--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Dense_Linear_Order.thy Wed Feb 27 14:39:58 2008 +0100
@@ -0,0 +1,879 @@
+(*
+ ID: $Id$
+ Author: Amine Chaieb, TU Muenchen
+*)
+
+header {* Dense linear order without endpoints
+ and a quantifier elimination procedure in Ferrante and Rackoff style *}
+
+theory Dense_Linear_Order
+imports Arith_Tools
+uses
+ "~~/src/HOL/Tools/Qelim/qelim.ML"
+ "~~/src/HOL/Tools/Qelim/langford_data.ML"
+ "~~/src/HOL/Tools/Qelim/ferrante_rackoff_data.ML"
+ ("~~/src/HOL/Tools/Qelim/langford.ML")
+ ("~~/src/HOL/Tools/Qelim/ferrante_rackoff.ML")
+begin
+
+setup Langford_Data.setup
+setup Ferrante_Rackoff_Data.setup
+
+context linorder
+begin
+
+lemma less_not_permute: "\<not> (x < y \<and> y < x)" by (simp add: not_less linear)
+
+lemma gather_simps:
+ shows
+ "(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> U. x < y) \<and> x < u \<and> P x) \<longleftrightarrow> (\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> (insert u U). x < y) \<and> P x)"
+ and "(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> U. x < y) \<and> l < x \<and> P x) \<longleftrightarrow> (\<exists>x. (\<forall>y \<in> (insert l L). y < x) \<and> (\<forall>y \<in> U. x < y) \<and> P x)"
+ "(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> U. x < y) \<and> x < u) \<longleftrightarrow> (\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> (insert u U). x < y))"
+ and "(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> U. x < y) \<and> l < x) \<longleftrightarrow> (\<exists>x. (\<forall>y \<in> (insert l L). y < x) \<and> (\<forall>y \<in> U. x < y))" by auto
+
+lemma
+ gather_start: "(\<exists>x. P x) \<equiv> (\<exists>x. (\<forall>y \<in> {}. y < x) \<and> (\<forall>y\<in> {}. x < y) \<and> P x)"
+ by simp
+
+text{* Theorems for @{text "\<exists>z. \<forall>x. x < z \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>-\<infinity>\<^esub>)"}*}
+lemma minf_lt: "\<exists>z . \<forall>x. x < z \<longrightarrow> (x < t \<longleftrightarrow> True)" by auto
+lemma minf_gt: "\<exists>z . \<forall>x. x < z \<longrightarrow> (t < x \<longleftrightarrow> False)"
+ by (simp add: not_less) (rule exI[where x="t"], auto simp add: less_le)
+
+lemma minf_le: "\<exists>z. \<forall>x. x < z \<longrightarrow> (x \<le> t \<longleftrightarrow> True)" by (auto simp add: less_le)
+lemma minf_ge: "\<exists>z. \<forall>x. x < z \<longrightarrow> (t \<le> x \<longleftrightarrow> False)"
+ by (auto simp add: less_le not_less not_le)
+lemma minf_eq: "\<exists>z. \<forall>x. x < z \<longrightarrow> (x = t \<longleftrightarrow> False)" by auto
+lemma minf_neq: "\<exists>z. \<forall>x. x < z \<longrightarrow> (x \<noteq> t \<longleftrightarrow> True)" by auto
+lemma minf_P: "\<exists>z. \<forall>x. x < z \<longrightarrow> (P \<longleftrightarrow> P)" by blast
+
+text{* Theorems for @{text "\<exists>z. \<forall>x. x < z \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>+\<infinity>\<^esub>)"}*}
+lemma pinf_gt: "\<exists>z . \<forall>x. z < x \<longrightarrow> (t < x \<longleftrightarrow> True)" by auto
+lemma pinf_lt: "\<exists>z . \<forall>x. z < x \<longrightarrow> (x < t \<longleftrightarrow> False)"
+ by (simp add: not_less) (rule exI[where x="t"], auto simp add: less_le)
+
+lemma pinf_ge: "\<exists>z. \<forall>x. z < x \<longrightarrow> (t \<le> x \<longleftrightarrow> True)" by (auto simp add: less_le)
+lemma pinf_le: "\<exists>z. \<forall>x. z < x \<longrightarrow> (x \<le> t \<longleftrightarrow> False)"
+ by (auto simp add: less_le not_less not_le)
+lemma pinf_eq: "\<exists>z. \<forall>x. z < x \<longrightarrow> (x = t \<longleftrightarrow> False)" by auto
+lemma pinf_neq: "\<exists>z. \<forall>x. z < x \<longrightarrow> (x \<noteq> t \<longleftrightarrow> True)" by auto
+lemma pinf_P: "\<exists>z. \<forall>x. z < x \<longrightarrow> (P \<longleftrightarrow> P)" by blast
+
+lemma nmi_lt: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> x < t \<longrightarrow> (\<exists> u\<in> U. u \<le> x)" by auto
+lemma nmi_gt: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> t < x \<longrightarrow> (\<exists> u\<in> U. u \<le> x)"
+ by (auto simp add: le_less)
+lemma nmi_le: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> x\<le> t \<longrightarrow> (\<exists> u\<in> U. u \<le> x)" by auto
+lemma nmi_ge: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> t\<le> x \<longrightarrow> (\<exists> u\<in> U. u \<le> x)" by auto
+lemma nmi_eq: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> x = t \<longrightarrow> (\<exists> u\<in> U. u \<le> x)" by auto
+lemma nmi_neq: "t \<in> U \<Longrightarrow>\<forall>x. \<not>True \<and> x \<noteq> t \<longrightarrow> (\<exists> u\<in> U. u \<le> x)" by auto
+lemma nmi_P: "\<forall> x. ~P \<and> P \<longrightarrow> (\<exists> u\<in> U. u \<le> x)" by auto
+lemma nmi_conj: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow> (\<exists> u\<in> U. u \<le> x) ;
+ \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow> (\<exists> u\<in> U. u \<le> x)\<rbrakk> \<Longrightarrow>
+ \<forall>x. \<not>(P1' \<and> P2') \<and> (P1 x \<and> P2 x) \<longrightarrow> (\<exists> u\<in> U. u \<le> x)" by auto
+lemma nmi_disj: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow> (\<exists> u\<in> U. u \<le> x) ;
+ \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow> (\<exists> u\<in> U. u \<le> x)\<rbrakk> \<Longrightarrow>
+ \<forall>x. \<not>(P1' \<or> P2') \<and> (P1 x \<or> P2 x) \<longrightarrow> (\<exists> u\<in> U. u \<le> x)" by auto
+
