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
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) ⋯
w : Cover (precoverage P) X.toComma (P := Q) (Q := ⊤) (W := ⊤).left
w₁ : Cover.Over S (P := P) 𝒰
⊢ 𝒰.toPresieveOverProp (P := P) (Q := Q) ⋯ ∈
    Precoverage.coverings (Pretopology.toPrecoverage (smallPretopology P Q : Pretopology (Q.Over ⊤ S))) X
have w₂ : ∀ (j : 𝒰.I₀), Q (𝒰.X j ↘ S)
Tactic: use 𝒰, inferInstance, 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) ⋯
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) ⋯ ∈
    Precoverage.coverings (Pretopology.toPrecoverage (smallPretopology P Q : Pretopology (Q.Over ⊤ S))) X
have h : 𝒰.toPresieveOverProp (P := P) (Q := Q) sorry = 𝒰.toPresieveOverProp (P := P) (Q := Q) sorry
Tactic: use 𝒰, inferInstance, fun i ↦ MorphismProperty.Comma.prop _
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.Small (goal)
Goal:
P : MorphismProperty Scheme
S : Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
inst_2 : P.RespectsIso
inst_3 : P.HasOfPostcompProperty P
X : P.Over ⊤ S
R : Sieve X
⊢ R ∈ (smallGrothendieckTopology P : (a : P.Over ⊤ S) → Set (Sieve a)) X ↔
    ∃ (𝒰 : Cover (precoverage P) X.toComma (P := P) (Q := ⊤) (W := ⊤).left) (x : Cover.Over S (P := P) 𝒰) (h :
      ∀ (j : 𝒰.I₀), P (𝒰.X j ↘ S)), 𝒰.toPresieveOverProp (P := P) (Q := P) h ≤ R.arrows
have rw : R ∈
    (Pretopology.toGrothendieck (smallPretopology P P : Pretopology (P.Over ⊤ S)) : (a : P.Over ⊤ S) → Set (Sieve a))
      X ↔
  ∃ (𝒰 : Cover (precoverage P) X.toComma (P := P) (Q := ⊤) (W := ⊤).left) (x : Cover.Over S (P := P) 𝒰) (h :
    ∀ (j : 𝒰.I₀), P (𝒰.X j ↘ S)), 𝒰.toPresieveOverProp (P := P) (Q := P) h ≤ R.arrows
Tactic: rw [smallGrothendieckTopology_eq_toGrothendieck_smallPretopology]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.Small (goal)
Goal:
P : MorphismProperty Scheme
S : Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
inst_2 : P.RespectsIso
inst_3 : P.HasOfPostcompProperty P
X : P.Over ⊤ S
R : Sieve X
⊢ R ∈
      (Pretopology.toGrothendieck (smallPretopology P P : Pretopology (P.Over ⊤ S)) : (a : P.Over ⊤ S) → Set (Sieve a))
        X ↔
    ∃ (𝒰 : Cover (precoverage P) X.toComma (P := P) (Q := ⊤) (W := ⊤).left) (x : Cover.Over S (P := P) 𝒰) (h :
      ∀ (j : 𝒰.I₀), P (𝒰.X j ↘ S)), 𝒰.toPresieveOverProp (P := P) (Q := P) h ≤ R.arrows
have mp : R ∈
    (Pretopology.toGrothendieck (smallPretopology P P : Pretopology (P.Over ⊤ S)) : (a : P.Over ⊤ S) → Set (Sieve a))
      X →
  ∃ (𝒰 : Cover (precoverage P) X.toComma (P := P) (Q := ⊤) (W := ⊤).left) (x : Cover.Over S (P := P) 𝒰) (h :
    ∀ (j : 𝒰.I₀), P (𝒰.X j ↘ S)), 𝒰.toPresieveOverProp (P := P) (Q := P) h ≤ R.arrows
Tactic: constructor
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.Small (goal)
Goal:
P : MorphismProperty Scheme
S : Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
inst_2 : P.RespectsIso
inst_3 : P.HasOfPostcompProperty P
X : P.Over ⊤ S
R : Sieve X
mp :
  R ∈
