--- a/src/HOL/Auth/Message.ML Fri Mar 02 13:14:37 2001 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,912 +0,0 @@
-(* 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()];
-
-AddIffs msg.inject;
-
-(*Equations hold because constructors are injective; cannot prove for all f*)
-Goal "(Friend x : Friend`A) = (x:A)";
-by Auto_tac;
-qed "Friend_image_eq";
-
-Goal "(Key x : Key`A) = (x:A)";
-by Auto_tac;
-qed "Key_image_eq";
-
-Goal "(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 "(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 [keysFor_def] "keysFor {} = {}";
-by (Blast_tac 1);
-qed "keysFor_empty";
-
-Goalw [keysFor_def] "keysFor (H Un H') = keysFor H Un keysFor H'";
-by (Blast_tac 1);
-qed "keysFor_Un";
-
-Goalw [keysFor_def] "keysFor (UN i:A. H i) = (UN i:A. keysFor (H i))";
-by (Blast_tac 1);
-qed "keysFor_UN";
-
-(*Monotonicity*)
-Goalw [keysFor_def] "G<=H ==> keysFor(G) <= keysFor(H)";
-by (Blast_tac 1);
-qed "keysFor_mono";
-
-Goalw [keysFor_def] "keysFor (insert (Agent A) H) = keysFor H";
-by Auto_tac;
-qed "keysFor_insert_Agent";
-
-Goalw [keysFor_def] "keysFor (insert (Nonce N) H) = keysFor H";
-by Auto_tac;
-qed "keysFor_insert_Nonce";
-
-Goalw [keysFor_def] "keysFor (insert (Number N) H) = keysFor H";
-by Auto_tac;
-qed "keysFor_insert_Number";
-
-Goalw [keysFor_def] "keysFor (insert (Key K) H) = keysFor H";
-by Auto_tac;
-qed "keysFor_insert_Key";
-
-Goalw [keysFor_def] "keysFor (insert (Hash X) H) = keysFor H";
-by Auto_tac;
-qed "keysFor_insert_Hash";
-
-Goalw [keysFor_def] "keysFor (insert {|X,Y|} H) = keysFor H";
-by Auto_tac;
-qed "keysFor_insert_MPair";
-
-Goalw [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 [keysFor_def] "keysFor (Key`E) = {}";
-by Auto_tac;
-qed "keysFor_image_Key";
-Addsimps [keysFor_image_Key];
-
-Goalw [keysFor_def] "Crypt K X : H ==> invKey K : keysFor H";
-by (Blast_tac 1);
-qed "Crypt_imp_invKey_keysFor";
-
-
-(**** Inductive relation "parts" ****)
-
-val major::prems =
-Goal "[| {|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];
-
-AddSEs [MPair_parts, make_elim parts.Body];
-(*NB These two rules are UNSAFE in the formal sense, as they discard the
- compound message. They work well on THIS FILE.
- MPair_parts is left as SAFE because it speeds up proofs.
- The Crypt rule is normally kept UNSAFE to avoid breaking up certificates.*)
-
-Goal "H <= parts(H)";
-by (Blast_tac 1);
-qed "parts_increasing";
-
-(*Monotonicity*)
-Goalw parts.defs "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 "parts{} = {}";
-by Safe_tac;
-by (etac parts.induct 1);
-by (ALLGOALS Blast_tac);
-qed "parts_empty";
-Addsimps [parts_empty];
-
-Goal "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 "X: parts H ==> EX Y:H. X: parts {Y}";
-by (etac parts.induct 1);
-by (ALLGOALS Blast_tac);
-qed "parts_singleton";
-
-
-(** Unions **)
-
-Goal "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 "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 "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 "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 "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 "(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 "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 "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 "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 "X: parts (parts H) ==> X: parts H";
-by (etac parts.induct 1);
-by (ALLGOALS Blast_tac);
-qed "parts_partsD";
-AddSDs [parts_partsD];
-
-Goal "parts (parts H) = parts H";
-by (Blast_tac 1);
-qed "parts_idem";
-Addsimps [parts_idem];
-
-Goal "[| X: parts G; G <= parts H |] ==> X: parts H";
-by (dtac parts_mono 1);
-by (Blast_tac 1);
-qed "parts_trans";
-
-(*Cut*)
-Goal "[| 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 "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,
- Auto_tac];
-
-Goal "parts (insert (Agent agt) H) = insert (Agent agt) (parts H)";
-by (parts_tac 1);
-qed "parts_insert_Agent";
-
-Goal "parts (insert (Nonce N) H) = insert (Nonce N) (parts H)";
-by (parts_tac 1);
-qed "parts_insert_Nonce";
-
-Goal "parts (insert (Number N) H) = insert (Number N) (parts H)";
