(* Title: HOL/Auth/Recur
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1996 University of Cambridge
Inductive relation "recur" for the Recursive Authentication protocol.
*)
open Recur;
proof_timing:=true;
HOL_quantifiers := false;
Pretty.setdepth 30;
(** Possibility properties: traces that reach the end
ONE theorem would be more elegant and faster!
By induction on a list of agents (no repetitions)
**)
(*Simplest case: Alice goes directly to the server*)
goal thy
"!!A. A ~= Server \
\ ==> EX K NA. EX evs: recur lost. \
\ Says Server A {|Crypt (shrK A) {|Key K, Agent Server, Nonce NA|}, \
\ Agent Server|} \
\ : set_of_list evs";
by (REPEAT (resolve_tac [exI,bexI] 1));
by (rtac (recur.Nil RS recur.RA1 RS
(respond.One RSN (4,recur.RA3))) 2);
by possibility_tac;
result();
(*Case two: Alice, Bob and the server*)
goal thy
"!!A B. [| A ~= B; A ~= Server; B ~= Server |] \
\ ==> EX K. EX NA. EX evs: recur lost. \
\ Says B A {|Crypt (shrK A) {|Key K, Agent B, Nonce NA|}, \
\ Agent Server|} \
\ : set_of_list evs";
by (cut_facts_tac [Nonce_supply2, Key_supply2] 1);
by (REPEAT (eresolve_tac [exE, conjE] 1));
by (REPEAT (resolve_tac [exI,bexI] 1));
by (rtac (recur.Nil RS recur.RA1 RS recur.RA2 RS
(respond.One RS respond.Cons RSN (4,recur.RA3)) RS
recur.RA4) 2);
by basic_possibility_tac;
by (DEPTH_SOLVE (eresolve_tac [asm_rl, less_not_refl2,
less_not_refl2 RS not_sym] 1));
result();
(*Case three: Alice, Bob, Charlie and the server
goal thy
"!!A B. [| A ~= B; B ~= C; A ~= Server; B ~= Server; C ~= Server |] \
\ ==> EX K. EX NA. EX evs: recur lost. \
\ Says B A {|Crypt (shrK A) {|Key K, Agent B, Nonce NA|}, \
\ Agent Server|} \
\ : set_of_list evs";
by (cut_facts_tac [Nonce_supply3, Key_supply3] 1);
by (REPEAT (eresolve_tac [exE, conjE] 1));
by (REPEAT (resolve_tac [exI,bexI] 1));
by (rtac (recur.Nil RS recur.RA1 RS recur.RA2 RS recur.RA2 RS
(respond.One RS respond.Cons RS respond.Cons RSN
(4,recur.RA3)) RS recur.RA4 RS recur.RA4) 2);
(*SLOW: 70 seconds*)
by basic_possibility_tac;
by (DEPTH_SOLVE (swap_res_tac [refl, conjI, disjCI] 1
ORELSE
eresolve_tac [asm_rl, less_not_refl2,
less_not_refl2 RS not_sym] 1));
result();
****************)
(**** Inductive proofs about recur ****)
(*Monotonicity*)
goal thy "!!evs. lost' <= lost ==> recur lost' <= recur lost";
by (rtac subsetI 1);
by (etac recur.induct 1);
by (REPEAT_FIRST
(best_tac (!claset addIs (impOfSubs (sees_mono RS analz_mono RS synth_mono)
:: recur.intrs))));
qed "recur_mono";
(*Nobody sends themselves messages*)
goal thy "!!evs. evs : recur lost ==> ALL A X. Says A A X ~: set_of_list evs";
by (etac recur.induct 1);
by (Auto_tac());
qed_spec_mp "not_Says_to_self";
Addsimps [not_Says_to_self];
AddSEs [not_Says_to_self RSN (2, rev_notE)];
goal thy "!!evs. (PA,RB,KAB) : respond evs ==> Key KAB : parts{RB}";
by (etac respond.induct 1);
by (ALLGOALS Simp_tac);
qed "respond_Key_in_parts";
goal thy "!!evs. (PA,RB,KAB) : respond evs ==> Key KAB ~: used evs";
by (etac respond.induct 1);
by (REPEAT (assume_tac 1));
qed "respond_imp_not_used";
goal thy
"!!evs. [| Key K : parts {RB}; (PB,RB,K') : respond evs |] \
\ ==> Key K ~: used evs";
by (etac rev_mp 1);
by (etac respond.induct 1);
by (auto_tac(!claset addDs [Key_not_used, respond_imp_not_used],
!simpset));
qed_spec_mp "Key_in_parts_respond";
(*Simple inductive reasoning about responses*)
