src/HOL/Metis_Examples/Message.thy
 author huffman Thu Aug 11 09:11:15 2011 -0700 (2011-08-11) changeset 44165 d26a45f3c835 parent 43197 c71657bbdbc0 child 45503 44790ec65f70 permissions -rw-r--r--
remove lemma stupid_ext
```     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 header {* Metis Example Featuring Message Authentication *}
```
```     9
```
```    10 theory Message
```
```    11 imports Main
```
```    12 begin
```
```    13
```
```    14 declare [[metis_new_skolemizer]]
```
```    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        --{*true if all keys are symmetric*}
```
```    23   invKey        :: "key=>key"  --{*inverse of a symmetric key*}
```
```    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{*The inverse of a symmetric key is itself; that of a public key
```
```    32       is the private key and vice versa*}
```
```    33
```
```    34 definition symKeys :: "key set" where
```
```    35   "symKeys == {K. invKey K = K}"
```
```    36
```
```    37 datatype  --{*We allow any number of friendly agents*}
```
```    38   agent = Server | Friend nat | Spy
```
```    39
```
```    40 datatype
```
```    41      msg = Agent  agent     --{*Agent names*}
```
```    42          | Number nat       --{*Ordinary integers, timestamps, ...*}
```
```    43          | Nonce  nat       --{*Unguessable nonces*}
```
```    44          | Key    key       --{*Crypto keys*}
```
```    45          | Hash   msg       --{*Hashing*}
```
```    46          | MPair  msg msg   --{*Compound messages*}
```
```    47          | Crypt  key msg   --{*Encryption, public- or shared-key*}
```
```    48
```
```    49
```
```    50 text{*Concrete syntax: messages appear as {|A,B,NA|}, etc...*}
```
```    51 syntax
```
```    52   "_MTuple"      :: "['a, args] => 'a * 'b"       ("(2{|_,/ _|})")
```
```    53
```
```    54 syntax (xsymbols)
```
```    55   "_MTuple"      :: "['a, args] => 'a * 'b"       ("(2\<lbrace>_,/ _\<rbrace>)")
```
```    56
```
```    57 translations
```
```    58   "{|x, y, z|}"   == "{|x, {|y, z|}|}"
```
```    59   "{|x, y|}"      == "CONST MPair x y"
```
```    60
```
```    61
```
```    62 definition HPair :: "[msg,msg] => msg" ("(4Hash[_] /_)" [0, 1000]) where
```
```    63     --{*Message Y paired with a MAC computed with the help of X*}
```
```    64     "Hash[X] Y == {| Hash{|X,Y|}, Y|}"
```
```    65
```
```    66 definition keysFor :: "msg set => key set" where
```
```    67     --{*Keys useful to decrypt elements of a message set*}
```
```    68   "keysFor H == invKey ` {K. \<exists>X. Crypt K X \<in> H}"
```
```    69
```
```    70
```
```    71 subsubsection{*Inductive Definition of All Parts" of a Message*}
```
```    72
```
```    73 inductive_set
```
```    74   parts :: "msg set => msg set"
```
```    75   for H :: "msg set"
```
```    76   where
```
```    77     Inj [intro]:               "X \<in> H ==> X \<in> parts H"
```
```    78   | Fst:         "{|X,Y|}   \<in> parts H ==> X \<in> parts H"
```
```    79   | Snd:         "{|X,Y|}   \<in> parts H ==> Y \<in> parts H"
```
```    80   | Body:        "Crypt K X \<in> parts H ==> X \<in> parts H"
```
```    81
```
```    82 lemma parts_mono: "G \<subseteq> H ==> parts(G) \<subseteq> parts(H)"
```
```    83 apply auto
```
```    84 apply (erule parts.induct)
```
```    85    apply (metis parts.Inj set_rev_mp)
```
```    86   apply (metis parts.Fst)
```
```    87  apply (metis parts.Snd)
```
```    88 by (metis parts.Body)
```
```    89
```
```    90 text{*Equations hold because constructors are injective.*}
```
```    91 lemma Friend_image_eq [simp]: "(Friend x \<in> Friend`A) = (x:A)"
```
```    92 by (metis agent.inject image_iff)
```
```    93
```
```    94 lemma Key_image_eq [simp]: "(Key x \<in> Key`A) = (x \<in> A)"
```
```    95 by (metis image_iff msg.inject(4))
```
```    96
```
```    97 lemma Nonce_Key_image_eq [simp]: "Nonce x \<notin> Key`A"
```
```    98 by (metis image_iff msg.distinct(23))
```
```    99
```
```   100
```
```   101 subsubsection{*Inverse of keys *}
```
```   102
```
```   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 lemma parts_insert2:
```
```   204      "parts (insert X (insert Y H)) = parts {X} \<union> parts {Y} \<union> parts H"
```
```   205 by (metis Un_commute Un_empty_left Un_empty_right Un_insert_left Un_insert_right parts_Un)
```
```   206
```
```   207
```
```   208 lemma parts_UN_subset1: "(\<Union>x\<in>A. parts(H x)) \<subseteq> parts(\<Union>x\<in>A. H x)"
```
```   209 by (intro UN_least parts_mono UN_upper)
```
```   210
```
```   211 lemma parts_UN_subset2: "parts(\<Union>x\<in>A. H x) \<subseteq> (\<Union>x\<in>A. parts(H x))"
```
```   212 apply (rule subsetI)
```
```   213 apply (erule parts.induct, blast+)
```
```   214 done
```
```   215
```
```   216 lemma parts_UN [simp]: "parts(\<Union>x\<in>A. H x) = (\<Union>x\<in>A. parts(H x))"
```
```   217 by (intro equalityI parts_UN_subset1 parts_UN_subset2)
```
```   218
```
```   219 text{*Added to simplify arguments to parts, analz and synth.
