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1 (* |
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2 ID: $Id$ |
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3 Author: Jeremy Dawson and Gerwin Klein, NICTA |
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4 |
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5 definition and basic theorems for bit-wise logical operations |
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6 for integers expressed using Pls, Min, BIT, |
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7 and converting them to and from lists of bools |
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8 *) |
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9 |
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10 theory BinOperations imports BinGeneral |
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11 |
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12 begin |
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13 |
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14 -- "bit-wise logical operations on the int type" |
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15 consts |
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16 int_and :: "int => int => int" |
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17 int_or :: "int => int => int" |
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18 bit_not :: "bit => bit" |
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19 bit_and :: "bit => bit => bit" |
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20 bit_or :: "bit => bit => bit" |
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21 bit_xor :: "bit => bit => bit" |
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22 int_not :: "int => int" |
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23 int_xor :: "int => int => int" |
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24 bin_sc :: "nat => bit => int => int" |
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25 |
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26 primrec |
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27 B0 : "bit_not bit.B0 = bit.B1" |
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28 B1 : "bit_not bit.B1 = bit.B0" |
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29 |
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30 primrec |
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31 B1 : "bit_xor bit.B1 x = bit_not x" |
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32 B0 : "bit_xor bit.B0 x = x" |
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33 |
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34 primrec |
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35 B1 : "bit_or bit.B1 x = bit.B1" |
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36 B0 : "bit_or bit.B0 x = x" |
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37 |
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38 primrec |
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39 B0 : "bit_and bit.B0 x = bit.B0" |
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40 B1 : "bit_and bit.B1 x = x" |
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41 |
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42 primrec |
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43 Z : "bin_sc 0 b w = bin_rest w BIT b" |
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44 Suc : |
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45 "bin_sc (Suc n) b w = bin_sc n b (bin_rest w) BIT bin_last w" |
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46 |
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47 defs |
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48 int_not_def : "int_not == bin_rec Numeral.Min Numeral.Pls |
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49 (%w b s. s BIT bit_not b)" |
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50 int_and_def : "int_and == bin_rec (%x. Numeral.Pls) (%y. y) |
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51 (%w b s y. s (bin_rest y) BIT (bit_and b (bin_last y)))" |
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52 int_or_def : "int_or == bin_rec (%x. x) (%y. Numeral.Min) |
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53 (%w b s y. s (bin_rest y) BIT (bit_or b (bin_last y)))" |
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54 int_xor_def : "int_xor == bin_rec (%x. x) int_not |
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55 (%w b s y. s (bin_rest y) BIT (bit_xor b (bin_last y)))" |
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56 |
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57 consts |
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58 bin_to_bl :: "nat => int => bool list" |
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59 bin_to_bl_aux :: "nat => int => bool list => bool list" |
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60 bl_to_bin :: "bool list => int" |
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61 bl_to_bin_aux :: "int => bool list => int" |
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62 |
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63 bl_of_nth :: "nat => (nat => bool) => bool list" |
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64 |
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65 primrec |
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66 Nil : "bl_to_bin_aux w [] = w" |
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67 Cons : "bl_to_bin_aux w (b # bs) = |
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68 bl_to_bin_aux (w BIT (if b then bit.B1 else bit.B0)) bs" |
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69 |
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70 primrec |
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71 Z : "bin_to_bl_aux 0 w bl = bl" |
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72 Suc : "bin_to_bl_aux (Suc n) w bl = |
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73 bin_to_bl_aux n (bin_rest w) ((bin_last w = bit.B1) # bl)" |
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74 |
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75 defs |
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76 bin_to_bl_def : "bin_to_bl n w == bin_to_bl_aux n w []" |
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77 bl_to_bin_def : "bl_to_bin bs == bl_to_bin_aux Numeral.Pls bs" |
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78 |
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79 primrec |
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80 Suc : "bl_of_nth (Suc n) f = f n # bl_of_nth n f" |
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81 Z : "bl_of_nth 0 f = []" |
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82 |
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83 consts |
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84 takefill :: "'a => nat => 'a list => 'a list" |
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85 app2 :: "('a => 'b => 'c) => 'a list => 'b list => 'c list" |
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86 |
