1 (* Title: HOL/SET-Protocol/MessageSET.thy |
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2 Author: Giampaolo Bella |
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3 Author: Fabio Massacci |
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4 Author: Lawrence C Paulson |
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5 *) |
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6 |
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7 header{*The Message Theory, Modified for SET*} |
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8 |
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9 theory MessageSET |
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10 imports Main Nat_Int_Bij |
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11 begin |
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12 |
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13 subsection{*General Lemmas*} |
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14 |
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15 text{*Needed occasionally with @{text spy_analz_tac}, e.g. in |
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16 @{text analz_insert_Key_newK}*} |
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17 |
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18 lemma Un_absorb3 [simp] : "A \<union> (B \<union> A) = B \<union> A" |
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19 by blast |
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20 |
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21 text{*Collapses redundant cases in the huge protocol proofs*} |
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22 lemmas disj_simps = disj_comms disj_left_absorb disj_assoc |
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23 |
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24 text{*Effective with assumptions like @{term "K \<notin> range pubK"} and |
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25 @{term "K \<notin> invKey`range pubK"}*} |
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26 lemma notin_image_iff: "(y \<notin> f`I) = (\<forall>i\<in>I. f i \<noteq> y)" |
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27 by blast |
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28 |
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29 text{*Effective with the assumption @{term "KK \<subseteq> - (range(invKey o pubK))"} *} |
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30 lemma disjoint_image_iff: "(A <= - (f`I)) = (\<forall>i\<in>I. f i \<notin> A)" |
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31 by blast |
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32 |
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33 |
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34 |
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35 types |
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36 key = nat |
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37 |
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38 consts |
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39 all_symmetric :: bool --{*true if all keys are symmetric*} |
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40 invKey :: "key=>key" --{*inverse of a symmetric key*} |
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41 |
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42 specification (invKey) |
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43 invKey [simp]: "invKey (invKey K) = K" |
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44 invKey_symmetric: "all_symmetric --> invKey = id" |
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45 by (rule exI [of _ id], auto) |
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46 |
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47 |
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48 text{*The inverse of a symmetric key is itself; that of a public key |
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49 is the private key and vice versa*} |
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50 |
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51 constdefs |
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52 symKeys :: "key set" |
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53 "symKeys == {K. invKey K = K}" |
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54 |
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55 text{*Agents. We allow any number of certification authorities, cardholders |
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56 merchants, and payment gateways.*} |
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57 datatype |
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58 agent = CA nat | Cardholder nat | Merchant nat | PG nat | Spy |
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59 |
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60 text{*Messages*} |
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61 datatype |
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62 msg = Agent agent --{*Agent names*} |
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63 | Number nat --{*Ordinary integers, timestamps, ...*} |
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64 | Nonce nat --{*Unguessable nonces*} |
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65 | Pan nat --{*Unguessable Primary Account Numbers (??)*} |
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66 | Key key --{*Crypto keys*} |
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67 | Hash msg --{*Hashing*} |
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68 | MPair msg msg --{*Compound messages*} |
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69 | Crypt key msg --{*Encryption, public- or shared-key*} |
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70 |
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71 |
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72 (*Concrete syntax: messages appear as {|A,B,NA|}, etc...*) |
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73 syntax |
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74 "@MTuple" :: "['a, args] => 'a * 'b" ("(2{|_,/ _|})") |
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75 |
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76 syntax (xsymbols) |
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77 "@MTuple" :: "['a, args] => 'a * 'b" ("(2\<lbrace>_,/ _\<rbrace>)") |
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78 |
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79 translations |
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80 "{|x, y, z|}" == "{|x, {|y, z|}|}" |
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81 "{|x, y|}" == "MPair x y" |
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82 |
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83 |
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84 constdefs |
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85 nat_of_agent :: "agent => nat" |
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86 "nat_of_agent == agent_case (curry nat2_to_nat 0) |
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87 (curry nat2_to_nat 1) |
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88 (curry nat2_to_nat 2) |
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89 (curry nat2_to_nat 3) |
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90 (nat2_to_nat (4,0))" |
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91 --{*maps each agent to a unique natural number, for specifications*} |
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92 |
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93 text{*The function is indeed injective*} |
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94 lemma inj_nat_of_agent: "inj nat_of_agent" |
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95 by (simp add: nat_of_agent_def inj_on_def curry_def |
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96 nat2_to_nat_inj [THEN inj_eq] split: agent.split) |
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97 |
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98 |
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99 constdefs |
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100 (*Keys useful to decrypt elements of a message set*) |
