ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- RingHom.finitePresentation_ofLocalizationSpanTarget_aux
lemma finitePresentation_ofLocalizationSpanTarget_aux
    {R S A : Type*} [CommRing R] [CommRing S] [CommRing A] [Algebra R S] [Algebra R A]
    [Algebra.FinitePresentation R A] (f : A →ₐ[R] S) (hf : Function.Surjective f)
    (t : Finset A) (ht : Ideal.span (t : Set A) = ⊤)
    (H : ∀ g : t, Algebra.FinitePresentation R (Localization.Away (f g))) :
    Algebra.FinitePresentation R S 
Arguments: {R : Type u_1} {S : Type u_2} {A : Type u_3} [CommRing R] [CommRing S] [CommRing A] [Algebra R S] [Algebra R A] [Algebra.FinitePresentation R A] (f : A →ₐ[R] S) (hf : Function.Surjective ⇑f) (t : Finset A) (ht : Ideal.span ↑t = ⊤) (H : ∀ (g : { x // x ∈ t }), Algebra.FinitePresentation R (Localization.Away (f ↑g)))
Goal: Algebra.FinitePresentation R S
-- RingHom.finitePresentation_ofLocalizationSpanTarget
theorem finitePresentation_ofLocalizationSpanTarget :
    OfLocalizationSpanTarget @FinitePresentation 
Goal: RingHom.OfLocalizationSpanTarget @RingHom.FinitePresentation
-- RingHom.finitePresentation_isLocal
theorem finitePresentation_isLocal : PropertyIsLocal @FinitePresentation 
Goal: RingHom.PropertyIsLocal @RingHom.FinitePresentation
-- RingHom.finitePresentation_respectsIso
theorem finitePresentation_respectsIso : RingHom.RespectsIso @RingHom.FinitePresentation 
Goal: RingHom.RespectsIso @RingHom.FinitePresentation
-- RingHom.finitePresentation_stableUnderBaseChange
theorem finitePresentation_stableUnderBaseChange : StableUnderBaseChange @FinitePresentation 
Goal: RingHom.StableUnderBaseChange @RingHom.FinitePresentation
-- WittVector.coe_mkUnit
@[simp]
theorem coe_mkUnit {a : Units k} {A : 𝕎 k} (hA : A.coeff 0 = a) : (mkUnit hA : 𝕎 k) = A 
Arguments: {p : ℕ} [hp : Fact (Nat.Prime p)] {k : Type u_1} [CommRing k] [CharP k p] {a : kˣ} {A : WittVector p k} (hA : A.coeff 0 = ↑a)
Goal: ↑(WittVector.mkUnit hA) = A
-- WittVector.isUnit_of_coeff_zero_ne_zero
theorem isUnit_of_coeff_zero_ne_zero (x : 𝕎 k) (hx : x.coeff 0 ≠ 0) : IsUnit x 
Arguments: {p : ℕ} [hp : Fact (Nat.Prime p)] {k : Type u_1} [Field k] [CharP k p] (x : WittVector p k) (hx : x.coeff 0 ≠ 0)
Goal: IsUnit x
-- WittVector.irreducible
theorem irreducible : Irreducible (p : 𝕎 k) 
Arguments: (p : ℕ) [hp : Fact (Nat.Prime p)] {k : Type u_1} [Field k] [CharP k p]
Goal: Irreducible ↑p
-- WittVector.exists_eq_pow_p_mul
theorem exists_eq_pow_p_mul (a : 𝕎 k) (ha : a ≠ 0) :
    ∃ (m : ℕ) (b : 𝕎 k), b.coeff 0 ≠ 0 ∧ a = (p : 𝕎 k) ^ m * b 
Arguments: {p : ℕ} [hp : Fact (Nat.Prime p)] {k : Type u_1} [CommRing k] [CharP k p] [PerfectRing k p] (a : WittVector p k) (ha : a ≠ 0)
Goal: ∃ m b, b.coeff 0 ≠ 0 ∧ a = ↑p ^ m * b
-- WittVector.exists_eq_pow_p_mul'
theorem exists_eq_pow_p_mul' (a : 𝕎 k) (ha : a ≠ 0) :
    ∃ (m : ℕ) (b : Units (𝕎 k)), a = (p : 𝕎 k) ^ m * b 
Arguments: {p : ℕ} [hp : Fact (Nat.Prime p)] {k : Type u_1} [Field k] [CharP k p] [PerfectRing k p] (a : WittVector p k) (ha : a ≠ 0)
Goal: ∃ m b, a = ↑p ^ m * ↑b
