l3lab/ntp-mathlib huggingface dataset:
theorem carrier.smul_mem (c x : A) (hx : x ∈ carrier f_deg q) : c • x ∈ carrier f_deg q := by
  revert c
  refine' DirectSum.Decomposition.inductionOn 𝒜 _ _ _
  · rw [zero_smul]; exact carrier.zero_mem f_deg hm _
  · rintro n ⟨a, ha⟩ i
    simp_rw [proj_apply, smul_eq_mul, coe_decompose_mul_of_left_mem 𝒜 i ha]
    -- Porting note: having trouble with Mul instance
    let product : A⁰_ f :=
      Mul.mul (Quotient.mk'' ⟨_, ⟨a ^ m, pow_mem_graded m ha⟩, ⟨_, ?_⟩, ⟨n, rfl⟩⟩ : A⁰_ f)
          (Quotient.mk'' ⟨_, ⟨proj 𝒜 (i - n) x ^ m, by mem_tac⟩, ⟨_, ?_⟩, ⟨i - n, rfl⟩⟩ : A⁰_ f)
    split_ifs with h
    · 

Proof state:
case pos
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toSheafedSpace.toPresheafedSpace
x : A
hx : x ∈ carrier f_deg q
n : ℕ
a : A
ha : a ∈ 𝒜 n
i : ℕ
product : A⁰_ f :=
  Mul.mul
    (Quotient.mk''
      { deg := m • n, num := { val := a ^ m, property := (_ : a ^ m ∈ 𝒜 (m • n)) },
        den := { val := f ^ n, property := ?refine'_2.mk.refine_1 },
        den_mem := (_ : ∃ y, (fun x => f ^ x) y = ↑{ val := f ^ n, property := ?refine'_2.mk.refine_1 }) })
    (Quotient.mk''
      { deg := m • (i - n),
        num := { val := (proj 𝒜 (i - n)) x ^ m, property := (_ : (proj 𝒜 (i - n)) x ^ m ∈ 𝒜 (m • (i - n))) },
        den := { val := f ^ (i - n), property := ?refine'_2.mk.refine_2 },
        den_mem := (_ : ∃ y, (fun x => f ^ x) y = ↑{ val := f ^ (i - n), property := ?refine'_2.mk.refine_2 }) })
h : n ≤ i
⊢ Quotient.mk''
      { deg := m * i,
        num :=
          { val := (a * ↑(((decompose 𝒜) x) (i - n))) ^ m,
            property := (_ : (fun x => x ∈ 𝒜 (m * i)) ((a * ↑(((decompose 𝒜) x) (i - n))) ^ m)) },
        den := { val := f ^ i, property := (_ : f ^ i ∈ 𝒜 (m * i)) },
        den_mem := (_ : ∃ y, (fun x => f ^ x) y = ↑{ val := f ^ i, property := (_ : f ^ i ∈ 𝒜 (m * i)) }) } ∈
    q.asIdeal

Next tactic:
convert_to product ∈ q.1
theorem carrier.smul_mem (c x : A) (hx : x ∈ carrier f_deg q) : c • x ∈ carrier f_deg q := by
  revert c
  refine' DirectSum.Decomposition.inductionOn 𝒜 _ _ _
  · rw [zero_smul]; exact carrier.zero_mem f_deg hm _
  · rintro n ⟨a, ha⟩ i
    simp_rw [proj_apply, smul_eq_mul, coe_decompose_mul_of_left_mem 𝒜 i ha]
    -- Porting note: having trouble with Mul instance
    let product : A⁰_ f :=
      Mul.mul (Quotient.mk'' ⟨_, ⟨a ^ m, pow_mem_graded m ha⟩, ⟨_, ?_⟩, ⟨n, rfl⟩⟩ : A⁰_ f)
          (Quotient.mk'' ⟨_, ⟨proj 𝒜 (i - n) x ^ m, by mem_tac⟩, ⟨_, ?_⟩, ⟨i - n, rfl⟩⟩ : A⁰_ f)
    split_ifs with h
    · convert_to product ∈ q.1
      · 

