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 | include alias "arithmetics/nat.ma". |
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5 | |
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6 | definition bit_elim_prop: ∀P: bool → Prop. Prop ≝ |
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7 | λP. |
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8 | P true ∧ P false. |
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9 | |
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10 | let rec bitvector_elim_prop_internal |
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11 | (n: nat) (P: BitVector n → Prop) (m: nat) |
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12 | on m: |
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13 | m ≤ n → BitVector (n - m) → Prop ≝ |
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14 | match m return λm. m ≤ n → BitVector (n - m) → Prop with |
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15 | [ O ⇒ λprf1. λprefix. P ? |
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16 | | S n' ⇒ λprf2. λprefix. |
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17 | bit_elim_prop (λbit. bitvector_elim_prop_internal n P n' …) |
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18 | ]. |
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19 | [1: |
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20 | applyS prefix |
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21 | |2: |
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22 | letin res ≝ (bit ::: prefix) |
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23 | <minus_S_S >minus_Sn_m |
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24 | try assumption @res |
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25 | |3: |
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26 | @le_S_to_le |
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27 | assumption |
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28 | ] |
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29 | qed. |
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30 | |
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31 | definition bitvector_elim_prop ≝ |
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32 | λn: nat. |
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33 | λP: BitVector n → Prop. |
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34 | bitvector_elim_prop_internal n P n ? ?. |
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35 | [ @(le_n ?) |
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36 | | <(minus_n_n ?) |
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37 | @[[ ]] |
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38 | ] |
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39 | qed. |
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40 | |
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41 | lemma bool_eq_internal_eq: |
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42 | ∀b, c. |
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43 | (λb. λc. (if b then c else (if c then false else true))) b c = true → b = c. |
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44 | #b #c |
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45 | cases b |
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46 | [1: |
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47 | normalize // |
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48 | |2: |
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49 | normalize |
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50 | cases c normalize // |
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51 | ] |
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52 | qed. |
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53 | |
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54 | definition bit_elim: ∀P: bool → bool. bool ≝ |
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55 | λP. |
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56 | P true ∧ P false. |
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57 | |
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58 | let rec bitvector_elim_internal |
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59 | (n: nat) (P: BitVector n → bool) (m: nat) |
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60 | on m: |
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61 | m ≤ n → BitVector (n - m) → bool ≝ |
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62 | match m return λm. m ≤ n → BitVector (n - m) → bool with |
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63 | [ O ⇒ λprf1. λprefix. P ? |
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64 | | S n' ⇒ λprf2. λprefix. bit_elim (λbit. bitvector_elim_internal n P n' ? ?) |
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65 | ]. |
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66 | [1: |
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67 | applyS prefix |
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68 | |2: |
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69 | letin res ≝ (bit ::: prefix) |
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70 | <(minus_S_S ? ?) >(minus_Sn_m ? ?) |
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71 | try @prf2 @res |
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72 | |3: |
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73 | /2/ |
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74 | ]. |
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75 | qed. |
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76 | |
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77 | definition bitvector_elim ≝ |
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78 | λn: nat. |
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79 | λP: BitVector n → bool. |
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80 | bitvector_elim_internal n P n ? ?. |
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81 | try @le_n |
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82 | <minus_n_n @[[]] |
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83 | qed. |
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84 | |
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85 | lemma super_rewrite2: |
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86 | ∀A:Type[0]. |
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87 | ∀n, m: nat. |
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88 | ∀v1: Vector A n. |
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89 | ∀v2: Vector A m. |
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90 | ∀P: ∀m. Vector A m → Prop. |
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91 | n = m → v1 ≃ v2 → P n v1 → P m v2. |
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92 | #A #n #m #v1 #v2 #P #eq #jmeq |
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93 | destruct #assm assumption |
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94 | qed. |
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95 | |
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96 | lemma vector_cons_append: |
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97 | ∀A: Type[0]. |
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98 | ∀n: nat. |
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99 | ∀e: A. |
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100 | ∀v: Vector A n. |
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101 | e ::: v = [[ e ]] @@ v. |
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102 | #A #n #e #v |
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103 | cases v try % |
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104 | #n' #hd #tl % |
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105 | qed. |
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106 | |
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107 | lemma vector_cons_append2: |
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108 | ∀A: Type[0]. |
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109 | ∀n, m: nat. |
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110 | ∀v: Vector A n. |
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111 | ∀q: Vector A m. |
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112 | ∀hd: A. |
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113 | hd:::(v@@q) = (hd:::v)@@q. |
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114 | #A #n #m #v #q |
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115 | elim v try (#hd %) |
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116 | #n' #hd' #tl' #ih #hd' |
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117 | <ih % |
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118 | qed. |
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119 | |
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120 | lemma jmeq_cons_vector_monotone: |
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121 | ∀A: Type[0]. |
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122 | ∀m, n: nat. |
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123 | ∀v: Vector A m. |
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124 | ∀q: Vector A n. |
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125 | ∀prf: m = n. |
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126 | ∀hd: A. |
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127 | v ≃ q → hd:::v ≃ hd:::q. |
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128 | #A #m #n #v #q #prf #hd #E |
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129 | @(super_rewrite2 A … E) |
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130 | try assumption % |
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131 | qed. |
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132 | |
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133 | lemma vector_associative_append: |
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134 | ∀A: Type[0]. |
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135 | ∀n, m, o: nat. |
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136 | ∀v: Vector A n. |
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137 | ∀q: Vector A m. |
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138 | ∀r: Vector A o. |
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139 | (v @@ q) @@ r ≃ v @@ (q @@ r). |
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140 | #A #n #m #o #v #q #r |
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141 | elim v try % |
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142 | #n' #hd #tl #ih |
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143 | <(vector_cons_append2 A … hd) |
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144 | @jmeq_cons_vector_monotone |
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145 | try assumption |
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146 | @associative_plus |
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147 | qed. |
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148 | |
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149 | lemma mem_middle_vector: |
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150 | ∀A: Type[0]. |
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151 | ∀m, o: nat. |
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152 | ∀eq: A → A → bool. |
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153 | ∀reflex: ∀a. eq a a = true. |
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154 | ∀p: Vector A m. |
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155 | ∀a: A. |
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156 | ∀r: Vector A o. |
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157 | mem A eq ? (p@@(a:::r)) a = true. |
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158 | #A #m #o #eq #reflex #p #a |
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159 | elim p try (normalize >reflex #H %) |
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160 | #m' #hd #tl #inductive_hypothesis |
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161 | normalize |
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162 | cases (eq ??) normalize nodelta |
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163 | try (#irrelevant %) |
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164 | @inductive_hypothesis |
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165 | qed. |
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166 | |
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167 | lemma mem_monotonic_wrt_append: |
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168 | ∀A: Type[0]. |
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169 | ∀m, o: nat. |
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170 | ∀eq: A → A → bool. |
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171 | ∀reflex: ∀a. eq a a = true. |
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172 | ∀p: Vector A m. |
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173 | ∀a: A. |
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174 | ∀r: Vector A o. |
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175 | mem A eq ? r a = true → mem A eq ? (p @@ r) a = true. |
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176 | #A #m #o #eq #reflex #p #a |
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177 | elim p try (#r #assm assumption) |
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178 | #m' #hd #tl #inductive_hypothesis #r #assm |
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179 | normalize |
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180 | cases (eq ??) try % |
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181 | @inductive_hypothesis assumption |
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182 | qed. |
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183 | |
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184 | lemma subvector_multiple_append: |
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185 | ∀A: Type[0]. |
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186 | ∀o, n: nat. |
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187 | ∀eq: A → A → bool. |
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188 | ∀refl: ∀a. eq a a = true. |
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189 | ∀h: Vector A o. |
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190 | ∀v: Vector A n. |
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191 | ∀m: nat. |
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192 | ∀q: Vector A m. |
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193 | bool_to_Prop (subvector_with A ? ? eq v (h @@ q @@ v)). |
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194 | #A #o #n #eq #reflex #h #v |
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195 | elim v try (normalize #m #irrelevant @I) |
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196 | #m' #hd #tl #inductive_hypothesis #m #q |
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197 | change with (bool_to_Prop (andb ??)) |
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198 | cut ((mem A eq (o + (m + S m')) (h@@q@@hd:::tl) hd) = true) |
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199 | [1: |
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200 | @mem_monotonic_wrt_append try assumption |
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201 | @mem_monotonic_wrt_append try assumption |
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202 | normalize >reflex % |
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203 | |2: |
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204 | #assm >assm |
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205 | >vector_cons_append |
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206 | change with (bool_to_Prop (subvector_with ??????)) |
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207 | @(super_rewrite2 … (vector_associative_append … q [[hd]] tl)) |
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208 | try @associative_plus |
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209 | @inductive_hypothesis |
