--- a/src/HOL/Auth/NS_Shared.ML Tue Oct 08 10:21:04 1996 +0200
+++ b/src/HOL/Auth/NS_Shared.ML Tue Oct 08 10:26:23 1996 +0200
@@ -55,8 +55,28 @@
addSDs [Says_imp_sees_Spy RS parts.Inj]) 1);
qed "NS3_msg_in_parts_sees_Spy";
+goal thy
+ "!!evs. Says S A (Crypt {|NA, B, K, X|} (shrK A)) : set_of_list evs ==> \
+\ K : parts (sees lost Spy evs)";
+by (fast_tac (!claset addSEs partsEs
+ addSDs [Says_imp_sees_Spy RS parts.Inj]) 1);
+qed "Reveal_parts_sees_Spy";
+
val parts_Fake_tac =
- dres_inst_tac [("lost","lost")] NS3_msg_in_parts_sees_Spy 5;
+ dres_inst_tac [("lost","lost")] NS3_msg_in_parts_sees_Spy 5 THEN
+ forw_inst_tac [("lost","lost")] Reveal_parts_sees_Spy 8;
+
+(*For proving the easier theorems about X ~: parts (sees lost Spy evs) *)
+fun parts_induct_tac i = SELECT_GOAL
+ (DETERM (etac ns_shared.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! **)
@@ -65,12 +85,8 @@
goal thy
"!!evs. [| evs : ns_shared lost; A ~: lost |] \
\ ==> Key (shrK A) ~: parts (sees lost Spy evs)";
-by (etac ns_shared.induct 1);
-by parts_Fake_tac;
+by (parts_induct_tac 1);
by (Auto_tac());
-(*Deals with Fake message*)
-by (best_tac (!claset addDs [impOfSubs analz_subset_parts,
- impOfSubs Fake_parts_insert]) 1);
qed "Spy_not_see_shrK";
bind_thm ("Spy_not_analz_shrK",
@@ -107,6 +123,7 @@
Addsimps [shrK_mem_analz];
+
(*** Future keys can't be seen or used! ***)
(*Nobody can have SEEN keys that will be generated in the future.
@@ -118,10 +135,7 @@
goal thy "!!evs. evs : ns_shared lost ==> \
\ length evs <= length evs' --> \
\ Key (newK evs') ~: (UN C. parts (sees lost C evs))";
-by (etac ns_shared.induct 1);
-by parts_Fake_tac;
-(*auto_tac does not work here, as it performs safe_tac first*)
-by (ALLGOALS Asm_simp_tac);
+by (parts_induct_tac 1);
by (ALLGOALS (fast_tac (!claset addDs [impOfSubs analz_subset_parts,
impOfSubs parts_insert_subset_Un,
Suc_leD]
@@ -149,28 +163,68 @@
qed "Says_imp_old_keys";
+
+(*** Future nonces can't be seen or used! [proofs resemble those above] ***)
+
+goal thy "!!evs. evs : ns_shared lost ==> \
+\ length evs <= length evt --> \
+\ Nonce (newN evt) ~: (UN C. parts (sees lost C evs))";
+by (etac ns_shared.induct 1);
+(*auto_tac does not work here, as it performs safe_tac first*)
+by (ALLGOALS (asm_simp_tac (!simpset addsimps [parts_insert2]
+ addcongs [disj_cong])));
+by (REPEAT_FIRST (fast_tac (!claset
+ addSEs partsEs
+ addSDs [Says_imp_sees_Spy RS parts.Inj]
+ addDs [impOfSubs analz_subset_parts,
+ impOfSubs parts_insert_subset_Un,
+ Suc_leD]
+ addss (!simpset))));
+val lemma = result();
+
+(*Variant needed below*)
+goal thy
+ "!!evs. [| evs : ns_shared lost; length evs <= length evs' |] \
+\ ==> Nonce (newN evs') ~: parts (sees lost C evs)";
+by (fast_tac (!claset addDs [lemma]) 1);
+qed "new_nonces_not_seen";
+Addsimps [new_nonces_not_seen];
+
+(*Another variant: old messages must contain old nonces!*)
+goal thy
+ "!!evs. [| Says A B X : set_of_list evs; \
+\ Nonce (newN evt) : parts {X}; \
+\ evs : ns_shared lost \
+\ |] ==> length evt < 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.
