src/HOL/Decision_Procs/DP_Library.thy
 author blanchet Thu Sep 11 18:54:36 2014 +0200 (2014-09-11) changeset 58306 117ba6cbe414 parent 55814 aefa1db74d9d permissions -rw-r--r--
renamed 'rep_datatype' to 'old_rep_datatype' (HOL)
1 theory DP_Library
2 imports Main
3 begin
5 primrec alluopairs:: "'a list \<Rightarrow> ('a \<times> 'a) list"
6 where
7   "alluopairs [] = []"
8 | "alluopairs (x # xs) = map (Pair x) (x # xs) @ alluopairs xs"
10 lemma alluopairs_set1: "set (alluopairs xs) \<le> {(x, y). x\<in> set xs \<and> y\<in> set xs}"
11   by (induct xs) auto
13 lemma alluopairs_set:
14   "x\<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> (x, y) \<in> set (alluopairs xs) \<or> (y, x) \<in> set (alluopairs xs)"
15   by (induct xs) auto
17 lemma alluopairs_bex:
18   assumes Pc: "\<forall>x \<in> set xs. \<forall>y \<in> set xs. P x y = P y x"
19   shows "(\<exists>x \<in> set xs. \<exists>y \<in> set xs. P x y) \<longleftrightarrow> (\<exists>(x, y) \<in> set (alluopairs xs). P x y)"
20 proof
21   assume "\<exists>x \<in> set xs. \<exists>y \<in> set xs. P x y"
22   then obtain x y where x: "x \<in> set xs" and y: "y \<in> set xs" and P: "P x y"
23     by blast
24   from alluopairs_set[OF x y] P Pc x y show "\<exists>(x, y) \<in> set (alluopairs xs). P x y"
25     by auto
26 next
27   assume "\<exists>(x, y) \<in> set (alluopairs xs). P x y"
28   then obtain x and y where xy: "(x, y) \<in> set (alluopairs xs)" and P: "P x y"
29     by blast+
30   from xy have "x \<in> set xs \<and> y \<in> set xs"
31     using alluopairs_set1 by blast
32   with P show "\<exists>x\<in>set xs. \<exists>y\<in>set xs. P x y" by blast
33 qed
35 lemma alluopairs_ex:
36   "\<forall>x y. P x y = P y x \<Longrightarrow>
37     (\<exists>x \<in> set xs. \<exists>y \<in> set xs. P x y) = (\<exists>(x, y) \<in> set (alluopairs xs). P x y)"
38   by (blast intro!: alluopairs_bex)
40 end