src/HOL/Metis_Examples/Message.thy
author haftmann
Mon Jun 05 15:59:41 2017 +0200 (2017-06-05)
changeset 66010 2f7d39285a1a
parent 63167 0909deb8059b
child 67443 3abf6a722518
permissions -rw-r--r--
executable domain membership checks
     1 (*  Title:      HOL/Metis_Examples/Message.thy
     2     Author:     Lawrence C. Paulson, Cambridge University Computer Laboratory
     3     Author:     Jasmin Blanchette, TU Muenchen
     4 
     5 Metis example featuring message authentication.
     6 *)
     7 
     8 section \<open>Metis Example Featuring Message Authentication\<close>
     9 
    10 theory Message
    11 imports Main
    12 begin
    13 
    14 declare [[metis_new_skolem]]
    15 
    16 lemma strange_Un_eq [simp]: "A \<union> (B \<union> A) = B \<union> A"
    17 by (metis Un_commute Un_left_absorb)
    18 
    19 type_synonym key = nat
    20 
    21 consts
    22   all_symmetric :: bool        \<comment>\<open>true if all keys are symmetric\<close>
    23   invKey        :: "key=>key"  \<comment>\<open>inverse of a symmetric key\<close>
    24 
    25 specification (invKey)
    26   invKey [simp]: "invKey (invKey K) = K"
    27   invKey_symmetric: "all_symmetric --> invKey = id"
    28 by (metis id_apply)
    29 
    30 
    31 text\<open>The inverse of a symmetric key is itself; that of a public key
    32       is the private key and vice versa\<close>
    33 
    34 definition symKeys :: "key set" where
    35   "symKeys == {K. invKey K = K}"
    36 
    37 datatype  \<comment>\<open>We allow any number of friendly agents\<close>
    38   agent = Server | Friend nat | Spy
    39 
    40 datatype
    41      msg = Agent  agent     \<comment>\<open>Agent names\<close>
    42          | Number nat       \<comment>\<open>Ordinary integers, timestamps, ...\<close>
    43          | Nonce  nat       \<comment>\<open>Unguessable nonces\<close>
    44          | Key    key       \<comment>\<open>Crypto keys\<close>
    45          | Hash   msg       \<comment>\<open>Hashing\<close>
    46          | MPair  msg msg   \<comment>\<open>Compound messages\<close>
    47          | Crypt  key msg   \<comment>\<open>Encryption, public- or shared-key\<close>
    48 
    49 
    50 text\<open>Concrete syntax: messages appear as \<open>\<lbrace>A,B,NA\<rbrace>\<close>, etc...\<close>
    51 syntax
    52   "_MTuple"      :: "['a, args] => 'a * 'b"       ("(2\<lbrace>_,/ _\<rbrace>)")
    53 translations
    54   "\<lbrace>x, y, z\<rbrace>"   == "\<lbrace>x, \<lbrace>y, z\<rbrace>\<rbrace>"
    55   "\<lbrace>x, y\<rbrace>"      == "CONST MPair x y"
    56 
    57 
    58 definition HPair :: "[msg,msg] => msg" ("(4Hash[_] /_)" [0, 1000]) where
    59     \<comment>\<open>Message Y paired with a MAC computed with the help of X\<close>
    60     "Hash[X] Y == \<lbrace> Hash\<lbrace>X,Y\<rbrace>, Y\<rbrace>"
    61 
    62 definition keysFor :: "msg set => key set" where
    63     \<comment>\<open>Keys useful to decrypt elements of a message set\<close>
    64   "keysFor H == invKey ` {K. \<exists>X. Crypt K X \<in> H}"
    65 
    66 
    67 subsubsection\<open>Inductive Definition of All Parts" of a Message\<close>
    68 
    69 inductive_set
    70   parts :: "msg set => msg set"
    71   for H :: "msg set"
    72   where
    73     Inj [intro]:               "X \<in> H ==> X \<in> parts H"
    74   | Fst:         "\<lbrace>X,Y\<rbrace>   \<in> parts H ==> X \<in> parts H"
    75   | Snd:         "\<lbrace>X,Y\<rbrace>   \<in> parts H ==> Y \<in> parts H"
    76   | Body:        "Crypt K X \<in> parts H ==> X \<in> parts H"
    77 
    78 lemma parts_mono: "G \<subseteq> H ==> parts(G) \<subseteq> parts(H)"
    79 apply auto
    80 apply (erule parts.induct)
    81    apply (metis parts.Inj set_rev_mp)
    82   apply (metis parts.Fst)
    83  apply (metis parts.Snd)
    84 by (metis parts.Body)
    85 
    86 text\<open>Equations hold because constructors are injective.\<close>
    87 lemma Friend_image_eq [simp]: "(Friend x \<in> Friend`A) = (x:A)"
    88 by (metis agent.inject image_iff)
    89 
    90 lemma Key_image_eq [simp]: "(Key x \<in> Key`A) = (x \<in> A)"
    91 by (metis image_iff msg.inject(4))
    92 
    93 lemma Nonce_Key_image_eq [simp]: "Nonce x \<notin> Key`A"
    94 by (metis image_iff msg.distinct(23))
    95 
    96 
    97 subsubsection\<open>Inverse of keys\<close>
    98 
    99 lemma invKey_eq [simp]: "(invKey K = invKey K') = (K = K')"
   100 by (metis invKey)
   101 
   102 
   103 subsection\<open>keysFor operator\<close>
   104 