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210 | ] |
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211 | qed. |
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212 | |
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213 | lemma vector_cons_empty: |
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214 | ∀A: Type[0]. |
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215 | ∀n: nat. |
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216 | ∀v: Vector A n. |
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217 | [[ ]] @@ v = v. |
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218 | #A #n #v |
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219 | cases v try % |
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220 | #n' #hd #tl % |
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221 | qed. |
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222 | |
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223 | corollary subvector_hd_tl: |
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224 | ∀A: Type[0]. |
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225 | ∀o: nat. |
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226 | ∀eq: A → A → bool. |
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227 | ∀refl: ∀a. eq a a = true. |
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228 | ∀h: A. |
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229 | ∀v: Vector A o. |
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230 | bool_to_Prop (subvector_with A ? ? eq v (h ::: v)). |
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231 | #A #o #eq #reflex #h #v |
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232 | >(vector_cons_append … h v) |
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233 | <(vector_cons_empty … ([[h]] @@ v)) |
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234 | @(subvector_multiple_append … eq reflex [[ ]] v ? [[h]]) |
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235 | qed. |
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236 | |
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237 | lemma eq_a_reflexive: |
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238 | ∀a. eq_a a a = true. |
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239 | #a cases a % |
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240 | qed. |
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241 | |
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242 | lemma is_in_monotonic_wrt_append: |
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243 | ∀m, n: nat. |
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244 | ∀p: Vector addressing_mode_tag m. |
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245 | ∀q: Vector addressing_mode_tag n. |
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246 | ∀to_search: addressing_mode. |
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247 | bool_to_Prop (is_in ? p to_search) → bool_to_Prop (is_in ? (q @@ p) to_search). |
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248 | #m #n #p #q #to_search #assm |
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249 | elim q try assumption |
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250 | #n' #hd #tl #inductive_hypothesis |
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251 | normalize |
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252 | cases (is_a ??) try @I |
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253 | >inductive_hypothesis @I |
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254 | qed. |
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255 | |
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256 | corollary is_in_hd_tl: |
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257 | ∀to_search: addressing_mode. |
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258 | ∀hd: addressing_mode_tag. |
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259 | ∀n: nat. |
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260 | ∀v: Vector addressing_mode_tag n. |
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261 | bool_to_Prop (is_in ? v to_search) → bool_to_Prop (is_in ? (hd:::v) to_search). |
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262 | #to_search #hd #n #v |
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263 | elim v |
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264 | [1: |
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265 | #absurd |
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266 | normalize in absurd; cases absurd |
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267 | |2: |
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268 | #n' #hd' #tl #inductive_hypothesis #assm |
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269 | >vector_cons_append >(vector_cons_append … hd' tl) |
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270 | @(is_in_monotonic_wrt_append … ([[hd']]@@tl) [[hd]] to_search) |
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271 | assumption |
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272 | ] |
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273 | qed. |
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274 | |
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275 | let rec list_addressing_mode_tags_elim |
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276 | (n: nat) (l: Vector addressing_mode_tag (S n)) |
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277 | on l: (l → bool) → bool ≝ |
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278 | match l return λx. |
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279 | match x with |
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280 | [ O ⇒ λl: Vector … O. bool |
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281 | | S x' ⇒ λl: Vector addressing_mode_tag (S x'). (l → bool) → bool |
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282 | ] with |
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283 | [ VEmpty ⇒ true |
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284 | | VCons len hd tl ⇒ λP. |
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285 | let process_hd ≝ |
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286 | match hd return λhd. ∀P: hd:::tl → bool. bool with |
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287 | [ direct ⇒ λP.bitvector_elim 8 (λx. P (DIRECT x)) |
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288 | | indirect ⇒ λP.bit_elim (λx. P (INDIRECT x)) |
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289 | | ext_indirect ⇒ λP.bit_elim (λx. P (EXT_INDIRECT x)) |
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290 | | registr ⇒ λP.bitvector_elim 3 (λx. P (REGISTER x)) |
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291 | | acc_a ⇒ λP.P ACC_A |
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292 | | acc_b ⇒ λP.P ACC_B |
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293 | | dptr ⇒ λP.P DPTR |
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294 | | data ⇒ λP.bitvector_elim 8 (λx. P (DATA x)) |
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295 | | data16 ⇒ λP.bitvector_elim 16 (λx. P (DATA16 x)) |
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296 | | acc_dptr ⇒ λP.P ACC_DPTR |
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297 | | acc_pc ⇒ λP.P ACC_PC |
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298 | | ext_indirect_dptr ⇒ λP.P EXT_INDIRECT_DPTR |
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299 | | indirect_dptr ⇒ λP.P INDIRECT_DPTR |
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300 | | carry ⇒ λP.P CARRY |
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301 | | bit_addr ⇒ λP.bitvector_elim 8 (λx. P (BIT_ADDR x)) |
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302 | | n_bit_addr ⇒ λP.bitvector_elim 8 (λx. P (N_BIT_ADDR x)) |
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303 | | relative ⇒ λP.bitvector_elim 8 (λx. P (RELATIVE x)) |
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304 | | addr11 ⇒ λP.bitvector_elim 11 (λx. P (ADDR11 x)) |
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305 | | addr16 ⇒ λP.bitvector_elim 16 (λx. P (ADDR16 x)) |
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306 | ] |
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307 | in |
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308 | andb (process_hd P) |
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309 | (match len return λx. x = len → bool with |
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310 | [ O ⇒ λprf. true |
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311 | | S y ⇒ λprf. list_addressing_mode_tags_elim y ? P ] (refl ? len)) |
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312 | ]. |
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313 | try % |
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314 | [2: |
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315 | cases (sym_eq ??? prf); assumption |
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316 | |1: |
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317 | generalize in match H; generalize in match tl; |
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318 | destruct #tl |
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319 | normalize in ⊢ (∀_: %. ?); |
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320 | #H |
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321 | whd normalize in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?]); |
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322 | cases (is_a hd (subaddressing_modeel y tl H)) |
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323 | whd try @I normalize nodelta // |
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324 | ] |
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325 | qed. |
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326 | |
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327 | definition product_elim ≝ |
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328 | λm, n: nat. |
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329 | λv: Vector addressing_mode_tag (S m). |
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330 | λq: Vector addressing_mode_tag (S n). |
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331 | λP: (v × q) → bool. |
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332 | list_addressing_mode_tags_elim ? v (λx. list_addressing_mode_tags_elim ? q (λy. P 〈x, y〉)). |
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333 | |
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334 | definition union_elim ≝ |
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335 | λA, B: Type[0]. |
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336 | λelimA: (A → bool) → bool. |
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337 | λelimB: (B → bool) → bool. |
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338 | λelimU: A ⊎ B → bool. |
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339 | elimA (λa. elimB (λb. elimU (inl ? ? a) ∧ elimU (inr ? ? b))). |
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340 | |
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341 | (* |
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342 | definition preinstruction_elim: ∀P: preinstruction [[ relative ]] → bool. bool ≝ |
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343 | λP. |
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344 | list_addressing_mode_tags_elim ? [[ registr ; direct ; indirect ; data ]] (λaddr. P (ADD ? ACC_A addr)) ∧ |
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345 | list_addressing_mode_tags_elim ? [[ registr ; direct ; indirect ; data ]] (λaddr. P (ADDC ? ACC_A addr)) ∧ |
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346 | list_addressing_mode_tags_elim ? [[ registr ; direct ; indirect ; data ]] (λaddr. P (SUBB ? ACC_A addr)) ∧ |
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347 | list_addressing_mode_tags_elim ? [[ acc_a ; registr ; direct ; indirect ; dptr ]] (λaddr. P (INC ? addr)) ∧ |
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348 | list_addressing_mode_tags_elim ? [[ acc_a ; registr ; direct ; indirect ]] (λaddr. P (DEC ? addr)) ∧ |
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349 | list_addressing_mode_tags_elim ? [[acc_b]] (λaddr. P (MUL ? ACC_A addr)) ∧ |
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350 | list_addressing_mode_tags_elim ? [[acc_b]] (λaddr. P (DIV ? ACC_A addr)) ∧ |
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351 | list_addressing_mode_tags_elim ? [[ registr ; direct ]] (λaddr. bitvector_elim 8 (λr. P (DJNZ ? addr (RELATIVE r)))) ∧ |
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352 | list_addressing_mode_tags_elim ? [[ acc_a ; carry ; bit_addr ]] (λaddr. P (CLR ? addr)) ∧ |
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353 | list_addressing_mode_tags_elim ? [[ acc_a ; carry ; bit_addr ]] (λaddr. P (CPL ? addr)) ∧ |
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354 | P (DA ? ACC_A) ∧ |
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355 | bitvector_elim 8 (λr. P (JC ? (RELATIVE r))) ∧ |
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356 | bitvector_elim 8 (λr. P (JNC ? (RELATIVE r))) ∧ |
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357 | bitvector_elim 8 (λr. P (JZ ? (RELATIVE r))) ∧ |
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358 | bitvector_elim 8 (λr. P (JNZ ? (RELATIVE r))) ∧ |
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359 | bitvector_elim 8 (λr. (bitvector_elim 8 (λb: BitVector 8. P (JB ? (BIT_ADDR b) (RELATIVE r))))) ∧ |
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360 | bitvector_elim 8 (λr. (bitvector_elim 8 (λb: BitVector 8. P (JNB ? (BIT_ADDR b) (RELATIVE r))))) ∧ |
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361 | bitvector_elim 8 (λr. (bitvector_elim 8 (λb: BitVector 8. P (JBC ? (BIT_ADDR b) (RELATIVE r))))) ∧ |
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362 | list_addressing_mode_tags_elim ? [[ registr; direct ]] (λaddr. bitvector_elim 8 (λr. P (DJNZ ? addr (RELATIVE r)))) ∧ |
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363 | P (RL ? ACC_A) ∧ |
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364 | P (RLC ? ACC_A) ∧ |
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365 | P (RR ? ACC_A) ∧ |
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366 | P (RRC ? ACC_A) ∧ |
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367 | P (SWAP ? ACC_A) ∧ |
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368 | P (RET ?) ∧ |
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369 | P (RETI ?) ∧ |
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370 | P (NOP ?) ∧ |
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371 | bit_elim (λb. P (XCHD ? ACC_A (INDIRECT b))) ∧ |
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372 | list_addressing_mode_tags_elim ? [[ carry; bit_addr ]] (λaddr. P (SETB ? addr)) ∧ |
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373 | bitvector_elim 8 (λaddr. P (PUSH ? (DIRECT addr))) ∧ |
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374 | bitvector_elim 8 (λaddr. P (POP ? (DIRECT addr))) ∧ |
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375 | union_elim ? ? (product_elim ? ? [[ acc_a ]] [[ direct; data ]]) |
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376 | (product_elim ? ? [[ registr; indirect ]] [[ data ]]) |
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377 | (λd. bitvector_elim 8 (λb. P (CJNE ? d (RELATIVE b)))) ∧ |
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378 | list_addressing_mode_tags_elim ? [[ registr; direct; indirect ]] (λaddr. P (XCH ? ACC_A addr)) ∧ |
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379 | union_elim ? ? (product_elim ? ? [[acc_a]] [[ data ; registr ; direct ; indirect ]]) |
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380 | (product_elim ? ? [[direct]] [[ acc_a ; data ]]) |
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381 | (λd. P (XRL ? d)) ∧ |
