ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- isSMulRegular_iff_smul_eq_zero_imp_eq_zero
lemma isSMulRegular_iff_smul_eq_zero_imp_eq_zero :
    IsSMulRegular M r ↔ ∀ x : M, r • x = 0 → x = 0 
Arguments: {R : Type u_1} (M : Type u_3) [Ring R] [AddCommGroup M] [Module R M] (r : R)
Goal: IsSMulRegular M r ↔ ∀ (x : M), r • x = 0 → x = 0
-- isSMulRegular_iff_mem_nonZeroSMulDivisors
lemma isSMulRegular_iff_mem_nonZeroSMulDivisors :
    IsSMulRegular M r ↔ r ∈ nonZeroSMulDivisors R M 
Arguments: {R : Type u_1} (M : Type u_3) [Ring R] [AddCommGroup M] [Module R M] (r : R)
Goal: IsSMulRegular M r ↔ r ∈ nonZeroSMulDivisors R M
-- isSMulRegular_of_smul_eq_zero_imp_eq_zero
lemma isSMulRegular_of_smul_eq_zero_imp_eq_zero
    (h : ∀ x : M, r • x = 0 → x = 0) : IsSMulRegular M r 
Arguments: {R : Type u_1} {M : Type u_3} [Ring R] [AddCommGroup M] [Module R M] {r : R} (h : ∀ (x : M), r • x = 0 → x = 0)
Goal: IsSMulRegular M r
-- isSMulRegular_on_submodule_iff_mem_imp_smul_eq_zero_imp_eq_zero
lemma isSMulRegular_on_submodule_iff_mem_imp_smul_eq_zero_imp_eq_zero :
    IsSMulRegular N r ↔ ∀ x ∈ N, r • x = 0 → x = 0 
Arguments: {R : Type u_1} {M : Type u_3} [Ring R] [AddCommGroup M] [Module R M] (N : Submodule R M) (r : R)
Goal: IsSMulRegular (↥N) r ↔ ∀ x ∈ N, r • x = 0 → x = 0
-- isSMulRegular_on_quot_iff_smul_mem_implies_mem
lemma isSMulRegular_on_quot_iff_smul_mem_implies_mem :
    IsSMulRegular (M ⧸ N) r ↔ ∀ x : M, r • x ∈ N → x ∈ N 
Arguments: {R : Type u_1} {M : Type u_3} [Ring R] [AddCommGroup M] [Module R M] (N : Submodule R M) (r : R)
Goal: IsSMulRegular (M ⧸ N) r ↔ ∀ (x : M), r • x ∈ N → x ∈ N
-- mem_of_isSMulRegular_on_quot_of_smul_mem
lemma mem_of_isSMulRegular_on_quot_of_smul_mem (h1 : IsSMulRegular (M ⧸ N) r)
    {x : M} (h2 : r • x ∈ N) : x ∈ N 
Arguments: {R : Type u_1} {M : Type u_3} [Ring R] [AddCommGroup M] [Module R M] {N : Submodule R M} {r : R} (h1 : IsSMulRegular (M ⧸ N) r) {x : M} (h2 : r • x ∈ N)
Goal: x ∈ N
-- isSMulRegular_of_range_eq_ker
lemma isSMulRegular_of_range_eq_ker {f : M →ₗ[R] M'} {g : M' →ₗ[R] M''}
    (hf : Function.Injective f) (hfg : LinearMap.range f = LinearMap.ker g)
    (h1 : IsSMulRegular M r) (h2 : IsSMulRegular M'' r) :
    IsSMulRegular M' r 
Arguments: {R : Type u_1} {M : Type u_3} {M' : Type u_4} {M'' : Type u_5} [Ring R] [AddCommGroup M] [Module R M] [AddCommGroup M'] [Module R M'] [AddCommGroup M''] [Module R M''] {r : R} {f : M →ₗ[R] M'} {g : M' →ₗ[R] M''} (hf : Function.Injective ⇑f) (hfg : LinearMap.range f = LinearMap.ker g) (h1 : IsSMulRegular M r) (h2 : IsSMulRegular M'' r)
Goal: IsSMulRegular M' r
-- isSMulRegular_of_isSMulRegular_on_submodule_on_quotient
lemma isSMulRegular_of_isSMulRegular_on_submodule_on_quotient
    (h1 : IsSMulRegular N r) (h2 : IsSMulRegular (M ⧸ N) r) : IsSMulRegular M r 
Arguments: {R : Type u_1} {M : Type u_3} [Ring R] [AddCommGroup M] [Module R M] {N : Submodule R M} {r : R} (h1 : IsSMulRegular (↥N) r) (h2 : IsSMulRegular (M ⧸ N) r)
