ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- IsLocalization.to_map_ne_zero_of_mem_nonZeroDivisors
protected theorem to_map_ne_zero_of_mem_nonZeroDivisors [Nontrivial R] (hM : M ≤ nonZeroDivisors R)
    {x : R} (hx : x ∈ nonZeroDivisors R) : algebraMap R S x ≠ 0 
Arguments: {R : Type u_1} [CommRing R] {M : Submonoid R} (S : Type u_2) [CommRing S] [Algebra R S] [IsLocalization M S] [Nontrivial R] (hM : M ≤ nonZeroDivisors R) {x : R} (hx : x ∈ nonZeroDivisors R)
Goal: (algebraMap R S) x ≠ 0
-- IsLocalization.sec_snd_ne_zero
theorem sec_snd_ne_zero [Nontrivial R] (hM : M ≤ nonZeroDivisors R) (x : S) :
    ((sec M x).snd : R) ≠ 0 
Arguments: {R : Type u_1} [CommRing R] {M : Submonoid R} {S : Type u_2} [CommRing S] [Algebra R S] [IsLocalization M S] [Nontrivial R] (hM : M ≤ nonZeroDivisors R) (x : S)
Goal: ↑(IsLocalization.sec M x).2 ≠ 0
-- IsLocalization.sec_fst_ne_zero
theorem sec_fst_ne_zero [Nontrivial R] [NoZeroDivisors S] (hM : M ≤ nonZeroDivisors R) {x : S}
    (hx : x ≠ 0) : (sec M x).fst ≠ 0 
Arguments: {R : Type u_1} [CommRing R] {M : Submonoid R} {S : Type u_2} [CommRing S] [Algebra R S] [IsLocalization M S] [Nontrivial R] [NoZeroDivisors S] (hM : M ≤ nonZeroDivisors R) {x : S} (hx : x ≠ 0)
Goal: (IsLocalization.sec M x).1 ≠ 0
-- IsLocalization.noZeroDivisors_of_le_nonZeroDivisors
theorem noZeroDivisors_of_le_nonZeroDivisors [Algebra A S] {M : Submonoid A} [IsLocalization M S]
    (hM : M ≤ nonZeroDivisors A) : NoZeroDivisors S 
Arguments: {S : Type u_2} [CommRing S] (A : Type u_6) [CommRing A] [IsDomain A] [Algebra A S] {M : Submonoid A} [IsLocalization M S] (hM : M ≤ nonZeroDivisors A)
Goal: NoZeroDivisors S
-- IsLocalization.isDomain_of_le_nonZeroDivisors
theorem isDomain_of_le_nonZeroDivisors [Algebra A S] {M : Submonoid A} [IsLocalization M S]
    (hM : M ≤ nonZeroDivisors A) : IsDomain S 
Arguments: {S : Type u_2} [CommRing S] (A : Type u_6) [CommRing A] [IsDomain A] [Algebra A S] {M : Submonoid A} [IsLocalization M S] (hM : M ≤ nonZeroDivisors A)
Goal: IsDomain S
-- IsLocalization.isDomain_localization
theorem isDomain_localization {M : Submonoid A} (hM : M ≤ nonZeroDivisors A) :
    IsDomain (Localization M) 
Arguments: {A : Type u_6} [CommRing A] [IsDomain A] {M : Submonoid A} (hM : M ≤ nonZeroDivisors A)
Goal: IsDomain (Localization M)
-- IsField.localization_map_bijective
theorem IsField.localization_map_bijective {R Rₘ : Type*} [CommRing R] [CommRing Rₘ]
    {M : Submonoid R} (hM : (0 : R) ∉ M) (hR : IsField R) [Algebra R Rₘ] [IsLocalization M Rₘ] :
    Function.Bijective (algebraMap R Rₘ) 
Arguments: {R : Type u_4} {Rₘ : Type u_5} [CommRing R] [CommRing Rₘ] {M : Submonoid R} (hM : 0 ∉ M) (hR : IsField R) [Algebra R Rₘ] [IsLocalization M Rₘ]
Goal: Function.Bijective ⇑(algebraMap R Rₘ)
-- Field.localization_map_bijective
theorem Field.localization_map_bijective {K Kₘ : Type*} [Field K] [CommRing Kₘ] {M : Submonoid K}
    (hM : (0 : K) ∉ M) [Algebra K Kₘ] [IsLocalization M Kₘ] :
    Function.Bijective (algebraMap K Kₘ) 
