsSheafFor (Functor.closedSieves J).toFunctor S.arrows
this : J.IsClosed (X := X) ⊤
Y : C
f : Y ⟶ X
hf : S.arrows f
⊢ Eq (α := Sieve.{v, u} Y)
    (↑((ConcreteCategory.hom (FC := TypeCat.Fun) ((Functor.closedSieves J).toFunctor.map f.op) :
            (a : { x : Sieve X // J.IsClosed x }) → { x : Sieve Y // J.IsClosed x })
          (Subtype.mk (p := J.IsClosed) (J.close S) ⋯)) :
      Sieve Y)
    (↑((ConcreteCategory.hom (FC := TypeCat.Fun) ((Functor.closedSieves J).toFunctor.map f.op) :
            (a : { x : Sieve X // J.IsClosed x }) → { x : Sieve Y // J.IsClosed x })
          (Subtype.mk (p := J.IsClosed) ⊤ this)) :
      Sieve Y)
have a : Sieve.pullback f (J.close S) = Sieve.pullback f ⊤
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Closed (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
X : C
S : Sieve.{v, u} X
H : Presieve.IsSheafFor (Functor.closedSieves J).toFunctor S.arrows
this : J.IsClosed (X := X) ⊤
Y : C
f : Y ⟶ X
hf : S.arrows f
⊢ Sieve.pullback f (J.close S) = Sieve.pullback f ⊤
have a : ⊤ ∈ (J : (a : C) → Set (Sieve a)) Y
Tactic: rw [Sieve.pullback_top, ← J.pullback_close, S.pullback_eq_top_of_mem hf,
    J.close_eq_top_iff_mem]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Closed (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J₁ J₂ : GrothendieckTopology.{v, u} C
h : Presieve.IsSheaf J₁ (Functor.closedSieves J₂).toFunctor
X : C
S : Sieve.{v, u} X
hS : S ∈ (J₁ : (a : C) → Set (Sieve a)) X
⊢ S ∈ (J₂ : (a : C) → Set (Sieve a)) X
have rw : Presieve.IsSheafFor (Functor.closedSieves J₂).toFunctor S.arrows
Tactic: rw [GrothendieckTopology.mem_iff_isSheafFor_closedSieves]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Closed (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J₁ J₂ : GrothendieckTopology.{v, u} C
⊢ J₁ = J₂ ↔ ∀ (P : Cᵒᵖ ⥤ Type (max v u)), Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₂ P
have mp : J₁ = J₂ → ∀ (P : Cᵒᵖ ⥤ Type (max v u)), Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₂ P
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Closed (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J₁ J₂ : GrothendieckTopology.{v, u} C
mp : J₁ = J₂ → ∀ (P : Cᵒᵖ ⥤ Type (max v u)), Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₂ P
⊢ J₁ = J₂ ↔ ∀ (P : Cᵒᵖ ⥤ Type (max v u)), Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₂ P
have mpr : (∀ (P : Cᵒᵖ ⥤ Type (max v u)), Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₂ P) → J₁ = J₂
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Closed (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J₁ J₂ : GrothendieckTopology.{v, u} C
h : ∀ (P : Cᵒᵖ ⥤ Type (max v u)), Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₂ P
⊢ J₁ = J₂
have mpr : J₁ ≤ J₂
Tactic: apply le_antisymm
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Closed (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J₁ J₂ : GrothendieckTopology.{v, u} C
h : ∀ (P : Cᵒᵖ ⥤ Type (max v u)), Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₂ P
mpr : J₁ ≤ J₂
⊢ J₁ = J₂
have mpr₁ : J₂ ≤ J₁
Tactic: apply le_antisymm
