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 | lemma mem_monotonic_wrt_append: |
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7 | ∀A: Type[0]. |
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8 | ∀m, o: nat. |
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9 | ∀eq: A → A → bool. |
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10 | ∀reflex: ∀a. eq a a = true. |
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11 | ∀p: Vector A m. |
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12 | ∀a: A. |
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13 | ∀r: Vector A o. |
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14 | mem A eq ? r a = true → mem A eq ? (p @@ r) a = true. |
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15 | #A #m #o #eq #reflex #p #a |
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16 | elim p try (#r #assm assumption) |
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17 | #m' #hd #tl #inductive_hypothesis #r #assm |
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18 | normalize |
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19 | cases (eq ??) try % |
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20 | @inductive_hypothesis assumption |
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21 | qed. |
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22 | |
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23 | lemma subvector_multiple_append: |
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24 | ∀A: Type[0]. |
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25 | ∀o, n: nat. |
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26 | ∀eq: A → A → bool. |
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27 | ∀refl: ∀a. eq a a = true. |
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28 | ∀h: Vector A o. |
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29 | ∀v: Vector A n. |
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30 | ∀m: nat. |
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31 | ∀q: Vector A m. |
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32 | bool_to_Prop (subvector_with A ? ? eq v (h @@ q @@ v)). |
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33 | #A #o #n #eq #reflex #h #v |
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34 | elim v try (normalize #m #irrelevant @I) |
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35 | #m' #hd #tl #inductive_hypothesis #m #q |
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36 | change with (bool_to_Prop (andb ??)) |
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37 | cut ((mem A eq (o + (m + S m')) (h@@q@@hd:::tl) hd) = true) |
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38 | [1: |
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39 | @mem_monotonic_wrt_append try assumption |
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40 | @mem_monotonic_wrt_append try assumption |
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41 | normalize >reflex % |
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42 | |2: |
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43 | #assm >assm |
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44 | >vector_cons_append |
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45 | change with (bool_to_Prop (subvector_with ??????)) |
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46 | @(dependent_rewrite_vectors … (vector_associative_append … q [[hd]] tl)) |
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47 | try @associative_plus |
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48 | @inductive_hypothesis |
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49 | ] |
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50 | qed. |
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51 | |
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52 | lemma vector_cons_empty: |
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53 | ∀A: Type[0]. |
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54 | ∀n: nat. |
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55 | ∀v: Vector A n. |
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56 | [[ ]] @@ v = v. |
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57 | #A #n #v |
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58 | cases v try % |
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59 | #n' #hd #tl % |
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60 | qed. |
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61 | |
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62 | lemma is_in_monotonic_wrt_append: |
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63 | ∀m, n: nat. |
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64 | ∀p: Vector addressing_mode_tag m. |
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65 | ∀q: Vector addressing_mode_tag n. |
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66 | ∀to_search: addressing_mode. |
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67 | bool_to_Prop (is_in ? p to_search) → bool_to_Prop (is_in ? (q @@ p) to_search). |
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68 | #m #n #p #q #to_search #assm |
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69 | elim q try assumption |
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70 | #n' #hd #tl #inductive_hypothesis |
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71 | normalize |
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72 | cases (is_a ??) try @I |
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73 | >inductive_hypothesis @I |
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74 | qed. |
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75 | |
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76 | corollary is_in_hd_tl: |
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77 | ∀to_search: addressing_mode. |
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78 | ∀hd: addressing_mode_tag. |
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79 | ∀n: nat. |
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80 | ∀v: Vector addressing_mode_tag n. |
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81 | bool_to_Prop (is_in ? v to_search) → bool_to_Prop (is_in ? (hd:::v) to_search). |
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82 | #to_search #hd #n #v |
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83 | elim v |
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84 | [1: |
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85 | #absurd |
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86 | normalize in absurd; cases absurd |
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87 | |2: |
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88 | #n' #hd' #tl #inductive_hypothesis #assm |
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89 | >vector_cons_append >(vector_cons_append … hd' tl) |
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90 | @(is_in_monotonic_wrt_append … ([[hd']]@@tl) [[hd]] to_search) |
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91 | assumption |
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92 | ] |
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93 | qed. |
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94 | |
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95 | definition eq_addressing_mode: addressing_mode → addressing_mode → bool ≝ |
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96 | λa, b: addressing_mode. |
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97 | match a with |
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98 | [ DIRECT d ⇒ |
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99 | match b with |
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100 | [ DIRECT e ⇒ eq_bv ? d e |
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101 | | _ ⇒ false |
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102 | ] |
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103 | | INDIRECT b' ⇒ |
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104 | match b with |
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105 | [ INDIRECT e ⇒ eq_b b' e |
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106 | | _ ⇒ false |
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107 | ] |
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108 | | EXT_INDIRECT b' ⇒ |
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109 | match b with |
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110 | [ EXT_INDIRECT e ⇒ eq_b b' e |
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111 | | _ ⇒ false |
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112 | ] |
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113 | | REGISTER bv ⇒ |
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114 | match b with |
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115 | [ REGISTER bv' ⇒ eq_bv ? bv bv' |
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116 | | _ ⇒ false |
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117 | ] |
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118 | | ACC_A ⇒ match b with [ ACC_A ⇒ true | _ ⇒ false ] |
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119 | | ACC_B ⇒ match b with [ ACC_B ⇒ true | _ ⇒ false ] |
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120 | | DPTR ⇒ match b with [ DPTR ⇒ true | _ ⇒ false ] |
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121 | | DATA b' ⇒ |
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122 | match b with |
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123 | [ DATA e ⇒ eq_bv ? b' e |
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124 | | _ ⇒ false |
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125 | ] |
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126 | | DATA16 w ⇒ |
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127 | match b with |
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128 | [ DATA16 e ⇒ eq_bv ? w e |
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129 | | _ ⇒ false |
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130 | ] |
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131 | | ACC_DPTR ⇒ match b with [ ACC_DPTR ⇒ true | _ ⇒ false ] |
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132 | | ACC_PC ⇒ match b with [ ACC_PC ⇒ true | _ ⇒ false ] |
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133 | | EXT_INDIRECT_DPTR ⇒ match b with [ EXT_INDIRECT_DPTR ⇒ true | _ ⇒ false ] |
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134 | | INDIRECT_DPTR ⇒ match b with [ INDIRECT_DPTR ⇒ true | _ ⇒ false ] |
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135 | | CARRY ⇒ match b with [ CARRY ⇒ true | _ ⇒ false ] |
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136 | | BIT_ADDR b' ⇒ |
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137 | match b with |
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138 | [ BIT_ADDR e ⇒ eq_bv ? b' e |
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139 | | _ ⇒ false |
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140 | ] |
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141 | | N_BIT_ADDR b' ⇒ |
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142 | match b with |
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143 | [ N_BIT_ADDR e ⇒ eq_bv ? b' e |
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144 | | _ ⇒ false |
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145 | ] |
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146 | | RELATIVE n ⇒ |
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147 | match b with |
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148 | [ RELATIVE e ⇒ eq_bv ? n e |
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149 | | _ ⇒ false |
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150 | ] |
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151 | | ADDR11 w ⇒ |
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152 | match b with |
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153 | [ ADDR11 e ⇒ eq_bv ? w e |
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154 | | _ ⇒ false |
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155 | ] |
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156 | | ADDR16 w ⇒ |
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157 | match b with |
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158 | [ ADDR16 e ⇒ eq_bv ? w e |
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159 | | _ ⇒ false |
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160 | ] |
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161 | ]. |
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162 | |
