1 | include "ASM/ASM.ma". |
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2 | include "ASM/Arithmetic.ma". |
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3 | include "ASM/Fetch.ma". |
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4 | include "ASM/Status.ma". |
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5 | include alias "basics/logic.ma". |
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6 | include alias "arithmetics/nat.ma". |
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7 | |
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8 | definition assembly_preinstruction ≝ |
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9 | λA: Type[0]. |
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10 | λaddr_of: A → Byte. (* relative *) |
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11 | λpre: preinstruction A. |
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12 | match pre with |
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13 | [ ADD addr1 addr2 ⇒ |
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14 | match addr2 return λx. bool_to_Prop (is_in ? [[registr;direct;indirect;data]] x) → ? with |
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15 | [ REGISTER r ⇒ λ_.[ ([[false;false;true;false;true]]) @@ r ] |
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16 | | DIRECT b1 ⇒ λ_.[ ([[false;false;true;false;false;true;false;true]]); b1 ] |
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17 | | INDIRECT i1 ⇒ λ_. [ ([[false;false;true;false;false;true;true;i1]]) ] |
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18 | | DATA b1 ⇒ λ_. [ ([[false;false;true;false;false;true;false;false]]) ; b1 ] |
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19 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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20 | | ADDC addr1 addr2 ⇒ |
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21 | match addr2 return λx. bool_to_Prop (is_in ? [[registr;direct;indirect;data]] x) → ? with |
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22 | [ REGISTER r ⇒ λ_.[ ([[false;false;true;true;true]]) @@ r ] |
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23 | | DIRECT b1 ⇒ λ_.[ ([[false;false;true;true;false;true;false;true]]); b1 ] |
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24 | | INDIRECT i1 ⇒ λ_. [ ([[false;false;true;true;false;true;true;i1]]) ] |
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25 | | DATA b1 ⇒ λ_. [ ([[false;false;true;true;false;true;false;false]]) ; b1 ] |
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26 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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27 | | ANL addrs ⇒ |
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28 | match addrs with |
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29 | [ inl addrs ⇒ match addrs with |
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30 | [ inl addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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31 | match addr2 return λx. bool_to_Prop (is_in ? [[registr;direct;indirect;data]] x) → ? with |
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32 | [ REGISTER r ⇒ λ_.[ ([[false;true;false;true;true]]) @@ r ] |
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33 | | DIRECT b1 ⇒ λ_.[ ([[false;true;false;true;false;true;false;true]]); b1 ] |
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34 | | INDIRECT i1 ⇒ λ_. [ ([[false;true;false;true;false;true;true;i1]]) ] |
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35 | | DATA b1 ⇒ λ_. [ ([[false;true;false;true;false;true;false;false]]) ; b1 ] |
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36 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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37 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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38 | let b1 ≝ |
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39 | match addr1 return λx. bool_to_Prop (is_in ? [[direct]] x) → ? with |
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40 | [ DIRECT b1 ⇒ λ_.b1 |
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41 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) in |
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42 | match addr2 return λx. bool_to_Prop (is_in ? [[acc_a;data]] x) → ? with |
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43 | [ ACC_A ⇒ λ_.[ ([[false;true;false;true;false;false;true;false]]) ; b1 ] |
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44 | | DATA b2 ⇒ λ_. [ ([[false;true;false;true;false;false;true;true]]) ; b1 ; b2 ] |
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45 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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46 | ] |
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47 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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48 | match addr2 return λx. bool_to_Prop (is_in ? [[bit_addr;n_bit_addr]] x) → ? with |
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49 | [ BIT_ADDR b1 ⇒ λ_.[ ([[true;false;false;false;false;false;true;false]]) ; b1 ] |
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50 | | N_BIT_ADDR b1 ⇒ λ_. [ ([[true;false;true;true;false;false;false;false]]) ; b1 ] |
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51 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2)] |
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52 | | CLR addr ⇒ |
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53 | match addr return λx. bool_to_Prop (is_in ? [[acc_a;carry;bit_addr]] x) → ? with |
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54 | [ ACC_A ⇒ λ_. |
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55 | [ ([[true; true; true; false; false; true; false; false]]) ] |
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56 | | CARRY ⇒ λ_. |
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57 | [ ([[true; true; false; false; false; false; true; true]]) ] |
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58 | | BIT_ADDR b1 ⇒ λ_. |
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59 | [ ([[true; true; false; false; false; false; true; false]]) ; b1 ] |
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60 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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61 | | CPL addr ⇒ |
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62 | match addr return λx. bool_to_Prop (is_in ? [[acc_a;carry;bit_addr]] x) → ? with |
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63 | [ ACC_A ⇒ λ_. |
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64 | [ ([[true; true; true; true; false; true; false; false]]) ] |
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65 | | CARRY ⇒ λ_. |
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66 | [ ([[true; false; true; true; false; false; true; true]]) ] |
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67 | | BIT_ADDR b1 ⇒ λ_. |
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68 | [ ([[true; false; true; true; false; false; true; false]]) ; b1 ] |
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69 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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70 | | DA addr ⇒ |
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71 | [ ([[true; true; false; true; false; true; false; false]]) ] |
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72 | | DEC addr ⇒ |
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73 | match addr return λx. bool_to_Prop (is_in ? [[acc_a;registr;direct;indirect]] x) → ? with |
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74 | [ ACC_A ⇒ λ_. |
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75 | [ ([[false; false; false; true; false; true; false; false]]) ] |
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76 | | REGISTER r ⇒ λ_. |
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77 | [ ([[false; false; false; true; true]]) @@ r ] |
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78 | | DIRECT b1 ⇒ λ_. |
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79 | [ ([[false; false; false; true; false; true; false; true]]); b1 ] |
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80 | | INDIRECT i1 ⇒ λ_. |
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81 | [ ([[false; false; false; true; false; true; true; i1]]) ] |
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82 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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83 | | DJNZ addr1 addr2 ⇒ |
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84 | let b2 ≝ addr_of addr2 in |
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85 | match addr1 return λx. bool_to_Prop (is_in ? [[registr;direct]] x) → ? with |
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86 | [ REGISTER r ⇒ λ_. |
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87 | [ ([[true; true; false; true; true]]) @@ r ; b2 ] |
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88 | | DIRECT b1 ⇒ λ_. |
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89 | [ ([[true; true; false; true; false; true; false; true]]); b1; b2 ] |
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90 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) |
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91 | | JC addr ⇒ |
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92 | let b1 ≝ addr_of addr in |
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93 | [ ([[false; true; false; false; false; false; false; false]]); b1 ] |
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94 | | JNC addr ⇒ |
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95 | let b1 ≝ addr_of addr in |
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96 | [ ([[false; true; false; true; false; false; false; false]]); b1 ] |
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97 | | JZ addr ⇒ |
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98 | let b1 ≝ addr_of addr in |
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99 | [ ([[false; true; true; false; false; false; false; false]]); b1 ] |
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100 | | JNZ addr ⇒ |
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101 | let b1 ≝ addr_of addr in |
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102 | [ ([[false; true; true; true; false; false; false; false]]); b1 ] |
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103 | | JB addr1 addr2 ⇒ |
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104 | let b2 ≝ addr_of addr2 in |
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105 | match addr1 return λx. bool_to_Prop (is_in ? [[bit_addr]] x) → ? with |
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106 | [ BIT_ADDR b1 ⇒ λ_. |
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107 | [ ([[false; false; true; false; false; false; false; false]]); b1; b2 ] |
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108 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) |
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109 | | JNB addr1 addr2 ⇒ |
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110 | let b2 ≝ addr_of addr2 in |
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111 | match addr1 return λx. bool_to_Prop (is_in ? [[bit_addr]] x) → ? with |
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112 | [ BIT_ADDR b1 ⇒ λ_. |
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113 | [ ([[false; false; true; true; false; false; false; false]]); b1; b2 ] |
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114 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) |
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115 | | JBC addr1 addr2 ⇒ |
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116 | let b2 ≝ addr_of addr2 in |
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117 | match addr1 return λx. bool_to_Prop (is_in ? [[bit_addr]] x) → ? with |
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118 | [ BIT_ADDR b1 ⇒ λ_. |
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119 | [ ([[false; false; false; true; false; false; false; false]]); b1; b2 ] |
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120 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) |
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121 | | CJNE addrs addr3 ⇒ |
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122 | let b3 ≝ addr_of addr3 in |
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123 | match addrs with |
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124 | [ inl addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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125 | match addr2 return λx. bool_to_Prop (is_in ? [[direct;data]] x) → ? with |
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126 | [ DIRECT b1 ⇒ λ_. |
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127 | [ ([[true; false; true; true; false; true; false; true]]); b1; b3 ] |
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128 | | DATA b1 ⇒ λ_. |
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129 | [ ([[true; false; true; true; false; true; false; false]]); b1; b3 ] |
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130 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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131 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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132 | let b2 ≝ |
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133 | match addr2 return λx. bool_to_Prop (is_in ? [[data]] x) → ? with |
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134 | [ DATA b2 ⇒ λ_. b2 |
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135 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) in |
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136 | match addr1 return λx. bool_to_Prop (is_in ? [[registr;indirect]] x) → list Byte with |
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137 | [ REGISTER r ⇒ λ_. |
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138 | [ ([[true; false; true; true; true]]) @@ r; b2; b3 ] |
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139 | | INDIRECT i1 ⇒ λ_. |
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140 | [ ([[true; false; true; true; false; true; true; i1]]); b2; b3 ] |
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141 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) |
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142 | ] |
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143 | | DIV addr1 addr2 ⇒ |
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144 | [ ([[true;false;false;false;false;true;false;false]]) ] |
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145 | | INC addr ⇒ |
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146 | match addr return λx. bool_to_Prop (is_in ? [[acc_a;registr;direct;indirect;dptr]] x) → ? with |
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147 | [ ACC_A ⇒ λ_. |
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148 | [ ([[false;false;false;false;false;true;false;false]]) ] |
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149 | | REGISTER r ⇒ λ_. |
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150 | [ ([[false;false;false;false;true]]) @@ r ] |
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151 | | DIRECT b1 ⇒ λ_. |
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152 | [ ([[false; false; false; false; false; true; false; true]]); b1 ] |
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153 | | INDIRECT i1 ⇒ λ_. |
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154 | [ ([[false; false; false; false; false; true; true; i1]]) ] |
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155 | | DPTR ⇒ λ_. |
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156 | [ ([[true;false;true;false;false;false;true;true]]) ] |
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157 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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158 | | MOV addrs ⇒ |
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159 | match addrs with |
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160 | [ inl addrs ⇒ |