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Closed (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J₁ J₂ : GrothendieckTopology.{v, u} C
h : ∀ (P : Cᵒᵖ ⥤ Type (max v u)), Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₂ P
⊢ J₁ ≤ J₂
have mpr : Presieve.IsSheaf J₁ (Functor.closedSieves J₂).toFunctor
Tactic: apply le_topology_of_closedSieves_isSheaf
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Closed (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J₁ J₂ : GrothendieckTopology.{v, u} C
h : ∀ (P : Cᵒᵖ ⥤ Type (max v u)), Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₂ P
⊢ Presieve.IsSheaf J₁ (Functor.closedSieves J₂).toFunctor
have mpr : Presieve.IsSheaf J₂ (Functor.closedSieves J₂).toFunctor
Tactic: rw [h]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Closed (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J₁ J₂ : GrothendieckTopology.{v, u} C
h : ∀ (P : Cᵒᵖ ⥤ Type (max v u)), Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₂ P
⊢ J₂ ≤ J₁
have mpr : Presieve.IsSheaf J₂ (Functor.closedSieves J₁).toFunctor
Tactic: apply le_topology_of_closedSieves_isSheaf
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Closed (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J₁ J₂ : GrothendieckTopology.{v, u} C
h : ∀ (P : Cᵒᵖ ⥤ Type (max v u)), Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₂ P
⊢ Presieve.IsSheaf J₂ (Functor.closedSieves J₁).toFunctor
have mpr : Presieve.IsSheaf J₁ (Functor.closedSieves J₁).toFunctor
Tactic: rw [← h]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Closed (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
J₁ J₂ : GrothendieckTopology.{v, u} C
c : (X : C) → ClosureOperator (Sieve.{v, u} X)
hc :
  ∀ ⦃X Y : C⦄ (f : Y ⟶ X) (S : Sieve.{v, u} X),
    (c Y : (a : Sieve Y) → Sieve Y) (Sieve.pullback f S) = Sieve.pullback f ((c X : (a : Sieve X) → Sieve X) S)
X Y : C
S : Sieve.{v, u} X
f : Y ⟶ X
hS :
  Membership.mem (γ := Set (Sieve.{v, u} X))
    {S : Sieve.{v, u} X |
      Eq (α := Sieve.{v, u} X) ((c X : (a : Sieve X) → Sieve X) S : Sieve.{v, u} X) (⊤ : Sieve.{v, u} X)}
    S
⊢ Membership.mem (γ := Set (Sieve.{v, u} Y))
    {S : Sieve.{v, u} Y |
      Eq (α := Sieve.{v, u} Y) ((c Y : (a : Sieve Y) → Sieve Y) S : Sieve.{v, u} Y) (⊤ : Sieve.{v, u} Y)}
    (Sieve.pullback f S)
have hS₁ : Eq (α := Sieve.{v, u} X) ((c X : (a : Sieve X) → Sieve X) S : Sieve.{v, u} X) (⊤ : Sieve.{v, u} X)
Tactic: rw [Set.mem_setOf_eq] at hS
Cleared hypotheses: hS
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Closed (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J₁ J₂ : GrothendieckTopology.{v, u} C
c : (X : C) → ClosureOperator (Sieve.{v, u} X)
hc :
  ∀ ⦃X Y : C⦄ (f : Y ⟶ X) (S : Sieve.{v, u} X),
    (c Y : (a : Sieve Y) → Sieve Y) (Sieve.pullback f S) = Sieve.pullback f ((c X : (a : Sieve X) → Sieve X) S)
X Y : C
S : Sieve.{v, u} X
f : Y ⟶ X
hS : Eq (α := Sieve.{v, u} X) ((c X : (a : Sieve X) → Sieve X) S : Sieve.{v, u} X) (⊤ : Sieve.{v, u} X)
⊢ Membership.mem (γ := Set (Sieve.{v, u} Y))
    {S : Sieve.{v, u} Y |
