diff --git a/thys/Inductive_Confidentiality/DolevYao/Message.thy b/thys/Inductive_Confidentiality/DolevYao/Message.thy
--- a/thys/Inductive_Confidentiality/DolevYao/Message.thy
+++ b/thys/Inductive_Confidentiality/DolevYao/Message.thy
@@ -1,936 +1,926 @@
 (*  Title:      HOL/Auth/Message.thy
     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     Copyright   1996  University of Cambridge
 
 Datatypes of agents and messages;
 Inductive relations "parts", "analz" and "synth"
 *)
 
 section\<open>Theory of Agents and Messages for Security Protocols against Dolev-Yao\<close>
 
 theory Message
 imports Main
 begin
 
 (*Needed occasionally with spy_analz_tac, e.g. in analz_insert_Key_newK*)
 lemma [simp] : "A \<union> (B \<union> A) = B \<union> A"
 by blast
 
 type_synonym
   key = nat
 
 consts
   all_symmetric :: bool        \<comment> \<open>true if all keys are symmetric\<close>
   invKey        :: "key=>key"  \<comment> \<open>inverse of a symmetric key\<close>
 
 specification (invKey)
   invKey [simp]: "invKey (invKey K) = K"
   invKey_symmetric: "all_symmetric --> invKey = id"
     by (rule exI [of _ id], auto)
 
 
 text\<open>The inverse of a symmetric key is itself; that of a public key
       is the private key and vice versa\<close>
 
 definition symKeys :: "key set" where
   "symKeys == {K. invKey K = K}"
 
 datatype  \<comment> \<open>We allow any number of friendly agents\<close>
   agent = Server | Friend nat | Spy
 
 datatype
      msg = Agent  agent     \<comment> \<open>Agent names\<close>
          | Number nat       \<comment> \<open>Ordinary integers, timestamps, ...\<close>
          | Nonce  nat       \<comment> \<open>Unguessable nonces\<close>
          | Key    key       \<comment> \<open>Crypto keys\<close>
          | Hash   msg       \<comment> \<open>Hashing\<close>
          | MPair  msg msg   \<comment> \<open>Compound messages\<close>
          | Crypt  key msg   \<comment> \<open>Encryption, public- or shared-key\<close>
 
 
 text\<open>Concrete syntax: messages appear as \<open>\<lbrace>A,B,NA\<rbrace>\<close>, etc...\<close>
 syntax
   "_MTuple" :: "['a, args] => 'a * 'b"  ("(2\<lbrace>_,/ _\<rbrace>)")
 translations
   "\<lbrace>x, y, z\<rbrace>"   == "\<lbrace>x, \<lbrace>y, z\<rbrace>\<rbrace>"
   "\<lbrace>x, y\<rbrace>"      == "CONST MPair x y"
 
 
 definition HPair :: "[msg,msg] => msg" ("(4Hash[_] /_)" [0, 1000]) where
     \<comment> \<open>Message Y paired with a MAC computed with the help of X\<close>
     "Hash[X] Y == \<lbrace> Hash\<lbrace>X,Y\<rbrace>, Y\<rbrace>"
 
 definition keysFor :: "msg set => key set" where
     \<comment> \<open>Keys useful to decrypt elements of a message set\<close>
   "keysFor H == invKey ` {K. \<exists>X. Crypt K X \<in> H}"
 
 
 subsection\<open>Inductive definition of all parts of a message\<close>
 
 inductive_set
   parts :: "msg set => msg set"
   for H :: "msg set"
   where
     Inj [intro]:               "X \<in> H ==> X \<in> parts H"
   | Fst:         "\<lbrace>X,Y\<rbrace>   \<in> parts H ==> X \<in> parts H"
   | Snd:         "\<lbrace>X,Y\<rbrace>   \<in> parts H ==> Y \<in> parts H"
   | Body:        "Crypt K X \<in> parts H ==> X \<in> parts H"
 
 
 text\<open>Monotonicity\<close>
 lemma parts_mono: "G \<subseteq> H ==> parts(G) \<subseteq> parts(H)"
 apply auto
-apply (erule parts.induct) 
+apply (erule parts.induct)
 apply (blast dest: parts.Fst parts.Snd parts.Body)+
 done
 
 
 text\<open>Equations hold because constructors are injective.\<close>
 lemma Friend_image_eq [simp]: "(Friend x \<in> Friend`A) = (x:A)"
 by auto
 
 lemma Key_image_eq [simp]: "(Key x \<in> Key`A) = (x\<in>A)"
 by auto
 
 lemma Nonce_Key_image_eq [simp]: "(Nonce x \<notin> Key`A)"
 by auto
 
 
 subsection\<open>Inverse of keys\<close>
 
 lemma invKey_eq [simp]: "(invKey K = invKey K') = (K=K')"
 by (metis invKey)
 
 
 subsection\<open>keysFor operator\<close>
 
 lemma keysFor_empty [simp]: "keysFor {} = {}"
 by (unfold keysFor_def, blast)
 
 lemma keysFor_Un [simp]: "keysFor (H \<union> H') = keysFor H \<union> keysFor H'"
 by (unfold keysFor_def, blast)
 
 lemma keysFor_UN [simp]: "keysFor (\<Union>i\<in>A. H i) = (\<Union>i\<in>A. keysFor (H i))"
 by (unfold keysFor_def, blast)
 
 text\<open>Monotonicity\<close>
 lemma keysFor_mono: "G \<subseteq> H ==> keysFor(G) \<subseteq> keysFor(H)"
 by (unfold keysFor_def, blast)
 
 lemma keysFor_insert_Agent [simp]: "keysFor (insert (Agent A) H) = keysFor H"
 by (unfold keysFor_def, auto)
 
 lemma keysFor_insert_Nonce [simp]: "keysFor (insert (Nonce N) H) = keysFor H"
 by (unfold keysFor_def, auto)
 
 lemma keysFor_insert_Number [simp]: "keysFor (insert (Number N) H) = keysFor H"
 by (unfold keysFor_def, auto)
 
 lemma keysFor_insert_Key [simp]: "keysFor (insert (Key K) H) = keysFor H"
 by (unfold keysFor_def, auto)
 
 lemma keysFor_insert_Hash [simp]: "keysFor (insert (Hash X) H) = keysFor H"
 by (unfold keysFor_def, auto)
 
 lemma keysFor_insert_MPair [simp]: "keysFor (insert \<lbrace>X,Y\<rbrace> H) = keysFor H"
 by (unfold keysFor_def, auto)
 
