awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme (goal)
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1)
x y : A
h : IsHomogeneousElem 𝒜 x
h_1 : IsHomogeneousElem 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm q
nx : ℕ
hnx : x ∈ 𝒜 nx
ny : ℕ
hny : y ∈ 𝒜 ny
n : ℕ
hn : n ≠ nx
⊢ HomogeneousLocalization.mk
      { deg := m * n,
        num := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * n)) (HPow.hPow (α := A) ((proj 𝒜 n : (a : A) → A) x : A) m) ⋯,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * n)) (f ^ n) ⋯, den_mem := ⋯ } ∈
    q.asIdeal
have h : Eq (α := HomogeneousLocalization 𝒜 (Submonoid.powers f))
  (HomogeneousLocalization.mk
    { deg := m * n,
      num := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * n)) (HPow.hPow (α := A) ((proj 𝒜 n : (a : A) → A) x : A) m) sorry,
      den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * n)) (f ^ n) sorry, den_mem := sorry })
  0
Tactic: convert! q.1.zero_mem using 1
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme (goal)
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1)
x y : A
h : IsHomogeneousElem 𝒜 x
h_1 : IsHomogeneousElem 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm q
nx : ℕ
hnx : x ∈ 𝒜 nx
ny : ℕ
hny : y ∈ 𝒜 ny
n : ℕ
hn : n ≠ nx
⊢ Eq (α := HomogeneousLocalization 𝒜 (Submonoid.powers f))
    (HomogeneousLocalization.mk
      { deg := m * n,
        num := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * n)) (HPow.hPow (α := A) ((proj 𝒜 n : (a : A) → A) x : A) m) ⋯,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * n)) (f ^ n) ⋯, den_mem := ⋯ })
    0
have h : Eq (α := Localization (Submonoid.powers f))
  (Localization.mk
    (↑(NumDenSameDeg.num
          { deg := m * n,
            num :=
              Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * n)) (HPow.hPow (α := A) ((proj 𝒜 n : (a : A) → A) x : A) m) sorry,
            den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * n)) (f ^ n) sorry, den_mem := sorry }) :
      A)
    (Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Submonoid A) (Submonoid.powers f) x)
      (↑(NumDenSameDeg.den
            { deg := m * n,
              num :=
                Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * n)) (HPow.hPow (α := A) ((proj 𝒜 n : (a : A) → A) x : A) m) sorry,
              den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * n)) (f ^ n) sorry, den_mem := sorry }) :
        A)
      sorry))
  0
Tactic: rw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk,
          HomogeneousLocalization.val_zero]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme (goal)
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1)
x y : A
h : IsHomogeneousElem 𝒜 x
h_1 : IsHomogeneousElem 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm q
nx : ℕ
hnx : x ∈ 𝒜 nx
ny : ℕ
hny : y ∈ 𝒜 ny
n : ℕ
hn : n ≠ nx
⊢ Eq (α := Localization (Submonoid.powers f))
    (Localization.mk
      (↑(NumDenSameDeg.num
            { deg := m * n,
              num :=
                Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * n)) (HPow.hPow (α := A) ((proj 𝒜 n : (a : A) → A) x : A) m) ⋯,
              den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * n)) (f ^ n) ⋯, den_mem := ⋯ }) :
        A)
      (Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Submonoid A) (Submonoid.powers f) x)
        (↑(NumDenSameDeg.den
              { deg := m * n,
                num :=
                  Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * n)) (HPow.hPow (α := A) ((proj 𝒜 n : (a : A) → A) x : A) m)
                    ⋯,
                den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * n)) (f ^ n) ⋯, den_mem := ⋯ }) :
          A)
        ⋯))
    0
have h : Eq (α := Localization (Submonoid.powers f))
  (Localization.mk
    ((↑((((decompose 𝒜 : (a : A) → ⨁ (i : ℕ), ↥(𝒜 i)) x : ⨁ (i : ℕ), ↥(𝒜 i)) : (a : ℕ) → ↥(𝒜 a)) n) : A) ^ m)
    (Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Submonoid A) (Submonoid.powers f) x) (f ^ n) sorry))
  0
