awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck
Goal:
C : Type u_3
inst : Category.{u_2, u_3} C
J : Precoverage.{u_2, u_3} C
P : Functor.{u_2, u_1, u_3, u_1 + 1} Cᵒᵖ (Type u_1)
H : ∀ {X Y : C} {f : Y ⟶ X}, ∀ R ∈ J.coverings X, Presieve.IsSheafFor P (Sieve.pullback f (Sieve.generate R)).arrows
X_1 : C
S_1 : Sieve.{u_2, u_3} X_1
X : C
R S : Sieve.{u_2, u_3} X
hS : J.Saturate X R
h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → J.Saturate Y (Sieve.pullback f S)
H1 : ∀ ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSheafFor P (Sieve.pullback f R).arrows
H2 :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    R.arrows f → ∀ ⦃Y_1 : C⦄ (f_1 : Y_1 ⟶ Y), Presieve.IsSheafFor P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows
Y : C
f : Y ⟶ X
mpr :
  ∀
    (H :
      ∀ {X Y : C} {f : Y ⟶ X},
        ∀ R ∈ J.coverings X,
          Presieve.IsSeparatedFor P (Sieve.pullback f (Sieve.generate R)).arrows ∧
            ∀ (x : Presieve.FamilyOfElements P (Sieve.pullback f (Sieve.generate R)).arrows),
              x.Compatible (R := (Sieve.pullback f (Sieve.generate R)).arrows) →
                ∃ (t : P.obj (Opposite.op Y)), x.IsAmalgamation (R := (Sieve.pullback f (Sieve.generate R)).arrows) t)
    (H1 :
      ∀ ⦃Y : C⦄ (f : Y ⟶ X),
        Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows ∧
          ∀ (x : Presieve.FamilyOfElements P (Sieve.pullback f R).arrows),
            x.Compatible (R := (Sieve.pullback f R).arrows) →
              ∃ (t : P.obj (Opposite.op Y)), x.IsAmalgamation (R := (Sieve.pullback f R).arrows) t)
    (H2 :
      ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
        R.arrows f →
          ∀ ⦃Y_1 : C⦄ (f_1 : Y_1 ⟶ Y),
            Presieve.IsSeparatedFor P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows ∧
              ∀ (x : Presieve.FamilyOfElements P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows),
                x.Compatible (R := (Sieve.pullback f_1 (Sieve.pullback f S)).arrows) →
                  ∃ (t : P.obj (Opposite.op Y_1)),
                    x.IsAmalgamation (R := (Sieve.pullback f_1 (Sieve.pullback f S)).arrows) t),
    Presieve.IsSeparatedFor P (Sieve.pullback f S).arrows ∧
      ∀ (x : Presieve.FamilyOfElements P (Sieve.pullback f S).arrows),
        x.Compatible (R := (Sieve.pullback f S).arrows) →
          ∃ (t : P.obj (Opposite.op Y)), x.IsAmalgamation (R := (Sieve.pullback f S).arrows) t
⊢ Presieve.IsSheafFor P (Sieve.pullback f S).arrows

Tactic:
simp only [← Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor] at *

Premises used: CategoryTheory.Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor
-- Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck
Goal:
C : Type u_3
inst : Category.{u_2, u_3} C
J : Precoverage.{u_2, u_3} C
P : Functor.{u_2, u_1, u_3, u_1 + 1} Cᵒᵖ (Type u_1)
X_1 : C
S_1 : Sieve.{u_2, u_3} X_1
X : C
R S : Sieve.{u_2, u_3} X
hS : J.Saturate X R
h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → J.Saturate Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
  ∀ {X Y : C} {f : Y ⟶ X},
    ∀ R ∈ J.coverings X,
      Presieve.IsSeparatedFor P (Sieve.pullback f (Sieve.generate R)).arrows ∧
        ∀ (x : Presieve.FamilyOfElements P (Sieve.pullback f (Sieve.generate R)).arrows),
          x.Compatible (R := (Sieve.pullback f (Sieve.generate R)).arrows) →
            ∃ (t : P.obj (Opposite.op Y)), x.IsAmalgamation (R := (Sieve.pullback f (Sieve.generate R)).arrows) t
H1 :
  ∀ ⦃Y : C⦄ (f : Y ⟶ X),
    Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows ∧
      ∀ (x : Presieve.FamilyOfElements P (Sieve.pullback f R).arrows),
        x.Compatible (R := (Sieve.pullback f R).arrows) →
          ∃ (t : P.obj (Opposite.op Y)), x.IsAmalgamation (R := (Sieve.pullback f R).arrows) t
H2 :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    R.arrows f →
      ∀ ⦃Y_1 : C⦄ (f_1 : Y_1 ⟶ Y),
        Presieve.IsSeparatedFor P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows ∧
          ∀ (x : Presieve.FamilyOfElements P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows),
            x.Compatible (R := (Sieve.pullback f_1 (Sieve.pullback f S)).arrows) →
              ∃ (t : P.obj (Opposite.op Y_1)),
                x.IsAmalgamation (R := (Sieve.pullback f_1 (Sieve.pullback f S)).arrows) t
⊢ ∀ ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows

Tactic:
choose H1 H1' using H1
-- Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck
Goal:
C : Type u_3
inst : Category.{u_2, u_3} C
J : Precoverage.{u_2, u_3} C
P : Functor.{u_2, u_1, u_3, u_1 + 1} Cᵒᵖ (Type u_1)
X_1 : C
S_1 : Sieve.{u_2, u_3} X_1
X : C
R S : Sieve.{u_2, u_3} X
hS : J.Saturate X R
h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → J.Saturate Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
  ∀ {X Y : C} {f : Y ⟶ X},
    ∀ R ∈ J.coverings X,
      Presieve.IsSeparatedFor P (Sieve.pullback f (Sieve.generate R)).arrows ∧
        ∀ (x : Presieve.FamilyOfElements P (Sieve.pullback f (Sieve.generate R)).arrows),
          x.Compatible (R := (Sieve.pullback f (Sieve.generate R)).arrows) →
            ∃ (t : P.obj (Opposite.op Y)), x.IsAmalgamation (R := (Sieve.pullback f (Sieve.generate R)).arrows) t
H1 :
  ∀ ⦃Y : C⦄ (f : Y ⟶ X),
    Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows ∧
      ∀ (x : Presieve.FamilyOfElements P (Sieve.pullback f R).arrows),
        x.Compatible (R := (Sieve.pullback f R).arrows) →
          ∃ (t : P.obj (Opposite.op Y)), x.IsAmalgamation (R := (Sieve.pullback f R).arrows) t
H2 :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    R.arrows f →
      ∀ ⦃Y_1 : C⦄ (f_1 : Y_1 ⟶ Y),
        Presieve.IsSeparatedFor P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows ∧
          ∀ (x : Presieve.FamilyOfElements P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows),
            x.Compatible (R := (Sieve.pullback f_1 (Sieve.pullback f S)).arrows) →
              ∃ (t : P.obj (Opposite.op Y_1)),
                x.IsAmalgamation (R := (Sieve.pullback f_1 (Sieve.pullback f S)).arrows) t
H1₁ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows
⊢ ∀ ⦃Y : C⦄ (f : Y ⟶ X) (x : Presieve.FamilyOfElements P (Sieve.pullback f R).arrows),
    x.Compatible (R := (Sieve.pullback f R).arrows) →
      ∃ (t : P.obj (Opposite.op Y)), x.IsAmalgamation (R := (Sieve.pullback f R).arrows) t

Tactic:
choose H1 H1' using H1
-- Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck
Goal:
C : Type u_3
inst : Category.{u_2, u_3} C
J : Precoverage.{u_2, u_3} C
P : Functor.{u_2, u_1, u_3, u_1 + 1} Cᵒᵖ (Type u_1)
X_1 : C
S_1 : Sieve.{u_2, u_3} X_1
X : C
R S : Sieve.{u_2, u_3} X
hS : J.Saturate X R
h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → J.Saturate Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
  ∀ {X Y : C} {f : Y ⟶ X},
    ∀ R ∈ J.coverings X,
      Presieve.IsSeparatedFor P (Sieve.pullback f (Sieve.generate R)).arrows ∧
        ∀ (x : Presieve.FamilyOfElements P (Sieve.pullback f (Sieve.generate R)).arrows),
          x.Compatible (R := (Sieve.pullback f (Sieve.generate R)).arrows) →
            ∃ (t : P.obj (Opposite.op Y)), x.IsAmalgamation (R := (Sieve.pullback f (Sieve.generate R)).arrows) t
H2 :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    R.arrows f →
      ∀ ⦃Y_1 : C⦄ (f_1 : Y_1 ⟶ Y),
        Presieve.IsSeparatedFor P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows ∧
          ∀ (x : Presieve.FamilyOfElements P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows),
            x.Compatible (R := (Sieve.pullback f_1 (Sieve.pullback f S)).arrows) →
              ∃ (t : P.obj (Opposite.op Y_1)),
                x.IsAmalgamation (R := (Sieve.pullback f_1 (Sieve.pullback f S)).arrows) t
H1 : ∀ ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows
H1' :
  ∀ ⦃Y : C⦄ (f : Y ⟶ X) (x : Presieve.FamilyOfElements P (Sieve.pullback f R).arrows),
    x.Compatible (R := (Sieve.pullback f R).arrows) →
      ∃ (t : P.obj (Opposite.op Y)), x.IsAmalgamation (R := (Sieve.pullback f R).arrows) t
⊢ ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    R.arrows f → ∀ ⦃Y_1 : C⦄ (f_1 : Y_1 ⟶ Y), Presieve.IsSeparatedFor P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows

Tactic:
choose H2 H2' using H2
-- Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck
Goal:
C : Type u_3
inst : Category.{u_2, u_3} C
J : Precoverage.{u_2, u_3} C
P : Functor.{u_2, u_1, u_3, u_1 + 1} Cᵒᵖ (Type u_1)
X_1 : C
S_1 : Sieve.{u_2, u_3} X_1
X : C
R S : Sieve.{u_2, u_3} X
hS : J.Saturate X R
h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → J.Saturate Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
  ∀ {X Y : C} {f : Y ⟶ X},
    ∀ R ∈ J.coverings X,
      Presieve.IsSeparatedFor P (Sieve.pullback f (Sieve.generate R)).arrows ∧
        ∀ (x : Presieve.FamilyOfElements P (Sieve.pullback f (Sieve.generate R)).arrows),
          x.Compatible (R := (Sieve.pullback f (Sieve.generate R)).arrows) →
            ∃ (t : P.obj (Opposite.op Y)), x.IsAmalgamation (R := (Sieve.pullback f (Sieve.generate R)).arrows) t
H2 :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    R.arrows f →
      ∀ ⦃Y_1 : C⦄ (f_1 : Y_1 ⟶ Y),
        Presieve.IsSeparatedFor P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows ∧
          ∀ (x : Presieve.FamilyOfElements P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows),
            x.Compatible (R := (Sieve.pullback f_1 (Sieve.pullback f S)).arrows) →
              ∃ (t : P.obj (Opposite.op Y_1)),
                x.IsAmalgamation (R := (Sieve.pullback f_1 (Sieve.pullback f S)).arrows) t
H1 : ∀ ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows
H1' :
  ∀ ⦃Y : C⦄ (f : Y ⟶ X) (x : Presieve.FamilyOfElements P (Sieve.pullback f R).arrows),
    x.Compatible (R := (Sieve.pullback f R).arrows) →
      ∃ (t : P.obj (Opposite.op Y)), x.IsAmalgamation (R := (Sieve.pullback f R).arrows) t
H2₁ :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    R.arrows f → ∀ ⦃Y_1 : C⦄ (f_1 : Y_1 ⟶ Y), Presieve.IsSeparatedFor P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows
⊢ ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    R.arrows f →
      ∀ ⦃Y_1 : C⦄ (f_1 : Y_1 ⟶ Y) (x : Presieve.FamilyOfElements P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows),
        x.Compatible (R := (Sieve.pullback f_1 (Sieve.pullback f S)).arrows) →
          ∃ (t : P.obj (Opposite.op Y_1)), x.IsAmalgamation (R := (Sieve.pullback f_1 (Sieve.pullback f S)).arrows) t