   105 lemma keysFor_empty [simp]: "keysFor {} = {}"
   106 by (unfold keysFor_def, blast)
   107 
   108 lemma keysFor_Un [simp]: "keysFor (H \<union> H') = keysFor H \<union> keysFor H'"
   109 by (unfold keysFor_def, blast)
   110 
   111 lemma keysFor_UN [simp]: "keysFor (\<Union>i\<in>A. H i) = (\<Union>i\<in>A. keysFor (H i))"
   112 by (unfold keysFor_def, blast)
   113 
   114 text\<open>Monotonicity\<close>
   115 lemma keysFor_mono: "G \<subseteq> H ==> keysFor(G) \<subseteq> keysFor(H)"
   116 by (unfold keysFor_def, blast)
   117 
   118 lemma keysFor_insert_Agent [simp]: "keysFor (insert (Agent A) H) = keysFor H"
   119 by (unfold keysFor_def, auto)
   120 
   121 lemma keysFor_insert_Nonce [simp]: "keysFor (insert (Nonce N) H) = keysFor H"
   122 by (unfold keysFor_def, auto)
   123 
   124 lemma keysFor_insert_Number [simp]: "keysFor (insert (Number N) H) = keysFor H"
   125 by (unfold keysFor_def, auto)
   126 
   127 lemma keysFor_insert_Key [simp]: "keysFor (insert (Key K) H) = keysFor H"
   128 by (unfold keysFor_def, auto)
   129 
   130 lemma keysFor_insert_Hash [simp]: "keysFor (insert (Hash X) H) = keysFor H"
   131 by (unfold keysFor_def, auto)
   132 
   133 lemma keysFor_insert_MPair [simp]: "keysFor (insert \<lbrace>X,Y\<rbrace> H) = keysFor H"
   134 by (unfold keysFor_def, auto)
   135 
   136 lemma keysFor_insert_Crypt [simp]:
   137     "keysFor (insert (Crypt K X) H) = insert (invKey K) (keysFor H)"
   138 by (unfold keysFor_def, auto)
   139 
   140 lemma keysFor_image_Key [simp]: "keysFor (Key`E) = {}"
   141 by (unfold keysFor_def, auto)
   142 
   143 lemma Crypt_imp_invKey_keysFor: "Crypt K X \<in> H ==> invKey K \<in> keysFor H"
   144 by (unfold keysFor_def, blast)
   145 
   146 
   147 subsection\<open>Inductive relation "parts"\<close>
   148 
   149 lemma MPair_parts:
   150      "[| \<lbrace>X,Y\<rbrace> \<in> parts H;
   151          [| X \<in> parts H; Y \<in> parts H |] ==> P |] ==> P"
   152 by (blast dest: parts.Fst parts.Snd)
   153 
   154 declare MPair_parts [elim!] parts.Body [dest!]
   155 text\<open>NB These two rules are UNSAFE in the formal sense, as they discard the
   156      compound message.  They work well on THIS FILE.
   157   \<open>MPair_parts\<close> is left as SAFE because it speeds up proofs.
   158   The Crypt rule is normally kept UNSAFE to avoid breaking up certificates.\<close>
   159 
   160 lemma parts_increasing: "H \<subseteq> parts(H)"
   161 by blast
   162 
   163 lemmas parts_insertI = subset_insertI [THEN parts_mono, THEN subsetD]
   164 
   165 lemma parts_empty [simp]: "parts{} = {}"
   166 apply safe
   167 apply (erule parts.induct)
   168 apply blast+
   169 done
   170 
   171 lemma parts_emptyE [elim!]: "X\<in> parts{} ==> P"
   172 by simp
   173 
   174 text\<open>WARNING: loops if H = {Y}, therefore must not be repeated!\<close>
   175 lemma parts_singleton: "X\<in> parts H ==> \<exists>Y\<in>H. X\<in> parts {Y}"
   176 apply (erule parts.induct)
   177 apply fast+
   178 done
   179 
   180 
   181 subsubsection\<open>Unions\<close>
   182 
   183 lemma parts_Un_subset1: "parts(G) \<union> parts(H) \<subseteq> parts(G \<union> H)"
   184 by (intro Un_least parts_mono Un_upper1 Un_upper2)
   185 
   186 lemma parts_Un_subset2: "parts(G \<union> H) \<subseteq> parts(G) \<union> parts(H)"
   187 apply (rule subsetI)
   188 apply (erule parts.induct, blast+)
   189 done
   190 
   191 lemma parts_Un [simp]: "parts(G \<union> H) = parts(G) \<union> parts(H)"
   192 by (intro equalityI parts_Un_subset1 parts_Un_subset2)
   193 
   194 lemma parts_insert: "parts (insert X H) = parts {X} \<union> parts H"
   195 apply (subst insert_is_Un [of _ H])
   196 apply (simp only: parts_Un)
   197 done
   198 
   199 lemma parts_insert2:
   200      "parts (insert X (insert Y H)) = parts {X} \<union> parts {Y} \<union> parts H"
   201 by (metis Un_commute Un_empty_left Un_empty_right Un_insert_left Un_insert_right parts_Un)
   202 
   203 
   204 lemma parts_UN_subset1: "(\<Union>x\<in>A. parts(H x)) \<subseteq> parts(\<Union>x\<in>A. H x)"
   205 by (intro UN_least parts_mono UN_upper)
   206 
   207 lemma parts_UN_subset2: "parts(\<Union>x\<in>A. H x) \<subseteq> (\<Union>x\<in>A. parts(H x))"
   208 apply (rule subsetI)
   209 apply (erule parts.induct, blast+)
   210 done
   211 
   212 lemma parts_UN [simp]: "parts(\<Union>x\<in>A. H x) = (\<Union>x\<in>A. parts(H x))"
   213 by (intro equalityI parts_UN_subset1 parts_UN_subset2)
   214 
   215 text\<open>Added to simplify arguments to parts, analz and synth.
