src/HOL/Auth/OtwayRees_Bad.ML

 \        ==> EX K. EX NA. EX evs: otway.          \
 \               Says B A {|Nonce NA, Crypt (shrK A) {|Nonce NA, Key K|}|} \
 \                 : set_of_list evs";
 by (REPEAT (resolve_tac [exI,bexI] 1));
 by (rtac (otway.Nil RS otway.OR1 RS otway.OR2 RS otway.OR3 RS otway.OR4) 2);
-by (ALLGOALS (simp_tac (!simpset setsolver safe_solver)));
-by (REPEAT_FIRST (resolve_tac [refl, conjI]));
-by (ALLGOALS (fast_tac (!claset addss (!simpset setsolver safe_solver))));
+by possibility_tac;
 result();
 
 
 (**** Inductive proofs about otway ****)
-
-(*The Spy can see more than anybody else, except for their initial state*)
-goal thy 
- "!!evs. evs : otway ==> \
-\     sees lost A evs <= initState lost A Un sees lost Spy evs";
-by (etac otway.induct 1);
-by (ALLGOALS (fast_tac (!claset addDs [sees_Says_subset_insert RS subsetD] 
-                                addss (!simpset))));
-qed "sees_agent_subset_sees_Spy";
-
 
 (*Nobody sends themselves messages*)
 goal thy "!!evs. evs : otway ==> ALL A X. Says A A X ~: set_of_list evs";
 by (etac otway.induct 1);
 by (Auto_tac());

 goal thy "!!evs. Says A' B {|N, Agent A, Agent B, X|} : set_of_list evs ==> \
 \                X : analz (sees lost Spy evs)";
 by (fast_tac (!claset addSDs [Says_imp_sees_Spy RS analz.Inj]) 1);
 qed "OR2_analz_sees_Spy";
 
-goal thy "!!evs. Says S B {|N, X, X'|} : set_of_list evs ==> \
+goal thy "!!evs. Says S B {|N, X, Crypt (shrK B) X'|} : set_of_list evs ==> \
 \                X : analz (sees lost Spy evs)";
 by (fast_tac (!claset addSDs [Says_imp_sees_Spy RS analz.Inj]) 1);
 qed "OR4_analz_sees_Spy";
 
 goal thy "!!evs. Says Server B {|NA, X, Crypt K' {|NB,K|}|} : set_of_list evs \
 \                 ==> K : parts (sees lost Spy evs)";
-by (fast_tac (!claset addSEs partsEs
-                      addSDs [Says_imp_sees_Spy RS parts.Inj]) 1);
+by (fast_tac (!claset addSEs sees_Spy_partsEs) 1);
 qed "Oops_parts_sees_Spy";
 
 (*OR2_analz... and OR4_analz... let us treat those cases using the same 
   argument as for the Fake case.  This is possible for most, but not all,
   proofs: Fake does not invent new nonces (as in OR2), and of course Fake

     forward_tac [Oops_parts_sees_Spy] 7;
 
 (*For proving the easier theorems about X ~: parts (sees lost Spy evs) *)
 fun parts_induct_tac i = SELECT_GOAL
     (DETERM (etac otway.induct 1 THEN parts_Fake_tac THEN
-	     (*Fake message*)
-	     TRY (best_tac (!claset addDs [impOfSubs analz_subset_parts,
-					   impOfSubs Fake_parts_insert]
+             (*Fake message*)
+             TRY (best_tac (!claset addDs [impOfSubs analz_subset_parts,
+                                           impOfSubs Fake_parts_insert]
                                     addss (!simpset)) 2)) THEN
      (*Base case*)
      fast_tac (!claset addss (!simpset)) 1 THEN
      ALLGOALS Asm_simp_tac) i;
 

 
 bind_thm ("Spy_analz_shrK_D", analz_subset_parts RS subsetD RS Spy_see_shrK_D);
 AddSDs [Spy_see_shrK_D, Spy_analz_shrK_D];
 
