Theory Char_ord

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theory Char_ord
imports Product_ord Char_nat
begin

(*  Title:      HOL/Library/Char_ord.thy
    ID:         $Id: Char_ord.thy,v 1.12 2008/01/02 14:14:17 haftmann Exp $
    Author:     Norbert Voelker, Florian Haftmann
*)

header {* Order on characters *}

theory Char_ord
imports Product_ord Char_nat
begin

instantiation nibble :: linorder
begin

definition
  nibble_less_eq_def: "n ≤ m <-> nat_of_nibble n ≤ nat_of_nibble m"

definition
  nibble_less_def: "n < m <-> nat_of_nibble n < nat_of_nibble m"

instance proof
  fix n :: nibble
  show "n ≤ n" unfolding nibble_less_eq_def nibble_less_def by auto
next
  fix n m q :: nibble
  assume "n ≤ m"
    and "m ≤ q"
  then show "n ≤ q" unfolding nibble_less_eq_def nibble_less_def by auto
next
  fix n m :: nibble
  assume "n ≤ m"
    and "m ≤ n"
  then show "n = m"
    unfolding nibble_less_eq_def nibble_less_def
    by (auto simp add: nat_of_nibble_eq)
next
  fix n m :: nibble
  show "n < m <-> n ≤ m ∧ n ≠ m"
    unfolding nibble_less_eq_def nibble_less_def less_le
    by (auto simp add: nat_of_nibble_eq)
next
  fix n m :: nibble
  show "n ≤ m ∨ m ≤ n"
    unfolding nibble_less_eq_def by auto
qed

end

instantiation nibble :: distrib_lattice
begin

definition
  "(inf :: nibble => _) = min"

definition
  "(sup :: nibble => _) = max"

instance by default (auto simp add:
    inf_nibble_def sup_nibble_def min_max.sup_inf_distrib1)

end

instantiation char :: linorder
begin

definition
  char_less_eq_def [code func del]: "c1 ≤ c2 <-> (case c1 of Char n1 m1 => case c2 of Char n2 m2 =>
    n1 < n2 ∨ n1 = n2 ∧ m1 ≤ m2)"

definition
  char_less_def [code func del]: "c1 < c2 <-> (case c1 of Char n1 m1 => case c2 of Char n2 m2 =>
    n1 < n2 ∨ n1 = n2 ∧ m1 < m2)"

instance
  by default (auto simp: char_less_eq_def char_less_def split: char.splits)

end

instantiation char :: distrib_lattice
begin

definition
  "(inf :: char => _) = min"

definition
  "(sup :: char => _) = max"

instance   by default (auto simp add:
    inf_char_def sup_char_def min_max.sup_inf_distrib1)

end

lemma [simp, code func]:
  shows char_less_eq_simp: "Char n1 m1 ≤ Char n2 m2 <-> n1 < n2 ∨ n1 = n2 ∧ m1 ≤ m2"
  and char_less_simp:      "Char n1 m1 < Char n2 m2 <-> n1 < n2 ∨ n1 = n2 ∧ m1 < m2"
  unfolding char_less_eq_def char_less_def by simp_all

end

lemma char_less_eq_simp:

  (Char n1.0 m1.0  Char n2.0 m2.0) = (n1.0 < n2.0n1.0 = n2.0m1.0  m2.0)

and char_less_simp:

  (Char n1.0 m1.0 < Char n2.0 m2.0) = (n1.0 < n2.0n1.0 = n2.0m1.0 < m2.0)