      (Pretopology.toGrothendieck (smallPretopology P P : Pretopology (P.Over ⊤ S)) : (a : P.Over ⊤ S) → Set (Sieve a))
        X →
    ∃ (𝒰 : Cover (precoverage P) X.toComma (P := P) (Q := ⊤) (W := ⊤).left) (x : Cover.Over S (P := P) 𝒰) (h :
      ∀ (j : 𝒰.I₀), P (𝒰.X j ↘ S)), 𝒰.toPresieveOverProp (P := P) (Q := P) h ≤ R.arrows
⊢ R ∈
      (Pretopology.toGrothendieck (smallPretopology P P : Pretopology (P.Over ⊤ S)) : (a : P.Over ⊤ S) → Set (Sieve a))
        X ↔
    ∃ (𝒰 : Cover (precoverage P) X.toComma (P := P) (Q := ⊤) (W := ⊤).left) (x : Cover.Over S (P := P) 𝒰) (h :
      ∀ (j : 𝒰.I₀), P (𝒰.X j ↘ S)), 𝒰.toPresieveOverProp (P := P) (Q := P) h ≤ R.arrows
have mpr : (∃ (𝒰 : Cover (precoverage P) X.toComma (P := P) (Q := ⊤) (W := ⊤).left) (x : Cover.Over S (P := P) 𝒰) (h :
    ∀ (j : 𝒰.I₀), P (𝒰.X j ↘ S)), 𝒰.toPresieveOverProp (P := P) (Q := P) h ≤ R.arrows) →
  R ∈
    (Pretopology.toGrothendieck (smallPretopology P P : Pretopology (P.Over ⊤ S)) : (a : P.Over ⊤ S) → Set (Sieve a)) X
Tactic: constructor
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
U V : X.AffineZariskiSite
h : (toOpensFunctor X).obj U ⟶ (toOpensFunctor X).obj V
x : ↥X
hx : x ∈ (toOpensFunctor X).obj U
⊢ ∃ (U_1 : TopologicalSpace.Opens ↥X) (f : U_1 ⟶ (toOpensFunctor X).obj U),
    (Sieve.functorPushforward (toOpensFunctor X) ((toOpensFunctor X).imageSieve h)).arrows f ∧ x ∈ U_1
have obtain : ∀ (f : (↑Γ(X, (↑V : X.Opens)) : Type u)) (hfU : X.basicOpen f ≤ (toOpensFunctor X).obj U)
  (hxf : (↑(Subtype.mk (p := fun (x : ↥X) ↦ x ∈ (toOpensFunctor X).obj U) x hx) : ↥X) ∈ X.basicOpen f),
  ∃ (U_1 : TopologicalSpace.Opens ↥X) (f : U_1 ⟶ (toOpensFunctor X).obj U),
    (Sieve.functorPushforward (toOpensFunctor X) ((toOpensFunctor X).imageSieve h)).arrows f ∧ x ∈ U_1
Tactic: obtain ⟨f, hfU, hxf⟩ := V.2.exists_basicOpen_le ⟨x, hx⟩ (h.le hx)
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
U : X.Opens
x : ↥X
hx : x ∈ U
⊢ ∃ (U_1 : TopologicalSpace.Opens ↥X) (f : U_1 ⟶ U), (Sieve.coverByImage (toOpensFunctor X) U).arrows f ∧ x ∈ U_1
have obtain : ∀ (V : TopologicalSpace.Opens ↥X) (hV : V ∈ X.affineOpens) (hxV : x ∈ (↑V : Set ↥X))
  (hVU : (↑V : Set ↥X) ⊆ (↑U : Set ↥X)),
  ∃ (U_1 : TopologicalSpace.Opens ↥X) (f : U_1 ⟶ U), (Sieve.coverByImage (toOpensFunctor X) U).arrows f ∧ x ∈ U_1
Tactic: obtain ⟨_, ⟨V, hV, rfl⟩, hxV, hVU⟩ := X.isBasis_affineOpens.exists_subset_of_mem_open hx U.2
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
U : X.AffineZariskiSite
S : Sieve U
⊢ S ∈ (grothendieckTopology X : (a : X.AffineZariskiSite) → Set (Sieve a)) U ↔
    ∀ x ∈ U.toOpens, ∃ (V : X.AffineZariskiSite) (f : V ⟶ U), S.arrows f ∧ x ∈ V.toOpens
have apply : ∀ (x : ↥X) (hxU : x ∈ (toOpensFunctor X).obj U),
  (∃ (U_1 : TopologicalSpace.Opens ↥X) (f : U_1 ⟶ (toOpensFunctor X).obj U),
      (Sieve.functorPushforward (toOpensFunctor X) S).arrows f ∧ x ∈ U_1) →
    ∃ (V : X.AffineZariskiSite) (f : V ⟶ U), S.arrows f ∧ x ∈ V.toOpens
Tactic: apply forall₂_congr fun x hxU ↦ ⟨?_, ?_⟩
