1 (* Title: HOL/Word/Reversed_Bit_Lists.thy |
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2 Author: Jeremy Dawson, NICTA |
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3 *) |
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4 |
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5 section \<open>Bit values as reversed lists of bools\<close> |
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6 |
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7 theory Reversed_Bit_Lists |
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8 imports Word Misc_Typedef |
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9 begin |
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10 |
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11 context comm_semiring_1 |
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12 begin |
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13 |
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14 lemma horner_sum_append: |
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15 \<open>horner_sum f a (xs @ ys) = horner_sum f a xs + a ^ length xs * horner_sum f a ys\<close> |
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16 using sum.atLeastLessThan_shift_bounds [of _ 0 \<open>length xs\<close> \<open>length ys\<close>] |
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17 atLeastLessThan_add_Un [of 0 \<open>length xs\<close> \<open>length ys\<close>] |
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18 by (simp add: horner_sum_eq_sum sum_distrib_left sum.union_disjoint ac_simps nth_append power_add) |
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19 |
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20 end |
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21 |
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22 lemma horner_sum_of_bool_2_concat: |
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23 \<open>horner_sum of_bool 2 (concat (map (\<lambda>x. map (bit x) [0..<LENGTH('a)]) ws)) = horner_sum uint (2 ^ LENGTH('a)) ws\<close> |
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24 for ws :: \<open>'a::len word list\<close> |
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25 proof (induction ws) |
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26 case Nil |
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27 then show ?case |
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28 by simp |
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29 next |
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30 case (Cons w ws) |
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31 moreover have \<open>horner_sum of_bool 2 (map (bit w) [0..<LENGTH('a)]) = uint w\<close> |
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32 proof transfer |
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33 fix k :: int |
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34 have \<open>map (\<lambda>n. n < LENGTH('a) \<and> bit k n) [0..<LENGTH('a)] |
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35 = map (bit k) [0..<LENGTH('a)]\<close> |
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36 by simp |
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37 then show \<open>horner_sum of_bool 2 (map (\<lambda>n. n < LENGTH('a) \<and> bit k n) [0..<LENGTH('a)]) |
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38 = take_bit LENGTH('a) k\<close> |
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39 by (simp only: horner_sum_bit_eq_take_bit) |
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40 qed |
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41 ultimately show ?case |
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42 by (simp add: horner_sum_append) |
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43 qed |
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44 |
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45 |
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46 subsection \<open>Implicit augmentation of list prefixes\<close> |
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47 |
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48 primrec takefill :: "'a \<Rightarrow> nat \<Rightarrow> 'a list \<Rightarrow> 'a list" |
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49 where |
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50 Z: "takefill fill 0 xs = []" |
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51 | Suc: "takefill fill (Suc n) xs = |
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52 (case xs of |
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53 [] \<Rightarrow> fill # takefill fill n xs |
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54 | y # ys \<Rightarrow> y # takefill fill n ys)" |
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55 |
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56 lemma nth_takefill: "m < n \<Longrightarrow> takefill fill n l ! m = (if m < length l then l ! m else fill)" |
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57 apply (induct n arbitrary: m l) |
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58 apply clarsimp |
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59 apply clarsimp |
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60 apply (case_tac m) |
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61 apply (simp split: list.split) |
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62 apply (simp split: list.split) |
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63 done |
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64 |
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65 lemma takefill_alt: "takefill fill n l = take n l @ replicate (n - length l) fill" |
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66 by (induct n arbitrary: l) (auto split: list.split) |
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67 |
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68 lemma takefill_replicate [simp]: "takefill fill n (replicate m fill) = replicate n fill" |
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69 by (simp add: takefill_alt replicate_add [symmetric]) |
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70 |
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71 lemma takefill_le': "n = m + k \<Longrightarrow> takefill x m (takefill x n l) = takefill x m l" |
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72 by (induct m arbitrary: l n) (auto split: list.split) |
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73 |
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74 lemma length_takefill [simp]: "length (takefill fill n l) = n" |
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75 by (simp add: takefill_alt) |
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76 |
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77 lemma take_takefill': "n = k + m \<Longrightarrow> take k (takefill fill n w) = takefill fill k w" |
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78 by (induct k arbitrary: w n) (auto split: list.split) |
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79 |
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80 lemma drop_takefill: "drop k (takefill fill (m + k) w) = takefill fill m (drop k w)" |
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81 by (induct k arbitrary: w) (auto split: list.split) |
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82 |
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83 lemma takefill_le [simp]: "m \<le> n \<Longrightarrow> takefill x m (takefill x n l) = takefill x m l" |
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84 by (auto simp: le_iff_add takefill_le') |
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85 |
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86 lemma take_takefill [simp]: "m \<le> n \<Longrightarrow> take m (takefill fill n w) = takefill fill m w" |
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87 by (auto simp: le_iff_add take_takefill') |
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88 |
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89 lemma takefill_append: "takefill fill (m + length xs) (xs @ w) = xs @ (takefill fill m w)" |
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90 by (induct xs) auto |
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91 |
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92 lemma takefill_same': "l = length xs \<Longrightarrow> takefill fill l xs = xs" |
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93 by (induct xs arbitrary: l) auto |
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94 |
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95 lemmas takefill_same [simp] = takefill_same' [OF refl] |
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96 |
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97 lemma tf_rev: |
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98 "n + k = m + length bl \<Longrightarrow> takefill x m (rev (takefill y n bl)) = |
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99 rev (takefill y m (rev (takefill x k (rev bl))))" |
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100 apply (rule nth_equalityI) |
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101 apply (auto simp add: nth_takefill rev_nth) |
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102 apply (rule_tac f = "\<lambda>n. bl ! n" in arg_cong) |
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103 apply arith |
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104 done |
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105 |
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106 lemma takefill_minus: "0 < n \<Longrightarrow> takefill fill (Suc (n - 1)) w = takefill fill n w" |
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107 by auto |
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108 |
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109 lemmas takefill_Suc_cases = |
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110 list.cases [THEN takefill.Suc [THEN trans]] |
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111 |
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112 lemmas takefill_Suc_Nil = takefill_Suc_cases (1) |
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113 lemmas takefill_Suc_Cons = takefill_Suc_cases (2) |
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114 |
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115 lemmas takefill_minus_simps = takefill_Suc_cases [THEN [2] |
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116 takefill_minus [symmetric, THEN trans]] |
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117 |
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118 lemma takefill_numeral_Nil [simp]: |
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119 "takefill fill (numeral k) [] = fill # takefill fill (pred_numeral k) []" |
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120 by (simp add: numeral_eq_Suc) |
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121 |
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122 lemma takefill_numeral_Cons [simp]: |
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123 "takefill fill (numeral k) (x # xs) = x # takefill fill (pred_numeral k) xs" |
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124 by (simp add: numeral_eq_Suc) |
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125 |
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126 |
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127 subsection \<open>Range projection\<close> |
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128 |
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129 definition bl_of_nth :: "nat \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> 'a list" |
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130 where "bl_of_nth n f = map f (rev [0..<n])" |
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131 |
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132 lemma bl_of_nth_simps [simp, code]: |
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133 "bl_of_nth 0 f = []" |
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134 "bl_of_nth (Suc n) f = f n # bl_of_nth n f" |
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135 by (simp_all add: bl_of_nth_def) |
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136 |
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137 lemma length_bl_of_nth [simp]: "length (bl_of_nth n f) = n" |
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138 by (simp add: bl_of_nth_def) |
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139 |
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140 lemma nth_bl_of_nth [simp]: "m < n \<Longrightarrow> rev (bl_of_nth n f) ! m = f m" |
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141 by (simp add: bl_of_nth_def rev_map) |
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142 |
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143 lemma bl_of_nth_inj: "(\<And>k. k < n \<Longrightarrow> f k = g k) \<Longrightarrow> bl_of_nth n f = bl_of_nth n g" |
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144 by (simp add: bl_of_nth_def) |
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145 |
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146 lemma bl_of_nth_nth_le: "n \<le> length xs \<Longrightarrow> bl_of_nth n (nth (rev xs)) = drop (length xs - n) xs" |
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147 apply (induct n arbitrary: xs) |
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148 apply clarsimp |
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149 apply clarsimp |
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150 apply (rule trans [OF _ hd_Cons_tl]) |
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151 apply (frule Suc_le_lessD) |
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152 apply (simp add: rev_nth trans [OF drop_Suc drop_tl, symmetric]) |
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153 apply (subst hd_drop_conv_nth) |
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154 apply force |
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155 apply simp_all |
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156 apply (rule_tac f = "\<lambda>n. drop n xs" in arg_cong) |
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157 apply simp |
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158 done |
