1 | include "ASM/Assembly.ma". |
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2 | include "ASM/Interpret.ma". |
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3 | include "ASM/StatusProofs.ma". |
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4 | |
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5 | definition bit_elim_prop: ∀P: bool → Prop. Prop ≝ |
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6 | λP. |
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7 | P true ∧ P false. |
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8 | |
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9 | let rec bitvector_elim_prop_internal |
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10 | (n: nat) (P: BitVector n → Prop) (m: nat) on m: m ≤ n → BitVector (n - m) → Prop ≝ |
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11 | match m return λm. m ≤ n → BitVector (n - m) → Prop with |
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12 | [ O ⇒ λprf1. λprefix. P ? |
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13 | | S n' ⇒ λprf2. λprefix. bit_elim_prop (λbit. bitvector_elim_prop_internal n P n' ? ?) |
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14 | ]. |
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15 | [ applyS prefix |
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16 | | letin res ≝ (bit ::: prefix) |
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17 | < (minus_S_S ? ?) |
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18 | > (minus_Sn_m ? ?) |
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19 | [ @ res |
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20 | | @ prf2 |
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21 | ] |
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22 | | /2/ |
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23 | ]. |
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24 | qed. |
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25 | |
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26 | definition bitvector_elim_prop ≝ |
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27 | λn: nat. |
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28 | λP: BitVector n → Prop. |
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29 | bitvector_elim_prop_internal n P n ? ?. |
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30 | [ @ (le_n ?) |
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31 | | < (minus_n_n ?) |
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32 | @ [[ ]] |
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33 | ] |
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34 | qed. |
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35 | |
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36 | lemma bool_eq_internal_eq: |
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37 | ∀b, c. |
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38 | (λb. λc. (if b then c else (if c then false else true))) b c = true → b = c. |
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39 | #b #c |
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40 | cases b |
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41 | [ normalize // |
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42 | | normalize |
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43 | cases c |
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44 | [ normalize // |
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45 | | normalize // |
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46 | ] |
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47 | ] |
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48 | qed. |
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49 | |
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50 | definition bit_elim: ∀P: bool → bool. bool ≝ |
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51 | λP. |
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52 | P true ∧ P false. |
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53 | |
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54 | let rec bitvector_elim_internal |
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55 | (n: nat) (P: BitVector n → bool) (m: nat) on m: m ≤ n → BitVector (n - m) → bool ≝ |
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56 | match m return λm. m ≤ n → BitVector (n - m) → bool with |
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57 | [ O ⇒ λprf1. λprefix. P ? |
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58 | | S n' ⇒ λprf2. λprefix. bit_elim (λbit. bitvector_elim_internal n P n' ? ?) |
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59 | ]. |
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60 | [ applyS prefix |
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61 | | letin res ≝ (bit ::: prefix) |
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62 | < (minus_S_S ? ?) |
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63 | > (minus_Sn_m ? ?) |
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64 | [ @ res |
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65 | | @ prf2 |
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66 | ] |
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67 | | /2/ |
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68 | ]. |
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69 | qed. |
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70 | |
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71 | definition bitvector_elim ≝ |
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72 | λn: nat. |
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73 | λP: BitVector n → bool. |
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74 | bitvector_elim_internal n P n ? ?. |
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75 | [ @ (le_n ?) |
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76 | | < (minus_n_n ?) |
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77 | @ [[ ]] |
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78 | ] |
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79 | qed. |
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80 | |
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81 | lemma super_rewrite2: |
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82 | ∀A:Type[0].∀n,m.∀v1: Vector A n.∀v2: Vector A m. |
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83 | ∀P: ∀m. Vector A m → Prop. |
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84 | n=m → v1 ≃ v2 → P n v1 → P m v2. |
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85 | #A #n #m #v1 #v2 #P #EQ <EQ in v2; #V #JMEQ >JMEQ // |
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86 | qed. |
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87 | |
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88 | lemma vector_cons_append: |
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89 | ∀A: Type[0]. |
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90 | ∀n: nat. |
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91 | ∀e: A. |
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92 | ∀v: Vector A n. |
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93 | e ::: v = [[ e ]] @@ v. |
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94 | # A # N # E # V |
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95 | elim V |
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96 | [ normalize % |
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97 | | # NN # AA # VV # IH |
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98 | normalize |
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99 | % |
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100 | qed. |
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101 | |
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102 | lemma vector_cons_append2: |
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103 | ∀A: Type[0]. |
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104 | ∀n, m: nat. |
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105 | ∀v: Vector A n. |
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106 | ∀q: Vector A m. |
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107 | ∀hd: A. |
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108 | hd:::(v@@q) = (hd:::v)@@q. |
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109 | #A #n #m #v #q |
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110 | elim v |
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111 | [ #hd % |
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112 | | #n' #hd' #tl' #ih #hd' <ih % |
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113 | ] |
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114 | qed. |
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115 | |
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116 | lemma jmeq_cons_vector_monotone: |
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117 | ∀A: Type[0]. |
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118 | ∀m, n: nat. |
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119 | ∀v: Vector A m. |
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120 | ∀q: Vector A n. |
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121 | ∀prf: m = n. |
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122 | ∀hd: A. |
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123 | v ≃ q → hd:::v ≃ hd:::q. |
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124 | #A #m #n #v #q #prf #hd #E |
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125 | @(super_rewrite2 A … E) |
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126 | [ assumption | @jmrefl ] |
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127 | qed. |
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128 | |
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129 | lemma vector_associative_append: |
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130 | ∀A: Type[0]. |
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131 | ∀n, m, o: nat. |
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132 | ∀v: Vector A n. |
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133 | ∀q: Vector A m. |
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134 | ∀r: Vector A o. |
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135 | ((v @@ q) @@ r) |
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136 | ≃ |
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137 | (v @@ (q @@ r)). |
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138 | #A #n #m #o #v #q #r |
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139 | elim v |
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140 | [ % |
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141 | | #n' #hd #tl #ih |
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142 | <(vector_cons_append2 A … hd) |
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143 | @jmeq_cons_vector_monotone |
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144 | // |
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145 | ] |
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146 | qed. |
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147 | |
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148 | lemma mem_middle_vector: |
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149 | ∀A: Type[0]. |
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150 | ∀m, o: nat. |
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151 | ∀eq: A → A → bool. |
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152 | ∀reflex: ∀a. eq a a = true. |
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153 | ∀p: Vector A m. |
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154 | ∀a: A. |
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155 | ∀r: Vector A o. |
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156 | mem A eq ? (p@@(a:::r)) a = true. |
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157 | # A # M # O # EQ # REFLEX # P # A |
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158 | elim P |
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159 | [ normalize |
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160 | > (REFLEX A) |
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161 | normalize |
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162 | # H |
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163 | % |
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164 | | # NN # AA # PP # IH |
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165 | normalize |
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166 | cases (EQ A AA) // |
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167 | @ IH |
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168 | ] |
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169 | qed. |
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170 | |
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171 | lemma mem_monotonic_wrt_append: |
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172 | ∀A: Type[0]. |
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173 | ∀m, o: nat. |
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174 | ∀eq: A → A → bool. |
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175 | ∀reflex: ∀a. eq a a = true. |
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176 | ∀p: Vector A m. |
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177 | ∀a: A. |
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178 | ∀r: Vector A o. |
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179 | mem A eq ? r a = true → mem A eq ? (p @@ r) a = true. |
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180 | # A # M # O # EQ # REFLEX # P # A |
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181 | elim P |
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182 | [ #R #H @H |
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183 | | #NN #AA # PP # IH #R #H |
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184 | normalize |
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185 | cases (EQ A AA) |
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186 | [ normalize % |
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187 | | @ IH @ H |
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188 | ] |
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189 | ] |
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190 | qed. |
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191 | |
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192 | lemma subvector_multiple_append: |
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193 | ∀A: Type[0]. |
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194 | ∀o, n: nat. |
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195 | ∀eq: A → A → bool. |
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196 | ∀refl: ∀a. eq a a = true. |
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197 | ∀h: Vector A o. |
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198 | ∀v: Vector A n. |
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199 | ∀m: nat. |
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200 | ∀q: Vector A m. |
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201 | bool_to_Prop (subvector_with A ? ? eq v (h @@ q @@ v)). |
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202 | # A # O # N # EQ # REFLEX # H # V |
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203 | elim V |
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204 | [ normalize |
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205 | # M # V % |
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206 | | # NN # AA # VV # IH # MM # QQ |
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207 | change with (bool_to_Prop (andb ??)) |
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208 | cut ((mem A EQ (O + (MM + S NN)) (H@@QQ@@AA:::VV) AA) = true) |
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209 | [ |
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210 | | # HH > HH |
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211 | > (vector_cons_append ? ? AA VV) |
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212 | change with (bool_to_Prop (subvector_with ??????)) |
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213 | @(super_rewrite2 A ((MM + 1)+ NN) (MM+S NN) ?? |
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214 | (λSS.λVS.bool_to_Prop (subvector_with ?? (O+SS) ?? (H@@VS))) |
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215 | ? |
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216 | (vector_associative_append A ? ? ? QQ [[AA]] VV)) |
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217 | [ >associative_plus // |
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218 | | @IH ] |
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219 | ] |
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220 | @(mem_monotonic_wrt_append) |
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221 | [ @ REFLEX |
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222 | | @(mem_monotonic_wrt_append) |
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223 | [ @ REFLEX |
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224 | | normalize |
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225 | > REFLEX |
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226 | normalize |
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227 | % |
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228 | ] |
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229 | ] |
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230 | qed. |
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231 | |
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232 | lemma vector_cons_empty: |
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233 | ∀A: Type[0]. |
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234 | ∀n: nat. |
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235 | ∀v: Vector A n. |
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236 | [[ ]] @@ v = v. |
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237 | # A # N # V |
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238 | elim V |
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239 | [ normalize % |
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240 | | # NN # HH # VV #H % |
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241 | ] |
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242 | qed. |
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243 | |
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244 | corollary subvector_hd_tl: |
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245 | ∀A: Type[0]. |
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246 | ∀o: nat. |
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247 | ∀eq: A → A → bool. |