-- WittVector.discreteValuationRing
theorem discreteValuationRing : DiscreteValuationRing (𝕎 k) 
Arguments: {p : ℕ} [hp : Fact (Nat.Prime p)] {k : Type u_1} [Field k] [CharP k p] [PerfectRing k p]
Goal: DiscreteValuationRing (WittVector p k)
-- ValuationRing.cond'
class ValuationRing (A : Type u) [CommRing A] [IsDomain A] : Prop where
  cond' : ∀ a b : A, ∃ c : A, a * c = b ∨ b * c = a
Arguments: {A : Type u} [CommRing A] [IsDomain A] [self : ValuationRing A] (a b : A)
Goal: ∃ c, a * c = b ∨ b * c = a
-- ValuationRing.cond
lemma ValuationRing.cond {A : Type u} [CommRing A] [IsDomain A] [ValuationRing A] (a b : A) :
    ∃ c : A, a * c = b ∨ b * c = a 
Arguments: {A : Type u} [CommRing A] [IsDomain A] [ValuationRing A] (a b : A)
Goal: ∃ c, a * c = b ∨ b * c = a
-- ValuationRing.le_total
protected theorem le_total (a b : ValueGroup A K) : a ≤ b ∨ b ≤ a 
Arguments: (A : Type u) [CommRing A] (K : Type v) [Field K] [Algebra A K] [IsDomain A] [ValuationRing A] [IsFractionRing A K] (a b : ValuationRing.ValueGroup A K)
Goal: a ≤ b ∨ b ≤ a
-- ValuationRing.mem_integer_iff
theorem mem_integer_iff (x : K) : x ∈ (valuation A K).integer ↔ ∃ a : A, algebraMap A K a = x 
Arguments: (A : Type u) [CommRing A] (K : Type v) [Field K] [Algebra A K] [IsDomain A] [ValuationRing A] [IsFractionRing A K] (x : K)
Goal: x ∈ (ValuationRing.valuation A K).integer ↔ ∃ a, (algebraMap A K) a = x
-- ValuationRing.coe_equivInteger_apply
@[simp]
theorem coe_equivInteger_apply (a : A) : (equivInteger A K a : K) = algebraMap A K a 
Arguments: (A : Type u) [CommRing A] (K : Type v) [Field K] [Algebra A K] [IsDomain A] [ValuationRing A] [IsFractionRing A K] (a : A)
Goal: ↑((ValuationRing.equivInteger A K) a) = (algebraMap A K) a
-- ValuationRing.range_algebraMap_eq
theorem range_algebraMap_eq : (valuation A K).integer = (algebraMap A K).range 
Arguments: (A : Type u) [CommRing A] (K : Type v) [Field K] [Algebra A K] [IsDomain A] [ValuationRing A] [IsFractionRing A K]
Goal: (ValuationRing.valuation A K).integer = (algebraMap A K).range
-- ValuationRing.iff_dvd_total
theorem iff_dvd_total : ValuationRing R ↔ IsTotal R (· ∣ ·) 
Arguments: {R : Type u_1} [CommRing R] [IsDomain R]
Goal: ValuationRing R ↔ IsTotal R fun x x_1 => x ∣ x_1
-- ValuationRing.iff_ideal_total
theorem iff_ideal_total : ValuationRing R ↔ IsTotal (Ideal R) (· ≤ ·) 
Arguments: {R : Type u_1} [CommRing R] [IsDomain R]
Goal: ValuationRing R ↔ IsTotal (Ideal R) fun x x_1 => x ≤ x_1
-- ValuationRing.dvd_total
theorem dvd_total [h : ValuationRing R] (x y : R) : x ∣ y ∨ y ∣ x 
Arguments: {R : Type u_1} [CommRing R] [IsDomain R] [h : ValuationRing R] (x y : R)
Goal: x ∣ y ∨ y ∣ x
-- ValuationRing.unique_irreducible
theorem unique_irreducible [ValuationRing R] ⦃p q : R⦄ (hp : Irreducible p) (hq : Irreducible q) :
    Associated p q 
Arguments: {R : Type u_1} [CommRing R] [IsDomain R] [ValuationRing R] ⦃p q : R⦄ (hp : Irreducible p) (hq : Irreducible q)
Goal: Associated p q
-- ValuationRing.iff_isInteger_or_isInteger
theorem iff_isInteger_or_isInteger :
