src/HOL/Decision_Procs/Dense_Linear_Order.thy
 author nipkow Tue Feb 23 16:25:08 2016 +0100 (2016-02-23) changeset 62390 842917225d56 parent 61586 5197a2ecb658 child 63198 c583ca33076a permissions -rw-r--r--
more canonical names
```     1 (*  Title       : HOL/Decision_Procs/Dense_Linear_Order.thy
```
```     2     Author      : Amine Chaieb, TU Muenchen
```
```     3 *)
```
```     4
```
```     5 section \<open>Dense linear order without endpoints
```
```     6   and a quantifier elimination procedure in Ferrante and Rackoff style\<close>
```
```     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\<open>Theorems for \<open>\<exists>z. \<forall>x. x < z \<longrightarrow> (P x \<longleftrightarrow> P\<^sub>-\<^sub>\<infinity>)\<close>\<close>
```
```    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\<open>Theorems for \<open>\<exists>z. \<forall>x. x < z \<longrightarrow> (P x \<longleftrightarrow> P\<^sub>+\<^sub>\<infinity>)\<close>\<close>
```
```    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"
```
```    92     and noU: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U"
```
```    93     and lx: "l < x"
```
```    94     and xu: "x < u"
```
```    95     and px: "x < t"
```
```    96     and ly: "l < y"
```
```    97     and yu: "y < u"
```
```    98   from tU noU ly yu have tny: "t \<noteq> y" by auto
```
```    99   have False if H: "t < y"
```
```   100   proof -
```
```   101     from less_trans[OF lx px] less_trans[OF H yu] have "l < t \<and> t < u"
```
```   102       by simp
```
```   103     with tU noU show ?thesis
```
```   104       by auto
```
```   105   qed
```
```   106   then have "\<not> t < y"
```
```   107     by auto
```
```   108   then have "y \<le> t"
```
```   109     by (simp add: not_less)
```
```   110   then show "y < t"
```
```   111     using tny by (simp add: less_le)
```
```   112 qed
```
```   113
```
```   114 lemma lin_dense_gt[no_atp]:
```
```   115   "t \<in> U \<Longrightarrow>
```
```   116     \<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)"
```
```   117 proof clarsimp
```
```   118   fix x l u y
```
```   119   assume tU: "t \<in> U"
```
```   120     and noU: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U"
```
```   121     and lx: "l < x"
```
```   122     and xu: "x < u"
```
```   123     and px: "t < x"
```
```   124     and ly: "l < y"
```
```   125     and yu: "y < u"
```
```   126   from tU noU ly yu have tny: "t \<noteq> y" by auto
```
```   127   have False if H: "y < t"
```
```   128   proof -
```
```   129     from less_trans[OF ly H] less_trans[OF px xu] have "l < t \<and> t < u"
```
```   130       by simp
```
```   131     with tU noU show ?thesis
```
```   132       by auto
```
```   133   qed
```
```   134   then have "\<not> y < t"
```
```   135     by auto
```
```   136   then have "t \<le> y"
```
```   137     by (auto simp add: not_less)
```
```   138   then show "t < y"
```
```   139     using tny by (simp add: less_le)
```
```   140 qed
```
```   141
```
```   142 lemma lin_dense_le[no_atp]:
```
```   143   "t \<in> U \<Longrightarrow>
```
```   144     \<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)"
```
```   145 proof clarsimp
```
```   146   fix x l u y
```
```   147   assume tU: "t \<in> U"
```
```   148     and noU: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U"
```
```   149     and lx: "l < x"
```
```   150     and xu: "x < u"
```
```   151     and px: "x \<le> t"
```
```   152     and ly: "l < y"
```
```   153     and yu: "y < u"
```
```   154   from tU noU ly yu have tny: "t \<noteq> y" by auto
```
```   155   have False if H: "t < y"
```
```   156   proof -
```
```   157     from less_le_trans[OF lx px] less_trans[OF H yu]
```
```   158     have "l < t \<and> t < u" by simp
```
```   159     with tU noU show ?thesis by auto
```
```   160   qed
```
```   161   then have "\<not> t < y" by auto
```
```   162   then show "y \<le> t" by (simp add: not_less)
```
```   163 qed
```
```   164
```
```   165 lemma lin_dense_ge[no_atp]:
```
```   166   "t \<in> U \<Longrightarrow>
```
```   167     \<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)"
```
```   168 proof clarsimp
```
```   169   fix x l u y
```
```   170   assume tU: "t \<in> U"
```
```   171     and noU: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U"
```
```   172     and lx: "l < x"
```
```   173     and xu: "x < u"
```
```   174     and px: "t \<le> x"
```
```   175     and ly: "l < y"
```
```   176     and yu: "y < u"
```
```   177   from tU noU ly yu have tny: "t \<noteq> y" by auto
```
```   178   have False if H: "y < t"
```
```   179   proof -
```
```   180     from less_trans[OF ly H] le_less_trans[OF px xu]
```
```   181     have "l < t \<and> t < u" by simp
```
```   182     with tU noU show ?thesis by auto
```
```   183   qed
```
```   184   then have "\<not> y < t" by auto
```
```   185   then show "t \<le> y" by (simp add: not_less)
```
```   186 qed
```
```   187
```
```   188 lemma lin_dense_eq[no_atp]:
```
```   189   "t \<in> U \<Longrightarrow>
```
```   190     \<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)"
```
```   191   by auto
```
```   192
```
```   193 lemma lin_dense_neq[no_atp]:
```
```   194   "t \<in> U \<Longrightarrow>
```
```   195     \<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)"
```
```   196   by auto
```
```   197
```
```   198 lemma lin_dense_P[no_atp]:
```
```   199   "\<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)"
```
```   200   by auto
```
```   201
```
```   202 lemma lin_dense_conj[no_atp]:
```
```   203   "\<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
```
```   204   \<longrightarrow> (\<forall>y. l < y \<and> y < u \<longrightarrow> P1 y) ;
```
```   205   \<forall>x l u. (\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U) \<and> l < x \<and> x < u \<and> P2 x
```
```   206   \<longrightarrow> (\<forall>y. l < y \<and> y < u \<longrightarrow> P2 y)\<rbrakk> \<Longrightarrow>
```
```   207   \<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)
```
```   208   \<longrightarrow> (\<forall>y. l < y \<and> y < u \<longrightarrow> (P1 y \<and> P2 y))"
```
```   209   by blast
```
```   210 lemma lin_dense_disj[no_atp]:
```
```   211   "\<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
```
```   212   \<longrightarrow> (\<forall>y. l < y \<and> y < u \<longrightarrow> P1 y) ;
```
```   213   \<forall>x l u. (\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U) \<and> l < x \<and> x < u \<and> P2 x
```
```   214   \<longrightarrow> (\<forall>y. l < y \<and> y < u \<longrightarrow> P2 y)\<rbrakk> \<Longrightarrow>
```
```   215   \<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)
```
```   216   \<longrightarrow> (\<forall>y. l < y \<and> y < u \<longrightarrow> (P1 y \<or> P2 y))"
```
```   217   by blast
```
```   218
```
```   219 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>
```
```   220   \<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')"
```
```   221   by auto
```
```   222
```
```   223 lemma finite_set_intervals[no_atp]:
```
```   224   assumes px: "P x"
```
```   225     and lx: "l \<le> x"
```
```   226     and xu: "x \<le> u"
```
```   227     and linS: "l\<in> S"
```
```   228     and uinS: "u \<in> S"
```
```   229     and fS:"finite S"
```
```   230     and lS: "\<forall>x\<in> S. l \<le> x"
```
```   231     and Su: "\<forall>x\<in> S. x \<le> u"
```
```   232   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"
```
```   233 proof -
```
```   234   let ?Mx = "{y. y\<in> S \<and> y \<le> x}"
```
