l3lab/ntp-mathlib huggingface dataset:
lemma isSheafFor_of_factorsThru
    {X : C} {S T : Presieve X}
    (P : Cᵒᵖ ⥤ Type w)
    (H : S.FactorsThru T) (hS : S.IsSheafFor P)
    (h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
      R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
    T.IsSheafFor P := by
  simp only [← Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor] at *
  choose W i e h1 h2 using H
  refine ⟨?_, fun x hx => ?_⟩
  · intro x y₁ y₂ h₁ h₂
    refine hS.1.ext (fun Y g hg => ?_)
    simp only [← h2 hg, op_comp, P.map_comp, types_comp_apply, h₁ _ (h1 _ ), h₂ _ (h1 _)]
  let y : S.FamilyOfElements P := fun Y g hg => P.map (i _).op (x (e hg) (h1 _))
  have hy : y.Compatible := by
    intro Y₁ Y₂ Z g₁ g₂ f₁ f₂ h₁ h₂ h
    rw [← types_comp_apply (P.map (i h₁).op) (P.map g₁.op),
      ← types_comp_apply (P.map (i h₂).op) (P.map g₂.op),
      ← P.map_comp, ← op_comp, ← P.map_comp, ← op_comp]
    apply hx
    simp only [h2, h, Category.assoc]
  let ⟨_, h2'⟩ := hS
  obtain ⟨z, hz⟩ := h2' y hy
  

Proof state:
case refine_2.intro
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X : C
S T : Presieve X
P : Cᵒᵖ ⥤ Type w
h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ R, IsSeparatedFor P R ∧ FactorsThruAlong R S f
hS :
  IsSeparatedFor P S ∧
    ∀ (x : FamilyOfElements P S), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
W : ⦃Z : C⦄ → ⦃g : Z ⟶ X⦄ → S g → C
i : ⦃Z : C⦄ → ⦃g : Z ⟶ X⦄ → (a : S g) → Z ⟶ W a
e : ⦃Z : C⦄ → ⦃g : Z ⟶ X⦄ → (a : S g) → W a ⟶ X
h1 : ∀ ⦃Z : C⦄ ⦃g : Z ⟶ X⦄ (a : S g), T (e a)
h2 : ∀ ⦃Z : C⦄ ⦃g : Z ⟶ X⦄ (a : S g), i a ≫ e a = g
x : FamilyOfElements P T
hx : FamilyOfElements.Compatible x
y : FamilyOfElements P S := fun Y g hg => P.map (i hg).op (x (e hg) (_ : T (e hg)))
hy : FamilyOfElements.Compatible y
left✝ : IsSeparatedFor P S
h2' : ∀ (x : FamilyOfElements P S), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
z : P.obj (Opposite.op X)
hz : FamilyOfElements.IsAmalgamation y z
⊢ ∃ t, FamilyOfElements.IsAmalgamation x t

Next tactic:
refine ⟨z, fun Y g hg => ?_⟩
lemma isSheafFor_of_factorsThru
    {X : C} {S T : Presieve X}
    (P : Cᵒᵖ ⥤ Type w)
    (H : S.FactorsThru T) (hS : S.IsSheafFor P)
    (h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
      R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
    T.IsSheafFor P := by
  simp only [← Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor] at *
  choose W i e h1 h2 using H
  refine ⟨?_, fun x hx => ?_⟩
  · intro x y₁ y₂ h₁ h₂
    refine hS.1.ext (fun Y g hg => ?_)
    simp only [← h2 hg, op_comp, P.map_comp, types_comp_apply, h₁ _ (h1 _ ), h₂ _ (h1 _)]
  let y : S.FamilyOfElements P := fun Y g hg => P.map (i _).op (x (e hg) (h1 _))
  have hy : y.Compatible := by
    intro Y₁ Y₂ Z g₁ g₂ f₁ f₂ h₁ h₂ h
    rw [← types_comp_apply (P.map (i h₁).op) (P.map g₁.op),
      ← types_comp_apply (P.map (i h₂).op) (P.map g₂.op),
      ← P.map_comp, ← op_comp, ← P.map_comp, ← op_comp]
    apply hx
    simp only [h2, h, Category.assoc]
  let ⟨_, h2'⟩ := hS
  obtain ⟨z, hz⟩ := h2' y hy
  refine ⟨z, fun Y g hg => ?_⟩
  