    ValuationRing R ↔ ∀ x : K, IsLocalization.IsInteger R x ∨ IsLocalization.IsInteger R x⁻¹ 
Arguments: (R : Type u_1) [CommRing R] [IsDomain R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K]
Goal: ValuationRing R ↔ ∀ (x : K), IsLocalization.IsInteger R x ∨ IsLocalization.IsInteger R x⁻¹
-- ValuationRing.isInteger_or_isInteger
theorem isInteger_or_isInteger [h : ValuationRing R] (x : K) :
    IsLocalization.IsInteger R x ∨ IsLocalization.IsInteger R x⁻¹ 
Arguments: (R : Type u_1) [CommRing R] [IsDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] [h : ValuationRing R] (x : K)
Goal: IsLocalization.IsInteger R x ∨ IsLocalization.IsInteger R x⁻¹
-- ValuationRing.iff_local_bezout_domain
theorem iff_local_bezout_domain : ValuationRing R ↔ LocalRing R ∧ IsBezout R 
Arguments: {R : Type u_1} [CommRing R] [IsDomain R]
Goal: ValuationRing R ↔ LocalRing R ∧ IsBezout R
-- ValuationRing.TFAE
protected theorem TFAE (R : Type u) [CommRing R] [IsDomain R] :
    List.TFAE
      [ValuationRing R,
        ∀ x : FractionRing R, IsLocalization.IsInteger R x ∨ IsLocalization.IsInteger R x⁻¹,
        IsTotal R (· ∣ ·), IsTotal (Ideal R) (· ≤ ·), LocalRing R ∧ IsBezout R] 
Arguments: (R : Type u) [CommRing R] [IsDomain R]
Goal: [ValuationRing R, ∀ (x : FractionRing R), IsLocalization.IsInteger R x ∨ IsLocalization.IsInteger R x⁻¹,
    IsTotal R fun x x_1 => x ∣ x_1, IsTotal (Ideal R) fun x x_1 => x ≤ x_1, LocalRing R ∧ IsBezout R].TFAE
-- Function.Surjective.valuationRing
theorem _root_.Function.Surjective.valuationRing {R S : Type*} [CommRing R] [IsDomain R]
    [ValuationRing R] [CommRing S] [IsDomain S] (f : R →+* S) (hf : Function.Surjective f) :
    ValuationRing S 
Arguments: {R : Type u_1} {S : Type u_2} [CommRing R] [IsDomain R] [ValuationRing R] [CommRing S] [IsDomain S] (f : R →+* S) (hf : Function.Surjective ⇑f)
Goal: ValuationRing S
-- ValuationRing.of_integers
theorem of_integers : ValuationRing 𝒪 
Arguments: {𝒪 : Type u} {K : Type v} {Γ : Type w} [CommRing 𝒪] [IsDomain 𝒪] [Field K] [Algebra 𝒪 K] [LinearOrderedCommGroupWithZero Γ] (v : Valuation K Γ) (hh : v.Integers 𝒪)
Goal: ValuationRing 𝒪
-- ValuationRing.isFractionRing_iff
theorem isFractionRing_iff [ValuationRing 𝒪] :
    IsFractionRing 𝒪 K ↔
      (∀ (x : K), ∃ a : 𝒪, x = algebraMap 𝒪 K a ∨ x⁻¹ = algebraMap 𝒪 K a) ∧
        Function.Injective (algebraMap 𝒪 K) 
Arguments: {𝒪 : Type u} {K : Type v} [CommRing 𝒪] [IsDomain 𝒪] [Field K] [Algebra 𝒪 K] [ValuationRing 𝒪]
Goal: IsFractionRing 𝒪 K ↔
  (∀ (x : K), ∃ a, x = (algebraMap 𝒪 K) a ∨ x⁻¹ = (algebraMap 𝒪 K) a) ∧ Function.Injective ⇑(algebraMap 𝒪 K)
-- WittVector.verschiebungFun_coeff
theorem verschiebungFun_coeff (x : 𝕎 R) (n : ℕ) :
    (verschiebungFun x).coeff n = if n = 0 then 0 else x.coeff (n - 1) 
Arguments: {p : ℕ} {R : Type u_1} [CommRing R] (x : WittVector p R) (n : ℕ)
Goal: x.verschiebungFun.coeff n = if n = 0 then 0 else x.coeff (n - 1)
-- WittVector.verschiebungFun_coeff_zero
theorem verschiebungFun_coeff_zero (x : 𝕎 R) : (verschiebungFun x).coeff 0 = 0 
