awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Sites.CartesianMonoidal
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
A : Type u₂
inst_1 : Category.{v₂, u₂} A
J : GrothendieckTopology.{v₁, u₁} C
inst_2 : CartesianMonoidalCategory.{v₂, u₂} A
X W : Sheaf.{v₁, v₂, u₁, u₂} J A
f : W ⟶ X
⊢ Eq (α :=
    (tensorObj (𝒞 := ObjectProperty.FullSubcategory.category (Presheaf.IsSheaf J)) W X).obj (P := Presheaf.IsSheaf J) ⟶
      (tensorObj (𝒞 := ObjectProperty.FullSubcategory.category (Presheaf.IsSheaf J)) X X).obj (P := Presheaf.IsSheaf J))
    (InducedCategory.Hom.hom (f ▷ X)) (f.hom ▷ X.obj (P := Presheaf.IsSheaf J))

Tactic:
rfl

Premises used: rfl
-- Module: Mathlib.CategoryTheory.Sites.CartesianMonoidal
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
A : Type u₂
inst_1 : Category.{v₂, u₂} A
J : GrothendieckTopology.{v₁, u₁} C
inst_2 : CartesianMonoidalCategory.{v₂, u₂} A
⊢ Eq (α :=
    Quiver.Hom (V := Functor.{v₁, v₂, u₁, u₂} Cᵒᵖ A) ((𝟙_ (Functor.{v₁, v₂, u₁, u₂} Cᵒᵖ A)) (𝒞 := Functor.category))
      (Functor.obj (sheafToPresheaf J A)
        ((𝟙_ (Sheaf.{v₁, v₂, u₁, u₂} J A)) (𝒞 := ObjectProperty.FullSubcategory.category (Presheaf.IsSheaf J)))))
    (ε (sheafToPresheaf J A)) (𝟙 ((𝟙_ (Functor.{v₁, v₂, u₁, u₂} Cᵒᵖ A)) (𝒞 := Functor.category)))

Tactic:
rfl

Premises used: CategoryTheory.NatTrans.id, rfl
-- Module: Mathlib.CategoryTheory.Sites.CartesianMonoidal
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
A : Type u₂
inst_1 : Category.{v₂, u₂} A
J : GrothendieckTopology.{v₁, u₁} C
inst_2 : CartesianMonoidalCategory.{v₂, u₂} A
⊢ Eq (α :=
    Quiver.Hom (V := Functor.{v₁, v₂, u₁, u₂} Cᵒᵖ A)
      (Functor.obj (sheafToPresheaf J A)
        ((𝟙_ (Sheaf.{v₁, v₂, u₁, u₂} J A)) (𝒞 := ObjectProperty.FullSubcategory.category (Presheaf.IsSheaf J))))
      ((𝟙_ (Functor.{v₁, v₂, u₁, u₂} Cᵒᵖ A)) (𝒞 := Functor.category)))
    (η (sheafToPresheaf J A))
    (𝟙
      (Functor.obj (sheafToPresheaf J A)
        ((𝟙_ (Sheaf.{v₁, v₂, u₁, u₂} J A)) (𝒞 := ObjectProperty.FullSubcategory.category (Presheaf.IsSheaf J)))))

Tactic:
rfl

Premises used: CategoryTheory.NatTrans.id, rfl
-- Module: Mathlib.CategoryTheory.Sites.CartesianMonoidal
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
A : Type u₂
inst_1 : Category.{v₂, u₂} A
J : GrothendieckTopology.{v₁, u₁} C
inst_2 : CartesianMonoidalCategory.{v₂, u₂} A
X Y : Sheaf.{v₁, v₂, u₁, u₂} J A
⊢ Eq (α :=
    tensorObj (𝒞 := Functor.category) (Functor.obj (sheafToPresheaf J A) X) (Functor.obj (sheafToPresheaf J A) Y) ⟶
      Functor.obj (sheafToPresheaf J A)
        (tensorObj (𝒞 := ObjectProperty.FullSubcategory.category (Presheaf.IsSheaf J)) X Y))
    (μ (sheafToPresheaf J A) X Y)
    (𝟙 (tensorObj (𝒞 := Functor.category) (Functor.obj (sheafToPresheaf J A) X) (Functor.obj (sheafToPresheaf J A) Y)))