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161 | match addrs with |
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162 | [ inl addrs ⇒ |
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163 | match addrs with |
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164 | [ inl addrs ⇒ |
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165 | match addrs with |
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166 | [ inl addrs ⇒ |
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167 | match addrs with |
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168 | [ inl addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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169 | match addr2 return λx. bool_to_Prop (is_in ? [[registr;direct;indirect;data]] x) → ? with |
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170 | [ REGISTER r ⇒ λ_.[ ([[true;true;true;false;true]]) @@ r ] |
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171 | | DIRECT b1 ⇒ λ_.[ ([[true;true;true;false;false;true;false;true]]); b1 ] |
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172 | | INDIRECT i1 ⇒ λ_. [ ([[true;true;true;false;false;true;true;i1]]) ] |
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173 | | DATA b1 ⇒ λ_. [ ([[false;true;true;true;false;true;false;false]]) ; b1 ] |
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174 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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175 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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176 | match addr1 return λx. bool_to_Prop (is_in ? [[registr;indirect]] x) → ? with |
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177 | [ REGISTER r ⇒ λ_. |
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178 | match addr2 return λx. bool_to_Prop (is_in ? [[acc_a;direct;data]] x) → ? with |
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179 | [ ACC_A ⇒ λ_.[ ([[true;true;true;true;true]]) @@ r ] |
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180 | | DIRECT b1 ⇒ λ_.[ ([[true;false;true;false;true]]) @@ r; b1 ] |
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181 | | DATA b1 ⇒ λ_. [ ([[false;true;true;true;true]]) @@ r; b1 ] |
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182 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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183 | | INDIRECT i1 ⇒ λ_. |
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184 | match addr2 return λx. bool_to_Prop (is_in ? [[acc_a;direct;data]] x) → ? with |
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185 | [ ACC_A ⇒ λ_.[ ([[true;true;true;true;false;true;true;i1]]) ] |
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186 | | DIRECT b1 ⇒ λ_.[ ([[true;false;true;false;false;true;true;i1]]); b1 ] |
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187 | | DATA b1 ⇒ λ_. [ ([[false;true;true;true;false;true;true;i1]]) ; b1 ] |
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188 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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189 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1)] |
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190 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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191 | let b1 ≝ |
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192 | match addr1 return λx. bool_to_Prop (is_in ? [[direct]] x) → ? with |
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193 | [ DIRECT b1 ⇒ λ_. b1 |
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194 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) in |
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195 | match addr2 return λx. bool_to_Prop (is_in ? [[acc_a;registr;direct;indirect;data]] x) → ? with |
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196 | [ ACC_A ⇒ λ_.[ ([[true;true;true;true;false;true;false;true]]); b1] |
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197 | | REGISTER r ⇒ λ_.[ ([[true;false;false;false;true]]) @@ r; b1 ] |
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198 | | DIRECT b2 ⇒ λ_.[ ([[true;false;false;false;false;true;false;true]]); b1; b2 ] |
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199 | | INDIRECT i1 ⇒ λ_. [ ([[true;false;false;false;false;true;true;i1]]); b1 ] |
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200 | | DATA b2 ⇒ λ_. [ ([[false;true;true;true;false;true;false;true]]); b1; b2 ] |
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201 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2)] |
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202 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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203 | match addr2 return λx. bool_to_Prop (is_in ? [[data16]] x) → ? with |
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204 | [ DATA16 w ⇒ λ_. |
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205 | let b1_b2 ≝ split ? 8 8 w in |
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206 | let b1 ≝ \fst b1_b2 in |
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207 | let b2 ≝ \snd b1_b2 in |
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208 | [ ([[true;false;false;true;false;false;false;false]]); b1; b2] |
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209 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2)] |
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210 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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211 | match addr2 return λx. bool_to_Prop (is_in ? [[bit_addr]] x) → ? with |
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212 | [ BIT_ADDR b1 ⇒ λ_. |
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213 | [ ([[true;false;true;false;false;false;true;false]]); b1 ] |
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214 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2)] |
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215 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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216 | match addr1 return λx. bool_to_Prop (is_in ? [[bit_addr]] x) → ? with |
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217 | [ BIT_ADDR b1 ⇒ λ_. |
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218 | [ ([[true;false;false;true;false;false;true;false]]); b1 ] |
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219 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1)] |
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220 | | MOVX addrs ⇒ |
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221 | match addrs with |
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222 | [ inl addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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223 | match addr2 return λx. bool_to_Prop (is_in ? [[ext_indirect;ext_indirect_dptr]] x) → ? with |
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224 | [ EXT_INDIRECT i1 ⇒ λ_. |
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225 | [ ([[true;true;true;false;false;false;true;i1]]) ] |
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226 | | EXT_INDIRECT_DPTR ⇒ λ_. |
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227 | [ ([[true;true;true;false;false;false;false;false]]) ] |
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228 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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229 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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230 | match addr1 return λx. bool_to_Prop (is_in ? [[ext_indirect;ext_indirect_dptr]] x) → ? with |
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231 | [ EXT_INDIRECT i1 ⇒ λ_. |
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232 | [ ([[true;true;true;true;false;false;true;i1]]) ] |
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233 | | EXT_INDIRECT_DPTR ⇒ λ_. |
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234 | [ ([[true;true;true;true;false;false;false;false]]) ] |
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235 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1)] |
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236 | | MUL addr1 addr2 ⇒ |
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237 | [ ([[true;false;true;false;false;true;false;false]]) ] |
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238 | | NOP ⇒ |
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239 | [ ([[false;false;false;false;false;false;false;false]]) ] |
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240 | | ORL addrs ⇒ |
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241 | match addrs with |
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242 | [ inl addrs ⇒ |
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243 | match addrs with |
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244 | [ inl addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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245 | match addr2 return λx. bool_to_Prop (is_in ? [[registr;data;direct;indirect]] x) → ? with |
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246 | [ REGISTER r ⇒ λ_.[ ([[false;true;false;false;true]]) @@ r ] |
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247 | | DIRECT b1 ⇒ λ_.[ ([[false;true;false;false;false;true;false;true]]); b1 ] |
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248 | | INDIRECT i1 ⇒ λ_. [ ([[false;true;false;false;false;true;true;i1]]) ] |
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249 | | DATA b1 ⇒ λ_. [ ([[false;true;false;false;false;true;false;false]]) ; b1 ] |
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250 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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251 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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252 | let b1 ≝ |
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253 | match addr1 return λx. bool_to_Prop (is_in ? [[direct]] x) → ? with |
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254 | [ DIRECT b1 ⇒ λ_. b1 |
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255 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) in |
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256 | match addr2 return λx. bool_to_Prop (is_in ? [[acc_a;data]] x) → ? with |
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257 | [ ACC_A ⇒ λ_. |
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258 | [ ([[false;true;false;false;false;false;true;false]]); b1 ] |
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259 | | DATA b2 ⇒ λ_. |
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260 | [ ([[false;true;false;false;false;false;true;true]]); b1; b2 ] |
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261 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2)] |
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262 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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263 | match addr2 return λx. bool_to_Prop (is_in ? [[bit_addr;n_bit_addr]] x) → ? with |
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264 | [ BIT_ADDR b1 ⇒ λ_. |
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265 | [ ([[false;true;true;true;false;false;true;false]]); b1 ] |
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266 | | N_BIT_ADDR b1 ⇒ λ_. |
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267 | [ ([[true;false;true;false;false;false;false;false]]); b1 ] |
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268 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2)] |
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269 | | POP addr ⇒ |
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270 | match addr return λx. bool_to_Prop (is_in ? [[direct]] x) → ? with |
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271 | [ DIRECT b1 ⇒ λ_. |
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272 | [ ([[true;true;false;true;false;false;false;false]]) ; b1 ] |
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273 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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274 | | PUSH addr ⇒ |
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275 | match addr return λx. bool_to_Prop (is_in ? [[direct]] x) → ? with |
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276 | [ DIRECT b1 ⇒ λ_. |
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277 | [ ([[true;true;false;false;false;false;false;false]]) ; b1 ] |
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278 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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279 | | RET ⇒ |
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280 | [ ([[false;false;true;false;false;false;true;false]]) ] |
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281 | | RETI ⇒ |
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282 | [ ([[false;false;true;true;false;false;true;false]]) ] |
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283 | | RL addr ⇒ |
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284 | [ ([[false;false;true;false;false;false;true;true]]) ] |
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285 | | RLC addr ⇒ |
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286 | [ ([[false;false;true;true;false;false;true;true]]) ] |
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287 | | RR addr ⇒ |
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288 | [ ([[false;false;false;false;false;false;true;true]]) ] |
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289 | | RRC addr ⇒ |
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290 | [ ([[false;false;false;true;false;false;true;true]]) ] |
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291 | | SETB addr ⇒ |
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292 | match addr return λx. bool_to_Prop (is_in ? [[carry;bit_addr]] x) → ? with |
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293 | [ CARRY ⇒ λ_. |
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294 | [ ([[true;true;false;true;false;false;true;true]]) ] |
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295 | | BIT_ADDR b1 ⇒ λ_. |
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296 | [ ([[true;true;false;true;false;false;true;false]]); b1 ] |
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297 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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298 | | SUBB addr1 addr2 ⇒ |
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299 | match addr2 return λx. bool_to_Prop (is_in ? [[registr;direct;indirect;data]] x) → ? with |
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300 | [ REGISTER r ⇒ λ_. |
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301 | [ ([[true;false;false;true;true]]) @@ r ] |
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302 | | DIRECT b1 ⇒ λ_. |
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303 | [ ([[true;false;false;true;false;true;false;true]]); b1] |
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304 | | INDIRECT i1 ⇒ λ_. |
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305 | [ ([[true;false;false;true;false;true;true;i1]]) ] |
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306 | | DATA b1 ⇒ λ_. |
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307 | [ ([[true;false;false;true;false;true;false;false]]); b1] |
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308 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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309 | | SWAP addr ⇒ |
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310 | [ ([[true;true;false;false;false;true;false;false]]) ] |
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311 | | XCH addr1 addr2 ⇒ |
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312 | match addr2 return λx. bool_to_Prop (is_in ? [[registr;direct;indirect]] x) → ? with |
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313 | [ REGISTER r ⇒ λ_. |
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314 | [ ([[true;true;false;false;true]]) @@ r ] |
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315 | | DIRECT b1 ⇒ λ_. |
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316 | [ ([[true;true;false;false;false;true;false;true]]); b1] |