Arguments: {p : ℕ} {R : Type u_1} [CommRing R] (x : WittVector p R)
Goal: x.verschiebungFun.coeff 0 = 0
-- WittVector.verschiebungFun_coeff_succ
@[simp]
theorem verschiebungFun_coeff_succ (x : 𝕎 R) (n : ℕ) :
    (verschiebungFun x).coeff n.succ = x.coeff n 
Arguments: {p : ℕ} {R : Type u_1} [CommRing R] (x : WittVector p R) (n : ℕ)
Goal: x.verschiebungFun.coeff n.succ = x.coeff n
-- WittVector.ghostComponent_zero_verschiebungFun
@[ghost_simps]
theorem ghostComponent_zero_verschiebungFun (x : 𝕎 R) :
    ghostComponent 0 (verschiebungFun x) = 0 
Arguments: {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] (x : WittVector p R)
Goal: (WittVector.ghostComponent 0) x.verschiebungFun = 0
-- WittVector.ghostComponent_verschiebungFun
@[ghost_simps]
theorem ghostComponent_verschiebungFun (x : 𝕎 R) (n : ℕ) :
    ghostComponent (n + 1) (verschiebungFun x) = p * ghostComponent n x 
Arguments: {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] (x : WittVector p R) (n : ℕ)
Goal: (WittVector.ghostComponent (n + 1)) x.verschiebungFun = ↑p * (WittVector.ghostComponent n) x
-- WittVector.verschiebungPoly_zero
@[simp]
theorem verschiebungPoly_zero : verschiebungPoly 0 = 0 
Goal: WittVector.verschiebungPoly 0 = 0
-- WittVector.aeval_verschiebung_poly'
theorem aeval_verschiebung_poly' (x : 𝕎 R) (n : ℕ) :
    aeval x.coeff (verschiebungPoly n) = (verschiebungFun x).coeff n 
Arguments: {p : ℕ} {R : Type u_1} [CommRing R] (x : WittVector p R) (n : ℕ)
Goal: (MvPolynomial.aeval x.coeff) (WittVector.verschiebungPoly n) = x.verschiebungFun.coeff n
-- WittVector.verschiebung_isPoly
@[is_poly]
theorem verschiebung_isPoly : IsPoly p fun R _Rcr => @verschiebung p R hp _Rcr 
Arguments: {p : ℕ} [hp : Fact (Nat.Prime p)]
Goal: WittVector.IsPoly p fun R _Rcr => ⇑WittVector.verschiebung
-- WittVector.map_verschiebung
@[simp]
theorem map_verschiebung (f : R →+* S) (x : 𝕎 R) :
    map f (verschiebung x) = verschiebung (map f x) 
Arguments: {p : ℕ} {R : Type u_1} {S : Type u_2} [hp : Fact (Nat.Prime p)] [CommRing R] [CommRing S] (f : R →+* S) (x : WittVector p R)
Goal: (WittVector.map f) (WittVector.verschiebung x) = WittVector.verschiebung ((WittVector.map f) x)
-- WittVector.ghostComponent_zero_verschiebung
@[ghost_simps]
theorem ghostComponent_zero_verschiebung (x : 𝕎 R) : ghostComponent 0 (verschiebung x) = 0 
Arguments: {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] (x : WittVector p R)
Goal: (WittVector.ghostComponent 0) (WittVector.verschiebung x) = 0
-- WittVector.ghostComponent_verschiebung
@[ghost_simps]
theorem ghostComponent_verschiebung (x : 𝕎 R) (n : ℕ) :
    ghostComponent (n + 1) (verschiebung x) = p * ghostComponent n x 
Arguments: {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] (x : WittVector p R) (n : ℕ)
Goal: (WittVector.ghostComponent (n + 1)) (WittVector.verschiebung x) = ↑p * (WittVector.ghostComponent n) x
-- WittVector.verschiebung_coeff_zero
@[simp]
theorem verschiebung_coeff_zero (x : 𝕎 R) : (verschiebung x).coeff 0 = 0 
