1 (* |
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2 Author: Jeremy Dawson and Gerwin Klein, NICTA |
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3 |
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4 Basic definition of word type and basic theorems following from |
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5 the definition of the word type |
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6 *) |
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7 |
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8 header {* Definition of Word Type *} |
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9 |
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10 theory WordDefinition |
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11 imports Size BinBoolList TdThs |
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12 begin |
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13 |
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14 subsection {* Type definition *} |
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15 |
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16 typedef (open word) 'a word = "{(0::int) ..< 2^len_of TYPE('a::len0)}" |
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17 morphisms uint Abs_word by auto |
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18 |
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19 definition word_of_int :: "int \<Rightarrow> 'a\<Colon>len0 word" where |
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20 -- {* representation of words using unsigned or signed bins, |
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21 only difference in these is the type class *} |
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22 "word_of_int w = Abs_word (bintrunc (len_of TYPE ('a)) w)" |
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23 |
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24 lemma uint_word_of_int [code]: "uint (word_of_int w \<Colon> 'a\<Colon>len0 word) = w mod 2 ^ len_of TYPE('a)" |
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25 by (auto simp add: word_of_int_def bintrunc_mod2p intro: Abs_word_inverse) |
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26 |
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27 code_datatype word_of_int |
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28 |
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29 notation fcomp (infixl "o>" 60) |
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30 notation scomp (infixl "o\<rightarrow>" 60) |
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31 |
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32 instantiation word :: ("{len0, typerep}") random |
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33 begin |
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34 |
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35 definition |
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36 "random_word i = Random.range (max i (2 ^ len_of TYPE('a))) o\<rightarrow> (\<lambda>k. Pair ( |
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37 let j = word_of_int (Code_Numeral.int_of k) :: 'a word |
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38 in (j, \<lambda>_::unit. Code_Evaluation.term_of j)))" |
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39 |
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40 instance .. |
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41 |
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42 end |
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43 |
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44 no_notation fcomp (infixl "o>" 60) |
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45 no_notation scomp (infixl "o\<rightarrow>" 60) |
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46 |
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47 |
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48 subsection {* Type conversions and casting *} |
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49 |
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50 definition sint :: "'a :: len word => int" where |
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51 -- {* treats the most-significant-bit as a sign bit *} |
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52 sint_uint: "sint w = sbintrunc (len_of TYPE ('a) - 1) (uint w)" |
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53 |
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54 definition unat :: "'a :: len0 word => nat" where |
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55 "unat w = nat (uint w)" |
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56 |
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57 definition uints :: "nat => int set" where |
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58 -- "the sets of integers representing the words" |
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59 "uints n = range (bintrunc n)" |
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60 |
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61 definition sints :: "nat => int set" where |
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62 "sints n = range (sbintrunc (n - 1))" |
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63 |
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64 definition unats :: "nat => nat set" where |
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65 "unats n = {i. i < 2 ^ n}" |
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66 |
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67 definition norm_sint :: "nat => int => int" where |
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68 "norm_sint n w = (w + 2 ^ (n - 1)) mod 2 ^ n - 2 ^ (n - 1)" |
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69 |
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70 definition scast :: "'a :: len word => 'b :: len word" where |
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71 -- "cast a word to a different length" |
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72 "scast w = word_of_int (sint w)" |
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73 |
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74 definition ucast :: "'a :: len0 word => 'b :: len0 word" where |
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75 "ucast w = word_of_int (uint w)" |
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76 |
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77 instantiation word :: (len0) size |
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78 begin |
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79 |
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80 definition |
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81 word_size: "size (w :: 'a word) = len_of TYPE('a)" |
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82 |
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83 instance .. |
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84 |
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85 end |
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86 |
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87 definition source_size :: "('a :: len0 word => 'b) => nat" where |
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88 -- "whether a cast (or other) function is to a longer or shorter length" |
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89 "source_size c = (let arb = undefined ; x = c arb in size arb)" |
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90 |
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91 definition target_size :: "('a => 'b :: len0 word) => nat" where |
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92 "target_size c = size (c undefined)" |
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93 |
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94 definition is_up :: "('a :: len0 word => 'b :: len0 word) => bool" where |
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95 "is_up c \<longleftrightarrow> source_size c <= target_size c" |
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96 |
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97 definition is_down :: "('a :: len0 word => 'b :: len0 word) => bool" where |
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98 "is_down c \<longleftrightarrow> target_size c <= source_size c" |
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99 |
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100 definition of_bl :: "bool list => 'a :: len0 word" where |
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101 "of_bl bl = word_of_int (bl_to_bin bl)" |
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102 |
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103 definition to_bl :: "'a :: len0 word => bool list" where |
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104 "to_bl w = bin_to_bl (len_of TYPE ('a)) (uint w)" |
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105 |
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106 definition word_reverse :: "'a :: len0 word => 'a word" where |
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107 "word_reverse w = of_bl (rev (to_bl w))" |
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108 |
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109 definition word_int_case :: "(int => 'b) => ('a :: len0 word) => 'b" where |
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110 "word_int_case f w = f (uint w)" |
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111 |