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382 | union_elim ? ? (union_elim ? ? (product_elim ? ? [[acc_a]] [[ registr ; direct ; indirect ; data ]]) |
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383 | (product_elim ? ? [[direct]] [[ acc_a ; data ]])) |
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384 | (product_elim ? ? [[carry]] [[ bit_addr ; n_bit_addr]]) |
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385 | (λd. P (ANL ? d)) ∧ |
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386 | union_elim ? ? (union_elim ? ? (product_elim ? ? [[acc_a]] [[ registr ; data ; direct ; indirect ]]) |
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387 | (product_elim ? ? [[direct]] [[ acc_a ; data ]])) |
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388 | (product_elim ? ? [[carry]] [[ bit_addr ; n_bit_addr]]) |
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389 | (λd. P (ORL ? d)) ∧ |
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390 | union_elim ? ? (product_elim ? ? [[acc_a]] [[ ext_indirect ; ext_indirect_dptr ]]) |
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391 | (product_elim ? ? [[ ext_indirect ; ext_indirect_dptr ]] [[acc_a]]) |
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392 | (λd. P (MOVX ? d)) ∧ |
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393 | union_elim ? ? ( |
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394 | union_elim ? ? ( |
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395 | union_elim ? ? ( |
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396 | union_elim ? ? ( |
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397 | union_elim ? ? (product_elim ? ? [[acc_a]] [[ registr ; direct ; indirect ; data ]]) |
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398 | (product_elim ? ? [[ registr ; indirect ]] [[ acc_a ; direct ; data ]])) |
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399 | (product_elim ? ? [[direct]] [[ acc_a ; registr ; direct ; indirect ; data ]])) |
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400 | (product_elim ? ? [[dptr]] [[data16]])) |
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401 | (product_elim ? ? [[carry]] [[bit_addr]])) |
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402 | (product_elim ? ? [[bit_addr]] [[carry]]) |
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403 | (λd. P (MOV ? d)). |
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404 | % |
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405 | qed. |
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406 | |
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407 | definition instruction_elim: ∀P: instruction → bool. bool ≝ |
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408 | λP. (* |
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409 | bitvector_elim 11 (λx. P (ACALL (ADDR11 x))) ∧ |
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410 | bitvector_elim 16 (λx. P (LCALL (ADDR16 x))) ∧ |
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411 | bitvector_elim 11 (λx. P (AJMP (ADDR11 x))) ∧ |
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412 | bitvector_elim 16 (λx. P (LJMP (ADDR16 x))) ∧ *) |
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413 | bitvector_elim 8 (λx. P (SJMP (RELATIVE x))). (* ∧ |
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414 | P (JMP INDIRECT_DPTR) ∧ |
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415 | list_addressing_mode_tags_elim ? [[ acc_dptr; acc_pc ]] (λa. P (MOVC ACC_A a)) ∧ |
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416 | preinstruction_elim (λp. P (RealInstruction p)). *) |
---|
417 | % |
---|
418 | qed. |
---|
419 | |
---|
420 | |
---|
421 | axiom instruction_elim_complete: |
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422 | ∀P. instruction_elim P = true → ∀i. P i = true. |
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423 | *) |
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424 | (*definition eq_instruction ≝ |
---|
425 | λi, j: instruction. |
---|
426 | true.*) |
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427 | |
---|
428 | definition eq_addressing_mode: addressing_mode → addressing_mode → bool ≝ |
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429 | λa, b: addressing_mode. |
---|
430 | match a with |
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431 | [ DIRECT d ⇒ |
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432 | match b with |
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433 | [ DIRECT e ⇒ eq_bv ? d e |
---|
434 | | _ ⇒ false |
---|
435 | ] |
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436 | | INDIRECT b' ⇒ |
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437 | match b with |
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438 | [ INDIRECT e ⇒ eq_b b' e |
---|
439 | | _ ⇒ false |
---|
440 | ] |
---|
441 | | EXT_INDIRECT b' ⇒ |
---|
442 | match b with |
---|
443 | [ EXT_INDIRECT e ⇒ eq_b b' e |
---|
444 | | _ ⇒ false |
---|
445 | ] |
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446 | | REGISTER bv ⇒ |
---|
447 | match b with |
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448 | [ REGISTER bv' ⇒ eq_bv ? bv bv' |
---|
449 | | _ ⇒ false |
---|
450 | ] |
---|
451 | | ACC_A ⇒ match b with [ ACC_A ⇒ true | _ ⇒ false ] |
---|
452 | | ACC_B ⇒ match b with [ ACC_B ⇒ true | _ ⇒ false ] |
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453 | | DPTR ⇒ match b with [ DPTR ⇒ true | _ ⇒ false ] |
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454 | | DATA b' ⇒ |
---|
455 | match b with |
---|
456 | [ DATA e ⇒ eq_bv ? b' e |
---|
457 | | _ ⇒ false |
---|
458 | ] |
---|
459 | | DATA16 w ⇒ |
---|
460 | match b with |
---|
461 | [ DATA16 e ⇒ eq_bv ? w e |
---|
462 | | _ ⇒ false |
---|
463 | ] |
---|
464 | | ACC_DPTR ⇒ match b with [ ACC_DPTR ⇒ true | _ ⇒ false ] |
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465 | | ACC_PC ⇒ match b with [ ACC_PC ⇒ true | _ ⇒ false ] |
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466 | | EXT_INDIRECT_DPTR ⇒ match b with [ EXT_INDIRECT_DPTR ⇒ true | _ ⇒ false ] |
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467 | | INDIRECT_DPTR ⇒ match b with [ INDIRECT_DPTR ⇒ true | _ ⇒ false ] |
---|
468 | | CARRY ⇒ match b with [ CARRY ⇒ true | _ ⇒ false ] |
---|
469 | | BIT_ADDR b' ⇒ |
---|
470 | match b with |
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471 | [ BIT_ADDR e ⇒ eq_bv ? b' e |
---|
472 | | _ ⇒ false |
---|
473 | ] |
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474 | | N_BIT_ADDR b' ⇒ |
---|
475 | match b with |
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476 | [ N_BIT_ADDR e ⇒ eq_bv ? b' e |
---|
477 | | _ ⇒ false |
---|
478 | ] |
---|
479 | | RELATIVE n ⇒ |
---|
480 | match b with |
---|
481 | [ RELATIVE e ⇒ eq_bv ? n e |
---|
482 | | _ ⇒ false |
---|
483 | ] |
---|
484 | | ADDR11 w ⇒ |
---|
485 | match b with |
---|
486 | [ ADDR11 e ⇒ eq_bv ? w e |
---|
487 | | _ ⇒ false |
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488 | ] |
---|
489 | | ADDR16 w ⇒ |
---|
490 | match b with |
---|
491 | [ ADDR16 e ⇒ eq_bv ? w e |
---|
492 | | _ ⇒ false |
---|
493 | ] |
---|
494 | ]. |
---|
495 | |
---|
496 | lemma eq_bv_refl: |
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497 | ∀n, b. |
---|
498 | eq_bv n b b = true. |
---|
499 | #n #b cases b // |
---|
500 | qed. |
---|
501 | |
---|
502 | lemma eq_b_refl: |
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503 | ∀b. |
---|
504 | eq_b b b = true. |
---|
505 | #b cases b // |
---|
506 | qed. |
---|
507 | |
---|
508 | lemma eq_addressing_mode_refl: |
---|
509 | ∀a. eq_addressing_mode a a = true. |
---|
510 | #a |
---|
511 | cases a try #arg1 try #arg2 |
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512 | try @eq_bv_refl try @eq_b_refl |
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513 | try normalize % |
---|
514 | qed. |
---|
515 | |
---|
516 | definition eq_sum: |
---|
517 | ∀A, B. (A → A → bool) → (B → B → bool) → (A ⊎ B) → (A ⊎ B) → bool ≝ |
---|
518 | λlt, rt, leq, req, left, right. |
---|
519 | match left with |
---|
520 | [ inl l ⇒ |
---|
521 | match right with |
---|
522 | [ inl l' ⇒ leq l l' |
---|
523 | | _ ⇒ false |
---|
524 | ] |
---|
525 | | inr r ⇒ |
---|
526 | match right with |
---|
527 | [ inr r' ⇒ req r r' |
---|
528 | | _ ⇒ false |
---|
529 | ] |
---|
530 | ]. |
---|
531 | |
---|
532 | definition eq_prod: ∀A, B. (A → A → bool) → (B → B → bool) → (A × B) → (A × B) → bool ≝ |
---|
533 | λlt, rt, leq, req, left, right. |
---|
534 | let 〈l, r〉 ≝ left in |
---|
535 | let 〈l', r'〉 ≝ right in |
---|
536 | leq l l' ∧ req r r'. |
---|
537 | |
---|
538 | definition eq_preinstruction: preinstruction [[relative]] → preinstruction [[relative]] → bool ≝ |
---|
539 | λi, j. |
---|
540 | match i with |
---|
541 | [ ADD arg1 arg2 ⇒ |
---|
542 | match j with |
---|
543 | [ ADD arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
---|
544 | | _ ⇒ false |
---|
545 | ] |
---|
546 | | ADDC arg1 arg2 ⇒ |
---|
547 | match j with |
---|
548 | [ ADDC arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
---|
549 | | _ ⇒ false |
---|
550 | ] |
---|
551 | | SUBB arg1 arg2 ⇒ |
---|
552 | match j with |
---|
553 | [ SUBB arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
---|
554 | | _ ⇒ false |
---|
555 | ] |
---|
556 | | INC arg ⇒ |
---|
557 | match j with |
---|
558 | [ INC arg' ⇒ eq_addressing_mode arg arg' |
---|
559 | | _ ⇒ false |
---|
560 | ] |
---|
561 | | DEC arg ⇒ |
---|
562 | match j with |
---|
563 | [ DEC arg' ⇒ eq_addressing_mode arg arg' |
---|
564 | | _ ⇒ false |
---|
565 | ] |
---|
566 | | MUL arg1 arg2 ⇒ |
---|
567 | match j with |
---|
568 | [ MUL arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
---|
569 | | _ ⇒ false |
---|
570 | ] |
---|
571 | | DIV arg1 arg2 ⇒ |
---|
572 | match j with |
---|
573 | [ DIV arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
---|
574 | | _ ⇒ false |
---|
575 | ] |
---|
576 | | DA arg ⇒ |
---|
577 | match j with |
---|
578 | [ DA arg' ⇒ eq_addressing_mode arg arg' |
---|
579 | | _ ⇒ false |
---|
580 | ] |
---|
581 | | JC arg ⇒ |
---|
582 | match j with |
---|
583 | [ JC arg' ⇒ eq_addressing_mode arg arg' |
---|
584 | | _ ⇒ false |
---|
585 | ] |
---|
586 | | JNC arg ⇒ |
---|
587 | match j with |
---|
588 | [ JNC arg' ⇒ eq_addressing_mode arg arg' |
---|
589 | | _ ⇒ false |
---|
590 | ] |
---|
591 | | JB arg1 arg2 ⇒ |
---|
592 | match j with |
---|
593 | [ JB arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
---|
594 | | _ ⇒ false |
---|
595 | ] |
---|
596 | | JNB arg1 arg2 ⇒ |
---|
597 | match j with |
---|
598 | [ JNB arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
---|
599 | | _ ⇒ false |
---|
600 | ] |
---|
601 | | JBC arg1 arg2 ⇒ |
---|
602 | match j with |
---|
603 | [ JBC arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
---|
604 | | _ ⇒ false |
---|
605 | ] |
---|
606 | | JZ arg ⇒ |
---|
607 | match j with |
---|
608 | [ JZ arg' ⇒ eq_addressing_mode arg arg' |
---|
609 | | _ ⇒ false |
---|
610 | ] |
---|
611 | | JNZ arg ⇒ |
---|
612 | match j with |
---|
613 | [ JNZ arg' ⇒ eq_addressing_mode arg arg' |
---|
614 | | _ ⇒ false |
---|
615 | ] |
---|
616 | | CJNE arg1 arg2 ⇒ |
---|
617 | match j with |
---|
618 | [ CJNE arg1' arg2' ⇒ |
---|
619 | let prod_eq_left ≝ eq_prod [[acc_a]] [[direct; data]] eq_addressing_mode eq_addressing_mode in |
---|
620 | let prod_eq_right ≝ eq_prod [[registr; indirect]] [[data]] eq_addressing_mode eq_addressing_mode in |
---|
621 | let arg1_eq ≝ eq_sum ? ? prod_eq_left prod_eq_right in |
---|
622 | arg1_eq arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
---|
623 | | _ ⇒ false |
---|
624 | ] |
---|
625 | | DJNZ arg1 arg2 ⇒ |
---|
626 | match j with |
---|
627 | [ DJNZ arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
---|
628 | | _ ⇒ false |
---|
629 | ] |
---|
630 | | CLR arg ⇒ |
---|
631 | match j with |
---|
632 | [ CLR arg' ⇒ eq_addressing_mode arg arg' |
---|
633 | | _ ⇒ false |
---|
634 | ] |
---|
635 | | CPL arg ⇒ |
---|
636 | match j with |
---|
637 | [ CPL arg' ⇒ eq_addressing_mode arg arg' |
---|
638 | | _ ⇒ false |
---|
639 | ] |
---|
640 | | RL arg ⇒ |
---|
641 | match j with |
---|
642 | [ RL arg' ⇒ eq_addressing_mode arg arg' |
---|
643 | | _ ⇒ false |
---|
644 | ] |
---|
645 | | RLC arg ⇒ |
---|
646 | match j with |
---|
647 | [ RLC arg' ⇒ eq_addressing_mode arg arg' |
---|
648 | | _ ⇒ false |
---|
649 | ] |
---|
650 | | RR arg ⇒ |
---|
651 | match j with |
---|
652 | [ RR arg' ⇒ eq_addressing_mode arg arg' |
---|
653 | | _ ⇒ false |
---|
654 | ] |
---|
655 | | RRC arg ⇒ |
---|
656 | match j with |
---|
657 | [ RRC arg' ⇒ eq_addressing_mode arg arg' |
---|
658 | | _ ⇒ false |
---|
659 | ] |
---|
660 | | SWAP arg ⇒ |
---|
661 | match j with |
---|
662 | [ SWAP arg' ⇒ eq_addressing_mode arg arg' |
---|
663 | | _ ⇒ false |
---|
664 | ] |
---|
665 | | SETB arg ⇒ |
---|
666 | match j with |
---|
667 | [ SETB arg' ⇒ eq_addressing_mode arg arg' |
---|
668 | | _ ⇒ false |
---|
669 | ] |
---|
670 | | PUSH arg ⇒ |
---|
671 | match j with |
---|
672 | [ PUSH arg' ⇒ eq_addressing_mode arg arg' |
---|
673 | | _ ⇒ false |
---|
674 | ] |
---|
675 | | POP arg ⇒ |
---|
676 | match j with |
---|
677 | [ POP arg' ⇒ eq_addressing_mode arg arg' |
---|
678 | | _ ⇒ false |
---|
679 | ] |
---|
680 | | XCH arg1 arg2 ⇒ |
---|
681 | match j with |
---|
682 | [ XCH arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
---|
683 | | _ ⇒ false |
---|
684 | ] |
---|
685 | | XCHD arg1 arg2 ⇒ |
---|
686 | match j with |
---|
687 | [ XCHD arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
---|
688 | | _ ⇒ false |
---|
689 | ] |
---|
690 | | RET ⇒ match j with [ RET ⇒ true | _ ⇒ false ] |
---|
691 | | RETI ⇒ match j with [ RETI ⇒ true | _ ⇒ false ] |
---|
692 | | NOP ⇒ match j with [ NOP ⇒ true | _ ⇒ false ] |
---|
693 | | MOVX arg ⇒ |
---|
694 | match j with |
---|
695 | [ MOVX arg' ⇒ |
---|
696 | let prod_eq_left ≝ eq_prod [[acc_a]] [[ext_indirect; ext_indirect_dptr]] eq_addressing_mode eq_addressing_mode in |
---|
697 | let prod_eq_right ≝ eq_prod [[ext_indirect; ext_indirect_dptr]] [[acc_a]] eq_addressing_mode eq_addressing_mode in |
---|
698 | let sum_eq ≝ eq_sum ? ? prod_eq_left prod_eq_right in |
---|
699 | sum_eq arg arg' |
---|
700 | | _ ⇒ false |
---|
701 | ] |
---|
702 | | XRL arg ⇒ |
---|
703 | match j with |
---|
704 | [ XRL arg' ⇒ |
---|
705 | let prod_eq_left ≝ eq_prod [[acc_a]] [[ data ; registr ; direct ; indirect ]] eq_addressing_mode eq_addressing_mode in |
---|
706 | let prod_eq_right ≝ eq_prod [[direct]] [[ acc_a ; data ]] eq_addressing_mode eq_addressing_mode in |
---|
707 | let sum_eq ≝ eq_sum ? ? prod_eq_left prod_eq_right in |
---|
708 | sum_eq arg arg' |
---|
709 | | _ ⇒ false |
---|
710 | ] |
---|
711 | | ORL arg ⇒ |
---|
712 | match j with |
---|
713 | [ ORL arg' ⇒ |
---|
714 | let prod_eq_left1 ≝ eq_prod [[acc_a]] [[ registr ; data ; direct ; indirect ]] eq_addressing_mode eq_addressing_mode in |
---|
715 | let prod_eq_left2 ≝ eq_prod [[direct]] [[ acc_a; data ]] eq_addressing_mode eq_addressing_mode in |
---|
716 | let prod_eq_left ≝ eq_sum ? ? prod_eq_left1 prod_eq_left2 in |
---|
717 | let prod_eq_right ≝ eq_prod [[carry]] [[ bit_addr ; n_bit_addr]] eq_addressing_mode eq_addressing_mode in |
---|
718 | let sum_eq ≝ eq_sum ? ? prod_eq_left prod_eq_right in |
---|
719 | sum_eq arg arg' |
---|
720 | | _ ⇒ false |
---|
721 | ] |
---|
722 | | ANL arg ⇒ |
---|
723 | match j with |
---|
724 | [ ANL arg' ⇒ |
---|
725 | let prod_eq_left1 ≝ eq_prod [[acc_a]] [[ registr ; direct ; indirect ; data ]] eq_addressing_mode eq_addressing_mode in |
---|
726 | let prod_eq_left2 ≝ eq_prod [[direct]] [[ acc_a; data ]] eq_addressing_mode eq_addressing_mode in |
---|
727 | let prod_eq_left ≝ eq_sum ? ? prod_eq_left1 prod_eq_left2 in |
---|
728 | let prod_eq_right ≝ eq_prod [[carry]] [[ bit_addr ; n_bit_addr]] eq_addressing_mode eq_addressing_mode in |
---|
729 | let sum_eq ≝ eq_sum ? ? prod_eq_left prod_eq_right in |
---|
730 | sum_eq arg arg' |
---|
731 | | _ ⇒ false |
---|
732 | ] |
---|
733 | | MOV arg ⇒ |
---|
734 | match j with |
---|
735 | [ MOV arg' ⇒ |
---|
736 | let prod_eq_6 ≝ eq_prod [[acc_a]] [[registr; direct; indirect; data]] eq_addressing_mode eq_addressing_mode in |
---|
737 | let prod_eq_5 ≝ eq_prod [[registr; indirect]] [[acc_a; direct; data]] eq_addressing_mode eq_addressing_mode in |
