l3lab/ntp-mathlib huggingface dataset:
theorem mem_carrier_iff' (q : Spec.T A⁰_ f) (a : A) :
    a ∈ carrier f_deg q ↔
      ∀ i, (Localization.mk (proj 𝒜 i a ^ m) ⟨f ^ i, ⟨i, rfl⟩⟩ : Localization.Away f) ∈
          algebraMap (HomogeneousLocalization.Away 𝒜 f) (Localization.Away f) '' { s | s ∈ q.1 } :=
  (mem_carrier_iff f_deg q a).trans
    (by
      constructor <;> intro h i <;> specialize h i
      · rw [Set.mem_image]; refine' ⟨_, h, rfl⟩
      · rw [Set.mem_image] at h; rcases h with ⟨x, h, hx⟩
        change x ∈ q.asIdeal at h
        convert h
        rw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'']
        

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
q : ↑↑(Spec A⁰_ f).toSheafedSpace.toPresheafedSpace
a : A
i : ℕ
x : A⁰_ f
hx :
  (algebraMap (A⁰_ f) (Away f)) x =
    Localization.mk ((proj 𝒜 i) a ^ m) { val := f ^ i, property := (_ : ∃ y, (fun x => f ^ x) y = f ^ i) }
h : x ∈ q.asIdeal
e_1✝ : Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding 𝒜 (Submonoid.powers f))) = A⁰_ f
⊢ Localization.mk
      ↑{ deg := m * i,
            num :=
              { val := (proj 𝒜 i) a ^ m,
                property :=
                  (_ :
                    ↑((LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap ↑(decomposeAlgEquiv 𝒜))) a) ^ 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)) }) }.num
      {
        val :=
          ↑{ deg := m * i,
                num :=
                  { val := (proj 𝒜 i) a ^ m,
                    property :=
                      (_ :
                        ↑((LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap ↑(decomposeAlgEquiv 𝒜))) a) ^ 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)) }) }.den,
        property :=
          (_ :
            ↑{ deg := m * i,
                    num :=
                      { val := (proj 𝒜 i) a ^ m,
                        property :=
                          (_ :
                            ↑((LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap ↑(decomposeAlgEquiv 𝒜))) a) ^ 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)) }) }.den ∈
              Submonoid.powers f) } =
    HomogeneousLocalization.val x

Next tactic:
dsimp only [Subtype.coe_mk]
theorem mem_carrier_iff' (q : Spec.T A⁰_ f) (a : A) :
    a ∈ carrier f_deg q ↔
      ∀ i, (Localization.mk (proj 𝒜 i a ^ m) ⟨f ^ i, ⟨i, rfl⟩⟩ : Localization.Away f) ∈
          algebraMap (HomogeneousLocalization.Away 𝒜 f) (Localization.Away f) '' { s | s ∈ q.1 } :=
  (mem_carrier_iff f_deg q a).trans
    (by
      constructor <;> intro h i <;> specialize h i
      · rw [Set.mem_image]; refine' ⟨_, h, rfl⟩
      · rw [Set.mem_image] at h; rcases h with ⟨x, h, hx⟩
        change x ∈ q.asIdeal at h
        convert h
        rw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'']
        dsimp only [Subtype.coe_mk]; 

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
q : ↑↑(Spec A⁰_ f).toSheafedSpace.toPresheafedSpace
a : A
i : ℕ
x : A⁰_ f
hx :
  (algebraMap (A⁰_ f) (Away f)) x =
    Localization.mk ((proj 𝒜 i) a ^ m) { val := f ^ i, property := (_ : ∃ y, (fun x => f ^ x) y = f ^ i) }
h : x ∈ q.asIdeal
e_1✝ : Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding 𝒜 (Submonoid.powers f))) = A⁰_ f
⊢ Localization.mk ((proj 𝒜 i) a ^ m) { val := f ^ i, property := (_ : ∃ y, f ^ y = f ^ i) } =
    HomogeneousLocalization.val x