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159 |
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160 lemma bl_of_nth_nth [simp]: "bl_of_nth (length xs) ((!) (rev xs)) = xs" |
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161 by (simp add: bl_of_nth_nth_le) |
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162 |
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163 |
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164 subsection \<open>More\<close> |
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165 |
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166 definition rotater1 :: "'a list \<Rightarrow> 'a list" |
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167 where "rotater1 ys = |
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168 (case ys of [] \<Rightarrow> [] | x # xs \<Rightarrow> last ys # butlast ys)" |
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169 |
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170 definition rotater :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" |
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171 where "rotater n = rotater1 ^^ n" |
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172 |
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173 lemmas rotater_0' [simp] = rotater_def [where n = "0", simplified] |
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174 |
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175 lemma rotate1_rl': "rotater1 (l @ [a]) = a # l" |
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176 by (cases l) (auto simp: rotater1_def) |
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177 |
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178 lemma rotate1_rl [simp] : "rotater1 (rotate1 l) = l" |
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179 apply (unfold rotater1_def) |
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180 apply (cases "l") |
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181 apply (case_tac [2] "list") |
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182 apply auto |
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183 done |
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184 |
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185 lemma rotate1_lr [simp] : "rotate1 (rotater1 l) = l" |
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186 by (cases l) (auto simp: rotater1_def) |
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187 |
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188 lemma rotater1_rev': "rotater1 (rev xs) = rev (rotate1 xs)" |
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189 by (cases "xs") (simp add: rotater1_def, simp add: rotate1_rl') |
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190 |
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191 lemma rotater_rev': "rotater n (rev xs) = rev (rotate n xs)" |
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192 by (induct n) (auto simp: rotater_def intro: rotater1_rev') |
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193 |
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194 lemma rotater_rev: "rotater n ys = rev (rotate n (rev ys))" |
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195 using rotater_rev' [where xs = "rev ys"] by simp |
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196 |
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197 lemma rotater_drop_take: |
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198 "rotater n xs = |
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199 drop (length xs - n mod length xs) xs @ |
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200 take (length xs - n mod length xs) xs" |
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201 by (auto simp: rotater_rev rotate_drop_take rev_take rev_drop) |
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202 |
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203 lemma rotater_Suc [simp]: "rotater (Suc n) xs = rotater1 (rotater n xs)" |
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204 unfolding rotater_def by auto |
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205 |
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206 lemma nth_rotater: |
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207 \<open>rotater m xs ! n = xs ! ((n + (length xs - m mod length xs)) mod length xs)\<close> if \<open>n < length xs\<close> |
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208 using that by (simp add: rotater_drop_take nth_append not_less less_diff_conv ac_simps le_mod_geq) |
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209 |
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210 lemma nth_rotater1: |
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211 \<open>rotater1 xs ! n = xs ! ((n + (length xs - 1)) mod length xs)\<close> if \<open>n < length xs\<close> |
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212 using that nth_rotater [of n xs 1] by simp |
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213 |
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214 lemma rotate_inv_plus [rule_format]: |
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215 "\<forall>k. k = m + n \<longrightarrow> rotater k (rotate n xs) = rotater m xs \<and> |
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216 rotate k (rotater n xs) = rotate m xs \<and> |
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217 rotater n (rotate k xs) = rotate m xs \<and> |
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218 rotate n (rotater k xs) = rotater m xs" |
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219 by (induct n) (auto simp: rotater_def rotate_def intro: funpow_swap1 [THEN trans]) |
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220 |
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221 lemmas rotate_inv_rel = le_add_diff_inverse2 [symmetric, THEN rotate_inv_plus] |
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222 |
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223 lemmas rotate_inv_eq = order_refl [THEN rotate_inv_rel, simplified] |
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224 |
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225 lemmas rotate_lr [simp] = rotate_inv_eq [THEN conjunct1] |
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226 lemmas rotate_rl [simp] = rotate_inv_eq [THEN conjunct2, THEN conjunct1] |
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227 |
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228 lemma rotate_gal: "rotater n xs = ys \<longleftrightarrow> rotate n ys = xs" |
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229 by auto |
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230 |
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231 lemma rotate_gal': "ys = rotater n xs \<longleftrightarrow> xs = rotate n ys" |
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232 by auto |
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233 |
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234 lemma length_rotater [simp]: "length (rotater n xs) = length xs" |
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235 by (simp add : rotater_rev) |
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236 |
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237 lemma rotate_eq_mod: "m mod length xs = n mod length xs \<Longrightarrow> rotate m xs = rotate n xs" |
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238 apply (rule box_equals) |
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239 defer |
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240 apply (rule rotate_conv_mod [symmetric])+ |
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241 apply simp |
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242 done |
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243 |
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244 lemma restrict_to_left: "x = y \<Longrightarrow> x = z \<longleftrightarrow> y = z" |
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245 by simp |
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246 |
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247 lemmas rotate_eqs = |
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248 trans [OF rotate0 [THEN fun_cong] id_apply] |
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249 rotate_rotate [symmetric] |
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250 rotate_id |
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251 rotate_conv_mod |
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252 rotate_eq_mod |
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253 |
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254 lemmas rrs0 = rotate_eqs [THEN restrict_to_left, |
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255 simplified rotate_gal [symmetric] rotate_gal' [symmetric]] |
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256 lemmas rrs1 = rrs0 [THEN refl [THEN rev_iffD1]] |
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257 lemmas rotater_eqs = rrs1 [simplified length_rotater] |
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258 lemmas rotater_0 = rotater_eqs (1) |
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259 lemmas rotater_add = rotater_eqs (2) |
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260 |
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261 lemma butlast_map: "xs \<noteq> [] \<Longrightarrow> butlast (map f xs) = map f (butlast xs)" |
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262 by (induct xs) auto |
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263 |
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264 lemma rotater1_map: "rotater1 (map f xs) = map f (rotater1 xs)" |
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265 by (cases xs) (auto simp: rotater1_def last_map butlast_map) |
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266 |
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267 lemma rotater_map: "rotater n (map f xs) = map f (rotater n xs)" |
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268 by (induct n) (auto simp: rotater_def rotater1_map) |
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269 |
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270 lemma but_last_zip [rule_format] : |
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271 "\<forall>ys. length xs = length ys \<longrightarrow> xs \<noteq> [] \<longrightarrow> |
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272 last (zip xs ys) = (last xs, last ys) \<and> |
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273 butlast (zip xs ys) = zip (butlast xs) (butlast ys)" |
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274 apply (induct xs) |
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275 apply auto |
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276 apply ((case_tac ys, auto simp: neq_Nil_conv)[1])+ |
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277 done |
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278 |
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279 lemma but_last_map2 [rule_format] : |
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280 "\<forall>ys. length xs = length ys \<longrightarrow> xs \<noteq> [] \<longrightarrow> |
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281 last (map2 f xs ys) = f (last xs) (last ys) \<and> |
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282 butlast (map2 f xs ys) = map2 f (butlast xs) (butlast ys)" |
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283 apply (induct xs) |
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284 apply auto |
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285 apply ((case_tac ys, auto simp: neq_Nil_conv)[1])+ |
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286 done |
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287 |
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288 lemma rotater1_zip: |
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289 "length xs = length ys \<Longrightarrow> |
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290 rotater1 (zip xs ys) = zip (rotater1 xs) (rotater1 ys)" |
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291 apply (unfold rotater1_def) |
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292 apply (cases xs) |
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293 apply auto |
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294 apply ((case_tac ys, auto simp: neq_Nil_conv but_last_zip)[1])+ |
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295 done |
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296 |
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297 lemma rotater1_map2: |
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298 "length xs = length ys \<Longrightarrow> |
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299 rotater1 (map2 f xs ys) = map2 f (rotater1 xs) (rotater1 ys)" |
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300 by (simp add: rotater1_map rotater1_zip) |
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301 |
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302 lemmas lrth = |
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303 box_equals [OF asm_rl length_rotater [symmetric] |
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304 length_rotater [symmetric], |
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305 THEN rotater1_map2] |
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306 |
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307 lemma rotater_map2: |
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308 "length xs = length ys \<Longrightarrow> |
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309 rotater n (map2 f xs ys) = map2 f (rotater n xs) (rotater n ys)" |
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310 by (induct n) (auto intro!: lrth) |
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311 |
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312 lemma rotate1_map2: |
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313 "length xs = length ys \<Longrightarrow> |
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314 rotate1 (map2 f xs ys) = map2 f (rotate1 xs) (rotate1 ys)" |
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315 by (cases xs; cases ys) auto |
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316 |