goal thy "!!evs. (PA,RB,KAB) : respond evs ==> RB : responses evs";
by (etac respond.induct 1);
by (REPEAT (ares_tac (respond_imp_not_used::responses.intrs) 1));
qed "respond_imp_responses";
(** For reasoning about the encrypted portion of messages **)
val RA2_analz_sees_Spy = Says_imp_sees_Spy RS analz.Inj |> standard;
goal thy "!!evs. Says C' B {|X, X', RA|} : set_of_list evs \
\ ==> RA : analz (sees lost Spy evs)";
by (fast_tac (!claset addSDs [Says_imp_sees_Spy RS analz.Inj]) 1);
qed "RA4_analz_sees_Spy";
(*RA2_analz... and RA4_analz... let us treat those cases using the same
argument as for the Fake case. This is possible for most, but not all,
proofs: Fake does not invent new nonces (as in RA2), and of course Fake
messages originate from the Spy. *)
bind_thm ("RA2_parts_sees_Spy",
RA2_analz_sees_Spy RS (impOfSubs analz_subset_parts));
bind_thm ("RA4_parts_sees_Spy",
RA4_analz_sees_Spy RS (impOfSubs analz_subset_parts));
(*We instantiate the variable to "lost". Leaving it as a Var makes proofs
harder to complete, since simplification does less for us.*)
val parts_Fake_tac =
let val tac = forw_inst_tac [("lost","lost")]
in tac RA2_parts_sees_Spy 4 THEN
etac subst 4 (*RA2: DELETE needless definition of PA!*) THEN
forward_tac [respond_imp_responses] 5 THEN
tac RA4_parts_sees_Spy 6
end;
(*For proving the easier theorems about X ~: parts (sees lost Spy evs) *)
fun parts_induct_tac i = SELECT_GOAL
(DETERM (etac recur.induct 1 THEN parts_Fake_tac THEN
(*Fake message*)
TRY (best_tac (!claset addDs [impOfSubs analz_subset_parts,
impOfSubs Fake_parts_insert]
addss (!simpset)) 2)) THEN
(*Base case*)
fast_tac (!claset addss (!simpset)) 1 THEN
ALLGOALS Asm_simp_tac) i;
(** Theorems of the form X ~: parts (sees lost Spy evs) imply that NOBODY
sends messages containing X! **)
(** Spy never sees another agent's long-term key (unless initially lost) **)
goal thy
"!!evs. evs : recur lost \
\ ==> (Key (shrK A) : parts (sees lost Spy evs)) = (A : lost)";
by (parts_induct_tac 1);
(*RA2*)
by (best_tac (!claset addSEs partsEs addSDs [parts_cut]
addss (!simpset)) 1);
(*RA3*)
by (fast_tac (!claset addDs [Key_in_parts_respond]
addss (!simpset addsimps [parts_insert_sees])) 1);
qed "Spy_see_shrK";
Addsimps [Spy_see_shrK];
goal thy
"!!evs. evs : recur lost \
\ ==> (Key (shrK A) : analz (sees lost Spy evs)) = (A : lost)";
by (auto_tac(!claset addDs [impOfSubs analz_subset_parts], !simpset));
qed "Spy_analz_shrK";
Addsimps [Spy_analz_shrK];
goal thy "!!A. [| Key (shrK A) : parts (sees lost Spy evs); \
\ evs : recur lost |] ==> A:lost";
by (fast_tac (!claset addDs [Spy_see_shrK]) 1);
qed "Spy_see_shrK_D";
bind_thm ("Spy_analz_shrK_D", analz_subset_parts RS subsetD RS Spy_see_shrK_D);
AddSDs [Spy_see_shrK_D, Spy_analz_shrK_D];
(** Nobody can have used non-existent keys! **)
goal thy
"!!evs. [| K : keysFor (parts {RB}); (PB,RB,K') : respond evs |] \
\ ==> K : range shrK";
by (etac rev_mp 1);
by (etac (respond_imp_responses RS responses.induct) 1);
by (Auto_tac());
qed_spec_mp "Key_in_keysFor_parts";
goal thy "!!evs. evs : recur lost ==> \
\ Key K ~: used evs --> K ~: keysFor (parts (sees lost Spy evs))";
by (parts_induct_tac 1);
(*RA3*)
by (best_tac (!claset addDs [Key_in_keysFor_parts]
addss (!simpset addsimps [parts_insert_sees])) 2);
(*Fake*)
by (best_tac
(!claset addIs [impOfSubs analz_subset_parts]
