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
⊢ (∀ (x_1 : ℕ),
      HomogeneousLocalization.mk
          { deg := m * x_1,
            num :=
              Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * x_1)) (HPow.hPow (α := A) ((proj 𝒜 x_1 : (a : A) → A) x : A) m)
                ⋯,
            den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * x_1)) (f ^ x_1) ⋯, den_mem := ⋯ } ∈
        q.asIdeal) ∨
    ∀ (x : ℕ),
      HomogeneousLocalization.mk
          { deg := m * x,
            num :=
              Subtype.mk (p := fun (x_1 : A) ↦ x_1 ∈ 𝒜 (m * x)) (HPow.hPow (α := A) ((proj 𝒜 x : (a : A) → A) y : A) m)
                ⋯,
            den := Subtype.mk (p := fun (x_1 : A) ↦ x_1 ∈ 𝒜 (m * x)) (f ^ x) ⋯, den_mem := ⋯ } ∈
        q.asIdeal
have rw : HomogeneousLocalization.mk
      { deg := m * nx,
        num :=
          Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx)) (HPow.hPow (α := A) ((proj 𝒜 nx : (a : A) → A) x : A) m) sorry,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx)) (f ^ nx) sorry, den_mem := sorry } ∈
    q.asIdeal ∨
  HomogeneousLocalization.mk
      { deg := m * ny,
        num :=
          Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny)) (HPow.hPow (α := A) ((proj 𝒜 ny : (a : A) → A) y : A) m) sorry,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny)) (f ^ ny) sorry, den_mem := sorry } ∈
    q.asIdeal
Tactic: rw [← and_forall_ne nx, and_iff_left, ← and_forall_ne ny, and_iff_left]
-- 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
rw :
  HomogeneousLocalization.mk
        { deg := m * nx,
          num :=
            Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx)) (HPow.hPow (α := A) ((proj 𝒜 nx : (a : A) → A) x : A) m) ⋯,
          den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx)) (f ^ nx) ⋯, den_mem := ⋯ } ∈
      q.asIdeal ∨
    HomogeneousLocalization.mk
        { deg := m * ny,
          num :=
            Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny)) (HPow.hPow (α := A) ((proj 𝒜 ny : (a : A) → A) y : A) m) ⋯,
          den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny)) (f ^ ny) ⋯, den_mem := ⋯ } ∈
      q.asIdeal
⊢ (∀ (x_1 : ℕ),
      HomogeneousLocalization.mk
          { deg := m * x_1,
            num :=
              Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * x_1)) (HPow.hPow (α := A) ((proj 𝒜 x_1 : (a : A) → A) x : A) m)
                ⋯,
            den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * x_1)) (f ^ x_1) ⋯, den_mem := ⋯ } ∈
        q.asIdeal) ∨
    ∀ (x : ℕ),
      HomogeneousLocalization.mk
          { deg := m * x,
            num :=
              Subtype.mk (p := fun (x_1 : A) ↦ x_1 ∈ 𝒜 (m * x)) (HPow.hPow (α := A) ((proj 𝒜 x : (a : A) → A) y : A) m)
                ⋯,
            den := Subtype.mk (p := fun (x_1 : A) ↦ x_1 ∈ 𝒜 (m * x)) (f ^ x) ⋯, den_mem := ⋯ } ∈
        q.asIdeal
have rw₁ : ∀ (b : ℕ),
  b ≠ ny →
    HomogeneousLocalization.mk
        { deg := m * b,
          num :=
            Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * b)) (HPow.hPow (α := A) ((proj 𝒜 b : (a : A) → A) y : A) m) sorry,
          den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * b)) (f ^ b) sorry, den_mem := sorry } ∈
      q.asIdeal
Tactic: rw [← and_forall_ne nx, and_iff_left, ← and_forall_ne ny, and_iff_left]
-- 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
rw :
  HomogeneousLocalization.mk
        { deg := m * nx,
          num :=
            Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx)) (HPow.hPow (α := A) ((proj 𝒜 nx : (a : A) → A) x : A) m) ⋯,
          den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx)) (f ^ nx) ⋯, den_mem := ⋯ } ∈
      q.asIdeal ∨
    HomogeneousLocalization.mk
        { deg := m * ny,
          num :=
            Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny)) (HPow.hPow (α := A) ((proj 𝒜 ny : (a : A) → A) y : A) m) ⋯,
          den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny)) (f ^ ny) ⋯, den_mem := ⋯ } ∈
      q.asIdeal
rw₁ :
  ∀ (b : ℕ),
    b ≠ ny →
      HomogeneousLocalization.mk
          { deg := m * b,
