ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- LocalRing.local_hom_TFAE
theorem local_hom_TFAE (f : R →+* S) :
    List.TFAE
      [IsLocalRingHom f, f '' (maximalIdeal R).1 ⊆ maximalIdeal S,
        (maximalIdeal R).map f ≤ maximalIdeal S, maximalIdeal R ≤ (maximalIdeal S).comap f,
        (maximalIdeal S).comap f = maximalIdeal R] 
Arguments: {R : Type u_1} {S : Type u_2} [CommSemiring R] [LocalRing R] [CommSemiring S] [LocalRing S] (f : R →+* S)
Goal: [IsLocalRingHom f, ⇑f '' ↑(LocalRing.maximalIdeal R).toAddSubmonoid ⊆ ↑(LocalRing.maximalIdeal S),
    Ideal.map f (LocalRing.maximalIdeal R) ≤ LocalRing.maximalIdeal S,
    LocalRing.maximalIdeal R ≤ Ideal.comap f (LocalRing.maximalIdeal S),
    Ideal.comap f (LocalRing.maximalIdeal S) = LocalRing.maximalIdeal R].TFAE
-- LocalRing.of_surjective
theorem of_surjective [CommSemiring R] [LocalRing R] [CommSemiring S] [Nontrivial S] (f : R →+* S)
    [IsLocalRingHom f] (hf : Function.Surjective f) : LocalRing S 
Arguments: {R : Type u_1} {S : Type u_2} [CommSemiring R] [LocalRing R] [CommSemiring S] [Nontrivial S] (f : R →+* S) [IsLocalRingHom f] (hf : Function.Surjective ⇑f)
Goal: LocalRing S
-- LocalRing.surjective_units_map_of_local_ringHom
theorem surjective_units_map_of_local_ringHom [CommRing R] [CommRing S] (f : R →+* S)
    (hf : Function.Surjective f) (h : IsLocalRingHom f) :
    Function.Surjective (Units.map <| f.toMonoidHom) 
Arguments: {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] (f : R →+* S) (hf : Function.Surjective ⇑f) (h : IsLocalRingHom f)
Goal: Function.Surjective ⇑(Units.map ↑f)
-- RingEquiv.localRing
protected theorem localRing {A B : Type*} [CommSemiring A] [LocalRing A] [CommSemiring B]
    (e : A ≃+* B) : LocalRing B 
Arguments: {A : Type u_4} {B : Type u_5} [CommSemiring A] [LocalRing A] [CommSemiring B] (e : A ≃+* B)
Goal: LocalRing B
-- HomogeneousLocalization.NumDenSameDeg.den_mem
-- Porting note(#5171): this linter isn't ported yet.
-- @[nolint has_nonempty_instance]
structure NumDenSameDeg where
  deg : ι
  (num den : 𝒜 deg)
  den_mem : (den : A) ∈ x
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} {x : Submonoid A} (self : HomogeneousLocalization.NumDenSameDeg 𝒜 x)
Goal: ↑self.den ∈ x
-- HomogeneousLocalization.NumDenSameDeg.ext
@[ext]
theorem ext {c1 c2 : NumDenSameDeg 𝒜 x} (hdeg : c1.deg = c2.deg) (hnum : (c1.num : A) = c2.num)
    (hden : (c1.den : A) = c2.den) : c1 = c2 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} (x : Submonoid A) {c1 c2 : HomogeneousLocalization.NumDenSameDeg 𝒜 x} (hdeg : c1.deg = c2.deg) (hnum : ↑c1.num = ↑c2.num) (hden : ↑c1.den = ↑c2.den)
Goal: c1 = c2
-- HomogeneousLocalization.NumDenSameDeg.deg_one
@[simp]
theorem deg_one : (1 : NumDenSameDeg 𝒜 x).deg = 0 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [AddCommMonoid ι] [DecidableEq ι] [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} [GradedAlgebra 𝒜] (x : Submonoid A)
Goal: HomogeneousLocalization.NumDenSameDeg.deg 1 = 0
-- HomogeneousLocalization.NumDenSameDeg.num_one
@[simp]
theorem num_one : ((1 : NumDenSameDeg 𝒜 x).num : A) = 1 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [AddCommMonoid ι] [DecidableEq ι] [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} [GradedAlgebra 𝒜] (x : Submonoid A)