addDs [impOfSubs (analz_subset_parts RS keysFor_mono),
impOfSubs (parts_insert_subset_Un RS keysFor_mono)]
addss (!simpset)) 1);
qed_spec_mp "new_keys_not_used";
bind_thm ("new_keys_not_analzd",
[analz_subset_parts RS keysFor_mono,
new_keys_not_used] MRS contra_subsetD);
Addsimps [new_keys_not_used, new_keys_not_analzd];
(*** Proofs involving analz ***)
(*For proofs involving analz. We again instantiate the variable to "lost".*)
val analz_Fake_tac =
etac subst 4 (*RA2: DELETE needless definition of PA!*) THEN
dres_inst_tac [("lost","lost")] RA2_analz_sees_Spy 4 THEN
forward_tac [respond_imp_responses] 5 THEN
dres_inst_tac [("lost","lost")] RA4_analz_sees_Spy 6;
(** Session keys are not used to encrypt other session keys **)
(*Version for "responses" relation. Handles case RA3 in the theorem below.
Note that it holds for *any* set H (not just "sees lost Spy evs")
satisfying the inductive hypothesis.*)
goal thy
"!!evs. [| RB : responses evs; \
\ ALL K KK. KK <= Compl (range shrK) --> \
\ (Key K : analz (Key``KK Un H)) = \
\ (K : KK | Key K : analz H) |] \
\ ==> ALL K KK. KK <= Compl (range shrK) --> \
\ (Key K : analz (insert RB (Key``KK Un H))) = \
\ (K : KK | Key K : analz (insert RB H))";
by (etac responses.induct 1);
by (ALLGOALS (asm_simp_tac analz_image_freshK_ss));
qed "resp_analz_image_freshK_lemma";
(*Version for the protocol. Proof is almost trivial, thanks to the lemma.*)
goal thy
"!!evs. evs : recur lost ==> \
\ ALL K KK. KK <= Compl (range shrK) --> \
\ (Key K : analz (Key``KK Un (sees lost Spy evs))) = \
\ (K : KK | Key K : analz (sees lost Spy evs))";
by (etac recur.induct 1);
by analz_Fake_tac;
by (REPEAT_FIRST (resolve_tac [allI, impI]));
by (REPEAT_FIRST (rtac analz_image_freshK_lemma ));
by (ALLGOALS
(asm_simp_tac
(analz_image_freshK_ss addsimps [resp_analz_image_freshK_lemma])));
(*Base*)
by (fast_tac (!claset addIs [image_eqI] addss (!simpset)) 1);
(*RA4, RA2, Fake*)
by (REPEAT (spy_analz_tac 1));
val raw_analz_image_freshK = result();
qed_spec_mp "analz_image_freshK";
(*Instance of the lemma with H replaced by (sees lost Spy evs):
[| RB : responses evs; evs : recur lost; |]
==> KK <= Compl (range shrK) -->
Key K : analz (insert RB (Key``KK Un sees lost Spy evs)) =
(K : KK | Key K : analz (insert RB (sees lost Spy evs)))
*)
bind_thm ("resp_analz_image_freshK",
raw_analz_image_freshK RSN
(2, resp_analz_image_freshK_lemma) RS spec RS spec);
goal thy
"!!evs. [| evs : recur lost; KAB ~: range shrK |] ==> \
\ Key K : analz (insert (Key KAB) (sees lost Spy evs)) = \
\ (K = KAB | Key K : analz (sees lost Spy evs))";
by (asm_simp_tac (analz_image_freshK_ss addsimps [analz_image_freshK]) 1);
qed "analz_insert_freshK";
(*Everything that's hashed is already in past traffic. *)
goal thy "!!i. [| evs : recur lost; A ~: lost |] ==> \
\ Hash {|Key(shrK A), X|} : parts (sees lost Spy evs) --> \
\ X : parts (sees lost Spy evs)";
by (etac recur.induct 1);
by parts_Fake_tac;
(*RA3 requires a further induction*)
by (etac responses.induct 5);
by (ALLGOALS Asm_simp_tac);
(*Fake*)
by (best_tac (!claset addDs [impOfSubs analz_subset_parts,
impOfSubs Fake_parts_insert]
addss (!simpset addsimps [parts_insert_sees])) 2);
(*Two others*)
by (REPEAT (fast_tac (!claset addss (!simpset)) 1));
bind_thm ("Hash_imp_body", result() RSN (2, rev_mp));
(** The Nonce NA uniquely identifies A's message.