-lemma keysFor_insert_Crypt [simp]: 
+lemma keysFor_insert_Crypt [simp]:
     "keysFor (insert (Crypt K X) H) = insert (invKey K) (keysFor H)"
 by (unfold keysFor_def, auto)
 
 lemma keysFor_image_Key [simp]: "keysFor (Key`E) = {}"
 by (unfold keysFor_def, auto)
 
 lemma Crypt_imp_invKey_keysFor: "Crypt K X \<in> H ==> invKey K \<in> keysFor H"
 by (unfold keysFor_def, blast)
 
 
 subsection\<open>Inductive relation "parts"\<close>
 
 lemma MPair_parts:
-     "[| \<lbrace>X,Y\<rbrace> \<in> parts H;        
+     "[| \<lbrace>X,Y\<rbrace> \<in> parts H;
          [| X \<in> parts H; Y \<in> parts H |] ==> P |] ==> P"
-by (blast dest: parts.Fst parts.Snd) 
+by (blast dest: parts.Fst parts.Snd)
 
 declare MPair_parts [elim!]  parts.Body [dest!]
 text\<open>NB These two rules are UNSAFE in the formal sense, as they discard the
-     compound message.  They work well on THIS FILE.  
+     compound message.  They work well on THIS FILE.
   \<open>MPair_parts\<close> is left as SAFE because it speeds up proofs.
   The Crypt rule is normally kept UNSAFE to avoid breaking up certificates.\<close>
 
 lemma parts_increasing: "H \<subseteq> parts(H)"
 by blast
 
 lemmas parts_insertI = subset_insertI [THEN parts_mono, THEN subsetD]
 
 lemma parts_empty [simp]: "parts{} = {}"
 apply safe
 apply (erule parts.induct, blast+)
 done
 
 lemma parts_emptyE [elim!]: "X\<in> parts{} ==> P"
 by simp
 
 text\<open>WARNING: loops if H = {Y}, therefore must not be repeated!\<close>
 lemma parts_singleton: "X\<in> parts H ==> \<exists>Y\<in>H. X\<in> parts {Y}"
 by (erule parts.induct, fast+)
 
 
 subsubsection\<open>Unions\<close>
 
 lemma parts_Un_subset1: "parts(G) \<union> parts(H) \<subseteq> parts(G \<union> H)"
 by (intro Un_least parts_mono Un_upper1 Un_upper2)
 
 lemma parts_Un_subset2: "parts(G \<union> H) \<subseteq> parts(G) \<union> parts(H)"
 apply (rule subsetI)
 apply (erule parts.induct, blast+)
 done
 
 lemma parts_Un [simp]: "parts(G \<union> H) = parts(G) \<union> parts(H)"
 by (intro equalityI parts_Un_subset1 parts_Un_subset2)
 
 lemma parts_insert: "parts (insert X H) = parts {X} \<union> parts H"
 by (metis insert_is_Un parts_Un)
 
 text\<open>TWO inserts to avoid looping.  This rewrite is better than nothing.
   Not suitable for Addsimps: its behaviour can be strange.\<close>
 lemma parts_insert2:
      "parts (insert X (insert Y H)) = parts {X} \<union> parts {Y} \<union> parts H"
 by (metis Un_commute Un_empty_right Un_insert_right insert_is_Un parts_Un)
 
 lemma parts_UN_subset1: "(\<Union>x\<in>A. parts(H x)) \<subseteq> parts(\<Union>x\<in>A. H x)"
 by (intro UN_least parts_mono UN_upper)
 
 lemma parts_UN_subset2: "parts(\<Union>x\<in>A. H x) \<subseteq> (\<Union>x\<in>A. parts(H x))"
 apply (rule subsetI)
 apply (erule parts.induct, blast+)
 done
 
 lemma parts_UN [simp]: "parts(\<Union>x\<in>A. H x) = (\<Union>x\<in>A. parts(H x))"
 by (intro equalityI parts_UN_subset1 parts_UN_subset2)
 
 text\<open>Added to simplify arguments to parts, analz and synth.
   NOTE: the UN versions are no longer used!\<close>
 
 
-text\<open>This allows \<open>blast\<close> to simplify occurrences of 
+text\<open>This allows \<open>blast\<close> to simplify occurrences of
   @{term "parts(G\<union>H)"} in the assumption.\<close>
-lemmas in_parts_UnE = parts_Un [THEN equalityD1, THEN subsetD, THEN UnE] 
+lemmas in_parts_UnE = parts_Un [THEN equalityD1, THEN subsetD, THEN UnE]
 declare in_parts_UnE [elim!]
 
 
 lemma parts_insert_subset: "insert X (parts H) \<subseteq> parts(insert X H)"
 by (blast intro: parts_mono [THEN [2] rev_subsetD])
 
 subsubsection\<open>Idempotence and transitivity\<close>
 
 lemma parts_partsD [dest!]: "X\<in> parts (parts H) ==> X\<in> parts H"
 by (erule parts.induct, blast+)
 
 lemma parts_idem [simp]: "parts (parts H) = parts H"
 by blast
 
 lemma parts_subset_iff [simp]: "(parts G \<subseteq> parts H) = (G \<subseteq> parts H)"
 by (metis parts_idem parts_increasing parts_mono subset_trans)
 
 lemma parts_trans: "[| X\<in> parts G;  G \<subseteq> parts H |] ==> X\<in> parts H"
 by (metis parts_subset_iff subsetD)
 
 text\<open>Cut\<close>
 lemma parts_cut:
-     "[| Y\<in> parts (insert X G);  X\<in> parts H |] ==> Y\<in> parts (G \<union> H)" 
-by (blast intro: parts_trans) 
+     "[| Y\<in> parts (insert X G);  X\<in> parts H |] ==> Y\<in> parts (G \<union> H)"
+by (blast intro: parts_trans)
 
 lemma parts_cut_eq [simp]: "X\<in> parts H ==> parts (insert X H) = parts H"
 by (metis insert_absorb parts_idem parts_insert)
 
 
 subsubsection\<open>Rewrite rules for pulling out atomic messages\<close>
 
 lemmas parts_insert_eq_I = equalityI [OF subsetI parts_insert_subset]
 
 
 lemma parts_insert_Agent [simp]:
      "parts (insert (Agent agt) H) = insert (Agent agt) (parts H)"
-apply (rule parts_insert_eq_I) 
-apply (erule parts.induct, auto) 
+apply (rule parts_insert_eq_I)
+apply (erule parts.induct, auto)
 done
 
 lemma parts_insert_Nonce [simp]:
      "parts (insert (Nonce N) H) = insert (Nonce N) (parts H)"
-apply (rule parts_insert_eq_I) 
-apply (erule parts.induct, auto) 
+apply (rule parts_insert_eq_I)
+apply (erule parts.induct, auto)
 done
 
