src/HOL/Dense_Linear_Order.thy
author wenzelm
Fri Mar 28 19:43:54 2008 +0100 (2008-03-28)
changeset 26462 dac4e2bce00d
parent 26040 08d52e2dba07
child 26733 47224a933c14
permissions -rw-r--r--
avoid rebinding of existing facts;
     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 < y \<and> y < x)" by (simp add: not_less linear)
    26 
    27 lemma gather_simps: 
    28   shows 
    29   "(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> U. x < y) \<and> x < u \<and> P x) \<longleftrightarrow> (\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> (insert u U). x < y) \<and> P x)"
    30   and "(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> U. x < y) \<and> l < x \<and> P x) \<longleftrightarrow> (\<exists>x. (\<forall>y \<in> (insert l L). y < x) \<and> (\<forall>y \<in> U. x < y) \<and> P x)"
    31   "(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> U. x < y) \<and> x < u) \<longleftrightarrow> (\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> (insert u U). x < y))"
    32   and "(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> U. x < y) \<and> l < x) \<longleftrightarrow> (\<exists>x. (\<forall>y \<in> (insert l L). y < x) \<and> (\<forall>y \<in> U. x < y))"  by auto
    33 
    34 lemma 
    35   gather_start: "(\<exists>x. P x) \<equiv> (\<exists>x. (\<forall>y \<in> {}. y < x) \<and> (\<forall>y\<in> {}. x < y) \<and> P x)" 
    36   by simp
    37 
    38 text{* Theorems for @{text "\<exists>z. \<forall>x. x < z \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>-\<infinity>\<^esub>)"}*}
    39 lemma minf_lt:  "\<exists>z . \<forall>x. x < z \<longrightarrow> (x < t \<longleftrightarrow> True)" by auto
    40 lemma minf_gt: "\<exists>z . \<forall>x. x < z \<longrightarrow>  (t < 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 < z \<longrightarrow> (x \<le> t \<longleftrightarrow> True)" by (auto simp add: less_le)
    44 lemma minf_ge: "\<exists>z. \<forall>x. x < z \<longrightarrow> (t \<le> x \<longleftrightarrow> False)"
    45   by (auto simp add: less_le not_less not_le)
    46 lemma minf_eq: "\<exists>z. \<forall>x. x < z \<longrightarrow> (x = t \<longleftrightarrow> False)" by auto
    47 lemma minf_neq: "\<exists>z. \<forall>x. x < z \<longrightarrow> (x \<noteq> t \<longleftrightarrow> True)" by auto
    48 lemma minf_P: "\<exists>z. \<forall>x. x < z \<longrightarrow> (P \<longleftrightarrow> P)" by blast
    49 
    50 text{* Theorems for @{text "\<exists>z. \<forall>x. x < z \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>+\<infinity>\<^esub>)"}*}
    51 lemma pinf_gt:  "\<exists>z . \<forall>x. z < x \<longrightarrow> (t < x \<longleftrightarrow> True)" by auto
    52 lemma pinf_lt: "\<exists>z . \<forall>x. z < x \<longrightarrow>  (x < 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 < x \<longrightarrow> (t \<le> x \<longleftrightarrow> True)" by (auto simp add: less_le)
    56 lemma pinf_le: "\<exists>z. \<forall>x. z < x \<longrightarrow> (x \<le> t \<longleftrightarrow> False)"
    57   by (auto simp add: less_le not_less not_le)
    58 lemma pinf_eq: "\<exists>z. \<forall>x. z < x \<longrightarrow> (x = t \<longleftrightarrow> False)" by auto
    59 lemma pinf_neq: "\<exists>z. \<forall>x. z < x \<longrightarrow> (x \<noteq> t \<longleftrightarrow> True)" by auto
    60 lemma pinf_P: "\<exists>z. \<forall>x. z < x \<longrightarrow> (P \<longleftrightarrow> P)" by blast
    61 
    62 lemma nmi_lt: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> x < t \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto
    63 lemma nmi_gt: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> t < x \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)"
    64   by (auto simp add: le_less)
    65 lemma  nmi_le: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> x\<le> t \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto
    66 lemma  nmi_ge: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> t\<le> x \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto
    67 lemma  nmi_eq: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and>  x = t \<longrightarrow>  (\<exists> u\<in> U. u \<le> 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 \<le> x)" by auto
    69 lemma  nmi_P: "\<forall> x. ~P \<and> P \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto
    70 lemma  nmi_conj: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow>  (\<exists> u\<in> U. u \<le> x) ;
    71   \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)\<rbrakk> \<Longrightarrow>
    72   \<forall>x. \<not>(P1' \<and> P2') \<and> (P1 x \<and> P2 x) \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto
    73 lemma  nmi_disj: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow>  (\<exists> u\<in> U. u \<le> x) ;
    74   \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)\<rbrakk> \<Longrightarrow>
    75   \<forall>x. \<not>(P1' \<or> P2') \<and> (P1 x \<or> P2 x) \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto
    76 
    77 lemma  npi_lt: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and>  x < t \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by (auto simp add: le_less)
    78 lemma  npi_gt: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> t < x \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by auto
    79 lemma  npi_le: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and>  x \<le> t \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by auto
    80 lemma  npi_ge: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> t \<le> x \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by auto
    81 lemma  npi_eq: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and>  x = t \<longrightarrow>  (\<exists> u\<in> U. x \<le> 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 \<le> u )" by auto
    83 lemma  npi_P: "\<forall> x. ~P \<and> P \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by auto
    84 lemma  npi_conj: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow>  (\<exists> u\<in> U. x \<le> u) ;  \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)\<rbrakk>
    85   \<Longrightarrow>  \<forall>x. \<not>(P1' \<and> P2') \<and> (P1 x \<and> P2 x) \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by auto
    86 lemma  npi_disj: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow>  (\<exists> u\<in> U. x \<le> u) ; \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)\<rbrakk>
    87   \<Longrightarrow> \<forall>x. \<not>(P1' \<or> P2') \<and> (P1 x \<or> P2 x) \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by auto
    88 
    89 lemma lin_dense_lt: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l < t \<and> t < u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> x < t \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> y < t)"
    90 proof(clarsimp)
    91   fix x l u y  assume tU: "t \<in> U" and noU: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U" and lx: "l < x"
    92     and xu: "x<u"  and px: "x < t" and ly: "l<y" and yu:"y < u"
    93   from tU noU ly yu have tny: "t\<noteq>y" by auto
    94   {assume H: "t < y"
    95     from less_trans[OF lx px] less_trans[OF H yu]
    96     have "l < t \<and> t < u"  by simp
    97     with tU noU have "False" by auto}
    98   hence "\<not> t < y"  by auto hence "y \<le> t" by (simp add: not_less)
    99   thus "y < 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 < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l < x \<and> x < u \<and> t < x \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> t < y)"
   103 proof(clarsimp)
   104   fix x l u y
   105   assume tU: "t \<in> U" and noU: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U" and lx: "l < x" and xu: "x<u"
   106   and px: "t < x" and ly: "l<y" and yu:"y < u"
   107   from tU noU ly yu have tny: "t\<noteq>y" by auto
   108   {assume H: "y< t"
   109     from less_trans[OF ly H] less_trans[OF px xu] have "l < t \<and> t < u" by simp
   110     with tU noU have "False" by auto}