Proof state:
case refine_2.intro
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X : C
S T : Presieve X
P : Cᵒᵖ ⥤ Type w
h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ R, IsSeparatedFor P R ∧ FactorsThruAlong R S f
hS :
  IsSeparatedFor P S ∧
    ∀ (x : FamilyOfElements P S), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
W : ⦃Z : C⦄ → ⦃g : Z ⟶ X⦄ → S g → C
i : ⦃Z : C⦄ → ⦃g : Z ⟶ X⦄ → (a : S g) → Z ⟶ W a
e : ⦃Z : C⦄ → ⦃g : Z ⟶ X⦄ → (a : S g) → W a ⟶ X
h1 : ∀ ⦃Z : C⦄ ⦃g : Z ⟶ X⦄ (a : S g), T (e a)
h2 : ∀ ⦃Z : C⦄ ⦃g : Z ⟶ X⦄ (a : S g), i a ≫ e a = g
x : FamilyOfElements P T
hx : FamilyOfElements.Compatible x
y : FamilyOfElements P S := fun Y g hg => P.map (i hg).op (x (e hg) (_ : T (e hg)))
hy : FamilyOfElements.Compatible y
left✝ : IsSeparatedFor P S
h2' : ∀ (x : FamilyOfElements P S), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
z : P.obj (Opposite.op X)
hz : FamilyOfElements.IsAmalgamation y z
Y : C
g : Y ⟶ X
hg : T g
⊢ P.map g.op z = x g hg

Next tactic:
obtain ⟨R, hR1, hR2⟩ := h hg
lemma isSheafFor_of_factorsThru
    {X : C} {S T : Presieve X}
    (P : Cᵒᵖ ⥤ Type w)
    (H : S.FactorsThru T) (hS : S.IsSheafFor P)
    (h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
      R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
    T.IsSheafFor P := by
  simp only [← Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor] at *
  choose W i e h1 h2 using H
  refine ⟨?_, fun x hx => ?_⟩
  · intro x y₁ y₂ h₁ h₂
    refine hS.1.ext (fun Y g hg => ?_)
    simp only [← h2 hg, op_comp, P.map_comp, types_comp_apply, h₁ _ (h1 _ ), h₂ _ (h1 _)]
  let y : S.FamilyOfElements P := fun Y g hg => P.map (i _).op (x (e hg) (h1 _))
  have hy : y.Compatible := by
    intro Y₁ Y₂ Z g₁ g₂ f₁ f₂ h₁ h₂ h
    rw [← types_comp_apply (P.map (i h₁).op) (P.map g₁.op),
      ← types_comp_apply (P.map (i h₂).op) (P.map g₂.op),
      ← P.map_comp, ← op_comp, ← P.map_comp, ← op_comp]
    apply hx
    simp only [h2, h, Category.assoc]
  let ⟨_, h2'⟩ := hS
  obtain ⟨z, hz⟩ := h2' y hy
  refine ⟨z, fun Y g hg => ?_⟩
  obtain ⟨R, hR1, hR2⟩ := h hg
  

Proof state:
case refine_2.intro.intro.intro
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X : C
S T : Presieve X
P : Cᵒᵖ ⥤ Type w
h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ R, IsSeparatedFor P R ∧ FactorsThruAlong R S f
hS :
  IsSeparatedFor P S ∧
    ∀ (x : FamilyOfElements P S), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
W : ⦃Z : C⦄ → ⦃g : Z ⟶ X⦄ → S g → C
i : ⦃Z : C⦄ → ⦃g : Z ⟶ X⦄ → (a : S g) → Z ⟶ W a
e : ⦃Z : C⦄ → ⦃g : Z ⟶ X⦄ → (a : S g) → W a ⟶ X
h1 : ∀ ⦃Z : C⦄ ⦃g : Z ⟶ X⦄ (a : S g), T (e a)
h2 : ∀ ⦃Z : C⦄ ⦃g : Z ⟶ X⦄ (a : S g), i a ≫ e a = g
x : FamilyOfElements P T
hx : FamilyOfElements.Compatible x
y : FamilyOfElements P S := fun Y g hg => P.map (i hg).op (x (e hg) (_ : T (e hg)))
hy : FamilyOfElements.Compatible y
left✝ : IsSeparatedFor P S
h2' : ∀ (x : FamilyOfElements P S), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
z : P.obj (Opposite.op X)
hz : FamilyOfElements.IsAmalgamation y z
Y : C
g : Y ⟶ X
hg : T g
R : Presieve Y
hR1 : IsSeparatedFor P R
hR2 : FactorsThruAlong R S g
⊢ P.map g.op z = x g hg