Tactic:
rfl

Premises used: CategoryTheory.NatTrans.id, rfl
-- Module: Mathlib.CategoryTheory.Sites.CartesianMonoidal
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
A : Type u₂
inst_1 : Category.{v₂, u₂} A
J : GrothendieckTopology.{v₁, u₁} C
inst_2 : CartesianMonoidalCategory.{v₂, u₂} A
X Y : Sheaf.{v₁, v₂, u₁, u₂} J A
⊢ δ (sheafToPresheaf J A) X Y =
    𝟙
      (Functor.obj (sheafToPresheaf J A)
        (tensorObj (𝒞 := ObjectProperty.FullSubcategory.category (Presheaf.IsSheaf J)) X Y))

Tactic:
rfl

Premises used: CategoryTheory.NatTrans.id, rfl
-- Module: Mathlib.CategoryTheory.Sites.Closed
Goal:
C : Type u
inst : Category.{v, u} C
J₁ : GrothendieckTopology.{v, u} C
X : C
S : Sieve.{v, u} X
⊢ S ≤ J₁.close S

Tactic:
fun _ _ hg => J₁.covering_of_eq_top (S.pullback_eq_top_of_mem hg)

Premises used: CategoryTheory.GrothendieckTopology.Covers, CategoryTheory.Sieve.pullback_eq_top_of_mem, CategoryTheory.GrothendieckTopology.covering_of_eq_top
-- Module: Mathlib.CategoryTheory.Sites.Closed
Goal:
C : Type u
inst : Category.{v, u} C
J₁ : GrothendieckTopology.{v, u} C
X : C
S : Sieve.{v, u} X
h : J₁.IsClosed S
Y : C
f : Y ⟶ X
⊢ J₁.Covers S f ↔ S.arrows f

Tactic:
⟨h _, J₁.arrow_max _ _⟩

Premises used: CategoryTheory.GrothendieckTopology.arrow_max, Iff.intro
-- Module: Mathlib.CategoryTheory.Sites.Closed
Goal:
C : Type u
inst : Category.{v, u} C
J₁ : GrothendieckTopology.{v, u} C
X Y : C
f : Y ⟶ X
S : Sieve.{v, u} X
hS : J₁.IsClosed S
Z : C
g : Z ⟶ Y
hg : J₁.Covers (Sieve.pullback f S) g
rwa : Sieve.pullback (g ≫ f) S ∈ (J₁ : (a : C) → Set (Sieve a)) Z
rwa₁ rwa₂ rwa₃ rwa₄ : Sieve.pullback g (Sieve.pullback f S) ∈ (J₁ : (a : C) → Set (Sieve a)) Z
⊢ J₁.Covers S (g ≫ f)

Tactic:
rwa [J₁.covers_iff, Sieve.pullback_comp]

Premises used: CategoryTheory.Sieve.pullback_comp, CategoryTheory.GrothendieckTopology.covers_iff
-- Module: Mathlib.CategoryTheory.Sites.Closed
Goal:
C : Type u
inst : Category.{v, u} C
J₁ : GrothendieckTopology.{v, u} C
X : C
S T : Sieve.{v, u} X
h : S ≤ T
hT : J₁.IsClosed T
⊢ J₁.close S ≤ T

Tactic:
fun _ f hf => hT _ (J₁.superset_covering (Sieve.pullback_monotone f h) hf)

Premises used: CategoryTheory.GrothendieckTopology.superset_covering, CategoryTheory.Sieve.pullback_monotone
-- Module: Mathlib.CategoryTheory.Sites.Closed
Goal:
C : Type u
inst : Category.{v, u} C
J₁ : GrothendieckTopology.{v, u} C
X : C
S : Sieve.{v, u} X
⊢ J₁.IsClosed (J₁.close S)

Tactic:
fun _ g hg => J₁.arrow_trans g _ S hg fun _ hS => hS

Premises used: CategoryTheory.GrothendieckTopology.Covers, CategoryTheory.GrothendieckTopology.IsClosed, CategoryTheory.GrothendieckTopology.close, CategoryTheory.GrothendieckTopology.arrow_trans
-- Module: Mathlib.CategoryTheory.Sites.Closed
Goal:
C : Type u
inst : Category.{v, u} C
J₁ : GrothendieckTopology.{v, u} C
X : C
S : Sieve.{v, u} X
⊢ J₁.IsClosed S ↔ J₁.close S = S