Arguments: {K : Type u_4} {Kₘ : Type u_5} [Field K] [CommRing Kₘ] {M : Submonoid K} (hM : 0 ∉ M) [Algebra K Kₘ] [IsLocalization M Kₘ]
Goal: Function.Bijective ⇑(algebraMap K Kₘ)
-- IsLocalization.map_units_map_submonoid
theorem IsLocalization.map_units_map_submonoid
    [IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ] (y : M) :
    IsUnit (algebraMap R Sₘ y) 
Arguments: {R : Type u_1} [CommRing R] {M : Submonoid R} (S : Type u_2) [CommRing S] [Algebra R S] (Sₘ : Type u_5) [CommRing Sₘ] [Algebra S Sₘ] [Algebra R Sₘ] [IsScalarTower R S Sₘ] [IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ] (y : ↥M)
Goal: IsUnit ((algebraMap R Sₘ) ↑y)
-- IsLocalization.algebraMap_mk'
theorem IsLocalization.algebraMap_mk' (x : R) (y : M) :
    algebraMap Rₘ Sₘ (IsLocalization.mk' Rₘ x y) =
      IsLocalization.mk' Sₘ (algebraMap R S x)
        ⟨algebraMap R S y, Algebra.mem_algebraMapSubmonoid_of_mem y⟩ 
Arguments: {R : Type u_1} [CommRing R] {M : Submonoid R} (S : Type u_2) [CommRing S] [Algebra R S] (Rₘ : Type u_4) (Sₘ : Type u_5) [CommRing Rₘ] [CommRing Sₘ] [Algebra R Rₘ] [IsLocalization M Rₘ] [Algebra S Sₘ] [i : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ] [Algebra Rₘ Sₘ] [Algebra R Sₘ] [IsScalarTower R Rₘ Sₘ] [IsScalarTower R S Sₘ] (x : R) (y : ↥M)
Goal: (algebraMap Rₘ Sₘ) (IsLocalization.mk' Rₘ x y) = IsLocalization.mk' Sₘ ((algebraMap R S) x) ⟨(algebraMap R S) ↑y, ⋯⟩
-- IsLocalization.algebraMap_eq_map_map_submonoid
theorem IsLocalization.algebraMap_eq_map_map_submonoid :
    algebraMap Rₘ Sₘ =
      map Sₘ (algebraMap R S)
        (show _ ≤ (Algebra.algebraMapSubmonoid S M).comap _ from M.le_comap_map) 
Arguments: {R : Type u_1} [CommRing R] (M : Submonoid R) (S : Type u_2) [CommRing S] [Algebra R S] (Rₘ : Type u_4) (Sₘ : Type u_5) [CommRing Rₘ] [CommRing Sₘ] [Algebra R Rₘ] [IsLocalization M Rₘ] [Algebra S Sₘ] [i : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ] [Algebra Rₘ Sₘ] [Algebra R Sₘ] [IsScalarTower R Rₘ Sₘ] [IsScalarTower R S Sₘ]
Goal: algebraMap Rₘ Sₘ = IsLocalization.map Sₘ (algebraMap R S) ⋯
-- IsLocalization.algebraMap_apply_eq_map_map_submonoid
theorem IsLocalization.algebraMap_apply_eq_map_map_submonoid (x) :
    algebraMap Rₘ Sₘ x =
      map Sₘ (algebraMap R S)
        (show _ ≤ (Algebra.algebraMapSubmonoid S M).comap _ from M.le_comap_map) x 
Arguments: {R : Type u_1} [CommRing R] (M : Submonoid R) (S : Type u_2) [CommRing S] [Algebra R S] (Rₘ : Type u_4) (Sₘ : Type u_5) [CommRing Rₘ] [CommRing Sₘ] [Algebra R Rₘ] [IsLocalization M Rₘ] [Algebra S Sₘ] [i : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ] [Algebra Rₘ Sₘ] [Algebra R Sₘ] [IsScalarTower R Rₘ Sₘ] [IsScalarTower R S Sₘ] (x : Rₘ)
Goal: (algebraMap Rₘ Sₘ) x = (IsLocalization.map Sₘ (algebraMap R S) ⋯) x
-- IsLocalization.lift_algebraMap_eq_algebraMap
theorem IsLocalization.lift_algebraMap_eq_algebraMap :
    IsLocalization.lift (M := M) (IsLocalization.map_units_map_submonoid S Sₘ) =
      algebraMap Rₘ Sₘ 
