src/HOL/Auth/Guard/Extensions.thy
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prefer symbols;
```     1 (*  Title:      HOL/Auth/Guard/Extensions.thy
```
```     2     Author:     Frederic Blanqui, University of Cambridge Computer Laboratory
```
```     3     Copyright   2001  University of Cambridge
```
```     4 *)
```
```     5
```
```     6 section {*Extensions to Standard Theories*}
```
```     7
```
```     8 theory Extensions
```
```     9 imports "../Event"
```
```    10 begin
```
```    11
```
```    12 subsection{*Extensions to Theory @{text Set}*}
```
```    13
```
```    14 lemma eq: "[| !!x. x:A ==> x:B; !!x. x:B ==> x:A |] ==> A=B"
```
```    15 by auto
```
```    16
```
```    17 lemma insert_Un: "P ({x} Un A) ==> P (insert x A)"
```
```    18 by simp
```
```    19
```
```    20 lemma in_sub: "x:A ==> {x}<=A"
```
```    21 by auto
```
```    22
```
```    23
```
```    24 subsection{*Extensions to Theory @{text List}*}
```
```    25
```
```    26 subsubsection{*"remove l x" erase the first element of "l" equal to "x"*}
```
```    27
```
```    28 primrec remove :: "'a list => 'a => 'a list" where
```
```    29 "remove [] y = []" |
```
```    30 "remove (x#xs) y = (if x=y then xs else x # remove xs y)"
```
```    31
```
```    32 lemma set_remove: "set (remove l x) <= set l"
```
```    33 by (induct l, auto)
```
```    34
```
```    35 subsection{*Extensions to Theory @{text Message}*}
```
```    36
```
```    37 subsubsection{*declarations for tactics*}
```
```    38
```
```    39 declare analz_subset_parts [THEN subsetD, dest]
```
```    40 declare parts_insert2 [simp]
```
```    41 declare analz_cut [dest]
```
```    42 declare split_if_asm [split]
```
```    43 declare analz_insertI [intro]
```
```    44 declare Un_Diff [simp]
```
```    45
```
```    46 subsubsection{*extract the agent number of an Agent message*}
```
```    47
```
```    48 primrec agt_nb :: "msg => agent" where
```
```    49 "agt_nb (Agent A) = A"
```
```    50
```
```    51 subsubsection{*messages that are pairs*}
```
```    52
```
```    53 definition is_MPair :: "msg => bool" where
```
```    54 "is_MPair X == EX Y Z. X = {|Y,Z|}"
```
```    55
```
```    56 declare is_MPair_def [simp]
```
```    57
```
```    58 lemma MPair_is_MPair [iff]: "is_MPair {|X,Y|}"
```
```    59 by simp
```
```    60
```
```    61 lemma Agent_isnt_MPair [iff]: "~ is_MPair (Agent A)"
```
```    62 by simp
```
```    63
```
```    64 lemma Number_isnt_MPair [iff]: "~ is_MPair (Number n)"
```
```    65 by simp
```
```    66
```
```    67 lemma Key_isnt_MPair [iff]: "~ is_MPair (Key K)"
```
```    68 by simp
```
```    69
```
```    70 lemma Nonce_isnt_MPair [iff]: "~ is_MPair (Nonce n)"
```
```    71 by simp
```
```    72
```
```    73 lemma Hash_isnt_MPair [iff]: "~ is_MPair (Hash X)"
```
```    74 by simp
```
```    75
```
```    76 lemma Crypt_isnt_MPair [iff]: "~ is_MPair (Crypt K X)"
```
```    77 by simp
```
```    78
```
```    79 abbreviation
```
```    80   not_MPair :: "msg => bool" where
```
```    81   "not_MPair X == ~ is_MPair X"
```
```    82
```
```    83 lemma is_MPairE: "[| is_MPair X ==> P; not_MPair X ==> P |] ==> P"
```
```    84 by auto
```
```    85
```
```    86 declare is_MPair_def [simp del]
```
```    87
```
```    88 definition has_no_pair :: "msg set => bool" where
```
```    89 "has_no_pair H == ALL X Y. {|X,Y|} ~:H"
```
```    90
```
```    91 declare has_no_pair_def [simp]
```
```    92
```
```    93 subsubsection{*well-foundedness of messages*}
```
```    94
```