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248 | ∀refl: ∀a. eq a a = true. |
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249 | ∀h: A. |
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250 | ∀v: Vector A o. |
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251 | bool_to_Prop (subvector_with A ? ? eq v (h ::: v)). |
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252 | # A # O # EQ # REFLEX # H # V |
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253 | > (vector_cons_append A ? H V) |
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254 | < (vector_cons_empty A ? ([[H]] @@ V)) |
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255 | @ (subvector_multiple_append A ? ? EQ REFLEX [[]] V ? [[ H ]]) |
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256 | qed. |
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257 | |
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258 | lemma eq_a_reflexive: |
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259 | ∀a. eq_a a a = true. |
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260 | # A |
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261 | cases A |
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262 | % |
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263 | qed. |
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264 | |
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265 | lemma is_in_monotonic_wrt_append: |
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266 | ∀m, n: nat. |
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267 | ∀p: Vector addressing_mode_tag m. |
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268 | ∀q: Vector addressing_mode_tag n. |
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269 | ∀to_search: addressing_mode. |
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270 | bool_to_Prop (is_in ? p to_search) → bool_to_Prop (is_in ? (q @@ p) to_search). |
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271 | # M # N # P # Q # TO_SEARCH |
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272 | # H |
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273 | elim Q |
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274 | [ normalize |
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275 | @ H |
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276 | | # NN # PP # QQ # IH |
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277 | normalize |
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278 | cases (is_a PP TO_SEARCH) |
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279 | [ normalize |
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280 | % |
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281 | | normalize |
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282 | normalize in IH |
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283 | @ IH |
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284 | ] |
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285 | ] |
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286 | qed. |
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287 | |
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288 | corollary is_in_hd_tl: |
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289 | ∀to_search: addressing_mode. |
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290 | ∀hd: addressing_mode_tag. |
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291 | ∀n: nat. |
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292 | ∀v: Vector addressing_mode_tag n. |
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293 | bool_to_Prop (is_in ? v to_search) → bool_to_Prop (is_in ? (hd:::v) to_search). |
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294 | # TO_SEARCH # HD # N # V |
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295 | elim V |
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296 | [ # H |
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297 | normalize in H; |
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298 | cases H |
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299 | | # NN # HHD # VV # IH # HH |
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300 | > vector_cons_append |
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301 | > (vector_cons_append ? ? HHD VV) |
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302 | @ (is_in_monotonic_wrt_append ? 1 ([[HHD]]@@VV) [[HD]] TO_SEARCH) |
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303 | @ HH |
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304 | ] |
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305 | qed. |
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306 | |
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307 | let rec list_addressing_mode_tags_elim |
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308 | (n: nat) (l: Vector addressing_mode_tag (S n)) on l: (l → bool) → bool ≝ |
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309 | match l return λx.match x with [O ⇒ λl: Vector … O. bool | S x' ⇒ λl: Vector addressing_mode_tag (S x'). |
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310 | (l → bool) → bool ] with |
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311 | [ VEmpty ⇒ true |
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312 | | VCons len hd tl ⇒ λP. |
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313 | let process_hd ≝ |
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314 | match hd return λhd. ∀P: hd:::tl → bool. bool with |
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315 | [ direct ⇒ λP.bitvector_elim 8 (λx. P (DIRECT x)) |
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316 | | indirect ⇒ λP.bit_elim (λx. P (INDIRECT x)) |
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317 | | ext_indirect ⇒ λP.bit_elim (λx. P (EXT_INDIRECT x)) |
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318 | | registr ⇒ λP.bitvector_elim 3 (λx. P (REGISTER x)) |
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319 | | acc_a ⇒ λP.P ACC_A |
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320 | | acc_b ⇒ λP.P ACC_B |
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321 | | dptr ⇒ λP.P DPTR |
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322 | | data ⇒ λP.bitvector_elim 8 (λx. P (DATA x)) |
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323 | | data16 ⇒ λP.bitvector_elim 16 (λx. P (DATA16 x)) |
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324 | | acc_dptr ⇒ λP.P ACC_DPTR |
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325 | | acc_pc ⇒ λP.P ACC_PC |
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326 | | ext_indirect_dptr ⇒ λP.P EXT_INDIRECT_DPTR |
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327 | | indirect_dptr ⇒ λP.P INDIRECT_DPTR |
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328 | | carry ⇒ λP.P CARRY |
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329 | | bit_addr ⇒ λP.bitvector_elim 8 (λx. P (BIT_ADDR x)) |
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330 | | n_bit_addr ⇒ λP.bitvector_elim 8 (λx. P (N_BIT_ADDR x)) |
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331 | | relative ⇒ λP.bitvector_elim 8 (λx. P (RELATIVE x)) |
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332 | | addr11 ⇒ λP.bitvector_elim 11 (λx. P (ADDR11 x)) |
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333 | | addr16 ⇒ λP.bitvector_elim 16 (λx. P (ADDR16 x)) |
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334 | ] |
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335 | in |
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336 | andb (process_hd P) |
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337 | (match len return λx. x = len → bool with |
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338 | [ O ⇒ λprf. true |
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339 | | S y ⇒ λprf. list_addressing_mode_tags_elim y ? P ] (refl ? len)) |
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340 | ]. |
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341 | try % |
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342 | [ 2: cases (sym_eq ??? prf); @tl |
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343 | | generalize in match H; generalize in match tl; cases prf; |
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344 | (* cases prf in tl H; : ??? WAS WORKING BEFORE *) |
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345 | #tl |
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346 | normalize in ⊢ (∀_: %. ?) |
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347 | # H |
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348 | whd |
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349 | normalize in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?]) |
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350 | cases (is_a hd (subaddressing_modeel y tl H)) whd // ] |
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351 | qed. |
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352 | |
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353 | definition product_elim ≝ |
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354 | λm, n: nat. |
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355 | λv: Vector addressing_mode_tag (S m). |
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356 | λq: Vector addressing_mode_tag (S n). |
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357 | λP: (v × q) → bool. |
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358 | list_addressing_mode_tags_elim ? v (λx. list_addressing_mode_tags_elim ? q (λy. P 〈x, y〉)). |
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359 | |
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360 | definition union_elim ≝ |
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361 | λA, B: Type[0]. |
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362 | λelimA: (A → bool) → bool. |
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363 | λelimB: (B → bool) → bool. |
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364 | λelimU: A ⊎ B → bool. |
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365 | elimA (λa. elimB (λb. elimU (inl ? ? a) ∧ elimU (inr ? ? b))). |
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366 | |
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367 | (* |
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368 | definition preinstruction_elim: ∀P: preinstruction [[ relative ]] → bool. bool ≝ |
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369 | λP. |
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370 | list_addressing_mode_tags_elim ? [[ registr ; direct ; indirect ; data ]] (λaddr. P (ADD ? ACC_A addr)) ∧ |
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371 | list_addressing_mode_tags_elim ? [[ registr ; direct ; indirect ; data ]] (λaddr. P (ADDC ? ACC_A addr)) ∧ |
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372 | list_addressing_mode_tags_elim ? [[ registr ; direct ; indirect ; data ]] (λaddr. P (SUBB ? ACC_A addr)) ∧ |
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373 | list_addressing_mode_tags_elim ? [[ acc_a ; registr ; direct ; indirect ; dptr ]] (λaddr. P (INC ? addr)) ∧ |
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374 | list_addressing_mode_tags_elim ? [[ acc_a ; registr ; direct ; indirect ]] (λaddr. P (DEC ? addr)) ∧ |
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375 | list_addressing_mode_tags_elim ? [[acc_b]] (λaddr. P (MUL ? ACC_A addr)) ∧ |
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376 | list_addressing_mode_tags_elim ? [[acc_b]] (λaddr. P (DIV ? ACC_A addr)) ∧ |
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377 | list_addressing_mode_tags_elim ? [[ registr ; direct ]] (λaddr. bitvector_elim 8 (λr. P (DJNZ ? addr (RELATIVE r)))) ∧ |
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378 | list_addressing_mode_tags_elim ? [[ acc_a ; carry ; bit_addr ]] (λaddr. P (CLR ? addr)) ∧ |
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379 | list_addressing_mode_tags_elim ? [[ acc_a ; carry ; bit_addr ]] (λaddr. P (CPL ? addr)) ∧ |
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380 | P (DA ? ACC_A) ∧ |
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381 | bitvector_elim 8 (λr. P (JC ? (RELATIVE r))) ∧ |
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382 | bitvector_elim 8 (λr. P (JNC ? (RELATIVE r))) ∧ |
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383 | bitvector_elim 8 (λr. P (JZ ? (RELATIVE r))) ∧ |
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384 | bitvector_elim 8 (λr. P (JNZ ? (RELATIVE r))) ∧ |
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385 | bitvector_elim 8 (λr. (bitvector_elim 8 (λb: BitVector 8. P (JB ? (BIT_ADDR b) (RELATIVE r))))) ∧ |
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386 | bitvector_elim 8 (λr. (bitvector_elim 8 (λb: BitVector 8. P (JNB ? (BIT_ADDR b) (RELATIVE r))))) ∧ |
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387 | bitvector_elim 8 (λr. (bitvector_elim 8 (λb: BitVector 8. P (JBC ? (BIT_ADDR b) (RELATIVE r))))) ∧ |
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388 | list_addressing_mode_tags_elim ? [[ registr; direct ]] (λaddr. bitvector_elim 8 (λr. P (DJNZ ? addr (RELATIVE r)))) ∧ |
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389 | P (RL ? ACC_A) ∧ |
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390 | P (RLC ? ACC_A) ∧ |
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391 | P (RR ? ACC_A) ∧ |
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392 | P (RRC ? ACC_A) ∧ |
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393 | P (SWAP ? ACC_A) ∧ |
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394 | P (RET ?) ∧ |
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395 | P (RETI ?) ∧ |
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396 | P (NOP ?) ∧ |
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397 | bit_elim (λb. P (XCHD ? ACC_A (INDIRECT b))) ∧ |
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398 | list_addressing_mode_tags_elim ? [[ carry; bit_addr ]] (λaddr. P (SETB ? addr)) ∧ |
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399 | bitvector_elim 8 (λaddr. P (PUSH ? (DIRECT addr))) ∧ |
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400 | bitvector_elim 8 (λaddr. P (POP ? (DIRECT addr))) ∧ |
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401 | union_elim ? ? (product_elim ? ? [[ acc_a ]] [[ direct; data ]]) |
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402 | (product_elim ? ? [[ registr; indirect ]] [[ data ]]) |
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403 | (λd. bitvector_elim 8 (λb. P (CJNE ? d (RELATIVE b)))) ∧ |
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404 | list_addressing_mode_tags_elim ? [[ registr; direct; indirect ]] (λaddr. P (XCH ? ACC_A addr)) ∧ |
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405 | union_elim ? ? (product_elim ? ? [[acc_a]] [[ data ; registr ; direct ; indirect ]]) |
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406 | (product_elim ? ? [[direct]] [[ acc_a ; data ]]) |
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407 | (λd. P (XRL ? d)) ∧ |
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408 | union_elim ? ? (union_elim ? ? (product_elim ? ? [[acc_a]] [[ registr ; direct ; indirect ; data ]]) |
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409 | (product_elim ? ? [[direct]] [[ acc_a ; data ]])) |
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410 | (product_elim ? ? [[carry]] [[ bit_addr ; n_bit_addr]]) |
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411 | (λd. P (ANL ? d)) ∧ |
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412 | union_elim ? ? (union_elim ? ? (product_elim ? ? [[acc_a]] [[ registr ; data ; direct ; indirect ]]) |
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413 | (product_elim ? ? [[direct]] [[ acc_a ; data ]])) |
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414 | (product_elim ? ? [[carry]] [[ bit_addr ; n_bit_addr]]) |
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415 | (λd. P (ORL ? d)) ∧ |
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416 | union_elim ? ? (product_elim ? ? [[acc_a]] [[ ext_indirect ; ext_indirect_dptr ]]) |
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417 | (product_elim ? ? [[ ext_indirect ; ext_indirect_dptr ]] [[acc_a]]) |
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418 | (λd. P (MOVX ? d)) ∧ |
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419 | union_elim ? ? ( |
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420 | union_elim ? ? ( |
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421 | union_elim ? ? ( |
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422 | union_elim ? ? ( |
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423 | union_elim ? ? (product_elim ? ? [[acc_a]] [[ registr ; direct ; indirect ; data ]]) |
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424 | (product_elim ? ? [[ registr ; indirect ]] [[ acc_a ; direct ; data ]])) |
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425 | (product_elim ? ? [[direct]] [[ acc_a ; registr ; direct ; indirect ; data ]])) |
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426 | (product_elim ? ? [[dptr]] [[data16]])) |
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427 | (product_elim ? ? [[carry]] [[bit_addr]])) |
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428 | (product_elim ? ? [[bit_addr]] [[carry]]) |
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429 | (λd. P (MOV ? d)). |
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430 | % |
---|
431 | qed. |
---|
432 | |
---|
433 | definition instruction_elim: ∀P: instruction → bool. bool ≝ |
---|
434 | λP. (* |
---|
435 | bitvector_elim 11 (λx. P (ACALL (ADDR11 x))) ∧ |
---|
436 | bitvector_elim 16 (λx. P (LCALL (ADDR16 x))) ∧ |
---|
437 | bitvector_elim 11 (λx. P (AJMP (ADDR11 x))) ∧ |
---|
438 | bitvector_elim 16 (λx. P (LJMP (ADDR16 x))) ∧ *) |
---|
439 | bitvector_elim 8 (λx. P (SJMP (RELATIVE x))). (* ∧ |
---|
440 | P (JMP INDIRECT_DPTR) ∧ |
---|
441 | list_addressing_mode_tags_elim ? [[ acc_dptr; acc_pc ]] (λa. P (MOVC ACC_A a)) ∧ |
---|
442 | preinstruction_elim (λp. P (RealInstruction p)). *) |
---|
443 | % |
---|
444 | qed. |
---|
445 | |
---|
446 | |
---|
447 | axiom instruction_elim_complete: |
---|
448 | ∀P. instruction_elim P = true → ∀i. P i = true. |
---|
449 | *) |
---|
450 | (*definition eq_instruction ≝ |
---|
451 | λi, j: instruction. |
---|
452 | true.*) |
---|
453 | |
---|
454 | definition eq_addressing_mode: addressing_mode → addressing_mode → bool ≝ |
---|
455 | λa, b: addressing_mode. |
---|
456 | match a with |
---|
457 | [ DIRECT d ⇒ |
---|
458 | match b with |
---|
459 | [ DIRECT e ⇒ eq_bv ? d e |
---|
460 | | _ ⇒ false |
---|
461 | ] |
---|
462 | | INDIRECT b' ⇒ |
