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(* Title: HOL/Auth/NS_Public
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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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Inductive relation "ns_public" for the Needham-Schroeder Public-Key protocol.
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Version incorporating Lowe's fix (inclusion of B's identity in round 2).
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*)
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theory NS_Public = Public:
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consts ns_public :: "event list set"
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inductive ns_public
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intros
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(*Initial trace is empty*)
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Nil: "[] \<in> ns_public"
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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: "\<lbrakk>evs \<in> ns_public; X \<in> synth (analz (knows Spy evs))\<rbrakk>
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\<Longrightarrow> Says Spy B X # evs \<in> ns_public"
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(*Alice initiates a protocol run, sending a nonce to Bob*)
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NS1: "\<lbrakk>evs1 \<in> ns_public; Nonce NA \<notin> used evs1\<rbrakk>
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\<Longrightarrow> Says A B (Crypt (pubK B) \<lbrace>Nonce NA, Agent A\<rbrace>)
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# evs1 \<in> ns_public"
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(*Bob responds to Alice's message with a further nonce*)
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NS2: "\<lbrakk>evs2 \<in> ns_public; Nonce NB \<notin> used evs2;
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Says A' B (Crypt (pubK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs2\<rbrakk>
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\<Longrightarrow> Says B A (Crypt (pubK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>)
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# evs2 \<in> ns_public"
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(*Alice proves her existence by sending NB back to Bob.*)
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NS3: "\<lbrakk>evs3 \<in> ns_public;
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Says A B (Crypt (pubK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs3;
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Says B' A (Crypt (pubK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>)
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\<in> set evs3\<rbrakk>
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\<Longrightarrow> Says A B (Crypt (pubK B) (Nonce NB)) # evs3 \<in> ns_public"
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declare knows_Spy_partsEs [elim]
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declare analz_subset_parts [THEN subsetD, dest]
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declare Fake_parts_insert [THEN subsetD, dest]
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declare image_eq_UN [simp] (*accelerates proofs involving nested images*)
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(*A "possibility property": there are traces that reach the end*)
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lemma "\<exists>NB. \<exists>evs \<in> ns_public. Says A B (Crypt (pubK B) (Nonce NB)) \<in> set evs"
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apply (intro exI bexI)
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apply (rule_tac [2] ns_public.Nil [THEN ns_public.NS1, THEN ns_public.NS2,
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THEN ns_public.NS3])
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by possibility
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(**** Inductive proofs about ns_public ****)
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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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(*Spy never sees another agent's private key! (unless it's bad at start)*)
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lemma Spy_see_priK [simp]:
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"evs \<in> ns_public \<Longrightarrow> (Key (priK A) \<in> parts (knows Spy evs)) = (A \<in> bad)"
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by (erule ns_public.induct, auto)
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lemma Spy_analz_priK [simp]:
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"evs \<in> ns_public \<Longrightarrow> (Key (priK A) \<in> analz (knows Spy evs)) = (A \<in> bad)"
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by auto
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(*** Authenticity properties obtained from NS2 ***)
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(*It is impossible to re-use a nonce in both NS1 and NS2, provided the nonce
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is secret. (Honest users generate fresh nonces.)*)
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lemma no_nonce_NS1_NS2:
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"\<lbrakk>Crypt (pubK C) \<lbrace>NA', Nonce NA, Agent D\<rbrace> \<in> parts (knows Spy evs);
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Crypt (pubK B) \<lbrace>Nonce NA, Agent A\<rbrace> \<in> parts (knows Spy evs);
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evs \<in> ns_public\<rbrakk>
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\<Longrightarrow> Nonce NA \<in> analz (knows Spy evs)"
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apply (erule rev_mp, erule rev_mp)
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apply (erule ns_public.induct, simp_all)
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apply (blast intro: analz_insertI)+
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done
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(*Unicity for NS1: nonce NA identifies agents A and B*)
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lemma unique_NA:
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"\<lbrakk>Crypt(pubK B) \<lbrace>Nonce NA, Agent A \<rbrace> \<in> parts(knows Spy evs);
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Crypt(pubK B') \<lbrace>Nonce NA, Agent A'\<rbrace> \<in> parts(knows Spy evs);
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Nonce NA \<notin> analz (knows Spy evs); evs \<in> ns_public\<rbrakk>
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\<Longrightarrow> A=A' \<and> B=B'"
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apply (erule rev_mp, erule rev_mp, erule rev_mp)
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apply (erule ns_public.induct, simp_all)
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(*Fake, NS1*)
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apply (blast intro: analz_insertI)+
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done
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(*Secrecy: Spy does not see the nonce sent in msg NS1 if A and B are secure
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The major premise "Says A B ..." makes it a dest-rule, so we use
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(erule rev_mp) rather than rule_format. *)
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theorem Spy_not_see_NA:
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"\<lbrakk>Says A B (Crypt(pubK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs;
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A \<notin> bad; B \<notin> bad; evs \<in> ns_public\<rbrakk>
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\<Longrightarrow> Nonce NA \<notin> analz (knows Spy evs)"
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apply (erule rev_mp)
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apply (erule ns_public.induct, simp_all)
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apply spy_analz
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apply (blast dest: unique_NA intro: no_nonce_NS1_NS2)+