Goal: IsSMulRegular M r
-- biUnion_associatedPrimes_eq_compl_regular
lemma biUnion_associatedPrimes_eq_compl_regular [IsNoetherianRing R] :
    ⋃ p ∈ associatedPrimes R M, p = { r : R | IsSMulRegular M r }ᶜ 
Arguments: (R : Type u_1) (M : Type u_3) [CommRing R] [AddCommGroup M] [Module R M] [IsNoetherianRing R]
Goal: ⋃ p ∈ associatedPrimes R M, ↑p = {r | IsSMulRegular M r}ᶜ
-- isSMulRegular_iff_ker_lsmul_eq_bot
lemma isSMulRegular_iff_ker_lsmul_eq_bot :
    IsSMulRegular M r ↔ LinearMap.ker (LinearMap.lsmul R M r) = ⊥ 
Arguments: {R : Type u_1} (M : Type u_3) [CommRing R] [AddCommGroup M] [Module R M] (r : R)
Goal: IsSMulRegular M r ↔ LinearMap.ker ((LinearMap.lsmul R M) r) = ⊥
-- isSMulRegular_on_submodule_iff_disjoint_ker_lsmul_submodule
lemma isSMulRegular_on_submodule_iff_disjoint_ker_lsmul_submodule :
    IsSMulRegular N r ↔ Disjoint (LinearMap.ker (LinearMap.lsmul R M r)) N 
Arguments: {R : Type u_1} {M : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] (N : Submodule R M) (r : R)
Goal: IsSMulRegular (↥N) r ↔ Disjoint (LinearMap.ker ((LinearMap.lsmul R M) r)) N
-- isSMulRegular_on_quot_iff_lsmul_comap_le
lemma isSMulRegular_on_quot_iff_lsmul_comap_le :
    IsSMulRegular (M ⧸ N) r ↔ N.comap (LinearMap.lsmul R M r) ≤ N 
Arguments: {R : Type u_1} {M : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] (N : Submodule R M) (r : R)
Goal: IsSMulRegular (M ⧸ N) r ↔ Submodule.comap ((LinearMap.lsmul R M) r) N ≤ N
-- isSMulRegular_on_quot_iff_lsmul_comap_eq
lemma isSMulRegular_on_quot_iff_lsmul_comap_eq :
    IsSMulRegular (M ⧸ N) r ↔ N.comap (LinearMap.lsmul R M r) = N 
Arguments: {R : Type u_1} {M : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] (N : Submodule R M) (r : R)
Goal: IsSMulRegular (M ⧸ N) r ↔ Submodule.comap ((LinearMap.lsmul R M) r) N = N
-- IsSMulRegular.isSMulRegular_on_quot_iff_smul_top_inf_eq_smul
lemma IsSMulRegular.isSMulRegular_on_quot_iff_smul_top_inf_eq_smul :
    IsSMulRegular M r → (IsSMulRegular (M ⧸ N) r ↔ r • ⊤ ⊓ N ≤ r • N) 
Arguments: {R : Type u_1} {M : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] (N : Submodule R M) {r : R}
Goal: IsSMulRegular M r → (IsSMulRegular (M ⧸ N) r ↔ r • ⊤ ⊓ N ≤ r • N)
-- isSMulRegular_of_ker_lsmul_eq_bot
lemma isSMulRegular_of_ker_lsmul_eq_bot
    (h : LinearMap.ker (LinearMap.lsmul R M r) = ⊥) :
    IsSMulRegular M r 
Arguments: {R : Type u_1} {M : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] {r : R} (h : LinearMap.ker ((LinearMap.lsmul R M) r) = ⊥)
Goal: IsSMulRegular M r
-- smul_top_inf_eq_smul_of_isSMulRegular_on_quot
lemma smul_top_inf_eq_smul_of_isSMulRegular_on_quot :
    IsSMulRegular (M ⧸ N) r → r • ⊤ ⊓ N ≤ r • N 
Arguments: {R : Type u_1} {M : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] {N : Submodule R M} {r : R}
Goal: IsSMulRegular (M ⧸ N) r → r • ⊤ ⊓ N ≤ r • N
-- QuotSMulTop.map_first_exact_on_four_term_exact_of_isSMulRegular_last
lemma QuotSMulTop.map_first_exact_on_four_term_exact_of_isSMulRegular_last