Tactic:
choose H2 H2' using H2
-- Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck
Goal:
C : Type u_3
inst : Category.{u_2, u_3} C
J : Precoverage.{u_2, u_3} C
P : Functor.{u_2, u_1, u_3, u_1 + 1} Cᵒᵖ (Type u_1)
X_1 : C
S_1 : Sieve.{u_2, u_3} X_1
X : C
R S : Sieve.{u_2, u_3} X
hS : J.Saturate X R
h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → J.Saturate Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
  ∀ {X Y : C} {f : Y ⟶ X},
    ∀ R ∈ J.coverings X,
      Presieve.IsSeparatedFor P (Sieve.pullback f (Sieve.generate R)).arrows ∧
        ∀ (x : Presieve.FamilyOfElements P (Sieve.pullback f (Sieve.generate R)).arrows),
          x.Compatible (R := (Sieve.pullback f (Sieve.generate R)).arrows) →
            ∃ (t : P.obj (Opposite.op Y)), x.IsAmalgamation (R := (Sieve.pullback f (Sieve.generate R)).arrows) t
H1 : ∀ ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows
H1' :
  ∀ ⦃Y : C⦄ (f : Y ⟶ X) (x : Presieve.FamilyOfElements P (Sieve.pullback f R).arrows),
    x.Compatible (R := (Sieve.pullback f R).arrows) →
      ∃ (t : P.obj (Opposite.op Y)), x.IsAmalgamation (R := (Sieve.pullback f R).arrows) t
H2 :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    R.arrows f → ∀ ⦃Y_1 : C⦄ (f_1 : Y_1 ⟶ Y), Presieve.IsSeparatedFor P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows
H2' :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    R.arrows f →
      ∀ ⦃Y_1 : C⦄ (f_1 : Y_1 ⟶ Y) (x : Presieve.FamilyOfElements P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows),
        x.Compatible (R := (Sieve.pullback f_1 (Sieve.pullback f S)).arrows) →
          ∃ (t : P.obj (Opposite.op Y_1)), x.IsAmalgamation (R := (Sieve.pullback f_1 (Sieve.pullback f S)).arrows) t
⊢ Presieve.IsSeparatedFor P (Sieve.pullback f S).arrows ∧
    ∀ (x : Presieve.FamilyOfElements P (Sieve.pullback f S).arrows),
      x.Compatible (R := (Sieve.pullback f S).arrows) →
        ∃ (t : P.obj (Opposite.op Y)), x.IsAmalgamation (R := (Sieve.pullback f S).arrows) t

Tactic:
⟨?_, fun x hx => ?_⟩

Premises used: And.intro
-- Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck
Goal:
C : Type u_3
inst : Category.{u_2, u_3} C
J : Precoverage.{u_2, u_3} C
P : Functor.{u_2, u_1, u_3, u_1 + 1} Cᵒᵖ (Type u_1)
X_1 : C
S_1 : Sieve.{u_2, u_3} X_1
X : C
R S : Sieve.{u_2, u_3} X
hS : J.Saturate X R
h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → J.Saturate Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
  ∀ {X Y : C} {f : Y ⟶ X},
    ∀ R ∈ J.coverings X,
      Presieve.IsSeparatedFor P (Sieve.pullback f (Sieve.generate R)).arrows ∧
        ∀ (x : Presieve.FamilyOfElements P (Sieve.pullback f (Sieve.generate R)).arrows),
          x.Compatible (R := (Sieve.pullback f (Sieve.generate R)).arrows) →
            ∃ (t : P.obj (Opposite.op Y)), x.IsAmalgamation (R := (Sieve.pullback f (Sieve.generate R)).arrows) t
H1 : ∀ ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows
H1' :
  ∀ ⦃Y : C⦄ (f : Y ⟶ X) (x : Presieve.FamilyOfElements P (Sieve.pullback f R).arrows),
    x.Compatible (R := (Sieve.pullback f R).arrows) →
      ∃ (t : P.obj (Opposite.op Y)), x.IsAmalgamation (R := (Sieve.pullback f R).arrows) t
H2 :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    R.arrows f → ∀ ⦃Y_1 : C⦄ (f_1 : Y_1 ⟶ Y), Presieve.IsSeparatedFor P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows
H2' :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    R.arrows f →
      ∀ ⦃Y_1 : C⦄ (f_1 : Y_1 ⟶ Y) (x : Presieve.FamilyOfElements P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows),
        x.Compatible (R := (Sieve.pullback f_1 (Sieve.pullback f S)).arrows) →
          ∃ (t : P.obj (Opposite.op Y_1)), x.IsAmalgamation (R := (Sieve.pullback f_1 (Sieve.pullback f S)).arrows) t
mpr : Presieve.IsSeparatedFor P (Sieve.pullback f S).arrows
mpr₁ :
  ∀ (x : Presieve.FamilyOfElements P (Sieve.pullback f S).arrows)
    (hx : x.Compatible (R := (Sieve.pullback f S).arrows)),
    ∃ (t : P.obj (Opposite.op Y)), x.IsAmalgamation (R := (Sieve.pullback f S).arrows) t
⊢ Presieve.IsSeparatedFor P (Sieve.pullback f S).arrows ∧
    ∀ (x : Presieve.FamilyOfElements P (Sieve.pullback f S).arrows),
      x.Compatible (R := (Sieve.pullback f S).arrows) →
        ∃ (t : P.obj (Opposite.op Y)), x.IsAmalgamation (R := (Sieve.pullback f S).arrows) t