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163 | lemma eq_bv_refl: |
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164 | ∀n, b. |
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165 | eq_bv n b b = true. |
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166 | #n #b cases b // |
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167 | qed. |
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168 | |
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169 | lemma eq_b_refl: |
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170 | ∀b. |
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171 | eq_b b b = true. |
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172 | #b cases b // |
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173 | qed. |
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174 | |
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175 | lemma eq_addressing_mode_refl: |
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176 | ∀a. eq_addressing_mode a a = true. |
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177 | #a |
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178 | cases a try #arg1 try #arg2 |
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179 | try @eq_bv_refl try @eq_b_refl |
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180 | try normalize % |
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181 | qed. |
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182 | |
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183 | definition eq_sum: |
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184 | ∀A, B. (A → A → bool) → (B → B → bool) → (A ⊎ B) → (A ⊎ B) → bool ≝ |
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185 | λlt, rt, leq, req, left, right. |
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186 | match left with |
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187 | [ inl l ⇒ |
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188 | match right with |
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189 | [ inl l' ⇒ leq l l' |
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190 | | _ ⇒ false |
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191 | ] |
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192 | | inr r ⇒ |
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193 | match right with |
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194 | [ inr r' ⇒ req r r' |
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195 | | _ ⇒ false |
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196 | ] |
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197 | ]. |
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198 | |
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199 | definition eq_prod: ∀A, B. (A → A → bool) → (B → B → bool) → (A × B) → (A × B) → bool ≝ |
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200 | λlt, rt, leq, req, left, right. |
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201 | let 〈l, r〉 ≝ left in |
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202 | let 〈l', r'〉 ≝ right in |
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203 | leq l l' ∧ req r r'. |
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204 | |
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205 | definition eq_preinstruction: preinstruction [[relative]] → preinstruction [[relative]] → bool ≝ |
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206 | λi, j. |
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207 | match i with |
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208 | [ ADD arg1 arg2 ⇒ |
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209 | match j with |
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210 | [ ADD arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
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211 | | _ ⇒ false |
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212 | ] |
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213 | | ADDC arg1 arg2 ⇒ |
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214 | match j with |
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215 | [ ADDC arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
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216 | | _ ⇒ false |
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217 | ] |
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218 | | SUBB arg1 arg2 ⇒ |
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219 | match j with |
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220 | [ SUBB arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
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221 | | _ ⇒ false |
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222 | ] |
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223 | | INC arg ⇒ |
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224 | match j with |
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225 | [ INC arg' ⇒ eq_addressing_mode arg arg' |
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226 | | _ ⇒ false |
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227 | ] |
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228 | | DEC arg ⇒ |
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229 | match j with |
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230 | [ DEC arg' ⇒ eq_addressing_mode arg arg' |
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231 | | _ ⇒ false |
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232 | ] |
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233 | | MUL arg1 arg2 ⇒ |
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234 | match j with |
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235 | [ MUL arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
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236 | | _ ⇒ false |
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237 | ] |
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238 | | DIV arg1 arg2 ⇒ |
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239 | match j with |
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240 | [ DIV arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
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241 | | _ ⇒ false |
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242 | ] |
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243 | | DA arg ⇒ |
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244 | match j with |
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245 | [ DA arg' ⇒ eq_addressing_mode arg arg' |
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246 | | _ ⇒ false |
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247 | ] |
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248 | | JC arg ⇒ |
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249 | match j with |
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250 | [ JC arg' ⇒ eq_addressing_mode arg arg' |
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251 | | _ ⇒ false |
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252 | ] |
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253 | | JNC arg ⇒ |
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254 | match j with |
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255 | [ JNC arg' ⇒ eq_addressing_mode arg arg' |
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256 | | _ ⇒ false |
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257 | ] |
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258 | | JB arg1 arg2 ⇒ |
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259 | match j with |
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260 | [ JB arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
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261 | | _ ⇒ false |
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262 | ] |
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263 | | JNB arg1 arg2 ⇒ |
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264 | match j with |
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265 | [ JNB arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
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266 | | _ ⇒ false |
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267 | ] |
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268 | | JBC arg1 arg2 ⇒ |
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269 | match j with |
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270 | [ JBC arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
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271 | | _ ⇒ false |
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272 | ] |
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273 | | JZ arg ⇒ |
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274 | match j with |
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275 | [ JZ arg' ⇒ eq_addressing_mode arg arg' |
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276 | | _ ⇒ false |
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277 | ] |
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278 | | JNZ arg ⇒ |
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279 | match j with |
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280 | [ JNZ arg' ⇒ eq_addressing_mode arg arg' |
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281 | | _ ⇒ false |
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282 | ] |
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283 | | CJNE arg1 arg2 ⇒ |
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284 | match j with |
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285 | [ CJNE arg1' arg2' ⇒ |
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286 | let prod_eq_left ≝ eq_prod [[acc_a]] [[direct; data]] eq_addressing_mode eq_addressing_mode in |
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287 | let prod_eq_right ≝ eq_prod [[registr; indirect]] [[data]] eq_addressing_mode eq_addressing_mode in |
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288 | let arg1_eq ≝ eq_sum ? ? prod_eq_left prod_eq_right in |
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289 | arg1_eq arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
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290 | | _ ⇒ false |
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291 | ] |
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292 | | DJNZ arg1 arg2 ⇒ |
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293 | match j with |
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294 | [ DJNZ arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
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295 | | _ ⇒ false |
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296 | ] |
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297 | | CLR arg ⇒ |
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298 | match j with |
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299 | [ CLR arg' ⇒ eq_addressing_mode arg arg' |
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300 | | _ ⇒ false |
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301 | ] |
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302 | | CPL arg ⇒ |
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303 | match j with |
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304 | [ CPL arg' ⇒ eq_addressing_mode arg arg' |
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305 | | _ ⇒ false |
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306 | ] |
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307 | | RL arg ⇒ |
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308 | match j with |
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309 | [ RL arg' ⇒ eq_addressing_mode arg arg' |
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310 | | _ ⇒ false |
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311 | ] |
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312 | | RLC arg ⇒ |
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313 | match j with |
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314 | [ RLC arg' ⇒ eq_addressing_mode arg arg' |
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315 | | _ ⇒ false |
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316 | ] |
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317 | | RR arg ⇒ |
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318 | match j with |
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319 | [ RR arg' ⇒ eq_addressing_mode arg arg' |
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320 | | _ ⇒ false |
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321 | ] |
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322 | | RRC arg ⇒ |