+lemma npi_lt: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> x < t \<longrightarrow> (\<exists> u\<in> U. x \<le> u)" by (auto simp add: le_less)
+lemma npi_gt: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> t < x \<longrightarrow> (\<exists> u\<in> U. x \<le> u)" by auto
+lemma npi_le: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> x \<le> t \<longrightarrow> (\<exists> u\<in> U. x \<le> u)" by auto
+lemma npi_ge: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> t \<le> x \<longrightarrow> (\<exists> u\<in> U. x \<le> u)" by auto
+lemma npi_eq: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> x = t \<longrightarrow> (\<exists> u\<in> U. x \<le> u)" by auto
+lemma npi_neq: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> x \<noteq> t \<longrightarrow> (\<exists> u\<in> U. x \<le> u )" by auto
+lemma npi_P: "\<forall> x. ~P \<and> P \<longrightarrow> (\<exists> u\<in> U. x \<le> u)" by auto
+lemma npi_conj: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow> (\<exists> u\<in> U. x \<le> u) ; \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow> (\<exists> u\<in> U. x \<le> u)\<rbrakk>
+ \<Longrightarrow> \<forall>x. \<not>(P1' \<and> P2') \<and> (P1 x \<and> P2 x) \<longrightarrow> (\<exists> u\<in> U. x \<le> u)" by auto
+lemma npi_disj: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow> (\<exists> u\<in> U. x \<le> u) ; \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow> (\<exists> u\<in> U. x \<le> u)\<rbrakk>
+ \<Longrightarrow> \<forall>x. \<not>(P1' \<or> P2') \<and> (P1 x \<or> P2 x) \<longrightarrow> (\<exists> u\<in> U. x \<le> u)" by auto
+
+lemma lin_dense_lt: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l < t \<and> t < u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> x < t \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> y < t)"
+proof(clarsimp)
+ fix x l u y assume tU: "t \<in> U" and noU: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U" and lx: "l < x"
+ and xu: "x<u" and px: "x < t" and ly: "l<y" and yu:"y < u"
+ from tU noU ly yu have tny: "t\<noteq>y" by auto
+ {assume H: "t < y"
+ from less_trans[OF lx px] less_trans[OF H yu]
+ have "l < t \<and> t < u" by simp
+ with tU noU have "False" by auto}
+ hence "\<not> t < y" by auto hence "y \<le> t" by (simp add: not_less)
+ thus "y < t" using tny by (simp add: less_le)
+qed
+
+lemma lin_dense_gt: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l < x \<and> x < u \<and> t < x \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> t < y)"
+proof(clarsimp)
+ fix x l u y
+ assume tU: "t \<in> U" and noU: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U" and lx: "l < x" and xu: "x<u"
+ and px: "t < x" and ly: "l<y" and yu:"y < u"
+ from tU noU ly yu have tny: "t\<noteq>y" by auto
+ {assume H: "y< t"
+ from less_trans[OF ly H] less_trans[OF px xu] have "l < t \<and> t < u" by simp
+ with tU noU have "False" by auto}
+ hence "\<not> y<t" by auto hence "t \<le> y" by (auto simp add: not_less)
+ thus "t < y" using tny by (simp add:less_le)
+qed
+
+lemma lin_dense_le: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> x \<le> t \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> y\<le> t)"
+proof(clarsimp)
+ fix x l u y
+ assume tU: "t \<in> U" and noU: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U" and lx: "l < x" and xu: "x<u"
+ and px: "x \<le> t" and ly: "l<y" and yu:"y < u"
+ from tU noU ly yu have tny: "t\<noteq>y" by auto
+ {assume H: "t < y"
+ from less_le_trans[OF lx px] less_trans[OF H yu]
+ have "l < t \<and> t < u" by simp
+ with tU noU have "False" by auto}
+ hence "\<not> t < y" by auto thus "y \<le> t" by (simp add: not_less)
+qed
+
+lemma lin_dense_ge: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> t \<le> x \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> t \<le> y)"
+proof(clarsimp)
+ fix x l u y
+ assume tU: "t \<in> U" and noU: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U" and lx: "l < x" and xu: "x<u"
+ and px: "t \<le> x" and ly: "l<y" and yu:"y < u"
+ from tU noU ly yu have tny: "t\<noteq>y" by auto
+ {assume H: "y< t"
+ from less_trans[OF ly H] le_less_trans[OF px xu]
+ have "l < t \<and> t < u" by simp
+ with tU noU have "False" by auto}
+ hence "\<not> y<t" by auto thus "t \<le> y" by (simp add: not_less)
+qed
+lemma lin_dense_eq: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> x = t \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> y= t)" by auto
+lemma lin_dense_neq: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> x \<noteq> t \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> y\<noteq> t)" by auto
+lemma lin_dense_P: "\<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> P \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> P)" by auto
+
+lemma lin_dense_conj:
+ "\<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
+ \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> P1 y) ;
+ \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> P2 x
+ \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> P2 y)\<rbrakk> \<Longrightarrow>
+ \<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)
+ \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> (P1 y \<and> P2 y))"
+ by blast
+lemma lin_dense_disj:
+ "\<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
+ \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> P1 y) ;
+ \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> P2 x
+ \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> P2 y)\<rbrakk> \<Longrightarrow>
