1 | include "ASM/ASM.ma". |
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2 | include "ASM/BitVectorTrie.ma". |
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3 | include "ASM/Arithmetic.ma". |
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4 | include "ASM/Fetch.ma". |
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5 | include "ASM/Status.ma". |
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6 | include alias "basics/logic.ma". |
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7 | include alias "arithmetics/nat.ma". |
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8 | |
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9 | definition assembly_preinstruction ≝ |
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10 | λA: Type[0]. |
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11 | λaddr_of: A → Byte. (* relative *) |
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12 | λpre: preinstruction A. |
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13 | match pre with |
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14 | [ ADD addr1 addr2 ⇒ |
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15 | match addr2 return λx. bool_to_Prop (is_in ? [[registr;direct;indirect;data]] x) → ? with |
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16 | [ REGISTER r ⇒ λ_.[ ([[false;false;true;false;true]]) @@ r ] |
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17 | | DIRECT b1 ⇒ λ_.[ ([[false;false;true;false;false;true;false;true]]); b1 ] |
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18 | | INDIRECT i1 ⇒ λ_. [ ([[false;false;true;false;false;true;true;i1]]) ] |
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19 | | DATA b1 ⇒ λ_. [ ([[false;false;true;false;false;true;false;false]]) ; b1 ] |
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20 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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21 | | ADDC addr1 addr2 ⇒ |
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22 | match addr2 return λx. bool_to_Prop (is_in ? [[registr;direct;indirect;data]] x) → ? with |
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23 | [ REGISTER r ⇒ λ_.[ ([[false;false;true;true;true]]) @@ r ] |
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24 | | DIRECT b1 ⇒ λ_.[ ([[false;false;true;true;false;true;false;true]]); b1 ] |
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25 | | INDIRECT i1 ⇒ λ_. [ ([[false;false;true;true;false;true;true;i1]]) ] |
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26 | | DATA b1 ⇒ λ_. [ ([[false;false;true;true;false;true;false;false]]) ; b1 ] |
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27 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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28 | | ANL addrs ⇒ |
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29 | match addrs with |
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30 | [ inl addrs ⇒ match addrs with |
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31 | [ inl addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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32 | match addr2 return λx. bool_to_Prop (is_in ? [[registr;direct;indirect;data]] x) → ? with |
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33 | [ REGISTER r ⇒ λ_.[ ([[false;true;false;true;true]]) @@ r ] |
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34 | | DIRECT b1 ⇒ λ_.[ ([[false;true;false;true;false;true;false;true]]); b1 ] |
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35 | | INDIRECT i1 ⇒ λ_. [ ([[false;true;false;true;false;true;true;i1]]) ] |
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36 | | DATA b1 ⇒ λ_. [ ([[false;true;false;true;false;true;false;false]]) ; b1 ] |
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37 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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38 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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39 | let b1 ≝ |
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40 | match addr1 return λx. bool_to_Prop (is_in ? [[direct]] x) → ? with |
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41 | [ DIRECT b1 ⇒ λ_.b1 |
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42 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) in |
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43 | match addr2 return λx. bool_to_Prop (is_in ? [[acc_a;data]] x) → ? with |
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44 | [ ACC_A ⇒ λ_.[ ([[false;true;false;true;false;false;true;false]]) ; b1 ] |
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45 | | DATA b2 ⇒ λ_. [ ([[false;true;false;true;false;false;true;true]]) ; b1 ; b2 ] |
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46 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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47 | ] |
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48 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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49 | match addr2 return λx. bool_to_Prop (is_in ? [[bit_addr;n_bit_addr]] x) → ? with |
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50 | [ BIT_ADDR b1 ⇒ λ_.[ ([[true;false;false;false;false;false;true;false]]) ; b1 ] |
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51 | | N_BIT_ADDR b1 ⇒ λ_. [ ([[true;false;true;true;false;false;false;false]]) ; b1 ] |
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52 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2)] |
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53 | | CLR addr ⇒ |
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54 | match addr return λx. bool_to_Prop (is_in ? [[acc_a;carry;bit_addr]] x) → ? with |
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55 | [ ACC_A ⇒ λ_. |
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56 | [ ([[true; true; true; false; false; true; false; false]]) ] |
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57 | | CARRY ⇒ λ_. |
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58 | [ ([[true; true; false; false; false; false; true; true]]) ] |
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59 | | BIT_ADDR b1 ⇒ λ_. |
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60 | [ ([[true; true; false; false; false; false; true; false]]) ; b1 ] |
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61 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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62 | | CPL addr ⇒ |
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63 | match addr return λx. bool_to_Prop (is_in ? [[acc_a;carry;bit_addr]] x) → ? with |
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64 | [ ACC_A ⇒ λ_. |
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65 | [ ([[true; true; true; true; false; true; false; false]]) ] |
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66 | | CARRY ⇒ λ_. |
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67 | [ ([[true; false; true; true; false; false; true; true]]) ] |
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68 | | BIT_ADDR b1 ⇒ λ_. |
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69 | [ ([[true; false; true; true; false; false; true; false]]) ; b1 ] |
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70 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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71 | | DA addr ⇒ |
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72 | [ ([[true; true; false; true; false; true; false; false]]) ] |
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73 | | DEC addr ⇒ |
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74 | match addr return λx. bool_to_Prop (is_in ? [[acc_a;registr;direct;indirect]] x) → ? with |
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75 | [ ACC_A ⇒ λ_. |
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76 | [ ([[false; false; false; true; false; true; false; false]]) ] |
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77 | | REGISTER r ⇒ λ_. |
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78 | [ ([[false; false; false; true; true]]) @@ r ] |
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79 | | DIRECT b1 ⇒ λ_. |
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80 | [ ([[false; false; false; true; false; true; false; true]]); b1 ] |
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81 | | INDIRECT i1 ⇒ λ_. |
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82 | [ ([[false; false; false; true; false; true; true; i1]]) ] |
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83 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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84 | | DJNZ addr1 addr2 ⇒ |
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85 | let b2 ≝ addr_of addr2 in |
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86 | match addr1 return λx. bool_to_Prop (is_in ? [[registr;direct]] x) → ? with |
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87 | [ REGISTER r ⇒ λ_. |
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88 | [ ([[true; true; false; true; true]]) @@ r ; b2 ] |
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89 | | DIRECT b1 ⇒ λ_. |
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90 | [ ([[true; true; false; true; false; true; false; true]]); b1; b2 ] |
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91 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) |
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92 | | JC addr ⇒ |
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93 | let b1 ≝ addr_of addr in |
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94 | [ ([[false; true; false; false; false; false; false; false]]); b1 ] |
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95 | | JNC addr ⇒ |
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96 | let b1 ≝ addr_of addr in |
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97 | [ ([[false; true; false; true; false; false; false; false]]); b1 ] |
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98 | | JZ addr ⇒ |
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99 | let b1 ≝ addr_of addr in |
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100 | [ ([[false; true; true; false; false; false; false; false]]); b1 ] |
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101 | | JNZ addr ⇒ |
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102 | let b1 ≝ addr_of addr in |
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103 | [ ([[false; true; true; true; false; false; false; false]]); b1 ] |
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104 | | JB addr1 addr2 ⇒ |
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105 | let b2 ≝ addr_of addr2 in |
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106 | match addr1 return λx. bool_to_Prop (is_in ? [[bit_addr]] x) → ? with |
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107 | [ BIT_ADDR b1 ⇒ λ_. |
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108 | [ ([[false; false; true; false; false; false; false; false]]); b1; b2 ] |
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109 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) |
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110 | | JNB addr1 addr2 ⇒ |
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111 | let b2 ≝ addr_of addr2 in |
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112 | match addr1 return λx. bool_to_Prop (is_in ? [[bit_addr]] x) → ? with |
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113 | [ BIT_ADDR b1 ⇒ λ_. |
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114 | [ ([[false; false; true; true; false; false; false; false]]); b1; b2 ] |
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115 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) |
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116 | | JBC addr1 addr2 ⇒ |
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117 | let b2 ≝ addr_of addr2 in |
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118 | match addr1 return λx. bool_to_Prop (is_in ? [[bit_addr]] x) → ? with |
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119 | [ BIT_ADDR b1 ⇒ λ_. |
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120 | [ ([[false; false; false; true; false; false; false; false]]); b1; b2 ] |
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121 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) |
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122 | | CJNE addrs addr3 ⇒ |
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123 | let b3 ≝ addr_of addr3 in |
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124 | match addrs with |