Tactic:
refine ⟨?_, fun x hx => ?_⟩
-- Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck
Goal:
C : Type u_3
inst : Category.{u_2, u_3} C
J : Precoverage.{u_2, u_3} C
P : Functor.{u_2, u_1, u_3, u_1 + 1} Cᵒᵖ (Type u_1)
X_1 : C
S_1 : Sieve.{u_2, u_3} X_1
X : C
R S : Sieve.{u_2, u_3} X
hS : J.Saturate X R
h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → J.Saturate Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
  ∀ {X Y : C} {f : Y ⟶ X},
    ∀ R ∈ J.coverings X,
      Presieve.IsSeparatedFor P (Sieve.pullback f (Sieve.generate R)).arrows ∧
        ∀ (x : Presieve.FamilyOfElements P (Sieve.pullback f (Sieve.generate R)).arrows),
          x.Compatible (R := (Sieve.pullback f (Sieve.generate R)).arrows) →
            ∃ (t : P.obj (Opposite.op Y)), x.IsAmalgamation (R := (Sieve.pullback f (Sieve.generate R)).arrows) t
H1 : ∀ ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows
H1' :
  ∀ ⦃Y : C⦄ (f : Y ⟶ X) (x : Presieve.FamilyOfElements P (Sieve.pullback f R).arrows),
    x.Compatible (R := (Sieve.pullback f R).arrows) →
      ∃ (t : P.obj (Opposite.op Y)), x.IsAmalgamation (R := (Sieve.pullback f R).arrows) t
H2 :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    R.arrows f → ∀ ⦃Y_1 : C⦄ (f_1 : Y_1 ⟶ Y), Presieve.IsSeparatedFor P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows
H2' :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    R.arrows f →
      ∀ ⦃Y_1 : C⦄ (f_1 : Y_1 ⟶ Y) (x : Presieve.FamilyOfElements P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows),
        x.Compatible (R := (Sieve.pullback f_1 (Sieve.pullback f S)).arrows) →
          ∃ (t : P.obj (Opposite.op Y_1)), x.IsAmalgamation (R := (Sieve.pullback f_1 (Sieve.pullback f S)).arrows) t
x : Presieve.FamilyOfElements P (Sieve.pullback f S).arrows
t₁ t₂ : P.obj (Opposite.op Y)
h₁ : x.IsAmalgamation (R := (Sieve.pullback f S).arrows) t₁
h₂ : x.IsAmalgamation (R := (Sieve.pullback f S).arrows) t₂
⊢ t₁ = t₂

Tactic:
(H1 f).ext (fun Z g hg => ?_)

