src/HOL/Decision_Procs/Dense_Linear_Order.thy
author blanchet
Thu Sep 11 18:54:36 2014 +0200 (2014-09-11)
changeset 58306 117ba6cbe414
parent 55848 1bfe72d14630
child 58889 5b7a9633cfa8
permissions -rw-r--r--
renamed 'rep_datatype' to 'old_rep_datatype' (HOL)
     1 (*  Title       : HOL/Decision_Procs/Dense_Linear_Order.thy
     2     Author      : Amine Chaieb, TU Muenchen
     3 *)
     4 
     5 header {* Dense linear order without endpoints
     6   and a quantifier elimination procedure in Ferrante and Rackoff style *}
     7 
     8 theory Dense_Linear_Order
     9 imports Main
    10 begin
    11 
    12 ML_file "langford_data.ML"
    13 ML_file "ferrante_rackoff_data.ML"
    14 
    15 context linorder
    16 begin
    17 
    18 lemma less_not_permute[no_atp]: "\<not> (x < y \<and> y < x)"
    19   by (simp add: not_less linear)
    20 
    21 lemma gather_simps[no_atp]: 
    22   "(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> U. x < y) \<and> x < u \<and> P x) \<longleftrightarrow>
    23     (\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> (insert u U). x < y) \<and> P x)"
    24   "(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> U. x < y) \<and> l < x \<and> P x) \<longleftrightarrow>
    25     (\<exists>x. (\<forall>y \<in> (insert l L). y < x) \<and> (\<forall>y \<in> U. x < y) \<and> P x)"
    26   "(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> U. x < y) \<and> x < u) \<longleftrightarrow>
    27     (\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> (insert u U). x < y))"
    28   "(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> U. x < y) \<and> l < x) \<longleftrightarrow>
    29     (\<exists>x. (\<forall>y \<in> (insert l L). y < x) \<and> (\<forall>y \<in> U. x < y))"
    30   by auto
    31 
    32 lemma gather_start [no_atp]: "(\<exists>x. P x) \<equiv> (\<exists>x. (\<forall>y \<in> {}. y < x) \<and> (\<forall>y\<in> {}. x < y) \<and> P x)" 
    33   by simp
    34 
    35 text{* Theorems for @{text "\<exists>z. \<forall>x. x < z \<longrightarrow> (P x \<longleftrightarrow> P\<^sub>-\<^sub>\<infinity>)"}*}
    36 lemma minf_lt[no_atp]:  "\<exists>z . \<forall>x. x < z \<longrightarrow> (x < t \<longleftrightarrow> True)" by auto
    37 lemma minf_gt[no_atp]: "\<exists>z . \<forall>x. x < z \<longrightarrow>  (t < x \<longleftrightarrow>  False)"
    38   by (simp add: not_less) (rule exI[where x="t"], auto simp add: less_le)
    39 
    40 lemma minf_le[no_atp]: "\<exists>z. \<forall>x. x < z \<longrightarrow> (x \<le> t \<longleftrightarrow> True)" by (auto simp add: less_le)
    41 lemma minf_ge[no_atp]: "\<exists>z. \<forall>x. x < z \<longrightarrow> (t \<le> x \<longleftrightarrow> False)"
    42   by (auto simp add: less_le not_less not_le)
    43 lemma minf_eq[no_atp]: "\<exists>z. \<forall>x. x < z \<longrightarrow> (x = t \<longleftrightarrow> False)" by auto
    44 lemma minf_neq[no_atp]: "\<exists>z. \<forall>x. x < z \<longrightarrow> (x \<noteq> t \<longleftrightarrow> True)" by auto
    45 lemma minf_P[no_atp]: "\<exists>z. \<forall>x. x < z \<longrightarrow> (P \<longleftrightarrow> P)" by blast
    46 
    47 text{* Theorems for @{text "\<exists>z. \<forall>x. x < z \<longrightarrow> (P x \<longleftrightarrow> P\<^sub>+\<^sub>\<infinity>)"}*}
    48 lemma pinf_gt[no_atp]:  "\<exists>z. \<forall>x. z < x \<longrightarrow> (t < x \<longleftrightarrow> True)" by auto
    49 lemma pinf_lt[no_atp]: "\<exists>z. \<forall>x. z < x \<longrightarrow>  (x < t \<longleftrightarrow>  False)"
    50   by (simp add: not_less) (rule exI[where x="t"], auto simp add: less_le)
    51 
    52 lemma pinf_ge[no_atp]: "\<exists>z. \<forall>x. z < x \<longrightarrow> (t \<le> x \<longleftrightarrow> True)" by (auto simp add: less_le)
    53 lemma pinf_le[no_atp]: "\<exists>z. \<forall>x. z < x \<longrightarrow> (x \<le> t \<longleftrightarrow> False)"
    54   by (auto simp add: less_le not_less not_le)
    55 lemma pinf_eq[no_atp]: "\<exists>z. \<forall>x. z < x \<longrightarrow> (x = t \<longleftrightarrow> False)" by auto
    56 lemma pinf_neq[no_atp]: "\<exists>z. \<forall>x. z < x \<longrightarrow> (x \<noteq> t \<longleftrightarrow> True)" by auto
    57 lemma pinf_P[no_atp]: "\<exists>z. \<forall>x. z < x \<longrightarrow> (P \<longleftrightarrow> P)" by blast
    58 
    59 lemma nmi_lt[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> x < t \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto
    60 lemma nmi_gt[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> t < x \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)"
    61   by (auto simp add: le_less)
    62 lemma  nmi_le[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> x\<le> t \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto
    63 lemma  nmi_ge[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> t\<le> x \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto
    64 lemma  nmi_eq[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and>  x = t \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto
    65 lemma  nmi_neq[no_atp]: "t \<in> U \<Longrightarrow>\<forall>x. \<not>True \<and> x \<noteq> t \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto
    66 lemma  nmi_P[no_atp]: "\<forall> x. ~P \<and> P \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto
    67 lemma  nmi_conj[no_atp]: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow>  (\<exists> u\<in> U. u \<le> x) ;
    68   \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)\<rbrakk> \<Longrightarrow>
    69   \<forall>x. \<not>(P1' \<and> P2') \<and> (P1 x \<and> P2 x) \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto
    70 lemma  nmi_disj[no_atp]: "\<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' \<or> P2') \<and> (P1 x \<or> P2 x) \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto
    73 
    74 lemma  npi_lt[no_atp]: "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)
    75 lemma  npi_gt[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> t < x \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by auto
    76 lemma  npi_le[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and>  x \<le> t \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by auto
    77 lemma  npi_ge[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> t \<le> x \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by auto
    78 lemma  npi_eq[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and>  x = t \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by auto
    79 lemma  npi_neq[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> x \<noteq> t \<longrightarrow>  (\<exists> u\<in> U. x \<le> u )" by auto
