(* 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