src/HOL/Dense_Linear_Order.thy
 author wenzelm Thu Oct 04 20:29:42 2007 +0200 (2007-10-04) changeset 24850 0cfd722ab579 parent 24748 ee0a0eb6b738 child 24914 95cda5dd58d5 permissions -rw-r--r--
Name.uu, Name.aT;
```     1 (*
```
```     2     ID:         \$Id\$
```
```     3     Author:     Amine Chaieb, TU Muenchen
```
```     4 *)
```
```     5
```
```     6 header {* Dense linear order without endpoints
```
```     7   and a quantifier elimination procedure in Ferrante and Rackoff style *}
```
```     8
```
```     9 theory Dense_Linear_Order
```
```    10 imports Finite_Set
```
```    11 uses
```
```    12   "Tools/Qelim/qelim.ML"
```
```    13   "Tools/Qelim/langford_data.ML"
```
```    14   "Tools/Qelim/ferrante_rackoff_data.ML"
```
```    15   ("Tools/Qelim/langford.ML")
```
```    16   ("Tools/Qelim/ferrante_rackoff.ML")
```
```    17 begin
```
```    18
```
```    19 setup Langford_Data.setup
```
```    20 setup Ferrante_Rackoff_Data.setup
```
```    21
```
```    22 context linorder
```
```    23 begin
```
```    24
```
```    25 lemma less_not_permute: "\<not> (x \<^loc>< y \<and> y \<^loc>< x)" by (simp add: not_less linear)
```
```    26
```
```    27 lemma gather_simps:
```
```    28   shows
```
```    29   "(\<exists>x. (\<forall>y \<in> L. y \<^loc>< x) \<and> (\<forall>y \<in> U. x \<^loc>< y) \<and> x \<^loc>< u \<and> P x) \<longleftrightarrow> (\<exists>x. (\<forall>y \<in> L. y \<^loc>< x) \<and> (\<forall>y \<in> (insert u U). x \<^loc>< y) \<and> P x)"
```
```    30   and "(\<exists>x. (\<forall>y \<in> L. y \<^loc>< x) \<and> (\<forall>y \<in> U. x \<^loc>< y) \<and> l \<^loc>< x \<and> P x) \<longleftrightarrow> (\<exists>x. (\<forall>y \<in> (insert l L). y \<^loc>< x) \<and> (\<forall>y \<in> U. x \<^loc>< y) \<and> P x)"
```
```    31   "(\<exists>x. (\<forall>y \<in> L. y \<^loc>< x) \<and> (\<forall>y \<in> U. x \<^loc>< y) \<and> x \<^loc>< u) \<longleftrightarrow> (\<exists>x. (\<forall>y \<in> L. y \<^loc>< x) \<and> (\<forall>y \<in> (insert u U). x \<^loc>< y))"
```
```    32   and "(\<exists>x. (\<forall>y \<in> L. y \<^loc>< x) \<and> (\<forall>y \<in> U. x \<^loc>< y) \<and> l \<^loc>< x) \<longleftrightarrow> (\<exists>x. (\<forall>y \<in> (insert l L). y \<^loc>< x) \<and> (\<forall>y \<in> U. x \<^loc>< y))"  by auto
```
```    33
```
```    34 lemma
```
```    35   gather_start: "(\<exists>x. P x) \<equiv> (\<exists>x. (\<forall>y \<in> {}. y \<^loc>< x) \<and> (\<forall>y\<in> {}. x \<^loc>< y) \<and> P x)"
```
```    36   by simp
```
```    37
```
```    38 text{* Theorems for @{text "\<exists>z. \<forall>x. x \<^loc>< z \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>-\<infinity>\<^esub>)"}*}
```
```    39 lemma minf_lt:  "\<exists>z . \<forall>x. x \<^loc>< z \<longrightarrow> (x \<^loc>< t \<longleftrightarrow> True)" by auto
```
```    40 lemma minf_gt: "\<exists>z . \<forall>x. x \<^loc>< z \<longrightarrow>  (t \<^loc>< x \<longleftrightarrow>  False)"
```
```    41   by (simp add: not_less) (rule exI[where x="t"], auto simp add: less_le)
```
```    42
```
```    43 lemma minf_le: "\<exists>z. \<forall>x. x \<^loc>< z \<longrightarrow> (x \<^loc>\<le> t \<longleftrightarrow> True)" by (auto simp add: less_le)
```
```    44 lemma minf_ge: "\<exists>z. \<forall>x. x \<^loc>< z \<longrightarrow> (t \<^loc>\<le> x \<longleftrightarrow> False)"
```
```    45   by (auto simp add: less_le not_less not_le)
```
```    46 lemma minf_eq: "\<exists>z. \<forall>x. x \<^loc>< z \<longrightarrow> (x = t \<longleftrightarrow> False)" by auto
```
```    47 lemma minf_neq: "\<exists>z. \<forall>x. x \<^loc>< z \<longrightarrow> (x \<noteq> t \<longleftrightarrow> True)" by auto
```
```    48 lemma minf_P: "\<exists>z. \<forall>x. x \<^loc>< z \<longrightarrow> (P \<longleftrightarrow> P)" by blast
```
```    49
```
```    50 text{* Theorems for @{text "\<exists>z. \<forall>x. x \<^loc>< z \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>+\<infinity>\<^esub>)"}*}
```
```    51 lemma pinf_gt:  "\<exists>z . \<forall>x. z \<^loc>< x \<longrightarrow> (t \<^loc>< x \<longleftrightarrow> True)" by auto
```
```    52 lemma pinf_lt: "\<exists>z . \<forall>x. z \<^loc>< x \<longrightarrow>  (x \<^loc>< t \<longleftrightarrow>  False)"
```
```    53   by (simp add: not_less) (rule exI[where x="t"], auto simp add: less_le)
```
```    54
```
```    55 lemma pinf_ge: "\<exists>z. \<forall>x. z \<^loc>< x \<longrightarrow> (t \<^loc>\<le> x \<longleftrightarrow> True)" by (auto simp add: less_le)
```
```    56 lemma pinf_le: "\<exists>z. \<forall>x. z \<^loc>< x \<longrightarrow> (x \<^loc>\<le> t \<longleftrightarrow> False)"
```
```    57   by (auto simp add: less_le not_less not_le)
```
```    58 lemma pinf_eq: "\<exists>z. \<forall>x. z \<^loc>< x \<longrightarrow> (x = t \<longleftrightarrow> False)" by auto
