(* Title: HOL/Library/List_lexord.thy ID: $Id: List_lexord.thy,v 1.11 2008/01/02 14:14:18 haftmann Exp $ Author: Norbert Voelker *) header {* Lexicographic order on lists *} theory List_lexord imports List begin instantiation list :: (ord) ord begin definition list_less_def [code func del]: "(xs::('a::ord) list) < ys <-> (xs, ys) ∈ lexord {(u,v). u < v}" definition list_le_def [code func del]: "(xs::('a::ord) list) ≤ ys <-> (xs < ys ∨ xs = ys)" instance .. end instance list :: (order) order apply (intro_classes, unfold list_less_def list_le_def) apply safe apply (rule_tac r1 = "{(a::'a,b). a < b}" in lexord_irreflexive [THEN notE]) apply simp apply assumption apply (blast intro: lexord_trans transI order_less_trans) apply (rule_tac r1 = "{(a::'a,b). a < b}" in lexord_irreflexive [THEN notE]) apply simp apply (blast intro: lexord_trans transI order_less_trans) done instance list :: (linorder) linorder apply (intro_classes, unfold list_le_def list_less_def, safe) apply (cut_tac x = x and y = y and r = "{(a,b). a < b}" in lexord_linear) apply force apply simp done instantiation list :: (linorder) distrib_lattice begin definition [code func del]: "(inf :: 'a list => _) = min" definition [code func del]: "(sup :: 'a list => _) = max" instance by intro_classes (auto simp add: inf_list_def sup_list_def min_max.sup_inf_distrib1) end lemma not_less_Nil [simp]: "¬ (x < [])" by (unfold list_less_def) simp lemma Nil_less_Cons [simp]: "[] < a # x" by (unfold list_less_def) simp lemma Cons_less_Cons [simp]: "a # x < b # y <-> a < b ∨ a = b ∧ x < y" by (unfold list_less_def) simp lemma le_Nil [simp]: "x ≤ [] <-> x = []" by (unfold list_le_def, cases x) auto lemma Nil_le_Cons [simp]: "[] ≤ x" by (unfold list_le_def, cases x) auto lemma Cons_le_Cons [simp]: "a # x ≤ b # y <-> a < b ∨ a = b ∧ x ≤ y" by (unfold list_le_def) auto lemma less_code [code func]: "xs < ([]::'a::{eq, order} list) <-> False" "[] < (x::'a::{eq, order}) # xs <-> True" "(x::'a::{eq, order}) # xs < y # ys <-> x < y ∨ x = y ∧ xs < ys" by simp_all lemma less_eq_code [code func]: "x # xs ≤ ([]::'a::{eq, order} list) <-> False" "[] ≤ (xs::'a::{eq, order} list) <-> True" "(x::'a::{eq, order}) # xs ≤ y # ys <-> x < y ∨ x = y ∧ xs ≤ ys" by simp_all end
lemma not_less_Nil:
¬ x < []
lemma Nil_less_Cons:
[] < a # x
lemma Cons_less_Cons:
(a # x < b # y) = (a < b ∨ a = b ∧ x < y)
lemma le_Nil:
(x ≤ []) = (x = [])
lemma Nil_le_Cons:
[] ≤ x
lemma Cons_le_Cons:
(a # x ≤ b # y) = (a < b ∨ a = b ∧ x ≤ y)
lemma less_code:
(xs < []) = False
([] < x # xs) = True
(x # xs < y # ys) = (x < y ∨ x = y ∧ xs < ys)
lemma less_eq_code:
(x # xs ≤ []) = False
([] ≤ xs) = True
(x # xs ≤ y # ys) = (x < y ∨ x = y ∧ xs ≤ ys)