moved lemmas that require the HOL-Complex logic image to Complex/ex/BigO_Complex.thy;
tuned presentation;
(* 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