(* Title: HOL/Library/Char_ord.thy
Author: Norbert Voelker, Florian Haftmann
*)
section \<open>Order on characters\<close>
theory Char_ord
imports Main
begin
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)
end
lemma less_eq_char_simps:
"(0 :: char) \<le> c"
"Char k \<le> 0 \<longleftrightarrow> numeral k mod 256 = (0 :: nat)"
"Char k \<le> Char l \<longleftrightarrow> numeral k mod 256 \<le> (numeral l mod 256 :: nat)"
by (simp_all add: Char_def less_eq_char_def)
lemma less_char_simps:
"\<not> c < (0 :: char)"
"0 < Char k \<longleftrightarrow> (0 :: nat) < numeral k mod 256"
"Char k < Char l \<longleftrightarrow> numeral k mod 256 < (numeral l mod 256 :: nat)"
by (simp_all add: Char_def less_char_def)
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
end