(* Title: HOL/Library/List_lexord.thy
Author: Norbert Voelker
*)
header {* Lexicographic order on lists *}
theory List_lexord
imports List 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" by (auto simp add: list_le_def list_less_def)
(rule lexord_trans, auto intro: transI)
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
[code del]: "(inf \<Colon> 'a list \<Rightarrow> _) = min"
definition
[code 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
instantiation list :: (order) bot
begin
definition
"bot = []"
instance proof
qed (simp add: bot_list_def)
end
lemma less_list_code [code]:
"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_list_code [code]:
"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