src/HOL/Auth/Yahalom.thy

    Nil:  "[] \<in> yahalom"
 
          (*The spy MAY say anything he CAN say.  We do not expect him to
            invent new nonces here, but he can also use NS1.  Common to
            all similar protocols.*)
- | Fake: "[| evsf \<in> yahalom;  X \<in> synth (analz (knows Spy evsf)) |]
-          ==> Says Spy B X  # evsf \<in> yahalom"
+ | Fake: "\<lbrakk>evsf \<in> yahalom;  X \<in> synth (analz (knows Spy evsf))\<rbrakk>
+          \<Longrightarrow> Says Spy B X  # evsf \<in> yahalom"
 
          (*A message that has been sent can be received by the
            intended recipient.*)
- | Reception: "[| evsr \<in> yahalom;  Says A B X \<in> set evsr |]
-               ==> Gets B X # evsr \<in> yahalom"
+ | Reception: "\<lbrakk>evsr \<in> yahalom;  Says A B X \<in> set evsr\<rbrakk>
+               \<Longrightarrow> Gets B X # evsr \<in> yahalom"
 
          (*Alice initiates a protocol run*)
- | YM1:  "[| evs1 \<in> yahalom;  Nonce NA \<notin> used evs1 |]
-          ==> Says A B \<lbrace>Agent A, Nonce NA\<rbrace> # evs1 \<in> yahalom"
+ | YM1:  "\<lbrakk>evs1 \<in> yahalom;  Nonce NA \<notin> used evs1\<rbrakk>
+          \<Longrightarrow> Says A B \<lbrace>Agent A, Nonce NA\<rbrace> # evs1 \<in> yahalom"
 
          (*Bob's response to Alice's message.*)
- | YM2:  "[| evs2 \<in> yahalom;  Nonce NB \<notin> used evs2;
-             Gets B \<lbrace>Agent A, Nonce NA\<rbrace> \<in> set evs2 |]
-          ==> Says B Server 
+ | YM2:  "\<lbrakk>evs2 \<in> yahalom;  Nonce NB \<notin> used evs2;
+             Gets B \<lbrace>Agent A, Nonce NA\<rbrace> \<in> set evs2\<rbrakk>
+          \<Longrightarrow> Says B Server 
                   \<lbrace>Agent B, Crypt (shrK B) \<lbrace>Agent A, Nonce NA, Nonce NB\<rbrace>\<rbrace>
                 # evs2 \<in> yahalom"
 
          (*The Server receives Bob's message.  He responds by sending a
             new session key to Alice, with a packet for forwarding to Bob.*)
- | YM3:  "[| evs3 \<in> yahalom;  Key KAB \<notin> used evs3;  KAB \<in> symKeys;
+ | YM3:  "\<lbrakk>evs3 \<in> yahalom;  Key KAB \<notin> used evs3;  KAB \<in> symKeys;
              Gets Server 
                   \<lbrace>Agent B, Crypt (shrK B) \<lbrace>Agent A, Nonce NA, Nonce NB\<rbrace>\<rbrace>
-               \<in> set evs3 |]
-          ==> Says Server A
+               \<in> set evs3\<rbrakk>
+          \<Longrightarrow> Says Server A
                    \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key KAB, Nonce NA, Nonce NB\<rbrace>,
                      Crypt (shrK B) \<lbrace>Agent A, Key KAB\<rbrace>\<rbrace>
                 # evs3 \<in> yahalom"
 
  | YM4:  
        \<comment>\<open>Alice receives the Server's (?) message, checks her Nonce, and
            uses the new session key to send Bob his Nonce.  The premise
            @{term "A \<noteq> Server"} is needed for \<open>Says_Server_not_range\<close>.
            Alice can check that K is symmetric by its length.\<close>
-         "[| evs4 \<in> yahalom;  A \<noteq> Server;  K \<in> symKeys;
+         "\<lbrakk>evs4 \<in> yahalom;  A \<noteq> Server;  K \<in> symKeys;
              Gets A \<lbrace>Crypt(shrK A) \<lbrace>Agent B, Key K, Nonce NA, Nonce NB\<rbrace>, X\<rbrace>
                 \<in> set evs4;
-             Says A B \<lbrace>Agent A, Nonce NA\<rbrace> \<in> set evs4 |]
-          ==> Says A B \<lbrace>X, Crypt K (Nonce NB)\<rbrace> # evs4 \<in> yahalom"
+             Says A B \<lbrace>Agent A, Nonce NA\<rbrace> \<in> set evs4\<rbrakk>
+          \<Longrightarrow> Says A B \<lbrace>X, Crypt K (Nonce NB)\<rbrace> # evs4 \<in> yahalom"
 
          (*This message models possible leaks of session keys.  The Nonces
            identify the protocol run.  Quoting Server here ensures they are
            correct.*)
- | Oops: "[| evso \<in> yahalom;  
+ | Oops: "\<lbrakk>evso \<in> yahalom;  
              Says Server A \<lbrace>Crypt (shrK A)
                                    \<lbrace>Agent B, Key K, Nonce NA, Nonce NB\<rbrace>,
-                             X\<rbrace>  \<in> set evso |]
-          ==> Notes Spy \<lbrace>Nonce NA, Nonce NB, Key K\<rbrace> # evso \<in> yahalom"
+                             X\<rbrace>  \<in> set evso\<rbrakk>
+          \<Longrightarrow> Notes Spy \<lbrace>Nonce NA, Nonce NB, Key K\<rbrace> # evso \<in> yahalom"
 
 
 definition KeyWithNonce :: "[key, nat, event list] => bool" where
   "KeyWithNonce K NB evs ==
      \<exists>A B na X. 

