l3lab/ntp-mathlib huggingface dataset:
/-- The presheaf of `J`-closed sieves is a `J`-sheaf.
The proof of this is adapted from [MM92], Chapter III, Section 7, Lemma 1.
-/
theorem classifier_isSheaf : Presieve.IsSheaf J₁ (Functor.closedSieves J₁) := by
  intro X S hS
  rw [← Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor]
  refine' ⟨_, _⟩
  · rintro x ⟨M, hM⟩ ⟨N, hN⟩ hM₂ hN₂
    simp only [Functor.closedSieves_obj]
    ext Y f
    dsimp only [Subtype.coe_mk]
    rw [← J₁.covers_iff_mem_of_isClosed hM, ← J₁.covers_iff_mem_of_isClosed hN]
    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)
    have MSNS : M ⊓ S = N ⊓ S := by
      ext Z g
      rw [Sieve.inter_apply, Sieve.inter_apply]
      simp only [and_comm]
      apply and_congr_right
      intro hg
      rw [Sieve.pullback_eq_top_iff_mem, Sieve.pullback_eq_top_iff_mem, q g hg]
    constructor
    · intro hf
      rw [J₁.covers_iff]
      apply J₁.superset_covering (Sieve.pullback_monotone f inf_le_left)
      rw [← MSNS]
      apply J₁.arrow_intersect f M S hf (J₁.pullback_stable _ hS)
    · intro hf
      rw [J₁.covers_iff]
      

Proof state:
case refine'_1.mk.mk.a.h.mpr
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ GrothendieckTopology.sieves J₁ X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows
M : Sieve (Opposite.op X).unop
hM : GrothendieckTopology.IsClosed J₁ M
N : Sieve (Opposite.op X).unop
hN : GrothendieckTopology.IsClosed J₁ N
hM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }
hN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }
Y : C
f : Y ⟶ (Opposite.op X).unop
q : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N
MSNS : M ⊓ S = N ⊓ S
hf : GrothendieckTopology.Covers J₁ N f
⊢ Sieve.pullback f M ∈ GrothendieckTopology.sieves J₁ Y

Next tactic:
apply J₁.superset_covering (Sieve.pullback_monotone f inf_le_left)
/-- The presheaf of `J`-closed sieves is a `J`-sheaf.
The proof of this is adapted from [MM92], Chapter III, Section 7, Lemma 1.
-/
theorem classifier_isSheaf : Presieve.IsSheaf J₁ (Functor.closedSieves J₁) := by
  intro X S hS
  rw [← Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor]
  refine' ⟨_, _⟩
  · rintro x ⟨M, hM⟩ ⟨N, hN⟩ hM₂ hN₂
    simp only [Functor.closedSieves_obj]
    ext Y f
    dsimp only [Subtype.coe_mk]
    rw [← J₁.covers_iff_mem_of_isClosed hM, ← J₁.covers_iff_mem_of_isClosed hN]
    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)
    have MSNS : M ⊓ S = N ⊓ S := by
      ext Z g
      rw [Sieve.inter_apply, Sieve.inter_apply]
      simp only [and_comm]
      apply and_congr_right
      intro hg
      rw [Sieve.pullback_eq_top_iff_mem, Sieve.pullback_eq_top_iff_mem, q g hg]
    constructor
    · intro hf
      rw [J₁.covers_iff]
      apply J₁.superset_covering (Sieve.pullback_monotone f inf_le_left)
      rw [← MSNS]
      apply J₁.arrow_intersect f M S hf (J₁.pullback_stable _ hS)
    · intro hf
      rw [J₁.covers_iff]
      apply J₁.superset_covering (Sieve.pullback_monotone f inf_le_left)
      