   111   hence "\<not> y<t"  by auto hence "t \<le> y" by (auto simp add: not_less)
   112   thus "t < 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 < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> x \<le> t \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> y\<le> t)"
   116 proof(clarsimp)
   117   fix x l u y
   118   assume tU: "t \<in> U" and noU: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U" and lx: "l < x" and xu: "x<u"
   119   and px: "x \<le> t" and ly: "l<y" and yu:"y < u"
   120   from tU noU ly yu have tny: "t\<noteq>y" by auto
   121   {assume H: "t < y"
   122     from less_le_trans[OF lx px] less_trans[OF H yu]
   123     have "l < t \<and> t < u" by simp
   124     with tU noU have "False" by auto}
   125   hence "\<not> t < y"  by auto thus "y \<le> 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 < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> t \<le> x \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> t \<le> y)"
   129 proof(clarsimp)
   130   fix x l u y
   131   assume tU: "t \<in> U" and noU: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U" and lx: "l < x" and xu: "x<u"
   132   and px: "t \<le> x" and ly: "l<y" and yu:"y < u"
   133   from tU noU ly yu have tny: "t\<noteq>y" by auto
   134   {assume H: "y< t"
   135     from less_trans[OF ly H] le_less_trans[OF px xu]
   136     have "l < t \<and> t < u" by simp
   137     with tU noU have "False" by auto}
   138   hence "\<not> y<t"  by auto thus "t \<le> y" by (simp add: not_less)
   139 qed
   140 lemma lin_dense_eq: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> x = t   \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> y= t)"  by auto
   141 lemma lin_dense_neq: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> x \<noteq> t   \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> y\<noteq> t)"  by auto
   142 lemma lin_dense_P: "\<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> P   \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> P)"  by auto
   143 
   144 lemma lin_dense_conj:
   145   "\<lbrakk>\<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> P1 x
   146   \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> P1 y) ;
   147   \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> P2 x
   148   \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> P2 y)\<rbrakk> \<Longrightarrow>
   149   \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> (P1 x \<and> P2 x)
   150   \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> (P1 y \<and> P2 y))"
   151   by blast
   152 lemma lin_dense_disj:
   153   "\<lbrakk>\<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> P1 x
   154   \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> P1 y) ;
   155   \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> P2 x
   156   \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> P2 y)\<rbrakk> \<Longrightarrow>
   157   \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> (P1 x \<or> P2 x)
   158   \<longrightarrow> (\<forall> y. l < y \<and> y < 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 \<le> x); \<forall>x. \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. x \<le> u)\<rbrakk>
   162   \<Longrightarrow> \<forall>x. \<not> MP \<and> \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. \<exists> u' \<in> U. u \<le> x \<and> x \<le> u')"
   163 by auto
   164 
   165 lemma finite_set_intervals:
   166   assumes px: "P x" and lx: "l \<le> x" and xu: "x \<le> u" and linS: "l\<in> S"
   167   and uinS: "u \<in> S" and fS:"finite S" and lS: "\<forall> x\<in> S. l \<le> x" and Su: "\<forall> x\<in> S. x \<le> u"
   168   shows "\<exists> a \<in> S. \<exists> b \<in> S. (\<forall> y. a < y \<and> y < b \<longrightarrow> y \<notin> S) \<and> a \<le> x \<and> x \<le> b \<and> P x"
   169 proof-
   170   let ?Mx = "{y. y\<in> S \<and> y \<le> x}"
   171   let ?xM = "{y. y\<in> S \<and> x \<le> 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 \<le> x" using Mxne fMx by auto