 declare parts.Body  [dest]
 declare Fake_parts_insert_in_Un  [dest]
 declare analz_into_parts [dest]
 
 text\<open>A "possibility property": there are traces that reach the end\<close>
-lemma "[| A \<noteq> Server; K \<in> symKeys; Key K \<notin> used [] |]
-      ==> \<exists>X NB. \<exists>evs \<in> yahalom.
+lemma "\<lbrakk>A \<noteq> Server; K \<in> symKeys; Key K \<notin> used []\<rbrakk>
+      \<Longrightarrow> \<exists>X NB. \<exists>evs \<in> yahalom.
              Says A B \<lbrace>X, Crypt K (Nonce NB)\<rbrace> \<in> set evs"
 apply (intro exI bexI)
 apply (rule_tac [2] yahalom.Nil
                     [THEN yahalom.YM1, THEN yahalom.Reception,
                      THEN yahalom.YM2, THEN yahalom.Reception,

 
 
 subsection\<open>Regularity Lemmas for Yahalom\<close>
 
 lemma Gets_imp_Says:
-     "[| Gets B X \<in> set evs; evs \<in> yahalom |] ==> \<exists>A. Says A B X \<in> set evs"
+     "\<lbrakk>Gets B X \<in> set evs; evs \<in> yahalom\<rbrakk> \<Longrightarrow> \<exists>A. Says A B X \<in> set evs"
 by (erule rev_mp, erule yahalom.induct, auto)
 
 text\<open>Must be proved separately for each protocol\<close>
 lemma Gets_imp_knows_Spy:
-     "[| Gets B X \<in> set evs; evs \<in> yahalom |]  ==> X \<in> knows Spy evs"
+     "\<lbrakk>Gets B X \<in> set evs; evs \<in> yahalom\<rbrakk>  \<Longrightarrow> X \<in> knows Spy evs"
 by (blast dest!: Gets_imp_Says Says_imp_knows_Spy)
 
 lemmas Gets_imp_analz_Spy = Gets_imp_knows_Spy [THEN analz.Inj]
 declare Gets_imp_analz_Spy [dest]
 
 
 text\<open>Lets us treat YM4 using a similar argument as for the Fake case.\<close>
 lemma YM4_analz_knows_Spy:
-     "[| Gets A \<lbrace>Crypt (shrK A) Y, X\<rbrace> \<in> set evs;  evs \<in> yahalom |]
-      ==> X \<in> analz (knows Spy evs)"
+     "\<lbrakk>Gets A \<lbrace>Crypt (shrK A) Y, X\<rbrace> \<in> set evs;  evs \<in> yahalom\<rbrakk>
+      \<Longrightarrow> X \<in> analz (knows Spy evs)"
 by blast
 
 lemmas YM4_parts_knows_Spy =
        YM4_analz_knows_Spy [THEN analz_into_parts]
 
 text\<open>For Oops\<close>
 lemma YM4_Key_parts_knows_Spy:
      "Says Server A \<lbrace>Crypt (shrK A) \<lbrace>B,K,NA,NB\<rbrace>, X\<rbrace> \<in> set evs
-      ==> K \<in> parts (knows Spy evs)"
+      \<Longrightarrow> K \<in> parts (knows Spy evs)"
   by (metis parts.Body parts.Fst parts.Snd  Says_imp_knows_Spy parts.Inj)
 
 text\<open>Theorems of the form @{term "X \<notin> parts (knows Spy evs)"} imply 
 that NOBODY sends messages containing X!\<close>
 
 text\<open>Spy never sees a good agent's shared key!\<close>
 lemma Spy_see_shrK [simp]:
-     "evs \<in> yahalom ==> (Key (shrK A) \<in> parts (knows Spy evs)) = (A \<in> bad)"
+     "evs \<in> yahalom \<Longrightarrow> (Key (shrK A) \<in> parts (knows Spy evs)) = (A \<in> bad)"
 by (erule yahalom.induct, force,
     drule_tac [6] YM4_parts_knows_Spy, simp_all, blast+)
 
 lemma Spy_analz_shrK [simp]:
-     "evs \<in> yahalom ==> (Key (shrK A) \<in> analz (knows Spy evs)) = (A \<in> bad)"
+     "evs \<in> yahalom \<Longrightarrow> (Key (shrK A) \<in> analz (knows Spy evs)) = (A \<in> bad)"
 by auto
 
 lemma Spy_see_shrK_D [dest!]:
-     "[|Key (shrK A) \<in> parts (knows Spy evs);  evs \<in> yahalom|] ==> A \<in> bad"
+     "\<lbrakk>Key (shrK A) \<in> parts (knows Spy evs);  evs \<in> yahalom\<rbrakk> \<Longrightarrow> A \<in> bad"
 by (blast dest: Spy_see_shrK)
 
 text\<open>Nobody can have used non-existent keys!
     Needed to apply \<open>analz_insert_Key\<close>\<close>
 lemma new_keys_not_used [simp]:
-    "[|Key K \<notin> used evs; K \<in> symKeys; evs \<in> yahalom|]
-     ==> K \<notin> keysFor (parts (spies evs))"
+    "\<lbrakk>Key K \<notin> used evs; K \<in> symKeys; evs \<in> yahalom\<rbrakk>
+     \<Longrightarrow> K \<notin> keysFor (parts (spies evs))"
 apply (erule rev_mp)
 apply (erule yahalom.induct, force,
        frule_tac [6] YM4_parts_knows_Spy, simp_all)
 txt\<open>Fake\<close>
 apply (force dest!: keysFor_parts_insert, auto)

