src/HOL/Auth/Message.thy
author paulson
Fri Sep 30 09:52:46 2005 +0200 (2005-09-30)
changeset 17729 d74d0b5052a0
parent 17689 a04b5b43625e
child 18492 b0fe60800623
permissions -rw-r--r--
theorems need names
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
haftmann@16417
    12
theory Message imports Main begin
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@16818
    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@16818
    51
text{*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@11189
    64
  HPair :: "[msg,msg] => msg"                       ("(4Hash[_] /_)" [0, 1000])
paulson@16818
    65
    --{*Message Y paired with a MAC computed with the help of X*}
paulson@2516
    66
    "Hash[X] Y == {| Hash{|X,Y|}, Y|}"
paulson@2484
    67
paulson@11189
    68
  keysFor :: "msg set => key set"
paulson@16818
    69
    --{*Keys useful to decrypt elements of a message set*}
paulson@11192
    70
  "keysFor H == invKey ` {K. \<exists>X. Crypt K X \<in> H}"
paulson@1839
    71
paulson@16818
    72
paulson@16818
    73
subsubsection{*Inductive Definition of All Parts" of a Message*}
paulson@1839
    74
paulson@11189
    75
consts  parts   :: "msg set => msg set"
paulson@1839
    76
inductive "parts H"
paulson@11189
    77
  intros 
paulson@11192
    78
    Inj [intro]:               "X \<in> H ==> X \<in> parts H"
paulson@11192
    79
    Fst:         "{|X,Y|}   \<in> parts H ==> X \<in> parts H"
paulson@11192
    80
    Snd:         "{|X,Y|}   \<in> parts H ==> Y \<in> parts H"
paulson@11192
    81
    Body:        "Crypt K X \<in> parts H ==> X \<in> parts H"
paulson@11189
    82
paulson@11189
    83
paulson@16818
    84
text{*Monotonicity*}
paulson@16818
    85
lemma parts_mono: "G \<subseteq> H ==> parts(G) \<subseteq> parts(H)"
paulson@11189
    86
apply auto
paulson@11189
    87
apply (erule parts.induct) 
paulson@16818
    88
apply (blast dest: parts.Fst parts.Snd parts.Body)+
paulson@11189
    89
done
paulson@1839
    90
paulson@1839
    91
paulson@16818
    92
text{*Equations hold because constructors are injective.*}
paulson@13926
    93
lemma Friend_image_eq [simp]: "(Friend x \<in> Friend`A) = (x:A)"
paulson@13926
    94
by auto
paulson@13926
    95
paulson@13926
    96
lemma Key_image_eq [simp]: "(Key x \<in> Key`A) = (x\<in>A)"
paulson@13926
    97
by auto
paulson@13926
    98
paulson@13926
    99
lemma Nonce_Key_image_eq [simp]: "(Nonce x \<notin> Key`A)"
paulson@13926
   100
by auto
paulson@13926
   101
paulson@13926
   102
paulson@14200
   103
subsubsection{*Inverse of keys *}
paulson@13926
   104
paulson@13926
   105
lemma invKey_eq [simp]: "(invKey K = invKey K') = (K=K')"
paulson@13926
   106
apply safe
paulson@13926
   107
apply (drule_tac f = invKey in arg_cong, simp)
paulson@13926
   108
done
paulson@13926
   109
paulson@13926
   110
paulson@13926
   111
subsection{*keysFor operator*}
paulson@13926
   112
paulson@13926
   113
lemma keysFor_empty [simp]: "keysFor {} = {}"
paulson@13926
   114
by (unfold keysFor_def, blast)
paulson@13926
   115
paulson@13926
   116
lemma keysFor_Un [simp]: "keysFor (H \<union> H') = keysFor H \<union> keysFor H'"
paulson@13926
   117
by (unfold keysFor_def, blast)
paulson@13926
   118
paulson@13926
   119
lemma keysFor_UN [simp]: "keysFor (\<Union>i\<in>A. H i) = (\<Union>i\<in>A. keysFor (H i))"
paulson@13926
   120
by (unfold keysFor_def, blast)
paulson@13926
   121
paulson@16818
   122
text{*Monotonicity*}
paulson@16818
   123
lemma keysFor_mono: "G \<subseteq> H ==> keysFor(G) \<subseteq> keysFor(H)"
paulson@13926
   124
by (unfold keysFor_def, blast)
paulson@13926
   125
paulson@13926
   126
lemma keysFor_insert_Agent [simp]: "keysFor (insert (Agent A) H) = keysFor H"
paulson@13926
   127
by (unfold keysFor_def, auto)
paulson@13926
   128
paulson@13926
   129
lemma keysFor_insert_Nonce [simp]: "keysFor (insert (Nonce N) H) = keysFor H"
paulson@13926
   130
by (unfold keysFor_def, auto)
paulson@13926
   131
paulson@13926
   132
lemma keysFor_insert_Number [simp]: "keysFor (insert (Number N) H) = keysFor H"
paulson@13926
   133
by (unfold keysFor_def, auto)
paulson@13926
   134
paulson@13926
   135
lemma keysFor_insert_Key [simp]: "keysFor (insert (Key K) H) = keysFor H"
paulson@13926
   136
by (unfold keysFor_def, auto)
paulson@13926
   137
paulson@13926
   138
lemma keysFor_insert_Hash [simp]: "keysFor (insert (Hash X) H) = keysFor H"
paulson@13926
   139
by (unfold keysFor_def, auto)
paulson@13926
   140
paulson@13926
   141
lemma keysFor_insert_MPair [simp]: "keysFor (insert {|X,Y|} H) = keysFor H"
paulson@13926
   142
by (unfold keysFor_def, auto)
paulson@13926
   143
paulson@13926
   144
lemma keysFor_insert_Crypt [simp]: 
paulson@13926
   145
    "keysFor (insert (Crypt K X) H) = insert (invKey K) (keysFor H)"
paulson@14200
   146
by (unfold keysFor_def, auto)
paulson@13926
   147
paulson@13926
   148
lemma keysFor_image_Key [simp]: "keysFor (Key`E) = {}"
paulson@13926
   149
by (unfold keysFor_def, auto)
paulson@13926
   150
paulson@13926
   151
lemma Crypt_imp_invKey_keysFor: "Crypt K X \<in> H ==> invKey K \<in> keysFor H"
paulson@13926
   152
by (unfold keysFor_def, blast)
paulson@13926
   153
paulson@13926
   154
paulson@13926
   155
subsection{*Inductive relation "parts"*}
paulson@13926
   156
paulson@13926
   157
lemma MPair_parts:
paulson@13926
   158
     "[| {|X,Y|} \<in> parts H;        
paulson@13926
   159
         [| X \<in> parts H; Y \<in> parts H |] ==> P |] ==> P"
paulson@13926
   160
by (blast dest: parts.Fst parts.Snd) 
paulson@13926
   161
paulson@13926
   162
declare MPair_parts [elim!]  parts.Body [dest!]
paulson@13926
   163
text{*NB These two rules are UNSAFE in the formal sense, as they discard the
paulson@13926
   164
     compound message.  They work well on THIS FILE.  
paulson@13926
   165
  @{text MPair_parts} is left as SAFE because it speeds up proofs.