```
```   220   NOTE: the UN versions are no longer used!*}
```
```   221
```
```   222
```
```   223 text{*This allows @{text blast} to simplify occurrences of
```
```   224   @{term "parts(G\<union>H)"} in the assumption.*}
```
```   225 lemmas in_parts_UnE = parts_Un [THEN equalityD1, THEN subsetD, THEN UnE]
```
```   226 declare in_parts_UnE [elim!]
```
```   227
```
```   228 lemma parts_insert_subset: "insert X (parts H) \<subseteq> parts(insert X H)"
```
```   229 by (blast intro: parts_mono [THEN [2] rev_subsetD])
```
```   230
```
```   231 subsubsection{*Idempotence and transitivity *}
```
```   232
```
```   233 lemma parts_partsD [dest!]: "X\<in> parts (parts H) ==> X\<in> parts H"
```
```   234 by (erule parts.induct, blast+)
```
```   235
```
```   236 lemma parts_idem [simp]: "parts (parts H) = parts H"
```
```   237 by blast
```
```   238
```
```   239 lemma parts_subset_iff [simp]: "(parts G \<subseteq> parts H) = (G \<subseteq> parts H)"
```
```   240 apply (rule iffI)
```
```   241 apply (metis Un_absorb1 Un_subset_iff parts_Un parts_increasing)
```
```   242 apply (metis parts_idem parts_mono)
```
```   243 done
```
```   244
```
```   245 lemma parts_trans: "[| X\<in> parts G;  G \<subseteq> parts H |] ==> X\<in> parts H"
```
```   246 by (blast dest: parts_mono);
```
```   247
```
```   248 lemma parts_cut: "[|Y\<in> parts (insert X G);  X\<in> parts H|] ==> Y\<in> parts(G \<union> H)"
```
```   249 by (metis Un_insert_left Un_insert_right insert_absorb mem_def parts_Un parts_idem sup1CI)
```
```   250
```
```   251
```
```   252 subsubsection{*Rewrite rules for pulling out atomic messages *}
```
```   253
```
```   254 lemmas parts_insert_eq_I = equalityI [OF subsetI parts_insert_subset]
```
```   255
```
```   256
```
```   257 lemma parts_insert_Agent [simp]:
```
```   258      "parts (insert (Agent agt) H) = insert (Agent agt) (parts H)"
```
```   259 apply (rule parts_insert_eq_I)
```
```   260 apply (erule parts.induct, auto)
```
```   261 done
```
```   262
```
```   263 lemma parts_insert_Nonce [simp]:
```
```   264      "parts (insert (Nonce N) H) = insert (Nonce N) (parts H)"
```
```   265 apply (rule parts_insert_eq_I)
```
```   266 apply (erule parts.induct, auto)
```
```   267 done
```
```   268
```
```   269 lemma parts_insert_Number [simp]:
```
```   270      "parts (insert (Number N) H) = insert (Number N) (parts H)"
```
```   271 apply (rule parts_insert_eq_I)
```
```   272 apply (erule parts.induct, auto)
```
```   273 done
```
```   274
```
```   275 lemma parts_insert_Key [simp]:
```
```   276      "parts (insert (Key K) H) = insert (Key K) (parts H)"
```
```   277 apply (rule parts_insert_eq_I)
```
```   278 apply (erule parts.induct, auto)
```
```   279 done
```
```   280
```
```   281 lemma parts_insert_Hash [simp]:
```
```   282      "parts (insert (Hash X) H) = insert (Hash X) (parts H)"
```
```   283 apply (rule parts_insert_eq_I)
```
```   284 apply (erule parts.induct, auto)
```
```   285 done
```
```   286
```
```   287 lemma parts_insert_Crypt [simp]:
```
```   288      "parts (insert (Crypt K X) H) =
```
```   289           insert (Crypt K X) (parts (insert X H))"
```
```   290 apply (rule equalityI)
```
```   291 apply (rule subsetI)
```
```   292 apply (erule parts.induct, auto)
```
```   293 apply (blast intro: parts.Body)
```
```   294 done
```
```   295
```
```   296 lemma parts_insert_MPair [simp]:
```
```   297      "parts (insert {|X,Y|} H) =
```
```   298           insert {|X,Y|} (parts (insert X (insert Y H)))"
```
```   299 apply (rule equalityI)
```
```   300 apply (rule subsetI)
```
```   301 apply (erule parts.induct, auto)
```
```   302 apply (blast intro: parts.Fst parts.Snd)+
```
```   303 done
```
```   304
```
```   305 lemma parts_image_Key [simp]: "parts (Key`N) = Key`N"
```
```   306 apply auto
```
```   307 apply (erule parts.induct, auto)
```
```   308 done
```
```   309
```
```   310 lemma msg_Nonce_supply: "\<exists>N. \<forall>n. N\<le>n --> Nonce n \<notin> parts {msg}"
