src/HOL/Dense_Linear_Order.thy
author haftmann
Wed Sep 26 20:27:55 2007 +0200 (2007-09-26)
changeset 24728 e2b3a1065676
parent 24679 5b168969ffe0
child 24748 ee0a0eb6b738
permissions -rw-r--r--
moved Finite_Set before Datatype
     1 (*
     2     ID:         $Id$
     3     Author:     Amine Chaieb, TU Muenchen
     4 *)
     5 
     6 header {* Dense linear order without endpoints
     7   and a quantifier elimination procedure in Ferrante and Rackoff style *}
     8 
     9 theory Dense_Linear_Order
    10 imports Finite_Set
    11 uses
    12   "Tools/Qelim/qelim.ML"
    13   "Tools/Qelim/langford_data.ML"
    14   "Tools/Qelim/ferrante_rackoff_data.ML"
    15   ("Tools/Qelim/langford.ML")
    16   ("Tools/Qelim/ferrante_rackoff.ML")
    17 begin
    18 
    19 setup Langford_Data.setup
    20 setup Ferrante_Rackoff_Data.setup
    21 
    22 context linorder
    23 begin
    24 
    25 lemma less_not_permute: "\<not> (x \<sqsubset> y \<and> y \<sqsubset> x)" by (simp add: not_less linear)
    26 
    27 lemma gather_simps: 
    28   shows 
    29   "(\<exists>x. (\<forall>y \<in> L. y \<sqsubset> x) \<and> (\<forall>y \<in> U. x \<sqsubset> y) \<and> x \<sqsubset> u \<and> P x) \<longleftrightarrow> (\<exists>x. (\<forall>y \<in> L. y \<sqsubset> x) \<and> (\<forall>y \<in> (insert u U). x \<sqsubset> y) \<and> P x)"
    30   and "(\<exists>x. (\<forall>y \<in> L. y \<sqsubset> x) \<and> (\<forall>y \<in> U. x \<sqsubset> y) \<and> l \<sqsubset> x \<and> P x) \<longleftrightarrow> (\<exists>x. (\<forall>y \<in> (insert l L). y \<sqsubset> x) \<and> (\<forall>y \<in> U. x \<sqsubset> y) \<and> P x)"
    31   "(\<exists>x. (\<forall>y \<in> L. y \<sqsubset> x) \<and> (\<forall>y \<in> U. x \<sqsubset> y) \<and> x \<sqsubset> u) \<longleftrightarrow> (\<exists>x. (\<forall>y \<in> L. y \<sqsubset> x) \<and> (\<forall>y \<in> (insert u U). x \<sqsubset> y))"
    32   and "(\<exists>x. (\<forall>y \<in> L. y \<sqsubset> x) \<and> (\<forall>y \<in> U. x \<sqsubset> y) \<and> l \<sqsubset> x) \<longleftrightarrow> (\<exists>x. (\<forall>y \<in> (insert l L). y \<sqsubset> x) \<and> (\<forall>y \<in> U. x \<sqsubset> y))"  by auto
    33 
    34 lemma 
    35   gather_start: "(\<exists>x. P x) \<equiv> (\<exists>x. (\<forall>y \<in> {}. y \<^loc>< x) \<and> (\<forall>y\<in> {}. x \<sqsubset> y) \<and> P x)" 
    36   by simp
    37 
    38 text{* Theorems for @{text "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>-\<infinity>\<^esub>)"}*}
    39 lemma minf_lt:  "\<exists>z . \<forall>x. x \<sqsubset> z \<longrightarrow> (x \<sqsubset> t \<longleftrightarrow> True)" by auto
    40 lemma minf_gt: "\<exists>z . \<forall>x. x \<sqsubset> z \<longrightarrow>  (t \<sqsubset> x \<longleftrightarrow>  False)"
    41   by (simp add: not_less) (rule exI[where x="t"], auto simp add: less_le)
    42 
    43 lemma minf_le: "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (x \<sqsubseteq> t \<longleftrightarrow> True)" by (auto simp add: less_le)
    44 lemma minf_ge: "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (t \<sqsubseteq> x \<longleftrightarrow> False)"
    45   by (auto simp add: less_le not_less not_le)
    46 lemma minf_eq: "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (x = t \<longleftrightarrow> False)" by auto
    47 lemma minf_neq: "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (x \<noteq> t \<longleftrightarrow> True)" by auto
    48 lemma minf_P: "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P \<longleftrightarrow> P)" by blast
    49 
    50 text{* Theorems for @{text "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>+\<infinity>\<^esub>)"}*}
    51 lemma pinf_gt:  "\<exists>z . \<forall>x. z \<sqsubset> x \<longrightarrow> (t \<sqsubset> x \<longleftrightarrow> True)" by auto
    52 lemma pinf_lt: "\<exists>z . \<forall>x. z \<sqsubset> x \<longrightarrow>  (x \<sqsubset> t \<longleftrightarrow>  False)"
    53   by (simp add: not_less) (rule exI[where x="t"], auto simp add: less_le)
    54 
    55 lemma pinf_ge: "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (t \<sqsubseteq> x \<longleftrightarrow> True)" by (auto simp add: less_le)
    56 lemma pinf_le: "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (x \<sqsubseteq> t \<longleftrightarrow> False)"
    57   by (auto simp add: less_le not_less not_le)
    58 lemma pinf_eq: "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (x = t \<longleftrightarrow> False)" by auto
