src/HOL/Metis_Examples/Message.thy
author hoelzl
Tue Mar 23 16:17:41 2010 +0100 (2010-03-23)
changeset 35928 d31f55f97663
parent 35416 d8d7d1b785af
child 36553 95bdfa572cee
permissions -rw-r--r--
Generate image for HOL-Probability
     1 (*  Title:      HOL/MetisTest/Message.thy
     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     3 
     4 Testing the metis method.
     5 *)
     6 
     7 theory Message imports Main begin
     8 
     9 (*Needed occasionally with spy_analz_tac, e.g. in analz_insert_Key_newK*)
    10 lemma strange_Un_eq [simp]: "A \<union> (B \<union> A) = B \<union> A"
    11 by blast
    12 
    13 types 
    14   key = nat
    15 
    16 consts
    17   all_symmetric :: bool        --{*true if all keys are symmetric*}
    18   invKey        :: "key=>key"  --{*inverse of a symmetric key*}
    19 
    20 specification (invKey)
    21   invKey [simp]: "invKey (invKey K) = K"
    22   invKey_symmetric: "all_symmetric --> invKey = id"
    23     by (rule exI [of _ id], auto)
    24 
    25 
    26 text{*The inverse of a symmetric key is itself; that of a public key
    27       is the private key and vice versa*}
    28 
    29 definition symKeys :: "key set" where
    30   "symKeys == {K. invKey K = K}"
    31 
    32 datatype  --{*We allow any number of friendly agents*}
    33   agent = Server | Friend nat | Spy
    34 
    35 datatype
    36      msg = Agent  agent     --{*Agent names*}
    37          | Number nat       --{*Ordinary integers, timestamps, ...*}
    38          | Nonce  nat       --{*Unguessable nonces*}
    39          | Key    key       --{*Crypto keys*}
    40          | Hash   msg       --{*Hashing*}
    41          | MPair  msg msg   --{*Compound messages*}
    42          | Crypt  key msg   --{*Encryption, public- or shared-key*}
    43 
    44 
    45 text{*Concrete syntax: messages appear as {|A,B,NA|}, etc...*}
    46 syntax
    47   "_MTuple"      :: "['a, args] => 'a * 'b"       ("(2{|_,/ _|})")
    48 
    49 syntax (xsymbols)
    50   "_MTuple"      :: "['a, args] => 'a * 'b"       ("(2\<lbrace>_,/ _\<rbrace>)")
    51 
    52 translations
    53   "{|x, y, z|}"   == "{|x, {|y, z|}|}"
    54   "{|x, y|}"      == "CONST MPair x y"
    55 
    56 
    57 definition HPair :: "[msg,msg] => msg" ("(4Hash[_] /_)" [0, 1000]) where
    58     --{*Message Y paired with a MAC computed with the help of X*}
    59     "Hash[X] Y == {| Hash{|X,Y|}, Y|}"
    60 
    61 definition keysFor :: "msg set => key set" where
    62     --{*Keys useful to decrypt elements of a message set*}
    63   "keysFor H == invKey ` {K. \<exists>X. Crypt K X \<in> H}"
    64 
    65 
    66 subsubsection{*Inductive Definition of All Parts" of a Message*}
    67 
    68 inductive_set
    69   parts :: "msg set => msg set"
    70   for H :: "msg set"
    71   where
    72     Inj [intro]:               "X \<in> H ==> X \<in> parts H"
    73   | Fst:         "{|X,Y|}   \<in> parts H ==> X \<in> parts H"
    74   | Snd:         "{|X,Y|}   \<in> parts H ==> Y \<in> parts H"
    75   | Body:        "Crypt K X \<in> parts H ==> X \<in> parts H"
    76 
    77 
    78 declare [[ atp_problem_prefix = "Message__parts_mono" ]]
    79 lemma parts_mono: "G \<subseteq> H ==> parts(G) \<subseteq> parts(H)"
    80 apply auto
    81 apply (erule parts.induct) 
    82 apply (metis Inj set_mp)
    83 apply (metis Fst)
    84 apply (metis Snd)
    85 apply (metis Body)
    86 done
    87 
    88 
    89 text{*Equations hold because constructors are injective.*}
    90 lemma Friend_image_eq [simp]: "(Friend x \<in> Friend`A) = (x:A)"
    91 by auto
    92 
    93 lemma Key_image_eq [simp]: "(Key x \<in> Key`A) = (x\<in>A)"
    94 by auto
    95 
    96 lemma Nonce_Key_image_eq [simp]: "(Nonce x \<notin> Key`A)"
    97 by auto
    98 
    99 
   100 subsubsection{*Inverse of keys *}
   101 
   102 declare [[ atp_problem_prefix = "Message__invKey_eq" ]]
   103 lemma invKey_eq [simp]: "(invKey K = invKey K') = (K=K')"
   104 by (metis invKey)
   105 
   106 
   107 subsection{*keysFor operator*}
   108 
   109 lemma keysFor_empty [simp]: "keysFor {} = {}"
   110 by (unfold keysFor_def, blast)
   111 
   112 lemma keysFor_Un [simp]: "keysFor (H \<union> H') = keysFor H \<union> keysFor H'"
   113 by (unfold keysFor_def, blast)
   114 
