l3lab/ntp-mathlib huggingface dataset:
theorem Subpresheaf.sheafify_isSheaf (hF : Presieve.IsSheaf J F) :
    Presieve.IsSheaf J (G.sheafify J).toPresheaf := by
  intro U S hS x hx
  let S' := Sieve.bind S fun Y f hf => G.sieveOfSection (x f hf).1
  have := fun (V) (i : V ⟶ U) (hi : S' i) => hi
  -- porting note: change to explicit variable so that `choose` can find the correct
  -- dependent functions. Thus everything follows need two additional explicit variables.
  choose W i₁ i₂ hi₂ h₁ h₂ using this
  dsimp [-Sieve.bind_apply] at *
  let x'' : Presieve.FamilyOfElements F S' := fun V i hi => F.map (i₁ V i hi).op (x _ (hi₂ V i hi))
  have H : ∀ s, x.IsAmalgamation s ↔ x''.IsAmalgamation s.1 := by
    intro s
    constructor
    · intro H V i hi
      dsimp only [show x'' = fun V i hi => F.map (i₁ V i hi).op (x _ (hi₂ V i hi)) from rfl]
      conv_lhs => rw [← h₂ _ _ hi]
      rw [← H _ (hi₂ _ _ hi)]
      exact FunctorToTypes.map_comp_apply F (i₂ _ _ hi).op (i₁ _ _ hi).op _
    · intro H V i hi
      refine Subtype.ext ?_
      apply (hF _ (x i hi).2).isSeparatedFor.ext
      intro V' i' hi'
      have hi'' : S' (i' ≫ i) := ⟨_, _, _, hi, hi', rfl⟩
      have := H _ hi''
      rw [op_comp, F.map_comp] at this
      refine' this.trans (congr_arg Subtype.val (hx _ _ (hi₂ _ _ hi'') hi (h₂ _ _ hi'')))
  

Proof state:
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F F' F'' : Cᵒᵖ ⥤ Type w
G G' : Subpresheaf F
hF : Presieve.IsSheaf J F
U : C
S : Sieve U
hS : S ∈ sieves J U
x : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows
hx : Presieve.FamilyOfElements.Compatible x
S' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G ↑(x f hf)
W : (V : C) → (i : V ⟶ U) → S'.arrows i → C
i₁ : (V : C) → (i : V ⟶ U) → (hi : S'.arrows i) → V ⟶ W V i hi
i₂ : (V : C) → (i : V ⟶ U) → (hi : S'.arrows i) → W V i hi ⟶ U
hi₂ : ∀ (V : C) (i : V ⟶ U) (hi : S'.arrows i), S.arrows (i₂ V i hi)
h₁ :
  ∀ (V : C) (i : V ⟶ U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G ↑(x f hf)).arrows i),
    F.map (i₁ V i hi).op ↑(x (i₂ V i hi) (_ : S.arrows (i₂ V i hi))) ∈ obj G (op V)
h₂ :
  ∀ (V : C) (i : V ⟶ U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G ↑(x f hf)).arrows i),
    i₁ V i hi ≫ i₂ V i hi = i
x'' : Presieve.FamilyOfElements F S'.arrows :=
  fun V i hi => F.map (i₁ V i hi).op ↑(x (i₂ V i hi) (_ : S.arrows (i₂ V i hi)))
H :
  ∀ (s : (toPresheaf (sheafify J G)).obj (op U)),
    Presieve.FamilyOfElements.IsAmalgamation x s ↔ Presieve.FamilyOfElements.IsAmalgamation x'' ↑s
⊢ ∃! t, Presieve.FamilyOfElements.IsAmalgamation x t

