src/HOL/Auth/OtwayRees_AN.ML
author paulson
Tue Nov 05 11:20:52 1996 +0100 (1996-11-05)
changeset 2160 ad4382e546fc
parent 2131 3106a99d30a5
child 2166 d276a806cc10
permissions -rw-r--r--
Simplified new_keys_not_seen, etc.: replaced the
union over all
agents by the Spy alone. Proofs run faster and they do not have to be
set up in terms of a previous lemma.
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(*  Title:      HOL/Auth/OtwayRees
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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 "otway" for the Otway-Rees protocol.
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Simplified version with minimal encryption but explicit messages
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From page 11 of
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  Abadi and Needham.  Prudent Engineering Practice for Cryptographic Protocols.
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  IEEE Trans. SE 22 (1), 1996
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*)
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open OtwayRees_AN;
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proof_timing:=true;
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HOL_quantifiers := false;
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(*Weak liveness: there are traces that reach the end*)
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goal thy 
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 "!!A B. [| A ~= B; A ~= Server; B ~= Server |]   \
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\        ==> EX K. EX NA. EX evs: otway lost.          \
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\             Says B A (Crypt {|Nonce NA, Agent A, Agent B, Key K|} (shrK A)) \
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\             : set_of_list evs";
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by (REPEAT (resolve_tac [exI,bexI] 1));
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by (rtac (otway.Nil RS otway.OR1 RS otway.OR2 RS otway.OR3 RS otway.OR4) 2);
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by (ALLGOALS (simp_tac (!simpset setsolver safe_solver)));
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by (REPEAT_FIRST (resolve_tac [refl, conjI]));
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by (REPEAT_FIRST (fast_tac (!claset addss (!simpset setsolver safe_solver))));
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result();
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(**** Inductive proofs about otway ****)
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(*Monotonicity*)
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goal thy "!!evs. lost' <= lost ==> otway lost' <= otway lost";
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by (rtac subsetI 1);
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by (etac otway.induct 1);
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by (REPEAT_FIRST
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    (best_tac (!claset addIs (impOfSubs (sees_mono RS analz_mono RS synth_mono)
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                              :: otway.intrs))));
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qed "otway_mono";
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(*Nobody sends themselves messages*)
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goal thy "!!evs. evs : otway lost ==> ALL A X. Says A A X ~: set_of_list evs";
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by (etac otway.induct 1);
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by (Auto_tac());
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qed_spec_mp "not_Says_to_self";
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Addsimps [not_Says_to_self];
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AddSEs   [not_Says_to_self RSN (2, rev_notE)];
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(** For reasoning about the encrypted portion of messages **)
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goal thy "!!evs. Says S B {|X, X'|} : set_of_list evs ==> \
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\                X : analz (sees lost Spy evs)";
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by (fast_tac (!claset addSDs [Says_imp_sees_Spy RS analz.Inj]) 1);
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qed "OR4_analz_sees_Spy";
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goal thy "!!evs. Says Server B {|X, Crypt {|NB, a, Agent B, K|} K'|} \
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\                  : set_of_list evs ==> K : parts (sees lost Spy evs)";
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by (fast_tac (!claset addSEs partsEs
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                      addSDs [Says_imp_sees_Spy RS parts.Inj]) 1);
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qed "Oops_parts_sees_Spy";
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(*OR4_analz_sees_Spy lets us treat those cases using the same 
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  argument as for the Fake case.  This is possible for most, but not all,
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  proofs, since Fake messages originate from the Spy. *)
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bind_thm ("OR4_parts_sees_Spy",
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          OR4_analz_sees_Spy RS (impOfSubs analz_subset_parts));
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(*We instantiate the variable to "lost".  Leaving it as a Var makes proofs
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  harder to complete, since simplification does less for us.*)
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val parts_Fake_tac = 
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    forw_inst_tac [("lost","lost")] OR4_parts_sees_Spy 6 THEN
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    forw_inst_tac [("lost","lost")] Oops_parts_sees_Spy 7;
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(*For proving the easier theorems about X ~: parts (sees lost Spy evs) *)
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fun parts_induct_tac i = SELECT_GOAL
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    (DETERM (etac otway.induct 1 THEN parts_Fake_tac THEN
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	     (*Fake message*)