Tactic:
(J₁.closureOperator _).isClosed_iff

Premises used: CategoryTheory.GrothendieckTopology.IsClosed, ClosureOperator.isClosed_iff, CategoryTheory.GrothendieckTopology.close, CategoryTheory.GrothendieckTopology.closureOperator
-- Module: Mathlib.CategoryTheory.Sites.Closed
Goal:
C : Type u
inst : Category.{v, u} C
J₁ : GrothendieckTopology.{v, u} C
X : C
S : Sieve.{v, u} X
hS : J₁.IsClosed S
⊢ J₁.close S = S

Tactic:
(J₁.isClosed_iff_close_eq_self S).1 hS

Premises used: Iff.mp, CategoryTheory.GrothendieckTopology.isClosed_iff_close_eq_self
-- Module: Mathlib.CategoryTheory.Sites.Closed
Goal:
C : Type u
inst : Category.{v, u} C
J₁ : GrothendieckTopology.{v, u} C
X Y : C
f : Y ⟶ X
S : Sieve.{v, u} X
a : J₁.close (Sieve.pullback f S) ≤ Sieve.pullback f (J₁.close S)
a₁ : Sieve.pullback f (J₁.close S) ≤ J₁.close (Sieve.pullback f S)
⊢ J₁.close (Sieve.pullback f S) = Sieve.pullback f (J₁.close S)

Tactic:
apply le_antisymm

Premises used: le_antisymm
-- Module: Mathlib.CategoryTheory.Sites.Closed
Goal:
C : Type u
inst : Category.{v, u} C
J₁ : GrothendieckTopology.{v, u} C
X Y : C
f : Y ⟶ X
S : Sieve.{v, u} X
⊢ J₁.close (Sieve.pullback f S) ≤ Sieve.pullback f (J₁.close S)

Tactic:
J₁.le_close_of_isClosed (Sieve.pullback_monotone _ (J₁.le_close S)) ?_

Premises used: CategoryTheory.Sieve.pullback_monotone, CategoryTheory.GrothendieckTopology.le_close_of_isClosed, CategoryTheory.GrothendieckTopology.le_close
-- Module: Mathlib.CategoryTheory.Sites.Closed
Goal:
C : Type u
inst : Category.{v, u} C
J₁ : GrothendieckTopology.{v, u} C
X Y : C
f : Y ⟶ X
S : Sieve.{v, u} X
a : J₁.IsClosed (Sieve.pullback f (J₁.close S))
⊢ J₁.close (Sieve.pullback f S) ≤ Sieve.pullback f (J₁.close S)

Tactic:
refine J₁.le_close_of_isClosed (Sieve.pullback_monotone _ (J₁.le_close S)) ?_

Premises used: CategoryTheory.Sieve.pullback_monotone, CategoryTheory.GrothendieckTopology.le_close_of_isClosed, CategoryTheory.GrothendieckTopology.le_close
-- Module: Mathlib.CategoryTheory.Sites.Closed
Goal:
C : Type u
inst : Category.{v, u} C
J₁ : GrothendieckTopology.{v, u} C
X Y : C
f : Y ⟶ X
S : Sieve.{v, u} X
⊢ J₁.IsClosed (Sieve.pullback f (J₁.close S))

Tactic:
apply J₁.isClosed_pullback _ _ (J₁.close_isClosed _)

Premises used: CategoryTheory.GrothendieckTopology.isClosed_pullback, CategoryTheory.GrothendieckTopology.close_isClosed
-- Module: Mathlib.CategoryTheory.Sites.Closed
Goal:
C : Type u
inst : Category.{v, u} C
J₁ : GrothendieckTopology.{v, u} C
X Y : C
f : Y ⟶ X
S : Sieve.{v, u} X
Z : C
g : Z ⟶ Y
hg : (Sieve.pullback f (J₁.close S)).arrows g
a : Sieve.pullback (g ≫ f) S ∈ (J₁ : (a : C) → Set (Sieve a)) Z
⊢ Sieve.pullback g (Sieve.pullback f S) ∈ (J₁ : (a : C) → Set (Sieve a)) Z

Tactic:
rw [← Sieve.pullback_comp]

Premises used: CategoryTheory.Sieve.pullback_comp
-- Module: Mathlib.CategoryTheory.Sites.Closed
Goal:
C : Type u
inst : Category.{v, u} C
J₁ : GrothendieckTopology.{v, u} C
X Y : C
f : Y ⟶ X
S : Sieve.{v, u} X
Z : C
g : Z ⟶ Y
hg : (Sieve.pullback f (J₁.close S)).arrows g
⊢ Sieve.pullback (g ≫ f) S ∈ (J₁ : (a : C) → Set (Sieve a)) Z