Proof state:
case refine'_1.mk.mk.a.h.mpr
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ GrothendieckTopology.sieves J₁ X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows
M : Sieve (Opposite.op X).unop
hM : GrothendieckTopology.IsClosed J₁ M
N : Sieve (Opposite.op X).unop
hN : GrothendieckTopology.IsClosed J₁ N
hM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }
hN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }
Y : C
f : Y ⟶ (Opposite.op X).unop
q : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N
MSNS : M ⊓ S = N ⊓ S
hf : GrothendieckTopology.Covers J₁ N f
⊢ Sieve.pullback f (M ⊓ ?m.16520) ∈ GrothendieckTopology.sieves J₁ Y
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ GrothendieckTopology.sieves J₁ X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows
M : Sieve (Opposite.op X).unop
hM : GrothendieckTopology.IsClosed J₁ M
N : Sieve (Opposite.op X).unop
hN : GrothendieckTopology.IsClosed J₁ N
hM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }
hN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }
Y : C
f : Y ⟶ (Opposite.op X).unop
q : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N
MSNS : M ⊓ S = N ⊓ S
hf : GrothendieckTopology.Covers J₁ N f
⊢ Sieve (Opposite.op X).unop

Next tactic:
rw [MSNS]
/-- The presheaf of `J`-closed sieves is a `J`-sheaf.
The proof of this is adapted from [MM92], Chapter III, Section 7, Lemma 1.
-/
theorem classifier_isSheaf : Presieve.IsSheaf J₁ (Functor.closedSieves J₁) := by
  intro X S hS
  rw [← Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor]
  refine' ⟨_, _⟩
  · rintro x ⟨M, hM⟩ ⟨N, hN⟩ hM₂ hN₂
    simp only [Functor.closedSieves_obj]
    ext Y f
    dsimp only [Subtype.coe_mk]
    rw [← J₁.covers_iff_mem_of_isClosed hM, ← J₁.covers_iff_mem_of_isClosed hN]
    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)
    have MSNS : M ⊓ S = N ⊓ S := by
      ext Z g
      rw [Sieve.inter_apply, Sieve.inter_apply]
      simp only [and_comm]
      apply and_congr_right
      intro hg
      rw [Sieve.pullback_eq_top_iff_mem, Sieve.pullback_eq_top_iff_mem, q g hg]
    constructor
    · intro hf
      rw [J₁.covers_iff]
      apply J₁.superset_covering (Sieve.pullback_monotone f inf_le_left)
      rw [← MSNS]
      apply J₁.arrow_intersect f M S hf (J₁.pullback_stable _ hS)
    · intro hf
      rw [J₁.covers_iff]
      apply J₁.superset_covering (Sieve.pullback_monotone f inf_le_left)
      rw [MSNS]
      

Proof state:
case refine'_1.mk.mk.a.h.mpr
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ GrothendieckTopology.sieves J₁ X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows
M : Sieve (Opposite.op X).unop
hM : GrothendieckTopology.IsClosed J₁ M
N : Sieve (Opposite.op X).unop
hN : GrothendieckTopology.IsClosed J₁ N
hM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }
hN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }
Y : C
f : Y ⟶ (Opposite.op X).unop
q : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N
MSNS : M ⊓ S = N ⊓ S
hf : GrothendieckTopology.Covers J₁ N f
⊢ Sieve.pullback f (N ⊓ S) ∈ GrothendieckTopology.sieves J₁ Y

Next tactic:
apply J₁.arrow_intersect f N S hf (J₁.pullback_stable _ hS)
/-- The presheaf of `J`-closed sieves is a `J`-sheaf.
The proof of this is adapted from [MM92], Chapter III, Section 7, Lemma 1.
-/
theorem classifier_isSheaf : Presieve.IsSheaf J₁ (Functor.closedSieves J₁) := by
  intro X S hS
  rw [← Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor]
  refine' ⟨_, _⟩
  · rintro x ⟨M, hM⟩ ⟨N, hN⟩ hM₂ hN₂
    simp only [Functor.closedSieves_obj]
    ext Y f
    dsimp only [Subtype.coe_mk]
    rw [← J₁.covers_iff_mem_of_isClosed hM, ← J₁.covers_iff_mem_of_isClosed hN]
    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)
    have MSNS : M ⊓ S = N ⊓ S := by
      ext Z g
      rw [Sieve.inter_apply, Sieve.inter_apply]
      simp only [and_comm]
      apply and_congr_right
      intro hg
      rw [Sieve.pullback_eq_top_iff_mem, Sieve.pullback_eq_top_iff_mem, q g hg]
    constructor
    · intro hf
      rw [J₁.covers_iff]
      apply J₁.superset_covering (Sieve.pullback_monotone f inf_le_left)
      rw [← MSNS]
      apply J₁.arrow_intersect f M S hf (J₁.pullback_stable _ hS)
    · intro hf
      rw [J₁.covers_iff]
      apply J₁.superset_covering (Sieve.pullback_monotone f inf_le_left)
      rw [MSNS]
      apply J₁.arrow_intersect f N S hf (J₁.pullback_stable _ hS)
  · 