 
-(*** Future keys can't be seen or used! ***)
-
-(*Nobody can have SEEN keys that will be generated in the future. *)
-goal thy "!!i. evs : otway ==>               \
-\              length evs <= i --> Key(newK i) ~: parts (sees lost Spy evs)";
+(*Nobody can have used non-existent keys!*)
+goal thy "!!evs. evs : otway ==>          \
+\         Key K ~: used evs --> K ~: keysFor (parts (sees lost Spy evs))";
 by (parts_induct_tac 1);
-by (REPEAT_FIRST (best_tac (!claset addEs [leD RS notE]
-				    addDs [impOfSubs analz_subset_parts,
-					   impOfSubs parts_insert_subset_Un,
-					   Suc_leD]
-                                addss (!simpset))));
-qed_spec_mp "new_keys_not_seen";
-Addsimps [new_keys_not_seen];
-
-(*Variant: old messages must contain old keys!*)
-goal thy 
- "!!evs. [| Says A B X : set_of_list evs;          \
-\           Key (newK i) : parts {X};    \
-\           evs : otway                            \
-\        |] ==> i < length evs";
-by (rtac ccontr 1);
-by (dtac leI 1);
-by (fast_tac (!claset addSDs [new_keys_not_seen, Says_imp_sees_Spy]
-                      addIs  [impOfSubs parts_mono]) 1);
-qed "Says_imp_old_keys";
-
-
-(*** Future nonces can't be seen or used! ***)
-
-goal thy "!!i. evs : otway ==>               \
-\              length evs <= i --> Nonce(newN i) ~: parts (sees lost Spy evs)";
-by (parts_induct_tac 1);
-by (REPEAT_FIRST
-    (fast_tac (!claset addSEs partsEs
-	               addSDs  [Says_imp_sees_Spy RS parts.Inj]
-		       addEs [leD RS notE]
-		       addDs  [impOfSubs analz_subset_parts,
-			       impOfSubs parts_insert_subset_Un, Suc_leD]
-		       addss (!simpset))));
-qed_spec_mp "new_nonces_not_seen";
-Addsimps [new_nonces_not_seen];
-
-(*Variant: old messages must contain old nonces!*)
-goal thy 
- "!!evs. [| Says A B X : set_of_list evs;            \
-\           Nonce (newN i) : parts {X};    \
-\           evs : otway                              \
-\        |] ==> i < length evs";
-by (rtac ccontr 1);
-by (dtac leI 1);
-by (fast_tac (!claset addSDs [new_nonces_not_seen, Says_imp_sees_Spy]
-                      addIs  [impOfSubs parts_mono]) 1);
-qed "Says_imp_old_nonces";
-
-
-(*Nobody can have USED keys that will be generated in the future.
-  ...very like new_keys_not_seen*)
-goal thy "!!i. evs : otway ==>               \
-\           length evs <= i --> newK i ~: keysFor (parts (sees lost Spy evs))";
-by (parts_induct_tac 1);
-(*OR1 and OR3*)
-by (EVERY (map (fast_tac (!claset addDs [Suc_leD] addss (!simpset))) [4,2]));
-(*Fake, OR2, OR4: these messages send unknown (X) components*)
-by (REPEAT
-    (best_tac
-      (!claset addDs [impOfSubs (analz_subset_parts RS keysFor_mono),
-                      impOfSubs (parts_insert_subset_Un RS keysFor_mono),
-                      Suc_leD]
-               addEs [new_keys_not_seen RS not_parts_not_analz RSN(2,rev_notE)]
-               addss (!simpset)) 1));
+(*Fake*)
+by (best_tac
+      (!claset addIs [impOfSubs analz_subset_parts]
+               addDs [impOfSubs (analz_subset_parts RS keysFor_mono),
+                      impOfSubs (parts_insert_subset_Un RS keysFor_mono)]
+               addss (!simpset)) 1);
+(*OR1-3*)
+by (REPEAT (fast_tac (!claset addss (!simpset)) 1));
 qed_spec_mp "new_keys_not_used";
 
 bind_thm ("new_keys_not_analzd",
           [analz_subset_parts RS keysFor_mono,
            new_keys_not_used] MRS contra_subsetD);

 (*** Proofs involving analz ***)
 
 (*Describes the form of K and NA when the Server sends this message.  Also
   for Oops case.*)
 goal thy 
- "!!evs. [| Says Server B \
-\            {|NA, X, Crypt (shrK B) {|NB, K|}|} : set_of_list evs;  \
-\           evs : otway |]                                           \
-\        ==> (EX n. K = Key(newK n)) &                               \
-\            (EX i. NA = Nonce i) & (EX j. NB = Nonce j)";
+ "!!evs. [| Says Server B                                                 \
+\            {|NA, X, Crypt (shrK B) {|NB, Key K|}|} : set_of_list evs;   \
+\           evs : otway |]                                                \
+\     ==> K ~: range shrK & (EX i. NA = Nonce i) & (EX j. NB = Nonce j)";
 by (etac rev_mp 1);
 by (etac otway.induct 1);
 by (ALLGOALS (fast_tac (!claset addss (!simpset))));
 qed "Says_Server_message_form";
 
