(*  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