Proof state:
case h.e'_4.h
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toSheafedSpace.toPresheafedSpace
x : A
hx : x ∈ carrier f_deg q
n : ℕ
a : A
ha : a ∈ 𝒜 n
i : ℕ
product : A⁰_ f :=
  Mul.mul
    (Quotient.mk''
      { deg := m • n, num := { val := a ^ m, property := (_ : a ^ m ∈ 𝒜 (m • n)) },
        den := { val := f ^ n, property := ?refine'_2.mk.refine_1 },
        den_mem := (_ : ∃ y, (fun x => f ^ x) y = ↑{ val := f ^ n, property := ?refine'_2.mk.refine_1 }) })
    (Quotient.mk''
      { deg := m • (i - n),
        num := { val := (proj 𝒜 (i - n)) x ^ m, property := (_ : (proj 𝒜 (i - n)) x ^ m ∈ 𝒜 (m • (i - n))) },
        den := { val := f ^ (i - n), property := ?refine'_2.mk.refine_2 },
        den_mem := (_ : ∃ y, (fun x => f ^ x) y = ↑{ val := f ^ (i - n), property := ?refine'_2.mk.refine_2 }) })
h : n ≤ i
e_1✝ : Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding 𝒜 (Submonoid.powers f))) = A⁰_ f
⊢ Quotient.mk''
      { deg := m * i,
        num :=
          { val := (a * ↑(((decompose 𝒜) x) (i - n))) ^ m,
            property := (_ : (fun x => x ∈ 𝒜 (m * i)) ((a * ↑(((decompose 𝒜) x) (i - n))) ^ m)) },
        den := { val := f ^ i, property := (_ : f ^ i ∈ 𝒜 (m * i)) },
        den_mem := (_ : ∃ y, (fun x => f ^ x) y = ↑{ val := f ^ i, property := (_ : f ^ i ∈ 𝒜 (m * i)) }) } =
    product

Next tactic:
dsimp
theorem carrier.smul_mem (c x : A) (hx : x ∈ carrier f_deg q) : c • x ∈ carrier f_deg q := by
  revert c
  refine' DirectSum.Decomposition.inductionOn 𝒜 _ _ _
  · rw [zero_smul]; exact carrier.zero_mem f_deg hm _
  · rintro n ⟨a, ha⟩ i
    simp_rw [proj_apply, smul_eq_mul, coe_decompose_mul_of_left_mem 𝒜 i ha]
    -- Porting note: having trouble with Mul instance
    let product : A⁰_ f :=
      Mul.mul (Quotient.mk'' ⟨_, ⟨a ^ m, pow_mem_graded m ha⟩, ⟨_, ?_⟩, ⟨n, rfl⟩⟩ : A⁰_ f)
          (Quotient.mk'' ⟨_, ⟨proj 𝒜 (i - n) x ^ m, by mem_tac⟩, ⟨_, ?_⟩, ⟨i - n, rfl⟩⟩ : A⁰_ f)
    split_ifs with h
    · convert_to product ∈ q.1
      · dsimp
        

