src/HOL/Auth/Message.thy
author webertj
Mon Mar 07 19:30:53 2005 +0100 (2005-03-07)
changeset 15584 3478bb4f93ff
parent 15032 02aed07e01bf
child 16417 9bc16273c2d4
permissions -rw-r--r--
refute_params: default value itself=1 added (for type classes)
paulson@1839
     1
(*  Title:      HOL/Auth/Message
paulson@1839
     2
    ID:         $Id$
paulson@1839
     3
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
paulson@1839
     4
    Copyright   1996  University of Cambridge
paulson@1839
     5
paulson@1839
     6
Datatypes of agents and messages;
paulson@1913
     7
Inductive relations "parts", "analz" and "synth"
paulson@1839
     8
*)
paulson@1839
     9
paulson@13956
    10
header{*Theory of Agents and Messages for Security Protocols*}
paulson@13956
    11
paulson@13926
    12
theory Message = Main:
paulson@11189
    13
paulson@11189
    14
(*Needed occasionally with spy_analz_tac, e.g. in analz_insert_Key_newK*)
paulson@13926
    15
lemma [simp] : "A \<union> (B \<union> A) = B \<union> A"
paulson@11189
    16
by blast
paulson@1839
    17
paulson@1839
    18
types 
paulson@1839
    19
  key = nat
paulson@1839
    20
paulson@1839
    21
consts
paulson@14126
    22
  all_symmetric :: bool        --{*true if all keys are symmetric*}
paulson@14126
    23
  invKey        :: "key=>key"  --{*inverse of a symmetric key*}
paulson@14126
    24
paulson@14126
    25
specification (invKey)
paulson@14181
    26
  invKey [simp]: "invKey (invKey K) = K"
paulson@14181
    27
  invKey_symmetric: "all_symmetric --> invKey = id"
paulson@14126
    28
    by (rule exI [of _ id], auto)
paulson@1839
    29
paulson@14126
    30
paulson@14126
    31
text{*The inverse of a symmetric key is itself; that of a public key
paulson@14126
    32
      is the private key and vice versa*}
paulson@1839
    33
paulson@1839
    34
constdefs
paulson@11230
    35
  symKeys :: "key set"
paulson@11230
    36
  "symKeys == {K. invKey K = K}"
paulson@1839
    37
paulson@1839
    38
datatype  (*We allow any number of friendly agents*)
paulson@2032
    39
  agent = Server | Friend nat | Spy
paulson@1839
    40
paulson@3668
    41
datatype
paulson@14200
    42
     msg = Agent  agent	    --{*Agent names*}
paulson@14200
    43
         | Number nat       --{*Ordinary integers, timestamps, ...*}
paulson@14200
    44
         | Nonce  nat       --{*Unguessable nonces*}
paulson@14200
    45
         | Key    key       --{*Crypto keys*}
paulson@14200
    46
	 | Hash   msg       --{*Hashing*}
paulson@14200
    47
	 | MPair  msg msg   --{*Compound messages*}
paulson@14200
    48
	 | Crypt  key msg   --{*Encryption, public- or shared-key*}
paulson@1839
    49
paulson@5234
    50
paulson@5234
    51
(*Concrete syntax: messages appear as {|A,B,NA|}, etc...*)
paulson@5234
    52
syntax
paulson@2516
    53
  "@MTuple"      :: "['a, args] => 'a * 'b"       ("(2{|_,/ _|})")
paulson@1839
    54
paulson@9686
    55
syntax (xsymbols)
paulson@11189
    56
  "@MTuple"      :: "['a, args] => 'a * 'b"       ("(2\<lbrace>_,/ _\<rbrace>)")
paulson@9686
    57
paulson@1839
    58
translations
paulson@1839
    59
  "{|x, y, z|}"   == "{|x, {|y, z|}|}"
paulson@1839
    60
  "{|x, y|}"      == "MPair x y"
paulson@1839
    61
paulson@1839
    62
paulson@2484
    63
constdefs
paulson@2484
    64
  (*Message Y, paired with a MAC computed with the help of X*)
paulson@11189
    65
  HPair :: "[msg,msg] => msg"                       ("(4Hash[_] /_)" [0, 1000])
paulson@2516
    66
    "Hash[X] Y == {| Hash{|X,Y|}, Y|}"
paulson@2484
    67
paulson@2484
    68
  (*Keys useful to decrypt elements of a message set*)
paulson@11189
    69
  keysFor :: "msg set => key set"
paulson@11192
    70
  "keysFor H == invKey ` {K. \<exists>X. Crypt K X \<in> H}"
paulson@1839
    71
paulson@14200
    72
subsubsection{*Inductive definition of all "parts" of a message.  *}
paulson@1839
    73
paulson@11189
    74
consts  parts   :: "msg set => msg set"
paulson@1839
    75
inductive "parts H"
paulson@11189
    76
  intros 
paulson@11192
    77
    Inj [intro]:               "X \<in> H ==> X \<in> parts H"
paulson@11192
    78
    Fst:         "{|X,Y|}   \<in> parts H ==> X \<in> parts H"
paulson@11192
    79
    Snd:         "{|X,Y|}   \<in> parts H ==> Y \<in> parts H"
paulson@11192
    80
    Body:        "Crypt K X \<in> parts H ==> X \<in> parts H"
paulson@11189
    81
paulson@11189
    82
paulson@11189
    83
(*Monotonicity*)
paulson@14200
    84
lemma parts_mono: "G\<subseteq>H ==> parts(G) \<subseteq> parts(H)"
paulson@11189
    85
apply auto
paulson@11189
    86
apply (erule parts.induct) 
paulson@11189
    87
apply (auto dest: Fst Snd Body) 
paulson@11189
    88
done
paulson@1839
    89
paulson@1839
    90
paulson@13926
    91
(*Equations hold because constructors are injective; cannot prove for all f*)
paulson@13926
    92
lemma Friend_image_eq [simp]: "(Friend x \<in> Friend`A) = (x:A)"
paulson@13926
    93
by auto
paulson@13926
    94
paulson@13926
    95
lemma Key_image_eq [simp]: "(Key x \<in> Key`A) = (x\<in>A)"
paulson@13926
    96
by auto
paulson@13926
    97
paulson@13926
    98
lemma Nonce_Key_image_eq [simp]: "(Nonce x \<notin> Key`A)"
paulson@13926
    99
by auto
paulson@13926
   100
paulson@13926
   101
paulson@14200
   102
subsubsection{*Inverse of keys *}
paulson@13926
   103
paulson@13926
   104
lemma invKey_eq [simp]: "(invKey K = invKey K') = (K=K')"
paulson@13926
   105
apply safe
paulson@13926
   106
apply (drule_tac f = invKey in arg_cong, simp)
paulson@13926
   107
done
paulson@13926
   108
paulson@13926
   109
paulson@13926
   110
subsection{*keysFor operator*}
paulson@13926
   111
paulson@13926
   112
lemma keysFor_empty [simp]: "keysFor {} = {}"
paulson@13926
   113
by (unfold keysFor_def, blast)
paulson@13926
   114
paulson@13926
   115
lemma keysFor_Un [simp]: "keysFor (H \<union> H') = keysFor H \<union> keysFor H'"
paulson@13926
   116
by (unfold keysFor_def, blast)
paulson@13926
   117
paulson@13926
   118
lemma keysFor_UN [simp]: "keysFor (\<Union>i\<in>A. H i) = (\<Union>i\<in>A. keysFor (H i))"