Next tactic:
rw [← hx]
theorem mem_carrier_iff' (q : Spec.T A⁰_ f) (a : A) :
    a ∈ carrier f_deg q ↔
      ∀ i, (Localization.mk (proj 𝒜 i a ^ m) ⟨f ^ i, ⟨i, rfl⟩⟩ : Localization.Away f) ∈
          algebraMap (HomogeneousLocalization.Away 𝒜 f) (Localization.Away f) '' { s | s ∈ q.1 } :=
  (mem_carrier_iff f_deg q a).trans
    (by
      constructor <;> intro h i <;> specialize h i
      · rw [Set.mem_image]; refine' ⟨_, h, rfl⟩
      · rw [Set.mem_image] at h; rcases h with ⟨x, h, hx⟩
        change x ∈ q.asIdeal at h
        convert h
        rw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'']
        dsimp only [Subtype.coe_mk]; rw [← hx]; 

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
q : ↑↑(Spec A⁰_ f).toSheafedSpace.toPresheafedSpace
a : A
i : ℕ
x : A⁰_ f
hx :
  (algebraMap (A⁰_ f) (Away f)) x =
    Localization.mk ((proj 𝒜 i) a ^ m) { val := f ^ i, property := (_ : ∃ y, (fun x => f ^ x) y = f ^ i) }
h : x ∈ q.asIdeal
e_1✝ : Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding 𝒜 (Submonoid.powers f))) = A⁰_ f
⊢ (algebraMap (A⁰_ f) (Away f)) x = HomogeneousLocalization.val x

Next tactic:
rfl
theorem carrier.add_mem (q : Spec.T A⁰_ f) {a b : A} (ha : a ∈ carrier f_deg q)
    (hb : b ∈ carrier f_deg q) : a + b ∈ carrier f_deg q := by
  

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
q : ↑↑(Spec A⁰_ f).toSheafedSpace.toPresheafedSpace
a b : A
ha : a ∈ carrier f_deg q
hb : b ∈ carrier f_deg q
⊢ a + b ∈ carrier f_deg q

Next tactic:
refine' fun i => (q.2.mem_or_mem _).elim id id
theorem carrier.add_mem (q : Spec.T A⁰_ f) {a b : A} (ha : a ∈ carrier f_deg q)
    (hb : b ∈ carrier f_deg q) : a + b ∈ carrier f_deg q := by
  refine' fun i => (q.2.mem_or_mem _).elim id id
  

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
q : ↑↑(Spec A⁰_ f).toSheafedSpace.toPresheafedSpace
a b : A
ha : a ∈ carrier f_deg q
hb : b ∈ carrier f_deg q
i : ℕ
⊢ Quotient.mk''
        { deg := m * i,
          num :=
            { val := (proj 𝒜 i) (a + b) ^ m,
              property :=
                (_ :
                  ↑((LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap ↑(decomposeAlgEquiv 𝒜))) (a + b)) ^ 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)) }) } *
      Quotient.mk''
        { deg := m * i,
          num :=
            { val := (proj 𝒜 i) (a + b) ^ m,
              property :=
                (_ :
                  ↑((LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap ↑(decomposeAlgEquiv 𝒜))) (a + b)) ^ 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)) }) } ∈
    q.asIdeal

Next tactic:
change (Quotient.mk'' ⟨_, _, _, _⟩ : A⁰_ f) ∈ q.1
theorem carrier.add_mem (q : Spec.T A⁰_ f) {a b : A} (ha : a ∈ carrier f_deg q)
    (hb : b ∈ carrier f_deg q) : a + b ∈ carrier f_deg q := by
  refine' fun i => (q.2.mem_or_mem _).elim id id
  change (Quotient.mk'' ⟨_, _, _, _⟩ : A⁰_ f) ∈ q.1; dsimp only [Subtype.coe_mk]
  simp_rw [← pow_add, map_add, add_pow, mul_comm, ← nsmul_eq_mul]
  

