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 "utilities/extralib.ma". |
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6 | include "ASM/Assembly.ma". |
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7 | |
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8 | include alias "basics/lists/list.ma". |
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9 | include alias "arithmetics/nat.ma". |
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10 | include alias "basics/logic.ma". |
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11 | |
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12 | (* Internal types *) |
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13 | |
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14 | (* ppc_pc_map: program length × (pseudo program counter ↦ 〈pc, jump_length〉) *) |
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15 | definition ppc_pc_map ≝ ℕ × (BitVectorTrie (ℕ × jump_length) 16). |
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16 | |
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17 | (* The different properties that we want/need to prove at some point *) |
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18 | (* Anything that's not in the program doesn't end up in the policy *) |
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19 | definition out_of_program_none: list labelled_instruction → ppc_pc_map → Prop ≝ |
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20 | λprefix.λsigma. |
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21 | ∀i:ℕ.i ≥ |prefix| → i < 2^16 → bvt_lookup_opt … (bitvector_of_nat 16 i) (\snd sigma) = None ?. |
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22 | |
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23 | (* If instruction i is a jump, then there will be something in the policy at |
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24 | * position i *) |
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25 | definition is_jump' ≝ |
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26 | λx:preinstruction Identifier. |
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27 | match x with |
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28 | [ JC _ ⇒ True |
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29 | | JNC _ ⇒ True |
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30 | | JZ _ ⇒ True |
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31 | | JNZ _ ⇒ True |
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32 | | JB _ _ ⇒ True |
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33 | | JNB _ _ ⇒ True |
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34 | | JBC _ _ ⇒ True |
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35 | | CJNE _ _ ⇒ True |
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36 | | DJNZ _ _ ⇒ True |
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37 | | _ ⇒ False |
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38 | ]. |
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39 | |
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40 | definition is_jump ≝ |
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41 | λinstr:pseudo_instruction. |
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42 | match instr with |
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43 | [ Instruction i ⇒ is_jump' i |
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44 | | Call _ ⇒ True |
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45 | | Jmp _ ⇒ True |
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46 | | _ ⇒ False |
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47 | ]. |
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48 | |
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49 | definition is_jump_to ≝ |
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50 | λx:pseudo_instruction.λd:Identifier. |
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51 | match x with |
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52 | [ Instruction i ⇒ match i with |
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53 | [ JC j ⇒ d = j |
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54 | | JNC j ⇒ d = j |
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55 | | JZ j ⇒ d = j |
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56 | | JNZ j ⇒ d = j |
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57 | | JB _ j ⇒ d = j |
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58 | | JNB _ j ⇒ d = j |
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59 | | CJNE _ j ⇒ d = j |
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60 | | DJNZ _ j ⇒ d = j |
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61 | | _ ⇒ False |
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62 | ] |
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63 | | Call c ⇒ d = c |
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64 | | Jmp j ⇒ d = j |
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65 | | _ ⇒ False |
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66 | ]. |
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67 | |
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68 | definition jump_not_in_policy: list labelled_instruction → ppc_pc_map → Prop ≝ |
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69 | λprefix.λsigma. |
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70 | ∀i:ℕ.i < |prefix| → |
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71 | ¬is_jump (\snd (nth i ? prefix 〈None ?, Comment []〉)) → |
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72 | \snd (bvt_lookup … (bitvector_of_nat 16 i) (\snd sigma) 〈0,short_jump〉) = short_jump. |
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73 | |
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74 | (* if the instruction 〈p,a〉 is a jump to label l, then label l is at address a *) |
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75 | (* definition labels_okay: label_map → ppc_pc_map → Prop ≝ |
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76 | λlabels.λsigma. |
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77 | bvt_forall ?? (\snd sigma) (λn.λx. |
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78 | let 〈pc,addr_nat〉 ≝ x in |
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79 | ∃id:Identifier.lookup_def … labels id 0 = addr_nat |
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80 | ). *) |
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81 | |
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82 | (* Between two policies, jumps cannot decrease *) |
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83 | definition jmpeqb: jump_length → jump_length → bool ≝ |
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84 | λj1.λj2. |
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85 | match j1 with |
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86 | [ short_jump ⇒ match j2 with [ short_jump ⇒ true | _ ⇒ false ] |
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87 | | medium_jump ⇒ match j2 with [ medium_jump ⇒ true | _ ⇒ false ] |
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88 | | long_jump ⇒ match j2 with [ long_jump ⇒ true | _ ⇒ false ] |
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89 | ]. |
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90 | |
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91 | lemma jmpeqb_to_eq: ∀j1,j2.jmpeqb j1 j2 → j1 = j2. |
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92 | #j1 #j2 cases j1 cases j2 |
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93 | [1,5,9: / by /] |
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94 | #H cases H |
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95 | qed. |
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96 | |
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97 | definition jmple: jump_length → jump_length → Prop ≝ |
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98 | λj1.λj2. |
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99 | match j1 with |
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100 | [ short_jump ⇒ |
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101 | match j2 with |
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102 | [ short_jump ⇒ False |
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103 | | _ ⇒ True |
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104 | ] |
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105 | | medium_jump ⇒ |
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106 | match j2 with |
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107 | [ long_jump ⇒ True |
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108 | | _ ⇒ False |
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109 | ] |
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110 | | long_jump ⇒ False |
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111 | ]. |
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112 | |
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113 | definition jmpleq: jump_length → jump_length → Prop ≝ |
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114 | λj1.λj2.jmple j1 j2 ∨ j1 = j2. |
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115 | |
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116 | definition policy_increase: list labelled_instruction → ppc_pc_map → |
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117 | ppc_pc_map → Prop ≝ |
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118 | λprogram.λop.λp. |
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119 | ∀i.i < |program| → |
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120 | let 〈opc,oj〉 ≝ bvt_lookup … (bitvector_of_nat 16 i) (\snd op) 〈0,short_jump〉 in |
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121 | let 〈pc,j〉 ≝ bvt_lookup … (bitvector_of_nat 16 i) (\snd p) 〈0,short_jump〉 in |
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122 | (*opc ≤ pc ∧*) jmpleq oj j. |
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123 | |
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124 | (* Policy safety *) |
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125 | (*definition policy_safe: list labelled_instruction → label_map → ppc_pc_map → Prop ≝ |
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126 | λprogram.λlabels.λsigma. |
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127 | ∀i.i < |program| → |
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128 | let 〈pc,j〉 ≝ bvt_lookup … (bitvector_of_nat 16 i) (\snd sigma) 〈0,false〉 in |
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129 | let 〈label,instr〉 ≝ nth i ? program 〈None ?, Comment [ ]〉 in |
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130 | ∀dest.is_jump_to instr dest → |
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131 | let paddr ≝ lookup_def … labels dest 0 in |
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132 | let addr ≝ \fst (bvt_lookup … (bitvector_of_nat 16 paddr) (\snd sigma) 〈0,false〉) in |
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133 | match j with |
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134 | [ None ⇒ True |
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135 | | Some j ⇒ match j with |
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136 | [ short_jump ⇒ |
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137 | if leb pc addr |
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138 | then le (addr - pc) 126 |
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139 | else le (pc - addr) 129 |
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140 | | medium_jump ⇒ |
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141 | let a ≝ bitvector_of_nat 16 addr in |
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142 | let p ≝ bitvector_of_nat 16 pc in |
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143 | let 〈fst_5_addr, rest_addr〉 ≝ split bool 5 11 a in |
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144 | let 〈fst_5_pc, rest_pc〉 ≝ split bool 5 11 p in |
