1 (* Title: HOL/Word/Word.thy |
1 (* Title: HOL/Word/Word.thy |
2 Author: Gerwin Klein, NICTA |
2 Author: Jeremy Dawson and Gerwin Klein, NICTA |
3 *) |
3 *) |
4 |
4 |
5 header {* Word Library interface *} |
5 header {* A type of finite bit strings *} |
6 |
6 |
7 theory Word |
7 theory Word |
8 imports WordGenLib |
8 imports Type_Length Misc_Typedef Boolean_Algebra Bool_List_Representation |
9 uses "~~/src/HOL/Tools/SMT/smt_word.ML" |
9 uses ("~~/src/HOL/Tools/SMT/smt_word.ML") |
10 begin |
10 begin |
11 |
11 |
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12 text {* see @{text "Examples/WordExamples.thy"} for examples *} |
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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 1 0) (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) = 1" |
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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 0)" |
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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) |
|
620 apply auto |
|
621 done |
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622 |
|
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 |
|
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) |
|
629 of_bl |
|
630 {bl. length bl = len_of TYPE('a)}" |
|
631 apply (unfold type_definition_def of_bl_def to_bl_def) |
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632 apply (simp add: word_ubin.eq_norm) |
|
633 apply safe |
|
634 apply (drule sym) |
|
635 apply simp |
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636 done |
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637 |
|
638 interpretation word_bl: |
|
639 type_definition "to_bl :: 'a::len0 word => bool list" |
|
640 of_bl |
|
641 "{bl. length bl = len_of TYPE('a::len0)}" |
|
642 by (rule td_bl) |
|
643 |
|
644 lemma word_size_bl: "size w == size (to_bl w)" |
|
645 unfolding word_size by auto |
|
646 |
|
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 |
|
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 |
|
657 lemma word_rev_gal: "word_reverse w = u ==> word_reverse u = w" |
|
658 by auto |
|
659 |
|
660 lemmas word_rev_gal' = sym [THEN word_rev_gal, symmetric, standard] |
|
661 |
|
662 lemmas length_bl_gt_0 [iff] = xtr1 [OF word_bl.Rep' len_gt_0, standard] |
|
663 lemmas bl_not_Nil [iff] = |
|
664 length_bl_gt_0 [THEN length_greater_0_conv [THEN iffD1], standard] |
|
665 lemmas length_bl_neq_0 [iff] = length_bl_gt_0 [THEN gr_implies_not0] |
|
666 |
|
667 lemma hd_bl_sign_sint: "hd (to_bl w) = (bin_sign (sint w) = Int.Min)" |
|
668 apply (unfold to_bl_def sint_uint) |
|
669 apply (rule trans [OF _ bl_sbin_sign]) |
|
670 apply simp |
|
671 done |
|
672 |
|
673 lemma of_bl_drop': |
|
674 "lend = length bl - len_of TYPE ('a :: len0) ==> |
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675 of_bl (drop lend bl) = (of_bl bl :: 'a word)" |
|
676 apply (unfold of_bl_def) |
|
677 apply (clarsimp simp add : trunc_bl2bin [symmetric]) |
|
678 done |
|
679 |
|
680 lemmas of_bl_no = of_bl_def [folded word_number_of_def] |
|
681 |
|
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)" |
|
684 apply (unfold of_bl_def word_test_bit_def) |
|
685 apply (auto simp add: word_size word_ubin.eq_norm nth_bintr bin_nth_of_bl) |
|
686 done |
|
687 |
|
688 lemma no_of_bl: |
|
689 "(number_of bin ::'a::len0 word) = of_bl (bin_to_bl (len_of TYPE ('a)) bin)" |
|
690 unfolding word_size of_bl_no by (simp add : word_number_of_def) |
|
691 |
|
692 lemma uint_bl: "to_bl w == bin_to_bl (size w) (uint w)" |
|
693 unfolding word_size to_bl_def by auto |
|
694 |
|
695 lemma to_bl_bin: "bl_to_bin (to_bl w) = uint w" |
|
696 unfolding uint_bl by (simp add : word_size) |
|
697 |
|
698 lemma to_bl_of_bin: |
|
699 "to_bl (word_of_int bin::'a::len0 word) = bin_to_bl (len_of TYPE('a)) bin" |
|
700 unfolding uint_bl by (clarsimp simp add: word_ubin.eq_norm word_size) |
|
701 |
|
702 lemmas to_bl_no_bin [simp] = to_bl_of_bin [folded word_number_of_def] |
|
703 |
|
704 lemma to_bl_to_bin [simp] : "bl_to_bin (to_bl w) = uint w" |
|
705 unfolding uint_bl by (simp add : word_size) |
|
706 |
|
707 lemmas uint_bl_bin [simp] = trans [OF bin_bl_bin word_ubin.norm_Rep, standard] |
|
708 |
|
709 lemmas num_AB_u [simp] = word_uint.Rep_inverse |
|
710 [unfolded o_def word_number_of_def [symmetric], standard] |
|
711 lemmas num_AB_s [simp] = word_sint.Rep_inverse |
|
712 [unfolded o_def word_number_of_def [symmetric], standard] |
|
713 |
|
714 (* naturals *) |
|
715 lemma uints_unats: "uints n = int ` unats n" |
|
716 apply (unfold unats_def uints_num) |
|
717 apply safe |
|
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" |
|
727 by (auto simp add : uints_unats image_iff) |
|
728 |
|
729 lemmas bintr_num = word_ubin.norm_eq_iff |
|
730 [symmetric, folded word_number_of_def, standard] |
|
731 lemmas sbintr_num = word_sbin.norm_eq_iff |
|
732 [symmetric, folded word_number_of_def, standard] |
|
733 |
|
734 lemmas num_of_bintr = word_ubin.Abs_norm [folded word_number_of_def, standard] |
|
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': |
|
741 "bintrunc (len_of TYPE('a :: len0)) a = b ==> |
|
742 number_of a = (number_of b :: 'a word)" |
|
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 ==> |
|
749 number_of a = (number_of b :: 'a word)" |
|
750 apply safe |
|
751 apply (rule_tac num_of_sbintr [symmetric]) |
|
752 done |
|
753 |
|
754 lemmas num_abs_bintr = sym [THEN trans, |
|
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] |
|
758 |
|
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 |
|
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) = |
|
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 |
|
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] |
|
900 |
|
901 lemmas isduu = is_up_down [where c = "ucast", THEN iffD2] |
|
902 lemmas isdus = is_up_down [where c = "scast", THEN iffD2] |
|
903 lemmas ucast_down_ucast_id = isduu [THEN ucast_up_ucast_id] |
|
904 lemmas scast_down_scast_id = isdus [THEN ucast_up_ucast_id] |
|
905 |
|
906 lemma up_ucast_surj: |
|
907 "is_up (ucast :: 'b::len0 word => 'a::len0 word) ==> |
|
908 surj (ucast :: 'a word => 'b word)" |
|
909 by (rule surjI, erule ucast_up_ucast_id) |
|
910 |
|
911 lemma up_scast_surj: |
|
912 "is_up (scast :: 'b::len word => 'a::len word) ==> |
|
913 surj (scast :: 'a word => 'b word)" |
|
914 by (rule surjI, erule scast_up_scast_id) |
|
915 |
|
916 lemma down_scast_inj: |
|
917 "is_down (scast :: 'b::len word => 'a::len word) ==> |
|
918 inj_on (ucast :: 'a word => 'b word) A" |
|
919 by (rule inj_on_inverseI, erule scast_down_scast_id) |
|
920 |
|
921 lemma down_ucast_inj: |
|
922 "is_down (ucast :: 'b::len0 word => 'a::len0 word) ==> |
|
923 inj_on (ucast :: 'a word => 'b word) A" |
|
924 by (rule inj_on_inverseI, erule ucast_down_ucast_id) |
|
925 |
|
926 lemma of_bl_append_same: "of_bl (X @ to_bl w) = w" |
|
927 by (rule word_bl.Rep_eqD) (simp add: word_rep_drop) |
|
928 |
|
929 lemma ucast_down_no': |
|
930 "uc = ucast ==> is_down uc ==> uc (number_of bin) = number_of bin" |
|
931 apply (unfold word_number_of_def is_down) |
|
932 apply (clarsimp simp add: ucast_def word_ubin.eq_norm) |
|
933 apply (rule word_ubin.norm_eq_iff [THEN iffD1]) |
|
934 apply (erule bintrunc_bintrunc_ge) |
|
935 done |
|
936 |
|
937 lemmas ucast_down_no = ucast_down_no' [OF refl] |
|
938 |
|
939 lemma ucast_down_bl': "uc = ucast ==> is_down uc ==> uc (of_bl bl) = of_bl bl" |
|
940 unfolding of_bl_no by clarify (erule ucast_down_no) |
|
941 |
|
942 lemmas ucast_down_bl = ucast_down_bl' [OF refl] |
|
943 |
|
944 lemmas slice_def' = slice_def [unfolded word_size] |
|
945 lemmas test_bit_def' = word_test_bit_def [THEN fun_cong] |
|
946 |
|
947 lemmas word_log_defs = word_and_def word_or_def word_xor_def word_not_def |
|
948 lemmas word_log_bin_defs = word_log_defs |
|
949 |
|
950 text {* Executable equality *} |
|
951 |
|
952 instantiation word :: ("{len0}") eq |
|
953 begin |
|
954 |
|
955 definition eq_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> bool" where |
|
956 "eq_word k l \<longleftrightarrow> HOL.eq (uint k) (uint l)" |
|
957 |
|
958 instance proof |
|
959 qed (simp add: eq eq_word_def) |
|
960 |
|
961 end |
|
962 |
|
963 |
|
964 subsection {* Word Arithmetic *} |
|
965 |
|
966 lemma word_less_alt: "(a < b) = (uint a < uint b)" |
|
967 unfolding word_less_def word_le_def |
|
968 by (auto simp del: word_uint.Rep_inject |
|
969 simp: word_uint.Rep_inject [symmetric]) |
|
970 |
|
971 lemma signed_linorder: "class.linorder word_sle word_sless" |
|
972 proof |
|
973 qed (unfold word_sle_def word_sless_def, auto) |
|
974 |
|
975 interpretation signed: linorder "word_sle" "word_sless" |
|
976 by (rule signed_linorder) |
|
977 |
|
978 lemmas word_arith_wis = |
|
979 word_add_def word_mult_def word_minus_def |
|
980 word_succ_def word_pred_def word_0_wi word_1_wi |
|
981 |
|
982 lemma udvdI: |
|
983 "0 \<le> n ==> uint b = n * uint a ==> a udvd b" |
|
984 by (auto simp: udvd_def) |
|
985 |
|
986 lemmas word_div_no [simp] = |
|
987 word_div_def [of "number_of a" "number_of b", standard] |
|
988 |
|
989 lemmas word_mod_no [simp] = |
|
990 word_mod_def [of "number_of a" "number_of b", standard] |
|
991 |
|
992 lemmas word_less_no [simp] = |
|
993 word_less_def [of "number_of a" "number_of b", standard] |
|
994 |
|
995 lemmas word_le_no [simp] = |
|
996 word_le_def [of "number_of a" "number_of b", standard] |
|
997 |
|
998 lemmas word_sless_no [simp] = |
|
999 word_sless_def [of "number_of a" "number_of b", standard] |
|
1000 |
|
1001 lemmas word_sle_no [simp] = |
|
1002 word_sle_def [of "number_of a" "number_of b", standard] |
|
1003 |
|
1004 (* following two are available in class number_ring, |
|
1005 but convenient to have them here here; |
|
1006 note - the number_ring versions, numeral_0_eq_0 and numeral_1_eq_1 |
|
1007 are in the default simpset, so to use the automatic simplifications for |
|
1008 (eg) sint (number_of bin) on sint 1, must do |
|
1009 (simp add: word_1_no del: numeral_1_eq_1) |
|
1010 *) |
|
1011 lemmas word_0_wi_Pls = word_0_wi [folded Pls_def] |
|
1012 lemmas word_0_no = word_0_wi_Pls [folded word_no_wi] |
|
1013 |
|
1014 lemma int_one_bin: "(1 :: int) == (Int.Pls BIT 1)" |
|
1015 unfolding Pls_def Bit_def by auto |
|
1016 |
|
1017 lemma word_1_no: |
|
1018 "(1 :: 'a :: len0 word) == number_of (Int.Pls BIT 1)" |
|
1019 unfolding word_1_wi word_number_of_def int_one_bin by auto |
|
1020 |
|
1021 lemma word_m1_wi: "-1 == word_of_int -1" |
|
1022 by (rule word_number_of_alt) |
|
1023 |
|
1024 lemma word_m1_wi_Min: "-1 = word_of_int Int.Min" |
|
1025 by (simp add: word_m1_wi number_of_eq) |
|
1026 |
|
1027 lemma word_0_bl: "of_bl [] = 0" |
|
1028 unfolding word_0_wi of_bl_def by (simp add : Pls_def) |
|
1029 |
|
1030 lemma word_1_bl: "of_bl [True] = 1" |
|
1031 unfolding word_1_wi of_bl_def |
|
1032 by (simp add : bl_to_bin_def Bit_def Pls_def) |
|
1033 |
|
1034 lemma uint_eq_0 [simp] : "(uint 0 = 0)" |
|
1035 unfolding word_0_wi |
|
1036 by (simp add: word_ubin.eq_norm Pls_def [symmetric]) |
|
1037 |
|
1038 lemma of_bl_0 [simp] : "of_bl (replicate n False) = 0" |
|
1039 by (simp add : word_0_wi of_bl_def bl_to_bin_rep_False Pls_def) |
|
1040 |
|
1041 lemma to_bl_0: |
|
1042 "to_bl (0::'a::len0 word) = replicate (len_of TYPE('a)) False" |
|
1043 unfolding uint_bl |
|
1044 by (simp add : word_size bin_to_bl_Pls Pls_def [symmetric]) |
|
1045 |
|
1046 lemma uint_0_iff: "(uint x = 0) = (x = 0)" |
|
1047 by (auto intro!: word_uint.Rep_eqD) |
|
1048 |
|
1049 lemma unat_0_iff: "(unat x = 0) = (x = 0)" |
|
1050 unfolding unat_def by (auto simp add : nat_eq_iff uint_0_iff) |
|
1051 |
|
1052 lemma unat_0 [simp]: "unat 0 = 0" |
|
1053 unfolding unat_def by auto |
|
1054 |
|
1055 lemma size_0_same': "size w = 0 ==> w = (v :: 'a :: len0 word)" |
|
1056 apply (unfold word_size) |
|
1057 apply (rule box_equals) |
|
1058 defer |
|
1059 apply (rule word_uint.Rep_inverse)+ |
|
1060 apply (rule word_ubin.norm_eq_iff [THEN iffD1]) |
|
1061 apply simp |
|
1062 done |
|
1063 |
|
1064 lemmas size_0_same = size_0_same' [folded word_size] |
|
1065 |
|
1066 lemmas unat_eq_0 = unat_0_iff |
|
1067 lemmas unat_eq_zero = unat_0_iff |
|
1068 |
|
1069 lemma unat_gt_0: "(0 < unat x) = (x ~= 0)" |
|
1070 by (auto simp: unat_0_iff [symmetric]) |
|
1071 |
|
1072 lemma ucast_0 [simp] : "ucast 0 = 0" |
|
1073 unfolding ucast_def |
|
1074 by simp (simp add: word_0_wi) |
|
1075 |
|
1076 lemma sint_0 [simp] : "sint 0 = 0" |
|
1077 unfolding sint_uint |
|
1078 by (simp add: Pls_def [symmetric]) |
|
1079 |
|
1080 lemma scast_0 [simp] : "scast 0 = 0" |
|
1081 apply (unfold scast_def) |
|
1082 apply simp |
|
1083 apply (simp add: word_0_wi) |
|
1084 done |
|
1085 |
|
1086 lemma sint_n1 [simp] : "sint -1 = -1" |
|
1087 apply (unfold word_m1_wi_Min) |
|
1088 apply (simp add: word_sbin.eq_norm) |
|
1089 apply (unfold Min_def number_of_eq) |
|
1090 apply simp |
|
1091 done |
|
1092 |
|
1093 lemma scast_n1 [simp] : "scast -1 = -1" |
|
1094 apply (unfold scast_def sint_n1) |
|
1095 apply (unfold word_number_of_alt) |
|
1096 apply (rule refl) |
|
1097 done |
|
1098 |
|
1099 lemma uint_1 [simp] : "uint (1 :: 'a :: len word) = 1" |
|
1100 unfolding word_1_wi |
|
1101 by (simp add: word_ubin.eq_norm int_one_bin bintrunc_minus_simps) |
|
1102 |
|
1103 lemma unat_1 [simp] : "unat (1 :: 'a :: len word) = 1" |
|
1104 by (unfold unat_def uint_1) auto |
|
1105 |
|
1106 lemma ucast_1 [simp] : "ucast (1 :: 'a :: len word) = 1" |
|
1107 unfolding ucast_def word_1_wi |
|
1108 by (simp add: word_ubin.eq_norm int_one_bin bintrunc_minus_simps) |
|
1109 |
|
1110 (* abstraction preserves the operations |
|
1111 (the definitions tell this for bins in range uint) *) |
|
1112 |
|
1113 lemmas arths = |
|
1114 bintr_ariths [THEN word_ubin.norm_eq_iff [THEN iffD1], |
|
1115 folded word_ubin.eq_norm, standard] |
|
1116 |
|
1117 lemma wi_homs: |
|
1118 shows |
|
1119 wi_hom_add: "word_of_int a + word_of_int b = word_of_int (a + b)" and |
|
1120 wi_hom_mult: "word_of_int a * word_of_int b = word_of_int (a * b)" and |
|
1121 wi_hom_neg: "- word_of_int a = word_of_int (- a)" and |
|
1122 wi_hom_succ: "word_succ (word_of_int a) = word_of_int (Int.succ a)" and |
|
1123 wi_hom_pred: "word_pred (word_of_int a) = word_of_int (Int.pred a)" |
|
1124 by (auto simp: word_arith_wis arths) |
|
1125 |
|
1126 lemmas wi_hom_syms = wi_homs [symmetric] |
|
1127 |
|
1128 lemma word_sub_def: "a - b == a + - (b :: 'a :: len0 word)" |
|
1129 unfolding word_sub_wi diff_def |
|
1130 by (simp only : word_uint.Rep_inverse wi_hom_syms) |
|
1131 |
|
1132 lemmas word_diff_minus = word_sub_def [THEN meta_eq_to_obj_eq, standard] |
|
1133 |
|
1134 lemma word_of_int_sub_hom: |
|
1135 "(word_of_int a) - word_of_int b = word_of_int (a - b)" |
|
1136 unfolding word_sub_def diff_def by (simp only : wi_homs) |
|
1137 |
|
1138 lemmas new_word_of_int_homs = |
|
1139 word_of_int_sub_hom wi_homs word_0_wi word_1_wi |
|
1140 |
|
1141 lemmas new_word_of_int_hom_syms = new_word_of_int_homs [symmetric, standard] |
|
1142 |
|
1143 lemmas word_of_int_hom_syms = |
|
1144 new_word_of_int_hom_syms [unfolded succ_def pred_def] |
|
1145 |
|
1146 lemmas word_of_int_homs = |
|
1147 new_word_of_int_homs [unfolded succ_def pred_def] |
|
1148 |
|
1149 lemmas word_of_int_add_hom = word_of_int_homs (2) |
|
1150 lemmas word_of_int_mult_hom = word_of_int_homs (3) |
|
1151 lemmas word_of_int_minus_hom = word_of_int_homs (4) |
|
1152 lemmas word_of_int_succ_hom = word_of_int_homs (5) |
|
1153 lemmas word_of_int_pred_hom = word_of_int_homs (6) |
|
1154 lemmas word_of_int_0_hom = word_of_int_homs (7) |
|
1155 lemmas word_of_int_1_hom = word_of_int_homs (8) |
|
1156 |
|
1157 (* now, to get the weaker results analogous to word_div/mod_def *) |
|
1158 |
|
1159 lemmas word_arith_alts = |
|
1160 word_sub_wi [unfolded succ_def pred_def, standard] |
|
1161 word_arith_wis [unfolded succ_def pred_def, standard] |
|
1162 |
|
1163 lemmas word_sub_alt = word_arith_alts (1) |
|
1164 lemmas word_add_alt = word_arith_alts (2) |
|
1165 lemmas word_mult_alt = word_arith_alts (3) |
|
1166 lemmas word_minus_alt = word_arith_alts (4) |
|
1167 lemmas word_succ_alt = word_arith_alts (5) |
|
1168 lemmas word_pred_alt = word_arith_alts (6) |
|
1169 lemmas word_0_alt = word_arith_alts (7) |
|
1170 lemmas word_1_alt = word_arith_alts (8) |
|
1171 |
|
1172 subsection "Transferring goals from words to ints" |
|
1173 |
|
1174 lemma word_ths: |
|
1175 shows |
|
1176 word_succ_p1: "word_succ a = a + 1" and |
|
1177 word_pred_m1: "word_pred a = a - 1" and |
|
1178 word_pred_succ: "word_pred (word_succ a) = a" and |
|
1179 word_succ_pred: "word_succ (word_pred a) = a" and |
|
1180 word_mult_succ: "word_succ a * b = b + a * b" |
|
1181 by (rule word_uint.Abs_cases [of b], |
|
1182 rule word_uint.Abs_cases [of a], |
|
1183 simp add: pred_def succ_def add_commute mult_commute |
|
1184 ring_distribs new_word_of_int_homs)+ |
|
1185 |
|
1186 lemmas uint_cong = arg_cong [where f = uint] |
|
1187 |
|
1188 lemmas uint_word_ariths = |
|
1189 word_arith_alts [THEN trans [OF uint_cong int_word_uint], standard] |
|
1190 |
|
1191 lemmas uint_word_arith_bintrs = uint_word_ariths [folded bintrunc_mod2p] |
|
1192 |
|
1193 (* similar expressions for sint (arith operations) *) |
|
1194 lemmas sint_word_ariths = uint_word_arith_bintrs |
|
1195 [THEN uint_sint [symmetric, THEN trans], |
|
1196 unfolded uint_sint bintr_arith1s bintr_ariths |
|
1197 len_gt_0 [THEN bin_sbin_eq_iff'] word_sbin.norm_Rep, standard] |
|
1198 |
|
1199 lemmas uint_div_alt = word_div_def |
|
1200 [THEN trans [OF uint_cong int_word_uint], standard] |
|
1201 lemmas uint_mod_alt = word_mod_def |
|
1202 [THEN trans [OF uint_cong int_word_uint], standard] |
|
1203 |
|
1204 lemma word_pred_0_n1: "word_pred 0 = word_of_int -1" |
|
1205 unfolding word_pred_def number_of_eq |
|
1206 by (simp add : pred_def word_no_wi) |
|
1207 |
|
1208 lemma word_pred_0_Min: "word_pred 0 = word_of_int Int.Min" |
|
1209 by (simp add: word_pred_0_n1 number_of_eq) |
|
1210 |
|
1211 lemma word_m1_Min: "- 1 = word_of_int Int.Min" |
|
1212 unfolding Min_def by (simp only: word_of_int_hom_syms) |
|
1213 |
|
1214 lemma succ_pred_no [simp]: |
|
1215 "word_succ (number_of bin) = number_of (Int.succ bin) & |
|
1216 word_pred (number_of bin) = number_of (Int.pred bin)" |
|
1217 unfolding word_number_of_def by (simp add : new_word_of_int_homs) |
|
1218 |
|
1219 lemma word_sp_01 [simp] : |
|
1220 "word_succ -1 = 0 & word_succ 0 = 1 & word_pred 0 = -1 & word_pred 1 = 0" |
|
1221 by (unfold word_0_no word_1_no) auto |
|
1222 |
|
1223 (* alternative approach to lifting arithmetic equalities *) |
|
1224 lemma word_of_int_Ex: |
|
1225 "\<exists>y. x = word_of_int y" |
|
1226 by (rule_tac x="uint x" in exI) simp |
|
1227 |
|
1228 lemma word_arith_eqs: |
|
1229 fixes a :: "'a::len0 word" |
|
1230 fixes b :: "'a::len0 word" |
|
1231 shows |
|
1232 word_add_0: "0 + a = a" and |
|
1233 word_add_0_right: "a + 0 = a" and |
|
1234 word_mult_1: "1 * a = a" and |
|
1235 word_mult_1_right: "a * 1 = a" and |
|
1236 word_add_commute: "a + b = b + a" and |
|
1237 word_add_assoc: "a + b + c = a + (b + c)" and |
|
1238 word_add_left_commute: "a + (b + c) = b + (a + c)" and |
|
1239 word_mult_commute: "a * b = b * a" and |
|
1240 word_mult_assoc: "a * b * c = a * (b * c)" and |
|
1241 word_mult_left_commute: "a * (b * c) = b * (a * c)" and |
|
1242 word_left_distrib: "(a + b) * c = a * c + b * c" and |
|
1243 word_right_distrib: "a * (b + c) = a * b + a * c" and |
|
1244 word_left_minus: "- a + a = 0" and |
|
1245 word_diff_0_right: "a - 0 = a" and |
|
1246 word_diff_self: "a - a = 0" |
|
1247 using word_of_int_Ex [of a] |
|
1248 word_of_int_Ex [of b] |
|
1249 word_of_int_Ex [of c] |
|
1250 by (auto simp: word_of_int_hom_syms [symmetric] |
|
1251 zadd_0_right add_commute add_assoc add_left_commute |
