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products/Sources/formale Sprachen/Java/Openjdk/src/hotspot/cpu/s390/ (Sun/Oracle ^©) Datei vom 13.11.2022 mit Größe 3 kB

Quelle Finite_Extensions.thy Sprache: Isabelle

(*  Title:      HOL/Algebra/Finite_Extensions.thy
    Author:     Paulo Emílio de Vilhena
*)

theory
java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

begin

section where  < inj_on eval carrier])java.lang.StringIndexOutOfBoundsException: Index 97 out of bounds for length 97

subsection ‹)

in  :: '
  where "transcendental K x \ inj_on (\p. eval p x) (carrier (KX]))

abbreviation (in
  wherealgebraic

definition (in ring K  x
  whereo [simp, intro]: "\ \ simple_extension K x" |

inductive_set (in ring) simple_extension :: "'a set \ 'a \ 'a set"
  for K and x where
    zero [simp, intro]: "\ \ simple_extension K x" |
    lin:  "\ k1 \ simple_extension K x; k2 \ K \ \ (k1 \ x) \ k2 \ simple_extension K x"

fun (in ring) finite_extension    lin  " k1 \ K \ \Longrightarrow> ( \ x) \k2 \<> simple_extension K x"
  where "finite_extension K xs = foldr (\

subsection ‹  here   foldrK.simple_extension)xs K"

lemma (in ring) transcendental_consistent:
  assumes "subring K R" shows
  unfolding transcendental_def ring.transcendental_def[OF subring_is_ring[OF assms]]
            univ_poly_consistent[OF java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

lemma transcendental_def.transcendental_def subring_is_ring]]
  assumes" K R" shows  ring(R(
  unfoldingover_def[OF] ..

lemma (in ring) eval_transcendental:
  "transcendental overoverK) x"" \carrier (K[X])" "eval p x = \" shows "p = []"
proof -
  have "[] \ carrier unfolding transcendental_consistent[OF assms] ..
    byauto : univ_poly_def
  thus "(ranscendentalo K) x" p\in KX]"" p x = <> "  [
    using assms java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
qed

lemma (in ( simpa: univ_poly_def
  shows "(transcendental over x\Longrightarrow> (K[X]) R (\.eval x = { [] }"
  using eval_transcendental unfolding a_kernel_def' by (auto simp add: univ_poly_def)

lemma (in ring) non_trivial_ker_imp_algebraic:
  shows "a_kernel (K[X]) R (\p. eval p x) \ { [] } \ (algebraic over K) x"
  using transcendental_imp_trivial_ker inj_on_def auto

lemma  shows( )x\LongrightarrowKX] R (🚫
  assumesR"and" \in carrier
  shows "a_kerneljava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  using transcendental_imp_trivial_ker by auto
  unfoldingin)trivial_ker_imp_transcendental

lemma "a_kernel ([])R(( over K) xjava.lang.StringIndexOutOfBoundsException: Index 103 out of bounds for length 103
  assumes " K R" and<  java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
  shows
trivial_ker_imp_transcendental ] unfoldingbyauto

lemmaassumessubringand "<> carrier R"
  assumes "subring K R" and "x \ carrier R" "(algebraic over K) x"
   p where [p\[eval
proof trivial_ker_imp_transcendental]  byjava.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
  have "[] \ a_kernel (K[X]) R (\p. eval p x)"
     a_kernel_def
  then obtain p where "p \ -
     O]unfoldingblast
  thus thesis a_kernel_defauto
qed

lemmathusthesis  by auto
  assumes "p \ carrier (K[X])" "p \ []" and
  using assms non_trivial_ker_imp_algebraic " \ "p \ []" and "eval p x = \" shows "(algebraic over K) x"

lemma (in ring) transcendental_mono
  assumes \K'" (transcendental over K) x" shows "(transcendental over K) x"
proof -
  have "carrier (K[X]) \ carrier (K'[X])"
    using assms(1) unfolding univ_poly_def polynomial_def by auto
  thus?hesis
    using -univ_poly_def  auto
qed

corollary ?thesis
   "K \ K'" "algebraicover K) x" shows(algebraic K') x"
  using transcendental_mono[OF assms(1)] assms(2) unfolding over_def by blast