```
```   311 apply (induct_tac "msg")
```
```   312 apply (simp_all add: parts_insert2)
```
```   313 apply (metis Suc_n_not_le_n)
```
```   314 apply (metis le_trans linorder_linear)
```
```   315 done
```
```   316
```
```   317 subsection{*Inductive relation "analz"*}
```
```   318
```
```   319 text{*Inductive definition of "analz" -- what can be broken down from a set of
```
```   320     messages, including keys.  A form of downward closure.  Pairs can
```
```   321     be taken apart; messages decrypted with known keys.  *}
```
```   322
```
```   323 inductive_set
```
```   324   analz :: "msg set => msg set"
```
```   325   for H :: "msg set"
```
```   326   where
```
```   327     Inj [intro,simp] :    "X \<in> H ==> X \<in> analz H"
```
```   328   | Fst:     "{|X,Y|} \<in> analz H ==> X \<in> analz H"
```
```   329   | Snd:     "{|X,Y|} \<in> analz H ==> Y \<in> analz H"
```
```   330   | Decrypt [dest]:
```
```   331              "[|Crypt K X \<in> analz H; Key(invKey K): analz H|] ==> X \<in> analz H"
```
```   332
```
```   333
```
```   334 text{*Monotonicity; Lemma 1 of Lowe's paper*}
```
```   335 lemma analz_mono: "G\<subseteq>H ==> analz(G) \<subseteq> analz(H)"
```
```   336 apply auto
```
```   337 apply (erule analz.induct)
```
```   338 apply (auto dest: analz.Fst analz.Snd)
```
```   339 done
```
```   340
```
```   341 text{*Making it safe speeds up proofs*}
```
```   342 lemma MPair_analz [elim!]:
```
```   343      "[| {|X,Y|} \<in> analz H;
```
```   344              [| X \<in> analz H; Y \<in> analz H |] ==> P
```
```   345           |] ==> P"
```
```   346 by (blast dest: analz.Fst analz.Snd)
```
```   347
```
```   348 lemma analz_increasing: "H \<subseteq> analz(H)"
```
```   349 by blast
```
```   350
```
```   351 lemma analz_subset_parts: "analz H \<subseteq> parts H"
```
```   352 apply (rule subsetI)
```
```   353 apply (erule analz.induct, blast+)
```
```   354 done
```
```   355
```
```   356 lemmas analz_into_parts = analz_subset_parts [THEN subsetD, standard]
```
```   357
```
```   358 lemmas not_parts_not_analz = analz_subset_parts [THEN contra_subsetD, standard]
```
```   359
```
```   360 lemma parts_analz [simp]: "parts (analz H) = parts H"
```
```   361 apply (rule equalityI)
```
```   362 apply (metis analz_subset_parts parts_subset_iff)
```
```   363 apply (metis analz_increasing parts_mono)
```
```   364 done
```
```   365
```
```   366
```
```   367 lemma analz_parts [simp]: "analz (parts H) = parts H"
```
```   368 apply auto
```
```   369 apply (erule analz.induct, auto)
```
```   370 done
```
```   371
```
```   372 lemmas analz_insertI = subset_insertI [THEN analz_mono, THEN [2] rev_subsetD, standard]
```
```   373
```
```   374 subsubsection{*General equational properties *}
```
```   375
```
```   376 lemma analz_empty [simp]: "analz{} = {}"
```
```   377 apply safe
```
```   378 apply (erule analz.induct, blast+)
```
```   379 done
```
```   380
```
```   381 text{*Converse fails: we can analz more from the union than from the
```
```   382   separate parts, as a key in one might decrypt a message in the other*}
```
```   383 lemma analz_Un: "analz(G) \<union> analz(H) \<subseteq> analz(G \<union> H)"
```
```   384 by (intro Un_least analz_mono Un_upper1 Un_upper2)
```
```   385
```
```   386 lemma analz_insert: "insert X (analz H) \<subseteq> analz(insert X H)"
```
```   387 by (blast intro: analz_mono [THEN [2] rev_subsetD])
```
```   388
```
```   389 subsubsection{*Rewrite rules for pulling out atomic messages *}
```
```   390
```
```   391 lemmas analz_insert_eq_I = equalityI [OF subsetI analz_insert]
```
```   392
```
```   393 lemma analz_insert_Agent [simp]:
```
```   394      "analz (insert (Agent agt) H) = insert (Agent agt) (analz H)"
```
```   395 apply (rule analz_insert_eq_I)
```
```   396 apply (erule analz.induct, auto)
```
```   397 done
```
```   398
```
```   399 lemma analz_insert_Nonce [simp]:
```
```   400      "analz (insert (Nonce N) H) = insert (Nonce N) (analz H)"
```
```   401 apply (rule analz_insert_eq_I)
```
```   402 apply (erule analz.induct, auto)
```
```   403 done
```
```   404
```
```   405 lemma analz_insert_Number [simp]:
```
```   406      "analz (insert (Number N) H) = insert (Number N) (analz H)"
```
```   407 apply (rule analz_insert_eq_I)
```
```   408 apply (erule analz.induct, auto)
```
```   409 done
```
```   410
```
```   411 lemma analz_insert_Hash [simp]:
```
```   412      "analz (insert (Hash X) H) = insert (Hash X) (analz H)"
```
```   413 apply (rule analz_insert_eq_I)
```
```   414 apply (erule analz.induct, auto)
```
```   415 done
```
```   416
```
```   417 text{*Can only pull out Keys if they are not needed to decrypt the rest*}
```
```   418 lemma analz_insert_Key [simp]:
```
```   419     "K \<notin> keysFor (analz H) ==>
```
```   420           analz (insert (Key K) H) = insert (Key K) (analz H)"
```
```   421 apply (unfold keysFor_def)
```
```   422 apply (rule analz_insert_eq_I)
```
```   423 apply (erule analz.induct, auto)
```
```   424 done
```
```   425
```
```   426 lemma analz_insert_MPair [simp]:
```
```   427      "analz (insert {|X,Y|} H) =
```
```   428           insert {|X,Y|} (analz (insert X (insert Y H)))"
```
```   429 apply (rule equalityI)
```
```   430 apply (rule subsetI)
```
```   431 apply (erule analz.induct, auto)
```
```   432 apply (erule analz.induct)
```
```   433 apply (blast intro: analz.Fst analz.Snd)+
```
```   434 done
```
```   435
```
```   436 text{*Can pull out enCrypted message if the Key is not known*}
```
```   437 lemma analz_insert_Crypt:
```
```   438      "Key (invKey K) \<notin> analz H
```
```   439       ==> analz (insert (Crypt K X) H) = insert (Crypt K X) (analz H)"
```
```   440 apply (rule analz_insert_eq_I)
```
```   441 apply (erule analz.induct, auto)
```
```   442
```
```   443 done
```
```   444
```
```   445 lemma lemma1: "Key (invKey K) \<in> analz H ==>
```
```   446                analz (insert (Crypt K X) H) \<subseteq>
```
```   447                insert (Crypt K X) (analz (insert X H))"
```
```   448 apply (rule subsetI)
```
```   449 apply (erule_tac x = x in analz.induct, auto)
```
```   450 done
```
```   451
```
```   452 lemma lemma2: "Key (invKey K) \<in> analz H ==>
```
```   453                insert (Crypt K X) (analz (insert X H)) \<subseteq>
```
```   454                analz (insert (Crypt K X) H)"
```
```   455 apply auto
```
```   456 apply (erule_tac x = x in analz.induct, auto)
```
```   457 apply (blast intro: analz_insertI analz.Decrypt)
```
```   458 done
```
```   459
```
```   460 lemma analz_insert_Decrypt:
```
```   461      "Key (invKey K) \<in> analz H ==>
```
```   462                analz (insert (Crypt K X) H) =
```
```   463                insert (Crypt K X) (analz (insert X H))"
```
```   464 by (intro equalityI lemma1 lemma2)
```
```   465
```
```   466 text{*Case analysis: either the message is secure, or it is not! Effective,
```
```   467 but can cause subgoals to blow up! Use with @{text "split_if"}; apparently
```
```   468 @{text "split_tac"} does not cope with patterns such as @{term"analz (insert
```
```   469 (Crypt K X) H)"} *}
```
```   470 lemma analz_Crypt_if [simp]:
```
```   471      "analz (insert (Crypt K X) H) =
```
```   472           (if (Key (invKey K) \<in> analz H)
```
```   473            then insert (Crypt K X) (analz (insert X H))
```
```   474            else insert (Crypt K X) (analz H))"
```
```   475 by (simp add: analz_insert_Crypt analz_insert_Decrypt)
```
```   476
```
```   477
```
```   478 text{*This rule supposes "for the sake of argument" that we have the key.*}
```
```   479 lemma analz_insert_Crypt_subset:
```
```   480      "analz (insert (Crypt K X) H) \<subseteq>
```
```   481            insert (Crypt K X) (analz (insert X H))"
```
```   482 apply (rule subsetI)
```
```   483 apply (erule analz.induct, auto)
