(* Title: HOL/Auth/Message
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1996 University of Cambridge
Datatypes of agents and messages;
Inductive relations "parts", "analz" and "synth"
*)
(*Eliminates a commonly-occurring expression*)
goal HOL.thy "~ (ALL x. x~=y)";
by (Blast_tac 1);
Addsimps [result()];
open Message;
AddIffs atomic.inject;
AddIffs msg.inject;
(*Equations hold because constructors are injective; cannot prove for all f*)
goal thy "(Friend x : Friend``A) = (x:A)";
by (Auto_tac());
qed "Friend_image_eq";
goal thy "(Key x : Key``A) = (x:A)";
by (Auto_tac());
qed "Key_image_eq";
goal thy "(Nonce x ~: Key``A)";
by (Auto_tac());
qed "Nonce_Key_image_eq";
Addsimps [Friend_image_eq, Key_image_eq, Nonce_Key_image_eq];
(** Inverse of keys **)
goal thy "!!K K'. (invKey K = invKey K') = (K=K')";
by Safe_tac;
by (rtac box_equals 1);
by (REPEAT (rtac invKey 2));
by (Asm_simp_tac 1);
qed "invKey_eq";
Addsimps [invKey, invKey_eq];
(**** keysFor operator ****)
goalw thy [keysFor_def] "keysFor {} = {}";
by (Blast_tac 1);
qed "keysFor_empty";
goalw thy [keysFor_def] "keysFor (H Un H') = keysFor H Un keysFor H'";
by (Blast_tac 1);
qed "keysFor_Un";
goalw thy [keysFor_def] "keysFor (UN i:A. H i) = (UN i:A. keysFor (H i))";
by (Blast_tac 1);
qed "keysFor_UN";
(*Monotonicity*)
goalw thy [keysFor_def] "!!G H. G<=H ==> keysFor(G) <= keysFor(H)";
by (Blast_tac 1);
qed "keysFor_mono";
goalw thy [keysFor_def] "keysFor (insert (Agent A) H) = keysFor H";
by (Blast_tac 1);
qed "keysFor_insert_Agent";
goalw thy [keysFor_def] "keysFor (insert (Nonce N) H) = keysFor H";
by (Blast_tac 1);
qed "keysFor_insert_Nonce";
goalw thy [keysFor_def] "keysFor (insert (Number N) H) = keysFor H";
by (Blast_tac 1);
qed "keysFor_insert_Number";
goalw thy [keysFor_def] "keysFor (insert (Key K) H) = keysFor H";
by (Blast_tac 1);
qed "keysFor_insert_Key";
goalw thy [keysFor_def] "keysFor (insert (Hash X) H) = keysFor H";
by (Blast_tac 1);
qed "keysFor_insert_Hash";
goalw thy [keysFor_def] "keysFor (insert {|X,Y|} H) = keysFor H";
by (Blast_tac 1);
qed "keysFor_insert_MPair";
goalw thy [keysFor_def]
"keysFor (insert (Crypt K X) H) = insert (invKey K) (keysFor H)";
by (Auto_tac());
qed "keysFor_insert_Crypt";
Addsimps [keysFor_empty, keysFor_Un, keysFor_UN,
keysFor_insert_Agent, keysFor_insert_Nonce,
keysFor_insert_Number, keysFor_insert_Key,
keysFor_insert_Hash, keysFor_insert_MPair, keysFor_insert_Crypt];
AddSEs [keysFor_Un RS equalityD1 RS subsetD RS UnE,
keysFor_UN RS equalityD1 RS subsetD RS UN_E];
goalw thy [keysFor_def] "keysFor (Key``E) = {}";
by (Auto_tac ());
qed "keysFor_image_Key";
Addsimps [keysFor_image_Key];
goalw thy [keysFor_def] "!!H. Crypt K X : H ==> invKey K : keysFor H";
by (Blast_tac 1);
qed "Crypt_imp_invKey_keysFor";
(**** Inductive relation "parts" ****)
val major::prems =
goal thy "[| {|X,Y|} : parts H; \
\ [| X : parts H; Y : parts H |] ==> P \
\ |] ==> P";
by (cut_facts_tac [major] 1);
by (resolve_tac prems 1);
by (REPEAT (eresolve_tac [asm_rl, parts.Fst, parts.Snd] 1));
qed "MPair_parts";
AddIs [parts.Inj];
val partsEs = [MPair_parts, make_elim parts.Body];
AddSEs partsEs;
(*NB These two rules are UNSAFE in the formal sense, as they discard the