Next tactic:
have : x''.Compatible := by
    intro V₁ V₂ V₃ g₁ g₂ g₃ g₄ S₁ S₂ e
    rw [← FunctorToTypes.map_comp_apply, ← FunctorToTypes.map_comp_apply]
    exact
      congr_arg Subtype.val
        (hx (g₁ ≫ i₁ _ _ S₁) (g₂ ≫ i₁ _ _ S₂) (hi₂ _ _ S₁) (hi₂ _ _ S₂)
        (by simp only [Category.assoc, h₂, e]))
theorem Subpresheaf.sheafify_isSheaf (hF : Presieve.IsSheaf J F) :
    Presieve.IsSheaf J (G.sheafify J).toPresheaf := by
  intro U S hS x hx
  let S' := Sieve.bind S fun Y f hf => G.sieveOfSection (x f hf).1
  have := fun (V) (i : V ⟶ U) (hi : S' i) => hi
  -- porting note: change to explicit variable so that `choose` can find the correct
  -- dependent functions. Thus everything follows need two additional explicit variables.
  choose W i₁ i₂ hi₂ h₁ h₂ using this
  dsimp [-Sieve.bind_apply] at *
  let x'' : Presieve.FamilyOfElements F S' := fun V i hi => F.map (i₁ V i hi).op (x _ (hi₂ V i hi))
  have H : ∀ s, x.IsAmalgamation s ↔ x''.IsAmalgamation s.1 := by
    intro s
    constructor
    · intro H V i hi
      dsimp only [show x'' = fun V i hi => F.map (i₁ V i hi).op (x _ (hi₂ V i hi)) from rfl]
      conv_lhs => rw [← h₂ _ _ hi]
      rw [← H _ (hi₂ _ _ hi)]
      exact FunctorToTypes.map_comp_apply F (i₂ _ _ hi).op (i₁ _ _ hi).op _
    · intro H V i hi
      refine Subtype.ext ?_
      apply (hF _ (x i hi).2).isSeparatedFor.ext
      intro V' i' hi'
      have hi'' : S' (i' ≫ i) := ⟨_, _, _, hi, hi', rfl⟩
      have := H _ hi''
      rw [op_comp, F.map_comp] at this
      refine' this.trans (congr_arg Subtype.val (hx _ _ (hi₂ _ _ hi'') hi (h₂ _ _ hi'')))
  have : x''.Compatible := by
    

Proof state:
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F F' F'' : Cᵒᵖ ⥤ Type w
G G' : Subpresheaf F
hF : Presieve.IsSheaf J F
U : C
S : Sieve U
hS : S ∈ sieves J U
x : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows
hx : Presieve.FamilyOfElements.Compatible x
S' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G ↑(x f hf)
W : (V : C) → (i : V ⟶ U) → S'.arrows i → C
i₁ : (V : C) → (i : V ⟶ U) → (hi : S'.arrows i) → V ⟶ W V i hi
i₂ : (V : C) → (i : V ⟶ U) → (hi : S'.arrows i) → W V i hi ⟶ U
hi₂ : ∀ (V : C) (i : V ⟶ U) (hi : S'.arrows i), S.arrows (i₂ V i hi)
h₁ :
  ∀ (V : C) (i : V ⟶ U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G ↑(x f hf)).arrows i),
    F.map (i₁ V i hi).op ↑(x (i₂ V i hi) (_ : S.arrows (i₂ V i hi))) ∈ obj G (op V)
h₂ :
  ∀ (V : C) (i : V ⟶ U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G ↑(x f hf)).arrows i),
    i₁ V i hi ≫ i₂ V i hi = i
x'' : Presieve.FamilyOfElements F S'.arrows :=
  fun V i hi => F.map (i₁ V i hi).op ↑(x (i₂ V i hi) (_ : S.arrows (i₂ V i hi)))
H :
  ∀ (s : (toPresheaf (sheafify J G)).obj (op U)),
    Presieve.FamilyOfElements.IsAmalgamation x s ↔ Presieve.FamilyOfElements.IsAmalgamation x'' ↑s
⊢ Presieve.FamilyOfElements.Compatible x''

