1 | |
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2 | include "ASM/BitVectorTrie.ma". |
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3 | include "ASM/Arithmetic.ma". |
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4 | (*include "ASM/UtilBranch.ma". *) |
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5 | |
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6 | include alias "arithmetics/nat.ma". |
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7 | |
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8 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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9 | (* Program Counters & Object Code: BitVectors, Nats and Lists of Bytes *) |
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10 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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11 | |
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12 | (* from Fetch.ma *) |
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13 | definition bitvector_max_nat ≝ λlength: nat. 2^length. |
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14 | |
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15 | definition size_ok ≝ |
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16 | λn.λsize. size ≤ bitvector_max_nat n. |
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17 | |
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18 | lemma size_ok_mono : |
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19 | ∀n. ∀s,s'. s' ≤ s → size_ok n s -> size_ok n s'. |
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20 | #n #s #s' #mono #ok @(transitive_le … ok) // |
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21 | qed. |
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22 | |
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23 | lemma size_okS : |
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24 | ∀n. ∀s. size_ok n (S s) → size_ok n s. |
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25 | #n #s #ok @size_ok_mono // |
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26 | qed. |
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27 | |
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28 | lemma size_okS' : |
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29 | ∀n. ∀s. size_ok n (s+1) → size_ok n s. |
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30 | #n #s #ok @size_ok_mono // |
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31 | qed. |
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32 | |
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33 | definition address_ok ≝ |
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34 | λn.λaddr. size_ok n (S addr). |
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35 | |
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36 | lemma size_okS_to_address_ok : |
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37 | ∀n. ∀s. size_ok n (S s) → address_ok n s. |
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38 | #n #s #ok assumption. |
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39 | qed. |
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40 | |
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41 | lemma address_ok_to_size_okS : |
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42 | ∀n. ∀pc. address_ok n pc → size_ok n (S pc). |
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43 | #n #pc #ok assumption. |
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44 | qed. |
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45 | |
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46 | lemma size_ok_neq_to_address_ok : |
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47 | ∀n. ∀s. size_ok n s → s ≠ bitvector_max_nat n → address_ok n s. |
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48 | #n #s #s_ok #s_neq |
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49 | @not_eq_to_le_to_lt assumption |
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50 | qed. |
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51 | |
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52 | lemma address_ok_mono : |
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53 | ∀n. ∀pc,pc'. pc' ≤ pc → address_ok n pc -> address_ok n pc'. |
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54 | #n #pc #pc' #mono #ok |
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55 | @size_okS_to_address_ok |
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56 | @(size_ok_mono … (le_S_S …)) |
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57 | assumption |
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58 | qed. |
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59 | |
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60 | lemma address_okS : |
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61 | ∀n. ∀pc. address_ok n (S pc) → address_ok n pc. |
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62 | #n #pc #ok @address_ok_mono // |
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63 | qed. |
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64 | |
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65 | lemma address_okS' : |
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66 | ∀n. ∀pc. address_ok n (pc+1) → address_ok n pc. |
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67 | #n #pc #ok @address_ok_mono // |
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68 | qed. |
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69 | |
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70 | definition addrS : ∀n. BitVector n → BitVector n ≝ (* can overflow back to 0 *) |
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71 | λn.λpc. add n pc (bitvector_of_nat n 1). |
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72 | |
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73 | lemma addrS_is_Saddr : ∀n. ∀pc. |
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74 | let npcS ≝ S (nat_of_bitvector n pc) in |
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75 | address_ok n npcS → |
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76 | nat_of_bitvector ? (addrS ? pc) = npcS. |
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77 | #n #pc #pcS_ok cases daemon |
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78 | (* XXX: needs UtilBranch.ma; needs re-proving! |
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79 | >succ_nat_of_bitvector_half_add_1 |
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80 | [2: |
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81 | @le_plus_to_minus_r |
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82 | change with (S ? ≤ ?) <plus_n_Sm <plus_n_O assumption |
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83 | |1: cases daemon |
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84 | ] *) |
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85 | qed. |
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86 | |
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87 | lemma addrS_useful : ∀n. ∀pc. |
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88 | let npcS ≝ S (nat_of_bitvector n pc) in |
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89 | ∀m. plus npcS (S m) = bitvector_max_nat n → |
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90 | nat_of_bitvector ? (addrS ? pc) = npcS. |
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91 | #n #pc #offset #npcS_ok |
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92 | @addrS_is_Saddr // |
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93 | qed. |
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94 | |
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95 | lemma addr_useful : ∀n. ∀pc. |
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96 | let npc ≝ nat_of_bitvector n pc in |
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97 | ∀k,m. plus npc m = bitvector_max_nat n → |
