1 (* Title: HOL/Library/Char_ord.thy |
1 (* Title: HOL/Library/Char_ord.thy |
2 ID: $Id$ |
2 ID: $Id$ |
3 Author: Norbert Voelker |
3 Author: Norbert Voelker, Florian Haftmann |
4 *) |
4 *) |
5 |
5 |
6 header {* Order on characters *} |
6 header {* Order on characters *} |
7 |
7 |
8 theory Char_ord |
8 theory Char_ord |
9 imports Product_ord |
9 imports Product_ord Char_nat |
10 begin |
10 begin |
11 |
11 |
12 text {* Conversions between nibbles and integers in [0..15]. *} |
12 instance nibble :: linorder |
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13 nibble_less_eq_def: "n \<le> m \<equiv> nat_of_nibble n \<le> nat_of_nibble m" |
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14 nibble_less_def: "n < m \<equiv> nat_of_nibble n < nat_of_nibble m" |
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15 proof |
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16 fix n :: nibble show "n \<le> n" unfolding nibble_less_eq_def nibble_less_def by auto |
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17 next |
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18 fix n m q :: nibble |
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19 assume "n \<le> m" |
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20 and "m \<le> q" |
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21 then show "n \<le> q" unfolding nibble_less_eq_def nibble_less_def by auto |
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22 next |
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23 fix n m :: nibble |
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24 assume "n \<le> m" |
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25 and "m \<le> n" |
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26 then show "n = m" unfolding nibble_less_eq_def nibble_less_def by (auto simp add: nat_of_nibble_eq) |
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27 next |
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28 fix n m :: nibble |
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29 show "n < m \<longleftrightarrow> n \<le> m \<and> n \<noteq> m" |
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30 unfolding nibble_less_eq_def nibble_less_def less_le by (auto simp add: nat_of_nibble_eq) |
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31 next |
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32 fix n m :: nibble |
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33 show "n \<le> m \<or> m \<le> n" |
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34 unfolding nibble_less_eq_def by auto |
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35 qed |
13 |
36 |
14 fun |
37 instance nibble :: distrib_lattice |
15 nibble_to_int:: "nibble \<Rightarrow> int" where |
38 "inf \<equiv> min" |
16 "nibble_to_int Nibble0 = 0" |
39 "sup \<equiv> max" |
17 | "nibble_to_int Nibble1 = 1" |
40 by default |
18 | "nibble_to_int Nibble2 = 2" |
41 (auto simp add: inf_nibble_def sup_nibble_def min_max.sup_inf_distrib1) |
19 | "nibble_to_int Nibble3 = 3" |
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20 | "nibble_to_int Nibble4 = 4" |
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21 | "nibble_to_int Nibble5 = 5" |
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22 | "nibble_to_int Nibble6 = 6" |
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23 | "nibble_to_int Nibble7 = 7" |
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24 | "nibble_to_int Nibble8 = 8" |
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25 | "nibble_to_int Nibble9 = 9" |
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26 | "nibble_to_int NibbleA = 10" |
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27 | "nibble_to_int NibbleB = 11" |
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28 | "nibble_to_int NibbleC = 12" |
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29 | "nibble_to_int NibbleD = 13" |
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30 | "nibble_to_int NibbleE = 14" |
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31 | "nibble_to_int NibbleF = 15" |
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32 |
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33 definition |
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34 int_to_nibble :: "int \<Rightarrow> nibble" where |
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35 "int_to_nibble x = (let y = x mod 16 in |
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36 if y = 0 then Nibble0 else |
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37 if y = 1 then Nibble1 else |
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38 if y = 2 then Nibble2 else |
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39 if y = 3 then Nibble3 else |
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40 if y = 4 then Nibble4 else |
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41 if y = 5 then Nibble5 else |
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42 if y = 6 then Nibble6 else |
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43 if y = 7 then Nibble7 else |
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44 if y = 8 then Nibble8 else |
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45 if y = 9 then Nibble9 else |
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46 if y = 10 then NibbleA else |
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47 if y = 11 then NibbleB else |
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48 if y = 12 then NibbleC else |
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49 if y = 13 then NibbleD else |
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50 if y = 14 then NibbleE else |
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51 NibbleF)" |
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52 |
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53 lemma int_to_nibble_nibble_to_int: "int_to_nibble (nibble_to_int x) = x" |
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54 by (cases x) (auto simp: int_to_nibble_def Let_def) |
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55 |
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56 lemma inj_nibble_to_int: "inj nibble_to_int" |
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57 by (rule inj_on_inverseI) (rule int_to_nibble_nibble_to_int) |
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58 |
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59 lemmas nibble_to_int_eq = inj_nibble_to_int [THEN inj_eq] |
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60 |
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61 lemma nibble_to_int_ge_0: "0 \<le> nibble_to_int x" |
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62 by (cases x) auto |
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63 |
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64 lemma nibble_to_int_less_16: "nibble_to_int x < 16" |
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65 by (cases x) auto |
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66 |
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67 text {* Conversion between chars and int pairs. *} |
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68 |
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69 fun |
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70 char_to_int_pair :: "char \<Rightarrow> int \<times> int" where |
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71 "char_to_int_pair (Char a b) = (nibble_to_int a, nibble_to_int b)" |
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72 |
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73 lemma inj_char_to_int_pair: "inj char_to_int_pair" |
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74 apply (rule inj_onI) |
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75 apply (case_tac x, case_tac y) |
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76 apply (auto simp: nibble_to_int_eq) |
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77 done |
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78 |
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79 lemmas char_to_int_pair_eq = inj_char_to_int_pair [THEN inj_eq] |
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80 |
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81 |
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82 text {* Instantiation of order classes *} |
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83 |
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84 instance char :: ord |
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85 char_le_def: "c \<le> d \<equiv> (char_to_int_pair c \<le> char_to_int_pair d)" |
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86 char_less_def: "c < d \<equiv> (char_to_int_pair c < char_to_int_pair d)" .. |
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87 |
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88 lemmas char_ord_defs = char_less_def char_le_def |
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89 |
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90 instance char :: order |
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91 by default (auto simp: char_ord_defs char_to_int_pair_eq order_less_le) |
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92 |
42 |
93 instance char :: linorder |
43 instance char :: linorder |
94 by default (auto simp: char_le_def) |
44 char_less_eq_def: "c1 \<le> c2 \<equiv> case c1 of Char n1 m1 \<Rightarrow> case c2 of Char n2 m2 \<Rightarrow> |
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45 n1 < n2 \<or> n1 = n2 \<and> m1 \<le> m2" |
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46 char_less_def: "c1 < c2 \<equiv> case c1 of Char n1 m1 \<Rightarrow> case c2 of Char n2 m2 \<Rightarrow> |
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47 n1 < n2 \<or> n1 = n2 \<and> m1 < m2" |
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48 by default (auto simp: char_less_eq_def char_less_def split: char.splits) |
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49 |
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50 lemmas [code nofunc] = char_less_eq_def char_less_def |
95 |
51 |
96 instance char :: distrib_lattice |
52 instance char :: distrib_lattice |
97 "inf \<equiv> min" |
53 "inf \<equiv> min" |
98 "sup \<equiv> max" |
54 "sup \<equiv> max" |
99 by intro_classes |
55 by default |
100 (auto simp add: inf_char_def sup_char_def min_max.sup_inf_distrib1) |
56 (auto simp add: inf_char_def sup_char_def min_max.sup_inf_distrib1) |
101 |
57 |
102 |
58 lemma [simp, code func]: |
103 text {* code generator setup *} |
59 shows char_less_eq_simp: "Char n1 m1 \<le> Char n2 m2 \<longleftrightarrow> n1 < n2 \<or> n1 = n2 \<and> m1 \<le> m2" |
104 |
60 and char_less_simp: "Char n1 m1 < Char n2 m2 \<longleftrightarrow> n1 < n2 \<or> n1 = n2 \<and> m1 < m2" |
105 code_const char_to_int_pair |
61 unfolding char_less_eq_def char_less_def by simp_all |
106 (SML "raise/ Fail/ \"char'_to'_int'_pair\"") |
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107 (OCaml "failwith \"char'_to'_int'_pair\"") |
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108 (Haskell "error/ \"char'_to'_int'_pair\"") |
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109 |
62 |
110 end |
63 end |