Auth proofs work up to the XXX...
(* Title: HOL/Auth/Message
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1996 University of Cambridge
Datatype of events;
inductive relation "traces" for Needham-Schroeder (shared keys)
Rewrites should not refer to initState (Friend i) -- not in normal form
*)
proof_timing:=true;
Delrules [less_SucE]; (*Perhaps avoid inserting this in Arith.ML????*)
(*FOR LIST.ML??*)
goal List.thy "x : setOfList (x#xs)";
by (Simp_tac 1);
qed "setOfList_I1";
goal List.thy "!!x. xs = x#xs' ==> x : setOfList xs";
by (Asm_simp_tac 1);
qed "setOfList_eqI1";
(*Not for Addsimps -- it can cause goals to blow up!*)
goal Set.thy "(a : (if Q then x else y)) = ((Q --> a:x) & (~Q --> a : y))";
by (simp_tac (!simpset setloop split_tac [expand_if]) 1);
qed "mem_if";
(*DELETE, AS ALREADY IN SET.ML*)
goalw Set.thy [Bex_def] "(? x:A. False) = False";
by (Simp_tac 1);
qed "bex_False";
Addsimps [bex_False];
goal Set.thy "!!x. x:A ==> insert (f x) (f``A) = f``A";
by (Fast_tac 1);
qed "insert_image";
Addsimps [insert_image];
(*FUN.ML?? WE NEED A NOTION OF INVERSE IMAGE!!*)
goal Set.thy "!!f. B <= range f = (B = f`` {x. f x: B})";
by (fast_tac (!claset addEs [equalityE]) 1);
val subset_range_iff = result();
open Event;
goal Set.thy "A Un (insert a B) = insert a (A Un B)";
by (Fast_tac 1);
qed "Un_insert_right";
Addsimps [Un_insert_left, Un_insert_right];
(*By default only o_apply is built-in. But in the presence of eta-expansion
this means that some terms displayed as (f o g) will be rewritten, and others
will not!*)
Addsimps [o_def];
(*** Basic properties of serverKey and newK ***)
(* invKey (serverKey A) = serverKey A *)
bind_thm ("invKey_serverKey", rewrite_rule [isSymKey_def] isSym_serverKey);
(* invKey (newK evs) = newK evs *)
bind_thm ("invKey_newK", rewrite_rule [isSymKey_def] isSym_newK);
Addsimps [invKey_serverKey, invKey_newK];
(*New keys and nonces are fresh*)
val serverKey_inject = inj_serverKey RS injD;
val newN_inject = inj_newN RS injD
and newK_inject = inj_newK RS injD;
AddSEs [serverKey_inject, newN_inject, newK_inject,
fresh_newK RS notE, fresh_newN RS notE];
Addsimps [inj_serverKey RS inj_eq, inj_newN RS inj_eq, inj_newK RS inj_eq];
Addsimps [fresh_newN, fresh_newK];
goal thy "newK evs ~= serverKey B";
by (subgoal_tac "newK evs = serverKey B --> \
\ Key (newK evs) : parts (initState B)" 1);
by (Fast_tac 1);
by (agent.induct_tac "B" 1);
by (auto_tac (!claset addIs [range_eqI], !simpset));
qed "newK_neq_serverKey";
Addsimps [newK_neq_serverKey, newK_neq_serverKey RS not_sym];
(*Good for talking about Server's initial state*)
goal thy "!!H. H <= Key``E ==> parts H = H";
by (Auto_tac ());
be parts.induct 1;
by (ALLGOALS (fast_tac (!claset addss (!simpset))));
qed "parts_image_subset";
bind_thm ("parts_image_Key", subset_refl RS parts_image_subset);
goal thy "!!H. H <= Key``E ==> analyze H = H";
by (Auto_tac ());
be analyze.induct 1;
by (ALLGOALS (fast_tac (!claset addss (!simpset))));
qed "analyze_image_subset";
bind_thm ("analyze_image_Key", subset_refl RS analyze_image_subset);
Addsimps [parts_image_Key, analyze_image_Key];
goalw thy [keysFor_def] "keysFor (parts (initState C)) = {}";
by (agent.induct_tac "C" 1);
by (auto_tac (!claset addIs [range_eqI], !simpset));
qed "keysFor_parts_initState";
Addsimps [keysFor_parts_initState];
goalw thy [keysFor_def] "keysFor (Key``E) = {}";
by (Auto_tac ());
qed "keysFor_image_Key";
Addsimps [keysFor_image_Key];
goal thy "serverKey A ~: newK``E";
by (agent.induct_tac "A" 1);
by (Auto_tac ());
qed "serverKey_notin_image_newK";
Addsimps [serverKey_notin_image_newK];
(*Agents see their own serverKeys!*)
goal thy "Key (serverKey A) : analyze (sees A evs)";
by (list.induct_tac "evs" 1);
by (asm_simp_tac (!simpset addsimps [impOfSubs(Un_upper2 RS analyze_mono)]) 2);
by (agent.induct_tac "A" 1);
by (auto_tac (!claset addIs [range_eqI], !simpset));
qed "analyze_own_serverKey";
bind_thm ("parts_own_serverKey",
[analyze_subset_parts, analyze_own_serverKey] MRS subsetD);
Addsimps [analyze_own_serverKey, parts_own_serverKey];
(**** Inductive relation "traces" -- basic properties ****)
val mk_cases = traces.mk_cases (list.simps @ agent.simps @ event.simps);
val Says_tracesE = mk_cases "Says A B X # evs: traces";
val Says_Server_tracesE = mk_cases "Says Server B X # evs: traces";
val Says_Enemy_tracesE = mk_cases "Says Enemy B X # evs: traces";
val Says_to_Server_tracesE = mk_cases "Says A Server X # evs: traces";
val Notes_tracesE = mk_cases "Notes A X # evs: traces";
(*The tail of a trace is a trace*)
goal thy "!!ev evs. ev#evs : traces ==> evs : traces";
by (fast_tac (!claset addEs [mk_cases "ev#evs:traces"]) 1);
qed "traces_ConsE";
goal thy "!!ev evs. [| evs = ev#evs'; evs : traces |] ==> evs' : traces";
by (fast_tac (!claset addEs [mk_cases "ev#evs:traces"]) 1);
qed "traces_eq_ConsE";
(** Specialized rewrite rules for (sees A (Says...#evs)) **)
goal thy "sees A (Says A B X # evs) = insert X (sees A evs)";
by (Simp_tac 1);
qed "sees_own";
goal thy "!!A. Server ~= A ==> \
\ sees Server (Says A B X # evs) = sees Server evs";
by (Asm_simp_tac 1);
qed "sees_Server";
goal thy "!!A. Friend i ~= A ==> \