    80 lemma  npi_P[no_atp]: "\<forall> x. ~P \<and> P \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by auto
    81 lemma  npi_conj[no_atp]: "\<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>
    82   \<Longrightarrow>  \<forall>x. \<not>(P1' \<and> P2') \<and> (P1 x \<and> P2 x) \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by auto
    83 lemma  npi_disj[no_atp]: "\<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>
    84   \<Longrightarrow> \<forall>x. \<not>(P1' \<or> P2') \<and> (P1 x \<or> P2 x) \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by auto
    85 
    86 lemma lin_dense_lt[no_atp]:
    87   "t \<in> U \<Longrightarrow>
    88     \<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)"
    89 proof(clarsimp)
    90   fix x l u y
    91   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   then have "\<not> t < y" by auto
    99   then have "y \<le> t" by (simp add: not_less)
   100   then show "y < t" using tny by (simp add: less_le)
   101 qed
   102 
   103 lemma lin_dense_gt[no_atp]:
   104   "t \<in> U \<Longrightarrow>
   105     \<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)"
   106 proof(clarsimp)
   107   fix x l u y
   108   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"
   109     and px: "t < x" and ly: "l<y" and yu:"y < u"
   110   from tU noU ly yu have tny: "t\<noteq>y" by auto
   111   { assume H: "y< t"
   112     from less_trans[OF ly H] less_trans[OF px xu] have "l < t \<and> t < u" by simp
   113     with tU noU have "False" by auto }
   114   then have "\<not> y<t" by auto
   115   then have "t \<le> y" by (auto simp add: not_less)
   116   then show "t < y" using tny by (simp add: less_le)
   117 qed
   118 
   119 lemma lin_dense_le[no_atp]:
   120   "t \<in> U \<Longrightarrow>
   121     \<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)"
   122 proof(clarsimp)
   123   fix x l u y
   124   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"
   125     and px: "x \<le> t" and ly: "l<y" and yu:"y < u"
   126   from tU noU ly yu have tny: "t\<noteq>y" by auto
   127   { assume H: "t < y"
   128     from less_le_trans[OF lx px] less_trans[OF H yu]
   129     have "l < t \<and> t < u" by simp
   130     with tU noU have "False" by auto }
   131   then have "\<not> t < y" by auto
   132   then show "y \<le> t" by (simp add: not_less)
   133 qed
   134 
   135 lemma lin_dense_ge[no_atp]:
   136   "t \<in> U \<Longrightarrow>
   137     \<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)"
   138 proof(clarsimp)
   139   fix x l u y
   140   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"
   141     and px: "t \<le> x" and ly: "l<y" and yu:"y < u"
   142   from tU noU ly yu have tny: "t\<noteq>y" by auto
   143   { assume H: "y< t"
   144     from less_trans[OF ly H] le_less_trans[OF px xu]
   145     have "l < t \<and> t < u" by simp
   146     with tU noU have "False" by auto }
   147   then have "\<not> y<t" by auto
   148   then show "t \<le> y" by (simp add: not_less)
   149 qed
   150 
   151 lemma lin_dense_eq[no_atp]:
   152   "t \<in> U \<Longrightarrow>
   153     \<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)"
   154   by auto
   155 
   156 lemma lin_dense_neq[no_atp]:
   157   "t \<in> U \<Longrightarrow>
   158     \<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)"
   159   by auto
   160 
   161 lemma lin_dense_P[no_atp]:
   162   "\<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)"
   163   by auto
   164 
   165 lemma lin_dense_conj[no_atp]:
   166   "\<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
   167   \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> P1 y) ;
   168   \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> P2 x
   169   \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> P2 y)\<rbrakk> \<Longrightarrow>
   170   \<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)
   171   \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> (P1 y \<and> P2 y))"
   172   by blast
   173 lemma lin_dense_disj[no_atp]:
   174   "\<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
   175   \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> P1 y) ;
   176   \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> P2 x
   177   \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> P2 y)\<rbrakk> \<Longrightarrow>
   178   \<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)
   179   \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> (P1 y \<or> P2 y))"
   180   by blast
   181 
   182 lemma npmibnd[no_atp]: "\<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>
   183   \<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')"
   184   by auto
   185 
   186 lemma finite_set_intervals[no_atp]:
   187   assumes px: "P x" and lx: "l \<le> x" and xu: "x \<le> u" and linS: "l\<in> S"
   188     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"
   189   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"
   190 proof -
   191   let ?Mx = "{y. y\<in> S \<and> y \<le> x}"
   192   let ?xM = "{y. y\<in> S \<and> x \<le> y}"
   193   let ?a = "Max ?Mx"
   194   let ?b = "Min ?xM"
   195   have MxS: "?Mx \<subseteq> S" by blast
   196   hence fMx: "finite ?Mx" using fS finite_subset by auto
   197   from lx linS have linMx: "l \<in> ?Mx" by blast
   198   hence Mxne: "?Mx \<noteq> {}" by blast
   199   have xMS: "?xM \<subseteq> S" by blast
   200   hence fxM: "finite ?xM" using fS finite_subset by auto
   201   from xu uinS have linxM: "u \<in> ?xM" by blast
   202   hence xMne: "?xM \<noteq> {}" by blast
   203   have ax:"?a \<le> x" using Mxne fMx by auto
   204   have xb:"x \<le> ?b" using xMne fxM by auto
   205   have "?a \<in> ?Mx" using Max_in[OF fMx Mxne] by simp hence ainS: "?a \<in> S" using MxS by blast
   206   have "?b \<in> ?xM" using Min_in[OF fxM xMne] by simp hence binS: "?b \<in> S" using xMS by blast
   207   have noy:"\<forall> y. ?a < y \<and> y < ?b \<longrightarrow> y \<notin> S"
   208   proof(clarsimp)
   209     fix y
   210     assume ay: "?a < y" and yb: "y < ?b" and yS: "y \<in> S"
   211     from yS have "y\<in> ?Mx \<or> y\<in> ?xM" by (auto simp add: linear)
   212     moreover {
   213       assume "y \<in> ?Mx"
   214       hence "y \<le> ?a" using Mxne fMx by auto
   215       with ay have "False" by (simp add: not_le[symmetric])
   216     }
   217     moreover {
   218       assume "y \<in> ?xM"
   219       hence "?b \<le> y" using xMne fxM by auto
   220       with yb have "False" by (simp add: not_le[symmetric])