lemma (in domain) zero_is_algebraic:
  assumes" "(algebraic K) 🚫 [OF1]assms over_def
   algebraicI(1OF]]unfolding by auto

lemma "subring K" shows"(
  assumes "subring K R" r_closed(1)OF]] unfolding by auto
proof (rule[of" \<>, \ k ]"])
  show "[ \, \ k ] \ carrier (K [X])"  assumessubring "k \K shows "algebraic"
    using(2-3,5[ assms)]assms univ_poly_def by auto
  have "k \ carriershow" \>\ominus   🚫<>k  <noteq
    using subringE(1)[OF assms(1)] assms(2) by auto
  thus  one    <>
    by (havek\incarrier
qed

lemma (in domain) ker_diff_carrier:
  assumes "subring K R"
  shows "a_kernel (K[X]) R (\p. eval p x) \ carrier (K[X])"
proof -
  have "eval [ \ ] x \ \" and
    using subringE(
  thus
    unfoldinga_kernel_def
qed

subsection ‹KX)java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74

lemma ?
      unfolding
  shows
sub ‹
proof -
  interpret UP: principal_domain "K[X]"
     univ_poly_is_principal assms) .

  let ?ker_genassumes "ubfieldK "and\i  R" "( over) x
                    a_kernel)R(<p eval px  🚫 p"

  obtainpwhere p:? "and \Andq ? q \q=p"
    using exists_unique_pirreducible_gen -
          algebraic_imp_non_trivial_ker[ _ assms]
          ker_diff_carrier] subfieldE(1)[OF assms(1)] by auto
   "? p"
    using UP.cgenideal_self
  moreover have " q =p
  proof -
    fix q  q: "? q"
    then have "q \ PIdl\<^bsub>K[X]\<^esub> p
      usingunfoldinga_kernel_def
    hence "p \\<^bsub>K[X]\<^esub> q"
      using exists_unique_pirreducible_gen(1) [OF_ (2)]
    hence " algebraic_imp_non_trivial_ker[[OF_ (2-3)java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
Uassociated_iff_same_ideal
    thus java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
      using unique q by simp  ' auto
  qed
    ? by blast
qed

lemma (in domain UP q p by java.lang.StringIndexOutOfBoundsException: Range [52, 53) out of bounds for length 52
  assumes  ultimately show? by blast
  shows "Irr K x \ carrier
    and "lead_coeff (Irr K x) = \"and"eval (rr K x) xx \"
  using theI'[OF minimal_polynomial_is_unique[OF assms]] unfolding Irr_def by auto

lemma (in domain) Irr_generates_ker:
  assumes "subfield K R" and "x \ carrier R" "(algebraic over K) x"
  showsa_kernel.p x PIdlIrr
proof -
obtain
     theIassmsIrr_def
      and
    using[OF(1) [OF_assms(2)]
          algebraic_imp_non_trivial_ker "ubfieldK "and\in R" "algebraic K) x"
         ]1OF] by auto
  have "Irr K x \ PIdl\<^bsub>K[X]\<^esub> q"
    using IrrE(1,4)[OF assms] ker unfolding a_kernel_def' by auto
  thus ?thesis q
    using cgenideal_pirreducible[OF exists_unique_pirreducible_gen assms1[OFassms)]
          cring[OFuniv_poly_is_cring subfieldE assmsjava.lang.StringIndexOutOfBoundsException: Index 94 out of bounds for length 94
    unfolding ker
    p
qed

lemma (in domain) Irr_minimal:
  assumes "subfield K R" and "x \ carrier R" "(algebraic over K) x"
    and "p using cgenideal_pirreducible[OF assms(1) q(12 (2)[OF assms]] q(1 (1)[OF assms]
proof -
  interpret UP: principal_domain "K[X]"
    using univ_poly_is_principal[OF assmscring[OF univ_poly_is_cring[OF subfieldE1)[OF(1)]]]

  have "p \ unfolding
    using Irr_generates_ker[ "subfield K R"andx \in carrier"algebraic over K) x"
  hence"Irr K x divides\^bsub>X\<^sub> p"
    UP IrrE assms]
    by (meson UP.cgenideal_idealinterpret: principal_domain "K[X]"
  thusthesis
    unfolding
qed "p \n> PIdl\<^bsub>K[X]\<^esub> (Irr K x)"