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	     TRY (best_tac (!claset addDs [impOfSubs analz_subset_parts,
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					   impOfSubs Fake_parts_insert]
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                                    addss (!simpset)) 2)) THEN
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     (*Base case*)
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     fast_tac (!claset addss (!simpset)) 1 THEN
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     ALLGOALS Asm_simp_tac) i;
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(** Theorems of the form X ~: parts (sees lost Spy evs) imply that NOBODY
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    sends messages containing X! **)
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(*Spy never sees another agent's shared key! (unless it's lost at start)*)
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goal thy 
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 "!!evs. evs : otway lost \
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\        ==> (Key (shrK A) : parts (sees lost Spy evs)) = (A : lost)";
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by (parts_induct_tac 1);
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by (Auto_tac());
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qed "Spy_see_shrK";
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Addsimps [Spy_see_shrK];
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goal thy 
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 "!!evs. evs : otway lost \
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\        ==> (Key (shrK A) : analz (sees lost Spy evs)) = (A : lost)";
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by (auto_tac(!claset addDs [impOfSubs analz_subset_parts], !simpset));
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qed "Spy_analz_shrK";
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Addsimps [Spy_analz_shrK];
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goal thy  "!!A. [| Key (shrK A) : parts (sees lost Spy evs);       \
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\                  evs : otway lost |] ==> A:lost";
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by (fast_tac (!claset addDs [Spy_see_shrK]) 1);
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qed "Spy_see_shrK_D";
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bind_thm ("Spy_analz_shrK_D", analz_subset_parts RS subsetD RS Spy_see_shrK_D);
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AddSDs [Spy_see_shrK_D, Spy_analz_shrK_D];
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(*** Future keys can't be seen or used! ***)
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(*Nobody can have SEEN keys that will be generated in the future. *)
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goal thy "!!evs. evs : otway lost ==> \
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\                length evs <= length evs' --> \
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\                Key (newK evs') ~: parts (sees lost Spy evs)";
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by (parts_induct_tac 1);
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by (REPEAT_FIRST (best_tac (!claset addDs [impOfSubs analz_subset_parts,
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                                           impOfSubs parts_insert_subset_Un,
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                                           Suc_leD]
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                                    addss (!simpset))));
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qed_spec_mp "new_keys_not_seen";
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Addsimps [new_keys_not_seen];
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(*Variant: old messages must contain old keys!*)
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goal thy 
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 "!!evs. [| Says A B X : set_of_list evs;  \
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\           Key (newK evt) : parts {X};    \
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\           evs : otway lost                 \
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\        |] ==> length evt < length evs";
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by (rtac ccontr 1);
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by (dtac leI 1);
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by (fast_tac (!claset addSDs [new_keys_not_seen, Says_imp_sees_Spy]
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                      addIs  [impOfSubs parts_mono]) 1);
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qed "Says_imp_old_keys";
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(*** Future nonces can't be seen or used! ***)
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goal thy "!!evs. evs : otway lost ==> \
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\                length evs <= length evt --> \
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\                Nonce (newN evt) ~: parts (sees lost Spy evs)";
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by (parts_induct_tac 1);
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by (REPEAT_FIRST (fast_tac (!claset 
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                              addSEs partsEs
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                              addSDs  [Says_imp_sees_Spy RS parts.Inj]
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                              addDs  [impOfSubs analz_subset_parts,
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                                      impOfSubs parts_insert_subset_Un,
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                                      Suc_leD]
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                              addss (!simpset))));
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qed_spec_mp "new_nonces_not_seen";
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Addsimps [new_nonces_not_seen];
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(*Nobody can have USED keys that will be generated in the future.
