Theory List_lexord

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theory List_lexord
imports Main

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

header {* Lexicographic order on lists *}

theory List_lexord
imports List Main
begin

instantiation list :: (ord) ord
begin

definition
  list_less_def [code del]: "(xs::('a::ord) list) < ys <-> (xs, ys) ∈ lexord {(u,v). u < v}"

definition
  list_le_def [code del]: "(xs::('a::ord) list) ≤ ys <-> (xs < ys ∨ xs = ys)"

instance ..

end

instance list :: (order) order
proof
  fix xs :: "'a list"
  show "xs ≤ xs" by (simp add: list_le_def)
next
  fix xs ys zs :: "'a list"
  assume "xs ≤ ys" and "ys ≤ zs"
  then show "xs ≤ zs" by (auto simp add: list_le_def list_less_def)
    (rule lexord_trans, auto intro: transI)
next
  fix xs ys :: "'a list"
  assume "xs ≤ ys" and "ys ≤ xs"
  then show "xs = ys" apply (auto simp add: list_le_def list_less_def)
  apply (rule lexord_irreflexive [THEN notE])
  defer
  apply (rule lexord_trans) apply (auto intro: transI) done
next
  fix xs ys :: "'a list"
  show "xs < ys <-> xs ≤ ys ∧ ¬ ys ≤ xs" 
  apply (auto simp add: list_less_def list_le_def)
  defer
  apply (rule lexord_irreflexive [THEN notE])
  apply auto
  apply (rule lexord_irreflexive [THEN notE])
  defer
  apply (rule lexord_trans) apply (auto intro: transI) done
qed

instance list :: (linorder) linorder
proof
  fix xs ys :: "'a list"
  have "(xs, ys) ∈ lexord {(u, v). u < v} ∨ xs = ys ∨ (ys, xs) ∈ lexord {(u, v). u < v}"
    by (rule lexord_linear) auto
  then show "xs ≤ ys ∨ ys ≤ xs" 
    by (auto simp add: list_le_def list_less_def)
qed

instantiation list :: (linorder) distrib_lattice
begin

definition
  [code del]: "(inf :: 'a list => _) = min"

definition
  [code 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]:
  "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]:
  "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