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101 keysFor :: "msg set => key set" |
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102 "keysFor H == invKey ` {K. \<exists>X. Crypt K X \<in> H}" |
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103 |
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104 subsubsection{*Inductive definition of all "parts" of a message.*} |
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105 |
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106 inductive_set |
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107 parts :: "msg set => msg set" |
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108 for H :: "msg set" |
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109 where |
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110 Inj [intro]: "X \<in> H ==> X \<in> parts H" |
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111 | Fst: "{|X,Y|} \<in> parts H ==> X \<in> parts H" |
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112 | Snd: "{|X,Y|} \<in> parts H ==> Y \<in> parts H" |
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113 | Body: "Crypt K X \<in> parts H ==> X \<in> parts H" |
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114 |
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115 |
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116 (*Monotonicity*) |
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117 lemma parts_mono: "G<=H ==> parts(G) <= parts(H)" |
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118 apply auto |
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119 apply (erule parts.induct) |
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120 apply (auto dest: Fst Snd Body) |
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121 done |
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122 |
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123 |
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124 subsubsection{*Inverse of keys*} |
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125 |
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126 (*Equations hold because constructors are injective; cannot prove for all f*) |
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127 lemma Key_image_eq [simp]: "(Key x \<in> Key`A) = (x\<in>A)" |
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128 by auto |
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129 |
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130 lemma Nonce_Key_image_eq [simp]: "(Nonce x \<notin> Key`A)" |
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131 by auto |
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132 |
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133 lemma Cardholder_image_eq [simp]: "(Cardholder x \<in> Cardholder`A) = (x \<in> A)" |
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134 by auto |
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135 |
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136 lemma CA_image_eq [simp]: "(CA x \<in> CA`A) = (x \<in> A)" |
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137 by auto |
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138 |
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139 lemma Pan_image_eq [simp]: "(Pan x \<in> Pan`A) = (x \<in> A)" |
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140 by auto |
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141 |
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142 lemma Pan_Key_image_eq [simp]: "(Pan x \<notin> Key`A)" |
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143 by auto |
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144 |
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145 lemma Nonce_Pan_image_eq [simp]: "(Nonce x \<notin> Pan`A)" |
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146 by auto |
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147 |
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148 lemma invKey_eq [simp]: "(invKey K = invKey K') = (K=K')" |
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149 apply safe |
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150 apply (drule_tac f = invKey in arg_cong, simp) |
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151 done |
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152 |
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153 |
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154 subsection{*keysFor operator*} |
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155 |
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156 lemma keysFor_empty [simp]: "keysFor {} = {}" |
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157 by (unfold keysFor_def, blast) |
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158 |
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159 lemma keysFor_Un [simp]: "keysFor (H \<union> H') = keysFor H \<union> keysFor H'" |
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160 by (unfold keysFor_def, blast) |
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161 |
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162 lemma keysFor_UN [simp]: "keysFor (\<Union>i\<in>A. H i) = (\<Union>i\<in>A. keysFor (H i))" |
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163 by (unfold keysFor_def, blast) |
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164 |
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165 (*Monotonicity*) |
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166 lemma keysFor_mono: "G\<subseteq>H ==> keysFor(G) \<subseteq> keysFor(H)" |
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167 by (unfold keysFor_def, blast) |
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168 |
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169 lemma keysFor_insert_Agent [simp]: "keysFor (insert (Agent A) H) = keysFor H" |
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170 by (unfold keysFor_def, auto) |
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171 |
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172 lemma keysFor_insert_Nonce [simp]: "keysFor (insert (Nonce N) H) = keysFor H" |
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173 by (unfold keysFor_def, auto) |
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174 |
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175 lemma keysFor_insert_Number [simp]: "keysFor (insert (Number N) H) = keysFor H" |
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176 by (unfold keysFor_def, auto) |
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177 |
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178 lemma keysFor_insert_Key [simp]: "keysFor (insert (Key K) H) = keysFor H" |
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179 by (unfold keysFor_def, auto) |
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180 |
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181 lemma keysFor_insert_Pan [simp]: "keysFor (insert (Pan A) H) = keysFor H" |
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182 by (unfold keysFor_def, auto) |
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183 |
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184 lemma keysFor_insert_Hash [simp]: "keysFor (insert (Hash X) H) = keysFor H" |
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185 by (unfold keysFor_def, auto) |
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186 |
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187 lemma keysFor_insert_MPair [simp]: "keysFor (insert {|X,Y|} H) = keysFor H" |
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188 by (unfold keysFor_def, auto) |
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189 |
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190 lemma keysFor_insert_Crypt [simp]: |
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191 "keysFor (insert (Crypt K X) H) = insert (invKey K) (keysFor H)" |
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192 by (unfold keysFor_def, auto) |
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193 |
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194 lemma keysFor_image_Key [simp]: "keysFor (Key`E) = {}" |
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195 by (unfold keysFor_def, auto) |
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196 |