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  ...very like new_keys_not_seen*)
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goal thy "!!evs. evs : otway lost ==> \
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\                length evs <= length evs' --> \
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\                newK evs' ~: keysFor (parts (sees lost Spy evs))";
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by (parts_induct_tac 1);
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(*OR1 and OR3*)
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by (EVERY (map (fast_tac (!claset addDs [Suc_leD] addss (!simpset))) [4,2]));
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(*Fake, OR2, OR4: these messages send unknown (X) components*)
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by (REPEAT
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    (best_tac
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      (!claset addDs [impOfSubs (analz_subset_parts RS keysFor_mono),
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                      impOfSubs (parts_insert_subset_Un RS keysFor_mono),
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                      Suc_leD]
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               addEs [new_keys_not_seen RS not_parts_not_analz RSN(2,rev_notE)]
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               addss (!simpset)) 1));
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qed_spec_mp "new_keys_not_used";
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bind_thm ("new_keys_not_analzd",
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          [analz_subset_parts RS keysFor_mono,
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           new_keys_not_used] MRS contra_subsetD);
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Addsimps [new_keys_not_used, new_keys_not_analzd];
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(*** Proofs involving analz ***)
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(*Describes the form of K and NA when the Server sends this message.*)
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goal thy 
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 "!!evs. [| Says Server B \
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\           {|Crypt {|NA, Agent A, Agent B, K|} (shrK A),                    \
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\             Crypt {|NB, Agent A, Agent B, K|} (shrK B)|} : set_of_list evs; \
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\           evs : otway lost |]                                        \
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\        ==> (EX evt: otway lost. K = Key(newK evt)) &                  \
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\            (EX i. NA = Nonce i) &                  \
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\            (EX j. NB = Nonce j)";
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by (etac rev_mp 1);
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by (etac otway.induct 1);
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by (ALLGOALS (fast_tac (!claset addss (!simpset))));
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qed "Says_Server_message_form";
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(*For proofs involving analz.  We again instantiate the variable to "lost".*)
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val analz_Fake_tac = 
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    dres_inst_tac [("lost","lost")] OR4_analz_sees_Spy 6 THEN
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    forw_inst_tac [("lost","lost")] Says_Server_message_form 7 THEN
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    assume_tac 7 THEN Full_simp_tac 7 THEN
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    REPEAT ((eresolve_tac [bexE, exE, conjE] ORELSE' hyp_subst_tac) 7);
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(****
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 The following is to prove theorems of the form
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          Key K : analz (insert (Key (newK evt)) (sees lost Spy evs)) ==>
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          Key K : analz (sees lost 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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(*The equality makes the induction hypothesis easier to apply*)
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goal thy  
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 "!!evs. evs : otway lost ==> \
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\  ALL K E. (Key K : analz (Key``(newK``E) Un (sees lost Spy evs))) = \
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\           (K : newK``E | Key K : analz (sees lost Spy evs))";
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by (etac otway.induct 1);
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by analz_Fake_tac;
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by (REPEAT_FIRST (ares_tac [allI, analz_image_newK_lemma]));
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by (ALLGOALS (*Takes 28 secs*)
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    (asm_simp_tac 
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     (!simpset addsimps ([insert_Key_singleton, insert_Key_image, pushKey_newK]
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                         @ pushes)
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               setloop split_tac [expand_if])));
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(** LEVEL 5 **)
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(*OR4, Fake*) 
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by (EVERY (map spy_analz_tac [4,2]));
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(*Oops, OR3, Base*)
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by (REPEAT (fast_tac (!claset addIs [image_eqI] addss (!simpset)) 1));
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qed_spec_mp "analz_image_newK";
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goal thy
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 "!!evs. evs : otway lost ==>                               \
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\        Key K : analz (insert (Key (newK evt)) (sees lost Spy evs)) = \
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\        (K = newK evt | Key K : analz (sees lost Spy evs))";
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by (asm_simp_tac (HOL_ss addsimps [pushKey_newK, analz_image_newK, 
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                                   insert_Key_singleton]) 1);
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by (Fast_tac 1);
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qed "analz_insert_Key_newK";
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(*** The Key K uniquely identifies the Server's  message. **)
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goal thy 
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 "!!evs. evs : otway lost ==>                      \
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\      EX A' B' NA' NB'. ALL A B NA NB.                    \
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\       Says Server B \
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\         {|Crypt {|NA, Agent A, Agent B, K|} (shrK A),                    \