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87 -- "takefill - like take but if argument list too short," |
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88 -- "extends result to get requested length" |
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89 primrec |
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90 Z : "takefill fill 0 xs = []" |
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91 Suc : "takefill fill (Suc n) xs = ( |
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92 case xs of [] => fill # takefill fill n xs |
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93 | y # ys => y # takefill fill n ys)" |
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94 |
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95 defs |
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96 app2_def : "app2 f as bs == map (split f) (zip as bs)" |
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97 |
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98 -- "rcat and rsplit" |
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99 consts |
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100 bin_rcat :: "nat => int list => int" |
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101 bin_rsplit_aux :: "nat * int list * nat * int => int list" |
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102 bin_rsplit :: "nat => (nat * int) => int list" |
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103 bin_rsplitl_aux :: "nat * int list * nat * int => int list" |
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104 bin_rsplitl :: "nat => (nat * int) => int list" |
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105 |
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106 recdef bin_rsplit_aux "measure (fst o snd o snd)" |
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107 "bin_rsplit_aux (n, bs, (m, c)) = |
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108 (if m = 0 | n = 0 then bs else |
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109 let (a, b) = bin_split n c |
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110 in bin_rsplit_aux (n, b # bs, (m - n, a)))" |
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111 |
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112 recdef bin_rsplitl_aux "measure (fst o snd o snd)" |
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113 "bin_rsplitl_aux (n, bs, (m, c)) = |
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114 (if m = 0 | n = 0 then bs else |
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115 let (a, b) = bin_split (min m n) c |
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116 in bin_rsplitl_aux (n, b # bs, (m - n, a)))" |
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117 |
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118 defs |
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119 bin_rcat_def : "bin_rcat n bs == foldl (%u v. bin_cat u n v) Numeral.Pls bs" |
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120 bin_rsplit_def : "bin_rsplit n w == bin_rsplit_aux (n, [], w)" |
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121 bin_rsplitl_def : "bin_rsplitl n w == bin_rsplitl_aux (n, [], w)" |
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122 |
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123 |
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124 lemma int_not_simps [simp]: |
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125 "int_not Numeral.Pls = Numeral.Min" |
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126 "int_not Numeral.Min = Numeral.Pls" |
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127 "int_not (w BIT b) = int_not w BIT bit_not b" |
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128 by (unfold int_not_def) (auto intro: bin_rec_simps) |
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129 |
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130 lemma bit_extra_simps [simp]: |
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131 "bit_and x bit.B0 = bit.B0" |
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132 "bit_and x bit.B1 = x" |
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133 "bit_or x bit.B1 = bit.B1" |
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134 "bit_or x bit.B0 = x" |
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135 "bit_xor x bit.B1 = bit_not x" |
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136 "bit_xor x bit.B0 = x" |
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137 by (cases x, auto)+ |
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138 |
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139 lemma bit_ops_comm: |
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140 "bit_and x y = bit_and y x" |
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141 "bit_or x y = bit_or y x" |
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142 "bit_xor x y = bit_xor y x" |
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143 by (cases y, auto)+ |
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144 |
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145 lemma bit_ops_same [simp]: |
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146 "bit_and x x = x" |
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147 "bit_or x x = x" |
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148 "bit_xor x x = bit.B0" |
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149 by (cases x, auto)+ |
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150 |
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151 lemma bit_not_not [simp]: "bit_not (bit_not x) = x" |
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152 by (cases x) auto |
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153 |
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154 lemma int_xor_Pls [simp]: |
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155 "int_xor Numeral.Pls x = x" |
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156 unfolding int_xor_def by (simp add: bin_rec_PM) |
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157 |
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158 lemma int_xor_Min [simp]: |
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159 "int_xor Numeral.Min x = int_not x" |
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160 unfolding int_xor_def by (simp add: bin_rec_PM) |
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161 |
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162 lemma int_xor_Bits [simp]: |
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163 "int_xor (x BIT b) (y BIT c) = int_xor x y BIT bit_xor b c" |
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164 apply (unfold int_xor_def) |
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165 apply (rule bin_rec_simps (1) [THEN fun_cong, THEN trans]) |
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166 apply (rule ext, simp) |
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167 prefer 2 |
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168 apply simp |
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169 apply (rule ext) |
