# Theory List_Lexorder

```(*  Title:      HOL/Library/List_Lexorder.thy
Author:     Norbert Voelker
*)

section ‹Lexicographic order on lists›

theory List_Lexorder
imports Main
begin

instantiation list :: (ord) ord
begin

definition
list_less_def: "xs < ys ⟷ (xs, ys) ∈ lexord {(u, v). u < v}"

definition
list_le_def: "(xs :: _ list) ≤ ys ⟷ xs < ys ∨ 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 §: False
if "(xs,ys) ∈ lexord ?r" "(ys,xs) ∈ lexord ?r" for xs ys :: "'a list"
proof -
have "(xs,xs) ∈ 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 ≤ xs" for xs :: "'a list" by (simp add: list_le_def)
show "xs ≤ zs" if "xs ≤ ys" and "ys ≤ 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 ≤ ys" "ys ≤ xs" for xs ys :: "'a list"
using § that list_le_def list_less_def by blast
show "xs < ys ⟷ xs ≤ ys ∧ ¬ ys ≤ xs" for xs ys :: "'a list"
by (auto simp add: list_less_def list_le_def dest: §)
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 ≤ ys ∨ ys ≤ 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 ⇒ _) = min"

definition "(sup :: 'a list ⇒ _) = max"

instance
by standard (auto simp add: inf_list_def sup_list_def max_min_distrib2)

end

lemma not_less_Nil [simp]: "¬ 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 ⟷ a < b ∨ a = b ∧ x < y"
by (simp add: list_less_def)

lemma le_Nil [simp]: "x ≤ [] ⟷ x = []"
unfolding list_le_def by (cases x) auto

lemma Nil_le_Cons [simp]: "[] ≤ x"
unfolding list_le_def by (cases x) auto

lemma Cons_le_Cons [simp]: "a # x ≤ b # y ⟷ a < b ∨ a = b ∧ x ≤ 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) ⟷ False"
"[] < (x::'a::{equal, order}) # xs ⟷ True"
"(x::'a::{equal, order}) # xs < y # ys ⟷ x < y ∨ x = y ∧ xs < ys"
by simp_all

lemma less_eq_list_code [code]:
"x # xs ≤ ([]::'a::{equal, order} list) ⟷ False"
"[] ≤ (xs::'a::{equal, order} list) ⟷ True"
"(x::'a::{equal, order}) # xs ≤ y # ys ⟷ x < y ∨ x = y ∧ xs ≤ ys"
by simp_all

end
```