l3lab/ntp-mathlib huggingface dataset:
theorem isSeparatedFor_iff_generate :
    IsSeparatedFor P R ↔ IsSeparatedFor P (generate R : Presieve X) := by
  constructor
  · intro h x t₁ t₂ ht₁ ht₂
    

Proof state:
case mp
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P Q U : Cᵒᵖ ⥤ Type w
X Y : C
S : Sieve X
R : Presieve X
h : IsSeparatedFor P R
x : FamilyOfElements P (generate R).arrows
t₁ t₂ : P.obj (op X)
ht₁ : FamilyOfElements.IsAmalgamation x t₁
ht₂ : FamilyOfElements.IsAmalgamation x t₂
⊢ t₁ = t₂

Next tactic:
apply h (x.restrict (le_generate R)) t₁ t₂ _ _
theorem isSeparatedFor_iff_generate :
    IsSeparatedFor P R ↔ IsSeparatedFor P (generate R : Presieve X) := by
  constructor
  · intro h x t₁ t₂ ht₁ ht₂
    apply h (x.restrict (le_generate R)) t₁ t₂ _ _
    · 

Proof state:
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P Q U : Cᵒᵖ ⥤ Type w
X Y : C
S : Sieve X
R : Presieve X
h : IsSeparatedFor P R
x : FamilyOfElements P (generate R).arrows
t₁ t₂ : P.obj (op X)
ht₁ : FamilyOfElements.IsAmalgamation x t₁
ht₂ : FamilyOfElements.IsAmalgamation x t₂
⊢ FamilyOfElements.IsAmalgamation (FamilyOfElements.restrict (_ : R ≤ (generate R).arrows) x) t₁

Next tactic:
exact isAmalgamation_restrict _ x t₁ ht₁
theorem isSeparatedFor_iff_generate :
    IsSeparatedFor P R ↔ IsSeparatedFor P (generate R : Presieve X) := by
  constructor
  · intro h x t₁ t₂ ht₁ ht₂
    apply h (x.restrict (le_generate R)) t₁ t₂ _ _
    · exact isAmalgamation_restrict _ x t₁ ht₁
    · 

Proof state:
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P Q U : Cᵒᵖ ⥤ Type w
X Y : C
S : Sieve X
R : Presieve X
h : IsSeparatedFor P R
x : FamilyOfElements P (generate R).arrows
t₁ t₂ : P.obj (op X)
ht₁ : FamilyOfElements.IsAmalgamation x t₁
ht₂ : FamilyOfElements.IsAmalgamation x t₂
⊢ FamilyOfElements.IsAmalgamation (FamilyOfElements.restrict (_ : R ≤ (generate R).arrows) x) t₂

Next tactic:
exact isAmalgamation_restrict _ x t₂ ht₂
theorem isSeparatedFor_iff_generate :
    IsSeparatedFor P R ↔ IsSeparatedFor P (generate R : Presieve X) := by
  constructor
  · intro h x t₁ t₂ ht₁ ht₂
    apply h (x.restrict (le_generate R)) t₁ t₂ _ _
    · exact isAmalgamation_restrict _ x t₁ ht₁
    · exact isAmalgamation_restrict _ x t₂ ht₂
  · 

Proof state:
case mpr
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P Q U : Cᵒᵖ ⥤ Type w
X Y : C
S : Sieve X
R : Presieve X
⊢ IsSeparatedFor P (generate R).arrows → IsSeparatedFor P R

Next tactic:
intro h x t₁ t₂ ht₁ ht₂
theorem isSeparatedFor_iff_generate :
    IsSeparatedFor P R ↔ IsSeparatedFor P (generate R : Presieve X) := by
  constructor
  · intro h x t₁ t₂ ht₁ ht₂
    apply h (x.restrict (le_generate R)) t₁ t₂ _ _
    · exact isAmalgamation_restrict _ x t₁ ht₁
    · exact isAmalgamation_restrict _ x t₂ ht₂
  · intro h x t₁ t₂ ht₁ ht₂
    

Proof state:
case mpr
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P Q U : Cᵒᵖ ⥤ Type w
X Y : C
S : Sieve X
R : Presieve X
h : IsSeparatedFor P (generate R).arrows
x : FamilyOfElements P R
t₁ t₂ : P.obj (op X)
ht₁ : FamilyOfElements.IsAmalgamation x t₁
ht₂ : FamilyOfElements.IsAmalgamation x t₂
⊢ t₁ = t₂

