src/HOL/Auth/Yahalom.thy

tuned parentheses in relational expressions;

(*  Title:      HOL/Auth/Yahalom
    ID:         $Id$
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1996  University of Cambridge

Inductive relation "yahalom" for the Yahalom protocol.

From page 257 of
  Burrows, Abadi and Needham.  A Logic of Authentication.
  Proc. Royal Soc. 426 (1989)

This theory has the prototypical example of a secrecy relation, KeyCryptNonce.
*)

theory Yahalom = Shared:

consts  yahalom   :: "event list set"
inductive "yahalom"
  intros 
         (*Initial trace is empty*)
   Nil:  "[] \<in> yahalom"

         (*The spy MAY say anything he CAN say.  We do not expect him to
           invent new nonces here, but he can also use NS1.  Common to
           all similar protocols.*)
   Fake: "[| evsf \<in> yahalom;  X \<in> synth (analz (knows Spy evsf)) |]
          ==> Says Spy B X  # evsf \<in> yahalom"

         (*A message that has been sent can be received by the
           intended recipient.*)
   Reception: "[| evsr \<in> yahalom;  Says A B X \<in> set evsr |]
               ==> Gets B X # evsr \<in> yahalom"

         (*Alice initiates a protocol run*)
   YM1:  "[| evs1 \<in> yahalom;  Nonce NA \<notin> used evs1 |]
          ==> Says A B {|Agent A, Nonce NA|} # evs1 \<in> yahalom"

         (*Bob's response to Alice's message.*)
   YM2:  "[| evs2 \<in> yahalom;  Nonce NB \<notin> used evs2;
             Gets B {|Agent A, Nonce NA|} \<in> set evs2 |]
          ==> Says B Server 
                  {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, Nonce NB|}|}
                # evs2 \<in> yahalom"

         (*The Server receives Bob's message.  He responds by sending a
            new session key to Alice, with a packet for forwarding to Bob.*)
   YM3:  "[| evs3 \<in> yahalom;  Key KAB \<notin> used evs3;
             Gets Server 
                  {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, Nonce NB|}|}
               \<in> set evs3 |]
          ==> Says Server A
                   {|Crypt (shrK A) {|Agent B, Key KAB, Nonce NA, Nonce NB|},
                     Crypt (shrK B) {|Agent A, Key KAB|}|}
                # evs3 \<in> yahalom"

         (*Alice receives the Server's (?) message, checks her Nonce, and
           uses the new session key to send Bob his Nonce.  The premise
           A \<noteq> Server is needed to prove Says_Server_not_range.*)
   YM4:  "[| evs4 \<in> yahalom;  A \<noteq> Server;
             Gets A {|Crypt(shrK A) {|Agent B, Key K, Nonce NA, Nonce NB|}, X|}
                \<in> set evs4;
             Says A B {|Agent A, Nonce NA|} \<in> set evs4 |]
          ==> Says A B {|X, Crypt K (Nonce NB)|} # evs4 \<in> yahalom"

         (*This message models possible leaks of session keys.  The Nonces
           identify the protocol run.  Quoting Server here ensures they are
           correct.*)
   Oops: "[| evso \<in> yahalom;  
             Says Server A {|Crypt (shrK A)
                                   {|Agent B, Key K, Nonce NA, Nonce NB|},
                             X|}  \<in> set evso |]
          ==> Notes Spy {|Nonce NA, Nonce NB, Key K|} # evso \<in> yahalom"


constdefs 
  KeyWithNonce :: "[key, nat, event list] => bool"
  "KeyWithNonce K NB evs ==
     \<exists>A B na X. 
       Says Server A {|Crypt (shrK A) {|Agent B, Key K, na, Nonce NB|}, X|} 
         \<in> set evs"


declare Says_imp_knows_Spy [THEN analz.Inj, dest]
declare parts.Body  [dest]
declare Fake_parts_insert_in_Un  [dest]
declare analz_into_parts [dest]

(*A "possibility property": there are traces that reach the end*)
lemma "A \<noteq> Server  
      ==> \<exists>X NB K. \<exists>evs \<in> yahalom.           
             Says A B {|X, Crypt K (Nonce NB)|} \<in> set evs"
apply (intro exI bexI)
apply (rule_tac [2] yahalom.Nil
                    [THEN yahalom.YM1, THEN yahalom.Reception, 
                     THEN yahalom.YM2, THEN yahalom.Reception, 
                     THEN yahalom.YM3, THEN yahalom.Reception, 
                     THEN yahalom.YM4])
apply possibility
done

lemma Gets_imp_Says:
     "[| Gets B X \<in> set evs; evs \<in> yahalom |] ==> \<exists>A. Says A B X \<in> set evs"
by (erule rev_mp, erule yahalom.induct, auto)