```
```    59 lemma pinf_neq: "\<exists>z. \<forall>x. z \<^loc>< x \<longrightarrow> (x \<noteq> t \<longleftrightarrow> True)" by auto
```
```    60 lemma pinf_P: "\<exists>z. \<forall>x. z \<^loc>< x \<longrightarrow> (P \<longleftrightarrow> P)" by blast
```
```    61
```
```    62 lemma nmi_lt: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> x \<^loc>< t \<longrightarrow>  (\<exists> u\<in> U. u \<^loc>\<le> x)" by auto
```
```    63 lemma nmi_gt: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> t \<^loc>< x \<longrightarrow>  (\<exists> u\<in> U. u \<^loc>\<le> x)"
```
```    64   by (auto simp add: le_less)
```
```    65 lemma  nmi_le: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> x\<^loc>\<le> t \<longrightarrow>  (\<exists> u\<in> U. u \<^loc>\<le> x)" by auto
```
```    66 lemma  nmi_ge: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> t\<^loc>\<le> x \<longrightarrow>  (\<exists> u\<in> U. u \<^loc>\<le> x)" by auto
```
```    67 lemma  nmi_eq: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and>  x = t \<longrightarrow>  (\<exists> u\<in> U. u \<^loc>\<le> x)" by auto
```
```    68 lemma  nmi_neq: "t \<in> U \<Longrightarrow>\<forall>x. \<not>True \<and> x \<noteq> t \<longrightarrow>  (\<exists> u\<in> U. u \<^loc>\<le> x)" by auto
```
```    69 lemma  nmi_P: "\<forall> x. ~P \<and> P \<longrightarrow>  (\<exists> u\<in> U. u \<^loc>\<le> x)" by auto
```
```    70 lemma  nmi_conj: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow>  (\<exists> u\<in> U. u \<^loc>\<le> x) ;
```
```    71   \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow>  (\<exists> u\<in> U. u \<^loc>\<le> x)\<rbrakk> \<Longrightarrow>
```
```    72   \<forall>x. \<not>(P1' \<and> P2') \<and> (P1 x \<and> P2 x) \<longrightarrow>  (\<exists> u\<in> U. u \<^loc>\<le> x)" by auto
```
```    73 lemma  nmi_disj: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow>  (\<exists> u\<in> U. u \<^loc>\<le> x) ;
```
```    74   \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow>  (\<exists> u\<in> U. u \<^loc>\<le> x)\<rbrakk> \<Longrightarrow>
```
```    75   \<forall>x. \<not>(P1' \<or> P2') \<and> (P1 x \<or> P2 x) \<longrightarrow>  (\<exists> u\<in> U. u \<^loc>\<le> x)" by auto
```
```    76
```
```    77 lemma  npi_lt: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and>  x \<^loc>< t \<longrightarrow>  (\<exists> u\<in> U. x \<^loc>\<le> u)" by (auto simp add: le_less)
```
```    78 lemma  npi_gt: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> t \<^loc>< x \<longrightarrow>  (\<exists> u\<in> U. x \<^loc>\<le> u)" by auto
```
```    79 lemma  npi_le: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and>  x \<^loc>\<le> t \<longrightarrow>  (\<exists> u\<in> U. x \<^loc>\<le> u)" by auto
```
```    80 lemma  npi_ge: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> t \<^loc>\<le> x \<longrightarrow>  (\<exists> u\<in> U. x \<^loc>\<le> u)" by auto
```
```    81 lemma  npi_eq: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and>  x = t \<longrightarrow>  (\<exists> u\<in> U. x \<^loc>\<le> u)" by auto
```
```    82 lemma  npi_neq: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> x \<noteq> t \<longrightarrow>  (\<exists> u\<in> U. x \<^loc>\<le> u )" by auto
```
```    83 lemma  npi_P: "\<forall> x. ~P \<and> P \<longrightarrow>  (\<exists> u\<in> U. x \<^loc>\<le> u)" by auto
```
```    84 lemma  npi_conj: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow>  (\<exists> u\<in> U. x \<^loc>\<le> u) ;  \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow>  (\<exists> u\<in> U. x \<^loc>\<le> u)\<rbrakk>
```
```    85   \<Longrightarrow>  \<forall>x. \<not>(P1' \<and> P2') \<and> (P1 x \<and> P2 x) \<longrightarrow>  (\<exists> u\<in> U. x \<^loc>\<le> u)" by auto
```
```    86 lemma  npi_disj: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow>  (\<exists> u\<in> U. x \<^loc>\<le> u) ; \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow>  (\<exists> u\<in> U. x \<^loc>\<le> u)\<rbrakk>
```
```    87   \<Longrightarrow> \<forall>x. \<not>(P1' \<or> P2') \<and> (P1 x \<or> P2 x) \<longrightarrow>  (\<exists> u\<in> U. x \<^loc>\<le> u)" by auto
```
```    88
```
```    89 lemma lin_dense_lt: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l \<^loc>< t \<and> t \<^loc>< u \<longrightarrow> t \<notin> U) \<and> l\<^loc>< x \<and> x \<^loc>< u \<and> x \<^loc>< t \<longrightarrow> (\<forall> y. l \<^loc>< y \<and> y \<^loc>< u \<longrightarrow> y \<^loc>< t)"
```
```    90 proof(clarsimp)
```
```    91   fix x l u y  assume tU: "t \<in> U" and noU: "\<forall>t. l \<^loc>< t \<and> t \<^loc>< u \<longrightarrow> t \<notin> U" and lx: "l \<^loc>< x"
```