Next tactic:
apply h x.sieveExtend
theorem isSeparatedFor_iff_generate :
    IsSeparatedFor P R ↔ IsSeparatedFor P (generate R : Presieve X) := by
  constructor
  · intro h x t₁ t₂ ht₁ ht₂
    apply h (x.restrict (le_generate R)) t₁ t₂ _ _
    · exact isAmalgamation_restrict _ x t₁ ht₁
    · exact isAmalgamation_restrict _ x t₂ ht₂
  · intro h x t₁ t₂ ht₁ ht₂
    apply h x.sieveExtend
    · 

Proof state:
case mpr.a
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P Q U : Cᵒᵖ ⥤ Type w
X Y : C
S : Sieve X
R : Presieve X
h : IsSeparatedFor P (generate R).arrows
x : FamilyOfElements P R
t₁ t₂ : P.obj (op X)
ht₁ : FamilyOfElements.IsAmalgamation x t₁
ht₂ : FamilyOfElements.IsAmalgamation x t₂
⊢ FamilyOfElements.IsAmalgamation (FamilyOfElements.sieveExtend x) t₁

Next tactic:
exact isAmalgamation_sieveExtend x t₁ ht₁
theorem isSeparatedFor_iff_generate :
    IsSeparatedFor P R ↔ IsSeparatedFor P (generate R : Presieve X) := by
  constructor
  · intro h x t₁ t₂ ht₁ ht₂
    apply h (x.restrict (le_generate R)) t₁ t₂ _ _
    · exact isAmalgamation_restrict _ x t₁ ht₁
    · exact isAmalgamation_restrict _ x t₂ ht₂
  · intro h x t₁ t₂ ht₁ ht₂
    apply h x.sieveExtend
    · exact isAmalgamation_sieveExtend x t₁ ht₁
    · 

Proof state:
case mpr.a
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P Q U : Cᵒᵖ ⥤ Type w
X Y : C
S : Sieve X
R : Presieve X
h : IsSeparatedFor P (generate R).arrows
x : FamilyOfElements P R
t₁ t₂ : P.obj (op X)
ht₁ : FamilyOfElements.IsAmalgamation x t₁
ht₂ : FamilyOfElements.IsAmalgamation x t₂
⊢ FamilyOfElements.IsAmalgamation (FamilyOfElements.sieveExtend x) t₂

Next tactic:
exact isAmalgamation_sieveExtend x t₂ ht₂
theorem isSeparatedFor_top (P : Cᵒᵖ ⥤ Type w) : IsSeparatedFor P (⊤ : Presieve X) :=
  fun x t₁ t₂ h₁ h₂ => by
  

Proof state:
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P✝ Q U : Cᵒᵖ ⥤ Type w
X Y : C
S : Sieve X
R : Presieve X
P : Cᵒᵖ ⥤ Type w
x : FamilyOfElements P ⊤
t₁ t₂ : P.obj (op X)
h₁ : FamilyOfElements.IsAmalgamation x t₁
h₂ : FamilyOfElements.IsAmalgamation x t₂
⊢ t₁ = t₂

Next tactic:
have q₁ := h₁ (𝟙 X) (by tauto)
theorem isSeparatedFor_top (P : Cᵒᵖ ⥤ Type w) : IsSeparatedFor P (⊤ : Presieve X) :=
  fun x t₁ t₂ h₁ h₂ => by
  have q₁ := h₁ (𝟙 X) (by 

Proof state:
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P✝ Q U : Cᵒᵖ ⥤ Type w
X Y : C
S : Sieve X
R : Presieve X
P : Cᵒᵖ ⥤ Type w
x : FamilyOfElements P ⊤
t₁ t₂ : P.obj (op X)
h₁ : FamilyOfElements.IsAmalgamation x t₁
h₂ : FamilyOfElements.IsAmalgamation x t₂
⊢ ⊤ (𝟙 X)

Next tactic:
tauto
theorem isSeparatedFor_top (P : Cᵒᵖ ⥤ Type w) : IsSeparatedFor P (⊤ : Presieve X) :=
  fun x t₁ t₂ h₁ h₂ => by
  have q₁ := h₁ (𝟙 X) (by tauto)
  

