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 : A
hx : x ∈ carrier f_deg q
n : ℕ
a : A
ha : a ∈ 𝒜 n
i : ℕ
product : A⁰_ f :=
  HMul.hMul (α := HomogeneousLocalization 𝒜 (Submonoid.powers f))
    (HomogeneousLocalization.mk
      { deg := m • n, num := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m • n)) (a ^ m) ⋯,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m • n)) (f ^ n) ⋯, den_mem := ⋯ })
    (HomogeneousLocalization.mk
      { deg := m • (i - n),
        num :=
          Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m • (i - n)))
            (HPow.hPow (α := A) ((proj 𝒜 (i - n) : (a : A) → A) x : A) m) ⋯,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m • (i - n))) (f ^ (i - n)) ⋯, den_mem := ⋯ })
pos :
  ∀ (h : n ≤ i),
    HomogeneousLocalization.mk
        { deg := m * i,
          num :=
            Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * i))
              ((a *
                  (↑((((decompose 𝒜 : (a : A) → ⨁ (i : ℕ), ↥(𝒜 i)) x : ⨁ (i : ℕ), ↥(𝒜 i)) : (a : ℕ) → ↥(𝒜 a)) (i - n)) :
                    A)) ^
                m)
              ⋯,
          den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * i)) (f ^ i) ⋯, den_mem := ⋯ } ∈
      q.asIdeal
⊢ HomogeneousLocalization.mk
      { deg := m * i,
        num :=
          Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * i))
            ((if n ≤ i then
                a *
                  (↑((((decompose 𝒜 : (a : A) → ⨁ (i : ℕ), ↥(𝒜 i)) x : ⨁ (i : ℕ), ↥(𝒜 i)) : (a : ℕ) → ↥(𝒜 a)) (i - n)) :
                    A)
              else 0) ^
              m)
            ⋯,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * i)) (f ^ i) ⋯, den_mem := ⋯ } ∈
    q.asIdeal
have neg : ∀ (h : ¬n ≤ i),
  HomogeneousLocalization.mk
      { deg := m * i, num := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * i)) (HPow.hPow (α := A) 0 m) sorry,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * i)) (f ^ i) sorry, den_mem := sorry } ∈
    q.asIdeal
Tactic: split_ifs with 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
q : (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1)
x : A
hx : x ∈ carrier f_deg q
n : ℕ
a : A
ha : a ∈ 𝒜 n
i : ℕ
product : A⁰_ f :=
  HMul.hMul (α := HomogeneousLocalization 𝒜 (Submonoid.powers f))
    (HomogeneousLocalization.mk
      { deg := m • n, num := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m • n)) (a ^ m) ⋯,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m • n)) (f ^ n) ⋯, den_mem := ⋯ })
    (HomogeneousLocalization.mk
      { deg := m • (i - n),
        num :=
          Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m • (i - n)))
            (HPow.hPow (α := A) ((proj 𝒜 (i - n) : (a : A) → A) x : A) m) ⋯,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m • (i - n))) (f ^ (i - n)) ⋯, den_mem := ⋯ })
h : n ≤ i
⊢ HomogeneousLocalization.mk
      { deg := m * i,
        num :=
          Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * i))
            ((a *
                (↑((((decompose 𝒜 : (a : A) → ⨁ (i : ℕ), ↥(𝒜 i)) x : ⨁ (i : ℕ), ↥(𝒜 i)) : (a : ℕ) → ↥(𝒜 a)) (i - n)) :
                  A)) ^
              m)
            ⋯,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * i)) (f ^ i) ⋯, den_mem := ⋯ } ∈
    q.asIdeal
have h₁ : HomogeneousLocalization.mk
    { deg := m * i,
      num :=
        Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * i))
          ((a *
              (↑((((decompose 𝒜 : (a : A) → ⨁ (i : ℕ), ↥(𝒜 i)) x : ⨁ (i : ℕ), ↥(𝒜 i)) : (a : ℕ) → ↥(𝒜 a)) (i - n)) :
                A)) ^
            m)
          sorry,
      den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * i)) (f ^ i) sorry, den_mem := sorry } =
  product
Tactic: convert_to product ∈ q.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 : A
hx : x ∈ carrier f_deg q
n : ℕ
a : A
ha : a ∈ 𝒜 n
i : ℕ
product : A⁰_ f :=
  HMul.hMul (α := HomogeneousLocalization 𝒜 (Submonoid.powers f))
    (HomogeneousLocalization.mk
      { deg := m • n, num := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m • n)) (a ^ m) ⋯,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m • n)) (f ^ n) ⋯, den_mem := ⋯ })
    (HomogeneousLocalization.mk
      { deg := m • (i - n),
        num :=
          Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m • (i - n)))
            (HPow.hPow (α := A) ((proj 𝒜 (i - n) : (a : A) → A) x : A) m) ⋯,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m • (i - n))) (f ^ (i - n)) ⋯, den_mem := ⋯ })
h : n ≤ i
h₁ :
  HomogeneousLocalization.mk
      { deg := m * i,
        num :=
          Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * i))
            ((a *
                (↑((((decompose 𝒜 : (a : A) → ⨁ (i : ℕ), ↥(𝒜 i)) x : ⨁ (i : ℕ), ↥(𝒜 i)) : (a : ℕ) → ↥(𝒜 a)) (i - n)) :
                  A)) ^
              m)
            ⋯,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * i)) (f ^ i) ⋯, den_mem := ⋯ } =
    product
⊢ HomogeneousLocalization.mk
      { deg := m * i,
        num :=
          Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * i))
            ((a *
                (↑((((decompose 𝒜 : (a : A) → ⨁ (i : ℕ), ↥(𝒜 i)) x : ⨁ (i : ℕ), ↥(𝒜 i)) : (a : ℕ) → ↥(𝒜 a)) (i - n)) :
                  A)) ^
              m)
            ⋯,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * i)) (f ^ i) ⋯, den_mem := ⋯ } ∈
    q.asIdeal
have pos : product ∈ q.asIdeal
Tactic: convert_to product ∈ q.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 : A
hx : x ∈ carrier f_deg q
n : ℕ
a : A
ha : a ∈ 𝒜 n
i : ℕ
product : A⁰_ f :=
  HMul.hMul (α := HomogeneousLocalization 𝒜 (Submonoid.powers f))
    (HomogeneousLocalization.mk
      { deg := m • n, num := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m • n)) (a ^ m) ⋯,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m • n)) (f ^ n) ⋯, den_mem := ⋯ })
    (HomogeneousLocalization.mk
      { deg := m • (i - n),
        num :=
          Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m • (i - n)))
            (HPow.hPow (α := A) ((proj 𝒜 (i - n) : (a : A) → A) x : A) m) ⋯,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m • (i - n))) (f ^ (i - n)) ⋯, den_mem := ⋯ })
h : n ≤ i
⊢ HomogeneousLocalization.mk
      { deg := m * i,
        num :=
          Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * i))
            ((a *
                (↑((((decompose 𝒜 : (a : A) → ⨁ (i : ℕ), ↥(𝒜 i)) x : ⨁ (i : ℕ), ↥(𝒜 i)) : (a : ℕ) → ↥(𝒜 a)) (i - n)) :
                  A)) ^
              m)
            ⋯,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * i)) (f ^ i) ⋯, den_mem := ⋯ } =
    product
have h₁ : HomogeneousLocalization.mk
    { deg := m * i,
      num :=
        Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * i))
          ((a *
              (↑((((decompose 𝒜 : (a : A) → ⨁ (i : ℕ), ↥(𝒜 i)) x : ⨁ (i : ℕ), ↥(𝒜 i)) : (a : ℕ) → ↥(𝒜 a)) (i - n)) :
                A)) ^
            m)
          sorry,
      den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * i)) (f ^ i) sorry, den_mem := sorry } =
  HMul.hMul (α := HomogeneousLocalization 𝒜 (Submonoid.powers f))
    (HomogeneousLocalization.mk
      { deg := m * n, num := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * n)) (a ^ m) sorry,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * n)) (f ^ n) sorry, den_mem := sorry :
        NumDenSameDeg 𝒜 (Submonoid.powers f) })
    (HomogeneousLocalization.mk
      { deg := m * (i - n),
        num :=
          Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * (i - n)))
            ((↑((((decompose 𝒜 : (a : A) → ⨁ (i : ℕ), ↥(𝒜 i)) x : ⨁ (i : ℕ), ↥(𝒜 i)) : (a : ℕ) → ↥(𝒜 a)) (i - n)) : A) ^
              m)
            sorry,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * (i - n))) (f ^ (i - n)) sorry, den_mem := sorry :
        NumDenSameDeg 𝒜 (Submonoid.powers f) })
Tactic: dsimp [product]
-- 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 : A
hx : x ∈ carrier f_deg q
n : ℕ
a : A
ha : a ∈ 𝒜 n
i : ℕ
product : A⁰_ f :=
  HMul.hMul (α := HomogeneousLocalization 𝒜 (Submonoid.powers f))
    (HomogeneousLocalization.mk
      { deg := m • n, num := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m • n)) (a ^ m) ⋯,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m • n)) (f ^ n) ⋯, den_mem := ⋯ })
    (HomogeneousLocalization.mk
      { deg := m • (i - n),
        num :=
          Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m • (i - n)))
            (HPow.hPow (α := A) ((proj 𝒜 (i - n) : (a : A) → A) x : A) m) ⋯,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m • (i - n))) (f ^ (i - n)) ⋯, den_mem := ⋯ })
h : n ≤ i
⊢ Localization.mk
      (a ^ m *
        (↑((((decompose 𝒜 : (a : A) → ⨁ (i : ℕ), ↥(𝒜 i)) x : ⨁ (i : ℕ), ↥(𝒜 i)) : (a : ℕ) → ↥(𝒜 a)) (i - n)) : A) ^ m)
      (Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Submonoid A) (Submonoid.powers f) x) (f ^ i) ⋯) =
    HMul.hMul (α := Localization (Submonoid.powers f))
      (Localization.mk (a ^ m)
        (Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Submonoid A) (Submonoid.powers f) x) (f ^ n) ⋯))
      (Localization.mk
        ((↑((((decompose 𝒜 : (a : A) → ⨁ (i : ℕ), ↥(𝒜 i)) x : ⨁ (i : ℕ), ↥(𝒜 i)) : (a : ℕ) → ↥(𝒜 a)) (i - n)) : A) ^ m)
        (Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Submonoid A) (Submonoid.powers f) x) (f ^ (i - n)) ⋯))
have h₁ : Eq (α := Localization (Submonoid.powers f))
  (Localization.mk
    (a ^ m *
      (↑((((decompose 𝒜 : (a : A) → ⨁ (i : ℕ), ↥(𝒜 i)) x : ⨁ (i : ℕ), ↥(𝒜 i)) : (a : ℕ) → ↥(𝒜 a)) (i - n)) : A) ^ m)
    (Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Submonoid A) (Submonoid.powers f) x) (f ^ i) sorry))
  (Localization.mk
    (a ^ m *
      (↑((((decompose 𝒜 : (a : A) → ⨁ (i : ℕ), ↥(𝒜 i)) x : ⨁ (i : ℕ), ↥(𝒜 i)) : (a : ℕ) → ↥(𝒜 a)) (i - n)) : A) ^ m)
    (HMul.hMul (α := ↥(Submonoid.powers f))
      (Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Submonoid A) (Submonoid.powers f) x) (f ^ n) sorry)
      (Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Submonoid A) (Submonoid.powers f) x) (f ^ (i - n)) sorry)))