```    92     and xu: "x\<^loc><u"  and px: "x \<^loc>< t" and ly: "l\<^loc><y" and yu:"y \<^loc>< u"
```
```    93   from tU noU ly yu have tny: "t\<noteq>y" by auto
```
```    94   {assume H: "t \<^loc>< y"
```
```    95     from less_trans[OF lx px] less_trans[OF H yu]
```
```    96     have "l \<^loc>< t \<and> t \<^loc>< u"  by simp
```
```    97     with tU noU have "False" by auto}
```
```    98   hence "\<not> t \<^loc>< y"  by auto hence "y \<^loc>\<le> t" by (simp add: not_less)
```
```    99   thus "y \<^loc>< t" using tny by (simp add: less_le)
```
```   100 qed
```
```   101
```
```   102 lemma lin_dense_gt: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l \<^loc>< t \<and> t\<^loc>< u \<longrightarrow> t \<notin> U) \<and> l \<^loc>< x \<and> x \<^loc>< u \<and> t \<^loc>< x \<longrightarrow> (\<forall> y. l \<^loc>< y \<and> y \<^loc>< u \<longrightarrow> t \<^loc>< y)"
```
```   103 proof(clarsimp)
```
```   104   fix x l u y
```
```   105   assume tU: "t \<in> U" and noU: "\<forall>t. l \<^loc>< t \<and> t \<^loc>< u \<longrightarrow> t \<notin> U" and lx: "l \<^loc>< x" and xu: "x\<^loc><u"
```
```   106   and px: "t \<^loc>< x" and ly: "l\<^loc><y" and yu:"y \<^loc>< u"
```
```   107   from tU noU ly yu have tny: "t\<noteq>y" by auto
```
```   108   {assume H: "y\<^loc>< t"
```
```   109     from less_trans[OF ly H] less_trans[OF px xu] have "l \<^loc>< t \<and> t \<^loc>< u" by simp
```
```   110     with tU noU have "False" by auto}
```
```   111   hence "\<not> y\<^loc><t"  by auto hence "t \<^loc>\<le> y" by (auto simp add: not_less)
```
```   112   thus "t \<^loc>< y" using tny by (simp add:less_le)
```
```   113 qed
```
```   114
```
```   115 lemma lin_dense_le: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l \<^loc>< t \<and> t\<^loc>< u \<longrightarrow> t \<notin> U) \<and> l\<^loc>< x \<and> x \<^loc>< u \<and> x \<^loc>\<le> t \<longrightarrow> (\<forall> y. l \<^loc>< y \<and> y \<^loc>< u \<longrightarrow> y\<^loc>\<le> t)"
```
```   116 proof(clarsimp)
```
```   117   fix x l u y
```
```   118   assume tU: "t \<in> U" and noU: "\<forall>t. l \<^loc>< t \<and> t \<^loc>< u \<longrightarrow> t \<notin> U" and lx: "l \<^loc>< x" and xu: "x\<^loc><u"
```
```   119   and px: "x \<^loc>\<le> t" and ly: "l\<^loc><y" and yu:"y \<^loc>< u"
```
```   120   from tU noU ly yu have tny: "t\<noteq>y" by auto
```
```   121   {assume H: "t \<^loc>< y"
```
```   122     from less_le_trans[OF lx px] less_trans[OF H yu]
```
```   123     have "l \<^loc>< t \<and> t \<^loc>< u" by simp
```
```   124     with tU noU have "False" by auto}
```
```   125   hence "\<not> t \<^loc>< y"  by auto thus "y \<^loc>\<le> t" by (simp add: not_less)
```
```   126 qed
```
```   127
```
```   128 lemma lin_dense_ge: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l \<^loc>< t \<and> t\<^loc>< u \<longrightarrow> t \<notin> U) \<and> l\<^loc>< x \<and> x \<^loc>< u \<and> t \<^loc>\<le> x \<longrightarrow> (\<forall> y. l \<^loc>< y \<and> y \<^loc>< u \<longrightarrow> t \<^loc>\<le> y)"
```
```   129 proof(clarsimp)
```
```   130   fix x l u y
```
```   131   assume tU: "t \<in> U" and noU: "\<forall>t. l \<^loc>< t \<and> t \<^loc>< u \<longrightarrow> t \<notin> U" and lx: "l \<^loc>< x" and xu: "x\<^loc><u"
```
```   132   and px: "t \<^loc>\<le> x" and ly: "l\<^loc><y" and yu:"y \<^loc>< u"
```
```   133   from tU noU ly yu have tny: "t\<noteq>y" by auto
```
```   134   {assume H: "y\<^loc>< t"
```
```   135     from less_trans[OF ly H] le_less_trans[OF px xu]
```
```   136     have "l \<^loc>< t \<and> t \<^loc>< u" by simp
```
```   137     with tU noU have "False" by auto}
```
```   138   hence "\<not> y\<^loc><t"  by auto thus "t \<^loc>\<le> y" by (simp add: not_less)
```
```   139 qed
```
```   140 lemma lin_dense_eq: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l \<^loc>< t \<and> t\<^loc>< u \<longrightarrow> t \<notin> U) \<and> l\<^loc>< x \<and> x \<^loc>< u \<and> x = t   \<longrightarrow> (\<forall> y. l \<^loc>< y \<and> y \<^loc>< u \<longrightarrow> y= t)"  by auto
```
```   141 lemma lin_dense_neq: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l \<^loc>< t \<and> t\<^loc>< u \<longrightarrow> t \<notin> U) \<and> l\<^loc>< x \<and> x \<^loc>< u \<and> x \<noteq> t   \<longrightarrow> (\<forall> y. l \<^loc>< y \<and> y \<^loc>< u \<longrightarrow> y\<noteq> t)"  by auto
```
```   142 lemma lin_dense_P: "\<forall>x l u. (\<forall> t. l \<^loc>< t \<and> t\<^loc>< u \<longrightarrow> t \<notin> U) \<and> l\<^loc>< x \<and> x \<^loc>< u \<and> P   \<longrightarrow> (\<forall> y. l \<^loc>< y \<and> y \<^loc>< u \<longrightarrow> P)"  by auto
```
```   143
```
```   144 lemma lin_dense_conj:
```
```   145   "\<lbrakk>\<forall>x l u. (\<forall> t. l \<^loc>< t \<and> t\<^loc>< u \<longrightarrow> t \<notin> U) \<and> l\<^loc>< x \<and> x \<^loc>< u \<and> P1 x