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323 | match j with |
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324 | [ RRC arg' ⇒ eq_addressing_mode arg arg' |
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325 | | _ ⇒ false |
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326 | ] |
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327 | | SWAP arg ⇒ |
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328 | match j with |
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329 | [ SWAP arg' ⇒ eq_addressing_mode arg arg' |
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330 | | _ ⇒ false |
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331 | ] |
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332 | | SETB arg ⇒ |
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333 | match j with |
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334 | [ SETB arg' ⇒ eq_addressing_mode arg arg' |
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335 | | _ ⇒ false |
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336 | ] |
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337 | | PUSH arg ⇒ |
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338 | match j with |
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339 | [ PUSH arg' ⇒ eq_addressing_mode arg arg' |
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340 | | _ ⇒ false |
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341 | ] |
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342 | | POP arg ⇒ |
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343 | match j with |
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344 | [ POP arg' ⇒ eq_addressing_mode arg arg' |
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345 | | _ ⇒ false |
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346 | ] |
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347 | | XCH arg1 arg2 ⇒ |
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348 | match j with |
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349 | [ XCH arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
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350 | | _ ⇒ false |
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351 | ] |
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352 | | XCHD arg1 arg2 ⇒ |
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353 | match j with |
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354 | [ XCHD arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
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355 | | _ ⇒ false |
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356 | ] |
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357 | | RET ⇒ match j with [ RET ⇒ true | _ ⇒ false ] |
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358 | | RETI ⇒ match j with [ RETI ⇒ true | _ ⇒ false ] |
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359 | | NOP ⇒ match j with [ NOP ⇒ true | _ ⇒ false ] |
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360 | | MOVX arg ⇒ |
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361 | match j with |
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362 | [ MOVX arg' ⇒ |
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363 | let prod_eq_left ≝ eq_prod [[acc_a]] [[ext_indirect; ext_indirect_dptr]] eq_addressing_mode eq_addressing_mode in |
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364 | let prod_eq_right ≝ eq_prod [[ext_indirect; ext_indirect_dptr]] [[acc_a]] eq_addressing_mode eq_addressing_mode in |
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365 | let sum_eq ≝ eq_sum ? ? prod_eq_left prod_eq_right in |
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366 | sum_eq arg arg' |
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367 | | _ ⇒ false |
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368 | ] |
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369 | | XRL arg ⇒ |
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370 | match j with |
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371 | [ XRL arg' ⇒ |
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372 | let prod_eq_left ≝ eq_prod [[acc_a]] [[ data ; registr ; direct ; indirect ]] eq_addressing_mode eq_addressing_mode in |
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373 | let prod_eq_right ≝ eq_prod [[direct]] [[ acc_a ; data ]] eq_addressing_mode eq_addressing_mode in |
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374 | let sum_eq ≝ eq_sum ? ? prod_eq_left prod_eq_right in |
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375 | sum_eq arg arg' |
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376 | | _ ⇒ false |
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377 | ] |
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378 | | ORL arg ⇒ |
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379 | match j with |
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380 | [ ORL arg' ⇒ |
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381 | let prod_eq_left1 ≝ eq_prod [[acc_a]] [[ registr ; data ; direct ; indirect ]] eq_addressing_mode eq_addressing_mode in |
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382 | let prod_eq_left2 ≝ eq_prod [[direct]] [[ acc_a; data ]] eq_addressing_mode eq_addressing_mode in |
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383 | let prod_eq_left ≝ eq_sum ? ? prod_eq_left1 prod_eq_left2 in |
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384 | let prod_eq_right ≝ eq_prod [[carry]] [[ bit_addr ; n_bit_addr]] eq_addressing_mode eq_addressing_mode in |
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385 | let sum_eq ≝ eq_sum ? ? prod_eq_left prod_eq_right in |
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386 | sum_eq arg arg' |
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387 | | _ ⇒ false |
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388 | ] |
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389 | | ANL arg ⇒ |
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390 | match j with |
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391 | [ ANL arg' ⇒ |
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392 | let prod_eq_left1 ≝ eq_prod [[acc_a]] [[ registr ; direct ; indirect ; data ]] eq_addressing_mode eq_addressing_mode in |
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393 | let prod_eq_left2 ≝ eq_prod [[direct]] [[ acc_a; data ]] eq_addressing_mode eq_addressing_mode in |
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394 | let prod_eq_left ≝ eq_sum ? ? prod_eq_left1 prod_eq_left2 in |
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395 | let prod_eq_right ≝ eq_prod [[carry]] [[ bit_addr ; n_bit_addr]] eq_addressing_mode eq_addressing_mode in |
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396 | let sum_eq ≝ eq_sum ? ? prod_eq_left prod_eq_right in |
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397 | sum_eq arg arg' |
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398 | | _ ⇒ false |
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399 | ] |
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400 | | MOV arg ⇒ |
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401 | match j with |
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402 | [ MOV arg' ⇒ |
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403 | let prod_eq_6 ≝ eq_prod [[acc_a]] [[registr; direct; indirect; data]] eq_addressing_mode eq_addressing_mode in |
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404 | let prod_eq_5 ≝ eq_prod [[registr; indirect]] [[acc_a; direct; data]] eq_addressing_mode eq_addressing_mode in |
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405 | let prod_eq_4 ≝ eq_prod [[direct]] [[acc_a; registr; direct; indirect; data]] eq_addressing_mode eq_addressing_mode in |
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406 | let prod_eq_3 ≝ eq_prod [[dptr]] [[data16]] eq_addressing_mode eq_addressing_mode in |
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407 | let prod_eq_2 ≝ eq_prod [[carry]] [[bit_addr]] eq_addressing_mode eq_addressing_mode in |
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408 | let prod_eq_1 ≝ eq_prod [[bit_addr]] [[carry]] eq_addressing_mode eq_addressing_mode in |
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409 | let sum_eq_1 ≝ eq_sum ? ? prod_eq_6 prod_eq_5 in |
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410 | let sum_eq_2 ≝ eq_sum ? ? sum_eq_1 prod_eq_4 in |
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411 | let sum_eq_3 ≝ eq_sum ? ? sum_eq_2 prod_eq_3 in |
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412 | let sum_eq_4 ≝ eq_sum ? ? sum_eq_3 prod_eq_2 in |
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413 | let sum_eq_5 ≝ eq_sum ? ? sum_eq_4 prod_eq_1 in |
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414 | sum_eq_5 arg arg' |
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415 | | _ ⇒ false |
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416 | ] |
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417 | ]. |
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418 | |
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419 | lemma eq_sum_refl: |
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420 | ∀A, B: Type[0]. |
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421 | ∀leq, req. |
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422 | ∀s. |
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423 | ∀leq_refl: (∀t. leq t t = true). |
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424 | ∀req_refl: (∀u. req u u = true). |
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425 | eq_sum A B leq req s s = true. |
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426 | #A #B #leq #req #s #leq_refl #req_refl |
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427 | cases s assumption |
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428 | qed. |
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429 | |
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430 | lemma eq_prod_refl: |
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431 | ∀A, B: Type[0]. |
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432 | ∀leq, req. |
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433 | ∀s. |
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434 | ∀leq_refl: (∀t. leq t t = true). |
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435 | ∀req_refl: (∀u. req u u = true). |
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436 | eq_prod A B leq req s s = true. |
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437 | #A #B #leq #req #s #leq_refl #req_refl |
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438 | cases s |
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439 | whd in ⊢ (? → ? → ??%?); |
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440 | #l #r |
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441 | >leq_refl @req_refl |
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442 | qed. |
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443 | |
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444 | lemma eq_preinstruction_refl: |
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445 | ∀i. |
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446 | eq_preinstruction i i = true. |
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447 | #i cases i try #arg1 try #arg2 |
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448 | try @eq_addressing_mode_refl |
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449 | [1,2,3,4,5,6,7,8,10,16,17,18,19,20: |
---|
450 | whd in ⊢ (??%?); try % |
---|
451 | >eq_addressing_mode_refl |
---|
452 | >eq_addressing_mode_refl % |
---|
453 | |13,15: |
---|
454 | whd in ⊢ (??%?); |
---|
455 | cases arg1 |
---|
456 | [*: |
---|
457 | #arg1_left normalize nodelta |
---|
458 | >eq_prod_refl [*: try % #argr @eq_addressing_mode_refl] |
---|
459 | ] |
---|
460 | |11,12: |
---|
461 | whd in ⊢ (??%?); |
---|
462 | cases arg1 |
---|
463 | [1: |
---|
464 | #arg1_left normalize nodelta |
---|
465 | >(eq_sum_refl …) |
---|
466 | [1: % | 2,3: #arg @eq_prod_refl ] |
---|
467 | @eq_addressing_mode_refl |
---|
468 | |2: |
---|
469 | #arg1_left normalize nodelta |
---|
470 | @eq_prod_refl [*: @eq_addressing_mode_refl ] |
---|
471 | |3: |
---|
472 | #arg1_left normalize nodelta |
---|
473 | >(eq_sum_refl …) |
---|
474 | [1: |
---|
475 | % |
---|
476 | |2,3: |
---|
477 | #arg @eq_prod_refl #arg @eq_addressing_mode_refl |
---|
478 | ] |
---|
479 | |4: |
---|
480 | #arg1_left normalize nodelta |
---|
481 | @eq_prod_refl [*: #arg @eq_addressing_mode_refl ] |
---|
482 | ] |
---|
483 | |14: |
---|
484 | whd in ⊢ (??%?); |
---|
485 | cases arg1 |
---|
486 | [1: |
---|
487 | #arg1_left normalize nodelta |
