src/HOL/Auth/Recur.ML
 author paulson Tue, 07 Jan 1997 16:29:43 +0100 changeset 2485 c4368c967c56 parent 2481 ee461c8bc9c3 child 2516 4d68fbe6378b permissions -rw-r--r--
Simplification of some proofs, especially by eliminating the equality in RA2
```
(*  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 25;

(** 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 {|Agent 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 (REPEAT
(ALLGOALS (asm_simp_tac (!simpset setsolver safe_solver))
THEN
REPEAT_FIRST (eq_assume_tac ORELSE' resolve_tac [refl, conjI])));
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 {|Agent A, Crypt (shrK A) {|Key K, Agent B, Nonce NA|}, \
\                       Agent Server|}                          \
\         : set_of_list evs";
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);
bw HPair_def;
by (REPEAT
(REPEAT_FIRST (eq_assume_tac ORELSE' resolve_tac [refl, conjI])
THEN
ALLGOALS (asm_simp_tac (!simpset setsolver safe_solver))));
result();

(*Case three: Alice, Bob, Charlie and the server*)
goal thy
"!!A B. [| A ~= B; A ~= Server; B ~= Server |]   \
\ ==> EX K. EX NA. EX evs: recur lost.          \
\       Says B A {|Agent A, Crypt (shrK A) {|Key K, Agent B, Nonce NA|}, \
\                       Agent Server|}                          \
\         : set_of_list evs";
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);
bw HPair_def;
by (REPEAT	(*SLOW: 37 seconds*)
(REPEAT_FIRST (eq_assume_tac ORELSE' resolve_tac [refl, conjI])
THEN
ALLGOALS (asm_simp_tac (!simpset setsolver safe_solver))));
by (ALLGOALS
(SELECT_GOAL (DEPTH_SOLVE
(swap_res_tac [refl, conjI, disjI1, disjI2] 1 APPEND
etac 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";
AddSEs   [not_Says_to_self RSN (2, rev_notE)];

(*Simple inductive reasoning about responses*)
goal thy "!!j. (j,PA,RB) : respond i ==> RB : responses i";
by (etac respond.induct 1);
by (REPEAT (ares_tac 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 {|Agent B, X, Agent B, 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. (j,PB,RB) : respond i \
\  ==> Key K : parts {RB} --> (EX j'. K = newK2(i,j') & j<=j')";
be respond.induct 1;
by (Auto_tac());
by (best_tac (!claset addDs [Suc_leD]) 1);
qed_spec_mp "Key_in_parts_respond";

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]
(*RA3*)
by (fast_tac (!claset addDs [Key_in_parts_respond]
qed "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";

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);

(*** Future keys can't be seen or used! ***)

(*Nobody can have SEEN keys that will be generated in the future. *)
goal thy "!!evs. evs : recur lost ==> \
\                length evs <= i -->   \
\                Key (newK2(i,j)) ~: parts (sees lost Spy evs)";
by (parts_induct_tac 1);
(*RA2*)
by (best_tac (!claset addSEs partsEs
(*Fake*)
by (best_tac (!claset addDs [impOfSubs analz_subset_parts,
impOfSubs parts_insert_subset_Un,
Suc_leD]
(*For RA3*)
by (asm_simp_tac (!simpset addsimps [parts_insert_sees]) 2);
(*RA1-RA4*)
by (REPEAT (best_tac (!claset addDs [Key_in_parts_respond, Suc_leD]
qed_spec_mp "new_keys_not_seen";

(*Variant: old messages must contain old keys!*)
goal thy
"!!evs. [| Says A B X : set_of_list evs;     \
\           Key (newK2(i,j)) : parts {X};     \
\           evs : recur lost                 \
\        |] ==> i < length evs";
by (rtac ccontr 1);
by (dtac leI 1);
by (fast_tac (!claset addSDs [new_keys_not_seen, Says_imp_sees_Spy]
addIs  [impOfSubs parts_mono]) 1);
qed "Says_imp_old_keys";

(*** Future nonces can't be seen or used! ***)

goal thy
"!!evs. (j, PB, RB) : respond i \
\        ==> Nonce N : parts {RB} --> Nonce N : parts {PB}";
be respond.induct 1;
by (Auto_tac());
qed_spec_mp "Nonce_in_parts_respond";

goal thy "!!i. evs : recur lost ==> \
\              length evs <= i --> Nonce(newN i) ~: parts (sees lost Spy evs)";
by (parts_induct_tac 1);
(*For RA3*)
by (asm_simp_tac (!simpset addsimps [parts_insert_sees]) 4);
by (deepen_tac (!claset addSDs [Says_imp_sees_Spy RS parts.Inj]
addDs  [Nonce_in_parts_respond, parts_cut, Suc_leD]
addss (!simpset)) 0 4);
(*Fake*)
by (fast_tac (!claset addDs  [impOfSubs analz_subset_parts,
impOfSubs parts_insert_subset_Un,
Suc_leD]
(*RA1, RA2, RA4*)
by (REPEAT_FIRST  (fast_tac (!claset
addEs [leD RS notE]
qed_spec_mp "new_nonces_not_seen";