   115 lemma keysFor_UN [simp]: "keysFor (\<Union>i\<in>A. H i) = (\<Union>i\<in>A. keysFor (H i))"
   116 by (unfold keysFor_def, blast)
   117 
   118 text{*Monotonicity*}
   119 lemma keysFor_mono: "G \<subseteq> H ==> keysFor(G) \<subseteq> keysFor(H)"
   120 by (unfold keysFor_def, blast)
   121 
   122 lemma keysFor_insert_Agent [simp]: "keysFor (insert (Agent A) H) = keysFor H"
   123 by (unfold keysFor_def, auto)
   124 
   125 lemma keysFor_insert_Nonce [simp]: "keysFor (insert (Nonce N) H) = keysFor H"
   126 by (unfold keysFor_def, auto)
   127 
   128 lemma keysFor_insert_Number [simp]: "keysFor (insert (Number N) H) = keysFor H"
   129 by (unfold keysFor_def, auto)
   130 
   131 lemma keysFor_insert_Key [simp]: "keysFor (insert (Key K) H) = keysFor H"
   132 by (unfold keysFor_def, auto)
   133 
   134 lemma keysFor_insert_Hash [simp]: "keysFor (insert (Hash X) H) = keysFor H"
   135 by (unfold keysFor_def, auto)
   136 
   137 lemma keysFor_insert_MPair [simp]: "keysFor (insert {|X,Y|} H) = keysFor H"
   138 by (unfold keysFor_def, auto)
   139 
   140 lemma keysFor_insert_Crypt [simp]: 
   141     "keysFor (insert (Crypt K X) H) = insert (invKey K) (keysFor H)"
   142 by (unfold keysFor_def, auto)
   143 
   144 lemma keysFor_image_Key [simp]: "keysFor (Key`E) = {}"
   145 by (unfold keysFor_def, auto)
   146 
   147 lemma Crypt_imp_invKey_keysFor: "Crypt K X \<in> H ==> invKey K \<in> keysFor H"
   148 by (unfold keysFor_def, blast)
   149 
   150 
   151 subsection{*Inductive relation "parts"*}
   152 
   153 lemma MPair_parts:
   154      "[| {|X,Y|} \<in> parts H;        
   155          [| X \<in> parts H; Y \<in> parts H |] ==> P |] ==> P"
   156 by (blast dest: parts.Fst parts.Snd) 
   157 
   158     declare MPair_parts [elim!]  parts.Body [dest!]
   159 text{*NB These two rules are UNSAFE in the formal sense, as they discard the
   160      compound message.  They work well on THIS FILE.  
   161   @{text MPair_parts} is left as SAFE because it speeds up proofs.
   162   The Crypt rule is normally kept UNSAFE to avoid breaking up certificates.*}
   163 
   164 lemma parts_increasing: "H \<subseteq> parts(H)"
   165 by blast
   166 
   167 lemmas parts_insertI = subset_insertI [THEN parts_mono, THEN subsetD, standard]
   168 
   169 lemma parts_empty [simp]: "parts{} = {}"
   170 apply safe
   171 apply (erule parts.induct)
   172 apply blast+
   173 done
   174 
   175 lemma parts_emptyE [elim!]: "X\<in> parts{} ==> P"
   176 by simp
   177 
   178 text{*WARNING: loops if H = {Y}, therefore must not be repeated!*}
   179 lemma parts_singleton: "X\<in> parts H ==> \<exists>Y\<in>H. X\<in> parts {Y}"
   180 apply (erule parts.induct)
   181 apply fast+
   182 done
   183 
   184 
   185 subsubsection{*Unions *}
   186 
   187 lemma parts_Un_subset1: "parts(G) \<union> parts(H) \<subseteq> parts(G \<union> H)"
   188 by (intro Un_least parts_mono Un_upper1 Un_upper2)
   189 
   190 lemma parts_Un_subset2: "parts(G \<union> H) \<subseteq> parts(G) \<union> parts(H)"
   191 apply (rule subsetI)
   192 apply (erule parts.induct, blast+)
   193 done
   194 
   195 lemma parts_Un [simp]: "parts(G \<union> H) = parts(G) \<union> parts(H)"
   196 by (intro equalityI parts_Un_subset1 parts_Un_subset2)
   197 
   198 lemma parts_insert: "parts (insert X H) = parts {X} \<union> parts H"
   199 apply (subst insert_is_Un [of _ H])
   200 apply (simp only: parts_Un)
   201 done
   202 
   203 declare [[ atp_problem_prefix = "Message__parts_insert_two" ]]
   204 lemma parts_insert2:
   205      "parts (insert X (insert Y H)) = parts {X} \<union> parts {Y} \<union> parts H"
   206 by (metis Un_commute Un_empty_left Un_empty_right Un_insert_left Un_insert_right parts_Un)
   207 
   208 
   209 lemma parts_UN_subset1: "(\<Union>x\<in>A. parts(H x)) \<subseteq> parts(\<Union>x\<in>A. H x)"
   210 by (intro UN_least parts_mono UN_upper)
   211 
   212 lemma parts_UN_subset2: "parts(\<Union>x\<in>A. H x) \<subseteq> (\<Union>x\<in>A. parts(H x))"
   213 apply (rule subsetI)
   214 apply (erule parts.induct, blast+)
   215 done
   216 
   217 lemma parts_UN [simp]: "parts(\<Union>x\<in>A. H x) = (\<Union>x\<in>A. parts(H x))"
   218 by (intro equalityI parts_UN_subset1 parts_UN_subset2)
   219 
   220 text{*Added to simplify arguments to parts, analz and synth.
