src/HOL/Dense_Linear_Order.thy
author haftmann
Fri Jan 02 08:12:46 2009 +0100 (2009-01-02)
changeset 29332 edc1e2a56398
parent 29252 ea97aa6aeba2
child 29509 1ff0f3f08a7b
permissions -rw-r--r--
named code theorem for Fract_norm
     1 (* Author: Amine Chaieb, TU Muenchen *)
     2 
     3 header {* Dense linear order without endpoints
     4   and a quantifier elimination procedure in Ferrante and Rackoff style *}
     5 
     6 theory Dense_Linear_Order
     7 imports Plain Groebner_Basis
     8 uses
     9   "~~/src/HOL/Tools/Qelim/langford_data.ML"
    10   "~~/src/HOL/Tools/Qelim/ferrante_rackoff_data.ML"
    11   ("~~/src/HOL/Tools/Qelim/langford.ML")
    12   ("~~/src/HOL/Tools/Qelim/ferrante_rackoff.ML")
    13 begin
    14 
    15 setup {* Langford_Data.setup #> Ferrante_Rackoff_Data.setup *}
    16 
    17 context linorder
    18 begin
    19 
    20 lemma less_not_permute: "\<not> (x < y \<and> y < x)" by (simp add: not_less linear)
    21 
    22 lemma gather_simps: 
    23   shows 
    24   "(\<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)"
    25   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)"
    26   "(\<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))"
    27   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
    28 
    29 lemma 
    30   gather_start: "(\<exists>x. P x) \<equiv> (\<exists>x. (\<forall>y \<in> {}. y < x) \<and> (\<forall>y\<in> {}. x < y) \<and> P x)" 
    31   by simp
    32 
    33 text{* Theorems for @{text "\<exists>z. \<forall>x. x < z \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>-\<infinity>\<^esub>)"}*}
    34 lemma minf_lt:  "\<exists>z . \<forall>x. x < z \<longrightarrow> (x < t \<longleftrightarrow> True)" by auto
    35 lemma minf_gt: "\<exists>z . \<forall>x. x < z \<longrightarrow>  (t < x \<longleftrightarrow>  False)"
    36   by (simp add: not_less) (rule exI[where x="t"], auto simp add: less_le)
    37 
    38 lemma minf_le: "\<exists>z. \<forall>x. x < z \<longrightarrow> (x \<le> t \<longleftrightarrow> True)" by (auto simp add: less_le)
    39 lemma minf_ge: "\<exists>z. \<forall>x. x < z \<longrightarrow> (t \<le> x \<longleftrightarrow> False)"
    40   by (auto simp add: less_le not_less not_le)
    41 lemma minf_eq: "\<exists>z. \<forall>x. x < z \<longrightarrow> (x = t \<longleftrightarrow> False)" by auto
    42 lemma minf_neq: "\<exists>z. \<forall>x. x < z \<longrightarrow> (x \<noteq> t \<longleftrightarrow> True)" by auto
    43 lemma minf_P: "\<exists>z. \<forall>x. x < z \<longrightarrow> (P \<longleftrightarrow> P)" by blast
    44 
    45 text{* Theorems for @{text "\<exists>z. \<forall>x. x < z \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>+\<infinity>\<^esub>)"}*}
    46 lemma pinf_gt:  "\<exists>z . \<forall>x. z < x \<longrightarrow> (t < x \<longleftrightarrow> True)" by auto
    47 lemma pinf_lt: "\<exists>z . \<forall>x. z < x \<longrightarrow>  (x < t \<longleftrightarrow>  False)"
    48   by (simp add: not_less) (rule exI[where x="t"], auto simp add: less_le)
    49 
    50 lemma pinf_ge: "\<exists>z. \<forall>x. z < x \<longrightarrow> (t \<le> x \<longleftrightarrow> True)" by (auto simp add: less_le)
    51 lemma pinf_le: "\<exists>z. \<forall>x. z < x \<longrightarrow> (x \<le> t \<longleftrightarrow> False)"
    52   by (auto simp add: less_le not_less not_le)
    53 lemma pinf_eq: "\<exists>z. \<forall>x. z < x \<longrightarrow> (x = t \<longleftrightarrow> False)" by auto
    54 lemma pinf_neq: "\<exists>z. \<forall>x. z < x \<longrightarrow> (x \<noteq> t \<longleftrightarrow> True)" by auto
    55 lemma pinf_P: "\<exists>z. \<forall>x. z < x \<longrightarrow> (P \<longleftrightarrow> P)" by blast
    56 
    57 lemma nmi_lt: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> x < t \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto
    58 lemma nmi_gt: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> t < x \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)"
    59   by (auto simp add: le_less)
    60 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
    61 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
    62 lemma  nmi_eq: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and>  x = t \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto
    63 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
    64 lemma  nmi_P: "\<forall> x. ~P \<and> P \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto
    65 lemma  nmi_conj: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow>  (\<exists> u\<in> U. u \<le> x) ;
    66   \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)\<rbrakk> \<Longrightarrow>
    67   \<forall>x. \<not>(P1' \<and> P2') \<and> (P1 x \<and> P2 x) \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto
    68 lemma  nmi_disj: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow>  (\<exists> u\<in> U. u \<le> x) ;
    69   \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)\<rbrakk> \<Longrightarrow>
    70   \<forall>x. \<not>(P1' \<or> P2') \<and> (P1 x \<or> P2 x) \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto
    71 
    72 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)
    73 lemma  npi_gt: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> t < x \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by auto
    74 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
    75 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
    76 lemma  npi_eq: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and>  x = t \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by auto
    77 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
    78 lemma  npi_P: "\<forall> x. ~P \<and> P \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by auto
    79 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>
    80   \<Longrightarrow>  \<forall>x. \<not>(P1' \<and> P2') \<and> (P1 x \<and> P2 x) \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by auto
    81 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>