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197 lemma Crypt_imp_invKey_keysFor: "Crypt K X \<in> H ==> invKey K \<in> keysFor H" |
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198 by (unfold keysFor_def, blast) |
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199 |
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200 |
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201 subsection{*Inductive relation "parts"*} |
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202 |
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203 lemma MPair_parts: |
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204 "[| {|X,Y|} \<in> parts H; |
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205 [| X \<in> parts H; Y \<in> parts H |] ==> P |] ==> P" |
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206 by (blast dest: parts.Fst parts.Snd) |
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207 |
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208 declare MPair_parts [elim!] parts.Body [dest!] |
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209 text{*NB These two rules are UNSAFE in the formal sense, as they discard the |
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210 compound message. They work well on THIS FILE. |
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211 @{text MPair_parts} is left as SAFE because it speeds up proofs. |
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212 The Crypt rule is normally kept UNSAFE to avoid breaking up certificates.*} |
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213 |
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214 lemma parts_increasing: "H \<subseteq> parts(H)" |
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215 by blast |
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216 |
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217 lemmas parts_insertI = subset_insertI [THEN parts_mono, THEN subsetD, standard] |
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218 |
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219 lemma parts_empty [simp]: "parts{} = {}" |
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220 apply safe |
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221 apply (erule parts.induct, blast+) |
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222 done |
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223 |
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224 lemma parts_emptyE [elim!]: "X\<in> parts{} ==> P" |
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225 by simp |
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226 |
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227 (*WARNING: loops if H = {Y}, therefore must not be repeated!*) |
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228 lemma parts_singleton: "X\<in> parts H ==> \<exists>Y\<in>H. X\<in> parts {Y}" |
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229 by (erule parts.induct, fast+) |
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230 |
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231 |
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232 subsubsection{*Unions*} |
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233 |
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234 lemma parts_Un_subset1: "parts(G) \<union> parts(H) \<subseteq> parts(G \<union> H)" |
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235 by (intro Un_least parts_mono Un_upper1 Un_upper2) |
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236 |
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237 lemma parts_Un_subset2: "parts(G \<union> H) \<subseteq> parts(G) \<union> parts(H)" |
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238 apply (rule subsetI) |
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239 apply (erule parts.induct, blast+) |
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240 done |
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241 |
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242 lemma parts_Un [simp]: "parts(G \<union> H) = parts(G) \<union> parts(H)" |
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243 by (intro equalityI parts_Un_subset1 parts_Un_subset2) |
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244 |
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245 lemma parts_insert: "parts (insert X H) = parts {X} \<union> parts H" |
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246 apply (subst insert_is_Un [of _ H]) |
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247 apply (simp only: parts_Un) |
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248 done |
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249 |
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250 (*TWO inserts to avoid looping. This rewrite is better than nothing. |
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251 Not suitable for Addsimps: its behaviour can be strange.*) |
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252 lemma parts_insert2: |
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253 "parts (insert X (insert Y H)) = parts {X} \<union> parts {Y} \<union> parts H" |
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254 apply (simp add: Un_assoc) |
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255 apply (simp add: parts_insert [symmetric]) |
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256 done |
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257 |
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258 lemma parts_UN_subset1: "(\<Union>x\<in>A. parts(H x)) \<subseteq> parts(\<Union>x\<in>A. H x)" |
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259 by (intro UN_least parts_mono UN_upper) |
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260 |
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261 lemma parts_UN_subset2: "parts(\<Union>x\<in>A. H x) \<subseteq> (\<Union>x\<in>A. parts(H x))" |
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262 apply (rule subsetI) |
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263 apply (erule parts.induct, blast+) |
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264 done |
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265 |
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266 lemma parts_UN [simp]: "parts(\<Union>x\<in>A. H x) = (\<Union>x\<in>A. parts(H x))" |
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267 by (intro equalityI parts_UN_subset1 parts_UN_subset2) |
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268 |
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269 (*Added to simplify arguments to parts, analz and synth. |
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270 NOTE: the UN versions are no longer used!*) |
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271 |
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272 |
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273 text{*This allows @{text blast} to simplify occurrences of |
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274 @{term "parts(G\<union>H)"} in the assumption.*} |
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275 declare parts_Un [THEN equalityD1, THEN subsetD, THEN UnE, elim!] |
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276 |
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277 |
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278 lemma parts_insert_subset: "insert X (parts H) \<subseteq> parts(insert X H)" |
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279 by (blast intro: parts_mono [THEN [2] rev_subsetD]) |
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280 |
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281 subsubsection{*Idempotence and transitivity*} |
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282 |
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283 lemma parts_partsD [dest!]: "X\<in> parts (parts H) ==> X\<in> parts H" |
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284 by (erule parts.induct, blast+) |
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285 |
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286 lemma parts_idem [simp]: "parts (parts H) = parts H" |
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287 by blast |
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288 |