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317 lemmas lth = box_equals [OF asm_rl length_rotate [symmetric] |
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318 length_rotate [symmetric], THEN rotate1_map2] |
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319 |
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320 lemma rotate_map2: |
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321 "length xs = length ys \<Longrightarrow> |
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322 rotate n (map2 f xs ys) = map2 f (rotate n xs) (rotate n ys)" |
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323 by (induct n) (auto intro!: lth) |
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324 |
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325 |
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326 subsection \<open>Explicit bit representation of \<^typ>\<open>int\<close>\<close> |
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327 |
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328 primrec bl_to_bin_aux :: "bool list \<Rightarrow> int \<Rightarrow> int" |
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329 where |
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330 Nil: "bl_to_bin_aux [] w = w" |
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331 | Cons: "bl_to_bin_aux (b # bs) w = bl_to_bin_aux bs (of_bool b + 2 * w)" |
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332 |
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333 definition bl_to_bin :: "bool list \<Rightarrow> int" |
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334 where "bl_to_bin bs = bl_to_bin_aux bs 0" |
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335 |
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336 primrec bin_to_bl_aux :: "nat \<Rightarrow> int \<Rightarrow> bool list \<Rightarrow> bool list" |
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337 where |
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338 Z: "bin_to_bl_aux 0 w bl = bl" |
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339 | Suc: "bin_to_bl_aux (Suc n) w bl = bin_to_bl_aux n (bin_rest w) ((bin_last w) # bl)" |
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340 |
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341 definition bin_to_bl :: "nat \<Rightarrow> int \<Rightarrow> bool list" |
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342 where "bin_to_bl n w = bin_to_bl_aux n w []" |
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343 |
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344 lemma bin_to_bl_aux_zero_minus_simp [simp]: |
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345 "0 < n \<Longrightarrow> bin_to_bl_aux n 0 bl = bin_to_bl_aux (n - 1) 0 (False # bl)" |
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346 by (cases n) auto |
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347 |
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348 lemma bin_to_bl_aux_minus1_minus_simp [simp]: |
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349 "0 < n \<Longrightarrow> bin_to_bl_aux n (- 1) bl = bin_to_bl_aux (n - 1) (- 1) (True # bl)" |
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350 by (cases n) auto |
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351 |
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352 lemma bin_to_bl_aux_one_minus_simp [simp]: |
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353 "0 < n \<Longrightarrow> bin_to_bl_aux n 1 bl = bin_to_bl_aux (n - 1) 0 (True # bl)" |
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354 by (cases n) auto |
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355 |
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356 lemma bin_to_bl_aux_Bit0_minus_simp [simp]: |
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357 "0 < n \<Longrightarrow> |
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358 bin_to_bl_aux n (numeral (Num.Bit0 w)) bl = bin_to_bl_aux (n - 1) (numeral w) (False # bl)" |
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359 by (cases n) simp_all |
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360 |
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361 lemma bin_to_bl_aux_Bit1_minus_simp [simp]: |
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362 "0 < n \<Longrightarrow> |
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363 bin_to_bl_aux n (numeral (Num.Bit1 w)) bl = bin_to_bl_aux (n - 1) (numeral w) (True # bl)" |
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364 by (cases n) simp_all |
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365 |
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366 lemma bl_to_bin_aux_append: "bl_to_bin_aux (bs @ cs) w = bl_to_bin_aux cs (bl_to_bin_aux bs w)" |
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367 by (induct bs arbitrary: w) auto |
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368 |
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369 lemma bin_to_bl_aux_append: "bin_to_bl_aux n w bs @ cs = bin_to_bl_aux n w (bs @ cs)" |
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370 by (induct n arbitrary: w bs) auto |
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371 |
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372 lemma bl_to_bin_append: "bl_to_bin (bs @ cs) = bl_to_bin_aux cs (bl_to_bin bs)" |
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373 unfolding bl_to_bin_def by (rule bl_to_bin_aux_append) |
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374 |
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375 lemma bin_to_bl_aux_alt: "bin_to_bl_aux n w bs = bin_to_bl n w @ bs" |
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376 by (simp add: bin_to_bl_def bin_to_bl_aux_append) |
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377 |
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378 lemma bin_to_bl_0 [simp]: "bin_to_bl 0 bs = []" |
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379 by (auto simp: bin_to_bl_def) |
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380 |
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381 lemma size_bin_to_bl_aux: "length (bin_to_bl_aux n w bs) = n + length bs" |
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382 by (induct n arbitrary: w bs) auto |
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383 |
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384 lemma size_bin_to_bl [simp]: "length (bin_to_bl n w) = n" |
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385 by (simp add: bin_to_bl_def size_bin_to_bl_aux) |
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386 |
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387 lemma bl_bin_bl': "bin_to_bl (n + length bs) (bl_to_bin_aux bs w) = bin_to_bl_aux n w bs" |
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388 apply (induct bs arbitrary: w n) |
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389 apply auto |
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390 apply (simp_all only: add_Suc [symmetric]) |
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391 apply (auto simp add: bin_to_bl_def) |
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392 done |
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393 |
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394 lemma bl_bin_bl [simp]: "bin_to_bl (length bs) (bl_to_bin bs) = bs" |
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395 unfolding bl_to_bin_def |
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396 apply (rule box_equals) |
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397 apply (rule bl_bin_bl') |
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398 prefer 2 |
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399 apply (rule bin_to_bl_aux.Z) |
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400 apply simp |
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401 done |
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402 |
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403 lemma bl_to_bin_inj: "bl_to_bin bs = bl_to_bin cs \<Longrightarrow> length bs = length cs \<Longrightarrow> bs = cs" |
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404 apply (rule_tac box_equals) |
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405 defer |
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406 apply (rule bl_bin_bl) |
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407 apply (rule bl_bin_bl) |
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408 apply simp |
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409 done |
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410 |
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411 lemma bl_to_bin_False [simp]: "bl_to_bin (False # bl) = bl_to_bin bl" |
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412 by (auto simp: bl_to_bin_def) |
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413 |
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414 lemma bl_to_bin_Nil [simp]: "bl_to_bin [] = 0" |
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415 by (auto simp: bl_to_bin_def) |
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416 |
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417 lemma bin_to_bl_zero_aux: "bin_to_bl_aux n 0 bl = replicate n False @ bl" |
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418 by (induct n arbitrary: bl) (auto simp: replicate_app_Cons_same) |
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419 |
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420 lemma bin_to_bl_zero: "bin_to_bl n 0 = replicate n False" |
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421 by (simp add: bin_to_bl_def bin_to_bl_zero_aux) |
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422 |
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423 lemma bin_to_bl_minus1_aux: "bin_to_bl_aux n (- 1) bl = replicate n True @ bl" |
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424 by (induct n arbitrary: bl) (auto simp: replicate_app_Cons_same) |
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425 |
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426 lemma bin_to_bl_minus1: "bin_to_bl n (- 1) = replicate n True" |
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427 by (simp add: bin_to_bl_def bin_to_bl_minus1_aux) |
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428 |
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429 |
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430 subsection \<open>Semantic interpretation of \<^typ>\<open>bool list\<close> as \<^typ>\<open>int\<close>\<close> |
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431 |
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432 lemma bin_bl_bin': "bl_to_bin (bin_to_bl_aux n w bs) = bl_to_bin_aux bs (bintrunc n w)" |
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433 by (induct n arbitrary: w bs) (auto simp: bl_to_bin_def take_bit_Suc ac_simps mod_2_eq_odd) |
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434 |
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435 lemma bin_bl_bin [simp]: "bl_to_bin (bin_to_bl n w) = bintrunc n w" |
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436 by (auto simp: bin_to_bl_def bin_bl_bin') |
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437 |
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438 lemma bl_to_bin_rep_F: "bl_to_bin (replicate n False @ bl) = bl_to_bin bl" |
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439 by (simp add: bin_to_bl_zero_aux [symmetric] bin_bl_bin') (simp add: bl_to_bin_def) |
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440 |
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441 lemma bin_to_bl_trunc [simp]: "n \<le> m \<Longrightarrow> bin_to_bl n (bintrunc m w) = bin_to_bl n w" |
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442 by (auto intro: bl_to_bin_inj) |
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443 |
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444 lemma bin_to_bl_aux_bintr: |
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445 "bin_to_bl_aux n (bintrunc m bin) bl = |
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446 replicate (n - m) False @ bin_to_bl_aux (min n m) bin bl" |
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447 apply (induct n arbitrary: m bin bl) |
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448 apply clarsimp |
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449 apply clarsimp |
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450 apply (case_tac "m") |
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451 apply (clarsimp simp: bin_to_bl_zero_aux) |
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452 apply (erule thin_rl) |
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453 apply (induct_tac n) |
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454 apply (auto simp add: take_bit_Suc) |
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455 done |
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456 |
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457 lemma bin_to_bl_bintr: |
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458 "bin_to_bl n (bintrunc m bin) = replicate (n - m) False @ bin_to_bl (min n m) bin" |
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459 unfolding bin_to_bl_def by (rule bin_to_bl_aux_bintr) |
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460 |
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461 lemma bl_to_bin_rep_False: "bl_to_bin (replicate n False) = 0" |
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462 by (induct n) auto |
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463 |
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464 lemma len_bin_to_bl_aux: "length (bin_to_bl_aux n w bs) = n + length bs" |
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465 by (fact size_bin_to_bl_aux) |
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466 |
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467 lemma len_bin_to_bl: "length (bin_to_bl n w) = n" |
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468 by (fact size_bin_to_bl) (* FIXME: duplicate *) |
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469 |
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470 lemma sign_bl_bin': "bin_sign (bl_to_bin_aux bs w) = bin_sign w" |