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170 apply (simp add: int_not_simps [symmetric]) |
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171 done |
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172 |
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173 lemma int_xor_x_simps': |
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174 "int_xor w (Numeral.Pls BIT bit.B0) = w" |
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175 "int_xor w (Numeral.Min BIT bit.B1) = int_not w" |
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176 apply (induct w rule: bin_induct) |
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177 apply simp_all[4] |
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178 apply (unfold int_xor_Bits) |
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179 apply clarsimp+ |
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180 done |
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181 |
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182 lemmas int_xor_extra_simps [simp] = int_xor_x_simps' [simplified arith_simps] |
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183 |
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184 lemma int_or_Pls [simp]: |
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185 "int_or Numeral.Pls x = x" |
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186 by (unfold int_or_def) (simp add: bin_rec_PM) |
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187 |
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188 lemma int_or_Min [simp]: |
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189 "int_or Numeral.Min x = Numeral.Min" |
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190 by (unfold int_or_def) (simp add: bin_rec_PM) |
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191 |
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192 lemma int_or_Bits [simp]: |
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193 "int_or (x BIT b) (y BIT c) = int_or x y BIT bit_or b c" |
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194 unfolding int_or_def by (simp add: bin_rec_simps) |
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195 |
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196 lemma int_or_x_simps': |
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197 "int_or w (Numeral.Pls BIT bit.B0) = w" |
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198 "int_or w (Numeral.Min BIT bit.B1) = Numeral.Min" |
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199 apply (induct w rule: bin_induct) |
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200 apply simp_all[4] |
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201 apply (unfold int_or_Bits) |
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202 apply clarsimp+ |
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203 done |
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204 |
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205 lemmas int_or_extra_simps [simp] = int_or_x_simps' [simplified arith_simps] |
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206 |
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207 |
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208 lemma int_and_Pls [simp]: |
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209 "int_and Numeral.Pls x = Numeral.Pls" |
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210 unfolding int_and_def by (simp add: bin_rec_PM) |
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211 |
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212 lemma int_and_Min [simp]: |
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213 "int_and Numeral.Min x = x" |
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214 unfolding int_and_def by (simp add: bin_rec_PM) |
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215 |
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216 lemma int_and_Bits [simp]: |
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217 "int_and (x BIT b) (y BIT c) = int_and x y BIT bit_and b c" |
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218 unfolding int_and_def by (simp add: bin_rec_simps) |
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219 |
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220 lemma int_and_x_simps': |
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221 "int_and w (Numeral.Pls BIT bit.B0) = Numeral.Pls" |
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222 "int_and w (Numeral.Min BIT bit.B1) = w" |
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223 apply (induct w rule: bin_induct) |
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224 apply simp_all[4] |
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225 apply (unfold int_and_Bits) |
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226 apply clarsimp+ |
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227 done |
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228 |
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229 lemmas int_and_extra_simps [simp] = int_and_x_simps' [simplified arith_simps] |
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230 |
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231 (* commutativity of the above *) |
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232 lemma bin_ops_comm: |
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233 shows |
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234 int_and_comm: "!!y. int_and x y = int_and y x" and |
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235 int_or_comm: "!!y. int_or x y = int_or y x" and |
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236 int_xor_comm: "!!y. int_xor x y = int_xor y x" |
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237 apply (induct x rule: bin_induct) |
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238 apply simp_all[6] |
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239 apply (case_tac y rule: bin_exhaust, simp add: bit_ops_comm)+ |
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240 done |
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241 |
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242 lemma bin_ops_same [simp]: |
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243 "int_and x x = x" |
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244 "int_or x x = x" |
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245 "int_xor x x = Numeral.Pls" |
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246 by (induct x rule: bin_induct) auto |
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247 |
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248 lemma int_not_not [simp]: "int_not (int_not x) = x" |
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249 by (induct x rule: bin_induct) auto |
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250 |
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251 lemmas bin_log_esimps = |
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252 int_and_extra_simps int_or_extra_simps int_xor_extra_simps |
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253 int_and_Pls int_and_Min int_or_Pls int_or_Min int_xor_Pls int_xor_Min |
