l3lab/ntp-mathlib huggingface dataset:
/-- Given sieve `S` and presheaf `P : Cᵒᵖ ⥤ A`, their natural associated cone admits at most one
    morphism from every cone in the same category (i.e. over the same diagram),
    iff `Hom (E, P -)`is separated for the sieve `S` and all `E : A`. -/
theorem subsingleton_iff_isSeparatedFor :
    (∀ c, Subsingleton (c ⟶ P.mapCone S.arrows.cocone.op)) ↔
      ∀ E : Aᵒᵖ, IsSeparatedFor (P ⋙ coyoneda.obj E) S.arrows := by
  constructor
  · intro hs E x t₁ t₂ h₁ h₂
    have hx := is_compatible_of_exists_amalgamation x ⟨t₁, h₁⟩
    rw [compatible_iff_sieveCompatible] at hx
    specialize hs hx.cone
    rcases hs with ⟨hs⟩
    

Proof state:
case mp.intro
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
A : Type u₂
inst✝ : Category.{v₂, u₂} A
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ A
X : C
S : Sieve X
R : Presieve X
E✝ : Aᵒᵖ
x✝ : FamilyOfElements (P ⋙ coyoneda.obj E✝) S.arrows
hx✝ : SieveCompatible x✝
E : Aᵒᵖ
x : FamilyOfElements (P ⋙ coyoneda.obj E) S.arrows
t₁ t₂ : (P ⋙ coyoneda.obj E).obj (op X)
h₁ : IsAmalgamation x t₁
h₂ : IsAmalgamation x t₂
hx : SieveCompatible x
hs : ∀ (a b : SieveCompatible.cone hx ⟶ P.mapCone (Cocone.op (cocone S.arrows))), a = b
⊢ t₁ = t₂

Next tactic:
simpa only [Subtype.mk.injEq] using (show Subtype.mk t₁ h₁ = ⟨t₂, h₂⟩ from
      (homEquivAmalgamation hx).symm.injective (hs _ _))
/-- Given sieve `S` and presheaf `P : Cᵒᵖ ⥤ A`, their natural associated cone admits at most one
    morphism from every cone in the same category (i.e. over the same diagram),
    iff `Hom (E, P -)`is separated for the sieve `S` and all `E : A`. -/
theorem subsingleton_iff_isSeparatedFor :
    (∀ c, Subsingleton (c ⟶ P.mapCone S.arrows.cocone.op)) ↔
      ∀ E : Aᵒᵖ, IsSeparatedFor (P ⋙ coyoneda.obj E) S.arrows := by
  constructor
  · intro hs E x t₁ t₂ h₁ h₂
    have hx := is_compatible_of_exists_amalgamation x ⟨t₁, h₁⟩
    rw [compatible_iff_sieveCompatible] at hx
    specialize hs hx.cone
    rcases hs with ⟨hs⟩
    simpa only [Subtype.mk.injEq] using (show Subtype.mk t₁ h₁ = ⟨t₂, h₂⟩ from
      (homEquivAmalgamation hx).symm.injective (hs _ _))
  · 

Proof state:
case mpr
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
A : Type u₂
inst✝ : Category.{v₂, u₂} A
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ A
X : C
S : Sieve X
R : Presieve X
E : Aᵒᵖ
x : FamilyOfElements (P ⋙ coyoneda.obj E) S.arrows
hx : SieveCompatible x
⊢ (∀ (E : Aᵒᵖ), IsSeparatedFor (P ⋙ coyoneda.obj E) S.arrows) →
    ∀ (c : Cone ((diagram S.arrows).op ⋙ P)), Subsingleton (c ⟶ P.mapCone (Cocone.op (cocone S.arrows)))

Next tactic:
rintro h ⟨E, π⟩
/-- Given sieve `S` and presheaf `P : Cᵒᵖ ⥤ A`, their natural associated cone admits at most one
    morphism from every cone in the same category (i.e. over the same diagram),
    iff `Hom (E, P -)`is separated for the sieve `S` and all `E : A`. -/
theorem subsingleton_iff_isSeparatedFor :
    (∀ c, Subsingleton (c ⟶ P.mapCone S.arrows.cocone.op)) ↔
      ∀ E : Aᵒᵖ, IsSeparatedFor (P ⋙ coyoneda.obj E) S.arrows := by
  constructor
  · intro hs E x t₁ t₂ h₁ h₂
    have hx := is_compatible_of_exists_amalgamation x ⟨t₁, h₁⟩
    rw [compatible_iff_sieveCompatible] at hx
    specialize hs hx.cone
    rcases hs with ⟨hs⟩
    simpa only [Subtype.mk.injEq] using (show Subtype.mk t₁ h₁ = ⟨t₂, h₂⟩ from
      (homEquivAmalgamation hx).symm.injective (hs _ _))
  · rintro h ⟨E, π⟩
    