This theorem applies to steps RA1 and RA2!
Unicity is not used in other proofs but is desirable in its own right.
**)
goal thy
"!!evs. [| evs : recur lost; A ~: lost |] \
\ ==> EX B' P'. ALL B P. \
\ Hash {|Key(shrK A), Agent A, Agent B, Nonce NA, P|} \
\ : parts (sees lost Spy evs) --> B=B' & P=P'";
by (parts_induct_tac 1);
by (etac responses.induct 3);
by (ALLGOALS (simp_tac (!simpset addsimps [all_conj_distrib])));
by (step_tac (!claset addSEs partsEs) 1);
(*RA1,2: creation of new Nonce. Move assertion into global context*)
by (ALLGOALS (expand_case_tac "NA = ?y"));
by (REPEAT_FIRST (ares_tac [exI]));
by (REPEAT (best_tac (!claset addSDs [Hash_imp_body]
addSEs sees_Spy_partsEs) 1));
val lemma = result();
goalw thy [HPair_def]
"!!evs.[| Hash[Key(shrK A)] {|Agent A, Agent B, Nonce NA, P|} \
\ : parts (sees lost Spy evs); \
\ Hash[Key(shrK A)] {|Agent A, Agent B', Nonce NA, P'|} \
\ : parts (sees lost Spy evs); \
\ evs : recur lost; A ~: lost |] \
\ ==> B=B' & P=P'";
by (REPEAT (eresolve_tac partsEs 1));
by (prove_unique_tac lemma 1);
qed "unique_NA";
(*** Lemmas concerning the Server's response
(relations "respond" and "responses")
***)
goal thy
"!!evs. [| RB : responses evs; evs : recur lost |] \
\ ==> (Key (shrK B) : analz (insert RB (sees lost Spy evs))) = (B:lost)";
by (etac responses.induct 1);
by (ALLGOALS
(asm_simp_tac
(analz_image_freshK_ss addsimps [Spy_analz_shrK,
resp_analz_image_freshK])));
qed "shrK_in_analz_respond";
Addsimps [shrK_in_analz_respond];
goal thy
"!!evs. [| RB : responses evs; \
\ ALL K KK. KK <= Compl (range shrK) --> \
\ (Key K : analz (Key``KK Un H)) = \
\ (K : KK | Key K : analz H) |] \
\ ==> (Key K : analz (insert RB H)) --> \
\ (Key K : parts{RB} | Key K : analz H)";
by (etac responses.induct 1);
by (ALLGOALS
(asm_simp_tac
(analz_image_freshK_ss addsimps [resp_analz_image_freshK_lemma])));
(*Simplification using two distinct treatments of "image"*)
by (simp_tac (!simpset addsimps [parts_insert2]) 1);
by (fast_tac (!claset delrules [allE]) 1);
qed "resp_analz_insert_lemma";
bind_thm ("resp_analz_insert",
raw_analz_image_freshK RSN
(2, resp_analz_insert_lemma) RSN(2, rev_mp));
(*The Server does not send such messages. This theorem lets us avoid
assuming B~=Server in RA4.*)
goal thy
"!!evs. evs : recur lost \
\ ==> ALL C X Y P. Says Server C {|X, Agent Server, Agent C, Y, P|} \
\ ~: set_of_list evs";
by (etac recur.induct 1);
by (etac (respond.induct) 5);
by (Auto_tac());
qed_spec_mp "Says_Server_not";
AddSEs [Says_Server_not RSN (2,rev_notE)];
(*The last key returned by respond indeed appears in a certificate*)
goal thy
"!!K. (Hash[Key(shrK A)] {|Agent A, B, NA, P|}, RA, K) : respond evs \
\ ==> Crypt (shrK A) {|Key K, B, NA|} : parts {RA}";
by (etac respond.elim 1);
by (ALLGOALS Asm_full_simp_tac);
qed "respond_certificate";
goal thy
"!!K'. (PB,RB,KXY) : respond evs \
\ ==> EX A' B'. ALL A B N. \
\ Crypt (shrK A) {|Key K, Agent B, N|} : parts {RB} \
\ --> (A'=A & B'=B) | (A'=B & B'=A)";
by (etac respond.induct 1);
by (ALLGOALS (asm_full_simp_tac (!simpset addsimps [all_conj_distrib])));