   221   NOTE: the UN versions are no longer used!*}
   222 
   223 
   224 text{*This allows @{text blast} to simplify occurrences of 
   225   @{term "parts(G\<union>H)"} in the assumption.*}
   226 lemmas in_parts_UnE = parts_Un [THEN equalityD1, THEN subsetD, THEN UnE] 
   227 declare in_parts_UnE [elim!]
   228 
   229 lemma parts_insert_subset: "insert X (parts H) \<subseteq> parts(insert X H)"
   230 by (blast intro: parts_mono [THEN [2] rev_subsetD])
   231 
   232 subsubsection{*Idempotence and transitivity *}
   233 
   234 lemma parts_partsD [dest!]: "X\<in> parts (parts H) ==> X\<in> parts H"
   235 by (erule parts.induct, blast+)
   236 
   237 lemma parts_idem [simp]: "parts (parts H) = parts H"
   238 by blast
   239 
   240 declare [[ atp_problem_prefix = "Message__parts_subset_iff" ]]
   241 lemma parts_subset_iff [simp]: "(parts G \<subseteq> parts H) = (G \<subseteq> parts H)"
   242 apply (rule iffI) 
   243 apply (metis Un_absorb1 Un_subset_iff parts_Un parts_increasing)
   244 apply (metis parts_idem parts_mono)
   245 done
   246 
   247 lemma parts_trans: "[| X\<in> parts G;  G \<subseteq> parts H |] ==> X\<in> parts H"
   248 by (blast dest: parts_mono); 
   249 
   250 
   251 declare [[ atp_problem_prefix = "Message__parts_cut" ]]
   252 lemma parts_cut: "[|Y\<in> parts(insert X G);  X\<in> parts H|] ==> Y\<in> parts(G \<union> H)"
   253 by (metis Un_insert_left Un_insert_right insert_absorb mem_def parts_Un parts_idem sup1CI)
   254 
   255 
   256 
   257 subsubsection{*Rewrite rules for pulling out atomic messages *}
   258 
   259 lemmas parts_insert_eq_I = equalityI [OF subsetI parts_insert_subset]
   260 
   261 
   262 lemma parts_insert_Agent [simp]:
   263      "parts (insert (Agent agt) H) = insert (Agent agt) (parts H)"
   264 apply (rule parts_insert_eq_I) 
   265 apply (erule parts.induct, auto) 
   266 done
   267 
   268 lemma parts_insert_Nonce [simp]:
   269      "parts (insert (Nonce N) H) = insert (Nonce N) (parts H)"
   270 apply (rule parts_insert_eq_I) 
   271 apply (erule parts.induct, auto) 
   272 done
   273 
   274 lemma parts_insert_Number [simp]:
   275      "parts (insert (Number N) H) = insert (Number N) (parts H)"
   276 apply (rule parts_insert_eq_I) 
   277 apply (erule parts.induct, auto) 
   278 done
   279 
   280 lemma parts_insert_Key [simp]:
   281      "parts (insert (Key K) H) = insert (Key K) (parts H)"
   282 apply (rule parts_insert_eq_I) 
   283 apply (erule parts.induct, auto) 
   284 done
   285 
   286 lemma parts_insert_Hash [simp]:
   287      "parts (insert (Hash X) H) = insert (Hash X) (parts H)"
   288 apply (rule parts_insert_eq_I) 
   289 apply (erule parts.induct, auto) 
   290 done
   291 
   292 lemma parts_insert_Crypt [simp]:
   293      "parts (insert (Crypt K X) H) =  
   294           insert (Crypt K X) (parts (insert X H))"
   295 apply (rule equalityI)
   296 apply (rule subsetI)
   297 apply (erule parts.induct, auto)
   298 apply (blast intro: parts.Body)
   299 done
   300 
   301 lemma parts_insert_MPair [simp]:
   302      "parts (insert {|X,Y|} H) =  
   303           insert {|X,Y|} (parts (insert X (insert Y H)))"
   304 apply (rule equalityI)
   305 apply (rule subsetI)
   306 apply (erule parts.induct, auto)
   307 apply (blast intro: parts.Fst parts.Snd)+
   308 done
   309 
   310 lemma parts_image_Key [simp]: "parts (Key`N) = Key`N"
   311 apply auto
   312 apply (erule parts.induct, auto)
   313 done
   314 
   315 
   316 declare [[ atp_problem_prefix = "Message__msg_Nonce_supply" ]]
   317 lemma msg_Nonce_supply: "\<exists>N. \<forall>n. N\<le>n --> Nonce n \<notin> parts {msg}"
   318 apply (induct_tac "msg") 
   319 apply (simp_all add: parts_insert2)
   320 apply (metis Suc_n_not_le_n)
   321 apply (metis le_trans linorder_linear)
   322 done
   323 
   324 subsection{*Inductive relation "analz"*}
   325 
   326 text{*Inductive definition of "analz" -- what can be broken down from a set of
   327     messages, including keys.  A form of downward closure.  Pairs can
   328     be taken apart; messages decrypted with known keys.  *}