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
U : X.AffineZariskiSite
S : Sieve U
apply :
  ∀ (x : ↥X) (hxU : x ∈ (toOpensFunctor X).obj U),
    (∃ (U_1 : TopologicalSpace.Opens ↥X) (f : U_1 ⟶ (toOpensFunctor X).obj U),
        (Sieve.functorPushforward (toOpensFunctor X) S).arrows f ∧ x ∈ U_1) →
      ∃ (V : X.AffineZariskiSite) (f : V ⟶ U), S.arrows f ∧ x ∈ V.toOpens
⊢ S ∈ (grothendieckTopology X : (a : X.AffineZariskiSite) → Set (Sieve a)) U ↔
    ∀ x ∈ U.toOpens, ∃ (V : X.AffineZariskiSite) (f : V ⟶ U), S.arrows f ∧ x ∈ V.toOpens
have apply₁ : ∀ (x : ↥X) (hxU : x ∈ (toOpensFunctor X).obj U),
  (∃ (V : X.AffineZariskiSite) (f : V ⟶ U), S.arrows f ∧ x ∈ V.toOpens) →
    ∃ (U_1 : TopologicalSpace.Opens ↥X) (f : U_1 ⟶ (toOpensFunctor X).obj U),
      (Sieve.functorPushforward (toOpensFunctor X) S).arrows f ∧ x ∈ U_1
Tactic: apply forall₂_congr fun x hxU ↦ ⟨?_, ?_⟩
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
U : X.AffineZariskiSite
P : Presieve U
⊢ U.presieveOfSections (sectionsOfPresieve P) = P
have refine_1 : ∀ (x : X.AffineZariskiSite) (V : X.Opens) (hV : IsAffineOpen V)
  (x : Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) V hV ⟶ U) (f : (↑Γ(X, U.toOpens) : Type u))
  (hf : X.basicOpen f = toOpens (Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) V hV)),
  U.presieveOfSections (sectionsOfPresieve P) (Y := Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) V hV)
      { down := { down := sorry } } →
    P (Y := Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) V hV) { down := { down := sorry } }
Tactic: refine funext₂ fun ⟨V, hV⟩ ⟨f, hf⟩ ↦ eq_iff_iff.mpr ⟨?_, ?_⟩
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
U : X.AffineZariskiSite
P : Presieve U
refine_1 :
  ∀ (x : X.AffineZariskiSite) (V : X.Opens) (hV : IsAffineOpen V)
    (x : Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) V hV ⟶ U) (f : (↑Γ(X, U.toOpens) : Type u))
    (hf : X.basicOpen f = toOpens (Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) V hV)),
    U.presieveOfSections (sectionsOfPresieve P) (Y := Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) V hV)
        { down := { down := ⋯ } } →
      P (Y := Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) V hV) { down := { down := ⋯ } }
⊢ U.presieveOfSections (sectionsOfPresieve P) = P
have refine_2 : ∀ (x : X.AffineZariskiSite) (V : X.Opens) (hV : IsAffineOpen V)
  (x : Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) V hV ⟶ U) (f : (↑Γ(X, U.toOpens) : Type u))
  (hf : X.basicOpen f = toOpens (Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) V hV)),
  P (Y := Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) V hV) { down := { down := sorry } } →
    U.presieveOfSections (sectionsOfPresieve P) (Y := Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) V hV)
      { down := { down := sorry } }
Tactic: refine funext₂ fun ⟨V, hV⟩ ⟨f, hf⟩ ↦ eq_iff_iff.mpr ⟨?_, ?_⟩
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
U : X.AffineZariskiSite
