Quelle  Dense_Linear_Order.thy   Sprache: Isabelle

(*  Title       : HOL/Decision_Procs/Dense_Linear_Order.thy    
    Author      : Amine Chaieb, TU Muenchen
*)

*)
  andsection🚫elimination     \close

theoryferrante_rackoff_data.ML›\>and 

begin

ML_filelangford_data.ML›>
ML_filejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

context linorder
begin

lemma less_not_permute[no_atp]: "\ (x < y \ y < x)"
  by (simp add: not_less linear)

[
  " \<>x.(y \ (insert l L). y < x)\<> (<>y \in> . < y)"
       auto)  <y\iny  < \forall< }   )<> )
  "(\x. (\ by simp
🚫 (insert l L).y <x)\and(<>y \in.x<y < P x)"
  "(\x. (\text\Theorems for \open>z \forall>x x< z ( \longleftrightarrow P\^sub>-\)\<>\<>
    (∃[]: "existsz." for 🚫
  "(\x. (\y \ Llemmapinf_gtno_atp:"
    ( auto
  by

lemma [}. x < y) ∧ P x)"
  by simp

text‹Theorems for ‹∃z. ∀x. x < z ⟶ (P x ⟷ P🚫-java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "macro" is null
lemma minf_lt[no_atp]: "\z. \x. x < z \ (x < t \ True)" 
  by auto
lemma minf_gt[no_atp]: "\z. \x. x < z \ (t < x \ False)"
  by (simp add: not_less) (rule exI[where x="t"], auto simp add: less_le)

lemma(<existsx (\forall ∈ (insert lL).y<x ∧ (∀y ∈ U. x < y) ∧ P x)"
  by (auto simp add: less_le)
lemma minf_ge[no_atp]: "\z. \x. x (t \ x \ False)"
  by (auto simp add: less_le not_less not_le
lemma minf_eq(🚫forall ∈ L. y < x) ∧(<>y ∈uU) x  ))
    (\exists.(forall\in .  )<>\forall \in>.x< <>
 minf_neq[]: "z. \.x (x \noteq> t \longleftrightarrow>True)
  by auto
lemma minf_P[no_atp]: "\z. \x. x < z \ (P byauto
  by

text simp
lemma text‹∃z.\forallx.   z  \longrightarrow>( x 🚫-🚫<>)<>\close>
  by auto
lemma pinf_lt[no_atp]: "\z. \x. z < x \ (x < t \ False)"
  by minf_lt[o_atp]: "x x z \longrightarrow>(x t \longleftrightarrow> )" 

lemma pinf_geno_atp]:"\ ( \x  
  by (auto simp add: less_lele minf_gt[no_atp:"\exists>. False"
lemma  bysimpaddnot_lessruleexIwherex""] auto add:less_le
autosimp add  not_lessnot_le
lemma (auto add:less_le
lemma minf_ge[]: "\exists>z.\x x z\ False)"
lemma pinf_P by ( simp add less_lenot_less)

lemma nmi_ltno_atp]:"t \x. \not>True \ x< t \ (\u\ U. u \byauto
lemma nmi_gt[no_atp]: "java.lang.StringIndexOutOfBoundsException: Range [0, 24) out of bounds for length 9
  byby

textopenTheorems for ‹x.   <> ( \> P🚫+java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "nextword" is null
  [] " \in U \Longrightarrow> \False \ x t u U. u \ x"by
lemma (simp add: not_less)add not_less) (rule exI[where""],auto add less_le)
lemma pinf_ge[no_atp]: "\z. \x. z < x \ (t \ x \ True)" 
lemma  uto  simpadd less_le)
  orall>x. 🚫 \>\rbrakk>\Longrightarrow
  \forallx \not>(P1\and> P2)∧ x \and  x)\longrightarrow  (<>u\in> U u \le x"byauto
lemma  nmi_disj]: "\. \\not>P1' \P1 x (u x ;
  ∀x.lemma pinf_neq[no_atp]: "\z. \<. \x.. z t \ True)" by auto
  ∀(P1' \ P2') ∧x \orP2)  (∃ U. u ≤)"by auto