paulson@13926
   119
by (unfold keysFor_def, blast)
paulson@13926
   120
paulson@13926
   121
(*Monotonicity*)
paulson@13926
   122
lemma keysFor_mono: "G\<subseteq>H ==> keysFor(G) \<subseteq> keysFor(H)"
paulson@13926
   123
by (unfold keysFor_def, blast)
paulson@13926
   124
paulson@13926
   125
lemma keysFor_insert_Agent [simp]: "keysFor (insert (Agent A) H) = keysFor H"
paulson@13926
   126
by (unfold keysFor_def, auto)
paulson@13926
   127
paulson@13926
   128
lemma keysFor_insert_Nonce [simp]: "keysFor (insert (Nonce N) H) = keysFor H"
paulson@13926
   129
by (unfold keysFor_def, auto)
paulson@13926
   130
paulson@13926
   131
lemma keysFor_insert_Number [simp]: "keysFor (insert (Number N) H) = keysFor H"
paulson@13926
   132
by (unfold keysFor_def, auto)
paulson@13926
   133
paulson@13926
   134
lemma keysFor_insert_Key [simp]: "keysFor (insert (Key K) H) = keysFor H"
paulson@13926
   135
by (unfold keysFor_def, auto)
paulson@13926
   136
paulson@13926
   137
lemma keysFor_insert_Hash [simp]: "keysFor (insert (Hash X) H) = keysFor H"
paulson@13926
   138
by (unfold keysFor_def, auto)
paulson@13926
   139
paulson@13926
   140
lemma keysFor_insert_MPair [simp]: "keysFor (insert {|X,Y|} H) = keysFor H"
paulson@13926
   141
by (unfold keysFor_def, auto)
paulson@13926
   142
paulson@13926
   143
lemma keysFor_insert_Crypt [simp]: 
paulson@13926
   144
    "keysFor (insert (Crypt K X) H) = insert (invKey K) (keysFor H)"
paulson@14200
   145
by (unfold keysFor_def, auto)
paulson@13926
   146
paulson@13926
   147
lemma keysFor_image_Key [simp]: "keysFor (Key`E) = {}"
paulson@13926
   148
by (unfold keysFor_def, auto)
paulson@13926
   149
paulson@13926
   150
lemma Crypt_imp_invKey_keysFor: "Crypt K X \<in> H ==> invKey K \<in> keysFor H"
paulson@13926
   151
by (unfold keysFor_def, blast)
paulson@13926
   152
paulson@13926
   153
paulson@13926
   154
subsection{*Inductive relation "parts"*}
paulson@13926
   155
paulson@13926
   156
lemma MPair_parts:
paulson@13926
   157
     "[| {|X,Y|} \<in> parts H;        
paulson@13926
   158
         [| X \<in> parts H; Y \<in> parts H |] ==> P |] ==> P"
paulson@13926
   159
by (blast dest: parts.Fst parts.Snd) 
paulson@13926
   160
paulson@13926
   161
declare MPair_parts [elim!]  parts.Body [dest!]
paulson@13926
   162
text{*NB These two rules are UNSAFE in the formal sense, as they discard the
paulson@13926
   163
     compound message.  They work well on THIS FILE.  
paulson@13926
   164
  @{text MPair_parts} is left as SAFE because it speeds up proofs.
paulson@13926
   165
  The Crypt rule is normally kept UNSAFE to avoid breaking up certificates.*}
paulson@13926
   166
paulson@13926
   167
lemma parts_increasing: "H \<subseteq> parts(H)"
paulson@13926
   168
by blast
paulson@13926
   169
paulson@13926
   170
lemmas parts_insertI = subset_insertI [THEN parts_mono, THEN subsetD, standard]
paulson@13926
   171
paulson@13926
   172
lemma parts_empty [simp]: "parts{} = {}"
paulson@13926
   173
apply safe
paulson@13926
   174
apply (erule parts.induct, blast+)
paulson@13926
   175
done
paulson@13926
   176
paulson@13926
   177
lemma parts_emptyE [elim!]: "X\<in> parts{} ==> P"
paulson@13926
   178
by simp
paulson@13926
   179
paulson@13926
   180
(*WARNING: loops if H = {Y}, therefore must not be repeated!*)
paulson@13926
   181
lemma parts_singleton: "X\<in> parts H ==> \<exists>Y\<in>H. X\<in> parts {Y}"
paulson@13926
   182
by (erule parts.induct, blast+)
paulson@13926
   183
paulson@13926
   184
paulson@14200
   185
subsubsection{*Unions *}
paulson@13926
   186
paulson@13926
   187
lemma parts_Un_subset1: "parts(G) \<union> parts(H) \<subseteq> parts(G \<union> H)"
paulson@13926
   188
by (intro Un_least parts_mono Un_upper1 Un_upper2)
paulson@13926
   189
paulson@13926
   190
lemma parts_Un_subset2: "parts(G \<union> H) \<subseteq> parts(G) \<union> parts(H)"
paulson@13926
   191
apply (rule subsetI)
paulson@13926
   192
apply (erule parts.induct, blast+)
paulson@13926
   193
done
paulson@13926
   194
paulson@13926
   195
lemma parts_Un [simp]: "parts(G \<union> H) = parts(G) \<union> parts(H)"
paulson@13926
   196
by (intro equalityI parts_Un_subset1 parts_Un_subset2)
paulson@13926
   197
paulson@13926
   198
lemma parts_insert: "parts (insert X H) = parts {X} \<union> parts H"
paulson@13926
   199
apply (subst insert_is_Un [of _ H])
paulson@13926
   200
apply (simp only: parts_Un)
paulson@13926
   201
done
paulson@13926
   202
paulson@13926
   203
(*TWO inserts to avoid looping.  This rewrite is better than nothing.
paulson@13926
   204
  Not suitable for Addsimps: its behaviour can be strange.*)
paulson@14200
   205
lemma parts_insert2:
paulson@14200
   206
     "parts (insert X (insert Y H)) = parts {X} \<union> parts {Y} \<union> parts H"
paulson@13926
   207
apply (simp add: Un_assoc)
paulson@13926
   208
apply (simp add: parts_insert [symmetric])
paulson@13926
   209
done
paulson@13926
   210
paulson@13926
   211
lemma parts_UN_subset1: "(\<Union>x\<in>A. parts(H x)) \<subseteq> parts(\<Union>x\<in>A. H x)"
paulson@13926
   212
by (intro UN_least parts_mono UN_upper)
paulson@13926
   213
paulson@13926
   214
lemma parts_UN_subset2: "parts(\<Union>x\<in>A. H x) \<subseteq> (\<Union>x\<in>A. parts(H x))"
paulson@13926
   215
apply (rule subsetI)
paulson@13926
   216
apply (erule parts.induct, blast+)
paulson@13926
   217
done
paulson@13926
   218
paulson@13926
   219
lemma parts_UN [simp]: "parts(\<Union>x\<in>A. H x) = (\<Union>x\<in>A. parts(H x))"
paulson@13926
   220
by (intro equalityI parts_UN_subset1 parts_UN_subset2)
paulson@13926
   221
paulson@13926
   222
(*Added to simplify arguments to parts, analz and synth.
paulson@13926
   223
  NOTE: the UN versions are no longer used!*)
paulson@13926
   224
paulson@13926
   225
paulson@13926
   226
text{*This allows @{text blast} to simplify occurrences of 
paulson@13926
   227
  @{term "parts(G\<union>H)"} in the assumption.*}
paulson@13926
   228
declare parts_Un [THEN equalityD1, THEN subsetD, THEN UnE, elim!] 