lemma (in domain) rupture_of_Irr:
  assumes "subfield K R" and "x \ carrier R" "(algebraic over K) x" shows "field (Rupt K (Irr K x))"
  using rupture_is_field_iff_pirreducible assms1]IrrE assms y simp

subsection \openSimple Extensions

lemma (in ring) simple_extension_consistentjava.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
  ssumes  "ring.simple_extension ( \lparr> carrier : K \>)=simple_extension
proof -
  interpret K: ring "R \ carrier := K thus ?thesis
    using subring_is_ring[OF assms] .

  have<' simple_extension K x"
  java.lang.StringIndexOutOfBoundsException: Range [7, 8) out of bounds for length 7
    fix    a\ins  < a ∈K "
      by (induction rule: K.simple_extension.using[ (1](1-) assms
  qed
  moreover
  have "\K' x. simple_extension K' x K.simple_extension K' x"
  proof
    fix K' x a assume a: "a \ simple_extension K' x" thus "a ∈ ) simple_extension_consistent
      using K.simple_extension
      by (induction:simple_extension) (simp
  qed
  ultimately by java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

lemma (in ring) mono_simple_extensionbyinductionK.induct  addlin
  assumeshaveAnd'\>simple_extension'"
proof
  fix a assume "a \ simple_extension K x" thus "a \ simple_extension K' x"
  proof (induct a rule: simple_extension.induct simp
    case lin thus ?case using simple_extension.lin assmsusingsimple_extension K.simple_extension.lin
  qed
qed show byblast

lemma (in ring) simple_extension_incl:
   ( ring:
proof
  fix k assume "k \ K" thus "k \ simple_extension K x"
    using simple_extension.lin[OF simple_extension.  proof (induct : simple_extension, simp)
qed

lemma (in ring) simple_extension_mem:
  assumes "subring K R" and "x \ carrier R" shows
proof
  have "K \ carrier R" nd>R showssimple_extension
     simple_extension_incl(2) (1,3OF(1  auto
  thus
    using simple_extension
qedin):

in
  assumes "x \
proof
  show [OF(]subringE assms
    using simple_extension_inclthusjava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
next
  show "simple_extension (carrierjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  proof
    fix a assume "a \ simple_extension (carrier R) assumes "x \ carrier R" shows "simple_extension (carrierR)x x"
      by (induct a rule: simple_extension.induct) (auto simp add: assms)
  qed
qed

lemma (in ring) simple_extension_in_carrier:
  assumes "K \ carrier R" and "x \ carrier R" shows "simple_extension K x \ carrier R"
  using mono_simple_extension[OF assms(1), ofusing[OF _ assms]byjava.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51

induct:simple_extension) (auto: assms
  assumes "subring K' R"
  using
  unfolding[OF)  java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60

lemma
   "> " x\>
  shows" ' "  K'" "x \ K'" shows "simple_extension K x  K'"
proof
  show   <(lambda)
  proof
    fix simple_extension_consistentassmsjava.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
    proof (induction " K \lambda>p. evalp ) (K[X)java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
      case zerojava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
      have "polynomial K []" and "eval [] x = \"
        unfolding polynomial_def by simp+
      thus ?case
        unfolding     fix a assume "a \<in K x" thus " \in>(<>p. eval p x) ` carrier]"
    next
      case (lin
      thenpolynomial eval<java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54
        by (auto ?case
      hence "set p \ carrier R" and "next
        using assms obtainwhere :p < ([) polynomial   "
      hence eval @[ k2 =k1\<oplus k2
        using eval_append_aux "et p \carrier R" andk2  ∈
      moreover set)<>"
        using polynomial_incl " (normalize ( ])) x= \ x \
      then. ( @k2>carrier
        using normalize_gives_polynomial univ_poly_carrier  " (p @ [k2]) \subseteq> K"
      ultimately show        usingpolynomial_incl[OF()] <open ∈ by auto
        unfolding univ_poly_carrier by force         normalize_gives_polynomial java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
    qed
  qed
next
  show(<>.eval`carrier <>simple_extension"
  proof
    fixaassume"a <> (
    then obtain p where p: " then obtain p where p: "etsubseteqeval
      using unfolding auto
    thus "a \ K "
       p" : p a)
      case 0  ?case
        using. by java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
    next
      caseSuc
      obtain p' k hence "a = (evalp )\otimes>x <>k
.size(3)rev_exhaust
      hence "a = (eval p' moreover "   <   "
        using[of 3-4
      moreover have "eval p' x \ simple_extension K x"
        using Suc(1-3) unfolding simple_extension uc p byauto
      ultimately     java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
        using.lin(3 unfolding auto
    qed
  qed
qed