    82   \<Longrightarrow> \<forall>x. \<not>(P1' \<or> P2') \<and> (P1 x \<or> P2 x) \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by auto
    83 
    84 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)"
    85 proof(clarsimp)
    86   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"
    87     and xu: "x<u"  and px: "x < t" and ly: "l<y" and yu:"y < u"
    88   from tU noU ly yu have tny: "t\<noteq>y" by auto
    89   {assume H: "t < y"
    90     from less_trans[OF lx px] less_trans[OF H yu]
    91     have "l < t \<and> t < u"  by simp
    92     with tU noU have "False" by auto}
    93   hence "\<not> t < y"  by auto hence "y \<le> t" by (simp add: not_less)
    94   thus "y < t" using tny by (simp add: less_le)
    95 qed
    96 
    97 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)"
    98 proof(clarsimp)
    99   fix x l u y
   100   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"
   101   and px: "t < x" and ly: "l<y" and yu:"y < u"
   102   from tU noU ly yu have tny: "t\<noteq>y" by auto
   103   {assume H: "y< t"
   104     from less_trans[OF ly H] less_trans[OF px xu] have "l < t \<and> t < u" by simp
   105     with tU noU have "False" by auto}
   106   hence "\<not> y<t"  by auto hence "t \<le> y" by (auto simp add: not_less)
   107   thus "t < y" using tny by (simp add:less_le)
   108 qed
   109 
   110 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)"
   111 proof(clarsimp)
   112   fix x l u y
   113   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"
   114   and px: "x \<le> t" and ly: "l<y" and yu:"y < u"
   115   from tU noU ly yu have tny: "t\<noteq>y" by auto
   116   {assume H: "t < y"
   117     from less_le_trans[OF lx px] less_trans[OF H yu]
   118     have "l < t \<and> t < u" by simp
   119     with tU noU have "False" by auto}
   120   hence "\<not> t < y"  by auto thus "y \<le> t" by (simp add: not_less)
   121 qed
   122 
   123 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)"
   124 proof(clarsimp)
   125   fix x l u y
   126   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"
   127   and px: "t \<le> x" and ly: "l<y" and yu:"y < u"
   128   from tU noU ly yu have tny: "t\<noteq>y" by auto
   129   {assume H: "y< t"
   130     from less_trans[OF ly H] le_less_trans[OF px xu]
   131     have "l < t \<and> t < u" by simp
   132     with tU noU have "False" by auto}
   133   hence "\<not> y<t"  by auto thus "t \<le> y" by (simp add: not_less)
   134 qed
   135 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
   136 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
   137 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
   138 
   139 lemma lin_dense_conj:
   140   "\<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
   141   \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> P1 y) ;
   142   \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> P2 x
   143   \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> P2 y)\<rbrakk> \<Longrightarrow>
   144   \<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)
   145   \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> (P1 y \<and> P2 y))"
   146   by blast
   147 lemma lin_dense_disj:
   148   "\<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
   149   \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> P1 y) ;
   150   \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> P2 x
   151   \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> P2 y)\<rbrakk> \<Longrightarrow>
   152   \<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)
   153   \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> (P1 y \<or> P2 y))"
   154   by blast
   155 
   156 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>
   157   \<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')"
   158 by auto
   159 
   160 lemma finite_set_intervals:
   161   assumes px: "P x" and lx: "l \<le> x" and xu: "x \<le> u" and linS: "l\<in> S"
   162   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"
   163   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"
   164 proof-
   165   let ?Mx = "{y. y\<in> S \<and> y \<le> x}"
   166   let ?xM = "{y. y\<in> S \<and> x \<le> y}"
   167   let ?a = "Max ?Mx"
   168   let ?b = "Min ?xM"
   169   have MxS: "?Mx \<subseteq> S" by blast
   170   hence fMx: "finite ?Mx" using fS finite_subset by auto
   171   from lx linS have linMx: "l \<in> ?Mx" by blast
   172   hence Mxne: "?Mx \<noteq> {}" by blast
   173   have xMS: "?xM \<subseteq> S" by blast
   174   hence fxM: "finite ?xM" using fS finite_subset by auto
   175   from xu uinS have linxM: "u \<in> ?xM" by blast
   176   hence xMne: "?xM \<noteq> {}" by blast
   177   have ax:"?a \<le> x" using Mxne fMx by auto
   178   have xb:"x \<le> ?b" using xMne fxM by auto
   179   have "?a \<in> ?Mx" using Max_in[OF fMx Mxne] by simp hence ainS: "?a \<in> S" using MxS by blast
   180   have "?b \<in> ?xM" using Min_in[OF fxM xMne] by simp hence binS: "?b \<in> S" using xMS by blast
   181   have noy:"\<forall> y. ?a < y \<and> y < ?b \<longrightarrow> y \<notin> S"
   182   proof(clarsimp)
   183     fix y   assume ay: "?a < y" and yb: "y < ?b" and yS: "y \<in> S"
   184     from yS have "y\<in> ?Mx \<or> y\<in> ?xM" by (auto simp add: linear)
   185     moreover {assume "y \<in> ?Mx" hence "y \<le> ?a" using Mxne fMx by auto with ay have "False" by (simp add: not_le[symmetric])}
   186     moreover {assume "y \<in> ?xM" hence "?b \<le> y" using xMne fxM by auto with yb have "False" by (simp add: not_le[symmetric])}
   187     ultimately show "False" by blast
   188   qed
   189   from ainS binS noy ax xb px show ?thesis by blast
   190 qed
   191 
   192 lemma finite_set_intervals2:
   193   assumes px: "P x" and lx: "l \<le> x" and xu: "x \<le> u" and linS: "l\<in> S"
   194   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"
   195   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)"
   196 proof-
   197   from finite_set_intervals[where P="P", OF px lx xu linS uinS fS lS Su]