Tactic:
apply hg
-- Module: Mathlib.CategoryTheory.Sites.Closed
Goal:
C : Type u
inst : Category.{v, u} C
J₁ : GrothendieckTopology.{v, u} C
X : C
⊢ Monotone (α := Sieve.{v, u} X) J₁.close

Tactic:
(J₁.closureOperator _).monotone

Premises used: CategoryTheory.GrothendieckTopology.close, ClosureOperator.monotone, CategoryTheory.GrothendieckTopology.closureOperator
-- Module: Mathlib.CategoryTheory.Sites.Closed
Goal:
C : Type u
inst : Category.{v, u} C
J₁ : GrothendieckTopology.{v, u} C
X : C
S : Sieve.{v, u} X
⊢ J₁.close (J₁.close S) = J₁.close S

Tactic:
(J₁.closureOperator _).idempotent _

Premises used: CategoryTheory.GrothendieckTopology.close, CategoryTheory.GrothendieckTopology.closureOperator, ClosureOperator.idempotent
-- Module: Mathlib.CategoryTheory.Sites.Closed
Goal:
C : Type u
inst : Category.{v, u} C
J₁ : GrothendieckTopology.{v, u} C
X : C
S : Sieve.{v, u} X
mp : J₁.close S = ⊤ → S ∈ (J₁ : (a : C) → Set (Sieve a)) X
mpr : S ∈ (J₁ : (a : C) → Set (Sieve a)) X → J₁.close S = ⊤
⊢ J₁.close S = ⊤ ↔ S ∈ (J₁ : (a : C) → Set (Sieve a)) X

Tactic:
constructor
-- Module: Mathlib.CategoryTheory.Sites.Closed
Goal:
C : Type u
inst : Category.{v, u} C
J₁ : GrothendieckTopology.{v, u} C
X : C
S : Sieve.{v, u} X
h : J₁.close S = ⊤
mp : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, Sieve.arrows ⊤ f → Sieve.pullback f S ∈ (J₁ : (a : C) → Set (Sieve a)) Y
⊢ S ∈ (J₁ : (a : C) → Set (Sieve a)) X

Tactic:
apply J₁.transitive (J₁.top_mem X)

Premises used: CategoryTheory.GrothendieckTopology.transitive, CategoryTheory.GrothendieckTopology.top_mem
-- Module: Mathlib.CategoryTheory.Sites.Closed
Goal:
C : Type u
inst : Category.{v, u} C
J₁ : GrothendieckTopology.{v, u} C
X : C
S : Sieve.{v, u} X
h : J₁.close S = ⊤
Y : C
f : Y ⟶ X
hf mp mp₁ mp₂ mp₃ : Sieve.arrows ⊤ f
⊢ (J₁.close S).arrows f

Tactic:
rwa [h]
-- Module: Mathlib.CategoryTheory.Sites.Closed
Goal:
C : Type u
inst : Category.{v, u} C
J₁ : GrothendieckTopology.{v, u} C
X : C
S : Sieve.{v, u} X
hS : S ∈ (J₁ : (a : C) → Set (Sieve a)) X
mpr : ⊤ ≤ J₁.close S
⊢ J₁.close S = ⊤

Tactic:
rw [_root_.eq_top_iff]

Premises used: eq_top_iff
-- Module: Mathlib.CategoryTheory.Sites.Closed
Goal:
C : Type u
inst : Category.{v, u} C
J₁ : GrothendieckTopology.{v, u} C
X : C
S : Sieve.{v, u} X
hS : S ∈ (J₁ : (a : C) → Set (Sieve a)) X
Y : C
f : Y ⟶ X
mpr : Sieve.arrows ⊤ f → (J₁.close S).arrows f
⊢ Sieve.arrows ⊤ f → (J₁.close S).arrows f

Tactic:
_
-- Module: Mathlib.CategoryTheory.Sites.Closed
Goal:
C : Type u
inst : Category.{v, u} C
J₁ : GrothendieckTopology.{v, u} C
X : C
S : Sieve.{v, u} X
hS : S ∈ (J₁ : (a : C) → Set (Sieve a)) X
Y : C
f : Y ⟶ X
h : Sieve.arrows ⊤ f
⊢ (J₁.close S).arrows f