```
```   484 done
```
```   485
```
```   486
```
```   487 lemma analz_image_Key [simp]: "analz (Key`N) = Key`N"
```
```   488 apply auto
```
```   489 apply (erule analz.induct, auto)
```
```   490 done
```
```   491
```
```   492
```
```   493 subsubsection{*Idempotence and transitivity *}
```
```   494
```
```   495 lemma analz_analzD [dest!]: "X\<in> analz (analz H) ==> X\<in> analz H"
```
```   496 by (erule analz.induct, blast+)
```
```   497
```
```   498 lemma analz_idem [simp]: "analz (analz H) = analz H"
```
```   499 by blast
```
```   500
```
```   501 lemma analz_subset_iff [simp]: "(analz G \<subseteq> analz H) = (G \<subseteq> analz H)"
```
```   502 apply (rule iffI)
```
```   503 apply (iprover intro: subset_trans analz_increasing)
```
```   504 apply (frule analz_mono, simp)
```
```   505 done
```
```   506
```
```   507 lemma analz_trans: "[| X\<in> analz G;  G \<subseteq> analz H |] ==> X\<in> analz H"
```
```   508 by (drule analz_mono, blast)
```
```   509
```
```   510
```
```   511 declare analz_trans[intro]
```
```   512
```
```   513 lemma analz_cut: "[| Y\<in> analz (insert X H);  X\<in> analz H |] ==> Y\<in> analz H"
```
```   514 (*TOO SLOW
```
```   515 by (metis analz_idem analz_increasing analz_mono insert_absorb insert_mono insert_subset) --{*317s*}
```
```   516 ??*)
```
```   517 by (erule analz_trans, blast)
```
```   518
```
```   519
```
```   520 text{*This rewrite rule helps in the simplification of messages that involve
```
```   521   the forwarding of unknown components (X).  Without it, removing occurrences
```
```   522   of X can be very complicated. *}
```
```   523 lemma analz_insert_eq: "X\<in> analz H ==> analz (insert X H) = analz H"
```
```   524 by (blast intro: analz_cut analz_insertI)
```
```   525
```
```   526
```
```   527 text{*A congruence rule for "analz" *}
```
```   528
```
```   529 lemma analz_subset_cong:
```
```   530      "[| analz G \<subseteq> analz G'; analz H \<subseteq> analz H' |]
```
```   531       ==> analz (G \<union> H) \<subseteq> analz (G' \<union> H')"
```
```   532 apply simp
```
```   533 apply (metis Un_absorb2 Un_commute Un_subset_iff Un_upper1 Un_upper2 analz_mono)
```
```   534 done
```
```   535
```
```   536
```
```   537 lemma analz_cong:
```
```   538      "[| analz G = analz G'; analz H = analz H'
```
```   539                |] ==> analz (G \<union> H) = analz (G' \<union> H')"
```
```   540 by (intro equalityI analz_subset_cong, simp_all)
```
```   541
```
```   542 lemma analz_insert_cong:
```
```   543      "analz H = analz H' ==> analz(insert X H) = analz(insert X H')"
```
```   544 by (force simp only: insert_def intro!: analz_cong)
```
```   545
```
```   546 text{*If there are no pairs or encryptions then analz does nothing*}
```
```   547 lemma analz_trivial:
```
```   548      "[| \<forall>X Y. {|X,Y|} \<notin> H;  \<forall>X K. Crypt K X \<notin> H |] ==> analz H = H"
```
```   549 apply safe
```
```   550 apply (erule analz.induct, blast+)
```
```   551 done
```
```   552
```
```   553 text{*These two are obsolete (with a single Spy) but cost little to prove...*}
```
```   554 lemma analz_UN_analz_lemma:
```
```   555      "X\<in> analz (\<Union>i\<in>A. analz (H i)) ==> X\<in> analz (\<Union>i\<in>A. H i)"
```
```   556 apply (erule analz.induct)
```
```   557 apply (blast intro: analz_mono [THEN [2] rev_subsetD])+
```
```   558 done
```
```   559
```
```   560 lemma analz_UN_analz [simp]: "analz (\<Union>i\<in>A. analz (H i)) = analz (\<Union>i\<in>A. H i)"
```
```   561 by (blast intro: analz_UN_analz_lemma analz_mono [THEN [2] rev_subsetD])
```
```   562
```
```   563
```
```   564 subsection{*Inductive relation "synth"*}
```
```   565
```
```   566 text{*Inductive definition of "synth" -- what can be built up from a set of
```
```   567     messages.  A form of upward closure.  Pairs can be built, messages
```
```   568     encrypted with known keys.  Agent names are public domain.