compound message. They work well on THIS FILE, perhaps because its
proofs concern only atomic messages.*)
goal thy "H <= parts(H)";
by (Blast_tac 1);
qed "parts_increasing";
(*Monotonicity*)
goalw thy parts.defs "!!G H. G<=H ==> parts(G) <= parts(H)";
by (rtac lfp_mono 1);
by (REPEAT (ares_tac basic_monos 1));
qed "parts_mono";
val parts_insertI = impOfSubs (subset_insertI RS parts_mono);
goal thy "parts{} = {}";
by Safe_tac;
by (etac parts.induct 1);
by (ALLGOALS Blast_tac);
qed "parts_empty";
Addsimps [parts_empty];
goal thy "!!X. X: parts{} ==> P";
by (Asm_full_simp_tac 1);
qed "parts_emptyE";
AddSEs [parts_emptyE];
(*WARNING: loops if H = {Y}, therefore must not be repeated!*)
goal thy "!!H. X: parts H ==> EX Y:H. X: parts {Y}";
by (etac parts.induct 1);
by (ALLGOALS Blast_tac);
qed "parts_singleton";
(** Unions **)
goal thy "parts(G) Un parts(H) <= parts(G Un H)";
by (REPEAT (ares_tac [Un_least, parts_mono, Un_upper1, Un_upper2] 1));
val parts_Un_subset1 = result();
goal thy "parts(G Un H) <= parts(G) Un parts(H)";
by (rtac subsetI 1);
by (etac parts.induct 1);
by (ALLGOALS Blast_tac);
val parts_Un_subset2 = result();
goal thy "parts(G Un H) = parts(G) Un parts(H)";
by (REPEAT (ares_tac [equalityI, parts_Un_subset1, parts_Un_subset2] 1));
qed "parts_Un";
goal thy "parts (insert X H) = parts {X} Un parts H";
by (stac (read_instantiate [("A","H")] insert_is_Un) 1);
by (simp_tac (HOL_ss addsimps [parts_Un]) 1);
qed "parts_insert";
(*TWO inserts to avoid looping. This rewrite is better than nothing.
Not suitable for Addsimps: its behaviour can be strange.*)
goal thy "parts (insert X (insert Y H)) = parts {X} Un parts {Y} Un parts H";
by (simp_tac (simpset() addsimps [Un_assoc]) 1);
by (simp_tac (simpset() addsimps [parts_insert RS sym]) 1);
qed "parts_insert2";
goal thy "(UN x:A. parts(H x)) <= parts(UN x:A. H x)";
by (REPEAT (ares_tac [UN_least, parts_mono, UN_upper] 1));
val parts_UN_subset1 = result();
goal thy "parts(UN x:A. H x) <= (UN x:A. parts(H x))";
by (rtac subsetI 1);
by (etac parts.induct 1);
by (ALLGOALS Blast_tac);
val parts_UN_subset2 = result();
goal thy "parts(UN x:A. H x) = (UN x:A. parts(H x))";
by (REPEAT (ares_tac [equalityI, parts_UN_subset1, parts_UN_subset2] 1));
qed "parts_UN";
(*Added to simplify arguments to parts, analz and synth.
NOTE: the UN versions are no longer used!*)
Addsimps [parts_Un, parts_UN];
AddSEs [parts_Un RS equalityD1 RS subsetD RS UnE,
parts_UN RS equalityD1 RS subsetD RS UN_E];
goal thy "insert X (parts H) <= parts(insert X H)";
by (blast_tac (claset() addIs [impOfSubs parts_mono]) 1);
qed "parts_insert_subset";
(** Idempotence and transitivity **)
goal thy "!!H. X: parts (parts H) ==> X: parts H";
by (etac parts.induct 1);
by (ALLGOALS Blast_tac);
qed "parts_partsD";
AddSDs [parts_partsD];
goal thy "parts (parts H) = parts H";
by (Blast_tac 1);
qed "parts_idem";
Addsimps [parts_idem];
goal thy "!!H. [| X: parts G; G <= parts H |] ==> X: parts H";
by (dtac parts_mono 1);
by (Blast_tac 1);
qed "parts_trans";
(*Cut*)
goal thy "!!H. [| Y: parts (insert X G); X: parts H |] \
\ ==> Y: parts (G Un H)";
by (etac parts_trans 1);
by (Auto_tac());
qed "parts_cut";
goal thy "!!H. X: parts H ==> parts (insert X H) = parts H";