```    95 lemma wf_Crypt1 [iff]: "Crypt K X ~= X"
```
```    96 by (induct X, auto)
```
```    97
```
```    98 lemma wf_Crypt2 [iff]: "X ~= Crypt K X"
```
```    99 by (induct X, auto)
```
```   100
```
```   101 lemma parts_size: "X:parts {Y} ==> X=Y | size X < size Y"
```
```   102 by (erule parts.induct, auto)
```
```   103
```
```   104 lemma wf_Crypt_parts [iff]: "Crypt K X ~:parts {X}"
```
```   105 by (auto dest: parts_size)
```
```   106
```
```   107 subsubsection{*lemmas on keysFor*}
```
```   108
```
```   109 definition usekeys :: "msg set => key set" where
```
```   110 "usekeys G == {K. EX Y. Crypt K Y:G}"
```
```   111
```
```   112 lemma finite_keysFor [intro]: "finite G ==> finite (keysFor G)"
```
```   113 apply (simp add: keysFor_def)
```
```   114 apply (rule finite_imageI)
```
```   115 apply (induct G rule: finite_induct)
```
```   116 apply auto
```
```   117 apply (case_tac "EX K X. x = Crypt K X", clarsimp)
```
```   118 apply (subgoal_tac "{Ka. EX Xa. (Ka=K & Xa=X) | Crypt Ka Xa:F}
```
```   119 = insert K (usekeys F)", auto simp: usekeys_def)
```
```   120 by (subgoal_tac "{K. EX X. Crypt K X = x | Crypt K X:F} = usekeys F",
```
```   121 auto simp: usekeys_def)
```
```   122
```
```   123 subsubsection{*lemmas on parts*}
```
```   124
```
```   125 lemma parts_sub: "[| X:parts G; G<=H |] ==> X:parts H"
```
```   126 by (auto dest: parts_mono)
```
```   127
```
```   128 lemma parts_Diff [dest]: "X:parts (G - H) ==> X:parts G"
```
```   129 by (erule parts_sub, auto)
```
```   130
```
```   131 lemma parts_Diff_notin: "[| Y ~:H; Nonce n ~:parts (H - {Y}) |]
```
```   132 ==> Nonce n ~:parts H"
```
```   133 by simp
```
```   134
```
```   135 lemmas parts_insert_substI = parts_insert [THEN ssubst]
```
```   136 lemmas parts_insert_substD = parts_insert [THEN sym, THEN ssubst]
```
```   137
```
```   138 lemma finite_parts_msg [iff]: "finite (parts {X})"
```
```   139 by (induct X, auto)
```
```   140
```
```   141 lemma finite_parts [intro]: "finite H ==> finite (parts H)"
```
```   142 apply (erule finite_induct, simp)
```
```   143 by (rule parts_insert_substI, simp)
```
```   144
```
```   145 lemma parts_parts: "[| X:parts {Y}; Y:parts G |] ==> X:parts G"
```
```   146 by (frule parts_cut, auto)
```
```   147
```
```   148
```
```   149 lemma parts_parts_parts: "[| X:parts {Y}; Y:parts {Z}; Z:parts G |] ==> X:parts G"
```
```   150 by (auto dest: parts_parts)
```
```   151
```
```   152 lemma parts_parts_Crypt: "[| Crypt K X:parts G; Nonce n:parts {X} |]
```
```   153 ==> Nonce n:parts G"
```
```   154 by (blast intro: parts.Body dest: parts_parts)
```
```   155
```
```   156 subsubsection{*lemmas on synth*}
```
```   157
```
```   158 lemma synth_sub: "[| X:synth G; G<=H |] ==> X:synth H"
```
```   159 by (auto dest: synth_mono)
```
```   160
```
```   161 lemma Crypt_synth [rule_format]: "[| X:synth G; Key K ~:G |] ==>
```
```   162 Crypt K Y:parts {X} --> Crypt K Y:parts G"
```
```   163 by (erule synth.induct, auto dest: parts_sub)
```
```   164
```
```   165 subsubsection{*lemmas on analz*}
```
```   166
```
```   167 lemma analz_UnI1 [intro]: "X:analz G ==> X:analz (G Un H)"
```
```   168   by (subgoal_tac "G <= G Un H") (blast dest: analz_mono)+
```
```   169
```
```   170 lemma analz_sub: "[| X:analz G; G <= H |] ==> X:analz H"
```
```   171 by (auto dest: analz_mono)
```
```   172
```
```   173 lemma analz_Diff [dest]: "X:analz (G - H) ==> X:analz G"
```
```   174 by (erule analz.induct, auto)
```
```   175
```
```   176 lemmas in_analz_subset_cong = analz_subset_cong [THEN subsetD]