    59 lemma pinf_neq: "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (x \<noteq> t \<longleftrightarrow> True)" by auto
    60 lemma pinf_P: "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (P \<longleftrightarrow> P)" by blast
    61 
    62 lemma nmi_lt: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> x \<sqsubset> t \<longrightarrow>  (\<exists> u\<in> U. u \<sqsubseteq> x)" by auto
    63 lemma nmi_gt: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> t \<sqsubset> x \<longrightarrow>  (\<exists> u\<in> U. u \<sqsubseteq> x)"
    64   by (auto simp add: le_less)
    65 lemma  nmi_le: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> x\<sqsubseteq> t \<longrightarrow>  (\<exists> u\<in> U. u \<sqsubseteq> x)" by auto
    66 lemma  nmi_ge: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> t\<sqsubseteq> x \<longrightarrow>  (\<exists> u\<in> U. u \<sqsubseteq> x)" by auto
    67 lemma  nmi_eq: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and>  x = t \<longrightarrow>  (\<exists> u\<in> U. u \<sqsubseteq> x)" by auto
    68 lemma  nmi_neq: "t \<in> U \<Longrightarrow>\<forall>x. \<not>True \<and> x \<noteq> t \<longrightarrow>  (\<exists> u\<in> U. u \<sqsubseteq> x)" by auto
    69 lemma  nmi_P: "\<forall> x. ~P \<and> P \<longrightarrow>  (\<exists> u\<in> U. u \<sqsubseteq> x)" by auto
    70 lemma  nmi_conj: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow>  (\<exists> u\<in> U. u \<sqsubseteq> x) ;
    71   \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow>  (\<exists> u\<in> U. u \<sqsubseteq> x)\<rbrakk> \<Longrightarrow>
    72   \<forall>x. \<not>(P1' \<and> P2') \<and> (P1 x \<and> P2 x) \<longrightarrow>  (\<exists> u\<in> U. u \<sqsubseteq> x)" by auto
    73 lemma  nmi_disj: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow>  (\<exists> u\<in> U. u \<sqsubseteq> x) ;
    74   \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow>  (\<exists> u\<in> U. u \<sqsubseteq> x)\<rbrakk> \<Longrightarrow>
    75   \<forall>x. \<not>(P1' \<or> P2') \<and> (P1 x \<or> P2 x) \<longrightarrow>  (\<exists> u\<in> U. u \<sqsubseteq> x)" by auto
    76 
    77 lemma  npi_lt: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and>  x \<sqsubset> t \<longrightarrow>  (\<exists> u\<in> U. x \<sqsubseteq> u)" by (auto simp add: le_less)
    78 lemma  npi_gt: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> t \<sqsubset> x \<longrightarrow>  (\<exists> u\<in> U. x \<sqsubseteq> u)" by auto
    79 lemma  npi_le: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and>  x \<sqsubseteq> t \<longrightarrow>  (\<exists> u\<in> U. x \<sqsubseteq> u)" by auto
    80 lemma  npi_ge: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> t \<sqsubseteq> x \<longrightarrow>  (\<exists> u\<in> U. x \<sqsubseteq> u)" by auto
    81 lemma  npi_eq: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and>  x = t \<longrightarrow>  (\<exists> u\<in> U. x \<sqsubseteq> u)" by auto
    82 lemma  npi_neq: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> x \<noteq> t \<longrightarrow>  (\<exists> u\<in> U. x \<sqsubseteq> u )" by auto
    83 lemma  npi_P: "\<forall> x. ~P \<and> P \<longrightarrow>  (\<exists> u\<in> U. x \<sqsubseteq> u)" by auto
    84 lemma  npi_conj: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow>  (\<exists> u\<in> U. x \<sqsubseteq> u) ;  \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow>  (\<exists> u\<in> U. x \<sqsubseteq> u)\<rbrakk>
    85   \<Longrightarrow>  \<forall>x. \<not>(P1' \<and> P2') \<and> (P1 x \<and> P2 x) \<longrightarrow>  (\<exists> u\<in> U. x \<sqsubseteq> u)" by auto
    86 lemma  npi_disj: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow>  (\<exists> u\<in> U. x \<sqsubseteq> u) ; \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow>  (\<exists> u\<in> U. x \<sqsubseteq> u)\<rbrakk>
    87   \<Longrightarrow> \<forall>x. \<not>(P1' \<or> P2') \<and> (P1 x \<or> P2 x) \<longrightarrow>  (\<exists> u\<in> U. x \<sqsubseteq> u)" by auto
    88 
    89 lemma lin_dense_lt: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l \<sqsubset> t \<and> t \<sqsubset> u \<longrightarrow> t \<notin> U) \<and> l\<sqsubset> x \<and> x \<sqsubset> u \<and> x \<sqsubset> t \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> y \<sqsubset> t)"
    90 proof(clarsimp)