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125 | [ inl addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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126 | match addr2 return λx. bool_to_Prop (is_in ? [[direct;data]] x) → ? with |
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127 | [ DIRECT b1 ⇒ λ_. |
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128 | [ ([[true; false; true; true; false; true; false; true]]); b1; b3 ] |
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129 | | DATA b1 ⇒ λ_. |
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130 | [ ([[true; false; true; true; false; true; false; false]]); b1; b3 ] |
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131 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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132 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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133 | let b2 ≝ |
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134 | match addr2 return λx. bool_to_Prop (is_in ? [[data]] x) → ? with |
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135 | [ DATA b2 ⇒ λ_. b2 |
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136 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) in |
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137 | match addr1 return λx. bool_to_Prop (is_in ? [[registr;indirect]] x) → list Byte with |
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138 | [ REGISTER r ⇒ λ_. |
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139 | [ ([[true; false; true; true; true]]) @@ r; b2; b3 ] |
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140 | | INDIRECT i1 ⇒ λ_. |
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141 | [ ([[true; false; true; true; false; true; true; i1]]); b2; b3 ] |
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142 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) |
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143 | ] |
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144 | | DIV addr1 addr2 ⇒ |
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145 | [ ([[true;false;false;false;false;true;false;false]]) ] |
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146 | | INC addr ⇒ |
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147 | match addr return λx. bool_to_Prop (is_in ? [[acc_a;registr;direct;indirect;dptr]] x) → ? with |
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148 | [ ACC_A ⇒ λ_. |
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149 | [ ([[false;false;false;false;false;true;false;false]]) ] |
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150 | | REGISTER r ⇒ λ_. |
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151 | [ ([[false;false;false;false;true]]) @@ r ] |
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152 | | DIRECT b1 ⇒ λ_. |
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153 | [ ([[false; false; false; false; false; true; false; true]]); b1 ] |
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154 | | INDIRECT i1 ⇒ λ_. |
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155 | [ ([[false; false; false; false; false; true; true; i1]]) ] |
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156 | | DPTR ⇒ λ_. |
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157 | [ ([[true;false;true;false;false;false;true;true]]) ] |
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158 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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159 | | MOV addrs ⇒ |
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160 | match addrs with |
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161 | [ inl addrs ⇒ |
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162 | match addrs with |
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163 | [ inl addrs ⇒ |
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164 | match addrs with |
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165 | [ inl addrs ⇒ |
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166 | match addrs with |
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167 | [ inl addrs ⇒ |
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168 | match addrs with |
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169 | [ inl addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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170 | match addr2 return λx. bool_to_Prop (is_in ? [[registr;direct;indirect;data]] x) → ? with |
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171 | [ REGISTER r ⇒ λ_.[ ([[true;true;true;false;true]]) @@ r ] |
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172 | | DIRECT b1 ⇒ λ_.[ ([[true;true;true;false;false;true;false;true]]); b1 ] |
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173 | | INDIRECT i1 ⇒ λ_. [ ([[true;true;true;false;false;true;true;i1]]) ] |
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174 | | DATA b1 ⇒ λ_. [ ([[false;true;true;true;false;true;false;false]]) ; b1 ] |
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175 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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176 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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177 | match addr1 return λx. bool_to_Prop (is_in ? [[registr;indirect]] x) → ? with |
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178 | [ REGISTER r ⇒ λ_. |
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179 | match addr2 return λx. bool_to_Prop (is_in ? [[acc_a;direct;data]] x) → ? with |
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180 | [ ACC_A ⇒ λ_.[ ([[true;true;true;true;true]]) @@ r ] |
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181 | | DIRECT b1 ⇒ λ_.[ ([[true;false;true;false;true]]) @@ r; b1 ] |
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182 | | DATA b1 ⇒ λ_. [ ([[false;true;true;true;true]]) @@ r; b1 ] |
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183 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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184 | | INDIRECT i1 ⇒ λ_. |
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185 | match addr2 return λx. bool_to_Prop (is_in ? [[acc_a;direct;data]] x) → ? with |
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186 | [ ACC_A ⇒ λ_.[ ([[true;true;true;true;false;true;true;i1]]) ] |
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187 | | DIRECT b1 ⇒ λ_.[ ([[true;false;true;false;false;true;true;i1]]); b1 ] |
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188 | | DATA b1 ⇒ λ_. [ ([[false;true;true;true;false;true;true;i1]]) ; b1 ] |
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189 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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190 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1)] |
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191 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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192 | let b1 ≝ |
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193 | match addr1 return λx. bool_to_Prop (is_in ? [[direct]] x) → ? with |
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194 | [ DIRECT b1 ⇒ λ_. b1 |
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195 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) in |
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196 | match addr2 return λx. bool_to_Prop (is_in ? [[acc_a;registr;direct;indirect;data]] x) → ? with |
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197 | [ ACC_A ⇒ λ_.[ ([[true;true;true;true;false;true;false;true]]); b1] |
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198 | | REGISTER r ⇒ λ_.[ ([[true;false;false;false;true]]) @@ r; b1 ] |
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199 | | DIRECT b2 ⇒ λ_.[ ([[true;false;false;false;false;true;false;true]]); b1; b2 ] |
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200 | | INDIRECT i1 ⇒ λ_. [ ([[true;false;false;false;false;true;true;i1]]); b1 ] |
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201 | | DATA b2 ⇒ λ_. [ ([[false;true;true;true;false;true;false;true]]); b1; b2 ] |
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202 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2)] |
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203 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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204 | match addr2 return λx. bool_to_Prop (is_in ? [[data16]] x) → ? with |
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205 | [ DATA16 w ⇒ λ_. |
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206 | let b1_b2 ≝ split ? 8 8 w in |
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207 | let b1 ≝ \fst b1_b2 in |
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208 | let b2 ≝ \snd b1_b2 in |
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209 | [ ([[true;false;false;true;false;false;false;false]]); b1; b2] |
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210 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2)] |
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211 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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212 | match addr2 return λx. bool_to_Prop (is_in ? [[bit_addr]] x) → ? with |
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213 | [ BIT_ADDR b1 ⇒ λ_. |
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214 | [ ([[true;false;true;false;false;false;true;false]]); b1 ] |
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215 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2)] |
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216 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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217 | match addr1 return λx. bool_to_Prop (is_in ? [[bit_addr]] x) → ? with |
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218 | [ BIT_ADDR b1 ⇒ λ_. |
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219 | [ ([[true;false;false;true;false;false;true;false]]); b1 ] |
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220 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1)] |
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221 | | MOVX addrs ⇒ |
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222 | match addrs with |
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223 | [ inl addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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224 | match addr2 return λx. bool_to_Prop (is_in ? [[ext_indirect;ext_indirect_dptr]] x) → ? with |
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225 | [ EXT_INDIRECT i1 ⇒ λ_. |
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226 | [ ([[true;true;true;false;false;false;true;i1]]) ] |
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227 | | EXT_INDIRECT_DPTR ⇒ λ_. |
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228 | [ ([[true;true;true;false;false;false;false;false]]) ] |
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229 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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230 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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231 | match addr1 return λx. bool_to_Prop (is_in ? [[ext_indirect;ext_indirect_dptr]] x) → ? with |
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232 | [ EXT_INDIRECT i1 ⇒ λ_. |
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233 | [ ([[true;true;true;true;false;false;true;i1]]) ] |
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234 | | EXT_INDIRECT_DPTR ⇒ λ_. |
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235 | [ ([[true;true;true;true;false;false;false;false]]) ] |
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236 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1)] |
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237 | | MUL addr1 addr2 ⇒ |
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238 | [ ([[true;false;true;false;false;true;false;false]]) ] |
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239 | | NOP ⇒ |
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240 | [ ([[false;false;false;false;false;false;false;false]]) ] |
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241 | | ORL addrs ⇒ |
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242 | match addrs with |
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243 | [ inl addrs ⇒ |
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244 | match addrs with |
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245 | [ inl addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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246 | match addr2 return λx. bool_to_Prop (is_in ? [[registr;data;direct;indirect]] x) → ? with |