```
```   177
```
```   178 lemma analz_eq: "A=A' ==> analz A = analz A'"
```
```   179 by auto
```
```   180
```
```   181 lemmas insert_commute_substI = insert_commute [THEN ssubst]
```
```   182
```
```   183 lemma analz_insertD:
```
```   184      "[| Crypt K Y:H; Key (invKey K):H |] ==> analz (insert Y H) = analz H"
```
```   185 by (blast intro: analz.Decrypt analz_insert_eq)
```
```   186
```
```   187 lemma must_decrypt [rule_format,dest]: "[| X:analz H; has_no_pair H |] ==>
```
```   188 X ~:H --> (EX K Y. Crypt K Y:H & Key (invKey K):H)"
```
```   189 by (erule analz.induct, auto)
```
```   190
```
```   191 lemma analz_needs_only_finite: "X:analz H ==> EX G. G <= H & finite G"
```
```   192 by (erule analz.induct, auto)
```
```   193
```
```   194 lemma notin_analz_insert: "X ~:analz (insert Y G) ==> X ~:analz G"
```
```   195 by auto
```
```   196
```
```   197 subsubsection{*lemmas on parts, synth and analz*}
```
```   198
```
```   199 lemma parts_invKey [rule_format,dest]:"X:parts {Y} ==>
```
```   200 X:analz (insert (Crypt K Y) H) --> X ~:analz H --> Key (invKey K):analz H"
```
```   201 by (erule parts.induct, auto dest: parts.Fst parts.Snd parts.Body)
```
```   202
```
```   203 lemma in_analz: "Y:analz H ==> EX X. X:H & Y:parts {X}"
```
```   204 by (erule analz.induct, auto intro: parts.Fst parts.Snd parts.Body)
```
```   205
```
```   206 lemmas in_analz_subset_parts = analz_subset_parts [THEN subsetD]
```
```   207
```
```   208 lemma Crypt_synth_insert: "[| Crypt K X:parts (insert Y H);
```
```   209 Y:synth (analz H); Key K ~:analz H |] ==> Crypt K X:parts H"
```
```   210 apply (drule parts_insert_substD, clarify)
```
```   211 apply (frule in_sub)
```
```   212 apply (frule parts_mono)
```
```   213 apply auto
```
```   214 done
```
```   215
```
```   216 subsubsection{*greatest nonce used in a message*}
```
```   217
```
```   218 fun greatest_msg :: "msg => nat"
```
```   219 where
```
```   220   "greatest_msg (Nonce n) = n"
```
```   221 | "greatest_msg {|X,Y|} = max (greatest_msg X) (greatest_msg Y)"
```
```   222 | "greatest_msg (Crypt K X) = greatest_msg X"
```
```   223 | "greatest_msg other = 0"
```
```   224
```
```   225 lemma greatest_msg_is_greatest: "Nonce n:parts {X} ==> n <= greatest_msg X"
```
```   226 by (induct X, auto)
```
```   227
```
```   228 subsubsection{*sets of keys*}
```
```   229
```
```   230 definition keyset :: "msg set => bool" where
```
```   231 "keyset G == ALL X. X:G --> (EX K. X = Key K)"
```
```   232
```
```   233 lemma keyset_in [dest]: "[| keyset G; X:G |] ==> EX K. X = Key K"
```
```   234 by (auto simp: keyset_def)
```
```   235
```
```   236 lemma MPair_notin_keyset [simp]: "keyset G ==> {|X,Y|} ~:G"
```
```   237 by auto
```
```   238
```
```   239 lemma Crypt_notin_keyset [simp]: "keyset G ==> Crypt K X ~:G"
```
```   240 by auto
```
```   241
```
```   242 lemma Nonce_notin_keyset [simp]: "keyset G ==> Nonce n ~:G"
```
```   243 by auto
```
```   244
```
```   245 lemma parts_keyset [simp]: "keyset G ==> parts G = G"
```
```   246 by (auto, erule parts.induct, auto)
```
```   247
```
```   248 subsubsection{*keys a priori necessary for decrypting the messages of G*}
```
```   249
```
```   250 definition keysfor :: "msg set => msg set" where
```
```   251 "keysfor G == Key ` keysFor (parts G)"
```
```   252
```
```   253 lemma keyset_keysfor [iff]: "keyset (keysfor G)"
```
```   254 by (simp add: keyset_def keysfor_def, blast)
```
```   255
```
```   256 lemma keyset_Diff_keysfor [simp]: "keyset H ==> keyset (H - keysfor G)"