Proof state:
case refine'_2
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ GrothendieckTopology.sieves J₁ X
⊢ ∀ (x : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows),
    Presieve.FamilyOfElements.Compatible x → ∃ t, Presieve.FamilyOfElements.IsAmalgamation x t

Next tactic:
intro x hx
/-- The presheaf of `J`-closed sieves is a `J`-sheaf.
The proof of this is adapted from [MM92], Chapter III, Section 7, Lemma 1.
-/
theorem classifier_isSheaf : Presieve.IsSheaf J₁ (Functor.closedSieves J₁) := by
  intro X S hS
  rw [← Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor]
  refine' ⟨_, _⟩
  · rintro x ⟨M, hM⟩ ⟨N, hN⟩ hM₂ hN₂
    simp only [Functor.closedSieves_obj]
    ext Y f
    dsimp only [Subtype.coe_mk]
    rw [← J₁.covers_iff_mem_of_isClosed hM, ← J₁.covers_iff_mem_of_isClosed hN]
    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)
    have MSNS : M ⊓ S = N ⊓ S := by
      ext Z g
      rw [Sieve.inter_apply, Sieve.inter_apply]
      simp only [and_comm]
      apply and_congr_right
      intro hg
      rw [Sieve.pullback_eq_top_iff_mem, Sieve.pullback_eq_top_iff_mem, q g hg]
    constructor
    · intro hf
      rw [J₁.covers_iff]
      apply J₁.superset_covering (Sieve.pullback_monotone f inf_le_left)
      rw [← MSNS]
      apply J₁.arrow_intersect f M S hf (J₁.pullback_stable _ hS)
    · intro hf
      rw [J₁.covers_iff]
      apply J₁.superset_covering (Sieve.pullback_monotone f inf_le_left)
      rw [MSNS]
      apply J₁.arrow_intersect f N S hf (J₁.pullback_stable _ hS)
  · intro x hx
    

Proof state:
case refine'_2
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ GrothendieckTopology.sieves J₁ X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows
hx : Presieve.FamilyOfElements.Compatible x
⊢ ∃ t, Presieve.FamilyOfElements.IsAmalgamation x t

Next tactic:
rw [Presieve.compatible_iff_sieveCompatible] at hx
/-- The presheaf of `J`-closed sieves is a `J`-sheaf.
The proof of this is adapted from [MM92], Chapter III, Section 7, Lemma 1.
-/
theorem classifier_isSheaf : Presieve.IsSheaf J₁ (Functor.closedSieves J₁) := by
  intro X S hS
  rw [← Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor]
  refine' ⟨_, _⟩
  · rintro x ⟨M, hM⟩ ⟨N, hN⟩ hM₂ hN₂
    simp only [Functor.closedSieves_obj]
    ext Y f
    dsimp only [Subtype.coe_mk]
    rw [← J₁.covers_iff_mem_of_isClosed hM, ← J₁.covers_iff_mem_of_isClosed hN]
    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)
    have MSNS : M ⊓ S = N ⊓ S := by
      ext Z g
      rw [Sieve.inter_apply, Sieve.inter_apply]
      simp only [and_comm]
      apply and_congr_right
      intro hg
      rw [Sieve.pullback_eq_top_iff_mem, Sieve.pullback_eq_top_iff_mem, q g hg]
    constructor
    · intro hf
      rw [J₁.covers_iff]
      apply J₁.superset_covering (Sieve.pullback_monotone f inf_le_left)
      rw [← MSNS]
      apply J₁.arrow_intersect f M S hf (J₁.pullback_stable _ hS)
    · intro hf
      rw [J₁.covers_iff]
      apply J₁.superset_covering (Sieve.pullback_monotone f inf_le_left)
      rw [MSNS]
      apply J₁.arrow_intersect f N S hf (J₁.pullback_stable _ hS)
  · intro x hx
    rw [Presieve.compatible_iff_sieveCompatible] at hx
    