Next tactic:
choose WW ii ee hh1 hh2 using hR2
lemma isSheafFor_of_factorsThru
    {X : C} {S T : Presieve X}
    (P : Cᵒᵖ ⥤ Type w)
    (H : S.FactorsThru T) (hS : S.IsSheafFor P)
    (h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
      R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
    T.IsSheafFor P := by
  simp only [← Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor] at *
  choose W i e h1 h2 using H
  refine ⟨?_, fun x hx => ?_⟩
  · intro x y₁ y₂ h₁ h₂
    refine hS.1.ext (fun Y g hg => ?_)
    simp only [← h2 hg, op_comp, P.map_comp, types_comp_apply, h₁ _ (h1 _ ), h₂ _ (h1 _)]
  let y : S.FamilyOfElements P := fun Y g hg => P.map (i _).op (x (e hg) (h1 _))
  have hy : y.Compatible := by
    intro Y₁ Y₂ Z g₁ g₂ f₁ f₂ h₁ h₂ h
    rw [← types_comp_apply (P.map (i h₁).op) (P.map g₁.op),
      ← types_comp_apply (P.map (i h₂).op) (P.map g₂.op),
      ← P.map_comp, ← op_comp, ← P.map_comp, ← op_comp]
    apply hx
    simp only [h2, h, Category.assoc]
  let ⟨_, h2'⟩ := hS
  obtain ⟨z, hz⟩ := h2' y hy
  refine ⟨z, fun Y g hg => ?_⟩
  obtain ⟨R, hR1, hR2⟩ := h hg
  choose WW ii ee hh1 hh2 using hR2
  

Proof state:
case refine_2.intro.intro.intro
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X : C
S T : Presieve X
P : Cᵒᵖ ⥤ Type w
h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ R, IsSeparatedFor P R ∧ FactorsThruAlong R S f
hS :
  IsSeparatedFor P S ∧
    ∀ (x : FamilyOfElements P S), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
W : ⦃Z : C⦄ → ⦃g : Z ⟶ X⦄ → S g → C
i : ⦃Z : C⦄ → ⦃g : Z ⟶ X⦄ → (a : S g) → Z ⟶ W a
e : ⦃Z : C⦄ → ⦃g : Z ⟶ X⦄ → (a : S g) → W a ⟶ X
h1 : ∀ ⦃Z : C⦄ ⦃g : Z ⟶ X⦄ (a : S g), T (e a)
h2 : ∀ ⦃Z : C⦄ ⦃g : Z ⟶ X⦄ (a : S g), i a ≫ e a = g
x : FamilyOfElements P T
hx : FamilyOfElements.Compatible x
y : FamilyOfElements P S := fun Y g hg => P.map (i hg).op (x (e hg) (_ : T (e hg)))
hy : FamilyOfElements.Compatible y
left✝ : IsSeparatedFor P S
h2' : ∀ (x : FamilyOfElements P S), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
z : P.obj (Opposite.op X)
hz : FamilyOfElements.IsAmalgamation y z
Y : C
g : Y ⟶ X
hg : T g
R : Presieve Y
hR1 : IsSeparatedFor P R
WW : ⦃Z : C⦄ → ⦃g : Z ⟶ Y⦄ → R g → C
ii : ⦃Z : C⦄ → ⦃g : Z ⟶ Y⦄ → (a : R g) → Z ⟶ WW a
ee : ⦃Z : C⦄ → ⦃g : Z ⟶ Y⦄ → (a : R g) → WW a ⟶ X
hh1 : ∀ ⦃Z : C⦄ ⦃g : Z ⟶ Y⦄ (a : R g), S (ee a)
hh2 : ∀ ⦃Z : C⦄ ⦃g_1 : Z ⟶ Y⦄ (a : R g_1), ii a ≫ ee a = g_1 ≫ g
⊢ P.map g.op z = x g hg