Arguments: {R : Type u_1} [CommRing R] (M : Submonoid R) (S : Type u_2) [CommRing S] [Algebra R S] (Rₘ : Type u_4) (Sₘ : Type u_5) [CommRing Rₘ] [CommRing Sₘ] [Algebra R Rₘ] [IsLocalization M Rₘ] [Algebra S Sₘ] [i : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ] [Algebra Rₘ Sₘ] [Algebra R Sₘ] [IsScalarTower R Rₘ Sₘ] [IsScalarTower R S Sₘ]
Goal: IsLocalization.lift ⋯ = algebraMap Rₘ Sₘ
-- localizationAlgebra_injective
theorem localizationAlgebra_injective (hRS : Function.Injective (algebraMap R S)) :
    Function.Injective (@algebraMap Rₘ Sₘ _ _ (localizationAlgebra M S)) 
Arguments: {R : Type u_1} [CommRing R] {M : Submonoid R} {S : Type u_2} [CommRing S] [Algebra R S] (Rₘ : Type u_4) (Sₘ : Type u_5) [CommRing Rₘ] [CommRing Sₘ] [Algebra R Rₘ] [IsLocalization M Rₘ] [Algebra S Sₘ] [i : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ] (hRS : Function.Injective ⇑(algebraMap R S))
Goal: Function.Injective ⇑(algebraMap Rₘ Sₘ)
-- IsLocalization.coeffIntegerNormalization_of_not_mem_support
theorem coeffIntegerNormalization_of_not_mem_support (p : S[X]) (i : ℕ) (h : coeff p i = 0) :
    coeffIntegerNormalization M p i = 0 
Arguments: {R : Type u_1} [CommRing R] (M : Submonoid R) {S : Type u_2} [CommRing S] [Algebra R S] [IsLocalization M S] (p : S[X]) (i : ℕ) (h : p.coeff i = 0)
Goal: IsLocalization.coeffIntegerNormalization M p i = 0
-- IsLocalization.coeffIntegerNormalization_mem_support
theorem coeffIntegerNormalization_mem_support (p : S[X]) (i : ℕ)
    (h : coeffIntegerNormalization M p i ≠ 0) : i ∈ p.support 
Arguments: {R : Type u_1} [CommRing R] (M : Submonoid R) {S : Type u_2} [CommRing S] [Algebra R S] [IsLocalization M S] (p : S[X]) (i : ℕ) (h : IsLocalization.coeffIntegerNormalization M p i ≠ 0)
Goal: i ∈ p.support
-- IsLocalization.integerNormalization_coeff
@[simp]
theorem integerNormalization_coeff (p : S[X]) (i : ℕ) :
    (integerNormalization M p).coeff i = coeffIntegerNormalization M p i 
Arguments: {R : Type u_1} [CommRing R] (M : Submonoid R) {S : Type u_2} [CommRing S] [Algebra R S] [IsLocalization M S] (p : S[X]) (i : ℕ)
Goal: (IsLocalization.integerNormalization M p).coeff i = IsLocalization.coeffIntegerNormalization M p i
-- IsLocalization.integerNormalization_spec
theorem integerNormalization_spec (p : S[X]) :
    ∃ b : M, ∀ i, algebraMap R S ((integerNormalization M p).coeff i) = (b : R) • p.coeff i 
Arguments: {R : Type u_1} [CommRing R] (M : Submonoid R) {S : Type u_2} [CommRing S] [Algebra R S] [IsLocalization M S] (p : S[X])
Goal: ∃ b, ∀ (i : ℕ), (algebraMap R S) ((IsLocalization.integerNormalization M p).coeff i) = ↑b • p.coeff i
-- IsLocalization.integerNormalization_map_to_map
theorem integerNormalization_map_to_map (p : S[X]) :
    ∃ b : M, (integerNormalization M p).map (algebraMap R S) = (b : R) • p 
Arguments: {R : Type u_1} [CommRing R] (M : Submonoid R) {S : Type u_2} [CommRing S] [Algebra R S] [IsLocalization M S] (p : S[X])
Goal: ∃ b, Polynomial.map (algebraMap R S) (IsLocalization.integerNormalization M p) = ↑b • p
-- IsLocalization.integerNormalization_eval₂_eq_zero
theorem integerNormalization_eval₂_eq_zero (g : S →+* R') (p : S[X]) {x : R'}