(*Must be proved separately for each protocol*)
lemma Gets_imp_knows_Spy:
     "[| Gets B X \<in> set evs; evs \<in> yahalom |]  ==> X \<in> knows Spy evs"
by (blast dest!: Gets_imp_Says Says_imp_knows_Spy)

declare Gets_imp_knows_Spy [THEN analz.Inj, dest]


(**** Inductive proofs about yahalom ****)

(*Lets us treat YM4 using a similar argument as for the Fake case.*)
lemma YM4_analz_knows_Spy:
     "[| Gets A {|Crypt (shrK A) Y, X|} \<in> set evs;  evs \<in> yahalom |]   
      ==> X \<in> analz (knows Spy evs)"
by blast

lemmas YM4_parts_knows_Spy = 
       YM4_analz_knows_Spy [THEN analz_into_parts, standard]

(*For Oops*)
lemma YM4_Key_parts_knows_Spy:
     "Says Server A {|Crypt (shrK A) {|B,K,NA,NB|}, X|} \<in> set evs  
      ==> K \<in> parts (knows Spy evs)"
by (blast dest!: parts.Body Says_imp_knows_Spy [THEN parts.Inj])


(** Theorems of the form X \<notin> parts (knows Spy evs) imply that NOBODY
    sends messages containing X! **)

(*Spy never sees a good agent's shared key!*)
lemma Spy_see_shrK [simp]:
     "evs \<in> yahalom ==> (Key (shrK A) \<in> parts (knows Spy evs)) = (A \<in> bad)"
apply (erule yahalom.induct, force, 
       drule_tac [6] YM4_parts_knows_Spy, simp_all)
apply blast+
done

lemma Spy_analz_shrK [simp]:
     "evs \<in> yahalom ==> (Key (shrK A) \<in> analz (knows Spy evs)) = (A \<in> bad)"
by auto

lemma Spy_see_shrK_D [dest!]:
     "[|Key (shrK A) \<in> parts (knows Spy evs);  evs \<in> yahalom|] ==> A \<in> bad"
by (blast dest: Spy_see_shrK)

(*Nobody can have used non-existent keys!  Needed to apply analz_insert_Key*)
lemma new_keys_not_used [rule_format, simp]:
 "evs \<in> yahalom ==> Key K \<notin> used evs --> K \<notin> keysFor (parts (knows Spy evs))"
apply (erule yahalom.induct, force, 
       frule_tac [6] YM4_parts_knows_Spy, simp_all)
(*Fake, YM3, YM4*)
apply (blast dest!: keysFor_parts_insert)+
done


(*Earlier, all protocol proofs declared this theorem.  
  But only a few proofs need it, e.g. Yahalom and Kerberos IV.*)
lemma new_keys_not_analzd:
 "[|evs \<in> yahalom; Key K \<notin> used evs|] ==> K \<notin> keysFor (analz (knows Spy evs))"
by (blast dest: new_keys_not_used intro: keysFor_mono [THEN subsetD]) 


(*Describes the form of K when the Server sends this message.  Useful for
  Oops as well as main secrecy property.*)
lemma Says_Server_not_range [simp]:
     "[| Says Server A {|Crypt (shrK A) {|Agent B, Key K, na, nb|}, X|}  
           \<in> set evs;   evs \<in> yahalom |]                                 
      ==> K \<notin> range shrK"
apply (erule rev_mp, erule yahalom.induct, simp_all)
apply blast
done


(*For proofs involving analz.
val analz_knows_Spy_tac = 
    ftac YM4_analz_knows_Spy 7 THEN assume_tac 7
*)

(****
 The following is to prove theorems of the form

  Key K \<in> analz (insert (Key KAB) (knows Spy evs)) ==>
  Key K \<in> analz (knows Spy evs)

 A more general formula must be proved inductively.
****)