Proof state:
case refine'_2
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ GrothendieckTopology.sieves J₁ X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows
hx : Presieve.FamilyOfElements.SieveCompatible x
⊢ ∃ t, Presieve.FamilyOfElements.IsAmalgamation x t

Next tactic:
let M := Sieve.bind S fun Y f hf => (x f hf).1
/-- The presheaf of `J`-closed sieves is a `J`-sheaf.
The proof of this is adapted from [MM92], Chapter III, Section 7, Lemma 1.
-/
theorem classifier_isSheaf : Presieve.IsSheaf J₁ (Functor.closedSieves J₁) := by
  intro X S hS
  rw [← Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor]
  refine' ⟨_, _⟩
  · rintro x ⟨M, hM⟩ ⟨N, hN⟩ hM₂ hN₂
    simp only [Functor.closedSieves_obj]
    ext Y f
    dsimp only [Subtype.coe_mk]
    rw [← J₁.covers_iff_mem_of_isClosed hM, ← J₁.covers_iff_mem_of_isClosed hN]
    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)
    have MSNS : M ⊓ S = N ⊓ S := by
      ext Z g
      rw [Sieve.inter_apply, Sieve.inter_apply]
      simp only [and_comm]
      apply and_congr_right
      intro hg
      rw [Sieve.pullback_eq_top_iff_mem, Sieve.pullback_eq_top_iff_mem, q g hg]
    constructor
    · intro hf
      rw [J₁.covers_iff]
      apply J₁.superset_covering (Sieve.pullback_monotone f inf_le_left)
      rw [← MSNS]
      apply J₁.arrow_intersect f M S hf (J₁.pullback_stable _ hS)
    · intro hf
      rw [J₁.covers_iff]
      apply J₁.superset_covering (Sieve.pullback_monotone f inf_le_left)
      rw [MSNS]
      apply J₁.arrow_intersect f N S hf (J₁.pullback_stable _ hS)
  · intro x hx
    rw [Presieve.compatible_iff_sieveCompatible] at hx
    let M := Sieve.bind S fun Y f hf => (x f hf).1
    

Proof state:
case refine'_2
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ GrothendieckTopology.sieves J₁ X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows
hx : Presieve.FamilyOfElements.SieveCompatible x
M : Sieve X := Sieve.bind S.arrows fun Y f hf => ↑(x f hf)
⊢ ∃ t, Presieve.FamilyOfElements.IsAmalgamation x t

Next 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 _, c⟩
        rw [Sieve.pullback_eq_top_iff_mem,
          ← 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
/-- The presheaf of `J`-closed sieves is a `J`-sheaf.
The proof of this is adapted from [MM92], Chapter III, Section 7, Lemma 1.
-/
theorem classifier_isSheaf : Presieve.IsSheaf J₁ (Functor.closedSieves J₁) := by
  intro X S hS
  rw [← Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor]
  refine' ⟨_, _⟩
  · rintro x ⟨M, hM⟩ ⟨N, hN⟩ hM₂ hN₂
    simp only [Functor.closedSieves_obj]
    ext Y f
    dsimp only [Subtype.coe_mk]
    rw [← J₁.covers_iff_mem_of_isClosed hM, ← J₁.covers_iff_mem_of_isClosed hN]
    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)
    have MSNS : M ⊓ S = N ⊓ S := by
      ext Z g
      rw [Sieve.inter_apply, Sieve.inter_apply]
      simp only [and_comm]
      apply and_congr_right
      intro hg
      rw [Sieve.pullback_eq_top_iff_mem, Sieve.pullback_eq_top_iff_mem, q g hg]
    constructor
    · intro hf
      rw [J₁.covers_iff]
      apply J₁.superset_covering (Sieve.pullback_monotone f inf_le_left)
      rw [← MSNS]
      apply J₁.arrow_intersect f M S hf (J₁.pullback_stable _ hS)
    · intro hf
      rw [J₁.covers_iff]
      apply J₁.superset_covering (Sieve.pullback_monotone f inf_le_left)
      rw [MSNS]
      apply J₁.arrow_intersect f N S hf (J₁.pullback_stable _ hS)
  · intro x hx
    rw [Presieve.compatible_iff_sieveCompatible] at hx
    let M := Sieve.bind S fun Y f hf => (x f hf).1
    have : ∀ ⦃Y⦄ (f : Y ⟶ X) (hf : S f), M.pullback f = (x f hf).1 := by
      