    (hx : eval₂ g x p = 0) : eval₂ (g.comp (algebraMap R S)) x (integerNormalization M p) = 0 
Arguments: {R : Type u_1} [CommRing R] (M : Submonoid R) {S : Type u_2} [CommRing S] [Algebra R S] [IsLocalization M S] {R' : Type u_4} [CommRing R'] (g : S →+* R') (p : S[X]) {x : R'} (hx : Polynomial.eval₂ g x p = 0)
Goal: Polynomial.eval₂ (g.comp (algebraMap R S)) x (IsLocalization.integerNormalization M p) = 0
-- IsLocalization.integerNormalization_aeval_eq_zero
theorem integerNormalization_aeval_eq_zero [Algebra R R'] [Algebra S R'] [IsScalarTower R S R']
    (p : S[X]) {x : R'} (hx : aeval x p = 0) : aeval x (integerNormalization M p) = 0 
Arguments: {R : Type u_1} [CommRing R] (M : Submonoid R) {S : Type u_2} [CommRing S] [Algebra R S] [IsLocalization M S] {R' : Type u_4} [CommRing R'] [Algebra R R'] [Algebra S R'] [IsScalarTower R S R'] (p : S[X]) {x : R'} (hx : (Polynomial.aeval x) p = 0)
Goal: (Polynomial.aeval x) (IsLocalization.integerNormalization M p) = 0
-- IsFractionRing.integerNormalization_eq_zero_iff
theorem integerNormalization_eq_zero_iff {p : K[X]} :
    integerNormalization (nonZeroDivisors A) p = 0 ↔ p = 0 
Arguments: {A : Type u_4} {K : Type u_5} [CommRing A] [IsDomain A] [Field K] [Algebra A K] [IsFractionRing A K] {p : K[X]}
Goal: IsLocalization.integerNormalization (nonZeroDivisors A) p = 0 ↔ p = 0
-- IsFractionRing.isAlgebraic_iff
theorem isAlgebraic_iff [Algebra A C] [Algebra K C] [IsScalarTower A K C] {x : C} :
    IsAlgebraic A x ↔ IsAlgebraic K x 
Arguments: (A : Type u_4) (K : Type u_5) (C : Type u_6) [CommRing A] [IsDomain A] [Field K] [Algebra A K] [IsFractionRing A K] [CommRing C] [Algebra A C] [Algebra K C] [IsScalarTower A K C] {x : C}
Goal: IsAlgebraic A x ↔ IsAlgebraic K x
-- IsFractionRing.comap_isAlgebraic_iff
theorem comap_isAlgebraic_iff [Algebra A C] [Algebra K C] [IsScalarTower A K C] :
    Algebra.IsAlgebraic A C ↔ Algebra.IsAlgebraic K C 
Arguments: {A : Type u_4} {K : Type u_5} {C : Type u_6} [CommRing A] [IsDomain A] [Field K] [Algebra A K] [IsFractionRing A K] [CommRing C] [Algebra A C] [Algebra K C] [IsScalarTower A K C]
Goal: Algebra.IsAlgebraic A C ↔ Algebra.IsAlgebraic K C
-- RingHom.isIntegralElem_localization_at_leadingCoeff
theorem RingHom.isIntegralElem_localization_at_leadingCoeff {R S : Type*} [CommRing R] [CommRing S]
    (f : R →+* S) (x : S) (p : R[X]) (hf : p.eval₂ f x = 0) (M : Submonoid R)
    (hM : p.leadingCoeff ∈ M) {Rₘ Sₘ : Type*} [CommRing Rₘ] [CommRing Sₘ] [Algebra R Rₘ]
    [IsLocalization M Rₘ] [Algebra S Sₘ] [IsLocalization (M.map f : Submonoid S) Sₘ] :
    (map Sₘ f M.le_comap_map : Rₘ →+* _).IsIntegralElem (algebraMap S Sₘ x) 
Arguments: {R : Type u_6} {S : Type u_7} [CommRing R] [CommRing S] (f : R →+* S) (x : S) (p : R[X]) (hf : Polynomial.eval₂ f x p = 0) (M : Submonoid R) (hM : p.leadingCoeff ∈ M) {Rₘ : Type u_8} {Sₘ : Type u_9} [CommRing Rₘ] [CommRing Sₘ] [Algebra R Rₘ] [IsLocalization M Rₘ] [Algebra S Sₘ] [IsLocalization (Submonoid.map f M) Sₘ]
Goal: (IsLocalization.map Sₘ f ⋯).IsIntegralElem ((algebraMap S Sₘ) x)
-- is_integral_localization_at_leadingCoeff