Next tactic:
intro V₁ V₂ V₃ g₁ g₂ g₃ g₄ S₁ S₂ e
theorem Subpresheaf.sheafify_isSheaf (hF : Presieve.IsSheaf J F) :
    Presieve.IsSheaf J (G.sheafify J).toPresheaf := by
  intro U S hS x hx
  let S' := Sieve.bind S fun Y f hf => G.sieveOfSection (x f hf).1
  have := fun (V) (i : V ⟶ U) (hi : S' i) => hi
  -- porting note: change to explicit variable so that `choose` can find the correct
  -- dependent functions. Thus everything follows need two additional explicit variables.
  choose W i₁ i₂ hi₂ h₁ h₂ using this
  dsimp [-Sieve.bind_apply] at *
  let x'' : Presieve.FamilyOfElements F S' := fun V i hi => F.map (i₁ V i hi).op (x _ (hi₂ V i hi))
  have H : ∀ s, x.IsAmalgamation s ↔ x''.IsAmalgamation s.1 := by
    intro s
    constructor
    · intro H V i hi
      dsimp only [show x'' = fun V i hi => F.map (i₁ V i hi).op (x _ (hi₂ V i hi)) from rfl]
      conv_lhs => rw [← h₂ _ _ hi]
      rw [← H _ (hi₂ _ _ hi)]
      exact FunctorToTypes.map_comp_apply F (i₂ _ _ hi).op (i₁ _ _ hi).op _
    · intro H V i hi
      refine Subtype.ext ?_
      apply (hF _ (x i hi).2).isSeparatedFor.ext
      intro V' i' hi'
      have hi'' : S' (i' ≫ i) := ⟨_, _, _, hi, hi', rfl⟩
      have := H _ hi''
      rw [op_comp, F.map_comp] at this
      refine' this.trans (congr_arg Subtype.val (hx _ _ (hi₂ _ _ hi'') hi (h₂ _ _ hi'')))
  have : x''.Compatible := by
    intro V₁ V₂ V₃ g₁ g₂ g₃ g₄ S₁ S₂ e
    

Proof state:
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F F' F'' : Cᵒᵖ ⥤ Type w
G G' : Subpresheaf F
hF : Presieve.IsSheaf J F
U : C
S : Sieve U
hS : S ∈ sieves J U
x : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows
hx : Presieve.FamilyOfElements.Compatible x
S' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G ↑(x f hf)
W : (V : C) → (i : V ⟶ U) → S'.arrows i → C
i₁ : (V : C) → (i : V ⟶ U) → (hi : S'.arrows i) → V ⟶ W V i hi
i₂ : (V : C) → (i : V ⟶ U) → (hi : S'.arrows i) → W V i hi ⟶ U
hi₂ : ∀ (V : C) (i : V ⟶ U) (hi : S'.arrows i), S.arrows (i₂ V i hi)
h₁ :
  ∀ (V : C) (i : V ⟶ U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G ↑(x f hf)).arrows i),
    F.map (i₁ V i hi).op ↑(x (i₂ V i hi) (_ : S.arrows (i₂ V i hi))) ∈ obj G (op V)
h₂ :
  ∀ (V : C) (i : V ⟶ U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G ↑(x f hf)).arrows i),
    i₁ V i hi ≫ i₂ V i hi = i
x'' : Presieve.FamilyOfElements F S'.arrows :=
  fun V i hi => F.map (i₁ V i hi).op ↑(x (i₂ V i hi) (_ : S.arrows (i₂ V i hi)))
H :
  ∀ (s : (toPresheaf (sheafify J G)).obj (op U)),
    Presieve.FamilyOfElements.IsAmalgamation x s ↔ Presieve.FamilyOfElements.IsAmalgamation x'' ↑s
V₁ V₂ V₃ : C
g₁ : V₃ ⟶ V₁
g₂ : V₃ ⟶ V₂
g₃ : V₁ ⟶ U
g₄ : V₂ ⟶ U
S₁ : S'.arrows g₃
S₂ : S'.arrows g₄
e : g₁ ≫ g₃ = g₂ ≫ g₄
⊢ F.map g₁.op (x'' g₃ S₁) = F.map g₂.op (x'' g₄ S₂)