---|
738 | let prod_eq_4 ≝ eq_prod [[direct]] [[acc_a; registr; direct; indirect; data]] eq_addressing_mode eq_addressing_mode in |
---|
739 | let prod_eq_3 ≝ eq_prod [[dptr]] [[data16]] eq_addressing_mode eq_addressing_mode in |
---|
740 | let prod_eq_2 ≝ eq_prod [[carry]] [[bit_addr]] eq_addressing_mode eq_addressing_mode in |
---|
741 | let prod_eq_1 ≝ eq_prod [[bit_addr]] [[carry]] eq_addressing_mode eq_addressing_mode in |
---|
742 | let sum_eq_1 ≝ eq_sum ? ? prod_eq_6 prod_eq_5 in |
---|
743 | let sum_eq_2 ≝ eq_sum ? ? sum_eq_1 prod_eq_4 in |
---|
744 | let sum_eq_3 ≝ eq_sum ? ? sum_eq_2 prod_eq_3 in |
---|
745 | let sum_eq_4 ≝ eq_sum ? ? sum_eq_3 prod_eq_2 in |
---|
746 | let sum_eq_5 ≝ eq_sum ? ? sum_eq_4 prod_eq_1 in |
---|
747 | sum_eq_5 arg arg' |
---|
748 | | _ ⇒ false |
---|
749 | ] |
---|
750 | ]. |
---|
751 | |
---|
752 | lemma eq_sum_refl: |
---|
753 | ∀A, B: Type[0]. |
---|
754 | ∀leq, req. |
---|
755 | ∀s. |
---|
756 | ∀leq_refl: (∀t. leq t t = true). |
---|
757 | ∀req_refl: (∀u. req u u = true). |
---|
758 | eq_sum A B leq req s s = true. |
---|
759 | #A #B #leq #req #s #leq_refl #req_refl |
---|
760 | cases s assumption |
---|
761 | qed. |
---|
762 | |
---|
763 | lemma eq_prod_refl: |
---|
764 | ∀A, B: Type[0]. |
---|
765 | ∀leq, req. |
---|
766 | ∀s. |
---|
767 | ∀leq_refl: (∀t. leq t t = true). |
---|
768 | ∀req_refl: (∀u. req u u = true). |
---|
769 | eq_prod A B leq req s s = true. |
---|
770 | #A #B #leq #req #s #leq_refl #req_refl |
---|
771 | cases s |
---|
772 | whd in ⊢ (? → ? → ??%?); |
---|
773 | #l #r |
---|
774 | >leq_refl @req_refl |
---|
775 | qed. |
---|
776 | |
---|
777 | lemma eq_preinstruction_refl: |
---|
778 | ∀i. |
---|
779 | eq_preinstruction i i = true. |
---|
780 | #i cases i try #arg1 try #arg2 |
---|
781 | try @eq_addressing_mode_refl |
---|
782 | [1,2,3,4,5,6,7,8,10,16,17,18,19,20: |
---|
783 | whd in ⊢ (??%?); try % |
---|
784 | >eq_addressing_mode_refl |
---|
785 | >eq_addressing_mode_refl % |
---|
786 | |13,15: |
---|
787 | whd in ⊢ (??%?); |
---|
788 | cases arg1 |
---|
789 | [*: |
---|
790 | #arg1_left normalize nodelta |
---|
791 | >eq_prod_refl [*: try % #argr @eq_addressing_mode_refl] |
---|
792 | ] |
---|
793 | |11,12: |
---|
794 | whd in ⊢ (??%?); |
---|
795 | cases arg1 |
---|
796 | [1: |
---|
797 | #arg1_left normalize nodelta |
---|
798 | >(eq_sum_refl …) |
---|
799 | [1: % | 2,3: #arg @eq_prod_refl ] |
---|
800 | @eq_addressing_mode_refl |
---|
801 | |2: |
---|
802 | #arg1_left normalize nodelta |
---|
803 | @eq_prod_refl [*: @eq_addressing_mode_refl ] |
---|
804 | |3: |
---|
805 | #arg1_left normalize nodelta |
---|
806 | >(eq_sum_refl …) |
---|
807 | [1: |
---|
808 | % |
---|
809 | |2,3: |
---|
810 | #arg @eq_prod_refl #arg @eq_addressing_mode_refl |
---|
811 | ] |
---|
812 | |4: |
---|
813 | #arg1_left normalize nodelta |
---|
814 | @eq_prod_refl [*: #arg @eq_addressing_mode_refl ] |
---|
815 | ] |
---|
816 | |14: |
---|
817 | whd in ⊢ (??%?); |
---|
818 | cases arg1 |
---|
819 | [1: |
---|
820 | #arg1_left normalize nodelta |
---|
821 | @eq_sum_refl |
---|
822 | [1: |
---|
823 | #arg @eq_sum_refl |
---|
824 | [1: |
---|
825 | #arg @eq_sum_refl |
---|
826 | [1: |
---|
827 | #arg @eq_sum_refl |
---|
828 | [1: |
---|
829 | #arg @eq_prod_refl |
---|
830 | [*: |
---|
831 | @eq_addressing_mode_refl |
---|
832 | ] |
---|
833 | |2: |
---|
834 | #arg @eq_prod_refl |
---|
835 | [*: |
---|
836 | #arg @eq_addressing_mode_refl |
---|
837 | ] |
---|
838 | ] |
---|
839 | |2: |
---|
840 | #arg @eq_prod_refl |
---|
841 | [*: |
---|
842 | #arg @eq_addressing_mode_refl |
---|
843 | ] |
---|
844 | ] |
---|
845 | |2: |
---|
846 | #arg @eq_prod_refl |
---|
847 | [*: |
---|
848 | #arg @eq_addressing_mode_refl |
---|
849 | ] |
---|
850 | ] |
---|
851 | |2: |
---|
852 | #arg @eq_prod_refl |
---|
853 | [*: |
---|
854 | #arg @eq_addressing_mode_refl |
---|
855 | ] |
---|
856 | ] |
---|
857 | |2: |
---|
858 | #arg1_right normalize nodelta |
---|
859 | @eq_prod_refl |
---|
860 | [*: |
---|
861 | #arg @eq_addressing_mode_refl |
---|
862 | ] |
---|
863 | ] |
---|
864 | |*: |
---|
865 | whd in ⊢ (??%?); |
---|
866 | cases arg1 |
---|
867 | [*: |
---|
868 | #arg1 >eq_sum_refl |
---|
869 | [1,4: |
---|
870 | @eq_addressing_mode_refl |
---|
871 | |2,3,5,6: |
---|
872 | #arg @eq_prod_refl |
---|
873 | [*: |
---|
874 | #arg @eq_addressing_mode_refl |
---|
875 | ] |
---|
876 | ] |
---|
877 | ] |
---|
878 | ] |
---|
879 | qed. |
---|
880 | |
---|
881 | definition eq_instruction: instruction → instruction → bool ≝ |
---|
882 | λi, j. |
---|
883 | match i with |
---|
884 | [ ACALL arg ⇒ |
---|
885 | match j with |
---|
886 | [ ACALL arg' ⇒ eq_addressing_mode arg arg' |
---|
887 | | _ ⇒ false |
---|
888 | ] |
---|
889 | | LCALL arg ⇒ |
---|
890 | match j with |
---|
891 | [ LCALL arg' ⇒ eq_addressing_mode arg arg' |
---|
892 | | _ ⇒ false |
---|
893 | ] |
---|
894 | | AJMP arg ⇒ |
---|
895 | match j with |
---|
896 | [ AJMP arg' ⇒ eq_addressing_mode arg arg' |
---|
897 | | _ ⇒ false |
---|
898 | ] |
---|
899 | | LJMP arg ⇒ |
---|
900 | match j with |
---|
901 | [ LJMP arg' ⇒ eq_addressing_mode arg arg' |
---|
902 | | _ ⇒ false |
---|
903 | ] |
---|
904 | | SJMP arg ⇒ |
---|
905 | match j with |
---|
906 | [ SJMP arg' ⇒ eq_addressing_mode arg arg' |
---|
907 | | _ ⇒ false |
---|
908 | ] |
---|
909 | | JMP arg ⇒ |
---|
910 | match j with |
---|
911 | [ JMP arg' ⇒ eq_addressing_mode arg arg' |
---|
912 | | _ ⇒ false |
---|
913 | ] |
---|
914 | | MOVC arg1 arg2 ⇒ |
---|
915 | match j with |
---|
916 | [ MOVC arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
---|
917 | | _ ⇒ false |
---|
918 | ] |
---|
919 | | RealInstruction instr ⇒ |
---|
920 | match j with |
---|
921 | [ RealInstruction instr' ⇒ eq_preinstruction instr instr' |
---|
922 | | _ ⇒ false |
---|
923 | ] |
---|
924 | ]. |
---|
925 | |
---|
926 | lemma eq_instruction_refl: |
---|
927 | ∀i. eq_instruction i i = true. |
---|
928 | #i cases i [*: #arg1 ] |
---|
929 | try @eq_addressing_mode_refl |
---|
930 | try @eq_preinstruction_refl |
---|
931 | #arg2 whd in ⊢ (??%?); |
---|
932 | >eq_addressing_mode_refl >eq_addressing_mode_refl % |
---|
933 | qed. |
---|
934 | |
---|
935 | let rec vect_member |
---|
936 | (A: Type[0]) (n: nat) (eq: A → A → bool) (v: Vector A n) (a: A) |
---|
937 | on v: bool ≝ |
---|
938 | match v with |
---|
939 | [ VEmpty ⇒ false |
---|
940 | | VCons len hd tl ⇒ |
---|
941 | eq hd a ∨ (vect_member A ? eq tl a) |
---|
942 | ]. |
---|
943 | |
---|
944 | let rec list_addressing_mode_tags_elim_prop |
---|
945 | (n: nat) |
---|
946 | (l: Vector addressing_mode_tag (S n)) |
---|
947 | on l: |
---|
948 | ∀P: l → Prop. |
---|
949 | ∀direct_a. ∀indirect_a. ∀ext_indirect_a. ∀register_a. ∀acc_a_a. |
---|
950 | ∀acc_b_a. ∀dptr_a. ∀data_a. ∀data16_a. ∀acc_dptr_a. ∀acc_pc_a. |
---|
951 | ∀ext_indirect_dptr_a. ∀indirect_dptr_a. ∀carry_a. ∀bit_addr_a. |
---|
952 | ∀n_bit_addr_a. ∀relative_a. ∀addr11_a. ∀addr16_a. |
---|
953 | ∀x: l. P x ≝ |
---|
954 | match l return |
---|
955 | λy. |
---|
956 | match y with |
---|
957 | [ O ⇒ λm: Vector addressing_mode_tag O. ∀prf: 0 = S n. True |
---|
958 | | S y' ⇒ λl: Vector addressing_mode_tag (S y'). ∀prf: S y' = S n.∀P:l → Prop. |
---|
959 | ∀direct_a: if vect_member … eq_a l direct then ∀x. P (DIRECT x) else True. |
---|
960 | ∀indirect_a: if vect_member … eq_a l indirect then ∀x. P (INDIRECT x) else True. |
---|
961 | ∀ext_indirect_a: if vect_member … eq_a l ext_indirect then ∀x. P (EXT_INDIRECT x) else True. |
---|
962 | ∀register_a: if vect_member … eq_a l registr then ∀x. P (REGISTER x) else True. |
---|
963 | ∀acc_a_a: if vect_member … eq_a l acc_a then P (ACC_A) else True. |
---|
964 | ∀acc_b_a: if vect_member … eq_a l acc_b then P (ACC_B) else True. |
---|
965 | ∀dptr_a: if vect_member … eq_a l dptr then P DPTR else True. |
---|
966 | ∀data_a: if vect_member … eq_a l data then ∀x. P (DATA x) else True. |
---|
967 | ∀data16_a: if vect_member … eq_a l data16 then ∀x. P (DATA16 x) else True. |
---|
968 | ∀acc_dptr_a: if vect_member … eq_a l acc_dptr then P ACC_DPTR else True. |
---|
969 | ∀acc_pc_a: if vect_member … eq_a l acc_pc then P ACC_PC else True. |
---|
970 | ∀ext_indirect_dptr_a: if vect_member … eq_a l ext_indirect_dptr then P EXT_INDIRECT_DPTR else True. |
---|
971 | ∀indirect_dptr_a: if vect_member … eq_a l indirect_dptr then P INDIRECT_DPTR else True. |
---|
972 | ∀carry_a: if vect_member … eq_a l carry then P CARRY else True. |
---|
973 | ∀bit_addr_a: if vect_member … eq_a l bit_addr then ∀x. P (BIT_ADDR x) else True. |
---|
974 | ∀n_bit_addr_a: if vect_member … eq_a l n_bit_addr then ∀x. P (N_BIT_ADDR x) else True. |
---|
975 | ∀relative_a: if vect_member … eq_a l relative then ∀x. P (RELATIVE x) else True. |
---|
976 | ∀addr11_a: if vect_member … eq_a l addr11 then ∀x. P (ADDR11 x) else True. |
---|
977 | ∀addr_16_a: if vect_member … eq_a l addr16 then ∀x. P (ADDR16 x) else True. |
---|
978 | ∀x:l. P x |
---|
979 | ] with |
---|
980 | [ VEmpty ⇒ λAbsurd. ⊥ |
---|
981 | | VCons len hd tl ⇒ λProof. ? |
---|
982 | ] (refl ? (S n)). cases daemon. qed. (* |
---|
983 | [ destruct(Absurd) |
---|
984 | | # A1 # A2 # A3 # A4 # A5 # A6 # A7 |
---|
985 | # A8 # A9 # A10 # A11 # A12 # A13 # A14 |
---|
986 | # A15 # A16 # A17 # A18 # A19 # X |
---|
987 | cases X |
---|
988 | # SUB cases daemon ] qed. |
---|
989 | cases SUB |
---|
990 | [ # BYTE |
---|
991 | normalize |
---|
992 | ]. |
---|
993 | |
---|
994 | |
---|
995 | (* let prepare_hd ≝ |
---|
996 | match hd with |
---|
997 | [ direct ⇒ λdirect_prf. ? |
---|
998 | | indirect ⇒ λindirect_prf. ? |
---|
999 | | ext_indirect ⇒ λext_indirect_prf. ? |
---|
1000 | | registr ⇒ λregistr_prf. ? |
---|
1001 | | acc_a ⇒ λacc_a_prf. ? |
---|
1002 | | acc_b ⇒ λacc_b_prf. ? |
---|
1003 | | dptr ⇒ λdptr_prf. ? |
---|
1004 | | data ⇒ λdata_prf. ? |
---|
1005 | | data16 ⇒ λdata16_prf. ? |
---|
1006 | | acc_dptr ⇒ λacc_dptr_prf. ? |
---|
1007 | | acc_pc ⇒ λacc_pc_prf. ? |
---|
1008 | | ext_indirect_dptr ⇒ λext_indirect_prf. ? |
---|
1009 | | indirect_dptr ⇒ λindirect_prf. ? |
---|
1010 | | carry ⇒ λcarry_prf. ? |
---|
1011 | | bit_addr ⇒ λbit_addr_prf. ? |
---|
1012 | | n_bit_addr ⇒ λn_bit_addr_prf. ? |
---|
1013 | | relative ⇒ λrelative_prf. ? |
---|
1014 | | addr11 ⇒ λaddr11_prf. ? |
---|
1015 | | addr16 ⇒ λaddr16_prf. ? |
---|
1016 | ] |
---|
1017 | in ? *) |
---|
1018 | ]. |
---|
1019 | [ 1: destruct(absd) |
---|
1020 | | 2: # A1 # A2 # A3 # A4 # A5 # A6 |
---|
1021 | # A7 # A8 # A9 # A10 # A11 # A12 |
---|
1022 | # A13 # A14 # A15 # A16 # A17 # A18 |
---|
1023 | # A19 * |
---|
1024 | ]. |
---|
1025 | |
---|
1026 | |
---|
1027 | match l return λx.match x with [O ⇒ λl: Vector … O. bool | S x' ⇒ λl: Vector addressing_mode_tag (S x'). |
---|
1028 | (l → bool) → bool ] with |
---|
1029 | [ VEmpty ⇒ true |
---|
1030 | | VCons len hd tl ⇒ λP. |
---|
1031 | let process_hd ≝ |
---|
1032 | match hd return λhd. ∀P: hd:::tl → bool. bool with |
---|
1033 | [ direct ⇒ λP.bitvector_elim 8 (λx. P (DIRECT x)) |
---|
1034 | | indirect ⇒ λP.bit_elim (λx. P (INDIRECT x)) |
---|
1035 | | ext_indirect ⇒ λP.bit_elim (λx. P (EXT_INDIRECT x)) |
---|
1036 | | registr ⇒ λP.bitvector_elim 3 (λx. P (REGISTER x)) |
---|
1037 | | acc_a ⇒ λP.P ACC_A |
---|
1038 | | acc_b ⇒ λP.P ACC_B |
---|
1039 | | dptr ⇒ λP.P DPTR |
---|
1040 | | data ⇒ λP.bitvector_elim 8 (λx. P (DATA x)) |
---|
1041 | | data16 ⇒ λP.bitvector_elim 16 (λx. P (DATA16 x)) |
---|
1042 | | acc_dptr ⇒ λP.P ACC_DPTR |
---|
1043 | | acc_pc ⇒ λP.P ACC_PC |
---|
1044 | | ext_indirect_dptr ⇒ λP.P EXT_INDIRECT_DPTR |
---|
1045 | | indirect_dptr ⇒ λP.P INDIRECT_DPTR |
---|
1046 | | carry ⇒ λP.P CARRY |
---|
1047 | | bit_addr ⇒ λP.bitvector_elim 8 (λx. P (BIT_ADDR x)) |
---|
1048 | | n_bit_addr ⇒ λP.bitvector_elim 8 (λx. P (N_BIT_ADDR x)) |
---|
1049 | | relative ⇒ λP.bitvector_elim 8 (λx. P (RELATIVE x)) |
---|
1050 | | addr11 ⇒ λP.bitvector_elim 11 (λx. P (ADDR11 x)) |
---|
1051 | | addr16 ⇒ λP.bitvector_elim 16 (λx. P (ADDR16 x)) |
---|
1052 | ] |
---|
1053 | in |
---|
1054 | andb (process_hd P) |
---|
1055 | (match len return λx. x = len → bool with |
---|
1056 | [ O ⇒ λprf. true |
---|
1057 | | S y ⇒ λprf. list_addressing_mode_tags_elim y ? P ] (refl ? len)) |
---|
1058 | ]. |
---|
1059 | try % |
---|
1060 | [ 2: cases (sym_eq ??? prf); @tl |
---|
1061 | | generalize in match H; generalize in match tl; cases prf; |
---|
1062 | (* cases prf in tl H; : ??? WAS WORKING BEFORE *) |
---|
1063 | #tl |
---|
1064 | normalize in ⊢ (∀_: %. ?) |
---|
1065 | # H |
---|
1066 | whd |
---|
1067 | normalize in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?]) |
---|
1068 | cases (is_a hd (subaddressing_modeel y tl H)) whd // ] |
---|
1069 | qed. |
---|
1070 | *) |
---|
1071 | |
---|
1072 | definition load_code_memory_aux ≝ |
---|
1073 | fold_left_i_aux … ( |
---|
1074 | λi, mem, v. |
---|
1075 | insert … (bitvector_of_nat … i) v mem) (Stub Byte 16). |
---|
1076 | |
---|
1077 | lemma split_zero: |
---|
1078 | ∀A,m. |
---|
1079 | ∀v: Vector A m. |
---|
1080 | 〈[[]], v〉 = split A 0 m v. |
---|
1081 | #A #m #v |
---|
1082 | cases v try % |
---|
1083 | #n #hd #tl % |
---|
1084 | qed. |
---|
1085 | |
---|
1086 | lemma Vector_O: |
---|
1087 | ∀A: Type[0]. |
---|
1088 | ∀v: Vector A 0. |
---|
1089 | v ≃ VEmpty A. |
---|
1090 | #A #v |
---|
1091 | generalize in match (refl … 0); |
---|
1092 | cases v in ⊢ (??%? → ?%%??); // |
---|
1093 | #n #hd #tl #absurd |
---|
1094 | destruct(absurd) |
---|
1095 | qed. |
---|
1096 | |
---|
1097 | lemma Vector_Sn: |
---|
1098 | ∀A: Type[0]. |
---|
1099 | ∀n: nat. |
---|
1100 | ∀v: Vector A (S n). |
---|
1101 | ∃hd: A. ∃tl: Vector A n. |
---|
1102 | v ≃ VCons A n hd tl. |
---|
1103 | #A #n #v |
---|
1104 | generalize in match (refl … (S n)); |
---|
1105 | cases v in ⊢ (??%? → ??(λ_.??(λ_.?%%??))); |
---|
1106 | [1: |
---|
1107 | #absurd destruct(absurd) |
---|
1108 | |2: |
---|
1109 | #m #hd #tl #eq |
---|
1110 | <(injective_S … eq) |
---|
1111 | %{hd} %{tl} % |
---|
1112 | ] |
---|
1113 | qed. |
---|
1114 | |
---|
1115 | lemma vector_append_zero: |
---|
1116 | ∀A,m. |
---|
1117 | ∀v: Vector A m. |
---|
1118 | ∀q: Vector A 0. |
---|
1119 | v = q@@v. |
---|
1120 | #A #m #v #q |
---|
1121 | >(Vector_O A q) % |
---|
1122 | qed. |
---|
1123 | |
---|
1124 | lemma prod_eq_left: |
---|
1125 | ∀A: Type[0]. |
---|
1126 | ∀p, q, r: A. |
---|
1127 | p = q → 〈p, r〉 = 〈q, r〉. |
---|
1128 | #A #p #q #r #hyp |
---|
1129 | destruct % |
---|
1130 | qed. |
---|
1131 | |
---|
1132 | lemma prod_eq_right: |
---|
1133 | ∀A: Type[0]. |
---|
1134 | ∀p, q, r: A. |
---|
1135 | p = q → 〈r, p〉 = 〈r, q〉. |
---|
1136 | #A #p #q #r #hyp |
---|
1137 | destruct % |
---|
1138 | qed. |
---|
1139 | |
---|
1140 | corollary prod_vector_zero_eq_left: |
---|
1141 | ∀A, n. |
---|
1142 | ∀q: Vector A O. |
---|
1143 | ∀r: Vector A n. |
---|
1144 | 〈q, r〉 = 〈[[ ]], r〉. |
---|
1145 | #A #n #q #r |
---|
1146 | generalize in match (Vector_O A q …); |
---|
1147 | #hyp destruct % |
---|
1148 | qed. |
---|
1149 | |
---|
1150 | lemma tail_head: |
---|
1151 | ∀a: Type[0]. |
---|
1152 | ∀m, n: nat. |
---|
1153 | ∀hd: a. |
---|
1154 | ∀l: Vector a m. |
---|
1155 | ∀r: Vector a n. |
---|
1156 | tail a ? (hd:::(l@@r)) = l@@r. |
---|
1157 | #a #m #n #hd #l #r |
---|
1158 | cases l try % |
---|
1159 | #m' #hd' #tl' % |
---|
1160 | qed. |
---|
1161 | |
---|
1162 | lemma head_head': |
---|
1163 | ∀a: Type[0]. |
---|
1164 | ∀m: nat. |
---|
1165 | ∀hd: a. |
---|
1166 | ∀l: Vector a m. |
---|
1167 | hd = head' … (hd:::l). |
---|
1168 | #a #m #hd #l cases l try % |
---|
1169 | #m' #hd' #tl % |
---|
1170 | qed. |
---|
1171 | |
---|
1172 | lemma split_succ: |
---|
1173 | ∀A: Type[0]. |
---|
1174 | ∀m, n: nat. |
---|
1175 | ∀l: Vector A m. |
---|
1176 | ∀r: Vector A n. |
---|
1177 | ∀v: Vector A (m + n). |
---|
1178 | ∀hd: A. |
---|
1179 | v = l@@r → (〈l, r〉 = split A m n v → 〈hd:::l, r〉 = split A (S m) n (hd:::v)). |
---|
1180 | #A #m |
---|
1181 | elim m |
---|
1182 | [1: |
---|
1183 | #n #l #r #v #hd #eq #hyp |
---|
1184 | destruct >(Vector_O … l) % |
---|
1185 | |2: |
---|
1186 | #m' #inductive_hypothesis #n #l #r #v #hd #equal #hyp |
---|
1187 | destruct |
---|
1188 | cases (Vector_Sn … l) #hd' #tl' |
---|
1189 | whd in ⊢ (???%); |
---|
1190 | >tail_head |
---|
1191 | <(? : split A (S m') n (l@@r) = split' A (S m') n (l@@r)) |