paulson@13926
   229
paulson@13926
   230
paulson@13926
   231
lemma parts_insert_subset: "insert X (parts H) \<subseteq> parts(insert X H)"
paulson@13926
   232
by (blast intro: parts_mono [THEN [2] rev_subsetD])
paulson@13926
   233
paulson@14200
   234
subsubsection{*Idempotence and transitivity *}
paulson@13926
   235
paulson@13926
   236
lemma parts_partsD [dest!]: "X\<in> parts (parts H) ==> X\<in> parts H"
paulson@13926
   237
by (erule parts.induct, blast+)
paulson@13926
   238
paulson@13926
   239
lemma parts_idem [simp]: "parts (parts H) = parts H"
paulson@13926
   240
by blast
paulson@13926
   241
paulson@13926
   242
lemma parts_trans: "[| X\<in> parts G;  G \<subseteq> parts H |] ==> X\<in> parts H"
paulson@13926
   243
by (drule parts_mono, blast)
paulson@13926
   244
paulson@13926
   245
(*Cut*)
paulson@14200
   246
lemma parts_cut:
paulson@14200
   247
     "[| Y\<in> parts (insert X G);  X\<in> parts H |] ==> Y\<in> parts (G \<union> H)"
paulson@14200
   248
by (erule parts_trans, auto)
paulson@13926
   249
paulson@13926
   250
lemma parts_cut_eq [simp]: "X\<in> parts H ==> parts (insert X H) = parts H"
paulson@13926
   251
by (force dest!: parts_cut intro: parts_insertI)
paulson@13926
   252
paulson@13926
   253
paulson@14200
   254
subsubsection{*Rewrite rules for pulling out atomic messages *}
paulson@13926
   255
paulson@13926
   256
lemmas parts_insert_eq_I = equalityI [OF subsetI parts_insert_subset]
paulson@13926
   257
paulson@13926
   258
paulson@14200
   259
lemma parts_insert_Agent [simp]:
paulson@14200
   260
     "parts (insert (Agent agt) H) = insert (Agent agt) (parts H)"
paulson@13926
   261
apply (rule parts_insert_eq_I) 
paulson@13926
   262
apply (erule parts.induct, auto) 
paulson@13926
   263
done
paulson@13926
   264
paulson@14200
   265
lemma parts_insert_Nonce [simp]:
paulson@14200
   266
     "parts (insert (Nonce N) H) = insert (Nonce N) (parts H)"
paulson@13926
   267
apply (rule parts_insert_eq_I) 
paulson@13926
   268
apply (erule parts.induct, auto) 
paulson@13926
   269
done
paulson@13926
   270
paulson@14200
   271
lemma parts_insert_Number [simp]:
paulson@14200
   272
     "parts (insert (Number N) H) = insert (Number N) (parts H)"
paulson@13926
   273
apply (rule parts_insert_eq_I) 
paulson@13926
   274
apply (erule parts.induct, auto) 
paulson@13926
   275
done
paulson@13926
   276
paulson@14200
   277
lemma parts_insert_Key [simp]:
paulson@14200
   278
     "parts (insert (Key K) H) = insert (Key K) (parts H)"
paulson@13926
   279
apply (rule parts_insert_eq_I) 
paulson@13926
   280
apply (erule parts.induct, auto) 
paulson@13926
   281
done
paulson@13926
   282
paulson@14200
   283
lemma parts_insert_Hash [simp]:
paulson@14200
   284
     "parts (insert (Hash X) H) = insert (Hash X) (parts H)"
paulson@13926
   285
apply (rule parts_insert_eq_I) 
paulson@13926
   286
apply (erule parts.induct, auto) 
paulson@13926
   287
done
paulson@13926
   288
paulson@14200
   289
lemma parts_insert_Crypt [simp]:
paulson@14200
   290
     "parts (insert (Crypt K X) H) =  
paulson@13926
   291
          insert (Crypt K X) (parts (insert X H))"
paulson@13926
   292
apply (rule equalityI)
paulson@13926
   293
apply (rule subsetI)
paulson@13926
   294
apply (erule parts.induct, auto)
paulson@13926
   295
apply (erule parts.induct)
paulson@13926
   296
apply (blast intro: parts.Body)+
paulson@13926
   297
done
paulson@13926
   298
paulson@14200
   299
lemma parts_insert_MPair [simp]:
paulson@14200
   300
     "parts (insert {|X,Y|} H) =  
paulson@13926
   301
          insert {|X,Y|} (parts (insert X (insert Y H)))"
paulson@13926
   302
apply (rule equalityI)
paulson@13926
   303
apply (rule subsetI)
paulson@13926
   304
apply (erule parts.induct, auto)
paulson@13926
   305
apply (erule parts.induct)
paulson@13926
   306
apply (blast intro: parts.Fst parts.Snd)+
paulson@13926
   307
done
paulson@13926
   308
paulson@13926
   309
lemma parts_image_Key [simp]: "parts (Key`N) = Key`N"
paulson@13926
   310
apply auto
paulson@13926
   311
apply (erule parts.induct, auto)
paulson@13926
   312
done
paulson@13926
   313
paulson@13926
   314
paulson@13926
   315
(*In any message, there is an upper bound N on its greatest nonce.*)
paulson@13926
   316
lemma msg_Nonce_supply: "\<exists>N. \<forall>n. N\<le>n --> Nonce n \<notin> parts {msg}"
paulson@13926
   317
apply (induct_tac "msg")
paulson@13926
   318
apply (simp_all (no_asm_simp) add: exI parts_insert2)
paulson@13926
   319
(*MPair case: blast_tac works out the necessary sum itself!*)
paulson@13926
   320
prefer 2 apply (blast elim!: add_leE)
paulson@13926
   321
(*Nonce case*)
paulson@13926
   322
apply (rule_tac x = "N + Suc nat" in exI)
paulson@13926
   323
apply (auto elim!: add_leE)
paulson@13926
   324
done
paulson@13926
   325
paulson@13926
   326
paulson@13926
   327
subsection{*Inductive relation "analz"*}
paulson@13926
   328
paulson@14200
   329
text{*Inductive definition of "analz" -- what can be broken down from a set of
paulson@1839
   330
    messages, including keys.  A form of downward closure.  Pairs can
paulson@14200
   331
    be taken apart; messages decrypted with known keys.  *}
paulson@1839
   332
paulson@11189
   333
consts  analz   :: "msg set => msg set"
paulson@1913
   334
inductive "analz H"
paulson@11189
   335
  intros 
paulson@11192
   336
    Inj [intro,simp] :    "X \<in> H ==> X \<in> analz H"
paulson@11192
   337
    Fst:     "{|X,Y|} \<in> analz H ==> X \<in> analz H"
paulson@11192
   338
    Snd:     "{|X,Y|} \<in> analz H ==> Y \<in> analz H"
paulson@11189
   339
    Decrypt [dest]: 
paulson@11192
   340
             "[|Crypt K X \<in> analz H; Key(invKey K): analz H|] ==> X \<in> analz H"
paulson@1839
   341
paulson@1839
   342
paulson@11189
   343
(*Monotonicity; Lemma 1 of Lowe's paper*)
paulson@14200
   344
lemma analz_mono: "G\<subseteq>H ==> analz(G) \<subseteq> analz(H)"
paulson@11189
   345
apply auto
paulson@11189
   346
apply (erule analz.induct) 
paulson@11189
   347
apply (auto dest: Fst Snd) 
paulson@11189
   348
done
paulson@11189
   349
paulson@13926
   350
text{*Making it safe speeds up proofs*}
paulson@13926
   351
lemma MPair_analz [elim!]:
paulson@13926
   352
     "[| {|X,Y|} \<in> analz H;        
paulson@13926
   353
             [| X \<in> analz H; Y \<in> analz H |] ==> P   
paulson@13926
   354
