(* Title: HOL/Library/List_Lexorder.thy
Author: Norbert Voelker
*)
section \<open>Lexicographic order on lists\<close>
theory List_Lexorder
imports Main
begin
instantiation list :: (ord) ord
begin
definition
list_less_def: "xs < ys \<longleftrightarrow> (xs, ys) \<in> lexord {(u, v). u < v}"
definition
list_le_def: "(xs :: _ list) \<le> ys \<longleftrightarrow> xs < ys \<or> xs = ys"
instance ..
end
instance list :: (order) order
proof
fix xs :: "'a list"
show "xs \<le> xs" by (simp add: list_le_def)
next
fix xs ys zs :: "'a list"
assume "xs \<le> ys" and "ys \<le> zs"
then show "xs \<le> zs"
apply (auto simp add: list_le_def list_less_def)
apply (rule lexord_trans)
apply (auto intro: transI)
done
next
fix xs ys :: "'a list"
assume "xs \<le> ys" and "ys \<le> xs"
then show "xs = ys"
apply (auto simp add: list_le_def list_less_def)
apply (rule lexord_irreflexive [THEN notE])
defer
apply (rule lexord_trans)
apply (auto intro: transI)
done
next
fix xs ys :: "'a list"
show "xs < ys \<longleftrightarrow> xs \<le> ys \<and> \<not> ys \<le> xs"
apply (auto simp add: list_less_def list_le_def)
defer
apply (rule lexord_irreflexive [THEN notE])
apply auto
apply (rule lexord_irreflexive [THEN notE])
defer
apply (rule lexord_trans)
apply (auto intro: transI)
done
qed
instance list :: (linorder) linorder
proof
fix xs ys :: "'a list"
have "(xs, ys) \<in> lexord {(u, v). u < v} \<or> xs = ys \<or> (ys, xs) \<in> lexord {(u, v). u < v}"
by (rule lexord_linear) auto
then show "xs \<le> ys \<or> ys \<le> xs"
by (auto simp add: list_le_def list_less_def)
qed
instantiation list :: (linorder) distrib_lattice
begin
definition "(inf :: 'a list \<Rightarrow> _) = min"
definition "(sup :: 'a list \<Rightarrow> _) = max"
instance
by standard (auto simp add: inf_list_def sup_list_def max_min_distrib2)
end
lemma not_less_Nil [simp]: "\<not> x < []"
by (simp add: list_less_def)
lemma Nil_less_Cons [simp]: "[] < a # x"
by (simp add: list_less_def)
lemma Cons_less_Cons [simp]: "a # x < b # y \<longleftrightarrow> a < b \<or> a = b \<and> x < y"
by (simp add: list_less_def)
lemma le_Nil [simp]: "x \<le> [] \<longleftrightarrow> x = []"
unfolding list_le_def by (cases x) auto
lemma Nil_le_Cons [simp]: "[] \<le> x"
unfolding list_le_def by (cases x) auto
lemma Cons_le_Cons [simp]: "a # x \<le> b # y \<longleftrightarrow> a < b \<or> a = b \<and> x \<le> y"
unfolding list_le_def by auto
instantiation list :: (order) order_bot
begin
definition "bot = []"
instance
by standard (simp add: bot_list_def)
end
lemma less_list_code [code]:
"xs < ([]::'a::{equal, order} list) \<longleftrightarrow> False"
"[] < (x::'a::{equal, order}) # xs \<longleftrightarrow> True"
"(x::'a::{equal, order}) # xs < y # ys \<longleftrightarrow> x < y \<or> x = y \<and> xs < ys"
by simp_all
lemma less_eq_list_code [code]:
"x # xs \<le> ([]::'a::{equal, order} list) \<longleftrightarrow> False"
"[] \<le> (xs::'a::{equal, order} list) \<longleftrightarrow> True"
"(x::'a::{equal, order}) # xs \<le> y # ys \<longleftrightarrow> x < y \<or> x = y \<and> xs \<le> ys"
by simp_all
end