Tactic: simp_rw [proj_apply]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme (goal)
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1)
x y : A
h : IsHomogeneousElem 𝒜 x
h_1 : IsHomogeneousElem 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm q
nx : ℕ
hnx : x ∈ 𝒜 nx
ny : ℕ
hny : y ∈ 𝒜 ny
n : ℕ
hn : n ≠ nx
⊢ Eq (α := Localization (Submonoid.powers f))
    (Localization.mk
      ((↑((((decompose 𝒜 : (a : A) → ⨁ (i : ℕ), ↥(𝒜 i)) x : ⨁ (i : ℕ), ↥(𝒜 i)) : (a : ℕ) → ↥(𝒜 a)) n) : A) ^ m)
      (Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Submonoid A) (Submonoid.powers f) x) (f ^ n) ⋯))
    0
have h : (↑((((decompose 𝒜 : (a : A) → ⨁ (i : ℕ), ↥(𝒜 i)) x : ⨁ (i : ℕ), ↥(𝒜 i)) : (a : ℕ) → ↥(𝒜 a)) n) : A) ^ m = 0
Tactic: convert! mk_zero (S := Submonoid.powers f) _
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme (goal)
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
x : (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1)
⊢ (ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := carrier) (X :=
          (↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                  ⋯).toPresheafedSpace :
            TopCat))
          (toSpec 𝒜 f) :
        (a : (↑(↑(Proj.restrict ⋯).toPresheafedSpace : TopCat) : Type u_1)) →
          (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1))
      (FromSpec.toFun f_deg hm x) =
    x
have asIdeal : PrimeSpectrum.asIdeal
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := carrier) (X :=
          (↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                  sorry).toPresheafedSpace :
            TopCat))
          (toSpec 𝒜 f) :
        (a : (↑(↑(Proj.restrict sorry).toPresheafedSpace : TopCat) : Type u_1)) →
          (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1))
      (FromSpec.toFun f_deg hm x)) =
  x.asIdeal
Tactic: apply PrimeSpectrum.ext
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme (goal)
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
x : (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1)
⊢ PrimeSpectrum.asIdeal
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := carrier)
            (X :=
            (↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                    ⋯).toPresheafedSpace :
              TopCat))
            (toSpec 𝒜 f) :
          (a : (↑(↑(Proj.restrict ⋯).toPresheafedSpace : TopCat) : Type u_1)) →
            (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1))
        (FromSpec.toFun f_deg hm x)) =
    x.asIdeal
have asIdeal : ∀ (z : (↑(CommRingCat.of (A⁰_ f)) : Type u_1)),
  Membership.mem (γ := Ideal (↑(CommRingCat.of (A⁰_ f)) : Type u_1))
      (PrimeSpectrum.asIdeal
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := carrier)
              (X :=
              (↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                      sorry).toPresheafedSpace :
                TopCat))
              (toSpec 𝒜 f) :
            (a : (↑(↑(Proj.restrict sorry).toPresheafedSpace : TopCat) : Type u_1)) →
              (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1))
          (FromSpec.toFun f_deg hm x)))
      z ↔
    z ∈ x.asIdeal
Tactic: ext z
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme (goal)
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
x : (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1)
z : (↑(CommRingCat.of (A⁰_ f)) : Type u_1)
⊢ Membership.mem (γ := Ideal (↑(CommRingCat.of (A⁰_ f)) : Type u_1))
      (PrimeSpectrum.asIdeal
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := carrier)
              (X :=
              (↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                      ⋯).toPresheafedSpace :
                TopCat))
              (toSpec 𝒜 f) :
            (a : (↑(↑(Proj.restrict ⋯).toPresheafedSpace : TopCat) : Type u_1)) →
              (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1))
          (FromSpec.toFun f_deg hm x)))
      z ↔
    z ∈ x.asIdeal
have asIdeal : ∀ (z : HomogeneousLocalization.NumDenSameDeg 𝒜 (Submonoid.powers f)),