   221     }
   222     ultimately show False by blast
   223   qed
   224   from ainS binS noy ax xb px show ?thesis by blast
   225 qed
   226 
   227 lemma finite_set_intervals2[no_atp]:
   228   assumes px: "P x" and lx: "l \<le> x" and xu: "x \<le> u" and linS: "l\<in> S"
   229     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"
   230   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)"
   231 proof-
   232   from finite_set_intervals[where P="P", OF px lx xu linS uinS fS lS Su]
   233   obtain a and b where as: "a\<in> S" and bs: "b\<in> S" and noS:"\<forall>y. a < y \<and> y < b \<longrightarrow> y \<notin> S"
   234     and axb: "a \<le> x \<and> x \<le> b \<and> P x" by auto
   235   from axb have "x= a \<or> x= b \<or> (a < x \<and> x < b)" by (auto simp add: le_less)
   236   thus ?thesis using px as bs noS by blast
   237 qed
   238 
   239 end
   240 
   241 
   242 section {* The classical QE after Langford for dense linear orders *}
   243 
   244 context unbounded_dense_linorder
   245 begin
   246 
   247 lemma interval_empty_iff: "{y. x < y \<and> y < z} = {} \<longleftrightarrow> \<not> x < z"
   248   by (auto dest: dense)
   249 
   250 lemma dlo_qe_bnds[no_atp]: 
   251   assumes ne: "L \<noteq> {}" and neU: "U \<noteq> {}" and fL: "finite L" and fU: "finite U"
   252   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)"
   253 proof (simp only: atomize_eq, rule iffI)
   254   assume H: "\<exists>x. (\<forall>y\<in>L. y < x) \<and> (\<forall>y\<in>U. x < y)"
   255   then obtain x where xL: "\<forall>y\<in>L. y < x" and xU: "\<forall>y\<in>U. x < y" by blast
   256   { fix l u assume l: "l \<in> L" and u: "u \<in> U"
   257     have "l < x" using xL l by blast
   258     also have "x < u" using xU u by blast
   259     finally (less_trans) have "l < u" . }
   260   then show "\<forall>l\<in>L. \<forall>u\<in>U. l < u" by blast
   261 next
   262   assume H: "\<forall>l\<in>L. \<forall>u\<in>U. l < u"
   263   let ?ML = "Max L"
   264   let ?MU = "Min U"  
   265   from fL ne have th1: "?ML \<in> L" and th1': "\<forall>l\<in>L. l \<le> ?ML" by auto
   266   from fU neU have th2: "?MU \<in> U" and th2': "\<forall>u\<in>U. ?MU \<le> u" by auto
   267   from th1 th2 H have "?ML < ?MU" by auto
   268   with dense obtain w where th3: "?ML < w" and th4: "w < ?MU" by blast
   269   from th3 th1' have "\<forall>l \<in> L. l < w" by auto
   270   moreover from th4 th2' have "\<forall>u \<in> U. w < u" by auto
   271   ultimately show "\<exists>x. (\<forall>y\<in>L. y < x) \<and> (\<forall>y\<in>U. x < y)" by auto
   272 qed
   273 
   274 lemma dlo_qe_noub[no_atp]: 
   275   assumes ne: "L \<noteq> {}" and fL: "finite L"
   276   shows "(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> {}. x < y)) \<equiv> True"
   277 proof(simp add: atomize_eq)
   278   from gt_ex[of "Max L"] obtain M where M: "Max L < M" by blast
   279   from ne fL have "\<forall>x \<in> L. x \<le> Max L" by simp
   280   with M have "\<forall>x\<in>L. x < M" by (auto intro: le_less_trans)
   281   thus "\<exists>x. \<forall>y\<in>L. y < x" by blast
   282 qed
   283 
   284 lemma dlo_qe_nolb[no_atp]: 
   285   assumes ne: "U \<noteq> {}" and fU: "finite U"
   286   shows "(\<exists>x. (\<forall>y \<in> {}. y < x) \<and> (\<forall>y \<in> U. x < y)) \<equiv> True"
   287 proof(simp add: atomize_eq)
   288   from lt_ex[of "Min U"] obtain M where M: "M < Min U" by blast
   289   from ne fU have "\<forall>x \<in> U. Min U \<le> x" by simp
   290   with M have "\<forall>x\<in>U. M < x" by (auto intro: less_le_trans)
   291   thus "\<exists>x. \<forall>y\<in>U. x < y" by blast
   292 qed
   293 
   294 lemma exists_neq[no_atp]: "\<exists>(x::'a). x \<noteq> t" "\<exists>(x::'a). t \<noteq> x" 
   295   using gt_ex[of t] by auto
   296 
   297 lemmas dlo_simps[no_atp] = order_refl less_irrefl not_less not_le exists_neq 
   298   le_less neq_iff linear less_not_permute
   299 
   300 lemma axiom[no_atp]: "class.unbounded_dense_linorder (op \<le>) (op <)"
   301   by (rule unbounded_dense_linorder_axioms)
   302 lemma atoms[no_atp]:
   303   shows "TERM (less :: 'a \<Rightarrow> _)"
   304     and "TERM (less_eq :: 'a \<Rightarrow> _)"
   305     and "TERM (op = :: 'a \<Rightarrow> _)" .
   306 
   307 declare axiom[langford qe: dlo_qe_bnds dlo_qe_nolb dlo_qe_noub gather: gather_start gather_simps atoms: atoms]
   308 declare dlo_simps[langfordsimp]
   309 
   310 end
   311 
   312 (* FIXME: Move to HOL -- together with the conj_aci_rule in langford.ML *)
   313 lemmas dnf[no_atp] = conj_disj_distribL conj_disj_distribR
   314 
   315 lemmas weak_dnf_simps[no_atp] = simp_thms dnf
   316 
   317 lemma nnf_simps[no_atp]:
   318     "(\<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)"
   319     "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not> \<not>(P)) = P"
   320   by blast+
   321 
   322 lemma ex_distrib[no_atp]: "(\<exists>x. P x \<or> Q x) \<longleftrightarrow> ((\<exists>x. P x) \<or> (\<exists>x. Q x))" by blast
   323 
   324 lemmas dnf_simps[no_atp] = weak_dnf_simps nnf_simps ex_distrib
   325 
   326 ML_file "langford.ML"
   327 method_setup dlo = {*
   328   Scan.succeed (SIMPLE_METHOD' o Langford.dlo_tac)
   329 *} "Langford's algorithm for quantifier elimination in dense linear orders"
   330 
   331 
   332 section {* Contructive dense linear orders yield QE for linear arithmetic over ordered Fields *}
   333 
   334 text {* Linear order without upper bounds *}
   335 
   336 locale linorder_stupid_syntax = linorder
   337 begin
   338 
   339 notation
   340   less_eq  ("op \<sqsubseteq>") and
   341   less_eq  ("(_/ \<sqsubseteq> _)" [51, 51] 50) and
   342   less  ("op \<sqsubset>") and
   343   less  ("(_/ \<sqsubset> _)"  [51, 51] 50)
   344 
   345 end
   346 
   347 locale linorder_no_ub = linorder_stupid_syntax +
   348   assumes gt_ex: "\<exists>y. less x y"
   349 begin
   350 
   351 lemma ge_ex[no_atp]: "\<exists>y. x \<sqsubseteq> y" using gt_ex by auto
   352 
   353 text {* Theorems for @{text "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (P x \<longleftrightarrow> P\<^sub>+\<^sub>\<infinity>)"} *}
   354 lemma pinf_conj[no_atp]:
   355   assumes ex1: "\<exists>z1. \<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
   356   and ex2: "\<exists>z2. \<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
   357   shows "\<exists>z. \<forall>x. z \<sqsubset>  x \<longrightarrow> ((P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2'))"
   358 proof-
   359   from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
