awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.Small (goal)
Goal:
P Q : MorphismProperty Scheme
S : Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
inst_2 : P.RespectsIso
inst_3 : Q.IsStableUnderComposition
inst_4 : Q.IsStableUnderBaseChange
inst_5 : Q.HasOfPostcompProperty Q
hPQ : P ≤ Q
X : Q.Over ⊤ S
R : Sieve X
⊢ R ∈ (S.smallGrothendieckTopologyOfLE (P := P) (Q := Q) hPQ : (a : Q.Over ⊤ S) → Set (Sieve a)) X ↔
    R ∈
      (Pretopology.toGrothendieck (smallPretopology P Q : Pretopology (Q.Over ⊤ S)) : (a : Q.Over ⊤ S) → Set (Sieve a))
        X
have h : (∃ (𝒰 : Cover (precoverage P) (Over.left (X.toComma (P := Q) (Q := ⊤) (W := ⊤)))) (x : Cover.Over S (P := P) 𝒰),
    𝒰.toPresieveOver (P := P) ≤ Sieve.arrows (Sieve.functorPushforward (MorphismProperty.Over.forget Q ⊤ S) R)) ↔
  ∃ R_1 ∈ Precoverage.coverings (Pretopology.toPrecoverage (smallPretopology P Q : Pretopology (Q.Over ⊤ S))) X,
    R_1 ≤ R.arrows
Tactic: simp only [Pretopology.mem_toGrothendieck, Functor.mem_inducedTopology_sieves_iff,
    MorphismProperty.Comma.forget_obj, mem_overGrothendieckTopology]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.Small (goal)
Goal:
P Q : MorphismProperty Scheme
S : Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
inst_2 : P.RespectsIso
inst_3 : Q.IsStableUnderComposition
inst_4 : Q.IsStableUnderBaseChange
inst_5 : Q.HasOfPostcompProperty Q
hPQ : P ≤ Q
X : Q.Over ⊤ S
R : Sieve X
⊢ (∃ (𝒰 : Cover (precoverage P) (Over.left (X.toComma (P := Q) (Q := ⊤) (W := ⊤)))) (x : Cover.Over S (P := P) 𝒰),
      𝒰.toPresieveOver (P := P) ≤ Sieve.arrows (Sieve.functorPushforward (MorphismProperty.Over.forget Q ⊤ S) R)) ↔
    ∃ R_1 ∈ Precoverage.coverings (Pretopology.toPrecoverage (smallPretopology P Q : Pretopology (Q.Over ⊤ S))) X,
      R_1 ≤ R.arrows
have h : (∃ (𝒰 : Cover (precoverage P) (Over.left (X.toComma (P := Q) (Q := ⊤) (W := ⊤)))) (x : Cover.Over S (P := P) 𝒰),
    𝒰.toPresieveOver (P := P) ≤ Sieve.arrows (Sieve.functorPushforward (MorphismProperty.Over.forget Q ⊤ S) R)) →
  ∃ R_1 ∈ Precoverage.coverings (Pretopology.toPrecoverage (smallPretopology P Q : Pretopology (Q.Over ⊤ S))) X,
    R_1 ≤ R.arrows
Tactic: constructor
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.Small (goal)
Goal:
P Q : MorphismProperty Scheme
S : Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
inst_2 : P.RespectsIso
inst_3 : Q.IsStableUnderComposition
inst_4 : Q.IsStableUnderBaseChange
inst_5 : Q.HasOfPostcompProperty Q
hPQ : P ≤ Q
X : Q.Over ⊤ S
R : Sieve X
h :
  (∃ (𝒰 : Cover (precoverage P) (Over.left (X.toComma (P := Q) (Q := ⊤) (W := ⊤)))) (x : Cover.Over S (P := P) 𝒰),
      𝒰.toPresieveOver (P := P) ≤ Sieve.arrows (Sieve.functorPushforward (MorphismProperty.Over.forget Q ⊤ S) R)) →
    ∃ R_1 ∈ Precoverage.coverings (Pretopology.toPrecoverage (smallPretopology P Q : Pretopology (Q.Over ⊤ S))) X,
      R_1 ≤ R.arrows
⊢ (∃ (𝒰 : Cover (precoverage P) (Over.left (X.toComma (P := Q) (Q := ⊤) (W := ⊤)))) (x : Cover.Over S (P := P) 𝒰),