Goal: ↑(HomogeneousLocalization.NumDenSameDeg.num 1) = 1
-- HomogeneousLocalization.NumDenSameDeg.den_one
@[simp]
theorem den_one : ((1 : NumDenSameDeg 𝒜 x).den : A) = 1 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [AddCommMonoid ι] [DecidableEq ι] [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} [GradedAlgebra 𝒜] (x : Submonoid A)
Goal: ↑(HomogeneousLocalization.NumDenSameDeg.den 1) = 1
-- HomogeneousLocalization.NumDenSameDeg.deg_zero
@[simp]
theorem deg_zero : (0 : NumDenSameDeg 𝒜 x).deg = 0 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [AddCommMonoid ι] [DecidableEq ι] [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} [GradedAlgebra 𝒜] (x : Submonoid A)
Goal: HomogeneousLocalization.NumDenSameDeg.deg 0 = 0
-- HomogeneousLocalization.NumDenSameDeg.num_zero
@[simp]
theorem num_zero : (0 : NumDenSameDeg 𝒜 x).num = 0 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [AddCommMonoid ι] [DecidableEq ι] [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} [GradedAlgebra 𝒜] (x : Submonoid A)
Goal: HomogeneousLocalization.NumDenSameDeg.num 0 = 0
-- HomogeneousLocalization.NumDenSameDeg.den_zero
@[simp]
theorem den_zero : ((0 : NumDenSameDeg 𝒜 x).den : A) = 1 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [AddCommMonoid ι] [DecidableEq ι] [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} [GradedAlgebra 𝒜] (x : Submonoid A)
Goal: ↑(HomogeneousLocalization.NumDenSameDeg.den 0) = 1
-- HomogeneousLocalization.NumDenSameDeg.deg_mul
@[simp]
theorem deg_mul (c1 c2 : NumDenSameDeg 𝒜 x) : (c1 * c2).deg = c1.deg + c2.deg 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [AddCommMonoid ι] [DecidableEq ι] [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} [GradedAlgebra 𝒜] (x : Submonoid A) (c1 c2 : HomogeneousLocalization.NumDenSameDeg 𝒜 x)
Goal: (c1 * c2).deg = c1.deg + c2.deg
-- HomogeneousLocalization.NumDenSameDeg.num_mul
@[simp]
theorem num_mul (c1 c2 : NumDenSameDeg 𝒜 x) : ((c1 * c2).num : A) = c1.num * c2.num 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [AddCommMonoid ι] [DecidableEq ι] [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} [GradedAlgebra 𝒜] (x : Submonoid A) (c1 c2 : HomogeneousLocalization.NumDenSameDeg 𝒜 x)
Goal: ↑(c1 * c2).num = ↑c1.num * ↑c2.num
-- HomogeneousLocalization.NumDenSameDeg.den_mul
@[simp]
theorem den_mul (c1 c2 : NumDenSameDeg 𝒜 x) : ((c1 * c2).den : A) = c1.den * c2.den 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [AddCommMonoid ι] [DecidableEq ι] [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} [GradedAlgebra 𝒜] (x : Submonoid A) (c1 c2 : HomogeneousLocalization.NumDenSameDeg 𝒜 x)
Goal: ↑(c1 * c2).den = ↑c1.den * ↑c2.den
-- HomogeneousLocalization.NumDenSameDeg.deg_add
@[simp]
theorem deg_add (c1 c2 : NumDenSameDeg 𝒜 x) : (c1 + c2).deg = c1.deg + c2.deg 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [AddCommMonoid ι] [DecidableEq ι] [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} [GradedAlgebra 𝒜] (x : Submonoid A) (c1 c2 : HomogeneousLocalization.NumDenSameDeg 𝒜 x)
Goal: (c1 + c2).deg = c1.deg + c2.deg
-- HomogeneousLocalization.NumDenSameDeg.num_add
@[simp]