```
```   146   \<longrightarrow> (\<forall> y. l \<^loc>< y \<and> y \<^loc>< u \<longrightarrow> P1 y) ;
```
```   147   \<forall>x l u. (\<forall> t. l \<^loc>< t \<and> t\<^loc>< u \<longrightarrow> t \<notin> U) \<and> l\<^loc>< x \<and> x \<^loc>< u \<and> P2 x
```
```   148   \<longrightarrow> (\<forall> y. l \<^loc>< y \<and> y \<^loc>< u \<longrightarrow> P2 y)\<rbrakk> \<Longrightarrow>
```
```   149   \<forall>x l u. (\<forall> t. l \<^loc>< t \<and> t\<^loc>< u \<longrightarrow> t \<notin> U) \<and> l\<^loc>< x \<and> x \<^loc>< u \<and> (P1 x \<and> P2 x)
```
```   150   \<longrightarrow> (\<forall> y. l \<^loc>< y \<and> y \<^loc>< u \<longrightarrow> (P1 y \<and> P2 y))"
```
```   151   by blast
```
```   152 lemma lin_dense_disj:
```
```   153   "\<lbrakk>\<forall>x l u. (\<forall> t. l \<^loc>< t \<and> t\<^loc>< u \<longrightarrow> t \<notin> U) \<and> l\<^loc>< x \<and> x \<^loc>< u \<and> P1 x
```
```   154   \<longrightarrow> (\<forall> y. l \<^loc>< y \<and> y \<^loc>< u \<longrightarrow> P1 y) ;
```
```   155   \<forall>x l u. (\<forall> t. l \<^loc>< t \<and> t\<^loc>< u \<longrightarrow> t \<notin> U) \<and> l\<^loc>< x \<and> x \<^loc>< u \<and> P2 x
```
```   156   \<longrightarrow> (\<forall> y. l \<^loc>< y \<and> y \<^loc>< u \<longrightarrow> P2 y)\<rbrakk> \<Longrightarrow>
```
```   157   \<forall>x l u. (\<forall> t. l \<^loc>< t \<and> t\<^loc>< u \<longrightarrow> t \<notin> U) \<and> l\<^loc>< x \<and> x \<^loc>< u \<and> (P1 x \<or> P2 x)
```
```   158   \<longrightarrow> (\<forall> y. l \<^loc>< y \<and> y \<^loc>< u \<longrightarrow> (P1 y \<or> P2 y))"
```
```   159   by blast
```
```   160
```
```   161 lemma npmibnd: "\<lbrakk>\<forall>x. \<not> MP \<and> P x \<longrightarrow> (\<exists> u\<in> U. u \<^loc>\<le> x); \<forall>x. \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. x \<^loc>\<le> u)\<rbrakk>
```
```   162   \<Longrightarrow> \<forall>x. \<not> MP \<and> \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. \<exists> u' \<in> U. u \<^loc>\<le> x \<and> x \<^loc>\<le> u')"
```
```   163 by auto
```
```   164
```
```   165 lemma finite_set_intervals:
```
```   166   assumes px: "P x" and lx: "l \<^loc>\<le> x" and xu: "x \<^loc>\<le> u" and linS: "l\<in> S"
```
```   167   and uinS: "u \<in> S" and fS:"finite S" and lS: "\<forall> x\<in> S. l \<^loc>\<le> x" and Su: "\<forall> x\<in> S. x \<^loc>\<le> u"
```
```   168   shows "\<exists> a \<in> S. \<exists> b \<in> S. (\<forall> y. a \<^loc>< y \<and> y \<^loc>< b \<longrightarrow> y \<notin> S) \<and> a \<^loc>\<le> x \<and> x \<^loc>\<le> b \<and> P x"
```
```   169 proof-
```
```   170   let ?Mx = "{y. y\<in> S \<and> y \<^loc>\<le> x}"
```
```   171   let ?xM = "{y. y\<in> S \<and> x \<^loc>\<le> y}"
```
```   172   let ?a = "Max ?Mx"
```
```   173   let ?b = "Min ?xM"
```
```   174   have MxS: "?Mx \<subseteq> S" by blast
```
```   175   hence fMx: "finite ?Mx" using fS finite_subset by auto
```
```   176   from lx linS have linMx: "l \<in> ?Mx" by blast
```
```   177   hence Mxne: "?Mx \<noteq> {}" by blast
```
```   178   have xMS: "?xM \<subseteq> S" by blast
```
```   179   hence fxM: "finite ?xM" using fS finite_subset by auto
```
```   180   from xu uinS have linxM: "u \<in> ?xM" by blast
```
```   181   hence xMne: "?xM \<noteq> {}" by blast
```
```   182   have ax:"?a \<^loc>\<le> x" using Mxne fMx by auto
```
```   183   have xb:"x \<^loc>\<le> ?b" using xMne fxM by auto
```
```   184   have "?a \<in> ?Mx" using Max_in[OF fMx Mxne] by simp hence ainS: "?a \<in> S" using MxS by blast
```
```   185   have "?b \<in> ?xM" using Min_in[OF fxM xMne] by simp hence binS: "?b \<in> S" using xMS by blast
```
```   186   have noy:"\<forall> y. ?a \<^loc>< y \<and> y \<^loc>< ?b \<longrightarrow> y \<notin> S"
```
```   187   proof(clarsimp)
```
```   188     fix y   assume ay: "?a \<^loc>< y" and yb: "y \<^loc>< ?b" and yS: "y \<in> S"
```
```   189     from yS have "y\<in> ?Mx \<or> y\<in> ?xM" by (auto simp add: linear)
```
```   190     moreover {assume "y \<in> ?Mx" hence "y \<^loc>\<le> ?a" using Mxne fMx by auto with ay have "False" by (simp add: not_le[symmetric])}
```
```   191     moreover {assume "y \<in> ?xM" hence "?b \<^loc>\<le> y" using xMne fxM by auto with yb have "False" by (simp add: not_le[symmetric])}
```
```   192     ultimately show "False" by blast
```
```   193   qed
```
```   194   from ainS binS noy ax xb px show ?thesis by blast
```
```   195 qed
```
```   196
```
```   197 lemma finite_set_intervals2:
```
```   198   assumes px: "P x" and lx: "l \<^loc>\<le> x" and xu: "x \<^loc>\<le> u" and linS: "l\<in> S"
```