      𝒰.toPresieveOver (P := P) ≤ Sieve.arrows (Sieve.functorPushforward (MorphismProperty.Over.forget Q ⊤ S) R)) ↔
    ∃ R_1 ∈ Precoverage.coverings (Pretopology.toPrecoverage (smallPretopology P Q : Pretopology (Q.Over ⊤ S))) X,
      R_1 ≤ R.arrows
have h₁ : (∃ R_1 ∈ Precoverage.coverings (Pretopology.toPrecoverage (smallPretopology P Q : Pretopology (Q.Over ⊤ S))) X,
    R_1 ≤ R.arrows) →
  ∃ (𝒰 : Cover (precoverage P) (Over.left (X.toComma (P := Q) (Q := ⊤) (W := ⊤)))) (x : Cover.Over S (P := P) 𝒰),
    𝒰.toPresieveOver (P := P) ≤ Sieve.arrows (Sieve.functorPushforward (MorphismProperty.Over.forget Q ⊤ S) R)
Tactic: constructor
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.Small (goal)
Goal:
P Q : MorphismProperty Scheme
S : Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
inst_2 : P.RespectsIso
inst_3 : Q.IsStableUnderComposition
inst_4 : Q.IsStableUnderBaseChange
inst_5 : Q.HasOfPostcompProperty Q
hPQ : P ≤ Q
X : Q.Over ⊤ S
R : Sieve X
𝒰 : Cover (precoverage P) (Over.left (X.toComma (P := Q) (Q := ⊤) (W := ⊤)))
h : Cover.Over S (P := P) 𝒰
le : 𝒰.toPresieveOver (P := P) ≤ Sieve.arrows (Sieve.functorPushforward (MorphismProperty.Over.forget Q ⊤ S) R)
j : 𝒰.I₀
⊢ Q (𝒰.X j ↘ S)
have rw : Q (𝒰.f j ≫ Over.left (X.toComma (P := Q) (Q := ⊤) (W := ⊤)) ↘ S)
Tactic: rw [← comp_over (𝒰.f j)]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.Small (hypothesis)
Goal:
P Q : MorphismProperty Scheme
S : Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
inst_2 : P.RespectsIso
inst_3 : Q.IsStableUnderComposition
inst_4 : Q.IsStableUnderBaseChange
inst_5 : Q.HasOfPostcompProperty Q
hPQ : P ≤ Q
X : Q.Over ⊤ S
R : Sieve X
𝒰 : Cover (precoverage P) (Over.left (X.toComma (P := Q) (Q := ⊤) (W := ⊤)))
h : Cover.Over S (P := P) 𝒰
le : 𝒰.toPresieveOver (P := P) ≤ Sieve.arrows (Sieve.functorPushforward (MorphismProperty.Over.forget Q ⊤ S) R)
⊢ ∃ R_1 ∈ Precoverage.coverings (Pretopology.toPrecoverage (smallPretopology P Q : Pretopology (Q.Over ⊤ S))) X,
    R_1 ≤ R.arrows
have hj : ∀ (j : 𝒰.I₀), Q (𝒰.X j ↘ S)
Tactic: have hj (j : 𝒰.I₀) : Q (𝒰.X j ↘ S) := by
      rw [← comp_over (𝒰.f j)]
      exact Q.comp_mem _ _ (hPQ _ <| 𝒰.map_prop _) X.prop
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.Small (goal)
Goal:
P Q : MorphismProperty Scheme
S : Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
inst_2 : P.RespectsIso
inst_3 : Q.IsStableUnderComposition
inst_4 : Q.IsStableUnderBaseChange
inst_5 : Q.HasOfPostcompProperty Q
hPQ : P ≤ Q
X : Q.Over ⊤ S
R : Sieve X
𝒰 : Cover (precoverage P) (Over.left (X.toComma (P := Q) (Q := ⊤) (W := ⊤)))
h : Cover.Over S (P := P) 𝒰
le : 𝒰.toPresieveOver (P := P) ≤ Sieve.arrows (Sieve.functorPushforward (MorphismProperty.Over.forget Q ⊤ S) R)
hj : ∀ (j : 𝒰.I₀), Q (𝒰.X j ↘ S)
⊢ ∃ R_1 ∈ Precoverage.coverings (Pretopology.toPrecoverage (smallPretopology P Q : Pretopology (Q.Over ⊤ S))) X,
    R_1 ≤ R.arrows
have h₁ : 𝒰.toPresieveOverProp (P := P) (Q := Q) hj ∈