   216   NOTE: the UN versions are no longer used!\<close>
   217 
   218 
   219 text\<open>This allows \<open>blast\<close> to simplify occurrences of
   220   @{term "parts(G\<union>H)"} in the assumption.\<close>
   221 lemmas in_parts_UnE = parts_Un [THEN equalityD1, THEN subsetD, THEN UnE]
   222 declare in_parts_UnE [elim!]
   223 
   224 lemma parts_insert_subset: "insert X (parts H) \<subseteq> parts(insert X H)"
   225 by (blast intro: parts_mono [THEN [2] rev_subsetD])
   226 
   227 subsubsection\<open>Idempotence and transitivity\<close>
   228 
   229 lemma parts_partsD [dest!]: "X\<in> parts (parts H) ==> X\<in> parts H"
   230 by (erule parts.induct, blast+)
   231 
   232 lemma parts_idem [simp]: "parts (parts H) = parts H"
   233 by blast
   234 
   235 lemma parts_subset_iff [simp]: "(parts G \<subseteq> parts H) = (G \<subseteq> parts H)"
   236 apply (rule iffI)
   237 apply (metis Un_absorb1 Un_subset_iff parts_Un parts_increasing)
   238 apply (metis parts_idem parts_mono)
   239 done
   240 
   241 lemma parts_trans: "[| X\<in> parts G;  G \<subseteq> parts H |] ==> X\<in> parts H"
   242 by (blast dest: parts_mono)
   243 
   244 lemma parts_cut: "[|Y\<in> parts (insert X G);  X\<in> parts H|] ==> Y\<in> parts(G \<union> H)"
   245 by (metis (no_types) Un_insert_left Un_insert_right insert_absorb le_supE
   246           parts_Un parts_idem parts_increasing parts_trans)
   247 
   248 subsubsection\<open>Rewrite rules for pulling out atomic messages\<close>
   249 
   250 lemmas parts_insert_eq_I = equalityI [OF subsetI parts_insert_subset]
   251 
   252 
   253 lemma parts_insert_Agent [simp]:
   254      "parts (insert (Agent agt) H) = insert (Agent agt) (parts H)"
   255 apply (rule parts_insert_eq_I)
   256 apply (erule parts.induct, auto)
   257 done
   258 
   259 lemma parts_insert_Nonce [simp]:
   260      "parts (insert (Nonce N) H) = insert (Nonce N) (parts H)"
   261 apply (rule parts_insert_eq_I)
   262 apply (erule parts.induct, auto)
   263 done
   264 
   265 lemma parts_insert_Number [simp]:
   266      "parts (insert (Number N) H) = insert (Number N) (parts H)"
   267 apply (rule parts_insert_eq_I)
   268 apply (erule parts.induct, auto)
   269 done
   270 
   271 lemma parts_insert_Key [simp]:
   272      "parts (insert (Key K) H) = insert (Key K) (parts H)"
   273 apply (rule parts_insert_eq_I)
   274 apply (erule parts.induct, auto)
   275 done
   276 
   277 lemma parts_insert_Hash [simp]:
   278      "parts (insert (Hash X) H) = insert (Hash X) (parts H)"
   279 apply (rule parts_insert_eq_I)
   280 apply (erule parts.induct, auto)
   281 done
   282 
   283 lemma parts_insert_Crypt [simp]:
   284      "parts (insert (Crypt K X) H) =
   285           insert (Crypt K X) (parts (insert X H))"
   286 apply (rule equalityI)
   287 apply (rule subsetI)
   288 apply (erule parts.induct, auto)
   289 apply (blast intro: parts.Body)
   290 done
   291 
   292 lemma parts_insert_MPair [simp]:
   293      "parts (insert \<lbrace>X,Y\<rbrace> H) =
   294           insert \<lbrace>X,Y\<rbrace> (parts (insert X (insert Y H)))"
   295 apply (rule equalityI)
   296 apply (rule subsetI)
   297 apply (erule parts.induct, auto)
   298 apply (blast intro: parts.Fst parts.Snd)+
   299 done
   300 
   301 lemma parts_image_Key [simp]: "parts (Key`N) = Key`N"
   302 apply auto
   303 apply (erule parts.induct, auto)
   304 done
   305 
   306 lemma msg_Nonce_supply: "\<exists>N. \<forall>n. N\<le>n --> Nonce n \<notin> parts {msg}"
   307 apply (induct_tac "msg")
   308 apply (simp_all add: parts_insert2)
   309 apply (metis Suc_n_not_le_n)
   310 apply (metis le_trans linorder_linear)
