awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- 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
⊢ M ⊓ S = N ⊓ S

Tactic:
have MSNS : M ⊓ S = N ⊓ S := by
      ext
      grind [Sieve.inter_apply, Sieve.mem_iff_pullback_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
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
MSNS : M ⊓ S = N ⊓ S
refine_1 : J₁.Covers M f → J₁.Covers N f
refine_1₁ : J₁.Covers N f → J₁.Covers M f
⊢ J₁.Covers M f ↔ J₁.Covers N f

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
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
MSNS : M ⊓ S = N ⊓ S
hf : J₁.Covers M f
refine_1 : Sieve.pullback f (N ⊓ S) ∈ (J₁ : (a : C) → Set (Sieve a)) Y
apply : Sieve.{v, u} X
⊢ Sieve.pullback f N ∈ (J₁ : (a : C) → Set (Sieve a)) Y

Tactic:
apply J₁.superset_covering (Sieve.pullback_monotone f inf_le_left)

Premises used: CategoryTheory.GrothendieckTopology.superset_covering, CategoryTheory.Sieve.pullback_monotone, inf_le_left
-- 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
MSNS : M ⊓ S = N ⊓ S
hf : J₁.Covers M f
⊢ Sieve.pullback f (M ⊓ S) ∈ (J₁ : (a : C) → Set (Sieve a)) Y

Tactic:
apply J₁.arrow_intersect f M S hf (J₁.pullback_stable _ hS)

Premises used: CategoryTheory.GrothendieckTopology.arrow_intersect, 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
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
MSNS : M ⊓ S = N ⊓ S
hf : J₁.Covers N f
refine_1 : Sieve.pullback f (M ⊓ S) ∈ (J₁ : (a : C) → Set (Sieve a)) Y
apply : Sieve.{v, u} X
⊢ Sieve.pullback f M ∈ (J₁ : (a : C) → Set (Sieve a)) Y

Tactic:
apply J₁.superset_covering (Sieve.pullback_monotone f inf_le_left)

Premises used: CategoryTheory.GrothendieckTopology.superset_covering, CategoryTheory.Sieve.pullback_monotone, inf_le_left
-- 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
MSNS : M ⊓ S = N ⊓ S
hf : J₁.Covers N f
⊢ Sieve.pullback f (N ⊓ S) ∈ (J₁ : (a : C) → Set (Sieve a)) Y

Tactic:
apply J₁.arrow_intersect f N S hf (J₁.pullback_stable _ hS)

Premises used: CategoryTheory.GrothendieckTopology.arrow_intersect, 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
x : Presieve.FamilyOfElements (Functor.closedSieves J₁).toFunctor S.arrows
hx : x.Compatible (R := S.arrows)
⊢ x.SieveCompatible

Tactic:
rw [Presieve.compatible_iff_sieveCompatible] at hx

Premises used: CategoryTheory.Presieve.compatible_iff_sieveCompatible
-- 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
hx : x.SieveCompatible
M : Sieve.{v, u} X :=
  Sieve.bind S.arrows fun (Y : C) (f : Y ⟶ X) (hf : S.arrows f) ↦ (↑(x f hf) : (Functor.sieves C).obj (Opposite.op Y))
Y : C
f : Y ⟶ X
hf : S.arrows f
a : Sieve.pullback f M ≤ (↑(x f hf) : (Functor.sieves C).obj (Opposite.op Y))
a₁ : (↑(x f hf) : (Functor.sieves C).obj (Opposite.op Y)) ≤ Sieve.pullback f M
⊢ Sieve.pullback f M = (↑(x f hf) : (Functor.sieves C).obj (Opposite.op Y))

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 : C
S : Sieve.{v, u} X
hS : S ∈ (J₁ : (a : C) → Set (Sieve a)) X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁).toFunctor S.arrows
hx : x.SieveCompatible
M : Sieve.{v, u} X :=
  Sieve.bind S.arrows fun (Y : C) (f : Y ⟶ X) (hf : S.arrows f) ↦ (↑(x f hf) : (Functor.sieves C).obj (Opposite.op Y))
Y : C
f : Y ⟶ X
hf : S.arrows f
Z : C
u : Z ⟶ Y
W : C
g : Z ⟶ W
f' : W ⟶ X
hf' : S.arrows f'
hg : (↑(x f' hf') : (Functor.sieves C).obj (Opposite.op W)).arrows g
c : g ≫ f' = u ≫ f
a : (↑(x (u ≫ f) ⋯) : (Functor.sieves C).obj (Opposite.op Z)) = ⊤
⊢ (↑(x f hf) : (Functor.sieves C).obj (Opposite.op Y)).arrows u

Tactic:
rw [Sieve.mem_iff_pullback_eq_top,
          ← show (x (u ≫ f) _).1 = (x f hf).1.pullback u from congr_arg Subtype.val (hx f u hf)]

