Global change: lost->bad and sees Spy->spies
First change just gives a more sensible name.
Second change eliminates the agent parameter of "sees" to simplify
definitions and theorems
(* Title: HOL/Auth/Shared
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1996 University of Cambridge
Theory of Shared Keys (common to all symmetric-key protocols)
Shared, long-term keys; initial states of agents
*)
open Shared;
(*** Basic properties of shrK ***)
(*Injectiveness: Agents' long-term keys are distinct.*)
AddIffs [inj_shrK RS inj_eq];
(* invKey(shrK A) = shrK A *)
Addsimps [rewrite_rule [isSymKey_def] isSym_keys];
(** Rewrites should not refer to initState(Friend i)
-- not in normal form! **)
goalw thy [keysFor_def] "keysFor (parts (initState C)) = {}";
by (induct_tac "C" 1);
by (Auto_tac ());
qed "keysFor_parts_initState";
Addsimps [keysFor_parts_initState];
(*** Function "spies" ***)
(*Spy sees shared keys of agents!*)
goal thy "!!A. A: bad ==> Key (shrK A) : spies evs";
by (induct_tac "evs" 1);
by (ALLGOALS (asm_simp_tac
(!simpset addsimps [imageI, spies_Cons]
setloop split_tac [expand_event_case, expand_if])));
qed "Spy_spies_bad";
AddSIs [Spy_spies_bad];
(*For not_bad_tac*)
goal thy "!!A. [| Crypt (shrK A) X : analz (spies evs); A: bad |] \
\ ==> X : analz (spies evs)";
by (fast_tac (!claset addSDs [analz.Decrypt] addss (!simpset)) 1);
qed "Crypt_Spy_analz_bad";
(*Prove that the agent is uncompromised by the confidentiality of
a component of a message she's said.*)
fun not_bad_tac s =
case_tac ("(" ^ s ^ ") : bad") THEN'
SELECT_GOAL
(REPEAT_DETERM (dtac (Says_imp_spies RS analz.Inj) 1) THEN
REPEAT_DETERM (etac MPair_analz 1) THEN
THEN_BEST_FIRST
(dres_inst_tac [("A", s)] Crypt_Spy_analz_bad 1 THEN assume_tac 1)
(has_fewer_prems 1, size_of_thm)
(Step_tac 1));
(** Fresh keys never clash with long-term shared keys **)
(*Agents see their own shared keys!*)
goal thy "Key (shrK A) : initState A";
by (induct_tac "A" 1);
by (Auto_tac());
qed "shrK_in_initState";
AddIffs [shrK_in_initState];
goal thy "Key (shrK A) : used evs";
br initState_into_used 1;
by (Blast_tac 1);
qed "shrK_in_used";
AddIffs [shrK_in_used];
(*Used in parts_induct_tac and analz_Fake_tac to distinguish session keys
from long-term shared keys*)
goal thy "!!K. Key K ~: used evs ==> K ~: range shrK";
by (Blast_tac 1);
qed "Key_not_used";
(*A session key cannot clash with a long-term shared key*)
goal thy "!!K. K ~: range shrK ==> shrK B ~= K";
by (Blast_tac 1);
qed "shrK_neq";
Addsimps [Key_not_used, shrK_neq, shrK_neq RS not_sym];
(*** Fresh nonces ***)
goal thy "Nonce N ~: parts (initState B)";
by (induct_tac "B" 1);
by (Auto_tac ());
qed "Nonce_notin_initState";
AddIffs [Nonce_notin_initState];
goal thy "Nonce N ~: used []";
by (simp_tac (!simpset addsimps [used_Nil]) 1);
qed "Nonce_notin_used_empty";
Addsimps [Nonce_notin_used_empty];
(*** Supply fresh nonces for possibility theorems. ***)
(*In any trace, there is an upper bound N on the greatest nonce in use.*)
goal thy "EX N. ALL n. N<=n --> Nonce n ~: used evs";
by (induct_tac "evs" 1);
by (res_inst_tac [("x","0")] exI 1);
by (ALLGOALS (asm_simp_tac
(!simpset addsimps [used_Cons]
setloop split_tac [expand_event_case, expand_if])));
by (Step_tac 1);
by (ALLGOALS (rtac (msg_Nonce_supply RS exE)));
by (ALLGOALS (blast_tac (!claset addSEs [add_leE])));
val lemma = result();
goal thy "EX N. Nonce N ~: used evs";
by (rtac (lemma RS exE) 1);
by (Blast_tac 1);
qed "Nonce_supply1";
goal thy "EX N N'. Nonce N ~: used evs & Nonce N' ~: used evs' & N ~= N'";
by (cut_inst_tac [("evs","evs")] lemma 1);
by (cut_inst_tac [("evs","evs'")] lemma 1);
by (Step_tac 1);
by (res_inst_tac [("x","N")] exI 1);
by (res_inst_tac [("x","Suc (N+Na)")] exI 1);
by (asm_simp_tac (!simpset addsimps [less_not_refl2 RS not_sym,
le_add2, le_add1,
le_eq_less_Suc RS sym]) 1);
qed "Nonce_supply2";