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145 | eq_bv 5 fst_5_addr fst_5_pc = true |
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146 | | long_jump ⇒ True |
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147 | ] |
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148 | ].*) |
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149 | |
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150 | (* this is the instruction size as determined by the distance from origin to destination *) |
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151 | (*definition instruction_size_sigma: label_map → ppc_pc_map → Word → pseudo_instruction → ℕ ≝ |
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152 | λlabels.λsigma.λpc.λi. |
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153 | \fst (assembly_1_pseudoinstruction |
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154 | (λid.bitvector_of_nat 16 (lookup_def … labels id 0)) |
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155 | (λi.bitvector_of_nat 16 (\fst (bvt_lookup ?? i (\snd sigma) 〈0,false〉))) pc |
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156 | (λx.zero 16) i).*) |
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157 | |
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158 | (* this is the instruction size as determined by the jump length given *) |
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159 | definition expand_relative_jump_internal_unsafe: |
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160 | jump_length → ([[relative]] → preinstruction [[relative]]) → list instruction ≝ |
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161 | λjmp_len:jump_length.λi. |
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162 | match jmp_len with |
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163 | [ short_jump ⇒ [ RealInstruction (i (RELATIVE (zero 8))) ] |
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164 | | medium_jump ⇒ [ ] (* this should not happen *) |
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165 | | long_jump ⇒ |
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166 | [ RealInstruction (i (RELATIVE (bitvector_of_nat ? 2))); |
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167 | SJMP (RELATIVE (bitvector_of_nat ? 3)); (* LJMP size? *) |
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168 | LJMP (ADDR16 (zero 16)) |
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169 | ] |
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170 | ]. |
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171 | @I |
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172 | qed. |
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173 | |
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174 | definition expand_relative_jump_unsafe: |
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175 | jump_length → preinstruction Identifier → list instruction ≝ |
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176 | λjmp_len:jump_length.λi. |
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177 | match i with |
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178 | [ JC jmp ⇒ expand_relative_jump_internal_unsafe jmp_len (JC ?) |
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179 | | JNC jmp ⇒ expand_relative_jump_internal_unsafe jmp_len (JNC ?) |
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180 | | JB baddr jmp ⇒ expand_relative_jump_internal_unsafe jmp_len (JB ? baddr) |
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181 | | JZ jmp ⇒ expand_relative_jump_internal_unsafe jmp_len (JZ ?) |
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182 | | JNZ jmp ⇒ expand_relative_jump_internal_unsafe jmp_len (JNZ ?) |
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183 | | JBC baddr jmp ⇒ expand_relative_jump_internal_unsafe jmp_len (JBC ? baddr) |
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184 | | JNB baddr jmp ⇒ expand_relative_jump_internal_unsafe jmp_len (JNB ? baddr) |
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185 | | CJNE addr jmp ⇒ expand_relative_jump_internal_unsafe jmp_len (CJNE ? addr) |
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186 | | DJNZ addr jmp ⇒ expand_relative_jump_internal_unsafe jmp_len (DJNZ ? addr) |
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187 | | ADD arg1 arg2 ⇒ [ ADD ? arg1 arg2 ] |
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188 | | ADDC arg1 arg2 ⇒ [ ADDC ? arg1 arg2 ] |
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189 | | SUBB arg1 arg2 ⇒ [ SUBB ? arg1 arg2 ] |
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190 | | INC arg ⇒ [ INC ? arg ] |
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191 | | DEC arg ⇒ [ DEC ? arg ] |
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192 | | MUL arg1 arg2 ⇒ [ MUL ? arg1 arg2 ] |
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193 | | DIV arg1 arg2 ⇒ [ DIV ? arg1 arg2 ] |
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194 | | DA arg ⇒ [ DA ? arg ] |
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195 | | ANL arg ⇒ [ ANL ? arg ] |
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196 | | ORL arg ⇒ [ ORL ? arg ] |
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197 | | XRL arg ⇒ [ XRL ? arg ] |
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198 | | CLR arg ⇒ [ CLR ? arg ] |
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199 | | CPL arg ⇒ [ CPL ? arg ] |
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200 | | RL arg ⇒ [ RL ? arg ] |
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201 | | RR arg ⇒ [ RR ? arg ] |
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202 | | RLC arg ⇒ [ RLC ? arg ] |
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203 | | RRC arg ⇒ [ RRC ? arg ] |
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204 | | SWAP arg ⇒ [ SWAP ? arg ] |
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205 | | MOV arg ⇒ [ MOV ? arg ] |
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206 | | MOVX arg ⇒ [ MOVX ? arg ] |
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207 | | SETB arg ⇒ [ SETB ? arg ] |
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208 | | PUSH arg ⇒ [ PUSH ? arg ] |
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209 | | POP arg ⇒ [ POP ? arg ] |
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210 | | XCH arg1 arg2 ⇒ [ XCH ? arg1 arg2 ] |
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211 | | XCHD arg1 arg2 ⇒ [ XCHD ? arg1 arg2 ] |
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212 | | RET ⇒ [ RET ? ] |
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213 | | RETI ⇒ [ RETI ? ] |
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214 | | NOP ⇒ [ RealInstruction (NOP ?) ] |
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215 | ]. |
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216 | |
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217 | definition instruction_size_jmplen: |
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218 | jump_length → pseudo_instruction → ℕ ≝ |
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219 | λjmp_len. |
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220 | λi. |
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221 | let pseudos ≝ match i with |
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222 | [ Cost cost ⇒ [ ] |
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223 | | Comment comment ⇒ [ ] |
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224 | | Call call ⇒ |
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225 | match jmp_len with |
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226 | [ short_jump ⇒ [ ] (* this should not happen *) |
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227 | | medium_jump ⇒ [ ACALL (ADDR11 (zero 11)) ] |
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228 | | long_jump ⇒ [ LCALL (ADDR16 (zero 16)) ] |
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229 | ] |
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230 | | Mov d trgt ⇒ |
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231 | [ RealInstruction (MOV ? (inl ? ? (inl ? ? (inr ? ? 〈DPTR, DATA16 (zero 16)〉))))] |
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232 | | Instruction instr ⇒ expand_relative_jump_unsafe jmp_len instr |
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233 | | Jmp jmp ⇒ |
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234 | match jmp_len with |
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235 | [ short_jump ⇒ [ SJMP (RELATIVE (zero 8)) ] |
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236 | | medium_jump ⇒ [ AJMP (ADDR11 (zero 11)) ] |
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237 | | long_jump ⇒ [ LJMP (ADDR16 (zero 16)) ] |
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238 | ] |
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239 | ] in |
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240 | let mapped ≝ map ? ? assembly1 pseudos in |
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241 | let flattened ≝ flatten ? mapped in |
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242 | let pc_len ≝ length ? flattened in |
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243 | pc_len. |
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244 | @I. |
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245 | qed. |
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246 | |
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247 | (* new safety condition: policy corresponds to program and resulting program is compact *) |
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248 | definition policy_compact: list labelled_instruction → label_map → ppc_pc_map → Prop ≝ |
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249 | λprogram.λlabels.λsigma. |
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250 | ∀n:ℕ.S n < |program| → |
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251 | match bvt_lookup_opt … (bitvector_of_nat ? n) (\snd sigma) with |
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252 | [ None ⇒ False |
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253 | | Some x ⇒ let 〈pc,j〉 ≝ x in |
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254 | match bvt_lookup_opt … (bitvector_of_nat ? (S n)) (\snd sigma) with |
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255 | [ None ⇒ False |
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256 | | Some x1 ⇒ let 〈pc1,j1〉 ≝ x1 in |
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257 | pc1 = instruction_size (λid.bitvector_of_nat ? (lookup_def ?? labels id 0)) |
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258 | (λppc.bitvector_of_nat ? (\fst (bvt_lookup ?? ppc (\snd sigma) 〈0,short_jump〉))) |
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259 | (λppc.jmpeqb long_jump (\snd (bvt_lookup ?? ppc (\snd sigma) 〈0,short_jump〉))) |
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260 | (bitvector_of_nat ? pc) (\snd (nth n ? program 〈None ?, Comment []〉)) |
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261 | ] |
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262 | ]. |
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263 | |
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264 | (* Definitions and theorems for the jump_length type (itself defined in Assembly) *) |
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265 | definition max_length: jump_length → jump_length → jump_length ≝ |
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266 | λj1.λj2. |
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267 | match j1 with |
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268 | [ long_jump ⇒ long_jump |
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269 | | medium_jump ⇒ |
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270 | match j2 with |
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271 | [ medium_jump ⇒ medium_jump |
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272 | | _ ⇒ long_jump |
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273 | ] |