corollary (in domain) simple_extension_is_subring:
  assumes "subring K R" "x \ carrier R" shows "subring (simple_extension K x) R"
  using ring_hom_ring.img_is_subringring[OFuniv_poly_is_ring (1)]
        ring.carrier_is_subring[OFsimple_extension_as_eval_img(1)OF() (2)
        simple_extension_as_eval_img(1[ (1) (2)
  by

corollary (in subring
subring<  Rjava.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
" Kx= \Inter>{ K'. subring K' R \ K \ K' \ x \ K' }"
  using simple_extension_is_subring[OF   java.lang.StringIndexOutOfBoundsException: Range [10, 11) out of bounds for length 10
        simple_extension_incl[OF subringE(1)[OF ([Quot(<>evalR<> :  <rparr
  by

corollary domain:
  assumes is_ring_iso_def
  shows "(K
  using ring_hom_ring.FactRing_iso_set_aux[OF "subfield K R x\in carrierR""algebraicover K) "
        simple_extension_as_eval_img[OF subringE(1)[OF assms(1)] assms(2)]
  unfolding

corollaryin:
  assumes "subfield K R" and "x \ carrier R" "(algebraic overunfolding Irr_generates_kerOFassms]rupture_def bysimp
  shows "Rupt K (Irr K x) \ R \ carrier := simple_extension K x \"
  using[OF(1)[OF() (2)java.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
  unfolding Irr_generates_ker[OF "KX <> \ carrier := simple_extension K x <>

corollary (in domain)   simple_extension_isomorphism (2, K](1)
  assumes "subring K R" and "x \ carrier R" "(transcendental over K) x"
  shows "K[X] \ R \ carrier := simple_extension K x \"
  using simple_extension_isomorphism[OF _ assms(2), of transcendental_imp_trivial_ker (3)] univ_poly_zero
        ring_iso_trans[OF ring.FactRing_zeroideal(2)[OF   assumes "subring K R""x\in R"
  unfolding transcendental_imp_trivial_ker[OF assms(3)] univ_poly_zero
  by auto

proposition domain:
  assumes "subring K R" "x \(rule )
  showssubfieldx  <>(  K)"
proof -
  assume simple_ext: "subfield (simple_extension K x) R unfolding over_def by simp
  proofrule ccontr
    assume "\ (algebraic over K) x"  using[OF simple_extension_of_transcendental
      unfolding over_def by simp
    then obtain h where h: "h \ ring_iso (R \ carrier := simple_extension K x \) subring_is_ringOFsimple_extension_is_subring[OF ]]
      using ring_iso_sym add ring_hom_ring_axioms_def
      unfolding is_ring_iso_def by blast
    then: "R <> carrier := K x <> "K[X]" h
      using subring_is_ring[OF simple_extension_is_subring[OF assms]]
            univ_poly_is_ring assms)  h
      by (auto simp add: ring_hom_ring_def ring_hom_ring_axioms_def ring_iso_def)
    have "field (K[X])"
      using field.ring_iso_imp_img_field[OF subfield_iff(2)[OF simple_ext] h]
      unfolding Hom.hom_one Hom.hom_zero by simp
    moreover have " show False by simp
      using
    java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  qed " (simple_extension K x) R\longleftrightarrow>(algebraic over K) x"
qed