(*Base case*)
by (Fast_tac 1);
by (Step_tac 1);
(*Case analysis on K=KBC*)
by (expand_case_tac "K = ?y" 1);
by (dtac respond_Key_in_parts 1);
by (best_tac (!claset addSIs [exI]
addSEs partsEs
addDs [Key_in_parts_respond]) 1);
(*Case analysis on K=KAB*)
by (expand_case_tac "K = ?y" 1);
by (REPEAT (ares_tac [exI] 2));
by (ex_strip_tac 1);
by (dtac respond_certificate 1);
by (Fast_tac 1);
val lemma = result();
goal thy
"!!RB. [| Crypt (shrK A) {|Key K, Agent B, N|} : parts {RB}; \
\ Crypt (shrK A') {|Key K, Agent B', N'|} : parts {RB}; \
\ (PB,RB,KXY) : respond evs |] \
\ ==> (A'=A & B'=B) | (A'=B & B'=A)";
by (prove_unique_tac lemma 1); (*50 seconds??, due to the disjunctions*)
qed "unique_session_keys";
(** Crucial secrecy property: Spy does not see the keys sent in msg RA3
Does not in itself guarantee security: an attack could violate
the premises, e.g. by having A=Spy **)
goal thy
"!!evs. [| (PB,RB,KAB) : respond evs; evs : recur lost |] \
\ ==> ALL A A' N. A ~: lost & A' ~: lost --> \
\ Crypt (shrK A) {|Key K, Agent A', N|} : parts{RB} --> \
\ Key K ~: analz (insert RB (sees lost Spy evs))";
by (etac respond.induct 1);
by (forward_tac [respond_imp_responses] 2);
by (forward_tac [respond_imp_not_used] 2);
by (ALLGOALS (*43 seconds*)
(asm_simp_tac
(analz_image_freshK_ss addsimps
[analz_image_freshK, not_parts_not_analz,
shrK_in_analz_respond,
read_instantiate [("H", "?ff``?xx")] parts_insert,
resp_analz_image_freshK, analz_insert_freshK])));
by (ALLGOALS Simp_tac);
by (fast_tac (!claset addIs [impOfSubs analz_subset_parts]) 1);
by (step_tac (!claset addSEs [MPair_parts]) 1);
(** LEVEL 7 **)
by (fast_tac (!claset addSDs [resp_analz_insert, Key_in_parts_respond]
addDs [impOfSubs analz_subset_parts]) 4);
by (fast_tac (!claset addSDs [respond_certificate]) 3);
by (best_tac (!claset addSEs partsEs
addDs [Key_in_parts_respond]
addss (!simpset)) 2);
by (dtac unique_session_keys 1);
by (etac respond_certificate 1);
by (assume_tac 1);
by (Fast_tac 1);
qed_spec_mp "respond_Spy_not_see_encrypted_key";
goal thy
"!!evs. [| A ~: lost; A' ~: lost; evs : recur lost |] \
\ ==> Says Server B RB : set_of_list evs --> \
\ Crypt (shrK A) {|Key K, Agent A', N|} : parts{RB} --> \
\ Key K ~: analz (sees lost Spy evs)";
by (etac recur.induct 1);
by analz_Fake_tac;
by (ALLGOALS
(asm_simp_tac
(!simpset addsimps [not_parts_not_analz, analz_insert_freshK]
setloop split_tac [expand_if])));
(*RA4*)
by (spy_analz_tac 4);
(*Fake*)
by (spy_analz_tac 1);
by (step_tac (!claset delrules [impCE]) 1);
(*RA2*)
by (spy_analz_tac 1);
(*RA3, case 2: K is an old key*)
by (best_tac (!claset addSDs [resp_analz_insert]
addSEs partsEs
addDs [Key_in_parts_respond,
Says_imp_sees_Spy RS parts.Inj RSN (2, parts_cut)]
addss (!simpset)) 2);
(*RA3, case 1: use lemma previously proved by induction*)
by (fast_tac (!claset addSEs [respond_Spy_not_see_encrypted_key RSN
(2,rev_notE)]) 1);
bind_thm ("Spy_not_see_encrypted_key", result() RS mp RSN (2, rev_mp));
goal thy
"!!evs. [| Says Server B RB : set_of_list evs; \