   360      and z2: "\<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')" by blast
   361   from gt_ex obtain z where z:"ord.max less_eq z1 z2 \<sqsubset> z" by blast
   362   from z have zz1: "z1 \<sqsubset> z" and zz2: "z2 \<sqsubset> z" by simp_all
   363   { fix x assume H: "z \<sqsubset> x"
   364     from less_trans[OF zz1 H] less_trans[OF zz2 H]
   365     have "(P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2')"  using z1 zz1 z2 zz2 by auto
   366   }
   367   thus ?thesis by blast
   368 qed
   369 
   370 lemma pinf_disj[no_atp]:
   371   assumes ex1: "\<exists>z1. \<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
   372     and ex2: "\<exists>z2. \<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
   373   shows "\<exists>z. \<forall>x. z \<sqsubset>  x \<longrightarrow> ((P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2'))"
   374 proof-
   375   from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
   376      and z2: "\<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')" by blast
   377   from gt_ex obtain z where z:"ord.max less_eq z1 z2 \<sqsubset> z" by blast
   378   from z have zz1: "z1 \<sqsubset> z" and zz2: "z2 \<sqsubset> z" by simp_all
   379   { fix x assume H: "z \<sqsubset> x"
   380     from less_trans[OF zz1 H] less_trans[OF zz2 H]
   381     have "(P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2')"  using z1 zz1 z2 zz2 by auto
   382   }
   383   thus ?thesis by blast
   384 qed
   385 
   386 lemma pinf_ex[no_atp]: assumes ex:"\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (P x \<longleftrightarrow> P1)" and p1: P1 shows "\<exists> x. P x"
   387 proof -
   388   from ex obtain z where z: "\<forall>x. z \<sqsubset> x \<longrightarrow> (P x \<longleftrightarrow> P1)" by blast
   389   from gt_ex obtain x where x: "z \<sqsubset> x" by blast
   390   from z x p1 show ?thesis by blast
   391 qed
   392 
   393 end
   394 
   395 text {* Linear order without upper bounds *}
   396 
   397 locale linorder_no_lb = linorder_stupid_syntax +
   398   assumes lt_ex: "\<exists>y. less y x"
   399 begin
   400 
   401 lemma le_ex[no_atp]: "\<exists>y. y \<sqsubseteq> x" using lt_ex by auto
   402 
   403 
   404 text {* Theorems for @{text "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P\<^sub>-\<^sub>\<infinity>)"} *}
   405 lemma minf_conj[no_atp]:
   406   assumes ex1: "\<exists>z1. \<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
   407     and ex2: "\<exists>z2. \<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
   408   shows "\<exists>z. \<forall>x. x \<sqsubset>  z \<longrightarrow> ((P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2'))"
   409 proof-
   410   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
   411   from lt_ex obtain z where z:"z \<sqsubset> ord.min less_eq z1 z2" by blast
   412   from z have zz1: "z \<sqsubset> z1" and zz2: "z \<sqsubset> z2" by simp_all
   413   { fix x assume H: "x \<sqsubset> z"
   414     from less_trans[OF H zz1] less_trans[OF H zz2]
   415     have "(P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2')"  using z1 zz1 z2 zz2 by auto
   416   }
   417   thus ?thesis by blast
   418 qed
   419 
   420 lemma minf_disj[no_atp]:
   421   assumes ex1: "\<exists>z1. \<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
   422     and ex2: "\<exists>z2. \<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
   423   shows "\<exists>z. \<forall>x. x \<sqsubset>  z \<longrightarrow> ((P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2'))"
   424 proof -
   425   from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
   426     and z2: "\<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')" by blast
   427   from lt_ex obtain z where z:"z \<sqsubset> ord.min less_eq z1 z2" by blast
   428   from z have zz1: "z \<sqsubset> z1" and zz2: "z \<sqsubset> z2" by simp_all
   429   { fix x assume H: "x \<sqsubset> z"
   430     from less_trans[OF H zz1] less_trans[OF H zz2]
   431     have "(P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2')"  using z1 zz1 z2 zz2 by auto
   432   }
   433   thus ?thesis by blast
   434 qed
   435 
   436 lemma minf_ex[no_atp]:
   437   assumes ex: "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P1)"
   438     and p1: P1
   439   shows "\<exists> x. P x"
   440 proof -
   441   from ex obtain z where z: "\<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P1)" by blast
   442   from lt_ex obtain x where x: "x \<sqsubset> z" by blast
   443   from z x p1 show ?thesis by blast
   444 qed
   445 
   446 end
   447 
   448 
   449 locale constr_dense_linorder = linorder_no_lb + linorder_no_ub +
   450   fixes between
   451   assumes between_less: "less x y \<Longrightarrow> less x (between x y) \<and> less (between x y) y"
   452     and between_same: "between x x = x"
   453 begin
   454 
   455 sublocale dlo: unbounded_dense_linorder 
   456   apply unfold_locales
   457   using gt_ex lt_ex between_less
   458   apply auto
   459   apply (rule_tac x="between x y" in exI)
   460   apply simp
   461   done
   462 
   463 lemma rinf_U[no_atp]:
   464   assumes fU: "finite U"
   465     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
   466       \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> P y )"
   467     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')"
   468     and nmi: "\<not> MP"  and npi: "\<not> PP"  and ex: "\<exists> x.  P x"
   469   shows "\<exists> u\<in> U. \<exists> u' \<in> U. P (between u u')"
   470 proof -
   471   from ex obtain x where px: "P x" by blast
   472   from px nmi npi nmpiU have "\<exists> u\<in> U. \<exists> u' \<in> U. u \<sqsubseteq> x \<and> x \<sqsubseteq> u'" by auto
   473   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
   474   from uU have Une: "U \<noteq> {}" by auto
   475   let ?l = "linorder.Min less_eq U"
   476   let ?u = "linorder.Max less_eq U"
   477   have linM: "?l \<in> U" using fU Une by simp
   478   have uinM: "?u \<in> U" using fU Une by simp
   479   have lM: "\<forall> t\<in> U. ?l \<sqsubseteq> t" using Une fU by auto
   480   have Mu: "\<forall> t\<in> U. t \<sqsubseteq> ?u" using Une fU by auto
   481   have th:"?l \<sqsubseteq> u" using uU Une lM by auto
   482   from order_trans[OF th ux] have lx: "?l \<sqsubseteq> x" .
   483   have th: "u' \<sqsubseteq> ?u" using uU' Une Mu by simp
   484   from order_trans[OF xu' th] have xu: "x \<sqsubseteq> ?u" .
   485   from finite_set_intervals2[where P="P",OF px lx xu linM uinM fU lM Mu]
   486   have "(\<exists> s\<in> U. P s) \<or>
   487       (\<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)" .