theorem is_integral_localization_at_leadingCoeff {x : S} (p : R[X]) (hp : aeval x p = 0)
    (hM : p.leadingCoeff ∈ M) :
    (map Sₘ (algebraMap R S)
            (show _ ≤ (Algebra.algebraMapSubmonoid S M).comap _ from M.le_comap_map) :
          Rₘ →+* _).IsIntegralElem
      (algebraMap S Sₘ x) 
Arguments: {R : Type u_1} [CommRing R] {M : Submonoid R} {S : Type u_2} [CommRing S] [Algebra R S] {Rₘ : Type u_4} {Sₘ : Type u_5} [CommRing Rₘ] [CommRing Sₘ] [Algebra R Rₘ] [IsLocalization M Rₘ] [Algebra S Sₘ] [IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ] {x : S} (p : R[X]) (hp : (Polynomial.aeval x) p = 0) (hM : p.leadingCoeff ∈ M)
Goal: (IsLocalization.map Sₘ (algebraMap R S) ⋯).IsIntegralElem ((algebraMap S Sₘ) x)
-- isIntegral_localization
theorem isIntegral_localization [Algebra.IsIntegral R S] :
    (map Sₘ (algebraMap R S)
          (show _ ≤ (Algebra.algebraMapSubmonoid S M).comap _ from M.le_comap_map) :
        Rₘ →+* _).IsIntegral 
Arguments: {R : Type u_1} [CommRing R] {M : Submonoid R} {S : Type u_2} [CommRing S] [Algebra R S] {Rₘ : Type u_4} {Sₘ : Type u_5} [CommRing Rₘ] [CommRing Sₘ] [Algebra R Rₘ] [IsLocalization M Rₘ] [Algebra S Sₘ] [IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ] [Algebra.IsIntegral R S]
Goal: (IsLocalization.map Sₘ (algebraMap R S) ⋯).IsIntegral
-- isIntegral_localization'
@[nolint unusedHavesSuffices] -- It claims the `have : IsLocalization` line is unnecessary,
                              -- but remove it and the proof won't work.
theorem isIntegral_localization' {R S : Type*} [CommRing R] [CommRing S] {f : R →+* S}
    (hf : f.IsIntegral) (M : Submonoid R) :
    (map (Localization (M.map (f : R →* S))) f
          (M.le_comap_map : _ ≤ Submonoid.comap (f : R →* S) _) :
        Localization M →+* _).IsIntegral 
Arguments: {R : Type u_6} {S : Type u_7} [CommRing R] [CommRing S] {f : R →+* S} (hf : f.IsIntegral) (M : Submonoid R)
Goal: (IsLocalization.map (Localization (Submonoid.map (↑f) M)) f ⋯).IsIntegral
-- IsLocalization.scaleRoots_commonDenom_mem_lifts
theorem IsLocalization.scaleRoots_commonDenom_mem_lifts (p : Rₘ[X])
    (hp : p.leadingCoeff ∈ (algebraMap R Rₘ).range) :
    p.scaleRoots (algebraMap R Rₘ <| IsLocalization.commonDenom M p.support p.coeff) ∈
      Polynomial.lifts (algebraMap R Rₘ) 
Arguments: {R : Type u_1} [CommRing R] (M : Submonoid R) {Rₘ : Type u_4} [CommRing Rₘ] [Algebra R Rₘ] [IsLocalization M Rₘ] (p : Rₘ[X]) (hp : p.leadingCoeff ∈ (algebraMap R Rₘ).range)
Goal: p.scaleRoots ((algebraMap R Rₘ) ↑(IsLocalization.commonDenom M p.support p.coeff)) ∈ Polynomial.lifts (algebraMap R Rₘ)
-- IsIntegral.exists_multiple_integral_of_isLocalization
theorem IsIntegral.exists_multiple_integral_of_isLocalization [Algebra Rₘ S] [IsScalarTower R Rₘ S]
    (x : S) (hx : IsIntegral Rₘ x) : ∃ m : M, IsIntegral R (m • x) 
Arguments: {R : Type u_1} [CommRing R] (M : Submonoid R) {S : Type u_2} [CommRing S] [Algebra R S] {Rₘ : Type u_4} [CommRing Rₘ] [Algebra R Rₘ] [IsLocalization M Rₘ] [Algebra Rₘ S] [IsScalarTower R Rₘ S] (x : S) (hx : IsIntegral Rₘ x)
