src/HOL/Auth/Yahalom2.thy
author paulson
Wed Apr 09 12:52:45 2003 +0200 (2003-04-09)
changeset 13907 2bc462b99e70
parent 13507 febb8e5d2a9d
child 13926 6e62e5357a10
permissions -rw-r--r--
tidying
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(*  Title:      HOL/Auth/Yahalom2
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1996  University of Cambridge
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This version trades encryption of NB for additional explicitness in YM3.
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Also in YM3, care is taken to make the two certificates distinct.
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From page 259 of
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  Burrows, Abadi and Needham.  A Logic of Authentication.
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  Proc. Royal Soc. 426 (1989)
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*)
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header{*Inductive Analysis of the Yahalom protocol, Variant 2*}
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theory Yahalom2 = Shared:
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consts  yahalom   :: "event list set"
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inductive "yahalom"
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  intros
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         (*Initial trace is empty*)
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   Nil:  "[] \<in> yahalom"
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         (*The spy MAY say anything he CAN say.  We do not expect him to
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           invent new nonces here, but he can also use NS1.  Common to
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           all similar protocols.*)
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   Fake: "[| evsf \<in> yahalom;  X \<in> synth (analz (knows Spy evsf)) |]
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          ==> Says Spy B X  # evsf \<in> yahalom"
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         (*A message that has been sent can be received by the
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           intended recipient.*)
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   Reception: "[| evsr \<in> yahalom;  Says A B X \<in> set evsr |]
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               ==> Gets B X # evsr \<in> yahalom"
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         (*Alice initiates a protocol run*)
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   YM1:  "[| evs1 \<in> yahalom;  Nonce NA \<notin> used evs1 |]
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          ==> Says A B {|Agent A, Nonce NA|} # evs1 \<in> yahalom"
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         (*Bob's response to Alice's message.*)
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   YM2:  "[| evs2 \<in> yahalom;  Nonce NB \<notin> used evs2;
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             Gets B {|Agent A, Nonce NA|} \<in> set evs2 |]
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          ==> Says B Server
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                  {|Agent B, Nonce NB, Crypt (shrK B) {|Agent A, Nonce NA|}|}
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                # evs2 \<in> yahalom"
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         (*The Server receives Bob's message.  He responds by sending a
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           new session key to Alice, with a certificate for forwarding to Bob.
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           Both agents are quoted in the 2nd certificate to prevent attacks!*)
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   YM3:  "[| evs3 \<in> yahalom;  Key KAB \<notin> used evs3;
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             Gets Server {|Agent B, Nonce NB,
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			   Crypt (shrK B) {|Agent A, Nonce NA|}|}
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               \<in> set evs3 |]
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          ==> Says Server A
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               {|Nonce NB,
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                 Crypt (shrK A) {|Agent B, Key KAB, Nonce NA|},
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                 Crypt (shrK B) {|Agent A, Agent B, Key KAB, Nonce NB|}|}
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                 # evs3 \<in> yahalom"
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         (*Alice receives the Server's (?) message, checks her Nonce, and
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           uses the new session key to send Bob his Nonce.*)
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   YM4:  "[| evs4 \<in> yahalom;
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             Gets A {|Nonce NB, Crypt (shrK A) {|Agent B, Key K, Nonce NA|},
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                      X|}  \<in> set evs4;
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             Says A B {|Agent A, Nonce NA|} \<in> set evs4 |]
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          ==> Says A B {|X, Crypt K (Nonce NB)|} # evs4 \<in> yahalom"
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         (*This message models possible leaks of session keys.  The nonces
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           identify the protocol run.  Quoting Server here ensures they are
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           correct. *)
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   Oops: "[| evso \<in> yahalom;
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             Says Server A {|Nonce NB,
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                             Crypt (shrK A) {|Agent B, Key K, Nonce NA|},
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                             X|}  \<in> set evso |]
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          ==> Notes Spy {|Nonce NA, Nonce NB, Key K|} # evso \<in> yahalom"
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declare Says_imp_knows_Spy [THEN analz.Inj, dest]
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declare parts.Body  [dest]
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declare Fake_parts_insert_in_Un  [dest]
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declare analz_into_parts [dest]
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text{*A "possibility property": there are traces that reach the end*}
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lemma "\<exists>X NB K. \<exists>evs \<in> yahalom.