---|
1192 | try (<hyp <head_head' %) |
---|
1193 | elim l normalize // |
---|
1194 | ] |
---|
1195 | qed. |
---|
1196 | |
---|
1197 | lemma split_prod: |
---|
1198 | ∀A: Type[0]. |
---|
1199 | ∀m, n: nat. |
---|
1200 | ∀p: Vector A (m + n). |
---|
1201 | ∀v: Vector A m. |
---|
1202 | ∀q: Vector A n. |
---|
1203 | p = v@@q → 〈v, q〉 = split A m n p. |
---|
1204 | #A #m elim m |
---|
1205 | [1: |
---|
1206 | #n #p #v #q #hyp |
---|
1207 | >hyp <(vector_append_zero A n q v) |
---|
1208 | >(prod_vector_zero_eq_left A …) |
---|
1209 | @split_zero |
---|
1210 | |2: |
---|
1211 | #r #ih #n #p #v #q #hyp |
---|
1212 | >hyp |
---|
1213 | cases (Vector_Sn A r v) #hd #exists |
---|
1214 | cases exists #tl #jmeq |
---|
1215 | >jmeq @split_succ try % |
---|
1216 | @ih % |
---|
1217 | ] |
---|
1218 | qed. |
---|
1219 | |
---|
1220 | (* |
---|
1221 | lemma split_prod_exists: |
---|
1222 | ∀A, m, n. |
---|
1223 | ∀p: Vector A (m + n). |
---|
1224 | ∃v: Vector A m. |
---|
1225 | ∃q: Vector A n. |
---|
1226 | 〈v, q〉 = split A m n p. |
---|
1227 | #A #m #n #p |
---|
1228 | elim m |
---|
1229 | @ex_intro |
---|
1230 | [1: |
---|
1231 | |2: @ex_intro |
---|
1232 | [1: |
---|
1233 | |2: |
---|
1234 | ] |
---|
1235 | ] |
---|
1236 | *) |
---|
1237 | |
---|
1238 | definition split_elim: |
---|
1239 | ∀A: Type[0]. |
---|
1240 | ∀l, m: nat. |
---|
1241 | ∀v: Vector A (l + m). |
---|
1242 | ∀P: (Vector A l) × (Vector A m) → Prop. |
---|
1243 | (∀vl: Vector A l. |
---|
1244 | ∀vm: Vector A m. |
---|
1245 | v = vl@@vm → P 〈vl,vm〉) → P (split A l m v) ≝ |
---|
1246 | λa: Type[0]. |
---|
1247 | λl, m: nat. |
---|
1248 | λv: Vector a (l + m). |
---|
1249 | λP. ?. |
---|
1250 | cases daemon |
---|
1251 | qed. |
---|
1252 | |
---|
1253 | (* XXX: this looks almost certainly wrong *) |
---|
1254 | example half_add_SO: |
---|
1255 | ∀pc. |
---|
1256 | \snd (half_add 16 (bitvector_of_nat … pc) (bitvector_of_nat … 1)) = bitvector_of_nat … (S pc). |
---|
1257 | cases daemon. |
---|
1258 | qed. |
---|
1259 | |
---|
1260 | (* |
---|
1261 | axiom not_eqvb_S: |
---|
1262 | ∀pc. |
---|
1263 | (¬eq_bv 16 (bitvector_of_nat 16 pc) (bitvector_of_nat 16 (S pc))). |
---|
1264 | |
---|
1265 | axiom not_eqvb_SS: |
---|
1266 | ∀pc. |
---|
1267 | (¬eq_bv 16 (bitvector_of_nat 16 pc) (bitvector_of_nat 16 (S (S pc)))). |
---|
1268 | |
---|
1269 | axiom bitvector_elim_complete: |
---|
1270 | ∀n,P. bitvector_elim n P = true → ∀bv. P bv. |
---|
1271 | |
---|
1272 | lemma bitvector_elim_complete': |
---|
1273 | ∀n,P. bitvector_elim n P = true → ∀bv. P bv = true. |
---|
1274 | #n #P #H generalize in match (bitvector_elim_complete … H) #K #bv |
---|
1275 | generalize in match (K bv) normalize cases (P bv) normalize // #abs @⊥ // |
---|
1276 | qed. |
---|
1277 | *) |
---|
1278 | |
---|
1279 | (* |
---|
1280 | lemma andb_elim': |
---|
1281 | ∀b1,b2. (b1 = true) → (b2 = true) → (b1 ∧ b2) = true. |
---|
1282 | #b1 #b2 #H1 #H2 @andb_elim cases b1 in H1; normalize // |
---|
1283 | qed. |
---|
1284 | *) |
---|
1285 | |
---|
1286 | let rec encoding_check |
---|
1287 | (code_memory: BitVectorTrie Byte 16) (pc: Word) (final_pc: Word) |
---|
1288 | (encoding: list Byte) |
---|
1289 | on encoding: Prop ≝ |
---|
1290 | match encoding with |
---|
1291 | [ nil ⇒ final_pc = pc |
---|
1292 | | cons hd tl ⇒ |
---|
1293 | let 〈new_pc, byte〉 ≝ next code_memory pc in |
---|
1294 | hd = byte ∧ encoding_check code_memory new_pc final_pc tl |
---|
1295 | ]. |
---|
1296 | |
---|
1297 | axiom add_bitvector_of_nat_Sm: |
---|
1298 | ∀n, m: nat. |
---|
1299 | add … (bitvector_of_nat … 1) (bitvector_of_nat … m) = |
---|
1300 | bitvector_of_nat n (S m). |
---|
1301 | |
---|
1302 | lemma encoding_check_append: |
---|
1303 | ∀code_memory: BitVectorTrie Byte 16. |
---|
1304 | ∀final_pc: Word. |
---|
1305 | ∀l1: list Byte. |
---|
1306 | ∀pc: Word. |
---|
1307 | ∀l2: list Byte. |
---|
1308 | encoding_check code_memory pc final_pc (l1@l2) → |
---|
1309 | let pc_plus_len ≝ add … pc (bitvector_of_nat … (length … l1)) in |
---|
1310 | encoding_check code_memory pc pc_plus_len l1 ∧ |
---|
1311 | encoding_check code_memory pc_plus_len final_pc l2. |
---|
1312 | #code_memory #final_pc #l1 elim l1 |
---|
1313 | [1: |
---|
1314 | #pc #l2 |
---|
1315 | whd in ⊢ (????% → ?); #H |
---|
1316 | <add_zero |
---|
1317 | whd whd in ⊢ (?%?); /2/ |
---|
1318 | |2: |
---|
1319 | #hd #tl #IH #pc #l2 * #H1 #H2 |
---|
1320 | (* >half_add_SO in H2; #H2 *) |
---|
1321 | cases (IH … H2) #E1 #E2 % |
---|
1322 | [1: |
---|
1323 | % try @H1 |
---|
1324 | <(add_bitvector_of_nat_Sm 16 (|tl|)) in E1; |
---|
1325 | <add_associative #assm assumption |
---|
1326 | |2: |
---|
1327 | <add_associative in E2; |
---|
1328 | <(add_bitvector_of_nat_Sm 16 (|tl|)) #assm |
---|
1329 | assumption |
---|
1330 | ] |
---|
1331 | ] |
---|
1332 | qed. |
---|
1333 | |
---|
1334 | axiom bitvector_3_cases: |
---|
1335 | ∀b: BitVector 3. |
---|
1336 | ∃l, c, r: bool. |
---|
1337 | b = [[l; c; r]]. |
---|
1338 | |
---|
1339 | lemma bitvector_3_elim_prop: |
---|
1340 | ∀P: BitVector 3 → Prop. |
---|
1341 | P [[false;false;false]] → P [[false;false;true]] → P [[false;true;false]] → |
---|
1342 | P [[false;true;true]] → P [[true;false;false]] → P [[true;false;true]] → |
---|
1343 | P [[true;true;false]] → P [[true;true;true]] → ∀v. P v. |
---|
1344 | #P #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9 |
---|
1345 | cases (bitvector_3_cases … H9) #l #assm cases assm |
---|
1346 | -assm #c #assm cases assm |
---|
1347 | -assm #r #assm cases assm destruct |
---|
1348 | cases l cases c cases r // |
---|
1349 | qed. |
---|
1350 | |
---|
1351 | definition ticks_of_instruction ≝ |
---|
1352 | λi. |
---|
1353 | let trivial_code_memory ≝ assembly1 i in |
---|
1354 | let trivial_status ≝ load_code_memory trivial_code_memory in |
---|
1355 | \snd (fetch trivial_status (zero ?)). |
---|
1356 | |
---|
1357 | lemma fetch_assembly: |
---|
1358 | ∀pc: Word. |
---|
1359 | ∀i: instruction. |
---|
1360 | ∀code_memory: BitVectorTrie Byte 16. |
---|
1361 | ∀assembled: list Byte. |
---|
1362 | assembled = assembly1 i → |
---|
1363 | let len ≝ length … assembled in |
---|
1364 | let pc_plus_len ≝ add … pc (bitvector_of_nat … len) in |
---|
1365 | encoding_check code_memory pc pc_plus_len assembled → |
---|
1366 | let 〈instr, pc', ticks〉 ≝ fetch code_memory pc in |
---|
1367 | (eq_instruction instr i ∧ eqb ticks (ticks_of_instruction instr) ∧ eq_bv … pc' pc_plus_len) = true. |
---|
1368 | #pc #i #code_memory #assembled cases i [8: *] |
---|
1369 | [16,20,29: * * |18,19: * * [1,2,4,5: *] |28: * * [1,2: * [1,2: * [1,2: * [1,2: *]]]]] |
---|
1370 | [47,48,49: |
---|
1371 | |*: #arg @(list_addressing_mode_tags_elim_prop … arg) whd try % -arg |
---|
1372 | [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, |
---|
1373 | 59,60,63,64,65,66,67: #ARG]] |
---|
1374 | [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, |
---|
1375 | 56,57,69,70,72,73,75: #arg2 @(list_addressing_mode_tags_elim_prop … arg2) whd try % -arg2 |
---|
1376 | [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, |
---|
1377 | 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, |
---|
1378 | 68,69,70,71: #ARG2]] |
---|
1379 | [1,2,19,20: #arg3 @(list_addressing_mode_tags_elim_prop … arg3) whd try % -arg3 #ARG3] |
---|
1380 | normalize in ⊢ (???% → ?); |
---|
1381 | [92,94,42,93,95: @split_elim #vl #vm #E >E -E; [2,4: @(bitvector_3_elim_prop … vl)] |
---|
1382 | normalize in ⊢ (???% → ?);] |
---|
1383 | #H >H * #H1 try (whd in ⊢ (% → ?); * #H2) |
---|
1384 | try (whd in ⊢ (% → ?); * #H3) whd in ⊢ (% → ?); #H4 |
---|
1385 | [ whd in match fetch; normalize nodelta <H1 ] cases daemon |
---|
1386 | (* |
---|
1387 | whd in ⊢ (let ? ≝ ??% in ?); <H1 whd in ⊢ (let fetched ≝ % in ?) |
---|
1388 | [17,18,19,20,21,22,23,24,25,26,31,34,35,36,37,38: <H3] |
---|
1389 | [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, |
---|
1390 | 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, |
---|
1391 | 69,70,73,74,78,80,81,84,85,95,98,101,102,103,104,105,106,107,108,109,110: <H2] |
---|
1392 | whd >eq_instruction_refl >H4 @eq_bv_refl |
---|
1393 | *) (* XXX: not working! *) |
---|
1394 | qed. |
---|
1395 | |
---|
1396 | let rec fetch_many |
---|
1397 | (code_memory: BitVectorTrie Byte 16) (final_pc: Word) (pc: Word) |
---|
1398 | (expected: list instruction) |
---|
1399 | on expected: Prop ≝ |
---|
1400 | match expected with |
---|
1401 | [ nil ⇒ eq_bv … pc final_pc = true |
---|
1402 | | cons i tl ⇒ |
---|
1403 | let fetched ≝ fetch code_memory pc in |
---|
1404 | let 〈instr_pc, ticks〉 ≝ fetched in |
---|
1405 | let 〈instr,pc'〉 ≝ instr_pc in |
---|
1406 | eq_instruction instr i = true ∧ |
---|
1407 | ticks = (ticks_of_instruction i) ∧ |
---|
1408 | fetch_many code_memory final_pc pc' tl |
---|
1409 | ]. |
---|
1410 | |
---|
1411 | lemma option_destruct_Some: |
---|
1412 | ∀A: Type[0]. |
---|
1413 | ∀a, b: A. |
---|
1414 | Some A a = Some A b → a = b. |
---|
1415 | #A #a #b #EQ |
---|
1416 | destruct % |
---|
1417 | qed. |
---|
1418 | |
---|
1419 | axiom eq_instruction_to_eq: |
---|
1420 | ∀i1, i2: instruction. |
---|
1421 | eq_instruction i1 i2 = true → i1 ≃ i2. |
---|
1422 | |
---|
1423 | lemma fetch_assembly_pseudo': |
---|
1424 | ∀lookup_labels. |
---|
1425 | ∀sigma: Word → Word × bool. |
---|
1426 | ∀ppc. |
---|
1427 | ∀lookup_datalabels. |
---|
1428 | ∀pi. |
---|
1429 | ∀code_memory. |
---|
1430 | ∀len. |
---|
1431 | ∀assembled. |
---|
1432 | ∀instructions. |
---|
1433 | let 〈pc, force_long_jump〉 ≝ sigma ppc in |
---|
1434 | instructions = expand_pseudo_instruction lookup_labels sigma ppc lookup_datalabels pi → |
---|
1435 | 〈len,assembled〉 = assembly_1_pseudoinstruction lookup_labels sigma ppc lookup_datalabels pi → |
---|
1436 | let pc_plus_len ≝ add … pc (bitvector_of_nat … len) in |
---|
1437 | encoding_check code_memory pc pc_plus_len assembled → |
---|
1438 | fetch_many code_memory pc_plus_len pc instructions. |
---|
1439 | #lookup_labels #sigma #ppc #lookup_datalabels #pi #code_memory #len #assembled #instructions |
---|
1440 | generalize in match (refl … (sigma ppc)); |
---|
1441 | cases (sigma ?) in ⊢ (??%? → %); #pc #force_long_jump #sigma_refl normalize nodelta |
---|
1442 | #instructions_refl whd in ⊢ (???% → ?); <instructions_refl whd in ⊢ (???% → ?); #assembled_refl |
---|
1443 | cases (pair_destruct ?????? assembled_refl) -assembled_refl #len_refl #assembled_refl |
---|
1444 | >len_refl >assembled_refl -len_refl |
---|
1445 | generalize in match (add 16 pc |
---|
1446 | (bitvector_of_nat 16 |
---|
1447 | (|flatten (Vector bool 8) |
---|
1448 | (map instruction (list (Vector bool 8)) assembly1 instructions)|))); |
---|
1449 | #final_pc |
---|
1450 | generalize in match pc; elim instructions |
---|
1451 | [1: |
---|
1452 | #pc whd in ⊢ (% → %); #H >H @eq_bv_refl |
---|
1453 | |2: |
---|
1454 | #i #tl #IH #pc #H whd |
---|
1455 | cases (encoding_check_append ????? H) -H #H1 #H2 |
---|
1456 | @pair_elim #instr_pc #ticks #fetch_refl normalize nodelta |
---|
1457 | @pair_elim #instr #pc' #instr_pc_refl normalize nodelta |
---|
1458 | lapply (fetch_assembly pc i code_memory (assembly1 i) (refl …)) whd in ⊢ (% → ?); |
---|
1459 | #H3 lapply (H3 H1) -H3 >fetch_refl >instr_pc_refl normalize nodelta |
---|
1460 | #H3 lapply (conjunction_true ?? H3) * #H4 #H5 % |
---|
1461 | [1: |
---|
1462 | lapply (conjunction_true … H4) * #B1 #B2 |
---|
1463 | % try assumption @eqb_true_to_eq |
---|
1464 | <(eq_instruction_to_eq … B1) assumption |
---|
1465 | |2: |
---|
1466 | >(eq_bv_eq … H5) @IH @H2 |
---|
1467 | ] |
---|
1468 | ] |
---|
1469 | qed. |
---|
1470 | |
---|
1471 | lemma fetch_assembly_pseudo: |
---|
1472 | ∀program: pseudo_assembly_program. |
---|
1473 | ∀sigma: Word → Word × bool. |
---|
1474 | let lookup_labels ≝ λx:Identifier.\fst (sigma (address_of_word_labels_code_mem (\snd program) x)) in |
---|
1475 | ∀ppc.∀code_memory. |
---|
1476 | let lookup_datalabels ≝ λx:Identifier.lookup_def … (construct_datalabels (\fst program)) x (zero 16) in |
---|
1477 | let pi ≝ \fst (fetch_pseudo_instruction (\snd program) ppc) in |
---|
1478 | let pc ≝ \fst (sigma ppc) in |
---|
1479 | let instructions ≝ expand_pseudo_instruction lookup_labels sigma ppc lookup_datalabels pi in |
---|
1480 | let 〈len,assembled〉 ≝ assembly_1_pseudoinstruction lookup_labels sigma ppc lookup_datalabels pi in |
---|
1481 | let pc_plus_len ≝ add … pc (bitvector_of_nat … len) in |
---|
1482 | encoding_check code_memory pc pc_plus_len assembled → |
---|
1483 | fetch_many code_memory pc_plus_len pc instructions. |
---|
1484 | #program #sigma letin lookup_labels ≝ (λx.?) #ppc #code_memory |
---|
1485 | letin lookup_datalabels ≝ (λx.?) |
---|
1486 | letin pi ≝ (fst ???) |
---|
1487 | letin pc ≝ (fst ???) |
---|
1488 | letin instructions ≝ (expand_pseudo_instruction ?????) |
---|
1489 | @pair_elim #len #assembled #assembled_refl normalize nodelta |
---|
1490 | #H |
---|
1491 | generalize in match |
---|
1492 | (fetch_assembly_pseudo' lookup_labels sigma ppc lookup_datalabels pi code_memory len assembled instructions) in ⊢ ?; |
---|
1493 | #X destruct |
---|
1494 | cases (sigma ppc) in H X; #pc #force_long_jump normalize nodelta |
---|
1495 | #H #X @X try % <assembled_refl try % assumption |
---|
1496 | qed. |
---|
1497 | |
---|
1498 | (* This is a trivial consequence of fetch_assembly_pseudo + the proof that the |
---|
1499 | function that load the code in memory is correct. The latter is based |
---|
1500 | on missing properties from the standard library on the BitVectorTrie |
---|
1501 | data structrure. |
---|
1502 | |
---|
1503 | Wrong at the moment, requires Jaap's precondition to ensure that the program |
---|
1504 | does not overflow when put into code memory (i.e. shorter than 2^16 bytes). |
---|
1505 | *) |
---|
1506 | axiom assembly_ok: |
---|
1507 | ∀program,sigma,assembled,costs'. |
---|
1508 | let 〈labels, costs〉 ≝ create_label_cost_map (\snd program) in |
---|
1509 | 〈assembled,costs'〉 = assembly program sigma → |
---|
1510 | costs = costs' ∧ |
---|
1511 | let code_memory ≝ load_code_memory assembled in |
---|
1512 | let datalabels ≝ construct_datalabels (\fst program) in |
---|
1513 | let lookup_labels ≝ λx.\fst (sigma (address_of_word_labels_code_mem (\snd program) x)) in |
---|
1514 | let lookup_datalabels ≝ λx. lookup_def ?? datalabels x (zero ?) in |
---|
1515 | ∀ppc. |
---|
1516 | let 〈pi, newppc〉 ≝ fetch_pseudo_instruction (\snd program) ppc in |
---|
1517 | let 〈len,assembled〉 ≝ assembly_1_pseudoinstruction lookup_labels sigma ppc lookup_datalabels pi in |
---|
1518 | let pc ≝ \fst (sigma ppc) in |
---|
1519 | let pc_plus_len ≝ add … pc (bitvector_of_nat … len) in |
---|
1520 | encoding_check code_memory pc pc_plus_len assembled ∧ |
---|
1521 | \fst (sigma newppc) = add … pc (bitvector_of_nat … len). |
---|
1522 | |
---|
1523 | (* XXX: should we add that costs = costs'? *) |
---|
1524 | lemma fetch_assembly_pseudo2: |
---|
1525 | ∀program,sigma,ppc. |
---|
1526 | let 〈labels, costs〉 ≝ create_label_cost_map (\snd program) in |
---|
1527 | let 〈assembled, costs'〉 ≝ assembly program sigma in |
---|
1528 | let code_memory ≝ load_code_memory assembled in |
---|
1529 | let data_labels ≝ construct_datalabels (\fst program) in |
---|
1530 | let lookup_labels ≝ λx.\fst (sigma (address_of_word_labels_code_mem (\snd program) x)) in |
---|
1531 | let lookup_datalabels ≝ λx. lookup_def ? ? data_labels x (zero ?) in |
---|
1532 | let 〈pi,newppc〉 ≝ fetch_pseudo_instruction (\snd program) ppc in |
---|
1533 | let instructions ≝ expand_pseudo_instruction lookup_labels sigma ppc lookup_datalabels pi in |
---|
1534 | fetch_many code_memory (\fst (sigma newppc)) (\fst (sigma ppc)) instructions. |
---|
1535 | * #preamble #instr_list #sigma #ppc |
---|
1536 | @pair_elim #labels #costs #create_label_map_refl |
---|
1537 | @pair_elim #assembled #costs' #assembled_refl |
---|
1538 | letin code_memory ≝ (load_code_memory ?) |
---|
1539 | letin data_labels ≝ (construct_datalabels ?) |
---|
1540 | letin lookup_labels ≝ (λx. ?) |
---|
1541 | letin lookup_datalabels ≝ (λx. ?) |
---|
1542 | @pair_elim #pi #newppc #fetch_pseudo_refl |
---|
1543 | lapply (assembly_ok 〈preamble, instr_list〉 sigma assembled costs') |
---|
1544 | @pair_elim #labels' #costs' #create_label_map_refl' #H |
---|
1545 | cases (H (sym_eq … assembled_refl)) |
---|
1546 | #_ |
---|