paulson@13926
   166
  The Crypt rule is normally kept UNSAFE to avoid breaking up certificates.*}
paulson@13926
   167
paulson@13926
   168
lemma parts_increasing: "H \<subseteq> parts(H)"
paulson@13926
   169
by blast
paulson@13926
   170
paulson@13926
   171
lemmas parts_insertI = subset_insertI [THEN parts_mono, THEN subsetD, standard]
paulson@13926
   172
paulson@13926
   173
lemma parts_empty [simp]: "parts{} = {}"
paulson@13926
   174
apply safe
paulson@13926
   175
apply (erule parts.induct, blast+)
paulson@13926
   176
done
paulson@13926
   177
paulson@13926
   178
lemma parts_emptyE [elim!]: "X\<in> parts{} ==> P"
paulson@13926
   179
by simp
paulson@13926
   180
paulson@16818
   181
text{*WARNING: loops if H = {Y}, therefore must not be repeated!*}
paulson@13926
   182
lemma parts_singleton: "X\<in> parts H ==> \<exists>Y\<in>H. X\<in> parts {Y}"
paulson@13926
   183
by (erule parts.induct, blast+)
paulson@13926
   184
paulson@13926
   185
paulson@14200
   186
subsubsection{*Unions *}
paulson@13926
   187
paulson@13926
   188
lemma parts_Un_subset1: "parts(G) \<union> parts(H) \<subseteq> parts(G \<union> H)"
paulson@13926
   189
by (intro Un_least parts_mono Un_upper1 Un_upper2)
paulson@13926
   190
paulson@13926
   191
lemma parts_Un_subset2: "parts(G \<union> H) \<subseteq> parts(G) \<union> parts(H)"
paulson@13926
   192
apply (rule subsetI)
paulson@13926
   193
apply (erule parts.induct, blast+)
paulson@13926
   194
done
paulson@13926
   195
paulson@13926
   196
lemma parts_Un [simp]: "parts(G \<union> H) = parts(G) \<union> parts(H)"
paulson@13926
   197
by (intro equalityI parts_Un_subset1 parts_Un_subset2)
paulson@13926
   198
paulson@13926
   199
lemma parts_insert: "parts (insert X H) = parts {X} \<union> parts H"
paulson@13926
   200
apply (subst insert_is_Un [of _ H])
paulson@13926
   201
apply (simp only: parts_Un)
paulson@13926
   202
done
paulson@13926
   203
paulson@16818
   204
text{*TWO inserts to avoid looping.  This rewrite is better than nothing.
paulson@16818
   205
  Not suitable for Addsimps: its behaviour can be strange.*}
paulson@14200
   206
lemma parts_insert2:
paulson@14200
   207
     "parts (insert X (insert Y H)) = parts {X} \<union> parts {Y} \<union> parts H"
paulson@13926
   208
apply (simp add: Un_assoc)
paulson@13926
   209
apply (simp add: parts_insert [symmetric])
paulson@13926
   210
done
paulson@13926
   211
paulson@13926
   212
lemma parts_UN_subset1: "(\<Union>x\<in>A. parts(H x)) \<subseteq> parts(\<Union>x\<in>A. H x)"
paulson@13926
   213
by (intro UN_least parts_mono UN_upper)
paulson@13926
   214
paulson@13926
   215
lemma parts_UN_subset2: "parts(\<Union>x\<in>A. H x) \<subseteq> (\<Union>x\<in>A. parts(H x))"
paulson@13926
   216
apply (rule subsetI)
paulson@13926
   217
apply (erule parts.induct, blast+)
paulson@13926
   218
done
paulson@13926
   219
paulson@13926
   220
lemma parts_UN [simp]: "parts(\<Union>x\<in>A. H x) = (\<Union>x\<in>A. parts(H x))"
paulson@13926
   221
by (intro equalityI parts_UN_subset1 parts_UN_subset2)
paulson@13926
   222
paulson@16818
   223
text{*Added to simplify arguments to parts, analz and synth.
paulson@16818
   224
  NOTE: the UN versions are no longer used!*}
paulson@13926
   225
paulson@13926
   226
paulson@13926
   227
text{*This allows @{text blast} to simplify occurrences of 
paulson@13926
   228
  @{term "parts(G\<union>H)"} in the assumption.*}
paulson@17729
   229
lemmas in_parts_UnE = parts_Un [THEN equalityD1, THEN subsetD, THEN UnE] 
paulson@17729
   230
declare in_parts_UnE [elim!]
paulson@13926
   231
paulson@13926
   232
paulson@13926
   233
lemma parts_insert_subset: "insert X (parts H) \<subseteq> parts(insert X H)"
paulson@13926
   234
by (blast intro: parts_mono [THEN [2] rev_subsetD])
paulson@13926
   235
paulson@14200
   236
subsubsection{*Idempotence and transitivity *}
paulson@13926
   237
paulson@13926
   238
lemma parts_partsD [dest!]: "X\<in> parts (parts H) ==> X\<in> parts H"
paulson@13926
   239
by (erule parts.induct, blast+)
paulson@13926
   240
paulson@13926
   241
lemma parts_idem [simp]: "parts (parts H) = parts H"
paulson@13926
   242
by blast
paulson@13926
   243
paulson@17689
   244
lemma parts_subset_iff [simp]: "(parts G \<subseteq> parts H) = (G \<subseteq> parts H)"
paulson@17689
   245
apply (rule iffI)
paulson@17689
   246
apply (iprover intro: subset_trans parts_increasing)  
paulson@17689
   247
apply (frule parts_mono, simp) 
paulson@17689
   248
done
paulson@17689
   249
paulson@13926
   250
lemma parts_trans: "[| X\<in> parts G;  G \<subseteq> parts H |] ==> X\<in> parts H"
paulson@13926
   251
by (drule parts_mono, blast)
paulson@13926
   252
paulson@16818
   253
text{*Cut*}
paulson@14200
   254
lemma parts_cut:
paulson@14200
   255
     "[| Y\<in> parts (insert X G);  X\<in> parts H |] ==> Y\<in> parts (G \<union> H)"
paulson@14200
   256
by (erule parts_trans, auto)
paulson@13926
   257
paulson@13926
   258
lemma parts_cut_eq [simp]: "X\<in> parts H ==> parts (insert X H) = parts H"
paulson@13926
   259
by (force dest!: parts_cut intro: parts_insertI)
paulson@13926
   260
paulson@13926
   261
paulson@14200
   262
subsubsection{*Rewrite rules for pulling out atomic messages *}
paulson@13926
   263
paulson@13926
   264
lemmas parts_insert_eq_I = equalityI [OF subsetI parts_insert_subset]
paulson@13926
   265
paulson@13926
   266
paulson@14200
   267
lemma parts_insert_Agent [simp]:
paulson@14200
   268
     "parts (insert (Agent agt) H) = insert (Agent agt) (parts H)"
paulson@13926
   269
apply (rule parts_insert_eq_I) 
paulson@13926
   270
apply (erule parts.induct, auto) 
paulson@13926
   271
done
paulson@13926
   272
paulson@14200
   273
lemma parts_insert_Nonce [simp]:
paulson@14200
   274
     "parts (insert (Nonce N) H) = insert (Nonce N) (parts H)"
paulson@13926
   275
apply (rule parts_insert_eq_I) 
paulson@13926
   276
apply (erule parts.induct, auto) 
paulson@13926
   277
done
paulson@13926
   278
paulson@14200
   279
lemma parts_insert_Number [simp]:
paulson@14200
   280
     "parts (insert (Number N) H) = insert (Number N) (parts H)"
paulson@13926
   281
apply (rule parts_insert_eq_I) 
paulson@13926
   282
apply (erule parts.induct, auto) 