(** Session keys are not used to encrypt other session keys **)

lemma analz_image_freshK [rule_format]:
 "evs \<in> yahalom ==>                                
   \<forall>K KK. KK <= - (range shrK) -->                  
          (Key K \<in> analz (Key`KK Un (knows Spy evs))) =   
          (K \<in> KK | Key K \<in> analz (knows Spy evs))"
apply (erule yahalom.induct, force, 
       drule_tac [6] YM4_analz_knows_Spy)
apply analz_freshK
apply spy_analz
apply (simp only: Says_Server_not_range analz_image_freshK_simps)
done

lemma analz_insert_freshK:
     "[| evs \<in> yahalom;  KAB \<notin> range shrK |] ==>      
      (Key K \<in> analz (insert (Key KAB) (knows Spy evs))) =
      (K = KAB | Key K \<in> analz (knows Spy evs))"
by (simp only: analz_image_freshK analz_image_freshK_simps)


(*** The Key K uniquely identifies the Server's  message. **)

lemma unique_session_keys:
     "[| Says Server A                                                  
          {|Crypt (shrK A) {|Agent B, Key K, na, nb|}, X|} \<in> set evs;  
        Says Server A'                                                 
          {|Crypt (shrK A') {|Agent B', Key K, na', nb'|}, X'|} \<in> set evs;  
        evs \<in> yahalom |]                                     
     ==> A=A' & B=B' & na=na' & nb=nb'"
apply (erule rev_mp, erule rev_mp)
apply (erule yahalom.induct, simp_all)
(*YM3, by freshness, and YM4*)
apply blast+
done


(** Crucial secrecy property: Spy does not see the keys sent in msg YM3 **)

lemma secrecy_lemma:
     "[| A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom |]                 
      ==> Says Server A                                         
            {|Crypt (shrK A) {|Agent B, Key K, na, nb|},        
              Crypt (shrK B) {|Agent A, Key K|}|}               
           \<in> set evs -->                                        
          Notes Spy {|na, nb, Key K|} \<notin> set evs -->            
          Key K \<notin> analz (knows Spy evs)"
apply (erule yahalom.induct, force, 
       drule_tac [6] YM4_analz_knows_Spy)
apply (simp_all add: pushes analz_insert_eq analz_insert_freshK)
apply spy_analz  (*Fake*)
apply (blast dest: unique_session_keys)+  (*YM3, Oops*)
done

(*Final version*)
lemma Spy_not_see_encrypted_key:
     "[| Says Server A                                          
            {|Crypt (shrK A) {|Agent B, Key K, na, nb|},        
              Crypt (shrK B) {|Agent A, Key K|}|}               
           \<in> set evs;                                           
         Notes Spy {|na, nb, Key K|} \<notin> set evs;                
         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom |]                 
      ==> Key K \<notin> analz (knows Spy evs)"
by (blast dest: secrecy_lemma)


(** Security Guarantee for A upon receiving YM3 **)

(*If the encrypted message appears then it originated with the Server*)
lemma A_trusts_YM3:
     "[| Crypt (shrK A) {|Agent B, Key K, na, nb|} \<in> parts (knows Spy evs);  
         A \<notin> bad;  evs \<in> yahalom |]                           
       ==> Says Server A                                             
            {|Crypt (shrK A) {|Agent B, Key K, na, nb|},             
              Crypt (shrK B) {|Agent A, Key K|}|}                    
           \<in> set evs"
apply (erule rev_mp)
apply (erule yahalom.induct, force, 
       frule_tac [6] YM4_parts_knows_Spy, simp_all)
(*Fake, YM3*)
apply blast+
done

(*The obvious combination of A_trusts_YM3 with Spy_not_see_encrypted_key*)
lemma A_gets_good_key:
     "[| Crypt (shrK A) {|Agent B, Key K, na, nb|} \<in> parts (knows Spy evs);  
         Notes Spy {|na, nb, Key K|} \<notin> set evs;                
         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom |]                 
      ==> Key K \<notin> analz (knows Spy evs)"
by (blast dest!: A_trusts_YM3 Spy_not_see_encrypted_key)

(** Security Guarantees for B upon receiving YM4 **)

