ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MvPolynomial.linearIndependent_X
theorem linearIndependent_X : LinearIndependent R (X : σ → MvPolynomial σ R) 
Arguments: (σ : Type u) (R : Type v) [CommSemiring R]
Goal: LinearIndependent R MvPolynomial.X
-- _private.Mathlib.RingTheory.MvPolynomial.Basic.0.MvPolynomial.finite_setOf_bounded
private lemma finite_setOf_bounded (α) [Finite α] (n : ℕ) : Finite {f : α →₀ ℕ | ∀ a, f a ≤ n} 
Arguments: (α : Type u_1) [Finite α] (n : ℕ)
Goal: Finite ↑{f | ∀ (a : α), f a ≤ n}
-- MvPolynomial.algebraMap_def
@[simp]
lemma algebraMap_def :
    algebraMap (MvPolynomial σ R) (MvPolynomial σ S) = MvPolynomial.map (algebraMap R S) 
Arguments: {R : Type u_1} {S : Type u_2} {σ : Type u_3} [CommSemiring R] [CommSemiring S] [Algebra R S]
Goal: algebraMap (MvPolynomial σ R) (MvPolynomial σ S) = MvPolynomial.map (algebraMap R S)
-- Polynomial.coe_basisMonomials
@[simp]
theorem coe_basisMonomials : (basisMonomials R : ℕ → R[X]) = fun s => monomial s 1 
Arguments: (R : Type v) [CommSemiring R]
Goal: ⇑(Polynomial.basisMonomials R) = fun s => (Polynomial.monomial s) 1
-- TwoSidedIdeal.sup_ringCon
lemma sup_ringCon (I J : TwoSidedIdeal R) : (I ⊔ J).ringCon = I.ringCon ⊔ J.ringCon 
Arguments: (R : Type u_1) [NonUnitalNonAssocRing R] (I J : TwoSidedIdeal R)
Goal: (I ⊔ J).ringCon = I.ringCon ⊔ J.ringCon
-- TwoSidedIdeal.mem_sup_left
lemma mem_sup_left {I J : TwoSidedIdeal R} {x : R} (h : x ∈ I) :
    x ∈ I ⊔ J 
Arguments: {R : Type u_1} [NonUnitalNonAssocRing R] {I J : TwoSidedIdeal R} {x : R} (h : x ∈ I)
Goal: x ∈ I ⊔ J
-- TwoSidedIdeal.mem_sup_right
lemma mem_sup_right {I J : TwoSidedIdeal R} {x : R} (h : x ∈ J) :
    x ∈ I ⊔ J 
Arguments: {R : Type u_1} [NonUnitalNonAssocRing R] {I J : TwoSidedIdeal R} {x : R} (h : x ∈ J)
Goal: x ∈ I ⊔ J
-- TwoSidedIdeal.mem_sup
lemma mem_sup {I J : TwoSidedIdeal R} {x : R} :
    x ∈ I ⊔ J ↔ ∃ y ∈ I, ∃ z ∈ J, y + z = x 
Arguments: {R : Type u_1} [NonUnitalNonAssocRing R] {I J : TwoSidedIdeal R} {x : R}
Goal: x ∈ I ⊔ J ↔ ∃ y ∈ I, ∃ z ∈ J, y + z = x
-- TwoSidedIdeal.inf_ringCon
lemma inf_ringCon (I J : TwoSidedIdeal R) : (I ⊓ J).ringCon = I.ringCon ⊓ J.ringCon 
Arguments: (R : Type u_1) [NonUnitalNonAssocRing R] (I J : TwoSidedIdeal R)
Goal: (I ⊓ J).ringCon = I.ringCon ⊓ J.ringCon
-- TwoSidedIdeal.mem_inf
lemma mem_inf {I J : TwoSidedIdeal R} {x : R} :
    x ∈ I ⊓ J ↔ x ∈ I ∧ x ∈ J 
Arguments: (R : Type u_1) [NonUnitalNonAssocRing R] {I J : TwoSidedIdeal R} {x : R}
Goal: x ∈ I ⊓ J ↔ x ∈ I ∧ x ∈ J
-- TwoSidedIdeal.sSup_ringCon
lemma sSup_ringCon (S : Set (TwoSidedIdeal R)) :
    (sSup S).ringCon = sSup (TwoSidedIdeal.ringCon '' S) 
Arguments: (R : Type u_1) [NonUnitalNonAssocRing R] (S : Set (TwoSidedIdeal R))
Goal: (sSup S).ringCon = sSup (TwoSidedIdeal.ringCon '' S)
-- TwoSidedIdeal.iSup_ringCon
lemma iSup_ringCon {ι : Type*} (I : ι → TwoSidedIdeal R) :
    (⨆ i, I i).ringCon = ⨆ i, (I i).ringCon 
Arguments: (R : Type u_1) [NonUnitalNonAssocRing R] {ι : Type u_2} (I : ι → TwoSidedIdeal R)
Goal: (⨆ i, I i).ringCon = ⨆ i, (I i).ringCon