by (fast_tac (claset() addSDs [parts_cut]
addIs [parts_insertI]
addss (simpset())) 1);
qed "parts_cut_eq";
Addsimps [parts_cut_eq];
(** Rewrite rules for pulling out atomic messages **)
fun parts_tac i =
EVERY [rtac ([subsetI, parts_insert_subset] MRS equalityI) i,
etac parts.induct i,
REPEAT (Blast_tac i)];
goal thy "parts (insert (Agent agt) H) = insert (Agent agt) (parts H)";
by (parts_tac 1);
qed "parts_insert_Agent";
goal thy "parts (insert (Nonce N) H) = insert (Nonce N) (parts H)";
by (parts_tac 1);
qed "parts_insert_Nonce";
goal thy "parts (insert (Number N) H) = insert (Number N) (parts H)";
by (parts_tac 1);
qed "parts_insert_Number";
goal thy "parts (insert (Key K) H) = insert (Key K) (parts H)";
by (parts_tac 1);
qed "parts_insert_Key";
goal thy "parts (insert (Hash X) H) = insert (Hash X) (parts H)";
by (parts_tac 1);
qed "parts_insert_Hash";
goal thy "parts (insert (Crypt K X) H) = \
\ insert (Crypt K X) (parts (insert X H))";
by (rtac equalityI 1);
by (rtac subsetI 1);
by (etac parts.induct 1);
by (Auto_tac());
by (etac parts.induct 1);
by (ALLGOALS (blast_tac (claset() addIs [parts.Body])));
qed "parts_insert_Crypt";
goal thy "parts (insert {|X,Y|} H) = \
\ insert {|X,Y|} (parts (insert X (insert Y H)))";
by (rtac equalityI 1);
by (rtac subsetI 1);
by (etac parts.induct 1);
by (Auto_tac());
by (etac parts.induct 1);
by (ALLGOALS (blast_tac (claset() addIs [parts.Fst, parts.Snd])));
qed "parts_insert_MPair";
Addsimps [parts_insert_Agent, parts_insert_Nonce,
parts_insert_Number, parts_insert_Key,
parts_insert_Hash, parts_insert_Crypt, parts_insert_MPair];
goal thy "parts (Key``N) = Key``N";
by (Auto_tac());
by (etac parts.induct 1);
by (Auto_tac());
qed "parts_image_Key";
Addsimps [parts_image_Key];
(*In any message, there is an upper bound N on its greatest nonce.*)
goal thy "EX N. ALL n. N<=n --> Nonce n ~: parts {msg}";
by (induct_tac "msg" 1);
by (induct_tac "atomic" 1);
by (ALLGOALS (asm_simp_tac (simpset() addsimps [exI, parts_insert2])));
(*MPair case: blast_tac works out the necessary sum itself!*)
by (blast_tac (claset() addSEs [add_leE]) 2);
(*Nonce case*)
by (res_inst_tac [("x","N + Suc nat")] exI 1);
by (fast_tac (claset() addSEs [add_leE] addaltern trans_tac) 1);
qed "msg_Nonce_supply";
(**** Inductive relation "analz" ****)
val major::prems =
goal thy "[| {|X,Y|} : analz H; \
\ [| X : analz H; Y : analz H |] ==> P \
\ |] ==> P";
by (cut_facts_tac [major] 1);
by (resolve_tac prems 1);
by (REPEAT (eresolve_tac [asm_rl, analz.Fst, analz.Snd] 1));
qed "MPair_analz";
AddIs [analz.Inj];
AddSEs [MPair_analz]; (*Perhaps it should NOT be deemed safe!*)
AddDs [analz.Decrypt];
Addsimps [analz.Inj];
goal thy "H <= analz(H)";
by (Blast_tac 1);
qed "analz_increasing";
goal thy "analz H <= parts H";
by (rtac subsetI 1);
by (etac analz.induct 1);
by (ALLGOALS Blast_tac);
qed "analz_subset_parts";
bind_thm ("not_parts_not_analz", analz_subset_parts RS contra_subsetD);
goal thy "parts (analz H) = parts H";
by (rtac equalityI 1);
by (rtac (analz_subset_parts RS parts_mono RS subset_trans) 1);
by (Simp_tac 1);
by (blast_tac (claset() addIs [analz_increasing RS parts_mono RS subsetD]) 1);
qed "parts_analz";