```   199   and uinS: "u \<in> S" and fS:"finite S" and lS: "\<forall> x\<in> S. l \<^loc>\<le> x" and Su: "\<forall> x\<in> S. x \<^loc>\<le> u"
```
```   200   shows "(\<exists> s\<in> S. P s) \<or> (\<exists> a \<in> S. \<exists> b \<in> S. (\<forall> y. a \<^loc>< y \<and> y \<^loc>< b \<longrightarrow> y \<notin> S) \<and> a \<^loc>< x \<and> x \<^loc>< b \<and> P x)"
```
```   201 proof-
```
```   202   from finite_set_intervals[where P="P", OF px lx xu linS uinS fS lS Su]
```
```   203   obtain a and b where
```
```   204     as: "a\<in> S" and bs: "b\<in> S" and noS:"\<forall>y. a \<^loc>< y \<and> y \<^loc>< b \<longrightarrow> y \<notin> S"
```
```   205     and axb: "a \<^loc>\<le> x \<and> x \<^loc>\<le> b \<and> P x"  by auto
```
```   206   from axb have "x= a \<or> x= b \<or> (a \<^loc>< x \<and> x \<^loc>< b)" by (auto simp add: le_less)
```
```   207   thus ?thesis using px as bs noS by blast
```
```   208 qed
```
```   209
```
```   210 end
```
```   211
```
```   212 section {* The classical QE after Langford for dense linear orders *}
```
```   213
```
```   214 context dense_linear_order
```
```   215 begin
```
```   216
```
```   217 lemma dlo_qe_bnds:
```
```   218   assumes ne: "L \<noteq> {}" and neU: "U \<noteq> {}" and fL: "finite L" and fU: "finite U"
```
```   219   shows "(\<exists>x. (\<forall>y \<in> L. y \<^loc>< x) \<and> (\<forall>y \<in> U. x \<^loc>< y)) \<equiv> (\<forall> l \<in> L. \<forall>u \<in> U. l \<^loc>< u)"
```
```   220 proof (simp only: atomize_eq, rule iffI)
```
```   221   assume H: "\<exists>x. (\<forall>y\<in>L. y \<^loc>< x) \<and> (\<forall>y\<in>U. x \<^loc>< y)"
```
```   222   then obtain x where xL: "\<forall>y\<in>L. y \<^loc>< x" and xU: "\<forall>y\<in>U. x \<^loc>< y" by blast
```
```   223   {fix l u assume l: "l \<in> L" and u: "u \<in> U"
```
```   224     from less_trans[OF xL[rule_format, OF l] xU[rule_format, OF u]]
```
```   225     have "l \<^loc>< u" .}
```
```   226   thus "\<forall>l\<in>L. \<forall>u\<in>U. l \<^loc>< u" by blast
```
```   227 next
```
```   228   assume H: "\<forall>l\<in>L. \<forall>u\<in>U. l \<^loc>< u"
```
```   229   let ?ML = "Max L"
```
```   230   let ?MU = "Min U"
```
```   231   from fL ne have th1: "?ML \<in> L" and th1': "\<forall>l\<in>L. l \<^loc>\<le> ?ML" by auto
```
```   232   from fU neU have th2: "?MU \<in> U" and th2': "\<forall>u\<in>U. ?MU \<^loc>\<le> u" by auto
```
```   233   from th1 th2 H have "?ML \<^loc>< ?MU" by auto
```
```   234   with dense obtain w where th3: "?ML \<^loc>< w" and th4: "w \<^loc>< ?MU" by blast
```
```   235   from th3 th1' have "\<forall>l \<in> L. l \<^loc>< w" by auto
```
```   236   moreover from th4 th2' have "\<forall>u \<in> U. w \<^loc>< u" by auto
```
```   237   ultimately show "\<exists>x. (\<forall>y\<in>L. y \<^loc>< x) \<and> (\<forall>y\<in>U. x \<^loc>< y)" by auto
```
```   238 qed
```
```   239
```
```   240 lemma dlo_qe_noub:
```
```   241   assumes ne: "L \<noteq> {}" and fL: "finite L"
```
```   242   shows "(\<exists>x. (\<forall>y \<in> L. y \<^loc>< x) \<and> (\<forall>y \<in> {}. x \<^loc>< y)) \<equiv> True"
```
```   243 proof(simp add: atomize_eq)
```
```   244   from gt_ex[rule_format, of "Max L"] obtain M where M: "Max L \<^loc>< M" by blast
```
```   245   from ne fL have "\<forall>x \<in> L. x \<^loc>\<le> Max L" by simp
```
```   246   with M have "\<forall>x\<in>L. x \<^loc>< M" by (auto intro: le_less_trans)
```
```   247   thus "\<exists>x. \<forall>y\<in>L. y \<^loc>< x" by blast
```
```   248 qed
```
```   249
```
```   250 lemma dlo_qe_nolb:
```
```   251   assumes ne: "U \<noteq> {}" and fU: "finite U"
```
```   252   shows "(\<exists>x. (\<forall>y \<in> {}. y \<^loc>< x) \<and> (\<forall>y \<in> U. x \<^loc>< y)) \<equiv> True"
```
```   253 proof(simp add: atomize_eq)
```
```   254   from lt_ex[rule_format, of "Min U"] obtain M where M: "M \<^loc>< Min U" by blast
```
```   255   from ne fU have "\<forall>x \<in> U. Min U \<^loc>\<le> x" by simp
```
```   256   with M have "\<forall>x\<in>U. M \<^loc>< x" by (auto intro: less_le_trans)
```
```   257   thus "\<exists>x. \<forall>y\<in>U. x \<^loc>< y" by blast
```
```   258 qed
```
```   259
```
```   260 lemma exists_neq: "\<exists>(x::'a). x \<noteq> t" "\<exists>(x::'a). t \<noteq> x"
```
```   261   using gt_ex[rule_format, of t] by auto
```
```   262
```
```   263 lemmas dlo_simps = order_refl less_irrefl not_less not_le exists_neq
```
```   264   le_less neq_iff linear less_not_permute
```
```   265
```
```   266 lemma axiom: "dense_linear_order (op \<^loc>\<le>) (op \<^loc><)" .
```
```   267 lemma atoms:
```
```   268   includes meta_term_syntax
```
```   269   shows "TERM (less :: 'a \<Rightarrow> _)"
```
```   270     and "TERM (less_eq :: 'a \<Rightarrow> _)"
```
```   271     and "TERM (op = :: 'a \<Rightarrow> _)" .