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317 | | INDIRECT i1 ⇒ λ_. |
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318 | [ ([[true;true;false;false;false;true;true;i1]]) ] |
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319 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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320 | | XCHD addr1 addr2 ⇒ |
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321 | match addr2 return λx. bool_to_Prop (is_in ? [[indirect]] x) → ? with |
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322 | [ INDIRECT i1 ⇒ λ_. |
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323 | [ ([[true;true;false;true;false;true;true;i1]]) ] |
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324 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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325 | | XRL addrs ⇒ |
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326 | match addrs with |
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327 | [ inl addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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328 | match addr2 return λx. bool_to_Prop (is_in ? [[data;registr;direct;indirect]] x) → ? with |
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329 | [ REGISTER r ⇒ λ_. |
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330 | [ ([[false;true;true;false;true]]) @@ r ] |
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331 | | DIRECT b1 ⇒ λ_. |
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332 | [ ([[false;true;true;false;false;true;false;true]]); b1] |
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333 | | INDIRECT i1 ⇒ λ_. |
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334 | [ ([[false;true;true;false;false;true;true;i1]]) ] |
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335 | | DATA b1 ⇒ λ_. |
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336 | [ ([[false;true;true;false;false;true;false;false]]); b1] |
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337 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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338 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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339 | let b1 ≝ |
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340 | match addr1 return λx. bool_to_Prop (is_in ? [[direct]] x) → ? with |
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341 | [ DIRECT b1 ⇒ λ_. b1 |
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342 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) in |
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343 | match addr2 return λx. bool_to_Prop (is_in ? [[acc_a;data]] x) → ? with |
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344 | [ ACC_A ⇒ λ_. |
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345 | [ ([[false;true;true;false;false;false;true;false]]); b1 ] |
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346 | | DATA b2 ⇒ λ_. |
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347 | [ ([[false;true;true;false;false;false;true;true]]); b1; b2 ] |
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348 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2)] |
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349 | ]. |
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350 | |
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351 | definition assembly1 ≝ |
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352 | λi: instruction. |
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353 | match i with |
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354 | [ ACALL addr ⇒ |
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355 | match addr return λx. bool_to_Prop (is_in ? [[addr11]] x) → ? with |
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356 | [ ADDR11 w ⇒ λ_. |
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357 | let v1_v2 ≝ split ? 3 8 w in |
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358 | let v1 ≝ \fst v1_v2 in |
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359 | let v2 ≝ \snd v1_v2 in |
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360 | [ (v1 @@ [[true; false; false; false; true]]) ; v2 ] |
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361 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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362 | | AJMP addr ⇒ |
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363 | match addr return λx. bool_to_Prop (is_in ? [[addr11]] x) → ? with |
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364 | [ ADDR11 w ⇒ λ_. |
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365 | let v1_v2 ≝ split ? 3 8 w in |
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366 | let v1 ≝ \fst v1_v2 in |
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367 | let v2 ≝ \snd v1_v2 in |
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368 | [ (v1 @@ [[false; false; false; false; true]]) ; v2 ] |
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369 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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370 | | JMP adptr ⇒ |
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371 | [ ([[false;true;true;true;false;false;true;true]]) ] |
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372 | | LCALL addr ⇒ |
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373 | match addr return λx. bool_to_Prop (is_in ? [[addr16]] x) → ? with |
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374 | [ ADDR16 w ⇒ λ_. |
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375 | let b1_b2 ≝ split ? 8 8 w in |
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376 | let b1 ≝ \fst b1_b2 in |
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377 | let b2 ≝ \snd b1_b2 in |
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378 | [ ([[false;false;false;true;false;false;true;false]]); b1; b2 ] |
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379 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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380 | | LJMP addr ⇒ |
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381 | match addr return λx. bool_to_Prop (is_in ? [[addr16]] x) → ? with |
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382 | [ ADDR16 w ⇒ λ_. |
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383 | let b1_b2 ≝ split ? 8 8 w in |
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384 | let b1 ≝ \fst b1_b2 in |
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385 | let b2 ≝ \snd b1_b2 in |
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386 | [ ([[false;false;false;false;false;false;true;false]]); b1; b2 ] |
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387 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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388 | | MOVC addr1 addr2 ⇒ |
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389 | match addr2 return λx. bool_to_Prop (is_in ? [[acc_dptr;acc_pc]] x) → ? with |
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390 | [ ACC_DPTR ⇒ λ_. |
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391 | [ ([[true;false;false;true;false;false;true;true]]) ] |
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392 | | ACC_PC ⇒ λ_. |
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393 | [ ([[true;false;false;false;false;false;true;true]]) ] |
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394 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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395 | | SJMP addr ⇒ |
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396 | match addr return λx. bool_to_Prop (is_in ? [[relative]] x) → ? with |
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397 | [ RELATIVE b1 ⇒ λ_. |
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398 | [ ([[true;false;false;false;false;false;false;false]]); b1 ] |
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399 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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400 | | RealInstruction instr ⇒ |
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401 | assembly_preinstruction [[ relative ]] |
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402 | (λx. |
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403 | match x return λs. bool_to_Prop (is_in ? [[ relative ]] s) → ? with |
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404 | [ RELATIVE r ⇒ λ_. r |
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405 | | _ ⇒ λabsd. ⊥ |
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406 | ] (subaddressing_modein … x)) instr |
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407 | ]. |
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408 | cases absd |
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409 | qed. |
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410 | |
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411 | inductive jump_length: Type[0] ≝ |
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412 | | short_jump: jump_length |
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413 | | medium_jump: jump_length |
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414 | | long_jump: jump_length. |
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415 | |
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416 | (* jump_expansion_policy: instruction number ↦ 〈pc, jump_length〉 *) |
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417 | definition jump_expansion_policy ≝ BitVectorTrie (ℕ × jump_length) 16. |
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418 | |
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419 | definition expand_relative_jump_internal: |
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420 | (Identifier → Word) → jump_length → Identifier → Word → ([[relative]] → preinstruction [[relative]]) → |
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421 | option (list instruction) |
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422 | ≝ |
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423 | λlookup_labels,jmp_len.λjmp:Identifier.λpc,i. |
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424 | match jmp_len with |
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425 | [ short_jump ⇒ |
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426 | let lookup_address ≝ lookup_labels jmp in |
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427 | let 〈result, flags〉 ≝ sub_16_with_carry pc lookup_address false in |
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428 | let 〈upper, lower〉 ≝ split ? 8 8 result in |
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429 | if eq_bv ? upper (zero 8) then |
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430 | let address ≝ RELATIVE lower in |
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431 | Some ? [ RealInstruction (i address) ] |
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432 | else |
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433 | None ? |
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434 | | medium_jump ⇒ None … |
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435 | | long_jump ⇒ |
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436 | Some ? |
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437 | [ RealInstruction (i (RELATIVE (bitvector_of_nat ? 2))); |
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438 | SJMP (RELATIVE (bitvector_of_nat ? 3)); (* LJMP size? *) |
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439 | LJMP (ADDR16 (lookup_labels jmp)) |
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440 | ] |
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441 | ]. |
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442 | @ I |
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443 | qed. |
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444 | |
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445 | definition expand_relative_jump: (Identifier → Word) → jump_length → Word → preinstruction Identifier → option (list instruction) ≝ |
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446 | λlookup_labels. |
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447 | λjmp_len: jump_length. |
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448 | λpc. |
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449 | λi: preinstruction Identifier. |
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450 | let rel_jmp ≝ RELATIVE (bitvector_of_nat ? 2) in |
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451 | match i with |
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452 | [ JC jmp ⇒ expand_relative_jump_internal lookup_labels jmp_len jmp pc (JC ?) |
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453 | | JNC jmp ⇒ expand_relative_jump_internal lookup_labels jmp_len jmp pc (JNC ?) |
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454 | | JB baddr jmp ⇒ expand_relative_jump_internal lookup_labels jmp_len jmp pc (JB ? baddr) |
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455 | | JZ jmp ⇒ expand_relative_jump_internal lookup_labels jmp_len jmp pc (JZ ?) |
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456 | | JNZ jmp ⇒ expand_relative_jump_internal lookup_labels jmp_len jmp pc (JNZ ?) |
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457 | | JBC baddr jmp ⇒ expand_relative_jump_internal lookup_labels jmp_len jmp pc (JBC ? baddr) |
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458 | | JNB baddr jmp ⇒ expand_relative_jump_internal lookup_labels jmp_len jmp pc (JNB ? baddr) |
---|
459 | | CJNE addr jmp ⇒ expand_relative_jump_internal lookup_labels jmp_len jmp pc (CJNE ? addr) |
---|
460 | | DJNZ addr jmp ⇒ expand_relative_jump_internal lookup_labels jmp_len jmp pc (DJNZ ? addr) |
---|
461 | | ADD arg1 arg2 ⇒ Some ? [ ADD ? arg1 arg2 ] |
---|
462 | | ADDC arg1 arg2 ⇒ Some ? [ ADDC ? arg1 arg2 ] |
---|
463 | | SUBB arg1 arg2 ⇒ Some ? [ SUBB ? arg1 arg2 ] |
---|
464 | | INC arg ⇒ Some ? [ INC ? arg ] |
---|
465 | | DEC arg ⇒ Some ? [ DEC ? arg ] |
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466 | | MUL arg1 arg2 ⇒ Some ? [ MUL ? arg1 arg2 ] |
---|
467 | | DIV arg1 arg2 ⇒ Some ? [ DIV ? arg1 arg2 ] |
---|
468 | | DA arg ⇒ Some ? [ DA ? arg ] |
---|
469 | | ANL arg ⇒ Some ? [ ANL ? arg ] |
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470 | | ORL arg ⇒ Some ? [ ORL ? arg ] |
---|
471 | | XRL arg ⇒ Some ? [ XRL ? arg ] |
---|
472 | | CLR arg ⇒ Some ? [ CLR ? arg ] |
---|
473 | | CPL arg ⇒ Some ? [ CPL ? arg ] |
---|
474 | | RL arg ⇒ Some ? [ RL ? arg ] |
---|
475 | | RR arg ⇒ Some ? [ RR ? arg ] |
---|
476 | | RLC arg ⇒ Some ? [ RLC ? arg ] |
---|
477 | | RRC arg ⇒ Some ? [ RRC ? arg ] |
---|
478 | | SWAP arg ⇒ Some ? [ SWAP ? arg ] |
---|
479 | | MOV arg ⇒ Some ? [ MOV ? arg ] |
---|
480 | | MOVX arg ⇒ Some ? [ MOVX ? arg ] |
---|
481 | | SETB arg ⇒ Some ? [ SETB ? arg ] |
---|
482 | | PUSH arg ⇒ Some ? [ PUSH ? arg ] |
---|
483 | | POP arg ⇒ Some ? [ POP ? arg ] |
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484 | | XCH arg1 arg2 ⇒ Some ? [ XCH ? arg1 arg2 ] |
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485 | | XCHD arg1 arg2 ⇒ Some ? [ XCHD ? arg1 arg2 ] |
---|
486 | | RET ⇒ Some ? [ RET ? ] |
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487 | | RETI ⇒ Some ? [ RETI ? ] |
---|
488 | | NOP ⇒ Some ? [ RealInstruction (NOP ?) ] |
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489 | ]. |
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490 | |
---|
491 | definition expand_pseudo_instruction_safe: ? → ? → Word → jump_length → pseudo_instruction → option (list instruction) ≝ |
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492 | λlookup_labels. |
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493 | λlookup_datalabels. |
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494 | λpc. |
---|
495 | λjmp_len. |
---|
496 | λi. |
---|
497 | match i with |
---|
498 | [ Cost cost ⇒ Some ? [ ] |
---|
499 | | Comment comment ⇒ Some ? [ ] |
---|
500 | | Call call ⇒ |
---|
501 | match jmp_len with |
---|
502 | [ short_jump ⇒ None ? |
---|
503 | | medium_jump ⇒ |
---|
504 | let 〈ignore, address〉 ≝ split ? 5 11 (lookup_labels call) in |
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505 | let 〈fst_5, rest〉 ≝ split ? 5 11 pc in |