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
q : ↑↑(Spec A⁰_ f).toSheafedSpace.toPresheafedSpace
a b : A
ha : a ∈ carrier f_deg q
hb : b ∈ carrier f_deg q
i : ℕ
⊢ Quotient.mk''
      { deg := m * i + m * i,
        num :=
          { val := ∑ x in range (m + m + 1), Nat.choose (m + m) x • ((proj 𝒜 i) a ^ x * (proj 𝒜 i) b ^ (m + m - x)),
            property :=
              (_ :
                (fun x => x ∈ 𝒜 (m * i + m * i))
                  (∑ x in range (m + m + 1), Nat.choose (m + m) x • ((proj 𝒜 i) a ^ x * (proj 𝒜 i) b ^ (m + m - x)))) },
        den := { val := f ^ (i + i), property := (_ : (fun x => x ∈ 𝒜 (m * i + m * i)) (f ^ (i + i))) },
        den_mem :=
          (_ :
            ↑{ val := f ^ (i + i), property := (_ : (fun x => x ∈ 𝒜 (m * i + m * i)) (f ^ (i + i))) } ∈
              Submonoid.powers f) } ∈
    q.asIdeal

Next tactic:
let g : ℕ → A⁰_ f := fun j => (m + m).choose j •
      if h2 : m + m < j then (0 : A⁰_ f)
      else
        -- Porting note: inlining `l`, `r` causes a "can't synth HMul A⁰_ f A⁰_ f ?" error
        if h1 : j ≤ m then
          letI l : A⁰_ f := Quotient.mk''
            ⟨m * i, ⟨proj 𝒜 i a ^ j * proj 𝒜 i b ^ (m - j), ?_⟩,
              ⟨_, by rw [mul_comm]; mem_tac⟩, ⟨i, rfl⟩⟩
          letI r : A⁰_ f := Quotient.mk''
            ⟨m * i, ⟨proj 𝒜 i b ^ m, by mem_tac⟩,
              ⟨_, by rw [mul_comm]; mem_tac⟩, ⟨i, rfl⟩⟩
          l * r
        else
          letI l : A⁰_ f := Quotient.mk''
            ⟨m * i, ⟨proj 𝒜 i a ^ m, by mem_tac⟩,
              ⟨_, by rw [mul_comm]; mem_tac⟩, ⟨i, rfl⟩⟩
          letI r : A⁰_ f := Quotient.mk''
            ⟨m * i, ⟨proj 𝒜 i a ^ (j - m) * proj 𝒜 i b ^ (m + m - j), ?_⟩,
              ⟨_, by rw [mul_comm]; mem_tac⟩, ⟨i, rfl⟩⟩
          l * r
theorem carrier.add_mem (q : Spec.T A⁰_ f) {a b : A} (ha : a ∈ carrier f_deg q)
    (hb : b ∈ carrier f_deg q) : a + b ∈ carrier f_deg q := by
  refine' fun i => (q.2.mem_or_mem _).elim id id
  change (Quotient.mk'' ⟨_, _, _, _⟩ : A⁰_ f) ∈ q.1; dsimp only [Subtype.coe_mk]
  simp_rw [← pow_add, map_add, add_pow, mul_comm, ← nsmul_eq_mul]
  let g : ℕ → A⁰_ f := fun j => (m + m).choose j •
      if h2 : m + m < j then (0 : A⁰_ f)
      else
        -- Porting note: inlining `l`, `r` causes a "can't synth HMul A⁰_ f A⁰_ f ?" error
        if h1 : j ≤ m then
          letI l : A⁰_ f := Quotient.mk''
            ⟨m * i, ⟨proj 𝒜 i a ^ j * proj 𝒜 i b ^ (m - j), ?_⟩,
              ⟨_, by 

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
q : ↑↑(Spec A⁰_ f).toSheafedSpace.toPresheafedSpace
a b : A
ha : a ∈ carrier f_deg q
hb : b ∈ carrier f_deg q
i j : ℕ
h2 : ¬m + m < j
h1 : j ≤ m
⊢ f ^ i ∈ 𝒜 (m * i)

