(* Title: HOL/Library/List_lexord.thy
ID: $Id$
Author: Norbert Voelker
*)
header {* Lexicographic order on lists *}
theory List_lexord
imports Main
begin
instance list :: (ord) ord
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
apply assumption
done
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)
apply force
apply simp
done
lemma not_less_Nil [simp, code func]: "~(x < [])"
by (unfold list_less_def) simp
lemma Nil_less_Cons [simp, code func]: "[] < a # x"
by (unfold list_less_def) simp
lemma Cons_less_Cons [simp, code func]: "(a # x < b # y) = (a < b | a = b & x < y)"
by (unfold list_less_def) simp
lemma le_Nil [simp, code func]: "(x <= []) = (x = [])"
by (unfold list_ord_defs, cases x) auto
lemma Nil_le_Cons [simp, code func]: "([] <= x)"
by (unfold list_ord_defs, cases x) auto
lemma Cons_le_Cons [simp, code func]: "(a # x <= b # y) = (a < b | a = b & x <= y)"
by (unfold list_ord_defs) auto
end