   311 done
   312 
   313 subsection\<open>Inductive relation "analz"\<close>
   314 
   315 text\<open>Inductive definition of "analz" -- what can be broken down from a set of
   316     messages, including keys.  A form of downward closure.  Pairs can
   317     be taken apart; messages decrypted with known keys.\<close>
   318 
   319 inductive_set
   320   analz :: "msg set => msg set"
   321   for H :: "msg set"
   322   where
   323     Inj [intro,simp] :    "X \<in> H ==> X \<in> analz H"
   324   | Fst:     "\<lbrace>X,Y\<rbrace> \<in> analz H ==> X \<in> analz H"
   325   | Snd:     "\<lbrace>X,Y\<rbrace> \<in> analz H ==> Y \<in> analz H"
   326   | Decrypt [dest]:
   327              "[|Crypt K X \<in> analz H; Key(invKey K): analz H|] ==> X \<in> analz H"
   328 
   329 
   330 text\<open>Monotonicity; Lemma 1 of Lowe's paper\<close>
   331 lemma analz_mono: "G\<subseteq>H ==> analz(G) \<subseteq> analz(H)"
   332 apply auto
   333 apply (erule analz.induct)
   334 apply (auto dest: analz.Fst analz.Snd)
   335 done
   336 
   337 text\<open>Making it safe speeds up proofs\<close>
   338 lemma MPair_analz [elim!]:
   339      "[| \<lbrace>X,Y\<rbrace> \<in> analz H;
   340              [| X \<in> analz H; Y \<in> analz H |] ==> P
   341           |] ==> P"
   342 by (blast dest: analz.Fst analz.Snd)
   343 
   344 lemma analz_increasing: "H \<subseteq> analz(H)"
   345 by blast
   346 
   347 lemma analz_subset_parts: "analz H \<subseteq> parts H"
   348 apply (rule subsetI)
   349 apply (erule analz.induct, blast+)
   350 done
   351 
   352 lemmas analz_into_parts = analz_subset_parts [THEN subsetD]
   353 
   354 lemmas not_parts_not_analz = analz_subset_parts [THEN contra_subsetD]
   355 
   356 lemma parts_analz [simp]: "parts (analz H) = parts H"
   357 apply (rule equalityI)
   358 apply (metis analz_subset_parts parts_subset_iff)
   359 apply (metis analz_increasing parts_mono)
   360 done
   361 
   362 
   363 lemma analz_parts [simp]: "analz (parts H) = parts H"
   364 apply auto
   365 apply (erule analz.induct, auto)
   366 done
   367 
   368 lemmas analz_insertI = subset_insertI [THEN analz_mono, THEN [2] rev_subsetD]
   369 
   370 subsubsection\<open>General equational properties\<close>
   371 
   372 lemma analz_empty [simp]: "analz{} = {}"
   373 apply safe
   374 apply (erule analz.induct, blast+)
   375 done
   376 
   377 text\<open>Converse fails: we can analz more from the union than from the
   378   separate parts, as a key in one might decrypt a message in the other\<close>
   379 lemma analz_Un: "analz(G) \<union> analz(H) \<subseteq> analz(G \<union> H)"
   380 by (intro Un_least analz_mono Un_upper1 Un_upper2)
   381 
   382 lemma analz_insert: "insert X (analz H) \<subseteq> analz(insert X H)"
   383 by (blast intro: analz_mono [THEN [2] rev_subsetD])
   384 
   385 subsubsection\<open>Rewrite rules for pulling out atomic messages\<close>
   386 
   387 lemmas analz_insert_eq_I = equalityI [OF subsetI analz_insert]
   388 
   389 lemma analz_insert_Agent [simp]:
   390      "analz (insert (Agent agt) H) = insert (Agent agt) (analz H)"
   391 apply (rule analz_insert_eq_I)
   392 apply (erule analz.induct, auto)
   393 done
   394 
   395 lemma analz_insert_Nonce [simp]:
   396      "analz (insert (Nonce N) H) = insert (Nonce N) (analz H)"
   397 apply (rule analz_insert_eq_I)
   398 apply (erule analz.induct, auto)
   399 done
   400 
   401 lemma analz_insert_Number [simp]:
   402      "analz (insert (Number N) H) = insert (Number N) (analz H)"
   403 apply (rule analz_insert_eq_I)
   404 apply (erule analz.induct, auto)
   405 done
   406 
   407 lemma analz_insert_Hash [simp]:
   408      "analz (insert (Hash X) H) = insert (Hash X) (analz H)"
   409 apply (rule analz_insert_eq_I)
   410 apply (erule analz.induct, auto)
   411 done
   412 
   413 text\<open>Can only pull out Keys if they are not needed to decrypt the rest\<close>
   414 lemma analz_insert_Key [simp]:
   415     "K \<notin> keysFor (analz H) ==>
   416           analz (insert (Key K) H) = insert (Key K) (analz H)"
   417 apply (unfold keysFor_def)
   418 apply (rule analz_insert_eq_I)
   419 apply (erule analz.induct, auto)
   420 done
   421 
   422 lemma analz_insert_MPair [simp]:
   423      "analz (insert \<lbrace>X,Y\<rbrace> H) =
   424           insert \<lbrace>X,Y\<rbrace> (analz (insert X (insert Y H)))"
   425 apply (rule equalityI)
   426 apply (rule subsetI)
   427 apply (erule analz.induct, auto)
   428 apply (erule analz.induct)
   429 apply (blast intro: analz.Fst analz.Snd)+
   430 done
   431 
   432 text\<open>Can pull out enCrypted message if the Key is not known\<close>
   433 lemma analz_insert_Crypt:
   434      "Key (invKey K) \<notin> analz H
   435       ==> analz (insert (Crypt K X) H) = insert (Crypt K X) (analz H)"
   436 apply (rule analz_insert_eq_I)
   437 apply (erule analz.induct, auto)
   438 
   439 done
   440 
   441 lemma lemma1: "Key (invKey K) \<in> analz H ==>
   442                analz (insert (Crypt K X) H) \<subseteq>
   443                insert (Crypt K X) (analz (insert X H))"
   444 apply (rule subsetI)
   445 apply (erule_tac x = x in analz.induct, auto)
   446 done
   447 
   448 lemma lemma2: "Key (invKey K) \<in> analz H ==>
   449                insert (Crypt K X) (analz (insert X H)) \<subseteq>
   450                analz (insert (Crypt K X) H)"
   451 apply auto
   452 apply (erule_tac x = x in analz.induct, auto)
   453 apply (blast intro: analz_insertI analz.Decrypt)
   454 done
   455 
   456 lemma analz_insert_Decrypt:
   457      "Key (invKey K) \<in> analz H ==>
   458                analz (insert (Crypt K X) H) =
   459                insert (Crypt K X) (analz (insert X H))"
   460 by (intro equalityI lemma1 lemma2)
   461 
   462 text\<open>Case analysis: either the message is secure, or it is not! Effective,
   463 but can cause subgoals to blow up! Use with \<open>if_split\<close>; apparently
   464 \<open>split_tac\<close> does not cope with patterns such as @{term"analz (insert
   465 (Crypt K X) H)"}\<close>
   466 lemma analz_Crypt_if [simp]:
   467      "analz (insert (Crypt K X) H) =
   468           (if (Key (invKey K) \<in> analz H)
   469            then insert (Crypt K X) (analz (insert X H))
   470            else insert (Crypt K X) (analz H))"
   471 by (simp add: analz_insert_Crypt analz_insert_Decrypt)
   472 
   473 
   474 text\<open>This rule supposes "for the sake of argument" that we have the key.\<close>
   475 lemma analz_insert_Crypt_subset:
   476      "analz (insert (Crypt K X) H) \<subseteq>
   477            insert (Crypt K X) (analz (insert X H))"
   478 apply (rule subsetI)
   479 apply (erule analz.induct, auto)
   480 done
   481 
   482 
   483 lemma analz_image_Key [simp]: "analz (Key`N) = Key`N"
   484 apply auto
   485 apply (erule analz.induct, auto)
   486 done
   487 
   488 
   489 subsubsection\<open>Idempotence and transitivity\<close>
   490 
   491 lemma analz_analzD [dest!]: "X\<in> analz (analz H) ==> X\<in> analz H"
   492 by (erule analz.induct, blast+)
   493 
   494 lemma analz_idem [simp]: "analz (analz H) = analz H"
   495 by blast
   496 
   497 lemma analz_subset_iff [simp]: "(analz G \<subseteq> analz H) = (G \<subseteq> analz H)"
   498 apply (rule iffI)