Proof state:
case mpr.mk
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
A : Type u₂
inst✝ : Category.{v₂, u₂} A
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ A
X : C
S : Sieve X
R : Presieve X
E✝ : Aᵒᵖ
x : FamilyOfElements (P ⋙ coyoneda.obj E✝) S.arrows
hx : SieveCompatible x
h : ∀ (E : Aᵒᵖ), IsSeparatedFor (P ⋙ coyoneda.obj E) S.arrows
E : A
π : (Functor.const (FullSubcategory fun f => S.arrows f.hom)ᵒᵖ).obj E ⟶ (diagram S.arrows).op ⋙ P
⊢ Subsingleton ({ pt := E, π := π } ⟶ P.mapCone (Cocone.op (cocone S.arrows)))

Next tactic:
let eqv := conesEquivSieveCompatibleFamily P S (op E)
/-- Given sieve `S` and presheaf `P : Cᵒᵖ ⥤ A`, their natural associated cone admits at most one
    morphism from every cone in the same category (i.e. over the same diagram),
    iff `Hom (E, P -)`is separated for the sieve `S` and all `E : A`. -/
theorem subsingleton_iff_isSeparatedFor :
    (∀ c, Subsingleton (c ⟶ P.mapCone S.arrows.cocone.op)) ↔
      ∀ E : Aᵒᵖ, IsSeparatedFor (P ⋙ coyoneda.obj E) S.arrows := by
  constructor
  · intro hs E x t₁ t₂ h₁ h₂
    have hx := is_compatible_of_exists_amalgamation x ⟨t₁, h₁⟩
    rw [compatible_iff_sieveCompatible] at hx
    specialize hs hx.cone
    rcases hs with ⟨hs⟩
    simpa only [Subtype.mk.injEq] using (show Subtype.mk t₁ h₁ = ⟨t₂, h₂⟩ from
      (homEquivAmalgamation hx).symm.injective (hs _ _))
  · rintro h ⟨E, π⟩
    let eqv := conesEquivSieveCompatibleFamily P S (op E)
    

Proof state:
case mpr.mk
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
A : Type u₂
inst✝ : Category.{v₂, u₂} A
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ A
X : C
S : Sieve X
R : Presieve X
E✝ : Aᵒᵖ
x : FamilyOfElements (P ⋙ coyoneda.obj E✝) S.arrows
hx : SieveCompatible x
h : ∀ (E : Aᵒᵖ), IsSeparatedFor (P ⋙ coyoneda.obj E) S.arrows
E : A
π : (Functor.const (FullSubcategory fun f => S.arrows f.hom)ᵒᵖ).obj E ⟶ (diagram S.arrows).op ⋙ P
eqv : (Functor.cones ((diagram S.arrows).op ⋙ P)).obj (op E) ≃ { x // SieveCompatible x } :=
  conesEquivSieveCompatibleFamily P S (op E)
⊢ Subsingleton ({ pt := E, π := π } ⟶ P.mapCone (Cocone.op (cocone S.arrows)))

Next tactic:
constructor
/-- Given sieve `S` and presheaf `P : Cᵒᵖ ⥤ A`, their natural associated cone admits at most one
    morphism from every cone in the same category (i.e. over the same diagram),
    iff `Hom (E, P -)`is separated for the sieve `S` and all `E : A`. -/
theorem subsingleton_iff_isSeparatedFor :
    (∀ c, Subsingleton (c ⟶ P.mapCone S.arrows.cocone.op)) ↔
      ∀ E : Aᵒᵖ, IsSeparatedFor (P ⋙ coyoneda.obj E) S.arrows := by
  constructor
  · intro hs E x t₁ t₂ h₁ h₂
    have hx := is_compatible_of_exists_amalgamation x ⟨t₁, h₁⟩
    rw [compatible_iff_sieveCompatible] at hx
    specialize hs hx.cone
    rcases hs with ⟨hs⟩
    simpa only [Subtype.mk.injEq] using (show Subtype.mk t₁ h₁ = ⟨t₂, h₂⟩ from
      (homEquivAmalgamation hx).symm.injective (hs _ _))
  · rintro h ⟨E, π⟩
    let eqv := conesEquivSieveCompatibleFamily P S (op E)
    constructor
    

