src/HOL/Library/Char_ord.thy

section headings for ROOT.ML;

(*  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