   329 
   330 inductive_set
   331   analz :: "msg set => msg set"
   332   for H :: "msg set"
   333   where
   334     Inj [intro,simp] :    "X \<in> H ==> X \<in> analz H"
   335   | Fst:     "{|X,Y|} \<in> analz H ==> X \<in> analz H"
   336   | Snd:     "{|X,Y|} \<in> analz H ==> Y \<in> analz H"
   337   | Decrypt [dest]: 
   338              "[|Crypt K X \<in> analz H; Key(invKey K): analz H|] ==> X \<in> analz H"
   339 
   340 
   341 text{*Monotonicity; Lemma 1 of Lowe's paper*}
   342 lemma analz_mono: "G\<subseteq>H ==> analz(G) \<subseteq> analz(H)"
   343 apply auto
   344 apply (erule analz.induct) 
   345 apply (auto dest: analz.Fst analz.Snd) 
   346 done
   347 
   348 text{*Making it safe speeds up proofs*}
   349 lemma MPair_analz [elim!]:
   350      "[| {|X,Y|} \<in> analz H;        
   351              [| X \<in> analz H; Y \<in> analz H |] ==> P   
   352           |] ==> P"
   353 by (blast dest: analz.Fst analz.Snd)
   354 
   355 lemma analz_increasing: "H \<subseteq> analz(H)"
   356 by blast
   357 
   358 lemma analz_subset_parts: "analz H \<subseteq> parts H"
   359 apply (rule subsetI)
   360 apply (erule analz.induct, blast+)
   361 done
   362 
   363 lemmas analz_into_parts = analz_subset_parts [THEN subsetD, standard]
   364 
   365 lemmas not_parts_not_analz = analz_subset_parts [THEN contra_subsetD, standard]
   366 
   367 
   368 declare [[ atp_problem_prefix = "Message__parts_analz" ]]
   369 lemma parts_analz [simp]: "parts (analz H) = parts H"
   370 apply (rule equalityI)
   371 apply (metis analz_subset_parts parts_subset_iff)
   372 apply (metis analz_increasing parts_mono)
   373 done
   374 
   375 
   376 lemma analz_parts [simp]: "analz (parts H) = parts H"
   377 apply auto
   378 apply (erule analz.induct, auto)
   379 done
   380 
   381 lemmas analz_insertI = subset_insertI [THEN analz_mono, THEN [2] rev_subsetD, standard]
   382 
   383 subsubsection{*General equational properties *}
   384 
   385 lemma analz_empty [simp]: "analz{} = {}"
   386 apply safe
   387 apply (erule analz.induct, blast+)
   388 done
   389 
   390 text{*Converse fails: we can analz more from the union than from the 
   391   separate parts, as a key in one might decrypt a message in the other*}
   392 lemma analz_Un: "analz(G) \<union> analz(H) \<subseteq> analz(G \<union> H)"
   393 by (intro Un_least analz_mono Un_upper1 Un_upper2)
   394 
   395 lemma analz_insert: "insert X (analz H) \<subseteq> analz(insert X H)"
   396 by (blast intro: analz_mono [THEN [2] rev_subsetD])
   397 
   398 subsubsection{*Rewrite rules for pulling out atomic messages *}
   399 
   400 lemmas analz_insert_eq_I = equalityI [OF subsetI analz_insert]
   401 
   402 lemma analz_insert_Agent [simp]:
   403      "analz (insert (Agent agt) H) = insert (Agent agt) (analz H)"
   404 apply (rule analz_insert_eq_I) 
   405 apply (erule analz.induct, auto) 
   406 done
   407 
   408 lemma analz_insert_Nonce [simp]:
   409      "analz (insert (Nonce N) H) = insert (Nonce N) (analz H)"
   410 apply (rule analz_insert_eq_I) 
   411 apply (erule analz.induct, auto) 
   412 done
   413 
   414 lemma analz_insert_Number [simp]:
   415      "analz (insert (Number N) H) = insert (Number N) (analz H)"
   416 apply (rule analz_insert_eq_I) 
   417 apply (erule analz.induct, auto) 
   418 done
   419 
   420 lemma analz_insert_Hash [simp]:
   421      "analz (insert (Hash X) H) = insert (Hash X) (analz H)"
   422 apply (rule analz_insert_eq_I) 
   423 apply (erule analz.induct, auto) 
   424 done
   425 
   426 text{*Can only pull out Keys if they are not needed to decrypt the rest*}
   427 lemma analz_insert_Key [simp]: 
   428     "K \<notin> keysFor (analz H) ==>   
   429           analz (insert (Key K) H) = insert (Key K) (analz H)"
   430 apply (unfold keysFor_def)
   431 apply (rule analz_insert_eq_I) 
   432 apply (erule analz.induct, auto) 
   433 done
   434 
   435 lemma analz_insert_MPair [simp]:
   436      "analz (insert {|X,Y|} H) =  
   437           insert {|X,Y|} (analz (insert X (insert Y H)))"