 lemma parts_insert_Number [simp]:
      "parts (insert (Number N) H) = insert (Number N) (parts H)"
-apply (rule parts_insert_eq_I) 
-apply (erule parts.induct, auto) 
+apply (rule parts_insert_eq_I)
+apply (erule parts.induct, auto)
 done
 
 lemma parts_insert_Key [simp]:
      "parts (insert (Key K) H) = insert (Key K) (parts H)"
-apply (rule parts_insert_eq_I) 
-apply (erule parts.induct, auto) 
+apply (rule parts_insert_eq_I)
+apply (erule parts.induct, auto)
 done
 
 lemma parts_insert_Hash [simp]:
      "parts (insert (Hash X) H) = insert (Hash X) (parts H)"
-apply (rule parts_insert_eq_I) 
-apply (erule parts.induct, auto) 
+apply (rule parts_insert_eq_I)
+apply (erule parts.induct, auto)
 done
 
 lemma parts_insert_Crypt [simp]:
      "parts (insert (Crypt K X) H) = insert (Crypt K X) (parts (insert X H))"
 apply (rule equalityI)
 apply (rule subsetI)
 apply (erule parts.induct, auto)
 apply (blast intro: parts.Body)
 done
 
 lemma parts_insert_MPair [simp]:
-     "parts (insert \<lbrace>X,Y\<rbrace> H) =  
+     "parts (insert \<lbrace>X,Y\<rbrace> H) =
           insert \<lbrace>X,Y\<rbrace> (parts (insert X (insert Y H)))"
 apply (rule equalityI)
 apply (rule subsetI)
 apply (erule parts.induct, auto)
 apply (blast intro: parts.Fst parts.Snd)+
 done
 
 lemma parts_image_Key [simp]: "parts (Key`N) = Key`N"
 apply auto
 apply (erule parts.induct, auto)
 done
 
 
 text\<open>In any message, there is an upper bound N on its greatest nonce.\<close>
 lemma msg_Nonce_supply: "\<exists>N. \<forall>n. N\<le>n --> Nonce n \<notin> parts {msg}"
 apply (induct msg)
 apply (simp_all (no_asm_simp) add: exI parts_insert2)
 txt\<open>Nonce case\<close>
 apply (metis Suc_n_not_le_n)
 txt\<open>MPair case: metis works out the necessary sum itself!\<close>
 apply (metis le_trans nat_le_linear)
 done
 
 
 subsection\<open>Inductive relation "analz"\<close>
 
 text\<open>Inductive definition of "analz" -- what can be broken down from a set of
     messages, including keys.  A form of downward closure.  Pairs can
     be taken apart; messages decrypted with known keys.\<close>
 
 inductive_set
   analz :: "msg set => msg set"
   for H :: "msg set"
   where
     Inj [intro,simp] :    "X \<in> H ==> X \<in> analz H"
   | Fst:     "\<lbrace>X,Y\<rbrace> \<in> analz H ==> X \<in> analz H"
   | Snd:     "\<lbrace>X,Y\<rbrace> \<in> analz H ==> Y \<in> analz H"
-  | Decrypt [dest]: 
+  | Decrypt [dest]:
              "[|Crypt K X \<in> analz H; Key(invKey K): analz H|] ==> X \<in> analz H"
 
 
 text\<open>Monotonicity; Lemma 1 of Lowe's paper\<close>
 lemma analz_mono: "G\<subseteq>H ==> analz(G) \<subseteq> analz(H)"
 apply auto
-apply (erule analz.induct) 
-apply (auto dest: analz.Fst analz.Snd) 
+apply (erule analz.induct)
+apply (auto dest: analz.Fst analz.Snd)
 done
 
 text\<open>Making it safe speeds up proofs\<close>
 lemma MPair_analz [elim!]:
-     "[| \<lbrace>X,Y\<rbrace> \<in> analz H;        
-             [| X \<in> analz H; Y \<in> analz H |] ==> P   
+     "[| \<lbrace>X,Y\<rbrace> \<in> analz H;
+             [| X \<in> analz H; Y \<in> analz H |] ==> P
           |] ==> P"
 by (blast dest: analz.Fst analz.Snd)
 
 lemma analz_increasing: "H \<subseteq> analz(H)"
 by blast
 
 lemma analz_subset_parts: "analz H \<subseteq> parts H"
 apply (rule subsetI)
 apply (erule analz.induct, blast+)
 done
 
 lemmas analz_into_parts = analz_subset_parts [THEN subsetD]
 
 lemmas not_parts_not_analz = analz_subset_parts [THEN contra_subsetD]
 
 
 lemma parts_analz [simp]: "parts (analz H) = parts H"
 by (metis analz_increasing analz_subset_parts equalityI parts_mono parts_subset_iff)
 
 lemma analz_parts [simp]: "analz (parts H) = parts H"
 apply auto
 apply (erule analz.induct, auto)
 done
 
 lemmas analz_insertI = subset_insertI [THEN analz_mono, THEN [2] rev_subsetD]
 
 subsubsection\<open>General equational properties\<close>
 
 lemma analz_empty [simp]: "analz{} = {}"
 apply safe
 apply (erule analz.induct, blast+)
 done
 
-text\<open>Converse fails: we can analz more from the union than from the 
+text\<open>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\<close>
 lemma analz_Un: "analz(G) \<union> analz(H) \<subseteq> analz(G \<union> H)"
 by (intro Un_least analz_mono Un_upper1 Un_upper2)
 
 lemma analz_insert: "insert X (analz H) \<subseteq> analz(insert X H)"
 by (blast intro: analz_mono [THEN [2] rev_subsetD])
 
 subsubsection\<open>Rewrite rules for pulling out atomic messages\<close>
 
 lemmas analz_insert_eq_I = equalityI [OF subsetI analz_insert]
 
 lemma analz_insert_Agent [simp]:
      "analz (insert (Agent agt) H) = insert (Agent agt) (analz H)"
-apply (rule analz_insert_eq_I) 
-apply (erule analz.induct, auto) 
+apply (rule analz_insert_eq_I)
+apply (erule analz.induct, auto)
 done
 
 lemma analz_insert_Nonce [simp]:
      "analz (insert (Nonce N) H) = insert (Nonce N) (analz H)"
-apply (rule analz_insert_eq_I) 
-apply (erule analz.induct, auto) 
+apply (rule analz_insert_eq_I)
+apply (erule analz.induct, auto)
 done
 