   198   obtain a and b where
   199     as: "a\<in> S" and bs: "b\<in> S" and noS:"\<forall>y. a < y \<and> y < b \<longrightarrow> y \<notin> S"
   200     and axb: "a \<le> x \<and> x \<le> b \<and> P x"  by auto
   201   from axb have "x= a \<or> x= b \<or> (a < x \<and> x < b)" by (auto simp add: le_less)
   202   thus ?thesis using px as bs noS by blast
   203 qed
   204 
   205 end
   206 
   207 section {* The classical QE after Langford for dense linear orders *}
   208 
   209 context dense_linear_order
   210 begin
   211 
   212 lemma interval_empty_iff:
   213   "{y. x < y \<and> y < z} = {} \<longleftrightarrow> \<not> x < z"
   214   by (auto dest: dense)
   215 
   216 lemma dlo_qe_bnds: 
   217   assumes ne: "L \<noteq> {}" and neU: "U \<noteq> {}" and fL: "finite L" and fU: "finite U"
   218   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)"
   219 proof (simp only: atomize_eq, rule iffI)
   220   assume H: "\<exists>x. (\<forall>y\<in>L. y < x) \<and> (\<forall>y\<in>U. x < y)"
   221   then obtain x where xL: "\<forall>y\<in>L. y < x" and xU: "\<forall>y\<in>U. x < y" by blast
   222   {fix l u assume l: "l \<in> L" and u: "u \<in> U"
   223     have "l < x" using xL l by blast
   224     also have "x < u" using xU u by blast
   225     finally (less_trans) have "l < u" .}
   226   thus "\<forall>l\<in>L. \<forall>u\<in>U. l < u" by blast
   227 next
   228   assume H: "\<forall>l\<in>L. \<forall>u\<in>U. l < u"
   229   let ?ML = "Max L"
   230   let ?MU = "Min U"  
   231   from fL ne have th1: "?ML \<in> L" and th1': "\<forall>l\<in>L. l \<le> ?ML" by auto
   232   from fU neU have th2: "?MU \<in> U" and th2': "\<forall>u\<in>U. ?MU \<le> u" by auto
   233   from th1 th2 H have "?ML < ?MU" by auto
   234   with dense obtain w where th3: "?ML < w" and th4: "w < ?MU" by blast
   235   from th3 th1' have "\<forall>l \<in> L. l < w" by auto
   236   moreover from th4 th2' have "\<forall>u \<in> U. w < u" by auto
   237   ultimately show "\<exists>x. (\<forall>y\<in>L. y < x) \<and> (\<forall>y\<in>U. x < y)" by auto
   238 qed
   239 
   240 lemma dlo_qe_noub: 
   241   assumes ne: "L \<noteq> {}" and fL: "finite L"
   242   shows "(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> {}. x < y)) \<equiv> True"
   243 proof(simp add: atomize_eq)
   244   from gt_ex[of "Max L"] obtain M where M: "Max L < M" by blast
   245   from ne fL have "\<forall>x \<in> L. x \<le> Max L" by simp
   246   with M have "\<forall>x\<in>L. x < M" by (auto intro: le_less_trans)
   247   thus "\<exists>x. \<forall>y\<in>L. y < x" by blast
   248 qed
   249 
   250 lemma dlo_qe_nolb: 
   251   assumes ne: "U \<noteq> {}" and fU: "finite U"
   252   shows "(\<exists>x. (\<forall>y \<in> {}. y < x) \<and> (\<forall>y \<in> U. x < y)) \<equiv> True"
   253 proof(simp add: atomize_eq)
   254   from lt_ex[of "Min U"] obtain M where M: "M < Min U" by blast
   255   from ne fU have "\<forall>x \<in> U. Min U \<le> x" by simp
   256   with M have "\<forall>x\<in>U. M < x" by (auto intro: less_le_trans)
   257   thus "\<exists>x. \<forall>y\<in>U. x < y" by blast
   258 qed
   259 
   260 lemma exists_neq: "\<exists>(x::'a). x \<noteq> t" "\<exists>(x::'a). t \<noteq> x" 
   261   using gt_ex[of t] by auto
   262 
   263 lemmas dlo_simps = order_refl less_irrefl not_less not_le exists_neq 
   264   le_less neq_iff linear less_not_permute
   265 
   266 lemma axiom: "dense_linear_order (op \<le>) (op <)" by (rule dense_linear_order_axioms)
   267 lemma atoms:
   268   shows "TERM (less :: 'a \<Rightarrow> _)"
   269     and "TERM (less_eq :: 'a \<Rightarrow> _)"
   270     and "TERM (op = :: 'a \<Rightarrow> _)" .
   271 
   272 declare axiom[langford qe: dlo_qe_bnds dlo_qe_nolb dlo_qe_noub gather: gather_start gather_simps atoms: atoms]
   273 declare dlo_simps[langfordsimp]
   274 
   275 end
   276 
   277 (* FIXME: Move to HOL -- together with the conj_aci_rule in langford.ML *)
   278 lemma dnf:
   279   "(P & (Q | R)) = ((P&Q) | (P&R))" 
   280   "((Q | R) & P) = ((Q&P) | (R&P))"
   281   by blast+
   282 
   283 lemmas weak_dnf_simps = simp_thms dnf
   284 
   285 lemma nnf_simps:
   286     "(\<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)"
   287     "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not> \<not>(P)) = P"
   288   by blast+
   289 
   290 lemma ex_distrib: "(\<exists>x. P x \<or> Q x) \<longleftrightarrow> ((\<exists>x. P x) \<or> (\<exists>x. Q x))" by blast
   291 
   292 lemmas dnf_simps = weak_dnf_simps nnf_simps ex_distrib
   293 
   294 use "~~/src/HOL/Tools/Qelim/langford.ML"
   295 method_setup dlo = {*
   296   Method.ctxt_args (Method.SIMPLE_METHOD' o LangfordQE.dlo_tac)
   297 *} "Langford's algorithm for quantifier elimination in dense linear orders"
   298 
   299 
   300 section {* Contructive dense linear orders yield QE for linear arithmetic over ordered Fields -- see @{text "Arith_Tools.thy"} *}
   301 
   302 text {* Linear order without upper bounds *}
   303 
   304 class_locale linorder_stupid_syntax = linorder
   305 begin
   306 notation
   307   less_eq  ("op \<sqsubseteq>") and
   308   less_eq  ("(_/ \<sqsubseteq> _)" [51, 51] 50) and
   309   less  ("op \<sqsubset>") and
   310   less  ("(_/ \<sqsubset> _)"  [51, 51] 50)
   311 
   312 end
   313 
   314 class_locale linorder_no_ub = linorder_stupid_syntax +
   315   assumes gt_ex: "\<exists>y. less x y"
   316 begin
   317 lemma ge_ex: "\<exists>y. x \<sqsubseteq> y" using gt_ex by auto
   318 
   319 text {* Theorems for @{text "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>+\<infinity>\<^esub>)"} *}
   320 lemma pinf_conj:
   321   assumes ex1: "\<exists>z1. \<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
   322   and ex2: "\<exists>z2. \<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
   323   shows "\<exists>z. \<forall>x. z \<sqsubset>  x \<longrightarrow> ((P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2'))"
   324 proof-
   325   from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