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             Says A B {|X, Crypt K (Nonce NB)|} \<in> set evs"
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apply (intro exI bexI)
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apply (rule_tac [2] yahalom.Nil
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                    [THEN yahalom.YM1, THEN yahalom.Reception,
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                     THEN yahalom.YM2, THEN yahalom.Reception,
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                     THEN yahalom.YM3, THEN yahalom.Reception,
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                     THEN yahalom.YM4], possibility)
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done
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lemma Gets_imp_Says:
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     "[| Gets B X \<in> set evs; evs \<in> yahalom |] ==> \<exists>A. Says A B X \<in> set evs"
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by (erule rev_mp, erule yahalom.induct, auto)
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text{*Must be proved separately for each protocol*}
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lemma Gets_imp_knows_Spy:
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     "[| Gets B X \<in> set evs; evs \<in> yahalom |]  ==> X \<in> knows Spy evs"
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by (blast dest!: Gets_imp_Says Says_imp_knows_Spy)
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declare Gets_imp_knows_Spy [THEN analz.Inj, dest]
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subsection{*Inductive Proofs*}
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text{*Result for reasoning about the encrypted portion of messages.
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Lets us treat YM4 using a similar argument as for the Fake case.*}
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lemma YM4_analz_knows_Spy:
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     "[| Gets A {|NB, Crypt (shrK A) Y, X|} \<in> set evs;  evs \<in> yahalom |]
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      ==> X \<in> analz (knows Spy evs)"
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by blast
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lemmas YM4_parts_knows_Spy =
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       YM4_analz_knows_Spy [THEN analz_into_parts, standard]
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(** Theorems of the form X \<notin> parts (knows Spy evs) imply that NOBODY
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    sends messages containing X! **)
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text{*Spy never sees a good agent's shared key!*}
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lemma Spy_see_shrK [simp]:
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     "evs \<in> yahalom ==> (Key (shrK A) \<in> parts (knows Spy evs)) = (A \<in> bad)"
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by (erule yahalom.induct, force,
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    drule_tac [6] YM4_parts_knows_Spy, simp_all, blast+)
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lemma Spy_analz_shrK [simp]:
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     "evs \<in> yahalom ==> (Key (shrK A) \<in> analz (knows Spy evs)) = (A \<in> bad)"
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by auto
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lemma Spy_see_shrK_D [dest!]:
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     "[|Key (shrK A) \<in> parts (knows Spy evs);  evs \<in> yahalom|] ==> A \<in> bad"
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by (blast dest: Spy_see_shrK)
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(*Nobody can have used non-existent keys!  Needed to apply analz_insert_Key*)
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lemma new_keys_not_used [rule_format, simp]:
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 "evs \<in> yahalom ==> Key K \<notin> used evs --> K \<notin> keysFor (parts (knows Spy evs))"
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apply (erule yahalom.induct, force,
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       frule_tac [6] YM4_parts_knows_Spy, simp_all)
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(*Fake, YM3, YM4*)
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apply (blast dest!: keysFor_parts_insert)+
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done
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(*Describes the form of K when the Server sends this message.  Useful for
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  Oops as well as main secrecy property.*)
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lemma Says_Server_message_form:
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     "[| Says Server A {|nb', Crypt (shrK A) {|Agent B, Key K, na|}, X|}
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          \<in> set evs;  evs \<in> yahalom |]
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      ==> K \<notin> range shrK"
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by (erule rev_mp, erule yahalom.induct, simp_all)
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(****
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 The following is to prove theorems of the form
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          Key K \<in> analz (insert (Key KAB) (knows Spy evs)) ==>
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          Key K \<in> analz (knows Spy evs)
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 A more general formula must be proved inductively.