  Precoverage.coverings (Pretopology.toPrecoverage (smallPretopology P Q : Pretopology (Q.Over ⊤ S))) X
Tactic: refine ⟨𝒰.toPresieveOverProp hj, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.Small (goal)
Goal:
P Q : MorphismProperty Scheme
S : Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
inst_2 : P.RespectsIso
inst_3 : Q.IsStableUnderComposition
inst_4 : Q.IsStableUnderBaseChange
inst_5 : Q.HasOfPostcompProperty Q
hPQ : P ≤ Q
X : Q.Over ⊤ S
R : Sieve X
𝒰 : Cover (precoverage P) (Over.left (X.toComma (P := Q) (Q := ⊤) (W := ⊤)))
h : Cover.Over S (P := P) 𝒰
le : 𝒰.toPresieveOver (P := P) ≤ Sieve.arrows (Sieve.functorPushforward (MorphismProperty.Over.forget Q ⊤ S) R)
hj : ∀ (j : 𝒰.I₀), Q (𝒰.X j ↘ S)
h₁ :
  𝒰.toPresieveOverProp (P := P) (Q := Q) hj ∈
    Precoverage.coverings (Pretopology.toPrecoverage (smallPretopology P Q : Pretopology (Q.Over ⊤ S))) X
⊢ ∃ R_1 ∈ Precoverage.coverings (Pretopology.toPrecoverage (smallPretopology P Q : Pretopology (Q.Over ⊤ S))) X,
    R_1 ≤ R.arrows
have h₂ : 𝒰.toPresieveOverProp (P := P) (Q := Q) hj ≤ R.arrows
Tactic: refine ⟨𝒰.toPresieveOverProp hj, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.Small (goal)
Goal:
P Q : MorphismProperty Scheme
S : Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
inst_2 : P.RespectsIso
inst_3 : Q.IsStableUnderComposition
inst_4 : Q.IsStableUnderBaseChange
inst_5 : Q.HasOfPostcompProperty Q
hPQ : P ≤ Q
X : Q.Over ⊤ S
R : Sieve X
𝒰 : Cover (precoverage P) (Over.left (X.toComma (P := Q) (Q := ⊤) (W := ⊤)))
h : Cover.Over S (P := P) 𝒰
le : 𝒰.toPresieveOver (P := P) ≤ Sieve.arrows (Sieve.functorPushforward (MorphismProperty.Over.forget Q ⊤ S) R)
hj : ∀ (j : 𝒰.I₀), Q (𝒰.X j ↘ S)
⊢ 𝒰.toPresieveOverProp (P := P) (Q := Q) hj ∈
    Precoverage.coverings (Pretopology.toPrecoverage (smallPretopology P Q : Pretopology (Q.Over ⊤ S))) X
have w : Cover (precoverage P) X.toComma (P := Q) (Q := ⊤) (W := ⊤).left
Tactic: use 𝒰, h, hj
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.Small (goal)
Goal:
P Q : MorphismProperty Scheme
S : Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
inst_2 : P.RespectsIso
inst_3 : Q.IsStableUnderComposition
inst_4 : Q.IsStableUnderBaseChange
inst_5 : Q.HasOfPostcompProperty Q
hPQ : P ≤ Q
X : Q.Over ⊤ S
R : Sieve X
𝒰 : Cover (precoverage P) (Over.left (X.toComma (P := Q) (Q := ⊤) (W := ⊤)))
h : Cover.Over S (P := P) 𝒰
le : 𝒰.toPresieveOver (P := P) ≤ Sieve.arrows (Sieve.functorPushforward (MorphismProperty.Over.forget Q ⊤ S) R)
hj : ∀ (j : 𝒰.I₀), Q (𝒰.X j ↘ S)
w : Cover (precoverage P) X.toComma (P := Q) (Q := ⊤) (W := ⊤).left
w₁ : Cover.Over S (P := P) 𝒰
w₂ : ∀ (j : 𝒰.I₀), Q (𝒰.X j ↘ S)
⊢ 𝒰.toPresieveOverProp (P := P) (Q := Q) hj ∈
    Precoverage.coverings (Pretopology.toPrecoverage (smallPretopology P Q : Pretopology (Q.Over ⊤ S))) X
have h₁ : 𝒰.toPresieveOverProp (P := P) (Q := Q) hj = 𝒰.toPresieveOverProp (P := P) (Q := Q) hj
Tactic: use 𝒰, h, hj