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247 | [ REGISTER r ⇒ λ_.[ ([[false;true;false;false;true]]) @@ r ] |
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248 | | DIRECT b1 ⇒ λ_.[ ([[false;true;false;false;false;true;false;true]]); b1 ] |
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249 | | INDIRECT i1 ⇒ λ_. [ ([[false;true;false;false;false;true;true;i1]]) ] |
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250 | | DATA b1 ⇒ λ_. [ ([[false;true;false;false;false;true;false;false]]) ; b1 ] |
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251 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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252 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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253 | let b1 ≝ |
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254 | match addr1 return λx. bool_to_Prop (is_in ? [[direct]] x) → ? with |
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255 | [ DIRECT b1 ⇒ λ_. b1 |
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256 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) in |
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257 | match addr2 return λx. bool_to_Prop (is_in ? [[acc_a;data]] x) → ? with |
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258 | [ ACC_A ⇒ λ_. |
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259 | [ ([[false;true;false;false;false;false;true;false]]); b1 ] |
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260 | | DATA b2 ⇒ λ_. |
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261 | [ ([[false;true;false;false;false;false;true;true]]); b1; b2 ] |
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262 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2)] |
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263 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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264 | match addr2 return λx. bool_to_Prop (is_in ? [[bit_addr;n_bit_addr]] x) → ? with |
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265 | [ BIT_ADDR b1 ⇒ λ_. |
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266 | [ ([[false;true;true;true;false;false;true;false]]); b1 ] |
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267 | | N_BIT_ADDR b1 ⇒ λ_. |
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268 | [ ([[true;false;true;false;false;false;false;false]]); b1 ] |
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269 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2)] |
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270 | | POP addr ⇒ |
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271 | match addr return λx. bool_to_Prop (is_in ? [[direct]] x) → ? with |
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272 | [ DIRECT b1 ⇒ λ_. |
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273 | [ ([[true;true;false;true;false;false;false;false]]) ; b1 ] |
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274 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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275 | | PUSH addr ⇒ |
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276 | match addr return λx. bool_to_Prop (is_in ? [[direct]] x) → ? with |
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277 | [ DIRECT b1 ⇒ λ_. |
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278 | [ ([[true;true;false;false;false;false;false;false]]) ; b1 ] |
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279 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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280 | | RET ⇒ |
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281 | [ ([[false;false;true;false;false;false;true;false]]) ] |
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282 | | RETI ⇒ |
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283 | [ ([[false;false;true;true;false;false;true;false]]) ] |
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284 | | RL addr ⇒ |
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285 | [ ([[false;false;true;false;false;false;true;true]]) ] |
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286 | | RLC addr ⇒ |
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287 | [ ([[false;false;true;true;false;false;true;true]]) ] |
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288 | | RR addr ⇒ |
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289 | [ ([[false;false;false;false;false;false;true;true]]) ] |
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290 | | RRC addr ⇒ |
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291 | [ ([[false;false;false;true;false;false;true;true]]) ] |
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292 | | SETB addr ⇒ |
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293 | match addr return λx. bool_to_Prop (is_in ? [[carry;bit_addr]] x) → ? with |
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294 | [ CARRY ⇒ λ_. |
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295 | [ ([[true;true;false;true;false;false;true;true]]) ] |
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296 | | BIT_ADDR b1 ⇒ λ_. |
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297 | [ ([[true;true;false;true;false;false;true;false]]); b1 ] |
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298 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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299 | | SUBB addr1 addr2 ⇒ |
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300 | match addr2 return λx. bool_to_Prop (is_in ? [[registr;direct;indirect;data]] x) → ? with |
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301 | [ REGISTER r ⇒ λ_. |
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302 | [ ([[true;false;false;true;true]]) @@ r ] |
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303 | | DIRECT b1 ⇒ λ_. |
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304 | [ ([[true;false;false;true;false;true;false;true]]); b1] |
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305 | | INDIRECT i1 ⇒ λ_. |
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306 | [ ([[true;false;false;true;false;true;true;i1]]) ] |
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307 | | DATA b1 ⇒ λ_. |
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308 | [ ([[true;false;false;true;false;true;false;false]]); b1] |
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309 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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310 | | SWAP addr ⇒ |
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311 | [ ([[true;true;false;false;false;true;false;false]]) ] |
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312 | | XCH addr1 addr2 ⇒ |
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313 | match addr2 return λx. bool_to_Prop (is_in ? [[registr;direct;indirect]] x) → ? with |
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314 | [ REGISTER r ⇒ λ_. |
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315 | [ ([[true;true;false;false;true]]) @@ r ] |
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316 | | DIRECT b1 ⇒ λ_. |
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317 | [ ([[true;true;false;false;false;true;false;true]]); b1] |
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318 | | INDIRECT i1 ⇒ λ_. |
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319 | [ ([[true;true;false;false;false;true;true;i1]]) ] |
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320 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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321 | | XCHD addr1 addr2 ⇒ |
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322 | match addr2 return λx. bool_to_Prop (is_in ? [[indirect]] x) → ? with |
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323 | [ INDIRECT i1 ⇒ λ_. |
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324 | [ ([[true;true;false;true;false;true;true;i1]]) ] |
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325 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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326 | | XRL addrs ⇒ |
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327 | match addrs with |
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328 | [ inl addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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329 | match addr2 return λx. bool_to_Prop (is_in ? [[data;registr;direct;indirect]] x) → ? with |
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330 | [ REGISTER r ⇒ λ_. |
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331 | [ ([[false;true;true;false;true]]) @@ r ] |
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332 | | DIRECT b1 ⇒ λ_. |
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333 | [ ([[false;true;true;false;false;true;false;true]]); b1] |
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334 | | INDIRECT i1 ⇒ λ_. |
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335 | [ ([[false;true;true;false;false;true;true;i1]]) ] |
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336 | | DATA b1 ⇒ λ_. |
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337 | [ ([[false;true;true;false;false;true;false;false]]); b1] |
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338 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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339 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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340 | let b1 ≝ |
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341 | match addr1 return λx. bool_to_Prop (is_in ? [[direct]] x) → ? with |
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342 | [ DIRECT b1 ⇒ λ_. b1 |
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343 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) in |
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344 | match addr2 return λx. bool_to_Prop (is_in ? [[acc_a;data]] x) → ? with |
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345 | [ ACC_A ⇒ λ_. |
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346 | [ ([[false;true;true;false;false;false;true;false]]); b1 ] |
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347 | | DATA b2 ⇒ λ_. |
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348 | [ ([[false;true;true;false;false;false;true;true]]); b1; b2 ] |
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349 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2)] |
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350 | ]. |
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351 | |
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352 | definition assembly1 ≝ |
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353 | λi: instruction. |
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354 | match i with |
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355 | [ ACALL addr ⇒ |
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356 | match addr return λx. bool_to_Prop (is_in ? [[addr11]] x) → ? with |
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357 | [ ADDR11 w ⇒ λ_. |
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358 | let v1_v2 ≝ split ? 3 8 w in |
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359 | let v1 ≝ \fst v1_v2 in |
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360 | let v2 ≝ \snd v1_v2 in |
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361 | [ (v1 @@ [[true; false; false; false; true]]) ; v2 ] |
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362 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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363 | | AJMP addr ⇒ |
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364 | match addr return λx. bool_to_Prop (is_in ? [[addr11]] x) → ? with |
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365 | [ ADDR11 w ⇒ λ_. |
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366 | let v1_v2 ≝ split ? 3 8 w in |
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367 | let v1 ≝ \fst v1_v2 in |
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368 | let v2 ≝ \snd v1_v2 in |
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369 | [ (v1 @@ [[false; false; false; false; true]]) ; v2 ] |
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370 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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371 | | JMP adptr ⇒ |
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372 | [ ([[false;true;true;true;false;false;true;true]]) ] |
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373 | | LCALL addr ⇒ |
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374 | match addr return λx. bool_to_Prop (is_in ? [[addr16]] x) → ? with |