Next tactic:
rw [mul_comm]
theorem carrier.add_mem (q : Spec.T A⁰_ f) {a b : A} (ha : a ∈ carrier f_deg q)
    (hb : b ∈ carrier f_deg q) : a + b ∈ carrier f_deg q := by
  refine' fun i => (q.2.mem_or_mem _).elim id id
  change (Quotient.mk'' ⟨_, _, _, _⟩ : A⁰_ f) ∈ q.1; dsimp only [Subtype.coe_mk]
  simp_rw [← pow_add, map_add, add_pow, mul_comm, ← nsmul_eq_mul]
  let g : ℕ → A⁰_ f := fun j => (m + m).choose j •
      if h2 : m + m < j then (0 : A⁰_ f)
      else
        -- Porting note: inlining `l`, `r` causes a "can't synth HMul A⁰_ f A⁰_ f ?" error
        if h1 : j ≤ m then
          letI l : A⁰_ f := Quotient.mk''
            ⟨m * i, ⟨proj 𝒜 i a ^ j * proj 𝒜 i b ^ (m - j), ?_⟩,
              ⟨_, by rw [mul_comm]; 

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
q : ↑↑(Spec A⁰_ f).toSheafedSpace.toPresheafedSpace
a b : A
ha : a ∈ carrier f_deg q
hb : b ∈ carrier f_deg q
i j : ℕ
h2 : ¬m + m < j
h1 : j ≤ m
⊢ f ^ i ∈ 𝒜 (i * m)

Next tactic:
mem_tac
theorem carrier.add_mem (q : Spec.T A⁰_ f) {a b : A} (ha : a ∈ carrier f_deg q)
    (hb : b ∈ carrier f_deg q) : a + b ∈ carrier f_deg q := by
  refine' fun i => (q.2.mem_or_mem _).elim id id
  change (Quotient.mk'' ⟨_, _, _, _⟩ : A⁰_ f) ∈ q.1; dsimp only [Subtype.coe_mk]
  simp_rw [← pow_add, map_add, add_pow, mul_comm, ← nsmul_eq_mul]
  let g : ℕ → A⁰_ f := fun j => (m + m).choose j •
      if h2 : m + m < j then (0 : A⁰_ f)
      else
        -- Porting note: inlining `l`, `r` causes a "can't synth HMul A⁰_ f A⁰_ f ?" error
        if h1 : j ≤ m then
          letI l : A⁰_ f := Quotient.mk''
            ⟨m * i, ⟨proj 𝒜 i a ^ j * proj 𝒜 i b ^ (m - j), ?_⟩,
              ⟨_, by rw [mul_comm]; mem_tac⟩, ⟨i, rfl⟩⟩
          letI r : A⁰_ f := Quotient.mk''
            ⟨m * i, ⟨proj 𝒜 i b ^ m, by 

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
q : ↑↑(Spec A⁰_ f).toSheafedSpace.toPresheafedSpace
a b : A
ha : a ∈ carrier f_deg q
hb : b ∈ carrier f_deg q
i j : ℕ
h2 : ¬m + m < j
h1 : j ≤ m
l : A⁰_ f :=
  Quotient.mk''
    { deg := m * i, num := { val := (proj 𝒜 i) a ^ j * (proj 𝒜 i) b ^ (m - j), property := ?m.1609416 },
      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)) }) }
⊢ (proj 𝒜 i) b ^ m ∈ 𝒜 (m * i)