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done
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(*Authentication for A: if she receives message 2 and has used NA
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to start a run, then B has sent message 2.*)
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lemma A_trusts_NS2_lemma [rule_format]:
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"\<lbrakk>A \<notin> bad; B \<notin> bad; evs \<in> ns_public\<rbrakk>
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\<Longrightarrow> Crypt (pubK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace> \<in> parts (knows Spy evs) \<longrightarrow>
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Says A B (Crypt(pubK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs \<longrightarrow>
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Says B A (Crypt(pubK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) \<in> set evs"
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apply (erule ns_public.induct, simp_all)
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(*Fake, NS1*)
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apply (blast dest: Spy_not_see_NA)+
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done
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theorem A_trusts_NS2:
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"\<lbrakk>Says A B (Crypt(pubK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs;
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Says B' A (Crypt(pubK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) \<in> set evs;
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A \<notin> bad; B \<notin> bad; evs \<in> ns_public\<rbrakk>
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\<Longrightarrow> Says B A (Crypt(pubK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) \<in> set evs"
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by (blast intro: A_trusts_NS2_lemma)
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(*If the encrypted message appears then it originated with Alice in NS1*)
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lemma B_trusts_NS1 [rule_format]:
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"evs \<in> ns_public
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\<Longrightarrow> Crypt (pubK B) \<lbrace>Nonce NA, Agent A\<rbrace> \<in> parts (knows Spy evs) \<longrightarrow>
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Nonce NA \<notin> analz (knows Spy evs) \<longrightarrow>
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Says A B (Crypt (pubK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs"
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apply (erule ns_public.induct, simp_all)
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(*Fake*)
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apply (blast intro!: analz_insertI)
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done
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(*** Authenticity properties obtained from NS2 ***)
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(*Unicity for NS2: nonce NB identifies nonce NA and agents A, B
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[unicity of B makes Lowe's fix work]
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[proof closely follows that for unique_NA] *)
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lemma unique_NB [dest]:
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"\<lbrakk>Crypt(pubK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace> \<in> parts(knows Spy evs);
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Crypt(pubK A') \<lbrace>Nonce NA', Nonce NB, Agent B'\<rbrace> \<in> parts(knows Spy evs);
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Nonce NB \<notin> analz (knows Spy evs); evs \<in> ns_public\<rbrakk>
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\<Longrightarrow> A=A' \<and> NA=NA' \<and> B=B'"
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apply (erule rev_mp, erule rev_mp, erule rev_mp)
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apply (erule ns_public.induct, simp_all)
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(*Fake, NS2*)
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apply (blast intro: analz_insertI)+
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done
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text{*
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@{thm[display] analz_Crypt_if[no_vars]}
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\rulename{analz_Crypt_if}
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*}
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(*Secrecy: Spy does not see the nonce sent in msg NS2 if A and B are secure*)
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theorem Spy_not_see_NB [dest]:
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"\<lbrakk>Says B A (Crypt (pubK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) \<in> set evs;
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A \<notin> bad; B \<notin> bad; evs \<in> ns_public\<rbrakk>
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\<Longrightarrow> Nonce NB \<notin> analz (knows Spy evs)"
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apply (erule rev_mp)
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apply (erule ns_public.induct, simp_all)
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apply spy_analz
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apply (blast intro: no_nonce_NS1_NS2)+
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done
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(*Authentication for B: if he receives message 3 and has used NB
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in message 2, then A has sent message 3.*)
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lemma B_trusts_NS3_lemma [rule_format]:
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"\<lbrakk>A \<notin> bad; B \<notin> bad; evs \<in> ns_public\<rbrakk> \<Longrightarrow>
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Crypt (pubK B) (Nonce NB) \<in> parts (knows Spy evs) \<longrightarrow>
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Says B A (Crypt (pubK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) \<in> set evs \<longrightarrow>
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Says A B (Crypt (pubK B) (Nonce NB)) \<in> set evs"
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by (erule ns_public.induct, auto)
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theorem B_trusts_NS3:
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"\<lbrakk>Says B A (Crypt (pubK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) \<in> set evs;
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Says A' B (Crypt (pubK B) (Nonce NB)) \<in> set evs;
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A \<notin> bad; B \<notin> bad; evs \<in> ns_public\<rbrakk>
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\<Longrightarrow> Says A B (Crypt (pubK B) (Nonce NB)) \<in> set evs"
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by (blast intro: B_trusts_NS3_lemma)
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(*** Overall guarantee for B ***)
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(*If NS3 has been sent and the nonce NB agrees with the nonce B joined with
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NA, then A initiated the run using NA.*)
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theorem B_trusts_protocol:
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"\<lbrakk>A \<notin> bad; B \<notin> bad; evs \<in> ns_public\<rbrakk> \<Longrightarrow>
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Crypt (pubK B) (Nonce NB) \<in> parts (knows Spy evs) \<longrightarrow>
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Says B A (Crypt (pubK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) \<in> set evs \<longrightarrow>
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Says A B (Crypt (pubK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs"
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by (erule ns_public.induct, auto)
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end
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