 
 (*For proofs involving analz.*)
 val analz_Fake_tac = 
     dtac OR2_analz_sees_Spy 4 THEN 
     dtac OR4_analz_sees_Spy 6 THEN
-    forward_tac [Says_Server_message_form] 7 THEN
-    assume_tac 7 THEN Full_simp_tac 7 THEN
+    forward_tac [Says_Server_message_form] 7 THEN assume_tac 7 THEN 
     REPEAT ((eresolve_tac [exE, conjE] ORELSE' hyp_subst_tac) 7);
 
 
 (****
  The following is to prove theorems of the form
 
-  Key K : analz (insert (Key (newK i)) (sees lost Spy evs)) ==>
+  Key K : analz (insert (Key KAB) (sees lost Spy evs)) ==>
   Key K : analz (sees lost Spy evs)
 
  A more general formula must be proved inductively.
-
 ****)
 
 
 (** Session keys are not used to encrypt other session keys **)
-
-(*Lemma for the trivial direction of the if-and-only-if*)
-goal thy  
- "!!evs. (Key K : analz (Key``nE Un sEe)) --> \
-\         (K : nE | Key K : analz sEe)  ==>     \
-\        (Key K : analz (Key``nE Un sEe)) = (K : nE | Key K : analz sEe)";
-by (fast_tac (!claset addSEs [impOfSubs analz_mono]) 1);
-val lemma = result();
-
 
 (*The equality makes the induction hypothesis easier to apply*)
 goal thy  
  "!!evs. evs : otway ==> \
-\  ALL K E. (Key K : analz (Key``(newK``E) Un (sees lost Spy evs))) = \
-\           (K : newK``E | Key K : analz (sees lost Spy evs))";
+\  ALL K KK. KK <= Compl (range shrK) -->                      \
+\            (Key K : analz (Key``KK Un (sees lost Spy evs))) = \
+\            (K : KK | Key K : analz (sees lost Spy evs))";
 by (etac otway.induct 1);
 by analz_Fake_tac;
-by (REPEAT_FIRST (ares_tac [allI, lemma]));
-by (ALLGOALS (*Takes 18 secs*)
-    (asm_simp_tac 
-     (!simpset addsimps [Un_assoc RS sym, 
-			 insert_Key_singleton, insert_Key_image, pushKey_newK]
-               setloop split_tac [expand_if])));
-(*OR4, OR2, Fake*) 
-by (EVERY (map spy_analz_tac [5,3,2]));
-(*Oops, OR3, Base*)
-by (REPEAT (fast_tac (!claset addIs [image_eqI] addss (!simpset)) 1));
-qed_spec_mp "analz_image_newK";
+by (REPEAT_FIRST (resolve_tac [allI, impI]));
+by (REPEAT_FIRST (rtac analz_image_freshK_lemma ));
+by (ALLGOALS (asm_simp_tac analz_image_freshK_ss));
+(*Base*)
+by (fast_tac (!claset addIs [image_eqI] addss (!simpset)) 1);
+(*Fake, OR2, OR4*) 
+by (REPEAT (spy_analz_tac 1));
+qed_spec_mp "analz_image_freshK";
 
 
 goal thy
- "!!evs. evs : otway ==>                               \
-\        Key K : analz (insert (Key(newK i)) (sees lost Spy evs)) = \
-\        (K = newK i | Key K : analz (sees lost Spy evs))";
-by (asm_simp_tac (HOL_ss addsimps [pushKey_newK, analz_image_newK, 
-                                   insert_Key_singleton]) 1);
-by (Fast_tac 1);
-qed "analz_insert_Key_newK";
+ "!!evs. [| evs : otway;  KAB ~: range shrK |] ==>              \
+\        Key K : analz (insert (Key KAB) (sees lost Spy evs)) = \
+\        (K = KAB | Key K : analz (sees lost Spy evs))";
+by (asm_simp_tac (analz_image_freshK_ss addsimps [analz_image_freshK]) 1);
+qed "analz_insert_freshK";
 
 
 (*** The Key K uniquely identifies the Server's  message. **)
 
 goal thy 

 (*Remaining cases: OR3 and OR4*)
 by (ex_strip_tac 2);
 by (Fast_tac 2);
 by (expand_case_tac "K = ?y" 1);
 by (REPEAT (ares_tac [refl,exI,impI,conjI] 2));
-(*...we assume X is a very new message, and handle this case by contradiction*)
-by (fast_tac (!claset addEs [Says_imp_old_keys RS less_irrefl]
+(*...we assume X is a recent message, and handle this case by contradiction*)
+by (fast_tac (!claset addSEs sees_Spy_partsEs
                       delrules [conjI]    (*prevent split-up into 4 subgoals*)
                       addss (!simpset addsimps [parts_insertI])) 1);
 val lemma = result();
 