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.Small (hypothesis)
Goal:
P Q : MorphismProperty Scheme
S : Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
inst_2 : P.RespectsIso
inst_3 : Q.IsStableUnderComposition
inst_4 : Q.IsStableUnderBaseChange
inst_5 : Q.HasOfPostcompProperty Q
hPQ : P ≤ Q
X : Q.Over ⊤ S
R : Sieve X
𝒰 : Cover (precoverage P) (Over.left (X.toComma (P := Q) (Q := ⊤) (W := ⊤)))
h : Cover.Over S (P := P) 𝒰
le : 𝒰.toPresieveOver (P := P) ≤ Sieve.arrows (Sieve.functorPushforward (MorphismProperty.Over.forget Q ⊤ S) R)
hj : ∀ (j : 𝒰.I₀), Q (𝒰.X j ↘ S)
Y : Q.Over ⊤ S
i : 𝒰.I₀
fi : (𝒰.X i).asOverProp (P := Q) S ⋯ ⟶ X := Hom.asOverProp (P := Q) (𝒰.f i) S
⊢ R.arrows (Y := (𝒰.X i).asOverProp (P := Q) S ⋯) (Hom.asOverProp (P := Q) (𝒰.f i) S)
have this : Sieve.arrows (X := Functor.obj (MorphismProperty.Over.forget Q ⊤ S) X)
  (Sieve.functorPushforward (MorphismProperty.Over.forget Q ⊤ S) R)
  (CategoryTheory.Functor.map (MorphismProperty.Over.forget Q ⊤ S) fi)
Tactic: have : R.functorPushforward _ ((MorphismProperty.Over.forget Q ⊤ S).map fi) := le _ _ ⟨i⟩
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.Small (goal)
Goal:
P Q : MorphismProperty Scheme
S : Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
inst_2 : P.RespectsIso
inst_3 : Q.IsStableUnderComposition
inst_4 : Q.IsStableUnderBaseChange
inst_5 : Q.HasOfPostcompProperty Q
hPQ : P ≤ Q
X : Q.Over ⊤ S
R : Sieve X
𝒰 : Cover (precoverage P) (Over.left (X.toComma (P := Q) (Q := ⊤) (W := ⊤)))
h : Cover.Over S (P := P) 𝒰
le : 𝒰.toPresieveOver (P := P) ≤ Sieve.arrows (Sieve.functorPushforward (MorphismProperty.Over.forget Q ⊤ S) R)
hj : ∀ (j : 𝒰.I₀), Q (𝒰.X j ↘ S)
Y : Q.Over ⊤ S
i : 𝒰.I₀
fi : (𝒰.X i).asOverProp (P := Q) S ⋯ ⟶ X := Hom.asOverProp (P := Q) (𝒰.f i) S
this :
  Sieve.arrows (X := Functor.obj (MorphismProperty.Over.forget Q ⊤ S) X)
    (Sieve.functorPushforward (MorphismProperty.Over.forget Q ⊤ S) R)
    (CategoryTheory.Functor.map (MorphismProperty.Over.forget Q ⊤ S) fi)
⊢ R.arrows (Y := (𝒰.X i).asOverProp (P := Q) S ⋯) (Hom.asOverProp (P := Q) (𝒰.f i) S)
have h₁ : ∀
  (this :
    Presieve.functorPushforward (MorphismProperty.Over.forget Q ⊤ S) R.arrows
      (CategoryTheory.Functor.map (MorphismProperty.Over.forget Q ⊤ S) fi)),
  R.arrows (Y := (𝒰.X i).asOverProp (P := Q) S sorry) (Hom.asOverProp (P := Q) (𝒰.f i) S)
Tactic: rwa [Sieve.functorPushforward_apply,
        Sieve.mem_functorPushforward_iff_of_full_of_faithful] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.Small (goal)
Goal:
P Q : MorphismProperty Scheme
S : Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
inst_2 : P.RespectsIso
inst_3 : Q.IsStableUnderComposition
inst_4 : Q.IsStableUnderBaseChange
inst_5 : Q.HasOfPostcompProperty Q
hPQ : P ≤ Q
X : Q.Over ⊤ S
R : Sieve X
𝒰 : Cover (precoverage P) (Over.left (X.toComma (P := Q) (Q := ⊤) (W := ⊤)))
h : Cover.Over S (P := P) 𝒰
le : 𝒰.toPresieveOver (P := P) ≤ Sieve.arrows (Sieve.functorPushforward (MorphismProperty.Over.forget Q ⊤ S) R)