    {M'''} [AddCommGroup M'''] [Module R M''']
    {r : R} {f₁ : M →ₗ[R] M'} {f₂ : M' →ₗ[R] M''} {f₃ : M'' →ₗ[R] M'''}
    (h₁₂ : Exact f₁ f₂) (h₂₃ : Exact f₂ f₃) (h : IsSMulRegular M''' r) :
    Exact (map r f₁) (map r f₂) 
Arguments: {R : Type u_1} {M : Type u_3} {M' : Type u_4} {M'' : Type u_5} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup M'] [Module R M'] [AddCommGroup M''] [Module R M''] {M''' : Type u_6} [AddCommGroup M'''] [Module R M'''] {r : R} {f₁ : M →ₗ[R] M'} {f₂ : M' →ₗ[R] M''} {f₃ : M'' →ₗ[R] M'''} (h₁₂ : Function.Exact ⇑f₁ ⇑f₂) (h₂₃ : Function.Exact ⇑f₂ ⇑f₃) (h : IsSMulRegular M''' r)
Goal: Function.Exact ⇑((QuotSMulTop.map r) f₁) ⇑((QuotSMulTop.map r) f₂)
-- Algebra.norm_one_add_smul
lemma Algebra.norm_one_add_smul {A B} [CommRing A] [CommRing B] [Algebra A B]
    [Module.Free A B] [Module.Finite A B] (a : A) (x : B) :
    ∃ r : A, Algebra.norm A (1 + a • x) = 1 + Algebra.trace A B x * a + r * a ^ 2 
Arguments: {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] [Algebra A B] [Module.Free A B] [Module.Finite A B] (a : A) (x : B)
Goal: ∃ r, (Algebra.norm A) (1 + a • x) = 1 + (Algebra.trace A B) x * a + r * a ^ 2
-- Ideal.IsHomogeneous.mem_iff
theorem Ideal.IsHomogeneous.mem_iff {I} (hI : Ideal.IsHomogeneous 𝒜 I) {x} :
    x ∈ I ↔ ∀ i, (decompose 𝒜 x i : A) ∈ I 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [SetLike σ A] [AddSubmonoidClass σ A] (𝒜 : ι → σ) [DecidableEq ι] [AddMonoid ι] [GradedRing 𝒜] {I : Ideal A} (hI : Ideal.IsHomogeneous 𝒜 I) {x : A}
Goal: x ∈ I ↔ ∀ (i : ι), ↑(((DirectSum.decompose 𝒜) x) i) ∈ I
-- HomogeneousIdeal.is_homogeneous'
structure HomogeneousIdeal extends Submodule A A where
  is_homogeneous' : Ideal.IsHomogeneous 𝒜 toSubmodule
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [SetLike σ A] [AddSubmonoidClass σ A] {𝒜 : ι → σ} [DecidableEq ι] [AddMonoid ι] [GradedRing 𝒜] (self : HomogeneousIdeal 𝒜)
Goal: Ideal.IsHomogeneous 𝒜 self.toSubmodule
-- HomogeneousIdeal.isHomogeneous
theorem HomogeneousIdeal.isHomogeneous (I : HomogeneousIdeal 𝒜) : I.toIdeal.IsHomogeneous 𝒜 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [SetLike σ A] [AddSubmonoidClass σ A] {𝒜 : ι → σ} [DecidableEq ι] [AddMonoid ι] [GradedRing 𝒜] (I : HomogeneousIdeal 𝒜)
Goal: Ideal.IsHomogeneous 𝒜 I.toIdeal
-- HomogeneousIdeal.toIdeal_injective
theorem HomogeneousIdeal.toIdeal_injective :
    Function.Injective (HomogeneousIdeal.toIdeal : HomogeneousIdeal 𝒜 → Ideal A) 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [SetLike σ A] [AddSubmonoidClass σ A] {𝒜 : ι → σ} [DecidableEq ι] [AddMonoid ι] [GradedRing 𝒜]
Goal: Function.Injective HomogeneousIdeal.toIdeal
-- HomogeneousIdeal.ext
@[ext]
theorem HomogeneousIdeal.ext {I J : HomogeneousIdeal 𝒜} (h : I.toIdeal = J.toIdeal) : I = J 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [SetLike σ A] [AddSubmonoidClass σ A] {𝒜 : ι → σ} [DecidableEq ι] [AddMonoid ι] [GradedRing 𝒜] {I J : HomogeneousIdeal 𝒜} (h : I.toIdeal = J.toIdeal)