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289 lemma parts_trans: "[| X\<in> parts G; G \<subseteq> parts H |] ==> X\<in> parts H" |
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290 by (drule parts_mono, blast) |
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291 |
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292 (*Cut*) |
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293 lemma parts_cut: |
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294 "[| Y\<in> parts (insert X G); X\<in> parts H |] ==> Y\<in> parts (G \<union> H)" |
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295 by (erule parts_trans, auto) |
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296 |
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297 lemma parts_cut_eq [simp]: "X\<in> parts H ==> parts (insert X H) = parts H" |
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298 by (force dest!: parts_cut intro: parts_insertI) |
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299 |
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300 |
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301 subsubsection{*Rewrite rules for pulling out atomic messages*} |
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302 |
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303 lemmas parts_insert_eq_I = equalityI [OF subsetI parts_insert_subset] |
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304 |
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305 |
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306 lemma parts_insert_Agent [simp]: |
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307 "parts (insert (Agent agt) H) = insert (Agent agt) (parts H)" |
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308 apply (rule parts_insert_eq_I) |
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309 apply (erule parts.induct, auto) |
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310 done |
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311 |
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312 lemma parts_insert_Nonce [simp]: |
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313 "parts (insert (Nonce N) H) = insert (Nonce N) (parts H)" |
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314 apply (rule parts_insert_eq_I) |
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315 apply (erule parts.induct, auto) |
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316 done |
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317 |
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318 lemma parts_insert_Number [simp]: |
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319 "parts (insert (Number N) H) = insert (Number N) (parts H)" |
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320 apply (rule parts_insert_eq_I) |
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321 apply (erule parts.induct, auto) |
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322 done |
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323 |
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324 lemma parts_insert_Key [simp]: |
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325 "parts (insert (Key K) H) = insert (Key K) (parts H)" |
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326 apply (rule parts_insert_eq_I) |
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327 apply (erule parts.induct, auto) |
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328 done |
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329 |
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330 lemma parts_insert_Pan [simp]: |
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331 "parts (insert (Pan A) H) = insert (Pan A) (parts H)" |
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332 apply (rule parts_insert_eq_I) |
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333 apply (erule parts.induct, auto) |
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334 done |
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335 |
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336 lemma parts_insert_Hash [simp]: |
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337 "parts (insert (Hash X) H) = insert (Hash X) (parts H)" |
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338 apply (rule parts_insert_eq_I) |
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339 apply (erule parts.induct, auto) |
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340 done |
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341 |
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342 lemma parts_insert_Crypt [simp]: |
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343 "parts (insert (Crypt K X) H) = |
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344 insert (Crypt K X) (parts (insert X H))" |
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345 apply (rule equalityI) |
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346 apply (rule subsetI) |
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347 apply (erule parts.induct, auto) |
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348 apply (erule parts.induct) |
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349 apply (blast intro: parts.Body)+ |
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350 done |
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351 |
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352 lemma parts_insert_MPair [simp]: |
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353 "parts (insert {|X,Y|} H) = |
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354 insert {|X,Y|} (parts (insert X (insert Y H)))" |
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355 apply (rule equalityI) |
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356 apply (rule subsetI) |
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357 apply (erule parts.induct, auto) |
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358 apply (erule parts.induct) |
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359 apply (blast intro: parts.Fst parts.Snd)+ |
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360 done |
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361 |
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362 lemma parts_image_Key [simp]: "parts (Key`N) = Key`N" |
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363 apply auto |
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364 apply (erule parts.induct, auto) |
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365 done |
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366 |
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367 lemma parts_image_Pan [simp]: "parts (Pan`A) = Pan`A" |
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368 apply auto |
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369 apply (erule parts.induct, auto) |
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370 done |
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371 |
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372 |
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373 (*In any message, there is an upper bound N on its greatest nonce.*) |
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374 lemma msg_Nonce_supply: "\<exists>N. \<forall>n. N\<le>n --> Nonce n \<notin> parts {msg}" |
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375 apply (induct_tac "msg") |
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376 apply (simp_all (no_asm_simp) add: exI parts_insert2) |
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377 (*MPair case: blast_tac works out the necessary sum itself!*) |
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378 prefer 2 apply (blast elim!: add_leE) |
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379 (*Nonce case*) |
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380 apply (rule_tac x = "N + Suc nat" in exI) |
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381 apply (auto elim!: add_leE) |
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382 done |
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383 |
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384 (* Ditto, for numbers.*) |
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385 lemma msg_Number_supply: "\<exists>N. \<forall>n. N<=n --> Number n \<notin> parts {msg}" |