|
1252 mult_commute mult_assoc mult_left_commute |
|
1253 left_distrib right_distrib) |
|
1254 |
|
1255 lemmas word_add_ac = word_add_commute word_add_assoc word_add_left_commute |
|
1256 lemmas word_mult_ac = word_mult_commute word_mult_assoc word_mult_left_commute |
|
1257 |
|
1258 lemmas word_plus_ac0 = word_add_0 word_add_0_right word_add_ac |
|
1259 lemmas word_times_ac1 = word_mult_1 word_mult_1_right word_mult_ac |
|
1260 |
|
1261 |
|
1262 subsection "Order on fixed-length words" |
|
1263 |
|
1264 lemma word_order_trans: "x <= y ==> y <= z ==> x <= (z :: 'a :: len0 word)" |
|
1265 unfolding word_le_def by auto |
|
1266 |
|
1267 lemma word_order_refl: "z <= (z :: 'a :: len0 word)" |
|
1268 unfolding word_le_def by auto |
|
1269 |
|
1270 lemma word_order_antisym: "x <= y ==> y <= x ==> x = (y :: 'a :: len0 word)" |
|
1271 unfolding word_le_def by (auto intro!: word_uint.Rep_eqD) |
|
1272 |
|
1273 lemma word_order_linear: |
|
1274 "y <= x | x <= (y :: 'a :: len0 word)" |
|
1275 unfolding word_le_def by auto |
|
1276 |
|
1277 lemma word_zero_le [simp] : |
|
1278 "0 <= (y :: 'a :: len0 word)" |
|
1279 unfolding word_le_def by auto |
|
1280 |
|
1281 instance word :: (len0) semigroup_add |
|
1282 by intro_classes (simp add: word_add_assoc) |
|
1283 |
|
1284 instance word :: (len0) linorder |
|
1285 by intro_classes (auto simp: word_less_def word_le_def) |
|
1286 |
|
1287 instance word :: (len0) ring |
|
1288 by intro_classes |
|
1289 (auto simp: word_arith_eqs word_diff_minus |
|
1290 word_diff_self [unfolded word_diff_minus]) |
|
1291 |
|
1292 lemma word_m1_ge [simp] : "word_pred 0 >= y" |
|
1293 unfolding word_le_def |
|
1294 by (simp only : word_pred_0_n1 word_uint.eq_norm m1mod2k) auto |
|
1295 |
|
1296 lemmas word_n1_ge [simp] = word_m1_ge [simplified word_sp_01] |
|
1297 |
|
1298 lemmas word_not_simps [simp] = |
|
1299 word_zero_le [THEN leD] word_m1_ge [THEN leD] word_n1_ge [THEN leD] |
|
1300 |
|
1301 lemma word_gt_0: "0 < y = (0 ~= (y :: 'a :: len0 word))" |
|
1302 unfolding word_less_def by auto |
|
1303 |
|
1304 lemmas word_gt_0_no [simp] = word_gt_0 [of "number_of y", standard] |
|
1305 |
|
1306 lemma word_sless_alt: "(a <s b) == (sint a < sint b)" |
|
1307 unfolding word_sle_def word_sless_def |
|
1308 by (auto simp add: less_le) |
|
1309 |
|
1310 lemma word_le_nat_alt: "(a <= b) = (unat a <= unat b)" |
|
1311 unfolding unat_def word_le_def |
|
1312 by (rule nat_le_eq_zle [symmetric]) simp |
|
1313 |
|
1314 lemma word_less_nat_alt: "(a < b) = (unat a < unat b)" |
|
1315 unfolding unat_def word_less_alt |
|
1316 by (rule nat_less_eq_zless [symmetric]) simp |
|
1317 |
|
1318 lemma wi_less: |
|
1319 "(word_of_int n < (word_of_int m :: 'a :: len0 word)) = |
|
1320 (n mod 2 ^ len_of TYPE('a) < m mod 2 ^ len_of TYPE('a))" |
|
1321 unfolding word_less_alt by (simp add: word_uint.eq_norm) |
|
1322 |
|
1323 lemma wi_le: |
|
1324 "(word_of_int n <= (word_of_int m :: 'a :: len0 word)) = |
|
1325 (n mod 2 ^ len_of TYPE('a) <= m mod 2 ^ len_of TYPE('a))" |
|
1326 unfolding word_le_def by (simp add: word_uint.eq_norm) |
|
1327 |
|
1328 lemma udvd_nat_alt: "a udvd b = (EX n>=0. unat b = n * unat a)" |
|
1329 apply (unfold udvd_def) |
|
1330 apply safe |
|
1331 apply (simp add: unat_def nat_mult_distrib) |
|
1332 apply (simp add: uint_nat int_mult) |
|
1333 apply (rule exI) |
|
1334 apply safe |
|
1335 prefer 2 |
|
1336 apply (erule notE) |
|
1337 apply (rule refl) |
|
1338 apply force |
|
1339 done |
|
1340 |
|
1341 lemma udvd_iff_dvd: "x udvd y <-> unat x dvd unat y" |
|
1342 unfolding dvd_def udvd_nat_alt by force |
|
1343 |
|
1344 lemmas unat_mono = word_less_nat_alt [THEN iffD1, standard] |
|
1345 |
|
1346 lemma word_zero_neq_one: "0 < len_of TYPE ('a :: len0) ==> (0 :: 'a word) ~= 1"; |
|
1347 unfolding word_arith_wis |
|
1348 by (auto simp add: word_ubin.norm_eq_iff [symmetric] gr0_conv_Suc) |
|
1349 |
|
1350 lemmas lenw1_zero_neq_one = len_gt_0 [THEN word_zero_neq_one] |
|
1351 |
|
1352 lemma no_no [simp] : "number_of (number_of b) = number_of b" |
|
1353 by (simp add: number_of_eq) |
|
1354 |
|
1355 lemma unat_minus_one: "x ~= 0 ==> unat (x - 1) = unat x - 1" |
|
1356 apply (unfold unat_def) |
|
1357 apply (simp only: int_word_uint word_arith_alts rdmods) |
|
1358 apply (subgoal_tac "uint x >= 1") |
|
1359 prefer 2 |
|
1360 apply (drule contrapos_nn) |
|
1361 apply (erule word_uint.Rep_inverse' [symmetric]) |
|
1362 apply (insert uint_ge_0 [of x])[1] |
|
1363 apply arith |
|
1364 apply (rule box_equals) |
|
1365 apply (rule nat_diff_distrib) |
|
1366 prefer 2 |
|
1367 apply assumption |
|
1368 apply simp |
|
1369 apply (subst mod_pos_pos_trivial) |
|
1370 apply arith |
|
1371 apply (insert uint_lt2p [of x])[1] |
|
1372 apply arith |
|
1373 apply (rule refl) |
|
1374 apply simp |
|
1375 done |
|
1376 |
|
1377 lemma measure_unat: "p ~= 0 ==> unat (p - 1) < unat p" |
|
1378 by (simp add: unat_minus_one) (simp add: unat_0_iff [symmetric]) |
|
1379 |
|
1380 lemmas uint_add_ge0 [simp] = |
|
1381 add_nonneg_nonneg [OF uint_ge_0 uint_ge_0, standard] |
|
1382 lemmas uint_mult_ge0 [simp] = |
|
1383 mult_nonneg_nonneg [OF uint_ge_0 uint_ge_0, standard] |
|
1384 |
|
1385 lemma uint_sub_lt2p [simp]: |
|
1386 "uint (x :: 'a :: len0 word) - uint (y :: 'b :: len0 word) < |
|
1387 2 ^ len_of TYPE('a)" |
|
1388 using uint_ge_0 [of y] uint_lt2p [of x] by arith |
|
1389 |
|
1390 |
|
1391 subsection "Conditions for the addition (etc) of two words to overflow" |
|
1392 |
|
1393 lemma uint_add_lem: |
|
1394 "(uint x + uint y < 2 ^ len_of TYPE('a)) = |
|
1395 (uint (x + y :: 'a :: len0 word) = uint x + uint y)" |
|
1396 by (unfold uint_word_ariths) (auto intro!: trans [OF _ int_mod_lem]) |
|
1397 |
|
1398 lemma uint_mult_lem: |
|
1399 "(uint x * uint y < 2 ^ len_of TYPE('a)) = |
|
1400 (uint (x * y :: 'a :: len0 word) = uint x * uint y)" |
|
1401 by (unfold uint_word_ariths) (auto intro!: trans [OF _ int_mod_lem]) |
|
1402 |
|
1403 lemma uint_sub_lem: |
|
1404 "(uint x >= uint y) = (uint (x - y) = uint x - uint y)" |
|
1405 by (unfold uint_word_ariths) (auto intro!: trans [OF _ int_mod_lem]) |
|
1406 |
|
1407 lemma uint_add_le: "uint (x + y) <= uint x + uint y" |
|
1408 unfolding uint_word_ariths by (auto simp: mod_add_if_z) |
|
1409 |
|
1410 lemma uint_sub_ge: "uint (x - y) >= uint x - uint y" |
|
1411 unfolding uint_word_ariths by (auto simp: mod_sub_if_z) |
|
1412 |
|
1413 lemmas uint_sub_if' = |
|
1414 trans [OF uint_word_ariths(1) mod_sub_if_z, simplified, standard] |
|
1415 lemmas uint_plus_if' = |
|
1416 trans [OF uint_word_ariths(2) mod_add_if_z, simplified, standard] |
|
1417 |
|
1418 |
|
1419 subsection {* Definition of uint\_arith *} |
|
1420 |
|
1421 lemma word_of_int_inverse: |
|
1422 "word_of_int r = a ==> 0 <= r ==> r < 2 ^ len_of TYPE('a) ==> |
|
1423 uint (a::'a::len0 word) = r" |
|
1424 apply (erule word_uint.Abs_inverse' [rotated]) |
|
1425 apply (simp add: uints_num) |
|
1426 done |
|
1427 |
|
1428 lemma uint_split: |
|
1429 fixes x::"'a::len0 word" |
|
1430 shows "P (uint x) = |
|
1431 (ALL i. word_of_int i = x & 0 <= i & i < 2^len_of TYPE('a) --> P i)" |
|
1432 apply (fold word_int_case_def) |
|
1433 apply (auto dest!: word_of_int_inverse simp: int_word_uint int_mod_eq' |
|
1434 split: word_int_split) |
|
1435 done |
|
1436 |
|
1437 lemma uint_split_asm: |
|
1438 fixes x::"'a::len0 word" |
|
1439 shows "P (uint x) = |
|
1440 (~(EX i. word_of_int i = x & 0 <= i & i < 2^len_of TYPE('a) & ~ P i))" |
|
1441 by (auto dest!: word_of_int_inverse |
|
1442 simp: int_word_uint int_mod_eq' |
|
1443 split: uint_split) |
|
1444 |
|
1445 lemmas uint_splits = uint_split uint_split_asm |
|
1446 |
|
1447 lemmas uint_arith_simps = |
|
1448 word_le_def word_less_alt |
|
1449 word_uint.Rep_inject [symmetric] |
|
1450 uint_sub_if' uint_plus_if' |
|
1451 |
|
1452 (* use this to stop, eg, 2 ^ len_of TYPE (32) being simplified *) |
|
1453 lemma power_False_cong: "False ==> a ^ b = c ^ d" |
|
1454 by auto |
|
1455 |
|
1456 (* uint_arith_tac: reduce to arithmetic on int, try to solve by arith *) |
|
1457 ML {* |
|
1458 fun uint_arith_ss_of ss = |
|
1459 ss addsimps @{thms uint_arith_simps} |
|
1460 delsimps @{thms word_uint.Rep_inject} |
|
1461 addsplits @{thms split_if_asm} |
|
1462 addcongs @{thms power_False_cong} |
|
1463 |
|
1464 fun uint_arith_tacs ctxt = |
|
1465 let |
|
1466 fun arith_tac' n t = |
|
1467 Arith_Data.verbose_arith_tac ctxt n t |
|
1468 handle Cooper.COOPER _ => Seq.empty; |
|
1469 val cs = claset_of ctxt; |
|
1470 val ss = simpset_of ctxt; |
|
1471 in |
|
1472 [ clarify_tac cs 1, |
|
1473 full_simp_tac (uint_arith_ss_of ss) 1, |
|
1474 ALLGOALS (full_simp_tac (HOL_ss addsplits @{thms uint_splits} |
|
1475 addcongs @{thms power_False_cong})), |
|
1476 rewrite_goals_tac @{thms word_size}, |
|
1477 ALLGOALS (fn n => REPEAT (resolve_tac [allI, impI] n) THEN |
|
1478 REPEAT (etac conjE n) THEN |
|
1479 REPEAT (dtac @{thm word_of_int_inverse} n |
|
1480 THEN atac n |
|
1481 THEN atac n)), |
|
1482 TRYALL arith_tac' ] |
|
1483 end |
|
1484 |
|
1485 fun uint_arith_tac ctxt = SELECT_GOAL (EVERY (uint_arith_tacs ctxt)) |
|
1486 *} |
|
1487 |
|
1488 method_setup uint_arith = |
|
1489 {* Scan.succeed (SIMPLE_METHOD' o uint_arith_tac) *} |
|
1490 "solving word arithmetic via integers and arith" |
|
1491 |
|
1492 |
|
1493 subsection "More on overflows and monotonicity" |
|
1494 |
|
1495 lemma no_plus_overflow_uint_size: |
|
1496 "((x :: 'a :: len0 word) <= x + y) = (uint x + uint y < 2 ^ size x)" |
|
1497 unfolding word_size by uint_arith |
|
1498 |
|
1499 lemmas no_olen_add = no_plus_overflow_uint_size [unfolded word_size] |
|
1500 |
|
1501 lemma no_ulen_sub: "((x :: 'a :: len0 word) >= x - y) = (uint y <= uint x)" |
|
1502 by uint_arith |
|
1503 |
|
1504 lemma no_olen_add': |
|
1505 fixes x :: "'a::len0 word" |
|
1506 shows "(x \<le> y + x) = (uint y + uint x < 2 ^ len_of TYPE('a))" |
|
1507 by (simp add: word_add_ac add_ac no_olen_add) |
|
1508 |
|
1509 lemmas olen_add_eqv = trans [OF no_olen_add no_olen_add' [symmetric], standard] |
|
1510 |
|
1511 lemmas uint_plus_simple_iff = trans [OF no_olen_add uint_add_lem, standard] |
|
1512 lemmas uint_plus_simple = uint_plus_simple_iff [THEN iffD1, standard] |
|
1513 lemmas uint_minus_simple_iff = trans [OF no_ulen_sub uint_sub_lem, standard] |
|
1514 lemmas uint_minus_simple_alt = uint_sub_lem [folded word_le_def] |
|
1515 lemmas word_sub_le_iff = no_ulen_sub [folded word_le_def] |
|
1516 lemmas word_sub_le = word_sub_le_iff [THEN iffD2, standard] |
|
1517 |
|
1518 lemma word_less_sub1: |
|
1519 "(x :: 'a :: len word) ~= 0 ==> (1 < x) = (0 < x - 1)" |
|
1520 by uint_arith |
|
1521 |
|
1522 lemma word_le_sub1: |
|
1523 "(x :: 'a :: len word) ~= 0 ==> (1 <= x) = (0 <= x - 1)" |
|
1524 by uint_arith |
|
1525 |
|
1526 lemma sub_wrap_lt: |
|
1527 "((x :: 'a :: len0 word) < x - z) = (x < z)" |
|
1528 by uint_arith |
|
1529 |
|
1530 lemma sub_wrap: |
|
1531 "((x :: 'a :: len0 word) <= x - z) = (z = 0 | x < z)" |
|
1532 by uint_arith |
|
1533 |
|
1534 lemma plus_minus_not_NULL_ab: |
|
1535 "(x :: 'a :: len0 word) <= ab - c ==> c <= ab ==> c ~= 0 ==> x + c ~= 0" |
|
1536 by uint_arith |
|
1537 |
|
1538 lemma plus_minus_no_overflow_ab: |
|
1539 "(x :: 'a :: len0 word) <= ab - c ==> c <= ab ==> x <= x + c" |
|
1540 by uint_arith |
|
1541 |
|
1542 lemma le_minus': |
|
1543 "(a :: 'a :: len0 word) + c <= b ==> a <= a + c ==> c <= b - a" |
|
1544 by uint_arith |
|
1545 |
|
1546 lemma le_plus': |
|
1547 "(a :: 'a :: len0 word) <= b ==> c <= b - a ==> a + c <= b" |
|
1548 by uint_arith |
|
1549 |
|
1550 lemmas le_plus = le_plus' [rotated] |
|
1551 |
|
1552 lemmas le_minus = leD [THEN thin_rl, THEN le_minus', standard] |
|
1553 |
|
1554 lemma word_plus_mono_right: |
|
1555 "(y :: 'a :: len0 word) <= z ==> x <= x + z ==> x + y <= x + z" |
|
1556 by uint_arith |
|
1557 |
|
1558 lemma word_less_minus_cancel: |
|
1559 "y - x < z - x ==> x <= z ==> (y :: 'a :: len0 word) < z" |
|
1560 by uint_arith |
|
1561 |
|
1562 lemma word_less_minus_mono_left: |
|
1563 "(y :: 'a :: len0 word) < z ==> x <= y ==> y - x < z - x" |
|
1564 by uint_arith |
|
1565 |
|
1566 lemma word_less_minus_mono: |
|
1567 "a < c ==> d < b ==> a - b < a ==> c - d < c |
|
1568 ==> a - b < c - (d::'a::len word)" |
|
1569 by uint_arith |
|
1570 |
|
1571 lemma word_le_minus_cancel: |
|
1572 "y - x <= z - x ==> x <= z ==> (y :: 'a :: len0 word) <= z" |
|
1573 by uint_arith |
|
1574 |
|
1575 lemma word_le_minus_mono_left: |
|
1576 "(y :: 'a :: len0 word) <= z ==> x <= y ==> y - x <= z - x" |
|
1577 by uint_arith |
|
1578 |
|
1579 lemma word_le_minus_mono: |
|
1580 "a <= c ==> d <= b ==> a - b <= a ==> c - d <= c |
|
1581 ==> a - b <= c - (d::'a::len word)" |
|
1582 by uint_arith |
|
1583 |
|
1584 lemma plus_le_left_cancel_wrap: |
|
1585 "(x :: 'a :: len0 word) + y' < x ==> x + y < x ==> (x + y' < x + y) = (y' < y)" |
|
1586 by uint_arith |
|
1587 |
|
1588 lemma plus_le_left_cancel_nowrap: |
|
1589 "(x :: 'a :: len0 word) <= x + y' ==> x <= x + y ==> |
|
1590 (x + y' < x + y) = (y' < y)" |
|
1591 by uint_arith |
|
1592 |
|
1593 lemma word_plus_mono_right2: |
|
1594 "(a :: 'a :: len0 word) <= a + b ==> c <= b ==> a <= a + c" |
|
1595 by uint_arith |
|
1596 |
|
1597 lemma word_less_add_right: |
|
1598 "(x :: 'a :: len0 word) < y - z ==> z <= y ==> x + z < y" |
|
1599 by uint_arith |
|
1600 |
|
1601 lemma word_less_sub_right: |
|
1602 "(x :: 'a :: len0 word) < y + z ==> y <= x ==> x - y < z" |
|
1603 by uint_arith |
|
1604 |
|
1605 lemma word_le_plus_either: |
|
1606 "(x :: 'a :: len0 word) <= y | x <= z ==> y <= y + z ==> x <= y + z" |
|
1607 by uint_arith |
|
1608 |
|
1609 lemma word_less_nowrapI: |
|
1610 "(x :: 'a :: len0 word) < z - k ==> k <= z ==> 0 < k ==> x < x + k" |
|
1611 by uint_arith |
|
1612 |
|
1613 lemma inc_le: "(i :: 'a :: len word) < m ==> i + 1 <= m" |
|
1614 by uint_arith |
|
1615 |
|
1616 lemma inc_i: |
|
1617 "(1 :: 'a :: len word) <= i ==> i < m ==> 1 <= (i + 1) & i + 1 <= m" |
|
1618 by uint_arith |
|
1619 |
|
1620 lemma udvd_incr_lem: |
|
1621 "up < uq ==> up = ua + n * uint K ==> |
|
1622 uq = ua + n' * uint K ==> up + uint K <= uq" |
|
1623 apply clarsimp |
|
1624 apply (drule less_le_mult) |
|
1625 apply safe |
|
1626 done |
|
1627 |
|
1628 lemma udvd_incr': |
|
1629 "p < q ==> uint p = ua + n * uint K ==> |
|
1630 uint q = ua + n' * uint K ==> p + K <= q" |
|
1631 apply (unfold word_less_alt word_le_def) |
|
1632 apply (drule (2) udvd_incr_lem) |
|
1633 apply (erule uint_add_le [THEN order_trans]) |
|
1634 done |
|
1635 |
|
1636 lemma udvd_decr': |
|
1637 "p < q ==> uint p = ua + n * uint K ==> |
|
1638 uint q = ua + n' * uint K ==> p <= q - K" |
|
1639 apply (unfold word_less_alt word_le_def) |
|
1640 apply (drule (2) udvd_incr_lem) |
|
1641 apply (drule le_diff_eq [THEN iffD2]) |
|
1642 apply (erule order_trans) |
|
1643 apply (rule uint_sub_ge) |
|
1644 done |
|
1645 |
|
1646 lemmas udvd_incr_lem0 = udvd_incr_lem [where ua=0, simplified] |
|
1647 lemmas udvd_incr0 = udvd_incr' [where ua=0, simplified] |
|
1648 lemmas udvd_decr0 = udvd_decr' [where ua=0, simplified] |
|
1649 |
|
1650 lemma udvd_minus_le': |
|
1651 "xy < k ==> z udvd xy ==> z udvd k ==> xy <= k - z" |
|
1652 apply (unfold udvd_def) |
|
1653 apply clarify |
|
1654 apply (erule (2) udvd_decr0) |
|
1655 done |
|
1656 |
|
1657 ML {* Delsimprocs Numeral_Simprocs.cancel_factors *} |
|
1658 |
|
1659 lemma udvd_incr2_K: |
|
1660 "p < a + s ==> a <= a + s ==> K udvd s ==> K udvd p - a ==> a <= p ==> |
|
1661 0 < K ==> p <= p + K & p + K <= a + s" |
|
1662 apply (unfold udvd_def) |
|
1663 apply clarify |
|
1664 apply (simp add: uint_arith_simps split: split_if_asm) |
|
1665 prefer 2 |
|
1666 apply (insert uint_range' [of s])[1] |
|
1667 apply arith |
|
1668 apply (drule add_commute [THEN xtr1]) |
|
1669 apply (simp add: diff_less_eq [symmetric]) |
|
1670 apply (drule less_le_mult) |
|
1671 apply arith |
|
1672 apply simp |
|
1673 done |
|
1674 |
|
1675 ML {* Addsimprocs Numeral_Simprocs.cancel_factors *} |
|
1676 |
|
1677 (* links with rbl operations *) |
|
1678 lemma word_succ_rbl: |
|
1679 "to_bl w = bl ==> to_bl (word_succ w) = (rev (rbl_succ (rev bl)))" |
|
1680 apply (unfold word_succ_def) |
|
1681 apply clarify |
|
1682 apply (simp add: to_bl_of_bin) |
|
1683 apply (simp add: to_bl_def rbl_succ) |
|
1684 done |
|
1685 |
|
1686 lemma word_pred_rbl: |
|
1687 "to_bl w = bl ==> to_bl (word_pred w) = (rev (rbl_pred (rev bl)))" |
|
1688 apply (unfold word_pred_def) |
|
1689 apply clarify |
|
1690 apply (simp add: to_bl_of_bin) |
|
1691 apply (simp add: to_bl_def rbl_pred) |
|
1692 done |
|
1693 |
|
1694 lemma word_add_rbl: |
|
1695 "to_bl v = vbl ==> to_bl w = wbl ==> |
|
1696 to_bl (v + w) = (rev (rbl_add (rev vbl) (rev wbl)))" |
|
1697 apply (unfold word_add_def) |
|
1698 apply clarify |
|
1699 apply (simp add: to_bl_of_bin) |
|
1700 apply (simp add: to_bl_def rbl_add) |
|
1701 done |
|
1702 |
|
1703 lemma word_mult_rbl: |
|
1704 "to_bl v = vbl ==> to_bl w = wbl ==> |
|
1705 to_bl (v * w) = (rev (rbl_mult (rev vbl) (rev wbl)))" |
|
1706 apply (unfold word_mult_def) |
|
1707 apply clarify |
|
1708 apply (simp add: to_bl_of_bin) |
|
1709 apply (simp add: to_bl_def rbl_mult) |
|
1710 done |
|
1711 |
|
1712 lemma rtb_rbl_ariths: |
|
1713 "rev (to_bl w) = ys \<Longrightarrow> rev (to_bl (word_succ w)) = rbl_succ ys" |
|
1714 |
|
1715 "rev (to_bl w) = ys \<Longrightarrow> rev (to_bl (word_pred w)) = rbl_pred ys" |
|
1716 |
|
1717 "[| rev (to_bl v) = ys; rev (to_bl w) = xs |] |
|
1718 ==> rev (to_bl (v * w)) = rbl_mult ys xs" |
|
1719 |
|
1720 "[| rev (to_bl v) = ys; rev (to_bl w) = xs |] |
|
1721 ==> rev (to_bl (v + w)) = rbl_add ys xs" |
|
1722 by (auto simp: rev_swap [symmetric] word_succ_rbl |
|
1723 word_pred_rbl word_mult_rbl word_add_rbl) |
|
1724 |
|
1725 |
|
1726 subsection "Arithmetic type class instantiations" |
|
1727 |
|
1728 instance word :: (len0) comm_monoid_add .. |
|
1729 |
|
1730 instance word :: (len0) comm_monoid_mult |
|
1731 apply (intro_classes) |
|
1732 apply (simp add: word_mult_commute) |
|
1733 apply (simp add: word_mult_1) |
|
1734 done |
|
1735 |
|
1736 instance word :: (len0) comm_semiring |
|
1737 by (intro_classes) (simp add : word_left_distrib) |
|
1738 |
|
1739 instance word :: (len0) ab_group_add .. |
|
1740 |
|
1741 instance word :: (len0) comm_ring .. |
|
1742 |
|
1743 instance word :: (len) comm_semiring_1 |
|
1744 by (intro_classes) (simp add: lenw1_zero_neq_one) |
|
1745 |
|
1746 instance word :: (len) comm_ring_1 .. |
|
1747 |
|
1748 instance word :: (len0) comm_semiring_0 .. |
|
1749 |
|
1750 instance word :: (len0) order .. |
|
1751 |
|
1752 (* note that iszero_def is only for class comm_semiring_1_cancel, |
|
1753 which requires word length >= 1, ie 'a :: len word *) |
|
1754 lemma zero_bintrunc: |
|
1755 "iszero (number_of x :: 'a :: len word) = |
|
1756 (bintrunc (len_of TYPE('a)) x = Int.Pls)" |
|
1757 apply (unfold iszero_def word_0_wi word_no_wi) |
|
1758 apply (rule word_ubin.norm_eq_iff [symmetric, THEN trans]) |
|
1759 apply (simp add : Pls_def [symmetric]) |
|
1760 done |
|
1761 |
|
1762 lemmas word_le_0_iff [simp] = |
|
1763 word_zero_le [THEN leD, THEN linorder_antisym_conv1] |
|
1764 |
|
1765 lemma word_of_nat: "of_nat n = word_of_int (int n)" |
|
1766 by (induct n) (auto simp add : word_of_int_hom_syms) |
|
1767 |
|
1768 lemma word_of_int: "of_int = word_of_int" |
|
1769 apply (rule ext) |
|
1770 apply (unfold of_int_def) |
|
1771 apply (rule contentsI) |
|
1772 apply safe |
|
1773 apply (simp_all add: word_of_nat word_of_int_homs) |
|
1774 defer |
|
1775 apply (rule Rep_Integ_ne [THEN nonemptyE]) |
|
1776 apply (rule bexI) |
|
1777 prefer 2 |
|
1778 apply assumption |
|
1779 apply (auto simp add: RI_eq_diff) |
|
1780 done |
|
1781 |
|
1782 lemma word_of_int_nat: |
|
1783 "0 <= x ==> word_of_int x = of_nat (nat x)" |
|
1784 by (simp add: of_nat_nat word_of_int) |
|
1785 |
|
1786 lemma word_number_of_eq: |
|
1787 "number_of w = (of_int w :: 'a :: len word)" |
|
1788 unfolding word_number_of_def word_of_int by auto |
|
1789 |
|
1790 instance word :: (len) number_ring |
|
1791 by (intro_classes) (simp add : word_number_of_eq) |
|
1792 |
|
1793 lemma iszero_word_no [simp] : |
|
1794 "iszero (number_of bin :: 'a :: len word) = |
|
1795 iszero (number_of (bintrunc (len_of TYPE('a)) bin) :: int)" |
|
1796 apply (simp add: zero_bintrunc number_of_is_id) |
|
1797 apply (unfold iszero_def Pls_def) |
|
1798 apply (rule refl) |
|
1799 done |
|
1800 |
|
1801 |
|
1802 subsection "Word and nat" |
|
1803 |
|
1804 lemma td_ext_unat': |
|
1805 "n = len_of TYPE ('a :: len) ==> |
|
1806 td_ext (unat :: 'a word => nat) of_nat |
|
1807 (unats n) (%i. i mod 2 ^ n)" |
|
1808 apply (unfold td_ext_def' unat_def word_of_nat unats_uints) |
|
1809 apply (auto intro!: imageI simp add : word_of_int_hom_syms) |
|
1810 apply (erule word_uint.Abs_inverse [THEN arg_cong]) |
|
1811 apply (simp add: int_word_uint nat_mod_distrib nat_power_eq) |
|
1812 done |
|
1813 |
|
1814 lemmas td_ext_unat = refl [THEN td_ext_unat'] |
|
1815 lemmas unat_of_nat = td_ext_unat [THEN td_ext.eq_norm, standard] |
|
1816 |
|
1817 interpretation word_unat: |
|
1818 td_ext "unat::'a::len word => nat" |
|
1819 of_nat |
|
1820 "unats (len_of TYPE('a::len))" |
|
1821 "%i. i mod 2 ^ len_of TYPE('a::len)" |
|
1822 by (rule td_ext_unat) |
|
1823 |
|
1824 lemmas td_unat = word_unat.td_thm |
|
1825 |
|
1826 lemmas unat_lt2p [iff] = word_unat.Rep [unfolded unats_def mem_Collect_eq] |
|
1827 |
|
1828 lemma unat_le: "y <= unat (z :: 'a :: len word) ==> y : unats (len_of TYPE ('a))" |
|
1829 apply (unfold unats_def) |
|
1830 apply clarsimp |
|
1831 apply (rule xtrans, rule unat_lt2p, assumption) |
|
1832 done |
|
1833 |
|
1834 lemma word_nchotomy: |
|
1835 "ALL w. EX n. (w :: 'a :: len word) = of_nat n & n < 2 ^ len_of TYPE ('a)" |
|
1836 apply (rule allI) |
|
1837 apply (rule word_unat.Abs_cases) |
|
1838 apply (unfold unats_def) |
|
1839 apply auto |
|
1840 done |
|
1841 |
|
1842 lemma of_nat_eq: |
|
1843 fixes w :: "'a::len word" |
|
1844 shows "(of_nat n = w) = (\<exists>q. n = unat w + q * 2 ^ len_of TYPE('a))" |
|
1845 apply (rule trans) |
|
1846 apply (rule word_unat.inverse_norm) |
|
1847 apply (rule iffI) |
|
1848 apply (rule mod_eqD) |
|
1849 apply simp |
|
1850 apply clarsimp |
|
1851 done |
|
1852 |
|
1853 lemma of_nat_eq_size: |
|
1854 "(of_nat n = w) = (EX q. n = unat w + q * 2 ^ size w)" |
|
1855 unfolding word_size by (rule of_nat_eq) |
|
1856 |
|
1857 lemma of_nat_0: |
|
1858 "(of_nat m = (0::'a::len word)) = (\<exists>q. m = q * 2 ^ len_of TYPE('a))" |
|
1859 by (simp add: of_nat_eq) |
|
1860 |
|
1861 lemmas of_nat_2p = mult_1 [symmetric, THEN iffD2 [OF of_nat_0 exI]] |
|
1862 |
|
1863 lemma of_nat_gt_0: "of_nat k ~= 0 ==> 0 < k" |
|
1864 by (cases k) auto |
|
1865 |
|
1866 lemma of_nat_neq_0: |
|
1867 "0 < k ==> k < 2 ^ len_of TYPE ('a :: len) ==> of_nat k ~= (0 :: 'a word)" |
|
1868 by (clarsimp simp add : of_nat_0) |
|
1869 |
|
1870 lemma Abs_fnat_hom_add: |
|
1871 "of_nat a + of_nat b = of_nat (a + b)" |
|
1872 by simp |
|
1873 |
|
1874 lemma Abs_fnat_hom_mult: |
|
1875 "of_nat a * of_nat b = (of_nat (a * b) :: 'a :: len word)" |
|
1876 by (simp add: word_of_nat word_of_int_mult_hom zmult_int) |
|
1877 |
|
1878 lemma Abs_fnat_hom_Suc: |
|
1879 "word_succ (of_nat a) = of_nat (Suc a)" |
|
1880 by (simp add: word_of_nat word_of_int_succ_hom add_ac) |
|
1881 |
|
1882 lemma Abs_fnat_hom_0: "(0::'a::len word) = of_nat 0" |
|
1883 by (simp add: word_of_nat word_0_wi) |
|
1884 |
|
1885 lemma Abs_fnat_hom_1: "(1::'a::len word) = of_nat (Suc 0)" |
|
1886 by (simp add: word_of_nat word_1_wi) |
|
1887 |
|
1888 lemmas Abs_fnat_homs = |
|
1889 Abs_fnat_hom_add Abs_fnat_hom_mult Abs_fnat_hom_Suc |
|
1890 Abs_fnat_hom_0 Abs_fnat_hom_1 |
|
1891 |
|
1892 lemma word_arith_nat_add: |
|
1893 "a + b = of_nat (unat a + unat b)" |
|
1894 by simp |
|
1895 |
|
1896 lemma word_arith_nat_mult: |
|
1897 "a * b = of_nat (unat a * unat b)" |
|
1898 by (simp add: Abs_fnat_hom_mult [symmetric]) |
|
1899 |
|
1900 lemma word_arith_nat_Suc: |
|
1901 "word_succ a = of_nat (Suc (unat a))" |
|
1902 by (subst Abs_fnat_hom_Suc [symmetric]) simp |
|
1903 |
|
1904 lemma word_arith_nat_div: |
|
1905 "a div b = of_nat (unat a div unat b)" |
|
1906 by (simp add: word_div_def word_of_nat zdiv_int uint_nat) |
|
1907 |
|
1908 lemma word_arith_nat_mod: |
|
1909 "a mod b = of_nat (unat a mod unat b)" |
|
1910 by (simp add: word_mod_def word_of_nat zmod_int uint_nat) |
|
1911 |
|
1912 lemmas word_arith_nat_defs = |
|
1913 word_arith_nat_add word_arith_nat_mult |
|
1914 word_arith_nat_Suc Abs_fnat_hom_0 |
|
1915 Abs_fnat_hom_1 word_arith_nat_div |
|
1916 word_arith_nat_mod |
|
1917 |
|
1918 lemmas unat_cong = arg_cong [where f = "unat"] |
|
1919 |
|
1920 lemmas unat_word_ariths = word_arith_nat_defs |
|
1921 [THEN trans [OF unat_cong unat_of_nat], standard] |
|
1922 |
|
1923 lemmas word_sub_less_iff = word_sub_le_iff |
|
1924 [simplified linorder_not_less [symmetric], simplified] |
|
1925 |
|
1926 lemma unat_add_lem: |
|
1927 "(unat x + unat y < 2 ^ len_of TYPE('a)) = |
|
1928 (unat (x + y :: 'a :: len word) = unat x + unat y)" |
|
1929 unfolding unat_word_ariths |
|
1930 by (auto intro!: trans [OF _ nat_mod_lem]) |
|
1931 |
|
1932 lemma unat_mult_lem: |
|
1933 "(unat x * unat y < 2 ^ len_of TYPE('a)) = |
|
1934 (unat (x * y :: 'a :: len word) = unat x * unat y)" |
|
1935 unfolding unat_word_ariths |
|
1936 by (auto intro!: trans [OF _ nat_mod_lem]) |
|
1937 |
|
1938 lemmas unat_plus_if' = |
|
1939 trans [OF unat_word_ariths(1) mod_nat_add, simplified, standard] |
|
1940 |
|
1941 lemma le_no_overflow: |
|
1942 "x <= b ==> a <= a + b ==> x <= a + (b :: 'a :: len0 word)" |
|
1943 apply (erule order_trans) |
|
1944 apply (erule olen_add_eqv [THEN iffD1]) |
|
1945 done |
|
1946 |
|
1947 lemmas un_ui_le = trans |
|
1948 [OF word_le_nat_alt [symmetric] |
|
1949 word_le_def, |
|
1950 standard] |
|
1951 |
|
1952 lemma unat_sub_if_size: |
|
1953 "unat (x - y) = (if unat y <= unat x |
|
1954 then unat x - unat y |
|
1955 else unat x + 2 ^ size x - unat y)" |
|
1956 apply (unfold word_size) |
|
1957 apply (simp add: un_ui_le) |
|
1958 apply (auto simp add: unat_def uint_sub_if') |
|
1959 apply (rule nat_diff_distrib) |
|
1960 prefer 3 |
|
1961 apply (simp add: algebra_simps) |
|
1962 apply (rule nat_diff_distrib [THEN trans]) |
|
1963 prefer 3 |
|
1964 apply (subst nat_add_distrib) |
|
1965 prefer 3 |
|
1966 apply (simp add: nat_power_eq) |
|
1967 apply auto |
|
1968 apply uint_arith |
|
1969 done |
|
1970 |
|
1971 lemmas unat_sub_if' = unat_sub_if_size [unfolded word_size] |
|
1972 |
|
1973 lemma unat_div: "unat ((x :: 'a :: len word) div y) = unat x div unat y" |
|
1974 apply (simp add : unat_word_ariths) |
|
1975 apply (rule unat_lt2p [THEN xtr7, THEN nat_mod_eq']) |
|
1976 apply (rule div_le_dividend) |
|
1977 done |
|
1978 |
|
1979 lemma unat_mod: "unat ((x :: 'a :: len word) mod y) = unat x mod unat y" |
|
1980 apply (clarsimp simp add : unat_word_ariths) |
|
1981 apply (cases "unat y") |
|
1982 prefer 2 |
|
1983 apply (rule unat_lt2p [THEN xtr7, THEN nat_mod_eq']) |
|
1984 apply (rule mod_le_divisor) |
|
1985 apply auto |
|
1986 done |
|
1987 |
|
1988 lemma uint_div: "uint ((x :: 'a :: len word) div y) = uint x div uint y" |
|
1989 unfolding uint_nat by (simp add : unat_div zdiv_int) |
|
1990 |
|
1991 lemma uint_mod: "uint ((x :: 'a :: len word) mod y) = uint x mod uint y" |
|
1992 unfolding uint_nat by (simp add : unat_mod zmod_int) |
|
1993 |
|
1994 |
|
1995 subsection {* Definition of unat\_arith tactic *} |
|
1996 |
|
1997 lemma unat_split: |
|
1998 fixes x::"'a::len word" |
|
1999 shows "P (unat x) = |
|
2000 (ALL n. of_nat n = x & n < 2^len_of TYPE('a) --> P n)" |
|
2001 by (auto simp: unat_of_nat) |
|
2002 |
|
2003 lemma unat_split_asm: |
|
2004 fixes x::"'a::len word" |
|
2005 shows "P (unat x) = |
|
2006 (~(EX n. of_nat n = x & n < 2^len_of TYPE('a) & ~ P n))" |
|
2007 by (auto simp: unat_of_nat) |
|
2008 |
|
2009 lemmas of_nat_inverse = |
|
2010 word_unat.Abs_inverse' [rotated, unfolded unats_def, simplified] |
|
2011 |
|
2012 lemmas unat_splits = unat_split unat_split_asm |
|
2013 |
|
2014 lemmas unat_arith_simps = |
|
2015 word_le_nat_alt word_less_nat_alt |
|
2016 word_unat.Rep_inject [symmetric] |
|
2017 unat_sub_if' unat_plus_if' unat_div unat_mod |
|
2018 |
|
2019 (* unat_arith_tac: tactic to reduce word arithmetic to nat, |
|
2020 try to solve via arith *) |
|
2021 ML {* |
|
2022 fun unat_arith_ss_of ss = |
|
2023 ss addsimps @{thms unat_arith_simps} |
|
2024 delsimps @{thms word_unat.Rep_inject} |
|
2025 addsplits @{thms split_if_asm} |
|
2026 addcongs @{thms power_False_cong} |
|
2027 |
|
2028 fun unat_arith_tacs ctxt = |
|
2029 let |
|
2030 fun arith_tac' n t = |
|
2031 Arith_Data.verbose_arith_tac ctxt n t |
|
2032 handle Cooper.COOPER _ => Seq.empty; |
|
2033 val cs = claset_of ctxt; |
|
2034 val ss = simpset_of ctxt; |
|
2035 in |
|
2036 [ clarify_tac cs 1, |
|
2037 full_simp_tac (unat_arith_ss_of ss) 1, |
|
2038 ALLGOALS (full_simp_tac (HOL_ss addsplits @{thms unat_splits} |
|
2039 addcongs @{thms power_False_cong})), |
|
2040 rewrite_goals_tac @{thms word_size}, |
|
2041 ALLGOALS (fn n => REPEAT (resolve_tac [allI, impI] n) THEN |
|
2042 REPEAT (etac conjE n) THEN |
|
2043 REPEAT (dtac @{thm of_nat_inverse} n THEN atac n)), |
|
2044 TRYALL arith_tac' ] |
|
2045 end |
|
2046 |
|
2047 fun unat_arith_tac ctxt = SELECT_GOAL (EVERY (unat_arith_tacs ctxt)) |
|
2048 *} |
|
2049 |
|
2050 method_setup unat_arith = |
|
2051 {* Scan.succeed (SIMPLE_METHOD' o unat_arith_tac) *} |
|
2052 "solving word arithmetic via natural numbers and arith" |
|
2053 |
|
2054 lemma no_plus_overflow_unat_size: |
|
2055 "((x :: 'a :: len word) <= x + y) = (unat x + unat y < 2 ^ size x)" |
|
2056 unfolding word_size by unat_arith |
|
2057 |
|
2058 lemmas no_olen_add_nat = no_plus_overflow_unat_size [unfolded word_size] |
|
2059 |
|
2060 lemmas unat_plus_simple = trans [OF no_olen_add_nat unat_add_lem, standard] |
|
2061 |
|
2062 lemma word_div_mult: |
|
2063 "(0 :: 'a :: len word) < y ==> unat x * unat y < 2 ^ len_of TYPE('a) ==> |
|
2064 x * y div y = x" |
|
2065 apply unat_arith |
|
2066 apply clarsimp |
|
2067 apply (subst unat_mult_lem [THEN iffD1]) |
|
2068 apply auto |
|
2069 done |
|
2070 |
|
2071 lemma div_lt': "(i :: 'a :: len word) <= k div x ==> |
|
2072 unat i * unat x < 2 ^ len_of TYPE('a)" |
|
2073 apply unat_arith |
|
2074 apply clarsimp |
|
2075 apply (drule mult_le_mono1) |
|
2076 apply (erule order_le_less_trans) |
|
2077 apply (rule xtr7 [OF unat_lt2p div_mult_le]) |
|
2078 done |
|
2079 |
|
2080 lemmas div_lt'' = order_less_imp_le [THEN div_lt'] |
|
2081 |
|
2082 lemma div_lt_mult: "(i :: 'a :: len word) < k div x ==> 0 < x ==> i * x < k" |
|
2083 apply (frule div_lt'' [THEN unat_mult_lem [THEN iffD1]]) |
|
2084 apply (simp add: unat_arith_simps) |
|
2085 apply (drule (1) mult_less_mono1) |
|
2086 apply (erule order_less_le_trans) |
|
2087 apply (rule div_mult_le) |
|
2088 done |
|
2089 |
|
2090 lemma div_le_mult: |
|
2091 "(i :: 'a :: len word) <= k div x ==> 0 < x ==> i * x <= k" |
|
2092 apply (frule div_lt' [THEN unat_mult_lem [THEN iffD1]]) |
|
2093 apply (simp add: unat_arith_simps) |
|
2094 apply (drule mult_le_mono1) |
|
2095 apply (erule order_trans) |
|
2096 apply (rule div_mult_le) |
|
2097 done |
|
2098 |
|
2099 lemma div_lt_uint': |
|
2100 "(i :: 'a :: len word) <= k div x ==> uint i * uint x < 2 ^ len_of TYPE('a)" |
|
2101 apply (unfold uint_nat) |
|
2102 apply (drule div_lt') |
|
2103 apply (simp add: zmult_int zless_nat_eq_int_zless [symmetric] |
|
2104 nat_power_eq) |
|
2105 done |
|
2106 |
|
2107 lemmas div_lt_uint'' = order_less_imp_le [THEN div_lt_uint'] |
|
2108 |
|
2109 lemma word_le_exists': |
|
2110 "(x :: 'a :: len0 word) <= y ==> |
|
2111 (EX z. y = x + z & uint x + uint z < 2 ^ len_of TYPE('a))" |
|
2112 apply (rule exI) |
|
2113 apply (rule conjI) |
|
2114 apply (rule zadd_diff_inverse) |
|
2115 apply uint_arith |
|
2116 done |
|
2117 |
|
2118 lemmas plus_minus_not_NULL = order_less_imp_le [THEN plus_minus_not_NULL_ab] |
|
2119 |
|
2120 lemmas plus_minus_no_overflow = |
|
2121 order_less_imp_le [THEN plus_minus_no_overflow_ab] |
|
2122 |
|
2123 lemmas mcs = word_less_minus_cancel word_less_minus_mono_left |
|
2124 word_le_minus_cancel word_le_minus_mono_left |
|
2125 |
|
2126 lemmas word_l_diffs = mcs [where y = "w + x", unfolded add_diff_cancel, standard] |
|
2127 lemmas word_diff_ls = mcs [where z = "w + x", unfolded add_diff_cancel, standard] |
|
2128 lemmas word_plus_mcs = word_diff_ls |
|
2129 [where y = "v + x", unfolded add_diff_cancel, standard] |
|
2130 |
|
2131 lemmas le_unat_uoi = unat_le [THEN word_unat.Abs_inverse] |
|
2132 |
|
2133 lemmas thd = refl [THEN [2] split_div_lemma [THEN iffD2], THEN conjunct1] |
|
2134 |
|
2135 lemma thd1: |
|
2136 "a div b * b \<le> (a::nat)" |
|
2137 using gt_or_eq_0 [of b] |
|
2138 apply (rule disjE) |
|
2139 apply (erule xtr4 [OF thd mult_commute]) |
|
2140 apply clarsimp |
|
2141 done |
|
2142 |
|
2143 lemmas uno_simps [THEN le_unat_uoi, standard] = |
|
2144 mod_le_divisor div_le_dividend thd1 |
|
2145 |
|
2146 lemma word_mod_div_equality: |
|
2147 "(n div b) * b + (n mod b) = (n :: 'a :: len word)" |
|
2148 apply (unfold word_less_nat_alt word_arith_nat_defs) |
|
2149 apply (cut_tac y="unat b" in gt_or_eq_0) |
|
2150 apply (erule disjE) |
|
2151 apply (simp add: mod_div_equality uno_simps) |
|
2152 apply simp |
|
2153 done |
|
2154 |
|
2155 lemma word_div_mult_le: "a div b * b <= (a::'a::len word)" |
|
2156 apply (unfold word_le_nat_alt word_arith_nat_defs) |
|
2157 apply (cut_tac y="unat b" in gt_or_eq_0) |
|
2158 apply (erule disjE) |
|
2159 apply (simp add: div_mult_le uno_simps) |
|
2160 apply simp |
|
2161 done |
|
2162 |
|
2163 lemma word_mod_less_divisor: "0 < n ==> m mod n < (n :: 'a :: len word)" |
|
2164 apply (simp only: word_less_nat_alt word_arith_nat_defs) |
|
2165 apply (clarsimp simp add : uno_simps) |
|
2166 done |
|
2167 |
|
2168 lemma word_of_int_power_hom: |
|
2169 "word_of_int a ^ n = (word_of_int (a ^ n) :: 'a :: len word)" |
|
2170 by (induct n) (simp_all add : word_of_int_hom_syms power_Suc) |
|
2171 |
|
2172 lemma word_arith_power_alt: |
|
2173 "a ^ n = (word_of_int (uint a ^ n) :: 'a :: len word)" |
|
2174 by (simp add : word_of_int_power_hom [symmetric]) |
|
2175 |
|
2176 lemma of_bl_length_less: |
|
2177 "length x = k ==> k < len_of TYPE('a) ==> (of_bl x :: 'a :: len word) < 2 ^ k" |
|
2178 apply (unfold of_bl_no [unfolded word_number_of_def] |
|
2179 word_less_alt word_number_of_alt) |
|
2180 apply safe |
|
2181 apply (simp (no_asm) add: word_of_int_power_hom word_uint.eq_norm |
|
2182 del: word_of_int_bin) |
|
2183 apply (simp add: mod_pos_pos_trivial) |
|
2184 apply (subst mod_pos_pos_trivial) |
|
2185 apply (rule bl_to_bin_ge0) |
|
2186 apply (rule order_less_trans) |
|
2187 apply (rule bl_to_bin_lt2p) |
|
2188 apply simp |
|
2189 apply (rule bl_to_bin_lt2p) |
|
2190 done |
|
2191 |
|
2192 |
|
2193 subsection "Cardinality, finiteness of set of words" |
|
2194 |
|
2195 lemmas card_lessThan' = card_lessThan [unfolded lessThan_def] |
|
2196 |
|
2197 lemmas card_eq = word_unat.Abs_inj_on [THEN card_image, |
|
2198 unfolded word_unat.image, unfolded unats_def, standard] |
|
2199 |
|
2200 lemmas card_word = trans [OF card_eq card_lessThan', standard] |
|
2201 |
|
2202 lemma finite_word_UNIV: "finite (UNIV :: 'a :: len word set)" |
|
2203 apply (rule contrapos_np) |
|
2204 prefer 2 |
|
2205 apply (erule card_infinite) |
|
2206 apply (simp add: card_word) |
|
2207 done |
|
2208 |
|
2209 lemma card_word_size: |
|
2210 "card (UNIV :: 'a :: len word set) = (2 ^ size (x :: 'a word))" |
|
2211 unfolding word_size by (rule card_word) |
|
2212 |
|
2213 |
|
2214 subsection {* Bitwise Operations on Words *} |
|
2215 |
|
2216 lemmas bin_log_bintrs = bin_trunc_not bin_trunc_xor bin_trunc_and bin_trunc_or |
|
2217 |
|
2218 (* following definitions require both arithmetic and bit-wise word operations *) |
|
2219 |
|
2220 (* to get word_no_log_defs from word_log_defs, using bin_log_bintrs *) |
|
2221 lemmas wils1 = bin_log_bintrs [THEN word_ubin.norm_eq_iff [THEN iffD1], |
|
2222 folded word_ubin.eq_norm, THEN eq_reflection, standard] |
|
2223 |
|
2224 (* the binary operations only *) |
|
2225 lemmas word_log_binary_defs = |
|
2226 word_and_def word_or_def word_xor_def |
|
2227 |
|
2228 lemmas word_no_log_defs [simp] = |
|
2229 word_not_def [where a="number_of a", |
|
2230 unfolded word_no_wi wils1, folded word_no_wi, standard] |
|
2231 word_log_binary_defs [where a="number_of a" and b="number_of b", |
|
2232 unfolded word_no_wi wils1, folded word_no_wi, standard] |
|
2233 |
|
2234 lemmas word_wi_log_defs = word_no_log_defs [unfolded word_no_wi] |
|
2235 |
|
2236 lemma uint_or: "uint (x OR y) = (uint x) OR (uint y)" |
|
2237 by (simp add: word_or_def word_no_wi [symmetric] number_of_is_id |
|
2238 bin_trunc_ao(2) [symmetric]) |
|
2239 |
|
2240 lemma uint_and: "uint (x AND y) = (uint x) AND (uint y)" |
|
2241 by (simp add: word_and_def number_of_is_id word_no_wi [symmetric] |
|
2242 bin_trunc_ao(1) [symmetric]) |
|
2243 |
|
2244 lemma word_ops_nth_size: |
|
2245 "n < size (x::'a::len0 word) ==> |
|
2246 (x OR y) !! n = (x !! n | y !! n) & |
|
2247 (x AND y) !! n = (x !! n & y !! n) & |
|
2248 (x XOR y) !! n = (x !! n ~= y !! n) & |
|
2249 (NOT x) !! n = (~ x !! n)" |
|
2250 unfolding word_size word_no_wi word_test_bit_def word_log_defs |
|
2251 by (clarsimp simp add : word_ubin.eq_norm nth_bintr bin_nth_ops) |
|
2252 |
|
2253 lemma word_ao_nth: |
|
2254 fixes x :: "'a::len0 word" |
|
2255 shows "(x OR y) !! n = (x !! n | y !! n) & |
|
2256 (x AND y) !! n = (x !! n & y !! n)" |
|
2257 apply (cases "n < size x") |
|
2258 apply (drule_tac y = "y" in word_ops_nth_size) |
|
2259 apply simp |
|
2260 apply (simp add : test_bit_bin word_size) |
|
2261 done |
|
2262 |
|
2263 (* get from commutativity, associativity etc of int_and etc |
|
2264 to same for word_and etc *) |
|
2265 |
|
2266 lemmas bwsimps = |
|
2267 word_of_int_homs(2) |
|
2268 word_0_wi_Pls |
|
2269 word_m1_wi_Min |
|
2270 word_wi_log_defs |
|
2271 |
|
2272 lemma word_bw_assocs: |
|
2273 fixes x :: "'a::len0 word" |
|
2274 shows |
|
2275 "(x AND y) AND z = x AND y AND z" |
|
2276 "(x OR y) OR z = x OR y OR z" |
|
2277 "(x XOR y) XOR z = x XOR y XOR z" |
|
2278 using word_of_int_Ex [where x=x] |
|
2279 word_of_int_Ex [where x=y] |
|
2280 word_of_int_Ex [where x=z] |
|
2281 by (auto simp: bwsimps bbw_assocs) |
|
2282 |
|
2283 lemma word_bw_comms: |
|
2284 fixes x :: "'a::len0 word" |
|
2285 shows |
|
2286 "x AND y = y AND x" |
|
2287 "x OR y = y OR x" |
|
2288 "x XOR y = y XOR x" |
|
2289 using word_of_int_Ex [where x=x] |
|
2290 word_of_int_Ex [where x=y] |
|
2291 by (auto simp: bwsimps bin_ops_comm) |
|
2292 |
|
2293 lemma word_bw_lcs: |
|
2294 fixes x :: "'a::len0 word" |
|
2295 shows |
|
2296 "y AND x AND z = x AND y AND z" |
|
2297 "y OR x OR z = x OR y OR z" |
|
2298 "y XOR x XOR z = x XOR y XOR z" |
|
2299 using word_of_int_Ex [where x=x] |
|
2300 word_of_int_Ex [where x=y] |
|
2301 word_of_int_Ex [where x=z] |
|
2302 by (auto simp: bwsimps) |
|
2303 |
|
2304 lemma word_log_esimps [simp]: |
|
2305 fixes x :: "'a::len0 word" |
|
2306 shows |
|
2307 "x AND 0 = 0" |
|
2308 "x AND -1 = x" |
|
2309 "x OR 0 = x" |
|
2310 "x OR -1 = -1" |
|
2311 "x XOR 0 = x" |
|
2312 "x XOR -1 = NOT x" |
|
2313 "0 AND x = 0" |
|
2314 "-1 AND x = x" |
|
2315 "0 OR x = x" |
|
2316 "-1 OR x = -1" |
|
2317 "0 XOR x = x" |
|
2318 "-1 XOR x = NOT x" |
|
2319 using word_of_int_Ex [where x=x] |
|
2320 by (auto simp: bwsimps) |
|
2321 |
|
2322 lemma word_not_dist: |
|
2323 fixes x :: "'a::len0 word" |
|
2324 shows |
|
2325 "NOT (x OR y) = NOT x AND NOT y" |
|
2326 "NOT (x AND y) = NOT x OR NOT y" |
|
2327 using word_of_int_Ex [where x=x] |
|
2328 word_of_int_Ex [where x=y] |
|
2329 by (auto simp: bwsimps bbw_not_dist) |
|
2330 |
|
2331 lemma word_bw_same: |
|
2332 fixes x :: "'a::len0 word" |
|
2333 shows |
|
2334 "x AND x = x" |
|
2335 "x OR x = x" |
|
2336 "x XOR x = 0" |
|
2337 using word_of_int_Ex [where x=x] |
|
2338 by (auto simp: bwsimps) |
|
2339 |
|
2340 lemma word_ao_absorbs [simp]: |
|
2341 fixes x :: "'a::len0 word" |
|
2342 shows |
|
2343 "x AND (y OR x) = x" |
|
2344 "x OR y AND x = x" |
|
2345 "x AND (x OR y) = x" |
|
2346 "y AND x OR x = x" |
|
2347 "(y OR x) AND x = x" |
|
2348 "x OR x AND y = x" |
|
2349 "(x OR y) AND x = x" |
|
2350 "x AND y OR x = x" |
|
2351 using word_of_int_Ex [where x=x] |
|
2352 word_of_int_Ex [where x=y] |
|
2353 by (auto simp: bwsimps) |
|
2354 |
|
2355 lemma word_not_not [simp]: |
|
2356 "NOT NOT (x::'a::len0 word) = x" |
|
2357 using word_of_int_Ex [where x=x] |
|
2358 by (auto simp: bwsimps) |
|
2359 |
|
2360 lemma word_ao_dist: |
|
2361 fixes x :: "'a::len0 word" |
|
2362 shows "(x OR y) AND z = x AND z OR y AND z" |
|
2363 using word_of_int_Ex [where x=x] |
|
2364 word_of_int_Ex [where x=y] |
|
2365 word_of_int_Ex [where x=z] |
|
2366 by (auto simp: bwsimps bbw_ao_dist simp del: bin_ops_comm) |
|
2367 |
|
2368 lemma word_oa_dist: |
|
2369 fixes x :: "'a::len0 word" |
|
2370 shows "x AND y OR z = (x OR z) AND (y OR z)" |
|
2371 using word_of_int_Ex [where x=x] |
|
2372 word_of_int_Ex [where x=y] |
|
2373 word_of_int_Ex [where x=z] |
|
2374 by (auto simp: bwsimps bbw_oa_dist simp del: bin_ops_comm) |
|
2375 |
|
2376 lemma word_add_not [simp]: |
|
2377 fixes x :: "'a::len0 word" |
|
2378 shows "x + NOT x = -1" |
|
2379 using word_of_int_Ex [where x=x] |
|
2380 by (auto simp: bwsimps bin_add_not) |
|
2381 |
|
2382 lemma word_plus_and_or [simp]: |
|
2383 fixes x :: "'a::len0 word" |
|
2384 shows "(x AND y) + (x OR y) = x + y" |
|
2385 using word_of_int_Ex [where x=x] |
|
2386 word_of_int_Ex [where x=y] |
|
2387 by (auto simp: bwsimps plus_and_or) |
|
2388 |
|
2389 lemma leoa: |
|
2390 fixes x :: "'a::len0 word" |
|
2391 shows "(w = (x OR y)) ==> (y = (w AND y))" by auto |
|
2392 lemma leao: |
|
2393 fixes x' :: "'a::len0 word" |
|
2394 shows "(w' = (x' AND y')) ==> (x' = (x' OR w'))" by auto |
|
2395 |
|
2396 lemmas word_ao_equiv = leao [COMP leoa [COMP iffI]] |
|
2397 |
|
2398 lemma le_word_or2: "x <= x OR (y::'a::len0 word)" |
|
2399 unfolding word_le_def uint_or |
|
2400 by (auto intro: le_int_or) |
|
2401 |
|
2402 lemmas le_word_or1 = xtr3 [OF word_bw_comms (2) le_word_or2, standard] |
|
2403 lemmas word_and_le1 = |
|
2404 xtr3 [OF word_ao_absorbs (4) [symmetric] le_word_or2, standard] |
|
2405 lemmas word_and_le2 = |
|
2406 xtr3 [OF word_ao_absorbs (8) [symmetric] le_word_or2, standard] |
|
2407 |
|
2408 lemma bl_word_not: "to_bl (NOT w) = map Not (to_bl w)" |
|
2409 unfolding to_bl_def word_log_defs |
|
2410 by (simp add: bl_not_bin number_of_is_id word_no_wi [symmetric] bin_to_bl_def[symmetric]) |
|
2411 |
|
2412 lemma bl_word_xor: "to_bl (v XOR w) = map2 op ~= (to_bl v) (to_bl w)" |
|
2413 unfolding to_bl_def word_log_defs bl_xor_bin |
|
2414 by (simp add: number_of_is_id word_no_wi [symmetric]) |
|
2415 |
|
2416 lemma bl_word_or: "to_bl (v OR w) = map2 op | (to_bl v) (to_bl w)" |
|
2417 unfolding to_bl_def word_log_defs |
|
2418 by (simp add: bl_or_bin number_of_is_id word_no_wi [symmetric]) |
|
2419 |
|
2420 lemma bl_word_and: "to_bl (v AND w) = map2 op & (to_bl v) (to_bl w)" |
|
2421 unfolding to_bl_def word_log_defs |
|
2422 by (simp add: bl_and_bin number_of_is_id word_no_wi [symmetric]) |
|
2423 |
|
2424 lemma word_lsb_alt: "lsb (w::'a::len0 word) = test_bit w 0" |
|
2425 by (auto simp: word_test_bit_def word_lsb_def) |
|
2426 |
|
2427 lemma word_lsb_1_0: "lsb (1::'a::len word) & ~ lsb (0::'b::len0 word)" |
|
2428 unfolding word_lsb_def word_1_no word_0_no by auto |
|
2429 |
|
2430 lemma word_lsb_last: "lsb (w::'a::len word) = last (to_bl w)" |
|
2431 apply (unfold word_lsb_def uint_bl bin_to_bl_def) |
|
2432 apply (rule_tac bin="uint w" in bin_exhaust) |
|
2433 apply (cases "size w") |
|
2434 apply auto |
|
2435 apply (auto simp add: bin_to_bl_aux_alt) |
|
2436 done |
|
2437 |
|
2438 lemma word_lsb_int: "lsb w = (uint w mod 2 = 1)" |
|
2439 unfolding word_lsb_def bin_last_mod by auto |
|
2440 |
|
2441 lemma word_msb_sint: "msb w = (sint w < 0)" |
|
2442 unfolding word_msb_def |
|
2443 by (simp add : sign_Min_lt_0 number_of_is_id) |
|
2444 |
|
2445 lemma word_msb_no': |
|
2446 "w = number_of bin ==> msb (w::'a::len word) = bin_nth bin (size w - 1)" |
|
2447 unfolding word_msb_def word_number_of_def |
|
2448 by (clarsimp simp add: word_sbin.eq_norm word_size bin_sign_lem) |
|
2449 |
|
2450 lemmas word_msb_no = refl [THEN word_msb_no', unfolded word_size] |
|
2451 |
|
2452 lemma word_msb_nth': "msb (w::'a::len word) = bin_nth (uint w) (size w - 1)" |
|
2453 apply (unfold word_size) |
|
2454 apply (rule trans [OF _ word_msb_no]) |
|
2455 apply (simp add : word_number_of_def) |
|
2456 done |
|
2457 |
|
2458 lemmas word_msb_nth = word_msb_nth' [unfolded word_size] |
|
2459 |
|
2460 lemma word_msb_alt: "msb (w::'a::len word) = hd (to_bl w)" |
|
2461 apply (unfold word_msb_nth uint_bl) |
|
2462 apply (subst hd_conv_nth) |
|
2463 apply (rule length_greater_0_conv [THEN iffD1]) |
|
2464 apply simp |
|
2465 apply (simp add : nth_bin_to_bl word_size) |
|
2466 done |
|
2467 |
|
2468 lemma word_set_nth: |
|
2469 "set_bit w n (test_bit w n) = (w::'a::len0 word)" |
|
2470 unfolding word_test_bit_def word_set_bit_def by auto |
|
2471 |
|
2472 lemma bin_nth_uint': |
|
2473 "bin_nth (uint w) n = (rev (bin_to_bl (size w) (uint w)) ! n & n < size w)" |
|
2474 apply (unfold word_size) |
|
2475 apply (safe elim!: bin_nth_uint_imp) |
|
2476 apply (frule bin_nth_uint_imp) |
|
2477 apply (fast dest!: bin_nth_bl)+ |
|
2478 done |
|
2479 |
|
2480 lemmas bin_nth_uint = bin_nth_uint' [unfolded word_size] |
|
2481 |
|
2482 lemma test_bit_bl: "w !! n = (rev (to_bl w) ! n & n < size w)" |
|
2483 unfolding to_bl_def word_test_bit_def word_size |
|
2484 by (rule bin_nth_uint) |
|
2485 |
|
2486 lemma to_bl_nth: "n < size w ==> to_bl w ! n = w !! (size w - Suc n)" |
|
2487 apply (unfold test_bit_bl) |
|
2488 apply clarsimp |
|
2489 apply (rule trans) |
|
2490 apply (rule nth_rev_alt) |
|
2491 apply (auto simp add: word_size) |
|
2492 done |
|
2493 |
|
2494 lemma test_bit_set: |
|
2495 fixes w :: "'a::len0 word" |
|
2496 shows "(set_bit w n x) !! n = (n < size w & x)" |
|
2497 unfolding word_size word_test_bit_def word_set_bit_def |
|
2498 by (clarsimp simp add : word_ubin.eq_norm nth_bintr) |
|
2499 |
|
2500 lemma test_bit_set_gen: |
|
2501 fixes w :: "'a::len0 word" |
|
2502 shows "test_bit (set_bit w n x) m = |
|
2503 (if m = n then n < size w & x else test_bit w m)" |
|
2504 apply (unfold word_size word_test_bit_def word_set_bit_def) |
|
2505 apply (clarsimp simp add: word_ubin.eq_norm nth_bintr bin_nth_sc_gen) |
|
2506 apply (auto elim!: test_bit_size [unfolded word_size] |
|
2507 simp add: word_test_bit_def [symmetric]) |
|
2508 done |
|
2509 |
|
2510 lemma of_bl_rep_False: "of_bl (replicate n False @ bs) = of_bl bs" |
|
2511 unfolding of_bl_def bl_to_bin_rep_F by auto |
|
2512 |
|
2513 lemma msb_nth': |
|
2514 fixes w :: "'a::len word" |
|
2515 shows "msb w = w !! (size w - 1)" |
|
2516 unfolding word_msb_nth' word_test_bit_def by simp |
|
2517 |
|
2518 lemmas msb_nth = msb_nth' [unfolded word_size] |
|
2519 |
|
2520 lemmas msb0 = len_gt_0 [THEN diff_Suc_less, THEN |
|
2521 word_ops_nth_size [unfolded word_size], standard] |
|
2522 lemmas msb1 = msb0 [where i = 0] |
|
2523 lemmas word_ops_msb = msb1 [unfolded msb_nth [symmetric, unfolded One_nat_def]] |
|
2524 |
|
2525 lemmas lsb0 = len_gt_0 [THEN word_ops_nth_size [unfolded word_size], standard] |
|
2526 lemmas word_ops_lsb = lsb0 [unfolded word_lsb_alt] |
|
2527 |
|
2528 lemma td_ext_nth': |
|
2529 "n = size (w::'a::len0 word) ==> ofn = set_bits ==> [w, ofn g] = l ==> |
|
2530 td_ext test_bit ofn {f. ALL i. f i --> i < n} (%h i. h i & i < n)" |
|
2531 apply (unfold word_size td_ext_def') |
|
2532 apply (safe del: subset_antisym) |
|
2533 apply (rule_tac [3] ext) |
|
2534 apply (rule_tac [4] ext) |
|
2535 apply (unfold word_size of_nth_def test_bit_bl) |
|
2536 apply safe |
|
2537 defer |
|
2538 apply (clarsimp simp: word_bl.Abs_inverse)+ |
|
2539 apply (rule word_bl.Rep_inverse') |
|
2540 apply (rule sym [THEN trans]) |
|
2541 apply (rule bl_of_nth_nth) |
|
2542 apply simp |
|
2543 apply (rule bl_of_nth_inj) |
|
2544 apply (clarsimp simp add : test_bit_bl word_size) |
|
2545 done |
|
2546 |
|
2547 lemmas td_ext_nth = td_ext_nth' [OF refl refl refl, unfolded word_size] |
|
2548 |
|
2549 interpretation test_bit: |
|
2550 td_ext "op !! :: 'a::len0 word => nat => bool" |
|
2551 set_bits |
|
2552 "{f. \<forall>i. f i \<longrightarrow> i < len_of TYPE('a::len0)}" |
|
2553 "(\<lambda>h i. h i \<and> i < len_of TYPE('a::len0))" |
|
2554 by (rule td_ext_nth) |
|
2555 |
|
2556 declare test_bit.Rep' [simp del] |
|
2557 declare test_bit.Rep' [rule del] |
|
2558 |
|
2559 lemmas td_nth = test_bit.td_thm |
|
2560 |
|
2561 lemma word_set_set_same: |
|
2562 fixes w :: "'a::len0 word" |
|
2563 shows "set_bit (set_bit w n x) n y = set_bit w n y" |
|
2564 by (rule word_eqI) (simp add : test_bit_set_gen word_size) |
|
2565 |
|
2566 lemma word_set_set_diff: |
|
2567 fixes w :: "'a::len0 word" |
|
2568 assumes "m ~= n" |
|
2569 shows "set_bit (set_bit w m x) n y = set_bit (set_bit w n y) m x" |
|
2570 by (rule word_eqI) (clarsimp simp add : test_bit_set_gen word_size prems) |
|
2571 |
|
2572 lemma test_bit_no': |
|
2573 fixes w :: "'a::len0 word" |
|
2574 shows "w = number_of bin ==> test_bit w n = (n < size w & bin_nth bin n)" |
|
2575 unfolding word_test_bit_def word_number_of_def word_size |
|
2576 by (simp add : nth_bintr [symmetric] word_ubin.eq_norm) |
|
2577 |
|
2578 lemmas test_bit_no = |
|
2579 refl [THEN test_bit_no', unfolded word_size, THEN eq_reflection, standard] |
|
2580 |
|
2581 lemma nth_0: "~ (0::'a::len0 word) !! n" |
|
2582 unfolding test_bit_no word_0_no by auto |
|
2583 |
|
2584 lemma nth_sint: |
|
2585 fixes w :: "'a::len word" |
|
2586 defines "l \<equiv> len_of TYPE ('a)" |
|
2587 shows "bin_nth (sint w) n = (if n < l - 1 then w !! n else w !! (l - 1))" |
|
2588 unfolding sint_uint l_def |
|
2589 by (clarsimp simp add: nth_sbintr word_test_bit_def [symmetric]) |
|
2590 |
|
2591 lemma word_lsb_no: |
|
2592 "lsb (number_of bin :: 'a :: len word) = (bin_last bin = 1)" |
|
2593 unfolding word_lsb_alt test_bit_no by auto |
|
2594 |
|
2595 lemma word_set_no: |
|
2596 "set_bit (number_of bin::'a::len0 word) n b = |
|
2597 number_of (bin_sc n (if b then 1 else 0) bin)" |
|
2598 apply (unfold word_set_bit_def word_number_of_def [symmetric]) |
|
2599 apply (rule word_eqI) |
|
2600 apply (clarsimp simp: word_size bin_nth_sc_gen number_of_is_id |
|
2601 test_bit_no nth_bintr) |
|
2602 done |
|
2603 |
|
2604 lemmas setBit_no = setBit_def [THEN trans [OF meta_eq_to_obj_eq word_set_no], |
|
2605 simplified if_simps, THEN eq_reflection, standard] |
|
2606 lemmas clearBit_no = clearBit_def [THEN trans [OF meta_eq_to_obj_eq word_set_no], |
|
2607 simplified if_simps, THEN eq_reflection, standard] |
|
2608 |
|
2609 lemma to_bl_n1: |
|
2610 "to_bl (-1::'a::len0 word) = replicate (len_of TYPE ('a)) True" |
|
2611 apply (rule word_bl.Abs_inverse') |
|
2612 apply simp |
|
2613 apply (rule word_eqI) |
|
2614 apply (clarsimp simp add: word_size test_bit_no) |
|
2615 apply (auto simp add: word_bl.Abs_inverse test_bit_bl word_size) |
|
2616 done |
|
2617 |
|
2618 lemma word_msb_n1: "msb (-1::'a::len word)" |
|
2619 unfolding word_msb_alt word_msb_alt to_bl_n1 by simp |
|
2620 |
|
2621 declare word_set_set_same [simp] word_set_nth [simp] |
|
2622 test_bit_no [simp] word_set_no [simp] nth_0 [simp] |
|
2623 setBit_no [simp] clearBit_no [simp] |
|
2624 word_lsb_no [simp] word_msb_no [simp] word_msb_n1 [simp] word_lsb_1_0 [simp] |
|
2625 |
|
2626 lemma word_set_nth_iff: |
|
2627 "(set_bit w n b = w) = (w !! n = b | n >= size (w::'a::len0 word))" |
|
2628 apply (rule iffI) |
|
2629 apply (rule disjCI) |
|
2630 apply (drule word_eqD) |
|
2631 apply (erule sym [THEN trans]) |
|
2632 apply (simp add: test_bit_set) |
|
2633 apply (erule disjE) |
|
2634 apply clarsimp |
|
2635 apply (rule word_eqI) |
|
2636 apply (clarsimp simp add : test_bit_set_gen) |
|
2637 apply (drule test_bit_size) |
|
2638 apply force |
|
2639 done |
|
2640 |
|
2641 lemma test_bit_2p': |
|
2642 "w = word_of_int (2 ^ n) ==> |
|
2643 w !! m = (m = n & m < size (w :: 'a :: len word))" |
|
2644 unfolding word_test_bit_def word_size |
|
2645 by (auto simp add: word_ubin.eq_norm nth_bintr nth_2p_bin) |
|
2646 |
|
2647 lemmas test_bit_2p = refl [THEN test_bit_2p', unfolded word_size] |
|
2648 |
|
2649 lemma nth_w2p: |
|
2650 "((2\<Colon>'a\<Colon>len word) ^ n) !! m \<longleftrightarrow> m = n \<and> m < len_of TYPE('a\<Colon>len)" |
|
2651 unfolding test_bit_2p [symmetric] word_of_int [symmetric] |
|
2652 by (simp add: of_int_power) |
|
2653 |
|
2654 lemma uint_2p: |
|
2655 "(0::'a::len word) < 2 ^ n ==> uint (2 ^ n::'a::len word) = 2 ^ n" |
|
2656 apply (unfold word_arith_power_alt) |
|
2657 apply (case_tac "len_of TYPE ('a)") |
|
2658 apply clarsimp |
|
2659 apply (case_tac "nat") |
|
2660 apply clarsimp |
|
2661 apply (case_tac "n") |
|
2662 apply (clarsimp simp add : word_1_wi [symmetric]) |
|
2663 apply (clarsimp simp add : word_0_wi [symmetric]) |
|
2664 apply (drule word_gt_0 [THEN iffD1]) |
|
2665 apply (safe intro!: word_eqI bin_nth_lem ext) |
|
2666 apply (auto simp add: test_bit_2p nth_2p_bin word_test_bit_def [symmetric]) |
|
2667 done |
|
2668 |
|
2669 lemma word_of_int_2p: "(word_of_int (2 ^ n) :: 'a :: len word) = 2 ^ n" |
|
2670 apply (unfold word_arith_power_alt) |
|
2671 apply (case_tac "len_of TYPE ('a)") |
|
2672 apply clarsimp |
|
2673 apply (case_tac "nat") |
|
2674 apply (rule word_ubin.norm_eq_iff [THEN iffD1]) |
|
2675 apply (rule box_equals) |
|
2676 apply (rule_tac [2] bintr_ariths (1))+ |
|
2677 apply (clarsimp simp add : number_of_is_id) |
|
2678 apply simp |
|
2679 done |
|
2680 |
|
2681 lemma bang_is_le: "x !! m ==> 2 ^ m <= (x :: 'a :: len word)" |
|
2682 apply (rule xtr3) |
|
2683 apply (rule_tac [2] y = "x" in le_word_or2) |
|
2684 apply (rule word_eqI) |
|
2685 apply (auto simp add: word_ao_nth nth_w2p word_size) |
|
2686 done |
|
2687 |
|
2688 lemma word_clr_le: |
|
2689 fixes w :: "'a::len0 word" |
|
2690 shows "w >= set_bit w n False" |
|
2691 apply (unfold word_set_bit_def word_le_def word_ubin.eq_norm) |
|
2692 apply simp |
|
2693 apply (rule order_trans) |
|
2694 apply (rule bintr_bin_clr_le) |
|
2695 apply simp |
|
2696 done |
|
2697 |
|
2698 lemma word_set_ge: |
|
2699 fixes w :: "'a::len word" |
|
2700 shows "w <= set_bit w n True" |
|
2701 apply (unfold word_set_bit_def word_le_def word_ubin.eq_norm) |
|
2702 apply simp |
|
2703 apply (rule order_trans [OF _ bintr_bin_set_ge]) |
|
2704 apply simp |
|
2705 done |
|
2706 |
|
2707 |
|
2708 subsection {* Shifting, Rotating, and Splitting Words *} |
|
2709 |
|
2710 lemma shiftl1_number [simp] : |
|
2711 "shiftl1 (number_of w) = number_of (w BIT 0)" |
|
2712 apply (unfold shiftl1_def word_number_of_def) |
|
2713 apply (simp add: word_ubin.norm_eq_iff [symmetric] word_ubin.eq_norm |
|
2714 del: BIT_simps) |
|
2715 apply (subst refl [THEN bintrunc_BIT_I, symmetric]) |
|
2716 apply (subst bintrunc_bintrunc_min) |
|
2717 apply simp |
|
2718 done |
|
2719 |
|
2720 lemma shiftl1_0 [simp] : "shiftl1 0 = 0" |
|
2721 unfolding word_0_no shiftl1_number by auto |
|
2722 |
|
2723 lemmas shiftl1_def_u = shiftl1_def [folded word_number_of_def] |
|
2724 |
|
2725 lemma shiftl1_def_s: "shiftl1 w = number_of (sint w BIT 0)" |
|
2726 by (rule trans [OF _ shiftl1_number]) simp |
|
2727 |
|
2728 lemma shiftr1_0 [simp] : "shiftr1 0 = 0" |
|
2729 unfolding shiftr1_def |
|
2730 by simp (simp add: word_0_wi) |
|
2731 |
|
2732 lemma sshiftr1_0 [simp] : "sshiftr1 0 = 0" |
|
2733 apply (unfold sshiftr1_def) |
|
2734 apply simp |
|
2735 apply (simp add : word_0_wi) |
|
2736 done |
|
2737 |
|
2738 lemma sshiftr1_n1 [simp] : "sshiftr1 -1 = -1" |
|
2739 unfolding sshiftr1_def by auto |
|
2740 |
|
2741 lemma shiftl_0 [simp] : "(0::'a::len0 word) << n = 0" |
|
2742 unfolding shiftl_def by (induct n) auto |
|
2743 |
|
2744 lemma shiftr_0 [simp] : "(0::'a::len0 word) >> n = 0" |
|
2745 unfolding shiftr_def by (induct n) auto |
|
2746 |
|
2747 lemma sshiftr_0 [simp] : "0 >>> n = 0" |
|
2748 unfolding sshiftr_def by (induct n) auto |
|
2749 |
|
2750 lemma sshiftr_n1 [simp] : "-1 >>> n = -1" |
|
2751 unfolding sshiftr_def by (induct n) auto |
|
2752 |
|
2753 lemma nth_shiftl1: "shiftl1 w !! n = (n < size w & n > 0 & w !! (n - 1))" |
|
2754 apply (unfold shiftl1_def word_test_bit_def) |
|
2755 apply (simp add: nth_bintr word_ubin.eq_norm word_size) |
|
2756 apply (cases n) |
|
2757 apply auto |
|
2758 done |
|
2759 |
|
2760 lemma nth_shiftl' [rule_format]: |
|
2761 "ALL n. ((w::'a::len0 word) << m) !! n = (n < size w & n >= m & w !! (n - m))" |
|
2762 apply (unfold shiftl_def) |
|
2763 apply (induct "m") |
|
2764 apply (force elim!: test_bit_size) |
|
2765 apply (clarsimp simp add : nth_shiftl1 word_size) |
|
2766 apply arith |
|
2767 done |
|
2768 |
|
2769 lemmas nth_shiftl = nth_shiftl' [unfolded word_size] |
|
2770 |
|
2771 lemma nth_shiftr1: "shiftr1 w !! n = w !! Suc n" |
|
2772 apply (unfold shiftr1_def word_test_bit_def) |
|
2773 apply (simp add: nth_bintr word_ubin.eq_norm) |
|
2774 apply safe |
|
2775 apply (drule bin_nth.Suc [THEN iffD2, THEN bin_nth_uint_imp]) |
|
2776 apply simp |
|
2777 done |
|
2778 |
|
2779 lemma nth_shiftr: |
|
2780 "\<And>n. ((w::'a::len0 word) >> m) !! n = w !! (n + m)" |
|
2781 apply (unfold shiftr_def) |
|
2782 apply (induct "m") |
|
2783 apply (auto simp add : nth_shiftr1) |
|
2784 done |
|
2785 |
|
2786 (* see paper page 10, (1), (2), shiftr1_def is of the form of (1), |
|
2787 where f (ie bin_rest) takes normal arguments to normal results, |
|
2788 thus we get (2) from (1) *) |
|
2789 |
|
2790 lemma uint_shiftr1: "uint (shiftr1 w) = bin_rest (uint w)" |
|
2791 apply (unfold shiftr1_def word_ubin.eq_norm bin_rest_trunc_i) |
|
2792 apply (subst bintr_uint [symmetric, OF order_refl]) |
|
2793 apply (simp only : bintrunc_bintrunc_l) |
|
2794 apply simp |
|
2795 done |
|
2796 |
|
2797 lemma nth_sshiftr1: |
|
2798 "sshiftr1 w !! n = (if n = size w - 1 then w !! n else w !! Suc n)" |
|
2799 apply (unfold sshiftr1_def word_test_bit_def) |
|
2800 apply (simp add: nth_bintr word_ubin.eq_norm |
|
2801 bin_nth.Suc [symmetric] word_size |
|
2802 del: bin_nth.simps) |
|
2803 apply (simp add: nth_bintr uint_sint del : bin_nth.simps) |
|
2804 apply (auto simp add: bin_nth_sint) |
|
2805 done |
|
2806 |
|
2807 lemma nth_sshiftr [rule_format] : |
|
2808 "ALL n. sshiftr w m !! n = (n < size w & |
|
2809 (if n + m >= size w then w !! (size w - 1) else w !! (n + m)))" |
|
2810 apply (unfold sshiftr_def) |
|
2811 apply (induct_tac "m") |
|
2812 apply (simp add: test_bit_bl) |
|
2813 apply (clarsimp simp add: nth_sshiftr1 word_size) |
|
2814 apply safe |
|
2815 apply arith |
|
2816 apply arith |
|
2817 apply (erule thin_rl) |
|
2818 apply (case_tac n) |
|
2819 apply safe |
|
2820 apply simp |
|
2821 apply simp |
|
2822 apply (erule thin_rl) |
|
2823 apply (case_tac n) |
|
2824 apply safe |
|
2825 apply simp |
|
2826 apply simp |
|
2827 apply arith+ |
|
2828 done |
|
2829 |
|
2830 lemma shiftr1_div_2: "uint (shiftr1 w) = uint w div 2" |
|
2831 apply (unfold shiftr1_def bin_rest_div) |
|
2832 apply (rule word_uint.Abs_inverse) |
|
2833 apply (simp add: uints_num pos_imp_zdiv_nonneg_iff) |
|
2834 apply (rule xtr7) |
|
2835 prefer 2 |
|
2836 apply (rule zdiv_le_dividend) |
|
2837 apply auto |
|
2838 done |
|
2839 |
|
2840 lemma sshiftr1_div_2: "sint (sshiftr1 w) = sint w div 2" |
|
2841 apply (unfold sshiftr1_def bin_rest_div [symmetric]) |
|
2842 apply (simp add: word_sbin.eq_norm) |
|
2843 apply (rule trans) |
|
2844 defer |
|
2845 apply (subst word_sbin.norm_Rep [symmetric]) |
|
2846 apply (rule refl) |
|
2847 apply (subst word_sbin.norm_Rep [symmetric]) |
|
2848 apply (unfold One_nat_def) |
|
2849 apply (rule sbintrunc_rest) |
|
2850 done |
|
2851 |
|
2852 lemma shiftr_div_2n: "uint (shiftr w n) = uint w div 2 ^ n" |
|
2853 apply (unfold shiftr_def) |
|
2854 apply (induct "n") |
|
2855 apply simp |
|
2856 apply (simp add: shiftr1_div_2 mult_commute |
|
2857 zdiv_zmult2_eq [symmetric]) |
|
2858 done |
|
2859 |
|
2860 lemma sshiftr_div_2n: "sint (sshiftr w n) = sint w div 2 ^ n" |
|
2861 apply (unfold sshiftr_def) |
|
2862 apply (induct "n") |
|
2863 apply simp |
|
2864 apply (simp add: sshiftr1_div_2 mult_commute |
|
2865 zdiv_zmult2_eq [symmetric]) |
|
2866 done |
|
2867 |
|
2868 subsubsection "shift functions in terms of lists of bools" |
|
2869 |
|
2870 lemmas bshiftr1_no_bin [simp] = |
|
2871 bshiftr1_def [where w="number_of w", unfolded to_bl_no_bin, standard] |
|
2872 |
|
2873 lemma bshiftr1_bl: "to_bl (bshiftr1 b w) = b # butlast (to_bl w)" |
|
2874 unfolding bshiftr1_def by (rule word_bl.Abs_inverse) simp |
|
2875 |
|
2876 lemma shiftl1_of_bl: "shiftl1 (of_bl bl) = of_bl (bl @ [False])" |
|
2877 unfolding uint_bl of_bl_no |
|
2878 by (simp add: bl_to_bin_aux_append bl_to_bin_def) |
|
2879 |
|
2880 lemma shiftl1_bl: "shiftl1 (w::'a::len0 word) = of_bl (to_bl w @ [False])" |
|
2881 proof - |
|
2882 have "shiftl1 w = shiftl1 (of_bl (to_bl w))" by simp |
|
2883 also have "\<dots> = of_bl (to_bl w @ [False])" by (rule shiftl1_of_bl) |
|
2884 finally show ?thesis . |
|
2885 qed |
|
2886 |
|
2887 lemma bl_shiftl1: |
|
2888 "to_bl (shiftl1 (w :: 'a :: len word)) = tl (to_bl w) @ [False]" |
|
2889 apply (simp add: shiftl1_bl word_rep_drop drop_Suc drop_Cons') |
|
2890 apply (fast intro!: Suc_leI) |
|
2891 done |
|
2892 |
|
2893 lemma shiftr1_bl: "shiftr1 w = of_bl (butlast (to_bl w))" |
|
2894 apply (unfold shiftr1_def uint_bl of_bl_def) |
|
2895 apply (simp add: butlast_rest_bin word_size) |
|
2896 apply (simp add: bin_rest_trunc [symmetric, unfolded One_nat_def]) |
|
2897 done |
|
2898 |
|
2899 lemma bl_shiftr1: |
|
2900 "to_bl (shiftr1 (w :: 'a :: len word)) = False # butlast (to_bl w)" |
|
2901 unfolding shiftr1_bl |
|
2902 by (simp add : word_rep_drop len_gt_0 [THEN Suc_leI]) |
|
2903 |
|
2904 |
|
2905 (* relate the two above : TODO - remove the :: len restriction on |
|
2906 this theorem and others depending on it *) |
|
2907 lemma shiftl1_rev: |
|
2908 "shiftl1 (w :: 'a :: len word) = word_reverse (shiftr1 (word_reverse w))" |
|
2909 apply (unfold word_reverse_def) |
|
2910 apply (rule word_bl.Rep_inverse' [symmetric]) |
|
2911 apply (simp add: bl_shiftl1 bl_shiftr1 word_bl.Abs_inverse) |
|
2912 apply (cases "to_bl w") |
|
2913 apply auto |
|
2914 done |
|
2915 |
|
2916 lemma shiftl_rev: |
|
2917 "shiftl (w :: 'a :: len word) n = word_reverse (shiftr (word_reverse w) n)" |
|
2918 apply (unfold shiftl_def shiftr_def) |
|
2919 apply (induct "n") |
|
2920 apply (auto simp add : shiftl1_rev) |
|
2921 done |
|
2922 |
|
2923 lemmas rev_shiftl = |
|
2924 shiftl_rev [where w = "word_reverse w", simplified, standard] |
|
2925 |
|
2926 lemmas shiftr_rev = rev_shiftl [THEN word_rev_gal', standard] |
|
2927 lemmas rev_shiftr = shiftl_rev [THEN word_rev_gal', standard] |
|
2928 |
|
2929 lemma bl_sshiftr1: |
|
2930 "to_bl (sshiftr1 (w :: 'a :: len word)) = hd (to_bl w) # butlast (to_bl w)" |
|
2931 apply (unfold sshiftr1_def uint_bl word_size) |
|
2932 apply (simp add: butlast_rest_bin word_ubin.eq_norm) |
|
2933 apply (simp add: sint_uint) |
|
2934 apply (rule nth_equalityI) |
|
2935 apply clarsimp |
|
2936 apply clarsimp |
|
2937 apply (case_tac i) |
|
2938 apply (simp_all add: hd_conv_nth length_0_conv [symmetric] |
|
2939 nth_bin_to_bl bin_nth.Suc [symmetric] |
|
2940 nth_sbintr |
|
2941 del: bin_nth.Suc) |
|
2942 apply force |
|
2943 apply (rule impI) |
|
2944 apply (rule_tac f = "bin_nth (uint w)" in arg_cong) |
|
2945 apply simp |
|
2946 done |
|
2947 |
|
2948 lemma drop_shiftr: |
|
2949 "drop n (to_bl ((w :: 'a :: len word) >> n)) = take (size w - n) (to_bl w)" |
|
2950 apply (unfold shiftr_def) |
|
2951 apply (induct n) |
|
2952 prefer 2 |
|
2953 apply (simp add: drop_Suc bl_shiftr1 butlast_drop [symmetric]) |
|
2954 apply (rule butlast_take [THEN trans]) |
|
2955 apply (auto simp: word_size) |
|
2956 done |
|
2957 |
|
2958 lemma drop_sshiftr: |
|
2959 "drop n (to_bl ((w :: 'a :: len word) >>> n)) = take (size w - n) (to_bl w)" |
|
2960 apply (unfold sshiftr_def) |
|
2961 apply (induct n) |
|
2962 prefer 2 |
|
2963 apply (simp add: drop_Suc bl_sshiftr1 butlast_drop [symmetric]) |
|
2964 apply (rule butlast_take [THEN trans]) |
|
2965 apply (auto simp: word_size) |
|
2966 done |
|
2967 |
|
2968 lemma take_shiftr [rule_format] : |
|
2969 "n <= size (w :: 'a :: len word) --> take n (to_bl (w >> n)) = |
|
2970 replicate n False" |
|
2971 apply (unfold shiftr_def) |
|
2972 apply (induct n) |
|
2973 prefer 2 |
|
2974 apply (simp add: bl_shiftr1) |
|
2975 apply (rule impI) |
|
2976 apply (rule take_butlast [THEN trans]) |
|
2977 apply (auto simp: word_size) |
|
2978 done |
|
2979 |
|
2980 lemma take_sshiftr' [rule_format] : |
|
2981 "n <= size (w :: 'a :: len word) --> hd (to_bl (w >>> n)) = hd (to_bl w) & |
|
2982 take n (to_bl (w >>> n)) = replicate n (hd (to_bl w))" |
|
2983 apply (unfold sshiftr_def) |
|
2984 apply (induct n) |
|
2985 prefer 2 |
|
2986 apply (simp add: bl_sshiftr1) |
|
2987 apply (rule impI) |
|
2988 apply (rule take_butlast [THEN trans]) |
|
2989 apply (auto simp: word_size) |
|
2990 done |
|
2991 |
|
2992 lemmas hd_sshiftr = take_sshiftr' [THEN conjunct1, standard] |
|
2993 lemmas take_sshiftr = take_sshiftr' [THEN conjunct2, standard] |
|
2994 |
|
2995 lemma atd_lem: "take n xs = t ==> drop n xs = d ==> xs = t @ d" |
|
2996 by (auto intro: append_take_drop_id [symmetric]) |
|
2997 |
|
2998 lemmas bl_shiftr = atd_lem [OF take_shiftr drop_shiftr] |
|
2999 lemmas bl_sshiftr = atd_lem [OF take_sshiftr drop_sshiftr] |
|
3000 |
|
3001 lemma shiftl_of_bl: "of_bl bl << n = of_bl (bl @ replicate n False)" |
|
3002 unfolding shiftl_def |
|
3003 by (induct n) (auto simp: shiftl1_of_bl replicate_app_Cons_same) |
|
3004 |
|
3005 lemma shiftl_bl: |
|
3006 "(w::'a::len0 word) << (n::nat) = of_bl (to_bl w @ replicate n False)" |
|
3007 proof - |
|
3008 have "w << n = of_bl (to_bl w) << n" by simp |
|
3009 also have "\<dots> = of_bl (to_bl w @ replicate n False)" by (rule shiftl_of_bl) |
|
3010 finally show ?thesis . |
|
3011 qed |
|
3012 |
|
3013 lemmas shiftl_number [simp] = shiftl_def [where w="number_of w", standard] |
|
3014 |
|
3015 lemma bl_shiftl: |
|
3016 "to_bl (w << n) = drop n (to_bl w) @ replicate (min (size w) n) False" |
|
3017 by (simp add: shiftl_bl word_rep_drop word_size) |
|
3018 |
|
3019 lemma shiftl_zero_size: |
|
3020 fixes x :: "'a::len0 word" |
|
3021 shows "size x <= n ==> x << n = 0" |
|
3022 apply (unfold word_size) |
|
3023 apply (rule word_eqI) |
|
3024 apply (clarsimp simp add: shiftl_bl word_size test_bit_of_bl nth_append) |
|
3025 done |
|
3026 |
|
3027 (* note - the following results use 'a :: len word < number_ring *) |
|
3028 |
|
3029 lemma shiftl1_2t: "shiftl1 (w :: 'a :: len word) = 2 * w" |
|
3030 apply (simp add: shiftl1_def_u) |
|
3031 apply (simp only: double_number_of_Bit0 [symmetric]) |
|
3032 apply simp |
|
3033 done |
|
3034 |
|
3035 lemma shiftl1_p: "shiftl1 (w :: 'a :: len word) = w + w" |
|
3036 apply (simp add: shiftl1_def_u) |
|
3037 apply (simp only: double_number_of_Bit0 [symmetric]) |
|
3038 apply simp |
|
3039 done |
|
3040 |
|
3041 lemma shiftl_t2n: "shiftl (w :: 'a :: len word) n = 2 ^ n * w" |
|
3042 unfolding shiftl_def |
|
3043 by (induct n) (auto simp: shiftl1_2t power_Suc) |
|
3044 |
|
3045 lemma shiftr1_bintr [simp]: |
|
3046 "(shiftr1 (number_of w) :: 'a :: len0 word) = |
|
3047 number_of (bin_rest (bintrunc (len_of TYPE ('a)) w))" |
|
3048 unfolding shiftr1_def word_number_of_def |
|
3049 by (simp add : word_ubin.eq_norm) |
|
3050 |
|
3051 lemma sshiftr1_sbintr [simp] : |
|
3052 "(sshiftr1 (number_of w) :: 'a :: len word) = |
|
3053 number_of (bin_rest (sbintrunc (len_of TYPE ('a) - 1) w))" |
|
3054 unfolding sshiftr1_def word_number_of_def |
|
3055 by (simp add : word_sbin.eq_norm) |
|
3056 |
|
3057 lemma shiftr_no': |
|
3058 "w = number_of bin ==> |
|
3059 (w::'a::len0 word) >> n = number_of ((bin_rest ^^ n) (bintrunc (size w) bin))" |
|
3060 apply clarsimp |
|
3061 apply (rule word_eqI) |
|
3062 apply (auto simp: nth_shiftr nth_rest_power_bin nth_bintr word_size) |
|
3063 done |
|
3064 |
|
3065 lemma sshiftr_no': |
|
3066 "w = number_of bin ==> w >>> n = number_of ((bin_rest ^^ n) |
|
3067 (sbintrunc (size w - 1) bin))" |
|
3068 apply clarsimp |
|
3069 apply (rule word_eqI) |
|
3070 apply (auto simp: nth_sshiftr nth_rest_power_bin nth_sbintr word_size) |
|
3071 apply (subgoal_tac "na + n = len_of TYPE('a) - Suc 0", simp, simp)+ |
|
3072 done |
|
3073 |
|
3074 lemmas sshiftr_no [simp] = |
|
3075 sshiftr_no' [where w = "number_of w", OF refl, unfolded word_size, standard] |
|
3076 |
|
3077 lemmas shiftr_no [simp] = |
|
3078 shiftr_no' [where w = "number_of w", OF refl, unfolded word_size, standard] |
|
3079 |
|
3080 lemma shiftr1_bl_of': |
|
3081 "us = shiftr1 (of_bl bl) ==> length bl <= size us ==> |
|
3082 us = of_bl (butlast bl)" |
|
3083 by (clarsimp simp: shiftr1_def of_bl_def word_size butlast_rest_bl2bin |
|
3084 word_ubin.eq_norm trunc_bl2bin) |
|
3085 |
|
3086 lemmas shiftr1_bl_of = refl [THEN shiftr1_bl_of', unfolded word_size] |
|
3087 |
|
3088 lemma shiftr_bl_of' [rule_format]: |
|
3089 "us = of_bl bl >> n ==> length bl <= size us --> |
|
3090 us = of_bl (take (length bl - n) bl)" |
|
3091 apply (unfold shiftr_def) |
|
3092 apply hypsubst |
|
3093 apply (unfold word_size) |
|
3094 apply (induct n) |
|
3095 apply clarsimp |
|
3096 apply clarsimp |
|
3097 apply (subst shiftr1_bl_of) |
|
3098 apply simp |
|
3099 apply (simp add: butlast_take) |
|
3100 done |
|
3101 |
|
3102 lemmas shiftr_bl_of = refl [THEN shiftr_bl_of', unfolded word_size] |
|
3103 |
|
3104 lemmas shiftr_bl = word_bl.Rep' [THEN eq_imp_le, THEN shiftr_bl_of, |
|
3105 simplified word_size, simplified, THEN eq_reflection, standard] |
|
3106 |
|
3107 lemma msb_shift': "msb (w::'a::len word) <-> (w >> (size w - 1)) ~= 0" |
|
3108 apply (unfold shiftr_bl word_msb_alt) |
|
3109 apply (simp add: word_size Suc_le_eq take_Suc) |
|
3110 apply (cases "hd (to_bl w)") |
|
3111 apply (auto simp: word_1_bl word_0_bl |
|
3112 of_bl_rep_False [where n=1 and bs="[]", simplified]) |
|
3113 done |
|
3114 |
|
3115 lemmas msb_shift = msb_shift' [unfolded word_size] |
|
3116 |
|
3117 lemma align_lem_or [rule_format] : |
|
3118 "ALL x m. length x = n + m --> length y = n + m --> |
|
3119 drop m x = replicate n False --> take m y = replicate m False --> |
|
3120 map2 op | x y = take m x @ drop m y" |
|
3121 apply (induct_tac y) |
|
3122 apply force |
|
3123 apply clarsimp |
|
3124 apply (case_tac x, force) |
|
3125 apply (case_tac m, auto) |
|
3126 apply (drule sym) |
|
3127 apply auto |
|
3128 apply (induct_tac list, auto) |
|
3129 done |
|
3130 |
|
3131 lemma align_lem_and [rule_format] : |
|
3132 "ALL x m. length x = n + m --> length y = n + m --> |
|
3133 drop m x = replicate n False --> take m y = replicate m False --> |
|
3134 map2 op & x y = replicate (n + m) False" |
|
3135 apply (induct_tac y) |
|
3136 apply force |
|
3137 apply clarsimp |
|
3138 apply (case_tac x, force) |
|
3139 apply (case_tac m, auto) |
|
3140 apply (drule sym) |
|
3141 apply auto |
|
3142 apply (induct_tac list, auto) |
|
3143 done |
|
3144 |
|
3145 lemma aligned_bl_add_size': |
|
3146 "size x - n = m ==> n <= size x ==> drop m (to_bl x) = replicate n False ==> |
|
3147 take m (to_bl y) = replicate m False ==> |
|
3148 to_bl (x + y) = take m (to_bl x) @ drop m (to_bl y)" |
|
3149 apply (subgoal_tac "x AND y = 0") |
|
3150 prefer 2 |
|
3151 apply (rule word_bl.Rep_eqD) |
|
3152 apply (simp add: bl_word_and to_bl_0) |
|
3153 apply (rule align_lem_and [THEN trans]) |
|
3154 apply (simp_all add: word_size)[5] |
|
3155 apply simp |
|
3156 apply (subst word_plus_and_or [symmetric]) |
|
3157 apply (simp add : bl_word_or) |
|
3158 apply (rule align_lem_or) |
|
3159 apply (simp_all add: word_size) |
|
3160 done |
|
3161 |
|
3162 lemmas aligned_bl_add_size = refl [THEN aligned_bl_add_size'] |
|
3163 |
|
3164 subsubsection "Mask" |
|
3165 |
|
3166 lemma nth_mask': "m = mask n ==> test_bit m i = (i < n & i < size m)" |
|
3167 apply (unfold mask_def test_bit_bl) |
|
3168 apply (simp only: word_1_bl [symmetric] shiftl_of_bl) |
|
3169 apply (clarsimp simp add: word_size) |
|
3170 apply (simp only: of_bl_no mask_lem number_of_succ add_diff_cancel2) |
|
3171 apply (fold of_bl_no) |
|
3172 apply (simp add: word_1_bl) |
|
3173 apply (rule test_bit_of_bl [THEN trans, unfolded test_bit_bl word_size]) |
|
3174 apply auto |
|
3175 done |
|
3176 |
|
3177 lemmas nth_mask [simp] = refl [THEN nth_mask'] |
|
3178 |
|
3179 lemma mask_bl: "mask n = of_bl (replicate n True)" |
|
3180 by (auto simp add : test_bit_of_bl word_size intro: word_eqI) |
|
3181 |
|
3182 lemma mask_bin: "mask n = number_of (bintrunc n Int.Min)" |
|
3183 by (auto simp add: nth_bintr word_size intro: word_eqI) |
|
3184 |
|
3185 lemma and_mask_bintr: "w AND mask n = number_of (bintrunc n (uint w))" |
|
3186 apply (rule word_eqI) |
|
3187 apply (simp add: nth_bintr word_size word_ops_nth_size) |
|
3188 apply (auto simp add: test_bit_bin) |
|
3189 done |
|
3190 |
|
3191 lemma and_mask_no: "number_of i AND mask n = number_of (bintrunc n i)" |
|
3192 by (auto simp add : nth_bintr word_size word_ops_nth_size intro: word_eqI) |
|
3193 |
|
3194 lemmas and_mask_wi = and_mask_no [unfolded word_number_of_def] |
|
3195 |
|
3196 lemma bl_and_mask': |
|
3197 "to_bl (w AND mask n :: 'a :: len word) = |
|
3198 replicate (len_of TYPE('a) - n) False @ |
|
3199 drop (len_of TYPE('a) - n) (to_bl w)" |
|
3200 apply (rule nth_equalityI) |
|
3201 apply simp |
|
3202 apply (clarsimp simp add: to_bl_nth word_size) |
|
3203 apply (simp add: word_size word_ops_nth_size) |
|
3204 apply (auto simp add: word_size test_bit_bl nth_append nth_rev) |
|
3205 done |
|
3206 |
|
3207 lemmas and_mask_mod_2p = |
|
3208 and_mask_bintr [unfolded word_number_of_alt no_bintr_alt] |
|
3209 |
|
3210 lemma and_mask_lt_2p: "uint (w AND mask n) < 2 ^ n" |
|
3211 apply (simp add : and_mask_bintr no_bintr_alt) |
|
3212 apply (rule xtr8) |
|
3213 prefer 2 |
|
3214 apply (rule pos_mod_bound) |
|
3215 apply auto |
|
3216 done |
|
3217 |
|
3218 lemmas eq_mod_iff = trans [symmetric, OF int_mod_lem eq_sym_conv] |
|
3219 |
|
3220 lemma mask_eq_iff: "(w AND mask n) = w <-> uint w < 2 ^ n" |
|
3221 apply (simp add: and_mask_bintr word_number_of_def) |
|
3222 apply (simp add: word_ubin.inverse_norm) |
|
3223 apply (simp add: eq_mod_iff bintrunc_mod2p min_def) |
|
3224 apply (fast intro!: lt2p_lem) |
|
3225 done |
|
3226 |
|
3227 lemma and_mask_dvd: "2 ^ n dvd uint w = (w AND mask n = 0)" |
|
3228 apply (simp add: dvd_eq_mod_eq_0 and_mask_mod_2p) |
|
3229 apply (simp add: word_uint.norm_eq_iff [symmetric] word_of_int_homs) |
|
3230 apply (subst word_uint.norm_Rep [symmetric]) |
|
3231 apply (simp only: bintrunc_bintrunc_min bintrunc_mod2p [symmetric] min_def) |
|
3232 apply auto |
|
3233 done |
|
3234 |
|
3235 lemma and_mask_dvd_nat: "2 ^ n dvd unat w = (w AND mask n = 0)" |
|
3236 apply (unfold unat_def) |
|
3237 apply (rule trans [OF _ and_mask_dvd]) |
|
3238 apply (unfold dvd_def) |
|
3239 apply auto |
|
3240 apply (drule uint_ge_0 [THEN nat_int.Abs_inverse' [simplified], symmetric]) |
|
3241 apply (simp add : int_mult int_power) |
|
3242 apply (simp add : nat_mult_distrib nat_power_eq) |
|
3243 done |
|
3244 |
|
3245 lemma word_2p_lem: |
|
3246 "n < size w ==> w < 2 ^ n = (uint (w :: 'a :: len word) < 2 ^ n)" |
|
3247 apply (unfold word_size word_less_alt word_number_of_alt) |
|
3248 apply (clarsimp simp add: word_of_int_power_hom word_uint.eq_norm |
|
3249 int_mod_eq' |
|
3250 simp del: word_of_int_bin) |
|
3251 done |
|
3252 |
|
3253 lemma less_mask_eq: "x < 2 ^ n ==> x AND mask n = (x :: 'a :: len word)" |
|
3254 apply (unfold word_less_alt word_number_of_alt) |
|
3255 apply (clarsimp simp add: and_mask_mod_2p word_of_int_power_hom |
|
3256 word_uint.eq_norm |
|
3257 simp del: word_of_int_bin) |
|
3258 apply (drule xtr8 [rotated]) |
|
3259 apply (rule int_mod_le) |
|
3260 apply (auto simp add : mod_pos_pos_trivial) |
|
3261 done |
|
3262 |
|
3263 lemmas mask_eq_iff_w2p = |
|
3264 trans [OF mask_eq_iff word_2p_lem [symmetric], standard] |
|
3265 |
|
3266 lemmas and_mask_less' = |
|
3267 iffD2 [OF word_2p_lem and_mask_lt_2p, simplified word_size, standard] |
|
3268 |
|
3269 lemma and_mask_less_size: "n < size x ==> x AND mask n < 2^n" |
|
3270 unfolding word_size by (erule and_mask_less') |
|
3271 |
|
3272 lemma word_mod_2p_is_mask': |
|
3273 "c = 2 ^ n ==> c > 0 ==> x mod c = (x :: 'a :: len word) AND mask n" |
|
3274 by (clarsimp simp add: word_mod_def uint_2p and_mask_mod_2p) |
|
3275 |
|
3276 lemmas word_mod_2p_is_mask = refl [THEN word_mod_2p_is_mask'] |
|
3277 |
|
3278 lemma mask_eqs: |
|
3279 "(a AND mask n) + b AND mask n = a + b AND mask n" |
|
3280 "a + (b AND mask n) AND mask n = a + b AND mask n" |
|
3281 "(a AND mask n) - b AND mask n = a - b AND mask n" |
|
3282 "a - (b AND mask n) AND mask n = a - b AND mask n" |
|
3283 "a * (b AND mask n) AND mask n = a * b AND mask n" |
|
3284 "(b AND mask n) * a AND mask n = b * a AND mask n" |
|
3285 "(a AND mask n) + (b AND mask n) AND mask n = a + b AND mask n" |
|
3286 "(a AND mask n) - (b AND mask n) AND mask n = a - b AND mask n" |
|
3287 "(a AND mask n) * (b AND mask n) AND mask n = a * b AND mask n" |
|
3288 "- (a AND mask n) AND mask n = - a AND mask n" |
|
3289 "word_succ (a AND mask n) AND mask n = word_succ a AND mask n" |
|
3290 "word_pred (a AND mask n) AND mask n = word_pred a AND mask n" |
|
3291 using word_of_int_Ex [where x=a] word_of_int_Ex [where x=b] |
|
3292 by (auto simp: and_mask_wi bintr_ariths bintr_arith1s new_word_of_int_homs) |
|
3293 |
|
3294 lemma mask_power_eq: |
|
3295 "(x AND mask n) ^ k AND mask n = x ^ k AND mask n" |
|
3296 using word_of_int_Ex [where x=x] |
|
3297 by (clarsimp simp: and_mask_wi word_of_int_power_hom bintr_ariths) |
|
3298 |
|
3299 |
|
3300 subsubsection "Revcast" |
|
3301 |
|
3302 lemmas revcast_def' = revcast_def [simplified] |
|
3303 lemmas revcast_def'' = revcast_def' [simplified word_size] |
|
3304 lemmas revcast_no_def [simp] = |
|
3305 revcast_def' [where w="number_of w", unfolded word_size, standard] |
|
3306 |
|
3307 lemma to_bl_revcast: |
|
3308 "to_bl (revcast w :: 'a :: len0 word) = |
|
3309 takefill False (len_of TYPE ('a)) (to_bl w)" |
|
3310 apply (unfold revcast_def' word_size) |
|
3311 apply (rule word_bl.Abs_inverse) |
|
3312 apply simp |
|
3313 done |
|
3314 |
|
3315 lemma revcast_rev_ucast': |
|
3316 "cs = [rc, uc] ==> rc = revcast (word_reverse w) ==> uc = ucast w ==> |
|
3317 rc = word_reverse uc" |
|
3318 apply (unfold ucast_def revcast_def' Let_def word_reverse_def) |
|
3319 apply (clarsimp simp add : to_bl_of_bin takefill_bintrunc) |
|
3320 apply (simp add : word_bl.Abs_inverse word_size) |
|
3321 done |
|
3322 |
|
3323 lemmas revcast_rev_ucast = revcast_rev_ucast' [OF refl refl refl] |
|
3324 |
|
3325 lemmas revcast_ucast = revcast_rev_ucast |
|
3326 [where w = "word_reverse w", simplified word_rev_rev, standard] |
|
3327 |
|
3328 lemmas ucast_revcast = revcast_rev_ucast [THEN word_rev_gal', standard] |
|
3329 lemmas ucast_rev_revcast = revcast_ucast [THEN word_rev_gal', standard] |
|
3330 |
|
3331 |
|
3332 -- "linking revcast and cast via shift" |
|
3333 |
|
3334 lemmas wsst_TYs = source_size target_size word_size |
|
3335 |
|
3336 lemma revcast_down_uu': |
|
3337 "rc = revcast ==> source_size rc = target_size rc + n ==> |
|
3338 rc (w :: 'a :: len word) = ucast (w >> n)" |
|
3339 apply (simp add: revcast_def') |
|
3340 apply (rule word_bl.Rep_inverse') |
|
3341 apply (rule trans, rule ucast_down_drop) |
|
3342 prefer 2 |
|
3343 apply (rule trans, rule drop_shiftr) |
|
3344 apply (auto simp: takefill_alt wsst_TYs) |
|
3345 done |
|
3346 |
|
3347 lemma revcast_down_us': |
|
3348 "rc = revcast ==> source_size rc = target_size rc + n ==> |
|
3349 rc (w :: 'a :: len word) = ucast (w >>> n)" |
|
3350 apply (simp add: revcast_def') |
|
3351 apply (rule word_bl.Rep_inverse') |
|
3352 apply (rule trans, rule ucast_down_drop) |
|
3353 prefer 2 |
|
3354 apply (rule trans, rule drop_sshiftr) |
|
3355 apply (auto simp: takefill_alt wsst_TYs) |
|
3356 done |
|
3357 |
|
3358 lemma revcast_down_su': |
|
3359 "rc = revcast ==> source_size rc = target_size rc + n ==> |
|
3360 rc (w :: 'a :: len word) = scast (w >> n)" |
|
3361 apply (simp add: revcast_def') |
|
3362 apply (rule word_bl.Rep_inverse') |
|
3363 apply (rule trans, rule scast_down_drop) |
|
3364 prefer 2 |
|
3365 apply (rule trans, rule drop_shiftr) |
|
3366 apply (auto simp: takefill_alt wsst_TYs) |
|
3367 done |
|
3368 |
|
3369 lemma revcast_down_ss': |
|
3370 "rc = revcast ==> source_size rc = target_size rc + n ==> |
|
3371 rc (w :: 'a :: len word) = scast (w >>> n)" |
|
3372 apply (simp add: revcast_def') |
|
3373 apply (rule word_bl.Rep_inverse') |
|
3374 apply (rule trans, rule scast_down_drop) |
|
3375 prefer 2 |
|
3376 apply (rule trans, rule drop_sshiftr) |
|
3377 apply (auto simp: takefill_alt wsst_TYs) |
|
3378 done |
|
3379 |
|
3380 lemmas revcast_down_uu = refl [THEN revcast_down_uu'] |
|
3381 lemmas revcast_down_us = refl [THEN revcast_down_us'] |
|
3382 lemmas revcast_down_su = refl [THEN revcast_down_su'] |
|
3383 lemmas revcast_down_ss = refl [THEN revcast_down_ss'] |
|
3384 |
|
3385 lemma cast_down_rev: |
|
3386 "uc = ucast ==> source_size uc = target_size uc + n ==> |
|
3387 uc w = revcast ((w :: 'a :: len word) << n)" |
|
3388 apply (unfold shiftl_rev) |
|
3389 apply clarify |
|
3390 apply (simp add: revcast_rev_ucast) |
|
3391 apply (rule word_rev_gal') |
|
3392 apply (rule trans [OF _ revcast_rev_ucast]) |
|
3393 apply (rule revcast_down_uu [symmetric]) |
|
3394 apply (auto simp add: wsst_TYs) |
|
3395 done |
|
3396 |
|
3397 lemma revcast_up': |
|
3398 "rc = revcast ==> source_size rc + n = target_size rc ==> |
|
3399 rc w = (ucast w :: 'a :: len word) << n" |
|
3400 apply (simp add: revcast_def') |
|
3401 apply (rule word_bl.Rep_inverse') |
|
3402 apply (simp add: takefill_alt) |
|
3403 apply (rule bl_shiftl [THEN trans]) |
|
3404 apply (subst ucast_up_app) |
|
3405 apply (auto simp add: wsst_TYs) |
|
3406 done |
|
3407 |
|
3408 lemmas revcast_up = refl [THEN revcast_up'] |
|
3409 |
|
3410 lemmas rc1 = revcast_up [THEN |
|
3411 revcast_rev_ucast [symmetric, THEN trans, THEN word_rev_gal, symmetric]] |
|
3412 lemmas rc2 = revcast_down_uu [THEN |
|
3413 revcast_rev_ucast [symmetric, THEN trans, THEN word_rev_gal, symmetric]] |
|
3414 |
|
3415 lemmas ucast_up = |
|
3416 rc1 [simplified rev_shiftr [symmetric] revcast_ucast [symmetric]] |
|
3417 lemmas ucast_down = |
|
3418 rc2 [simplified rev_shiftr revcast_ucast [symmetric]] |
|
3419 |
|
3420 |
|
3421 subsubsection "Slices" |
|
3422 |
|
3423 lemmas slice1_no_bin [simp] = |
|
3424 slice1_def [where w="number_of w", unfolded to_bl_no_bin, standard] |
|
3425 |
|
3426 lemmas slice_no_bin [simp] = |
|
3427 trans [OF slice_def [THEN meta_eq_to_obj_eq] |
|
3428 slice1_no_bin [THEN meta_eq_to_obj_eq], |
|
3429 unfolded word_size, standard] |
|
3430 |
|
3431 lemma slice1_0 [simp] : "slice1 n 0 = 0" |
|
3432 unfolding slice1_def by (simp add : to_bl_0) |
|
3433 |
|
3434 lemma slice_0 [simp] : "slice n 0 = 0" |
|
3435 unfolding slice_def by auto |
|
3436 |
|
3437 lemma slice_take': "slice n w = of_bl (take (size w - n) (to_bl w))" |
|
3438 unfolding slice_def' slice1_def |
|
3439 by (simp add : takefill_alt word_size) |
|
3440 |
|
3441 lemmas slice_take = slice_take' [unfolded word_size] |
|
3442 |
|
3443 -- "shiftr to a word of the same size is just slice, |
|
3444 slice is just shiftr then ucast" |
|
3445 lemmas shiftr_slice = trans |
|
3446 [OF shiftr_bl [THEN meta_eq_to_obj_eq] slice_take [symmetric], standard] |
|
3447 |
|
3448 lemma slice_shiftr: "slice n w = ucast (w >> n)" |
|
3449 apply (unfold slice_take shiftr_bl) |
|
3450 apply (rule ucast_of_bl_up [symmetric]) |
|
3451 apply (simp add: word_size) |
|
3452 done |
|
3453 |
|
3454 lemma nth_slice: |
|
3455 "(slice n w :: 'a :: len0 word) !! m = |
|
3456 (w !! (m + n) & m < len_of TYPE ('a))" |
|
3457 unfolding slice_shiftr |
|
3458 by (simp add : nth_ucast nth_shiftr) |
|
3459 |
|
3460 lemma slice1_down_alt': |
|
3461 "sl = slice1 n w ==> fs = size sl ==> fs + k = n ==> |
|
3462 to_bl sl = takefill False fs (drop k (to_bl w))" |
|
3463 unfolding slice1_def word_size of_bl_def uint_bl |
|
3464 by (clarsimp simp: word_ubin.eq_norm bl_bin_bl_rep_drop drop_takefill) |
|
3465 |
|
3466 lemma slice1_up_alt': |
|
3467 "sl = slice1 n w ==> fs = size sl ==> fs = n + k ==> |
|
3468 to_bl sl = takefill False fs (replicate k False @ (to_bl w))" |
|
3469 apply (unfold slice1_def word_size of_bl_def uint_bl) |
|
3470 apply (clarsimp simp: word_ubin.eq_norm bl_bin_bl_rep_drop |
|
3471 takefill_append [symmetric]) |
|
3472 apply (rule_tac f = "%k. takefill False (len_of TYPE('a)) |
|
3473 (replicate k False @ bin_to_bl (len_of TYPE('b)) (uint w))" in arg_cong) |
|
3474 apply arith |
|
3475 done |
|
3476 |
|
3477 lemmas sd1 = slice1_down_alt' [OF refl refl, unfolded word_size] |
|
3478 lemmas su1 = slice1_up_alt' [OF refl refl, unfolded word_size] |
|
3479 lemmas slice1_down_alt = le_add_diff_inverse [THEN sd1] |
|
3480 lemmas slice1_up_alts = |
|
3481 le_add_diff_inverse [symmetric, THEN su1] |
|
3482 le_add_diff_inverse2 [symmetric, THEN su1] |
|
3483 |
|
3484 lemma ucast_slice1: "ucast w = slice1 (size w) w" |
|
3485 unfolding slice1_def ucast_bl |
|
3486 by (simp add : takefill_same' word_size) |
|
3487 |
|
3488 lemma ucast_slice: "ucast w = slice 0 w" |
|
3489 unfolding slice_def by (simp add : ucast_slice1) |
|
3490 |
|
3491 lemmas slice_id = trans [OF ucast_slice [symmetric] ucast_id] |
|
3492 |
|
3493 lemma revcast_slice1': |
|
3494 "rc = revcast w ==> slice1 (size rc) w = rc" |
|
3495 unfolding slice1_def revcast_def' by (simp add : word_size) |
|
3496 |
|
3497 lemmas revcast_slice1 = refl [THEN revcast_slice1'] |
|
3498 |
|
3499 lemma slice1_tf_tf': |
|
3500 "to_bl (slice1 n w :: 'a :: len0 word) = |
|
3501 rev (takefill False (len_of TYPE('a)) (rev (takefill False n (to_bl w))))" |
|
3502 unfolding slice1_def by (rule word_rev_tf) |
|
3503 |
|
3504 lemmas slice1_tf_tf = slice1_tf_tf' |
|
3505 [THEN word_bl.Rep_inverse', symmetric, standard] |
|
3506 |
|
3507 lemma rev_slice1: |
|
3508 "n + k = len_of TYPE('a) + len_of TYPE('b) \<Longrightarrow> |
|
3509 slice1 n (word_reverse w :: 'b :: len0 word) = |
|
3510 word_reverse (slice1 k w :: 'a :: len0 word)" |
|
3511 apply (unfold word_reverse_def slice1_tf_tf) |
|
3512 apply (rule word_bl.Rep_inverse') |
|
3513 apply (rule rev_swap [THEN iffD1]) |
|
3514 apply (rule trans [symmetric]) |
|
3515 apply (rule tf_rev) |
|
3516 apply (simp add: word_bl.Abs_inverse) |
|
3517 apply (simp add: word_bl.Abs_inverse) |
|
3518 done |
|
3519 |
|
3520 lemma rev_slice': |
|
3521 "res = slice n (word_reverse w) ==> n + k + size res = size w ==> |
|
3522 res = word_reverse (slice k w)" |
|
3523 apply (unfold slice_def word_size) |
|
3524 apply clarify |
|
3525 apply (rule rev_slice1) |
|
3526 apply arith |
|
3527 done |
|
3528 |
|
3529 lemmas rev_slice = refl [THEN rev_slice', unfolded word_size] |
|
3530 |
|
3531 lemmas sym_notr = |
|
3532 not_iff [THEN iffD2, THEN not_sym, THEN not_iff [THEN iffD1]] |
|
3533 |
|
3534 -- {* problem posed by TPHOLs referee: |
|
3535 criterion for overflow of addition of signed integers *} |
|
3536 |
|
3537 lemma sofl_test: |
|
3538 "(sint (x :: 'a :: len word) + sint y = sint (x + y)) = |
|
3539 ((((x+y) XOR x) AND ((x+y) XOR y)) >> (size x - 1) = 0)" |
|
3540 apply (unfold word_size) |
|
3541 apply (cases "len_of TYPE('a)", simp) |
|
3542 apply (subst msb_shift [THEN sym_notr]) |
|
3543 apply (simp add: word_ops_msb) |
|
3544 apply (simp add: word_msb_sint) |
|
3545 apply safe |
|
3546 apply simp_all |
|
3547 apply (unfold sint_word_ariths) |
|
3548 apply (unfold word_sbin.set_iff_norm [symmetric] sints_num) |
|
3549 apply safe |
|
3550 apply (insert sint_range' [where x=x]) |
|
3551 apply (insert sint_range' [where x=y]) |
|
3552 defer |
|
3553 apply (simp (no_asm), arith) |
|
3554 apply (simp (no_asm), arith) |
|
3555 defer |
|
3556 defer |
|
3557 apply (simp (no_asm), arith) |
|
3558 apply (simp (no_asm), arith) |
|
3559 apply (rule notI [THEN notnotD], |
|
3560 drule leI not_leE, |
|
3561 drule sbintrunc_inc sbintrunc_dec, |
|
3562 simp)+ |
|
3563 done |
|
3564 |
|
3565 |
|
3566 subsection "Split and cat" |
|
3567 |
|
3568 lemmas word_split_bin' = word_split_def [THEN meta_eq_to_obj_eq, standard] |
|
3569 lemmas word_cat_bin' = word_cat_def [THEN meta_eq_to_obj_eq, standard] |
|
3570 |
|
3571 lemma word_rsplit_no: |
|
3572 "(word_rsplit (number_of bin :: 'b :: len0 word) :: 'a word list) = |
|
3573 map number_of (bin_rsplit (len_of TYPE('a :: len)) |
|
3574 (len_of TYPE('b), bintrunc (len_of TYPE('b)) bin))" |
|
3575 apply (unfold word_rsplit_def word_no_wi) |
|
3576 apply (simp add: word_ubin.eq_norm) |
|
3577 done |
|
3578 |
|
3579 lemmas word_rsplit_no_cl [simp] = word_rsplit_no |
|
3580 [unfolded bin_rsplitl_def bin_rsplit_l [symmetric]] |
|
3581 |
|
3582 lemma test_bit_cat: |
|
3583 "wc = word_cat a b ==> wc !! n = (n < size wc & |
|
3584 (if n < size b then b !! n else a !! (n - size b)))" |
|
3585 apply (unfold word_cat_bin' test_bit_bin) |
|
3586 apply (auto simp add : word_ubin.eq_norm nth_bintr bin_nth_cat word_size) |
|
3587 apply (erule bin_nth_uint_imp) |
|
3588 done |
|
3589 |
|
3590 lemma word_cat_bl: "word_cat a b = of_bl (to_bl a @ to_bl b)" |
|
3591 apply (unfold of_bl_def to_bl_def word_cat_bin') |
|
3592 apply (simp add: bl_to_bin_app_cat) |
|
3593 done |
|
3594 |
|
3595 lemma of_bl_append: |
|
3596 "(of_bl (xs @ ys) :: 'a :: len word) = of_bl xs * 2^(length ys) + of_bl ys" |
|
3597 apply (unfold of_bl_def) |
|
3598 apply (simp add: bl_to_bin_app_cat bin_cat_num) |
|
3599 apply (simp add: word_of_int_power_hom [symmetric] new_word_of_int_hom_syms) |
|
3600 done |
|
3601 |
|
3602 lemma of_bl_False [simp]: |
|
3603 "of_bl (False#xs) = of_bl xs" |
|
3604 by (rule word_eqI) |
|
3605 (auto simp add: test_bit_of_bl nth_append) |
|
3606 |
|
3607 lemma of_bl_True: |
|
3608 "(of_bl (True#xs)::'a::len word) = 2^length xs + of_bl xs" |
|
3609 by (subst of_bl_append [where xs="[True]", simplified]) |
|
3610 (simp add: word_1_bl) |
|
3611 |
|
3612 lemma of_bl_Cons: |
|
3613 "of_bl (x#xs) = of_bool x * 2^length xs + of_bl xs" |
|
3614 by (cases x) (simp_all add: of_bl_True) |
|
3615 |
|
3616 lemma split_uint_lem: "bin_split n (uint (w :: 'a :: len0 word)) = (a, b) ==> |
|
3617 a = bintrunc (len_of TYPE('a) - n) a & b = bintrunc (len_of TYPE('a)) b" |
|
3618 apply (frule word_ubin.norm_Rep [THEN ssubst]) |
|
3619 apply (drule bin_split_trunc1) |
|
3620 apply (drule sym [THEN trans]) |
|
3621 apply assumption |
|
3622 apply safe |
|
3623 done |
|
3624 |
|
3625 lemma word_split_bl': |
|
3626 "std = size c - size b ==> (word_split c = (a, b)) ==> |
|
3627 (a = of_bl (take std (to_bl c)) & b = of_bl (drop std (to_bl c)))" |
|
3628 apply (unfold word_split_bin') |
|
3629 apply safe |
|
3630 defer |
|
3631 apply (clarsimp split: prod.splits) |
|
3632 apply (drule word_ubin.norm_Rep [THEN ssubst]) |
|
3633 apply (drule split_bintrunc) |
|
3634 apply (simp add : of_bl_def bl2bin_drop word_size |
|
3635 word_ubin.norm_eq_iff [symmetric] min_def del : word_ubin.norm_Rep) |
|
3636 apply (clarsimp split: prod.splits) |
|
3637 apply (frule split_uint_lem [THEN conjunct1]) |
|
3638 apply (unfold word_size) |
|
3639 apply (cases "len_of TYPE('a) >= len_of TYPE('b)") |
|
3640 defer |
|
3641 apply (simp add: word_0_bl word_0_wi_Pls) |
|
3642 apply (simp add : of_bl_def to_bl_def) |
|
3643 apply (subst bin_split_take1 [symmetric]) |
|
3644 prefer 2 |
|
3645 apply assumption |
|
3646 apply simp |
|
3647 apply (erule thin_rl) |
|
3648 apply (erule arg_cong [THEN trans]) |
|
3649 apply (simp add : word_ubin.norm_eq_iff [symmetric]) |
|
3650 done |
|
3651 |
|
3652 lemma word_split_bl: "std = size c - size b ==> |
|
3653 (a = of_bl (take std (to_bl c)) & b = of_bl (drop std (to_bl c))) <-> |
|
3654 word_split c = (a, b)" |
|
3655 apply (rule iffI) |
|
3656 defer |
|
3657 apply (erule (1) word_split_bl') |
|
3658 apply (case_tac "word_split c") |
|
3659 apply (auto simp add : word_size) |
|
3660 apply (frule word_split_bl' [rotated]) |
|
3661 apply (auto simp add : word_size) |
|
3662 done |
|
3663 |
|
3664 lemma word_split_bl_eq: |
|
3665 "(word_split (c::'a::len word) :: ('c :: len0 word * 'd :: len0 word)) = |
|
3666 (of_bl (take (len_of TYPE('a::len) - len_of TYPE('d::len0)) (to_bl c)), |
|
3667 of_bl (drop (len_of TYPE('a) - len_of TYPE('d)) (to_bl c)))" |
|
3668 apply (rule word_split_bl [THEN iffD1]) |
|
3669 apply (unfold word_size) |
|
3670 apply (rule refl conjI)+ |
|
3671 done |
|
3672 |
|
3673 -- "keep quantifiers for use in simplification" |
|
3674 lemma test_bit_split': |
|
3675 "word_split c = (a, b) --> (ALL n m. b !! n = (n < size b & c !! n) & |
|
3676 a !! m = (m < size a & c !! (m + size b)))" |
|
3677 apply (unfold word_split_bin' test_bit_bin) |
|
3678 apply (clarify) |
|
3679 apply (clarsimp simp: word_ubin.eq_norm nth_bintr word_size split: prod.splits) |
|
3680 apply (drule bin_nth_split) |
|
3681 apply safe |
|
3682 apply (simp_all add: add_commute) |
|
3683 apply (erule bin_nth_uint_imp)+ |
|
3684 done |
|
3685 |
|
3686 lemma test_bit_split: |
|
3687 "word_split c = (a, b) \<Longrightarrow> |
|
3688 (\<forall>n\<Colon>nat. b !! n \<longleftrightarrow> n < size b \<and> c !! n) \<and> (\<forall>m\<Colon>nat. a !! m \<longleftrightarrow> m < size a \<and> c !! (m + size b))" |
|
3689 by (simp add: test_bit_split') |
|
3690 |
|
3691 lemma test_bit_split_eq: "word_split c = (a, b) <-> |
|
3692 ((ALL n::nat. b !! n = (n < size b & c !! n)) & |
|
3693 (ALL m::nat. a !! m = (m < size a & c !! (m + size b))))" |
|
3694 apply (rule_tac iffI) |
|
3695 apply (rule_tac conjI) |
|
3696 apply (erule test_bit_split [THEN conjunct1]) |
|
3697 apply (erule test_bit_split [THEN conjunct2]) |
|
3698 apply (case_tac "word_split c") |
|
3699 apply (frule test_bit_split) |
|
3700 apply (erule trans) |
|
3701 apply (fastsimp intro ! : word_eqI simp add : word_size) |
|
3702 done |
|
3703 |
|
3704 -- {* this odd result is analogous to @{text "ucast_id"}, |
|
3705 result to the length given by the result type *} |
|
3706 |
|
3707 lemma word_cat_id: "word_cat a b = b" |
|
3708 unfolding word_cat_bin' by (simp add: word_ubin.inverse_norm) |
|
3709 |
|
3710 -- "limited hom result" |
|
3711 lemma word_cat_hom: |
|
3712 "len_of TYPE('a::len0) <= len_of TYPE('b::len0) + len_of TYPE ('c::len0) |
|
3713 ==> |
|
3714 (word_cat (word_of_int w :: 'b word) (b :: 'c word) :: 'a word) = |
|
3715 word_of_int (bin_cat w (size b) (uint b))" |
|
3716 apply (unfold word_cat_def word_size) |
|
3717 apply (clarsimp simp add: word_ubin.norm_eq_iff [symmetric] |
|
3718 word_ubin.eq_norm bintr_cat min_max.inf_absorb1) |
|
3719 done |
|
3720 |
|
3721 lemma word_cat_split_alt: |
|
3722 "size w <= size u + size v ==> word_split w = (u, v) ==> word_cat u v = w" |
|
3723 apply (rule word_eqI) |
|
3724 apply (drule test_bit_split) |
|
3725 apply (clarsimp simp add : test_bit_cat word_size) |
|
3726 apply safe |
|
3727 apply arith |
|
3728 done |
|
3729 |
|
3730 lemmas word_cat_split_size = |
|
3731 sym [THEN [2] word_cat_split_alt [symmetric], standard] |
|
3732 |
|
3733 |
|
3734 subsubsection "Split and slice" |
|
3735 |
|
3736 lemma split_slices: |
|
3737 "word_split w = (u, v) ==> u = slice (size v) w & v = slice 0 w" |
|
3738 apply (drule test_bit_split) |
|
3739 apply (rule conjI) |
|
3740 apply (rule word_eqI, clarsimp simp: nth_slice word_size)+ |
|
3741 done |
|
3742 |
|
3743 lemma slice_cat1': |
|
3744 "wc = word_cat a b ==> size wc >= size a + size b ==> slice (size b) wc = a" |
|
3745 apply safe |
|
3746 apply (rule word_eqI) |
|
3747 apply (simp add: nth_slice test_bit_cat word_size) |
|
3748 done |
|
3749 |
|
3750 lemmas slice_cat1 = refl [THEN slice_cat1'] |
|
3751 lemmas slice_cat2 = trans [OF slice_id word_cat_id] |
|
3752 |
|
3753 lemma cat_slices: |
|
3754 "a = slice n c ==> b = slice 0 c ==> n = size b ==> |
|
3755 size a + size b >= size c ==> word_cat a b = c" |
|
3756 apply safe |
|
3757 apply (rule word_eqI) |
|
3758 apply (simp add: nth_slice test_bit_cat word_size) |
|
3759 apply safe |
|
3760 apply arith |
|
3761 done |
|
3762 |
|
3763 lemma word_split_cat_alt: |
|
3764 "w = word_cat u v ==> size u + size v <= size w ==> word_split w = (u, v)" |
|
3765 apply (case_tac "word_split ?w") |
|
3766 apply (rule trans, assumption) |
|
3767 apply (drule test_bit_split) |
|
3768 apply safe |
|
3769 apply (rule word_eqI, clarsimp simp: test_bit_cat word_size)+ |
|
3770 done |
|
3771 |
|
3772 lemmas word_cat_bl_no_bin [simp] = |
|
3773 word_cat_bl [where a="number_of a" |
|
3774 and b="number_of b", |
|
3775 unfolded to_bl_no_bin, standard] |
|
3776 |
|
3777 lemmas word_split_bl_no_bin [simp] = |
|
3778 word_split_bl_eq [where c="number_of c", unfolded to_bl_no_bin, standard] |
|
3779 |
|
3780 -- {* this odd result arises from the fact that the statement of the |
|
3781 result implies that the decoded words are of the same type, |
|
3782 and therefore of the same length, as the original word *} |
|
3783 |
|
3784 lemma word_rsplit_same: "word_rsplit w = [w]" |
|
3785 unfolding word_rsplit_def by (simp add : bin_rsplit_all) |
|
3786 |
|
3787 lemma word_rsplit_empty_iff_size: |
|
3788 "(word_rsplit w = []) = (size w = 0)" |
|
3789 unfolding word_rsplit_def bin_rsplit_def word_size |
|
3790 by (simp add: bin_rsplit_aux_simp_alt Let_def split: Product_Type.split_split) |
|
3791 |
|
3792 lemma test_bit_rsplit: |
|
3793 "sw = word_rsplit w ==> m < size (hd sw :: 'a :: len word) ==> |
|
3794 k < length sw ==> (rev sw ! k) !! m = (w !! (k * size (hd sw) + m))" |
|
3795 apply (unfold word_rsplit_def word_test_bit_def) |
|
3796 apply (rule trans) |
|
3797 apply (rule_tac f = "%x. bin_nth x m" in arg_cong) |
|
3798 apply (rule nth_map [symmetric]) |
|
3799 apply simp |
|
3800 apply (rule bin_nth_rsplit) |
|
3801 apply simp_all |
|
3802 apply (simp add : word_size rev_map) |
|
3803 apply (rule trans) |
|
3804 defer |
|
3805 apply (rule map_ident [THEN fun_cong]) |
|
3806 apply (rule refl [THEN map_cong]) |
|
3807 apply (simp add : word_ubin.eq_norm) |
|
3808 apply (erule bin_rsplit_size_sign [OF len_gt_0 refl]) |
|
3809 done |
|
3810 |
|
3811 lemma word_rcat_bl: "word_rcat wl == of_bl (concat (map to_bl wl))" |
|
3812 unfolding word_rcat_def to_bl_def' of_bl_def |
|
3813 by (clarsimp simp add : bin_rcat_bl) |
|
3814 |
|
3815 lemma size_rcat_lem': |
|
3816 "size (concat (map to_bl wl)) = length wl * size (hd wl)" |
|
3817 unfolding word_size by (induct wl) auto |
|
3818 |
|
3819 lemmas size_rcat_lem = size_rcat_lem' [unfolded word_size] |
|
3820 |
|
3821 lemmas td_gal_lt_len = len_gt_0 [THEN td_gal_lt, standard] |
|
3822 |
|
3823 lemma nth_rcat_lem' [rule_format] : |
|
3824 "sw = size (hd wl :: 'a :: len word) ==> (ALL n. n < size wl * sw --> |
|
3825 rev (concat (map to_bl wl)) ! n = |
|
3826 rev (to_bl (rev wl ! (n div sw))) ! (n mod sw))" |
|
3827 apply (unfold word_size) |
|
3828 apply (induct "wl") |
|
3829 apply clarsimp |
|
3830 apply (clarsimp simp add : nth_append size_rcat_lem) |
|
3831 apply (simp (no_asm_use) only: mult_Suc [symmetric] |
|
3832 td_gal_lt_len less_Suc_eq_le mod_div_equality') |
|
3833 apply clarsimp |
|
3834 done |
|
3835 |
|
3836 lemmas nth_rcat_lem = refl [THEN nth_rcat_lem', unfolded word_size] |
|
3837 |
|
3838 lemma test_bit_rcat: |
|
3839 "sw = size (hd wl :: 'a :: len word) ==> rc = word_rcat wl ==> rc !! n = |
|
3840 (n < size rc & n div sw < size wl & (rev wl) ! (n div sw) !! (n mod sw))" |
|
3841 apply (unfold word_rcat_bl word_size) |
|
3842 apply (clarsimp simp add : |
|
3843 test_bit_of_bl size_rcat_lem word_size td_gal_lt_len) |
|
3844 apply safe |
|
3845 apply (auto simp add : |
|
3846 test_bit_bl word_size td_gal_lt_len [THEN iffD2, THEN nth_rcat_lem]) |
|
3847 done |
|
3848 |
|
3849 lemma foldl_eq_foldr [rule_format] : |
|
3850 "ALL x. foldl op + x xs = foldr op + (x # xs) (0 :: 'a :: comm_monoid_add)" |
|
3851 by (induct xs) (auto simp add : add_assoc) |
|
3852 |
|
3853 lemmas test_bit_cong = arg_cong [where f = "test_bit", THEN fun_cong] |
|
3854 |
|
3855 lemmas test_bit_rsplit_alt = |
|
3856 trans [OF nth_rev_alt [THEN test_bit_cong] |
|
3857 test_bit_rsplit [OF refl asm_rl diff_Suc_less]] |
|
3858 |
|
3859 -- "lazy way of expressing that u and v, and su and sv, have same types" |
|
3860 lemma word_rsplit_len_indep': |
|
3861 "[u,v] = p ==> [su,sv] = q ==> word_rsplit u = su ==> |
|
3862 word_rsplit v = sv ==> length su = length sv" |
|
3863 apply (unfold word_rsplit_def) |
|
3864 apply (auto simp add : bin_rsplit_len_indep) |
|
3865 done |
|
3866 |
|
3867 lemmas word_rsplit_len_indep = word_rsplit_len_indep' [OF refl refl refl refl] |
|
3868 |
|
3869 lemma length_word_rsplit_size: |
|
3870 "n = len_of TYPE ('a :: len) ==> |
|
3871 (length (word_rsplit w :: 'a word list) <= m) = (size w <= m * n)" |
|
3872 apply (unfold word_rsplit_def word_size) |
|
3873 apply (clarsimp simp add : bin_rsplit_len_le) |
|
3874 done |
|
3875 |
|
3876 lemmas length_word_rsplit_lt_size = |
|
3877 length_word_rsplit_size [unfolded Not_eq_iff linorder_not_less [symmetric]] |
|
3878 |
|
3879 lemma length_word_rsplit_exp_size: |
|
3880 "n = len_of TYPE ('a :: len) ==> |
|
3881 length (word_rsplit w :: 'a word list) = (size w + n - 1) div n" |
|
3882 unfolding word_rsplit_def by (clarsimp simp add : word_size bin_rsplit_len) |
|
3883 |
|
3884 lemma length_word_rsplit_even_size: |
|
3885 "n = len_of TYPE ('a :: len) ==> size w = m * n ==> |
|
3886 length (word_rsplit w :: 'a word list) = m" |
|
3887 by (clarsimp simp add : length_word_rsplit_exp_size given_quot_alt) |
|
3888 |
|
3889 lemmas length_word_rsplit_exp_size' = refl [THEN length_word_rsplit_exp_size] |
|
3890 |
|
3891 (* alternative proof of word_rcat_rsplit *) |
|
3892 lemmas tdle = iffD2 [OF split_div_lemma refl, THEN conjunct1] |
|
3893 lemmas dtle = xtr4 [OF tdle mult_commute] |
|
3894 |
|
3895 lemma word_rcat_rsplit: "word_rcat (word_rsplit w) = w" |
|
3896 apply (rule word_eqI) |
|
3897 apply (clarsimp simp add : test_bit_rcat word_size) |
|
3898 apply (subst refl [THEN test_bit_rsplit]) |
|
3899 apply (simp_all add: word_size |
|
3900 refl [THEN length_word_rsplit_size [simplified not_less [symmetric], simplified]]) |
|
3901 apply safe |
|
3902 apply (erule xtr7, rule len_gt_0 [THEN dtle])+ |
|
3903 done |
|
3904 |
|
3905 lemma size_word_rsplit_rcat_size': |
|
3906 "word_rcat (ws :: 'a :: len word list) = frcw ==> |
|
3907 size frcw = length ws * len_of TYPE ('a) ==> |
|
3908 size (hd [word_rsplit frcw, ws]) = size ws" |
|
3909 apply (clarsimp simp add : word_size length_word_rsplit_exp_size') |
|
3910 apply (fast intro: given_quot_alt) |
|
3911 done |
|
3912 |
|
3913 lemmas size_word_rsplit_rcat_size = |
|
3914 size_word_rsplit_rcat_size' [simplified] |
|
3915 |
|
3916 lemma msrevs: |
|
3917 fixes n::nat |
|
3918 shows "0 < n \<Longrightarrow> (k * n + m) div n = m div n + k" |
|
3919 and "(k * n + m) mod n = m mod n" |
|
3920 by (auto simp: add_commute) |
|
3921 |
|
3922 lemma word_rsplit_rcat_size': |
|
3923 "word_rcat (ws :: 'a :: len word list) = frcw ==> |
|
3924 size frcw = length ws * len_of TYPE ('a) ==> word_rsplit frcw = ws" |
|
3925 apply (frule size_word_rsplit_rcat_size, assumption) |
|
3926 apply (clarsimp simp add : word_size) |
|
3927 apply (rule nth_equalityI, assumption) |
|
3928 apply clarsimp |
|
3929 apply (rule word_eqI) |
|
3930 apply (rule trans) |
|
3931 apply (rule test_bit_rsplit_alt) |
|
3932 apply (clarsimp simp: word_size)+ |
|
3933 apply (rule trans) |
|
3934 apply (rule test_bit_rcat [OF refl refl]) |
|
3935 apply (simp add : word_size msrevs) |
|
3936 apply (subst nth_rev) |
|
3937 apply arith |
|
3938 apply (simp add : le0 [THEN [2] xtr7, THEN diff_Suc_less]) |
|
3939 apply safe |
|
3940 apply (simp add : diff_mult_distrib) |
|
3941 apply (rule mpl_lem) |
|
3942 apply (cases "size ws") |
|
3943 apply simp_all |
|
3944 done |
|
3945 |
|
3946 lemmas word_rsplit_rcat_size = refl [THEN word_rsplit_rcat_size'] |
|
3947 |
|
3948 |
|
3949 subsection "Rotation" |
|
3950 |
|
3951 lemmas rotater_0' [simp] = rotater_def [where n = "0", simplified] |
|
3952 |
|
3953 lemmas word_rot_defs = word_roti_def word_rotr_def word_rotl_def |
|
3954 |
|
3955 lemma rotate_eq_mod: |
|
3956 "m mod length xs = n mod length xs ==> rotate m xs = rotate n xs" |
|
3957 apply (rule box_equals) |
|
3958 defer |
|
3959 apply (rule rotate_conv_mod [symmetric])+ |
|
3960 apply simp |
|
3961 done |
|
3962 |
|
3963 lemmas rotate_eqs [standard] = |
|
3964 trans [OF rotate0 [THEN fun_cong] id_apply] |
|
3965 rotate_rotate [symmetric] |
|
3966 rotate_id |
|
3967 rotate_conv_mod |
|
3968 rotate_eq_mod |
|
3969 |
|
3970 |
|
3971 subsubsection "Rotation of list to right" |
|
3972 |
|
3973 lemma rotate1_rl': "rotater1 (l @ [a]) = a # l" |
|
3974 unfolding rotater1_def by (cases l) auto |
|
3975 |
|
3976 lemma rotate1_rl [simp] : "rotater1 (rotate1 l) = l" |
|
3977 apply (unfold rotater1_def) |
|
3978 apply (cases "l") |
|
3979 apply (case_tac [2] "list") |
|
3980 apply auto |
|
3981 done |
|
3982 |
|
3983 lemma rotate1_lr [simp] : "rotate1 (rotater1 l) = l" |
|
3984 unfolding rotater1_def by (cases l) auto |
|
3985 |
|
3986 lemma rotater1_rev': "rotater1 (rev xs) = rev (rotate1 xs)" |
|
3987 apply (cases "xs") |
|
3988 apply (simp add : rotater1_def) |
|
3989 apply (simp add : rotate1_rl') |
|
3990 done |
|
3991 |
|
3992 lemma rotater_rev': "rotater n (rev xs) = rev (rotate n xs)" |
|
3993 unfolding rotater_def by (induct n) (auto intro: rotater1_rev') |
|
3994 |
|
3995 lemmas rotater_rev = rotater_rev' [where xs = "rev ys", simplified, standard] |
|
3996 |
|
3997 lemma rotater_drop_take: |
|
3998 "rotater n xs = |
|
3999 drop (length xs - n mod length xs) xs @ |
|
4000 take (length xs - n mod length xs) xs" |
|
4001 by (clarsimp simp add : rotater_rev rotate_drop_take rev_take rev_drop) |
|
4002 |
|
4003 lemma rotater_Suc [simp] : |
|
4004 "rotater (Suc n) xs = rotater1 (rotater n xs)" |
|
4005 unfolding rotater_def by auto |
|
4006 |
|
4007 lemma rotate_inv_plus [rule_format] : |
|
4008 "ALL k. k = m + n --> rotater k (rotate n xs) = rotater m xs & |
|
4009 rotate k (rotater n xs) = rotate m xs & |
|
4010 rotater n (rotate k xs) = rotate m xs & |
|
4011 rotate n (rotater k xs) = rotater m xs" |
|
4012 unfolding rotater_def rotate_def |
|
4013 by (induct n) (auto intro: funpow_swap1 [THEN trans]) |
|
4014 |
|
4015 lemmas rotate_inv_rel = le_add_diff_inverse2 [symmetric, THEN rotate_inv_plus] |
|
4016 |
|
4017 lemmas rotate_inv_eq = order_refl [THEN rotate_inv_rel, simplified] |
|
4018 |
|
4019 lemmas rotate_lr [simp] = rotate_inv_eq [THEN conjunct1, standard] |
|
4020 lemmas rotate_rl [simp] = |
|
4021 rotate_inv_eq [THEN conjunct2, THEN conjunct1, standard] |
|
4022 |
|
4023 lemma rotate_gal: "(rotater n xs = ys) = (rotate n ys = xs)" |
|
4024 by auto |
|
4025 |
|
4026 lemma rotate_gal': "(ys = rotater n xs) = (xs = rotate n ys)" |
|
4027 by auto |
|
4028 |
|
4029 lemma length_rotater [simp]: |
|
4030 "length (rotater n xs) = length xs" |
|
4031 by (simp add : rotater_rev) |
|
4032 |
|
4033 lemmas rrs0 = rotate_eqs [THEN restrict_to_left, |
|
4034 simplified rotate_gal [symmetric] rotate_gal' [symmetric], standard] |
|
4035 lemmas rrs1 = rrs0 [THEN refl [THEN rev_iffD1]] |
|
4036 lemmas rotater_eqs = rrs1 [simplified length_rotater, standard] |
|
4037 lemmas rotater_0 = rotater_eqs (1) |
|
4038 lemmas rotater_add = rotater_eqs (2) |
|
4039 |
|
4040 |
|
4041 subsubsection "map, map2, commuting with rotate(r)" |
|
4042 |
|
4043 lemma last_map: "xs ~= [] ==> last (map f xs) = f (last xs)" |
|
4044 by (induct xs) auto |
|
4045 |
|
4046 lemma butlast_map: |
|
4047 "xs ~= [] ==> butlast (map f xs) = map f (butlast xs)" |
|
4048 by (induct xs) auto |
|
4049 |
|
4050 lemma rotater1_map: "rotater1 (map f xs) = map f (rotater1 xs)" |
|
4051 unfolding rotater1_def |
|
4052 by (cases xs) (auto simp add: last_map butlast_map) |
|
4053 |
|
4054 lemma rotater_map: |
|
4055 "rotater n (map f xs) = map f (rotater n xs)" |
|
4056 unfolding rotater_def |
|
4057 by (induct n) (auto simp add : rotater1_map) |
|
4058 |
|
4059 lemma but_last_zip [rule_format] : |
|
4060 "ALL ys. length xs = length ys --> xs ~= [] --> |
|
4061 last (zip xs ys) = (last xs, last ys) & |
|
4062 butlast (zip xs ys) = zip (butlast xs) (butlast ys)" |
|
4063 apply (induct "xs") |
|
4064 apply auto |
|
4065 apply ((case_tac ys, auto simp: neq_Nil_conv)[1])+ |
|
4066 done |
|
4067 |
|
4068 lemma but_last_map2 [rule_format] : |
|
4069 "ALL ys. length xs = length ys --> xs ~= [] --> |
|
4070 last (map2 f xs ys) = f (last xs) (last ys) & |
|
4071 butlast (map2 f xs ys) = map2 f (butlast xs) (butlast ys)" |
|
4072 apply (induct "xs") |
|
4073 apply auto |
|
4074 apply (unfold map2_def) |
|
4075 apply ((case_tac ys, auto simp: neq_Nil_conv)[1])+ |
|
4076 done |
|
4077 |
|
4078 lemma rotater1_zip: |
|
4079 "length xs = length ys ==> |
|
4080 rotater1 (zip xs ys) = zip (rotater1 xs) (rotater1 ys)" |
|
4081 apply (unfold rotater1_def) |
|
4082 apply (cases "xs") |
|
4083 apply auto |
|
4084 apply ((case_tac ys, auto simp: neq_Nil_conv but_last_zip)[1])+ |
|
4085 done |
|
4086 |
|
4087 lemma rotater1_map2: |
|
4088 "length xs = length ys ==> |
|
4089 rotater1 (map2 f xs ys) = map2 f (rotater1 xs) (rotater1 ys)" |
|
4090 unfolding map2_def by (simp add: rotater1_map rotater1_zip) |
|
4091 |
|
4092 lemmas lrth = |
|
4093 box_equals [OF asm_rl length_rotater [symmetric] |
|
4094 length_rotater [symmetric], |
|
4095 THEN rotater1_map2] |
|
4096 |
|
4097 lemma rotater_map2: |
|
4098 "length xs = length ys ==> |
|
4099 rotater n (map2 f xs ys) = map2 f (rotater n xs) (rotater n ys)" |
|
4100 by (induct n) (auto intro!: lrth) |
|
4101 |
|
4102 lemma rotate1_map2: |
|
4103 "length xs = length ys ==> |
|
4104 rotate1 (map2 f xs ys) = map2 f (rotate1 xs) (rotate1 ys)" |
|
4105 apply (unfold map2_def) |
|
4106 apply (cases xs) |
|
4107 apply (cases ys, auto simp add : rotate1_def)+ |
|
4108 done |
|
4109 |
|
4110 lemmas lth = box_equals [OF asm_rl length_rotate [symmetric] |
|
4111 length_rotate [symmetric], THEN rotate1_map2] |
|
4112 |
|
4113 lemma rotate_map2: |
|
4114 "length xs = length ys ==> |
|
4115 rotate n (map2 f xs ys) = map2 f (rotate n xs) (rotate n ys)" |
|
4116 by (induct n) (auto intro!: lth) |
|
4117 |
|
4118 |
|
4119 -- "corresponding equalities for word rotation" |
|
4120 |
|
4121 lemma to_bl_rotl: |
|
4122 "to_bl (word_rotl n w) = rotate n (to_bl w)" |
|
4123 by (simp add: word_bl.Abs_inverse' word_rotl_def) |
|
4124 |
|
4125 lemmas blrs0 = rotate_eqs [THEN to_bl_rotl [THEN trans]] |
|
4126 |
|
4127 lemmas word_rotl_eqs = |
|
4128 blrs0 [simplified word_bl.Rep' word_bl.Rep_inject to_bl_rotl [symmetric]] |
|
4129 |
|
4130 lemma to_bl_rotr: |
|
4131 "to_bl (word_rotr n w) = rotater n (to_bl w)" |
|
4132 by (simp add: word_bl.Abs_inverse' word_rotr_def) |
|
4133 |
|
4134 lemmas brrs0 = rotater_eqs [THEN to_bl_rotr [THEN trans]] |
|
4135 |
|
4136 lemmas word_rotr_eqs = |
|
4137 brrs0 [simplified word_bl.Rep' word_bl.Rep_inject to_bl_rotr [symmetric]] |
|
4138 |
|
4139 declare word_rotr_eqs (1) [simp] |
|
4140 declare word_rotl_eqs (1) [simp] |
|
4141 |
|
4142 lemma |
|
4143 word_rot_rl [simp]: |
|
4144 "word_rotl k (word_rotr k v) = v" and |
|
4145 word_rot_lr [simp]: |
|
4146 "word_rotr k (word_rotl k v) = v" |
|
4147 by (auto simp add: to_bl_rotr to_bl_rotl word_bl.Rep_inject [symmetric]) |
|
4148 |
|
4149 lemma |
|
4150 word_rot_gal: |
|
4151 "(word_rotr n v = w) = (word_rotl n w = v)" and |
|
4152 word_rot_gal': |
|
4153 "(w = word_rotr n v) = (v = word_rotl n w)" |
|
4154 by (auto simp: to_bl_rotr to_bl_rotl word_bl.Rep_inject [symmetric] |
|
4155 dest: sym) |
|
4156 |
|
4157 lemma word_rotr_rev: |
|
4158 "word_rotr n w = word_reverse (word_rotl n (word_reverse w))" |
|
4159 by (simp add: word_bl.Rep_inject [symmetric] to_bl_word_rev |
|
4160 to_bl_rotr to_bl_rotl rotater_rev) |
|
4161 |
|
4162 lemma word_roti_0 [simp]: "word_roti 0 w = w" |
|
4163 by (unfold word_rot_defs) auto |
|
4164 |
|
4165 lemmas abl_cong = arg_cong [where f = "of_bl"] |
|
4166 |
|
4167 lemma word_roti_add: |
|
4168 "word_roti (m + n) w = word_roti m (word_roti n w)" |
|
4169 proof - |
|
4170 have rotater_eq_lem: |
|
4171 "\<And>m n xs. m = n ==> rotater m xs = rotater n xs" |
|
4172 by auto |
|
4173 |
|
4174 have rotate_eq_lem: |
|
4175 "\<And>m n xs. m = n ==> rotate m xs = rotate n xs" |
|
4176 by auto |
|
4177 |
|
4178 note rpts [symmetric, standard] = |
|
4179 rotate_inv_plus [THEN conjunct1] |
|
4180 rotate_inv_plus [THEN conjunct2, THEN conjunct1] |
|
4181 rotate_inv_plus [THEN conjunct2, THEN conjunct2, THEN conjunct1] |
|
4182 rotate_inv_plus [THEN conjunct2, THEN conjunct2, THEN conjunct2] |
|
4183 |
|
4184 note rrp = trans [symmetric, OF rotate_rotate rotate_eq_lem] |
|
4185 note rrrp = trans [symmetric, OF rotater_add [symmetric] rotater_eq_lem] |
|
4186 |
|
4187 show ?thesis |
|
4188 apply (unfold word_rot_defs) |
|
4189 apply (simp only: split: split_if) |
|
4190 apply (safe intro!: abl_cong) |
|
4191 apply (simp_all only: to_bl_rotl [THEN word_bl.Rep_inverse'] |
|
4192 to_bl_rotl |
|
4193 to_bl_rotr [THEN word_bl.Rep_inverse'] |
|
4194 to_bl_rotr) |
|
4195 apply (rule rrp rrrp rpts, |
|
4196 simp add: nat_add_distrib [symmetric] |
|
4197 nat_diff_distrib [symmetric])+ |
|
4198 done |
|
4199 qed |
|
4200 |
|
4201 lemma word_roti_conv_mod': "word_roti n w = word_roti (n mod int (size w)) w" |
|
4202 apply (unfold word_rot_defs) |
|
4203 apply (cut_tac y="size w" in gt_or_eq_0) |
|
4204 apply (erule disjE) |
|
4205 apply simp_all |
|
4206 apply (safe intro!: abl_cong) |
|
4207 apply (rule rotater_eqs) |
|
4208 apply (simp add: word_size nat_mod_distrib) |
|
4209 apply (simp add: rotater_add [symmetric] rotate_gal [symmetric]) |
|
4210 apply (rule rotater_eqs) |
|
4211 apply (simp add: word_size nat_mod_distrib) |
|
4212 apply (rule int_eq_0_conv [THEN iffD1]) |
|
4213 apply (simp only: zmod_int zadd_int [symmetric]) |
|
4214 apply (simp add: rdmods) |
|
4215 done |
|
4216 |
|
4217 lemmas word_roti_conv_mod = word_roti_conv_mod' [unfolded word_size] |
|
4218 |
|
4219 |
|
4220 subsubsection "Word rotation commutes with bit-wise operations" |
|
4221 |
|
4222 (* using locale to not pollute lemma namespace *) |
|
4223 locale word_rotate |
|
4224 |
|
4225 context word_rotate |
|
4226 begin |
|
4227 |
|
4228 lemmas word_rot_defs' = to_bl_rotl to_bl_rotr |
|
4229 |
|
4230 lemmas blwl_syms [symmetric] = bl_word_not bl_word_and bl_word_or bl_word_xor |
|
4231 |
|
4232 lemmas lbl_lbl = trans [OF word_bl.Rep' word_bl.Rep' [symmetric]] |
|
4233 |
|
4234 lemmas ths_map2 [OF lbl_lbl] = rotate_map2 rotater_map2 |
|
4235 |
|
4236 lemmas ths_map [where xs = "to_bl v", standard] = rotate_map rotater_map |
|
4237 |
|
4238 lemmas th1s [simplified word_rot_defs' [symmetric]] = ths_map2 ths_map |
|
4239 |
|
4240 lemma word_rot_logs: |
|
4241 "word_rotl n (NOT v) = NOT word_rotl n v" |
|
4242 "word_rotr n (NOT v) = NOT word_rotr n v" |
|
4243 "word_rotl n (x AND y) = word_rotl n x AND word_rotl n y" |
|
4244 "word_rotr n (x AND y) = word_rotr n x AND word_rotr n y" |
|
4245 "word_rotl n (x OR y) = word_rotl n x OR word_rotl n y" |
|
4246 "word_rotr n (x OR y) = word_rotr n x OR word_rotr n y" |
|
4247 "word_rotl n (x XOR y) = word_rotl n x XOR word_rotl n y" |
|
4248 "word_rotr n (x XOR y) = word_rotr n x XOR word_rotr n y" |
|
4249 by (rule word_bl.Rep_eqD, |
|
4250 rule word_rot_defs' [THEN trans], |
|
4251 simp only: blwl_syms [symmetric], |
|
4252 rule th1s [THEN trans], |
|
4253 rule refl)+ |
|
4254 end |
|
4255 |
|
4256 lemmas word_rot_logs = word_rotate.word_rot_logs |
|
4257 |
|
4258 lemmas bl_word_rotl_dt = trans [OF to_bl_rotl rotate_drop_take, |
|
4259 simplified word_bl.Rep', standard] |
|
4260 |
|
4261 lemmas bl_word_rotr_dt = trans [OF to_bl_rotr rotater_drop_take, |
|
4262 simplified word_bl.Rep', standard] |
|
4263 |
|
4264 lemma bl_word_roti_dt': |
|
4265 "n = nat ((- i) mod int (size (w :: 'a :: len word))) ==> |
|
4266 to_bl (word_roti i w) = drop n (to_bl w) @ take n (to_bl w)" |
|
4267 apply (unfold word_roti_def) |
|
4268 apply (simp add: bl_word_rotl_dt bl_word_rotr_dt word_size) |
|
4269 apply safe |
|
4270 apply (simp add: zmod_zminus1_eq_if) |
|
4271 apply safe |
|
4272 apply (simp add: nat_mult_distrib) |
|
4273 apply (simp add: nat_diff_distrib [OF pos_mod_sign pos_mod_conj |
|
4274 [THEN conjunct2, THEN order_less_imp_le]] |
|
4275 nat_mod_distrib) |
|
4276 apply (simp add: nat_mod_distrib) |
|
4277 done |
|
4278 |
|
4279 lemmas bl_word_roti_dt = bl_word_roti_dt' [unfolded word_size] |
|
4280 |
|
4281 lemmas word_rotl_dt = bl_word_rotl_dt |
|
4282 [THEN word_bl.Rep_inverse' [symmetric], standard] |
|
4283 lemmas word_rotr_dt = bl_word_rotr_dt |
|
4284 [THEN word_bl.Rep_inverse' [symmetric], standard] |
|
4285 lemmas word_roti_dt = bl_word_roti_dt |
|
4286 [THEN word_bl.Rep_inverse' [symmetric], standard] |
|
4287 |
|
4288 lemma word_rotx_0 [simp] : "word_rotr i 0 = 0 & word_rotl i 0 = 0" |
|
4289 by (simp add : word_rotr_dt word_rotl_dt to_bl_0 replicate_add [symmetric]) |
|
4290 |
|
4291 lemma word_roti_0' [simp] : "word_roti n 0 = 0" |
|
4292 unfolding word_roti_def by auto |
|
4293 |
|
4294 lemmas word_rotr_dt_no_bin' [simp] = |
|
4295 word_rotr_dt [where w="number_of w", unfolded to_bl_no_bin, standard] |
|
4296 |
|
4297 lemmas word_rotl_dt_no_bin' [simp] = |
|
4298 word_rotl_dt [where w="number_of w", unfolded to_bl_no_bin, standard] |
|
4299 |
|
4300 declare word_roti_def [simp] |
|
4301 |
|
4302 |
|
4303 subsection {* Miscellaneous *} |
|
4304 |
|
4305 declare of_nat_2p [simp] |
|
4306 |
|
4307 lemma word_int_cases: |
|
4308 "\<lbrakk>\<And>n. \<lbrakk>(x ::'a::len0 word) = word_of_int n; 0 \<le> n; n < 2^len_of TYPE('a)\<rbrakk> \<Longrightarrow> P\<rbrakk> |
|
4309 \<Longrightarrow> P" |
|
4310 by (cases x rule: word_uint.Abs_cases) (simp add: uints_num) |
|
4311 |
|
4312 lemma word_nat_cases [cases type: word]: |
|
4313 "\<lbrakk>\<And>n. \<lbrakk>(x ::'a::len word) = of_nat n; n < 2^len_of TYPE('a)\<rbrakk> \<Longrightarrow> P\<rbrakk> |
|
4314 \<Longrightarrow> P" |
|
4315 by (cases x rule: word_unat.Abs_cases) (simp add: unats_def) |
|
4316 |
|
4317 lemma max_word_eq: |
|
4318 "(max_word::'a::len word) = 2^len_of TYPE('a) - 1" |
|
4319 by (simp add: max_word_def word_of_int_hom_syms word_of_int_2p) |
|
4320 |
|
4321 lemma max_word_max [simp,intro!]: |
|
4322 "n \<le> max_word" |
|
4323 by (cases n rule: word_int_cases) |
|
4324 (simp add: max_word_def word_le_def int_word_uint int_mod_eq') |
|
4325 |
|
4326 lemma word_of_int_2p_len: |
|
4327 "word_of_int (2 ^ len_of TYPE('a)) = (0::'a::len0 word)" |
|
4328 by (subst word_uint.Abs_norm [symmetric]) |
|
4329 (simp add: word_of_int_hom_syms) |
|
4330 |
|
4331 lemma word_pow_0: |
|
4332 "(2::'a::len word) ^ len_of TYPE('a) = 0" |
|
4333 proof - |
|
4334 have "word_of_int (2 ^ len_of TYPE('a)) = (0::'a word)" |
|
4335 by (rule word_of_int_2p_len) |
|
4336 thus ?thesis by (simp add: word_of_int_2p) |
|
4337 qed |
|
4338 |
|
4339 lemma max_word_wrap: "x + 1 = 0 \<Longrightarrow> x = max_word" |
|
4340 apply (simp add: max_word_eq) |
|
4341 apply uint_arith |
|
4342 apply auto |
|
4343 apply (simp add: word_pow_0) |
|
4344 done |
|
4345 |
|
4346 lemma max_word_minus: |
|
4347 "max_word = (-1::'a::len word)" |
|
4348 proof - |
|
4349 have "-1 + 1 = (0::'a word)" by simp |
|
4350 thus ?thesis by (rule max_word_wrap [symmetric]) |
|
4351 qed |
|
4352 |
|
4353 lemma max_word_bl [simp]: |
|
4354 "to_bl (max_word::'a::len word) = replicate (len_of TYPE('a)) True" |
|
4355 by (subst max_word_minus to_bl_n1)+ simp |
|
4356 |
|
4357 lemma max_test_bit [simp]: |
|
4358 "(max_word::'a::len word) !! n = (n < len_of TYPE('a))" |
|
4359 by (auto simp add: test_bit_bl word_size) |
|
4360 |
|
4361 lemma word_and_max [simp]: |
|
4362 "x AND max_word = x" |
|
4363 by (rule word_eqI) (simp add: word_ops_nth_size word_size) |
|
4364 |
|
4365 lemma word_or_max [simp]: |
|
4366 "x OR max_word = max_word" |
|
4367 by (rule word_eqI) (simp add: word_ops_nth_size word_size) |
|
4368 |
|
4369 lemma word_ao_dist2: |
|
4370 "x AND (y OR z) = x AND y OR x AND (z::'a::len0 word)" |
|
4371 by (rule word_eqI) (auto simp add: word_ops_nth_size word_size) |
|
4372 |
|
4373 lemma word_oa_dist2: |
|
4374 "x OR y AND z = (x OR y) AND (x OR (z::'a::len0 word))" |
|
4375 by (rule word_eqI) (auto simp add: word_ops_nth_size word_size) |
|
4376 |
|
4377 lemma word_and_not [simp]: |
|
4378 "x AND NOT x = (0::'a::len0 word)" |
|
4379 by (rule word_eqI) (auto simp add: word_ops_nth_size word_size) |
|
4380 |
|
4381 lemma word_or_not [simp]: |
|
4382 "x OR NOT x = max_word" |
|
4383 by (rule word_eqI) (auto simp add: word_ops_nth_size word_size) |
|
4384 |
|
4385 lemma word_boolean: |
|
4386 "boolean (op AND) (op OR) bitNOT 0 max_word" |
|
4387 apply (rule boolean.intro) |
|
4388 apply (rule word_bw_assocs) |
|
4389 apply (rule word_bw_assocs) |
|
4390 apply (rule word_bw_comms) |
|
4391 apply (rule word_bw_comms) |
|
4392 apply (rule word_ao_dist2) |
|
4393 apply (rule word_oa_dist2) |
|
4394 apply (rule word_and_max) |
|
4395 apply (rule word_log_esimps) |
|
4396 apply (rule word_and_not) |
|
4397 apply (rule word_or_not) |
|
4398 done |
|
4399 |
|
4400 interpretation word_bool_alg: |
|
4401 boolean "op AND" "op OR" bitNOT 0 max_word |
|
4402 by (rule word_boolean) |
|
4403 |
|
4404 lemma word_xor_and_or: |
|
4405 "x XOR y = x AND NOT y OR NOT x AND (y::'a::len0 word)" |
|
4406 by (rule word_eqI) (auto simp add: word_ops_nth_size word_size) |
|
4407 |
|
4408 interpretation word_bool_alg: |
|
4409 boolean_xor "op AND" "op OR" bitNOT 0 max_word "op XOR" |
|
4410 apply (rule boolean_xor.intro) |
|
4411 apply (rule word_boolean) |
|
4412 apply (rule boolean_xor_axioms.intro) |
|
4413 apply (rule word_xor_and_or) |
|
4414 done |
|
4415 |
|
4416 lemma shiftr_x_0 [iff]: |
|
4417 "(x::'a::len0 word) >> 0 = x" |
|
4418 by (simp add: shiftr_bl) |
|
4419 |
|
4420 lemma shiftl_x_0 [simp]: |
|
4421 "(x :: 'a :: len word) << 0 = x" |
|
4422 by (simp add: shiftl_t2n) |
|
4423 |
|
4424 lemma shiftl_1 [simp]: |
|
4425 "(1::'a::len word) << n = 2^n" |
|
4426 by (simp add: shiftl_t2n) |
|
4427 |
|
4428 lemma uint_lt_0 [simp]: |
|
4429 "uint x < 0 = False" |
|
4430 by (simp add: linorder_not_less) |
|
4431 |
|
4432 lemma shiftr1_1 [simp]: |
|
4433 "shiftr1 (1::'a::len word) = 0" |
|
4434 by (simp add: shiftr1_def word_0_alt) |
|
4435 |
|
4436 lemma shiftr_1[simp]: |
|
4437 "(1::'a::len word) >> n = (if n = 0 then 1 else 0)" |
|
4438 by (induct n) (auto simp: shiftr_def) |
|
4439 |
|
4440 lemma word_less_1 [simp]: |
|
4441 "((x::'a::len word) < 1) = (x = 0)" |
|
4442 by (simp add: word_less_nat_alt unat_0_iff) |
|
4443 |
|
4444 lemma to_bl_mask: |
|
4445 "to_bl (mask n :: 'a::len word) = |
|
4446 replicate (len_of TYPE('a) - n) False @ |
|
4447 replicate (min (len_of TYPE('a)) n) True" |
|
4448 by (simp add: mask_bl word_rep_drop min_def) |
|
4449 |
|
4450 lemma map_replicate_True: |
|
4451 "n = length xs ==> |
|
4452 map (\<lambda>(x,y). x & y) (zip xs (replicate n True)) = xs" |
|
4453 by (induct xs arbitrary: n) auto |
|
4454 |
|
4455 lemma map_replicate_False: |
|
4456 "n = length xs ==> map (\<lambda>(x,y). x & y) |
|
4457 (zip xs (replicate n False)) = replicate n False" |
|
4458 by (induct xs arbitrary: n) auto |
|
4459 |
|
4460 lemma bl_and_mask: |
|
4461 fixes w :: "'a::len word" |
|
4462 fixes n |
|
4463 defines "n' \<equiv> len_of TYPE('a) - n" |
|
4464 shows "to_bl (w AND mask n) = replicate n' False @ drop n' (to_bl w)" |
|
4465 proof - |
|
4466 note [simp] = map_replicate_True map_replicate_False |
|
4467 have "to_bl (w AND mask n) = |
|
4468 map2 op & (to_bl w) (to_bl (mask n::'a::len word))" |
|
4469 by (simp add: bl_word_and) |
|
4470 also |
|
4471 have "to_bl w = take n' (to_bl w) @ drop n' (to_bl w)" by simp |
|
4472 also |
|
4473 have "map2 op & \<dots> (to_bl (mask n::'a::len word)) = |
|
4474 replicate n' False @ drop n' (to_bl w)" |
|
4475 unfolding to_bl_mask n'_def map2_def |
|
4476 by (subst zip_append) auto |
|
4477 finally |
|
4478 show ?thesis . |
|
4479 qed |
|
4480 |
|
4481 lemma drop_rev_takefill: |
|
4482 "length xs \<le> n ==> |
|
4483 drop (n - length xs) (rev (takefill False n (rev xs))) = xs" |
|
4484 by (simp add: takefill_alt rev_take) |
|
4485 |
|
4486 lemma map_nth_0 [simp]: |
|
4487 "map (op !! (0::'a::len0 word)) xs = replicate (length xs) False" |
|
4488 by (induct xs) auto |
|
4489 |
|
4490 lemma uint_plus_if_size: |
|
4491 "uint (x + y) = |
|
4492 (if uint x + uint y < 2^size x then |
|
4493 uint x + uint y |
|
4494 else |
|
4495 uint x + uint y - 2^size x)" |
|
4496 by (simp add: word_arith_alts int_word_uint mod_add_if_z |
|
4497 word_size) |
|
4498 |
|
4499 lemma unat_plus_if_size: |
|
4500 "unat (x + (y::'a::len word)) = |
|
4501 (if unat x + unat y < 2^size x then |
|
4502 unat x + unat y |
|
4503 else |
|
4504 unat x + unat y - 2^size x)" |
|
4505 apply (subst word_arith_nat_defs) |
|
4506 apply (subst unat_of_nat) |
|
4507 apply (simp add: mod_nat_add word_size) |
|
4508 done |
|
4509 |
|
4510 lemma word_neq_0_conv [simp]: |
|
4511 fixes w :: "'a :: len word" |
|
4512 shows "(w \<noteq> 0) = (0 < w)" |
|
4513 proof - |
|
4514 have "0 \<le> w" by (rule word_zero_le) |
|
4515 thus ?thesis by (auto simp add: word_less_def) |
|
4516 qed |
|
4517 |
|
4518 lemma max_lt: |
|
4519 "unat (max a b div c) = unat (max a b) div unat (c:: 'a :: len word)" |
|
4520 apply (subst word_arith_nat_defs) |
|
4521 apply (subst word_unat.eq_norm) |
|
4522 apply (subst mod_if) |
|
4523 apply clarsimp |
|
4524 apply (erule notE) |
|
4525 apply (insert div_le_dividend [of "unat (max a b)" "unat c"]) |
|
4526 apply (erule order_le_less_trans) |
|
4527 apply (insert unat_lt2p [of "max a b"]) |
|
4528 apply simp |
|
4529 done |
|
4530 |
|
4531 lemma uint_sub_if_size: |
|
4532 "uint (x - y) = |
|
4533 (if uint y \<le> uint x then |
|
4534 uint x - uint y |
|
4535 else |
|
4536 uint x - uint y + 2^size x)" |
|
4537 by (simp add: word_arith_alts int_word_uint mod_sub_if_z |
|
4538 word_size) |
|
4539 |
|
4540 lemma unat_sub: |
|
4541 "b <= a ==> unat (a - b) = unat a - unat b" |
|
4542 by (simp add: unat_def uint_sub_if_size word_le_def nat_diff_distrib) |
|
4543 |
|
4544 lemmas word_less_sub1_numberof [simp] = |
|
4545 word_less_sub1 [of "number_of w", standard] |
|
4546 lemmas word_le_sub1_numberof [simp] = |
|
4547 word_le_sub1 [of "number_of w", standard] |
|
4548 |
|
4549 lemma word_of_int_minus: |
|
4550 "word_of_int (2^len_of TYPE('a) - i) = (word_of_int (-i)::'a::len word)" |
|
4551 proof - |
|
4552 have x: "2^len_of TYPE('a) - i = -i + 2^len_of TYPE('a)" by simp |
|
4553 show ?thesis |
|
4554 apply (subst x) |
|
4555 apply (subst word_uint.Abs_norm [symmetric], subst mod_add_self2) |
|
4556 apply simp |
|
4557 done |
|
4558 qed |
|
4559 |
|
4560 lemmas word_of_int_inj = |
|
4561 word_uint.Abs_inject [unfolded uints_num, simplified] |
|
4562 |
|
4563 lemma word_le_less_eq: |
|
4564 "(x ::'z::len word) \<le> y = (x = y \<or> x < y)" |
|
4565 by (auto simp add: word_less_def) |
|
4566 |
|
4567 lemma mod_plus_cong: |
|
4568 assumes 1: "(b::int) = b'" |
|
4569 and 2: "x mod b' = x' mod b'" |
|
4570 and 3: "y mod b' = y' mod b'" |
|
4571 and 4: "x' + y' = z'" |
|
4572 shows "(x + y) mod b = z' mod b'" |
|
4573 proof - |
|
4574 from 1 2[symmetric] 3[symmetric] have "(x + y) mod b = (x' mod b' + y' mod b') mod b'" |
|
4575 by (simp add: mod_add_eq[symmetric]) |
|
4576 also have "\<dots> = (x' + y') mod b'" |
|
4577 by (simp add: mod_add_eq[symmetric]) |
|
4578 finally show ?thesis by (simp add: 4) |
|
4579 qed |
|
4580 |
|
4581 lemma mod_minus_cong: |
|
4582 assumes 1: "(b::int) = b'" |
|
4583 and 2: "x mod b' = x' mod b'" |
|
4584 and 3: "y mod b' = y' mod b'" |
|
4585 and 4: "x' - y' = z'" |
|
4586 shows "(x - y) mod b = z' mod b'" |
|
4587 using assms |
|
4588 apply (subst zmod_zsub_left_eq) |
|
4589 apply (subst zmod_zsub_right_eq) |
|
4590 apply (simp add: zmod_zsub_left_eq [symmetric] zmod_zsub_right_eq [symmetric]) |
|
4591 done |
|
4592 |
|
4593 lemma word_induct_less: |
|
4594 "\<lbrakk>P (0::'a::len word); \<And>n. \<lbrakk>n < m; P n\<rbrakk> \<Longrightarrow> P (1 + n)\<rbrakk> \<Longrightarrow> P m" |
|
4595 apply (cases m) |
|
4596 apply atomize |
|
4597 apply (erule rev_mp)+ |
|
4598 apply (rule_tac x=m in spec) |
|
4599 apply (induct_tac n) |
|
4600 apply simp |
|
4601 apply clarsimp |
|
4602 apply (erule impE) |
|
4603 apply clarsimp |
|
4604 apply (erule_tac x=n in allE) |
|
4605 apply (erule impE) |
|
4606 apply (simp add: unat_arith_simps) |
|
4607 apply (clarsimp simp: unat_of_nat) |
|
4608 apply simp |
|
4609 apply (erule_tac x="of_nat na" in allE) |
|
4610 apply (erule impE) |
|
4611 apply (simp add: unat_arith_simps) |
|
4612 apply (clarsimp simp: unat_of_nat) |
|
4613 apply simp |
|
4614 done |
|
4615 |
|
4616 lemma word_induct: |
|
4617 "\<lbrakk>P (0::'a::len word); \<And>n. P n \<Longrightarrow> P (1 + n)\<rbrakk> \<Longrightarrow> P m" |
|
4618 by (erule word_induct_less, simp) |
|
4619 |
|
4620 lemma word_induct2 [induct type]: |
|
4621 "\<lbrakk>P 0; \<And>n. \<lbrakk>1 + n \<noteq> 0; P n\<rbrakk> \<Longrightarrow> P (1 + n)\<rbrakk> \<Longrightarrow> P (n::'b::len word)" |
|
4622 apply (rule word_induct, simp) |
|
4623 apply (case_tac "1+n = 0", auto) |
|
4624 done |
|
4625 |
|
4626 definition word_rec :: "'a \<Rightarrow> ('b::len word \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'b word \<Rightarrow> 'a" where |
|
4627 "word_rec forZero forSuc n \<equiv> nat_rec forZero (forSuc \<circ> of_nat) (unat n)" |
|
4628 |
|
4629 lemma word_rec_0: "word_rec z s 0 = z" |
|
4630 by (simp add: word_rec_def) |
|
4631 |
|
4632 lemma word_rec_Suc: |
|
4633 "1 + n \<noteq> (0::'a::len word) \<Longrightarrow> word_rec z s (1 + n) = s n (word_rec z s n)" |
|
4634 apply (simp add: word_rec_def unat_word_ariths) |
|
4635 apply (subst nat_mod_eq') |
|
4636 apply (cut_tac x=n in unat_lt2p) |
|
4637 apply (drule Suc_mono) |
|
4638 apply (simp add: less_Suc_eq_le) |
|
4639 apply (simp only: order_less_le, simp) |
|
4640 apply (erule contrapos_pn, simp) |
|
4641 apply (drule arg_cong[where f=of_nat]) |
|
4642 apply simp |
|
4643 apply (subst (asm) word_unat.Rep_inverse[of n]) |
|
4644 apply simp |
|
4645 apply simp |
|
4646 done |
|
4647 |
|
4648 lemma word_rec_Pred: |
|
4649 "n \<noteq> 0 \<Longrightarrow> word_rec z s n = s (n - 1) (word_rec z s (n - 1))" |
|
4650 apply (rule subst[where t="n" and s="1 + (n - 1)"]) |
|
4651 apply simp |
|
4652 apply (subst word_rec_Suc) |
|
4653 apply simp |
|
4654 apply simp |
|
4655 done |
|
4656 |
|
4657 lemma word_rec_in: |
|
4658 "f (word_rec z (\<lambda>_. f) n) = word_rec (f z) (\<lambda>_. f) n" |
|
4659 by (induct n) (simp_all add: word_rec_0 word_rec_Suc) |
|
4660 |
|
4661 lemma word_rec_in2: |
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4662 "f n (word_rec z f n) = word_rec (f 0 z) (f \<circ> op + 1) n" |
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4663 by (induct n) (simp_all add: word_rec_0 word_rec_Suc) |
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4664 |
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4665 lemma word_rec_twice: |
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4666 "m \<le> n \<Longrightarrow> word_rec z f n = word_rec (word_rec z f (n - m)) (f \<circ> op + (n - m)) m" |
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4667 apply (erule rev_mp) |
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4668 apply (rule_tac x=z in spec) |
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4669 apply (rule_tac x=f in spec) |
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4670 apply (induct n) |
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4671 apply (simp add: word_rec_0) |
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4672 apply clarsimp |
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4673 apply (rule_tac t="1 + n - m" and s="1 + (n - m)" in subst) |
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4674 apply simp |
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4675 apply (case_tac "1 + (n - m) = 0") |
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4676 apply (simp add: word_rec_0) |
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4677 apply (rule_tac f = "word_rec ?a ?b" in arg_cong) |
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4678 apply (rule_tac t="m" and s="m + (1 + (n - m))" in subst) |
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4679 apply simp |
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4680 apply (simp (no_asm_use)) |
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4681 apply (simp add: word_rec_Suc word_rec_in2) |
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4682 apply (erule impE) |
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4683 apply uint_arith |
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4684 apply (drule_tac x="x \<circ> op + 1" in spec) |
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4685 apply (drule_tac x="x 0 xa" in spec) |
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4686 apply simp |
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4687 apply (rule_tac t="\<lambda>a. x (1 + (n - m + a))" and s="\<lambda>a. x (1 + (n - m) + a)" |
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4688 in subst) |
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4689 apply (clarsimp simp add: expand_fun_eq) |
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4690 apply (rule_tac t="(1 + (n - m + xb))" and s="1 + (n - m) + xb" in subst) |
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4691 apply simp |
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4692 apply (rule refl) |
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4693 apply (rule refl) |
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4694 done |
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4695 |
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4696 lemma word_rec_id: "word_rec z (\<lambda>_. id) n = z" |
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4697 by (induct n) (auto simp add: word_rec_0 word_rec_Suc) |
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4698 |
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4699 lemma word_rec_id_eq: "\<forall>m < n. f m = id \<Longrightarrow> word_rec z f n = z" |
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4700 apply (erule rev_mp) |
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4701 apply (induct n) |
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4702 apply (auto simp add: word_rec_0 word_rec_Suc) |
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4703 apply (drule spec, erule mp) |
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4704 apply uint_arith |
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4705 apply (drule_tac x=n in spec, erule impE) |
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4706 apply uint_arith |
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4707 apply simp |
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4708 done |
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4709 |
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4710 lemma word_rec_max: |
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4711 "\<forall>m\<ge>n. m \<noteq> -1 \<longrightarrow> f m = id \<Longrightarrow> word_rec z f -1 = word_rec z f n" |
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4712 apply (subst word_rec_twice[where n="-1" and m="-1 - n"]) |
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4713 apply simp |
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4714 apply simp |
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4715 apply (rule word_rec_id_eq) |
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4716 apply clarsimp |
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4717 apply (drule spec, rule mp, erule mp) |
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4718 apply (rule word_plus_mono_right2[OF _ order_less_imp_le]) |
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4719 prefer 2 |
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4720 apply assumption |
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4721 apply simp |
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4722 apply (erule contrapos_pn) |
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4723 apply simp |
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4724 apply (drule arg_cong[where f="\<lambda>x. x - n"]) |
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4725 apply simp |
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4726 done |
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4727 |
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4728 lemma unatSuc: |
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4729 "1 + n \<noteq> (0::'a::len word) \<Longrightarrow> unat (1 + n) = Suc (unat n)" |
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4730 by unat_arith |
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4731 |
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4732 |
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4733 lemmas word_no_1 [simp] = word_1_no [symmetric, unfolded BIT_simps] |
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4734 lemmas word_no_0 [simp] = word_0_no [symmetric] |
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4735 |
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4736 declare word_0_bl [simp] |
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4737 declare bin_to_bl_def [simp] |
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4738 declare to_bl_0 [simp] |
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4739 declare of_bl_True [simp] |
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4740 |
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4741 |
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4742 use "~~/src/HOL/Tools/SMT/smt_word.ML" |
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4743 |
12 setup {* SMT_Word.setup *} |
4744 setup {* SMT_Word.setup *} |
13 |
4745 |
14 text {* see @{text "Examples/WordExamples.thy"} for examples *} |
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15 |
|
16 end |
4746 end |