-by (parts_tac 1);
-qed "parts_insert_Number";
-
-Goal "parts (insert (Key K) H) = insert (Key K) (parts H)";
-by (parts_tac 1);
-qed "parts_insert_Key";
-
-Goal "parts (insert (Hash X) H) = insert (Hash X) (parts H)";
-by (parts_tac 1);
-qed "parts_insert_Hash";
-
-Goal "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 "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 "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 "EX N. ALL n. N<=n --> Nonce n ~: parts {msg}";
-by (induct_tac "msg" 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 (auto_tac (claset() addSEs [add_leE], simpset()));
-qed "msg_Nonce_supply";
-
-
-(**** Inductive relation "analz" ****)
-
-val major::prems =
-Goal "[| {|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";
-
-AddSEs [MPair_analz]; (*Making it safe speeds up proofs*)
-AddDs [analz.Decrypt]; (*Unsafe: we don't want to split up certificates!*)
-AddIs [analz.Inj];
-
-Addsimps [analz.Inj];
-
-Goal "H <= analz(H)";
-by (Blast_tac 1);
-qed "analz_increasing";
-
-Goal "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 "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 "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 analz.defs "G<=H ==> analz(G) <= analz(H)";
-by (rtac lfp_mono 1);
-by (REPEAT (ares_tac basic_monos 1));
-qed "analz_mono";
-
-bind_thm ("analz_insertI", impOfSubs (subset_insertI RS analz_mono));
-
-(** General equational properties **)
-
-Goal "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 "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 "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,
- Auto_tac];
-
-Goal "analz (insert (Agent agt) H) = insert (Agent agt) (analz H)";
-by (analz_tac 1);
-qed "analz_insert_Agent";
-
-Goal "analz (insert (Nonce N) H) = insert (Nonce N) (analz H)";
-by (analz_tac 1);
-qed "analz_insert_Nonce";
-
-Goal "analz (insert (Number N) H) = insert (Number N) (analz H)";
-by (analz_tac 1);
-qed "analz_insert_Number";
-
-Goal "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 [keysFor_def]
- "K ~: keysFor (analz H) ==> \
-\ analz (insert (Key K) H) = insert (Key K) (analz H)";
-by (analz_tac 1);
-qed "analz_insert_Key";
-
-Goal "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 "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 "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 [("xa","x")] analz.induct 1);
-by Auto_tac;
-val lemma1 = result();
-
-Goal "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 [("xa","x")] analz.induct 1);
-by Auto_tac;
-by (blast_tac (claset() addIs [analz_insertI, analz.Decrypt]) 1);
-val lemma2 = result();
-
-Goal "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 split_if; apparently split_tac does not cope with patterns
- such as "analz (insert (Crypt K X) H)" *)
-Goal "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 "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 "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 "X: analz (analz H) ==> X: analz H";
-by (etac analz.induct 1);
-by (ALLGOALS Blast_tac);
-qed "analz_analzD";
-AddSDs [analz_analzD];
-
-Goal "analz (analz H) = analz H";
-by (Blast_tac 1);
-qed "analz_idem";
-Addsimps [analz_idem];
-
-Goal "[| 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 "[| 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
- "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 "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 "[| 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 "[| 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 "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 "[| 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 "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 "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;
-
-(*NO Agent_synth, as any Agent name can be synthesized. Ditto for Number*)
-val Nonce_synth = synth.mk_cases "Nonce n : synth H";
-val Key_synth = synth.mk_cases "Key K : synth H";
-val Hash_synth = synth.mk_cases "Hash X : synth H";
-val MPair_synth = synth.mk_cases "{|X,Y|} : synth H";
-val Crypt_synth = synth.mk_cases "Crypt K X : synth H";