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386 apply (induct_tac "msg") |
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387 apply (simp_all (no_asm_simp) add: exI parts_insert2) |
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388 prefer 2 apply (blast elim!: add_leE) |
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389 apply (rule_tac x = "N + Suc nat" in exI, auto) |
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390 done |
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391 |
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392 subsection{*Inductive relation "analz"*} |
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393 |
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394 text{*Inductive definition of "analz" -- what can be broken down from a set of |
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395 messages, including keys. A form of downward closure. Pairs can |
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396 be taken apart; messages decrypted with known keys.*} |
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397 |
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398 inductive_set |
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399 analz :: "msg set => msg set" |
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400 for H :: "msg set" |
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401 where |
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402 Inj [intro,simp] : "X \<in> H ==> X \<in> analz H" |
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403 | Fst: "{|X,Y|} \<in> analz H ==> X \<in> analz H" |
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404 | Snd: "{|X,Y|} \<in> analz H ==> Y \<in> analz H" |
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405 | Decrypt [dest]: |
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406 "[|Crypt K X \<in> analz H; Key(invKey K): analz H|] ==> X \<in> analz H" |
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407 |
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408 |
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409 (*Monotonicity; Lemma 1 of Lowe's paper*) |
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410 lemma analz_mono: "G<=H ==> analz(G) <= analz(H)" |
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411 apply auto |
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412 apply (erule analz.induct) |
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413 apply (auto dest: Fst Snd) |
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414 done |
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415 |
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416 text{*Making it safe speeds up proofs*} |
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417 lemma MPair_analz [elim!]: |
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418 "[| {|X,Y|} \<in> analz H; |
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419 [| X \<in> analz H; Y \<in> analz H |] ==> P |
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420 |] ==> P" |
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421 by (blast dest: analz.Fst analz.Snd) |
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422 |
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423 lemma analz_increasing: "H \<subseteq> analz(H)" |
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424 by blast |
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425 |
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426 lemma analz_subset_parts: "analz H \<subseteq> parts H" |
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427 apply (rule subsetI) |
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428 apply (erule analz.induct, blast+) |
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429 done |
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430 |
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431 lemmas analz_into_parts = analz_subset_parts [THEN subsetD, standard] |
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432 |
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433 lemmas not_parts_not_analz = analz_subset_parts [THEN contra_subsetD, standard] |
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434 |
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435 |
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436 lemma parts_analz [simp]: "parts (analz H) = parts H" |
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437 apply (rule equalityI) |
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438 apply (rule analz_subset_parts [THEN parts_mono, THEN subset_trans], simp) |
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439 apply (blast intro: analz_increasing [THEN parts_mono, THEN subsetD]) |
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440 done |
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441 |
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442 lemma analz_parts [simp]: "analz (parts H) = parts H" |
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443 apply auto |
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444 apply (erule analz.induct, auto) |
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445 done |
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446 |
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447 lemmas analz_insertI = subset_insertI [THEN analz_mono, THEN [2] rev_subsetD, standard] |
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448 |
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449 subsubsection{*General equational properties*} |
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450 |
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451 lemma analz_empty [simp]: "analz{} = {}" |
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452 apply safe |
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453 apply (erule analz.induct, blast+) |
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454 done |
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455 |
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456 (*Converse fails: we can analz more from the union than from the |
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457 separate parts, as a key in one might decrypt a message in the other*) |
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458 lemma analz_Un: "analz(G) \<union> analz(H) \<subseteq> analz(G \<union> H)" |
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459 by (intro Un_least analz_mono Un_upper1 Un_upper2) |
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460 |
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461 lemma analz_insert: "insert X (analz H) \<subseteq> analz(insert X H)" |
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462 by (blast intro: analz_mono [THEN [2] rev_subsetD]) |
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463 |
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464 subsubsection{*Rewrite rules for pulling out atomic messages*} |
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465 |
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466 lemmas analz_insert_eq_I = equalityI [OF subsetI analz_insert] |
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467 |
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468 lemma analz_insert_Agent [simp]: |
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469 "analz (insert (Agent agt) H) = insert (Agent agt) (analz H)" |
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470 apply (rule analz_insert_eq_I) |
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471 apply (erule analz.induct, auto) |
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472 done |
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473 |
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474 lemma analz_insert_Nonce [simp]: |
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475 "analz (insert (Nonce N) H) = insert (Nonce N) (analz H)" |
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476 apply (rule analz_insert_eq_I) |
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477 apply (erule analz.induct, auto) |
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478 done |
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479 |
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480 lemma analz_insert_Number [simp]: |