lemma  npi_lt[no_atp]: "t \ by (simp add: not_less (rule exI[where x="t", auto simp add: less_le)
lemmanpi_gtno_atp t <in <>\forall  auto
lemma  npi_le   ( simp : less_le)
lemma[no_atp: t<  \Longrightarrowforall\not \and  ≤ x ⟶  
  no_atpt<ULongrightarrow\\ \exists>uuin> u)" by auto
lemma  npi_neq pinf_neq[no_atp] "z. . (x True)"  auto
lemmano_atp\forall P\and   <u< . le  auto
lemma  npi_conj[no_atp]: "\\x. \P1' \ P1 x \ (\u\ U. x \ u) ; \x. \P2' \ P2 x \ (\u\ U. x \ u)\
  ==> nmi_ltno_atp: "t \in>U \Longrightarrow> x \not>rue \<> x < t \ (\exists>u\in> U. byauto
lemma nmi_gtno_atp " \in U \forallx \not>False \ (\
  (simp:)

lemma]
  "<><
      [:t\>Longrightarrow> notFalse     \><uin  auto
by  order)

lemma[]
  "tanmi_conj[]:"lbrakk∀P1' \and> x\longrightarrow>\U.u <> x);
    \forall  .\forallt.l   <>t<u\longrightarrowt\notin>U <>   <>x<  < t < x ⟶y. l   <>y<u\longrightarrow t<"
   antisym_conv3strict_trans

lemma lin_dense_le  nmi_disjno_atp \forall <P1   longrightarrow<u<> >)
  "t\in \Longrightarrow>
    ∀x.\not' P2')\and (P1 x \or>P2 \longrightarrow>(exists>u∈ ) by
  by local.less_le_trans local.less_trans local.not_less

lemma lin_dense_ge[no_atp:
  "
    ∀:t\>🚫x\le u)  auto
  bymetis. local<and> x ≠>< . le "auto

lemma[:"lbrakk\x]: "\brakk∀P1 \and P1 x ⟶> .x\le )<xnotandx >u .x<> <

    🚫 <>.\notP1\or>' <> P1 or 2 \longrightarrow \>\in U. x\le> u" by auto
   

lemmalin_dense_neq]
  \>lu >l<t<><u⟶ U <><x <> x  u\and> t longrightarrow\forall>. l<y\and>y<\longrightarrow> yt
java.lang.StringIndexOutOfBoundsException: Range [6, 4) out of bounds for length 212
  auto

lemma lin_dense_P[no_atp
  mma  nmi_neq]" \in> U \x.\True\ x \ t \\ \u\ U. u \le>x"byauto
  "
    \forall> llu.(\foralltl t\and    \>t<> U)) <  < x and> x  u\and t  x⟶>y.   y <and  u <> t   y"
lemma[
lemmano_atp:\>x. notP1 <>>>(<>in.\le);
  ⟶ (∀y. l < y ∧ y < u ⟶ P1 y) ;
  ∀ lin_dense_le[no_atp]
  ⟶t in U Longrightarrow
  ∀ l u.(<>t. <t ∧ t < u ⟶>U) ∧ l < x ∧ x < u ∧ (P1 x ∧ P2 x)
  < (<.l<y\and y  ul> P1y < P2y)
  by blast

lemma lin_dense_disj[  \\<forall>.¬ <>P2 <>(  by( localess_le_trans local.less_translocal)
  "\"<><xlu(<t.l<t \andt< u <> t \notin )∧ x \and   u\and> P1x
  ∀u (forallt l<t < t  u < t\> )\and l  x\andx u <>tlex\longrightarrow(<> l<y <>y< longrightarrowt le y"
  ∀
  ⟶[no_atp:
  <> <existsinU. <x)<>\Longrightarrow
  \    forallx l u. (∀  t\andt<u\longrightarrow t \notin   forall<>P1 \or> ')\and>(P1  orP2)<(<< U ule) by
  by

lemmanpmibndno_atp:\lbrakk><>x. \\¬andP x  \longrightarrow(<xistsu<inU. <>)\forallx\notPP🚫>(<u\>U x \> u)rbrakk
  Longrightarrow <forall\>x.\not \and>\notPP\and>  < \exists>u∈U <>u'<>.u le )
  by autolemma  npi_ge[no_atp:" > U >x. \True \ t \ x \ (\ujava.lang.StringIndexOutOfBoundsException: Range [126, 4) out of bounds for length 212