theorem num_add (c1 c2 : NumDenSameDeg 𝒜 x) :
    ((c1 + c2).num : A) = c1.den * c2.num + c2.den * c1.num 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [AddCommMonoid ι] [DecidableEq ι] [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} [GradedAlgebra 𝒜] (x : Submonoid A) (c1 c2 : HomogeneousLocalization.NumDenSameDeg 𝒜 x)
Goal: ↑(c1 + c2).num = ↑c1.den * ↑c2.num + ↑c2.den * ↑c1.num
-- HomogeneousLocalization.NumDenSameDeg.den_add
@[simp]
theorem den_add (c1 c2 : NumDenSameDeg 𝒜 x) : ((c1 + c2).den : A) = c1.den * c2.den 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [AddCommMonoid ι] [DecidableEq ι] [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} [GradedAlgebra 𝒜] (x : Submonoid A) (c1 c2 : HomogeneousLocalization.NumDenSameDeg 𝒜 x)
Goal: ↑(c1 + c2).den = ↑c1.den * ↑c2.den
-- HomogeneousLocalization.NumDenSameDeg.deg_neg
@[simp]
theorem deg_neg (c : NumDenSameDeg 𝒜 x) : (-c).deg = c.deg 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} (x : Submonoid A) (c : HomogeneousLocalization.NumDenSameDeg 𝒜 x)
Goal: (-c).deg = c.deg
-- HomogeneousLocalization.NumDenSameDeg.num_neg
@[simp]
theorem num_neg (c : NumDenSameDeg 𝒜 x) : ((-c).num : A) = -c.num 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} (x : Submonoid A) (c : HomogeneousLocalization.NumDenSameDeg 𝒜 x)
Goal: ↑(-c).num = -↑c.num
-- HomogeneousLocalization.NumDenSameDeg.den_neg
@[simp]
theorem den_neg (c : NumDenSameDeg 𝒜 x) : ((-c).den : A) = c.den 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} (x : Submonoid A) (c : HomogeneousLocalization.NumDenSameDeg 𝒜 x)
Goal: ↑(-c).den = ↑c.den
-- HomogeneousLocalization.NumDenSameDeg.deg_pow
@[simp]
theorem deg_pow (c : NumDenSameDeg 𝒜 x) (n : ℕ) : (c ^ n).deg = n • c.deg 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [AddCommMonoid ι] [DecidableEq ι] [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} [GradedAlgebra 𝒜] (x : Submonoid A) (c : HomogeneousLocalization.NumDenSameDeg 𝒜 x) (n : ℕ)
Goal: (c ^ n).deg = n • c.deg
-- HomogeneousLocalization.NumDenSameDeg.num_pow
@[simp]
theorem num_pow (c : NumDenSameDeg 𝒜 x) (n : ℕ) : ((c ^ n).num : A) = (c.num : A) ^ n 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [AddCommMonoid ι] [DecidableEq ι] [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} [GradedAlgebra 𝒜] (x : Submonoid A) (c : HomogeneousLocalization.NumDenSameDeg 𝒜 x) (n : ℕ)
Goal: ↑(c ^ n).num = ↑c.num ^ n
-- HomogeneousLocalization.NumDenSameDeg.den_pow
@[simp]
theorem den_pow (c : NumDenSameDeg 𝒜 x) (n : ℕ) : ((c ^ n).den : A) = (c.den : A) ^ n 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [AddCommMonoid ι] [DecidableEq ι] [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} [GradedAlgebra 𝒜] (x : Submonoid A) (c : HomogeneousLocalization.NumDenSameDeg 𝒜 x) (n : ℕ)
Goal: ↑(c ^ n).den = ↑c.den ^ n
-- HomogeneousLocalization.NumDenSameDeg.deg_smul
@[simp]
theorem deg_smul (c : NumDenSameDeg 𝒜 x) (m : α) : (m • c).deg = c.deg 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} (x : Submonoid A) {α : Type u_4} [SMul α R] [SMul α A] [IsScalarTower α R A] (c : HomogeneousLocalization.NumDenSameDeg 𝒜 x) (m : α)
Goal: (m • c).deg = c.deg
-- HomogeneousLocalization.NumDenSameDeg.num_smul