   326      and z2: "\<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')" by blast
   327   from gt_ex obtain z where z:"ord.max less_eq z1 z2 \<sqsubset> z" by blast
   328   from z have zz1: "z1 \<sqsubset> z" and zz2: "z2 \<sqsubset> z" by simp_all
   329   {fix x assume H: "z \<sqsubset> x"
   330     from less_trans[OF zz1 H] less_trans[OF zz2 H]
   331     have "(P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2')"  using z1 zz1 z2 zz2 by auto
   332   }
   333   thus ?thesis by blast
   334 qed
   335 
   336 lemma pinf_disj:
   337   assumes ex1: "\<exists>z1. \<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
   338   and ex2: "\<exists>z2. \<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
   339   shows "\<exists>z. \<forall>x. z \<sqsubset>  x \<longrightarrow> ((P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2'))"
   340 proof-
   341   from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
   342      and z2: "\<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')" by blast
   343   from gt_ex obtain z where z:"ord.max less_eq z1 z2 \<sqsubset> z" by blast
   344   from z have zz1: "z1 \<sqsubset> z" and zz2: "z2 \<sqsubset> z" by simp_all
   345   {fix x assume H: "z \<sqsubset> x"
   346     from less_trans[OF zz1 H] less_trans[OF zz2 H]
   347     have "(P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2')"  using z1 zz1 z2 zz2 by auto
   348   }
   349   thus ?thesis by blast
   350 qed
   351 
   352 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"
   353 proof-
   354   from ex obtain z where z: "\<forall>x. z \<sqsubset> x \<longrightarrow> (P x \<longleftrightarrow> P1)" by blast
   355   from gt_ex obtain x where x: "z \<sqsubset> x" by blast
   356   from z x p1 show ?thesis by blast
   357 qed
   358 
   359 end
   360 
   361 text {* Linear order without upper bounds *}
   362 
   363 class_locale linorder_no_lb = linorder_stupid_syntax +
   364   assumes lt_ex: "\<exists>y. less y x"
   365 begin
   366 lemma le_ex: "\<exists>y. y \<sqsubseteq> x" using lt_ex by auto
   367 
   368 
   369 text {* Theorems for @{text "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>-\<infinity>\<^esub>)"} *}
   370 lemma minf_conj:
   371   assumes ex1: "\<exists>z1. \<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
   372   and ex2: "\<exists>z2. \<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
   373   shows "\<exists>z. \<forall>x. x \<sqsubset>  z \<longrightarrow> ((P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2'))"
   374 proof-
   375   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
   376   from lt_ex obtain z where z:"z \<sqsubset> ord.min less_eq z1 z2" by blast
   377   from z have zz1: "z \<sqsubset> z1" and zz2: "z \<sqsubset> z2" by simp_all
   378   {fix x assume H: "x \<sqsubset> z"
   379     from less_trans[OF H zz1] less_trans[OF H zz2]
   380     have "(P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2')"  using z1 zz1 z2 zz2 by auto
   381   }
   382   thus ?thesis by blast
   383 qed
   384 
   385 lemma minf_disj:
   386   assumes ex1: "\<exists>z1. \<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
   387   and ex2: "\<exists>z2. \<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
   388   shows "\<exists>z. \<forall>x. x \<sqsubset>  z \<longrightarrow> ((P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2'))"
   389 proof-
   390   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
   391   from lt_ex obtain z where z:"z \<sqsubset> ord.min less_eq z1 z2" by blast
   392   from z have zz1: "z \<sqsubset> z1" and zz2: "z \<sqsubset> z2" by simp_all
   393   {fix x assume H: "x \<sqsubset> z"
   394     from less_trans[OF H zz1] less_trans[OF H zz2]
   395     have "(P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2')"  using z1 zz1 z2 zz2 by auto
   396   }
   397   thus ?thesis by blast
   398 qed
   399 
   400 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"
   401 proof-
   402   from ex obtain z where z: "\<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P1)" by blast
   403   from lt_ex obtain x where x: "x \<sqsubset> z" by blast
   404   from z x p1 show ?thesis by blast
   405 qed
   406 
   407 end
   408 
   409 
   410 class_locale constr_dense_linear_order = linorder_no_lb + linorder_no_ub +
   411   fixes between
   412   assumes between_less: "less x y \<Longrightarrow> less x (between x y) \<and> less (between x y) y"
   413      and  between_same: "between x x = x"
   414 
   415 class_interpretation  constr_dense_linear_order < dense_linear_order 
   416   apply unfold_locales
   417   using gt_ex lt_ex between_less
   418     by (auto, rule_tac x="between x y" in exI, simp)
   419 
   420 context  constr_dense_linear_order
   421 begin
   422 
   423 lemma rinf_U:
   424   assumes fU: "finite U"
   425   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
   426   \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> P y )"
   427   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')"
   428   and nmi: "\<not> MP"  and npi: "\<not> PP"  and ex: "\<exists> x.  P x"
   429   shows "\<exists> u\<in> U. \<exists> u' \<in> U. P (between u u')"
   430 proof-
   431   from ex obtain x where px: "P x" by blast
   432   from px nmi npi nmpiU have "\<exists> u\<in> U. \<exists> u' \<in> U. u \<sqsubseteq> x \<and> x \<sqsubseteq> u'" by auto
   433   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
   434   from uU have Une: "U \<noteq> {}" by auto
   435   term "linorder.Min less_eq"
   436   let ?l = "linorder.Min less_eq U"
   437   let ?u = "linorder.Max less_eq U"
   438   have linM: "?l \<in> U" using fU Une by simp
   439   have uinM: "?u \<in> U" using fU Une by simp
   440   have lM: "\<forall> t\<in> U. ?l \<sqsubseteq> t" using Une fU by auto
   441   have Mu: "\<forall> t\<in> U. t \<sqsubseteq> ?u" using Une fU by auto
   442   have th:"?l \<sqsubseteq> u" using uU Une lM by auto
   443   from order_trans[OF th ux] have lx: "?l \<sqsubseteq> x" .