 lemma analz_insert_Number [simp]:
      "analz (insert (Number N) H) = insert (Number N) (analz H)"
-apply (rule analz_insert_eq_I) 
-apply (erule analz.induct, auto) 
+apply (rule analz_insert_eq_I)
+apply (erule analz.induct, auto)
 done
 
 lemma analz_insert_Hash [simp]:
      "analz (insert (Hash X) H) = insert (Hash X) (analz H)"
-apply (rule analz_insert_eq_I) 
-apply (erule analz.induct, auto) 
+apply (rule analz_insert_eq_I)
+apply (erule analz.induct, auto)
 done
 
 text\<open>Can only pull out Keys if they are not needed to decrypt the rest\<close>
-lemma analz_insert_Key [simp]: 
-    "K \<notin> keysFor (analz H) ==>   
+lemma analz_insert_Key [simp]:
+    "K \<notin> keysFor (analz H) ==>
           analz (insert (Key K) H) = insert (Key K) (analz H)"
 apply (unfold keysFor_def)
-apply (rule analz_insert_eq_I) 
-apply (erule analz.induct, auto) 
+apply (rule analz_insert_eq_I)
+apply (erule analz.induct, auto)
 done
 
 lemma analz_insert_MPair [simp]:
-     "analz (insert \<lbrace>X,Y\<rbrace> H) =  
+     "analz (insert \<lbrace>X,Y\<rbrace> H) =
           insert \<lbrace>X,Y\<rbrace> (analz (insert X (insert Y H)))"
 apply (rule equalityI)
 apply (rule subsetI)
 apply (erule analz.induct, auto)
 apply (erule analz.induct)
 apply (blast intro: analz.Fst analz.Snd)+
 done
 
 text\<open>Can pull out enCrypted message if the Key is not known\<close>
 lemma analz_insert_Crypt:
-     "Key (invKey K) \<notin> analz H 
+     "Key (invKey K) \<notin> analz H
       ==> analz (insert (Crypt K X) H) = insert (Crypt K X) (analz H)"
-apply (rule analz_insert_eq_I) 
-apply (erule analz.induct, auto) 
+apply (rule analz_insert_eq_I)
+apply (erule analz.induct, auto)
 
 done
 
-lemma lemma1: "Key (invKey K) \<in> analz H ==>   
-               analz (insert (Crypt K X) H) \<subseteq>  
+lemma lemma1: "Key (invKey K) \<in> analz H ==>
+               analz (insert (Crypt K X) H) \<subseteq>
                insert (Crypt K X) (analz (insert X H))"
 apply (rule subsetI)
 apply (erule_tac x = x in analz.induct, auto)
 done
 
-lemma lemma2: "Key (invKey K) \<in> analz H ==>   
-               insert (Crypt K X) (analz (insert X H)) \<subseteq>  
+lemma lemma2: "Key (invKey K) \<in> analz H ==>
+               insert (Crypt K X) (analz (insert X H)) \<subseteq>
                analz (insert (Crypt K X) H)"
 apply auto
 apply (erule_tac x = x in analz.induct, auto)
 apply (blast intro: analz_insertI analz.Decrypt)
 done
 
 lemma analz_insert_Decrypt:
-     "Key (invKey K) \<in> analz H ==>   
-               analz (insert (Crypt K X) H) =  
+     "Key (invKey K) \<in> analz H ==>
+               analz (insert (Crypt K X) H) =
                insert (Crypt K X) (analz (insert X H))"
 by (intro equalityI lemma1 lemma2)
 
 text\<open>Case analysis: either the message is secure, or it is not! Effective,
 but can cause subgoals to blow up! Use with \<open>if_split\<close>; apparently
 \<open>split_tac\<close> does not cope with patterns such as @{term"analz (insert
-(Crypt K X) H)"}\<close> 
+(Crypt K X) H)"}\<close>
 lemma analz_Crypt_if [simp]:
-     "analz (insert (Crypt K X) H) =                 
-          (if (Key (invKey K) \<in> analz H)                 
-           then insert (Crypt K X) (analz (insert X H))  
+     "analz (insert (Crypt K X) H) =
+          (if (Key (invKey K) \<in> analz H)
+           then insert (Crypt K X) (analz (insert X H))
            else insert (Crypt K X) (analz H))"
 by (simp add: analz_insert_Crypt analz_insert_Decrypt)
 
 
 text\<open>This rule supposes "for the sake of argument" that we have the key.\<close>
 lemma analz_insert_Crypt_subset:
-     "analz (insert (Crypt K X) H) \<subseteq>   
+     "analz (insert (Crypt K X) H) \<subseteq>
            insert (Crypt K X) (analz (insert X H))"
 apply (rule subsetI)
 apply (erule analz.induct, auto)
 done
 
 
 lemma analz_image_Key [simp]: "analz (Key`N) = Key`N"
 apply auto
 apply (erule analz.induct, auto)
 done
 
 
 subsubsection\<open>Idempotence and transitivity\<close>
 
 lemma analz_analzD [dest!]: "X\<in> analz (analz H) ==> X\<in> analz H"
 by (erule analz.induct, blast+)
 
 lemma analz_idem [simp]: "analz (analz H) = analz H"
 by blast
 
 lemma analz_subset_iff [simp]: "(analz G \<subseteq> analz H) = (G \<subseteq> analz H)"
 by (metis analz_idem analz_increasing analz_mono subset_trans)
 
 lemma analz_trans: "[| X\<in> analz G;  G \<subseteq> analz H |] ==> X\<in> analz H"
 by (drule analz_mono, blast)
 
 text\<open>Cut; Lemma 2 of Lowe\<close>
 lemma analz_cut: "[| Y\<in> analz (insert X H);  X\<in> analz H |] ==> Y\<in> analz H"
 by (erule analz_trans, blast)
 
 (*Cut can be proved easily by induction on
    "Y: analz (insert X H) ==> X: analz H --> Y: analz H"
 *)
 
 text\<open>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.\<close>
 lemma analz_insert_eq: "X\<in> analz H ==> analz (insert X H) = analz H"
 by (metis analz_cut analz_insert_eq_I insert_absorb)
 
 
 text\<open>A congruence rule for "analz"\<close>
 
 lemma analz_subset_cong:
-     "[| analz G \<subseteq> analz G'; analz H \<subseteq> analz H' |] 
+     "[| analz G \<subseteq> analz G'; analz H \<subseteq> analz H' |]
       ==> analz (G \<union> H) \<subseteq> analz (G' \<union> H')"
 by (metis Un_mono analz_Un analz_subset_iff subset_trans)
 
 lemma analz_cong:
-     "[| analz G = analz G'; analz H = analz H' |] 
+     "[| analz G = analz G'; analz H = analz H' |]
       ==> analz (G \<union> H) = analz (G' \<union> H')"
-by (intro equalityI analz_subset_cong, simp_all) 
+by (intro equalityI analz_subset_cong, simp_all)
 
 lemma analz_insert_cong:
      "analz H = analz H' ==> analz(insert X H) = analz(insert X H')"
 by (force simp only: insert_def intro!: analz_cong)
 
 text\<open>If there are no pairs or encryptions then analz does nothing\<close>
 lemma analz_trivial:
      "[| \<forall>X Y. \<lbrace>X,Y\<rbrace> \<notin> H;  \<forall>X K. Crypt K X \<notin> H |] ==> analz H = H"
 apply safe
 apply (erule analz.induct, blast+)
 done
 