...very like new_keys_not_seen*)
goal thy "!!evs. evs : ns_shared lost ==> \
\ length evs <= length evs' --> \
\ newK evs' ~: keysFor (UN C. parts (sees lost C evs))";
-by (etac ns_shared.induct 1);
-by parts_Fake_tac;
-by (ALLGOALS Asm_simp_tac);
+by (parts_induct_tac 1);
(*NS1 and NS2*)
-map (by o fast_tac (!claset addDs [Suc_leD] addss (!simpset))) [3,2];
+by (EVERY (map (fast_tac (!claset addDs [Suc_leD] addss (!simpset))) [3,2]));
(*Fake and NS3*)
-map (by o 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)]
- addss (!simpset)))
- [2,1];
+by (EVERY
+ (map
+ (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)]
+ addss (!simpset)))
+ [2,1]));
(*NS4 and NS5: nonce exchange*)
by (ALLGOALS (deepen_tac (!claset addSDs [Says_imp_old_keys]
addIs [less_SucI, impOfSubs keysFor_mono]
- addss (!simpset addsimps [le_def])) 0));
+ addss (!simpset addsimps [le_def])) 1));
val lemma = result();
goal thy
@@ -202,16 +256,17 @@
qed "Says_Server_message_form";
-(*Describes the form of X when the following message is sent. The use of
- "parts" strengthens the induction hyp for proving the Fake case. The
- assumptions on A are needed to prevent its being a Faked message.*)
+(*If the encrypted message appears then it originated with the Server*)
goal thy
- "!!evs. evs : ns_shared lost ==> \
-\ Crypt {|Nonce NA, Agent B, Key K, X|} (shrK A) \
-\ : parts (sees lost Spy evs) & \
-\ A ~: lost --> \
-\ (EX evt: ns_shared lost. K = newK evt & \
-\ X = (Crypt {|Key K, Agent A|} (shrK B)))";
+ "!!evs. [| Crypt {|NA, Agent B, Key K, X|} (shrK A) \
+\ : parts (sees lost Spy evs); \
+\ A ~: lost; evs : ns_shared lost |] \
+\ ==> X = (Crypt {|Key K, Agent A|} (shrK B)) & \
+\ Says Server A \
+\ (Crypt {|NA, Agent B, Key K, \
+\ Crypt {|Key K, Agent A|} (shrK B)|} (shrK A)) \
+\ : set_of_list evs";
+by (etac rev_mp 1);
by (etac ns_shared.induct 1);
by parts_Fake_tac;
(*Fake case*)
@@ -219,30 +274,27 @@
addDs [impOfSubs analz_subset_parts]
addss (!simpset)) 2);
by (Auto_tac());
-val lemma = result() RS mp;
-
-
-(*The following theorem is proved by cases. If the message was sent with a
- bad key then the Spy reads it -- even if he didn't send it in the first
- place.*)
+qed "A_trust_NS2";
(*EITHER describes the form of X when the following message is sent,
OR reduces it to the Fake case.
Use Says_Server_message_form if applicable.*)
goal thy
- "!!evs. [| Says S A (Crypt {|Nonce NA, Agent B, Key K, X|} (shrK A)) \
-\ : set_of_list evs; evs : ns_shared lost |] \
-\ ==> (EX evt: ns_shared lost. K = newK evt & length evt < length evs & \
-\ X = (Crypt {|Key K, Agent A|} (shrK B))) | \
-\ X : analz (sees lost Spy evs)";
+ "!!evs. [| Says S A (Crypt {|Nonce NA, Agent B, Key K, X|} (shrK A)) \
+\ : set_of_list evs; evs : ns_shared lost |] \
+\ ==> (EX evt: ns_shared lost. K = newK evt & \
+\ length evt < length evs & \
+\ X = (Crypt {|Key K, Agent A|} (shrK B))) \
+\ | X : analz (sees lost Spy evs)";
by (excluded_middle_tac "A : lost" 1);