Proof state:
case mpr.mk.allEq
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
A : Type u₂
inst✝ : Category.{v₂, u₂} A
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ A
X : C
S : Sieve X
R : Presieve X
E✝ : Aᵒᵖ
x : FamilyOfElements (P ⋙ coyoneda.obj E✝) S.arrows
hx : SieveCompatible x
h : ∀ (E : Aᵒᵖ), IsSeparatedFor (P ⋙ coyoneda.obj E) S.arrows
E : A
π : (Functor.const (FullSubcategory fun f => S.arrows f.hom)ᵒᵖ).obj E ⟶ (diagram S.arrows).op ⋙ P
eqv : (Functor.cones ((diagram S.arrows).op ⋙ P)).obj (op E) ≃ { x // SieveCompatible x } :=
  conesEquivSieveCompatibleFamily P S (op E)
⊢ ∀ (a b : { pt := E, π := π } ⟶ P.mapCone (Cocone.op (cocone S.arrows))), a = b

Next tactic:
rw [← eqv.left_inv π]
/-- Given sieve `S` and presheaf `P : Cᵒᵖ ⥤ A`, their natural associated cone admits at most one
    morphism from every cone in the same category (i.e. over the same diagram),
    iff `Hom (E, P -)`is separated for the sieve `S` and all `E : A`. -/
theorem subsingleton_iff_isSeparatedFor :
    (∀ c, Subsingleton (c ⟶ P.mapCone S.arrows.cocone.op)) ↔
      ∀ E : Aᵒᵖ, IsSeparatedFor (P ⋙ coyoneda.obj E) S.arrows := by
  constructor
  · intro hs E x t₁ t₂ h₁ h₂
    have hx := is_compatible_of_exists_amalgamation x ⟨t₁, h₁⟩
    rw [compatible_iff_sieveCompatible] at hx
    specialize hs hx.cone
    rcases hs with ⟨hs⟩
    simpa only [Subtype.mk.injEq] using (show Subtype.mk t₁ h₁ = ⟨t₂, h₂⟩ from
      (homEquivAmalgamation hx).symm.injective (hs _ _))
  · rintro h ⟨E, π⟩
    let eqv := conesEquivSieveCompatibleFamily P S (op E)
    constructor
    rw [← eqv.left_inv π]
    

Proof state:
case mpr.mk.allEq
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
A : Type u₂
inst✝ : Category.{v₂, u₂} A
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ A
X : C
S : Sieve X
R : Presieve X
E✝ : Aᵒᵖ
x : FamilyOfElements (P ⋙ coyoneda.obj E✝) S.arrows
hx : SieveCompatible x
h : ∀ (E : Aᵒᵖ), IsSeparatedFor (P ⋙ coyoneda.obj E) S.arrows
E : A
π : (Functor.const (FullSubcategory fun f => S.arrows f.hom)ᵒᵖ).obj E ⟶ (diagram S.arrows).op ⋙ P
eqv : (Functor.cones ((diagram S.arrows).op ⋙ P)).obj (op E) ≃ { x // SieveCompatible x } :=
  conesEquivSieveCompatibleFamily P S (op E)
⊢ ∀ (a b : { pt := E, π := Equiv.invFun eqv (Equiv.toFun eqv π) } ⟶ P.mapCone (Cocone.op (cocone S.arrows))), a = b

Next tactic:
intro f₁ f₂
/-- Given sieve `S` and presheaf `P : Cᵒᵖ ⥤ A`, their natural associated cone admits at most one
    morphism from every cone in the same category (i.e. over the same diagram),
    iff `Hom (E, P -)`is separated for the sieve `S` and all `E : A`. -/
theorem subsingleton_iff_isSeparatedFor :
    (∀ c, Subsingleton (c ⟶ P.mapCone S.arrows.cocone.op)) ↔
      ∀ E : Aᵒᵖ, IsSeparatedFor (P ⋙ coyoneda.obj E) S.arrows := by
  constructor
  · intro hs E x t₁ t₂ h₁ h₂
    have hx := is_compatible_of_exists_amalgamation x ⟨t₁, h₁⟩
    rw [compatible_iff_sieveCompatible] at hx
    specialize hs hx.cone
    rcases hs with ⟨hs⟩
    simpa only [Subtype.mk.injEq] using (show Subtype.mk t₁ h₁ = ⟨t₂, h₂⟩ from
      (homEquivAmalgamation hx).symm.injective (hs _ _))
  · rintro h ⟨E, π⟩
    let eqv := conesEquivSieveCompatibleFamily P S (op E)
    constructor
    rw [← eqv.left_inv π]
    intro f₁ f₂
    