Next tactic:
refine hR1.ext (fun Q t ht => ?_)
lemma isSheafFor_of_factorsThru
    {X : C} {S T : Presieve X}
    (P : Cᵒᵖ ⥤ Type w)
    (H : S.FactorsThru T) (hS : S.IsSheafFor P)
    (h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
      R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
    T.IsSheafFor P := by
  simp only [← Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor] at *
  choose W i e h1 h2 using H
  refine ⟨?_, fun x hx => ?_⟩
  · intro x y₁ y₂ h₁ h₂
    refine hS.1.ext (fun Y g hg => ?_)
    simp only [← h2 hg, op_comp, P.map_comp, types_comp_apply, h₁ _ (h1 _ ), h₂ _ (h1 _)]
  let y : S.FamilyOfElements P := fun Y g hg => P.map (i _).op (x (e hg) (h1 _))
  have hy : y.Compatible := by
    intro Y₁ Y₂ Z g₁ g₂ f₁ f₂ h₁ h₂ h
    rw [← types_comp_apply (P.map (i h₁).op) (P.map g₁.op),
      ← types_comp_apply (P.map (i h₂).op) (P.map g₂.op),
      ← P.map_comp, ← op_comp, ← P.map_comp, ← op_comp]
    apply hx
    simp only [h2, h, Category.assoc]
  let ⟨_, h2'⟩ := hS
  obtain ⟨z, hz⟩ := h2' y hy
  refine ⟨z, fun Y g hg => ?_⟩
  obtain ⟨R, hR1, hR2⟩ := h hg
  choose WW ii ee hh1 hh2 using hR2
  refine hR1.ext (fun Q t ht => ?_)
  

Proof state:
case refine_2.intro.intro.intro
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X : C
S T : Presieve X
P : Cᵒᵖ ⥤ Type w
h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ R, IsSeparatedFor P R ∧ FactorsThruAlong R S f
hS :
  IsSeparatedFor P S ∧
    ∀ (x : FamilyOfElements P S), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
W : ⦃Z : C⦄ → ⦃g : Z ⟶ X⦄ → S g → C
i : ⦃Z : C⦄ → ⦃g : Z ⟶ X⦄ → (a : S g) → Z ⟶ W a
e : ⦃Z : C⦄ → ⦃g : Z ⟶ X⦄ → (a : S g) → W a ⟶ X
h1 : ∀ ⦃Z : C⦄ ⦃g : Z ⟶ X⦄ (a : S g), T (e a)
h2 : ∀ ⦃Z : C⦄ ⦃g : Z ⟶ X⦄ (a : S g), i a ≫ e a = g
x : FamilyOfElements P T
hx : FamilyOfElements.Compatible x
y : FamilyOfElements P S := fun Y g hg => P.map (i hg).op (x (e hg) (_ : T (e hg)))
hy : FamilyOfElements.Compatible y
left✝ : IsSeparatedFor P S
h2' : ∀ (x : FamilyOfElements P S), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
z : P.obj (Opposite.op X)
hz : FamilyOfElements.IsAmalgamation y z
Y : C
g : Y ⟶ X
hg : T g
R : Presieve Y
hR1 : IsSeparatedFor P R
WW : ⦃Z : C⦄ → ⦃g : Z ⟶ Y⦄ → R g → C
ii : ⦃Z : C⦄ → ⦃g : Z ⟶ Y⦄ → (a : R g) → Z ⟶ WW a
ee : ⦃Z : C⦄ → ⦃g : Z ⟶ Y⦄ → (a : R g) → WW a ⟶ X
hh1 : ∀ ⦃Z : C⦄ ⦃g : Z ⟶ Y⦄ (a : R g), S (ee a)
hh2 : ∀ ⦃Z : C⦄ ⦃g_1 : Z ⟶ Y⦄ (a : R g_1), ii a ≫ ee a = g_1 ≫ g
Q : C
t : Q ⟶ Y
ht : R t
⊢ P.map t.op (P.map g.op z) = P.map t.op (x g hg)