```
```   569     Numbers can be guessed, but Nonces cannot be.  *}
```
```   570
```
```   571 inductive_set
```
```   572   synth :: "msg set => msg set"
```
```   573   for H :: "msg set"
```
```   574   where
```
```   575     Inj    [intro]:   "X \<in> H ==> X \<in> synth H"
```
```   576   | Agent  [intro]:   "Agent agt \<in> synth H"
```
```   577   | Number [intro]:   "Number n  \<in> synth H"
```
```   578   | Hash   [intro]:   "X \<in> synth H ==> Hash X \<in> synth H"
```
```   579   | MPair  [intro]:   "[|X \<in> synth H;  Y \<in> synth H|] ==> {|X,Y|} \<in> synth H"
```
```   580   | Crypt  [intro]:   "[|X \<in> synth H;  Key(K) \<in> H|] ==> Crypt K X \<in> synth H"
```
```   581
```
```   582 text{*Monotonicity*}
```
```   583 lemma synth_mono: "G\<subseteq>H ==> synth(G) \<subseteq> synth(H)"
```
```   584   by (auto, erule synth.induct, auto)
```
```   585
```
```   586 text{*NO @{text Agent_synth}, as any Agent name can be synthesized.
```
```   587   The same holds for @{term Number}*}
```
```   588 inductive_cases Nonce_synth [elim!]: "Nonce n \<in> synth H"
```
```   589 inductive_cases Key_synth   [elim!]: "Key K \<in> synth H"
```
```   590 inductive_cases Hash_synth  [elim!]: "Hash X \<in> synth H"
```
```   591 inductive_cases MPair_synth [elim!]: "{|X,Y|} \<in> synth H"
```
```   592 inductive_cases Crypt_synth [elim!]: "Crypt K X \<in> synth H"
```
```   593
```
```   594
```
```   595 lemma synth_increasing: "H \<subseteq> synth(H)"
```
```   596 by blast
```
```   597
```
```   598 subsubsection{*Unions *}
```
```   599
```
```   600 text{*Converse fails: we can synth more from the union than from the
```
```   601   separate parts, building a compound message using elements of each.*}
```
```   602 lemma synth_Un: "synth(G) \<union> synth(H) \<subseteq> synth(G \<union> H)"
```
```   603 by (intro Un_least synth_mono Un_upper1 Un_upper2)
```
```   604
```
```   605 lemma synth_insert: "insert X (synth H) \<subseteq> synth(insert X H)"
```
```   606 by (metis insert_iff insert_subset subset_insertI synth.Inj synth_mono)
```
```   607
```
```   608 subsubsection{*Idempotence and transitivity *}
```
```   609
```
```   610 lemma synth_synthD [dest!]: "X\<in> synth (synth H) ==> X\<in> synth H"
```
```   611 by (erule synth.induct, blast+)
```
```   612
```
```   613 lemma synth_idem: "synth (synth H) = synth H"
```
```   614 by blast
```
```   615
```
```   616 lemma synth_subset_iff [simp]: "(synth G \<subseteq> synth H) = (G \<subseteq> synth H)"
```
```   617 apply (rule iffI)
```
```   618 apply (iprover intro: subset_trans synth_increasing)
```
```   619 apply (frule synth_mono, simp add: synth_idem)
```
```   620 done
```
```   621
```
```   622 lemma synth_trans: "[| X\<in> synth G;  G \<subseteq> synth H |] ==> X\<in> synth H"
```
```   623 by (drule synth_mono, blast)
```
```   624
```
```   625 lemma synth_cut: "[| Y\<in> synth (insert X H);  X\<in> synth H |] ==> Y\<in> synth H"
```
```   626 (*TOO SLOW
```
```   627 by (metis insert_absorb insert_mono insert_subset synth_idem synth_increasing synth_mono)
```
```   628 *)
```
```   629 by (erule synth_trans, blast)
```
```   630
```
```   631
```
```   632 lemma Agent_synth [simp]: "Agent A \<in> synth H"
```
```   633 by blast
```
```   634
```
```   635 lemma Number_synth [simp]: "Number n \<in> synth H"
```
```   636 by blast
```
```   637
```
```   638 lemma Nonce_synth_eq [simp]: "(Nonce N \<in> synth H) = (Nonce N \<in> H)"
```
```   639 by blast
```
```   640
```
```   641 lemma Key_synth_eq [simp]: "(Key K \<in> synth H) = (Key K \<in> H)"
```
```   642 by blast
```
```   643
```
```   644 lemma Crypt_synth_eq [simp]:
```