 
 
 text\<open>Earlier, all protocol proofs declared this theorem.
   But only a few proofs need it, e.g. Yahalom and Kerberos IV.\<close>
 lemma new_keys_not_analzd:
- "[|K \<in> symKeys; evs \<in> yahalom; Key K \<notin> used evs|]
-  ==> K \<notin> keysFor (analz (knows Spy evs))"
+ "\<lbrakk>K \<in> symKeys; evs \<in> yahalom; Key K \<notin> used evs\<rbrakk>
+  \<Longrightarrow> K \<notin> keysFor (analz (knows Spy evs))"
 by (blast dest: new_keys_not_used intro: keysFor_mono [THEN subsetD])
 
 
 text\<open>Describes the form of K when the Server sends this message.  Useful for
   Oops as well as main secrecy property.\<close>
 lemma Says_Server_not_range [simp]:
-     "[| Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, na, nb\<rbrace>, X\<rbrace>
-           \<in> set evs;   evs \<in> yahalom |]
-      ==> K \<notin> range shrK"
+     "\<lbrakk>Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, na, nb\<rbrace>, X\<rbrace>
+           \<in> set evs;   evs \<in> yahalom\<rbrakk>
+      \<Longrightarrow> K \<notin> range shrK"
 by (erule rev_mp, erule yahalom.induct, simp_all)
 
 
 subsection\<open>Secrecy Theorems\<close>
 
 (****
  The following is to prove theorems of the form
 
-  Key K \<in> analz (insert (Key KAB) (knows Spy evs)) ==>
+  Key K \<in> analz (insert (Key KAB) (knows Spy evs)) \<Longrightarrow>
   Key K \<in> analz (knows Spy evs)
 
  A more general formula must be proved inductively.
 ****)
 
 text\<open>Session keys are not used to encrypt other session keys\<close>
 
 lemma analz_image_freshK [rule_format]:
- "evs \<in> yahalom ==>
+ "evs \<in> yahalom \<Longrightarrow>
    \<forall>K KK. KK <= - (range shrK) -->
           (Key K \<in> analz (Key`KK Un (knows Spy evs))) =
           (K \<in> KK | Key K \<in> analz (knows Spy evs))"
 apply (erule yahalom.induct,
        drule_tac [7] YM4_analz_knows_Spy, analz_freshK, spy_analz, blast)
 apply (simp only: Says_Server_not_range analz_image_freshK_simps)
 apply safe
 done
 
 lemma analz_insert_freshK:
-     "[| evs \<in> yahalom;  KAB \<notin> range shrK |] ==>
+     "\<lbrakk>evs \<in> yahalom;  KAB \<notin> range shrK\<rbrakk> \<Longrightarrow>
       (Key K \<in> analz (insert (Key KAB) (knows Spy evs))) =
       (K = KAB | Key K \<in> analz (knows Spy evs))"
 by (simp only: analz_image_freshK analz_image_freshK_simps)
 
 
 text\<open>The Key K uniquely identifies the Server's  message.\<close>
 lemma unique_session_keys:
-     "[| Says Server A
+     "\<lbrakk>Says Server A
           \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, na, nb\<rbrace>, X\<rbrace> \<in> set evs;
         Says Server A'
           \<lbrace>Crypt (shrK A') \<lbrace>Agent B', Key K, na', nb'\<rbrace>, X'\<rbrace> \<in> set evs;
-        evs \<in> yahalom |]
-     ==> A=A' & B=B' & na=na' & nb=nb'"
+        evs \<in> yahalom\<rbrakk>
+     \<Longrightarrow> A=A' & B=B' & na=na' & nb=nb'"
 apply (erule rev_mp, erule rev_mp)
 apply (erule yahalom.induct, simp_all)
 txt\<open>YM3, by freshness, and YM4\<close>
 apply blast+
 done
 
 
 text\<open>Crucial secrecy property: Spy does not see the keys sent in msg YM3\<close>
 lemma secrecy_lemma:
-     "[| A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom |]
-      ==> Says Server A
+     "\<lbrakk>A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom\<rbrakk>
+      \<Longrightarrow> Says Server A
             \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, na, nb\<rbrace>,
               Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>\<rbrace>
            \<in> set evs -->
           Notes Spy \<lbrace>na, nb, Key K\<rbrace> \<notin> set evs -->
           Key K \<notin> analz (knows Spy evs)"
 apply (erule yahalom.induct, force,
        drule_tac [6] YM4_analz_knows_Spy)
-apply (simp_all add: pushes analz_insert_eq analz_insert_freshK, spy_analz)   \<comment>\<open>Fake\<close>
-apply (blast dest: unique_session_keys)+  \<comment>\<open>YM3, Oops\<close>
+apply (simp_all add: pushes analz_insert_eq analz_insert_freshK) 
+  subgoal --\<open>Fake\<close> by spy_analz
+  subgoal --\<open>YM3\<close> by blast   
+  subgoal --\<open>Oops\<close> by  (blast dest: unique_session_keys)   
 done
 
 text\<open>Final version\<close>
 lemma Spy_not_see_encrypted_key:
-     "[| Says Server A
+     "\<lbrakk>Says Server A
             \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, na, nb\<rbrace>,
               Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>\<rbrace>
            \<in> set evs;
          Notes Spy \<lbrace>na, nb, Key K\<rbrace> \<notin> set evs;
-         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom |]
-      ==> Key K \<notin> analz (knows Spy evs)"
+         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom\<rbrakk>
+      \<Longrightarrow> Key K \<notin> analz (knows Spy evs)"
 by (blast dest: secrecy_lemma)
 