   488   moreover {
   489     fix u assume um: "u\<in>U" and pu: "P u"
   490     have "between u u = u" by (simp add: between_same)
   491     with um pu have "P (between u u)" by simp
   492     with um have ?thesis by blast }
   493   moreover {
   494     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"
   495     then obtain t1 and t2 where t1M: "t1 \<in> U" and t2M: "t2\<in> U"
   496       and noM: "\<forall> y. t1 \<sqsubset> y \<and> y \<sqsubset> t2 \<longrightarrow> y \<notin> U"
   497       and t1x: "t1 \<sqsubset> x" and xt2: "x \<sqsubset> t2" and px: "P x" by blast
   498     from less_trans[OF t1x xt2] have t1t2: "t1 \<sqsubset> t2" .
   499     let ?u = "between t1 t2"
   500     from between_less t1t2 have t1lu: "t1 \<sqsubset> ?u" and ut2: "?u \<sqsubset> t2" by auto
   501     from lin_dense noM t1x xt2 px t1lu ut2 have "P ?u" by blast
   502     with t1M t2M have ?thesis by blast
   503   }
   504   ultimately show ?thesis by blast
   505 qed
   506 
   507 theorem fr_eq[no_atp]:
   508   assumes fU: "finite U"
   509     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
   510      \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> P y )"
   511     and nmibnd: "\<forall>x. \<not> MP \<and> P x \<longrightarrow> (\<exists> u\<in> U. u \<sqsubseteq> x)"
   512     and npibnd: "\<forall>x. \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. x \<sqsubseteq> u)"
   513     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)"
   514   shows "(\<exists> x. P x) \<equiv> (MP \<or> PP \<or> (\<exists> u \<in> U. \<exists> u'\<in> U. P (between u u')))"
   515   (is "_ \<equiv> (_ \<or> _ \<or> ?F)" is "?E \<equiv> ?D")
   516 proof -
   517   { assume px: "\<exists> x. P x"
   518     have "MP \<or> PP \<or> (\<not> MP \<and> \<not> PP)" by blast
   519     moreover { assume "MP \<or> PP" hence "?D" by blast }
   520     moreover {
   521       assume nmi: "\<not> MP" and npi: "\<not> PP"
   522       from npmibnd[OF nmibnd npibnd]
   523       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')" .
   524       from rinf_U[OF fU lin_dense nmpiU nmi npi px] have "?D" by blast }
   525     ultimately have "?D" by blast }
   526   moreover
   527   { assume "?D"
   528     moreover { assume m:"MP" from minf_ex[OF mi m] have "?E" . }
   529     moreover { assume p: "PP" from pinf_ex[OF pi p] have "?E" . }
   530     moreover { assume f:"?F" hence "?E" by blast }
   531     ultimately have "?E" by blast }
   532   ultimately have "?E = ?D" by blast thus "?E \<equiv> ?D" by simp
   533 qed
   534 
   535 lemmas minf_thms[no_atp] = minf_conj minf_disj minf_eq minf_neq minf_lt minf_le minf_gt minf_ge minf_P
   536 lemmas pinf_thms[no_atp] = pinf_conj pinf_disj pinf_eq pinf_neq pinf_lt pinf_le pinf_gt pinf_ge pinf_P
   537 
   538 lemmas nmi_thms[no_atp] = nmi_conj nmi_disj nmi_eq nmi_neq nmi_lt nmi_le nmi_gt nmi_ge nmi_P
   539 lemmas npi_thms[no_atp] = npi_conj npi_disj npi_eq npi_neq npi_lt npi_le npi_gt npi_ge npi_P
   540 lemmas lin_dense_thms[no_atp] = 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
   541 
   542 lemma ferrack_axiom[no_atp]: "constr_dense_linorder less_eq less between"
   543   by (rule constr_dense_linorder_axioms)
   544 
   545 lemma atoms[no_atp]:
   546   shows "TERM (less :: 'a \<Rightarrow> _)"
   547     and "TERM (less_eq :: 'a \<Rightarrow> _)"
   548     and "TERM (op = :: 'a \<Rightarrow> _)" .
   549 
   550 declare ferrack_axiom [ferrack minf: minf_thms pinf: pinf_thms
   551     nmi: nmi_thms npi: npi_thms lindense:
   552     lin_dense_thms qe: fr_eq atoms: atoms]
   553 
   554 declaration {*
   555 let
   556   fun simps phi = map (Morphism.thm phi) [@{thm "not_less"}, @{thm "not_le"}]
   557   fun generic_whatis phi =
   558     let
   559       val [lt, le] = map (Morphism.term phi) [@{term "op \<sqsubset>"}, @{term "op \<sqsubseteq>"}]
   560       fun h x t =
   561         case term_of t of
   562           Const(@{const_name HOL.eq}, _)$y$z =>
   563             if term_of x aconv y then Ferrante_Rackoff_Data.Eq
   564             else Ferrante_Rackoff_Data.Nox
   565        | @{term "Not"}$(Const(@{const_name HOL.eq}, _)$y$z) =>
   566             if term_of x aconv y then Ferrante_Rackoff_Data.NEq
   567             else Ferrante_Rackoff_Data.Nox
   568        | b$y$z => if Term.could_unify (b, lt) then
   569                      if term_of x aconv y then Ferrante_Rackoff_Data.Lt
   570                      else if term_of x aconv z then Ferrante_Rackoff_Data.Gt
   571                      else Ferrante_Rackoff_Data.Nox
   572                  else if Term.could_unify (b, le) then
   573                      if term_of x aconv y then Ferrante_Rackoff_Data.Le
   574                      else if term_of x aconv z then Ferrante_Rackoff_Data.Ge
   575                      else Ferrante_Rackoff_Data.Nox
   576                  else Ferrante_Rackoff_Data.Nox
   577        | _ => Ferrante_Rackoff_Data.Nox
   578   in h end
   579   fun ss phi =
   580     simpset_of (put_simpset HOL_ss @{context} addsimps (simps phi))
   581 in
   582   Ferrante_Rackoff_Data.funs  @{thm "ferrack_axiom"}
   583     {isolate_conv = K (K (K Thm.reflexive)), whatis = generic_whatis, simpset = ss}
   584 end
   585 *}
   586 
   587 end
   588 
   589 ML_file "ferrante_rackoff.ML"
   590 
   591 method_setup ferrack = {*
   592   Scan.succeed (SIMPLE_METHOD' o FerranteRackoff.dlo_tac)
   593 *} "Ferrante and Rackoff's algorithm for quantifier elimination in dense linear orders"
   594 
   595 
   596 subsection {* Ferrante and Rackoff algorithm over ordered fields *}
   597 
   598 lemma neg_prod_lt:"(c\<Colon>'a\<Colon>linordered_field) < 0 \<Longrightarrow> ((c*x < 0) == (x > 0))"
   599 proof -
   600   assume H: "c < 0"
   601   have "c*x < 0 = (0/c < x)"
   602     by (simp only: neg_divide_less_eq[OF H] algebra_simps)
   603   also have "\<dots> = (0 < x)" by simp
   604   finally show  "(c*x < 0) == (x > 0)" by simp
   605 qed
   606 