Proof state:
case mpr.mk.allEq
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
A : Type u₂
inst✝ : Category.{v₂, u₂} A
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ A
X : C
S : Sieve X
R : Presieve X
E✝ : Aᵒᵖ
x : FamilyOfElements (P ⋙ coyoneda.obj E✝) S.arrows
hx : SieveCompatible x
h : ∀ (E : Aᵒᵖ), IsSeparatedFor (P ⋙ coyoneda.obj E) S.arrows
E : A
π : (Functor.const (FullSubcategory fun f => S.arrows f.hom)ᵒᵖ).obj E ⟶ (diagram S.arrows).op ⋙ P
eqv : (Functor.cones ((diagram S.arrows).op ⋙ P)).obj (op E) ≃ { x // SieveCompatible x } :=
  conesEquivSieveCompatibleFamily P S (op E)
f₁ f₂ : { pt := E, π := Equiv.invFun eqv (Equiv.toFun eqv π) } ⟶ P.mapCone (Cocone.op (cocone S.arrows))
⊢ f₁ = f₂

Next tactic:
let eqv' := homEquivAmalgamation (eqv π).2
/-- Given sieve `S` and presheaf `P : Cᵒᵖ ⥤ A`, their natural associated cone admits at most one
    morphism from every cone in the same category (i.e. over the same diagram),
    iff `Hom (E, P -)`is separated for the sieve `S` and all `E : A`. -/
theorem subsingleton_iff_isSeparatedFor :
    (∀ c, Subsingleton (c ⟶ P.mapCone S.arrows.cocone.op)) ↔
      ∀ E : Aᵒᵖ, IsSeparatedFor (P ⋙ coyoneda.obj E) S.arrows := by
  constructor
  · intro hs E x t₁ t₂ h₁ h₂
    have hx := is_compatible_of_exists_amalgamation x ⟨t₁, h₁⟩
    rw [compatible_iff_sieveCompatible] at hx
    specialize hs hx.cone
    rcases hs with ⟨hs⟩
    simpa only [Subtype.mk.injEq] using (show Subtype.mk t₁ h₁ = ⟨t₂, h₂⟩ from
      (homEquivAmalgamation hx).symm.injective (hs _ _))
  · rintro h ⟨E, π⟩
    let eqv := conesEquivSieveCompatibleFamily P S (op E)
    constructor
    rw [← eqv.left_inv π]
    intro f₁ f₂
    let eqv' := homEquivAmalgamation (eqv π).2
    

Proof state:
case mpr.mk.allEq
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
A : Type u₂
inst✝ : Category.{v₂, u₂} A
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ A
X : C
S : Sieve X
R : Presieve X
E✝ : Aᵒᵖ
x : FamilyOfElements (P ⋙ coyoneda.obj E✝) S.arrows
hx : SieveCompatible x
h : ∀ (E : Aᵒᵖ), IsSeparatedFor (P ⋙ coyoneda.obj E) S.arrows
E : A
π : (Functor.const (FullSubcategory fun f => S.arrows f.hom)ᵒᵖ).obj E ⟶ (diagram S.arrows).op ⋙ P
eqv : (Functor.cones ((diagram S.arrows).op ⋙ P)).obj (op E) ≃ { x // SieveCompatible x } :=
  conesEquivSieveCompatibleFamily P S (op E)
f₁ f₂ : { pt := E, π := Equiv.invFun eqv (Equiv.toFun eqv π) } ⟶ P.mapCone (Cocone.op (cocone S.arrows))
eqv' : (SieveCompatible.cone (_ : SieveCompatible ↑(eqv π)) ⟶ P.mapCone (Cocone.op (cocone S.arrows))) ≃
  { t // IsAmalgamation (↑(eqv π)) t } :=
  homEquivAmalgamation (_ : SieveCompatible ↑(eqv π))
⊢ f₁ = f₂