   444   have th: "u' \<sqsubseteq> ?u" using uU' Une Mu by simp
   445   from order_trans[OF xu' th] have xu: "x \<sqsubseteq> ?u" .
   446   from finite_set_intervals2[where P="P",OF px lx xu linM uinM fU lM Mu]
   447   have "(\<exists> s\<in> U. P s) \<or>
   448       (\<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)" .
   449   moreover { fix u assume um: "u\<in>U" and pu: "P u"
   450     have "between u u = u" by (simp add: between_same)
   451     with um pu have "P (between u u)" by simp
   452     with um have ?thesis by blast}
   453   moreover{
   454     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"
   455       then obtain t1 and t2 where t1M: "t1 \<in> U" and t2M: "t2\<in> U"
   456         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"
   457         by blast
   458       from less_trans[OF t1x xt2] have t1t2: "t1 \<sqsubset> t2" .
   459       let ?u = "between t1 t2"
   460       from between_less t1t2 have t1lu: "t1 \<sqsubset> ?u" and ut2: "?u \<sqsubset> t2" by auto
   461       from lin_dense noM t1x xt2 px t1lu ut2 have "P ?u" by blast
   462       with t1M t2M have ?thesis by blast}
   463     ultimately show ?thesis by blast
   464   qed
   465 
   466 theorem fr_eq:
   467   assumes fU: "finite U"
   468   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
   469    \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> P y )"
   470   and nmibnd: "\<forall>x. \<not> MP \<and> P x \<longrightarrow> (\<exists> u\<in> U. u \<sqsubseteq> x)"
   471   and npibnd: "\<forall>x. \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. x \<sqsubseteq> u)"
   472   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)"
   473   shows "(\<exists> x. P x) \<equiv> (MP \<or> PP \<or> (\<exists> u \<in> U. \<exists> u'\<in> U. P (between u u')))"
   474   (is "_ \<equiv> (_ \<or> _ \<or> ?F)" is "?E \<equiv> ?D")
   475 proof-
   476  {
   477    assume px: "\<exists> x. P x"
   478    have "MP \<or> PP \<or> (\<not> MP \<and> \<not> PP)" by blast
   479    moreover {assume "MP \<or> PP" hence "?D" by blast}
   480    moreover {assume nmi: "\<not> MP" and npi: "\<not> PP"
   481      from npmibnd[OF nmibnd npibnd]
   482      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')" .
   483      from rinf_U[OF fU lin_dense nmpiU nmi npi px] have "?D" by blast}
   484    ultimately have "?D" by blast}
   485  moreover
   486  { assume "?D"
   487    moreover {assume m:"MP" from minf_ex[OF mi m] have "?E" .}
   488    moreover {assume p: "PP" from pinf_ex[OF pi p] have "?E" . }
   489    moreover {assume f:"?F" hence "?E" by blast}
   490    ultimately have "?E" by blast}
   491  ultimately have "?E = ?D" by blast thus "?E \<equiv> ?D" by simp
   492 qed
   493 
   494 lemmas minf_thms = minf_conj minf_disj minf_eq minf_neq minf_lt minf_le minf_gt minf_ge minf_P
   495 lemmas pinf_thms = pinf_conj pinf_disj pinf_eq pinf_neq pinf_lt pinf_le pinf_gt pinf_ge pinf_P
   496 
   497 lemmas nmi_thms = nmi_conj nmi_disj nmi_eq nmi_neq nmi_lt nmi_le nmi_gt nmi_ge nmi_P
   498 lemmas npi_thms = npi_conj npi_disj npi_eq npi_neq npi_lt npi_le npi_gt npi_ge npi_P
   499 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
   500 
   501 lemma ferrack_axiom: "constr_dense_linear_order less_eq less between"
   502   by (rule constr_dense_linear_order_axioms)
   503 lemma atoms:
   504   shows "TERM (less :: 'a \<Rightarrow> _)"
   505     and "TERM (less_eq :: 'a \<Rightarrow> _)"
   506     and "TERM (op = :: 'a \<Rightarrow> _)" .
   507 
   508 declare ferrack_axiom [ferrack minf: minf_thms pinf: pinf_thms
   509     nmi: nmi_thms npi: npi_thms lindense:
   510     lin_dense_thms qe: fr_eq atoms: atoms]
   511 
   512 declaration {*
   513 let
   514 fun simps phi = map (Morphism.thm phi) [@{thm "not_less"}, @{thm "not_le"}]
   515 fun generic_whatis phi =
   516  let
   517   val [lt, le] = map (Morphism.term phi) [@{term "op \<sqsubset>"}, @{term "op \<sqsubseteq>"}]
   518   fun h x t =
   519    case term_of t of
   520      Const("op =", _)$y$z => if term_of x aconv y then Ferrante_Rackoff_Data.Eq
   521                             else Ferrante_Rackoff_Data.Nox
   522    | @{term "Not"}$(Const("op =", _)$y$z) => if term_of x aconv y then Ferrante_Rackoff_Data.NEq
   523                             else Ferrante_Rackoff_Data.Nox
   524    | b$y$z => if Term.could_unify (b, lt) then
   525                  if term_of x aconv y then Ferrante_Rackoff_Data.Lt
   526                  else if term_of x aconv z then Ferrante_Rackoff_Data.Gt
   527                  else Ferrante_Rackoff_Data.Nox
   528              else if Term.could_unify (b, le) then
   529                  if term_of x aconv y then Ferrante_Rackoff_Data.Le
   530                  else if term_of x aconv z then Ferrante_Rackoff_Data.Ge
   531                  else Ferrante_Rackoff_Data.Nox
   532              else Ferrante_Rackoff_Data.Nox
   533    | _ => Ferrante_Rackoff_Data.Nox
   534  in h end
   535  fun ss phi = HOL_ss addsimps (simps phi)
   536 in
   537  Ferrante_Rackoff_Data.funs  @{thm "ferrack_axiom"}
   538   {isolate_conv = K (K (K Thm.reflexive)), whatis = generic_whatis, simpset = ss}
   539 end
   540 *}
   541 
   542 end
   543 
   544 use "~~/src/HOL/Tools/Qelim/ferrante_rackoff.ML"
   545 
   546 method_setup ferrack = {*
   547   Method.ctxt_args (Method.SIMPLE_METHOD' o FerranteRackoff.dlo_tac)
