awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck (goal)
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 : C
S : Sieve.{u_2, u_3} X
hS : S ∈ (J.toGrothendieck : (a : C) → Set (Sieve a)) X
⊢ Presieve.IsSheafFor P S.arrows
have mpr : ∀ ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSheafFor P (Sieve.pullback f S).arrows
Tactic: suffices ∀ ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSheafFor P (S.pullback f).arrows by
      simpa using this (f := 𝟙 _)
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck (goal)
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
S : Sieve.{u_2, u_3} X
hS : J.Saturate X S
Y_1 : C
f_1 : Y_1 ⟶ X
ih : ∀ ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSheafFor P (Sieve.pullback f S).arrows
Y : C
f : Y ⟶ Y_1
⊢ Presieve.IsSheafFor P (Sieve.pullback f (Sieve.pullback f_1 S)).arrows
have mpr : Presieve.IsSheafFor P (Sieve.pullback (f ≫ f_1) S).arrows
Tactic: rw [← S.pullback_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck (goal)
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
⊢ Presieve.IsSheafFor P (Sieve.pullback f S).arrows
have 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
Tactic: simp only [← Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor] at *
Cleared hypotheses: H, H1, H2
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck (hypothesis)
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
⊢ 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
have H1₁ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows
Tactic: choose H1 H1' using H1
Cleared hypotheses: H1
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck (hypothesis)
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
⊢ 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
have 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
Tactic: choose H1 H1' using H1
Cleared hypotheses: H1
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck (hypothesis)
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
⊢ 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
have 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
Tactic: choose H2 H2' using H2
Cleared hypotheses: H2
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck (hypothesis)
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
⊢ 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
have 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
Tactic: choose H2 H2' using H2
Cleared hypotheses: H2
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck (goal)
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
have mpr : Presieve.IsSeparatedFor P (Sieve.pullback f S).arrows
Tactic: refine ⟨?_, fun x hx => ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck (goal)
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
⊢ 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
have 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
Tactic: refine ⟨?_, fun x hx => ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck (goal)
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
have 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)))
Tactic: refine (H1 f).ext (fun Z g hg => ?_)
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck (hypothesis)
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
⊢ Eq (α := (fun (X : Type u_1) ↦ X) (P.obj (Opposite.op ZZ)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map gg.op) :
          (a : (fun (X : Type u_1) ↦ X) (P.obj (Opposite.op Z))) → (fun (X : Type u_1) ↦ X) (P.obj (Opposite.op ZZ)))
        ((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))) :
      (fun (X : Type u_1) ↦ X) (P.obj (Opposite.op ZZ)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map gg.op) :
          (a : (fun (X : Type u_1) ↦ X) (P.obj (Opposite.op Z))) → (fun (X : Type u_1) ↦ X) (P.obj (Opposite.op ZZ)))
        ((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))) :
      (fun (X : Type u_1) ↦ X) (P.obj (Opposite.op ZZ)))
have hgg₁ : S.arrows (gg ≫ g ≫ f)
Tactic: simp only [Sieve.pullback_id, Sieve.pullback_apply] at hgg
Cleared hypotheses: hgg