Next tactic:
apply eqv'.injective
/-- Given sieve `S` and presheaf `P : Cᵒᵖ ⥤ A`, their natural associated cone admits at most one
    morphism from every cone in the same category (i.e. over the same diagram),
    iff `Hom (E, P -)`is separated for the sieve `S` and all `E : A`. -/
theorem subsingleton_iff_isSeparatedFor :
    (∀ c, Subsingleton (c ⟶ P.mapCone S.arrows.cocone.op)) ↔
      ∀ E : Aᵒᵖ, IsSeparatedFor (P ⋙ coyoneda.obj E) S.arrows := by
  constructor
  · intro hs E x t₁ t₂ h₁ h₂
    have hx := is_compatible_of_exists_amalgamation x ⟨t₁, h₁⟩
    rw [compatible_iff_sieveCompatible] at hx
    specialize hs hx.cone
    rcases hs with ⟨hs⟩
    simpa only [Subtype.mk.injEq] using (show Subtype.mk t₁ h₁ = ⟨t₂, h₂⟩ from
      (homEquivAmalgamation hx).symm.injective (hs _ _))
  · rintro h ⟨E, π⟩
    let eqv := conesEquivSieveCompatibleFamily P S (op E)
    constructor
    rw [← eqv.left_inv π]
    intro f₁ f₂
    let eqv' := homEquivAmalgamation (eqv π).2
    apply eqv'.injective
    

Proof state:
case mpr.mk.allEq.a
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
A : Type u₂
inst✝ : Category.{v₂, u₂} A
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ A
X : C
S : Sieve X
R : Presieve X
E✝ : Aᵒᵖ
x : FamilyOfElements (P ⋙ coyoneda.obj E✝) S.arrows
hx : SieveCompatible x
h : ∀ (E : Aᵒᵖ), IsSeparatedFor (P ⋙ coyoneda.obj E) S.arrows
E : A
π : (Functor.const (FullSubcategory fun f => S.arrows f.hom)ᵒᵖ).obj E ⟶ (diagram S.arrows).op ⋙ P
eqv : (Functor.cones ((diagram S.arrows).op ⋙ P)).obj (op E) ≃ { x // SieveCompatible x } :=
  conesEquivSieveCompatibleFamily P S (op E)
f₁ f₂ : { pt := E, π := Equiv.invFun eqv (Equiv.toFun eqv π) } ⟶ P.mapCone (Cocone.op (cocone S.arrows))
eqv' : (SieveCompatible.cone (_ : SieveCompatible ↑(eqv π)) ⟶ P.mapCone (Cocone.op (cocone S.arrows))) ≃
  { t // IsAmalgamation (↑(eqv π)) t } :=
  homEquivAmalgamation (_ : SieveCompatible ↑(eqv π))
⊢ eqv' f₁ = eqv' f₂

Next tactic:
ext
/-- Given sieve `S` and presheaf `P : Cᵒᵖ ⥤ A`, their natural associated cone admits at most one
    morphism from every cone in the same category (i.e. over the same diagram),
    iff `Hom (E, P -)`is separated for the sieve `S` and all `E : A`. -/
theorem subsingleton_iff_isSeparatedFor :
    (∀ c, Subsingleton (c ⟶ P.mapCone S.arrows.cocone.op)) ↔
      ∀ E : Aᵒᵖ, IsSeparatedFor (P ⋙ coyoneda.obj E) S.arrows := by
  constructor
  · intro hs E x t₁ t₂ h₁ h₂
    have hx := is_compatible_of_exists_amalgamation x ⟨t₁, h₁⟩
    rw [compatible_iff_sieveCompatible] at hx
    specialize hs hx.cone
    rcases hs with ⟨hs⟩
    simpa only [Subtype.mk.injEq] using (show Subtype.mk t₁ h₁ = ⟨t₂, h₂⟩ from
      (homEquivAmalgamation hx).symm.injective (hs _ _))
  · rintro h ⟨E, π⟩
    let eqv := conesEquivSieveCompatibleFamily P S (op E)
    constructor
    rw [← eqv.left_inv π]
    intro f₁ f₂
    let eqv' := homEquivAmalgamation (eqv π).2
    apply eqv'.injective
    ext
    