Proof state:
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P✝ Q U : Cᵒᵖ ⥤ Type w
X Y : C
S : Sieve X
R : Presieve X
P : Cᵒᵖ ⥤ Type w
x : FamilyOfElements P ⊤
t₁ t₂ : P.obj (op X)
h₁ : FamilyOfElements.IsAmalgamation x t₁
h₂ : FamilyOfElements.IsAmalgamation x t₂
q₁ : P.map (𝟙 X).op t₁ = x (𝟙 X) (_ : True)
⊢ t₁ = t₂

Next tactic:
have q₂ := h₂ (𝟙 X) (by tauto)
theorem isSeparatedFor_top (P : Cᵒᵖ ⥤ Type w) : IsSeparatedFor P (⊤ : Presieve X) :=
  fun x t₁ t₂ h₁ h₂ => by
  have q₁ := h₁ (𝟙 X) (by tauto)
  have q₂ := h₂ (𝟙 X) (by 

Proof state:
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P✝ Q U : Cᵒᵖ ⥤ Type w
X Y : C
S : Sieve X
R : Presieve X
P : Cᵒᵖ ⥤ Type w
x : FamilyOfElements P ⊤
t₁ t₂ : P.obj (op X)
h₁ : FamilyOfElements.IsAmalgamation x t₁
h₂ : FamilyOfElements.IsAmalgamation x t₂
q₁ : P.map (𝟙 X).op t₁ = x (𝟙 X) (_ : True)
⊢ ⊤ (𝟙 X)

Next tactic:
tauto
theorem isSeparatedFor_top (P : Cᵒᵖ ⥤ Type w) : IsSeparatedFor P (⊤ : Presieve X) :=
  fun x t₁ t₂ h₁ h₂ => by
  have q₁ := h₁ (𝟙 X) (by tauto)
  have q₂ := h₂ (𝟙 X) (by tauto)
  

Proof state:
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P✝ Q U : Cᵒᵖ ⥤ Type w
X Y : C
S : Sieve X
R : Presieve X
P : Cᵒᵖ ⥤ Type w
x : FamilyOfElements P ⊤
t₁ t₂ : P.obj (op X)
h₁ : FamilyOfElements.IsAmalgamation x t₁
h₂ : FamilyOfElements.IsAmalgamation x t₂
q₁ : P.map (𝟙 X).op t₁ = x (𝟙 X) (_ : True)
q₂ : P.map (𝟙 X).op t₂ = x (𝟙 X) (_ : True)
⊢ t₁ = t₂

Next tactic:
simp only [op_id, FunctorToTypes.map_id_apply] at q₁ q₂
theorem isSeparatedFor_top (P : Cᵒᵖ ⥤ Type w) : IsSeparatedFor P (⊤ : Presieve X) :=
  fun x t₁ t₂ h₁ h₂ => by
  have q₁ := h₁ (𝟙 X) (by tauto)
  have q₂ := h₂ (𝟙 X) (by tauto)
  simp only [op_id, FunctorToTypes.map_id_apply] at q₁ q₂
  

Proof state:
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P✝ Q U : Cᵒᵖ ⥤ Type w
X Y : C
S : Sieve X
R : Presieve X
P : Cᵒᵖ ⥤ Type w
x : FamilyOfElements P ⊤
t₁ t₂ : P.obj (op X)
h₁ : FamilyOfElements.IsAmalgamation x t₁
h₂ : FamilyOfElements.IsAmalgamation x t₂
q₁ : t₁ = x (𝟙 X) (_ : True)
q₂ : t₂ = x (𝟙 X) (_ : True)
⊢ t₁ = t₂

Next tactic:
rw [q₁, q₂]
/-- (Implementation). This is a (primarily internal) equivalence between natural transformations
and compatible families.

Cf the discussion after Lemma 7.47.10 in <https://stacks.math.columbia.edu/tag/00YW>. See also
the proof of C2.1.4 of [Elephant], and the discussion in [MM92], Chapter III, Section 4.
-/
def natTransEquivCompatibleFamily {P : Cᵒᵖ ⥤ Type v₁} :
    (S.functor ⟶ P) ≃ { x : FamilyOfElements P (S : Presieve X) // x.Compatible } where
  toFun α := by
    

Proof state:
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P✝ Q U : Cᵒᵖ ⥤ Type w
X Y : C
S : Sieve X
R : Presieve X
P : Cᵒᵖ ⥤ Type v₁
α : functor S ⟶ P
⊢ { x // FamilyOfElements.Compatible x }

Next tactic:
refine' ⟨fun Y f hf => _, _⟩
/-- (Implementation). This is a (primarily internal) equivalence between natural transformations
and compatible families.