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254 |
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255 (* potential for looping *) |
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256 declare bin_rsplit_aux.simps [simp del] |
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257 declare bin_rsplitl_aux.simps [simp del] |
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258 |
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259 |
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260 lemma bin_sign_cat: |
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261 "!!y. bin_sign (bin_cat x n y) = bin_sign x" |
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262 by (induct n) auto |
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263 |
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264 lemma bin_cat_Suc_Bit: |
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265 "bin_cat w (Suc n) (v BIT b) = bin_cat w n v BIT b" |
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266 by auto |
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267 |
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268 lemma bin_nth_cat: |
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269 "!!n y. bin_nth (bin_cat x k y) n = |
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270 (if n < k then bin_nth y n else bin_nth x (n - k))" |
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271 apply (induct k) |
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272 apply clarsimp |
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273 apply (case_tac n, auto) |
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274 done |
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275 |
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276 lemma bin_nth_split: |
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277 "!!b c. bin_split n c = (a, b) ==> |
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278 (ALL k. bin_nth a k = bin_nth c (n + k)) & |
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279 (ALL k. bin_nth b k = (k < n & bin_nth c k))" |
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280 apply (induct n) |
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281 apply clarsimp |
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282 apply (clarsimp simp: Let_def split: ls_splits) |
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283 apply (case_tac k) |
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284 apply auto |
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285 done |
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286 |
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287 lemma bin_cat_assoc: |
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288 "!!z. bin_cat (bin_cat x m y) n z = bin_cat x (m + n) (bin_cat y n z)" |
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289 by (induct n) auto |
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290 |
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291 lemma bin_cat_assoc_sym: "!!z m. |
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292 bin_cat x m (bin_cat y n z) = bin_cat (bin_cat x (m - n) y) (min m n) z" |
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293 apply (induct n, clarsimp) |
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294 apply (case_tac m, auto) |
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295 done |
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296 |
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297 lemma bin_cat_Pls [simp]: |
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298 "!!w. bin_cat Numeral.Pls n w = bintrunc n w" |
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299 by (induct n) auto |
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300 |
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301 lemma bintr_cat1: |
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302 "!!b. bintrunc (k + n) (bin_cat a n b) = bin_cat (bintrunc k a) n b" |
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303 by (induct n) auto |
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304 |
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305 lemma bintr_cat: "bintrunc m (bin_cat a n b) = |
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306 bin_cat (bintrunc (m - n) a) n (bintrunc (min m n) b)" |
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307 by (rule bin_eqI) (auto simp: bin_nth_cat nth_bintr) |
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308 |
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309 lemma bintr_cat_same [simp]: |
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310 "bintrunc n (bin_cat a n b) = bintrunc n b" |
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311 by (auto simp add : bintr_cat) |
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312 |
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313 lemma cat_bintr [simp]: |
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314 "!!b. bin_cat a n (bintrunc n b) = bin_cat a n b" |
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315 by (induct n) auto |
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316 |
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317 lemma split_bintrunc: |
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318 "!!b c. bin_split n c = (a, b) ==> b = bintrunc n c" |
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319 by (induct n) (auto simp: Let_def split: ls_splits) |
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320 |
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321 lemma bin_cat_split: |
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322 "!!v w. bin_split n w = (u, v) ==> w = bin_cat u n v" |
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323 by (induct n) (auto simp: Let_def split: ls_splits) |
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324 |
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325 lemma bin_split_cat: |
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326 "!!w. bin_split n (bin_cat v n w) = (v, bintrunc n w)" |
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327 by (induct n) auto |
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328 |
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329 lemma bin_split_Pls [simp]: |
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330 "bin_split n Numeral.Pls = (Numeral.Pls, Numeral.Pls)" |
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331 by (induct n) (auto simp: Let_def split: ls_splits) |
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332 |
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333 lemma bin_split_Min [simp]: |
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334 "bin_split n Numeral.Min = (Numeral.Min, bintrunc n Numeral.Min)" |
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335 by (induct n) (auto simp: Let_def split: ls_splits) |
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336 |
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337 lemma bin_split_trunc: |