   548 *} "Ferrante and Rackoff's algorithm for quantifier elimination in dense linear orders"
   549 
   550 subsection {* Ferrante and Rackoff algorithm over ordered fields *}
   551 
   552 lemma neg_prod_lt:"(c\<Colon>'a\<Colon>ordered_field) < 0 \<Longrightarrow> ((c*x < 0) == (x > 0))"
   553 proof-
   554   assume H: "c < 0"
   555   have "c*x < 0 = (0/c < x)" by (simp only: neg_divide_less_eq[OF H] ring_simps)
   556   also have "\<dots> = (0 < x)" by simp
   557   finally show  "(c*x < 0) == (x > 0)" by simp
   558 qed
   559 
   560 lemma pos_prod_lt:"(c\<Colon>'a\<Colon>ordered_field) > 0 \<Longrightarrow> ((c*x < 0) == (x < 0))"
   561 proof-
   562   assume H: "c > 0"
   563   hence "c*x < 0 = (0/c > x)" by (simp only: pos_less_divide_eq[OF H] ring_simps)
   564   also have "\<dots> = (0 > x)" by simp
   565   finally show  "(c*x < 0) == (x < 0)" by simp
   566 qed
   567 
   568 lemma neg_prod_sum_lt: "(c\<Colon>'a\<Colon>ordered_field) < 0 \<Longrightarrow> ((c*x + t< 0) == (x > (- 1/c)*t))"
   569 proof-
   570   assume H: "c < 0"
   571   have "c*x + t< 0 = (c*x < -t)" by (subst less_iff_diff_less_0 [of "c*x" "-t"], simp)
   572   also have "\<dots> = (-t/c < x)" by (simp only: neg_divide_less_eq[OF H] ring_simps)
   573   also have "\<dots> = ((- 1/c)*t < x)" by simp
   574   finally show  "(c*x + t < 0) == (x > (- 1/c)*t)" by simp
   575 qed
   576 
   577 lemma pos_prod_sum_lt:"(c\<Colon>'a\<Colon>ordered_field) > 0 \<Longrightarrow> ((c*x + t < 0) == (x < (- 1/c)*t))"
   578 proof-
   579   assume H: "c > 0"
   580   have "c*x + t< 0 = (c*x < -t)"  by (subst less_iff_diff_less_0 [of "c*x" "-t"], simp)
   581   also have "\<dots> = (-t/c > x)" by (simp only: pos_less_divide_eq[OF H] ring_simps)
   582   also have "\<dots> = ((- 1/c)*t > x)" by simp
   583   finally show  "(c*x + t < 0) == (x < (- 1/c)*t)" by simp
   584 qed
   585 
   586 lemma sum_lt:"((x::'a::pordered_ab_group_add) + t < 0) == (x < - t)"
   587   using less_diff_eq[where a= x and b=t and c=0] by simp
   588 
   589 lemma neg_prod_le:"(c\<Colon>'a\<Colon>ordered_field) < 0 \<Longrightarrow> ((c*x <= 0) == (x >= 0))"
   590 proof-
   591   assume H: "c < 0"
   592   have "c*x <= 0 = (0/c <= x)" by (simp only: neg_divide_le_eq[OF H] ring_simps)
   593   also have "\<dots> = (0 <= x)" by simp
   594   finally show  "(c*x <= 0) == (x >= 0)" by simp
   595 qed
   596 
   597 lemma pos_prod_le:"(c\<Colon>'a\<Colon>ordered_field) > 0 \<Longrightarrow> ((c*x <= 0) == (x <= 0))"
   598 proof-
   599   assume H: "c > 0"
   600   hence "c*x <= 0 = (0/c >= x)" by (simp only: pos_le_divide_eq[OF H] ring_simps)
   601   also have "\<dots> = (0 >= x)" by simp
   602   finally show  "(c*x <= 0) == (x <= 0)" by simp
   603 qed
   604 
   605 lemma neg_prod_sum_le: "(c\<Colon>'a\<Colon>ordered_field) < 0 \<Longrightarrow> ((c*x + t <= 0) == (x >= (- 1/c)*t))"
   606 proof-
   607   assume H: "c < 0"
   608   have "c*x + t <= 0 = (c*x <= -t)"  by (subst le_iff_diff_le_0 [of "c*x" "-t"], simp)
   609   also have "\<dots> = (-t/c <= x)" by (simp only: neg_divide_le_eq[OF H] ring_simps)
   610   also have "\<dots> = ((- 1/c)*t <= x)" by simp
   611   finally show  "(c*x + t <= 0) == (x >= (- 1/c)*t)" by simp
   612 qed
   613 
   614 lemma pos_prod_sum_le:"(c\<Colon>'a\<Colon>ordered_field) > 0 \<Longrightarrow> ((c*x + t <= 0) == (x <= (- 1/c)*t))"
   615 proof-
   616   assume H: "c > 0"
   617   have "c*x + t <= 0 = (c*x <= -t)" by (subst le_iff_diff_le_0 [of "c*x" "-t"], simp)
   618   also have "\<dots> = (-t/c >= x)" by (simp only: pos_le_divide_eq[OF H] ring_simps)
   619   also have "\<dots> = ((- 1/c)*t >= x)" by simp
   620   finally show  "(c*x + t <= 0) == (x <= (- 1/c)*t)" by simp
   621 qed
   622 
   623 lemma sum_le:"((x::'a::pordered_ab_group_add) + t <= 0) == (x <= - t)"
   624   using le_diff_eq[where a= x and b=t and c=0] by simp
   625 
   626 lemma nz_prod_eq:"(c\<Colon>'a\<Colon>ordered_field) \<noteq> 0 \<Longrightarrow> ((c*x = 0) == (x = 0))" by simp
   627 lemma nz_prod_sum_eq: "(c\<Colon>'a\<Colon>ordered_field) \<noteq> 0 \<Longrightarrow> ((c*x + t = 0) == (x = (- 1/c)*t))"
   628 proof-
   629   assume H: "c \<noteq> 0"
   630   have "c*x + t = 0 = (c*x = -t)" by (subst eq_iff_diff_eq_0 [of "c*x" "-t"], simp)
   631   also have "\<dots> = (x = -t/c)" by (simp only: nonzero_eq_divide_eq[OF H] ring_simps)
   632   finally show  "(c*x + t = 0) == (x = (- 1/c)*t)" by simp
   633 qed
   634 lemma sum_eq:"((x::'a::pordered_ab_group_add) + t = 0) == (x = - t)"
   635   using eq_diff_eq[where a= x and b=t and c=0] by simp
   636 
   637 
   638 class_interpretation class_ordered_field_dense_linear_order: constr_dense_linear_order
   639  ["op <=" "op <"
   640    "\<lambda> x y. 1/2 * ((x::'a::{ordered_field,recpower,number_ring}) + y)"]
   641 proof (unfold_locales, dlo, dlo, auto)
   642   fix x y::'a assume lt: "x < y"
   643   from  less_half_sum[OF lt] show "x < (x + y) /2" by simp
   644 next
   645   fix x y::'a assume lt: "x < y"
   646   from  gt_half_sum[OF lt] show "(x + y) /2 < y" by simp
   647 qed
   648 
   649 declaration{*
   650 let
   651 fun earlier [] x y = false
   652         | earlier (h::t) x y =
   653     if h aconvc y then false else if h aconvc x then true else earlier t x y;
   654 
   655 fun dest_frac ct = case term_of ct of
   656    Const (@{const_name "HOL.divide"},_) $ a $ b=>