---|
488 | @eq_sum_refl |
---|
489 | [1: |
---|
490 | #arg @eq_sum_refl |
---|
491 | [1: |
---|
492 | #arg @eq_sum_refl |
---|
493 | [1: |
---|
494 | #arg @eq_sum_refl |
---|
495 | [1: |
---|
496 | #arg @eq_prod_refl |
---|
497 | [*: |
---|
498 | @eq_addressing_mode_refl |
---|
499 | ] |
---|
500 | |2: |
---|
501 | #arg @eq_prod_refl |
---|
502 | [*: |
---|
503 | #arg @eq_addressing_mode_refl |
---|
504 | ] |
---|
505 | ] |
---|
506 | |2: |
---|
507 | #arg @eq_prod_refl |
---|
508 | [*: |
---|
509 | #arg @eq_addressing_mode_refl |
---|
510 | ] |
---|
511 | ] |
---|
512 | |2: |
---|
513 | #arg @eq_prod_refl |
---|
514 | [*: |
---|
515 | #arg @eq_addressing_mode_refl |
---|
516 | ] |
---|
517 | ] |
---|
518 | |2: |
---|
519 | #arg @eq_prod_refl |
---|
520 | [*: |
---|
521 | #arg @eq_addressing_mode_refl |
---|
522 | ] |
---|
523 | ] |
---|
524 | |2: |
---|
525 | #arg1_right normalize nodelta |
---|
526 | @eq_prod_refl |
---|
527 | [*: |
---|
528 | #arg @eq_addressing_mode_refl |
---|
529 | ] |
---|
530 | ] |
---|
531 | |*: |
---|
532 | whd in ⊢ (??%?); |
---|
533 | cases arg1 |
---|
534 | [*: |
---|
535 | #arg1 >eq_sum_refl |
---|
536 | [1,4: |
---|
537 | @eq_addressing_mode_refl |
---|
538 | |2,3,5,6: |
---|
539 | #arg @eq_prod_refl |
---|
540 | [*: |
---|
541 | #arg @eq_addressing_mode_refl |
---|
542 | ] |
---|
543 | ] |
---|
544 | ] |
---|
545 | ] |
---|
546 | qed. |
---|
547 | |
---|
548 | definition eq_instruction: instruction → instruction → bool ≝ |
---|
549 | λi, j. |
---|
550 | match i with |
---|
551 | [ ACALL arg ⇒ |
---|
552 | match j with |
---|
553 | [ ACALL arg' ⇒ eq_addressing_mode arg arg' |
---|
554 | | _ ⇒ false |
---|
555 | ] |
---|
556 | | LCALL arg ⇒ |
---|
557 | match j with |
---|
558 | [ LCALL arg' ⇒ eq_addressing_mode arg arg' |
---|
559 | | _ ⇒ false |
---|
560 | ] |
---|
561 | | AJMP arg ⇒ |
---|
562 | match j with |
---|
563 | [ AJMP arg' ⇒ eq_addressing_mode arg arg' |
---|
564 | | _ ⇒ false |
---|
565 | ] |
---|
566 | | LJMP arg ⇒ |
---|
567 | match j with |
---|
568 | [ LJMP arg' ⇒ eq_addressing_mode arg arg' |
---|
569 | | _ ⇒ false |
---|
570 | ] |
---|
571 | | SJMP arg ⇒ |
---|
572 | match j with |
---|
573 | [ SJMP arg' ⇒ eq_addressing_mode arg arg' |
---|
574 | | _ ⇒ false |
---|
575 | ] |
---|
576 | | JMP arg ⇒ |
---|
577 | match j with |
---|
578 | [ JMP arg' ⇒ eq_addressing_mode arg arg' |
---|
579 | | _ ⇒ false |
---|
580 | ] |
---|
581 | | MOVC arg1 arg2 ⇒ |
---|
582 | match j with |
---|
583 | [ MOVC arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
---|
584 | | _ ⇒ false |
---|
585 | ] |
---|
586 | | RealInstruction instr ⇒ |
---|
587 | match j with |
---|
588 | [ RealInstruction instr' ⇒ eq_preinstruction instr instr' |
---|
589 | | _ ⇒ false |
---|
590 | ] |
---|
591 | ]. |
---|
592 | |
---|
593 | lemma eq_instruction_refl: |
---|
594 | ∀i. eq_instruction i i = true. |
---|
595 | #i cases i [*: #arg1 ] |
---|
596 | try @eq_addressing_mode_refl |
---|
597 | try @eq_preinstruction_refl |
---|
598 | #arg2 whd in ⊢ (??%?); |
---|
599 | >eq_addressing_mode_refl >eq_addressing_mode_refl % |
---|
600 | qed. |
---|
601 | |
---|
602 | (*CSC: in is_a_to_mem_to_is_in use vect_member in place of mem … *) |
---|
603 | definition vect_member ≝ |
---|
604 | λA,n,eq,v,a. mem A eq (S n) v a. |
---|
605 | |
---|
606 | definition tech_if_vect_member ≝ |
---|
607 | λn,l,am,H. |
---|
608 | bool_inv_rect_Type0 (vect_member … n … eq_a l am) ? H (λ_.True). |
---|
609 | |
---|
610 | definition is_in_cons_elim: |
---|
611 | ∀len.∀hd,tl.∀m:addressing_mode |
---|
612 | .is_in (S len) (hd:::tl) m → |
---|
613 | (bool_to_Prop (is_a hd m)) ∨ (bool_to_Prop (is_in len tl m)). |
---|
614 | #len #hd #tl #am #ISIN whd in match (is_in ???) in ISIN; |
---|
615 | cases (is_a hd am) in ISIN; /2/ |
---|
616 | qed. |
---|
617 | |
---|
618 | definition load_code_memory_aux ≝ |
---|
619 | fold_left_i_aux … ( |
---|
620 | λi, mem, v. |
---|
621 | insert … (bitvector_of_nat … i) v mem) (Stub Byte 16). |
---|
622 | |
---|
623 | lemma vsplit_zero: |
---|
624 | ∀A,m. |
---|
625 | ∀v: Vector A m. |
---|
626 | 〈[[]], v〉 = vsplit A 0 m v. |
---|
627 | #A #m #v |
---|
628 | cases v try % |
---|
629 | #n #hd #tl % |
---|
630 | qed. |
---|
631 | |
---|
632 | lemma Vector_O: |
---|
633 | ∀A: Type[0]. |
---|
634 | ∀v: Vector A 0. |
---|
635 | v ≃ VEmpty A. |
---|
636 | #A #v |
---|
637 | generalize in match (refl … 0); |
---|
638 | cases v in ⊢ (??%? → ?%%??); // |
---|
639 | #n #hd #tl #absurd |
---|
640 | destruct(absurd) |
---|
641 | qed. |
---|
642 | |
---|
643 | lemma Vector_Sn: |
---|
644 | ∀A: Type[0]. |
---|
645 | ∀n: nat. |
---|
646 | ∀v: Vector A (S n). |
---|
647 | ∃hd: A. ∃tl: Vector A n. |
---|
648 | v ≃ VCons A n hd tl. |
---|
649 | #A #n #v |
---|
650 | generalize in match (refl … (S n)); |
---|
651 | cases v in ⊢ (??%? → ??(λ_.??(λ_.?%%??))); |
---|
652 | [1: |
---|
653 | #absurd destruct(absurd) |
---|
654 | |2: |
---|
655 | #m #hd #tl #eq |
---|
656 | <(injective_S … eq) |
---|
657 | %{hd} %{tl} % |
---|
658 | ] |
---|
659 | qed. |
---|
660 | |
---|
661 | lemma vector_append_zero: |
---|
662 | ∀A,m. |
---|
663 | ∀v: Vector A m. |
---|
664 | ∀q: Vector A 0. |
---|
665 | v = q@@v. |
---|
666 | #A #m #v #q |
---|
667 | >(Vector_O A q) % |
---|
668 | qed. |
---|
669 | |
---|
670 | lemma prod_eq_left: |
---|
671 | ∀A: Type[0]. |
---|
672 | ∀p, q, r: A. |
---|
673 | p = q → 〈p, r〉 = 〈q, r〉. |
---|
674 | #A #p #q #r #hyp |
---|
675 | destruct % |
---|
676 | qed. |
---|
677 | |
---|
678 | lemma prod_eq_right: |
---|
679 | ∀A: Type[0]. |
---|
680 | ∀p, q, r: A. |
---|
681 | p = q → 〈r, p〉 = 〈r, q〉. |
---|
682 | #A #p #q #r #hyp |
---|
683 | destruct % |
---|
684 | qed. |
---|
685 | |
---|
686 | corollary prod_vector_zero_eq_left: |
---|
687 | ∀A, n. |
---|
688 | ∀q: Vector A O. |
---|
689 | ∀r: Vector A n. |
---|
690 | 〈q, r〉 = 〈[[ ]], r〉. |
---|
691 | #A #n #q #r |
---|
692 | generalize in match (Vector_O A q …); |
---|
693 | #hyp destruct % |
---|
694 | qed. |
---|
695 | |
---|
696 | lemma tail_head: |
---|
697 | ∀a: Type[0]. |
---|
698 | ∀m, n: nat. |
---|
699 | ∀hd: a. |
---|
700 | ∀l: Vector a m. |
---|
701 | ∀r: Vector a n. |
---|
702 | tail a ? (hd:::(l@@r)) = l@@r. |
---|
703 | #a #m #n #hd #l #r |
---|
704 | cases l try % |
---|
705 | #m' #hd' #tl' % |
---|
706 | qed. |
---|
707 | |
---|
708 | lemma head_head': |
---|
709 | ∀a: Type[0]. |
---|
710 | ∀m: nat. |
---|
711 | ∀hd: a. |
---|
712 | ∀l: Vector a m. |
---|
713 | hd = head' … (hd:::l). |
---|
714 | #a #m #hd #l cases l try % |
---|
715 | #m' #hd' #tl % |
---|
716 | qed. |
---|
717 | |
---|
718 | lemma vsplit_succ: |
---|
719 | ∀A: Type[0]. |
---|
720 | ∀m, n: nat. |
---|
721 | ∀l: Vector A m. |
---|
722 | ∀r: Vector A n. |
---|
723 | ∀v: Vector A (m + n). |
---|
724 | ∀hd: A. |
---|
725 | v = l@@r → (〈l, r〉 = vsplit A m n v → 〈hd:::l, r〉 = vsplit A (S m) n (hd:::v)). |
---|
726 | #A #m |
---|
727 | elim m |
---|
728 | [1: |
---|
729 | #n #l #r #v #hd #eq #hyp |
---|
730 | destruct >(Vector_O … l) % |
---|
731 | |2: |
---|
732 | #m' #inductive_hypothesis #n #l #r #v #hd #equal #hyp |
---|
733 | destruct |
---|
734 | cases (Vector_Sn … l) #hd' #tl' |
---|
735 | whd in ⊢ (???%); |
---|
736 | >tail_head |
---|
737 | <(? : vsplit A (S m') n (l@@r) = vsplit' A (S m') n (l@@r)) |
---|
738 | try (<hyp <head_head' %) |
---|
739 | elim l normalize // |
---|
740 | ] |
---|
741 | qed. |
---|
742 | |
---|
743 | lemma vsplit_prod: |
---|
744 | ∀A: Type[0]. |
---|
745 | ∀m, n: nat. |
---|
746 | ∀p: Vector A (m + n). |
---|
747 | ∀v: Vector A m. |
---|
748 | ∀q: Vector A n. |
---|
749 | p = v@@q → 〈v, q〉 = vsplit A m n p. |
---|
750 | #A #m elim m |
---|
751 | [1: |
---|
752 | #n #p #v #q #hyp |
---|
753 | >hyp <(vector_append_zero A n q v) |
---|
754 | >(prod_vector_zero_eq_left A …) |
---|
755 | @vsplit_zero |
---|
756 | |2: |
---|
757 | #r #ih #n #p #v #q #hyp |
---|
758 | >hyp |
---|
759 | cases (Vector_Sn A r v) #hd #exists |
---|
760 | cases exists #tl #jmeq |
---|
761 | >jmeq @vsplit_succ try % |
---|
762 | @ih % |
---|
763 | ] |
---|
764 | qed. |
---|
765 | |
---|
766 | definition vsplit_elim: |
---|
767 | ∀A: Type[0]. |
---|
768 | ∀l, m: nat. |
---|
769 | ∀v: Vector A (l + m). |
---|
770 | ∀P: (Vector A l) × (Vector A m) → Prop. |
---|
771 | (∀vl: Vector A l. |
---|
772 | ∀vm: Vector A m. |
---|
773 | v = vl@@vm → P 〈vl,vm〉) → P (vsplit A l m v) ≝ |
---|
774 | λa: Type[0]. |
---|
775 | λl, m: nat. |
---|
776 | λv: Vector a (l + m). |
---|
777 | λP. ?. |
---|
778 | cases daemon |
---|
779 | qed. |
---|
780 | |
---|
781 | let rec encoding_check |
---|
782 | (code_memory: BitVectorTrie Byte 16) (pc: Word) (final_pc: Word) |
---|
783 | (encoding: list Byte) |
---|
784 | on encoding: Prop ≝ |
---|
785 | match encoding with |
---|
786 | [ nil ⇒ final_pc = pc |
---|
787 | | cons hd tl ⇒ |
---|
788 | let 〈new_pc, byte〉 ≝ next code_memory pc in |
---|
789 | hd = byte ∧ encoding_check code_memory new_pc final_pc tl |
---|
790 | ]. |
---|
791 | |
---|
792 | lemma add_bitvector_of_nat_Sm: |
---|
793 | ∀n, m: nat. |
---|
794 | add … (bitvector_of_nat … 1) (bitvector_of_nat … m) = |
---|
795 | bitvector_of_nat n (S m). |
---|
796 | #n #m @add_bitvector_of_nat_plus |
---|
797 | qed. |
---|
798 | |
---|
799 | lemma encoding_check_append: |
---|
800 | ∀code_memory: BitVectorTrie Byte 16. |
---|
801 | ∀final_pc: Word. |
---|
802 | ∀l1: list Byte. |
---|
803 | ∀pc: Word. |
---|
804 | ∀l2: list Byte. |
---|
805 | encoding_check code_memory pc final_pc (l1@l2) → |
---|
806 | let pc_plus_len ≝ add … pc (bitvector_of_nat … (length … l1)) in |
---|
807 | encoding_check code_memory pc pc_plus_len l1 ∧ |
---|
808 | encoding_check code_memory pc_plus_len final_pc l2. |
---|
809 | #code_memory #final_pc #l1 elim l1 |
---|
810 | [1: |
---|
811 | #pc #l2 |
---|
812 | whd in ⊢ (????% → ?); #H |
---|
813 | <add_zero |
---|
814 | whd whd in ⊢ (?%?); /2/ |
---|
815 | |2: |
---|
816 | #hd #tl #IH #pc #l2 * #H1 #H2 |
---|
817 | (* >add_SO in H2; #H2 *) |
---|
818 | cases (IH … H2) #E1 #E2 % |
---|
819 | [1: |
---|
820 | % try @H1 |
---|
821 | <(add_bitvector_of_nat_Sm 16 (|tl|)) in E1; |
---|
822 | <add_associative #assm assumption |
---|
823 | |2: |
---|
824 | <add_associative in E2; |
---|
825 | <(add_bitvector_of_nat_Sm 16 (|tl|)) #assm |
---|
826 | assumption |
---|
827 | ] |
---|
828 | ] |
---|
829 | qed. |
---|
830 | |
---|
831 | lemma destruct_bug_fix_1: |
---|
832 | ∀n: nat. |
---|
833 | S n = 0 → False. |
---|
834 | #n #absurd destruct(absurd) |
---|
835 | qed. |
---|
836 | |
---|
837 | lemma destruct_bug_fix_2: |
---|
838 | ∀m, n: nat. |
---|
839 | S m = S n → m = n. |
---|
840 | #m #n #refl destruct % |
---|
841 | qed. |
---|
842 | |
---|
843 | definition bitvector_3_cases: |
---|
844 | ∀b: BitVector 3. |
---|
845 | ∃l, c, r: bool. |
---|
846 | b ≃ [[l; c; r]]. |
---|
847 | #b |
---|
848 | @(Vector_inv_ind bool 3 b (λn: nat. λv: Vector bool n. ∃l:bool.∃c:bool.∃r:bool. v ≃ [[l; c; r]])) |
---|
849 | [1: |
---|
850 | #absurd @⊥ -b @(destruct_bug_fix_1 2) |
---|
851 | >absurd % |
---|
852 | |2: |
---|
853 | #n #hd #tl #_ #size_refl #hd_tl_refl %{hd} |
---|
854 | cut (n = 2) |
---|
855 | [1: |
---|
856 | @destruct_bug_fix_2 |
---|
857 | >size_refl % |
---|
858 | |2: |
---|
859 | #n_refl >n_refl in tl; #V |
---|
860 | @(Vector_inv_ind bool 2 V (λn: nat. λv: Vector bool n. ∃c:bool. ∃r:bool. hd:::v ≃ [[hd; c; r]])) |
---|
861 | [1: |
---|
862 | #absurd @⊥ -V @(destruct_bug_fix_1 1) |
---|
863 | >absurd % |
---|
864 | |2: |
---|
865 | #n' #hd' #tl' #_ #size_refl' #hd_tl_refl' %{hd'} |
---|
866 | cut (n' = 1) |
---|
867 | [1: |
---|
868 | @destruct_bug_fix_2 >size_refl' % |
---|
869 | |2: |
---|
870 | #n_refl' >n_refl' in tl'; #V' |
---|
871 | @(Vector_inv_ind bool 1 V' (λn: nat. λv: Vector bool n. ∃r: bool. hd:::hd':::v ≃ [[hd; hd'; r]])) |
---|
872 | [1: |
---|
873 | #absurd @⊥ -V' @(destruct_bug_fix_1 0) |
---|
874 | >absurd % |
---|
875 | |2: |
---|
876 | #n'' #hd'' #tl'' #_ #size_refl'' #hd_tl_refl'' %{hd''} |
---|
877 | cut (n'' = 0) |
---|
878 | [1: |
---|
879 | @destruct_bug_fix_2 >size_refl'' % |
---|
880 | |2: |
---|
881 | #n_refl'' >n_refl'' in tl''; #tl''' |
---|
882 | >(Vector_O … tl''') % |
---|
883 | ] |
---|
884 | ] |
---|
885 | ] |
---|
886 | ] |
---|
887 | ] |
---|
888 | ] |
---|
889 | qed. |
---|
890 | |
---|
891 | lemma bitvector_3_elim_prop: |
---|
892 | ∀P: BitVector 3 → Prop. |
---|
893 | P [[false;false;false]] → P [[false;false;true]] → P [[false;true;false]] → |
---|
894 | P [[false;true;true]] → P [[true;false;false]] → P [[true;false;true]] → |
---|
895 | P [[true;true;false]] → P [[true;true;true]] → ∀v. P v. |
---|
896 | #P #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9 |
---|
897 | cases (bitvector_3_cases … H9) #l #assm cases assm |
---|
898 | -assm #c #assm cases assm |
---|
899 | -assm #r #assm cases assm destruct |
---|
900 | cases l cases c cases r assumption |
---|