paulson@13926
   283
done
paulson@13926
   284
paulson@14200
   285
lemma parts_insert_Key [simp]:
paulson@14200
   286
     "parts (insert (Key K) H) = insert (Key K) (parts H)"
paulson@13926
   287
apply (rule parts_insert_eq_I) 
paulson@13926
   288
apply (erule parts.induct, auto) 
paulson@13926
   289
done
paulson@13926
   290
paulson@14200
   291
lemma parts_insert_Hash [simp]:
paulson@14200
   292
     "parts (insert (Hash X) H) = insert (Hash X) (parts H)"
paulson@13926
   293
apply (rule parts_insert_eq_I) 
paulson@13926
   294
apply (erule parts.induct, auto) 
paulson@13926
   295
done
paulson@13926
   296
paulson@14200
   297
lemma parts_insert_Crypt [simp]:
paulson@17689
   298
     "parts (insert (Crypt K X) H) = insert (Crypt K X) (parts (insert X H))"
paulson@13926
   299
apply (rule equalityI)
paulson@13926
   300
apply (rule subsetI)
paulson@13926
   301
apply (erule parts.induct, auto)
paulson@17689
   302
apply (blast intro: parts.Body)
paulson@13926
   303
done
paulson@13926
   304
paulson@14200
   305
lemma parts_insert_MPair [simp]:
paulson@14200
   306
     "parts (insert {|X,Y|} H) =  
paulson@13926
   307
          insert {|X,Y|} (parts (insert X (insert Y H)))"
paulson@13926
   308
apply (rule equalityI)
paulson@13926
   309
apply (rule subsetI)
paulson@13926
   310
apply (erule parts.induct, auto)
paulson@13926
   311
apply (blast intro: parts.Fst parts.Snd)+
paulson@13926
   312
done
paulson@13926
   313
paulson@13926
   314
lemma parts_image_Key [simp]: "parts (Key`N) = Key`N"
paulson@13926
   315
apply auto
paulson@13926
   316
apply (erule parts.induct, auto)
paulson@13926
   317
done
paulson@13926
   318
paulson@13926
   319
paulson@16818
   320
text{*In any message, there is an upper bound N on its greatest nonce.*}
paulson@13926
   321
lemma msg_Nonce_supply: "\<exists>N. \<forall>n. N\<le>n --> Nonce n \<notin> parts {msg}"
paulson@13926
   322
apply (induct_tac "msg")
paulson@13926
   323
apply (simp_all (no_asm_simp) add: exI parts_insert2)
paulson@16818
   324
 txt{*MPair case: blast works out the necessary sum itself!*}
paulson@16818
   325
 prefer 2 apply (blast elim!: add_leE)
paulson@16818
   326
txt{*Nonce case*}
paulson@16818
   327
apply (rule_tac x = "N + Suc nat" in exI, auto) 
paulson@13926
   328
done
paulson@13926
   329
paulson@13926
   330
paulson@13926
   331
subsection{*Inductive relation "analz"*}
paulson@13926
   332
paulson@14200
   333
text{*Inductive definition of "analz" -- what can be broken down from a set of
paulson@1839
   334
    messages, including keys.  A form of downward closure.  Pairs can
paulson@14200
   335
    be taken apart; messages decrypted with known keys.  *}
paulson@1839
   336
paulson@11189
   337
consts  analz   :: "msg set => msg set"
paulson@1913
   338
inductive "analz H"
paulson@11189
   339
  intros 
paulson@11192
   340
    Inj [intro,simp] :    "X \<in> H ==> X \<in> analz H"
paulson@11192
   341
    Fst:     "{|X,Y|} \<in> analz H ==> X \<in> analz H"
paulson@11192
   342
    Snd:     "{|X,Y|} \<in> analz H ==> Y \<in> analz H"
paulson@11189
   343
    Decrypt [dest]: 
paulson@11192
   344
             "[|Crypt K X \<in> analz H; Key(invKey K): analz H|] ==> X \<in> analz H"
paulson@1839
   345
paulson@1839
   346
paulson@16818
   347
text{*Monotonicity; Lemma 1 of Lowe's paper*}
paulson@14200
   348
lemma analz_mono: "G\<subseteq>H ==> analz(G) \<subseteq> analz(H)"
paulson@11189
   349
apply auto
paulson@11189
   350
apply (erule analz.induct) 
paulson@16818
   351
apply (auto dest: analz.Fst analz.Snd) 
paulson@11189
   352
done
paulson@11189
   353
paulson@13926
   354
text{*Making it safe speeds up proofs*}
paulson@13926
   355
lemma MPair_analz [elim!]:
paulson@13926
   356
     "[| {|X,Y|} \<in> analz H;        
paulson@13926
   357
             [| X \<in> analz H; Y \<in> analz H |] ==> P   
paulson@13926
   358
          |] ==> P"
paulson@13926
   359
by (blast dest: analz.Fst analz.Snd)
paulson@13926
   360
paulson@13926
   361
lemma analz_increasing: "H \<subseteq> analz(H)"
paulson@13926
   362
by blast
paulson@13926
   363
paulson@13926
   364
lemma analz_subset_parts: "analz H \<subseteq> parts H"
paulson@13926
   365
apply (rule subsetI)
paulson@13926
   366
apply (erule analz.induct, blast+)
paulson@13926
   367
done
paulson@13926
   368
paulson@14200
   369
lemmas analz_into_parts = analz_subset_parts [THEN subsetD, standard]
paulson@14200
   370
paulson@13926
   371
lemmas not_parts_not_analz = analz_subset_parts [THEN contra_subsetD, standard]
paulson@13926
   372
paulson@13926
   373
paulson@13926
   374
lemma parts_analz [simp]: "parts (analz H) = parts H"
paulson@13926
   375
apply (rule equalityI)
paulson@13926
   376
apply (rule analz_subset_parts [THEN parts_mono, THEN subset_trans], simp)
paulson@13926
   377
apply (blast intro: analz_increasing [THEN parts_mono, THEN subsetD])
paulson@13926
   378
done
paulson@13926
   379
paulson@13926
   380
lemma analz_parts [simp]: "analz (parts H) = parts H"
paulson@13926
   381
apply auto
paulson@13926
   382
apply (erule analz.induct, auto)
paulson@13926
   383
done
paulson@13926
   384
paulson@13926
   385
lemmas analz_insertI = subset_insertI [THEN analz_mono, THEN [2] rev_subsetD, standard]
paulson@13926
   386
paulson@14200
   387
subsubsection{*General equational properties *}
paulson@13926
   388
paulson@13926
   389
lemma analz_empty [simp]: "analz{} = {}"
paulson@13926
   390
apply safe
paulson@13926
   391
apply (erule analz.induct, blast+)
paulson@13926
   392
done
paulson@13926
   393
paulson@16818
   394
text{*Converse fails: we can analz more from the union than from the 
paulson@16818
   395
  separate parts, as a key in one might decrypt a message in the other*}
paulson@13926
   396
lemma analz_Un: "analz(G) \<union> analz(H) \<subseteq> analz(G \<union> H)"
paulson@13926
   397
by (intro Un_least analz_mono Un_upper1 Un_upper2)
paulson@13926
   398
paulson@13926
   399
lemma analz_insert: "insert X (analz H) \<subseteq> analz(insert X H)"
paulson@13926
   400
by (blast intro: analz_mono [THEN [2] rev_subsetD])
paulson@13926
   401
paulson@14200
   402
subsubsection{*Rewrite rules for pulling out atomic messages *}
paulson@13926
