ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Submodule.le_traceDual_comm
lemma le_traceDual_comm {I J : Submodule B L} :
    I ≤ Jᵛ ↔ J ≤ Iᵛ 
Arguments: {A : Type u_1} {K : Type u_2} {L : Type u} {B : Type u_3} [CommRing A] [Field K] [CommRing B] [Field L] [Algebra A K] [Algebra B L] [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L] {I J : Submodule B L}
Goal: I ≤ Submodule.traceDual A K J ↔ J ≤ Submodule.traceDual A K I
-- Submodule.le_traceDual_traceDual
lemma le_traceDual_traceDual {I : Submodule B L} :
    I ≤ Iᵛᵛ 
Arguments: {A : Type u_1} {K : Type u_2} {L : Type u} {B : Type u_3} [CommRing A] [Field K] [CommRing B] [Field L] [Algebra A K] [Algebra B L] [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L] {I : Submodule B L}
Goal: I ≤ Submodule.traceDual A K (Submodule.traceDual A K I)
-- Submodule.traceDual_bot
@[simp]
lemma traceDual_bot :
    (⊥ : Submodule B L)ᵛ = ⊤ 
Arguments: {A : Type u_1} {K : Type u_2} {L : Type u} {B : Type u_3} [CommRing A] [Field K] [CommRing B] [Field L] [Algebra A K] [Algebra B L] [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L]
Goal: Submodule.traceDual A K ⊥ = ⊤
-- Submodule.traceDual_top'
lemma traceDual_top' :
    (⊤ : Submodule B L)ᵛ =
      if ((LinearMap.range (Algebra.trace K L)).restrictScalars A ≤ 1) then ⊤ else ⊥ 
Arguments: {A : Type u_1} {K : Type u_2} {L : Type u} {B : Type u_3} [CommRing A] [Field K] [CommRing B] [Field L] [Algebra A K] [Algebra B L] [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L]
Goal: Submodule.traceDual A K ⊤ = if Submodule.restrictScalars A (LinearMap.range (Algebra.trace K L)) ≤ 1 then ⊤ else ⊥
-- Submodule.traceDual_top
lemma traceDual_top [Decidable (IsField A)] :
    (⊤ : Submodule B L)ᵛ = if IsField A then ⊤ else ⊥ 
Arguments: {A : Type u_1} {K : Type u_2} {L : Type u} {B : Type u_3} [CommRing A] [Field K] [CommRing B] [Field L] [Algebra A K] [Algebra B L] [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L] [IsDomain A] [IsFractionRing A K] [FiniteDimensional K L] [Algebra.IsSeparable K L] [Decidable (IsField A)]
Goal: Submodule.traceDual A K ⊤ = if IsField A then ⊤ else ⊥
-- Submodule.mem_traceDual_iff_isIntegral
lemma Submodule.mem_traceDual_iff_isIntegral {I : Submodule B L} {x} :
    x ∈ Iᵛ ↔ ∀ a ∈ I, IsIntegral A (traceForm K L x a) 
Arguments: {A : Type u_1} {K : Type u_2} {L : Type u} {B : Type u_3} [CommRing A] [Field K] [CommRing B] [Field L] [Algebra A K] [Algebra B L] [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L] [IsFractionRing A K] [IsIntegrallyClosed A] {I : Submodule B L} {x : L}
Goal: x ∈ Submodule.traceDual A K I ↔ ∀ a ∈ I, IsIntegral A (((Algebra.traceForm K L) x) a)
-- Submodule.one_le_traceDual_one
lemma Submodule.one_le_traceDual_one :
    (1 : Submodule B L) ≤ 1ᵛ 
Arguments: {A : Type u_1} {K : Type u_2} {L : Type u} {B : Type u_3} [CommRing A] [Field K] [CommRing B] [Field L] [Algebra A K] [Algebra B L] [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L] [IsFractionRing A K] [IsIntegralClosure B A L] [FiniteDimensional K L] [IsIntegrallyClosed A]
Goal: 1 ≤ Submodule.traceDual A K 1
-- isIntegral_discr_mul_of_mem_traceDual
lemma isIntegral_discr_mul_of_mem_traceDual
    (I : Submodule B L) {ι} [DecidableEq ι] [Fintype ι]
    {b : Basis ι K L} (hb : ∀ i, IsIntegral A (b i))
    {a x : L} (ha : a ∈ I) (hx : x ∈ Iᵛ) :
    IsIntegral A ((discr K b) • a * x) 
