--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Auth/Shared_lemmas.ML Tue Feb 13 16:02:53 2001 +0100
@@ -0,0 +1,260 @@
+(* 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
+*)
+
+val inj_shrK = thm "inj_shrK";
+val isSym_keys = thm "isSym_keys";
+val Key_supply_ax = thm "Key_supply_ax";
+
+(*** 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 [keysFor_def] "keysFor (parts (initState C)) = {}";
+by (induct_tac "C" 1);
+by Auto_tac;
+qed "keysFor_parts_initState";
+Addsimps [keysFor_parts_initState];
+
+(*Specialized to shared-key model: no need for addss in protocol proofs*)
+Goal "[| K: keysFor (parts (insert X H)); X : synth (analz H) |] \
+\ ==> K : keysFor (parts H) | Key K : parts H";
+by (force_tac
+ (claset() addSDs [impOfSubs (parts_insert_subset_Un RS keysFor_mono),
+ impOfSubs (analz_subset_parts RS keysFor_mono)]
+ addIs [impOfSubs analz_subset_parts],
+ simpset()) 1);
+qed "keysFor_parts_insert";
+
+Goal "Crypt K X : H ==> K : keysFor H";
+by (dtac Crypt_imp_invKey_keysFor 1);
+by (Asm_full_simp_tac 1);
+qed "Crypt_imp_keysFor";
+
+
+(*** Function "knows" ***)
+
+(*Spy sees shared keys of agents!*)
+Goal "A: bad ==> Key (shrK A) : knows Spy evs";
+by (induct_tac "evs" 1);
+by (ALLGOALS (asm_simp_tac
+ (simpset() addsimps [imageI, knows_Cons]
+ addsplits [expand_event_case])));
+qed "Spy_knows_Spy_bad";
+AddSIs [Spy_knows_Spy_bad];
+
+(*For not_bad_tac*)
+Goal "[| Crypt (shrK A) X : analz (knows Spy evs); A: bad |] \
+\ ==> X : analz (knows Spy evs)";
+by (force_tac (claset() addSDs [analz.Decrypt], 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 (etac exE 1) THEN
+ 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 "Key (shrK A) : initState A";
+by (induct_tac "A" 1);
+by Auto_tac;
+qed "shrK_in_initState";
+AddIffs [shrK_in_initState];
+
+Goal "Key (shrK A) : used evs";
+by (rtac 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 "Key K ~: used evs ==> K ~: range shrK";
+by (Blast_tac 1);
+qed "Key_not_used";
+
+Goal "Key K ~: used evs ==> shrK B ~= K";
+by (Blast_tac 1);
+qed "shrK_neq";
+
+Addsimps [Key_not_used, shrK_neq, shrK_neq RS not_sym];
+
+
+(*** 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 "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 (Clarify_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_refl3, le_add1, le_add2,
+ less_Suc_eq_le]) 1);
+qed "Nonce_supply2";
+
+Goal "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 (Clarify_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_refl3, le_add1, le_add2,
+ less_Suc_eq_le]) 1);
+qed "Nonce_supply3";
+
+Goal "Nonce (@ N. Nonce N ~: used evs) ~: used evs";
+by (rtac (lemma RS exE) 1);
+by (rtac someI 1);
+by (Blast_tac 1);
+qed "Nonce_supply";
+
+(*** Supply fresh keys for possibility theorems. ***)
+
+Goal "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 "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 (Asm_full_simp_tac 1 THEN depth_tac (claset()) 2 1); (* replaces Auto_tac *)
+qed "Key_supply2";
+
+Goal "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);
+(*Blast_tac requires instantiation of the keys for some reason*)
+by (cut_inst_tac [("evs","evs'"), ("a1","K")]
+ (Finites.emptyI RS Finites.insertI RS Key_supply_ax) 1);
+by (etac exE 1);
+by (cut_inst_tac [("evs","evs''"), ("a1","Ka"), ("a2","K")]
+ (Finites.emptyI RS Finites.insertI RS Finites.insertI RS Key_supply_ax) 1);
+by (Blast_tac 1);
+qed "Key_supply3";
+
+Goal "Key (@ K. Key K ~: used evs) ~: used evs";
+by (rtac (Finites.emptyI RS Key_supply_ax RS exE) 1);
+by (rtac someI 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, used_Notes, used_Gets]
+ 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 "A <= - (range shrK) ==> shrK x ~: A";
+by (Blast_tac 1);
+qed "subset_Compl_range";
+
+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. Use analz_insert_eq with (Un_upper2 RS analz_mono) to
+ erase occurrences of forwarded message components (X). **)
+
+bind_thms ("analz_image_freshK_simps",
+ simp_thms @ mem_simps @ (*these two allow its use with only:*)
+ disj_comms @
+ [image_insert RS sym, image_Un RS sym, empty_subsetI, insert_subset,
+ 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]);
+
+val analz_image_freshK_ss =
+ simpset() delsimps [image_insert, image_Un]
+ delsimps [imp_disjL] (*reduces blow-up*)
+ addsimps analz_image_freshK_simps;
+
+(*Lemma for the trivial direction of the if-and-only-if*)
+Goal "(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";
+