src/HOL/Decision_Procs/DP_Library.thy

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Decision_Procs/DP_Library.thy	Fri Feb 25 20:07:48 2011 +0100
@@ -0,0 +1,36 @@
+theory DP_Library
+imports Main
+begin
+
+primrec alluopairs:: "'a list \<Rightarrow> ('a \<times> 'a) list" where
+  "alluopairs [] = []"
+| "alluopairs (x#xs) = (map (Pair x) (x#xs))@(alluopairs xs)"
+
+lemma alluopairs_set1: "set (alluopairs xs) \<le> {(x,y). x\<in> set xs \<and> y\<in> set xs}"
+by (induct xs, auto)
+
+lemma alluopairs_set:
+  "\<lbrakk>x\<in> set xs ; y \<in> set xs\<rbrakk> \<Longrightarrow> (x,y) \<in> set (alluopairs xs) \<or> (y,x) \<in> set (alluopairs xs) "
+by (induct xs, auto)
+
+lemma alluopairs_bex:
+  assumes Pc: "\<forall> x \<in> set xs. \<forall>y\<in> set xs. P x y = P y x"
+  shows "(\<exists> x \<in> set xs. \<exists> y \<in> set xs. P x y) = (\<exists> (x,y) \<in> set (alluopairs xs). P x y)"
+proof
+  assume "\<exists>x\<in>set xs. \<exists>y\<in>set xs. P x y"
+  then obtain x y where x: "x \<in> set xs" and y:"y \<in> set xs" and P: "P x y"  by blast
+  from alluopairs_set[OF x y] P Pc x y show"\<exists>(x, y)\<in>set (alluopairs xs). P x y" 
+    by auto
+next
+  assume "\<exists>(x, y)\<in>set (alluopairs xs). P x y"
+  then obtain "x" and "y"  where xy:"(x,y) \<in> set (alluopairs xs)" and P: "P x y" by blast+
+  from xy have "x \<in> set xs \<and> y\<in> set xs" using alluopairs_set1 by blast
+  with P show "\<exists>x\<in>set xs. \<exists>y\<in>set xs. P x y" by blast
+qed
+
+lemma alluopairs_ex:
+  "\<forall> x y. P x y = P y x \<Longrightarrow>
+  (\<exists> x \<in> set xs. \<exists> y \<in> set xs. P x y) = (\<exists> (x,y) \<in> set (alluopairs xs). P x y)"
+by(blast intro!: alluopairs_bex)
+
+end