   183   have xb:"x \<le> ?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 < y \<and> y < ?b \<longrightarrow> y \<notin> S"
   187   proof(clarsimp)
   188     fix y   assume ay: "?a < y" and yb: "y < ?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 \<le> ?a" using Mxne fMx by auto with ay have "False" by (simp add: not_le[symmetric])}
   191     moreover {assume "y \<in> ?xM" hence "?b \<le> 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 \<le> x" and xu: "x \<le> u" and linS: "l\<in> S"
   199   and uinS: "u \<in> S" and fS:"finite S" and lS: "\<forall> x\<in> S. l \<le> x" and Su: "\<forall> x\<in> S. x \<le> u"
   200   shows "(\<exists> s\<in> S. P s) \<or> (\<exists> a \<in> S. \<exists> b \<in> S. (\<forall> y. a < y \<and> y < b \<longrightarrow> y \<notin> S) \<and> a < x \<and> x < 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 < y \<and> y < b \<longrightarrow> y \<notin> S"
   205     and axb: "a \<le> x \<and> x \<le> b \<and> P x"  by auto
   206   from axb have "x= a \<or> x= b \<or> (a < x \<and> x < 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 < x) \<and> (\<forall>y \<in> U. x < y)) \<equiv> (\<forall> l \<in> L. \<forall>u \<in> U. l < u)"
   220 proof (simp only: atomize_eq, rule iffI)
   221   assume H: "\<exists>x. (\<forall>y\<in>L. y < x) \<and> (\<forall>y\<in>U. x < y)"
   222   then obtain x where xL: "\<forall>y\<in>L. y < x" and xU: "\<forall>y\<in>U. x < y" by blast
   223   {fix l u assume l: "l \<in> L" and u: "u \<in> U"
   224     have "l < x" using xL l by blast
   225     also have "x < u" using xU u by blast
   226     finally (less_trans) have "l < u" .}
   227   thus "\<forall>l\<in>L. \<forall>u\<in>U. l < u" by blast
   228 next
   229   assume H: "\<forall>l\<in>L. \<forall>u\<in>U. l < u"
   230   let ?ML = "Max L"
   231   let ?MU = "Min U"  
   232   from fL ne have th1: "?ML \<in> L" and th1': "\<forall>l\<in>L. l \<le> ?ML" by auto
   233   from fU neU have th2: "?MU \<in> U" and th2': "\<forall>u\<in>U. ?MU \<le> u" by auto
   234   from th1 th2 H have "?ML < ?MU" by auto
   235   with dense obtain w where th3: "?ML < w" and th4: "w < ?MU" by blast
   236   from th3 th1' have "\<forall>l \<in> L. l < w" by auto
   237   moreover from th4 th2' have "\<forall>u \<in> U. w < u" by auto
   238   ultimately show "\<exists>x. (\<forall>y\<in>L. y < x) \<and> (\<forall>y\<in>U. x < y)" by auto
   239 qed
   240 
   241 lemma dlo_qe_noub: 
   242   assumes ne: "L \<noteq> {}" and fL: "finite L"
   243   shows "(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> {}. x < y)) \<equiv> True"
   244 proof(simp add: atomize_eq)
   245   from gt_ex[of "Max L"] obtain M where M: "Max L < M" by blast
   246   from ne fL have "\<forall>x \<in> L. x \<le> Max L" by simp
   247   with M have "\<forall>x\<in>L. x < M" by (auto intro: le_less_trans)
   248   thus "\<exists>x. \<forall>y\<in>L. y < x" by blast
   249 qed
   250 
   251 lemma dlo_qe_nolb: 
   252   assumes ne: "U \<noteq> {}" and fU: "finite U"
   253   shows "(\<exists>x. (\<forall>y \<in> {}. y < x) \<and> (\<forall>y \<in> U. x < y)) \<equiv> True"
   254 proof(simp add: atomize_eq)
   255   from lt_ex[of "Min U"] obtain M where M: "M < Min U" by blast
   256   from ne fU have "\<forall>x \<in> U. Min U \<le> x" by simp
   257   with M have "\<forall>x\<in>U. M < x" by (auto intro: less_le_trans)
   258   thus "\<exists>x. \<forall>y\<in>U. x < y" by blast
   259 qed
   260 
   261 lemma exists_neq: "\<exists>(x::'a). x \<noteq> t" "\<exists>(x::'a). t \<noteq> x" 
   262   using gt_ex[of t] by auto
   263 
   264 lemmas dlo_simps = order_refl less_irrefl not_less not_le exists_neq 
   265   le_less neq_iff linear less_not_permute
   266 
   267 lemma axiom: "dense_linear_order (op \<le>) (op <)" by fact
   268 lemma atoms:
   269   includes meta_term_syntax
   270   shows "TERM (less :: 'a \<Rightarrow> _)"
   271     and "TERM (less_eq :: 'a \<Rightarrow> _)"
   272     and "TERM (op = :: 'a \<Rightarrow> _)" .