-- TwoSidedIdeal.sInf_ringCon
lemma sInf_ringCon (S : Set (TwoSidedIdeal R)) :
    (sInf S).ringCon = sInf (TwoSidedIdeal.ringCon '' S) 
Arguments: (R : Type u_1) [NonUnitalNonAssocRing R] (S : Set (TwoSidedIdeal R))
Goal: (sInf S).ringCon = sInf (TwoSidedIdeal.ringCon '' S)
-- TwoSidedIdeal.iInf_ringCon
lemma iInf_ringCon {ι : Type*} (I : ι → TwoSidedIdeal R) :
    (⨅ i, I i).ringCon = ⨅ i, (I i).ringCon 
Arguments: (R : Type u_1) [NonUnitalNonAssocRing R] {ι : Type u_2} (I : ι → TwoSidedIdeal R)
Goal: (⨅ i, I i).ringCon = ⨅ i, (I i).ringCon
-- TwoSidedIdeal.mem_iInf
lemma mem_iInf {ι : Type*} {I : ι → TwoSidedIdeal R} {x : R} :
    x ∈ iInf I ↔ ∀ i, x ∈ I i 
Arguments: (R : Type u_1) [NonUnitalNonAssocRing R] {ι : Type u_2} {I : ι → TwoSidedIdeal R} {x : R}
Goal: x ∈ iInf I ↔ ∀ (i : ι), x ∈ I i
-- TwoSidedIdeal.mem_sInf
lemma mem_sInf {S : Set (TwoSidedIdeal R)} {x : R} :
    x ∈ sInf S ↔ ∀ I ∈ S, x ∈ I 
Arguments: (R : Type u_1) [NonUnitalNonAssocRing R] {S : Set (TwoSidedIdeal R)} {x : R}
Goal: x ∈ sInf S ↔ ∀ I ∈ S, x ∈ I
-- TwoSidedIdeal.top_ringCon
lemma top_ringCon : (⊤ : TwoSidedIdeal R).ringCon = ⊤ 
Arguments: (R : Type u_1) [NonUnitalNonAssocRing R]
Goal: ⊤.ringCon = ⊤
-- TwoSidedIdeal.bot_ringCon
lemma bot_ringCon : (⊥ : TwoSidedIdeal R).ringCon = ⊥ 
Arguments: (R : Type u_1) [NonUnitalNonAssocRing R]
Goal: ⊥.ringCon = ⊥
-- TwoSidedIdeal.mem_bot
lemma mem_bot {x : R} : x ∈ (⊥ : TwoSidedIdeal R) ↔ x = 0 
Arguments: (R : Type u_1) [NonUnitalNonAssocRing R] {x : R}
Goal: x ∈ ⊥ ↔ x = 0
-- MvPolynomial.aeval_map_algebraMap
theorem aeval_map_algebraMap (x : σ → B) (p : MvPolynomial σ R) :
    aeval x (map (algebraMap R A) p) = aeval x p 
Arguments: {R : Type u_1} (A : Type u_2) {B : Type u_3} {σ : Type u_4} [CommSemiring R] [CommSemiring A] [CommSemiring B] [Algebra R A] [Algebra A B] [Algebra R B] [IsScalarTower R A B] (x : σ → B) (p : MvPolynomial σ R)
Goal: (MvPolynomial.aeval x) ((MvPolynomial.map (algebraMap R A)) p) = (MvPolynomial.aeval x) p
-- MvPolynomial.aeval_algebraMap_apply
theorem aeval_algebraMap_apply (x : σ → A) (p : MvPolynomial σ R) :
    aeval (algebraMap A B ∘ x) p = algebraMap A B (MvPolynomial.aeval x p) 
Arguments: {R : Type u_1} {A : Type u_2} (B : Type u_3) {σ : Type u_4} [CommSemiring R] [CommSemiring A] [CommSemiring B] [Algebra R A] [Algebra A B] [Algebra R B] [IsScalarTower R A B] (x : σ → A) (p : MvPolynomial σ R)
Goal: (MvPolynomial.aeval (⇑(algebraMap A B) ∘ x)) p = (algebraMap A B) ((MvPolynomial.aeval x) p)
-- MvPolynomial.aeval_algebraMap_eq_zero_iff
theorem aeval_algebraMap_eq_zero_iff [NoZeroSMulDivisors A B] [Nontrivial B] (x : σ → A)
    (p : MvPolynomial σ R) : aeval (algebraMap A B ∘ x) p = 0 ↔ aeval x p = 0 
Arguments: {R : Type u_1} {A : Type u_2} (B : Type u_3) {σ : Type u_4} [CommSemiring R] [CommSemiring A] [CommSemiring B] [Algebra R A] [Algebra A B] [Algebra R B] [IsScalarTower R A B] [NoZeroSMulDivisors A B] [Nontrivial B] (x : σ → A) (p : MvPolynomial σ R)