```
```   257 by (auto simp: keyset_def)
```
```   258
```
```   259 lemma keysfor_Crypt: "Crypt K X:parts G ==> Key (invKey K):keysfor G"
```
```   260 by (auto simp: keysfor_def Crypt_imp_invKey_keysFor)
```
```   261
```
```   262 lemma no_key_no_Crypt: "Key K ~:keysfor G ==> Crypt (invKey K) X ~:parts G"
```
```   263 by (auto dest: keysfor_Crypt)
```
```   264
```
```   265 lemma finite_keysfor [intro]: "finite G ==> finite (keysfor G)"
```
```   266 by (auto simp: keysfor_def intro: finite_UN_I)
```
```   267
```
```   268 subsubsection{*only the keys necessary for G are useful in analz*}
```
```   269
```
```   270 lemma analz_keyset: "keyset H ==>
```
```   271 analz (G Un H) = H - keysfor G Un (analz (G Un (H Int keysfor G)))"
```
```   272 apply (rule eq)
```
```   273 apply (erule analz.induct, blast)
```
```   274 apply (simp, blast)
```
```   275 apply (simp, blast)
```
```   276 apply (case_tac "Key (invKey K):H - keysfor G", clarsimp)
```
```   277 apply (drule_tac X=X in no_key_no_Crypt)
```
```   278 by (auto intro: analz_sub)
```
```   279
```
```   280 lemmas analz_keyset_substD = analz_keyset [THEN sym, THEN ssubst]
```
```   281
```
```   282
```
```   283 subsection{*Extensions to Theory @{text Event}*}
```
```   284
```
```   285
```
```   286 subsubsection{*general protocol properties*}
```
```   287
```
```   288 definition is_Says :: "event => bool" where
```
```   289 "is_Says ev == (EX A B X. ev = Says A B X)"
```
```   290
```
```   291 lemma is_Says_Says [iff]: "is_Says (Says A B X)"
```
```   292 by (simp add: is_Says_def)
```
```   293
```
```   294 (* one could also require that Gets occurs after Says
```
```   295 but this is sufficient for our purpose *)
```
```   296 definition Gets_correct :: "event list set => bool" where
```
```   297 "Gets_correct p == ALL evs B X. evs:p --> Gets B X:set evs
```
```   298 --> (EX A. Says A B X:set evs)"
```
```   299
```
```   300 lemma Gets_correct_Says: "[| Gets_correct p; Gets B X # evs:p |]
```
```   301 ==> EX A. Says A B X:set evs"
```
```   302 apply (simp add: Gets_correct_def)
```
```   303 by (drule_tac x="Gets B X # evs" in spec, auto)
```
```   304
```
```   305 definition one_step :: "event list set => bool" where
```
```   306 "one_step p == ALL evs ev. ev#evs:p --> evs:p"
```
```   307
```
```   308 lemma one_step_Cons [dest]: "[| one_step p; ev#evs:p |] ==> evs:p"
```
```   309 by (unfold one_step_def, blast)
```
```   310
```
```   311 lemma one_step_app: "[| evs@evs':p; one_step p; []:p |] ==> evs':p"
```
```   312 by (induct evs, auto)
```
```   313
```
```   314 lemma trunc: "[| evs @ evs':p; one_step p |] ==> evs':p"
```
```   315 by (induct evs, auto)
```
```   316
```
```   317 definition has_only_Says :: "event list set => bool" where
```
```   318 "has_only_Says p == ALL evs ev. evs:p --> ev:set evs
```
```   319 --> (EX A B X. ev = Says A B X)"
```
```   320
```
```   321 lemma has_only_SaysD: "[| ev:set evs; evs:p; has_only_Says p |]
```
```   322 ==> EX A B X. ev = Says A B X"
```
```   323 by (unfold has_only_Says_def, blast)
```
```   324
```
```   325 lemma in_has_only_Says [dest]: "[| has_only_Says p; evs:p; ev:set evs |]
```
```   326 ==> EX A B X. ev = Says A B X"
```
```   327 by (auto simp: has_only_Says_def)
```
```   328
```
```   329 lemma has_only_Says_imp_Gets_correct [simp]: "has_only_Says p
```
```   330 ==> Gets_correct p"
```
```   331 by (auto simp: has_only_Says_def Gets_correct_def)
```
```   332
```
```   333 subsubsection{*lemma on knows*}
```
```   334