          |] ==> P"
paulson@13926
   355
by (blast dest: analz.Fst analz.Snd)
paulson@13926
   356
paulson@13926
   357
lemma analz_increasing: "H \<subseteq> analz(H)"
paulson@13926
   358
by blast
paulson@13926
   359
paulson@13926
   360
lemma analz_subset_parts: "analz H \<subseteq> parts H"
paulson@13926
   361
apply (rule subsetI)
paulson@13926
   362
apply (erule analz.induct, blast+)
paulson@13926
   363
done
paulson@13926
   364
paulson@14200
   365
lemmas analz_into_parts = analz_subset_parts [THEN subsetD, standard]
paulson@14200
   366
paulson@13926
   367
lemmas not_parts_not_analz = analz_subset_parts [THEN contra_subsetD, standard]
paulson@13926
   368
paulson@13926
   369
paulson@13926
   370
lemma parts_analz [simp]: "parts (analz H) = parts H"
paulson@13926
   371
apply (rule equalityI)
paulson@13926
   372
apply (rule analz_subset_parts [THEN parts_mono, THEN subset_trans], simp)
paulson@13926
   373
apply (blast intro: analz_increasing [THEN parts_mono, THEN subsetD])
paulson@13926
   374
done
paulson@13926
   375
paulson@13926
   376
lemma analz_parts [simp]: "analz (parts H) = parts H"
paulson@13926
   377
apply auto
paulson@13926
   378
apply (erule analz.induct, auto)
paulson@13926
   379
done
paulson@13926
   380
paulson@13926
   381
lemmas analz_insertI = subset_insertI [THEN analz_mono, THEN [2] rev_subsetD, standard]
paulson@13926
   382
paulson@14200
   383
subsubsection{*General equational properties *}
paulson@13926
   384
paulson@13926
   385
lemma analz_empty [simp]: "analz{} = {}"
paulson@13926
   386
apply safe
paulson@13926
   387
apply (erule analz.induct, blast+)
paulson@13926
   388
done
paulson@13926
   389
paulson@13926
   390
(*Converse fails: we can analz more from the union than from the 
paulson@13926
   391
  separate parts, as a key in one might decrypt a message in the other*)
paulson@13926
   392
lemma analz_Un: "analz(G) \<union> analz(H) \<subseteq> analz(G \<union> H)"
paulson@13926
   393
by (intro Un_least analz_mono Un_upper1 Un_upper2)
paulson@13926
   394
paulson@13926
   395
lemma analz_insert: "insert X (analz H) \<subseteq> analz(insert X H)"
paulson@13926
   396
by (blast intro: analz_mono [THEN [2] rev_subsetD])
paulson@13926
   397
paulson@14200
   398
subsubsection{*Rewrite rules for pulling out atomic messages *}
paulson@13926
   399
paulson@13926
   400
lemmas analz_insert_eq_I = equalityI [OF subsetI analz_insert]
paulson@13926
   401
paulson@14200
   402
lemma analz_insert_Agent [simp]:
paulson@14200
   403
     "analz (insert (Agent agt) H) = insert (Agent agt) (analz H)"
paulson@13926
   404
apply (rule analz_insert_eq_I) 
paulson@13926
   405
apply (erule analz.induct, auto) 
paulson@13926
   406
done
paulson@13926
   407
paulson@14200
   408
lemma analz_insert_Nonce [simp]:
paulson@14200
   409
     "analz (insert (Nonce N) H) = insert (Nonce N) (analz H)"
paulson@13926
   410
apply (rule analz_insert_eq_I) 
paulson@13926
   411
apply (erule analz.induct, auto) 
paulson@13926
   412
done
paulson@13926
   413
paulson@14200
   414
lemma analz_insert_Number [simp]:
paulson@14200
   415
     "analz (insert (Number N) H) = insert (Number N) (analz H)"
paulson@13926
   416
apply (rule analz_insert_eq_I) 
paulson@13926
   417
apply (erule analz.induct, auto) 
paulson@13926
   418
done
paulson@13926
   419
paulson@14200
   420
lemma analz_insert_Hash [simp]:
paulson@14200
   421
     "analz (insert (Hash X) H) = insert (Hash X) (analz H)"
paulson@13926
   422
apply (rule analz_insert_eq_I) 
paulson@13926
   423
apply (erule analz.induct, auto) 
paulson@13926
   424
done
paulson@13926
   425
paulson@13926
   426
(*Can only pull out Keys if they are not needed to decrypt the rest*)
paulson@13926
   427
lemma analz_insert_Key [simp]: 
paulson@13926
   428
    "K \<notin> keysFor (analz H) ==>   
paulson@13926
   429
          analz (insert (Key K) H) = insert (Key K) (analz H)"
paulson@13926
   430
apply (unfold keysFor_def)
paulson@13926
   431
apply (rule analz_insert_eq_I) 
paulson@13926
   432
apply (erule analz.induct, auto) 
paulson@13926
   433
done
paulson@13926
   434
paulson@14200
   435
lemma analz_insert_MPair [simp]:
paulson@14200
   436
     "analz (insert {|X,Y|} H) =  
paulson@13926
   437
          insert {|X,Y|} (analz (insert X (insert Y H)))"
paulson@13926
   438
apply (rule equalityI)
paulson@13926
   439
apply (rule subsetI)
paulson@13926
   440
apply (erule analz.induct, auto)
paulson@13926
   441
apply (erule analz.induct)
paulson@13926
   442
apply (blast intro: analz.Fst analz.Snd)+
paulson@13926
   443
done
paulson@13926
   444
paulson@13926
   445
(*Can pull out enCrypted message if the Key is not known*)
paulson@13926
   446
lemma analz_insert_Crypt:
paulson@13926
   447
     "Key (invKey K) \<notin> analz H 
paulson@13926
   448
      ==> analz (insert (Crypt K X) H) = insert (Crypt K X) (analz H)"
paulson@13926
   449
apply (rule analz_insert_eq_I) 
paulson@13926
   450
apply (erule analz.induct, auto) 
paulson@13926
   451
paulson@13926
   452
done
paulson@13926
   453
paulson@13926
   454
lemma lemma1: "Key (invKey K) \<in> analz H ==>   
paulson@13926
   455
               analz (insert (Crypt K X) H) \<subseteq>  
paulson@13926
   456
               insert (Crypt K X) (analz (insert X H))"
paulson@13926
   457
apply (rule subsetI)
paulson@13926
   458
apply (erule_tac xa = x in analz.induct, auto)
paulson@13926
   459
done
paulson@13926
   460
paulson@13926
   461
lemma lemma2: "Key (invKey K) \<in> analz H ==>   
paulson@13926
   462
               insert (Crypt K X) (analz (insert X H)) \<subseteq>  
paulson@13926
   463
               analz (insert (Crypt K X) H)"
paulson@13926
   464
apply auto
paulson@13926
   465
apply (erule_tac xa = x in analz.induct, auto)
paulson@13926
   466
apply (blast intro: analz_insertI analz.Decrypt)
paulson@13926
   467
done
paulson@13926
   468
paulson@14200
   469
lemma analz_insert_Decrypt:
paulson@14200
   470
     "Key (invKey K) \<in> analz H ==>   
paulson@13926
   471
               analz (insert (Crypt K X) H) =  
paulson@13926
   472
               insert (Crypt K X) (analz (insert X H))"
paulson@13926
   473
by (intro equalityI lemma1 lemma2)
paulson@13926
   474
paulson@13926
   475
(*Case analysis: either the message is secure, or it is not!
paulson@13926
   476
  Effective, but can cause subgoals to blow up!