Goal: (MvPolynomial.aeval (⇑(algebraMap A B) ∘ x)) p = 0 ↔ (MvPolynomial.aeval x) p = 0
-- MvPolynomial.aeval_algebraMap_eq_zero_iff_of_injective
theorem aeval_algebraMap_eq_zero_iff_of_injective {x : σ → A} {p : MvPolynomial σ R}
    (h : Function.Injective (algebraMap A B)) :
    aeval (algebraMap A B ∘ x) p = 0 ↔ aeval x p = 0 
Arguments: {R : Type u_1} {A : Type u_2} (B : Type u_3) {σ : Type u_4} [CommSemiring R] [CommSemiring A] [CommSemiring B] [Algebra R A] [Algebra A B] [Algebra R B] [IsScalarTower R A B] {x : σ → A} {p : MvPolynomial σ R} (h : Function.Injective ⇑(algebraMap A B))
Goal: (MvPolynomial.aeval (⇑(algebraMap A B) ∘ x)) p = 0 ↔ (MvPolynomial.aeval x) p = 0
-- Subalgebra.mvPolynomial_aeval_coe
@[simp]
theorem mvPolynomial_aeval_coe (S : Subalgebra R A) (x : σ → S) (p : MvPolynomial σ R) :
    aeval (fun i => (x i : A)) p = aeval x p 
Arguments: {R : Type u_1} {A : Type u_2} {σ : Type u_4} [CommSemiring R] [CommSemiring A] [Algebra R A] (S : Subalgebra R A) (x : σ → ↥S) (p : MvPolynomial σ R)
Goal: (MvPolynomial.aeval fun i => ↑(x i)) p = ↑((MvPolynomial.aeval x) p)
-- RingCon.listSum
protected lemma listSum {ι S : Type*} [AddMonoid S] [Mul S]
    (t : RingCon S) (l : List ι) {f g : ι → S} (h : ∀ i ∈ l, t (f i) (g i)) :
    t (l.map f).sum (l.map g).sum 
Arguments: {ι : Type u_1} {S : Type u_2} [AddMonoid S] [Mul S] (t : RingCon S) (l : List ι) {f g : ι → S} (h : ∀ i ∈ l, t (f i) (g i))
Goal: t (List.map f l).sum (List.map g l).sum
-- RingCon.multisetSum
protected lemma multisetSum {ι S : Type*} [AddCommMonoid S] [Mul S] (t : RingCon S)
    (s : Multiset ι) {f g : ι → S} (h : ∀ i ∈ s, t (f i) (g i)) :
    t (s.map f).sum (s.map g).sum 
Arguments: {ι : Type u_1} {S : Type u_2} [AddCommMonoid S] [Mul S] (t : RingCon S) (s : Multiset ι) {f g : ι → S} (h : ∀ i ∈ s, t (f i) (g i))
Goal: t (Multiset.map f s).sum (Multiset.map g s).sum
-- RingCon.finsetSum
protected lemma finsetSum {ι S : Type*} [AddCommMonoid S] [Mul S] (t : RingCon S) (s : Finset ι)
    {f g : ι → S} (h : ∀ i ∈ s, t (f i) (g i)) :
    t (s.sum f) (s.sum g) 
Arguments: {ι : Type u_1} {S : Type u_2} [AddCommMonoid S] [Mul S] (t : RingCon S) (s : Finset ι) {f g : ι → S} (h : ∀ i ∈ s, t (f i) (g i))
Goal: t (s.sum f) (s.sum g)
-- Ideal.IsJacobson.out'
class IsJacobson (R : Type*) [CommRing R] : Prop where
  out' : ∀ I : Ideal R, I.IsRadical → I.jacobson = I
Arguments: {R : Type u_3} [CommRing R] [self : Ideal.IsJacobson R] (I : Ideal R)
Goal: I.IsRadical → I.jacobson = I
-- Ideal.isJacobson_iff
theorem isJacobson_iff {R} [CommRing R] :
    IsJacobson R ↔ ∀ I : Ideal R, I.IsRadical → I.jacobson = I 
Arguments: {R : Type u_3} [CommRing R]
Goal: Ideal.IsJacobson R ↔ ∀ (I : Ideal R), I.IsRadical → I.jacobson = I
-- Ideal.IsJacobson.out
theorem IsJacobson.out {R} [CommRing R] :
    IsJacobson R → ∀ {I : Ideal R}, I.IsRadical → I.jacobson = I 
Arguments: {R : Type u_3} [CommRing R]
Goal: Ideal.IsJacobson R → ∀ {I : Ideal R}, I.IsRadical → I.jacobson = I
-- Ideal.isJacobson_iff_prime_eq
theorem isJacobson_iff_prime_eq : IsJacobson R ↔ ∀ P : Ideal R, IsPrime P → P.jacobson = P 