            num :=
              Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * b)) (HPow.hPow (α := A) ((proj 𝒜 b : (a : A) → A) y : A) m) ⋯,
            den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * b)) (f ^ b) ⋯, den_mem := ⋯ } ∈
        q.asIdeal
⊢ (∀ (x_1 : ℕ),
      HomogeneousLocalization.mk
          { deg := m * x_1,
            num :=
              Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * x_1)) (HPow.hPow (α := A) ((proj 𝒜 x_1 : (a : A) → A) x : A) m)
                ⋯,
            den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * x_1)) (f ^ x_1) ⋯, den_mem := ⋯ } ∈
        q.asIdeal) ∨
    ∀ (x : ℕ),
      HomogeneousLocalization.mk
          { deg := m * x,
            num :=
              Subtype.mk (p := fun (x_1 : A) ↦ x_1 ∈ 𝒜 (m * x)) (HPow.hPow (α := A) ((proj 𝒜 x : (a : A) → A) y : A) m)
                ⋯,
            den := Subtype.mk (p := fun (x_1 : A) ↦ x_1 ∈ 𝒜 (m * x)) (f ^ x) ⋯, den_mem := ⋯ } ∈
        q.asIdeal
have rw₂ : ∀ (b : ℕ),
  b ≠ nx →
    HomogeneousLocalization.mk
        { deg := m * b,
          num :=
            Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * b)) (HPow.hPow (α := A) ((proj 𝒜 b : (a : A) → A) x : A) m) sorry,
          den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * b)) (f ^ b) sorry, den_mem := sorry } ∈
      q.asIdeal
Tactic: rw [← and_forall_ne nx, and_iff_left, ← and_forall_ne ny, and_iff_left]
-- 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
⊢ HomogeneousLocalization.mk
        { deg := m * nx,
          num :=
            Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx)) (HPow.hPow (α := A) ((proj 𝒜 nx : (a : A) → A) x : A) m) ⋯,
          den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx)) (f ^ nx) ⋯, den_mem := ⋯ } ∈
      q.asIdeal ∨
    HomogeneousLocalization.mk
        { deg := m * ny,
          num :=
            Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny)) (HPow.hPow (α := A) ((proj 𝒜 ny : (a : A) → A) y : A) m) ⋯,
          den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny)) (f ^ ny) ⋯, den_mem := ⋯ } ∈
      q.asIdeal
have apply : HMul.hMul (α := (↑(CommRingCat.of (A⁰_ f)) : Type u_1))
    (HomogeneousLocalization.mk
      { deg := m * nx,
        num :=
          Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx)) (HPow.hPow (α := A) ((proj 𝒜 nx : (a : A) → A) x : A) m) sorry,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx)) (f ^ nx) sorry, den_mem := sorry :
        NumDenSameDeg 𝒜 (Submonoid.powers f) })
    (HomogeneousLocalization.mk
      { deg := m * ny,
        num :=
          Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny)) (HPow.hPow (α := A) ((proj 𝒜 ny : (a : A) → A) y : A) m) sorry,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny)) (f ^ ny) sorry, den_mem := sorry :
        NumDenSameDeg 𝒜 (Submonoid.powers f) }) ∈
  q.asIdeal
Tactic: apply q.2.mem_or_mem
-- 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
⊢ HMul.hMul (α := (↑(CommRingCat.of (A⁰_ f)) : Type u_1))
      (HomogeneousLocalization.mk
        { deg := m * nx,
          num :=
            Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx)) (HPow.hPow (α := A) ((proj 𝒜 nx : (a : A) → A) x : A) m) ⋯,
          den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx)) (f ^ nx) ⋯, den_mem := ⋯ :
          NumDenSameDeg 𝒜 (Submonoid.powers f) })
      (HomogeneousLocalization.mk
        { deg := m * ny,
          num :=
            Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny)) (HPow.hPow (α := A) ((proj 𝒜 ny : (a : A) → A) y : A) m) ⋯,
          den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny)) (f ^ ny) ⋯, den_mem := ⋯ :
          NumDenSameDeg 𝒜 (Submonoid.powers f) }) ∈
    q.asIdeal
have h : HMul.hMul (α := (↑(CommRingCat.of (A⁰_ f)) : Type u_1))
    (HomogeneousLocalization.mk
      { deg := m * nx,
        num :=
          Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx)) (HPow.hPow (α := A) ((proj 𝒜 nx : (a : A) → A) x : A) m) sorry,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx)) (f ^ nx) sorry, den_mem := sorry :