-text\<open>These two are obsolete (with a single Spy) but cost little to prove...\<close>
-lemma analz_UN_analz_lemma:
-     "X\<in> analz (\<Union>i\<in>A. analz (H i)) ==> X\<in> analz (\<Union>i\<in>A. H i)"
-apply (erule analz.induct)
-apply (blast intro: analz_mono [THEN [2] rev_subsetD])+
-done
-
-lemma analz_UN_analz [simp]: "analz (\<Union>i\<in>A. analz (H i)) = analz (\<Union>i\<in>A. H i)"
-by (blast intro: analz_UN_analz_lemma analz_mono [THEN [2] rev_subsetD])
-
 
 subsection\<open>Inductive relation "synth"\<close>
 
 text\<open>Inductive definition of "synth" -- what can be built up from a set of
     messages.  A form of upward closure.  Pairs can be built, messages
     encrypted with known keys.  Agent names are public domain.
     Numbers can be guessed, but Nonces cannot be.\<close>
 
 inductive_set
   synth :: "msg set => msg set"
   for H :: "msg set"
   where
     Inj    [intro]:   "X \<in> H ==> X \<in> synth H"
   | Agent  [intro]:   "Agent agt \<in> synth H"
   | Number [intro]:   "Number n  \<in> synth H"
   | Hash   [intro]:   "X \<in> synth H ==> Hash X \<in> synth H"
   | MPair  [intro]:   "[|X \<in> synth H;  Y \<in> synth H|] ==> \<lbrace>X,Y\<rbrace> \<in> synth H"
   | Crypt  [intro]:   "[|X \<in> synth H;  Key(K) \<in> H|] ==> Crypt K X \<in> synth H"
 
 text\<open>Monotonicity\<close>
 lemma synth_mono: "G\<subseteq>H ==> synth(G) \<subseteq> synth(H)"
-  by (auto, erule synth.induct, auto)  
+  by (auto, erule synth.induct, auto)
 
-text\<open>NO \<open>Agent_synth\<close>, as any Agent name can be synthesized.  
+text\<open>NO \<open>Agent_synth\<close>, as any Agent name can be synthesized.
   The same holds for @{term Number}\<close>
 
 inductive_simps synth_simps [iff]:
  "Nonce n \<in> synth H"
  "Key K \<in> synth H"
  "Hash X \<in> synth H"
  "\<lbrace>X,Y\<rbrace> \<in> synth H"
  "Crypt K X \<in> synth H"
 
 lemma synth_increasing: "H \<subseteq> synth(H)"
 by blast
 
 subsubsection\<open>Unions\<close>
 
-text\<open>Converse fails: we can synth more from the union than from the 
+text\<open>Converse fails: we can synth more from the union than from the
   separate parts, building a compound message using elements of each.\<close>
 lemma synth_Un: "synth(G) \<union> synth(H) \<subseteq> synth(G \<union> H)"
 by (intro Un_least synth_mono Un_upper1 Un_upper2)
 
 lemma synth_insert: "insert X (synth H) \<subseteq> synth(insert X H)"
 by (blast intro: synth_mono [THEN [2] rev_subsetD])
 
 subsubsection\<open>Idempotence and transitivity\<close>
 
 lemma synth_synthD [dest!]: "X\<in> synth (synth H) ==> X\<in> synth H"
 by (erule synth.induct, auto)
 
 lemma synth_idem: "synth (synth H) = synth H"
 by blast
 
 lemma synth_subset_iff [simp]: "(synth G \<subseteq> synth H) = (G \<subseteq> synth H)"
 by (metis subset_trans synth_idem synth_increasing synth_mono)
 
 lemma synth_trans: "[| X\<in> synth G;  G \<subseteq> synth H |] ==> X\<in> synth H"
 by (drule synth_mono, blast)
 
 text\<open>Cut; Lemma 2 of Lowe\<close>
 lemma synth_cut: "[| Y\<in> synth (insert X H);  X\<in> synth H |] ==> Y\<in> synth H"
 by (erule synth_trans, blast)
 
 lemma Agent_synth [simp]: "Agent A \<in> synth H"
 by blast
 
 lemma Number_synth [simp]: "Number n \<in> synth H"
 by blast
 
 lemma Nonce_synth_eq [simp]: "(Nonce N \<in> synth H) = (Nonce N \<in> H)"
 by blast
 
 lemma Key_synth_eq [simp]: "(Key K \<in> synth H) = (Key K \<in> H)"
 by blast
 
 lemma Crypt_synth_eq [simp]:
      "Key K \<notin> H ==> (Crypt K X \<in> synth H) = (Crypt K X \<in> H)"
 by blast
 
 
-lemma keysFor_synth [simp]: 
+lemma keysFor_synth [simp]:
     "keysFor (synth H) = keysFor H \<union> invKey`{K. Key K \<in> H}"
 by (unfold keysFor_def, blast)
 
 
 subsubsection\<open>Combinations of parts, analz and synth\<close>
 
 lemma parts_synth [simp]: "parts (synth H) = parts H \<union> synth H"
 apply (rule equalityI)
 apply (rule subsetI)
 apply (erule parts.induct)
-apply (blast intro: synth_increasing [THEN parts_mono, THEN subsetD] 
+apply (blast intro: synth_increasing [THEN parts_mono, THEN subsetD]
                     parts.Fst parts.Snd parts.Body)+
 done
 
 lemma analz_analz_Un [simp]: "analz (analz G \<union> H) = analz (G \<union> H)"
 apply (intro equalityI analz_subset_cong)+
 apply simp_all
 done
 
 lemma analz_synth_Un [simp]: "analz (synth G \<union> H) = analz (G \<union> H) \<union> synth G"
 apply (rule equalityI)
 apply (rule subsetI)
 apply (erule analz.induct)
 prefer 5 apply (blast intro: analz_mono [THEN [2] rev_subsetD])
 apply (blast intro: analz.Fst analz.Snd analz.Decrypt)+
 done
 