Arguments: {R : Type u_1} [CommRing R]
Goal: Ideal.IsJacobson R ↔ ∀ (P : Ideal R), P.IsPrime → P.jacobson = P
-- Ideal.isJacobson_iff_sInf_maximal
theorem isJacobson_iff_sInf_maximal : IsJacobson R ↔ ∀ {I : Ideal R}, I.IsPrime →
    ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M 
Arguments: {R : Type u_1} [CommRing R]
Goal: Ideal.IsJacobson R ↔ ∀ {I : Ideal R}, I.IsPrime → ∃ M, (∀ J ∈ M, J.IsMaximal ∨ J = ⊤) ∧ I = sInf M
-- Ideal.isJacobson_iff_sInf_maximal'
theorem isJacobson_iff_sInf_maximal' : IsJacobson R ↔ ∀ {I : Ideal R}, I.IsPrime →
    ∃ M : Set (Ideal R), (∀ J ∈ M, ∀ (K : Ideal R), J < K → K = ⊤) ∧ I = sInf M 
Arguments: {R : Type u_1} [CommRing R]
Goal: Ideal.IsJacobson R ↔ ∀ {I : Ideal R}, I.IsPrime → ∃ M, (∀ J ∈ M, ∀ (K : Ideal R), J < K → K = ⊤) ∧ I = sInf M
-- Ideal.radical_eq_jacobson
theorem radical_eq_jacobson [H : IsJacobson R] (I : Ideal R) : I.radical = I.jacobson 
Arguments: {R : Type u_1} [CommRing R] [H : Ideal.IsJacobson R] (I : Ideal R)
Goal: I.radical = I.jacobson
-- Ideal.isJacobson_of_surjective
theorem isJacobson_of_surjective [H : IsJacobson R] :
    (∃ f : R →+* S, Function.Surjective ↑f) → IsJacobson S 
Arguments: {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [H : Ideal.IsJacobson R]
Goal: (∃ f, Function.Surjective ⇑f) → Ideal.IsJacobson S
-- Ideal.isJacobson_iso
theorem isJacobson_iso (e : R ≃+* S) : IsJacobson R ↔ IsJacobson S 
Arguments: {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] (e : R ≃+* S)
Goal: Ideal.IsJacobson R ↔ Ideal.IsJacobson S
-- Ideal.isJacobson_of_isIntegral
theorem isJacobson_of_isIntegral [Algebra R S] [Algebra.IsIntegral R S] (hR : IsJacobson R) :
    IsJacobson S 
Arguments: {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] [Algebra.IsIntegral R S] (hR : Ideal.IsJacobson R)
Goal: Ideal.IsJacobson S
-- Ideal.isJacobson_of_isIntegral'
theorem isJacobson_of_isIntegral' (f : R →+* S) (hf : f.IsIntegral) (hR : IsJacobson R) :
    IsJacobson S 
Arguments: {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] (f : R →+* S) (hf : f.IsIntegral) (hR : Ideal.IsJacobson R)
Goal: Ideal.IsJacobson S
-- Ideal.isMaximal_iff_isMaximal_disjoint
theorem isMaximal_iff_isMaximal_disjoint [H : IsJacobson R] (J : Ideal S) :
    J.IsMaximal ↔ (comap (algebraMap R S) J).IsMaximal ∧ y ∉ Ideal.comap (algebraMap R S) J 
Arguments: {R : Type u_1} (S : Type u_2) [CommRing R] [CommRing S] (y : R) [Algebra R S] [IsLocalization.Away y S] [H : Ideal.IsJacobson R] (J : Ideal S)
Goal: J.IsMaximal ↔ (Ideal.comap (algebraMap R S) J).IsMaximal ∧ y ∉ Ideal.comap (algebraMap R S) J
-- Ideal.isMaximal_of_isMaximal_disjoint
theorem isMaximal_of_isMaximal_disjoint [IsJacobson R] (I : Ideal R) (hI : I.IsMaximal)
    (hy : y ∉ I) : (map (algebraMap R S) I).IsMaximal 
Arguments: {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] (y : R) [Algebra R S] [IsLocalization.Away y S] [Ideal.IsJacobson R] (I : Ideal R) (hI : I.IsMaximal) (hy : y ∉ I)
Goal: (Ideal.map (algebraMap R S) I).IsMaximal