   438 apply (rule equalityI)
   439 apply (rule subsetI)
   440 apply (erule analz.induct, auto)
   441 apply (erule analz.induct)
   442 apply (blast intro: analz.Fst analz.Snd)+
   443 done
   444 
   445 text{*Can pull out enCrypted message if the Key is not known*}
   446 lemma analz_insert_Crypt:
   447      "Key (invKey K) \<notin> analz H 
   448       ==> analz (insert (Crypt K X) H) = insert (Crypt K X) (analz H)"
   449 apply (rule analz_insert_eq_I) 
   450 apply (erule analz.induct, auto) 
   451 
   452 done
   453 
   454 lemma lemma1: "Key (invKey K) \<in> analz H ==>   
   455                analz (insert (Crypt K X) H) \<subseteq>  
   456                insert (Crypt K X) (analz (insert X H))" 
   457 apply (rule subsetI)
   458 apply (erule_tac x = x in analz.induct, auto)
   459 done
   460 
   461 lemma lemma2: "Key (invKey K) \<in> analz H ==>   
   462                insert (Crypt K X) (analz (insert X H)) \<subseteq>  
   463                analz (insert (Crypt K X) H)"
   464 apply auto
   465 apply (erule_tac x = x in analz.induct, auto)
   466 apply (blast intro: analz_insertI analz.Decrypt)
   467 done
   468 
   469 lemma analz_insert_Decrypt:
   470      "Key (invKey K) \<in> analz H ==>   
   471                analz (insert (Crypt K X) H) =  
   472                insert (Crypt K X) (analz (insert X H))"
   473 by (intro equalityI lemma1 lemma2)
   474 
   475 text{*Case analysis: either the message is secure, or it is not! Effective,
   476 but can cause subgoals to blow up! Use with @{text "split_if"}; apparently
   477 @{text "split_tac"} does not cope with patterns such as @{term"analz (insert
   478 (Crypt K X) H)"} *} 
   479 lemma analz_Crypt_if [simp]:
   480      "analz (insert (Crypt K X) H) =                 
   481           (if (Key (invKey K) \<in> analz H)                 
   482            then insert (Crypt K X) (analz (insert X H))  
   483            else insert (Crypt K X) (analz H))"
   484 by (simp add: analz_insert_Crypt analz_insert_Decrypt)
   485 
   486 
   487 text{*This rule supposes "for the sake of argument" that we have the key.*}
   488 lemma analz_insert_Crypt_subset:
   489      "analz (insert (Crypt K X) H) \<subseteq>   
   490            insert (Crypt K X) (analz (insert X H))"
   491 apply (rule subsetI)
   492 apply (erule analz.induct, auto)
   493 done
   494 
   495 
   496 lemma analz_image_Key [simp]: "analz (Key`N) = Key`N"
   497 apply auto
   498 apply (erule analz.induct, auto)
   499 done
   500 
   501 
   502 subsubsection{*Idempotence and transitivity *}
   503 
   504 lemma analz_analzD [dest!]: "X\<in> analz (analz H) ==> X\<in> analz H"
   505 by (erule analz.induct, blast+)
   506 
   507 lemma analz_idem [simp]: "analz (analz H) = analz H"
   508 by blast
   509 
   510 lemma analz_subset_iff [simp]: "(analz G \<subseteq> analz H) = (G \<subseteq> analz H)"
   511 apply (rule iffI)
   512 apply (iprover intro: subset_trans analz_increasing)  
   513 apply (frule analz_mono, simp) 
   514 done
   515 
   516 lemma analz_trans: "[| X\<in> analz G;  G \<subseteq> analz H |] ==> X\<in> analz H"
   517 by (drule analz_mono, blast)
   518 
   519 
   520 declare [[ atp_problem_prefix = "Message__analz_cut" ]]
   521     declare analz_trans[intro]
   522 lemma analz_cut: "[| Y\<in> analz (insert X H);  X\<in> analz H |] ==> Y\<in> analz H"
   523 (*TOO SLOW
   524 by (metis analz_idem analz_increasing analz_mono insert_absorb insert_mono insert_subset) --{*317s*}
   525 ??*)
   526 by (erule analz_trans, blast)
   527 
   528 
   529 text{*This rewrite rule helps in the simplification of messages that involve
   530   the forwarding of unknown components (X).  Without it, removing occurrences
   531   of X can be very complicated. *}
   532 lemma analz_insert_eq: "X\<in> analz H ==> analz (insert X H) = analz H"
   533 by (blast intro: analz_cut analz_insertI)
   534 
   535 
   536 text{*A congruence rule for "analz" *}
   537 
   538 declare [[ atp_problem_prefix = "Message__analz_subset_cong" ]]
   539 lemma analz_subset_cong:
   540      "[| analz G \<subseteq> analz G'; analz H \<subseteq> analz H' |] 
   541       ==> analz (G \<union> H) \<subseteq> analz (G' \<union> H')"
   542 apply simp
   543 apply (metis Un_absorb2 Un_commute Un_subset_iff Un_upper1 Un_upper2 analz_mono)
   544 done
   545 
   546 
   547 lemma analz_cong:
   548      "[| analz G = analz G'; analz H = analz H'  
   549                |] ==> analz (G \<union> H) = analz (G' \<union> H')"
   550 by (intro equalityI analz_subset_cong, simp_all) 
   551 
   552 lemma analz_insert_cong:
   553      "analz H = analz H' ==> analz(insert X H) = analz(insert X H')"
   554 by (force simp only: insert_def intro!: analz_cong)
   555 
   556 text{*If there are no pairs or encryptions then analz does nothing*}
   557 lemma analz_trivial:
   558      "[| \<forall>X Y. {|X,Y|} \<notin> H;  \<forall>X K. Crypt K X \<notin> H |] ==> analz H = H"
   559 apply safe
   560 apply (erule analz.induct, blast+)
   561 done
   562 
   563 text{*These two are obsolete (with a single Spy) but cost little to prove...*}
   564 lemma analz_UN_analz_lemma:
   565      "X\<in> analz (\<Union>i\<in>A. analz (H i)) ==> X\<in> analz (\<Union>i\<in>A. H i)"
   566 apply (erule analz.induct)
   567 apply (blast intro: analz_mono [THEN [2] rev_subsetD])+
   568 done
   569 
   570 lemma analz_UN_analz [simp]: "analz (\<Union>i\<in>A. analz (H i)) = analz (\<Union>i\<in>A. H i)"
   571 by (blast intro: analz_UN_analz_lemma analz_mono [THEN [2] rev_subsetD])
   572 
   573 
   574 subsection{*Inductive relation "synth"*}
   575 
   576 text{*Inductive definition of "synth" -- what can be built up from a set of
   577     messages.  A form of upward closure.  Pairs can be built, messages
   578     encrypted with known keys.  Agent names are public domain.
   579     Numbers can be guessed, but Nonces cannot be.  *}
   580 
   581 inductive_set
   582   synth :: "msg set => msg set"
   583   for H :: "msg set"
   584   where
   585     Inj    [intro]:   "X \<in> H ==> X \<in> synth H"
   586   | Agent  [intro]:   "Agent agt \<in> synth H"
   587   | Number [intro]:   "Number n  \<in> synth H"
   588   | Hash   [intro]:   "X \<in> synth H ==> Hash X \<in> synth H"
   589   | MPair  [intro]:   "[|X \<in> synth H;  Y \<in> synth H|] ==> {|X,Y|} \<in> synth H"
   590   | Crypt  [intro]:   "[|X \<in> synth H;  Key(K) \<in> H|] ==> Crypt K X \<in> synth H"
   591 
   592 text{*Monotonicity*}
   593 lemma synth_mono: "G\<subseteq>H ==> synth(G) \<subseteq> synth(H)"
   594   by (auto, erule synth.induct, auto)  
   595 
   596 text{*NO @{text Agent_synth}, as any Agent name can be synthesized.  
   597   The same holds for @{term Number}*}
   598 inductive_cases Nonce_synth [elim!]: "Nonce n \<in> synth H"
   599 inductive_cases Key_synth   [elim!]: "Key K \<in> synth H"
   600 inductive_cases Hash_synth  [elim!]: "Hash X \<in> synth H"
   601 inductive_cases MPair_synth [elim!]: "{|X,Y|} \<in> synth H"
   602 inductive_cases Crypt_synth [elim!]: "Crypt K X \<in> synth H"
   603 
   604 
   605 lemma synth_increasing: "H \<subseteq> synth(H)"
   606 by blast
   607 
   608 subsubsection{*Unions *}
   609 
   610 text{*Converse fails: we can synth more from the union than from the 
   611   separate parts, building a compound message using elements of each.*}
   612 lemma synth_Un: "synth(G) \<union> synth(H) \<subseteq> synth(G \<union> H)"
   613 by (intro Un_least synth_mono Un_upper1 Un_upper2)
   614 
   615 
   616 declare [[ atp_problem_prefix = "Message__synth_insert" ]]
   617  
   618 lemma synth_insert: "insert X (synth H) \<subseteq> synth(insert X H)"
   619 by (metis insert_iff insert_subset subset_insertI synth.Inj synth_mono)
   620 
   621 subsubsection{*Idempotence and transitivity *}
   622 
   623 lemma synth_synthD [dest!]: "X\<in> synth (synth H) ==> X\<in> synth H"
   624 by (erule synth.induct, blast+)
   625 
   626 lemma synth_idem: "synth (synth H) = synth H"
   627 by blast
   628 
   629 lemma synth_subset_iff [simp]: "(synth G \<subseteq> synth H) = (G \<subseteq> synth H)"