Goal: I = J
-- HomogeneousIdeal.ext'
theorem HomogeneousIdeal.ext' {I J : HomogeneousIdeal 𝒜} (h : ∀ i, ∀ x ∈ 𝒜 i, x ∈ I ↔ x ∈ J) :
    I = J 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [SetLike σ A] [AddSubmonoidClass σ A] {𝒜 : ι → σ} [DecidableEq ι] [AddMonoid ι] [GradedRing 𝒜] {I J : HomogeneousIdeal 𝒜} (h : ∀ (i : ι), ∀ x ∈ 𝒜 i, x ∈ I ↔ x ∈ J)
Goal: I = J
-- HomogeneousIdeal.mem_iff
@[simp]
theorem HomogeneousIdeal.mem_iff {I : HomogeneousIdeal 𝒜} {x : A} : x ∈ I.toIdeal ↔ x ∈ I 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [SetLike σ A] [AddSubmonoidClass σ A] {𝒜 : ι → σ} [DecidableEq ι] [AddMonoid ι] [GradedRing 𝒜] {I : HomogeneousIdeal 𝒜} {x : A}
Goal: x ∈ I.toIdeal ↔ x ∈ I
-- Ideal.homogeneousCore'_mono
theorem Ideal.homogeneousCore'_mono : Monotone (Ideal.homogeneousCore' 𝒜) 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [SetLike σ A] (𝒜 : ι → σ)
Goal: Monotone (Ideal.homogeneousCore' 𝒜)
-- Ideal.homogeneousCore'_le
theorem Ideal.homogeneousCore'_le : I.homogeneousCore' 𝒜 ≤ I 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [SetLike σ A] (𝒜 : ι → σ) (I : Ideal A)
Goal: Ideal.homogeneousCore' 𝒜 I ≤ I
-- Ideal.isHomogeneous_iff_forall_subset
theorem Ideal.isHomogeneous_iff_forall_subset :
    I.IsHomogeneous 𝒜 ↔ ∀ i, (I : Set A) ⊆ GradedRing.proj 𝒜 i ⁻¹' I 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [SetLike σ A] [AddSubmonoidClass σ A] (𝒜 : ι → σ) [DecidableEq ι] [AddMonoid ι] [GradedRing 𝒜] (I : Ideal A)
Goal: Ideal.IsHomogeneous 𝒜 I ↔ ∀ (i : ι), ↑I ⊆ ⇑(GradedRing.proj 𝒜 i) ⁻¹' ↑I
-- Ideal.isHomogeneous_iff_subset_iInter
theorem Ideal.isHomogeneous_iff_subset_iInter :
    I.IsHomogeneous 𝒜 ↔ (I : Set A) ⊆ ⋂ i, GradedRing.proj 𝒜 i ⁻¹' ↑I 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [SetLike σ A] [AddSubmonoidClass σ A] (𝒜 : ι → σ) [DecidableEq ι] [AddMonoid ι] [GradedRing 𝒜] (I : Ideal A)
Goal: Ideal.IsHomogeneous 𝒜 I ↔ ↑I ⊆ ⋂ i, ⇑(GradedRing.proj 𝒜 i) ⁻¹' ↑I
-- Ideal.mul_homogeneous_element_mem_of_mem
theorem Ideal.mul_homogeneous_element_mem_of_mem {I : Ideal A} (r x : A) (hx₁ : Homogeneous 𝒜 x)
    (hx₂ : x ∈ I) (j : ι) : GradedRing.proj 𝒜 j (r * x) ∈ I 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [SetLike σ A] [AddSubmonoidClass σ A] (𝒜 : ι → σ) [DecidableEq ι] [AddMonoid ι] [GradedRing 𝒜] {I : Ideal A} (r x : A) (hx₁ : SetLike.Homogeneous 𝒜 x) (hx₂ : x ∈ I) (j : ι)
Goal: (GradedRing.proj 𝒜 j) (r * x) ∈ I
-- Ideal.homogeneous_span
theorem Ideal.homogeneous_span (s : Set A) (h : ∀ x ∈ s, Homogeneous 𝒜 x) :
    (Ideal.span s).IsHomogeneous 𝒜 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [SetLike σ A] [AddSubmonoidClass σ A] (𝒜 : ι → σ) [DecidableEq ι] [AddMonoid ι] [GradedRing 𝒜] (s : Set A) (h : ∀ x ∈ s, SetLike.Homogeneous 𝒜 x)
Goal: Ideal.IsHomogeneous 𝒜 (Ideal.span s)
-- Ideal.homogeneousCore_mono