Next tactic:
mem_tac
theorem carrier.add_mem (q : Spec.T A⁰_ f) {a b : A} (ha : a ∈ carrier f_deg q)
    (hb : b ∈ carrier f_deg q) : a + b ∈ carrier f_deg q := by
  refine' fun i => (q.2.mem_or_mem _).elim id id
  change (Quotient.mk'' ⟨_, _, _, _⟩ : A⁰_ f) ∈ q.1; dsimp only [Subtype.coe_mk]
  simp_rw [← pow_add, map_add, add_pow, mul_comm, ← nsmul_eq_mul]
  let g : ℕ → A⁰_ f := fun j => (m + m).choose j •
      if h2 : m + m < j then (0 : A⁰_ f)
      else
        -- Porting note: inlining `l`, `r` causes a "can't synth HMul A⁰_ f A⁰_ f ?" error
        if h1 : j ≤ m then
          letI l : A⁰_ f := Quotient.mk''
            ⟨m * i, ⟨proj 𝒜 i a ^ j * proj 𝒜 i b ^ (m - j), ?_⟩,
              ⟨_, by rw [mul_comm]; mem_tac⟩, ⟨i, rfl⟩⟩
          letI r : A⁰_ f := Quotient.mk''
            ⟨m * i, ⟨proj 𝒜 i b ^ m, by mem_tac⟩,
              ⟨_, by 

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
q : ↑↑(Spec A⁰_ f).toSheafedSpace.toPresheafedSpace
a b : A
ha : a ∈ carrier f_deg q
hb : b ∈ carrier f_deg q
i j : ℕ
h2 : ¬m + m < j
h1 : j ≤ m
l : A⁰_ f :=
  Quotient.mk''
    { deg := m * i, num := { val := (proj 𝒜 i) a ^ j * (proj 𝒜 i) b ^ (m - j), property := ?m.1609416 },
      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)) }) }
⊢ f ^ i ∈ 𝒜 (m * i)

Next tactic:
rw [mul_comm]
theorem carrier.add_mem (q : Spec.T A⁰_ f) {a b : A} (ha : a ∈ carrier f_deg q)
    (hb : b ∈ carrier f_deg q) : a + b ∈ carrier f_deg q := by
  refine' fun i => (q.2.mem_or_mem _).elim id id
  change (Quotient.mk'' ⟨_, _, _, _⟩ : A⁰_ f) ∈ q.1; dsimp only [Subtype.coe_mk]
  simp_rw [← pow_add, map_add, add_pow, mul_comm, ← nsmul_eq_mul]
  let g : ℕ → A⁰_ f := fun j => (m + m).choose j •
      if h2 : m + m < j then (0 : A⁰_ f)
      else
        -- Porting note: inlining `l`, `r` causes a "can't synth HMul A⁰_ f A⁰_ f ?" error
        if h1 : j ≤ m then
          letI l : A⁰_ f := Quotient.mk''
            ⟨m * i, ⟨proj 𝒜 i a ^ j * proj 𝒜 i b ^ (m - j), ?_⟩,
              ⟨_, by rw [mul_comm]; mem_tac⟩, ⟨i, rfl⟩⟩
          letI r : A⁰_ f := Quotient.mk''
            ⟨m * i, ⟨proj 𝒜 i b ^ m, by mem_tac⟩,
              ⟨_, by rw [mul_comm]; 

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
q : ↑↑(Spec A⁰_ f).toSheafedSpace.toPresheafedSpace
a b : A
ha : a ∈ carrier f_deg q
hb : b ∈ carrier f_deg q
i j : ℕ
h2 : ¬m + m < j
h1 : j ≤ m
l : A⁰_ f :=
  Quotient.mk''
    { deg := m * i, num := { val := (proj 𝒜 i) a ^ j * (proj 𝒜 i) b ^ (m - j), property := ?m.1609416 },
      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)) }) }
⊢ f ^ i ∈ 𝒜 (i * m)