  Membership.mem (γ := Ideal (↑(CommRingCat.of (A⁰_ f)) : Type u_1))
      (PrimeSpectrum.asIdeal
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := carrier)
              (X :=
              (↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                      sorry).toPresheafedSpace :
                TopCat))
              (toSpec 𝒜 f) :
            (a : (↑(↑(Proj.restrict sorry).toPresheafedSpace : TopCat) : Type u_1)) →
              (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1))
          (FromSpec.toFun f_deg hm x)))
      (HomogeneousLocalization.mk z) ↔
    HomogeneousLocalization.mk z ∈ x.asIdeal
Tactic: obtain ⟨z, rfl⟩ := HomogeneousLocalization.mk_surjective z
Cleared hypotheses: z
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme (goal)
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
x : (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1)
z : HomogeneousLocalization.NumDenSameDeg 𝒜 (Submonoid.powers f)
⊢ Membership.mem (γ := Ideal (↑(CommRingCat.of (A⁰_ f)) : Type u_1))
      (PrimeSpectrum.asIdeal
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := carrier)
              (X :=
              (↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                      ⋯).toPresheafedSpace :
                TopCat))
              (toSpec 𝒜 f) :
            (a : (↑(↑(Proj.restrict ⋯).toPresheafedSpace : TopCat) : Type u_1)) →
              (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1))
          (FromSpec.toFun f_deg hm x)))
      (HomogeneousLocalization.mk z) ↔
    HomogeneousLocalization.mk z ∈ x.asIdeal
have asIdeal : Membership.mem (γ := Ideal (↑(CommRingCat.of (A⁰_ f)) : Type u_1))
    (PrimeSpectrum.asIdeal
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := carrier)
            (X :=
            (↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                    sorry).toPresheafedSpace :
              TopCat))
            (toSpec 𝒜 f) :
          (a : (↑(↑(Proj.restrict sorry).toPresheafedSpace : TopCat) : Type u_1)) →
            (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1))
        (FromSpec.toFun f_deg hm x)))
    (HomogeneousLocalization.mk z) ↔
  (↑z.num : A) ∈ FromSpec.carrier f_deg x
Tactic: rw [← FromSpec.num_mem_carrier_iff f_deg hm x]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme (goal)
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
x :
  (↑(↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
              ⋯).toPresheafedSpace :
        TopCat) :
    Type u_1)
⊢ FromSpec.toFun f_deg hm
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := carrier)
            (X :=
            (↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                    ⋯).toPresheafedSpace :
              TopCat))
            (toSpec 𝒜 f) :
          (a : (↑(↑(Proj.restrict ⋯).toPresheafedSpace : TopCat) : Type u_1)) →
            (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1))
        x) =
    x
have refine : ∀ (i : ℕ),
  ∀ x_1 ∈ 𝒜 i,
    Membership.mem (γ := HomogeneousIdeal 𝒜)
        (ProjectiveSpectrum.asHomogeneousIdeal
            (↑(FromSpec.toFun f_deg hm
                  ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
                        carrier) (X :=
                        (↑(Proj.restrict (U :=
                                Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                                sorry).toPresheafedSpace :
                          TopCat))
                        (toSpec 𝒜 f) :
                      (a : (↑(↑(Proj.restrict sorry).toPresheafedSpace : TopCat) : Type u_1)) →
                        (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1))
                    x)) :
              (↑Proj.T : Type u_1)) :
          HomogeneousIdeal 𝒜)
        x_1 ↔
      x_1 ∈ (↑x : (↑Proj.T : Type u_1)).asHomogeneousIdeal
Tactic: refine Subtype.ext <| ProjectiveSpectrum.ext <| HomogeneousIdeal.ext' ?_
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme (goal)