lemmafinite_set_intervals[no_atp]:
  assumes l> l u. (🚫[no_atp] t\  Longrightarrow \x not >x t<><\\in .x leu byy auto
     lemma[]
    "<[no_atp] \forallx.otP1 \and P1 x (>\U x P2 (\existsin Ux le> u>
    and linS: "\ S"
    and uinS: "u \P2 x uin>>U x <> ) by auto
    and (forally l < y\and y<u < P2 y)<>\Longrightarrow
    and  \Longrightarrow   \<forallx l u.\ltand<\>tnotin)<> l  x < x  <and P1  \  x)
nd  \\longrightarrow \forallyl y\>   u\> P1y\and> y)
    showsin U <Longrightarrow>
proof -
  let?Mx  { y\> S and  < x"
  let? = {y\in>S \and>x ≤
  let?= Maxrallxlu \forallt.l< t \and>  u⟶ \notin> U <nd>l  x \and> x   < P1x
M"
  have: " by (metis antisym_conv3 orderstrict_trans
    by allx lu \>. l< <>t<u \>t \notin U \and> l < \> x  <>P2 x
  have" Mx"
     fSfinite_subset byauto
  from lx linShave linMx "l \ ?Mx"
by blast
    then
    byblast
  have xMS ?\subseteqS
    java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
finitexM
    using
  from : x\ java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23

  then have xMne    <> lu \forallt   t\and t      and"u <>S
    blast
\x\in .l<lex
    usingfMx byauto
   
lemma 
    \forall". \<>S x}
      by auto
  thenlin_dense_conj]
     MxS blast
  have ? <>?"
    using  \forall>  xM
  thenhavebinShaveMxS:"Mx \<> S
    usingby  \forall>xl .\>.l  t <>t<u⟶notin U and <x\and  <u\and P1x < P2x)
 \> a yand ? longrightarrowy<otinS"
    usingusing finite_subset by auto
  from binS noyax pxshow
        by blast
qed

finite_set_intervals2no_atp]:
sumes   blast
   lx l >
    byforall>x lu 🚫 u⟶ <>U)<> <x <nd x<u 🚫
     linS:l\>S
    and\forall>    usingfS by auto
     fS finite"
    and lS:"
    and
  shows"\existss by blast
    <>\forall>\not MP\>\not> <>  longrightarrow\<u\in> U. ∃\in>U u <e x \and>x \le>u'"
  by metislocalneq_le_trans

  lemmafinite_set_intervalsno_atp:


section \\<openThe classicalQE afterafter Langford fordense linear ordersorders\close>

context    usingOF  Mxnebysimp
beginandlinS:"\ S

lemma fS"finite S""
  by autodest:dense

lemma [? in?"
  proof
    and <>   then havebinS "b\S""
    andusing xMSby 
    and ?a ="Max?Mx
" using Mxne fxM local.linear xMne by auto
proof
  assume H: "\x. (\y\L. y < x) \ (\y\U. x < y)"
  thenthen obtain x where: "\forall>y\inL.. y < x" and xU: "\y\U. x < y"
    by 

    .dual_order 1 uxL xU by blast
  then    fS  auto
      andlx: "l x"
      then haveMxne:"?Mx \
  let  ""
  let :finite
    and :  \forall<inSl\> 
    andSu ">xin .x
  shows(\in)or<exists\in.. <<exists <in.(forall<a<  < y < b <longrightarrowy <> S\  <x and b\ Px)
    by auto
  from   from uinSwhereP""OF pxlxxu uinSfSlS Su
        byblast
  with
      haveax
  fromusingMxne fMx  auto
    by
  moreover fromth4 th2th2 have"\forall>u > U. w< uh2 have \forallu
    [fMxMxne by
    then
    lemma: y x  y <>y <z}={ \longleftrightarrow<>x <z"
qedby  Min_inOF fxMxMne] simp

lemma xMSby 
  assumes
    andfL:finite
  shows ">x. (y )\and>> (y \in> {. x < y))) = Truejava.lang.StringIndexOutOfBoundsException: Index 94 out of bounds for length 94
usinglocal. by

lemmashows\>\forall Ly<)\and(forall>U   )=\foralll ∈ )
  assumes ne: U <noteqjava.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
    and fU finite
  showsbinSnoy  px showthesis
proof" u if l "lin" and u " \>"forl u
  fromassumes:"
by
  from e   \>> in U Min le x
     simp
th have\>\inU M x
    by (auto: less_le_trans
  thenML Max
    by
qed

mma[:\>ax noteqnoteq> t \exists>a)t<x
  usingfrom fU neUhave: "MU \in>U"and th2:"\ "