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471 by (induction bs arbitrary: w) (simp_all add: bin_sign_def) |
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472 |
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473 lemma sign_bl_bin: "bin_sign (bl_to_bin bs) = 0" |
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474 by (simp add: bl_to_bin_def sign_bl_bin') |
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475 |
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476 lemma bl_sbin_sign_aux: "hd (bin_to_bl_aux (Suc n) w bs) = (bin_sign (sbintrunc n w) = -1)" |
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477 by (induction n arbitrary: w bs) (auto simp add: bin_sign_def even_iff_mod_2_eq_zero bit_Suc) |
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478 |
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479 lemma bl_sbin_sign: "hd (bin_to_bl (Suc n) w) = (bin_sign (sbintrunc n w) = -1)" |
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480 unfolding bin_to_bl_def by (rule bl_sbin_sign_aux) |
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481 |
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482 lemma bin_nth_of_bl_aux: |
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483 "bin_nth (bl_to_bin_aux bl w) n = |
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484 (n < size bl \<and> rev bl ! n \<or> n \<ge> length bl \<and> bin_nth w (n - size bl))" |
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485 apply (induction bl arbitrary: w) |
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486 apply simp_all |
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487 apply safe |
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488 apply (simp_all add: not_le nth_append bit_double_iff even_bit_succ_iff split: if_splits) |
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489 done |
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490 |
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491 lemma bin_nth_of_bl: "bin_nth (bl_to_bin bl) n = (n < length bl \<and> rev bl ! n)" |
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492 by (simp add: bl_to_bin_def bin_nth_of_bl_aux) |
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493 |
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494 lemma bin_nth_bl: "n < m \<Longrightarrow> bin_nth w n = nth (rev (bin_to_bl m w)) n" |
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495 apply (induct n arbitrary: m w) |
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496 apply clarsimp |
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497 apply (case_tac m, clarsimp) |
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498 apply (clarsimp simp: bin_to_bl_def) |
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499 apply (simp add: bin_to_bl_aux_alt) |
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500 apply (case_tac m, clarsimp) |
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501 apply (clarsimp simp: bin_to_bl_def) |
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502 apply (simp add: bin_to_bl_aux_alt bit_Suc) |
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503 done |
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504 |
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505 lemma nth_bin_to_bl_aux: |
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506 "n < m + length bl \<Longrightarrow> (bin_to_bl_aux m w bl) ! n = |
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507 (if n < m then bit w (m - 1 - n) else bl ! (n - m))" |
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508 apply (induction bl arbitrary: w) |
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509 apply simp_all |
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510 apply (simp add: bin_nth_bl [of \<open>m - Suc n\<close> m] rev_nth flip: bin_to_bl_def) |
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511 apply (metis One_nat_def Suc_pred add_diff_cancel_left' |
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512 add_diff_cancel_right' bin_to_bl_aux_alt bin_to_bl_def |
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513 diff_Suc_Suc diff_is_0_eq diff_zero less_Suc_eq_0_disj |
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514 less_antisym less_imp_Suc_add list.size(3) nat_less_le nth_append size_bin_to_bl_aux) |
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515 done |
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516 |
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517 lemma nth_bin_to_bl: "n < m \<Longrightarrow> (bin_to_bl m w) ! n = bin_nth w (m - Suc n)" |
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518 by (simp add: bin_to_bl_def nth_bin_to_bl_aux) |
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519 |
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520 lemma takefill_bintrunc: "takefill False n bl = rev (bin_to_bl n (bl_to_bin (rev bl)))" |
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521 apply (rule nth_equalityI) |
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522 apply simp |
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523 apply (clarsimp simp: nth_takefill rev_nth nth_bin_to_bl bin_nth_of_bl) |
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524 done |
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525 |
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526 lemma bl_bin_bl_rtf: "bin_to_bl n (bl_to_bin bl) = rev (takefill False n (rev bl))" |
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527 by (simp add: takefill_bintrunc) |
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528 |
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529 lemma bl_to_bin_lt2p_aux: "bl_to_bin_aux bs w < (w + 1) * (2 ^ length bs)" |
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530 proof (induction bs arbitrary: w) |
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531 case Nil |
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532 then show ?case |
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533 by simp |
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534 next |
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535 case (Cons b bs) |
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536 from Cons.IH [of \<open>1 + 2 * w\<close>] Cons.IH [of \<open>2 * w\<close>] |
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537 show ?case |
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538 apply (auto simp add: algebra_simps) |
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539 apply (subst mult_2 [of \<open>2 ^ length bs\<close>]) |
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540 apply (simp only: add.assoc) |
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541 apply (rule pos_add_strict) |
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542 apply simp_all |
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543 done |
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544 qed |
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545 |
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546 lemma bl_to_bin_lt2p_drop: "bl_to_bin bs < 2 ^ length (dropWhile Not bs)" |
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547 proof (induct bs) |
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548 case Nil |
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549 then show ?case by simp |
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550 next |
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551 case (Cons b bs) |
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552 with bl_to_bin_lt2p_aux[where w=1] show ?case |
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553 by (simp add: bl_to_bin_def) |
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554 qed |
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555 |
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556 lemma bl_to_bin_lt2p: "bl_to_bin bs < 2 ^ length bs" |
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557 by (metis bin_bl_bin bintr_lt2p bl_bin_bl) |
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558 |
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559 lemma bl_to_bin_ge2p_aux: "bl_to_bin_aux bs w \<ge> w * (2 ^ length bs)" |
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560 proof (induction bs arbitrary: w) |
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561 case Nil |
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562 then show ?case |
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563 by simp |
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564 next |
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565 case (Cons b bs) |
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566 from Cons.IH [of \<open>1 + 2 * w\<close>] Cons.IH [of \<open>2 * w\<close>] |
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567 show ?case |
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568 apply (auto simp add: algebra_simps) |
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569 apply (rule add_le_imp_le_left [of \<open>2 ^ length bs\<close>]) |
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570 apply (rule add_increasing) |
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571 apply simp_all |
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572 done |
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573 qed |
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574 |
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575 lemma bl_to_bin_ge0: "bl_to_bin bs \<ge> 0" |
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576 apply (unfold bl_to_bin_def) |
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577 apply (rule xtrans(4)) |
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578 apply (rule bl_to_bin_ge2p_aux) |
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579 apply simp |
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580 done |
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581 |
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582 lemma butlast_rest_bin: "butlast (bin_to_bl n w) = bin_to_bl (n - 1) (bin_rest w)" |
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583 apply (unfold bin_to_bl_def) |
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584 apply (cases n, clarsimp) |
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585 apply clarsimp |
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586 apply (auto simp add: bin_to_bl_aux_alt) |
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587 done |
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588 |
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589 lemma butlast_bin_rest: "butlast bl = bin_to_bl (length bl - Suc 0) (bin_rest (bl_to_bin bl))" |
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590 using butlast_rest_bin [where w="bl_to_bin bl" and n="length bl"] by simp |
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591 |
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592 lemma butlast_rest_bl2bin_aux: |
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593 "bl \<noteq> [] \<Longrightarrow> bl_to_bin_aux (butlast bl) w = bin_rest (bl_to_bin_aux bl w)" |
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594 by (induct bl arbitrary: w) auto |
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595 |
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596 lemma butlast_rest_bl2bin: "bl_to_bin (butlast bl) = bin_rest (bl_to_bin bl)" |
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597 by (cases bl) (auto simp: bl_to_bin_def butlast_rest_bl2bin_aux) |
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598 |
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599 lemma trunc_bl2bin_aux: |
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600 "bintrunc m (bl_to_bin_aux bl w) = |
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601 bl_to_bin_aux (drop (length bl - m) bl) (bintrunc (m - length bl) w)" |
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602 proof (induct bl arbitrary: w) |
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603 case Nil |
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604 show ?case by simp |
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605 next |
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606 case (Cons b bl) |
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607 show ?case |
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608 proof (cases "m - length bl") |
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609 case 0 |
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610 then have "Suc (length bl) - m = Suc (length bl - m)" by simp |
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611 with Cons show ?thesis by simp |
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612 next |
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613 case (Suc n) |
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614 then have "m - Suc (length bl) = n" by simp |
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615 with Cons Suc show ?thesis by (simp add: take_bit_Suc ac_simps) |
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616 qed |
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617 qed |
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618 |
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619 lemma trunc_bl2bin: "bintrunc m (bl_to_bin bl) = bl_to_bin (drop (length bl - m) bl)" |
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620 by (simp add: bl_to_bin_def trunc_bl2bin_aux) |
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621 |
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622 lemma trunc_bl2bin_len [simp]: "bintrunc (length bl) (bl_to_bin bl) = bl_to_bin bl" |
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623 by (simp add: trunc_bl2bin) |
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624 |
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625 lemma bl2bin_drop: "bl_to_bin (drop k bl) = bintrunc (length bl - k) (bl_to_bin bl)" |
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626 apply (rule trans) |
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627 prefer 2 |
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628 apply (rule trunc_bl2bin [symmetric]) |
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629 apply (cases "k \<le> length bl") |
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630 apply auto |