   657     Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b))
   658  | t => Rat.rat_of_int (snd (HOLogic.dest_number t))
   659 
   660 fun mk_frac phi cT x =
   661  let val (a, b) = Rat.quotient_of_rat x
   662  in if b = 1 then Numeral.mk_cnumber cT a
   663     else Thm.capply
   664          (Thm.capply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"})
   665                      (Numeral.mk_cnumber cT a))
   666          (Numeral.mk_cnumber cT b)
   667  end
   668 
   669 fun whatis x ct = case term_of ct of
   670   Const(@{const_name "HOL.plus"}, _)$(Const(@{const_name "HOL.times"},_)$_$y)$_ =>
   671      if y aconv term_of x then ("c*x+t",[(funpow 2 Thm.dest_arg1) ct, Thm.dest_arg ct])
   672      else ("Nox",[])
   673 | Const(@{const_name "HOL.plus"}, _)$y$_ =>
   674      if y aconv term_of x then ("x+t",[Thm.dest_arg ct])
   675      else ("Nox",[])
   676 | Const(@{const_name "HOL.times"}, _)$_$y =>
   677      if y aconv term_of x then ("c*x",[Thm.dest_arg1 ct])
   678      else ("Nox",[])
   679 | t => if t aconv term_of x then ("x",[]) else ("Nox",[]);
   680 
   681 fun xnormalize_conv ctxt [] ct = reflexive ct
   682 | xnormalize_conv ctxt (vs as (x::_)) ct =
   683    case term_of ct of
   684    Const(@{const_name HOL.less},_)$_$Const(@{const_name "HOL.zero"},_) =>
   685     (case whatis x (Thm.dest_arg1 ct) of
   686     ("c*x+t",[c,t]) =>
   687        let
   688         val cr = dest_frac c
   689         val clt = Thm.dest_fun2 ct
   690         val cz = Thm.dest_arg ct
   691         val neg = cr </ Rat.zero
   692         val cthp = Simplifier.rewrite (local_simpset_of ctxt)
   693                (Thm.capply @{cterm "Trueprop"}
   694                   (if neg then Thm.capply (Thm.capply clt c) cz
   695                     else Thm.capply (Thm.capply clt cz) c))
   696         val cth = equal_elim (symmetric cthp) TrueI
   697         val th = implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x,t])
   698              (if neg then @{thm neg_prod_sum_lt} else @{thm pos_prod_sum_lt})) cth
   699         val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
   700                    (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
   701       in rth end
   702     | ("x+t",[t]) =>
   703        let
   704         val T = ctyp_of_term x
   705         val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_lt"}
   706         val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
   707               (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
   708        in  rth end
   709     | ("c*x",[c]) =>
   710        let
   711         val cr = dest_frac c
   712         val clt = Thm.dest_fun2 ct
   713         val cz = Thm.dest_arg ct
   714         val neg = cr </ Rat.zero
   715         val cthp = Simplifier.rewrite (local_simpset_of ctxt)
   716                (Thm.capply @{cterm "Trueprop"}
   717                   (if neg then Thm.capply (Thm.capply clt c) cz
   718                     else Thm.capply (Thm.capply clt cz) c))
   719         val cth = equal_elim (symmetric cthp) TrueI
   720         val th = implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x])
   721              (if neg then @{thm neg_prod_lt} else @{thm pos_prod_lt})) cth
   722         val rth = th
   723       in rth end
   724     | _ => reflexive ct)
   725 
   726 
   727 |  Const(@{const_name HOL.less_eq},_)$_$Const(@{const_name "HOL.zero"},_) =>
   728    (case whatis x (Thm.dest_arg1 ct) of
   729     ("c*x+t",[c,t]) =>
   730        let
   731         val T = ctyp_of_term x
   732         val cr = dest_frac c
   733         val clt = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat "op <"}
   734         val cz = Thm.dest_arg ct
   735         val neg = cr </ Rat.zero
   736         val cthp = Simplifier.rewrite (local_simpset_of ctxt)
   737                (Thm.capply @{cterm "Trueprop"}
   738                   (if neg then Thm.capply (Thm.capply clt c) cz
   739                     else Thm.capply (Thm.capply clt cz) c))
   740         val cth = equal_elim (symmetric cthp) TrueI
   741         val th = implies_elim (instantiate' [SOME T] (map SOME [c,x,t])
   742              (if neg then @{thm neg_prod_sum_le} else @{thm pos_prod_sum_le})) cth
   743         val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
   744                    (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
   745       in rth end
   746     | ("x+t",[t]) =>
   747        let
   748         val T = ctyp_of_term x
   749         val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_le"}
   750         val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
   751               (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
   752        in  rth end
   753     | ("c*x",[c]) =>
   754        let
   755         val T = ctyp_of_term x
   756         val cr = dest_frac c
   757         val clt = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat "op <"}
   758         val cz = Thm.dest_arg ct
   759         val neg = cr </ Rat.zero
   760         val cthp = Simplifier.rewrite (local_simpset_of ctxt)
   761                (Thm.capply @{cterm "Trueprop"}
   762                   (if neg then Thm.capply (Thm.capply clt c) cz
   763                     else Thm.capply (Thm.capply clt cz) c))
   764         val cth = equal_elim (symmetric cthp) TrueI