Goal: ∃ m, IsIntegral R (m • x)
-- IsIntegralClosure.isFractionRing_of_algebraic
theorem isFractionRing_of_algebraic [Algebra.IsAlgebraic A L]
    (inj : ∀ x, algebraMap A L x = 0 → x = 0) : IsFractionRing C L 
Arguments: (A : Type u_4) [CommRing A] [IsDomain A] {L : Type u_6} [Field L] [Algebra A L] (C : Type u_7) [CommRing C] [IsDomain C] [Algebra C L] [IsIntegralClosure C A L] [Algebra A C] [IsScalarTower A C L] [Algebra.IsAlgebraic A L] (inj : ∀ (x : A), (algebraMap A L) x = 0 → x = 0)
Goal: IsFractionRing C L
-- IsIntegralClosure.isFractionRing_of_finite_extension
theorem isFractionRing_of_finite_extension [Algebra K L] [IsScalarTower A K L]
    [FiniteDimensional K L] : IsFractionRing C L 
Arguments: (A : Type u_4) (K : Type u_5) [CommRing A] [IsDomain A] (L : Type u_6) [Field K] [Field L] [Algebra A K] [Algebra A L] [IsFractionRing A K] (C : Type u_7) [CommRing C] [IsDomain C] [Algebra C L] [IsIntegralClosure C A L] [Algebra A C] [IsScalarTower A C L] [Algebra K L] [IsScalarTower A K L] [FiniteDimensional K L]
Goal: IsFractionRing C L
-- integralClosure.isFractionRing_of_algebraic
theorem isFractionRing_of_algebraic [Algebra A L] [Algebra.IsAlgebraic A L]
    (inj : ∀ x, algebraMap A L x = 0 → x = 0) : IsFractionRing (integralClosure A L) L 
Arguments: {A : Type u_4} [CommRing A] [IsDomain A] {L : Type u_6} [Field L] [Algebra A L] [Algebra.IsAlgebraic A L] (inj : ∀ (x : A), (algebraMap A L) x = 0 → x = 0)
Goal: IsFractionRing (↥(integralClosure A L)) L
-- integralClosure.isFractionRing_of_finite_extension
theorem isFractionRing_of_finite_extension [Algebra A L] [Algebra K L] [IsScalarTower A K L]
    [FiniteDimensional K L] : IsFractionRing (integralClosure A L) L 
Arguments: {A : Type u_4} (K : Type u_5) [CommRing A] [IsDomain A] (L : Type u_6) [Field K] [Field L] [Algebra A K] [IsFractionRing A K] [Algebra A L] [Algebra K L] [IsScalarTower A K L] [FiniteDimensional K L]
Goal: IsFractionRing (↥(integralClosure A L)) L
-- IsFractionRing.isAlgebraic_iff'
theorem isAlgebraic_iff' [Field K] [IsDomain R] [IsDomain S] [Algebra R K] [Algebra S K]
    [NoZeroSMulDivisors R K] [IsFractionRing S K] [IsScalarTower R S K] :
    Algebra.IsAlgebraic R S ↔ Algebra.IsAlgebraic R K 
Arguments: (R : Type u_1) [CommRing R] (S : Type u_2) [CommRing S] [Algebra R S] (K : Type u_5) [Field K] [IsDomain R] [IsDomain S] [Algebra R K] [Algebra S K] [NoZeroSMulDivisors R K] [IsFractionRing S K] [IsScalarTower R S K]
Goal: Algebra.IsAlgebraic R S ↔ Algebra.IsAlgebraic R K
-- IsFractionRing.ideal_span_singleton_map_subset
theorem ideal_span_singleton_map_subset {L : Type*} [IsDomain R] [IsDomain S] [Field K] [Field L]
    [Algebra R K] [Algebra R L] [Algebra S L] [IsIntegralClosure S R L] [IsFractionRing S L]
    [Algebra K L] [IsScalarTower R S L] [IsScalarTower R K L] {a : S} {b : Set S}
    [Algebra.IsAlgebraic R L] (inj : Function.Injective (algebraMap R L))
    (h : (Ideal.span ({a} : Set S) : Set S) ⊆ Submodule.span R b) :
    (Ideal.span ({algebraMap S L a} : Set L) : Set L) ⊆ Submodule.span K (algebraMap S L '' b) 