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631 done |
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632 |
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633 lemma take_rest_power_bin: "m \<le> n \<Longrightarrow> take m (bin_to_bl n w) = bin_to_bl m ((bin_rest ^^ (n - m)) w)" |
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634 apply (rule nth_equalityI) |
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635 apply simp |
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636 apply (clarsimp simp add: nth_bin_to_bl nth_rest_power_bin) |
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637 done |
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638 |
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639 lemma last_bin_last': "size xs > 0 \<Longrightarrow> last xs \<longleftrightarrow> bin_last (bl_to_bin_aux xs w)" |
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640 by (induct xs arbitrary: w) auto |
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641 |
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642 lemma last_bin_last: "size xs > 0 \<Longrightarrow> last xs \<longleftrightarrow> bin_last (bl_to_bin xs)" |
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643 unfolding bl_to_bin_def by (erule last_bin_last') |
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644 |
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645 lemma bin_last_last: "bin_last w \<longleftrightarrow> last (bin_to_bl (Suc n) w)" |
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646 by (simp add: bin_to_bl_def) (auto simp: bin_to_bl_aux_alt) |
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647 |
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648 lemma drop_bin2bl_aux: |
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649 "drop m (bin_to_bl_aux n bin bs) = |
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650 bin_to_bl_aux (n - m) bin (drop (m - n) bs)" |
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651 apply (induction n arbitrary: m bin bs) |
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652 apply auto |
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653 apply (case_tac "m \<le> n") |
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654 apply (auto simp add: not_le Suc_diff_le) |
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655 apply (case_tac "m - n") |
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656 apply auto |
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657 apply (use Suc_diff_Suc in fastforce) |
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658 done |
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659 |
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660 lemma drop_bin2bl: "drop m (bin_to_bl n bin) = bin_to_bl (n - m) bin" |
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661 by (simp add: bin_to_bl_def drop_bin2bl_aux) |
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662 |
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663 lemma take_bin2bl_lem1: "take m (bin_to_bl_aux m w bs) = bin_to_bl m w" |
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664 apply (induct m arbitrary: w bs) |
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665 apply clarsimp |
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666 apply clarsimp |
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667 apply (simp add: bin_to_bl_aux_alt) |
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668 apply (simp add: bin_to_bl_def) |
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669 apply (simp add: bin_to_bl_aux_alt) |
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670 done |
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671 |
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672 lemma take_bin2bl_lem: "take m (bin_to_bl_aux (m + n) w bs) = take m (bin_to_bl (m + n) w)" |
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673 by (induct n arbitrary: w bs) (simp_all (no_asm) add: bin_to_bl_def take_bin2bl_lem1, simp) |
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674 |
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675 lemma bin_split_take: "bin_split n c = (a, b) \<Longrightarrow> bin_to_bl m a = take m (bin_to_bl (m + n) c)" |
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676 apply (induct n arbitrary: b c) |
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677 apply clarsimp |
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678 apply (clarsimp simp: Let_def split: prod.split_asm) |
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679 apply (simp add: bin_to_bl_def) |
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680 apply (simp add: take_bin2bl_lem drop_bit_Suc) |
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681 done |
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682 |
|
683 lemma bin_to_bl_drop_bit: |
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684 "k = m + n \<Longrightarrow> bin_to_bl m (drop_bit n c) = take m (bin_to_bl k c)" |
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685 using bin_split_take by simp |
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686 |
|
687 lemma bin_split_take1: |
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688 "k = m + n \<Longrightarrow> bin_split n c = (a, b) \<Longrightarrow> bin_to_bl m a = take m (bin_to_bl k c)" |
|
689 using bin_split_take by simp |
|
690 |
|
691 lemma bl_bin_bl_rep_drop: |
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692 "bin_to_bl n (bl_to_bin bl) = |
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693 replicate (n - length bl) False @ drop (length bl - n) bl" |
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694 by (simp add: bl_to_bin_inj bl_to_bin_rep_F trunc_bl2bin) |
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695 |
|
696 lemma bl_to_bin_aux_cat: |
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697 "bl_to_bin_aux bs (bin_cat w nv v) = |
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698 bin_cat w (nv + length bs) (bl_to_bin_aux bs v)" |
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699 by (rule bit_eqI) |
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700 (auto simp add: bin_nth_of_bl_aux bin_nth_cat algebra_simps) |
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701 |
|
702 lemma bin_to_bl_aux_cat: |
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703 "bin_to_bl_aux (nv + nw) (bin_cat v nw w) bs = |
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704 bin_to_bl_aux nv v (bin_to_bl_aux nw w bs)" |
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705 by (induction nw arbitrary: w bs) (simp_all add: concat_bit_Suc) |
|
706 |
|
707 lemma bl_to_bin_aux_alt: "bl_to_bin_aux bs w = bin_cat w (length bs) (bl_to_bin bs)" |
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708 using bl_to_bin_aux_cat [where nv = "0" and v = "0"] |
|
709 by (simp add: bl_to_bin_def [symmetric]) |
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710 |
|
711 lemma bin_to_bl_cat: |
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712 "bin_to_bl (nv + nw) (bin_cat v nw w) = |
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713 bin_to_bl_aux nv v (bin_to_bl nw w)" |
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714 by (simp add: bin_to_bl_def bin_to_bl_aux_cat) |
|
715 |
|
716 lemmas bl_to_bin_aux_app_cat = |
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717 trans [OF bl_to_bin_aux_append bl_to_bin_aux_alt] |
|
718 |
|
719 lemmas bin_to_bl_aux_cat_app = |
|
720 trans [OF bin_to_bl_aux_cat bin_to_bl_aux_alt] |
|
721 |
|
722 lemma bl_to_bin_app_cat: |
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723 "bl_to_bin (bsa @ bs) = bin_cat (bl_to_bin bsa) (length bs) (bl_to_bin bs)" |
|
724 by (simp only: bl_to_bin_aux_app_cat bl_to_bin_def) |
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725 |
|
726 lemma bin_to_bl_cat_app: |
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727 "bin_to_bl (n + nw) (bin_cat w nw wa) = bin_to_bl n w @ bin_to_bl nw wa" |
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728 by (simp only: bin_to_bl_def bin_to_bl_aux_cat_app) |
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729 |
|
730 text \<open>\<open>bl_to_bin_app_cat_alt\<close> and \<open>bl_to_bin_app_cat\<close> are easily interderivable.\<close> |
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731 lemma bl_to_bin_app_cat_alt: "bin_cat (bl_to_bin cs) n w = bl_to_bin (cs @ bin_to_bl n w)" |
|
732 by (simp add: bl_to_bin_app_cat) |
|
733 |
|
734 lemma mask_lem: "(bl_to_bin (True # replicate n False)) = bl_to_bin (replicate n True) + 1" |
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735 apply (unfold bl_to_bin_def) |
|
736 apply (induct n) |
|
737 apply simp |
|
738 apply (simp only: Suc_eq_plus1 replicate_add append_Cons [symmetric] bl_to_bin_aux_append) |
|
739 apply simp |
|
740 done |
|
741 |
|
742 lemma bin_exhaust: |
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743 "(\<And>x b. bin = of_bool b + 2 * x \<Longrightarrow> Q) \<Longrightarrow> Q" for bin :: int |
|
744 apply (cases \<open>even bin\<close>) |
|
745 apply (auto elim!: evenE oddE) |
|
746 apply fastforce |
|
747 apply fastforce |
|
748 done |
|
749 |
|
750 primrec rbl_succ :: "bool list \<Rightarrow> bool list" |
|
751 where |
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752 Nil: "rbl_succ Nil = Nil" |
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753 | Cons: "rbl_succ (x # xs) = (if x then False # rbl_succ xs else True # xs)" |
|
754 |
|
755 primrec rbl_pred :: "bool list \<Rightarrow> bool list" |
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756 where |
|
757 Nil: "rbl_pred Nil = Nil" |
|
758 | Cons: "rbl_pred (x # xs) = (if x then False # xs else True # rbl_pred xs)" |
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759 |
|
760 primrec rbl_add :: "bool list \<Rightarrow> bool list \<Rightarrow> bool list" |
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761 where \<comment> \<open>result is length of first arg, second arg may be longer\<close> |
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762 Nil: "rbl_add Nil x = Nil" |
|
763 | Cons: "rbl_add (y # ys) x = |
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764 (let ws = rbl_add ys (tl x) |
|
765 in (y \<noteq> hd x) # (if hd x \<and> y then rbl_succ ws else ws))" |
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766 |
|
767 primrec rbl_mult :: "bool list \<Rightarrow> bool list \<Rightarrow> bool list" |
|
768 where \<comment> \<open>result is length of first arg, second arg may be longer\<close> |
|
769 Nil: "rbl_mult Nil x = Nil" |
|
770 | Cons: "rbl_mult (y # ys) x = |
|
771 (let ws = False # rbl_mult ys x |
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772 in if y then rbl_add ws x else ws)" |
|
773 |
|
774 lemma size_rbl_pred: "length (rbl_pred bl) = length bl" |
|
775 by (induct bl) auto |
|
776 |
|
777 lemma size_rbl_succ: "length (rbl_succ bl) = length bl" |
|
778 by (induct bl) auto |
|
779 |
|
780 lemma size_rbl_add: "length (rbl_add bl cl) = length bl" |
|
781 by (induct bl arbitrary: cl) (auto simp: Let_def size_rbl_succ) |
|
782 |
|
783 lemma size_rbl_mult: "length (rbl_mult bl cl) = length bl" |
|
784 by (induct bl arbitrary: cl) (auto simp add: Let_def size_rbl_add) |
|
785 |
|
786 lemmas rbl_sizes [simp] = |
|
787 size_rbl_pred size_rbl_succ size_rbl_add size_rbl_mult |
|
788 |
|
789 lemmas rbl_Nils = |
|
790 rbl_pred.Nil rbl_succ.Nil rbl_add.Nil rbl_mult.Nil |
|
791 |
|
792 lemma rbl_add_app2: "length blb \<ge> length bla \<Longrightarrow> rbl_add bla (blb @ blc) = rbl_add bla blb" |
|
793 apply (induct bla arbitrary: blb) |
|
794 apply simp |
|
795 apply clarsimp |
|
796 apply (case_tac blb, clarsimp) |
|
797 apply (clarsimp simp: Let_def) |
|
798 done |
|
799 |
|
800 lemma rbl_add_take2: |
|
801 "length blb \<ge> length bla \<Longrightarrow> rbl_add bla (take (length bla) blb) = rbl_add bla blb" |
|
802 apply (induct bla arbitrary: blb) |
|
803 apply simp |
|
804 apply clarsimp |
|
805 apply (case_tac blb, clarsimp) |
|
806 apply (clarsimp simp: Let_def) |
|
807 done |
|
808 |
|
809 lemma rbl_mult_app2: "length blb \<ge> length bla \<Longrightarrow> rbl_mult bla (blb @ blc) = rbl_mult bla blb" |
|
810 apply (induct bla arbitrary: blb) |
|
811 apply simp |
|
812 apply clarsimp |
|
813 apply (case_tac blb, clarsimp) |
|
814 apply (clarsimp simp: Let_def rbl_add_app2) |
|
815 done |
|
816 |
|
817 lemma rbl_mult_take2: |
|
818 "length blb \<ge> length bla \<Longrightarrow> rbl_mult bla (take (length bla) blb) = rbl_mult bla blb" |
|
819 apply (rule trans) |
|
820 apply (rule rbl_mult_app2 [symmetric]) |
|
821 apply simp |
|
822 apply (rule_tac f = "rbl_mult bla" in arg_cong) |
|
823 apply (rule append_take_drop_id) |
|
824 done |
|
825 |
|
826 lemma rbl_add_split: |
|
827 "P (rbl_add (y # ys) (x # xs)) = |
|
828 (\<forall>ws. length ws = length ys \<longrightarrow> ws = rbl_add ys xs \<longrightarrow> |
|
829 (y \<longrightarrow> ((x \<longrightarrow> P (False # rbl_succ ws)) \<and> (\<not> x \<longrightarrow> P (True # ws)))) \<and> |
|
830 (\<not> y \<longrightarrow> P (x # ws)))" |
|
831 by (cases y) (auto simp: Let_def) |
|
832 |
|
833 lemma rbl_mult_split: |
|
834 "P (rbl_mult (y # ys) xs) = |
|
835 (\<forall>ws. length ws = Suc (length ys) \<longrightarrow> ws = False # rbl_mult ys xs \<longrightarrow> |
|
836 (y \<longrightarrow> P (rbl_add ws xs)) \<and> (\<not> y \<longrightarrow> P ws))" |
|
837 by (auto simp: Let_def) |
|
838 |
|
839 lemma rbl_pred: "rbl_pred (rev (bin_to_bl n bin)) = rev (bin_to_bl n (bin - 1))" |
|
840 proof (unfold bin_to_bl_def, induction n arbitrary: bin) |
|
841 case 0 |
|
842 then show ?case |
|
843 by simp |
|
844 next |
|
845 case (Suc n) |
|
846 obtain b k where \<open>bin = of_bool b + 2 * k\<close> |
|
847 using bin_exhaust by blast |
|
848 moreover have \<open>(2 * k - 1) div 2 = k - 1\<close> |
|
849 using even_succ_div_2 [of \<open>2 * (k - 1)\<close>] |
|
850 by simp |
|
851 ultimately show ?case |
|
852 using Suc [of \<open>bin div 2\<close>] |
|
853 by simp (simp add: bin_to_bl_aux_alt) |
|
854 qed |
|
855 |
|
856 lemma rbl_succ: "rbl_succ (rev (bin_to_bl n bin)) = rev (bin_to_bl n (bin + 1))" |
|
857 apply (unfold bin_to_bl_def) |
|
858 apply (induction n arbitrary: bin) |
|
859 apply simp_all |
|
860 apply (case_tac bin rule: bin_exhaust) |
|
861 apply simp |
|
862 apply (simp add: bin_to_bl_aux_alt ac_simps) |
|
863 done |
|
864 |
|
865 lemma rbl_add: |