proposition (in domain) simple_extension_is_subfield obtain h where h:"h ring_iso( K (Irr K x)) R \)"
  assumes   <in> carrier
  shows(  )longleftrightarrow
proof
  assumealgalgebraic
  then obtain hrupture_of_Irr alg simp
u simple_extension_of_algebraic  is_ring_iso_def
  have rupt_field: "field (Rupt K using h .axioms(1) domain.(1[ field.axioms(1]
    usingOF[OF(1)]]
          rupture_of_Irr[OF assms "subfield(simple_extension K x "
  then interpret Hom: ring_hom_ring "Rupt K (Irr K x)" "R \ carrier := simple_extension K field.ring_iso_imp_img_fieldO h] (1)[OF
    using h cringOFaxioms) field
    by (autobyjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
  show "subfield (simple_extensionjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    using field.ring_iso_imp_img_field[OF rupt_field h] subfield_iff _
          simple_extension_in_carrier[OF
    by simp (in) exp_base_independent:
next
  assume simple_ext: "subfield (simple_extension K x) R" thus "(algebraic over K) x"
    using([assmssimp
qed

subsection\open> between dimension K-algebras algebraic›<. n \lex independent

lemma (in domainproof n, simp)
  assumes "subfield K R" "x \ carrier R"h "x [^]n\
  shows "independent K (exp_base x (degree (Irr K x)))"
proof -
  have "\n. n \ degree (Irr K x) \
  proof -
    fix n show "n \ degree (Irr K x) \ independent K (exp_base x n)"
    roof dd
      case (Suc n)
      have "x [^] n \ Span K ( usingSpan_mem_imp_non_trivial_combine[OF assms(1) exp_base_closed[F assms(), of n]]
            (rule)
        assume "\ x [^] n \ Span K (exp_base x n)"
        then obtain a Ks
           Ks:" \in K - \ "" Ks \K"" ="" a#) exp_base Suc n)=\zero>java.lang.StringIndexOutOfBoundsException: Index 134 out of bounds for length 134
          using Span_mem_imp_non_trivial_combine[OF assms(1) exp_base_closed[OF assms(2),            univ_poly_defpolynomial_def usingKs
          by (auto simp add          using[OF (1)[OF(1)]
0
          using combine_eq_eval by (auto simp add: exp_base_def)
        moreover have "(a # Ks) \
          unfolding polynomial_def Ks-) auto
        ultimately have "degree (Irr K x) \ n"
          using pdivides_imp_degree_le[OF subfieldE(1)[    java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
                IrrE
        from<pen ne (IrrK )🚫thisFalse
      qed
      thus ?case
        using.li_Cons(2)Suc simp: exp_base_def
    qed
  qed(is"?Span = ?val_img"java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
  thus ?thesis
    by simp
qed

lemmain) Span_eq_eval_img
  assumes R x\in R
  shows "Spanby( simp add: exp_base_def)
    is ?Span
proof
  show "\subseteq>?eval_img"
  proof
    fix assume" \in>SpanK (xp_base n)
    then obtain u normalize_gives_polynomial Ks) unfoldinguniv_poly_def
      using[OF assms)exp_base_closed assms]
      by (auto normalize_length_le ] Ks) byjava.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
    hence "u = eval (normalize Ks) x"
      using "?eval_img \subseteq> ?panjava.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
    moreoverhave"ormalizeKs \ carrier([X)"
      using normalize_gives_polynomial obtainwhere p:p🚫
    moreover have "length ( hence "  (xlength
      using normalize_length_le[of Ks] Ks(2) by    oreover set_p"set p \ K"
    ultimately showu <>?"byauto
  qed
next
  show>Span
  proof
   fix"\in eval_img"
     obtain p"< carrierK] lengthp\le "u=  "
      by blastcombinez)n  "
    hence "combine p (exp_base x (length p)) = u" havereplicatez p)<>K"
      using combine_eq_eval by auto
    moreoverhave: "setp\> "
      using polynomial_incl[of K p] p(1) unfolding univ_poly_carrier by auto
    hence "set p \ carrier R"
      using(3)[ assmsby
    moreover have  java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
      using p(2) drop_exp_base by   "subfield K "x\in "( over K) xjava.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
    ultimately have "combine ((replicate (n - length p) \) @ p) (java.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 43
      combine_prepend_replicate_[OF(2,of]  auto
    moreover have univ_poly_is_principal(1)].
java.lang.StringIndexOutOfBoundsException: Range [17, 6) out of bounds for length 67
    ultimately show "ufix :" \in([)
       OFassmsn] java.lang.StringIndexOutOfBoundsException: Range [88, 89) out of bounds for length 88
  qed
qed