 
 subsubsection\<open>Security Guarantee for A upon receiving YM3\<close>
 
 text\<open>If the encrypted message appears then it originated with the Server\<close>
 lemma A_trusts_YM3:
-     "[| Crypt (shrK A) \<lbrace>Agent B, Key K, na, nb\<rbrace> \<in> parts (knows Spy evs);
-         A \<notin> bad;  evs \<in> yahalom |]
-       ==> Says Server A
+     "\<lbrakk>Crypt (shrK A) \<lbrace>Agent B, Key K, na, nb\<rbrace> \<in> parts (knows Spy evs);
+         A \<notin> bad;  evs \<in> yahalom\<rbrakk>
+       \<Longrightarrow> Says Server A
             \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, na, nb\<rbrace>,
               Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>\<rbrace>
            \<in> set evs"
 apply (erule rev_mp)
 apply (erule yahalom.induct, force,

 done
 
 text\<open>The obvious combination of \<open>A_trusts_YM3\<close> with
   \<open>Spy_not_see_encrypted_key\<close>\<close>
 lemma A_gets_good_key:
-     "[| Crypt (shrK A) \<lbrace>Agent B, Key K, na, nb\<rbrace> \<in> parts (knows Spy evs);
+     "\<lbrakk>Crypt (shrK A) \<lbrace>Agent B, Key K, na, nb\<rbrace> \<in> parts (knows Spy evs);
          Notes Spy \<lbrace>na, nb, Key K\<rbrace> \<notin> set evs;
-         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom |]
-      ==> Key K \<notin> analz (knows Spy evs)"
+         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom\<rbrakk>
+      \<Longrightarrow> Key K \<notin> analz (knows Spy evs)"
   by (metis A_trusts_YM3 secrecy_lemma)
 
 
 subsubsection\<open>Security Guarantees for B upon receiving YM4\<close>
 
 text\<open>B knows, by the first part of A's message, that the Server distributed
   the key for A and B.  But this part says nothing about nonces.\<close>
 lemma B_trusts_YM4_shrK:
-     "[| Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace> \<in> parts (knows Spy evs);
-         B \<notin> bad;  evs \<in> yahalom |]
-      ==> \<exists>NA NB. Says Server A
+     "\<lbrakk>Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace> \<in> parts (knows Spy evs);
+         B \<notin> bad;  evs \<in> yahalom\<rbrakk>
+      \<Longrightarrow> \<exists>NA NB. Says Server A
                       \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K,
                                          Nonce NA, Nonce NB\<rbrace>,
                         Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>\<rbrace>
                      \<in> set evs"
 apply (erule rev_mp)

   agent names.  Secrecy of NB is crucial.  Note that @{term "Nonce NB
   \<notin> analz(knows Spy evs)"} must be the FIRST antecedent of the
   induction formula.\<close>
 
 lemma B_trusts_YM4_newK [rule_format]:
-     "[|Crypt K (Nonce NB) \<in> parts (knows Spy evs);
-        Nonce NB \<notin> analz (knows Spy evs);  evs \<in> yahalom|]
-      ==> \<exists>A B NA. Says Server A
+     "\<lbrakk>Crypt K (Nonce NB) \<in> parts (knows Spy evs);
+        Nonce NB \<notin> analz (knows Spy evs);  evs \<in> yahalom\<rbrakk>
+      \<Longrightarrow> \<exists>A B NA. Says Server A
                       \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, Nonce NA, Nonce NB\<rbrace>,
                         Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>\<rbrace>
                      \<in> set evs"
 apply (erule rev_mp, erule rev_mp)
 apply (erule yahalom.induct, force,
        frule_tac [6] YM4_parts_knows_Spy)
-apply (analz_mono_contra, simp_all)
-txt\<open>Fake, YM3\<close>
-apply blast
-apply blast
+         apply (analz_mono_contra, simp_all)
+  subgoal --\<open>Fake\<close> by blast
+  subgoal --\<open>YM3\<close> by blast   
 txt\<open>YM4.  A is uncompromised because NB is secure
   A's certificate guarantees the existence of the Server message\<close>
 apply (blast dest!: Gets_imp_Says Crypt_Spy_analz_bad
              dest: Says_imp_spies
                    parts.Inj [THEN parts.Fst, THEN A_trusts_YM3])

 text\<open>Lemmas about the predicate KeyWithNonce\<close>
 
 lemma KeyWithNonceI:
  "Says Server A
           \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, na, Nonce NB\<rbrace>, X\<rbrace>
-        \<in> set evs ==> KeyWithNonce K NB evs"
+        \<in> set evs \<Longrightarrow> KeyWithNonce K NB evs"
 by (unfold KeyWithNonce_def, blast)
 
 lemma KeyWithNonce_Says [simp]:
    "KeyWithNonce K NB (Says S A X # evs) =
       (Server = S &

 by (simp add: KeyWithNonce_def)
 
 text\<open>A fresh key cannot be associated with any nonce
   (with respect to a given trace).\<close>
 lemma fresh_not_KeyWithNonce:
-     "Key K \<notin> used evs ==> ~ KeyWithNonce K NB evs"
+     "Key K \<notin> used evs \<Longrightarrow> ~ KeyWithNonce K NB evs"
 by (unfold KeyWithNonce_def, blast)
 
 text\<open>The Server message associates K with NB' and therefore not with any
   other nonce NB.\<close>
 lemma Says_Server_KeyWithNonce:
- "[| Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, na, Nonce NB'\<rbrace>, X\<rbrace>
+ "\<lbrakk>Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, na, Nonce NB'\<rbrace>, X\<rbrace>
        \<in> set evs;
-     NB \<noteq> NB';  evs \<in> yahalom |]
-  ==> ~ KeyWithNonce K NB evs"
+     NB \<noteq> NB';  evs \<in> yahalom\<rbrakk>
+  \<Longrightarrow> ~ KeyWithNonce K NB evs"
 by (unfold KeyWithNonce_def, blast dest: unique_session_keys)
 
 
 text\<open>The only nonces that can be found with the help of session keys are
   those distributed as nonce NB by the Server.  The form of the theorem