      Eq (α := Sieve.{v, u} Y) ((c Y : (a : Sieve Y) → Sieve Y) S : Sieve.{v, u} Y) (⊤ : Sieve.{v, u} Y)}
    (Sieve.pullback f S)
have rw : Eq (α := Sieve.{v, u} Y) ((c Y : (a : Sieve Y) → Sieve Y) (Sieve.pullback f S) : Sieve.{v, u} Y) (⊤ : Sieve.{v, u} Y)
Tactic: rw [Set.mem_setOf_eq, hc, hS, Sieve.pullback_top]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Closed (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J₁ J₂ : GrothendieckTopology.{v, u} C
c : (X : C) → ClosureOperator (Sieve.{v, u} X)
hc :
  ∀ ⦃X Y : C⦄ (f : Y ⟶ X) (S : Sieve.{v, u} X),
    (c Y : (a : Sieve Y) → Sieve Y) (Sieve.pullback f S) = Sieve.pullback f ((c X : (a : Sieve X) → Sieve X) S)
X Y : C
S : Sieve.{v, u} X
f : Y ⟶ X
hS : Eq (α := Sieve.{v, u} X) ((c X : (a : Sieve X) → Sieve X) S : Sieve.{v, u} X) (⊤ : Sieve.{v, u} X)
rw :
  Eq (α := Sieve.{v, u} Y) ((c Y : (a : Sieve Y) → Sieve Y) (Sieve.pullback f S) : Sieve.{v, u} Y) (⊤ : Sieve.{v, u} Y)
⊢ Membership.mem (γ := Set (Sieve.{v, u} Y))
    {S : Sieve.{v, u} Y |
      Eq (α := Sieve.{v, u} Y) ((c Y : (a : Sieve Y) → Sieve Y) S : Sieve.{v, u} Y) (⊤ : Sieve.{v, u} Y)}
    (Sieve.pullback f S)
have rw₁ : Sieve.pullback f ((c X : (a : Sieve X) → Sieve X) S) = ⊤
Tactic: rw [Set.mem_setOf_eq, hc, hS, Sieve.pullback_top]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Closed (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J₁ J₂ : GrothendieckTopology.{v, u} C
c : (X : C) → ClosureOperator (Sieve.{v, u} X)
hc :
  ∀ ⦃X Y : C⦄ (f : Y ⟶ X) (S : Sieve.{v, u} X),
    (c Y : (a : Sieve Y) → Sieve Y) (Sieve.pullback f S) = Sieve.pullback f ((c X : (a : Sieve X) → Sieve X) S)
X Y : C
S : Sieve.{v, u} X
f : Y ⟶ X
hS : Eq (α := Sieve.{v, u} X) ((c X : (a : Sieve X) → Sieve X) S : Sieve.{v, u} X) (⊤ : Sieve.{v, u} X)
rw :
  Eq (α := Sieve.{v, u} Y) ((c Y : (a : Sieve Y) → Sieve Y) (Sieve.pullback f S) : Sieve.{v, u} Y) (⊤ : Sieve.{v, u} Y)
rw₁ : Sieve.pullback f ((c X : (a : Sieve X) → Sieve X) S) = ⊤
⊢ Membership.mem (γ := Set (Sieve.{v, u} Y))
    {S : Sieve.{v, u} Y |
      Eq (α := Sieve.{v, u} Y) ((c Y : (a : Sieve Y) → Sieve Y) S : Sieve.{v, u} Y) (⊤ : Sieve.{v, u} Y)}
    (Sieve.pullback f S)
have rw₂ : Sieve.pullback f ⊤ = ⊤
Tactic: rw [Set.mem_setOf_eq, hc, hS, Sieve.pullback_top]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Closed (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J₁ J₂ : GrothendieckTopology.{v, u} C
c : (X : C) → ClosureOperator (Sieve.{v, u} X)
hc :
  ∀ ⦃X Y : C⦄ (f : Y ⟶ X) (S : Sieve.{v, u} X),
    (c Y : (a : Sieve Y) → Sieve Y) (Sieve.pullback f S) = Sieve.pullback f ((c X : (a : Sieve X) → Sieve X) S)
X Y : C
S : Sieve.{v, u} X
f : Y ⟶ X
hS : Eq (α := Sieve.{v, u} X) ((c X : (a : Sieve X) → Sieve X) S : Sieve.{v, u} X) (⊤ : Sieve.{v, u} X)
rw :
  Eq (α := Sieve.{v, u} Y) ((c Y : (a : Sieve Y) → Sieve Y) (Sieve.pullback f S) : Sieve.{v, u} Y) (⊤ : Sieve.{v, u} Y)
rw₁ : Sieve.pullback f ((c X : (a : Sieve X) → Sieve X) S) = ⊤
rw₂ : Sieve.pullback f ⊤ = ⊤
⊢ Membership.mem (γ := Set (Sieve.{v, u} Y))
    {S : Sieve.{v, u} Y |