  fromth1th2 Hhave "L< ?MU"
  le_less  less_not_permute

lemma
    from  th1 have>in  w"

atp
   show(\L  and\\>inUx<y
    qed
    and:

declare fL L
declare dlo_simps "<>x. (\forall>y \ L y (

end

Move HOL - with theconj_aci_rulein. *
lemmas    fU" Ujava.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22

lemmas weak_dnf_simps[no_atp] = simp_thms 

lemma simp
  "(<> ( \andQ <>\notP< not
  (not   l  "if l: "l\in andu " U or l u
  P<Q\(notthenshowthesis
  "P\longleftrightarrowjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
   [ t byauto
  by dlo_simps] order_refl not_less java.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 76

lemma auto
  mmaatoms[]:

as[    andless_eq:  

langfordclose
method_setup
    axiomlangfordqe dlo_qe_bndsdlo_qe_bnds dlo_qe_nolb  gather:gather_startgather_simps: 
declareare[java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31


sectionopendenselinearordersyieldQE  arithmeticoverordered\>



localelinorder_stupid_syntax=linorder
begin

notation
_\open\ P<and Q)) ⟷ (\not> P\r ¬
l  \"P <Q <>\not P\ )
  <qed\:"\exists>(x::'a). t x"
  lessbyblast+

end

locale linorder_no_ub = linorder_stupid_syntax +
   blast


lemma
usingbyauto

text Scan SIMPLE_METHODoLangford
lemmapinf_conj:
  assumes:"x. z1 \ x \longrightarrow>> (P1 x \longleftrightarrow
     " ((=): a \
  shows<.forallxz<sqsubset  x longrightarrow(P1x \\and x))\longleftrightarrow P1🚫begin
    \'(>)

lemma
  assumes linorder_no_ub linorder_stupid_syntaxjava.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
    and:"<>z2.\forall>x.z2 \ x \ ((P2 x \longleftrightarrow>>P2java.lang.StringIndexOutOfBoundsException: Index 107 out of bounds for length 107
shows\.<> z \>\longrightarrowx( x<orP2x \longleftrightarrow>P1 orP2)
  by (metis ex1 ex2 local.max.strict_boundedE  (<notusinggt_ex auto

lemma pinf_exno_atp]
  assumes blast
    andlemmas[]] nnf_simps
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    gt_exp1byauto

end

text<Linear without bounds<java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53

locale
  and "existsz2 .z2\<> <> (P2 x\longleftrightarrow P2)
begin

java.lang.StringIndexOutOfBoundsException: Range [0, 5) out of bounds for length 0
    showsexistsPx


text
lemma[:
mesex1 \existsz1. \forallxx \sqsubsetz1>(x\longleftrightarrowP1"
     ex2existsz2. ∀ z2 ⟶ (P2 x ⟷ P2')"
  shows linorder_no_lb = linorder_stupid_syntax +
  by (is ex1 ex2 locallocal.min_less_iff_conj

lemma minf_disjbegin
  
    andex2\exists.\forallx<>longrightarrow  longleftrightarrowjava.lang.StringIndexOutOfBoundsException: Index 107 out of bounds for length 107
  showsexists.<  sqsubsetz longrightarrow x\orP2 > P1 P2"
  by ( ex1 .min_less_iff_conj

lemmao_atp
  assumes ex"<existsz \forallx x \sqsubsetz \longrightarrow (  ongleftrightarrow"
   :
    bymetisex1 ..)
  using lemmaminf_disj]]

end      ex2 \>.<forall. x🚫 '


locale[no_atp:
   ex "> <>x z
   between_less " x lessx betweenx y andless x y "
     between_same" x = x"
begin

sublocale
proofunfold_localestextopen orderwithout upperbounds<
locale  lessx 🚫   java.lang.StringIndexOutOfBoundsException: Index 101 out of bounds for length 101
e
  13

java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
   lin_denseforalluu<   andtsqsubsetlongrightarrownotin\>x<x\sqsubset u <>P x
      < \forally  sqsubset>y \and>y<>u longrightarrow y "
   3
  then minf_ex]
qed  shows<uin<> <in.Pbetween uu"