   499 apply (iprover intro: subset_trans analz_increasing)
   500 apply (frule analz_mono, simp)
   501 done
   502 
   503 lemma analz_trans: "[| X\<in> analz G;  G \<subseteq> analz H |] ==> X\<in> analz H"
   504 by (drule analz_mono, blast)
   505 
   506 
   507 declare analz_trans[intro]
   508 
   509 lemma analz_cut: "[| Y\<in> analz (insert X H);  X\<in> analz H |] ==> Y\<in> analz H"
   510 by (metis analz_idem analz_increasing analz_mono insert_absorb insert_mono insert_subset)
   511 
   512 text\<open>This rewrite rule helps in the simplification of messages that involve
   513   the forwarding of unknown components (X).  Without it, removing occurrences
   514   of X can be very complicated.\<close>
   515 lemma analz_insert_eq: "X\<in> analz H ==> analz (insert X H) = analz H"
   516 by (blast intro: analz_cut analz_insertI)
   517 
   518 
   519 text\<open>A congruence rule for "analz"\<close>
   520 
   521 lemma analz_subset_cong:
   522      "[| analz G \<subseteq> analz G'; analz H \<subseteq> analz H' |]
   523       ==> analz (G \<union> H) \<subseteq> analz (G' \<union> H')"
   524 apply simp
   525 apply (metis Un_absorb2 Un_commute Un_subset_iff Un_upper1 Un_upper2 analz_mono)
   526 done
   527 
   528 
   529 lemma analz_cong:
   530      "[| analz G = analz G'; analz H = analz H'
   531                |] ==> analz (G \<union> H) = analz (G' \<union> H')"
   532 by (intro equalityI analz_subset_cong, simp_all)
   533 
   534 lemma analz_insert_cong:
   535      "analz H = analz H' ==> analz(insert X H) = analz(insert X H')"
   536 by (force simp only: insert_def intro!: analz_cong)
   537 
   538 text\<open>If there are no pairs or encryptions then analz does nothing\<close>
   539 lemma analz_trivial:
   540      "[| \<forall>X Y. \<lbrace>X,Y\<rbrace> \<notin> H;  \<forall>X K. Crypt K X \<notin> H |] ==> analz H = H"
   541 apply safe
   542 apply (erule analz.induct, blast+)
   543 done
   544 
   545 text\<open>These two are obsolete (with a single Spy) but cost little to prove...\<close>
   546 lemma analz_UN_analz_lemma:
   547      "X\<in> analz (\<Union>i\<in>A. analz (H i)) ==> X\<in> analz (\<Union>i\<in>A. H i)"
   548 apply (erule analz.induct)
   549 apply (blast intro: analz_mono [THEN [2] rev_subsetD])+
   550 done
   551 
   552 lemma analz_UN_analz [simp]: "analz (\<Union>i\<in>A. analz (H i)) = analz (\<Union>i\<in>A. H i)"
   553 by (blast intro: analz_UN_analz_lemma analz_mono [THEN [2] rev_subsetD])
   554 
   555 
   556 subsection\<open>Inductive relation "synth"\<close>
   557 
   558 text\<open>Inductive definition of "synth" -- what can be built up from a set of
   559     messages.  A form of upward closure.  Pairs can be built, messages
   560     encrypted with known keys.  Agent names are public domain.
   561     Numbers can be guessed, but Nonces cannot be.\<close>
   562 
   563 inductive_set
   564   synth :: "msg set => msg set"
   565   for H :: "msg set"
   566   where
   567     Inj    [intro]:   "X \<in> H ==> X \<in> synth H"
   568   | Agent  [intro]:   "Agent agt \<in> synth H"
   569   | Number [intro]:   "Number n  \<in> synth H"
   570   | Hash   [intro]:   "X \<in> synth H ==> Hash X \<in> synth H"
   571   | MPair  [intro]:   "[|X \<in> synth H;  Y \<in> synth H|] ==> \<lbrace>X,Y\<rbrace> \<in> synth H"
   572   | Crypt  [intro]:   "[|X \<in> synth H;  Key(K) \<in> H|] ==> Crypt K X \<in> synth H"
   573 
   574 text\<open>Monotonicity\<close>
   575 lemma synth_mono: "G\<subseteq>H ==> synth(G) \<subseteq> synth(H)"
   576   by (auto, erule synth.induct, auto)
   577 
   578 text\<open>NO \<open>Agent_synth\<close>, as any Agent name can be synthesized.
