(* Title: HOL/Library/Char_ord.thy ID: $Id: Char_ord.thy,v 1.2 2005/08/31 13:46:37 wenzelm Exp $ Author: Norbert Voelker *) header {* Order on characters *} theory Char_ord imports Product_ord begin text {* Conversions between nibbles and integers in [0..15]. *} consts nibble_to_int:: "nibble => int" int_to_nibble:: "int => nibble" primrec "nibble_to_int Nibble0 = 0" "nibble_to_int Nibble1 = 1" "nibble_to_int Nibble2 = 2" "nibble_to_int Nibble3 = 3" "nibble_to_int Nibble4 = 4" "nibble_to_int Nibble5 = 5" "nibble_to_int Nibble6 = 6" "nibble_to_int Nibble7 = 7" "nibble_to_int Nibble8 = 8" "nibble_to_int Nibble9 = 9" "nibble_to_int NibbleA = 10" "nibble_to_int NibbleB = 11" "nibble_to_int NibbleC = 12" "nibble_to_int NibbleD = 13" "nibble_to_int NibbleE = 14" "nibble_to_int NibbleF = 15" defs int_to_nibble_def: "int_to_nibble x ≡ (let y = x mod 16 in if y = 0 then Nibble0 else if y = 1 then Nibble1 else if y = 2 then Nibble2 else if y = 3 then Nibble3 else if y = 4 then Nibble4 else if y = 5 then Nibble5 else if y = 6 then Nibble6 else if y = 7 then Nibble7 else if y = 8 then Nibble8 else if y = 9 then Nibble9 else if y = 10 then NibbleA else if y = 11 then NibbleB else if y = 12 then NibbleC else if y = 13 then NibbleD else if y = 14 then NibbleE else NibbleF)" lemma int_to_nibble_nibble_to_int: "int_to_nibble(nibble_to_int x) = x" by (cases x) (auto simp: int_to_nibble_def Let_def) lemma inj_nibble_to_int: "inj nibble_to_int" by (rule inj_on_inverseI) (rule int_to_nibble_nibble_to_int) lemmas nibble_to_int_eq = inj_nibble_to_int [THEN inj_eq] lemma nibble_to_int_ge_0: "0 ≤ nibble_to_int x" by (cases x) auto lemma nibble_to_int_less_16: "nibble_to_int x < 16" by (cases x) auto text {* Conversion between chars and int pairs. *} consts char_to_int_pair :: "char => int × int" primrec "char_to_int_pair (Char a b) = (nibble_to_int a, nibble_to_int b)" lemma inj_char_to_int_pair: "inj char_to_int_pair" apply (rule inj_onI) apply (case_tac x, case_tac y) apply (auto simp: nibble_to_int_eq) done lemmas char_to_int_pair_eq = inj_char_to_int_pair [THEN inj_eq] text {* Instantiation of order classes *} instance char :: ord .. defs (overloaded) char_le_def: "c ≤ d ≡ (char_to_int_pair c ≤ char_to_int_pair d)" char_less_def: "c < d ≡ (char_to_int_pair c < char_to_int_pair d)" lemmas char_ord_defs = char_less_def char_le_def instance char :: order apply intro_classes apply (unfold char_ord_defs) apply (auto simp: char_to_int_pair_eq order_less_le) done instance char::linorder apply intro_classes apply (unfold char_le_def) apply auto done end
lemma int_to_nibble_nibble_to_int:
int_to_nibble (nibble_to_int x) = x
lemma inj_nibble_to_int:
inj nibble_to_int
lemmas nibble_to_int_eq:
(nibble_to_int x = nibble_to_int y) = (x = y)
lemmas nibble_to_int_eq:
(nibble_to_int x = nibble_to_int y) = (x = y)
lemma nibble_to_int_ge_0:
0 ≤ nibble_to_int x
lemma nibble_to_int_less_16:
nibble_to_int x < 16
lemma inj_char_to_int_pair:
inj char_to_int_pair
lemmas char_to_int_pair_eq:
(char_to_int_pair x = char_to_int_pair y) = (x = y)
lemmas char_to_int_pair_eq:
(char_to_int_pair x = char_to_int_pair y) = (x = y)
lemmas char_ord_defs:
c < d == char_to_int_pair c < char_to_int_pair d
c ≤ d == char_to_int_pair c ≤ char_to_int_pair d
lemmas char_ord_defs:
c < d == char_to_int_pair c < char_to_int_pair d
c ≤ d == char_to_int_pair c ≤ char_to_int_pair d