```   235   let ?xM = "{y. y\<in> S \<and> x \<le> y}"
```
```   236   let ?a = "Max ?Mx"
```
```   237   let ?b = "Min ?xM"
```
```   238   have MxS: "?Mx \<subseteq> S"
```
```   239     by blast
```
```   240   then have fMx: "finite ?Mx"
```
```   241     using fS finite_subset by auto
```
```   242   from lx linS have linMx: "l \<in> ?Mx"
```
```   243     by blast
```
```   244   then have Mxne: "?Mx \<noteq> {}"
```
```   245     by blast
```
```   246   have xMS: "?xM \<subseteq> S"
```
```   247     by blast
```
```   248   then have fxM: "finite ?xM"
```
```   249     using fS finite_subset by auto
```
```   250   from xu uinS have linxM: "u \<in> ?xM"
```
```   251     by blast
```
```   252   then have xMne: "?xM \<noteq> {}"
```
```   253     by blast
```
```   254   have ax: "?a \<le> x"
```
```   255     using Mxne fMx by auto
```
```   256   have xb: "x \<le> ?b"
```
```   257     using xMne fxM by auto
```
```   258   have "?a \<in> ?Mx"
```
```   259     using Max_in[OF fMx Mxne] by simp
```
```   260   then have ainS: "?a \<in> S"
```
```   261     using MxS by blast
```
```   262   have "?b \<in> ?xM"
```
```   263     using Min_in[OF fxM xMne] by simp
```
```   264   then have binS: "?b \<in> S"
```
```   265     using xMS by blast
```
```   266   have noy: "\<forall>y. ?a < y \<and> y < ?b \<longrightarrow> y \<notin> S"
```
```   267   proof clarsimp
```
```   268     fix y
```
```   269     assume ay: "?a < y" and yb: "y < ?b" and yS: "y \<in> S"
```
```   270     from yS have "y \<in> ?Mx \<or> y \<in> ?xM"
```
```   271       by (auto simp add: linear)
```
```   272     then show False
```
```   273     proof
```
```   274       assume "y \<in> ?Mx"
```
```   275       then have "y \<le> ?a"
```
```   276         using Mxne fMx by auto
```
```   277       with ay show ?thesis
```
```   278         by (simp add: not_le[symmetric])
```
```   279     next
```
```   280       assume "y \<in> ?xM"
```
```   281       then have "?b \<le> y"
```
```   282         using xMne fxM by auto
```
```   283       with yb show ?thesis
```
```   284         by (simp add: not_le[symmetric])
```
```   285     qed
```
```   286   qed
```
```   287   from ainS binS noy ax xb px show ?thesis
```
```   288     by blast
```
```   289 qed
```
```   290
```
```   291 lemma finite_set_intervals2[no_atp]:
```
```   292   assumes px: "P x"
```
```   293     and lx: "l \<le> x"
```
```   294     and xu: "x \<le> u"
```
```   295     and linS: "l\<in> S"
```
```   296     and uinS: "u \<in> S"
```
```   297     and fS: "finite S"
```
```   298     and lS: "\<forall>x\<in> S. l \<le> x"
```
```   299     and Su: "\<forall>x\<in> S. x \<le> u"
```
```   300   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)"
```
```   301 proof -
```
```   302   from finite_set_intervals[where P="P", OF px lx xu linS uinS fS lS Su]
```
```   303   obtain a and b where as: "a \<in> S" and bs: "b \<in> S"
```
```   304     and noS: "\<forall>y. a < y \<and> y < b \<longrightarrow> y \<notin> S"
```
```   305     and axb: "a \<le> x \<and> x \<le> b \<and> P x"
```
```   306     by auto
```
```   307   from axb have "x = a \<or> x = b \<or> (a < x \<and> x < b)"
```
```   308     by (auto simp add: le_less)
```
```   309   then show ?thesis
```
```   310     using px as bs noS by blast
```
```   311 qed
```
```   312
```
```   313 end
```
```   314
```
```   315
```
```   316 section \<open>The classical QE after Langford for dense linear orders\<close>
```
```   317
```
```   318 context unbounded_dense_linorder
```
```   319 begin
```
```   320
```
```   321 lemma interval_empty_iff: "{y. x < y \<and> y < z} = {} \<longleftrightarrow> \<not> x < z"
```
```   322   by (auto dest: dense)
```
```   323
```
```   324 lemma dlo_qe_bnds[no_atp]:
```
```   325   assumes ne: "L \<noteq> {}"
```
```   326     and neU: "U \<noteq> {}"
```
```   327     and fL: "finite L"
```
```   328     and fU: "finite U"
```
```   329   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)"
```
```   330 proof (simp only: atomize_eq, rule iffI)
```
```   331   assume H: "\<exists>x. (\<forall>y\<in>L. y < x) \<and> (\<forall>y\<in>U. x < y)"
```
```   332   then obtain x where xL: "\<forall>y\<in>L. y < x" and xU: "\<forall>y\<in>U. x < y"
```
```   333     by blast
```
```   334   have "l < u" if l: "l \<in> L" and u: "u \<in> U" for l u
```
```   335   proof -
```
```   336     have "l < x" using xL l by blast
```
```   337     also have "x < u" using xU u by blast
```
```   338     finally show ?thesis .
```
```   339   qed
```
```   340   then show "\<forall>l\<in>L. \<forall>u\<in>U. l < u" by blast
```
```   341 next
```
```   342   assume H: "\<forall>l\<in>L. \<forall>u\<in>U. l < u"
```
```   343   let ?ML = "Max L"
```
```   344   let ?MU = "Min U"
```
```   345   from fL ne have th1: "?ML \<in> L" and th1': "\<forall>l\<in>L. l \<le> ?ML"
```
```   346     by auto
```
```   347   from fU neU have th2: "?MU \<in> U" and th2': "\<forall>u\<in>U. ?MU \<le> u"
```
```   348     by auto
```
```   349   from th1 th2 H have "?ML < ?MU"
```
```   350     by auto
```
```   351   with dense obtain w where th3: "?ML < w" and th4: "w < ?MU"
```
```   352     by blast
```
```   353   from th3 th1' have "\<forall>l \<in> L. l < w"
```
```   354     by auto
```
```   355   moreover from th4 th2' have "\<forall>u \<in> U. w < u"
```
```   356     by auto
```
```   357   ultimately show "\<exists>x. (\<forall>y\<in>L. y < x) \<and> (\<forall>y\<in>U. x < y)"
```
```   358     by auto
```
```   359 qed
```
```   360
```
```   361 lemma dlo_qe_noub[no_atp]:
```
```   362   assumes ne: "L \<noteq> {}"
```
```   363     and fL: "finite L"
```
```   364   shows "(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> {}. x < y)) \<equiv> True"
```
```   365 proof (simp add: atomize_eq)
```
```   366   from gt_ex[of "Max L"] obtain M where M: "Max L < M"
```
```   367     by blast
```
```   368   from ne fL have "\<forall>x \<in> L. x \<le> Max L"
```
```   369     by simp
```
```   370   with M have "\<forall>x\<in>L. x < M"
```
```   371     by (auto intro: le_less_trans)
```
```   372   then show "\<exists>x. \<forall>y\<in>L. y < x"
```
```   373     by blast
```
```   374 qed
```
```   375
```
```   376 lemma dlo_qe_nolb[no_atp]:
```
```   377   assumes ne: "U \<noteq> {}"
```
```   378     and fU: "finite U"
```
```   379   shows "(\<exists>x. (\<forall>y \<in> {}. y < x) \<and> (\<forall>y \<in> U. x < y)) \<equiv> True"
```
```   380 proof (simp add: atomize_eq)
```
```   381   from lt_ex[of "Min U"] obtain M where M: "M < Min U"
```
```   382     by blast
```
```   383   from ne fU have "\<forall>x \<in> U. Min U \<le> x"
```
```   384     by simp
```
```   385   with M have "\<forall>x\<in>U. M < x"
```
```   386     by (auto intro: less_le_trans)
```
```   387   then show "\<exists>x. \<forall>y\<in>U. x < y"
```
```   388     by blast
```
```   389 qed
```
```   390
```
```   391 lemma exists_neq[no_atp]: "\<exists>(x::'a). x \<noteq> t" "\<exists>(x::'a). t \<noteq> x"
```
```   392   using gt_ex[of t] by auto
```
```   393
```
```   394 lemmas dlo_simps[no_atp] = order_refl less_irrefl not_less not_le exists_neq
```
```   395   le_less neq_iff linear less_not_permute
```
```   396
```
```   397 lemma axiom[no_atp]: "class.unbounded_dense_linorder (op \<le>) (op <)"
```
```   398   by (rule unbounded_dense_linorder_axioms)
```
```   399 lemma atoms[no_atp]:
```
```   400   shows "TERM (less :: 'a \<Rightarrow> _)"
```
```   401     and "TERM (less_eq :: 'a \<Rightarrow> _)"
```
```   402     and "TERM (op = :: 'a \<Rightarrow> _)" .