Premises used: CategoryTheory.Presieve.IsSeparatedFor.ext
-- Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck
Goal:
C : Type u_3
inst : Category.{u_2, u_3} C
J : Precoverage.{u_2, u_3} C
P : Functor.{u_2, u_1, u_3, u_1 + 1} Cᵒᵖ (Type u_1)
X_1 : C
S_1 : Sieve.{u_2, u_3} X_1
X : C
R S : Sieve.{u_2, u_3} X
hS : J.Saturate X R
h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → J.Saturate Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
  ∀ {X Y : C} {f : Y ⟶ X},
    ∀ R ∈ J.coverings X,
      Presieve.IsSeparatedFor P (Sieve.pullback f (Sieve.generate R)).arrows ∧
        ∀ (x : Presieve.FamilyOfElements P (Sieve.pullback f (Sieve.generate R)).arrows),
          x.Compatible (R := (Sieve.pullback f (Sieve.generate R)).arrows) →
            ∃ (t : P.obj (Opposite.op Y)), x.IsAmalgamation (R := (Sieve.pullback f (Sieve.generate R)).arrows) t
H1 : ∀ ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows
H1' :
  ∀ ⦃Y : C⦄ (f : Y ⟶ X) (x : Presieve.FamilyOfElements P (Sieve.pullback f R).arrows),
    x.Compatible (R := (Sieve.pullback f R).arrows) →
      ∃ (t : P.obj (Opposite.op Y)), x.IsAmalgamation (R := (Sieve.pullback f R).arrows) t
H2 :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    R.arrows f → ∀ ⦃Y_1 : C⦄ (f_1 : Y_1 ⟶ Y), Presieve.IsSeparatedFor P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows
H2' :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    R.arrows f →
      ∀ ⦃Y_1 : C⦄ (f_1 : Y_1 ⟶ Y) (x : Presieve.FamilyOfElements P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows),
        x.Compatible (R := (Sieve.pullback f_1 (Sieve.pullback f S)).arrows) →
          ∃ (t : P.obj (Opposite.op Y_1)), x.IsAmalgamation (R := (Sieve.pullback f_1 (Sieve.pullback f S)).arrows) t
x : Presieve.FamilyOfElements P (Sieve.pullback f S).arrows
t₁ t₂ : P.obj (Opposite.op Y)
h₁ : x.IsAmalgamation (R := (Sieve.pullback f S).arrows) t₁
h₂ : x.IsAmalgamation (R := (Sieve.pullback f S).arrows) t₂
mpr :
  ∀ (Z : C) (g : Z ⟶ Y) (hg : (Sieve.pullback f R).arrows g),
    Eq (α := (fun (X : Type u_1) ↦ X) (P.obj (Opposite.op Z)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map g.op) :
            (a : (fun (X : Type u_1) ↦ X) (P.obj (Opposite.op Y))) → (fun (X : Type u_1) ↦ X) (P.obj (Opposite.op Z)))
          t₁ :
        (fun (X : Type u_1) ↦ X) (P.obj (Opposite.op Z)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map g.op) :
            (a : (fun (X : Type u_1) ↦ X) (P.obj (Opposite.op Y))) → (fun (X : Type u_1) ↦ X) (P.obj (Opposite.op Z)))
          t₂ :
        (fun (X : Type u_1) ↦ X) (P.obj (Opposite.op Z)))
⊢ t₁ = t₂

Tactic:
refine (H1 f).ext (fun Z g hg => ?_)

Premises used: CategoryTheory.Presieve.IsSeparatedFor.ext
-- Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck
Goal:
C : Type u_3
inst : Category.{u_2, u_3} C
J : Precoverage.{u_2, u_3} C
P : Functor.{u_2, u_1, u_3, u_1 + 1} Cᵒᵖ (Type u_1)
X_1 : C
S_1 : Sieve.{u_2, u_3} X_1
X : C
R S : Sieve.{u_2, u_3} X
hS : J.Saturate X R
h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → J.Saturate Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
  ∀ {X Y : C} {f : Y ⟶ X},
    ∀ R ∈ J.coverings X,
      Presieve.IsSeparatedFor P (Sieve.pullback f (Sieve.generate R)).arrows ∧
        ∀ (x : Presieve.FamilyOfElements P (Sieve.pullback f (Sieve.generate R)).arrows),
          x.Compatible (R := (Sieve.pullback f (Sieve.generate R)).arrows) →
            ∃ (t : P.obj (Opposite.op Y)), x.IsAmalgamation (R := (Sieve.pullback f (Sieve.generate R)).arrows) t
H1 : ∀ ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows
H1' :
  ∀ ⦃Y : C⦄ (f : Y ⟶ X) (x : Presieve.FamilyOfElements P (Sieve.pullback f R).arrows),
    x.Compatible (R := (Sieve.pullback f R).arrows) →
      ∃ (t : P.obj (Opposite.op Y)), x.IsAmalgamation (R := (Sieve.pullback f R).arrows) t
H2 :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    R.arrows f → ∀ ⦃Y_1 : C⦄ (f_1 : Y_1 ⟶ Y), Presieve.IsSeparatedFor P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows
H2' :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    R.arrows f →
      ∀ ⦃Y_1 : C⦄ (f_1 : Y_1 ⟶ Y) (x : Presieve.FamilyOfElements P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows),
        x.Compatible (R := (Sieve.pullback f_1 (Sieve.pullback f S)).arrows) →
          ∃ (t : P.obj (Opposite.op Y_1)), x.IsAmalgamation (R := (Sieve.pullback f_1 (Sieve.pullback f S)).arrows) t
x : Presieve.FamilyOfElements P (Sieve.pullback f S).arrows
t₁ t₂ : P.obj (Opposite.op Y)
h₁ : x.IsAmalgamation (R := (Sieve.pullback f S).arrows) t₁
h₂ : x.IsAmalgamation (R := (Sieve.pullback f S).arrows) t₂
Z : C
g : Z ⟶ Y
hg : (Sieve.pullback f R).arrows g
⊢ Eq (α := (fun (X : Type u_1) ↦ X) (P.obj (Opposite.op Z)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map g.op) :
          (a : (fun (X : Type u_1) ↦ X) (P.obj (Opposite.op Y))) → (fun (X : Type u_1) ↦ X) (P.obj (Opposite.op Z)))
        t₁ :
      (fun (X : Type u_1) ↦ X) (P.obj (Opposite.op Z)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map g.op) :
          (a : (fun (X : Type u_1) ↦ X) (P.obj (Opposite.op Y))) → (fun (X : Type u_1) ↦ X) (P.obj (Opposite.op Z)))
        t₂ :
      (fun (X : Type u_1) ↦ X) (P.obj (Opposite.op Z)))