 lemma analz_synth [simp]: "analz (synth H) = analz H \<union> synth H"
 by (metis Un_empty_right analz_synth_Un)
 
 
 subsubsection\<open>For reasoning about the Fake rule in traces\<close>
 
 lemma parts_insert_subset_Un: "X\<in> G ==> parts(insert X H) \<subseteq> parts G \<union> parts H"
 by (metis UnCI Un_upper2 insert_subset parts_Un parts_mono)
 
 text\<open>More specifically for Fake. See also \<open>Fake_parts_sing\<close> below\<close>
 lemma Fake_parts_insert:
-     "X \<in> synth (analz H) ==>  
+     "X \<in> synth (analz H) ==>
       parts (insert X H) \<subseteq> synth (analz H) \<union> parts H"
-by (metis Un_commute analz_increasing insert_subset parts_analz parts_mono 
+by (metis Un_commute analz_increasing insert_subset parts_analz parts_mono
           parts_synth synth_mono synth_subset_iff)
 
 lemma Fake_parts_insert_in_Un:
-     "[|Z \<in> parts (insert X H);  X: synth (analz H)|] 
+     "[|Z \<in> parts (insert X H);  X: synth (analz H)|]
       ==> Z \<in>  synth (analz H) \<union> parts H"
 by (metis Fake_parts_insert subsetD)
 
-text\<open>@{term H} is sometimes @{term"Key ` KK \<union> spies evs"}, so can't put 
+text\<open>@{term H} is sometimes @{term"Key ` KK \<union> spies evs"}, so can't put
   @{term "G=H"}.\<close>
 lemma Fake_analz_insert:
-     "X\<in> synth (analz G) ==>  
+     "X\<in> synth (analz G) ==>
       analz (insert X H) \<subseteq> synth (analz G) \<union> analz (G \<union> H)"
 apply (rule subsetI)
 apply (subgoal_tac "x \<in> analz (synth (analz G) \<union> H)", force)
 apply (blast intro: analz_mono [THEN [2] rev_subsetD] analz_mono [THEN synth_mono, THEN [2] rev_subsetD])
 done
 
 lemma analz_conj_parts [simp]:
      "(X \<in> analz H \<and> X \<in> parts H) = (X \<in> analz H)"
 by (blast intro: analz_subset_parts [THEN subsetD])
 
 lemma analz_disj_parts [simp]:
      "(X \<in> analz H | X \<in> parts H) = (X \<in> parts H)"
 by (blast intro: analz_subset_parts [THEN subsetD])
 
 text\<open>Without this equation, other rules for synth and analz would yield
   redundant cases\<close>
 lemma MPair_synth_analz [iff]:
-     "(\<lbrace>X,Y\<rbrace> \<in> synth (analz H)) =  
+     "(\<lbrace>X,Y\<rbrace> \<in> synth (analz H)) =
       (X \<in> synth (analz H) \<and> Y \<in> synth (analz H))"
 by blast
 
 lemma Crypt_synth_analz:
-     "[| Key K \<in> analz H;  Key (invKey K) \<in> analz H |]  
+     "[| Key K \<in> analz H;  Key (invKey K) \<in> analz H |]
        ==> (Crypt K X \<in> synth (analz H)) = (X \<in> synth (analz H))"
 by blast
 
 
 lemma Hash_synth_analz [simp]:
-     "X \<notin> synth (analz H)  
+     "X \<notin> synth (analz H)
       ==> (Hash\<lbrace>X,Y\<rbrace> \<in> synth (analz H)) = (Hash\<lbrace>X,Y\<rbrace> \<in> analz H)"
 by blast
 
 
 subsection\<open>HPair: a combination of Hash and MPair\<close>
 
 subsubsection\<open>Freeness\<close>
 
 lemma Agent_neq_HPair: "Agent A ~= Hash[X] Y"
 by (unfold HPair_def, simp)
 
 lemma Nonce_neq_HPair: "Nonce N ~= Hash[X] Y"
 by (unfold HPair_def, simp)
 
 lemma Number_neq_HPair: "Number N ~= Hash[X] Y"
 by (unfold HPair_def, simp)
 
 lemma Key_neq_HPair: "Key K ~= Hash[X] Y"
 by (unfold HPair_def, simp)
 
 lemma Hash_neq_HPair: "Hash Z ~= Hash[X] Y"
 by (unfold HPair_def, simp)
 
 lemma Crypt_neq_HPair: "Crypt K X' ~= Hash[X] Y"
 by (unfold HPair_def, simp)
 
-lemmas HPair_neqs = Agent_neq_HPair Nonce_neq_HPair Number_neq_HPair 
+lemmas HPair_neqs = Agent_neq_HPair Nonce_neq_HPair Number_neq_HPair
                     Key_neq_HPair Hash_neq_HPair Crypt_neq_HPair
 
 declare HPair_neqs [iff]
 declare HPair_neqs [symmetric, iff]
 
 lemma HPair_eq [iff]: "(Hash[X'] Y' = Hash[X] Y) = (X' = X \<and> Y'=Y)"
 by (simp add: HPair_def)
 
 lemma MPair_eq_HPair [iff]:
      "(\<lbrace>X',Y'\<rbrace> = Hash[X] Y) = (X' = Hash\<lbrace>X,Y\<rbrace> \<and> Y'=Y)"
 by (simp add: HPair_def)
 
 lemma HPair_eq_MPair [iff]:
      "(Hash[X] Y = \<lbrace>X',Y'\<rbrace>) = (X' = Hash\<lbrace>X,Y\<rbrace> \<and> Y'=Y)"
 by (auto simp add: HPair_def)
 
 
 subsubsection\<open>Specialized laws, proved in terms of those for Hash and MPair\<close>
 
 lemma keysFor_insert_HPair [simp]: "keysFor (insert (Hash[X] Y) H) = keysFor H"
 by (simp add: HPair_def)
 
-lemma parts_insert_HPair [simp]: 
-    "parts (insert (Hash[X] Y) H) =  
+lemma parts_insert_HPair [simp]:
+    "parts (insert (Hash[X] Y) H) =
      insert (Hash[X] Y) (insert (Hash\<lbrace>X,Y\<rbrace>) (parts (insert Y H)))"
 by (simp add: HPair_def)
 