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98 | k < m → address_ok n (npc + k). |
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99 | #n #pc #k #m #max #lt change with (? < ?) |
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100 | <max @monotonic_lt_plus_r assumption |
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101 | qed. |
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102 | |
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103 | lemma addr_max : ∀n. ∀pc. |
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104 | let npc ≝ nat_of_bitvector n pc in |
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105 | address_ok n npc -> address_ok n (S npc) ∨ S npc = bitvector_max_nat n. |
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106 | #n #pc #ok @(le_to_or_lt_eq … (lt_nat_of_bitvector … pc)) |
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107 | qed. |
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108 | |
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109 | lemma addr_overflow_offset : ∀n.∀pc. |
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110 | let npc ≝ nat_of_bitvector n pc in |
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111 | ∀m. npc + m = bitvector_max_nat n → add n pc (bitvector_of_nat n m) = (zero n). |
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112 | #n #pc #m #eq <(add_overflow … eq) >bitvector_of_nat_inverse_nat_of_bitvector // |
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113 | qed. |
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114 | |
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115 | lemma addrS_overflow : ∀n. ∀pc. |
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116 | let npc ≝ nat_of_bitvector n pc in |
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117 | S npc = bitvector_max_nat n → addrS n pc = (zero n). |
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118 | #n #pc #max change with (add ? ? ? = ?) |
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119 | @addr_overflow_offset >commutative_plus assumption |
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120 | qed. |
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121 | |
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122 | (* now we fix the magic number 16 *) |
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123 | |
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124 | definition ADDRESS_WIDTH ≝ 16. |
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125 | |
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126 | definition PC ≝ BitVector ADDRESS_WIDTH. |
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127 | |
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128 | definition pc0 : PC ≝ zero ?. |
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129 | |
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130 | definition nat_of_PC : PC → nat ≝ nat_of_bitvector ?. |
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131 | |
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132 | definition PC_of_nat : nat → PC ≝ bitvector_of_nat ?. |
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133 | |
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134 | definition pcS : PC → PC ≝ addrS ?. |
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135 | |
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136 | definition max_program_size ≝ bitvector_max_nat ADDRESS_WIDTH. |
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137 | |
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138 | definition program_size_ok ≝ size_ok ADDRESS_WIDTH. |
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139 | |
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140 | definition program_size_ok_mono ≝ size_ok_mono ADDRESS_WIDTH. |
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141 | |
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142 | definition program_size_okS_to_program_counter_ok ≝ size_okS_to_address_ok ADDRESS_WIDTH. |
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143 | |
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144 | definition program_size_okS ≝ size_okS ADDRESS_WIDTH. |
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145 | |
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146 | definition program_size_okS' ≝ size_okS' ADDRESS_WIDTH. |
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147 | |
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148 | definition program_counter_ok ≝ address_ok ADDRESS_WIDTH. |
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149 | |
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150 | definition program_counter_ok_mono ≝ address_ok_mono ADDRESS_WIDTH. |
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151 | |
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152 | definition program_counter_okS ≝ address_okS ADDRESS_WIDTH. |
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153 | |
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154 | definition program_counter_okS' ≝ address_okS' ADDRESS_WIDTH. |
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155 | |
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156 | definition program_size_ok_neq_to_program_counter_ok: |
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157 | ∀s. program_size_ok s → s ≠ max_program_size → program_counter_ok s ≝ |
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158 | size_ok_neq_to_address_ok ?. |
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159 | |
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160 | lemma nat_of_PC_inverse_PC_of_nat : |
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161 | ∀n. program_counter_ok n → |
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162 | nat_of_PC (PC_of_nat n) = n. |
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163 | #n #n_ok @nat_of_bitvector_bitvector_of_nat_inverse assumption |
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164 | qed. |
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165 | |
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166 | lemma PC_of_nat_inverse_nat_of_PC : |
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167 | ∀pc. PC_of_nat (nat_of_PC pc) = pc. |
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168 | #pc @bitvector_of_nat_inverse_nat_of_bitvector |
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169 | qed. |
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170 | |
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171 | lemma nat_of_PC_offset : ∀n.∀pc: PC. |
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172 | program_counter_ok (n + nat_of_PC pc) → |
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173 | nat_of_PC (add … (PC_of_nat n) pc) = n + nat_of_PC pc. |
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174 | #n #pc #n_pc_ok |
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175 | cut (program_counter_ok n) |
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176 | [@program_counter_ok_mono //] |
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177 | #n_ok change with (nat_of_bitvector … (add ? ? ?) = ?) |
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178 | >nat_of_bitvector_add |
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179 | [1: change with (((nat_of_PC ?) + (nat_of_PC ?)) = ?) |
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180 | |2: change with (program_counter_ok ((nat_of_PC ?) + (nat_of_PC ?))) |
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181 | ] |
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182 | >nat_of_PC_inverse_PC_of_nat // |
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183 | qed. |
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184 | |
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185 | lemma nat_of_PC_offset' : ∀n.∀pc: PC. |
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186 | program_counter_ok (nat_of_PC pc + n) → |
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187 | nat_of_PC (add … pc (PC_of_nat n)) = nat_of_PC pc + n. |