Proof state:
case h.e'_4.h
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toSheafedSpace.toPresheafedSpace
x : A
hx : x ∈ carrier f_deg q
n : ℕ
a : A
ha : a ∈ 𝒜 n
i : ℕ
product : A⁰_ f :=
  Mul.mul
    (Quotient.mk''
      { deg := m • n, num := { val := a ^ m, property := (_ : a ^ m ∈ 𝒜 (m • n)) },
        den := { val := f ^ n, property := ?refine'_2.mk.refine_1 },
        den_mem := (_ : ∃ y, (fun x => f ^ x) y = ↑{ val := f ^ n, property := ?refine'_2.mk.refine_1 }) })
    (Quotient.mk''
      { deg := m • (i - n),
        num := { val := (proj 𝒜 (i - n)) x ^ m, property := (_ : (proj 𝒜 (i - n)) x ^ m ∈ 𝒜 (m • (i - n))) },
        den := { val := f ^ (i - n), property := ?refine'_2.mk.refine_2 },
        den_mem := (_ : ∃ y, (fun x => f ^ x) y = ↑{ val := f ^ (i - n), property := ?refine'_2.mk.refine_2 }) })
h : n ≤ i
e_1✝ : Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding 𝒜 (Submonoid.powers f))) = A⁰_ f
⊢ Quotient.mk''
      { deg := m * i,
        num :=
          { val := (a * ↑(((decompose 𝒜) x) (i - n))) ^ m,
            property := (_ : (a * ↑(((decompose 𝒜) x) (i - n))) ^ m ∈ 𝒜 (m * i)) },
        den := { val := f ^ i, property := (_ : f ^ i ∈ 𝒜 (m * i)) },
        den_mem := (_ : ∃ y, (fun x => f ^ x) y = ↑{ val := f ^ i, property := (_ : f ^ i ∈ 𝒜 (m * i)) }) } =
    Mul.mul
      (Quotient.mk''
        { deg := m * n, num := { val := a ^ m, property := (_ : a ^ m ∈ 𝒜 (m • n)) },
          den := { val := f ^ n, property := ?refine'_2.mk.refine_1 }, den_mem := (_ : ∃ y, f ^ y = f ^ n) })
      (Quotient.mk''
        { deg := m * (i - n),
          num :=
            { val := ↑(((decompose 𝒜) x) (i - n)) ^ m,
              property := (_ : ↑(((decompose 𝒜) x) (i - n)) ^ m ∈ 𝒜 (m • (i - n))) },
          den := { val := f ^ (i - n), property := ?refine'_2.mk.refine_2 },
          den_mem := (_ : ∃ y, f ^ y = f ^ (i - n)) })

Next tactic:
erw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'',
          HomogeneousLocalization.mul_val, HomogeneousLocalization.val_mk'',
          HomogeneousLocalization.val_mk'']
theorem carrier.smul_mem (c x : A) (hx : x ∈ carrier f_deg q) : c • x ∈ carrier f_deg q := by
  revert c
  refine' DirectSum.Decomposition.inductionOn 𝒜 _ _ _
  · rw [zero_smul]; exact carrier.zero_mem f_deg hm _
  · rintro n ⟨a, ha⟩ i
    simp_rw [proj_apply, smul_eq_mul, coe_decompose_mul_of_left_mem 𝒜 i ha]
    -- Porting note: having trouble with Mul instance
    let product : A⁰_ f :=
      Mul.mul (Quotient.mk'' ⟨_, ⟨a ^ m, pow_mem_graded m ha⟩, ⟨_, ?_⟩, ⟨n, rfl⟩⟩ : A⁰_ f)
          (Quotient.mk'' ⟨_, ⟨proj 𝒜 (i - n) x ^ m, by mem_tac⟩, ⟨_, ?_⟩, ⟨i - n, rfl⟩⟩ : A⁰_ f)
    split_ifs with h
    · convert_to product ∈ q.1
      · dsimp
        erw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'',
          HomogeneousLocalization.mul_val, HomogeneousLocalization.val_mk'',
          HomogeneousLocalization.val_mk'']
        simp_rw [mul_pow]; rw [Localization.mk_mul]
        congr; erw [← pow_add, Nat.add_sub_of_le h]
        · 

Proof state:
case refine'_2.mk.refine_1
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toSheafedSpace.toPresheafedSpace
x : A
hx : x ∈ carrier f_deg q
n : ℕ
a : A
ha : a ∈ 𝒜 n
i : ℕ
⊢ f ^ n ∈ 𝒜 (m • n)