   765         val th = implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x])
   766              (if neg then @{thm neg_prod_le} else @{thm pos_prod_le})) cth
   767         val rth = th
   768       in rth end
   769     | _ => reflexive ct)
   770 
   771 |  Const("op =",_)$_$Const(@{const_name "HOL.zero"},_) =>
   772    (case whatis x (Thm.dest_arg1 ct) of
   773     ("c*x+t",[c,t]) =>
   774        let
   775         val T = ctyp_of_term x
   776         val cr = dest_frac c
   777         val ceq = Thm.dest_fun2 ct
   778         val cz = Thm.dest_arg ct
   779         val cthp = Simplifier.rewrite (local_simpset_of ctxt)
   780             (Thm.capply @{cterm "Trueprop"}
   781              (Thm.capply @{cterm "Not"} (Thm.capply (Thm.capply ceq c) cz)))
   782         val cth = equal_elim (symmetric cthp) TrueI
   783         val th = implies_elim
   784                  (instantiate' [SOME T] (map SOME [c,x,t]) @{thm nz_prod_sum_eq}) cth
   785         val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
   786                    (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
   787       in rth end
   788     | ("x+t",[t]) =>
   789        let
   790         val T = ctyp_of_term x
   791         val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_eq"}
   792         val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
   793               (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
   794        in  rth end
   795     | ("c*x",[c]) =>
   796        let
   797         val T = ctyp_of_term x
   798         val cr = dest_frac c
   799         val ceq = Thm.dest_fun2 ct
   800         val cz = Thm.dest_arg ct
   801         val cthp = Simplifier.rewrite (local_simpset_of ctxt)
   802             (Thm.capply @{cterm "Trueprop"}
   803              (Thm.capply @{cterm "Not"} (Thm.capply (Thm.capply ceq c) cz)))
   804         val cth = equal_elim (symmetric cthp) TrueI
   805         val rth = implies_elim
   806                  (instantiate' [SOME T] (map SOME [c,x]) @{thm nz_prod_eq}) cth
   807       in rth end
   808     | _ => reflexive ct);
   809 
   810 local
   811   val less_iff_diff_less_0 = mk_meta_eq @{thm "less_iff_diff_less_0"}
   812   val le_iff_diff_le_0 = mk_meta_eq @{thm "le_iff_diff_le_0"}
   813   val eq_iff_diff_eq_0 = mk_meta_eq @{thm "eq_iff_diff_eq_0"}
   814 in
   815 fun field_isolate_conv phi ctxt vs ct = case term_of ct of
   816   Const(@{const_name HOL.less},_)$a$b =>
   817    let val (ca,cb) = Thm.dest_binop ct
   818        val T = ctyp_of_term ca
   819        val th = instantiate' [SOME T] [SOME ca, SOME cb] less_iff_diff_less_0
   820        val nth = Conv.fconv_rule
   821          (Conv.arg_conv (Conv.arg1_conv
   822               (Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th
   823        val rth = transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
   824    in rth end
   825 | Const(@{const_name HOL.less_eq},_)$a$b =>
   826    let val (ca,cb) = Thm.dest_binop ct
   827        val T = ctyp_of_term ca
   828        val th = instantiate' [SOME T] [SOME ca, SOME cb] le_iff_diff_le_0
   829        val nth = Conv.fconv_rule
   830          (Conv.arg_conv (Conv.arg1_conv
   831               (Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th
   832        val rth = transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
   833    in rth end
   834 
   835 | Const("op =",_)$a$b =>
   836    let val (ca,cb) = Thm.dest_binop ct
   837        val T = ctyp_of_term ca
   838        val th = instantiate' [SOME T] [SOME ca, SOME cb] eq_iff_diff_eq_0
   839        val nth = Conv.fconv_rule
   840          (Conv.arg_conv (Conv.arg1_conv
   841               (Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th
   842        val rth = transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
   843    in rth end
   844 | @{term "Not"} $(Const("op =",_)$a$b) => Conv.arg_conv (field_isolate_conv phi ctxt vs) ct
   845 | _ => reflexive ct
   846 end;
   847 
   848 fun classfield_whatis phi =
   849  let
   850   fun h x t =
   851    case term_of t of
   852      Const("op =", _)$y$z => if term_of x aconv y then Ferrante_Rackoff_Data.Eq
   853                             else Ferrante_Rackoff_Data.Nox
   854    | @{term "Not"}$(Const("op =", _)$y$z) => if term_of x aconv y then Ferrante_Rackoff_Data.NEq
   855                             else Ferrante_Rackoff_Data.Nox
   856    | Const(@{const_name HOL.less},_)$y$z =>
   857        if term_of x aconv y then Ferrante_Rackoff_Data.Lt
   858         else if term_of x aconv z then Ferrante_Rackoff_Data.Gt
   859         else Ferrante_Rackoff_Data.Nox
   860    | Const (@{const_name HOL.less_eq},_)$y$z =>
   861          if term_of x aconv y then Ferrante_Rackoff_Data.Le
   862          else if term_of x aconv z then Ferrante_Rackoff_Data.Ge
   863          else Ferrante_Rackoff_Data.Nox
   864    | _ => Ferrante_Rackoff_Data.Nox
   865  in h end;
   866 fun class_field_ss phi =
   867    HOL_basic_ss addsimps ([@{thm "linorder_not_less"}, @{thm "linorder_not_le"}])
   868    addsplits [@{thm "abs_split"},@{thm "split_max"}, @{thm "split_min"}]
   869 
   870 in
   871 Ferrante_Rackoff_Data.funs @{thm "class_ordered_field_dense_linear_order.ferrack_axiom"}
   872   {isolate_conv = field_isolate_conv, whatis = classfield_whatis, simpset = class_field_ss}
   873 end
   874 *}
   875 
   876 
   877 end