Arguments: (R : Type u_1) [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] {K : Type u_5} {L : Type u_6} [IsDomain R] [IsDomain S] [Field K] [Field L] [Algebra R K] [Algebra R L] [Algebra S L] [IsIntegralClosure S R L] [IsFractionRing S L] [Algebra K L] [IsScalarTower R S L] [IsScalarTower R K L] {a : S} {b : Set S} [Algebra.IsAlgebraic R L] (inj : Function.Injective ⇑(algebraMap R L)) (h : ↑(Ideal.span {a}) ⊆ ↑(Submodule.span R b))
Goal: ↑(Ideal.span {(algebraMap S L) a}) ⊆ ↑(Submodule.span K (⇑(algebraMap S L) '' b))
-- isAlgebraic_of_isLocalization
lemma isAlgebraic_of_isLocalization {R} [CommRing R] (M : Submonoid R) (S) [CommRing S]
    [Nontrivial R] [Algebra R S] [IsLocalization M S] : Algebra.IsAlgebraic R S 
Arguments: {R : Type u_6} [CommRing R] (M : Submonoid R) (S : Type u_7) [CommRing S] [Nontrivial R] [Algebra R S] [IsLocalization M S]
Goal: Algebra.IsAlgebraic R S
-- isAlgebraic_of_isFractionRing
lemma isAlgebraic_of_isFractionRing {R S} (K L) [CommRing R] [CommRing S] [Field K] [CommRing L]
    [Algebra R S] [Algebra R K] [Algebra R L] [Algebra S L] [Algebra K L] [IsScalarTower R S L]
    [IsScalarTower R K L] [IsFractionRing S L]
    [Algebra.IsIntegral R S] : Algebra.IsAlgebraic K L 
Arguments: {R : Type u_6} {S : Type u_7} (K : Type u_8) (L : Type u_9) [CommRing R] [CommRing S] [Field K] [CommRing L] [Algebra R S] [Algebra R K] [Algebra R L] [Algebra S L] [Algebra K L] [IsScalarTower R S L] [IsScalarTower R K L] [IsFractionRing S L] [Algebra.IsIntegral R S]
Goal: Algebra.IsAlgebraic K L
-- PrimeSpectrum.isPrime
@[ext]
structure PrimeSpectrum [CommSemiring R] where
  asIdeal : Ideal R
  isPrime : asIdeal.IsPrime
Arguments: {R : Type u} [CommSemiring R] (self : PrimeSpectrum R)
Goal: self.asIdeal.IsPrime
-- PrimeSpectrum.primeSpectrumProd_symm_inl_asIdeal
@[simp]
theorem primeSpectrumProd_symm_inl_asIdeal (x : PrimeSpectrum R) :
    ((primeSpectrumProd R S).symm <| Sum.inl x).asIdeal = Ideal.prod x.asIdeal ⊤ 
Arguments: {R : Type u} {S : Type v} [CommSemiring R] [CommSemiring S] (x : PrimeSpectrum R)
Goal: ((PrimeSpectrum.primeSpectrumProd R S).symm (Sum.inl x)).asIdeal = x.asIdeal.prod ⊤
-- PrimeSpectrum.primeSpectrumProd_symm_inr_asIdeal
@[simp]
theorem primeSpectrumProd_symm_inr_asIdeal (x : PrimeSpectrum S) :
    ((primeSpectrumProd R S).symm <| Sum.inr x).asIdeal = Ideal.prod ⊤ x.asIdeal 
Arguments: {R : Type u} {S : Type v} [CommSemiring R] [CommSemiring S] (x : PrimeSpectrum S)
Goal: ((PrimeSpectrum.primeSpectrumProd R S).symm (Sum.inr x)).asIdeal = ⊤.prod x.asIdeal
-- PrimeSpectrum.mem_zeroLocus
@[simp]
theorem mem_zeroLocus (x : PrimeSpectrum R) (s : Set R) : x ∈ zeroLocus s ↔ s ⊆ x.asIdeal 
Arguments: {R : Type u} [CommSemiring R] (x : PrimeSpectrum R) (s : Set R)
Goal: x ∈ PrimeSpectrum.zeroLocus s ↔ s ⊆ ↑x.asIdeal
-- PrimeSpectrum.zeroLocus_span
@[simp]
theorem zeroLocus_span (s : Set R) : zeroLocus (Ideal.span s : Set R) = zeroLocus s 
Arguments: {R : Type u} [CommSemiring R] (s : Set R)