theorem Ideal.homogeneousCore_mono : Monotone (Ideal.homogeneousCore 𝒜) 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [SetLike σ A] [AddSubmonoidClass σ A] (𝒜 : ι → σ) [DecidableEq ι] [AddMonoid ι] [GradedRing 𝒜]
Goal: Monotone (Ideal.homogeneousCore 𝒜)
-- Ideal.toIdeal_homogeneousCore_le
theorem Ideal.toIdeal_homogeneousCore_le : (I.homogeneousCore 𝒜).toIdeal ≤ I 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [SetLike σ A] [AddSubmonoidClass σ A] (𝒜 : ι → σ) [DecidableEq ι] [AddMonoid ι] [GradedRing 𝒜] (I : Ideal A)
Goal: (Ideal.homogeneousCore 𝒜 I).toIdeal ≤ I
-- Ideal.mem_homogeneousCore_of_homogeneous_of_mem
theorem Ideal.mem_homogeneousCore_of_homogeneous_of_mem {x : A} (h : SetLike.Homogeneous 𝒜 x)
    (hmem : x ∈ I) : x ∈ I.homogeneousCore 𝒜 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [SetLike σ A] [AddSubmonoidClass σ A] {𝒜 : ι → σ} [DecidableEq ι] [AddMonoid ι] [GradedRing 𝒜] {I : Ideal A} {x : A} (h : SetLike.Homogeneous 𝒜 x) (hmem : x ∈ I)
Goal: x ∈ Ideal.homogeneousCore 𝒜 I
-- Ideal.IsHomogeneous.toIdeal_homogeneousCore_eq_self
theorem Ideal.IsHomogeneous.toIdeal_homogeneousCore_eq_self (h : I.IsHomogeneous 𝒜) :
    (I.homogeneousCore 𝒜).toIdeal = I 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [SetLike σ A] [AddSubmonoidClass σ A] {𝒜 : ι → σ} [DecidableEq ι] [AddMonoid ι] [GradedRing 𝒜] {I : Ideal A} (h : Ideal.IsHomogeneous 𝒜 I)
Goal: (Ideal.homogeneousCore 𝒜 I).toIdeal = I
-- HomogeneousIdeal.toIdeal_homogeneousCore_eq_self
@[simp]
theorem HomogeneousIdeal.toIdeal_homogeneousCore_eq_self (I : HomogeneousIdeal 𝒜) :
    I.toIdeal.homogeneousCore 𝒜 = I 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [SetLike σ A] [AddSubmonoidClass σ A] {𝒜 : ι → σ} [DecidableEq ι] [AddMonoid ι] [GradedRing 𝒜] (I : HomogeneousIdeal 𝒜)
Goal: Ideal.homogeneousCore 𝒜 I.toIdeal = I
-- Ideal.IsHomogeneous.iff_eq
theorem Ideal.IsHomogeneous.iff_eq : I.IsHomogeneous 𝒜 ↔ (I.homogeneousCore 𝒜).toIdeal = I 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [SetLike σ A] [AddSubmonoidClass σ A] (𝒜 : ι → σ) [DecidableEq ι] [AddMonoid ι] [GradedRing 𝒜] (I : Ideal A)
Goal: Ideal.IsHomogeneous 𝒜 I ↔ (Ideal.homogeneousCore 𝒜 I).toIdeal = I
-- Ideal.IsHomogeneous.iff_exists
theorem Ideal.IsHomogeneous.iff_exists :
    I.IsHomogeneous 𝒜 ↔ ∃ S : Set (homogeneousSubmonoid 𝒜), I = Ideal.span ((↑) '' S) 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [SetLike σ A] [AddSubmonoidClass σ A] (𝒜 : ι → σ) [DecidableEq ι] [AddMonoid ι] [GradedRing 𝒜] (I : Ideal A)
Goal: Ideal.IsHomogeneous 𝒜 I ↔ ∃ S, I = Ideal.span (Subtype.val '' S)
-- Ideal.IsHomogeneous.bot
theorem bot : Ideal.IsHomogeneous 𝒜 ⊥ 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [DecidableEq ι] [AddMonoid ι] [SetLike σ A] [AddSubmonoidClass σ A] (𝒜 : ι → σ) [GradedRing 𝒜]
Goal: Ideal.IsHomogeneous 𝒜 ⊥
-- Ideal.IsHomogeneous.top
theorem top : Ideal.IsHomogeneous 𝒜 ⊤ 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [DecidableEq ι] [AddMonoid ι] [SetLike σ A] [AddSubmonoidClass σ A] (𝒜 : ι → σ) [GradedRing 𝒜]
