1 | include "ASM/PolicyStep.ma". |
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2 | |
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3 | include alias "basics/lists/list.ma". |
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4 | include alias "arithmetics/nat.ma". |
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5 | include alias "basics/logic.ma". |
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6 | |
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7 | let rec jump_expansion_internal (program: Σl:list labelled_instruction.lt (S (|l|)) 2^16) (n: ℕ) |
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8 | on n:(Σx:bool × (option ppc_pc_map). |
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9 | let 〈c,pol〉 ≝ x in |
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10 | And |
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11 | (match pol with |
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12 | [ None ⇒ True |
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13 | | Some x ⇒ |
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14 | And (And (And (And |
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15 | (out_of_program_none program x) |
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16 | (jump_not_in_policy program x)) |
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17 | (n > 0 → policy_compact program (create_label_map program) x)) |
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18 | (\fst (bvt_lookup … (bitvector_of_nat ? 0) (\snd x) 〈0,short_jump〉) = 0)) |
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19 | (\fst x < 2^16) |
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20 | ]) (n = 0 → c = true)) ≝ |
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21 | let labels ≝ create_label_map program in |
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22 | match n return λx.n = x → Σa:bool × (option ppc_pc_map).? with |
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23 | [ O ⇒ λp.〈true,pi1 ?? (jump_expansion_start program labels)〉 |
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24 | | S m ⇒ λp.let 〈ch,z〉 as p1 ≝ (pi1 ?? (jump_expansion_internal program m)) in |
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25 | match z return λx. z=x → Σa:bool × (option ppc_pc_map).? with |
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26 | [ None ⇒ λp2.〈false,None ?〉 |
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27 | | Some op ⇒ λp2.if ch |
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28 | then pi1 ?? (jump_expansion_step program labels «op,?») |
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29 | else pi1 ?? (jump_expansion_internal program m) |
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30 | ] (refl … z) |
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31 | ] (refl … n). |
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32 | [ normalize nodelta cases (jump_expansion_start program (create_label_map program)) |
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33 | #x cases x -x |
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34 | [ #H @conj / by I/ |
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35 | | #sigma normalize nodelta #H @conj [ @conj [ @conj [ @conj |
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36 | [ @(proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? H))))) |
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37 | | >p #H @⊥ @(absurd ? H) @le_to_not_lt @le_n |
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38 | ] |
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39 | | @(proj2 ?? (proj1 ?? (proj1 ?? (proj1 ?? H)))) |
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40 | ] |
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41 | | @(proj2 ?? H) |
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42 | ] |
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43 | | / by / |
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44 | ] |
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45 | ] |
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46 | | normalize nodelta @conj [ / by I/ | >p #H destruct (H) ] |
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47 | | cases ch in p1; #p1 |
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48 | [ normalize nodelta |
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49 | cases (jump_expansion_step program (create_label_map program) «op,?») |
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50 | #x cases x -x #ch2 #z2 cases z2 normalize nodelta |
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51 | [ #_ @conj [ / by I/ | >p #H2 destruct (H2) ] |
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52 | | #j2 #H2 @conj [ @conj [ @conj [ @conj |
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53 | [ @(proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? H2)))))) |
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54 | | #_ @(proj2 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? H2))))) |
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55 | ] |
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56 | | @(proj2 ?? (proj1 ?? (proj1 ?? (proj1 ?? H2)))) |
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57 | ] |
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58 | | @(proj2 ?? H2) |
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59 | ] |
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60 | | >p #H3 destruct (H3) |
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61 | ] |
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62 | ] |
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63 | | normalize nodelta lapply (pi2 ?? (jump_expansion_internal program m)) |
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64 | <p1 >p2 normalize nodelta #H @conj [ @conj [ @conj [ @conj |
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65 | [ @(proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? H)))) |
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66 | | >p #Hm cases (le_to_or_lt_eq … Hm) -Hm #Hm |
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67 | [ @(proj2 ?? (proj1 ?? (proj1 ?? (proj1 ?? H)))) @(le_S_S_to_le … Hm) |
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68 | | lapply ((proj2 ?? H) (sym_eq … (injective_S … Hm))) #H2 destruct (H2) |
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69 | ] |
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70 | ] |
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71 | | @(proj2 ?? (proj1 ?? (proj1 ?? H))) |
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72 | ] |
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73 | | @(proj2 ?? (proj1 ?? H)) |
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74 | ] |
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75 | | >p #H2 destruct (H2) |
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76 | ] |
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77 | ] |
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78 | | cases (jump_expansion_internal program m) in p1; |
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79 | #p cases p -p #p #r cases r normalize nodelta |
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80 | [ #_ >p2 #ABS destruct (ABS) |
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81 | | #map >p2 normalize nodelta |
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82 | #H #eq destruct (eq) @conj [ @conj |