---|
463 | match b with |
---|
464 | [ INDIRECT e ⇒ eq_b b' e |
---|
465 | | _ ⇒ false |
---|
466 | ] |
---|
467 | | EXT_INDIRECT b' ⇒ |
---|
468 | match b with |
---|
469 | [ EXT_INDIRECT e ⇒ eq_b b' e |
---|
470 | | _ ⇒ false |
---|
471 | ] |
---|
472 | | REGISTER bv ⇒ |
---|
473 | match b with |
---|
474 | [ REGISTER bv' ⇒ eq_bv ? bv bv' |
---|
475 | | _ ⇒ false |
---|
476 | ] |
---|
477 | | ACC_A ⇒ match b with [ ACC_A ⇒ true | _ ⇒ false ] |
---|
478 | | ACC_B ⇒ match b with [ ACC_B ⇒ true | _ ⇒ false ] |
---|
479 | | DPTR ⇒ match b with [ DPTR ⇒ true | _ ⇒ false ] |
---|
480 | | DATA b' ⇒ |
---|
481 | match b with |
---|
482 | [ DATA e ⇒ eq_bv ? b' e |
---|
483 | | _ ⇒ false |
---|
484 | ] |
---|
485 | | DATA16 w ⇒ |
---|
486 | match b with |
---|
487 | [ DATA16 e ⇒ eq_bv ? w e |
---|
488 | | _ ⇒ false |
---|
489 | ] |
---|
490 | | ACC_DPTR ⇒ match b with [ ACC_DPTR ⇒ true | _ ⇒ false ] |
---|
491 | | ACC_PC ⇒ match b with [ ACC_PC ⇒ true | _ ⇒ false ] |
---|
492 | | EXT_INDIRECT_DPTR ⇒ match b with [ EXT_INDIRECT_DPTR ⇒ true | _ ⇒ false ] |
---|
493 | | INDIRECT_DPTR ⇒ match b with [ INDIRECT_DPTR ⇒ true | _ ⇒ false ] |
---|
494 | | CARRY ⇒ match b with [ CARRY ⇒ true | _ ⇒ false ] |
---|
495 | | BIT_ADDR b' ⇒ |
---|
496 | match b with |
---|
497 | [ BIT_ADDR e ⇒ eq_bv ? b' e |
---|
498 | | _ ⇒ false |
---|
499 | ] |
---|
500 | | N_BIT_ADDR b' ⇒ |
---|
501 | match b with |
---|
502 | [ N_BIT_ADDR e ⇒ eq_bv ? b' e |
---|
503 | | _ ⇒ false |
---|
504 | ] |
---|
505 | | RELATIVE n ⇒ |
---|
506 | match b with |
---|
507 | [ RELATIVE e ⇒ eq_bv ? n e |
---|
508 | | _ ⇒ false |
---|
509 | ] |
---|
510 | | ADDR11 w ⇒ |
---|
511 | match b with |
---|
512 | [ ADDR11 e ⇒ eq_bv ? w e |
---|
513 | | _ ⇒ false |
---|
514 | ] |
---|
515 | | ADDR16 w ⇒ |
---|
516 | match b with |
---|
517 | [ ADDR16 e ⇒ eq_bv ? w e |
---|
518 | | _ ⇒ false |
---|
519 | ] |
---|
520 | ]. |
---|
521 | |
---|
522 | lemma eq_bv_refl: |
---|
523 | ∀n, b. |
---|
524 | eq_bv n b b = true. |
---|
525 | #n #b cases b // |
---|
526 | qed. |
---|
527 | |
---|
528 | lemma eq_b_refl: |
---|
529 | ∀b. |
---|
530 | eq_b b b = true. |
---|
531 | #b cases b // |
---|
532 | qed. |
---|
533 | |
---|
534 | lemma eq_addressing_mode_refl: |
---|
535 | ∀a. eq_addressing_mode a a = true. |
---|
536 | #a cases a try #arg1 try #arg2 try @eq_bv_refl try @eq_b_refl |
---|
537 | try normalize % |
---|
538 | qed. |
---|
539 | |
---|
540 | definition eq_sum: ∀A, B. (A → A → bool) → (B → B → bool) → (A ⊎ B) → (A ⊎ B) → bool ≝ |
---|
541 | λlt, rt, leq, req, left, right. |
---|
542 | match left with |
---|
543 | [ inl l ⇒ |
---|
544 | match right with |
---|
545 | [ inl l' ⇒ leq l l' |
---|
546 | | _ ⇒ false |
---|
547 | ] |
---|
548 | | inr r ⇒ |
---|
549 | match right with |
---|
550 | [ inr r' ⇒ req r r' |
---|
551 | | _ ⇒ false |
---|
552 | ] |
---|
553 | ]. |
---|
554 | |
---|
555 | definition eq_prod: ∀A, B. (A → A → bool) → (B → B → bool) → (A × B) → (A × B) → bool ≝ |
---|
556 | λlt, rt, leq, req, left, right. |
---|
557 | let 〈l, r〉 ≝ left in |
---|
558 | let 〈l', r'〉 ≝ right in |
---|
559 | leq l l' ∧ req r r'. |
---|
560 | |
---|
561 | definition eq_preinstruction: preinstruction [[relative]] → preinstruction [[relative]] → bool ≝ |
---|
562 | λi, j. |
---|
563 | match i with |
---|
564 | [ ADD arg1 arg2 ⇒ |
---|
565 | match j with |
---|
566 | [ ADD arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
---|
567 | | _ ⇒ false |
---|
568 | ] |
---|
569 | | ADDC arg1 arg2 ⇒ |
---|
570 | match j with |
---|
571 | [ ADDC arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
---|
572 | | _ ⇒ false |
---|
573 | ] |
---|
574 | | SUBB arg1 arg2 ⇒ |
---|
575 | match j with |
---|
576 | [ SUBB arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
---|
577 | | _ ⇒ false |
---|
578 | ] |
---|
579 | | INC arg ⇒ |
---|
580 | match j with |
---|
581 | [ INC arg' ⇒ eq_addressing_mode arg arg' |
---|
582 | | _ ⇒ false |
---|
583 | ] |
---|
584 | | DEC arg ⇒ |
---|
585 | match j with |
---|
586 | [ DEC arg' ⇒ eq_addressing_mode arg arg' |
---|
587 | | _ ⇒ false |
---|
588 | ] |
---|
589 | | MUL arg1 arg2 ⇒ |
---|
590 | match j with |
---|
591 | [ MUL arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
---|
592 | | _ ⇒ false |
---|
593 | ] |
---|
594 | | DIV arg1 arg2 ⇒ |
---|
595 | match j with |
---|
596 | [ DIV arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
---|
597 | | _ ⇒ false |
---|
598 | ] |
---|
599 | | DA arg ⇒ |
---|
600 | match j with |
---|
601 | [ DA arg' ⇒ eq_addressing_mode arg arg' |
---|
602 | | _ ⇒ false |
---|
603 | ] |
---|
604 | | JC arg ⇒ |
---|
605 | match j with |
---|
606 | [ JC arg' ⇒ eq_addressing_mode arg arg' |
---|
607 | | _ ⇒ false |
---|
608 | ] |
---|
609 | | JNC arg ⇒ |
---|
610 | match j with |
---|
611 | [ JNC arg' ⇒ eq_addressing_mode arg arg' |
---|
612 | | _ ⇒ false |
---|
613 | ] |
---|
614 | | JB arg1 arg2 ⇒ |
---|
615 | match j with |
---|
616 | [ JB arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
---|
617 | | _ ⇒ false |
---|
618 | ] |
---|
619 | | JNB arg1 arg2 ⇒ |
---|
620 | match j with |
---|
621 | [ JNB arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
---|
622 | | _ ⇒ false |
---|
623 | ] |
---|
624 | | JBC arg1 arg2 ⇒ |
---|
625 | match j with |
---|
626 | [ JBC arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
---|
627 | | _ ⇒ false |
---|
628 | ] |
---|
629 | | JZ arg ⇒ |
---|
630 | match j with |
---|
631 | [ JZ arg' ⇒ eq_addressing_mode arg arg' |
---|
632 | | _ ⇒ false |
---|
633 | ] |
---|
634 | | JNZ arg ⇒ |
---|
635 | match j with |
---|
636 | [ JNZ arg' ⇒ eq_addressing_mode arg arg' |
---|
637 | | _ ⇒ false |
---|
638 | ] |
---|
639 | | CJNE arg1 arg2 ⇒ |
---|
640 | match j with |
---|
641 | [ CJNE arg1' arg2' ⇒ |
---|
642 | let prod_eq_left ≝ eq_prod [[acc_a]] [[direct; data]] eq_addressing_mode eq_addressing_mode in |
---|
643 | let prod_eq_right ≝ eq_prod [[registr; indirect]] [[data]] eq_addressing_mode eq_addressing_mode in |
---|
644 | let arg1_eq ≝ eq_sum ? ? prod_eq_left prod_eq_right in |
---|
645 | arg1_eq arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
---|
646 | | _ ⇒ false |
---|
647 | ] |
---|
648 | | DJNZ arg1 arg2 ⇒ |
---|
649 | match j with |
---|
650 | [ DJNZ arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
---|
651 | | _ ⇒ false |
---|
652 | ] |
---|
653 | | CLR arg ⇒ |
---|
654 | match j with |
---|
655 | [ CLR arg' ⇒ eq_addressing_mode arg arg' |
---|
656 | | _ ⇒ false |
---|
657 | ] |
---|
658 | | CPL arg ⇒ |
---|
659 | match j with |
---|
660 | [ CPL arg' ⇒ eq_addressing_mode arg arg' |
---|
661 | | _ ⇒ false |
---|
662 | ] |
---|
663 | | RL arg ⇒ |
---|
664 | match j with |
---|
665 | [ RL arg' ⇒ eq_addressing_mode arg arg' |
---|
666 | | _ ⇒ false |
---|
667 | ] |
---|
668 | | RLC arg ⇒ |
---|
669 | match j with |
---|
670 | [ RLC arg' ⇒ eq_addressing_mode arg arg' |
---|
671 | | _ ⇒ false |
---|
672 | ] |
---|
673 | | RR arg ⇒ |
---|
674 | match j with |
---|
675 | [ RR arg' ⇒ eq_addressing_mode arg arg' |
---|
676 | | _ ⇒ false |
---|
677 | ] |
---|
678 | | RRC arg ⇒ |
---|
679 | match j with |
---|
680 | [ RRC arg' ⇒ eq_addressing_mode arg arg' |
---|
681 | | _ ⇒ false |
---|
682 | ] |
---|
683 | | SWAP arg ⇒ |
---|
684 | match j with |
---|
685 | [ SWAP arg' ⇒ eq_addressing_mode arg arg' |
---|
686 | | _ ⇒ false |
---|
687 | ] |
---|
688 | | SETB arg ⇒ |
---|
689 | match j with |
---|
690 | [ SETB arg' ⇒ eq_addressing_mode arg arg' |
---|
691 | | _ ⇒ false |
---|
692 | ] |
---|
693 | | PUSH arg ⇒ |
---|
694 | match j with |
---|
695 | [ PUSH arg' ⇒ eq_addressing_mode arg arg' |
---|
696 | | _ ⇒ false |
---|
697 | ] |
---|
698 | | POP arg ⇒ |
---|
699 | match j with |
---|
700 | [ POP arg' ⇒ eq_addressing_mode arg arg' |
---|
701 | | _ ⇒ false |
---|
702 | ] |
---|
703 | | XCH arg1 arg2 ⇒ |
---|
704 | match j with |
---|
705 | [ XCH arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
---|
706 | | _ ⇒ false |
---|
707 | ] |
---|
708 | | XCHD arg1 arg2 ⇒ |
---|
709 | match j with |
---|
710 | [ XCHD arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
---|
711 | | _ ⇒ false |
---|
712 | ] |
---|
713 | | RET ⇒ match j with [ RET ⇒ true | _ ⇒ false ] |
---|
714 | | RETI ⇒ match j with [ RETI ⇒ true | _ ⇒ false ] |
---|
715 | | NOP ⇒ match j with [ NOP ⇒ true | _ ⇒ false ] |
---|
716 | | MOVX arg ⇒ |
---|
717 | match j with |
---|
718 | [ MOVX arg' ⇒ |
---|
719 | let prod_eq_left ≝ eq_prod [[acc_a]] [[ext_indirect; ext_indirect_dptr]] eq_addressing_mode eq_addressing_mode in |
---|
720 | let prod_eq_right ≝ eq_prod [[ext_indirect; ext_indirect_dptr]] [[acc_a]] eq_addressing_mode eq_addressing_mode in |
---|
721 | let sum_eq ≝ eq_sum ? ? prod_eq_left prod_eq_right in |
---|
722 | sum_eq arg arg' |
---|
723 | | _ ⇒ false |
---|
724 | ] |
---|
725 | | XRL arg ⇒ |
---|
726 | match j with |
---|
727 | [ XRL arg' ⇒ |
---|
728 | let prod_eq_left ≝ eq_prod [[acc_a]] [[ data ; registr ; direct ; indirect ]] eq_addressing_mode eq_addressing_mode in |
---|
729 | let prod_eq_right ≝ eq_prod [[direct]] [[ acc_a ; data ]] eq_addressing_mode eq_addressing_mode in |
---|
730 | let sum_eq ≝ eq_sum ? ? prod_eq_left prod_eq_right in |
---|
731 | sum_eq arg arg' |
---|
732 | | _ ⇒ false |
---|
733 | ] |
---|
734 | | ORL arg ⇒ |
---|
735 | match j with |
---|
736 | [ ORL arg' ⇒ |
---|
737 | let prod_eq_left1 ≝ eq_prod [[acc_a]] [[ registr ; data ; direct ; indirect ]] eq_addressing_mode eq_addressing_mode in |
---|
738 | let prod_eq_left2 ≝ eq_prod [[direct]] [[ acc_a; data ]] eq_addressing_mode eq_addressing_mode in |
---|
739 | let prod_eq_left ≝ eq_sum ? ? prod_eq_left1 prod_eq_left2 in |
---|
740 | let prod_eq_right ≝ eq_prod [[carry]] [[ bit_addr ; n_bit_addr]] eq_addressing_mode eq_addressing_mode in |
---|
741 | let sum_eq ≝ eq_sum ? ? prod_eq_left prod_eq_right in |
---|
742 | sum_eq arg arg' |
---|
743 | | _ ⇒ false |
---|
744 | ] |
---|
745 | | ANL arg ⇒ |
---|
746 | match j with |
---|
747 | [ ANL arg' ⇒ |
---|
748 | let prod_eq_left1 ≝ eq_prod [[acc_a]] [[ registr ; direct ; indirect ; data ]] eq_addressing_mode eq_addressing_mode in |
---|
749 | let prod_eq_left2 ≝ eq_prod [[direct]] [[ acc_a; data ]] eq_addressing_mode eq_addressing_mode in |
---|
750 | let prod_eq_left ≝ eq_sum ? ? prod_eq_left1 prod_eq_left2 in |
---|
751 | let prod_eq_right ≝ eq_prod [[carry]] [[ bit_addr ; n_bit_addr]] eq_addressing_mode eq_addressing_mode in |
---|
752 | let sum_eq ≝ eq_sum ? ? prod_eq_left prod_eq_right in |
---|
753 | sum_eq arg arg' |
---|
754 | | _ ⇒ false |
---|
755 | ] |
---|
756 | | MOV arg ⇒ |
---|
757 | match j with |
---|
758 | [ MOV arg' ⇒ |
---|
759 | let prod_eq_6 ≝ eq_prod [[acc_a]] [[registr; direct; indirect; data]] eq_addressing_mode eq_addressing_mode in |
---|
760 | let prod_eq_5 ≝ eq_prod [[registr; indirect]] [[acc_a; direct; data]] eq_addressing_mode eq_addressing_mode in |
---|
761 | let prod_eq_4 ≝ eq_prod [[direct]] [[acc_a; registr; direct; indirect; data]] eq_addressing_mode eq_addressing_mode in |
---|
762 | let prod_eq_3 ≝ eq_prod [[dptr]] [[data16]] eq_addressing_mode eq_addressing_mode in |
---|
763 | let prod_eq_2 ≝ eq_prod [[carry]] [[bit_addr]] eq_addressing_mode eq_addressing_mode in |
---|
764 | let prod_eq_1 ≝ eq_prod [[bit_addr]] [[carry]] eq_addressing_mode eq_addressing_mode in |
---|
765 | let sum_eq_1 ≝ eq_sum ? ? prod_eq_6 prod_eq_5 in |
---|
766 | let sum_eq_2 ≝ eq_sum ? ? sum_eq_1 prod_eq_4 in |
---|
767 | let sum_eq_3 ≝ eq_sum ? ? sum_eq_2 prod_eq_3 in |
---|
768 | let sum_eq_4 ≝ eq_sum ? ? sum_eq_3 prod_eq_2 in |
---|
769 | let sum_eq_5 ≝ eq_sum ? ? sum_eq_4 prod_eq_1 in |
---|
770 | sum_eq_5 arg arg' |
---|
771 | | _ ⇒ false |
---|
772 | ] |
---|
773 | ]. |
---|
774 | |
---|
775 | lemma eq_sum_refl: |
---|
776 | ∀A, B: Type[0]. |
---|
777 | ∀leq, req. |
---|
778 | ∀s. |
---|
779 | ∀leq_refl: (∀t. leq t t = true). |
---|
780 | ∀req_refl: (∀u. req u u = true). |
---|
781 | eq_sum A B leq req s s = true. |
---|
782 | #A #B #leq #req #s #leq_refl #req_refl |
---|
783 | cases s whd in ⊢ (? → ??%?) // |
---|
784 | qed. |
---|
785 | |
---|
786 | lemma eq_prod_refl: |
---|
787 | ∀A, B: Type[0]. |
---|
788 | ∀leq, req. |
---|
789 | ∀s. |
---|
790 | ∀leq_refl: (∀t. leq t t = true). |
---|
791 | ∀req_refl: (∀u. req u u = true). |
---|
792 | eq_prod A B leq req s s = true. |
---|
793 | #A #B #leq #req #s #leq_refl #req_refl |
---|
794 | cases s whd in ⊢ (? → ? → ??%?) #l #r >leq_refl normalize @req_refl |
---|
795 | qed. |
---|
796 | |
---|
797 | lemma eq_preinstruction_refl: |
---|
798 | ∀i. |
---|
799 | eq_preinstruction i i = true. |
---|
800 | #i cases i try #arg1 try #arg2 |
---|
801 | try @eq_addressing_mode_refl |
---|
802 | [1,2,3,4,5,6,7,8,10,16,17,18,19,20: |
---|
803 | whd in ⊢ (??%?) |
---|
804 | try % |
---|
805 | >eq_addressing_mode_refl |
---|
806 | >eq_addressing_mode_refl % |
---|
807 | |13,15: |
---|
808 | whd in ⊢ (??%?) |
---|
809 | cases arg1 |
---|
810 | [*: |
---|
811 | #arg1_left normalize nodelta |
---|
812 | >eq_prod_refl [*: try % #argr @eq_addressing_mode_refl] |
---|
813 | ] |
---|
814 | |11,12: |
---|
815 | whd in ⊢ (??%?) |
---|
816 | cases arg1 |
---|
817 | [1: |
---|
818 | #arg1_left normalize nodelta |
---|
819 | >(eq_sum_refl …) |
---|
820 | [1: % | 2,3: #arg @eq_prod_refl ] |
---|
821 | @eq_addressing_mode_refl |
---|
822 | |2: |
---|
823 | #arg1_left normalize nodelta |
---|
824 | @eq_prod_refl [*: @eq_addressing_mode_refl ] |
---|
825 | |3: |
---|
826 | #arg1_left normalize nodelta |
---|
827 | >(eq_sum_refl …) [1: % |
---|
828 | |2,3: #arg @eq_prod_refl #arg @eq_addressing_mode_refl ] |
---|
829 | |4: |
---|
830 | #arg1_left normalize nodelta |
---|
831 | @eq_prod_refl [*: #arg @eq_addressing_mode_refl ] |
---|
832 | ] |
---|
833 | |14: |
---|
834 | whd in ⊢ (??%?) |
---|
835 | cases arg1 |
---|
836 | [ #arg1_left normalize nodelta |
---|
837 | @eq_sum_refl |
---|
838 | [1: #arg @eq_sum_refl |
---|
839 | [1: #arg @eq_sum_refl |
---|
840 | [1: #arg @eq_sum_refl |
---|
841 | [1: #arg @eq_prod_refl |
---|
842 | [*: @eq_addressing_mode_refl ] |
---|
843 | |2: #arg @eq_prod_refl |
---|
844 | [*: #arg @eq_addressing_mode_refl ] |
---|
845 | ] |
---|
846 | |2: #arg @eq_prod_refl |
---|
847 | [*: #arg @eq_addressing_mode_refl ] |
---|
848 | ] |
---|
849 | |2: #arg @eq_prod_refl |
---|
850 | [*: #arg @eq_addressing_mode_refl ] |
---|
851 | ] |
---|
852 | |2: #arg @eq_prod_refl |
---|
853 | [*: #arg @eq_addressing_mode_refl ] |
---|
854 | ] |
---|
855 | |2: #arg1_right normalize nodelta |
---|
856 | @eq_prod_refl [*: #arg @eq_addressing_mode_refl ] |
---|
857 | ] |
---|
858 | |*: |
---|
859 | whd in ⊢ (??%?) |
---|
860 | cases arg1 |
---|
861 | [*: #arg1 >eq_sum_refl |
---|
862 | [1,4: normalize @eq_addressing_mode_refl |
---|
863 | |2,3,5,6: #arg @eq_prod_refl |
---|
864 | [*: #arg @eq_addressing_mode_refl ] |
---|
865 | ] |
---|
866 | ] |
---|
867 | ] |
---|
868 | qed. |
---|
869 | |
---|
870 | definition eq_instruction: instruction → instruction → bool ≝ |
---|
871 | λi, j. |
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872 | match i with |
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873 | [ ACALL arg ⇒ |
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874 | match j with |
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875 | [ ACALL arg' ⇒ eq_addressing_mode arg arg' |
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876 | | _ ⇒ false |
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877 | ] |
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878 | | LCALL arg ⇒ |
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879 | match j with |
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880 | [ LCALL arg' ⇒ eq_addressing_mode arg arg' |
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881 | | _ ⇒ false |
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882 | ] |
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883 | | AJMP arg ⇒ |
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884 | match j with |
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885 | [ AJMP arg' ⇒ eq_addressing_mode arg arg' |
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886 | | _ ⇒ false |
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887 | ] |
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888 | | LJMP arg ⇒ |
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889 | match j with |
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890 | [ LJMP arg' ⇒ eq_addressing_mode arg arg' |
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891 | | _ ⇒ false |
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892 | ] |
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893 | | SJMP arg ⇒ |
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894 | match j with |
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895 | [ SJMP arg' ⇒ eq_addressing_mode arg arg' |
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896 | | _ ⇒ false |
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897 | ] |
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898 | | JMP arg ⇒ |
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899 | match j with |
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900 | [ JMP arg' ⇒ eq_addressing_mode arg arg' |
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901 | | _ ⇒ false |
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902 | ] |
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903 | | MOVC arg1 arg2 ⇒ |
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904 | match j with |
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905 | [ MOVC arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
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906 | | _ ⇒ false |
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907 | ] |
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908 | | RealInstruction instr ⇒ |
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909 | match j with |
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910 | [ RealInstruction instr' ⇒ eq_preinstruction instr instr' |
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911 | | _ ⇒ false |
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912 | ] |
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913 | ]. |
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914 | |
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915 | lemma eq_instruction_refl: |
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916 | ∀i. eq_instruction i i = true. |
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917 | #i cases i |
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918 | [1,2,3,4,5,6: #arg1 @eq_addressing_mode_refl |
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919 | |7: #arg1 #arg2 |
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920 | whd in ⊢ (??%?) |
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921 | >eq_addressing_mode_refl |
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922 | >eq_addressing_mode_refl |
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923 | % |