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
x :
  (↑(↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
              ⋯).toPresheafedSpace :
        TopCat) :
    Type u_1)
i : ℕ
z : A
hzi : z ∈ 𝒜 i
⊢ Membership.mem (γ := HomogeneousIdeal 𝒜)
      (ProjectiveSpectrum.asHomogeneousIdeal
          (↑(FromSpec.toFun f_deg hm
                ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
                      carrier) (X :=
                      (↑(Proj.restrict (U :=
                              Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                              ⋯).toPresheafedSpace :
                        TopCat))
                      (toSpec 𝒜 f) :
                    (a : (↑(↑(Proj.restrict ⋯).toPresheafedSpace : TopCat) : Type u_1)) →
                      (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1))
                  x)) :
            (↑Proj.T : Type u_1)) :
        HomogeneousIdeal 𝒜)
      z ↔
    z ∈ (↑x : (↑Proj.T : Type u_1)).asHomogeneousIdeal
have refine : Membership.mem (γ := Ideal (↑(CommRingCat.of (A⁰_ f)) : Type u_1))
    (PrimeSpectrum.asIdeal
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := carrier)
            (X :=
            (↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                    sorry).toPresheafedSpace :
              TopCat))
            (toSpec 𝒜 f) :
          (a : (↑(↑(Proj.restrict sorry).toPresheafedSpace : TopCat) : Type u_1)) →
            (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1))
        x))
    (HomogeneousLocalization.mk
      { deg := m * i, num := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * i)) (z ^ m) sorry,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * i)) (f ^ i) sorry, den_mem := sorry }) ↔
  z ∈ (↑x : (↑Proj.T : Type u_1)).asHomogeneousIdeal
Tactic: refine (FromSpec.mem_carrier_iff_of_mem f_deg hm _ _ hzi).trans ?_
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme (hypothesis)
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
x₁ x₂ :
  (↑(↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
              ⋯).toPresheafedSpace :
        TopCat) :
    Type u_1)
h :
  Eq (α := (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := carrier) (X :=
            (↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                    ⋯).toPresheafedSpace :
              TopCat))
            (toSpec 𝒜 f) :
          (a : (↑(↑(Proj.restrict ⋯).toPresheafedSpace : TopCat) : Type u_1)) →
            (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1))
        x₁ :
      (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := carrier) (X :=
            (↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                    ⋯).toPresheafedSpace :
              TopCat))
            (toSpec 𝒜 f) :
          (a : (↑(↑(Proj.restrict ⋯).toPresheafedSpace : TopCat) : Type u_1)) →
            (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1))
        x₂ :
      (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1))
⊢ x₁ = x₂
have this : Eq (α :=
  (↑(↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
              sorry).toPresheafedSpace :
        TopCat) :
    Type u_1))
  (FromSpec.toFun f_deg hm
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := carrier) (X :=
          (↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                  sorry).toPresheafedSpace :
            TopCat))
          (toSpec 𝒜 f) :
        (a : (↑(↑(Proj.restrict sorry).toPresheafedSpace : TopCat) : Type u_1)) →
          (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1))
      x₁))
  (FromSpec.toFun f_deg hm
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := carrier) (X :=
          (↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                  sorry).toPresheafedSpace :
            TopCat))
          (toSpec 𝒜 f) :
        (a : (↑(↑(Proj.restrict sorry).toPresheafedSpace : TopCat) : Type u_1)) →
          (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1))
