(* Title: HOL/Library/Char_ord.thy
Author: Norbert Voelker, Florian Haftmann
*)
header {* Order on characters *}
theory Char_ord
imports Main
begin
instantiation nibble :: linorder
begin
definition
"n \<le> m \<longleftrightarrow> nat_of_nibble n \<le> nat_of_nibble m"
definition
"n < m \<longleftrightarrow> nat_of_nibble n < nat_of_nibble m"
instance proof
qed (auto simp add: less_eq_nibble_def less_nibble_def not_le nat_of_nibble_eq_iff)
end
instantiation nibble :: distrib_lattice
begin
definition
"(inf \<Colon> nibble \<Rightarrow> _) = min"
definition
"(sup \<Colon> nibble \<Rightarrow> _) = max"
instance proof
qed (auto simp add: inf_nibble_def sup_nibble_def min_max.sup_inf_distrib1)
end
instantiation char :: linorder
begin
definition
"c1 \<le> c2 \<longleftrightarrow> nat_of_char c1 \<le> nat_of_char c2"
definition
"c1 < c2 \<longleftrightarrow> nat_of_char c1 < nat_of_char c2"
instance proof
qed (auto simp add: less_eq_char_def less_char_def nat_of_char_eq_iff)
end
lemma less_eq_char_Char:
"Char n1 m1 \<le> Char n2 m2 \<longleftrightarrow> n1 < n2 \<or> n1 = n2 \<and> m1 \<le> m2"
proof -
{
assume "nat_of_nibble n1 * 16 + nat_of_nibble m1
\<le> nat_of_nibble n2 * 16 + nat_of_nibble m2"
then have "nat_of_nibble n1 \<le> nat_of_nibble n2"
using nat_of_nibble_less_16 [of m1] nat_of_nibble_less_16 [of m2] by auto
}
note * = this
show ?thesis
using nat_of_nibble_less_16 [of m1] nat_of_nibble_less_16 [of m2]
by (auto simp add: less_eq_char_def nat_of_char_Char less_eq_nibble_def less_nibble_def not_less nat_of_nibble_eq_iff dest: *)
qed
lemma less_char_Char:
"Char n1 m1 < Char n2 m2 \<longleftrightarrow> n1 < n2 \<or> n1 = n2 \<and> m1 < m2"
proof -
{
assume "nat_of_nibble n1 * 16 + nat_of_nibble m1
< nat_of_nibble n2 * 16 + nat_of_nibble m2"
then have "nat_of_nibble n1 \<le> nat_of_nibble n2"
using nat_of_nibble_less_16 [of m1] nat_of_nibble_less_16 [of m2] by auto
}
note * = this
show ?thesis
using nat_of_nibble_less_16 [of m1] nat_of_nibble_less_16 [of m2]
by (auto simp add: less_char_def nat_of_char_Char less_eq_nibble_def less_nibble_def not_less nat_of_nibble_eq_iff dest: *)
qed
instantiation char :: distrib_lattice
begin
definition
"(inf \<Colon> char \<Rightarrow> _) = min"
definition
"(sup \<Colon> char \<Rightarrow> _) = max"
instance proof
qed (auto simp add: inf_char_def sup_char_def min_max.sup_inf_distrib1)
end
text {* Legacy aliasses *}
lemmas nibble_less_eq_def = less_eq_nibble_def
lemmas nibble_less_def = less_nibble_def
lemmas char_less_eq_def = less_eq_char_def
lemmas char_less_def = less_char_def
lemmas char_less_eq_simp = less_eq_char_Char
lemmas char_less_simp = less_char_Char
end