Next tactic:
rw [← types_comp_apply (P.map g.op) (P.map t.op), ← P.map_comp, ← op_comp, ← hh2 ht,
    op_comp, P.map_comp, types_comp_apply, hz _ (hh1 _),
    ← types_comp_apply _ (P.map (ii ht).op), ← P.map_comp, ← op_comp]
lemma isSheafFor_of_factorsThru
    {X : C} {S T : Presieve X}
    (P : Cᵒᵖ ⥤ Type w)
    (H : S.FactorsThru T) (hS : S.IsSheafFor P)
    (h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
      R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
    T.IsSheafFor P := by
  simp only [← Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor] at *
  choose W i e h1 h2 using H
  refine ⟨?_, fun x hx => ?_⟩
  · intro x y₁ y₂ h₁ h₂
    refine hS.1.ext (fun Y g hg => ?_)
    simp only [← h2 hg, op_comp, P.map_comp, types_comp_apply, h₁ _ (h1 _ ), h₂ _ (h1 _)]
  let y : S.FamilyOfElements P := fun Y g hg => P.map (i _).op (x (e hg) (h1 _))
  have hy : y.Compatible := by
    intro Y₁ Y₂ Z g₁ g₂ f₁ f₂ h₁ h₂ h
    rw [← types_comp_apply (P.map (i h₁).op) (P.map g₁.op),
      ← types_comp_apply (P.map (i h₂).op) (P.map g₂.op),
      ← P.map_comp, ← op_comp, ← P.map_comp, ← op_comp]
    apply hx
    simp only [h2, h, Category.assoc]
  let ⟨_, h2'⟩ := hS
  obtain ⟨z, hz⟩ := h2' y hy
  refine ⟨z, fun Y g hg => ?_⟩
  obtain ⟨R, hR1, hR2⟩ := h hg
  choose WW ii ee hh1 hh2 using hR2
  refine hR1.ext (fun Q t ht => ?_)
  rw [← types_comp_apply (P.map g.op) (P.map t.op), ← P.map_comp, ← op_comp, ← hh2 ht,
    op_comp, P.map_comp, types_comp_apply, hz _ (hh1 _),
    ← types_comp_apply _ (P.map (ii ht).op), ← P.map_comp, ← op_comp]
  

Proof state:
case refine_2.intro.intro.intro
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X : C
S T : Presieve X
P : Cᵒᵖ ⥤ Type w
h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ R, IsSeparatedFor P R ∧ FactorsThruAlong R S f
hS :
  IsSeparatedFor P S ∧
    ∀ (x : FamilyOfElements P S), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
W : ⦃Z : C⦄ → ⦃g : Z ⟶ X⦄ → S g → C
i : ⦃Z : C⦄ → ⦃g : Z ⟶ X⦄ → (a : S g) → Z ⟶ W a
e : ⦃Z : C⦄ → ⦃g : Z ⟶ X⦄ → (a : S g) → W a ⟶ X
h1 : ∀ ⦃Z : C⦄ ⦃g : Z ⟶ X⦄ (a : S g), T (e a)
h2 : ∀ ⦃Z : C⦄ ⦃g : Z ⟶ X⦄ (a : S g), i a ≫ e a = g
x : FamilyOfElements P T
hx : FamilyOfElements.Compatible x
y : FamilyOfElements P S := fun Y g hg => P.map (i hg).op (x (e hg) (_ : T (e hg)))
hy : FamilyOfElements.Compatible y
left✝ : IsSeparatedFor P S
h2' : ∀ (x : FamilyOfElements P S), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
z : P.obj (Opposite.op X)
hz : FamilyOfElements.IsAmalgamation y z
Y : C
g : Y ⟶ X
hg : T g
R : Presieve Y
hR1 : IsSeparatedFor P R
WW : ⦃Z : C⦄ → ⦃g : Z ⟶ Y⦄ → R g → C
ii : ⦃Z : C⦄ → ⦃g : Z ⟶ Y⦄ → (a : R g) → Z ⟶ WW a
ee : ⦃Z : C⦄ → ⦃g : Z ⟶ Y⦄ → (a : R g) → WW a ⟶ X
hh1 : ∀ ⦃Z : C⦄ ⦃g : Z ⟶ Y⦄ (a : R g), S (ee a)
hh2 : ∀ ⦃Z : C⦄ ⦃g_1 : Z ⟶ Y⦄ (a : R g_1), ii a ≫ ee a = g_1 ≫ g
Q : C
t : Q ⟶ Y
ht : R t
⊢ P.map (ii ht ≫ i (_ : S (ee ht))).op (x (e (_ : S (ee ht))) (_ : T (e (_ : S (ee ht))))) = P.map t.op (x g hg)