```
```   335 lemma Says_imp_spies2: "Says A B {|X,Y|}:set evs ==> Y:parts (spies evs)"
```
```   336 by (drule Says_imp_spies, drule parts.Inj, drule parts.Snd, simp)
```
```   337
```
```   338 lemma Says_not_parts: "[| Says A B X:set evs; Y ~:parts (spies evs) |]
```
```   339 ==> Y ~:parts {X}"
```
```   340 by (auto dest: Says_imp_spies parts_parts)
```
```   341
```
```   342 subsubsection{*knows without initState*}
```
```   343
```
```   344 primrec knows' :: "agent => event list => msg set" where
```
```   345   knows'_Nil: "knows' A [] = {}" |
```
```   346   knows'_Cons0:
```
```   347  "knows' A (ev # evs) = (
```
```   348    if A = Spy then (
```
```   349      case ev of
```
```   350        Says A' B X => insert X (knows' A evs)
```
```   351      | Gets A' X => knows' A evs
```
```   352      | Notes A' X => if A':bad then insert X (knows' A evs) else knows' A evs
```
```   353    ) else (
```
```   354      case ev of
```
```   355        Says A' B X => if A=A' then insert X (knows' A evs) else knows' A evs
```
```   356      | Gets A' X => if A=A' then insert X (knows' A evs) else knows' A evs
```
```   357      | Notes A' X => if A=A' then insert X (knows' A evs) else knows' A evs
```
```   358    ))"
```
```   359
```
```   360 abbreviation
```
```   361   spies' :: "event list => msg set" where
```
```   362   "spies' == knows' Spy"
```
```   363
```
```   364 subsubsection{*decomposition of knows into knows' and initState*}
```
```   365
```
```   366 lemma knows_decomp: "knows A evs = knows' A evs Un (initState A)"
```
```   367 by (induct evs, auto split: event.split simp: knows.simps)
```
```   368
```
```   369 lemmas knows_decomp_substI = knows_decomp [THEN ssubst]
```
```   370 lemmas knows_decomp_substD = knows_decomp [THEN sym, THEN ssubst]
```
```   371
```
```   372 lemma knows'_sub_knows: "knows' A evs <= knows A evs"
```
```   373 by (auto simp: knows_decomp)
```
```   374
```
```   375 lemma knows'_Cons: "knows' A (ev#evs) = knows' A [ev] Un knows' A evs"
```
```   376 by (induct ev, auto)
```
```   377
```
```   378 lemmas knows'_Cons_substI = knows'_Cons [THEN ssubst]
```
```   379 lemmas knows'_Cons_substD = knows'_Cons [THEN sym, THEN ssubst]
```
```   380
```
```   381 lemma knows_Cons: "knows A (ev#evs) = initState A Un knows' A [ev]
```
```   382 Un knows A evs"
```
```   383 apply (simp only: knows_decomp)
```
```   384 apply (rule_tac s="(knows' A [ev] Un knows' A evs) Un initState A" in trans)
```
```   385 apply (simp only: knows'_Cons [of A ev evs] Un_ac)
```
```   386 apply blast
```
```   387 done
```
```   388
```
```   389
```
```   390 lemmas knows_Cons_substI = knows_Cons [THEN ssubst]
```
```   391 lemmas knows_Cons_substD = knows_Cons [THEN sym, THEN ssubst]
```
```   392
```
```   393 lemma knows'_sub_spies': "[| evs:p; has_only_Says p; one_step p |]
```
```   394 ==> knows' A evs <= spies' evs"
```
```   395 by (induct evs, auto split: event.splits)
```
```   396
```
```   397 subsubsection{*knows' is finite*}
```
```   398
```
```   399 lemma finite_knows' [iff]: "finite (knows' A evs)"
```
```   400 by (induct evs, auto split: event.split simp: knows.simps)
```
```   401
```
```   402 subsubsection{*monotonicity of knows*}
```
```   403
```
```   404 lemma knows_sub_Cons: "knows A evs <= knows A (ev#evs)"
```
```   405 by(cases A, induct evs, auto simp: knows.simps split:event.split)
```
```   406
```
```   407 lemma knows_ConsI: "X:knows A evs ==> X:knows A (ev#evs)"
```
```   408 by (auto dest: knows_sub_Cons [THEN subsetD])
```