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338 "!!m b c. bin_split (min m n) c = (a, b) ==> |
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339 bin_split n (bintrunc m c) = (bintrunc (m - n) a, b)" |
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340 apply (induct n, clarsimp) |
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341 apply (simp add: bin_rest_trunc Let_def split: ls_splits) |
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342 apply (case_tac m) |
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343 apply (auto simp: Let_def split: ls_splits) |
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344 done |
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345 |
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346 lemma bin_split_trunc1: |
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347 "!!m b c. bin_split n c = (a, b) ==> |
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348 bin_split n (bintrunc m c) = (bintrunc (m - n) a, bintrunc m b)" |
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349 apply (induct n, clarsimp) |
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350 apply (simp add: bin_rest_trunc Let_def split: ls_splits) |
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351 apply (case_tac m) |
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352 apply (auto simp: Let_def split: ls_splits) |
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353 done |
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354 |
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355 lemma bin_cat_num: |
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356 "!!b. bin_cat a n b = a * 2 ^ n + bintrunc n b" |
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357 apply (induct n, clarsimp) |
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358 apply (simp add: Bit_def cong: number_of_False_cong) |
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359 done |
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360 |
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361 lemma bin_split_num: |
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362 "!!b. bin_split n b = (b div 2 ^ n, b mod 2 ^ n)" |
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363 apply (induct n, clarsimp) |
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364 apply (simp add: bin_rest_div zdiv_zmult2_eq) |
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365 apply (case_tac b rule: bin_exhaust) |
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366 apply simp |
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367 apply (simp add: Bit_def zmod_zmult_zmult1 p1mod22k |
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368 split: bit.split |
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369 cong: number_of_False_cong) |
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370 done |
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371 |
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372 |
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373 (* basic properties of logical (bit-wise) operations *) |
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374 |
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375 lemma bbw_ao_absorb: |
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376 "!!y. int_and x (int_or y x) = x & int_or x (int_and y x) = x" |
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377 apply (induct x rule: bin_induct) |
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378 apply auto |
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379 apply (case_tac [!] y rule: bin_exhaust) |
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380 apply auto |
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381 apply (case_tac [!] bit) |
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382 apply auto |
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383 done |
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384 |
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385 lemma bbw_ao_absorbs_other: |
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386 "int_and x (int_or x y) = x \<and> int_or (int_and y x) x = x" |
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387 "int_and (int_or y x) x = x \<and> int_or x (int_and x y) = x" |
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388 "int_and (int_or x y) x = x \<and> int_or (int_and x y) x = x" |
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389 apply (auto simp: bbw_ao_absorb int_or_comm) |
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390 apply (subst int_or_comm) |
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391 apply (simp add: bbw_ao_absorb) |
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392 apply (subst int_and_comm) |
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393 apply (subst int_or_comm) |
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394 apply (simp add: bbw_ao_absorb) |
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395 apply (subst int_and_comm) |
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396 apply (simp add: bbw_ao_absorb) |
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397 done |
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398 |
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399 lemmas bbw_ao_absorbs [simp] = bbw_ao_absorb bbw_ao_absorbs_other |
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400 |
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401 lemma int_xor_not: |
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402 "!!y. int_xor (int_not x) y = int_not (int_xor x y) & |
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403 int_xor x (int_not y) = int_not (int_xor x y)" |
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404 apply (induct x rule: bin_induct) |
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405 apply auto |
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406 apply (case_tac y rule: bin_exhaust, auto, |
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407 case_tac b, auto)+ |
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408 done |
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409 |
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410 lemma bbw_assocs': |
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411 "!!y z. int_and (int_and x y) z = int_and x (int_and y z) & |
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412 int_or (int_or x y) z = int_or x (int_or y z) & |
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413 int_xor (int_xor x y) z = int_xor x (int_xor y z)" |
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414 apply (induct x rule: bin_induct) |
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415 apply (auto simp: int_xor_not) |
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416 apply (case_tac [!] y rule: bin_exhaust) |
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417 apply (case_tac [!] z rule: bin_exhaust) |
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418 apply (case_tac [!] bit) |
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419 apply (case_tac [!] b) |