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83 | [ @(proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? H)))) |
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84 | | @(proj2 ?? (proj1 ?? (proj1 ?? H))) |
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85 | ] |
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86 | | @(proj2 ?? (proj1 ?? H)) |
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87 | ] |
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88 | ] |
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89 | ] |
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90 | qed. |
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91 | |
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92 | lemma pe_int_refl: ∀program.reflexive ? (policy_equal program). |
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93 | #program whd #x whd #n #Hn |
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94 | cases (bvt_lookup … (bitvector_of_nat 16 n) (\snd x) 〈0,short_jump〉) |
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95 | #y #z normalize nodelta @refl |
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96 | qed. |
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97 | |
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98 | lemma pe_int_sym: ∀program.symmetric ? (policy_equal program). |
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99 | #program whd #x #y #Hxy whd #n #Hn |
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100 | lapply (Hxy n Hn) cases (bvt_lookup … (bitvector_of_nat ? n) (\snd x) 〈0,short_jump〉) |
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101 | #x1 #x2 |
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102 | cases (bvt_lookup … (bitvector_of_nat ? n) (\snd y) 〈0,short_jump〉) |
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103 | #y1 #y2 normalize nodelta // |
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104 | qed. |
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105 | |
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106 | lemma pe_int_trans: ∀program.transitive ? (policy_equal program). |
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107 | #program whd #x #y #z whd in match (policy_equal ???); whd in match (policy_equal ?y ?); |
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108 | whd in match (policy_equal ? x z); #Hxy #Hyz #n #Hn lapply (Hxy n Hn) -Hxy |
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109 | lapply (Hyz n Hn) -Hyz cases (bvt_lookup … (bitvector_of_nat ? n) (\snd x) 〈0,short_jump〉) |
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110 | #x1 #x2 |
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111 | cases (bvt_lookup … (bitvector_of_nat ? n) (\snd y) 〈0,short_jump〉) #y1 #y2 |
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112 | cases (bvt_lookup … (bitvector_of_nat ? n) (\snd z) 〈0,short_jump〉) #z1 #z2 |
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113 | normalize nodelta // |
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114 | qed. |
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115 | |
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116 | definition policy_equal_opt ≝ |
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117 | λprogram:list labelled_instruction.λp1,p2:option ppc_pc_map. |
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118 | match p1 with |
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119 | [ Some x1 ⇒ match p2 with |
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120 | [ Some x2 ⇒ policy_equal program x1 x2 |
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121 | | _ ⇒ False |
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122 | ] |
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123 | | None ⇒ p2 = None ? |
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124 | ]. |
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125 | |
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126 | lemma pe_refl: ∀program.reflexive ? (policy_equal_opt program). |
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127 | #program whd #x whd cases x |
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128 | [ // |
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129 | | #y @pe_int_refl |
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130 | ] |
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131 | qed. |
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132 | |
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133 | lemma pe_sym: ∀program.symmetric ? (policy_equal_opt program). |
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134 | #program whd #x #y #Hxy whd cases y in Hxy; |
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135 | [ cases x |
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136 | [ // |
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137 | | #x' #H @⊥ @(absurd ? H) /2 by nmk/ |
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138 | ] |
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139 | | #y' cases x |
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140 | [ #H @⊥ @(absurd ? H) whd in match (policy_equal_opt ???); @nmk #H destruct (H) |
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141 | | #x' #H @pe_int_sym @H |
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142 | ] |
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143 | ] |
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144 | qed. |
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145 | |
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146 | lemma pe_trans: ∀program.transitive ? (policy_equal_opt program). |
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147 | #program whd #x #y #z cases x |
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148 | [ #Hxy #Hyz >Hxy in Hyz; // |
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149 | | #x' cases y |
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150 | [ #H @⊥ @(absurd ? H) /2 by nmk/ |
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151 | | #y' cases z |
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152 | [ #_ #H @⊥ @(absurd ? H) /2 by nmk/ |
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153 | | #z' @pe_int_trans |
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154 | ] |
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155 | ] |
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156 | ] |
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157 | qed. |
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158 | |
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159 | definition step_none: ∀program.∀n. |
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160 | (\snd (pi1 ?? (jump_expansion_internal program n))) = None ? → |
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161 | ∀k.(\snd (pi1 ?? (jump_expansion_internal program (n+k)))) = None ?. |
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162 | #program #n lapply (refl ? (jump_expansion_internal program n)) |
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163 | cases (jump_expansion_internal program n) in ⊢ (???% → %); |
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164 | #x1 cases x1 #p1 #j1 -x1; #H1 #Heqj #Hj #k elim k |
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165 | [ <plus_n_O >Heqj @Hj |
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166 | | #k' -k <plus_n_Sm (*whd in match (jump_expansion_internal program (S (n+k')));*) |