```
```   403
```
```   404 declare axiom[langford qe: dlo_qe_bnds dlo_qe_nolb dlo_qe_noub gather: gather_start gather_simps atoms: atoms]
```
```   405 declare dlo_simps[langfordsimp]
```
```   406
```
```   407 end
```
```   408
```
```   409 (* FIXME: Move to HOL -- together with the conj_aci_rule in langford.ML *)
```
```   410 lemmas dnf[no_atp] = conj_disj_distribL conj_disj_distribR
```
```   411
```
```   412 lemmas weak_dnf_simps[no_atp] = simp_thms dnf
```
```   413
```
```   414 lemma nnf_simps[no_atp]:
```
```   415   "(\<not> (P \<and> Q)) \<longleftrightarrow> (\<not> P \<or> \<not> Q)"
```
```   416   "(\<not> (P \<or> Q)) \<longleftrightarrow> (\<not> P \<and> \<not> Q)"
```
```   417   "(P \<longrightarrow> Q) \<longleftrightarrow> (\<not> P \<or> Q)"
```
```   418   "(P \<longleftrightarrow> Q) \<longleftrightarrow> ((P \<and> Q) \<or> (\<not> P \<and> \<not> Q))"
```
```   419   "(\<not> \<not> P) \<longleftrightarrow> P"
```
```   420   by blast+
```
```   421
```
```   422 lemma ex_distrib[no_atp]: "(\<exists>x. P x \<or> Q x) \<longleftrightarrow> ((\<exists>x. P x) \<or> (\<exists>x. Q x))"
```
```   423   by blast
```
```   424
```
```   425 lemmas dnf_simps[no_atp] = weak_dnf_simps nnf_simps ex_distrib
```
```   426
```
```   427 ML_file "langford.ML"
```
```   428 method_setup dlo = \<open>
```
```   429   Scan.succeed (SIMPLE_METHOD' o Langford.dlo_tac)
```
```   430 \<close> "Langford's algorithm for quantifier elimination in dense linear orders"
```
```   431
```
```   432
```
```   433 section \<open>Contructive dense linear orders yield QE for linear arithmetic over ordered Fields\<close>
```
```   434
```
```   435 text \<open>Linear order without upper bounds\<close>
```
```   436
```
```   437 locale linorder_stupid_syntax = linorder
```
```   438 begin
```
```   439
```
```   440 notation
```
```   441   less_eq  ("op \<sqsubseteq>") and
```
```   442   less_eq  ("(_/ \<sqsubseteq> _)" [51, 51] 50) and
```
```   443   less  ("op \<sqsubset>") and
```
```   444   less  ("(_/ \<sqsubset> _)"  [51, 51] 50)
```
```   445
```
```   446 end
```
```   447
```
```   448 locale linorder_no_ub = linorder_stupid_syntax +
```
```   449   assumes gt_ex: "\<exists>y. less x y"
```
```   450 begin
```
```   451
```
```   452 lemma ge_ex[no_atp]: "\<exists>y. x \<sqsubseteq> y"
```
```   453   using gt_ex by auto
```
```   454
```
```   455 text \<open>Theorems for \<open>\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (P x \<longleftrightarrow> P\<^sub>+\<^sub>\<infinity>)\<close>\<close>
```
```   456 lemma pinf_conj[no_atp]:
```
```   457   assumes ex1: "\<exists>z1. \<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
```
```   458     and ex2: "\<exists>z2. \<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
```
```   459   shows "\<exists>z. \<forall>x. z \<sqsubset>  x \<longrightarrow> ((P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2'))"
```
```   460 proof -
```
```   461   from ex1 ex2 obtain z1 and z2
```
```   462     where z1: "\<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
```
```   463     and z2: "\<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
```
```   464     by blast
```
```   465   from gt_ex obtain z where z:"ord.max less_eq z1 z2 \<sqsubset> z"
```
```   466     by blast
```
```   467   from z have zz1: "z1 \<sqsubset> z" and zz2: "z2 \<sqsubset> z"
```
```   468     by simp_all
```
```   469   have "(P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2')" if H: "z \<sqsubset> x" for x
```
```   470     using less_trans[OF zz1 H] less_trans[OF zz2 H] z1 zz1 z2 zz2 by auto
```
```   471   then show ?thesis
```
```   472     by blast
```
```   473 qed
```
```   474
```
```   475 lemma pinf_disj[no_atp]:
```
```   476   assumes ex1: "\<exists>z1. \<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
```
```   477     and ex2: "\<exists>z2. \<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
```
```   478   shows "\<exists>z. \<forall>x. z \<sqsubset>  x \<longrightarrow> ((P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2'))"
```
```   479 proof-
```
```   480   from ex1 ex2 obtain z1 and z2
```
```   481     where z1: "\<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
```
```   482     and z2: "\<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
```
```   483     by blast
```
```   484   from gt_ex obtain z where z: "ord.max less_eq z1 z2 \<sqsubset> z"
```
```   485     by blast
```
```   486   from z have zz1: "z1 \<sqsubset> z" and zz2: "z2 \<sqsubset> z"
```
```   487     by simp_all
```
```   488   have "(P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2')" if H: "z \<sqsubset> x" for x
```
```   489     using less_trans[OF zz1 H] less_trans[OF zz2 H] z1 zz1 z2 zz2 by auto
```
```   490   then show ?thesis
```
```   491     by blast
```
```   492 qed
```
```   493
```
```   494 lemma pinf_ex[no_atp]:
```
```   495   assumes ex: "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (P x \<longleftrightarrow> P1)"
```
```   496     and p1: P1
```
```   497   shows "\<exists>x. P x"
```
```   498 proof -
```
```   499   from ex obtain z where z: "\<forall>x. z \<sqsubset> x \<longrightarrow> (P x \<longleftrightarrow> P1)"
```
```   500     by blast
```
```   501   from gt_ex obtain x where x: "z \<sqsubset> x"
```
```   502     by blast
```
```   503   from z x p1 show ?thesis
```
```   504     by blast
```
```   505 qed
```
```   506
```
```   507 end
```
```   508
```
```   509 text \<open>Linear order without upper bounds\<close>
```
```   510
```
```   511 locale linorder_no_lb = linorder_stupid_syntax +
```
```   512   assumes lt_ex: "\<exists>y. less y x"
```
```   513 begin
```
```   514
```
```   515 lemma le_ex[no_atp]: "\<exists>y. y \<sqsubseteq> x"
```
```   516   using lt_ex by auto
```
```   517
```
```   518
```
```   519 text \<open>Theorems for \<open>\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P\<^sub>-\<^sub>\<infinity>)\<close>\<close>
```
```   520 lemma minf_conj[no_atp]:
```
```   521   assumes ex1: "\<exists>z1. \<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
```
```   522     and ex2: "\<exists>z2. \<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
```