   403
paulson@13926
   404
lemmas analz_insert_eq_I = equalityI [OF subsetI analz_insert]
paulson@13926
   405
paulson@14200
   406
lemma analz_insert_Agent [simp]:
paulson@14200
   407
     "analz (insert (Agent agt) H) = insert (Agent agt) (analz H)"
paulson@13926
   408
apply (rule analz_insert_eq_I) 
paulson@13926
   409
apply (erule analz.induct, auto) 
paulson@13926
   410
done
paulson@13926
   411
paulson@14200
   412
lemma analz_insert_Nonce [simp]:
paulson@14200
   413
     "analz (insert (Nonce N) H) = insert (Nonce N) (analz H)"
paulson@13926
   414
apply (rule analz_insert_eq_I) 
paulson@13926
   415
apply (erule analz.induct, auto) 
paulson@13926
   416
done
paulson@13926
   417
paulson@14200
   418
lemma analz_insert_Number [simp]:
paulson@14200
   419
     "analz (insert (Number N) H) = insert (Number N) (analz H)"
paulson@13926
   420
apply (rule analz_insert_eq_I) 
paulson@13926
   421
apply (erule analz.induct, auto) 
paulson@13926
   422
done
paulson@13926
   423
paulson@14200
   424
lemma analz_insert_Hash [simp]:
paulson@14200
   425
     "analz (insert (Hash X) H) = insert (Hash X) (analz H)"
paulson@13926
   426
apply (rule analz_insert_eq_I) 
paulson@13926
   427
apply (erule analz.induct, auto) 
paulson@13926
   428
done
paulson@13926
   429
paulson@16818
   430
text{*Can only pull out Keys if they are not needed to decrypt the rest*}
paulson@13926
   431
lemma analz_insert_Key [simp]: 
paulson@13926
   432
    "K \<notin> keysFor (analz H) ==>   
paulson@13926
   433
          analz (insert (Key K) H) = insert (Key K) (analz H)"
paulson@13926
   434
apply (unfold keysFor_def)
paulson@13926
   435
apply (rule analz_insert_eq_I) 
paulson@13926
   436
apply (erule analz.induct, auto) 
paulson@13926
   437
done
paulson@13926
   438
paulson@14200
   439
lemma analz_insert_MPair [simp]:
paulson@14200
   440
     "analz (insert {|X,Y|} H) =  
paulson@13926
   441
          insert {|X,Y|} (analz (insert X (insert Y H)))"
paulson@13926
   442
apply (rule equalityI)
paulson@13926
   443
apply (rule subsetI)
paulson@13926
   444
apply (erule analz.induct, auto)
paulson@13926
   445
apply (erule analz.induct)
paulson@13926
   446
apply (blast intro: analz.Fst analz.Snd)+
paulson@13926
   447
done
paulson@13926
   448
paulson@16818
   449
text{*Can pull out enCrypted message if the Key is not known*}
paulson@13926
   450
lemma analz_insert_Crypt:
paulson@13926
   451
     "Key (invKey K) \<notin> analz H 
paulson@13926
   452
      ==> analz (insert (Crypt K X) H) = insert (Crypt K X) (analz H)"
paulson@13926
   453
apply (rule analz_insert_eq_I) 
paulson@13926
   454
apply (erule analz.induct, auto) 
paulson@13926
   455
paulson@13926
   456
done
paulson@13926
   457
paulson@13926
   458
lemma lemma1: "Key (invKey K) \<in> analz H ==>   
paulson@13926
   459
               analz (insert (Crypt K X) H) \<subseteq>  
paulson@13926
   460
               insert (Crypt K X) (analz (insert X H))"
paulson@13926
   461
apply (rule subsetI)
paulson@13926
   462
apply (erule_tac xa = x in analz.induct, auto)
paulson@13926
   463
done
paulson@13926
   464
paulson@13926
   465
lemma lemma2: "Key (invKey K) \<in> analz H ==>   
paulson@13926
   466
               insert (Crypt K X) (analz (insert X H)) \<subseteq>  
paulson@13926
   467
               analz (insert (Crypt K X) H)"
paulson@13926
   468
apply auto
paulson@13926
   469
apply (erule_tac xa = x in analz.induct, auto)
paulson@13926
   470
apply (blast intro: analz_insertI analz.Decrypt)
paulson@13926
   471
done
paulson@13926
   472
paulson@14200
   473
lemma analz_insert_Decrypt:
paulson@14200
   474
     "Key (invKey K) \<in> analz H ==>   
paulson@13926
   475
               analz (insert (Crypt K X) H) =  
paulson@13926
   476
               insert (Crypt K X) (analz (insert X H))"
paulson@13926
   477
by (intro equalityI lemma1 lemma2)
paulson@13926
   478
paulson@16818
   479
text{*Case analysis: either the message is secure, or it is not! Effective,
paulson@16818
   480
but can cause subgoals to blow up! Use with @{text "split_if"}; apparently
paulson@16818
   481
@{text "split_tac"} does not cope with patterns such as @{term"analz (insert
paulson@16818
   482
(Crypt K X) H)"} *} 
paulson@13926
   483
lemma analz_Crypt_if [simp]:
paulson@13926
   484
     "analz (insert (Crypt K X) H) =                 
paulson@13926
   485
          (if (Key (invKey K) \<in> analz H)                 
paulson@13926
   486
           then insert (Crypt K X) (analz (insert X H))  
paulson@13926
   487
           else insert (Crypt K X) (analz H))"
paulson@13926
   488
by (simp add: analz_insert_Crypt analz_insert_Decrypt)
paulson@13926
   489
paulson@13926
   490
paulson@16818
   491
text{*This rule supposes "for the sake of argument" that we have the key.*}
paulson@14200
   492
lemma analz_insert_Crypt_subset:
paulson@14200
   493
     "analz (insert (Crypt K X) H) \<subseteq>   
paulson@13926
   494
           insert (Crypt K X) (analz (insert X H))"
paulson@13926
   495
apply (rule subsetI)
paulson@13926
   496
apply (erule analz.induct, auto)
paulson@13926
   497
done
paulson@13926
   498
paulson@13926
   499
paulson@13926
   500
lemma analz_image_Key [simp]: "analz (Key`N) = Key`N"
paulson@13926
   501
apply auto
paulson@13926
   502
apply (erule analz.induct, auto)
paulson@13926
   503
done
paulson@13926
   504
paulson@13926
   505
paulson@14200
   506
subsubsection{*Idempotence and transitivity *}
paulson@13926
   507
paulson@13926
   508
lemma analz_analzD [dest!]: "X\<in> analz (analz H) ==> X\<in> analz H"
paulson@13926
   509
by (erule analz.induct, blast+)
paulson@13926
   510
paulson@13926
   511
lemma analz_idem [simp]: "analz (analz H) = analz H"
paulson@13926
   512
by blast
paulson@13926
   513
paulson@17689
   514
lemma analz_subset_iff [simp]: "(analz G \<subseteq> analz H) = (G \<subseteq> analz H)"
paulson@17689
   515
apply (rule iffI)
paulson@17689
   516
apply (iprover intro: subset_trans analz_increasing)  
paulson@17689
   517
apply (frule analz_mono, simp) 
paulson@17689
   518
done
paulson@17689
   519
paulson@13926
   520
lemma analz_trans: "[| X\<in> analz G;  G \<subseteq> analz H |] ==> X\<in> analz H"
paulson@13926
