1 | include "Universes.ma". |
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2 | include "Plogic/equality.ma". |
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3 | include "Connectives.ma". |
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4 | include "Nat.ma". |
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5 | include "Exponential.ma". |
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6 | include "Bool.ma". |
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7 | include "BitVector.ma". |
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8 | include "List.ma". |
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9 | |
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10 | ndefinition one ≝ S Z. |
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11 | ndefinition two ≝ (S(S(Z))). |
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12 | ndefinition three ≝ two + one. |
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13 | ndefinition four ≝ two + two. |
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14 | ndefinition five ≝ three + two. |
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15 | ndefinition six ≝ three + three. |
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16 | ndefinition seven ≝ three + four. |
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17 | ndefinition eight ≝ four + four. |
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18 | ndefinition nine ≝ five + four. |
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19 | ndefinition ten ≝ five + five. |
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20 | ndefinition eleven ≝ six + five. |
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21 | ndefinition twelve ≝ six + six. |
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22 | ndefinition thirteen ≝ seven + six. |
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23 | ndefinition fourteen ≝ seven + seven. |
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24 | ndefinition fifteen ≝ eight + seven. |
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25 | ndefinition sixteen ≝ eight + eight. |
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26 | ndefinition seventeen ≝ nine + eight. |
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27 | ndefinition eighteen ≝ nine + nine. |
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28 | ndefinition nineteen ≝ ten + nine. |
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29 | ndefinition twenty_four ≝ sixteen + eight. |
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30 | ndefinition thirty_two ≝ sixteen + sixteen. |
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31 | ndefinition one_hundred ≝ ten * ten. |
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32 | ndefinition one_hundred_and_twenty_eight ≝ sixteen * eight. |
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33 | ndefinition one_hundred_and_twenty_nine ≝ one_hundred_and_twenty_eight + one. |
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34 | ndefinition one_hundred_and_thirty ≝ one_hundred_and_twenty_nine + one. |
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35 | ndefinition one_hundred_and_thirty_one ≝ one_hundred_and_thirty + one. |
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36 | ndefinition one_hundred_and_thirty_five ≝ one_hundred_and_twenty_eight + seven. |
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37 | ndefinition one_hundred_and_thirty_six ≝ one_hundred_and_thirty_five + one. |
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38 | ndefinition one_hundred_and_thirty_seven ≝ one_hundred_and_thirty_six + one. |
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39 | ndefinition one_hundred_and_thirty_eight ≝ one_hundred_and_twenty_eight + ten. |
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40 | ndefinition one_hundred_and_thirty_nine ≝ one_hundred_and_thirty_eight + one. |
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41 | ndefinition one_hundred_and_forty ≝ one_hundred_and_thirty_nine + one. |
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42 | ndefinition one_hundred_and_forty_one ≝ one_hundred_and_forty + one. |
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43 | ndefinition one_hundred_and_forty_four ≝ one_hundred_and_twenty_eight + sixteen. |
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44 | ndefinition one_hundred_and_fifty_two ≝ one_hundred_and_forty_four + eight. |
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45 | ndefinition one_hundred_and_fifty_three ≝ one_hundred_and_forty_four + nine. |
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46 | ndefinition one_hundred_and_sixty ≝ one_hundred_and_forty_four + sixteen. |
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47 | ndefinition one_hundred_and_sixty_eight ≝ one_hundred_and_sixty + eight. |
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48 | ndefinition one_hundred_and_seventy_six ≝ one_hundred_and_sixty + sixteen. |
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49 | ndefinition one_hundred_and_eighty_four ≝ one_hundred_and_seventy_six + eight. |
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50 | ndefinition two_hundred ≝ one_hundred + one_hundred. |
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51 | ndefinition two_hundred_and_two ≝ two_hundred + two. |
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52 | ndefinition two_hundred_and_three ≝ two_hundred_and_two + one. |
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53 | ndefinition two_hundred_and_four ≝ two_hundred_and_three + one. |
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54 | ndefinition two_hundred_and_five ≝ two_hundred_and_four + one. |
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55 | ndefinition two_hundred_and_eight ≝ two_hundred_and_five + three. |
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56 | ndefinition two_hundred_and_twenty_four ≝ two_hundred_and_eight + sixteen. |
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57 | ndefinition two_hundred_and_forty ≝ two_hundred_and_twenty_four + sixteen. |
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58 | ndefinition two_hundred_and_fifty_six ≝ |
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59 | one_hundred_and_twenty_eight + one_hundred_and_twenty_eight. |
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60 | |
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61 | ndefinition nat_of_bool ≝ |
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62 | λb: Bool. |
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63 | match b with |
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64 | [ false ⇒ Z |
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65 | | true ⇒ S Z |
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66 | ]. |
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67 | |
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68 | ndefinition add_n_with_carry: |
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69 | ∀n: Nat. ∀b, c: BitVector n. ∀carry: Bool. Cartesian (BitVector n) (List Bool) ≝ |
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70 | λn: Nat. |
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71 | λb: BitVector n. |
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72 | λc: BitVector n. |
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73 | λcarry: Bool. |
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74 | let b_as_nat ≝ nat_of_bitvector n b in |
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75 | let c_as_nat ≝ nat_of_bitvector n c in |
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76 | let carry_as_nat ≝ nat_of_bool carry in |
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77 | let result_old ≝ b_as_nat + c_as_nat + carry_as_nat in |
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78 | let ac_flag ≝ ((modulus b_as_nat ((S (S Z)) * n)) + |
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79 | (modulus c_as_nat ((S (S Z)) * n)) + |
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80 | c_as_nat) ≥ ((S (S Z)) * n) in |