```   523   shows "\<exists>z. \<forall>x. x \<sqsubset>  z \<longrightarrow> ((P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2'))"
```
```   524 proof -
```
```   525   from ex1 ex2 obtain z1 and z2
```
```   526     where z1: "\<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
```
```   527     and z2: "\<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
```
```   528     by blast
```
```   529   from lt_ex obtain z where z: "z \<sqsubset> ord.min less_eq z1 z2"
```
```   530     by blast
```
```   531   from z have zz1: "z \<sqsubset> z1" and zz2: "z \<sqsubset> z2"
```
```   532     by simp_all
```
```   533   have "(P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2')" if H: "x \<sqsubset> z" for x
```
```   534     using less_trans[OF H zz1] less_trans[OF H zz2] z1 zz1 z2 zz2 by auto
```
```   535   then show ?thesis
```
```   536     by blast
```
```   537 qed
```
```   538
```
```   539 lemma minf_disj[no_atp]:
```
```   540   assumes ex1: "\<exists>z1. \<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
```
```   541     and ex2: "\<exists>z2. \<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
```
```   542   shows "\<exists>z. \<forall>x. x \<sqsubset>  z \<longrightarrow> ((P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2'))"
```
```   543 proof -
```
```   544   from ex1 ex2 obtain z1 and z2
```
```   545     where z1: "\<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
```
```   546     and z2: "\<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
```
```   547     by blast
```
```   548   from lt_ex obtain z where z: "z \<sqsubset> ord.min less_eq z1 z2"
```
```   549     by blast
```
```   550   from z have zz1: "z \<sqsubset> z1" and zz2: "z \<sqsubset> z2"
```
```   551     by simp_all
```
```   552   have "(P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2')" if H: "x \<sqsubset> z" for x
```
```   553     using less_trans[OF H zz1] less_trans[OF H zz2] z1 zz1 z2 zz2 by auto
```
```   554   then show ?thesis
```
```   555     by blast
```
```   556 qed
```
```   557
```
```   558 lemma minf_ex[no_atp]:
```
```   559   assumes ex: "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P1)"
```
```   560     and p1: P1
```
```   561   shows "\<exists>x. P x"
```
```   562 proof -
```
```   563   from ex obtain z where z: "\<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P1)"
```
```   564     by blast
```
```   565   from lt_ex obtain x where x: "x \<sqsubset> z"
```
```   566     by blast
```
```   567   from z x p1 show ?thesis
```
```   568     by blast
```
```   569 qed
```
```   570
```
```   571 end
```
```   572
```
```   573
```
```   574 locale constr_dense_linorder = linorder_no_lb + linorder_no_ub +
```
```   575   fixes between
```
```   576   assumes between_less: "less x y \<Longrightarrow> less x (between x y) \<and> less (between x y) y"
```
```   577     and between_same: "between x x = x"
```
```   578 begin
```
```   579
```
```   580 sublocale dlo: unbounded_dense_linorder
```
```   581 proof (unfold_locales, goal_cases)
```
```   582   case (1 x y)
```
```   583   then show ?case
```
```   584     using between_less [of x y] by auto
```
```   585 next
```
```   586   case 2
```
```   587   then show ?case by (rule lt_ex)
```
```   588 next
```
```   589   case 3
```
```   590   then show ?case by (rule gt_ex)
```
```   591 qed
```
```   592
```
```   593 lemma rinf_U[no_atp]:
```
```   594   assumes fU: "finite U"
```
```   595     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
```
```   596       \<longrightarrow> (\<forall>y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> P y )"
```
```   597     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')"
```
```   598     and nmi: "\<not> MP"  and npi: "\<not> PP"  and ex: "\<exists>x.  P x"
```
```   599   shows "\<exists>u\<in> U. \<exists>u' \<in> U. P (between u u')"
```
```   600 proof -
```
```   601   from ex obtain x where px: "P x"
```
```   602     by blast
```
```   603   from px nmi npi nmpiU have "\<exists>u\<in> U. \<exists>u' \<in> U. u \<sqsubseteq> x \<and> x \<sqsubseteq> u'"
```
```   604     by auto
```
```   605   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'"
```
```   606     by auto
```
```   607   from uU have Une: "U \<noteq> {}"
```
```   608     by auto
```
```   609   let ?l = "linorder.Min less_eq U"
```
```   610   let ?u = "linorder.Max less_eq U"
```
```   611   have linM: "?l \<in> U"
```
```   612     using fU Une by simp
```
```   613   have uinM: "?u \<in> U"
```
```   614     using fU Une by simp
```
```   615   have lM: "\<forall>t\<in> U. ?l \<sqsubseteq> t"
```
```   616     using Une fU by auto
```
```   617   have Mu: "\<forall>t\<in> U. t \<sqsubseteq> ?u"
```
```   618     using Une fU by auto
```
```   619   have th: "?l \<sqsubseteq> u"
```
```   620     using uU Une lM by auto
```
```   621   from order_trans[OF th ux] have lx: "?l \<sqsubseteq> x" .
```
```   622   have th: "u' \<sqsubseteq> ?u"
```
```   623     using uU' Une Mu by simp
```
```   624   from order_trans[OF xu' th] have xu: "x \<sqsubseteq> ?u" .
```
```   625   from finite_set_intervals2[where P="P",OF px lx xu linM uinM fU lM Mu]
```
```   626   consider u where "u \<in> U" "P u" |
```
```   627     t1 t2 where "t1 \<in> U" "t2 \<in> U" "\<forall>y. t1 \<sqsubset> y \<and> y \<sqsubset> t2 \<longrightarrow> y \<notin> U" "t1 \<sqsubset> x" "x \<sqsubset> t2" "P x"
```
```   628     by blast
```
```   629   then show ?thesis
```
```   630   proof cases
```
```   631     case u: 1
```
```   632     have "between u u = u" by (simp add: between_same)
```
```   633     with u have "P (between u u)" by simp
```
```   634     with u show ?thesis by blast
```
```   635   next
```
```   636     case 2
```
```   637     note t1M = \<open>t1 \<in> U\<close> and t2M = \<open>t2\<in> U\<close>
```
```   638       and noM = \<open>\<forall>y. t1 \<sqsubset> y \<and> y \<sqsubset> t2 \<longrightarrow> y \<notin> U\<close>
```
```   639       and t1x = \<open>t1 \<sqsubset> x\<close> and xt2 = \<open>x \<sqsubset> t2\<close>
```
```   640       and px = \<open>P x\<close>
```
```   641     from less_trans[OF t1x xt2] have t1t2: "t1 \<sqsubset> t2" .