   521
by (drule analz_mono, blast)
paulson@13926
   522
paulson@16818
   523
text{*Cut; Lemma 2 of Lowe*}
paulson@13926
   524
lemma analz_cut: "[| Y\<in> analz (insert X H);  X\<in> analz H |] ==> Y\<in> analz H"
paulson@13926
   525
by (erule analz_trans, blast)
paulson@13926
   526
paulson@13926
   527
(*Cut can be proved easily by induction on
paulson@13926
   528
   "Y: analz (insert X H) ==> X: analz H --> Y: analz H"
paulson@13926
   529
*)
paulson@13926
   530
paulson@16818
   531
text{*This rewrite rule helps in the simplification of messages that involve
paulson@13926
   532
  the forwarding of unknown components (X).  Without it, removing occurrences
paulson@16818
   533
  of X can be very complicated. *}
paulson@13926
   534
lemma analz_insert_eq: "X\<in> analz H ==> analz (insert X H) = analz H"
paulson@13926
   535
by (blast intro: analz_cut analz_insertI)
paulson@13926
   536
paulson@13926
   537
paulson@14200
   538
text{*A congruence rule for "analz" *}
paulson@13926
   539
paulson@14200
   540
lemma analz_subset_cong:
paulson@17689
   541
     "[| analz G \<subseteq> analz G'; analz H \<subseteq> analz H' |] 
paulson@17689
   542
      ==> analz (G \<union> H) \<subseteq> analz (G' \<union> H')"
paulson@17689
   543
apply simp
paulson@17689
   544
apply (iprover intro: conjI subset_trans analz_mono Un_upper1 Un_upper2) 
paulson@13926
   545
done
paulson@13926
   546
paulson@14200
   547
lemma analz_cong:
paulson@17689
   548
     "[| analz G = analz G'; analz H = analz H' |] 
paulson@17689
   549
      ==> analz (G \<union> H) = analz (G' \<union> H')"
paulson@14200
   550
by (intro equalityI analz_subset_cong, simp_all) 
paulson@13926
   551
paulson@14200
   552
lemma analz_insert_cong:
paulson@14200
   553
     "analz H = analz H' ==> analz(insert X H) = analz(insert X H')"
paulson@13926
   554
by (force simp only: insert_def intro!: analz_cong)
paulson@13926
   555
paulson@16818
   556
text{*If there are no pairs or encryptions then analz does nothing*}
paulson@14200
   557
lemma analz_trivial:
paulson@14200
   558
     "[| \<forall>X Y. {|X,Y|} \<notin> H;  \<forall>X K. Crypt K X \<notin> H |] ==> analz H = H"
paulson@13926
   559
apply safe
paulson@13926
   560
apply (erule analz.induct, blast+)
paulson@13926
   561
done
paulson@13926
   562
paulson@16818
   563
text{*These two are obsolete (with a single Spy) but cost little to prove...*}
paulson@14200
   564
lemma analz_UN_analz_lemma:
paulson@14200
   565
     "X\<in> analz (\<Union>i\<in>A. analz (H i)) ==> X\<in> analz (\<Union>i\<in>A. H i)"
paulson@13926
   566
apply (erule analz.induct)
paulson@13926
   567
apply (blast intro: analz_mono [THEN [2] rev_subsetD])+
paulson@13926
   568
done
paulson@13926
   569
paulson@13926
   570
lemma analz_UN_analz [simp]: "analz (\<Union>i\<in>A. analz (H i)) = analz (\<Union>i\<in>A. H i)"
paulson@13926
   571
by (blast intro: analz_UN_analz_lemma analz_mono [THEN [2] rev_subsetD])
paulson@13926
   572
paulson@13926
   573
paulson@13926
   574
subsection{*Inductive relation "synth"*}
paulson@13926
   575
paulson@14200
   576
text{*Inductive definition of "synth" -- what can be built up from a set of
paulson@1839
   577
    messages.  A form of upward closure.  Pairs can be built, messages
paulson@3668
   578
    encrypted with known keys.  Agent names are public domain.
paulson@14200
   579
    Numbers can be guessed, but Nonces cannot be.  *}
paulson@1839
   580
paulson@11189
   581
consts  synth   :: "msg set => msg set"
paulson@1913
   582
inductive "synth H"
paulson@11189
   583
  intros 
paulson@11192
   584
    Inj    [intro]:   "X \<in> H ==> X \<in> synth H"
paulson@11192
   585
    Agent  [intro]:   "Agent agt \<in> synth H"
paulson@11192
   586
    Number [intro]:   "Number n  \<in> synth H"
paulson@11192
   587
    Hash   [intro]:   "X \<in> synth H ==> Hash X \<in> synth H"
paulson@11192
   588
    MPair  [intro]:   "[|X \<in> synth H;  Y \<in> synth H|] ==> {|X,Y|} \<in> synth H"
paulson@11192
   589
    Crypt  [intro]:   "[|X \<in> synth H;  Key(K) \<in> H|] ==> Crypt K X \<in> synth H"
paulson@11189
   590
paulson@16818
   591
text{*Monotonicity*}
paulson@14200
   592
lemma synth_mono: "G\<subseteq>H ==> synth(G) \<subseteq> synth(H)"
paulson@16818
   593
  by (auto, erule synth.induct, auto)  
paulson@11189
   594
paulson@16818
   595
text{*NO @{text Agent_synth}, as any Agent name can be synthesized.  
paulson@16818
   596
  The same holds for @{term Number}*}
paulson@11192
   597
inductive_cases Nonce_synth [elim!]: "Nonce n \<in> synth H"
paulson@11192
   598
inductive_cases Key_synth   [elim!]: "Key K \<in> synth H"
paulson@11192
   599
inductive_cases Hash_synth  [elim!]: "Hash X \<in> synth H"
paulson@11192
   600
inductive_cases MPair_synth [elim!]: "{|X,Y|} \<in> synth H"
paulson@11192
   601
inductive_cases Crypt_synth [elim!]: "Crypt K X \<in> synth H"
paulson@11189
   602
paulson@13926
   603
paulson@13926
   604
lemma synth_increasing: "H \<subseteq> synth(H)"
paulson@13926
   605
by blast
paulson@13926
   606
paulson@14200
   607
subsubsection{*Unions *}
paulson@13926
   608
paulson@16818
   609
text{*Converse fails: we can synth more from the union than from the 
paulson@16818
   610
  separate parts, building a compound message using elements of each.*}
paulson@13926
   611
lemma synth_Un: "synth(G) \<union> synth(H) \<subseteq> synth(G \<union> H)"
paulson@13926
   612
by (intro Un_least synth_mono Un_upper1 Un_upper2)
paulson@13926
   613
paulson@13926
   614
lemma synth_insert: "insert X (synth H) \<subseteq> synth(insert X H)"
paulson@13926
   615
by (blast intro: synth_mono [THEN [2] rev_subsetD])
paulson@13926
   616
paulson@14200
   617
subsubsection{*Idempotence and transitivity *}
paulson@13926
   618
paulson@13926
   619
lemma synth_synthD [dest!]: "X\<in> synth (synth H) ==> X\<in> synth H"
paulson@13926
   620
by (erule synth.induct, blast+)
paulson@13926
   621
paulson@13926
   622
lemma synth_idem: "synth (synth H) = synth H"
paulson@13926
   623
by blast
paulson@13926
   624
paulson@17689
   625
lemma synth_subset_iff [simp]: "(synth G \<subseteq> synth H) = (G \<subseteq> synth H)"
paulson@17689
   626
apply (rule iffI)
paulson@17689