P : Presieve U
x : X.AffineZariskiSite
V : X.Opens
hV : IsAffineOpen V
x_1 : Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) V hV ⟶ U
f : (↑Γ(X, U.toOpens) : Type u)
hf : X.basicOpen f = toOpens (Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) V hV)
H : P (Y := Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) V hV) { down := { down := ⋯ } }
⊢ U.presieveOfSections (sectionsOfPresieve P) (Y := Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) V hV)
    { down := { down := ⋯ } }
have refine_2 : ∀ (hV : IsAffineOpen (X.basicOpen f)) (x : Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) (X.basicOpen f) hV ⟶ U)
  (H : P (Y := Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) (X.basicOpen f) hV) { down := { down := sorry } }),
  U.presieveOfSections (sectionsOfPresieve P) (Y :=
    Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) (X.basicOpen f) hV) { down := { down := sorry } }
Tactic: obtain rfl : _ = V := hf
Cleared hypotheses: V, hV, x_1, hf, H
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
U : X.AffineZariskiSite
s : Set (↑Γ(X, U.toOpens) : Type u)
⊢ U.presieveOfSections s =
    Presieve.ofArrows (fun (i : (↑s : Type u)) ↦ U.basicOpen (↑i : (↑Γ(X, U.toOpens) : Type u)))
      fun (i : (↑s : Type u)) ↦ homOfLE ⋯
have refine_1 : ∀ (x : X.AffineZariskiSite) (V : X.Opens) (hV : IsAffineOpen V)
  (x : Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) V hV ⟶ U) (f : (↑Γ(X, U.toOpens) : Type u))
  (hf : X.basicOpen f = toOpens (Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) V hV)),
  U.presieveOfSections s (Y := Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) V hV) { down := { down := sorry } } →
    Presieve.ofArrows (C := X.AffineZariskiSite) (X := U)
      (fun (i : (↑s : Type u)) ↦ U.basicOpen (↑i : (↑Γ(X, U.toOpens) : Type u))) fun (i : (↑s : Type u)) ↦ homOfLE sorry
      (Y := Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) V hV) { down := { down := sorry } }
Tactic: refine funext₂ fun ⟨V, hV⟩ ⟨f, hf⟩ ↦ eq_iff_iff.mpr ⟨?_, ?_⟩
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
U : X.AffineZariskiSite
s : Set (↑Γ(X, U.toOpens) : Type u)
refine_1 :
  ∀ (x : X.AffineZariskiSite) (V : X.Opens) (hV : IsAffineOpen V)
    (x : Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) V hV ⟶ U) (f : (↑Γ(X, U.toOpens) : Type u))
    (hf : X.basicOpen f = toOpens (Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) V hV)),
    U.presieveOfSections s (Y := Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) V hV) { down := { down := ⋯ } } →
      Presieve.ofArrows (C := X.AffineZariskiSite) (X := U)
        (fun (i : (↑s : Type u)) ↦ U.basicOpen (↑i : (↑Γ(X, U.toOpens) : Type u))) fun (i : (↑s : Type u)) ↦ homOfLE ⋯
        (Y := Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) V hV) { down := { down := ⋯ } }
⊢ U.presieveOfSections s =
    Presieve.ofArrows (fun (i : (↑s : Type u)) ↦ U.basicOpen (↑i : (↑Γ(X, U.toOpens) : Type u)))
      fun (i : (↑s : Type u)) ↦ homOfLE ⋯