hj : ∀ (j : 𝒰.I₀), Q (𝒰.X j ↘ S)
Y : Q.Over ⊤ S
i : 𝒰.I₀
fi : (𝒰.X i).asOverProp (P := Q) S ⋯ ⟶ X := Hom.asOverProp (P := Q) (𝒰.f i) S
this :
  Sieve.arrows (X := Functor.obj (MorphismProperty.Over.forget Q ⊤ S) X)
    (Sieve.functorPushforward (MorphismProperty.Over.forget Q ⊤ S) R)
    (CategoryTheory.Functor.map (MorphismProperty.Over.forget Q ⊤ S) fi)
h₁ :
  ∀
    (this :
      Presieve.functorPushforward (MorphismProperty.Over.forget Q ⊤ S) R.arrows
        (CategoryTheory.Functor.map (MorphismProperty.Over.forget Q ⊤ S) fi)),
    R.arrows (Y := (𝒰.X i).asOverProp (P := Q) S ⋯) (Hom.asOverProp (P := Q) (𝒰.f i) S)
⊢ R.arrows (Y := (𝒰.X i).asOverProp (P := Q) S ⋯) (Hom.asOverProp (P := Q) (𝒰.f i) S)
have h₂ : ∀ (this : R.arrows fi), R.arrows (Y := (𝒰.X i).asOverProp (P := Q) S sorry) (Hom.asOverProp (P := Q) (𝒰.f i) S)
Tactic: rwa [Sieve.functorPushforward_apply,
        Sieve.mem_functorPushforward_iff_of_full_of_faithful] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.Small (goal)
Goal:
P Q : MorphismProperty Scheme
S : Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
inst_2 : P.RespectsIso
inst_3 : Q.IsStableUnderComposition
inst_4 : Q.IsStableUnderBaseChange
inst_5 : Q.HasOfPostcompProperty Q
hPQ : P ≤ Q
X : Q.Over ⊤ S
R : Sieve X
𝒰 : Cover (precoverage P) X.toComma (P := Q) (Q := ⊤) (W := ⊤).left
h : Cover.Over S (P := P) 𝒰
p : ∀ (j : 𝒰.I₀), Q (𝒰.X j ↘ S)
le : 𝒰.toPresieveOverProp (P := P) (Q := Q) p ≤ R.arrows
⊢ ∃ (𝒰 : Cover (precoverage P) (Over.left (X.toComma (P := Q) (Q := ⊤) (W := ⊤)))) (x : Cover.Over S (P := P) 𝒰),
    𝒰.toPresieveOver (P := P) ≤ Sieve.arrows (Sieve.functorPushforward (MorphismProperty.Over.forget Q ⊤ S) R)
have h₁ : 𝒰.toPresieveOver (P := P) ≤ Sieve.arrows (Sieve.functorPushforward (MorphismProperty.Over.forget Q ⊤ S) R)
Tactic: use 𝒰, h
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.Small (goal)
Goal:
P Q : MorphismProperty Scheme
S : Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
inst_2 : P.RespectsIso
inst_3 : Q.IsStableUnderComposition
inst_4 : Q.IsStableUnderBaseChange
inst_5 : Q.HasOfPostcompProperty Q
X : Q.Over ⊤ S
R : Sieve X
⊢ (∃ R_1 ∈ Precoverage.coverings (Pretopology.toPrecoverage (smallPretopology P Q : Pretopology (Q.Over ⊤ S))) X,
      R_1 ≤ R.arrows) ↔
    ∀ (x : ↥X.toComma (P := Q) (Q := ⊤) (W := ⊤).left),
      ∃ (Y : Q.Over ⊤ S) (f : Y ⟶ X) (y : ↥Y.toComma (P := Q) (Q := ⊤) (W := ⊤).left),
        R.arrows f ∧
          P f.toCommaMorphism (P := Q) (Q := ⊤) (W := ⊤).left ∧ f.toCommaMorphism (P := Q) (Q := ⊤) (W := ⊤).left y = x
have refine_1 : (∃ R_1 ∈ Precoverage.coverings (Pretopology.toPrecoverage (smallPretopology P Q : Pretopology (Q.Over ⊤ S))) X,
    R_1 ≤ R.arrows) →
  ∀ (x : ↥X.toComma (P := Q) (Q := ⊤) (W := ⊤).left),
    ∃ (Y : Q.Over ⊤ S) (f : Y ⟶ X) (y : ↥Y.toComma (P := Q) (Q := ⊤) (W := ⊤).left),
      R.arrows f ∧
        P f.toCommaMorphism (P := Q) (Q := ⊤) (W := ⊤).left ∧ f.toCommaMorphism (P := Q) (Q := ⊤) (W := ⊤).left y = x