Arguments: {A : Type u_1} {K : Type u_2} {L : Type u} {B : Type u_3} [CommRing A] [Field K] [CommRing B] [Field L] [Algebra A K] [Algebra B L] [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L] [IsFractionRing A K] [IsIntegralClosure B A L] [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsIntegrallyClosed A] (I : Submodule B L) {ι : Type u_4} [DecidableEq ι] [Fintype ι] {b : Basis ι K L} (hb : ∀ (i : ι), IsIntegral A (b i)) {a x : L} (ha : a ∈ I) (hx : x ∈ Submodule.traceDual A K I)
Goal: IsIntegral A (Algebra.discr K ⇑b • a * x)
-- map_equiv_traceDual
lemma map_equiv_traceDual [NoZeroSMulDivisors A B] (I : Submodule B (FractionRing B)) :
    (traceDual A (FractionRing A) I).map (FractionRing.algEquiv B L) =
      traceDual A K (I.map (FractionRing.algEquiv B L)) 
Arguments: (A : Type u_1) (K : Type u_2) {L : Type u} {B : Type u_3} [CommRing A] [Field K] [CommRing B] [Field L] [Algebra A K] [Algebra B L] [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L] [IsDomain A] [IsFractionRing A K] [IsFractionRing B L] [IsDomain B] [NoZeroSMulDivisors A B] (I : Submodule B (FractionRing B))
Goal: Submodule.map (FractionRing.algEquiv B L) (Submodule.traceDual A (FractionRing A) I) =
  Submodule.traceDual A K (Submodule.map (FractionRing.algEquiv B L) I)
-- FractionalIdeal.coe_dual
lemma coe_dual :
    (dual A K I : Submodule B L) = Iᵛ 
Arguments: (A : Type u_1) (K : Type u_2) {L : Type u} {B : Type u_3} [CommRing A] [Field K] [CommRing B] [Field L] [Algebra A K] [Algebra B L] [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L] [IsDomain A] [IsFractionRing A K] [IsIntegralClosure B A L] [IsFractionRing B L] [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsIntegrallyClosed A] [IsDedekindDomain B] {I : FractionalIdeal B⁰ L} (hI : I ≠ 0)
Goal: ↑(FractionalIdeal.dual A K I) = Submodule.traceDual A K ↑I
-- FractionalIdeal.coe_dual_one
@[simp]
lemma coe_dual_one :
    (dual A K (1 : FractionalIdeal B⁰ L) : Submodule B L) = 1ᵛ 
Arguments: (A : Type u_1) (K : Type u_2) (L : Type u) (B : Type u_3) [CommRing A] [Field K] [CommRing B] [Field L] [Algebra A K] [Algebra B L] [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L] [IsDomain A] [IsFractionRing A K] [IsIntegralClosure B A L] [IsFractionRing B L] [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsIntegrallyClosed A] [IsDedekindDomain B]
Goal: ↑(FractionalIdeal.dual A K 1) = Submodule.traceDual A K 1
-- FractionalIdeal.dual_zero
@[simp]
lemma dual_zero :
    dual A K (0 : FractionalIdeal B⁰ L) = 0 
Arguments: (A : Type u_1) (K : Type u_2) (L : Type u) (B : Type u_3) [CommRing A] [Field K] [CommRing B] [Field L] [Algebra A K] [Algebra B L] [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L] [IsDomain A] [IsFractionRing A K] [IsIntegralClosure B A L] [IsFractionRing B L] [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsIntegrallyClosed A] [IsDedekindDomain B]
Goal: FractionalIdeal.dual A K 0 = 0
-- FractionalIdeal.mem_dual
lemma mem_dual {x} :
    x ∈ dual A K I ↔ ∀ a ∈ I, traceForm K L x a ∈ (algebraMap A K).range 
Arguments: {A : Type u_1} {K : Type u_2} {L : Type u} {B : Type u_3} [CommRing A] [Field K] [CommRing B] [Field L] [Algebra A K] [Algebra B L] [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L] [IsDomain A] [IsFractionRing A K] [IsIntegralClosure B A L] [IsFractionRing B L] [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsIntegrallyClosed A] [IsDedekindDomain B] {I : FractionalIdeal B⁰ L} (hI : I ≠ 0) {x : L}