---|
506 | if eq_bv ? ignore fst_5 then |
---|
507 | let address ≝ ADDR11 address in |
---|
508 | Some ? ([ ACALL address ]) |
---|
509 | else |
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510 | None ? |
---|
511 | | long_jump ⇒ |
---|
512 | let address ≝ ADDR16 (lookup_labels call) in |
---|
513 | Some ? [ LCALL address ] |
---|
514 | ] |
---|
515 | | Mov d trgt ⇒ |
---|
516 | let address ≝ DATA16 (lookup_datalabels trgt) in |
---|
517 | Some ? [ RealInstruction (MOV ? (inl ? ? (inl ? ? (inr ? ? 〈DPTR, address〉))))] |
---|
518 | | Instruction instr ⇒ expand_relative_jump lookup_labels jmp_len pc instr |
---|
519 | | Jmp jmp ⇒ |
---|
520 | match jmp_len with |
---|
521 | [ short_jump ⇒ |
---|
522 | let lookup_address ≝ lookup_labels jmp in |
---|
523 | let 〈result, flags〉 ≝ sub_16_with_carry pc lookup_address false in |
---|
524 | let 〈upper, lower〉 ≝ split ? 8 8 result in |
---|
525 | if eq_bv ? upper (zero 8) then |
---|
526 | let address ≝ RELATIVE lower in |
---|
527 | Some ? [ SJMP address ] |
---|
528 | else |
---|
529 | None ? |
---|
530 | | medium_jump ⇒ |
---|
531 | let address ≝ lookup_labels jmp in |
---|
532 | let 〈fst_5_addr, rest_addr〉 ≝ split ? 5 11 address in |
---|
533 | let 〈fst_5_pc, rest_pc〉 ≝ split ? 5 11 pc in |
---|
534 | if eq_bv ? fst_5_addr fst_5_pc then |
---|
535 | let address ≝ ADDR11 rest_addr in |
---|
536 | Some ? ([ AJMP address ]) |
---|
537 | else |
---|
538 | None ? |
---|
539 | | long_jump ⇒ |
---|
540 | let address ≝ ADDR16 (lookup_labels jmp) in |
---|
541 | Some ? [ LJMP address ] |
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542 | ] |
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543 | ]. |
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544 | @ I |
---|
545 | qed. |
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546 | |
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547 | (* label_map: identifier ↦ 〈instruction number, address〉 *) |
---|
548 | definition label_map ≝ identifier_map ASMTag (nat × nat). |
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549 | |
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550 | definition add_instruction_size: ℕ → jump_length → pseudo_instruction → ℕ ≝ |
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551 | λpc.λjmp_len.λinstr. |
---|
552 | let bv_pc ≝ bitvector_of_nat 16 pc in |
---|
553 | let ilist ≝ expand_pseudo_instruction_safe (λx.bv_pc) (λx.bv_pc) bv_pc jmp_len instr in |
---|
554 | let bv: list (BitVector 8) ≝ match ilist with |
---|
555 | [ None ⇒ (* hmm, this shouldn't happen *) [ ] |
---|
556 | | Some l ⇒ flatten … (map … assembly1 l) |
---|
557 | ] in |
---|
558 | pc + (|bv|). |
---|
559 | |
---|
560 | definition is_label ≝ |
---|
561 | λx:labelled_instruction.λl:Identifier. |
---|
562 | let 〈lbl,instr〉 ≝ x in |
---|
563 | match lbl with |
---|
564 | [ Some l' ⇒ l' = l |
---|
565 | | _ ⇒ False |
---|
566 | ]. |
---|
567 | |
---|
568 | lemma label_does_not_occur: |
---|
569 | ∀i,p,l. |
---|
570 | is_label (nth i ? p 〈None ?, Comment [ ]〉) l → does_not_occur l p = false. |
---|
571 | #i #p #l generalize in match i; elim p |
---|
572 | [ #i >nth_nil #H @⊥ @H |
---|
573 | | #h #t #IH #i cases i -i |
---|
574 | [ cases h #hi #hp cases hi |
---|
575 | [ normalize #H @⊥ @H |
---|
576 | | #l' #Heq whd in ⊢ (??%?); change with (eq_identifier ? l' l) in match (instruction_matches_identifier ??); |
---|
577 | whd in Heq; >Heq |
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578 | >eq_identifier_refl // |
---|
579 | ] |
---|
580 | | #i #H whd in match (does_not_occur ??); |
---|
581 | whd in match (instruction_matches_identifier ??); |
---|
582 | cases h #hi #hp cases hi normalize nodelta |
---|
583 | [ @(IH i) @H |
---|
584 | | #l' @eq_identifier_elim |
---|
585 | [ normalize // |
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586 | | normalize #_ @(IH i) @H |
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587 | ] |
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588 | ] |
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589 | ] |
---|
590 | ] |
---|
591 | qed. |
---|
592 | |
---|
593 | lemma coerc_pair_sigma: |
---|
594 | ∀A,B,P. ∀p:A × B. P (\snd p) → A × (Σx:B.P x). |
---|
595 | #A #B #P * #a #b #p % [@a | /2/] |
---|
596 | qed. |
---|
597 | coercion coerc_pair_sigma:∀A,B,P. ∀p:A × B. P (\snd p) → A × (Σx:B.P x) |
---|
598 | ≝ coerc_pair_sigma on p: (? × ?) to (? × (Sig ??)). |
---|
599 | |
---|
600 | definition create_label_map: ∀program:list labelled_instruction. |
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601 | ∀policy:jump_expansion_policy. |
---|
602 | (Σlabels:label_map. |
---|
603 | ∀i:ℕ.lt i (|program|) → |
---|
604 | ∀l.occurs_exactly_once l program → |
---|
605 | is_label (nth i ? program 〈None ?, Comment [ ]〉) l → |
---|
606 | ∃a.lookup … labels l = Some ? 〈i,a〉 |
---|
607 | ) ≝ |
---|
608 | λprogram.λpolicy. |
---|
609 | let 〈final_pc, final_labels〉 ≝ |
---|
610 | foldl_strong (option Identifier × pseudo_instruction) |
---|
611 | (λprefix.ℕ × (Σlabels. |
---|
612 | ∀i:ℕ.lt i (|prefix|) → |
---|
613 | ∀l.occurs_exactly_once l prefix → |
---|
614 | is_label (nth i ? prefix 〈None ?, Comment [ ]〉) l → |
---|
615 | ∃a.lookup … labels l = Some ? 〈i,a〉) |
---|
616 | ) |
---|
617 | program |
---|
618 | (λprefix.λx.λtl.λprf.λacc. |
---|
619 | let 〈pc,labels〉 ≝ acc in |
---|
620 | let 〈label,instr〉 ≝ x in |
---|
621 | let new_labels ≝ |
---|
622 | match label with |
---|
623 | [ None ⇒ labels |
---|
624 | | Some l ⇒ add … labels l 〈|prefix|, pc〉 |
---|
625 | ] in |
---|
626 | let jmp_len ≝ \snd (bvt_lookup ?? (bitvector_of_nat 16 (|prefix|)) policy 〈pc, long_jump〉) in |
---|
627 | 〈add_instruction_size pc jmp_len instr, new_labels〉 |
---|
628 | ) 〈0, empty_map …〉 in |
---|
629 | final_labels. |
---|
630 | [ #i >append_length >commutative_plus #Hi normalize in Hi; cases (le_to_or_lt_eq … Hi) -Hi; |
---|
631 | [ #Hi #l normalize nodelta; cases label normalize nodelta |
---|
632 | [ >occurs_exactly_once_None #Hocc >(nth_append_first ? ? prefix ? ? (le_S_S_to_le ? ? Hi)) #Hl |
---|
633 | lapply (sig2 … labels) #Hacc elim (Hacc i (le_S_S_to_le … Hi) l Hocc Hl) #addr #Haddr |
---|
634 | % [ @addr | @Haddr ] |
---|
635 | | #l' #Hocc #Hl lapply (occurs_exactly_once_Some_stronger … Hocc) -Hocc; |
---|
636 | @eq_identifier_elim #Heq #Hocc |
---|
637 | [ normalize in Hocc; |
---|
638 | >(nth_append_first ? ? prefix ? ? (le_S_S_to_le … Hi)) in Hl; #Hl |
---|
639 | @⊥ @(absurd … Hocc) |
---|
640 | | normalize nodelta lapply (sig2 … labels) #Hacc elim (Hacc i (le_S_S_to_le … Hi) l Hocc ?) |
---|
641 | [ #addr #Haddr % [ @addr | <Haddr @lookup_add_miss /2/ ] |
---|
642 | | >(nth_append_first ? ? prefix ? ? (le_S_S_to_le … Hi)) in Hl; // |
---|
643 | ] |
---|
644 | ] |
---|
645 | >(label_does_not_occur i … Hl) normalize nodelta @nmk // |
---|
646 | ] |
---|
647 | | #Hi #l #Hocc >(injective_S … Hi) >nth_append_second |
---|
648 | [ <minus_n_n #Hl normalize in Hl; normalize nodelta cases label in Hl; |
---|
649 | [ normalize nodelta #H @⊥ @H |
---|
650 | | #l' normalize nodelta #Heq % [ @pc | <Heq normalize nodelta @lookup_add_hit ] |
---|
651 | ] |
---|
652 | | @le_n |
---|
653 | ] |
---|
654 | ] |
---|
655 | | #i #Hi #l #Hl @⊥ @Hl |
---|
656 | ] |
---|
657 | qed. |
---|
658 | |
---|
659 | definition select_reljump_length: label_map → ℕ → Identifier → jump_length ≝ |
---|
660 | λlabels.λpc.λlbl. |
---|
661 | let 〈n, addr〉 ≝ lookup_def … labels lbl 〈0, pc〉 in |
---|
662 | if leb pc addr (* forward jump *) |
---|
663 | then if leb (addr - pc) 126 |
---|
664 | then short_jump |
---|
665 | else long_jump |
---|
666 | else if leb (pc - addr) 129 |
---|
667 | then short_jump |
---|
668 | else long_jump. |
---|
669 | |
---|
670 | definition select_call_length: label_map → ℕ → Identifier → jump_length ≝ |
---|
671 | λlabels.λpc_nat.λlbl. |
---|
672 | let pc ≝ bitvector_of_nat 16 pc_nat in |
---|
673 | let addr ≝ bitvector_of_nat 16 (\snd (lookup_def ? ? labels lbl 〈0, pc_nat〉)) in |
---|
674 | let 〈fst_5_addr, rest_addr〉 ≝ split ? 5 11 addr in |
---|
675 | let 〈fst_5_pc, rest_pc〉 ≝ split ? 5 11 pc in |
---|
676 | if eq_bv ? fst_5_addr fst_5_pc |
---|
677 | then medium_jump |
---|
678 | else long_jump. |
---|
679 | |
---|
680 | definition select_jump_length: label_map → ℕ → Identifier → jump_length ≝ |
---|
681 | λlabels.λpc.λlbl. |
---|
682 | let 〈n, addr〉 ≝ lookup_def … labels lbl 〈0, pc〉 in |
---|
683 | if leb pc addr |
---|
684 | then if leb (addr - pc) 126 |
---|
685 | then short_jump |
---|
686 | else select_call_length labels pc lbl |
---|
687 | else if leb (pc - addr) 129 |
---|
688 | then short_jump |
---|
689 | else select_call_length labels pc lbl. |
---|
690 | |
---|
691 | definition jump_expansion_step_instruction: label_map → ℕ → |
---|
692 | preinstruction Identifier → option jump_length ≝ |
---|
693 | λlabels.λpc.λi. |
---|
694 | match i with |
---|
695 | [ JC j ⇒ Some ? (select_reljump_length labels pc j) |
---|
696 | | JNC j ⇒ Some ? (select_reljump_length labels pc j) |
---|
697 | | JZ j ⇒ Some ? (select_reljump_length labels pc j) |
---|
698 | | JNZ j ⇒ Some ? (select_reljump_length labels pc j) |
---|
699 | | JB _ j ⇒ Some ? (select_reljump_length labels pc j) |
---|
700 | | JBC _ j ⇒ Some ? (select_reljump_length labels pc j) |
---|
701 | | JNB _ j ⇒ Some ? (select_reljump_length labels pc j) |
---|
702 | | CJNE _ j ⇒ Some ? (select_reljump_length labels pc j) |
---|
703 | | DJNZ _ j ⇒ Some ? (select_reljump_length labels pc j) |
---|
704 | | _ ⇒ None ? |
---|
705 | ]. |
---|
706 | |
---|
707 | definition max_length: jump_length → jump_length → jump_length ≝ |
---|
708 | λj1.λj2. |
---|
709 | match j1 with |
---|
710 | [ long_jump ⇒ long_jump |
---|
711 | | medium_jump ⇒ |
---|
712 | match j2 with |
---|
713 | [ long_jump ⇒ long_jump |
---|
714 | | _ ⇒ medium_jump |
---|
715 | ] |
---|
716 | | short_jump ⇒ j2 |
---|
717 | ]. |
---|
718 | |
---|
719 | definition jmple: jump_length → jump_length → Prop ≝ |
---|
720 | λj1.λj2. |
---|
721 | match j1 with |
---|
722 | [ short_jump ⇒ |
---|
723 | match j2 with |
---|
724 | [ short_jump ⇒ False |
---|
725 | | _ ⇒ True |
---|
726 | ] |
---|
727 | | medium_jump ⇒ |
---|
728 | match j2 with |
---|
729 | [ long_jump ⇒ True |
---|
730 | | _ ⇒ False |
---|
731 | ] |
---|
732 | | long_jump ⇒ False |
---|
733 | ]. |
---|
734 | |
---|
735 | definition jmpleq: jump_length → jump_length → Prop ≝ |
---|
736 | λj1.λj2.jmple j1 j2 ∨ j1 = j2. |
---|
737 | |
---|
738 | lemma dec_jmple: ∀x,y:jump_length.jmple x y + ¬(jmple x y). |
---|
739 | #x #y cases x cases y /3 by inl, inr, nmk, I/ |
---|
740 | qed. |
---|
741 | |
---|
742 | lemma jmpleq_max_length: ∀ol,nl. |
---|
743 | jmpleq ol (max_length ol nl). |
---|
744 | #ol #nl cases ol cases nl |
---|
745 | /2 by or_introl, or_intror, I/ |
---|
746 | qed. |
---|
747 | |
---|
748 | definition is_jump' ≝ |
---|
749 | λx:preinstruction Identifier. |
---|
750 | match x with |
---|
751 | [ JC _ ⇒ True |
---|
752 | | JNC _ ⇒ True |
---|
753 | | JZ _ ⇒ True |
---|
754 | | JNZ _ ⇒ True |
---|
755 | | JB _ _ ⇒ True |
---|
756 | | JNB _ _ ⇒ True |
---|
757 | | JBC _ _ ⇒ True |
---|
758 | | CJNE _ _ ⇒ True |
---|
759 | | DJNZ _ _ ⇒ True |
---|
760 | | _ ⇒ False |
---|
761 | ]. |
---|
762 | |
---|
763 | definition is_jump ≝ |
---|
764 | λx:labelled_instruction. |
---|
765 | let 〈label,instr〉 ≝ x in |
---|
766 | match instr with |
---|
767 | [ Instruction i ⇒ is_jump' i |
---|
768 | | Call _ ⇒ True |
---|
769 | | Jmp _ ⇒ True |
---|
770 | | _ ⇒ False |
---|
771 | ]. |
---|
772 | |
---|
773 | definition jump_in_policy ≝ |
---|
774 | λprefix:list labelled_instruction.λpolicy:jump_expansion_policy. |
---|
775 | ∀i:ℕ.i < |prefix| → |
---|
776 | (is_jump (nth i ? prefix 〈None ?, Comment []〉) ↔ |
---|
777 | ∃p,j.lookup_opt … (bitvector_of_nat 16 i) policy = Some ? 〈p,j〉). |
---|
778 | |
---|
779 | axiom bitvector_of_nat_abs: |
---|
780 | ∀x,y:ℕ.x ≠ y → ¬eq_bv 16 (bitvector_of_nat 16 x) (bitvector_of_nat 16 y). |
---|
781 | |
---|
782 | lemma le_S_to_le: ∀n,m:ℕ.S n ≤ m → n ≤ m. |
---|
783 | /2/ qed. |
---|
784 | |
---|
785 | lemma jump_not_in_policy: ∀prefix:list labelled_instruction. |
---|
786 | ∀policy:(Σp:jump_expansion_policy. |
---|
787 | (∀i.i ≥ |prefix| → lookup_opt … (bitvector_of_nat ? i) p = None ?) ∧ |
---|
788 | jump_in_policy prefix p). |
---|
789 | ∀i:ℕ.i < |prefix| → |
---|
790 | ¬is_jump (nth i ? prefix 〈None ?, Comment []〉) ↔ |
---|
791 | lookup_opt … (bitvector_of_nat 16 i) policy = None ?. |
---|
792 | #prefix #policy #i #Hi @conj |
---|
793 | [ #Hnotjmp lapply (refl ? (lookup_opt … (bitvector_of_nat 16 i) policy)) |
---|
794 | cases (lookup_opt … (bitvector_of_nat 16 i) policy) in ⊢ (???% → ?); |
---|
795 | [ #H @H |
---|
796 | | #x cases x #y #z #H @⊥ @(absurd ? ? Hnotjmp) @(proj2 ?? (proj2 ?? (sig2 ?? policy) i Hi)) |
---|
797 | @(ex_intro … y (ex_intro … z H)) |
---|
798 | ] |
---|
799 | | #Hnone @nmk #Hj lapply (proj1 ?? (proj2 ?? (sig2 ?? policy) i Hi) Hj) |
---|
800 | #H elim H -H; #x #H elim H -H; #y #H >H in Hnone; #abs destruct (abs) |
---|
801 | ] |
---|
802 | qed. |
---|
803 | |
---|
804 | definition jump_expansion_start: ∀program:list labelled_instruction. |
---|
805 | Σpolicy:jump_expansion_policy. |
---|
806 | (∀i.i ≥ |program| → lookup_opt … (bitvector_of_nat 16 i) policy = None ?) ∧ |
---|
807 | jump_in_policy program policy ∧ |
---|
808 | ∀i.i < |program| → is_jump (nth i ? program 〈None ?, Comment []〉) → |
---|
809 | lookup_opt … (bitvector_of_nat 16 i) policy = Some ? 〈0,short_jump〉 ≝ |
---|
810 | λprogram. |
---|
811 | foldl_strong (option Identifier × pseudo_instruction) |
---|
812 | (λprefix.Σpolicy:jump_expansion_policy. |
---|
813 | (∀i.i ≥ |prefix| → lookup_opt … (bitvector_of_nat 16 i) policy = None ?) ∧ |
---|
814 | jump_in_policy prefix policy ∧ |
---|
815 | ∀i.i < |prefix| → is_jump (nth i ? prefix 〈None ?, Comment []〉) → |
---|
816 | lookup_opt … (bitvector_of_nat 16 i) policy = Some ? 〈0,short_jump〉) |
---|
817 | program |
---|
818 | (λprefix.λx.λtl.λprf.λpolicy. |
---|
819 | let 〈label,instr〉 ≝ x in |
---|
820 | match instr with |
---|
821 | [ Instruction i ⇒ match i with |
---|
822 | [ JC _ ⇒ bvt_insert … (bitvector_of_nat 16 (|prefix|)) 〈0,short_jump〉 policy |
---|
823 | | JNC _ ⇒ bvt_insert … (bitvector_of_nat 16 (|prefix|)) 〈0,short_jump〉 policy |
---|
824 | | JZ _ ⇒ bvt_insert … (bitvector_of_nat 16 (|prefix|)) 〈0,short_jump〉 policy |
---|
825 | | JNZ _ ⇒ bvt_insert … (bitvector_of_nat 16 (|prefix|)) 〈0,short_jump〉 policy |
---|
826 | | JB _ _ ⇒ bvt_insert … (bitvector_of_nat 16 (|prefix|)) 〈0,short_jump〉 policy |
---|
827 | | JNB _ _ ⇒ bvt_insert … (bitvector_of_nat 16 (|prefix|)) 〈0,short_jump〉 policy |
---|
828 | | JBC _ _ ⇒ bvt_insert … (bitvector_of_nat 16 (|prefix|)) 〈0,short_jump〉 policy |
---|
829 | | CJNE _ _ ⇒ bvt_insert … (bitvector_of_nat 16 (|prefix|)) 〈0,short_jump〉 policy |
---|
830 | | DJNZ _ _ ⇒ bvt_insert … (bitvector_of_nat 16 (|prefix|)) 〈0,short_jump〉 policy |
---|
831 | | _ ⇒ (eject … policy) |
---|
832 | ] |
---|
833 | | Call c ⇒ bvt_insert … (bitvector_of_nat 16 (|prefix|)) 〈0,short_jump〉 policy |
---|
834 | | Jmp j ⇒ bvt_insert … (bitvector_of_nat 16 (|prefix|)) 〈0,short_jump〉 policy |
---|
835 | | _ ⇒ (eject … policy) |
---|
836 | ] |
---|
837 | ) (Stub ? ?). |
---|
838 | [1,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,35,36,37,38,39,40,41,42: |
---|
839 | @conj |
---|
840 | [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: |
---|
841 | @conj |
---|
842 | #i >append_length <commutative_plus #Hi normalize in Hi; |
---|
843 | [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: |
---|
844 | cases (le_to_or_lt_eq … Hi) -Hi; #Hi @(proj1 ?? (proj1 ?? (sig2 ?? policy)) i) |
---|
845 | [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: |
---|
846 | <Hi @le_n_Sn |
---|
847 | ] |
---|
848 | @le_S_to_le @le_S_to_le @Hi |
---|
849 | ] |
---|
850 | cases (le_to_or_lt_eq … Hi) -Hi; #Hi |
---|
851 | [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: |
---|
852 | >(nth_append_first ? ? prefix ? ? (le_S_S_to_le … Hi)) |
---|
853 | @(proj2 ?? (proj1 ?? (sig2 ?? policy)) i (le_S_S_to_le … Hi)) |
---|
854 | ] |
---|
855 | @conj >(injective_S … Hi) |
---|
856 | [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: |
---|
857 | >(nth_append_second ? ? prefix ? ? (le_n (|prefix|))) <minus_n_n #H @⊥ @H |
---|
858 | ] |
---|
859 | #H elim H; -H; #t1 #H elim H; -H #t2 #H |
---|
860 | lapply (proj1 ?? (proj1 ?? (sig2 ?? policy)) (|prefix|) (le_n (|prefix|))) |
---|
861 | #H2 >H2 in H; #H destruct (H) |
---|
862 | ] |