+ \<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)
+ \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> (P1 y \<or> P2 y))"
+ by blast
+
+lemma npmibnd: "\<lbrakk>\<forall>x. \<not> MP \<and> P x \<longrightarrow> (\<exists> u\<in> U. u \<le> x); \<forall>x. \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. x \<le> u)\<rbrakk>
+ \<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')"
+by auto
+
+lemma finite_set_intervals:
+ assumes px: "P x" and lx: "l \<le> x" and xu: "x \<le> u" and linS: "l\<in> S"
+ and uinS: "u \<in> S" and fS:"finite S" and lS: "\<forall> x\<in> S. l \<le> x" and Su: "\<forall> x\<in> S. x \<le> u"
+ 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"
+proof-
+ let ?Mx = "{y. y\<in> S \<and> y \<le> x}"
+ let ?xM = "{y. y\<in> S \<and> x \<le> y}"
+ let ?a = "Max ?Mx"
+ let ?b = "Min ?xM"
+ have MxS: "?Mx \<subseteq> S" by blast
+ hence fMx: "finite ?Mx" using fS finite_subset by auto
+ from lx linS have linMx: "l \<in> ?Mx" by blast
+ hence Mxne: "?Mx \<noteq> {}" by blast
+ have xMS: "?xM \<subseteq> S" by blast
+ hence fxM: "finite ?xM" using fS finite_subset by auto
+ from xu uinS have linxM: "u \<in> ?xM" by blast
+ hence xMne: "?xM \<noteq> {}" by blast
+ have ax:"?a \<le> x" using Mxne fMx by auto
+ have xb:"x \<le> ?b" using xMne fxM by auto
+ have "?a \<in> ?Mx" using Max_in[OF fMx Mxne] by simp hence ainS: "?a \<in> S" using MxS by blast
+ have "?b \<in> ?xM" using Min_in[OF fxM xMne] by simp hence binS: "?b \<in> S" using xMS by blast
+ have noy:"\<forall> y. ?a < y \<and> y < ?b \<longrightarrow> y \<notin> S"
+ proof(clarsimp)
+ fix y assume ay: "?a < y" and yb: "y < ?b" and yS: "y \<in> S"
+ from yS have "y\<in> ?Mx \<or> y\<in> ?xM" by (auto simp add: linear)
+ moreover {assume "y \<in> ?Mx" hence "y \<le> ?a" using Mxne fMx by auto with ay have "False" by (simp add: not_le[symmetric])}
+ moreover {assume "y \<in> ?xM" hence "?b \<le> y" using xMne fxM by auto with yb have "False" by (simp add: not_le[symmetric])}
+ ultimately show "False" by blast
+ qed
+ from ainS binS noy ax xb px show ?thesis by blast
+qed
+
+lemma finite_set_intervals2:
+ assumes px: "P x" and lx: "l \<le> x" and xu: "x \<le> u" and linS: "l\<in> S"
+ and uinS: "u \<in> S" and fS:"finite S" and lS: "\<forall> x\<in> S. l \<le> x" and Su: "\<forall> x\<in> S. x \<le> u"
+ 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)"
+proof-
+ from finite_set_intervals[where P="P", OF px lx xu linS uinS fS lS Su]
+ obtain a and b where
+ as: "a\<in> S" and bs: "b\<in> S" and noS:"\<forall>y. a < y \<and> y < b \<longrightarrow> y \<notin> S"
+ and axb: "a \<le> x \<and> x \<le> b \<and> P x" by auto
+ from axb have "x= a \<or> x= b \<or> (a < x \<and> x < b)" by (auto simp add: le_less)
+ thus ?thesis using px as bs noS by blast
+qed
+
+end
+
+section {* The classical QE after Langford for dense linear orders *}
+
+context dense_linear_order
+begin
+
+lemma dlo_qe_bnds:
+ assumes ne: "L \<noteq> {}" and neU: "U \<noteq> {}" and fL: "finite L" and fU: "finite U"
+ 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)"
+proof (simp only: atomize_eq, rule iffI)
+ assume H: "\<exists>x. (\<forall>y\<in>L. y < x) \<and> (\<forall>y\<in>U. x < y)"
+ then obtain x where xL: "\<forall>y\<in>L. y < x" and xU: "\<forall>y\<in>U. x < y" by blast
+ {fix l u assume l: "l \<in> L" and u: "u \<in> U"
+ have "l < x" using xL l by blast
+ also have "x < u" using xU u by blast
+ finally (less_trans) have "l < u" .}
+ thus "\<forall>l\<in>L. \<forall>u\<in>U. l < u" by blast
+next
+ assume H: "\<forall>l\<in>L. \<forall>u\<in>U. l < u"
+ let ?ML = "Max L"
+ let ?MU = "Min U"
+ from fL ne have th1: "?ML \<in> L" and th1': "\<forall>l\<in>L. l \<le> ?ML" by auto
+ from fU neU have th2: "?MU \<in> U" and th2': "\<forall>u\<in>U. ?MU \<le> u" by auto
+ from th1 th2 H have "?ML < ?MU" by auto
+ with dense obtain w where th3: "?ML < w" and th4: "w < ?MU" by blast
+ from th3 th1' have "\<forall>l \<in> L. l < w" by auto
+ moreover from th4 th2' have "\<forall>u \<in> U. w < u" by auto
+ ultimately show "\<exists>x. (\<forall>y\<in>L. y < x) \<and> (\<forall>y\<in>U. x < y)" by auto
+qed
+
+lemma dlo_qe_noub:
+ assumes ne: "L \<noteq> {}" and fL: "finite L"
+ shows "(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> {}. x < y)) \<equiv> True"
+proof(simp add: atomize_eq)
+ from gt_ex[of "Max L"] obtain M where M: "Max L < M" by blast
+ from ne fL have "\<forall>x \<in> L. x \<le> Max L" by simp
+ with M have "\<forall>x\<in>L. x < M" by (auto intro: le_less_trans)
+ thus "\<exists>x. \<forall>y\<in>L. y < x" by blast
+qed
+
+lemma dlo_qe_nolb:
+ assumes ne: "U \<noteq> {}" and fU: "finite U"
+ shows "(\<exists>x. (\<forall>y \<in> {}. y < x) \<and> (\<forall>y \<in> U. x < y)) \<equiv> True"
+proof(simp add: atomize_eq)
+ from lt_ex[of "Min U"] obtain M where M: "M < Min U" by blast
+ from ne fU have "\<forall>x \<in> U. Min U \<le> x" by simp
+ with M have "\<forall>x\<in>U. M < x" by (auto intro: less_le_trans)
+ thus "\<exists>x. \<forall>y\<in>U. x < y" by blast
+qed
+
+lemma exists_neq: "\<exists>(x::'a). x \<noteq> t" "\<exists>(x::'a). t \<noteq> x"
+ using gt_ex[of t] by auto
+
+lemmas dlo_simps = order_refl less_irrefl not_less not_le exists_neq
+ le_less neq_iff linear less_not_permute
+
+lemma axiom: "dense_linear_order (op \<le>) (op <)" by fact
+lemma atoms:
+ includes meta_term_syntax
+ shows "TERM (less :: 'a \<Rightarrow> _)"
+ and "TERM (less_eq :: 'a \<Rightarrow> _)"
+ and "TERM (op = :: 'a \<Rightarrow> _)" .