   627
apply (iprover intro: subset_trans synth_increasing)  
paulson@17689
   628
apply (frule synth_mono, simp add: synth_idem) 
paulson@17689
   629
done
paulson@17689
   630
paulson@13926
   631
lemma synth_trans: "[| X\<in> synth G;  G \<subseteq> synth H |] ==> X\<in> synth H"
paulson@13926
   632
by (drule synth_mono, blast)
paulson@13926
   633
paulson@16818
   634
text{*Cut; Lemma 2 of Lowe*}
paulson@13926
   635
lemma synth_cut: "[| Y\<in> synth (insert X H);  X\<in> synth H |] ==> Y\<in> synth H"
paulson@13926
   636
by (erule synth_trans, blast)
paulson@13926
   637
paulson@13926
   638
lemma Agent_synth [simp]: "Agent A \<in> synth H"
paulson@13926
   639
by blast
paulson@13926
   640
paulson@13926
   641
lemma Number_synth [simp]: "Number n \<in> synth H"
paulson@13926
   642
by blast
paulson@13926
   643
paulson@13926
   644
lemma Nonce_synth_eq [simp]: "(Nonce N \<in> synth H) = (Nonce N \<in> H)"
paulson@13926
   645
by blast
paulson@13926
   646
paulson@13926
   647
lemma Key_synth_eq [simp]: "(Key K \<in> synth H) = (Key K \<in> H)"
paulson@13926
   648
by blast
paulson@13926
   649
paulson@14200
   650
lemma Crypt_synth_eq [simp]:
paulson@14200
   651
     "Key K \<notin> H ==> (Crypt K X \<in> synth H) = (Crypt K X \<in> H)"
paulson@13926
   652
by blast
paulson@13926
   653
paulson@13926
   654
paulson@13926
   655
lemma keysFor_synth [simp]: 
paulson@13926
   656
    "keysFor (synth H) = keysFor H \<union> invKey`{K. Key K \<in> H}"
paulson@14200
   657
by (unfold keysFor_def, blast)
paulson@13926
   658
paulson@13926
   659
paulson@14200
   660
subsubsection{*Combinations of parts, analz and synth *}
paulson@13926
   661
paulson@13926
   662
lemma parts_synth [simp]: "parts (synth H) = parts H \<union> synth H"
paulson@13926
   663
apply (rule equalityI)
paulson@13926
   664
apply (rule subsetI)
paulson@13926
   665
apply (erule parts.induct)
paulson@13926
   666
apply (blast intro: synth_increasing [THEN parts_mono, THEN subsetD] 
paulson@13926
   667
                    parts.Fst parts.Snd parts.Body)+
paulson@13926
   668
done
paulson@13926
   669
paulson@13926
   670
lemma analz_analz_Un [simp]: "analz (analz G \<union> H) = analz (G \<union> H)"
paulson@13926
   671
apply (intro equalityI analz_subset_cong)+
paulson@13926
   672
apply simp_all
paulson@13926
   673
done
paulson@13926
   674
paulson@13926
   675
lemma analz_synth_Un [simp]: "analz (synth G \<union> H) = analz (G \<union> H) \<union> synth G"
paulson@13926
   676
apply (rule equalityI)
paulson@13926
   677
apply (rule subsetI)
paulson@13926
   678
apply (erule analz.induct)
paulson@13926
   679
prefer 5 apply (blast intro: analz_mono [THEN [2] rev_subsetD])
paulson@13926
   680
apply (blast intro: analz.Fst analz.Snd analz.Decrypt)+
paulson@13926
   681
done
paulson@13926
   682
paulson@13926
   683
lemma analz_synth [simp]: "analz (synth H) = analz H \<union> synth H"
paulson@13926
   684
apply (cut_tac H = "{}" in analz_synth_Un)
paulson@13926
   685
apply (simp (no_asm_use))
paulson@13926
   686
done
paulson@13926
   687
paulson@13926
   688
paulson@14200
   689
subsubsection{*For reasoning about the Fake rule in traces *}
paulson@13926
   690
paulson@13926
   691
lemma parts_insert_subset_Un: "X\<in> G ==> parts(insert X H) \<subseteq> parts G \<union> parts H"
paulson@13926
   692
by (rule subset_trans [OF parts_mono parts_Un_subset2], blast)
paulson@13926
   693
paulson@16818
   694
text{*More specifically for Fake.  Very occasionally we could do with a version
paulson@16818
   695
  of the form  @{term"parts{X} \<subseteq> synth (analz H) \<union> parts H"} *}
paulson@14200
   696
lemma Fake_parts_insert:
paulson@14200
   697
     "X \<in> synth (analz H) ==>  
paulson@13926
   698
      parts (insert X H) \<subseteq> synth (analz H) \<union> parts H"
paulson@13926
   699
apply (drule parts_insert_subset_Un)
paulson@13926
   700
apply (simp (no_asm_use))
paulson@13926
   701
apply blast
paulson@13926
   702
done
paulson@13926
   703
paulson@14200
   704
lemma Fake_parts_insert_in_Un:
paulson@14200
   705
     "[|Z \<in> parts (insert X H);  X: synth (analz H)|] 
paulson@14200
   706
      ==> Z \<in>  synth (analz H) \<union> parts H";
paulson@14200
   707
by (blast dest: Fake_parts_insert  [THEN subsetD, dest])
paulson@14200
   708
paulson@16818
   709
text{*@{term H} is sometimes @{term"Key ` KK \<union> spies evs"}, so can't put 
paulson@16818
   710
  @{term "G=H"}.*}
paulson@14200
   711
lemma Fake_analz_insert:
paulson@14200
   712
     "X\<in> synth (analz G) ==>  
paulson@13926
   713
      analz (insert X H) \<subseteq> synth (analz G) \<union> analz (G \<union> H)"
paulson@13926
   714
apply (rule subsetI)
paulson@13926
   715
apply (subgoal_tac "x \<in> analz (synth (analz G) \<union> H) ")
paulson@13926
   716
prefer 2 apply (blast intro: analz_mono [THEN [2] rev_subsetD] analz_mono [THEN synth_mono, THEN [2] rev_subsetD])
paulson@13926
   717
apply (simp (no_asm_use))
paulson@13926
   718
apply blast
paulson@13926
   719
done
paulson@13926
   720
paulson@14200
   721
lemma analz_conj_parts [simp]:
paulson@14200
   722
     "(X \<in> analz H & X \<in> parts H) = (X \<in> analz H)"
paulson@14145
   723
by (blast intro: analz_subset_parts [THEN subsetD])
paulson@13926
   724
paulson@14200
   725
lemma analz_disj_parts [simp]:
paulson@14200
   726
     "(X \<in> analz H | X \<in> parts H) = (X \<in> parts H)"
paulson@14145
   727
by (blast intro: analz_subset_parts [THEN subsetD])
paulson@13926
   728
paulson@16818
   729
text{*Without this equation, other rules for synth and analz would yield
paulson@16818
   730
  redundant cases*}
paulson@13926
   731
lemma MPair_synth_analz [iff]:
paulson@13926
   732
     "({|X,Y|} \<in> synth (analz H)) =  
paulson@13926
   733
      (X \<in> synth (analz H) & Y \<in> synth (analz H))"
paulson@13926
   734
by blast
paulson@13926
   735
paulson@14200
   736
lemma Crypt_synth_analz:
paulson@14200
   737
     "[| Key K \<in> analz H;  Key (invKey K) \<in> analz H |]  
paulson@13926
   738
       ==> (Crypt K X \<in> synth (analz H)) = (X \<in> synth (analz H))"
paulson@13926
   739
by blast
paulson@13926
   740
paulson@13926
   741
paulson@14200
   742