Premises used: CategoryTheory.Sieve.mem_iff_pullback_eq_top, congr_arg, Subtype.val
-- 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
hx : x.SieveCompatible
M : Sieve.{v, u} X :=
  Sieve.bind S.arrows fun (Y : C) (f : Y ⟶ X) (hf : S.arrows f) ↦ (↑(x f hf) : (Functor.sieves C).obj (Opposite.op Y))
Y : C
f : Y ⟶ X
hf : S.arrows f
Z : C
u : Z ⟶ Y
W : C
g : Z ⟶ W
f' : W ⟶ X
hf' : S.arrows f'
hg : (↑(x f' hf') : (Functor.sieves C).obj (Opposite.op W)).arrows g
c : g ≫ f' = u ≫ f
self : x (u ≫ f) ⋯
| (↑(x (u ≫ f) ⋯) : (Functor.sieves C).obj (Opposite.op Z))

Tactic:
congr
-- 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
hx : x.SieveCompatible
M : Sieve.{v, u} X :=
  Sieve.bind S.arrows fun (Y : C) (f : Y ⟶ X) (hf : S.arrows f) ↦ (↑(x f hf) : (Functor.sieves C).obj (Opposite.op Y))
Y : C
f : Y ⟶ X
hf : S.arrows f
Z : C
u : Z ⟶ Y
W : C
g : Z ⟶ W
f' : W ⟶ X
hf' : S.arrows f'
hg : (↑(x f' hf') : (Functor.sieves C).obj (Opposite.op W)).arrows g
c : g ≫ f' = u ≫ f
self : u ≫ f
| x (u ≫ f) ⋯

Tactic:
congr
-- 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
hx : x.SieveCompatible
M : Sieve.{v, u} X :=
  Sieve.bind S.arrows fun (Y : C) (f : Y ⟶ X) (hf : S.arrows f) ↦ (↑(x f hf) : (Functor.sieves C).obj (Opposite.op Y))
Y : C
f : Y ⟶ X
hf : S.arrows f
Z : C
u : Z ⟶ Y
W : C
g : Z ⟶ W
f' : W ⟶ X
hf' : S.arrows f'
hg : (↑(x f' hf') : (Functor.sieves C).obj (Opposite.op W)).arrows g
c : g ≫ f' = u ≫ f
self : g ≫ f'
| u ≫ f

Tactic:
rw [← c]
-- 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
hx : x.SieveCompatible
M : Sieve.{v, u} X :=
  Sieve.bind S.arrows fun (Y : C) (f : Y ⟶ X) (hf : S.arrows f) ↦ (↑(x f hf) : (Functor.sieves C).obj (Opposite.op Y))
Y : C
f : Y ⟶ X
hf : S.arrows f
Z : C
u : Z ⟶ Y
W : C
g : Z ⟶ W
f' : W ⟶ X
hf' : S.arrows f'
hg : (↑(x f' hf') : (Functor.sieves C).obj (Opposite.op W)).arrows g
c : g ≫ f' = u ≫ f
self : g ≫ f'
| (↑(x (u ≫ f) ⋯) : (Functor.sieves C).obj (Opposite.op Z))

Tactic:
congr; congr; rw [← c]
-- 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
hx : x.SieveCompatible
M : Sieve.{v, u} X :=
  Sieve.bind S.arrows fun (Y : C) (f : Y ⟶ X) (hf : S.arrows f) ↦ (↑(x f hf) : (Functor.sieves C).obj (Opposite.op Y))
Y : C
f : Y ⟶ X
hf : S.arrows f
Z : C
u : Z ⟶ Y
W : C
g : Z ⟶ W
f' : W ⟶ X
hf' : S.arrows f'
hg : (↑(x f' hf') : (Functor.sieves C).obj (Opposite.op W)).arrows g
c : g ≫ f' = u ≫ f
a : (↑(x (g ≫ f') ⋯) : (Functor.sieves C).obj (Opposite.op Z)) = ⊤
⊢ (↑(x (u ≫ f) ⋯) : (Functor.sieves C).obj (Opposite.op Z)) = ⊤