@[simp]
theorem num_smul (c : NumDenSameDeg 𝒜 x) (m : α) : ((m • c).num : A) = m • c.num 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} (x : Submonoid A) {α : Type u_4} [SMul α R] [SMul α A] [IsScalarTower α R A] (c : HomogeneousLocalization.NumDenSameDeg 𝒜 x) (m : α)
Goal: ↑(m • c).num = m • ↑c.num
-- HomogeneousLocalization.NumDenSameDeg.den_smul
@[simp]
theorem den_smul (c : NumDenSameDeg 𝒜 x) (m : α) : ((m • c).den : A) = c.den 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} (x : Submonoid A) {α : Type u_4} [SMul α R] [SMul α A] [IsScalarTower α R A] (c : HomogeneousLocalization.NumDenSameDeg 𝒜 x) (m : α)
Goal: ↑(m • c).den = ↑c.den
-- HomogeneousLocalization.mk_surjective
lemma mk_surjective : Function.Surjective (mk (𝒜 := 𝒜) (x := x)) 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} {x : Submonoid A}
Goal: Function.Surjective HomogeneousLocalization.mk
-- HomogeneousLocalization.val_mk
@[simp]
theorem val_mk (i : NumDenSameDeg 𝒜 x) :
    val (mk i) = Localization.mk (i.num : A) ⟨i.den, i.den_mem⟩ 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} {x : Submonoid A} (i : HomogeneousLocalization.NumDenSameDeg 𝒜 x)
Goal: (HomogeneousLocalization.mk i).val = Localization.mk ↑i.num ⟨↑i.den, ⋯⟩
-- HomogeneousLocalization.val_injective
@[ext]
theorem val_injective : Function.Injective (HomogeneousLocalization.val (𝒜 := 𝒜) (x := x)) 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} (x : Submonoid A)
Goal: Function.Injective HomogeneousLocalization.val
-- HomogeneousLocalization.subsingleton
lemma subsingleton (hx : 0 ∈ x) : Subsingleton (HomogeneousLocalization 𝒜 x) 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [CommRing R] [CommRing A] [Algebra R A] (𝒜 : ι → Submodule R A) {x : Submonoid A} (hx : 0 ∈ x)
Goal: Subsingleton (HomogeneousLocalization 𝒜 x)
-- HomogeneousLocalization.mk_pow
@[simp] lemma mk_pow (i : NumDenSameDeg 𝒜 x) (n : ℕ) : mk (i ^ n) = mk i ^ n 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [AddCommMonoid ι] [DecidableEq ι] [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} [GradedAlgebra 𝒜] (x : Submonoid A) (i : HomogeneousLocalization.NumDenSameDeg 𝒜 x) (n : ℕ)
Goal: HomogeneousLocalization.mk (i ^ n) = HomogeneousLocalization.mk i ^ n
-- HomogeneousLocalization.mk_smul
@[simp] lemma mk_smul (i : NumDenSameDeg 𝒜 x) (m : α) : mk (m • i) = m • mk i 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} (x : Submonoid A) {α : Type u_4} [SMul α R] [SMul α A] [IsScalarTower α R A] [IsScalarTower α A A] (i : HomogeneousLocalization.NumDenSameDeg 𝒜 x) (m : α)
Goal: HomogeneousLocalization.mk (m • i) = m • HomogeneousLocalization.mk i
-- HomogeneousLocalization.val_smul
@[simp]
theorem val_smul (n : α) : ∀ y : HomogeneousLocalization 𝒜 x, (n • y).val = n • y.val 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} (x : Submonoid A) {α : Type u_4} [SMul α R] [SMul α A] [IsScalarTower α R A] [IsScalarTower α A A] (n : α) (y : HomogeneousLocalization 𝒜 x)
Goal: (n • y).val = n • y.val
-- HomogeneousLocalization.val_nsmul
theorem val_nsmul (n : ℕ) (y : HomogeneousLocalization 𝒜 x) : (n • y).val = n • y.val 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} (x : Submonoid A) (n : ℕ) (y : HomogeneousLocalization 𝒜 x)