   607 lemma pos_prod_lt:"(c\<Colon>'a\<Colon>linordered_field) > 0 \<Longrightarrow> ((c*x < 0) == (x < 0))"
   608 proof -
   609   assume H: "c > 0"
   610   then have "c*x < 0 = (0/c > x)"
   611     by (simp only: pos_less_divide_eq[OF H] algebra_simps)
   612   also have "\<dots> = (0 > x)" by simp
   613   finally show  "(c*x < 0) == (x < 0)" by simp
   614 qed
   615 
   616 lemma neg_prod_sum_lt: "(c\<Colon>'a\<Colon>linordered_field) < 0 \<Longrightarrow> ((c*x + t< 0) == (x > (- 1/c)*t))"
   617 proof -
   618   assume H: "c < 0"
   619   have "c*x + t< 0 = (c*x < -t)"
   620     by (subst less_iff_diff_less_0 [of "c*x" "-t"], simp)
   621   also have "\<dots> = (-t/c < x)"
   622     by (simp only: neg_divide_less_eq[OF H] algebra_simps)
   623   also have "\<dots> = ((- 1/c)*t < x)" by simp
   624   finally show "(c*x + t < 0) == (x > (- 1/c)*t)" by simp
   625 qed
   626 
   627 lemma pos_prod_sum_lt:"(c\<Colon>'a\<Colon>linordered_field) > 0 \<Longrightarrow> ((c*x + t < 0) == (x < (- 1/c)*t))"
   628 proof -
   629   assume H: "c > 0"
   630   have "c*x + t< 0 = (c*x < -t)"
   631     by (subst less_iff_diff_less_0 [of "c*x" "-t"], simp)
   632   also have "\<dots> = (-t/c > x)"
   633     by (simp only: pos_less_divide_eq[OF H] algebra_simps)
   634   also have "\<dots> = ((- 1/c)*t > x)" by simp
   635   finally show  "(c*x + t < 0) == (x < (- 1/c)*t)" by simp
   636 qed
   637 
   638 lemma sum_lt:"((x::'a::ordered_ab_group_add) + t < 0) == (x < - t)"
   639   using less_diff_eq[where a= x and b=t and c=0] by simp
   640 
   641 lemma neg_prod_le:"(c\<Colon>'a\<Colon>linordered_field) < 0 \<Longrightarrow> ((c*x <= 0) == (x >= 0))"
   642 proof -
   643   assume H: "c < 0"
   644   have "c*x <= 0 = (0/c <= x)"
   645     by (simp only: neg_divide_le_eq[OF H] algebra_simps)
   646   also have "\<dots> = (0 <= x)" by simp
   647   finally show  "(c*x <= 0) == (x >= 0)" by simp
   648 qed
   649 
   650 lemma pos_prod_le:"(c\<Colon>'a\<Colon>linordered_field) > 0 \<Longrightarrow> ((c*x <= 0) == (x <= 0))"
   651 proof -
   652   assume H: "c > 0"
   653   hence "c*x <= 0 = (0/c >= x)"
   654     by (simp only: pos_le_divide_eq[OF H] algebra_simps)
   655   also have "\<dots> = (0 >= x)" by simp
   656   finally show  "(c*x <= 0) == (x <= 0)" by simp
   657 qed
   658 
   659 lemma neg_prod_sum_le: "(c\<Colon>'a\<Colon>linordered_field) < 0 \<Longrightarrow> ((c*x + t <= 0) == (x >= (- 1/c)*t))"
   660 proof -
   661   assume H: "c < 0"
   662   have "c*x + t <= 0 = (c*x <= -t)"
   663     by (subst le_iff_diff_le_0 [of "c*x" "-t"], simp)
   664   also have "\<dots> = (-t/c <= x)"
   665     by (simp only: neg_divide_le_eq[OF H] algebra_simps)
   666   also have "\<dots> = ((- 1/c)*t <= x)" by simp
   667   finally show "(c*x + t <= 0) == (x >= (- 1/c)*t)" by simp
   668 qed
   669 
   670 lemma pos_prod_sum_le:"(c\<Colon>'a\<Colon>linordered_field) > 0 \<Longrightarrow> ((c*x + t <= 0) == (x <= (- 1/c)*t))"
   671 proof -
   672   assume H: "c > 0"
   673   have "c*x + t <= 0 = (c*x <= -t)"
   674     by (subst le_iff_diff_le_0 [of "c*x" "-t"], simp)
   675   also have "\<dots> = (-t/c >= x)"
   676     by (simp only: pos_le_divide_eq[OF H] algebra_simps)
   677   also have "\<dots> = ((- 1/c)*t >= x)" by simp
   678   finally show "(c*x + t <= 0) == (x <= (- 1/c)*t)" by simp
   679 qed
   680 
   681 lemma sum_le:"((x::'a::ordered_ab_group_add) + t <= 0) == (x <= - t)"
   682   using le_diff_eq[where a= x and b=t and c=0] by simp
   683 
   684 lemma nz_prod_eq:"(c\<Colon>'a\<Colon>linordered_field) \<noteq> 0 \<Longrightarrow> ((c*x = 0) == (x = 0))" by simp
   685 
   686 lemma nz_prod_sum_eq: "(c\<Colon>'a\<Colon>linordered_field) \<noteq> 0 \<Longrightarrow> ((c*x + t = 0) == (x = (- 1/c)*t))"
   687 proof -
   688   assume H: "c \<noteq> 0"
   689   have "c*x + t = 0 = (c*x = -t)"
   690     by (subst eq_iff_diff_eq_0 [of "c*x" "-t"], simp)
   691   also have "\<dots> = (x = -t/c)"
   692     by (simp only: nonzero_eq_divide_eq[OF H] algebra_simps)
   693   finally show  "(c*x + t = 0) == (x = (- 1/c)*t)" by simp
   694 qed
   695 
   696 lemma sum_eq:"((x::'a::ordered_ab_group_add) + t = 0) == (x = - t)"
   697   using eq_diff_eq[where a= x and b=t and c=0] by simp
   698 
   699 
   700 interpretation class_dense_linordered_field: constr_dense_linorder
   701  "op <=" "op <"
   702    "\<lambda> x y. 1/2 * ((x::'a::{linordered_field}) + y)"
   703   by unfold_locales (dlo, dlo, auto)
   704 
   705 declaration{*
   706 let
   707   fun earlier [] x y = false
   708     | earlier (h::t) x y =
   709         if h aconvc y then false else if h aconvc x then true else earlier t x y;
   710 
   711 fun dest_frac ct =
   712   case term_of ct of
   713     Const (@{const_name Fields.divide},_) $ a $ b=>
   714       Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b))
   715   | Const(@{const_name inverse}, _)$a => Rat.rat_of_quotient(1, HOLogic.dest_number a |> snd)
   716   | t => Rat.rat_of_int (snd (HOLogic.dest_number t))
   717 
   718 fun mk_frac phi cT x =
   719   let val (a, b) = Rat.quotient_of_rat x
   720   in if b = 1 then Numeral.mk_cnumber cT a
   721     else Thm.apply
   722          (Thm.apply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"})
   723                      (Numeral.mk_cnumber cT a))
   724          (Numeral.mk_cnumber cT b)
   725  end
   726 
   727 fun whatis x ct = case term_of ct of
   728   Const(@{const_name Groups.plus}, _)$(Const(@{const_name Groups.times},_)$_$y)$_ =>
   729      if y aconv term_of x then ("c*x+t",[(funpow 2 Thm.dest_arg1) ct, Thm.dest_arg ct])
   730      else ("Nox",[])
   731 | Const(@{const_name Groups.plus}, _)$y$_ =>
   732      if y aconv term_of x then ("x+t",[Thm.dest_arg ct])
   733      else ("Nox",[])
   734 | Const(@{const_name Groups.times}, _)$_$y =>
   735      if y aconv term_of x then ("c*x",[Thm.dest_arg1 ct])
   736      else ("Nox",[])
   737 | t => if t aconv term_of x then ("x",[]) else ("Nox",[]);
   738 
   739 fun xnormalize_conv ctxt [] ct = Thm.reflexive ct
   740 | xnormalize_conv ctxt (vs as (x::_)) ct =