Tactic: rw [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 : A
hx : x ∈ carrier f_deg q
n : ℕ
a : A
ha : a ∈ 𝒜 n
i : ℕ
product : A⁰_ f :=
  HMul.hMul (α := HomogeneousLocalization 𝒜 (Submonoid.powers f))
    (HomogeneousLocalization.mk
      { deg := m • n, num := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m • n)) (a ^ m) ⋯,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m • n)) (f ^ n) ⋯, den_mem := ⋯ })
    (HomogeneousLocalization.mk
      { deg := m • (i - n),
        num :=
          Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m • (i - n)))
            (HPow.hPow (α := A) ((proj 𝒜 (i - n) : (a : A) → A) x : A) m) ⋯,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m • (i - n))) (f ^ (i - n)) ⋯, den_mem := ⋯ })
h : n ≤ i
⊢ Eq (α := Localization (Submonoid.powers f))
    (Localization.mk
      (a ^ m *
        (↑((((decompose 𝒜 : (a : A) → ⨁ (i : ℕ), ↥(𝒜 i)) x : ⨁ (i : ℕ), ↥(𝒜 i)) : (a : ℕ) → ↥(𝒜 a)) (i - n)) : A) ^ m)
      (Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Submonoid A) (Submonoid.powers f) x) (f ^ i) ⋯))
    (Localization.mk
      (a ^ m *
        (↑((((decompose 𝒜 : (a : A) → ⨁ (i : ℕ), ↥(𝒜 i)) x : ⨁ (i : ℕ), ↥(𝒜 i)) : (a : ℕ) → ↥(𝒜 a)) (i - n)) : A) ^ m)
      (HMul.hMul (α := ↥(Submonoid.powers f))
        (Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Submonoid A) (Submonoid.powers f) x) (f ^ n) ⋯)
        (Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Submonoid A) (Submonoid.powers f) x) (f ^ (i - n)) ⋯)))
have h₁ : f ^ i =
  (↑(Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Submonoid A) (Submonoid.powers f) x) (f ^ n) sorry) : A) *
    (↑(Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Submonoid A) (Submonoid.powers f) x) (f ^ (i - n)) sorry) : A)
Tactic: congr
-- 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 : A
hx : x ∈ carrier f_deg q
n : ℕ
a : A
ha : a ∈ 𝒜 n
i : ℕ
product : A⁰_ f :=
  HMul.hMul (α := HomogeneousLocalization 𝒜 (Submonoid.powers f))
    (HomogeneousLocalization.mk
      { deg := m • n, num := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m • n)) (a ^ m) ⋯,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m • n)) (f ^ n) ⋯, den_mem := ⋯ })
    (HomogeneousLocalization.mk
      { deg := m • (i - n),
        num :=
          Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m • (i - n)))
            (HPow.hPow (α := A) ((proj 𝒜 (i - n) : (a : A) → A) x : A) m) ⋯,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m • (i - n))) (f ^ (i - n)) ⋯, den_mem := ⋯ })
h : n ≤ i
⊢ f ^ i =
    (↑(Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Submonoid A) (Submonoid.powers f) x) (f ^ n) ⋯) : A) *
      (↑(Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Submonoid A) (Submonoid.powers f) x) (f ^ (i - n)) ⋯) : A)
have h₁ : f ^ i = f ^ (n + (i - n))
Tactic: rw [← pow_add, Nat.add_sub_of_le 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
q : (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1)
x : A
hx : x ∈ carrier f_deg q
n : ℕ
a : A
ha : a ∈ 𝒜 n
i : ℕ
product : A⁰_ f :=
  HMul.hMul (α := HomogeneousLocalization 𝒜 (Submonoid.powers f))
    (HomogeneousLocalization.mk
      { deg := m • n, num := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m • n)) (a ^ m) ⋯,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m • n)) (f ^ n) ⋯, den_mem := ⋯ })