\ sees (Friend i) (Says A B X # evs) = sees (Friend i) evs";
by (Asm_simp_tac 1);
qed "sees_Friend";
goal thy "sees Enemy (Says A B X # evs) = insert X (sees Enemy evs)";
by (Simp_tac 1);
qed "sees_Enemy";
goal thy "sees A (Says A' B X # evs) <= insert X (sees A evs)";
by (simp_tac (!simpset setloop split_tac [expand_if]) 1);
by (Fast_tac 1);
qed "sees_Says_subset_insert";
goal thy "sees A evs <= sees A (Says A' B X # evs)";
by (simp_tac (!simpset setloop split_tac [expand_if]) 1);
by (Fast_tac 1);
qed "sees_subset_sees_Says";
(*Pushing Unions into parts; one of the A's equals B, and thus sees Y*)
goal thy "(UN A. parts (sees A (Says B C Y # evs))) = \
\ parts {Y} Un (UN A. parts (sees A evs))";
by (Step_tac 1);
be rev_mp 1; (*for some reason, split_tac does not work on assumptions*)
val ss = (!simpset addsimps [parts_Un, sees_Cons]
setloop split_tac [expand_if]);
by (ALLGOALS (fast_tac (!claset addss ss)));
qed "UN_parts_sees_Says";
goal thy "Says A B X : setOfList evs --> X : sees Enemy evs";
by (list.induct_tac "evs" 1);
by (Auto_tac ());
qed_spec_mp "Says_imp_sees_Enemy";
Addsimps [Says_imp_sees_Enemy];
AddIs [Says_imp_sees_Enemy];
Addsimps [UN_parts_sees_Says, sees_own, sees_Server, sees_Friend, sees_Enemy];
Delsimps [sees_Cons]; (**** NOTE REMOVAL -- laws above are cleaner ****)
(**** Inductive proofs about traces ****)
(*The Enemy can see more than anybody else, except for their initial state*)
goal thy
"!!evs. evs : traces ==> \
\ sees A evs <= initState A Un sees Enemy evs";
be traces.induct 1;
be subst 4; (*NS2: discard evsa = Says A Server... as irrelevant!*)
be subst 5; (*NS3: discard evsa = Says S Aa... as irrelevant!*)
by (ALLGOALS (fast_tac (!claset addDs [sees_Says_subset_insert RS subsetD]
addss (!simpset))));
qed "sees_agent_subset_sees_Enemy";
(*Nobody sends themselves messages*)
goal thy "!!evs. evs : traces ==> ALL A X. Says A A X ~: setOfList evs";
be traces.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)];
(*** Server keys are not betrayed ***)
(*Enemy never sees another agent's server key!*)
goal thy
"!!evs. [| evs : traces; A ~= Enemy |] ==> \
\ Key (serverKey A) ~: parts (sees Enemy evs)";
be traces.induct 1;
be subst 4; (*NS2: discard evsa = Says A Server...#evs'a as irrelevant!*)
by (auto_tac(!claset, !simpset addsimps [parts_insert2]));
(*Deals with Faked messages*)
by (best_tac (!claset addDs [impOfSubs analyze_subset_parts,
impOfSubs parts_insert_subset_Un]
addss (!simpset)) 1);
qed "Enemy_not_see_serverKey";
bind_thm ("Enemy_not_analyze_serverKey",
[analyze_subset_parts, Enemy_not_see_serverKey] MRS contra_subsetD);
Addsimps [Enemy_not_see_serverKey,
not_sym RSN (2, Enemy_not_see_serverKey),
Enemy_not_analyze_serverKey,
not_sym RSN (2, Enemy_not_analyze_serverKey)];
(*No Friend will ever see another agent's server key
(excluding the Enemy, who might transmit his).
The Server, of course, knows all server keys.*)
goal thy
"!!evs. [| evs : traces; A ~= Enemy; A ~= Friend j |] ==> \
\ Key (serverKey A) ~: parts (sees (Friend j) evs)";
br (sees_agent_subset_sees_Enemy RS parts_mono RS contra_subsetD) 1;
by (ALLGOALS Asm_simp_tac);
qed "Friend_not_see_serverKey";
(*Not for Addsimps -- it can cause goals to blow up!*)
goal thy
"!!evs. evs : traces ==> \
\ (Key (serverKey A) \
\ : analyze (insert (Key (serverKey B)) (sees Enemy evs))) = \
\ (A=B | A=Enemy)";
by (best_tac (!claset addDs [impOfSubs analyze_subset_parts]
addIs [impOfSubs (subset_insertI RS analyze_mono)]
addss (!simpset)) 1);
qed "serverKey_mem_analyze";
(*** Future keys can't be seen or used! ***)
(*Nobody can have SEEN keys that will be generated in the future.
This has to be proved anew for each protocol description,
but should go by similar reasoning every time. Hardest case is the
standard Fake rule.
The length comparison, and Union over C, are essential for the
induction! *)
goal thy "!!evs. evs : traces ==> \
\ length evs <= length evs' --> \
\ Key (newK evs') ~: (UN C. parts (sees C evs))";
be traces.induct 1;
be subst 4; (*NS2: discard evsa = Says A Server...#evs'a as irrelevant!*)
by (ALLGOALS (asm_full_simp_tac
(!simpset addsimps [de_Morgan_disj,Suc_le_eq])));
(*Initial state? New keys cannot clash*)
by (Fast_tac 1);
(*Rule NS1 in protocol*)
by (fast_tac (!claset addDs [less_imp_le]) 2);
(*Rule NS2 in protocol*)
by (fast_tac (!claset addIs [less_imp_le] addss (!simpset)) 2);
(*Rule NS3 in protocol*)
by (fast_tac (!claset addIs [less_imp_le] addss (!simpset)) 2);
(*Rule Fake: where an Enemy agent can say practically anything*)
by (best_tac (!claset addDs [impOfSubs analyze_subset_parts,
impOfSubs parts_insert_subset_Un,
less_imp_le]
addss (!simpset)) 1);
val lemma = result();
(*Variant needed for the main theorem below*)
goal thy
"!!evs. [| evs : traces; length evs <= length evs' |] ==> \
\ Key (newK evs') ~: parts (sees C evs)";
by (fast_tac (!claset addDs [lemma]) 1);
qed "new_keys_not_seen";
Addsimps [new_keys_not_seen];
goal thy "!!K. newK evs = invKey K ==> newK evs = K";
br (invKey_eq RS iffD1) 1;
by (Simp_tac 1);
val newK_invKey = result();
val keysFor_parts_mono = parts_mono RS keysFor_mono;
(*Nobody can have USED keys that will be generated in the future.