   579   The same holds for @{term Number}\<close>
   580 inductive_cases Nonce_synth [elim!]: "Nonce n \<in> synth H"
   581 inductive_cases Key_synth   [elim!]: "Key K \<in> synth H"
   582 inductive_cases Hash_synth  [elim!]: "Hash X \<in> synth H"
   583 inductive_cases MPair_synth [elim!]: "\<lbrace>X,Y\<rbrace> \<in> synth H"
   584 inductive_cases Crypt_synth [elim!]: "Crypt K X \<in> synth H"
   585 
   586 
   587 lemma synth_increasing: "H \<subseteq> synth(H)"
   588 by blast
   589 
   590 subsubsection\<open>Unions\<close>
   591 
   592 text\<open>Converse fails: we can synth more from the union than from the
   593   separate parts, building a compound message using elements of each.\<close>
   594 lemma synth_Un: "synth(G) \<union> synth(H) \<subseteq> synth(G \<union> H)"
   595 by (intro Un_least synth_mono Un_upper1 Un_upper2)
   596 
   597 lemma synth_insert: "insert X (synth H) \<subseteq> synth(insert X H)"
   598 by (metis insert_iff insert_subset subset_insertI synth.Inj synth_mono)
   599 
   600 subsubsection\<open>Idempotence and transitivity\<close>
   601 
   602 lemma synth_synthD [dest!]: "X\<in> synth (synth H) ==> X\<in> synth H"
   603 by (erule synth.induct, blast+)
   604 
   605 lemma synth_idem: "synth (synth H) = synth H"
   606 by blast
   607 
   608 lemma synth_subset_iff [simp]: "(synth G \<subseteq> synth H) = (G \<subseteq> synth H)"
   609 apply (rule iffI)
   610 apply (iprover intro: subset_trans synth_increasing)
   611 apply (frule synth_mono, simp add: synth_idem)
   612 done
   613 
   614 lemma synth_trans: "[| X\<in> synth G;  G \<subseteq> synth H |] ==> X\<in> synth H"
   615 by (drule synth_mono, blast)
   616 
   617 lemma synth_cut: "[| Y\<in> synth (insert X H);  X\<in> synth H |] ==> Y\<in> synth H"
   618 by (metis insert_absorb insert_mono insert_subset synth_idem synth_increasing synth_mono)
   619 
   620 lemma Agent_synth [simp]: "Agent A \<in> synth H"
   621 by blast
   622 
   623 lemma Number_synth [simp]: "Number n \<in> synth H"
   624 by blast
   625 
   626 lemma Nonce_synth_eq [simp]: "(Nonce N \<in> synth H) = (Nonce N \<in> H)"
   627 by blast
   628 
   629 lemma Key_synth_eq [simp]: "(Key K \<in> synth H) = (Key K \<in> H)"
   630 by blast
   631 
   632 lemma Crypt_synth_eq [simp]:
   633      "Key K \<notin> H ==> (Crypt K X \<in> synth H) = (Crypt K X \<in> H)"
   634 by blast
   635 
   636 
   637 lemma keysFor_synth [simp]:
   638     "keysFor (synth H) = keysFor H \<union> invKey`{K. Key K \<in> H}"
   639 by (unfold keysFor_def, blast)
   640 
   641 
   642 subsubsection\<open>Combinations of parts, analz and synth\<close>
   643 
   644 lemma parts_synth [simp]: "parts (synth H) = parts H \<union> synth H"
   645 apply (rule equalityI)
   646 apply (rule subsetI)
   647 apply (erule parts.induct)
   648 apply (metis UnCI)
   649 apply (metis MPair_synth UnCI UnE insert_absorb insert_subset parts.Fst parts_increasing)
   650 apply (metis MPair_synth UnCI UnE insert_absorb insert_subset parts.Snd parts_increasing)
   651 apply (metis Body Crypt_synth UnCI UnE insert_absorb insert_subset parts_increasing)
   652 apply (metis Un_subset_iff parts_increasing parts_mono synth_increasing)
   653 done
   654 
   655 lemma analz_analz_Un [simp]: "analz (analz G \<union> H) = analz (G \<union> H)"
   656 apply (rule equalityI)
   657 apply (metis analz_idem analz_subset_cong order_eq_refl)
   658 apply (metis analz_increasing analz_subset_cong order_eq_refl)
   659 done
   660 
   661 declare analz_mono [intro] analz.Fst [intro] analz.Snd [intro] Un_least [intro]
   662 
   663 lemma analz_synth_Un [simp]: "analz (synth G \<union> H) = analz (G \<union> H) \<union> synth G"
   664 apply (rule equalityI)
   665 apply (rule subsetI)
   666 apply (erule analz.induct)
   667 apply (metis UnCI UnE Un_commute analz.Inj)
   668 apply (metis MPair_synth UnCI UnE Un_commute analz.Fst analz.Inj)
   669 apply (metis MPair_synth UnCI UnE Un_commute analz.Inj analz.Snd)
   670 apply (blast intro: analz.Decrypt)
   671 apply blast
   672 done
   673 
   674 lemma analz_synth [simp]: "analz (synth H) = analz H \<union> synth H"
   675 proof -
   676   have "\<forall>x\<^sub>2 x\<^sub>1. synth x\<^sub>1 \<union> analz (x\<^sub>1 \<union> x\<^sub>2) = analz (synth x\<^sub>1 \<union> x\<^sub>2)" by (metis Un_commute analz_synth_Un)