Addsimps [parts_analz];
goal thy "analz (parts H) = parts H";
by (Auto_tac());
by (etac analz.induct 1);
by (Auto_tac());
qed "analz_parts";
Addsimps [analz_parts];
(*Monotonicity; Lemma 1 of Lowe*)
goalw thy analz.defs "!!G H. G<=H ==> analz(G) <= analz(H)";
by (rtac lfp_mono 1);
by (REPEAT (ares_tac basic_monos 1));
qed "analz_mono";
val analz_insertI = impOfSubs (subset_insertI RS analz_mono);
(** General equational properties **)
goal thy "analz{} = {}";
by Safe_tac;
by (etac analz.induct 1);
by (ALLGOALS Blast_tac);
qed "analz_empty";
Addsimps [analz_empty];
(*Converse fails: we can analz more from the union than from the
separate parts, as a key in one might decrypt a message in the other*)
goal thy "analz(G) Un analz(H) <= analz(G Un H)";
by (REPEAT (ares_tac [Un_least, analz_mono, Un_upper1, Un_upper2] 1));
qed "analz_Un";
goal thy "insert X (analz H) <= analz(insert X H)";
by (blast_tac (claset() addIs [impOfSubs analz_mono]) 1);
qed "analz_insert";
(** Rewrite rules for pulling out atomic messages **)
fun analz_tac i =
EVERY [rtac ([subsetI, analz_insert] MRS equalityI) i,
etac analz.induct i,
REPEAT (Blast_tac i)];
goal thy "analz (insert (Agent agt) H) = insert (Agent agt) (analz H)";
by (analz_tac 1);
qed "analz_insert_Agent";
goal thy "analz (insert (Nonce N) H) = insert (Nonce N) (analz H)";
by (analz_tac 1);
qed "analz_insert_Nonce";
goal thy "analz (insert (Number N) H) = insert (Number N) (analz H)";
by (analz_tac 1);
qed "analz_insert_Number";
goal thy "analz (insert (Hash X) H) = insert (Hash X) (analz H)";
by (analz_tac 1);
qed "analz_insert_Hash";
(*Can only pull out Keys if they are not needed to decrypt the rest*)
goalw thy [keysFor_def]
"!!K. K ~: keysFor (analz H) ==> \
\ analz (insert (Key K) H) = insert (Key K) (analz H)";
by (analz_tac 1);
qed "analz_insert_Key";
goal thy "analz (insert {|X,Y|} H) = \
\ insert {|X,Y|} (analz (insert X (insert Y H)))";
by (rtac equalityI 1);
by (rtac subsetI 1);
by (etac analz.induct 1);
by (Auto_tac());
by (etac analz.induct 1);
by (ALLGOALS (blast_tac (claset() addIs [analz.Fst, analz.Snd])));
qed "analz_insert_MPair";
(*Can pull out enCrypted message if the Key is not known*)
goal thy "!!H. Key (invKey K) ~: analz H ==> \
\ analz (insert (Crypt K X) H) = \
\ insert (Crypt K X) (analz H)";
by (analz_tac 1);
qed "analz_insert_Crypt";
goal thy "!!H. Key (invKey K) : analz H ==> \
\ analz (insert (Crypt K X) H) <= \
\ insert (Crypt K X) (analz (insert X H))";
by (rtac subsetI 1);
by (eres_inst_tac [("za","x")] analz.induct 1);
by (ALLGOALS (Blast_tac));
val lemma1 = result();
goal thy "!!H. Key (invKey K) : analz H ==> \
\ insert (Crypt K X) (analz (insert X H)) <= \
\ analz (insert (Crypt K X) H)";
by (Auto_tac());
by (eres_inst_tac [("za","x")] analz.induct 1);
by (Auto_tac());
by (blast_tac (claset() addIs [analz_insertI, analz.Decrypt]) 1);
val lemma2 = result();
goal thy "!!H. Key (invKey K) : analz H ==> \
\ analz (insert (Crypt K X) H) = \
\ insert (Crypt K X) (analz (insert X H))";
by (REPEAT (ares_tac [equalityI, lemma1, lemma2] 1));
qed "analz_insert_Decrypt";
(*Case analysis: either the message is secure, or it is not!
Effective, but can cause subgoals to blow up!