1547 | lapply (refl … (assembly_1_pseudoinstruction lookup_labels sigma ppc lookup_datalabels pi)) |
---|
1548 | cases (assembly_1_pseudoinstruction ?????) in ⊢ (???% → ?); |
---|
1549 | #len #assembledi #EQ4 #H |
---|
1550 | lapply (H ppc) >fetch_pseudo_refl #H |
---|
1551 | lapply (fetch_assembly_pseudo 〈preamble,instr_list〉 sigma ppc (load_code_memory assembled)) |
---|
1552 | >EQ4 #H1 cases H #H2 #H3 >H3 normalize nodelta in H1; normalize nodelta |
---|
1553 | >fetch_pseudo_refl in H1; #assm @assm assumption |
---|
1554 | qed. |
---|
1555 | |
---|
1556 | (* OLD? |
---|
1557 | definition assembly_specification: |
---|
1558 | ∀assembly_program: pseudo_assembly_program. |
---|
1559 | ∀code_mem: BitVectorTrie Byte 16. Prop ≝ |
---|
1560 | λpseudo_assembly_program. |
---|
1561 | λcode_mem. |
---|
1562 | ∀pc: Word. |
---|
1563 | let 〈preamble, instr_list〉 ≝ pseudo_assembly_program in |
---|
1564 | let 〈pre_instr, pre_new_pc〉 ≝ fetch_pseudo_instruction instr_list pc in |
---|
1565 | let labels ≝ λx. sigma' pseudo_assembly_program (address_of_word_labels_code_mem instr_list x) in |
---|
1566 | let datalabels ≝ λx. sigma' pseudo_assembly_program (lookup ? ? x (construct_datalabels preamble) (zero ?)) in |
---|
1567 | let pre_assembled ≝ assembly_1_pseudoinstruction pseudo_assembly_program |
---|
1568 | (sigma' pseudo_assembly_program pc) labels datalabels pre_instr in |
---|
1569 | match pre_assembled with |
---|
1570 | [ None ⇒ True |
---|
1571 | | Some pc_code ⇒ |
---|
1572 | let 〈new_pc,code〉 ≝ pc_code in |
---|
1573 | encoding_check code_mem pc (sigma' pseudo_assembly_program pre_new_pc) code ]. |
---|
1574 | |
---|
1575 | axiom assembly_meets_specification: |
---|
1576 | ∀pseudo_assembly_program. |
---|
1577 | match assembly pseudo_assembly_program with |
---|
1578 | [ None ⇒ True |
---|
1579 | | Some code_mem_cost ⇒ |
---|
1580 | let 〈code_mem, cost〉 ≝ code_mem_cost in |
---|
1581 | assembly_specification pseudo_assembly_program (load_code_memory code_mem) |
---|
1582 | ]. |
---|
1583 | (* |
---|
1584 | # PROGRAM |
---|
1585 | [ cases PROGRAM |
---|
1586 | # PREAMBLE |
---|
1587 | # INSTR_LIST |
---|
1588 | elim INSTR_LIST |
---|
1589 | [ whd |
---|
1590 | whd in ⊢ (∀_. %) |
---|
1591 | # PC |
---|
1592 | whd |
---|
1593 | | # INSTR |
---|
1594 | # INSTR_LIST_TL |
---|
1595 | # H |
---|
1596 | whd |
---|
1597 | whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?]) |
---|
1598 | ] |
---|
1599 | | cases not_implemented |
---|
1600 | ] *) |
---|
1601 | *) |
---|
1602 | |
---|
1603 | definition internal_pseudo_address_map ≝ list (BitVector 8). |
---|
1604 | |
---|
1605 | axiom low_internal_ram_of_pseudo_low_internal_ram: |
---|
1606 | ∀M:internal_pseudo_address_map.∀ram:BitVectorTrie Byte 7.BitVectorTrie Byte 7. |
---|
1607 | |
---|
1608 | axiom high_internal_ram_of_pseudo_high_internal_ram: |
---|
1609 | ∀M:internal_pseudo_address_map.∀ram:BitVectorTrie Byte 7.BitVectorTrie Byte 7. |
---|
1610 | |
---|
1611 | axiom low_internal_ram_of_pseudo_internal_ram_hit: |
---|
1612 | ∀M:internal_pseudo_address_map.∀cm.∀s:PseudoStatus cm.∀sigma:Word → Word × bool.∀addr:BitVector 7. |
---|
1613 | member ? (eq_bv 8) (false:::addr) M = true → |
---|
1614 | let ram ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … s) in |
---|
1615 | let pbl ≝ lookup ? 7 addr (low_internal_ram … s) (zero 8) in |
---|
1616 | let pbu ≝ lookup ? 7 (\snd (half_add ? addr (bitvector_of_nat 7 1))) (low_internal_ram … s) (zero 8) in |
---|
1617 | let bl ≝ lookup ? 7 addr ram (zero 8) in |
---|
1618 | let bu ≝ lookup ? 7 (\snd (half_add ? addr (bitvector_of_nat 7 1))) ram (zero 8) in |
---|
1619 | bu@@bl = \fst (sigma (pbu@@pbl)). |
---|
1620 | |
---|
1621 | (* changed from add to sub *) |
---|
1622 | axiom low_internal_ram_of_pseudo_internal_ram_miss: |
---|
1623 | ∀T.∀M:internal_pseudo_address_map.∀cm.∀s:PreStatus T cm.∀addr:BitVector 7. |
---|
1624 | let ram ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … s) in |
---|
1625 | let 〈Saddr,flags〉 ≝ sub_7_with_carry addr (bitvector_of_nat 7 1) false in |
---|
1626 | let carr ≝ get_index_v ? ? flags 1 ? in |
---|
1627 | carr = false → |
---|
1628 | member ? (eq_bv 8) (false:::Saddr) M = false → |
---|
1629 | member ? (eq_bv 8) (false:::addr) M = false → |
---|
1630 | lookup ? 7 addr ram (zero ?) = lookup ? 7 addr (low_internal_ram … s) (zero ?). |
---|
1631 | // |
---|
1632 | qed. |
---|
1633 | |
---|
1634 | definition addressing_mode_ok ≝ |
---|
1635 | λT.λM:internal_pseudo_address_map.λcm.λs:PreStatus T cm. |
---|
1636 | λaddr:addressing_mode. |
---|
1637 | match addr with |
---|
1638 | [ DIRECT d ⇒ |
---|
1639 | ¬(member ? (eq_bv 8) d M) ∧ |
---|
1640 | ¬(member ? (eq_bv 8) (\fst (sub_8_with_carry d (bitvector_of_nat 8 1) false)) M) |
---|
1641 | | INDIRECT i ⇒ |
---|
1642 | let d ≝ get_register … s [[false;false;i]] in |
---|
1643 | ¬(member ? (eq_bv 8) d M) ∧ |
---|
1644 | ¬(member ? (eq_bv 8) (\fst (sub_8_with_carry d (bitvector_of_nat 8 1) false)) M) |
---|
1645 | | EXT_INDIRECT _ ⇒ true |
---|
1646 | | REGISTER _ ⇒ true |
---|
1647 | | ACC_A ⇒ true |
---|
1648 | | ACC_B ⇒ true |
---|
1649 | | DPTR ⇒ true |
---|
1650 | | DATA _ ⇒ true |
---|
1651 | | DATA16 _ ⇒ true |
---|
1652 | | ACC_DPTR ⇒ true |
---|
1653 | | ACC_PC ⇒ true |
---|
1654 | | EXT_INDIRECT_DPTR ⇒ true |
---|
1655 | | INDIRECT_DPTR ⇒ true |
---|
1656 | | CARRY ⇒ true |
---|
1657 | | BIT_ADDR _ ⇒ ¬true (* TO BE COMPLETED *) |
---|
1658 | | N_BIT_ADDR _ ⇒ ¬true (* TO BE COMPLETED *) |
---|
1659 | | RELATIVE _ ⇒ true |
---|
1660 | | ADDR11 _ ⇒ true |
---|
1661 | | ADDR16 _ ⇒ true ]. |
---|
1662 | |
---|
1663 | definition next_internal_pseudo_address_map0 ≝ |
---|
1664 | λT. |
---|
1665 | λfetched. |
---|
1666 | λM: internal_pseudo_address_map. |
---|
1667 | λcm:T. |
---|
1668 | λs: PreStatus T cm. |
---|
1669 | match fetched with |
---|
1670 | [ Comment _ ⇒ Some ? M |
---|
1671 | | Cost _ ⇒ Some … M |
---|
1672 | | Jmp _ ⇒ Some … M |
---|
1673 | | Call _ ⇒ |
---|
1674 | Some … (\snd (half_add ? (get_8051_sfr … s SFR_SP) (bitvector_of_nat 8 1))::M) |
---|
1675 | | Mov _ _ ⇒ Some … M |
---|
1676 | | Instruction instr ⇒ |
---|
1677 | match instr with |
---|
1678 | [ ADD addr1 addr2 ⇒ |
---|
1679 | if addressing_mode_ok T M … s addr1 ∧ addressing_mode_ok T M … s addr2 then |
---|
1680 | Some ? M |
---|
1681 | else |
---|
1682 | None ? |
---|
1683 | | ADDC addr1 addr2 ⇒ |
---|
1684 | if addressing_mode_ok T M … s addr1 ∧ addressing_mode_ok T M … s addr2 then |
---|
1685 | Some ? M |
---|
1686 | else |
---|
1687 | None ? |
---|
1688 | | SUBB addr1 addr2 ⇒ |
---|
1689 | if addressing_mode_ok T M … s addr1 ∧ addressing_mode_ok T M … s addr2 then |
---|
1690 | Some ? M |
---|
1691 | else |
---|
1692 | None ? |
---|
1693 | | _ ⇒ (* TO BE COMPLETED *) Some ? M ]]. |
---|
1694 | |
---|
1695 | |
---|
1696 | definition next_internal_pseudo_address_map ≝ |
---|
1697 | λM:internal_pseudo_address_map. |
---|
1698 | λcm. |
---|
1699 | λs:PseudoStatus cm. |
---|
1700 | next_internal_pseudo_address_map0 ? |
---|
1701 | (\fst (fetch_pseudo_instruction (\snd cm) (program_counter … s))) M cm s. |
---|
1702 | |
---|
1703 | definition code_memory_of_pseudo_assembly_program: |
---|
1704 | ∀pap:pseudo_assembly_program. (Word → Word × bool) → BitVectorTrie Byte 16 |
---|
1705 | ≝ λpap,sigma. |
---|
1706 | let p ≝ assembly pap sigma in |
---|
1707 | load_code_memory (\fst p). |
---|
1708 | |
---|
1709 | definition status_of_pseudo_status: |
---|
1710 | internal_pseudo_address_map → ∀pap.∀ps:PseudoStatus pap. ∀sigma: Word → Word × bool. |
---|
1711 | Status (code_memory_of_pseudo_assembly_program pap sigma) ≝ |
---|
1712 | λM,pap,ps,sigma. |
---|
1713 | let cm ≝ code_memory_of_pseudo_assembly_program … sigma in |
---|
1714 | let pc ≝ \fst (sigma (program_counter … ps)) in |
---|
1715 | let lir ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … ps) in |
---|
1716 | let hir ≝ high_internal_ram_of_pseudo_high_internal_ram M (high_internal_ram … ps) in |
---|
1717 | mk_PreStatus (BitVectorTrie Byte 16) |
---|
1718 | cm |
---|
1719 | lir |
---|
1720 | hir |
---|
1721 | (external_ram … ps) |
---|
1722 | pc |
---|
1723 | (special_function_registers_8051 … ps) |
---|
1724 | (special_function_registers_8052 … ps) |
---|
1725 | (p1_latch … ps) |
---|
1726 | (p3_latch … ps) |
---|
1727 | (clock … ps). |
---|
1728 | |
---|
1729 | (* |
---|
1730 | definition write_at_stack_pointer': |
---|
1731 | ∀M. ∀ps: PreStatus M. Byte → Σps':PreStatus M.(code_memory … ps = code_memory … ps') ≝ |
---|
1732 | λM: Type[0]. |
---|
1733 | λs: PreStatus M. |
---|
1734 | λv: Byte. |
---|
1735 | let 〈 nu, nl 〉 ≝ split … 4 4 (get_8051_sfr ? s SFR_SP) in |
---|
1736 | let bit_zero ≝ get_index_v… nu O ? in |
---|
1737 | let bit_1 ≝ get_index_v… nu 1 ? in |
---|
1738 | let bit_2 ≝ get_index_v… nu 2 ? in |
---|
1739 | let bit_3 ≝ get_index_v… nu 3 ? in |
---|
1740 | if bit_zero then |
---|
1741 | let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl) |
---|
1742 | v (low_internal_ram ? s) in |
---|
1743 | set_low_internal_ram ? s memory |
---|
1744 | else |
---|
1745 | let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl) |
---|
1746 | v (high_internal_ram ? s) in |
---|
1747 | set_high_internal_ram ? s memory. |
---|
1748 | [ cases l0 % |
---|
1749 | |2,3,4,5: normalize repeat (@ le_S_S) @ le_O_n ] |
---|
1750 | qed. |
---|
1751 | |
---|
1752 | definition execute_1_pseudo_instruction': (Word → nat) → ∀ps:PseudoStatus. |
---|
1753 | Σps':PseudoStatus.(code_memory … ps = code_memory … ps') |
---|
1754 | ≝ |
---|
1755 | λticks_of. |
---|
1756 | λs. |
---|
1757 | let 〈instr, pc〉 ≝ fetch_pseudo_instruction (\snd (code_memory ? s)) (program_counter ? s) in |
---|
1758 | let ticks ≝ ticks_of (program_counter ? s) in |
---|
1759 | let s ≝ set_clock ? s (clock ? s + ticks) in |
---|
1760 | let s ≝ set_program_counter ? s pc in |
---|
1761 | match instr with |
---|
1762 | [ Instruction instr ⇒ |
---|
1763 | execute_1_preinstruction … (λx, y. address_of_word_labels y x) instr s |
---|
1764 | | Comment cmt ⇒ s |
---|
1765 | | Cost cst ⇒ s |
---|
1766 | | Jmp jmp ⇒ set_program_counter ? s (address_of_word_labels s jmp) |
---|
1767 | | Call call ⇒ |
---|
1768 | let a ≝ address_of_word_labels s call in |
---|
1769 | let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in |
---|
1770 | let s ≝ set_8051_sfr ? s SFR_SP new_sp in |
---|
1771 | let 〈pc_bu, pc_bl〉 ≝ split ? 8 8 (program_counter ? s) in |
---|
1772 | let s ≝ write_at_stack_pointer' ? s pc_bl in |
---|
1773 | let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in |
---|
1774 | let s ≝ set_8051_sfr ? s SFR_SP new_sp in |
---|
1775 | let s ≝ write_at_stack_pointer' ? s pc_bu in |
---|
1776 | set_program_counter ? s a |
---|
1777 | | Mov dptr ident ⇒ |
---|
1778 | set_arg_16 ? s (get_arg_16 ? s (DATA16 (address_of_word_labels s ident))) dptr |
---|
1779 | ]. |
---|
1780 | [ |
---|
1781 | |2,3,4: % |
---|
1782 | | <(sig2 … l7) whd in ⊢ (??? (??%)) <(sig2 … l5) % |
---|
1783 | | |
---|
1784 | | % |
---|
1785 | ] |
---|
1786 | cases not_implemented |
---|
1787 | qed. |
---|
1788 | *) |
---|
1789 | |
---|
1790 | (* |
---|
1791 | lemma execute_code_memory_unchanged: |
---|
1792 | ∀ticks_of,ps. code_memory ? ps = code_memory ? (execute_1_pseudo_instruction ticks_of ps). |
---|
1793 | #ticks #ps whd in ⊢ (??? (??%)) |
---|
1794 | cases (fetch_pseudo_instruction (\snd (code_memory pseudo_assembly_program ps)) |
---|
1795 | (program_counter pseudo_assembly_program ps)) #instr #pc |
---|
1796 | whd in ⊢ (??? (??%)) cases instr |
---|
1797 | [ #pre cases pre |
---|
1798 | [ #a1 #a2 whd in ⊢ (??? (??%)) cases (add_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%)) |
---|
1799 | cases (split ????) #z1 #z2 % |
---|
1800 | | #a1 #a2 whd in ⊢ (??? (??%)) cases (add_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%)) |
---|
1801 | cases (split ????) #z1 #z2 % |
---|
1802 | | #a1 #a2 whd in ⊢ (??? (??%)) cases (sub_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%)) |
---|
1803 | cases (split ????) #z1 #z2 % |
---|
1804 | | #a1 whd in ⊢ (??? (??%)) cases a1 #x #H whd in ⊢ (??? (??%)) cases x |
---|
1805 | [ #x1 whd in ⊢ (??? (??%)) |
---|
1806 | | *: cases not_implemented |
---|
1807 | ] |
---|
1808 | | #comment % |
---|
1809 | | #cost % |
---|
1810 | | #label % |
---|
1811 | | #label whd in ⊢ (??? (??%)) cases (half_add ???) #x1 #x2 whd in ⊢ (??? (??%)) |
---|
1812 | cases (split ????) #y1 #y2 whd in ⊢ (??? (??%)) cases (half_add ???) #z1 #z2 |
---|
1813 | whd in ⊢ (??? (??%)) whd in ⊢ (??? (??%)) cases (split ????) #w1 #w2 |
---|
1814 | whd in ⊢ (??? (??%)) cases (get_index_v bool ????) whd in ⊢ (??? (??%)) |
---|
1815 | (* CSC: ??? *) |
---|
1816 | | #dptr #label (* CSC: ??? *) |
---|
1817 | ] |
---|
1818 | cases not_implemented |
---|
1819 | qed. |
---|
1820 | *) |
---|
1821 | |
---|
1822 | (* DEAD CODE? |
---|
1823 | lemma status_of_pseudo_status_failure_depends_only_on_code_memory: |
---|
1824 | ∀M:internal_pseudo_address_map. |
---|
1825 | ∀ps,ps': PseudoStatus. |
---|
1826 | ∀pol. |
---|
1827 | ∀prf:code_memory … ps = code_memory … ps'. |
---|
1828 | let pol' ≝ ? in |
---|
1829 | match status_of_pseudo_status M ps pol with |
---|
1830 | [ None ⇒ status_of_pseudo_status M ps' pol' = None … |
---|
1831 | | Some _ ⇒ ∃w. status_of_pseudo_status M ps' pol' = Some … w |
---|
1832 | ]. |
---|
1833 | [2: <prf @pol] |
---|
1834 | #M #ps #ps' #pol #H normalize nodelta; whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ? ]) |
---|
1835 | generalize in match (refl … (assembly (code_memory … ps) pol)) |
---|
1836 | cases (assembly ??) in ⊢ (???% → %) |
---|
1837 | [ #K whd whd in ⊢ (??%?) <H >K % |
---|
1838 | | #x #K whd whd in ⊢ (?? (λ_.??%?)) <H >K % [2: % ] ] |
---|
1839 | qed. |
---|
1840 | *) |
---|
1841 | |
---|
1842 | definition ticks_of0: ∀p:pseudo_assembly_program. (Word → Word × bool) → Word → pseudo_instruction → nat × nat ≝ |
---|
1843 | λprogram: pseudo_assembly_program.λsigma. |
---|
1844 | λppc: Word. |
---|
1845 | λfetched. ?. |
---|
1846 | cases daemon(* |
---|
1847 | match fetched with |
---|
1848 | [ Instruction instr ⇒ |
---|
1849 | match instr with |
---|
1850 | [ JC lbl ⇒ |
---|
1851 | match pol lookup_labels ppc with |
---|
1852 | [ short_jump ⇒ 〈2, 2〉 |
---|
1853 | | medium_jump ⇒ ? |
---|
1854 | | long_jump ⇒ 〈4, 4〉 |
---|
1855 | ] |
---|
1856 | | JNC lbl ⇒ |
---|
1857 | match pol lookup_labels ppc with |
---|
1858 | [ short_jump ⇒ 〈2, 2〉 |
---|
1859 | | medium_jump ⇒ ? |
---|
1860 | | long_jump ⇒ 〈4, 4〉 |
---|
1861 | ] |
---|
1862 | | JB bit lbl ⇒ |
---|
1863 | match pol lookup_labels ppc with |
---|
1864 | [ short_jump ⇒ 〈2, 2〉 |
---|
1865 | | medium_jump ⇒ ? |
---|
1866 | | long_jump ⇒ 〈4, 4〉 |
---|
1867 | ] |
---|
1868 | | JNB bit lbl ⇒ |
---|
1869 | match pol lookup_labels ppc with |
---|
1870 | [ short_jump ⇒ 〈2, 2〉 |
---|
1871 | | medium_jump ⇒ ? |
---|
1872 | | long_jump ⇒ 〈4, 4〉 |
---|
1873 | ] |
---|
1874 | | JBC bit lbl ⇒ |
---|
1875 | match pol lookup_labels ppc with |
---|
1876 | [ short_jump ⇒ 〈2, 2〉 |
---|
1877 | | medium_jump ⇒ ? |
---|
1878 | | long_jump ⇒ 〈4, 4〉 |
---|
1879 | ] |
---|
1880 | | JZ lbl ⇒ |
---|
1881 | match pol lookup_labels ppc with |
---|
1882 | [ short_jump ⇒ 〈2, 2〉 |
---|
1883 | | medium_jump ⇒ ? |
---|
1884 | | long_jump ⇒ 〈4, 4〉 |
---|
1885 | ] |
---|
1886 | | JNZ lbl ⇒ |
---|
1887 | match pol lookup_labels ppc with |
---|
1888 | [ short_jump ⇒ 〈2, 2〉 |
---|
1889 | | medium_jump ⇒ ? |
---|
1890 | | long_jump ⇒ 〈4, 4〉 |
---|
1891 | ] |
---|
1892 | | CJNE arg lbl ⇒ |
---|
1893 | match pol lookup_labels ppc with |
---|
1894 | [ short_jump ⇒ 〈2, 2〉 |
---|
1895 | | medium_jump ⇒ ? |
---|
1896 | | long_jump ⇒ 〈4, 4〉 |
---|
1897 | ] |
---|
1898 | | DJNZ arg lbl ⇒ |
---|
1899 | match pol lookup_labels ppc with |
---|
1900 | [ short_jump ⇒ 〈2, 2〉 |
---|
1901 | | medium_jump ⇒ ? |
---|
1902 | | long_jump ⇒ 〈4, 4〉 |
---|
1903 | ] |
---|
1904 | | ADD arg1 arg2 ⇒ |
---|
1905 | let ticks ≝ ticks_of_instruction (ADD ? arg1 arg2) in |
---|
1906 | 〈ticks, ticks〉 |
---|
1907 | | ADDC arg1 arg2 ⇒ |
---|
1908 | let ticks ≝ ticks_of_instruction (ADDC ? arg1 arg2) in |
---|
1909 | 〈ticks, ticks〉 |
---|
1910 | | SUBB arg1 arg2 ⇒ |
---|
1911 | let ticks ≝ ticks_of_instruction (SUBB ? arg1 arg2) in |
---|
1912 | 〈ticks, ticks〉 |
---|
1913 | | INC arg ⇒ |
---|
1914 | let ticks ≝ ticks_of_instruction (INC ? arg) in |
---|