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\           Crypt {|NB, Agent A, Agent B, K|} (shrK B)|} : set_of_list evs  \
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\       --> A=A' & B=B' & NA=NA' & NB=NB'";
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by (etac otway.induct 1);
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by (ALLGOALS (asm_simp_tac (!simpset addsimps [all_conj_distrib])));
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by (Step_tac 1);
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(*Remaining cases: OR3 and OR4*)
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by (ex_strip_tac 2);
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by (Fast_tac 2);
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by (expand_case_tac "K = ?y" 1);
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by (REPEAT (ares_tac [refl,exI,impI,conjI] 2));
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(*...we assume X is a very new message, and handle this case by contradiction*)
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by (fast_tac (!claset addEs [Says_imp_old_keys RS less_irrefl]
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                      delrules [conjI]    (*prevent split-up into 4 subgoals*)
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                      addss (!simpset addsimps [parts_insertI])) 1);
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val lemma = result();
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goal thy 
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"!!evs. [| Says Server B                                           \
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\            {|Crypt {|NA, Agent A, Agent B, K|} (shrK A),         \
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\              Crypt {|NB, Agent A, Agent B, K|} (shrK B)|}        \
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\           : set_of_list evs;                                     \
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\          Says Server B'                                          \
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\            {|Crypt {|NA', Agent A', Agent B', K|} (shrK A'),     \
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\              Crypt {|NB', Agent A', Agent B', K|} (shrK B')|}    \
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\           : set_of_list evs;                                     \
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\          evs : otway lost |]                                     \
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\       ==> A=A' & B=B' & NA=NA' & NB=NB'";
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by (dtac lemma 1);
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by (REPEAT (etac exE 1));
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(*Duplicate the assumption*)
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by (forw_inst_tac [("psi", "ALL C.?P(C)")] asm_rl 1);
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by (fast_tac (!claset addSDs [spec]) 1);
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qed "unique_session_keys";
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(**** Authenticity properties relating to NA ****)
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(*If the encrypted message appears then it originated with the Server!*)
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goal thy 
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 "!!evs. [| A ~: lost;  evs : otway lost |]                 \
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\ ==> Crypt {|NA, Agent A, Agent B, Key K|} (shrK A)        \
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\      : parts (sees lost Spy evs)                          \
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\     --> (EX NB. Says Server B                                     \
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\                  {|Crypt {|NA, Agent A, Agent B, Key K|} (shrK A),     \
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\                    Crypt {|NB, Agent A, Agent B, Key K|} (shrK B)|}    \
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\                  : set_of_list evs)";
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by (parts_induct_tac 1);
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by (ALLGOALS (asm_simp_tac (!simpset addsimps [ex_disj_distrib])));
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(*OR3*)
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by (Fast_tac 1);
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qed_spec_mp "NA_Crypt_imp_Server_msg";
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(*Corollary: if A receives B's OR4 message and the nonce NA agrees
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  then the key really did come from the Server!  CANNOT prove this of the
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  bad form of this protocol, even though we can prove
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  Spy_not_see_encrypted_key.  (We can implicitly infer freshness of
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  the Server's message from its nonce NA.)*)
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goal thy 
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 "!!evs. [| Says B' A (Crypt {|NA, Agent A, Agent B, Key K|} (shrK A))  \
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\            : set_of_list evs;                                         \
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\           A ~: lost;  evs : otway lost |]                             \
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\        ==> EX NB. Says Server B                                       \
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\                    {|Crypt {|NA, Agent A, Agent B, Key K|} (shrK A),  \
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\                      Crypt {|NB, Agent A, Agent B, Key K|} (shrK B)|} \
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\                   : set_of_list evs";
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by (fast_tac (!claset addSIs [NA_Crypt_imp_Server_msg]
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                      addEs  partsEs
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                      addDs  [Says_imp_sees_Spy RS parts.Inj]) 1);
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qed "A_trust_OR4";
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(** Crucial secrecy property: Spy does not see the keys sent in msg OR3
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    Does not in itself guarantee security: an attack could violate 
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    the premises, e.g. by having A=Spy **)
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goal thy 
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 "!!evs. [| A ~: lost;  B ~: lost;  evs : otway lost;  evt : otway lost |] \
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\        ==> Says Server B                                                 \
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\             {|Crypt {|NA, Agent A, Agent B, Key K|} (shrK A),            \
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\               Crypt {|NB, Agent A, Agent B, Key K|} (shrK B)|}           \