Next tactic:
rw [← FunctorToTypes.map_comp_apply, ← FunctorToTypes.map_comp_apply]
theorem Subpresheaf.sheafify_isSheaf (hF : Presieve.IsSheaf J F) :
    Presieve.IsSheaf J (G.sheafify J).toPresheaf := by
  intro U S hS x hx
  let S' := Sieve.bind S fun Y f hf => G.sieveOfSection (x f hf).1
  have := fun (V) (i : V ⟶ U) (hi : S' i) => hi
  -- porting note: change to explicit variable so that `choose` can find the correct
  -- dependent functions. Thus everything follows need two additional explicit variables.
  choose W i₁ i₂ hi₂ h₁ h₂ using this
  dsimp [-Sieve.bind_apply] at *
  let x'' : Presieve.FamilyOfElements F S' := fun V i hi => F.map (i₁ V i hi).op (x _ (hi₂ V i hi))
  have H : ∀ s, x.IsAmalgamation s ↔ x''.IsAmalgamation s.1 := by
    intro s
    constructor
    · intro H V i hi
      dsimp only [show x'' = fun V i hi => F.map (i₁ V i hi).op (x _ (hi₂ V i hi)) from rfl]
      conv_lhs => rw [← h₂ _ _ hi]
      rw [← H _ (hi₂ _ _ hi)]
      exact FunctorToTypes.map_comp_apply F (i₂ _ _ hi).op (i₁ _ _ hi).op _
    · intro H V i hi
      refine Subtype.ext ?_
      apply (hF _ (x i hi).2).isSeparatedFor.ext
      intro V' i' hi'
      have hi'' : S' (i' ≫ i) := ⟨_, _, _, hi, hi', rfl⟩
      have := H _ hi''
      rw [op_comp, F.map_comp] at this
      refine' this.trans (congr_arg Subtype.val (hx _ _ (hi₂ _ _ hi'') hi (h₂ _ _ hi'')))
  have : x''.Compatible := by
    intro V₁ V₂ V₃ g₁ g₂ g₃ g₄ S₁ S₂ e
    rw [← FunctorToTypes.map_comp_apply, ← FunctorToTypes.map_comp_apply]
    

Proof state:
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F F' F'' : Cᵒᵖ ⥤ Type w
G G' : Subpresheaf F
hF : Presieve.IsSheaf J F
U : C
S : Sieve U
hS : S ∈ sieves J U
x : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows
hx : Presieve.FamilyOfElements.Compatible x
S' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G ↑(x f hf)
W : (V : C) → (i : V ⟶ U) → S'.arrows i → C
i₁ : (V : C) → (i : V ⟶ U) → (hi : S'.arrows i) → V ⟶ W V i hi
i₂ : (V : C) → (i : V ⟶ U) → (hi : S'.arrows i) → W V i hi ⟶ U
hi₂ : ∀ (V : C) (i : V ⟶ U) (hi : S'.arrows i), S.arrows (i₂ V i hi)
h₁ :
  ∀ (V : C) (i : V ⟶ U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G ↑(x f hf)).arrows i),
    F.map (i₁ V i hi).op ↑(x (i₂ V i hi) (_ : S.arrows (i₂ V i hi))) ∈ obj G (op V)
h₂ :
  ∀ (V : C) (i : V ⟶ U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G ↑(x f hf)).arrows i),
    i₁ V i hi ≫ i₂ V i hi = i
x'' : Presieve.FamilyOfElements F S'.arrows :=
  fun V i hi => F.map (i₁ V i hi).op ↑(x (i₂ V i hi) (_ : S.arrows (i₂ V i hi)))
H :
  ∀ (s : (toPresheaf (sheafify J G)).obj (op U)),
    Presieve.FamilyOfElements.IsAmalgamation x s ↔ Presieve.FamilyOfElements.IsAmalgamation x'' ↑s
V₁ V₂ V₃ : C
g₁ : V₃ ⟶ V₁
g₂ : V₃ ⟶ V₂
g₃ : V₁ ⟶ U
g₄ : V₂ ⟶ U
S₁ : S'.arrows g₃
S₂ : S'.arrows g₄
e : g₁ ≫ g₃ = g₂ ≫ g₄
⊢ F.map ((i₁ V₁ g₃ S₁).op ≫ g₁.op) ↑(x (i₂ V₁ g₃ S₁) (_ : S.arrows (i₂ V₁ g₃ S₁))) =
    F.map ((i₁ V₂ g₄ S₂).op ≫ g₂.op) ↑(x (i₂ V₂ g₄ S₂) (_ : S.arrows (i₂ V₂ g₄ S₂)))

