l3lab/ntp-mathlib huggingface dataset:
theorem Subpresheaf.eq_sheafify (h : Presieve.IsSheaf J F) (hG : Presieve.IsSheaf J G.toPresheaf) :
    G = G.sheafify J := by
  apply (G.le_sheafify J).antisymm
  intro U s hs
  suffices ((hG _ hs).amalgamate _ (G.family_of_elements_compatible s)).1 = s by
    rw [← this]
    exact ((hG _ hs).amalgamate _ (G.family_of_elements_compatible s)).2
  

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
hG : Presieve.IsSheaf J (toPresheaf G)
U : Cᵒᵖ
s : F.obj U
hs : s ∈ obj (sheafify J G) U
⊢ ↑(Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor (toPresheaf G) (sieveOfSection G s).arrows)
        (familyOfElementsOfSection G s) (_ : Presieve.FamilyOfElements.Compatible (familyOfElementsOfSection G s))) =
    s

Next tactic:
apply (h _ hs).isSeparatedFor.ext
theorem Subpresheaf.eq_sheafify (h : Presieve.IsSheaf J F) (hG : Presieve.IsSheaf J G.toPresheaf) :
    G = G.sheafify J := by
  apply (G.le_sheafify J).antisymm
  intro U s hs
  suffices ((hG _ hs).amalgamate _ (G.family_of_elements_compatible s)).1 = s by
    rw [← this]
    exact ((hG _ hs).amalgamate _ (G.family_of_elements_compatible s)).2
  apply (h _ hs).isSeparatedFor.ext
  

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
hG : Presieve.IsSheaf J (toPresheaf G)
U : Cᵒᵖ
s : F.obj U
hs : s ∈ obj (sheafify J G) U
⊢ ∀ ⦃Y : C⦄ ⦃f : Y ⟶ U.unop⦄,
    (sieveOfSection G s).arrows f →
      F.map f.op
          ↑(Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor (toPresheaf G) (sieveOfSection G s).arrows)
              (familyOfElementsOfSection G s)
              (_ : Presieve.FamilyOfElements.Compatible (familyOfElementsOfSection G s))) =
        F.map f.op s

Next tactic:
intro V i hi
theorem Subpresheaf.eq_sheafify (h : Presieve.IsSheaf J F) (hG : Presieve.IsSheaf J G.toPresheaf) :
    G = G.sheafify J := by
  apply (G.le_sheafify J).antisymm
  intro U s hs
  suffices ((hG _ hs).amalgamate _ (G.family_of_elements_compatible s)).1 = s by
    rw [← this]
    exact ((hG _ hs).amalgamate _ (G.family_of_elements_compatible s)).2
  apply (h _ hs).isSeparatedFor.ext
  intro V i hi
  

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
hG : Presieve.IsSheaf J (toPresheaf G)
U : Cᵒᵖ
s : F.obj U
hs : s ∈ obj (sheafify J G) U
V : C
i : V ⟶ U.unop
hi : (sieveOfSection G s).arrows i
⊢ F.map i.op
      ↑(Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor (toPresheaf G) (sieveOfSection G s).arrows)
          (familyOfElementsOfSection G s) (_ : Presieve.FamilyOfElements.Compatible (familyOfElementsOfSection G s))) =
    F.map i.op s

Next tactic:
exact (congr_arg Subtype.val ((hG _ hs).valid_glue (G.family_of_elements_compatible s) _ hi) : _)
theorem Subpresheaf.sheafify_isSheaf (hF : Presieve.IsSheaf J F) :
    Presieve.IsSheaf J (G.sheafify J).toPresheaf := 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
⊢ Presieve.IsSheaf J (toPresheaf (sheafify J G))

Next tactic:
intro U S hS x hx
theorem Subpresheaf.sheafify_isSheaf (hF : Presieve.IsSheaf J F) :
    Presieve.IsSheaf J (G.sheafify J).toPresheaf := by
  intro U S hS x hx
  

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
⊢ ∃! t, Presieve.FamilyOfElements.IsAmalgamation x t

Next tactic:
let S' := Sieve.bind S fun Y f hf => G.sieveOfSection (x f hf).1
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
  

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)
⊢ ∃! t, Presieve.FamilyOfElements.IsAmalgamation x t

Next tactic:
have := fun (V) (i : V ⟶ U) (hi : S' i) => hi
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.
  

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)
this : ∀ (V : C) (i : V ⟶ U), S'.arrows i → S'.arrows i
⊢ ∃! t, Presieve.FamilyOfElements.IsAmalgamation x t

Next tactic:
choose W i₁ i₂ hi₂ h₁ h₂ using this
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
  

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 : S'.arrows i),
    (fun Y f h => ((fun Y f hf => sieveOfSection G ↑(x f hf)) Y f h).arrows) (W V i hi) (i₂ V i hi)
      (_ : S.arrows (i₂ V i hi)) (i₁ V i hi)
h₂ : ∀ (V : C) (i : V ⟶ U) (hi : S'.arrows i), i₁ V i hi ≫ i₂ V i hi = i
⊢ ∃! t, Presieve.FamilyOfElements.IsAmalgamation x t

Next tactic:
dsimp [-Sieve.bind_apply] at *
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 *
  

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
⊢ ∃! t, Presieve.FamilyOfElements.IsAmalgamation x t

Next tactic:
let x'' : Presieve.FamilyOfElements F S' := fun V i hi => F.map (i₁ V i hi).op (x _ (hi₂ V i hi))
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))
  

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)))
⊢ ∃! t, Presieve.FamilyOfElements.IsAmalgamation x t

Next tactic:
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'')))
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
    

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)))
⊢ ∀ (s : (toPresheaf (sheafify J G)).obj (op U)),
    Presieve.FamilyOfElements.IsAmalgamation x s ↔ Presieve.FamilyOfElements.IsAmalgamation x'' ↑s

Next tactic:
intro s
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
    

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)))
s : (toPresheaf (sheafify J G)).obj (op U)
⊢ Presieve.FamilyOfElements.IsAmalgamation x s ↔ Presieve.FamilyOfElements.IsAmalgamation x'' ↑s

Next tactic:
constructor
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
    · 

Proof state:
case mp
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)))
s : (toPresheaf (sheafify J G)).obj (op U)
⊢ Presieve.FamilyOfElements.IsAmalgamation x s → Presieve.FamilyOfElements.IsAmalgamation x'' ↑s

Next tactic:
intro H V i hi
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
      

Proof state:
case mp
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)))
s : (toPresheaf (sheafify J G)).obj (op U)
H : Presieve.FamilyOfElements.IsAmalgamation x s
V : C
i : V ⟶ U
hi : S'.arrows i
⊢ F.map i.op ↑s = x'' i hi

Next tactic:
dsimp only [show x'' = fun V i hi => F.map (i₁ V i hi).op (x _ (hi₂ V i hi)) from rfl]
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]
      

Proof state:
case mp
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)))
s : (toPresheaf (sheafify J G)).obj (op U)
H : Presieve.FamilyOfElements.IsAmalgamation x s
V : C
i : V ⟶ U
hi : S'.arrows i
⊢ F.map i.op ↑s = F.map (i₁ V i hi).op ↑(x (i₂ V i hi) (_ : S.arrows (i₂ V i hi)))

Next tactic:
conv_lhs => rw [← h₂ _ _ hi]
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 => 

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)))
s : (toPresheaf (sheafify J G)).obj (op U)
H : Presieve.FamilyOfElements.IsAmalgamation x s
V : C
i : V ⟶ U
hi : S'.arrows i
| F.map i.op ↑s

Next tactic:
rw [← h₂ _ _ hi]