 
 
 text\<open>As with \<open>analz_image_freshK\<close>, we take some pains to express the 
   property as a logical equivalence so that the simplifier can apply it.\<close>
 lemma Nonce_secrecy_lemma:
-     "P --> (X \<in> analz (G Un H)) --> (X \<in> analz H)  ==>
+     "P --> (X \<in> analz (G Un H)) --> (X \<in> analz H)  \<Longrightarrow>
       P --> (X \<in> analz (G Un H)) = (X \<in> analz H)"
 by (blast intro: analz_mono [THEN subsetD])
 
 lemma Nonce_secrecy:
-     "evs \<in> yahalom ==>
+     "evs \<in> yahalom \<Longrightarrow>
       (\<forall>KK. KK <= - (range shrK) -->
            (\<forall>K \<in> KK. K \<in> symKeys --> ~ KeyWithNonce K NB evs)   -->
            (Nonce NB \<in> analz (Key`KK Un (knows Spy evs))) =
            (Nonce NB \<in> analz (knows Spy evs)))"
 apply (erule yahalom.induct,

             Says_Server_KeyWithNonce)
 txt\<open>For Oops, simplification proves @{prop "NBa\<noteq>NB"}.  By
   @{term Says_Server_KeyWithNonce}, we get @{prop "~ KeyWithNonce K NB
   evs"}; then simplification can apply the induction hypothesis with
   @{term "KK = {K}"}.\<close>
-txt\<open>Fake\<close>
-apply spy_analz
-txt\<open>YM2\<close>
-apply blast
-txt\<open>YM3\<close>
-apply blast
-txt\<open>YM4\<close>
-apply (erule_tac V = "\<forall>KK. P KK" for P in thin_rl, clarify)
-txt\<open>If @{prop "A \<in> bad"} then @{term NBa} is known, therefore
-  @{prop "NBa \<noteq> NB"}.  Previous two steps make the next step
-  faster.\<close>
-apply (metis A_trusts_YM3 Gets_imp_analz_Spy Gets_imp_knows_Spy KeyWithNonce_def
-      Spy_analz_shrK analz.Fst analz.Snd analz_shrK_Decrypt parts.Fst parts.Inj)
+  subgoal --\<open>Fake\<close> by spy_analz
+  subgoal --\<open>YM2\<close> by blast
+  subgoal --\<open>YM3\<close> by blast
+  subgoal --\<open>YM4: If @{prop "A \<in> bad"} then @{term NBa} is known, therefore @{prop "NBa \<noteq> NB"}.\<close>
+    by (metis A_trusts_YM3 Gets_imp_analz_Spy Gets_imp_knows_Spy KeyWithNonce_def
+        Spy_analz_shrK analz.Fst analz.Snd analz_shrK_Decrypt parts.Fst parts.Inj)
 done
 
 
 text\<open>Version required below: if NB can be decrypted using a session key then
    it was distributed with that key.  The more general form above is required
    for the induction to carry through.\<close>
 lemma single_Nonce_secrecy:
-     "[| Says Server A
+     "\<lbrakk>Says Server A
           \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key KAB, na, Nonce NB'\<rbrace>, X\<rbrace>
          \<in> set evs;
-         NB \<noteq> NB';  KAB \<notin> range shrK;  evs \<in> yahalom |]
-      ==> (Nonce NB \<in> analz (insert (Key KAB) (knows Spy evs))) =
+         NB \<noteq> NB';  KAB \<notin> range shrK;  evs \<in> yahalom\<rbrakk>
+      \<Longrightarrow> (Nonce NB \<in> analz (insert (Key KAB) (knows Spy evs))) =
           (Nonce NB \<in> analz (knows Spy evs))"
 by (simp_all del: image_insert image_Un imp_disjL
              add: analz_image_freshK_simps split_ifs
                   Nonce_secrecy Says_Server_KeyWithNonce)
 
 
 subsubsection\<open>The Nonce NB uniquely identifies B's message.\<close>
 
 lemma unique_NB:
-     "[| Crypt (shrK B) \<lbrace>Agent A, Nonce NA, nb\<rbrace> \<in> parts (knows Spy evs);
+     "\<lbrakk>Crypt (shrK B) \<lbrace>Agent A, Nonce NA, nb\<rbrace> \<in> parts (knows Spy evs);
          Crypt (shrK B') \<lbrace>Agent A', Nonce NA', nb\<rbrace> \<in> parts (knows Spy evs);
-        evs \<in> yahalom;  B \<notin> bad;  B' \<notin> bad |]
-      ==> NA' = NA & A' = A & B' = B"
+        evs \<in> yahalom;  B \<notin> bad;  B' \<notin> bad\<rbrakk>
+      \<Longrightarrow> NA' = NA & A' = A & B' = B"
 apply (erule rev_mp, erule rev_mp)
 apply (erule yahalom.induct, force,
        frule_tac [6] YM4_parts_knows_Spy, simp_all)
 txt\<open>Fake, and YM2 by freshness\<close>
 apply blast+