-- Ideal.isJacobson_localization
theorem isJacobson_localization [H : IsJacobson R] : IsJacobson S 
Arguments: {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] (y : R) [Algebra R S] [IsLocalization.Away y S] [H : Ideal.IsJacobson R]
Goal: Ideal.IsJacobson S
-- Ideal.Polynomial.Subring.mem_closure_image_of
lemma Subring.mem_closure_image_of {S T : Type*} [CommRing S] [CommRing T] (g : S →+* T)
    (u : Set S) (x : S) (hx : x ∈ Subring.closure u) : g x ∈ Subring.closure (g '' u) 
Arguments: {S : Type u_1} {T : Type u_2} [CommRing S] [CommRing T] (g : S →+* T) (u : Set S) (x : S) (hx : x ∈ Subring.closure u)
Goal: g x ∈ Subring.closure (⇑g '' u)
-- Ideal.Polynomial.mem_closure_X_union_C
lemma mem_closure_X_union_C {R : Type*} [Ring R] (p : R[X]) :
    p ∈ Subring.closure (insert X {f | f.degree ≤ 0} : Set R[X]) 
Arguments: {R : Type u_1} [Ring R] (p : Polynomial R)
Goal: p ∈ Subring.closure (insert Polynomial.X {f | f.degree ≤ 0})
-- Ideal.Polynomial.isIntegral_isLocalization_polynomial_quotient
theorem isIntegral_isLocalization_polynomial_quotient
    (P : Ideal R[X]) (pX : R[X]) (hpX : pX ∈ P) [Algebra (R ⧸ P.comap (C : R →+* R[X])) Rₘ]
    [IsLocalization.Away (pX.map (Quotient.mk (P.comap (C : R →+* R[X])))).leadingCoeff Rₘ]
    [Algebra (R[X] ⧸ P) Sₘ] [IsLocalization ((Submonoid.powers (pX.map (Quotient.mk (P.comap
      (C : R →+* R[X])))).leadingCoeff).map (quotientMap P C le_rfl) : Submonoid (R[X] ⧸ P)) Sₘ] :
    (IsLocalization.map Sₘ (quotientMap P C le_rfl) (Submonoid.powers (pX.map (Quotient.mk (P.comap
      (C : R →+* R[X])))).leadingCoeff).le_comap_map : Rₘ →+* Sₘ).IsIntegral 
Arguments: {R : Type u_1} [CommRing R] {Rₘ : Type u_3} {Sₘ : Type u_4} [CommRing Rₘ] [CommRing Sₘ] (P : Ideal (Polynomial R)) (pX : Polynomial R) (hpX : pX ∈ P) [Algebra (R ⧸ Ideal.comap Polynomial.C P) Rₘ] [IsLocalization.Away (Polynomial.map (Ideal.Quotient.mk (Ideal.comap Polynomial.C P)) pX).leadingCoeff Rₘ] [Algebra (Polynomial R ⧸ P) Sₘ] [IsLocalization
    (Submonoid.map (Ideal.quotientMap P Polynomial.C ⋯)
      (Submonoid.powers (Polynomial.map (Ideal.Quotient.mk (Ideal.comap Polynomial.C P)) pX).leadingCoeff))
    Sₘ]
Goal: (IsLocalization.map Sₘ (Ideal.quotientMap P Polynomial.C ⋯) ⋯).IsIntegral
-- Ideal.Polynomial.jacobson_bot_of_integral_localization
theorem jacobson_bot_of_integral_localization {R : Type*} [CommRing R] [IsDomain R] [IsJacobson R]
    (Rₘ Sₘ : Type*) [CommRing Rₘ] [CommRing Sₘ] (φ : R →+* S) (hφ : Function.Injective ↑φ) (x : R)
    (hx : x ≠ 0) [Algebra R Rₘ] [IsLocalization.Away x Rₘ] [Algebra S Sₘ]
    [IsLocalization ((Submonoid.powers x).map φ : Submonoid S) Sₘ]
    (hφ' :
      RingHom.IsIntegral (IsLocalization.map Sₘ φ (Submonoid.powers x).le_comap_map : Rₘ →+* Sₘ)) :
    (⊥ : Ideal S).jacobson = (⊥ : Ideal S) 
Arguments: {S : Type u_2} [CommRing S] [IsDomain S] {R : Type u_5} [CommRing R] [IsDomain R] [Ideal.IsJacobson R] (Rₘ : Type u_6) (Sₘ : Type u_7) [CommRing Rₘ] [CommRing Sₘ] (φ : R →+* S) (hφ : Function.Injective ⇑φ) (x : R) (hx : x ≠ 0) [Algebra R Rₘ] [IsLocalization.Away x Rₘ] [Algebra S Sₘ] [IsLocalization (Submonoid.map φ (Submonoid.powers x)) Sₘ] (hφ' : (IsLocalization.map Sₘ φ ⋯).IsIntegral)