Goal: (n • y).val = n • y.val
-- HomogeneousLocalization.val_zsmul
theorem val_zsmul (n : ℤ) (y : HomogeneousLocalization 𝒜 x) : (n • y).val = n • y.val 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} (x : Submonoid A) (n : ℤ) (y : HomogeneousLocalization 𝒜 x)
Goal: (n • y).val = n • y.val
-- HomogeneousLocalization.mk_neg
@[simp] lemma mk_neg (i : NumDenSameDeg 𝒜 x) : mk (-i) = -mk i 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} (x : Submonoid A) (i : HomogeneousLocalization.NumDenSameDeg 𝒜 x)
Goal: HomogeneousLocalization.mk (-i) = -HomogeneousLocalization.mk i
-- HomogeneousLocalization.mk_add
@[simp] lemma mk_add (i j : NumDenSameDeg 𝒜 x) : mk (i + j) = mk i + mk j 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [AddCommMonoid ι] [DecidableEq ι] [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} [GradedAlgebra 𝒜] (x : Submonoid A) (i j : HomogeneousLocalization.NumDenSameDeg 𝒜 x)
Goal: HomogeneousLocalization.mk (i + j) = HomogeneousLocalization.mk i + HomogeneousLocalization.mk j
-- HomogeneousLocalization.mk_mul
@[simp] lemma mk_mul (i j : NumDenSameDeg 𝒜 x) : mk (i * j) = mk i * mk j 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [AddCommMonoid ι] [DecidableEq ι] [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} [GradedAlgebra 𝒜] (x : Submonoid A) (i j : HomogeneousLocalization.NumDenSameDeg 𝒜 x)
Goal: HomogeneousLocalization.mk (i * j) = HomogeneousLocalization.mk i * HomogeneousLocalization.mk j
-- HomogeneousLocalization.mk_one
@[simp] lemma mk_one : mk (1 : NumDenSameDeg 𝒜 x) = 1 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [AddCommMonoid ι] [DecidableEq ι] [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} [GradedAlgebra 𝒜] (x : Submonoid A)
Goal: HomogeneousLocalization.mk 1 = 1
-- HomogeneousLocalization.mk_zero
@[simp] lemma mk_zero : mk (0 : NumDenSameDeg 𝒜 x) = 0 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [AddCommMonoid ι] [DecidableEq ι] [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} [GradedAlgebra 𝒜] (x : Submonoid A)
Goal: HomogeneousLocalization.mk 0 = 0
-- HomogeneousLocalization.zero_eq
theorem zero_eq : (0 : HomogeneousLocalization 𝒜 x) = Quotient.mk'' 0 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [AddCommMonoid ι] [DecidableEq ι] [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} [GradedAlgebra 𝒜] (x : Submonoid A)
Goal: 0 = Quotient.mk'' 0
-- HomogeneousLocalization.one_eq
theorem one_eq : (1 : HomogeneousLocalization 𝒜 x) = Quotient.mk'' 1 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [AddCommMonoid ι] [DecidableEq ι] [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} [GradedAlgebra 𝒜] (x : Submonoid A)
Goal: 1 = Quotient.mk'' 1
-- HomogeneousLocalization.val_zero
@[simp]
theorem val_zero : (0 : HomogeneousLocalization 𝒜 x).val = 0 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [AddCommMonoid ι] [DecidableEq ι] [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} [GradedAlgebra 𝒜] {x : Submonoid A}
Goal: HomogeneousLocalization.val 0 = 0
-- HomogeneousLocalization.val_one
@[simp]
theorem val_one : (1 : HomogeneousLocalization 𝒜 x).val = 1 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [AddCommMonoid ι] [DecidableEq ι] [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} [GradedAlgebra 𝒜] {x : Submonoid A}