---|
863 | #i >append_length <commutative_plus #Hi normalize in Hi; cases (le_to_or_lt_eq … Hi) |
---|
864 | -Hi; #Hi |
---|
865 | [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: |
---|
866 | #Hj @(proj2 ?? (sig2 ?? policy) i (le_S_S_to_le … Hi)) |
---|
867 | >(nth_append_first ?? prefix ?? (le_S_S_to_le ?? Hi)) in Hj; // |
---|
868 | ] |
---|
869 | >(injective_S … Hi) >(nth_append_second ?? prefix ?? (le_n (|prefix|))) <minus_n_n |
---|
870 | #H @⊥ @H |
---|
871 | |2,3,26,27,28,29,30,31,32,33,34: @conj |
---|
872 | [1,3,5,7,9,11,13,15,17,19,21: @conj |
---|
873 | [1,3,5,7,9,11,13,15,17,19,21: |
---|
874 | #i >append_length <commutative_plus #Hi normalize in Hi; >lookup_opt_insert_miss |
---|
875 | [1,3,5,7,9,11,13,15,17,19,21: |
---|
876 | @(proj1 ?? (proj1 ?? (sig2 ?? policy)) i (le_S_to_le … Hi)) |
---|
877 | ] |
---|
878 | >eq_bv_sym @bitvector_of_nat_abs @lt_to_not_eq @Hi |
---|
879 | ] |
---|
880 | #i >append_length <commutative_plus #Hi normalize in Hi; cases (le_to_or_lt_eq … Hi) |
---|
881 | -Hi #Hi |
---|
882 | [1,3,5,7,9,11,13,15,17,19,21: |
---|
883 | >(nth_append_first ?? prefix ?? (le_S_S_to_le … Hi)) >lookup_opt_insert_miss |
---|
884 | [1,3,5,7,9,11,13,15,17,19,21: |
---|
885 | @(proj2 ?? (proj1 ?? (sig2 ?? policy)) i (le_S_S_to_le … Hi)) |
---|
886 | ] |
---|
887 | @bitvector_of_nat_abs @(lt_to_not_eq … (le_S_S_to_le … Hi)) |
---|
888 | ] |
---|
889 | @conj >(injective_S … Hi) #H |
---|
890 | [2,4,6,8,10,12,14,16,18,20,22: |
---|
891 | >(nth_append_second ?? prefix ?? (le_n (|prefix|))) <minus_n_n // |
---|
892 | ] |
---|
893 | @(ex_intro ?? 0 (ex_intro ?? short_jump (lookup_opt_insert_hit ?? 16 ? policy))) |
---|
894 | ] |
---|
895 | #i >append_length <commutative_plus #Hi normalize in Hi; cases (le_to_or_lt_eq … Hi) |
---|
896 | -Hi #Hi |
---|
897 | [1,3,5,7,9,11,13,15,17,19,21: |
---|
898 | >(nth_append_first ?? prefix ?? (le_S_S_to_le … Hi)) #Hj >lookup_opt_insert_miss |
---|
899 | [1,3,5,7,9,11,13,15,17,19,21: |
---|
900 | @(proj2 ?? (sig2 ?? policy) i (le_S_S_to_le … Hi) Hj) |
---|
901 | ] |
---|
902 | @bitvector_of_nat_abs @(lt_to_not_eq … (le_S_S_to_le … Hi)) |
---|
903 | ] |
---|
904 | #_ >(injective_S … Hi) @lookup_opt_insert_hit |
---|
905 | |@conj |
---|
906 | [@conj |
---|
907 | [ #i #Hi // |
---|
908 | | whd #i #Hi @⊥ @(absurd (i < 0) Hi (not_le_Sn_O ?)) |
---|
909 | ] |
---|
910 | | #i #Hi >nth_nil #Hj @⊥ @Hj |
---|
911 | ] |
---|
912 | qed. |
---|
913 | |
---|
914 | definition policy_increase: list labelled_instruction → jump_expansion_policy → |
---|
915 | jump_expansion_policy → Prop ≝ |
---|
916 | λprogram.λop.λp. |
---|
917 | (* (∀i:ℕ.i < |program| → |
---|
918 | lookup_opt … (bitvector_of_nat ? i) op = lookup_opt … (bitvector_of_nat ? i) p) ∨ *) |
---|
919 | (∀i:ℕ.i < |program| → |
---|
920 | jmpleq |
---|
921 | (\snd (bvt_lookup … (bitvector_of_nat ? i) op 〈0,short_jump〉)) |
---|
922 | (\snd (bvt_lookup … (bitvector_of_nat ? i) p 〈0,short_jump〉))). |
---|
923 | |
---|
924 | definition jump_expansion_step: ∀program:list labelled_instruction. |
---|
925 | ∀old_policy:(Σpolicy. |
---|
926 | (∀i.i ≥ |program| → lookup_opt … (bitvector_of_nat 16 i) policy = None ?) ∧ |
---|
927 | jump_in_policy program policy). |
---|
928 | (Σpolicy. |
---|
929 | (∀i.i ≥ |program| → lookup_opt … (bitvector_of_nat 16 i) policy = None ?) ∧ |
---|
930 | jump_in_policy program policy ∧ |
---|
931 | policy_increase program old_policy policy) |
---|
932 | ≝ |
---|
933 | λprogram.λold_policy. |
---|
934 | let labels ≝ create_label_map program old_policy in |
---|
935 | let 〈final_pc, final_policy〉 ≝ |
---|
936 | foldl_strong (option Identifier × pseudo_instruction) |
---|
937 | (λprefix.ℕ × Σpolicy. |
---|
938 | (∀i.i ≥ |prefix| → lookup_opt … (bitvector_of_nat 16 i) policy = None ?) ∧ |
---|
939 | jump_in_policy prefix policy ∧ |
---|
940 | policy_increase prefix old_policy policy |
---|
941 | ) |
---|
942 | program |
---|
943 | (λprefix.λx.λtl.λprf.λacc. |
---|
944 | let 〈pc, policy〉 ≝ acc in |
---|
945 | let 〈label,instr〉 ≝ x in |
---|
946 | let old_jump_length ≝ lookup_opt ? ? (bitvector_of_nat 16 (|prefix|)) old_policy in |
---|
947 | let add_instr ≝ |
---|
948 | match instr with |
---|
949 | [ Instruction i ⇒ jump_expansion_step_instruction labels pc i |
---|
950 | | Call c ⇒ Some ? (select_call_length labels pc c) |
---|
951 | | Jmp j ⇒ Some ? (select_jump_length labels pc j) |
---|
952 | | _ ⇒ None ? |
---|
953 | ] in |
---|
954 | let 〈new_pc, new_policy〉 ≝ |
---|
955 | let 〈ignore,old_length〉 ≝ lookup … (bitvector_of_nat 16 (|prefix|)) old_policy 〈0, short_jump〉 in |
---|
956 | match add_instr with |
---|
957 | [ None ⇒ (* i.e. it's not a jump *) |
---|
958 | 〈add_instruction_size pc long_jump instr, policy〉 |
---|
959 | | Some ai ⇒ |
---|
960 | let new_length ≝ max_length old_length ai in |
---|
961 | 〈add_instruction_size pc new_length instr, insert … (bitvector_of_nat 16 (|prefix|)) 〈pc, new_length〉 policy〉 |
---|
962 | ] in |
---|
963 | 〈new_pc, new_policy〉 |
---|
964 | ) 〈0, Stub ? ?〉 in |
---|
965 | final_policy. |
---|
966 | [ @conj [ @conj #i >append_length <commutative_plus #Hi normalize in Hi; |
---|
967 | [ cases (lookup ??? old_policy ?) #h #n cases add_instr |
---|
968 | [ @(proj1 ?? (proj1 ?? (sig2 ?? policy)) i (le_S_to_le … Hi)) |
---|
969 | | #z normalize nodelta >lookup_opt_insert_miss |
---|
970 | [ @(proj1 ?? (proj1 ?? (sig2 ?? policy)) i (le_S_to_le … Hi)) |
---|
971 | | >eq_bv_sym @bitvector_of_nat_abs @lt_to_not_eq @Hi |
---|
972 | ] |
---|
973 | ] |
---|
974 | | cases (le_to_or_lt_eq … Hi) -Hi; |
---|
975 | [ #Hi; >(nth_append_first ? ? prefix ? ? (le_S_S_to_le … Hi)) @conj |
---|
976 | [ #Hj lapply (proj2 ?? (proj1 ?? (sig2 ?? policy)) i (le_S_S_to_le … Hi)) #Hacc |
---|
977 | cases add_instr cases (lookup ??? old_policy ?) normalize nodelta #x #y |
---|
978 | [ @(proj1 ?? Hacc Hj) |
---|
979 | | #z elim (proj1 ?? Hacc Hj) #h #n elim n -n #n #Hn |
---|
980 | % [ @h | % [ @n | <Hn @lookup_opt_insert_miss @bitvector_of_nat_abs |
---|
981 | @(lt_to_not_eq i (|prefix|)) @(le_S_S_to_le … Hi) ] ] |
---|
982 | ] |
---|
983 | | lapply (proj2 ?? (proj1 ?? (sig2 ?? policy)) i (le_S_S_to_le … Hi)) #Hacc |
---|
984 | #H elim H -H; #h #H elim H -H; #n cases add_instr cases (lookup ??? old_policy ?) |
---|
985 | normalize nodelta #x #y [2: #z] |
---|
986 | #Hl @(proj2 ?? Hacc) @(ex_intro ?? h (ex_intro ?? n ?)) |
---|
987 | [ <Hl @sym_eq @lookup_opt_insert_miss @bitvector_of_nat_abs @lt_to_not_eq @(le_S_S_to_le … Hi) |
---|
988 | | @Hl |
---|
989 | ] |
---|
990 | ] |
---|
991 | | #Hi >(injective_S … Hi) >(nth_append_second ? ? prefix ? ? (le_n (|prefix|))) |
---|
992 | <minus_n_n whd in match (nth ????); whd in match (add_instr); cases instr |
---|
993 | [1: #pi | 2,3: #x | 4,5: #id | 6: #x #y] @conj normalize nodelta |
---|
994 | [3,5,11: #H @⊥ @H (* instr is not a jump *) |
---|
995 | |4,6,12: #H elim H -H; #h #H elim H -H #n cases (lookup ??? old_policy ?) |
---|
996 | #x #y normalize nodelta >(proj1 ?? (proj1 ?? (sig2 ?? policy)) (|prefix|) (le_n (|prefix|))) |
---|
997 | #H destruct (H) |
---|
998 | |7,9: (* instr is a jump *) #_ cases (lookup ??? old_policy ?) #h #n |
---|
999 | whd in match (snd ???); @(ex_intro ?? pc) |
---|
1000 | [ @(ex_intro ?? (max_length n (select_jump_length (create_label_map program old_policy) pc id))) |
---|
1001 | | @(ex_intro ?? (max_length n (select_call_length (create_label_map program old_policy) pc id))) |
---|
1002 | ] @lookup_opt_insert_hit |
---|
1003 | |8,10: #_ // |
---|
1004 | |1,2: cases pi |
---|
1005 | [35,36,37: #H @⊥ @H |
---|
1006 | |4,5,8,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32: #x #H @⊥ @H |
---|
1007 | |1,2,3,6,7,33,34: #x #y #H @⊥ @H |
---|
1008 | |9,10,14,15: #id #_ cases (lookup ??? old_policy ?) #h #n |
---|
1009 | whd in match (snd ???); |
---|
1010 | @(ex_intro ?? pc (ex_intro ?? (max_length n (select_reljump_length (create_label_map program old_policy) pc id)) ?)) |
---|
1011 | @lookup_opt_insert_hit |
---|
1012 | |11,12,13,16,17: #x #id #_ cases (lookup ??? old_policy ?) #h #n |
---|
1013 | whd in match (snd ???); |
---|
1014 | @(ex_intro ?? pc (ex_intro ?? (max_length n (select_reljump_length (create_label_map program old_policy) pc id)) ?)) |
---|
1015 | @lookup_opt_insert_hit |
---|
1016 | |72,73,74: #H elim H -H; #h #H elim H -H #n cases (lookup ??? old_policy ?) |
---|
1017 | #x #y normalize nodelta |
---|
1018 | >(proj1 ?? (proj1 ?? (sig2 ?? policy)) (|prefix|) (le_n (|prefix|))) #H destruct (H) |
---|
1019 | |41,42,45,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69: #x |
---|
1020 | #H elim H -H; #h #H elim H -H #n cases (lookup ??? old_policy ?) |
---|
1021 | #x #y normalize nodelta |
---|
1022 | >(proj1 ?? (proj1 ?? (sig2 ?? policy)) (|prefix|) (le_n (|prefix|))) #H destruct (H) |
---|
1023 | |38,39,40,43,44,70,71: #x #y #H elim H -H; #h #H elim H -H #n |
---|
1024 | cases (lookup ??? old_policy ?) #x #y normalize nodelta |
---|
1025 | >(proj1 ?? (proj1 ?? (sig2 ?? policy)) (|prefix|) (le_n (|prefix|))) #H destruct (H) |
---|
1026 | |46,47,51,52: #id #_ // |
---|
1027 | |48,49,50,53,54: #x #id #_ // |
---|
1028 | ] |
---|
1029 | ] |
---|
1030 | ] |
---|
1031 | ] |
---|
1032 | | lapply (refl ? add_instr) cases add_instr in ⊢ (???% → %); |
---|
1033 | [ #Ha #i >append_length >commutative_plus #Hi normalize in Hi; |
---|
1034 | cases (le_to_or_lt_eq … Hi) -Hi; #Hi |
---|
1035 | [ cases (lookup … (bitvector_of_nat ? (|prefix|)) old_policy 〈0,short_jump〉) |
---|
1036 | #x #y @((proj2 ?? (sig2 ?? policy)) i (le_S_S_to_le … Hi)) |
---|
1037 | | normalize nodelta >(injective_S … Hi) |
---|
1038 | >lookup_opt_lookup_miss |
---|
1039 | [ >lookup_opt_lookup_miss |
---|
1040 | [ // |
---|
1041 | | cases (lookup ?? (bitvector_of_nat ? (|prefix|)) old_policy 〈0,short_jump〉) |
---|
1042 | #x #y normalize nodelta |
---|
1043 | >(proj1 ?? (proj1 ?? (sig2 ?? policy)) (|prefix|) (le_n (|prefix|))) // |
---|
1044 | ] |
---|
1045 | | >(proj1 ?? (jump_not_in_policy program old_policy (|prefix|) ?)) |
---|
1046 | [ // |
---|
1047 | | >prf >p1 >nth_append_second [ <minus_n_n |
---|
1048 | generalize in match Ha; normalize nodelta cases instr normalize nodelta |
---|
1049 | [1: #pi cases pi |
---|
1050 | [1,2,3,6,7,33,34: #x #y #H normalize /2 by nmk/ |
---|
1051 | |4,5,8,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32: #x #H normalize /2 by nmk/ |
---|
1052 | |35,36,37: #H normalize /2 by nmk/ |
---|
1053 | |9,10,14,15: #id whd in match (jump_expansion_step_instruction ???); |
---|
1054 | #H destruct (H) |
---|
1055 | |11,12,13,16,17: #x #id whd in match (jump_expansion_step_instruction ???); |
---|
1056 | #H destruct (H) |
---|
1057 | ] |
---|
1058 | |2,3: #x #H normalize /2 by nmk/ |
---|
1059 | |6: #x #y #H normalize /2 by nmk/ |
---|
1060 | |4,5: #id #H destruct (H) |
---|
1061 | ] | @le_n ] |
---|
1062 | | >prf >append_length normalize <plus_n_Sm @le_plus_n_r |
---|
1063 | ] |
---|
1064 | ] |
---|
1065 | ] |
---|
1066 | | #x #Ha #i >append_length >commutative_plus #Hi normalize in Hi; |
---|
1067 | cases (le_to_or_lt_eq … Hi) -Hi; #Hi |
---|
1068 | [ cases (lookup … (bitvector_of_nat ? (|prefix|)) old_policy 〈0,short_jump〉) |
---|
1069 | #y #z normalize nodelta normalize nodelta >lookup_insert_miss |
---|
1070 | [ @((proj2 ?? (sig2 ?? policy)) i (le_S_S_to_le … Hi)) |
---|
1071 | | @bitvector_of_nat_abs @lt_to_not_eq @(le_S_S_to_le … Hi) |
---|
1072 | ] |
---|
1073 | | >(injective_S … Hi) elim (proj1 ?? (proj2 ?? (sig2 ?? old_policy) (|prefix|) ?) ?) |
---|
1074 | [ #a #H elim H -H; #b #H >H >(lookup_opt_lookup_hit … 〈a,b〉 H) |
---|
1075 | normalize nodelta >lookup_insert_hit @jmpleq_max_length |
---|
1076 | | >prf >p1 >nth_append_second |
---|
1077 | [ <minus_n_n generalize in match Ha; normalize nodelta cases instr normalize nodelta |
---|
1078 | [1: #pi cases pi |
---|
1079 | [1,2,3,6,7,33,34: #x #y #H normalize in H; destruct (H) |
---|
1080 | |4,5,8,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32: #x #H normalize in H; destruct (H) |
---|
1081 | |35,36,37: #H normalize in H; destruct (H) |
---|
1082 | |9,10,14,15: #id #H // |
---|
1083 | |11,12,13,16,17: #x #id #H // |
---|
1084 | ] |
---|
1085 | |2,3: #x #H normalize in H; destruct (H) |
---|
1086 | |6: #x #y #H normalize in H; destruct (H) |
---|
1087 | |4,5: #id #H // |
---|
1088 | ] |
---|
1089 | | @le_n ] |
---|
1090 | | >prf >append_length normalize <plus_n_Sm @le_plus_n_r |
---|
1091 | ] |
---|
1092 | ] |
---|
1093 | ] ] |
---|
1094 | | @conj [ @conj |
---|
1095 | [ #i #Hi // |
---|
1096 | | #i #Hi @conj [ >nth_nil #H @⊥ @H | #H elim H #x #H1 elim H1 #y #H2 |
---|
1097 | normalize in H2; destruct (H2) ] |
---|
1098 | ] |
---|
1099 | | #i #Hi @⊥ @(absurd (i<0)) [ @Hi | @(not_le_Sn_O) ] |
---|
1100 | ] |
---|
1101 | qed. |
---|
1102 | |
---|
1103 | let rec jump_expansion_internal (program: list labelled_instruction) |
---|
1104 | (n: ℕ) on n: (Σpolicy:jump_expansion_policy. |
---|
1105 | And |
---|
1106 | (∀i:ℕ.i ≥ |program| → lookup_opt ? 16 (bitvector_of_nat ? i) policy = None ?) |
---|
1107 | (jump_in_policy program policy)) ≝ |
---|
1108 | match n with |
---|
1109 | [ O ⇒ jump_expansion_start program |
---|
1110 | | S m ⇒ jump_expansion_step program (jump_expansion_internal program m) |
---|
1111 | ]. |
---|
1112 | [ @(proj1 ?? (sig2 ?? (jump_expansion_start program))) |
---|
1113 | | @(proj1 ?? (sig2 ?? (jump_expansion_step program (jump_expansion_internal program m)))) |
---|
1114 | ] |
---|
1115 | qed. |
---|
1116 | |
---|
1117 | definition policy_equal ≝ |
---|
1118 | λprogram:list labelled_instruction.λp1,p2:jump_expansion_policy. |
---|
1119 | ∀n:ℕ.n < |program| → |
---|
1120 | (\snd (bvt_lookup … (bitvector_of_nat 16 n) p1 〈0,short_jump〉)) = |
---|
1121 | (\snd (bvt_lookup … (bitvector_of_nat 16 n) p2 〈0,short_jump〉)). |
---|
1122 | |
---|
1123 | lemma pe_refl: |
---|
1124 | ∀program.reflexive ? (policy_equal program). |
---|
1125 | #program whd #x whd #n #Hn @refl |
---|
1126 | qed. |
---|
1127 | |
---|
1128 | lemma pe_sym: |
---|
1129 | ∀program.symmetric ? (policy_equal program). |
---|
1130 | #program whd #x #y #Hxy whd #n #Hn |
---|
1131 | >(Hxy n Hn) @refl |
---|
1132 | qed. |
---|
1133 | |
---|
1134 | lemma pe_trans: |
---|
1135 | ∀program.transitive ? (policy_equal program). |
---|
1136 | #program whd #x #y #z #Hxy #Hyz whd #n #Hn |
---|
1137 | >(Hxy n Hn) @(Hyz n Hn) |
---|
1138 | qed. |
---|
1139 | |
---|
1140 | lemma le_plus: |
---|
1141 | ∀n,m:ℕ.n ≤ m → ∃k:ℕ.m = n + k. |
---|
1142 | #n #m elim m -m; |
---|
1143 | [ #Hn % [ @O | <(le_n_O_to_eq n Hn) // ] |
---|
1144 | | #m #Hind #Hn cases (le_to_or_lt_eq … Hn) -Hn; #Hn |
---|
1145 | [ elim (Hind (le_S_S_to_le … Hn)) #k #Hk % [ @(S k) | >Hk // ] |
---|
1146 | | % [ @O | <Hn // ] |
---|
1147 | ] |
---|
1148 | ] |
---|
1149 | qed. |
---|
1150 | |
---|
1151 | theorem plus_Sn_m1: ∀n,m:nat. S m + n = m + S n. |
---|
1152 | #n (elim n) normalize /2 by S_pred/ qed. |
---|
1153 | |
---|
1154 | lemma pe_step: ∀program:list labelled_instruction. |
---|
1155 | ∀p1,p2:Σpolicy. |
---|
1156 | (∀i:ℕ.i ≥ |program| → lookup_opt … (bitvector_of_nat ? i) policy = None ?) |
---|
1157 | ∧jump_in_policy program policy. |
---|
1158 | policy_equal program p1 p2 → |
---|
1159 | policy_equal program (jump_expansion_step program p1) (jump_expansion_step program p2). |
---|
1160 | #program #p1 #p2 #Heq whd #n #Hn lapply (Heq n Hn) #H |
---|
1161 | lapply (refl ? (lookup_opt … (bitvector_of_nat ? n) p1)) |
---|
1162 | cases (lookup_opt … (bitvector_of_nat ? n) p1) in ⊢ (???% → ?); |
---|
1163 | [ #Hl lapply ((proj2 ?? (jump_not_in_policy program p1 n Hn)) Hl) |
---|
1164 | #Hnotjmp >lookup_opt_lookup_miss |
---|
1165 | [ >lookup_opt_lookup_miss |
---|
1166 | [ @refl |
---|
1167 | | @(proj1 ?? (jump_not_in_policy program (eject … (jump_expansion_step program p2)) n Hn)) |
---|
1168 | [ @(proj1 ?? (sig2 … (jump_expansion_step program p2))) |
---|
1169 | | @Hnotjmp |
---|
1170 | ] |
---|
1171 | ] |
---|
1172 | | @(proj1 ?? (jump_not_in_policy program (eject … (jump_expansion_step program p1)) n Hn)) |
---|
1173 | [ @(proj1 ?? (sig2 ?? (jump_expansion_step program p1))) |
---|
1174 | | @Hnotjmp |
---|
1175 | ] |
---|
1176 | ] |
---|
1177 | | #x #Hl cases daemon |
---|
1178 | ] |
---|
1179 | qed. |
---|
1180 | |
---|
1181 | lemma equal_remains_equal: ∀program:list labelled_instruction.∀n:ℕ. |
---|
1182 | policy_equal program (jump_expansion_internal program n) (jump_expansion_internal program (S n)) → |
---|
1183 | ∀k.k ≥ n → policy_equal program (jump_expansion_internal program n) (jump_expansion_internal program k). |