Next tactic:
mem_tac
theorem carrier.add_mem (q : Spec.T A⁰_ f) {a b : A} (ha : a ∈ carrier f_deg q)
    (hb : b ∈ carrier f_deg q) : a + b ∈ carrier f_deg q := by
  refine' fun i => (q.2.mem_or_mem _).elim id id
  change (Quotient.mk'' ⟨_, _, _, _⟩ : A⁰_ f) ∈ q.1; dsimp only [Subtype.coe_mk]
  simp_rw [← pow_add, map_add, add_pow, mul_comm, ← nsmul_eq_mul]
  let g : ℕ → A⁰_ f := fun j => (m + m).choose j •
      if h2 : m + m < j then (0 : A⁰_ f)
      else
        -- Porting note: inlining `l`, `r` causes a "can't synth HMul A⁰_ f A⁰_ f ?" error
        if h1 : j ≤ m then
          letI l : A⁰_ f := Quotient.mk''
            ⟨m * i, ⟨proj 𝒜 i a ^ j * proj 𝒜 i b ^ (m - j), ?_⟩,
              ⟨_, by rw [mul_comm]; mem_tac⟩, ⟨i, rfl⟩⟩
          letI r : A⁰_ f := Quotient.mk''
            ⟨m * i, ⟨proj 𝒜 i b ^ m, by mem_tac⟩,
              ⟨_, by rw [mul_comm]; mem_tac⟩, ⟨i, rfl⟩⟩
          l * r
        else
          letI l : A⁰_ f := Quotient.mk''
            ⟨m * i, ⟨proj 𝒜 i a ^ m, by 

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
q : ↑↑(Spec A⁰_ f).toSheafedSpace.toPresheafedSpace
a b : A
ha : a ∈ carrier f_deg q
hb : b ∈ carrier f_deg q
i j : ℕ
h2 : ¬m + m < j
h1 : ¬j ≤ m
⊢ (proj 𝒜 i) a ^ m ∈ 𝒜 (m * i)

Next tactic:
mem_tac
theorem carrier.add_mem (q : Spec.T A⁰_ f) {a b : A} (ha : a ∈ carrier f_deg q)
    (hb : b ∈ carrier f_deg q) : a + b ∈ carrier f_deg q := by
  refine' fun i => (q.2.mem_or_mem _).elim id id
  change (Quotient.mk'' ⟨_, _, _, _⟩ : A⁰_ f) ∈ q.1; dsimp only [Subtype.coe_mk]
  simp_rw [← pow_add, map_add, add_pow, mul_comm, ← nsmul_eq_mul]
  let g : ℕ → A⁰_ f := fun j => (m + m).choose j •
      if h2 : m + m < j then (0 : A⁰_ f)
      else
        -- Porting note: inlining `l`, `r` causes a "can't synth HMul A⁰_ f A⁰_ f ?" error
        if h1 : j ≤ m then
          letI l : A⁰_ f := Quotient.mk''
            ⟨m * i, ⟨proj 𝒜 i a ^ j * proj 𝒜 i b ^ (m - j), ?_⟩,
              ⟨_, by rw [mul_comm]; mem_tac⟩, ⟨i, rfl⟩⟩
          letI r : A⁰_ f := Quotient.mk''
            ⟨m * i, ⟨proj 𝒜 i b ^ m, by mem_tac⟩,
              ⟨_, by rw [mul_comm]; mem_tac⟩, ⟨i, rfl⟩⟩
          l * r
        else
          letI l : A⁰_ f := Quotient.mk''
            ⟨m * i, ⟨proj 𝒜 i a ^ m, by mem_tac⟩,
              ⟨_, by 

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
q : ↑↑(Spec A⁰_ f).toSheafedSpace.toPresheafedSpace
a b : A
ha : a ∈ carrier f_deg q
hb : b ∈ carrier f_deg q
i j : ℕ
h2 : ¬m + m < j
h1 : ¬j ≤ m
⊢ f ^ i ∈ 𝒜 (m * i)