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274 | | short_jump ⇒ |
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275 | match j2 with |
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276 | [ short_jump ⇒ short_jump |
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277 | | _ ⇒ long_jump |
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278 | ] |
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279 | ]. |
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280 | |
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281 | lemma dec_jmple: ∀x,y:jump_length.Sum (jmple x y) (¬(jmple x y)). |
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282 | #x #y cases x cases y /3 by inl, inr, nmk, I/ |
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283 | qed. |
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284 | |
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285 | lemma jmpleq_max_length: ∀ol,nl. |
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286 | jmpleq ol (max_length ol nl). |
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287 | #ol #nl cases ol cases nl |
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288 | /2 by or_introl, or_intror, I/ |
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289 | qed. |
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290 | |
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291 | lemma dec_eq_jump_length: ∀a,b:jump_length.Sum (a = b) (a ≠ b). |
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292 | #a #b cases a cases b /2/ |
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293 | %2 @nmk #H destruct (H) |
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294 | qed. |
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295 | |
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296 | (* definition policy_isize_sum ≝ |
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297 | λprefix:list labelled_instruction.λlabels:label_map.λsigma:ppc_pc_map. |
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298 | (\fst sigma) = foldl_strong (option Identifier × pseudo_instruction) |
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299 | (λacc.ℕ) |
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300 | prefix |
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301 | (λhd.λx.λtl.λp.λacc. |
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302 | acc + (instruction_size (λid.bitvector_of_nat ? (lookup_def ?? labels id 0)) |
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303 | (λppc.bitvector_of_nat ? (\fst (bvt_lookup ?? ppc (\snd sigma) 〈0,short_jump〉))) |
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304 | (λppc.jmpeqb long_jump (\snd (bvt_lookup ?? ppc (\snd sigma) 〈0,short_jump〉))) |
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305 | (bitvector_of_nat 16 (\fst sigma)) (\snd x))) |
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306 | 0. *) |
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307 | |
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308 | (* The function that creates the label-to-address map *) |
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309 | definition create_label_map: ∀program:list labelled_instruction. |
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310 | (Σlabels:label_map. |
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311 | ∀l.occurs_exactly_once ?? l program → |
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312 | bitvector_of_nat ? (lookup_def ?? labels l 0) = |
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313 | address_of_word_labels_code_mem program l |
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314 | ) ≝ |
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315 | λprogram. |
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316 | \fst (create_label_cost_map program). |
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317 | #l #Hl lapply (pi2 ?? (create_label_cost_map0 program)) @pair_elim |
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318 | #labels #costs #EQ normalize nodelta #H whd in match create_label_cost_map; |
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319 | normalize nodelta >EQ @(H l Hl) |
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320 | qed. |
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321 | |
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322 | definition select_reljump_length: label_map → ppc_pc_map → ppc_pc_map → ℕ → ℕ → |
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323 | Identifier → jump_length ≝ |
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324 | λlabels.λold_sigma.λinc_sigma.λadded.λppc.λlbl. |
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325 | let paddr ≝ lookup_def … labels lbl 0 in |
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326 | if leb ppc paddr (* forward jump *) |
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327 | then |
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328 | let addr ≝ \fst (bvt_lookup … (bitvector_of_nat 16 paddr) (\snd old_sigma) 〈0,short_jump〉) |
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329 | + added in |
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330 | if leb (addr - \fst inc_sigma) 126 |
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331 | then short_jump |
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332 | else long_jump |
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333 | else |
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334 | let addr ≝ \fst (bvt_lookup … (bitvector_of_nat 16 paddr) (\snd inc_sigma) 〈0,short_jump〉) in |
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335 | if leb (\fst inc_sigma - addr) 129 |
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336 | then short_jump |
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337 | else long_jump. |
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338 | |
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339 | definition select_call_length: label_map → ppc_pc_map → ppc_pc_map → ℕ → ℕ → |
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340 | Identifier → jump_length ≝ |
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341 | λlabels.λold_sigma.λinc_sigma.λadded.λppc.λlbl. |
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342 | let paddr ≝ lookup_def ? ? labels lbl 0 in |
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343 | let addr ≝ |
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344 | if leb ppc paddr (* forward jump *) |
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345 | then \fst (bvt_lookup … (bitvector_of_nat ? paddr) (\snd old_sigma) 〈0,short_jump〉) |
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346 | + added |
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347 | else \fst (bvt_lookup … (bitvector_of_nat ? paddr) (\snd inc_sigma) 〈0,short_jump〉) in |
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348 | let 〈fst_5_addr, rest_addr〉 ≝ split ? 5 11 (bitvector_of_nat ? addr) in |
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349 | let 〈fst_5_pc, rest_pc〉 ≝ split ? 5 11 (bitvector_of_nat ? (\fst inc_sigma)) in |
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350 | if eq_bv ? fst_5_addr fst_5_pc |
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351 | then medium_jump |
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352 | else long_jump. |
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353 | |
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354 | definition select_jump_length: label_map → ppc_pc_map → ppc_pc_map → ℕ → ℕ → |
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355 | Identifier → jump_length ≝ |
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356 | λlabels.λold_sigma.λinc_sigma.λadded.λppc.λlbl. |
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357 | let paddr ≝ lookup_def … labels lbl 0 in |
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358 | if leb ppc paddr (* forward jump *) |
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359 | then |
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360 | let addr ≝ \fst (bvt_lookup … (bitvector_of_nat 16 paddr) (\snd old_sigma) 〈0,short_jump〉) |
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361 | + added in |
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362 | if leb (addr - \fst inc_sigma) 126 |
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363 | then short_jump |
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364 | else select_call_length labels old_sigma inc_sigma added ppc lbl |
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365 | else |
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366 | let addr ≝ \fst (bvt_lookup … (bitvector_of_nat 16 paddr) (\snd inc_sigma) 〈0,short_jump〉) in |
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367 | if leb (\fst inc_sigma - addr) 129 |
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368 | then short_jump |
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369 | else select_call_length labels old_sigma inc_sigma added ppc lbl. |
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370 | |
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371 | definition jump_expansion_step_instruction: label_map → ppc_pc_map → ppc_pc_map → |
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372 | ℕ → ℕ → preinstruction Identifier → option jump_length ≝ |
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373 | λlabels.λold_sigma.λinc_sigma.λadded.λppc.λi. |
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374 | match i with |
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375 | [ JC j ⇒ Some ? (select_reljump_length labels old_sigma inc_sigma added ppc j) |
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376 | | JNC j ⇒ Some ? (select_reljump_length labels old_sigma inc_sigma added ppc j) |
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377 | | JZ j ⇒ Some ? (select_reljump_length labels old_sigma inc_sigma added ppc j) |
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378 | | JNZ j ⇒ Some ? (select_reljump_length labels old_sigma inc_sigma added ppc j) |
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379 | | JB _ j ⇒ Some ? (select_reljump_length labels old_sigma inc_sigma added ppc j) |
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380 | | JBC _ j ⇒ Some ? (select_reljump_length labels old_sigma inc_sigma added ppc j) |
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381 | | JNB _ j ⇒ Some ? (select_reljump_length labels old_sigma inc_sigma added ppc j) |
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382 | | CJNE _ j ⇒ Some ? (select_reljump_length labels old_sigma inc_sigma added ppc j) |
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383 | | DJNZ _ j ⇒ Some ? (select_reljump_length labels old_sigma inc_sigma added ppc j) |
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384 | | _ ⇒ None ? |
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385 | ]. |
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386 | |
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387 | lemma dec_is_jump: ∀x.Sum (is_jump x) (¬is_jump x). |
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388 | #i cases i |
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389 | [#id cases id |
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390 | [1,2,3,6,7,33,34: |
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391 | #x #y %2 whd in match (is_jump ?); /2 by nmk/ |
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392 | |4,5,8,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32: |
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393 | #x %2 whd in match (is_jump ?); /2 by nmk/ |
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394 | |35,36,37: %2 whd in match (is_jump ?); /2 by nmk/ |
---|
395 | |9,10,14,15: #x %1 / by I/ |
---|
396 | |11,12,13,16,17: #x #y %1 / by I/ |
---|
397 | ] |
---|
398 | |2,3: #x %2 /2 by nmk/ |
---|
399 | |4,5: #x %1 / by I/ |
---|
400 | |6: #x #y %2 /2 by nmk/ |
---|
401 | ] |
---|
402 | qed. |
---|
403 | |
---|