Goal: HomogeneousLocalization.val 1 = 1
-- HomogeneousLocalization.val_add
@[simp]
theorem val_add : ∀ y1 y2 : HomogeneousLocalization 𝒜 x, (y1 + y2).val = y1.val + y2.val 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [AddCommMonoid ι] [DecidableEq ι] [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} [GradedAlgebra 𝒜] {x : Submonoid A} (y1 y2 : HomogeneousLocalization 𝒜 x)
Goal: (y1 + y2).val = y1.val + y2.val
-- HomogeneousLocalization.val_mul
@[simp]
theorem val_mul : ∀ y1 y2 : HomogeneousLocalization 𝒜 x, (y1 * y2).val = y1.val * y2.val 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [AddCommMonoid ι] [DecidableEq ι] [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} [GradedAlgebra 𝒜] {x : Submonoid A} (y1 y2 : HomogeneousLocalization 𝒜 x)
Goal: (y1 * y2).val = y1.val * y2.val
-- HomogeneousLocalization.val_neg
@[simp]
theorem val_neg : ∀ y : HomogeneousLocalization 𝒜 x, (-y).val = -y.val 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} {x : Submonoid A} (y : HomogeneousLocalization 𝒜 x)
Goal: (-y).val = -y.val
-- HomogeneousLocalization.val_sub
@[simp]
theorem val_sub (y1 y2 : HomogeneousLocalization 𝒜 x) : (y1 - y2).val = y1.val - y2.val 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [AddCommMonoid ι] [DecidableEq ι] [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} [GradedAlgebra 𝒜] {x : Submonoid A} (y1 y2 : HomogeneousLocalization 𝒜 x)
Goal: (y1 - y2).val = y1.val - y2.val
-- HomogeneousLocalization.val_pow
@[simp]
theorem val_pow : ∀ (y : HomogeneousLocalization 𝒜 x) (n : ℕ), (y ^ n).val = y.val ^ n 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [AddCommMonoid ι] [DecidableEq ι] [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} [GradedAlgebra 𝒜] {x : Submonoid A} (y : HomogeneousLocalization 𝒜 x) (n : ℕ)
Goal: (y ^ n).val = y.val ^ n
-- HomogeneousLocalization.val_natCast
@[simp]
theorem val_natCast (n : ℕ) : (n : HomogeneousLocalization 𝒜 x).val = n 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [AddCommMonoid ι] [DecidableEq ι] [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} [GradedAlgebra 𝒜] {x : Submonoid A} (n : ℕ)
Goal: (↑n).val = ↑n
-- HomogeneousLocalization.val_intCast
@[simp]
theorem val_intCast (n : ℤ) : (n : HomogeneousLocalization 𝒜 x).val = n 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [AddCommMonoid ι] [DecidableEq ι] [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} [GradedAlgebra 𝒜] {x : Submonoid A} (n : ℤ)
Goal: (↑n).val = ↑n
-- HomogeneousLocalization.algebraMap_apply
@[simp] lemma algebraMap_apply (y) :
    algebraMap (HomogeneousLocalization 𝒜 x) (Localization x) y = y.val 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [AddCommMonoid ι] [DecidableEq ι] [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} [GradedAlgebra 𝒜] {x : Submonoid A} (y : HomogeneousLocalization 𝒜 x)
Goal: (algebraMap (HomogeneousLocalization 𝒜 x) (at x)) y = y.val
-- HomogeneousLocalization.mk_eq_zero_of_num