```
```   642     let ?u = "between t1 t2"
```
```   643     from between_less t1t2 have t1lu: "t1 \<sqsubset> ?u" and ut2: "?u \<sqsubset> t2" by auto
```
```   644     from lin_dense noM t1x xt2 px t1lu ut2 have "P ?u" by blast
```
```   645     with t1M t2M show ?thesis by blast
```
```   646   qed
```
```   647 qed
```
```   648
```
```   649 theorem fr_eq[no_atp]:
```
```   650   assumes fU: "finite U"
```
```   651     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
```
```   652      \<longrightarrow> (\<forall>y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> P y )"
```
```   653     and nmibnd: "\<forall>x. \<not> MP \<and> P x \<longrightarrow> (\<exists>u\<in> U. u \<sqsubseteq> x)"
```
```   654     and npibnd: "\<forall>x. \<not>PP \<and> P x \<longrightarrow> (\<exists>u\<in> U. x \<sqsubseteq> u)"
```
```   655     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)"
```
```   656   shows "(\<exists>x. P x) \<equiv> (MP \<or> PP \<or> (\<exists>u \<in> U. \<exists>u'\<in> U. P (between u u')))"
```
```   657   (is "_ \<equiv> (_ \<or> _ \<or> ?F)" is "?E \<equiv> ?D")
```
```   658 proof -
```
```   659   have "?E \<longleftrightarrow> ?D"
```
```   660   proof
```
```   661     show ?D if px: ?E
```
```   662     proof -
```
```   663       consider "MP \<or> PP" | "\<not> MP" "\<not> PP" by blast
```
```   664       then show ?thesis
```
```   665       proof cases
```
```   666         case 1
```
```   667         then show ?thesis by blast
```
```   668       next
```
```   669         case 2
```
```   670         from npmibnd[OF nmibnd npibnd]
```
```   671         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')" .
```
```   672         from rinf_U[OF fU lin_dense nmpiU \<open>\<not> MP\<close> \<open>\<not> PP\<close> px] show ?thesis
```
```   673           by blast
```
```   674       qed
```
```   675     qed
```
```   676     show ?E if ?D
```
```   677     proof -
```
```   678       from that consider MP | PP | ?F by blast
```
```   679       then show ?thesis
```
```   680       proof cases
```
```   681         case 1
```
```   682         from minf_ex[OF mi this] show ?thesis .
```
```   683       next
```
```   684         case 2
```
```   685         from pinf_ex[OF pi this] show ?thesis .
```
```   686       next
```
```   687         case 3
```
```   688         then show ?thesis by blast
```
```   689       qed
```
```   690     qed
```
```   691   qed
```
```   692   then show "?E \<equiv> ?D" by simp
```
```   693 qed
```
```   694
```
```   695 lemmas minf_thms[no_atp] = minf_conj minf_disj minf_eq minf_neq minf_lt minf_le minf_gt minf_ge minf_P
```
```   696 lemmas pinf_thms[no_atp] = pinf_conj pinf_disj pinf_eq pinf_neq pinf_lt pinf_le pinf_gt pinf_ge pinf_P
```
```   697
```
```   698 lemmas nmi_thms[no_atp] = nmi_conj nmi_disj nmi_eq nmi_neq nmi_lt nmi_le nmi_gt nmi_ge nmi_P
```
```   699 lemmas npi_thms[no_atp] = npi_conj npi_disj npi_eq npi_neq npi_lt npi_le npi_gt npi_ge npi_P
```
```   700 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
```
```   701
```
```   702 lemma ferrack_axiom[no_atp]: "constr_dense_linorder less_eq less between"
```
```   703   by (rule constr_dense_linorder_axioms)
```
```   704
```
```   705 lemma atoms[no_atp]:
```
```   706   shows "TERM (less :: 'a \<Rightarrow> _)"
```
```   707     and "TERM (less_eq :: 'a \<Rightarrow> _)"
```
```   708     and "TERM (op = :: 'a \<Rightarrow> _)" .
```
```   709
```
```   710 declare ferrack_axiom [ferrack minf: minf_thms pinf: pinf_thms
```
```   711     nmi: nmi_thms npi: npi_thms lindense:
```
```   712     lin_dense_thms qe: fr_eq atoms: atoms]
```
```   713
```
```   714 declaration \<open>
```
```   715 let
```
```   716   fun simps phi = map (Morphism.thm phi) [@{thm "not_less"}, @{thm "not_le"}]
```
```   717   fun generic_whatis phi =
```
```   718     let
```
```   719       val [lt, le] = map (Morphism.term phi) [@{term "op \<sqsubset>"}, @{term "op \<sqsubseteq>"}]
```
```   720       fun h x t =
```
```   721         case Thm.term_of t of
```
```   722           Const(@{const_name HOL.eq}, _)\$y\$z =>
```
```   723             if Thm.term_of x aconv y then Ferrante_Rackoff_Data.Eq
```
```   724             else Ferrante_Rackoff_Data.Nox
```
```   725        | @{term "Not"}\$(Const(@{const_name HOL.eq}, _)\$y\$z) =>
```
```   726             if Thm.term_of x aconv y then Ferrante_Rackoff_Data.NEq
```
```   727             else Ferrante_Rackoff_Data.Nox
```
```   728        | b\$y\$z => if Term.could_unify (b, lt) then
```
```   729                      if Thm.term_of x aconv y then Ferrante_Rackoff_Data.Lt
```
```   730                      else if Thm.term_of x aconv z then Ferrante_Rackoff_Data.Gt
```
```   731                      else Ferrante_Rackoff_Data.Nox
```
```   732                  else if Term.could_unify (b, le) then
```
```   733                      if Thm.term_of x aconv y then Ferrante_Rackoff_Data.Le
```
```   734                      else if Thm.term_of x aconv z then Ferrante_Rackoff_Data.Ge
```
```   735                      else Ferrante_Rackoff_Data.Nox
```
```   736                  else Ferrante_Rackoff_Data.Nox
```
```   737        | _ => Ferrante_Rackoff_Data.Nox
```
```   738   in h end
```
```   739   fun ss phi ctxt =
```
```   740     simpset_of (put_simpset HOL_ss ctxt addsimps (simps phi))
```
```   741 in
```
```   742   Ferrante_Rackoff_Data.funs  @{thm "ferrack_axiom"}
```
```   743     {isolate_conv = K (K (K Thm.reflexive)), whatis = generic_whatis, simpset = ss}
```
```   744 end
```
```   745 \<close>
```
```   746
```
```   747 end
```
```   748
```
```   749 ML_file "ferrante_rackoff.ML"
```
```   750
```
```   751 method_setup ferrack = \<open>
```
```   752   Scan.succeed (SIMPLE_METHOD' o FerranteRackoff.dlo_tac)
```
```   753 \<close> "Ferrante and Rackoff's algorithm for quantifier elimination in dense linear orders"
```
```   754
```
```   755
```
```   756 subsection \<open>Ferrante and Rackoff algorithm over ordered fields\<close>
```
```   757
```
```   758 lemma neg_prod_lt:
```
```   759   fixes c :: "'a::linordered_field"
```
```   760   assumes "c < 0"
```
```   761   shows "c * x < 0 \<equiv> x > 0"
```
```   762 proof -
```
```   763   have "c * x < 0 \<longleftrightarrow> 0 / c < x"
```
```   764     by (simp only: neg_divide_less_eq[OF \<open>c < 0\<close>] algebra_simps)
```