Use with expand_if; apparently split_tac does not cope with patterns
such as "analz (insert (Crypt K X) H)" *)
goal thy "analz (insert (Crypt K X) H) = \
\ (if (Key (invKey K) : analz H) \
\ then insert (Crypt K X) (analz (insert X H)) \
\ else insert (Crypt K X) (analz H))";
by (case_tac "Key (invKey K) : analz H " 1);
by (ALLGOALS (asm_simp_tac (simpset() addsimps [analz_insert_Crypt,
analz_insert_Decrypt])));
qed "analz_Crypt_if";
Addsimps [analz_insert_Agent, analz_insert_Nonce,
analz_insert_Number, analz_insert_Key,
analz_insert_Hash, analz_insert_MPair, analz_Crypt_if];
(*This rule supposes "for the sake of argument" that we have the key.*)
goal thy "analz (insert (Crypt K X) H) <= \
\ insert (Crypt K X) (analz (insert X H))";
by (rtac subsetI 1);
by (etac analz.induct 1);
by (Auto_tac());
qed "analz_insert_Crypt_subset";
goal thy "analz (Key``N) = Key``N";
by (Auto_tac());
by (etac analz.induct 1);
by (Auto_tac());
qed "analz_image_Key";
Addsimps [analz_image_Key];
(** Idempotence and transitivity **)
goal thy "!!H. X: analz (analz H) ==> X: analz H";
by (etac analz.induct 1);
by (ALLGOALS Blast_tac);
qed "analz_analzD";
AddSDs [analz_analzD];
goal thy "analz (analz H) = analz H";
by (Blast_tac 1);
qed "analz_idem";
Addsimps [analz_idem];
goal thy "!!H. [| X: analz G; G <= analz H |] ==> X: analz H";
by (dtac analz_mono 1);
by (Blast_tac 1);
qed "analz_trans";
(*Cut; Lemma 2 of Lowe*)
goal thy "!!H. [| Y: analz (insert X H); X: analz H |] ==> Y: analz H";
by (etac analz_trans 1);
by (Blast_tac 1);
qed "analz_cut";
(*Cut can be proved easily by induction on
"!!H. Y: analz (insert X H) ==> X: analz H --> Y: analz H"
*)
(*This rewrite rule helps in the simplification of messages that involve
the forwarding of unknown components (X). Without it, removing occurrences
of X can be very complicated. *)
goal thy "!!H. X: analz H ==> analz (insert X H) = analz H";
by (blast_tac (claset() addIs [analz_cut, analz_insertI]) 1);
qed "analz_insert_eq";
(** A congruence rule for "analz" **)
goal thy "!!H. [| analz G <= analz G'; analz H <= analz H' \
\ |] ==> analz (G Un H) <= analz (G' Un H')";
by (Clarify_tac 1);
by (etac analz.induct 1);
by (ALLGOALS (best_tac (claset() addIs [analz_mono RS subsetD])));
qed "analz_subset_cong";
goal thy "!!H. [| analz G = analz G'; analz H = analz H' \
\ |] ==> analz (G Un H) = analz (G' Un H')";
by (REPEAT_FIRST (ares_tac [equalityI, analz_subset_cong]
ORELSE' etac equalityE));
qed "analz_cong";
goal thy "!!H. analz H = analz H' ==> analz(insert X H) = analz(insert X H')";
by (asm_simp_tac (simpset() addsimps [insert_def] delsimps [singleton_conv]
setloop (rtac analz_cong)) 1);
qed "analz_insert_cong";
(*If there are no pairs or encryptions then analz does nothing*)
goal thy "!!H. [| ALL X Y. {|X,Y|} ~: H; ALL X K. Crypt K X ~: H |] ==> \
\ analz H = H";
by Safe_tac;
by (etac analz.induct 1);
by (ALLGOALS Blast_tac);
qed "analz_trivial";
(*These two are obsolete (with a single Spy) but cost little to prove...*)
goal thy "!!X. X: analz (UN i:A. analz (H i)) ==> X: analz (UN i:A. H i)";
by (etac analz.induct 1);
by (ALLGOALS (blast_tac (claset() addIs [impOfSubs analz_mono])));
val lemma = result();
goal thy "analz (UN i:A. analz (H i)) = analz (UN i:A. H i)";
by (blast_tac (claset() addIs [lemma, impOfSubs analz_mono]) 1);
qed "analz_UN_analz";
Addsimps [analz_UN_analz];
(**** Inductive relation "synth" ****)