Cf the discussion after Lemma 7.47.10 in <https://stacks.math.columbia.edu/tag/00YW>. See also
the proof of C2.1.4 of [Elephant], and the discussion in [MM92], Chapter III, Section 4.
-/
def natTransEquivCompatibleFamily {P : Cᵒᵖ ⥤ Type v₁} :
    (S.functor ⟶ P) ≃ { x : FamilyOfElements P (S : Presieve X) // x.Compatible } where
  toFun α := by
    refine' ⟨fun Y f hf => _, _⟩
    · 

Proof state:
case refine'_1
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P✝ Q U : Cᵒᵖ ⥤ Type w
X Y✝ : C
S : Sieve X
R : Presieve X
P : Cᵒᵖ ⥤ Type v₁
α : functor S ⟶ P
Y : C
f : Y ⟶ X
hf : S.arrows f
⊢ P.obj (op Y)

Next tactic:
apply α.app (op Y) ⟨_, hf⟩
/-- (Implementation). This is a (primarily internal) equivalence between natural transformations
and compatible families.

Cf the discussion after Lemma 7.47.10 in <https://stacks.math.columbia.edu/tag/00YW>. See also
the proof of C2.1.4 of [Elephant], and the discussion in [MM92], Chapter III, Section 4.
-/
def natTransEquivCompatibleFamily {P : Cᵒᵖ ⥤ Type v₁} :
    (S.functor ⟶ P) ≃ { x : FamilyOfElements P (S : Presieve X) // x.Compatible } where
  toFun α := by
    refine' ⟨fun Y f hf => _, _⟩
    · apply α.app (op Y) ⟨_, hf⟩
    · 

Proof state:
case refine'_2
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P✝ Q U : Cᵒᵖ ⥤ Type w
X Y : C
S : Sieve X
R : Presieve X
P : Cᵒᵖ ⥤ Type v₁
α : functor S ⟶ P
⊢ FamilyOfElements.Compatible fun Y f hf => α.app (op Y) { val := f, property := hf }

Next tactic:
rw [compatible_iff_sieveCompatible]
/-- (Implementation). This is a (primarily internal) equivalence between natural transformations
and compatible families.

Cf the discussion after Lemma 7.47.10 in <https://stacks.math.columbia.edu/tag/00YW>. See also
the proof of C2.1.4 of [Elephant], and the discussion in [MM92], Chapter III, Section 4.
-/
def natTransEquivCompatibleFamily {P : Cᵒᵖ ⥤ Type v₁} :
    (S.functor ⟶ P) ≃ { x : FamilyOfElements P (S : Presieve X) // x.Compatible } where
  toFun α := by
    refine' ⟨fun Y f hf => _, _⟩
    · apply α.app (op Y) ⟨_, hf⟩
    · rw [compatible_iff_sieveCompatible]
      

Proof state:
case refine'_2
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P✝ Q U : Cᵒᵖ ⥤ Type w
X Y : C
S : Sieve X
R : Presieve X
P : Cᵒᵖ ⥤ Type v₁
α : functor S ⟶ P
⊢ FamilyOfElements.SieveCompatible fun Y f hf => α.app (op Y) { val := f, property := hf }

Next tactic:
intro Y Z f g hf
/-- (Implementation). This is a (primarily internal) equivalence between natural transformations
and compatible families.

Cf the discussion after Lemma 7.47.10 in <https://stacks.math.columbia.edu/tag/00YW>. See also
the proof of C2.1.4 of [Elephant], and the discussion in [MM92], Chapter III, Section 4.
-/
def natTransEquivCompatibleFamily {P : Cᵒᵖ ⥤ Type v₁} :
    (S.functor ⟶ P) ≃ { x : FamilyOfElements P (S : Presieve X) // x.Compatible } where
  toFun α := by
    refine' ⟨fun Y f hf => _, _⟩
    · apply α.app (op Y) ⟨_, hf⟩
    · rw [compatible_iff_sieveCompatible]
      intro Y Z f g hf
      

Proof state:
case refine'_2
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P✝ Q U : Cᵒᵖ ⥤ Type w
X Y✝ : C
S : Sieve X
R : Presieve X
P : Cᵒᵖ ⥤ Type v₁
α : functor S ⟶ P
Y Z : C
f : Y ⟶ X
g : Z ⟶ Y
hf : S.arrows f
⊢ (fun Y f hf => α.app (op Y) { val := f, property := hf }) Z (g ≫ f) (_ : S.arrows (g ≫ f)) =
    P.map g.op ((fun Y f hf => α.app (op Y) { val := f, property := hf }) Y f hf)

Next tactic:
dsimp
/-- (Implementation). This is a (primarily internal) equivalence between natural transformations
and compatible families.