Goal: x ∈ FractionalIdeal.dual A K I ↔ ∀ a ∈ I, ((Algebra.traceForm K L) x) a ∈ (algebraMap A K).range
-- FractionalIdeal.dual_ne_zero
lemma dual_ne_zero :
    dual A K I ≠ 0 
Arguments: (A : Type u_1) (K : Type u_2) {L : Type u} {B : Type u_3} [CommRing A] [Field K] [CommRing B] [Field L] [Algebra A K] [Algebra B L] [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L] [IsDomain A] [IsFractionRing A K] [IsIntegralClosure B A L] [IsFractionRing B L] [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsIntegrallyClosed A] [IsDedekindDomain B] {I : FractionalIdeal B⁰ L} (hI : I ≠ 0)
Goal: FractionalIdeal.dual A K I ≠ 0
-- FractionalIdeal.dual_eq_zero_iff
@[simp]
lemma dual_eq_zero_iff :
    dual A K I = 0 ↔ I = 0 
Arguments: {A : Type u_1} {K : Type u_2} {L : Type u} {B : Type u_3} [CommRing A] [Field K] [CommRing B] [Field L] [Algebra A K] [Algebra B L] [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L] [IsDomain A] [IsFractionRing A K] [IsIntegralClosure B A L] [IsFractionRing B L] [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsIntegrallyClosed A] [IsDedekindDomain B] {I : FractionalIdeal B⁰ L}
Goal: FractionalIdeal.dual A K I = 0 ↔ I = 0
-- FractionalIdeal.dual_ne_zero_iff
lemma dual_ne_zero_iff :
    dual A K I ≠ 0 ↔ I ≠ 0 
Arguments: {A : Type u_1} {K : Type u_2} {L : Type u} {B : Type u_3} [CommRing A] [Field K] [CommRing B] [Field L] [Algebra A K] [Algebra B L] [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L] [IsDomain A] [IsFractionRing A K] [IsIntegralClosure B A L] [IsFractionRing B L] [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsIntegrallyClosed A] [IsDedekindDomain B] {I : FractionalIdeal B⁰ L}
Goal: FractionalIdeal.dual A K I ≠ 0 ↔ I ≠ 0
-- FractionalIdeal.le_dual_inv_aux
lemma le_dual_inv_aux (hIJ : I * J ≤ 1) :
    J ≤ dual A K I 
Arguments: (A : Type u_1) (K : Type u_2) {L : Type u} {B : Type u_3} [CommRing A] [Field K] [CommRing B] [Field L] [Algebra A K] [Algebra B L] [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L] [IsDomain A] [IsFractionRing A K] [IsIntegralClosure B A L] [IsFractionRing B L] [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsIntegrallyClosed A] [IsDedekindDomain B] {I J : FractionalIdeal B⁰ L} (hI : I ≠ 0) (hIJ : I * J ≤ 1)
Goal: J ≤ FractionalIdeal.dual A K I
-- FractionalIdeal.one_le_dual_one
lemma one_le_dual_one :
    1 ≤ dual A K (1 : FractionalIdeal B⁰ L) 
Arguments: (A : Type u_1) (K : Type u_2) {L : Type u} {B : Type u_3} [CommRing A] [Field K] [CommRing B] [Field L] [Algebra A K] [Algebra B L] [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L] [IsDomain A] [IsFractionRing A K] [IsIntegralClosure B A L] [IsFractionRing B L] [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsIntegrallyClosed A] [IsDedekindDomain B]
Goal: 1 ≤ FractionalIdeal.dual A K 1
-- FractionalIdeal.le_dual_iff
lemma le_dual_iff :
    I ≤ dual A K J ↔ I * J ≤ dual A K 1 
Arguments: (A : Type u_1) (K : Type u_2) {L : Type u} {B : Type u_3} [CommRing A] [Field K] [CommRing B] [Field L] [Algebra A K] [Algebra B L] [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L] [IsDomain A] [IsFractionRing A K] [IsIntegralClosure B A L] [IsFractionRing B L] [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsIntegrallyClosed A] [IsDedekindDomain B] {I J : FractionalIdeal B⁰ L} (hJ : J ≠ 0)