    91   fix x l u y  assume tU: "t \<in> U" and noU: "\<forall>t. l \<sqsubset> t \<and> t \<sqsubset> u \<longrightarrow> t \<notin> U" and lx: "l \<sqsubset> x"
    92     and xu: "x\<sqsubset>u"  and px: "x \<sqsubset> t" and ly: "l\<sqsubset>y" and yu:"y \<sqsubset> u"
    93   from tU noU ly yu have tny: "t\<noteq>y" by auto
    94   {assume H: "t \<sqsubset> y"
    95     from less_trans[OF lx px] less_trans[OF H yu]
    96     have "l \<sqsubset> t \<and> t \<sqsubset> u"  by simp
    97     with tU noU have "False" by auto}
    98   hence "\<not> t \<sqsubset> y"  by auto hence "y \<sqsubseteq> t" by (simp add: not_less)
    99   thus "y \<sqsubset> t" using tny by (simp add: less_le)
   100 qed
   101 
   102 lemma lin_dense_gt: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l \<sqsubset> t \<and> t\<sqsubset> u \<longrightarrow> t \<notin> U) \<and> l \<sqsubset> x \<and> x \<sqsubset> u \<and> t \<sqsubset> x \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> t \<sqsubset> y)"
   103 proof(clarsimp)
   104   fix x l u y
   105   assume tU: "t \<in> U" and noU: "\<forall>t. l \<sqsubset> t \<and> t \<sqsubset> u \<longrightarrow> t \<notin> U" and lx: "l \<sqsubset> x" and xu: "x\<sqsubset>u"
   106   and px: "t \<sqsubset> x" and ly: "l\<sqsubset>y" and yu:"y \<sqsubset> u"
   107   from tU noU ly yu have tny: "t\<noteq>y" by auto
   108   {assume H: "y\<sqsubset> t"
   109     from less_trans[OF ly H] less_trans[OF px xu] have "l \<sqsubset> t \<and> t \<sqsubset> u" by simp
   110     with tU noU have "False" by auto}
   111   hence "\<not> y\<sqsubset>t"  by auto hence "t \<sqsubseteq> y" by (auto simp add: not_less)
   112   thus "t \<sqsubset> y" using tny by (simp add:less_le)
   113 qed
   114 
   115 lemma lin_dense_le: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l \<sqsubset> t \<and> t\<sqsubset> u \<longrightarrow> t \<notin> U) \<and> l\<sqsubset> x \<and> x \<sqsubset> u \<and> x \<sqsubseteq> t \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> y\<sqsubseteq> t)"
   116 proof(clarsimp)
   117   fix x l u y
   118   assume tU: "t \<in> U" and noU: "\<forall>t. l \<sqsubset> t \<and> t \<sqsubset> u \<longrightarrow> t \<notin> U" and lx: "l \<sqsubset> x" and xu: "x\<sqsubset>u"
   119   and px: "x \<sqsubseteq> t" and ly: "l\<sqsubset>y" and yu:"y \<sqsubset> u"
   120   from tU noU ly yu have tny: "t\<noteq>y" by auto
   121   {assume H: "t \<sqsubset> y"
   122     from less_le_trans[OF lx px] less_trans[OF H yu]
   123     have "l \<sqsubset> t \<and> t \<sqsubset> u" by simp
   124     with tU noU have "False" by auto}
   125   hence "\<not> t \<sqsubset> y"  by auto thus "y \<sqsubseteq> t" by (simp add: not_less)
   126 qed
   127 
   128 lemma lin_dense_ge: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l \<sqsubset> t \<and> t\<sqsubset> u \<longrightarrow> t \<notin> U) \<and> l\<sqsubset> x \<and> x \<sqsubset> u \<and> t \<sqsubseteq> x \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> t \<sqsubseteq> y)"
   129 proof(clarsimp)
   130   fix x l u y
   131   assume tU: "t \<in> U" and noU: "\<forall>t. l \<sqsubset> t \<and> t \<sqsubset> u \<longrightarrow> t \<notin> U" and lx: "l \<sqsubset> x" and xu: "x\<sqsubset>u"
   132   and px: "t \<sqsubseteq> x" and ly: "l\<sqsubset>y" and yu:"y \<sqsubset> u"
   133   from tU noU ly yu have tny: "t\<noteq>y" by auto
   134   {assume H: "y\<sqsubset> t"
   135     from less_trans[OF ly H] le_less_trans[OF px xu]
   136     have "l \<sqsubset> t \<and> t \<sqsubset> u" by simp
   137     with tU noU have "False" by auto}
   138   hence "\<not> y\<sqsubset>t"  by auto thus "t \<sqsubseteq> y" by (simp add: not_less)
   139 qed
   140 lemma lin_dense_eq: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l \<sqsubset> t \<and> t\<sqsubset> u \<longrightarrow> t \<notin> U) \<and> l\<sqsubset> x \<and> x \<sqsubset> u \<and> x = t   \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> y= t)"  by auto
   141 lemma lin_dense_neq: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l \<sqsubset> t \<and> t\<sqsubset> u \<longrightarrow> t \<notin> U) \<and> l\<sqsubset> x \<and> x \<sqsubset> u \<and> x \<noteq> t   \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> y\<noteq> t)"  by auto
   142 lemma lin_dense_P: "\<forall>x l u. (\<forall> t. l \<sqsubset> t \<and> t\<sqsubset> u \<longrightarrow> t \<notin> U) \<and> l\<sqsubset> x \<and> x \<sqsubset> u \<and> P   \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> P)"  by auto