Tactic:
apply J₁.pullback_stable _ hS

Premises used: CategoryTheory.GrothendieckTopology.pullback_stable
-- Module: Mathlib.CategoryTheory.Sites.Closed
Goal:
C : Type u
inst : Category.{v, u} C
J₁ : GrothendieckTopology.{v, u} C
X : C
S : Sieve.{v, u} X
hS : S ∈ (J₁ : (a : C) → Set (Sieve a)) X
rw :
  Presieve.IsSeparatedFor (Functor.closedSieves J₁).toFunctor S.arrows ∧
    ∀ (x : Presieve.FamilyOfElements (Functor.closedSieves J₁).toFunctor S.arrows),
      x.Compatible (R := S.arrows) →
        ∃ (t : (Functor.closedSieves J₁).toFunctor.obj (Opposite.op X)), x.IsAmalgamation (R := S.arrows) t
⊢ Presieve.IsSheafFor (Functor.closedSieves J₁).toFunctor S.arrows

Tactic:
rw [← Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor]

Premises used: CategoryTheory.Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor
-- Module: Mathlib.CategoryTheory.Sites.Closed
Goal:
C : Type u
inst : Category.{v, u} C
J₁ : GrothendieckTopology.{v, u} C
X : C
S : Sieve.{v, u} X
hS : S ∈ (J₁ : (a : C) → Set (Sieve a)) X
⊢ Presieve.IsSeparatedFor (Functor.closedSieves J₁).toFunctor S.arrows ∧
    ∀ (x : Presieve.FamilyOfElements (Functor.closedSieves J₁).toFunctor S.arrows),
      x.Compatible (R := S.arrows) →
        ∃ (t : (Functor.closedSieves J₁).toFunctor.obj (Opposite.op X)), x.IsAmalgamation (R := S.arrows) t

Tactic:
⟨?_, ?_⟩

Premises used: And.intro
-- Module: Mathlib.CategoryTheory.Sites.Closed
Goal:
C : Type u
inst : Category.{v, u} C
J₁ : GrothendieckTopology.{v, u} C
X : C
S : Sieve.{v, u} X
hS : S ∈ (J₁ : (a : C) → Set (Sieve a)) X
refine_1 : Presieve.IsSeparatedFor (Functor.closedSieves J₁).toFunctor S.arrows
refine_2 :
  ∀ (x : Presieve.FamilyOfElements (Functor.closedSieves J₁).toFunctor S.arrows),
    x.Compatible (R := S.arrows) →
      ∃ (t : (Functor.closedSieves J₁).toFunctor.obj (Opposite.op X)), x.IsAmalgamation (R := S.arrows) t
⊢ Presieve.IsSeparatedFor (Functor.closedSieves J₁).toFunctor S.arrows ∧
    ∀ (x : Presieve.FamilyOfElements (Functor.closedSieves J₁).toFunctor S.arrows),
      x.Compatible (R := S.arrows) →
        ∃ (t : (Functor.closedSieves J₁).toFunctor.obj (Opposite.op X)), x.IsAmalgamation (R := S.arrows) t

Tactic:
refine ⟨?_, ?_⟩
-- Module: Mathlib.CategoryTheory.Sites.Closed
Goal:
C : Type u
inst : Category.{v, u} C
J₁ : GrothendieckTopology.{v, u} C
X : C
S : Sieve.{v, u} X
hS : S ∈ (J₁ : (a : C) → Set (Sieve a)) X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁).toFunctor S.arrows
M : Functor.obj (Functor.sieves C) (Opposite.op X)
hM :
  Membership.mem (γ := Set (Functor.obj (Functor.sieves C) (Opposite.op X)))
    ((Functor.closedSieves J₁).obj (Opposite.op X)) M
N : Functor.obj (Functor.sieves C) (Opposite.op X)
hN :
  Membership.mem (γ := Set (Functor.obj (Functor.sieves C) (Opposite.op X)))
    ((Functor.closedSieves J₁).obj (Opposite.op X)) N
hM₂ :
  x.IsAmalgamation (R := S.arrows)
    (Subtype.mk (p := fun (x : Functor.obj (Functor.sieves C) (Opposite.op X)) ↦
      Membership.mem (γ := Set (Functor.obj (Functor.sieves C) (Opposite.op X)))
        ((Functor.closedSieves J₁).obj (Opposite.op X)) x)
      M hM)
hN₂ :
  x.IsAmalgamation (R := S.arrows)
    (Subtype.mk (p := fun (x : Functor.obj (Functor.sieves C) (Opposite.op X)) ↦
      Membership.mem (γ := Set (Functor.obj (Functor.sieves C) (Opposite.op X)))
        ((Functor.closedSieves J₁).obj (Opposite.op X)) x)
      N hN)
refine_1 :
  Subtype.mk (p := fun (x : Sieve.{v, u} X) ↦ J₁.IsClosed x) M hM =
    Subtype.mk (p := fun (x : Sieve.{v, u} X) ↦ J₁.IsClosed x) N hN
⊢ Subtype.mk (p := fun (x : Functor.obj (Functor.sieves C) (Opposite.op X)) ↦
      Membership.mem (γ := Set (Functor.obj (Functor.sieves C) (Opposite.op X)))
        ((Functor.closedSieves J₁).obj (Opposite.op X)) x)
      M hM =
    Subtype.mk (p := fun (x : Functor.obj (Functor.sieves C) (Opposite.op X)) ↦
      Membership.mem (γ := Set (Functor.obj (Functor.sieves C) (Opposite.op X)))
        ((Functor.closedSieves J₁).obj (Opposite.op X)) x)
      N hN