    (HomogeneousLocalization.mk
      { deg := m • (i - n),
        num :=
          Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m • (i - n)))
            (HPow.hPow (α := A) ((proj 𝒜 (i - n) : (a : A) → A) x : A) m) ⋯,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m • (i - n))) (f ^ (i - n)) ⋯, den_mem := ⋯ })
h : n ≤ i
h₁ : f ^ i = f ^ (n + (i - n))
⊢ f ^ i =
    (↑(Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Submonoid A) (Submonoid.powers f) x) (f ^ n) ⋯) : A) *
      (↑(Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Submonoid A) (Submonoid.powers f) x) (f ^ (i - n)) ⋯) : A)
have h₂ : f ^ i = f ^ i
Tactic: rw [← pow_add, Nat.add_sub_of_le 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
q : (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1)
x : A
hx : x ∈ carrier f_deg q
n : ℕ
a : A
ha : a ∈ 𝒜 n
i : ℕ
product : A⁰_ f :=
  HMul.hMul (α := HomogeneousLocalization 𝒜 (Submonoid.powers f))
    (HomogeneousLocalization.mk
      { deg := m • n, num := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m • n)) (a ^ m) ⋯,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m • n)) (f ^ n) ⋯, den_mem := ⋯ })
    (HomogeneousLocalization.mk
      { deg := m • (i - n),
        num :=
          Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m • (i - n)))
            (HPow.hPow (α := A) ((proj 𝒜 (i - n) : (a : A) → A) x : A) m) ⋯,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m • (i - n))) (f ^ (i - n)) ⋯, den_mem := ⋯ })
h : n ≤ i
⊢ product ∈ q.asIdeal
have refine_2 : f ^ n ∈ 𝒜 (m • n)
Tactic: apply Ideal.mul_mem_left (α := A⁰_ f) _ _ (hx _)
Cleared hypotheses: product, 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
q : (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1)
x : A
hx : x ∈ carrier f_deg q
n : ℕ
a : A
ha : a ∈ 𝒜 n
i : ℕ
⊢ f ^ n ∈ 𝒜 (m • n)
have refine_2 : f ^ n ∈ 𝒜 (n • m)
Tactic: rw [(_ : m • n = _)]
-- 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 : A
hx : x ∈ carrier f_deg q
n : ℕ
a : A
ha : a ∈ 𝒜 n
i : ℕ
refine_2 : f ^ n ∈ 𝒜 (n • m)
⊢ f ^ n ∈ 𝒜 (m • n)
have rw : m • n = n • m
Tactic: rw [(_ : m • n = _)]
-- 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 : A
hx : x ∈ carrier f_deg q
n : ℕ
a : A
ha : a ∈ 𝒜 n
i : ℕ
⊢ f ^ (i - n) ∈ 𝒜 (m • (i - n))
have refine_2 : f ^ (i - n) ∈ 𝒜 ((i - n) • m)
Tactic: rw [(_ : m • (i - n) = _)]
-- 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 : A
hx : x ∈ carrier f_deg q
n : ℕ
a : A
ha : a ∈ 𝒜 n
i : ℕ
refine_2 : f ^ (i - n) ∈ 𝒜 ((i - n) • m)
⊢ f ^ (i - n) ∈ 𝒜 (m • (i - n))
have rw : m • (i - n) = (i - n) • m
Tactic: rw [(_ : m • (i - n) = _)]
-- 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 : A
hx : x ∈ carrier f_deg q
⊢ ∀ (m_1 m' : A), m_1 • x ∈ carrier f_deg q → m' • x ∈ carrier f_deg q → (m_1 + m') • x ∈ carrier f_deg q
have refine_3 : ∀ (m_1 m' : A), m_1 • x ∈ carrier f_deg q → m' • x ∈ carrier f_deg q → m_1 • x + m' • x ∈ carrier f_deg q
Tactic: simp_rw [add_smul]
-- 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)
rid : f ∈ asIdeal f_deg hm q
⊢ 1 ∈ q.asIdeal
have h : Eq (α := (↑(CommRingCat.of (A⁰_ f)) : Type u_1)) 1
  (HomogeneousLocalization.mk
    { deg := m * m,
      num := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * m)) (HPow.hPow (α := A) ((proj 𝒜 m : (a : A) → A) f : A) m) sorry,