404 | lemma geb_to_leb: ∀a,b:ℕ.geb a b = leb b a. |
---|
405 | #a #b / by refl/ |
---|
406 | qed. |
---|
407 | |
---|
408 | (* The first step of the jump expansion: everything to short. *) |
---|
409 | definition jump_expansion_start: |
---|
410 | ∀program:(Σl:list labelled_instruction.|l| < 2^16). |
---|
411 | ∀labels:label_map. |
---|
412 | Σpolicy:option ppc_pc_map. |
---|
413 | match policy with |
---|
414 | [ None ⇒ True |
---|
415 | | Some p ⇒ |
---|
416 | And (And (And (And (out_of_program_none (pi1 ?? program) p) |
---|
417 | (jump_not_in_policy (pi1 ?? program) p)) |
---|
418 | (policy_compact program labels p)) |
---|
419 | (∀i.i < |program| → |
---|
420 | \snd (bvt_lookup … (bitvector_of_nat ? i) (\snd p) 〈0,short_jump〉) = short_jump)) |
---|
421 | (\fst p < 2^16) |
---|
422 | ] ≝ |
---|
423 | λprogram.λlabels. |
---|
424 | let final_policy ≝ foldl_strong (option Identifier × pseudo_instruction) |
---|
425 | (λprefix.Σpolicy:ppc_pc_map. |
---|
426 | And (And (And (out_of_program_none prefix policy) |
---|
427 | (jump_not_in_policy prefix policy)) |
---|
428 | (policy_compact prefix labels policy)) |
---|
429 | (∀i.i < |prefix| → |
---|
430 | \snd (bvt_lookup … (bitvector_of_nat ? i) (\snd policy) 〈0,short_jump〉) = short_jump)) |
---|
431 | program |
---|
432 | (λprefix.λx.λtl.λprf.λp. |
---|
433 | let 〈pc,sigma〉 ≝ p in |
---|
434 | let 〈label,instr〉 ≝ x in |
---|
435 | let isize ≝ instruction_size_jmplen short_jump instr in |
---|
436 | 〈pc + isize, bvt_insert … (bitvector_of_nat 16 (|prefix|)) 〈pc,short_jump〉 sigma〉 |
---|
437 | ) 〈0, Stub ? ?〉 in |
---|
438 | if geb (\fst final_policy) 2^16 then |
---|
439 | None ? |
---|
440 | else |
---|
441 | Some ? (pi1 ?? final_policy). |
---|
442 | [ / by I/ |
---|
443 | | lapply p -p generalize in match (foldl_strong ?????); * #p #Hp #hg |
---|
444 | @conj [ @Hp | @not_le_to_lt @leb_false_to_not_le <geb_to_leb @hg ] |
---|
445 | | @conj [ @conj [ @conj |
---|
446 | [ (* out_of_program_none *) |
---|
447 | #i >append_length <commutative_plus #Hi normalize in Hi; #Hi2 |
---|
448 | cases (le_to_or_lt_eq … Hi) -Hi #Hi |
---|
449 | cases p -p #p cases p -p #pc #p #Hp cases x -x #l #pi |
---|
450 | [ >lookup_opt_insert_miss |
---|
451 | [ @(proj1 ?? (proj1 ?? (proj1 ?? Hp)) i ? Hi2) @le_S_to_le @le_S_to_le @Hi |
---|
452 | | @bitvector_of_nat_abs |
---|
453 | [ @Hi2 |
---|
454 | | @(transitive_lt … Hi2) @le_S_to_le @Hi |
---|
455 | | @sym_neq @lt_to_not_eq @le_S_to_le @Hi |
---|
456 | ] |
---|
457 | ] |
---|
458 | | >lookup_opt_insert_miss |
---|
459 | [ <Hi @(proj1 ?? (proj1 ?? (proj1 ?? Hp)) (S (|prefix|)) (le_S ?? (le_n (|prefix|))) ?) |
---|
460 | >Hi @Hi2 |
---|
461 | | @bitvector_of_nat_abs |
---|
462 | [ @Hi2 |
---|
463 | | @(transitive_lt … Hi2) <Hi @le_n |
---|
464 | | @sym_neq @lt_to_not_eq <Hi @le_n |
---|
465 | ] |
---|
466 | ] |
---|
467 | ] |
---|
468 | | (* jump_not_in_policy *) #i >append_length <commutative_plus |
---|
469 | #Hi normalize in Hi; cases (le_to_or_lt_eq … (le_S_S_to_le … Hi)) -Hi #Hi |
---|
470 | [ cases p -p #p cases p -p #pc #sigma #Hp cases x #l #ins >lookup_insert_miss |
---|
471 | [ >(nth_append_first ? i prefix ?? Hi) @((proj2 ?? (proj1 ?? (proj1 ?? Hp))) i Hi) |
---|
472 | | @bitvector_of_nat_abs |
---|
473 | [ @(transitive_lt … (pi2 ?? program)) >prf >append_length >commutative_plus |
---|
474 | @le_plus_a @Hi |
---|
475 | | @(transitive_lt … (pi2 ?? program)) >prf >append_length <plus_n_Sm @le_S_S |
---|
476 | @le_plus_n_r |
---|
477 | | @lt_to_not_eq @Hi |
---|
478 | ] |
---|
479 | ] |
---|
480 | | >Hi >nth_append_second [2: @le_n] <minus_n_n whd in match (nth ????); |
---|
481 | cases p -p #p cases p -p #pc #sigma #Hp cases x #lbl #ins |
---|
482 | >lookup_insert_hit #_ / by / |
---|
483 | ] |
---|
484 | ] |
---|
485 | | (* policy_compact *) cases daemon |
---|
486 | ] |
---|
487 | | (* lookup = short_jump *) #i >append_length <commutative_plus #Hi normalize in Hi; |
---|
488 | cases p -p #p cases p -p #pc #sigma #Hp cases x #lbl #instr normalize nodelta |
---|
489 | cases (le_to_or_lt_eq … (le_S_S_to_le … Hi)) -Hi #Hi |
---|
490 | [ >lookup_insert_miss |
---|
491 | [ @((proj2 ?? Hp) i Hi) |
---|
492 | | @bitvector_of_nat_abs |
---|
493 | [ @(transitive_lt … (pi2 ?? program)) >prf >append_length >commutative_plus |
---|
494 | @le_plus_a @Hi |
---|
495 | | @(transitive_lt … (pi2 ?? program)) >prf >append_length <plus_n_Sm @le_S_S |
---|
496 | @le_plus_n_r |
---|
497 | | @lt_to_not_eq @Hi |
---|
498 | ] |
---|
499 | ] |
---|
500 | | >Hi >lookup_insert_hit @refl |
---|
501 | ] |
---|
502 | ] |
---|
503 | | @conj [ @conj [ @conj |
---|
504 | [ #i #_ #Hi2 / by refl/ |
---|
505 | ]]] |
---|
506 | #i #H @⊥ @(absurd … H) @not_le_Sn_O |
---|
507 | ] |
---|
508 | qed. |
---|
509 | |
---|
510 | definition policy_equal ≝ |
---|
511 | λprogram:list labelled_instruction.λp1,p2:ppc_pc_map. |
---|
512 | (* \fst p1 = \fst p2 ∧ *) |
---|
513 | (∀n:ℕ.n < |program| → |
---|
514 | let pc1 ≝ bvt_lookup … (bitvector_of_nat 16 n) (\snd p1) 〈0,short_jump〉 in |
---|
515 | let pc2 ≝ bvt_lookup … (bitvector_of_nat 16 n) (\snd p2) 〈0,short_jump〉 in |
---|
516 | \snd pc1 = \snd pc2). |
---|
517 | |
---|
518 | definition nec_plus_ultra ≝ |
---|
519 | λprogram:list labelled_instruction.λp:ppc_pc_map. |
---|
520 | ¬(∀i.i < |program| → is_jump (\snd (nth i ? program 〈None ?, Comment []〉)) → |
---|
521 | \snd (bvt_lookup … (bitvector_of_nat 16 i) (\snd p) 〈0,short_jump〉) = long_jump). |
---|
522 | |
---|
523 | (*include alias "common/Identifiers.ma".*) |
---|
524 | include alias "ASM/BitVector.ma". |
---|
525 | include alias "basics/lists/list.ma". |
---|
526 | include alias "arithmetics/nat.ma". |
---|
527 | include alias "basics/logic.ma". |
---|
528 | |
---|
529 | lemma blerpque: ∀a,b,i. |
---|
530 | is_jump i → instruction_size_jmplen (max_length a b) i = instruction_size_jmplen a i → |
---|
531 | (max_length a b) = a. |
---|
532 | #a #b #i cases i |
---|
533 | [1: #pi cases pi |
---|
534 | [1,2,3,4,5,6,7,8,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37: |
---|
535 | [1,2,3,6,7,24,25: #x #y |
---|
536 | |4,5,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23: #x] |
---|
537 | #H cases H |
---|
538 | |9,10,11,12,13,14,15,16,17: #x [3,4,5,8,9: #y] |
---|
539 | #_ cases a cases b |
---|
540 | [1,5,7,8,9: #_ / by refl/ |
---|
541 | |10,14,16,17,18: #_ / by refl/ |
---|
542 | |19,23,25,26,27: #_ / by refl/ |
---|
543 | |28,32,34,35,36: #_ / by refl/ |
---|
544 | |37,41,43,44,45: #_ / by refl/ |
---|
545 | |46,50,52,53,54: #_ / by refl/ |
---|
546 | |55,59,61,62,63: #_ / by refl/ |
---|
547 | |64,68,70,71,72: #_ / by refl/ |
---|
548 | |73,77,79,80,81: #_ / by refl/ |
---|
549 | |2,3,4,6: cases x #a cases a |
---|
550 | [1,2,3,4,8,9,16,17,18,19: #b #Hb cases Hb |
---|
551 | |20,21,22,23,27,28,35,36,37,38: #b #Hb cases Hb |
---|
552 | |39,40,41,42,46,47,54,55,56,57: #b #Hb cases Hb |
---|
553 | |58,59,60,61,65,66,73,74,75,76: #b #Hb cases Hb |
---|
554 | |5,6,7,10,11,12,13,14: #Hb cases Hb |
---|
555 | |24,25,26,29,30,31,32,33: #Hb cases Hb |
---|
556 | |43,44,45,48,49,50,51,52: #Hb cases Hb |
---|
557 | |62,63,64,67,68,69,70,71: #Hb cases Hb |
---|
558 | |15,34,53,72: #b #Hb #H normalize in H; destruct (H) |
---|
559 | ] |
---|
560 | |11,12,13,15: cases x #a cases a |
---|
561 | [1,2,3,4,8,9,16,17,18,19: #b #Hb cases Hb |
---|
562 | |20,21,22,23,27,28,35,36,37,38: #b #Hb cases Hb |
---|
563 | |39,40,41,42,46,47,54,55,56,57: #b #Hb cases Hb |
---|
564 | |58,59,60,61,65,66,73,74,75,76: #b #Hb cases Hb |
---|
565 | |5,6,7,10,11,12,13,14: #Hb cases Hb |
---|
566 | |24,25,26,29,30,31,32,33: #Hb cases Hb |
---|
567 | |43,44,45,48,49,50,51,52: #Hb cases Hb |
---|
568 | |62,63,64,67,68,69,70,71: #Hb cases Hb |
---|
569 | |15,34,53,72: #b #Hb #H normalize in H; destruct (H) |
---|
570 | ] |
---|
571 | |20,21,22,24: cases x #a cases a |
---|
572 | [1,2,3,4,8,9,16,17,18,19: #b #Hb cases Hb |
---|
573 | |20,21,22,23,27,28,35,36,37,38: #b #Hb cases Hb |
---|
574 | |39,40,41,42,46,47,54,55,56,57: #b #Hb cases Hb |
---|
575 | |58,59,60,61,65,66,73,74,75,76: #b #Hb cases Hb |
---|
576 | |5,6,7,10,11,12,13,14: #Hb cases Hb |
---|
577 | |24,25,26,29,30,31,32,33: #Hb cases Hb |
---|
578 | |43,44,45,48,49,50,51,52: #Hb cases Hb |
---|
579 | |62,63,64,67,68,69,70,71: #Hb cases Hb |
---|
580 | |15,34,53,72: #b #Hb #H normalize in H; destruct (H) |
---|
581 | ] |
---|
582 | |29,30,31,33: cases x #a cases a #a1 #a2 |
---|
583 | [1,3,5,7: cases a2 #b cases b |
---|
584 | [2,3,4,9,15,16,17,18,19: #b #Hb cases Hb |
---|
585 | |21,22,23,28,34,35,36,37,38: #b #Hb cases Hb |
---|
586 | |40,41,42,47,53,54,55,56,57: #b #Hb cases Hb |
---|
587 | |59,60,61,66,72,73,74,75,76: #b #Hb cases Hb |
---|
588 | |5,6,7,10,11,12,13,14: #Hb cases Hb |
---|
589 | |24,25,26,29,30,31,32,33: #Hb cases Hb |
---|
590 | |43,44,45,48,49,50,51,52: #Hb cases Hb |
---|
591 | |62,63,64,67,68,69,70,71: #Hb cases Hb |
---|
592 | |1,8: #b #Hb #H normalize in H; destruct (H) |
---|
593 | |20,27: #b #Hb #H normalize in H; destruct (H) |
---|
594 | |39,46: #b #Hb #H normalize in H; destruct (H) |
---|
595 | |58,65: #b #Hb #H normalize in H; destruct (H) |
---|
596 | ] |
---|
597 | |2,4,6,8: cases a1 #b cases b |
---|
598 | [1,3,8,9,15,16,17,18,19: #b #Hb cases Hb |
---|
599 | |20,22,27,28,34,35,36,37,38: #b #Hb cases Hb |
---|
600 | |39,41,46,47,53,54,55,56,57: #b #Hb cases Hb |
---|
601 | |58,60,65,66,72,73,74,75,76: #b #Hb cases Hb |
---|
602 | |5,6,7,10,11,12,13,14: #Hb cases Hb |
---|
603 | |24,25,26,29,30,31,32,33: #Hb cases Hb |
---|
604 | |43,44,45,48,49,50,51,52: #Hb cases Hb |
---|
605 | |62,63,64,67,68,69,70,71: #Hb cases Hb |
---|
606 | |2,4: #b #Hb #H normalize in H; destruct (H) |
---|
607 | |21,23: #b #Hb #H normalize in H; destruct (H) |
---|
608 | |40,42: #b #Hb #H normalize in H; destruct (H) |
---|
609 | |59,61: #b #Hb #H normalize in H; destruct (H) |
---|
610 | ] |
---|
611 | ] |
---|
612 | |38,39,40,42: cases x #a cases a |
---|
613 | [2,3,8,9,15,16,17,18,19: #b #Hb cases Hb |
---|
614 | |21,22,27,28,34,35,36,37,38: #b #Hb cases Hb |
---|
615 | |40,41,46,47,53,54,55,56,57: #b #Hb cases Hb |
---|
616 | |59,60,65,66,72,73,74,75,76: #b #Hb cases Hb |
---|
617 | |5,6,7,10,11,12,13,14: #Hb cases Hb |
---|
618 | |24,25,26,29,30,31,32,33: #Hb cases Hb |
---|
619 | |43,44,45,48,49,50,51,52: #Hb cases Hb |
---|
620 | |62,63,64,67,68,69,70,71: #Hb cases Hb |
---|
621 | |1,4: #b #Hb #H normalize in H; destruct (H) |
---|
622 | |20,23: #b #Hb #H normalize in H; destruct (H) |
---|
623 | |39,42: #b #Hb #H normalize in H; destruct (H) |
---|
624 | |58,61: #b #Hb #H normalize in H; destruct (H) |
---|
625 | ] |
---|
626 | |47,48,49,51: cases x #a #H normalize in H; destruct (H) |
---|
627 | |56,57,58,60: cases x #a #H normalize in H; destruct (H) |
---|
628 | |65,66,67,69: cases x #a #H normalize in H; destruct (H) |
---|
629 | |74,75,76,78: cases x #a #H normalize in H; destruct (H) |
---|
630 | ] |
---|
631 | ] |
---|
632 | |2,3,6: #x [3: #y] #H cases H |
---|
633 | |4,5: #id #_ cases a cases b |
---|
634 | [2,3,4,6,11,12,13,15: normalize #H destruct (H) |
---|
635 | |1,5,7,8,9,10,14,16,17,18: #H / by refl/ |
---|
636 | ] |
---|
637 | ] |
---|
638 | qed. |
---|
639 | |
---|
640 | lemma etblorp: ∀a,b,i.is_jump i → |
---|
641 | instruction_size_jmplen a i ≤ instruction_size_jmplen (max_length a b) i. |
---|
642 | #a #b #i cases i |
---|
643 | [2,3,6: #x [3: #y] #H cases H |
---|
644 | |4,5: #id #_ cases a cases b / by le_n/ |
---|
645 | |1: #pi cases pi |
---|
646 | [1,2,3,4,5,6,7,8,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37: |
---|
647 | [1,2,3,6,7,24,25: #x #y |
---|
648 | |4,5,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23: #x] |
---|
649 | #H cases H |
---|
650 | |9,10,11,12,13,14,15,16,17: #x [3,4,5,8,9: #y] |
---|
651 | #_ cases a cases b |
---|
652 | [2,3: cases x #ad cases ad |
---|
653 | [15,34: #b #Hb / by le_n/ |
---|
654 | |1,2,3,4,8,9,16,17,18,19,20,21,22,23,27,28,35,36,37,38: #b] #Hb cases Hb |
---|
655 | |1,4,5,6,7,8,9: / by le_n/ |
---|
656 | |11,12: cases x #ad cases ad |
---|
657 | [15,34: #b #Hb / by le_n/ |
---|
658 | |1,2,3,4,8,9,16,17,18,19,20,21,22,23,27,28,35,36,37,38: #b] #Hb cases Hb |
---|
659 | |10,13,14,15,16,17,18: / by le_n/ |
---|