Goal: ⊥.jacobson = ⊥
-- _private.Mathlib.RingTheory.Jacobson.0.Ideal.Polynomial.isJacobson_polynomial_of_domain
private theorem isJacobson_polynomial_of_domain (R : Type*) [CommRing R] [IsDomain R]
    [hR : IsJacobson R] (P : Ideal R[X]) [IsPrime P] (hP : ∀ x : R, C x ∈ P → x = 0) :
    P.jacobson = P 
Arguments: (R : Type u_5) [CommRing R] [IsDomain R] [hR : Ideal.IsJacobson R] (P : Ideal (Polynomial R)) [P.IsPrime] (hP : ∀ (x : R), Polynomial.C x ∈ P → x = 0)
Goal: P.jacobson = P
-- Ideal.Polynomial.isJacobson_polynomial_of_isJacobson
theorem isJacobson_polynomial_of_isJacobson (hR : IsJacobson R) : IsJacobson R[X] 
Arguments: {R : Type u_1} [CommRing R] (hR : Ideal.IsJacobson R)
Goal: Ideal.IsJacobson (Polynomial R)
-- Ideal.Polynomial.isJacobson_polynomial_iff_isJacobson
theorem isJacobson_polynomial_iff_isJacobson : IsJacobson R[X] ↔ IsJacobson R 
Arguments: {R : Type u_1} [CommRing R]
Goal: Ideal.IsJacobson (Polynomial R) ↔ Ideal.IsJacobson R
-- Ideal.Polynomial.isMaximal_comap_C_of_isMaximal
theorem isMaximal_comap_C_of_isMaximal [Nontrivial R] (hP' : ∀ x : R, C x ∈ P → x = 0) :
    IsMaximal (comap (C : R →+* R[X]) P : Ideal R) 
Arguments: {R : Type u_1} [CommRing R] [Ideal.IsJacobson R] (P : Ideal (Polynomial R)) [hP : P.IsMaximal] [Nontrivial R] (hP' : ∀ (x : R), Polynomial.C x ∈ P → x = 0)
Goal: (Ideal.comap Polynomial.C P).IsMaximal
-- _private.Mathlib.RingTheory.Jacobson.0.Ideal.Polynomial.quotient_mk_comp_C_isIntegral_of_jacobson'
private theorem quotient_mk_comp_C_isIntegral_of_jacobson' [Nontrivial R] (hR : IsJacobson R)
    (hP' : ∀ x : R, C x ∈ P → x = 0) : ((Quotient.mk P).comp C : R →+* R[X] ⧸ P).IsIntegral 
Arguments: {R : Type u_1} [CommRing R] (P : Ideal (Polynomial R)) [hP : P.IsMaximal] [Nontrivial R] (hR : Ideal.IsJacobson R) (hP' : ∀ (x : R), Polynomial.C x ∈ P → x = 0)
Goal: ((Ideal.Quotient.mk P).comp Polynomial.C).IsIntegral
-- Ideal.Polynomial.quotient_mk_comp_C_isIntegral_of_jacobson
theorem quotient_mk_comp_C_isIntegral_of_jacobson :
    ((Quotient.mk P).comp C : R →+* R[X] ⧸ P).IsIntegral 
Arguments: {R : Type u_1} [CommRing R] [Ideal.IsJacobson R] (P : Ideal (Polynomial R)) [hP : P.IsMaximal]
Goal: ((Ideal.Quotient.mk P).comp Polynomial.C).IsIntegral
-- Ideal.Polynomial.isMaximal_comap_C_of_isJacobson
theorem isMaximal_comap_C_of_isJacobson : (P.comap (C : R →+* R[X])).IsMaximal 
Arguments: {R : Type u_1} [CommRing R] [Ideal.IsJacobson R] (P : Ideal (Polynomial R)) [hP : P.IsMaximal]
Goal: (Ideal.comap Polynomial.C P).IsMaximal
-- Ideal.Polynomial.isMaximal_comap_C_of_isJacobson'
lemma isMaximal_comap_C_of_isJacobson' {P : Ideal R[X]} (hP : IsMaximal P) :
    (P.comap (C : R →+* R[X])).IsMaximal 
Arguments: {R : Type u_1} [CommRing R] [Ideal.IsJacobson R] {P : Ideal (Polynomial R)} (hP : P.IsMaximal)
Goal: (Ideal.comap Polynomial.C P).IsMaximal
-- Ideal.Polynomial.comp_C_integral_of_surjective_of_jacobson