Goal: I ≤ FractionalIdeal.dual A K J ↔ I * J ≤ FractionalIdeal.dual A K 1
-- FractionalIdeal.inv_le_dual
lemma inv_le_dual :
    I⁻¹ ≤ dual A K I 
Arguments: (A : Type u_1) (K : Type u_2) {L : Type u} {B : Type u_3} [CommRing A] [Field K] [CommRing B] [Field L] [Algebra A K] [Algebra B L] [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L] [IsDomain A] [IsFractionRing A K] [IsIntegralClosure B A L] [IsFractionRing B L] [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsIntegrallyClosed A] [IsDedekindDomain B] (I : FractionalIdeal B⁰ L)
Goal: I⁻¹ ≤ FractionalIdeal.dual A K I
-- FractionalIdeal.dual_inv_le
lemma dual_inv_le :
    (dual A K I)⁻¹ ≤ I 
Arguments: (A : Type u_1) (K : Type u_2) {L : Type u} {B : Type u_3} [CommRing A] [Field K] [CommRing B] [Field L] [Algebra A K] [Algebra B L] [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L] [IsDomain A] [IsFractionRing A K] [IsIntegralClosure B A L] [IsFractionRing B L] [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsIntegrallyClosed A] [IsDedekindDomain B] (I : FractionalIdeal B⁰ L)
Goal: (FractionalIdeal.dual A K I)⁻¹ ≤ I
-- FractionalIdeal.dual_eq_mul_inv
lemma dual_eq_mul_inv :
    dual A K I = dual A K 1 * I⁻¹ 
Arguments: (A : Type u_1) (K : Type u_2) {L : Type u} {B : Type u_3} [CommRing A] [Field K] [CommRing B] [Field L] [Algebra A K] [Algebra B L] [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L] [IsDomain A] [IsFractionRing A K] [IsIntegralClosure B A L] [IsFractionRing B L] [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsIntegrallyClosed A] [IsDedekindDomain B] (I : FractionalIdeal B⁰ L)
Goal: FractionalIdeal.dual A K I = FractionalIdeal.dual A K 1 * I⁻¹
-- FractionalIdeal.dual_div_dual
lemma dual_div_dual :
    dual A K J / dual A K I = I / J 
Arguments: (A : Type u_1) (K : Type u_2) {L : Type u} {B : Type u_3} [CommRing A] [Field K] [CommRing B] [Field L] [Algebra A K] [Algebra B L] [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L] [IsDomain A] [IsFractionRing A K] [IsIntegralClosure B A L] [IsFractionRing B L] [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsIntegrallyClosed A] [IsDedekindDomain B] {I J : FractionalIdeal B⁰ L}
Goal: FractionalIdeal.dual A K J / FractionalIdeal.dual A K I = I / J
-- FractionalIdeal.dual_mul_self
lemma dual_mul_self :
    dual A K I * I = dual A K 1 
Arguments: (A : Type u_1) (K : Type u_2) {L : Type u} {B : Type u_3} [CommRing A] [Field K] [CommRing B] [Field L] [Algebra A K] [Algebra B L] [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L] [IsDomain A] [IsFractionRing A K] [IsIntegralClosure B A L] [IsFractionRing B L] [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsIntegrallyClosed A] [IsDedekindDomain B] {I : FractionalIdeal B⁰ L} (hI : I ≠ 0)
Goal: FractionalIdeal.dual A K I * I = FractionalIdeal.dual A K 1
-- FractionalIdeal.self_mul_dual
lemma self_mul_dual :
    I * dual A K I = dual A K 1 
Arguments: (A : Type u_1) (K : Type u_2) {L : Type u} {B : Type u_3} [CommRing A] [Field K] [CommRing B] [Field L] [Algebra A K] [Algebra B L] [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L] [IsDomain A] [IsFractionRing A K] [IsIntegralClosure B A L] [IsFractionRing B L] [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsIntegrallyClosed A] [IsDedekindDomain B] {I : FractionalIdeal B⁰ L} (hI : I ≠ 0)