Tactic:
(H2 hg (𝟙 _)).ext (fun ZZ gg hgg => ?_)

Premises used: CategoryTheory.CategoryStruct.id, CategoryTheory.Presieve.IsSeparatedFor.ext
-- Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck
Goal:
C : Type u_3
inst : Category.{u_2, u_3} C
J : Precoverage.{u_2, u_3} C
P : Functor.{u_2, u_1, u_3, u_1 + 1} Cᵒᵖ (Type u_1)
X_1 : C
S_1 : Sieve.{u_2, u_3} X_1
X : C
R S : Sieve.{u_2, u_3} X
hS : J.Saturate X R
h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → J.Saturate Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
  ∀ {X Y : C} {f : Y ⟶ X},
    ∀ R ∈ J.coverings X,
      Presieve.IsSeparatedFor P (Sieve.pullback f (Sieve.generate R)).arrows ∧
        ∀ (x : Presieve.FamilyOfElements P (Sieve.pullback f (Sieve.generate R)).arrows),
          x.Compatible (R := (Sieve.pullback f (Sieve.generate R)).arrows) →
            ∃ (t : P.obj (Opposite.op Y)), x.IsAmalgamation (R := (Sieve.pullback f (Sieve.generate R)).arrows) t
H1 : ∀ ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows
H1' :
  ∀ ⦃Y : C⦄ (f : Y ⟶ X) (x : Presieve.FamilyOfElements P (Sieve.pullback f R).arrows),
    x.Compatible (R := (Sieve.pullback f R).arrows) →
      ∃ (t : P.obj (Opposite.op Y)), x.IsAmalgamation (R := (Sieve.pullback f R).arrows) t
H2 :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    R.arrows f → ∀ ⦃Y_1 : C⦄ (f_1 : Y_1 ⟶ Y), Presieve.IsSeparatedFor P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows
H2' :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    R.arrows f →
      ∀ ⦃Y_1 : C⦄ (f_1 : Y_1 ⟶ Y) (x : Presieve.FamilyOfElements P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows),
        x.Compatible (R := (Sieve.pullback f_1 (Sieve.pullback f S)).arrows) →
          ∃ (t : P.obj (Opposite.op Y_1)), x.IsAmalgamation (R := (Sieve.pullback f_1 (Sieve.pullback f S)).arrows) t
x : Presieve.FamilyOfElements P (Sieve.pullback f S).arrows
t₁ t₂ : P.obj (Opposite.op Y)
h₁ : x.IsAmalgamation (R := (Sieve.pullback f S).arrows) t₁
h₂ : x.IsAmalgamation (R := (Sieve.pullback f S).arrows) t₂
Z : C
g : Z ⟶ Y
hg : (Sieve.pullback f R).arrows g
ZZ : C
gg : ZZ ⟶ Z
hgg : (Sieve.pullback (𝟙 Z) (Sieve.pullback (g ≫ f) S)).arrows gg
⊢ S.arrows (gg ≫ g ≫ f)

Tactic:
simp only [Sieve.pullback_id, Sieve.pullback_apply] at hgg

Premises used: CategoryTheory.Sieve.pullback_apply, CategoryTheory.Sieve.pullback_id