901 | qed. |
---|
902 | |
---|
903 | definition ticks_of_instruction ≝ |
---|
904 | λi. |
---|
905 | let trivial_code_memory ≝ assembly1 i in |
---|
906 | let trivial_status ≝ load_code_memory trivial_code_memory in |
---|
907 | \snd (fetch trivial_status (zero ?)). |
---|
908 | |
---|
909 | lemma fetch_assembly: |
---|
910 | ∀pc: Word. |
---|
911 | ∀i: instruction. |
---|
912 | ∀code_memory: BitVectorTrie Byte 16. |
---|
913 | ∀assembled: list Byte. |
---|
914 | assembled = assembly1 i → |
---|
915 | let len ≝ length … assembled in |
---|
916 | let pc_plus_len ≝ add … pc (bitvector_of_nat … len) in |
---|
917 | encoding_check code_memory pc pc_plus_len assembled → |
---|
918 | let 〈instr, pc', ticks〉 ≝ fetch code_memory pc in |
---|
919 | (eq_instruction instr i ∧ eqb ticks (ticks_of_instruction instr) ∧ eq_bv … pc' pc_plus_len) = true. |
---|
920 | #pc #i #code_memory #assembled cases i [8: *] |
---|
921 | [16,20,29: * * |18,19: * * [1,2,4,5: *] |28: * * [1,2: * [1,2: * [1,2: * [1,2: *]]]]] |
---|
922 | [47,48,49: |
---|
923 | |*: #arg @(subaddressing_mode_elim … arg) |
---|
924 | [2,3,5,7,10,12,16,17,18,21,26,27,28,31,32,33,37,38,39,40,41,42,43,44,45,48,51,58, |
---|
925 | 59,60,63,64,65,66,67: #ARG]] |
---|
926 | [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, |
---|
927 | 56,57,69,70,72: #arg2 @(subaddressing_mode_elim … arg2) |
---|
928 | [1,2,4,7,9,11,12,14,15,17,18,19,20,22,23,24,25,26,27,28,29,30,31,32,33,36,37,38, |
---|
929 | 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, |
---|
930 | 68,69,70,71: #ARG2]] |
---|
931 | [1,2,19,20: #arg3 @(subaddressing_mode_elim … arg3) #ARG3] |
---|
932 | normalize in ⊢ (???% → ?); |
---|
933 | [92,94,42,93,95: @vsplit_elim #vl #vm #E >E -E; [2,4: @(bitvector_3_elim_prop … vl)] |
---|
934 | normalize in ⊢ (???% → ?);] |
---|
935 | #H >H * #H1 try (whd in ⊢ (% → ?); * #H2) |
---|
936 | try (whd in ⊢ (% → ?); * #H3) whd in ⊢ (% → ?); #H4 |
---|
937 | whd in match fetch; normalize nodelta <H1 whd in ⊢ (match % with [ _ ⇒ ? ]); |
---|
938 | [17,18,19,20,21,22,23,24,25,26,32,34,35,36,37,39: <H3] |
---|
939 | try <H2 whd >eq_instruction_refl >H4 @eq_bv_refl |
---|
940 | qed. |
---|
941 | |
---|
942 | let rec fetch_many |
---|
943 | (code_memory: BitVectorTrie Byte 16) (final_pc: Word) (pc: Word) |
---|
944 | (expected: list instruction) |
---|
945 | on expected: Prop ≝ |
---|
946 | match expected with |
---|
947 | [ nil ⇒ eq_bv … pc final_pc = true |
---|
948 | | cons i tl ⇒ |
---|
949 | let pc' ≝ add 16 pc (bitvector_of_nat 16 (|assembly1 … i|)) in |
---|
950 | (〈i, pc', ticks_of_instruction i〉 = fetch code_memory pc ∧ |
---|
951 | fetch_many code_memory final_pc pc' tl) |
---|
952 | ]. |
---|
953 | |
---|
954 | lemma option_destruct_Some: |
---|
955 | ∀A: Type[0]. |
---|
956 | ∀a, b: A. |
---|
957 | Some A a = Some A b → a = b. |
---|
958 | #A #a #b #EQ |
---|
959 | destruct % |
---|
960 | qed. |
---|
961 | |
---|
962 | lemma eq_instruction_to_eq: |
---|
963 | ∀i1, i2: instruction. |
---|
964 | eq_instruction i1 i2 = true → i1 ≃ i2. |
---|
965 | #i1 #i2 |
---|
966 | cases i1 cases i2 cases daemon (* |
---|
967 | [1,10,19,28,37,46: |
---|
968 | #arg1 #arg2 |
---|
969 | whd in match (eq_instruction ??); |
---|
970 | cases arg1 #subaddressing_mode |
---|
971 | cases subaddressing_mode |
---|
972 | try (#arg1' #arg2' normalize in ⊢ (% → ?); #absurd cases absurd @I) |
---|
973 | try (#arg1' normalize in ⊢ (% → ?); #absurd cases absurd @I) |
---|
974 | try (normalize in ⊢ (% → ?); #absurd cases absurd @I) |
---|
975 | #word11 #irrelevant |
---|
976 | cases arg2 #subaddressing_mode |
---|
977 | cases subaddressing_mode |
---|
978 | try (#arg1' #arg2' normalize in ⊢ (% → ?); #absurd cases absurd @I) |
---|
979 | try (#arg1' normalize in ⊢ (% → ?); #absurd cases absurd @I) |
---|
980 | try (normalize in ⊢ (% → ?); #absurd cases absurd @I) |
---|
981 | #word11' #irrelevant normalize nodelta |
---|
982 | #eq_bv_assm cases (eq_bv_eq … eq_bv_assm) % *) |
---|
983 | qed. |
---|
984 | |
---|
985 | lemma fetch_assembly_pseudo': |
---|
986 | ∀lookup_labels. |
---|
987 | ∀sigma: Word → Word. |
---|
988 | ∀policy: Word → bool. |
---|
989 | ∀ppc. |
---|
990 | ∀lookup_datalabels. |
---|
991 | ∀pi. |
---|
992 | ∀code_memory. |
---|
993 | ∀len. |
---|
994 | ∀assembled. |
---|
995 | ∀instructions. |
---|
996 | let pc ≝ sigma ppc in |
---|
997 | instructions = expand_pseudo_instruction lookup_labels sigma policy ppc lookup_datalabels pi → |
---|
998 | 〈len,assembled〉 = assembly_1_pseudoinstruction lookup_labels sigma policy ppc lookup_datalabels pi → |
---|
999 | let pc_plus_len ≝ add … pc (bitvector_of_nat … len) in |
---|
1000 | encoding_check code_memory pc pc_plus_len assembled → |
---|
1001 | fetch_many code_memory pc_plus_len pc instructions. |
---|
1002 | #lookup_labels #sigma #policy #ppc #lookup_datalabels #pi #code_memory #len #assembled #instructions |
---|
1003 | normalize nodelta #instructions_refl whd in ⊢ (???% → ?); <instructions_refl whd in ⊢ (???% → ?); #assembled_refl |
---|
1004 | cases (pair_destruct ?????? assembled_refl) -assembled_refl #len_refl #assembled_refl |
---|
1005 | >len_refl >assembled_refl -len_refl |
---|
1006 | generalize in match (add 16 (sigma ppc) |
---|
1007 | (bitvector_of_nat 16 |
---|
1008 | (|flatten (Vector bool 8) |
---|
1009 | (map instruction (list (Vector bool 8)) assembly1 instructions)|))); |
---|
1010 | #final_pc |
---|
1011 | generalize in match (sigma ppc); elim instructions |
---|
1012 | [1: |
---|
1013 | #pc whd in ⊢ (% → %); #H >H @eq_bv_refl |
---|
1014 | |2: |
---|
1015 | #i #tl #IH #pc #H whd |
---|
1016 | cases (encoding_check_append ????? H) -H #H1 #H2 |
---|
1017 | lapply (fetch_assembly pc i code_memory (assembly1 i) (refl …)) whd in ⊢ (% → ?); |
---|
1018 | cases (fetch ??) * #instr #pc' #ticks |
---|
1019 | #H3 lapply (H3 H1) -H3 normalize nodelta #H3 |
---|
1020 | lapply (conjunction_true ?? H3) * #H4 #H5 |
---|
1021 | lapply (conjunction_true … H4) * #B1 #B2 |
---|
1022 | >(eq_instruction_to_eq … B1) >(eq_bv_eq … H5) % |
---|
1023 | >(eqb_true_to_refl … B2) >(eq_instruction_to_eq … B1) try % @IH @H2 |
---|
1024 | ] |
---|
1025 | qed. |
---|
1026 | |
---|
1027 | lemma fetch_assembly_pseudo: |
---|
1028 | ∀program: pseudo_assembly_program. |
---|
1029 | ∀sigma: Word → Word. |
---|
1030 | ∀policy: Word → bool. |
---|
1031 | let lookup_labels ≝ λx:Identifier. address_of_word_labels_code_mem (\snd program) x in |
---|
1032 | ∀ppc.∀ppc_ok. |
---|
1033 | ∀code_memory. |
---|
1034 | let lookup_datalabels ≝ λx:Identifier. lookup_def … (construct_datalabels (\fst program)) x (zero 16) in |
---|
1035 | let pi ≝ \fst (fetch_pseudo_instruction (\snd program) ppc ppc_ok) in |
---|
1036 | let pc ≝ sigma ppc in |
---|
1037 | let instructions ≝ expand_pseudo_instruction lookup_labels sigma policy ppc lookup_datalabels pi in |
---|
1038 | let 〈len,assembled〉 ≝ assembly_1_pseudoinstruction lookup_labels sigma policy ppc lookup_datalabels pi in |
---|
1039 | let pc_plus_len ≝ add … pc (bitvector_of_nat … len) in |
---|
1040 | encoding_check code_memory pc pc_plus_len assembled → |
---|
1041 | fetch_many code_memory pc_plus_len pc instructions. |
---|
1042 | #program #sigma #policy letin lookup_labels ≝ (λx.?) #ppc #ppc_ok #code_memory |
---|
1043 | letin lookup_datalabels ≝ (λx.?) |
---|
1044 | letin pi ≝ (fst ???) |
---|
1045 | letin pc ≝ (sigma ?) |
---|
1046 | letin instructions ≝ (expand_pseudo_instruction ??????) |
---|
1047 | @pair_elim #len #assembled #assembled_refl normalize nodelta |
---|
1048 | #H |
---|
1049 | generalize in match |
---|
1050 | (fetch_assembly_pseudo' lookup_labels sigma policy ppc lookup_datalabels pi code_memory len assembled instructions) in ⊢ ?; |
---|
1051 | #X destruct normalize nodelta @X try % <assembled_refl try % assumption |
---|
1052 | qed. |
---|
1053 | |
---|
1054 | definition is_present_in_machine_code_image_p: ∀pseudo_instruction. Prop ≝ |
---|
1055 | λpseudo_instruction. |
---|
1056 | match pseudo_instruction with |
---|
1057 | [ Comment c ⇒ False |
---|
1058 | | Cost c ⇒ False |
---|
1059 | | _ ⇒ True |
---|
1060 | ]. |
---|
1061 | |
---|
1062 | (* This is a trivial consequence of fetch_assembly_pseudo + the proof that the |
---|
1063 | function that load the code in memory is correct. The latter is based |
---|
1064 | on missing properties from the standard library on the BitVectorTrie |
---|
1065 | data structrure. |
---|
1066 | |
---|
1067 | Wrong at the moment, requires Jaap's precondition to ensure that the program |
---|
1068 | does not overflow when put into code memory (i.e. shorter than 2^16 bytes). |
---|
1069 | *) |
---|
1070 | |
---|
1071 | lemma load_code_memory_ok: |
---|
1072 | ∀program. |
---|
1073 | let program_size ≝ |program| in |
---|
1074 | program_size ≤ 2^16 → |
---|
1075 | ∀pc. ∀pc_ok: pc < program_size. |
---|
1076 | nth_safe ? pc program pc_ok = \snd (next (load_code_memory program) (bitvector_of_nat … pc)). |
---|
1077 | #program elim program |
---|
1078 | [ #_ #pc #abs normalize in abs; @⊥ /2/ |
---|
1079 | | #hd #tl #IH #size_ok * |
---|
1080 | [ #pc_ok whd in ⊢ (??%?); whd in match (load_code_memory ?); |
---|
1081 | whd in match next; normalize nodelta |
---|
1082 | | #pc' #pc_ok' whd in ⊢ (??%?); whd in match (load_code_memory ?); |
---|
1083 | whd in match next; normalize nodelta |
---|
1084 | ] |
---|
1085 | cases daemon (*CSC: complete! *) |
---|
1086 | qed. |
---|
1087 | (*NO! Prima dimostrare tipo Russell di assembly in Assembly.ma |
---|
1088 | Poi dimostrare che per ogni i, se fetcho l'i-esimo bit di program |
---|
1089 | e' come fetchare l'i-esimo bit dalla memoria. |
---|
1090 | Concludere assembly_ok come semplice corollario. |
---|
1091 | *) |
---|
1092 | lemma assembly_ok: |
---|
1093 | ∀program. |
---|
1094 | ∀length_proof: |\snd program| ≤ 2^16. |
---|
1095 | ∀sigma: Word → Word. |
---|
1096 | ∀policy: Word → bool. |
---|
1097 | ∀sigma_policy_witness: sigma_policy_specification program sigma policy. |
---|
1098 | ∀assembled. |
---|
1099 | ∀costs': BitVectorTrie costlabel 16. |
---|
1100 | let 〈preamble, instr_list〉 ≝ program in |
---|
1101 | let 〈labels, costs〉 ≝ create_label_cost_map instr_list in |
---|
1102 | let datalabels ≝ construct_datalabels preamble in |
---|
1103 | let lookup_datalabels ≝ λx. lookup_def … datalabels x (zero …) in |
---|
1104 | 〈assembled,costs'〉 = assembly program sigma policy → |
---|
1105 | (* costs = costs' ∧ CSC: costs != costs', costs' = sigma costs! *) |
---|
1106 | let code_memory ≝ load_code_memory assembled in |
---|
1107 | let lookup_labels ≝ λx. address_of_word_labels_code_mem instr_list x in |
---|
1108 | ∀ppc.∀ppc_ok. |
---|
1109 | let 〈pi, newppc〉 ≝ fetch_pseudo_instruction (\snd program) ppc ppc_ok in |
---|
1110 | let 〈len,assembled〉 ≝ assembly_1_pseudoinstruction lookup_labels sigma policy ppc lookup_datalabels pi in |
---|
1111 | let pc ≝ sigma ppc in |
---|
1112 | let pc_plus_len ≝ add … pc (bitvector_of_nat … len) in |
---|
1113 | encoding_check code_memory pc pc_plus_len assembled ∧ |
---|
1114 | sigma newppc = add … pc (bitvector_of_nat … len). |
---|
1115 | #program #length_proof #sigma #policy #sigma_policy_witness #assembled #costs' |
---|
1116 | cases (assembly program sigma policy) * #assembled' #costs'' |
---|
1117 | @pair_elim #preamble #instr_list #EQprogram whd in ⊢ (% → %); |
---|
1118 | cut (|instr_list| ≤ 2^16) [ >EQprogram in length_proof; // ] #instr_list_ok |
---|
1119 | #H lapply (H sigma_policy_witness instr_list_ok) -H whd in ⊢ (% → ?); |
---|
1120 | @pair_elim #labels #costs #create_label_cost_refl whd in ⊢ (% → %); |
---|
1121 | * #assembly_ok1 #assembly_ok2 #Pair_eq destruct(Pair_eq) whd |
---|
1122 | #ppc #ppc_ok @pair_elim #pi #newppc #eq_fetch_pseudo_instruction |
---|
1123 | @pair_elim #len #assembled #eq_assembly_1_pseudoinstruction whd |
---|
1124 | lapply (assembly_ok2 ppc ?) try // -assembly_ok2 |
---|
1125 | >eq_fetch_pseudo_instruction |
---|
1126 | change with ((let 〈len0,assembledi〉 ≝ assembly_1_pseudoinstruction ????? pi in ∀j.∀H:j<|assembledi|.?) → ?) |
---|
1127 | > (?:((λx.bitvector_of_nat ? (lookup_def … labels x 0)) = |
---|
1128 | (λx.address_of_word_labels_code_mem instr_list x))) |
---|
1129 | [2: lapply (create_label_cost_map_ok 〈preamble,instr_list〉) >create_label_cost_refl |
---|
1130 | #H (*CSC: REQUIRES FUNCTIONAL EXTENSIONALITY; REPHRASE THE LEMMA *) cases daemon ] |
---|
1131 | >eq_assembly_1_pseudoinstruction |
---|
1132 | whd in ⊢ (% → ?); #assembly_ok |
---|
1133 | % |
---|
1134 | [2: >(pair_destruct_2 ????? (sym_eq … eq_fetch_pseudo_instruction)) |
---|
1135 | >snd_fetch_pseudo_instruction |
---|
1136 | cases sigma_policy_witness #_ >EQprogram #H cases (H ? ppc_ok) -H |
---|
1137 | #spw1 #_ >spw1 -spw1 [2: @(transitive_le … ppc_ok) // ] @eq_f @eq_f |
---|
1138 | >eq_fetch_pseudo_instruction whd in match instruction_size; |
---|
1139 | normalize nodelta (*CSC: TRUE, NEEDS LEMMA AND FUNCTIONAL EXTENSIONALITY *) |