...very like new_keys_not_seen*)
goal thy "!!evs. evs : traces ==> \
\ length evs <= length evs' --> \
\ newK evs' ~: keysFor (UN C. parts (sees C evs))";
be traces.induct 1;
be subst 4; (*NS2: discard evsa = Says A Server...#evs'a as irrelevant!*)
by (ALLGOALS (asm_full_simp_tac
(!simpset addsimps [de_Morgan_disj,Suc_le_eq])));
(*Rule NS1 in protocol*)
by (fast_tac (!claset addDs [less_imp_le]) 2);
(*Rule NS2 in protocol*)
by (fast_tac (!claset addDs [less_imp_le]) 2);
(*Rule Fake: where an Enemy agent can say practically anything*)
by (best_tac
(!claset addSDs [newK_invKey]
addDs [impOfSubs (analyze_subset_parts RS keysFor_mono),
impOfSubs (parts_insert_subset_Un RS keysFor_mono),
less_imp_le]
addEs [new_keys_not_seen RS not_parts_not_analyze RSN (2,rev_notE)]
addss (!simpset)) 1);
(*Rule NS3: quite messy...*)
by (hyp_subst_tac 1);
by (full_simp_tac (!simpset addcongs [conj_cong]) 1);
br impI 1;
bd mp 1;
by (fast_tac (!claset addDs [less_imp_le]) 1);
by (best_tac (!claset addIs
[impOfSubs (sees_subset_sees_Says RS keysFor_parts_mono),
impOfSubs (sees_own RS equalityD2 RS keysFor_parts_mono),
impOfSubs (empty_subsetI RS insert_mono RS keysFor_parts_mono)]
addss (!simpset)) 1);
val lemma = result();
goal thy
"!!evs. [| evs : traces; length evs <= length evs' |] ==> \
\ newK evs' ~: keysFor (parts (sees C evs))";
by (fast_tac (!claset addSDs [lemma] addss (!simpset)) 1);
qed "new_keys_not_used";
bind_thm ("new_keys_not_analyzed",
[analyze_subset_parts RS keysFor_mono,
new_keys_not_used] MRS contra_subsetD);
Addsimps [new_keys_not_used, new_keys_not_analyzed];
(** Rewrites to push in Key and Crypt messages, so that other messages can
be pulled out using the analyze_insert rules **)
fun insComm x y = read_instantiate_sg (sign_of thy) [("x",x), ("y",y)]
insert_commute;
val pushKeys = map (insComm "Key ?K")
["Agent ?C", "Nonce ?N", "MPair ?X ?Y", "Crypt ?X ?K'"];
val pushCrypts = map (insComm "Crypt ?X ?K")
["Agent ?C", "Nonce ?N", "MPair ?X' ?Y"];
(*Cannot be added with Addsimps -- we don't always want to re-order messages*)
val pushes = pushKeys@pushCrypts;
val pushKey_newK = insComm "Key (newK ?evs)" "Key (serverKey ?C)";
(** Lemmas concerning the form of items passed in messages **)
(*Describes the form *and age* of K, and the form of X,
when the following message is sent*)
goal thy
"!!evs. [| Says Server A (Crypt {|N, Agent B, K, X|} K') : setOfList evs; \
\ evs : traces \
\ |] ==> (EX evs':traces. K = Key(newK evs') & length evs' < length evs & \
\ X = (Crypt {|K, Agent A|} (serverKey B)))";
be rev_mp 1;
be traces.induct 1;
be subst 4; (*NS2: discard evsa = Says A Server...#evs'a as irrelevant!*)
by (ALLGOALS (fast_tac (!claset addIs [less_SucI] addss (!simpset))));
qed "Says_Server_message_form";
(* c ~: keysFor (parts H) ==> c ~: keysFor (analyze H) *)
bind_thm ("not_parts_not_keysFor_analyze",
analyze_subset_parts RS keysFor_mono RS contra_subsetD);
(*USELESS for NS3, case 1, as the ind hyp says the same thing!
goal thy
"!!evs. [| evs = Says Server (Friend i) \
\ (Crypt {|N, Agent(Friend j), K, X|} K') # evs'; \
\ evs : traces; i~=k \
\ |] ==> \
\ K ~: analyze (insert (Key (serverKey (Friend k))) (sees Enemy evs))";
be rev_mp 1;
be traces.induct 1;
be subst 4; (*NS2: discard evsa = Says A Server... as irrelevant!*)
by (ALLGOALS (asm_full_simp_tac (!simpset addsimps pushes)));
by (Step_tac 1);
by (asm_full_simp_tac (!simpset addsimps[analyze_subset_parts RS contra_subsetD]) 1);
val Enemy_not_see_encrypted_key_lemma = result();
*)
(*Describes the form of X when the following message is sent*)
goal thy
"!!evs. evs : traces ==> \
\ ALL A NA B K X. \
\ (Crypt {|Nonce NA, Agent B, Key K, X|} (serverKey A)) \
\ : parts (sees Enemy evs) & A ~= Enemy --> \
\ (EX evt:traces. K = newK evt & \
\ X = (Crypt {|Key K, Agent A|} (serverKey B)))";
be traces.induct 1;
be subst 4; (*NS2: discard evsa = Says A Server...#evs'a as irrelevant!*)
by (Step_tac 1);
by (ALLGOALS Asm_full_simp_tac);
(*Remaining cases are Fake, NS2 and NS3*)
by (fast_tac (!claset addSDs [spec]) 2);
(*The NS3 case needs the induction hypothesis twice!
One application is to the X component of the most recent message.*)
by (full_simp_tac (!simpset addsimps [conj_disj_distribR, all_conj_distrib]) 2);
be conjE 2;
by (subgoal_tac "? evs':traces. K = newK evs' & \
\ X = Crypt {|Key K, Agent (Friend i)|} (serverKey B)" 2);
(*DELETE the second quantified formula for speed*)
by (eres_inst_tac [("V","ALL A NA B K Xa. ?f A NA B K Xa : ?H X & ?P(A) --> \
\ ?Q K Xa A B")] thin_rl 3);
by (fast_tac (!claset addSEs [allE, mp] addss (!simpset)) 3);
(*DELETE the first quantified formula: it's now useless*)
by (eres_inst_tac [("V","ALL S.?P(S)")] thin_rl 2);
by (fast_tac (!claset addss (!simpset)) 2);
(*Now for the Fake case*)
by (subgoal_tac
"Crypt {|Nonce NA, Agent Ba, Key K, Xa|} (serverKey A) : \
\ parts (synthesize (analyze (sees Enemy evsa)) Un (sees Enemy evsa))" 1);
by (fast_tac (!claset addSEs [impOfSubs parts_mono]) 2);
by (best_tac (!claset addDs [impOfSubs analyze_subset_parts]
addss (!simpset)) 1);
val encrypted_form = result();
(*For eliminating the A ~= Enemy condition from the previous result*)
goal thy
"!!evs. evs : traces ==> \
\ ALL S A NA B K X. \
\ Says S A (Crypt {|Nonce NA, Agent B, Key K, X|} (serverKey A)) \
\ : setOfList evs --> \
\ S = Server | S = Enemy";
be traces.induct 1;
be subst 4; (*NS2: discard evsa = Says A Server...#evs'a as irrelevant!*)
by (ALLGOALS Asm_full_simp_tac);
(*We are left with NS3*)
by (subgoal_tac "S = Server | S = Enemy" 1);
(*First justify this assumption!*)
by (fast_tac (!claset addSEs [allE, mp] addss (!simpset)) 2);
by (Step_tac 1);
bd (setOfList_I1 RS Says_Server_message_form) 1;
by (ALLGOALS Full_simp_tac);
(*Final case. Clear out needless quantifiers to speed the following step*)
by (eres_inst_tac [("V","ALL x. ?P(x)")] thin_rl 1);
bd (normalize_thm [RSspec,RSmp] encrypted_form) 1;
br (parts.Inj RS conjI) 1;
auto();
qed_spec_mp "Server_or_Enemy";
(*Describes the form of X when the following message is sent;
use Says_Server_message_form if applicable*)
goal thy
"!!evs. [| Says S A (Crypt {|Nonce NA, Agent B, Key K, X|} (serverKey A)) \
\ : setOfList evs; \
\ evs : traces \
\ |] ==> (EX evt:traces. K = newK evt & \
\ X = (Crypt {|Key K, Agent A|} (serverKey B)))";
by (forward_tac [Server_or_Enemy] 1);
ba 1;
by (Step_tac 1);
by (fast_tac (!claset addSDs [Says_Server_message_form] addss (!simpset)) 1);
by (forward_tac [normalize_thm [RSspec,RSmp] encrypted_form] 1);
br (parts.Inj RS conjI) 1;
by (Auto_tac());
qed "Says_S_message_form";
goal thy
"!!evs. [| evs = Says S A (Crypt {|Nonce NA, Agent B, Key K, X|} \
\ (serverKey A)) # evs'; \
\ evs : traces \
\ |] ==> (EX evt:traces. evs' : traces & \
\ K = newK evt & \
\ X = (Crypt {|Key K, Agent A|} (serverKey B)))";
by (forward_tac [traces_eq_ConsE] 1);
by (dtac (setOfList_eqI1 RS Says_S_message_form) 2);
by (Auto_tac());
qed "Says_S_message_form_eq";
(****
The following is to prove theorems of the form
Key K : analyze (insert (Key (newK evt))
(insert (Key (serverKey C)) (sees Enemy evs))) ==>
Key K : analyze (insert (Key (serverKey C)) (sees Enemy evs))
A more general formula must be proved inductively.