Cf the discussion after Lemma 7.47.10 in <https://stacks.math.columbia.edu/tag/00YW>. See also
the proof of C2.1.4 of [Elephant], and the discussion in [MM92], Chapter III, Section 4.
-/
def natTransEquivCompatibleFamily {P : Cᵒᵖ ⥤ Type v₁} :
    (S.functor ⟶ P) ≃ { x : FamilyOfElements P (S : Presieve X) // x.Compatible } where
  toFun α := by
    refine' ⟨fun Y f hf => _, _⟩
    · apply α.app (op Y) ⟨_, hf⟩
    · rw [compatible_iff_sieveCompatible]
      intro Y Z f g hf
      dsimp
      

Proof state:
case refine'_2
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P✝ Q U : Cᵒᵖ ⥤ Type w
X Y✝ : C
S : Sieve X
R : Presieve X
P : Cᵒᵖ ⥤ Type v₁
α : functor S ⟶ P
Y Z : C
f : Y ⟶ X
g : Z ⟶ Y
hf : S.arrows f
⊢ α.app (op Z) { val := g ≫ f, property := (_ : S.arrows (g ≫ f)) } =
    P.map g.op (α.app (op Y) { val := f, property := hf })

Next tactic:
rw [← FunctorToTypes.naturality _ _ α g.op]
/-- (Implementation). This is a (primarily internal) equivalence between natural transformations
and compatible families.

Cf the discussion after Lemma 7.47.10 in <https://stacks.math.columbia.edu/tag/00YW>. See also
the proof of C2.1.4 of [Elephant], and the discussion in [MM92], Chapter III, Section 4.
-/
def natTransEquivCompatibleFamily {P : Cᵒᵖ ⥤ Type v₁} :
    (S.functor ⟶ P) ≃ { x : FamilyOfElements P (S : Presieve X) // x.Compatible } where
  toFun α := by
    refine' ⟨fun Y f hf => _, _⟩
    · apply α.app (op Y) ⟨_, hf⟩
    · rw [compatible_iff_sieveCompatible]
      intro Y Z f g hf
      dsimp
      rw [← FunctorToTypes.naturality _ _ α g.op]
      

Proof state:
case refine'_2
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P✝ Q U : Cᵒᵖ ⥤ Type w
X Y✝ : C
S : Sieve X
R : Presieve X
P : Cᵒᵖ ⥤ Type v₁
α : functor S ⟶ P
Y Z : C
f : Y ⟶ X
g : Z ⟶ Y
hf : S.arrows f
⊢ α.app (op Z) { val := g ≫ f, property := (_ : S.arrows (g ≫ f)) } =
    α.app (op Z) ((functor S).map g.op { val := f, property := hf })

Next tactic:
rfl
/-- (Implementation). This is a (primarily internal) equivalence between natural transformations
and compatible families.

Cf the discussion after Lemma 7.47.10 in <https://stacks.math.columbia.edu/tag/00YW>. See also
the proof of C2.1.4 of [Elephant], and the discussion in [MM92], Chapter III, Section 4.
-/
def natTransEquivCompatibleFamily {P : Cᵒᵖ ⥤ Type v₁} :
    (S.functor ⟶ P) ≃ { x : FamilyOfElements P (S : Presieve X) // x.Compatible } where
  toFun α := by
    refine' ⟨fun Y f hf => _, _⟩
    · apply α.app (op Y) ⟨_, hf⟩
    · rw [compatible_iff_sieveCompatible]
      intro Y Z f g hf
      dsimp
      rw [← FunctorToTypes.naturality _ _ α g.op]
      rfl
  invFun t :=
    { app := fun Y f => t.1 _ f.2
      naturality := fun Y Z g => by
        

Proof state:
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P✝ Q U : Cᵒᵖ ⥤ Type w
X Y✝ : C
S : Sieve X
R : Presieve X
P : Cᵒᵖ ⥤ Type v₁
t : { x // FamilyOfElements.Compatible x }
Y Z : Cᵒᵖ
g : Y ⟶ Z
⊢ (functor S).map g ≫ (fun Y f => ↑t ↑f (_ : S.arrows ↑f)) Z = (fun Y f => ↑t ↑f (_ : S.arrows ↑f)) Y ≫ P.map g

Next tactic:
ext ⟨f, hf⟩
/-- (Implementation). This is a (primarily internal) equivalence between natural transformations
and compatible families.