(*B knows, by the first part of A's message, that the Server distributed 
  the key for A and B.  But this part says nothing about nonces.*)
lemma B_trusts_YM4_shrK:
     "[| Crypt (shrK B) {|Agent A, Key K|} \<in> parts (knows Spy evs);       
         B \<notin> bad;  evs \<in> yahalom |]                                  
      ==> \<exists>NA NB. Says Server A                                     
                      {|Crypt (shrK A) {|Agent B, Key K,              
                                         Nonce NA, Nonce NB|},        
                        Crypt (shrK B) {|Agent A, Key K|}|}           
                     \<in> set evs"
apply (erule rev_mp)
apply (erule yahalom.induct, force, 
       frule_tac [6] YM4_parts_knows_Spy, simp_all)
(*Fake, YM3*)
apply blast+
done

(*B knows, by the second part of A's message, that the Server distributed 
  the key quoting nonce NB.  This part says nothing about agent names. 
  Secrecy of NB is crucial.  Note that  Nonce NB \<notin> analz(knows Spy evs)  must
  be the FIRST antecedent of the induction formula.*)
lemma B_trusts_YM4_newK[rule_format]:
     "[|Crypt K (Nonce NB) \<in> parts (knows Spy evs);
        Nonce NB \<notin> analz (knows Spy evs);  evs \<in> yahalom|]
      ==> \<exists>A B NA. Says Server A                           
                      {|Crypt (shrK A) {|Agent B, Key K, Nonce NA, Nonce NB|},
                        Crypt (shrK B) {|Agent A, Key K|}|}   
                     \<in> set evs"
apply (erule rev_mp, erule rev_mp)
apply (erule yahalom.induct, force, 
       frule_tac [6] YM4_parts_knows_Spy)
apply (analz_mono_contra, simp_all)
(*Fake, YM3*)
apply blast
apply blast
(*YM4*)
(*A is uncompromised because NB is secure
  A's certificate guarantees the existence of the Server message*)
apply (blast dest!: Gets_imp_Says Crypt_Spy_analz_bad 
             dest: Says_imp_spies 
                   parts.Inj [THEN parts.Fst, THEN A_trusts_YM3])
done


(**** Towards proving secrecy of Nonce NB ****)

(** Lemmas about the predicate KeyWithNonce **)

lemma KeyWithNonceI: 
 "Says Server A                                               
          {|Crypt (shrK A) {|Agent B, Key K, na, Nonce NB|}, X|}  
        \<in> set evs ==> KeyWithNonce K NB evs"
by (unfold KeyWithNonce_def, blast)

lemma KeyWithNonce_Says [simp]: 
   "KeyWithNonce K NB (Says S A X # evs) =                                     
      (Server = S &
       (\<exists>B n X'. X = {|Crypt (shrK A) {|Agent B, Key K, n, Nonce NB|}, X'|})  
      | KeyWithNonce K NB evs)"
by (simp add: KeyWithNonce_def, blast)


lemma KeyWithNonce_Notes [simp]: 
   "KeyWithNonce K NB (Notes A X # evs) = KeyWithNonce K NB evs"
by (simp add: KeyWithNonce_def)

lemma KeyWithNonce_Gets [simp]: 
   "KeyWithNonce K NB (Gets A X # evs) = KeyWithNonce K NB evs"
by (simp add: KeyWithNonce_def)

(*A fresh key cannot be associated with any nonce 
  (with respect to a given trace). *)
lemma fresh_not_KeyWithNonce: 
 "Key K \<notin> used evs ==> ~ KeyWithNonce K NB evs"
by (unfold KeyWithNonce_def, blast)

(*The Server message associates K with NB' and therefore not with any 
  other nonce NB.*)
lemma Says_Server_KeyWithNonce: 
 "[| Says Server A {|Crypt (shrK A) {|Agent B, Key K, na, Nonce NB'|}, X|}  
       \<in> set evs;                                                  
     NB \<noteq> NB';  evs \<in> yahalom |]                                  
  ==> ~ KeyWithNonce K NB evs"
by (unfold KeyWithNonce_def, blast dest: unique_session_keys)


(*The only nonces that can be found with the help of session keys are
  those distributed as nonce NB by the Server.  The form of the theorem
  recalls analz_image_freshK, but it is much more complicated.*)