Tactic:
dsimp at S M N ⊢
-- Module: Mathlib.CategoryTheory.Sites.Closed
Goal:
C : Type u
inst : Category.{v, u} C
J₁ : GrothendieckTopology.{v, u} C
X : C
S : Sieve.{v, u} X
hS : S ∈ (J₁ : (a : C) → Set (Sieve a)) X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁).toFunctor S.arrows
M : Sieve.{v, u} X
hM :
  Membership.mem (γ := Set (Functor.obj (Functor.sieves C) (Opposite.op X)))
    ((Functor.closedSieves J₁).obj (Opposite.op X)) M
N : Sieve.{v, u} X
hN :
  Membership.mem (γ := Set (Functor.obj (Functor.sieves C) (Opposite.op X)))
    ((Functor.closedSieves J₁).obj (Opposite.op X)) N
hM₂ :
  x.IsAmalgamation (R := S.arrows)
    (Subtype.mk (p := fun (x : Functor.obj (Functor.sieves C) (Opposite.op X)) ↦
      Membership.mem (γ := Set (Functor.obj (Functor.sieves C) (Opposite.op X)))
        ((Functor.closedSieves J₁).obj (Opposite.op X)) x)
      M hM)
hN₂ :
  x.IsAmalgamation (R := S.arrows)
    (Subtype.mk (p := fun (x : Functor.obj (Functor.sieves C) (Opposite.op X)) ↦
      Membership.mem (γ := Set (Functor.obj (Functor.sieves C) (Opposite.op X)))
        ((Functor.closedSieves J₁).obj (Opposite.op X)) x)
      N hN)
refine_1 :
  ∀ ⦃Y : C⦄ (f : Y ⟶ X),
    (↑(Subtype.mk (p := fun (x : Sieve.{v, u} X) ↦ J₁.IsClosed x) M hM) : Sieve X).arrows f ↔
      (↑(Subtype.mk (p := fun (x : Sieve.{v, u} X) ↦ J₁.IsClosed x) N hN) : Sieve X).arrows f
⊢ Subtype.mk (p := fun (x : Sieve.{v, u} X) ↦ J₁.IsClosed x) M hM =
    Subtype.mk (p := fun (x : Sieve.{v, u} X) ↦ J₁.IsClosed x) N hN