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924 | |8: #arg @eq_preinstruction_refl |
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925 | ] |
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926 | qed. |
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927 | |
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928 | let rec vect_member |
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929 | (A: Type[0]) (n: nat) (eq: A → A → bool) |
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930 | (v: Vector A n) (a: A) on v: bool ≝ |
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931 | match v with |
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932 | [ VEmpty ⇒ false |
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933 | | VCons len hd tl ⇒ |
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934 | eq hd a ∨ (vect_member A ? eq tl a) |
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935 | ]. |
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936 | |
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937 | let rec list_addressing_mode_tags_elim_prop |
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938 | (n: nat) |
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939 | (l: Vector addressing_mode_tag (S n)) |
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940 | on l: |
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941 | ∀P: l → Prop. |
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942 | ∀direct_a. ∀indirect_a. ∀ext_indirect_a. ∀register_a. ∀acc_a_a. |
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943 | ∀acc_b_a. ∀dptr_a. ∀data_a. ∀data16_a. ∀acc_dptr_a. ∀acc_pc_a. |
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944 | ∀ext_indirect_dptr_a. ∀indirect_dptr_a. ∀carry_a. ∀bit_addr_a. |
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945 | ∀n_bit_addr_a. ∀relative_a. ∀addr11_a. ∀addr16_a. |
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946 | ∀x: l. P x ≝ |
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947 | match l return |
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948 | λy. |
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949 | match y with |
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950 | [ O ⇒ λm: Vector addressing_mode_tag O. ∀prf: 0 = S n. True |
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951 | | S y' ⇒ λl: Vector addressing_mode_tag (S y'). ∀prf: S y' = S n.∀P:l → Prop. |
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952 | ∀direct_a: if vect_member … eq_a l direct then ∀x. P (DIRECT x) else True. |
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953 | ∀indirect_a: if vect_member … eq_a l indirect then ∀x. P (INDIRECT x) else True. |
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954 | ∀ext_indirect_a: if vect_member … eq_a l ext_indirect then ∀x. P (EXT_INDIRECT x) else True. |
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955 | ∀register_a: if vect_member … eq_a l registr then ∀x. P (REGISTER x) else True. |
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956 | ∀acc_a_a: if vect_member … eq_a l acc_a then P (ACC_A) else True. |
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957 | ∀acc_b_a: if vect_member … eq_a l acc_b then P (ACC_B) else True. |
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958 | ∀dptr_a: if vect_member … eq_a l dptr then P DPTR else True. |
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959 | ∀data_a: if vect_member … eq_a l data then ∀x. P (DATA x) else True. |
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960 | ∀data16_a: if vect_member … eq_a l data16 then ∀x. P (DATA16 x) else True. |
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961 | ∀acc_dptr_a: if vect_member … eq_a l acc_dptr then P ACC_DPTR else True. |
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962 | ∀acc_pc_a: if vect_member … eq_a l acc_pc then P ACC_PC else True. |
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963 | ∀ext_indirect_dptr_a: if vect_member … eq_a l ext_indirect_dptr then P EXT_INDIRECT_DPTR else True. |
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964 | ∀indirect_dptr_a: if vect_member … eq_a l indirect_dptr then P INDIRECT_DPTR else True. |
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965 | ∀carry_a: if vect_member … eq_a l carry then P CARRY else True. |
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966 | ∀bit_addr_a: if vect_member … eq_a l bit_addr then ∀x. P (BIT_ADDR x) else True. |
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967 | ∀n_bit_addr_a: if vect_member … eq_a l n_bit_addr then ∀x. P (N_BIT_ADDR x) else True. |
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968 | ∀relative_a: if vect_member … eq_a l relative then ∀x. P (RELATIVE x) else True. |
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969 | ∀addr11_a: if vect_member … eq_a l addr11 then ∀x. P (ADDR11 x) else True. |
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970 | ∀addr_16_a: if vect_member … eq_a l addr16 then ∀x. P (ADDR16 x) else True. |
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971 | ∀x:l. P x |
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972 | ] with |
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973 | [ VEmpty ⇒ λAbsurd. ⊥ |
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974 | | VCons len hd tl ⇒ λProof. ? |
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975 | ] (refl ? (S n)). cases daemon. qed. (* |
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976 | [ destruct(Absurd) |
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977 | | # A1 # A2 # A3 # A4 # A5 # A6 # A7 |
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978 | # A8 # A9 # A10 # A11 # A12 # A13 # A14 |
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979 | # A15 # A16 # A17 # A18 # A19 # X |
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980 | cases X |
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981 | # SUB cases daemon ] qed. |
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982 | cases SUB |
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983 | [ # BYTE |
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984 | normalize |
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985 | ]. |
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986 | |
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987 | |
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988 | (* let prepare_hd ≝ |
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989 | match hd with |
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990 | [ direct ⇒ λdirect_prf. ? |
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991 | | indirect ⇒ λindirect_prf. ? |
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992 | | ext_indirect ⇒ λext_indirect_prf. ? |
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993 | | registr ⇒ λregistr_prf. ? |
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994 | | acc_a ⇒ λacc_a_prf. ? |
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995 | | acc_b ⇒ λacc_b_prf. ? |
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996 | | dptr ⇒ λdptr_prf. ? |
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997 | | data ⇒ λdata_prf. ? |
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998 | | data16 ⇒ λdata16_prf. ? |
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999 | | acc_dptr ⇒ λacc_dptr_prf. ? |
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1000 | | acc_pc ⇒ λacc_pc_prf. ? |
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1001 | | ext_indirect_dptr ⇒ λext_indirect_prf. ? |
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1002 | | indirect_dptr ⇒ λindirect_prf. ? |
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1003 | | carry ⇒ λcarry_prf. ? |
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1004 | | bit_addr ⇒ λbit_addr_prf. ? |
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1005 | | n_bit_addr ⇒ λn_bit_addr_prf. ? |
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1006 | | relative ⇒ λrelative_prf. ? |
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1007 | | addr11 ⇒ λaddr11_prf. ? |
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1008 | | addr16 ⇒ λaddr16_prf. ? |
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1009 | ] |
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1010 | in ? *) |
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1011 | ]. |
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1012 | [ 1: destruct(absd) |
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1013 | | 2: # A1 # A2 # A3 # A4 # A5 # A6 |
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1014 | # A7 # A8 # A9 # A10 # A11 # A12 |
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1015 | # A13 # A14 # A15 # A16 # A17 # A18 |
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1016 | # A19 * |
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1017 | ]. |
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1018 | |
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1019 | |
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1020 | match l return λx.match x with [O ⇒ λl: Vector … O. bool | S x' ⇒ λl: Vector addressing_mode_tag (S x'). |
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1021 | (l → bool) → bool ] with |
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1022 | [ VEmpty ⇒ true |
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1023 | | VCons len hd tl ⇒ λP. |
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1024 | let process_hd ≝ |
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1025 | match hd return λhd. ∀P: hd:::tl → bool. bool with |
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1026 | [ direct ⇒ λP.bitvector_elim 8 (λx. P (DIRECT x)) |
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1027 | | indirect ⇒ λP.bit_elim (λx. P (INDIRECT x)) |
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1028 | | ext_indirect ⇒ λP.bit_elim (λx. P (EXT_INDIRECT x)) |
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1029 | | registr ⇒ λP.bitvector_elim 3 (λx. P (REGISTER x)) |
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1030 | | acc_a ⇒ λP.P ACC_A |
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1031 | | acc_b ⇒ λP.P ACC_B |
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1032 | | dptr ⇒ λP.P DPTR |
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1033 | | data ⇒ λP.bitvector_elim 8 (λx. P (DATA x)) |
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1034 | | data16 ⇒ λP.bitvector_elim 16 (λx. P (DATA16 x)) |
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1035 | | acc_dptr ⇒ λP.P ACC_DPTR |
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1036 | | acc_pc ⇒ λP.P ACC_PC |
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1037 | | ext_indirect_dptr ⇒ λP.P EXT_INDIRECT_DPTR |
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1038 | | indirect_dptr ⇒ λP.P INDIRECT_DPTR |
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1039 | | carry ⇒ λP.P CARRY |
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1040 | | bit_addr ⇒ λP.bitvector_elim 8 (λx. P (BIT_ADDR x)) |
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1041 | | n_bit_addr ⇒ λP.bitvector_elim 8 (λx. P (N_BIT_ADDR x)) |
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1042 | | relative ⇒ λP.bitvector_elim 8 (λx. P (RELATIVE x)) |
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1043 | | addr11 ⇒ λP.bitvector_elim 11 (λx. P (ADDR11 x)) |
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1044 | | addr16 ⇒ λP.bitvector_elim 16 (λx. P (ADDR16 x)) |
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1045 | ] |
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1046 | in |
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1047 | andb (process_hd P) |
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1048 | (match len return λx. x = len → bool with |
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1049 | [ O ⇒ λprf. true |
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1050 | | S y ⇒ λprf. list_addressing_mode_tags_elim y ? P ] (refl ? len)) |
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1051 | ]. |
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1052 | try % |
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1053 | [ 2: cases (sym_eq ??? prf); @tl |
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1054 | | generalize in match H; generalize in match tl; cases prf; |
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1055 | (* cases prf in tl H; : ??? WAS WORKING BEFORE *) |
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1056 | #tl |
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1057 | normalize in ⊢ (∀_: %. ?) |
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1058 | # H |
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1059 | whd |
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1060 | normalize in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?]) |
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1061 | cases (is_a hd (subaddressing_modeel y tl H)) whd // ] |
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1062 | qed. |
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1063 | *) |
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1064 | |
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1065 | definition load_code_memory_aux ≝ |
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1066 | fold_left_i_aux … ( |
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1067 | λi, mem, v. |
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1068 | insert … (bitvector_of_nat … i) v mem) (Stub Byte 16). |
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1069 | |
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1070 | axiom split_elim: |
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1071 | ∀A,l,m,v.∀P: (Vector A l) × (Vector A m) → Prop. |
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1072 | (∀vl,vm. v = vl@@vm → P 〈vl,vm〉) → P (split A l m v). |
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1073 | |
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1074 | example half_add_SO: |
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1075 | ∀pc. |
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1076 | \snd (half_add 16 (bitvector_of_nat … pc) (bitvector_of_nat … 1)) = bitvector_of_nat … (S pc). |
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1077 | cases daemon. |
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1078 | qed. |
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1079 | |
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1080 | (* |
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1081 | axiom not_eqvb_S: |
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1082 | ∀pc. |
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1083 | (¬eq_bv 16 (bitvector_of_nat 16 pc) (bitvector_of_nat 16 (S pc))). |
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1084 | |
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1085 | axiom not_eqvb_SS: |
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1086 | ∀pc. |
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1087 | (¬eq_bv 16 (bitvector_of_nat 16 pc) (bitvector_of_nat 16 (S (S pc)))). |
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1088 | |
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1089 | axiom bitvector_elim_complete: |
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1090 | ∀n,P. bitvector_elim n P = true → ∀bv. P bv. |
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1091 | |
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1092 | lemma bitvector_elim_complete': |
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1093 | ∀n,P. bitvector_elim n P = true → ∀bv. P bv = true. |
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1094 | #n #P #H generalize in match (bitvector_elim_complete … H) #K #bv |
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1095 | generalize in match (K bv) normalize cases (P bv) normalize // #abs @⊥ // |
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1096 | qed. |
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1097 | *) |
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1098 | |
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1099 | (* |
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1100 | lemma andb_elim': |
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1101 | ∀b1,b2. (b1 = true) → (b2 = true) → (b1 ∧ b2) = true. |
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1102 | #b1 #b2 #H1 #H2 @andb_elim cases b1 in H1; normalize // |
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1103 | qed. |
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1104 | *) |
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1105 | |
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1106 | let rec encoding_check (code_memory: BitVectorTrie Byte 16) (pc: Word) (final_pc: Word) |
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1107 | (encoding: list Byte) on encoding: Prop ≝ |
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1108 | match encoding with |
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1109 | [ nil ⇒ final_pc = pc |
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1110 | | cons hd tl ⇒ |
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1111 | let 〈new_pc, byte〉 ≝ next code_memory pc in |
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1112 | hd = byte ∧ encoding_check code_memory new_pc final_pc tl |
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1113 | ]. |
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1114 | |
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1115 | lemma encoding_check_append: ∀code_memory,final_pc,l1,pc,l2. |
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1116 | encoding_check code_memory (bitvector_of_nat … pc) (bitvector_of_nat … final_pc) (l1@l2) → |
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1117 | let intermediate_pc ≝ pc + length … l1 in |
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1118 | encoding_check code_memory (bitvector_of_nat … pc) (bitvector_of_nat … intermediate_pc) l1 ∧ |
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1119 | encoding_check code_memory (bitvector_of_nat … intermediate_pc) (bitvector_of_nat … final_pc) l2. |
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1120 | #code_memory #final_pc #l1 elim l1 |
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1121 | [ #pc #l2 whd in ⊢ (????% → ?) #H <plus_n_O whd whd in ⊢ (?%?) /2/ |
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1122 | | #hd #tl #IH #pc #l2 * #H1 #H2 >half_add_SO in H2; #H2 cases (IH … H2) <plus_n_Sm |
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1123 | #K1 #K2 % [2:@K2] whd % // >half_add_SO @K1 ] |
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1124 | qed. |
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1125 | |