theorem comp_C_integral_of_surjective_of_jacobson {S : Type*} [Field S] (f : R[X] →+* S)
    (hf : Function.Surjective ↑f) : (f.comp C).IsIntegral 
Arguments: {R : Type u_1} [CommRing R] [Ideal.IsJacobson R] {S : Type u_2} [Field S] (f : Polynomial R →+* S) (hf : Function.Surjective ⇑f)
Goal: (f.comp Polynomial.C).IsIntegral
-- Ideal.MvPolynomial.isJacobson_MvPolynomial_fin
theorem isJacobson_MvPolynomial_fin {R : Type u} [CommRing R] [H : IsJacobson R] :
    ∀ n : ℕ, IsJacobson (MvPolynomial (Fin n) R)
  
Arguments: {R : Type u} [CommRing R] [H : Ideal.IsJacobson R] (n : ℕ)
Goal: Ideal.IsJacobson (MvPolynomial (Fin n) R)
-- _private.Mathlib.RingTheory.Jacobson.0.Ideal.MvPolynomial.aux_IH
private lemma aux_IH {R : Type u} {S : Type v} {T : Type w}
  [CommRing R] [CommRing S] [CommRing T] [IsJacobson S] [Algebra R S] [Algebra R T]
  (IH : ∀ (Q : Ideal S), (IsMaximal Q) → RingHom.IsIntegral (algebraMap R (S ⧸ Q)))
  (v : S[X] ≃ₐ[R] T) (P : Ideal T) (hP : P.IsMaximal) :
  RingHom.IsIntegral (algebraMap R (T ⧸ P)) 
Arguments: {R : Type u} {S : Type v} {T : Type w} [CommRing R] [CommRing S] [CommRing T] [Ideal.IsJacobson S] [Algebra R S] [Algebra R T] (IH : ∀ (Q : Ideal S), Q.IsMaximal → (algebraMap R (S ⧸ Q)).IsIntegral) (v : Polynomial S ≃ₐ[R] T) (P : Ideal T) (hP : P.IsMaximal)
Goal: (algebraMap R (T ⧸ P)).IsIntegral
-- _private.Mathlib.RingTheory.Jacobson.0.Ideal.MvPolynomial.quotient_mk_comp_C_isIntegral_of_jacobson'
private theorem quotient_mk_comp_C_isIntegral_of_jacobson' {R : Type*} [CommRing R] [IsJacobson R]
    (P : Ideal (MvPolynomial (Fin n) R)) (hP : P.IsMaximal) :
    RingHom.IsIntegral (algebraMap R (MvPolynomial (Fin n) R ⧸ P)) 
Arguments: {n : ℕ} {R : Type u_1} [CommRing R] [Ideal.IsJacobson R] (P : Ideal (MvPolynomial (Fin n) R)) (hP : P.IsMaximal)
Goal: (algebraMap R (MvPolynomial (Fin n) R ⧸ P)).IsIntegral
-- Ideal.MvPolynomial.quotient_mk_comp_C_isIntegral_of_jacobson
theorem quotient_mk_comp_C_isIntegral_of_jacobson {R : Type*} [CommRing R] [IsJacobson R]
    (P : Ideal (MvPolynomial (Fin n) R)) [hP : P.IsMaximal] :
    RingHom.IsIntegral (RingHom.comp (Quotient.mk P) (MvPolynomial.C)) 
Arguments: {n : ℕ} {R : Type u_1} [CommRing R] [Ideal.IsJacobson R] (P : Ideal (MvPolynomial (Fin n) R)) [hP : P.IsMaximal]
Goal: ((Ideal.Quotient.mk P).comp MvPolynomial.C).IsIntegral
-- Ideal.MvPolynomial.comp_C_integral_of_surjective_of_jacobson
theorem comp_C_integral_of_surjective_of_jacobson {R : Type*} [CommRing R] [IsJacobson R]
    {σ : Type*} [Finite σ] {S : Type*} [Field S] (f : MvPolynomial σ R →+* S)
    (hf : Function.Surjective ↑f) : (f.comp C).IsIntegral 
Arguments: {R : Type u_1} [CommRing R] [Ideal.IsJacobson R] {σ : Type u_2} [Finite σ] {S : Type u_3} [Field S] (f : MvPolynomial σ R →+* S) (hf : Function.Surjective ⇑f)
Goal: (f.comp MvPolynomial.C).IsIntegral
-- IsLocalization.exists_of_eq
@[mk_iff] class IsLocalization : Prop where
  -- Porting note: add ' to fields, and made new versions of these with either `S` or `M` explicit.