      Eq (α := Sieve.{v, u} Y) ((c Y : (a : Sieve Y) → Sieve Y) S : Sieve.{v, u} Y) (⊤ : Sieve.{v, u} Y)}
    (Sieve.pullback f S)
have rw₃ : Eq (α := Sieve.{v, u} Y) ⊤ ⊤
Tactic: rw [Set.mem_setOf_eq, hc, hS, Sieve.pullback_top]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Closed (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J₁ J₂ : GrothendieckTopology.{v, u} C
c : (X : C) → ClosureOperator (Sieve.{v, u} X)
hc :
  ∀ ⦃X Y : C⦄ (f : Y ⟶ X) (S : Sieve.{v, u} X),
    (c Y : (a : Sieve Y) → Sieve Y) (Sieve.pullback f S) = Sieve.pullback f ((c X : (a : Sieve X) → Sieve X) S)
X : C
S : Sieve.{v, u} X
hS :
  Membership.mem (γ := Set (Sieve.{v, u} X))
    {S : Sieve.{v, u} X |
      Eq (α := Sieve.{v, u} X) ((c X : (a : Sieve X) → Sieve X) S : Sieve.{v, u} X) (⊤ : Sieve.{v, u} X)}
    S
R : Sieve.{v, u} X
hR :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    S.arrows f →
      Membership.mem (γ := Set (Sieve.{v, u} Y))
        {S : Sieve.{v, u} Y |
          Eq (α := Sieve.{v, u} Y) ((c Y : (a : Sieve Y) → Sieve Y) S : Sieve.{v, u} Y) (⊤ : Sieve.{v, u} Y)}
        (Sieve.pullback f R)
⊢ Membership.mem (γ := Set (Sieve.{v, u} X))
    {S : Sieve.{v, u} X |
      Eq (α := Sieve.{v, u} X) ((c X : (a : Sieve X) → Sieve X) S : Sieve.{v, u} X) (⊤ : Sieve.{v, u} X)}
    R
have rw : ∀ (hS : Eq (α := Sieve.{v, u} X) ((c X : (a : Sieve X) → Sieve X) S : Sieve.{v, u} X) (⊤ : Sieve.{v, u} X))
  (R : Sieve.{v, u} X)
  (hR :
    ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
      S.arrows f →
        Membership.mem (γ := Set (Sieve.{v, u} Y))
          {S : Sieve.{v, u} Y |
            Eq (α := Sieve.{v, u} Y) ((c Y : (a : Sieve Y) → Sieve Y) S : Sieve.{v, u} Y) (⊤ : Sieve.{v, u} Y)}
          (Sieve.pullback f R)),
  Membership.mem (γ := Set (Sieve.{v, u} X))
    {S : Sieve.{v, u} X |
      Eq (α := Sieve.{v, u} X) ((c X : (a : Sieve X) → Sieve X) S : Sieve.{v, u} X) (⊤ : Sieve.{v, u} X)}
    R
Tactic: rw [Set.mem_setOf_eq] at hS
Cleared hypotheses: hS, R, hR
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Closed (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J₁ J₂ : GrothendieckTopology.{v, u} C
c : (X : C) → ClosureOperator (Sieve.{v, u} X)
hc :
  ∀ ⦃X Y : C⦄ (f : Y ⟶ X) (S : Sieve.{v, u} X),
    (c Y : (a : Sieve Y) → Sieve Y) (Sieve.pullback f S) = Sieve.pullback f ((c X : (a : Sieve X) → Sieve X) S)
X : C
S : Sieve.{v, u} X
hS : Eq (α := Sieve.{v, u} X) ((c X : (a : Sieve X) → Sieve X) S : Sieve.{v, u} X) (⊤ : Sieve.{v, u} X)
R : Sieve.{v, u} X
hR :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    S.arrows f →
      Membership.mem (γ := Set (Sieve.{v, u} Y))
        {S : Sieve.{v, u} Y |
          Eq (α := Sieve.{v, u} Y) ((c Y : (a : Sieve Y) → Sieve Y) S : Sieve.{v, u} Y) (⊤ : Sieve.{v, u} Y)}
        (Sieve.pullback f R)
⊢ Membership.mem (γ := Set (Sieve.{v, u} X))
    {S : Sieve.{v, u} X |
      Eq (α := Sieve.{v, u} X) ((c X : (a : Sieve X) → Sieve X) S : Sieve.{v, u} X) (⊤ : Sieve.{v, u} X)}
    R
have rw : LE.le (α := Sieve.{v, u} X) ((c X : (a : Sieve X) → Sieve X) S : Sieve.{v, u} X)