****)
(*Not useful in this form, but it says that session keys are not used
to encrypt messages containing other keys, in the actual protocol.
We require that agents should behave like this subsequently also.*)
goal thy
"!!evs. evs : traces ==> \
\ (Crypt X (newK evt)) : parts (sees Enemy evs) & \
\ Key K : parts {X} --> Key K : parts (sees Enemy evs)";
be traces.induct 1;
be subst 4; (*NS2: discard evsa = Says A Server...#evs'a as irrelevant!*)
by (hyp_subst_tac 5); (*NS3: apply evsa = Says S A (Crypt ...) # ... *)
by (ALLGOALS (asm_full_simp_tac (!simpset addsimps pushes)));
(*Case NS3*)
by (res_inst_tac [("x1","Xa")] (insert_commute RS ssubst) 2);
by (Asm_full_simp_tac 2);
(*Deals with Faked messages*)
by (best_tac (!claset addDs [impOfSubs analyze_subset_parts,
impOfSubs parts_insert_subset_Un]
addss (!simpset)) 1);
result();
(** Specialized rewrites for this proof **)
Delsimps [image_insert];
Addsimps [image_insert RS sym];
goal thy "insert (Key (newK x)) (sees A evs) = \
\ Key `` (newK``{x}) Un (sees A evs)";
by (Fast_tac 1);
val insert_Key_singleton = result();
goal thy "insert (Key (f x)) (Key``(f``E) Un C) = \
\ Key `` (f `` (insert x E)) Un C";
by (Fast_tac 1);
val insert_Key_image = result();
goal thy
"!!evs. evs : traces ==> \
\ ALL K E. (Key K : analyze (insert (Key (serverKey C)) \
\ (Key``(newK``E) Un (sees Enemy evs)))) = \
\ (K : newK``E | \
\ Key K : analyze (insert (Key (serverKey C)) \
\ (sees Enemy evs)))";
be traces.induct 1;
be subst 4; (*NS2: discard evsa = Says A Server...#evs'a as irrelevant!*)
by (dtac Says_S_message_form_eq 5 THEN assume_tac 5);
by (REPEAT ((eresolve_tac [bexE, conjE] ORELSE' hyp_subst_tac) 5));
by (ALLGOALS
(asm_full_simp_tac
(!simpset addsimps ([insert_Key_singleton, insert_Key_image, pushKey_newK]
@ pushes)
setloop split_tac [expand_if])));
(*Cases NS2 and NS3!! Simple, thanks to auto case splits*)
by (REPEAT (Fast_tac 3));
(*Base case*)
by (fast_tac (!claset addIs [image_eqI] addss (!simpset)) 1);
(** LEVEL 7 **)
(*Fake case*)
by (REPEAT (Safe_step_tac 1));
by (fast_tac (!claset addSEs [impOfSubs (analyze_mono)]) 2);
by (subgoal_tac
"Key K : analyze \
\ (synthesize \
\ (analyze (insert (Key (serverKey C)) \
\ (Key``(newK``E) Un (sees Enemy evsa)))))" 1);
(*First, justify this subgoal*)
(*Discard the quantified formula for better speed*)
by (eres_inst_tac [("V","ALL S.?P(S)")] thin_rl 2);
by (eres_inst_tac [("V","?Q ~: ?QQ")] thin_rl 2);
by (best_tac (!claset addIs [impOfSubs (analyze_mono RS synthesize_mono)]
addSEs [impOfSubs analyze_mono]) 2);
by (Asm_full_simp_tac 1);
by (deepen_tac (!claset addIs [impOfSubs analyze_mono]) 0 1);
qed_spec_mp "analyze_image_newK";
goal thy
"!!evs. evs : traces ==> \
\ Key K : analyze (insert (Key (newK evt)) \
\ (insert (Key (serverKey C)) \
\ (sees Enemy evs))) = \
\ (K = newK evt | \
\ Key K : analyze (insert (Key (serverKey C)) \
\ (sees Enemy evs)))";
by (asm_simp_tac (HOL_ss addsimps [pushKey_newK, analyze_image_newK,
insert_Key_singleton]) 1);
by (Fast_tac 1);
qed "analyze_insert_Key_newK";
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX;
goal thy
"!!evs. [| evs = Says Server (Friend i) \
\ (Crypt {|N, Agent(Friend j), K, X|} K') # evs'; \
\ evs : traces; i~=k \
\ |] ==> \
\ K = newK evs'"; ????????????????
goals_limit:=2;
goal thy
"!!evs. [| Says S A \
\ (Crypt {|N, Agent B, Key K, X|} (serverKey A)) : setOfList evs; \
\ evs : traces \
\ |] ==> \
\ ALL A' N' B' X'. \
\ Says Server A' \
\ (Crypt {|N', Agent B', Key K, X'|} (serverKey A')) : \
\ setOfList evs --> X = X'";
be rev_mp 1;
be traces.induct 1;
be subst 4; (*NS2: discard evsa = Says A Server... as irrelevant!*)
by (ALLGOALS (asm_full_simp_tac (!simpset addsimps pushes)));
by (REPEAT_FIRST (resolve_tac [conjI, impI, allI] ORELSE' etac conjE
ORELSE' hyp_subst_tac));
by (ALLGOALS (full_simp_tac (!simpset addsimps [not_Cons_self])));
by (Step_tac 1);
fe Crypt_synthesize;
by (fast_tac (!claset addss (!simpset)) 2);
by (Step_tac 1);
by (asm_full_simp_tac (!simpset addsimps[analyze_subset_parts RS contra_subsetD]) 1);
val Enemy_not_see_encrypted_key_lemma = result();
AddSEs [less_irrefl];
(*Crucial security property: Enemy does not see the keys sent in msg NS2
-- even if another key is compromised*)
goal thy
"!!evs. [| Says Server (Friend i) \
\ (Crypt {|N, Agent(Friend j), K, X|} K') : setOfList evs; \
\ evs : traces; Friend i ~= C; Friend j ~= C \
\ |] ==> \
\ K ~: analyze (insert (Key (serverKey C)) (sees Enemy evs))";
be rev_mp 1;
be traces.induct 1;
be subst 4; (*NS2: discard evsa = Says A Server... as irrelevant!*)
by (ALLGOALS (asm_full_simp_tac (!simpset addsimps pushes)));
(*Next 3 steps infer that K has the form "Key (newK evs'" ... *)
by (REPEAT_FIRST (resolve_tac [conjI, impI]));
by (TRYALL (forward_tac [Says_Server_message_form] THEN' assume_tac));
by (REPEAT_FIRST (eresolve_tac [bexE, conjE] ORELSE' hyp_subst_tac));
by (ALLGOALS
(asm_full_simp_tac
(!simpset addsimps ([analyze_subset_parts RS contra_subsetD,
analyze_insert_Key_newK] @ pushes)
setloop split_tac [expand_if])));
(*NS2*)
by (Fast_tac 2);
(** LEVEL 9 **)
(*Now for the Fake case*)
br notI 1;
by (subgoal_tac
"Key (newK evs') : \
\ analyze (synthesize (analyze (insert (Key (serverKey C)) \
\ (sees Enemy evsa))))" 1);
be (impOfSubs analyze_mono) 2;
by (deepen_tac (!claset addIs [analyze_mono RS synthesize_mono RSN
(2,rev_subsetD),
impOfSubs synthesize_increasing,
impOfSubs analyze_increasing]) 0 2);
(*Proves the Fake goal*)
by (fast_tac (!claset addss (!simpset)) 1);
(**LEVEL 16**)
(*Now for NS3*)
(*NS3, case 1: that message from the Server was just sent*)
by (ALLGOALS (forward_tac [traces_ConsE]));
by (forward_tac [setOfList_I1 RS Says_Server_message_form] 1);
by (Full_simp_tac 1);
by (REPEAT_FIRST (eresolve_tac [bexE, conjE] ORELSE' hyp_subst_tac));
(*Can simplify the Crypt messages: their key is secure*)
by (dres_inst_tac [("psi", "?X ~: ?H")] asm_rl 1);