 
 
 text\<open>Variant useful for proving secrecy of NB.  Because nb is assumed to be
   secret, we no longer must assume B, B' not bad.\<close>
 lemma Says_unique_NB:
-     "[| Says C S   \<lbrace>X,  Crypt (shrK B) \<lbrace>Agent A, Nonce NA, nb\<rbrace>\<rbrace>
+     "\<lbrakk>Says C S   \<lbrace>X,  Crypt (shrK B) \<lbrace>Agent A, Nonce NA, nb\<rbrace>\<rbrace>
            \<in> set evs;
          Gets S' \<lbrace>X', Crypt (shrK B') \<lbrace>Agent A', Nonce NA', nb\<rbrace>\<rbrace>
            \<in> set evs;
-         nb \<notin> analz (knows Spy evs);  evs \<in> yahalom |]
-      ==> NA' = NA & A' = A & B' = B"
+         nb \<notin> analz (knows Spy evs);  evs \<in> yahalom\<rbrakk>
+      \<Longrightarrow> NA' = NA & A' = A & B' = B"
 by (blast dest!: Gets_imp_Says Crypt_Spy_analz_bad
           dest: Says_imp_spies unique_NB parts.Inj analz.Inj)
 
 
 subsubsection\<open>A nonce value is never used both as NA and as NB\<close>
 
 lemma no_nonce_YM1_YM2:
-     "[|Crypt (shrK B') \<lbrace>Agent A', Nonce NB, nb'\<rbrace> \<in> parts(knows Spy evs);
-        Nonce NB \<notin> analz (knows Spy evs);  evs \<in> yahalom|]
-  ==> Crypt (shrK B)  \<lbrace>Agent A, na, Nonce NB\<rbrace> \<notin> parts(knows Spy evs)"
+     "\<lbrakk>Crypt (shrK B') \<lbrace>Agent A', Nonce NB, nb'\<rbrace> \<in> parts(knows Spy evs);
+        Nonce NB \<notin> analz (knows Spy evs);  evs \<in> yahalom\<rbrakk>
+  \<Longrightarrow> Crypt (shrK B)  \<lbrace>Agent A, na, Nonce NB\<rbrace> \<notin> parts(knows Spy evs)"
 apply (erule rev_mp, erule rev_mp)
 apply (erule yahalom.induct, force,
        frule_tac [6] YM4_parts_knows_Spy)
 apply (analz_mono_contra, simp_all)
 txt\<open>Fake, YM2\<close>
 apply blast+
 done
 
 text\<open>The Server sends YM3 only in response to YM2.\<close>
 lemma Says_Server_imp_YM2:
-     "[| Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, k, na, nb\<rbrace>, X\<rbrace> \<in> set evs;
-         evs \<in> yahalom |]
-      ==> Gets Server \<lbrace>Agent B, Crypt (shrK B) \<lbrace>Agent A, na, nb\<rbrace>\<rbrace>
+     "\<lbrakk>Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, k, na, nb\<rbrace>, X\<rbrace> \<in> set evs;
+         evs \<in> yahalom\<rbrakk>
+      \<Longrightarrow> Gets Server \<lbrace>Agent B, Crypt (shrK B) \<lbrace>Agent A, na, nb\<rbrace>\<rbrace>
              \<in> set evs"
 by (erule rev_mp, erule yahalom.induct, auto)
 