Proof state:
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ GrothendieckTopology.sieves J₁ X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows
hx : Presieve.FamilyOfElements.SieveCompatible x
M : Sieve X := Sieve.bind S.arrows fun Y f hf => ↑(x f hf)
⊢ ∀ ⦃Y : C⦄ (f : Y ⟶ X) (hf : S.arrows f), Sieve.pullback f M = ↑(x f hf)

Next tactic:
intro Y f hf
/-- The presheaf of `J`-closed sieves is a `J`-sheaf.
The proof of this is adapted from [MM92], Chapter III, Section 7, Lemma 1.
-/
theorem classifier_isSheaf : Presieve.IsSheaf J₁ (Functor.closedSieves J₁) := by
  intro X S hS
  rw [← Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor]
  refine' ⟨_, _⟩
  · rintro x ⟨M, hM⟩ ⟨N, hN⟩ hM₂ hN₂
    simp only [Functor.closedSieves_obj]
    ext Y f
    dsimp only [Subtype.coe_mk]
    rw [← J₁.covers_iff_mem_of_isClosed hM, ← J₁.covers_iff_mem_of_isClosed hN]
    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)
    have MSNS : M ⊓ S = N ⊓ S := by
      ext Z g
      rw [Sieve.inter_apply, Sieve.inter_apply]
      simp only [and_comm]
      apply and_congr_right
      intro hg
      rw [Sieve.pullback_eq_top_iff_mem, Sieve.pullback_eq_top_iff_mem, q g hg]
    constructor
    · intro hf
      rw [J₁.covers_iff]
      apply J₁.superset_covering (Sieve.pullback_monotone f inf_le_left)
      rw [← MSNS]
      apply J₁.arrow_intersect f M S hf (J₁.pullback_stable _ hS)
    · intro hf
      rw [J₁.covers_iff]
      apply J₁.superset_covering (Sieve.pullback_monotone f inf_le_left)
      rw [MSNS]
      apply J₁.arrow_intersect f N S hf (J₁.pullback_stable _ hS)
  · intro x hx
    rw [Presieve.compatible_iff_sieveCompatible] at hx
    let M := Sieve.bind S fun Y f hf => (x f hf).1
    have : ∀ ⦃Y⦄ (f : Y ⟶ X) (hf : S f), M.pullback f = (x f hf).1 := by
      intro Y f hf
      

Proof state:
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ GrothendieckTopology.sieves J₁ X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows
hx : Presieve.FamilyOfElements.SieveCompatible x
M : Sieve X := Sieve.bind S.arrows fun Y f hf => ↑(x f hf)
Y : C
f : Y ⟶ X
hf : S.arrows f
⊢ Sieve.pullback f M = ↑(x f hf)