   143 
   144 lemma lin_dense_conj:
   145   "\<lbrakk>\<forall>x l u. (\<forall> t. l \<sqsubset> t \<and> t\<sqsubset> u \<longrightarrow> t \<notin> U) \<and> l\<sqsubset> x \<and> x \<sqsubset> u \<and> P1 x
   146   \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> P1 y) ;
   147   \<forall>x l u. (\<forall> t. l \<sqsubset> t \<and> t\<sqsubset> u \<longrightarrow> t \<notin> U) \<and> l\<sqsubset> x \<and> x \<sqsubset> u \<and> P2 x
   148   \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> P2 y)\<rbrakk> \<Longrightarrow>
   149   \<forall>x l u. (\<forall> t. l \<sqsubset> t \<and> t\<sqsubset> u \<longrightarrow> t \<notin> U) \<and> l\<sqsubset> x \<and> x \<sqsubset> u \<and> (P1 x \<and> P2 x)
   150   \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> (P1 y \<and> P2 y))"
   151   by blast
   152 lemma lin_dense_disj:
   153   "\<lbrakk>\<forall>x l u. (\<forall> t. l \<sqsubset> t \<and> t\<sqsubset> u \<longrightarrow> t \<notin> U) \<and> l\<sqsubset> x \<and> x \<sqsubset> u \<and> P1 x
   154   \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> P1 y) ;
   155   \<forall>x l u. (\<forall> t. l \<sqsubset> t \<and> t\<sqsubset> u \<longrightarrow> t \<notin> U) \<and> l\<sqsubset> x \<and> x \<sqsubset> u \<and> P2 x
   156   \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> P2 y)\<rbrakk> \<Longrightarrow>
   157   \<forall>x l u. (\<forall> t. l \<sqsubset> t \<and> t\<sqsubset> u \<longrightarrow> t \<notin> U) \<and> l\<sqsubset> x \<and> x \<sqsubset> u \<and> (P1 x \<or> P2 x)
   158   \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> (P1 y \<or> P2 y))"
   159   by blast
   160 
   161 lemma npmibnd: "\<lbrakk>\<forall>x. \<not> MP \<and> P x \<longrightarrow> (\<exists> u\<in> U. u \<sqsubseteq> x); \<forall>x. \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. x \<sqsubseteq> u)\<rbrakk>
   162   \<Longrightarrow> \<forall>x. \<not> MP \<and> \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. \<exists> u' \<in> U. u \<sqsubseteq> x \<and> x \<sqsubseteq> u')"
   163 by auto
   164 
   165 lemma finite_set_intervals:
   166   assumes px: "P x" and lx: "l \<sqsubseteq> x" and xu: "x \<sqsubseteq> u" and linS: "l\<in> S"
   167   and uinS: "u \<in> S" and fS:"finite S" and lS: "\<forall> x\<in> S. l \<sqsubseteq> x" and Su: "\<forall> x\<in> S. x \<sqsubseteq> u"
   168   shows "\<exists> a \<in> S. \<exists> b \<in> S. (\<forall> y. a \<sqsubset> y \<and> y \<sqsubset> b \<longrightarrow> y \<notin> S) \<and> a \<sqsubseteq> x \<and> x \<sqsubseteq> b \<and> P x"
   169 proof-
   170   let ?Mx = "{y. y\<in> S \<and> y \<sqsubseteq> x}"
   171   let ?xM = "{y. y\<in> S \<and> x \<sqsubseteq> y}"
   172   let ?a = "Max ?Mx"
   173   let ?b = "Min ?xM"
   174   have MxS: "?Mx \<subseteq> S" by blast
   175   hence fMx: "finite ?Mx" using fS finite_subset by auto
   176   from lx linS have linMx: "l \<in> ?Mx" by blast
   177   hence Mxne: "?Mx \<noteq> {}" by blast
   178   have xMS: "?xM \<subseteq> S" by blast
   179   hence fxM: "finite ?xM" using fS finite_subset by auto
   180   from xu uinS have linxM: "u \<in> ?xM" by blast
   181   hence xMne: "?xM \<noteq> {}" by blast
   182   have ax:"?a \<sqsubseteq> x" using Mxne fMx by auto
   183   have xb:"x \<sqsubseteq> ?b" using xMne fxM by auto
   184   have "?a \<in> ?Mx" using Max_in[OF fMx Mxne] by simp hence ainS: "?a \<in> S" using MxS by blast
   185   have "?b \<in> ?xM" using Min_in[OF fxM xMne] by simp hence binS: "?b \<in> S" using xMS by blast
   186   have noy:"\<forall> y. ?a \<sqsubset> y \<and> y \<sqsubset> ?b \<longrightarrow> y \<notin> S"
   187   proof(clarsimp)
   188     fix y   assume ay: "?a \<sqsubset> y" and yb: "y \<sqsubset> ?b" and yS: "y \<in> S"
   189     from yS have "y\<in> ?Mx \<or> y\<in> ?xM" by (auto simp add: linear)
   190     moreover {assume "y \<in> ?Mx" hence "y \<sqsubseteq> ?a" using Mxne fMx by auto with ay have "False" by (simp add: not_le[symmetric])}
   191     moreover {assume "y \<in> ?xM" hence "?b \<sqsubseteq> y" using xMne fxM by auto with yb have "False" by (simp add: not_le[symmetric])}
   192     ultimately show "False" by blast
   193   qed
   194   from ainS binS noy ax xb px show ?thesis by blast
   195 qed
   196 
   197 lemma finite_set_intervals2:
   198   assumes px: "P x" and lx: "l \<sqsubseteq> x" and xu: "x \<sqsubseteq> u" and linS: "l\<in> S"