---|
1184 | #program #n #Heq #k #Hk elim (le_plus … Hk); #z #H >H -H -Hk -k; |
---|
1185 | lapply Heq -Heq; lapply n -n; elim z -z; |
---|
1186 | [ #n #Heq <plus_n_O @pe_refl |
---|
1187 | | #z #Hind #n #Heq <plus_Sn_m1 whd in match (plus (S n) z); @(pe_trans … (jump_expansion_internal program (S n))) |
---|
1188 | [ @Heq |
---|
1189 | | @pe_step @Hind @Heq |
---|
1190 | ] |
---|
1191 | ] |
---|
1192 | qed. |
---|
1193 | |
---|
1194 | lemma dec_bounded_forall: |
---|
1195 | ∀P:ℕ → Prop.(∀n.(P n) + (¬P n)) → ∀k.(∀n.n < k → P n) + ¬(∀n.n < k → P n). |
---|
1196 | #P #HP_dec #k elim k -k |
---|
1197 | [ %1 #n #Hn @⊥ @(absurd (n < 0) Hn) @not_le_Sn_O |
---|
1198 | | #k #Hind cases Hind |
---|
1199 | [ #Hk cases (HP_dec k) |
---|
1200 | [ #HPk %1 #n #Hn cases (le_to_or_lt_eq … Hn) |
---|
1201 | [ #H @(Hk … (le_S_S_to_le … H)) |
---|
1202 | | #H >(injective_S … H) @HPk |
---|
1203 | ] |
---|
1204 | | #HPk %2 @nmk #Habs @(absurd (P k)) [ @(Habs … (le_n (S k))) | @HPk ] |
---|
1205 | ] |
---|
1206 | | #Hk %2 @nmk #Habs @(absurd (∀n.n<k→P n)) [ #n' #Hn' @(Habs … (le_S … Hn')) | @Hk ] |
---|
1207 | ] |
---|
1208 | ] |
---|
1209 | qed. |
---|
1210 | |
---|
1211 | lemma dec_bounded_exists: |
---|
1212 | ∀P:ℕ→Prop.(∀n.(P n) + (¬P n)) → ∀k.(∃n.n < k ∧ P n) + ¬(∃n.n < k ∧ P n). |
---|
1213 | #P #HP_dec #k elim k -k |
---|
1214 | [ %2 @nmk #Habs elim Habs #n #Hn @(absurd (n < 0) (proj1 … Hn)) @not_le_Sn_O |
---|
1215 | | #k #Hind cases Hind |
---|
1216 | [ #Hk %1 elim Hk #n #Hn @(ex_intro … n) @conj [ @le_S @(proj1 … Hn) | @(proj2 … Hn) ] |
---|
1217 | | #Hk cases (HP_dec k) |
---|
1218 | [ #HPk %1 @(ex_intro … k) @conj [ @le_n | @HPk ] |
---|
1219 | | #HPk %2 @nmk #Habs elim Habs #n #Hn cases (le_to_or_lt_eq … (proj1 … Hn)) |
---|
1220 | [ #H @(absurd (∃n.n < k ∧ P n)) [ @(ex_intro … n) @conj |
---|
1221 | [ @(le_S_S_to_le … H) | @(proj2 … Hn) ] | @Hk ] |
---|
1222 | | #H @(absurd (P k)) [ <(injective_S … H) @(proj2 … Hn) | @HPk ] |
---|
1223 | ] |
---|
1224 | ] |
---|
1225 | ] |
---|
1226 | ] |
---|
1227 | qed. |
---|
1228 | |
---|
1229 | lemma not_exists_forall: |
---|
1230 | ∀k:ℕ.∀P:ℕ → Prop.¬(∃x.x < k ∧ P x) → ∀x.x < k → ¬P x. |
---|
1231 | #k #P #Hex #x #Hx @nmk #Habs @(absurd ? ? Hex) @(ex_intro … x) |
---|
1232 | @conj [ @Hx | @Habs ] |
---|
1233 | qed. |
---|
1234 | |
---|
1235 | lemma not_forall_exists: |
---|
1236 | ∀k:ℕ.∀P:ℕ → Prop.(∀n.(P n) + (¬P n)) → ¬(∀x.x < k → P x) → ∃x.x < k ∧ ¬P x. |
---|
1237 | #k #P #Hdec elim k |
---|
1238 | [ #Hfa @⊥ @(absurd ?? Hfa) #z #Hz @⊥ @(absurd ? Hz) @not_le_Sn_O |
---|
1239 | | -k #k #Hind #Hfa cases (Hdec k) |
---|
1240 | [ #HP elim (Hind ?) |
---|
1241 | [ -Hind; #x #Hx @(ex_intro ?? x) @conj [ @le_S @(proj1 ?? Hx) | @(proj2 ?? Hx) ] |
---|
1242 | | @nmk #H @(absurd ?? Hfa) #x #Hx cases (le_to_or_lt_eq ?? Hx) |
---|
1243 | [ #H2 @H @(le_S_S_to_le … H2) |
---|
1244 | | #H2 >(injective_S … H2) @HP |
---|
1245 | ] |
---|
1246 | ] |
---|
1247 | | #HP @(ex_intro … k) @conj [ @le_n | @HP ] |
---|
1248 | ] |
---|
1249 | ] |
---|
1250 | qed. |
---|
1251 | |
---|
1252 | (* lemma de_morgan1: |
---|
1253 | ∀A,B:Prop.¬(A ∧ ¬B) → A → ¬¬B. |
---|
1254 | #A #B #Hnot #HA @nmk #H @(absurd (A∧¬B)) [ % [ @HA | @H ] | @Hnot ] |
---|
1255 | qed. *) |
---|
1256 | |
---|
1257 | lemma thingie: |
---|
1258 | ∀A:Prop.(A + ¬A) → (¬¬A) → A. |
---|
1259 | #A #Adec #nnA cases Adec |
---|
1260 | [ // |
---|
1261 | | #H @⊥ @(absurd (¬A) H nnA) |
---|
1262 | ] |
---|
1263 | qed. |
---|
1264 | |
---|
1265 | lemma dec_eq_jump_length: ∀a,b:jump_length.(a = b) + (a ≠ b). |
---|
1266 | #a #b cases a cases b /2/ |
---|
1267 | %2 @nmk #H destruct (H) |
---|
1268 | qed. |
---|
1269 | |
---|
1270 | (* lemma incr_or_equal: ∀program:list labelled_instruction. |
---|
1271 | ∀policy:(Σp:jump_expansion_policy. |
---|
1272 | (∀i.i ≥ |program| → lookup_opt … (bitvector_of_nat ? i) p = None ?) ∧ |
---|
1273 | jump_in_policy program p). |
---|
1274 | policy_equal program policy (jump_expansion_step program policy) ∨ |
---|
1275 | ∃n:ℕ.n < (|program|) ∧ jmple |
---|
1276 | (\snd (bvt_lookup … (bitvector_of_nat ? n) policy 〈0,short_jump〉)) |
---|
1277 | (\snd (bvt_lookup … (bitvector_of_nat ? n) (jump_expansion_step program policy) 〈0,short_jump〉)). |
---|
1278 | #program #policy cases (dec_bounded_exists |
---|
1279 | (λk. |
---|
1280 | \snd (bvt_lookup ?? (bitvector_of_nat ? k) policy 〈0,short_jump〉) ≠ |
---|
1281 | \snd (bvt_lookup ?? (bitvector_of_nat ? k) (jump_expansion_step program policy) 〈0,short_jump〉)) |
---|
1282 | ? (|program|)) |
---|
1283 | [ #H %2 elim H -H; #i #Hi |
---|
1284 | cases (proj2 ?? (sig2 ?? (jump_expansion_step program policy)) i (proj1 ?? Hi)) |
---|
1285 | [ #H @(ex_intro ?? i (conj ?? (proj1 ?? Hi) H)) |
---|
1286 | | #H @⊥ @(absurd ? H (proj2 ?? Hi)) |
---|
1287 | ] |
---|
1288 | | #H %1 whd #i #Hi @(thingie ? (dec_eq_jump_length ??)) elim H -H #H; @nmk #H2 @H |
---|
1289 | @(ex_intro … i) @conj [ @Hi | @H2 ] |
---|
1290 | | #n cases (dec_eq_jump_length (\snd (lookup ?? (bitvector_of_nat ? n) policy 〈0,short_jump〉)) |
---|
1291 | (\snd (lookup ?? (bitvector_of_nat ? n) (jump_expansion_step program policy) 〈0,short_jump〉))) |
---|
1292 | [ #H %2 @nmk #H1 elim H1 #H2 @H2 @H |
---|
1293 | | #H %1 @H |
---|
1294 | ] |
---|
1295 | ] |
---|
1296 | qed. *) |
---|
1297 | |
---|
1298 | lemma policy_not_equal_incr: ∀program:list labelled_instruction. |
---|
1299 | ∀policy:(Σp:jump_expansion_policy. |
---|
1300 | (∀i.i ≥ |program| → lookup_opt … (bitvector_of_nat ? i) p = None ?) ∧ |
---|
1301 | jump_in_policy program p). |
---|
1302 | ¬policy_equal program policy (jump_expansion_step program policy) → |
---|
1303 | ∃n:ℕ.n < (|program|) ∧ jmple |
---|
1304 | (\snd (bvt_lookup … (bitvector_of_nat ? n) policy 〈0,short_jump〉)) |
---|
1305 | (\snd (bvt_lookup … (bitvector_of_nat ? n) (jump_expansion_step program policy) 〈0,short_jump〉)). |
---|
1306 | #program #policy #Hp |
---|
1307 | cases (dec_bounded_exists (λn.jmple |
---|
1308 | (\snd (bvt_lookup ?? (bitvector_of_nat ? n) policy 〈0,short_jump〉)) |
---|
1309 | (\snd (bvt_lookup ?? (bitvector_of_nat ? n) (jump_expansion_step program policy) 〈0,short_jump〉))) ? (|program|)) |
---|
1310 | [ #H elim H; -H #i #Hi @(ex_intro ?? i) @Hi |
---|
1311 | | #abs @⊥ @(absurd ?? Hp) #n #Hn cases (proj2 ?? (sig2 ?? (jump_expansion_step program policy)) n Hn) |
---|
1312 | [ #Hj @⊥ @(absurd ?? abs) @(ex_intro ?? n) @conj [ @Hn | @Hj ] |
---|
1313 | | #H @H |
---|
1314 | ] |
---|
1315 | | #n @dec_jmple |
---|
1316 | ] |
---|
1317 | qed. |
---|
1318 | |
---|
1319 | lemma stupid: |
---|
1320 | ∀program,n. |
---|
1321 | eject … (jump_expansion_step program (jump_expansion_internal program n)) = |
---|
1322 | eject … (jump_expansion_internal program (S n)). |
---|
1323 | // |
---|
1324 | qed. |
---|
1325 | |
---|
1326 | let rec measure_int (program: list labelled_instruction) (policy: jump_expansion_policy) (acc: ℕ) |
---|
1327 | on program: ℕ ≝ |
---|
1328 | match program with |
---|
1329 | [ nil ⇒ acc |
---|
1330 | | cons h t ⇒ match (\snd (bvt_lookup ?? (bitvector_of_nat ? (|t|)) policy 〈0,short_jump〉)) with |
---|
1331 | [ long_jump ⇒ measure_int t policy (acc + 2) |
---|
1332 | | medium_jump ⇒ measure_int t policy (acc + 1) |
---|
1333 | | _ ⇒ measure_int t policy acc |
---|
1334 | ] |
---|
1335 | ]. |
---|
1336 | |
---|
1337 | definition measure ≝ |
---|
1338 | λprogram.λpolicy.measure_int program policy 0. |
---|
1339 | |
---|
1340 | (* lemma measure_le: ∀program.∀policy.∀x,y:ℕ. |
---|
1341 | x ≤ y → measure_int program policy x ≤ measure_int program policy y. |
---|
1342 | #program #policy |
---|
1343 | elim program |
---|
1344 | [ // |
---|
1345 | | #h #t #Hind #x #y #Hxy whd in match (measure_int ??x); whd in match (measure_int ??y); |
---|
1346 | cases (\snd (lookup … (bitvector_of_nat ? (|t|)) policy 〈0,short_jump〉)) |
---|
1347 | [ @Hind @Hxy |
---|
1348 | |2,3: @Hind @monotonic_le_plus_l @Hxy |
---|
1349 | ] |
---|
1350 | ] |
---|
1351 | qed. *) |
---|
1352 | |
---|
1353 | lemma measure_plus: ∀program.∀policy.∀x,d:ℕ. |
---|
1354 | measure_int program policy (x+d) = measure_int program policy x + d. |
---|
1355 | #program #policy #x #d generalize in match x; -x elim d |
---|
1356 | [ // |
---|
1357 | | -d; #d #Hind elim program |
---|
1358 | [ // |
---|
1359 | | #h #t #Hd #x whd in match (measure_int ???); whd in match (measure_int ?? x); |
---|
1360 | cases (\snd (lookup … (bitvector_of_nat ? (|t|)) policy 〈0,short_jump〉)) |
---|
1361 | [ normalize nodelta @Hd |
---|
1362 | |2,3: normalize nodelta >associative_plus >(commutative_plus (S d) ?) <associative_plus |
---|
1363 | @Hd |
---|
1364 | ] |
---|
1365 | ] |
---|
1366 | ] |
---|
1367 | qed. |
---|
1368 | |
---|
1369 | lemma measure_incr_or_equal: ∀program.∀policy:Σp:jump_expansion_policy. |
---|
1370 | (∀i.i ≥ |program| → lookup_opt … (bitvector_of_nat ? i) p = None ?) ∧ |
---|
1371 | jump_in_policy program p.∀l.|l| ≤ |program| → ∀acc:ℕ. |
---|
1372 | measure_int l policy acc ≤ measure_int l (jump_expansion_step program policy) acc. |
---|
1373 | #program #policy #l (* lapply (le_n (|program|)) *) elim l -l; |
---|
1374 | [ #Hp #acc normalize @le_n |
---|
1375 | | #h #t #Hind #Hp #acc |
---|
1376 | cases (proj2 ?? (sig2 ?? (jump_expansion_step program policy)) (|t|) ?) |
---|
1377 | [ whd in match (measure_int ???); whd in match (measure_int ?(jump_expansion_step ??)?); |
---|
1378 | cases (\snd (bvt_lookup ?? (bitvector_of_nat ? (|t|)) policy 〈0,short_jump〉)) |
---|
1379 | cases (\snd (bvt_lookup ?? (bitvector_of_nat ? (|t|)) (jump_expansion_step program policy) 〈0,short_jump〉)) |
---|
1380 | [1,4,5,7,8,9: #H @⊥ @H |
---|
1381 | |2,3,6: #_ normalize nodelta |
---|
1382 | [1,2: @(transitive_le ? (measure_int t (jump_expansion_step program policy) acc)) |
---|
1383 | |3: @(transitive_le ? (measure_int t (jump_expansion_step program policy) (acc+1))) |
---|
1384 | ] |
---|
1385 | [1,3,5: @Hind @(transitive_le … Hp) @le_n_Sn |
---|
1386 | |2,4,6: >measure_plus [1,2: @le_plus_n_r] >measure_plus @le_plus [ @le_n | //] |
---|
1387 | ] |
---|
1388 | ] |
---|
1389 | | #Heq whd in match (measure_int ???); whd in match (measure_int ?(jump_expansion_step ??)?); |
---|
1390 | >Heq cases (\snd (lookup … (bitvector_of_nat ? (|t|)) ? 〈0,short_jump〉)) |
---|
1391 | [ normalize nodelta @Hind @(transitive_le … Hp) @le_n_Sn |
---|
1392 | | normalize nodelta @Hind @(transitive_le … Hp) @le_n_Sn |
---|
1393 | | normalize nodelta @Hind @(transitive_le … Hp) @le_n_Sn |
---|
1394 | ] |
---|
1395 | | @Hp |
---|
1396 | ] |
---|
1397 | ] |
---|
1398 | qed. |
---|
1399 | |
---|
1400 | lemma measure_le: ∀program.∀policy. |
---|
1401 | measure_int program policy 0 ≤ 2*|program|. |
---|
1402 | #program #policy elim program |
---|
1403 | [ normalize @le_n |
---|
1404 | | #h #t #Hind whd in match (measure_int ???); |
---|
1405 | cases (\snd (lookup ?? (bitvector_of_nat ? (|t|)) policy 〈0,short_jump〉)) |
---|
1406 | [ normalize nodelta @(transitive_le ??? Hind) /2 by monotonic_le_times_r/ |
---|
1407 | |2,3: normalize nodelta >measure_plus <times_n_Sm >(commutative_plus 2 ?) |
---|
1408 | @le_plus [1,3: @Hind |2,4: // ] |
---|
1409 | ] |
---|
1410 | ] |
---|
1411 | qed. |
---|
1412 | |
---|
1413 | lemma bla: ∀a,b:ℕ.a + a = b + b → a = b. |
---|
1414 | #a elim a |
---|
1415 | [ normalize #b // |
---|
1416 | | -a #a #Hind #b cases b [ /2 by le_n_O_to_eq/ | -b #b normalize |
---|
1417 | <plus_n_Sm <plus_n_Sm #H |
---|
1418 | >(Hind b (injective_S ?? (injective_S ?? H))) // ] |
---|
1419 | ] |
---|
1420 | qed. |
---|
1421 | |
---|
1422 | lemma sth_not_s: ∀x.x ≠ S x. |
---|
1423 | #x cases x |
---|
1424 | [ // | #y // ] |
---|
1425 | qed. |
---|
1426 | |
---|
1427 | lemma measure_full: ∀program.∀policy. |
---|
1428 | measure_int program policy 0 = 2*|program| → ∀i.i<|program| → |
---|
1429 | (\snd (bvt_lookup ?? (bitvector_of_nat ? i) policy 〈0,short_jump〉)) = long_jump. |
---|
1430 | #program #policy elim program |
---|
1431 | [ #Hm #i #Hi @⊥ @(absurd … Hi) @not_le_Sn_O |
---|
1432 | | #h #t #Hind #Hm #i #Hi cut (measure_int t policy 0 = 2*|t|) |
---|
1433 | [ whd in match (measure_int ???) in Hm; |
---|
1434 | cases (\snd (lookup … (bitvector_of_nat ? (|t|)) policy 〈0,short_jump〉)) in Hm; |
---|
1435 | normalize nodelta |
---|
1436 | [ #H @⊥ @(absurd ? (measure_le t policy)) >H @lt_to_not_le /2 by lt_plus, le_n/ |
---|
1437 | | >measure_plus >commutative_plus #H @⊥ @(absurd ? (measure_le t policy)) |
---|
1438 | <(plus_to_minus … (sym_eq … H)) @lt_to_not_le normalize |
---|
1439 | >(commutative_plus (|t|) 0) <plus_O_n <minus_n_O |
---|
1440 | >plus_n_Sm @le_n |
---|
1441 | | >measure_plus <times_n_Sm >commutative_plus #H lapply (injective_plus_r … H) // |
---|
1442 | ] |
---|
1443 | | #Hmt cases (le_to_or_lt_eq … Hi) -Hi; |
---|
1444 | [ #Hi @(Hind Hmt i (le_S_S_to_le … Hi)) |
---|
1445 | | #Hi >(injective_S … Hi) whd in match (measure_int ???) in Hm; |
---|
1446 | cases (\snd (lookup … (bitvector_of_nat ? (|t|)) policy 〈0,short_jump〉)) in Hm; |
---|
1447 | normalize nodelta |
---|
1448 | [ >Hmt normalize <plus_n_O >(commutative_plus (|t|) (S (|t|))) |
---|
1449 | >plus_n_Sm #H @⊥ @(absurd ? (bla ?? H)) @sth_not_s |
---|
1450 | | >measure_plus >Hmt normalize <plus_n_O >commutative_plus normalize |
---|
1451 | #H @⊥ @(absurd ? (injective_plus_r … (injective_S ?? H))) @sth_not_s |
---|
1452 | | #_ // |
---|
1453 | ] |
---|
1454 | ]] |
---|
1455 | ] |
---|
1456 | qed. |
---|
1457 | |
---|
1458 | lemma eq_plus_S_to_lt: |
---|
1459 | ∀n,m,p:ℕ.n = m + (S p) → m < n. |
---|
1460 | // |
---|
1461 | qed. |
---|
1462 | |
---|
1463 | lemma measure_special: ∀program.∀policy:Σp:jump_expansion_policy. |
---|
1464 | (∀i.i ≥ |program| → lookup_opt … (bitvector_of_nat ? i) p = None ?) ∧ |
---|
1465 | jump_in_policy program p. |
---|
1466 | measure_int program policy 0 = measure_int program (jump_expansion_step program policy) 0 → |
---|
1467 | policy_equal program policy (jump_expansion_step program policy). |
---|
1468 | #program lapply (le_n (|program|)) elim program in ⊢ (?%? → ∀policy.??(?%??)(?%??) → ?%??); |
---|
1469 | [ #_ #policy #Hm #i #Hi @⊥ @(absurd ? Hi) @not_le_Sn_O |
---|
1470 | | #h #t #Hind #Hp #policy #Hm #i #Hi cases (le_to_or_lt_eq … Hi) -Hi; |
---|
1471 | [ #Hi @Hind |
---|
1472 | [ @(transitive_le … Hp) // |
---|
1473 | | whd in match (measure_int ???) in Hm; whd in match (measure_int ?(jump_expansion_step ??)?) in Hm; |
---|
1474 | lapply (proj2 ?? (sig2 ?? (jump_expansion_step program policy)) (|t|) ?) |
---|
1475 | [ @(lt_to_le_to_lt … (|h::t|)) [ // | @Hp ] |
---|
1476 | | cases (\snd (bvt_lookup ?? (bitvector_of_nat ? (|t|)) policy 〈0,short_jump〉)) in Hm; |
---|
1477 | cases (\snd (bvt_lookup ?? (bitvector_of_nat ? (|t|)) (jump_expansion_step program policy) 〈0,short_jump〉)); |
---|
1478 | normalize nodelta |
---|
1479 | [1: #H #_ @H |
---|
1480 | |2,3: >measure_plus #H #_ @⊥ @(absurd ? (eq_plus_S_to_lt … H)) @le_to_not_lt |
---|
1481 | @measure_incr_or_equal @(transitive_le … Hp) @le_n_Sn |
---|
1482 | |4,7,8: #_ #H elim H #H2 [1,3,5: @⊥ @H2 |2,4,6: destruct (H2) ] |
---|
1483 | |5: >measure_plus >measure_plus >commutative_plus >(commutative_plus ? 1) |
---|
1484 | #H #_ @(injective_plus_r … H) |
---|
1485 | |6: >measure_plus >measure_plus |
---|
1486 | change with (1+1) in match (2); >assoc_plus1 >(commutative_plus 1 (measure_int ???)) |
---|
1487 | #H #_ @⊥ @(absurd ? (eq_plus_S_to_lt … H)) @le_to_not_lt @monotonic_le_plus_l |
---|
1488 | @measure_incr_or_equal @(transitive_le … Hp) @le_n_Sn |
---|
1489 | |9: >measure_plus >measure_plus >commutative_plus >(commutative_plus ? 2) |
---|
1490 | #H #_ @(injective_plus_r … H) |
---|
1491 | ] |
---|
1492 | ] |
---|
1493 | | @(le_S_S_to_le … Hi) |
---|
1494 | ] |
---|
1495 | | #Hi >(injective_S … Hi) whd in match (measure_int ???) in Hm; |
---|
1496 | whd in match (measure_int ?(jump_expansion_step ??)?) in Hm; |
---|
1497 | lapply (proj2 ?? (sig2 ?? (jump_expansion_step program policy)) (|t|) ?) |
---|
1498 | [ @(lt_to_le_to_lt … (|h::t|)) [ // | @Hp ] |
---|
1499 | | cases (\snd (bvt_lookup ?? (bitvector_of_nat ? (|t|)) policy 〈0,short_jump〉)) in Hm; |
---|
1500 | cases (\snd (bvt_lookup ?? (bitvector_of_nat ? (|t|)) (jump_expansion_step program policy) 〈0,short_jump〉)); |
---|
1501 | normalize nodelta |
---|
1502 | [1,5,9: #_ #_ // |
---|
1503 | |4,7,8: #_ #H elim H #H2 [1,3,5: @⊥ @H2 |2,4,6: destruct (H2) ] |
---|
1504 | |2,3: >measure_plus #H #_ @⊥ @(absurd ? (eq_plus_S_to_lt … H)) @le_to_not_lt |
---|
1505 | @measure_incr_or_equal @(transitive_le … Hp) @le_n_Sn |
---|
1506 | |6: >measure_plus >measure_plus |
