# Theory List_Lenlexorder

```(*  Title:      HOL/Library/List_Lenlexorder.thy
*)

section ‹Lexicographic order on lists›

text ‹This version prioritises length and can yield wellorderings›

theory List_Lenlexorder
imports Main
begin

instantiation list :: (ord) ord
begin

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

definition
list_le_def: "(xs :: _ list) ≤ ys ⟷ xs < ys ∨ xs = ys"

instance ..

end

instance list :: (order) order
proof
have tr: "trans {(u, v::'a). u < v}"
using trans_def by fastforce
have §: False
if "(xs,ys) ∈ lenlex {(u, v). u < v}" "(ys,xs) ∈ lenlex {(u, v). u < v}" for xs ys :: "'a list"
proof -
have "(xs,xs) ∈ lenlex {(u, v). u < v}"
using that transD [OF lenlex_transI [OF tr]] by blast
then show False
by (meson case_prodD lenlex_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 lenlex_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 (lenlex {(u, v::'a). u < v})"
by (rule total_lenlex) (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

instance list :: (wellorder) wellorder
proof
fix P :: "'a list ⇒ bool" and a
assume "⋀x. (⋀y. y < x ⟹ P y) ⟹ P x"
then show "P a"
unfolding list_less_def by (metis wf_lenlex wf_induct wf_lenlex wf)
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 < []"

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

lemma Cons_less_Cons: "a # x < b # y ⟷ length x < length y ∨ length x = length y ∧ (a < b ∨ a = b ∧ x < y)"
using lenlex_length
by (fastforce simp: list_less_def Cons_lenlex_iff)

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: "a # x ≤ b # y ⟷ length x < length y ∨ length x = length y ∧ (a < b ∨ a = b ∧ x ≤ y)"
by (auto simp: list_le_def Cons_less_Cons)

instantiation list :: (order) order_bot
begin

definition "bot = []"

instance