```   765   also have "\<dots> \<longleftrightarrow> 0 < x" by simp
```
```   766   finally show "PROP ?thesis" by simp
```
```   767 qed
```
```   768
```
```   769 lemma pos_prod_lt:
```
```   770   fixes c :: "'a::linordered_field"
```
```   771   assumes "c > 0"
```
```   772   shows "c * x < 0 \<equiv> x < 0"
```
```   773 proof -
```
```   774   have "c * x < 0 \<longleftrightarrow> 0 /c > x"
```
```   775     by (simp only: pos_less_divide_eq[OF \<open>c > 0\<close>] algebra_simps)
```
```   776   also have "\<dots> \<longleftrightarrow> 0 > x" by simp
```
```   777   finally show "PROP ?thesis" by simp
```
```   778 qed
```
```   779
```
```   780 lemma neg_prod_sum_lt:
```
```   781   fixes c :: "'a::linordered_field"
```
```   782   assumes "c < 0"
```
```   783   shows "c * x + t < 0 \<equiv> x > (- 1 / c) * t"
```
```   784 proof -
```
```   785   have "c * x + t < 0 \<longleftrightarrow> c * x < - t"
```
```   786     by (subst less_iff_diff_less_0 [of "c * x" "- t"]) simp
```
```   787   also have "\<dots> \<longleftrightarrow> - t / c < x"
```
```   788     by (simp only: neg_divide_less_eq[OF \<open>c < 0\<close>] algebra_simps)
```
```   789   also have "\<dots> \<longleftrightarrow> (- 1 / c) * t < x" by simp
```
```   790   finally show "PROP ?thesis" by simp
```
```   791 qed
```
```   792
```
```   793 lemma pos_prod_sum_lt:
```
```   794   fixes c :: "'a::linordered_field"
```
```   795   assumes "c > 0"
```
```   796   shows "c * x + t < 0 \<equiv> x < (- 1 / c) * t"
```
```   797 proof -
```
```   798   have "c * x + t < 0 \<longleftrightarrow> c * x < - t"
```
```   799     by (subst less_iff_diff_less_0 [of "c * x" "- t"]) simp
```
```   800   also have "\<dots> \<longleftrightarrow> - t / c > x"
```
```   801     by (simp only: pos_less_divide_eq[OF \<open>c > 0\<close>] algebra_simps)
```
```   802   also have "\<dots> \<longleftrightarrow> (- 1 / c) * t > x" by simp
```
```   803   finally show "PROP ?thesis" by simp
```
```   804 qed
```
```   805
```
```   806 lemma sum_lt:
```
```   807   fixes x :: "'a::ordered_ab_group_add"
```
```   808   shows "x + t < 0 \<equiv> x < - t"
```
```   809   using less_diff_eq[where a= x and b=t and c=0] by simp
```
```   810
```
```   811 lemma neg_prod_le:
```
```   812   fixes c :: "'a::linordered_field"
```
```   813   assumes "c < 0"
```
```   814   shows "c * x \<le> 0 \<equiv> x \<ge> 0"
```
```   815 proof -
```
```   816   have "c * x \<le> 0 \<longleftrightarrow> 0 / c \<le> x"
```
```   817     by (simp only: neg_divide_le_eq[OF \<open>c < 0\<close>] algebra_simps)
```
```   818   also have "\<dots> \<longleftrightarrow> 0 \<le> x" by simp
```
```   819   finally show "PROP ?thesis" by simp
```
```   820 qed
```
```   821
```
```   822 lemma pos_prod_le:
```
```   823   fixes c :: "'a::linordered_field"
```
```   824   assumes "c > 0"
```
```   825   shows "c * x \<le> 0 \<equiv> x \<le> 0"
```
```   826 proof -
```
```   827   have "c * x \<le> 0 \<longleftrightarrow> 0 / c \<ge> x"
```
```   828     by (simp only: pos_le_divide_eq[OF \<open>c > 0\<close>] algebra_simps)
```
```   829   also have "\<dots> \<longleftrightarrow> 0 \<ge> x" by simp
```
```   830   finally show "PROP ?thesis" by simp
```
```   831 qed
```
```   832
```
```   833 lemma neg_prod_sum_le:
```
```   834   fixes c :: "'a::linordered_field"
```
```   835   assumes "c < 0"
```
```   836   shows "c * x + t \<le> 0 \<equiv> x \<ge> (- 1 / c) * t"
```
```   837 proof -
```
```   838   have "c * x + t \<le> 0 \<longleftrightarrow> c * x \<le> -t"
```
```   839     by (subst le_iff_diff_le_0 [of "c*x" "-t"]) simp
```
```   840   also have "\<dots> \<longleftrightarrow> - t / c \<le> x"
```
```   841     by (simp only: neg_divide_le_eq[OF \<open>c < 0\<close>] algebra_simps)
```
```   842   also have "\<dots> \<longleftrightarrow> (- 1 / c) * t \<le> x" by simp
```
```   843   finally show "PROP ?thesis" by simp
```
```   844 qed
```
```   845
```
```   846 lemma pos_prod_sum_le:
```
```   847   fixes c :: "'a::linordered_field"
```
```   848   assumes "c > 0"
```
```   849   shows "c * x + t \<le> 0 \<equiv> x \<le> (- 1 / c) * t"
```
```   850 proof -
```
```   851   have "c * x + t \<le> 0 \<longleftrightarrow> c * x \<le> - t"
```
```   852     by (subst le_iff_diff_le_0 [of "c*x" "-t"]) simp
```
```   853   also have "\<dots> \<longleftrightarrow> - t / c \<ge> x"
```
```   854     by (simp only: pos_le_divide_eq[OF \<open>c > 0\<close>] algebra_simps)
```
```   855   also have "\<dots> \<longleftrightarrow> (- 1 / c) * t \<ge> x" by simp
```
```   856   finally show "PROP ?thesis" by simp
```
```   857 qed
```
```   858
```
```   859 lemma sum_le:
```
```   860   fixes x :: "'a::ordered_ab_group_add"
```
```   861   shows "x + t \<le> 0 \<equiv> x \<le> - t"
```
```   862   using le_diff_eq[where a= x and b=t and c=0] by simp
```
```   863
```
```   864 lemma nz_prod_eq:
```
```   865   fixes c :: "'a::linordered_field"
```
```   866   assumes "c \<noteq> 0"
```
```   867   shows "c * x = 0 \<equiv> x = 0"
```
```   868   using assms by simp
```
```   869
```
```   870 lemma nz_prod_sum_eq:
```
```   871   fixes c :: "'a::linordered_field"
```
```   872   assumes "c \<noteq> 0"
```
```   873   shows "c * x + t = 0 \<equiv> x = (- 1/c) * t"
```
```   874 proof -
```
```   875   have "c * x + t = 0 \<longleftrightarrow> c * x = - t"
```
```   876     by (subst eq_iff_diff_eq_0 [of "c*x" "-t"]) simp
```
```   877   also have "\<dots> \<longleftrightarrow> x = - t / c"
```
```   878     by (simp only: nonzero_eq_divide_eq[OF \<open>c \<noteq> 0\<close>] algebra_simps)
```
```   879   finally show "PROP ?thesis" by simp
```
```   880 qed
```
```   881
```
```   882 lemma sum_eq:
```
```   883   fixes x :: "'a::ordered_ab_group_add"
```
```   884   shows "x + t = 0 \<equiv> x = - t"
```
```   885   using eq_diff_eq[where a= x and b=t and c=0] by simp
```
```   886
```
```   887 interpretation class_dense_linordered_field: constr_dense_linorder
```
```   888   "op \<le>" "op <" "\<lambda>x y. 1/2 * ((x::'a::linordered_field) + y)"
```
```   889   by unfold_locales (dlo, dlo, auto)
```
```   890
```
```   891 declaration \<open>
```
```   892 let
```
```   893   fun earlier [] x y = false
```
```   894     | earlier (h::t) x y =
```
```   895         if h aconvc y then false else if h aconvc x then true else earlier t x y;
```
```   896
```
```   897 fun dest_frac ct =
```