 goal thy 

 \            Says B Spy {|NA, NB, Key K|} ~: set_of_list evs -->      \
 \            Key K ~: analz (sees lost Spy evs)";
 by (etac otway.induct 1);
 by analz_Fake_tac;
 by (ALLGOALS
-    (asm_simp_tac (!simpset addsimps [not_parts_not_analz,
-				      analz_insert_Key_newK]
-		            setloop split_tac [expand_if])));
+    (asm_simp_tac (!simpset addcongs [conj_cong] 
+                            addsimps [not_parts_not_analz, analz_insert_freshK]
+                            setloop split_tac [expand_if])));
 (*OR3*)
-by (fast_tac (!claset addEs [Says_imp_old_keys RS less_irrefl]
+by (fast_tac (!claset delrules [impCE]
+                      addSEs sees_Spy_partsEs
+                      addIs [impOfSubs analz_subset_parts]
                       addss (!simpset addsimps [parts_insert2])) 3);
 (*OR4, OR2, Fake*) 
 by (REPEAT_FIRST spy_analz_tac);
 (*Oops*) (** LEVEL 5 **)
 by (fast_tac (!claset delrules [disjE]
-		      addDs [unique_session_keys] addss (!simpset)) 1);
+                      addDs [unique_session_keys] addss (!simpset)) 1);
 val lemma = result() RS mp RS mp RSN(2,rev_notE);
 
-
-goal thy 
- "!!evs. [| Says Server B                                         \
-\            {|NA, Crypt (shrK A) {|NA, K|},                      \
-\                  Crypt (shrK B) {|NB, K|}|} : set_of_list evs;  \
-\            Says B Spy {|NA, NB, K|} ~: set_of_list evs;         \
+goal thy 
+ "!!evs. [| Says Server B                                                \
+\            {|NA, Crypt (shrK A) {|NA, Key K|},                         \
+\                  Crypt (shrK B) {|NB, Key K|}|} : set_of_list evs;     \
+\           Says B Spy {|NA, NB, Key K|} ~: set_of_list evs;             \
 \           A ~: lost;  B ~: lost;  evs : otway |]                \
-\        ==> K ~: analz (sees lost Spy evs)";
+\        ==> Key K ~: analz (sees lost Spy evs)";
 by (forward_tac [Says_Server_message_form] 1 THEN assume_tac 1);
 by (fast_tac (!claset addSEs [lemma]) 1);
 qed "Spy_not_see_encrypted_key";
 
 
 (*** Attempting to prove stronger properties ***)
 
 (*Only OR1 can have caused such a part of a message to appear.
   I'm not sure why A ~= B premise is needed: OtwayRees.ML doesn't need it.
   Perhaps it's because OR2 has two similar-looking encrypted messages in
-	this version.*)
-goal thy 
- "!!evs. [| A ~: lost;  A ~= B; evs : otway |]                 \
+        this version.*)
+goal thy 
+ "!!evs. [| A ~: lost;  A ~= B;  evs : otway |]                \
 \        ==> Crypt (shrK A) {|NA, Agent A, Agent B|}           \
 \             : parts (sees lost Spy evs) -->                  \
 \            Says A B {|NA, Agent A, Agent B,                  \
 \                       Crypt (shrK A) {|NA, Agent A, Agent B|}|}  \
 \             : set_of_list evs";

 \                   Crypt (shrK B) {|NB, Key K|}|}             \
 \                   : set_of_list evs)";
 by (parts_induct_tac 1);
 (*OR1: it cannot be a new Nonce, contradiction.*)
 by (fast_tac (!claset addSIs [parts_insertI]
-                      addSEs partsEs
-                      addEs [Says_imp_old_nonces RS less_irrefl]
+                      addSEs sees_Spy_partsEs
                       addss (!simpset)) 1);
 (*OR4*)
 by (REPEAT (Safe_step_tac 2));
 by (REPEAT (best_tac (!claset addSDs [parts_cut]) 3));
 by (fast_tac (!claset addSIs [Crypt_imp_OR1]
-                      addEs  partsEs
-                      addDs [Says_imp_sees_Spy RS parts.Inj]) 2);
+                      addEs  sees_Spy_partsEs) 2);
 (*OR3*)  (** LEVEL 5 **)
 by (ALLGOALS (asm_simp_tac (!simpset addsimps [ex_disj_distrib])));
 by (step_tac (!claset delrules [disjCI, impCE]) 1);
 (*The hypotheses at this point suggest an attack in which nonce NA is used
   in two different roles: