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1 (* Author: Florian Haftmann, TU Muenchen |
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2 *) |
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3 |
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4 header {* A HOL random engine *} |
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5 |
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6 theory Random |
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7 imports State_Monad Code_Index |
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8 begin |
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9 |
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10 subsection {* Auxiliary functions *} |
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11 |
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12 definition |
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13 inc_shift :: "index \<Rightarrow> index \<Rightarrow> index" |
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14 where |
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15 "inc_shift v k = (if v = k then 1 else k + 1)" |
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16 |
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17 definition |
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18 minus_shift :: "index \<Rightarrow> index \<Rightarrow> index \<Rightarrow> index" |
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19 where |
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20 "minus_shift r k l = (if k < l then r + k - l else k - l)" |
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21 |
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22 fun |
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23 log :: "index \<Rightarrow> index \<Rightarrow> index" |
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24 where |
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25 "log b i = (if b \<le> 1 \<or> i < b then 1 else 1 + log b (i div b))" |
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26 |
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27 subsection {* Random seeds *} |
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28 |
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29 types seed = "index \<times> index" |
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30 |
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31 primrec |
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32 "next" :: "seed \<Rightarrow> index \<times> seed" |
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33 where |
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34 "next (v, w) = (let |
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35 k = v div 53668; |
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36 v' = minus_shift 2147483563 (40014 * (v mod 53668)) (k * 12211); |
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37 l = w div 52774; |
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38 w' = minus_shift 2147483399 (40692 * (w mod 52774)) (l * 3791); |
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39 z = minus_shift 2147483562 v' (w' + 1) + 1 |
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40 in (z, (v', w')))" |
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41 |
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42 lemma next_not_0: |
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43 "fst (next s) \<noteq> 0" |
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44 apply (cases s) |
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45 apply (auto simp add: minus_shift_def Let_def) |
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46 done |
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47 |
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48 primrec |
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49 seed_invariant :: "seed \<Rightarrow> bool" |
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50 where |
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51 "seed_invariant (v, w) \<longleftrightarrow> 0 < v \<and> v < 9438322952 \<and> 0 < w \<and> True" |
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52 |
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53 lemma if_same: |
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54 "(if b then f x else f y) = f (if b then x else y)" |
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55 by (cases b) simp_all |
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56 |
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57 definition |
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58 split_seed :: "seed \<Rightarrow> seed \<times> seed" |
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59 where |
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60 "split_seed s = (let |
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61 (v, w) = s; |
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62 (v', w') = snd (next s); |
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63 v'' = inc_shift 2147483562 v; |
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64 s'' = (v'', w'); |
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65 w'' = inc_shift 2147483398 w; |
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66 s''' = (v', w'') |
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67 in (s'', s'''))" |
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68 |
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69 |
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70 subsection {* Base selectors *} |
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71 |
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72 function |
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73 range_aux :: "index \<Rightarrow> index \<Rightarrow> seed \<Rightarrow> index \<times> seed" |
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74 where |
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75 "range_aux k l s = (if k = 0 then (l, s) else |
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76 let (v, s') = next s |
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77 in range_aux (k - 1) (v + l * 2147483561) s')" |
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78 by pat_completeness auto |
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79 termination |
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80 by (relation "measure (nat_of_index o fst)") |
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81 (auto simp add: index) |
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82 |
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83 definition |
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84 range :: "index \<Rightarrow> seed \<Rightarrow> index \<times> seed" |
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85 where |
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86 "range k = (do |
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87 v \<leftarrow> range_aux (log 2147483561 k) 1; |
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88 return (v mod k) |
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89 done)" |
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90 |
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91 lemma range: |
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92 assumes "k > 0" |
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93 shows "fst (range k s) < k" |
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94 proof - |
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95 obtain v w where range_aux: |
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96 "range_aux (log 2147483561 k) 1 s = (v, w)" |
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97 by (cases "range_aux (log 2147483561 k) 1 s") |
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98 with assms show ?thesis |
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99 by (simp add: monad_collapse range_def del: range_aux.simps log.simps) |
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100 qed |
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101 |
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102 definition |
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103 select :: "'a list \<Rightarrow> seed \<Rightarrow> 'a \<times> seed" |
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104 where |
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105 "select xs = (do |
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106 k \<leftarrow> range (index_of_nat (length xs)); |
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107 return (nth xs (nat_of_index k)) |
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108 done)" |
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109 |
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110 lemma select: |
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111 assumes "xs \<noteq> []" |
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112 shows "fst (select xs s) \<in> set xs" |
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113 proof - |
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114 from assms have "index_of_nat (length xs) > 0" by simp |
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115 with range have |
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116 "fst (range (index_of_nat (length xs)) s) < index_of_nat (length xs)" by best |
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117 then have |
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118 "nat_of_index (fst (range (index_of_nat (length xs)) s)) < length xs" by simp |
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119 then show ?thesis |
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120 by (auto simp add: monad_collapse select_def) |
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121 qed |
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122 |
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123 definition |
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124 select_default :: "index \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> seed \<Rightarrow> 'a \<times> seed" |
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125 where |
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126 [code del]: "select_default k x y = (do |
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127 l \<leftarrow> range k; |
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128 return (if l + 1 < k then x else y) |
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129 done)" |
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130 |
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131 lemma select_default_zero: |
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132 "fst (select_default 0 x y s) = y" |
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133 by (simp add: monad_collapse select_default_def) |
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134 |
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135 lemma select_default_code [code]: |
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136 "select_default k x y = (if k = 0 then do |
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137 _ \<leftarrow> range 1; |
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138 return y |
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139 done else do |
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140 l \<leftarrow> range k; |
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141 return (if l + 1 < k then x else y) |
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142 done)" |
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143 proof (cases "k = 0") |
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144 case False then show ?thesis by (simp add: select_default_def) |
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145 next |
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146 case True then show ?thesis |
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147 by (simp add: monad_collapse select_default_def range_def) |
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148 qed |
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149 |
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150 |
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151 subsection {* @{text ML} interface *} |
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152 |
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153 ML {* |
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154 structure Random_Engine = |
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155 struct |
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156 |
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157 type seed = int * int; |
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158 |
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159 local |
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160 |
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161 val seed = ref |
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162 (let |
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163 val now = Time.toMilliseconds (Time.now ()); |
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164 val (q, s1) = IntInf.divMod (now, 2147483562); |
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165 val s2 = q mod 2147483398; |
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166 in (s1 + 1, s2 + 1) end); |
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167 |
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168 in |
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169 |
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170 fun run f = |
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171 let |
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172 val (x, seed') = f (! seed); |
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173 val _ = seed := seed' |
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174 in x end; |
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175 |
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176 end; |
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177 |
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178 end; |
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179 *} |
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180 |
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181 end |
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182 |