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****)
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(** Session keys are not used to encrypt other session keys **)
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lemma analz_image_freshK [rule_format]:
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 "evs \<in> yahalom ==>
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   \<forall>K KK. KK <= - (range shrK) -->
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          (Key K \<in> analz (Key`KK Un (knows Spy evs))) =
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          (K \<in> KK | Key K \<in> analz (knows Spy evs))"
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apply (erule yahalom.induct, force, frule_tac [7] Says_Server_message_form,
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       drule_tac [6] YM4_analz_knows_Spy, analz_freshK, spy_analz)
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done
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lemma analz_insert_freshK:
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     "[| evs \<in> yahalom;  KAB \<notin> range shrK |] ==>
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      (Key K \<in> analz (insert (Key KAB) (knows Spy evs))) =
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      (K = KAB | Key K \<in> analz (knows Spy evs))"
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by (simp only: analz_image_freshK analz_image_freshK_simps)
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text{*The Key K uniquely identifies the Server's  message*}
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lemma unique_session_keys:
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     "[| Says Server A
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          {|nb, Crypt (shrK A) {|Agent B, Key K, na|}, X|} \<in> set evs;
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        Says Server A'
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          {|nb', Crypt (shrK A') {|Agent B', Key K, na'|}, X'|} \<in> set evs;
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        evs \<in> yahalom |]
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     ==> A=A' & B=B' & na=na' & nb=nb'"
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apply (erule rev_mp, erule rev_mp)
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apply (erule yahalom.induct, simp_all)
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txt{*YM3, by freshness*}
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apply blast
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done
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subsection{*Crucial Secrecy Property: Spy Does Not See Key @{term KAB}*}
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lemma secrecy_lemma:
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     "[| A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom |]
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      ==> Says Server A
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            {|nb, Crypt (shrK A) {|Agent B, Key K, na|},
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                  Crypt (shrK B) {|Agent A, Agent B, Key K, nb|}|}
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           \<in> set evs -->
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          Notes Spy {|na, nb, Key K|} \<notin> set evs -->
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          Key K \<notin> analz (knows Spy evs)"
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apply (erule yahalom.induct, force, frule_tac [7] Says_Server_message_form,
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       drule_tac [6] YM4_analz_knows_Spy)
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apply (simp_all add: pushes analz_insert_eq analz_insert_freshK, spy_analz)
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apply (blast dest: unique_session_keys)+  (*YM3, Oops*)
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done
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(*Final version*)
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lemma Spy_not_see_encrypted_key:
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     "[| Says Server A
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            {|nb, Crypt (shrK A) {|Agent B, Key K, na|},
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                  Crypt (shrK B) {|Agent A, Agent B, Key K, nb|}|}
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         \<in> set evs;
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         Notes Spy {|na, nb, Key K|} \<notin> set evs;
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         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom |]
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      ==> Key K \<notin> analz (knows Spy evs)"
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by (blast dest: secrecy_lemma Says_Server_message_form)
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text{*This form is an immediate consequence of the previous result.  It is 
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similar to the assertions established by other methods.  It is equivalent
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to the previous result in that the Spy already has @{term analz} and
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@{term synth} at his disposal.  However, the conclusion 
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@{term "Key K \<notin> knows Spy evs"} appears not to be inductive: all the cases
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other than Fake are trivial, while Fake requires 
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@{term "Key K \<notin> analz (knows Spy evs)"}. *}
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lemma Spy_not_know_encrypted_key:
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     "[| Says Server A
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            {|nb, Crypt (shrK A) {|Agent B, Key K, na|},
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                  Crypt (shrK B) {|Agent A, Agent B, Key K, nb|}|}
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         \<in> set evs;
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         Notes Spy {|na, nb, Key K|} \<notin> set evs;
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         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom |]
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      ==> Key K \<notin> knows Spy evs"
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by (blast dest: Spy_not_see_encrypted_key)
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subsection{*Security Guarantee for A upon receiving YM3*}
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(*If the encrypted message appears then it originated with the Server.