+
+declare axiom[langford qe: dlo_qe_bnds dlo_qe_nolb dlo_qe_noub gather: gather_start gather_simps atoms: atoms]
+declare dlo_simps[langfordsimp]
+
+end
+
+(* FIXME: Move to HOL -- together with the conj_aci_rule in langford.ML *)
+lemma dnf:
+ "(P & (Q | R)) = ((P&Q) | (P&R))"
+ "((Q | R) & P) = ((Q&P) | (R&P))"
+ by blast+
+
+lemmas weak_dnf_simps = simp_thms dnf
+
+lemma nnf_simps:
+ "(\<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)"
+ "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not> \<not>(P)) = P"
+ by blast+
+
+lemma ex_distrib: "(\<exists>x. P x \<or> Q x) \<longleftrightarrow> ((\<exists>x. P x) \<or> (\<exists>x. Q x))" by blast
+
+lemmas dnf_simps = weak_dnf_simps nnf_simps ex_distrib
+
+use "~~/src/HOL/Tools/Qelim/langford.ML"
+method_setup dlo = {*
+ Method.ctxt_args (Method.SIMPLE_METHOD' o LangfordQE.dlo_tac)
+*} "Langford's algorithm for quantifier elimination in dense linear orders"
+
+
+section {* Contructive dense linear orders yield QE for linear arithmetic over ordered Fields -- see @{text "Arith_Tools.thy"} *}
+
+text {* Linear order without upper bounds *}
+
+locale linorder_stupid_syntax = linorder
+begin
+notation
+ less_eq ("op \<sqsubseteq>") and
+ less_eq ("(_/ \<sqsubseteq> _)" [51, 51] 50) and
+ less ("op \<sqsubset>") and
+ less ("(_/ \<sqsubset> _)" [51, 51] 50)
+
+end
+
+locale linorder_no_ub = linorder_stupid_syntax +
+ assumes gt_ex: "\<exists>y. less x y"
+begin
+lemma ge_ex: "\<exists>y. x \<sqsubseteq> y" using gt_ex by auto
+
+text {* Theorems for @{text "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>+\<infinity>\<^esub>)"} *}
+lemma pinf_conj:
+ assumes ex1: "\<exists>z1. \<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
+ and ex2: "\<exists>z2. \<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
+ shows "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> ((P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2'))"
+proof-
+ from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
+ and z2: "\<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')" by blast
+ from gt_ex obtain z where z:"ord.max less_eq z1 z2 \<sqsubset> z" by blast
+ from z have zz1: "z1 \<sqsubset> z" and zz2: "z2 \<sqsubset> z" by simp_all
+ {fix x assume H: "z \<sqsubset> x"
+ from less_trans[OF zz1 H] less_trans[OF zz2 H]
+ have "(P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2')" using z1 zz1 z2 zz2 by auto
+ }
+ thus ?thesis by blast
+qed
+
+lemma pinf_disj:
+ assumes ex1: "\<exists>z1. \<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
+ and ex2: "\<exists>z2. \<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
+ shows "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> ((P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2'))"
+proof-
+ from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
+ and z2: "\<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')" by blast
+ from gt_ex obtain z where z:"ord.max less_eq z1 z2 \<sqsubset> z" by blast
+ from z have zz1: "z1 \<sqsubset> z" and zz2: "z2 \<sqsubset> z" by simp_all
+ {fix x assume H: "z \<sqsubset> x"
+ from less_trans[OF zz1 H] less_trans[OF zz2 H]
+ have "(P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2')" using z1 zz1 z2 zz2 by auto
+ }
+ thus ?thesis by blast
+qed
+
+lemma pinf_ex: assumes ex:"\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (P x \<longleftrightarrow> P1)" and p1: P1 shows "\<exists> x. P x"
+proof-
+ from ex obtain z where z: "\<forall>x. z \<sqsubset> x \<longrightarrow> (P x \<longleftrightarrow> P1)" by blast
+ from gt_ex obtain x where x: "z \<sqsubset> x" by blast
+ from z x p1 show ?thesis by blast
+qed
+
+end
+
+text {* Linear order without upper bounds *}
+
+locale linorder_no_lb = linorder_stupid_syntax +
+ assumes lt_ex: "\<exists>y. less y x"
+begin
+lemma le_ex: "\<exists>y. y \<sqsubseteq> x" using lt_ex by auto
+
+
+text {* Theorems for @{text "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>-\<infinity>\<^esub>)"} *}
+lemma minf_conj:
+ assumes ex1: "\<exists>z1. \<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
+ and ex2: "\<exists>z2. \<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
+ shows "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> ((P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2'))"
+proof-
+ from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"and z2: "\<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')" by blast
+ from lt_ex obtain z where z:"z \<sqsubset> ord.min less_eq z1 z2" by blast
+ from z have zz1: "z \<sqsubset> z1" and zz2: "z \<sqsubset> z2" by simp_all
+ {fix x assume H: "x \<sqsubset> z"
+ from less_trans[OF H zz1] less_trans[OF H zz2]
+ have "(P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2')" using z1 zz1 z2 zz2 by auto
+ }
+ thus ?thesis by blast
+qed
+
+lemma minf_disj:
+ assumes ex1: "\<exists>z1. \<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
+ and ex2: "\<exists>z2. \<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
+ shows "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> ((P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2'))"
+proof-
+ from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"and z2: "\<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')" by blast
+ from lt_ex obtain z where z:"z \<sqsubset> ord.min less_eq z1 z2" by blast
+ from z have zz1: "z \<sqsubset> z1" and zz2: "z \<sqsubset> z2" by simp_all
+ {fix x assume H: "x \<sqsubset> z"
+ from less_trans[OF H zz1] less_trans[OF H zz2]
+ have "(P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2')" using z1 zz1 z2 zz2 by auto
+ }
+ thus ?thesis by blast
+qed
+
+lemma minf_ex: assumes ex:"\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P1)" and p1: P1 shows "\<exists> x. P x"
+proof-
+ from ex obtain z where z: "\<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P1)" by blast
+ from lt_ex obtain x where x: "x \<sqsubset> z" by blast
+ from z x p1 show ?thesis by blast
+qed
+
+end
+
+
+locale constr_dense_linear_order = linorder_no_lb + linorder_no_ub +
+ fixes between
+ assumes between_less: "less x y \<Longrightarrow> less x (between x y) \<and> less (between x y) y"
+ and between_same: "between x x = x"
+
+interpretation constr_dense_linear_order < dense_linear_order
+ apply unfold_locales
+ using gt_ex lt_ex between_less
+ by (auto, rule_tac x="between x y" in exI, simp)
+
+context constr_dense_linear_order
+begin
+
+lemma rinf_U:
+ assumes fU: "finite U"
+ 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
+ \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> P y )"
+ 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')"
+ and nmi: "\<not> MP" and npi: "\<not> PP" and ex: "\<exists> x. P x"
+ shows "\<exists> u\<in> U. \<exists> u' \<in> U. P (between u u')"
+proof-
+ from ex obtain x where px: "P x" by blast
+ from px nmi npi nmpiU have "\<exists> u\<in> U. \<exists> u' \<in> U. u \<sqsubseteq> x \<and> x \<sqsubseteq> u'" by auto
+ then obtain u and u' where uU:"u\<in> U" and uU': "u' \<in> U" and ux:"u \<sqsubseteq> x" and xu':"x \<sqsubseteq> u'" by auto
+ from uU have Une: "U \<noteq> {}" by auto
+ term "linorder.Min less_eq"
+ let ?l = "linorder.Min less_eq U"
+ let ?u = "linorder.Max less_eq U"
+ have linM: "?l \<in> U" using fU Une by simp
+ have uinM: "?u \<in> U" using fU Une by simp
+ have lM: "\<forall> t\<in> U. ?l \<sqsubseteq> t" using Une fU by auto
+ have Mu: "\<forall> t\<in> U. t \<sqsubseteq> ?u" using Une fU by auto
+ have th:"?l \<sqsubseteq> u" using uU Une lM by auto
+ from order_trans[OF th ux] have lx: "?l \<sqsubseteq> x" .