AddIs synth.intrs;
(*Can only produce a nonce or key if it is already known,
but can synth a pair or encryption from its components...*)
val mk_cases = synth.mk_cases (atomic.simps @ msg.simps);
(*NO Agent_synth, as any Agent name can be synthesized. Ditto for Number*)
val Nonce_synth = mk_cases "Nonce n : synth H";
val Key_synth = mk_cases "Key K : synth H";
val Hash_synth = mk_cases "Hash X : synth H";
val MPair_synth = mk_cases "{|X,Y|} : synth H";
val Crypt_synth = mk_cases "Crypt K X : synth H";
AddSEs [Nonce_synth, Key_synth, Hash_synth, MPair_synth, Crypt_synth];
goal thy "H <= synth(H)";
by (Blast_tac 1);
qed "synth_increasing";
(*Monotonicity*)
goalw thy synth.defs "!!G H. G<=H ==> synth(G) <= synth(H)";
by (rtac lfp_mono 1);
by (REPEAT (ares_tac basic_monos 1));
qed "synth_mono";
(** Unions **)
(*Converse fails: we can synth more from the union than from the
separate parts, building a compound message using elements of each.*)
goal thy "synth(G) Un synth(H) <= synth(G Un H)";
by (REPEAT (ares_tac [Un_least, synth_mono, Un_upper1, Un_upper2] 1));
qed "synth_Un";
goal thy "insert X (synth H) <= synth(insert X H)";
by (blast_tac (claset() addIs [impOfSubs synth_mono]) 1);
qed "synth_insert";
(** Idempotence and transitivity **)
goal thy "!!H. X: synth (synth H) ==> X: synth H";
by (etac synth.induct 1);
by (ALLGOALS Blast_tac);
qed "synth_synthD";
AddSDs [synth_synthD];
goal thy "synth (synth H) = synth H";
by (Blast_tac 1);
qed "synth_idem";
goal thy "!!H. [| X: synth G; G <= synth H |] ==> X: synth H";
by (dtac synth_mono 1);
by (Blast_tac 1);
qed "synth_trans";
(*Cut; Lemma 2 of Lowe*)
goal thy "!!H. [| Y: synth (insert X H); X: synth H |] ==> Y: synth H";
by (etac synth_trans 1);
by (Blast_tac 1);
qed "synth_cut";
goal thy "Agent A : synth H";
by (Blast_tac 1);
qed "Agent_synth";
goal thy "Number n : synth H";
by (Blast_tac 1);
qed "Number_synth";
goal thy "(Nonce N : synth H) = (Nonce N : H)";
by (Blast_tac 1);
qed "Nonce_synth_eq";
goal thy "(Key K : synth H) = (Key K : H)";
by (Blast_tac 1);
qed "Key_synth_eq";
goal thy "!!K. Key K ~: H ==> (Crypt K X : synth H) = (Crypt K X : H)";
by (Blast_tac 1);
qed "Crypt_synth_eq";
Addsimps [Agent_synth, Number_synth,
Nonce_synth_eq, Key_synth_eq, Crypt_synth_eq];
goalw thy [keysFor_def]
"keysFor (synth H) = keysFor H Un invKey``{K. Key K : H}";
by (Blast_tac 1);
qed "keysFor_synth";
Addsimps [keysFor_synth];
(*** Combinations of parts, analz and synth ***)
goal thy "parts (synth H) = parts H Un synth H";
by (rtac equalityI 1);
by (rtac subsetI 1);
by (etac parts.induct 1);
by (ALLGOALS
(blast_tac (claset() addIs ((synth_increasing RS parts_mono RS subsetD)
::parts.intrs))));
qed "parts_synth";
Addsimps [parts_synth];
goal thy "analz (analz G Un H) = analz (G Un H)";
by (REPEAT_FIRST (resolve_tac [equalityI, analz_subset_cong]));
by (ALLGOALS Simp_tac);
qed "analz_analz_Un";
goal thy "analz (synth G Un H) = analz (G Un H) Un synth G";
by (rtac equalityI 1);
by (rtac subsetI 1);
by (etac analz.induct 1);
by (blast_tac (claset() addIs [impOfSubs analz_mono]) 5);
by (ALLGOALS (blast_tac (claset() addIs analz.intrs)));
qed "analz_synth_Un";
goal thy "analz (synth H) = analz H Un synth H";
by (cut_inst_tac [("H","{}")] analz_synth_Un 1);
by (Full_simp_tac 1);
qed "analz_synth";
Addsimps [analz_analz_Un, analz_synth_Un, analz_synth];
(** For reasoning about the Fake rule in traces **)