Tactic: refine ⟨?_, fun h ↦ ?_⟩
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.Small (goal)
Goal:
P Q : MorphismProperty Scheme
S : Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
inst_2 : P.RespectsIso
inst_3 : Q.IsStableUnderComposition
inst_4 : Q.IsStableUnderBaseChange
inst_5 : Q.HasOfPostcompProperty Q
X : Q.Over ⊤ S
R : Sieve X
refine_1 :
  (∃ R_1 ∈ Precoverage.coverings (Pretopology.toPrecoverage (smallPretopology P Q : Pretopology (Q.Over ⊤ S))) X,
      R_1 ≤ R.arrows) →
    ∀ (x : ↥X.toComma (P := Q) (Q := ⊤) (W := ⊤).left),
      ∃ (Y : Q.Over ⊤ S) (f : Y ⟶ X) (y : ↥Y.toComma (P := Q) (Q := ⊤) (W := ⊤).left),
        R.arrows f ∧
          P f.toCommaMorphism (P := Q) (Q := ⊤) (W := ⊤).left ∧ f.toCommaMorphism (P := Q) (Q := ⊤) (W := ⊤).left y = x
⊢ (∃ R_1 ∈ Precoverage.coverings (Pretopology.toPrecoverage (smallPretopology P Q : Pretopology (Q.Over ⊤ S))) X,
      R_1 ≤ R.arrows) ↔
    ∀ (x : ↥X.toComma (P := Q) (Q := ⊤) (W := ⊤).left),
      ∃ (Y : Q.Over ⊤ S) (f : Y ⟶ X) (y : ↥Y.toComma (P := Q) (Q := ⊤) (W := ⊤).left),
        R.arrows f ∧
          P f.toCommaMorphism (P := Q) (Q := ⊤) (W := ⊤).left ∧ f.toCommaMorphism (P := Q) (Q := ⊤) (W := ⊤).left y = x
have refine_2 : ∀
  (h :
    ∀ (x : ↥X.toComma (P := Q) (Q := ⊤) (W := ⊤).left),
      ∃ (Y : Q.Over ⊤ S) (f : Y ⟶ X) (y : ↥Y.toComma (P := Q) (Q := ⊤) (W := ⊤).left),
        R.arrows f ∧
          P f.toCommaMorphism (P := Q) (Q := ⊤) (W := ⊤).left ∧
            f.toCommaMorphism (P := Q) (Q := ⊤) (W := ⊤).left y = x),
  ∃ R_1 ∈ Precoverage.coverings (Pretopology.toPrecoverage (smallPretopology P Q : Pretopology (Q.Over ⊤ S))) X,
    R_1 ≤ R.arrows
Tactic: refine ⟨?_, fun h ↦ ?_⟩
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.Small (goal)
Goal:
P Q : MorphismProperty Scheme
S : Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
inst_2 : P.RespectsIso
inst_3 : Q.IsStableUnderComposition
inst_4 : Q.IsStableUnderBaseChange
inst_5 : Q.HasOfPostcompProperty Q
X : Q.Over ⊤ S
R : Sieve X
𝒰 : Cover (precoverage P) X.toComma (P := Q) (Q := ⊤) (W := ⊤).left
h : Cover.Over S (P := P) 𝒰
p : ∀ (j : 𝒰.I₀), Q (𝒰.X j ↘ S)
hle : 𝒰.toPresieveOverProp (P := P) (Q := Q) p ≤ R.arrows
x : ↥X.toComma (P := Q) (Q := ⊤) (W := ⊤).left
⊢ ∃ (Y : Q.Over ⊤ S) (f : Y ⟶ X) (y : ↥Y.toComma (P := Q) (Q := ⊤) (W := ⊤).left),
    R.arrows f ∧
      P f.toCommaMorphism (P := Q) (Q := ⊤) (W := ⊤).left ∧ f.toCommaMorphism (P := Q) (Q := ⊤) (W := ⊤).left y = x
have refine_1 : ∀ (y : ↥(𝒰.X (𝒰.idx x))) (hy : (𝒰.f (𝒰.idx x)) y = x),
  ∃ (Y : Q.Over ⊤ S) (f : Y ⟶ X) (y : ↥Y.toComma (P := Q) (Q := ⊤) (W := ⊤).left),
    R.arrows f ∧
      P f.toCommaMorphism (P := Q) (Q := ⊤) (W := ⊤).left ∧ f.toCommaMorphism (P := Q) (Q := ⊤) (W := ⊤).left y = x
Tactic: obtain ⟨y, hy⟩ := 𝒰.covers x
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.Small (goal)
Goal:
P Q : MorphismProperty Scheme
S : Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
inst_2 : P.RespectsIso
inst_3 : Q.IsStableUnderComposition
inst_4 : Q.IsStableUnderBaseChange
inst_5 : Q.HasOfPostcompProperty Q
X : Q.Over ⊤ S
R : Sieve X
𝒰 : Cover (precoverage P) X.toComma (P := Q) (Q := ⊤) (W := ⊤).left
h : Cover.Over S (P := P) 𝒰
p : ∀ (j : 𝒰.I₀), Q (𝒰.X j ↘ S)
hle : 𝒰.toPresieveOverProp (P := P) (Q := Q) p ≤ R.arrows
x : ↥X.toComma (P := Q) (Q := ⊤) (W := ⊤).left
y : ↥(𝒰.X (𝒰.idx x))
hy : (𝒰.f (𝒰.idx x)) y = x
⊢ ∃ (Y : Q.Over ⊤ S) (f : Y ⟶ X) (y : ↥Y.toComma (P := Q) (Q := ⊤) (W := ⊤).left),
    R.arrows f ∧
      P f.toCommaMorphism (P := Q) (Q := ⊤) (W := ⊤).left ∧ f.toCommaMorphism (P := Q) (Q := ⊤) (W := ⊤).left y = x
have refine_1 : 𝒰.toPresieveOverProp (P := P) (Q := Q) p (Y := (𝒰.X (𝒰.idx x)).asOverProp (P := Q) S sorry)
  (Hom.asOverProp (P := Q) (𝒰.f (𝒰.idx x)) S)
Tactic: refine ⟨(𝒰.X (𝒰.idx x)).asOverProp S (p _), (𝒰.f (𝒰.idx x)).asOverProp S, y, hle _ _ ?_,
      𝒰.map_prop _, hy⟩
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.Small (goal)
Goal:
P Q : MorphismProperty Scheme
S : Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
inst_2 : P.RespectsIso
inst_3 : Q.IsStableUnderComposition
inst_4 : Q.IsStableUnderBaseChange
inst_5 : Q.HasOfPostcompProperty Q
X : Q.Over ⊤ S
R : Sieve X
h :
  ∀ (x : ↥X.toComma (P := Q) (Q := ⊤) (W := ⊤).left),
    ∃ (Y : Q.Over ⊤ S) (f : Y ⟶ X) (y : ↥Y.toComma (P := Q) (Q := ⊤) (W := ⊤).left),
      R.arrows f ∧
        P f.toCommaMorphism (P := Q) (Q := ⊤) (W := ⊤).left ∧ f.toCommaMorphism (P := Q) (Q := ⊤) (W := ⊤).left y = x
⊢ ∃ R_1 ∈ Precoverage.coverings (Pretopology.toPrecoverage (smallPretopology P Q : Pretopology (Q.Over ⊤ S))) X,
    R_1 ≤ R.arrows
have refine_2 : ∀ (Y : (x : ↥X.toComma (P := Q) (Q := ⊤) (W := ⊤).left) → Q.Over ⊤ S)
  (f : (x : ↥X.toComma (P := Q) (Q := ⊤) (W := ⊤).left) → Y x ⟶ X)
  (y : (x : ↥X.toComma (P := Q) (Q := ⊤) (W := ⊤).left) → ↥(Y x).toComma (P := Q) (Q := ⊤) (W := ⊤).left)
  (hf : ∀ (x : ↥X.toComma (P := Q) (Q := ⊤) (W := ⊤).left), R.arrows (f x))
  (hP : ∀ (x : ↥X.toComma (P := Q) (Q := ⊤) (W := ⊤).left), P (f x).toCommaMorphism (P := Q) (Q := ⊤) (W := ⊤).left)
  (hy :
    ∀ (x : ↥X.toComma (P := Q) (Q := ⊤) (W := ⊤).left),
      (f x).toCommaMorphism (P := Q) (Q := ⊤) (W := ⊤).left (y x) = x),
  ∃ R_1 ∈ Precoverage.coverings (Pretopology.toPrecoverage (smallPretopology P Q : Pretopology (Q.Over ⊤ S))) X,
    R_1 ≤ R.arrows
Tactic: choose Y f y hf hP hy using h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.Small (goal)
Goal:
P Q : MorphismProperty Scheme
S : Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
inst_2 : P.RespectsIso
inst_3 : Q.IsStableUnderComposition
inst_4 : Q.IsStableUnderBaseChange
inst_5 : Q.HasOfPostcompProperty Q
X : Q.Over ⊤ S
R : Sieve X
Y : (x : ↥X.toComma (P := Q) (Q := ⊤) (W := ⊤).left) → Q.Over ⊤ S
f : (x : ↥X.toComma (P := Q) (Q := ⊤) (W := ⊤).left) → Y x ⟶ X
y : (x : ↥X.toComma (P := Q) (Q := ⊤) (W := ⊤).left) → ↥(Y x).toComma (P := Q) (Q := ⊤) (W := ⊤).left
