(* Title: HOL/Library/List_lexord.thy
ID: $Id$
Author: Norbert Voelker
*)
header {* Lexicographic order on lists *}
theory List_lexord
imports Plain "~~/src/HOL/List"
begin
instantiation list :: (ord) ord
begin
definition
list_less_def [code func del]: "(xs::('a::ord) list) < ys \<longleftrightarrow> (xs, ys) \<in> lexord {(u,v). u < v}"
definition
list_le_def [code func del]: "(xs::('a::ord) list) \<le> ys \<longleftrightarrow> (xs < ys \<or> xs = ys)"
instance ..
end
instance list :: (order) order
apply (intro_classes, unfold list_less_def list_le_def)
apply safe
apply (rule_tac r1 = "{(a::'a,b). a < b}" in lexord_irreflexive [THEN notE])
apply simp
apply assumption
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)
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
instantiation list :: (linorder) distrib_lattice
begin
definition
[code func del]: "(inf \<Colon> 'a list \<Rightarrow> _) = min"
definition
[code func del]: "(sup \<Colon> 'a list \<Rightarrow> _) = max"
instance
by intro_classes
(auto simp add: inf_list_def sup_list_def min_max.sup_inf_distrib1)
end
lemma not_less_Nil [simp]: "\<not> (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 \<longleftrightarrow> a < b \<or> a = b \<and> x < y"
by (unfold list_less_def) simp
lemma le_Nil [simp]: "x \<le> [] \<longleftrightarrow> x = []"
by (unfold list_le_def, cases x) auto
lemma Nil_le_Cons [simp]: "[] \<le> x"
by (unfold list_le_def, cases x) auto
lemma Cons_le_Cons [simp]: "a # x \<le> b # y \<longleftrightarrow> a < b \<or> a = b \<and> x \<le> y"
by (unfold list_le_def) auto
lemma less_code [code func]:
"xs < ([]\<Colon>'a\<Colon>{eq, order} list) \<longleftrightarrow> False"
"[] < (x\<Colon>'a\<Colon>{eq, order}) # xs \<longleftrightarrow> True"
"(x\<Colon>'a\<Colon>{eq, order}) # xs < y # ys \<longleftrightarrow> x < y \<or> x = y \<and> xs < ys"
by simp_all
lemma less_eq_code [code func]:
"x # xs \<le> ([]\<Colon>'a\<Colon>{eq, order} list) \<longleftrightarrow> False"
"[] \<le> (xs\<Colon>'a\<Colon>{eq, order} list) \<longleftrightarrow> True"
"(x\<Colon>'a\<Colon>{eq, order}) # xs \<le> y # ys \<longleftrightarrow> x < y \<or> x = y \<and> xs \<le> ys"
by simp_all
end