(*As with analz_image_freshK, we take some pains to express the property
  as a logical equivalence so that the simplifier can apply it.*)
lemma Nonce_secrecy_lemma:
     "P --> (X \<in> analz (G Un H)) --> (X \<in> analz H)  ==>  
      P --> (X \<in> analz (G Un H)) = (X \<in> analz H)"
by (blast intro: analz_mono [THEN subsetD])

lemma Nonce_secrecy:
     "evs \<in> yahalom ==>                                       
      (\<forall>KK. KK <= - (range shrK) -->                       
           (\<forall>K \<in> KK. ~ KeyWithNonce K NB evs)   -->         
           (Nonce NB \<in> analz (Key`KK Un (knows Spy evs))) =      
           (Nonce NB \<in> analz (knows Spy evs)))"
apply (erule yahalom.induct, force, 
       frule_tac [6] YM4_analz_knows_Spy)
apply (safe del: allI impI intro!: Nonce_secrecy_lemma [THEN impI, THEN allI])
apply (simp_all del: image_insert image_Un 
       add: analz_image_freshK_simps split_ifs
            all_conj_distrib ball_conj_distrib 
            analz_image_freshK fresh_not_KeyWithNonce
            imp_disj_not1               (*Moves NBa\<noteq>NB to the front*)
            Says_Server_KeyWithNonce)
(*For Oops, simplification proves NBa\<noteq>NB.  By Says_Server_KeyWithNonce,
  we get (~ KeyWithNonce K NB evs); then simplification can apply the
  induction hypothesis with KK = {K}.*)
(*Fake*) 
apply spy_analz
(*YM4*)  (** LEVEL 6 **)
apply (erule_tac V = "\<forall>KK. ?P KK" in thin_rl)
apply clarify
(*If A \<in> bad then NBa is known, therefore NBa \<noteq> NB.  Previous two steps make
  the next step faster.*)
apply (blast dest!: Gets_imp_Says Says_imp_spies Crypt_Spy_analz_bad
         dest: analz.Inj
           parts.Inj [THEN parts.Fst, THEN A_trusts_YM3, THEN KeyWithNonceI])
done


(*Version required below: if NB can be decrypted using a session key then it
  was distributed with that key.  The more general form above is required
  for the induction to carry through.*)
lemma single_Nonce_secrecy:
     "[| Says Server A                                                
          {|Crypt (shrK A) {|Agent B, Key KAB, na, Nonce NB'|}, X|}   
         \<in> set evs;                                                   
         NB \<noteq> NB';  KAB \<notin> range shrK;  evs \<in> yahalom |]             
      ==> (Nonce NB \<in> analz (insert (Key KAB) (knows Spy evs))) =         
          (Nonce NB \<in> analz (knows Spy evs))"
by (simp_all del: image_insert image_Un imp_disjL
             add: analz_image_freshK_simps split_ifs
                  Nonce_secrecy Says_Server_KeyWithNonce);


(*** The Nonce NB uniquely identifies B's message. ***)

lemma unique_NB:
     "[| Crypt (shrK B) {|Agent A, Nonce NA, nb|} \<in> parts (knows Spy evs);     
         Crypt (shrK B') {|Agent A', Nonce NA', nb|} \<in> parts (knows Spy evs);  
        evs \<in> yahalom;  B \<notin> bad;  B' \<notin> bad |]   
      ==> NA' = NA & A' = A & B' = B"
apply (erule rev_mp, erule rev_mp)
apply (erule yahalom.induct, force, 
       frule_tac [6] YM4_parts_knows_Spy, simp_all)
(*Fake, and YM2 by freshness*)
apply blast+
done


(*Variant useful for proving secrecy of NB.  Because nb is assumed to be 
  secret, we no longer must assume B, B' not bad.*)
lemma Says_unique_NB:
     "[| Says C S   {|X,  Crypt (shrK B) {|Agent A, Nonce NA, nb|}|}     
           \<in> set evs;                           
         Gets S' {|X', Crypt (shrK B') {|Agent A', Nonce NA', nb|}|}     
           \<in> set evs;                                                    
         nb \<notin> analz (knows Spy evs);  evs \<in> yahalom |]                  
      ==> NA' = NA & A' = A & B' = B"
by (blast dest!: Gets_imp_Says Crypt_Spy_analz_bad 
          dest: Says_imp_spies unique_NB parts.Inj analz.Inj)


(** A nonce value is never used both as NA and as NB **)

lemma no_nonce_YM1_YM2:
     "[|Crypt (shrK B') {|Agent A', Nonce NB, nb'|} \<in> parts(knows Spy evs);
        Nonce NB \<notin> analz (knows Spy evs);  evs \<in> yahalom|]
  ==> Crypt (shrK B)  {|Agent A, na, Nonce NB|} \<notin> parts(knows Spy evs)"
apply (erule rev_mp, erule rev_mp)
apply (erule yahalom.induct, force, 
       frule_tac [6] YM4_parts_knows_Spy)
apply (analz_mono_contra, simp_all)
(*Fake, YM2*)
apply blast+
done