mmarinf_Uno_atp
    byjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
t <sqsubsett<and u   U \> lsqsubset \andx 🚫
\longrightarrow\forall> l < y\and  sqsubset u<longrightarrowP  java.lang.StringIndexOutOfBoundsException: Index 100 out of bounds for length 100
    and "forall>x \ \exists\in>U.\exists>'\in>U.u\sqsubseteq>x \and>x \u'"
    :    using by
    haveuinM"\in> UU"
proof
  from     fUUneUnejava.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24

  fromjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
  using fUby
by
  from uUfromorder_transOFth ux lx " \ x" 
     autouU by
     linorder  U
  let  nmpiU"forall>x.\not> \ P \longrightarrow ( U.\exists>' \ Uu u))
  have:"l < and nmi: "not" and npi: "notPP  and\exists>  Px"
    shows"u\t1\in U  >U" "U <forall t1s> \> yy\sqsubset \longrightarrow\notinU""t1 " x
  haveproofbyblast
    using Unebyjava.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
    obtainuu uU\U and' " \in\in U ux:u"andsqsubsetequ
    using  
have 
    usingthen have t1t2 " \ t2"
  have:"l\sqsubseteq u"
    using?  "linorder.Min less_eq U"
  from    from between_lesslet u=. less_eq
  then thesis
    usinguU'Une Muby simp
  
  from[[ P", px lx xulx xu linM uinM fU lM Mu]
  consider u where "u\in U l
    t1 wheret1
    byhaveMu:"t\in> U. t\ ?u"
    assumes:" using Une fU byauto
  prooflin_dense: <x  .. sqsubset andt<u\longrightarrow><sqsubsetx<  <u\> 
1 show
       and  \x \>\and>Px <>\existsu<inu \sqsubseteq xjava.lang.StringIndexOutOfBoundsException: Index 107 out of bounds for length 107
  next
    shows "<>x P x)= (MP \or> PP \\ ( U. \u'\ U. P (between u u')))"
    :         (isE D)
   order 
    between t1 "
from  have :"t1 \
    w?  px E
       2  proof
  qed
qed

theorem fr_eq
  assumes     case show byblast
     andlin_dense <x   <tlsqsubsett\andtsqsubsetu ongrightarrowt\>U\and \  andx\u∧x
     \longrightarrow> (<forallby order 
 nmibnd \forallx. ¬MP\and>P  <longrightarrow> (∃      fromOFnmibnd ]
    and"> \not\notPP > (
    and      from rinf_UOFfU nmpiU<opennmpiUopennot>MP>\pen\not>PPclose ] show ?thesis
  s "(\exists>x>x x))= (MP \(\exists>u\in U.existsu
         is    qed
 
  fr_eqno_atp
  usinggt_ex. mi 
    consider
    then show ?
    
      and<x<>MP\and x <> \U. \sqsubseteqx)
  ? 
    
       
         "\ PP< \existsu \in>U.\lemmas nmi_thms[no_atp = nmi_disj nmi_neq nmi_lt nmi_le nmi_gt nmi_genmi_ge nmi_P
      have nmpiU("E =lemmas npi_thms[no_atp npi_conj npi_disj npi_eqnpi_neq npi_lt npi_le npi_le npi_gt npi_genpi_P
      from lin_dense_thms[]proof 
        byblast
    qed
  qed
  showlemma ferrack_axiomno_atp:"constr_dense_linorder less_eq less between"
    ( constr_dense_linorder_axiomscase
qed

lemmas atoms      case2
lemmasshows "(less : 'a \ _)"

lemmas[  nmi_conjnmi_disj nmi_eqnmi_neqnmi_ltnmi_lenmi_gt nmi_P
lemmas 
lemmas ferrack_axiom[ minf  pinf pinf_thms

lemma :fr_eq:]
using localgt_exlocal.lt_ex mipi thatby blast

lemma atoms minf_thmsno_atp] 
  shows" less :a <> _
  fun generic_whatis phi=
    and "TERM ((=) : 'a \

declare[no_atp =lin_dense_conj lin_dense_disjlin_dense_eq  lin_dense_le lin_dense_gelin_dense_P
nmi_thms:npi_thms:
         Thm tof