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375 | [ ADDR16 w ⇒ λ_. |
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376 | let b1_b2 ≝ split ? 8 8 w in |
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377 | let b1 ≝ \fst b1_b2 in |
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378 | let b2 ≝ \snd b1_b2 in |
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379 | [ ([[false;false;false;true;false;false;true;false]]); b1; b2 ] |
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380 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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381 | | LJMP addr ⇒ |
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382 | match addr return λx. bool_to_Prop (is_in ? [[addr16]] x) → ? with |
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383 | [ ADDR16 w ⇒ λ_. |
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384 | let b1_b2 ≝ split ? 8 8 w in |
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385 | let b1 ≝ \fst b1_b2 in |
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386 | let b2 ≝ \snd b1_b2 in |
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387 | [ ([[false;false;false;false;false;false;true;false]]); b1; b2 ] |
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388 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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389 | | MOVC addr1 addr2 ⇒ |
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390 | match addr2 return λx. bool_to_Prop (is_in ? [[acc_dptr;acc_pc]] x) → ? with |
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391 | [ ACC_DPTR ⇒ λ_. |
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392 | [ ([[true;false;false;true;false;false;true;true]]) ] |
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393 | | ACC_PC ⇒ λ_. |
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394 | [ ([[true;false;false;false;false;false;true;true]]) ] |
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395 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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396 | | SJMP addr ⇒ |
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397 | match addr return λx. bool_to_Prop (is_in ? [[relative]] x) → ? with |
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398 | [ RELATIVE b1 ⇒ λ_. |
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399 | [ ([[true;false;false;false;false;false;false;false]]); b1 ] |
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400 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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401 | | RealInstruction instr ⇒ |
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402 | assembly_preinstruction [[ relative ]] |
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403 | (λx. |
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404 | match x return λs. bool_to_Prop (is_in ? [[ relative ]] s) → ? with |
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405 | [ RELATIVE r ⇒ λ_. r |
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406 | | _ ⇒ λabsd. ⊥ |
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407 | ] (subaddressing_modein … x)) instr |
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408 | ]. |
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409 | cases absd |
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410 | qed. |
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411 | |
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412 | inductive jump_length: Type[0] ≝ |
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413 | | short_jump: jump_length |
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414 | | medium_jump: jump_length |
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415 | | long_jump: jump_length. |
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416 | |
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417 | (* jump_expansion_policy: instruction number ↦ 〈pc, jump_length〉 *) |
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418 | definition jump_expansion_policy ≝ BitVectorTrie (ℕ × jump_length) 16. |
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419 | |
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420 | definition expand_relative_jump_internal: |
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421 | (Identifier → Word) → jump_length → Identifier → Word → ([[relative]] → preinstruction [[relative]]) → |
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422 | option (list instruction) |
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423 | ≝ |
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424 | λlookup_labels,jmp_len.λjmp:Identifier.λpc,i. |
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425 | match jmp_len with |
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426 | [ short_jump ⇒ |
---|
427 | let lookup_address ≝ lookup_labels jmp in |
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428 | let 〈result, flags〉 ≝ sub_16_with_carry pc lookup_address false in |
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429 | let 〈upper, lower〉 ≝ split ? 8 8 result in |
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430 | if eq_bv ? upper (zero 8) then |
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431 | let address ≝ RELATIVE lower in |
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432 | Some ? [ RealInstruction (i address) ] |
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433 | else |
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434 | None ? |
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435 | | medium_jump ⇒ None … |
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436 | | long_jump ⇒ |
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437 | Some ? |
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438 | [ RealInstruction (i (RELATIVE (bitvector_of_nat ? 2))); |
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439 | SJMP (RELATIVE (bitvector_of_nat ? 3)); (* LJMP size? *) |
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440 | LJMP (ADDR16 (lookup_labels jmp)) |
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441 | ] |
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442 | ]. |
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443 | @ I |
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444 | qed. |
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445 | |
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446 | definition expand_relative_jump: (Identifier → Word) → jump_length → Word → preinstruction Identifier → option (list instruction) ≝ |
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447 | λlookup_labels. |
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448 | λjmp_len: jump_length. |
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449 | λpc. |
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450 | λi: preinstruction Identifier. |
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451 | let rel_jmp ≝ RELATIVE (bitvector_of_nat ? 2) in |
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452 | match i with |
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453 | [ JC jmp ⇒ expand_relative_jump_internal lookup_labels jmp_len jmp pc (JC ?) |
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454 | | JNC jmp ⇒ expand_relative_jump_internal lookup_labels jmp_len jmp pc (JNC ?) |
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455 | | JB baddr jmp ⇒ expand_relative_jump_internal lookup_labels jmp_len jmp pc (JB ? baddr) |
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456 | | JZ jmp ⇒ expand_relative_jump_internal lookup_labels jmp_len jmp pc (JZ ?) |
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457 | | JNZ jmp ⇒ expand_relative_jump_internal lookup_labels jmp_len jmp pc (JNZ ?) |
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458 | | JBC baddr jmp ⇒ expand_relative_jump_internal lookup_labels jmp_len jmp pc (JBC ? baddr) |
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459 | | JNB baddr jmp ⇒ expand_relative_jump_internal lookup_labels jmp_len jmp pc (JNB ? baddr) |
---|
460 | | CJNE addr jmp ⇒ expand_relative_jump_internal lookup_labels jmp_len jmp pc (CJNE ? addr) |
---|
461 | | DJNZ addr jmp ⇒ expand_relative_jump_internal lookup_labels jmp_len jmp pc (DJNZ ? addr) |
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462 | | ADD arg1 arg2 ⇒ Some ? [ ADD ? arg1 arg2 ] |
---|
463 | | ADDC arg1 arg2 ⇒ Some ? [ ADDC ? arg1 arg2 ] |
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464 | | SUBB arg1 arg2 ⇒ Some ? [ SUBB ? arg1 arg2 ] |
---|
465 | | INC arg ⇒ Some ? [ INC ? arg ] |
---|
466 | | DEC arg ⇒ Some ? [ DEC ? arg ] |
---|
467 | | MUL arg1 arg2 ⇒ Some ? [ MUL ? arg1 arg2 ] |
---|
468 | | DIV arg1 arg2 ⇒ Some ? [ DIV ? arg1 arg2 ] |
---|
469 | | DA arg ⇒ Some ? [ DA ? arg ] |
---|
470 | | ANL arg ⇒ Some ? [ ANL ? arg ] |
---|
471 | | ORL arg ⇒ Some ? [ ORL ? arg ] |
---|
472 | | XRL arg ⇒ Some ? [ XRL ? arg ] |
---|
473 | | CLR arg ⇒ Some ? [ CLR ? arg ] |
---|
474 | | CPL arg ⇒ Some ? [ CPL ? arg ] |
---|
475 | | RL arg ⇒ Some ? [ RL ? arg ] |
---|
476 | | RR arg ⇒ Some ? [ RR ? arg ] |
---|
477 | | RLC arg ⇒ Some ? [ RLC ? arg ] |
---|
478 | | RRC arg ⇒ Some ? [ RRC ? arg ] |
---|
479 | | SWAP arg ⇒ Some ? [ SWAP ? arg ] |
---|
480 | | MOV arg ⇒ Some ? [ MOV ? arg ] |
---|
481 | | MOVX arg ⇒ Some ? [ MOVX ? arg ] |
---|
482 | | SETB arg ⇒ Some ? [ SETB ? arg ] |
---|
483 | | PUSH arg ⇒ Some ? [ PUSH ? arg ] |
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484 | | POP arg ⇒ Some ? [ POP ? arg ] |
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485 | | XCH arg1 arg2 ⇒ Some ? [ XCH ? arg1 arg2 ] |
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486 | | XCHD arg1 arg2 ⇒ Some ? [ XCHD ? arg1 arg2 ] |
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487 | | RET ⇒ Some ? [ RET ? ] |
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488 | | RETI ⇒ Some ? [ RETI ? ] |
---|
489 | | NOP ⇒ Some ? [ RealInstruction (NOP ?) ] |
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490 | ]. |
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491 | |
---|
492 | definition expand_pseudo_instruction_safe: ? → ? → Word → jump_length → pseudo_instruction → option (list instruction) ≝ |
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493 | λlookup_labels. |
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494 | λlookup_datalabels. |
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495 | λpc. |
---|
496 | λjmp_len. |
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497 | λi. |
---|
498 | match i with |
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499 | [ Cost cost ⇒ Some ? [ ] |
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500 | | Comment comment ⇒ Some ? [ ] |
---|
501 | | Call call ⇒ |
---|
502 | match jmp_len with |
---|
503 | [ short_jump ⇒ None ? |
---|
504 | | medium_jump ⇒ |
---|
505 | let 〈ignore, address〉 ≝ split ? 5 11 (lookup_labels call) in |
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506 | let 〈fst_5, rest〉 ≝ split ? 5 11 pc in |
---|
507 | if eq_bv ? ignore fst_5 then |
---|
508 | let address ≝ ADDR11 address in |
---|
509 | Some ? ([ ACALL address ]) |
---|
510 | else |
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511 | None ? |
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512 | | long_jump ⇒ |
---|
513 | let address ≝ ADDR16 (lookup_labels call) in |
---|
514 | Some ? [ LCALL address ] |
---|
515 | ] |
---|
516 | | Mov d trgt ⇒ |
---|
517 | let address ≝ DATA16 (lookup_datalabels trgt) in |
---|
518 | Some ? [ RealInstruction (MOV ? (inl ? ? (inl ? ? (inr ? ? 〈DPTR, address〉))))] |
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519 | | Instruction instr ⇒ expand_relative_jump lookup_labels jmp_len pc instr |
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520 | | Jmp jmp ⇒ |
---|
521 | match jmp_len with |
---|
522 | [ short_jump ⇒ |
---|
523 | let lookup_address ≝ lookup_labels jmp in |
---|
524 | let 〈result, flags〉 ≝ sub_16_with_carry pc lookup_address false in |
---|
525 | let 〈upper, lower〉 ≝ split ? 8 8 result in |
---|
526 | if eq_bv ? upper (zero 8) then |
---|
527 | let address ≝ RELATIVE lower in |
---|
528 | Some ? [ SJMP address ] |
---|
529 | else |
---|
530 | None ? |
---|
531 | | medium_jump ⇒ |
---|
532 | let address ≝ lookup_labels jmp in |
---|
533 | let 〈fst_5_addr, rest_addr〉 ≝ split ? 5 11 address in |
---|
534 | let 〈fst_5_pc, rest_pc〉 ≝ split ? 5 11 pc in |
---|
535 | if eq_bv ? fst_5_addr fst_5_pc then |
---|
536 | let address ≝ ADDR11 rest_addr in |
---|
537 | Some ? ([ AJMP address ]) |
---|
538 | else |
---|
539 | None ? |
---|
540 | | long_jump ⇒ |
---|
541 | let address ≝ ADDR16 (lookup_labels jmp) in |
---|