Goal: PrimeSpectrum.zeroLocus ↑(Ideal.span s) = PrimeSpectrum.zeroLocus s
-- PrimeSpectrum.coe_vanishingIdeal
theorem coe_vanishingIdeal (t : Set (PrimeSpectrum R)) :
    (vanishingIdeal t : Set R) = { f : R | ∀ x ∈ t, f ∈ x.asIdeal } 
Arguments: {R : Type u} [CommSemiring R] (t : Set (PrimeSpectrum R))
Goal: ↑(PrimeSpectrum.vanishingIdeal t) = {f | ∀ x ∈ t, f ∈ x.asIdeal}
-- PrimeSpectrum.mem_vanishingIdeal
theorem mem_vanishingIdeal (t : Set (PrimeSpectrum R)) (f : R) :
    f ∈ vanishingIdeal t ↔ ∀ x ∈ t, f ∈ x.asIdeal 
Arguments: {R : Type u} [CommSemiring R] (t : Set (PrimeSpectrum R)) (f : R)
Goal: f ∈ PrimeSpectrum.vanishingIdeal t ↔ ∀ x ∈ t, f ∈ x.asIdeal
-- PrimeSpectrum.vanishingIdeal_singleton
@[simp]
theorem vanishingIdeal_singleton (x : PrimeSpectrum R) :
    vanishingIdeal ({x} : Set (PrimeSpectrum R)) = x.asIdeal 
Arguments: {R : Type u} [CommSemiring R] (x : PrimeSpectrum R)
Goal: PrimeSpectrum.vanishingIdeal {x} = x.asIdeal
-- PrimeSpectrum.subset_zeroLocus_iff_le_vanishingIdeal
theorem subset_zeroLocus_iff_le_vanishingIdeal (t : Set (PrimeSpectrum R)) (I : Ideal R) :
    t ⊆ zeroLocus I ↔ I ≤ vanishingIdeal t 
Arguments: {R : Type u} [CommSemiring R] (t : Set (PrimeSpectrum R)) (I : Ideal R)
Goal: t ⊆ PrimeSpectrum.zeroLocus ↑I ↔ I ≤ PrimeSpectrum.vanishingIdeal t
-- PrimeSpectrum.gc
theorem gc :
    @GaloisConnection (Ideal R) (Set (PrimeSpectrum R))ᵒᵈ _ _ (fun I => zeroLocus I) fun t =>
      vanishingIdeal t 
Arguments: (R : Type u) [CommSemiring R]
Goal: GaloisConnection (fun I => PrimeSpectrum.zeroLocus ↑I) fun t => PrimeSpectrum.vanishingIdeal t
-- PrimeSpectrum.gc_set
theorem gc_set :
    @GaloisConnection (Set R) (Set (PrimeSpectrum R))ᵒᵈ _ _ (fun s => zeroLocus s) fun t =>
      vanishingIdeal t 
Arguments: (R : Type u) [CommSemiring R]
Goal: GaloisConnection (fun s => PrimeSpectrum.zeroLocus s) fun t => ↑(PrimeSpectrum.vanishingIdeal t)
-- PrimeSpectrum.subset_zeroLocus_iff_subset_vanishingIdeal
theorem subset_zeroLocus_iff_subset_vanishingIdeal (t : Set (PrimeSpectrum R)) (s : Set R) :
    t ⊆ zeroLocus s ↔ s ⊆ vanishingIdeal t 
Arguments: (R : Type u) [CommSemiring R] (t : Set (PrimeSpectrum R)) (s : Set R)
Goal: t ⊆ PrimeSpectrum.zeroLocus s ↔ s ⊆ ↑(PrimeSpectrum.vanishingIdeal t)
-- PrimeSpectrum.subset_vanishingIdeal_zeroLocus
theorem subset_vanishingIdeal_zeroLocus (s : Set R) : s ⊆ vanishingIdeal (zeroLocus s) 
Arguments: {R : Type u} [CommSemiring R] (s : Set R)
Goal: s ⊆ ↑(PrimeSpectrum.vanishingIdeal (PrimeSpectrum.zeroLocus s))
-- PrimeSpectrum.le_vanishingIdeal_zeroLocus
theorem le_vanishingIdeal_zeroLocus (I : Ideal R) : I ≤ vanishingIdeal (zeroLocus I) 
Arguments: {R : Type u} [CommSemiring R] (I : Ideal R)
Goal: I ≤ PrimeSpectrum.vanishingIdeal (PrimeSpectrum.zeroLocus ↑I)
-- PrimeSpectrum.vanishingIdeal_zeroLocus_eq_radical
@[simp]
theorem vanishingIdeal_zeroLocus_eq_radical (I : Ideal R) :
    vanishingIdeal (zeroLocus (I : Set R)) = I.radical 
Arguments: {R : Type u} [CommSemiring R] (I : Ideal R)
Goal: PrimeSpectrum.vanishingIdeal (PrimeSpectrum.zeroLocus ↑I) = I.radical