declaration<
let
  fun simps phi  :: aifThmterm_ofxaconvy thenEq
  fungeneric_whatis phi=
    let
      declare .term_ofx   Ferrante_Rackoff_Data
      elseFerrante_Rackoff_DataNox
     Thmif .could_unify(,lt 
          🍋 le] Thmterm_ofx  ythen.
            if.term_of x aconv y thenFerrante_Rackoff_DataEq
            Thmterm_of  Ferrante_Rackoff_DataFerrante_Rackoff_Data.Nox
       |\^Const_\open> java.lang.StringIndexOutOfBoundsException: Range [33, 17) out of bounds for length 54
            if.term_of xaconvy thenFerrante_Rackoff_Data.NEq
             elseifThm.term_ofx aconv thenFerrante_Rackoff_Datajava.lang.StringIndexOutOfBoundsException: Index 80 out of bounds for length 80
        byz =ifTermcould_unifyb  
                      x  aconvy  Ferrante_Rackoff_DataNEq
                 else unify(,) 
                      .simpset_ofput_simpset   .simpsphi)
                 else TermelseFerrante_Rackoff_Data
                     Ferrante_Rackoff_Datafuns{ ferrack_axiom
                     else if.term_of aconv z  Ferrante_Rackoff_DataGe
                     elseelseif Thmterm_of aconvzthen.
                 else
       
end
  fun ss phi       <>
    simpset_ofput_simpsetctxt|. simpsphi)
in<> "and Rackoff's algorithm for quantifier elimination inalgorithm for quantifier elimination in dense linear orders"
  Ferrante_Rackoff_Data
    
sub <penFerrante   Rackoffalgorithmoverordered java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
›

endlemma:neg_prod_sum_lt:

ML_file"< 0

method_setup=\open
  Scan.lemma pos_prod_sum_lt
›  assumes "fixesc : "shows   0 \longleftrightarrow   -/)*java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63


subsectionFerrante and Rackoff algorithm 

lemmalemma lemmaneg_prod_le
 c:':"
    assumesc <
  showsshows " ** x \le>0\longleftrightarrow x \ge>>"
    metisassms  mult_neg_negzero_less_mult_pos

lemma: ::
  fixes c : "'a:linordered_field"java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
   c > "
  shows "c * x < 0 der_not_less mult_le_0_iff by auto
  

lemmaneg_prod_sum_lt:
  fixes c :: "a::linordered_field"
     "c < 0"
  shows  fixesfixes :"a:ordered_ab_group_add"
  using  auto  : multcommutedivide_simps)

lemma
    c ':inordered_field"
  assumes fixes c ::':linordered_field
x  0\longleftrightarrow< -/ t
automult)

lemma:
  fixes :
  shows":linordered_field"
  less_diff_eqwhere xand  c0  simp

     z_prod_eqjava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
  fixes : ""'a:linordered_fieldjava.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
   "c 0
  showsusingassms bysimp
  using

lemma
  fixesc  "a:linordered_field""
  assumesfixesc :a:"
  shows   c 0
  usingshows" * + \longleftrightarrow x=(-/c* t"

lemma
  fixes:
  assumesc<
  shows "c shows "x +  0\longleftrightarrow=- t"
   assmsby( simp:  shows"+t\le>0< \

lemma pos_prod_sum_leinterpretation:
  fixes : "'a::linordered_field"
c  0
ghtarrowunfold_localesauto simpadd  lt_ex
  using  usingusing \open

lemmasum_le
 rlier []_=false
   er(: xy 
using[ a= x andbt  c=]=]by

 nz_prod_eq
  
  assumes "c \ red_ab_group_add"
  shows"c x = 0 = 00
   else

lemma nz_prod_sum_eqinterpretationdest_fracct=
  fixes
   "
  showsRatmake>
   assms byfun  _ false
   t=Rat. (sndHOLogicdest_number t)
lemma java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  fixesx :'a:ordered_ab_group_add
  shows "x Thm.term_of x ("*t(  .),     y 
   where  andbtand]by

interpretation>fun ctjava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
  "else (else ("",[]
  byunfold_localesautosimp add gt_ex

 <java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19

    [   
    | earlierh: x y =
        
val thm
funearlier_ord x,y)java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
    Nox]
   earliervs( y thenLESS
    else
val     else Nox[
fun<Const_<times fory< 
  case Thm  if aconvterm_ofthencx[Thmdest_arg1ct
    \^><>.divide for a bclose
st_number, (.dest_number
  |t= ifaconvThmterm_ofx then(x[)else ""[)
  | t => xnormalize_conv ] =Thmreflexive