lemma Hash_synth_analz [simp]:
paulson@14200
   743
     "X \<notin> synth (analz H)  
paulson@13926
   744
      ==> (Hash{|X,Y|} \<in> synth (analz H)) = (Hash{|X,Y|} \<in> analz H)"
paulson@13926
   745
by blast
paulson@13926
   746
paulson@13926
   747
paulson@13926
   748
subsection{*HPair: a combination of Hash and MPair*}
paulson@13926
   749
paulson@14200
   750
subsubsection{*Freeness *}
paulson@13926
   751
paulson@13926
   752
lemma Agent_neq_HPair: "Agent A ~= Hash[X] Y"
paulson@13926
   753
by (unfold HPair_def, simp)
paulson@13926
   754
paulson@13926
   755
lemma Nonce_neq_HPair: "Nonce N ~= Hash[X] Y"
paulson@13926
   756
by (unfold HPair_def, simp)
paulson@13926
   757
paulson@13926
   758
lemma Number_neq_HPair: "Number N ~= Hash[X] Y"
paulson@13926
   759
by (unfold HPair_def, simp)
paulson@13926
   760
paulson@13926
   761
lemma Key_neq_HPair: "Key K ~= Hash[X] Y"
paulson@13926
   762
by (unfold HPair_def, simp)
paulson@13926
   763
paulson@13926
   764
lemma Hash_neq_HPair: "Hash Z ~= Hash[X] Y"
paulson@13926
   765
by (unfold HPair_def, simp)
paulson@13926
   766
paulson@13926
   767
lemma Crypt_neq_HPair: "Crypt K X' ~= Hash[X] Y"
paulson@13926
   768
by (unfold HPair_def, simp)
paulson@13926
   769
paulson@13926
   770
lemmas HPair_neqs = Agent_neq_HPair Nonce_neq_HPair Number_neq_HPair 
paulson@13926
   771
                    Key_neq_HPair Hash_neq_HPair Crypt_neq_HPair
paulson@13926
   772
paulson@13926
   773
declare HPair_neqs [iff]
paulson@13926
   774
declare HPair_neqs [symmetric, iff]
paulson@13926
   775
paulson@13926
   776
lemma HPair_eq [iff]: "(Hash[X'] Y' = Hash[X] Y) = (X' = X & Y'=Y)"
paulson@13926
   777
by (simp add: HPair_def)
paulson@13926
   778
paulson@14200
   779
lemma MPair_eq_HPair [iff]:
paulson@14200
   780
     "({|X',Y'|} = Hash[X] Y) = (X' = Hash{|X,Y|} & Y'=Y)"
paulson@13926
   781
by (simp add: HPair_def)
paulson@13926
   782
paulson@14200
   783
lemma HPair_eq_MPair [iff]:
paulson@14200
   784
     "(Hash[X] Y = {|X',Y'|}) = (X' = Hash{|X,Y|} & Y'=Y)"
paulson@13926
   785
by (auto simp add: HPair_def)
paulson@13926
   786
paulson@13926
   787
paulson@14200
   788
subsubsection{*Specialized laws, proved in terms of those for Hash and MPair *}
paulson@13926
   789
paulson@13926
   790
lemma keysFor_insert_HPair [simp]: "keysFor (insert (Hash[X] Y) H) = keysFor H"
paulson@13926
   791
by (simp add: HPair_def)
paulson@13926
   792
paulson@13926
   793
lemma parts_insert_HPair [simp]: 
paulson@13926
   794
    "parts (insert (Hash[X] Y) H) =  
paulson@13926
   795
     insert (Hash[X] Y) (insert (Hash{|X,Y|}) (parts (insert Y H)))"
paulson@13926
   796
by (simp add: HPair_def)
paulson@13926
   797
paulson@13926
   798
lemma analz_insert_HPair [simp]: 
paulson@13926
   799
    "analz (insert (Hash[X] Y) H) =  
paulson@13926
   800
     insert (Hash[X] Y) (insert (Hash{|X,Y|}) (analz (insert Y H)))"
paulson@13926
   801
by (simp add: HPair_def)
paulson@13926
   802
paulson@13926
   803
lemma HPair_synth_analz [simp]:
paulson@13926
   804
     "X \<notin> synth (analz H)  
paulson@13926
   805
    ==> (Hash[X] Y \<in> synth (analz H)) =  
paulson@13926
   806
        (Hash {|X, Y|} \<in> analz H & Y \<in> synth (analz H))"
paulson@13926
   807
by (simp add: HPair_def)
paulson@13926
   808
paulson@13926
   809
paulson@16818
   810
text{*We do NOT want Crypt... messages broken up in protocols!!*}
paulson@13926
   811
declare parts.Body [rule del]
paulson@13926
   812
paulson@13926
   813
paulson@14200
   814
text{*Rewrites to push in Key and Crypt messages, so that other messages can
paulson@14200
   815
    be pulled out using the @{text analz_insert} rules*}
paulson@13926
   816
ML
paulson@13926
   817
{*
paulson@13926
   818
fun insComm x y = inst "x" x (inst "y" y insert_commute);
paulson@13926
   819
paulson@13926
   820
bind_thms ("pushKeys",
paulson@13926
   821
           map (insComm "Key ?K") 
paulson@13926
   822
                   ["Agent ?C", "Nonce ?N", "Number ?N", 
paulson@13926
   823
		    "Hash ?X", "MPair ?X ?Y", "Crypt ?X ?K'"]);
paulson@13926
   824
paulson@13926
   825
bind_thms ("pushCrypts",
paulson@13926
   826
           map (insComm "Crypt ?X ?K") 
paulson@13926
   827
                     ["Agent ?C", "Nonce ?N", "Number ?N", 
paulson@13926
   828
		      "Hash ?X'", "MPair ?X' ?Y"]);
paulson@13926
   829
*}
paulson@13926
   830
paulson@13926
   831
text{*Cannot be added with @{text "[simp]"} -- messages should not always be
paulson@13926
   832
  re-ordered. *}
paulson@13926
   833
lemmas pushes = pushKeys pushCrypts
paulson@13926
   834
paulson@13926
   835
paulson@13926
   836
subsection{*Tactics useful for many protocol proofs*}
paulson@13926
   837
ML
paulson@13926
   838
{*
paulson@14200
   839
val invKey = thm "invKey"
paulson@14200
   840
val keysFor_def = thm "keysFor_def"
paulson@14200
   841
val HPair_def = thm "HPair_def"
paulson@14200
   842
val symKeys_def = thm "symKeys_def"
paulson@13926
   843
val parts_mono = thm "parts_mono";
paulson@13926
   844
val analz_mono = thm "analz_mono";
paulson@14200
   845
val synth_mono = thm "synth_mono";
paulson@13926
   846
val analz_increasing = thm "analz_increasing";
paulson@14200
   847
paulson@14200
   848
val analz_insertI = thm "analz_insertI";
paulson@13926
   849
val analz_subset_parts = thm "analz_subset_parts";
paulson@13926
   850
val Fake_parts_insert = thm "Fake_parts_insert";
paulson@13926
   851
val Fake_analz_insert = thm "Fake_analz_insert";
paulson@13926
   852
val pushes = thms "pushes";
paulson@13926
   853
paulson@13926
   854
paulson@13926
   855
(*Prove base case (subgoal i) and simplify others.  A typical base case
paulson@13926
   856
  concerns  Crypt K X \<notin> Key`shrK`bad  and cannot be proved by rewriting
paulson@13926
   857
  alone.*)
paulson@13926
   858
fun prove_simple_subgoals_tac i = 
paulson@13926
   859
    force_tac (claset(), simpset() addsimps [image_eq_UN]) i THEN
paulson@13926
   860
    ALLGOALS Asm_simp_tac
paulson@13926
   861
paulson@13926
   862
(*Analysis of Fake cases.  Also works for messages that forward unknown parts,
paulson@13926
   863
  but this application is no longer necessary if analz_insert_eq is used.