542 | Some ? [ LJMP address ] |
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543 | ] |
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544 | ]. |
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545 | @ I |
---|
546 | qed. |
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547 | |
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548 | (* label_map: identifier ↦ 〈instruction number, address〉 *) |
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549 | definition label_map ≝ identifier_map ASMTag (nat × nat). |
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550 | |
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551 | definition add_instruction_size: ℕ → jump_length → pseudo_instruction → ℕ ≝ |
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552 | λpc.λjmp_len.λinstr. |
---|
553 | let bv_pc ≝ bitvector_of_nat 16 pc in |
---|
554 | let ilist ≝ expand_pseudo_instruction_safe (λx.bv_pc) (λx.bv_pc) bv_pc jmp_len instr in |
---|
555 | let bv: list (BitVector 8) ≝ match ilist with |
---|
556 | [ None ⇒ (* hmm, this shouldn't happen *) [ ] |
---|
557 | | Some l ⇒ flatten … (map … assembly1 l) |
---|
558 | ] in |
---|
559 | pc + (|bv|). |
---|
560 | |
---|
561 | definition is_label ≝ |
---|
562 | λx:labelled_instruction.λl:Identifier. |
---|
563 | let 〈lbl,instr〉 ≝ x in |
---|
564 | match lbl with |
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565 | [ Some l' ⇒ l' = l |
---|
566 | | _ ⇒ False |
---|
567 | ]. |
---|
568 | |
---|
569 | lemma label_does_not_occur: |
---|
570 | ∀i,p,l. |
---|
571 | is_label (nth i ? p 〈None ?, Comment [ ]〉) l → does_not_occur l p = false. |
---|
572 | #i #p #l generalize in match i; elim p |
---|
573 | [ #i >nth_nil #H @⊥ @H |
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574 | | #h #t #IH #i cases i -i |
---|
575 | [ cases h #hi #hp cases hi |
---|
576 | [ normalize #H @⊥ @H |
---|
577 | | #l' #Heq whd in ⊢ (??%?); change with (eq_identifier ? l' l) in match (instruction_matches_identifier ??); |
---|
578 | whd in Heq; >Heq |
---|
579 | >eq_identifier_refl // |
---|
580 | ] |
---|
581 | | #i #H whd in match (does_not_occur ??); |
---|
582 | whd in match (instruction_matches_identifier ??); |
---|
583 | cases h #hi #hp cases hi normalize nodelta |
---|
584 | [ @(IH i) @H |
---|
585 | | #l' @eq_identifier_elim |
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586 | [ normalize // |
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587 | | normalize #_ @(IH i) @H |
---|
588 | ] |
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589 | ] |
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590 | ] |
---|
591 | ] |
---|
592 | qed. |
---|
593 | |
---|
594 | (* lemma coerc_sigma: |
---|
595 | ∀A,P.∀p:A.P p → Σx:A.P x. |
---|
596 | #A #P #a #p % [ @a | /2/] |
---|
597 | qed. |
---|
598 | coercion coerc_sigma:∀A,P.∀p:A.P p → Σx:A.P x |
---|
599 | ≝ coerc_sigma on p: ? to (Sig ??). *) |
---|
600 | |
---|
601 | lemma coerc_pair_sigma: |
---|
602 | ∀A,B,P. ∀p:A × B. P (\snd p) → A × (Σx:B.P x). |
---|
603 | #A #B #P * #a #b #p % [@a | /2/] |
---|
604 | qed. |
---|
605 | coercion coerc_pair_sigma:∀A,B,P. ∀p:A × B. P (\snd p) → A × (Σx:B.P x) |
---|
606 | ≝ coerc_pair_sigma on p: (? × ?) to (? × (Sig ??)). |
---|
607 | |
---|
608 | let rec create_label_map (program: list labelled_instruction) |
---|
609 | (policy: jump_expansion_policy): |
---|
610 | (Σlabels:label_map. |
---|
611 | ∀i:ℕ.lt i (|program|) (* XXX using < causes (false?) ambiguity *) → |
---|
612 | ∀l.occurs_exactly_once l program → |
---|
613 | is_label (nth i ? program 〈None ?, Comment [ ]〉) l → |
---|
614 | ∃a.lookup … labels l = Some ? 〈i,a〉 |
---|
615 | ) ≝ |
---|
616 | let 〈final_pc, final_labels〉 ≝ |
---|
617 | foldl_strong (option Identifier × pseudo_instruction) |
---|
618 | (λprefix.ℕ × (Σlabels. |
---|
619 | ∀i:ℕ.i < |prefix| → |
---|
620 | ∀l.occurs_exactly_once l prefix → |
---|
621 | is_label (nth i ? prefix 〈None ?, Comment [ ]〉) l → |
---|
622 | ∃a.lookup … labels l = Some ? 〈i,a〉) |
---|
623 | ) |
---|
624 | program |
---|
625 | (λprefix.λx.λtl.λprf.λacc. |
---|
626 | let 〈pc,labels〉 ≝ acc in |
---|
627 | let 〈label,instr〉 ≝ x in |
---|
628 | let new_labels ≝ |
---|
629 | match label with |
---|
630 | [ None ⇒ labels |
---|
631 | | Some l ⇒ add … labels l 〈|prefix|, pc〉 |
---|
632 | ] in |
---|
633 | let jmp_len ≝ \snd (bvt_lookup ?? (bitvector_of_nat 16 (|prefix|)) policy 〈pc, long_jump〉) in |
---|
634 | 〈add_instruction_size pc jmp_len instr, new_labels〉 |
---|
635 | ) 〈0, empty_map …〉 in |
---|
636 | final_labels. |
---|
637 | [ #i >append_length >commutative_plus #Hi normalize in Hi; cases (le_to_or_lt_eq … Hi) -Hi; |
---|
638 | [ #Hi #l normalize nodelta; cases label normalize nodelta |
---|
639 | [ >occurs_exactly_once_None #Hocc >(nth_append_first ? ? prefix ? ? (le_S_S_to_le ? ? Hi)) #Hl |
---|
640 | lapply (sig2 … labels) #Hacc elim (Hacc i (le_S_S_to_le … Hi) l Hocc Hl) #addr #Haddr |
---|
641 | % [ @addr | @Haddr ] |
---|
642 | | #l' #Hocc #Hl lapply (occurs_exactly_once_Some_stronger … Hocc) -Hocc; |
---|
643 | @eq_identifier_elim #Heq #Hocc |
---|
644 | [ normalize in Hocc; |
---|
645 | >(nth_append_first ? ? prefix ? ? (le_S_S_to_le … Hi)) in Hl; #Hl |
---|
646 | @⊥ @(absurd … Hocc) |
---|
647 | | normalize nodelta lapply (sig2 … labels) #Hacc elim (Hacc i (le_S_S_to_le … Hi) l Hocc ?) |
---|
648 | [ #addr #Haddr % [ @addr | <Haddr @lookup_add_miss /2/ ] |
---|
649 | | >(nth_append_first ? ? prefix ? ? (le_S_S_to_le … Hi)) in Hl; // |
---|
650 | ] |
---|
651 | ] |
---|
652 | >(label_does_not_occur i … Hl) normalize nodelta @nmk // |
---|
653 | ] |
---|
654 | | #Hi #l #Hocc >(injective_S … Hi) >nth_append_second |
---|
655 | [ <minus_n_n #Hl normalize in Hl; normalize nodelta cases label in Hl; |
---|
656 | [ normalize nodelta #H @⊥ @H |
---|
657 | | #l' normalize nodelta #Heq % [ @pc | <Heq normalize nodelta @lookup_add_hit ] |
---|
658 | ] |
---|
659 | | @le_n |
---|
660 | ] |
---|
661 | ] |
---|
662 | | #i #Hi #l #Hl @⊥ @Hl |
---|
663 | ] |
---|
664 | qed. |
---|
665 | |
---|
666 | definition select_reljump_length: label_map → ℕ → Identifier → jump_length ≝ |
---|
667 | λlabels.λpc.λlbl. |
---|
668 | let 〈n, addr〉 ≝ lookup_def … labels lbl 〈0, pc〉 in |
---|
669 | if leb pc addr (* forward jump *) |
---|
670 | then if leb (addr - pc) 126 |
---|
671 | then short_jump |
---|
672 | else long_jump |
---|
673 | else if leb (pc - addr) 129 |
---|
674 | then short_jump |
---|
675 | else long_jump. |
---|
676 | |
---|
677 | definition select_call_length: label_map → ℕ → Identifier → jump_length ≝ |
---|
678 | λlabels.λpc_nat.λlbl. |
---|
679 | let pc ≝ bitvector_of_nat 16 pc_nat in |
---|
680 | let addr ≝ bitvector_of_nat 16 (\snd (lookup_def ? ? labels lbl 〈0, pc_nat〉)) in |
---|
681 | let 〈fst_5_addr, rest_addr〉 ≝ split ? 5 11 addr in |
---|
682 | let 〈fst_5_pc, rest_pc〉 ≝ split ? 5 11 pc in |
---|
683 | if eq_bv ? fst_5_addr fst_5_pc |
---|
684 | then medium_jump |
---|
685 | else long_jump. |
---|
686 | |
---|
687 | definition select_jump_length: label_map → ℕ → Identifier → jump_length ≝ |
---|
688 | λlabels.λpc.λlbl. |
---|
689 | let 〈n, addr〉 ≝ lookup_def … labels lbl 〈0, pc〉 in |
---|
690 | if leb pc addr |
---|
691 | then if leb (addr - pc) 126 |
---|
692 | then short_jump |
---|
693 | else select_call_length labels pc lbl |
---|
694 | else if leb (pc - addr) 129 |
---|
695 | then short_jump |
---|
696 | else select_call_length labels pc lbl. |
---|
697 | |
---|
698 | definition jump_expansion_step_instruction: label_map → ℕ → |
---|
699 | preinstruction Identifier → option jump_length ≝ |
---|
700 | λlabels.λpc.λi. |
---|
701 | match i with |
---|
702 | [ JC j ⇒ Some ? (select_reljump_length labels pc j) |
---|
703 | | JNC j ⇒ Some ? (select_reljump_length labels pc j) |
---|
704 | | JZ j ⇒ Some ? (select_reljump_length labels pc j) |
---|
705 | | JNZ j ⇒ Some ? (select_reljump_length labels pc j) |
---|
706 | | JB _ j ⇒ Some ? (select_reljump_length labels pc j) |
---|
707 | | JBC _ j ⇒ Some ? (select_reljump_length labels pc j) |
---|
708 | | JNB _ j ⇒ Some ? (select_reljump_length labels pc j) |
---|
709 | | CJNE _ j ⇒ Some ? (select_reljump_length labels pc j) |
---|
710 | | DJNZ _ j ⇒ Some ? (select_reljump_length labels pc j) |
---|
711 | | _ ⇒ None ? |
---|
712 | ]. |
---|
713 | |
---|
714 | definition max_length: jump_length → jump_length → jump_length ≝ |
---|
715 | λj1.λj2. |
---|
716 | match j1 with |
---|
717 | [ long_jump ⇒ long_jump |
---|
718 | | medium_jump ⇒ |
---|
719 | match j2 with |
---|
720 | [ long_jump ⇒ long_jump |
---|
721 | | _ ⇒ medium_jump |
---|
722 | ] |
---|
723 | | short_jump ⇒ j2 |
---|
724 | ]. |
---|
725 | |
---|
726 | definition jmpleq: jump_length → jump_length → Prop ≝ |
---|
727 | λj1.λj2. |
---|
728 | match j1 with |
---|
729 | [ short_jump ⇒ True |
---|
730 | | medium_jump ⇒ |
---|
731 | match j2 with |
---|
732 | [ short_jump ⇒ False |
---|
733 | | _ ⇒ True |
---|
734 | ] |
---|
735 | | long_jump ⇒ |
---|
736 | match j2 with |
---|
737 | [ long_jump ⇒ True |
---|
738 | | _ ⇒ False |
---|
739 | ] |
---|
740 | ]. |
---|
741 | |
---|
742 | lemma jmpleq_max_length: ∀x,y.jmpleq y (max_length y x). |
---|
743 | #x #y |
---|
744 | cases y |
---|
745 | [ // |
---|
746 | | cases x // |
---|
747 | | // |
---|
748 | ] |
---|
749 | qed. |
---|
750 | |
---|
751 | definition is_jump' ≝ |
---|
752 | λx:preinstruction Identifier. |
---|
753 | match x with |
---|
754 | [ JC _ ⇒ True |
---|
755 | | JNC _ ⇒ True |
---|
756 | | JZ _ ⇒ True |
---|
757 | | JNZ _ ⇒ True |
---|
758 | | JB _ _ ⇒ True |
---|
759 | | JNB _ _ ⇒ True |
---|
760 | | JBC _ _ ⇒ True |
---|
761 | | CJNE _ _ ⇒ True |
---|
762 | | DJNZ _ _ ⇒ True |
---|
763 | | _ ⇒ False |
---|
764 | ]. |
---|
765 | |
---|
766 | definition is_jump ≝ |
---|
767 | λx:labelled_instruction. |
---|
768 | let 〈label,instr〉 ≝ x in |
---|
769 | match instr with |
---|
770 | [ Instruction i ⇒ is_jump' i |
---|
771 | | Call _ ⇒ True |
---|
772 | | Jmp _ ⇒ True |
---|
773 | | _ ⇒ False |
---|
774 | ]. |
---|
775 | |
---|
776 | definition jump_in_policy ≝ |
---|
777 | λprefix:list labelled_instruction.λpolicy:jump_expansion_policy. |
---|
778 | ∀i:ℕ.i < |prefix| → is_jump (nth i ? prefix 〈None ?, Comment []〉) → |
---|
779 | ∃p,j.lookup_opt … (bitvector_of_nat 16 i) policy = Some ? 〈p,j〉. |
---|
780 | |
---|
781 | axiom bitvector_of_nat_abs: |
---|
782 | ∀x,y:ℕ.x ≠ y → ¬eq_bv 16 (bitvector_of_nat 16 x) (bitvector_of_nat 16 y). |
---|
783 | |
---|
784 | let rec jump_expansion_start (program: list labelled_instruction): |
---|
785 | (Σpolicy.jump_in_policy program policy) ≝ |
---|
786 | foldl_strong (option Identifier × pseudo_instruction) |
---|
787 | (λprefix.Σpolicy.jump_in_policy prefix policy) |
---|
788 | program |
---|
789 | (λprefix.λx.λtl.λprf.λpolicy. |
---|
790 | let 〈label,instr〉 ≝ x in |
---|
791 | match instr with |
---|
792 | [ Instruction i ⇒ match i with |
---|
793 | [ JC _ ⇒ insert … (bitvector_of_nat 16 (|prefix|)) 〈0,short_jump〉 policy |
---|
794 | | JNC _ ⇒ insert … (bitvector_of_nat 16 (|prefix|)) 〈0,short_jump〉 policy |
---|
795 | | JZ _ ⇒ insert … (bitvector_of_nat 16 (|prefix|)) 〈0,short_jump〉 policy |
---|
796 | | JNZ _ ⇒ insert … (bitvector_of_nat 16 (|prefix|)) 〈0,short_jump〉 policy |
---|
797 | | JB _ _ ⇒ insert … (bitvector_of_nat 16 (|prefix|)) 〈0,short_jump〉 policy |
---|
798 | | JNB _ _ ⇒ insert … (bitvector_of_nat 16 (|prefix|)) 〈0,short_jump〉 policy |
---|
799 | | JBC _ _ ⇒ insert … (bitvector_of_nat 16 (|prefix|)) 〈0,short_jump〉 policy |
---|
800 | | CJNE _ _ ⇒ insert … (bitvector_of_nat 16 (|prefix|)) 〈0,short_jump〉 policy |
---|
801 | | DJNZ _ _ ⇒ insert … (bitvector_of_nat 16 (|prefix|)) 〈0,short_jump〉 policy |
---|
802 | | _ ⇒ (eject … policy) |
---|
803 | ] |
---|
804 | | Call c ⇒ insert … (bitvector_of_nat 16 (|prefix|)) 〈0,short_jump〉 policy |
---|
805 | | Jmp j ⇒ insert … (bitvector_of_nat 16 (|prefix|)) 〈0,short_jump〉 policy |
---|
806 | | _ ⇒ (eject … policy) |
---|
807 | ] |
---|
808 | ) (Stub ? ?). |
---|
809 | [43: normalize #i #Hi @⊥ @(absurd (i < 0)) [ @Hi | @not_le_Sn_O ] ] |
---|
810 | whd #i >append_length <commutative_plus #Hi normalize in Hi; cases (le_to_or_lt_eq … Hi) -Hi; |
---|
811 | #Hi |
---|
812 | [2,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,70,72,74,76,78,80,82,84: |
---|
813 | >nth_append_second >(injective_S … Hi) |
---|
814 | [2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62: @le_n] |
---|
815 | <minus_n_n #Hj normalize in Hj; @⊥ @Hj |
---|
816 | |4,6,52,54,56,58,60,62,64,66,68: >nth_append_second >(injective_S … Hi) |
---|
817 | [2,4,6,8,10,12,14,16,18,20,22: @le_n] |
---|
818 | <minus_n_n #Hj % [1,3,5,7,9,11,13,15,17,19,21: @O ] % |
---|
819 | [1,3,5,7,9,11,13,15,17,19,21: @short_jump ] @lookup_opt_insert_hit |
---|
820 | ] |
---|
821 | >(nth_append_first ? ? prefix ? ? (le_S_S_to_le … Hi)) #Hj |
---|
822 | lapply (sig2 … policy) #Hpolicy elim (Hpolicy i (le_S_S_to_le … Hi) Hj) |