 text\<open>A vital theorem for B, that nonce NB remains secure from the Spy.\<close>
-lemma Spy_not_see_NB :
-     "[| Says B Server
+theorem Spy_not_see_NB :
+     "\<lbrakk>Says B Server
                 \<lbrace>Agent B, Crypt (shrK B) \<lbrace>Agent A, Nonce NA, Nonce NB\<rbrace>\<rbrace>
            \<in> set evs;
          (\<forall>k. Notes Spy \<lbrace>Nonce NA, Nonce NB, k\<rbrace> \<notin> set evs);
-         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom |]
-      ==> Nonce NB \<notin> analz (knows Spy evs)"
+         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom\<rbrakk>
+      \<Longrightarrow> Nonce NB \<notin> analz (knows Spy evs)"
 apply (erule rev_mp, erule rev_mp)
 apply (erule yahalom.induct, force,
        frule_tac [6] YM4_analz_knows_Spy)
 apply (simp_all add: split_ifs pushes new_keys_not_analzd analz_insert_eq
                      analz_insert_freshK)
-txt\<open>Fake\<close>
-apply spy_analz
-txt\<open>YM1: NB=NA is impossible anyway, but NA is secret because it is fresh!\<close>
-apply blast
-txt\<open>YM2\<close>
-apply blast
-txt\<open>Prove YM3 by showing that no NB can also be an NA\<close>
-apply (blast dest!: no_nonce_YM1_YM2 dest: Gets_imp_Says Says_unique_NB)
-txt\<open>LEVEL 7: YM4 and Oops remain\<close>
-apply (clarify, simp add: all_conj_distrib)
-txt\<open>YM4: key K is visible to Spy, contradicting session key secrecy theorem\<close>
-txt\<open>Case analysis on Aa:bad; PROOF FAILED problems
-  use \<open>Says_unique_NB\<close> to identify message components: @{term "Aa=A"}, @{term "Ba=B"}\<close>
-apply (blast dest!: Says_unique_NB analz_shrK_Decrypt
-                    parts.Inj [THEN parts.Fst, THEN A_trusts_YM3]
-             dest: Gets_imp_Says Says_imp_spies Says_Server_imp_YM2
-                   Spy_not_see_encrypted_key)
-txt\<open>Oops case: if the nonce is betrayed now, show that the Oops event is
-  covered by the quantified Oops assumption.\<close>
-apply (clarify, simp add: all_conj_distrib)
-apply (frule Says_Server_imp_YM2, assumption)
-apply (metis Gets_imp_Says Says_Server_not_range Says_unique_NB no_nonce_YM1_YM2 parts.Snd single_Nonce_secrecy spies_partsEs(1))
-done
-
+  subgoal --\<open>Fake\<close> by spy_analz
+  subgoal --\<open>YM1: NB=NA is impossible anyway, but NA is secret because it is fresh!\<close> by blast
+  subgoal --\<open>YM2\<close> by blast
+  subgoal --\<open>YM3: because no NB can also be an NA\<close> 
+    by (blast dest!: no_nonce_YM1_YM2 dest: Gets_imp_Says Says_unique_NB)
+  subgoal --\<open>YM4: key K is visible to Spy, contradicting session key secrecy theorem\<close>
+    --\<open>Case analysis on whether Aa is bad;
+            use \<open>Says_unique_NB\<close> to identify message components: @{term "Aa=A"}, @{term "Ba=B"}\<close>
+    apply clarify
+    apply (blast dest!: Says_unique_NB analz_shrK_Decrypt
+                        parts.Inj [THEN parts.Fst, THEN A_trusts_YM3]
+                 dest: Gets_imp_Says Says_imp_spies Says_Server_imp_YM2
+                       Spy_not_see_encrypted_key)
+    done
+  subgoal --\<open>Oops case: if the nonce is betrayed now, show that the Oops event is
+    covered by the quantified Oops assumption.\<close>
+    apply clarsimp
+    apply (metis Says_Server_imp_YM2 Gets_imp_Says Says_Server_not_range Says_unique_NB no_nonce_YM1_YM2 parts.Snd single_Nonce_secrecy spies_partsEs(1))
+    done
+done
 
 text\<open>B's session key guarantee from YM4.  The two certificates contribute to a
   single conclusion about the Server's message.  Note that the "Notes Spy"
   assumption must quantify over \<open>\<forall>\<close> POSSIBLE keys instead of our particular K.
   If this run is broken and the spy substitutes a certificate containing an
   old key, B has no means of telling.\<close>
 lemma B_trusts_YM4:
-     "[| Gets B \<lbrace>Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>,
+     "\<lbrakk>Gets B \<lbrace>Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>,
                   Crypt K (Nonce NB)\<rbrace> \<in> set evs;
          Says B Server
            \<lbrace>Agent B, Crypt (shrK B) \<lbrace>Agent A, Nonce NA, Nonce NB\<rbrace>\<rbrace>
            \<in> set evs;
          \<forall>k. Notes Spy \<lbrace>Nonce NA, Nonce NB, k\<rbrace> \<notin> set evs;
-         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom |]
-       ==> Says Server A
+         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom\<rbrakk>
+       \<Longrightarrow> Says Server A
                    \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K,
                              Nonce NA, Nonce NB\<rbrace>,
                      Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>\<rbrace>
              \<in> set evs"
 by (blast dest: Spy_not_see_NB Says_unique_NB

 
 
 text\<open>The obvious combination of \<open>B_trusts_YM4\<close> with 
   \<open>Spy_not_see_encrypted_key\<close>\<close>
 lemma B_gets_good_key:
-     "[| Gets B \<lbrace>Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>,
+     "\<lbrakk>Gets B \<lbrace>Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>,
                   Crypt K (Nonce NB)\<rbrace> \<in> set evs;
          Says B Server
            \<lbrace>Agent B, Crypt (shrK B) \<lbrace>Agent A, Nonce NA, Nonce NB\<rbrace>\<rbrace>
            \<in> set evs;
          \<forall>k. Notes Spy \<lbrace>Nonce NA, Nonce NB, k\<rbrace> \<notin> set evs;
-         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom |]
-      ==> Key K \<notin> analz (knows Spy evs)"
+         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom\<rbrakk>
+      \<Longrightarrow> Key K \<notin> analz (knows Spy evs)"
   by (metis B_trusts_YM4 Spy_not_see_encrypted_key)
 
 
 subsection\<open>Authenticating B to A\<close>
 
 text\<open>The encryption in message YM2 tells us it cannot be faked.\<close>
 lemma B_Said_YM2 [rule_format]:
-     "[|Crypt (shrK B) \<lbrace>Agent A, Nonce NA, nb\<rbrace> \<in> parts (knows Spy evs);
-        evs \<in> yahalom|]
-      ==> B \<notin> bad -->
+     "\<lbrakk>Crypt (shrK B) \<lbrace>Agent A, Nonce NA, nb\<rbrace> \<in> parts (knows Spy evs);
+        evs \<in> yahalom\<rbrakk>
+      \<Longrightarrow> B \<notin> bad -->
           Says B Server \<lbrace>Agent B, Crypt (shrK B) \<lbrace>Agent A, Nonce NA, nb\<rbrace>\<rbrace>
             \<in> set evs"
 apply (erule rev_mp, erule yahalom.induct, force,
        frule_tac [6] YM4_parts_knows_Spy, simp_all)
 txt\<open>Fake\<close>
 apply blast
 done
 