Arguments: {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] (x : WittVector p R)
Goal: (WittVector.verschiebung x).coeff 0 = 0
-- WittVector.verschiebung_coeff_add_one
theorem verschiebung_coeff_add_one (x : 𝕎 R) (n : ℕ) :
    (verschiebung x).coeff (n + 1) = x.coeff n 
Arguments: {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] (x : WittVector p R) (n : ℕ)
Goal: (WittVector.verschiebung x).coeff (n + 1) = x.coeff n
-- WittVector.verschiebung_coeff_succ
@[simp]
theorem verschiebung_coeff_succ (x : 𝕎 R) (n : ℕ) : (verschiebung x).coeff n.succ = x.coeff n 
Arguments: {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] (x : WittVector p R) (n : ℕ)
Goal: (WittVector.verschiebung x).coeff n.succ = x.coeff n
-- WittVector.aeval_verschiebungPoly
theorem aeval_verschiebungPoly (x : 𝕎 R) (n : ℕ) :
    aeval x.coeff (verschiebungPoly n) = (verschiebung x).coeff n 
Arguments: {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] (x : WittVector p R) (n : ℕ)
Goal: (MvPolynomial.aeval x.coeff) (WittVector.verschiebungPoly n) = (WittVector.verschiebung x).coeff n
-- WittVector.bind₁_verschiebungPoly_wittPolynomial
@[simp]
theorem bind₁_verschiebungPoly_wittPolynomial (n : ℕ) :
    bind₁ verschiebungPoly (wittPolynomial p ℤ n) =
      if n = 0 then 0 else p * wittPolynomial p ℤ (n - 1) 
Arguments: {p : ℕ} [hp : Fact (Nat.Prime p)] (n : ℕ)
Goal: (MvPolynomial.bind₁ WittVector.verschiebungPoly) (wittPolynomial p ℤ n) =
  if n = 0 then 0 else ↑p * wittPolynomial p ℤ (n - 1)
-- is_transcendental_of_subsingleton
theorem is_transcendental_of_subsingleton [Subsingleton R] (x : A) : Transcendental R x 
Arguments: (R : Type u) {A : Type v} [CommRing R] [Ring A] [Algebra R A] [Subsingleton R] (x : A)
Goal: Transcendental R x
-- Algebra.IsAlgebraic.isAlgebraic
protected class Algebra.IsAlgebraic : Prop :=
  isAlgebraic : ∀ x : A, IsAlgebraic R x
Arguments: {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A] [self : Algebra.IsAlgebraic R A] (x : A)
Goal: IsAlgebraic R x
-- Algebra.Transcendental.transcendental
protected class Algebra.Transcendental : Prop :=
  transcendental : ∃ x : A, Transcendental R x
Arguments: {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A] [self : Algebra.Transcendental R A]
Goal: ∃ x, Transcendental R x
-- Algebra.isAlgebraic_def
lemma Algebra.isAlgebraic_def : Algebra.IsAlgebraic R A ↔ ∀ x : A, IsAlgebraic R x 
Arguments: {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]
Goal: Algebra.IsAlgebraic R A ↔ ∀ (x : A), IsAlgebraic R x
-- Algebra.transcendental_def
lemma Algebra.transcendental_def : Algebra.Transcendental R A ↔ ∃ x : A, Transcendental R x 
Arguments: {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]
Goal: Algebra.Transcendental R A ↔ ∃ x, Transcendental R x
-- Algebra.transcendental_iff_not_isAlgebraic
theorem Algebra.transcendental_iff_not_isAlgebraic :
    Algebra.Transcendental R A ↔ ¬ Algebra.IsAlgebraic R A 
Arguments: {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]
Goal: Algebra.Transcendental R A ↔ ¬Algebra.IsAlgebraic R A