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1126 | axiom bitvector_3_elim_prop: |
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1127 | ∀P: BitVector 3 → Prop. |
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1128 | P [[false;false;false]] → P [[false;false;true]] → P [[false;true;false]] → |
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1129 | P [[false;true;true]] → P [[true;false;false]] → P [[true;false;true]] → |
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1130 | P [[true;true;false]] → P [[true;true;true]] → ∀v. P v. |
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1131 | |
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1132 | definition ticks_of_instruction ≝ |
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1133 | λi. |
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1134 | let trivial_code_memory ≝ assembly1 i in |
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1135 | let trivial_status ≝ load_code_memory trivial_code_memory in |
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1136 | \snd (fetch trivial_status (zero ?)). |
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1137 | |
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1138 | axiom fetch_assembly: |
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1139 | ∀pc,i,code_memory,assembled. |
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1140 | assembled = assembly1 i → |
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1141 | let len ≝ length … assembled in |
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1142 | encoding_check code_memory (bitvector_of_nat … pc) (bitvector_of_nat … (pc + len)) assembled → |
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1143 | let fetched ≝ fetch code_memory (bitvector_of_nat … pc) in |
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1144 | let 〈instr_pc, ticks〉 ≝ fetched in |
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1145 | let 〈instr,pc'〉 ≝ instr_pc in |
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1146 | (eq_instruction instr i ∧ eqb ticks (ticks_of_instruction instr) ∧ eq_bv … pc' (bitvector_of_nat … (pc + len))) = true. |
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1147 | (* #pc #i #code_memory #assembled cases i [8: *] |
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1148 | [16,20,29: * * |18,19: * * [1,2,4,5: *] |28: * * [1,2: * [1,2: * [1,2: * [1,2: *]]]]] |
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1149 | [47,48,49: |
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1150 | |*: #arg @(list_addressing_mode_tags_elim_prop … arg) whd try % -arg |
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1151 | [2,3,5,7,10,12,16,17,18,21,25,26,27,30,31,32,37,38,39,40,41,42,43,44,45,48,51,58, |
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1152 | 59,60,63,64,65,66,67: #ARG]] |
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1153 | [4,5,6,7,8,9,10,11,12,13,22,23,24,27,28,39,40,41,42,43,44,45,46,47,48,49,50,51,52, |
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1154 | 56,57,69,70,72,73,75: #arg2 @(list_addressing_mode_tags_elim_prop … arg2) whd try % -arg2 |
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1155 | [1,2,4,7,9,10,12,13,15,16,17,18,20,22,23,24,25,26,27,28,29,30,31,32,33,36,37,38, |
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1156 | 39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65, |
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1157 | 68,69,70,71: #ARG2]] |
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1158 | [1,2,19,20: #arg3 @(list_addressing_mode_tags_elim_prop … arg3) whd try % -arg3 #ARG3] |
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1159 | normalize in ⊢ (???% → ?) |
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1160 | [92,94,42,93,95: @split_elim #vl #vm #E >E -E; [2,4: @(bitvector_3_elim_prop … vl)] |
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1161 | normalize in ⊢ (???% → ?)] |
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1162 | #H >H * #H1 try (change in ⊢ (% → ?) with (? ∧ ?) * #H2) |
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1163 | try (change in ⊢ (% → ?) with (? ∧ ?) * #H3) whd in ⊢ (% → ?) #H4 |
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1164 | change in ⊢ (let fetched ≝ % in ?) with (fetch0 ??) |
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1165 | whd in ⊢ (let fetched ≝ ??% in ?) <H1 whd in ⊢ (let fetched ≝ % in ?) |
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1166 | [17,18,19,20,21,22,23,24,25,26,31,34,35,36,37,38: <H3] |
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1167 | [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29, |
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1168 | 30,31,32,33,34,35,36,37,38,39,40,43,45,48,49,52,53,54,55,56,57,60,61,62,65,66, |
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1169 | 69,70,73,74,78,80,81,84,85,95,98,101,102,103,104,105,106,107,108,109,110: <H2] |
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1170 | whd >eq_instruction_refl >H4 @eq_bv_refl |
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1171 | qed. *) |
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1172 | |
---|
1173 | let rec fetch_many code_memory final_pc pc expected on expected: Prop ≝ |
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1174 | match expected with |
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1175 | [ nil ⇒ eq_bv … pc final_pc = true |
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1176 | | cons i tl ⇒ |
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1177 | let fetched ≝ fetch code_memory pc in |
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1178 | let 〈instr_pc, ticks〉 ≝ fetched in |
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1179 | let 〈instr,pc'〉 ≝ instr_pc in |
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1180 | eq_instruction instr i = true ∧ |
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1181 | ticks = (ticks_of_instruction i) ∧ |
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1182 | fetch_many code_memory final_pc pc' tl]. |
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1183 | |
---|
1184 | lemma option_destruct_Some: ∀A,a,b. Some A a = Some A b → a=b. |
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1185 | #A #a #b #EQ destruct // |
---|
1186 | qed. |
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1187 | |
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1188 | axiom eq_instruction_to_eq: |
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1189 | ∀i1,i2. eq_instruction i1 i2 = true → i1 = i2. |
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1190 | |
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1191 | |
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1192 | notation < "❰ x ❱" with precedence 90 for @{'dp $x $y }. |
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1193 | interpretation "dp" 'dp x y = (dp x y). |
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1194 | |
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1195 | lemma fetch_assembly_pseudo': |
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1196 | ∀program.∀pol:policy program.∀ppc,lookup_labels,lookup_datalabels. |
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1197 | ∀pi,code_memory,len,assembled,instructions,pc. |
---|
1198 | let expansion ≝ pol ppc in |
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1199 | Some ? instructions = expand_pseudo_instruction_safe lookup_labels lookup_datalabels (bitvector_of_nat ? pc) expansion pi → |
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1200 | Some … 〈len,assembled〉 = assembly_1_pseudoinstruction_safe program pol ppc (bitvector_of_nat ? pc) lookup_labels lookup_datalabels pi → |
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1201 | encoding_check code_memory (bitvector_of_nat … pc) (bitvector_of_nat … (pc + len)) assembled → |
---|
1202 | fetch_many code_memory (bitvector_of_nat … (pc + len)) (bitvector_of_nat … pc) instructions. |
---|
1203 | #program #pol #ppc #lookup_labels #lookup_datalabels #pi #code_memory #len #assembled #instructions #pc |
---|
1204 | #EQ1 whd in ⊢ (???% → ?) <EQ1 whd in ⊢ (???% → ?) #EQ2 |
---|
1205 | cases (pair_destruct ?????? (option_destruct_Some … EQ2)) -EQ2; #EQ2a #EQ2b |
---|
1206 | >EQ2a >EQ2b -EQ2a EQ2b; |
---|
1207 | generalize in match (pc + |flatten … (map … assembly1 instructions)|); #final_pc |
---|
1208 | generalize in match pc elim instructions |
---|
1209 | [ #pc whd in ⊢ (% → %) #H >H @eq_bv_refl |
---|
1210 | | #i #tl #IH #pc #H whd cases (encoding_check_append … H); -H; #H1 #H2 whd |
---|
1211 | generalize in match (fetch_assembly pc i code_memory … (refl …) H1) |
---|
1212 | cases (fetch code_memory (bitvector_of_nat … pc)) #newi_pc #ticks whd in ⊢ (% → %) |
---|
1213 | cases newi_pc #newi #newpc whd in ⊢ (% → %) #K cases (conjunction_true … K) -K; #K1 |
---|
1214 | cases (conjunction_true … K1) -K1; #K1 #K2 #K3 % try % |
---|
1215 | [ @K1 | @eqb_true_to_eq <(eq_instruction_to_eq … K1) @K2 | >(eq_bv_eq … K3) @IH @H2 ] |
---|
1216 | qed. |
---|
1217 | |
---|
1218 | axiom bitvector_of_nat_nat_of_bitvector: |
---|
1219 | ∀n,v. |
---|
1220 | bitvector_of_nat n (nat_of_bitvector n v) = v. |
---|
1221 | |
---|
1222 | (* CSC: soo long next one; can we merge with previous one now? *) |
---|
1223 | lemma fetch_assembly_pseudo: |
---|
1224 | ∀program.∀pol:policy program.∀ppc. |
---|
1225 | ∀code_memory. |
---|
1226 | let lookup_labels ≝ ? in |
---|
1227 | let lookup_datalabels ≝ ? in |
---|
1228 | let pc ≝ ? in |
---|
1229 | let pi ≝ \fst (fetch_pseudo_instruction (\snd program) ppc) in |
---|
1230 | let instructions ≝ expand_pseudo_instruction program pol ppc lookup_labels lookup_datalabels pc (refl …) (refl …) (refl …) in |
---|
1231 | let 〈len,assembled〉 ≝ assembly_1_pseudoinstruction program pol ppc lookup_labels lookup_datalabels pi (refl …) (refl …) (refl …) in |
---|
1232 | encoding_check code_memory pc (bitvector_of_nat … (nat_of_bitvector ? pc + len)) assembled → |
---|
1233 | fetch_many code_memory (bitvector_of_nat … (nat_of_bitvector ? pc + len)) pc instructions. |
---|
1234 | #program #pol #ppc #code_memory |
---|
1235 | letin lookup_labels ≝ (λx:Identifier |
---|
1236 | .sigma program pol (address_of_word_labels_code_mem (\snd program) x)) |
---|
1237 | letin lookup_datalabels ≝ |
---|
1238 | (λx:BitVector 16 |
---|
1239 | .lookup Identifier 16 x (construct_datalabels (\fst program)) (zero 16)) |
---|
1240 | letin pc ≝ (sigma program pol ppc) |
---|
1241 | letin pi ≝ (\fst (fetch_pseudo_instruction (\snd program) ppc)) |
---|
1242 | letin instructions ≝ (expand_pseudo_instruction program pol ppc lookup_labels lookup_datalabels pc (refl …) (refl …) (refl …)) |
---|
1243 | @pair_elim' #len #assembled #EQ1 #H |
---|
1244 | generalize in match |
---|
1245 | (fetch_assembly_pseudo' program pol ppc lookup_labels lookup_datalabels pi |
---|
1246 | code_memory len assembled instructions (nat_of_bitvector ? pc) ???) in ⊢ ?; |
---|
1247 | [ >bitvector_of_nat_nat_of_bitvector // |
---|
1248 | | >bitvector_of_nat_nat_of_bitvector normalize nodelta >(sig2 ?? (expand_pseudo_instruction …)) % |
---|
1249 | | >bitvector_of_nat_nat_of_bitvector <EQ1 @assembly_1_pseudoinstruction_ok1 |
---|
1250 | | //] |
---|
1251 | qed. |
---|
1252 | |
---|
1253 | axiom assembly_ok: |
---|
1254 | ∀program,pol,assembled. |
---|
1255 | let 〈labels,costs〉 ≝ build_maps program pol in |
---|
1256 | 〈assembled,costs〉 = assembly program pol → |
---|
1257 | let code_memory ≝ load_code_memory assembled in |
---|
1258 | let preamble ≝ \fst program in |
---|
1259 | let datalabels ≝ construct_datalabels preamble in |
---|
1260 | let lookup_labels ≝ λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x) in |
---|
1261 | let lookup_datalabels ≝ λx. lookup ?? x datalabels (zero ?) in |
---|
1262 | ∀ppc. |
---|
1263 | let pi_newppc ≝ fetch_pseudo_instruction (\snd program) ppc in |
---|
1264 | ∀len,assembledi. |
---|
1265 | 〈len,assembledi〉 = assembly_1_pseudoinstruction program pol ppc lookup_labels lookup_datalabels (\fst pi_newppc) (refl …) (refl …) (refl …) → |
---|
1266 | encoding_check code_memory (sigma program pol ppc) (bitvector_of_nat … (nat_of_bitvector … (sigma program pol ppc) + len)) assembledi ∧ |
---|
1267 | sigma program pol (\snd pi_newppc) = bitvector_of_nat … (nat_of_bitvector … (sigma program pol ppc) + len). |
---|
1268 | |
---|
1269 | axiom assembly_ok_to_expand_pseudo_instruction_ok: |
---|
1270 | ∀program,pol,assembled,costs. |
---|
1271 | 〈assembled,costs〉 = assembly program pol → |
---|
1272 | ∀ppc. |
---|
1273 | let code_memory ≝ load_code_memory assembled in |
---|
1274 | let preamble ≝ \fst program in |
---|
1275 | let data_labels ≝ construct_datalabels preamble in |
---|
1276 | let lookup_labels ≝ λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x) in |
---|
1277 | let lookup_datalabels ≝ λx. lookup ? ? x data_labels (zero ?) in |
---|
1278 | let expansion ≝ pol ppc in |
---|
1279 | let 〈pi,newppc〉 ≝ fetch_pseudo_instruction (\snd program) ppc in |
---|
1280 | ∃instructions. |
---|
1281 | Some ? instructions = expand_pseudo_instruction_safe lookup_labels lookup_datalabels (sigma program pol ppc) expansion pi. |
---|
1282 | |
---|
1283 | (*CSC: repair me!*) |
---|
1284 | axiom fetch_assembly_pseudo2: |
---|
1285 | ∀program,pol,ppc. |
---|
1286 | let 〈labels,costs〉 ≝ build_maps program pol in |
---|
1287 | let assembled ≝ \fst (assembly program pol) in |
---|
1288 | let code_memory ≝ load_code_memory assembled in |
---|
1289 | let data_labels ≝ construct_datalabels (\fst program) in |
---|
1290 | let lookup_labels ≝ λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x) in |
---|
1291 | let lookup_datalabels ≝ λx. lookup ? ? x data_labels (zero ?) in |
---|
1292 | let 〈pi,newppc〉 ≝ fetch_pseudo_instruction (\snd program) ppc in |
---|
1293 | let instructions ≝ expand_pseudo_instruction program pol ppc lookup_labels lookup_datalabels (sigma program pol ppc) (refl …) (refl …) (refl …) in |
---|
1294 | fetch_many code_memory (sigma program pol newppc) (sigma program pol ppc) instructions. |
---|
1295 | (* |
---|
1296 | #program #pol #assembled generalize in match (assembly_ok program pol assembled) |
---|
1297 | @pair_elim' #labels #costs #BUILD_MAPS |
---|
1298 | #H #ASSEMBLY #ppc #pi #newppc |
---|
1299 | letin code_memory ≝ (load_code_memory assembled) |
---|
1300 | letin preamble ≝ (\fst program) |
---|
1301 | letin data_labels ≝ (construct_datalabels preamble) |
---|
1302 | letin lookup_labels ≝ (λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x)) |
---|
1303 | letin lookup_datalabels ≝ (λx. lookup ? ? x data_labels (zero ?)) |
---|
1304 | normalize nodelta #EQ generalize in match (H ASSEMBLY ppc) -H; |
---|
1305 | generalize in match (fetch_assembly_pseudo program pol ppc code_memory) in ⊢ ? normalize nodelta |
---|
1306 | @pair_elim' #len #assembledi #ASSEMBLE1 #H1 #H2 |
---|
1307 | generalize in match (H1 ?) [2: cases (H2 len assembledi (refl …)) #H1 #_ @H1] |
---|
1308 | >bitvector_of_nat_nat_of_bitvector |
---|
1309 | #K |
---|
1310 | |
---|
1311 | |
---|
1312 | cases |
---|
1313 | (assembly_1_pseudoinstruction program pol ppc lookup_labels lookup_datalabels ????) |
---|
1314 | @split_elim' #len #assembledi #ASSEMBLY1 |
---|
1315 | |
---|
1316 | @pair_elim' |
---|
1317 | |
---|
1318 | normalize nodelta; |
---|
1319 | whd in ⊢ (% → %) generalize in match (assembly_ok program pol assembled) >BUILD_MAPS #XX generalize in match (XX ASSEMBLY ppc) -XX; |
---|
1320 | cases (fetch_pseudo_instruction (\snd program) ppc) #pi #newppc |
---|
1321 | generalize in match (fetch_assembly_pseudo program pol ppc |
---|
1322 | (λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x)) (λx. lookup ?? x data_labels (zero ?)) pi |
---|
1323 | (load_code_memory assembled)); |
---|
1324 | whd in ⊢ ((∀_.∀_.∀_.∀_.%) → (∀_.∀_.%) → % → %) #H1 #H2 * #instructions #EXPAND |
---|
1325 | whd in H1:(∀_.∀_.∀_.∀_.? → ???% → ?) H2:(∀_.∀_.???% → ?); |
---|
1326 | normalize nodelta in EXPAND; (* HERE *) |
---|
1327 | generalize in match (λlen, assembled.H1 len assembled instructions (nat_of_bitvector … (sigma program pol ppc))) -H1; #H1 |
---|
1328 | >bitvector_of_nat_nat_of_bitvector in H1; #H1 |
---|
1329 | <EXPAND in H1 H2; whd in ⊢ ((∀_.∀_.? → ???% → ?) → (∀_.∀_.???% → ?) → ?) |
---|
1330 | #H1 #H2 |
---|
1331 | cases (H2 ?? (refl …)) -H2; #K1 #K2 >K2 |
---|
1332 | generalize in match (H1 ?? (refl …) (refl …) ?) -H1; |
---|
1333 | [ #K3 % [2: % [% | @K3]] | @K1 ] |
---|
1334 | qed. |
---|
1335 | *) |
---|
1336 | |
---|
1337 | (* |
---|
1338 | lemma fetch_assembly_pseudo2: |
---|
1339 | ∀program,pol,assembled. |
---|
1340 | let 〈labels,costs〉 ≝ build_maps program pol in |
---|
1341 | 〈assembled,costs〉 = assembly program pol → |
---|
1342 | ∀ppc. |
---|
1343 | let code_memory ≝ load_code_memory assembled in |
---|
1344 | let preamble ≝ \fst program in |
---|
1345 | let data_labels ≝ construct_datalabels preamble in |
---|
1346 | let lookup_labels ≝ λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x) in |
---|
1347 | let lookup_datalabels ≝ λx. lookup ? ? x data_labels (zero ?) in |
---|
1348 | let expansion ≝ pol ppc in |
---|
1349 | let 〈pi,newppc〉 ≝ fetch_pseudo_instruction (\snd program) ppc in |
---|
1350 | ∃instructions. |
---|
1351 | Some ? instructions = expand_pseudo_instruction_safe lookup_labels lookup_datalabels (sigma program pol ppc) expansion pi ∧ |
---|
1352 | fetch_many code_memory (sigma program pol newppc) (sigma program pol ppc) instructions. |
---|
1353 | #program #pol #assembled @pair_elim' #labels #costs #BUILD_MAPS #ASSEMBLY #ppc |
---|
1354 | generalize in match (assembly_ok_to_expand_pseudo_instruction_ok program pol assembled costs ASSEMBLY ppc) |
---|
1355 | letin code_memory ≝ (load_code_memory assembled) |
---|
1356 | letin preamble ≝ (\fst program) |
---|
1357 | letin data_labels ≝ (construct_datalabels preamble) |
---|
1358 | letin lookup_labels ≝ (λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x)) |
---|
1359 | letin lookup_datalabels ≝ (λx. lookup ? ? x data_labels (zero ?)) |
---|
1360 | whd in ⊢ (% → %) generalize in match (assembly_ok program pol assembled) >BUILD_MAPS #XX generalize in match (XX ASSEMBLY ppc) -XX; |
---|
1361 | cases (fetch_pseudo_instruction (\snd program) ppc) #pi #newppc |
---|
1362 | generalize in match (fetch_assembly_pseudo program pol ppc |
---|
1363 | (λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x)) (λx. lookup ?? x data_labels (zero ?)) |
---|
1364 | (load_code_memory assembled)); |
---|
1365 | whd in ⊢ ((∀_.∀_.∀_.∀_.%) → (∀_.∀_.%) → % → %) #H1 #H2 * #instructions #EXPAND |
---|
1366 | whd in H1:(∀_.∀_.∀_.∀_.? → ???% → ?) H2:(∀_.∀_.???% → ?); |
---|
1367 | normalize nodelta in EXPAND; (* HERE *) |
---|
1368 | generalize in match (λlen, assembled.H1 len assembled instructions (nat_of_bitvector … (sigma program pol ppc))) -H1; #H1 |
---|
1369 | >bitvector_of_nat_nat_of_bitvector in H1; #H1 |
---|
1370 | <EXPAND in H1 H2; whd in ⊢ ((∀_.∀_.? → ???% → ?) → (∀_.∀_.???% → ?) → ?) |