 text\<open>If the server sends YM3 then B sent YM2\<close>
 lemma YM3_auth_B_to_A_lemma:
-     "[|Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, Nonce NA, nb\<rbrace>, X\<rbrace>
-       \<in> set evs;  evs \<in> yahalom|]
-      ==> B \<notin> bad -->
+     "\<lbrakk>Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, Nonce NA, nb\<rbrace>, X\<rbrace>
+       \<in> set evs;  evs \<in> yahalom\<rbrakk>
+      \<Longrightarrow> B \<notin> bad -->
           Says B Server \<lbrace>Agent B, Crypt (shrK B) \<lbrace>Agent A, Nonce NA, nb\<rbrace>\<rbrace>
             \<in> set evs"
 apply (erule rev_mp, erule yahalom.induct, simp_all)
 txt\<open>YM3, YM4\<close>
 apply (blast dest!: B_Said_YM2)+
 done
 
 text\<open>If A receives YM3 then B has used nonce NA (and therefore is alive)\<close>
-lemma YM3_auth_B_to_A:
-     "[| Gets A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, Nonce NA, nb\<rbrace>, X\<rbrace>
-           \<in> set evs;
-         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom |]
-      ==> Says B Server \<lbrace>Agent B, Crypt (shrK B) \<lbrace>Agent A, Nonce NA, nb\<rbrace>\<rbrace>
+theorem YM3_auth_B_to_A:
+     "\<lbrakk>Gets A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, Nonce NA, nb\<rbrace>, X\<rbrace>
+           \<in> set evs;
+         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom\<rbrakk>
+      \<Longrightarrow> Says B Server \<lbrace>Agent B, Crypt (shrK B) \<lbrace>Agent A, Nonce NA, nb\<rbrace>\<rbrace>
        \<in> set evs"
   by (metis A_trusts_YM3 Gets_imp_analz_Spy YM3_auth_B_to_A_lemma analz.Fst
          not_parts_not_analz)
 
 

   @{term "Crypt K (Nonce NB)"}\<close>
 
 text\<open>Assuming the session key is secure, if both certificates are present then
   A has said NB.  We can't be sure about the rest of A's message, but only
   NB matters for freshness.\<close>
-lemma A_Said_YM3_lemma [rule_format]:
+theorem A_Said_YM3_lemma [rule_format]:
      "evs \<in> yahalom
-      ==> Key K \<notin> analz (knows Spy evs) -->
+      \<Longrightarrow> Key K \<notin> analz (knows Spy evs) -->
           Crypt K (Nonce NB) \<in> parts (knows Spy evs) -->
           Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace> \<in> parts (knows Spy evs) -->
           B \<notin> bad -->
           (\<exists>X. Says A B \<lbrace>X, Crypt K (Nonce NB)\<rbrace> \<in> set evs)"
 apply (erule yahalom.induct, force,
        frule_tac [6] YM4_parts_knows_Spy)
 apply (analz_mono_contra, simp_all)
-txt\<open>Fake\<close>
-apply blast
-txt\<open>YM3: by \<open>new_keys_not_used\<close>, the message
-   @{term "Crypt K (Nonce NB)"} could not exist\<close>
-apply (force dest!: Crypt_imp_keysFor)
-txt\<open>YM4: was @{term "Crypt K (Nonce NB)"} the very last message?
-    If not, use the induction hypothesis\<close>
-apply (simp add: ex_disj_distrib)
-txt\<open>yes: apply unicity of session keys\<close>
-apply (blast dest!: Gets_imp_Says A_trusts_YM3 B_trusts_YM4_shrK
-                    Crypt_Spy_analz_bad
+  subgoal --\<open>Fake\<close> by blast
+  subgoal --\<open>YM3 because the message @{term "Crypt K (Nonce NB)"} could not exist\<close>
+     by (force dest!: Crypt_imp_keysFor)
+   subgoal --\<open>YM4: was @{term "Crypt K (Nonce NB)"} the very last message? If not, use the induction hypothesis,
+               otherwise by unicity of session keys\<close>
+     by (blast dest!: Gets_imp_Says A_trusts_YM3 B_trusts_YM4_shrK Crypt_Spy_analz_bad
              dest: Says_imp_knows_Spy [THEN parts.Inj] unique_session_keys)
 done
 
 text\<open>If B receives YM4 then A has used nonce NB (and therefore is alive).
   Moreover, A associates K with NB (thus is talking about the same run).
   Other premises guarantee secrecy of K.\<close>
-lemma YM4_imp_A_Said_YM3 [rule_format]:
-     "[| Gets B \<lbrace>Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>,
+theorem YM4_imp_A_Said_YM3 [rule_format]:
+     "\<lbrakk>Gets B \<lbrace>Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>,
                   Crypt K (Nonce NB)\<rbrace> \<in> set evs;
          Says B Server
            \<lbrace>Agent B, Crypt (shrK B) \<lbrace>Agent A, Nonce NA, Nonce NB\<rbrace>\<rbrace>
            \<in> set evs;
          (\<forall>NA k. Notes Spy \<lbrace>Nonce NA, Nonce NB, k\<rbrace> \<notin> set evs);
-         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom |]
-      ==> \<exists>X. Says A B \<lbrace>X, Crypt K (Nonce NB)\<rbrace> \<in> set evs"
+         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom\<rbrakk>
+      \<Longrightarrow> \<exists>X. Says A B \<lbrace>X, Crypt K (Nonce NB)\<rbrace> \<in> set evs"
 by (metis A_Said_YM3_lemma B_gets_good_key Gets_imp_analz_Spy YM4_parts_knows_Spy analz.Fst not_parts_not_analz)
+
 end