goal thy "!!Y. X: G ==> parts(insert X H) <= parts G Un parts H";
by (rtac ([parts_mono, parts_Un_subset2] MRS subset_trans) 1);
by (Blast_tac 1);
qed "parts_insert_subset_Un";
(*More specifically for Fake*)
goal thy "!!H. X: synth (analz G) ==> \
\ parts (insert X H) <= synth (analz G) Un parts G Un parts H";
by (dtac parts_insert_subset_Un 1);
by (Full_simp_tac 1);
by (Blast_tac 1);
qed "Fake_parts_insert";
goal thy
"!!H. [| Crypt K Y : parts (insert X H); X: synth (analz G); \
\ Key K ~: analz G |] \
\ ==> Crypt K Y : parts G Un parts H";
by (dtac (impOfSubs Fake_parts_insert) 1);
by (assume_tac 1);
by (blast_tac (claset() addDs [impOfSubs analz_subset_parts]) 1);
qed "Crypt_Fake_parts_insert";
goal thy "!!H. X: synth (analz G) ==> \
\ analz (insert X H) <= synth (analz G) Un analz (G Un H)";
by (rtac subsetI 1);
by (subgoal_tac "x : analz (synth (analz G) Un H)" 1);
by (blast_tac (claset() addIs [impOfSubs analz_mono,
impOfSubs (analz_mono RS synth_mono)]) 2);
by (Full_simp_tac 1);
by (Blast_tac 1);
qed "Fake_analz_insert";
goal thy "(X: analz H & X: parts H) = (X: analz H)";
by (blast_tac (claset() addIs [impOfSubs analz_subset_parts]) 1);
val analz_conj_parts = result();
goal thy "(X: analz H | X: parts H) = (X: parts H)";
by (blast_tac (claset() addIs [impOfSubs analz_subset_parts]) 1);
val analz_disj_parts = result();
AddIffs [analz_conj_parts, analz_disj_parts];
(*Without this equation, other rules for synth and analz would yield
redundant cases*)
goal thy "({|X,Y|} : synth (analz H)) = \
\ (X : synth (analz H) & Y : synth (analz H))";
by (Blast_tac 1);
qed "MPair_synth_analz";
AddIffs [MPair_synth_analz];
goal thy "!!K. [| Key K : analz H; Key (invKey K) : analz H |] \
\ ==> (Crypt K X : synth (analz H)) = (X : synth (analz H))";
by (Blast_tac 1);
qed "Crypt_synth_analz";
goal thy "!!K. X ~: synth (analz H) \
\ ==> (Hash{|X,Y|} : synth (analz H)) = (Hash{|X,Y|} : analz H)";
by (Blast_tac 1);
qed "Hash_synth_analz";
Addsimps [Hash_synth_analz];
(**** HPair: a combination of Hash and MPair ****)
(*** Freeness ***)
goalw thy [HPair_def] "Agent A ~= Hash[X] Y";
by (Simp_tac 1);
qed "Agent_neq_HPair";
goalw thy [HPair_def] "Nonce N ~= Hash[X] Y";
by (Simp_tac 1);
qed "Nonce_neq_HPair";
goalw thy [HPair_def] "Number N ~= Hash[X] Y";
by (Simp_tac 1);
qed "Number_neq_HPair";
goalw thy [HPair_def] "Key K ~= Hash[X] Y";
by (Simp_tac 1);
qed "Key_neq_HPair";
goalw thy [HPair_def] "Hash Z ~= Hash[X] Y";
by (Simp_tac 1);
qed "Hash_neq_HPair";
goalw thy [HPair_def] "Crypt K X' ~= Hash[X] Y";
by (Simp_tac 1);
qed "Crypt_neq_HPair";
val HPair_neqs = [Agent_neq_HPair, Nonce_neq_HPair, Number_neq_HPair,
Key_neq_HPair, Hash_neq_HPair, Crypt_neq_HPair];
AddIffs HPair_neqs;
AddIffs (HPair_neqs RL [not_sym]);
goalw thy [HPair_def] "(Hash[X'] Y' = Hash[X] Y) = (X' = X & Y'=Y)";
by (Simp_tac 1);
qed "HPair_eq";
goalw thy [HPair_def] "({|X',Y'|} = Hash[X] Y) = (X' = Hash{|X,Y|} & Y'=Y)";
by (Simp_tac 1);
qed "MPair_eq_HPair";
goalw thy [HPair_def] "(Hash[X] Y = {|X',Y'|}) = (X' = Hash{|X,Y|} & Y'=Y)";
by (Auto_tac());
qed "HPair_eq_MPair";
AddIffs [HPair_eq, MPair_eq_HPair, HPair_eq_MPair];
(*** Specialized laws, proved in terms of those for Hash and MPair ***)
goalw thy [HPair_def] "keysFor (insert (Hash[X] Y) H) = keysFor H";
by (Simp_tac 1);
qed "keysFor_insert_HPair";
goalw thy [HPair_def]
"parts (insert (Hash[X] Y) H) = \