-lemma analz_insert_HPair [simp]: 
-    "analz (insert (Hash[X] Y) H) =  
+lemma analz_insert_HPair [simp]:
+    "analz (insert (Hash[X] Y) H) =
      insert (Hash[X] Y) (insert (Hash\<lbrace>X,Y\<rbrace>) (analz (insert Y H)))"
 by (simp add: HPair_def)
 
 lemma HPair_synth_analz [simp]:
-     "X \<notin> synth (analz H)  
-    ==> (Hash[X] Y \<in> synth (analz H)) =  
+     "X \<notin> synth (analz H)
+    ==> (Hash[X] Y \<in> synth (analz H)) =
         (Hash \<lbrace>X, Y\<rbrace> \<in> analz H \<and> Y \<in> synth (analz H))"
 by (auto simp add: HPair_def)
 
 
 text\<open>We do NOT want Crypt... messages broken up in protocols!!\<close>
 declare parts.Body [rule del]
 
 
 text\<open>Rewrites to push in Key and Crypt messages, so that other messages can
     be pulled out using the \<open>analz_insert\<close> rules\<close>
 
 lemmas pushKeys =
   insert_commute [of "Key K" "Agent C"]
   insert_commute [of "Key K" "Nonce N"]
   insert_commute [of "Key K" "Number N"]
   insert_commute [of "Key K" "Hash X"]
   insert_commute [of "Key K" "MPair X Y"]
   insert_commute [of "Key K" "Crypt X K'"]
   for K C N X Y K'
 
 lemmas pushCrypts =
   insert_commute [of "Crypt X K" "Agent C"]
   insert_commute [of "Crypt X K" "Agent C"]
   insert_commute [of "Crypt X K" "Nonce N"]
   insert_commute [of "Crypt X K" "Number N"]
   insert_commute [of "Crypt X K" "Hash X'"]
   insert_commute [of "Crypt X K" "MPair X' Y"]
   for X K C N X' Y
 
 text\<open>Cannot be added with \<open>[simp]\<close> -- messages should not always be
   re-ordered.\<close>
 lemmas pushes = pushKeys pushCrypts
 
 
 subsection\<open>The set of key-free messages\<close>
 
 (*Note that even the encryption of a key-free message remains key-free.
   This concept is valuable because of the theorem analz_keyfree_into_Un, proved below. *)
 
 inductive_set
   keyfree :: "msg set"
   where
     Agent:  "Agent A \<in> keyfree"
   | Number: "Number N \<in> keyfree"
   | Nonce:  "Nonce N \<in> keyfree"
   | Hash:   "Hash X \<in> keyfree"
   | MPair:  "[|X \<in> keyfree;  Y \<in> keyfree|] ==> \<lbrace>X,Y\<rbrace> \<in> keyfree"
   | Crypt:  "[|X \<in> keyfree|] ==> Crypt K X \<in> keyfree"
 
 
-declare keyfree.intros [intro] 
+declare keyfree.intros [intro]
 
 inductive_cases keyfree_KeyE: "Key K \<in> keyfree"
 inductive_cases keyfree_MPairE: "\<lbrace>X,Y\<rbrace> \<in> keyfree"
 inductive_cases keyfree_CryptE: "Crypt K X \<in> keyfree"
 
 lemma parts_keyfree: "parts (keyfree) \<subseteq> keyfree"
   by (clarify, erule parts.induct, auto elim!: keyfree_KeyE keyfree_MPairE keyfree_CryptE)
 
 (*The key-free part of a set of messages can be removed from the scope of the analz operator.*)
 lemma analz_keyfree_into_Un: "\<lbrakk>X \<in> analz (G \<union> H); G \<subseteq> keyfree\<rbrakk> \<Longrightarrow> X \<in> parts G \<union> analz H"
 apply (erule analz.induct, auto)
 apply (blast dest:parts.Body)
 apply (blast dest: parts.Body)
 apply (metis Un_absorb2 keyfree_KeyE parts_Un parts_keyfree UnI2)
 done
 
 subsection\<open>Tactics useful for many protocol proofs\<close>
 ML
 \<open>
 (*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.
   DEPENDS UPON "X" REFERRING TO THE FRADULENT MESSAGE *)
 
 fun impOfSubs th = th RSN (2, @{thm rev_subsetD})
 
 (*Apply rules to break down assumptions of the form
   Y \<in> parts(insert X H)  and  Y \<in> analz(insert X H)
 *)
-fun Fake_insert_tac ctxt = 
+fun Fake_insert_tac ctxt =
     dresolve_tac ctxt [impOfSubs @{thm Fake_analz_insert},
                   impOfSubs @{thm Fake_parts_insert}] THEN'
     eresolve_tac ctxt [asm_rl, @{thm synth.Inj}];
 
-fun Fake_insert_simp_tac ctxt i = 
+fun Fake_insert_simp_tac ctxt i =
   REPEAT (Fake_insert_tac ctxt i) THEN asm_full_simp_tac ctxt i;
 
 fun atomic_spy_analz_tac ctxt =
   SELECT_GOAL
    (Fake_insert_simp_tac ctxt 1 THEN
     IF_UNSOLVED
       (Blast.depth_tac
         (ctxt addIs [@{thm analz_insertI}, impOfSubs @{thm analz_subset_parts}]) 4 1));
 
 fun spy_analz_tac ctxt i =
   DETERM
    (SELECT_GOAL
-     (EVERY 
+     (EVERY
       [  (*push in occurrences of X...*)
        (REPEAT o CHANGED)
          (Rule_Insts.res_inst_tac ctxt [((("x", 1), Position.none), "X")] []
           (insert_commute RS ssubst) 1),
        (*...allowing further simplifications*)
        simp_tac ctxt 1,
        REPEAT (FIRSTGOAL (resolve_tac ctxt [allI,impI,notI,conjI,iffI])),
        DEPTH_SOLVE (atomic_spy_analz_tac ctxt 1)]) i);
 \<close>
 
 text\<open>By default only \<open>o_apply\<close> is built-in.  But in the presence of
 eta-expansion this means that some terms displayed as @{term "f o g"} will be
 rewritten, and others will not!\<close>
 declare o_def [simp]
 
 
 lemma Crypt_notin_image_Key [simp]: "Crypt K X \<notin> Key ` A"
 by auto
 
 lemma Hash_notin_image_Key [simp] :"Hash X \<notin> Key ` A"
 by auto
 
 lemma synth_analz_mono: "G\<subseteq>H ==> synth (analz(G)) \<subseteq> synth (analz(H))"
-by (iprover intro: synth_mono analz_mono) 
+by (iprover intro: synth_mono analz_mono)
 
 lemma Fake_analz_eq [simp]:
      "X \<in> synth(analz H) ==> synth (analz (insert X H)) = synth (analz H)"
-by (metis Fake_analz_insert Un_absorb Un_absorb1 Un_commute 
+by (metis Fake_analz_insert Un_absorb Un_absorb1 Un_commute
           subset_insertI synth_analz_mono synth_increasing synth_subset_iff)
 
