src/HOL/Library/List_lexord.thy

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/List_lexord.thy	Fri Apr 15 14:14:24 2005 +0200
@@ -0,0 +1,53 @@
+(*  Title:      HOL/Library/List_lexord.thy
+    ID:         $Id$
+    Author:     Norbert Voelker
+*)
+
+header {* Instantiation of order classes for lexord on lists *}
+
+theory List_lexord
+imports Main
+begin
+
+instance list :: (ord) ord ..
+defs(overloaded)
+  list_le_def:  "(xs::('a::ord) list) \<le> ys \<equiv> (xs < ys \<or> xs = ys)" 
+  list_less_def: "(xs::('a::ord) list) < ys \<equiv> (xs,ys) \<in> lexord {(u,v). u < v}"
+
+lemmas list_ord_defs = list_less_def list_le_def
+
+instance list::(order)order
+  apply (intro_classes, unfold list_ord_defs)
+  apply (rule disjI2, safe)
+  apply (blast intro: lexord_trans transI order_less_trans)
+  apply (rule_tac r1 = "{(a::'a,b). a < b}" in lexord_irreflexive [THEN notE])
+  apply simp
+  apply (blast intro: lexord_trans transI order_less_trans)
+  apply (rule_tac r1 = "{(a::'a,b). a < b}" in lexord_irreflexive [THEN notE])
+  apply simp
+  by assumption
+
+instance list::(linorder)linorder
+  apply (intro_classes, unfold list_le_def list_less_def, safe)
+  apply (cut_tac x="x" and y="y" and  r = "{(a,b). a < b}"  in lexord_linear)
+  by (force, simp)
+
+lemma not_less_Nil[simp]: "~(x < [])";
+  by (unfold list_less_def, simp);
+
+lemma Nil_less_Cons[simp]: "[] < a # x";
+  by (unfold list_less_def, simp);
+
+lemma Cons_less_Cons[simp]: "(a # x < b # y) = (a < b | a = b & x < y)";
+  by (unfold list_less_def, simp);
+
+lemma le_Nil[simp]: "(x <= [])   = (x = [])";
+  by (unfold list_ord_defs, case_tac x, auto);
+
+lemma Nil_le_Cons[simp]: "([] <= x)";
+  by (unfold list_ord_defs, case_tac x, auto);
+
+lemma Cons_le_Cons[simp]: "(a # x <= b # y) = (a < b | a = b & x <= y)";
+  by (unfold list_ord_defs, auto);
+
+end
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