---|
1371 | #H1 #H2 |
---|
1372 | cases (H2 ?? (refl …)) -H2; #K1 #K2 >K2 |
---|
1373 | generalize in match (H1 ?? (refl …) (refl …) ?) -H1; |
---|
1374 | [ #K3 % [2: % [% | @K3]] | @K1 ] |
---|
1375 | qed. |
---|
1376 | *) |
---|
1377 | |
---|
1378 | (* OLD? |
---|
1379 | definition assembly_specification: |
---|
1380 | ∀assembly_program: pseudo_assembly_program. |
---|
1381 | ∀code_mem: BitVectorTrie Byte 16. Prop ≝ |
---|
1382 | λpseudo_assembly_program. |
---|
1383 | λcode_mem. |
---|
1384 | ∀pc: Word. |
---|
1385 | let 〈preamble, instr_list〉 ≝ pseudo_assembly_program in |
---|
1386 | let 〈pre_instr, pre_new_pc〉 ≝ fetch_pseudo_instruction instr_list pc in |
---|
1387 | let labels ≝ λx. sigma' pseudo_assembly_program (address_of_word_labels_code_mem instr_list x) in |
---|
1388 | let datalabels ≝ λx. sigma' pseudo_assembly_program (lookup ? ? x (construct_datalabels preamble) (zero ?)) in |
---|
1389 | let pre_assembled ≝ assembly_1_pseudoinstruction pseudo_assembly_program |
---|
1390 | (sigma' pseudo_assembly_program pc) labels datalabels pre_instr in |
---|
1391 | match pre_assembled with |
---|
1392 | [ None ⇒ True |
---|
1393 | | Some pc_code ⇒ |
---|
1394 | let 〈new_pc,code〉 ≝ pc_code in |
---|
1395 | encoding_check code_mem pc (sigma' pseudo_assembly_program pre_new_pc) code ]. |
---|
1396 | |
---|
1397 | axiom assembly_meets_specification: |
---|
1398 | ∀pseudo_assembly_program. |
---|
1399 | match assembly pseudo_assembly_program with |
---|
1400 | [ None ⇒ True |
---|
1401 | | Some code_mem_cost ⇒ |
---|
1402 | let 〈code_mem, cost〉 ≝ code_mem_cost in |
---|
1403 | assembly_specification pseudo_assembly_program (load_code_memory code_mem) |
---|
1404 | ]. |
---|
1405 | (* |
---|
1406 | # PROGRAM |
---|
1407 | [ cases PROGRAM |
---|
1408 | # PREAMBLE |
---|
1409 | # INSTR_LIST |
---|
1410 | elim INSTR_LIST |
---|
1411 | [ whd |
---|
1412 | whd in ⊢ (∀_. %) |
---|
1413 | # PC |
---|
1414 | whd |
---|
1415 | | # INSTR |
---|
1416 | # INSTR_LIST_TL |
---|
1417 | # H |
---|
1418 | whd |
---|
1419 | whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?]) |
---|
1420 | ] |
---|
1421 | | cases not_implemented |
---|
1422 | ] *) |
---|
1423 | *) |
---|
1424 | |
---|
1425 | definition internal_pseudo_address_map ≝ list (BitVector 8). |
---|
1426 | |
---|
1427 | axiom low_internal_ram_of_pseudo_low_internal_ram: |
---|
1428 | ∀M:internal_pseudo_address_map.∀ram:BitVectorTrie Byte 7.BitVectorTrie Byte 7. |
---|
1429 | |
---|
1430 | axiom high_internal_ram_of_pseudo_high_internal_ram: |
---|
1431 | ∀M:internal_pseudo_address_map.∀ram:BitVectorTrie Byte 7.BitVectorTrie Byte 7. |
---|
1432 | |
---|
1433 | axiom low_internal_ram_of_pseudo_internal_ram_hit: |
---|
1434 | ∀M:internal_pseudo_address_map.∀s:PseudoStatus.∀pol:policy (code_memory … s).∀addr:BitVector 7. |
---|
1435 | member ? (eq_bv 8) (false:::addr) M = true → |
---|
1436 | let ram ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … s) in |
---|
1437 | let pbl ≝ lookup ? 7 addr (low_internal_ram ? s) (zero 8) in |
---|
1438 | let pbu ≝ lookup ? 7 (\snd (half_add ? addr (bitvector_of_nat 7 1))) (low_internal_ram ? s) (zero 8) in |
---|
1439 | let bl ≝ lookup ? 7 addr ram (zero 8) in |
---|
1440 | let bu ≝ lookup ? 7 (\snd (half_add ? addr (bitvector_of_nat 7 1))) ram (zero 8) in |
---|
1441 | bu@@bl = sigma (code_memory … s) pol (pbu@@pbl). |
---|
1442 | |
---|
1443 | (* changed from add to sub *) |
---|
1444 | axiom low_internal_ram_of_pseudo_internal_ram_miss: |
---|
1445 | ∀T.∀M:internal_pseudo_address_map.∀s:PreStatus T.∀addr:BitVector 7. |
---|
1446 | let ram ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … s) in |
---|
1447 | let 〈Saddr,flags〉 ≝ sub_7_with_carry addr (bitvector_of_nat 7 1) false in |
---|
1448 | let carr ≝ get_index_v ? ? flags 1 ? in |
---|
1449 | carr = false → |
---|
1450 | member ? (eq_bv 8) (false:::Saddr) M = false → |
---|
1451 | member ? (eq_bv 8) (false:::addr) M = false → |
---|
1452 | lookup ? 7 addr ram (zero ?) = lookup ? 7 addr (low_internal_ram … s) (zero ?). |
---|
1453 | // |
---|
1454 | qed. |
---|
1455 | |
---|
1456 | definition addressing_mode_ok ≝ |
---|
1457 | λT.λM:internal_pseudo_address_map.λs:PreStatus T. |
---|
1458 | λaddr:addressing_mode. |
---|
1459 | match addr with |
---|
1460 | [ DIRECT d ⇒ |
---|
1461 | ¬(member ? (eq_bv 8) d M) ∧ |
---|
1462 | ¬(member ? (eq_bv 8) (\fst (sub_8_with_carry d (bitvector_of_nat 8 1) false)) M) |
---|
1463 | | INDIRECT i ⇒ |
---|
1464 | let d ≝ get_register ? s [[false;false;i]] in |
---|
1465 | ¬(member ? (eq_bv 8) d M) ∧ |
---|
1466 | ¬(member ? (eq_bv 8) (\fst (sub_8_with_carry d (bitvector_of_nat 8 1) false)) M) |
---|
1467 | | EXT_INDIRECT _ ⇒ true |
---|
1468 | | REGISTER _ ⇒ true |
---|
1469 | | ACC_A ⇒ true |
---|
1470 | | ACC_B ⇒ true |
---|
1471 | | DPTR ⇒ true |
---|
1472 | | DATA _ ⇒ true |
---|
1473 | | DATA16 _ ⇒ true |
---|
1474 | | ACC_DPTR ⇒ true |
---|
1475 | | ACC_PC ⇒ true |
---|
1476 | | EXT_INDIRECT_DPTR ⇒ true |
---|
1477 | | INDIRECT_DPTR ⇒ true |
---|
1478 | | CARRY ⇒ true |
---|
1479 | | BIT_ADDR _ ⇒ ¬true (* TO BE COMPLETED *) |
---|
1480 | | N_BIT_ADDR _ ⇒ ¬true (* TO BE COMPLETED *) |
---|
1481 | | RELATIVE _ ⇒ true |
---|
1482 | | ADDR11 _ ⇒ true |
---|
1483 | | ADDR16 _ ⇒ true ]. |
---|
1484 | |
---|
1485 | definition next_internal_pseudo_address_map0 ≝ |
---|
1486 | λT. |
---|
1487 | λfetched. |
---|
1488 | λM: internal_pseudo_address_map. |
---|
1489 | λs: PreStatus T. |
---|
1490 | match fetched with |
---|
1491 | [ Comment _ ⇒ Some ? M |
---|
1492 | | Cost _ ⇒ Some … M |
---|
1493 | | Jmp _ ⇒ Some … M |
---|
1494 | | Call _ ⇒ |
---|
1495 | Some … (\snd (half_add ? (get_8051_sfr … s SFR_SP) (bitvector_of_nat 8 1))::M) |
---|
1496 | | Mov _ _ ⇒ Some … M |
---|
1497 | | Instruction instr ⇒ |
---|
1498 | match instr with |
---|
1499 | [ ADD addr1 addr2 ⇒ |
---|
1500 | if addressing_mode_ok T M s addr1 ∧ addressing_mode_ok T M s addr2 then |
---|
1501 | Some ? M |
---|
1502 | else |
---|
1503 | None ? |
---|
1504 | | ADDC addr1 addr2 ⇒ |
---|
1505 | if addressing_mode_ok T M s addr1 ∧ addressing_mode_ok T M s addr2 then |
---|
1506 | Some ? M |
---|
1507 | else |
---|
1508 | None ? |
---|
1509 | | SUBB addr1 addr2 ⇒ |
---|
1510 | if addressing_mode_ok T M s addr1 ∧ addressing_mode_ok T M s addr2 then |
---|
1511 | Some ? M |
---|
1512 | else |
---|
1513 | None ? |
---|
1514 | | _ ⇒ (* TO BE COMPLETED *) Some ? M ]]. |
---|
1515 | |
---|
1516 | |
---|
1517 | definition next_internal_pseudo_address_map ≝ |
---|
1518 | λM:internal_pseudo_address_map. |
---|
1519 | λs:PseudoStatus. |
---|
1520 | next_internal_pseudo_address_map0 ? |
---|
1521 | (\fst (fetch_pseudo_instruction (\snd (code_memory ? s)) (program_counter ? s))) M s. |
---|
1522 | |
---|
1523 | definition status_of_pseudo_status: |
---|
1524 | internal_pseudo_address_map → ∀ps:PseudoStatus. policy (code_memory … ps) → Status ≝ |
---|
1525 | λM,ps,pol. |
---|
1526 | let pap ≝ code_memory … ps in |
---|
1527 | let p ≝ assembly pap pol in |
---|
1528 | let cm ≝ load_code_memory (\fst p) in |
---|
1529 | let pc ≝ sigma pap pol (program_counter ? ps) in |
---|
1530 | let lir ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … ps) in |
---|
1531 | let hir ≝ high_internal_ram_of_pseudo_high_internal_ram M (high_internal_ram … ps) in |
---|
1532 | mk_PreStatus (BitVectorTrie Byte 16) |
---|
1533 | cm |
---|
1534 | lir |
---|
1535 | hir |
---|
1536 | (external_ram … ps) |
---|
1537 | pc |
---|
1538 | (special_function_registers_8051 … ps) |
---|
1539 | (special_function_registers_8052 … ps) |
---|
1540 | (p1_latch … ps) |
---|
1541 | (p3_latch … ps) |
---|
1542 | (clock … ps). |
---|
1543 | |
---|
1544 | (* |
---|
1545 | definition write_at_stack_pointer': |
---|
1546 | ∀M. ∀ps: PreStatus M. Byte → Σps':PreStatus M.(code_memory … ps = code_memory … ps') ≝ |
---|
1547 | λM: Type[0]. |
---|
1548 | λs: PreStatus M. |
---|
1549 | λv: Byte. |
---|
1550 | let 〈 nu, nl 〉 ≝ split … 4 4 (get_8051_sfr ? s SFR_SP) in |
---|
1551 | let bit_zero ≝ get_index_v… nu O ? in |
---|
1552 | let bit_1 ≝ get_index_v… nu 1 ? in |
---|
1553 | let bit_2 ≝ get_index_v… nu 2 ? in |
---|
1554 | let bit_3 ≝ get_index_v… nu 3 ? in |
---|
1555 | if bit_zero then |
---|
1556 | let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl) |
---|
1557 | v (low_internal_ram ? s) in |
---|
1558 | set_low_internal_ram ? s memory |
---|
1559 | else |
---|
1560 | let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl) |
---|
1561 | v (high_internal_ram ? s) in |
---|
1562 | set_high_internal_ram ? s memory. |
---|
1563 | [ cases l0 % |
---|
1564 | |2,3,4,5: normalize repeat (@ le_S_S) @ le_O_n ] |
---|
1565 | qed. |
---|
1566 | |
---|
1567 | definition execute_1_pseudo_instruction': (Word → nat) → ∀ps:PseudoStatus. |
---|
1568 | Σps':PseudoStatus.(code_memory … ps = code_memory … ps') |
---|
1569 | ≝ |
---|
1570 | λticks_of. |
---|
1571 | λs. |
---|
1572 | let 〈instr, pc〉 ≝ fetch_pseudo_instruction (\snd (code_memory ? s)) (program_counter ? s) in |
---|
1573 | let ticks ≝ ticks_of (program_counter ? s) in |
---|
1574 | let s ≝ set_clock ? s (clock ? s + ticks) in |
---|
1575 | let s ≝ set_program_counter ? s pc in |
---|
1576 | match instr with |
---|
1577 | [ Instruction instr ⇒ |
---|
1578 | execute_1_preinstruction … (λx, y. address_of_word_labels y x) instr s |
---|
1579 | | Comment cmt ⇒ s |
---|
1580 | | Cost cst ⇒ s |
---|
1581 | | Jmp jmp ⇒ set_program_counter ? s (address_of_word_labels s jmp) |
---|
1582 | | Call call ⇒ |
---|
1583 | let a ≝ address_of_word_labels s call in |
---|
1584 | let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in |
---|
1585 | let s ≝ set_8051_sfr ? s SFR_SP new_sp in |
---|
1586 | let 〈pc_bu, pc_bl〉 ≝ split ? 8 8 (program_counter ? s) in |
---|
1587 | let s ≝ write_at_stack_pointer' ? s pc_bl in |
---|
1588 | let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in |
---|
1589 | let s ≝ set_8051_sfr ? s SFR_SP new_sp in |
---|
1590 | let s ≝ write_at_stack_pointer' ? s pc_bu in |
---|
1591 | set_program_counter ? s a |
---|
1592 | | Mov dptr ident ⇒ |
---|
1593 | set_arg_16 ? s (get_arg_16 ? s (DATA16 (address_of_word_labels s ident))) dptr |
---|
1594 | ]. |
---|
1595 | [ |
---|
1596 | |2,3,4: % |
---|
1597 | | <(sig2 … l7) whd in ⊢ (??? (??%)) <(sig2 … l5) % |
---|
1598 | | |
---|
1599 | | % |
---|
1600 | ] |
---|
1601 | cases not_implemented |
---|
1602 | qed. |
---|
1603 | *) |
---|
1604 | |
---|
1605 | axiom execute_1_pseudo_instruction_preserves_code_memory: |
---|
1606 | ∀ticks_of,ps. |
---|
1607 | code_memory … (execute_1_pseudo_instruction ticks_of ps) = code_memory … ps. |
---|
1608 | |
---|
1609 | (* |
---|
1610 | lemma execute_code_memory_unchanged: |
---|
1611 | ∀ticks_of,ps. code_memory ? ps = code_memory ? (execute_1_pseudo_instruction ticks_of ps). |
---|
1612 | #ticks #ps whd in ⊢ (??? (??%)) |
---|
1613 | cases (fetch_pseudo_instruction (\snd (code_memory pseudo_assembly_program ps)) |
---|
1614 | (program_counter pseudo_assembly_program ps)) #instr #pc |
---|
1615 | whd in ⊢ (??? (??%)) cases instr |
---|
1616 | [ #pre cases pre |
---|
1617 | [ #a1 #a2 whd in ⊢ (??? (??%)) cases (add_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%)) |
---|
1618 | cases (split ????) #z1 #z2 % |
---|
1619 | | #a1 #a2 whd in ⊢ (??? (??%)) cases (add_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%)) |
---|
1620 | cases (split ????) #z1 #z2 % |
---|
1621 | | #a1 #a2 whd in ⊢ (??? (??%)) cases (sub_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%)) |
---|
1622 | cases (split ????) #z1 #z2 % |
---|
1623 | | #a1 whd in ⊢ (??? (??%)) cases a1 #x #H whd in ⊢ (??? (??%)) cases x |
---|
1624 | [ #x1 whd in ⊢ (??? (??%)) |
---|
1625 | | *: cases not_implemented |
---|
1626 | ] |
---|
1627 | | #comment % |
---|
1628 | | #cost % |
---|
1629 | | #label % |
---|
1630 | | #label whd in ⊢ (??? (??%)) cases (half_add ???) #x1 #x2 whd in ⊢ (??? (??%)) |
---|
1631 | cases (split ????) #y1 #y2 whd in ⊢ (??? (??%)) cases (half_add ???) #z1 #z2 |
---|
1632 | whd in ⊢ (??? (??%)) whd in ⊢ (??? (??%)) cases (split ????) #w1 #w2 |
---|
1633 | whd in ⊢ (??? (??%)) cases (get_index_v bool ????) whd in ⊢ (??? (??%)) |
---|
1634 | (* CSC: ??? *) |
---|
1635 | | #dptr #label (* CSC: ??? *) |
---|
1636 | ] |
---|
1637 | cases not_implemented |
---|
1638 | qed. |
---|
1639 | *) |
---|
1640 | |
---|
1641 | (* DEAD CODE? |
---|
1642 | lemma status_of_pseudo_status_failure_depends_only_on_code_memory: |
---|
1643 | ∀M:internal_pseudo_address_map. |
---|
1644 | ∀ps,ps': PseudoStatus. |
---|
1645 | ∀pol. |
---|
1646 | ∀prf:code_memory … ps = code_memory … ps'. |
---|
1647 | let pol' ≝ ? in |
---|
1648 | match status_of_pseudo_status M ps pol with |
---|
1649 | [ None ⇒ status_of_pseudo_status M ps' pol' = None … |
---|
1650 | | Some _ ⇒ ∃w. status_of_pseudo_status M ps' pol' = Some … w |
---|
1651 | ]. |
---|
1652 | [2: <prf @pol] |
---|
1653 | #M #ps #ps' #pol #H normalize nodelta; whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ? ]) |
---|
1654 | generalize in match (refl … (assembly (code_memory … ps) pol)) |
---|
1655 | cases (assembly ??) in ⊢ (???% → %) |
---|
1656 | [ #K whd whd in ⊢ (??%?) <H >K % |
---|
1657 | | #x #K whd whd in ⊢ (?? (λ_.??%?)) <H >K % [2: % ] ] |
---|
1658 | qed. |
---|
1659 | *) |
---|
1660 | |
---|
1661 | definition ticks_of0: ∀p:pseudo_assembly_program. policy p → Word → ? → nat × nat ≝ |
---|
1662 | λprogram: pseudo_assembly_program.λpol. |
---|
1663 | λppc: Word. |
---|
1664 | λfetched. |
---|
1665 | match fetched with |
---|
1666 | [ Instruction instr ⇒ |
---|
1667 | match instr with |
---|
1668 | [ JC lbl ⇒ |
---|
1669 | match pol ppc with |
---|
1670 | [ short_jump ⇒ 〈2, 2〉 |
---|
1671 | | medium_jump ⇒ ? |
---|
1672 | | long_jump ⇒ 〈4, 4〉 |
---|
1673 | ] |
---|
1674 | | JNC lbl ⇒ |
---|
1675 | match pol ppc with |
---|
1676 | [ short_jump ⇒ 〈2, 2〉 |
---|
1677 | | medium_jump ⇒ ? |
---|
1678 | | long_jump ⇒ 〈4, 4〉 |
---|
1679 | ] |
---|
1680 | | JB bit lbl ⇒ |
---|
1681 | match pol ppc with |
---|
1682 | [ short_jump ⇒ 〈2, 2〉 |
---|
1683 | | medium_jump ⇒ ? |
---|
1684 | | long_jump ⇒ 〈4, 4〉 |
---|
1685 | ] |
---|
1686 | | JNB bit lbl ⇒ |
---|
1687 | match pol ppc with |
---|
1688 | [ short_jump ⇒ 〈2, 2〉 |
---|
1689 | | medium_jump ⇒ ? |
---|
1690 | | long_jump ⇒ 〈4, 4〉 |
---|
1691 | ] |
---|
1692 | | JBC bit lbl ⇒ |
---|
1693 | match pol ppc with |
---|
1694 | [ short_jump ⇒ 〈2, 2〉 |
---|
1695 | | medium_jump ⇒ ? |
---|
1696 | | long_jump ⇒ 〈4, 4〉 |
---|
1697 | ] |
---|
1698 | | JZ lbl ⇒ |
---|
1699 | match pol ppc with |
---|
1700 | [ short_jump ⇒ 〈2, 2〉 |
---|
1701 | | medium_jump ⇒ ? |
---|
1702 | | long_jump ⇒ 〈4, 4〉 |
---|
1703 | ] |
---|
1704 | | JNZ lbl ⇒ |
---|
1705 | match pol ppc with |
---|
1706 | [ short_jump ⇒ 〈2, 2〉 |
---|
1707 | | medium_jump ⇒ ? |
---|
1708 | | long_jump ⇒ 〈4, 4〉 |
---|
1709 | ] |
---|
1710 | | CJNE arg lbl ⇒ |
---|
1711 | match pol ppc with |
---|
1712 | [ short_jump ⇒ 〈2, 2〉 |
---|
1713 | | medium_jump ⇒ ? |
---|
1714 | | long_jump ⇒ 〈4, 4〉 |
---|
1715 | ] |
---|
1716 | | DJNZ arg lbl ⇒ |
---|
1717 | match pol ppc with |
---|
1718 | [ short_jump ⇒ 〈2, 2〉 |
---|
1719 | | medium_jump ⇒ ? |
---|
1720 | | long_jump ⇒ 〈4, 4〉 |
---|
1721 | ] |
---|
1722 | | ADD arg1 arg2 ⇒ |
---|
1723 | let ticks ≝ ticks_of_instruction (ADD ? arg1 arg2) in |
---|
1724 | 〈ticks, ticks〉 |
---|
1725 | | ADDC arg1 arg2 ⇒ |
---|
1726 | let ticks ≝ ticks_of_instruction (ADDC ? arg1 arg2) in |
---|
1727 | 〈ticks, ticks〉 |
---|
1728 | | SUBB arg1 arg2 ⇒ |
---|
1729 | let ticks ≝ ticks_of_instruction (SUBB ? arg1 arg2) in |
---|
1730 | 〈ticks, ticks〉 |
---|
1731 | | INC arg ⇒ |
---|
1732 | let ticks ≝ ticks_of_instruction (INC ? arg) in |
---|
1733 | 〈ticks, ticks〉 |
---|
1734 | | DEC arg ⇒ |
---|
1735 | let ticks ≝ ticks_of_instruction (DEC ? arg) in |
---|
1736 | 〈ticks, ticks〉 |
---|
1737 | | MUL arg1 arg2 ⇒ |
---|
1738 | let ticks ≝ ticks_of_instruction (MUL ? arg1 arg2) in |
---|
1739 | 〈ticks, ticks〉 |
---|
1740 | | DIV arg1 arg2 ⇒ |
---|
1741 | let ticks ≝ ticks_of_instruction (DIV ? arg1 arg2) in |
---|
1742 | 〈ticks, ticks〉 |
---|
1743 | | DA arg ⇒ |
---|
1744 | let ticks ≝ ticks_of_instruction (DA ? arg) in |
---|
1745 | 〈ticks, ticks〉 |
---|
1746 | | ANL arg ⇒ |
---|
1747 | let ticks ≝ ticks_of_instruction (ANL ? arg) in |
---|
1748 | 〈ticks, ticks〉 |
---|
1749 | | ORL arg ⇒ |
---|
1750 | let ticks ≝ ticks_of_instruction (ORL ? arg) in |
---|
1751 | 〈ticks, ticks〉 |
---|
1752 | | XRL arg ⇒ |
---|
1753 | let ticks ≝ ticks_of_instruction (XRL ? arg) in |
---|
1754 | 〈ticks, ticks〉 |
---|
1755 | | CLR arg ⇒ |
---|
1756 | let ticks ≝ ticks_of_instruction (CLR ? arg) in |
---|
1757 | 〈ticks, ticks〉 |
---|
1758 | | CPL arg ⇒ |
---|
1759 | let ticks ≝ ticks_of_instruction (CPL ? arg) in |
---|
1760 | 〈ticks, ticks〉 |
---|
1761 | | RL arg ⇒ |
---|
1762 | let ticks ≝ ticks_of_instruction (RL ? arg) in |
---|
1763 | 〈ticks, ticks〉 |
---|
1764 | | RLC arg ⇒ |
---|
1765 | let ticks ≝ ticks_of_instruction (RLC ? arg) in |
---|
1766 | 〈ticks, ticks〉 |
---|
1767 | | RR arg ⇒ |
---|
1768 | let ticks ≝ ticks_of_instruction (RR ? arg) in |
---|
1769 | 〈ticks, ticks〉 |
---|
1770 | | RRC arg ⇒ |
---|
1771 | let ticks ≝ ticks_of_instruction (RRC ? arg) in |
---|
1772 | 〈ticks, ticks〉 |
---|
1773 | | SWAP arg ⇒ |
---|
1774 | let ticks ≝ ticks_of_instruction (SWAP ? arg) in |
---|
1775 | 〈ticks, ticks〉 |
---|
1776 | | MOV arg ⇒ |
---|
1777 | let ticks ≝ ticks_of_instruction (MOV ? arg) in |
---|
1778 | 〈ticks, ticks〉 |
---|
1779 | | MOVX arg ⇒ |
---|
1780 | let ticks ≝ ticks_of_instruction (MOVX ? arg) in |
---|
1781 | 〈ticks, ticks〉 |
---|
1782 | | SETB arg ⇒ |
---|
1783 | let ticks ≝ ticks_of_instruction (SETB ? arg) in |
---|
1784 | 〈ticks, ticks〉 |
---|
1785 | | PUSH arg ⇒ |
---|
1786 | let ticks ≝ ticks_of_instruction (PUSH ? arg) in |
---|
1787 | 〈ticks, ticks〉 |
---|
1788 | | POP arg ⇒ |
---|
1789 | let ticks ≝ ticks_of_instruction (POP ? arg) in |
---|
1790 | 〈ticks, ticks〉 |
---|
1791 | | XCH arg1 arg2 ⇒ |
---|
1792 | let ticks ≝ ticks_of_instruction (XCH ? arg1 arg2) in |
---|
1793 | 〈ticks, ticks〉 |
---|
1794 | | XCHD arg1 arg2 ⇒ |
---|
1795 | let ticks ≝ ticks_of_instruction (XCHD ? arg1 arg2) in |
---|
1796 | 〈ticks, ticks〉 |
---|
1797 | | RET ⇒ |
---|
1798 | let ticks ≝ ticks_of_instruction (RET ?) in |
---|
1799 | 〈ticks, ticks〉 |
---|
1800 | | RETI ⇒ |
---|
1801 | let ticks ≝ ticks_of_instruction (RETI ?) in |
---|
1802 | 〈ticks, ticks〉 |
---|
1803 | | NOP ⇒ |
---|
1804 | let ticks ≝ ticks_of_instruction (NOP ?) in |
---|
1805 | 〈ticks, ticks〉 |
---|
1806 | ] |
---|
1807 | | Comment comment ⇒ 〈0, 0〉 |
---|
1808 | | Cost cost ⇒ 〈0, 0〉 |
---|
1809 | | Jmp jmp ⇒ 〈2, 2〉 |
---|
1810 | | Call call ⇒ 〈2, 2〉 |
---|
1811 | | Mov dptr tgt ⇒ 〈2, 2〉 |
---|
1812 | ]. |
---|
1813 | cases not_implemented (* policy returned medium_jump for conditional jumping = impossible *) |
---|
1814 | qed. |
---|
1815 | |
---|
1816 | definition ticks_of: ∀p:pseudo_assembly_program. policy p → Word → nat × nat ≝ |
---|
1817 | λprogram: pseudo_assembly_program.λpol. |