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481 "analz (insert (Number N) H) = insert (Number N) (analz H)" |
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482 apply (rule analz_insert_eq_I) |
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483 apply (erule analz.induct, auto) |
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484 done |
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485 |
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486 lemma analz_insert_Hash [simp]: |
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487 "analz (insert (Hash X) H) = insert (Hash X) (analz H)" |
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488 apply (rule analz_insert_eq_I) |
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489 apply (erule analz.induct, auto) |
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490 done |
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491 |
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492 (*Can only pull out Keys if they are not needed to decrypt the rest*) |
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493 lemma analz_insert_Key [simp]: |
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494 "K \<notin> keysFor (analz H) ==> |
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495 analz (insert (Key K) H) = insert (Key K) (analz H)" |
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496 apply (unfold keysFor_def) |
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497 apply (rule analz_insert_eq_I) |
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498 apply (erule analz.induct, auto) |
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499 done |
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500 |
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501 lemma analz_insert_MPair [simp]: |
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502 "analz (insert {|X,Y|} H) = |
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503 insert {|X,Y|} (analz (insert X (insert Y H)))" |
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504 apply (rule equalityI) |
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505 apply (rule subsetI) |
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506 apply (erule analz.induct, auto) |
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507 apply (erule analz.induct) |
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508 apply (blast intro: analz.Fst analz.Snd)+ |
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509 done |
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510 |
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511 (*Can pull out enCrypted message if the Key is not known*) |
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512 lemma analz_insert_Crypt: |
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513 "Key (invKey K) \<notin> analz H |
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514 ==> analz (insert (Crypt K X) H) = insert (Crypt K X) (analz H)" |
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515 apply (rule analz_insert_eq_I) |
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516 apply (erule analz.induct, auto) |
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517 done |
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518 |
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519 lemma analz_insert_Pan [simp]: |
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520 "analz (insert (Pan A) H) = insert (Pan A) (analz H)" |
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521 apply (rule analz_insert_eq_I) |
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522 apply (erule analz.induct, auto) |
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523 done |
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524 |
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525 lemma lemma1: "Key (invKey K) \<in> analz H ==> |
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526 analz (insert (Crypt K X) H) \<subseteq> |
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527 insert (Crypt K X) (analz (insert X H))" |
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528 apply (rule subsetI) |
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529 apply (erule_tac x = x in analz.induct, auto) |
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530 done |
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531 |
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532 lemma lemma2: "Key (invKey K) \<in> analz H ==> |
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533 insert (Crypt K X) (analz (insert X H)) \<subseteq> |
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534 analz (insert (Crypt K X) H)" |
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535 apply auto |
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536 apply (erule_tac x = x in analz.induct, auto) |
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537 apply (blast intro: analz_insertI analz.Decrypt) |
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538 done |
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539 |
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540 lemma analz_insert_Decrypt: |
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541 "Key (invKey K) \<in> analz H ==> |
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542 analz (insert (Crypt K X) H) = |
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543 insert (Crypt K X) (analz (insert X H))" |
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544 by (intro equalityI lemma1 lemma2) |
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545 |
|
546 (*Case analysis: either the message is secure, or it is not! |
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547 Effective, but can cause subgoals to blow up! |
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548 Use with split_if; apparently split_tac does not cope with patterns |
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549 such as "analz (insert (Crypt K X) H)" *) |
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550 lemma analz_Crypt_if [simp]: |
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551 "analz (insert (Crypt K X) H) = |
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552 (if (Key (invKey K) \<in> analz H) |
|
553 then insert (Crypt K X) (analz (insert X H)) |
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554 else insert (Crypt K X) (analz H))" |
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555 by (simp add: analz_insert_Crypt analz_insert_Decrypt) |
|
556 |
|
557 |
|
558 (*This rule supposes "for the sake of argument" that we have the key.*) |
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559 lemma analz_insert_Crypt_subset: |
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560 "analz (insert (Crypt K X) H) \<subseteq> |
|
561 insert (Crypt K X) (analz (insert X H))" |
|
562 apply (rule subsetI) |
|
563 apply (erule analz.induct, auto) |
|
564 done |
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565 |
|
566 lemma analz_image_Key [simp]: "analz (Key`N) = Key`N" |
|
567 apply auto |
|
568 apply (erule analz.induct, auto) |
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569 done |
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570 |
|
571 lemma analz_image_Pan [simp]: "analz (Pan`A) = Pan`A" |
|
572 apply auto |
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573 apply (erule analz.induct, auto) |
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574 done |
|
575 |
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576 |
|
577 subsubsection{*Idempotence and transitivity*} |
|
578 |
|
579 lemma analz_analzD [dest!]: "X\<in> analz (analz H) ==> X\<in> analz H" |
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580 by (erule analz.induct, blast+) |
|
581 |
|
582 lemma analz_idem [simp]: "analz (analz H) = analz H" |
|
583 by blast |
|
584 |
|
585 lemma analz_trans: "[| X\<in> analz G; G \<subseteq> analz H |] ==> X\<in> analz H" |
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586 by (drule analz_mono, blast) |
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587 |
|
588 (*Cut; Lemma 2 of Lowe*) |
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589 lemma analz_cut: "[| Y\<in> analz (insert X H); X\<in> analz H |] ==> Y\<in> analz H" |