paulson@13926
   477
  Use with split_if;  apparently split_tac does not cope with patterns
paulson@13926
   478
  such as "analz (insert (Crypt K X) H)" *)
paulson@13926
   479
lemma analz_Crypt_if [simp]:
paulson@13926
   480
     "analz (insert (Crypt K X) H) =                 
paulson@13926
   481
          (if (Key (invKey K) \<in> analz H)                 
paulson@13926
   482
           then insert (Crypt K X) (analz (insert X H))  
paulson@13926
   483
           else insert (Crypt K X) (analz H))"
paulson@13926
   484
by (simp add: analz_insert_Crypt analz_insert_Decrypt)
paulson@13926
   485
paulson@13926
   486
paulson@13926
   487
(*This rule supposes "for the sake of argument" that we have the key.*)
paulson@14200
   488
lemma analz_insert_Crypt_subset:
paulson@14200
   489
     "analz (insert (Crypt K X) H) \<subseteq>   
paulson@13926
   490
           insert (Crypt K X) (analz (insert X H))"
paulson@13926
   491
apply (rule subsetI)
paulson@13926
   492
apply (erule analz.induct, auto)
paulson@13926
   493
done
paulson@13926
   494
paulson@13926
   495
paulson@13926
   496
lemma analz_image_Key [simp]: "analz (Key`N) = Key`N"
paulson@13926
   497
apply auto
paulson@13926
   498
apply (erule analz.induct, auto)
paulson@13926
   499
done
paulson@13926
   500
paulson@13926
   501
paulson@14200
   502
subsubsection{*Idempotence and transitivity *}
paulson@13926
   503
paulson@13926
   504
lemma analz_analzD [dest!]: "X\<in> analz (analz H) ==> X\<in> analz H"
paulson@13926
   505
by (erule analz.induct, blast+)
paulson@13926
   506
paulson@13926
   507
lemma analz_idem [simp]: "analz (analz H) = analz H"
paulson@13926
   508
by blast
paulson@13926
   509
paulson@13926
   510
lemma analz_trans: "[| X\<in> analz G;  G \<subseteq> analz H |] ==> X\<in> analz H"
paulson@13926
   511
by (drule analz_mono, blast)
paulson@13926
   512
paulson@13926
   513
(*Cut; Lemma 2 of Lowe*)
paulson@13926
   514
lemma analz_cut: "[| Y\<in> analz (insert X H);  X\<in> analz H |] ==> Y\<in> analz H"
paulson@13926
   515
by (erule analz_trans, blast)
paulson@13926
   516
paulson@13926
   517
(*Cut can be proved easily by induction on
paulson@13926
   518
   "Y: analz (insert X H) ==> X: analz H --> Y: analz H"
paulson@13926
   519
*)
paulson@13926
   520
paulson@13926
   521
(*This rewrite rule helps in the simplification of messages that involve
paulson@13926
   522
  the forwarding of unknown components (X).  Without it, removing occurrences
paulson@13926
   523
  of X can be very complicated. *)
paulson@13926
   524
lemma analz_insert_eq: "X\<in> analz H ==> analz (insert X H) = analz H"
paulson@13926
   525
by (blast intro: analz_cut analz_insertI)
paulson@13926
   526
paulson@13926
   527
paulson@14200
   528
text{*A congruence rule for "analz" *}
paulson@13926
   529
paulson@14200
   530
lemma analz_subset_cong:
paulson@14200
   531
     "[| analz G \<subseteq> analz G'; analz H \<subseteq> analz H'  
paulson@13926
   532
               |] ==> analz (G \<union> H) \<subseteq> analz (G' \<union> H')"
paulson@13926
   533
apply clarify
paulson@13926
   534
apply (erule analz.induct)
paulson@13926
   535
apply (best intro: analz_mono [THEN subsetD])+
paulson@13926
   536
done
paulson@13926
   537
paulson@14200
   538
lemma analz_cong:
paulson@14200
   539
     "[| analz G = analz G'; analz H = analz H'  
paulson@13926
   540
               |] ==> analz (G \<union> H) = analz (G' \<union> H')"
paulson@14200
   541
by (intro equalityI analz_subset_cong, simp_all) 
paulson@13926
   542
paulson@14200
   543
lemma analz_insert_cong:
paulson@14200
   544
     "analz H = analz H' ==> analz(insert X H) = analz(insert X H')"
paulson@13926
   545
by (force simp only: insert_def intro!: analz_cong)
paulson@13926
   546
paulson@13926
   547
(*If there are no pairs or encryptions then analz does nothing*)
paulson@14200
   548
lemma analz_trivial:
paulson@14200
   549
     "[| \<forall>X Y. {|X,Y|} \<notin> H;  \<forall>X K. Crypt K X \<notin> H |] ==> analz H = H"
paulson@13926
   550
apply safe
paulson@13926
   551
apply (erule analz.induct, blast+)
paulson@13926
   552
done
paulson@13926
   553
paulson@13926
   554
(*These two are obsolete (with a single Spy) but cost little to prove...*)
paulson@14200
   555
lemma analz_UN_analz_lemma:
paulson@14200
   556
     "X\<in> analz (\<Union>i\<in>A. analz (H i)) ==> X\<in> analz (\<Union>i\<in>A. H i)"
paulson@13926
   557
apply (erule analz.induct)
paulson@13926
   558
apply (blast intro: analz_mono [THEN [2] rev_subsetD])+
paulson@13926
   559
done
paulson@13926
   560
paulson@13926
   561
lemma analz_UN_analz [simp]: "analz (\<Union>i\<in>A. analz (H i)) = analz (\<Union>i\<in>A. H i)"
paulson@13926
   562
by (blast intro: analz_UN_analz_lemma analz_mono [THEN [2] rev_subsetD])
paulson@13926
   563
paulson@13926
   564
paulson@13926
   565
subsection{*Inductive relation "synth"*}
paulson@13926
   566
paulson@14200
   567
text{*Inductive definition of "synth" -- what can be built up from a set of
paulson@1839
   568
    messages.  A form of upward closure.  Pairs can be built, messages
paulson@3668
   569
    encrypted with known keys.  Agent names are public domain.
paulson@14200
   570
    Numbers can be guessed, but Nonces cannot be.  *}
paulson@1839
   571
paulson@11189
   572
consts  synth   :: "msg set => msg set"
paulson@1913
   573
inductive "synth H"
paulson@11189
   574
  intros 
paulson@11192
   575
    Inj    [intro]:   "X \<in> H ==> X \<in> synth H"
paulson@11192
   576
    Agent  [intro]:   "Agent agt \<in> synth H"
paulson@11192
   577
    Number [intro]:   "Number n  \<in> synth H"
paulson@11192
   578
    Hash   [intro]:   "X \<in> synth H ==> Hash X \<in> synth H"
paulson@11192
   579
    MPair  [intro]:   "[|X \<in> synth H;  Y \<in> synth H|] ==> {|X,Y|} \<in> synth H"
paulson@11192
   580
    Crypt  [intro]:   "[|X \<in> synth H;  Key(K) \<in> H|] ==> Crypt K X \<in> synth H"
paulson@11189
   581
paulson@11189
   582
(*Monotonicity*)
paulson@14200
   583
lemma synth_mono: "G\<subseteq>H ==> synth(G) \<subseteq> synth(H)"
paulson@11189
   584
apply auto
paulson@11189
   585
apply (erule synth.induct) 
paulson@11189
   586
apply (auto dest: Fst Snd Body) 
paulson@11189
   587
done
paulson@11189
   588
paulson@11189
   589
(*NO Agent_synth, as any Agent name can be synthesized.  Ditto for Number*)
paulson@11192
   590
inductive_cases Nonce_synth [elim!]: "Nonce n \<in> synth H"
paulson@11192
   591
inductive_cases Key_synth   [elim!]: "Key K \<in> synth H"
paulson@11192
   592
inductive_cases Hash_synth  [elim!]: "Hash X \<in> synth H"
paulson@11192
   593
inductive_cases MPair_synth [elim!]: "{|X,Y|} \<in> synth H"
paulson@11192