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167 | lapply (refl ? (jump_expansion_internal program (n+k'))) |
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168 | cases (jump_expansion_internal program (n+k')) in ⊢ (???% → %); |
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169 | #x2 cases x2 -x2 #c2 #p2 normalize nodelta #H #Heqj2 |
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170 | cases p2 in H Heqj2; |
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171 | [ #H #Heqj2 #_ whd in match (jump_expansion_internal ??); |
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172 | >Heqj2 normalize nodelta @refl |
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173 | | #x #H #Heqj2 #abs destruct (abs) |
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174 | ] |
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175 | ] |
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176 | qed. |
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177 | |
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178 | lemma pe_step: ∀program:(Σl:list labelled_instruction.S (|l|) < 2^16). |
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179 | ∀n.policy_equal_opt program (\snd (pi1 ?? (jump_expansion_internal program n))) |
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180 | (\snd (pi1 ?? (jump_expansion_internal program (S n)))) → |
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181 | policy_equal_opt program (\snd (pi1 ?? (jump_expansion_internal program (S n)))) |
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182 | (\snd (pi1 ?? (jump_expansion_internal program (S (S n))))). |
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183 | #program #n #Heq |
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184 | cases daemon (* XXX *) |
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185 | qed. |
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186 | |
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187 | (* this is in the stdlib, but commented out, why? *) |
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188 | theorem plus_Sn_m1: ∀n,m:nat. S m + n = m + S n. |
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189 | #n (elim n) normalize /2 by S_pred/ qed. |
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190 | |
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191 | lemma equal_remains_equal: ∀program:(Σl:list labelled_instruction.S (|l|) < 2^16).∀n:ℕ. |
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192 | policy_equal_opt program (\snd (pi1 … (jump_expansion_internal program n))) |
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193 | (\snd (pi1 … (jump_expansion_internal program (S n)))) → |
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194 | ∀k.k ≥ n → policy_equal_opt program (\snd (pi1 … (jump_expansion_internal program n))) |
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195 | (\snd (pi1 … (jump_expansion_internal program k))). |
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196 | #program #n #Heq #k #Hk elim (le_plus_k … Hk); #z #H >H -H -Hk -k; |
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197 | lapply Heq -Heq; lapply n -n; elim z -z; |
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198 | [ #n #Heq <plus_n_O @pe_refl |
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199 | | #z #Hind #n #Heq <plus_Sn_m1 whd in match (plus (S n) z); |
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200 | @(pe_trans … (\snd (pi1 … (jump_expansion_internal program (S n))))) |
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201 | [ @Heq |
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202 | | @Hind @pe_step @Heq |
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203 | ] |
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204 | ] |
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205 | qed. |
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206 | |
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207 | (* this number monotonically increases over iterations, maximum 2*|program| *) |
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208 | let rec measure_int (program: list labelled_instruction) (policy: ppc_pc_map) (acc: ℕ) |
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209 | on program: ℕ ≝ |
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210 | match program with |
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211 | [ nil ⇒ acc |
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212 | | cons h t ⇒ match (\snd (bvt_lookup ?? (bitvector_of_nat ? (|t|)) (\snd policy) 〈0,short_jump〉)) with |
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213 | [ long_jump ⇒ measure_int t policy (acc + 2) |
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214 | | medium_jump ⇒ measure_int t policy (acc + 1) |
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215 | | _ ⇒ measure_int t policy acc |
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216 | ] |
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217 | ]. |
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218 | |
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219 | lemma measure_plus: ∀program.∀policy.∀x,d:ℕ. |
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220 | measure_int program policy (x+d) = measure_int program policy x + d. |
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221 | #program #policy #x #d generalize in match x; -x elim d |
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222 | [ // |
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223 | | -d; #d #Hind elim program |
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224 | [ / by refl/ |
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225 | | #h #t #Hd #x whd in match (measure_int ???); whd in match (measure_int ?? x); |
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226 | cases (\snd (bvt_lookup … (bitvector_of_nat ? (|t|)) (\snd policy) 〈0,short_jump〉)) |
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227 | [ normalize nodelta @Hd |
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228 | |2,3: normalize nodelta >associative_plus >(commutative_plus (S d) ?) <associative_plus |
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229 | @Hd |
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230 | ] |
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231 | ] |
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232 | ] |
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233 | qed. |
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234 | |
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235 | lemma measure_le: ∀program.∀policy. |
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236 | measure_int program policy 0 ≤ 2*|program|. |
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237 | #program #policy elim program |
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238 | [ normalize @le_n |
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239 | | #h #t #Hind whd in match (measure_int ???); |
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240 | cases (\snd (lookup ?? (bitvector_of_nat ? (|t|)) (\snd policy) 〈0,short_jump〉)) |
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241 | [ normalize nodelta @(transitive_le ??? Hind) /2 by monotonic_le_times_r/ |
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242 | |2,3: normalize nodelta >measure_plus <times_n_Sm >(commutative_plus 2 ?) |