```
```   272
```
```   273 declare axiom[langford qe: dlo_qe_bnds dlo_qe_nolb dlo_qe_noub gather: gather_start gather_simps atoms: atoms]
```
```   274 declare dlo_simps[langfordsimp]
```
```   275
```
```   276 end
```
```   277
```
```   278 (* FIXME: Move to HOL -- together with the conj_aci_rule in langford.ML *)
```
```   279 lemma dnf:
```
```   280   "(P & (Q | R)) = ((P&Q) | (P&R))"
```
```   281   "((Q | R) & P) = ((Q&P) | (R&P))"
```
```   282   by blast+
```
```   283
```
```   284 lemmas weak_dnf_simps = simp_thms dnf
```
```   285
```
```   286 lemma nnf_simps:
```
```   287     "(\<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)"
```
```   288     "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not> \<not>(P)) = P"
```
```   289   by blast+
```
```   290
```
```   291 lemma ex_distrib: "(\<exists>x. P x \<or> Q x) \<longleftrightarrow> ((\<exists>x. P x) \<or> (\<exists>x. Q x))" by blast
```
```   292
```
```   293 lemmas dnf_simps = weak_dnf_simps nnf_simps ex_distrib
```
```   294
```
```   295 use "Tools/Qelim/langford.ML"
```
```   296 method_setup dlo = {*
```
```   297   Method.ctxt_args (Method.SIMPLE_METHOD' o LangfordQE.dlo_tac)
```
```   298 *} "Langford's algorithm for quantifier elimination in dense linear orders"
```
```   299
```
```   300
```
```   301 section {* Contructive dense linear orders yield QE for linear arithmetic over ordered Fields -- see @{text "Arith_Tools.thy"} *}
```
```   302
```
```   303 text {* Linear order without upper bounds *}
```
```   304
```
```   305 class linorder_no_ub = linorder +
```
```   306   assumes gt_ex: "\<exists>y. x \<^loc>< y"
```
```   307 begin
```
```   308
```
```   309 lemma ge_ex: "\<exists>y. x \<^loc>\<le> y" using gt_ex by auto
```
```   310
```
```   311 text {* Theorems for @{text "\<exists>z. \<forall>x. z \<^loc>< x \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>+\<infinity>\<^esub>)"} *}
```
```   312 lemma pinf_conj:
```
```   313   assumes ex1: "\<exists>z1. \<forall>x. z1 \<^loc>< x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
```
```   314   and ex2: "\<exists>z2. \<forall>x. z2 \<^loc>< x \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
```
```   315   shows "\<exists>z. \<forall>x. z \<^loc><  x \<longrightarrow> ((P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2'))"
```
```   316 proof-
```
```   317   from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. z1 \<^loc>< x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
```
```   318      and z2: "\<forall>x. z2 \<^loc>< x \<longrightarrow> (P2 x \<longleftrightarrow> P2')" by blast
```
```   319   from gt_ex obtain z where z:"max z1 z2 \<^loc>< z" by blast
```
```   320   from z have zz1: "z1 \<^loc>< z" and zz2: "z2 \<^loc>< z" by simp_all
```
```   321   {fix x assume H: "z \<^loc>< x"
```
```   322     from less_trans[OF zz1 H] less_trans[OF zz2 H]
```
```   323     have "(P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2')"  using z1 zz1 z2 zz2 by auto
```
```   324   }
```
```   325   thus ?thesis by blast
```
```   326 qed
```
```   327
```
```   328 lemma pinf_disj:
```
```   329   assumes ex1: "\<exists>z1. \<forall>x. z1 \<^loc>< x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
```
```   330   and ex2: "\<exists>z2. \<forall>x. z2 \<^loc>< x \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
```
```   331   shows "\<exists>z. \<forall>x. z \<^loc><  x \<longrightarrow> ((P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2'))"
```
```   332 proof-
```
```   333   from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. z1 \<^loc>< x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
```
```   334      and z2: "\<forall>x. z2 \<^loc>< x \<longrightarrow> (P2 x \<longleftrightarrow> P2')" by blast
```
```   335   from gt_ex obtain z where z:"max z1 z2 \<^loc>< z" by blast
```
```   336   from z have zz1: "z1 \<^loc>< z" and zz2: "z2 \<^loc>< z" by simp_all
```
```   337   {fix x assume H: "z \<^loc>< x"
```
```   338     from less_trans[OF zz1 H] less_trans[OF zz2 H]
```
```   339     have "(P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2')"  using z1 zz1 z2 zz2 by auto
```
```   340   }
```
```   341   thus ?thesis by blast
```
```   342 qed
```
```   343
```
```   344 lemma pinf_ex: assumes ex:"\<exists>z. \<forall>x. z \<^loc>< x \<longrightarrow> (P x \<longleftrightarrow> P1)" and p1: P1 shows "\<exists> x. P x"
```
```   345 proof-
```
```   346   from ex obtain z where z: "\<forall>x. z \<^loc>< x \<longrightarrow> (P x \<longleftrightarrow> P1)" by blast
```
```   347   from gt_ex obtain x where x: "z \<^loc>< x" by blast
```
```   348   from z x p1 show ?thesis by blast
```
```   349 qed
```
```   350
```
```   351 end
```
```   352
```
```   353 text {* Linear order without upper bounds *}
```
```   354
```
```   355 class linorder_no_lb = linorder +
```
```   356   assumes lt_ex: "\<exists>y. y \<^loc>< x"
```
```   357 begin
```
```   358
```
```   359 lemma le_ex: "\<exists>y. y \<^loc>\<le> x" using lt_ex by auto
```
```   360
```
```   361
```
```   362 text {* Theorems for @{text "\<exists>z. \<forall>x. x \<^loc>< z \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>-\<infinity>\<^esub>)"} *}
```
```   363 lemma minf_conj:
```
```   364   assumes ex1: "\<exists>z1. \<forall>x. x \<^loc>< z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
```
```   365   and ex2: "\<exists>z2. \<forall>x. x \<^loc>< z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
```
```   366   shows "\<exists>z. \<forall>x. x \<^loc><  z \<longrightarrow> ((P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2'))"
```
```   367 proof-
```
```   368   from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. x \<^loc>< z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"and z2: "\<forall>x. x \<^loc>< z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')" by blast
```