---|
1507 | change with (1+1) in match (2); >assoc_plus1 >(commutative_plus 1 (measure_int ???)) |
---|
1508 | #H #_ @⊥ @(absurd ? (eq_plus_S_to_lt … H)) @le_to_not_lt @monotonic_le_plus_l |
---|
1509 | @measure_incr_or_equal @(transitive_le … Hp) @le_n_Sn |
---|
1510 | ] |
---|
1511 | ] |
---|
1512 | ] |
---|
1513 | ] |
---|
1514 | qed. |
---|
1515 | |
---|
1516 | lemma dec_is_jump: ∀x.(is_jump x) + (¬is_jump x). |
---|
1517 | #x cases x #l #i cases i |
---|
1518 | [#id cases id |
---|
1519 | [1,2,3,6,7,33,34: |
---|
1520 | #x #y %2 whd in match (is_jump ?); /2 by nmk/ |
---|
1521 | |4,5,8,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32: |
---|
1522 | #x %2 whd in match (is_jump ?); /2 by nmk/ |
---|
1523 | |35,36,37: %2 whd in match (is_jump ?); /2 by nmk/ |
---|
1524 | |9,10,14,15: #x %1 // |
---|
1525 | |11,12,13,16,17: #x #y %1 // |
---|
1526 | ] |
---|
1527 | |2,3: #x %2 /2 by nmk/ |
---|
1528 | |4,5: #x %1 // |
---|
1529 | |6: #x #y %2 /2 by nmk/ |
---|
1530 | ] |
---|
1531 | qed. |
---|
1532 | |
---|
1533 | lemma measure_zero: ∀l.∀program. |
---|
1534 | |l| ≤ |program| → measure_int l (jump_expansion_internal program 0) 0 = 0. |
---|
1535 | #l #program elim l (* lapply (le_n (|program|)) elim program in ⊢ (?%? → ?%(?%??)?); *) |
---|
1536 | [ // |
---|
1537 | | #h #t #Hind #Hp whd in match (measure_int ???); |
---|
1538 | cases (dec_is_jump (nth (|t|) ? program 〈None ?, Comment []〉)) #Hj |
---|
1539 | [ >(lookup_opt_lookup_hit … (proj2 ?? (sig2 ?? (jump_expansion_start program)) (|t|) ? Hj) 〈0,short_jump〉) |
---|
1540 | [ normalize nodelta @Hind @le_S_to_le ] |
---|
1541 | @Hp |
---|
1542 | | >(lookup_opt_lookup_miss … (proj1 ?? (jump_not_in_policy program (jump_expansion_internal program 0) (|t|) ?) Hj) 〈0,short_jump〉) |
---|
1543 | [ normalize nodelta @Hind @le_S_to_le ] |
---|
1544 | @Hp |
---|
1545 | ] |
---|
1546 | ] |
---|
1547 | qed. |
---|
1548 | |
---|
1549 | definition je_fixpoint: ∀program:list labelled_instruction. |
---|
1550 | Σp:jump_expansion_policy.∃n.∀k.n < k → policy_equal program (jump_expansion_internal program k) p. |
---|
1551 | #program @(dp … (jump_expansion_internal program (2*|program|))) |
---|
1552 | @(ex_intro … (2*|program|)) #k #Hk |
---|
1553 | cases (dec_bounded_exists (λk.policy_equal program (jump_expansion_internal program k) |
---|
1554 | (jump_expansion_internal program (S k))) ? (2*|program|)) |
---|
1555 | [ #H elim H -H #x #Hx @pe_trans |
---|
1556 | [ @(jump_expansion_internal program x) |
---|
1557 | | @pe_sym @equal_remains_equal |
---|
1558 | [ @(proj2 ?? Hx) |
---|
1559 | | @(transitive_le ? (2*|program|)) |
---|
1560 | [ @le_S_S_to_le @le_S @(proj1 ?? Hx) |
---|
1561 | | @le_S_S_to_le @le_S @Hk |
---|
1562 | ] |
---|
1563 | ] |
---|
1564 | | @equal_remains_equal |
---|
1565 | [ @(proj2 ?? Hx) |
---|
1566 | | @le_S_S_to_le @le_S @(proj1 ?? Hx) |
---|
1567 | ] |
---|
1568 | ] |
---|
1569 | | #Hnex lapply (not_exists_forall … Hnex) -Hnex; #Hfa @pe_sym @equal_remains_equal |
---|
1570 | [ lapply (measure_full program (jump_expansion_internal program (2*|program|))) |
---|
1571 | #Hfull #i #Hi |
---|
1572 | lapply (proj2 ?? (sig2 ?? (jump_expansion_step program (jump_expansion_internal program (2*|program|)))) i Hi) |
---|
1573 | >(Hfull ? i Hi) |
---|
1574 | [ cases (\snd (bvt_lookup ?? (bitvector_of_nat 16 i) (jump_expansion_step program (jump_expansion_internal program (2*|program|))) 〈0,short_jump〉)) |
---|
1575 | [1,2: #H elim H #H2 [1,3: @⊥ @H2 |2,4: destruct (H2) ] |
---|
1576 | | #_ // |
---|
1577 | ] |
---|
1578 | | -i @le_to_le_to_eq |
---|
1579 | [ @measure_le |
---|
1580 | | lapply (le_n (2*|program|)) elim (2*|program|) in ⊢ (?%? → %); |
---|
1581 | [ #_ >measure_zero @le_n |
---|
1582 | | #x #Hind #Hx |
---|
1583 | cut (measure_int program (jump_expansion_internal program x) 0 < |
---|
1584 | measure_int program (jump_expansion_internal program (S x)) 0) |
---|
1585 | [ elim (le_to_or_lt_eq … |
---|
1586 | (measure_incr_or_equal program (jump_expansion_internal program x) program (le_n (|program|)) 0)) |
---|
1587 | [ // |
---|
1588 | | #H @⊥ @(absurd ?? (Hfa x Hx)) @measure_special @H |
---|
1589 | ] |
---|
1590 | | #H lapply (Hind (le_S_to_le … Hx)) #H2 @(le_to_lt_to_lt … H) @H2 |
---|
1591 | ] |
---|
1592 | ] |
---|
1593 | ] |
---|
1594 | ] |
---|
1595 | | @le_S_to_le @Hk |
---|
1596 | ] |
---|
1597 | | #n @dec_bounded_forall #m @dec_eq_jump_length |
---|
1598 | ] |
---|
1599 | qed. |
---|
1600 | |
---|
1601 | (**************************************** START OF POLICY ABSTRACTION ********************) |
---|
1602 | |
---|
1603 | definition policy_type≝ Word → jump_length. |
---|
1604 | |
---|
1605 | definition jump_expansion': pseudo_assembly_program → policy_type ≝ |
---|
1606 | λprogram.λpc. |
---|
1607 | let policy ≝ jump_expansion_internal (\snd program) (|\snd program|) in |
---|
1608 | let 〈n,j〉 ≝ lookup ? ? pc policy 〈0, long_jump〉 in |
---|
1609 | j. |
---|
1610 | |
---|
1611 | definition assembly_1_pseudoinstruction_safe ≝ |
---|
1612 | λprogram: pseudo_assembly_program. |
---|
1613 | λjump_expansion: policy_type. |
---|
1614 | λppc: Word. |
---|
1615 | λpc: Word. |
---|
1616 | λlookup_labels. |
---|
1617 | λlookup_datalabels. |
---|
1618 | λi. |
---|
1619 | let expansion ≝ jump_expansion ppc in |
---|
1620 | match expand_pseudo_instruction_safe lookup_labels lookup_datalabels pc expansion i with |
---|
1621 | [ None ⇒ None ? |
---|
1622 | | Some pseudos ⇒ |
---|
1623 | let mapped ≝ map ? ? assembly1 pseudos in |
---|
1624 | let flattened ≝ flatten ? mapped in |
---|
1625 | let pc_len ≝ length ? flattened in |
---|
1626 | Some ? (〈pc_len, flattened〉) |
---|
1627 | ]. |
---|
1628 | |
---|
1629 | definition construct_costs_safe ≝ |
---|
1630 | λprogram. |
---|
1631 | λjump_expansion: policy_type. |
---|
1632 | λpseudo_program_counter: nat. |
---|
1633 | λprogram_counter: nat. |
---|
1634 | λcosts: BitVectorTrie costlabel 16. |
---|
1635 | λi. |
---|
1636 | match i with |
---|
1637 | [ Cost cost ⇒ |
---|
1638 | let program_counter_bv ≝ bitvector_of_nat ? program_counter in |
---|
1639 | Some ? 〈program_counter, (insert … program_counter_bv cost costs)〉 |
---|
1640 | | _ ⇒ |
---|
1641 | let pc_bv ≝ bitvector_of_nat ? program_counter in |
---|
1642 | let ppc_bv ≝ bitvector_of_nat ? pseudo_program_counter in |
---|
1643 | let lookup_labels ≝ λx.pc_bv in |
---|
1644 | let lookup_datalabels ≝ λx.zero ? in |
---|
1645 | let pc_delta_assembled ≝ |
---|
1646 | assembly_1_pseudoinstruction_safe program jump_expansion ppc_bv pc_bv |
---|
1647 | lookup_labels lookup_datalabels i |
---|
1648 | in |
---|
1649 | match pc_delta_assembled with |
---|
1650 | [ None ⇒ None ? |
---|
1651 | | Some pc_delta_assembled ⇒ |
---|
1652 | let 〈pc_delta, assembled〉 ≝ pc_delta_assembled in |
---|
1653 | Some ? 〈pc_delta + program_counter, costs〉 |
---|
1654 | ] |
---|
1655 | ]. |
---|
1656 | |
---|
1657 | (* This establishes the correspondence between pseudo program counters and |
---|
1658 | program counters. It is at the heart of the proof. *) |
---|
1659 | (*CSC: code taken from build_maps *) |
---|
1660 | definition sigma00: pseudo_assembly_program → policy_type → list ? → ? → option (nat × (nat × (BitVectorTrie Word 16))) ≝ |
---|
1661 | λinstr_list. |
---|
1662 | λjump_expansion: policy_type. |
---|
1663 | λl:list labelled_instruction. |
---|
1664 | λacc. |
---|
1665 | foldl … |
---|
1666 | (λt,i. |
---|
1667 | match t with |
---|
1668 | [ None ⇒ None … |
---|
1669 | | Some ppc_pc_map ⇒ |
---|
1670 | let 〈ppc,pc_map〉 ≝ ppc_pc_map in |
---|
1671 | let 〈program_counter, sigma_map〉 ≝ pc_map in |
---|
1672 | let 〈label, i〉 ≝ i in |
---|
1673 | match construct_costs_safe instr_list jump_expansion ppc program_counter (Stub …) i with |
---|
1674 | [ None ⇒ None ? |
---|
1675 | | Some pc_ignore ⇒ |
---|
1676 | let 〈pc,ignore〉 ≝ pc_ignore in |
---|
1677 | Some … 〈S ppc, 〈pc, insert ?? (bitvector_of_nat 16 ppc) (bitvector_of_nat 16 pc) sigma_map〉〉 ] |
---|
1678 | ]) acc l. |
---|
1679 | |
---|
1680 | definition sigma0: pseudo_assembly_program → policy_type → option (nat × (nat × (BitVectorTrie Word 16))) ≝ |
---|
1681 | λprog. |
---|
1682 | λjump_expansion. |
---|
1683 | sigma00 prog jump_expansion (\snd prog) (Some ? 〈0, 〈0, Stub …〉〉). |
---|
1684 | |
---|
1685 | definition tech_pc_sigma00: pseudo_assembly_program → policy_type → list labelled_instruction → option (nat × nat) ≝ |
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1686 | λprogram,jump_expansion,instr_list. |
---|
1687 | match sigma00 program jump_expansion instr_list (Some ? 〈0, 〈0, (Stub ? ?)〉〉) (* acc copied from sigma0 *) with |
---|
1688 | [ None ⇒ None … |
---|
1689 | | Some result ⇒ |
---|
1690 | let 〈ppc,pc_sigma_map〉 ≝ result in |
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1691 | let 〈pc,map〉 ≝ pc_sigma_map in |
---|
1692 | Some … 〈ppc,pc〉 ]. |
---|
1693 | |
---|
1694 | definition sigma_safe: pseudo_assembly_program → policy_type → option (Word → Word) ≝ |
---|
1695 | λinstr_list,jump_expansion. |
---|
1696 | match sigma0 instr_list jump_expansion with |
---|
1697 | [ None ⇒ None ? |
---|
1698 | | Some result ⇒ |
---|
1699 | let 〈ppc,pc_sigma_map〉 ≝ result in |
---|
1700 | let 〈pc, sigma_map〉 ≝ pc_sigma_map in |
---|
1701 | if gtb pc (2^16) then |
---|
1702 | None ? |
---|
1703 | else |
---|
1704 | Some ? (λx. lookup … x sigma_map (zero …)) ]. |
---|
1705 | |
---|
1706 | (* stuff about policy *) |
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1707 | |
---|
1708 | definition policy_ok ≝ λjump_expansion,p. sigma_safe p jump_expansion ≠ None …. |
---|
1709 | |
---|
1710 | definition policy ≝ λp. Σjump_expansion:policy_type. policy_ok jump_expansion p. |
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1711 | |
---|
1712 | lemma eject_policy: ∀p. policy p → policy_type. |
---|
1713 | #p #pol @(eject … pol) |
---|
1714 | qed. |
---|
1715 | |
---|
1716 | coercion eject_policy nocomposites: ∀p.∀pol:policy p. policy_type ≝ eject_policy on _pol:(policy ?) to policy_type. |
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1717 | |
---|
1718 | definition sigma: ∀p:pseudo_assembly_program. policy p → Word → Word ≝ |
---|
1719 | λp,policy. |
---|
1720 | match sigma_safe p (eject … policy) return λr:option (Word → Word). r ≠ None … → Word → Word with |
---|
1721 | [ None ⇒ λabs. ⊥ |
---|
1722 | | Some r ⇒ λ_.r] (sig2 … policy). |
---|
1723 | cases abs /2/ |
---|
1724 | qed. |
---|
1725 | |
---|
1726 | example sigma_0: ∀p,pol. sigma p pol (bitvector_of_nat ? 0) = bitvector_of_nat ? 0. |
---|
1727 | cases daemon. |
---|
1728 | qed. |
---|
1729 | |
---|
1730 | axiom fetch_pseudo_instruction_split: |
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1731 | ∀instr_list,ppc. |
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1732 | ∃pre,suff,lbl. |
---|
1733 | (pre @ [〈lbl,\fst (fetch_pseudo_instruction instr_list ppc)〉]) @ suff = instr_list. |
---|
1734 | |
---|
1735 | lemma sigma00_append: |
---|
1736 | ∀instr_list,jump_expansion,l1,l2,acc. |
---|
1737 | sigma00 instr_list jump_expansion (l1@l2) acc = |
---|
1738 | sigma00 instr_list jump_expansion |
---|
1739 | l2 (sigma00 instr_list jump_expansion l1 acc). |
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1740 | whd in match sigma00; normalize nodelta; |
---|
1741 | #instr_list #jump_expansion #l1 #l2 #acc @foldl_append |
---|
1742 | qed. |
---|
1743 | |
---|
1744 | lemma sigma00_strict: |
---|
1745 | ∀instr_list,jump_expansion,l,acc. acc = None ? → |
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1746 | sigma00 instr_list jump_expansion l acc = None …. |
---|
1747 | #instr_list #jump_expansion #l elim l |
---|
1748 | [ #acc #H >H % |
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1749 | | #hd #tl #IH #acc #H >H change with (sigma00 ?? tl ? = ?) @IH % ] |
---|
1750 | qed. |
---|
1751 | |
---|
1752 | lemma policy_ok_prefix_ok: |
---|
1753 | ∀program.∀pol:policy program.∀suffix,prefix. |
---|
1754 | prefix@suffix = \snd program → |
---|
1755 | sigma00 program pol prefix (Some … 〈0, 〈0, Stub …〉〉) ≠ None …. |
---|
1756 | * #preamble #instr_list #pol #suffix #prefix #prf whd in prf:(???%); |
---|
1757 | generalize in match (sig2 ?? pol); whd in prf:(???%); <prf in pol; #pol |
---|
1758 | whd in match policy_ok; whd in match sigma_safe; whd in match sigma0; |
---|
1759 | normalize nodelta >sigma00_append |
---|
1760 | cases (sigma00 ?? prefix ?) |
---|
1761 | [2: #x #_ % #abs destruct(abs) |
---|
1762 | | * #abs @⊥ @abs >sigma00_strict % ] |
---|
1763 | qed. |
---|
1764 | |
---|
1765 | lemma policy_ok_prefix_hd_ok: |
---|
1766 | ∀program.∀pol:policy program.∀suffix,hd,prefix,ppc_pc_map. |
---|
1767 | (prefix@[hd])@suffix = \snd program → |
---|
1768 | Some ? ppc_pc_map = sigma00 program pol prefix (Some … 〈0, 〈0, Stub …〉〉) → |
---|
1769 | let 〈ppc,pc_map〉 ≝ ppc_pc_map in |
---|
1770 | let 〈program_counter, sigma_map〉 ≝ pc_map in |
---|
1771 | let 〈label, i〉 ≝ hd in |
---|
1772 | construct_costs_safe program pol ppc program_counter (Stub …) i ≠ None …. |
---|
1773 | * #preamble #instr_list #pol #suffix #hd #prefix #ppc_pc_map #EQ1 #EQ2 |
---|
1774 | generalize in match (policy_ok_prefix_ok 〈preamble,instr_list〉 pol suffix |
---|
1775 | (prefix@[hd]) EQ1) in ⊢ ?; >sigma00_append <EQ2 whd in ⊢ (?(??%?) → ?); |
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1776 | @pair_elim' #ppc #pc_map #EQ3 normalize nodelta |
---|
1777 | @pair_elim' #pc #map #EQ4 normalize nodelta |
---|
1778 | @pair_elim' #l' #i' #EQ5 normalize nodelta |
---|
1779 | cases (construct_costs_safe ??????) normalize |
---|
1780 | [* #abs @⊥ @abs % | #X #_ % #abs destruct(abs)] |
---|
1781 | qed. |
---|
1782 | |
---|
1783 | definition expand_pseudo_instruction: |
---|
1784 | ∀program:pseudo_assembly_program.∀pol: policy program. |
---|
1785 | ∀ppc:Word.∀lookup_labels,lookup_datalabels,pc. |
---|
1786 | lookup_labels = (λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x)) → |
---|
1787 | lookup_datalabels = (λx. lookup_def … (construct_datalabels (\fst program)) x (zero ?)) → |
---|
1788 | let pi ≝ \fst (fetch_pseudo_instruction (\snd program) ppc) in |
---|
1789 | pc = sigma program pol ppc → |
---|
1790 | Σres:list instruction. Some … res = expand_pseudo_instruction_safe lookup_labels lookup_datalabels pc (pol ppc) pi |
---|
1791 | ≝ λprogram,pol,ppc,lookup_labels,lookup_datalabels,pc,prf1,prf2,prf3. |
---|
1792 | match expand_pseudo_instruction_safe lookup_labels lookup_datalabels pc (pol ppc) (\fst (fetch_pseudo_instruction (\snd program) ppc)) with |
---|
1793 | [ None ⇒ let dummy ≝ [ ] in dummy |
---|
1794 | | Some res ⇒ res ]. |
---|
1795 | [ @⊥ whd in p:(??%??); |
---|
1796 | generalize in match (sig2 ?? pol); whd in ⊢ (% → ?); |
---|
1797 | whd in ⊢ (?(??%?) → ?); change with (sigma00 ????) in ⊢ (?(??(match % with [_ ⇒ ? | _ ⇒ ?])?) → ?); |
---|
1798 | generalize in match (refl … (sigma00 program pol (\snd program) (Some ? 〈O,〈O,Stub (BitVector 16) 16〉〉))); |
---|
1799 | cases (sigma00 ????) in ⊢ (??%? → %); normalize nodelta [#_ * #abs @abs %] |
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1800 | #res #K |
---|
1801 | cases (fetch_pseudo_instruction_split (\snd program) ppc) #pre * #suff * #lbl #EQ1 |
---|
1802 | generalize in match (policy_ok_prefix_hd_ok program pol … EQ1 ?) in ⊢ ?; |
---|
1803 | cases daemon (* CSC: XXXXXXXX Ero qui |
---|
1804 | |
---|
1805 | [3: @policy_ok_prefix_ok ] |
---|
1806 | | sigma00 program pol pre |
---|
1807 | |
---|
1808 | |
---|
1809 | |
---|
1810 | QUA USARE LEMMA policy_ok_prefix_hd_ok combinato a lemma da fare che |
---|
1811 | fetch ppc = hd sse program = pre @ [hd] @ tl e |pre| = ppc |
---|
1812 | per concludere construct_costs_safe ≠ None *) |
---|
1813 | | >p %] |
---|
1814 | qed. |
---|
1815 | |
---|
1816 | (* MAIN AXIOM HERE, HIDDEN USING cases daemon *) |
---|
1817 | definition assembly_1_pseudoinstruction': |
---|
1818 | ∀program:pseudo_assembly_program.∀pol: policy program. |
---|
1819 | ∀ppc:Word.∀lookup_labels,lookup_datalabels,pi. |
---|
1820 | lookup_labels = (λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x)) → |
---|
1821 | lookup_datalabels = (λx. lookup_def … (construct_datalabels (\fst program)) x (zero ?)) → |
---|
1822 | \fst (fetch_pseudo_instruction (\snd program) ppc) = pi → |
---|
1823 | Σres:(nat × (list Byte)). |
---|
1824 | Some … res = assembly_1_pseudoinstruction_safe program pol ppc (sigma program pol ppc) lookup_labels lookup_datalabels pi ∧ |
---|
1825 | let 〈len,code〉 ≝ res in |
---|
1826 | sigma program pol (\snd (half_add ? ppc (bitvector_of_nat ? 1))) = |
---|