by (fast_tac (!claset addSDs [Says_imp_sees_Spy RS analz.Inj]
addss (!simpset)) 2);
-by (forward_tac [lemma] 1);
+by (forward_tac [Says_imp_sees_Spy RS parts.Inj] 1);
by (fast_tac (!claset addEs partsEs
- addSDs [Says_imp_sees_Spy RS parts.Inj]) 1);
-by (fast_tac (!claset addIs [Says_imp_old_keys] addss (!simpset)) 1);
+ addSDs [A_trust_NS2, Says_Server_message_form]
+ addIs [Says_imp_old_keys]
+ addss (!simpset)) 1);
qed "Says_S_message_form";
@@ -267,7 +319,7 @@
\ Key K : parts {X} --> Key K : parts (sees lost Spy evs)";
by (etac ns_shared.induct 1);
by parts_Fake_tac;
-by (ALLGOALS (asm_simp_tac (!simpset addsimps pushes)));
+by (ALLGOALS Asm_simp_tac);
(*Deals with Faked messages*)
by (best_tac (!claset addSEs partsEs
addDs [impOfSubs parts_insert_subset_Un]
@@ -279,6 +331,36 @@
(** Session keys are not used to encrypt other session keys **)
+(*Describes the form of Key K when the following message is sent. The use of
+ "parts" strengthens the induction hyp for proving the Fake case. The
+ assumption A ~: lost prevents its being a Faked message. *)
+goal thy
+ "!!evs. evs: ns_shared lost ==> \
+\ Crypt {|NA, Agent B, Key K, X|} (shrK A) \
+\ : parts (sees lost Spy evs) & A ~: lost \
+\ --> (EX evt: ns_shared lost. K = newK evt)";
+by (parts_induct_tac 1);
+by (Auto_tac());
+val lemma = result() RS mp;
+
+
+(*EITHER describes the form of Key K when the following message is sent,
+ OR reduces it to the Fake case.*)
+goal thy
+ "!!evs. [| Says S A (Crypt {|NA, Agent B, Key K, X|} (shrK A)) \
+\ : set_of_list evs; \
+\ evs : ns_shared lost |] \
+\ ==> (EX evt: ns_shared lost. K = newK evt) \
+\ | Key K : analz (sees lost Spy evs)";
+by (excluded_middle_tac "A : lost" 1);
+by (fast_tac (!claset addSDs [Says_imp_sees_Spy RS analz.Inj]
+ addss (!simpset)) 2);
+by (forward_tac [lemma] 1);
+by (fast_tac (!claset addEs partsEs
+ addSDs [Says_imp_sees_Spy RS parts.Inj]) 1);
+by (Fast_tac 1);
+qed "Reveal_message_form";
+
(*The equality makes the induction hypothesis easier to apply*)
goal thy
"!!evs. evs : ns_shared lost ==> \
@@ -286,22 +368,20 @@
\ (K : newK``E | Key K : analz (sees lost Spy evs))";
by (etac ns_shared.induct 1);
by (forward_tac [Says_S_message_form] 5 THEN assume_tac 5);
-by (REPEAT ((eresolve_tac [bexE, conjE, disjE] ORELSE' hyp_subst_tac) 5));
+by (dtac Reveal_message_form 8);
+by (REPEAT_FIRST (eresolve_tac [bexE, conjE, disjE] ORELSE' hyp_subst_tac));
by (REPEAT_FIRST (resolve_tac [allI, analz_image_newK_lemma]));
by (ALLGOALS
(asm_simp_tac
(!simpset addsimps ([insert_Key_singleton, insert_Key_image, pushKey_newK]
@ pushes)
setloop split_tac [expand_if])));
-(** LEVEL 5 **)
-(*NS3, Fake subcase*)
-by (spy_analz_tac 5);
-(*Cases NS2 and NS3!! Simple, thanks to auto case splits*)
+(** LEVEL 6 **)
+(*Reveal case 2, NS3, Fake*)
+by (EVERY (map spy_analz_tac [7,5,2]));
+by (REPEAT (fast_tac (!claset addIs [image_eqI] addss (!simpset)) 1));
+(*NS3, NS2, Base*)
by (REPEAT (Fast_tac 3));
-(*Fake case*) (** LEVEL 7 **)
-by (spy_analz_tac 2);
-(*Base case*)