      x₂))
Tactic: have := congr_arg (FromSpec.toFun f_deg hm) h
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme (goal)
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
a : A
i : ℕ
⊢ f ^ i ∈ 𝒜 (m * i)
have rw : f ^ i ∈ 𝒜 (i * m)
Tactic: rw [mul_comm]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme (goal)
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
⊢ Continuous (X := PrimeSpectrum (↑(CommRingCat.of (A⁰_ f)) : Type u_1)) (FromSpec.toFun f_deg hm)
have rw : ∀ (s : Set ↥(pbo f)),
  Membership.mem (γ := Set (Set ↥(pbo f)))
      (Set.preimage
          (Subtype.val (p := fun (x : ProjectiveSpectrum 𝒜) ↦
            Membership.mem (γ := Opens (ProjectiveSpectrum 𝒜)) (pbo f) x)) ''
        Set.range fun (r : A) ↦ (↑(pbo r) : Set (ProjectiveSpectrum 𝒜)))
      s →
    IsOpen (X := PrimeSpectrum (↑(CommRingCat.of (A⁰_ f)) : Type u_1)) (FromSpec.toFun f_deg hm ⁻¹' s)
Tactic: rw [isTopologicalBasis_subtype (ProjectiveSpectrum.isTopologicalBasis_basic_opens 𝒜)
        (· ∈ pbo f) |>.continuous_iff]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme (hypothesis)
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
a : A
⊢ IsOpen (X := PrimeSpectrum (↑(CommRingCat.of (A⁰_ f)) : Type u_1))
    (FromSpec.toFun f_deg hm ⁻¹'
      Subtype.val (p := fun (x : ProjectiveSpectrum 𝒜) ↦
          Membership.mem (γ := Opens (ProjectiveSpectrum 𝒜)) (pbo f) x) ⁻¹'
        (fun (r : A) ↦ (↑(pbo r) : Set (ProjectiveSpectrum 𝒜))) a)
have h₁ : Eq (α := Set ↥(pbo f))
  (Subtype.val (p := fun (x : ProjectiveSpectrum 𝒜) ↦ Membership.mem (γ := Opens (ProjectiveSpectrum 𝒜)) (pbo f) x) ⁻¹'
    (↑(pbo a) : Set (ProjectiveSpectrum 𝒜)))
  (⋃ (i : ℕ),
    Subtype.val (p := fun (x : ProjectiveSpectrum 𝒜) ↦ Membership.mem (γ := Opens (ProjectiveSpectrum 𝒜)) (pbo f) x) ⁻¹'
      (↑(pbo (↑((((decompose 𝒜 : (a : A) → ⨁ (i : ℕ), ↥(𝒜 i)) a : ⨁ (i : ℕ), ↥(𝒜 i)) : (a : ℕ) → ↥(𝒜 a)) i) : A)) :
        Set (ProjectiveSpectrum 𝒜)))
Tactic: have h₁ : Subtype.val (p := (· ∈ pbo f)) ⁻¹' (pbo a) =
          ⋃ i : ℕ, Subtype.val (p := (· ∈ pbo f)) ⁻¹' (pbo (decompose 𝒜 a i)) := by
        simp [ProjectiveSpectrum.basicOpen_eq_union_of_projection 𝒜 a]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme (goal)
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
a : A
h₁ :
  Eq (α := Set ↥(pbo f))
    (Subtype.val (p := fun (x : ProjectiveSpectrum 𝒜) ↦
        Membership.mem (γ := Opens (ProjectiveSpectrum 𝒜)) (pbo f) x) ⁻¹'
      (↑(pbo a) : Set (ProjectiveSpectrum 𝒜)))
    (⋃ (i : ℕ),
      Subtype.val (p := fun (x : ProjectiveSpectrum 𝒜) ↦
          Membership.mem (γ := Opens (ProjectiveSpectrum 𝒜)) (pbo f) x) ⁻¹'
        (↑(pbo (↑((((decompose 𝒜 : (a : A) → ⨁ (i : ℕ), ↥(𝒜 i)) a : ⨁ (i : ℕ), ↥(𝒜 i)) : (a : ℕ) → ↥(𝒜 a)) i) : A)) :
          Set (ProjectiveSpectrum 𝒜)))
e : (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1) ≃
  (↑(↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
              ⋯).toPresheafedSpace :
        TopCat) :
    Type u_1) :=
  { toFun := FromSpec.toFun f_deg hm, invFun := ToSpec.toFun f, left_inv := ⋯, right_inv := ⋯ }
⊢ IsOpen (X := PrimeSpectrum (↑(CommRingCat.of (A⁰_ f)) : Type u_1))
    (FromSpec.toFun f_deg hm ⁻¹'
      Subtype.val (p := fun (x : ProjectiveSpectrum 𝒜) ↦
          Membership.mem (γ := Opens (ProjectiveSpectrum 𝒜)) (pbo f) x) ⁻¹'
        (fun (r : A) ↦ (↑(pbo r) : Set (ProjectiveSpectrum 𝒜))) a)
have change : IsOpen (X := (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1))
  (Set.preimage (β :=
    (↑(↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                sorry).toPresheafedSpace :
          TopCat) :
      Type u_1))
    (⇑e :
      (a : (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1)) →
        (↑(↑(Proj.restrict sorry).toPresheafedSpace : TopCat) : Type u_1))
    (Subtype.val (p := fun (x : ProjectiveSpectrum 𝒜) ↦
        Membership.mem (γ := Opens (ProjectiveSpectrum 𝒜)) (pbo f) x) ⁻¹'
      (fun (r : A) ↦ (↑(pbo r) : Set (ProjectiveSpectrum 𝒜))) a))
Tactic: change IsOpen <| e ⁻¹' _