+ have th: "u' \<sqsubseteq> ?u" using uU' Une Mu by simp
+ from order_trans[OF xu' th] have xu: "x \<sqsubseteq> ?u" .
+ from finite_set_intervals2[where P="P",OF px lx xu linM uinM fU lM Mu]
+ have "(\<exists> s\<in> U. P s) \<or>
+ (\<exists> t1\<in> U. \<exists> t2 \<in> U. (\<forall> y. t1 \<sqsubset> y \<and> y \<sqsubset> t2 \<longrightarrow> y \<notin> U) \<and> t1 \<sqsubset> x \<and> x \<sqsubset> t2 \<and> P x)" .
+ moreover { fix u assume um: "u\<in>U" and pu: "P u"
+ have "between u u = u" by (simp add: between_same)
+ with um pu have "P (between u u)" by simp
+ with um have ?thesis by blast}
+ moreover{
+ assume "\<exists> t1\<in> U. \<exists> t2 \<in> U. (\<forall> y. t1 \<sqsubset> y \<and> y \<sqsubset> t2 \<longrightarrow> y \<notin> U) \<and> t1 \<sqsubset> x \<and> x \<sqsubset> t2 \<and> P x"
+ then obtain t1 and t2 where t1M: "t1 \<in> U" and t2M: "t2\<in> U"
+ and noM: "\<forall> y. t1 \<sqsubset> y \<and> y \<sqsubset> t2 \<longrightarrow> y \<notin> U" and t1x: "t1 \<sqsubset> x" and xt2: "x \<sqsubset> t2" and px: "P x"
+ by blast
+ from less_trans[OF t1x xt2] have t1t2: "t1 \<sqsubset> t2" .
+ let ?u = "between t1 t2"
+ from between_less t1t2 have t1lu: "t1 \<sqsubset> ?u" and ut2: "?u \<sqsubset> t2" by auto
+ from lin_dense noM t1x xt2 px t1lu ut2 have "P ?u" by blast
+ with t1M t2M have ?thesis by blast}
+ ultimately show ?thesis by blast
+ qed
+
+theorem fr_eq:
+ assumes fU: "finite U"
+ 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
+ \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> P y )"
+ and nmibnd: "\<forall>x. \<not> MP \<and> P x \<longrightarrow> (\<exists> u\<in> U. u \<sqsubseteq> x)"
+ and npibnd: "\<forall>x. \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. x \<sqsubseteq> u)"
+ 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)"
+ shows "(\<exists> x. P x) \<equiv> (MP \<or> PP \<or> (\<exists> u \<in> U. \<exists> u'\<in> U. P (between u u')))"
+ (is "_ \<equiv> (_ \<or> _ \<or> ?F)" is "?E \<equiv> ?D")
+proof-
+ {
+ assume px: "\<exists> x. P x"
+ have "MP \<or> PP \<or> (\<not> MP \<and> \<not> PP)" by blast
+ moreover {assume "MP \<or> PP" hence "?D" by blast}
+ moreover {assume nmi: "\<not> MP" and npi: "\<not> PP"
+ from npmibnd[OF nmibnd npibnd]
+ 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')" .
+ from rinf_U[OF fU lin_dense nmpiU nmi npi px] have "?D" by blast}
+ ultimately have "?D" by blast}
+ moreover
+ { assume "?D"
+ moreover {assume m:"MP" from minf_ex[OF mi m] have "?E" .}
+ moreover {assume p: "PP" from pinf_ex[OF pi p] have "?E" . }
+ moreover {assume f:"?F" hence "?E" by blast}
+ ultimately have "?E" by blast}
+ ultimately have "?E = ?D" by blast thus "?E \<equiv> ?D" by simp
+qed
+
+lemmas minf_thms = minf_conj minf_disj minf_eq minf_neq minf_lt minf_le minf_gt minf_ge minf_P
+lemmas pinf_thms = pinf_conj pinf_disj pinf_eq pinf_neq pinf_lt pinf_le pinf_gt pinf_ge pinf_P
+
+lemmas nmi_thms = nmi_conj nmi_disj nmi_eq nmi_neq nmi_lt nmi_le nmi_gt nmi_ge nmi_P
+lemmas npi_thms = npi_conj npi_disj npi_eq npi_neq npi_lt npi_le npi_gt npi_ge npi_P
+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
+
+lemma ferrack_axiom: "constr_dense_linear_order less_eq less between" by fact
+lemma atoms:
+ includes meta_term_syntax
+ shows "TERM (less :: 'a \<Rightarrow> _)"
+ and "TERM (less_eq :: 'a \<Rightarrow> _)"
+ and "TERM (op = :: 'a \<Rightarrow> _)" .