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420 apply auto |
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421 done |
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422 |
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423 lemma int_and_assoc: |
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424 "int_and (int_and x y) z = int_and x (int_and y z)" |
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425 by (simp add: bbw_assocs') |
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426 |
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427 lemma int_or_assoc: |
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428 "int_or (int_or x y) z = int_or x (int_or y z)" |
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429 by (simp add: bbw_assocs') |
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430 |
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431 lemma int_xor_assoc: |
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432 "int_xor (int_xor x y) z = int_xor x (int_xor y z)" |
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433 by (simp add: bbw_assocs') |
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434 |
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435 lemmas bbw_assocs = int_and_assoc int_or_assoc int_xor_assoc |
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436 |
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437 lemma bbw_lcs [simp]: |
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438 "int_and y (int_and x z) = int_and x (int_and y z)" |
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439 "int_or y (int_or x z) = int_or x (int_or y z)" |
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440 "int_xor y (int_xor x z) = int_xor x (int_xor y z)" |
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441 apply (auto simp: bbw_assocs [symmetric]) |
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442 apply (auto simp: bin_ops_comm) |
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443 done |
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444 |
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445 lemma bbw_not_dist: |
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446 "!!y. int_not (int_or x y) = int_and (int_not x) (int_not y)" |
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447 "!!y. int_not (int_and x y) = int_or (int_not x) (int_not y)" |
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448 apply (induct x rule: bin_induct) |
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449 apply auto |
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450 apply (case_tac [!] y rule: bin_exhaust) |
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451 apply (case_tac [!] bit, auto) |
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452 done |
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453 |
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454 lemma bbw_oa_dist: |
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455 "!!y z. int_or (int_and x y) z = |
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456 int_and (int_or x z) (int_or y z)" |
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457 apply (induct x rule: bin_induct) |
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458 apply auto |
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459 apply (case_tac y rule: bin_exhaust) |
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460 apply (case_tac z rule: bin_exhaust) |
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461 apply (case_tac ba, auto) |
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462 done |
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463 |
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464 lemma bbw_ao_dist: |
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465 "!!y z. int_and (int_or x y) z = |
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466 int_or (int_and x z) (int_and y z)" |
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467 apply (induct x rule: bin_induct) |
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468 apply auto |
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469 apply (case_tac y rule: bin_exhaust) |
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470 apply (case_tac z rule: bin_exhaust) |
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471 apply (case_tac ba, auto) |
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472 done |
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473 |
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474 declare bin_ops_comm [simp] bbw_assocs [simp] |
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475 |
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476 lemma plus_and_or [rule_format]: |
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477 "ALL y. int_and x y + int_or x y = x + y" |
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478 apply (induct x rule: bin_induct) |
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479 apply clarsimp |
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480 apply clarsimp |
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481 apply clarsimp |
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482 apply (case_tac y rule: bin_exhaust) |
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483 apply clarsimp |
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484 apply (unfold Bit_def) |
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485 apply clarsimp |
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486 apply (erule_tac x = "x" in allE) |
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487 apply (simp split: bit.split) |
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488 done |
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489 |
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490 lemma le_int_or: |
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491 "!!x. bin_sign y = Numeral.Pls ==> x <= int_or x y" |
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492 apply (induct y rule: bin_induct) |
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493 apply clarsimp |
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494 apply clarsimp |
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495 apply (case_tac x rule: bin_exhaust) |
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496 apply (case_tac b) |
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497 apply (case_tac [!] bit) |
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498 apply (auto simp: less_eq_numeral_code) |
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499 done |
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500 |
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501 lemmas int_and_le = |
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502 xtr3 [OF bbw_ao_absorbs (2) [THEN conjunct2, symmetric] le_int_or] ; |
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503 |
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504 (** nth bit, set/clear **) |
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505 |
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506 lemma bin_nth_sc [simp]: |
