(* Title: HOL/Library/Char_ord.thy
Author: Norbert Voelker, Florian Haftmann
*)
section \<open>Order on characters\<close>
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
by standard (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 :: nibble \<Rightarrow> _) = min"
definition "(sup :: nibble \<Rightarrow> _) = max"
instance
by standard (auto simp add: inf_nibble_def sup_nibble_def max_min_distrib2)
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
by standard (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 :: char \<Rightarrow> _) = min"
definition "(sup :: char \<Rightarrow> _) = max"
instance
by standard (auto simp add: inf_char_def sup_char_def max_min_distrib2)
end
instantiation String.literal :: linorder
begin
context includes literal.lifting begin
lift_definition less_literal :: "String.literal \<Rightarrow> String.literal \<Rightarrow> bool" is "ord.lexordp op <" .
lift_definition less_eq_literal :: "String.literal \<Rightarrow> String.literal \<Rightarrow> bool" is "ord.lexordp_eq op <" .
instance
proof -
interpret linorder "ord.lexordp_eq op <" "ord.lexordp op < :: string \<Rightarrow> string \<Rightarrow> bool"
by(rule linorder.lexordp_linorder[where less_eq="op \<le>"])(unfold_locales)
show "PROP ?thesis"
by(intro_classes)(transfer, simp add: less_le_not_le linear)+
qed
end
end
lemma less_literal_code [code]:
"op < = (\<lambda>xs ys. ord.lexordp op < (String.explode xs) (String.explode ys))"
by(simp add: less_literal.rep_eq fun_eq_iff)
lemma less_eq_literal_code [code]:
"op \<le> = (\<lambda>xs ys. ord.lexordp_eq op < (String.explode xs) (String.explode ys))"
by(simp add: less_eq_literal.rep_eq fun_eq_iff)
lifting_update literal.lifting
lifting_forget literal.lifting
text \<open>Legacy aliasses\<close>
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