Proof state:
case mpr.mk.allEq.a.a
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
A : Type u₂
inst✝ : Category.{v₂, u₂} A
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ A
X : C
S : Sieve X
R : Presieve X
E✝ : Aᵒᵖ
x : FamilyOfElements (P ⋙ coyoneda.obj E✝) S.arrows
hx : SieveCompatible x
h : ∀ (E : Aᵒᵖ), IsSeparatedFor (P ⋙ coyoneda.obj E) S.arrows
E : A
π : (Functor.const (FullSubcategory fun f => S.arrows f.hom)ᵒᵖ).obj E ⟶ (diagram S.arrows).op ⋙ P
eqv : (Functor.cones ((diagram S.arrows).op ⋙ P)).obj (op E) ≃ { x // SieveCompatible x } :=
  conesEquivSieveCompatibleFamily P S (op E)
f₁ f₂ : { pt := E, π := Equiv.invFun eqv (Equiv.toFun eqv π) } ⟶ P.mapCone (Cocone.op (cocone S.arrows))
eqv' : (SieveCompatible.cone (_ : SieveCompatible ↑(eqv π)) ⟶ P.mapCone (Cocone.op (cocone S.arrows))) ≃
  { t // IsAmalgamation (↑(eqv π)) t } :=
  homEquivAmalgamation (_ : SieveCompatible ↑(eqv π))
⊢ ↑(eqv' f₁) = ↑(eqv' f₂)

Next tactic:
apply h _ (eqv π).1
/-- Given sieve `S` and presheaf `P : Cᵒᵖ ⥤ A`, their natural associated cone admits at most one
    morphism from every cone in the same category (i.e. over the same diagram),
    iff `Hom (E, P -)`is separated for the sieve `S` and all `E : A`. -/
theorem subsingleton_iff_isSeparatedFor :
    (∀ c, Subsingleton (c ⟶ P.mapCone S.arrows.cocone.op)) ↔
      ∀ E : Aᵒᵖ, IsSeparatedFor (P ⋙ coyoneda.obj E) S.arrows := by
  constructor
  · intro hs E x t₁ t₂ h₁ h₂
    have hx := is_compatible_of_exists_amalgamation x ⟨t₁, h₁⟩
    rw [compatible_iff_sieveCompatible] at hx
    specialize hs hx.cone
    rcases hs with ⟨hs⟩
    simpa only [Subtype.mk.injEq] using (show Subtype.mk t₁ h₁ = ⟨t₂, h₂⟩ from
      (homEquivAmalgamation hx).symm.injective (hs _ _))
  · rintro h ⟨E, π⟩
    let eqv := conesEquivSieveCompatibleFamily P S (op E)
    constructor
    rw [← eqv.left_inv π]
    intro f₁ f₂
    let eqv' := homEquivAmalgamation (eqv π).2
    apply eqv'.injective
    ext
    apply h _ (eqv π).1 <;> 

Proof state:
case mpr.mk.allEq.a.a.a
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
A : Type u₂
inst✝ : Category.{v₂, u₂} A
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ A
X : C
S : Sieve X
R : Presieve X
E✝ : Aᵒᵖ
x : FamilyOfElements (P ⋙ coyoneda.obj E✝) S.arrows
hx : SieveCompatible x
h : ∀ (E : Aᵒᵖ), IsSeparatedFor (P ⋙ coyoneda.obj E) S.arrows
E : A
π : (Functor.const (FullSubcategory fun f => S.arrows f.hom)ᵒᵖ).obj E ⟶ (diagram S.arrows).op ⋙ P
eqv : (Functor.cones ((diagram S.arrows).op ⋙ P)).obj (op E) ≃ { x // SieveCompatible x } :=
  conesEquivSieveCompatibleFamily P S (op E)
f₁ f₂ : { pt := E, π := Equiv.invFun eqv (Equiv.toFun eqv π) } ⟶ P.mapCone (Cocone.op (cocone S.arrows))
eqv' : (SieveCompatible.cone (_ : SieveCompatible ↑(eqv π)) ⟶ P.mapCone (Cocone.op (cocone S.arrows))) ≃
  { t // IsAmalgamation (↑(eqv π)) t } :=
  homEquivAmalgamation (_ : SieveCompatible ↑(eqv π))
⊢ IsAmalgamation ↑(eqv π) ↑(eqv' f₁)

