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