   273 
   274 declare axiom[langford qe: dlo_qe_bnds dlo_qe_nolb dlo_qe_noub gather: gather_start gather_simps atoms: atoms]
   275 declare dlo_simps[langfordsimp]
   276 
   277 end
   278 
   279 (* FIXME: Move to HOL -- together with the conj_aci_rule in langford.ML *)
   280 lemma dnf:
   281   "(P & (Q | R)) = ((P&Q) | (P&R))" 
   282   "((Q | R) & P) = ((Q&P) | (R&P))"
   283   by blast+
   284 
   285 lemmas weak_dnf_simps = simp_thms dnf
   286 
   287 lemma nnf_simps:
   288     "(\<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)"
   289     "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not> \<not>(P)) = P"
   290   by blast+
   291 
   292 lemma ex_distrib: "(\<exists>x. P x \<or> Q x) \<longleftrightarrow> ((\<exists>x. P x) \<or> (\<exists>x. Q x))" by blast
   293 
   294 lemmas dnf_simps = weak_dnf_simps nnf_simps ex_distrib
   295 
   296 use "Tools/Qelim/langford.ML"
   297 method_setup dlo = {*
   298   Method.ctxt_args (Method.SIMPLE_METHOD' o LangfordQE.dlo_tac)
   299 *} "Langford's algorithm for quantifier elimination in dense linear orders"
   300 
   301 
   302 section {* Contructive dense linear orders yield QE for linear arithmetic over ordered Fields -- see @{text "Arith_Tools.thy"} *}
   303 
   304 text {* Linear order without upper bounds *}
   305 
   306 locale linorder_stupid_syntax = linorder
   307 begin
   308 notation
   309   less_eq  ("op \<sqsubseteq>") and
   310   less_eq  ("(_/ \<sqsubseteq> _)" [51, 51] 50) and
   311   less  ("op \<sqsubset>") and
   312   less  ("(_/ \<sqsubset> _)"  [51, 51] 50)
   313 
   314 end
   315 
   316 locale linorder_no_ub = linorder_stupid_syntax +
   317   assumes gt_ex: "\<exists>y. less x y"
   318 begin
   319 lemma ge_ex: "\<exists>y. x \<sqsubseteq> y" using gt_ex by auto
   320 
   321 text {* Theorems for @{text "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>+\<infinity>\<^esub>)"} *}
   322 lemma pinf_conj:
   323   assumes ex1: "\<exists>z1. \<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
   324   and ex2: "\<exists>z2. \<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
   325   shows "\<exists>z. \<forall>x. z \<sqsubset>  x \<longrightarrow> ((P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2'))"
   326 proof-
   327   from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
   328      and z2: "\<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')" by blast
   329   from gt_ex obtain z where z:"ord.max less_eq z1 z2 \<sqsubset> z" by blast
   330   from z have zz1: "z1 \<sqsubset> z" and zz2: "z2 \<sqsubset> z" by simp_all
   331   {fix x assume H: "z \<sqsubset> x"
   332     from less_trans[OF zz1 H] less_trans[OF zz2 H]
   333     have "(P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2')"  using z1 zz1 z2 zz2 by auto
   334   }
   335   thus ?thesis by blast
   336 qed
   337 
   338 lemma pinf_disj:
   339   assumes ex1: "\<exists>z1. \<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
   340   and ex2: "\<exists>z2. \<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
   341   shows "\<exists>z. \<forall>x. z \<sqsubset>  x \<longrightarrow> ((P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2'))"
   342 proof-
   343   from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
   344      and z2: "\<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')" by blast
   345   from gt_ex obtain z where z:"ord.max less_eq z1 z2 \<sqsubset> z" by blast
   346   from z have zz1: "z1 \<sqsubset> z" and zz2: "z2 \<sqsubset> z" by simp_all