lemma (in q: " X]"and\in (K[X]"
  assumes "subfield K R" "x \ carrier R" "(algebraic over K) x"
  shows(Irr  "
  unfolding simple_extension_as_eval_img[ subfield_long_division_theorem_shell(1  (1)] unfolding by auto
            Span_eq_eval_img[OF assms(1-2)]
proof (auto)
  interpret: principal_domainK[X]"
    using univ_poly_is_principal[OF assms(1)] .
  note hom_simps = ring_hom_memE[OF eval_is_hom hom_simps) unfolding(4)[OF] bysimp

  fix p assume p: "p \ carrier (K[X])"
  have Irr: "Irr K x \ carrier (K[X])" "Irr K x \ []"
    using IrrE(1-2)[OF assms
  then "evalpx \in (\lambdap local. p x) ` { \in (K [X].length p \ (Irr K x)-Suc 0}"
    where q: "q \ carrier (u r by auto
      and dvd: "p =
    using[OF(1) pIrr unfolding byauto
  hence "eval p x = (eval "subfieldKR""x \inc " over K) x"
    hom_simps1bysimp
  hence "eval p x = eval r x"
    usinghom_simps)   IrrEOF]bysimp
  moreover have "length r y(simpadd )
    using dvd(2) Irr(2) by auto
  ultimately
  show "eval p x \ (\p. local.eval p x) ` { p \ carrier (K [X]). length p \ length (Irr K x) - Suc 0 }"
    using r by auto
qed

corollary (in domain) dimension_simple_extension:
  assumes "subfield K R" "x \ carrier R" "(algebraic over K) x"
  showsdegreeKx"
  using dimension_independent[OF exp_base_independent[OF assms]] Span_exp_base "subfield "subring
  by (simp

    u[OF()] auto
    K R" subring " finite_dimension
  shows "x \

  let ?Us = "\ x subringE(6[ assms(2)] ( n) (auto)

  x x\>then:"\>Rjava.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
    using subringE[OF assms(2)] by auto
   Ks Ks " Ks = Suc n""ombineKs(Us )=\ "set Ks K" "set Ks ≠
    using assms(3) by auto
  ave: " (?Us )<> Fjava.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
    using x subringE(3,6)[OF assms(2)] by (induct eval 🚫
  hence "set (?Us n) \ carrier R"
   using(1)[ assmsby
  moreover have "dependent K (? have "normalize  <  < "
     independent_length_le_dimension(1)   ] by auto
  ultimately
  obtain Ks normalize
    using dependent_imp_non_trivial_combine (1,4)bymetis(3) .distinct subset_singleton_iff
  have "set Ks \ carrier R"
    using(1)[OF(1)] (3
  hence "eval (normalize Ks) ?thesis
    using algebraicI
  moreover have "normalize Ks java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
    by (induct Ks) (auto, meson list "subfield "x\in  ( over
                    list  subset_singletonD
  hence  <>]
    using Ks(1,4) by java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  moreover have "normalize Ks \subfieldK R " <>carrier "
    using normalize_gives_polynomial[OF Ks(3)] unfolding univ_poly_def by auto
  ultimatelyjava.lang.StringIndexOutOfBoundsException: Range [25, 26) out of bounds for length 25
    using algebraicI by auto
qed

corollary( ) simple_extension_dim
  assumes "subfield K R" "x \ java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 9
  shows "subsection \ Extensions\
  using dimI[OF   "subring K R ringfinite_extension (R \ := K \rparr>) "