Next tactic:
rw [(_ : m • n = _)]
theorem carrier.smul_mem (c x : A) (hx : x ∈ carrier f_deg q) : c • x ∈ carrier f_deg q := by
  revert c
  refine' DirectSum.Decomposition.inductionOn 𝒜 _ _ _
  · rw [zero_smul]; exact carrier.zero_mem f_deg hm _
  · rintro n ⟨a, ha⟩ i
    simp_rw [proj_apply, smul_eq_mul, coe_decompose_mul_of_left_mem 𝒜 i ha]
    -- Porting note: having trouble with Mul instance
    let product : A⁰_ f :=
      Mul.mul (Quotient.mk'' ⟨_, ⟨a ^ m, pow_mem_graded m ha⟩, ⟨_, ?_⟩, ⟨n, rfl⟩⟩ : A⁰_ f)
          (Quotient.mk'' ⟨_, ⟨proj 𝒜 (i - n) x ^ m, by mem_tac⟩, ⟨_, ?_⟩, ⟨i - n, rfl⟩⟩ : A⁰_ f)
    split_ifs with h
    · convert_to product ∈ q.1
      · dsimp
        erw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'',
          HomogeneousLocalization.mul_val, HomogeneousLocalization.val_mk'',
          HomogeneousLocalization.val_mk'']
        simp_rw [mul_pow]; rw [Localization.mk_mul]
        congr; erw [← pow_add, Nat.add_sub_of_le h]
        · rw [(_ : m • n = _)]; 

Proof state:
case refine'_2.mk.refine_1
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toSheafedSpace.toPresheafedSpace
x : A
hx : x ∈ carrier f_deg q
n : ℕ
a : A
ha : a ∈ 𝒜 n
i : ℕ
⊢ f ^ n ∈ 𝒜 ?m.1791833
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toSheafedSpace.toPresheafedSpace
x : A
hx : x ∈ carrier f_deg q
n : ℕ
a : A
ha : a ∈ 𝒜 n
i : ℕ
⊢ ℕ
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toSheafedSpace.toPresheafedSpace
x : A
hx : x ∈ carrier f_deg q
n : ℕ
a : A
ha : a ∈ 𝒜 n
i : ℕ
⊢ m • n = ?m.1791833

Next tactic:
mem_tac
theorem carrier.smul_mem (c x : A) (hx : x ∈ carrier f_deg q) : c • x ∈ carrier f_deg q := by
  revert c
  refine' DirectSum.Decomposition.inductionOn 𝒜 _ _ _
  · rw [zero_smul]; exact carrier.zero_mem f_deg hm _
  · rintro n ⟨a, ha⟩ i
    simp_rw [proj_apply, smul_eq_mul, coe_decompose_mul_of_left_mem 𝒜 i ha]
    -- Porting note: having trouble with Mul instance
    let product : A⁰_ f :=
      Mul.mul (Quotient.mk'' ⟨_, ⟨a ^ m, pow_mem_graded m ha⟩, ⟨_, ?_⟩, ⟨n, rfl⟩⟩ : A⁰_ f)
          (Quotient.mk'' ⟨_, ⟨proj 𝒜 (i - n) x ^ m, by mem_tac⟩, ⟨_, ?_⟩, ⟨i - n, rfl⟩⟩ : A⁰_ f)
    split_ifs with h
    · convert_to product ∈ q.1
      · dsimp
        erw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'',
          HomogeneousLocalization.mul_val, HomogeneousLocalization.val_mk'',
          HomogeneousLocalization.val_mk'']
        simp_rw [mul_pow]; rw [Localization.mk_mul]
        congr; erw [← pow_add, Nat.add_sub_of_le h]
        · rw [(_ : m • n = _)]; mem_tac; 

Proof state:
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toSheafedSpace.toPresheafedSpace
x : A
hx : x ∈ carrier f_deg q
n : ℕ
a : A
ha : a ∈ 𝒜 n
i : ℕ
⊢ m • n = n • m