Next tactic:
rw [mul_comm]
theorem carrier.add_mem (q : Spec.T A⁰_ f) {a b : A} (ha : a ∈ carrier f_deg q)
    (hb : b ∈ carrier f_deg q) : a + b ∈ carrier f_deg q := by
  refine' fun i => (q.2.mem_or_mem _).elim id id
  change (Quotient.mk'' ⟨_, _, _, _⟩ : A⁰_ f) ∈ q.1; dsimp only [Subtype.coe_mk]
  simp_rw [← pow_add, map_add, add_pow, mul_comm, ← nsmul_eq_mul]
  let g : ℕ → A⁰_ f := fun j => (m + m).choose j •
      if h2 : m + m < j then (0 : A⁰_ f)
      else
        -- Porting note: inlining `l`, `r` causes a "can't synth HMul A⁰_ f A⁰_ f ?" error
        if h1 : j ≤ m then
          letI l : A⁰_ f := Quotient.mk''
            ⟨m * i, ⟨proj 𝒜 i a ^ j * proj 𝒜 i b ^ (m - j), ?_⟩,
              ⟨_, by rw [mul_comm]; mem_tac⟩, ⟨i, rfl⟩⟩
          letI r : A⁰_ f := Quotient.mk''
            ⟨m * i, ⟨proj 𝒜 i b ^ m, by mem_tac⟩,
              ⟨_, by rw [mul_comm]; mem_tac⟩, ⟨i, rfl⟩⟩
          l * r
        else
          letI l : A⁰_ f := Quotient.mk''
            ⟨m * i, ⟨proj 𝒜 i a ^ m, by mem_tac⟩,
              ⟨_, by rw [mul_comm]; 

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
q : ↑↑(Spec A⁰_ f).toSheafedSpace.toPresheafedSpace
a b : A
ha : a ∈ carrier f_deg q
hb : b ∈ carrier f_deg q
i j : ℕ
h2 : ¬m + m < j
h1 : ¬j ≤ m
⊢ f ^ i ∈ 𝒜 (i * m)

Next tactic:
mem_tac
theorem carrier.add_mem (q : Spec.T A⁰_ f) {a b : A} (ha : a ∈ carrier f_deg q)
    (hb : b ∈ carrier f_deg q) : a + b ∈ carrier f_deg q := by
  refine' fun i => (q.2.mem_or_mem _).elim id id
  change (Quotient.mk'' ⟨_, _, _, _⟩ : A⁰_ f) ∈ q.1; dsimp only [Subtype.coe_mk]
  simp_rw [← pow_add, map_add, add_pow, mul_comm, ← nsmul_eq_mul]
  let g : ℕ → A⁰_ f := fun j => (m + m).choose j •
      if h2 : m + m < j then (0 : A⁰_ f)
      else
        -- Porting note: inlining `l`, `r` causes a "can't synth HMul A⁰_ f A⁰_ f ?" error
        if h1 : j ≤ m then
          letI l : A⁰_ f := Quotient.mk''
            ⟨m * i, ⟨proj 𝒜 i a ^ j * proj 𝒜 i b ^ (m - j), ?_⟩,
              ⟨_, by rw [mul_comm]; mem_tac⟩, ⟨i, rfl⟩⟩
          letI r : A⁰_ f := Quotient.mk''
            ⟨m * i, ⟨proj 𝒜 i b ^ m, by mem_tac⟩,
              ⟨_, by rw [mul_comm]; mem_tac⟩, ⟨i, rfl⟩⟩
          l * r
        else
          letI l : A⁰_ f := Quotient.mk''
            ⟨m * i, ⟨proj 𝒜 i a ^ m, by mem_tac⟩,
              ⟨_, by rw [mul_comm]; mem_tac⟩, ⟨i, rfl⟩⟩
          letI r : A⁰_ f := Quotient.mk''
            ⟨m * i, ⟨proj 𝒜 i a ^ (j - m) * proj 𝒜 i b ^ (m + m - j), ?_⟩,
              ⟨_, by 

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
q : ↑↑(Spec A⁰_ f).toSheafedSpace.toPresheafedSpace
a b : A
ha : a ∈ carrier f_deg q
hb : b ∈ carrier f_deg q
i j : ℕ
h2 : ¬m + m < j
h1 : ¬j ≤ m
l : A⁰_ f :=
  Quotient.mk''
    { deg := m * i,
      num :=
        { val := (proj 𝒜 i) a ^ m,
          property :=
            (_ :
              ↑((LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap ↑(decomposeAlgEquiv 𝒜))) a) ^ 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)) }) }
⊢ f ^ i ∈ 𝒜 (m * i)

Next tactic:
rw [mul_comm]