```   409
```
```   410 lemma knows_sub_app: "knows A evs <= knows A (evs @ evs')"
```
```   411 apply (induct evs, auto)
```
```   412 apply (simp add: knows_decomp)
```
```   413 apply (rename_tac a b c)
```
```   414 by (case_tac a, auto simp: knows.simps)
```
```   415
```
```   416 subsubsection{*maximum knowledge an agent can have
```
```   417 includes messages sent to the agent*}
```
```   418
```
```   419 primrec knows_max' :: "agent => event list => msg set" where
```
```   420 knows_max'_def_Nil: "knows_max' A [] = {}" |
```
```   421 knows_max'_def_Cons: "knows_max' A (ev # evs) = (
```
```   422   if A=Spy then (
```
```   423     case ev of
```
```   424       Says A' B X => insert X (knows_max' A evs)
```
```   425     | Gets A' X => knows_max' A evs
```
```   426     | Notes A' X =>
```
```   427       if A':bad then insert X (knows_max' A evs) else knows_max' A evs
```
```   428   ) else (
```
```   429     case ev of
```
```   430       Says A' B X =>
```
```   431       if A=A' | A=B then insert X (knows_max' A evs) else knows_max' A evs
```
```   432     | Gets A' X =>
```
```   433       if A=A' then insert X (knows_max' A evs) else knows_max' A evs
```
```   434     | Notes A' X =>
```
```   435       if A=A' then insert X (knows_max' A evs) else knows_max' A evs
```
```   436   ))"
```
```   437
```
```   438 definition knows_max :: "agent => event list => msg set" where
```
```   439 "knows_max A evs == knows_max' A evs Un initState A"
```
```   440
```
```   441 abbreviation
```
```   442   spies_max :: "event list => msg set" where
```
```   443   "spies_max evs == knows_max Spy evs"
```
```   444
```
```   445 subsubsection{*basic facts about @{term knows_max}*}
```
```   446
```
```   447 lemma spies_max_spies [iff]: "spies_max evs = spies evs"
```
```   448 by (induct evs, auto simp: knows_max_def split: event.splits)
```
```   449
```
```   450 lemma knows_max'_Cons: "knows_max' A (ev#evs)
```
```   451 = knows_max' A [ev] Un knows_max' A evs"
```
```   452 by (auto split: event.splits)
```
```   453
```
```   454 lemmas knows_max'_Cons_substI = knows_max'_Cons [THEN ssubst]
```
```   455 lemmas knows_max'_Cons_substD = knows_max'_Cons [THEN sym, THEN ssubst]
```
```   456
```
```   457 lemma knows_max_Cons: "knows_max A (ev#evs)
```
```   458 = knows_max' A [ev] Un knows_max A evs"
```
```   459 apply (simp add: knows_max_def del: knows_max'_def_Cons)
```
```   460 apply (rule_tac evs=evs in knows_max'_Cons_substI)
```
```   461 by blast
```
```   462
```
```   463 lemmas knows_max_Cons_substI = knows_max_Cons [THEN ssubst]
```
```   464 lemmas knows_max_Cons_substD = knows_max_Cons [THEN sym, THEN ssubst]
```
```   465
```
```   466 lemma finite_knows_max' [iff]: "finite (knows_max' A evs)"
```
```   467 by (induct evs, auto split: event.split)
```
```   468
```
```   469 lemma knows_max'_sub_spies': "[| evs:p; has_only_Says p; one_step p |]
```
```   470 ==> knows_max' A evs <= spies' evs"
```
```   471 by (induct evs, auto split: event.splits)
```
```   472
```
```   473 lemma knows_max'_in_spies' [dest]: "[| evs:p; X:knows_max' A evs;
```
```   474 has_only_Says p; one_step p |] ==> X:spies' evs"
```
```   475 by (rule knows_max'_sub_spies' [THEN subsetD], auto)
```
```   476
```
```   477 lemma knows_max'_app: "knows_max' A (evs @ evs')
```
```   478 = knows_max' A evs Un knows_max' A evs'"
```
```   479 by (induct evs, auto split: event.splits)
```
```   480
```
```   481 lemma Says_to_knows_max': "Says A B X:set evs ==> X:knows_max' B evs"
```