1915 | 〈ticks, ticks〉 |
---|
1916 | | DEC arg ⇒ |
---|
1917 | let ticks ≝ ticks_of_instruction (DEC ? arg) in |
---|
1918 | 〈ticks, ticks〉 |
---|
1919 | | MUL arg1 arg2 ⇒ |
---|
1920 | let ticks ≝ ticks_of_instruction (MUL ? arg1 arg2) in |
---|
1921 | 〈ticks, ticks〉 |
---|
1922 | | DIV arg1 arg2 ⇒ |
---|
1923 | let ticks ≝ ticks_of_instruction (DIV ? arg1 arg2) in |
---|
1924 | 〈ticks, ticks〉 |
---|
1925 | | DA arg ⇒ |
---|
1926 | let ticks ≝ ticks_of_instruction (DA ? arg) in |
---|
1927 | 〈ticks, ticks〉 |
---|
1928 | | ANL arg ⇒ |
---|
1929 | let ticks ≝ ticks_of_instruction (ANL ? arg) in |
---|
1930 | 〈ticks, ticks〉 |
---|
1931 | | ORL arg ⇒ |
---|
1932 | let ticks ≝ ticks_of_instruction (ORL ? arg) in |
---|
1933 | 〈ticks, ticks〉 |
---|
1934 | | XRL arg ⇒ |
---|
1935 | let ticks ≝ ticks_of_instruction (XRL ? arg) in |
---|
1936 | 〈ticks, ticks〉 |
---|
1937 | | CLR arg ⇒ |
---|
1938 | let ticks ≝ ticks_of_instruction (CLR ? arg) in |
---|
1939 | 〈ticks, ticks〉 |
---|
1940 | | CPL arg ⇒ |
---|
1941 | let ticks ≝ ticks_of_instruction (CPL ? arg) in |
---|
1942 | 〈ticks, ticks〉 |
---|
1943 | | RL arg ⇒ |
---|
1944 | let ticks ≝ ticks_of_instruction (RL ? arg) in |
---|
1945 | 〈ticks, ticks〉 |
---|
1946 | | RLC arg ⇒ |
---|
1947 | let ticks ≝ ticks_of_instruction (RLC ? arg) in |
---|
1948 | 〈ticks, ticks〉 |
---|
1949 | | RR arg ⇒ |
---|
1950 | let ticks ≝ ticks_of_instruction (RR ? arg) in |
---|
1951 | 〈ticks, ticks〉 |
---|
1952 | | RRC arg ⇒ |
---|
1953 | let ticks ≝ ticks_of_instruction (RRC ? arg) in |
---|
1954 | 〈ticks, ticks〉 |
---|
1955 | | SWAP arg ⇒ |
---|
1956 | let ticks ≝ ticks_of_instruction (SWAP ? arg) in |
---|
1957 | 〈ticks, ticks〉 |
---|
1958 | | MOV arg ⇒ |
---|
1959 | let ticks ≝ ticks_of_instruction (MOV ? arg) in |
---|
1960 | 〈ticks, ticks〉 |
---|
1961 | | MOVX arg ⇒ |
---|
1962 | let ticks ≝ ticks_of_instruction (MOVX ? arg) in |
---|
1963 | 〈ticks, ticks〉 |
---|
1964 | | SETB arg ⇒ |
---|
1965 | let ticks ≝ ticks_of_instruction (SETB ? arg) in |
---|
1966 | 〈ticks, ticks〉 |
---|
1967 | | PUSH arg ⇒ |
---|
1968 | let ticks ≝ ticks_of_instruction (PUSH ? arg) in |
---|
1969 | 〈ticks, ticks〉 |
---|
1970 | | POP arg ⇒ |
---|
1971 | let ticks ≝ ticks_of_instruction (POP ? arg) in |
---|
1972 | 〈ticks, ticks〉 |
---|
1973 | | XCH arg1 arg2 ⇒ |
---|
1974 | let ticks ≝ ticks_of_instruction (XCH ? arg1 arg2) in |
---|
1975 | 〈ticks, ticks〉 |
---|
1976 | | XCHD arg1 arg2 ⇒ |
---|
1977 | let ticks ≝ ticks_of_instruction (XCHD ? arg1 arg2) in |
---|
1978 | 〈ticks, ticks〉 |
---|
1979 | | RET ⇒ |
---|
1980 | let ticks ≝ ticks_of_instruction (RET ?) in |
---|
1981 | 〈ticks, ticks〉 |
---|
1982 | | RETI ⇒ |
---|
1983 | let ticks ≝ ticks_of_instruction (RETI ?) in |
---|
1984 | 〈ticks, ticks〉 |
---|
1985 | | NOP ⇒ |
---|
1986 | let ticks ≝ ticks_of_instruction (NOP ?) in |
---|
1987 | 〈ticks, ticks〉 |
---|
1988 | ] |
---|
1989 | | Comment comment ⇒ 〈0, 0〉 |
---|
1990 | | Cost cost ⇒ 〈0, 0〉 |
---|
1991 | | Jmp jmp ⇒ 〈2, 2〉 |
---|
1992 | | Call call ⇒ 〈2, 2〉 |
---|
1993 | | Mov dptr tgt ⇒ 〈2, 2〉 |
---|
1994 | ]. |
---|
1995 | cases not_implemented (* policy returned medium_jump for conditional jumping = impossible *) |
---|
1996 | *)qed. |
---|
1997 | |
---|
1998 | definition ticks_of: ∀p:pseudo_assembly_program. (Word → Word × bool) → Word → nat × nat ≝ |
---|
1999 | λprogram: pseudo_assembly_program.λsigma. |
---|
2000 | λppc: Word. |
---|
2001 | let 〈preamble, pseudo〉 ≝ program in |
---|
2002 | let 〈fetched, new_ppc〉 ≝ fetch_pseudo_instruction pseudo ppc in |
---|
2003 | ticks_of0 program sigma ppc fetched. |
---|
2004 | |
---|
2005 | lemma eq_rect_Type1_r: |
---|
2006 | ∀A: Type[1]. |
---|
2007 | ∀a:A. |
---|
2008 | ∀P: ∀x:A. eq ? x a → Type[1]. P a (refl A a) → ∀x: A.∀p:eq ? x a. P x p. |
---|
2009 | #A #a #P #H #x #p |
---|
2010 | generalize in match H; |
---|
2011 | generalize in match P; |
---|
2012 | cases p |
---|
2013 | // |
---|
2014 | qed. |
---|
2015 | |
---|
2016 | axiom split_append: |
---|
2017 | ∀A: Type[0]. |
---|
2018 | ∀m, n: nat. |
---|
2019 | ∀v, v': Vector A m. |
---|
2020 | ∀q, q': Vector A n. |
---|
2021 | let 〈v', q'〉 ≝ split A m n (v@@q) in |
---|
2022 | v = v' ∧ q = q'. |
---|
2023 | |
---|
2024 | axiom split_vector_singleton: |
---|
2025 | ∀A: Type[0]. |
---|
2026 | ∀n: nat. |
---|
2027 | ∀v: Vector A (S n). |
---|
2028 | ∀rest: Vector A n. |
---|
2029 | ∀s: Vector A 1. |
---|
2030 | ∀prf. |
---|
2031 | v = s @@ rest → |
---|
2032 | ((get_index_v A ? v 0 prf) ::: rest) = v. |
---|
2033 | |
---|
2034 | example sub_minus_one_seven_eight: |
---|
2035 | ∀v: BitVector 7. |
---|
2036 | false ::: (\fst (sub_7_with_carry v (bitvector_of_nat ? 1) false)) = |
---|
2037 | \fst (sub_8_with_carry (false ::: v) (bitvector_of_nat ? 1) false). |
---|
2038 | cases daemon. |
---|
2039 | qed. |
---|
2040 | |
---|
2041 | (* |
---|
2042 | lemma blah: |
---|
2043 | ∀m: internal_pseudo_address_map. |
---|
2044 | ∀s: PseudoStatus. |
---|
2045 | ∀arg: Byte. |
---|
2046 | ∀b: bool. |
---|
2047 | addressing_mode_ok m s (DIRECT arg) = true → |
---|
2048 | get_arg_8 ? (set_low_internal_ram ? s (low_internal_ram_of_pseudo_low_internal_ram m (low_internal_ram ? s))) b (DIRECT arg) = |
---|
2049 | get_arg_8 ? s b (DIRECT arg). |
---|
2050 | [2, 3: normalize % ] |
---|
2051 | #m #s #arg #b #hyp |
---|
2052 | whd in ⊢ (??%%) |
---|
2053 | @split_elim'' |
---|
2054 | #nu' #nl' #arg_nu_nl_eq |
---|
2055 | normalize nodelta |
---|
2056 | generalize in match (refl ? (get_index_v bool 4 nu' ? ?)) |
---|
2057 | cases (get_index_v bool 4 nu' ? ?) in ⊢ (??%? → %) |
---|
2058 | #get_index_v_eq |
---|
2059 | normalize nodelta |
---|
2060 | [2: |
---|
2061 | normalize nodelta |
---|
2062 | @split_elim'' |
---|
2063 | #bit_one' #three_bits' #bit_one_three_bit_eq |
---|
2064 | generalize in match (low_internal_ram_of_pseudo_internal_ram_miss m s (three_bits'@@nl')) |
---|
2065 | normalize nodelta |
---|
2066 | generalize in match (refl ? (sub_7_with_carry ? ? ?)) |
---|
2067 | cases (sub_7_with_carry ? ? ?) in ⊢ (??%? → %) |
---|
2068 | #Saddr #carr' #Saddr_carr_eq |
---|
2069 | normalize nodelta |
---|
2070 | #carr_hyp' |
---|
2071 | @carr_hyp' |
---|
2072 | [1: |
---|
2073 | |2: whd in hyp:(??%?); generalize in match hyp; -hyp; |
---|
2074 | generalize in match (refl ? (¬(member (BitVector 8) ? arg m))) |
---|
2075 | cases (¬(member (BitVector 8) ? arg m)) in ⊢ (??%? → %) |
---|
2076 | #member_eq |
---|
2077 | normalize nodelta |
---|
2078 | [2: #destr destruct(destr) |
---|
2079 | |1: -carr_hyp'; |
---|
2080 | >arg_nu_nl_eq |
---|
2081 | <(split_vector_singleton ? ? nu' ? ? ? bit_one_three_bit_eq) |
---|
2082 | [1: >get_index_v_eq in ⊢ (??%? → ?) |
---|
2083 | |2: @le_S @le_S @le_S @le_n |
---|
2084 | ] |
---|
2085 | cases (member (BitVector 8) ? (\fst ?) ?) |
---|
2086 | [1: #destr normalize in destr; destruct(destr) |
---|
2087 | |2: |
---|
2088 | ] |
---|
2089 | ] |
---|
2090 | |3: >get_index_v_eq in ⊢ (??%?) |
---|
2091 | change in ⊢ (??(???%?)?) with ((? ::: three_bits') @@ nl') |
---|
2092 | >(split_vector_singleton … bit_one_three_bit_eq) |
---|
2093 | <arg_nu_nl_eq |
---|
2094 | whd in hyp:(??%?) |
---|
2095 | cases (member (BitVector 8) (eq_bv 8) arg m) in hyp |
---|
2096 | normalize nodelta [*: #ignore @sym_eq ] |
---|
2097 | ] |
---|
2098 | | |
---|
2099 | ]. |
---|
2100 | *) |
---|
2101 | (* |
---|
2102 | map_address0 ... (DIRECT arg) = Some .. → |
---|
2103 | get_arg_8 (map_address0 ... (internal_ram ...) (DIRECT arg) = |
---|
2104 | get_arg_8 (internal_ram ...) (DIRECT arg) |
---|
2105 | |
---|
2106 | ((if addressing_mode_ok M ps ACC_A∧addressing_mode_ok M ps (DIRECT ARG2) |
---|
2107 | then Some internal_pseudo_address_map M |
---|
2108 | else None internal_pseudo_address_map ) |
---|
2109 | =Some internal_pseudo_address_map M') |
---|
2110 | *) |
---|
2111 | |
---|
2112 | axiom low_internal_ram_write_at_stack_pointer: |
---|
2113 | ∀T1,T2,M,cm1,s1,cm2,s2,cm3,s3.∀sigma: Word → Word × bool.∀pbu,pbl,bu,bl,sp1,sp2:BitVector 8. |
---|
2114 | get_8051_sfr T2 cm2 s2 SFR_SP = get_8051_sfr ? cm3 s3 SFR_SP → |
---|
2115 | low_internal_ram ? cm2 s2 = low_internal_ram T2 cm3 s3 → |
---|
2116 | sp1 = \snd (half_add ? (get_8051_sfr … cm1 s1 SFR_SP) (bitvector_of_nat 8 1)) → |
---|
2117 | sp2 = \snd (half_add ? sp1 (bitvector_of_nat 8 1)) → |
---|
2118 | bu@@bl = \fst (sigma (pbu@@pbl)) → |
---|
2119 | low_internal_ram T1 cm1 |
---|
2120 | (write_at_stack_pointer … |
---|
2121 | (set_8051_sfr … |
---|
2122 | (write_at_stack_pointer … |
---|
2123 | (set_8051_sfr … |
---|
2124 | (set_low_internal_ram … s1 |
---|
2125 | (low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … s2))) |
---|
2126 | SFR_SP sp1) |
---|
2127 | bl) |
---|
2128 | SFR_SP sp2) |
---|
2129 | bu) |
---|
2130 | = low_internal_ram_of_pseudo_low_internal_ram (sp1::M) |
---|
2131 | (low_internal_ram … |
---|
2132 | (write_at_stack_pointer … |
---|
2133 | (set_8051_sfr … |
---|
2134 | (write_at_stack_pointer … (set_8051_sfr … s3 SFR_SP sp1) pbl) |
---|
2135 | SFR_SP sp2) |
---|
2136 | pbu)). |
---|
2137 | |
---|
2138 | axiom high_internal_ram_write_at_stack_pointer: |
---|
2139 | ∀T1,T2,M,cm1,s1,cm2,s2,cm3,s3.∀sigma:Word → Word × bool.∀pbu,pbl,bu,bl,sp1,sp2:BitVector 8. |
---|
2140 | get_8051_sfr T2 cm2 s2 SFR_SP = get_8051_sfr ? cm3 s3 SFR_SP → |
---|
2141 | high_internal_ram ?? s2 = high_internal_ram T2 cm3 s3 → |
---|
2142 | sp1 = \snd (half_add ? (get_8051_sfr ? cm1 s1 SFR_SP) (bitvector_of_nat 8 1)) → |
---|
2143 | sp2 = \snd (half_add ? sp1 (bitvector_of_nat 8 1)) → |
---|
2144 | bu@@bl = \fst (sigma (pbu@@pbl)) → |
---|
2145 | high_internal_ram T1 cm1 |
---|
2146 | (write_at_stack_pointer … |
---|
2147 | (set_8051_sfr … |
---|
2148 | (write_at_stack_pointer … |
---|
2149 | (set_8051_sfr … |
---|
2150 | (set_high_internal_ram … s1 |
---|
2151 | (high_internal_ram_of_pseudo_high_internal_ram M (high_internal_ram … s2))) |
---|
2152 | SFR_SP sp1) |
---|
2153 | bl) |
---|
2154 | SFR_SP sp2) |
---|
2155 | bu) |
---|
2156 | = high_internal_ram_of_pseudo_high_internal_ram (sp1::M) |
---|
2157 | (high_internal_ram … |
---|
2158 | (write_at_stack_pointer … |
---|
2159 | (set_8051_sfr … |
---|
2160 | (write_at_stack_pointer … (set_8051_sfr … s3 SFR_SP sp1) pbl) |
---|
2161 | SFR_SP sp2) |
---|
2162 | pbu)). |
---|
2163 | |
---|
2164 | lemma Some_Some_elim: |
---|
2165 | ∀T:Type[0].∀x,y:T.∀P:Type[2]. (x=y → P) → Some T x = Some T y → P. |
---|
2166 | #T #x #y #P #H #K @H @option_destruct_Some // |
---|
2167 | qed. |
---|
2168 | |
---|
2169 | (*CSC: ???*) |
---|
2170 | axiom snd_assembly_1_pseudoinstruction_ok: |
---|
2171 | ∀program:pseudo_assembly_program.∀sigma. |
---|
2172 | ∀ppc:Word.∀pi,lookup_labels,lookup_datalabels. |
---|
2173 | lookup_labels = (λx. \fst (sigma (address_of_word_labels_code_mem (\snd program) x))) → |
---|
2174 | lookup_datalabels = (λx. lookup_def … (construct_datalabels (\fst program)) x (zero ?)) → |
---|
2175 | \fst (fetch_pseudo_instruction (\snd program) ppc) = pi → |
---|
2176 | let len ≝ \fst (assembly_1_pseudoinstruction lookup_labels sigma (\fst (sigma ppc)) lookup_datalabels pi) in |
---|
2177 | \fst (sigma (add … ppc (bitvector_of_nat ? 1))) = |
---|
2178 | add … (\fst (sigma ppc)) (bitvector_of_nat ? len). |
---|
2179 | |
---|
2180 | lemma pose: ∀A:Type[0].∀B:A → Type[0].∀a:A. (∀a':A. a'=a → B a') → B a. |
---|
2181 | /2/ |
---|
2182 | qed. |
---|
2183 | |
---|
2184 | (* To be moved in ProofStatus *) |
---|
2185 | lemma program_counter_set_program_counter: |
---|
2186 | ∀T,cm,s,x. program_counter T cm (set_program_counter T cm s x) = x. |
---|
2187 | // |
---|
2188 | qed. |
---|
2189 | |
---|
2190 | theorem main_thm: |
---|
2191 | ∀M,M',cm,ps,sigma. |
---|
2192 | next_internal_pseudo_address_map M cm ps = Some … M' → |
---|
2193 | ∃n. |
---|
2194 | execute n … (status_of_pseudo_status M … ps sigma) |
---|
2195 | = status_of_pseudo_status M' … (execute_1_pseudo_instruction (ticks_of cm sigma) cm ps) sigma. |
---|
2196 | #M #M' * #preamble #instr_list #ps #sigma |
---|
2197 | change with (next_internal_pseudo_address_map0 ????? = ? → ?) |
---|
2198 | @(pose … (program_counter ?? ps)) #ppc #EQppc |
---|
2199 | generalize in match (fetch_assembly_pseudo2 〈preamble,instr_list〉 sigma ppc) in ⊢ ?; |
---|
2200 | @pair_elim #labels #costs #H0 normalize nodelta |
---|
2201 | @pair_elim #assembled #costs' #EQ1 normalize nodelta |
---|
2202 | @pair_elim #pi #newppc #EQ2 normalize nodelta |
---|
2203 | @(pose … (λx. \fst (sigma (address_of_word_labels_code_mem instr_list x)))) #lookup_labels #EQlookup_labels |
---|
2204 | @(pose … (λx. lookup_def … (construct_datalabels preamble) x (zero 16))) #lookup_datalabels #EQlookup_datalabels |
---|
2205 | whd in match execute_1_pseudo_instruction; normalize nodelta |
---|
2206 | whd in match ticks_of; normalize nodelta <EQppc #H2 >EQ2 normalize nodelta |
---|
2207 | lapply (snd_fetch_pseudo_instruction instr_list ppc) >EQ2 #EQnewppc >EQnewppc |
---|
2208 | lapply (snd_assembly_1_pseudoinstruction_ok 〈preamble,instr_list〉 … ppc pi … EQlookup_labels EQlookup_datalabels ?) |
---|
2209 | [>EQ2 %] |
---|
2210 | inversion pi |
---|
2211 | [2,3: (* Comment, Cost *) #ARG #EQ |
---|
2212 | #H3 normalize nodelta in H3; normalize in match (assembly_1_pseudoinstruction ?????) in H3; |
---|
2213 | whd in ⊢ (??%? → ?); @Some_Some_elim #MAP <MAP |
---|
2214 | whd in match (execute_1_pseudo_instruction0 ?????); |
---|
2215 | %{0} @split_eq_status CSC:STOP HERE // |
---|
2216 | |4: (* Jmp *) #label whd in ⊢ (??%? → ???% → ?) |
---|
2217 | @Some_Some_elim #MAP cases (pol ?) normalize nodelta |
---|
2218 | [3: (* long *) #EQ3 @(Some_Some_elim ????? EQ3) #EQ3' |
---|
2219 | whd in match eject normalize nodelta >EQ3' in ⊢ (% → ?) whd in ⊢ (% → ?) |
---|
2220 | @pair_elim' * #instr #newppc' #ticks #EQ4 |
---|
2221 | * * #H2a #H2b whd in ⊢ (% → ?) #H2 |
---|
2222 | >H2b >(eq_instruction_to_eq … H2a) |
---|
2223 | #EQ %[@1] |
---|
2224 | <MAP >(eq_bv_eq … H2) >EQ |
---|
2225 | whd in ⊢ (??%?) >EQ4 whd in ⊢ (??%?) |
---|
2226 | cases ps in EQ4; #A1 #A2 #A3 #A4 #A5 #A6 #A7 #A8 #A9 #A10 #XXX >XXX % |
---|
2227 | whd in ⊢ (??%?) |
---|
2228 | whd in ⊢ (??(match ?%? with [_ ⇒ ?])?) |
---|
2229 | cases ps in EQ0 ⊢ %; #A1 #A2 #A3 #A4 #A5 #A6 #A7 #A8 #A9 #A10 #XXXX >XXXX % |
---|
2230 | |6: (* Mov *) #arg1 #arg2 |
---|
2231 | #H1 #H2 #EQ %[@1] |
---|
2232 | normalize in H1; generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?) |
---|
2233 | change in ⊢ (? → ??%?) with (execute_1_0 ??) |
---|
2234 | cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta; |
---|
2235 | * * #H2a #H2b whd in ⊢ (% → ?) #H2c |
---|
2236 | >H2b >(eq_instruction_to_eq … H2a) |
---|
2237 | generalize in match EQ; -EQ; whd in ⊢ (???% → ??%?); |
---|
2238 | @(list_addressing_mode_tags_elim_prop … arg1) whd try % -arg1; whd in ⊢ (???% → ??%?) |
---|
2239 | @(list_addressing_mode_tags_elim_prop … arg2) whd try % -arg2; #ARG2 |
---|
2240 | normalize nodelta; |
---|
2241 | [1,2,3,4,5,6,7,8: cases (add_8_with_carry ???) |*: cases (sub_8_with_carry ???)] |
---|
2242 | #result #flags |
---|
2243 | #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) % |
---|
2244 | |5: (* Call *) #label #MAP |
---|
2245 | generalize in match (option_destruct_Some ??? MAP) -MAP; #MAP <MAP -MAP; |
---|
2246 | whd in ⊢ (???% → ?) cases (pol ?) normalize nodelta; |
---|
2247 | [ (* short *) #abs @⊥ destruct (abs) |
---|
2248 | |3: (* long *) #H1 #H2 #EQ %[@1] |
---|
2249 | (* normalize in H1; !!!*) generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?) |
---|
2250 | change in ⊢ (? → ??%?) with (execute_1_0 ??) |
---|
2251 | cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta; |
---|
2252 | * * #H2a #H2b whd in ⊢ (% → ?) #H2c |
---|
2253 | >H2b >(eq_instruction_to_eq … H2a) |
---|
2254 | generalize in match EQ; -EQ; |
---|
2255 | whd in ⊢ (???% → ??%?); |
---|
2256 | 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; |
---|
2257 | >(eq_bv_eq … H2c) |
---|
2258 | change with |
---|
2259 | ((?=let 〈ppc_bu,ppc_bl〉 ≝ split bool 8 8 newppc in ?) → |
---|
2260 | (let 〈pc_bu,pc_bl〉 ≝ split bool 8 8 (sigma 〈preamble,instr_list〉 pol newppc) in ?)=?) |
---|
2261 | generalize in match (refl … (split … 8 8 newppc)) cases (split bool 8 8 newppc) in ⊢ (??%? → %) #ppc_bu #ppc_bl #EQppc |
---|
2262 | 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; |
---|