Cf the discussion after Lemma 7.47.10 in <https://stacks.math.columbia.edu/tag/00YW>. See also
the proof of C2.1.4 of [Elephant], and the discussion in [MM92], Chapter III, Section 4.
-/
def natTransEquivCompatibleFamily {P : Cᵒᵖ ⥤ Type v₁} :
    (S.functor ⟶ P) ≃ { x : FamilyOfElements P (S : Presieve X) // x.Compatible } where
  toFun α := by
    refine' ⟨fun Y f hf => _, _⟩
    · apply α.app (op Y) ⟨_, hf⟩
    · rw [compatible_iff_sieveCompatible]
      intro Y Z f g hf
      dsimp
      rw [← FunctorToTypes.naturality _ _ α g.op]
      rfl
  invFun t :=
    { app := fun Y f => t.1 _ f.2
      naturality := fun Y Z g => by
        ext ⟨f, hf⟩
        

Proof state:
case h.mk
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P✝ Q U : Cᵒᵖ ⥤ Type w
X Y✝ : C
S : Sieve X
R : Presieve X
P : Cᵒᵖ ⥤ Type v₁
t : { x // FamilyOfElements.Compatible x }
Y Z : Cᵒᵖ
g : Y ⟶ Z
f : Y.unop ⟶ X
hf : S.arrows f
⊢ ((functor S).map g ≫ (fun Y f => ↑t ↑f (_ : S.arrows ↑f)) Z) { val := f, property := hf } =
    ((fun Y f => ↑t ↑f (_ : S.arrows ↑f)) Y ≫ P.map g) { val := f, property := hf }

Next tactic:
apply t.2.to_sieveCompatible _
/-- (Implementation). This is a (primarily internal) equivalence between natural transformations
and compatible families.

Cf the discussion after Lemma 7.47.10 in <https://stacks.math.columbia.edu/tag/00YW>. See also
the proof of C2.1.4 of [Elephant], and the discussion in [MM92], Chapter III, Section 4.
-/
def natTransEquivCompatibleFamily {P : Cᵒᵖ ⥤ Type v₁} :
    (S.functor ⟶ P) ≃ { x : FamilyOfElements P (S : Presieve X) // x.Compatible } where
  toFun α := by
    refine' ⟨fun Y f hf => _, _⟩
    · apply α.app (op Y) ⟨_, hf⟩
    · rw [compatible_iff_sieveCompatible]
      intro Y Z f g hf
      dsimp
      rw [← FunctorToTypes.naturality _ _ α g.op]
      rfl
  invFun t :=
    { app := fun Y f => t.1 _ f.2
      naturality := fun Y Z g => by
        ext ⟨f, hf⟩
        apply t.2.to_sieveCompatible _ }
  left_inv α := by
    

Proof state:
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P✝ Q U : Cᵒᵖ ⥤ Type w
X Y : C
S : Sieve X
R : Presieve X
P : Cᵒᵖ ⥤ Type v₁
α : functor S ⟶ P
⊢ (fun t => NatTrans.mk fun Y f => ↑t ↑f (_ : S.arrows ↑f))
      ((fun α =>
          { val := fun Y f hf => α.app (op Y) { val := f, property := hf },
            property := (_ : FamilyOfElements.Compatible fun Y f hf => α.app (op Y) { val := f, property := hf }) })
        α) =
    α

Next tactic:
ext X ⟨_, _⟩
/-- (Implementation). This is a (primarily internal) equivalence between natural transformations
and compatible families.