|
590 by (erule analz_trans, blast) |
|
591 |
|
592 (*Cut can be proved easily by induction on |
|
593 "Y: analz (insert X H) ==> X: analz H --> Y: analz H" |
|
594 *) |
|
595 |
|
596 (*This rewrite rule helps in the simplification of messages that involve |
|
597 the forwarding of unknown components (X). Without it, removing occurrences |
|
598 of X can be very complicated. *) |
|
599 lemma analz_insert_eq: "X\<in> analz H ==> analz (insert X H) = analz H" |
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600 by (blast intro: analz_cut analz_insertI) |
|
601 |
|
602 |
|
603 text{*A congruence rule for "analz"*} |
|
604 |
|
605 lemma analz_subset_cong: |
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606 "[| analz G \<subseteq> analz G'; analz H \<subseteq> analz H' |
|
607 |] ==> analz (G \<union> H) \<subseteq> analz (G' \<union> H')" |
|
608 apply clarify |
|
609 apply (erule analz.induct) |
|
610 apply (best intro: analz_mono [THEN subsetD])+ |
|
611 done |
|
612 |
|
613 lemma analz_cong: |
|
614 "[| analz G = analz G'; analz H = analz H' |
|
615 |] ==> analz (G \<union> H) = analz (G' \<union> H')" |
|
616 by (intro equalityI analz_subset_cong, simp_all) |
|
617 |
|
618 lemma analz_insert_cong: |
|
619 "analz H = analz H' ==> analz(insert X H) = analz(insert X H')" |
|
620 by (force simp only: insert_def intro!: analz_cong) |
|
621 |
|
622 (*If there are no pairs or encryptions then analz does nothing*) |
|
623 lemma analz_trivial: |
|
624 "[| \<forall>X Y. {|X,Y|} \<notin> H; \<forall>X K. Crypt K X \<notin> H |] ==> analz H = H" |
|
625 apply safe |
|
626 apply (erule analz.induct, blast+) |
|
627 done |
|
628 |
|
629 (*These two are obsolete (with a single Spy) but cost little to prove...*) |
|
630 lemma analz_UN_analz_lemma: |
|
631 "X\<in> analz (\<Union>i\<in>A. analz (H i)) ==> X\<in> analz (\<Union>i\<in>A. H i)" |
|
632 apply (erule analz.induct) |
|
633 apply (blast intro: analz_mono [THEN [2] rev_subsetD])+ |
|
634 done |
|
635 |
|
636 lemma analz_UN_analz [simp]: "analz (\<Union>i\<in>A. analz (H i)) = analz (\<Union>i\<in>A. H i)" |
|
637 by (blast intro: analz_UN_analz_lemma analz_mono [THEN [2] rev_subsetD]) |
|
638 |
|
639 |
|
640 subsection{*Inductive relation "synth"*} |
|
641 |
|
642 text{*Inductive definition of "synth" -- what can be built up from a set of |
|
643 messages. A form of upward closure. Pairs can be built, messages |
|
644 encrypted with known keys. Agent names are public domain. |
|
645 Numbers can be guessed, but Nonces cannot be.*} |
|
646 |
|
647 inductive_set |
|
648 synth :: "msg set => msg set" |
|
649 for H :: "msg set" |
|
650 where |
|
651 Inj [intro]: "X \<in> H ==> X \<in> synth H" |
|
652 | Agent [intro]: "Agent agt \<in> synth H" |
|
653 | Number [intro]: "Number n \<in> synth H" |
|
654 | Hash [intro]: "X \<in> synth H ==> Hash X \<in> synth H" |
|
655 | MPair [intro]: "[|X \<in> synth H; Y \<in> synth H|] ==> {|X,Y|} \<in> synth H" |
|
656 | Crypt [intro]: "[|X \<in> synth H; Key(K) \<in> H|] ==> Crypt K X \<in> synth H" |
|
657 |
|
658 (*Monotonicity*) |
|
659 lemma synth_mono: "G<=H ==> synth(G) <= synth(H)" |
|
660 apply auto |
|
661 apply (erule synth.induct) |
|
662 apply (auto dest: Fst Snd Body) |
|
663 done |
|
664 |
|
665 (*NO Agent_synth, as any Agent name can be synthesized. Ditto for Number*) |
|
666 inductive_cases Nonce_synth [elim!]: "Nonce n \<in> synth H" |
|
667 inductive_cases Key_synth [elim!]: "Key K \<in> synth H" |
|
668 inductive_cases Hash_synth [elim!]: "Hash X \<in> synth H" |
|
669 inductive_cases MPair_synth [elim!]: "{|X,Y|} \<in> synth H" |
|
670 inductive_cases Crypt_synth [elim!]: "Crypt K X \<in> synth H" |
|
671 inductive_cases Pan_synth [elim!]: "Pan A \<in> synth H" |
|
672 |
|
673 |
|
674 lemma synth_increasing: "H \<subseteq> synth(H)" |
|
675 by blast |
|
676 |
|
677 subsubsection{*Unions*} |
|
678 |
|
679 (*Converse fails: we can synth more from the union than from the |
|
680 separate parts, building a compound message using elements of each.*) |
|
681 lemma synth_Un: "synth(G) \<union> synth(H) \<subseteq> synth(G \<union> H)" |
|
682 by (intro Un_least synth_mono Un_upper1 Un_upper2) |
|
683 |
|
684 lemma synth_insert: "insert X (synth H) \<subseteq> synth(insert X H)" |
|
685 by (blast intro: synth_mono [THEN [2] rev_subsetD]) |
|
686 |
|
687 subsubsection{*Idempotence and transitivity*} |
|
688 |
|
689 lemma synth_synthD [dest!]: "X\<in> synth (synth H) ==> X\<in> synth H" |
|
690 by (erule synth.induct, blast+) |
|
691 |
|
692 lemma synth_idem: "synth (synth H) = synth H" |
|
693 by blast |
|
694 |
|
695 lemma synth_trans: "[| X\<in> synth G; G \<subseteq> synth H |] ==> X\<in> synth H" |
|
696 by (drule synth_mono, blast) |
|
697 |
|
698 (*Cut; Lemma 2 of Lowe*) |
|
699 lemma synth_cut: "[| Y\<in> synth (insert X H); X\<in> synth H |] ==> Y\<in> synth H" |
|
700 by (erule synth_trans, blast) |
|
701 |
|
702 lemma Agent_synth [simp]: "Agent A \<in> synth H" |
|
703 by blast |
|
704 |
|
705 lemma Number_synth [simp]: "Number n \<in> synth H" |
|
706 by blast |
|
707 |
|
708 lemma Nonce_synth_eq [simp]: "(Nonce N \<in> synth H) = (Nonce N \<in> H)" |
|
709 by blast |
|
710 |
|
711 lemma Key_synth_eq [simp]: "(Key K \<in> synth H) = (Key K \<in> H)" |
|
712 by blast |
|
713 |
|
714 lemma Crypt_synth_eq [simp]: "Key K \<notin> H ==> (Crypt K X \<in> synth H) = (Crypt K X \<in> H)" |
|
715 by blast |
|
716 |
|
717 lemma Pan_synth_eq [simp]: "(Pan A \<in> synth H) = (Pan A \<in> H)" |
|
718 by blast |
|
719 |
|
720 lemma keysFor_synth [simp]: |
|
721 "keysFor (synth H) = keysFor H \<union> invKey`{K. Key K \<in> H}" |
|
722 by (unfold keysFor_def, blast) |
|
723 |
|
724 |
|
725 subsubsection{*Combinations of parts, analz and synth*} |
|
726 |
|
727 lemma parts_synth [simp]: "parts (synth H) = parts H \<union> synth H" |
|
728 apply (rule equalityI) |
|
729 apply (rule subsetI) |
|
730 apply (erule parts.induct) |
|
731 apply (blast intro: synth_increasing [THEN parts_mono, THEN subsetD] |
|
732 parts.Fst parts.Snd parts.Body)+ |
|
733 done |
|
734 |
|
735 lemma analz_analz_Un [simp]: "analz (analz G \<union> H) = analz (G \<union> H)" |
|
736 apply (intro equalityI analz_subset_cong)+ |
|
737 apply simp_all |
|
738 done |
|
739 |
|
740 lemma analz_synth_Un [simp]: "analz (synth G \<union> H) = analz (G \<union> H) \<union> synth G" |
|
741 apply (rule equalityI) |
|
742 apply (rule subsetI) |
|
743 apply (erule analz.induct) |
|
744 prefer 5 apply (blast intro: analz_mono [THEN [2] rev_subsetD]) |
|
745 apply (blast intro: analz.Fst analz.Snd analz.Decrypt)+ |
|
746 done |
|
747 |
|
748 lemma analz_synth [simp]: "analz (synth H) = analz H \<union> synth H" |
|
749 apply (cut_tac H = "{}" in analz_synth_Un) |
|
750 apply (simp (no_asm_use)) |
|
751 done |
|
752 |
|
753 |
|
754 subsubsection{*For reasoning about the Fake rule in traces*} |
|
755 |
|
756 lemma parts_insert_subset_Un: "X\<in> G ==> parts(insert X H) \<subseteq> parts G \<union> parts H" |
|
757 by (rule subset_trans [OF parts_mono parts_Un_subset2], blast) |
|
758 |
|
759 (*More specifically for Fake. Very occasionally we could do with a version |
|
760 of the form parts{X} \<subseteq> synth (analz H) \<union> parts H *) |
|
761 lemma Fake_parts_insert: "X \<in> synth (analz H) ==> |
|
762 parts (insert X H) \<subseteq> synth (analz H) \<union> parts H" |
|
763 apply (drule parts_insert_subset_Un) |
|
764 apply (simp (no_asm_use)) |
|
765 apply blast |
|
766 done |
|
767 |
|
768 lemma Fake_parts_insert_in_Un: |
|
769 "[|Z \<in> parts (insert X H); X: synth (analz H)|] |
|
770 ==> Z \<in> synth (analz H) \<union> parts H"; |
|
771 by (blast dest: Fake_parts_insert [THEN subsetD, dest]) |
|
772 |
|
773 (*H is sometimes (Key ` KK \<union> spies evs), so can't put G=H*) |
|
774 lemma Fake_analz_insert: |
|
775 "X\<in> synth (analz G) ==> |
|
776 analz (insert X H) \<subseteq> synth (analz G) \<union> analz (G \<union> H)" |
|
777 apply (rule subsetI) |
|
778 apply (subgoal_tac "x \<in> analz (synth (analz G) \<union> H) ") |
|
779 prefer 2 apply (blast intro: analz_mono [THEN [2] rev_subsetD] analz_mono [THEN synth_mono, THEN [2] rev_subsetD]) |
|
780 apply (simp (no_asm_use)) |
|
781 apply blast |
|
782 done |
|
783 |
|
784 lemma analz_conj_parts [simp]: |
|
785 "(X \<in> analz H & X \<in> parts H) = (X \<in> analz H)" |
|
786 by (blast intro: analz_subset_parts [THEN subsetD]) |