```   369   from lt_ex obtain z where z:"z \<^loc>< min z1 z2" by blast
```
```   370   from z have zz1: "z \<^loc>< z1" and zz2: "z \<^loc>< z2" by simp_all
```
```   371   {fix x assume H: "x \<^loc>< z"
```
```   372     from less_trans[OF H zz1] less_trans[OF H zz2]
```
```   373     have "(P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2')"  using z1 zz1 z2 zz2 by auto
```
```   374   }
```
```   375   thus ?thesis by blast
```
```   376 qed
```
```   377
```
```   378 lemma minf_disj:
```
```   379   assumes ex1: "\<exists>z1. \<forall>x. x \<^loc>< z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
```
```   380   and ex2: "\<exists>z2. \<forall>x. x \<^loc>< z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
```
```   381   shows "\<exists>z. \<forall>x. x \<^loc><  z \<longrightarrow> ((P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2'))"
```
```   382 proof-
```
```   383   from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. x \<^loc>< z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"and z2: "\<forall>x. x \<^loc>< z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')" by blast
```
```   384   from lt_ex obtain z where z:"z \<^loc>< min z1 z2" by blast
```
```   385   from z have zz1: "z \<^loc>< z1" and zz2: "z \<^loc>< z2" by simp_all
```
```   386   {fix x assume H: "x \<^loc>< z"
```
```   387     from less_trans[OF H zz1] less_trans[OF H zz2]
```
```   388     have "(P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2')"  using z1 zz1 z2 zz2 by auto
```
```   389   }
```
```   390   thus ?thesis by blast
```
```   391 qed
```
```   392
```
```   393 lemma minf_ex: assumes ex:"\<exists>z. \<forall>x. x \<^loc>< z \<longrightarrow> (P x \<longleftrightarrow> P1)" and p1: P1 shows "\<exists> x. P x"
```
```   394 proof-
```
```   395   from ex obtain z where z: "\<forall>x. x \<^loc>< z \<longrightarrow> (P x \<longleftrightarrow> P1)" by blast
```
```   396   from lt_ex obtain x where x: "x \<^loc>< z" by blast
```
```   397   from z x p1 show ?thesis by blast
```
```   398 qed
```
```   399
```
```   400 end
```
```   401
```
```   402
```
```   403 class constr_dense_linear_order = linorder_no_lb + linorder_no_ub +
```
```   404   fixes between
```
```   405   assumes between_less: "x \<^loc>< y \<Longrightarrow> x \<^loc>< between x y \<and> between x y \<^loc>< y"
```
```   406      and  between_same: "between x x = x"
```
```   407
```
```   408 instance advanced constr_dense_linear_order < dense_linear_order
```
```   409   apply unfold_locales
```
```   410   using gt_ex lt_ex between_less
```
```   411     by (auto, rule_tac x="between x y" in exI, simp)
```
```   412 (*FIXME*)
```
```   413 lemmas gt_ex = dense_linear_order_class.less_eq_less.gt_ex
```
```   414 lemmas lt_ex = dense_linear_order_class.less_eq_less.lt_ex
```
```   415 lemmas dense = dense_linear_order_class.less_eq_less.dense
```
```   416
```
```   417 context constr_dense_linear_order
```
```   418 begin
```
```   419
```
```   420 lemma rinf_U:
```
```   421   assumes fU: "finite U"
```
```   422   and lin_dense: "\<forall>x l u. (\<forall> t. l \<^loc>< t \<and> t\<^loc>< u \<longrightarrow> t \<notin> U) \<and> l\<^loc>< x \<and> x \<^loc>< u \<and> P x
```
```   423   \<longrightarrow> (\<forall> y. l \<^loc>< y \<and> y \<^loc>< u \<longrightarrow> P y )"
```
```   424   and nmpiU: "\<forall>x. \<not> MP \<and> \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. \<exists> u' \<in> U. u \<^loc>\<le> x \<and> x \<^loc>\<le> u')"
```
```   425   and nmi: "\<not> MP"  and npi: "\<not> PP"  and ex: "\<exists> x.  P x"
```
```   426   shows "\<exists> u\<in> U. \<exists> u' \<in> U. P (between u u')"
```
```   427 proof-
```
```   428   from ex obtain x where px: "P x" by blast
```
```   429   from px nmi npi nmpiU have "\<exists> u\<in> U. \<exists> u' \<in> U. u \<^loc>\<le> x \<and> x \<^loc>\<le> u'" by auto
```
```   430   then obtain u and u' where uU:"u\<in> U" and uU': "u' \<in> U" and ux:"u \<^loc>\<le> x" and xu':"x \<^loc>\<le> u'" by auto
```
```   431   from uU have Une: "U \<noteq> {}" by auto
```
```   432   let ?l = "Min U"
```
```   433   let ?u = "Max U"
```
```   434   have linM: "?l \<in> U" using fU Une by simp
```
```   435   have uinM: "?u \<in> U" using fU Une by simp
```
```   436   have lM: "\<forall> t\<in> U. ?l \<^loc>\<le> t" using Une fU by auto
```
```   437   have Mu: "\<forall> t\<in> U. t \<^loc>\<le> ?u" using Une fU by auto
```
```   438   have th:"?l \<^loc>\<le> u" using uU Une lM by auto
```
```   439   from order_trans[OF th ux] have lx: "?l \<^loc>\<le> x" .
```
```   440   have th: "u' \<^loc>\<le> ?u" using uU' Une Mu by simp
```
```   441   from order_trans[OF xu' th] have xu: "x \<^loc>\<le> ?u" .
```
```   442   from finite_set_intervals2[where P="P",OF px lx xu linM uinM fU lM Mu]
```
```   443   have "(\<exists> s\<in> U. P s) \<or>
```
```   444       (\<exists> t1\<in> U. \<exists> t2 \<in> U. (\<forall> y. t1 \<^loc>< y \<and> y \<^loc>< t2 \<longrightarrow> y \<notin> U) \<and> t1 \<^loc>< x \<and> x \<^loc>< t2 \<and> P x)" .
```
```   445   moreover { fix u assume um: "u\<in>U" and pu: "P u"
```
```   446     have "between u u = u" by (simp add: between_same)
```
```   447     with um pu have "P (between u u)" by simp
```
```   448     with um have ?thesis by blast}
```
```   449   moreover{
```
```   450     assume "\<exists> t1\<in> U. \<exists> t2 \<in> U. (\<forall> y. t1 \<^loc>< y \<and> y \<^loc>< t2 \<longrightarrow> y \<notin> U) \<and> t1 \<^loc>< x \<and> x \<^loc>< t2 \<and> P x"
```
```   451       then obtain t1 and t2 where t1M: "t1 \<in> U" and t2M: "t2\<in> U"
```
```   452         and noM: "\<forall> y. t1 \<^loc>< y \<and> y \<^loc>< t2 \<longrightarrow> y \<notin> U" and t1x: "t1 \<^loc>< x" and xt2: "x \<^loc>< t2" and px: "P x"
```
```   453         by blast
```
```   454       from less_trans[OF t1x xt2] have t1t2: "t1 \<^loc>< t2" .