   741    case term_of ct of
   742    Const(@{const_name Orderings.less},_)$_$Const(@{const_name Groups.zero},_) =>
   743     (case whatis x (Thm.dest_arg1 ct) of
   744     ("c*x+t",[c,t]) =>
   745        let
   746         val cr = dest_frac c
   747         val clt = Thm.dest_fun2 ct
   748         val cz = Thm.dest_arg ct
   749         val neg = cr </ Rat.zero
   750         val cthp = Simplifier.rewrite ctxt
   751                (Thm.apply @{cterm "Trueprop"}
   752                   (if neg then Thm.apply (Thm.apply clt c) cz
   753                     else Thm.apply (Thm.apply clt cz) c))
   754         val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI
   755         val th = Thm.implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x,t])
   756              (if neg then @{thm neg_prod_sum_lt} else @{thm pos_prod_sum_lt})) cth
   757         val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
   758                    (Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
   759       in rth end
   760     | ("x+t",[t]) =>
   761        let
   762         val T = ctyp_of_term x
   763         val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_lt"}
   764         val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
   765               (Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
   766        in  rth end
   767     | ("c*x",[c]) =>
   768        let
   769         val cr = dest_frac c
   770         val clt = Thm.dest_fun2 ct
   771         val cz = Thm.dest_arg ct
   772         val neg = cr </ Rat.zero
   773         val cthp = Simplifier.rewrite ctxt
   774                (Thm.apply @{cterm "Trueprop"}
   775                   (if neg then Thm.apply (Thm.apply clt c) cz
   776                     else Thm.apply (Thm.apply clt cz) c))
   777         val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI
   778         val th = Thm.implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x])
   779              (if neg then @{thm neg_prod_lt} else @{thm pos_prod_lt})) cth
   780         val rth = th
   781       in rth end
   782     | _ => Thm.reflexive ct)
   783 
   784 
   785 |  Const(@{const_name Orderings.less_eq},_)$_$Const(@{const_name Groups.zero},_) =>
   786    (case whatis x (Thm.dest_arg1 ct) of
   787     ("c*x+t",[c,t]) =>
   788        let
   789         val T = ctyp_of_term x
   790         val cr = dest_frac c
   791         val clt = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat "op <"}
   792         val cz = Thm.dest_arg ct
   793         val neg = cr </ Rat.zero
   794         val cthp = Simplifier.rewrite ctxt
   795                (Thm.apply @{cterm "Trueprop"}
   796                   (if neg then Thm.apply (Thm.apply clt c) cz
   797                     else Thm.apply (Thm.apply clt cz) c))
   798         val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI
   799         val th = Thm.implies_elim (instantiate' [SOME T] (map SOME [c,x,t])
   800              (if neg then @{thm neg_prod_sum_le} else @{thm pos_prod_sum_le})) cth
   801         val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
   802                    (Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
   803       in rth end
   804     | ("x+t",[t]) =>
   805        let
   806         val T = ctyp_of_term x
   807         val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_le"}
   808         val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
   809               (Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
   810        in  rth end
   811     | ("c*x",[c]) =>
   812        let
   813         val T = ctyp_of_term x
   814         val cr = dest_frac c
   815         val clt = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat "op <"}
   816         val cz = Thm.dest_arg ct
   817         val neg = cr </ Rat.zero
   818         val cthp = Simplifier.rewrite ctxt
   819                (Thm.apply @{cterm "Trueprop"}
   820                   (if neg then Thm.apply (Thm.apply clt c) cz
   821                     else Thm.apply (Thm.apply clt cz) c))
   822         val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI
   823         val th = Thm.implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x])
   824              (if neg then @{thm neg_prod_le} else @{thm pos_prod_le})) cth
   825         val rth = th
   826       in rth end
   827     | _ => Thm.reflexive ct)
   828 
   829 |  Const(@{const_name HOL.eq},_)$_$Const(@{const_name Groups.zero},_) =>
   830    (case whatis x (Thm.dest_arg1 ct) of
   831     ("c*x+t",[c,t]) =>
   832        let
   833         val T = ctyp_of_term x
   834         val cr = dest_frac c
   835         val ceq = Thm.dest_fun2 ct
   836         val cz = Thm.dest_arg ct
   837         val cthp = Simplifier.rewrite ctxt
   838             (Thm.apply @{cterm "Trueprop"}
   839              (Thm.apply @{cterm "Not"} (Thm.apply (Thm.apply ceq c) cz)))
   840         val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI
   841         val th = Thm.implies_elim
   842                  (instantiate' [SOME T] (map SOME [c,x,t]) @{thm nz_prod_sum_eq}) cth
   843         val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
   844                    (Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
   845       in rth end
   846     | ("x+t",[t]) =>
   847        let
   848         val T = ctyp_of_term x
   849         val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_eq"}
   850         val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
   851               (Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
   852        in  rth end
   853     | ("c*x",[c]) =>
   854        let
   855         val T = ctyp_of_term x
   856         val cr = dest_frac c
   857         val ceq = Thm.dest_fun2 ct
   858         val cz = Thm.dest_arg ct
   859         val cthp = Simplifier.rewrite ctxt
   860             (Thm.apply @{cterm "Trueprop"}
   861              (Thm.apply @{cterm "Not"} (Thm.apply (Thm.apply ceq c) cz)))
