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1 (* Title: HOL/Auth/Public_lemmas |
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2 ID: $Id$ |
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3 Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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4 Copyright 1996 University of Cambridge |
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5 |
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6 Theory of Public Keys (common to all symmetric-key protocols) |
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7 |
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8 Server keys; initial states of agents; new nonces and keys; function "sees" |
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9 *) |
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10 |
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11 val inj_pubK = thm "inj_pubK"; |
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12 val priK_neq_pubK = thm "priK_neq_pubK"; |
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13 |
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14 (*** Basic properties of pubK & priK ***) |
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15 |
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16 AddIffs [inj_pubK RS inj_eq]; |
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17 |
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18 Goal "(priK A = priK B) = (A=B)"; |
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19 by Safe_tac; |
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20 by (dres_inst_tac [("f","invKey")] arg_cong 1); |
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21 by (Full_simp_tac 1); |
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22 qed "priK_inj_eq"; |
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23 |
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24 AddIffs [priK_inj_eq]; |
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25 AddIffs [priK_neq_pubK, priK_neq_pubK RS not_sym]; |
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26 |
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27 Goalw [isSymKey_def] "~ isSymKey (pubK A)"; |
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28 by (Simp_tac 1); |
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29 qed "not_isSymKey_pubK"; |
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30 |
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31 Goalw [isSymKey_def] "~ isSymKey (priK A)"; |
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32 by (Simp_tac 1); |
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33 qed "not_isSymKey_priK"; |
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34 |
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35 AddIffs [not_isSymKey_pubK, not_isSymKey_priK]; |
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36 |
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37 |
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38 (** "Image" equations that hold for injective functions **) |
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39 |
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40 Goal "(invKey x : invKey`A) = (x:A)"; |
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41 by Auto_tac; |
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42 qed "invKey_image_eq"; |
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43 |
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44 (*holds because invKey is injective*) |
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45 Goal "(pubK x : pubK`A) = (x:A)"; |
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46 by Auto_tac; |
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47 qed "pubK_image_eq"; |
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48 |
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49 Goal "(priK x ~: pubK`A)"; |
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50 by Auto_tac; |
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51 qed "priK_pubK_image_eq"; |
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52 Addsimps [invKey_image_eq, pubK_image_eq, priK_pubK_image_eq]; |
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53 |
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54 |
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55 (** Rewrites should not refer to initState(Friend i) |
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56 -- not in normal form! **) |
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57 |
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58 Goalw [keysFor_def] "keysFor (parts (initState C)) = {}"; |
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59 by (induct_tac "C" 1); |
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60 by (auto_tac (claset() addIs [range_eqI], simpset())); |
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61 qed "keysFor_parts_initState"; |
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62 Addsimps [keysFor_parts_initState]; |
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63 |
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64 |
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65 (*** Function "spies" ***) |
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66 |
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67 (*Agents see their own private keys!*) |
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68 Goal "Key (priK A) : initState A"; |
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69 by (induct_tac "A" 1); |
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70 by Auto_tac; |
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71 qed "priK_in_initState"; |
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72 AddIffs [priK_in_initState]; |
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73 |
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74 (*All public keys are visible*) |
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75 Goal "Key (pubK A) : spies evs"; |
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76 by (induct_tac "evs" 1); |
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77 by (ALLGOALS (asm_simp_tac |
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78 (simpset() addsimps [imageI, knows_Cons] |
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79 addsplits [expand_event_case]))); |
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80 qed_spec_mp "spies_pubK"; |
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81 |
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82 (*Spy sees private keys of bad agents!*) |
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83 Goal "A: bad ==> Key (priK A) : spies evs"; |
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84 by (induct_tac "evs" 1); |
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85 by (ALLGOALS (asm_simp_tac |
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86 (simpset() addsimps [imageI, knows_Cons] |
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87 addsplits [expand_event_case]))); |
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88 qed "Spy_spies_bad"; |
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89 |
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90 AddIffs [spies_pubK, spies_pubK RS analz.Inj]; |
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91 AddSIs [Spy_spies_bad]; |
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92 |
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93 |