Next tactic:
apply hx
lemma isSheafFor_of_factorsThru
    {X : C} {S T : Presieve X}
    (P : Cᵒᵖ ⥤ Type w)
    (H : S.FactorsThru T) (hS : S.IsSheafFor P)
    (h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
      R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
    T.IsSheafFor P := by
  simp only [← Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor] at *
  choose W i e h1 h2 using H
  refine ⟨?_, fun x hx => ?_⟩
  · intro x y₁ y₂ h₁ h₂
    refine hS.1.ext (fun Y g hg => ?_)
    simp only [← h2 hg, op_comp, P.map_comp, types_comp_apply, h₁ _ (h1 _ ), h₂ _ (h1 _)]
  let y : S.FamilyOfElements P := fun Y g hg => P.map (i _).op (x (e hg) (h1 _))
  have hy : y.Compatible := by
    intro Y₁ Y₂ Z g₁ g₂ f₁ f₂ h₁ h₂ h
    rw [← types_comp_apply (P.map (i h₁).op) (P.map g₁.op),
      ← types_comp_apply (P.map (i h₂).op) (P.map g₂.op),
      ← P.map_comp, ← op_comp, ← P.map_comp, ← op_comp]
    apply hx
    simp only [h2, h, Category.assoc]
  let ⟨_, h2'⟩ := hS
  obtain ⟨z, hz⟩ := h2' y hy
  refine ⟨z, fun Y g hg => ?_⟩
  obtain ⟨R, hR1, hR2⟩ := h hg
  choose WW ii ee hh1 hh2 using hR2
  refine hR1.ext (fun Q t ht => ?_)
  rw [← types_comp_apply (P.map g.op) (P.map t.op), ← P.map_comp, ← op_comp, ← hh2 ht,
    op_comp, P.map_comp, types_comp_apply, hz _ (hh1 _),
    ← types_comp_apply _ (P.map (ii ht).op), ← P.map_comp, ← op_comp]
  apply hx
  

Proof state:
case refine_2.intro.intro.intro.a
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X : C
S T : Presieve X
P : Cᵒᵖ ⥤ Type w
h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ R, IsSeparatedFor P R ∧ FactorsThruAlong R S f
hS :
  IsSeparatedFor P S ∧
    ∀ (x : FamilyOfElements P S), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
W : ⦃Z : C⦄ → ⦃g : Z ⟶ X⦄ → S g → C
i : ⦃Z : C⦄ → ⦃g : Z ⟶ X⦄ → (a : S g) → Z ⟶ W a
e : ⦃Z : C⦄ → ⦃g : Z ⟶ X⦄ → (a : S g) → W a ⟶ X
h1 : ∀ ⦃Z : C⦄ ⦃g : Z ⟶ X⦄ (a : S g), T (e a)
h2 : ∀ ⦃Z : C⦄ ⦃g : Z ⟶ X⦄ (a : S g), i a ≫ e a = g
x : FamilyOfElements P T
hx : FamilyOfElements.Compatible x
y : FamilyOfElements P S := fun Y g hg => P.map (i hg).op (x (e hg) (_ : T (e hg)))
hy : FamilyOfElements.Compatible y
left✝ : IsSeparatedFor P S
h2' : ∀ (x : FamilyOfElements P S), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
z : P.obj (Opposite.op X)
hz : FamilyOfElements.IsAmalgamation y z
Y : C
g : Y ⟶ X
hg : T g
R : Presieve Y
hR1 : IsSeparatedFor P R
WW : ⦃Z : C⦄ → ⦃g : Z ⟶ Y⦄ → R g → C
ii : ⦃Z : C⦄ → ⦃g : Z ⟶ Y⦄ → (a : R g) → Z ⟶ WW a
ee : ⦃Z : C⦄ → ⦃g : Z ⟶ Y⦄ → (a : R g) → WW a ⟶ X
hh1 : ∀ ⦃Z : C⦄ ⦃g : Z ⟶ Y⦄ (a : R g), S (ee a)
hh2 : ∀ ⦃Z : C⦄ ⦃g_1 : Z ⟶ Y⦄ (a : R g_1), ii a ≫ ee a = g_1 ≫ g
Q : C
t : Q ⟶ Y
ht : R t
⊢ (ii ht ≫ i (_ : S (ee ht))) ≫ e (_ : S (ee ht)) = t ≫ g

Next tactic:
simp only [Category.assoc, h2, hh2]
variable (C) in
/--
Associate a coverage to any Grothendieck topology.
If `J` is a Grothendieck topology, and `K` is the associated coverage, then a presieve
`S` is a covering presieve for `K` if and only if the sieve that it generates is a
covering sieve for `J`.
-/
def ofGrothendieck (J : GrothendieckTopology C) : Coverage C where
  covering X := { S | Sieve.generate S ∈ J X }
  pullback := by
    

Proof state:
C : Type ?u.23525
inst✝ : Category.{?u.23529, ?u.23525} C
J : GrothendieckTopology C
⊢ ∀ ⦃X Y : C⦄ (f : Y ⟶ X),
    ∀ S ∈ (fun X => {S | Sieve.generate S ∈ GrothendieckTopology.sieves J X}) X,
      ∃ T ∈ (fun X => {S | Sieve.generate S ∈ GrothendieckTopology.sieves J X}) Y, Presieve.FactorsThruAlong T S f