   199   and uinS: "u \<in> S" and fS:"finite S" and lS: "\<forall> x\<in> S. l \<sqsubseteq> x" and Su: "\<forall> x\<in> S. x \<sqsubseteq> u"
   200   shows "(\<exists> s\<in> S. P s) \<or> (\<exists> a \<in> S. \<exists> b \<in> S. (\<forall> y. a \<sqsubset> y \<and> y \<sqsubset> b \<longrightarrow> y \<notin> S) \<and> a \<sqsubset> x \<and> x \<sqsubset> b \<and> P x)"
   201 proof-
   202   from finite_set_intervals[where P="P", OF px lx xu linS uinS fS lS Su]
   203   obtain a and b where
   204     as: "a\<in> S" and bs: "b\<in> S" and noS:"\<forall>y. a \<sqsubset> y \<and> y \<sqsubset> b \<longrightarrow> y \<notin> S"
   205     and axb: "a \<sqsubseteq> x \<and> x \<sqsubseteq> b \<and> P x"  by auto
   206   from axb have "x= a \<or> x= b \<or> (a \<sqsubset> x \<and> x \<sqsubset> b)" by (auto simp add: le_less)
   207   thus ?thesis using px as bs noS by blast
   208 qed
   209 
   210 end
   211 
   212 section {* The classical QE after Langford for dense linear orders *}
   213 
   214 context dense_linear_order
   215 begin
   216 
   217 lemma dlo_qe_bnds: 
   218   assumes ne: "L \<noteq> {}" and neU: "U \<noteq> {}" and fL: "finite L" and fU: "finite U"
   219   shows "(\<exists>x. (\<forall>y \<in> L. y \<sqsubset> x) \<and> (\<forall>y \<in> U. x \<sqsubset> y)) \<equiv> (\<forall> l \<in> L. \<forall>u \<in> U. l \<sqsubset> u)"
   220 proof (simp only: atomize_eq, rule iffI)
   221   assume H: "\<exists>x. (\<forall>y\<in>L. y \<^loc>< x) \<and> (\<forall>y\<in>U. x \<^loc>< y)"
   222   then obtain x where xL: "\<forall>y\<in>L. y \<^loc>< x" and xU: "\<forall>y\<in>U. x \<^loc>< y" by blast
   223   {fix l u assume l: "l \<in> L" and u: "u \<in> U"
   224     from less_trans[OF xL[rule_format, OF l] xU[rule_format, OF u]]
   225     have "l \<sqsubset> u" .}
   226   thus "\<forall>l\<in>L. \<forall>u\<in>U. l \<^loc>< u" by blast
   227 next
   228   assume H: "\<forall>l\<in>L. \<forall>u\<in>U. l \<^loc>< u"
   229   let ?ML = "Max L"
   230   let ?MU = "Min U"  
   231   from fL ne have th1: "?ML \<in> L" and th1': "\<forall>l\<in>L. l \<sqsubseteq> ?ML" by auto
   232   from fU neU have th2: "?MU \<in> U" and th2': "\<forall>u\<in>U. ?MU \<sqsubseteq> u" by auto
   233   from th1 th2 H have "?ML \<sqsubset> ?MU" by auto
   234   with dense obtain w where th3: "?ML \<sqsubset> w" and th4: "w \<sqsubset> ?MU" by blast
   235   from th3 th1' have "\<forall>l \<in> L. l \<sqsubset> w" by auto
   236   moreover from th4 th2' have "\<forall>u \<in> U. w \<sqsubset> u" by auto
   237   ultimately show "\<exists>x. (\<forall>y\<in>L. y \<^loc>< x) \<and> (\<forall>y\<in>U. x \<^loc>< y)" by auto
   238 qed
   239 
   240 lemma dlo_qe_noub: 
   241   assumes ne: "L \<noteq> {}" and fL: "finite L"
   242   shows "(\<exists>x. (\<forall>y \<in> L. y \<sqsubset> x) \<and> (\<forall>y \<in> {}. x \<sqsubset> y)) \<equiv> True"
   243 proof(simp add: atomize_eq)
   244   from gt_ex[rule_format, of "Max L"] obtain M where M: "Max L \<sqsubset> M" by blast
   245   from ne fL have "\<forall>x \<in> L. x \<sqsubseteq> Max L" by simp
   246   with M have "\<forall>x\<in>L. x \<sqsubset> M" by (auto intro: le_less_trans)
   247   thus "\<exists>x. \<forall>y\<in>L. y \<^loc>< x" by blast
   248 qed
   249 
   250 lemma dlo_qe_nolb: 
   251   assumes ne: "U \<noteq> {}" and fU: "finite U"
   252   shows "(\<exists>x. (\<forall>y \<in> {}. y \<sqsubset> x) \<and> (\<forall>y \<in> U. x \<sqsubset> y)) \<equiv> True"
   253 proof(simp add: atomize_eq)
   254   from lt_ex[rule_format, of "Min U"] obtain M where M: "M \<sqsubset> Min U" by blast
   255   from ne fU have "\<forall>x \<in> U. Min U \<sqsubseteq> x" by simp
   256   with M have "\<forall>x\<in>U. M \<sqsubset> x" by (auto intro: less_le_trans)
   257   thus "\<exists>x. \<forall>y\<in>U. x \<^loc>< y" by blast
   258 qed
   259 
   260 lemma exists_neq: "\<exists>(x::'a). x \<noteq> t" "\<exists>(x::'a). t \<noteq> x" 
   261   using gt_ex[rule_format, of t] by auto
   262 
   263 lemmas dlo_simps = order_refl less_irrefl not_less not_le exists_neq 
   264   le_less neq_iff linear less_not_permute
   265 
   266 lemma axiom: "dense_linear_order (op \<sqsubseteq>) (op \<sqsubset>)" .
   267 lemma atoms: includes meta_term_syntax
   268   shows "TERM (op \<sqsubset> :: 'a \<Rightarrow> _)" and "TERM (op \<sqsubseteq>)" and "TERM (op = :: 'a \<Rightarrow> _)" .