Next tactic:
exact (eqv' _).2
/-- Given sieve `S` and presheaf `P : Cᵒᵖ ⥤ A`, their natural associated cone admits at most one
    morphism from every cone in the same category (i.e. over the same diagram),
    iff `Hom (E, P -)`is separated for the sieve `S` and all `E : A`. -/
theorem subsingleton_iff_isSeparatedFor :
    (∀ c, Subsingleton (c ⟶ P.mapCone S.arrows.cocone.op)) ↔
      ∀ E : Aᵒᵖ, IsSeparatedFor (P ⋙ coyoneda.obj E) S.arrows := by
  constructor
  · intro hs E x t₁ t₂ h₁ h₂
    have hx := is_compatible_of_exists_amalgamation x ⟨t₁, h₁⟩
    rw [compatible_iff_sieveCompatible] at hx
    specialize hs hx.cone
    rcases hs with ⟨hs⟩
    simpa only [Subtype.mk.injEq] using (show Subtype.mk t₁ h₁ = ⟨t₂, h₂⟩ from
      (homEquivAmalgamation hx).symm.injective (hs _ _))
  · rintro h ⟨E, π⟩
    let eqv := conesEquivSieveCompatibleFamily P S (op E)
    constructor
    rw [← eqv.left_inv π]
    intro f₁ f₂
    let eqv' := homEquivAmalgamation (eqv π).2
    apply eqv'.injective
    ext
    apply h _ (eqv π).1 <;> 

Proof state:
case mpr.mk.allEq.a.a.a
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
A : Type u₂
inst✝ : Category.{v₂, u₂} A
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ A
X : C
S : Sieve X
R : Presieve X
E✝ : Aᵒᵖ
x : FamilyOfElements (P ⋙ coyoneda.obj E✝) S.arrows
hx : SieveCompatible x
h : ∀ (E : Aᵒᵖ), IsSeparatedFor (P ⋙ coyoneda.obj E) S.arrows
E : A
π : (Functor.const (FullSubcategory fun f => S.arrows f.hom)ᵒᵖ).obj E ⟶ (diagram S.arrows).op ⋙ P
eqv : (Functor.cones ((diagram S.arrows).op ⋙ P)).obj (op E) ≃ { x // SieveCompatible x } :=
  conesEquivSieveCompatibleFamily P S (op E)
f₁ f₂ : { pt := E, π := Equiv.invFun eqv (Equiv.toFun eqv π) } ⟶ P.mapCone (Cocone.op (cocone S.arrows))
eqv' : (SieveCompatible.cone (_ : SieveCompatible ↑(eqv π)) ⟶ P.mapCone (Cocone.op (cocone S.arrows))) ≃
  { t // IsAmalgamation (↑(eqv π)) t } :=
  homEquivAmalgamation (_ : SieveCompatible ↑(eqv π))
⊢ IsAmalgamation ↑(eqv π) ↑(eqv' f₂)

Next tactic:
exact (eqv' _).2
/-- A presheaf `P` is a sheaf for the Grothendieck topology generated by a pretopology `K`
    iff for every covering presieve `R` of `K`, the natural cone associated to `P` and
    `Sieve.generate R` is a limit cone. -/
theorem isSheaf_iff_isLimit_pretopology [HasPullbacks C] (K : Pretopology C) :
    IsSheaf (K.toGrothendieck C) P ↔
      ∀ ⦃X : C⦄ (R : Presieve X),
        R ∈ K X → Nonempty (IsLimit (P.mapCone (generate R).arrows.cocone.op)) := by
  

Proof state:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
A : Type u₂
inst✝¹ : Category.{v₂, u₂} A
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ A
X : C
S : Sieve X
R : Presieve X
E : Aᵒᵖ
x : FamilyOfElements (P ⋙ coyoneda.obj E) S.arrows
hx : SieveCompatible x
inst✝ : HasPullbacks C
K : Pretopology C
⊢ IsSheaf (Pretopology.toGrothendieck C K) P ↔
    ∀ ⦃X : C⦄, ∀ R ∈ Pretopology.coverings K X, Nonempty (IsLimit (P.mapCone (Cocone.op (cocone (generate R).arrows))))