660 | |20,21: cases x #ad cases ad |
---|
661 | [15,34: #b #Hb / by le_n/ |
---|
662 | |1,2,3,4,8,9,16,17,18,19,20,21,22,23,27,28,35,36,37,38: #b] #Hb cases Hb |
---|
663 | |19,22,23,24,25,26,27: / by le_n/ |
---|
664 | |29,30: cases x #ad cases ad #a1 #a2 |
---|
665 | [1,3: cases a2 #ad2 cases ad2 |
---|
666 | [1,8,20,27: #b #Hb / by le_n/ |
---|
667 | |2,3,4,9,15,16,17,18,19,21,22,23,28,34,35,36,37,38: #b] #Hb cases Hb |
---|
668 | |2,4: cases a1 #ad1 cases ad1 |
---|
669 | [2,4,21,23: #b #Hb / by le_n/ |
---|
670 | |1,3,8,9,15,16,17,18,19,20,22,27,28,34,35,36,37,38: #b] #Hb cases Hb |
---|
671 | ] |
---|
672 | |28,31,32,33,34,35,36: / by le_n/ |
---|
673 | |38,39: cases x #ad cases ad |
---|
674 | [1,4,20,23: #b #Hb / by le_n/ |
---|
675 | |2,3,8,9,15,16,17,18,19,21,22,27,28,34,35,36,37,38: #b] #Hb cases Hb |
---|
676 | |37,40,41,42,43,44,45: / by le_n/ |
---|
677 | |46,47,48,49,50,51,52,53,54: / by le_n/ |
---|
678 | |55,56,57,58,59,60,61,62,63: / by le_n/ |
---|
679 | |64,65,66,67,68,69,70,71,72: / by le_n/ |
---|
680 | |73,74,75,76,77,78,79,80,81: / by le_n/ |
---|
681 | ] |
---|
682 | ] |
---|
683 | ] |
---|
684 | qed. |
---|
685 | |
---|
686 | lemma minus_zero_to_le: ∀n,m:ℕ.n - m = 0 → n ≤ m. |
---|
687 | #n |
---|
688 | elim n |
---|
689 | [ #m #_ @le_O_n |
---|
690 | | #n' #Hind #m cases m |
---|
691 | [ #H -n whd in match (minus ??) in H; >H @le_n |
---|
692 | | #m' -m #H whd in match (minus ??) in H; @le_S_S @Hind @H |
---|
693 | ] |
---|
694 | ] |
---|
695 | qed. |
---|
696 | |
---|
697 | lemma plus_zero_zero: ∀n,m:ℕ.n + m = 0 → m = 0. |
---|
698 | #n #m #Hn @sym_eq @le_n_O_to_eq <Hn >commutative_plus @le_plus_n_r |
---|
699 | qed. |
---|
700 | |
---|
701 | (* One step in the search for a jump expansion fixpoint. *) |
---|
702 | definition jump_expansion_step: ∀program:(Σl:list labelled_instruction.|l| < 2^16). |
---|
703 | ∀labels:(Σlm:label_map.∀l. |
---|
704 | occurs_exactly_once ?? l program → |
---|
705 | bitvector_of_nat ? (lookup_def … lm l 0) = |
---|
706 | address_of_word_labels_code_mem program l). |
---|
707 | ∀old_policy:(Σpolicy:ppc_pc_map. |
---|
708 | And (And (out_of_program_none program policy) |
---|
709 | (jump_not_in_policy program policy)) |
---|
710 | (\fst policy < 2^16)). |
---|
711 | (Σx:bool × (option ppc_pc_map). |
---|
712 | let 〈no_ch,y〉 ≝ x in |
---|
713 | match y with |
---|
714 | [ None ⇒ nec_plus_ultra program old_policy |
---|
715 | | Some p ⇒ And (And (And (And (And (out_of_program_none program p) |
---|
716 | (jump_not_in_policy program p)) |
---|
717 | (policy_increase program old_policy p)) |
---|
718 | (policy_compact program labels p)) |
---|
719 | (no_ch = true → policy_equal program old_policy p)) |
---|
720 | (\fst p < 2^16) |
---|
721 | ]) |
---|
722 | ≝ |
---|
723 | λprogram.λlabels.λold_sigma. |
---|
724 | let 〈final_added, final_policy〉 ≝ |
---|
725 | foldl_strong (option Identifier × pseudo_instruction) |
---|
726 | (λprefix.Σx:ℕ × ppc_pc_map. |
---|
727 | let 〈added,policy〉 ≝ x in |
---|
728 | And (And (And (And (out_of_program_none prefix policy) |
---|
729 | (jump_not_in_policy prefix policy)) |
---|
730 | (policy_increase prefix old_sigma policy)) |
---|
731 | (policy_compact prefix labels policy)) |
---|
732 | (added = 0 → policy_equal prefix old_sigma policy)) |
---|
733 | program |
---|
734 | (λprefix.λx.λtl.λprf.λacc. |
---|
735 | let 〈inc_added, inc_pc_sigma〉 ≝ (pi1 ?? acc) in |
---|
736 | let 〈label,instr〉 ≝ x in |
---|
737 | (* Now, we must add the current ppc and its pc translation. |
---|
738 | * Three possibilities: |
---|
739 | * - Instruction is not a jump; i.e. constant size whatever the sigma we use; |
---|
740 | * - Instruction is a backward jump; we can use the sigma we're constructing, |
---|
741 | * since it will already know the translation of its destination; |
---|
742 | * - Instruction is a forward jump; we must use the old sigma (the new sigma |
---|
743 | * does not know the translation yet), but compensate for the jumps we |
---|
744 | * have lengthened. |
---|
745 | *) |
---|
746 | let add_instr ≝ match instr with |
---|
747 | [ Jmp j ⇒ Some ? (select_jump_length labels old_sigma inc_pc_sigma inc_added (|prefix|) j) |
---|
748 | | Call c ⇒ Some ? (select_call_length labels old_sigma inc_pc_sigma inc_added (|prefix|) c) |
---|
749 | | Instruction i ⇒ jump_expansion_step_instruction labels old_sigma inc_pc_sigma inc_added (|prefix|) i |
---|
750 | | _ ⇒ None ? |
---|
751 | ] in |
---|
752 | let 〈inc_pc, inc_sigma〉 ≝ inc_pc_sigma in |
---|
753 | let 〈old_pc,old_length〉 ≝ bvt_lookup … (bitvector_of_nat ? (|prefix|)) (\snd old_sigma) 〈0,short_jump〉 in |
---|
754 | let old_size ≝ instruction_size_jmplen old_length instr in |
---|
755 | let 〈new_length, isize〉 ≝ match add_instr with |
---|
756 | [ None ⇒ 〈short_jump, instruction_size_jmplen short_jump instr〉 |
---|
757 | | Some pl ⇒ 〈max_length old_length pl, instruction_size_jmplen (max_length old_length pl) instr〉 |
---|
758 | ] in |
---|
759 | let new_added ≝ match add_instr with |
---|
760 | [ None ⇒ inc_added |
---|
761 | | Some x ⇒ plus inc_added (minus isize old_size) |
---|
762 | ] in |
---|
763 | 〈new_added, 〈plus inc_pc isize, bvt_insert … (bitvector_of_nat ? (|prefix|)) 〈inc_pc, new_length〉 inc_sigma〉〉 |
---|
764 | ) 〈0, 〈0, Stub ??〉〉 in |
---|
765 | if geb (\fst final_policy) 2^16 then |
---|
766 | 〈eqb final_added 0, None ?〉 |
---|
767 | else |
---|
768 | 〈eqb final_added 0, Some ? final_policy〉. |
---|
769 | [ normalize nodelta cases daemon (* XXX nec_plus_ultra *) |
---|
770 | | normalize nodelta lapply p generalize in match (foldl_strong ?????); * #x #H #H2 |
---|
771 | >H2 in H; normalize nodelta -H2 -x #H @conj |
---|
772 | [ @conj |
---|
773 | [ @(proj1 ?? H) |
---|
774 | | #H2 @(proj2 ?? H) @eqb_true_to_eq @H2 |
---|
775 | ] |
---|
776 | | @not_le_to_lt @leb_false_to_not_le <geb_to_leb @p1 |
---|
777 | ] |
---|
778 | | lapply (pi2 ?? acc) >p cases inc_pc_sigma #inc_pc #inc_sigma |
---|
779 | lapply (refl ? (bvt_lookup … (bitvector_of_nat ? (|prefix|)) (\snd old_sigma) 〈0,short_jump〉)) |
---|
780 | cases (bvt_lookup … (bitvector_of_nat ? (|prefix|)) (\snd old_sigma) 〈0,short_jump〉) in ⊢ (???% → %); |
---|
781 | #old_pc #old_length #Holdeq #Hpolicy @pair_elim #added #policy normalize nodelta |
---|
782 | @pair_elim #new_length #isize normalize nodelta #Heq1 #Heq2 |
---|
783 | @conj [ @conj [ @conj [ @conj |
---|
784 | [ (* out_of_program_none *) #i >append_length <commutative_plus #Hi normalize in Hi; #Hi2 |
---|
785 | cases instr in Heq2; normalize nodelta |
---|
786 | #x [6: #y] #H <(proj2 ?? (pair_destruct ?????? H)) >lookup_opt_insert_miss |
---|
787 | [1,3,5,7,9,11: @(proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? Hpolicy))) i ? Hi2) |
---|
788 | @le_S_to_le @Hi |
---|
789 | |2,4,6,8,10,12: @bitvector_of_nat_abs |
---|
790 | [1,4,7,10,13,16: @Hi2 |
---|
791 | |2,5,8,11,14,17: @(transitive_lt … Hi2) @Hi |
---|
792 | |3,6,9,12,15,18: @sym_neq @lt_to_not_eq @Hi |
---|
793 | ] |
---|
794 | ] |
---|
795 | | (* jump_not_in_policy *) #i >append_length <commutative_plus #Hi normalize in Hi; |
---|
796 | cases (le_to_or_lt_eq … (le_S_S_to_le … Hi)) -Hi #Hi |
---|
797 | [ <(proj2 ?? (pair_destruct ?????? Heq2)) >lookup_insert_miss |
---|
798 | [ >(nth_append_first ? i prefix ?? Hi) |
---|
799 | @(proj2 ?? (proj1 ?? (proj1 ?? (proj1 ?? Hpolicy))) i Hi) |
---|
800 | | @bitvector_of_nat_abs |
---|
801 | [ @(transitive_lt … (pi2 ?? program)) >prf >append_length >commutative_plus |
---|
802 | @le_plus_a @Hi |
---|
803 | | @(transitive_lt … (pi2 ?? program)) >prf >append_length <plus_n_Sm @le_S_S |
---|
804 | @le_plus_n_r |
---|
805 | | @lt_to_not_eq @Hi |
---|
806 | ] |
---|
807 | ] |
---|
808 | | >Hi <(proj2 ?? (pair_destruct ?????? Heq2)) >lookup_insert_hit |
---|
809 | >nth_append_second |
---|
810 | [ <minus_n_n whd in match (nth ????); cases instr in Heq1; |
---|
811 | [4,5: #x #_ #H cases H #H2 @⊥ @H2 / by I/ |
---|
812 | |2,3,6: #x [3: #y] #Heq1 <(proj1 ?? (pair_destruct ?????? Heq1)) #_ / by / |
---|
813 | |1: #pi cases pi |
---|
814 | [1,2,3,4,5,6,7,8,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37: |
---|
815 | [1,2,3,6,7,24,25: #x #y |
---|
816 | |4,5,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23: #x] #Heq1 |
---|
817 | <(proj1 ?? (pair_destruct ?????? Heq1)) #_ / by / |
---|
818 | |9,10,11,12,13,14,15,16,17: #x [3,4,5,8,9: #y] |
---|
819 | #_ #H @⊥ cases H #H2 @H2 / by I/ |
---|
820 | ] |
---|
821 | ] |
---|
822 | | @le_n |
---|
823 | ] |
---|
824 | ] |
---|
825 | ] |
---|
826 | | (* policy_increase *) #i >append_length >commutative_plus #Hi normalize in Hi; |
---|
827 | cases (le_to_or_lt_eq … (le_S_S_to_le … Hi)) -Hi; #Hi |
---|
828 | [ lapply (proj2 ?? (proj1 ?? (proj1 ?? Hpolicy)) i Hi) |
---|
829 | <(proj2 ?? (pair_destruct ?????? Heq2)) |
---|
830 | @pair_elim #opc #oj #EQ1 >lookup_insert_miss |
---|
831 | [ @pair_elim #pc #j #EQ2 / by / |
---|
832 | | @bitvector_of_nat_abs |
---|
833 | [ @(transitive_lt … (pi2 ?? program)) >prf >append_length >commutative_plus @le_plus_a |
---|
834 | @Hi |
---|
835 | | @(transitive_lt … (pi2 ?? program)) >prf >append_length <plus_n_Sm @le_S_S @le_plus_n_r |
---|
836 | | @lt_to_not_eq @Hi |
---|
837 | ] |
---|
838 | ] |
---|
839 | | >Hi <(proj2 ?? (pair_destruct ?????? Heq2)) >lookup_insert_hit |
---|
840 | cases (dec_is_jump instr) |
---|
841 | [ cases instr in Heq1; normalize nodelta |
---|
842 | [ #pi cases pi |
---|
843 | [1,2,3,4,5,6,7,8,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37: |
---|
844 | [1,2,3,6,7,24,25: #x #y |
---|
845 | |4,5,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23: #x] #_ #Hj cases Hj |
---|
846 | |9,10,11,12,13,14,15,16,17: #x [3,4,5,8,9: #y] |
---|
847 | whd in match jump_expansion_step_instruction; normalize nodelta #Heq1 |
---|
848 | <(proj1 ?? (pair_destruct ?????? Heq1)) #_ >Holdeq normalize nodelta |
---|
849 | @jmpleq_max_length |
---|
850 | ] |
---|
851 | |2,3,6: #x [3: #y] #_ #Hj cases Hj |
---|
852 | |4,5: #x #Heq1 #_ <(proj1 ?? (pair_destruct ?????? Heq1)) >Holdeq normalize nodelta |
---|
853 | @jmpleq_max_length |
---|
854 | ] |
---|
855 | | lapply Heq1 -Heq1; lapply (refl ? instr); cases instr in ⊢ (???% → %); normalize nodelta |
---|
856 | [ #pi cases pi |
---|
857 | [1,2,3,4,5,6,7,8,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37: |
---|
858 | [1,2,3,6,7,24,25: #x #y |
---|
859 | |4,5,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23: #x] |
---|
860 | whd in match jump_expansion_step_instruction; normalize nodelta #Heqi #Heq1 |
---|
861 | #Hj <(proj1 ?? (pair_destruct ?????? Heq1)) |
---|
862 | lapply (proj2 ?? (proj1 ?? (pi2 ?? old_sigma)) (|prefix|) ??) |
---|
863 | [1,4,7,10,13,16,19,22,25,28,31,34,37,40,43,46,49,52,55,58,61,64,67,70,73,76,79,82: |
---|
864 | >prf >nth_append_second |
---|
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: |
---|
866 | <minus_n_n whd in match (nth ????); >p1 >Heqi @Hj |
---|
867 | |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: |
---|
868 | @le_n |
---|
869 | ] |
---|
870 | |2,5,8,11,14,17,20,23,26,29,32,35,38,41,44,47,50,53,56,59,62,65,68,71,74,77,80,83: |
---|
871 | >prf >append_length <plus_n_Sm @le_S_S @le_plus_n_r |
---|
872 | |3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,60,63,66,69,72,75,78,81,84: |
---|
873 | cases (lookup ?? (bitvector_of_nat ? (|prefix|)) (\snd old_sigma) 〈0,short_jump〉) |
---|
874 | #a #b #H >H normalize nodelta %2 @refl |
---|
875 | ] |
---|
876 | |9,10,11,12,13,14,15,16,17: #x [3,4,5,8,9: #y] |
---|
877 | #_ #_ #abs cases abs #ABS @⊥ @ABS / by I/ |
---|
878 | ] |
---|
879 | |2,3,6: #x [3: #y] #Heqi #Heq1 #Hj <(proj1 ?? (pair_destruct ?????? Heq1)) |
---|
880 | lapply (proj2 ?? (proj1 ?? (pi2 ?? old_sigma)) (|prefix|) ??) |
---|
881 | [1,4,7: >prf >nth_append_second |
---|
882 | [1,3,5: <minus_n_n whd in match (nth ????); >p1 >Heqi @Hj |
---|
883 | |2,4,6: @le_n |
---|
884 | ] |
---|
885 | |2,5,8: >prf >append_length <plus_n_Sm @le_S_S @le_plus_n_r |
---|
886 | |3,6,9: cases (lookup ?? (bitvector_of_nat ? (|prefix|)) (\snd old_sigma) 〈0,short_jump〉) |
---|
887 | #a #b #H >H normalize nodelta %2 @refl |
---|
888 | ] |
---|
889 | |4,5: #x #_ #_ #abs cases abs #ABS @⊥ @ABS / by I/ |
---|
890 | ] |
---|
891 | ] |
---|
892 | ] |