   862         val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI
   863         val rth = Thm.implies_elim
   864                  (instantiate' [SOME T] (map SOME [c,x]) @{thm nz_prod_eq}) cth
   865       in rth end
   866     | _ => Thm.reflexive ct);
   867 
   868 local
   869   val less_iff_diff_less_0 = mk_meta_eq @{thm "less_iff_diff_less_0"}
   870   val le_iff_diff_le_0 = mk_meta_eq @{thm "le_iff_diff_le_0"}
   871   val eq_iff_diff_eq_0 = mk_meta_eq @{thm "eq_iff_diff_eq_0"}
   872   val ss = simpset_of @{context}
   873 in
   874 fun field_isolate_conv phi ctxt vs ct = case term_of ct of
   875   Const(@{const_name Orderings.less},_)$a$b =>
   876    let val (ca,cb) = Thm.dest_binop ct
   877        val T = ctyp_of_term ca
   878        val th = instantiate' [SOME T] [SOME ca, SOME cb] less_iff_diff_less_0
   879        val nth = Conv.fconv_rule
   880          (Conv.arg_conv (Conv.arg1_conv
   881               (Semiring_Normalizer.semiring_normalize_ord_conv (put_simpset ss ctxt) (earlier vs)))) th
   882        val rth = Thm.transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
   883    in rth end
   884 | Const(@{const_name Orderings.less_eq},_)$a$b =>
   885    let val (ca,cb) = Thm.dest_binop ct
   886        val T = ctyp_of_term ca
   887        val th = instantiate' [SOME T] [SOME ca, SOME cb] le_iff_diff_le_0
   888        val nth = Conv.fconv_rule
   889          (Conv.arg_conv (Conv.arg1_conv
   890               (Semiring_Normalizer.semiring_normalize_ord_conv (put_simpset ss ctxt) (earlier vs)))) th
   891        val rth = Thm.transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
   892    in rth end
   893 
   894 | Const(@{const_name HOL.eq},_)$a$b =>
   895    let val (ca,cb) = Thm.dest_binop ct
   896        val T = ctyp_of_term ca
   897        val th = instantiate' [SOME T] [SOME ca, SOME cb] eq_iff_diff_eq_0
   898        val nth = Conv.fconv_rule
   899          (Conv.arg_conv (Conv.arg1_conv
   900               (Semiring_Normalizer.semiring_normalize_ord_conv (put_simpset ss ctxt) (earlier vs)))) th
   901        val rth = Thm.transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
   902    in rth end
   903 | @{term "Not"} $(Const(@{const_name HOL.eq},_)$a$b) => Conv.arg_conv (field_isolate_conv phi ctxt vs) ct
   904 | _ => Thm.reflexive ct
   905 end;
   906 
   907 fun classfield_whatis phi =
   908  let
   909   fun h x t =
   910    case term_of t of
   911      Const(@{const_name HOL.eq}, _)$y$z => if term_of x aconv y then Ferrante_Rackoff_Data.Eq
   912                             else Ferrante_Rackoff_Data.Nox
   913    | @{term "Not"}$(Const(@{const_name HOL.eq}, _)$y$z) => if term_of x aconv y then Ferrante_Rackoff_Data.NEq
   914                             else Ferrante_Rackoff_Data.Nox
   915    | Const(@{const_name Orderings.less},_)$y$z =>
   916        if term_of x aconv y then Ferrante_Rackoff_Data.Lt
   917         else if term_of x aconv z then Ferrante_Rackoff_Data.Gt
   918         else Ferrante_Rackoff_Data.Nox
   919    | Const (@{const_name Orderings.less_eq},_)$y$z =>
   920          if term_of x aconv y then Ferrante_Rackoff_Data.Le
   921          else if term_of x aconv z then Ferrante_Rackoff_Data.Ge
   922          else Ferrante_Rackoff_Data.Nox
   923    | _ => Ferrante_Rackoff_Data.Nox
   924  in h end;
   925 fun class_field_ss phi =
   926   simpset_of (put_simpset HOL_basic_ss @{context}
   927     addsimps ([@{thm "linorder_not_less"}, @{thm "linorder_not_le"}])
   928     |> fold Splitter.add_split [@{thm "abs_split"}, @{thm "split_max"}, @{thm "split_min"}])
   929 
   930 in
   931 Ferrante_Rackoff_Data.funs @{thm "class_dense_linordered_field.ferrack_axiom"}
   932   {isolate_conv = field_isolate_conv, whatis = classfield_whatis, simpset = class_field_ss}
   933 end
   934 *}
   935 (*
   936 lemma upper_bound_finite_set:
   937   assumes fS: "finite S"
   938   shows "\<exists>(a::'a::linorder). \<forall>x \<in> S. f x \<le> a"
   939 proof(induct rule: finite_induct[OF fS])
   940   case 1 thus ?case by simp
   941 next
   942   case (2 x F)
   943   from "2.hyps" obtain a where a:"\<forall>x \<in>F. f x \<le> a" by blast
   944   let ?a = "max a (f x)"
   945   have m: "a \<le> ?a" "f x \<le> ?a" by simp_all
   946   {fix y assume y: "y \<in> insert x F"
   947     {assume "y = x" hence "f y \<le> ?a" using m by simp}
   948     moreover
   949     {assume yF: "y\<in> F" from a[rule_format, OF yF] m have "f y \<le> ?a" by (simp add: max_def)}
   950     ultimately have "f y \<le> ?a" using y by blast}
   951   then show ?case by blast
   952 qed
   953 
   954 lemma lower_bound_finite_set:
   955   assumes fS: "finite S"
   956   shows "\<exists>(a::'a::linorder). \<forall>x \<in> S. f x \<ge> a"
   957 proof(induct rule: finite_induct[OF fS])
   958   case 1 thus ?case by simp
   959 next
   960   case (2 x F)
   961   from "2.hyps" obtain a where a:"\<forall>x \<in>F. f x \<ge> a" by blast
   962   let ?a = "min a (f x)"
   963   have m: "a \<ge> ?a" "f x \<ge> ?a" by simp_all
   964   {fix y assume y: "y \<in> insert x F"
   965     {assume "y = x" hence "f y \<ge> ?a" using m by simp}
   966     moreover
   967     {assume yF: "y\<in> F" from a[rule_format, OF yF] m have "f y \<ge> ?a" by (simp add: min_def)}
   968     ultimately have "f y \<ge> ?a" using y by blast}
   969   then show ?case by blast
   970 qed
   971 
   972 lemma bound_finite_set: assumes f: "finite S"
   973   shows "\<exists>a. \<forall>x \<in>S. (f x:: 'a::linorder) \<le> a"
   974 proof-
   975   let ?F = "f ` S"
   976   from f have fF: "finite ?F" by simp
   977   let ?a = "Max ?F"
   978   {assume "S = {}" hence ?thesis by blast}
   979   moreover
   980   {assume Se: "S \<noteq> {}" hence Fe: "?F \<noteq> {}" by simp
   981   {fix x assume x: "x \<in> S"
   982     hence th0: "f x \<in> ?F" by simp
   983     hence "f x \<le> ?a" using Max_ge[OF fF th0] ..}
   984   hence ?thesis by blast}
   985 ultimately show ?thesis by blast
   986 qed
   987 *)
   988 
   989 end