-by (fast_tac (!claset addIs [image_eqI] addss (!simpset)) 1);
qed_spec_mp "analz_image_newK";
@@ -321,63 +401,59 @@
fun ex_strip_tac i = REPEAT (ares_tac [exI, conjI] i) THEN assume_tac (i+1);
goal thy
- "!!evs. evs : ns_shared lost ==> \
-\ EX X'. ALL A X N B. \
-\ A ~: lost --> \
-\ Crypt {|N, Agent B, Key K, X|} (shrK A) : parts (sees lost Spy evs) --> \
-\ X=X'";
-by (Simp_tac 1); (*Miniscoping*)
+ "!!evs. evs : ns_shared lost ==> \
+\ EX A' NA' B' X'. ALL A NA B X. \
+\ Says Server A (Crypt {|NA, Agent B, Key K, X|} (shrK A)) \
+\ : set_of_list evs --> A=A' & NA=NA' & B=B' & X=X'";
by (etac ns_shared.induct 1);
by (forward_tac [Says_S_message_form] 5 THEN assume_tac 5);
-by (ALLGOALS
- (asm_simp_tac (!simpset addsimps [all_conj_distrib, ex_disj_distrib,
- imp_conj_distrib, parts_insert_sees])));
-by (REPEAT_FIRST (eresolve_tac [exE,disjE]));
-(*NS2: Cextraction of K = newK evsa to global context...*)
-(** LEVEL 5 **)
-by (excluded_middle_tac "K = newK evsa" 3);
-by (Asm_simp_tac 3);
-by (etac exI 3);
-(*...we assume X is a very new message, and handle this case by contradiction*)
-by (fast_tac (!claset addSEs partsEs
- addEs [Says_imp_old_keys RS less_irrefl]
- addss (!simpset)) 3);
-(*Base, Fake, NS3*) (** LEVEL 9 **)
-by (REPEAT_FIRST ex_strip_tac);
-by (dtac synth.Inj 4);
-by (REPEAT_FIRST (best_tac (!claset addDs [impOfSubs Fake_parts_insert]
- addss (!simpset))));
+by (REPEAT_FIRST (eresolve_tac [conjE, bexE, disjE] ORELSE' hyp_subst_tac));
+by (ALLGOALS (asm_simp_tac (!simpset addsimps [all_conj_distrib])));
+by (step_tac (!claset delrules [conjI]) 1);
+(*NS3*)
+by (ex_strip_tac 2);
+by (Fast_tac 2);
+(*NS2: it can't be a new key*)
+by (expand_case_tac "K = ?y" 1);
+by (REPEAT (ares_tac [refl,exI,impI,conjI] 2));
+by (fast_tac (!claset addEs [Says_imp_old_keys RS less_irrefl]
+ delrules [conjI] (*prevent split-up into 4 subgoals*)
+ addss (!simpset addsimps [parts_insertI])) 1);
val lemma = result();
(*In messages of this form, the session key uniquely identifies the rest*)
goal thy
- "!!evs. [| Says S A \
-\ (Crypt {|N, Agent B, Key K, X|} (shrK C)) \
-\ : set_of_list evs; \
-\ Says S' A' \
-\ (Crypt {|N', Agent B', Key K, X'|} (shrK C')) \
-\ : set_of_list evs; \
-\ evs : ns_shared lost; C ~: lost; C' ~: lost |] ==> X = X'";
+ "!!evs. [| Says Server A \
+\ (Crypt {|NA, Agent B, Key K, X|} (shrK A)) \
+\ : set_of_list evs; \
+\ Says Server A' \
+\ (Crypt {|NA', Agent B', Key K, X'|} (shrK A')) \
+\ : set_of_list evs; \
+\ evs : ns_shared lost |] ==> A=A' & NA=NA' & B=B' & X = X'";
by (dtac lemma 1);
-by (etac exE 1);
+by (REPEAT (etac exE 1));
(*Duplicate the assumption*)
-by (forw_inst_tac [("psi", "ALL C.?P(C)")] asm_rl 1);
-by (fast_tac (!claset addSDs [Says_imp_sees_Spy RS parts.Inj]) 1);
+by (forw_inst_tac [("psi", "ALL A.?P(A)")] asm_rl 1);
+by (fast_tac (!claset addSDs [ spec]
+ delrules [conjI] addss (!simpset)) 1);
qed "unique_session_keys";
-
(** Crucial secrecy property: Spy does not see the keys sent in msg NS2 **)