Next tactic:
exact
      congr_arg Subtype.val
        (hx (g₁ ≫ i₁ _ _ S₁) (g₂ ≫ i₁ _ _ S₂) (hi₂ _ _ S₁) (hi₂ _ _ S₂)
        (by simp only [Category.assoc, h₂, e]))
theorem Subpresheaf.sheafify_isSheaf (hF : Presieve.IsSheaf J F) :
    Presieve.IsSheaf J (G.sheafify J).toPresheaf := by
  intro U S hS x hx
  let S' := Sieve.bind S fun Y f hf => G.sieveOfSection (x f hf).1
  have := fun (V) (i : V ⟶ U) (hi : S' i) => hi
  -- porting note: change to explicit variable so that `choose` can find the correct
  -- dependent functions. Thus everything follows need two additional explicit variables.
  choose W i₁ i₂ hi₂ h₁ h₂ using this
  dsimp [-Sieve.bind_apply] at *
  let x'' : Presieve.FamilyOfElements F S' := fun V i hi => F.map (i₁ V i hi).op (x _ (hi₂ V i hi))
  have H : ∀ s, x.IsAmalgamation s ↔ x''.IsAmalgamation s.1 := by
    intro s
    constructor
    · intro H V i hi
      dsimp only [show x'' = fun V i hi => F.map (i₁ V i hi).op (x _ (hi₂ V i hi)) from rfl]
      conv_lhs => rw [← h₂ _ _ hi]
      rw [← H _ (hi₂ _ _ hi)]
      exact FunctorToTypes.map_comp_apply F (i₂ _ _ hi).op (i₁ _ _ hi).op _
    · intro H V i hi
      refine Subtype.ext ?_
      apply (hF _ (x i hi).2).isSeparatedFor.ext
      intro V' i' hi'
      have hi'' : S' (i' ≫ i) := ⟨_, _, _, hi, hi', rfl⟩
      have := H _ hi''
      rw [op_comp, F.map_comp] at this
      refine' this.trans (congr_arg Subtype.val (hx _ _ (hi₂ _ _ hi'') hi (h₂ _ _ hi'')))
  have : x''.Compatible := by
    intro V₁ V₂ V₃ g₁ g₂ g₃ g₄ S₁ S₂ e
    rw [← FunctorToTypes.map_comp_apply, ← FunctorToTypes.map_comp_apply]
    exact
      congr_arg Subtype.val
        (hx (g₁ ≫ i₁ _ _ S₁) (g₂ ≫ i₁ _ _ S₂) (hi₂ _ _ S₁) (hi₂ _ _ S₂)
        (by 