---|
893 | ] |
---|
894 | | (* policy_compact *) (*XXX*) cases daemon |
---|
895 | ] |
---|
896 | | (* added = 0 → policy_equal *) lapply (proj2 ?? Hpolicy) |
---|
897 | lapply Heq2 -Heq2 lapply Heq1 -Heq1 lapply (refl ? instr) |
---|
898 | cases instr in ⊢ (???% → % → % → %); normalize nodelta |
---|
899 | [ #pi cases pi normalize nodelta |
---|
900 | [1,2,3,4,5,6,7,8,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37: |
---|
901 | [1,2,3,6,7,24,25: #x #y |
---|
902 | |4,5,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23: #x] |
---|
903 | #Hins #Heq1 #Heq2 #Hold <(proj1 ?? (pair_destruct ?????? Heq2)) #Hadded |
---|
904 | #i >append_length >commutative_plus #Hi normalize in Hi; |
---|
905 | cases (le_to_or_lt_eq … (le_S_S_to_le … Hi)) -Hi #Hi |
---|
906 | [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: |
---|
907 | <(proj2 ?? (pair_destruct ?????? Heq2)) >lookup_insert_miss |
---|
908 | [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: |
---|
909 | @(Hold Hadded i Hi) |
---|
910 | |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: |
---|
911 | @bitvector_of_nat_abs |
---|
912 | [1,4,7,10,13,16,19,22,25,28,31,34,37,40,43,46,49,52,55,58,61,64,67,70,73,76,79,82: |
---|
913 | @(transitive_lt … (pi2 ?? program)) >prf >append_length >commutative_plus |
---|
914 | @le_plus_a @Hi |
---|
915 | |2,5,8,11,14,17,20,23,26,29,32,35,38,41,44,47,50,53,56,59,62,65,68,71,74,77,80,83: |
---|
916 | @(transitive_lt … (pi2 ?? program)) >prf >append_length <plus_n_Sm @le_S_S |
---|
917 | @le_plus_n_r |
---|
918 | |3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,60,63,66,69,72,75,78,81,84: |
---|
919 | @lt_to_not_eq @Hi |
---|
920 | ] |
---|
921 | ] |
---|
922 | |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: |
---|
923 | <(proj2 ?? (pair_destruct ?????? Heq2)) >Hi >lookup_insert_hit |
---|
924 | lapply (proj2 ?? (proj1 ?? (pi2 ?? old_sigma)) (|prefix|) ??) |
---|
925 | [1,4,7,10,13,16,19,22,25,28,31,34,37,40,43,46,49,52,55,58,61,64,67,70,73,76,79,82: |
---|
926 | >prf >nth_append_second |
---|
927 | [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: |
---|
928 | <minus_n_n whd in match (nth ????); >p1 >Hins @nmk #H @H |
---|
929 | |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: |
---|
930 | @le_n |
---|
931 | ] |
---|
932 | |2,5,8,11,14,17,20,23,26,29,32,35,38,41,44,47,50,53,56,59,62,65,68,71,74,77,80,83: |
---|
933 | >prf >append_length <plus_n_Sm @le_S_S @le_plus_n_r |
---|
934 | |3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,60,63,66,69,72,75,78,81,84: |
---|
935 | cases (bvt_lookup … (bitvector_of_nat ? (|prefix|)) (\snd old_sigma) 〈0,short_jump〉) |
---|
936 | #a #b #H >H <(proj1 ?? (pair_destruct ?????? Heq1)) normalize nodelta @refl |
---|
937 | ] |
---|
938 | ] |
---|
939 | |9,10,11,12,13,14,15,16,17: #x [3,4,5,8,9: #y] #Hins #Heq1 #Heq2 #Hold |
---|
940 | <(proj1 ?? (pair_destruct ?????? Heq2)) <(proj2 ?? (pair_destruct ?????? Heq1)) |
---|
941 | #H #i >append_length >commutative_plus #Hi normalize in Hi; |
---|
942 | cases (le_to_or_lt_eq … (le_S_S_to_le … Hi)) -Hi #Hi |
---|
943 | [1,3,5,7,9,11,13,15,17: <(proj2 ?? (pair_destruct ?????? Heq2)) |
---|
944 | >lookup_insert_miss |
---|
945 | [1,3,5,7,9,11,13,15,17: @(Hold ? i Hi) |
---|
946 | [1,2,3,4,5,6,7,8,9: @sym_eq @le_n_O_to_eq <H @le_plus_n_r] |
---|
947 | ] |
---|
948 | @bitvector_of_nat_abs |
---|
949 | [1,4,7,10,13,16,19,22,25: @(transitive_lt … (pi2 ?? program)) >prf |
---|
950 | >append_length >commutative_plus @le_plus_a @Hi |
---|
951 | |2,5,8,11,14,17,20,23,26: @(transitive_lt … (pi2 ?? program)) >prf |
---|
952 | >append_length <plus_n_Sm @le_S_S |
---|
953 | |3,6,9,12,15,18,21,24,27: @lt_to_not_eq @Hi |
---|
954 | ] @le_plus_n_r |
---|
955 | |2,4,6,8,10,12,14,16,18: <(proj2 ?? (pair_destruct ?????? Heq2)) >Hi |
---|
956 | >lookup_insert_hit <(proj1 ?? (pair_destruct ?????? Heq1)) |
---|
957 | >Holdeq normalize nodelta @sym_eq @blerpque |
---|
958 | [3,6,9,12,15,18,21,24,27: |
---|
959 | elim (le_to_or_lt_eq … (minus_zero_to_le … (plus_zero_zero … H))) |
---|
960 | [1,3,5,7,9,11,13,15,17: #H @⊥ @(absurd ? H) @le_to_not_lt @etblorp |
---|
961 | |2,4,6,8,10,12,14,16,18: #H @H |
---|
962 | ] |
---|
963 | / by I/ |
---|
964 | |2,5,8,11,14,17,20,23,26: / by I/ |
---|
965 | ] |
---|
966 | ] |
---|
967 | ] |
---|
968 | |2,3,6: #x [3: #y] #Hins #Heq1 #Heq2 #Hold <(proj1 ?? (pair_destruct ?????? Heq2)) |
---|
969 | #Hadded #i >append_length >commutative_plus #Hi normalize in Hi; |
---|
970 | cases (le_to_or_lt_eq … (le_S_S_to_le … Hi)) -Hi #Hi |
---|
971 | [1,3,5: <(proj2 ?? (pair_destruct ?????? Heq2)) >lookup_insert_miss |
---|
972 | [1,3,5: @(Hold Hadded i Hi) |
---|
973 | |2,4,6: @bitvector_of_nat_abs |
---|
974 | [1,4,7: @(transitive_lt … (pi2 ?? program)) >prf >append_length >commutative_plus |
---|
975 | @le_plus_a @Hi |
---|
976 | |2,5,8: @(transitive_lt … (pi2 ?? program)) >prf >append_length <plus_n_Sm @le_S_S |
---|
977 | @le_plus_n_r |
---|
978 | |3,6,9: @lt_to_not_eq @Hi |
---|
979 | ] |
---|
980 | ] |
---|
981 | |2,4,6: <(proj2 ?? (pair_destruct ?????? Heq2)) >Hi >lookup_insert_hit |
---|
982 | lapply (proj2 ?? (proj1 ?? (pi2 ?? old_sigma)) (|prefix|) ??) |
---|
983 | [1,4,7: >prf >nth_append_second |
---|
984 | [1,3,5: <minus_n_n whd in match (nth ????); >p1 >Hins @nmk #H @H |
---|
985 | |2,4,6: @le_n |
---|
986 | ] |
---|
987 | |2,5,8: >prf >append_length <plus_n_Sm @le_S_S @le_plus_n_r |
---|
988 | |3,6,9: cases (bvt_lookup … (bitvector_of_nat ? (|prefix|)) (\snd old_sigma) 〈0,short_jump〉) |
---|
989 | #a #b #H >H <(proj1 ?? (pair_destruct ?????? Heq1)) normalize nodelta @refl |
---|
990 | ] |
---|
991 | ] |
---|
992 | |4,5: #x #Hins #Heq1 #Heq2 #Hold |
---|
993 | <(proj1 ?? (pair_destruct ?????? Heq2)) <(proj2 ?? (pair_destruct ?????? Heq1)) |
---|
994 | #H #i >append_length >commutative_plus #Hi normalize in Hi; |
---|
995 | cases (le_to_or_lt_eq … (le_S_S_to_le … Hi)) -Hi #Hi |
---|
996 | [1,3: <(proj2 ?? (pair_destruct ?????? Heq2)) >lookup_insert_miss |
---|
997 | [1,3: @(Hold ? i Hi) |
---|
998 | [1,2: @sym_eq @le_n_O_to_eq <H @le_plus_n_r] |
---|
999 | ] |
---|
1000 | @bitvector_of_nat_abs |
---|
1001 | [1,4: @(transitive_lt … (pi2 ?? program)) >prf |
---|
1002 | >append_length >commutative_plus @le_plus_a @Hi |
---|
1003 | |2,5: @(transitive_lt … (pi2 ?? program)) >prf |
---|
1004 | >append_length <plus_n_Sm @le_S_S |
---|
1005 | |3,6: @lt_to_not_eq @Hi |
---|
1006 | ] @le_plus_n_r |
---|
1007 | |2,4: <(proj2 ?? (pair_destruct ?????? Heq2)) >Hi >lookup_insert_hit |
---|
1008 | <(proj1 ?? (pair_destruct ?????? Heq1))>Holdeq normalize nodelta |
---|
1009 | @sym_eq @blerpque |
---|
1010 | [3,6: elim (le_to_or_lt_eq … (minus_zero_to_le … (plus_zero_zero … H))) |
---|
1011 | [1,3: #H @⊥ @(absurd ? H) @le_to_not_lt @etblorp |
---|
1012 | |2,4: #H @H |
---|
1013 | ] |
---|
1014 | / by I/ |
---|
1015 | |2,5: / by I/ |
---|
1016 | ] |
---|
1017 | ] |
---|
1018 | ] |
---|
1019 | ] |
---|
1020 | | normalize nodelta @conj [ @conj [ @conj [ @conj |
---|
1021 | [ #i #Hi / by refl/ |
---|
1022 | | / by refl/ |
---|
1023 | ]]]] |
---|
1024 | [3: #_] |
---|
1025 | #i #Hi @⊥ @(absurd ? Hi) @not_le_Sn_O |
---|
1026 | ] |
---|
1027 | qed. |
---|
1028 | |
---|
1029 | let rec jump_expansion_internal (program: Σl:list labelled_instruction.lt (length ? l) 2^16) (n: ℕ) |
---|
1030 | on n:(Σx:bool × (option ppc_pc_map). |
---|
1031 | let 〈c,pol〉 ≝ x in |
---|
1032 | match pol with |
---|
1033 | [ None ⇒ True |
---|
1034 | | Some x ⇒ |
---|
1035 | And (And (And |
---|
1036 | (out_of_program_none program x) |
---|
1037 | (jump_not_in_policy program x)) |
---|
1038 | (policy_compact program (create_label_map program) x)) |
---|
1039 | (\fst x < 2^16) |
---|
1040 | ]) ≝ |
---|
1041 | let labels ≝ create_label_map program in |
---|
1042 | match n with |
---|
1043 | [ O ⇒ 〈true,pi1 ?? (jump_expansion_start program labels)〉 |
---|
1044 | | S m ⇒ let 〈ch,z〉 as p1 ≝ (pi1 ?? (jump_expansion_internal program m)) in |
---|
1045 | match z return λx. z=x → Σa:bool × (option ppc_pc_map).? with |
---|
1046 | [ None ⇒ λp2.〈false,None ?〉 |
---|
1047 | | Some op ⇒ λp2.if ch |
---|
1048 | then pi1 ?? (jump_expansion_step program labels «op,?») |
---|
1049 | else (jump_expansion_internal program m) |
---|
1050 | ] (refl … z) |
---|
1051 | ]. |
---|
1052 | [ normalize nodelta cases (jump_expansion_start program (create_label_map program)) |
---|
1053 | #p cases p |
---|
1054 | [ / by I/ |
---|
1055 | | #pm normalize nodelta #H @conj [ @(proj1 ?? (proj1 ?? H)) | @(proj2 ?? H) ] |
---|
1056 | ] |
---|
1057 | | lapply (pi2 ?? (jump_expansion_internal program m)) <p1 >p2 normalize nodelta / by / |
---|
1058 | | lapply (pi2 ?? (jump_expansion_internal program m)) <p1 >p2 normalize nodelta |
---|
1059 | #H @conj [ @(proj1 ?? (proj1 ?? H)) | @(proj2 ?? H) ] |
---|
1060 | | normalize nodelta cases (jump_expansion_step program labels «op,?») |
---|
1061 | #p cases p -p #p #r cases r normalize nodelta |
---|
1062 | [ #H / by I/ |
---|
1063 | | #j #H @conj |
---|
1064 | [ @conj |
---|
1065 | [ @(proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? H)))) |
---|
1066 | | @(proj2 ?? (proj1 ?? (proj1 ?? H))) |
---|
1067 | ] |
---|
1068 | | @(proj2 ?? H) |
---|
1069 | ] |
---|
1070 | ] |
---|
1071 | ] |
---|
1072 | qed. |
---|
1073 | |
---|
1074 | lemma pe_int_refl: ∀program.reflexive ? (policy_equal program). |
---|
1075 | #program whd #x whd #n #Hn |
---|
1076 | cases (bvt_lookup … (bitvector_of_nat 16 n) (\snd x) 〈0,short_jump〉) |
---|
1077 | #y #z normalize nodelta @refl |
---|
1078 | qed. |
---|
1079 | |
---|
1080 | lemma pe_int_sym: ∀program.symmetric ? (policy_equal program). |
---|
1081 | #program whd #x #y #Hxy whd #n #Hn |
---|
1082 | lapply (Hxy n Hn) cases (bvt_lookup … (bitvector_of_nat ? n) (\snd x) 〈0,short_jump〉) |
---|
1083 | #x1 #x2 |
---|
1084 | cases (bvt_lookup … (bitvector_of_nat ? n) (\snd y) 〈0,short_jump〉) |
---|
1085 | #y1 #y2 normalize nodelta // |
---|
1086 | qed. |
---|
1087 | |
---|
1088 | lemma pe_int_trans: ∀program.transitive ? (policy_equal program). |
---|
1089 | #program whd #x #y #z whd in match (policy_equal ???); whd in match (policy_equal ?y ?); |
---|
1090 | whd in match (policy_equal ? x z); #Hxy #Hyz #n #Hn lapply (Hxy n Hn) -Hxy |
---|
1091 | lapply (Hyz n Hn) -Hyz cases (bvt_lookup … (bitvector_of_nat ? n) (\snd x) 〈0,short_jump〉) |
---|
1092 | #x1 #x2 |
---|
1093 | cases (bvt_lookup … (bitvector_of_nat ? n) (\snd y) 〈0,short_jump〉) #y1 #y2 |
---|
1094 | cases (bvt_lookup … (bitvector_of_nat ? n) (\snd z) 〈0,short_jump〉) #z1 #z2 |
---|
1095 | normalize nodelta // |
---|
1096 | qed. |
---|
1097 | |
---|
1098 | definition policy_equal_opt ≝ |
---|
1099 | λprogram:list labelled_instruction.λp1,p2:option ppc_pc_map. |
---|
1100 | match p1 with |
---|
1101 | [ Some x1 ⇒ match p2 with |
---|
1102 | [ Some x2 ⇒ policy_equal program x1 x2 |
---|
1103 | | _ ⇒ False |
---|
1104 | ] |
---|
1105 | | None ⇒ p2 = None ? |
---|
1106 | ]. |
---|
1107 | |
---|
1108 | lemma pe_refl: ∀program.reflexive ? (policy_equal_opt program). |
---|
1109 | #program whd #x whd cases x |
---|
1110 | [ // |
---|
1111 | | #y @pe_int_refl |
---|
1112 | ] |
---|
1113 | qed. |
---|
1114 | |
---|
1115 | lemma pe_sym: ∀program.symmetric ? (policy_equal_opt program). |
---|
1116 | #program whd #x #y #Hxy whd cases y in Hxy; |
---|
1117 | [ cases x |
---|
1118 | [ // |
---|
1119 | | #x' #H @⊥ @(absurd ? H) /2 by nmk/ |
---|
1120 | ] |
---|
1121 | | #y' cases x |
---|
1122 | [ #H @⊥ @(absurd ? H) whd in match (policy_equal_opt ???); @nmk #H destruct (H) |
---|
1123 | | #x' #H @pe_int_sym @H |
---|
1124 | ] |
---|
1125 | ] |
---|
1126 | qed. |
---|
1127 | |
---|
1128 | lemma pe_trans: ∀program.transitive ? (policy_equal_opt program). |
---|
1129 | #program whd #x #y #z cases x |
---|
1130 | [ #Hxy #Hyz >Hxy in Hyz; // |
---|
1131 | | #x' cases y |
---|
1132 | [ #H @⊥ @(absurd ? H) /2 by nmk/ |
---|
1133 | | #y' cases z |
---|
1134 | [ #_ #H @⊥ @(absurd ? H) /2 by nmk/ |
---|
1135 | | #z' @pe_int_trans |
---|
1136 | ] |
---|
1137 | ] |
---|
1138 | ] |
---|
1139 | qed. |
---|
1140 | |
---|
1141 | definition step_none: ∀program.∀n. |
---|
1142 | (\snd (pi1 ?? (jump_expansion_internal program n))) = None ? → |
---|