   347   {fix x assume H: "z \<sqsubset> x"
   348     from less_trans[OF zz1 H] less_trans[OF zz2 H]
   349     have "(P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2')"  using z1 zz1 z2 zz2 by auto
   350   }
   351   thus ?thesis by blast
   352 qed
   353 
   354 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"
   355 proof-
   356   from ex obtain z where z: "\<forall>x. z \<sqsubset> x \<longrightarrow> (P x \<longleftrightarrow> P1)" by blast
   357   from gt_ex obtain x where x: "z \<sqsubset> x" by blast
   358   from z x p1 show ?thesis by blast
   359 qed
   360 
   361 end
   362 
   363 text {* Linear order without upper bounds *}
   364 
   365 locale linorder_no_lb = linorder_stupid_syntax +
   366   assumes lt_ex: "\<exists>y. less y x"
   367 begin
   368 lemma le_ex: "\<exists>y. y \<sqsubseteq> x" using lt_ex by auto
   369 
   370 
   371 text {* Theorems for @{text "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>-\<infinity>\<^esub>)"} *}
   372 lemma minf_conj:
   373   assumes ex1: "\<exists>z1. \<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
   374   and ex2: "\<exists>z2. \<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
   375   shows "\<exists>z. \<forall>x. x \<sqsubset>  z \<longrightarrow> ((P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2'))"
   376 proof-
   377   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
   378   from lt_ex obtain z where z:"z \<sqsubset> ord.min less_eq z1 z2" by blast
   379   from z have zz1: "z \<sqsubset> z1" and zz2: "z \<sqsubset> z2" by simp_all
   380   {fix x assume H: "x \<sqsubset> z"
   381     from less_trans[OF H zz1] less_trans[OF H zz2]
   382     have "(P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2')"  using z1 zz1 z2 zz2 by auto
   383   }
   384   thus ?thesis by blast
   385 qed
   386 
   387 lemma minf_disj:
   388   assumes ex1: "\<exists>z1. \<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
   389   and ex2: "\<exists>z2. \<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
   390   shows "\<exists>z. \<forall>x. x \<sqsubset>  z \<longrightarrow> ((P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2'))"
   391 proof-
   392   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
   393   from lt_ex obtain z where z:"z \<sqsubset> ord.min less_eq z1 z2" by blast
   394   from z have zz1: "z \<sqsubset> z1" and zz2: "z \<sqsubset> z2" by simp_all
   395   {fix x assume H: "x \<sqsubset> z"
   396     from less_trans[OF H zz1] less_trans[OF H zz2]
   397     have "(P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2')"  using z1 zz1 z2 zz2 by auto
   398   }
   399   thus ?thesis by blast
   400 qed
   401 
   402 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"
   403 proof-
   404   from ex obtain z where z: "\<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P1)" by blast
   405   from lt_ex obtain x where x: "x \<sqsubset> z" by blast
   406   from z x p1 show ?thesis by blast
   407 qed
   408 
   409 end
   410 
   411 
   412 locale constr_dense_linear_order = linorder_no_lb + linorder_no_ub +
   413   fixes between
   414   assumes between_less: "less x y \<Longrightarrow> less x (between x y) \<and> less (between x y) y"
   415      and  between_same: "between x x = x"
   416 
   417 interpretation  constr_dense_linear_order < dense_linear_order 
   418   apply unfold_locales
   419   using gt_ex lt_ex between_less
   420     by (auto, rule_tac x="between x y" in exI, simp)
   421 
   422 context  constr_dense_linear_order