goal thy
"!!evs. [| A ~: lost; B ~: lost; \
\ evs : ns_shared lost; evt: ns_shared lost |] \
\ ==> Says Server A \
-\ (Crypt {|N, Agent B, Key K, \
-\ Crypt {|Key K, Agent A|} (shrK B)|} (shrK A)) \
-\ : set_of_list evs --> \
+\ (Crypt {|NA, Agent B, Key K, \
+\ Crypt {|Key K, Agent A|} (shrK B)|} (shrK A)) \
+\ : set_of_list evs --> \
+\ (ALL NB. Says A Spy {|NA, NB, Key K|} ~: set_of_list evs) --> \
\ Key K ~: analz (sees lost Spy evs)";
by (etac ns_shared.induct 1);
+by (forward_tac [Reveal_message_form] 8);
+by (forward_tac [Says_S_message_form] 5 THEN assume_tac 5);
+by (REPEAT_FIRST (eresolve_tac [asm_rl, bexE, disjE] ORELSE' hyp_subst_tac));
by (ALLGOALS
(asm_simp_tac
(!simpset addsimps ([analz_subset_parts RS contra_subsetD,
@@ -387,48 +463,117 @@
by (fast_tac (!claset addIs [parts_insertI]
addEs [Says_imp_old_keys RS less_irrefl]
addss (!simpset)) 2);
-(*Fake case*)
-by (spy_analz_tac 1);
-(*NS3: that message from the Server was sent earlier*)
-by (forward_tac [Says_S_message_form] 1 THEN assume_tac 1);
-by (Step_tac 1);
-by (REPEAT_FIRST assume_tac);
-by (spy_analz_tac 2); (*Prove the Fake subcase*)
-by (asm_full_simp_tac
- (!simpset addsimps (mem_if::analz_insert_Key_newK::pushes)) 1);
-by (Step_tac 1);
-(**LEVEL 10 **)
-by (excluded_middle_tac "Aa : lost" 1);
-(*But this contradicts Key(newK evt) ~: analz (sees lost Spy evsa) *)
-by (dtac (Says_imp_sees_Spy RS analz.Inj) 2);
-by (fast_tac (!claset addSDs [analz.Decrypt]
- addss (!simpset)) 2);
-(*So now we have Aa ~: lost *)
-by (dtac unique_session_keys 1);
-by (Auto_tac ());
-val lemma = result() RS mp RSN(2,rev_notE);
+(*Revl case 2, OR4, OR2, Fake*)
+by (REPEAT_FIRST (resolve_tac [conjI, impI] ORELSE' spy_analz_tac));
+(*NS3 and Revl subcases*) (**LEVEL 7 **)
+by (step_tac (!claset delrules [impCE]) 1);
+by (full_simp_tac (!simpset addsimps [all_conj_distrib]) 2);
+be conjE 2;
+by (mp_tac 2);
+(**LEVEL 11 **)
+by (forward_tac [Says_imp_sees_Spy RS parts.Inj RS A_trust_NS2] 2);
+ba 3;
+by (fast_tac (!claset addSEs [Says_Crypt_not_lost]) 2);
+by (fast_tac (!claset addDs [unique_session_keys] addss (!simpset)) 2);
+(*NS3*)
+by (forward_tac [Says_imp_sees_Spy RS parts.Inj RS A_trust_NS2] 1);
+ba 2;
+by (fast_tac (!claset addSEs [Says_Crypt_not_lost]) 1);
+by (fast_tac (!claset addDs [unique_session_keys] addss (!simpset)) 1);
+val lemma = result() RS mp RS mp RSN(2,rev_notE);
(*Final version: Server's message in the most abstract form*)
goal thy
- "!!evs. [| Says Server A \
-\ (Crypt {|N, Agent B, K, X|} K') : set_of_list evs; \
-\ A ~: lost; B ~: lost; evs : ns_shared lost \
-\ |] ==> \
-\ K ~: analz (sees lost Spy evs)";
+ "!!evs. [| Says Server A \
+\ (Crypt {|NA, Agent B, K, X|} K') : set_of_list evs; \
+\ (ALL NB. Says A Spy {|NA, NB, K|} ~: set_of_list evs); \
+\ A ~: lost; B ~: lost; evs : ns_shared lost \
+\ |] ==> K ~: analz (sees lost Spy evs)";
by (forward_tac [Says_Server_message_form] 1 THEN assume_tac 1);
by (fast_tac (!claset addSEs [lemma]) 1);