Proof state:
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F F' F'' : Cᵒᵖ ⥤ Type w
G G' : Subpresheaf F
hF : Presieve.IsSheaf J F
U : C
S : Sieve U
hS : S ∈ sieves J U
x : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows
hx : Presieve.FamilyOfElements.Compatible x
S' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G ↑(x f hf)
W : (V : C) → (i : V ⟶ U) → S'.arrows i → C
i₁ : (V : C) → (i : V ⟶ U) → (hi : S'.arrows i) → V ⟶ W V i hi
i₂ : (V : C) → (i : V ⟶ U) → (hi : S'.arrows i) → W V i hi ⟶ U
hi₂ : ∀ (V : C) (i : V ⟶ U) (hi : S'.arrows i), S.arrows (i₂ V i hi)
h₁ :
  ∀ (V : C) (i : V ⟶ U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G ↑(x f hf)).arrows i),
    F.map (i₁ V i hi).op ↑(x (i₂ V i hi) (_ : S.arrows (i₂ V i hi))) ∈ obj G (op V)
h₂ :
  ∀ (V : C) (i : V ⟶ U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G ↑(x f hf)).arrows i),
    i₁ V i hi ≫ i₂ V i hi = i
x'' : Presieve.FamilyOfElements F S'.arrows :=
  fun V i hi => F.map (i₁ V i hi).op ↑(x (i₂ V i hi) (_ : S.arrows (i₂ V i hi)))
H :
  ∀ (s : (toPresheaf (sheafify J G)).obj (op U)),
    Presieve.FamilyOfElements.IsAmalgamation x s ↔ Presieve.FamilyOfElements.IsAmalgamation x'' ↑s
V₁ V₂ V₃ : C
g₁ : V₃ ⟶ V₁
g₂ : V₃ ⟶ V₂
g₃ : V₁ ⟶ U
g₄ : V₂ ⟶ U
S₁ : S'.arrows g₃
S₂ : S'.arrows g₄
e : g₁ ≫ g₃ = g₂ ≫ g₄
⊢ (g₁ ≫ i₁ V₁ g₃ S₁) ≫ i₂ V₁ g₃ S₁ = (g₂ ≫ i₁ V₂ g₄ S₂) ≫ i₂ V₂ g₄ S₂

Next tactic:
simp only [Category.assoc, h₂, e]
theorem Subpresheaf.sheafify_isSheaf (hF : Presieve.IsSheaf J F) :
    Presieve.IsSheaf J (G.sheafify J).toPresheaf := by
  intro U S hS x hx
  let S' := Sieve.bind S fun Y f hf => G.sieveOfSection (x f hf).1
  have := fun (V) (i : V ⟶ U) (hi : S' i) => hi
  -- porting note: change to explicit variable so that `choose` can find the correct
  -- dependent functions. Thus everything follows need two additional explicit variables.
  choose W i₁ i₂ hi₂ h₁ h₂ using this
  dsimp [-Sieve.bind_apply] at *
  let x'' : Presieve.FamilyOfElements F S' := fun V i hi => F.map (i₁ V i hi).op (x _ (hi₂ V i hi))
  have H : ∀ s, x.IsAmalgamation s ↔ x''.IsAmalgamation s.1 := by
    intro s
    constructor
    · intro H V i hi
      dsimp only [show x'' = fun V i hi => F.map (i₁ V i hi).op (x _ (hi₂ V i hi)) from rfl]
      conv_lhs => rw [← h₂ _ _ hi]
      rw [← H _ (hi₂ _ _ hi)]
      exact FunctorToTypes.map_comp_apply F (i₂ _ _ hi).op (i₁ _ _ hi).op _
    · intro H V i hi
      refine Subtype.ext ?_
      apply (hF _ (x i hi).2).isSeparatedFor.ext
      intro V' i' hi'
      have hi'' : S' (i' ≫ i) := ⟨_, _, _, hi, hi', rfl⟩
      have := H _ hi''
      rw [op_comp, F.map_comp] at this
      refine' this.trans (congr_arg Subtype.val (hx _ _ (hi₂ _ _ hi'') hi (h₂ _ _ hi'')))
  have : x''.Compatible := by
    intro V₁ V₂ V₃ g₁ g₂ g₃ g₄ S₁ S₂ e
    rw [← FunctorToTypes.map_comp_apply, ← FunctorToTypes.map_comp_apply]
    exact
      congr_arg Subtype.val
        (hx (g₁ ≫ i₁ _ _ S₁) (g₂ ≫ i₁ _ _ S₂) (hi₂ _ _ S₁) (hi₂ _ _ S₂)
        (by simp only [Category.assoc, h₂, e]))
  

Proof state:
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F F' F'' : Cᵒᵖ ⥤ Type w
G G' : Subpresheaf F
hF : Presieve.IsSheaf J F
U : C
S : Sieve U
hS : S ∈ sieves J U
x : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows
hx : Presieve.FamilyOfElements.Compatible x
S' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G ↑(x f hf)
W : (V : C) → (i : V ⟶ U) → S'.arrows i → C
i₁ : (V : C) → (i : V ⟶ U) → (hi : S'.arrows i) → V ⟶ W V i hi
i₂ : (V : C) → (i : V ⟶ U) → (hi : S'.arrows i) → W V i hi ⟶ U
hi₂ : ∀ (V : C) (i : V ⟶ U) (hi : S'.arrows i), S.arrows (i₂ V i hi)
h₁ :
  ∀ (V : C) (i : V ⟶ U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G ↑(x f hf)).arrows i),
    F.map (i₁ V i hi).op ↑(x (i₂ V i hi) (_ : S.arrows (i₂ V i hi))) ∈ obj G (op V)
h₂ :
  ∀ (V : C) (i : V ⟶ U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G ↑(x f hf)).arrows i),
    i₁ V i hi ≫ i₂ V i hi = i
x'' : Presieve.FamilyOfElements F S'.arrows :=
  fun V i hi => F.map (i₁ V i hi).op ↑(x (i₂ V i hi) (_ : S.arrows (i₂ V i hi)))
H :
  ∀ (s : (toPresheaf (sheafify J G)).obj (op U)),
    Presieve.FamilyOfElements.IsAmalgamation x s ↔ Presieve.FamilyOfElements.IsAmalgamation x'' ↑s
this : Presieve.FamilyOfElements.Compatible x''
⊢ ∃! t, Presieve.FamilyOfElements.IsAmalgamation x t

Next tactic:
obtain ⟨t, ht, ht'⟩ := hF _ (J.bind_covering hS fun V i hi => (x i hi).2) _ this
theorem Subpresheaf.isSheaf_iff (h : Presieve.IsSheaf J F) :
    Presieve.IsSheaf J G.toPresheaf ↔
      ∀ (U) (s : F.obj U), G.sieveOfSection s ∈ J (unop U) → s ∈ G.obj U := by
  

Proof state:
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F F' F'' : Cᵒᵖ ⥤ Type w
G G' : Subpresheaf F
h : Presieve.IsSheaf J F
⊢ Presieve.IsSheaf J (toPresheaf G) ↔ ∀ (U : Cᵒᵖ) (s : F.obj U), sieveOfSection G s ∈ sieves J U.unop → s ∈ obj G U

Next tactic:
rw [← G.eq_sheafify_iff h]
theorem Subpresheaf.isSheaf_iff (h : Presieve.IsSheaf J F) :
    Presieve.IsSheaf J G.toPresheaf ↔
      ∀ (U) (s : F.obj U), G.sieveOfSection s ∈ J (unop U) → s ∈ G.obj U := by
  rw [← G.eq_sheafify_iff h]
  

Proof state:
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F F' F'' : Cᵒᵖ ⥤ Type w
G G' : Subpresheaf F
h : Presieve.IsSheaf J F
⊢ G = sheafify J G ↔ ∀ (U : Cᵒᵖ) (s : F.obj U), sieveOfSection G s ∈ sieves J U.unop → s ∈ obj G U

Next tactic:
change _ ↔ G.sheafify J ≤ G
theorem Subpresheaf.isSheaf_iff (h : Presieve.IsSheaf J F) :
    Presieve.IsSheaf J G.toPresheaf ↔
      ∀ (U) (s : F.obj U), G.sieveOfSection s ∈ J (unop U) → s ∈ G.obj U := by
  rw [← G.eq_sheafify_iff h]
  change _ ↔ G.sheafify J ≤ G
  

Proof state:
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F F' F'' : Cᵒᵖ ⥤ Type w
G G' : Subpresheaf F
h : Presieve.IsSheaf J F
⊢ G = sheafify J G ↔ sheafify J G ≤ G

