(* Title: HOL/Auth/Public_lemmas
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1996 University of Cambridge
Theory of Public Keys (common to all symmetric-key protocols)
Server keys; initial states of agents; new nonces and keys; function "sees"
*)
val inj_pubK = thm "inj_pubK";
val priK_neq_pubK = thm "priK_neq_pubK";
(*** Basic properties of pubK & priK ***)
AddIffs [inj_pubK RS inj_eq];
Goal "(priK A = priK B) = (A=B)";
by Safe_tac;
by (dres_inst_tac [("f","invKey")] arg_cong 1);
by (Full_simp_tac 1);
qed "priK_inj_eq";
AddIffs [priK_inj_eq];
AddIffs [priK_neq_pubK, priK_neq_pubK RS not_sym];
Goalw [isSymKey_def] "~ isSymKey (pubK A)";
by (Simp_tac 1);
qed "not_isSymKey_pubK";
Goalw [isSymKey_def] "~ isSymKey (priK A)";
by (Simp_tac 1);
qed "not_isSymKey_priK";
AddIffs [not_isSymKey_pubK, not_isSymKey_priK];
Goal "[| Crypt K X \\<in> analz H; isSymKey K; Key K \\<in> analz H |] \
\ ==> X \\<in> analz H";
by (auto_tac(claset(), simpset() addsimps [isSymKey_def]));
qed "analz_isSymKey_Decrypt";
(** "Image" equations that hold for injective functions **)
Goal "(invKey x : invKey`A) = (x:A)";
by Auto_tac;
qed "invKey_image_eq";
(*holds because invKey is injective*)
Goal "(pubK x : pubK`A) = (x:A)";
by Auto_tac;
qed "pubK_image_eq";
Goal "(priK x ~: pubK`A)";
by Auto_tac;
qed "priK_pubK_image_eq";
Addsimps [invKey_image_eq, pubK_image_eq, priK_pubK_image_eq];
(** Rewrites should not refer to initState(Friend i)
-- not in normal form! **)
Goalw [keysFor_def] "keysFor (parts (initState C)) = {}";
by (induct_tac "C" 1);
by (auto_tac (claset() addIs [range_eqI], simpset()));
qed "keysFor_parts_initState";
Addsimps [keysFor_parts_initState];
(*** Function "spies" ***)
(*Agents see their own private keys!*)
Goal "Key (priK A) : initState A";
by (induct_tac "A" 1);
by Auto_tac;
qed "priK_in_initState";
AddIffs [priK_in_initState];
(*All public keys are visible*)
Goal "Key (pubK A) : spies evs";
by (induct_tac "evs" 1);
by (ALLGOALS (asm_simp_tac
(simpset() addsimps [imageI, knows_Cons]
addsplits [expand_event_case])));
qed_spec_mp "spies_pubK";
(*Spy sees private keys of bad agents!*)
Goal "A: bad ==> Key (priK A) : spies evs";
by (induct_tac "evs" 1);
by (ALLGOALS (asm_simp_tac
(simpset() addsimps [imageI, knows_Cons]
addsplits [expand_event_case])));
qed "Spy_spies_bad";
AddIffs [spies_pubK, spies_pubK RS analz.Inj];
AddSIs [Spy_spies_bad];
(*** Fresh nonces ***)
Goal "Nonce N ~: parts (initState B)";
by (induct_tac "B" 1);
by Auto_tac;
qed "Nonce_notin_initState";
AddIffs [Nonce_notin_initState];
Goal "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 "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]
addsplits [expand_event_case])));
by Safe_tac;
by (ALLGOALS (rtac (msg_Nonce_supply RS exE)));
by (ALLGOALS (blast_tac (claset() addSEs [add_leE])));
val lemma = result();
Goal "EX N. Nonce N ~: used evs";
by (rtac (lemma RS exE) 1);
by (Blast_tac 1);
qed "Nonce_supply1";
Goal "Nonce (@ N. Nonce N ~: used evs) ~: used evs";
by (rtac (lemma RS exE) 1);
by (rtac someI 1);
by (Fast_tac 1);
qed "Nonce_supply";
(*Tactic for possibility theorems*)
fun possibility_tac st = st |>
REPEAT (*omit used_Says so that Nonces start from different traces!*)
(ALLGOALS (simp_tac (simpset() delsimps [used_Says]))
THEN
REPEAT_FIRST (eq_assume_tac ORELSE'
resolve_tac [refl, conjI, Nonce_supply]));
(*** Specialized rewriting for the analz_image_... theorems ***)
Goal "insert (Key K) H = Key ` {K} Un H";
by (Blast_tac 1);
qed "insert_Key_singleton";
Goal "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. Based on analz_image_freshK_ss, but simpler.*)
val analz_image_keys_ss =
simpset() delsimps [image_insert, image_Un]
delsimps [imp_disjL] (*reduces blow-up*)
addsimps [image_insert RS sym, image_Un RS sym,
rangeI,
insert_Key_singleton,
insert_Key_image, Un_assoc RS sym];