goal thy "EX N N' N''. Nonce N ~: used evs & Nonce N' ~: used evs' & \
\ Nonce N'' ~: used evs'' & N ~= N' & N' ~= N'' & N ~= N''";
by (cut_inst_tac [("evs","evs")] lemma 1);
by (cut_inst_tac [("evs","evs'")] lemma 1);
by (cut_inst_tac [("evs","evs''")] lemma 1);
by (Step_tac 1);
by (res_inst_tac [("x","N")] exI 1);
by (res_inst_tac [("x","Suc (N+Na)")] exI 1);
by (res_inst_tac [("x","Suc (Suc (N+Na+Nb))")] exI 1);
by (asm_simp_tac (!simpset addsimps [less_not_refl2 RS not_sym,
le_add2, le_add1,
le_eq_less_Suc RS sym]) 1);
by (rtac (less_trans RS less_not_refl2 RS not_sym) 1);
by (stac (le_eq_less_Suc RS sym) 1);
by (asm_simp_tac (!simpset addsimps [le_eq_less_Suc RS sym]) 2);
by (REPEAT (rtac le_add1 1));
qed "Nonce_supply3";
goal thy "Nonce (@ N. Nonce N ~: used evs) ~: used evs";
by (rtac (lemma RS exE) 1);
by (rtac selectI 1);
by (Blast_tac 1);
qed "Nonce_supply";
(*** Supply fresh keys for possibility theorems. ***)
goal thy "EX K. Key K ~: used evs";
by (rtac (Finites.emptyI RS Key_supply_ax RS exE) 1);
by (Blast_tac 1);
qed "Key_supply1";
goal thy "EX K K'. Key K ~: used evs & Key K' ~: used evs' & K ~= K'";
by (cut_inst_tac [("evs","evs")] (Finites.emptyI RS Key_supply_ax) 1);
by (etac exE 1);
by (cut_inst_tac [("evs","evs'")]
(Finites.emptyI RS Finites.insertI RS Key_supply_ax) 1);
by (Auto_tac());
qed "Key_supply2";
goal thy "EX K K' K''. Key K ~: used evs & Key K' ~: used evs' & \
\ Key K'' ~: used evs'' & K ~= K' & K' ~= K'' & K ~= K''";
by (cut_inst_tac [("evs","evs")] (Finites.emptyI RS Key_supply_ax) 1);
by (etac exE 1);
by (cut_inst_tac [("evs","evs'")]
(Finites.emptyI RS Finites.insertI RS Key_supply_ax) 1);
by (etac exE 1);
by (cut_inst_tac [("evs","evs''")]
(Finites.emptyI RS Finites.insertI RS Finites.insertI RS Key_supply_ax) 1);
by (Step_tac 1);
by (Full_simp_tac 1);
by (fast_tac (!claset addSEs [allE]) 1);
qed "Key_supply3";
goal thy "Key (@ K. Key K ~: used evs) ~: used evs";
by (rtac (Finites.emptyI RS Key_supply_ax RS exE) 1);
by (rtac selectI 1);
by (Blast_tac 1);
qed "Key_supply";
(*** Tactics for possibility theorems ***)
(*Omitting used_Says makes the tactic much faster: it leaves expressions
such as Nonce ?N ~: used evs that match Nonce_supply*)
fun possibility_tac st = st |>
(REPEAT
(ALLGOALS (simp_tac (!simpset delsimps [used_Says] setSolver safe_solver))
THEN
REPEAT_FIRST (eq_assume_tac ORELSE'
resolve_tac [refl, conjI, Nonce_supply, Key_supply])));
(*For harder protocols (such as Recur) where we have to set up some
nonces and keys initially*)
fun basic_possibility_tac st = st |>
REPEAT
(ALLGOALS (asm_simp_tac (!simpset setSolver safe_solver))
THEN
REPEAT_FIRST (resolve_tac [refl, conjI]));
(*** Specialized rewriting for analz_insert_freshK ***)
goal thy "!!A. A <= Compl (range shrK) ==> shrK x ~: A";
by (Blast_tac 1);
qed "subset_Compl_range";
goal thy "insert (Key K) H = Key `` {K} Un H";
by (Blast_tac 1);
qed "insert_Key_singleton";
goal thy "insert (Key K) (Key``KK Un C) = Key `` (insert K KK) Un C";
by (Blast_tac 1);
qed "insert_Key_image";
(*Reverse the normal simplification of "image" to build up (not break down)
the set of keys. Use analz_insert_eq with (Un_upper2 RS analz_mono) to
erase occurrences of forwarded message components (X).*)
val analz_image_freshK_ss =
!simpset addcongs [if_weak_cong]
delsimps [image_insert, image_Un]
delsimps [imp_disjL] (*reduces blow-up*)
addsimps ([image_insert RS sym, image_Un RS sym,
analz_insert_eq, impOfSubs (Un_upper2 RS analz_mono),
insert_Key_singleton, subset_Compl_range,
Key_not_used, insert_Key_image, Un_assoc RS sym]
@disj_comms)
setloop split_tac [expand_if];
(*Lemma for the trivial direction of the if-and-only-if*)
goal thy
"!!evs. (Key K : analz (Key``nE Un H)) --> (K : nE | Key K : analz H) ==> \
\ (Key K : analz (Key``nE Un H)) = (K : nE | Key K : analz H)";
by (blast_tac (!claset addIs [impOfSubs analz_mono]) 1);
qed "analz_image_freshK_lemma";