---|
1818 | λppc: Word. |
---|
1819 | let 〈preamble, pseudo〉 ≝ program in |
---|
1820 | let 〈fetched, new_ppc〉 ≝ fetch_pseudo_instruction pseudo ppc in |
---|
1821 | ticks_of0 program pol ppc fetched. |
---|
1822 | |
---|
1823 | lemma eq_rect_Type1_r: |
---|
1824 | ∀A: Type[1]. |
---|
1825 | ∀a:A. |
---|
1826 | ∀P: ∀x:A. eq ? x a → Type[1]. P a (refl A a) → ∀x: A.∀p:eq ? x a. P x p. |
---|
1827 | #A #a #P #H #x #p |
---|
1828 | generalize in match H |
---|
1829 | generalize in match P |
---|
1830 | cases p |
---|
1831 | // |
---|
1832 | qed. |
---|
1833 | |
---|
1834 | axiom split_append: |
---|
1835 | ∀A: Type[0]. |
---|
1836 | ∀m, n: nat. |
---|
1837 | ∀v, v': Vector A m. |
---|
1838 | ∀q, q': Vector A n. |
---|
1839 | let 〈v', q'〉 ≝ split A m n (v@@q) in |
---|
1840 | v = v' ∧ q = q'. |
---|
1841 | |
---|
1842 | axiom split_vector_singleton: |
---|
1843 | ∀A: Type[0]. |
---|
1844 | ∀n: nat. |
---|
1845 | ∀v: Vector A (S n). |
---|
1846 | ∀rest: Vector A n. |
---|
1847 | ∀s: Vector A 1. |
---|
1848 | ∀prf. |
---|
1849 | v = s @@ rest → |
---|
1850 | ((get_index_v A ? v 0 prf) ::: rest) = v. |
---|
1851 | |
---|
1852 | example sub_minus_one_seven_eight: |
---|
1853 | ∀v: BitVector 7. |
---|
1854 | false ::: (\fst (sub_7_with_carry v (bitvector_of_nat ? 1) false)) = |
---|
1855 | \fst (sub_8_with_carry (false ::: v) (bitvector_of_nat ? 1) false). |
---|
1856 | cases daemon. |
---|
1857 | qed. |
---|
1858 | |
---|
1859 | (* |
---|
1860 | lemma blah: |
---|
1861 | ∀m: internal_pseudo_address_map. |
---|
1862 | ∀s: PseudoStatus. |
---|
1863 | ∀arg: Byte. |
---|
1864 | ∀b: bool. |
---|
1865 | addressing_mode_ok m s (DIRECT arg) = true → |
---|
1866 | get_arg_8 ? (set_low_internal_ram ? s (low_internal_ram_of_pseudo_low_internal_ram m (low_internal_ram ? s))) b (DIRECT arg) = |
---|
1867 | get_arg_8 ? s b (DIRECT arg). |
---|
1868 | [2, 3: normalize % ] |
---|
1869 | #m #s #arg #b #hyp |
---|
1870 | whd in ⊢ (??%%) |
---|
1871 | @split_elim'' |
---|
1872 | #nu' #nl' #arg_nu_nl_eq |
---|
1873 | normalize nodelta |
---|
1874 | generalize in match (refl ? (get_index_v bool 4 nu' ? ?)) |
---|
1875 | cases (get_index_v bool 4 nu' ? ?) in ⊢ (??%? → %) |
---|
1876 | #get_index_v_eq |
---|
1877 | normalize nodelta |
---|
1878 | [2: |
---|
1879 | normalize nodelta |
---|
1880 | @split_elim'' |
---|
1881 | #bit_one' #three_bits' #bit_one_three_bit_eq |
---|
1882 | generalize in match (low_internal_ram_of_pseudo_internal_ram_miss m s (three_bits'@@nl')) |
---|
1883 | normalize nodelta |
---|
1884 | generalize in match (refl ? (sub_7_with_carry ? ? ?)) |
---|
1885 | cases (sub_7_with_carry ? ? ?) in ⊢ (??%? → %) |
---|
1886 | #Saddr #carr' #Saddr_carr_eq |
---|
1887 | normalize nodelta |
---|
1888 | #carr_hyp' |
---|
1889 | @carr_hyp' |
---|
1890 | [1: |
---|
1891 | |2: whd in hyp:(??%?); generalize in match hyp; -hyp; |
---|
1892 | generalize in match (refl ? (¬(member (BitVector 8) ? arg m))) |
---|
1893 | cases (¬(member (BitVector 8) ? arg m)) in ⊢ (??%? → %) |
---|
1894 | #member_eq |
---|
1895 | normalize nodelta |
---|
1896 | [2: #destr destruct(destr) |
---|
1897 | |1: -carr_hyp'; |
---|
1898 | >arg_nu_nl_eq |
---|
1899 | <(split_vector_singleton ? ? nu' ? ? ? bit_one_three_bit_eq) |
---|
1900 | [1: >get_index_v_eq in ⊢ (??%? → ?) |
---|
1901 | |2: @le_S @le_S @le_S @le_n |
---|
1902 | ] |
---|
1903 | cases (member (BitVector 8) ? (\fst ?) ?) |
---|
1904 | [1: #destr normalize in destr; destruct(destr) |
---|
1905 | |2: |
---|
1906 | ] |
---|
1907 | ] |
---|
1908 | |3: >get_index_v_eq in ⊢ (??%?) |
---|
1909 | change in ⊢ (??(???%?)?) with ((? ::: three_bits') @@ nl') |
---|
1910 | >(split_vector_singleton … bit_one_three_bit_eq) |
---|
1911 | <arg_nu_nl_eq |
---|
1912 | whd in hyp:(??%?) |
---|
1913 | cases (member (BitVector 8) (eq_bv 8) arg m) in hyp |
---|
1914 | normalize nodelta [*: #ignore @sym_eq ] |
---|
1915 | ] |
---|
1916 | | |
---|
1917 | ]. |
---|
1918 | *) |
---|
1919 | (* |
---|
1920 | map_address0 ... (DIRECT arg) = Some .. → |
---|
1921 | get_arg_8 (map_address0 ... (internal_ram ...) (DIRECT arg) = |
---|
1922 | get_arg_8 (internal_ram ...) (DIRECT arg) |
---|
1923 | |
---|
1924 | ((if addressing_mode_ok M ps ACC_A∧addressing_mode_ok M ps (DIRECT ARG2) |
---|
1925 | then Some internal_pseudo_address_map M |
---|
1926 | else None internal_pseudo_address_map ) |
---|
1927 | =Some internal_pseudo_address_map M') |
---|
1928 | *) |
---|
1929 | |
---|
1930 | axiom low_internal_ram_write_at_stack_pointer: |
---|
1931 | ∀T1,T2,M,s1,s2,s3.∀pol.∀pbu,pbl,bu,bl,sp1,sp2:BitVector 8. |
---|
1932 | get_8051_sfr ? s2 SFR_SP = get_8051_sfr ? s3 SFR_SP → |
---|
1933 | low_internal_ram ? s2 = low_internal_ram T2 s3 → |
---|
1934 | sp1 = \snd (half_add ? (get_8051_sfr ? s1 SFR_SP) (bitvector_of_nat 8 1)) → |
---|
1935 | sp2 = \snd (half_add ? sp1 (bitvector_of_nat 8 1)) → |
---|
1936 | bu@@bl = sigma (code_memory … s2) pol (pbu@@pbl) → |
---|
1937 | low_internal_ram T1 |
---|
1938 | (write_at_stack_pointer ? |
---|
1939 | (set_8051_sfr ? |
---|
1940 | (write_at_stack_pointer ? |
---|
1941 | (set_8051_sfr ? |
---|
1942 | (set_low_internal_ram ? s1 |
---|
1943 | (low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram ? s2))) |
---|
1944 | SFR_SP sp1) |
---|
1945 | bl) |
---|
1946 | SFR_SP sp2) |
---|
1947 | bu) |
---|
1948 | = low_internal_ram_of_pseudo_low_internal_ram (sp1::M) |
---|
1949 | (low_internal_ram ? |
---|
1950 | (write_at_stack_pointer ? |
---|
1951 | (set_8051_sfr ? |
---|
1952 | (write_at_stack_pointer ? (set_8051_sfr ? s3 SFR_SP sp1) pbl) |
---|
1953 | SFR_SP sp2) |
---|
1954 | pbu)). |
---|
1955 | |
---|
1956 | axiom high_internal_ram_write_at_stack_pointer: |
---|
1957 | ∀T1,T2,M,s1,s2,s3,pol.∀pbu,pbl,bu,bl,sp1,sp2:BitVector 8. |
---|
1958 | get_8051_sfr ? s2 SFR_SP = get_8051_sfr ? s3 SFR_SP → |
---|
1959 | high_internal_ram ? s2 = high_internal_ram T2 s3 → |
---|
1960 | sp1 = \snd (half_add ? (get_8051_sfr ? s1 SFR_SP) (bitvector_of_nat 8 1)) → |
---|
1961 | sp2 = \snd (half_add ? sp1 (bitvector_of_nat 8 1)) → |
---|
1962 | bu@@bl = sigma (code_memory … s2) pol (pbu@@pbl) → |
---|
1963 | high_internal_ram T1 |
---|
1964 | (write_at_stack_pointer ? |
---|
1965 | (set_8051_sfr ? |
---|
1966 | (write_at_stack_pointer ? |
---|
1967 | (set_8051_sfr ? |
---|
1968 | (set_high_internal_ram ? s1 |
---|
1969 | (high_internal_ram_of_pseudo_high_internal_ram M (high_internal_ram ? s2))) |
---|
1970 | SFR_SP sp1) |
---|
1971 | bl) |
---|
1972 | SFR_SP sp2) |
---|
1973 | bu) |
---|
1974 | = high_internal_ram_of_pseudo_high_internal_ram (sp1::M) |
---|
1975 | (high_internal_ram ? |
---|
1976 | (write_at_stack_pointer ? |
---|
1977 | (set_8051_sfr ? |
---|
1978 | (write_at_stack_pointer ? (set_8051_sfr ? s3 SFR_SP sp1) pbl) |
---|
1979 | SFR_SP sp2) |
---|
1980 | pbu)). |
---|
1981 | |
---|
1982 | theorem main_thm: |
---|
1983 | ∀M,M',ps,pol. |
---|
1984 | next_internal_pseudo_address_map M ps = Some … M' → |
---|
1985 | ∃n. |
---|
1986 | execute n (status_of_pseudo_status M ps pol) |
---|
1987 | = status_of_pseudo_status M' (execute_1_pseudo_instruction (ticks_of (code_memory … ps) pol) ps) ?. |
---|
1988 | [2: >execute_1_pseudo_instruction_preserves_code_memory @pol] |
---|
1989 | #M #M' #ps #pol #SAFE |
---|
1990 | cut |
---|
1991 | (∀ps'. |
---|
1992 | ∀prf:ps'=execute_1_pseudo_instruction (ticks_of (code_memory … ps) pol) ps. |
---|
1993 | ∃n. execute n (status_of_pseudo_status M ps pol) = status_of_pseudo_status M' ps' ?) |
---|
1994 | [ >prf >execute_1_pseudo_instruction_preserves_code_memory @pol |3: #K @(K ? (refl …))] |
---|
1995 | #ps' #EQ |
---|
1996 | whd in ⊢ (??(λ_.??(??%)?)) |
---|
1997 | change with |
---|
1998 | (∃n. |
---|
1999 | execute n |
---|
2000 | (set_low_internal_ram ? |
---|
2001 | (set_high_internal_ram ? |
---|
2002 | (set_program_counter ? |
---|
2003 | (set_code_memory ?? ps (load_code_memory ?)) |
---|
2004 | (sigma ? pol (program_counter ? ps))) |
---|
2005 | (high_internal_ram_of_pseudo_high_internal_ram M ?)) |
---|
2006 | (low_internal_ram_of_pseudo_low_internal_ram M ?)) |
---|
2007 | = set_low_internal_ram ? |
---|
2008 | (set_high_internal_ram ? |
---|
2009 | (set_program_counter ? |
---|
2010 | (set_code_memory ?? ? (load_code_memory ?)) |
---|
2011 | (sigma ???)) ?) ?) |
---|
2012 | >EQ whd in match eq_rect_Type0_r normalize nodelta |
---|
2013 | >execute_1_pseudo_instruction_preserves_code_memory normalize nodelta |
---|
2014 | generalize in match EQ -EQ; |
---|
2015 | generalize in match (refl … (code_memory pseudo_assembly_program ps)) |
---|
2016 | generalize in match pol generalize in ⊢ (∀_.??%? → ?) |
---|
2017 | * #preamble #instr_list #pol #EQ1 generalize in match pol -pol <EQ1 #pol #EQps' <EQps' |
---|
2018 | |
---|
2019 | |
---|
2020 | |
---|
2021 | lemma main_thm0: |
---|
2022 | ∀M,M',ps,preamble,instr_list,pol. |
---|
2023 | ∀EQ0 : (code_memory pseudo_assembly_program ps=〈preamble,instr_list〉). |
---|
2024 | ∀assembled. |
---|
2025 | ∀H: |
---|
2026 | (∀ppc. |
---|
2027 | let 〈pi,newppc〉 ≝ fetch_pseudo_instruction instr_list ppc in |
---|
2028 | let instructions ≝ |
---|
2029 | expand_pseudo_instruction |
---|
2030 | 〈preamble,instr_list〉 pol ppc |
---|
2031 | (λx:Identifier |
---|
2032 | .sigma 〈preamble,instr_list〉 pol |
---|
2033 | (address_of_word_labels_code_mem instr_list x)) |
---|
2034 | (λx:BitVector 16 |
---|
2035 | .lookup Identifier 16 x (construct_datalabels preamble) (zero 16)) |
---|
2036 | (sigma 〈preamble,instr_list〉 pol ppc) |
---|
2037 | (refl …) (refl …) (refl …) |
---|
2038 | in |
---|
2039 | fetch_many (load_code_memory assembled) |
---|
2040 | (sigma 〈preamble,instr_list〉 pol newppc) (sigma 〈preamble,instr_list〉 pol ppc) |
---|
2041 | instructions). |
---|
2042 | ∀MAP:(next_internal_pseudo_address_map0 pseudo_assembly_program |
---|
2043 | (\fst (fetch_pseudo_instruction instr_list |
---|
2044 | (program_counter pseudo_assembly_program ps))) M ps |
---|
2045 | =Some internal_pseudo_address_map M'). |
---|
2046 | ∀ps'. |
---|
2047 | (ps'=execute_1_pseudo_instruction (ticks_of 〈preamble,instr_list〉 pol) ps |
---|
2048 | →∃n:ℕ |
---|
2049 | .execute n |
---|
2050 | (set_low_internal_ram (BitVectorTrie Byte 16) |
---|
2051 | (set_high_internal_ram (BitVectorTrie Byte 16) |
---|
2052 | (set_program_counter (BitVectorTrie Byte 16) |
---|
2053 | (set_code_memory pseudo_assembly_program (BitVectorTrie Byte 16) ps |
---|
2054 | (load_code_memory assembled)) |
---|
2055 | (sigma 〈preamble,instr_list〉 pol (program_counter pseudo_assembly_program ps))) |
---|
2056 | (high_internal_ram_of_pseudo_high_internal_ram M |
---|
2057 | (high_internal_ram pseudo_assembly_program ps))) |
---|
2058 | (low_internal_ram_of_pseudo_low_internal_ram M |
---|
2059 | (low_internal_ram pseudo_assembly_program ps))) |
---|
2060 | =set_low_internal_ram (BitVectorTrie Byte 16) |
---|
2061 | (set_high_internal_ram (BitVectorTrie Byte 16) |
---|
2062 | (set_program_counter (BitVectorTrie Byte 16) |
---|
2063 | (set_code_memory pseudo_assembly_program (BitVectorTrie Byte 16) ps' |
---|
2064 | (load_code_memory assembled)) |
---|
2065 | (sigma 〈preamble,instr_list〉 pol |
---|
2066 | (program_counter pseudo_assembly_program ps'))) |
---|
2067 | (high_internal_ram_of_pseudo_high_internal_ram M' |
---|
2068 | (high_internal_ram pseudo_assembly_program ps'))) |
---|
2069 | (low_internal_ram_of_pseudo_low_internal_ram M' |
---|
2070 | (low_internal_ram pseudo_assembly_program ps'))). |
---|
2071 | #M #M' #ps #preamble #instr_list #pol #EQ0 #assembled #H #MAP #ps' |
---|
2072 | whd in ⊢ (???(?%?) → ?) normalize nodelta |
---|
2073 | generalize in match (H (program_counter … ps)) -H; >EQ0 normalize nodelta; |
---|
2074 | |
---|
2075 | whd in match expand_pseudo_instruction normalize nodelta; |
---|
2076 | whd in match execute_1_pseudo_instruction (*!!! USARE 0 !!!*) normalize nodelta; >EQ0 |
---|
2077 | normalize in match (\snd 〈preamble,instr_list〉); |
---|
2078 | cases (fetch_pseudo_instruction instr_list (program_counter pseudo_assembly_program ps)) in MAP ⊢ % |
---|
2079 | #pi #newppc normalize nodelta; #MAP |
---|
2080 | cases pi in MAP; normalize nodelta; |
---|
2081 | [2,3: (* Comment, Cost *) #ARG #MAP whd in ⊢ (% → ?) #H2 #EQ %[1,3:@0] |
---|
2082 | generalize in match (option_destruct_Some ??? MAP) -MAP; #MAP <MAP -MAP M'; |
---|
2083 | (* |
---|
2084 | normalize in H1; generalize in match (option_destruct_Some ??? H1) #K1 >K1 i |
---|
2085 | n H2; whd in ⊢ (% → ?) |
---|
2086 | #H2*) >(eq_bv_eq … H2) >EQ % |
---|
2087 | |
---|
2088 | |
---|
2089 | |
---|
2090 | lemma main_thm0: |
---|
2091 | ∀M,M',ps,s,s'',pol. |
---|
2092 | next_internal_pseudo_address_map M ps = Some … M' → |
---|
2093 | status_of_pseudo_status M ps pol = s → |
---|
2094 | status_of_pseudo_status M' (execute_1_pseudo_instruction (ticks_of (code_memory … ps) pol) ps) ? = s'' → |
---|
2095 | ∃n. execute n s = s''. |
---|
2096 | [2: >execute_1_pseudo_instruction_preserves_code_memory @pol] |
---|
2097 | #M #M' #ps #s #s'' #pol |
---|
2098 | generalize in match (fetch_assembly_pseudo2 (code_memory … ps) pol) |
---|
2099 | >execute_1_pseudo_instruction_preserves_code_memory |
---|
2100 | generalize in match (refl … (assembly (code_memory … ps) pol)) |
---|
2101 | generalize in match (assembly (code_memory … ps) pol) in ⊢ (??%? → ?) #ASS #K <K generalize in match K; -K; |
---|
2102 | |
---|
2103 | |
---|
2104 | whd in ⊢ (? → ? → ??%? → ??%? → ?) |
---|
2105 | >execute_1_pseudo_instruction_preserves_code_memory |
---|
2106 | generalize in match (refl … (assembly (code_memory … ps) pol)) |
---|
2107 | generalize in match (assembly (code_memory … ps) pol) in ⊢ (??%? → ?) #ASS #K <K generalize in match K; -K; |
---|
2108 | whd in ⊢ (???% → ?) |
---|
2109 | cases (build_maps (code_memory … ps) pol) |
---|
2110 | #labels #costs change in ⊢ (? → ? → ??%? → ?) with (next_internal_pseudo_address_map0 ? ? ? ?) |
---|
2111 | cases (code_memory … ps) in pol ⊢ %; |
---|
2112 | generalize in match pol; -pol; |
---|
2113 | @(match code_memory … ps return λx. ∀e:code_memory … ps = x.? with [pair preamble instr_list ⇒ ?]) [%] |
---|
2114 | #EQ0 #pol normalize nodelta; |
---|
2115 | |
---|
2116 | |
---|
2117 | |
---|
2118 | lemma main_thm0: |
---|
2119 | ∀M,M',ps,ps',pol. |
---|
2120 | next_internal_pseudo_address_map M ps = Some … M' → |
---|
2121 | ∀H:ps' = execute_1_pseudo_instruction (ticks_of (code_memory … ps) pol) ps. |
---|
2122 | ∃n. |
---|
2123 | execute n (status_of_pseudo_status M ps pol) |
---|
2124 | = status_of_pseudo_status M' ps' ?. |
---|
2125 | [2: >H >execute_1_pseudo_instruction_preserves_code_memory @pol] |
---|
2126 | #M #M' #ps #ps' #pol |
---|
2127 | change with (next_internal_pseudo_address_map0 ???? = ? → ?) |
---|
2128 | generalize in match (fetch_assembly_pseudo2 (code_memory … ps) pol (program_counter … ps)) |
---|
2129 | @pair_elim' #labels #costs #EQ1 normalize nodelta |
---|
2130 | whd in match execute_1_pseudo_instruction |
---|
2131 | @pair_elim' #pi #newppc #EQ2 normalize nodelta |
---|
2132 | cases pi in EQ2; normalize nodelta |
---|
2133 | [2: #ARG #MAP #H1 #H2 #EQ >MAP in EQ; |
---|
2134 | |
---|
2135 | |
---|
2136 | cases (execute_1_pseudo_instruction_preserves_code_memory ??) XX |
---|
2137 | cases (assembly ??) #assembled #costs normalize nodelta |
---|
2138 | cases (build_maps (code_memory … ps) pol) * #labels #costs whd in match eject; normalize nodelta; |
---|
2139 | cases ps in pol ⊢ %; #code_mem #lir #hir #er #pc #sfr1 #sfr2 #p1l #p3l #clock #pol |
---|
2140 | #MAP |
---|
2141 | cases (fetch_pseudo_instruction (\snd code_mem) ppc) |
---|
2142 | |
---|
2143 | change with |
---|
2144 | (? → ∃n. |
---|
2145 | execute n |
---|
2146 | (set_low_internal_ram ? |
---|
2147 | (set_high_internal_ram ? |
---|
2148 | (set_program_counter ? |
---|
2149 | (set_code_memory ?? ps (load_code_memory ?)) |
---|
2150 | (sigma ? pol (program_counter ? ps))) |
---|
2151 | (high_internal_ram_of_pseudo_high_internal_ram M ?)) |
---|
2152 | (low_internal_ram_of_pseudo_low_internal_ram M ?)) |
---|
2153 | = set_low_internal_ram ? |
---|
2154 | (set_high_internal_ram ? |
---|
2155 | (set_program_counter ? |
---|
2156 | (set_code_memory ?? (execute_1_pseudo_instruction ? ps) (load_code_memory ?)) |
---|
2157 | (sigma ???)) |
---|
2158 | ?) |
---|
2159 | ?) |
---|
2160 | change with (next_internal_pseudo_address_map0 ???? = ? → ?) |
---|
2161 | generalize in match (fetch_assembly_pseudo2 (code_memory … ps) pol (program_counter … ps)) |
---|
2162 | @pair_elim' #labels #costs #EQ1 normalize nodelta |
---|
2163 | @pair_elim' #pi #newppc #EQ2 |
---|
2164 | >execute_1_pseudo_instruction_preserves_code_memory |
---|
2165 | cases (assembly ??) #assembled #costs normalize nodelta |
---|
2166 | cases (build_maps (code_memory … ps) pol) * #labels #costs whd in match eject; normalize nodelta; |
---|
2167 | cases ps in pol ⊢ %; #code_mem #lir #hir #er #pc #sfr1 #sfr2 #p1l #p3l #clock #pol |
---|
2168 | #MAP |
---|
2169 | cases (fetch_pseudo_instruction (\snd code_mem) ppc) |
---|
2170 | |
---|
2171 | |
---|
2172 | (code_memory pseudo_assembly_program ps) |
---|
2173 | #MAP |
---|
2174 | |
---|
2175 | |
---|
2176 | |
---|
2177 | whd in ⊢ (? → ? → ??%? → ??%? → ?) |
---|
2178 | generalize in match (assembly (code_memory … ps) pol) in ⊢ (??%? → %) #ASS whd in ⊢ (???% → ?) |
---|
2179 | cases (build_maps (code_memory … ps) pol) |
---|
2180 | #labels #costs change in ⊢ (? → ? → ??%? → ?) with (next_internal_pseudo_address_map0 ? ? ? ?) |
---|
2181 | generalize in match pol; -pol; |
---|
2182 | @(match code_memory … ps return λx. ∀e:code_memory … ps = x.? with [pair preamble instr_list ⇒ ?]) [%] |
---|
2183 | #EQ0 #pol normalize nodelta; |
---|
2184 | generalize in ⊢ (???(match % with [_ ⇒ ? | _ ⇒ ?]) → ?) *; normalize nodelta; |
---|
2185 | [ #EQ >EQ #_ #_ #abs @⊥ normalize in abs; destruct (abs) ] |
---|
2186 | * #final_ppc * #final_pc #assembled #EQ >EQ -EQ ASS; normalize nodelta; |
---|
2187 | #H generalize in match (H ? (refl …)) -H; #H; |
---|
2188 | #MAP |
---|
2189 | #H1 generalize in match (option_destruct_Some ??? H1) -H1; #H1 <H1 -H1; |
---|
2190 | #H2 generalize in match (option_destruct_Some ??? H2) -H2; #H2 <H2 -H2; |
---|
2191 | change with |
---|
2192 | (∃n. |
---|
2193 | execute n |
---|
2194 | (set_low_internal_ram ? |
---|
2195 | (set_high_internal_ram ? |
---|
2196 | (set_program_counter ? |
---|
2197 | (set_code_memory ?? ps (load_code_memory assembled)) |
---|
2198 | (sigma 〈preamble,instr_list〉 pol (program_counter ? ps))) |
---|
2199 | (high_internal_ram_of_pseudo_high_internal_ram M ?)) |
---|
2200 | (low_internal_ram_of_pseudo_low_internal_ram M ?)) |
---|
2201 | = set_low_internal_ram ? |
---|
2202 | (set_high_internal_ram ? |
---|
2203 | (set_program_counter ? |
---|
2204 | (set_code_memory ?? (execute_1_pseudo_instruction ? ps) (load_code_memory assembled)) |
---|
2205 | (sigma ???)) |
---|
2206 | ?) |
---|
2207 | ?) |
---|
2208 | whd in match (\snd 〈preamble,instr_list〉) in H; |
---|
2209 | whd in match (\fst 〈preamble,instr_list〉) in H; |
---|
2210 | whd in match (\snd 〈final_pc,assembled〉) in H; |
---|