Next tactic:
intro X Y f S (hS : Sieve.generate S ∈ J X)
variable (C) in
/--
Associate a coverage to any Grothendieck topology.
If `J` is a Grothendieck topology, and `K` is the associated coverage, then a presieve
`S` is a covering presieve for `K` if and only if the sieve that it generates is a
covering sieve for `J`.
-/
def ofGrothendieck (J : GrothendieckTopology C) : Coverage C where
  covering X := { S | Sieve.generate S ∈ J X }
  pullback := by
    intro X Y f S (hS : Sieve.generate S ∈ J X)
    

Proof state:
C : Type ?u.23525
inst✝ : Category.{?u.23529, ?u.23525} C
J : GrothendieckTopology C
X Y : C
f : Y ⟶ X
S : Presieve X
hS : Sieve.generate S ∈ GrothendieckTopology.sieves J X
⊢ ∃ T ∈ (fun X => {S | Sieve.generate S ∈ GrothendieckTopology.sieves J X}) Y, Presieve.FactorsThruAlong T S f

Next tactic:
refine ⟨(Sieve.generate S).pullback f, ?_, fun Z g h => h⟩
variable (C) in
/--
Associate a coverage to any Grothendieck topology.
If `J` is a Grothendieck topology, and `K` is the associated coverage, then a presieve
`S` is a covering presieve for `K` if and only if the sieve that it generates is a
covering sieve for `J`.
-/
def ofGrothendieck (J : GrothendieckTopology C) : Coverage C where
  covering X := { S | Sieve.generate S ∈ J X }
  pullback := by
    intro X Y f S (hS : Sieve.generate S ∈ J X)
    refine ⟨(Sieve.generate S).pullback f, ?_, fun Z g h => h⟩
    

Proof state:
C : Type ?u.23525
inst✝ : Category.{?u.23529, ?u.23525} C
J : GrothendieckTopology C
X Y : C
f : Y ⟶ X
S : Presieve X
hS : Sieve.generate S ∈ GrothendieckTopology.sieves J X
⊢ (Sieve.pullback f (Sieve.generate S)).arrows ∈ (fun X => {S | Sieve.generate S ∈ GrothendieckTopology.sieves J X}) Y

Next tactic:
dsimp
variable (C) in
/--
Associate a coverage to any Grothendieck topology.
If `J` is a Grothendieck topology, and `K` is the associated coverage, then a presieve
`S` is a covering presieve for `K` if and only if the sieve that it generates is a
covering sieve for `J`.
-/
def ofGrothendieck (J : GrothendieckTopology C) : Coverage C where
  covering X := { S | Sieve.generate S ∈ J X }
  pullback := by
    intro X Y f S (hS : Sieve.generate S ∈ J X)
    refine ⟨(Sieve.generate S).pullback f, ?_, fun Z g h => h⟩
    dsimp
    

Proof state:
C : Type ?u.23525
inst✝ : Category.{?u.23529, ?u.23525} C
J : GrothendieckTopology C
X Y : C
f : Y ⟶ X
S : Presieve X
hS : Sieve.generate S ∈ GrothendieckTopology.sieves J X
⊢ Sieve.generate (Sieve.pullback f (Sieve.generate S)).arrows ∈ GrothendieckTopology.sieves J Y

Next tactic:
rw [Sieve.generate_sieve]
variable (C) in
/--
Associate a coverage to any Grothendieck topology.
If `J` is a Grothendieck topology, and `K` is the associated coverage, then a presieve
`S` is a covering presieve for `K` if and only if the sieve that it generates is a
covering sieve for `J`.
-/
def ofGrothendieck (J : GrothendieckTopology C) : Coverage C where
  covering X := { S | Sieve.generate S ∈ J X }
  pullback := by
    intro X Y f S (hS : Sieve.generate S ∈ J X)
    refine ⟨(Sieve.generate S).pullback f, ?_, fun Z g h => h⟩
    dsimp
    rw [Sieve.generate_sieve]
    

Proof state:
C : Type ?u.23525
inst✝ : Category.{?u.23529, ?u.23525} C
J : GrothendieckTopology C
X Y : C
f : Y ⟶ X
S : Presieve X
hS : Sieve.generate S ∈ GrothendieckTopology.sieves J X
⊢ Sieve.pullback f (Sieve.generate S) ∈ GrothendieckTopology.sieves J Y

Next tactic:
exact J.pullback_stable _ hS
lemma eq_top_pullback {X Y : C} {S T : Sieve X} (h : S ≤ T) (f : Y ⟶ X) (hf : S f) :
    T.pullback f = ⊤ := by
  