-- Subalgebra.isAlgebraic_iff
theorem Subalgebra.isAlgebraic_iff (S : Subalgebra R A) :
    S.IsAlgebraic ↔ @Algebra.IsAlgebraic R S _ _ S.algebra 
Arguments: {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A] (S : Subalgebra R A)
Goal: S.IsAlgebraic ↔ Algebra.IsAlgebraic R ↥S
-- Algebra.isAlgebraic_iff
theorem Algebra.isAlgebraic_iff : Algebra.IsAlgebraic R A ↔ (⊤ : Subalgebra R A).IsAlgebraic 
Arguments: {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]
Goal: Algebra.IsAlgebraic R A ↔ ⊤.IsAlgebraic
-- isAlgebraic_iff_not_injective
theorem isAlgebraic_iff_not_injective {x : A} :
    IsAlgebraic R x ↔ ¬Function.Injective (Polynomial.aeval x : R[X] →ₐ[R] A) 
Arguments: {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A] {x : A}
Goal: IsAlgebraic R x ↔ ¬Function.Injective ⇑(Polynomial.aeval x)
-- transcendental_iff_injective
theorem transcendental_iff_injective {x : A} :
    Transcendental R x ↔ Function.Injective (Polynomial.aeval x : R[X] →ₐ[R] A) 
Arguments: {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A] {x : A}
Goal: Transcendental R x ↔ Function.Injective ⇑(Polynomial.aeval x)
-- IsIntegral.isAlgebraic
theorem IsIntegral.isAlgebraic [Nontrivial R] {x : A} : IsIntegral R x → IsAlgebraic R x 
Arguments: {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A] [Nontrivial R] {x : A}
Goal: IsIntegral R x → IsAlgebraic R x
-- isAlgebraic_zero
theorem isAlgebraic_zero [Nontrivial R] : IsAlgebraic R (0 : A) 
Arguments: {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A] [Nontrivial R]
Goal: IsAlgebraic R 0
-- isAlgebraic_algebraMap
theorem isAlgebraic_algebraMap [Nontrivial R] (x : R) : IsAlgebraic R (algebraMap R A x) 
Arguments: {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A] [Nontrivial R] (x : R)
Goal: IsAlgebraic R ((algebraMap R A) x)
-- isAlgebraic_one
theorem isAlgebraic_one [Nontrivial R] : IsAlgebraic R (1 : A) 
Arguments: {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A] [Nontrivial R]
Goal: IsAlgebraic R 1
-- isAlgebraic_nat
theorem isAlgebraic_nat [Nontrivial R] (n : ℕ) : IsAlgebraic R (n : A) 
Arguments: {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A] [Nontrivial R] (n : ℕ)
Goal: IsAlgebraic R ↑n
-- isAlgebraic_int
theorem isAlgebraic_int [Nontrivial R] (n : ℤ) : IsAlgebraic R (n : A) 
Arguments: {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A] [Nontrivial R] (n : ℤ)
Goal: IsAlgebraic R ↑n
-- isAlgebraic_rat
theorem isAlgebraic_rat (R : Type u) {A : Type v} [DivisionRing A] [Field R] [Algebra R A] (n : ℚ) :
    IsAlgebraic R (n : A) 
Arguments: (R : Type u) {A : Type v} [DivisionRing A] [Field R] [Algebra R A] (n : ℚ)
Goal: IsAlgebraic R ↑n
-- isAlgebraic_of_mem_rootSet
theorem isAlgebraic_of_mem_rootSet {R : Type u} {A : Type v} [Field R] [Field A] [Algebra R A]
    {p : R[X]} {x : A} (hx : x ∈ p.rootSet A) : IsAlgebraic R x 
Arguments: {R : Type u} {A : Type v} [Field R] [Field A] [Algebra R A] {p : R[X]} {x : A} (hx : x ∈ p.rootSet A)
Goal: IsAlgebraic R x
-- IsAlgebraic.algebraMap