paulson@13926
   864
  Abstraction over i is ESSENTIAL: it delays the dereferencing of claset
paulson@13926
   865
  DEPENDS UPON "X" REFERRING TO THE FRADULENT MESSAGE *)
paulson@13926
   866
paulson@13926
   867
(*Apply rules to break down assumptions of the form
paulson@13926
   868
  Y \<in> parts(insert X H)  and  Y \<in> analz(insert X H)
paulson@13926
   869
*)
paulson@13926
   870
val Fake_insert_tac = 
paulson@13926
   871
    dresolve_tac [impOfSubs Fake_analz_insert,
paulson@13926
   872
                  impOfSubs Fake_parts_insert] THEN'
paulson@13926
   873
    eresolve_tac [asm_rl, thm"synth.Inj"];
paulson@13926
   874
paulson@13926
   875
fun Fake_insert_simp_tac ss i = 
paulson@13926
   876
    REPEAT (Fake_insert_tac i) THEN asm_full_simp_tac ss i;
paulson@13926
   877
paulson@13926
   878
fun atomic_spy_analz_tac (cs,ss) = SELECT_GOAL
paulson@13926
   879
    (Fake_insert_simp_tac ss 1
paulson@13926
   880
     THEN
paulson@13926
   881
     IF_UNSOLVED (Blast.depth_tac
paulson@13926
   882
		  (cs addIs [analz_insertI,
paulson@13926
   883
				   impOfSubs analz_subset_parts]) 4 1))
paulson@13926
   884
paulson@13926
   885
(*The explicit claset and simpset arguments help it work with Isar*)
paulson@13926
   886
fun gen_spy_analz_tac (cs,ss) i =
paulson@13926
   887
  DETERM
paulson@13926
   888
   (SELECT_GOAL
paulson@13926
   889
     (EVERY 
paulson@13926
   890
      [  (*push in occurrences of X...*)
paulson@13926
   891
       (REPEAT o CHANGED)
paulson@13926
   892
           (res_inst_tac [("x1","X")] (insert_commute RS ssubst) 1),
paulson@13926
   893
       (*...allowing further simplifications*)
paulson@13926
   894
       simp_tac ss 1,
paulson@13926
   895
       REPEAT (FIRSTGOAL (resolve_tac [allI,impI,notI,conjI,iffI])),
paulson@13926
   896
       DEPTH_SOLVE (atomic_spy_analz_tac (cs,ss) 1)]) i)
paulson@13926
   897
paulson@13926
   898
fun spy_analz_tac i = gen_spy_analz_tac (claset(), simpset()) i
paulson@13926
   899
*}
paulson@13926
   900
paulson@16818
   901
text{*By default only @{text o_apply} is built-in.  But in the presence of
paulson@16818
   902
eta-expansion this means that some terms displayed as @{term "f o g"} will be
paulson@16818
   903
rewritten, and others will not!*}
paulson@13926
   904
declare o_def [simp]
paulson@13926
   905
paulson@11189
   906
paulson@13922
   907
lemma Crypt_notin_image_Key [simp]: "Crypt K X \<notin> Key ` A"
paulson@13922
   908
by auto
paulson@13922
   909
paulson@13922
   910
lemma Hash_notin_image_Key [simp] :"Hash X \<notin> Key ` A"
paulson@13922
   911
by auto
paulson@13922
   912
paulson@14200
   913
lemma synth_analz_mono: "G\<subseteq>H ==> synth (analz(G)) \<subseteq> synth (analz(H))"
paulson@17689
   914
by (iprover intro: synth_mono analz_mono) 
paulson@13922
   915
paulson@13922
   916
lemma Fake_analz_eq [simp]:
paulson@13922
   917
     "X \<in> synth(analz H) ==> synth (analz (insert X H)) = synth (analz H)"
paulson@13922
   918
apply (drule Fake_analz_insert[of _ _ "H"])
paulson@13922
   919
apply (simp add: synth_increasing[THEN Un_absorb2])
paulson@13922
   920
apply (drule synth_mono)
paulson@13922
   921
apply (simp add: synth_idem)
paulson@17689
   922
apply (rule equalityI)
paulson@17689
   923
apply (simp add: );
paulson@17689
   924
apply (rule synth_analz_mono, blast)   
paulson@13922
   925
done
paulson@13922
   926
paulson@13922
   927
text{*Two generalizations of @{text analz_insert_eq}*}
paulson@13922
   928
lemma gen_analz_insert_eq [rule_format]:
paulson@13922
   929
     "X \<in> analz H ==> ALL G. H \<subseteq> G --> analz (insert X G) = analz G";
paulson@13922
   930
by (blast intro: analz_cut analz_insertI analz_mono [THEN [2] rev_subsetD])
paulson@13922
   931
paulson@13922
   932
lemma synth_analz_insert_eq [rule_format]:
paulson@13922
   933
     "X \<in> synth (analz H) 
paulson@13922
   934
      ==> ALL G. H \<subseteq> G --> (Key K \<in> analz (insert X G)) = (Key K \<in> analz G)";
paulson@13922
   935
apply (erule synth.induct) 
paulson@13922
   936
apply (simp_all add: gen_analz_insert_eq subset_trans [OF _ subset_insertI]) 
paulson@13922
   937
done
paulson@13922
   938
paulson@13922
   939
lemma Fake_parts_sing:
paulson@13926
   940
     "X \<in> synth (analz H) ==> parts{X} \<subseteq> synth (analz H) \<union> parts H";
paulson@13922
   941
apply (rule subset_trans) 
paulson@17689
   942
 apply (erule_tac [2] Fake_parts_insert)
paulson@17689
   943
apply (rule parts_mono)  
paulson@17689
   944
apply (blast intro: elim:); 
paulson@13922
   945
done
paulson@13922
   946
paulson@14145
   947
lemmas Fake_parts_sing_imp_Un = Fake_parts_sing [THEN [2] rev_subsetD]
paulson@14145
   948
paulson@11189
   949
method_setup spy_analz = {*
paulson@11270
   950
    Method.ctxt_args (fn ctxt =>
paulson@11270
   951
        Method.METHOD (fn facts => 
wenzelm@15032
   952
            gen_spy_analz_tac (local_clasimpset_of ctxt) 1)) *}
paulson@11189
   953
    "for proving the Fake case when analz is involved"
paulson@1839
   954
paulson@11264
   955
method_setup atomic_spy_analz = {*
paulson@11270
   956
    Method.ctxt_args (fn ctxt =>
paulson@11270
   957
        Method.METHOD (fn facts => 
wenzelm@15032
   958
            atomic_spy_analz_tac (local_clasimpset_of ctxt) 1)) *}
paulson@11264
   959
    "for debugging spy_analz"
paulson@11264
   960
paulson@11264
   961
method_setup Fake_insert_simp = {*
paulson@11270
   962
    Method.ctxt_args (fn ctxt =>
paulson@11270
   963
        Method.METHOD (fn facts =>
wenzelm@15032
   964
            Fake_insert_simp_tac (local_simpset_of ctxt) 1)) *}
paulson@11264
   965
    "for debugging spy_analz"
paulson@11264
   966
paulson@13926
   967
paulson@1839
   968
end