(* 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 < []" by (simp add: list_less_def) lemma Nil_less_Cons [simp]: "[] < a # x" by (simp add: list_less_def) 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 by standard (simp add: bot_list_def) end end