protected theorem IsAlgebraic.algebraMap {a : S} :
    IsAlgebraic R a → IsAlgebraic R (algebraMap S A a) 
Arguments: {R : Type u} {S : Type u_1} {A : Type v} [CommRing R] [CommRing S] [Ring A] [Algebra R A] [Algebra R S] [Algebra S A] [IsScalarTower R S A] {a : S}
Goal: IsAlgebraic R a → IsAlgebraic R ((algebraMap S A) a)
-- IsAlgebraic.algHom
protected theorem IsAlgebraic.algHom (f : A →ₐ[R] B) {a : A}
    (h : IsAlgebraic R a) : IsAlgebraic R (f a) 
Arguments: {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A] {B : Type u_2} [Ring B] [Algebra R B] (f : A →ₐ[R] B) {a : A} (h : IsAlgebraic R a)
Goal: IsAlgebraic R (f a)
-- isAlgebraic_algHom_iff
theorem isAlgebraic_algHom_iff (f : A →ₐ[R] B) (hf : Function.Injective f)
    {a : A} : IsAlgebraic R (f a) ↔ IsAlgebraic R a 
Arguments: {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A] {B : Type u_2} [Ring B] [Algebra R B] (f : A →ₐ[R] B) (hf : Function.Injective ⇑f) {a : A}
Goal: IsAlgebraic R (f a) ↔ IsAlgebraic R a
-- Algebra.IsAlgebraic.of_injective
theorem Algebra.IsAlgebraic.of_injective (f : A →ₐ[R] B) (hf : Function.Injective f)
    [Algebra.IsAlgebraic R B] : Algebra.IsAlgebraic R A 
Arguments: {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A] {B : Type u_2} [Ring B] [Algebra R B] (f : A →ₐ[R] B) (hf : Function.Injective ⇑f) [Algebra.IsAlgebraic R B]
Goal: Algebra.IsAlgebraic R A
-- AlgEquiv.isAlgebraic
theorem AlgEquiv.isAlgebraic (e : A ≃ₐ[R] B)
    [Algebra.IsAlgebraic R A] : Algebra.IsAlgebraic R B 
Arguments: {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A] {B : Type u_2} [Ring B] [Algebra R B] (e : A ≃ₐ[R] B) [Algebra.IsAlgebraic R A]
Goal: Algebra.IsAlgebraic R B
-- AlgEquiv.isAlgebraic_iff
theorem AlgEquiv.isAlgebraic_iff (e : A ≃ₐ[R] B) :
    Algebra.IsAlgebraic R A ↔ Algebra.IsAlgebraic R B 
Arguments: {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A] {B : Type u_2} [Ring B] [Algebra R B] (e : A ≃ₐ[R] B)
Goal: Algebra.IsAlgebraic R A ↔ Algebra.IsAlgebraic R B
-- isAlgebraic_algebraMap_iff
theorem isAlgebraic_algebraMap_iff {a : S} (h : Function.Injective (algebraMap S A)) :
    IsAlgebraic R (algebraMap S A a) ↔ IsAlgebraic R a 
Arguments: {R : Type u} {S : Type u_1} {A : Type v} [CommRing R] [CommRing S] [Ring A] [Algebra R A] [Algebra R S] [Algebra S A] [IsScalarTower R S A] {a : S} (h : Function.Injective ⇑(algebraMap S A))
Goal: IsAlgebraic R ((algebraMap S A) a) ↔ IsAlgebraic R a
-- IsAlgebraic.of_pow
theorem IsAlgebraic.of_pow {r : A} {n : ℕ} (hn : 0 < n) (ht : IsAlgebraic R (r ^ n)) :
    IsAlgebraic R r 
Arguments: {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A] {r : A} {n : ℕ} (hn : 0 < n) (ht : IsAlgebraic R (r ^ n))
Goal: IsAlgebraic R r
-- Transcendental.pow
theorem Transcendental.pow {r : A} (ht : Transcendental R r) {n : ℕ} (hn : 0 < n) :
    Transcendental R (r ^ n) 
Arguments: {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A] {r : A} (ht : Transcendental R r) {n : ℕ} (hn : 0 < n)
Goal: Transcendental R (r ^ n)
-- IsAlgebraic.invOf