|
866 "\<And>bina binb. rbl_add (rev (bin_to_bl n bina)) (rev (bin_to_bl n binb)) = |
|
867 rev (bin_to_bl n (bina + binb))" |
|
868 apply (unfold bin_to_bl_def) |
|
869 apply (induct n) |
|
870 apply simp |
|
871 apply clarsimp |
|
872 apply (case_tac bina rule: bin_exhaust) |
|
873 apply (case_tac binb rule: bin_exhaust) |
|
874 apply (case_tac b) |
|
875 apply (case_tac [!] "ba") |
|
876 apply (auto simp: rbl_succ bin_to_bl_aux_alt Let_def ac_simps) |
|
877 done |
|
878 |
|
879 lemma rbl_add_long: |
|
880 "m \<ge> n \<Longrightarrow> rbl_add (rev (bin_to_bl n bina)) (rev (bin_to_bl m binb)) = |
|
881 rev (bin_to_bl n (bina + binb))" |
|
882 apply (rule box_equals [OF _ rbl_add_take2 rbl_add]) |
|
883 apply (rule_tac f = "rbl_add (rev (bin_to_bl n bina))" in arg_cong) |
|
884 apply (rule rev_swap [THEN iffD1]) |
|
885 apply (simp add: rev_take drop_bin2bl) |
|
886 apply simp |
|
887 done |
|
888 |
|
889 lemma rbl_mult_gt1: |
|
890 "m \<ge> length bl \<Longrightarrow> |
|
891 rbl_mult bl (rev (bin_to_bl m binb)) = |
|
892 rbl_mult bl (rev (bin_to_bl (length bl) binb))" |
|
893 apply (rule trans) |
|
894 apply (rule rbl_mult_take2 [symmetric]) |
|
895 apply simp_all |
|
896 apply (rule_tac f = "rbl_mult bl" in arg_cong) |
|
897 apply (rule rev_swap [THEN iffD1]) |
|
898 apply (simp add: rev_take drop_bin2bl) |
|
899 done |
|
900 |
|
901 lemma rbl_mult_gt: |
|
902 "m > n \<Longrightarrow> |
|
903 rbl_mult (rev (bin_to_bl n bina)) (rev (bin_to_bl m binb)) = |
|
904 rbl_mult (rev (bin_to_bl n bina)) (rev (bin_to_bl n binb))" |
|
905 by (auto intro: trans [OF rbl_mult_gt1]) |
|
906 |
|
907 lemmas rbl_mult_Suc = lessI [THEN rbl_mult_gt] |
|
908 |
|
909 lemma rbbl_Cons: "b # rev (bin_to_bl n x) = rev (bin_to_bl (Suc n) (of_bool b + 2 * x))" |
|
910 by (simp add: bin_to_bl_def) (simp add: bin_to_bl_aux_alt) |
|
911 |
|
912 lemma rbl_mult: |
|
913 "rbl_mult (rev (bin_to_bl n bina)) (rev (bin_to_bl n binb)) = |
|
914 rev (bin_to_bl n (bina * binb))" |
|
915 apply (induct n arbitrary: bina binb) |
|
916 apply simp_all |
|
917 apply (unfold bin_to_bl_def) |
|
918 apply clarsimp |
|
919 apply (case_tac bina rule: bin_exhaust) |
|
920 apply (case_tac binb rule: bin_exhaust) |
|
921 apply simp |
|
922 apply (simp add: bin_to_bl_aux_alt) |
|
923 apply (simp add: rbbl_Cons rbl_mult_Suc rbl_add algebra_simps) |
|
924 done |
|
925 |
|
926 lemma sclem: "size (concat (map (bin_to_bl n) xs)) = length xs * n" |
|
927 by (induct xs) auto |
|
928 |
|
929 lemma bin_cat_foldl_lem: |
|
930 "foldl (\<lambda>u. bin_cat u n) x xs = |
|
931 bin_cat x (size xs * n) (foldl (\<lambda>u. bin_cat u n) y xs)" |
|
932 apply (induct xs arbitrary: x) |
|
933 apply simp |
|
934 apply (simp (no_asm)) |
|
935 apply (frule asm_rl) |
|
936 apply (drule meta_spec) |
|
937 apply (erule trans) |
|
938 apply (drule_tac x = "bin_cat y n a" in meta_spec) |
|
939 apply (simp add: bin_cat_assoc_sym min.absorb2) |
|
940 done |
|
941 |
|
942 lemma bin_rcat_bl: "bin_rcat n wl = bl_to_bin (concat (map (bin_to_bl n) wl))" |
|
943 apply (unfold bin_rcat_eq_foldl) |
|
944 apply (rule sym) |
|
945 apply (induct wl) |
|
946 apply (auto simp add: bl_to_bin_append) |
|
947 apply (simp add: bl_to_bin_aux_alt sclem) |
|
948 apply (simp add: bin_cat_foldl_lem [symmetric]) |
|
949 done |
|
950 |
|
951 lemma bin_last_bl_to_bin: "bin_last (bl_to_bin bs) \<longleftrightarrow> bs \<noteq> [] \<and> last bs" |
|
952 by(cases "bs = []")(auto simp add: bl_to_bin_def last_bin_last'[where w=0]) |
|
953 |
|
954 lemma bin_rest_bl_to_bin: "bin_rest (bl_to_bin bs) = bl_to_bin (butlast bs)" |
|
955 by(cases "bs = []")(simp_all add: bl_to_bin_def butlast_rest_bl2bin_aux) |
|
956 |
|
957 lemma bl_xor_aux_bin: |
|
958 "map2 (\<lambda>x y. x \<noteq> y) (bin_to_bl_aux n v bs) (bin_to_bl_aux n w cs) = |
|
959 bin_to_bl_aux n (v XOR w) (map2 (\<lambda>x y. x \<noteq> y) bs cs)" |
|
960 apply (induction n arbitrary: v w bs cs) |
|
961 apply auto |
|
962 apply (case_tac v rule: bin_exhaust) |
|
963 apply (case_tac w rule: bin_exhaust) |
|
964 apply clarsimp |
|
965 done |
|
966 |
|
967 lemma bl_or_aux_bin: |
|
968 "map2 (\<or>) (bin_to_bl_aux n v bs) (bin_to_bl_aux n w cs) = |
|
969 bin_to_bl_aux n (v OR w) (map2 (\<or>) bs cs)" |
|
970 by (induct n arbitrary: v w bs cs) simp_all |
|
971 |
|
972 lemma bl_and_aux_bin: |
|
973 "map2 (\<and>) (bin_to_bl_aux n v bs) (bin_to_bl_aux n w cs) = |
|
974 bin_to_bl_aux n (v AND w) (map2 (\<and>) bs cs)" |
|
975 by (induction n arbitrary: v w bs cs) simp_all |
|
976 |
|
977 lemma bl_not_aux_bin: "map Not (bin_to_bl_aux n w cs) = bin_to_bl_aux n (NOT w) (map Not cs)" |
|
978 by (induct n arbitrary: w cs) auto |
|
979 |
|
980 lemma bl_not_bin: "map Not (bin_to_bl n w) = bin_to_bl n (NOT w)" |
|
981 by (simp add: bin_to_bl_def bl_not_aux_bin) |
|
982 |
|
983 lemma bl_and_bin: "map2 (\<and>) (bin_to_bl n v) (bin_to_bl n w) = bin_to_bl n (v AND w)" |
|
984 by (simp add: bin_to_bl_def bl_and_aux_bin) |
|
985 |
|
986 lemma bl_or_bin: "map2 (\<or>) (bin_to_bl n v) (bin_to_bl n w) = bin_to_bl n (v OR w)" |
|
987 by (simp add: bin_to_bl_def bl_or_aux_bin) |
|
988 |
|
989 lemma bl_xor_bin: "map2 (\<noteq>) (bin_to_bl n v) (bin_to_bl n w) = bin_to_bl n (v XOR w)" |
|
990 using bl_xor_aux_bin by (simp add: bin_to_bl_def) |
|
991 |
|
992 |
|
993 subsection \<open>Type \<^typ>\<open>'a word\<close>\<close> |
|
994 |
|
995 lift_definition of_bl :: \<open>bool list \<Rightarrow> 'a::len word\<close> |
|
996 is bl_to_bin . |
|
997 |
|
998 lift_definition to_bl :: \<open>'a::len word \<Rightarrow> bool list\<close> |
|
999 is \<open>bin_to_bl LENGTH('a)\<close> |
|
1000 by (simp add: bl_to_bin_inj) |
|
1001 |
|
1002 lemma to_bl_eq: |
|
1003 \<open>to_bl w = bin_to_bl (LENGTH('a)) (uint w)\<close> |
|
1004 for w :: \<open>'a::len word\<close> |
|
1005 by transfer simp |
|
1006 |
|
1007 lemma bit_of_bl_iff: |
|
1008 \<open>bit (of_bl bs :: 'a word) n \<longleftrightarrow> rev bs ! n \<and> n < LENGTH('a::len) \<and> n < length bs\<close> |
|
1009 by transfer (simp add: bin_nth_of_bl ac_simps) |
|
1010 |
|
1011 lemma rev_to_bl_eq: |
|
1012 \<open>rev (to_bl w) = map (bit w) [0..<LENGTH('a)]\<close> |
|
1013 for w :: \<open>'a::len word\<close> |
|
1014 apply (rule nth_equalityI) |
|
1015 apply (simp add: to_bl.rep_eq) |
|
1016 apply (simp add: bin_nth_bl bit_word.rep_eq to_bl.rep_eq) |
|
1017 done |
|
1018 |
|
1019 lemma to_bl_eq_rev: |
|
1020 \<open>to_bl w = map (bit w) (rev [0..<LENGTH('a)])\<close> |
|
1021 for w :: \<open>'a::len word\<close> |
|
1022 using rev_to_bl_eq [of w] |
|
1023 apply (subst rev_is_rev_conv [symmetric]) |
|
1024 apply (simp add: rev_map) |
|
1025 done |
|
1026 |
|
1027 lemma of_bl_rev_eq: |
|
1028 \<open>of_bl (rev bs) = horner_sum of_bool 2 bs\<close> |
|
1029 apply (rule bit_word_eqI) |
|
1030 apply (simp add: bit_of_bl_iff) |
|
1031 apply transfer |
|
1032 apply (simp add: bit_horner_sum_bit_iff ac_simps) |
|
1033 done |
|
1034 |
|
1035 lemma of_bl_eq: |
|
1036 \<open>of_bl bs = horner_sum of_bool 2 (rev bs)\<close> |
|
1037 using of_bl_rev_eq [of \<open>rev bs\<close>] by simp |
|
1038 |
|
1039 lemma bshiftr1_eq: |
|
1040 \<open>bshiftr1 b w = of_bl (b # butlast (to_bl w))\<close> |
|
1041 apply transfer |
|
1042 apply auto |
|
1043 apply (subst bl_to_bin_app_cat [of \<open>[True]\<close>, simplified]) |
|
1044 apply simp |
|
1045 apply (metis One_nat_def add.commute bin_bl_bin bin_last_bl_to_bin bin_rest_bl_to_bin butlast_bin_rest concat_bit_eq last.simps list.distinct(1) list.size(3) list.size(4) odd_iff_mod_2_eq_one plus_1_eq_Suc power_Suc0_right push_bit_of_1 size_bin_to_bl take_bit_eq_mod trunc_bl2bin_len) |
|
1046 apply (simp add: butlast_rest_bl2bin) |
|
1047 done |
|
1048 |
|
1049 lemma length_to_bl_eq: |
|
1050 \<open>length (to_bl w) = LENGTH('a)\<close> |
|
1051 for w :: \<open>'a::len word\<close> |
|
1052 by transfer simp |
|
1053 |
|
1054 lemma word_rotr_eq: |
|
1055 \<open>word_rotr n w = of_bl (rotater n (to_bl w))\<close> |
|
1056 apply (rule bit_word_eqI) |
|
1057 subgoal for n |
|
1058 apply (cases \<open>n < LENGTH('a)\<close>) |
|
1059 apply (simp_all add: bit_word_rotr_iff bit_of_bl_iff rotater_rev length_to_bl_eq nth_rotate rev_to_bl_eq ac_simps) |
|
1060 done |
|
1061 done |
|
1062 |
|
1063 lemma word_rotl_eq: |
|
1064 \<open>word_rotl n w = of_bl (rotate n (to_bl w))\<close> |
|
1065 proof - |
|
1066 have \<open>rotate n (to_bl w) = rev (rotater n (rev (to_bl w)))\<close> |
|
1067 by (simp add: rotater_rev') |
|
1068 then show ?thesis |
|
1069 apply (simp add: word_rotl_eq_word_rotr bit_of_bl_iff length_to_bl_eq rev_to_bl_eq) |
|
1070 apply (rule bit_word_eqI) |
|
1071 subgoal for n |
|
1072 apply (cases \<open>n < LENGTH('a)\<close>) |
|
1073 apply (simp_all add: bit_word_rotr_iff bit_of_bl_iff nth_rotater) |
|
1074 done |
|
1075 done |
|
1076 qed |
|
1077 |
|
1078 lemma to_bl_def': "(to_bl :: 'a::len word \<Rightarrow> bool list) = bin_to_bl (LENGTH('a)) \<circ> uint" |
|
1079 by transfer (simp add: fun_eq_iff) |
|
1080 |
|
1081 \<comment> \<open>type definitions theorem for in terms of equivalent bool list\<close> |
|
1082 lemma td_bl: |
|
1083 "type_definition |
|
1084 (to_bl :: 'a::len word \<Rightarrow> bool list) |
|
1085 of_bl |
|
1086 {bl. length bl = LENGTH('a)}" |
|
1087 apply (standard; transfer) |
|
1088 apply (auto dest: sym) |
|
1089 done |
|
1090 |
|
1091 interpretation word_bl: |
|
1092 type_definition |
|
1093 "to_bl :: 'a::len word \<Rightarrow> bool list" |
|
1094 of_bl |
|
1095 "{bl. length bl = LENGTH('a::len)}" |
|
1096 by (fact td_bl) |
|
1097 |
|
1098 lemmas word_bl_Rep' = word_bl.Rep [unfolded mem_Collect_eq, iff] |
|
1099 |
|
1100 lemma word_size_bl: "size w = size (to_bl w)" |
|
1101 by (auto simp: word_size) |
|
1102 |
|
1103 lemma to_bl_use_of_bl: "to_bl w = bl \<longleftrightarrow> w = of_bl bl \<and> length bl = length (to_bl w)" |
|
1104 by (fastforce elim!: word_bl.Abs_inverse [unfolded mem_Collect_eq]) |
|
1105 |
|
1106 lemma length_bl_gt_0 [iff]: "0 < length (to_bl x)" |
|
1107 for x :: "'a::len word" |
|
1108 unfolding word_bl_Rep' by (rule len_gt_0) |
|
1109 |
|
1110 lemma bl_not_Nil [iff]: "to_bl x \<noteq> []" |
|
1111 for x :: "'a::len word" |
|
1112 by (fact length_bl_gt_0 [unfolded length_greater_0_conv]) |
|
1113 |
|
1114 lemma length_bl_neq_0 [iff]: "length (to_bl x) \<noteq> 0" |
|
1115 for x :: "'a::len word" |
|
1116 by (fact length_bl_gt_0 [THEN gr_implies_not0]) |
|
1117 |
|
1118 lemma hd_bl_sign_sint: "hd (to_bl w) = (bin_sign (sint w) = -1)" |
|
1119 apply transfer |
|
1120 apply (auto simp add: bin_sign_def) |
|
1121 using bin_sign_lem bl_sbin_sign apply fastforce |
|
1122 using bin_sign_lem bl_sbin_sign apply force |
|
1123 done |
|
1124 |
|
1125 lemma of_bl_drop': |
|
1126 "lend = length bl - LENGTH('a::len) \<Longrightarrow> |
|
1127 of_bl (drop lend bl) = (of_bl bl :: 'a word)" |
|
1128 by transfer (simp flip: trunc_bl2bin) |
|
1129 |
|
1130 lemma test_bit_of_bl: |
|
1131 "(of_bl bl::'a::len word) !! n = (rev bl ! n \<and> n < LENGTH('a) \<and> n < length bl)" |
|
1132 by transfer (simp add: bin_nth_of_bl ac_simps) |
|
1133 |
|
1134 lemma no_of_bl: "(numeral bin ::'a::len word) = of_bl (bin_to_bl (LENGTH('a)) (numeral bin))" |
|
1135 by transfer simp |
|
1136 |
|
1137 lemma uint_bl: "to_bl w = bin_to_bl (size w) (uint w)" |
|
1138 by transfer simp |
|
1139 |
|
1140 lemma to_bl_bin: "bl_to_bin (to_bl w) = uint w" |
|
1141 by (simp add: uint_bl word_size) |
|
1142 |
|
1143 lemma to_bl_of_bin: "to_bl (word_of_int bin::'a::len word) = bin_to_bl (LENGTH('a)) bin" |
|
1144 by (auto simp: uint_bl word_ubin.eq_norm word_size) |
|
1145 |
|
1146 lemma to_bl_numeral [simp]: |
|
1147 "to_bl (numeral bin::'a::len word) = |
|
1148 bin_to_bl (LENGTH('a)) (numeral bin)" |
|
1149 unfolding word_numeral_alt by (rule to_bl_of_bin) |
|
1150 |
|
1151 lemma to_bl_neg_numeral [simp]: |
|
1152 "to_bl (- numeral bin::'a::len word) = |
|
1153 bin_to_bl (LENGTH('a)) (- numeral bin)" |
|
1154 unfolding word_neg_numeral_alt by (rule to_bl_of_bin) |
|
1155 |
|
1156 lemma to_bl_to_bin [simp] : "bl_to_bin (to_bl w) = uint w" |
|
1157 by (simp add: uint_bl word_size) |
|
1158 |
|
1159 lemma uint_bl_bin: "bl_to_bin (bin_to_bl (LENGTH('a)) (uint x)) = uint x" |
|
1160 for x :: "'a::len word" |
|
1161 by (rule trans [OF bin_bl_bin word_ubin.norm_Rep]) |
|
1162 |
|
1163 lemma ucast_bl: "ucast w = of_bl (to_bl w)" |
|
1164 by transfer simp |
|
1165 |
|
1166 lemma ucast_down_bl: |
|
1167 \<open>(ucast :: 'a::len word \<Rightarrow> 'b::len word) (of_bl bl) = of_bl bl\<close> |
|
1168 if \<open>is_down (ucast :: 'a::len word \<Rightarrow> 'b::len word)\<close> |
|
1169 using that by transfer simp |
|
1170 |
|
1171 lemma of_bl_append_same: "of_bl (X @ to_bl w) = w" |
|
1172 by transfer (simp add: bl_to_bin_app_cat) |
|
1173 |
|
1174 lemma ucast_of_bl_up: |
|
1175 \<open>ucast (of_bl bl :: 'a::len word) = of_bl bl\<close> |
|
1176 if \<open>size bl \<le> size (of_bl bl :: 'a::len word)\<close> |
|
1177 using that |
|
1178 apply transfer |
|
1179 apply (rule bit_eqI) |
|
1180 apply (auto simp add: bit_take_bit_iff) |
|
1181 apply (subst (asm) trunc_bl2bin_len [symmetric]) |
|
1182 apply (auto simp only: bit_take_bit_iff) |
|
1183 done |
|
1184 |
|
1185 lemma word_rev_tf: |
|
1186 "to_bl (of_bl bl::'a::len word) = |
|
1187 rev (takefill False (LENGTH('a)) (rev bl))" |
|
1188 by transfer (simp add: bl_bin_bl_rtf) |
|
1189 |
|
1190 lemma word_rep_drop: |
|
1191 "to_bl (of_bl bl::'a::len word) = |
|
1192 replicate (LENGTH('a) - length bl) False @ |
|
1193 drop (length bl - LENGTH('a)) bl" |
|
1194 by (simp add: word_rev_tf takefill_alt rev_take) |
|
1195 |
|
1196 lemma to_bl_ucast: |
|
1197 "to_bl (ucast (w::'b::len word) ::'a::len word) = |
|
1198 replicate (LENGTH('a) - LENGTH('b)) False @ |
|
1199 drop (LENGTH('b) - LENGTH('a)) (to_bl w)" |
|
1200 apply (unfold ucast_bl) |
|
1201 apply (rule trans) |
|
1202 apply (rule word_rep_drop) |
|
1203 apply simp |
|
1204 done |
|
1205 |
|
1206 lemma ucast_up_app: |
|
1207 \<open>to_bl (ucast w :: 'b::len word) = replicate n False @ (to_bl w)\<close> |
|
1208 if \<open>source_size (ucast :: 'a word \<Rightarrow> 'b word) + n = target_size (ucast :: 'a word \<Rightarrow> 'b word)\<close> |
|
1209 for w :: \<open>'a::len word\<close> |
|
1210 using that |
|
1211 by (auto simp add : source_size target_size to_bl_ucast) |
|
1212 |
|
1213 lemma ucast_down_drop [OF refl]: |
|
1214 "uc = ucast \<Longrightarrow> source_size uc = target_size uc + n \<Longrightarrow> |
|
1215 to_bl (uc w) = drop n (to_bl w)" |
|
1216 by (auto simp add : source_size target_size to_bl_ucast) |
|
1217 |
|
1218 lemma scast_down_drop [OF refl]: |
|
1219 "sc = scast \<Longrightarrow> source_size sc = target_size sc + n \<Longrightarrow> |
|
1220 to_bl (sc w) = drop n (to_bl w)" |
|
1221 apply (subgoal_tac "sc = ucast") |
|
1222 apply safe |
|
1223 apply simp |
|
1224 apply (erule ucast_down_drop) |
|
1225 apply (rule down_cast_same [symmetric]) |
|
1226 apply (simp add : source_size target_size is_down) |
|
1227 done |
|
1228 |
|
1229 lemma word_0_bl [simp]: "of_bl [] = 0" |
|
1230 by transfer simp |
|
1231 |
|
1232 lemma word_1_bl: "of_bl [True] = 1" |
|
1233 by transfer (simp add: bl_to_bin_def) |
|
1234 |
|
1235 lemma of_bl_0 [simp]: "of_bl (replicate n False) = 0" |
|
1236 by transfer (simp add: bl_to_bin_rep_False) |
|
1237 |
|
1238 lemma to_bl_0 [simp]: "to_bl (0::'a::len word) = replicate (LENGTH('a)) False" |
|
1239 by (simp add: uint_bl word_size bin_to_bl_zero) |
|
1240 |
|
1241 \<comment> \<open>links with \<open>rbl\<close> operations\<close> |
|
1242 lemma word_succ_rbl: "to_bl w = bl \<Longrightarrow> to_bl (word_succ w) = rev (rbl_succ (rev bl))" |
|
1243 by transfer (simp add: rbl_succ) |
|
1244 |
|
1245 lemma word_pred_rbl: "to_bl w = bl \<Longrightarrow> to_bl (word_pred w) = rev (rbl_pred (rev bl))" |
|
1246 by transfer (simp add: rbl_pred) |
|
1247 |
|
1248 lemma word_add_rbl: |
|
1249 "to_bl v = vbl \<Longrightarrow> to_bl w = wbl \<Longrightarrow> |
|
1250 to_bl (v + w) = rev (rbl_add (rev vbl) (rev wbl))" |
|
1251 apply transfer |
|
1252 apply (drule sym) |
|
1253 apply (drule sym) |
|
1254 apply (simp add: rbl_add) |
|
1255 done |
|
1256 |
|
1257 lemma word_mult_rbl: |
|
1258 "to_bl v = vbl \<Longrightarrow> to_bl w = wbl \<Longrightarrow> |
|
1259 to_bl (v * w) = rev (rbl_mult (rev vbl) (rev wbl))" |
|
1260 apply transfer |
|
1261 apply (drule sym) |
|
1262 apply (drule sym) |
|
1263 apply (simp add: rbl_mult) |
|
1264 done |
|
1265 |
|
1266 lemma rtb_rbl_ariths: |
|
1267 "rev (to_bl w) = ys \<Longrightarrow> rev (to_bl (word_succ w)) = rbl_succ ys" |
|
1268 "rev (to_bl w) = ys \<Longrightarrow> rev (to_bl (word_pred w)) = rbl_pred ys" |
|
1269 "rev (to_bl v) = ys \<Longrightarrow> rev (to_bl w) = xs \<Longrightarrow> rev (to_bl (v * w)) = rbl_mult ys xs" |
|
1270 "rev (to_bl v) = ys \<Longrightarrow> rev (to_bl w) = xs \<Longrightarrow> rev (to_bl (v + w)) = rbl_add ys xs" |
|
1271 by (auto simp: rev_swap [symmetric] word_succ_rbl word_pred_rbl word_mult_rbl word_add_rbl) |
|
1272 |
|
1273 lemma of_bl_length_less: |
|
1274 \<open>(of_bl x :: 'a::len word) < 2 ^ k\<close> |
|
1275 if \<open>length x = k\<close> \<open>k < LENGTH('a)\<close> |
|
1276 proof - |
|
1277 from that have \<open>length x < LENGTH('a)\<close> |
|
1278 by simp |
|
1279 then have \<open>(of_bl x :: 'a::len word) < 2 ^ length x\<close> |
|
1280 apply (simp add: of_bl_eq) |
|
1281 apply transfer |
|
1282 apply (simp add: take_bit_horner_sum_bit_eq) |
|
1283 apply (subst length_rev [symmetric]) |
|
1284 apply (simp only: horner_sum_of_bool_2_less) |
|
1285 done |
|
1286 with that show ?thesis |
|
1287 by simp |
|
1288 qed |
|
1289 |
|
1290 lemma word_eq_rbl_eq: "x = y \<longleftrightarrow> rev (to_bl x) = rev (to_bl y)" |
|
1291 by simp |
|
1292 |
|
1293 lemma bl_word_not: "to_bl (NOT w) = map Not (to_bl w)" |
|
1294 by transfer (simp add: bl_not_bin) |
|
1295 |
|
1296 lemma bl_word_xor: "to_bl (v XOR w) = map2 (\<noteq>) (to_bl v) (to_bl w)" |
|
1297 by transfer (simp flip: bl_xor_bin) |
|
1298 |
|
1299 lemma bl_word_or: "to_bl (v OR w) = map2 (\<or>) (to_bl v) (to_bl w)" |
|
1300 by transfer (simp flip: bl_or_bin) |
|
1301 |
|
1302 lemma bl_word_and: "to_bl (v AND w) = map2 (\<and>) (to_bl v) (to_bl w)" |
|
1303 by transfer (simp flip: bl_and_bin) |
|
1304 |
|
1305 lemma bin_nth_uint': "bin_nth (uint w) n \<longleftrightarrow> rev (bin_to_bl (size w) (uint w)) ! n \<and> n < size w" |
|
1306 apply (unfold word_size) |
|
1307 apply (safe elim!: bin_nth_uint_imp) |
|
1308 apply (frule bin_nth_uint_imp) |
|
1309 apply (fast dest!: bin_nth_bl)+ |
|
1310 done |
|
1311 |
|
1312 lemmas bin_nth_uint = bin_nth_uint' [unfolded word_size] |
|
1313 |
|
1314 lemma test_bit_bl: "w !! n \<longleftrightarrow> rev (to_bl w) ! n \<and> n < size w" |
|
1315 by transfer (auto simp add: bin_nth_bl) |
|
1316 |
|
1317 lemma to_bl_nth: "n < size w \<Longrightarrow> to_bl w ! n = w !! (size w - Suc n)" |
|
1318 by (simp add: word_size rev_nth test_bit_bl) |
|
1319 |
|
1320 lemma map_bit_interval_eq: |
|
1321 \<open>map (bit w) [0..<n] = takefill False n (rev (to_bl w))\<close> for w :: \<open>'a::len word\<close> |
|
1322 proof (rule nth_equalityI) |
|
1323 show \<open>length (map (bit w) [0..<n]) = |
|
1324 length (takefill False n (rev (to_bl w)))\<close> |
|
1325 by simp |
|
1326 fix m |
|
1327 assume \<open>m < length (map (bit w) [0..<n])\<close> |
|
1328 then have \<open>m < n\<close> |
|
1329 by simp |
|
1330 then have \<open>bit w m \<longleftrightarrow> takefill False n (rev (to_bl w)) ! m\<close> |
|
1331 by (auto simp add: nth_takefill not_less rev_nth to_bl_nth word_size test_bit_word_eq dest: bit_imp_le_length) |
|
1332 with \<open>m < n \<close>show \<open>map (bit w) [0..<n] ! m \<longleftrightarrow> takefill False n (rev (to_bl w)) ! m\<close> |
|
1333 by simp |
|
1334 qed |
|
1335 |
|
1336 lemma to_bl_unfold: |
|
1337 \<open>to_bl w = rev (map (bit w) [0..<LENGTH('a)])\<close> for w :: \<open>'a::len word\<close> |
|
1338 by (simp add: map_bit_interval_eq takefill_bintrunc to_bl_def flip: bin_to_bl_def) |
|
1339 |
|
1340 lemma nth_rev_to_bl: |
|
1341 \<open>rev (to_bl w) ! n \<longleftrightarrow> bit w n\<close> |
|
1342 if \<open>n < LENGTH('a)\<close> for w :: \<open>'a::len word\<close> |
|
1343 using that by (simp add: to_bl_unfold) |
|
1344 |
|
1345 lemma nth_to_bl: |
|
1346 \<open>to_bl w ! n \<longleftrightarrow> bit w (LENGTH('a) - Suc n)\<close> |
|
1347 if \<open>n < LENGTH('a)\<close> for w :: \<open>'a::len word\<close> |
|
1348 using that by (simp add: to_bl_unfold rev_nth) |
|
1349 |
|
1350 lemma of_bl_rep_False: "of_bl (replicate n False @ bs) = of_bl bs" |
|
1351 by (auto simp: of_bl_def bl_to_bin_rep_F) |
|
1352 |
|
1353 lemma [code abstract]: |
|
1354 \<open>uint (of_bl bs :: 'a word) = horner_sum of_bool 2 (take LENGTH('a::len) (rev bs))\<close> |
|
1355 apply (simp add: of_bl_eq flip: take_bit_horner_sum_bit_eq) |
|
1356 apply transfer |
|
1357 apply simp |
|
1358 done |
|
1359 |
|
1360 lemma [code]: |
|
1361 \<open>to_bl w = map (bit w) (rev [0..<LENGTH('a::len)])\<close> |
|
1362 for w :: \<open>'a::len word\<close> |
|
1363 by (simp add: to_bl_unfold rev_map) |
|
1364 |
|
1365 lemma word_reverse_eq_of_bl_rev_to_bl: |
|
1366 \<open>word_reverse w = of_bl (rev (to_bl w))\<close> |
|
1367 by (rule bit_word_eqI) |
|
1368 (auto simp add: bit_word_reverse_iff bit_of_bl_iff nth_to_bl) |
|
1369 |
|
1370 lemmas word_reverse_no_def [simp] = |
|
1371 word_reverse_eq_of_bl_rev_to_bl [of "numeral w"] for w |
|
1372 |
|
1373 lemma to_bl_word_rev: "to_bl (word_reverse w) = rev (to_bl w)" |
|
1374 by (rule nth_equalityI) (simp_all add: nth_rev_to_bl word_reverse_def word_rep_drop flip: of_bl_eq) |
|
1375 |
|
1376 lemma to_bl_n1 [simp]: "to_bl (-1::'a::len word) = replicate (LENGTH('a)) True" |
|
1377 apply (rule word_bl.Abs_inverse') |
|
1378 apply simp |
|
1379 apply (rule word_eqI) |
|
1380 apply (clarsimp simp add: word_size) |
|
1381 apply (auto simp add: word_bl.Abs_inverse test_bit_bl word_size) |
|
1382 done |
|
1383 |
|
1384 lemma rbl_word_or: "rev (to_bl (x OR y)) = map2 (\<or>) (rev (to_bl x)) (rev (to_bl y))" |
|
1385 by (simp add: zip_rev bl_word_or rev_map) |
|
1386 |
|
1387 lemma rbl_word_and: "rev (to_bl (x AND y)) = map2 (\<and>) (rev (to_bl x)) (rev (to_bl y))" |
|
1388 by (simp add: zip_rev bl_word_and rev_map) |
|
1389 |
|
1390 lemma rbl_word_xor: "rev (to_bl (x XOR y)) = map2 (\<noteq>) (rev (to_bl x)) (rev (to_bl y))" |
|
1391 by (simp add: zip_rev bl_word_xor rev_map) |
|
1392 |
|
1393 lemma rbl_word_not: "rev (to_bl (NOT x)) = map Not (rev (to_bl x))" |
|
1394 by (simp add: bl_word_not rev_map) |
|
1395 |
|
1396 lemma bshiftr1_numeral [simp]: |
|
1397 \<open>bshiftr1 b (numeral w :: 'a word) = of_bl (b # butlast (bin_to_bl LENGTH('a::len) (numeral w)))\<close> |
|
1398 by (simp add: bshiftr1_eq) |
|
1399 |
|
1400 lemma bshiftr1_bl: "to_bl (bshiftr1 b w) = b # butlast (to_bl w)" |
|
1401 unfolding bshiftr1_eq by (rule word_bl.Abs_inverse) simp |
|
1402 |
|
1403 lemma shiftl1_of_bl: "shiftl1 (of_bl bl) = of_bl (bl @ [False])" |
|
1404 by transfer (simp add: bl_to_bin_append) |
|
1405 |
|
1406 lemma shiftl1_bl: "shiftl1 w = of_bl (to_bl w @ [False])" |
|
1407 for w :: "'a::len word" |
|
1408 proof - |
|
1409 have "shiftl1 w = shiftl1 (of_bl (to_bl w))" |
|
1410 by simp |
|
1411 also have "\<dots> = of_bl (to_bl w @ [False])" |
|
1412 by (rule shiftl1_of_bl) |
|
1413 finally show ?thesis . |
|
1414 qed |
|
1415 |
|
1416 lemma bl_shiftl1: "to_bl (shiftl1 w) = tl (to_bl w) @ [False]" |
|
1417 for w :: "'a::len word" |
|
1418 by (simp add: shiftl1_bl word_rep_drop drop_Suc drop_Cons') (fast intro!: Suc_leI) |
|
1419 |
|
1420 \<comment> \<open>Generalized version of \<open>bl_shiftl1\<close>. Maybe this one should replace it?\<close> |
|
1421 lemma bl_shiftl1': "to_bl (shiftl1 w) = tl (to_bl w @ [False])" |
|
1422 by (simp add: shiftl1_bl word_rep_drop drop_Suc del: drop_append) |
|
1423 |
|
1424 lemma shiftr1_bl: |
|
1425 \<open>shiftr1 w = of_bl (butlast (to_bl w))\<close> |
|
1426 proof (rule bit_word_eqI) |
|
1427 fix n |
|
1428 assume \<open>n < LENGTH('a)\<close> |
|
1429 show \<open>bit (shiftr1 w) n \<longleftrightarrow> bit (of_bl (butlast (to_bl w)) :: 'a word) n\<close> |
|
1430 proof (cases \<open>n = LENGTH('a) - 1\<close>) |
|
1431 case True |
|
1432 then show ?thesis |
|
1433 by (simp add: bit_shiftr1_iff bit_of_bl_iff) |
|
1434 next |
|
1435 case False |
|
1436 with \<open>n < LENGTH('a)\<close> |
|
1437 have \<open>n < LENGTH('a) - 1\<close> |
|
1438 by simp |
|
1439 with \<open>n < LENGTH('a)\<close> show ?thesis |
|
1440 by (simp add: bit_shiftr1_iff bit_of_bl_iff rev_nth nth_butlast |
|
1441 word_size test_bit_word_eq to_bl_nth) |
|
1442 qed |
|
1443 qed |
|
1444 |
|
1445 lemma bl_shiftr1: "to_bl (shiftr1 w) = False # butlast (to_bl w)" |
|
1446 for w :: "'a::len word" |
|
1447 by (simp add: shiftr1_bl word_rep_drop len_gt_0 [THEN Suc_leI]) |
|
1448 |
|
1449 \<comment> \<open>Generalized version of \<open>bl_shiftr1\<close>. Maybe this one should replace it?\<close> |
|
1450 lemma bl_shiftr1': "to_bl (shiftr1 w) = butlast (False # to_bl w)" |
|
1451 apply (rule word_bl.Abs_inverse') |
|
1452 apply (simp del: butlast.simps) |
|
1453 apply (simp add: shiftr1_bl of_bl_def) |
|
1454 done |
|
1455 |
|
1456 lemma bl_sshiftr1: "to_bl (sshiftr1 w) = hd (to_bl w) # butlast (to_bl w)" |
|
1457 for w :: "'a::len word" |
|
1458 proof (rule nth_equalityI) |
|
1459 fix n |
|
1460 assume \<open>n < length (to_bl (sshiftr1 w))\<close> |
|
1461 then have \<open>n < LENGTH('a)\<close> |
|
1462 by simp |
|
1463 then show \<open>to_bl (sshiftr1 w) ! n \<longleftrightarrow> (hd (to_bl w) # butlast (to_bl w)) ! n\<close> |
|
1464 apply (cases n) |
|
1465 apply (simp_all add: to_bl_nth word_size hd_conv_nth test_bit_eq_bit bit_sshiftr1_iff nth_butlast Suc_diff_Suc nth_to_bl) |
|
1466 done |
|
1467 qed simp |
|
1468 |
|
1469 lemma drop_shiftr: "drop n (to_bl (w >> n)) = take (size w - n) (to_bl w)" |
|
1470 for w :: "'a::len word" |
|
1471 apply (unfold shiftr_def) |
|
1472 apply (induct n) |
|
1473 prefer 2 |
|
1474 apply (simp add: drop_Suc bl_shiftr1 butlast_drop [symmetric]) |
|
1475 apply (rule butlast_take [THEN trans]) |
|
1476 apply (auto simp: word_size) |
|
1477 done |
|
1478 |
|
1479 lemma drop_sshiftr: "drop n (to_bl (w >>> n)) = take (size w - n) (to_bl w)" |
|
1480 for w :: "'a::len word" |
|
1481 apply (simp_all add: word_size sshiftr_eq) |
|
1482 apply (rule nth_equalityI) |
|
1483 apply (simp_all add: word_size nth_to_bl bit_signed_drop_bit_iff) |
|
1484 done |
|
1485 |
|
1486 lemma take_shiftr: "n \<le> size w \<Longrightarrow> take n (to_bl (w >> n)) = replicate n False" |
|
1487 apply (unfold shiftr_def) |
|
1488 apply (induct n) |
|
1489 prefer 2 |
|
1490 apply (simp add: bl_shiftr1' length_0_conv [symmetric] word_size) |
|
1491 apply (rule take_butlast [THEN trans]) |
|
1492 apply (auto simp: word_size) |
|
1493 done |
|
1494 |
|
1495 lemma take_sshiftr': |
|
1496 "n \<le> size w \<Longrightarrow> hd (to_bl (w >>> n)) = hd (to_bl w) \<and> |
|
1497 take n (to_bl (w >>> n)) = replicate n (hd (to_bl w))" |
|
1498 for w :: "'a::len word" |
|
1499 apply (auto simp add: sshiftr_eq hd_bl_sign_sint bin_sign_def not_le word_size sint_signed_drop_bit_eq) |
|
1500 apply (rule nth_equalityI) |
|
1501 apply (auto simp add: nth_to_bl bit_signed_drop_bit_iff bit_last_iff) |
|
1502 apply (rule nth_equalityI) |
|
1503 apply (auto simp add: nth_to_bl bit_signed_drop_bit_iff bit_last_iff) |
|
1504 done |
|
1505 |
|
1506 lemmas hd_sshiftr = take_sshiftr' [THEN conjunct1] |
|
1507 lemmas take_sshiftr = take_sshiftr' [THEN conjunct2] |
|
1508 |
|
1509 lemma atd_lem: "take n xs = t \<Longrightarrow> drop n xs = d \<Longrightarrow> xs = t @ d" |
|
1510 by (auto intro: append_take_drop_id [symmetric]) |
|
1511 |
|
1512 lemmas bl_shiftr = atd_lem [OF take_shiftr drop_shiftr] |
|
1513 lemmas bl_sshiftr = atd_lem [OF take_sshiftr drop_sshiftr] |
|
1514 |
|
1515 lemma shiftl_of_bl: "of_bl bl << n = of_bl (bl @ replicate n False)" |
|
1516 by (induct n) (auto simp: shiftl_def shiftl1_of_bl replicate_app_Cons_same) |
|
1517 |
|
1518 lemma shiftl_bl: "w << n = of_bl (to_bl w @ replicate n False)" |
|
1519 for w :: "'a::len word" |
|
1520 proof - |
|
1521 have "w << n = of_bl (to_bl w) << n" |
|
1522 by simp |
|
1523 also have "\<dots> = of_bl (to_bl w @ replicate n False)" |
|
1524 by (rule shiftl_of_bl) |
|
1525 finally show ?thesis . |
|
1526 qed |
|
1527 |
|
1528 lemma bl_shiftl: "to_bl (w << n) = drop n (to_bl w) @ replicate (min (size w) n) False" |
|
1529 by (simp add: shiftl_bl word_rep_drop word_size) |
|
1530 |
|
1531 lemma shiftr1_bl_of: |
|
1532 "length bl \<le> LENGTH('a) \<Longrightarrow> |
|
1533 shiftr1 (of_bl bl::'a::len word) = of_bl (butlast bl)" |
|
1534 by transfer (simp add: butlast_rest_bl2bin trunc_bl2bin) |
|
1535 |
|
1536 lemma shiftr_bl_of: |
|
1537 "length bl \<le> LENGTH('a) \<Longrightarrow> |
|
1538 (of_bl bl::'a::len word) >> n = of_bl (take (length bl - n) bl)" |
|
1539 apply (unfold shiftr_def) |
|