Goal: Ideal.IsHomogeneous 𝒜 ⊤
-- Ideal.IsHomogeneous.inf
theorem inf {I J : Ideal A} (HI : I.IsHomogeneous 𝒜) (HJ : J.IsHomogeneous 𝒜) :
    (I ⊓ J).IsHomogeneous 𝒜 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [DecidableEq ι] [AddMonoid ι] [SetLike σ A] [AddSubmonoidClass σ A] {𝒜 : ι → σ} [GradedRing 𝒜] {I J : Ideal A} (HI : Ideal.IsHomogeneous 𝒜 I) (HJ : Ideal.IsHomogeneous 𝒜 J)
Goal: Ideal.IsHomogeneous 𝒜 (I ⊓ J)
-- Ideal.IsHomogeneous.sup
theorem sup {I J : Ideal A} (HI : I.IsHomogeneous 𝒜) (HJ : J.IsHomogeneous 𝒜) :
    (I ⊔ J).IsHomogeneous 𝒜 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [DecidableEq ι] [AddMonoid ι] [SetLike σ A] [AddSubmonoidClass σ A] {𝒜 : ι → σ} [GradedRing 𝒜] {I J : Ideal A} (HI : Ideal.IsHomogeneous 𝒜 I) (HJ : Ideal.IsHomogeneous 𝒜 J)
Goal: Ideal.IsHomogeneous 𝒜 (I ⊔ J)
-- Ideal.IsHomogeneous.iSup
protected theorem iSup {κ : Sort*} {f : κ → Ideal A} (h : ∀ i, (f i).IsHomogeneous 𝒜) :
    (⨆ i, f i).IsHomogeneous 𝒜 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [DecidableEq ι] [AddMonoid ι] [SetLike σ A] [AddSubmonoidClass σ A] {𝒜 : ι → σ} [GradedRing 𝒜] {κ : Sort u_5} {f : κ → Ideal A} (h : ∀ (i : κ), Ideal.IsHomogeneous 𝒜 (f i))
Goal: Ideal.IsHomogeneous 𝒜 (⨆ i, f i)
-- Ideal.IsHomogeneous.iInf
protected theorem iInf {κ : Sort*} {f : κ → Ideal A} (h : ∀ i, (f i).IsHomogeneous 𝒜) :
    (⨅ i, f i).IsHomogeneous 𝒜 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [DecidableEq ι] [AddMonoid ι] [SetLike σ A] [AddSubmonoidClass σ A] {𝒜 : ι → σ} [GradedRing 𝒜] {κ : Sort u_5} {f : κ → Ideal A} (h : ∀ (i : κ), Ideal.IsHomogeneous 𝒜 (f i))
Goal: Ideal.IsHomogeneous 𝒜 (⨅ i, f i)
-- Ideal.IsHomogeneous.iSup₂
theorem iSup₂ {κ : Sort*} {κ' : κ → Sort*} {f : ∀ i, κ' i → Ideal A}
    (h : ∀ i j, (f i j).IsHomogeneous 𝒜) : (⨆ (i) (j), f i j).IsHomogeneous 𝒜 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [DecidableEq ι] [AddMonoid ι] [SetLike σ A] [AddSubmonoidClass σ A] {𝒜 : ι → σ} [GradedRing 𝒜] {κ : Sort u_5} {κ' : κ → Sort u_6} {f : (i : κ) → κ' i → Ideal A} (h : ∀ (i : κ) (j : κ' i), Ideal.IsHomogeneous 𝒜 (f i j))
Goal: Ideal.IsHomogeneous 𝒜 (⨆ i, ⨆ j, f i j)
-- Ideal.IsHomogeneous.iInf₂
theorem iInf₂ {κ : Sort*} {κ' : κ → Sort*} {f : ∀ i, κ' i → Ideal A}
    (h : ∀ i j, (f i j).IsHomogeneous 𝒜) : (⨅ (i) (j), f i j).IsHomogeneous 𝒜 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [DecidableEq ι] [AddMonoid ι] [SetLike σ A] [AddSubmonoidClass σ A] {𝒜 : ι → σ} [GradedRing 𝒜] {κ : Sort u_5} {κ' : κ → Sort u_6} {f : (i : κ) → κ' i → Ideal A} (h : ∀ (i : κ) (j : κ' i), Ideal.IsHomogeneous 𝒜 (f i j))
Goal: Ideal.IsHomogeneous 𝒜 (⨅ i, ⨅ j, f i j)
-- Ideal.IsHomogeneous.sSup
theorem sSup {ℐ : Set (Ideal A)} (h : ∀ I ∈ ℐ, Ideal.IsHomogeneous 𝒜 I) :
    (sSup ℐ).IsHomogeneous 𝒜 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [DecidableEq ι] [AddMonoid ι] [SetLike σ A] [AddSubmonoidClass σ A] {𝒜 : ι → σ} [GradedRing 𝒜] {ℐ : Set (Ideal A)} (h : ∀ I ∈ ℐ, Ideal.IsHomogeneous 𝒜 I)