1143 | ∀k.(\snd (pi1 ?? (jump_expansion_internal program (n+k)))) = None ?. |
---|
1144 | #program #n lapply (refl ? (jump_expansion_internal program n)) |
---|
1145 | cases (jump_expansion_internal program n) in ⊢ (???% → %); |
---|
1146 | #x1 cases x1 #p1 #j1 -x1; #H1 #Heqj #Hj #k elim k |
---|
1147 | [ <plus_n_O >Heqj @Hj |
---|
1148 | | #k' -k <plus_n_Sm whd in match (jump_expansion_internal program (S (n+k'))); |
---|
1149 | lapply (refl ? (jump_expansion_internal program (n+k'))) |
---|
1150 | cases (jump_expansion_internal program (n+k')) in ⊢ (???% → % → %); |
---|
1151 | #x2 cases x2 -x2 #c2 #p2 normalize nodelta #H #Heqj2 |
---|
1152 | cases p2 in H Heqj2; |
---|
1153 | [ #H #Heqj2 #_ whd in match (jump_expansion_internal ??); |
---|
1154 | >Heqj2 normalize nodelta @refl |
---|
1155 | | #x #H #Heqj2 #abs destruct (abs) |
---|
1156 | ] |
---|
1157 | ] |
---|
1158 | qed. |
---|
1159 | |
---|
1160 | lemma pe_step: ∀program:(Σl:list labelled_instruction.|l| < 2^16). |
---|
1161 | ∀n.policy_equal_opt program (\snd (pi1 ?? (jump_expansion_internal program n))) |
---|
1162 | (\snd (pi1 ?? (jump_expansion_internal program (S n)))) → |
---|
1163 | policy_equal_opt program (\snd (pi1 ?? (jump_expansion_internal program (S n)))) |
---|
1164 | (\snd (pi1 ?? (jump_expansion_internal program (S (S n))))). |
---|
1165 | #program #n #Heq |
---|
1166 | cases daemon (* XXX *) |
---|
1167 | qed. |
---|
1168 | |
---|
1169 | (* this is in the stdlib, but commented out, why? *) |
---|
1170 | theorem plus_Sn_m1: ∀n,m:nat. S m + n = m + S n. |
---|
1171 | #n (elim n) normalize /2 by S_pred/ qed. |
---|
1172 | |
---|
1173 | lemma equal_remains_equal: ∀program:(Σl:list labelled_instruction.|l| < 2^16).∀n:ℕ. |
---|
1174 | policy_equal_opt program (\snd (pi1 … (jump_expansion_internal program n))) |
---|
1175 | (\snd (pi1 … (jump_expansion_internal program (S n)))) → |
---|
1176 | ∀k.k ≥ n → policy_equal_opt program (\snd (pi1 … (jump_expansion_internal program n))) |
---|
1177 | (\snd (pi1 … (jump_expansion_internal program k))). |
---|
1178 | #program #n #Heq #k #Hk elim (le_plus_k … Hk); #z #H >H -H -Hk -k; |
---|
1179 | lapply Heq -Heq; lapply n -n; elim z -z; |
---|
1180 | [ #n #Heq <plus_n_O @pe_refl |
---|
1181 | | #z #Hind #n #Heq <plus_Sn_m1 whd in match (plus (S n) z); |
---|
1182 | @(pe_trans … (\snd (pi1 … (jump_expansion_internal program (S n))))) |
---|
1183 | [ @Heq |
---|
1184 | | @Hind @pe_step @Heq |
---|
1185 | ] |
---|
1186 | ] |
---|
1187 | qed. |
---|
1188 | |
---|
1189 | (* this number monotonically increases over iterations, maximum 2*|program| *) |
---|
1190 | let rec measure_int (program: list labelled_instruction) (policy: ppc_pc_map) (acc: ℕ) |
---|
1191 | on program: ℕ ≝ |
---|
1192 | match program with |
---|
1193 | [ nil ⇒ acc |
---|
1194 | | cons h t ⇒ match (\snd (bvt_lookup ?? (bitvector_of_nat ? (|t|)) (\snd policy) 〈0,short_jump〉)) with |
---|
1195 | [ long_jump ⇒ measure_int t policy (acc + 2) |
---|
1196 | | medium_jump ⇒ measure_int t policy (acc + 1) |
---|
1197 | | _ ⇒ measure_int t policy acc |
---|
1198 | ] |
---|
1199 | ]. |
---|
1200 | |
---|
1201 | lemma measure_plus: ∀program.∀policy.∀x,d:ℕ. |
---|
1202 | measure_int program policy (x+d) = measure_int program policy x + d. |
---|
1203 | #program #policy #x #d generalize in match x; -x elim d |
---|
1204 | [ // |
---|
1205 | | -d; #d #Hind elim program |
---|
1206 | [ / by refl/ |
---|
1207 | | #h #t #Hd #x whd in match (measure_int ???); whd in match (measure_int ?? x); |
---|
1208 | cases (\snd (bvt_lookup … (bitvector_of_nat ? (|t|)) (\snd policy) 〈0,short_jump〉)) |
---|
1209 | [ normalize nodelta @Hd |
---|
1210 | |2,3: normalize nodelta >associative_plus >(commutative_plus (S d) ?) <associative_plus |
---|
1211 | @Hd |
---|
1212 | ] |
---|
1213 | ] |
---|
1214 | ] |
---|
1215 | qed. |
---|
1216 | |
---|
1217 | lemma measure_le: ∀program.∀policy. |
---|
1218 | measure_int program policy 0 ≤ 2*|program|. |
---|
1219 | #program #policy elim program |
---|
1220 | [ normalize @le_n |
---|
1221 | | #h #t #Hind whd in match (measure_int ???); |
---|
1222 | cases (\snd (lookup ?? (bitvector_of_nat ? (|t|)) (\snd policy) 〈0,short_jump〉)) |
---|
1223 | [ normalize nodelta @(transitive_le ??? Hind) /2 by monotonic_le_times_r/ |
---|
1224 | |2,3: normalize nodelta >measure_plus <times_n_Sm >(commutative_plus 2 ?) |
---|
1225 | @le_plus [1,3: @Hind |2,4: / by le_n/ ] |
---|
1226 | ] |
---|
1227 | ] |
---|
1228 | qed. |
---|
1229 | |
---|
1230 | (* uses the second part of policy_increase *) |
---|
1231 | lemma measure_incr_or_equal: ∀program:Σl:list labelled_instruction.|l|<2^16. |
---|
1232 | ∀policy:Σp:ppc_pc_map. |
---|
1233 | out_of_program_none program p ∧ |
---|
1234 | jump_not_in_policy program p ∧ \fst p < 2^16. |
---|
1235 | ∀l.|l| ≤ |program| → ∀acc:ℕ. |
---|
1236 | match \snd (jump_expansion_step program (create_label_map program) policy) with |
---|
1237 | [ None ⇒ True |
---|
1238 | | Some p ⇒ measure_int l policy acc ≤ measure_int l p acc |
---|
1239 | ]. |
---|
1240 | #program #policy #l elim l -l; |
---|
1241 | [ #Hp #acc cases (jump_expansion_step ???) #pi1 cases pi1 #p #q -pi1; cases q [ // | #x #_ @le_n ] |
---|
1242 | | #h #t #Hind #Hp #acc |
---|
1243 | lapply (refl ? (jump_expansion_step program (create_label_map program) policy)) |
---|
1244 | cases (jump_expansion_step ???) in ⊢ (???% → %); #pi1 cases pi1 -pi1 #c #r cases r |
---|
1245 | [ / by I/ |
---|
1246 | | #x normalize nodelta #Hx #Hjeq lapply (proj2 ?? (proj1 ?? (proj1 ?? (proj1 ?? Hx))) (|t|) Hp) |
---|
1247 | whd in match (measure_int ???); whd in match (measure_int ? x ?); |
---|
1248 | cases (bvt_lookup ?? (bitvector_of_nat ? (|t|)) (\snd policy) 〈0,short_jump〉) |
---|
1249 | #x1 #x2 cases (bvt_lookup ?? (bitvector_of_nat ? (|t|)) (\snd x) 〈0,short_jump〉) |
---|
1250 | #y1 #y2 normalize nodelta #Hblerp cases Hblerp cases x2 cases y2 |
---|
1251 | [1,4,5,7,8,9: #H cases H |
---|
1252 | |2,3,6: #_ normalize nodelta |
---|
1253 | [1,2: @(transitive_le ? (measure_int t x acc)) |
---|
1254 | |3: @(transitive_le ? (measure_int t x (acc+1))) |
---|
1255 | ] |
---|
1256 | [2,4,5,6: >measure_plus [1,2: @le_plus_n_r] >measure_plus @le_plus / by le_n/] |
---|
1257 | >Hjeq in Hind; #Hind @Hind @(transitive_le … Hp) @le_n_Sn |
---|
1258 | |11,12,13,15,16,17: #H destruct (H) |
---|
1259 | |10,14,18: normalize nodelta #_ >Hjeq in Hind; #Hind @Hind @(transitive_le … Hp) @le_n_Sn |
---|
1260 | ] |
---|
1261 | ] |
---|
1262 | ] |
---|
1263 | qed. |
---|
1264 | |
---|
1265 | (* these lemmas seem superfluous, but not sure how *) |
---|
1266 | lemma bla: ∀a,b:ℕ.a + a = b + b → a = b. |
---|
1267 | #a elim a |
---|
1268 | [ normalize #b // |
---|
1269 | | -a #a #Hind #b cases b [ /2 by le_n_O_to_eq/ | -b #b normalize |
---|
1270 | <plus_n_Sm <plus_n_Sm #H |
---|
1271 | >(Hind b (injective_S ?? (injective_S ?? H))) // ] |
---|
1272 | ] |
---|
1273 | qed. |
---|
1274 | |
---|
1275 | lemma sth_not_s: ∀x.x ≠ S x. |
---|
1276 | #x cases x |
---|
1277 | [ // | #y // ] |
---|
1278 | qed. |
---|
1279 | |
---|
1280 | lemma measure_full: ∀program.∀policy. |
---|
1281 | measure_int program policy 0 = 2*|program| → ∀i.i<|program| → |
---|
1282 | is_jump (\snd (nth i ? program 〈None ?,Comment []〉)) → |
---|
1283 | (\snd (bvt_lookup ?? (bitvector_of_nat ? i) (\snd policy) 〈0,short_jump〉)) = long_jump. |
---|
1284 | #program #policy elim program in ⊢ (% → ∀i.% → ? → %); |
---|
1285 | [ #Hm #i #Hi @⊥ @(absurd … Hi) @not_le_Sn_O |
---|
1286 | | #h #t #Hind #Hm #i #Hi #Hj |
---|
1287 | cases (le_to_or_lt_eq … Hi) -Hi |
---|
1288 | [ #Hi @Hind |
---|
1289 | [ whd in match (measure_int ???) in Hm; |
---|
1290 | cases (\snd (bvt_lookup … (bitvector_of_nat ? (|t|)) (\snd policy) 〈0,short_jump〉)) in Hm; |
---|
1291 | normalize nodelta |
---|
1292 | [ #H @⊥ @(absurd ? (measure_le t policy)) >H @lt_to_not_le /2 by lt_plus, le_n/ |
---|
1293 | | >measure_plus >commutative_plus #H @⊥ @(absurd ? (measure_le t policy)) |
---|
1294 | <(plus_to_minus … (sym_eq … H)) @lt_to_not_le normalize /2 by le_n/ |
---|
1295 | | >measure_plus <times_n_Sm >commutative_plus /2 by injective_plus_r/ |
---|
1296 | ] |
---|
1297 | | @(le_S_S_to_le … Hi) |
---|
1298 | | @Hj |
---|
1299 | ] |
---|
1300 | | #Hi >(injective_S … Hi) whd in match (measure_int ???) in Hm; |
---|
1301 | cases (\snd (bvt_lookup … (bitvector_of_nat ? (|t|)) (\snd policy) 〈0,short_jump〉)) in Hm; |
---|
1302 | normalize nodelta |
---|
1303 | [ #Hm @⊥ @(absurd ? (measure_le t policy)) >Hm @lt_to_not_le /2 by lt_plus, le_n/ |
---|
1304 | | >measure_plus >commutative_plus #H @⊥ @(absurd ? (measure_le t policy)) |
---|
1305 | <(plus_to_minus … (sym_eq … H)) @lt_to_not_le normalize /2 by le_n/ |
---|
1306 | | >measure_plus <times_n_Sm >commutative_plus /2 by injective_plus_r/ |
---|
1307 | ] |
---|
1308 | ] |
---|
1309 | ] |
---|
1310 | qed. |
---|
1311 | |
---|
1312 | (* uses second part of policy_increase *) |
---|
1313 | lemma measure_special: ∀program:(Σl:list labelled_instruction.|l| < 2^16). |
---|
1314 | ∀policy:Σp:ppc_pc_map. |
---|
1315 | out_of_program_none program p ∧ jump_not_in_policy program p ∧ \fst p < 2^16. |
---|
1316 | match (\snd (pi1 ?? (jump_expansion_step program (create_label_map program) policy))) with |
---|
1317 | [ None ⇒ True |
---|
1318 | | Some p ⇒ measure_int program policy 0 = measure_int program p 0 → policy_equal program policy p ]. |
---|
1319 | #program #policy lapply (refl ? (pi1 ?? (jump_expansion_step program (create_label_map program) policy))) |
---|
1320 | cases (jump_expansion_step program (create_label_map program) policy) in ⊢ (???% → %); |
---|
1321 | #p cases p -p #ch #pol normalize nodelta cases pol |
---|
1322 | [ / by I/ |
---|
1323 | | #p normalize nodelta #Hpol #eqpol lapply (le_n (|program|)) |
---|
1324 | @(list_ind ? (λx.|x| ≤ |pi1 ?? program| → |
---|
1325 | measure_int x policy 0 = measure_int x p 0 → |
---|
1326 | policy_equal x policy p) ?? (pi1 ?? program)) |
---|
1327 | [ #_ #_ #i #Hi @⊥ @(absurd ? Hi) @not_le_Sn_O |
---|
1328 | | #h #t #Hind #Hp #Hm #i #Hi cases (le_to_or_lt_eq … Hi) -Hi; |
---|
1329 | [ #Hi @Hind |
---|
1330 | [ @(transitive_le … Hp) / by / |
---|
1331 | | whd in match (measure_int ???) in Hm; whd in match (measure_int ? p ?) in Hm; |
---|
1332 | lapply (proj2 ?? (proj1 ?? (proj1 ?? (proj1 ?? Hpol))) (|t|) Hp) #Hinc |
---|
1333 | cases (bvt_lookup ?? (bitvector_of_nat ? (|t|)) ? 〈0,short_jump〉) in Hm Hinc; #x1 #x2 |
---|
1334 | cases (bvt_lookup ?? (bitvector_of_nat ? (|t|)) ? 〈0,short_jump〉); #y1 #y2 |
---|
1335 | #Hm #Hinc lapply Hm -Hm; lapply Hinc -Hinc; normalize nodelta |
---|
1336 | cases x2 cases y2 normalize nodelta |
---|
1337 | [1: / by / |
---|
1338 | |2,3: >measure_plus #_ #H @⊥ @(absurd ? (eq_plus_S_to_lt … H)) @le_to_not_lt |
---|
1339 | lapply (measure_incr_or_equal program policy t ? 0) |
---|
1340 | [1,3: @(transitive_le … Hp) @le_n_Sn ] >eqpol / by / |
---|
1341 | |4,7,8: #H elim H #H2 [1,3,5: cases H2 |2,4,6: destruct (H2) ] |
---|
1342 | |5: >measure_plus >measure_plus >commutative_plus >(commutative_plus ? 1) |
---|
1343 | #_ #H @(injective_plus_r … H) |
---|
1344 | |6: >measure_plus >measure_plus |
---|
1345 | change with (1+1) in match (2); >assoc_plus1 >(commutative_plus 1 (measure_int ???)) |
---|
1346 | #_ #H @⊥ @(absurd ? (eq_plus_S_to_lt … H)) @le_to_not_lt @monotonic_le_plus_l |
---|
1347 | lapply (measure_incr_or_equal program policy t ? 0) |
---|
1348 | [ @(transitive_le … Hp) @le_n_Sn ] >eqpol / by / |
---|
1349 | |9: >measure_plus >measure_plus >commutative_plus >(commutative_plus ? 2) |
---|
1350 | #_ #H @(injective_plus_r … H) |
---|
1351 | ] |
---|
1352 | | @(le_S_S_to_le … Hi) |
---|
1353 | ] |
---|
1354 | | #Hi >(injective_S … Hi) whd in match (measure_int ???) in Hm; |
---|
1355 | whd in match (measure_int ? p ?) in Hm; |
---|
1356 | lapply (proj2 ?? (proj1 ?? (proj1 ?? (proj1 ?? Hpol))) (|t|) Hp) |
---|
1357 | cases (bvt_lookup ?? (bitvector_of_nat ? (|t|)) ? 〈0,short_jump〉) in Hm; |
---|
1358 | #x1 #x2 |
---|
1359 | cases (bvt_lookup ?? (bitvector_of_nat ? (|t|)) ? 〈0,short_jump〉); |
---|
1360 | #y1 #y2 |
---|
1361 | normalize nodelta cases x2 cases y2 normalize nodelta |
---|
1362 | [1,5,9: #_ #_ @refl |
---|