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243 | @le_plus [1,3: @Hind |2,4: / by le_n/ ] |
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244 | ] |
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245 | ] |
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246 | qed. |
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247 | |
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248 | (* uses the second part of policy_increase *) |
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249 | lemma measure_incr_or_equal: ∀program:Σl:list labelled_instruction.S (|l|) <2^16. |
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250 | ∀policy:Σp:ppc_pc_map. |
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251 | out_of_program_none program p ∧ |
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252 | jump_not_in_policy program p ∧ |
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253 | \fst (bvt_lookup … (bitvector_of_nat ? 0) (\snd p) 〈0,short_jump〉) = 0 ∧ |
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254 | \fst p < 2^16. |
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255 | ∀l.|l| ≤ |program| → ∀acc:ℕ. |
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256 | match \snd (pi1 ?? (jump_expansion_step program (create_label_map program) policy)) with |
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257 | [ None ⇒ True |
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258 | | Some p ⇒ measure_int l policy acc ≤ measure_int l p acc |
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259 | ]. |
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260 | #program #policy #l elim l -l; |
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261 | [ #Hp #acc cases (jump_expansion_step ???) #pi1 cases pi1 #p #q -pi1; cases q [ // | #x #_ @le_n ] |
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262 | | #h #t #Hind #Hp #acc |
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263 | lapply (refl ? (jump_expansion_step program (create_label_map program) policy)) |
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264 | cases (jump_expansion_step ???) in ⊢ (???% → %); #pi1 cases pi1 -pi1 #c #r cases r |
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265 | [ / by I/ |
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266 | | #x normalize nodelta #Hx #Hjeq |
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267 | lapply (proj2 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? Hx))))) (|t|) Hp) |
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268 | whd in match (measure_int ???); whd in match (measure_int ? x ?); |
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269 | cases (bvt_lookup ?? (bitvector_of_nat ? (|t|)) (\snd (pi1 ?? policy)) 〈0,short_jump〉) |
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270 | #x1 #x2 cases (bvt_lookup ?? (bitvector_of_nat ? (|t|)) (\snd x) 〈0,short_jump〉) |
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271 | #y1 #y2 normalize nodelta #Hblerp cases Hblerp cases x2 cases y2 |
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272 | [1,4,5,7,8,9: #H cases H |
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273 | |2,3,6: #_ normalize nodelta |
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274 | [1,2: @(transitive_le ? (measure_int t x acc)) |
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275 | |3: @(transitive_le ? (measure_int t x (acc+1))) |
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276 | ] |
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277 | [2,4,5,6: >measure_plus [1,2: @le_plus_n_r] >measure_plus @le_plus / by le_n/] |
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278 | >Hjeq in Hind; #Hind @Hind @(transitive_le … Hp) @le_n_Sn |
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279 | |11,12,13,15,16,17: #H destruct (H) |
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280 | |10,14,18: normalize nodelta #_ >Hjeq in Hind; #Hind @Hind @(transitive_le … Hp) @le_n_Sn |
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281 | ] |
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282 | ] |
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283 | ] |
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284 | qed. |
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285 | |
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286 | (* these lemmas seem superfluous, but not sure how *) |
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287 | lemma bla: ∀a,b:ℕ.a + a = b + b → a = b. |
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288 | #a elim a |
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289 | [ normalize #b // |
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290 | | -a #a #Hind #b cases b [ /2 by le_n_O_to_eq/ | -b #b normalize |
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291 | <plus_n_Sm <plus_n_Sm #H |
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292 | >(Hind b (injective_S ?? (injective_S ?? H))) // ] |
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293 | ] |
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294 | qed. |
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295 | |
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296 | lemma sth_not_s: ∀x.x ≠ S x. |
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297 | #x cases x |
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298 | [ // | #y // ] |
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299 | qed. |
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300 | |
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301 | lemma measure_full: ∀program.∀policy. |
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302 | measure_int program policy 0 = 2*|program| → ∀i.i<|program| → |
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303 | is_jump (\snd (nth i ? program 〈None ?,Comment []〉)) → |
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304 | (\snd (bvt_lookup ?? (bitvector_of_nat ? i) (\snd policy) 〈0,short_jump〉)) = long_jump. |
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305 | #program #policy elim program in ⊢ (% → ∀i.% → ? → %); |
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306 | [ #Hm #i #Hi @⊥ @(absurd … Hi) @not_le_Sn_O |
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307 | | #h #t #Hind #Hm #i #Hi #Hj |
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308 | cases (le_to_or_lt_eq … Hi) -Hi |
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309 | [ #Hi @Hind |
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310 | [ whd in match (measure_int ???) in Hm; |
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311 | cases (\snd (bvt_lookup … (bitvector_of_nat ? (|t|)) (\snd policy) 〈0,short_jump〉)) in Hm; |
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312 | normalize nodelta |
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313 | [ #H @⊥ @(absurd ? (measure_le t policy)) >H @lt_to_not_le /2 by lt_plus, le_n/ |
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314 | | >measure_plus >commutative_plus #H @⊥ @(absurd ? (measure_le t policy)) |
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315 | <(plus_to_minus … (sym_eq … H)) @lt_to_not_le normalize /2 by le_n/ |
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316 | | >measure_plus <times_n_Sm >commutative_plus /2 by injective_plus_r/ |
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317 | ] |
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318 | | @(le_S_S_to_le … Hi) |