```   645      "Key K \<notin> H ==> (Crypt K X \<in> synth H) = (Crypt K X \<in> H)"
```
```   646 by blast
```
```   647
```
```   648
```
```   649 lemma keysFor_synth [simp]:
```
```   650     "keysFor (synth H) = keysFor H \<union> invKey`{K. Key K \<in> H}"
```
```   651 by (unfold keysFor_def, blast)
```
```   652
```
```   653
```
```   654 subsubsection{*Combinations of parts, analz and synth *}
```
```   655
```
```   656 lemma parts_synth [simp]: "parts (synth H) = parts H \<union> synth H"
```
```   657 apply (rule equalityI)
```
```   658 apply (rule subsetI)
```
```   659 apply (erule parts.induct)
```
```   660 apply (metis UnCI)
```
```   661 apply (metis MPair_synth UnCI UnE insert_absorb insert_subset parts.Fst parts_increasing)
```
```   662 apply (metis MPair_synth UnCI UnE insert_absorb insert_subset parts.Snd parts_increasing)
```
```   663 apply (metis Body Crypt_synth UnCI UnE insert_absorb insert_subset parts_increasing)
```
```   664 apply (metis Un_subset_iff parts_increasing parts_mono synth_increasing)
```
```   665 done
```
```   666
```
```   667 lemma analz_analz_Un [simp]: "analz (analz G \<union> H) = analz (G \<union> H)"
```
```   668 apply (rule equalityI);
```
```   669 apply (metis analz_idem analz_subset_cong order_eq_refl)
```
```   670 apply (metis analz_increasing analz_subset_cong order_eq_refl)
```
```   671 done
```
```   672
```
```   673 declare analz_mono [intro] analz.Fst [intro] analz.Snd [intro] Un_least [intro]
```
```   674
```
```   675 lemma analz_synth_Un [simp]: "analz (synth G \<union> H) = analz (G \<union> H) \<union> synth G"
```
```   676 apply (rule equalityI)
```
```   677 apply (rule subsetI)
```
```   678 apply (erule analz.induct)
```
```   679 apply (metis UnCI UnE Un_commute analz.Inj)
```
```   680 apply (metis MPair_synth UnCI UnE Un_commute analz.Fst analz.Inj mem_def)
```
```   681 apply (metis MPair_synth UnCI UnE Un_commute analz.Inj analz.Snd mem_def)
```
```   682 apply (blast intro: analz.Decrypt)
```
```   683 apply blast
```
```   684 done
```
```   685
```
```   686 lemma analz_synth [simp]: "analz (synth H) = analz H \<union> synth H"
```
```   687 proof -
```
```   688   have "\<forall>x\<^isub>2 x\<^isub>1. synth x\<^isub>1 \<union> analz (x\<^isub>1 \<union> x\<^isub>2) = analz (synth x\<^isub>1 \<union> x\<^isub>2)" by (metis Un_commute analz_synth_Un)
```
```   689   hence "\<forall>x\<^isub>1. synth x\<^isub>1 \<union> analz x\<^isub>1 = analz (synth x\<^isub>1 \<union> {})" by (metis Un_empty_right)
```
```   690   hence "\<forall>x\<^isub>1. synth x\<^isub>1 \<union> analz x\<^isub>1 = analz (synth x\<^isub>1)" by (metis Un_empty_right)
```
```   691   hence "\<forall>x\<^isub>1. analz x\<^isub>1 \<union> synth x\<^isub>1 = analz (synth x\<^isub>1)" by (metis Un_commute)
```
```   692   thus "analz (synth H) = analz H \<union> synth H" by metis
```
```   693 qed
```
```   694
```
```   695
```
```   696 subsubsection{*For reasoning about the Fake rule in traces *}
```
```   697
```
```   698 lemma parts_insert_subset_Un: "X\<in> G ==> parts(insert X H) \<subseteq> parts G \<union> parts H"
```
```   699 proof -
```
```   700   assume "X \<in> G"
```
```   701   hence "G X" by (metis mem_def)
```
```   702   hence "\<forall>x\<^isub>1. G \<subseteq> x\<^isub>1 \<longrightarrow> x\<^isub>1 X" by (metis predicate1D)
```
```   703   hence "\<forall>x\<^isub>1. (G \<union> x\<^isub>1) X" by (metis Un_upper1)
```
```   704   hence "\<forall>x\<^isub>1. X \<in> G \<union> x\<^isub>1" by (metis mem_def)
```
```   705   hence "insert X H \<subseteq> G \<union> H" by (metis Un_upper2 insert_subset)
```
```   706   hence "parts (insert X H) \<subseteq> parts (G \<union> H)" by (metis parts_mono)
```