---|
823 | #p #H elim H #j #Hpj % |
---|
824 | [1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51,53,55,57,59,61,63,65,67,69,71,73,75,77,79,81,83: @p] % |
---|
825 | [1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51,53,55,57,59,61,63,65,67,69,71,73,75,77,79,81,83: @j] |
---|
826 | [2,3,26,27,28,29,30,31,32,33,34: >lookup_opt_insert_miss] |
---|
827 | [2,4,6,8,10,12,14,16,18,20,22: @bitvector_of_nat_abs @lt_to_not_eq @(le_S_S_to_le … Hi) ] @Hpj |
---|
828 | qed. |
---|
829 | |
---|
830 | (* not really recursive *) |
---|
831 | let rec jump_expansion_step (program: list labelled_instruction) |
---|
832 | (old_policy: Σpolicy.jump_in_policy program policy): |
---|
833 | (Σpolicy. |
---|
834 | And (jump_in_policy program policy) (* XXX ∧ causes ambiguity *) |
---|
835 | (jump_in_policy program policy → |
---|
836 | (∀i.lt i (|program|) (* XXX < causes ambiguity *) → is_jump (nth i ? program 〈None ?, Comment []〉) → |
---|
837 | jmpleq |
---|
838 | (\snd (lookup … (bitvector_of_nat 16 i) old_policy 〈0,short_jump〉)) |
---|
839 | (\snd (lookup … (bitvector_of_nat 16 i) policy 〈0,short_jump〉))) |
---|
840 | ) |
---|
841 | ) ≝ |
---|
842 | let labels ≝ create_label_map program old_policy in |
---|
843 | let 〈final_pc, final_policy〉 ≝ |
---|
844 | foldl_strong (option Identifier × pseudo_instruction) |
---|
845 | (λprefix.ℕ × Σpolicy. |
---|
846 | jump_in_policy prefix policy ∧ |
---|
847 | (jump_in_policy prefix policy → |
---|
848 | (∀i.i < |prefix| → is_jump (nth i ? prefix 〈None ?, Comment []〉) → |
---|
849 | jmpleq |
---|
850 | (\snd (lookup … (bitvector_of_nat 16 i) old_policy 〈0,short_jump〉)) |
---|
851 | (\snd (lookup … (bitvector_of_nat 16 i) policy 〈0,short_jump〉))) |
---|
852 | ) |
---|
853 | ) |
---|
854 | program |
---|
855 | (λprefix.λx.λtl.λprf.λacc. |
---|
856 | let 〈pc, policy〉 ≝ acc in |
---|
857 | let 〈label,instr〉 ≝ x in |
---|
858 | let old_jump_length ≝ lookup_opt ? ? (bitvector_of_nat 16 (|prefix|)) old_policy in |
---|
859 | let add_instr ≝ |
---|
860 | match instr with |
---|
861 | [ Instruction i ⇒ jump_expansion_step_instruction labels pc i |
---|
862 | | Call c ⇒ Some ? (select_call_length labels pc c) |
---|
863 | | Jmp j ⇒ Some ? (select_jump_length labels pc j) |
---|
864 | | _ ⇒ None ? |
---|
865 | ] in |
---|
866 | let 〈new_pc, new_policy〉 ≝ |
---|
867 | let 〈ignore,old_length〉 ≝ lookup … (bitvector_of_nat 16 (|prefix|)) old_policy 〈0, short_jump〉 in |
---|
868 | match add_instr with |
---|
869 | [ None ⇒ (* i.e. it's not a jump *) |
---|
870 | 〈add_instruction_size pc long_jump instr, policy〉 |
---|
871 | | Some ai ⇒ |
---|
872 | let new_length ≝ max_length old_length ai in |
---|
873 | 〈add_instruction_size pc new_length instr, insert … (bitvector_of_nat 16 (|prefix|)) 〈pc, new_length〉 policy〉 |
---|
874 | ] in |
---|
875 | 〈new_pc, new_policy〉 |
---|
876 | ) 〈0, Stub ? ?〉 in |
---|
877 | final_policy. |
---|
878 | [ @conj |
---|
879 | [ #i >append_length <commutative_plus #Hi normalize in Hi; cases (le_to_or_lt_eq … Hi) -Hi; |
---|
880 | [ #Hi; >(nth_append_first ? ? prefix ? ? (le_S_S_to_le … Hi)) #Hj |
---|
881 | lapply (sig2 … policy) #Hacc elim ((proj1 … Hacc) i (le_S_S_to_le … Hi) Hj) #h #n elim n |
---|
882 | -n #n #Hn |
---|
883 | % [ @h | % [ @n | normalize nodelta cases instr normalize nodelta |
---|
884 | [2,3: #x cases (lookup ??? old_policy ?) #y #z @Hn |
---|
885 | |6: #x #y cases (lookup ??? old_policy ?) #w #z @Hn |
---|
886 | |1: #pi cases (lookup ??? old_policy ?) #x #y cases (jump_expansion_step_instruction ???) |
---|
887 | [ @Hn |
---|
888 | | #z normalize nodelta <Hn @lookup_opt_insert_miss |
---|
889 | @bitvector_of_nat_abs @(lt_to_not_eq i (|prefix|)) @(le_S_S_to_le … Hi) |
---|
890 | ] |
---|
891 | |4,5: #id cases (lookup ??? old_policy ?) #x #y normalize nodelta <Hn |
---|
892 | @lookup_opt_insert_miss @bitvector_of_nat_abs |
---|
893 | @(lt_to_not_eq i (|prefix|)) @(le_S_S_to_le … Hi) |
---|
894 | ] ] ] |
---|
895 | | #Hi >nth_append_second >(injective_S … Hi) |
---|
896 | [ <minus_n_n #Hj normalize in Hj; normalize nodelta cases instr in Hj; |
---|
897 | [2,3: #x #Hj @⊥ @Hj |
---|
898 | |6: #x #y #Hj @⊥ @Hj |
---|
899 | |1: #pi cases pi |
---|
900 | [35,36,37: #Hj @⊥ @Hj |
---|
901 | |4,5,8,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32: #x #Hj @⊥ @Hj |
---|
902 | |1,2,3,6,7,33,34: #x #y #Hj @⊥ @Hj |
---|
903 | |9,10,14,15: #j normalize nodelta #_ |
---|
904 | % [1,3,5,7: @pc |
---|
905 | |2,4,6,8: lapply (refl ? (lookup … (bitvector_of_nat ? (|prefix|)) old_policy 〈0,short_jump〉)) |
---|
906 | cases (lookup … (bitvector_of_nat ? (|prefix|)) old_policy 〈0,short_jump〉) in ⊢ (???% → %); |
---|
907 | #x #y #Hl normalize nodelta |
---|
908 | % [1,3,5,7: @(max_length y (select_reljump_length (create_label_map program old_policy) pc j)) |
---|
909 | |2,4,6,8: @lookup_opt_insert_hit |
---|
910 | ] ] |
---|
911 | |11,12,13,16,17: #x #j normalize nodelta #_ |
---|
912 | % [1,3,5,7,9: @pc |
---|
913 | |2,4,6,8,10: lapply (refl ? (lookup … (bitvector_of_nat ? (|prefix|)) old_policy 〈0,short_jump〉)) |
---|
914 | cases (lookup … (bitvector_of_nat ? (|prefix|)) old_policy 〈0,short_jump〉) in ⊢ (???% → %); |
---|
915 | #x #y #Hl normalize nodelta |
---|
916 | % [1,3,5,7,9: @(max_length y (select_reljump_length (create_label_map program old_policy) pc j)) |
---|
917 | |2,4,6,8,10: @lookup_opt_insert_hit |
---|
918 | ] ] |
---|
919 | ] |
---|
920 | |4,5: #j normalize nodelta #_ |
---|
921 | % [1,3: @pc |
---|
922 | |2,4: cases (lookup … (bitvector_of_nat ? (|prefix|)) old_policy 〈0,short_jump〉) |
---|
923 | #x #y normalize nodelta |
---|
924 | % [1: @(max_length y (select_jump_length (create_label_map program old_policy) pc j)) |
---|
925 | |3: @(max_length y (select_call_length (create_label_map program old_policy) pc j)) |
---|
926 | |2,4: @lookup_opt_insert_hit |
---|
927 | ] |
---|
928 | ] |
---|
929 | ] |
---|
930 | | @le_n |
---|
931 | ] |
---|
932 | ] |
---|
933 | | #Hjip #i >append_length <commutative_plus #Hi normalize in Hi; cases (le_to_or_lt_eq … Hi) -Hi; |
---|
934 | [ #Hi >(nth_append_first ? ? prefix ? ? (le_S_S_to_le … Hi)) #Hj |
---|
935 | lapply (sig2 … policy) #Hpolicy normalize nodelta cases instr normalize nodelta |
---|
936 | [2,3: #x cases (lookup ? 16 (bitvector_of_nat 16 (|prefix|)) old_policy ?) #y #z |
---|
937 | normalize nodelta @(proj2 ? ? Hpolicy (proj1 ? ? Hpolicy) ? (le_S_S_to_le … Hi) Hj) |
---|
938 | |6: #x #y cases (lookup ? 16 (bitvector_of_nat 16 (|prefix|)) old_policy ?) #w #z |
---|
939 | normalize nodelta @(proj2 ? ? Hpolicy (proj1 ? ? Hpolicy) ? (le_S_S_to_le … Hi) Hj) |
---|
940 | |1: #pi cases (lookup ? 16 (bitvector_of_nat 16 (|prefix|)) old_policy ?) #x #y |
---|
941 | cases (jump_expansion_step_instruction ???) |
---|
942 | [ @(proj2 ? ? Hpolicy (proj1 ? ? Hpolicy) ? (le_S_S_to_le … Hi) Hj) |
---|
943 | | #z normalize nodelta |
---|
944 | whd in match (snd ℕ jump_expansion_policy ?); >lookup_insert_miss |
---|
945 | [ @(proj2 … Hpolicy (proj1 ? ? Hpolicy) ? (le_S_S_to_le … Hi) Hj) |
---|
946 | | @bitvector_of_nat_abs @(lt_to_not_eq i (|prefix|)) @(le_S_S_to_le … Hi) |
---|
947 | ] |
---|
948 | ] |
---|
949 | |4,5: #id cases (lookup ? 16 (bitvector_of_nat 16 (|prefix|)) old_policy ?) #x #y |
---|
950 | normalize nodelta >lookup_insert_miss |
---|
951 | [1,3: @(proj2 … Hpolicy (proj1 ? ? Hpolicy) ? (le_S_S_to_le … Hi) Hj) |
---|
952 | |2,4: @bitvector_of_nat_abs @(lt_to_not_eq i (|prefix|)) @(le_S_S_to_le … Hi) |
---|
953 | ] |
---|
954 | ] |
---|
955 | | #Hi >nth_append_second >(injective_S … Hi) |
---|
956 | [ <minus_n_n #Hj normalize in Hj; normalize nodelta cases instr in Hj; |
---|
957 | [2,3: #x #Hj @⊥ @Hj |
---|
958 | |6: #x #y #Hj @⊥ @Hj |
---|
959 | |1: #pi cases pi |
---|
960 | [35,36,37: #Hj @⊥ @Hj |
---|
961 | |4,5,8,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32: #x #Hj @⊥ @Hj |
---|
962 | |1,2,3,6,7,33,34: #x #y #Hj @⊥ @Hj |
---|
963 | |9,10,14,15: #j normalize nodelta #_ |
---|
964 | cases (lookup ???? 〈0,short_jump〉) #x #y |
---|
965 | whd in match (snd ℕ jump_expansion_policy ?); |
---|
966 | >lookup_insert_hit normalize @jmpleq_max_length |
---|
967 | |11,12,13,16,17: #z #j normalize nodelta #_ |
---|
968 | cases (lookup ???? 〈0,short_jump〉) #x #y |
---|
969 | whd in match (snd ℕ jump_expansion_policy ?); |
---|
970 | >lookup_insert_hit normalize @jmpleq_max_length |
---|
971 | ] |
---|
972 | |4,5: #id #_ cases (lookup ???? 〈0,short_jump〉) #x #y |
---|
973 | whd in match (snd ℕ jump_expansion_policy ?); |
---|
974 | >lookup_insert_hit normalize @jmpleq_max_length |
---|
975 | ] |
---|
976 | | @le_n |
---|
977 | ] |
---|
978 | ] |
---|
979 | ] |
---|
980 | | @conj |
---|
981 | [ #i #Hi @⊥ @(absurd (i<0)) [ @Hi | @(not_le_Sn_O) ] |
---|
982 | | #H #i #Hi @⊥ @(absurd (i<0)) [ @Hi | @(not_le_Sn_O) ] |
---|
983 | ] |
---|
984 | ] |
---|
985 | qed. |
---|
986 | |
---|
987 | let rec jump_expansion_internal (program: list labelled_instruction) |
---|
988 | (n: ℕ) on n: (Σpolicy:jump_expansion_policy.jump_in_policy program policy) ≝ |
---|
989 | match n with |
---|
990 | [ O ⇒ jump_expansion_start program |
---|
991 | | S m ⇒ jump_expansion_step program (jump_expansion_internal program m) |
---|
992 | ]. |
---|
993 | [ @(sig2 … (jump_expansion_start program)) |
---|
994 | | @(proj1 … (sig2 … (jump_expansion_step program (jump_expansion_internal program m)))) |
---|
995 | ] |
---|
996 | qed. |
---|
997 | |
---|
998 | (**************************************** START OF POLICY ABSTRACTION ********************) |
---|
999 | |
---|
1000 | definition policy_type ≝ Word → jump_length. |
---|
1001 | |
---|
1002 | definition jump_expansion': pseudo_assembly_program → policy_type ≝ |
---|
1003 | λprogram.λpc. |
---|
1004 | let policy ≝ jump_expansion_internal (\snd program) (|\snd program|) in |
---|
1005 | let 〈n,j〉 ≝ lookup ? ? pc policy 〈0, long_jump〉 in |
---|
1006 | j. |
---|
1007 | |
---|
1008 | definition assembly_1_pseudoinstruction_safe ≝ |
---|
1009 | λprogram: pseudo_assembly_program. |
---|
1010 | λjump_expansion: policy_type. |
---|
1011 | λppc: Word. |
---|
1012 | λpc: Word. |
---|
1013 | λlookup_labels. |
---|
1014 | λlookup_datalabels. |
---|
1015 | λi. |
---|
1016 | let expansion ≝ jump_expansion ppc in |
---|
1017 | match expand_pseudo_instruction_safe lookup_labels lookup_datalabels pc expansion i with |
---|
1018 | [ None ⇒ None ? |
---|
1019 | | Some pseudos ⇒ |
---|
1020 | let mapped ≝ map ? ? assembly1 pseudos in |
---|
1021 | let flattened ≝ flatten ? mapped in |
---|
1022 | let pc_len ≝ length ? flattened in |
---|
1023 | Some ? (〈pc_len, flattened〉) |
---|
1024 | ]. |
---|
1025 | |
---|
1026 | definition construct_costs_safe ≝ |
---|
1027 | λprogram. |
---|
1028 | λjump_expansion: policy_type. |
---|
1029 | λpseudo_program_counter: nat. |
---|
1030 | λprogram_counter: nat. |
---|
1031 | λcosts: BitVectorTrie costlabel 16. |
---|
1032 | λi. |
---|
1033 | match i with |
---|
1034 | [ Cost cost ⇒ |
---|
1035 | let program_counter_bv ≝ bitvector_of_nat ? program_counter in |
---|
1036 | Some ? 〈program_counter, (insert … program_counter_bv cost costs)〉 |
---|
1037 | | _ ⇒ |
---|
1038 | let pc_bv ≝ bitvector_of_nat ? program_counter in |
---|
1039 | let ppc_bv ≝ bitvector_of_nat ? pseudo_program_counter in |
---|
1040 | let lookup_labels ≝ λx.pc_bv in |
---|
1041 | let lookup_datalabels ≝ λx.zero ? in |
---|
1042 | let pc_delta_assembled ≝ |
---|
1043 | assembly_1_pseudoinstruction_safe program jump_expansion ppc_bv pc_bv |
---|
1044 | lookup_labels lookup_datalabels i |
---|
1045 | in |
---|
1046 | match pc_delta_assembled with |
---|
1047 | [ None ⇒ None ? |
---|
1048 | | Some pc_delta_assembled ⇒ |
---|
1049 | let 〈pc_delta, assembled〉 ≝ pc_delta_assembled in |
---|
1050 | Some ? 〈pc_delta + program_counter, costs〉 |
---|
1051 | ] |
---|
1052 | ]. |
---|
1053 | |
---|
1054 | (* This establishes the correspondence between pseudo program counters and |
---|
1055 | program counters. It is at the heart of the proof. *) |
---|
1056 | (*CSC: code taken from build_maps *) |
---|
1057 | definition sigma00: pseudo_assembly_program → policy_type → list ? → ? → option (nat × (nat × (BitVectorTrie Word 16))) ≝ |
---|
1058 | λinstr_list. |
---|
1059 | λjump_expansion: policy_type. |
---|
1060 | λl:list labelled_instruction. |
---|
1061 | λacc. |
---|
1062 | foldl ?? |
---|
1063 | (λt,i. |
---|
1064 | match t with |
---|
1065 | [ None ⇒ None ? |
---|
1066 | | Some ppc_pc_map ⇒ |
---|
1067 | let 〈ppc,pc_map〉 ≝ ppc_pc_map in |
---|
1068 | let 〈program_counter, sigma_map〉 ≝ pc_map in |
---|
1069 | let 〈label, i〉 ≝ i in |
---|
1070 | match construct_costs_safe instr_list jump_expansion ppc program_counter (Stub …) i with |
---|
1071 | [ None ⇒ None ? |
---|
1072 | | Some pc_ignore ⇒ |
---|
1073 | let 〈pc,ignore〉 ≝ pc_ignore in |
---|
1074 | Some … 〈S ppc,〈pc, insert ? ? (bitvector_of_nat ? ppc) (bitvector_of_nat ? pc) sigma_map〉〉 ] |