Proof state:
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X Y : C
S T : Sieve X
h : S ≤ T
f : Y ⟶ X
hf : S.arrows f
⊢ Sieve.pullback f T = ⊤

Next tactic:
ext Z g
lemma eq_top_pullback {X Y : C} {S T : Sieve X} (h : S ≤ T) (f : Y ⟶ X) (hf : S f) :
    T.pullback f = ⊤ := by
  ext Z g
  

Proof state:
case h
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X Y : C
S T : Sieve X
h : S ≤ T
f : Y ⟶ X
hf : S.arrows f
Z : C
g : Z ⟶ Y
⊢ (Sieve.pullback f T).arrows g ↔ ⊤.arrows g

Next tactic:
simp only [Sieve.pullback_apply, Sieve.top_apply, iff_true]
lemma eq_top_pullback {X Y : C} {S T : Sieve X} (h : S ≤ T) (f : Y ⟶ X) (hf : S f) :
    T.pullback f = ⊤ := by
  ext Z g
  simp only [Sieve.pullback_apply, Sieve.top_apply, iff_true]
  

Proof state:
case h
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X Y : C
S T : Sieve X
h : S ≤ T
f : Y ⟶ X
hf : S.arrows f
Z : C
g : Z ⟶ Y
⊢ T.arrows (g ≫ f)

Next tactic:
apply h
lemma eq_top_pullback {X Y : C} {S T : Sieve X} (h : S ≤ T) (f : Y ⟶ X) (hf : S f) :
    T.pullback f = ⊤ := by
  ext Z g
  simp only [Sieve.pullback_apply, Sieve.top_apply, iff_true]
  apply h
  

Proof state:
case h.a
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X Y : C
S T : Sieve X
h : S ≤ T
f : Y ⟶ X
hf : S.arrows f
Z : C
g : Z ⟶ Y
⊢ S.arrows (g ≫ f)

Next tactic:
apply S.downward_closed
lemma eq_top_pullback {X Y : C} {S T : Sieve X} (h : S ≤ T) (f : Y ⟶ X) (hf : S f) :
    T.pullback f = ⊤ := by
  ext Z g
  simp only [Sieve.pullback_apply, Sieve.top_apply, iff_true]
  apply h
  apply S.downward_closed
  

Proof state:
case h.a.x
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X Y : C
S T : Sieve X
h : S ≤ T
f : Y ⟶ X
hf : S.arrows f
Z : C
g : Z ⟶ Y
⊢ S.arrows f

Next tactic:
exact hf
lemma saturate_of_superset (K : Coverage C) {X : C} {S T : Sieve X} (h : S ≤ T)
    (hS : saturate K X S) : saturate K X T := by
  

Proof state:
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
X : C
S T : Sieve X
h : S ≤ T
hS : saturate K X S
⊢ saturate K X T

Next tactic:
apply saturate.transitive _ _ _ hS
lemma saturate_of_superset (K : Coverage C) {X : C} {S T : Sieve X} (h : S ≤ T)
    (hS : saturate K X S) : saturate K X T := by
  apply saturate.transitive _ _ _ hS
  

Proof state:
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
X : C
S T : Sieve X
h : S ≤ T
hS : saturate K X S
⊢ ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → saturate K Y (Sieve.pullback f T)

Next tactic:
intro Y g hg
lemma saturate_of_superset (K : Coverage C) {X : C} {S T : Sieve X} (h : S ≤ T)
    (hS : saturate K X S) : saturate K X T := by
  apply saturate.transitive _ _ _ hS
  intro Y g hg
  

Proof state:
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
X : C
S T : Sieve X
h : S ≤ T
hS : saturate K X S
Y : C
g : Y ⟶ X
hg : S.arrows g
⊢ saturate K Y (Sieve.pullback g T)

Next tactic:
rw [eq_top_pullback (h := h)]
lemma saturate_of_superset (K : Coverage C) {X : C} {S T : Sieve X} (h : S ≤ T)
    (hS : saturate K X S) : saturate K X T := by
  apply saturate.transitive _ _ _ hS
  intro Y g hg
  rw [eq_top_pullback (h := h)]
  · 

Proof state:
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
X : C
S T : Sieve X
h : S ≤ T
hS : saturate K X S
Y : C
g : Y ⟶ X
hg : S.arrows g
⊢ saturate K Y ⊤

Next tactic:
apply saturate.top