\ insert (Hash[X] Y) (insert (Hash{|X,Y|}) (parts (insert Y H)))";
by (Simp_tac 1);
qed "parts_insert_HPair";
goalw thy [HPair_def]
"analz (insert (Hash[X] Y) H) = \
\ insert (Hash[X] Y) (insert (Hash{|X,Y|}) (analz (insert Y H)))";
by (Simp_tac 1);
qed "analz_insert_HPair";
goalw thy [HPair_def] "!!H. X ~: synth (analz H) \
\ ==> (Hash[X] Y : synth (analz H)) = \
\ (Hash {|X, Y|} : analz H & Y : synth (analz H))";
by (Simp_tac 1);
by (Blast_tac 1);
qed "HPair_synth_analz";
Addsimps [keysFor_insert_HPair, parts_insert_HPair, analz_insert_HPair,
HPair_synth_analz, HPair_synth_analz];
(*We do NOT want Crypt... messages broken up in protocols!!*)
Delrules partsEs;
(** Rewrites to push in Key and Crypt messages, so that other messages can
be pulled out using the analz_insert rules **)
fun insComm thy x y = read_instantiate_sg (sign_of thy) [("x",x), ("y",y)]
insert_commute;
val pushKeys = map (insComm thy "Key ?K")
["Agent ?C", "Nonce ?N", "Number ?N",
"Hash ?X", "MPair ?X ?Y", "Crypt ?X ?K'"];
val pushCrypts = map (insComm thy "Crypt ?X ?K")
["Agent ?C", "Nonce ?N", "Number ?N",
"Hash ?X'", "MPair ?X' ?Y"];
(*Cannot be added with Addsimps -- we don't always want to re-order messages*)
val pushes = pushKeys@pushCrypts;
(*** Tactics useful for many protocol proofs ***)
(*Prove base case (subgoal i) and simplify others. A typical base case
concerns Crypt K X ~: Key``shrK``bad and cannot be proved by rewriting
alone.*)
fun prove_simple_subgoals_tac i =
fast_tac (claset() addss (simpset())) i THEN
ALLGOALS Asm_simp_tac;
fun Fake_parts_insert_tac i =
blast_tac (claset() addDs [impOfSubs analz_subset_parts,
impOfSubs Fake_parts_insert]) i;
(*Apply rules to break down assumptions of the form
Y : parts(insert X H) and Y : analz(insert X H)
*)
val Fake_insert_tac =
dresolve_tac [impOfSubs Fake_analz_insert,
impOfSubs Fake_parts_insert] THEN'
eresolve_tac [asm_rl, synth.Inj];
fun Fake_insert_simp_tac i =
REPEAT (Fake_insert_tac i) THEN Asm_full_simp_tac i;
(*Analysis of Fake cases. Also works for messages that forward unknown parts,
but this application is no longer necessary if analz_insert_eq is used.
Abstraction over i is ESSENTIAL: it delays the dereferencing of claset
DEPENDS UPON "X" REFERRING TO THE FRADULENT MESSAGE *)
fun spy_analz_tac i =
DETERM
(SELECT_GOAL
(EVERY
[ (*push in occurrences of X...*)
(REPEAT o CHANGED)
(res_inst_tac [("x1","X")] (insert_commute RS ssubst) 1),
(*...allowing further simplifications*)
simp_tac (simpset() addsplits [expand_if]) 1,
REPEAT (FIRSTGOAL (resolve_tac [allI,impI,notI,conjI,iffI])),
DEPTH_SOLVE
(Fake_insert_simp_tac 1
THEN
IF_UNSOLVED (Blast.depth_tac
(claset() addIs [analz_insertI,
impOfSubs analz_subset_parts]) 4 1))
]) i);
(** Useful in many uniqueness proofs **)
fun ex_strip_tac i = REPEAT (swap_res_tac [exI, conjI] i) THEN
assume_tac (i+1);
(*Apply the EX-ALL quantifification to prove uniqueness theorems in
their standard form*)
fun prove_unique_tac lemma =
EVERY' [dtac lemma,
REPEAT o (mp_tac ORELSE' eresolve_tac [asm_rl,exE]),
(*Duplicate the assumption*)
forw_inst_tac [("psi", "ALL C.?P(C)")] asm_rl,
Blast.depth_tac (claset() addSDs [spec]) 0];
(*Needed occasionally with spy_analz_tac, e.g. in analz_insert_Key_newK*)
goal Set.thy "A Un (B Un A) = B Un A";
by (Blast_tac 1);
val Un_absorb3 = result();
Addsimps [Un_absorb3];
(*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];