```   898   case Thm.term_of ct of
```
```   899     Const (@{const_name Rings.divide},_) \$ a \$ b=>
```
```   900       Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b))
```
```   901   | Const(@{const_name inverse}, _)\$a => Rat.rat_of_quotient(1, HOLogic.dest_number a |> snd)
```
```   902   | t => Rat.rat_of_int (snd (HOLogic.dest_number t))
```
```   903
```
```   904 fun whatis x ct = case Thm.term_of ct of
```
```   905   Const(@{const_name Groups.plus}, _)\$(Const(@{const_name Groups.times},_)\$_\$y)\$_ =>
```
```   906      if y aconv Thm.term_of x then ("c*x+t",[(funpow 2 Thm.dest_arg1) ct, Thm.dest_arg ct])
```
```   907      else ("Nox",[])
```
```   908 | Const(@{const_name Groups.plus}, _)\$y\$_ =>
```
```   909      if y aconv Thm.term_of x then ("x+t",[Thm.dest_arg ct])
```
```   910      else ("Nox",[])
```
```   911 | Const(@{const_name Groups.times}, _)\$_\$y =>
```
```   912      if y aconv Thm.term_of x then ("c*x",[Thm.dest_arg1 ct])
```
```   913      else ("Nox",[])
```
```   914 | t => if t aconv Thm.term_of x then ("x",[]) else ("Nox",[]);
```
```   915
```
```   916 fun xnormalize_conv ctxt [] ct = Thm.reflexive ct
```
```   917 | xnormalize_conv ctxt (vs as (x::_)) ct =
```
```   918    case Thm.term_of ct of
```
```   919    Const(@{const_name Orderings.less},_)\$_\$Const(@{const_name Groups.zero},_) =>
```
```   920     (case whatis x (Thm.dest_arg1 ct) of
```
```   921     ("c*x+t",[c,t]) =>
```
```   922        let
```
```   923         val cr = dest_frac c
```
```   924         val clt = Thm.dest_fun2 ct
```
```   925         val cz = Thm.dest_arg ct
```
```   926         val neg = cr </ Rat.zero
```
```   927         val cthp = Simplifier.rewrite ctxt
```
```   928                (Thm.apply @{cterm "Trueprop"}
```
```   929                   (if neg then Thm.apply (Thm.apply clt c) cz
```
```   930                     else Thm.apply (Thm.apply clt cz) c))
```
```   931         val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI
```
```   932         val th = Thm.implies_elim (Thm.instantiate' [SOME (Thm.ctyp_of_cterm x)] (map SOME [c,x,t])
```
```   933              (if neg then @{thm neg_prod_sum_lt} else @{thm pos_prod_sum_lt})) cth
```
```   934         val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
```
```   935                    (Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
```
```   936       in rth end
```
```   937     | ("x+t",[t]) =>
```
```   938        let
```
```   939         val T = Thm.ctyp_of_cterm x
```
```   940         val th = Thm.instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_lt"}
```
```   941         val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
```
```   942               (Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
```
```   943        in  rth end
```
```   944     | ("c*x",[c]) =>
```
```   945        let
```
```   946         val cr = dest_frac c
```
```   947         val clt = Thm.dest_fun2 ct
```
```   948         val cz = Thm.dest_arg ct
```
```   949         val neg = cr </ Rat.zero
```
```   950         val cthp = Simplifier.rewrite ctxt
```
```   951                (Thm.apply @{cterm "Trueprop"}
```
```   952                   (if neg then Thm.apply (Thm.apply clt c) cz
```
```   953                     else Thm.apply (Thm.apply clt cz) c))
```
```   954         val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI
```
```   955         val th = Thm.implies_elim (Thm.instantiate' [SOME (Thm.ctyp_of_cterm x)] (map SOME [c,x])
```
```   956              (if neg then @{thm neg_prod_lt} else @{thm pos_prod_lt})) cth
```
```   957         val rth = th
```
```   958       in rth end
```
```   959     | _ => Thm.reflexive ct)
```
```   960
```
```   961
```
```   962 |  Const(@{const_name Orderings.less_eq},_)\$_\$Const(@{const_name Groups.zero},_) =>
```
```   963    (case whatis x (Thm.dest_arg1 ct) of
```
```   964     ("c*x+t",[c,t]) =>
```
```   965        let
```
```   966         val T = Thm.typ_of_cterm x
```
```   967         val cT = Thm.ctyp_of_cterm x
```
```   968         val cr = dest_frac c
```
```   969         val clt = Thm.cterm_of ctxt (Const (@{const_name ord_class.less}, T --> T --> @{typ bool}))
```
```   970         val cz = Thm.dest_arg ct
```
```   971         val neg = cr </ Rat.zero
```
```   972         val cthp = Simplifier.rewrite ctxt
```
```   973                (Thm.apply @{cterm "Trueprop"}
```
```   974                   (if neg then Thm.apply (Thm.apply clt c) cz
```
```   975                     else Thm.apply (Thm.apply clt cz) c))
```
```   976         val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI
```
```   977         val th = Thm.implies_elim (Thm.instantiate' [SOME cT] (map SOME [c,x,t])
```
```   978              (if neg then @{thm neg_prod_sum_le} else @{thm pos_prod_sum_le})) cth
```
```   979         val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
```
```   980                    (Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
```
```   981       in rth end
```
```   982     | ("x+t",[t]) =>
```
```   983        let
```
```   984         val T = Thm.ctyp_of_cterm x
```
```   985         val th = Thm.instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_le"}
```
```   986         val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
```
```   987               (Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
```
```   988        in  rth end
```
```   989     | ("c*x",[c]) =>
```
```   990        let
```
```   991         val T = Thm.typ_of_cterm x
```
```   992         val cT = Thm.ctyp_of_cterm x
```
```   993         val cr = dest_frac c
```
```   994         val clt = Thm.cterm_of ctxt (Const (@{const_name ord_class.less}, T --> T --> @{typ bool}))
```
```   995         val cz = Thm.dest_arg ct
```
```   996         val neg = cr </ Rat.zero
```
```   997         val cthp = Simplifier.rewrite ctxt
```
```   998                (Thm.apply @{cterm "Trueprop"}
```
```   999                   (if neg then Thm.apply (Thm.apply clt c) cz
```
```  1000                     else Thm.apply (Thm.apply clt cz) c))
```
```  1001         val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI
```
```  1002         val th = Thm.implies_elim (Thm.instantiate' [SOME (Thm.ctyp_of_cterm x)] (map SOME [c,x])
```
```  1003              (if neg then @{thm neg_prod_le} else @{thm pos_prod_le})) cth
```
```  1004         val rth = th
```
```  1005       in rth end
```
```  1006     | _ => Thm.reflexive ct)
```
```  1007
```
```  1008 |  Const(@{const_name HOL.eq},_)\$_\$Const(@{const_name Groups.zero},_) =>
```
```  1009    (case whatis x (Thm.dest_arg1 ct) of
```
```  1010     ("c*x+t",[c,t]) =>
```
```  1011        let
```
```  1012         val T = Thm.ctyp_of_cterm x
```
```  1013         val cr = dest_frac c
```
```  1014         val ceq = Thm.dest_fun2 ct
```
```  1015         val cz = Thm.dest_arg ct
```
```  1016         val cthp = Simplifier.rewrite ctxt
```
```  1017             (Thm.apply @{cterm "Trueprop"}
```
```  1018              (Thm.apply @{cterm "Not"} (Thm.apply (Thm.apply ceq c) cz)))
```
```  1019         val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI
```
```  1020         val th = Thm.implies_elim
```
```  1021                  (Thm.instantiate' [SOME T] (map SOME [c,x,t]) @{thm nz_prod_sum_eq}) cth
```
```  1022         val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
```
```  1023                    (Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
```
```  1024       in rth end
```
```  1025     | ("x+t",[t]) =>
```
```  1026        let
```
```  1027         val T = Thm.ctyp_of_cterm x
```
```  1028         val th = Thm.instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_eq"}
```
```  1029         val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
```
```  1030               (Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
```
```  1031        in  rth end
```
```  1032     | ("c*x",[c]) =>
```
```  1033        let
```
```  1034         val T = Thm.ctyp_of_cterm x
```
```  1035         val cr = dest_frac c
```
```  1036         val ceq = Thm.dest_fun2 ct
```
```  1037         val cz = Thm.dest_arg ct
```
```  1038         val cthp = Simplifier.rewrite ctxt
```
```  1039             (Thm.apply @{cterm "Trueprop"}
```
```  1040              (Thm.apply @{cterm "Not"} (Thm.apply (Thm.apply ceq c) cz)))
```
```  1041         val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI
```
```  1042         val rth = Thm.implies_elim
```
```  1043                  (Thm.instantiate' [SOME T] (map SOME [c,x]) @{thm nz_prod_eq}) cth
```
```  1044       in rth end
```
```  1045     | _ => Thm.reflexive ct);
```
```  1046
```
```  1047 local
```
```  1048   val less_iff_diff_less_0 = mk_meta_eq @{thm "less_iff_diff_less_0"}
```
```  1049   val le_iff_diff_le_0 = mk_meta_eq @{thm "le_iff_diff_le_0"}
```
```  1050   val eq_iff_diff_eq_0 = mk_meta_eq @{thm "eq_iff_diff_eq_0"}
```
```  1051   val ss = simpset_of @{context}
```
```  1052 in
```
```  1053 fun field_isolate_conv phi ctxt vs ct = case Thm.term_of ct of
```
```  1054   Const(@{const_name Orderings.less},_)\$a\$b =>
```
```  1055    let val (ca,cb) = Thm.dest_binop ct
```
```  1056        val T = Thm.ctyp_of_cterm ca
```
```  1057        val th = Thm.instantiate' [SOME T] [SOME ca, SOME cb] less_iff_diff_less_0
```
```  1058        val nth = Conv.fconv_rule
```
```  1059          (Conv.arg_conv (Conv.arg1_conv
```
```  1060               (Semiring_Normalizer.semiring_normalize_ord_conv (put_simpset ss ctxt) (earlier vs)))) th
```
```  1061        val rth = Thm.transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
```
```  1062    in rth end
```
```  1063 | Const(@{const_name Orderings.less_eq},_)\$a\$b =>
```
```  1064    let val (ca,cb) = Thm.dest_binop ct
```
```  1065        val T = Thm.ctyp_of_cterm ca
```
```  1066        val th = Thm.instantiate' [SOME T] [SOME ca, SOME cb] le_iff_diff_le_0
```
```  1067        val nth = Conv.fconv_rule
```
```  1068          (Conv.arg_conv (Conv.arg1_conv
```
```  1069               (Semiring_Normalizer.semiring_normalize_ord_conv (put_simpset ss ctxt) (earlier vs)))) th
```
```  1070        val rth = Thm.transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
```
```  1071    in rth end
```
```  1072
```
```  1073 | Const(@{const_name HOL.eq},_)\$a\$b =>
```
```  1074    let val (ca,cb) = Thm.dest_binop ct
```
```  1075        val T = Thm.ctyp_of_cterm ca
```
```  1076        val th = Thm.instantiate' [SOME T] [SOME ca, SOME cb] eq_iff_diff_eq_0
```
```  1077        val nth = Conv.fconv_rule
```
```  1078          (Conv.arg_conv (Conv.arg1_conv
```
```  1079               (Semiring_Normalizer.semiring_normalize_ord_conv (put_simpset ss ctxt) (earlier vs)))) th
```
```  1080        val rth = Thm.transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
```
```  1081    in rth end
```
```  1082 | @{term "Not"} \$(Const(@{const_name HOL.eq},_)\$a\$b) => Conv.arg_conv (field_isolate_conv phi ctxt vs) ct
```
```  1083 | _ => Thm.reflexive ct
```
```  1084 end;
```
```  1085
```
```  1086 fun classfield_whatis phi =
```
```  1087  let
```
```  1088   fun h x t =
```
```  1089    case Thm.term_of t of
```
```  1090      Const(@{const_name HOL.eq}, _)\$y\$z =>
```
```  1091       if Thm.term_of x aconv y then Ferrante_Rackoff_Data.Eq
```
```  1092       else Ferrante_Rackoff_Data.Nox
```
```  1093    | @{term "Not"}\$(Const(@{const_name HOL.eq}, _)\$y\$z) =>
```
```  1094       if Thm.term_of x aconv y then Ferrante_Rackoff_Data.NEq
```
```  1095       else Ferrante_Rackoff_Data.Nox
```
```  1096    | Const(@{const_name Orderings.less},_)\$y\$z =>
```
```  1097        if Thm.term_of x aconv y then Ferrante_Rackoff_Data.Lt
```
```  1098        else if Thm.term_of x aconv z then Ferrante_Rackoff_Data.Gt
```
```  1099        else Ferrante_Rackoff_Data.Nox
```
```  1100    | Const (@{const_name Orderings.less_eq},_)\$y\$z =>
```
```  1101        if Thm.term_of x aconv y then Ferrante_Rackoff_Data.Le
```
```  1102        else if Thm.term_of x aconv z then Ferrante_Rackoff_Data.Ge
```
```  1103        else Ferrante_Rackoff_Data.Nox
```
```  1104    | _ => Ferrante_Rackoff_Data.Nox
```
```  1105  in h end;
```
```  1106 fun class_field_ss phi ctxt =
```
```  1107   simpset_of (put_simpset HOL_basic_ss ctxt
```
```  1108     addsimps ([@{thm "linorder_not_less"}, @{thm "linorder_not_le"}])
```
```  1109     |> fold Splitter.add_split [@{thm "abs_split"}, @{thm "split_max"}, @{thm "split_min"}])
```
```  1110
```
```  1111 in
```
```  1112 Ferrante_Rackoff_Data.funs @{thm "class_dense_linordered_field.ferrack_axiom"}
```
```  1113   {isolate_conv = field_isolate_conv, whatis = classfield_whatis, simpset = class_field_ss}
```
```  1114 end
```
```  1115 \<close>
```
```  1116
```
```  1117 end
```