lemma mk_eq_zero_of_num (f : NumDenSameDeg 𝒜 x) (h : f.num = 0) : mk f = 0 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [AddCommMonoid ι] [DecidableEq ι] [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} [GradedAlgebra 𝒜] {x : Submonoid A} (f : HomogeneousLocalization.NumDenSameDeg 𝒜 x) (h : f.num = 0)
Goal: HomogeneousLocalization.mk f = 0
-- HomogeneousLocalization.mk_eq_zero_of_den
lemma mk_eq_zero_of_den (f : NumDenSameDeg 𝒜 x) (h : f.den = 0) : mk f = 0 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [AddCommMonoid ι] [DecidableEq ι] [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} [GradedAlgebra 𝒜] {x : Submonoid A} (f : HomogeneousLocalization.NumDenSameDeg 𝒜 x) (h : f.den = 0)
Goal: HomogeneousLocalization.mk f = 0
-- HomogeneousLocalization.den_mem
theorem den_mem (f : HomogeneousLocalization 𝒜 x) : f.den ∈ x 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} {x : Submonoid A} (f : HomogeneousLocalization 𝒜 x)
Goal: f.den ∈ x
-- HomogeneousLocalization.num_mem_deg
theorem num_mem_deg (f : HomogeneousLocalization 𝒜 x) : f.num ∈ 𝒜 f.deg 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} {x : Submonoid A} (f : HomogeneousLocalization 𝒜 x)
Goal: f.num ∈ 𝒜 f.deg
-- HomogeneousLocalization.den_mem_deg
theorem den_mem_deg (f : HomogeneousLocalization 𝒜 x) : f.den ∈ 𝒜 f.deg 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} {x : Submonoid A} (f : HomogeneousLocalization 𝒜 x)
Goal: f.den ∈ 𝒜 f.deg
-- HomogeneousLocalization.eq_num_div_den
theorem eq_num_div_den (f : HomogeneousLocalization 𝒜 x) :
    f.val = Localization.mk f.num ⟨f.den, f.den_mem⟩ 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} {x : Submonoid A} (f : HomogeneousLocalization 𝒜 x)
Goal: f.val = Localization.mk f.num ⟨f.den, ⋯⟩
-- HomogeneousLocalization.den_smul_val
theorem den_smul_val (f : HomogeneousLocalization 𝒜 x) :
    f.den • f.val = algebraMap _ _ f.num 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} {x : Submonoid A} (f : HomogeneousLocalization 𝒜 x)
Goal: f.den • f.val = (algebraMap A (at x)) f.num
-- HomogeneousLocalization.ext_iff_val
theorem ext_iff_val (f g : HomogeneousLocalization 𝒜 x) : f = g ↔ f.val = g.val 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} {x : Submonoid A} (f g : HomogeneousLocalization 𝒜 x)
Goal: f = g ↔ f.val = g.val
-- HomogeneousLocalization.isUnit_iff_isUnit_val
theorem isUnit_iff_isUnit_val (f : HomogeneousLocalization.AtPrime 𝒜 𝔭) :
    IsUnit f.val ↔ IsUnit f 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [AddCommMonoid ι] [DecidableEq ι] [CommRing R] [CommRing A] [Algebra R A] (𝒜 : ι → Submodule R A) [GradedAlgebra 𝒜] (𝔭 : Ideal A) [𝔭.IsPrime] (f : HomogeneousLocalization.AtPrime 𝒜 𝔭)
Goal: IsUnit (HomogeneousLocalization.val f) ↔ IsUnit f
-- HomogeneousLocalization.Away.eventually_smul_mem
theorem Away.eventually_smul_mem {m} (hf : f ∈ 𝒜 m) (z : Away 𝒜 f) :
    ∀ᶠ n in Filter.atTop, f ^ n • z.val ∈ algebraMap _ _ '' (𝒜 (n • m) : Set A) 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [AddCommMonoid ι] [DecidableEq ι] [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ι → Submodule R A} [GradedAlgebra 𝒜] {f : A} {m : ι} (hf : f ∈ 𝒜 m) (z : HomogeneousLocalization.Away 𝒜 f)
Goal: ∀ᶠ (n : ℕ) in Filter.atTop,
  f ^ n • HomogeneousLocalization.val z ∈ ⇑(algebraMap A (at Submonoid.powers f)) '' ↑(𝒜 (n • m))