```   707   thus "parts (insert X H) \<subseteq> parts G \<union> parts H" by (metis parts_Un)
```
```   708 qed
```
```   709
```
```   710 lemma Fake_parts_insert:
```
```   711      "X \<in> synth (analz H) ==>
```
```   712       parts (insert X H) \<subseteq> synth (analz H) \<union> parts H"
```
```   713 proof -
```
```   714   assume A1: "X \<in> synth (analz H)"
```
```   715   have F1: "\<forall>x\<^isub>1. analz x\<^isub>1 \<union> synth (analz x\<^isub>1) = analz (synth (analz x\<^isub>1))"
```
```   716     by (metis analz_idem analz_synth)
```
```   717   have F2: "\<forall>x\<^isub>1. parts x\<^isub>1 \<union> synth (analz x\<^isub>1) = parts (synth (analz x\<^isub>1))"
```
```   718     by (metis parts_analz parts_synth)
```
```   719   have F3: "synth (analz H) X" using A1 by (metis mem_def)
```
```   720   have "\<forall>x\<^isub>2 x\<^isub>1\<Colon>msg set. x\<^isub>1 \<le> sup x\<^isub>1 x\<^isub>2" by (metis inf_sup_ord(3))
```
```   721   hence F4: "\<forall>x\<^isub>1. analz x\<^isub>1 \<subseteq> analz (synth x\<^isub>1)" by (metis analz_synth)
```
```   722   have F5: "X \<in> synth (analz H)" using F3 by (metis mem_def)
```
```   723   have "\<forall>x\<^isub>1. analz x\<^isub>1 \<subseteq> synth (analz x\<^isub>1)
```
```   724          \<longrightarrow> analz (synth (analz x\<^isub>1)) = synth (analz x\<^isub>1)"
```
```   725     using F1 by (metis subset_Un_eq)
```
```   726   hence F6: "\<forall>x\<^isub>1. analz (synth (analz x\<^isub>1)) = synth (analz x\<^isub>1)"
```
```   727     by (metis synth_increasing)
```
```   728   have "\<forall>x\<^isub>1. x\<^isub>1 \<subseteq> analz (synth x\<^isub>1)" using F4 by (metis analz_subset_iff)
```
```   729   hence "\<forall>x\<^isub>1. x\<^isub>1 \<subseteq> analz (synth (analz x\<^isub>1))" by (metis analz_subset_iff)
```
```   730   hence "\<forall>x\<^isub>1. x\<^isub>1 \<subseteq> synth (analz x\<^isub>1)" using F6 by metis
```
```   731   hence "H \<subseteq> synth (analz H)" by metis
```
```   732   hence "H \<subseteq> synth (analz H) \<and> X \<in> synth (analz H)" using F5 by metis
```
```   733   hence "insert X H \<subseteq> synth (analz H)" by (metis insert_subset)
```
```   734   hence "parts (insert X H) \<subseteq> parts (synth (analz H))" by (metis parts_mono)
```
```   735   hence "parts (insert X H) \<subseteq> parts H \<union> synth (analz H)" using F2 by metis
```
```   736   thus "parts (insert X H) \<subseteq> synth (analz H) \<union> parts H" by (metis Un_commute)
```
```   737 qed
```
```   738
```
```   739 lemma Fake_parts_insert_in_Un:
```
```   740      "[|Z \<in> parts (insert X H);  X: synth (analz H)|]
```
```   741       ==> Z \<in>  synth (analz H) \<union> parts H";
```
```   742 by (blast dest: Fake_parts_insert [THEN subsetD, dest])
```
```   743
```
```   744 declare analz_mono [intro] synth_mono [intro]
```
```   745
```
```   746 lemma Fake_analz_insert:
```
```   747      "X \<in> synth (analz G) ==>
```
```   748       analz (insert X H) \<subseteq> synth (analz G) \<union> analz (G \<union> H)"
```
```   749 by (metis Un_commute Un_insert_left Un_insert_right Un_upper1 analz_analz_Un
```
```   750           analz_mono analz_synth_Un insert_absorb)
```
```   751
```
```   752 lemma Fake_analz_insert_simpler:
```
```   753      "X \<in> synth (analz G) ==>
```
```   754       analz (insert X H) \<subseteq> synth (analz G) \<union> analz (G \<union> H)"
```
```   755 apply (rule subsetI)
```
```   756 apply (subgoal_tac "x \<in> analz (synth (analz G) \<union> H) ")
```
```   757 apply (metis Un_commute analz_analz_Un analz_synth_Un)
```
```   758 by (metis Un_upper1 Un_upper2 analz_mono insert_absorb insert_subset)
```
```   759
```
```   760 end
```