    {s}
   m { sum_eq
      y aconvThmx then(cx+[ .dest_arg1 Thmct
     else" ("]
|^<> _ for 🚫valpos_prod_sum_le mk_meta_eq{ }
if .term_of Thmx  "x+t"[Thm ct
     elseNox]
|^><times for y<>>
     else. Thm  cz cjava.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
     z_prod_sum_eq = ( negthenelsepos_prod_sum_ltcth
 t=if t  Thmterm_ofx then(x[ else"ox"]

java.lang.StringIndexOutOfBoundsException: Range [5, 6) out of bounds for length 5
val   {sum_lt
val sum_le = mk_meta_eq|"+,[] (case whatis x (Thmest_arg1 ct)of
valsum_eq  thm
val  {thmneg_prod_sum_lt
val        dest_fracc
prod_sum_le mk_meta_eq{ neg_prod_sum_le
pos_prod_sum_le=mk_meta_eqthm}
val   cr @
@{ pos_prod_lt}
al =mk_meta_eq @{ neg_prod_le
val              Semiring_Normalizersemiring_normalize_ord_conv ctxtearlier_ord)))
_ 
val  =Thmequal_elimThm
in   java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
functxt]   .ct
|       in
   |         mk_judgment
   \open>less   ^>\openzero_class_closeclose>
    case  (.dest_arg1 
    (*+,[,)java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
       let
                val =.implies_elim Thm' (Thm.ctyp_of_cterm x] map [c,]
        valclt .dest_fun2
        val cz =val =th
        val =cr 0
        val        . ct              Semiring_Normalizer ctxtearlier_ord)))th
               
        (if neg Thm  .apply c cz
                   (casewhatis xThm ctof
         Thm.cthpTrueI
val  Thm x
 neg neg_prod_sum_lt pos_prod_sum_lt 
        val rth = Conv.fconv_rule (Conv        val cr  dest_fracc
                   Semiring_Normalizersemiring_normalize_ord_convctxtearlier_ord
      inend
    |(x+               .java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
       let
           Thmapplyapplycltccz
         th=instantiateSOME]SOMEx  ]sum_lt
        valcthThm Thmsymmetric)TrueI
                                           .  ())java.lang.StringIndexOutOfBoundsException: Index 95 out of bounds for length 95
       in
    |(c*c]=
       valrthConv arg_conv(Convbinop_conv
         cr =dest_frac
        val clt = Thm  =.       in 
        val        val    | c*[)=
        valneg cr 0
        val  Thm 
               valcz   dest_argct
                  ifnegthen. (.apply c cz
                    else. Thm clt ))
        valvalczThm ct
         th=Thmimplies_eliminstantiate SOMEctyp_of_cterm   c
             val=Simplifier ctxt
val negthen. (hmapplycltc 

    valcth  Thm( neg thenneg_prod_sum_le  )cth


|valend
tisThm ctof
    "xt,[,t >
       let
        val Thm.  
         Thm java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
        val cr = dest_frac    |c"[]) >
  .  ^>\openless<>
val  =Thm ct
        valneg=  @
val          Srewrite
ic
     ifthenThmapply(Thm cltc
            elseThm(.apply  )
          Thm(.  
        valThminstantiate[                  ((if neg (ifthen Thm. (. clt 
                 (ifnegthen neg_prod_sum_leelsepos_prod_sum_le cth
           . Conv (.
   .(Thm'[ Thmctyp_of_cterm x)](map SOME [cx)
       +,t 
    |  val  java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
       
                   . Conv^\openHOL.   forConst_\open>zero_class._close›
        valth Thm[SOME][OMExSOMEt 
        val end
  . ctxtearlier_ord) 
         rthend
            val ceq  Thm.dest_fun2 ct
       
           Thm Thmtyp_of_cterm
                 ceq Thmdest_fun2ct
        val  dest_fracc
          Thmcterm_of\Const<lessT<
        val cz Thm..dest_argct
         cr <0
        val cthp=Simplifier. ctxt
               (.mk_judgment
                  ( negval  . ThmcthpTrueI
      applyThm cltcz )
         cth=Thmequal_elimThmsymmetriccthp 
vrth
                  if          T Thm x
        valend
      in rth endlocal
_ . ct)

|<Const_HOLval T = java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 35
   (   Thmctof
        valopen  a b<>=
    ca)= .dest_binop
        val T =       val(apply^‹ (.apply(Thmapply c cz)
 val  Thm. (Thmsymmetriccthp)
valceq         nth=Conv
                 .
        val   Simplifierrewritectxt
            (th Thmnth ctxt  (Thmrhs_ofnth
             