   269 
   270 declare axiom[langford qe: dlo_qe_bnds dlo_qe_nolb dlo_qe_noub gather: gather_start gather_simps atoms: atoms]
   271 declare dlo_simps[langfordsimp]
   272 
   273 end
   274 
   275 (* FIXME: Move to HOL -- together with the conj_aci_rule in langford.ML *)
   276 lemma dnf:
   277   "(P & (Q | R)) = ((P&Q) | (P&R))" 
   278   "((Q | R) & P) = ((Q&P) | (R&P))"
   279   by blast+
   280 
   281 lemmas weak_dnf_simps = simp_thms dnf
   282 
   283 lemma nnf_simps:
   284     "(\<not>(P \<and> Q)) = (\<not>P \<or> \<not>Q)" "(\<not>(P \<or> Q)) = (\<not>P \<and> \<not>Q)" "(P \<longrightarrow> Q) = (\<not>P \<or> Q)"
   285     "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not> \<not>(P)) = P"
   286   by blast+
   287 
   288 lemma ex_distrib: "(\<exists>x. P x \<or> Q x) \<longleftrightarrow> ((\<exists>x. P x) \<or> (\<exists>x. Q x))" by blast
   289 
   290 lemmas dnf_simps = weak_dnf_simps nnf_simps ex_distrib
   291 
   292 use "Tools/Qelim/langford.ML"
   293 method_setup dlo = {*
   294   Method.ctxt_args (Method.SIMPLE_METHOD' o LangfordQE.dlo_tac)
   295 *} "Langford's algorithm for quantifier elimination in dense linear orders"
   296 
   297 
   298 section {* Contructive dense linear orders yield QE for linear arithmetic over ordered Fields -- see @{text "Arith_Tools.thy"} *}
   299 
   300 text {* Linear order without upper bounds *}
   301 
   302 class linorder_no_ub = linorder +
   303   assumes gt_ex: "\<exists>y. x \<sqsubset> y"
   304 begin
   305 
   306 lemma ge_ex: "\<exists>y. x \<sqsubseteq> y" using gt_ex by auto
   307 
   308 text {* Theorems for @{text "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>+\<infinity>\<^esub>)"} *}
   309 lemma pinf_conj:
   310   assumes ex1: "\<exists>z1. \<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
   311   and ex2: "\<exists>z2. \<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
   312   shows "\<exists>z. \<forall>x. z \<sqsubset>  x \<longrightarrow> ((P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2'))"
   313 proof-
   314   from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
   315      and z2: "\<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')" by blast
   316   from gt_ex obtain z where z:"max z1 z2 \<sqsubset> z" by blast
   317   from z have zz1: "z1 \<sqsubset> z" and zz2: "z2 \<sqsubset> z" by simp_all
   318   {fix x assume H: "z \<sqsubset> x"
   319     from less_trans[OF zz1 H] less_trans[OF zz2 H]
   320     have "(P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2')"  using z1 zz1 z2 zz2 by auto
   321   }
   322   thus ?thesis by blast
   323 qed
   324 
   325 lemma pinf_disj:
   326   assumes ex1: "\<exists>z1. \<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
   327   and ex2: "\<exists>z2. \<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
   328   shows "\<exists>z. \<forall>x. z \<sqsubset>  x \<longrightarrow> ((P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2'))"
   329 proof-
   330   from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
   331      and z2: "\<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')" by blast
   332   from gt_ex obtain z where z:"max z1 z2 \<sqsubset> z" by blast
   333   from z have zz1: "z1 \<sqsubset> z" and zz2: "z2 \<sqsubset> z" by simp_all
   334   {fix x assume H: "z \<sqsubset> x"
   335     from less_trans[OF zz1 H] less_trans[OF zz2 H]
   336     have "(P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2')"  using z1 zz1 z2 zz2 by auto
   337   }
   338   thus ?thesis by blast
   339 qed
   340 
   341 lemma pinf_ex: assumes ex:"\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (P x \<longleftrightarrow> P1)" and p1: P1 shows "\<exists> x. P x"
   342 proof-
   343   from ex obtain z where z: "\<forall>x. z \<sqsubset> x \<longrightarrow> (P x \<longleftrightarrow> P1)" by blast
   344   from gt_ex obtain x where x: "z \<sqsubset> x" by blast
   345   from z x p1 show ?thesis by blast
   346 qed
   347 
   348 end
   349 
   350 text {* Linear order without upper bounds *}
   351 
   352 class linorder_no_lb = linorder +
   353   assumes lt_ex: "\<exists>y. y \<sqsubset> x"
   354 begin
   355 
   356 lemma le_ex: "\<exists>y. y \<sqsubseteq> x" using lt_ex by auto
   357 
   358 
   359 text {* Theorems for @{text "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>-\<infinity>\<^esub>)"} *}
   360 lemma minf_conj:
   361   assumes ex1: "\<exists>z1. \<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
   362   and ex2: "\<exists>z2. \<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
   363   shows "\<exists>z. \<forall>x. x \<sqsubset>  z \<longrightarrow> ((P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2'))"
   364 proof-
   365   from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"and z2: "\<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')" by blast
   366   from lt_ex obtain z where z:"z \<sqsubset> min z1 z2" by blast
   367   from z have zz1: "z \<sqsubset> z1" and zz2: "z \<sqsubset> z2" by simp_all
   368   {fix x assume H: "x \<sqsubset> z"