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\            : set_of_list evs -->                                         \
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\            Says B Spy {|NA, NB, Key K|} ~: set_of_list evs -->           \
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\            Key K ~: analz (sees lost Spy evs)";
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by (etac otway.induct 1);
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by analz_Fake_tac;
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by (ALLGOALS
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    (asm_full_simp_tac 
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     (!simpset addsimps ([analz_subset_parts RS contra_subsetD,
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                          analz_insert_Key_newK] @ pushes)
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               setloop split_tac [expand_if])));
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(*OR3*)
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by (fast_tac (!claset addSIs [parts_insertI]
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                      addEs [Says_imp_old_keys RS less_irrefl]
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                      addss (!simpset addsimps [parts_insert2])) 2);
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(*OR4, Fake*) 
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by (REPEAT_FIRST (resolve_tac [conjI, impI] ORELSE' spy_analz_tac));
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(*Oops*) 
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by (fast_tac (!claset addDs [unique_session_keys] addss (!simpset)) 1);
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val lemma = result() RS mp RS mp RSN(2,rev_notE);
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goal thy 
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 "!!evs. [| Says Server B                                                     \
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\            {|Crypt {|NA, Agent A, Agent B, K|} (shrK A),                    \
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\              Crypt {|NB, Agent A, Agent B, K|} (shrK B)|} : set_of_list evs;\
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\           Says B Spy {|NA, NB, K|} ~: set_of_list evs;                      \
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\           A ~: lost;  B ~: lost;  evs : otway lost |]                       \
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\        ==> K ~: analz (sees lost Spy evs)";
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by (forward_tac [Says_Server_message_form] 1 THEN assume_tac 1);
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by (fast_tac (!claset addSEs [lemma]) 1);
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qed "Spy_not_see_encrypted_key";
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goal thy 
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 "!!evs. [| C ~: {A,B,Server};                                                \
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\           Says Server B                                                     \
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\            {|Crypt {|NA, Agent A, Agent B, K|} (shrK A),                    \
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\              Crypt {|NB, Agent A, Agent B, K|} (shrK B)|} : set_of_list evs;\
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\           Says B Spy {|NA, NB, K|} ~: set_of_list evs;                 \
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\           A ~: lost;  B ~: lost;  evs : otway lost |]                  \
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\        ==> K ~: analz (sees lost C evs)";
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by (rtac (subset_insertI RS sees_mono RS analz_mono RS contra_subsetD) 1);
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by (rtac (sees_lost_agent_subset_sees_Spy RS analz_mono RS contra_subsetD) 1);
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by (FIRSTGOAL (rtac Spy_not_see_encrypted_key));
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by (REPEAT_FIRST (fast_tac (!claset addIs [otway_mono RS subsetD])));
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qed "Agent_not_see_encrypted_key";
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(**** Authenticity properties relating to NB ****)
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(*If the encrypted message appears then it originated with the Server!*)
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goal thy 
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 "!!evs. [| B ~: lost;  evs : otway lost |]                                 \
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\    ==> Crypt {|NB, Agent A, Agent B, Key K|} (shrK B)                     \
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\         : parts (sees lost Spy evs)                                       \
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\        --> (EX NA. Says Server B                                          \
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\                     {|Crypt {|NA, Agent A, Agent B, Key K|} (shrK A),     \
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\                       Crypt {|NB, Agent A, Agent B, Key K|} (shrK B)|}    \
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\                     : set_of_list evs)";
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by (parts_induct_tac 1);
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   408
by (ALLGOALS (asm_simp_tac (!simpset addsimps [ex_disj_distrib])));
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(*OR3*)
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by (Fast_tac 1);
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qed_spec_mp "NB_Crypt_imp_Server_msg";
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(*Guarantee for B: if it gets a message with matching NB then the Server
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  has sent the correct message.*)
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goal thy 
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 "!!evs. [| B ~: lost;  evs : otway lost;                                   \
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\           Says S B {|X, Crypt {|NB, Agent A, Agent B, Key K|} (shrK B)|}  \
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\            : set_of_list evs |]                                           \
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\        ==> EX NA. Says Server B                                           \
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\                     {|Crypt {|NA, Agent A, Agent B, Key K|} (shrK A),     \
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\                       Crypt {|NB, Agent A, Agent B, Key K|} (shrK B)|}    \
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\                     : set_of_list evs";
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by (fast_tac (!claset addSIs [NB_Crypt_imp_Server_msg]
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                      addEs  partsEs
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                      addDs  [Says_imp_sees_Spy RS parts.Inj]) 1);
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qed "B_trust_OR3";