src/HOL/Library/List_Lexorder.thy
 author paulson Fri, 17 Apr 2020 20:55:53 +0100 changeset 71766 1249b998e377 parent 68312 e9b5f25f6712 child 72166 bb37571139bf permissions -rw-r--r--
New theory Library/List_Lenlexorder.thy, a type class instantiation for well-ordering lists
```
(*  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
let ?r = "{(u, v::'a). u < v}"
have tr: "trans ?r"
using trans_def by fastforce
have \<section>: False
if "(xs,ys) \<in> lexord ?r" "(ys,xs) \<in> lexord ?r" for xs ys :: "'a list"
proof -
have "(xs,xs) \<in> lexord ?r"
using that transD [OF lexord_transI [OF tr]] by blast
then show False
by (meson case_prodD lexord_irreflexive less_irrefl mem_Collect_eq)
qed
show "xs \<le> xs" for xs :: "'a list" by (simp add: list_le_def)
show "xs \<le> zs" if "xs \<le> ys" and "ys \<le> zs" for xs ys zs :: "'a list"
using that transD [OF lexord_transI [OF tr]] by (auto simp add: list_le_def list_less_def)
show "xs = ys" if "xs \<le> ys" "ys \<le> xs" for xs ys :: "'a list"
using \<section> that list_le_def list_less_def by blast
show "xs < ys \<longleftrightarrow> xs \<le> ys \<and> \<not> ys \<le> xs" for xs ys :: "'a list"
by (auto simp add: list_less_def list_le_def dest: \<section>)
qed

instance list :: (linorder) linorder
proof
fix xs ys :: "'a list"
have "total (lexord {(u, v::'a). u < v})"
by (rule total_lexord) (auto simp: total_on_def)
then show "xs \<le> ys \<or> ys \<le> xs"
by (auto simp add: total_on_def 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 < []"

lemma Nil_less_Cons [simp]: "[] < a # x"

lemma Cons_less_Cons [simp]: "a # x < b # y \<longleftrightarrow> a < b \<or> a = b \<and> x < y"

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

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
```