        val cth    less_iff_diff_less_0  @ "
         th .
       valT Thmctyp_of_cterm ca
       valss=simpset_of<>       valnth.fconv_rule
                   (.semiring_normalize_ord_conv 
                    Semiring_Normalizersemiring_normalize_ord_conv put_simpset  vs)th
     "+"[)=
       
        java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
        th Thm[T] ,SOME sum_eq
         =Thm.ctyp_of_ctermca
              (Semiring_Normalizer.       val  Thmtransitiventh T SOME caSOMEcb]eq_iff_diff_eq_0
       rthjava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
    | ("c*x",[c]letSemiring_Normalizer (ssctxtearlier_ord) th
       let
       val th=Thm SOMET [ ca cb le_iff_diff_le_0
        val (. Conv<.   \closeclose=> Convarg_conv (field_isolate_conv  ctxt)
         ceq Thmdest_fun2
          Thmdest_arg ct
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
            HOLogic
             (Thm val  fun    =
cthThm. (symmetriccthp)
        val rth = Thm        th     <<Const_open.  for y close >
                 (Thm'(.rg_conv Conv.arg1_conv
       Ferrante_Rackoff_DataNox
     _ Thm. ct;
end

local
  if.funclassfield_whatis 
  le_iff_diff_le_0  thm"
  elseFerrante_Rackoff_DataNox
       ^<>. yz\close=
inThm xaconv y then.
funelse Thmterm_of x zthen.Ge
  🚫 Ferrante_Rackoff_Data.Nox
   let      = Ferrante_Rackoff_DataNox
       val =hm ca
        phictxt 
       valnth=.
  arg_convConvarg1_conv
              (>fold Ferrante_Rackoff_DataNox
       val
   in
|. thmclass_dense_linordered_fieldjava.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78
   let
\close
       val th funend
       valnth  Convfconv_rule
         (Conv.arg_conv(Convarg1_conv
              (    > fold Splitteradd_split[{thm""} @thmsplit_max,@thm"])
       val  =Thm(  Thmrhs_of)
   in end

| <Const_>\openHOL.  forab<> =
   let val java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
       val T = Thm.ctyp_of_cterm ca
       val th = Thm.instantiate' [SOME T] [SOME ca, SOME cb] eq_iff_diff_eq_0
       val nth = Conv.fconv_rule
         (Conv.arg_conv (Conv.arg1_conv
              (Semiring_Normalizer.semiring_normalize_ord_conv (put_simpset ss ctxt) (earlier_ord vs)))) th
       val rth = Thm.transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
   in rth end
| 🍋‹Not for 🍋‹HOL.eq _ for a b›› => Conv.arg_conv (field_isolate_conv phi ctxt vs) ct
| _ => Thm.reflexive ct
end;

fun classfield_whatis phi =
 let
  fun h x t =
   case Thm.term_of t of
     🍋‹HOL.eq _ for y z› =>
      if Thm.term_of x aconv y then Ferrante_Rackoff_Data.Eq
      else Ferrante_Rackoff_Data.Nox
   | 🍋‹Not for 🍋‹HOL.eq _ for y z›› =>
      if Thm.term_of x aconv y then Ferrante_Rackoff_Data.NEq
      else Ferrante_Rackoff_Data.Nox
   | 🍋‹less _ for y z› =>
       if Thm.term_of x aconv y then Ferrante_Rackoff_Data.Lt
       else if Thm.term_of x aconv z then Ferrante_Rackoff_Data.Gt
       else Ferrante_Rackoff_Data.Nox
   | 🍋‹less_eq _ for y z› =>
       if Thm.term_of x aconv y then Ferrante_Rackoff_Data.Le
       else if Thm.term_of x aconv z then Ferrante_Rackoff_Data.Ge
       else Ferrante_Rackoff_Data.Nox
   | _ => Ferrante_Rackoff_Data.Nox
 in h end;
fun class_field_ss phi ctxt =
  simpset_of (put_simpset HOL_basic_ss ctxt
    |> Simplifier.add_simps ([@{thm "linorder_not_less"}, @{thm "linorder_not_le"}])
    |> fold Splitter.add_split [@{thm "abs_split"}, @{thm "split_max"}, @{thm "split_min"}])

in
Ferrante_Rackoff_Data.funs @{thm".ferrack_axiom"}
  {isolate_convs = classfield_whatis simpset=}
end
<close

end