---|
1140 | cases daemon |
---|
1141 | | lapply (load_code_memory_ok assembled' ?) [ assumption ] |
---|
1142 | #load_code_memory_ok |
---|
1143 | cut (len=|assembled|) |
---|
1144 | [1: (*CSC: provable before cleaning *) |
---|
1145 | cases daemon |
---|
1146 | ] |
---|
1147 | #EQlen |
---|
1148 | (* Nice statement here *) |
---|
1149 | cut (∀j. ∀H: j < |assembled|. |
---|
1150 | nth_safe Byte j assembled H = |
---|
1151 | \snd (next (load_code_memory assembled') |
---|
1152 | (bitvector_of_nat 16 |
---|
1153 | (nat_of_bitvector … |
---|
1154 | (add … (sigma ppc) (bitvector_of_nat … j)))))) |
---|
1155 | [1: |
---|
1156 | cases daemon |
---|
1157 | |2: |
---|
1158 | -assembly_ok -load_code_memory_ok generalize in match (sigma ppc); >EQlen -len |
---|
1159 | elim assembled |
---|
1160 | [1: |
---|
1161 | #pc #_ whd <add_zero % |
---|
1162 | | #hd #tl #IH #pc #H % |
---|
1163 | [ lapply (H 0 ?) [ normalize @le_S_S @le_O_n ] whd in ⊢ (??%? → ?); -H #H |
---|
1164 | >H -H whd in ⊢ (??%?); <add_zero // |
---|
1165 | | >(?: add … pc (bitvector_of_nat … (S (|tl|))) = add … (add … pc (bitvector_of_nat … 1)) (bitvector_of_nat … (|tl|))) |
---|
1166 | [2: <add_bitvector_of_nat_Sm @add_associative ] |
---|
1167 | @IH -IH #j #LTj lapply (H (S j) ?) [ @le_S_S @LTj ] |
---|
1168 | <(nth_safe_prepend … [hd] … LTj) #IH >IH <add_bitvector_of_nat_Sm |
---|
1169 | >add_associative % ]] |
---|
1170 | ]] |
---|
1171 | qed. |
---|
1172 | |
---|
1173 | (* XXX: should we add that costs = costs'? *) |
---|
1174 | lemma fetch_assembly_pseudo2: |
---|
1175 | ∀program. |
---|
1176 | ∀length_proof: |\snd program| ≤ 2^16. |
---|
1177 | ∀sigma. |
---|
1178 | ∀policy. |
---|
1179 | ∀sigma_policy_specification_witness: sigma_policy_specification program sigma policy. |
---|
1180 | ∀ppc.∀ppc_ok. |
---|
1181 | let 〈labels, costs〉 ≝ create_label_cost_map (\snd program) in |
---|
1182 | let 〈assembled, costs'〉 ≝ pi1 … (assembly program sigma policy) in |
---|
1183 | let code_memory ≝ load_code_memory assembled in |
---|
1184 | let data_labels ≝ construct_datalabels (\fst program) in |
---|
1185 | let lookup_labels ≝ λx. address_of_word_labels_code_mem (\snd program) x in |
---|
1186 | let lookup_datalabels ≝ λx. lookup_def ? ? data_labels x (zero ?) in |
---|
1187 | let 〈pi,newppc〉 ≝ fetch_pseudo_instruction (\snd program) ppc ppc_ok in |
---|
1188 | let instructions ≝ expand_pseudo_instruction lookup_labels sigma policy ppc lookup_datalabels pi in |
---|
1189 | fetch_many code_memory (sigma newppc) (sigma ppc) instructions. |
---|
1190 | * #preamble #instr_list #length_proof #sigma #policy #sigma_policy_specification_witness #ppc #ppc_ok |
---|
1191 | @pair_elim #labels #costs #create_label_map_refl |
---|
1192 | @pair_elim #assembled #costs' #assembled_refl |
---|
1193 | letin code_memory ≝ (load_code_memory ?) |
---|
1194 | letin data_labels ≝ (construct_datalabels ?) |
---|
1195 | letin lookup_labels ≝ (λx. ?) |
---|
1196 | letin lookup_datalabels ≝ (λx. ?) |
---|
1197 | @pair_elim #pi #newppc #fetch_pseudo_refl |
---|
1198 | lapply (assembly_ok 〈preamble, instr_list〉 ? sigma policy sigma_policy_specification_witness assembled costs') |
---|
1199 | normalize nodelta try assumption |
---|
1200 | @pair_elim #labels' #costs' #create_label_map_refl' #H |
---|
1201 | lapply (H (sym_eq … assembled_refl)) -H |
---|
1202 | lapply (refl … (assembly_1_pseudoinstruction lookup_labels sigma policy ppc lookup_datalabels pi)) |
---|
1203 | cases (assembly_1_pseudoinstruction ??????) in ⊢ (???% → ?); |
---|
1204 | #len #assembledi #EQ4 #H |
---|
1205 | lapply (H ppc) >fetch_pseudo_refl #H |
---|
1206 | lapply (fetch_assembly_pseudo 〈preamble,instr_list〉 sigma policy ppc ppc_ok (load_code_memory assembled)) |
---|
1207 | >EQ4 #H1 cases (H ppc_ok) |
---|
1208 | #H2 #H3 >H3 normalize nodelta in H1; normalize nodelta |
---|
1209 | >fetch_pseudo_refl in H1; #assm @assm assumption |
---|
1210 | qed. |
---|
1211 | |
---|
1212 | (* OLD? |
---|
1213 | definition assembly_specification: |
---|
1214 | ∀assembly_program: pseudo_assembly_program. |
---|
1215 | ∀code_mem: BitVectorTrie Byte 16. Prop ≝ |
---|
1216 | λpseudo_assembly_program. |
---|
1217 | λcode_mem. |
---|
1218 | ∀pc: Word. |
---|
1219 | let 〈preamble, instr_list〉 ≝ pseudo_assembly_program in |
---|
1220 | let 〈pre_instr, pre_new_pc〉 ≝ fetch_pseudo_instruction instr_list pc in |
---|
1221 | let labels ≝ λx. sigma' pseudo_assembly_program (address_of_word_labels_code_mem instr_list x) in |
---|
1222 | let datalabels ≝ λx. sigma' pseudo_assembly_program (lookup ? ? x (construct_datalabels preamble) (zero ?)) in |
---|
1223 | let pre_assembled ≝ assembly_1_pseudoinstruction pseudo_assembly_program |
---|
1224 | (sigma' pseudo_assembly_program pc) labels datalabels pre_instr in |
---|
1225 | match pre_assembled with |
---|
1226 | [ None ⇒ True |
---|
1227 | | Some pc_code ⇒ |
---|
1228 | let 〈new_pc,code〉 ≝ pc_code in |
---|
1229 | encoding_check code_mem pc (sigma' pseudo_assembly_program pre_new_pc) code ]. |
---|
1230 | |
---|
1231 | axiom assembly_meets_specification: |
---|
1232 | ∀pseudo_assembly_program. |
---|
1233 | match assembly pseudo_assembly_program with |
---|
1234 | [ None ⇒ True |
---|
1235 | | Some code_mem_cost ⇒ |
---|
1236 | let 〈code_mem, cost〉 ≝ code_mem_cost in |
---|
1237 | assembly_specification pseudo_assembly_program (load_code_memory code_mem) |
---|
1238 | ]. |
---|
1239 | (* |
---|
1240 | # PROGRAM |
---|
1241 | [ cases PROGRAM |
---|
1242 | # PREAMBLE |
---|
1243 | # INSTR_LIST |
---|
1244 | elim INSTR_LIST |
---|
1245 | [ whd |
---|
1246 | whd in ⊢ (∀_. %) |
---|
1247 | # PC |
---|
1248 | whd |
---|
1249 | | # INSTR |
---|
1250 | # INSTR_LIST_TL |
---|
1251 | # H |
---|
1252 | whd |
---|
1253 | whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?]) |
---|
1254 | ] |
---|
1255 | | cases not_implemented |
---|
1256 | ] *) |
---|
1257 | *) |
---|
1258 | |
---|
1259 | (* XXX: changed this type. Bool specifies whether byte is first or second |
---|
1260 | component of an address, and the Word is the pseudoaddress that it |
---|
1261 | corresponds to. Second component is the same principle for the accumulator |
---|
1262 | A. |
---|
1263 | *) |
---|
1264 | definition internal_pseudo_address_map ≝ list ((BitVector 8) × (bool × Word)) × (option (bool × Word)). |
---|
1265 | |
---|
1266 | include alias "ASM/BitVectorTrie.ma". |
---|
1267 | |
---|
1268 | include "common/AssocList.ma". |
---|
1269 | |
---|
1270 | axiom low_internal_ram_of_pseudo_low_internal_ram: |
---|
1271 | ∀M:internal_pseudo_address_map.∀ram:BitVectorTrie Byte 7.BitVectorTrie Byte 7. |
---|
1272 | |
---|
1273 | axiom high_internal_ram_of_pseudo_high_internal_ram: |
---|
1274 | ∀M:internal_pseudo_address_map.∀ram:BitVectorTrie Byte 7.BitVectorTrie Byte 7. |
---|
1275 | |
---|
1276 | axiom low_internal_ram_of_pseudo_internal_ram_hit: |
---|
1277 | ∀M:internal_pseudo_address_map.∀cm.∀s:PseudoStatus cm.∀sigma:Word → Word × bool.∀high: bool. ∀points_to: Word. ∀addr:BitVector 7. |
---|
1278 | assoc_list_lookup ?? (false:::addr) (eq_bv 8) (\fst M) = Some … (〈high, points_to〉) → |
---|
1279 | let ram ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … s) in |
---|
1280 | let pbl ≝ lookup ? 7 addr (low_internal_ram … s) (zero 8) in |
---|
1281 | let bl ≝ lookup ? 7 addr ram (zero 8) in |
---|
1282 | let 〈lower, higher〉 ≝ vsplit ? 8 8 points_to in |
---|
1283 | let 〈plower, phigher〉 ≝ vsplit ? 8 8 (\fst (sigma points_to)) in |
---|
1284 | if high then |
---|
1285 | (pbl = higher) ∧ (bl = phigher) |
---|
1286 | else |
---|
1287 | (pbl = lower) ∧ (bl = plower). |
---|
1288 | |
---|
1289 | (* changed from add to sub *) |
---|
1290 | axiom low_internal_ram_of_pseudo_internal_ram_miss: |
---|
1291 | ∀T.∀M:internal_pseudo_address_map.∀cm.∀s:PreStatus T cm.∀addr:BitVector 7. |
---|
1292 | let ram ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … s) in |
---|
1293 | assoc_list_exists ?? (false:::addr) (eq_bv 8) (\fst M) = false → |
---|
1294 | lookup ? 7 addr ram (zero ?) = lookup ? 7 addr (low_internal_ram … s) (zero ?). |
---|
1295 | |
---|
1296 | definition addressing_mode_ok ≝ |
---|
1297 | λT.λM:internal_pseudo_address_map.λcm.λs:PreStatus T cm. |
---|
1298 | λaddr:addressing_mode. |
---|
1299 | match addr with |
---|
1300 | [ DIRECT d ⇒ |
---|
1301 | ¬(assoc_list_exists ?? d (eq_bv 8) (\fst M)) ∧ |
---|
1302 | ¬(assoc_list_exists ?? (\fst (sub_8_with_carry d (bitvector_of_nat 8 1) false)) (eq_bv 8) (\fst M)) |
---|
1303 | | INDIRECT i ⇒ |
---|
1304 | let d ≝ get_register … s [[false;false;i]] in |
---|
1305 | ¬(assoc_list_exists ?? d (eq_bv 8) (\fst M)) ∧ |
---|
1306 | ¬(assoc_list_exists ?? (\fst (sub_8_with_carry d (bitvector_of_nat 8 1) false)) (eq_bv 8) (\fst M)) |
---|
1307 | | EXT_INDIRECT _ ⇒ true |
---|
1308 | | REGISTER _ ⇒ true |
---|
1309 | | ACC_A ⇒ match \snd M with [ None ⇒ true | _ ⇒ false ] |
---|
1310 | | ACC_B ⇒ true |
---|
1311 | | DPTR ⇒ true |
---|
1312 | | DATA _ ⇒ true |
---|
1313 | | DATA16 _ ⇒ true |
---|
1314 | | ACC_DPTR ⇒ true |
---|
1315 | | ACC_PC ⇒ true |
---|
1316 | | EXT_INDIRECT_DPTR ⇒ true |
---|
1317 | | INDIRECT_DPTR ⇒ true |
---|
1318 | | CARRY ⇒ true |
---|
1319 | | BIT_ADDR _ ⇒ ¬true (* TO BE COMPLETED *) |
---|
1320 | | N_BIT_ADDR _ ⇒ ¬true (* TO BE COMPLETED *) |
---|
1321 | | RELATIVE _ ⇒ true |
---|
1322 | | ADDR11 _ ⇒ true |
---|
1323 | | ADDR16 _ ⇒ true ]. |
---|
1324 | |
---|
1325 | definition next_internal_pseudo_address_map0 ≝ |
---|
1326 | λT. |
---|
1327 | λcm:T. |
---|
1328 | λaddr_of: Identifier → PreStatus T cm → Word. |
---|
1329 | λfetched. |
---|
1330 | λM: internal_pseudo_address_map. |
---|
1331 | λs: PreStatus T cm. |
---|
1332 | match fetched with |
---|
1333 | [ Comment _ ⇒ Some ? M |
---|
1334 | | Cost _ ⇒ Some … M |
---|
1335 | | Jmp _ ⇒ Some … M |
---|
1336 | | Call a ⇒ |
---|
1337 | let a' ≝ addr_of a s in |
---|
1338 | let 〈callM, accM〉 ≝ M in |
---|
1339 | Some … 〈〈(add 8 (get_8051_sfr ?? s SFR_SP) (bitvector_of_nat 8 1)), 〈false, a'〉〉:: |
---|
1340 | 〈(add 8 (get_8051_sfr ?? s SFR_SP) (bitvector_of_nat 8 2)), 〈true, a'〉〉::callM, accM〉 |
---|
1341 | | Mov _ _ ⇒ Some … M |
---|
1342 | | Instruction instr ⇒ |
---|
1343 | match instr with |
---|
1344 | [ ADD addr1 addr2 ⇒ |
---|
1345 | if addressing_mode_ok T M … s addr1 ∧ addressing_mode_ok T M … s addr2 then |
---|
1346 | Some ? M |
---|
1347 | else |
---|
1348 | None ? |
---|
1349 | | ADDC addr1 addr2 ⇒ |
---|
1350 | if addressing_mode_ok T M … s addr1 ∧ addressing_mode_ok T M … s addr2 then |
---|
1351 | Some ? M |
---|
1352 | else |
---|
1353 | None ? |
---|
1354 | | SUBB addr1 addr2 ⇒ |
---|
1355 | if addressing_mode_ok T M … s addr1 ∧ addressing_mode_ok T M … s addr2 then |
---|
1356 | Some ? M |
---|
1357 | else |
---|
1358 | None ? |
---|
1359 | | _ ⇒ (* TO BE COMPLETED *) Some ? M ]]. |
---|
1360 | |
---|
1361 | definition next_internal_pseudo_address_map ≝ |
---|
1362 | λM:internal_pseudo_address_map. |
---|
1363 | λcm. |
---|
1364 | λaddr_of. |
---|
1365 | λs:PseudoStatus cm. |
---|
1366 | λppc_ok. |
---|
1367 | next_internal_pseudo_address_map0 ? cm addr_of |
---|
1368 | (\fst (fetch_pseudo_instruction (\snd cm) (program_counter … s) ppc_ok)) M s. |
---|
1369 | |
---|
1370 | definition code_memory_of_pseudo_assembly_program: |
---|
1371 | ∀pap:pseudo_assembly_program. |
---|
1372 | (Word → Word) → (Word → bool) → BitVectorTrie Byte 16 ≝ |
---|
1373 | λpap. |
---|
1374 | λsigma. |
---|
1375 | λpolicy. |
---|
1376 | let p ≝ pi1 … (assembly pap sigma policy) in |
---|
1377 | load_code_memory (\fst p). |
---|
1378 | |
---|
1379 | definition sfr_8051_of_pseudo_sfr_8051 ≝ |
---|
1380 | λM: internal_pseudo_address_map. |
---|
1381 | λsfrs: Vector Byte 19. |
---|
1382 | λsigma: Word → Word. |
---|
1383 | match \snd M with |
---|
1384 | [ None ⇒ sfrs |
---|
1385 | | Some s ⇒ |
---|
1386 | let 〈high, address〉 ≝ s in |
---|
1387 | let index ≝ sfr_8051_index SFR_ACC_A in |
---|
1388 | let 〈upper, lower〉 ≝ vsplit ? 8 8 (sigma address) in |
---|
1389 | if high then |
---|
1390 | set_index Byte 19 sfrs index upper ? |
---|
1391 | else |
---|
1392 | set_index Byte 19 sfrs index lower ? |
---|
1393 | ]. |
---|
1394 | // |
---|
1395 | qed. |
---|
1396 | |
---|
1397 | definition status_of_pseudo_status: |
---|
1398 | internal_pseudo_address_map → ∀pap. ∀ps: PseudoStatus pap. |
---|
1399 | ∀sigma: Word → Word. ∀policy: Word → bool. |
---|
1400 | Status (code_memory_of_pseudo_assembly_program pap sigma policy) ≝ |
---|
1401 | λM. |
---|
1402 | λpap. |
---|
1403 | λps. |
---|
1404 | λsigma. |
---|
1405 | λpolicy. |
---|
1406 | let cm ≝ code_memory_of_pseudo_assembly_program … sigma policy in |
---|
1407 | let pc ≝ sigma (program_counter … ps) in |
---|
1408 | let lir ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … ps) in |
---|
1409 | let hir ≝ high_internal_ram_of_pseudo_high_internal_ram M (high_internal_ram … ps) in |