   677   hence "\<forall>x\<^sub>1. synth x\<^sub>1 \<union> analz x\<^sub>1 = analz (synth x\<^sub>1 \<union> {})" by (metis Un_empty_right)
   678   hence "\<forall>x\<^sub>1. synth x\<^sub>1 \<union> analz x\<^sub>1 = analz (synth x\<^sub>1)" by (metis Un_empty_right)
   679   hence "\<forall>x\<^sub>1. analz x\<^sub>1 \<union> synth x\<^sub>1 = analz (synth x\<^sub>1)" by (metis Un_commute)
   680   thus "analz (synth H) = analz H \<union> synth H" by metis
   681 qed
   682 
   683 
   684 subsubsection\<open>For reasoning about the Fake rule in traces\<close>
   685 
   686 lemma parts_insert_subset_Un: "X \<in> G ==> parts(insert X H) \<subseteq> parts G \<union> parts H"
   687 proof -
   688   assume "X \<in> G"
   689   hence "\<forall>x\<^sub>1. G \<subseteq> x\<^sub>1 \<longrightarrow> X \<in> x\<^sub>1 " by auto
   690   hence "\<forall>x\<^sub>1. X \<in> G \<union> x\<^sub>1" by (metis Un_upper1)
   691   hence "insert X H \<subseteq> G \<union> H" by (metis Un_upper2 insert_subset)
   692   hence "parts (insert X H) \<subseteq> parts (G \<union> H)" by (metis parts_mono)
   693   thus "parts (insert X H) \<subseteq> parts G \<union> parts H" by (metis parts_Un)
   694 qed
   695 
   696 lemma Fake_parts_insert:
   697      "X \<in> synth (analz H) ==>
   698       parts (insert X H) \<subseteq> synth (analz H) \<union> parts H"
   699 proof -
   700   assume A1: "X \<in> synth (analz H)"
   701   have F1: "\<forall>x\<^sub>1. analz x\<^sub>1 \<union> synth (analz x\<^sub>1) = analz (synth (analz x\<^sub>1))"
   702     by (metis analz_idem analz_synth)
   703   have F2: "\<forall>x\<^sub>1. parts x\<^sub>1 \<union> synth (analz x\<^sub>1) = parts (synth (analz x\<^sub>1))"
   704     by (metis parts_analz parts_synth)
   705   have F3: "X \<in> synth (analz H)" using A1 by metis
   706   have "\<forall>x\<^sub>2 x\<^sub>1::msg set. x\<^sub>1 \<le> sup x\<^sub>1 x\<^sub>2" by (metis inf_sup_ord(3))
   707   hence F4: "\<forall>x\<^sub>1. analz x\<^sub>1 \<subseteq> analz (synth x\<^sub>1)" by (metis analz_synth)
   708   have F5: "X \<in> synth (analz H)" using F3 by metis
   709   have "\<forall>x\<^sub>1. analz x\<^sub>1 \<subseteq> synth (analz x\<^sub>1)
   710          \<longrightarrow> analz (synth (analz x\<^sub>1)) = synth (analz x\<^sub>1)"
   711     using F1 by (metis subset_Un_eq)
   712   hence F6: "\<forall>x\<^sub>1. analz (synth (analz x\<^sub>1)) = synth (analz x\<^sub>1)"
   713     by (metis synth_increasing)
   714   have "\<forall>x\<^sub>1. x\<^sub>1 \<subseteq> analz (synth x\<^sub>1)" using F4 by (metis analz_subset_iff)
   715   hence "\<forall>x\<^sub>1. x\<^sub>1 \<subseteq> analz (synth (analz x\<^sub>1))" by (metis analz_subset_iff)
   716   hence "\<forall>x\<^sub>1. x\<^sub>1 \<subseteq> synth (analz x\<^sub>1)" using F6 by metis
   717   hence "H \<subseteq> synth (analz H)" by metis
   718   hence "H \<subseteq> synth (analz H) \<and> X \<in> synth (analz H)" using F5 by metis
   719   hence "insert X H \<subseteq> synth (analz H)" by (metis insert_subset)
   720   hence "parts (insert X H) \<subseteq> parts (synth (analz H))" by (metis parts_mono)
   721   hence "parts (insert X H) \<subseteq> parts H \<union> synth (analz H)" using F2 by metis
   722   thus "parts (insert X H) \<subseteq> synth (analz H) \<union> parts H" by (metis Un_commute)
   723 qed
   724 
   725 lemma Fake_parts_insert_in_Un:
   726      "[|Z \<in> parts (insert X H);  X: synth (analz H)|]
   727       ==> Z \<in>  synth (analz H) \<union> parts H"
   728 by (blast dest: Fake_parts_insert [THEN subsetD, dest])
   729 
   730 declare synth_mono [intro]
   731 
   732 lemma Fake_analz_insert:
   733      "X \<in> synth (analz G) ==>
   734       analz (insert X H) \<subseteq> synth (analz G) \<union> analz (G \<union> H)"
   735 by (metis Un_commute Un_insert_left Un_insert_right Un_upper1 analz_analz_Un
   736           analz_mono analz_synth_Un insert_absorb)
   737 
   738 lemma Fake_analz_insert_simpler:
   739      "X \<in> synth (analz G) ==>
   740       analz (insert X H) \<subseteq> synth (analz G) \<union> analz (G \<union> H)"
   741 apply (rule subsetI)
   742 apply (subgoal_tac "x \<in> analz (synth (analz G) \<union> H) ")
   743 apply (metis Un_commute analz_analz_Un analz_synth_Un)
   744 by (metis Un_upper1 Un_upper2 analz_mono insert_absorb insert_subset)
   745 
   746 end