Tactic:
ext Y f
-- Module: Mathlib.CategoryTheory.Sites.Closed
Goal:
C : Type u
inst : Category.{v, u} C
J₁ : GrothendieckTopology.{v, u} C
X : C
S : Sieve.{v, u} X
hS : S ∈ (J₁ : (a : C) → Set (Sieve a)) X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁).toFunctor S.arrows
M : Sieve.{v, u} X
hM :
  Membership.mem (γ := Set (Functor.obj (Functor.sieves C) (Opposite.op X)))
    ((Functor.closedSieves J₁).obj (Opposite.op X)) M
N : Sieve.{v, u} X
hN :
  Membership.mem (γ := Set (Functor.obj (Functor.sieves C) (Opposite.op X)))
    ((Functor.closedSieves J₁).obj (Opposite.op X)) N
hM₂ :
  x.IsAmalgamation (R := S.arrows)
    (Subtype.mk (p := fun (x : Functor.obj (Functor.sieves C) (Opposite.op X)) ↦
      Membership.mem (γ := Set (Functor.obj (Functor.sieves C) (Opposite.op X)))
        ((Functor.closedSieves J₁).obj (Opposite.op X)) x)
      M hM)
hN₂ :
  x.IsAmalgamation (R := S.arrows)
    (Subtype.mk (p := fun (x : Functor.obj (Functor.sieves C) (Opposite.op X)) ↦
      Membership.mem (γ := Set (Functor.obj (Functor.sieves C) (Opposite.op X)))
        ((Functor.closedSieves J₁).obj (Opposite.op X)) x)
      N hN)
Y : C
f : Y ⟶ X
refine_1 : M.arrows f ↔ N.arrows f
⊢ (↑(Subtype.mk (p := fun (x : Sieve.{v, u} X) ↦ J₁.IsClosed x) M hM) : Sieve X).arrows f ↔
    (↑(Subtype.mk (p := fun (x : Sieve.{v, u} X) ↦ J₁.IsClosed x) N hN) : Sieve X).arrows f

Tactic:
dsimp only [Subtype.coe_mk]

Premises used: Subtype.coe_mk
-- Module: Mathlib.CategoryTheory.Sites.Closed
Goal:
C : Type u
inst : Category.{v, u} C
J₁ : GrothendieckTopology.{v, u} C
X : C
S : Sieve.{v, u} X
hS : S ∈ (J₁ : (a : C) → Set (Sieve a)) X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁).toFunctor S.arrows
M : Sieve.{v, u} X
hM :
  Membership.mem (γ := Set (Functor.obj (Functor.sieves C) (Opposite.op X)))
    ((Functor.closedSieves J₁).obj (Opposite.op X)) M
N : Sieve.{v, u} X
hN :
  Membership.mem (γ := Set (Functor.obj (Functor.sieves C) (Opposite.op X)))
    ((Functor.closedSieves J₁).obj (Opposite.op X)) N
hM₂ :
  x.IsAmalgamation (R := S.arrows)
    (Subtype.mk (p := fun (x : Functor.obj (Functor.sieves C) (Opposite.op X)) ↦
      Membership.mem (γ := Set (Functor.obj (Functor.sieves C) (Opposite.op X)))
        ((Functor.closedSieves J₁).obj (Opposite.op X)) x)
      M hM)
hN₂ :
  x.IsAmalgamation (R := S.arrows)
    (Subtype.mk (p := fun (x : Functor.obj (Functor.sieves C) (Opposite.op X)) ↦
      Membership.mem (γ := Set (Functor.obj (Functor.sieves C) (Opposite.op X)))
        ((Functor.closedSieves J₁).obj (Opposite.op X)) x)
      N hN)
Y : C
f : Y ⟶ X
refine_1 : J₁.Covers M f ↔ J₁.Covers N f
⊢ M.arrows f ↔ N.arrows f

Tactic:
rw [← J₁.covers_iff_mem_of_isClosed hM, ← J₁.covers_iff_mem_of_isClosed hN]

Premises used: CategoryTheory.GrothendieckTopology.covers_iff_mem_of_isClosed
-- Module: Mathlib.CategoryTheory.Sites.Closed
Goal:
C : Type u
inst : Category.{v, u} C
J₁ : GrothendieckTopology.{v, u} C
X : C
S : Sieve.{v, u} X
hS : S ∈ (J₁ : (a : C) → Set (Sieve a)) X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁).toFunctor S.arrows
M : Sieve.{v, u} X
hM :
  Membership.mem (γ := Set (Functor.obj (Functor.sieves C) (Opposite.op X)))
    ((Functor.closedSieves J₁).obj (Opposite.op X)) M
N : Sieve.{v, u} X
hN :
  Membership.mem (γ := Set (Functor.obj (Functor.sieves C) (Opposite.op X)))
    ((Functor.closedSieves J₁).obj (Opposite.op X)) N
hM₂ :
  x.IsAmalgamation (R := S.arrows)
    (Subtype.mk (p := fun (x : Functor.obj (Functor.sieves C) (Opposite.op X)) ↦
      Membership.mem (γ := Set (Functor.obj (Functor.sieves C) (Opposite.op X)))
        ((Functor.closedSieves J₁).obj (Opposite.op X)) x)
      M hM)
hN₂ :
  x.IsAmalgamation (R := S.arrows)
    (Subtype.mk (p := fun (x : Functor.obj (Functor.sieves C) (Opposite.op X)) ↦
      Membership.mem (γ := Set (Functor.obj (Functor.sieves C) (Opposite.op X)))
        ((Functor.closedSieves J₁).obj (Opposite.op X)) x)
      N hN)
Y : C
f : Y ⟶ X
⊢ ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N