Next tactic:
apply le_antisymm
/-- The presheaf of `J`-closed sieves is a `J`-sheaf.
The proof of this is adapted from [MM92], Chapter III, Section 7, Lemma 1.
-/
theorem classifier_isSheaf : Presieve.IsSheaf J₁ (Functor.closedSieves J₁) := by
  intro X S hS
  rw [← Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor]
  refine' ⟨_, _⟩
  · rintro x ⟨M, hM⟩ ⟨N, hN⟩ hM₂ hN₂
    simp only [Functor.closedSieves_obj]
    ext Y f
    dsimp only [Subtype.coe_mk]
    rw [← J₁.covers_iff_mem_of_isClosed hM, ← J₁.covers_iff_mem_of_isClosed hN]
    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)
    have MSNS : M ⊓ S = N ⊓ S := by
      ext Z g
      rw [Sieve.inter_apply, Sieve.inter_apply]
      simp only [and_comm]
      apply and_congr_right
      intro hg
      rw [Sieve.pullback_eq_top_iff_mem, Sieve.pullback_eq_top_iff_mem, q g hg]
    constructor
    · intro hf
      rw [J₁.covers_iff]
      apply J₁.superset_covering (Sieve.pullback_monotone f inf_le_left)
      rw [← MSNS]
      apply J₁.arrow_intersect f M S hf (J₁.pullback_stable _ hS)
    · intro hf
      rw [J₁.covers_iff]
      apply J₁.superset_covering (Sieve.pullback_monotone f inf_le_left)
      rw [MSNS]
      apply J₁.arrow_intersect f N S hf (J₁.pullback_stable _ hS)
  · intro x hx
    rw [Presieve.compatible_iff_sieveCompatible] at hx
    let M := Sieve.bind S fun Y f hf => (x f hf).1
    have : ∀ ⦃Y⦄ (f : Y ⟶ X) (hf : S f), M.pullback f = (x f hf).1 := by
      intro Y f hf
      apply le_antisymm
      · 

Proof state:
case a
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ GrothendieckTopology.sieves J₁ X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows
hx : Presieve.FamilyOfElements.SieveCompatible x
M : Sieve X := Sieve.bind S.arrows fun Y f hf => ↑(x f hf)
Y : C
f : Y ⟶ X
hf : S.arrows f
⊢ Sieve.pullback f M ≤ ↑(x f hf)

Next tactic:
rintro Z u ⟨W, g, f', hf', hg : (x f' hf').1 _, c⟩
/-- The presheaf of `J`-closed sieves is a `J`-sheaf.
The proof of this is adapted from [MM92], Chapter III, Section 7, Lemma 1.
-/
theorem classifier_isSheaf : Presieve.IsSheaf J₁ (Functor.closedSieves J₁) := by
  intro X S hS
  rw [← Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor]
  refine' ⟨_, _⟩
  · rintro x ⟨M, hM⟩ ⟨N, hN⟩ hM₂ hN₂
    simp only [Functor.closedSieves_obj]
    ext Y f
    dsimp only [Subtype.coe_mk]
    rw [← J₁.covers_iff_mem_of_isClosed hM, ← J₁.covers_iff_mem_of_isClosed hN]
    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)
    have MSNS : M ⊓ S = N ⊓ S := by
      ext Z g
      rw [Sieve.inter_apply, Sieve.inter_apply]
      simp only [and_comm]
      apply and_congr_right
      intro hg
      rw [Sieve.pullback_eq_top_iff_mem, Sieve.pullback_eq_top_iff_mem, q g hg]
    constructor
    · intro hf
      rw [J₁.covers_iff]
      apply J₁.superset_covering (Sieve.pullback_monotone f inf_le_left)
      rw [← MSNS]
      apply J₁.arrow_intersect f M S hf (J₁.pullback_stable _ hS)
    · intro hf
      rw [J₁.covers_iff]
      apply J₁.superset_covering (Sieve.pullback_monotone f inf_le_left)
      rw [MSNS]
      apply J₁.arrow_intersect f N S hf (J₁.pullback_stable _ hS)
  · intro x hx
    rw [Presieve.compatible_iff_sieveCompatible] at hx
    let M := Sieve.bind S fun Y f hf => (x f hf).1
    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 _, c⟩
        

Proof state:
case a.intro.intro.intro.intro.intro
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ GrothendieckTopology.sieves J₁ X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows
hx : Presieve.FamilyOfElements.SieveCompatible x
M : Sieve X := Sieve.bind S.arrows fun Y f hf => ↑(x f hf)
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')).arrows g
c : g ≫ f' = u ≫ f
⊢ (↑(x f hf)).arrows u

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