2211 | whd in match (\snd 〈preamble,instr_list〉) in MAP; |
---|
2212 | -s s'' labels costs final_ppc final_pc; |
---|
2213 | letin ps' ≝ (execute_1_pseudo_instruction (ticks_of 〈preamble,instr_list〉 pol) ps) |
---|
2214 | (* NICE STATEMENT HERE *) |
---|
2215 | generalize in match (refl … ps') generalize in ⊢ (??%? → ?) normalize nodelta; -ps'; #ps' |
---|
2216 | #K <K generalize in match K; -K; |
---|
2217 | (* STATEMENT WITH EQUALITY HERE *) |
---|
2218 | whd in ⊢ (???(?%?) → ?) |
---|
2219 | whd in ⊢ (???% → ?) generalize in match (H (program_counter … ps)) -H; >EQ0 normalize nodelta; |
---|
2220 | cases (fetch_pseudo_instruction instr_list (program_counter … ps)) in MAP ⊢ % |
---|
2221 | #pi #newppc normalize nodelta; #MAP * #instructions *; |
---|
2222 | cases pi in MAP; normalize nodelta; |
---|
2223 | [2,3: (* Comment, Cost *) #ARG #MAP #H1 #H2 #EQ %[1,3:@0] |
---|
2224 | generalize in match (option_destruct_Some ??? MAP) -MAP; #MAP <MAP -MAP M'; |
---|
2225 | normalize in H1; generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?) |
---|
2226 | #H2 >(eq_bv_eq … H2) >EQ % |
---|
2227 | (* |6: (* Mov *) #arg1 #arg2 |
---|
2228 | #H1 #H2 #EQ %[@1] |
---|
2229 | normalize in H1; generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?) |
---|
2230 | change in ⊢ (? → ??%?) with (execute_1_0 ??) |
---|
2231 | cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta; |
---|
2232 | * * #H2a #H2b whd in ⊢ (% → ?) #H2c |
---|
2233 | >H2b >(eq_instruction_to_eq … H2a) |
---|
2234 | generalize in match EQ; -EQ; whd in ⊢ (???% → ??%?); |
---|
2235 | @(list_addressing_mode_tags_elim_prop … arg1) whd try % -arg1; whd in ⊢ (???% → ??%?) |
---|
2236 | @(list_addressing_mode_tags_elim_prop … arg2) whd try % -arg2; #ARG2 |
---|
2237 | normalize nodelta; |
---|
2238 | [1,2,3,4,5,6,7,8: cases (add_8_with_carry ???) |*: cases (sub_8_with_carry ???)] |
---|
2239 | #result #flags |
---|
2240 | #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) % *) |
---|
2241 | (* |5: (* Call *) #label #MAP |
---|
2242 | generalize in match (option_destruct_Some ??? MAP) -MAP; #MAP <MAP -MAP; |
---|
2243 | whd in ⊢ (???% → ?) cases (pol ?) normalize nodelta; |
---|
2244 | [ (* short *) #abs @⊥ destruct (abs) |
---|
2245 | |3: (* long *) #H1 #H2 #EQ %[@1] |
---|
2246 | (* normalize in H1; !!!*) generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?) |
---|
2247 | change in ⊢ (? → ??%?) with (execute_1_0 ??) |
---|
2248 | cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta; |
---|
2249 | * * #H2a #H2b whd in ⊢ (% → ?) #H2c |
---|
2250 | >H2b >(eq_instruction_to_eq … H2a) |
---|
2251 | generalize in match EQ; -EQ; |
---|
2252 | whd in ⊢ (???% → ??%?); |
---|
2253 | generalize in match (refl … (half_add 8 (get_8051_sfr ? ps SFR_SP) (bitvector_of_nat ? 1))) cases (half_add ???) in ⊢ (??%? → %) #carry #new_sp #EQ1 normalize nodelta; |
---|
2254 | >(eq_bv_eq … H2c) |
---|
2255 | change with |
---|
2256 | ((?=let 〈ppc_bu,ppc_bl〉 ≝ split bool 8 8 newppc in ?) → |
---|
2257 | (let 〈pc_bu,pc_bl〉 ≝ split bool 8 8 (sigma 〈preamble,instr_list〉 pol newppc) in ?)=?) |
---|
2258 | generalize in match (refl … (split … 8 8 newppc)) cases (split bool 8 8 newppc) in ⊢ (??%? → %) #ppc_bu #ppc_bl #EQppc |
---|
2259 | generalize in match (refl … (split … 8 8 (sigma 〈preamble,instr_list〉 pol newppc))) cases (split bool 8 8 (sigma 〈preamble,instr_list〉 pol newppc)) in ⊢ (??%? → %) #pc_bu #pc_bl #EQpc normalize nodelta; |
---|
2260 | >get_8051_sfr_write_at_stack_pointer >get_8051_sfr_write_at_stack_pointer |
---|
2261 | >get_8051_sfr_set_8051_sfr >get_8051_sfr_set_8051_sfr |
---|
2262 | generalize in match (refl … (half_add ? new_sp (bitvector_of_nat ? 1))) cases (half_add ???) in ⊢ (??%? → %) #carry' #new_sp' #EQ2 normalize nodelta; |
---|
2263 | #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) |
---|
2264 | @split_eq_status; |
---|
2265 | [ >code_memory_write_at_stack_pointer whd in ⊢ (??%?) |
---|
2266 | >code_memory_write_at_stack_pointer % |
---|
2267 | | >set_program_counter_set_low_internal_ram |
---|
2268 | >set_clock_set_low_internal_ram |
---|
2269 | @low_internal_ram_write_at_stack_pointer |
---|
2270 | [ >EQ0 @pol | % | % |
---|
2271 | | @(pair_destruct_2 … EQ1) |
---|
2272 | | @(pair_destruct_2 … EQ2) |
---|
2273 | | >(pair_destruct_1 ????? EQpc) |
---|
2274 | >(pair_destruct_2 ????? EQpc) |
---|
2275 | @split_elim #x #y #H <H -x y H; |
---|
2276 | >(pair_destruct_1 ????? EQppc) |
---|
2277 | >(pair_destruct_2 ????? EQppc) |
---|
2278 | @split_elim #x #y #H <H -x y H; |
---|
2279 | >EQ0 % ] |
---|
2280 | | >set_low_internal_ram_set_high_internal_ram |
---|
2281 | >set_program_counter_set_high_internal_ram |
---|
2282 | >set_clock_set_high_internal_ram |
---|
2283 | @high_internal_ram_write_at_stack_pointer |
---|
2284 | [ >EQ0 @pol | % | % |
---|
2285 | | @(pair_destruct_2 … EQ1) |
---|
2286 | | @(pair_destruct_2 … EQ2) |
---|
2287 | | >(pair_destruct_1 ????? EQpc) |
---|
2288 | >(pair_destruct_2 ????? EQpc) |
---|
2289 | @split_elim #x #y #H <H -x y H; |
---|
2290 | >(pair_destruct_1 ????? EQppc) |
---|
2291 | >(pair_destruct_2 ????? EQppc) |
---|
2292 | @split_elim #x #y #H <H -x y H; |
---|
2293 | >EQ0 % ] |
---|
2294 | | >external_ram_write_at_stack_pointer whd in ⊢ (??%?) |
---|
2295 | >external_ram_write_at_stack_pointer whd in ⊢ (???%) |
---|
2296 | >external_ram_write_at_stack_pointer whd in ⊢ (???%) |
---|
2297 | >external_ram_write_at_stack_pointer % |
---|
2298 | | change with (? = sigma ?? (address_of_word_labels_code_mem (\snd (code_memory ? ps)) ?)) |
---|
2299 | >EQ0 % |
---|
2300 | | >special_function_registers_8051_write_at_stack_pointer whd in ⊢ (??%?) |
---|
2301 | >special_function_registers_8051_write_at_stack_pointer whd in ⊢ (???%) |
---|
2302 | >special_function_registers_8051_write_at_stack_pointer whd in ⊢ (???%) |
---|
2303 | >special_function_registers_8051_write_at_stack_pointer % |
---|
2304 | | >special_function_registers_8052_write_at_stack_pointer whd in ⊢ (??%?) |
---|
2305 | >special_function_registers_8052_write_at_stack_pointer whd in ⊢ (???%) |
---|
2306 | >special_function_registers_8052_write_at_stack_pointer whd in ⊢ (???%) |
---|
2307 | >special_function_registers_8052_write_at_stack_pointer % |
---|
2308 | | >p1_latch_write_at_stack_pointer whd in ⊢ (??%?) |
---|
2309 | >p1_latch_write_at_stack_pointer whd in ⊢ (???%) |
---|
2310 | >p1_latch_write_at_stack_pointer whd in ⊢ (???%) |
---|
2311 | >p1_latch_write_at_stack_pointer % |
---|
2312 | | >p3_latch_write_at_stack_pointer whd in ⊢ (??%?) |
---|
2313 | >p3_latch_write_at_stack_pointer whd in ⊢ (???%) |
---|
2314 | >p3_latch_write_at_stack_pointer whd in ⊢ (???%) |
---|
2315 | >p3_latch_write_at_stack_pointer % |
---|
2316 | | >clock_write_at_stack_pointer whd in ⊢ (??%?) |
---|
2317 | >clock_write_at_stack_pointer whd in ⊢ (???%) |
---|
2318 | >clock_write_at_stack_pointer whd in ⊢ (???%) |
---|
2319 | >clock_write_at_stack_pointer %] |
---|
2320 | (*| (* medium *) #H1 #H2 #EQ %[@1] generalize in match H1; -H1; |
---|
2321 | @pair_elim' #fst_5_addr #rest_addr #EQ1 |
---|
2322 | @pair_elim' #fst_5_pc #rest_pc #EQ2 |
---|
2323 | generalize in match (refl … (eq_bv … fst_5_addr fst_5_pc)) |
---|
2324 | cases (eq_bv ???) in ⊢ (??%? → %) normalize nodelta; #EQ3 #TEQ [2: destruct (TEQ)] |
---|
2325 | generalize in match (option_destruct_Some ??? TEQ) -TEQ; #K1 >K1 in H2; whd in ⊢ (% → ?) |
---|
2326 | change in ⊢ (? →??%?) with (execute_1_0 ??) |
---|
2327 | @pair_elim' * #instr #newppc' #ticks #EQn |
---|
2328 | * * #H2a #H2b whd in ⊢ (% → ?) #H2c >H2b >(eq_instruction_to_eq … H2a) whd in ⊢ (??%?) |
---|
2329 | generalize in match EQ; -EQ; normalize nodelta; >(eq_bv_eq … H2c) |
---|
2330 | @pair_elim' #carry #new_sp change with (half_add ? (get_8051_sfr ? ps ?) ? = ? → ?) #EQ4 |
---|
2331 | @split_elim' #pc_bu #pc_bl >program_counter_set_8051_sfr XXX change with (newppc = ?) #EQ5 |
---|
2332 | @pair_elim' #carry' #new_sp' #EQ6 normalize nodelta; #EQx >EQx -EQx; |
---|
2333 | change in ⊢ (??(match ????% with [_ ⇒ ?])?) with (sigma … newppc) |
---|
2334 | @split_elim' #pc_bu' #pc_bl' #EQ7 change with (newppc' = ? → ?) |
---|
2335 | >get_8051_sfr_set_8051_sfr |
---|
2336 | |
---|
2337 | whd in EQ:(???%) ⊢ ? >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) whd in ⊢ (??%?) |
---|
2338 | change with ((let 〈pc_bu,pc_bl〉 ≝ split bool 8 8 (sigma 〈preamble,instr_list〉 newppc) in ?)=?) |
---|
2339 | generalize in match (refl … (split bool 8 8 (sigma 〈preamble,instr_list〉 newppc))) |
---|
2340 | cases (split ????) in ⊢ (??%? → %) #pc_bu #pc_bl normalize nodelta; #EQ4 |
---|
2341 | generalize in match (refl … (split bool 4 4 pc_bu)) |
---|
2342 | cases (split ????) in ⊢ (??%? → %) #nu #nl normalize nodelta; #EQ5 |
---|
2343 | generalize in match (refl … (split bool 3 8 rest_addr)) |
---|
2344 | cases (split ????) in ⊢ (??%? → %) #relevant1 #relevant2 normalize nodelta; #EQ6 |
---|
2345 | change with ((let 〈carry,new_pc〉 ≝ half_add ? (sigma … newppc) ? in ?) = ?) |
---|
2346 | generalize in match |
---|
2347 | (refl … |
---|
2348 | (half_add 16 (sigma 〈preamble,instr_list〉 newppc) |
---|
2349 | ((nu@@get_index' bool 0 3 nl:::relevant1)@@relevant2))) |
---|
2350 | cases (half_add ???) in ⊢ (??%? → %) #carry #new_pc normalize nodelta; #EQ7 |
---|
2351 | @split_eq_status try % |
---|
2352 | [ change with (? = sigma ? (address_of_word_labels ps label)) |
---|
2353 | (* ARITHMETICS, BUT THE GOAL SEEMS FALSE *) |
---|
2354 | | whd in ⊢ (??%%) whd in ⊢ (??(?%?)?) whd in ⊢ (??(?(match ?(?%)? with [_ ⇒ ?])?)?) |
---|
2355 | @(bitvector_3_elim_prop … (\fst (split bool 3 8 rest_addr))) %]] *)] |
---|
2356 | |4: (* Jmp *) #label #MAP |
---|
2357 | generalize in match (option_destruct_Some ??? MAP) -MAP; #MAP >MAP -MAP; |
---|
2358 | whd in ⊢ (???% → ?) cases (pol ?) normalize nodelta; |
---|
2359 | [3: (* long *) #H1 #H2 #EQ %[@1] |
---|
2360 | (* normalize in H1; !!!*) generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?) |
---|
2361 | change in ⊢ (? → ??%?) with (execute_1_0 ??) |
---|
2362 | cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta; |
---|
2363 | * * #H2a #H2b whd in ⊢ (% → ?) #H2c |
---|
2364 | >H2b >(eq_instruction_to_eq … H2a) |
---|
2365 | generalize in match EQ; -EQ; |
---|
2366 | #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) |
---|
2367 | cases ps in EQ0 ⊢ %; #A1 #A2 #A3 #A4 #A5 #A6 #A7 #A8 #A9 #A10 #XXXX >XXXX % |
---|
2368 | |1: (* short *) #H1 #H2 #EQ %[@1] generalize in match H1; -H1; |
---|
2369 | generalize in match |
---|
2370 | (refl ? |
---|
2371 | (sub_16_with_carry |
---|
2372 | (sigma 〈preamble,instr_list〉 pol (program_counter … ps)) |
---|
2373 | (sigma 〈preamble,instr_list〉 pol (address_of_word_labels_code_mem instr_list label)) |
---|
2374 | false)) |
---|
2375 | cases (sub_16_with_carry ???) in ⊢ (??%? → %); #results #flags normalize nodelta; |
---|
2376 | generalize in match (refl … (split … 8 8 results)) cases (split ????) in ⊢ (??%? → %) #upper #lower normalize nodelta; |
---|
2377 | generalize in match (refl … (eq_bv … upper (zero 8))) cases (eq_bv ???) in ⊢ (??%? → %) normalize nodelta; |
---|
2378 | #EQ1 #EQ2 #EQ3 #H1 [2: @⊥ destruct (H1)] |
---|
2379 | generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?) |
---|
2380 | change in ⊢ (? → ??%?) with (execute_1_0 ??) |
---|
2381 | cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta; |
---|
2382 | * * #H2a #H2b whd in ⊢ (% → ?) #H2c |
---|
2383 | >H2b >(eq_instruction_to_eq … H2a) |
---|
2384 | generalize in match EQ; -EQ; |
---|
2385 | whd in ⊢ (???% → ?); |
---|
2386 | #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) |
---|
2387 | change with ((let 〈carry,new_pc〉 ≝ half_add ? (sigma ???) ? in ?) = ?) |
---|
2388 | generalize in match (refl … (half_add 16 (sigma 〈preamble,instr_list〉 pol newppc) (sign_extension lower))) |
---|
2389 | cases (half_add ???) in ⊢ (??%? → %) #carry #newpc normalize nodelta #EQ4 |
---|
2390 | @split_eq_status try % change with (newpc = sigma ?? (address_of_word_labels ps label)) |
---|
2391 | (* ARITHMETICS, BUT THE GOAL SEEMS FALSE *) |
---|
2392 | | (* medium *) #H1 #H2 #EQ %[@1] generalize in match H1; -H1; |
---|
2393 | generalize in match |
---|
2394 | (refl … |
---|
2395 | (split … 5 11 (sigma 〈preamble,instr_list〉 pol (address_of_word_labels_code_mem instr_list label)))) |
---|
2396 | cases (split ????) in ⊢ (??%? → %) #fst_5_addr #rest_addr normalize nodelta; #EQ1 |
---|
2397 | generalize in match |
---|
2398 | (refl … |
---|
2399 | (split … 5 11 (sigma 〈preamble,instr_list〉 pol (program_counter … ps)))) |
---|
2400 | cases (split ????) in ⊢ (??%? → %) #fst_5_pc #rest_pc normalize nodelta; #EQ2 |
---|
2401 | generalize in match (refl … (eq_bv … fst_5_addr fst_5_pc)) |
---|
2402 | cases (eq_bv ???) in ⊢ (??%? → %) normalize nodelta; #EQ3 #TEQ [2: destruct (TEQ)] |
---|
2403 | generalize in match (option_destruct_Some ??? TEQ) -TEQ; #K1 >K1 in H2; whd in ⊢ (% → ?) |
---|
2404 | change in ⊢ (? →??%?) with (execute_1_0 ??) |
---|
2405 | cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta; |
---|
2406 | * * #H2a #H2b whd in ⊢ (% → ?) #H2c |
---|
2407 | >H2b >(eq_instruction_to_eq … H2a) |
---|
2408 | generalize in match EQ; -EQ; |
---|
2409 | whd in ⊢ (???% → ?); |
---|
2410 | #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) whd in ⊢ (??%?) |
---|
2411 | change with ((let 〈pc_bu,pc_bl〉 ≝ split bool 8 8 (sigma 〈preamble,instr_list〉 pol newppc) in ?)=?) |
---|
2412 | generalize in match (refl … (split bool 8 8 (sigma 〈preamble,instr_list〉 pol newppc))) |
---|
2413 | cases (split ????) in ⊢ (??%? → %) #pc_bu #pc_bl normalize nodelta; #EQ4 |
---|
2414 | generalize in match (refl … (split bool 4 4 pc_bu)) |
---|
2415 | cases (split ????) in ⊢ (??%? → %) #nu #nl normalize nodelta; #EQ5 |
---|
2416 | generalize in match (refl … (split bool 3 8 rest_addr)) |
---|
2417 | cases (split ????) in ⊢ (??%? → %) #relevant1 #relevant2 normalize nodelta; #EQ6 |
---|
2418 | change with ((let 〈carry,new_pc〉 ≝ half_add ? (sigma … newppc) ? in ?) = ?) |
---|
2419 | generalize in match |
---|
2420 | (refl … |
---|
2421 | (half_add 16 (sigma 〈preamble,instr_list〉 pol newppc) |
---|
2422 | ((nu@@get_index' bool 0 3 nl:::relevant1)@@relevant2))) |
---|
2423 | cases (half_add ???) in ⊢ (??%? → %) #carry #new_pc normalize nodelta; #EQ7 |
---|
2424 | @split_eq_status try % |
---|
2425 | [ change with (? = sigma ?? (address_of_word_labels ps label)) |
---|
2426 | (* ARITHMETICS, BUT THE GOAL SEEMS FALSE *) |
---|
2427 | | whd in ⊢ (??%%) whd in ⊢ (??(?%?)?) whd in ⊢ (??(?(match ?(?%)? with [_ ⇒ ?])?)?) |
---|
2428 | @(bitvector_3_elim_prop … (\fst (split bool 3 8 rest_addr))) %]] |
---|
2429 | *) | (* Instruction *) -pi; whd in ⊢ (? → ??%? → ?) *; normalize nodelta; |
---|
2430 | [1,2,3: (* ADD, ADDC, SUBB *) #arg1 #arg2 #MAP #H1 #H2 #EQ %[1,3,5:@1] |
---|
2431 | normalize in H1; generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?) |
---|
2432 | change in ⊢ (? → ??%?) with (execute_1_0 ??) |
---|
2433 | cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta; |
---|
2434 | * * #H2a #H2b whd in ⊢ (% → ?) #H2c |
---|
2435 | >H2b >(eq_instruction_to_eq … H2a) |
---|
2436 | generalize in match EQ; -EQ; whd in ⊢ (???% → ??%?); generalize in match MAP; -MAP; |
---|
2437 | @(list_addressing_mode_tags_elim_prop … arg1) whd try % -arg1; |
---|
2438 | @(list_addressing_mode_tags_elim_prop … arg2) whd try % -arg2; #ARG2 |
---|
2439 | normalize nodelta; #MAP; (* |
---|
2440 | [1: change in ⊢ (? → %) with |
---|
2441 | ((let 〈result,flags〉 ≝ |
---|
2442 | add_8_with_carry |
---|
2443 | (get_arg_8 ? ps false ACC_A) |
---|
2444 | (get_arg_8 ? |
---|
2445 | (set_low_internal_ram ? ps (low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … ps))) |
---|
2446 | false (DIRECT ARG2)) |
---|
2447 | ? in ?) = ?) |
---|
2448 | [2,3: %] |
---|
2449 | change in ⊢ (???% → ?) with |
---|
2450 | (let 〈result,flags〉 ≝ add_8_with_carry ?(*(get_arg_8 ? ps false ACC_A)*) ?? in ?) |
---|
2451 | >get_arg_8_set_clock *) |
---|
2452 | [1,2: cases (addressing_mode_ok ???? ∧ addressing_mode_ok ????) in MAP ⊢ ? |
---|
2453 | [2,4: #abs @⊥ normalize in abs; destruct (abs) |
---|
2454 | |*:whd in ⊢ (??%? → ?) #H <(option_destruct_Some ??? H)] |
---|
2455 | [ change in ⊢ (? → %) with |
---|
2456 | ((let 〈result,flags〉 ≝ |
---|
2457 | add_8_with_carry |
---|
2458 | (get_arg_8 ? ps false ACC_A) |
---|
2459 | (get_arg_8 ? |
---|
2460 | (set_low_internal_ram ? ps (low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … ps))) |
---|
2461 | false (DIRECT ARG2)) |
---|
2462 | ? in ?) = ?) |
---|
2463 | >get_arg_8_set_low_internal_ram |
---|
2464 | |
---|
2465 | cases (add_8_with_carry ???) |
---|
2466 | |
---|
2467 | [1,2,3,4,5,6,7,8: cases (add_8_with_carry ???) |*: cases (sub_8_with_carry ???)] |
---|
2468 | #result #flags |
---|
2469 | #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) % |
---|
2470 | (*| (* INC *) #arg1 #H1 #H2 #EQ %[@1] |
---|
2471 | normalize in H1; generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?) |
---|
2472 | change in ⊢ (? → ??%?) with (execute_1_0 ??) |
---|
2473 | cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta; |
---|
2474 | * * #H2a #H2b whd in ⊢ (% → ?) #H2c |
---|
2475 | >H2b >(eq_instruction_to_eq … H2a) |
---|
2476 | generalize in match EQ; -EQ; whd in ⊢ (???% → ??%?); |
---|
2477 | @(list_addressing_mode_tags_elim_prop … arg1) whd try % -arg1; normalize nodelta; [1,2,3: #ARG] |
---|
2478 | [1,2,3,4: cases (half_add ???) #carry #result |
---|
2479 | | cases (half_add ???) #carry #bl normalize nodelta; |
---|
2480 | cases (full_add ????) #carry' #bu normalize nodelta ] |
---|
2481 | #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) -newppc'; |
---|
2482 | [5: % |
---|
2483 | |1: <(set_arg_8_set_code_memory 0 [[direct]] ? ? ? (set_clock pseudo_assembly_program |
---|
2484 | (set_program_counter pseudo_assembly_program ps newppc) |
---|
2485 | (\fst (ticks_of0 〈preamble,instr_list〉 |
---|
2486 | (program_counter pseudo_assembly_program ps) |
---|
2487 | (Instruction (INC Identifier (DIRECT ARG)))) |
---|
2488 | +clock pseudo_assembly_program |
---|
2489 | (set_program_counter pseudo_assembly_program ps newppc))) (load_code_memory assembled) result (DIRECT ARG)) |
---|
2490 | [2,3: // ] |
---|
2491 | <(set_arg_8_set_program_counter 0 [[direct]] ? ? ? ? ?) [2://] |
---|
2492 | whd in ⊢ (??%%) |
---|
2493 | cases (split bool 4 4 ARG) |
---|
2494 | #nu' #nl' |
---|
2495 | normalize nodelta |
---|
2496 | cases (split bool 1 3 nu') |
---|
2497 | #bit_1' #ignore' |
---|
2498 | normalize nodelta |
---|
2499 | cases (get_index_v bool 4 nu' ? ?) |
---|
2500 | [ normalize nodelta (* HERE *) whd in ⊢ (??%%) % |
---|
2501 | | |
---|
2502 | ] *) |
---|
2503 | |
---|
2504 | lemma main_thm: |
---|
2505 | ∀M,M',ps,pol. |
---|
2506 | next_internal_pseudo_address_map M ps = Some … M' → |
---|
2507 | ∃n. |
---|
2508 | execute n (status_of_pseudo_status M ps pol) |
---|
2509 | = status_of_pseudo_status M' (execute_1_pseudo_instruction (ticks_of (code_memory … ps) pol) ps) ?. |
---|
2510 | [2: >execute_1_pseudo_instruction_preserves_code_memory @pol] |
---|