have refine_2 : ∀ (x : X.AffineZariskiSite) (V : X.Opens) (hV : IsAffineOpen V)
  (x : Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) V hV ⟶ U) (f : (↑Γ(X, U.toOpens) : Type u))
  (hf : X.basicOpen f = toOpens (Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) V hV)),
  Presieve.ofArrows (C := X.AffineZariskiSite) (X := U)
      (fun (i : (↑s : Type u)) ↦ U.basicOpen (↑i : (↑Γ(X, U.toOpens) : Type u))) fun (i : (↑s : Type u)) ↦ homOfLE sorry
      (Y := Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) V hV) { down := { down := sorry } } →
    U.presieveOfSections s (Y := Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) V hV) { down := { down := sorry } }
Tactic: refine funext₂ fun ⟨V, hV⟩ ⟨f, hf⟩ ↦ eq_iff_iff.mpr ⟨?_, ?_⟩
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
U V : X.AffineZariskiSite
s : Set (↑Γ(X, U.toOpens) : Type u)
f : V ⟶ U
⊢ (Sieve.generate (U.presieveOfSections s)).arrows f ↔
    ∃ f ∈ s, ∃ (g : (↑Γ(X, U.toOpens) : Type u)), X.basicOpen (f * g) = V.toOpens
have obtain : ∀ (V : X.Opens) (hV : IsAffineOpen V) {f : Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) V hV ⟶ U},
  (Sieve.generate (U.presieveOfSections s)).arrows f ↔
    ∃ f ∈ s,
      ∃ (g : (↑Γ(X, U.toOpens) : Type u)),
        X.basicOpen (f * g) = toOpens (Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) V hV)
Tactic: obtain ⟨V, hV⟩ := V
Cleared hypotheses: V, f
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
U : X.AffineZariskiSite
s : Set (↑Γ(X, U.toOpens) : Type u)
V : X.Opens
hV : IsAffineOpen V
f : Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) V hV ⟶ U
⊢ (Sieve.generate (U.presieveOfSections s)).arrows f ↔
    ∃ f ∈ s,
      ∃ (g : (↑Γ(X, U.toOpens) : Type u)),
        X.basicOpen (f * g) = toOpens (Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) V hV)
have mp : (Sieve.generate (U.presieveOfSections s)).arrows f →
  ∃ f ∈ s,
    ∃ (g : (↑Γ(X, U.toOpens) : Type u)),
      X.basicOpen (f * g) = toOpens (Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) V hV)
Tactic: constructor
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
U : X.AffineZariskiSite
s : Set (↑Γ(X, U.toOpens) : Type u)
V : X.Opens
hV : IsAffineOpen V
f : Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) V hV ⟶ U
mp :
  (Sieve.generate (U.presieveOfSections s)).arrows f →
    ∃ f ∈ s,
      ∃ (g : (↑Γ(X, U.toOpens) : Type u)),
        X.basicOpen (f * g) = toOpens (Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) V hV)
⊢ (Sieve.generate (U.presieveOfSections s)).arrows f ↔
    ∃ f ∈ s,
      ∃ (g : (↑Γ(X, U.toOpens) : Type u)),
        X.basicOpen (f * g) = toOpens (Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) V hV)
have mpr : (∃ f ∈ s,
    ∃ (g : (↑Γ(X, U.toOpens) : Type u)),
      X.basicOpen (f * g) = toOpens (Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) V hV)) →
  (Sieve.generate (U.presieveOfSections s)).arrows f
Tactic: constructor
