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1 (* |
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2 ID: $Id$ |
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3 Author: Gerwin Klein, NICTA |
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4 |
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5 Examples demonstrating and testing various word operations. |
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6 *) |
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7 |
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8 theory WordExamples imports WordMain |
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9 begin |
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10 |
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11 -- "modulus" |
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12 |
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13 lemma "(27 :: 4 word) = -5" by simp |
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14 |
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15 lemma "(27 :: 4 word) = 11" by simp |
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16 |
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17 lemma "27 \<noteq> (11 :: 6 word)" by simp |
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18 |
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19 -- "signed" |
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20 lemma "(127 :: 6 word) = -1" by simp |
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21 |
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22 -- "number ring simps" |
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23 lemma |
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24 "27 + 11 = (38::'a::finite word)" |
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25 "27 + 11 = (6::5 word)" |
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26 "7 * 3 = (21::'a::finite word)" |
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27 "11 - 27 = (-16::'a::finite word)" |
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28 "- -11 = (11::'a::finite word)" |
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29 "-40 + 1 = (-39::'a::finite word)" |
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30 by simp_all |
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31 |
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32 lemma "word_pred 2 = 1" by simp |
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33 |
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34 lemma "word_succ -3 = -2" by simp |
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35 |
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36 lemma "23 < (27::8 word)" by simp |
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37 lemma "23 \<le> (27::8 word)" by simp |
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38 lemma "\<not> 23 < (27::2 word)" by simp |
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39 lemma "0 < (4::3 word)" by simp |
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40 |
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41 -- "ring operations" |
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42 |
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43 lemma "a + 2 * b + c - b = (b + c) + (a :: 32 word)" by simp |
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44 |
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45 -- "casting" |
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46 |
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47 lemma "uint (234567 :: 10 word) = 71" by simp |
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48 lemma "uint (-234567 :: 10 word) = 953" by simp |
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49 lemma "sint (234567 :: 10 word) = 71" by simp |
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50 lemma "sint (-234567 :: 10 word) = -71" by simp |
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51 |
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52 lemma "unat (-234567 :: 10 word) = 953" by simp |
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53 |
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54 lemma "ucast (0b1010 :: 4 word) = (0b10 :: 2 word)" by simp |
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55 lemma "ucast (0b1010 :: 4 word) = (0b1010 :: 10 word)" by simp |
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56 lemma "scast (0b1010 :: 4 word) = (0b111010 :: 6 word)" by simp |
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57 |
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58 -- "reducing goals to nat or int and arith:" |
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59 lemma "i < x ==> i < (i + 1 :: 'a :: finite word)" by unat_arith |
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60 lemma "i < x ==> i < (i + 1 :: 'a :: finite word)" by uint_arith |
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61 |
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62 -- "bool lists" |
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63 |
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64 lemma "of_bl [True, False, True, True] = (0b1011::'a::finite word)" by simp |
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65 |
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66 lemma "to_bl (0b110::4 word) = [False, True, True, False]" by simp |
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67 |
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68 -- "this is not exactly fast, but bearable" |
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69 lemma "of_bl (replicate 32 True) = (0xFFFFFFFF::32 word)" by simp |
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70 |
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71 -- "this works only for replicate n True" |
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72 lemma "of_bl (replicate 32 True) = (0xFFFFFFFF::32 word)" |
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73 by (unfold mask_bl [symmetric]) (simp add: mask_def) |
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74 |
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75 |
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76 -- "bit operations" |
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77 |
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78 lemma "0b110 AND 0b101 = (0b100 :: 32 word)" by simp |
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79 |
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80 lemma "0b110 OR 0b011 = (0b111 :: 8 word)" by simp |
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81 |
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82 lemma "0xF0 XOR 0xFF = (0x0F :: byte)" by simp |
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83 |
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84 lemma "NOT (0xF0 :: 16 word) = 0xFF0F" by simp |
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85 |
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86 lemma "(-1 :: 32 word) = 0xFFFFFFFF" by simp |
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87 |
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88 lemma "(0b0010 :: 4 word) !! 1" by simp |
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89 lemma "\<not> (0b0010 :: 4 word) !! 0" by simp |
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90 lemma "\<not> (0b1000 :: 3 word) !! 4" by simp |
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91 |
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92 lemma "(0b11000 :: 10 word) !! n = (n = 4 \<or> n = 3)" |
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93 by (auto simp add: bin_nth_Bit) |
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94 |
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95 lemma "set_bit 55 7 True = (183::'a word)" by simp |
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96 lemma "set_bit 0b0010 7 True = (0b10000010::'a word)" by simp |
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97 lemma "set_bit 0b0010 1 False = (0::'a word)" by simp |
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98 |
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99 lemma "lsb (0b0101::'a::finite word)" by simp |
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100 lemma "\<not> lsb (0b1000::'a::finite word)" by simp |
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101 |
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102 lemma "\<not> msb (0b0101::4 word)" by simp |
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103 lemma "msb (0b1000::4 word)" by simp |
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104 |
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105 lemma "word_cat (27::4 word) (27::8 word) = (2843::'a::finite word)" by simp |
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106 lemma "word_cat (0b0011::4 word) (0b1111::6word) = (0b0011001111 :: 10 word)" |
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107 by simp |
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108 |
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109 lemma "0b1011 << 2 = (0b101100::'a word)" by simp |
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110 lemma "0b1011 >> 2 = (0b10::8 word)" by simp |
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111 lemma "0b1011 >>> 2 = (0b10::8 word)" by simp |
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112 |
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113 lemma "slice 3 (0b101111::6 word) = (0b101::3 word)" by simp |
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114 |
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115 lemma "word_rotr 2 0b0110 = (0b1001::4 word)" by simp |
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116 lemma "word_rotl 1 0b1110 = (0b1101::4 word)" by simp |
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117 lemma "word_roti 2 0b1110 = (0b1011::4 word)" by simp |
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118 lemma "word_roti -2 0b0110 = (0b1001::4 word)" by simp |
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119 |
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120 lemma "(x AND 0xff00) OR (x AND 0x00ff) = (x::16 word)" |
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121 proof - |
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122 have "(x AND 0xff00) OR (x AND 0x00ff) = x AND (0xff00 OR 0x00ff)" |
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123 by (simp only: word_ao_dist2) |
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124 also have "0xff00 OR 0x00ff = (-1::16 word)" |
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125 by simp |
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126 also have "x AND -1 = x" |
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127 by simp |
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128 finally show ?thesis . |
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129 qed |
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130 |
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131 end |