Next tactic:
dsimp [IsSheaf]
/-- A presheaf `P` is a sheaf for the Grothendieck topology generated by a pretopology `K`
    iff for every covering presieve `R` of `K`, the natural cone associated to `P` and
    `Sieve.generate R` is a limit cone. -/
theorem isSheaf_iff_isLimit_pretopology [HasPullbacks C] (K : Pretopology C) :
    IsSheaf (K.toGrothendieck C) P ↔
      ∀ ⦃X : C⦄ (R : Presieve X),
        R ∈ K X → Nonempty (IsLimit (P.mapCone (generate R).arrows.cocone.op)) := by
  dsimp [IsSheaf]
  

Proof state:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
A : Type u₂
inst✝¹ : Category.{v₂, u₂} A
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ A
X : C
S : Sieve X
R : Presieve X
E : Aᵒᵖ
x : FamilyOfElements (P ⋙ coyoneda.obj E) S.arrows
hx : SieveCompatible x
inst✝ : HasPullbacks C
K : Pretopology C
⊢ (∀ (E : A), Presieve.IsSheaf (Pretopology.toGrothendieck C K) (P ⋙ coyoneda.obj (op E))) ↔
    ∀ ⦃X : C⦄, ∀ R ∈ Pretopology.coverings K X, Nonempty (IsLimit (P.mapCone (Cocone.op (cocone (generate R).arrows))))

Next tactic:
simp_rw [isSheaf_pretopology]
/-- A presheaf `P` is a sheaf for the Grothendieck topology generated by a pretopology `K`
    iff for every covering presieve `R` of `K`, the natural cone associated to `P` and
    `Sieve.generate R` is a limit cone. -/
theorem isSheaf_iff_isLimit_pretopology [HasPullbacks C] (K : Pretopology C) :
    IsSheaf (K.toGrothendieck C) P ↔
      ∀ ⦃X : C⦄ (R : Presieve X),
        R ∈ K X → Nonempty (IsLimit (P.mapCone (generate R).arrows.cocone.op)) := by
  dsimp [IsSheaf]
  simp_rw [isSheaf_pretopology]
  

Proof state:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
A : Type u₂
inst✝¹ : Category.{v₂, u₂} A
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ A
X : C
S : Sieve X
R : Presieve X
E : Aᵒᵖ
x : FamilyOfElements (P ⋙ coyoneda.obj E) S.arrows
hx : SieveCompatible x
inst✝ : HasPullbacks C
K : Pretopology C
⊢ (∀ (E : A) {X : C}, ∀ R ∈ Pretopology.coverings K X, IsSheafFor (P ⋙ coyoneda.obj (op E)) R) ↔
    ∀ ⦃X : C⦄, ∀ R ∈ Pretopology.coverings K X, Nonempty (IsLimit (P.mapCone (Cocone.op (cocone (generate R).arrows))))

Next tactic:
exact
    ⟨fun h X R hR => (isLimit_iff_isSheafFor_presieve P R).2 fun E => h E.unop R hR,
      fun h E X R hR => (isLimit_iff_isSheafFor_presieve P R).1 (h R hR) (op E)⟩
@[reassoc (attr := simp)]
theorem IsSheaf.amalgamate_map {A : Type u₂} [Category.{v₂} A] {E : A} {X : C} {P : Cᵒᵖ ⥤ A}
    (hP : Presheaf.IsSheaf J P) (S : J.Cover X) (x : ∀ I : S.Arrow, E ⟶ P.obj (op I.Y))
    (hx : ∀ I : S.Relation, x I.fst ≫ P.map I.g₁.op = x I.snd ≫ P.map I.g₂.op) (I : S.Arrow) :
    hP.amalgamate S x hx ≫ P.map I.f.op = x _ := by
  

Proof state:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
A✝ : Type u₂
inst✝¹ : Category.{v₂, u₂} A✝
J : GrothendieckTopology C
A : Type u₂
inst✝ : Category.{v₂, u₂} A
E : A
X : C
P : Cᵒᵖ ⥤ A
hP : IsSheaf J P
S : GrothendieckTopology.Cover J X
x : (I : GrothendieckTopology.Cover.Arrow S) → E ⟶ P.obj (op I.Y)
hx :
  ∀ (I : GrothendieckTopology.Cover.Relation S),
    x (GrothendieckTopology.Cover.Relation.fst I) ≫ P.map I.g₁.op =
      x (GrothendieckTopology.Cover.Relation.snd I) ≫ P.map I.g₂.op
I : GrothendieckTopology.Cover.Arrow S
⊢ amalgamate hP S x hx ≫ P.map I.f.op = x I

Next tactic:
rcases I with ⟨Y, f, hf⟩