Next tactic:
simp only [smul_eq_mul, mul_comm]
theorem carrier.smul_mem (c x : A) (hx : x ∈ carrier f_deg q) : c • x ∈ carrier f_deg q := by
  revert c
  refine' DirectSum.Decomposition.inductionOn 𝒜 _ _ _
  · rw [zero_smul]; exact carrier.zero_mem f_deg hm _
  · rintro n ⟨a, ha⟩ i
    simp_rw [proj_apply, smul_eq_mul, coe_decompose_mul_of_left_mem 𝒜 i ha]
    -- Porting note: having trouble with Mul instance
    let product : A⁰_ f :=
      Mul.mul (Quotient.mk'' ⟨_, ⟨a ^ m, pow_mem_graded m ha⟩, ⟨_, ?_⟩, ⟨n, rfl⟩⟩ : A⁰_ f)
          (Quotient.mk'' ⟨_, ⟨proj 𝒜 (i - n) x ^ m, by mem_tac⟩, ⟨_, ?_⟩, ⟨i - n, rfl⟩⟩ : A⁰_ f)
    split_ifs with h
    · convert_to product ∈ q.1
      · dsimp
        erw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'',
          HomogeneousLocalization.mul_val, HomogeneousLocalization.val_mk'',
          HomogeneousLocalization.val_mk'']
        simp_rw [mul_pow]; rw [Localization.mk_mul]
        congr; erw [← pow_add, Nat.add_sub_of_le h]
        · rw [(_ : m • n = _)]; mem_tac; simp only [smul_eq_mul, mul_comm]
        · 

Proof state:
case refine'_2.mk.refine_2
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toSheafedSpace.toPresheafedSpace
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))

Next tactic:
rw [(_ : m • (i - n) = _)]
theorem carrier.smul_mem (c x : A) (hx : x ∈ carrier f_deg q) : c • x ∈ carrier f_deg q := by
  revert c
  refine' DirectSum.Decomposition.inductionOn 𝒜 _ _ _
  · rw [zero_smul]; exact carrier.zero_mem f_deg hm _
  · rintro n ⟨a, ha⟩ i
    simp_rw [proj_apply, smul_eq_mul, coe_decompose_mul_of_left_mem 𝒜 i ha]
    -- Porting note: having trouble with Mul instance
    let product : A⁰_ f :=
      Mul.mul (Quotient.mk'' ⟨_, ⟨a ^ m, pow_mem_graded m ha⟩, ⟨_, ?_⟩, ⟨n, rfl⟩⟩ : A⁰_ f)
          (Quotient.mk'' ⟨_, ⟨proj 𝒜 (i - n) x ^ m, by mem_tac⟩, ⟨_, ?_⟩, ⟨i - n, rfl⟩⟩ : A⁰_ f)
    split_ifs with h
    · convert_to product ∈ q.1
      · dsimp
        erw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'',
          HomogeneousLocalization.mul_val, HomogeneousLocalization.val_mk'',
          HomogeneousLocalization.val_mk'']
        simp_rw [mul_pow]; rw [Localization.mk_mul]
        congr; erw [← pow_add, Nat.add_sub_of_le h]
        · rw [(_ : m • n = _)]; mem_tac; simp only [smul_eq_mul, mul_comm]
        · rw [(_ : m • (i - n) = _)]; 

Proof state:
case refine'_2.mk.refine_2
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toSheafedSpace.toPresheafedSpace
x : A
hx : x ∈ carrier f_deg q
n : ℕ
a : A
ha : a ∈ 𝒜 n
i : ℕ
⊢ f ^ (i - n) ∈ 𝒜 ?m.1794362
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toSheafedSpace.toPresheafedSpace
x : A
hx : x ∈ carrier f_deg q
n : ℕ
a : A
ha : a ∈ 𝒜 n
i : ℕ
⊢ ℕ
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toSheafedSpace.toPresheafedSpace
x : A
hx : x ∈ carrier f_deg q
n : ℕ
a : A
ha : a ∈ 𝒜 n
i : ℕ
⊢ m • (i - n) = ?m.1794362

