--- a/src/HOL/Library/Char_ord.thy Thu Apr 26 13:33:12 2007 +0200
+++ b/src/HOL/Library/Char_ord.thy Thu Apr 26 13:33:15 2007 +0200
@@ -1,110 +1,63 @@
(* Title: HOL/Library/Char_ord.thy
ID: $Id$
- Author: Norbert Voelker
+ Author: Norbert Voelker, Florian Haftmann
*)
header {* Order on characters *}
theory Char_ord
-imports Product_ord
+imports Product_ord Char_nat
begin
-text {* Conversions between nibbles and integers in [0..15]. *}
-
-fun
- nibble_to_int:: "nibble \<Rightarrow> int" where
- "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"
+instance nibble :: linorder
+ nibble_less_eq_def: "n \<le> m \<equiv> nat_of_nibble n \<le> nat_of_nibble m"
+ nibble_less_def: "n < m \<equiv> nat_of_nibble n < nat_of_nibble m"
+proof
+ fix n :: nibble show "n \<le> n" unfolding nibble_less_eq_def nibble_less_def by auto
+next
+ fix n m q :: nibble
+ assume "n \<le> m"
+ and "m \<le> q"
+ then show "n \<le> q" unfolding nibble_less_eq_def nibble_less_def by auto
+next
+ fix n m :: nibble
+ assume "n \<le> m"
+ and "m \<le> n"
+ then show "n = m" unfolding nibble_less_eq_def nibble_less_def by (auto simp add: nat_of_nibble_eq)
+next
+ fix n m :: nibble
+ show "n < m \<longleftrightarrow> n \<le> m \<and> n \<noteq> m"
+ unfolding nibble_less_eq_def nibble_less_def less_le by (auto simp add: nat_of_nibble_eq)
+next
+ fix n m :: nibble
+ show "n \<le> m \<or> m \<le> n"
+ unfolding nibble_less_eq_def by auto
+qed
-definition
- int_to_nibble :: "int \<Rightarrow> nibble" where
- "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 \<le> 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. *}
-
-fun
- char_to_int_pair :: "char \<Rightarrow> int \<times> int" where
- "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
- char_le_def: "c \<le> d \<equiv> (char_to_int_pair c \<le> char_to_int_pair d)"
- char_less_def: "c < d \<equiv> (char_to_int_pair c < char_to_int_pair d)" ..
-
-lemmas char_ord_defs = char_less_def char_le_def
-
-instance char :: order
- by default (auto simp: char_ord_defs char_to_int_pair_eq order_less_le)
+instance nibble :: distrib_lattice
+ "inf \<equiv> min"
+ "sup \<equiv> max"
+ by default
+ (auto simp add: inf_nibble_def sup_nibble_def min_max.sup_inf_distrib1)
instance char :: linorder
- by default (auto simp: char_le_def)
+ char_less_eq_def: "c1 \<le> c2 \<equiv> case c1 of Char n1 m1 \<Rightarrow> case c2 of Char n2 m2 \<Rightarrow>
+ n1 < n2 \<or> n1 = n2 \<and> m1 \<le> m2"
+ char_less_def: "c1 < c2 \<equiv> case c1 of Char n1 m1 \<Rightarrow> case c2 of Char n2 m2 \<Rightarrow>
+ n1 < n2 \<or> n1 = n2 \<and> m1 < m2"
+ by default (auto simp: char_less_eq_def char_less_def split: char.splits)
+
+lemmas [code nofunc] = char_less_eq_def char_less_def
instance char :: distrib_lattice
"inf \<equiv> min"
"sup \<equiv> max"
- by intro_classes
+ by default
(auto simp add: inf_char_def sup_char_def min_max.sup_inf_distrib1)
-
-text {* code generator setup *}
-
-code_const char_to_int_pair
- (SML "raise/ Fail/ \"char'_to'_int'_pair\"")
- (OCaml "failwith \"char'_to'_int'_pair\"")
- (Haskell "error/ \"char'_to'_int'_pair\"")
+lemma [simp, code func]:
+ shows char_less_eq_simp: "Char n1 m1 \<le> Char n2 m2 \<longleftrightarrow> n1 < n2 \<or> n1 = n2 \<and> m1 \<le> m2"
+ and char_less_simp: "Char n1 m1 < Char n2 m2 \<longleftrightarrow> n1 < n2 \<or> n1 = n2 \<and> m1 < m2"
+ unfolding char_less_eq_def char_less_def by simp_all
end