\ Crypt (shrK A) {|Key K, Agent A', N|} : parts{RB}; \
\ C ~: {A,A',Server}; \
\ A ~: lost; A' ~: lost; evs : recur lost |] \
\ ==> Key K ~: analz (sees lost C evs)";
by (rtac (subset_insertI RS sees_mono RS analz_mono RS contra_subsetD) 1);
by (rtac (sees_lost_agent_subset_sees_Spy RS analz_mono RS contra_subsetD) 1);
by (FIRSTGOAL (rtac Spy_not_see_encrypted_key));
by (REPEAT_FIRST (fast_tac (!claset addIs [recur_mono RS subsetD])));
qed "Agent_not_see_encrypted_key";
(**** Authenticity properties for Agents ****)
(*The response never contains Hashes*)
goal thy
"!!evs. (PB,RB,K) : respond evs \
\ ==> Hash {|Key (shrK B), M|} : parts (insert RB H) --> \
\ Hash {|Key (shrK B), M|} : parts H";
by (etac (respond_imp_responses RS responses.induct) 1);
by (Auto_tac());
bind_thm ("Hash_in_parts_respond", result() RSN (2, rev_mp));
(*Only RA1 or RA2 can have caused such a part of a message to appear.*)
goalw thy [HPair_def]
"!!evs. [| Hash {|Key(shrK A), Agent A, Agent B, NA, P|} \
\ : parts (sees lost Spy evs); \
\ A ~: lost; evs : recur lost |] \
\ ==> Says A B (Hash[Key(shrK A)] {|Agent A, Agent B, NA, P|}) \
\ : set_of_list evs";
by (etac rev_mp 1);
by (parts_induct_tac 1);
(*RA3*)
by (fast_tac (!claset addSDs [Hash_in_parts_respond]) 1);
qed_spec_mp "Hash_auth_sender";
(** These two results subsume (for all agents) the guarantees proved
separately for A and B in the Otway-Rees protocol.
**)
(*Encrypted messages can only originate with the Server.*)
goal thy
"!!evs. [| A ~: lost; A ~= Spy; evs : recur lost |] \
\ ==> Crypt (shrK A) Y : parts (sees lost Spy evs) \
\ --> (EX C RC. Says Server C RC : set_of_list evs & \
\ Crypt (shrK A) Y : parts {RC})";
by (parts_induct_tac 1);
(*RA4*)
by (Fast_tac 4);
(*RA3*)
by (full_simp_tac (!simpset addsimps [parts_insert_sees]) 3
THEN Fast_tac 3);
(*RA1*)
by (Fast_tac 1);
(*RA2: it cannot be a new Nonce, contradiction.*)
by (deepen_tac (!claset delrules [impCE]
addSIs [disjI2]
addSEs [MPair_parts]
addDs [parts_cut, parts.Body]
addss (!simpset)) 0 1);
qed_spec_mp "Crypt_imp_Server_msg";
(*Corollary: if A receives B's message then the key came from the Server*)
goal thy
"!!evs. [| Says B' A RA : set_of_list evs; \
\ Crypt (shrK A) {|Key K, Agent A', NA|} : parts {RA}; \
\ A ~: lost; A ~= Spy; evs : recur lost |] \
\ ==> EX C RC. Says Server C RC : set_of_list evs & \
\ Crypt (shrK A) {|Key K, Agent A', NA|} : parts {RC}";
by (best_tac (!claset addSIs [Crypt_imp_Server_msg]
addDs [Says_imp_sees_Spy RS parts.Inj RSN (2,parts_cut)]
addss (!simpset)) 1);
qed "Agent_trust";
(*Overall guarantee: if A receives a certificant mentioning A'
then the only other agent who knows the key is A'.*)
goal thy
"!!evs. [| Says B' A RA : set_of_list evs; \
\ Crypt (shrK A) {|Key K, Agent A', NA|} : parts {RA}; \
\ C ~: {A,A',Server}; \
\ A ~: lost; A' ~: lost; A ~= Spy; evs : recur lost |] \
\ ==> Key K ~: analz (sees lost C evs)";
by (dtac Agent_trust 1 THEN REPEAT_FIRST assume_tac);
by (fast_tac (!claset addSEs [Agent_not_see_encrypted_key RSN(2,rev_notE)]) 1);
qed "Agent_secrecy";