(*The Server sends YM3 only in response to YM2.*)
lemma Says_Server_imp_YM2:
     "[| Says Server A {|Crypt (shrK A) {|Agent B, k, na, nb|}, X|} \<in> set evs;
         evs \<in> yahalom |]                                              
      ==> Gets Server {| Agent B, Crypt (shrK B) {|Agent A, na, nb|} |}  
             \<in> set evs"
apply (erule rev_mp, erule yahalom.induct)
apply auto
done


(*A vital theorem for B, that nonce NB remains secure from the Spy.*)
lemma Spy_not_see_NB :
     "[| Says B Server                                                     
	        {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, Nonce NB|}|}  
	   \<in> set evs;
	 (\<forall>k. Notes Spy {|Nonce NA, Nonce NB, k|} \<notin> set evs);
         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom |]   
      ==> Nonce NB \<notin> analz (knows Spy evs)"
apply (erule rev_mp, erule rev_mp)
apply (erule yahalom.induct, force, 
       frule_tac [6] YM4_analz_knows_Spy)
apply (simp_all add: split_ifs pushes new_keys_not_analzd analz_insert_eq
                     analz_insert_freshK)
(*Fake*)
apply spy_analz
(*YM1: NB=NA is impossible anyway, but NA is secret because it is fresh!*)
apply blast
(*YM2*)
apply blast
(*Prove YM3 by showing that no NB can also be an NA*)
apply (blast dest!: no_nonce_YM1_YM2 dest: Gets_imp_Says Says_unique_NB)
(** LEVEL 7: YM4 and Oops remain **)
apply (clarify, simp add: all_conj_distrib)
(*YM4: key K is visible to Spy, contradicting session key secrecy theorem*) 
(*Case analysis on Aa:bad; PROOF FAILED problems
  use Says_unique_NB to identify message components: Aa=A, Ba=B*)  
apply (blast dest!: Says_unique_NB 
                    parts.Inj [THEN parts.Fst, THEN A_trusts_YM3] 
             dest: Gets_imp_Says Says_imp_spies Says_Server_imp_YM2
                   Spy_not_see_encrypted_key)
(*Oops case: if the nonce is betrayed now, show that the Oops event is 
  covered by the quantified Oops assumption.*)
apply (clarify, simp add: all_conj_distrib)
apply (frule Says_Server_imp_YM2, assumption)
apply (case_tac "NB = NBa")
(*If NB=NBa then all other components of the Oops message agree*)
apply (blast dest: Says_unique_NB)
(*case NB \<noteq> NBa*)
apply (simp add: single_Nonce_secrecy)
apply (blast dest!: no_nonce_YM1_YM2 (*to prove NB\<noteq>NAa*))
done


(*B's session key guarantee from YM4.  The two certificates contribute to a
  single conclusion about the Server's message.  Note that the "Notes Spy"
  assumption must quantify over \<forall>POSSIBLE keys instead of our particular K.
  If this run is broken and the spy substitutes a certificate containing an
  old key, B has no means of telling.*)
lemma B_trusts_YM4:
     "[| Gets B {|Crypt (shrK B) {|Agent A, Key K|},                   
                  Crypt K (Nonce NB)|} \<in> set evs;                      
         Says B Server                                                    
           {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, Nonce NB|}|}    
           \<in> set evs;                                                     
         \<forall>k. Notes Spy {|Nonce NA, Nonce NB, k|} \<notin> set evs;           
         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom |]        
       ==> Says Server A                                                  
                   {|Crypt (shrK A) {|Agent B, Key K,                     
                             Nonce NA, Nonce NB|},                        
                     Crypt (shrK B) {|Agent A, Key K|}|}                  
             \<in> set evs"
by (blast dest: Spy_not_see_NB Says_unique_NB 
                Says_Server_imp_YM2 B_trusts_YM4_newK)