Goal: I * FractionalIdeal.dual A K I = FractionalIdeal.dual A K 1
-- FractionalIdeal.dual_inv
lemma dual_inv :
    dual A K I⁻¹ = dual A K 1 * I 
Arguments: (A : Type u_1) (K : Type u_2) {L : Type u} {B : Type u_3} [CommRing A] [Field K] [CommRing B] [Field L] [Algebra A K] [Algebra B L] [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L] [IsDomain A] [IsFractionRing A K] [IsIntegralClosure B A L] [IsFractionRing B L] [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsIntegrallyClosed A] [IsDedekindDomain B] {I : FractionalIdeal B⁰ L}
Goal: FractionalIdeal.dual A K I⁻¹ = FractionalIdeal.dual A K 1 * I
-- FractionalIdeal.dual_dual
@[simp]
lemma dual_dual :
    dual A K (dual A K I) = I 
Arguments: (A : Type u_1) (K : Type u_2) {L : Type u} {B : Type u_3} [CommRing A] [Field K] [CommRing B] [Field L] [Algebra A K] [Algebra B L] [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L] [IsDomain A] [IsFractionRing A K] [IsIntegralClosure B A L] [IsFractionRing B L] [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsIntegrallyClosed A] [IsDedekindDomain B] (I : FractionalIdeal B⁰ L)
Goal: FractionalIdeal.dual A K (FractionalIdeal.dual A K I) = I
-- FractionalIdeal.dual_le_dual
@[simp]
lemma dual_le_dual :
    dual A K I ≤ dual A K J ↔ J ≤ I 
Arguments: (A : Type u_1) (K : Type u_2) {L : Type u} {B : Type u_3} [CommRing A] [Field K] [CommRing B] [Field L] [Algebra A K] [Algebra B L] [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L] [IsDomain A] [IsFractionRing A K] [IsIntegralClosure B A L] [IsFractionRing B L] [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsIntegrallyClosed A] [IsDedekindDomain B] {I J : FractionalIdeal B⁰ L} (hI : I ≠ 0) (hJ : J ≠ 0)
Goal: FractionalIdeal.dual A K I ≤ FractionalIdeal.dual A K J ↔ J ≤ I
-- FractionalIdeal.dual_involutive
lemma dual_involutive :
    Function.Involutive (dual A K : FractionalIdeal B⁰ L → FractionalIdeal B⁰ L) 
Arguments: {A : Type u_1} {K : Type u_2} {L : Type u} {B : Type u_3} [CommRing A] [Field K] [CommRing B] [Field L] [Algebra A K] [Algebra B L] [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L] [IsDomain A] [IsFractionRing A K] [IsIntegralClosure B A L] [IsFractionRing B L] [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsIntegrallyClosed A] [IsDedekindDomain B]
Goal: Function.Involutive (FractionalIdeal.dual A K)
-- FractionalIdeal.dual_injective
lemma dual_injective :
    Function.Injective (dual A K : FractionalIdeal B⁰ L → FractionalIdeal B⁰ L) 
Arguments: {A : Type u_1} {K : Type u_2} {L : Type u} {B : Type u_3} [CommRing A] [Field K] [CommRing B] [Field L] [Algebra A K] [Algebra B L] [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L] [IsDomain A] [IsFractionRing A K] [IsIntegralClosure B A L] [IsFractionRing B L] [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsIntegrallyClosed A] [IsDedekindDomain B]
Goal: Function.Injective (FractionalIdeal.dual A K)
-- coeSubmodule_differentIdeal_fractionRing
lemma coeSubmodule_differentIdeal_fractionRing
    [NoZeroSMulDivisors A B] [Algebra.IsIntegral A B]
    [Algebra.IsSeparable (FractionRing A) (FractionRing B)]
    [FiniteDimensional (FractionRing A) (FractionRing B)] :
    coeSubmodule (FractionRing B) (differentIdeal A B) =
      1 / Submodule.traceDual A (FractionRing A) 1 