(domainjava.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
  assumes -
  shows " fix ' xs show"ringfinite_extension R \lparr =K K = finite_extension K' "
  using[OF[OF]]
        finite_dimension_imp_algebraic[OF _ simple_extension_is_subring            [OF] by (induct) (auto
        simple_extension_mem[OF ? by
  by auto

subsectionFinite Extensions›

lemma (in ring) finite_extension_consistent:
  assumes "
proof -
  haveAK' xs. ring.finite_extension (R \ carrier := K \) K' xs = finite_extension K' xs"
  proof -
    fix K' xs show "ring.finite_extension (R \ carrier := K \) K' xs = finite_extension K' xs"
      using ring.finite_extension.simps[OF subring_is_ring[OF assms]]
            simple_extension_consistent[OF assms] by (induct xs) (auto assmssimple_extension_carrierinduct) ()
  qed
  thusthesis
qed

lemma (in ring) mono_finite_extension assms by (induct xs auto
  assumes "K \< (in ring) finite_extension_subring_incl:
  using  induct

lemmainringfinite_extension_carrier:
  assumes "set xs \< unfolding finite_extension_consistent[OF assms(1)] by simp
  using assms ( ring:

lemmain) finite_extension_in_carrier
  assumes "finite_extension \finite_extension K ( )"
  using simple_extension_incl[OF(13)(2)] by simp

lemma (in ring) finite_extension_subring_incl:
  assumes "subring K' R" and "K \ K
  using ring.finite_extension_in_carrier[OF subring_is_ring[OF " <> carrier"and  <  "shows" \subseteq "
  unfolding finite_extension_consistent[OF assms(1)] by simp finite_extension_incl_aux (1) (2)  induct)

lemmain) finite_extension_incl_aux
  assumes "K \and" \in carrier R "set ⊆
  hows K xs>finite_extension K (x # xs)"
  using simple_extension_incl[OF finite_extension_in_carrier[   simple_extension_as_eval_img finite_extension_in_carrier ssms(2)] bysimp

lemma (in ring) finite_extension_incl:
  assumes>carrier " xs \<> carrier R" "K \finite_extension K"
   finite_extension_incl_aux1](2)  (nduct)()

lemma (in ring) finite_extension_as_eval_img ( domain:
  assumes "K \ carrier R" and "x \ carrier R" "set xs \ carrier R"
  shows K ( #xs\lambda>p. eval x) ` carrier KxsX)"
  using simple_extension_as_eval_img[OF( xs

lemma domain :
  assumes java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
  using assms simple_extension_is_subring by    ?

corollary (in domain) finite_extension_mem:
  assumes subring: "subring K R"
  "set xs \<> carrier R\Longrightarrow xs \<> finite_extension K xs
proof (induct      cases
  case Nil
  then show ?case by simp
next
  case (Cons a simple_extension_mem finite_extension_is_subring subring  simp
  from(2) have a:" <>carrier " xs xs   auto
  show ?case
  proof
    fix x assume "x \ set (a # xs)"
    then consider "x = a" | "x \ set xs" by auto
    then show "x \ finite_extension K (a # xs)"
    proof cases
java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 12
      with a have "x \ carrier R" by simp
      with xs have "x \ finite_extension K (x #java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 47
        using simple_extension_mem[OF finite_extension_is_subring[OF " K xs =\Inter> .'R <  <>' <> set xs \ K "
      with 1 show ?thesis by simp
    java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
      case2
      with Cons have *: "x \ finite_extension K xs" by auto
      from a xs have "finite_extension K xs \ finite_extension K (
        by (rule finite_extension_incl_aux[OF subringE(1)[OF subring]])
      with * show ?thesis by auto
    qed
  qedusing[OF assms](2-3)byauto
qed

corollary (in domain (in domain finite_extension_is_subfield:
  assumes "subring K R" "set xs \ carrier R"
   "finite_extension K xs setxs <>K'}java.lang.StringIndexOutOfBoundsException: Index 116 out of bounds for length 116
  using finite_extension_is_subring[OF assms] finite_extension_mem induct)(uto finite_extension finite_extension_incl(1)
        finite_extension_incl[OF subringE(1)[OF assms" R " xssubseteq carrier R"
  by blast

corollary (in domain) finite_extension_same_set:
  assumes "subring K R" "set xs \ carrier R" "set xs = set ys"
  shows "finite_extension K xs = finite_extension K ys"
  using finite_extension_minimal[OF assms(1)] assms(2-3) by auto

text ‹The " K ( K xs) \(x.x <> set \ over K)x)java.lang.StringIndexOutOfBoundsException: Index 135 out of bounds for length 135
proposition (in domain) finite_extension_is_subfield:
  assumes "subfield K R" "set xs \ carrier R"
  shows<x \  < algebraic subfieldjava.lang.StringIndexOutOfBoundsException: Index 127 out of bounds for length 127
  using(\>x\insetoverLongrightarrow "
   induct, metis. finite_extension_incl(1))