-
-AddSEs [Nonce_synth, Key_synth, Hash_synth, MPair_synth, Crypt_synth];
-
-Goal "H <= synth(H)";
-by (Blast_tac 1);
-qed "synth_increasing";
-
-(*Monotonicity*)
-Goalw synth.defs "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 "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 "insert X (synth H) <= synth(insert X H)";
-by (blast_tac (claset() addIs [impOfSubs synth_mono]) 1);
-qed "synth_insert";
-
-(** Idempotence and transitivity **)
-
-Goal "X: synth (synth H) ==> X: synth H";
-by (etac synth.induct 1);
-by (ALLGOALS Blast_tac);
-qed "synth_synthD";
-AddSDs [synth_synthD];
-
-Goal "synth (synth H) = synth H";
-by (Blast_tac 1);
-qed "synth_idem";
-
-Goal "[| 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 "[| Y: synth (insert X H); X: synth H |] ==> Y: synth H";
-by (etac synth_trans 1);
-by (Blast_tac 1);
-qed "synth_cut";
-
-Goal "Agent A : synth H";
-by (Blast_tac 1);
-qed "Agent_synth";
-
-Goal "Number n : synth H";
-by (Blast_tac 1);
-qed "Number_synth";
-
-Goal "(Nonce N : synth H) = (Nonce N : H)";
-by (Blast_tac 1);
-qed "Nonce_synth_eq";
-
-Goal "(Key K : synth H) = (Key K : H)";
-by (Blast_tac 1);
-qed "Key_synth_eq";
-
-Goal "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 [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 "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 "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 "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 "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 "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. Very occasionally we could do with a version
- of the form parts{X} <= synth (analz H) Un parts H *)
-Goal "X: synth (analz H) ==> \
-\ parts (insert X H) <= synth (analz H) Un parts H";
-by (dtac parts_insert_subset_Un 1);
-by (Full_simp_tac 1);
-by (Blast_tac 1);
-qed "Fake_parts_insert";
-
-(*H is sometimes (Key ` KK Un spies evs), so can't put G=H*)
-Goal "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 "(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 "(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 "({|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 "[| 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 "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 [HPair_def] "Agent A ~= Hash[X] Y";
-by (Simp_tac 1);
-qed "Agent_neq_HPair";
-
-Goalw [HPair_def] "Nonce N ~= Hash[X] Y";
-by (Simp_tac 1);
-qed "Nonce_neq_HPair";
-
-Goalw [HPair_def] "Number N ~= Hash[X] Y";
-by (Simp_tac 1);
-qed "Number_neq_HPair";
-
-Goalw [HPair_def] "Key K ~= Hash[X] Y";
-by (Simp_tac 1);
-qed "Key_neq_HPair";
-
-Goalw [HPair_def] "Hash Z ~= Hash[X] Y";
-by (Simp_tac 1);
-qed "Hash_neq_HPair";
-
-Goalw [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 [HPair_def] "(Hash[X'] Y' = Hash[X] Y) = (X' = X & Y'=Y)";
-by (Simp_tac 1);
-qed "HPair_eq";
-
-Goalw [HPair_def] "({|X',Y'|} = Hash[X] Y) = (X' = Hash{|X,Y|} & Y'=Y)";
-by (Simp_tac 1);
-qed "MPair_eq_HPair";
-
-Goalw [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 [HPair_def] "keysFor (insert (Hash[X] Y) H) = keysFor H";
-by (Simp_tac 1);
-qed "keysFor_insert_HPair";
-
-Goalw [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 [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 [HPair_def] "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 [make_elim parts.Body];
-
-
-(** 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*)
-bind_thms ("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 =
- force_tac (claset(), simpset() addsimps [image_eq_UN]) i THEN
- ALLGOALS Asm_simp_tac;
-
-fun Fake_parts_insert_tac i =
- blast_tac (claset() addIs [parts_insertI]
- 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 *)
-
-val atomic_spy_analz_tac = SELECT_GOAL
- (Fake_insert_simp_tac 1
- THEN
- IF_UNSOLVED (Blast.depth_tac
- (claset() addIs [analz_insertI,
- impOfSubs analz_subset_parts]) 4 1));
-
-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 1,
- REPEAT (FIRSTGOAL (resolve_tac [allI,impI,notI,conjI,iffI])),
- DEPTH_SOLVE (atomic_spy_analz_tac 1)]) i);
-
-
-(*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];