+
+declare ferrack_axiom [ferrack minf: minf_thms pinf: pinf_thms
+ nmi: nmi_thms npi: npi_thms lindense:
+ lin_dense_thms qe: fr_eq atoms: atoms]
+
+declaration {*
+let
+fun simps phi = map (Morphism.thm phi) [@{thm "not_less"}, @{thm "not_le"}]
+fun generic_whatis phi =
+ let
+ val [lt, le] = map (Morphism.term phi) [@{term "op \<sqsubset>"}, @{term "op \<sqsubseteq>"}]
+ fun h x t =
+ case term_of t of
+ Const("op =", _)$y$z => if term_of x aconv y then Ferrante_Rackoff_Data.Eq
+ else Ferrante_Rackoff_Data.Nox
+ | @{term "Not"}$(Const("op =", _)$y$z) => if term_of x aconv y then Ferrante_Rackoff_Data.NEq
+ else Ferrante_Rackoff_Data.Nox
+ | b$y$z => if Term.could_unify (b, lt) then
+ if term_of x aconv y then Ferrante_Rackoff_Data.Lt
+ else if term_of x aconv z then Ferrante_Rackoff_Data.Gt
+ else Ferrante_Rackoff_Data.Nox
+ else if Term.could_unify (b, le) then
+ if term_of x aconv y then Ferrante_Rackoff_Data.Le
+ else if term_of x aconv z then Ferrante_Rackoff_Data.Ge
+ else Ferrante_Rackoff_Data.Nox
+ else Ferrante_Rackoff_Data.Nox
+ | _ => Ferrante_Rackoff_Data.Nox
+ in h end
+ fun ss phi = HOL_ss addsimps (simps phi)
+in
+ Ferrante_Rackoff_Data.funs @{thm "ferrack_axiom"}
+ {isolate_conv = K (K (K Thm.reflexive)), whatis = generic_whatis, simpset = ss}
+end
+*}
+
+end
+
+use "~~/src/HOL/Tools/Qelim/ferrante_rackoff.ML"
+
+method_setup ferrack = {*
+ Method.ctxt_args (Method.SIMPLE_METHOD' o FerranteRackoff.dlo_tac)
+*} "Ferrante and Rackoff's algorithm for quantifier elimination in dense linear orders"
+
+subsection {* Ferrante and Rackoff algorithm over ordered fields *}
+
+lemma neg_prod_lt:"(c\<Colon>'a\<Colon>ordered_field) < 0 \<Longrightarrow> ((c*x < 0) == (x > 0))"
+proof-
+ assume H: "c < 0"
+ have "c*x < 0 = (0/c < x)" by (simp only: neg_divide_less_eq[OF H] ring_simps)
+ also have "\<dots> = (0 < x)" by simp
+ finally show "(c*x < 0) == (x > 0)" by simp
+qed
+
+lemma pos_prod_lt:"(c\<Colon>'a\<Colon>ordered_field) > 0 \<Longrightarrow> ((c*x < 0) == (x < 0))"
+proof-
+ assume H: "c > 0"
+ hence "c*x < 0 = (0/c > x)" by (simp only: pos_less_divide_eq[OF H] ring_simps)
+ also have "\<dots> = (0 > x)" by simp
+ finally show "(c*x < 0) == (x < 0)" by simp
+qed
+
+lemma neg_prod_sum_lt: "(c\<Colon>'a\<Colon>ordered_field) < 0 \<Longrightarrow> ((c*x + t< 0) == (x > (- 1/c)*t))"
+proof-
+ assume H: "c < 0"
+ have "c*x + t< 0 = (c*x < -t)" by (subst less_iff_diff_less_0 [of "c*x" "-t"], simp)
+ also have "\<dots> = (-t/c < x)" by (simp only: neg_divide_less_eq[OF H] ring_simps)
+ also have "\<dots> = ((- 1/c)*t < x)" by simp
+ finally show "(c*x + t < 0) == (x > (- 1/c)*t)" by simp
+qed
+
+lemma pos_prod_sum_lt:"(c\<Colon>'a\<Colon>ordered_field) > 0 \<Longrightarrow> ((c*x + t < 0) == (x < (- 1/c)*t))"
+proof-
+ assume H: "c > 0"
+ have "c*x + t< 0 = (c*x < -t)" by (subst less_iff_diff_less_0 [of "c*x" "-t"], simp)
+ also have "\<dots> = (-t/c > x)" by (simp only: pos_less_divide_eq[OF H] ring_simps)
+ also have "\<dots> = ((- 1/c)*t > x)" by simp
+ finally show "(c*x + t < 0) == (x < (- 1/c)*t)" by simp
+qed
+
+lemma sum_lt:"((x::'a::pordered_ab_group_add) + t < 0) == (x < - t)"
+ using less_diff_eq[where a= x and b=t and c=0] by simp
+
+lemma neg_prod_le:"(c\<Colon>'a\<Colon>ordered_field) < 0 \<Longrightarrow> ((c*x <= 0) == (x >= 0))"
+proof-
+ assume H: "c < 0"
+ have "c*x <= 0 = (0/c <= x)" by (simp only: neg_divide_le_eq[OF H] ring_simps)
+ also have "\<dots> = (0 <= x)" by simp
+ finally show "(c*x <= 0) == (x >= 0)" by simp
+qed
+
+lemma pos_prod_le:"(c\<Colon>'a\<Colon>ordered_field) > 0 \<Longrightarrow> ((c*x <= 0) == (x <= 0))"
+proof-
+ assume H: "c > 0"
+ hence "c*x <= 0 = (0/c >= x)" by (simp only: pos_le_divide_eq[OF H] ring_simps)
+ also have "\<dots> = (0 >= x)" by simp
+ finally show "(c*x <= 0) == (x <= 0)" by simp
+qed
+
+lemma neg_prod_sum_le: "(c\<Colon>'a\<Colon>ordered_field) < 0 \<Longrightarrow> ((c*x + t <= 0) == (x >= (- 1/c)*t))"
+proof-
+ assume H: "c < 0"
+ have "c*x + t <= 0 = (c*x <= -t)" by (subst le_iff_diff_le_0 [of "c*x" "-t"], simp)
+ also have "\<dots> = (-t/c <= x)" by (simp only: neg_divide_le_eq[OF H] ring_simps)
+ also have "\<dots> = ((- 1/c)*t <= x)" by simp
+ finally show "(c*x + t <= 0) == (x >= (- 1/c)*t)" by simp
+qed
+
+lemma pos_prod_sum_le:"(c\<Colon>'a\<Colon>ordered_field) > 0 \<Longrightarrow> ((c*x + t <= 0) == (x <= (- 1/c)*t))"
+proof-
+ assume H: "c > 0"
+ have "c*x + t <= 0 = (c*x <= -t)" by (subst le_iff_diff_le_0 [of "c*x" "-t"], simp)
+ also have "\<dots> = (-t/c >= x)" by (simp only: pos_le_divide_eq[OF H] ring_simps)
+ also have "\<dots> = ((- 1/c)*t >= x)" by simp
+ finally show "(c*x + t <= 0) == (x <= (- 1/c)*t)" by simp
+qed
+
+lemma sum_le:"((x::'a::pordered_ab_group_add) + t <= 0) == (x <= - t)"
+ using le_diff_eq[where a= x and b=t and c=0] by simp
+
+lemma nz_prod_eq:"(c\<Colon>'a\<Colon>ordered_field) \<noteq> 0 \<Longrightarrow> ((c*x = 0) == (x = 0))" by simp
+lemma nz_prod_sum_eq: "(c\<Colon>'a\<Colon>ordered_field) \<noteq> 0 \<Longrightarrow> ((c*x + t = 0) == (x = (- 1/c)*t))"
+proof-
+ assume H: "c \<noteq> 0"
+ have "c*x + t = 0 = (c*x = -t)" by (subst eq_iff_diff_eq_0 [of "c*x" "-t"], simp)
+ also have "\<dots> = (x = -t/c)" by (simp only: nonzero_eq_divide_eq[OF H] ring_simps)
+ finally show "(c*x + t = 0) == (x = (- 1/c)*t)" by simp
+qed
+lemma sum_eq:"((x::'a::pordered_ab_group_add) + t = 0) == (x = - t)"
+ using eq_diff_eq[where a= x and b=t and c=0] by simp
+
+
+interpretation class_ordered_field_dense_linear_order: constr_dense_linear_order
+ ["op <=" "op <"
+ "\<lambda> x y. 1/2 * ((x::'a::{ordered_field,recpower,number_ring}) + y)"]
+proof (unfold_locales, dlo, dlo, auto)
+ fix x y::'a assume lt: "x < y"
+ from less_half_sum[OF lt] show "x < (x + y) /2" by simp
+next
+ fix x y::'a assume lt: "x < y"
+ from gt_half_sum[OF lt] show "(x + y) /2 < y" by simp
+qed
+
+declaration{*
+let
+fun earlier [] x y = false
+ | earlier (h::t) x y =
+ if h aconvc y then false else if h aconvc x then true else earlier t x y;
+
+fun dest_frac ct = case term_of ct of
+ Const (@{const_name "HOL.divide"},_) $ a $ b=>
+ Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b))
+ | t => Rat.rat_of_int (snd (HOLogic.dest_number t))
+
+fun mk_frac phi cT x =
+ let val (a, b) = Rat.quotient_of_rat x
+ in if b = 1 then Numeral.mk_cnumber cT a
+ else Thm.capply
+ (Thm.capply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"})
+ (Numeral.mk_cnumber cT a))
+ (Numeral.mk_cnumber cT b)
+ end
+
+fun whatis x ct = case term_of ct of
+ Const(@{const_name "HOL.plus"}, _)$(Const(@{const_name "HOL.times"},_)$_$y)$_ =>
+ if y aconv term_of x then ("c*x+t",[(funpow 2 Thm.dest_arg1) ct, Thm.dest_arg ct])
+ else ("Nox",[])
+| Const(@{const_name "HOL.plus"}, _)$y$_ =>
+ if y aconv term_of x then ("x+t",[Thm.dest_arg ct])
+ else ("Nox",[])
+| Const(@{const_name "HOL.times"}, _)$_$y =>
+ if y aconv term_of x then ("c*x",[Thm.dest_arg1 ct])
+ else ("Nox",[])
+| t => if t aconv term_of x then ("x",[]) else ("Nox",[]);
+
+fun xnormalize_conv ctxt [] ct = reflexive ct
+| xnormalize_conv ctxt (vs as (x::_)) ct =
+ case term_of ct of
+ Const(@{const_name HOL.less},_)$_$Const(@{const_name "HOL.zero"},_) =>
+ (case whatis x (Thm.dest_arg1 ct) of
+ ("c*x+t",[c,t]) =>
+ let
+ val cr = dest_frac c
+ val clt = Thm.dest_fun2 ct
+ val cz = Thm.dest_arg ct
+ val neg = cr </ Rat.zero
+ val cthp = Simplifier.rewrite (local_simpset_of ctxt)
+ (Thm.capply @{cterm "Trueprop"}
+ (if neg then Thm.capply (Thm.capply clt c) cz
+ else Thm.capply (Thm.capply clt cz) c))
+ val cth = equal_elim (symmetric cthp) TrueI
+ val th = implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x,t])
+ (if neg then @{thm neg_prod_sum_lt} else @{thm pos_prod_sum_lt})) cth
+ val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
+ (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
+ in rth end
+ | ("x+t",[t]) =>
+ let
+ val T = ctyp_of_term x
+ val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_lt"}
+ val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
+ (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
+ in rth end
+ | ("c*x",[c]) =>
+ let
+ val cr = dest_frac c
+ val clt = Thm.dest_fun2 ct
+ val cz = Thm.dest_arg ct
+ val neg = cr </ Rat.zero
+ val cthp = Simplifier.rewrite (local_simpset_of ctxt)
+ (Thm.capply @{cterm "Trueprop"}
+ (if neg then Thm.capply (Thm.capply clt c) cz
+ else Thm.capply (Thm.capply clt cz) c))
+ val cth = equal_elim (symmetric cthp) TrueI
+ val th = implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x])
+ (if neg then @{thm neg_prod_lt} else @{thm pos_prod_lt})) cth
+ val rth = th
+ in rth end
+ | _ => reflexive ct)
+
+
+| Const(@{const_name HOL.less_eq},_)$_$Const(@{const_name "HOL.zero"},_) =>
+ (case whatis x (Thm.dest_arg1 ct) of
+ ("c*x+t",[c,t]) =>
+ let
+ val T = ctyp_of_term x
+ val cr = dest_frac c
+ val clt = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat "op <"}
+ val cz = Thm.dest_arg ct
+ val neg = cr </ Rat.zero
+ val cthp = Simplifier.rewrite (local_simpset_of ctxt)
+ (Thm.capply @{cterm "Trueprop"}
+ (if neg then Thm.capply (Thm.capply clt c) cz
+ else Thm.capply (Thm.capply clt cz) c))
+ val cth = equal_elim (symmetric cthp) TrueI
+ val th = implies_elim (instantiate' [SOME T] (map SOME [c,x,t])
+ (if neg then @{thm neg_prod_sum_le} else @{thm pos_prod_sum_le})) cth
+ val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
+ (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
+ in rth end
+ | ("x+t",[t]) =>
+ let
+ val T = ctyp_of_term x