(*The obvious combination of B_trusts_YM4 with Spy_not_see_encrypted_key*)
lemma B_gets_good_key:
     "[| Gets B {|Crypt (shrK B) {|Agent A, Key K|},
                  Crypt K (Nonce NB)|} \<in> set evs;
         Says B Server                                                    
           {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, Nonce NB|}|}    
           \<in> set evs;                                                     
         \<forall>k. Notes Spy {|Nonce NA, Nonce NB, k|} \<notin> set evs;           
         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom |]                 
      ==> Key K \<notin> analz (knows Spy evs)"
by (blast dest!: B_trusts_YM4 Spy_not_see_encrypted_key)


(*** Authenticating B to A ***)

(*The encryption in message YM2 tells us it cannot be faked.*)
lemma B_Said_YM2 [rule_format]:
     "[|Crypt (shrK B) {|Agent A, Nonce NA, nb|} \<in> parts (knows Spy evs);
        evs \<in> yahalom|]
      ==> B \<notin> bad -->
          Says B Server {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, nb|}|}
            \<in> set evs"
apply (erule rev_mp, erule yahalom.induct, force, 
       frule_tac [6] YM4_parts_knows_Spy, simp_all)
(*Fake*)
apply blast
done

(*If the server sends YM3 then B sent YM2*)
lemma YM3_auth_B_to_A_lemma:
     "[|Says Server A {|Crypt (shrK A) {|Agent B, Key K, Nonce NA, nb|}, X|}  
       \<in> set evs;  evs \<in> yahalom|]
      ==> B \<notin> bad -->                                                         
          Says B Server {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, nb|}|}
            \<in> set evs"
apply (erule rev_mp, erule yahalom.induct, simp_all)
(*YM3, YM4*)
apply (blast dest!: B_Said_YM2)+
done

(*If A receives YM3 then B has used nonce NA (and therefore is alive)*)
lemma YM3_auth_B_to_A:
     "[| Gets A {|Crypt (shrK A) {|Agent B, Key K, Nonce NA, nb|}, X|}  
           \<in> set evs;                                                     
         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom |]                         
      ==> Says B Server {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, nb|}|}  
       \<in> set evs"
by (blast dest!: A_trusts_YM3 YM3_auth_B_to_A_lemma elim: knows_Spy_partsEs)


(*** Authenticating A to B using the certificate Crypt K (Nonce NB) ***)

(*Assuming the session key is secure, if both certificates are present then
  A has said NB.  We can't be sure about the rest of A's message, but only
  NB matters for freshness.*)  
lemma A_Said_YM3_lemma [rule_format]:
     "evs \<in> yahalom
      ==> Key K \<notin> analz (knows Spy evs) -->
          Crypt K (Nonce NB) \<in> parts (knows Spy evs) -->
          Crypt (shrK B) {|Agent A, Key K|} \<in> parts (knows Spy evs) -->
          B \<notin> bad -->
          (\<exists>X. Says A B {|X, Crypt K (Nonce NB)|} \<in> set evs)"
apply (erule yahalom.induct, force, 
       frule_tac [6] YM4_parts_knows_Spy)
apply (analz_mono_contra, simp_all)
(*Fake*)
apply blast
(*YM3: by new_keys_not_used we note that Crypt K (Nonce NB) could not exist*)
apply (force dest!: Crypt_imp_keysFor)
(*YM4: was Crypt K (Nonce NB) the very last message?  If not, use ind. hyp.*)
apply (simp add: ex_disj_distrib)
(*yes: apply unicity of session keys*)
apply (blast dest!: Gets_imp_Says A_trusts_YM3 B_trusts_YM4_shrK
                    Crypt_Spy_analz_bad 
             dest: Says_imp_knows_Spy [THEN parts.Inj] unique_session_keys)
done

(*If B receives YM4 then A has used nonce NB (and therefore is alive).
  Moreover, A associates K with NB (thus is talking about the same run).
  Other premises guarantee secrecy of K.*)
lemma YM4_imp_A_Said_YM3 [rule_format]:
     "[| Gets B {|Crypt (shrK B) {|Agent A, Key K|},
                  Crypt K (Nonce NB)|} \<in> set evs;
         Says B Server
           {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, Nonce NB|}|}
           \<in> set evs;
         (\<forall>NA k. Notes Spy {|Nonce NA, Nonce NB, k|} \<notin> set evs);
         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom |]
      ==> \<exists>X. Says A B {|X, Crypt K (Nonce NB)|} \<in> set evs"
by (blast intro!: A_Said_YM3_lemma 
          dest: Spy_not_see_encrypted_key B_trusts_YM4 Gets_imp_Says)

end