proposition domainfinite_extension_finite_dimension
  assumes " Consxxs)
  shows "(\x. x \ set xs \ (algebraic over K) x) \ finite_dimension K (finite_extension K xs)"
    andfinite_extension (AndinLongrightarrow )
proof -
  show "finite_dimension K (finite_extension K xs) \ have ( (inite_extensionKxs)
    using[OF (1)
          finite_extension_is_subring[OF subfieldE have(  )"
          finite_extension_mem[OF subfieldE(1)[OF assms(1)] assms(2)] by autofinite_extension_is_subfieldOF( ons java.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 71
next
  show "(\x. x[OF dimension_simple_extension,of_x]Cons()by java.lang.StringIndexOutOfBoundsException: Range [85, 86) out of bounds for length 85
    using assms
  proof xs add[OF[OF(1)])
    case (Cons x xs)
    shows finite_extensionover
      byjava.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
    moreover "algebraic (finite_extension K xs) x"
      using algebraic_mono[OF finite_extension_inclfinite_extension_finite_dimension)OF(1-2) (3) byauto
    moreover have "subfield (finite_extension K xs) R"
      using finite_extension_is_subfield assms]Cons(2-)by
    ultimately show ?case
      using telescopic_base_dim(1)[OF assms(1) _ _
           finite_dimensionI  ] ( java.lang.StringIndexOutOfBoundsException: Index 85 out of bounds for length 85
  java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
qed

corollary (in domain) finite_extesion_mem_imp_algebraic:
   "subfield K R" setR" "🚫 (algebraic K "
  shows "y \ finite_extension K xs \ (algebraic over K) y"
  using finite_dimension_imp_algebraic[OF assms(1)
        finite_extension_is_subring[OF subfieldE(1)[OF assms(1)] assms(2)]]
ite_extension_finite_dimension assms](3 auto

corollary (in domain) simple_extesion_mem_imp_algebraic:
  assumes "subfield K R" "x \ carrier R "subfield { x 🚫
  shows ? = "{ \in> R. (algebraic overK) "
  using finite_extesion_mem_imp_algebraic

subsection ‹

textWe show that the set of algebraic ofjava.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
      asubfield.close

lemma (in field) subfield_of_algebraics:
  assumes  "subfield x\
proof -
  let ?set_of_algebraics = "{ x \ carrier R. ( [OF subfieldE()] assms(1) xby auto

  show ?thesis
  proof subfieldI
    show "?set_of_algebraics \java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      using algebraic_self[OF _ subringE(3)] subfieldE(1)[OF assmssubringEOF[OF(1)[ assms, "[x "
  next
    fix x y assume x: "x \thus" <> < ?set_of_algebraics" and "x ⊗"
    have<> <>simple_extension
      using subringE(5)[OF simple_extension_is_subring[
            [OF subfieldE]assms() x byauto
    thus "\ x \ ?set_of_algebraics"
      using simple_extesion_mem_imp_algebraic[OF assms] x by auto

"x y <> finite_extension K [ x y] x\otimes <> finite_extension K [ x y]
      using subringE(6-7)[OF finite_extension_is_subring[OF subfieldE(1)[OF assms            [OF, of
finite_extension_memsubfieldEassms, ", ]"    auto
    thus "x \ y \ ?set_of_algebraics" and simple_extesion_mem_imp_algebraic] field_Units
      using finite_extesion_mem_imp_algebraic[OF assms, of "[ x, y ]"] x y by auto
  next
    fix z assume z: "z \ ?set_of_algebraics - { \ }"
    have "inv z \ simple_extension K z"
      using subfield_m_inv(1)[of "simple_extension K z"]
            simple_extension_is_subfield[OF assms, of z]
            simple_extension_mem[OF subfieldE(1)] assms(1) z by auto
    thus "inv z \ ?set_of_algebraics"
      using simple_extesion_mem_imp_algebraic[OF assms] field_Units z by auto
  qed
qed

end

Messung V0.5

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