   594
inductive_cases Crypt_synth [elim!]: "Crypt K X \<in> synth H"
paulson@11189
   595
paulson@13926
   596
paulson@13926
   597
lemma synth_increasing: "H \<subseteq> synth(H)"
paulson@13926
   598
by blast
paulson@13926
   599
paulson@14200
   600
subsubsection{*Unions *}
paulson@13926
   601
paulson@13926
   602
(*Converse fails: we can synth more from the union than from the 
paulson@13926
   603
  separate parts, building a compound message using elements of each.*)
paulson@13926
   604
lemma synth_Un: "synth(G) \<union> synth(H) \<subseteq> synth(G \<union> H)"
paulson@13926
   605
by (intro Un_least synth_mono Un_upper1 Un_upper2)
paulson@13926
   606
paulson@13926
   607
lemma synth_insert: "insert X (synth H) \<subseteq> synth(insert X H)"
paulson@13926
   608
by (blast intro: synth_mono [THEN [2] rev_subsetD])
paulson@13926
   609
paulson@14200
   610
subsubsection{*Idempotence and transitivity *}
paulson@13926
   611
paulson@13926
   612
lemma synth_synthD [dest!]: "X\<in> synth (synth H) ==> X\<in> synth H"
paulson@13926
   613
by (erule synth.induct, blast+)
paulson@13926
   614
paulson@13926
   615
lemma synth_idem: "synth (synth H) = synth H"
paulson@13926
   616
by blast
paulson@13926
   617
paulson@13926
   618
lemma synth_trans: "[| X\<in> synth G;  G \<subseteq> synth H |] ==> X\<in> synth H"
paulson@13926
   619
by (drule synth_mono, blast)
paulson@13926
   620
paulson@13926
   621
(*Cut; Lemma 2 of Lowe*)
paulson@13926
   622
lemma synth_cut: "[| Y\<in> synth (insert X H);  X\<in> synth H |] ==> Y\<in> synth H"
paulson@13926
   623
by (erule synth_trans, blast)
paulson@13926
   624
paulson@13926
   625
lemma Agent_synth [simp]: "Agent A \<in> synth H"
paulson@13926
   626
by blast
paulson@13926
   627
paulson@13926
   628
lemma Number_synth [simp]: "Number n \<in> synth H"
paulson@13926
   629
by blast
paulson@13926
   630
paulson@13926
   631
lemma Nonce_synth_eq [simp]: "(Nonce N \<in> synth H) = (Nonce N \<in> H)"
paulson@13926
   632
by blast
paulson@13926
   633
paulson@13926
   634
lemma Key_synth_eq [simp]: "(Key K \<in> synth H) = (Key K \<in> H)"
paulson@13926
   635
by blast
paulson@13926
   636
paulson@14200
   637
lemma Crypt_synth_eq [simp]:
paulson@14200
   638
     "Key K \<notin> H ==> (Crypt K X \<in> synth H) = (Crypt K X \<in> H)"
paulson@13926
   639
by blast
paulson@13926
   640
paulson@13926
   641
paulson@13926
   642
lemma keysFor_synth [simp]: 
paulson@13926
   643
    "keysFor (synth H) = keysFor H \<union> invKey`{K. Key K \<in> H}"
paulson@14200
   644
by (unfold keysFor_def, blast)
paulson@13926
   645
paulson@13926
   646
paulson@14200
   647
subsubsection{*Combinations of parts, analz and synth *}
paulson@13926
   648
paulson@13926
   649
lemma parts_synth [simp]: "parts (synth H) = parts H \<union> synth H"
paulson@13926
   650
apply (rule equalityI)
paulson@13926
   651
apply (rule subsetI)
paulson@13926
   652
apply (erule parts.induct)
paulson@13926
   653
apply (blast intro: synth_increasing [THEN parts_mono, THEN subsetD] 
paulson@13926
   654
                    parts.Fst parts.Snd parts.Body)+
paulson@13926
   655
done
paulson@13926
   656
paulson@13926
   657
lemma analz_analz_Un [simp]: "analz (analz G \<union> H) = analz (G \<union> H)"
paulson@13926
   658
apply (intro equalityI analz_subset_cong)+
paulson@13926
   659
apply simp_all
paulson@13926
   660
done
paulson@13926
   661
paulson@13926
   662
lemma analz_synth_Un [simp]: "analz (synth G \<union> H) = analz (G \<union> H) \<union> synth G"
paulson@13926
   663
apply (rule equalityI)
paulson@13926
   664
apply (rule subsetI)
paulson@13926
   665
apply (erule analz.induct)
paulson@13926
   666
prefer 5 apply (blast intro: analz_mono [THEN [2] rev_subsetD])
paulson@13926
   667
apply (blast intro: analz.Fst analz.Snd analz.Decrypt)+
paulson@13926
   668
done
paulson@13926
   669
paulson@13926
   670
lemma analz_synth [simp]: "analz (synth H) = analz H \<union> synth H"
paulson@13926
   671
apply (cut_tac H = "{}" in analz_synth_Un)
paulson@13926
   672
apply (simp (no_asm_use))
paulson@13926
   673
done
paulson@13926
   674
paulson@13926
   675
paulson@14200
   676
subsubsection{*For reasoning about the Fake rule in traces *}
paulson@13926
   677
paulson@13926
   678
lemma parts_insert_subset_Un: "X\<in> G ==> parts(insert X H) \<subseteq> parts G \<union> parts H"
paulson@13926
   679
by (rule subset_trans [OF parts_mono parts_Un_subset2], blast)
paulson@13926
   680
paulson@13926
   681
(*More specifically for Fake.  Very occasionally we could do with a version
paulson@13926
   682
  of the form  parts{X} \<subseteq> synth (analz H) \<union> parts H *)
paulson@14200
   683
lemma Fake_parts_insert:
paulson@14200
   684
     "X \<in> synth (analz H) ==>  
paulson@13926
   685
      parts (insert X H) \<subseteq> synth (analz H) \<union> parts H"
paulson@13926
   686
apply (drule parts_insert_subset_Un)
paulson@13926
   687
apply (simp (no_asm_use))
paulson@13926
   688
apply blast
paulson@13926
   689
done
paulson@13926
   690
paulson@14200
   691
lemma Fake_parts_insert_in_Un:
paulson@14200
   692
     "[|Z \<in> parts (insert X H);  X: synth (analz H)|] 
paulson@14200
   693
      ==> Z \<in>  synth (analz H) \<union> parts H";
paulson@14200
   694
by (blast dest: Fake_parts_insert  [THEN subsetD, dest])
paulson@14200
   695
paulson@13926
   696
(*H is sometimes (Key ` KK \<union> spies evs), so can't put G=H*)
paulson@14200
   697
lemma Fake_analz_insert:
paulson@14200
   698
     "X\<in> synth (analz G) ==>  
paulson@13926
   699
      analz (insert X H) \<subseteq> synth (analz G) \<union> analz (G \<union> H)"
paulson@13926
   700
apply (rule subsetI)
paulson@13926
   701
apply (subgoal_tac "x \<in> analz (synth (analz G) \<union> H) ")
paulson@13926
   702
prefer 2 apply (blast intro: analz_mono [THEN [2] rev_subsetD] analz_mono [THEN synth_mono, THEN [2] rev_subsetD])
paulson@13926
   703
apply (simp (no_asm_use))
paulson@13926
   704
apply blast
paulson@13926
   705
done
paulson@13926
   706
paulson@14200
   707
lemma analz_conj_parts [simp]:
paulson@14200
   708
     "(X \<in> analz H & X \<in> parts H) = (X \<in> analz H)"
paulson@14145
   709
by (blast intro: analz_subset_parts [THEN subsetD])
paulson@13926
   710
paulson@14200
   711
lemma analz_disj_parts [simp]:
paulson@14200
   712
     "(X \<in> analz H | X \<in> parts H) = (X \<in> parts H)"
paulson@14145
   713
by (blast intro: analz_subset_parts [THEN subsetD])
paulson@13926
   714
paulson@13926
   715
(*Without this equation, other rules for synth and analz would yield
paulson@13926
   716
  redundant cases*)
paulson@13926
   717
lemma MPair_synth_analz [iff]:
paulson@13926
   718