Next tactic:
mem_tac
theorem carrier.smul_mem (c x : A) (hx : x ∈ carrier f_deg q) : c • x ∈ carrier f_deg q := by
  revert c
  refine' DirectSum.Decomposition.inductionOn 𝒜 _ _ _
  · rw [zero_smul]; exact carrier.zero_mem f_deg hm _
  · rintro n ⟨a, ha⟩ i
    simp_rw [proj_apply, smul_eq_mul, coe_decompose_mul_of_left_mem 𝒜 i ha]
    -- Porting note: having trouble with Mul instance
    let product : A⁰_ f :=
      Mul.mul (Quotient.mk'' ⟨_, ⟨a ^ m, pow_mem_graded m ha⟩, ⟨_, ?_⟩, ⟨n, rfl⟩⟩ : A⁰_ f)
          (Quotient.mk'' ⟨_, ⟨proj 𝒜 (i - n) x ^ m, by mem_tac⟩, ⟨_, ?_⟩, ⟨i - n, rfl⟩⟩ : A⁰_ f)
    split_ifs with h
    · convert_to product ∈ q.1
      · dsimp
        erw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'',
          HomogeneousLocalization.mul_val, HomogeneousLocalization.val_mk'',
          HomogeneousLocalization.val_mk'']
        simp_rw [mul_pow]; rw [Localization.mk_mul]
        congr; erw [← pow_add, Nat.add_sub_of_le h]
        · rw [(_ : m • n = _)]; mem_tac; simp only [smul_eq_mul, mul_comm]
        · rw [(_ : m • (i - n) = _)]; mem_tac; 

Proof state:
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toSheafedSpace.toPresheafedSpace
x : A
hx : x ∈ carrier f_deg q
n : ℕ
a : A
ha : a ∈ 𝒜 n
i : ℕ
⊢ m • (i - n) = (i - n) • m

Next tactic:
simp only [smul_eq_mul, mul_comm]
theorem carrier.smul_mem (c x : A) (hx : x ∈ carrier f_deg q) : c • x ∈ carrier f_deg q := by
  revert c
  refine' DirectSum.Decomposition.inductionOn 𝒜 _ _ _
  · rw [zero_smul]; exact carrier.zero_mem f_deg hm _
  · rintro n ⟨a, ha⟩ i
    simp_rw [proj_apply, smul_eq_mul, coe_decompose_mul_of_left_mem 𝒜 i ha]
    -- Porting note: having trouble with Mul instance
    let product : A⁰_ f :=
      Mul.mul (Quotient.mk'' ⟨_, ⟨a ^ m, pow_mem_graded m ha⟩, ⟨_, ?_⟩, ⟨n, rfl⟩⟩ : A⁰_ f)
          (Quotient.mk'' ⟨_, ⟨proj 𝒜 (i - n) x ^ m, by mem_tac⟩, ⟨_, ?_⟩, ⟨i - n, rfl⟩⟩ : A⁰_ f)
    split_ifs with h
    · convert_to product ∈ q.1
      · dsimp
        erw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'',
          HomogeneousLocalization.mul_val, HomogeneousLocalization.val_mk'',
          HomogeneousLocalization.val_mk'']
        simp_rw [mul_pow]; rw [Localization.mk_mul]
        congr; erw [← pow_add, Nat.add_sub_of_le h]
        · rw [(_ : m • n = _)]; mem_tac; simp only [smul_eq_mul, mul_comm]
        · rw [(_ : m • (i - n) = _)]; mem_tac; simp only [smul_eq_mul, mul_comm]
      · 

Proof state:
case pos
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toSheafedSpace.toPresheafedSpace
x : A
hx : x ∈ carrier f_deg q
n : ℕ
a : A
ha : a ∈ 𝒜 n
i : ℕ
product : A⁰_ f :=
  Mul.mul
    (Quotient.mk''
      { deg := m • n, num := { val := a ^ m, property := (_ : a ^ m ∈ 𝒜 (m • n)) },
        den := { val := f ^ n, property := (_ : f ^ n ∈ 𝒜 (m • n)) },
        den_mem := (_ : ∃ y, (fun x => f ^ x) y = ↑{ val := f ^ n, property := (_ : f ^ n ∈ 𝒜 (m • n)) }) })
    (Quotient.mk''
      { deg := m • (i - n),
        num := { val := (proj 𝒜 (i - n)) x ^ m, property := (_ : (proj 𝒜 (i - n)) x ^ m ∈ 𝒜 (m • (i - n))) },
        den := { val := f ^ (i - n), property := (_ : f ^ (i - n) ∈ 𝒜 (m • (i - n))) },
        den_mem :=
          (_ : ∃ y, (fun x => f ^ x) y = ↑{ val := f ^ (i - n), property := (_ : f ^ (i - n) ∈ 𝒜 (m • (i - n))) }) })
h : n ≤ i
⊢ product ∈ q.asIdeal