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507 "!!w. bin_nth (bin_sc n b w) n = (b = bit.B1)" |
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508 by (induct n) auto |
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509 |
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510 lemma bin_sc_sc_same [simp]: |
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511 "!!w. bin_sc n c (bin_sc n b w) = bin_sc n c w" |
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512 by (induct n) auto |
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513 |
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514 lemma bin_sc_sc_diff: |
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515 "!!w m. m ~= n ==> |
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516 bin_sc m c (bin_sc n b w) = bin_sc n b (bin_sc m c w)" |
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517 apply (induct n) |
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518 apply safe |
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519 apply (case_tac [!] m) |
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520 apply auto |
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521 done |
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522 |
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523 lemma bin_nth_sc_gen: |
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524 "!!w m. bin_nth (bin_sc n b w) m = (if m = n then b = bit.B1 else bin_nth w m)" |
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525 by (induct n) (case_tac [!] m, auto) |
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526 |
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527 lemma bin_sc_nth [simp]: |
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528 "!!w. (bin_sc n (If (bin_nth w n) bit.B1 bit.B0) w) = w" |
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529 by (induct n) auto |
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530 |
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531 lemma bin_sign_sc [simp]: |
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532 "!!w. bin_sign (bin_sc n b w) = bin_sign w" |
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533 by (induct n) auto |
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534 |
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535 lemma bin_sc_bintr [simp]: |
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536 "!!w m. bintrunc m (bin_sc n x (bintrunc m (w))) = bintrunc m (bin_sc n x w)" |
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537 apply (induct n) |
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538 apply (case_tac [!] w rule: bin_exhaust) |
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539 apply (case_tac [!] m, auto) |
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540 done |
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541 |
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542 lemma bin_clr_le: |
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543 "!!w. bin_sc n bit.B0 w <= w" |
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544 apply (induct n) |
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545 apply (case_tac [!] w rule: bin_exhaust) |
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546 apply auto |
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547 apply (unfold Bit_def) |
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548 apply (simp_all split: bit.split) |
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549 done |
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550 |
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551 lemma bin_set_ge: |
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552 "!!w. bin_sc n bit.B1 w >= w" |
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553 apply (induct n) |
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554 apply (case_tac [!] w rule: bin_exhaust) |
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555 apply auto |
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556 apply (unfold Bit_def) |
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557 apply (simp_all split: bit.split) |
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558 done |
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559 |
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560 lemma bintr_bin_clr_le: |
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561 "!!w m. bintrunc n (bin_sc m bit.B0 w) <= bintrunc n w" |
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562 apply (induct n) |
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563 apply simp |
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564 apply (case_tac w rule: bin_exhaust) |
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565 apply (case_tac m) |
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566 apply auto |
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567 apply (unfold Bit_def) |
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568 apply (simp_all split: bit.split) |
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569 done |
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570 |
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571 lemma bintr_bin_set_ge: |
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572 "!!w m. bintrunc n (bin_sc m bit.B1 w) >= bintrunc n w" |
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573 apply (induct n) |
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574 apply simp |
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575 apply (case_tac w rule: bin_exhaust) |
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576 apply (case_tac m) |
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577 apply auto |
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578 apply (unfold Bit_def) |
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579 apply (simp_all split: bit.split) |
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580 done |
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581 |
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582 lemma bin_nth_ops: |
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583 "!!x y. bin_nth (int_and x y) n = (bin_nth x n & bin_nth y n)" |
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584 "!!x y. bin_nth (int_or x y) n = (bin_nth x n | bin_nth y n)" |
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585 "!!x y. bin_nth (int_xor x y) n = (bin_nth x n ~= bin_nth y n)" |
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586 "!!x. bin_nth (int_not x) n = (~ bin_nth x n)" |
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587 apply (induct n) |
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588 apply safe |
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589 apply (case_tac [!] x rule: bin_exhaust) |
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590 apply simp_all |
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591 apply (case_tac [!] y rule: bin_exhaust) |
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592 apply simp_all |