   630 apply (rule iffI)
   631 apply (iprover intro: subset_trans synth_increasing)  
   632 apply (frule synth_mono, simp add: synth_idem) 
   633 done
   634 
   635 lemma synth_trans: "[| X\<in> synth G;  G \<subseteq> synth H |] ==> X\<in> synth H"
   636 by (drule synth_mono, blast)
   637 
   638 declare [[ atp_problem_prefix = "Message__synth_cut" ]]
   639 lemma synth_cut: "[| Y\<in> synth (insert X H);  X\<in> synth H |] ==> Y\<in> synth H"
   640 (*TOO SLOW
   641 by (metis insert_absorb insert_mono insert_subset synth_idem synth_increasing synth_mono)
   642 *)
   643 by (erule synth_trans, blast)
   644 
   645 
   646 lemma Agent_synth [simp]: "Agent A \<in> synth H"
   647 by blast
   648 
   649 lemma Number_synth [simp]: "Number n \<in> synth H"
   650 by blast
   651 
   652 lemma Nonce_synth_eq [simp]: "(Nonce N \<in> synth H) = (Nonce N \<in> H)"
   653 by blast
   654 
   655 lemma Key_synth_eq [simp]: "(Key K \<in> synth H) = (Key K \<in> H)"
   656 by blast
   657 
   658 lemma Crypt_synth_eq [simp]:
   659      "Key K \<notin> H ==> (Crypt K X \<in> synth H) = (Crypt K X \<in> H)"
   660 by blast
   661 
   662 
   663 lemma keysFor_synth [simp]: 
   664     "keysFor (synth H) = keysFor H \<union> invKey`{K. Key K \<in> H}"
   665 by (unfold keysFor_def, blast)
   666 
   667 
   668 subsubsection{*Combinations of parts, analz and synth *}
   669 
   670 declare [[ atp_problem_prefix = "Message__parts_synth" ]]
   671 lemma parts_synth [simp]: "parts (synth H) = parts H \<union> synth H"
   672 apply (rule equalityI)
   673 apply (rule subsetI)
   674 apply (erule parts.induct)
   675 apply (metis UnCI)
   676 apply (metis MPair_synth UnCI UnE insert_absorb insert_subset parts.Fst parts_increasing)
   677 apply (metis MPair_synth UnCI UnE insert_absorb insert_subset parts.Snd parts_increasing)
   678 apply (metis Body Crypt_synth UnCI UnE insert_absorb insert_subset parts_increasing)
   679 apply (metis Un_subset_iff parts_increasing parts_mono synth_increasing)
   680 done
   681 
   682 
   683 
   684 
   685 declare [[ atp_problem_prefix = "Message__analz_analz_Un" ]]
   686 lemma analz_analz_Un [simp]: "analz (analz G \<union> H) = analz (G \<union> H)"
   687 apply (rule equalityI);
   688 apply (metis analz_idem analz_subset_cong order_eq_refl)
   689 apply (metis analz_increasing analz_subset_cong order_eq_refl)
   690 done
   691 
   692 declare [[ atp_problem_prefix = "Message__analz_synth_Un" ]]
   693     declare analz_mono [intro] analz.Fst [intro] analz.Snd [intro] Un_least [intro]
   694 lemma analz_synth_Un [simp]: "analz (synth G \<union> H) = analz (G \<union> H) \<union> synth G"
   695 apply (rule equalityI)
   696 apply (rule subsetI)
   697 apply (erule analz.induct)
   698 apply (metis UnCI UnE Un_commute analz.Inj)
   699 apply (metis MPair_synth UnCI UnE Un_commute analz.Fst analz.Inj mem_def)
   700 apply (metis MPair_synth UnCI UnE Un_commute analz.Inj analz.Snd mem_def)
   701 apply (blast intro: analz.Decrypt)
   702 apply blast
   703 done
   704 
   705 declare [[ atp_problem_prefix = "Message__analz_synth" ]]
   706 lemma analz_synth [simp]: "analz (synth H) = analz H \<union> synth H"
   707 proof (neg_clausify)
   708 assume 0: "analz (synth H) \<noteq> analz H \<union> synth H"
   709 have 1: "\<And>X1 X3. sup (analz (sup X3 X1)) (synth X3) = analz (sup (synth X3) X1)"
   710   by (metis analz_synth_Un)
   711 have 2: "sup (analz H) (synth H) \<noteq> analz (synth H)"
   712   by (metis 0)
   713 have 3: "\<And>X1 X3. sup (synth X3) (analz (sup X3 X1)) = analz (sup (synth X3) X1)"
   714   by (metis 1 Un_commute)
   715 have 4: "\<And>X3. sup (synth X3) (analz X3) = analz (sup (synth X3) {})"
   716   by (metis 3 Un_empty_right)
   717 have 5: "\<And>X3. sup (synth X3) (analz X3) = analz (synth X3)"
   718   by (metis 4 Un_empty_right)
   719 have 6: "\<And>X3. sup (analz X3) (synth X3) = analz (synth X3)"
   720   by (metis 5 Un_commute)
   721 show "False"
   722   by (metis 2 6)
   723 qed
   724 
   725 
   726 subsubsection{*For reasoning about the Fake rule in traces *}