1363 | |4,7,8: #_ #H elim H #H2 [1,3,5: cases H2 |2,4,6: destruct (H2) ] |
---|
1364 | |2,3: >measure_plus #H #_ @⊥ @(absurd ? (eq_plus_S_to_lt … H)) @le_to_not_lt |
---|
1365 | lapply (measure_incr_or_equal program policy t ? 0) |
---|
1366 | [1,3: @(transitive_le … Hp) @le_n_Sn ] >eqpol / by / |
---|
1367 | |6: >measure_plus >measure_plus |
---|
1368 | change with (1+1) in match (2); >assoc_plus1 >(commutative_plus 1 (measure_int ???)) |
---|
1369 | #H #_ @⊥ @(absurd ? (eq_plus_S_to_lt … H)) @le_to_not_lt @monotonic_le_plus_l |
---|
1370 | lapply (measure_incr_or_equal program policy t ? 0) |
---|
1371 | [ @(transitive_le … Hp) @le_n_Sn ] >eqpol / by / |
---|
1372 | ] |
---|
1373 | ] |
---|
1374 | ] |
---|
1375 | qed. |
---|
1376 | |
---|
1377 | lemma le_to_eq_plus: ∀n,z. |
---|
1378 | n ≤ z → ∃k.z = n + k. |
---|
1379 | #n #z elim z |
---|
1380 | [ #H cases (le_to_or_lt_eq … H) |
---|
1381 | [ #H2 @⊥ @(absurd … H2) @not_le_Sn_O |
---|
1382 | | #H2 @(ex_intro … 0) >H2 // |
---|
1383 | ] |
---|
1384 | | #z' #Hind #H cases (le_to_or_lt_eq … H) |
---|
1385 | [ #H' elim (Hind (le_S_S_to_le … H')) #k' #H2 @(ex_intro … (S k')) |
---|
1386 | >H2 >plus_n_Sm // |
---|
1387 | | #H' @(ex_intro … 0) >H' // |
---|
1388 | ] |
---|
1389 | ] |
---|
1390 | qed. |
---|
1391 | |
---|
1392 | lemma measure_zero: ∀l.∀program:Σl:list labelled_instruction.|l| < 2^16. |
---|
1393 | match jump_expansion_start program (create_label_map program) with |
---|
1394 | [ None ⇒ True |
---|
1395 | | Some p ⇒ |l| ≤ |program| → measure_int l p 0 = 0 |
---|
1396 | ]. |
---|
1397 | #l #program lapply (refl ? (jump_expansion_start program (create_label_map program))) |
---|
1398 | cases (jump_expansion_start program (create_label_map program)) in ⊢ (???% → %); #p #Hp #EQ |
---|
1399 | cases p in Hp EQ; |
---|
1400 | [ / by I/ |
---|
1401 | | #pl normalize nodelta #Hpl #EQ elim l |
---|
1402 | [ / by refl/ |
---|
1403 | | #h #t #Hind #Hp whd in match (measure_int ???); |
---|
1404 | >(proj2 ?? (proj1 ?? Hpl) (|t|)) |
---|
1405 | [ normalize nodelta @Hind ] |
---|
1406 | @(transitive_le … Hp) [ @le_n_Sn | @ le_n ] |
---|
1407 | ] |
---|
1408 | ] |
---|
1409 | qed. |
---|
1410 | |
---|
1411 | (* the actual computation of the fixpoint *) |
---|
1412 | definition je_fixpoint: ∀program:(Σl:list labelled_instruction.|l| < 2^16). |
---|
1413 | Σp:option ppc_pc_map. |
---|
1414 | And (match p with |
---|
1415 | [ None ⇒ True |
---|
1416 | | Some pol ⇒ And (out_of_program_none program pol) |
---|
1417 | (policy_compact program (create_label_map program) pol) |
---|
1418 | ]) |
---|
1419 | (∃n.∀k.n < k → |
---|
1420 | policy_equal_opt program (\snd (pi1 ?? (jump_expansion_internal program k))) p). |
---|
1421 | #program @(\snd (pi1 ?? (jump_expansion_internal program (2*|program|)))) @conj |
---|
1422 | [ lapply (pi2 ?? (jump_expansion_internal program (2*|program|))) |
---|
1423 | cases (jump_expansion_internal program (2*|program|)) #p cases p -p |
---|
1424 | #c #pol #Hp cases pol |
---|
1425 | [ normalize nodelta // |
---|
1426 | | #x normalize nodelta #H @conj [ @(proj1 ?? (proj1 ?? (proj1 ?? H))) |
---|
1427 | | cases daemon ] ] |
---|
1428 | | cases (dec_bounded_exists (λk.policy_equal_opt (pi1 ?? program) |
---|
1429 | (\snd (pi1 ?? (jump_expansion_internal program k))) |
---|
1430 | (\snd (pi1 ?? (jump_expansion_internal program (S k))))) ? (2*|program|)) |
---|
1431 | [ #Hex elim Hex -Hex #x #Hx @(ex_intro … x) #k #Hk |
---|
1432 | @pe_trans |
---|
1433 | [ @(\snd (pi1 ?? (jump_expansion_internal program x))) |
---|
1434 | | @pe_sym @equal_remains_equal |
---|
1435 | [ @(proj2 ?? Hx) |
---|
1436 | | @le_S_S_to_le @le_S @Hk |
---|
1437 | ] |
---|
1438 | | @equal_remains_equal |
---|
1439 | [ @(proj2 ?? Hx) |
---|
1440 | | @le_S_S_to_le @le_S @(proj1 ?? Hx) |
---|
1441 | ] |
---|
1442 | ] |
---|
1443 | | #Hnex lapply (not_exists_forall … Hnex) -Hnex; #Hfa |
---|
1444 | @(ex_intro … (2*|program|)) #k #Hk @pe_sym @equal_remains_equal |
---|
1445 | [ lapply (refl ? (jump_expansion_internal program (2*|program|))) |
---|
1446 | cases (jump_expansion_internal program (2*|program|)) in ⊢ (???% → %); |
---|
1447 | #x cases x -x #Fch #Fpol normalize nodelta #HFpol cases Fpol in HFpol; normalize nodelta |
---|
1448 | [ (* if we're at None in 2*|program|, we're at None in S 2*|program| too *) |
---|
1449 | #HFpol #EQ whd in match (jump_expansion_internal ??); >EQ |
---|
1450 | normalize nodelta / by / |
---|
1451 | | #Fp #HFp #EQ whd in match (jump_expansion_internal ??); |
---|
1452 | >EQ normalize nodelta |
---|
1453 | lapply (refl ? (jump_expansion_step program (create_label_map program) «Fp,?»)) |
---|
1454 | [ @conj [ @(proj1 ?? (proj1 ?? HFp)) | @(proj2 ?? HFp) ] |
---|
1455 | | lapply (measure_full program Fp ?) |
---|
1456 | [ @le_to_le_to_eq |
---|
1457 | [ @measure_le |
---|
1458 | | cut (∀x:ℕ.x ≤ 2*|program| → |
---|
1459 | ∃p.(\snd (pi1 ?? (jump_expansion_internal program x)) = Some ? p ∧ |
---|
1460 | x ≤ measure_int program p 0)) |
---|
1461 | [ #x elim x |
---|
1462 | [ #Hx lapply (refl ? (jump_expansion_start program (create_label_map program))) |
---|
1463 | cases (jump_expansion_start program (create_label_map program)) in ⊢ (???% → %); |
---|
1464 | #z cases z -z normalize nodelta |
---|
1465 | [ #Waar #Heqn @⊥ elim (le_to_eq_plus ?? Hx) #k #Hk |
---|
1466 | @(absurd … (step_none program 0 ? k)) |
---|
1467 | [ whd in match (jump_expansion_internal ??); >Heqn @refl |
---|
1468 | | <Hk >EQ @nmk #H destruct (H) |
---|
1469 | ] |
---|
1470 | | #pol #Hpol #Heqpol @(ex_intro ?? pol) @conj |
---|
1471 | [ whd in match (jump_expansion_internal ??); >Heqpol @refl |
---|
1472 | | @le_O_n |
---|
1473 | ] |
---|
1474 | ] |
---|
1475 | | -x #x #Hind #Hx |
---|
1476 | lapply (refl ? (jump_expansion_internal program (S x))) |
---|
1477 | cases (jump_expansion_internal program (S x)) in ⊢ (???% → %); |
---|
1478 | #z cases z -z #Sxch #Sxpol cases Sxpol -Sxpol normalize nodelta |
---|
1479 | [ #H #HeqSxpol @⊥ elim (le_to_eq_plus ?? Hx) #k #Hk |
---|
1480 | @(absurd … (step_none program (S x) ? k)) |
---|
1481 | [ >HeqSxpol / by / |
---|
1482 | | <Hk >EQ @nmk #H destruct (H) |
---|
1483 | ] |
---|
1484 | | #Sxpol #HSxpol #HeqSxpol @(ex_intro ?? Sxpol) @conj |
---|
1485 | [ @refl |
---|
1486 | | elim (Hind (transitive_le … (le_n_Sn x) Hx)) |
---|
1487 | #xpol #Hxpol @(le_to_lt_to_lt … (proj2 ?? Hxpol)) |
---|
1488 | lapply (measure_incr_or_equal program xpol program (le_n (|program|)) 0) |
---|
1489 | [ cases (jump_expansion_internal program x) in Hxpol; |
---|
1490 | #z cases z -z #xch #xpol normalize nodelta #H #H2 >(proj1 ?? H2) in H; |
---|
1491 | normalize nodelta #H @conj [ @(proj1 ?? (proj1 ?? H)) | @(proj2 ?? H) ] |
---|
1492 | | lapply (Hfa x Hx) lapply HeqSxpol -HeqSxpol |
---|
1493 | whd in match (jump_expansion_internal program (S x)); |
---|
1494 | lapply (refl ? (jump_expansion_internal program x)) |
---|
1495 | lapply Hxpol -Hxpol cases (jump_expansion_internal program x) in ⊢ (% → ???% → %); |
---|
1496 | #z cases z -z #xch #b normalize nodelta #H #Heq >(proj1 ?? Heq) in H; |
---|
1497 | #H #Heq cases xch in Heq; #Heq normalize nodelta |
---|
1498 | [ lapply (refl ? (jump_expansion_step program (create_label_map (pi1 ?? program)) «xpol,?»)) |
---|
1499 | [ @conj [ @(proj1 ?? (proj1 ?? H)) | @(proj2 ?? H) ] |
---|
1500 | | cases (jump_expansion_step ???) in ⊢ (???% → %); #z cases z -z #a #c |
---|
1501 | normalize nodelta cases c normalize nodelta |
---|
1502 | [ #H1 #Heq #H2 destruct (H2) |
---|
1503 | | #d #H1 #Heq #H2 destruct (H2) #Hfull #H2 elim (le_to_or_lt_eq … H2) |
---|
1504 | [ / by / |
---|
1505 | | #H3 lapply (measure_special program «xpol,?») |
---|
1506 | [ @conj [ @(proj1 ?? (proj1 ?? H)) | @(proj2 ?? H) ] |
---|
1507 | | >Heq normalize nodelta #H4 @⊥ @(absurd … (H4 H3)) @Hfull |
---|
1508 | ] |
---|
1509 | ] |
---|
1510 | ] |
---|
1511 | ] |
---|
1512 | | lapply (refl ? (jump_expansion_step program (create_label_map (pi1 ?? program)) «xpol,?»)) |
---|
1513 | [ @conj [ @(proj1 ?? (proj1 ?? H)) | @(proj2 ?? H) ] |
---|
1514 | | cases (jump_expansion_step ???) in ⊢ (???% → %); #z cases z -z #a #c |
---|
1515 | normalize nodelta cases c normalize nodelta |
---|
1516 | [ #H1 #Heq #H2 #H3 #_ @⊥ @(absurd ?? H3) @pe_refl |
---|
1517 | | #d #H1 #Heq #H2 #H3 @⊥ @(absurd ?? H3) @pe_refl |
---|
1518 | ] |
---|
1519 | ] |
---|
1520 | ] |
---|
1521 | ] |
---|
1522 | ] |
---|
1523 | ] |
---|
1524 | ] |
---|
1525 | | #H elim (H (2*|program|) (le_n ?)) #plp >EQ #Hplp |
---|
1526 | >(Some_eq ??? (proj1 ?? Hplp)) @(proj2 ?? Hplp) |
---|
1527 | ] |
---|
1528 | ] |
---|
1529 | | #Hfull cases (jump_expansion_step program (create_label_map program) «Fp,?») in ⊢ (???% → %); |
---|
1530 | #x cases x -x #Gch #Gpol cases Gpol normalize nodelta |
---|
1531 | [ #H #EQ2 @⊥ @(absurd ?? H) @Hfull |
---|
1532 | | #Gp #HGp #EQ2 cases Fch |
---|
1533 | [ normalize nodelta #i #Hi |
---|
1534 | cases (dec_is_jump (\snd (nth i ? program 〈None ?, Comment []〉))) #Hj |
---|
1535 | [ lapply (proj2 ?? (proj1 ?? (proj1 ?? (proj1 ?? HGp))) i Hi) |
---|
1536 | lapply (Hfull i Hi Hj) cases (bvt_lookup … (bitvector_of_nat ? i) (\snd Fp) 〈0,short_jump〉) |
---|
1537 | #fp #fj #Hfj >Hfj normalize nodelta |
---|
1538 | cases (bvt_lookup … (bitvector_of_nat ? i) (\snd Gp) 〈0,short_jump〉) #gp #gj |
---|
1539 | cases gj normalize nodelta |
---|
1540 | [1,2: #H cases H #H2 cases H2 destruct (H2) |
---|
1541 | |3: #_ @refl |
---|
1542 | ] |
---|
1543 | | >(proj2 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? HGp)))) i Hi Hj) |
---|
1544 | >(proj2 ?? (proj1 ?? (proj1 ?? HFp)) i Hi Hj) @refl |
---|
1545 | ] |
---|
1546 | | normalize nodelta /2 by pe_int_refl/ |
---|
1547 | ] |
---|
1548 | ] |
---|
1549 | ] |
---|
1550 | ] |
---|
1551 | ] |
---|
1552 | | @le_S_S_to_le @le_S @Hk |
---|
1553 | ] |
---|
1554 | | #n cases (jump_expansion_internal program n) cases (jump_expansion_internal program (S n)) |
---|
1555 | #x cases x -x #nch #npol normalize nodelta #Hnpol |
---|
1556 | #x cases x -x #Sch #Spol normalize nodelta #HSpol |
---|
1557 | cases npol in Hnpol; cases Spol in HSpol; |
---|
1558 | [ #Hnpol #HSpol %1 // |
---|
1559 | |2,3: #x #Hnpol #HSpol %2 @nmk whd in match (policy_equal ???); // |
---|
1560 | #H destruct (H) |
---|
1561 | |4: #np #Hnp #Sp #HSp whd in match (policy_equal ???); @dec_bounded_forall #m |
---|
1562 | cases (bvt_lookup ?? (bitvector_of_nat 16 m) ? 〈0,short_jump〉) |
---|
1563 | #x1 #x2 |
---|
1564 | cases (bvt_lookup ?? (bitvector_of_nat ? m) ? 〈0,short_jump〉) |
---|
1565 | #y1 #y2 normalize nodelta |
---|
1566 | @dec_eq_jump_length |
---|
1567 | ] |
---|
1568 | ] |
---|
1569 | qed. |
---|
1570 | |
---|
1571 | include alias "arithmetics/nat.ma". |
---|
1572 | include alias "basics/logic.ma". |
---|
1573 | |
---|
1574 | (* The glue between Policy and Assembly. *) |
---|
1575 | definition jump_expansion': |
---|
1576 | ∀program:preamble × (Σl:list labelled_instruction.|l| < 2^16). |
---|
1577 | option (Σsigma:Word → Word × bool. |
---|
1578 | ∀ppc: Word. |
---|
1579 | let pc ≝ \fst (sigma ppc) in |
---|
1580 | let labels ≝ \fst (create_label_cost_map (\snd program)) in |
---|
1581 | let lookup_labels ≝ λx. bitvector_of_nat ? (lookup_def ?? labels x 0) in |
---|
1582 | let instruction ≝ \fst (fetch_pseudo_instruction (\snd program) ppc) in |
---|
1583 | let next_pc ≝ \fst (sigma (add ? ppc (bitvector_of_nat ? 1))) in |
---|
1584 | And (nat_of_bitvector … ppc ≤ |\snd program| → |
---|
1585 | next_pc = add ? pc (bitvector_of_nat … |
---|
1586 | (instruction_size lookup_labels (λx.\fst (sigma x)) (λx.\snd (sigma x)) ppc instruction))) |
---|
1587 | (Or (nat_of_bitvector … ppc < |\snd program| → |
---|
1588 | nat_of_bitvector … pc < nat_of_bitvector … next_pc) |
---|
1589 | (nat_of_bitvector … ppc = |\snd program| → next_pc = (zero …)))) ≝ |
---|
1590 | λprogram. |
---|
1591 | let policy ≝ pi1 … (je_fixpoint (\snd program)) in |
---|
1592 | match policy with |
---|
1593 | [ None ⇒ None ? |
---|
1594 | | Some x ⇒ Some ? |
---|
1595 | «λppc.let 〈pc,jl〉 ≝ bvt_lookup ?? ppc (\snd x) 〈0,short_jump〉 in |
---|
1596 | 〈bitvector_of_nat 16 pc,jmpeqb jl long_jump〉,?» |
---|
1597 | ]. |
---|
1598 | #ppc normalize nodelta cases daemon |
---|
1599 | qed. |
---|