1540 apply (induct n) |
|
1541 apply clarsimp |
|
1542 apply clarsimp |
|
1543 apply (subst shiftr1_bl_of) |
|
1544 apply simp |
|
1545 apply (simp add: butlast_take) |
|
1546 done |
|
1547 |
|
1548 lemma shiftr_bl: "x >> n \<equiv> of_bl (take (LENGTH('a) - n) (to_bl x))" |
|
1549 for x :: "'a::len word" |
|
1550 using shiftr_bl_of [where 'a='a, of "to_bl x"] by simp |
|
1551 |
|
1552 lemma aligned_bl_add_size [OF refl]: |
|
1553 "size x - n = m \<Longrightarrow> n \<le> size x \<Longrightarrow> drop m (to_bl x) = replicate n False \<Longrightarrow> |
|
1554 take m (to_bl y) = replicate m False \<Longrightarrow> |
|
1555 to_bl (x + y) = take m (to_bl x) @ drop m (to_bl y)" for x :: \<open>'a::len word\<close> |
|
1556 apply (subgoal_tac "x AND y = 0") |
|
1557 prefer 2 |
|
1558 apply (rule word_bl.Rep_eqD) |
|
1559 apply (simp add: bl_word_and) |
|
1560 apply (rule align_lem_and [THEN trans]) |
|
1561 apply (simp_all add: word_size)[5] |
|
1562 apply simp |
|
1563 apply (subst word_plus_and_or [symmetric]) |
|
1564 apply (simp add : bl_word_or) |
|
1565 apply (rule align_lem_or) |
|
1566 apply (simp_all add: word_size) |
|
1567 done |
|
1568 |
|
1569 lemma mask_bl: "mask n = of_bl (replicate n True)" |
|
1570 by (auto simp add : test_bit_of_bl word_size intro: word_eqI) |
|
1571 |
|
1572 lemma bl_and_mask': |
|
1573 "to_bl (w AND mask n :: 'a::len word) = |
|
1574 replicate (LENGTH('a) - n) False @ |
|
1575 drop (LENGTH('a) - n) (to_bl w)" |
|
1576 apply (rule nth_equalityI) |
|
1577 apply simp |
|
1578 apply (clarsimp simp add: to_bl_nth word_size) |
|
1579 apply (simp add: word_size word_ops_nth_size) |
|
1580 apply (auto simp add: word_size test_bit_bl nth_append rev_nth) |
|
1581 done |
|
1582 |
|
1583 lemma slice1_eq_of_bl: |
|
1584 \<open>(slice1 n w :: 'b::len word) = of_bl (takefill False n (to_bl w))\<close> |
|
1585 for w :: \<open>'a::len word\<close> |
|
1586 proof (rule bit_word_eqI) |
|
1587 fix m |
|
1588 assume \<open>m < LENGTH('b)\<close> |
|
1589 show \<open>bit (slice1 n w :: 'b::len word) m \<longleftrightarrow> bit (of_bl (takefill False n (to_bl w)) :: 'b word) m\<close> |
|
1590 by (cases \<open>m \<ge> n\<close>; cases \<open>LENGTH('a) \<ge> n\<close>) |
|
1591 (auto simp add: bit_slice1_iff bit_of_bl_iff not_less rev_nth not_le nth_takefill nth_to_bl algebra_simps) |
|
1592 qed |
|
1593 |
|
1594 lemma slice1_no_bin [simp]: |
|
1595 "slice1 n (numeral w :: 'b word) = of_bl (takefill False n (bin_to_bl (LENGTH('b::len)) (numeral w)))" |
|
1596 by (simp add: slice1_eq_of_bl) (* TODO: neg_numeral *) |
|
1597 |
|
1598 lemma slice_no_bin [simp]: |
|
1599 "slice n (numeral w :: 'b word) = of_bl (takefill False (LENGTH('b::len) - n) |
|
1600 (bin_to_bl (LENGTH('b::len)) (numeral w)))" |
|
1601 by (simp add: slice_def) (* TODO: neg_numeral *) |
|
1602 |
|
1603 lemma slice_take': "slice n w = of_bl (take (size w - n) (to_bl w))" |
|
1604 by (simp add: slice_def word_size slice1_eq_of_bl takefill_alt) |
|
1605 |
|
1606 lemmas slice_take = slice_take' [unfolded word_size] |
|
1607 |
|
1608 \<comment> \<open>shiftr to a word of the same size is just slice, |
|
1609 slice is just shiftr then ucast\<close> |
|
1610 lemmas shiftr_slice = trans [OF shiftr_bl [THEN meta_eq_to_obj_eq] slice_take [symmetric]] |
|
1611 |
|
1612 lemma slice1_down_alt': |
|
1613 "sl = slice1 n w \<Longrightarrow> fs = size sl \<Longrightarrow> fs + k = n \<Longrightarrow> |
|
1614 to_bl sl = takefill False fs (drop k (to_bl w))" |
|
1615 apply (simp add: slice1_eq_of_bl) |
|
1616 apply transfer |
|
1617 apply (simp add: bl_bin_bl_rep_drop) |
|
1618 using drop_takefill |
|
1619 apply force |
|
1620 done |
|
1621 |
|
1622 lemma slice1_up_alt': |
|
1623 "sl = slice1 n w \<Longrightarrow> fs = size sl \<Longrightarrow> fs = n + k \<Longrightarrow> |
|
1624 to_bl sl = takefill False fs (replicate k False @ (to_bl w))" |
|
1625 apply (simp add: slice1_eq_of_bl) |
|
1626 apply transfer |
|
1627 apply (simp add: bl_bin_bl_rep_drop flip: takefill_append) |
|
1628 apply (metis diff_add_inverse) |
|
1629 done |
|
1630 |
|
1631 lemmas sd1 = slice1_down_alt' [OF refl refl, unfolded word_size] |
|
1632 lemmas su1 = slice1_up_alt' [OF refl refl, unfolded word_size] |
|
1633 lemmas slice1_down_alt = le_add_diff_inverse [THEN sd1] |
|
1634 lemmas slice1_up_alts = |
|
1635 le_add_diff_inverse [symmetric, THEN su1] |
|
1636 le_add_diff_inverse2 [symmetric, THEN su1] |
|
1637 |
|
1638 lemma slice1_tf_tf': |
|
1639 "to_bl (slice1 n w :: 'a::len word) = |
|
1640 rev (takefill False (LENGTH('a)) (rev (takefill False n (to_bl w))))" |
|
1641 unfolding slice1_eq_of_bl by (rule word_rev_tf) |
|
1642 |
|
1643 lemmas slice1_tf_tf = slice1_tf_tf' [THEN word_bl.Rep_inverse', symmetric] |
|
1644 |
|
1645 lemma revcast_eq_of_bl: |
|
1646 \<open>(revcast w :: 'b::len word) = of_bl (takefill False (LENGTH('b)) (to_bl w))\<close> |
|
1647 for w :: \<open>'a::len word\<close> |
|
1648 by (simp add: revcast_def slice1_eq_of_bl) |
|
1649 |
|
1650 lemmas revcast_no_def [simp] = revcast_eq_of_bl [where w="numeral w", unfolded word_size] for w |
|
1651 |
|
1652 lemma to_bl_revcast: |
|
1653 "to_bl (revcast w :: 'a::len word) = takefill False (LENGTH('a)) (to_bl w)" |
|
1654 apply (rule nth_equalityI) |
|
1655 apply simp |
|
1656 apply (cases \<open>LENGTH('a) \<le> LENGTH('b)\<close>) |
|
1657 apply (auto simp add: nth_to_bl nth_takefill bit_revcast_iff) |
|
1658 done |
|
1659 |
|
1660 lemma word_cat_bl: "word_cat a b = of_bl (to_bl a @ to_bl b)" |
|
1661 apply (rule bit_word_eqI) |
|
1662 apply (simp add: bit_word_cat_iff bit_of_bl_iff nth_append not_less nth_rev_to_bl) |
|
1663 apply (meson bit_word.rep_eq less_diff_conv2 nth_rev_to_bl) |
|
1664 done |
|
1665 |
|
1666 lemma of_bl_append: |
|
1667 "(of_bl (xs @ ys) :: 'a::len word) = of_bl xs * 2^(length ys) + of_bl ys" |
|
1668 apply transfer |
|
1669 apply (simp add: bl_to_bin_app_cat bin_cat_num) |
|
1670 done |
|
1671 |
|
1672 lemma of_bl_False [simp]: "of_bl (False#xs) = of_bl xs" |
|
1673 by (rule word_eqI) (auto simp: test_bit_of_bl nth_append) |
|
1674 |
|
1675 lemma of_bl_True [simp]: "(of_bl (True # xs) :: 'a::len word) = 2^length xs + of_bl xs" |
|
1676 by (subst of_bl_append [where xs="[True]", simplified]) (simp add: word_1_bl) |
|
1677 |
|
1678 lemma of_bl_Cons: "of_bl (x#xs) = of_bool x * 2^length xs + of_bl xs" |
|
1679 by (cases x) simp_all |
|
1680 |
|
1681 lemma word_split_bl': |
|
1682 "std = size c - size b \<Longrightarrow> (word_split c = (a, b)) \<Longrightarrow> |
|
1683 (a = of_bl (take std (to_bl c)) \<and> b = of_bl (drop std (to_bl c)))" |
|
1684 apply (simp add: word_split_def) |
|
1685 apply transfer |
|
1686 apply (cases \<open>LENGTH('b) \<le> LENGTH('a)\<close>) |
|
1687 apply (auto simp add: drop_bit_take_bit drop_bin2bl bin_to_bl_drop_bit [symmetric, of \<open>LENGTH('a)\<close> \<open>LENGTH('a) - LENGTH('b)\<close> \<open>LENGTH('b)\<close>] min_absorb2) |
|
1688 done |
|
1689 |
|
1690 lemma word_split_bl: "std = size c - size b \<Longrightarrow> |
|
1691 (a = of_bl (take std (to_bl c)) \<and> b = of_bl (drop std (to_bl c))) \<longleftrightarrow> |
|
1692 word_split c = (a, b)" |
|
1693 apply (rule iffI) |
|
1694 defer |
|
1695 apply (erule (1) word_split_bl') |
|
1696 apply (case_tac "word_split c") |
|
1697 apply (auto simp add: word_size) |
|
1698 apply (frule word_split_bl' [rotated]) |
|
1699 apply (auto simp add: word_size) |
|
1700 done |
|
1701 |
|
1702 lemma word_split_bl_eq: |
|
1703 "(word_split c :: ('c::len word \<times> 'd::len word)) = |
|
1704 (of_bl (take (LENGTH('a::len) - LENGTH('d::len)) (to_bl c)), |
|
1705 of_bl (drop (LENGTH('a) - LENGTH('d)) (to_bl c)))" |
|
1706 for c :: "'a::len word" |
|
1707 apply (rule word_split_bl [THEN iffD1]) |
|
1708 apply (unfold word_size) |
|
1709 apply (rule refl conjI)+ |
|
1710 done |
|
1711 |
|
1712 lemma word_rcat_bl: |
|
1713 \<open>word_rcat wl = of_bl (concat (map to_bl wl))\<close> |
|
1714 proof - |
|
1715 define ws where \<open>ws = rev wl\<close> |
|
1716 moreover have \<open>word_rcat (rev ws) = of_bl (concat (map to_bl (rev ws)))\<close> |
|
1717 apply (simp add: word_rcat_def of_bl_eq rev_concat rev_map comp_def rev_to_bl_eq flip: horner_sum_of_bool_2_concat) |
|
1718 apply transfer |
|
1719 apply simp |
|
1720 done |
|
1721 ultimately show ?thesis |
|
1722 by simp |
|
1723 qed |
|
1724 |
|
1725 lemma size_rcat_lem': "size (concat (map to_bl wl)) = length wl * size (hd wl)" |
|
1726 by (induct wl) (auto simp: word_size) |
|
1727 |
|
1728 lemmas size_rcat_lem = size_rcat_lem' [unfolded word_size] |
|
1729 |
|
1730 lemma nth_rcat_lem: |
|
1731 "n < length (wl::'a word list) * LENGTH('a::len) \<Longrightarrow> |
|
1732 rev (concat (map to_bl wl)) ! n = |
|
1733 rev (to_bl (rev wl ! (n div LENGTH('a)))) ! (n mod LENGTH('a))" |
|
1734 apply (induct wl) |
|
1735 apply clarsimp |
|
1736 apply (clarsimp simp add : nth_append size_rcat_lem) |
|
1737 apply (simp flip: mult_Suc minus_div_mult_eq_mod add: less_Suc_eq_le not_less) |
|
1738 apply (metis (no_types, lifting) diff_is_0_eq div_le_mono len_not_eq_0 less_Suc_eq less_mult_imp_div_less nonzero_mult_div_cancel_right not_le nth_Cons_0) |
|
1739 done |
|
1740 |
|
1741 lemma foldl_eq_foldr: "foldl (+) x xs = foldr (+) (x # xs) 0" |
|
1742 for x :: "'a::comm_monoid_add" |
|
1743 by (induct xs arbitrary: x) (auto simp: add.assoc) |
|
1744 |
|
1745 lemmas word_cat_bl_no_bin [simp] = |
|
1746 word_cat_bl [where a="numeral a" and b="numeral b", unfolded to_bl_numeral] |
|
1747 for a b (* FIXME: negative numerals, 0 and 1 *) |
|
1748 |
|
1749 lemmas word_split_bl_no_bin [simp] = |
|
1750 word_split_bl_eq [where c="numeral c", unfolded to_bl_numeral] for c |
|
1751 |
|
1752 lemmas word_rot_defs = word_roti_eq_word_rotr_word_rotl word_rotr_eq word_rotl_eq |
|
1753 |
|
1754 lemma to_bl_rotl: "to_bl (word_rotl n w) = rotate n (to_bl w)" |
|
1755 by (simp add: word_rotl_eq to_bl_use_of_bl) |
|
1756 |
|
1757 lemmas blrs0 = rotate_eqs [THEN to_bl_rotl [THEN trans]] |
|
1758 |
|
1759 lemmas word_rotl_eqs = |
|
1760 blrs0 [simplified word_bl_Rep' word_bl.Rep_inject to_bl_rotl [symmetric]] |
|
1761 |
|
1762 lemma to_bl_rotr: "to_bl (word_rotr n w) = rotater n (to_bl w)" |
|
1763 by (simp add: word_rotr_eq to_bl_use_of_bl) |
|
1764 |
|
1765 lemmas brrs0 = rotater_eqs [THEN to_bl_rotr [THEN trans]] |
|
1766 |
|
1767 lemmas word_rotr_eqs = |
|
1768 brrs0 [simplified word_bl_Rep' word_bl.Rep_inject to_bl_rotr [symmetric]] |
|
1769 |
|
1770 declare word_rotr_eqs (1) [simp] |
|
1771 declare word_rotl_eqs (1) [simp] |
|
1772 |
|
1773 lemmas abl_cong = arg_cong [where f = "of_bl"] |
|
1774 |
|
1775 locale word_rotate |
|
1776 begin |
|
1777 |
|
1778 lemmas word_rot_defs' = to_bl_rotl to_bl_rotr |
|
1779 |
|
1780 lemmas blwl_syms [symmetric] = bl_word_not bl_word_and bl_word_or bl_word_xor |
|
1781 |
|
1782 lemmas lbl_lbl = trans [OF word_bl_Rep' word_bl_Rep' [symmetric]] |
|
1783 |
|
1784 lemmas ths_map2 [OF lbl_lbl] = rotate_map2 rotater_map2 |
|
1785 |
|
1786 lemmas ths_map [where xs = "to_bl v"] = rotate_map rotater_map for v |
|
1787 |
|
1788 lemmas th1s [simplified word_rot_defs' [symmetric]] = ths_map2 ths_map |
|
1789 |
|
1790 end |
|
1791 |
|
1792 lemmas bl_word_rotl_dt = trans [OF to_bl_rotl rotate_drop_take, |
|
1793 simplified word_bl_Rep'] |
|
1794 |
|
1795 lemmas bl_word_rotr_dt = trans [OF to_bl_rotr rotater_drop_take, |
|
1796 simplified word_bl_Rep'] |
|
1797 |
|
1798 lemma bl_word_roti_dt': |
|
1799 "n = nat ((- i) mod int (size (w :: 'a::len word))) \<Longrightarrow> |
|
1800 to_bl (word_roti i w) = drop n (to_bl w) @ take n (to_bl w)" |
|
1801 apply (unfold word_roti_eq_word_rotr_word_rotl) |
|
1802 apply (simp add: bl_word_rotl_dt bl_word_rotr_dt word_size) |
|
1803 apply safe |
|
1804 apply (simp add: zmod_zminus1_eq_if) |
|
1805 apply safe |
|
1806 apply (simp add: nat_mult_distrib) |
|
1807 apply (simp add: nat_diff_distrib [OF pos_mod_sign pos_mod_conj |
|
1808 [THEN conjunct2, THEN order_less_imp_le]] |
|
1809 nat_mod_distrib) |
|
1810 apply (simp add: nat_mod_distrib) |
|
1811 done |
|
1812 |
|
1813 lemmas bl_word_roti_dt = bl_word_roti_dt' [unfolded word_size] |
|
1814 |
|
1815 lemmas word_rotl_dt = bl_word_rotl_dt [THEN word_bl.Rep_inverse' [symmetric]] |
|
1816 lemmas word_rotr_dt = bl_word_rotr_dt [THEN word_bl.Rep_inverse' [symmetric]] |
|
1817 lemmas word_roti_dt = bl_word_roti_dt [THEN word_bl.Rep_inverse' [symmetric]] |
|
1818 |
|
1819 lemmas word_rotr_dt_no_bin' [simp] = |
|
1820 word_rotr_dt [where w="numeral w", unfolded to_bl_numeral] for w |
|
1821 (* FIXME: negative numerals, 0 and 1 *) |
|
1822 |
|
1823 lemmas word_rotl_dt_no_bin' [simp] = |
|
1824 word_rotl_dt [where w="numeral w", unfolded to_bl_numeral] for w |
|
1825 (* FIXME: negative numerals, 0 and 1 *) |
|
1826 |
|
1827 lemma max_word_bl: "to_bl (max_word::'a::len word) = replicate LENGTH('a) True" |
|
1828 by (fact to_bl_n1) |
|
1829 |
|
1830 lemma to_bl_mask: |
|
1831 "to_bl (mask n :: 'a::len word) = |
|
1832 replicate (LENGTH('a) - n) False @ |
|
1833 replicate (min (LENGTH('a)) n) True" |
|
1834 by (simp add: mask_bl word_rep_drop min_def) |
|
1835 |
|
1836 lemma map_replicate_True: |
|
1837 "n = length xs \<Longrightarrow> |
|
1838 map (\<lambda>(x,y). x \<and> y) (zip xs (replicate n True)) = xs" |
|
1839 by (induct xs arbitrary: n) auto |
|
1840 |
|
1841 lemma map_replicate_False: |
|
1842 "n = length xs \<Longrightarrow> map (\<lambda>(x,y). x \<and> y) |
|
1843 (zip xs (replicate n False)) = replicate n False" |
|
1844 by (induct xs arbitrary: n) auto |
|
1845 |
|
1846 lemma bl_and_mask: |
|
1847 fixes w :: "'a::len word" |
|
1848 and n :: nat |
|
1849 defines "n' \<equiv> LENGTH('a) - n" |
|
1850 shows "to_bl (w AND mask n) = replicate n' False @ drop n' (to_bl w)" |
|
1851 proof - |
|
1852 note [simp] = map_replicate_True map_replicate_False |
|
1853 have "to_bl (w AND mask n) = map2 (\<and>) (to_bl w) (to_bl (mask n::'a::len word))" |
|
1854 by (simp add: bl_word_and) |
|
1855 also have "to_bl w = take n' (to_bl w) @ drop n' (to_bl w)" |
|
1856 by simp |
|
1857 also have "map2 (\<and>) \<dots> (to_bl (mask n::'a::len word)) = |
|
1858 replicate n' False @ drop n' (to_bl w)" |
|
1859 unfolding to_bl_mask n'_def by (subst zip_append) auto |
|
1860 finally show ?thesis . |
|
1861 qed |
|
1862 |
|
1863 lemma drop_rev_takefill: |
|
1864 "length xs \<le> n \<Longrightarrow> |
|
1865 drop (n - length xs) (rev (takefill False n (rev xs))) = xs" |
|
1866 by (simp add: takefill_alt rev_take) |
|
1867 |
|
1868 declare bin_to_bl_def [simp] |
|
1869 |
|
1870 end |
|