        NumDenSameDeg 𝒜 (Submonoid.powers f) })
    (HomogeneousLocalization.mk
      { deg := m * ny,
        num :=
          Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny)) (HPow.hPow (α := A) ((proj 𝒜 ny : (a : A) → A) y : A) m) sorry,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny)) (f ^ ny) sorry, den_mem := sorry :
        NumDenSameDeg 𝒜 (Submonoid.powers f) }) =
  HomogeneousLocalization.mk
    { deg := m * (nx + ny),
      num :=
        Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * (nx + ny)))
          (HPow.hPow (α := A) ((proj 𝒜 (nx + ny) : (a : A) → A) (x * y) : A) m) sorry,
      den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * (nx + ny))) (f ^ (nx + ny)) sorry, den_mem := sorry }
Tactic: convert! hxy (nx + ny) 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
⊢ HMul.hMul (α := (↑(CommRingCat.of (A⁰_ f)) : Type u_1))
      (HomogeneousLocalization.mk
        { deg := m * nx,
          num :=
            Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx)) (HPow.hPow (α := A) ((proj 𝒜 nx : (a : A) → A) x : A) m) ⋯,
          den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx)) (f ^ nx) ⋯, den_mem := ⋯ :
          NumDenSameDeg 𝒜 (Submonoid.powers f) })
      (HomogeneousLocalization.mk
        { deg := m * ny,
          num :=
            Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny)) (HPow.hPow (α := A) ((proj 𝒜 ny : (a : A) → A) y : A) m) ⋯,
          den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny)) (f ^ ny) ⋯, den_mem := ⋯ :
          NumDenSameDeg 𝒜 (Submonoid.powers f) }) =
    HomogeneousLocalization.mk
      { deg := m * (nx + ny),
        num :=
          Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * (nx + ny)))
            (HPow.hPow (α := A) ((proj 𝒜 (nx + ny) : (a : A) → A) (x * y) : A) m) ⋯,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * (nx + ny))) (f ^ (nx + ny)) ⋯, den_mem := ⋯ }
have h : HMul.hMul (α := A⁰_ f)
    (HomogeneousLocalization.mk
      { deg := m * nx,
        num :=
          Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx))
            ((↑((((decompose 𝒜 : (a : A) → ⨁ (i : ℕ), ↥(𝒜 i)) x : ⨁ (i : ℕ), ↥(𝒜 i)) : (a : ℕ) → ↥(𝒜 a)) nx) : A) ^ m)
            sorry,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx)) (f ^ nx) sorry, den_mem := sorry :
        NumDenSameDeg 𝒜 (Submonoid.powers f) })
    (HomogeneousLocalization.mk
      { deg := m * ny,
        num :=
          Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny))
            ((↑((((decompose 𝒜 : (a : A) → ⨁ (i : ℕ), ↥(𝒜 i)) y : ⨁ (i : ℕ), ↥(𝒜 i)) : (a : ℕ) → ↥(𝒜 a)) ny) : A) ^ m)
            sorry,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny)) (f ^ ny) sorry, den_mem := sorry :
        NumDenSameDeg 𝒜 (Submonoid.powers f) }) =
  HomogeneousLocalization.mk
    { deg := m * (nx + ny),
      num :=
        Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * (nx + ny)))
          ((↑((((decompose 𝒜 : (a : A) → ⨁ (i : ℕ), ↥(𝒜 i)) (x * y) : ⨁ (i : ℕ), ↥(𝒜 i)) : (a : ℕ) → ↥(𝒜 a))
                  (nx + ny)) :
              A) ^
            m)
          sorry,
      den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * (nx + ny))) (f ^ (nx + ny)) sorry, den_mem := sorry }
Tactic: dsimp
-- 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
⊢ HMul.hMul (α := A⁰_ f)
      (HomogeneousLocalization.mk
        { deg := m * nx,
          num :=
            Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx))
              ((↑((((decompose 𝒜 : (a : A) → ⨁ (i : ℕ), ↥(𝒜 i)) x : ⨁ (i : ℕ), ↥(𝒜 i)) : (a : ℕ) → ↥(𝒜 a)) nx) : A) ^ m)
              ⋯,
          den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx)) (f ^ nx) ⋯, den_mem := ⋯ :
          NumDenSameDeg 𝒜 (Submonoid.powers f) })
      (HomogeneousLocalization.mk
        { deg := m * ny,
          num :=
            Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny))
              ((↑((((decompose 𝒜 : (a : A) → ⨁ (i : ℕ), ↥(𝒜 i)) y : ⨁ (i : ℕ), ↥(𝒜 i)) : (a : ℕ) → ↥(𝒜 a)) ny) : A) ^ m)
              ⋯,
          den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny)) (f ^ ny) ⋯, den_mem := ⋯ :