```
```   455       let ?u = "between t1 t2"
```
```   456       from between_less t1t2 have t1lu: "t1 \<^loc>< ?u" and ut2: "?u \<^loc>< t2" by auto
```
```   457       from lin_dense[rule_format, OF] noM t1x xt2 px t1lu ut2 have "P ?u" by blast
```
```   458       with t1M t2M have ?thesis by blast}
```
```   459     ultimately show ?thesis by blast
```
```   460   qed
```
```   461
```
```   462 theorem fr_eq:
```
```   463   assumes fU: "finite U"
```
```   464   and lin_dense: "\<forall>x l u. (\<forall> t. l \<^loc>< t \<and> t\<^loc>< u \<longrightarrow> t \<notin> U) \<and> l\<^loc>< x \<and> x \<^loc>< u \<and> P x
```
```   465    \<longrightarrow> (\<forall> y. l \<^loc>< y \<and> y \<^loc>< u \<longrightarrow> P y )"
```
```   466   and nmibnd: "\<forall>x. \<not> MP \<and> P x \<longrightarrow> (\<exists> u\<in> U. u \<^loc>\<le> x)"
```
```   467   and npibnd: "\<forall>x. \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. x \<^loc>\<le> u)"
```
```   468   and mi: "\<exists>z. \<forall>x. x \<^loc>< z \<longrightarrow> (P x = MP)"  and pi: "\<exists>z. \<forall>x. z \<^loc>< x \<longrightarrow> (P x = PP)"
```
```   469   shows "(\<exists> x. P x) \<equiv> (MP \<or> PP \<or> (\<exists> u \<in> U. \<exists> u'\<in> U. P (between u u')))"
```
```   470   (is "_ \<equiv> (_ \<or> _ \<or> ?F)" is "?E \<equiv> ?D")
```
```   471 proof-
```
```   472  {
```
```   473    assume px: "\<exists> x. P x"
```
```   474    have "MP \<or> PP \<or> (\<not> MP \<and> \<not> PP)" by blast
```
```   475    moreover {assume "MP \<or> PP" hence "?D" by blast}
```
```   476    moreover {assume nmi: "\<not> MP" and npi: "\<not> PP"
```
```   477      from npmibnd[OF nmibnd npibnd]
```
```   478      have nmpiU: "\<forall>x. \<not> MP \<and> \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. \<exists> u' \<in> U. u \<^loc>\<le> x \<and> x \<^loc>\<le> u')" .
```
```   479      from rinf_U[OF fU lin_dense nmpiU nmi npi px] have "?D" by blast}
```
```   480    ultimately have "?D" by blast}
```
```   481  moreover
```
```   482  { assume "?D"
```
```   483    moreover {assume m:"MP" from minf_ex[OF mi m] have "?E" .}
```
```   484    moreover {assume p: "PP" from pinf_ex[OF pi p] have "?E" . }
```
```   485    moreover {assume f:"?F" hence "?E" by blast}
```
```   486    ultimately have "?E" by blast}
```
```   487  ultimately have "?E = ?D" by blast thus "?E \<equiv> ?D" by simp
```
```   488 qed
```
```   489
```
```   490 lemmas minf_thms = minf_conj minf_disj minf_eq minf_neq minf_lt minf_le minf_gt minf_ge minf_P
```
```   491 lemmas pinf_thms = pinf_conj pinf_disj pinf_eq pinf_neq pinf_lt pinf_le pinf_gt pinf_ge pinf_P
```
```   492
```
```   493 lemmas nmi_thms = nmi_conj nmi_disj nmi_eq nmi_neq nmi_lt nmi_le nmi_gt nmi_ge nmi_P
```
```   494 lemmas npi_thms = npi_conj npi_disj npi_eq npi_neq npi_lt npi_le npi_gt npi_ge npi_P
```
```   495 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
```
```   496
```
```   497 lemma ferrack_axiom: "constr_dense_linear_order less_eq less between" by fact
```
```   498 lemma atoms:
```
```   499   includes meta_term_syntax
```
```   500   shows "TERM (less :: 'a \<Rightarrow> _)"
```
```   501     and "TERM (less_eq :: 'a \<Rightarrow> _)"
```
```   502     and "TERM (op = :: 'a \<Rightarrow> _)" .
```
```   503
```
```   504 declare ferrack_axiom [ferrack minf: minf_thms pinf: pinf_thms
```
```   505     nmi: nmi_thms npi: npi_thms lindense:
```
```   506     lin_dense_thms qe: fr_eq atoms: atoms]
```
```   507
```
```   508 declaration {*
```
```   509 let
```
```   510 fun simps phi = map (Morphism.thm phi) [@{thm "not_less"}, @{thm "not_le"}]
```
```   511 fun generic_whatis phi =
```
```   512  let
```
```   513   val [lt, le] = map (Morphism.term phi) [@{term "op \<^loc><"}, @{term "op \<^loc>\<le>"}]
```
```   514   fun h x t =
```
```   515    case term_of t of
```
```   516      Const("op =", _)\$y\$z => if term_of x aconv y then Ferrante_Rackoff_Data.Eq
```
```   517                             else Ferrante_Rackoff_Data.Nox
```
```   518    | @{term "Not"}\$(Const("op =", _)\$y\$z) => if term_of x aconv y then Ferrante_Rackoff_Data.NEq
```
```   519                             else Ferrante_Rackoff_Data.Nox
```
```   520    | b\$y\$z => if Term.could_unify (b, lt) then
```
```   521                  if term_of x aconv y then Ferrante_Rackoff_Data.Lt
```
```   522                  else if term_of x aconv z then Ferrante_Rackoff_Data.Gt
```
```   523                  else Ferrante_Rackoff_Data.Nox
```
```   524              else if Term.could_unify (b, le) then
```
```   525                  if term_of x aconv y then Ferrante_Rackoff_Data.Le
```
```   526                  else if term_of x aconv z then Ferrante_Rackoff_Data.Ge
```
```   527                  else Ferrante_Rackoff_Data.Nox
```
```   528              else Ferrante_Rackoff_Data.Nox
```
```   529    | _ => Ferrante_Rackoff_Data.Nox
```
```   530  in h end
```
```   531  fun ss phi = HOL_ss addsimps (simps phi)
```
```   532 in
```
```   533  Ferrante_Rackoff_Data.funs  @{thm "ferrack_axiom"}
```
```   534   {isolate_conv = K (K (K Thm.reflexive)), whatis = generic_whatis, simpset = ss}
```
```   535 end
```
```   536 *}
```
```   537
```
```   538 end
```
```   539
```
```   540 use "Tools/Qelim/ferrante_rackoff.ML"
```
```   541
```
```   542 method_setup ferrack = {*
```
```   543   Method.ctxt_args (Method.SIMPLE_METHOD' o FerranteRackoff.dlo_tac)
```
```   544 *} "Ferrante and Rackoff's algorithm for quantifier elimination in dense linear orders"
```
```   545
```
```   546 end
```