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81 | let bit_xxx ≝ (((modulus b_as_nat ((S (S Z))^(n - (S Z)))) + |
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82 | (modulus c_as_nat ((S (S Z))^(n - (S Z)))) + |
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83 | c_as_nat) ≥ ((S (S Z))^(n - (S Z)))) in |
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84 | let result ≝ modulus result_old ((S (S Z))^n) in |
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85 | let cy_flag ≝ (result_old ≥ ((S (S Z))^n)) in |
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86 | let ov_flag ≝ exclusive_disjunction cy_flag bit_xxx in |
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87 | ? (mk_Cartesian (BitVector n) ? (? (bitvector_of_nat n result)) |
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88 | (cy_flag :: ac_flag :: ov_flag :: Empty Bool)). |
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89 | #H; nassumption; |
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90 | nqed. |
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91 | |
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92 | ndefinition sub_8_with_carry: ∀b,c: BitVector eight. ∀carry: Bool. Cartesian (BitVector eight) (List Bool) ≝ |
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93 | λb: BitVector eight. |
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94 | λc: BitVector eight. |
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95 | λcarry: Bool. |
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96 | let b_as_nat ≝ nat_of_bitvector eight b in |
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97 | let c_as_nat ≝ nat_of_bitvector eight c in |
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98 | let carry_as_nat ≝ nat_of_bool carry in |
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99 | let temporary ≝ b_as_nat mod sixteen - c_as_nat mod sixteen in |
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100 | let ac_flag ≝ negation (conjunction ((b_as_nat mod sixteen) ≤ (c_as_nat mod sixteen)) (temporary ≤ carry_as_nat)) in |
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101 | let bit_six ≝ negation (conjunction ((b_as_nat mod one_hundred_and_twenty_eight) ≤ (c_as_nat mod one_hundred_and_twenty_eight)) (temporary ≤ carry_as_nat)) in |
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102 | let old_result_1 ≝ b_as_nat - c_as_nat in |
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103 | let old_result_2 ≝ old_result_1 - carry_as_nat in |
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104 | let ov_flag ≝ exclusive_disjunction carry bit_six in |
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105 | match conjunction (b_as_nat ≤ c_as_nat) (old_result_1 ≤ carry_as_nat) with |
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106 | [ false ⇒ |
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107 | let cy_flag ≝ false in |
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108 | 〈 bitvector_of_nat eight old_result_2, [cy_flag ; ac_flag ; ov_flag ] 〉 |
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109 | | true ⇒ |
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110 | let cy_flag ≝ true in |
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111 | let new_result ≝ b_as_nat + two_hundred_and_fifty_six - c_as_nat - carry_as_nat in |
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112 | 〈 bitvector_of_nat eight new_result, [ cy_flag ; ac_flag ; ov_flag ] 〉 |
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113 | ]. |
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114 | |
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115 | ndefinition add_8_with_carry ≝ add_n_with_carry eight. |
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116 | ndefinition add_16_with_carry ≝ add_n_with_carry sixteen. |
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117 | |
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118 | ndefinition increment ≝ |
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119 | λn: Nat. |
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120 | λb: BitVector n. |
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121 | let b_as_nat ≝ (nat_of_bitvector n b) + (S Z) in |
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122 | let overflow ≝ b_as_nat ≥ (S (S Z))^n in |
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123 | match overflow with |
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124 | [ false ⇒ bitvector_of_nat n b_as_nat |
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125 | | true ⇒ zero n |
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126 | ]. |
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127 | |
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128 | ndefinition decrement ≝ |
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129 | λn: Nat. |
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130 | λb: BitVector n. |
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131 | let b_as_nat ≝ nat_of_bitvector n b in |
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132 | match b_as_nat with |
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133 | [ Z ⇒ max n |
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134 | | S o ⇒ bitvector_of_nat n o |
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135 | ]. |
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136 | |
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137 | alias symbol "greater_than_or_equal" (instance 1) = "Nat greater than or equal prop". |
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138 | |
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139 | ndefinition bitvector_of_bool: |
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140 | ∀n: Nat. ∀b: Bool. BitVector n ≝ |
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141 | λn: Nat. |
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142 | λb: Bool. |
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143 | ? (pad (n - (S Z)) (S Z) (Cons Bool ? b (Empty Bool))). |
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144 | nrewrite > (plus_minus_inverse_right n ?); |
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145 | #H; |
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146 | nassumption; |
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147 | nqed. |
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148 | |
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149 | ndefinition full_add: |
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150 | ∀n: Nat. |
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151 | ∀b, c: BitVector n. |
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152 | ∀d: Bit. |
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153 | fold_left_i ? ? ( |
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154 | λb1, b2: Bool. |
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155 | λd. |
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156 | let 〈c1,r〉 ≝ d in |
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157 | 〈inclusive_disjunction (conjunction b1 b2) |
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158 | (conjunction c1 (inclusive_disjunction b1 b2)), |
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159 | (exclusive_disjunction (exclusive_disjunction b1 b2) c1) :: r〉) |
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160 | b c 〈c, [ ]〉. |
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161 | |
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162 | (fun b1 b2 (c,r) -> b1 & b2 || c & (b1 || b2),xor (xor b1 b2) c::r) l r (c,[]) |
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163 | |
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164 | ndefinition half_add ≝ |
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165 | λn: Nat. |
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166 | λb, c: BitVector n. |
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167 | full_add n b c false. |
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168 | |
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