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319 | | @Hj |
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320 | ] |
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321 | | #Hi >(injective_S … Hi) whd in match (measure_int ???) in Hm; |
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322 | cases (\snd (bvt_lookup … (bitvector_of_nat ? (|t|)) (\snd policy) 〈0,short_jump〉)) in Hm; |
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323 | normalize nodelta |
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324 | [ #Hm @⊥ @(absurd ? (measure_le t policy)) >Hm @lt_to_not_le /2 by lt_plus, le_n/ |
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325 | | >measure_plus >commutative_plus #H @⊥ @(absurd ? (measure_le t policy)) |
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326 | <(plus_to_minus … (sym_eq … H)) @lt_to_not_le normalize /2 by le_n/ |
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327 | | >measure_plus <times_n_Sm >commutative_plus /2 by injective_plus_r/ |
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328 | ] |
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329 | ] |
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330 | ] |
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331 | qed. |
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332 | |
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333 | (* uses second part of policy_increase *) |
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334 | lemma measure_special: ∀program:(Σl:list labelled_instruction.(S (|l|)) < 2^16). |
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335 | ∀policy:Σp:ppc_pc_map. |
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336 | out_of_program_none program p ∧ jump_not_in_policy program p ∧ |
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337 | \fst (bvt_lookup … (bitvector_of_nat ? 0) (\snd p) 〈0,short_jump〉) = 0 ∧ |
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338 | \fst p < 2^16. |
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339 | match (\snd (pi1 ?? (jump_expansion_step program (create_label_map program) policy))) with |
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340 | [ None ⇒ True |
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341 | | Some p ⇒ measure_int program policy 0 = measure_int program p 0 → policy_equal program policy p ]. |
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342 | #program #policy lapply (refl ? (pi1 ?? (jump_expansion_step program (create_label_map program) policy))) |
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343 | cases (jump_expansion_step program (create_label_map program) policy) in ⊢ (???% → %); |
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344 | #p cases p -p #ch #pol normalize nodelta cases pol |
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345 | [ / by I/ |
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346 | | #p normalize nodelta #Hpol #eqpol lapply (le_n (|program|)) |
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347 | @(list_ind ? (λx.|x| ≤ |pi1 ?? program| → |
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348 | measure_int x policy 0 = measure_int x p 0 → |
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349 | policy_equal x policy p) ?? (pi1 ?? program)) |
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350 | [ #_ #_ #i #Hi @⊥ @(absurd ? Hi) @le_to_not_lt @le_O_n |
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351 | | #h #t #Hind #Hp #Hm #i #Hi cases (le_to_or_lt_eq … (le_S_S_to_le … Hi)) -Hi #Hi |
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352 | [ @Hind |
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353 | [ @(transitive_le … Hp) / by / |
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354 | | whd in match (measure_int ???) in Hm; whd in match (measure_int ? p ?) in Hm; |
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355 | lapply (proj2 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? Hpol))))) (|t|) Hp) #Hinc |
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356 | cases (bvt_lookup ?? (bitvector_of_nat ? (|t|)) ? 〈0,short_jump〉) in Hm Hinc; #x1 #x2 |
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357 | cases (bvt_lookup ?? (bitvector_of_nat ? (|t|)) ? 〈0,short_jump〉); #y1 #y2 |
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358 | #Hm #Hinc lapply Hm -Hm; lapply Hinc -Hinc; normalize nodelta |
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359 | cases x2 cases y2 normalize nodelta |
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360 | [1: / by / |
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361 | |2,3: >measure_plus #_ #H @⊥ @(absurd ? (eq_plus_S_to_lt … H)) @le_to_not_lt |
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362 | lapply (measure_incr_or_equal program policy t ? 0) |
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363 | [1,3: @(transitive_le … Hp) @le_n_Sn ] >eqpol / by / |
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364 | |4,7,8: #H elim H #H2 [1,3,5: cases H2 |2,4,6: destruct (H2) ] |
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365 | |5: >measure_plus >measure_plus >commutative_plus >(commutative_plus ? 1) |
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366 | #_ #H @(injective_plus_r … H) |
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367 | |6: >measure_plus >measure_plus |
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368 | change with (1+1) in match (2); >assoc_plus1 >(commutative_plus 1 (measure_int ???)) |
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369 | #_ #H @⊥ @(absurd ? (eq_plus_S_to_lt … H)) @le_to_not_lt @monotonic_le_plus_l |
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370 | lapply (measure_incr_or_equal program policy t ? 0) |
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371 | [ @(transitive_le … Hp) @le_n_Sn ] >eqpol / by / |
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372 | |9: >measure_plus >measure_plus >commutative_plus >(commutative_plus ? 2) |
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373 | #_ #H @(injective_plus_r … H) |
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374 | ] |
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375 | | @Hi |
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376 | ] |
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377 | | >Hi whd in match (measure_int ???) in Hm; whd in match (measure_int ? p ?) in Hm; |
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378 | lapply (proj2 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? Hpol))))) (|t|) Hp) |
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379 | cases (bvt_lookup ?? (bitvector_of_nat ? (|t|)) ? 〈0,short_jump〉) in Hm; |
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380 | #x1 #x2 |
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381 | cases (bvt_lookup ?? (bitvector_of_nat ? (|t|)) ? 〈0,short_jump〉); #y1 #y2 |
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382 | normalize nodelta cases x2 cases y2 normalize nodelta |
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383 | [1,5,9: #_ #_ @refl |
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384 | |4,7,8: #_ #H elim H #H2 [1,3,5: cases H2 |2,4,6: destruct (H2) ] |
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385 | |2,3: >measure_plus #H #_ @⊥ @(absurd ? (eq_plus_S_to_lt … H)) @le_to_not_lt |
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386 | lapply (measure_incr_or_equal program policy t ? 0) |
