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
      

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
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
⊢ M ⊓ S = N ⊓ S

Next tactic:
ext Z g
/-- 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
      

Proof state:
case h
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
Z : C
g : Z ⟶ (Opposite.op X).unop
⊢ (M ⊓ S).arrows g ↔ (N ⊓ S).arrows g

Next tactic:
rw [Sieve.inter_apply, Sieve.inter_apply]
/-- 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]
      

Proof state:
case h
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
Z : C
g : Z ⟶ (Opposite.op X).unop
⊢ M.arrows g ∧ S.arrows g ↔ N.arrows g ∧ S.arrows g

Next tactic:
simp only [and_comm]
/-- 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]
      

Proof state:
case h
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
Z : C
g : Z ⟶ (Opposite.op X).unop
⊢ S.arrows g ∧ M.arrows g ↔ S.arrows g ∧ N.arrows g

Next tactic:
apply and_congr_right
/-- 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
      

Proof state:
case h.h
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
Z : C
g : Z ⟶ (Opposite.op X).unop
⊢ S.arrows g → (M.arrows g ↔ N.arrows g)

Next tactic:
intro hg
/-- 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
      

Proof state:
case h.h
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
Z : C
g : Z ⟶ (Opposite.op X).unop
hg : S.arrows g
⊢ M.arrows g ↔ N.arrows g

Next tactic:
rw [Sieve.pullback_eq_top_iff_mem, Sieve.pullback_eq_top_iff_mem, q g hg]
/-- 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]
    

Proof state:
case refine'_1.mk.mk.a.h
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
⊢ GrothendieckTopology.Covers J₁ M f ↔ GrothendieckTopology.Covers J₁ N f

Next tactic:
constructor
/-- 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
    · 

Proof state:
case refine'_1.mk.mk.a.h.mp
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
⊢ GrothendieckTopology.Covers J₁ M f → GrothendieckTopology.Covers J₁ N f

Next tactic:
intro 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
      

Proof state:
case refine'_1.mk.mk.a.h.mp
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 M f
⊢ GrothendieckTopology.Covers J₁ N f

Next tactic:
rw [J₁.covers_iff]
/-- 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]
      

Proof state:
case refine'_1.mk.mk.a.h.mp
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 M f
⊢ Sieve.pullback f N ∈ 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)
      

Proof state:
case refine'_1.mk.mk.a.h.mp
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 M f
⊢ Sieve.pullback f (N ⊓ ?m.16294) ∈ 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 M 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]
      

Proof state:
case refine'_1.mk.mk.a.h.mp
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 M f
⊢ Sieve.pullback f (M ⊓ S) ∈ GrothendieckTopology.sieves J₁ Y

Next tactic:
apply J₁.arrow_intersect f M 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)
    · 

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
⊢ GrothendieckTopology.Covers J₁ N f → GrothendieckTopology.Covers J₁ M f

Next tactic:
intro 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
      

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
⊢ GrothendieckTopology.Covers J₁ M f

Next tactic:
rw [J₁.covers_iff]