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112 syntax |
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113 of_int :: "int => 'a" |
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114 translations |
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115 "case x of CONST of_int y => b" == "CONST word_int_case (%y. b) x" |
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116 |
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117 |
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118 subsection "Arithmetic operations" |
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119 |
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120 instantiation word :: (len0) "{number, uminus, minus, plus, one, zero, times, Divides.div, ord, bit}" |
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121 begin |
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122 |
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123 definition |
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124 word_0_wi: "0 = word_of_int 0" |
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125 |
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126 definition |
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127 word_1_wi: "1 = word_of_int 1" |
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128 |
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129 definition |
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130 word_add_def: "a + b = word_of_int (uint a + uint b)" |
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131 |
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132 definition |
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133 word_sub_wi: "a - b = word_of_int (uint a - uint b)" |
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134 |
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135 definition |
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136 word_minus_def: "- a = word_of_int (- uint a)" |
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137 |
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138 definition |
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139 word_mult_def: "a * b = word_of_int (uint a * uint b)" |
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140 |
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141 definition |
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142 word_div_def: "a div b = word_of_int (uint a div uint b)" |
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143 |
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144 definition |
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145 word_mod_def: "a mod b = word_of_int (uint a mod uint b)" |
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146 |
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147 definition |
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148 word_number_of_def: "number_of w = word_of_int w" |
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149 |
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150 definition |
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151 word_le_def: "a \<le> b \<longleftrightarrow> uint a \<le> uint b" |
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152 |
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153 definition |
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154 word_less_def: "x < y \<longleftrightarrow> x \<le> y \<and> x \<noteq> (y \<Colon> 'a word)" |
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155 |
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156 definition |
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157 word_and_def: |
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158 "(a::'a word) AND b = word_of_int (uint a AND uint b)" |
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159 |
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160 definition |
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161 word_or_def: |
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162 "(a::'a word) OR b = word_of_int (uint a OR uint b)" |
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163 |
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164 definition |
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165 word_xor_def: |
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166 "(a::'a word) XOR b = word_of_int (uint a XOR uint b)" |
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167 |
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168 definition |
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169 word_not_def: |
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170 "NOT (a::'a word) = word_of_int (NOT (uint a))" |
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171 |
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172 instance .. |
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173 |
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174 end |
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175 |
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176 definition |
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177 word_succ :: "'a :: len0 word => 'a word" |
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178 where |
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179 "word_succ a = word_of_int (Int.succ (uint a))" |
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180 |
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181 definition |
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182 word_pred :: "'a :: len0 word => 'a word" |
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183 where |
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184 "word_pred a = word_of_int (Int.pred (uint a))" |
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185 |
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186 definition udvd :: "'a::len word => 'a::len word => bool" (infixl "udvd" 50) where |
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187 "a udvd b == EX n>=0. uint b = n * uint a" |
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188 |
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189 definition word_sle :: "'a :: len word => 'a word => bool" ("(_/ <=s _)" [50, 51] 50) where |
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190 "a <=s b == sint a <= sint b" |
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191 |
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192 definition word_sless :: "'a :: len word => 'a word => bool" ("(_/ <s _)" [50, 51] 50) where |
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193 "(x <s y) == (x <=s y & x ~= y)" |
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194 |
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195 |
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196 |
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197 subsection "Bit-wise operations" |
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198 |
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199 instantiation word :: (len0) bits |
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200 begin |
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201 |
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202 definition |
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203 word_test_bit_def: "test_bit a = bin_nth (uint a)" |
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204 |
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205 definition |
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206 word_set_bit_def: "set_bit a n x = |
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207 word_of_int (bin_sc n (If x bit.B1 bit.B0) (uint a))" |
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208 |
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209 definition |
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210 word_set_bits_def: "(BITS n. f n) = of_bl (bl_of_nth (len_of TYPE ('a)) f)" |
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211 |
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212 definition |
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213 word_lsb_def: "lsb a \<longleftrightarrow> bin_last (uint a) = bit.B1" |
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214 |
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215 definition shiftl1 :: "'a word \<Rightarrow> 'a word" where |
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216 "shiftl1 w = word_of_int (uint w BIT bit.B0)" |
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217 |
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218 definition shiftr1 :: "'a word \<Rightarrow> 'a word" where |
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219 -- "shift right as unsigned or as signed, ie logical or arithmetic" |
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220 "shiftr1 w = word_of_int (bin_rest (uint w))" |
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221 |
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222 definition |
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223 shiftl_def: "w << n = (shiftl1 ^^ n) w" |
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224 |
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225 definition |
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226 shiftr_def: "w >> n = (shiftr1 ^^ n) w" |
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227 |
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228 instance .. |
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229 |
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230 end |
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231 |
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232 instantiation word :: (len) bitss |
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233 begin |
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234 |
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235 definition |
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236 word_msb_def: |
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237 "msb a \<longleftrightarrow> bin_sign (sint a) = Int.Min" |
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238 |
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239 instance .. |
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240 |
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241 end |
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242 |
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243 definition setBit :: "'a :: len0 word => nat => 'a word" where |
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244 "setBit w n == set_bit w n True" |
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245 |
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246 definition clearBit :: "'a :: len0 word => nat => 'a word" where |
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247 "clearBit w n == set_bit w n False" |
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248 |
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249 |
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250 subsection "Shift operations" |
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251 |
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252 definition sshiftr1 :: "'a :: len word => 'a word" where |
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253 "sshiftr1 w == word_of_int (bin_rest (sint w))" |
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254 |
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255 definition bshiftr1 :: "bool => 'a :: len word => 'a word" where |
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256 "bshiftr1 b w == of_bl (b # butlast (to_bl w))" |
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257 |
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258 definition sshiftr :: "'a :: len word => nat => 'a word" (infixl ">>>" 55) where |
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259 "w >>> n == (sshiftr1 ^^ n) w" |
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260 |
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261 definition mask :: "nat => 'a::len word" where |
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262 "mask n == (1 << n) - 1" |
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263 |
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264 definition revcast :: "'a :: len0 word => 'b :: len0 word" where |
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265 "revcast w == of_bl (takefill False (len_of TYPE('b)) (to_bl w))" |
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266 |
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267 definition slice1 :: "nat => 'a :: len0 word => 'b :: len0 word" where |
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268 "slice1 n w == of_bl (takefill False n (to_bl w))" |
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269 |
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270 definition slice :: "nat => 'a :: len0 word => 'b :: len0 word" where |
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271 "slice n w == slice1 (size w - n) w" |
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272 |
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273 |
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274 subsection "Rotation" |
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275 |
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276 definition rotater1 :: "'a list => 'a list" where |
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277 "rotater1 ys == |
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278 case ys of [] => [] | x # xs => last ys # butlast ys" |
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279 |
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280 definition rotater :: "nat => 'a list => 'a list" where |
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281 "rotater n == rotater1 ^^ n" |
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282 |
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283 definition word_rotr :: "nat => 'a :: len0 word => 'a :: len0 word" where |
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284 "word_rotr n w == of_bl (rotater n (to_bl w))" |
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285 |
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286 definition word_rotl :: "nat => 'a :: len0 word => 'a :: len0 word" where |
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287 "word_rotl n w == of_bl (rotate n (to_bl w))" |
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288 |
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289 definition word_roti :: "int => 'a :: len0 word => 'a :: len0 word" where |
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290 "word_roti i w == if i >= 0 then word_rotr (nat i) w |
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291 else word_rotl (nat (- i)) w" |
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292 |
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293 |
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294 subsection "Split and cat operations" |
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295 |
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296 definition word_cat :: "'a :: len0 word => 'b :: len0 word => 'c :: len0 word" where |
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297 "word_cat a b == word_of_int (bin_cat (uint a) (len_of TYPE ('b)) (uint b))" |
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298 |
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299 definition word_split :: "'a :: len0 word => ('b :: len0 word) * ('c :: len0 word)" where |
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300 "word_split a == |
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301 case bin_split (len_of TYPE ('c)) (uint a) of |
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302 (u, v) => (word_of_int u, word_of_int v)" |
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303 |
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304 definition word_rcat :: "'a :: len0 word list => 'b :: len0 word" where |
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305 "word_rcat ws == |
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306 word_of_int (bin_rcat (len_of TYPE ('a)) (map uint ws))" |
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307 |
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308 definition word_rsplit :: "'a :: len0 word => 'b :: len word list" where |
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309 "word_rsplit w == |
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310 map word_of_int (bin_rsplit (len_of TYPE ('b)) (len_of TYPE ('a), uint w))" |
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311 |
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312 definition max_word :: "'a::len word" -- "Largest representable machine integer." where |
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313 "max_word \<equiv> word_of_int (2 ^ len_of TYPE('a) - 1)" |
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314 |
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315 primrec of_bool :: "bool \<Rightarrow> 'a::len word" where |
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316 "of_bool False = 0" |
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317 | "of_bool True = 1" |
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318 |
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319 |
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320 lemmas of_nth_def = word_set_bits_def |
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321 |
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322 lemmas word_size_gt_0 [iff] = |
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323 xtr1 [OF word_size len_gt_0, standard] |
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324 lemmas lens_gt_0 = word_size_gt_0 len_gt_0 |
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325 lemmas lens_not_0 [iff] = lens_gt_0 [THEN gr_implies_not0, standard] |
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326 |
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327 lemma uints_num: "uints n = {i. 0 \<le> i \<and> i < 2 ^ n}" |
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328 by (simp add: uints_def range_bintrunc) |
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329 |
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330 lemma sints_num: "sints n = {i. - (2 ^ (n - 1)) \<le> i \<and> i < 2 ^ (n - 1)}" |
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331 by (simp add: sints_def range_sbintrunc) |
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332 |
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333 lemmas atLeastLessThan_alt = atLeastLessThan_def [unfolded |
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334 atLeast_def lessThan_def Collect_conj_eq [symmetric]] |
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335 |
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336 lemma mod_in_reps: "m > 0 ==> y mod m : {0::int ..< m}" |
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337 unfolding atLeastLessThan_alt by auto |
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338 |
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339 lemma |
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340 uint_0:"0 <= uint x" and |
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341 uint_lt: "uint (x::'a::len0 word) < 2 ^ len_of TYPE('a)" |
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342 by (auto simp: uint [simplified]) |
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343 |
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344 lemma uint_mod_same: |
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345 "uint x mod 2 ^ len_of TYPE('a) = uint (x::'a::len0 word)" |
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346 by (simp add: int_mod_eq uint_lt uint_0) |
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347 |
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348 lemma td_ext_uint: |
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349 "td_ext (uint :: 'a word => int) word_of_int (uints (len_of TYPE('a::len0))) |
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350 (%w::int. w mod 2 ^ len_of TYPE('a))" |
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351 apply (unfold td_ext_def') |
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352 apply (simp add: uints_num word_of_int_def bintrunc_mod2p) |
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353 apply (simp add: uint_mod_same uint_0 uint_lt |
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354 word.uint_inverse word.Abs_word_inverse int_mod_lem) |
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355 done |
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356 |
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357 lemmas int_word_uint = td_ext_uint [THEN td_ext.eq_norm, standard] |
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358 |
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359 interpretation word_uint: |
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360 td_ext "uint::'a::len0 word \<Rightarrow> int" |
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361 word_of_int |
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362 "uints (len_of TYPE('a::len0))" |
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363 "\<lambda>w. w mod 2 ^ len_of TYPE('a::len0)" |
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364 by (rule td_ext_uint) |
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365 |
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366 lemmas td_uint = word_uint.td_thm |
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367 |
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368 lemmas td_ext_ubin = td_ext_uint |
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369 [simplified len_gt_0 no_bintr_alt1 [symmetric]] |
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370 |
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371 interpretation word_ubin: |
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372 td_ext "uint::'a::len0 word \<Rightarrow> int" |
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373 word_of_int |
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374 "uints (len_of TYPE('a::len0))" |
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375 "bintrunc (len_of TYPE('a::len0))" |
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376 by (rule td_ext_ubin) |
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377 |
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378 lemma sint_sbintrunc': |
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379 "sint (word_of_int bin :: 'a word) = |
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380 (sbintrunc (len_of TYPE ('a :: len) - 1) bin)" |
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381 unfolding sint_uint |
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382 by (auto simp: word_ubin.eq_norm sbintrunc_bintrunc_lt) |
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383 |
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384 lemma uint_sint: |
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385 "uint w = bintrunc (len_of TYPE('a)) (sint (w :: 'a :: len word))" |
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386 unfolding sint_uint by (auto simp: bintrunc_sbintrunc_le) |
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387 |
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388 lemma bintr_uint': |
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389 "n >= size w ==> bintrunc n (uint w) = uint w" |
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390 apply (unfold word_size) |
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391 apply (subst word_ubin.norm_Rep [symmetric]) |
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392 apply (simp only: bintrunc_bintrunc_min word_size) |
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393 apply (simp add: min_max.inf_absorb2) |
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394 done |
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395 |
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396 lemma wi_bintr': |
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397 "wb = word_of_int bin ==> n >= size wb ==> |
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398 word_of_int (bintrunc n bin) = wb" |
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399 unfolding word_size |
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400 by (clarsimp simp add: word_ubin.norm_eq_iff [symmetric] min_max.inf_absorb1) |
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401 |
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402 lemmas bintr_uint = bintr_uint' [unfolded word_size] |
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403 lemmas wi_bintr = wi_bintr' [unfolded word_size] |
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404 |
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405 lemma td_ext_sbin: |
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406 "td_ext (sint :: 'a word => int) word_of_int (sints (len_of TYPE('a::len))) |
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407 (sbintrunc (len_of TYPE('a) - 1))" |
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408 apply (unfold td_ext_def' sint_uint) |
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409 apply (simp add : word_ubin.eq_norm) |
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410 apply (cases "len_of TYPE('a)") |
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411 apply (auto simp add : sints_def) |
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412 apply (rule sym [THEN trans]) |
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413 apply (rule word_ubin.Abs_norm) |
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414 apply (simp only: bintrunc_sbintrunc) |
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415 apply (drule sym) |
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416 apply simp |
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417 done |
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418 |
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419 lemmas td_ext_sint = td_ext_sbin |
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420 [simplified len_gt_0 no_sbintr_alt2 Suc_pred' [symmetric]] |
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421 |
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422 (* We do sint before sbin, before sint is the user version |
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423 and interpretations do not produce thm duplicates. I.e. |
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424 we get the name word_sint.Rep_eqD, but not word_sbin.Req_eqD, |
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425 because the latter is the same thm as the former *) |
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426 interpretation word_sint: |
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427 td_ext "sint ::'a::len word => int" |
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428 word_of_int |
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429 "sints (len_of TYPE('a::len))" |
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430 "%w. (w + 2^(len_of TYPE('a::len) - 1)) mod 2^len_of TYPE('a::len) - |
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431 2 ^ (len_of TYPE('a::len) - 1)" |
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432 by (rule td_ext_sint) |
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433 |
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434 interpretation word_sbin: |
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435 td_ext "sint ::'a::len word => int" |
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436 word_of_int |
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437 "sints (len_of TYPE('a::len))" |
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438 "sbintrunc (len_of TYPE('a::len) - 1)" |
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439 by (rule td_ext_sbin) |
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440 |
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441 lemmas int_word_sint = td_ext_sint [THEN td_ext.eq_norm, standard] |
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442 |
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443 lemmas td_sint = word_sint.td |
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444 |
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445 lemma word_number_of_alt: "number_of b == word_of_int (number_of b)" |
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446 unfolding word_number_of_def by (simp add: number_of_eq) |
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447 |
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448 lemma word_no_wi: "number_of = word_of_int" |
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449 by (auto simp: word_number_of_def intro: ext) |
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450 |
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451 lemma to_bl_def': |
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452 "(to_bl :: 'a :: len0 word => bool list) = |
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453 bin_to_bl (len_of TYPE('a)) o uint" |
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454 by (auto simp: to_bl_def intro: ext) |
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455 |
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456 lemmas word_reverse_no_def [simp] = word_reverse_def [of "number_of w", standard] |
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457 |
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458 lemmas uints_mod = uints_def [unfolded no_bintr_alt1] |
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459 |
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460 lemma uint_bintrunc: "uint (number_of bin :: 'a word) = |
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461 number_of (bintrunc (len_of TYPE ('a :: len0)) bin)" |
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462 unfolding word_number_of_def number_of_eq |
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463 by (auto intro: word_ubin.eq_norm) |
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464 |
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465 lemma sint_sbintrunc: "sint (number_of bin :: 'a word) = |
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466 number_of (sbintrunc (len_of TYPE ('a :: len) - 1) bin)" |
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467 unfolding word_number_of_def number_of_eq |
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468 by (subst word_sbin.eq_norm) simp |
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469 |
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470 lemma unat_bintrunc: |
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471 "unat (number_of bin :: 'a :: len0 word) = |
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472 number_of (bintrunc (len_of TYPE('a)) bin)" |
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473 unfolding unat_def nat_number_of_def |
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474 by (simp only: uint_bintrunc) |
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475 |
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476 (* WARNING - these may not always be helpful *) |
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477 declare |
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478 uint_bintrunc [simp] |
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479 sint_sbintrunc [simp] |
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480 unat_bintrunc [simp] |
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481 |
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482 lemma size_0_eq: "size (w :: 'a :: len0 word) = 0 ==> v = w" |
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483 apply (unfold word_size) |
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484 apply (rule word_uint.Rep_eqD) |
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485 apply (rule box_equals) |
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486 defer |
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487 apply (rule word_ubin.norm_Rep)+ |
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488 apply simp |
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489 done |
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490 |
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491 lemmas uint_lem = word_uint.Rep [unfolded uints_num mem_Collect_eq] |
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492 lemmas sint_lem = word_sint.Rep [unfolded sints_num mem_Collect_eq] |
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493 lemmas uint_ge_0 [iff] = uint_lem [THEN conjunct1, standard] |
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494 lemmas uint_lt2p [iff] = uint_lem [THEN conjunct2, standard] |
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495 lemmas sint_ge = sint_lem [THEN conjunct1, standard] |
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496 lemmas sint_lt = sint_lem [THEN conjunct2, standard] |
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497 |
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498 lemma sign_uint_Pls [simp]: |
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499 "bin_sign (uint x) = Int.Pls" |
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500 by (simp add: sign_Pls_ge_0 number_of_eq) |
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501 |
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502 lemmas uint_m2p_neg = iffD2 [OF diff_less_0_iff_less uint_lt2p, standard] |
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503 lemmas uint_m2p_not_non_neg = |
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504 iffD2 [OF linorder_not_le uint_m2p_neg, standard] |
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505 |
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506 lemma lt2p_lem: |
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507 "len_of TYPE('a) <= n ==> uint (w :: 'a :: len0 word) < 2 ^ n" |
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508 by (rule xtr8 [OF _ uint_lt2p]) simp |
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509 |
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510 lemmas uint_le_0_iff [simp] = |
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511 uint_ge_0 [THEN leD, THEN linorder_antisym_conv1, standard] |
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512 |
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513 lemma uint_nat: "uint w == int (unat w)" |
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514 unfolding unat_def by auto |
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515 |
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516 lemma uint_number_of: |
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517 "uint (number_of b :: 'a :: len0 word) = number_of b mod 2 ^ len_of TYPE('a)" |
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518 unfolding word_number_of_alt |
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519 by (simp only: int_word_uint) |
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520 |
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521 lemma unat_number_of: |
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522 "bin_sign b = Int.Pls ==> |
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523 unat (number_of b::'a::len0 word) = number_of b mod 2 ^ len_of TYPE ('a)" |
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524 apply (unfold unat_def) |
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525 apply (clarsimp simp only: uint_number_of) |
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526 apply (rule nat_mod_distrib [THEN trans]) |
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527 apply (erule sign_Pls_ge_0 [THEN iffD1]) |
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528 apply (simp_all add: nat_power_eq) |
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529 done |
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530 |
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531 lemma sint_number_of: "sint (number_of b :: 'a :: len word) = (number_of b + |
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532 2 ^ (len_of TYPE('a) - 1)) mod 2 ^ len_of TYPE('a) - |
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533 2 ^ (len_of TYPE('a) - 1)" |
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534 unfolding word_number_of_alt by (rule int_word_sint) |
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535 |
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536 lemma word_of_int_bin [simp] : |
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537 "(word_of_int (number_of bin) :: 'a :: len0 word) = (number_of bin)" |
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538 unfolding word_number_of_alt by auto |
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539 |
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540 lemma word_int_case_wi: |
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541 "word_int_case f (word_of_int i :: 'b word) = |
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542 f (i mod 2 ^ len_of TYPE('b::len0))" |
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543 unfolding word_int_case_def by (simp add: word_uint.eq_norm) |
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544 |
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545 lemma word_int_split: |
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546 "P (word_int_case f x) = |
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547 (ALL i. x = (word_of_int i :: 'b :: len0 word) & |
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548 0 <= i & i < 2 ^ len_of TYPE('b) --> P (f i))" |
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549 unfolding word_int_case_def |
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550 by (auto simp: word_uint.eq_norm int_mod_eq') |
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551 |
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552 lemma word_int_split_asm: |
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553 "P (word_int_case f x) = |
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554 (~ (EX n. x = (word_of_int n :: 'b::len0 word) & |
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555 0 <= n & n < 2 ^ len_of TYPE('b::len0) & ~ P (f n)))" |
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556 unfolding word_int_case_def |
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557 by (auto simp: word_uint.eq_norm int_mod_eq') |
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558 |
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559 lemmas uint_range' = |
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560 word_uint.Rep [unfolded uints_num mem_Collect_eq, standard] |
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561 lemmas sint_range' = word_sint.Rep [unfolded One_nat_def |
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562 sints_num mem_Collect_eq, standard] |
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563 |
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564 lemma uint_range_size: "0 <= uint w & uint w < 2 ^ size w" |
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565 unfolding word_size by (rule uint_range') |
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566 |
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567 lemma sint_range_size: |
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568 "- (2 ^ (size w - Suc 0)) <= sint w & sint w < 2 ^ (size w - Suc 0)" |
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569 unfolding word_size by (rule sint_range') |
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570 |
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571 lemmas sint_above_size = sint_range_size |
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572 [THEN conjunct2, THEN [2] xtr8, folded One_nat_def, standard] |
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573 |
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574 lemmas sint_below_size = sint_range_size |
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575 [THEN conjunct1, THEN [2] order_trans, folded One_nat_def, standard] |
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576 |
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577 lemma test_bit_eq_iff: "(test_bit (u::'a::len0 word) = test_bit v) = (u = v)" |
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578 unfolding word_test_bit_def by (simp add: bin_nth_eq_iff) |
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579 |
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580 lemma test_bit_size [rule_format] : "(w::'a::len0 word) !! n --> n < size w" |
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581 apply (unfold word_test_bit_def) |
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582 apply (subst word_ubin.norm_Rep [symmetric]) |
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583 apply (simp only: nth_bintr word_size) |
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584 apply fast |
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585 done |
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586 |
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587 lemma word_eqI [rule_format] : |
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588 fixes u :: "'a::len0 word" |
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589 shows "(ALL n. n < size u --> u !! n = v !! n) ==> u = v" |
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590 apply (rule test_bit_eq_iff [THEN iffD1]) |
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591 apply (rule ext) |
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592 apply (erule allE) |
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593 apply (erule impCE) |
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594 prefer 2 |
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595 apply assumption |
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596 apply (auto dest!: test_bit_size simp add: word_size) |
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597 done |
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598 |
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599 lemmas word_eqD = test_bit_eq_iff [THEN iffD2, THEN fun_cong, standard] |
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600 |
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601 lemma test_bit_bin': "w !! n = (n < size w & bin_nth (uint w) n)" |
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602 unfolding word_test_bit_def word_size |
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603 by (simp add: nth_bintr [symmetric]) |
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604 |
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605 lemmas test_bit_bin = test_bit_bin' [unfolded word_size] |
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606 |
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607 lemma bin_nth_uint_imp': "bin_nth (uint w) n --> n < size w" |
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608 apply (unfold word_size) |
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609 apply (rule impI) |
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610 apply (rule nth_bintr [THEN iffD1, THEN conjunct1]) |
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611 apply (subst word_ubin.norm_Rep) |
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612 apply assumption |
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613 done |
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614 |
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615 lemma bin_nth_sint': |
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616 "n >= size w --> bin_nth (sint w) n = bin_nth (sint w) (size w - 1)" |
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617 apply (rule impI) |
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618 apply (subst word_sbin.norm_Rep [symmetric]) |
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619 apply (simp add : nth_sbintr word_size) |
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620 apply auto |
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621 done |
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622 |
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623 lemmas bin_nth_uint_imp = bin_nth_uint_imp' [rule_format, unfolded word_size] |
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624 lemmas bin_nth_sint = bin_nth_sint' [rule_format, unfolded word_size] |
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625 |
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626 (* type definitions theorem for in terms of equivalent bool list *) |
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627 lemma td_bl: |
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628 "type_definition (to_bl :: 'a::len0 word => bool list) |
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629 of_bl |
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630 {bl. length bl = len_of TYPE('a)}" |
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631 apply (unfold type_definition_def of_bl_def to_bl_def) |
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632 apply (simp add: word_ubin.eq_norm) |
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633 apply safe |
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634 apply (drule sym) |
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635 apply simp |
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636 done |
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637 |
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638 interpretation word_bl: |
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639 type_definition "to_bl :: 'a::len0 word => bool list" |
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640 of_bl |
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641 "{bl. length bl = len_of TYPE('a::len0)}" |
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642 by (rule td_bl) |
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643 |
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644 lemma word_size_bl: "size w == size (to_bl w)" |
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645 unfolding word_size by auto |
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646 |
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647 lemma to_bl_use_of_bl: |
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648 "(to_bl w = bl) = (w = of_bl bl \<and> length bl = length (to_bl w))" |
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649 by (fastsimp elim!: word_bl.Abs_inverse [simplified]) |
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650 |
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651 lemma to_bl_word_rev: "to_bl (word_reverse w) = rev (to_bl w)" |
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652 unfolding word_reverse_def by (simp add: word_bl.Abs_inverse) |
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653 |
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654 lemma word_rev_rev [simp] : "word_reverse (word_reverse w) = w" |
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655 unfolding word_reverse_def by (simp add : word_bl.Abs_inverse) |
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656 |
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657 lemma word_rev_gal: "word_reverse w = u ==> word_reverse u = w" |
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658 by auto |
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659 |
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660 lemmas word_rev_gal' = sym [THEN word_rev_gal, symmetric, standard] |
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661 |
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662 lemmas length_bl_gt_0 [iff] = xtr1 [OF word_bl.Rep' len_gt_0, standard] |
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663 lemmas bl_not_Nil [iff] = |
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664 length_bl_gt_0 [THEN length_greater_0_conv [THEN iffD1], standard] |
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665 lemmas length_bl_neq_0 [iff] = length_bl_gt_0 [THEN gr_implies_not0] |
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666 |
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667 lemma hd_bl_sign_sint: "hd (to_bl w) = (bin_sign (sint w) = Int.Min)" |
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668 apply (unfold to_bl_def sint_uint) |
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669 apply (rule trans [OF _ bl_sbin_sign]) |
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670 apply simp |
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671 done |
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672 |
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673 lemma of_bl_drop': |
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674 "lend = length bl - len_of TYPE ('a :: len0) ==> |
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675 of_bl (drop lend bl) = (of_bl bl :: 'a word)" |
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676 apply (unfold of_bl_def) |
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677 apply (clarsimp simp add : trunc_bl2bin [symmetric]) |
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678 done |
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679 |
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680 lemmas of_bl_no = of_bl_def [folded word_number_of_def] |
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681 |
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682 lemma test_bit_of_bl: |
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683 "(of_bl bl::'a::len0 word) !! n = (rev bl ! n \<and> n < len_of TYPE('a) \<and> n < length bl)" |
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684 apply (unfold of_bl_def word_test_bit_def) |
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685 apply (auto simp add: word_size word_ubin.eq_norm nth_bintr bin_nth_of_bl) |
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686 done |
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687 |
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688 lemma no_of_bl: |
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689 "(number_of bin ::'a::len0 word) = of_bl (bin_to_bl (len_of TYPE ('a)) bin)" |
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690 unfolding word_size of_bl_no by (simp add : word_number_of_def) |
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691 |
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692 lemma uint_bl: "to_bl w == bin_to_bl (size w) (uint w)" |
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693 unfolding word_size to_bl_def by auto |
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694 |
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695 lemma to_bl_bin: "bl_to_bin (to_bl w) = uint w" |
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696 unfolding uint_bl by (simp add : word_size) |
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697 |
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698 lemma to_bl_of_bin: |
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699 "to_bl (word_of_int bin::'a::len0 word) = bin_to_bl (len_of TYPE('a)) bin" |
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700 unfolding uint_bl by (clarsimp simp add: word_ubin.eq_norm word_size) |
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701 |
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702 lemmas to_bl_no_bin [simp] = to_bl_of_bin [folded word_number_of_def] |
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703 |
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704 lemma to_bl_to_bin [simp] : "bl_to_bin (to_bl w) = uint w" |
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705 unfolding uint_bl by (simp add : word_size) |
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706 |
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707 lemmas uint_bl_bin [simp] = trans [OF bin_bl_bin word_ubin.norm_Rep, standard] |
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708 |
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709 lemmas num_AB_u [simp] = word_uint.Rep_inverse |
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710 [unfolded o_def word_number_of_def [symmetric], standard] |
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711 lemmas num_AB_s [simp] = word_sint.Rep_inverse |
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712 [unfolded o_def word_number_of_def [symmetric], standard] |
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713 |
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714 (* naturals *) |
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715 lemma uints_unats: "uints n = int ` unats n" |
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716 apply (unfold unats_def uints_num) |
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717 apply safe |
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718 apply (rule_tac image_eqI) |
|
719 apply (erule_tac nat_0_le [symmetric]) |
|
720 apply auto |
|
721 apply (erule_tac nat_less_iff [THEN iffD2]) |
|
722 apply (rule_tac [2] zless_nat_eq_int_zless [THEN iffD1]) |
|
723 apply (auto simp add : nat_power_eq int_power) |
|
724 done |
|
725 |
|
726 lemma unats_uints: "unats n = nat ` uints n" |
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727 by (auto simp add : uints_unats image_iff) |
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728 |
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729 lemmas bintr_num = word_ubin.norm_eq_iff |
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730 [symmetric, folded word_number_of_def, standard] |
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731 lemmas sbintr_num = word_sbin.norm_eq_iff |
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732 [symmetric, folded word_number_of_def, standard] |
|
733 |
|
734 lemmas num_of_bintr = word_ubin.Abs_norm [folded word_number_of_def, standard] |
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735 lemmas num_of_sbintr = word_sbin.Abs_norm [folded word_number_of_def, standard]; |
|
736 |
|
737 (* don't add these to simpset, since may want bintrunc n w to be simplified; |
|
738 may want these in reverse, but loop as simp rules, so use following *) |
|
739 |
|
740 lemma num_of_bintr': |
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741 "bintrunc (len_of TYPE('a :: len0)) a = b ==> |
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742 number_of a = (number_of b :: 'a word)" |
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743 apply safe |
|
744 apply (rule_tac num_of_bintr [symmetric]) |
|
745 done |
|
746 |
|
747 lemma num_of_sbintr': |
|
748 "sbintrunc (len_of TYPE('a :: len) - 1) a = b ==> |
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749 number_of a = (number_of b :: 'a word)" |
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750 apply safe |
|
751 apply (rule_tac num_of_sbintr [symmetric]) |
|
752 done |
|
753 |
|
754 lemmas num_abs_bintr = sym [THEN trans, |
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755 OF num_of_bintr word_number_of_def, standard] |
|
756 lemmas num_abs_sbintr = sym [THEN trans, |
|
757 OF num_of_sbintr word_number_of_def, standard] |
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758 |
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759 (** cast - note, no arg for new length, as it's determined by type of result, |
|
760 thus in "cast w = w, the type means cast to length of w! **) |
|
761 |
|
762 lemma ucast_id: "ucast w = w" |
|
763 unfolding ucast_def by auto |
|
764 |
|
765 lemma scast_id: "scast w = w" |
|
766 unfolding scast_def by auto |
|
767 |
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768 lemma ucast_bl: "ucast w == of_bl (to_bl w)" |
|
769 unfolding ucast_def of_bl_def uint_bl |
|
770 by (auto simp add : word_size) |
|
771 |
|
772 lemma nth_ucast: |
|
773 "(ucast w::'a::len0 word) !! n = (w !! n & n < len_of TYPE('a))" |
|
774 apply (unfold ucast_def test_bit_bin) |
|
775 apply (simp add: word_ubin.eq_norm nth_bintr word_size) |
|
776 apply (fast elim!: bin_nth_uint_imp) |
|
777 done |
|
778 |
|
779 (* for literal u(s)cast *) |
|
780 |
|
781 lemma ucast_bintr [simp]: |
|
782 "ucast (number_of w ::'a::len0 word) = |
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783 number_of (bintrunc (len_of TYPE('a)) w)" |
|
784 unfolding ucast_def by simp |
|
785 |
|
786 lemma scast_sbintr [simp]: |
|
787 "scast (number_of w ::'a::len word) = |
|
788 number_of (sbintrunc (len_of TYPE('a) - Suc 0) w)" |
|
789 unfolding scast_def by simp |
|
790 |
|
791 lemmas source_size = source_size_def [unfolded Let_def word_size] |
|
792 lemmas target_size = target_size_def [unfolded Let_def word_size] |
|
793 lemmas is_down = is_down_def [unfolded source_size target_size] |
|
794 lemmas is_up = is_up_def [unfolded source_size target_size] |
|
795 |
|
796 lemmas is_up_down = trans [OF is_up is_down [symmetric], standard] |
|
797 |
|
798 lemma down_cast_same': "uc = ucast ==> is_down uc ==> uc = scast" |
|
799 apply (unfold is_down) |
|
800 apply safe |
|
801 apply (rule ext) |
|
802 apply (unfold ucast_def scast_def uint_sint) |
|
803 apply (rule word_ubin.norm_eq_iff [THEN iffD1]) |
|
804 apply simp |
|
805 done |
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806 |
|
807 lemma word_rev_tf': |
|
808 "r = to_bl (of_bl bl) ==> r = rev (takefill False (length r) (rev bl))" |
|
809 unfolding of_bl_def uint_bl |
|
810 by (clarsimp simp add: bl_bin_bl_rtf word_ubin.eq_norm word_size) |
|
811 |
|
812 lemmas word_rev_tf = refl [THEN word_rev_tf', unfolded word_bl.Rep', standard] |
|
813 |
|
814 lemmas word_rep_drop = word_rev_tf [simplified takefill_alt, |
|
815 simplified, simplified rev_take, simplified] |
|
816 |
|
817 lemma to_bl_ucast: |
|
818 "to_bl (ucast (w::'b::len0 word) ::'a::len0 word) = |
|
819 replicate (len_of TYPE('a) - len_of TYPE('b)) False @ |
|
820 drop (len_of TYPE('b) - len_of TYPE('a)) (to_bl w)" |
|
821 apply (unfold ucast_bl) |
|
822 apply (rule trans) |
|
823 apply (rule word_rep_drop) |
|
824 apply simp |
|
825 done |
|
826 |
|
827 lemma ucast_up_app': |
|
828 "uc = ucast ==> source_size uc + n = target_size uc ==> |
|
829 to_bl (uc w) = replicate n False @ (to_bl w)" |
|
830 by (auto simp add : source_size target_size to_bl_ucast) |
|
831 |
|
832 lemma ucast_down_drop': |
|
833 "uc = ucast ==> source_size uc = target_size uc + n ==> |
|
834 to_bl (uc w) = drop n (to_bl w)" |
|
835 by (auto simp add : source_size target_size to_bl_ucast) |
|
836 |
|
837 lemma scast_down_drop': |
|
838 "sc = scast ==> source_size sc = target_size sc + n ==> |
|
839 to_bl (sc w) = drop n (to_bl w)" |
|
840 apply (subgoal_tac "sc = ucast") |
|
841 apply safe |
|
842 apply simp |
|
843 apply (erule refl [THEN ucast_down_drop']) |
|
844 apply (rule refl [THEN down_cast_same', symmetric]) |
|
845 apply (simp add : source_size target_size is_down) |
|
846 done |
|
847 |
|
848 lemma sint_up_scast': |
|
849 "sc = scast ==> is_up sc ==> sint (sc w) = sint w" |
|
850 apply (unfold is_up) |
|
851 apply safe |
|
852 apply (simp add: scast_def word_sbin.eq_norm) |
|
853 apply (rule box_equals) |
|
854 prefer 3 |
|
855 apply (rule word_sbin.norm_Rep) |
|
856 apply (rule sbintrunc_sbintrunc_l) |
|
857 defer |
|
858 apply (subst word_sbin.norm_Rep) |
|
859 apply (rule refl) |
|
860 apply simp |
|
861 done |
|
862 |
|
863 lemma uint_up_ucast': |
|
864 "uc = ucast ==> is_up uc ==> uint (uc w) = uint w" |
|
865 apply (unfold is_up) |
|
866 apply safe |
|
867 apply (rule bin_eqI) |
|
868 apply (fold word_test_bit_def) |
|
869 apply (auto simp add: nth_ucast) |
|
870 apply (auto simp add: test_bit_bin) |
|
871 done |
|
872 |
|
873 lemmas down_cast_same = refl [THEN down_cast_same'] |
|
874 lemmas ucast_up_app = refl [THEN ucast_up_app'] |
|
875 lemmas ucast_down_drop = refl [THEN ucast_down_drop'] |
|
876 lemmas scast_down_drop = refl [THEN scast_down_drop'] |
|
877 lemmas uint_up_ucast = refl [THEN uint_up_ucast'] |
|
878 lemmas sint_up_scast = refl [THEN sint_up_scast'] |
|
879 |
|
880 lemma ucast_up_ucast': "uc = ucast ==> is_up uc ==> ucast (uc w) = ucast w" |
|
881 apply (simp (no_asm) add: ucast_def) |
|
882 apply (clarsimp simp add: uint_up_ucast) |
|
883 done |
|
884 |
|
885 lemma scast_up_scast': "sc = scast ==> is_up sc ==> scast (sc w) = scast w" |
|
886 apply (simp (no_asm) add: scast_def) |
|
887 apply (clarsimp simp add: sint_up_scast) |
|
888 done |
|
889 |
|
890 lemma ucast_of_bl_up': |
|
891 "w = of_bl bl ==> size bl <= size w ==> ucast w = of_bl bl" |
|
892 by (auto simp add : nth_ucast word_size test_bit_of_bl intro!: word_eqI) |
|
893 |
|
894 lemmas ucast_up_ucast = refl [THEN ucast_up_ucast'] |
|
895 lemmas scast_up_scast = refl [THEN scast_up_scast'] |
|
896 lemmas ucast_of_bl_up = refl [THEN ucast_of_bl_up'] |
|
897 |
|
898 lemmas ucast_up_ucast_id = trans [OF ucast_up_ucast ucast_id] |
|
899 lemmas scast_up_scast_id = trans [OF scast_up_scast scast_id] |
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900 |
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901 lemmas isduu = is_up_down [where c = "ucast", THEN iffD2] |
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902 lemmas isdus = is_up_down [where c = "scast", THEN iffD2] |
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903 lemmas ucast_down_ucast_id = isduu [THEN ucast_up_ucast_id] |
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904 lemmas scast_down_scast_id = isdus [THEN ucast_up_ucast_id] |
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905 |
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906 lemma up_ucast_surj: |
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907 "is_up (ucast :: 'b::len0 word => 'a::len0 word) ==> |
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908 surj (ucast :: 'a word => 'b word)" |
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909 by (rule surjI, erule ucast_up_ucast_id) |
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910 |
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911 lemma up_scast_surj: |
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912 "is_up (scast :: 'b::len word => 'a::len word) ==> |
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913 surj (scast :: 'a word => 'b word)" |
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914 by (rule surjI, erule scast_up_scast_id) |
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915 |
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916 lemma down_scast_inj: |
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917 "is_down (scast :: 'b::len word => 'a::len word) ==> |
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918 inj_on (ucast :: 'a word => 'b word) A" |
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919 by (rule inj_on_inverseI, erule scast_down_scast_id) |
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920 |
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921 lemma down_ucast_inj: |
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922 "is_down (ucast :: 'b::len0 word => 'a::len0 word) ==> |
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923 inj_on (ucast :: 'a word => 'b word) A" |
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924 by (rule inj_on_inverseI, erule ucast_down_ucast_id) |
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925 |
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926 lemma of_bl_append_same: "of_bl (X @ to_bl w) = w" |
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927 by (rule word_bl.Rep_eqD) (simp add: word_rep_drop) |
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928 |
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929 lemma ucast_down_no': |
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930 "uc = ucast ==> is_down uc ==> uc (number_of bin) = number_of bin" |
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931 apply (unfold word_number_of_def is_down) |
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932 apply (clarsimp simp add: ucast_def word_ubin.eq_norm) |
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933 apply (rule word_ubin.norm_eq_iff [THEN iffD1]) |
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934 apply (erule bintrunc_bintrunc_ge) |
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935 done |
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936 |
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937 lemmas ucast_down_no = ucast_down_no' [OF refl] |
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938 |
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939 lemma ucast_down_bl': "uc = ucast ==> is_down uc ==> uc (of_bl bl) = of_bl bl" |
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940 unfolding of_bl_no by clarify (erule ucast_down_no) |
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941 |
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942 lemmas ucast_down_bl = ucast_down_bl' [OF refl] |
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943 |
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944 lemmas slice_def' = slice_def [unfolded word_size] |
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945 lemmas test_bit_def' = word_test_bit_def [THEN fun_cong] |
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946 |
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947 lemmas word_log_defs = word_and_def word_or_def word_xor_def word_not_def |
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948 lemmas word_log_bin_defs = word_log_defs |
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949 |
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950 text {* Executable equality *} |
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951 |
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952 instantiation word :: ("{len0}") eq |
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953 begin |
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954 |
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955 definition eq_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> bool" where |
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956 "eq_word k l \<longleftrightarrow> HOL.eq (uint k) (uint l)" |
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957 |
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958 instance proof |
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959 qed (simp add: eq eq_word_def) |
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960 |
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961 end |
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962 |
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963 end |
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