Tactic:
conv_lhs => congr; congr; rw [← c]
-- 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
hx : x.SieveCompatible
M : Sieve.{v, u} X :=
  Sieve.bind S.arrows fun (Y : C) (f : Y ⟶ X) (hf : S.arrows f) ↦ (↑(x f hf) : (Functor.sieves C).obj (Opposite.op Y))
Y : C
f : Y ⟶ X
hf : S.arrows f
Z : C
u : Z ⟶ Y
W : C
g : Z ⟶ W
f' : W ⟶ X
hf' : S.arrows f'
hg : (↑(x f' hf') : (Functor.sieves C).obj (Opposite.op W)).arrows g
c : g ≫ f' = u ≫ f
a :
  Eq (α := Sieve.{v, u} Z)
    (↑((ConcreteCategory.hom (FC := TypeCat.Fun) ((Functor.closedSieves J₁).toFunctor.map g.op) :
            (a : (fun (X : Type (max u v)) ↦ X) ((Functor.closedSieves J₁).toFunctor.obj (Opposite.op W))) →
              (fun (X : Type (max u v)) ↦ X) ((Functor.closedSieves J₁).toFunctor.obj (Opposite.op Z)))
          (x f' hf')) :
      (Functor.sieves C).obj (Opposite.op Z))
    ⊤
⊢ (↑(x (g ≫ f') ⋯) : (Functor.sieves C).obj (Opposite.op Z)) = ⊤

Tactic:
rw [show (x (g ≫ f') _).1 = _ from congr_arg Subtype.val (hx f' g hf')]

Premises used: congr_arg, Subtype.val
-- 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
hx : x.SieveCompatible
M : Sieve.{v, u} X :=
  Sieve.bind S.arrows fun (Y : C) (f : Y ⟶ X) (hf : S.arrows f) ↦ (↑(x f hf) : (Functor.sieves C).obj (Opposite.op Y))
Y : C
f : Y ⟶ X
hf : S.arrows f
Z : C
u : Z ⟶ Y
W : C
g : Z ⟶ W
f' : W ⟶ X
hf' : S.arrows f'
hg : (↑(x f' hf') : (Functor.sieves C).obj (Opposite.op W)).arrows g
c : g ≫ f' = u ≫ f
⊢ Eq (α := Sieve.{v, u} Z)
    (↑((ConcreteCategory.hom (FC := TypeCat.Fun) ((Functor.closedSieves J₁).toFunctor.map g.op) :
            (a : (fun (X : Type (max u v)) ↦ X) ((Functor.closedSieves J₁).toFunctor.obj (Opposite.op W))) →
              (fun (X : Type (max u v)) ↦ X) ((Functor.closedSieves J₁).toFunctor.obj (Opposite.op Z)))
          (x f' hf')) :
      (Functor.sieves C).obj (Opposite.op Z))
    ⊤

Tactic:
apply Sieve.pullback_eq_top_of_mem _ hg

Premises used: CategoryTheory.Sieve.pullback_eq_top_of_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
hS : S ∈ (J₁ : (a : C) → Set (Sieve a)) X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁).toFunctor S.arrows
hx : x.SieveCompatible
M : Sieve.{v, u} X :=
  Sieve.bind S.arrows fun (Y : C) (f : Y ⟶ X) (hf : S.arrows f) ↦ (↑(x f hf) : (Functor.sieves C).obj (Opposite.op Y))
Y : C
f : Y ⟶ X
hf : S.arrows f
⊢ (↑(x f hf) : (Functor.sieves C).obj (Opposite.op Y)) ≤ Sieve.pullback f M

Tactic:
apply Sieve.le_pullback_bind S fun Y f hf => (x f hf).1

Premises used: CategoryTheory.Sieve.le_pullback_bind, Subtype.val
-- 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
hx : x.SieveCompatible
M : Sieve.{v, u} X :=
  Sieve.bind S.arrows fun (Y : C) (f : Y ⟶ X) (hf : S.arrows f) ↦ (↑(x f hf) : (Functor.sieves C).obj (Opposite.op Y))
⊢ ∀ ⦃Y : C⦄ (f : Y ⟶ X) (hf : S.arrows f), Sieve.pullback f M = (↑(x f hf) : (Functor.sieves C).obj (Opposite.op Y))

Tactic:
have : ∀ ⦃Y⦄ (f : Y ⟶ X) (hf : S f), M.pullback f = (x f hf).1 := by
      intro Y f hf
      apply le_antisymm
      · rintro Z u ⟨W, g, f', hf', hg : (x f' hf').1.1 _, c⟩
        rw [Sieve.mem_iff_pullback_eq_top,
          ← show (x (u ≫ f) _).1 = (x f hf).1.pullback u from congr_arg Subtype.val (hx f u hf)]
        conv_lhs => congr; congr; rw [← c] -- Porting note: Originally `simp_rw [← c]`
        rw [show (x (g ≫ f') _).1 = _ from congr_arg Subtype.val (hx f' g hf')]
        apply Sieve.pullback_eq_top_of_mem _ hg
      · apply Sieve.le_pullback_bind S fun Y f hf => (x f hf).1
-- 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
hx : x.SieveCompatible
M : Sieve.{v, u} X :=
  Sieve.bind S.arrows fun (Y : C) (f : Y ⟶ X) (hf : S.arrows f) ↦ (↑(x f hf) : (Functor.sieves C).obj (Opposite.op Y))
this :
  ∀ ⦃Y : C⦄ (f : Y ⟶ X) (hf : S.arrows f), Sieve.pullback f M = (↑(x f hf) : (Functor.sieves C).obj (Opposite.op Y))
⊢ ∃ (t : (Functor.closedSieves J₁).toFunctor.obj (Opposite.op X)), x.IsAmalgamation (R := S.arrows) t