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188 | #n #pc >commutative_plus >add_commutative @nat_of_PC_offset |
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189 | qed. |
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190 | |
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191 | definition pcS_useful : ∀pc. |
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192 | let npcS ≝ S (nat_of_PC pc) in |
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193 | ∀m. plus npcS (S m) = max_program_size → |
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194 | nat_of_PC (pcS pc) = npcS ≝ |
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195 | addrS_useful ?. |
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196 | |
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197 | lemma pc_useful : ∀pc. |
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198 | let npc ≝ nat_of_PC pc in |
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199 | ∀k,m. plus npc k = max_program_size → |
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200 | m < k → address_ok ? (npc + m) ≝ |
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201 | λpc.λk,m.λEQ.λLT.(addr_useful … EQ LT). |
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202 | |
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203 | definition pcS_is_Spc : ∀pc: PC. |
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204 | let npcS ≝ S (nat_of_PC pc) in |
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205 | address_ok ? npcS → |
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206 | nat_of_PC (pcS pc) = npcS ≝ |
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207 | addrS_is_Saddr ?. |
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208 | |
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209 | definition pc_max : ∀pc. |
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210 | let npc ≝ nat_of_PC pc in |
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211 | program_counter_ok npc -> |
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212 | program_counter_ok (S npc) ∨ S npc = max_program_size ≝ |
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213 | addr_max ?. |
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214 | |
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215 | definition pcS_overflow : ∀pc. |
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216 | let npc ≝ nat_of_PC pc in |
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217 | S npc = max_program_size -> pcS pc = pc0 ≝ addrS_overflow ?. |
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218 | |
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219 | definition pc_overflow_offset : ∀m.∀pc. |
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220 | let npc ≝ nat_of_PC pc in |
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221 | npc + m = max_program_size -> add ? pc (PC_of_nat m) = pc0. |
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222 | #m #pc #max @addr_overflow_offset assumption |
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223 | qed. |
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224 | |
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225 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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226 | (* Memory: one of two distinguished instances of BitVectorTrie. *) |
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227 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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228 | |
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229 | definition byte0 : Byte ≝ zero ?. |
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230 | |
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231 | definition RAM ≝ BitVectorTrie Byte. |
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232 | |
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233 | definition ram0 : ∀n. RAM n ≝ |
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234 | λn. Stub …. |
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235 | |
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236 | definition memory_lookup : ∀n. BitVector n → RAM n → Byte ≝ |
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237 | λn,addr,mem. lookup ? n addr mem byte0. |
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238 | |
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239 | (* instances *) |
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240 | |
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241 | definition code_memory ≝ RAM ADDRESS_WIDTH. |
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242 | |
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243 | definition code_memory0 : code_memory ≝ ram0 ?. |
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244 | |
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245 | definition code_memory_lookup : PC → code_memory → Byte ≝ memory_lookup ?. |
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246 | |
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247 | (* ***** Object-code ***** JHM: push elsewherein ASM/*.ma? *) |
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248 | |
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249 | inductive nat_bounded : nat → nat → Type[0] ≝ |
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250 | | nat_ok : ∀n,m:nat. nat_bounded n (n+m) |
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251 | | nat_overflow : ∀n,m:nat. nat_bounded (n+S m) n. |
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252 | |
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253 | let rec nat_bound (n:nat) (m:nat) : nat_bounded n m ≝ |
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254 | match n return λx. nat_bounded x m with |
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255 | [ O ⇒ nat_ok ?? |
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256 | | S n' ⇒ |
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257 | match m return λy. nat_bounded (S n') y with |
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258 | [ O ⇒ nat_overflow O ? |
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259 | | S m' ⇒ match nat_bound n' m' return λx,y.λ_. nat_bounded (S x) (S y) with |
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260 | [ nat_ok x y ⇒ nat_ok ?? |
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261 | | nat_overflow x y ⇒ nat_overflow (S x) y |
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262 | ] |
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263 | ] |
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264 | ]. |
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265 | |
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266 | lemma nat_bound_opt : ∀N,n:nat. option (n ≤ N). |
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267 | #N #n elim (nat_bound n N) -n #n #offset |
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268 | [ @Some // |
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269 | | @None |
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270 | ] |
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271 | qed. |
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272 | |
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273 | definition program_ok_opt : ∀A. ∀instrs : list A. option (program_size_ok (S(S(|instrs|)))) |
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274 | ≝ λA.λinstrs. nat_bound_opt max_program_size (S(S(|instrs|))). (* for Policy.ma etc. *) |
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275 | |
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276 | |
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277 | |
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278 | |
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279 | |
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280 | |
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