2263 | >get_8051_sfr_write_at_stack_pointer >get_8051_sfr_write_at_stack_pointer |
---|
2264 | >get_8051_sfr_set_8051_sfr >get_8051_sfr_set_8051_sfr |
---|
2265 | generalize in match (refl … (half_add ? new_sp (bitvector_of_nat ? 1))) cases (half_add ???) in ⊢ (??%? → %) #carry' #new_sp' #EQ2 normalize nodelta; |
---|
2266 | #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) |
---|
2267 | @split_eq_status; |
---|
2268 | [ >code_memory_write_at_stack_pointer whd in ⊢ (??%?) |
---|
2269 | >code_memory_write_at_stack_pointer % |
---|
2270 | | >set_program_counter_set_low_internal_ram |
---|
2271 | >set_clock_set_low_internal_ram |
---|
2272 | @low_internal_ram_write_at_stack_pointer |
---|
2273 | [ >EQ0 @pol | % | % |
---|
2274 | | @(pair_destruct_2 … EQ1) |
---|
2275 | | @(pair_destruct_2 … EQ2) |
---|
2276 | | >(pair_destruct_1 ????? EQpc) |
---|
2277 | >(pair_destruct_2 ????? EQpc) |
---|
2278 | @split_elim #x #y #H <H -x y H; |
---|
2279 | >(pair_destruct_1 ????? EQppc) |
---|
2280 | >(pair_destruct_2 ????? EQppc) |
---|
2281 | @split_elim #x #y #H <H -x y H; |
---|
2282 | >EQ0 % ] |
---|
2283 | | >set_low_internal_ram_set_high_internal_ram |
---|
2284 | >set_program_counter_set_high_internal_ram |
---|
2285 | >set_clock_set_high_internal_ram |
---|
2286 | @high_internal_ram_write_at_stack_pointer |
---|
2287 | [ >EQ0 @pol | % | % |
---|
2288 | | @(pair_destruct_2 … EQ1) |
---|
2289 | | @(pair_destruct_2 … EQ2) |
---|
2290 | | >(pair_destruct_1 ????? EQpc) |
---|
2291 | >(pair_destruct_2 ????? EQpc) |
---|
2292 | @split_elim #x #y #H <H -x y H; |
---|
2293 | >(pair_destruct_1 ????? EQppc) |
---|
2294 | >(pair_destruct_2 ????? EQppc) |
---|
2295 | @split_elim #x #y #H <H -x y H; |
---|
2296 | >EQ0 % ] |
---|
2297 | | >external_ram_write_at_stack_pointer whd in ⊢ (??%?) |
---|
2298 | >external_ram_write_at_stack_pointer whd in ⊢ (???%) |
---|
2299 | >external_ram_write_at_stack_pointer whd in ⊢ (???%) |
---|
2300 | >external_ram_write_at_stack_pointer % |
---|
2301 | | change with (? = sigma ?? (address_of_word_labels_code_mem (\snd (code_memory ? ps)) ?)) |
---|
2302 | >EQ0 % |
---|
2303 | | >special_function_registers_8051_write_at_stack_pointer whd in ⊢ (??%?) |
---|
2304 | >special_function_registers_8051_write_at_stack_pointer whd in ⊢ (???%) |
---|
2305 | >special_function_registers_8051_write_at_stack_pointer whd in ⊢ (???%) |
---|
2306 | >special_function_registers_8051_write_at_stack_pointer % |
---|
2307 | | >special_function_registers_8052_write_at_stack_pointer whd in ⊢ (??%?) |
---|
2308 | >special_function_registers_8052_write_at_stack_pointer whd in ⊢ (???%) |
---|
2309 | >special_function_registers_8052_write_at_stack_pointer whd in ⊢ (???%) |
---|
2310 | >special_function_registers_8052_write_at_stack_pointer % |
---|
2311 | | >p1_latch_write_at_stack_pointer whd in ⊢ (??%?) |
---|
2312 | >p1_latch_write_at_stack_pointer whd in ⊢ (???%) |
---|
2313 | >p1_latch_write_at_stack_pointer whd in ⊢ (???%) |
---|
2314 | >p1_latch_write_at_stack_pointer % |
---|
2315 | | >p3_latch_write_at_stack_pointer whd in ⊢ (??%?) |
---|
2316 | >p3_latch_write_at_stack_pointer whd in ⊢ (???%) |
---|
2317 | >p3_latch_write_at_stack_pointer whd in ⊢ (???%) |
---|
2318 | >p3_latch_write_at_stack_pointer % |
---|
2319 | | >clock_write_at_stack_pointer whd in ⊢ (??%?) |
---|
2320 | >clock_write_at_stack_pointer whd in ⊢ (???%) |
---|
2321 | >clock_write_at_stack_pointer whd in ⊢ (???%) |
---|
2322 | >clock_write_at_stack_pointer %] |
---|
2323 | (*| (* medium *) #H1 #H2 #EQ %[@1] generalize in match H1; -H1; |
---|
2324 | @pair_elim' #fst_5_addr #rest_addr #EQ1 |
---|
2325 | @pair_elim' #fst_5_pc #rest_pc #EQ2 |
---|
2326 | generalize in match (refl … (eq_bv … fst_5_addr fst_5_pc)) |
---|
2327 | cases (eq_bv ???) in ⊢ (??%? → %) normalize nodelta; #EQ3 #TEQ [2: destruct (TEQ)] |
---|
2328 | generalize in match (option_destruct_Some ??? TEQ) -TEQ; #K1 >K1 in H2; whd in ⊢ (% → ?) |
---|
2329 | change in ⊢ (? →??%?) with (execute_1_0 ??) |
---|
2330 | @pair_elim' * #instr #newppc' #ticks #EQn |
---|
2331 | * * #H2a #H2b whd in ⊢ (% → ?) #H2c >H2b >(eq_instruction_to_eq … H2a) whd in ⊢ (??%?) |
---|
2332 | generalize in match EQ; -EQ; normalize nodelta; >(eq_bv_eq … H2c) |
---|
2333 | @pair_elim' #carry #new_sp change with (half_add ? (get_8051_sfr ? ps ?) ? = ? → ?) #EQ4 |
---|
2334 | @split_elim' #pc_bu #pc_bl >program_counter_set_8051_sfr XXX change with (newppc = ?) #EQ5 |
---|
2335 | @pair_elim' #carry' #new_sp' #EQ6 normalize nodelta; #EQx >EQx -EQx; |
---|
2336 | change in ⊢ (??(match ????% with [_ ⇒ ?])?) with (sigma … newppc) |
---|
2337 | @split_elim' #pc_bu' #pc_bl' #EQ7 change with (newppc' = ? → ?) |
---|
2338 | >get_8051_sfr_set_8051_sfr |
---|
2339 | |
---|
2340 | whd in EQ:(???%) ⊢ ? >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) whd in ⊢ (??%?) |
---|
2341 | change with ((let 〈pc_bu,pc_bl〉 ≝ split bool 8 8 (sigma 〈preamble,instr_list〉 newppc) in ?)=?) |
---|
2342 | generalize in match (refl … (split bool 8 8 (sigma 〈preamble,instr_list〉 newppc))) |
---|
2343 | cases (split ????) in ⊢ (??%? → %) #pc_bu #pc_bl normalize nodelta; #EQ4 |
---|
2344 | generalize in match (refl … (split bool 4 4 pc_bu)) |
---|
2345 | cases (split ????) in ⊢ (??%? → %) #nu #nl normalize nodelta; #EQ5 |
---|
2346 | generalize in match (refl … (split bool 3 8 rest_addr)) |
---|
2347 | cases (split ????) in ⊢ (??%? → %) #relevant1 #relevant2 normalize nodelta; #EQ6 |
---|
2348 | change with ((let 〈carry,new_pc〉 ≝ half_add ? (sigma … newppc) ? in ?) = ?) |
---|
2349 | generalize in match |
---|
2350 | (refl … |
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2351 | (half_add 16 (sigma 〈preamble,instr_list〉 newppc) |
---|
2352 | ((nu@@get_index' bool 0 3 nl:::relevant1)@@relevant2))) |
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2353 | cases (half_add ???) in ⊢ (??%? → %) #carry #new_pc normalize nodelta; #EQ7 |
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2354 | @split_eq_status try % |
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2355 | [ change with (? = sigma ? (address_of_word_labels ps label)) |
---|
2356 | (* ARITHMETICS, BUT THE GOAL SEEMS FALSE *) |
---|
2357 | | whd in ⊢ (??%%) whd in ⊢ (??(?%?)?) whd in ⊢ (??(?(match ?(?%)? with [_ ⇒ ?])?)?) |
---|
2358 | @(bitvector_3_elim_prop … (\fst (split bool 3 8 rest_addr))) %]] *)] |
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2359 | |4: (* Jmp *) #label #MAP |
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2360 | generalize in match (option_destruct_Some ??? MAP) -MAP; #MAP >MAP -MAP; |
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2361 | whd in ⊢ (???% → ?) cases (pol ?) normalize nodelta; |
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2362 | [3: (* long *) #H1 #H2 #EQ %[@1] |
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2363 | (* normalize in H1; !!!*) generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?) |
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2364 | change in ⊢ (? → ??%?) with (execute_1_0 ??) |
---|
2365 | cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta; |
---|
2366 | * * #H2a #H2b whd in ⊢ (% → ?) #H2c |
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2367 | >H2b >(eq_instruction_to_eq … H2a) |
---|
2368 | generalize in match EQ; -EQ; |
---|
2369 | #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) |
---|
2370 | cases ps in EQ0 ⊢ %; #A1 #A2 #A3 #A4 #A5 #A6 #A7 #A8 #A9 #A10 #XXXX >XXXX % |
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2371 | |1: (* short *) #H1 #H2 #EQ %[@1] generalize in match H1; -H1; |
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2372 | generalize in match |
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2373 | (refl ? |
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2374 | (sub_16_with_carry |
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2375 | (sigma 〈preamble,instr_list〉 pol (program_counter … ps)) |
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2376 | (sigma 〈preamble,instr_list〉 pol (address_of_word_labels_code_mem instr_list label)) |
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2377 | false)) |
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2378 | cases (sub_16_with_carry ???) in ⊢ (??%? → %); #results #flags normalize nodelta; |
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2379 | generalize in match (refl … (split … 8 8 results)) cases (split ????) in ⊢ (??%? → %) #upper #lower normalize nodelta; |
---|
2380 | generalize in match (refl … (eq_bv … upper (zero 8))) cases (eq_bv ???) in ⊢ (??%? → %) normalize nodelta; |
---|
2381 | #EQ1 #EQ2 #EQ3 #H1 [2: @⊥ destruct (H1)] |
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2382 | generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?) |
---|
2383 | change in ⊢ (? → ??%?) with (execute_1_0 ??) |
---|
2384 | cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta; |
---|
2385 | * * #H2a #H2b whd in ⊢ (% → ?) #H2c |
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2386 | >H2b >(eq_instruction_to_eq … H2a) |
---|
2387 | generalize in match EQ; -EQ; |
---|
2388 | whd in ⊢ (???% → ?); |
---|
2389 | #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) |
---|
2390 | change with ((let 〈carry,new_pc〉 ≝ half_add ? (sigma ???) ? in ?) = ?) |
---|
2391 | generalize in match (refl … (half_add 16 (sigma 〈preamble,instr_list〉 pol newppc) (sign_extension lower))) |
---|
2392 | cases (half_add ???) in ⊢ (??%? → %) #carry #newpc normalize nodelta #EQ4 |
---|
2393 | @split_eq_status try % change with (newpc = sigma ?? (address_of_word_labels ps label)) |
---|
2394 | (* ARITHMETICS, BUT THE GOAL SEEMS FALSE *) |
---|
2395 | | (* medium *) #H1 #H2 #EQ %[@1] generalize in match H1; -H1; |
---|
2396 | generalize in match |
---|
2397 | (refl … |
---|
2398 | (split … 5 11 (sigma 〈preamble,instr_list〉 pol (address_of_word_labels_code_mem instr_list label)))) |
---|
2399 | cases (split ????) in ⊢ (??%? → %) #fst_5_addr #rest_addr normalize nodelta; #EQ1 |
---|
2400 | generalize in match |
---|
2401 | (refl … |
---|
2402 | (split … 5 11 (sigma 〈preamble,instr_list〉 pol (program_counter … ps)))) |
---|
2403 | cases (split ????) in ⊢ (??%? → %) #fst_5_pc #rest_pc normalize nodelta; #EQ2 |
---|
2404 | generalize in match (refl … (eq_bv … fst_5_addr fst_5_pc)) |
---|
2405 | cases (eq_bv ???) in ⊢ (??%? → %) normalize nodelta; #EQ3 #TEQ [2: destruct (TEQ)] |
---|
2406 | generalize in match (option_destruct_Some ??? TEQ) -TEQ; #K1 >K1 in H2; whd in ⊢ (% → ?) |
---|
2407 | change in ⊢ (? →??%?) with (execute_1_0 ??) |
---|
2408 | cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta; |
---|
2409 | * * #H2a #H2b whd in ⊢ (% → ?) #H2c |
---|
2410 | >H2b >(eq_instruction_to_eq … H2a) |
---|
2411 | generalize in match EQ; -EQ; |
---|
2412 | whd in ⊢ (???% → ?); |
---|
2413 | #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) whd in ⊢ (??%?) |
---|
2414 | change with ((let 〈pc_bu,pc_bl〉 ≝ split bool 8 8 (sigma 〈preamble,instr_list〉 pol newppc) in ?)=?) |
---|
2415 | generalize in match (refl … (split bool 8 8 (sigma 〈preamble,instr_list〉 pol newppc))) |
---|
2416 | cases (split ????) in ⊢ (??%? → %) #pc_bu #pc_bl normalize nodelta; #EQ4 |
---|
2417 | generalize in match (refl … (split bool 4 4 pc_bu)) |
---|
2418 | cases (split ????) in ⊢ (??%? → %) #nu #nl normalize nodelta; #EQ5 |
---|
2419 | generalize in match (refl … (split bool 3 8 rest_addr)) |
---|
2420 | cases (split ????) in ⊢ (??%? → %) #relevant1 #relevant2 normalize nodelta; #EQ6 |
---|
2421 | change with ((let 〈carry,new_pc〉 ≝ half_add ? (sigma … newppc) ? in ?) = ?) |
---|
2422 | generalize in match |
---|
2423 | (refl … |
---|
2424 | (half_add 16 (sigma 〈preamble,instr_list〉 pol newppc) |
---|
2425 | ((nu@@get_index' bool 0 3 nl:::relevant1)@@relevant2))) |
---|
2426 | cases (half_add ???) in ⊢ (??%? → %) #carry #new_pc normalize nodelta; #EQ7 |
---|
2427 | @split_eq_status try % |
---|
2428 | [ change with (? = sigma ?? (address_of_word_labels ps label)) |
---|
2429 | (* ARITHMETICS, BUT THE GOAL SEEMS FALSE *) |
---|
2430 | | whd in ⊢ (??%%) whd in ⊢ (??(?%?)?) whd in ⊢ (??(?(match ?(?%)? with [_ ⇒ ?])?)?) |
---|
2431 | @(bitvector_3_elim_prop … (\fst (split bool 3 8 rest_addr))) %]] |
---|
2432 | | (* Instruction *) -pi; whd in ⊢ (? → ??%? → ?) *; normalize nodelta; |
---|
2433 | [1,2,3: (* ADD, ADDC, SUBB *) #arg1 #arg2 #MAP #H1 #H2 #EQ %[1,3,5:@1] |
---|
2434 | normalize in H1; generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?) |
---|
2435 | change in ⊢ (? → ??%?) with (execute_1_0 ??) |
---|
2436 | cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta; |
---|
2437 | * * #H2a #H2b whd in ⊢ (% → ?) #H2c |
---|
2438 | >H2b >(eq_instruction_to_eq … H2a) |
---|
2439 | generalize in match EQ; -EQ; whd in ⊢ (???% → ??%?); generalize in match MAP; -MAP; |
---|
2440 | @(list_addressing_mode_tags_elim_prop … arg1) whd try % -arg1; |
---|
2441 | @(list_addressing_mode_tags_elim_prop … arg2) whd try % -arg2; #ARG2 |
---|
2442 | normalize nodelta; #MAP; |
---|
2443 | [1: change in ⊢ (? → %) with |
---|
2444 | ((let 〈result,flags〉 ≝ |
---|
2445 | add_8_with_carry |
---|
2446 | (get_arg_8 ? ps false ACC_A) |
---|
2447 | (get_arg_8 ? |
---|
2448 | (set_low_internal_ram ? ps (low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … ps))) |
---|
2449 | false (DIRECT ARG2)) |
---|
2450 | ? in ?) = ?) |
---|
2451 | [2,3: %] |
---|
2452 | change in ⊢ (???% → ?) with |
---|
2453 | (let 〈result,flags〉 ≝ add_8_with_carry ?(*(get_arg_8 ? ps false ACC_A)*) ?? in ?) |
---|
2454 | >get_arg_8_set_clock |
---|
2455 | [1,2: cases (addressing_mode_ok ???? ∧ addressing_mode_ok ????) in MAP ⊢ ? |
---|
2456 | [2,4: #abs @⊥ normalize in abs; destruct (abs) |
---|
2457 | |*:whd in ⊢ (??%? → ?) #H <(option_destruct_Some ??? H)] |
---|
2458 | [ change in ⊢ (? → %) with |
---|
2459 | ((let 〈result,flags〉 ≝ |
---|
2460 | add_8_with_carry |
---|
2461 | (get_arg_8 ? ps false ACC_A) |
---|
2462 | (get_arg_8 ? |
---|
2463 | (set_low_internal_ram ? ps (low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … ps))) |
---|
2464 | false (DIRECT ARG2)) |
---|
2465 | ? in ?) = ?) |
---|
2466 | >get_arg_8_set_low_internal_ram |
---|
2467 | |
---|
2468 | cases (add_8_with_carry ???) |
---|
2469 | |
---|
2470 | [1,2,3,4,5,6,7,8: cases (add_8_with_carry ???) |*: cases (sub_8_with_carry ???)] |
---|
2471 | #result #flags |
---|
2472 | #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) % |
---|
2473 | | (* INC *) #arg1 #H1 #H2 #EQ %[@1] |
---|
2474 | normalize in H1; generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?) |
---|
2475 | change in ⊢ (? → ??%?) with (execute_1_0 ??) |
---|
2476 | cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta; |
---|
2477 | * * #H2a #H2b whd in ⊢ (% → ?) #H2c |
---|
2478 | >H2b >(eq_instruction_to_eq … H2a) |
---|
2479 | generalize in match EQ; -EQ; whd in ⊢ (???% → ??%?); |
---|
2480 | @(list_addressing_mode_tags_elim_prop … arg1) whd try % -arg1; normalize nodelta; [1,2,3: #ARG] |
---|
2481 | [1,2,3,4: cases (half_add ???) #carry #result |
---|
2482 | | cases (half_add ???) #carry #bl normalize nodelta; |
---|
2483 | cases (full_add ????) #carry' #bu normalize nodelta ] |
---|
2484 | #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) -newppc'; |
---|
2485 | [5: % |
---|
2486 | |1: <(set_arg_8_set_code_memory 0 [[direct]] ? ? ? (set_clock pseudo_assembly_program |
---|
2487 | (set_program_counter pseudo_assembly_program ps newppc) |
---|
2488 | (\fst (ticks_of0 〈preamble,instr_list〉 |
---|
2489 | (program_counter pseudo_assembly_program ps) |
---|
2490 | (Instruction (INC Identifier (DIRECT ARG)))) |
---|
2491 | +clock pseudo_assembly_program |
---|
2492 | (set_program_counter pseudo_assembly_program ps newppc))) (load_code_memory assembled) result (DIRECT ARG)) |
---|
2493 | [2,3: // ] |
---|
2494 | <(set_arg_8_set_program_counter 0 [[direct]] ? ? ? ? ?) [2://] |
---|
2495 | whd in ⊢ (??%%) |
---|
2496 | cases (split bool 4 4 ARG) |
---|
2497 | #nu' #nl' |
---|
2498 | normalize nodelta |
---|
2499 | cases (split bool 1 3 nu') |
---|
2500 | #bit_1' #ignore' |
---|
2501 | normalize nodelta |
---|
2502 | cases (get_index_v bool 4 nu' ? ?) |
---|
2503 | [ normalize nodelta (* HERE *) whd in ⊢ (??%%) % |
---|
2504 | | |
---|
2505 | ] |
---|
2506 | cases daemon (* EASY CASES TO BE COMPLETED *) |
---|
2507 | qed. |
---|