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593 apply (auto dest: not_B1_is_B0 intro: B1_ass_B0) |
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594 done |
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595 |
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596 lemma bin_sc_FP [simp]: "bin_sc n bit.B0 Numeral.Pls = Numeral.Pls" |
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597 by (induct n) auto |
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598 |
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599 lemma bin_sc_TM [simp]: "bin_sc n bit.B1 Numeral.Min = Numeral.Min" |
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600 by (induct n) auto |
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601 |
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602 lemmas bin_sc_simps = bin_sc.Z bin_sc.Suc bin_sc_TM bin_sc_FP |
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603 |
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604 lemma bin_sc_minus: |
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605 "0 < n ==> bin_sc (Suc (n - 1)) b w = bin_sc n b w" |
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606 by auto |
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607 |
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608 lemmas bin_sc_Suc_minus = |
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609 trans [OF bin_sc_minus [symmetric] bin_sc.Suc, standard] |
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610 |
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611 lemmas bin_sc_Suc_pred [simp] = |
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612 bin_sc_Suc_minus [of "number_of bin", simplified nobm1, standard] |
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613 |
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614 (* interaction between bit-wise and arithmetic *) |
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615 (* good example of bin_induction *) |
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616 lemma bin_add_not: "x + int_not x = Numeral.Min" |
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617 apply (induct x rule: bin_induct) |
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618 apply clarsimp |
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619 apply clarsimp |
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620 apply (case_tac bit, auto) |
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621 done |
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622 |
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623 (* truncating results of bit-wise operations *) |
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624 lemma bin_trunc_ao: |
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625 "!!x y. int_and (bintrunc n x) (bintrunc n y) = bintrunc n (int_and x y)" |
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626 "!!x y. int_or (bintrunc n x) (bintrunc n y) = bintrunc n (int_or x y)" |
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627 apply (induct n) |
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628 apply auto |
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629 apply (case_tac [!] x rule: bin_exhaust) |
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630 apply (case_tac [!] y rule: bin_exhaust) |
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631 apply auto |
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632 done |
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633 |
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634 lemma bin_trunc_xor: |
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635 "!!x y. bintrunc n (int_xor (bintrunc n x) (bintrunc n y)) = |
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636 bintrunc n (int_xor x y)" |
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637 apply (induct n) |
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638 apply auto |
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639 apply (case_tac [!] x rule: bin_exhaust) |
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640 apply (case_tac [!] y rule: bin_exhaust) |
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641 apply auto |
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642 done |
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643 |
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644 lemma bin_trunc_not: |
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645 "!!x. bintrunc n (int_not (bintrunc n x)) = bintrunc n (int_not x)" |
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646 apply (induct n) |
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647 apply auto |
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648 apply (case_tac [!] x rule: bin_exhaust) |
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649 apply auto |
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650 done |
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651 |
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652 (* want theorems of the form of bin_trunc_xor *) |
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653 lemma bintr_bintr_i: |
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654 "x = bintrunc n y ==> bintrunc n x = bintrunc n y" |
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655 by auto |
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656 |
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657 lemmas bin_trunc_and = bin_trunc_ao(1) [THEN bintr_bintr_i] |
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658 lemmas bin_trunc_or = bin_trunc_ao(2) [THEN bintr_bintr_i] |
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659 |
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660 lemma nth_2p_bin: |
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661 "!!m. bin_nth (2 ^ n) m = (m = n)" |
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662 apply (induct n) |
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663 apply clarsimp |
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664 apply safe |
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665 apply (case_tac m) |
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666 apply (auto simp: trans [OF numeral_1_eq_1 [symmetric] number_of_eq]) |
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667 apply (case_tac m) |
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668 apply (auto simp: Bit_B0_2t [symmetric]) |
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669 done |
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670 |
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671 (* for use when simplifying with bin_nth_Bit *) |
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672 |
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673 lemma ex_eq_or: |
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674 "(EX m. n = Suc m & (m = k | P m)) = (n = Suc k | (EX m. n = Suc m & P m))" |
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675 by auto |
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676 |
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677 end |
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678 |