|
787 |
|
788 lemma analz_disj_parts [simp]: |
|
789 "(X \<in> analz H | X \<in> parts H) = (X \<in> parts H)" |
|
790 by (blast intro: analz_subset_parts [THEN subsetD]) |
|
791 |
|
792 (*Without this equation, other rules for synth and analz would yield |
|
793 redundant cases*) |
|
794 lemma MPair_synth_analz [iff]: |
|
795 "({|X,Y|} \<in> synth (analz H)) = |
|
796 (X \<in> synth (analz H) & Y \<in> synth (analz H))" |
|
797 by blast |
|
798 |
|
799 lemma Crypt_synth_analz: |
|
800 "[| Key K \<in> analz H; Key (invKey K) \<in> analz H |] |
|
801 ==> (Crypt K X \<in> synth (analz H)) = (X \<in> synth (analz H))" |
|
802 by blast |
|
803 |
|
804 |
|
805 lemma Hash_synth_analz [simp]: |
|
806 "X \<notin> synth (analz H) |
|
807 ==> (Hash{|X,Y|} \<in> synth (analz H)) = (Hash{|X,Y|} \<in> analz H)" |
|
808 by blast |
|
809 |
|
810 |
|
811 (*We do NOT want Crypt... messages broken up in protocols!!*) |
|
812 declare parts.Body [rule del] |
|
813 |
|
814 |
|
815 text{*Rewrites to push in Key and Crypt messages, so that other messages can |
|
816 be pulled out using the @{text analz_insert} rules*} |
|
817 |
|
818 lemmas pushKeys [standard] = |
|
819 insert_commute [of "Key K" "Agent C"] |
|
820 insert_commute [of "Key K" "Nonce N"] |
|
821 insert_commute [of "Key K" "Number N"] |
|
822 insert_commute [of "Key K" "Pan PAN"] |
|
823 insert_commute [of "Key K" "Hash X"] |
|
824 insert_commute [of "Key K" "MPair X Y"] |
|
825 insert_commute [of "Key K" "Crypt X K'"] |
|
826 |
|
827 lemmas pushCrypts [standard] = |
|
828 insert_commute [of "Crypt X K" "Agent C"] |
|
829 insert_commute [of "Crypt X K" "Nonce N"] |
|
830 insert_commute [of "Crypt X K" "Number N"] |
|
831 insert_commute [of "Crypt X K" "Pan PAN"] |
|
832 insert_commute [of "Crypt X K" "Hash X'"] |
|
833 insert_commute [of "Crypt X K" "MPair X' Y"] |
|
834 |
|
835 text{*Cannot be added with @{text "[simp]"} -- messages should not always be |
|
836 re-ordered.*} |
|
837 lemmas pushes = pushKeys pushCrypts |
|
838 |
|
839 |
|
840 subsection{*Tactics useful for many protocol proofs*} |
|
841 (*<*) |
|
842 ML |
|
843 {* |
|
844 structure MessageSET = |
|
845 struct |
|
846 |
|
847 (*Prove base case (subgoal i) and simplify others. A typical base case |
|
848 concerns Crypt K X \<notin> Key`shrK`bad and cannot be proved by rewriting |
|
849 alone.*) |
|
850 fun prove_simple_subgoals_tac (cs, ss) i = |
|
851 force_tac (cs, ss addsimps [@{thm image_eq_UN}]) i THEN |
|
852 ALLGOALS (asm_simp_tac ss) |
|
853 |
|
854 (*Analysis of Fake cases. Also works for messages that forward unknown parts, |
|
855 but this application is no longer necessary if analz_insert_eq is used. |
|
856 Abstraction over i is ESSENTIAL: it delays the dereferencing of claset |
|
857 DEPENDS UPON "X" REFERRING TO THE FRADULENT MESSAGE *) |
|
858 |
|
859 fun impOfSubs th = th RSN (2, @{thm rev_subsetD}) |
|
860 |
|
861 (*Apply rules to break down assumptions of the form |
|
862 Y \<in> parts(insert X H) and Y \<in> analz(insert X H) |
|
863 *) |
|
864 val Fake_insert_tac = |
|
865 dresolve_tac [impOfSubs @{thm Fake_analz_insert}, |
|
866 impOfSubs @{thm Fake_parts_insert}] THEN' |
|
867 eresolve_tac [asm_rl, @{thm synth.Inj}]; |
|
868 |
|
869 fun Fake_insert_simp_tac ss i = |
|
870 REPEAT (Fake_insert_tac i) THEN asm_full_simp_tac ss i; |
|
871 |
|
872 fun atomic_spy_analz_tac (cs,ss) = SELECT_GOAL |
|
873 (Fake_insert_simp_tac ss 1 |
|
874 THEN |
|
875 IF_UNSOLVED (Blast.depth_tac |
|
876 (cs addIs [@{thm analz_insertI}, |
|
877 impOfSubs @{thm analz_subset_parts}]) 4 1)) |
|
878 |
|
879 fun spy_analz_tac (cs,ss) i = |
|
880 DETERM |
|
881 (SELECT_GOAL |
|
882 (EVERY |
|
883 [ (*push in occurrences of X...*) |
|
884 (REPEAT o CHANGED) |
|
885 (res_inst_tac (Simplifier.the_context ss) |
|
886 [(("x", 1), "X")] (insert_commute RS ssubst) 1), |
|
887 (*...allowing further simplifications*) |
|
888 simp_tac ss 1, |
|
889 REPEAT (FIRSTGOAL (resolve_tac [allI,impI,notI,conjI,iffI])), |
|
890 DEPTH_SOLVE (atomic_spy_analz_tac (cs,ss) 1)]) i) |
|
891 |
|
892 end |
|
893 *} |
|
894 (*>*) |
|
895 |
|
896 |
|
897 (*By default only o_apply is built-in. But in the presence of eta-expansion |
|
898 this means that some terms displayed as (f o g) will be rewritten, and others |
|
899 will not!*) |
|
900 declare o_def [simp] |
|
901 |
|
902 |
|
903 lemma Crypt_notin_image_Key [simp]: "Crypt K X \<notin> Key ` A" |
|
904 by auto |
|
905 |
|
906 lemma Hash_notin_image_Key [simp] :"Hash X \<notin> Key ` A" |
|
907 by auto |
|
908 |
|
909 lemma synth_analz_mono: "G<=H ==> synth (analz(G)) <= synth (analz(H))" |
|
910 by (simp add: synth_mono analz_mono) |
|
911 |
|
912 lemma Fake_analz_eq [simp]: |
|
913 "X \<in> synth(analz H) ==> synth (analz (insert X H)) = synth (analz H)" |
|
914 apply (drule Fake_analz_insert[of _ _ "H"]) |
|
915 apply (simp add: synth_increasing[THEN Un_absorb2]) |
|
916 apply (drule synth_mono) |
|
917 apply (simp add: synth_idem) |
|
918 apply (blast intro: synth_analz_mono [THEN [2] rev_subsetD]) |
|
919 done |
|
920 |
|
921 text{*Two generalizations of @{text analz_insert_eq}*} |
|
922 lemma gen_analz_insert_eq [rule_format]: |
|
923 "X \<in> analz H ==> ALL G. H \<subseteq> G --> analz (insert X G) = analz G"; |
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924 by (blast intro: analz_cut analz_insertI analz_mono [THEN [2] rev_subsetD]) |
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925 |
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926 lemma synth_analz_insert_eq [rule_format]: |
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927 "X \<in> synth (analz H) |
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928 ==> ALL G. H \<subseteq> G --> (Key K \<in> analz (insert X G)) = (Key K \<in> analz G)"; |
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929 apply (erule synth.induct) |
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930 apply (simp_all add: gen_analz_insert_eq subset_trans [OF _ subset_insertI]) |
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931 done |
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932 |
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933 lemma Fake_parts_sing: |
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934 "X \<in> synth (analz H) ==> parts{X} \<subseteq> synth (analz H) \<union> parts H"; |
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935 apply (rule subset_trans) |
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936 apply (erule_tac [2] Fake_parts_insert) |
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937 apply (simp add: parts_mono) |
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938 done |
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939 |
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940 lemmas Fake_parts_sing_imp_Un = Fake_parts_sing [THEN [2] rev_subsetD] |
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941 |
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942 method_setup spy_analz = {* |
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943 Scan.succeed (fn ctxt => |
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944 SIMPLE_METHOD' (MessageSET.spy_analz_tac (clasimpset_of ctxt))) *} |
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945 "for proving the Fake case when analz is involved" |
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946 |
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947 method_setup atomic_spy_analz = {* |
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948 Scan.succeed (fn ctxt => |
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949 SIMPLE_METHOD' (MessageSET.atomic_spy_analz_tac (clasimpset_of ctxt))) *} |
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950 "for debugging spy_analz" |
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951 |
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952 method_setup Fake_insert_simp = {* |
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953 Scan.succeed (fn ctxt => |
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954 SIMPLE_METHOD' (MessageSET.Fake_insert_simp_tac (simpset_of ctxt))) *} |
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955 "for debugging spy_analz" |
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956 |
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957 end |
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