qed "Spy_not_see_encrypted_key";
goal thy
- "!!evs. [| C ~: {A,B,Server}; \
-\ Says Server A \
-\ (Crypt {|N, Agent B, K, X|} K') : set_of_list evs; \
-\ A ~: lost; B ~: lost; evs : ns_shared lost |] \
+ "!!evs. [| C ~: {A,B,Server}; \
+\ Says Server A \
+\ (Crypt {|NA, Agent B, K, X|} K') : set_of_list evs; \
+\ (ALL NB. Says A Spy {|NA, NB, K|} ~: set_of_list evs); \
+\ A ~: lost; B ~: lost; evs : ns_shared lost |] \
\ ==> 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 [ns_shared_mono RS subsetD])));
qed "Agent_not_see_encrypted_key";
+
+
+
+(**** Guarantees available at various stages of protocol ***)
+
+A_trust_NS2 RS conjunct2 RS Spy_not_see_encrypted_key;
+
+
+(*If the encrypted message appears then it originated with the Server*)
+goal thy
+ "!!evs. [| Crypt {|Key K, Agent A|} (shrK B) : parts (sees lost Spy evs); \
+\ B ~: lost; evs : ns_shared lost |] \
+\ ==> EX NA. Says Server A \
+\ (Crypt {|NA, Agent B, Key K, \
+\ Crypt {|Key K, Agent A|} (shrK B)|} (shrK A)) \
+\ : set_of_list evs";
+by (etac rev_mp 1);
+by (etac ns_shared.induct 1);
+by parts_Fake_tac;
+(*Fake case*)
+by (best_tac (!claset addSDs [impOfSubs Fake_parts_insert]
+ addDs [impOfSubs analz_subset_parts]
+ addss (!simpset)) 2);
+by (Auto_tac());
+qed "B_trust_NS3";
+
+
+goal thy
+ "!!evs. [| Crypt (Nonce NB) K : parts (sees lost Spy evs); \
+\ A ~: lost; B ~: lost; evs : ns_shared lost |] \
+\ ==> Says Server A \
+\ (Crypt {|NA, Agent B, Key K, X|} (shrK A)) \
+\ : set_of_list evs \
+\ --> (ALL NB. Says A Spy {|NA, NB, Key K|} ~: set_of_list evs) \
+\ --> Says B A (Crypt (Nonce NB) K) : set_of_list evs";
+by (etac rev_mp 1);
+by (etac ns_shared.induct 1);
+by (forward_tac [Says_S_message_form] 5 THEN assume_tac 5);
+by parts_Fake_tac;
+by (ALLGOALS
+ (asm_simp_tac (!simpset addsimps [de_Morgan_disj, all_conj_distrib])));
+(**LEVEL 5**)
+by (REPEAT_FIRST (rtac impI));
+by (subgoal_tac "Key K ~: analz (sees lost Spy evsa)" 1);
+by (fast_tac (!claset addSEs [Spy_not_see_encrypted_key RSN (2, rev_notE)]) 2);
+by (best_tac (!claset addSDs [Crypt_Fake_parts_insert]
+ addSIs [disjI2,
+ impOfSubs (sees_subset_sees_Says RS analz_mono)]
+ addss (!simpset)) 1);
+by (fast_tac (!claset addSIs [impOfSubs (sees_subset_sees_Says RS analz_mono)]
+ addss (!simpset)) 2);
+(**LEVEL 10**)
+(*Contradiction from the assumption
+ Crypt (Nonce NB) (newK evsa) : parts (sees lost Spy evsa) *)
+bd Crypt_imp_invKey_keysFor 1;
+by (Asm_full_simp_tac 1);
+
+fr disjI1;
+by (REPEAT_FIRST (eresolve_tac [asm_rl, conjE] ORELSE' hyp_subst_tac));
+by (thin_tac "?PP-->?QQ" 1);
+by (subgoal_tac "Key K ~: analz (sees lost Spy evsa)" 1);
+by (fast_tac (!claset addSEs [Spy_not_see_encrypted_key RSN (2, rev_notE)]) 2);
+by (case_tac "Ba : lost" 1);
+by (dtac Says_Crypt_lost 1 THEN assume_tac 1 THEN Fast_tac 1);
+by (dtac (Says_imp_sees_Spy RS parts.Inj RS B_trust_NS3) 1 THEN
+ REPEAT (assume_tac 1));
+be exE 1;
+by (dtac unique_session_keys 1 THEN REPEAT (assume_tac 1));
+by (Fast_tac 1);
+qed "A_trust_NS4";