```   482 by (simp add: in_set_conv_decomp, clarify, simp add: knows_max'_app)
```
```   483
```
```   484 lemma Says_from_knows_max': "Says A B X:set evs ==> X:knows_max' A evs"
```
```   485 by (simp add: in_set_conv_decomp, clarify, simp add: knows_max'_app)
```
```   486
```
```   487 subsubsection{*used without initState*}
```
```   488
```
```   489 primrec used' :: "event list => msg set" where
```
```   490 "used' [] = {}" |
```
```   491 "used' (ev # evs) = (
```
```   492   case ev of
```
```   493     Says A B X => parts {X} Un used' evs
```
```   494     | Gets A X => used' evs
```
```   495     | Notes A X => parts {X} Un used' evs
```
```   496   )"
```
```   497
```
```   498 definition init :: "msg set" where
```
```   499 "init == used []"
```
```   500
```
```   501 lemma used_decomp: "used evs = init Un used' evs"
```
```   502 by (induct evs, auto simp: init_def split: event.split)
```
```   503
```
```   504 lemma used'_sub_app: "used' evs <= used' (evs@evs')"
```
```   505 by (induct evs, auto split: event.split)
```
```   506
```
```   507 lemma used'_parts [rule_format]: "X:used' evs ==> Y:parts {X} --> Y:used' evs"
```
```   508 apply (induct evs, simp)
```
```   509 apply (rename_tac a b)
```
```   510 apply (case_tac a, simp_all)
```
```   511 apply (blast dest: parts_trans)+
```
```   512 done
```
```   513
```
```   514 subsubsection{*monotonicity of used*}
```
```   515
```
```   516 lemma used_sub_Cons: "used evs <= used (ev#evs)"
```
```   517 by (induct evs, (induct ev, auto)+)
```
```   518
```
```   519 lemma used_ConsI: "X:used evs ==> X:used (ev#evs)"
```
```   520 by (auto dest: used_sub_Cons [THEN subsetD])
```
```   521
```
```   522 lemma notin_used_ConsD: "X ~:used (ev#evs) ==> X ~:used evs"
```
```   523 by (auto dest: used_sub_Cons [THEN subsetD])
```
```   524
```
```   525 lemma used_appD [dest]: "X:used (evs @ evs') ==> X:used evs | X:used evs'"
```
```   526 by (induct evs, auto, rename_tac a b, case_tac a, auto)
```
```   527
```
```   528 lemma used_ConsD: "X:used (ev#evs) ==> X:used [ev] | X:used evs"
```
```   529 by (case_tac ev, auto)
```
```   530
```
```   531 lemma used_sub_app: "used evs <= used (evs@evs')"
```
```   532 by (auto simp: used_decomp dest: used'_sub_app [THEN subsetD])
```
```   533
```
```   534 lemma used_appIL: "X:used evs ==> X:used (evs' @ evs)"
```
```   535 by (induct evs', auto intro: used_ConsI)
```
```   536
```
```   537 lemma used_appIR: "X:used evs ==> X:used (evs @ evs')"
```
```   538 by (erule used_sub_app [THEN subsetD])
```
```   539
```
```   540 lemma used_parts: "[| X:parts {Y}; Y:used evs |] ==> X:used evs"
```
```   541 apply (auto simp: used_decomp dest: used'_parts)
```
```   542 by (auto simp: init_def used_Nil dest: parts_trans)
```
```   543
```
```   544 lemma parts_Says_used: "[| Says A B X:set evs; Y:parts {X} |] ==> Y:used evs"
```
```   545 by (induct evs, simp_all, safe, auto intro: used_ConsI)
```
```   546
```
```   547 lemma parts_used_app: "X:parts {Y} ==> X:used (evs @ Says A B Y # evs')"
```
```   548 apply (drule_tac evs="[Says A B Y]" in used_parts, simp, blast)
```
```   549 apply (drule_tac evs'=evs' in used_appIR)
```
```   550 apply (drule_tac evs'=evs in used_appIL)
```
```   551 by simp
```
```   552
```
```   553 subsubsection{*lemmas on used and knows*}
```
```   554
```
```   555 lemma initState_used: "X:parts (initState A) ==> X:used evs"
```
```   556 by (induct evs, auto simp: used.simps split: event.split)
```
```   557
```
```   558 lemma Says_imp_used: "Says A B X:set evs ==> parts {X} <= used evs"