   727 
   728 declare [[ atp_problem_prefix = "Message__parts_insert_subset_Un" ]]
   729 lemma parts_insert_subset_Un: "X\<in> G ==> parts(insert X H) \<subseteq> parts G \<union> parts H"
   730 proof (neg_clausify)
   731 assume 0: "X \<in> G"
   732 assume 1: "\<not> parts (insert X H) \<subseteq> parts G \<union> parts H"
   733 have 2: "\<not> parts (insert X H) \<subseteq> parts (G \<union> H)"
   734   by (metis 1 parts_Un)
   735 have 3: "\<not> insert X H \<subseteq> G \<union> H"
   736   by (metis 2 parts_mono)
   737 have 4: "X \<notin> G \<union> H \<or> \<not> H \<subseteq> G \<union> H"
   738   by (metis 3 insert_subset)
   739 have 5: "X \<notin> G \<union> H"
   740   by (metis 4 Un_upper2)
   741 have 6: "X \<notin> G"
   742   by (metis 5 UnCI)
   743 show "False"
   744   by (metis 6 0)
   745 qed
   746 
   747 declare [[ atp_problem_prefix = "Message__Fake_parts_insert" ]]
   748 lemma Fake_parts_insert:
   749      "X \<in> synth (analz H) ==>  
   750       parts (insert X H) \<subseteq> synth (analz H) \<union> parts H"
   751 proof (neg_clausify)
   752 assume 0: "X \<in> synth (analz H)"
   753 assume 1: "\<not> parts (insert X H) \<subseteq> synth (analz H) \<union> parts H"
   754 have 2: "\<And>X3. parts X3 \<union> synth (analz X3) = parts (synth (analz X3))"
   755   by (metis parts_synth parts_analz)
   756 have 3: "\<And>X3. analz X3 \<union> synth (analz X3) = analz (synth (analz X3))"
   757   by (metis analz_synth analz_idem)
   758 have 4: "\<And>X3. analz X3 \<subseteq> analz (synth X3)"
   759   by (metis Un_upper1 analz_synth)
   760 have 5: "\<not> parts (insert X H) \<subseteq> parts H \<union> synth (analz H)"
   761   by (metis 1 Un_commute)
   762 have 6: "\<not> parts (insert X H) \<subseteq> parts (synth (analz H))"
   763   by (metis 5 2)
   764 have 7: "\<not> insert X H \<subseteq> synth (analz H)"
   765   by (metis 6 parts_mono)
   766 have 8: "X \<notin> synth (analz H) \<or> \<not> H \<subseteq> synth (analz H)"
   767   by (metis 7 insert_subset)
   768 have 9: "\<not> H \<subseteq> synth (analz H)"
   769   by (metis 8 0)
   770 have 10: "\<And>X3. X3 \<subseteq> analz (synth X3)"
   771   by (metis analz_subset_iff 4)
   772 have 11: "\<And>X3. X3 \<subseteq> analz (synth (analz X3))"
   773   by (metis analz_subset_iff 10)
   774 have 12: "\<And>X3. analz (synth (analz X3)) = synth (analz X3) \<or>
   775      \<not> analz X3 \<subseteq> synth (analz X3)"
   776   by (metis Un_absorb1 3)
   777 have 13: "\<And>X3. analz (synth (analz X3)) = synth (analz X3)"
   778   by (metis 12 synth_increasing)
   779 have 14: "\<And>X3. X3 \<subseteq> synth (analz X3)"
   780   by (metis 11 13)
   781 show "False"
   782   by (metis 9 14)
   783 qed
   784 
   785 lemma Fake_parts_insert_in_Un:
   786      "[|Z \<in> parts (insert X H);  X: synth (analz H)|] 
   787       ==> Z \<in>  synth (analz H) \<union> parts H";
   788 by (blast dest: Fake_parts_insert  [THEN subsetD, dest])
   789 
   790 declare [[ atp_problem_prefix = "Message__Fake_analz_insert" ]]
   791     declare analz_mono [intro] synth_mono [intro] 
   792 lemma Fake_analz_insert:
   793      "X\<in> synth (analz G) ==>  
   794       analz (insert X H) \<subseteq> synth (analz G) \<union> analz (G \<union> H)"
   795 by (metis Un_commute Un_insert_left Un_insert_right Un_upper1 analz_analz_Un analz_mono analz_synth_Un equalityE insert_absorb order_le_less xt1(12))
   796 
   797 declare [[ atp_problem_prefix = "Message__Fake_analz_insert_simpler" ]]
   798 (*simpler problems?  BUT METIS CAN'T PROVE
   799 lemma Fake_analz_insert_simpler:
   800      "X\<in> synth (analz G) ==>  
   801       analz (insert X H) \<subseteq> synth (analz G) \<union> analz (G \<union> H)"
   802 apply (rule subsetI)
   803 apply (subgoal_tac "x \<in> analz (synth (analz G) \<union> H) ")
   804 apply (metis Un_commute analz_analz_Un analz_synth_Un)
   805 apply (metis Un_commute Un_upper1 Un_upper2 analz_cut analz_increasing analz_mono insert_absorb insert_mono insert_subset)
   806 done
   807 *)
   808 
   809 end