---|
1075 | ]) acc l. |
---|
1076 | |
---|
1077 | definition sigma0: pseudo_assembly_program → policy_type → option (nat × (nat × (BitVectorTrie Word 16))) |
---|
1078 | ≝ λprog.λjump_expansion.sigma00 prog jump_expansion (\snd prog) (Some ? 〈0, 〈0, (Stub ? ?)〉〉). |
---|
1079 | |
---|
1080 | definition tech_pc_sigma00: pseudo_assembly_program → policy_type → list labelled_instruction → option (nat × nat) ≝ |
---|
1081 | λprogram,jump_expansion,instr_list. |
---|
1082 | match sigma00 program jump_expansion instr_list (Some ? 〈0, 〈0, (Stub ? ?)〉〉) (* acc copied from sigma0 *) with |
---|
1083 | [ None ⇒ None … |
---|
1084 | | Some result ⇒ |
---|
1085 | let 〈ppc,pc_sigma_map〉 ≝ result in |
---|
1086 | let 〈pc,map〉 ≝ pc_sigma_map in |
---|
1087 | Some … 〈ppc,pc〉 ]. |
---|
1088 | |
---|
1089 | definition sigma_safe: pseudo_assembly_program → policy_type → option (Word → Word) ≝ |
---|
1090 | λinstr_list,jump_expansion. |
---|
1091 | match sigma0 instr_list jump_expansion with |
---|
1092 | [ None ⇒ None ? |
---|
1093 | | Some result ⇒ |
---|
1094 | let 〈ppc,pc_sigma_map〉 ≝ result in |
---|
1095 | let 〈pc, sigma_map〉 ≝ pc_sigma_map in |
---|
1096 | if gtb pc (2^16) then |
---|
1097 | None ? |
---|
1098 | else |
---|
1099 | Some ? (λx.lookup ?? x sigma_map (zero …)) ]. |
---|
1100 | |
---|
1101 | (* stuff about policy *) |
---|
1102 | |
---|
1103 | definition policy_ok ≝ λjump_expansion,p. sigma_safe p jump_expansion ≠ None …. |
---|
1104 | |
---|
1105 | definition policy ≝ λp. Σjump_expansion:policy_type. policy_ok jump_expansion p. |
---|
1106 | |
---|
1107 | lemma eject_policy: ∀p. policy p → policy_type. |
---|
1108 | #p #pol @(eject … pol) |
---|
1109 | qed. |
---|
1110 | |
---|
1111 | coercion eject_policy nocomposites: ∀p.∀pol:policy p. policy_type ≝ eject_policy on _pol:(policy ?) to policy_type. |
---|
1112 | |
---|
1113 | definition sigma: ∀p:pseudo_assembly_program. policy p → Word → Word ≝ |
---|
1114 | λp,policy. |
---|
1115 | match sigma_safe p (eject … policy) return λr:option (Word → Word). r ≠ None … → Word → Word with |
---|
1116 | [ None ⇒ λabs. ⊥ |
---|
1117 | | Some r ⇒ λ_.r] (sig2 … policy). |
---|
1118 | cases abs /2/ |
---|
1119 | qed. |
---|
1120 | |
---|
1121 | example sigma_0: ∀p,pol. sigma p pol (bitvector_of_nat ? 0) = bitvector_of_nat ? 0. |
---|
1122 | cases daemon. |
---|
1123 | qed. |
---|
1124 | |
---|
1125 | axiom fetch_pseudo_instruction_split: |
---|
1126 | ∀instr_list,ppc. |
---|
1127 | ∃pre,suff,lbl. |
---|
1128 | (pre @ [〈lbl,\fst (fetch_pseudo_instruction instr_list ppc)〉]) @ suff = instr_list. |
---|
1129 | |
---|
1130 | lemma sigma00_append: |
---|
1131 | ∀instr_list,jump_expansion,l1,l2,acc. |
---|
1132 | sigma00 instr_list jump_expansion (l1@l2) acc = |
---|
1133 | sigma00 instr_list jump_expansion |
---|
1134 | l2 (sigma00 instr_list jump_expansion l1 acc). |
---|
1135 | whd in match sigma00; normalize nodelta; |
---|
1136 | #instr_list #jump_expansion #l1 #l2 #acc @foldl_append |
---|
1137 | qed. |
---|
1138 | |
---|
1139 | lemma sigma00_strict: |
---|
1140 | ∀instr_list,jump_expansion,l,acc. acc = None ? → |
---|
1141 | sigma00 instr_list jump_expansion l acc = None …. |
---|
1142 | #instr_list #jump_expansion #l elim l |
---|
1143 | [ #acc #H >H % |
---|
1144 | | #hd #tl #IH #acc #H >H change with (sigma00 ?? tl ? = ?) @IH % ] |
---|
1145 | qed. |
---|
1146 | |
---|
1147 | lemma policy_ok_prefix_ok: |
---|
1148 | ∀program.∀pol:policy program.∀suffix,prefix. |
---|
1149 | prefix@suffix = \snd program → |
---|
1150 | sigma00 program pol prefix (Some … 〈0, 〈0, Stub …〉〉) ≠ None …. |
---|
1151 | * #preamble #instr_list #pol #suffix #prefix #prf whd in prf:(???%); |
---|
1152 | generalize in match (sig2 ?? pol); whd in prf:(???%); <prf in pol; #pol |
---|
1153 | whd in match policy_ok; whd in match sigma_safe; whd in match sigma0; |
---|
1154 | normalize nodelta >sigma00_append |
---|
1155 | cases (sigma00 ?? prefix ?) |
---|
1156 | [2: #x #_ % #abs destruct(abs) |
---|
1157 | | * #abs @⊥ @abs >sigma00_strict % ] |
---|
1158 | qed. |
---|
1159 | |
---|
1160 | lemma policy_ok_prefix_hd_ok: |
---|
1161 | ∀program.∀pol:policy program.∀suffix,hd,prefix,ppc_pc_map. |
---|
1162 | (prefix@[hd])@suffix = \snd program → |
---|
1163 | Some ? ppc_pc_map = sigma00 program pol prefix (Some … 〈0, 〈0, Stub …〉〉) → |
---|
1164 | let 〈ppc,pc_map〉 ≝ ppc_pc_map in |
---|
1165 | let 〈program_counter, sigma_map〉 ≝ pc_map in |
---|
1166 | let 〈label, i〉 ≝ hd in |
---|
1167 | construct_costs_safe program pol ppc program_counter (Stub …) i ≠ None …. |
---|
1168 | * #preamble #instr_list #pol #suffix #hd #prefix #ppc_pc_map #EQ1 #EQ2 |
---|
1169 | generalize in match (policy_ok_prefix_ok 〈preamble,instr_list〉 pol suffix |
---|
1170 | (prefix@[hd]) EQ1) in ⊢ ?; >sigma00_append <EQ2 whd in ⊢ (?(??%?) → ?); |
---|
1171 | @pair_elim' #ppc #pc_map #EQ3 normalize nodelta |
---|
1172 | @pair_elim' #pc #map #EQ4 normalize nodelta |
---|
1173 | @pair_elim' #l' #i' #EQ5 normalize nodelta |
---|
1174 | cases (construct_costs_safe ??????) normalize |
---|
1175 | [* #abs @⊥ @abs % | #X #_ % #abs destruct(abs)] |
---|
1176 | qed. |
---|
1177 | |
---|
1178 | definition expand_pseudo_instruction: |
---|
1179 | ∀program:pseudo_assembly_program.∀pol: policy program. |
---|
1180 | ∀ppc:Word.∀lookup_labels,lookup_datalabels,pc. |
---|
1181 | lookup_labels = (λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x)) → |
---|
1182 | lookup_datalabels = (λx. lookup ?? x (construct_datalabels (\fst program)) (zero ?)) → |
---|
1183 | let pi ≝ \fst (fetch_pseudo_instruction (\snd program) ppc) in |
---|
1184 | pc = sigma program pol ppc → |
---|
1185 | Σres:list instruction. Some … res = expand_pseudo_instruction_safe lookup_labels lookup_datalabels pc (pol ppc) pi |
---|
1186 | ≝ λprogram,pol,ppc,lookup_labels,lookup_datalabels,pc,prf1,prf2,prf3. |
---|
1187 | match expand_pseudo_instruction_safe lookup_labels lookup_datalabels pc (pol ppc) (\fst (fetch_pseudo_instruction (\snd program) ppc)) with |
---|
1188 | [ None ⇒ let dummy ≝ [ ] in dummy |
---|
1189 | | Some res ⇒ res ]. |
---|
1190 | [ @⊥ whd in p:(??%??); |
---|
1191 | generalize in match (sig2 ?? pol) whd in ⊢ (% → ?) |
---|
1192 | whd in ⊢ (?(??%?) → ?) change in ⊢ (?(??(match % with [_ ⇒ ? | _ ⇒ ?])?) → ?) with (sigma00 ????) |
---|
1193 | generalize in match (refl … (sigma00 program pol (\snd program) (Some ? 〈O,〈O,Stub (BitVector 16) 16〉〉))) |
---|
1194 | cases (sigma00 ????) in ⊢ (??%? → %) normalize nodelta [#_ * #abs @abs %] |
---|
1195 | #res #K |
---|
1196 | cases (fetch_pseudo_instruction_split (\snd program) ppc) #pre * #suff * #lbl #EQ1 |
---|
1197 | generalize in match (policy_ok_prefix_hd_ok program pol … EQ1 ?) in ⊢ ? |
---|
1198 | cases daemon (* CSC: XXXXXXXX Ero qui |
---|
1199 | |
---|
1200 | [3: @policy_ok_prefix_ok ] |
---|
1201 | | sigma00 program pol pre |
---|
1202 | |
---|
1203 | |
---|
1204 | |
---|
1205 | QUA USARE LEMMA policy_ok_prefix_hd_ok combinato a lemma da fare che |
---|
1206 | fetch ppc = hd sse program = pre @ [hd] @ tl e |pre| = ppc |
---|
1207 | per concludere construct_costs_safe ≠ None *) |
---|
1208 | | >p %] |
---|
1209 | qed. |
---|
1210 | |
---|
1211 | (* MAIN AXIOM HERE, HIDDEN USING cases daemon *) |
---|
1212 | definition assembly_1_pseudoinstruction': |
---|
1213 | ∀program:pseudo_assembly_program.∀pol: policy program. |
---|
1214 | ∀ppc:Word.∀lookup_labels,lookup_datalabels,pi. |
---|
1215 | lookup_labels = (λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x)) → |
---|
1216 | lookup_datalabels = (λx. lookup ?? x (construct_datalabels (\fst program)) (zero ?)) → |
---|
1217 | \fst (fetch_pseudo_instruction (\snd program) ppc) = pi → |
---|
1218 | Σres:(nat × (list Byte)). |
---|
1219 | Some … res = assembly_1_pseudoinstruction_safe program pol ppc (sigma program pol ppc) lookup_labels lookup_datalabels pi ∧ |
---|
1220 | let 〈len,code〉 ≝ res in |
---|
1221 | sigma program pol (\snd (half_add ? ppc (bitvector_of_nat ? 1))) = |
---|
1222 | bitvector_of_nat … (nat_of_bitvector … (sigma program pol ppc) + len) |
---|
1223 | ≝ λprogram: pseudo_assembly_program. |
---|
1224 | λpol: policy program. |
---|
1225 | λppc: Word. |
---|
1226 | λlookup_labels. |
---|
1227 | λlookup_datalabels. |
---|
1228 | λpi. |
---|
1229 | λprf1,prf2,prf3. |
---|
1230 | match assembly_1_pseudoinstruction_safe program pol ppc (sigma program pol ppc) lookup_labels lookup_datalabels pi with |
---|
1231 | [ None ⇒ let dummy ≝ 〈0,[ ]〉 in dummy |
---|
1232 | | Some res ⇒ res ]. |
---|
1233 | [ @⊥ elim pi in p [*] |
---|
1234 | try (#ARG1 #ARG2 #ARG3 #abs) try (#ARG1 #ARG2 #abs) try (#ARG1 #abs) try #abs |
---|
1235 | generalize in match (jmeq_to_eq ??? abs) -abs; #abs whd in abs:(??%?); try destruct(abs) |
---|
1236 | whd in abs:(??match % with [_ ⇒ ? | _ ⇒ ?]?); |
---|
1237 | (* WRONG HERE, NEEDS LEMMA SAYING THAT THE POLICY DOES NOT RETURN MEDIUM! *) |
---|
1238 | cases daemon |
---|
1239 | | % [ >p %] |
---|
1240 | cases res in p ⊢ %; -res; #len #code #EQ normalize nodelta; |
---|
1241 | (* THIS SHOULD BE TRUE INSTEAD *) |
---|
1242 | cases daemon] |
---|
1243 | qed. |
---|
1244 | |
---|
1245 | definition assembly_1_pseudoinstruction: |
---|
1246 | ∀program:pseudo_assembly_program.∀pol: policy program. |
---|
1247 | ∀ppc:Word.∀lookup_labels,lookup_datalabels,pi. |
---|
1248 | lookup_labels = (λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x)) → |
---|
1249 | lookup_datalabels = (λx. lookup ?? x (construct_datalabels (\fst program)) (zero ?)) → |
---|
1250 | \fst (fetch_pseudo_instruction (\snd program) ppc) = pi → |
---|
1251 | nat × (list Byte) |
---|
1252 | ≝ λprogram,pol,ppc,lookup_labels,lookup_datalabels,pi,prf1,prf2,prf3. |
---|
1253 | assembly_1_pseudoinstruction' program pol ppc lookup_labels lookup_datalabels pi prf1 |
---|
1254 | prf2 prf3. |
---|
1255 | |
---|
1256 | lemma assembly_1_pseudoinstruction_ok1: |
---|
1257 | ∀program:pseudo_assembly_program.∀pol: policy program. |
---|
1258 | ∀ppc:Word.∀lookup_labels,lookup_datalabels,pi. |
---|
1259 | ∀prf1:lookup_labels = (λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x)). |
---|
1260 | ∀prf2:lookup_datalabels = (λx. lookup ?? x (construct_datalabels (\fst program)) (zero ?)). |
---|
1261 | ∀prf3:\fst (fetch_pseudo_instruction (\snd program) ppc) = pi. |
---|
1262 | Some … (assembly_1_pseudoinstruction program pol ppc lookup_labels lookup_datalabels pi prf1 prf2 prf3) |
---|
1263 | = assembly_1_pseudoinstruction_safe program pol ppc (sigma program pol ppc) lookup_labels lookup_datalabels pi. |
---|
1264 | #program #pol #ppc #lookup_labels #lookup_datalabels #pi #prf1 #prf2 #prf3 |
---|
1265 | cases (sig2 … (assembly_1_pseudoinstruction' program pol ppc lookup_labels lookup_datalabels pi prf1 prf2 prf3)) |
---|
1266 | #H1 #_ @H1 |
---|
1267 | qed. |
---|
1268 | |
---|
1269 | (* MAIN AXIOM HERE, HIDDEN USING cases daemon *) |
---|
1270 | definition construct_costs': |
---|
1271 | ∀program. ∀pol:policy program. ∀ppc,pc,costs,i. |
---|
1272 | Σres:(nat × (BitVectorTrie Word 16)). Some … res = construct_costs_safe program pol ppc pc costs i |
---|
1273 | ≝ |
---|
1274 | λprogram.λpol: policy program.λppc,pc,costs,i. |
---|
1275 | match construct_costs_safe program pol ppc pc costs i with |
---|
1276 | [ None ⇒ let dummy ≝ 〈0, Stub ??〉 in dummy |
---|
1277 | | Some res ⇒ res ]. |
---|
1278 | [ cases daemon |
---|
1279 | | >p %] |
---|
1280 | qed. |
---|
1281 | |
---|
1282 | definition construct_costs ≝ |
---|
1283 | λprogram,pol,ppc,pc,costs,i. eject … (construct_costs' program pol ppc pc costs i). |
---|
1284 | |
---|
1285 | (* |
---|
1286 | axiom suffix_of: ∀A:Type[0]. ∀l,prefix:list A. list A. |
---|
1287 | axiom suffix_of_ok: ∀A,l,prefix. prefix @ suffix_of A l prefix = l. |
---|
1288 | |
---|
1289 | axiom foldl_strong_step: |
---|
1290 | ∀A:Type[0]. |
---|
1291 | ∀P: list A → Type[0]. |
---|
1292 | ∀l: list A. |
---|
1293 | ∀H: ∀prefix,hd,tl. l = prefix @ [hd] @ tl → P prefix → P (prefix @ [hd]). |
---|
1294 | ∀acc: P [ ]. |
---|
1295 | ∀Q: ∀prefix. P prefix → Prop. |
---|
1296 | ∀HQ: ∀prefix,hd,tl.∀prf: l = prefix @ [hd] @ tl. |
---|
1297 | ∀acc: P prefix. Q prefix acc → Q (prefix @ [hd]) (H prefix hd tl prf acc). |
---|
1298 | Q [ ] acc → |
---|
1299 | Q l (foldl_strong A P l H acc). |
---|
1300 | (* |
---|
1301 | #A #P #l #H #acc #Q #HQ #Hacc normalize; |
---|
1302 | generalize in match |
---|
1303 | (foldl_strong ? |
---|
1304 | (λpre. Q pre (foldl_strong_internal A P l (suffix_of A l pre) ? [ ] pre acc ?)) |
---|
1305 | l ? Hacc) |
---|
1306 | [3: >suffix_of_ok % | 2: #prefix #hd #tl #EQ @(H prefix hd (tl@suffix_of A l pre) EQ) ] |
---|
1307 | [2: #prefix #hd #tl #prf #X whd in ⊢ (??%) |
---|
1308 | #K |
---|
1309 | |
---|
1310 | generalize in match |
---|
1311 | (foldl_strong ? |
---|
1312 | (λpre. Q pre (foldl_strong_internal A P l H pre (suffix_of A l pre) acc (suffix_of_ok A l pre)))) |
---|
1313 | [2: @H |