     "({|X,Y|} \<in> synth (analz H)) =  
paulson@13926
   719
      (X \<in> synth (analz H) & Y \<in> synth (analz H))"
paulson@13926
   720
by blast
paulson@13926
   721
paulson@14200
   722
lemma Crypt_synth_analz:
paulson@14200
   723
     "[| Key K \<in> analz H;  Key (invKey K) \<in> analz H |]  
paulson@13926
   724
       ==> (Crypt K X \<in> synth (analz H)) = (X \<in> synth (analz H))"
paulson@13926
   725
by blast
paulson@13926
   726
paulson@13926
   727
paulson@14200
   728
lemma Hash_synth_analz [simp]:
paulson@14200
   729
     "X \<notin> synth (analz H)  
paulson@13926
   730
      ==> (Hash{|X,Y|} \<in> synth (analz H)) = (Hash{|X,Y|} \<in> analz H)"
paulson@13926
   731
by blast
paulson@13926
   732
paulson@13926
   733
paulson@13926
   734
subsection{*HPair: a combination of Hash and MPair*}
paulson@13926
   735
paulson@14200
   736
subsubsection{*Freeness *}
paulson@13926
   737
paulson@13926
   738
lemma Agent_neq_HPair: "Agent A ~= Hash[X] Y"
paulson@13926
   739
by (unfold HPair_def, simp)
paulson@13926
   740
paulson@13926
   741
lemma Nonce_neq_HPair: "Nonce N ~= Hash[X] Y"
paulson@13926
   742
by (unfold HPair_def, simp)
paulson@13926
   743
paulson@13926
   744
lemma Number_neq_HPair: "Number N ~= Hash[X] Y"
paulson@13926
   745
by (unfold HPair_def, simp)
paulson@13926
   746
paulson@13926
   747
lemma Key_neq_HPair: "Key K ~= Hash[X] Y"
paulson@13926
   748
by (unfold HPair_def, simp)
paulson@13926
   749
paulson@13926
   750
lemma Hash_neq_HPair: "Hash Z ~= Hash[X] Y"
paulson@13926
   751
by (unfold HPair_def, simp)
paulson@13926
   752
paulson@13926
   753
lemma Crypt_neq_HPair: "Crypt K X' ~= Hash[X] Y"
paulson@13926
   754
by (unfold HPair_def, simp)
paulson@13926
   755
paulson@13926
   756
lemmas HPair_neqs = Agent_neq_HPair Nonce_neq_HPair Number_neq_HPair 
paulson@13926
   757
                    Key_neq_HPair Hash_neq_HPair Crypt_neq_HPair
paulson@13926
   758
paulson@13926
   759
declare HPair_neqs [iff]
paulson@13926
   760
declare HPair_neqs [symmetric, iff]
paulson@13926
   761
paulson@13926
   762
lemma HPair_eq [iff]: "(Hash[X'] Y' = Hash[X] Y) = (X' = X & Y'=Y)"
paulson@13926
   763
by (simp add: HPair_def)
paulson@13926
   764
paulson@14200
   765
lemma MPair_eq_HPair [iff]:
paulson@14200
   766
     "({|X',Y'|} = Hash[X] Y) = (X' = Hash{|X,Y|} & Y'=Y)"
paulson@13926
   767
by (simp add: HPair_def)
paulson@13926
   768
paulson@14200
   769
lemma HPair_eq_MPair [iff]:
paulson@14200
   770
     "(Hash[X] Y = {|X',Y'|}) = (X' = Hash{|X,Y|} & Y'=Y)"
paulson@13926
   771
by (auto simp add: HPair_def)
paulson@13926
   772
paulson@13926
   773
paulson@14200
   774
subsubsection{*Specialized laws, proved in terms of those for Hash and MPair *}
paulson@13926
   775
paulson@13926
   776
lemma keysFor_insert_HPair [simp]: "keysFor (insert (Hash[X] Y) H) = keysFor H"
paulson@13926
   777
by (simp add: HPair_def)
paulson@13926
   778
paulson@13926
   779
lemma parts_insert_HPair [simp]: 
paulson@13926
   780
    "parts (insert (Hash[X] Y) H) =  
paulson@13926
   781
     insert (Hash[X] Y) (insert (Hash{|X,Y|}) (parts (insert Y H)))"
paulson@13926
   782
by (simp add: HPair_def)
paulson@13926
   783
paulson@13926
   784
lemma analz_insert_HPair [simp]: 
paulson@13926
   785
    "analz (insert (Hash[X] Y) H) =  
paulson@13926
   786
     insert (Hash[X] Y) (insert (Hash{|X,Y|}) (analz (insert Y H)))"
paulson@13926
   787
by (simp add: HPair_def)
paulson@13926
   788
paulson@13926
   789
lemma HPair_synth_analz [simp]:
paulson@13926
   790
     "X \<notin> synth (analz H)  
paulson@13926
   791
    ==> (Hash[X] Y \<in> synth (analz H)) =  
paulson@13926
   792
        (Hash {|X, Y|} \<in> analz H & Y \<in> synth (analz H))"
paulson@13926
   793
by (simp add: HPair_def)
paulson@13926
   794
paulson@13926
   795
paulson@13926
   796
(*We do NOT want Crypt... messages broken up in protocols!!*)
paulson@13926
   797
declare parts.Body [rule del]
paulson@13926
   798
paulson@13926
   799
paulson@14200
   800
text{*Rewrites to push in Key and Crypt messages, so that other messages can
paulson@14200
   801
    be pulled out using the @{text analz_insert} rules*}
paulson@13926
   802
ML
paulson@13926
   803
{*
paulson@13926
   804
fun insComm x y = inst "x" x (inst "y" y insert_commute);
paulson@13926
   805
paulson@13926
   806
bind_thms ("pushKeys",
paulson@13926
   807
           map (insComm "Key ?K") 
paulson@13926
   808
                   ["Agent ?C", "Nonce ?N", "Number ?N", 
paulson@13926
   809
		    "Hash ?X", "MPair ?X ?Y", "Crypt ?X ?K'"]);
paulson@13926
   810
paulson@13926
   811
bind_thms ("pushCrypts",
paulson@13926
   812
           map (insComm "Crypt ?X ?K") 
paulson@13926
   813
                     ["Agent ?C", "Nonce ?N", "Number ?N", 
paulson@13926
   814
		      "Hash ?X'", "MPair ?X' ?Y"]);
paulson@13926
   815
*}
paulson@13926
   816
paulson@13926
   817
text{*Cannot be added with @{text "[simp]"} -- messages should not always be
paulson@13926
   818
  re-ordered. *}
paulson@13926
   819
lemmas pushes = pushKeys pushCrypts
paulson@13926
   820
paulson@13926
   821
paulson@13926
   822
subsection{*Tactics useful for many protocol proofs*}
paulson@13926
   823
ML
paulson@13926
   824
{*
paulson@14200
   825
val invKey = thm "invKey"
paulson@14200
   826
val keysFor_def = thm "keysFor_def"
paulson@14200
   827
val HPair_def = thm "HPair_def"
paulson@14200
   828
val symKeys_def = thm "symKeys_def"
paulson@13926
   829
val parts_mono = thm "parts_mono";
paulson@13926
   830
val analz_mono = thm "analz_mono";
paulson@14200
   831
val synth_mono = thm "synth_mono";
paulson@13926
   832
val analz_increasing = thm "analz_increasing";
paulson@14200
   833
paulson@14200
   834
val analz_insertI = thm "analz_insertI";
paulson@13926
   835
val analz_subset_parts = thm "analz_subset_parts";
paulson@13926
   836
val Fake_parts_insert = thm "Fake_parts_insert";
paulson@13926
   837
val Fake_analz_insert = thm "Fake_analz_insert";
paulson@13926
   838
val pushes = thms "pushes";
paulson@13926
   839
paulson@13926
   840
paulson@13926
   841
(*Prove base case (subgoal i) and simplify others.  A typical base case
paulson@13926
   842
  concerns  Crypt K X \<notin> Key`shrK`bad  and cannot be proved by rewriting
paulson@13926
   843
  alone.*)
paulson@13926
   844
fun prove_simple_subgoals_tac i = 
paulson@13926
   845
    force_tac (claset(), simpset() addsimps [image_eq_UN]) i THEN
paulson@13926
   846
    ALLGOALS Asm_simp_tac
paulson@13926
   847
paulson@13926
   848
(*Analysis of Fake cases.  Also works for messages that forward unknown parts,
paulson@13926
   849
  but this application is no longer necessary if analz_insert_eq is used.