by (asm_full_simp_tac
(!simpset addsimps (pushes @ [analyze_insert_Crypt,
analyze_subset_parts RS contra_subsetD])) 1);
(**LEVEL 20, one subgoal left! **)
(*NS3, case 2: that message from the Server was sent earlier*)
(*simplify the good implication*)
by (Asm_full_simp_tac 1);
by (eres_inst_tac [("V","?P-->?Q")] thin_rl 1); (*delete the bad implication*)
by ((forward_tac [Says_Server_message_form] THEN' assume_tac) 1);
by (REPEAT_FIRST (eresolve_tac [bexE, conjE] ORELSE' hyp_subst_tac));
by (Full_simp_tac 1);
(**LEVEL 25 **)
by ((forward_tac [setOfList_I1 RS Says_S_message_form]) 1);
by (REPEAT_FIRST (eresolve_tac [bexE, conjE] ORELSE' hyp_subst_tac));
by (asm_full_simp_tac (!simpset addsimps (analyze_insert_Key_newK::pushes)) 1);
by (step_tac (!claset delrules [impCE]) 1);
by (ALLGOALS (asm_full_simp_tac (!simpset addsimps pushes)));
by (Step_tac 1);
by (Fast_tac 1);
(*May do this once we have proved that the keys coincide*)
by (subgoal_tac "i = ia & N = Nonce NA & Friend j = C & K' = serverKey (Friend ia)" 1);
by (REPEAT_FIRST (eresolve_tac [bexE, conjE] ORELSE' hyp_subst_tac));
(**LEVEL 29 **)
by (asm_full_simp_tac (!simpset addsimps (pushes)) 1);
(*ALL THAT REMAINS IS TO PROVE THE SUBGOAL!*)
by (Step_tac 1);
by (Fast_tac 1);
by (forward_tac [setOfList_I1 RSN (2, Server_or_Enemy)] 1);
(!simpset addsimps ([analyze_subset_parts RS keysFor_mono RS contra_subsetD])) 1);
by (excluded_middle_tac "evs'a=evt" 1);
(*Keys agree? Then should have B = Friend j, so encrypted msg is secure*)
by (subgoal_tac "B = Friend j" 2);
by (REPEAT_FIRST hyp_subst_tac);
by (asm_full_simp_tac
(!simpset addsimps ([analyze_subset_parts RS contra_subsetD])) 2);
(*Keys differ? Then they cannot clash*)
br notI 1;
bd (impOfSubs analyze_insert_Crypt_subset) 1;
????????????????????????????????????????????????????????????????;
(**LEVEL 35 **)
(*In this case, the Key in X (newK evs') is younger than
the Key we are worried about (newK evs'a). And nobody has used the
new Key yet, so it can be pulled out of the "analyze".*)
by (asm_full_simp_tac
(!simpset addsimps ([analyze_subset_parts RS keysFor_mono RS contra_subsetD])) 1);
by (Fast_tac 1);
(*In this case, Enemy has seen the (encrypted!) message at hand...*)
by (dres_inst_tac [("psi", "?X ~: ?H")] asm_rl 1);
by (asm_full_simp_tac (!simpset addsimps ([insert_absorb]@pushes)) 1);
(**LEVEL 39 **)
br notI 1;
bd (impOfSubs analyze_insert_Crypt_subset) 1;
by (asm_full_simp_tac
(!simpset addsimps ([analyze_subset_parts RS keysFor_mono RS contra_subsetD])) 1);
by (excluded_middle_tac "evs'a=evt" 1);
(*Keys agree? Then should have B = Friend j, so encrypted msg is secure*)
by (subgoal_tac "B = Friend j" 2);
by (REPEAT_FIRST hyp_subst_tac);
by (asm_full_simp_tac
(!simpset addsimps ([analyze_subset_parts RS contra_subsetD])) 2);
(*Keys differ? Then they cannot clash*)
br notI 1;
bd (impOfSubs analyze_insert_Crypt_subset) 1;
(**LEVEL 28 OLD VERSION, with extra case split on ia=k **)
by (res_inst_tac [("x1","X")] (insert_commute RS ssubst) 1);
by (excluded_middle_tac "ia=k" 1);
(*Case where the key is compromised*)
by (hyp_subst_tac 2);
by (asm_full_simp_tac (!simpset addsimps pushes) 2);
by (full_simp_tac (!simpset addsimps [insert_commute]) 2);
(**LEVEL 33 **)
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX;
proof_timing:=true;
AddSEs [less_irrefl];
(*Case where the key is secure*)
by (dres_inst_tac [("psi", "?X ~: ?H")] asm_rl 1);
(*pretend we have the right lemma!*)
(*EITHER the message was just sent by the Server, OR is a replay from the Enemy
-- in either case it has the right form, only the age differs
*)
by (subgoal_tac "EX evt. K = newK evt & X = (Crypt {|Key K, Agent(Friend ia)|} (serverKey B)) & (evt = evs' | (Crypt {|Nonce NA, Agent B, Key K, X|} (serverKey (Friend ia))) : sees Enemy evs')" 1);
by (REPEAT_FIRST (eresolve_tac [exE, disjE, conjE] ORELSE' hyp_subst_tac));
by (asm_full_simp_tac (!simpset addsimps pushes) 1);
by (res_inst_tac [("x1","Crypt ?U (?K ia)")] (insert_commute RS ssubst) 1);
br notI 1;
bd (impOfSubs analyze_insert_Crypt_subset) 1;
(**LEVEL 40 **)
(*In this case, the Key in X (newK evs') is younger than
the Key we are worried about (newK evs'a). And nobody has used the
new Key yet, so it can be pulled out of the "analyze".*)
by (asm_full_simp_tac
(!simpset addsimps ([analyze_subset_parts RS keysFor_mono RS contra_subsetD])) 1);
by (Fast_tac 1);
(*In this case, Enemy already knows about the message at hand...*)
by (dres_inst_tac [("psi", "?X ~: ?H")] asm_rl 1);
by (asm_full_simp_tac (!simpset addsimps ([insert_absorb]@pushes)) 1);
(**LEVEL 44 **)
by (excluded_middle_tac "evs'a=evt" 1);
(*Keys agree? Then should have B = Friend j, so encrypted msg is secure*)
by (subgoal_tac "B = Friend j" 2);
by (REPEAT_FIRST hyp_subst_tac);
by (asm_full_simp_tac
(!simpset addsimps ([analyze_subset_parts RS contra_subsetD])) 2);
(*Keys differ? Then they cannot clash*)
br notI 1;
bd (impOfSubs analyze_insert_Crypt_subset) 1;
by (asm_full_simp_tac
(!simpset addsimps (pushes@[analyze_subset_parts RS keysFor_mono RS contra_subsetD])) 1);
by (Fast_tac 1);
by (excluded_middle_tac "evs'a=evt" 1);
(*Keys agree? Then should have B = Friend j, so encrypted msg is secure*)
by (subgoal_tac "B = Friend j & ia=i" 2);
by (REPEAT_FIRST (etac conjE ORELSE' hyp_subst_tac));
by (asm_full_simp_tac
(!simpset addsimps ([analyze_subset_parts RS contra_subsetD])) 2);
(*Keys differ? Then they cannot clash*)
br notI 1;
bd (impOfSubs analyze_insert_Crypt_subset) 1;
by (asm_full_simp_tac
(!simpset addsimps (pushes@[analyze_subset_parts RS keysFor_mono RS contra_subsetD])) 1);
by (Fast_tac 1);
Level 51
!!evs. [| Says Server (Friend i) (Crypt {|N, Agent (Friend j), K, X|} K') :
setOfList evs;
evs : traces; i ~= k; j ~= k |] ==>
K ~: analyze (insert (Key (serverKey (Friend k))) (sees Enemy evs))
1. !!evs A B K NA S X evsa evs' ia evs'a evt.
[| i ~= k; j ~= k;
Says S (Friend ia)
(Crypt
{|Nonce NA, Agent B, Key (newK evt),
Crypt {|Key (newK evt), Agent (Friend ia)|} (serverKey B)|}
(serverKey (Friend ia))) #
evs' :
traces;
Friend ia ~= B;
Says (Friend ia) Server {|Agent (Friend ia), Agent B, Nonce NA|} :
setOfList evs';
Says Server (Friend i)
(Crypt
{|N, Agent (Friend j), Key (newK evs'a),
Crypt {|Key (newK evs'a), Agent (Friend i)|}
(serverKey (Friend j))|}
K') :
setOfList evs';
evs' : traces; length evs'a < length evs'; ia ~= k;
Crypt
{|Nonce NA, Agent B, Key (newK evt),
Crypt {|Key (newK evt), Agent (Friend ia)|} (serverKey B)|}
(serverKey (Friend ia)) :
sees Enemy evs';
Key (newK evs'a) ~:
analyze (insert (Key (serverKey (Friend k))) (sees Enemy evs'));
evs'a ~= evt;
Key (newK evs'a) :
analyze
(insert (Key (newK evt))
(insert (Key (serverKey (Friend k))) (sees Enemy evs'))) |] ==>
False
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY;
push_proof();
goal thy
"!!evs. [| evs = Says S (Friend i) \
\ (Crypt {|N, Agent(Friend j), K, X|} K') # evs'; \
\ evs : traces; i~=k \
\ |] ==> \
\ K ~: analyze (insert (Key (serverKey (Friend k))) (sees Enemy evs))";
be rev_mp 1;
be traces.induct 1;
be subst 4; (*NS2: discard evsa = Says A Server... as irrelevant!*)
by (ALLGOALS (asm_full_simp_tac (!simpset addsimps pushes)));
by (Step_tac 1);
by (asm_full_simp_tac (!simpset addsimps[analyze_subset_parts RS contra_subsetD]) 1);
val Enemy_not_see_encrypted_key_lemma = result();
by (asm_full_simp_tac
(!simpset addsimps (pushes @
[analyze_subset_parts RS contra_subsetD,
traces_ConsE RS Enemy_not_see_serverKey])) 1);
by (asm_full_simp_tac
(!simpset addsimps [analyze_subset_parts RS keysFor_mono RS contra_subsetD,
traces_ConsE RS new_keys_not_used]) 1);
by (Fast_tac 1);
(*Precisely formulated as needed below. Perhaps redundant, as simplification
with the aid of analyze_subset_parts RS contra_subsetD might do the
same thing.*)
goal thy
"!!evs. [| evs : traces; A ~= Enemy; A ~= Friend j |] ==> \
\ Key (serverKey A) ~: \
\ analyze (insert (Key (serverKey (Friend j))) (sees Enemy evs))";
br (analyze_subset_parts RS contra_subsetD) 1;
by (Asm_simp_tac 1);
qed "insert_not_analyze_serverKey";
by (asm_full_simp_tac
(!simpset addsimps (pushes @
[insert_not_analyze_serverKey,
traces_ConsE RS insert_not_analyze_serverKey])) 1);
by (stac analyze_insert_Crypt 1);
(*NS3, case 1: that message from the Server was just sent*)
by (asm_full_simp_tac (!simpset addsimps pushes) 1);
(*NS3, case 2: that message from the Server was sent earlier*)
by (REPEAT_FIRST (eresolve_tac [exE, conjE] ORELSE' hyp_subst_tac));
by (mp_tac 1);
by (asm_full_simp_tac (!simpset addsimps pushes) 1);
by (Step_tac 1);
by (asm_full_simp_tac
(!simpset addsimps [traces_ConsE RS insert_not_analyze_serverKey]) 1);
(*pretend we have the right lemma!*)
by (subgoal_tac "(EX evt. X = (Crypt {|Key(newK evt), Agent(Friend ia)|} (serverKey B)))" 1);
by (REPEAT_FIRST (eresolve_tac [exE, conjE] ORELSE' hyp_subst_tac));
by (asm_full_simp_tac (!simpset addsimps pushes) 1);
by (excluded_middle_tac "ia=k" 1);
(*Case where the key is compromised*)
by (hyp_subst_tac 2);
by (stac insert_commute 2); (*Pushes in the "insert X" for simplification*)
by (Asm_full_simp_tac 2);
by (asm_full_simp_tac (!simpset addsimps pushes) 1);
by (REPEAT_FIRST Safe_step_tac);
by (ALLGOALS (asm_full_simp_tac (!simpset addsimps pushes)));
by (REPEAT (Safe_step_tac 1));
by (ALLGOALS (asm_full_simp_tac (!simpset addsimps pushes)));
by (REPEAT_FIRST (eresolve_tac [exE, conjE] ORELSE' hyp_subst_tac));
by (eres_inst_tac [("V","?P-->?Q")] thin_rl 1); (*deletes the bad implication*)
by ((forward_tac [Says_Server_message_form] THEN'
fast_tac (!claset addSEs [traces_ConsE])) 1);
by (REPEAT_FIRST (eresolve_tac [exE, conjE] ORELSE' hyp_subst_tac));
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX;
proof_timing:=true;
(*Case where the key is secure*)
by (dres_inst_tac [("psi", "?X ~: ?H")] asm_rl 1);
by (asm_full_simp_tac
(!simpset addsimps [traces_ConsE RS insert_not_analyze_serverKey]) 1);
by (full_simp_tac (!simpset addsimps [insert_Key_Agent_delay,
insert_Key_Crypt_delay]) 1);
by (forward_tac [setOfList_I1 RS Says_Server_message_form] 1);
by (REPEAT_FIRST (eresolve_tac [exE, conjE] ORELSE' hyp_subst_tac));
by (asm_full_simp_tac
(!simpset addsimps [traces_ConsE RS insert_not_analyze_serverKey]) 1);
by (stac insert_commute 1); (*Pushes in the "insert X" for simplification*)
by (stac analyze_insert_Crypt 1);
by (asm_full_simp_tac
(!simpset addsimps [insert_not_analyze_serverKey,
traces_ConsE RS insert_not_analyze_serverKey]) 1);
1. !!evs A B K NA S X evsa evs' ia evs'a.
[| i ~= k; j ~= k;
Says S (Friend ia)
(Crypt {|Nonce NA, Agent B, Key K, X|} (serverKey (Friend ia))) #
evs' :
traces;
Friend ia ~= B;
Says (Friend ia) Server {|Agent (Friend ia), Agent B, Nonce NA|} :
setOfList evs';
Says Server (Friend i)
(Crypt
{|N, Agent (Friend j), Key (newK evs'a),
Crypt {|Key (newK evs'a), Agent (Friend i)|}
(serverKey (Friend j))|}
K') :
setOfList evs';
Key (newK evs'a) ~:
analyze
(insert
(Crypt {|Nonce NA, Agent B, Key K, X|} (serverKey (Friend ia)))
(insert (Key (serverKey (Friend k))) (sees Enemy evs')));
length evs'a < length evs' |] ==>
Key (newK evs'a) ~:
analyze
(insert X
(insert
(Crypt {|Nonce NA, Agent B, Key K, X|} (serverKey (Friend ia)))
(insert (Key (serverKey (Friend k))) (sees Enemy evs'))))
by (Asm_full_simp_tac 1);
by (Simp_tac 2);
by (Simp_tac 2);
by (simp_tac (HOL_ss addsimps [insert_commute]) 2);
by (simp_tac (HOL_ss addsimps [analyze_insert_insert]) 2);
by (Asm_full_simp_tac 2);
by (simp_tac (!simpset addsimps [insert_absorb]) 2);
by (stac analyze_insert_Decrypt 2);
goal thy "analyze (insert X (insert Y H)) = analyze (insert X (analyze (insert Y H)))";
br analyze_insert_cong 1;
by (Simp_tac 1);
qed "analyze_insert_insert";
1. !!evs A B K NA S X evsa evs' ia evs'a.
[| i ~= k; j ~= k;
Says S (Friend ia)
(Crypt {|Nonce NA, Agent B, Key K, X|} (serverKey (Friend ia))) #
evs' :
traces;
Friend ia ~= B;
Says (Friend ia) Server {|Agent (Friend ia), Agent B, Nonce NA|} :
setOfList evs';
Says Server (Friend i)
(Crypt
{|N, Agent (Friend j), Key (newK evs'a),
Crypt {|Key (newK evs'a), Agent (Friend i)|}
(serverKey (Friend j))|}
K') :
setOfList evs';
length evs'a < length evs'; ia ~= k;
Key (newK evs'a) ~:
analyze (insert (Key (serverKey (Friend k))) (sees Enemy evs')) |] ==>
Key (newK evs'a) ~:
analyze
(insert X
(insert
(Crypt {|Nonce NA, Agent B, Key K, X|} (serverKey (Friend ia)))
(insert (Key (serverKey (Friend k))) (sees Enemy evs'))))
by (ALLGOALS Asm_full_simp_tac);
by (Asm_full_simp_tac 1);
by (asm_simp_tac (!simpset addsimps [insert_not_analyze_serverKey]) 1);
fr insert_not_analyze_serverKey;
by (fast_tac (!claset addSEs [traces_ConsE]) 1);
by (forward_tac [traces_ConsE] 1);
by (asm_full_simp_tac (!simpset addsimps [analyze_subset_parts RS contra_subsetD]) 1);
auto();
by (REPEAT_FIRST (resolve_tac [conjI, impI] (*DELETE NEXT TWO LINES??*)
ORELSE' eresolve_tac [conjE]
ORELSE' hyp_subst_tac));
by (forward_tac [Says_Server_message_form] 2);
bd Says_Server_message_form 2;
by (fast_tac (!claset addSEs [traces_ConsE]) 2);
auto();
by (ALLGOALS (full_simp_tac (!simpset addsimps [insert_Key_delay])));
(*SUBGOAL 1: use analyze_insert_Crypt to pull out
Crypt{|...|} (serverKey (Friend i))
SUBGOAL 2: cannot do this, as we don't seem to have ia~=j
*)
qed "Enemy_not_see_encrypted_key";
goal thy
"!!evs. [| Says Server (Friend i) \
\ (Crypt {|N, Agent B, K|} K') : setOfList evs; \
\ evs : traces; i~=j \
\ |] ==> K ~: analyze (sees (Friend j) evs)";
br (sees_agent_subset_sees_Enemy RS analyze_mono RS contra_subsetD) 1;
by (ALLGOALS (asm_simp_tac (!simpset addsimps [Enemy_not_see_encrypted_key])));
qed "Friend_not_see_encrypted_key";
goal thy
"!!evs. [| Says Server (Friend i) \
\ (Crypt {|N, Agent B, K|} K') : setOfList evs; \
\ A ~: {Friend i, Server}; \
\ evs : traces \
\ |] ==> K ~: analyze (sees A evs)";
by (eres_inst_tac [("P", "A~:?X")] rev_mp 1);
by (agent.induct_tac "A" 1);
by (ALLGOALS Simp_tac);
by (asm_simp_tac (!simpset addsimps [eq_sym_conv,
Friend_not_see_encrypted_key]) 1);
br ([analyze_mono, Enemy_not_see_encrypted_key] MRS contra_subsetD) 1;
(* hyp_subst_tac would deletes the equality assumption... *)
by (ALLGOALS (rtac n_not_Suc_n ORELSE' Fast_tac));
qed "Agent_not_see_encrypted_key";
(*Neatly packaged, perhaps in a useless form*)
goalw thy [knownBy_def]
"!!evs. [| Says Server (Friend i) \
\ (Crypt {|N, Agent B, K|} K') : setOfList evs; \
\ evs : traces \
\ |] ==> knownBy evs K <= {Friend i, Server}";
by (Step_tac 1);
br ccontr 1;
by (fast_tac (!claset addDs [Agent_not_see_encrypted_key]) 1);
qed "knownBy_encrypted_key";
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX;
ZZZZZZZZZZZZZZZZ;
(*Fake case below may need something like this...*)
goal thy
"!!evs. evs : traces ==> \
\ ALL B X. Crypt X (serverKey B) : analyze (sees Enemy evs) --> \
\ (EX A A'. Says A A' (Crypt X (serverKey B)) : setOfList evs";
goal thy
"!!evs. evs : traces ==> \
\ ALL NA B K X. Crypt {|Nonce NA, Agent B, Key K, X|} (serverKey B) :
analyze (sees Enemy evs) --> \
\ (EX A A'. Says A A' (Crypt {|Nonce NA, Agent B, Key K, X|} (serverKey B)) : setOfList evs";
(*Describes the form of X when the following message is sent*)
goal thy
"!!evs. evs : traces ==> \
\ ALL S A NA B K X. \
\ Says S A (Crypt {|Nonce NA, Agent B, Key K, X|} (serverKey A)) \
\ : setOfList evs --> \
\ (EX evs'. X = (Crypt {|Key(newK evs'), Agent A|} (serverKey B)))";
be traces.induct 1;
be subst 4; (*NS2: discard evsa = Says A Server...#evs'a as irrelevant!*)
by (Step_tac 1);
by (ALLGOALS Asm_full_simp_tac);
(*Remaining cases are Fake, NS2 and NS3*)
by (Fast_tac 2); (*Solves the NS2 case*)
(*The NS3 case needs the induction hypothesis twice!