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
U : X.AffineZariskiSite
s : Set (↑Γ(X, U.toOpens) : Type u)
V : X.Opens
hV : IsAffineOpen V
f₂ : (↑Γ(X, U.toOpens) : Type u)
hf₂s : f₂ ∈ s
hW : IsAffineOpen (X.basicOpen f₂)
f₁ : (↑Γ(X, toOpens (Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) (X.basicOpen f₂) hW)) : Type u)
hf₁ : X.basicOpen f₁ = toOpens (Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) V hV)
⊢ ∃ f ∈ s,
    ∃ (g : (↑Γ(X, U.toOpens) : Type u)),
      X.basicOpen (f * g) = toOpens (Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) V hV)
have mp : ∀ (hV : IsAffineOpen (X.basicOpen f₁)),
  ∃ f ∈ s,
    ∃ (g : (↑Γ(X, U.toOpens) : Type u)),
      X.basicOpen (f * g) = toOpens (Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) (X.basicOpen f₁) hV)
Tactic: subst hf₁
Cleared hypotheses: V, hV, hf₁
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
U : X.AffineZariskiSite
s : Set (↑Γ(X, U.toOpens) : Type u)
f₂ : (↑Γ(X, U.toOpens) : Type u)
hf₂s : f₂ ∈ s
hW : IsAffineOpen (X.basicOpen f₂)
f₁ : (↑Γ(X, toOpens (Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) (X.basicOpen f₂) hW)) : Type u)
hV : IsAffineOpen (X.basicOpen f₁)
⊢ ∃ f ∈ s,
    ∃ (g : (↑Γ(X, U.toOpens) : Type u)),
      X.basicOpen (f * g) = toOpens (Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) (X.basicOpen f₁) hV)
have mp : ∀ (f₃ : (↑Γ(X, (↑U : X.Opens)) : Type u)) (hf₃ : X.basicOpen f₃ = X.basicOpen f₁),
  ∃ f ∈ s,
    ∃ (g : (↑Γ(X, U.toOpens) : Type u)),
      X.basicOpen (f * g) = toOpens (Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) (X.basicOpen f₁) hV)
Tactic: obtain ⟨f₃, hf₃⟩ := U.2.basicOpen_basicOpen_is_basicOpen f₂ f₁
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
U : X.AffineZariskiSite
s : Set (↑Γ(X, U.toOpens) : Type u)
f₂ : (↑Γ(X, U.toOpens) : Type u)
hf₂s : f₂ ∈ s
hW : IsAffineOpen (X.basicOpen f₂)
f₁ : (↑Γ(X, toOpens (Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) (X.basicOpen f₂) hW)) : Type u)
hV : IsAffineOpen (X.basicOpen f₁)
f₃ : (↑Γ(X, (↑U : X.Opens)) : Type u)
hf₃ : X.basicOpen f₃ = X.basicOpen f₁
⊢ ∃ f ∈ s,
    ∃ (g : (↑Γ(X, U.toOpens) : Type u)),
      X.basicOpen (f * g) = toOpens (Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) (X.basicOpen f₁) hV)
have mp : X.basicOpen (f₂ * f₃) = toOpens (Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) (X.basicOpen f₁) hV)
Tactic: refine ⟨f₂, hf₂s, f₃, ?_⟩
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
U : X.AffineZariskiSite
s : Set (↑Γ(X, U.toOpens) : Type u)
f₂ : (↑Γ(X, U.toOpens) : Type u)
hf₂s : f₂ ∈ s
hW : IsAffineOpen (X.basicOpen f₂)
f₁ : (↑Γ(X, toOpens (Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) (X.basicOpen f₂) hW)) : Type u)
hV : IsAffineOpen (X.basicOpen f₁)
f₃ : (↑Γ(X, (↑U : X.Opens)) : Type u)
hf₃ : X.basicOpen f₃ = X.basicOpen f₁
⊢ X.basicOpen (f₂ * f₃) = toOpens (Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) (X.basicOpen f₁) hV)
have mp : X.basicOpen f₁ ≤ X.basicOpen f₂
Tactic: rw [X.basicOpen_mul, hf₃, inf_eq_right]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
U : X.AffineZariskiSite
s : Set (↑Γ(X, U.toOpens) : Type u)
f₁ : (↑Γ(X, U.toOpens) : Type u)
hf₁s : f₁ ∈ s
f₂ : (↑Γ(X, U.toOpens) : Type u)
hV : IsAffineOpen (X.basicOpen (f₁ * f₂))
f : Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) (X.basicOpen (f₁ * f₂)) hV ⟶ U
⊢ (Sieve.generate (U.presieveOfSections s)).arrows f
have mpr : X.basicOpen
    (TopCat.Presheaf.restrictOpen (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
      CommRingCat.carrier) f₂ (U.basicOpen f₁).toOpens sorry) =
  toOpens (Subtype.mk (p := fun (U : X.Opens) ↦ IsAffineOpen U) (X.basicOpen (f₁ * f₂)) hV)
Tactic: refine ⟨U.basicOpen f₁, ⟨f₂ |_ _, ?_⟩, ⟨f₁, rfl⟩, ⟨f₁, hf₁s, rfl⟩, rfl⟩
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
U : X.AffineZariskiSite
s : Set (↑Γ(X, U.toOpens) : Type u)
⊢ Sieve.generate (U.presieveOfSections s) ∈ (grothendieckTopology X : (a : X.AffineZariskiSite) → Set (Sieve a)) U ↔
    Eq (α := Ideal (↑Γ(X, U.toOpens) : Type u)) (Ideal.span s) ⊤
have rw : (∀ x ∈ U.toOpens,
    ∃ (V : X.AffineZariskiSite) (f : V ⟶ U), (Sieve.generate (U.presieveOfSections s)).arrows f ∧ x ∈ V.toOpens) ↔
  ∀ ⦃x : ↥X⦄,
    x ∈ (↑U : X.Opens) →
      Membership.mem (γ := X.Opens) (⨆ (f : (↑s : Type u)), X.basicOpen (↑f : (↑Γ(X, (↑U : X.Opens)) : Type u))) x
Tactic: rw [← U.2.self_le_iSup_basicOpen_iff, mem_grothendieckTopology, SetLike.le_def]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
U : X.AffineZariskiSite
s : Set (↑Γ(X, U.toOpens) : Type u)
⊢ (∀ x ∈ U.toOpens,
      ∃ (V : X.AffineZariskiSite) (f : V ⟶ U), (Sieve.generate (U.presieveOfSections s)).arrows f ∧ x ∈ V.toOpens) ↔
    ∀ ⦃x : ↥X⦄,
      x ∈ (↑U : X.Opens) →
        Membership.mem (γ := X.Opens) (⨆ (f : (↑s : Type u)), X.basicOpen (↑f : (↑Γ(X, (↑U : X.Opens)) : Type u))) x
have refine : ∀ (x : ↥X) (hx : x ∈ U.toOpens),
  (∃ (V : X.AffineZariskiSite) (f : V ⟶ U), (Sieve.generate (U.presieveOfSections s)).arrows f ∧ x ∈ V.toOpens) ↔
    Membership.mem (γ := X.Opens) (⨆ (f : (↑s : Type u)), X.basicOpen (↑f : (↑Γ(X, (↑U : X.Opens)) : Type u))) x
Tactic: refine forall₂_congr fun x hx ↦ ?_
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
U : X.AffineZariskiSite
s : Set (↑Γ(X, U.toOpens) : Type u)
x : ↥X
hx : x ∈ U.toOpens
⊢ (∃ (V : X.AffineZariskiSite) (f : V ⟶ U), (Sieve.generate (U.presieveOfSections s)).arrows f ∧ x ∈ V.toOpens) ↔
    Membership.mem (γ := X.Opens) (⨆ (f : (↑s : Type u)), X.basicOpen (↑f : (↑Γ(X, (↑U : X.Opens)) : Type u))) x
have simp : (∃ (V : X.AffineZariskiSite),
    (∃ f ∈ s, ∃ (g : (↑Γ(X, U.toOpens) : Type u)), X.basicOpen (f * g) = V.toOpens) ∧ ∃ (x_1 : V ⟶ U), x ∈ V.toOpens) ↔
  ∃ i ∈ s, x ∈ X.basicOpen i
Tactic: simp only [exists_and_left, TopologicalSpace.Opens.iSup_mk,
    TopologicalSpace.Opens.carrier_eq_coe, Set.iUnion_coe_set, TopologicalSpace.Opens.mem_mk,
    Set.mem_iUnion, SetLike.mem_coe, exists_prop, generate_presieveOfSections]