1827 | bitvector_of_nat … (nat_of_bitvector … (sigma program pol ppc) + len) |
---|
1828 | ≝ λprogram: pseudo_assembly_program. |
---|
1829 | λpol: policy program. |
---|
1830 | λppc: Word. |
---|
1831 | λlookup_labels. |
---|
1832 | λlookup_datalabels. |
---|
1833 | λpi. |
---|
1834 | λprf1,prf2,prf3. |
---|
1835 | match assembly_1_pseudoinstruction_safe program pol ppc (sigma program pol ppc) lookup_labels lookup_datalabels pi with |
---|
1836 | [ None ⇒ let dummy ≝ 〈0,[ ]〉 in dummy |
---|
1837 | | Some res ⇒ res ]. |
---|
1838 | [ @⊥ elim pi in p; [*] |
---|
1839 | try (#ARG1 #ARG2 #ARG3 #abs) try (#ARG1 #ARG2 #abs) try (#ARG1 #abs) try #abs |
---|
1840 | generalize in match (jmeq_to_eq ??? abs); -abs; #abs whd in abs:(??%?); try destruct(abs) |
---|
1841 | whd in abs:(??match % with [_ ⇒ ? | _ ⇒ ?]?); |
---|
1842 | (* WRONG HERE, NEEDS LEMMA SAYING THAT THE POLICY DOES NOT RETURN MEDIUM! *) |
---|
1843 | cases daemon |
---|
1844 | | % [ >p %] |
---|
1845 | cases res in p ⊢ %; -res; #len #code #EQ normalize nodelta; |
---|
1846 | (* THIS SHOULD BE TRUE INSTEAD *) |
---|
1847 | cases daemon] |
---|
1848 | qed. |
---|
1849 | |
---|
1850 | definition assembly_1_pseudoinstruction: |
---|
1851 | ∀program:pseudo_assembly_program.∀pol: policy program. |
---|
1852 | ∀ppc:Word.∀lookup_labels,lookup_datalabels,pi. |
---|
1853 | lookup_labels = (λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x)) → |
---|
1854 | lookup_datalabels = (λx. lookup_def … (construct_datalabels (\fst program)) x (zero ?)) → |
---|
1855 | \fst (fetch_pseudo_instruction (\snd program) ppc) = pi → |
---|
1856 | nat × (list Byte) |
---|
1857 | ≝ λprogram,pol,ppc,lookup_labels,lookup_datalabels,pi,prf1,prf2,prf3. |
---|
1858 | assembly_1_pseudoinstruction' program pol ppc lookup_labels lookup_datalabels pi prf1 |
---|
1859 | prf2 prf3. |
---|
1860 | |
---|
1861 | lemma assembly_1_pseudoinstruction_ok1: |
---|
1862 | ∀program:pseudo_assembly_program.∀pol: policy program. |
---|
1863 | ∀ppc:Word.∀lookup_labels,lookup_datalabels,pi. |
---|
1864 | ∀prf1:lookup_labels = (λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x)). |
---|
1865 | ∀prf2:lookup_datalabels = (λx. lookup_def … (construct_datalabels (\fst program)) x (zero ?)). |
---|
1866 | ∀prf3:\fst (fetch_pseudo_instruction (\snd program) ppc) = pi. |
---|
1867 | Some … (assembly_1_pseudoinstruction program pol ppc lookup_labels lookup_datalabels pi prf1 prf2 prf3) |
---|
1868 | = assembly_1_pseudoinstruction_safe program pol ppc (sigma program pol ppc) lookup_labels lookup_datalabels pi. |
---|
1869 | #program #pol #ppc #lookup_labels #lookup_datalabels #pi #prf1 #prf2 #prf3 |
---|
1870 | cases (sig2 … (assembly_1_pseudoinstruction' program pol ppc lookup_labels lookup_datalabels pi prf1 prf2 prf3)) |
---|
1871 | #H1 #_ @H1 |
---|
1872 | qed. |
---|
1873 | |
---|
1874 | (* MAIN AXIOM HERE, HIDDEN USING cases daemon *) |
---|
1875 | definition construct_costs': |
---|
1876 | ∀program. ∀pol:policy program. ∀ppc,pc,costs,i. |
---|
1877 | Σres:(nat × (BitVectorTrie costlabel 16)). Some … res = construct_costs_safe program pol ppc pc costs i |
---|
1878 | ≝ |
---|
1879 | λprogram.λpol: policy program.λppc,pc,costs,i. |
---|
1880 | match construct_costs_safe program pol ppc pc costs i with |
---|
1881 | [ None ⇒ let dummy ≝ 〈0, Stub costlabel 16〉 in dummy |
---|
1882 | | Some res ⇒ res ]. |
---|
1883 | [ cases daemon |
---|
1884 | | >p %] |
---|
1885 | qed. |
---|
1886 | |
---|
1887 | definition construct_costs ≝ |
---|
1888 | λprogram,pol,ppc,pc,costs,i. eject … (construct_costs' program pol ppc pc costs i). |
---|
1889 | |
---|
1890 | (* |
---|
1891 | axiom suffix_of: ∀A:Type[0]. ∀l,prefix:list A. list A. |
---|
1892 | axiom suffix_of_ok: ∀A,l,prefix. prefix @ suffix_of A l prefix = l. |
---|
1893 | |
---|
1894 | axiom foldl_strong_step: |
---|
1895 | ∀A:Type[0]. |
---|
1896 | ∀P: list A → Type[0]. |
---|
1897 | ∀l: list A. |
---|
1898 | ∀H: ∀prefix,hd,tl. l = prefix @ [hd] @ tl → P prefix → P (prefix @ [hd]). |
---|
1899 | ∀acc: P [ ]. |
---|
1900 | ∀Q: ∀prefix. P prefix → Prop. |
---|
1901 | ∀HQ: ∀prefix,hd,tl.∀prf: l = prefix @ [hd] @ tl. |
---|
1902 | ∀acc: P prefix. Q prefix acc → Q (prefix @ [hd]) (H prefix hd tl prf acc). |
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1903 | Q [ ] acc → |
---|
1904 | Q l (foldl_strong A P l H acc). |
---|
1905 | (* |
---|
1906 | #A #P #l #H #acc #Q #HQ #Hacc normalize; |
---|
1907 | generalize in match |
---|
1908 | (foldl_strong ? |
---|
1909 | (λpre. Q pre (foldl_strong_internal A P l (suffix_of A l pre) ? [ ] pre acc ?)) |
---|
1910 | l ? Hacc) |
---|
1911 | [3: >suffix_of_ok % | 2: #prefix #hd #tl #EQ @(H prefix hd (tl@suffix_of A l pre) EQ) ] |
---|
1912 | [2: #prefix #hd #tl #prf #X whd in ⊢ (??%) |
---|
1913 | #K |
---|
1914 | |
---|
1915 | generalize in match |
---|
1916 | (foldl_strong ? |
---|
1917 | (λpre. Q pre (foldl_strong_internal A P l H pre (suffix_of A l pre) acc (suffix_of_ok A l pre)))) |
---|
1918 | [2: @H |
---|
1919 | *) |
---|
1920 | |
---|
1921 | axiom foldl_elim: |
---|
1922 | ∀A:Type[0]. |
---|
1923 | ∀B: Type[0]. |
---|
1924 | ∀H: A → B → A. |
---|
1925 | ∀acc: A. |
---|
1926 | ∀l: list B. |
---|
1927 | ∀Q: A → Prop. |
---|
1928 | (∀acc:A.∀b:B. Q acc → Q (H acc b)) → |
---|
1929 | Q acc → |
---|
1930 | Q (foldl A B H acc l). |
---|
1931 | *) |
---|
1932 | |
---|
1933 | lemma option_destruct_Some: ∀A,a,b. Some A a = Some A b → a=b. |
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1934 | #A #a #b #EQ destruct // |
---|
1935 | qed. |
---|
1936 | |
---|
1937 | (* |
---|
1938 | lemma tech_pc_sigma00_append_Some: |
---|
1939 | ∀program.∀pol:policy program.∀prefix,costs,label,i,ppc,pc. |
---|
1940 | tech_pc_sigma00 program pol prefix = Some … 〈ppc,pc〉 → |
---|
1941 | tech_pc_sigma00 program pol (prefix@[〈label,i〉]) = Some … 〈S ppc,\fst (construct_costs program pol … ppc pc costs i)〉. |
---|
1942 | #program #pol #prefix #costs #label #i #ppc #pc #H |
---|
1943 | whd in match tech_pc_sigma00 in ⊢ %; normalize nodelta; |
---|
1944 | whd in match sigma00 in ⊢ %; normalize nodelta in ⊢ %; |
---|
1945 | generalize in match (sig2 … pol) whd in ⊢ (% → ?) whd in ⊢ (?(??%?) → ?) |
---|
1946 | whd in match sigma0; normalize nodelta; |
---|
1947 | >foldl_step |
---|
1948 | change with (? → match match sigma00 program pol prefix with [None ⇒ ? | Some res ⇒ ?] with [ None ⇒ ? | Some res ⇒ ? ] = ?) |
---|
1949 | whd in match tech_pc_sigma00 in H; normalize nodelta in H; |
---|
1950 | cases (sigma00 program pol prefix) in H ⊢ % |
---|
1951 | [ whd in ⊢ (??%% → ?) #abs destruct(abs) |
---|
1952 | | * #ppc' * #pc' #sigma_map normalize nodelta; #H generalize in match (option_destruct_Some ??? H) |
---|
1953 | |
---|
1954 | normalize nodelta; -H; |
---|
1955 | |
---|
1956 | |
---|
1957 | generalize in match H; -H; |
---|
1958 | generalize in match (foldl ?????); in H ⊢ (??match match % with [_ ⇒ ? | _ ⇒ ?] with [_ ⇒ ? | _ ⇒ ?]?) |
---|
1959 | [2: whd in ⊢ (??%%) |
---|
1960 | XXX |
---|
1961 | *) |
---|
1962 | |
---|
1963 | axiom construct_costs_sigma: |
---|
1964 | ∀p.∀pol:policy p.∀ppc,pc,costs,i. |
---|
1965 | bitvector_of_nat ? pc = sigma p pol (bitvector_of_nat ? ppc) → |
---|
1966 | bitvector_of_nat ? (\fst (construct_costs p pol ppc pc costs i)) = sigma p pol (bitvector_of_nat 16 (S ppc)). |
---|
1967 | |
---|
1968 | axiom tech_pc_sigma00_append_Some: |
---|
1969 | ∀program.∀pol:policy program.∀prefix,costs,label,i,ppc,pc. |
---|
1970 | tech_pc_sigma00 program pol prefix = Some … 〈ppc,pc〉 → |
---|
1971 | tech_pc_sigma00 program pol (prefix@[〈label,i〉]) = Some … 〈S ppc,\fst (construct_costs program pol … ppc pc costs i)〉. |
---|
1972 | |
---|
1973 | axiom eq_identifier_eq: |
---|
1974 | ∀tag: String. |
---|
1975 | ∀l. |
---|
1976 | ∀r. |
---|
1977 | eq_identifier tag l r = true → l = r. |
---|
1978 | |
---|
1979 | axiom neq_identifier_neq: |
---|
1980 | ∀tag: String. |
---|
1981 | ∀l, r: identifier tag. |
---|
1982 | eq_identifier tag l r = false → (l = r → False). |
---|
1983 | |
---|
1984 | definition build_maps: |
---|
1985 | ∀pseudo_program.∀pol:policy pseudo_program. |
---|
1986 | Σres:((identifier_map ASMTag Word) × (BitVectorTrie costlabel 16)). |
---|
1987 | let 〈labels, costs〉 ≝ res in |
---|
1988 | ∀id. occurs_exactly_once id (\snd pseudo_program) → |
---|
1989 | lookup_def … labels id (zero ?) = sigma pseudo_program pol (address_of_word_labels_code_mem (\snd pseudo_program) id) ≝ |
---|
1990 | λpseudo_program. |
---|
1991 | λpol:policy pseudo_program. |
---|
1992 | let result ≝ |
---|
1993 | foldl_strong |
---|
1994 | (option Identifier × pseudo_instruction) |
---|
1995 | (λpre. Σres:((identifier_map ASMTag Word) × (nat × (nat × (BitVectorTrie costlabel 16)))). |
---|
1996 | let 〈labels,ppc_pc_costs〉 ≝ res in |
---|
1997 | let 〈ppc',pc_costs〉 ≝ ppc_pc_costs in |
---|
1998 | let 〈pc',costs〉 ≝ pc_costs in |
---|
1999 | And ( And ( |
---|
2000 | And (bitvector_of_nat ? pc' = sigma pseudo_program pol (bitvector_of_nat ? ppc')) (*∧*) |
---|
2001 | (ppc' = length … pre)) (*∧ *) |
---|
2002 | (tech_pc_sigma00 pseudo_program (eject … pol) pre = Some ? 〈ppc',pc'〉)) (*∧*) |
---|
2003 | (∀id. occurs_exactly_once id pre → |
---|
2004 | lookup_def … labels id (zero …) = sigma pseudo_program pol (address_of_word_labels_code_mem pre id))) |
---|
2005 | (\snd pseudo_program) |
---|
2006 | (λprefix,i,tl,prf,t. |
---|
2007 | let 〈labels, ppc_pc_costs〉 ≝ t in |
---|
2008 | let 〈ppc, pc_costs〉 ≝ ppc_pc_costs in |
---|
2009 | let 〈pc,costs〉 ≝ pc_costs in |
---|
2010 | let 〈label, i'〉 ≝ i in |
---|
2011 | let labels ≝ |
---|
2012 | match label with |
---|
2013 | [ None ⇒ labels |
---|
2014 | | Some label ⇒ |
---|
2015 | let program_counter_bv ≝ bitvector_of_nat ? pc in |
---|
2016 | add ? ? labels label program_counter_bv |
---|
2017 | ] |
---|
2018 | in |
---|
2019 | let construct ≝ construct_costs pseudo_program pol ppc pc costs i' in |
---|
2020 | 〈labels, 〈S ppc,construct〉〉) 〈empty_map …, 〈0, 〈0, Stub ? ?〉〉〉 |
---|
2021 | in |
---|
2022 | let 〈labels, ppc_pc_costs〉 ≝ result in |
---|
2023 | let 〈ppc,pc_costs〉 ≝ ppc_pc_costs in |
---|
2024 | let 〈pc, costs〉 ≝ pc_costs in |
---|
2025 | 〈labels, costs〉. |
---|
2026 | [2: whd generalize in match (sig2 … t); >p >p1 >p2 >p3 * * * #IHn1 #IH0 #IH1 #IH2 |
---|
2027 | generalize in match (refl … construct); cases construct in ⊢ (???% → %); #PC #CODE #JMEQ % [% [%]] |
---|
2028 | [ <(construct_costs_sigma … costs i1 IHn1) change with (? = ?(\fst construct)) >JMEQ % |
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2029 | | >append_length <IH0 normalize; -IHn1; (*CSC: otherwise it diverges!*) // |
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2030 | | >(tech_pc_sigma00_append_Some … costs … IH1) change with (Some … 〈S ppc,\fst construct〉 = ?) >JMEQ % |
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2031 | | #id normalize nodelta; -labels1; cases label; normalize nodelta |
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2032 | [ #K <address_of_word_labels_code_mem_None [2:@K] @IH2 -IHn1; (*CSC: otherwise it diverges!*) // |
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2033 | | #l #H generalize in match (occurs_exactly_once_Some ???? H) in ⊢ ?; |
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2034 | generalize in match (refl … (eq_identifier ? l id)); cases (eq_identifier … l id) in ⊢ (???% → %); |
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2035 | [ #EQ #_ <(eq_identifier_eq … EQ) >lookup_def_add_hit >address_of_word_labels_code_mem_Some_hit |
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2036 | <IH0 [@IHn1 | <(eq_identifier_eq … EQ) in H; #H @H] |
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2037 | | #EQ change with (occurs_exactly_once ?? → ?) #K >lookup_def_add_miss [2: -IHn1; |
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2038 | (*Andrea:XXXX used to work /2/*)@nmk #IDL lapply (sym_eq ? ? ? IDL) |
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2039 | lapply (neq_identifier_neq ? ? ? EQ) #ASSM assumption ] |
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2040 | <(address_of_word_labels_code_mem_Some_miss … EQ … H) @IH2 assumption ]]] |
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2041 | |3: whd % [% [%]] [>sigma_0 % | % | % | #id normalize in ⊢ (% → ?); #abs @⊥ // ] |
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2042 | | generalize in match (sig2 … result) in ⊢ ?; |
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2043 | normalize nodelta in p ⊢ %; -result; >p in ⊢ (match % with [_ ⇒ ?] → ?); |
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2044 | normalize nodelta; >p1 normalize nodelta; >p2; normalize nodelta; * #_; #H @H] |
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2045 | qed. |
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2046 | |
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2047 | definition assembly: |
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2048 | ∀p:pseudo_assembly_program. policy p → list Byte × (BitVectorTrie costlabel 16) ≝ |
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2049 | λp.let 〈preamble, instr_list〉 ≝ p in |
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2050 | λpol. |
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2051 | let 〈labels,costs〉 ≝ build_maps 〈preamble,instr_list〉 pol in |
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2052 | let datalabels ≝ construct_datalabels preamble in |
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2053 | let lookup_labels ≝ λx. lookup_def … labels x (zero ?) in |
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2054 | let lookup_datalabels ≝ λx. lookup_def … datalabels x (zero ?) in |
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2055 | let result ≝ |
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2056 | foldl_strong |
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2057 | (option Identifier × pseudo_instruction) |
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2058 | (λpre. Σpc_ppc_code:(Word × (Word × (list Byte))). |
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2059 | let 〈pc,ppc_code〉 ≝ pc_ppc_code in |
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2060 | let 〈ppc,code〉 ≝ ppc_code in |
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2061 | ∀ppc'. |
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2062 | let 〈pi,newppc〉 ≝ fetch_pseudo_instruction instr_list ppc' in |
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2063 | let 〈len,assembledi〉 ≝ |
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2064 | assembly_1_pseudoinstruction 〈preamble,instr_list〉 pol ppc' lookup_labels lookup_datalabels pi ??? in |
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2065 | True) |
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2066 | instr_list |
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2067 | (λprefix,hd,tl,prf,pc_ppc_code. |
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2068 | let 〈pc, ppc_code〉 ≝ pc_ppc_code in |
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2069 | let 〈ppc, code〉 ≝ ppc_code in |
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2070 | let 〈pc_delta, program〉 ≝ assembly_1_pseudoinstruction 〈preamble,instr_list〉 pol ppc lookup_labels lookup_datalabels (\snd hd) ??? in |
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2071 | let 〈new_pc, flags〉 ≝ add_16_with_carry pc (bitvector_of_nat ? pc_delta) false in |
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2072 | let new_ppc ≝ \snd (half_add ? ppc (bitvector_of_nat ? 1)) in |
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2073 | 〈new_pc, 〈new_ppc, (code @ program)〉〉) |
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2074 | 〈(zero ?), 〈(zero ?), [ ]〉〉 |
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2075 | in |
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2076 | 〈\snd (\snd result), costs〉. |
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2077 | [2,5: % |
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2078 | |*: cases daemon ] |
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2079 | qed. |
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2080 | |
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2081 | definition assembly_unlabelled_program: assembly_program → option (list Byte × (BitVectorTrie Identifier 16)) ≝ |
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2082 | λp. Some ? (〈foldr ? ? (λi,l. assembly1 i @ l) [ ] p, Stub …〉). |
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