(*Variant: old messages must contain old nonces!*)
goal thy "!!evs. [| Says A B X : set_of_list evs;    \
\                   Nonce (newN i) : parts {X};      \
\                   evs : recur lost                 \
\                |] ==> i < length evs";
by (rtac ccontr 1);
by (dtac leI 1);
by (fast_tac (!claset addSDs [new_nonces_not_seen, Says_imp_sees_Spy]
addIs  [impOfSubs parts_mono]) 1);
qed "Says_imp_old_nonces";

(** Nobody can have USED keys that will be generated in the future. **)

goal thy
"!!evs. (j,PB,RB) : respond i \
\        ==> K : keysFor (parts {RB}) --> (EX A. K = shrK A)";
be (respond_imp_responses RS responses.induct) 1;
by (Auto_tac());
qed_spec_mp "Key_in_keysFor_parts_respond";

goal thy "!!i. evs : recur lost ==>          \
\       length evs <= i --> newK2(i,j) ~: keysFor (parts (sees lost Spy evs))";
by (parts_induct_tac 1);
(*RA3*)
by (fast_tac (!claset addDs  [Key_in_keysFor_parts_respond, Suc_leD]
(*RA2*)
by (fast_tac (!claset addSEs partsEs
(*Fake, RA1, RA4*)
by (REPEAT
(best_tac
(!claset addDs [impOfSubs (analz_subset_parts RS keysFor_mono),
impOfSubs (parts_insert_subset_Un RS keysFor_mono),
Suc_leD]
addEs [new_keys_not_seen RS not_parts_not_analz RSN(2,rev_notE)]
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);

(*** 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;

Delsimps [image_insert];

(** 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 i;                             \
\           ALL K I. (Key K : analz (Key``(newK``I) Un H)) = \
\           (K : newK``I | Key K : analz H) |]                \
\       ==> ALL K I. (Key K : analz (insert RB (Key``(newK``I) Un H))) = \
\           (K : newK``I | Key K : analz (insert RB H))";
be responses.induct 1;
by (ALLGOALS
(asm_simp_tac
(!simpset addsimps [insert_Key_singleton, insert_Key_image,
Un_assoc RS sym, pushKey_newK]
setloop split_tac [expand_if])));
by (fast_tac (!claset addIs [image_eqI] addss (!simpset)) 1);
qed "resp_analz_image_newK_lemma";

(*Version for the protocol.  Proof is almost trivial, thanks to the lemma.*)
goal thy
"!!evs. evs : recur lost ==>                                            \
\  ALL K I. (Key K : analz (Key``(newK``I) Un (sees lost Spy evs))) = \
\           (K : newK``I | Key K : analz (sees lost Spy evs))";
by (etac recur.induct 1);
by analz_Fake_tac;
by (REPEAT_FIRST (ares_tac [allI, analz_image_newK_lemma]));
by (ALLGOALS (asm_simp_tac (!simpset addsimps [resp_analz_image_newK_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_newK = result();
qed_spec_mp "analz_image_newK";

(*Instance of the lemma with H replaced by (sees lost Spy evs):
[| RB : responses i;  evs : recur lost |]
==> Key xa : analz (insert RB (Key``newK``x Un sees lost Spy evs)) =
(xa : newK``x | Key xa : analz (insert RB (sees lost Spy evs)))
*)
bind_thm ("resp_analz_image_newK",
raw_analz_image_newK RSN
(2, resp_analz_image_newK_lemma) RS spec RS spec);

goal thy
"!!evs. evs : recur lost ==>                               \
\        Key K : analz (insert (Key (newK x)) (sees lost Spy evs)) = \
\        (K = newK x | Key K : analz (sees lost Spy evs))";
by (asm_simp_tac (HOL_ss addsimps [pushKey_newK, analz_image_newK,
insert_Key_singleton]) 1);
by (Fast_tac 1);
qed "analz_insert_Key_newK";

(*This is more general than proving Hash_new_nonces_not_seen: we don't prove
that "future nonces" can't be hashed, but that 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)";
be recur.induct 1;
by parts_Fake_tac;
(*RA3 requires a further induction*)
be responses.induct 5;
by (ALLGOALS Asm_simp_tac);
(*Fake*)
by (best_tac (!claset addDs [impOfSubs analz_subset_parts,
impOfSubs Fake_parts_insert]
(*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 rounds 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);
be responses.induct 3;
by (ALLGOALS (simp_tac (!simpset addsimps [all_conj_distrib])));
by (step_tac (!claset addSEs partsEs) 1);
(*RA1: creation of new Nonce.  Move assertion into global context*)
by (expand_case_tac "NA = ?y" 1);
by (best_tac (!claset addSIs [exI]
by (best_tac (!claset addss  (!simpset)) 1);
(*RA2: creation of new Nonce*)
by (expand_case_tac "NA = ?y" 1);
by (best_tac (!claset addSIs [exI]
by (best_tac (!claset addss  (!simpset)) 1);
val lemma = result();

goalw thy [HPair_def]
"!!evs.[| HPair (Key(shrK A)) {|Agent A, Agent B, Nonce NA, P|}   \