    map_units' : ∀ y : M, IsUnit (algebraMap R S y)
    surj' : ∀ z : S, ∃ x : R × M, z * algebraMap R S x.2 = algebraMap R S x.1
    exists_of_eq : ∀ {x y}, algebraMap R S x = algebraMap R S y → ∃ c : M, ↑c * x = ↑c * y
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} {S : Type u_2} [CommSemiring S] [Algebra R S] [self : IsLocalization M S] {x y : R}
Goal: (algebraMap R S) x = (algebraMap R S) y → ∃ c, ↑c * x = ↑c * y
-- isLocalization_iff
@[mk_iff] class IsLocalization : Prop where
  -- Porting note: add ' to fields, and made new versions of these with either `S` or `M` explicit.
    map_units' : ∀ y : M, IsUnit (algebraMap R S y)
    surj' : ∀ z : S, ∃ x : R × M, z * algebraMap R S x.2 = algebraMap R S x.1
    exists_of_eq : ∀ {x y}, algebraMap R S x = algebraMap R S y → ∃ c : M, ↑c * x = ↑c * y
Arguments: {R : Type u_1} [CommSemiring R] (M : Submonoid R) (S : Type u_2) [CommSemiring S] [Algebra R S]
Goal: IsLocalization M S ↔
  (∀ (y : ↥M), IsUnit ((algebraMap R S) ↑y)) ∧
    (∀ (z : S), ∃ x, z * (algebraMap R S) ↑x.2 = (algebraMap R S) x.1) ∧
      ∀ {x y : R}, (algebraMap R S) x = (algebraMap R S) y → ∃ c, ↑c * x = ↑c * y
-- IsLocalization.map_units'
@[mk_iff] class IsLocalization : Prop where
  -- Porting note: add ' to fields, and made new versions of these with either `S` or `M` explicit.
    map_units' : ∀ y : M, IsUnit (algebraMap R S y)
    surj' : ∀ z : S, ∃ x : R × M, z * algebraMap R S x.2 = algebraMap R S x.1
    exists_of_eq : ∀ {x y}, algebraMap R S x = algebraMap R S y → ∃ c : M, ↑c * x = ↑c * y
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} {S : Type u_2} [CommSemiring S] [Algebra R S] [self : IsLocalization M S] (y : ↥M)
Goal: IsUnit ((algebraMap R S) ↑y)
-- IsLocalization.surj'
@[mk_iff] class IsLocalization : Prop where
  -- Porting note: add ' to fields, and made new versions of these with either `S` or `M` explicit.
    map_units' : ∀ y : M, IsUnit (algebraMap R S y)
    surj' : ∀ z : S, ∃ x : R × M, z * algebraMap R S x.2 = algebraMap R S x.1
    exists_of_eq : ∀ {x y}, algebraMap R S x = algebraMap R S y → ∃ c : M, ↑c * x = ↑c * y
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} {S : Type u_2} [CommSemiring S] [Algebra R S] [self : IsLocalization M S] (z : S)
Goal: ∃ x, z * (algebraMap R S) ↑x.2 = (algebraMap R S) x.1
-- IsLocalization.map_units
@[inherit_doc IsLocalization.map_units']
theorem map_units : ∀ y : M, IsUnit (algebraMap R S y) 
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} (S : Type u_2) [CommSemiring S] [Algebra R S] [IsLocalization M S] (y : ↥M)
Goal: IsUnit ((algebraMap R S) ↑y)
-- IsLocalization.surj
@[inherit_doc IsLocalization.surj']
theorem surj : ∀ z : S, ∃ x : R × M, z * algebraMap R S x.2 = algebraMap R S x.1 
Arguments: {R : Type u_1} [CommSemiring R] (M : Submonoid R) {S : Type u_2} [CommSemiring S] [Algebra R S] [IsLocalization M S] (z : S)
Goal: ∃ x, z * (algebraMap R S) ↑x.2 = (algebraMap R S) x.1