Arguments: (A : Type u_1) (B : Type u_3) [CommRing A] [CommRing B] [Algebra A B] [IsDomain A] [IsIntegrallyClosed A] [IsDedekindDomain B] [NoZeroSMulDivisors A B] [Algebra.IsIntegral A B] [Algebra.IsSeparable (FractionRing A) (FractionRing B)] [FiniteDimensional (FractionRing A) (FractionRing B)]
Goal: IsLocalization.coeSubmodule (FractionRing B) (differentIdeal A B) = 1 / Submodule.traceDual A (FractionRing A) 1
-- coeSubmodule_differentIdeal
lemma coeSubmodule_differentIdeal [NoZeroSMulDivisors A B] :
    coeSubmodule L (differentIdeal A B) = 1 / Submodule.traceDual A K 1 
Arguments: (A : Type u_1) (K : Type u_2) {L : Type u} (B : Type u_3) [CommRing A] [Field K] [CommRing B] [Field L] [Algebra A K] [Algebra B L] [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L] [IsDomain A] [IsFractionRing A K] [IsIntegralClosure B A L] [IsFractionRing B L] [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsIntegrallyClosed A] [IsDedekindDomain B] [NoZeroSMulDivisors A B]
Goal: IsLocalization.coeSubmodule L (differentIdeal A B) = 1 / Submodule.traceDual A K 1
-- coeIdeal_differentIdeal
lemma coeIdeal_differentIdeal [NoZeroSMulDivisors A B] :
    ↑(differentIdeal A B) = (FractionalIdeal.dual A K (1 : FractionalIdeal B⁰ L))⁻¹ 
Arguments: (A : Type u_1) (K : Type u_2) (L : Type u) (B : Type u_3) [CommRing A] [Field K] [CommRing B] [Field L] [Algebra A K] [Algebra B L] [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L] [IsDomain A] [IsFractionRing A K] [IsIntegralClosure B A L] [IsFractionRing B L] [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsIntegrallyClosed A] [IsDedekindDomain B] [NoZeroSMulDivisors A B]
Goal: ↑(differentIdeal A B) = (FractionalIdeal.dual A K 1)⁻¹
-- differentialIdeal_le_fractionalIdeal_iff
lemma differentialIdeal_le_fractionalIdeal_iff
    {I : FractionalIdeal B⁰ L} (hI : I ≠ 0) [NoZeroSMulDivisors A B] :
    differentIdeal A B ≤ I ↔ (((I⁻¹ : _) : Submodule B L).restrictScalars A).map
      ((Algebra.trace K L).restrictScalars A) ≤ 1 
Arguments: {A : Type u_1} {K : Type u_2} {L : Type u} {B : Type u_3} [CommRing A] [Field K] [CommRing B] [Field L] [Algebra A K] [Algebra B L] [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L] [IsDomain A] [IsFractionRing A K] [IsIntegralClosure B A L] [IsFractionRing B L] [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsIntegrallyClosed A] [IsDedekindDomain B] {I : FractionalIdeal B⁰ L} (hI : I ≠ 0) [NoZeroSMulDivisors A B]
Goal: ↑(differentIdeal A B) ≤ I ↔ Submodule.map (↑A (Algebra.trace K L)) (Submodule.restrictScalars A ↑I⁻¹) ≤ 1
-- differentialIdeal_le_iff
lemma differentialIdeal_le_iff {I : Ideal B} (hI : I ≠ ⊥) [NoZeroSMulDivisors A B] :
    differentIdeal A B ≤ I ↔ (((I⁻¹ : FractionalIdeal B⁰ L) : Submodule B L).restrictScalars A).map
      ((Algebra.trace K L).restrictScalars A) ≤ 1 
Arguments: {A : Type u_1} {K : Type u_2} {L : Type u} {B : Type u_3} [CommRing A] [Field K] [CommRing B] [Field L] [Algebra A K] [Algebra B L] [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L] [IsDomain A] [IsFractionRing A K] [IsIntegralClosure B A L] [IsFractionRing B L] [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsIntegrallyClosed A] [IsDedekindDomain B] {I : Ideal B} (hI : I ≠ ⊥) [NoZeroSMulDivisors A B]
Goal: differentIdeal A B ≤ I ↔ Submodule.map (↑A (Algebra.trace K L)) (Submodule.restrictScalars A ↑(↑I)⁻¹) ≤ 1