Next tactic:
apply Ideal.mul_mem_left (α := A⁰_ f) _ _ (hx _)
theorem carrier.smul_mem (c x : A) (hx : x ∈ carrier f_deg q) : c • x ∈ carrier f_deg q := by
  revert c
  refine' DirectSum.Decomposition.inductionOn 𝒜 _ _ _
  · rw [zero_smul]; exact carrier.zero_mem f_deg hm _
  · rintro n ⟨a, ha⟩ i
    simp_rw [proj_apply, smul_eq_mul, coe_decompose_mul_of_left_mem 𝒜 i ha]
    -- Porting note: having trouble with Mul instance
    let product : A⁰_ f :=
      Mul.mul (Quotient.mk'' ⟨_, ⟨a ^ m, pow_mem_graded m ha⟩, ⟨_, ?_⟩, ⟨n, rfl⟩⟩ : A⁰_ f)
          (Quotient.mk'' ⟨_, ⟨proj 𝒜 (i - n) x ^ m, by mem_tac⟩, ⟨_, ?_⟩, ⟨i - n, rfl⟩⟩ : A⁰_ f)
    split_ifs with h
    · convert_to product ∈ q.1
      · dsimp
        erw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'',
          HomogeneousLocalization.mul_val, HomogeneousLocalization.val_mk'',
          HomogeneousLocalization.val_mk'']
        simp_rw [mul_pow]; rw [Localization.mk_mul]
        congr; erw [← pow_add, Nat.add_sub_of_le h]
        · rw [(_ : m • n = _)]; mem_tac; simp only [smul_eq_mul, mul_comm]
        · rw [(_ : m • (i - n) = _)]; mem_tac; simp only [smul_eq_mul, mul_comm]
      · apply Ideal.mul_mem_left (α := A⁰_ f) _ _ (hx _)
    · 

Proof state:
case neg
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toSheafedSpace.toPresheafedSpace
x : A
hx : x ∈ carrier f_deg q
n : ℕ
a : A
ha : a ∈ 𝒜 n
i : ℕ
product : A⁰_ f :=
  Mul.mul
    (Quotient.mk''
      { deg := m • n, num := { val := a ^ m, property := (_ : a ^ m ∈ 𝒜 (m • n)) },
        den := { val := f ^ n, property := (_ : f ^ n ∈ 𝒜 (m • n)) },
        den_mem := (_ : ∃ y, (fun x => f ^ x) y = ↑{ val := f ^ n, property := (_ : f ^ n ∈ 𝒜 (m • n)) }) })
    (Quotient.mk''
      { deg := m • (i - n),
        num := { val := (proj 𝒜 (i - n)) x ^ m, property := (_ : (proj 𝒜 (i - n)) x ^ m ∈ 𝒜 (m • (i - n))) },
        den := { val := f ^ (i - n), property := (_ : f ^ (i - n) ∈ 𝒜 (m • (i - n))) },
        den_mem :=
          (_ : ∃ y, (fun x => f ^ x) y = ↑{ val := f ^ (i - n), property := (_ : f ^ (i - n) ∈ 𝒜 (m • (i - n))) }) })
h : ¬n ≤ i
⊢ Quotient.mk''
      { deg := m * i, num := { val := 0 ^ m, property := (_ : (fun x => x ∈ 𝒜 (m * i)) (0 ^ m)) },
        den := { val := f ^ i, property := (_ : f ^ i ∈ 𝒜 (m * i)) },
        den_mem := (_ : ∃ y, (fun x => f ^ x) y = ↑{ val := f ^ i, property := (_ : f ^ i ∈ 𝒜 (m * i)) }) } ∈
    q.asIdeal