Cf the discussion after Lemma 7.47.10 in <https://stacks.math.columbia.edu/tag/00YW>. See also
the proof of C2.1.4 of [Elephant], and the discussion in [MM92], Chapter III, Section 4.
-/
def natTransEquivCompatibleFamily {P : Cᵒᵖ ⥤ Type v₁} :
    (S.functor ⟶ P) ≃ { x : FamilyOfElements P (S : Presieve X) // x.Compatible } where
  toFun α := by
    refine' ⟨fun Y f hf => _, _⟩
    · apply α.app (op Y) ⟨_, hf⟩
    · rw [compatible_iff_sieveCompatible]
      intro Y Z f g hf
      dsimp
      rw [← FunctorToTypes.naturality _ _ α g.op]
      rfl
  invFun t :=
    { app := fun Y f => t.1 _ f.2
      naturality := fun Y Z g => by
        ext ⟨f, hf⟩
        apply t.2.to_sieveCompatible _ }
  left_inv α := by
    ext X ⟨_, _⟩
    

Proof state:
case w.h.h.mk
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P✝ Q U : Cᵒᵖ ⥤ Type w
X✝ Y : C
S : Sieve X✝
R : Presieve X✝
P : Cᵒᵖ ⥤ Type v₁
α : functor S ⟶ P
X : Cᵒᵖ
val✝ : X.unop ⟶ X✝
property✝ : S.arrows val✝
⊢ ((fun t => NatTrans.mk fun Y f => ↑t ↑f (_ : S.arrows ↑f))
          ((fun α =>
              { val := fun Y f hf => α.app (op Y) { val := f, property := hf },
                property := (_ : FamilyOfElements.Compatible fun Y f hf => α.app (op Y) { val := f, property := hf }) })
            α)).app
      X { val := val✝, property := property✝ } =
    α.app X { val := val✝, property := property✝ }

Next tactic:
rfl
/-- (Implementation). This is a (primarily internal) equivalence between natural transformations
and compatible families.

Cf the discussion after Lemma 7.47.10 in <https://stacks.math.columbia.edu/tag/00YW>. See also
the proof of C2.1.4 of [Elephant], and the discussion in [MM92], Chapter III, Section 4.
-/
def natTransEquivCompatibleFamily {P : Cᵒᵖ ⥤ Type v₁} :
    (S.functor ⟶ P) ≃ { x : FamilyOfElements P (S : Presieve X) // x.Compatible } where
  toFun α := by
    refine' ⟨fun Y f hf => _, _⟩
    · apply α.app (op Y) ⟨_, hf⟩
    · rw [compatible_iff_sieveCompatible]
      intro Y Z f g hf
      dsimp
      rw [← FunctorToTypes.naturality _ _ α g.op]
      rfl
  invFun t :=
    { app := fun Y f => t.1 _ f.2
      naturality := fun Y Z g => by
        ext ⟨f, hf⟩
        apply t.2.to_sieveCompatible _ }
  left_inv α := by
    ext X ⟨_, _⟩
    rfl
  right_inv := by
    

Proof state:
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P✝ Q U : Cᵒᵖ ⥤ Type w
X Y : C
S : Sieve X
R : Presieve X
P : Cᵒᵖ ⥤ Type v₁
⊢ Function.RightInverse (fun t => NatTrans.mk fun Y f => ↑t ↑f (_ : S.arrows ↑f)) fun α =>
    { val := fun Y f hf => α.app (op Y) { val := f, property := hf },
      property := (_ : FamilyOfElements.Compatible fun Y f hf => α.app (op Y) { val := f, property := hf }) }

Next tactic:
rintro ⟨x, hx⟩
/-- (Implementation). This is a (primarily internal) equivalence between natural transformations
and compatible families.

Cf the discussion after Lemma 7.47.10 in <https://stacks.math.columbia.edu/tag/00YW>. See also
the proof of C2.1.4 of [Elephant], and the discussion in [MM92], Chapter III, Section 4.
-/
def natTransEquivCompatibleFamily {P : Cᵒᵖ ⥤ Type v₁} :
    (S.functor ⟶ P) ≃ { x : FamilyOfElements P (S : Presieve X) // x.Compatible } where
  toFun α := by
    refine' ⟨fun Y f hf => _, _⟩
    · apply α.app (op Y) ⟨_, hf⟩
    · rw [compatible_iff_sieveCompatible]
      intro Y Z f g hf
      dsimp
      rw [← FunctorToTypes.naturality _ _ α g.op]
      rfl
  invFun t :=
    { app := fun Y f => t.1 _ f.2
      naturality := fun Y Z g => by
        ext ⟨f, hf⟩
        apply t.2.to_sieveCompatible _ }
  left_inv α := by
    ext X ⟨_, _⟩
    rfl
  right_inv := by
    rintro ⟨x, hx⟩
    