---|
1410 | mk_PreStatus (BitVectorTrie Byte 16) |
---|
1411 | cm |
---|
1412 | lir |
---|
1413 | hir |
---|
1414 | (external_ram … ps) |
---|
1415 | pc |
---|
1416 | (special_function_registers_8051 … ps) |
---|
1417 | (special_function_registers_8052 … ps) |
---|
1418 | (p1_latch … ps) |
---|
1419 | (p3_latch … ps) |
---|
1420 | (clock … ps). |
---|
1421 | |
---|
1422 | (* |
---|
1423 | definition write_at_stack_pointer': |
---|
1424 | ∀M. ∀ps: PreStatus M. Byte → Σps':PreStatus M.(code_memory … ps = code_memory … ps') ≝ |
---|
1425 | λM: Type[0]. |
---|
1426 | λs: PreStatus M. |
---|
1427 | λv: Byte. |
---|
1428 | let 〈 nu, nl 〉 ≝ vsplit … 4 4 (get_8051_sfr ? s SFR_SP) in |
---|
1429 | let bit_zero ≝ get_index_v… nu O ? in |
---|
1430 | let bit_1 ≝ get_index_v… nu 1 ? in |
---|
1431 | let bit_2 ≝ get_index_v… nu 2 ? in |
---|
1432 | let bit_3 ≝ get_index_v… nu 3 ? in |
---|
1433 | if bit_zero then |
---|
1434 | let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl) |
---|
1435 | v (low_internal_ram ? s) in |
---|
1436 | set_low_internal_ram ? s memory |
---|
1437 | else |
---|
1438 | let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl) |
---|
1439 | v (high_internal_ram ? s) in |
---|
1440 | set_high_internal_ram ? s memory. |
---|
1441 | [ cases l0 % |
---|
1442 | |2,3,4,5: normalize repeat (@ le_S_S) @ le_O_n ] |
---|
1443 | qed. |
---|
1444 | |
---|
1445 | definition execute_1_pseudo_instruction': (Word → nat) → ∀ps:PseudoStatus. |
---|
1446 | Σps':PseudoStatus.(code_memory … ps = code_memory … ps') |
---|
1447 | ≝ |
---|
1448 | λticks_of. |
---|
1449 | λs. |
---|
1450 | let 〈instr, pc〉 ≝ fetch_pseudo_instruction (\snd (code_memory ? s)) (program_counter ? s) in |
---|
1451 | let ticks ≝ ticks_of (program_counter ? s) in |
---|
1452 | let s ≝ set_clock ? s (clock ? s + ticks) in |
---|
1453 | let s ≝ set_program_counter ? s pc in |
---|
1454 | match instr with |
---|
1455 | [ Instruction instr ⇒ |
---|
1456 | execute_1_preinstruction … (λx, y. address_of_word_labels y x) instr s |
---|
1457 | | Comment cmt ⇒ s |
---|
1458 | | Cost cst ⇒ s |
---|
1459 | | Jmp jmp ⇒ set_program_counter ? s (address_of_word_labels s jmp) |
---|
1460 | | Call call ⇒ |
---|
1461 | let a ≝ address_of_word_labels s call in |
---|
1462 | let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in |
---|
1463 | let s ≝ set_8051_sfr ? s SFR_SP new_sp in |
---|
1464 | let 〈pc_bu, pc_bl〉 ≝ vsplit ? 8 8 (program_counter ? s) in |
---|
1465 | let s ≝ write_at_stack_pointer' ? s pc_bl in |
---|
1466 | let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in |
---|
1467 | let s ≝ set_8051_sfr ? s SFR_SP new_sp in |
---|
1468 | let s ≝ write_at_stack_pointer' ? s pc_bu in |
---|
1469 | set_program_counter ? s a |
---|
1470 | | Mov dptr ident ⇒ |
---|
1471 | set_arg_16 ? s (get_arg_16 ? s (DATA16 (address_of_word_labels s ident))) dptr |
---|
1472 | ]. |
---|
1473 | [ |
---|
1474 | |2,3,4: % |
---|
1475 | | <(sig2 … l7) whd in ⊢ (??? (??%)) <(sig2 … l5) % |
---|
1476 | | |
---|
1477 | | % |
---|
1478 | ] |
---|
1479 | cases not_implemented |
---|
1480 | qed. |
---|
1481 | *) |
---|
1482 | |
---|
1483 | (* |
---|
1484 | lemma execute_code_memory_unchanged: |
---|
1485 | ∀ticks_of,ps. code_memory ? ps = code_memory ? (execute_1_pseudo_instruction ticks_of ps). |
---|
1486 | #ticks #ps whd in ⊢ (??? (??%)) |
---|
1487 | cases (fetch_pseudo_instruction (\snd (code_memory pseudo_assembly_program ps)) |
---|
1488 | (program_counter pseudo_assembly_program ps)) #instr #pc |
---|
1489 | whd in ⊢ (??? (??%)) cases instr |
---|
1490 | [ #pre cases pre |
---|
1491 | [ #a1 #a2 whd in ⊢ (??? (??%)) cases (add_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%)) |
---|
1492 | cases (vsplit ????) #z1 #z2 % |
---|
1493 | | #a1 #a2 whd in ⊢ (??? (??%)) cases (add_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%)) |
---|
1494 | cases (vsplit ????) #z1 #z2 % |
---|
1495 | | #a1 #a2 whd in ⊢ (??? (??%)) cases (sub_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%)) |
---|
1496 | cases (vsplit ????) #z1 #z2 % |
---|
1497 | | #a1 whd in ⊢ (??? (??%)) cases a1 #x #H whd in ⊢ (??? (??%)) cases x |
---|
1498 | [ #x1 whd in ⊢ (??? (??%)) |
---|
1499 | | *: cases not_implemented |
---|
1500 | ] |
---|
1501 | | #comment % |
---|
1502 | | #cost % |
---|
1503 | | #label % |
---|
1504 | | #label whd in ⊢ (??? (??%)) cases (half_add ???) #x1 #x2 whd in ⊢ (??? (??%)) |
---|
1505 | cases (vsplit ????) #y1 #y2 whd in ⊢ (??? (??%)) cases (half_add ???) #z1 #z2 |
---|
1506 | whd in ⊢ (??? (??%)) whd in ⊢ (??? (??%)) cases (vsplit ????) #w1 #w2 |
---|
1507 | whd in ⊢ (??? (??%)) cases (get_index_v bool ????) whd in ⊢ (??? (??%)) |
---|
1508 | (* CSC: ??? *) |
---|
1509 | | #dptr #label (* CSC: ??? *) |
---|
1510 | ] |
---|
1511 | cases not_implemented |
---|
1512 | qed. |
---|
1513 | *) |
---|
1514 | |
---|
1515 | (* DEAD CODE? |
---|
1516 | lemma status_of_pseudo_status_failure_depends_only_on_code_memory: |
---|
1517 | ∀M:internal_pseudo_address_map. |
---|
1518 | ∀ps,ps': PseudoStatus. |
---|
1519 | ∀pol. |
---|
1520 | ∀prf:code_memory … ps = code_memory … ps'. |
---|
1521 | let pol' ≝ ? in |
---|
1522 | match status_of_pseudo_status M ps pol with |
---|
1523 | [ None ⇒ status_of_pseudo_status M ps' pol' = None … |
---|
1524 | | Some _ ⇒ ∃w. status_of_pseudo_status M ps' pol' = Some … w |
---|
1525 | ]. |
---|
1526 | [2: <prf @pol] |
---|
1527 | #M #ps #ps' #pol #H normalize nodelta; whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ? ]) |
---|
1528 | generalize in match (refl … (assembly (code_memory … ps) pol)) |
---|
1529 | cases (assembly ??) in ⊢ (???% → %) |
---|
1530 | [ #K whd whd in ⊢ (??%?) <H >K % |
---|
1531 | | #x #K whd whd in ⊢ (?? (λ_.??%?)) <H >K % [2: % ] ] |
---|
1532 | qed. |
---|
1533 | *) |
---|
1534 | |
---|
1535 | definition ticks_of0: |
---|
1536 | ∀p:pseudo_assembly_program. |
---|
1537 | (Word → Word) → (Word → bool) → Word → pseudo_instruction → nat × nat ≝ |
---|
1538 | λprogram: pseudo_assembly_program. |
---|
1539 | λsigma. |
---|
1540 | λpolicy. |
---|
1541 | λppc: Word. |
---|
1542 | λfetched. |
---|
1543 | match fetched with |
---|
1544 | [ Instruction instr ⇒ |
---|
1545 | match instr with |
---|
1546 | [ JC lbl ⇒ ? (* |
---|
1547 | match pol lookup_labels ppc with |
---|
1548 | [ short_jump ⇒ 〈2, 2〉 |
---|
1549 | | absolute_jump ⇒ ? |
---|
1550 | | long_jump ⇒ 〈4, 4〉 |
---|
1551 | ] *) |
---|
1552 | | JNC lbl ⇒ ? (* |
---|
1553 | match pol lookup_labels ppc with |
---|
1554 | [ short_jump ⇒ 〈2, 2〉 |
---|
1555 | | absolute_jump ⇒ ? |
---|
1556 | | long_jump ⇒ 〈4, 4〉 |
---|
1557 | ] *) |
---|
1558 | | JB bit lbl ⇒ ? (* |
---|
1559 | match pol lookup_labels ppc with |
---|
1560 | [ short_jump ⇒ 〈2, 2〉 |
---|
1561 | | absolute_jump ⇒ ? |
---|
1562 | | long_jump ⇒ 〈4, 4〉 |
---|
1563 | ] *) |
---|
1564 | | JNB bit lbl ⇒ ? (* |
---|
1565 | match pol lookup_labels ppc with |
---|
1566 | [ short_jump ⇒ 〈2, 2〉 |
---|
1567 | | absolute_jump ⇒ ? |
---|
1568 | | long_jump ⇒ 〈4, 4〉 |
---|
1569 | ] *) |
---|
1570 | | JBC bit lbl ⇒ ? (* |
---|
1571 | match pol lookup_labels ppc with |
---|
1572 | [ short_jump ⇒ 〈2, 2〉 |
---|
1573 | | absolute_jump ⇒ ? |
---|
1574 | | long_jump ⇒ 〈4, 4〉 |
---|
1575 | ] *) |
---|
1576 | | JZ lbl ⇒ ? (* |
---|
1577 | match pol lookup_labels ppc with |
---|
1578 | [ short_jump ⇒ 〈2, 2〉 |
---|
1579 | | absolute_jump ⇒ ? |
---|
1580 | | long_jump ⇒ 〈4, 4〉 |
---|
1581 | ] *) |
---|
1582 | | JNZ lbl ⇒ ? (* |
---|
1583 | match pol lookup_labels ppc with |
---|
1584 | [ short_jump ⇒ 〈2, 2〉 |
---|
1585 | | absolute_jump ⇒ ? |
---|
1586 | | long_jump ⇒ 〈4, 4〉 |
---|
1587 | ] *) |
---|
1588 | | CJNE arg lbl ⇒ ? (* |
---|
1589 | match pol lookup_labels ppc with |
---|
1590 | [ short_jump ⇒ 〈2, 2〉 |
---|
1591 | | absolute_jump ⇒ ? |
---|
1592 | | long_jump ⇒ 〈4, 4〉 |
---|
1593 | ] *) |
---|
1594 | | DJNZ arg lbl ⇒ ? (* |
---|
1595 | match pol lookup_labels ppc with |
---|
1596 | [ short_jump ⇒ 〈2, 2〉 |
---|
1597 | | absolute_jump ⇒ ? |
---|
1598 | | long_jump ⇒ 〈4, 4〉 |
---|
1599 | ] *) |
---|
1600 | | ADD arg1 arg2 ⇒ |
---|
1601 | let ticks ≝ ticks_of_instruction (ADD ? arg1 arg2) in |
---|
1602 | 〈ticks, ticks〉 |
---|
1603 | | ADDC arg1 arg2 ⇒ |
---|
1604 | let ticks ≝ ticks_of_instruction (ADDC ? arg1 arg2) in |
---|
1605 | 〈ticks, ticks〉 |
---|
1606 | | SUBB arg1 arg2 ⇒ |
---|
1607 | let ticks ≝ ticks_of_instruction (SUBB ? arg1 arg2) in |
---|
1608 | 〈ticks, ticks〉 |
---|
1609 | | INC arg ⇒ |
---|
1610 | let ticks ≝ ticks_of_instruction (INC ? arg) in |
---|
1611 | 〈ticks, ticks〉 |
---|
1612 | | DEC arg ⇒ |
---|
1613 | let ticks ≝ ticks_of_instruction (DEC ? arg) in |
---|
1614 | 〈ticks, ticks〉 |
---|
1615 | | MUL arg1 arg2 ⇒ |
---|
1616 | let ticks ≝ ticks_of_instruction (MUL ? arg1 arg2) in |
---|
1617 | 〈ticks, ticks〉 |
---|
1618 | | DIV arg1 arg2 ⇒ |
---|
1619 | let ticks ≝ ticks_of_instruction (DIV ? arg1 arg2) in |
---|
1620 | 〈ticks, ticks〉 |
---|
1621 | | DA arg ⇒ |
---|
1622 | let ticks ≝ ticks_of_instruction (DA ? arg) in |
---|
1623 | 〈ticks, ticks〉 |
---|
1624 | | ANL arg ⇒ |
---|
1625 | let ticks ≝ ticks_of_instruction (ANL ? arg) in |
---|
1626 | 〈ticks, ticks〉 |
---|
1627 | | ORL arg ⇒ |
---|
1628 | let ticks ≝ ticks_of_instruction (ORL ? arg) in |
---|
1629 | 〈ticks, ticks〉 |
---|
1630 | | XRL arg ⇒ |
---|
1631 | let ticks ≝ ticks_of_instruction (XRL ? arg) in |
---|
1632 | 〈ticks, ticks〉 |
---|
1633 | | CLR arg ⇒ |
---|
1634 | let ticks ≝ ticks_of_instruction (CLR ? arg) in |
---|
1635 | 〈ticks, ticks〉 |
---|
1636 | | CPL arg ⇒ |
---|
1637 | let ticks ≝ ticks_of_instruction (CPL ? arg) in |
---|
1638 | 〈ticks, ticks〉 |
---|
1639 | | RL arg ⇒ |
---|
1640 | let ticks ≝ ticks_of_instruction (RL ? arg) in |
---|
1641 | 〈ticks, ticks〉 |
---|
1642 | | RLC arg ⇒ |
---|
1643 | let ticks ≝ ticks_of_instruction (RLC ? arg) in |
---|
1644 | 〈ticks, ticks〉 |
---|
1645 | | RR arg ⇒ |
---|
1646 | let ticks ≝ ticks_of_instruction (RR ? arg) in |
---|
1647 | 〈ticks, ticks〉 |
---|
1648 | | RRC arg ⇒ |
---|
1649 | let ticks ≝ ticks_of_instruction (RRC ? arg) in |
---|
1650 | 〈ticks, ticks〉 |
---|
1651 | | SWAP arg ⇒ |
---|
1652 | let ticks ≝ ticks_of_instruction (SWAP ? arg) in |
---|
1653 | 〈ticks, ticks〉 |
---|
1654 | | MOV arg ⇒ |
---|
1655 | let ticks ≝ ticks_of_instruction (MOV ? arg) in |
---|
1656 | 〈ticks, ticks〉 |
---|
1657 | | MOVX arg ⇒ |
---|
1658 | let ticks ≝ ticks_of_instruction (MOVX ? arg) in |
---|
1659 | 〈ticks, ticks〉 |
---|
1660 | | SETB arg ⇒ |
---|
1661 | let ticks ≝ ticks_of_instruction (SETB ? arg) in |
---|
1662 | 〈ticks, ticks〉 |
---|
1663 | | PUSH arg ⇒ |
---|
1664 | let ticks ≝ ticks_of_instruction (PUSH ? arg) in |
---|
1665 | 〈ticks, ticks〉 |
---|
1666 | | POP arg ⇒ |
---|
1667 | let ticks ≝ ticks_of_instruction (POP ? arg) in |
---|
1668 | 〈ticks, ticks〉 |
---|
1669 | | XCH arg1 arg2 ⇒ |
---|
1670 | let ticks ≝ ticks_of_instruction (XCH ? arg1 arg2) in |
---|
1671 | 〈ticks, ticks〉 |
---|
1672 | | XCHD arg1 arg2 ⇒ |
---|
1673 | let ticks ≝ ticks_of_instruction (XCHD ? arg1 arg2) in |
---|
1674 | 〈ticks, ticks〉 |
---|
1675 | | RET ⇒ |
---|
1676 | let ticks ≝ ticks_of_instruction (RET ?) in |
---|
1677 | 〈ticks, ticks〉 |
---|
1678 | | RETI ⇒ |
---|
1679 | let ticks ≝ ticks_of_instruction (RETI ?) in |
---|
1680 | 〈ticks, ticks〉 |
---|
1681 | | NOP ⇒ |
---|
1682 | let ticks ≝ ticks_of_instruction (NOP ?) in |
---|
1683 | 〈ticks, ticks〉 |
---|
1684 | ] |
---|
1685 | | Comment comment ⇒ 〈0, 0〉 |
---|
1686 | | Cost cost ⇒ 〈0, 0〉 |
---|
1687 | | Jmp jmp ⇒ 〈2, 2〉 |
---|
1688 | | Call call ⇒ 〈2, 2〉 |
---|
1689 | | Mov dptr tgt ⇒ 〈2, 2〉 |
---|
1690 | ]. |
---|
1691 | cases daemon |
---|
1692 | qed. |
---|
1693 | |
---|
1694 | definition ticks_of: |
---|
1695 | ∀p:pseudo_assembly_program. |
---|
1696 | (Word → Word) → (Word → bool) → ∀ppc:Word. |
---|
1697 | nat_of_bitvector … ppc < |\snd p| → nat × nat ≝ |
---|
1698 | λprogram: pseudo_assembly_program. |
---|
1699 | λsigma. |
---|
1700 | λpolicy. |
---|
1701 | λppc: Word. λppc_ok. |
---|
1702 | let pseudo ≝ \snd program in |
---|
1703 | let 〈fetched, new_ppc〉 ≝ fetch_pseudo_instruction pseudo ppc ppc_ok in |
---|
1704 | ticks_of0 program sigma policy ppc fetched. |
---|
1705 | |
---|
1706 | lemma eq_rect_Type1_r: |
---|
1707 | ∀A: Type[1]. |
---|
1708 | ∀a: A. |
---|
1709 | ∀P: ∀x:A. eq ? x a → Type[1]. P a (refl A a) → ∀x: A.∀p:eq ? x a. P x p. |
---|
1710 | #A #a #P #H #x #p |
---|
1711 | generalize in match H; |
---|
1712 | generalize in match P; |
---|
1713 | cases p // |
---|
1714 | qed. |
---|
1715 | |
---|
1716 | axiom vsplit_append: |
---|
1717 | ∀A: Type[0]. |
---|
1718 | ∀m, n: nat. |
---|
1719 | ∀v, v': Vector A m. |
---|
1720 | ∀q, q': Vector A n. |
---|
1721 | let 〈v', q'〉 ≝ vsplit A m n (v@@q) in |
---|
1722 | v = v' ∧ q = q'. |
---|
1723 | |
---|
1724 | lemma vsplit_vector_singleton: |
---|
1725 | ∀A: Type[0]. |
---|