\            : parts (sees lost Spy evs);                          \
\          HPair (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 i;  evs : recur lost |] \
\ ==> (Key (shrK B) : analz (insert RB (sees lost Spy evs))) = (B:lost)";
be responses.induct 1;
by (ALLGOALS
(asm_simp_tac
(!simpset addsimps [resp_analz_image_newK, insert_Key_singleton]
setloop split_tac [expand_if])));
qed "shrK_in_analz_respond";

goal thy
"!!evs. [| RB : responses i;                             \
\           ALL K I. (Key K : analz (Key``(newK``I) Un H)) = \
\           (K : newK``I | Key K : analz H) |]                \
\       ==> (Key K : analz (insert RB H)) --> \
\                  (Key K : parts{RB} | Key K : analz H)";
be responses.induct 1;
by (ALLGOALS
(asm_simp_tac
resp_analz_image_newK_lemma,
insert_Key_singleton, insert_Key_image,
Un_assoc RS sym, pushKey_newK]
setloop split_tac [expand_if])));
(*The "Message" simpset gives the standard treatment of "image"*)
by (simp_tac (simpset_of "Message") 1);
by (fast_tac (!claset delrules [allE]) 1);
qed "resp_analz_insert_lemma";

bind_thm ("resp_analz_insert",
raw_analz_image_newK 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);
be (respond.induct) 5;
by (Auto_tac());
qed_spec_mp "Says_Server_not";
AddSEs [Says_Server_not RSN (2,rev_notE)];

goal thy
"!!i. (j,PB,RB) : respond i               \
\  ==> 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)";
be respond.induct 1;
by (ALLGOALS (asm_full_simp_tac (!simpset addsimps [all_conj_distrib])));
(*Base case*)
by (Fast_tac 1);
by (Step_tac 1);
by (expand_case_tac "K = ?y" 1);
by (best_tac (!claset addSIs [exI]
by (expand_case_tac "K = ?y" 1);
by (REPEAT (ares_tac [exI] 2));
by (ex_strip_tac 1);
be respond.elim 1;
by (REPEAT_FIRST (etac Pair_inject ORELSE' hyp_subst_tac));
by (ALLGOALS (asm_full_simp_tac
(!simpset addsimps [all_conj_distrib, ex_disj_distrib])));
by (Fast_tac 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};   \
\          (j,PB,RB) : respond i |]               \
\ ==>   (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
"!!j. (j, HPair (Key(shrK A)) {|Agent A, B, NA, P|}, RA) : respond i \
\ ==> Crypt (shrK A) {|Key (newK2 (i,j)), B, NA|} : parts {RA}";
be respond.elim 1;
by (ALLGOALS Asm_full_simp_tac);
qed "newK2_respond_lemma";

goal thy
"!!evs. [| (j,PB,RB) : respond (length 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))";
be respond.induct 1;
by (forward_tac [respond_imp_responses] 2);
by (ALLGOALS (*4 MINUTES???*)
(asm_simp_tac
([analz_image_newK, not_parts_not_analz,
read_instantiate [("H", "?ff``?xx")] parts_insert,
Un_assoc RS sym, resp_analz_image_newK,
insert_Key_singleton, analz_insert_Key_newK])
setloop split_tac [expand_if])));
by (ALLGOALS (simp_tac (simpset_of "Message")));
by (Fast_tac 1);
by (step_tac (!claset addSEs [MPair_parts]) 1);
(** LEVEL 6 **)
by (fast_tac (!claset addDs [resp_analz_insert, Key_in_parts_respond]
addSEs [new_keys_not_seen RS not_parts_not_analz
RSN(2,rev_notE)]
by (fast_tac (!claset addSDs [newK2_respond_lemma]) 3);
by (best_tac (!claset addSEs partsEs
by (thin_tac "ALL x.?P(x)" 1);
be respond.elim 1;
by (fast_tac (!claset addss (!simpset)) 1);
by (step_tac (!claset addss (!simpset)) 1);
by (best_tac (!claset addSEs partsEs
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_Key_newK]
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 (fast_tac (!claset addSDs [resp_analz_insert]
addEs [Says_imp_old_keys RS less_irrefl]) 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*)
(*NEEDED????????????????*)
goal thy
"!!evs. (j,PB,RB) : respond i \
\        ==> Hash {|Key (shrK B), M|} : parts (insert RB H) --> \
\            Hash {|Key (shrK B), M|} : parts H";
be (respond_imp_responses RS responses.induct) 1;
by (Auto_tac());
bind_thm ("Hash_in_parts_respond", result() RSN (2, rev_mp));

(*NEEDED????????????????*)
(*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 (HPair (Key(shrK A)) {|Agent A, Agent B, NA, P|})  \
\             : set_of_list evs";
be 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";

val nonce_not_seen_now = le_refl RSN (2, new_nonces_not_seen) RSN (2,rev_notE);

(** These two results should 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]
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)]
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";

```