```
```   559 by (induct evs, auto intro: used_ConsI)
```
```   560
```
```   561 lemma not_used_not_spied: "X ~:used evs ==> X ~:parts (spies evs)"
```
```   562 by (induct evs, auto simp: used_Nil)
```
```   563
```
```   564 lemma not_used_not_parts: "[| Y ~:used evs; Says A B X:set evs |]
```
```   565 ==> Y ~:parts {X}"
```
```   566 by (induct evs, auto intro: used_ConsI)
```
```   567
```
```   568 lemma not_used_parts_false: "[| X ~:used evs; Y:parts (spies evs) |]
```
```   569 ==> X ~:parts {Y}"
```
```   570 by (auto dest: parts_parts)
```
```   571
```
```   572 lemma known_used [rule_format]: "[| evs:p; Gets_correct p; one_step p |]
```
```   573 ==> X:parts (knows A evs) --> X:used evs"
```
```   574 apply (case_tac "A=Spy", blast)
```
```   575 apply (induct evs)
```
```   576 apply (simp add: used.simps, blast)
```
```   577 apply (rename_tac a evs)
```
```   578 apply (frule_tac ev=a and evs=evs in one_step_Cons, simp, clarify)
```
```   579 apply (drule_tac P="%G. X:parts G" in knows_Cons_substD, safe)
```
```   580 apply (erule initState_used)
```
```   581 apply (case_tac a, auto)
```
```   582 apply (rename_tac msg)
```
```   583 apply (drule_tac B=A and X=msg and evs=evs in Gets_correct_Says)
```
```   584 by (auto dest: Says_imp_used intro: used_ConsI)
```
```   585
```
```   586 lemma known_max_used [rule_format]: "[| evs:p; Gets_correct p; one_step p |]
```
```   587 ==> X:parts (knows_max A evs) --> X:used evs"
```
```   588 apply (case_tac "A=Spy")
```
```   589 apply force
```
```   590 apply (induct evs)
```
```   591 apply (simp add: knows_max_def used.simps, blast)
```
```   592 apply (rename_tac a evs)
```
```   593 apply (frule_tac ev=a and evs=evs in one_step_Cons, simp, clarify)
```
```   594 apply (drule_tac P="%G. X:parts G" in knows_max_Cons_substD, safe)
```
```   595 apply (case_tac a, auto)
```
```   596 apply (rename_tac msg)
```
```   597 apply (drule_tac B=A and X=msg and evs=evs in Gets_correct_Says)
```
```   598 by (auto simp: knows_max'_Cons dest: Says_imp_used intro: used_ConsI)
```
```   599
```
```   600 lemma not_used_not_known: "[| evs:p; X ~:used evs;
```
```   601 Gets_correct p; one_step p |] ==> X ~:parts (knows A evs)"
```
```   602 by (case_tac "A=Spy", auto dest: not_used_not_spied known_used)
```
```   603
```
```   604 lemma not_used_not_known_max: "[| evs:p; X ~:used evs;
```
```   605 Gets_correct p; one_step p |] ==> X ~:parts (knows_max A evs)"
```
```   606 by (case_tac "A=Spy", auto dest: not_used_not_spied known_max_used)
```
```   607
```
```   608 subsubsection{*a nonce or key in a message cannot equal a fresh nonce or key*}
```
```   609
```
```   610 lemma Nonce_neq [dest]: "[| Nonce n' ~:used evs;
```
```   611 Says A B X:set evs; Nonce n:parts {X} |] ==> n ~= n'"
```
```   612 by (drule not_used_not_spied, auto dest: Says_imp_knows_Spy parts_sub)
```
```   613
```
```   614 lemma Key_neq [dest]: "[| Key n' ~:used evs;
```
```   615 Says A B X:set evs; Key n:parts {X} |] ==> n ~= n'"
```
```   616 by (drule not_used_not_spied, auto dest: Says_imp_knows_Spy parts_sub)
```
```   617
```
```   618 subsubsection{*message of an event*}
```
```   619
```
```   620 primrec msg :: "event => msg"
```
```   621 where
```
```   622   "msg (Says A B X) = X"
```
```   623 | "msg (Gets A X) = X"
```
```   624 | "msg (Notes A X) = X"
```
```   625
```
```   626 lemma used_sub_parts_used: "X:used (ev # evs) ==> X:parts {msg ev} Un used evs"
```
```   627 by (induct ev, auto)
```
```   628
```
```   629 end
```