lemma IsAlgebraic.invOf {x : S} [Invertible x] (h : IsAlgebraic R x) : IsAlgebraic R (⅟ x) 
Arguments: {R : Type u} {S : Type u_1} [CommRing R] [CommRing S] [Algebra R S] {x : S} [Invertible x] (h : IsAlgebraic R x)
Goal: IsAlgebraic R ⅟x
-- IsAlgebraic.invOf_iff
lemma IsAlgebraic.invOf_iff {x : S} [Invertible x] :
    IsAlgebraic R (⅟ x) ↔ IsAlgebraic R x 
Arguments: {R : Type u} {S : Type u_1} [CommRing R] [CommRing S] [Algebra R S] {x : S} [Invertible x]
Goal: IsAlgebraic R ⅟x ↔ IsAlgebraic R x
-- IsAlgebraic.inv_iff
lemma IsAlgebraic.inv_iff {K} [Field K] [Algebra R K] {x : K} :
    IsAlgebraic R (x⁻¹) ↔ IsAlgebraic R x 
Arguments: {R : Type u} [CommRing R] {K : Type u_2} [Field K] [Algebra R K] {x : K}
Goal: IsAlgebraic R x⁻¹ ↔ IsAlgebraic R x
-- isAlgebraic_iff_isIntegral
theorem isAlgebraic_iff_isIntegral {x : A} : IsAlgebraic K x ↔ IsIntegral K x 
Arguments: {K : Type u} {A : Type v} [Field K] [Ring A] [Algebra K A] {x : A}
Goal: IsAlgebraic K x ↔ IsIntegral K x
-- Algebra.isAlgebraic_iff_isIntegral
protected theorem Algebra.isAlgebraic_iff_isIntegral :
    Algebra.IsAlgebraic K A ↔ Algebra.IsIntegral K A 
Arguments: {K : Type u} {A : Type v} [Field K] [Ring A] [Algebra K A]
Goal: Algebra.IsAlgebraic K A ↔ Algebra.IsIntegral K A
-- IsAlgebraic.tower_top_of_injective
theorem IsAlgebraic.tower_top_of_injective
    (hinj : Function.Injective (algebraMap R S)) {x : A}
    (A_alg : IsAlgebraic R x) : IsAlgebraic S x 
Arguments: {R : Type u_3} {S : Type u_4} {A : Type u_5} [CommRing R] [CommRing S] [Ring A] [Algebra R S] [Algebra S A] [Algebra R A] [IsScalarTower R S A] (hinj : Function.Injective ⇑(algebraMap R S)) {x : A} (A_alg : IsAlgebraic R x)
Goal: IsAlgebraic S x
-- Algebra.IsAlgebraic.tower_top_of_injective
theorem Algebra.IsAlgebraic.tower_top_of_injective (hinj : Function.Injective (algebraMap R S))
    [Algebra.IsAlgebraic R A] : Algebra.IsAlgebraic S A 
Arguments: {R : Type u_3} {S : Type u_4} {A : Type u_5} [CommRing R] [CommRing S] [Ring A] [Algebra R S] [Algebra S A] [Algebra R A] [IsScalarTower R S A] (hinj : Function.Injective ⇑(algebraMap R S)) [Algebra.IsAlgebraic R A]
Goal: Algebra.IsAlgebraic S A
-- IsAlgebraic.tower_top
theorem IsAlgebraic.tower_top {x : A} (A_alg : IsAlgebraic K x) :
    IsAlgebraic L x 
Arguments: {K : Type u_1} (L : Type u_2) {A : Type u_5} [Field K] [Field L] [Ring A] [Algebra K L] [Algebra L A] [Algebra K A] [IsScalarTower K L A] {x : A} (A_alg : IsAlgebraic K x)
Goal: IsAlgebraic L x
-- Algebra.IsAlgebraic.tower_top
theorem Algebra.IsAlgebraic.tower_top [Algebra.IsAlgebraic K A] : Algebra.IsAlgebraic L A 
Arguments: {K : Type u_1} (L : Type u_2) {A : Type u_5} [Field K] [Field L] [Ring A] [Algebra K L] [Algebra L A] [Algebra K A] [IsScalarTower K L A] [Algebra.IsAlgebraic K A]
Goal: Algebra.IsAlgebraic L A
-- IsAlgebraic.of_finite
theorem IsAlgebraic.of_finite (e : A) [FiniteDimensional K A] : IsAlgebraic K e 
Arguments: (K : Type u_1) {A : Type u_5} [Field K] [Ring A] [Algebra K A] (e : A) [FiniteDimensional K A]
Goal: IsAlgebraic K e