-- traceForm_dualSubmodule_adjoin
lemma traceForm_dualSubmodule_adjoin
    {x : L} (hx : Algebra.adjoin K {x} = ⊤) (hAx : IsIntegral A x) :
    (traceForm K L).dualSubmodule (Subalgebra.toSubmodule (Algebra.adjoin A {x})) =
      (aeval x (derivative <| minpoly K x) : L)⁻¹ •
        (Subalgebra.toSubmodule (Algebra.adjoin A {x})) 
Arguments: (A : Type u_1) (K : Type u_2) {L : Type u} [CommRing A] [Field K] [Field L] [Algebra A K] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsDomain A] [IsFractionRing A K] [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsIntegrallyClosed A] {x : L} (hx : Algebra.adjoin K {x} = ⊤) (hAx : IsIntegral A x)
Goal: (Algebra.traceForm K L).dualSubmodule (Subalgebra.toSubmodule (Algebra.adjoin A {x})) =
  ((Polynomial.aeval x) (Polynomial.derivative (minpoly K x)))⁻¹ • Subalgebra.toSubmodule (Algebra.adjoin A {x})
-- conductor_mul_differentIdeal
lemma conductor_mul_differentIdeal [NoZeroSMulDivisors A B]
    (x : B) (hx : Algebra.adjoin K {algebraMap B L x} = ⊤) :
    (conductor A x) * differentIdeal A B = Ideal.span {aeval x (derivative (minpoly A x))} 
Arguments: (A : Type u_1) (K : Type u_2) (L : Type u) {B : Type u_3} [CommRing A] [Field K] [CommRing B] [Field L] [Algebra A K] [Algebra B L] [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L] [IsDomain A] [IsFractionRing A K] [IsIntegralClosure B A L] [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsIntegrallyClosed A] [IsDedekindDomain B] [NoZeroSMulDivisors A B] (x : B) (hx : Algebra.adjoin K {(algebraMap B L) x} = ⊤)
Goal: conductor A x * differentIdeal A B = Ideal.span {(Polynomial.aeval x) (Polynomial.derivative (minpoly A x))}
-- aeval_derivative_mem_differentIdeal
lemma aeval_derivative_mem_differentIdeal [NoZeroSMulDivisors A B]
    (x : B) (hx : Algebra.adjoin K {algebraMap B L x} = ⊤) :
    aeval x (derivative (minpoly A x)) ∈ differentIdeal A B 
Arguments: (A : Type u_1) (K : Type u_2) (L : Type u) {B : Type u_3} [CommRing A] [Field K] [CommRing B] [Field L] [Algebra A K] [Algebra B L] [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L] [IsDomain A] [IsFractionRing A K] [IsIntegralClosure B A L] [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsIntegrallyClosed A] [IsDedekindDomain B] [NoZeroSMulDivisors A B] (x : B) (hx : Algebra.adjoin K {(algebraMap B L) x} = ⊤)
Goal: (Polynomial.aeval x) (Polynomial.derivative (minpoly A x)) ∈ differentIdeal A B
-- Matrix.J_transpose
@[simp]
theorem J_transpose : (J l R)ᵀ = -J l R 
Arguments: (l : Type u_1) (R : Type u_2) [DecidableEq l] [CommRing R]
Goal: (Matrix.J l R)ᵀ = -Matrix.J l R
-- Matrix.J_squared
theorem J_squared : J l R * J l R = -1 
Arguments: (l : Type u_1) (R : Type u_2) [DecidableEq l] [CommRing R] [Fintype l]
Goal: Matrix.J l R * Matrix.J l R = -1
-- Matrix.J_inv
theorem J_inv : (J l R)⁻¹ = -J l R 
Arguments: (l : Type u_1) (R : Type u_2) [DecidableEq l] [CommRing R] [Fintype l]
Goal: (Matrix.J l R)⁻¹ = -Matrix.J l R
-- Matrix.J_det_mul_J_det
theorem J_det_mul_J_det : det (J l R) * det (J l R) = 1 
Arguments: (l : Type u_1) (R : Type u_2) [DecidableEq l] [CommRing R] [Fintype l]
Goal: (Matrix.J l R).det * (Matrix.J l R).det = 1
-- Matrix.isUnit_det_J
theorem isUnit_det_J : IsUnit (det (J l R)) 
Arguments: (l : Type u_1) (R : Type u_2) [DecidableEq l] [CommRing R] [Fintype l]
Goal: IsUnit (Matrix.J l R).det