   423 begin
   424 
   425 lemma rinf_U:
   426   assumes fU: "finite U"
   427   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
   428   \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> P y )"
   429   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')"
   430   and nmi: "\<not> MP"  and npi: "\<not> PP"  and ex: "\<exists> x.  P x"
   431   shows "\<exists> u\<in> U. \<exists> u' \<in> U. P (between u u')"
   432 proof-
   433   from ex obtain x where px: "P x" by blast
   434   from px nmi npi nmpiU have "\<exists> u\<in> U. \<exists> u' \<in> U. u \<sqsubseteq> x \<and> x \<sqsubseteq> u'" by auto
   435   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
   436   from uU have Une: "U \<noteq> {}" by auto
   437   term "linorder.Min less_eq"
   438   let ?l = "linorder.Min less_eq U"
   439   let ?u = "linorder.Max less_eq U"
   440   have linM: "?l \<in> U" using fU Une by simp
   441   have uinM: "?u \<in> U" using fU Une by simp
   442   have lM: "\<forall> t\<in> U. ?l \<sqsubseteq> t" using Une fU by auto
   443   have Mu: "\<forall> t\<in> U. t \<sqsubseteq> ?u" using Une fU by auto
   444   have th:"?l \<sqsubseteq> u" using uU Une lM by auto
   445   from order_trans[OF th ux] have lx: "?l \<sqsubseteq> x" .
   446   have th: "u' \<sqsubseteq> ?u" using uU' Une Mu by simp
   447   from order_trans[OF xu' th] have xu: "x \<sqsubseteq> ?u" .
   448   from finite_set_intervals2[where P="P",OF px lx xu linM uinM fU lM Mu]
   449   have "(\<exists> s\<in> U. P s) \<or>
   450       (\<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)" .
   451   moreover { fix u assume um: "u\<in>U" and pu: "P u"
   452     have "between u u = u" by (simp add: between_same)
   453     with um pu have "P (between u u)" by simp
   454     with um have ?thesis by blast}
   455   moreover{
   456     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"
   457       then obtain t1 and t2 where t1M: "t1 \<in> U" and t2M: "t2\<in> U"
   458         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"
   459         by blast
   460       from less_trans[OF t1x xt2] have t1t2: "t1 \<sqsubset> t2" .
   461       let ?u = "between t1 t2"
   462       from between_less t1t2 have t1lu: "t1 \<sqsubset> ?u" and ut2: "?u \<sqsubset> t2" by auto
   463       from lin_dense noM t1x xt2 px t1lu ut2 have "P ?u" by blast
   464       with t1M t2M have ?thesis by blast}
   465     ultimately show ?thesis by blast
   466   qed
   467 
   468 theorem fr_eq:
   469   assumes fU: "finite U"
   470   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
   471    \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> P y )"
   472   and nmibnd: "\<forall>x. \<not> MP \<and> P x \<longrightarrow> (\<exists> u\<in> U. u \<sqsubseteq> x)"
   473   and npibnd: "\<forall>x. \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. x \<sqsubseteq> u)"
   474   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)"
   475   shows "(\<exists> x. P x) \<equiv> (MP \<or> PP \<or> (\<exists> u \<in> U. \<exists> u'\<in> U. P (between u u')))"
   476   (is "_ \<equiv> (_ \<or> _ \<or> ?F)" is "?E \<equiv> ?D")
   477 proof-
   478  {
   479    assume px: "\<exists> x. P x"
   480    have "MP \<or> PP \<or> (\<not> MP \<and> \<not> PP)" by blast
   481    moreover {assume "MP \<or> PP" hence "?D" by blast}
   482    moreover {assume nmi: "\<not> MP" and npi: "\<not> PP"
   483      from npmibnd[OF nmibnd npibnd]
   484      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')" .