Next tactic:
simp_rw [zero_pow hm]
theorem carrier.smul_mem (c x : A) (hx : x ∈ carrier f_deg q) : c • x ∈ carrier f_deg q := by
  revert c
  refine' DirectSum.Decomposition.inductionOn 𝒜 _ _ _
  · rw [zero_smul]; exact carrier.zero_mem f_deg hm _
  · rintro n ⟨a, ha⟩ i
    simp_rw [proj_apply, smul_eq_mul, coe_decompose_mul_of_left_mem 𝒜 i ha]
    -- Porting note: having trouble with Mul instance
    let product : A⁰_ f :=
      Mul.mul (Quotient.mk'' ⟨_, ⟨a ^ m, pow_mem_graded m ha⟩, ⟨_, ?_⟩, ⟨n, rfl⟩⟩ : A⁰_ f)
          (Quotient.mk'' ⟨_, ⟨proj 𝒜 (i - n) x ^ m, by mem_tac⟩, ⟨_, ?_⟩, ⟨i - n, rfl⟩⟩ : A⁰_ f)
    split_ifs with h
    · convert_to product ∈ q.1
      · dsimp
        erw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'',
          HomogeneousLocalization.mul_val, HomogeneousLocalization.val_mk'',
          HomogeneousLocalization.val_mk'']
        simp_rw [mul_pow]; rw [Localization.mk_mul]
        congr; erw [← pow_add, Nat.add_sub_of_le h]
        · rw [(_ : m • n = _)]; mem_tac; simp only [smul_eq_mul, mul_comm]
        · rw [(_ : m • (i - n) = _)]; mem_tac; simp only [smul_eq_mul, mul_comm]
      · apply Ideal.mul_mem_left (α := A⁰_ f) _ _ (hx _)
    · simp_rw [zero_pow hm]; 

Proof state:
case neg
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toSheafedSpace.toPresheafedSpace
x : A
hx : x ∈ carrier f_deg q
n : ℕ
a : A
ha : a ∈ 𝒜 n
i : ℕ
product : A⁰_ f :=
  Mul.mul
    (Quotient.mk''
      { deg := m • n, num := { val := a ^ m, property := (_ : a ^ m ∈ 𝒜 (m • n)) },
        den := { val := f ^ n, property := (_ : f ^ n ∈ 𝒜 (m • n)) },
        den_mem := (_ : ∃ y, (fun x => f ^ x) y = ↑{ val := f ^ n, property := (_ : f ^ n ∈ 𝒜 (m • n)) }) })
    (Quotient.mk''
      { deg := m • (i - n),
        num := { val := (proj 𝒜 (i - n)) x ^ m, property := (_ : (proj 𝒜 (i - n)) x ^ m ∈ 𝒜 (m • (i - n))) },
        den := { val := f ^ (i - n), property := (_ : f ^ (i - n) ∈ 𝒜 (m • (i - n))) },
        den_mem :=
          (_ : ∃ y, (fun x => f ^ x) y = ↑{ val := f ^ (i - n), property := (_ : f ^ (i - n) ∈ 𝒜 (m • (i - n))) }) })
h : ¬n ≤ i
⊢ Quotient.mk''
      { deg := m * i, num := { val := 0, property := (_ : (fun x => x ∈ 𝒜 (m * i)) 0) },
        den := { val := f ^ i, property := (_ : f ^ i ∈ 𝒜 (m * i)) },
        den_mem := (_ : ∃ y, (fun x => f ^ x) y = ↑{ val := f ^ i, property := (_ : f ^ i ∈ 𝒜 (m * i)) }) } ∈
    q.asIdeal

Next tactic:
convert carrier.zero_mem f_deg hm q i