Proof state:
case mk
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P✝ Q U : Cᵒᵖ ⥤ Type w
X Y : C
S : Sieve X
R : Presieve X
P : Cᵒᵖ ⥤ Type v₁
x : FamilyOfElements P S.arrows
hx : FamilyOfElements.Compatible x
⊢ (fun α =>
        { val := fun Y f hf => α.app (op Y) { val := f, property := hf },
          property := (_ : FamilyOfElements.Compatible fun Y f hf => α.app (op Y) { val := f, property := hf }) })
      ((fun t => NatTrans.mk fun Y f => ↑t ↑f (_ : S.arrows ↑f)) { val := x, property := hx }) =
    { val := x, property := hx }

Next tactic:
rfl
/-- (Implementation). A lemma useful to prove `isSheafFor_iff_yonedaSheafCondition`. -/
theorem extension_iff_amalgamation {P : Cᵒᵖ ⥤ Type v₁} (x : S.functor ⟶ P) (g : yoneda.obj X ⟶ P) :
    S.functorInclusion ≫ g = x ↔
      (natTransEquivCompatibleFamily x).1.IsAmalgamation (yonedaEquiv g) := by
  

Proof state:
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P✝ Q U : Cᵒᵖ ⥤ Type w
X Y : C
S : Sieve X
R : Presieve X
P : Cᵒᵖ ⥤ Type v₁
x : functor S ⟶ P
g : yoneda.obj X ⟶ P
⊢ functorInclusion S ≫ g = x ↔ FamilyOfElements.IsAmalgamation (↑(natTransEquivCompatibleFamily x)) (yonedaEquiv g)

Next tactic:
change _ ↔ ∀ ⦃Y : C⦄ (f : Y ⟶ X) (h : S f), P.map f.op (yonedaEquiv g) = x.app (op Y) ⟨f, h⟩
/-- (Implementation). A lemma useful to prove `isSheafFor_iff_yonedaSheafCondition`. -/
theorem extension_iff_amalgamation {P : Cᵒᵖ ⥤ Type v₁} (x : S.functor ⟶ P) (g : yoneda.obj X ⟶ P) :
    S.functorInclusion ≫ g = x ↔
      (natTransEquivCompatibleFamily x).1.IsAmalgamation (yonedaEquiv g) := by
  change _ ↔ ∀ ⦃Y : C⦄ (f : Y ⟶ X) (h : S f), P.map f.op (yonedaEquiv g) = x.app (op Y) ⟨f, h⟩
  

Proof state:
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P✝ Q U : Cᵒᵖ ⥤ Type w
X Y : C
S : Sieve X
R : Presieve X
P : Cᵒᵖ ⥤ Type v₁
x : functor S ⟶ P
g : yoneda.obj X ⟶ P
⊢ functorInclusion S ≫ g = x ↔
    ∀ ⦃Y : C⦄ (f : Y ⟶ X) (h : S.arrows f), P.map f.op (yonedaEquiv g) = x.app (op Y) { val := f, property := h }

Next tactic:
constructor
/-- (Implementation). A lemma useful to prove `isSheafFor_iff_yonedaSheafCondition`. -/
theorem extension_iff_amalgamation {P : Cᵒᵖ ⥤ Type v₁} (x : S.functor ⟶ P) (g : yoneda.obj X ⟶ P) :
    S.functorInclusion ≫ g = x ↔
      (natTransEquivCompatibleFamily x).1.IsAmalgamation (yonedaEquiv g) := by
  change _ ↔ ∀ ⦃Y : C⦄ (f : Y ⟶ X) (h : S f), P.map f.op (yonedaEquiv g) = x.app (op Y) ⟨f, h⟩
  constructor
  · 

Proof state:
case mp
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P✝ Q U : Cᵒᵖ ⥤ Type w
X Y : C
S : Sieve X
R : Presieve X
P : Cᵒᵖ ⥤ Type v₁
x : functor S ⟶ P
g : yoneda.obj X ⟶ P
⊢ functorInclusion S ≫ g = x →
    ∀ ⦃Y : C⦄ (f : Y ⟶ X) (h : S.arrows f), P.map f.op (yonedaEquiv g) = x.app (op Y) { val := f, property := h }

Next tactic:
rintro rfl Y f hf