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387 | [1,3: @(transitive_le … Hp) @le_n_Sn ] >eqpol / by / |
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388 | |6: >measure_plus >measure_plus |
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389 | change with (1+1) in match (2); >assoc_plus1 >(commutative_plus 1 (measure_int ???)) |
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390 | #H #_ @⊥ @(absurd ? (eq_plus_S_to_lt … H)) @le_to_not_lt @monotonic_le_plus_l |
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391 | lapply (measure_incr_or_equal program policy t ? 0) |
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392 | [ @(transitive_le … Hp) @le_n_Sn ] >eqpol / by / |
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393 | ] |
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394 | ] |
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395 | ] |
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396 | qed. |
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397 | |
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398 | lemma le_to_eq_plus: ∀n,z. |
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399 | n ≤ z → ∃k.z = n + k. |
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400 | #n #z elim z |
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401 | [ #H cases (le_to_or_lt_eq … H) |
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402 | [ #H2 @⊥ @(absurd … H2) @not_le_Sn_O |
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403 | | #H2 @(ex_intro … 0) >H2 // |
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404 | ] |
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405 | | #z' #Hind #H cases (le_to_or_lt_eq … H) |
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406 | [ #H' elim (Hind (le_S_S_to_le … H')) #k' #H2 @(ex_intro … (S k')) |
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407 | >H2 >plus_n_Sm // |
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408 | | #H' @(ex_intro … 0) >H' // |
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409 | ] |
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410 | ] |
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411 | qed. |
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412 | |
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413 | lemma measure_zero: ∀l.∀program:Σl:list labelled_instruction.S (|l|) < 2^16. |
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414 | match jump_expansion_start program (create_label_map program) with |
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415 | [ None ⇒ True |
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416 | | Some p ⇒ |l| ≤ |program| → measure_int l p 0 = 0 |
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417 | ]. |
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418 | #l #program lapply (refl ? (jump_expansion_start program (create_label_map program))) |
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419 | cases (jump_expansion_start program (create_label_map program)) in ⊢ (???% → %); #p #Hp #EQ |
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420 | cases p in Hp EQ; |
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421 | [ / by I/ |
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422 | | #pl normalize nodelta #Hpl #EQ elim l |
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423 | [ / by refl/ |
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424 | | #h #t #Hind #Hp whd in match (measure_int ???); |
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425 | elim (proj2 ?? (proj1 ?? (proj1 ?? Hpl)) (|t|) (le_S_to_le … Hp)) |
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426 | #pc #Hpc >(lookup_opt_lookup_hit … Hpc 〈0,short_jump〉) normalize nodelta @Hind |
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427 | @(transitive_le … Hp) @le_n_Sn |
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428 | ] |
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429 | ] |
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430 | qed. |
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431 | |
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432 | (* the actual computation of the fixpoint *) |
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433 | definition je_fixpoint: ∀program:(Σl:list labelled_instruction.S (|l|) < 2^16). |
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434 | Σp:option ppc_pc_map. |
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435 | And (match p with |
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436 | [ None ⇒ True |
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437 | | Some pol ⇒ And (out_of_program_none program pol) |
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438 | ((pi1 ?? program) ≠ [] → policy_compact program (create_label_map program) pol) |
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439 | ]) |
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440 | (∃n.∀k.n < k → |
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441 | policy_equal_opt program (\snd (pi1 ?? (jump_expansion_internal program k))) p). |
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442 | #program @(\snd (pi1 ?? (jump_expansion_internal program (2*|program|)))) @conj |
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443 | [ lapply (pi2 ?? (jump_expansion_internal program (2*|program|))) |
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444 | cases (jump_expansion_internal program (2*|program|)) #p cases p -p |
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445 | #c #pol #Hp cases pol |
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446 | [ normalize nodelta // |
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447 | | #x normalize nodelta #H @conj [ @(proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? H))))) |
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448 | | #Hneq @(proj2 ?? (proj1 ?? (proj1 ?? (proj1 ?? H)))) cases (pi1 ?? program) in Hneq; |
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449 | [ #H cases H #H @⊥ @H @refl |
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450 | | #h #t #_ / by / |
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451 | ] ] |
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452 | ] |
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453 | | cases (dec_bounded_exists (λk.policy_equal_opt (pi1 ?? program) |
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454 | (\snd (pi1 ?? (jump_expansion_internal program k))) |
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455 | (\snd (pi1 ?? (jump_expansion_internal program (S k))))) ? (2*|program|)) |
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456 | [ #Hex elim Hex -Hex #x #Hx @(ex_intro … x) #k #Hk |
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457 | @pe_trans |
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458 | [ @(\snd (pi1 ?? (jump_expansion_internal program x))) |
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459 | | @pe_sym @equal_remains_equal |
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460 | [ @(proj2 ?? Hx) |
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461 | | @le_S_S_to_le @le_S @Hk |
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462 | ] |
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463 | | @equal_remains_equal |
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464 | [ @(proj2 ?? Hx) |
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465 | | @(proj1 ?? Hx) |
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466 | ] |
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467 | ] |
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468 | | #Hnex lapply (not_exists_forall … Hnex) -Hnex; #Hfa |
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469 | @(ex_intro … (2*|program|)) #k #Hk @pe_sym @equal_remains_equal |
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470 | [ lapply (refl ? (jump_expansion_internal program (2*|program|))) |
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471 | cases (jump_expansion_internal program (2*|program|)) in ⊢ (???% → %); |
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472 | #x cases x -x #Fch #Fpol normalize nodelta #HFpol cases Fpol in HFpol; normalize nodelta |
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473 | [ (* if we're at None in 2*|program|, we're at None in S 2*|program| too *) |
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474 | #HFpol #EQ whd in match (jump_expansion_internal ??); >EQ |
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475 | normalize nodelta / by / |
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476 | | #Fp #HFp #EQ whd in match (jump_expansion_internal ??); |
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477 | >EQ normalize nodelta |
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478 | lapply (refl ? (jump_expansion_step program (create_label_map program) «Fp,?»)) |
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479 | [ @conj [ @conj |
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480 | [ @(proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? HFp)))) |
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481 | | @(proj2 ?? (proj1 ?? (proj1 ?? HFp))) ] |
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482 | | @(proj2 ?? (proj1 ?? HFp)) ] |
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483 | | lapply (measure_full program Fp ?) |
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484 | [ @le_to_le_to_eq |
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485 | [ @measure_le |
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486 | | cut (∀x:ℕ.x ≤ 2*|program| → |
---|
487 | ∃p.(\snd (pi1 ?? (jump_expansion_internal program x)) = Some ? p ∧ |
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488 | x ≤ measure_int program p 0)) |
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489 | [ #x elim x |
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490 | [ #Hx lapply (refl ? (jump_expansion_start program (create_label_map program))) |
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491 | cases (jump_expansion_start program (create_label_map program)) in ⊢ (???% → %); |
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492 | #z cases z -z normalize nodelta |
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493 | [ #Waar #Heqn @⊥ elim (le_to_eq_plus ?? Hx) #k #Hk |
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494 | @(absurd … (step_none program 0 ? k)) |
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495 | [ whd in match (jump_expansion_internal ??); >Heqn @refl |
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496 | | <Hk >EQ @nmk #H destruct (H) |
---|
497 | ] |
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498 | | #pol #Hpol #Heqpol @(ex_intro ?? pol) @conj |
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499 | [ whd in match (jump_expansion_internal ??); >Heqpol @refl |
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500 | | @le_O_n |
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501 | ] |
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502 | ] |
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503 | | -x #x #Hind #Hx |
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504 | lapply (refl ? (jump_expansion_internal program (S x))) |
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505 | cases (jump_expansion_internal program (S x)) in ⊢ (???% → %); |
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506 | #z cases z -z #Sxch #Sxpol cases Sxpol -Sxpol normalize nodelta |
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507 | [ #H #HeqSxpol @⊥ elim (le_to_eq_plus ?? Hx) #k #Hk |
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508 | @(absurd … (step_none program (S x) ? k)) |
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509 | [ >HeqSxpol / by / |
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510 | | <Hk >EQ @nmk #H destruct (H) |
---|
511 | ] |
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512 | | #Sxpol #HSxpol #HeqSxpol @(ex_intro ?? Sxpol) @conj |
---|
513 | [ @refl |
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514 | | elim (Hind (transitive_le … (le_n_Sn x) Hx)) |
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515 | #xpol #Hxpol @(le_to_lt_to_lt … (proj2 ?? Hxpol)) |
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516 | lapply (measure_incr_or_equal program xpol program (le_n (|program|)) 0) |
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517 | [ cases (jump_expansion_internal program x) in Hxpol; |
---|
518 | #z cases z -z #xch #xpol normalize nodelta #H #H2 >(proj1 ?? H2) in H; |
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519 | normalize nodelta #H @conj [ @conj |
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520 | [ @(proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? H)))) |
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521 | | @(proj2 ?? (proj1 ?? (proj1 ?? H))) ] |
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522 | | @(proj2 ?? (proj1 ?? H)) ] |
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523 | | lapply (Hfa x (le_S_to_le … Hx)) lapply HeqSxpol -HeqSxpol |
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524 | lapply (refl ? (jump_expansion_internal program x)) |
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525 | lapply Hxpol -Hxpol |
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526 | whd in match (jump_expansion_internal program (S x)); |
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527 | cases (jump_expansion_internal program x) in ⊢ (% → ???% → %); |
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528 | #z cases z -z #xch #b normalize nodelta #H #Heq >(proj1 ?? Heq) in H; |
---|
529 | #H #Heq2 cases xch in H Heq2; #H #Heq2 normalize nodelta |
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530 | [ lapply (refl ? (jump_expansion_step program (create_label_map (pi1 ?? program)) «xpol,?»)) |
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531 | [ @conj [ @conj |
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532 | [ @(proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? H)))) |
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533 | | @(proj2 ?? (proj1 ?? (proj1 ?? H))) ] |
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534 | | @(proj2 ?? (proj1 ?? H)) ] |
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535 | | cases (jump_expansion_step ???) in ⊢ (???% → %); #z cases z -z #a #c |
---|
536 | normalize nodelta cases c normalize nodelta |
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537 | [ #H1 #Heq #H2 destruct (H2) |
---|
538 | | #d #H1 #Heq #H2 destruct (H2) #Hfull #H2 elim (le_to_or_lt_eq … H2) |
---|
539 | [ / by / |
---|
540 | | #H3 lapply (measure_special program «xpol,?») |
---|
541 | [ @conj [ @conj |
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542 | [ @(proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? H)))) |
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543 | | @(proj2 ?? (proj1 ?? (proj1 ?? H))) ] |
---|
544 | | @(proj2 ?? (proj1 ?? H)) ] |
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545 | | >Heq normalize nodelta #H4 @⊥ @(absurd … (H4 H3)) @Hfull |
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546 | ] |
---|
547 | ] |
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548 | ] |
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549 | ] |
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550 | | lapply (refl ? (jump_expansion_step program (create_label_map (pi1 ?? program)) «xpol,?»)) |
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551 | [ @conj [ @conj |
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552 | [ @(proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? H)))) |
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553 | | @(proj2 ?? (proj1 ?? (proj1 ?? H))) ] |
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554 | | @(proj2 ?? (proj1 ?? H)) ] |
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555 | | cases (jump_expansion_step ???) in ⊢ (???% → %); #z cases z -z #a #c |
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556 | normalize nodelta cases c normalize nodelta |
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557 | [ #H1 #Heq #H2 #H3 #_ @⊥ @(absurd ?? H3) @pe_refl |
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558 | | #d #H1 #Heq #H2 #H3 @⊥ @(absurd ?? H3) @pe_refl |
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559 | ] |
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560 | ] |
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561 | ] |
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562 | ] |
---|
563 | ] |
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564 | ] |
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565 | ] |
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566 | | #H elim (H (2*|program|) (le_n ?)) #plp >EQ #Hplp |
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567 | >(Some_eq ??? (proj1 ?? Hplp)) @(proj2 ?? Hplp) |
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568 | ] |
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569 | ] |
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570 | | #Hfull cases (jump_expansion_step program (create_label_map program) «Fp,?») in ⊢ (???% → %); |
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571 | #x cases x -x #Gch #Gpol cases Gpol normalize nodelta |
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572 | [ #H #EQ2 @⊥ @(absurd ?? H) @Hfull |
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573 | | #Gp #HGp #EQ2 cases Fch |
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574 | [ normalize nodelta #i #Hi |
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575 | cases (dec_is_jump (\snd (nth i ? program 〈None ?, Comment []〉))) #Hj |
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576 | [ lapply (proj2 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? HGp))))) i Hi) |
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577 | lapply (Hfull i Hi Hj) |
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578 | cases (bvt_lookup … (bitvector_of_nat ? i) (\snd Fp) 〈0,short_jump〉) |
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579 | #fp #fj #Hfj >Hfj normalize nodelta |
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580 | cases (bvt_lookup … (bitvector_of_nat ? i) (\snd Gp) 〈0,short_jump〉) |
---|
581 | #gp #gj cases gj normalize nodelta |
---|
582 | [1,2: #H cases H #H2 cases H2 destruct (H2) |
---|
583 | |3: #_ @refl |
---|
584 | ] |
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585 | | >(proj2 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? HGp)))))) i Hi Hj) |
---|
586 | >(proj2 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? HFp)))) i Hi Hj) @refl |
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587 | ] |
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588 | | normalize nodelta /2 by pe_int_refl/ |
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589 | ] |
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590 | ] |
---|
591 | ] |
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592 | ] |
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593 | ] |
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594 | | @le_S_S_to_le @le_S @Hk |
---|
595 | ] |
---|
596 | | #n cases (jump_expansion_internal program n) cases (jump_expansion_internal program (S n)) |
---|
597 | #x cases x -x #nch #npol normalize nodelta #Hnpol |
---|
598 | #x cases x -x #Sch #Spol normalize nodelta #HSpol |
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599 | cases npol in Hnpol; cases Spol in HSpol; |
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600 | [ #Hnpol #HSpol %1 // |
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601 | |2,3: #x #Hnpol #HSpol %2 @nmk whd in match (policy_equal ???); // |
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602 | #H destruct (H) |
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603 | |4: #np #Hnp #Sp #HSp whd in match (policy_equal ???); cases program #p #_ cases p |
---|
604 | [ %1 #m #Hm @⊥ @(absurd ? Hm) @le_to_not_lt @le_O_n |
---|
605 | | #h #t elim (dec_bounded_forall ?? (|t|)) |
---|
606 | [1: #Hyp %1 #m #Hm @(Hyp m) @(le_S_S_to_le … Hm) |
---|
607 | |2: #Hyp %2 @nmk #H @(absurd ?? Hyp) #m #Hm @(H m) @(le_S_S … Hm) |
---|
608 | | #m cases (bvt_lookup ?? (bitvector_of_nat 16 m) ? 〈0,short_jump〉) |
---|
609 | #x1 #x2 |
---|
610 | cases (bvt_lookup ?? (bitvector_of_nat ? m) ? 〈0,short_jump〉) |
---|
611 | #y1 #y2 normalize nodelta |
---|
612 | @dec_eq_jump_length |
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613 | ] |
---|
614 | |
---|
615 | ] |
---|
616 | ] |
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617 | ] |
---|
618 | qed. |
---|
619 | |
---|
620 | include alias "arithmetics/nat.ma". |
---|
621 | include alias "basics/logic.ma". |
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622 | |
---|
623 | (* The glue between Policy and Assembly. *) |
---|
624 | definition jump_expansion': |
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625 | ∀program:preamble × (Σl:list labelled_instruction.S (|l|) < 2^16). |
---|
626 | option (Σsigma:Word → Word × bool. |
---|
627 | ∀ppc: Word. |
---|
628 | let pc ≝ \fst (sigma ppc) in |
---|
629 | let labels ≝ \fst (create_label_cost_map (\snd program)) in |
---|
630 | let lookup_labels ≝ λx. bitvector_of_nat ? (lookup_def ?? labels x 0) in |
---|
631 | let instruction ≝ \fst (fetch_pseudo_instruction (\snd program) ppc) in |
---|
632 | let next_pc ≝ \fst (sigma (add ? ppc (bitvector_of_nat ? 1))) in |
---|
633 | And (nat_of_bitvector … ppc ≤ |\snd program| → |
---|
634 | next_pc = add ? pc (bitvector_of_nat … |
---|
635 | (instruction_size lookup_labels (λx.\fst (sigma x)) (λx.\snd (sigma x)) ppc instruction))) |
---|
636 | (Or (nat_of_bitvector … ppc < |\snd program| → |
---|
637 | nat_of_bitvector … pc < nat_of_bitvector … next_pc) |
---|
638 | (nat_of_bitvector … ppc = |\snd program| → next_pc = (zero …)))) ≝ |
---|
639 | λprogram. |
---|
640 | let policy ≝ pi1 … (je_fixpoint (\snd program)) in |
---|
641 | match policy with |
---|
642 | [ None ⇒ None ? |
---|
643 | | Some x ⇒ Some ? |
---|
644 | «λppc.let 〈pc,jl〉 ≝ bvt_lookup ?? ppc (\snd x) 〈0,short_jump〉 in |
---|
645 | 〈bitvector_of_nat 16 pc,jmpeqb jl long_jump〉,?» |
---|
646 | ]. |
---|
647 | #ppc normalize nodelta cases daemon |
---|
648 | qed. |
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