1726 | ∀n: nat. |
---|
1727 | ∀v: Vector A (S n). |
---|
1728 | ∀rest: Vector A n. |
---|
1729 | ∀s: Vector A 1. |
---|
1730 | v = s @@ rest → |
---|
1731 | ((get_index_v A ? v 0 ?) ::: rest) = v. |
---|
1732 | [1: |
---|
1733 | #A #n #v cases daemon (* XXX: !!! *) |
---|
1734 | |2: |
---|
1735 | @le_S_S @le_O_n |
---|
1736 | ] |
---|
1737 | qed. |
---|
1738 | |
---|
1739 | example sub_minus_one_seven_eight: |
---|
1740 | ∀v: BitVector 7. |
---|
1741 | false ::: (\fst (sub_7_with_carry v (bitvector_of_nat ? 1) false)) = |
---|
1742 | \fst (sub_8_with_carry (false ::: v) (bitvector_of_nat ? 1) false). |
---|
1743 | cases daemon. |
---|
1744 | qed. |
---|
1745 | |
---|
1746 | (* |
---|
1747 | lemma blah: |
---|
1748 | ∀m: internal_pseudo_address_map. |
---|
1749 | ∀s: PseudoStatus. |
---|
1750 | ∀arg: Byte. |
---|
1751 | ∀b: bool. |
---|
1752 | addressing_mode_ok m s (DIRECT arg) = true → |
---|
1753 | get_arg_8 ? (set_low_internal_ram ? s (low_internal_ram_of_pseudo_low_internal_ram m (low_internal_ram ? s))) b (DIRECT arg) = |
---|
1754 | get_arg_8 ? s b (DIRECT arg). |
---|
1755 | [2, 3: normalize % ] |
---|
1756 | #m #s #arg #b #hyp |
---|
1757 | whd in ⊢ (??%%) |
---|
1758 | @vsplit_elim'' |
---|
1759 | #nu' #nl' #arg_nu_nl_eq |
---|
1760 | normalize nodelta |
---|
1761 | generalize in match (refl ? (get_index_v bool 4 nu' ? ?)) |
---|
1762 | cases (get_index_v bool 4 nu' ? ?) in ⊢ (??%? → %) |
---|
1763 | #get_index_v_eq |
---|
1764 | normalize nodelta |
---|
1765 | [2: |
---|
1766 | normalize nodelta |
---|
1767 | @vsplit_elim'' |
---|
1768 | #bit_one' #three_bits' #bit_one_three_bit_eq |
---|
1769 | generalize in match (low_internal_ram_of_pseudo_internal_ram_miss m s (three_bits'@@nl')) |
---|
1770 | normalize nodelta |
---|
1771 | generalize in match (refl ? (sub_7_with_carry ? ? ?)) |
---|
1772 | cases (sub_7_with_carry ? ? ?) in ⊢ (??%? → %) |
---|
1773 | #Saddr #carr' #Saddr_carr_eq |
---|
1774 | normalize nodelta |
---|
1775 | #carr_hyp' |
---|
1776 | @carr_hyp' |
---|
1777 | [1: |
---|
1778 | |2: whd in hyp:(??%?); generalize in match hyp; -hyp; |
---|
1779 | generalize in match (refl ? (¬(member (BitVector 8) ? arg m))) |
---|
1780 | cases (¬(member (BitVector 8) ? arg m)) in ⊢ (??%? → %) |
---|
1781 | #member_eq |
---|
1782 | normalize nodelta |
---|
1783 | [2: #destr destruct(destr) |
---|
1784 | |1: -carr_hyp'; |
---|
1785 | >arg_nu_nl_eq |
---|
1786 | <(vsplit_vector_singleton ? ? nu' ? ? ? bit_one_three_bit_eq) |
---|
1787 | [1: >get_index_v_eq in ⊢ (??%? → ?) |
---|
1788 | |2: @le_S @le_S @le_S @le_n |
---|
1789 | ] |
---|
1790 | cases (member (BitVector 8) ? (\fst ?) ?) |
---|
1791 | [1: #destr normalize in destr; destruct(destr) |
---|
1792 | |2: |
---|
1793 | ] |
---|
1794 | ] |
---|
1795 | |3: >get_index_v_eq in ⊢ (??%?) |
---|
1796 | change in ⊢ (??(???%?)?) with ((? ::: three_bits') @@ nl') |
---|
1797 | >(vsplit_vector_singleton … bit_one_three_bit_eq) |
---|
1798 | <arg_nu_nl_eq |
---|
1799 | whd in hyp:(??%?) |
---|
1800 | cases (member (BitVector 8) (eq_bv 8) arg m) in hyp |
---|
1801 | normalize nodelta [*: #ignore @sym_eq ] |
---|
1802 | ] |
---|
1803 | | |
---|
1804 | ]. |
---|
1805 | *) |
---|
1806 | (* |
---|
1807 | map_address0 ... (DIRECT arg) = Some .. → |
---|
1808 | get_arg_8 (map_address0 ... (internal_ram ...) (DIRECT arg) = |
---|
1809 | get_arg_8 (internal_ram ...) (DIRECT arg) |
---|
1810 | |
---|
1811 | ((if addressing_mode_ok M ps ACC_A∧addressing_mode_ok M ps (DIRECT ARG2) |
---|
1812 | then Some internal_pseudo_address_map M |
---|
1813 | else None internal_pseudo_address_map ) |
---|
1814 | =Some internal_pseudo_address_map M') |
---|
1815 | |
---|
1816 | axiom low_internal_ram_write_at_stack_pointer: |
---|
1817 | ∀T1,T2,M,cm1,s1,cm2,s2,cm3,s3.∀sigma: Word → Word.∀policy: Word → bool. |
---|
1818 | ∀pbu,pbl,bu,bl,sp1,sp2:BitVector 8. |
---|
1819 | get_8051_sfr T2 cm2 s2 SFR_SP = get_8051_sfr ? cm3 s3 SFR_SP → |
---|
1820 | low_internal_ram ? cm2 s2 = low_internal_ram T2 cm3 s3 → |
---|
1821 | sp1 = add ? (get_8051_sfr … cm1 s1 SFR_SP) (bitvector_of_nat 8 1) → |
---|
1822 | sp2 = add ? sp1 (bitvector_of_nat 8 1) → |
---|
1823 | bu@@bl = sigma (pbu@@pbl) → |
---|
1824 | low_internal_ram T1 cm1 |
---|
1825 | (write_at_stack_pointer … |
---|
1826 | (set_8051_sfr … |
---|
1827 | (write_at_stack_pointer … |
---|
1828 | (set_8051_sfr … |
---|
1829 | (set_low_internal_ram … s1 |
---|
1830 | (low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … s2))) |
---|
1831 | SFR_SP sp1) |
---|
1832 | bl) |
---|
1833 | SFR_SP sp2) |
---|
1834 | bu) |
---|
1835 | = low_internal_ram_of_pseudo_low_internal_ram (sp1::M) |
---|
1836 | (low_internal_ram … |
---|
1837 | (write_at_stack_pointer … |
---|
1838 | (set_8051_sfr … |
---|
1839 | (write_at_stack_pointer … (set_8051_sfr … s3 SFR_SP sp1) pbl) |
---|
1840 | SFR_SP sp2) |
---|
1841 | pbu)). |
---|
1842 | |
---|
1843 | lemma high_internal_ram_write_at_stack_pointer: |
---|
1844 | ∀T1,T2,M,cm1,s1,cm2,s2,cm3,s3.∀sigma:Word → Word.∀policy: Word → bool. |
---|
1845 | ∀pbu,pbl,bu,bl,sp1,sp2:BitVector 8. |
---|
1846 | get_8051_sfr T2 cm2 s2 SFR_SP = get_8051_sfr ? cm3 s3 SFR_SP → |
---|
1847 | high_internal_ram ?? s2 = high_internal_ram T2 cm3 s3 → |
---|
1848 | sp1 = add ? (get_8051_sfr ? cm1 s1 SFR_SP) (bitvector_of_nat 8 1) → |
---|
1849 | sp2 = add ? sp1 (bitvector_of_nat 8 1) → |
---|
1850 | bu@@bl = sigma (pbu@@pbl) → |
---|
1851 | high_internal_ram T1 cm1 |
---|
1852 | (write_at_stack_pointer … |
---|
1853 | (set_8051_sfr … |
---|
1854 | (write_at_stack_pointer … |
---|
1855 | (set_8051_sfr … |
---|
1856 | (set_high_internal_ram … s1 |
---|
1857 | (high_internal_ram_of_pseudo_high_internal_ram M (high_internal_ram … s2))) |
---|
1858 | SFR_SP sp1) |
---|
1859 | bl) |
---|
1860 | SFR_SP sp2) |
---|
1861 | bu) |
---|
1862 | = high_internal_ram_of_pseudo_high_internal_ram (sp1::M) |
---|
1863 | (high_internal_ram … |
---|
1864 | (write_at_stack_pointer … |
---|
1865 | (set_8051_sfr … |
---|
1866 | (write_at_stack_pointer … (set_8051_sfr … s3 SFR_SP sp1) pbl) |
---|
1867 | SFR_SP sp2) |
---|
1868 | pbu)). |
---|
1869 | #T1 #T2 #M #cm1 #s1 #cm2 #s2 #cm3 #s3 #sigma #policy #pbu #pbl #bu #bl #sp1 #sp2 |
---|
1870 | #get_8051_sfr_refl #high_internal_ram_refl #sp1_refl #sp2_refl #sigma_refl |
---|
1871 | cases daemon (* XXX: !!! *) |
---|
1872 | qed. |
---|
1873 | *) |
---|
1874 | |
---|
1875 | lemma Some_Some_elim: |
---|
1876 | ∀T:Type[0].∀x,y:T.∀P:Type[2]. (x=y → P) → Some T x = Some T y → P. |
---|
1877 | #T #x #y #P #H #K @H @option_destruct_Some // |
---|
1878 | qed. |
---|
1879 | |
---|
1880 | lemma pair_destruct_right: |
---|
1881 | ∀A: Type[0]. |
---|
1882 | ∀B: Type[0]. |
---|
1883 | ∀a, c: A. |
---|
1884 | ∀b, d: B. |
---|
1885 | 〈a, b〉 = 〈c, d〉 → b = d. |
---|
1886 | #A #B #a #b #c #d // |
---|
1887 | qed. |
---|
1888 | |
---|
1889 | (*CSC: ???*) |
---|
1890 | (* XXX: we need a precondition here stating that the PPC is within the |
---|
1891 | bounds of the instruction list in order for Jaap's specification to |
---|
1892 | apply. |
---|
1893 | *) |
---|
1894 | lemma snd_assembly_1_pseudoinstruction_ok: |
---|
1895 | ∀program: pseudo_assembly_program. |
---|
1896 | ∀program_length_proof: |\snd program| ≤ 2^16. |
---|
1897 | ∀sigma: Word → Word. |
---|
1898 | ∀policy: Word → bool. |
---|
1899 | ∀sigma_policy_specification_witness: sigma_policy_specification program sigma policy. |
---|
1900 | ∀ppc: Word. |
---|
1901 | ∀ppc_in_bounds: nat_of_bitvector 16 ppc < |\snd program|. |
---|
1902 | ∀pi. |
---|
1903 | ∀lookup_labels. |
---|
1904 | ∀lookup_datalabels. |
---|
1905 | lookup_labels = (λx. (address_of_word_labels_code_mem (\snd program) x)) → |
---|
1906 | lookup_datalabels = (λx. lookup_def … (construct_datalabels (\fst program)) x (zero ?)) → |
---|
1907 | \fst (fetch_pseudo_instruction (\snd program) ppc ppc_in_bounds) = pi → |
---|
1908 | let len ≝ \fst (assembly_1_pseudoinstruction lookup_labels sigma policy (*(sigma ppc)*) ppc lookup_datalabels pi) in |
---|
1909 | sigma (add … ppc (bitvector_of_nat ? 1)) = add … (sigma ppc) (bitvector_of_nat ? len). |
---|
1910 | #program #program_length_proof #sigma #policy #sigma_policy_specification_witness #ppc #ppc_in_bounds #pi |
---|
1911 | #lookup_labels #lookup_datalabels |
---|
1912 | #lookup_labels_refl #lookup_datalabels_refl #fetch_pseudo_refl |
---|
1913 | normalize nodelta |
---|
1914 | generalize in match fetch_pseudo_refl; -fetch_pseudo_refl |
---|
1915 | #fetch_pseudo_refl |
---|
1916 | letin assembled ≝ (\fst (pi1 … (assembly program sigma policy))) |
---|
1917 | letin costs ≝ (\snd (pi1 … (assembly program sigma policy))) |
---|
1918 | lapply (assembly_ok program program_length_proof sigma policy sigma_policy_specification_witness assembled costs) |
---|
1919 | @pair_elim #preamble #instr_list #program_refl |
---|
1920 | @pair_elim #labels #costs' #create_label_cost_map_refl |
---|
1921 | <eq_pair_fst_snd #H lapply (H (refl …)) -H #H |
---|
1922 | lapply (H ppc ppc_in_bounds) -H |
---|
1923 | @pair_elim #pi' #newppc #fetch_pseudo_refl' |
---|
1924 | @pair_elim #len #assembled #assembly1_refl #H |
---|
1925 | cases H |
---|
1926 | #encoding_check_assm #sigma_newppc_refl |
---|
1927 | >fetch_pseudo_refl' in fetch_pseudo_refl; #pi_refl' |
---|
1928 | >pi_refl' in assembly1_refl; #assembly1_refl |
---|
1929 | >lookup_labels_refl >lookup_datalabels_refl |
---|
1930 | >program_refl normalize nodelta |
---|
1931 | >assembly1_refl |
---|
1932 | <sigma_newppc_refl |
---|
1933 | generalize in match fetch_pseudo_refl'; |
---|
1934 | whd in match (fetch_pseudo_instruction ???); |
---|
1935 | @pair_elim #lbl #instr #nth_refl normalize nodelta |
---|
1936 | #G cases (pair_destruct_right ?????? G) % |
---|
1937 | qed. |
---|
1938 | |
---|
1939 | lemma pose: ∀A:Type[0].∀B:A → Type[0].∀a:A. (∀a':A. a'=a → B a') → B a. |
---|
1940 | /2/ |
---|
1941 | qed. |
---|
1942 | |
---|
1943 | (* To be moved in ProofStatus *) |
---|
1944 | lemma program_counter_set_program_counter: |
---|
1945 | ∀T. |
---|
1946 | ∀cm. |
---|
1947 | ∀s. |
---|
1948 | ∀x. |
---|
1949 | program_counter T cm (set_program_counter T cm s x) = x. |
---|
1950 | // |
---|
1951 | qed. |
---|
1952 | |
---|
1953 | (* XXX: easy but tedious *) |
---|
1954 | lemma assembly1_lt_128: |
---|
1955 | ∀i: instruction. |
---|
1956 | |(assembly1 i)| < 128. |
---|
1957 | #i cases i |
---|
1958 | try (#assm1 #assm2) try #assm1 |
---|
1959 | [8: |
---|
1960 | cases assm1 |
---|
1961 | try (#assm1 #assm2) try #assm1 |
---|
1962 | whd in match assembly1; normalize nodelta |
---|
1963 | whd in match assembly_preinstruction; normalize nodelta |
---|
1964 | try @(subaddressing_mode_elim … assm2) |
---|
1965 | try @(subaddressing_mode_elim … assm1) try #w try #w' normalize nodelta |
---|
1966 | [32: |
---|
1967 | cases assm1 -assm1 #assm1 normalize nodelta |
---|
1968 | cases assm1 #addr1 #addr2 normalize nodelta |
---|
1969 | [1: |
---|
1970 | @(subaddressing_mode_elim … addr2) |
---|
1971 | |2: |
---|
1972 | @(subaddressing_mode_elim … addr1) |
---|
1973 | ] |
---|
1974 | #w |
---|
1975 | |35,36,37: |
---|
1976 | cases assm1 -assm1 #assm1 normalize nodelta |
---|
1977 | [1,3: |
---|
1978 | cases assm1 -assm1 #assm1 normalize nodelta |
---|
1979 | ] |
---|
1980 | cases assm1 #addr1 #addr2 normalize nodelta |
---|
1981 | @(subaddressing_mode_elim … addr2) try #w |
---|
1982 | |49: |
---|
1983 | cases assm1 -assm1 #assm1 normalize nodelta |
---|
1984 | [1: |
---|
1985 | cases assm1 -assm1 #assm1 normalize nodelta |
---|
1986 | [1: |
---|
1987 | cases assm1 -assm1 #assm1 normalize nodelta |
---|
1988 | [1: |
---|
1989 | cases assm1 -assm1 #assm1 normalize nodelta |
---|
1990 | [1: |
---|
1991 | cases assm1 -assm1 #assm1 normalize nodelta |
---|
1992 | ] |
---|
1993 | ] |
---|
1994 | ] |
---|
1995 | ] |
---|
1996 | cases assm1 #addr1 #addr2 normalize nodelta |
---|
1997 | [1,3,4,5: |
---|
1998 | @(subaddressing_mode_elim … addr2) try #w |
---|
1999 | |*: |
---|
2000 | @(subaddressing_mode_elim … addr1) try #w |
---|
2001 | normalize nodelta |
---|
2002 | [1,2: |
---|
2003 | @(subaddressing_mode_elim … addr2) try #w |
---|
2004 | ] |
---|
2005 | ] |
---|
2006 | |50: |
---|
2007 | cases assm1 -assm1 #assm1 normalize nodelta |
---|
2008 | cases assm1 #addr1 #addr2 normalize nodelta |
---|
2009 | [1: |
---|
2010 | @(subaddressing_mode_elim … addr2) try #w |
---|
2011 | |2: |
---|
2012 | @(subaddressing_mode_elim … addr1) try #w |
---|
2013 | ] |
---|
2014 | ] |
---|
2015 | normalize repeat @le_S_S @le_O_n |
---|
2016 | ] |
---|
2017 | whd in match assembly1; normalize nodelta |
---|
2018 | [6: |
---|
2019 | normalize repeat @le_S_S @le_O_n |
---|
2020 | |7: |
---|
2021 | @(subaddressing_mode_elim … assm2) normalize repeat @le_S_S @le_O_n |
---|
2022 | |*: |
---|
2023 | @(subaddressing_mode_elim … assm1) #w normalize nodelta repeat @le_S_S @le_O_n |
---|
2024 | ] |
---|
2025 | qed. |
---|