Next tactic:
exact ⟨Eq.ge, (G.le_sheafify J).antisymm⟩
/-- The lift of a presheaf morphism onto the sheafification subpresheaf.  -/
noncomputable def Subpresheaf.sheafifyLift (f : G.toPresheaf ⟶ F') (h : Presieve.IsSheaf J F') :
    (G.sheafify J).toPresheaf ⟶ F' where
  app U s := (h (G.sieveOfSection s.1) s.prop).amalgamate
    (_) ((G.family_of_elements_compatible s.1).compPresheafMap f)
  naturality := by
    

Proof state:
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F F' F'' : Cᵒᵖ ⥤ Type w
G G' : Subpresheaf F
f : toPresheaf G ⟶ F'
h : Presieve.IsSheaf J F'
⊢ ∀ ⦃X Y : Cᵒᵖ⦄ (f_1 : X ⟶ Y),
    (toPresheaf (sheafify J G)).map f_1 ≫
        (fun U s =>
            Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor F' (sieveOfSection G ↑s).arrows)
              (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G ↑s))
              (_ :
                Presieve.FamilyOfElements.Compatible
                  (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G ↑s))))
          Y =
      (fun U s =>
            Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor F' (sieveOfSection G ↑s).arrows)
              (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G ↑s))
              (_ :
                Presieve.FamilyOfElements.Compatible
                  (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G ↑s))))
          X ≫
        F'.map f_1

Next tactic:
intro U V i
/-- The lift of a presheaf morphism onto the sheafification subpresheaf.  -/
noncomputable def Subpresheaf.sheafifyLift (f : G.toPresheaf ⟶ F') (h : Presieve.IsSheaf J F') :
    (G.sheafify J).toPresheaf ⟶ F' where
  app U s := (h (G.sieveOfSection s.1) s.prop).amalgamate
    (_) ((G.family_of_elements_compatible s.1).compPresheafMap f)
  naturality := by
    intro U V i
    

Proof state:
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F F' F'' : Cᵒᵖ ⥤ Type w
G G' : Subpresheaf F
f : toPresheaf G ⟶ F'
h : Presieve.IsSheaf J F'
U V : Cᵒᵖ
i : U ⟶ V
⊢ (toPresheaf (sheafify J G)).map i ≫
      (fun U s =>
          Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor F' (sieveOfSection G ↑s).arrows)
            (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G ↑s))
            (_ :
              Presieve.FamilyOfElements.Compatible
                (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G ↑s))))
        V =
    (fun U s =>
          Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor F' (sieveOfSection G ↑s).arrows)
            (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G ↑s))
            (_ :
              Presieve.FamilyOfElements.Compatible
                (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G ↑s))))
        U ≫
      F'.map i

Next tactic:
ext s
/-- The lift of a presheaf morphism onto the sheafification subpresheaf.  -/
noncomputable def Subpresheaf.sheafifyLift (f : G.toPresheaf ⟶ F') (h : Presieve.IsSheaf J F') :
    (G.sheafify J).toPresheaf ⟶ F' where
  app U s := (h (G.sieveOfSection s.1) s.prop).amalgamate
    (_) ((G.family_of_elements_compatible s.1).compPresheafMap f)
  naturality := by
    intro U V i
    ext s
    

Proof state:
case h
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F F' F'' : Cᵒᵖ ⥤ Type w
G G' : Subpresheaf F
f : toPresheaf G ⟶ F'
h : Presieve.IsSheaf J F'
U V : Cᵒᵖ
i : U ⟶ V
s : (toPresheaf (sheafify J G)).obj U
⊢ ((toPresheaf (sheafify J G)).map i ≫
        (fun U s =>
            Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor F' (sieveOfSection G ↑s).arrows)
              (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G ↑s))
              (_ :
                Presieve.FamilyOfElements.Compatible
                  (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G ↑s))))
          V)
      s =
    ((fun U s =>
            Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor F' (sieveOfSection G ↑s).arrows)
              (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G ↑s))
              (_ :
                Presieve.FamilyOfElements.Compatible
                  (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G ↑s))))
          U ≫
        F'.map i)
      s

Next tactic:
apply (h _ ((Subpresheaf.sheafify J G).toPresheaf.map i s).prop).isSeparatedFor.ext