   485      from rinf_U[OF fU lin_dense nmpiU nmi npi px] have "?D" by blast}
   486    ultimately have "?D" by blast}
   487  moreover
   488  { assume "?D"
   489    moreover {assume m:"MP" from minf_ex[OF mi m] have "?E" .}
   490    moreover {assume p: "PP" from pinf_ex[OF pi p] have "?E" . }
   491    moreover {assume f:"?F" hence "?E" by blast}
   492    ultimately have "?E" by blast}
   493  ultimately have "?E = ?D" by blast thus "?E \<equiv> ?D" by simp
   494 qed
   495 
   496 lemmas minf_thms = minf_conj minf_disj minf_eq minf_neq minf_lt minf_le minf_gt minf_ge minf_P
   497 lemmas pinf_thms = pinf_conj pinf_disj pinf_eq pinf_neq pinf_lt pinf_le pinf_gt pinf_ge pinf_P
   498 
   499 lemmas nmi_thms = nmi_conj nmi_disj nmi_eq nmi_neq nmi_lt nmi_le nmi_gt nmi_ge nmi_P
   500 lemmas npi_thms = npi_conj npi_disj npi_eq npi_neq npi_lt npi_le npi_gt npi_ge npi_P
   501 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
   502 
   503 lemma ferrack_axiom: "constr_dense_linear_order less_eq less between" by fact
   504 lemma atoms:
   505   includes meta_term_syntax
   506   shows "TERM (less :: 'a \<Rightarrow> _)"
   507     and "TERM (less_eq :: 'a \<Rightarrow> _)"
   508     and "TERM (op = :: 'a \<Rightarrow> _)" .
   509 
   510 declare ferrack_axiom [ferrack minf: minf_thms pinf: pinf_thms
   511     nmi: nmi_thms npi: npi_thms lindense:
   512     lin_dense_thms qe: fr_eq atoms: atoms]
   513 
   514 declaration {*
   515 let
   516 fun simps phi = map (Morphism.thm phi) [@{thm "not_less"}, @{thm "not_le"}]
   517 fun generic_whatis phi =
   518  let
   519   val [lt, le] = map (Morphism.term phi) [@{term "op \<sqsubset>"}, @{term "op \<sqsubseteq>"}]
   520   fun h x t =
   521    case term_of t of
   522      Const("op =", _)$y$z => if term_of x aconv y then Ferrante_Rackoff_Data.Eq
   523                             else Ferrante_Rackoff_Data.Nox
   524    | @{term "Not"}$(Const("op =", _)$y$z) => if term_of x aconv y then Ferrante_Rackoff_Data.NEq
   525                             else Ferrante_Rackoff_Data.Nox
   526    | b$y$z => if Term.could_unify (b, lt) then
   527                  if term_of x aconv y then Ferrante_Rackoff_Data.Lt
   528                  else if term_of x aconv z then Ferrante_Rackoff_Data.Gt
   529                  else Ferrante_Rackoff_Data.Nox
   530              else if Term.could_unify (b, le) then
   531                  if term_of x aconv y then Ferrante_Rackoff_Data.Le
   532                  else if term_of x aconv z then Ferrante_Rackoff_Data.Ge
   533                  else Ferrante_Rackoff_Data.Nox
   534              else Ferrante_Rackoff_Data.Nox
   535    | _ => Ferrante_Rackoff_Data.Nox
   536  in h end
   537  fun ss phi = HOL_ss addsimps (simps phi)
   538 in
   539  Ferrante_Rackoff_Data.funs  @{thm "ferrack_axiom"}
   540   {isolate_conv = K (K (K Thm.reflexive)), whatis = generic_whatis, simpset = ss}
   541 end
   542 *}
   543 
   544 end
   545 
   546 use "Tools/Qelim/ferrante_rackoff.ML"
   547 
   548 method_setup ferrack = {*
   549   Method.ctxt_args (Method.SIMPLE_METHOD' o FerranteRackoff.dlo_tac)
   550 *} "Ferrante and Rackoff's algorithm for quantifier elimination in dense linear orders"
   551 
   552 end