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94 (*For not_bad_tac*) |
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95 Goal "[| Crypt (pubK A) X : analz (spies evs); A: bad |] \ |
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96 \ ==> X : analz (spies evs)"; |
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97 by (blast_tac (claset() addSDs [analz.Decrypt]) 1); |
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98 qed "Crypt_Spy_analz_bad"; |
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99 |
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100 (*Prove that the agent is uncompromised by the confidentiality of |
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101 a component of a message she's said.*) |
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102 fun not_bad_tac s = |
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103 case_tac ("(" ^ s ^ ") : bad") THEN' |
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104 SELECT_GOAL |
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105 (REPEAT_DETERM (dtac (Says_imp_spies RS analz.Inj) 1) THEN |
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106 REPEAT_DETERM (etac MPair_analz 1) THEN |
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107 THEN_BEST_FIRST |
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108 (dres_inst_tac [("A", s)] Crypt_Spy_analz_bad 1 THEN assume_tac 1) |
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109 (has_fewer_prems 1, size_of_thm) |
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110 Safe_tac); |
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111 |
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112 |
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113 (*** Fresh nonces ***) |
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114 |
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115 Goal "Nonce N ~: parts (initState B)"; |
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116 by (induct_tac "B" 1); |
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117 by Auto_tac; |
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118 qed "Nonce_notin_initState"; |
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119 AddIffs [Nonce_notin_initState]; |
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120 |
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121 Goal "Nonce N ~: used []"; |
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122 by (simp_tac (simpset() addsimps [used_Nil]) 1); |
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123 qed "Nonce_notin_used_empty"; |
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124 Addsimps [Nonce_notin_used_empty]; |
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125 |
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126 |
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127 (*** Supply fresh nonces for possibility theorems. ***) |
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128 |
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129 (*In any trace, there is an upper bound N on the greatest nonce in use.*) |
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130 Goal "EX N. ALL n. N<=n --> Nonce n ~: used evs"; |
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131 by (induct_tac "evs" 1); |
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132 by (res_inst_tac [("x","0")] exI 1); |
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133 by (ALLGOALS (asm_simp_tac |
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134 (simpset() addsimps [used_Cons] |
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135 addsplits [expand_event_case]))); |
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136 by Safe_tac; |
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137 by (ALLGOALS (rtac (msg_Nonce_supply RS exE))); |
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138 by (ALLGOALS (blast_tac (claset() addSEs [add_leE]))); |
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139 val lemma = result(); |
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140 |
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141 Goal "EX N. Nonce N ~: used evs"; |
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142 by (rtac (lemma RS exE) 1); |
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143 by (Blast_tac 1); |
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144 qed "Nonce_supply1"; |
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145 |
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146 Goal "Nonce (@ N. Nonce N ~: used evs) ~: used evs"; |
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147 by (rtac (lemma RS exE) 1); |
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148 by (rtac someI 1); |
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149 by (Fast_tac 1); |
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150 qed "Nonce_supply"; |
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151 |
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152 (*Tactic for possibility theorems*) |
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153 fun possibility_tac st = st |> |
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154 REPEAT (*omit used_Says so that Nonces start from different traces!*) |
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155 (ALLGOALS (simp_tac (simpset() delsimps [used_Says])) |
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156 THEN |
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157 REPEAT_FIRST (eq_assume_tac ORELSE' |
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158 resolve_tac [refl, conjI, Nonce_supply])); |
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159 |
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160 |
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161 (*** Specialized rewriting for the analz_image_... theorems ***) |
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162 |
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163 Goal "insert (Key K) H = Key ` {K} Un H"; |
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164 by (Blast_tac 1); |
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165 qed "insert_Key_singleton"; |
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166 |
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167 Goal "insert (Key K) (Key`KK Un C) = Key ` (insert K KK) Un C"; |
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168 by (Blast_tac 1); |
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169 qed "insert_Key_image"; |
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170 |
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171 (*Reverse the normal simplification of "image" to build up (not break down) |
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172 the set of keys. Based on analz_image_freshK_ss, but simpler.*) |
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173 val analz_image_keys_ss = |
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174 simpset() delsimps [image_insert, image_Un] |
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175 delsimps [imp_disjL] (*reduces blow-up*) |
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176 addsimps [image_insert RS sym, image_Un RS sym, |
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177 rangeI, |
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178 insert_Key_singleton, |
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179 insert_Key_image, Un_assoc RS sym]; |
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180 |