Goal: Ideal.IsHomogeneous 𝒜 (sSup ℐ)
-- Ideal.IsHomogeneous.sInf
theorem sInf {ℐ : Set (Ideal A)} (h : ∀ I ∈ ℐ, Ideal.IsHomogeneous 𝒜 I) :
    (sInf ℐ).IsHomogeneous 𝒜 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [DecidableEq ι] [AddMonoid ι] [SetLike σ A] [AddSubmonoidClass σ A] {𝒜 : ι → σ} [GradedRing 𝒜] {ℐ : Set (Ideal A)} (h : ∀ I ∈ ℐ, Ideal.IsHomogeneous 𝒜 I)
Goal: Ideal.IsHomogeneous 𝒜 (sInf ℐ)
-- HomogeneousIdeal.coe_top
@[simp]
theorem coe_top : ((⊤ : HomogeneousIdeal 𝒜) : Set A) = univ 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [DecidableEq ι] [AddMonoid ι] [SetLike σ A] [AddSubmonoidClass σ A] {𝒜 : ι → σ} [GradedRing 𝒜]
Goal: ↑⊤ = Set.univ
-- HomogeneousIdeal.coe_bot
@[simp]
theorem coe_bot : ((⊥ : HomogeneousIdeal 𝒜) : Set A) = 0 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [DecidableEq ι] [AddMonoid ι] [SetLike σ A] [AddSubmonoidClass σ A] {𝒜 : ι → σ} [GradedRing 𝒜]
Goal: ↑⊥ = 0
-- HomogeneousIdeal.coe_sup
@[simp]
theorem coe_sup (I J : HomogeneousIdeal 𝒜) : ↑(I ⊔ J) = (I + J : Set A) 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [DecidableEq ι] [AddMonoid ι] [SetLike σ A] [AddSubmonoidClass σ A] {𝒜 : ι → σ} [GradedRing 𝒜] (I J : HomogeneousIdeal 𝒜)
Goal: ↑(I ⊔ J) = ↑I + ↑J
-- HomogeneousIdeal.coe_inf
@[simp]
theorem coe_inf (I J : HomogeneousIdeal 𝒜) : (↑(I ⊓ J) : Set A) = ↑I ∩ ↑J 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [DecidableEq ι] [AddMonoid ι] [SetLike σ A] [AddSubmonoidClass σ A] {𝒜 : ι → σ} [GradedRing 𝒜] (I J : HomogeneousIdeal 𝒜)
Goal: ↑(I ⊓ J) = ↑I ∩ ↑J
-- HomogeneousIdeal.toIdeal_top
@[simp]
theorem toIdeal_top : (⊤ : HomogeneousIdeal 𝒜).toIdeal = (⊤ : Ideal A) 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [DecidableEq ι] [AddMonoid ι] [SetLike σ A] [AddSubmonoidClass σ A] {𝒜 : ι → σ} [GradedRing 𝒜]
Goal: ⊤.toIdeal = ⊤
-- HomogeneousIdeal.toIdeal_bot
@[simp]
theorem toIdeal_bot : (⊥ : HomogeneousIdeal 𝒜).toIdeal = (⊥ : Ideal A) 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [DecidableEq ι] [AddMonoid ι] [SetLike σ A] [AddSubmonoidClass σ A] {𝒜 : ι → σ} [GradedRing 𝒜]
Goal: ⊥.toIdeal = ⊥
-- HomogeneousIdeal.toIdeal_sup
@[simp]
theorem toIdeal_sup (I J : HomogeneousIdeal 𝒜) : (I ⊔ J).toIdeal = I.toIdeal ⊔ J.toIdeal 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [DecidableEq ι] [AddMonoid ι] [SetLike σ A] [AddSubmonoidClass σ A] {𝒜 : ι → σ} [GradedRing 𝒜] (I J : HomogeneousIdeal 𝒜)
Goal: (I ⊔ J).toIdeal = I.toIdeal ⊔ J.toIdeal
-- HomogeneousIdeal.toIdeal_inf
@[simp]
theorem toIdeal_inf (I J : HomogeneousIdeal 𝒜) : (I ⊓ J).toIdeal = I.toIdeal ⊓ J.toIdeal 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [DecidableEq ι] [AddMonoid ι] [SetLike σ A] [AddSubmonoidClass σ A] {𝒜 : ι → σ} [GradedRing 𝒜] (I J : HomogeneousIdeal 𝒜)
Goal: (I ⊓ J).toIdeal = I.toIdeal ⊓ J.toIdeal
-- HomogeneousIdeal.toIdeal_sSup
@[simp]