Tactic:
⟨⟨_, J₁.close_isClosed M⟩, ?_⟩

Premises used: Subtype.mk, CategoryTheory.GrothendieckTopology.close_isClosed, Exists.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
x : Presieve.FamilyOfElements (Functor.closedSieves J₁).toFunctor S.arrows
hx : x.SieveCompatible
M : Sieve.{v, u} X :=
  Sieve.bind S.arrows fun (Y : C) (f : Y ⟶ X) (hf : S.arrows f) ↦ (↑(x f hf) : (Functor.sieves C).obj (Opposite.op Y))
this :
  ∀ ⦃Y : C⦄ (f : Y ⟶ X) (hf : S.arrows f), Sieve.pullback f M = (↑(x f hf) : (Functor.sieves C).obj (Opposite.op Y))
refine_2 :
  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)
      (J₁.close M) ⋯)
⊢ ∃ (t : (Functor.closedSieves J₁).toFunctor.obj (Opposite.op X)), x.IsAmalgamation (R := S.arrows) t

Tactic:
refine ⟨⟨_, J₁.close_isClosed M⟩, ?_⟩
-- 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
hx : x.SieveCompatible
M : Sieve.{v, u} X :=
  Sieve.bind S.arrows fun (Y : C) (f : Y ⟶ X) (hf : S.arrows f) ↦ (↑(x f hf) : (Functor.sieves C).obj (Opposite.op Y))
this :
  ∀ ⦃Y : C⦄ (f : Y ⟶ X) (hf : S.arrows f), Sieve.pullback f M = (↑(x f hf) : (Functor.sieves C).obj (Opposite.op Y))
Y : C
f : Y ⟶ X
hf : S.arrows f
refine_2 : Subtype.mk (p := fun (x : Sieve.{v, u} Y) ↦ J₁.IsClosed x) (Sieve.pullback f (J₁.close M)) ⋯ = x f hf
⊢ (ConcreteCategory.hom (FC := TypeCat.Fun) ((Functor.closedSieves J₁).toFunctor.map f.op) :
        (a : (fun (X : Type (max u v)) ↦ X) ((Functor.closedSieves J₁).toFunctor.obj (Opposite.op X))) →
          (fun (X : Type (max u v)) ↦ X) ((Functor.closedSieves J₁).toFunctor.obj (Opposite.op Y)))
      (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)
        (J₁.close M) ⋯) =
    x f hf

Tactic:
dsimp
-- 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
hx : x.SieveCompatible
M : Sieve.{v, u} X :=
  Sieve.bind S.arrows fun (Y : C) (f : Y ⟶ X) (hf : S.arrows f) ↦ (↑(x f hf) : (Functor.sieves C).obj (Opposite.op Y))
this :
  ∀ ⦃Y : C⦄ (f : Y ⟶ X) (hf : S.arrows f), Sieve.pullback f M = (↑(x f hf) : (Functor.sieves C).obj (Opposite.op Y))
Y : C
f : Y ⟶ X
hf : S.arrows f
refine_2 :
  (↑(Subtype.mk (p := fun (x : Sieve.{v, u} Y) ↦ J₁.IsClosed x) (Sieve.pullback f (J₁.close M)) ⋯) : Sieve Y) =
    (↑(x f hf) : Sieve Y)
⊢ Subtype.mk (p := fun (x : Sieve.{v, u} Y) ↦ J₁.IsClosed x) (Sieve.pullback f (J₁.close M)) ⋯ = x f hf

Tactic:
ext1
-- 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
hx : x.SieveCompatible
M : Sieve.{v, u} X :=
  Sieve.bind S.arrows fun (Y : C) (f : Y ⟶ X) (hf : S.arrows f) ↦ (↑(x f hf) : (Functor.sieves C).obj (Opposite.op Y))
this :
  ∀ ⦃Y : C⦄ (f : Y ⟶ X) (hf : S.arrows f), Sieve.pullback f M = (↑(x f hf) : (Functor.sieves C).obj (Opposite.op Y))
Y : C
f : Y ⟶ X
hf : S.arrows f
refine_2 : J₁.close (↑(x f hf) : (Functor.sieves C).obj (Opposite.op Y)) = (↑(x f hf) : Sieve Y)
⊢ Sieve.pullback f (J₁.close M) = (↑(x f hf) : Sieve Y)

Tactic:
rw [← J₁.pullback_close, this _ hf]

Premises used: CategoryTheory.GrothendieckTopology.pullback_close