  ((c X : (a : Sieve X) → Sieve X) ((c X : (a : Sieve X) → Sieve X) R) : Sieve.{v, u} X)
Tactic: rw [Set.mem_setOf_eq, ← (c X).idempotent, eq_top_iff, ← hS]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Closed (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J₁ J₂ : GrothendieckTopology.{v, u} C
c : (X : C) → ClosureOperator (Sieve.{v, u} X)
hc :
  ∀ ⦃X Y : C⦄ (f : Y ⟶ X) (S : Sieve.{v, u} X),
    (c Y : (a : Sieve Y) → Sieve Y) (Sieve.pullback f S) = Sieve.pullback f ((c X : (a : Sieve X) → Sieve X) S)
X : C
S : Sieve.{v, u} X
hS : Eq (α := Sieve.{v, u} X) ((c X : (a : Sieve X) → Sieve X) S : Sieve.{v, u} X) (⊤ : Sieve.{v, u} X)
R : Sieve.{v, u} X
hR :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    S.arrows f →
      Membership.mem (γ := Set (Sieve.{v, u} Y))
        {S : Sieve.{v, u} Y |
          Eq (α := Sieve.{v, u} Y) ((c Y : (a : Sieve Y) → Sieve Y) S : Sieve.{v, u} Y) (⊤ : Sieve.{v, u} Y)}
        (Sieve.pullback f R)
⊢ LE.le (α := Sieve.{v, u} X) ((c X : (a : Sieve X) → Sieve X) S : Sieve.{v, u} X)
    ((c X : (a : Sieve X) → Sieve X) ((c X : (a : Sieve X) → Sieve X) R) : Sieve.{v, u} X)
have apply : ∀ (Y : C) (f : Y ⟶ X) (hf : S.arrows f), Sieve.arrows ((c X : (a : Sieve X) → Sieve X) R) f
Tactic: apply (c X).monotone fun Y f hf => _
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Closed (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J₁ J₂ : GrothendieckTopology.{v, u} C
c : (X : C) → ClosureOperator (Sieve.{v, u} X)
hc :
  ∀ ⦃X Y : C⦄ (f : Y ⟶ X) (S : Sieve.{v, u} X),
    (c Y : (a : Sieve Y) → Sieve Y) (Sieve.pullback f S) = Sieve.pullback f ((c X : (a : Sieve X) → Sieve X) S)
X : C
S : Sieve.{v, u} X
hS : Eq (α := Sieve.{v, u} X) ((c X : (a : Sieve X) → Sieve X) S : Sieve.{v, u} X) (⊤ : Sieve.{v, u} X)
R : Sieve.{v, u} X
hR :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    S.arrows f →
      Membership.mem (γ := Set (Sieve.{v, u} Y))
        {S : Sieve.{v, u} Y |
          Eq (α := Sieve.{v, u} Y) ((c Y : (a : Sieve Y) → Sieve Y) S : Sieve.{v, u} Y) (⊤ : Sieve.{v, u} Y)}
        (Sieve.pullback f R)
Y : C
f : Y ⟶ X
hf : S.arrows f
⊢ Sieve.arrows ((c X : (a : Sieve X) → Sieve X) R) f
have rw : Eq (α := Sieve.{v, u} Y) ((c Y : (a : Sieve Y) → Sieve Y) (Sieve.pullback f R) : Sieve.{v, u} Y) (⊤ : Sieve.{v, u} Y)
Tactic: rw [Sieve.mem_iff_pullback_eq_top, ← hc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Closed (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J₁ : GrothendieckTopology.{v, u} C
⊢ topologyOfClosureOperator J₁.closureOperator ⋯ = J₁
have h : ∀ (X : C) (S : Sieve.{v, u} X),
  S ∈ (topologyOfClosureOperator J₁.closureOperator sorry : (a : C) → Set (Sieve a)) X ↔
    S ∈ (J₁ : (a : C) → Set (Sieve a)) X
Tactic: ext X S
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Closed (goal)
Goal:
C : Type u
inst : Category.{v, u} C
c : (X : C) → ClosureOperator (Sieve.{v, u} X)
pb :
  ∀ ⦃X Y : C⦄ (f : Y ⟶ X) (S : Sieve.{v, u} X),
    (c Y : (a : Sieve Y) → Sieve Y) (Sieve.pullback f S) = Sieve.pullback f ((c X : (a : Sieve X) → Sieve X) S)