theorem toIdeal_sSup (ℐ : Set (HomogeneousIdeal 𝒜)) : (sSup ℐ).toIdeal = ⨆ s ∈ ℐ, toIdeal s 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [DecidableEq ι] [AddMonoid ι] [SetLike σ A] [AddSubmonoidClass σ A] {𝒜 : ι → σ} [GradedRing 𝒜] (ℐ : Set (HomogeneousIdeal 𝒜))
Goal: (sSup ℐ).toIdeal = ⨆ s ∈ ℐ, s.toIdeal
-- HomogeneousIdeal.toIdeal_sInf
@[simp]
theorem toIdeal_sInf (ℐ : Set (HomogeneousIdeal 𝒜)) : (sInf ℐ).toIdeal = ⨅ s ∈ ℐ, toIdeal s 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [DecidableEq ι] [AddMonoid ι] [SetLike σ A] [AddSubmonoidClass σ A] {𝒜 : ι → σ} [GradedRing 𝒜] (ℐ : Set (HomogeneousIdeal 𝒜))
Goal: (sInf ℐ).toIdeal = ⨅ s ∈ ℐ, s.toIdeal
-- HomogeneousIdeal.toIdeal_iSup
@[simp]
theorem toIdeal_iSup {κ : Sort*} (s : κ → HomogeneousIdeal 𝒜) :
    (⨆ i, s i).toIdeal = ⨆ i, (s i).toIdeal 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [DecidableEq ι] [AddMonoid ι] [SetLike σ A] [AddSubmonoidClass σ A] {𝒜 : ι → σ} [GradedRing 𝒜] {κ : Sort u_5} (s : κ → HomogeneousIdeal 𝒜)
Goal: (⨆ i, s i).toIdeal = ⨆ i, (s i).toIdeal
-- HomogeneousIdeal.toIdeal_iInf
@[simp]
theorem toIdeal_iInf {κ : Sort*} (s : κ → HomogeneousIdeal 𝒜) :
    (⨅ i, s i).toIdeal = ⨅ i, (s i).toIdeal 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [DecidableEq ι] [AddMonoid ι] [SetLike σ A] [AddSubmonoidClass σ A] {𝒜 : ι → σ} [GradedRing 𝒜] {κ : Sort u_5} (s : κ → HomogeneousIdeal 𝒜)
Goal: (⨅ i, s i).toIdeal = ⨅ i, (s i).toIdeal
-- HomogeneousIdeal.toIdeal_iSup₂
theorem toIdeal_iSup₂ {κ : Sort*} {κ' : κ → Sort*} (s : ∀ i, κ' i → HomogeneousIdeal 𝒜) :
    (⨆ (i) (j), s i j).toIdeal = ⨆ (i) (j), (s i j).toIdeal 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [DecidableEq ι] [AddMonoid ι] [SetLike σ A] [AddSubmonoidClass σ A] {𝒜 : ι → σ} [GradedRing 𝒜] {κ : Sort u_5} {κ' : κ → Sort u_6} (s : (i : κ) → κ' i → HomogeneousIdeal 𝒜)
Goal: (⨆ i, ⨆ j, s i j).toIdeal = ⨆ i, ⨆ j, (s i j).toIdeal
-- HomogeneousIdeal.toIdeal_iInf₂
theorem toIdeal_iInf₂ {κ : Sort*} {κ' : κ → Sort*} (s : ∀ i, κ' i → HomogeneousIdeal 𝒜) :
    (⨅ (i) (j), s i j).toIdeal = ⨅ (i) (j), (s i j).toIdeal 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [DecidableEq ι] [AddMonoid ι] [SetLike σ A] [AddSubmonoidClass σ A] {𝒜 : ι → σ} [GradedRing 𝒜] {κ : Sort u_5} {κ' : κ → Sort u_6} (s : (i : κ) → κ' i → HomogeneousIdeal 𝒜)
Goal: (⨅ i, ⨅ j, s i j).toIdeal = ⨅ i, ⨅ j, (s i j).toIdeal
-- HomogeneousIdeal.eq_top_iff
@[simp]
theorem eq_top_iff (I : HomogeneousIdeal 𝒜) : I = ⊤ ↔ I.toIdeal = ⊤ 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [DecidableEq ι] [AddMonoid ι] [SetLike σ A] [AddSubmonoidClass σ A] {𝒜 : ι → σ} [GradedRing 𝒜] (I : HomogeneousIdeal 𝒜)
Goal: I = ⊤ ↔ I.toIdeal = ⊤
-- HomogeneousIdeal.eq_bot_iff
@[simp]
theorem eq_bot_iff (I : HomogeneousIdeal 𝒜) : I = ⊥ ↔ I.toIdeal = ⊥ 
Arguments: {ι : Type u_1} {σ : Type u_2} {A : Type u_4} [Semiring A] [DecidableEq ι] [AddMonoid ι] [SetLike σ A] [AddSubmonoidClass σ A] {𝒜 : ι → σ} [GradedRing 𝒜] (I : HomogeneousIdeal 𝒜)
Goal: I = ⊥ ↔ I.toIdeal = ⊥