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  May now apply Spy_not_see_encrypted_key, subject to its conditions.*)
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lemma A_trusts_YM3:
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     "[| Crypt (shrK A) {|Agent B, Key K, na|} \<in> parts (knows Spy evs);
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         A \<notin> bad;  evs \<in> yahalom |]
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      ==> \<exists>nb. Says Server A
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                    {|nb, Crypt (shrK A) {|Agent B, Key K, na|},
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                          Crypt (shrK B) {|Agent A, Agent B, Key K, nb|}|}
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                  \<in> set evs"
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apply (erule rev_mp)
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apply (erule yahalom.induct, force,
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       frule_tac [6] YM4_parts_knows_Spy, simp_all)
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txt{*Fake, YM3*}
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apply blast+
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done
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(*The obvious combination of A_trusts_YM3 with Spy_not_see_encrypted_key*)
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theorem A_gets_good_key:
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     "[| Crypt (shrK A) {|Agent B, Key K, na|} \<in> parts (knows Spy evs);
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         \<forall>nb. Notes Spy {|na, nb, Key K|} \<notin> set evs;
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         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom |]
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      ==> Key K \<notin> analz (knows Spy evs)"
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by (blast dest!: A_trusts_YM3 Spy_not_see_encrypted_key)
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subsection{*Security Guarantee for B upon receiving YM4*}
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(*B knows, by the first part of A's message, that the Server distributed
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  the key for A and B, and has associated it with NB.*)
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lemma B_trusts_YM4_shrK:
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     "[| Crypt (shrK B) {|Agent A, Agent B, Key K, Nonce NB|}
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           \<in> parts (knows Spy evs);
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         B \<notin> bad;  evs \<in> yahalom |]
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  ==> \<exists>NA. Says Server A
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             {|Nonce NB,
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               Crypt (shrK A) {|Agent B, Key K, Nonce NA|},
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               Crypt (shrK B) {|Agent A, Agent B, Key K, Nonce NB|}|}
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             \<in> set evs"
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apply (erule rev_mp)
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apply (erule yahalom.induct, force,
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       frule_tac [6] YM4_parts_knows_Spy, simp_all)
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(*Fake, YM3*)
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apply blast+
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done
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(*With this protocol variant, we don't need the 2nd part of YM4 at all:
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  Nonce NB is available in the first part.*)
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(*What can B deduce from receipt of YM4?  Stronger and simpler than Yahalom
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  because we do not have to show that NB is secret. *)
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lemma B_trusts_YM4:
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     "[| Gets B {|Crypt (shrK B) {|Agent A, Agent B, Key K, Nonce NB|},  X|}
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           \<in> set evs;
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         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom |]
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  ==> \<exists>NA. Says Server A
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             {|Nonce NB,
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               Crypt (shrK A) {|Agent B, Key K, Nonce NA|},
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               Crypt (shrK B) {|Agent A, Agent B, Key K, Nonce NB|}|}
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            \<in> set evs"
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by (blast dest!: B_trusts_YM4_shrK)
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(*The obvious combination of B_trusts_YM4 with Spy_not_see_encrypted_key*)
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theorem B_gets_good_key:
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     "[| Gets B {|Crypt (shrK B) {|Agent A, Agent B, Key K, Nonce NB|}, X|}
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           \<in> set evs;
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         \<forall>na. Notes Spy {|na, Nonce NB, Key K|} \<notin> set evs;
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         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom |]
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      ==> Key K \<notin> analz (knows Spy evs)"
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by (blast dest!: B_trusts_YM4 Spy_not_see_encrypted_key)
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subsection{*Authenticating B to A*}
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(*The encryption in message YM2 tells us it cannot be faked.*)
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lemma B_Said_YM2:
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     "[| Crypt (shrK B) {|Agent A, Nonce NA|} \<in> parts (knows Spy evs);
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         B \<notin> bad;  evs \<in> yahalom |]
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      ==> \<exists>NB. Says B Server {|Agent B, Nonce NB,
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                               Crypt (shrK B) {|Agent A, Nonce NA|}|}
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                      \<in> set evs"
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apply (erule rev_mp)
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apply (erule yahalom.induct, force,
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       frule_tac [6] YM4_parts_knows_Spy, simp_all)