   369     from less_trans[OF H zz1] less_trans[OF H zz2]
   370     have "(P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2')"  using z1 zz1 z2 zz2 by auto
   371   }
   372   thus ?thesis by blast
   373 qed
   374 
   375 lemma minf_disj:
   376   assumes ex1: "\<exists>z1. \<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
   377   and ex2: "\<exists>z2. \<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
   378   shows "\<exists>z. \<forall>x. x \<sqsubset>  z \<longrightarrow> ((P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2'))"
   379 proof-
   380   from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"and z2: "\<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')" by blast
   381   from lt_ex obtain z where z:"z \<sqsubset> min z1 z2" by blast
   382   from z have zz1: "z \<sqsubset> z1" and zz2: "z \<sqsubset> z2" by simp_all
   383   {fix x assume H: "x \<sqsubset> z"
   384     from less_trans[OF H zz1] less_trans[OF H zz2]
   385     have "(P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2')"  using z1 zz1 z2 zz2 by auto
   386   }
   387   thus ?thesis by blast
   388 qed
   389 
   390 lemma minf_ex: assumes ex:"\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P1)" and p1: P1 shows "\<exists> x. P x"
   391 proof-
   392   from ex obtain z where z: "\<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P1)" by blast
   393   from lt_ex obtain x where x: "x \<sqsubset> z" by blast
   394   from z x p1 show ?thesis by blast
   395 qed
   396 
   397 end
   398 
   399 
   400 class constr_dense_linear_order = linorder_no_lb + linorder_no_ub +
   401   fixes between
   402   assumes between_less: "x \<sqsubset> y \<Longrightarrow> x \<sqsubset> between x y \<and> between x y \<sqsubset> y"
   403      and  between_same: "between x x = x"
   404 
   405 instance advanced constr_dense_linear_order < dense_linear_order
   406   apply unfold_locales
   407   using gt_ex lt_ex between_less
   408     by (auto, rule_tac x="between x y" in exI, simp)
   409 (*FIXME*)
   410 lemmas gt_ex = dense_linear_order_class.less_eq_less.gt_ex
   411 lemmas lt_ex = dense_linear_order_class.less_eq_less.lt_ex
   412 lemmas dense = dense_linear_order_class.less_eq_less.dense
   413 
   414 context constr_dense_linear_order
   415 begin
   416 
   417 lemma rinf_U:
   418   assumes fU: "finite U"
   419   and lin_dense: "\<forall>x l u. (\<forall> t. l \<sqsubset> t \<and> t\<sqsubset> u \<longrightarrow> t \<notin> U) \<and> l\<sqsubset> x \<and> x \<sqsubset> u \<and> P x
   420   \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> P y )"
   421   and nmpiU: "\<forall>x. \<not> MP \<and> \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. \<exists> u' \<in> U. u \<sqsubseteq> x \<and> x \<sqsubseteq> u')"
   422   and nmi: "\<not> MP"  and npi: "\<not> PP"  and ex: "\<exists> x.  P x"
   423   shows "\<exists> u\<in> U. \<exists> u' \<in> U. P (between u u')"
   424 proof-
   425   from ex obtain x where px: "P x" by blast
   426   from px nmi npi nmpiU have "\<exists> u\<in> U. \<exists> u' \<in> U. u \<sqsubseteq> x \<and> x \<sqsubseteq> u'" by auto
   427   then obtain u and u' where uU:"u\<in> U" and uU': "u' \<in> U" and ux:"u \<sqsubseteq> x" and xu':"x \<sqsubseteq> u'" by auto
   428   from uU have Une: "U \<noteq> {}" by auto
   429   let ?l = "Min U"
   430   let ?u = "Max U"
   431   have linM: "?l \<in> U" using fU Une by simp
   432   have uinM: "?u \<in> U" using fU Une by simp
   433   have lM: "\<forall> t\<in> U. ?l \<sqsubseteq> t" using Une fU by auto
   434   have Mu: "\<forall> t\<in> U. t \<sqsubseteq> ?u" using Une fU by auto
   435   have th:"?l \<sqsubseteq> u" using uU Une lM by auto
   436   from order_trans[OF th ux] have lx: "?l \<sqsubseteq> x" .
   437   have th: "u' \<sqsubseteq> ?u" using uU' Une Mu by simp
   438   from order_trans[OF xu' th] have xu: "x \<sqsubseteq> ?u" .
   439   from finite_set_intervals2[where P="P",OF px lx xu linM uinM fU lM Mu]
   440   have "(\<exists> s\<in> U. P s) \<or>
   441       (\<exists> t1\<in> U. \<exists> t2 \<in> U. (\<forall> y. t1 \<sqsubset> y \<and> y \<sqsubset> t2 \<longrightarrow> y \<notin> U) \<and> t1 \<sqsubset> x \<and> x \<sqsubset> t2 \<and> P x)" .
   442   moreover { fix u assume um: "u\<in>U" and pu: "P u"
   443     have "between u u = u" by (simp add: between_same)
   444     with um pu have "P (between u u)" by simp
   445     with um have ?thesis by blast}
   446   moreover{
   447     assume "\<exists> t1\<in> U. \<exists> t2 \<in> U. (\<forall> y. t1 \<sqsubset> y \<and> y \<sqsubset> t2 \<longrightarrow> y \<notin> U) \<and> t1 \<sqsubset> x \<and> x \<sqsubset> t2 \<and> P x"
   448       then obtain t1 and t2 where t1M: "t1 \<in> U" and t2M: "t2\<in> U"
   449         and noM: "\<forall> y. t1 \<sqsubset> y \<and> y \<sqsubset> t2 \<longrightarrow> y \<notin> U" and t1x: "t1 \<sqsubset> x" and xt2: "x \<sqsubset> t2" and px: "P x"
   450         by blast
   451       from less_trans[OF t1x xt2] have t1t2: "t1 \<sqsubset> t2" .