---|
1314 | *) |
---|
1315 | |
---|
1316 | axiom foldl_elim: |
---|
1317 | ∀A:Type[0]. |
---|
1318 | ∀B: Type[0]. |
---|
1319 | ∀H: A → B → A. |
---|
1320 | ∀acc: A. |
---|
1321 | ∀l: list B. |
---|
1322 | ∀Q: A → Prop. |
---|
1323 | (∀acc:A.∀b:B. Q acc → Q (H acc b)) → |
---|
1324 | Q acc → |
---|
1325 | Q (foldl A B H acc l). |
---|
1326 | *) |
---|
1327 | |
---|
1328 | lemma option_destruct_Some: ∀A,a,b. Some A a = Some A b → a=b. |
---|
1329 | #A #a #b #EQ destruct // |
---|
1330 | qed. |
---|
1331 | |
---|
1332 | (* |
---|
1333 | lemma tech_pc_sigma00_append_Some: |
---|
1334 | ∀program.∀pol:policy program.∀prefix,costs,label,i,ppc,pc. |
---|
1335 | tech_pc_sigma00 program pol prefix = Some … 〈ppc,pc〉 → |
---|
1336 | tech_pc_sigma00 program pol (prefix@[〈label,i〉]) = Some … 〈S ppc,\fst (construct_costs program pol … ppc pc costs i)〉. |
---|
1337 | #program #pol #prefix #costs #label #i #ppc #pc #H |
---|
1338 | whd in match tech_pc_sigma00 in ⊢ %; normalize nodelta; |
---|
1339 | whd in match sigma00 in ⊢ %; normalize nodelta in ⊢ %; |
---|
1340 | generalize in match (sig2 … pol) whd in ⊢ (% → ?) whd in ⊢ (?(??%?) → ?) |
---|
1341 | whd in match sigma0; normalize nodelta; |
---|
1342 | >foldl_step |
---|
1343 | change with (? → match match sigma00 program pol prefix with [None ⇒ ? | Some res ⇒ ?] with [ None ⇒ ? | Some res ⇒ ? ] = ?) |
---|
1344 | whd in match tech_pc_sigma00 in H; normalize nodelta in H; |
---|
1345 | cases (sigma00 program pol prefix) in H ⊢ % |
---|
1346 | [ whd in ⊢ (??%% → ?) #abs destruct(abs) |
---|
1347 | | * #ppc' * #pc' #sigma_map normalize nodelta; #H generalize in match (option_destruct_Some ??? H) |
---|
1348 | |
---|
1349 | normalize nodelta; -H; |
---|
1350 | |
---|
1351 | |
---|
1352 | generalize in match H; -H; |
---|
1353 | generalize in match (foldl ?????); in H ⊢ (??match match % with [_ ⇒ ? | _ ⇒ ?] with [_ ⇒ ? | _ ⇒ ?]?) |
---|
1354 | [2: whd in ⊢ (??%%) |
---|
1355 | XXX |
---|
1356 | *) |
---|
1357 | |
---|
1358 | axiom construct_costs_sigma: |
---|
1359 | ∀p.∀pol:policy p.∀ppc,pc,costs,i. |
---|
1360 | bitvector_of_nat ? pc = sigma p pol (bitvector_of_nat ? ppc) → |
---|
1361 | bitvector_of_nat ? (\fst (construct_costs p pol ppc pc costs i)) = sigma p pol (bitvector_of_nat 16 (S ppc)). |
---|
1362 | |
---|
1363 | axiom tech_pc_sigma00_append_Some: |
---|
1364 | ∀program.∀pol:policy program.∀prefix,costs,label,i,ppc,pc. |
---|
1365 | tech_pc_sigma00 program pol prefix = Some … 〈ppc,pc〉 → |
---|
1366 | tech_pc_sigma00 program pol (prefix@[〈label,i〉]) = Some … 〈S ppc,\fst (construct_costs program pol … ppc pc costs i)〉. |
---|
1367 | |
---|
1368 | definition build_maps: |
---|
1369 | ∀pseudo_program.∀pol:policy pseudo_program. |
---|
1370 | Σres:((BitVectorTrie Word 16) × (BitVectorTrie Word 16)). |
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1371 | let 〈labels,costs〉 ≝ res in |
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1372 | ∀id. occurs_exactly_once id (\snd pseudo_program) → |
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1373 | lookup ?? id labels (zero …) = sigma pseudo_program pol (address_of_word_labels_code_mem (\snd pseudo_program) id) |
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1374 | ≝ |
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1375 | λpseudo_program.λpol:policy pseudo_program. |
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1376 | let result ≝ |
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1377 | foldl_strong |
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1378 | (option Identifier × pseudo_instruction) |
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1379 | (λpre. Σres:((BitVectorTrie Word 16) × (nat × (nat × (BitVectorTrie Word 16)))). |
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1380 | let 〈labels,ppc_pc_costs〉 ≝ res in |
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1381 | let 〈ppc',pc_costs〉 ≝ ppc_pc_costs in |
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1382 | let 〈pc',costs〉 ≝ pc_costs in |
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1383 | And ( And ( |
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1384 | And (bitvector_of_nat ? pc' = sigma pseudo_program pol (bitvector_of_nat ? ppc')) (*∧*) |
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1385 | (ppc' = length … pre)) (*∧ *) |
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1386 | (tech_pc_sigma00 pseudo_program (eject … pol) pre = Some ? 〈ppc',pc'〉)) (*∧*) |
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1387 | (∀id. occurs_exactly_once id pre → |
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1388 | lookup ?? id labels (zero …) = sigma pseudo_program pol (address_of_word_labels_code_mem pre id))) |
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1389 | (\snd pseudo_program) |
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1390 | (λprefix,i,tl,prf,t. |
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1391 | let 〈labels, ppc_pc_costs〉 ≝ t in |
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1392 | let 〈ppc, pc_costs〉 ≝ ppc_pc_costs in |
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1393 | let 〈pc,costs〉 ≝ pc_costs in |
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1394 | let 〈label, i'〉 ≝ i in |
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1395 | let labels ≝ |
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1396 | match label with |
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1397 | [ None ⇒ labels |
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1398 | | Some label ⇒ |
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1399 | let program_counter_bv ≝ bitvector_of_nat ? pc in |
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1400 | insert ? ? label program_counter_bv labels] |
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1401 | in |
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1402 | let construct ≝ construct_costs pseudo_program pol ppc pc costs i' in |
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1403 | 〈labels, 〈S ppc,construct〉〉 |
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1404 | ) 〈(Stub ? ?), 〈0, 〈0, Stub ? ?〉〉〉 |
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1405 | in |
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1406 | let 〈labels, ppc_pc_costs〉 ≝ result in |
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1407 | let 〈ppc,pc_costs〉 ≝ ppc_pc_costs in |
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1408 | let 〈pc, costs〉 ≝ pc_costs in |
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1409 | 〈labels, costs〉. |
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1410 | [2: whd generalize in match (sig2 … t) >p >p1 >p2 >p3 * * * #IHn1 #IH0 #IH1 #IH2 |
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1411 | generalize in match (refl … construct); cases construct in ⊢ (???% → %) #PC #CODE #JMEQ % [% [%]] |
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1412 | [ <(construct_costs_sigma … costs i1 IHn1) change with (? = ?(\fst construct)) >JMEQ % |
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1413 | | >append_length <IH0 normalize; -IHn1; (*CSC: otherwise it diverges!*) // |
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1414 | | >(tech_pc_sigma00_append_Some … costs … IH1) change with (Some … 〈S ppc,\fst construct〉 = ?) >JMEQ % |
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1415 | | #id normalize nodelta; -labels1; cases label; normalize nodelta |
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1416 | [ #K <address_of_word_labels_code_mem_None [2:@K] @IH2 -IHn1; (*CSC: otherwise it diverges!*) // |
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1417 | | #l #H generalize in match (occurs_exactly_once_Some ???? H) in ⊢ ?; |
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1418 | generalize in match (refl … (eq_bv ? l id)); cases (eq_bv … l id) in ⊢ (???% → %) |
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1419 | [ #EQ #_ <(eq_bv_eq … EQ) >lookup_insert_hit >address_of_word_labels_code_mem_Some_hit |
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1420 | <IH0 [@IHn1 | <(eq_bv_eq … EQ) in H #H @H] |
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1421 | | #EQ change with (occurs_exactly_once ?? → ?) #K >lookup_insert_miss [2: -IHn1; (*Andrea:XXXX used to work /2/*)>eq_bv_sym >EQ // ] |
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1422 | <(address_of_word_labels_code_mem_Some_miss … EQ … H) @IH2 -IHn1; //]]] |
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1423 | |3: whd % [% [%]] // [>sigma_0//] #id normalize in ⊢ (% → ?) #abs @⊥ // |
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1424 | | generalize in match (sig2 … result) in ⊢ ?; |
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1425 | normalize nodelta in p ⊢ %; -result; >p in ⊢ (match % with [_ ⇒ ?] → ?) |
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1426 | normalize nodelta; >p1 normalize nodelta; >p2; normalize nodelta; * #_; #H @H] |
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1427 | qed. |
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1428 | |
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1429 | definition assembly: |
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1430 | ∀p:pseudo_assembly_program. policy p → list Byte × (BitVectorTrie Identifier 16) ≝ |
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1431 | λp.let 〈preamble, instr_list〉 ≝ p in |
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1432 | λpol. |
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1433 | let 〈labels,costs〉 ≝ build_maps 〈preamble,instr_list〉 pol in |
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1434 | let datalabels ≝ construct_datalabels preamble in |
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1435 | let lookup_labels ≝ λx. lookup ? ? x labels (zero ?) in |
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1436 | let lookup_datalabels ≝ λx. lookup ? ? x datalabels (zero ?) in |
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1437 | let result ≝ |
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1438 | foldl_strong |
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1439 | (option Identifier × pseudo_instruction) |
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1440 | (λpre. Σpc_ppc_code:(Word × (Word × (list Byte))). |
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1441 | let 〈pc,ppc_code〉 ≝ pc_ppc_code in |
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1442 | let 〈ppc,code〉 ≝ ppc_code in |
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1443 | ∀ppc'. |
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1444 | let 〈pi,newppc〉 ≝ fetch_pseudo_instruction instr_list ppc' in |
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1445 | let 〈len,assembledi〉 ≝ |
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1446 | assembly_1_pseudoinstruction 〈preamble,instr_list〉 pol ppc' lookup_labels lookup_datalabels pi ??? in |
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1447 | True) |
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1448 | instr_list |
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1449 | (λprefix,hd,tl,prf,pc_ppc_code. |
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1450 | let 〈pc, ppc_code〉 ≝ pc_ppc_code in |
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1451 | let 〈ppc, code〉 ≝ ppc_code in |
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1452 | let 〈pc_delta, program〉 ≝ assembly_1_pseudoinstruction 〈preamble,instr_list〉 pol ppc lookup_labels lookup_datalabels (\snd hd) ??? in |
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1453 | let 〈new_pc, flags〉 ≝ add_16_with_carry pc (bitvector_of_nat ? pc_delta) false in |
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1454 | let new_ppc ≝ \snd (half_add ? ppc (bitvector_of_nat ? 1)) in |
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1455 | 〈new_pc, 〈new_ppc, (code @ program)〉〉) |
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1456 | 〈(zero ?), 〈(zero ?), [ ]〉〉 |
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1457 | in |
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1458 | 〈\snd (\snd result), costs〉. |
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1459 | [2,5: % |
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1460 | |*: cases daemon ] |
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1461 | qed. |
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1462 | |
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1463 | definition assembly_unlabelled_program: assembly_program → option (list Byte × (BitVectorTrie Identifier 16)) ≝ |
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1464 | λp. Some ? (〈foldr ? ? (λi,l. assembly1 i @ l) [ ] p, Stub …〉). |
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