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(*Fake, YM2*)
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apply blast+
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done
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(*If the server sends YM3 then B sent YM2, perhaps with a different NB*)
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lemma YM3_auth_B_to_A_lemma:
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     "[| Says Server A {|nb, Crypt (shrK A) {|Agent B, Key K, Nonce NA|}, X|}
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           \<in> set evs;
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         B \<notin> bad;  evs \<in> yahalom |]
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      ==> \<exists>nb'. Says B Server {|Agent B, nb',
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                                   Crypt (shrK B) {|Agent A, Nonce NA|}|}
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                       \<in> set evs"
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apply (erule rev_mp)
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apply (erule yahalom.induct, simp_all)
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(*Fake, YM2, YM3*)
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apply (blast dest!: B_Said_YM2)+
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done
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text{*If A receives YM3 then B has used nonce NA (and therefore is alive)*}
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theorem YM3_auth_B_to_A:
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     "[| Gets A {|nb, Crypt (shrK A) {|Agent B, Key K, Nonce NA|}, X|}
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           \<in> set evs;
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         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom |]
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 ==> \<exists>nb'. Says B Server
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                  {|Agent B, nb', Crypt (shrK B) {|Agent A, Nonce NA|}|}
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               \<in> set evs"
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by (blast dest!: A_trusts_YM3 YM3_auth_B_to_A_lemma)
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subsection{*Authenticating A to B*}
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text{*using the certificate @{term "Crypt K (Nonce NB)"}*}
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(*Assuming the session key is secure, if both certificates are present then
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  A has said NB.  We can't be sure about the rest of A's message, but only
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  NB matters for freshness.  Note that  Key K \<notin> analz (knows Spy evs)  must be
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  the FIRST antecedent of the induction formula.*)
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(*This lemma allows a use of unique_session_keys in the next proof,
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  which otherwise is extremely slow.*)
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lemma secure_unique_session_keys:
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     "[| Crypt (shrK A) {|Agent B, Key K, na|} \<in> analz (spies evs);
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         Crypt (shrK A') {|Agent B', Key K, na'|} \<in> analz (spies evs);
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         Key K \<notin> analz (knows Spy evs);  evs \<in> yahalom |]
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     ==> A=A' & B=B'"
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by (blast dest!: A_trusts_YM3 dest: unique_session_keys Crypt_Spy_analz_bad)
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lemma Auth_A_to_B_lemma [rule_format]:
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     "evs \<in> yahalom
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      ==> Key K \<notin> analz (knows Spy evs) -->
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          Crypt K (Nonce NB) \<in> parts (knows Spy evs) -->
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          Crypt (shrK B) {|Agent A, Agent B, Key K, Nonce NB|}
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            \<in> parts (knows Spy evs) -->
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          B \<notin> bad -->
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          (\<exists>X. Says A B {|X, Crypt K (Nonce NB)|} \<in> set evs)"
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apply (erule yahalom.induct, force,
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       frule_tac [6] YM4_parts_knows_Spy)
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apply (analz_mono_contra, simp_all)
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(*Fake*)
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apply blast
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(*YM3: by new_keys_not_used we note that Crypt K (Nonce NB) could not exist*)
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apply (force dest!: Crypt_imp_keysFor)
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(*YM4: was Crypt K (Nonce NB) the very last message?  If so, apply unicity
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  of session keys; if not, use ind. hyp.*)
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apply (blast dest!: B_trusts_YM4_shrK dest: secure_unique_session_keys  )
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done
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text{*If B receives YM4 then A has used nonce NB (and therefore is alive).
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  Moreover, A associates K with NB (thus is talking about the same run).
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  Other premises guarantee secrecy of K.*}
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theorem YM4_imp_A_Said_YM3 [rule_format]:
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     "[| Gets B {|Crypt (shrK B) {|Agent A, Agent B, Key K, Nonce NB|},
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                  Crypt K (Nonce NB)|} \<in> set evs;
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         (\<forall>NA. Notes Spy {|Nonce NA, Nonce NB, Key K|} \<notin> set evs);
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         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom |]
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      ==> \<exists>X. Says A B {|X, Crypt K (Nonce NB)|} \<in> set evs"
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by (blast intro: Auth_A_to_B_lemma
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          dest: Spy_not_see_encrypted_key B_trusts_YM4_shrK)
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   412
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end