X : C
S : Sieve.{v, u} X
⊢ (topologyOfClosureOperator c pb).close S = (c X : (a : Sieve X) → Sieve X) S
have h : ∀ ⦃Y : C⦄ (f : Y ⟶ X),
  ((topologyOfClosureOperator c pb).close S).arrows f ↔ Sieve.arrows ((c X : (a : Sieve X) → Sieve X) S) f
Tactic: ext Y f
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Closed (goal)
Goal:
C : Type u
inst : Category.{v, u} C
c : (X : C) → ClosureOperator (Sieve.{v, u} X)
pb :
  ∀ ⦃X Y : C⦄ (f : Y ⟶ X) (S : Sieve.{v, u} X),
    (c Y : (a : Sieve Y) → Sieve Y) (Sieve.pullback f S) = Sieve.pullback f ((c X : (a : Sieve X) → Sieve X) S)
X : C
S : Sieve.{v, u} X
Y : C
f : Y ⟶ X
⊢ Eq (α := Sieve.{v, u} Y) ((c Y : (a : Sieve Y) → Sieve Y) (Sieve.pullback f S) : Sieve.{v, u} Y)
      (⊤ : Sieve.{v, u} Y) ↔
    Sieve.arrows ((c X : (a : Sieve X) → Sieve X) S) f
have h : Sieve.pullback f ((c X : (a : Sieve X) → Sieve X) S) = ⊤ ↔ Sieve.arrows ((c X : (a : Sieve X) → Sieve X) S) f
Tactic: rw [pb, Sieve.mem_iff_pullback_eq_top]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Closed (goal)
Goal:
C : Type u
inst : Category.{v, u} C
c : (X : C) → ClosureOperator (Sieve.{v, u} X)
pb :
  ∀ ⦃X Y : C⦄ (f : Y ⟶ X) (S : Sieve.{v, u} X),
    (c Y : (a : Sieve Y) → Sieve Y) (Sieve.pullback f S) = Sieve.pullback f ((c X : (a : Sieve X) → Sieve X) S)
X : C
S : Sieve.{v, u} X
Y : C
f : Y ⟶ X
h : Sieve.pullback f ((c X : (a : Sieve X) → Sieve X) S) = ⊤ ↔ Sieve.arrows ((c X : (a : Sieve X) → Sieve X) S) f
⊢ Eq (α := Sieve.{v, u} Y) ((c Y : (a : Sieve Y) → Sieve Y) (Sieve.pullback f S) : Sieve.{v, u} Y)
      (⊤ : Sieve.{v, u} Y) ↔
    Sieve.arrows ((c X : (a : Sieve X) → Sieve X) S) f
have h₁ : Sieve.pullback f ((c X : (a : Sieve X) → Sieve X) S) = ⊤ ↔ Sieve.pullback f ((c X : (a : Sieve X) → Sieve X) S) = ⊤
Tactic: rw [pb, Sieve.mem_iff_pullback_eq_top]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.Basic (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Precoherent.{v_1, u_1} C
B₁ B₂ : C
f : B₂ ⟶ B₁
α : Type
w : Finite α
X₁ : α → C
π₁ : (a : α) → X₁ a ⟶ B₁
hS : EffectiveEpiFamily X₁ π₁
⊢ ∃ (T : Presieve.{v_1, u_1} B₂),
    Membership.mem (γ := Set (Presieve.{v_1, u_1} B₂))
        {S : Presieve.{v_1, u_1} B₂ |
          ∃ (α : Type) (_ : Finite α) (X : α → C) (π : (a : α) → X a ⟶ B₂),
            S = Presieve.ofArrows X π ∧ EffectiveEpiFamily X π}
        T ∧
      T.FactorsThruAlong (Presieve.ofArrows X₁ π₁) f
have obtain : ∀ (β : Type),
  Finite β →
    ∀ (X₂ : β → C) (π₂ : (b : β) → X₂ b ⟶ B₂) (h : EffectiveEpiFamily X₂ π₂) (i : β → α) (ι : (b : β) → X₂ b ⟶ X₁ (i b))
      (hh : ∀ (b : β), ι b ≫ π₁ (i b) = π₂ b ≫ f),
      ∃ (T : Presieve.{v_1, u_1} B₂),
        Membership.mem (γ := Set (Presieve.{v_1, u_1} B₂))
            {S : Presieve.{v_1, u_1} B₂ |
              ∃ (α : Type) (_ : Finite α) (X : α → C) (π : (a : α) → X a ⟶ B₂),
                S = Presieve.ofArrows X π ∧ EffectiveEpiFamily X π}
            T ∧
          T.FactorsThruAlong (Presieve.ofArrows X₁ π₁) f
Tactic: obtain ⟨β, _, X₂, π₂, h, i, ι, hh⟩ := Precoherent.pullback f α X₁ π₁ hS