   452       let ?u = "between t1 t2"
   453       from between_less t1t2 have t1lu: "t1 \<sqsubset> ?u" and ut2: "?u \<sqsubset> t2" by auto
   454       from lin_dense[rule_format, OF] noM t1x xt2 px t1lu ut2 have "P ?u" by blast
   455       with t1M t2M have ?thesis by blast}
   456     ultimately show ?thesis by blast
   457   qed
   458 
   459 theorem fr_eq:
   460   assumes fU: "finite U"
   461   and lin_dense: "\<forall>x l u. (\<forall> t. l \<sqsubset> t \<and> t\<sqsubset> u \<longrightarrow> t \<notin> U) \<and> l\<sqsubset> x \<and> x \<sqsubset> u \<and> P x
   462    \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> P y )"
   463   and nmibnd: "\<forall>x. \<not> MP \<and> P x \<longrightarrow> (\<exists> u\<in> U. u \<sqsubseteq> x)"
   464   and npibnd: "\<forall>x. \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. x \<sqsubseteq> u)"
   465   and mi: "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P x = MP)"  and pi: "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (P x = PP)"
   466   shows "(\<exists> x. P x) \<equiv> (MP \<or> PP \<or> (\<exists> u \<in> U. \<exists> u'\<in> U. P (between u u')))"
   467   (is "_ \<equiv> (_ \<or> _ \<or> ?F)" is "?E \<equiv> ?D")
   468 proof-
   469  {
   470    assume px: "\<exists> x. P x"
   471    have "MP \<or> PP \<or> (\<not> MP \<and> \<not> PP)" by blast
   472    moreover {assume "MP \<or> PP" hence "?D" by blast}
   473    moreover {assume nmi: "\<not> MP" and npi: "\<not> PP"
   474      from npmibnd[OF nmibnd npibnd]
   475      have nmpiU: "\<forall>x. \<not> MP \<and> \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. \<exists> u' \<in> U. u \<sqsubseteq> x \<and> x \<sqsubseteq> u')" .
   476      from rinf_U[OF fU lin_dense nmpiU nmi npi px] have "?D" by blast}
   477    ultimately have "?D" by blast}
   478  moreover
   479  { assume "?D"
   480    moreover {assume m:"MP" from minf_ex[OF mi m] have "?E" .}
   481    moreover {assume p: "PP" from pinf_ex[OF pi p] have "?E" . }
   482    moreover {assume f:"?F" hence "?E" by blast}
   483    ultimately have "?E" by blast}
   484  ultimately have "?E = ?D" by blast thus "?E \<equiv> ?D" by simp
   485 qed
   486 
   487 lemmas minf_thms = minf_conj minf_disj minf_eq minf_neq minf_lt minf_le minf_gt minf_ge minf_P
   488 lemmas pinf_thms = pinf_conj pinf_disj pinf_eq pinf_neq pinf_lt pinf_le pinf_gt pinf_ge pinf_P
   489 
   490 lemmas nmi_thms = nmi_conj nmi_disj nmi_eq nmi_neq nmi_lt nmi_le nmi_gt nmi_ge nmi_P
   491 lemmas npi_thms = npi_conj npi_disj npi_eq npi_neq npi_lt npi_le npi_gt npi_ge npi_P
   492 lemmas lin_dense_thms = lin_dense_conj lin_dense_disj lin_dense_eq lin_dense_neq lin_dense_lt lin_dense_le lin_dense_gt lin_dense_ge lin_dense_P
   493 
   494 lemma ferrack_axiom: "constr_dense_linear_order less_eq less between" by fact
   495 lemma atoms: includes meta_term_syntax
   496   shows "TERM (op \<sqsubset> :: 'a \<Rightarrow> _)" and "TERM (op \<sqsubseteq>)" and "TERM (op = :: 'a \<Rightarrow> _)" .
   497 
   498 declare ferrack_axiom [ferrack minf: minf_thms pinf: pinf_thms
   499     nmi: nmi_thms npi: npi_thms lindense:
   500     lin_dense_thms qe: fr_eq atoms: atoms]
   501 
   502 declaration {*
   503 let
   504 fun simps phi = map (Morphism.thm phi) [@{thm "not_less"}, @{thm "not_le"}]
   505 fun generic_whatis phi =
   506  let
   507   val [lt, le] = map (Morphism.term phi) [@{term "op \<sqsubset>"}, @{term "op \<sqsubseteq>"}]
   508   fun h x t =
   509    case term_of t of
   510      Const("op =", _)$y$z => if term_of x aconv y then Ferrante_Rackoff_Data.Eq
   511                             else Ferrante_Rackoff_Data.Nox
   512    | @{term "Not"}$(Const("op =", _)$y$z) => if term_of x aconv y then Ferrante_Rackoff_Data.NEq
   513                             else Ferrante_Rackoff_Data.Nox
   514    | b$y$z => if Term.could_unify (b, lt) then
   515                  if term_of x aconv y then Ferrante_Rackoff_Data.Lt
   516                  else if term_of x aconv z then Ferrante_Rackoff_Data.Gt
   517                  else Ferrante_Rackoff_Data.Nox
   518              else if Term.could_unify (b, le) then
   519                  if term_of x aconv y then Ferrante_Rackoff_Data.Le
   520                  else if term_of x aconv z then Ferrante_Rackoff_Data.Ge
   521                  else Ferrante_Rackoff_Data.Nox
   522              else Ferrante_Rackoff_Data.Nox
   523    | _ => Ferrante_Rackoff_Data.Nox
   524  in h end
   525  fun ss phi = HOL_ss addsimps (simps phi)
   526 in
   527  Ferrante_Rackoff_Data.funs  @{thm "ferrack_axiom"}
   528   {isolate_conv = K (K (K Thm.reflexive)), whatis = generic_whatis, simpset = ss}
   529 end
   530 *}
   531 
   532 end
   533 
   534 use "Tools/Qelim/ferrante_rackoff.ML"
   535 
   536 method_setup ferrack = {*
   537   Method.ctxt_args (Method.SIMPLE_METHOD' o FerranteRackoff.dlo_tac)
   538 *} "Ferrante and Rackoff's algorithm for quantifier elimination in dense linear orders"
   539 
   540 end