Tactic:
have q : ∀ ⦃Z : C⦄ (g : Z ⟶ X) (_ : S g), M.pullback g = N.pullback g :=
      fun Z g hg => congr_arg Subtype.val ((hM₂ g hg).trans (hN₂ g hg).symm)
-- Module: Mathlib.CategoryTheory.Sites.Closed
Goal:
C : Type u
inst : Category.{v, u} C
J₁ : GrothendieckTopology.{v, u} C
X : C
S : Sieve.{v, u} X
hS : S ∈ (J₁ : (a : C) → Set (Sieve a)) X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁).toFunctor S.arrows
M : Sieve.{v, u} X
hM :
  Membership.mem (γ := Set (Functor.obj (Functor.sieves C) (Opposite.op X)))
    ((Functor.closedSieves J₁).obj (Opposite.op X)) M
N : Sieve.{v, u} X
hN :
  Membership.mem (γ := Set (Functor.obj (Functor.sieves C) (Opposite.op X)))
    ((Functor.closedSieves J₁).obj (Opposite.op X)) N
hM₂ :
  x.IsAmalgamation (R := S.arrows)
    (Subtype.mk (p := fun (x : Functor.obj (Functor.sieves C) (Opposite.op X)) ↦
      Membership.mem (γ := Set (Functor.obj (Functor.sieves C) (Opposite.op X)))
        ((Functor.closedSieves J₁).obj (Opposite.op X)) x)
      M hM)
hN₂ :
  x.IsAmalgamation (R := S.arrows)
    (Subtype.mk (p := fun (x : Functor.obj (Functor.sieves C) (Opposite.op X)) ↦
      Membership.mem (γ := Set (Functor.obj (Functor.sieves C) (Opposite.op X)))
        ((Functor.closedSieves J₁).obj (Opposite.op X)) x)
      N hN)
Y : C
f : Y ⟶ X
q : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N
h : ∀ ⦃Y : C⦄ (f : Y ⟶ X), (M ⊓ S).arrows f ↔ (N ⊓ S).arrows f
⊢ M ⊓ S = N ⊓ S

Tactic:
ext
-- Module: Mathlib.CategoryTheory.Sites.Closed
Goal:
C : Type u
inst : Category.{v, u} C
J₁ : GrothendieckTopology.{v, u} C
X : C
S : Sieve.{v, u} X
hS : S ∈ (J₁ : (a : C) → Set (Sieve a)) X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁).toFunctor S.arrows
M : Sieve.{v, u} X
hM :
  Membership.mem (γ := Set (Functor.obj (Functor.sieves C) (Opposite.op X)))
    ((Functor.closedSieves J₁).obj (Opposite.op X)) M
N : Sieve.{v, u} X
hN :
  Membership.mem (γ := Set (Functor.obj (Functor.sieves C) (Opposite.op X)))
    ((Functor.closedSieves J₁).obj (Opposite.op X)) N
hM₂ :
  x.IsAmalgamation (R := S.arrows)
    (Subtype.mk (p := fun (x : Functor.obj (Functor.sieves C) (Opposite.op X)) ↦
      Membership.mem (γ := Set (Functor.obj (Functor.sieves C) (Opposite.op X)))
        ((Functor.closedSieves J₁).obj (Opposite.op X)) x)
      M hM)
hN₂ :
  x.IsAmalgamation (R := S.arrows)
    (Subtype.mk (p := fun (x : Functor.obj (Functor.sieves C) (Opposite.op X)) ↦
      Membership.mem (γ := Set (Functor.obj (Functor.sieves C) (Opposite.op X)))
        ((Functor.closedSieves J₁).obj (Opposite.op X)) x)
      N hN)
Y : C
f : Y ⟶ X
q : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N
Y_1 : C
f_1 : Y_1 ⟶ X
⊢ (M ⊓ S).arrows f_1 ↔ (N ⊓ S).arrows f_1

Tactic:
grind [Sieve.inter_apply, Sieve.mem_iff_pullback_eq_top]

Premises used: CategoryTheory.Sieve.mem_iff_pullback_eq_top, CategoryTheory.Sieve.inter_apply