      den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * m)) (f ^ m) sorry, den_mem := sorry })
Tactic: convert! rid m
-- 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)
rid : f ∈ asIdeal f_deg hm q
⊢ Eq (α := (↑(CommRingCat.of (A⁰_ f)) : Type u_1)) 1
    (HomogeneousLocalization.mk
      { deg := m * m,
        num := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * m)) (HPow.hPow (α := A) ((proj 𝒜 m : (a : A) → A) f : A) m) ⋯,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * m)) (f ^ m) ⋯, den_mem := ⋯ })
have h : Eq (α := Localization (Submonoid.powers f)) 1
  (Localization.mk
    (↑(NumDenSameDeg.num
          { deg := m * m,
            num :=
              Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * m)) (HPow.hPow (α := A) ((proj 𝒜 m : (a : A) → A) f : A) m) sorry,
            den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * m)) (f ^ m) sorry, den_mem := sorry }) :
      A)
    (Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Submonoid A) (Submonoid.powers f) x)
      (↑(NumDenSameDeg.den
            { deg := m * m,
              num :=
                Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * m)) (HPow.hPow (α := A) ((proj 𝒜 m : (a : A) → A) f : A) m) sorry,
              den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * m)) (f ^ m) sorry, den_mem := sorry }) :
        A)
      sorry))
Tactic: rw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_one,
          HomogeneousLocalization.val_mk]
-- 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)
rid : f ∈ asIdeal f_deg hm q
⊢ Eq (α := Localization (Submonoid.powers f)) 1
    (Localization.mk
      (↑(NumDenSameDeg.num
            { deg := m * m,
              num :=
                Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * m)) (HPow.hPow (α := A) ((proj 𝒜 m : (a : A) → A) f : A) m) ⋯,
              den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * m)) (f ^ m) ⋯, den_mem := ⋯ }) :
        A)
      (Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Submonoid A) (Submonoid.powers f) x)
        (↑(NumDenSameDeg.den
              { deg := m * m,
                num :=
                  Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * m)) (HPow.hPow (α := A) ((proj 𝒜 m : (a : A) → A) f : A) m)
                    ⋯,
                den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * m)) (f ^ m) ⋯, den_mem := ⋯ }) :
          A)
        ⋯))
have h : Eq (α := Localization (Submonoid.powers f)) 1
  (Localization.mk
    ((↑((((decompose 𝒜 : (a : A) → ⨁ (i : ℕ), ↥(𝒜 i)) f : ⨁ (i : ℕ), ↥(𝒜 i)) : (a : ℕ) → ↥(𝒜 a)) m) : A) ^ m)
    (Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Submonoid A) (Submonoid.powers f) x) (f ^ m) 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)
rid : f ∈ asIdeal f_deg hm q
⊢ Eq (α := Localization (Submonoid.powers f)) 1
    (Localization.mk
      ((↑((((decompose 𝒜 : (a : A) → ⨁ (i : ℕ), ↥(𝒜 i)) f : ⨁ (i : ℕ), ↥(𝒜 i)) : (a : ℕ) → ↥(𝒜 a)) m) : A) ^ m)
      (Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Submonoid A) (Submonoid.powers f) x) (f ^ m) ⋯))
have h : Eq (α := Localization (Submonoid.powers f)) 1
  (Localization.mk (f ^ m)
    (Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Submonoid A) (Submonoid.powers f) x) (f ^ m) sorry))
Tactic: simp_rw [decompose_of_mem_same _ f_deg]