+ val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_le"}
+ val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
+ (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
+ in rth end
+ | ("c*x",[c]) =>
+ let
+ val T = ctyp_of_term x
+ val cr = dest_frac c
+ val clt = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat "op <"}
+ val cz = Thm.dest_arg ct
+ val neg = cr </ Rat.zero
+ val cthp = Simplifier.rewrite (local_simpset_of ctxt)
+ (Thm.capply @{cterm "Trueprop"}
+ (if neg then Thm.capply (Thm.capply clt c) cz
+ else Thm.capply (Thm.capply clt cz) c))
+ val cth = equal_elim (symmetric cthp) TrueI
+ val th = implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x])
+ (if neg then @{thm neg_prod_le} else @{thm pos_prod_le})) cth
+ val rth = th
+ in rth end
+ | _ => reflexive ct)
+
+| Const("op =",_)$_$Const(@{const_name "HOL.zero"},_) =>
+ (case whatis x (Thm.dest_arg1 ct) of
+ ("c*x+t",[c,t]) =>
+ let
+ val T = ctyp_of_term x
+ val cr = dest_frac c
+ val ceq = Thm.dest_fun2 ct
+ val cz = Thm.dest_arg ct
+ val cthp = Simplifier.rewrite (local_simpset_of ctxt)
+ (Thm.capply @{cterm "Trueprop"}
+ (Thm.capply @{cterm "Not"} (Thm.capply (Thm.capply ceq c) cz)))
+ val cth = equal_elim (symmetric cthp) TrueI
+ val th = implies_elim
+ (instantiate' [SOME T] (map SOME [c,x,t]) @{thm nz_prod_sum_eq}) cth
+ val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
+ (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
+ in rth end
+ | ("x+t",[t]) =>
+ let
+ val T = ctyp_of_term x
+ val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_eq"}
+ val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
+ (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
+ in rth end
+ | ("c*x",[c]) =>
+ let
+ val T = ctyp_of_term x
+ val cr = dest_frac c
+ val ceq = Thm.dest_fun2 ct
+ val cz = Thm.dest_arg ct
+ val cthp = Simplifier.rewrite (local_simpset_of ctxt)
+ (Thm.capply @{cterm "Trueprop"}
+ (Thm.capply @{cterm "Not"} (Thm.capply (Thm.capply ceq c) cz)))
+ val cth = equal_elim (symmetric cthp) TrueI
+ val rth = implies_elim
+ (instantiate' [SOME T] (map SOME [c,x]) @{thm nz_prod_eq}) cth
+ in rth end
+ | _ => reflexive ct);
+
+local
+ val less_iff_diff_less_0 = mk_meta_eq @{thm "less_iff_diff_less_0"}
+ val le_iff_diff_le_0 = mk_meta_eq @{thm "le_iff_diff_le_0"}
+ val eq_iff_diff_eq_0 = mk_meta_eq @{thm "eq_iff_diff_eq_0"}
+in
+fun field_isolate_conv phi ctxt vs ct = case term_of ct of
+ Const(@{const_name HOL.less},_)$a$b =>
+ let val (ca,cb) = Thm.dest_binop ct
+ val T = ctyp_of_term ca
+ val th = instantiate' [SOME T] [SOME ca, SOME cb] less_iff_diff_less_0
+ val nth = Conv.fconv_rule
+ (Conv.arg_conv (Conv.arg1_conv
+ (Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th
+ val rth = transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
+ in rth end
+| Const(@{const_name HOL.less_eq},_)$a$b =>
+ let val (ca,cb) = Thm.dest_binop ct
+ val T = ctyp_of_term ca
+ val th = instantiate' [SOME T] [SOME ca, SOME cb] le_iff_diff_le_0
+ val nth = Conv.fconv_rule
+ (Conv.arg_conv (Conv.arg1_conv
+ (Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th
+ val rth = transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
+ in rth end
+
+| Const("op =",_)$a$b =>
+ let val (ca,cb) = Thm.dest_binop ct
+ val T = ctyp_of_term ca
+ val th = instantiate' [SOME T] [SOME ca, SOME cb] eq_iff_diff_eq_0
+ val nth = Conv.fconv_rule
+ (Conv.arg_conv (Conv.arg1_conv
+ (Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th
+ val rth = transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
+ in rth end
+| @{term "Not"} $(Const("op =",_)$a$b) => Conv.arg_conv (field_isolate_conv phi ctxt vs) ct
+| _ => reflexive ct
+end;
+
+fun classfield_whatis phi =
+ let
+ fun h x t =
+ case term_of t of
+ Const("op =", _)$y$z => if term_of x aconv y then Ferrante_Rackoff_Data.Eq
+ else Ferrante_Rackoff_Data.Nox
+ | @{term "Not"}$(Const("op =", _)$y$z) => if term_of x aconv y then Ferrante_Rackoff_Data.NEq
+ else Ferrante_Rackoff_Data.Nox
+ | Const(@{const_name HOL.less},_)$y$z =>
+ if term_of x aconv y then Ferrante_Rackoff_Data.Lt
+ else if term_of x aconv z then Ferrante_Rackoff_Data.Gt
+ else Ferrante_Rackoff_Data.Nox
+ | Const (@{const_name HOL.less_eq},_)$y$z =>
+ if term_of x aconv y then Ferrante_Rackoff_Data.Le
+ else if term_of x aconv z then Ferrante_Rackoff_Data.Ge
+ else Ferrante_Rackoff_Data.Nox
+ | _ => Ferrante_Rackoff_Data.Nox
+ in h end;
+fun class_field_ss phi =
+ HOL_basic_ss addsimps ([@{thm "linorder_not_less"}, @{thm "linorder_not_le"}])
+ addsplits [@{thm "abs_split"},@{thm "split_max"}, @{thm "split_min"}]
+
+in
+Ferrante_Rackoff_Data.funs @{thm "class_ordered_field_dense_linear_order.ferrack_axiom"}
+ {isolate_conv = field_isolate_conv, whatis = classfield_whatis, simpset = class_field_ss}
+end
+*}
+
+
+end