          NumDenSameDeg 𝒜 (Submonoid.powers f) }) =
    HomogeneousLocalization.mk
      { deg := m * (nx + ny),
        num :=
          Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * (nx + ny)))
            ((↑((((decompose 𝒜 : (a : A) → ⨁ (i : ℕ), ↥(𝒜 i)) (x * y) : ⨁ (i : ℕ), ↥(𝒜 i)) : (a : ℕ) → ↥(𝒜 a))
                    (nx + ny)) :
                A) ^
              m)
            ⋯,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * (nx + ny))) (f ^ (nx + ny)) ⋯, den_mem := ⋯ }
have h : HMul.hMul (α := A⁰_ f)
    (HomogeneousLocalization.mk
      { deg := m * nx, num := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx)) (x ^ m) sorry,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx)) (f ^ nx) sorry, den_mem := sorry :
        NumDenSameDeg 𝒜 (Submonoid.powers f) })
    (HomogeneousLocalization.mk
      { deg := m * ny, num := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny)) (y ^ m) sorry,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny)) (f ^ ny) sorry, den_mem := sorry :
        NumDenSameDeg 𝒜 (Submonoid.powers f) }) =
  HomogeneousLocalization.mk
    { deg := m * (nx + ny), num := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * (nx + ny))) (x ^ m * y ^ m) sorry,
      den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * (nx + ny))) (f ^ nx * f ^ ny) sorry, den_mem := sorry }
Tactic: simp_rw [decompose_of_mem_same 𝒜 hnx, decompose_of_mem_same 𝒜 hny,
          decompose_of_mem_same 𝒜 (SetLike.GradedMonoid.toGradedMul.mul_mem hnx hny),
          mul_pow, pow_add]
-- 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
⊢ HMul.hMul (α := A⁰_ f)
      (HomogeneousLocalization.mk
        { deg := m * nx, num := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx)) (x ^ m) ⋯,
          den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx)) (f ^ nx) ⋯, den_mem := ⋯ :
          NumDenSameDeg 𝒜 (Submonoid.powers f) })
      (HomogeneousLocalization.mk
        { deg := m * ny, num := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny)) (y ^ m) ⋯,
          den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny)) (f ^ ny) ⋯, den_mem := ⋯ :
          NumDenSameDeg 𝒜 (Submonoid.powers f) }) =
    HomogeneousLocalization.mk
      { deg := m * (nx + ny), num := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * (nx + ny))) (x ^ m * y ^ m) ⋯,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * (nx + ny))) (f ^ nx * f ^ ny) ⋯, den_mem := ⋯ }
have h : Eq (α := Localization (Submonoid.powers f))
  (Localization.mk (x ^ m * y ^ m)
    (HMul.hMul (α := ↥(Submonoid.powers f))
      (Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Submonoid A) (Submonoid.powers f) x) (f ^ nx) sorry)
      (Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Submonoid A) (Submonoid.powers f) x) (f ^ ny) sorry)))
  (Localization.mk (x ^ m * y ^ m)
    (Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Submonoid A) (Submonoid.powers f) x) (f ^ nx * f ^ ny) sorry))
Tactic: simp only [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk,
          HomogeneousLocalization.val_mul, Localization.mk_mul]
-- 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 ≠ ny
⊢ 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) y : 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) y : 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 ≠ ny
⊢ 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) y : 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) y : 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) y : 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 ≠ ny
⊢ 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) y : 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) y : 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)) y : ⨁ (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 ≠ ny
⊢ Eq (α := Localization (Submonoid.powers f))
    (Localization.mk
      ((↑((((decompose 𝒜 : (a : A) → ⨁ (i : ℕ), ↥(𝒜 i)) y : ⨁ (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)) y : ⨁ (i : ℕ), ↥(𝒜 i)) : (a : ℕ) → ↥(𝒜 a)) n) : A) ^ m = 0
Tactic: convert! mk_zero (S := Submonoid.powers f) _