One application is to the X component of the most recent message.*)
by (full_simp_tac (!simpset addsimps [all_conj_distrib]) 2);
be conjE 2;
by (subgoal_tac "? evs'. X = Crypt {|Key (newK evs'), Agent (Friend i)|} (serverKey B)" 2);
by (eres_inst_tac [("V","?P|?Q")] thin_rl 3); (*speeds the following step*)
by (Fast_tac 3);
(*DELETE the first quantified formula: it's now useless*)
by (eres_inst_tac [("V","ALL S.?P(S)")] thin_rl 2);
by (fast_tac (!claset addss (!simpset)) 2);
(*Now for the Fake case*)
be disjE 1;
(*The subcase of Fake, where the message in question is NOT the most recent*)
by (Best_tac 2);
(*The subcase of Fake proved because server keys are kept secret*)
by (REPEAT_FIRST (etac conjE ORELSE' hyp_subst_tac));
be Crypt_synthesize 1;
be Key_synthesize 2;
by (Asm_full_simp_tac 2);
1. !!evs B X evsa S A NA Ba K Xa.
[| evsa : traces;
! S A NA B K X.
Says S A (Crypt {|Nonce NA, Agent B, Key K, X|} (serverKey A)) :
setOfList evsa -->
(? evs'. X = Crypt {|Key (newK evs'), Agent A|} (serverKey B));
B ~= Enemy;
Crypt {|Nonce NA, Agent Ba, Key K, Xa|} (serverKey B) :
analyze (sees Enemy evsa) |] ==>
? evs'. Xa = Crypt {|Key (newK evs'), Agent B|} (serverKey Ba)
(*Split up the possibilities of that message being synthesized...*)
by (Step_tac 1);
by (Best_tac 6);
by (Safe_step_tac 1);
by (Safe_step_tac 2);
by (ALLGOALS Asm_full_simp_tac);
by (REPEAT_FIRST (etac disjE));
by (REPEAT_FIRST (etac conjE ORELSE' hyp_subst_tac));
by (hyp_subst_tac 5);
by (REPEAT (dtac spec 3));
fe conjE;
fe mp ;
by (REPEAT (resolve_tac [refl, conjI] 3));
by (REPEAT_FIRST (etac conjE));
by (REPEAT_FIRST (etac conjE ORELSE' hyp_subst_tac));
by (subgoal_tac "? evs'. Xa = Crypt {|Key (newK evs'), Agent Aa|} (serverKey Ba)" 2);
by (Fast_tac 3);
be subst 5; (*NS3: discard evsa = Says S Aa... as irrelevant!*)
auto();
by (ALLGOALS
(asm_full_simp_tac (!simpset addsimps [de_Morgan_disj,de_Morgan_conj])));
by (REPEAT_FIRST (etac conjE ORELSE' hyp_subst_tac));
by (REPEAT (Safe_step_tac 1));
qed "Says_Crypt_Nonce_imp_msg_Crypt";
goal thy
"!!evs. [| Says S A (Crypt {|Nonce NA, Agent B, Key K, X|} (serverKey A)) \
\ : setOfList evs; \
\ evs : traces \
\ |] ==> (EX evs'. X = (Crypt {|Key(newK evs'), Agent A|} (serverKey B)))";
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY;
WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW;
(*If a message is sent, encrypted with a Friend's server key, then either
that Friend or the Server originally sent it.*)
goal thy
"!!evs. evs : traces ==> \
\ ALL A B X i. Says A B (Crypt X (serverKey (Friend i))) \
\ : setOfList evs --> \
\ (EX B'. Says Server B' (Crypt X (serverKey (Friend i))) : setOfList evs | \
\ Says (Friend i) B' (Crypt X (serverKey (Friend i))) : setOfList evs)";
be traces.induct 1;
be subst 4; (*NS2: discard evsa = Says A Server...#evs'a as irrelevant!*)
by (Step_tac 1);
by (ALLGOALS Asm_full_simp_tac);
(*Remaining cases are Fake, NS2 and NS3*)
by (Fast_tac 2); (*Proves the NS2 case*)
by (REPEAT (dtac spec 2));
fe conjE;
bd mp 2;
by (REPEAT (resolve_tac [refl, conjI] 2));
by (ALLGOALS Asm_full_simp_tac);
by (full_simp_tac (!simpset addsimps [all_conj_distrib]) 2);
be conjE 2;
by ((dtac spec THEN' dtac spec THEN' dtac spec THEN' dtac spec)2);
(*The NS3 case needs the induction hypothesis twice!
One application is to the X component of the most recent message.*)
by (subgoal_tac "? evs'. X = Crypt {|Key (newK evs'), Agent A|} (serverKey B)" 2);
by (Fast_tac 3);
by (full_simp_tac (!simpset addsimps [all_conj_distrib]) 2);
be conjE 2;
(*DELETE the first quantified formula: it's now useless*)
by (eres_inst_tac [("V","ALL S.?P(S)")] thin_rl 2);
by (fast_tac (!claset addss (!simpset)) 2);
(*Now for the Fake case*)
be disjE 1;
(*The subcase of Fake, where the message in question is NOT the most recent*)
by (Best_tac 2);
by (REPEAT_FIRST (etac conjE ORELSE' hyp_subst_tac));
be Crypt_synthesize 1;
be Key_synthesize 2;
(*Split up the possibilities of that message being synthesized...*)
by (Step_tac 1);
by (Best_tac 6);
(*The NS3 case needs the induction hypothesis twice!
One application is to the X component of the most recent message.*)
????????????????????????????????????????????????????????????????
by (subgoal_tac "? evs'. X = Crypt {|Key (newK evs'), Agent A|} (serverKey B)" 1);
by (Fast_tac 2);
by (full_simp_tac (!simpset addsimps [all_conj_distrib]) 1);
be conjE 1;
(*DELETE the first quantified formula: it's now useless*)
by (eres_inst_tac [("V","ALL S.?P(S)")] thin_rl 1);
by (fast_tac (!claset addss (!simpset)) 1);
(*Now for the Fake case*)
be disjE 1;
(*The subcase of Fake, where the message in question is NOT the most recent*)
by (Best_tac 2);
WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW;
goal thy
"!!evs. evs : traces ==> \
\ Says Server A \
\ (Crypt {|Agent A, Agent B, K, N|} K') : setOfList evs \
\ --> (EX evs'. N = Nonce (newN evs'))";
be traces.induct 1;
by (ALLGOALS Asm_simp_tac);
by (Fast_tac 1);
val Says_Server_lemma = result();
(*NEEDED??*)
goal thy "!!A. X ~= Y ==> (X : sees A (Says B C Y # evs)) = (X : sees A evs)";
by (asm_simp_tac (!simpset addsimps [sees_Cons]
setloop split_tac [expand_if]) 1);
qed "in_sees_Says";
goal thy "!!A. X ~: parts {Y} ==> \
\ (X : parts (sees A (Says B C Y # evs))) = \
\ (X : parts (sees A evs))";
by (asm_simp_tac (!simpset addsimps [sees_Cons]
setloop split_tac [expand_if]) 1);
by (asm_simp_tac (!simpset addsimps [parts_Un]) 1);
qed "in_parts_sees_Says";
goal thy "!!evs. length evs < length evs' ==> newK evs ~= newK (A',evs')";
by (fast_tac (!claset addSEs [less_irrefl]) 1);
qed "length_less_newK_neq";
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY;
goalw thy [keysFor_def]
"!!X. Crypt X K: H ==> invKey K : keysFor H";
by (Fast_tac 1);
qed "keysFor_I";
AddSIs [keysFor_I];
goalw thy [keysFor_def]
"!!K. K : keysFor H ==> EX X K'. K=invKey K' & Crypt X K': H";
by (Fast_tac 1);
qed "keysFor_D";
AddSDs [keysFor_D];