hf : ∀ (x : ↥X.toComma (P := Q) (Q := ⊤) (W := ⊤).left), R.arrows (f x)
hP : ∀ (x : ↥X.toComma (P := Q) (Q := ⊤) (W := ⊤).left), P (f x).toCommaMorphism (P := Q) (Q := ⊤) (W := ⊤).left
hy : ∀ (x : ↥X.toComma (P := Q) (Q := ⊤) (W := ⊤).left), (f x).toCommaMorphism (P := Q) (Q := ⊤) (W := ⊤).left (y x) = x
𝒰 : Cover (precoverage P) X.toComma (P := Q) (Q := ⊤) (W := ⊤).left :=
  { I₀ := ↥X.toComma (P := Q) (Q := ⊤) (W := ⊤).left,
    X := fun (i : ↥X.toComma (P := Q) (Q := ⊤) (W := ⊤).left) ↦ (Y i).toComma (P := Q) (Q := ⊤) (W := ⊤).left,
    f := fun (i : ↥X.toComma (P := Q) (Q := ⊤) (W := ⊤).left) ↦ (f i).toCommaMorphism (P := Q) (Q := ⊤) (W := ⊤).left,
    mem₀ := ⋯ }
this : Cover.Over S (P := P) 𝒰 := Cover.Over.mk (P := P) (fun (i : 𝒰.I₀) ↦ inferInstance) ⋯
⊢ ∃ R_1 ∈ Precoverage.coverings (Pretopology.toPrecoverage (smallPretopology P Q : Pretopology (Q.Over ⊤ S))) X,
    R_1 ≤ R.arrows
have refine_2 : 𝒰.toPresieveOverProp (P := P) (Q := Q) sorry ∈
  Precoverage.coverings (Pretopology.toPrecoverage (smallPretopology P Q : Pretopology (Q.Over ⊤ S))) X
Tactic: refine ⟨𝒰.toPresieveOverProp fun i ↦ MorphismProperty.Comma.prop _, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.Small (goal)
Goal:
P Q : MorphismProperty Scheme
S : Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
inst_2 : P.RespectsIso
inst_3 : Q.IsStableUnderComposition
inst_4 : Q.IsStableUnderBaseChange
inst_5 : Q.HasOfPostcompProperty Q
X : Q.Over ⊤ S
R : Sieve X
Y : (x : ↥X.toComma (P := Q) (Q := ⊤) (W := ⊤).left) → Q.Over ⊤ S
f : (x : ↥X.toComma (P := Q) (Q := ⊤) (W := ⊤).left) → Y x ⟶ X
y : (x : ↥X.toComma (P := Q) (Q := ⊤) (W := ⊤).left) → ↥(Y x).toComma (P := Q) (Q := ⊤) (W := ⊤).left
hf : ∀ (x : ↥X.toComma (P := Q) (Q := ⊤) (W := ⊤).left), R.arrows (f x)
hP : ∀ (x : ↥X.toComma (P := Q) (Q := ⊤) (W := ⊤).left), P (f x).toCommaMorphism (P := Q) (Q := ⊤) (W := ⊤).left
hy : ∀ (x : ↥X.toComma (P := Q) (Q := ⊤) (W := ⊤).left), (f x).toCommaMorphism (P := Q) (Q := ⊤) (W := ⊤).left (y x) = x
𝒰 : Cover (precoverage P) X.toComma (P := Q) (Q := ⊤) (W := ⊤).left :=
  { I₀ := ↥X.toComma (P := Q) (Q := ⊤) (W := ⊤).left,
    X := fun (i : ↥X.toComma (P := Q) (Q := ⊤) (W := ⊤).left) ↦ (Y i).toComma (P := Q) (Q := ⊤) (W := ⊤).left,
    f := fun (i : ↥X.toComma (P := Q) (Q := ⊤) (W := ⊤).left) ↦ (f i).toCommaMorphism (P := Q) (Q := ⊤) (W := ⊤).left,
    mem₀ := ⋯ }
this : Cover.Over S (P := P) 𝒰 := Cover.Over.mk (P := P) (fun (i : 𝒰.I₀) ↦ inferInstance) ⋯
refine_2 :
  𝒰.toPresieveOverProp (P := P) (Q := Q) ⋯ ∈
    Precoverage.coverings (Pretopology.toPrecoverage (smallPretopology P Q : Pretopology (Q.Over ⊤ S))) X
⊢ ∃ R_1 ∈ Precoverage.coverings (Pretopology.toPrecoverage (smallPretopology P Q : Pretopology (Q.Over ⊤ S))) X,
    R_1 ≤ R.arrows
have refine_2₁ : 𝒰.toPresieveOverProp (P := P) (Q := Q) sorry ≤ R.arrows
Tactic: refine ⟨𝒰.toPresieveOverProp fun i ↦ MorphismProperty.Comma.prop _, ?_, ?_⟩