paulson@13926
   850
  Abstraction over i is ESSENTIAL: it delays the dereferencing of claset
paulson@13926
   851
  DEPENDS UPON "X" REFERRING TO THE FRADULENT MESSAGE *)
paulson@13926
   852
paulson@13926
   853
(*Apply rules to break down assumptions of the form
paulson@13926
   854
  Y \<in> parts(insert X H)  and  Y \<in> analz(insert X H)
paulson@13926
   855
*)
paulson@13926
   856
val Fake_insert_tac = 
paulson@13926
   857
    dresolve_tac [impOfSubs Fake_analz_insert,
paulson@13926
   858
                  impOfSubs Fake_parts_insert] THEN'
paulson@13926
   859
    eresolve_tac [asm_rl, thm"synth.Inj"];
paulson@13926
   860
paulson@13926
   861
fun Fake_insert_simp_tac ss i = 
paulson@13926
   862
    REPEAT (Fake_insert_tac i) THEN asm_full_simp_tac ss i;
paulson@13926
   863
paulson@13926
   864
fun atomic_spy_analz_tac (cs,ss) = SELECT_GOAL
paulson@13926
   865
    (Fake_insert_simp_tac ss 1
paulson@13926
   866
     THEN
paulson@13926
   867
     IF_UNSOLVED (Blast.depth_tac
paulson@13926
   868
		  (cs addIs [analz_insertI,
paulson@13926
   869
				   impOfSubs analz_subset_parts]) 4 1))
paulson@13926
   870
paulson@13926
   871
(*The explicit claset and simpset arguments help it work with Isar*)
paulson@13926
   872
fun gen_spy_analz_tac (cs,ss) i =
paulson@13926
   873
  DETERM
paulson@13926
   874
   (SELECT_GOAL
paulson@13926
   875
     (EVERY 
paulson@13926
   876
      [  (*push in occurrences of X...*)
paulson@13926
   877
       (REPEAT o CHANGED)
paulson@13926
   878
           (res_inst_tac [("x1","X")] (insert_commute RS ssubst) 1),
paulson@13926
   879
       (*...allowing further simplifications*)
paulson@13926
   880
       simp_tac ss 1,
paulson@13926
   881
       REPEAT (FIRSTGOAL (resolve_tac [allI,impI,notI,conjI,iffI])),
paulson@13926
   882
       DEPTH_SOLVE (atomic_spy_analz_tac (cs,ss) 1)]) i)
paulson@13926
   883
paulson@13926
   884
fun spy_analz_tac i = gen_spy_analz_tac (claset(), simpset()) i
paulson@13926
   885
*}
paulson@13926
   886
paulson@13926
   887
(*By default only o_apply is built-in.  But in the presence of eta-expansion
paulson@13926
   888
  this means that some terms displayed as (f o g) will be rewritten, and others
paulson@13926
   889
  will not!*)
paulson@13926
   890
declare o_def [simp]
paulson@13926
   891
paulson@11189
   892
paulson@13922
   893
lemma Crypt_notin_image_Key [simp]: "Crypt K X \<notin> Key ` A"
paulson@13922
   894
by auto
paulson@13922
   895
paulson@13922
   896
lemma Hash_notin_image_Key [simp] :"Hash X \<notin> Key ` A"
paulson@13922
   897
by auto
paulson@13922
   898
paulson@14200
   899
lemma synth_analz_mono: "G\<subseteq>H ==> synth (analz(G)) \<subseteq> synth (analz(H))"
paulson@13922
   900
by (simp add: synth_mono analz_mono) 
paulson@13922
   901
paulson@13922
   902
lemma Fake_analz_eq [simp]:
paulson@13922
   903
     "X \<in> synth(analz H) ==> synth (analz (insert X H)) = synth (analz H)"
paulson@13922
   904
apply (drule Fake_analz_insert[of _ _ "H"])
paulson@13922
   905
apply (simp add: synth_increasing[THEN Un_absorb2])
paulson@13922
   906
apply (drule synth_mono)
paulson@13922
   907
apply (simp add: synth_idem)
paulson@13922
   908
apply (blast intro: synth_analz_mono [THEN [2] rev_subsetD]) 
paulson@13922
   909
done
paulson@13922
   910
paulson@13922
   911
text{*Two generalizations of @{text analz_insert_eq}*}
paulson@13922
   912
lemma gen_analz_insert_eq [rule_format]:
paulson@13922
   913
     "X \<in> analz H ==> ALL G. H \<subseteq> G --> analz (insert X G) = analz G";
paulson@13922
   914
by (blast intro: analz_cut analz_insertI analz_mono [THEN [2] rev_subsetD])
paulson@13922
   915
paulson@13922
   916
lemma synth_analz_insert_eq [rule_format]:
paulson@13922
   917
     "X \<in> synth (analz H) 
paulson@13922
   918
      ==> ALL G. H \<subseteq> G --> (Key K \<in> analz (insert X G)) = (Key K \<in> analz G)";
paulson@13922
   919
apply (erule synth.induct) 
paulson@13922
   920
apply (simp_all add: gen_analz_insert_eq subset_trans [OF _ subset_insertI]) 
paulson@13922
   921
done
paulson@13922
   922
paulson@13922
   923
lemma Fake_parts_sing:
paulson@13926
   924
     "X \<in> synth (analz H) ==> parts{X} \<subseteq> synth (analz H) \<union> parts H";
paulson@13922
   925
apply (rule subset_trans) 
paulson@13922
   926
 apply (erule_tac [2] Fake_parts_insert) 
paulson@13922
   927
apply (simp add: parts_mono) 
paulson@13922
   928
done
paulson@13922
   929
paulson@14145
   930
lemmas Fake_parts_sing_imp_Un = Fake_parts_sing [THEN [2] rev_subsetD]
paulson@14145
   931
paulson@11189
   932
method_setup spy_analz = {*
paulson@11270
   933
    Method.ctxt_args (fn ctxt =>
paulson@11270
   934
        Method.METHOD (fn facts => 
wenzelm@15032
   935
            gen_spy_analz_tac (local_clasimpset_of ctxt) 1)) *}
paulson@11189
   936
    "for proving the Fake case when analz is involved"
paulson@1839
   937
paulson@11264
   938
method_setup atomic_spy_analz = {*
paulson@11270
   939
    Method.ctxt_args (fn ctxt =>
paulson@11270
   940
        Method.METHOD (fn facts => 
wenzelm@15032
   941
            atomic_spy_analz_tac (local_clasimpset_of ctxt) 1)) *}
paulson@11264
   942
    "for debugging spy_analz"
paulson@11264
   943
paulson@11264
   944
method_setup Fake_insert_simp = {*
paulson@11270
   945
    Method.ctxt_args (fn ctxt =>
paulson@11270
   946
        Method.METHOD (fn facts =>
wenzelm@15032
   947
            Fake_insert_simp_tac (local_simpset_of ctxt) 1)) *}
paulson@11264
   948
    "for debugging spy_analz"
paulson@11264
   949
paulson@13926
   950
paulson@1839
   951
end