 text\<open>Two generalizations of \<open>analz_insert_eq\<close>\<close>
 lemma gen_analz_insert_eq [rule_format]:
      "X \<in> analz H ==> \<forall>G. H \<subseteq> G --> analz (insert X G) = analz G"
 by (blast intro: analz_cut analz_insertI analz_mono [THEN [2] rev_subsetD])
 
 lemma synth_analz_insert_eq [rule_format]:
-     "X \<in> synth (analz H) 
+     "X \<in> synth (analz H)
       ==> \<forall>G. H \<subseteq> G --> (Key K \<in> analz (insert X G)) = (Key K \<in> analz G)"
-apply (erule synth.induct) 
-apply (simp_all add: gen_analz_insert_eq subset_trans [OF _ subset_insertI]) 
+apply (erule synth.induct)
+apply (simp_all add: gen_analz_insert_eq subset_trans [OF _ subset_insertI])
 done
 
 lemma Fake_parts_sing:
      "X \<in> synth (analz H) ==> parts{X} \<subseteq> synth (analz H) \<union> parts H"
 by (metis Fake_parts_insert empty_subsetI insert_mono parts_mono subset_trans)
 
 lemmas Fake_parts_sing_imp_Un = Fake_parts_sing [THEN [2] rev_subsetD]
 
 method_setup spy_analz = \<open>
     Scan.succeed (SIMPLE_METHOD' o spy_analz_tac)\<close>
     "for proving the Fake case when analz is involved"
 
 method_setup atomic_spy_analz = \<open>
     Scan.succeed (SIMPLE_METHOD' o atomic_spy_analz_tac)\<close>
     "for debugging spy_analz"
 
 method_setup Fake_insert_simp = \<open>
     Scan.succeed (SIMPLE_METHOD' o Fake_insert_simp_tac)\<close>
     "for debugging spy_analz"
 
 end
diff --git a/thys/Inductive_Confidentiality/GeneralAttacker/MessageGA.thy b/thys/Inductive_Confidentiality/GeneralAttacker/MessageGA.thy
--- a/thys/Inductive_Confidentiality/GeneralAttacker/MessageGA.thy
+++ b/thys/Inductive_Confidentiality/GeneralAttacker/MessageGA.thy
@@ -1,929 +1,919 @@
 section\<open>Theory of Agents and Messages for Security Protocols against the General Attacker\<close>
 
 theory MessageGA imports Main begin
 
 (*Needed occasionally with spy_analz_tac, e.g. in analz_insert_Key_newK*)
 lemma [simp] : "A \<union> (B \<union> A) = B \<union> A"
 by blast
 
 type_synonym
   key = nat
 
 consts
   all_symmetric :: bool        \<comment> \<open>true if all keys are symmetric\<close>
   invKey        :: "key=>key"  \<comment> \<open>inverse of a symmetric key\<close>
 
 specification (invKey)
   invKey [simp]: "invKey (invKey K) = K"
   invKey_symmetric: "all_symmetric \<longrightarrow> invKey = id"
     by (rule exI [of _ id], auto)
 
 
 text\<open>The inverse of a symmetric key is itself; that of a public key
       is the private key and vice versa\<close>
 
 definition symKeys :: "key set" where
   "symKeys == {K. invKey K = K}"
 
 datatype  \<comment> \<open>We only allow for any number of friendly agents\<close>
   agent = Friend nat
 
 datatype
      msg = Agent  agent     \<comment> \<open>Agent names\<close>
          | Number nat       \<comment> \<open>Ordinary integers, timestamps, ...\<close>
          | Nonce  nat       \<comment> \<open>Unguessable nonces\<close>
          | Key    key       \<comment> \<open>Crypto keys\<close>
          | Hash   msg       \<comment> \<open>Hashing\<close>
          | MPair  msg msg   \<comment> \<open>Compound messages\<close>
          | Crypt  key msg   \<comment> \<open>Encryption, public- or shared-key\<close>
 
 
 text\<open>Concrete syntax: messages appear as \<open>\<lbrace>A,B,NA\<rbrace>\<close>, etc...\<close>
 syntax
   "_MTuple"      :: "['a, args] => 'a * 'b"       ("(2\<lbrace>_,/ _\<rbrace>)")
 translations
   "\<lbrace>x, y, z\<rbrace>"   == "\<lbrace>x, \<lbrace>y, z\<rbrace>\<rbrace>"
   "\<lbrace>x, y\<rbrace>"      == "CONST MPair x y"
 
 
 definition HPair :: "[msg,msg] => msg" ("(4Hash[_] /_)" [0, 1000]) where
     \<comment> \<open>Message Y paired with a MAC computed with the help of X\<close>
     "Hash[X] Y == \<lbrace> Hash\<lbrace>X,Y\<rbrace>, Y\<rbrace>"
 
 definition keysFor :: "msg set => key set" where
     \<comment> \<open>Keys useful to decrypt elements of a message set\<close>
   "keysFor H == invKey ` {K. \<exists>X. Crypt K X \<in> H}"
 
 
 subsection\<open>Inductive definition of all parts of a message\<close>
 
 inductive_set
   parts :: "msg set => msg set"
   for H :: "msg set"
   where
     Inj [intro]:               "X \<in> H \<Longrightarrow> X \<in> parts H"
   | Fst:         "\<lbrace>X,Y\<rbrace>   \<in> parts H \<Longrightarrow> X \<in> parts H"
   | Snd:         "\<lbrace>X,Y\<rbrace>   \<in> parts H \<Longrightarrow> Y \<in> parts H"
   | Body:        "Crypt K X \<in> parts H \<Longrightarrow> X \<in> parts H"
 
 
 text\<open>Monotonicity\<close>
 lemma parts_mono: "G \<subseteq> H \<Longrightarrow> parts(G) \<subseteq> parts(H)"
 apply auto
-apply (erule parts.induct) 
+apply (erule parts.induct)
 apply (blast dest: parts.Fst parts.Snd parts.Body)+
 done
 
 
 text\<open>Equations hold because constructors are injective.\<close>
 lemma Friend_image_eq [simp]: "(Friend x \<in> Friend`A) = (x:A)"
 by auto
 
 lemma Key_image_eq [simp]: "(Key x \<in> Key`A) = (x\<in>A)"
 by auto
 
 lemma Nonce_Key_image_eq [simp]: "(Nonce x \<notin> Key`A)"
 by auto
 
 
 subsection\<open>Inverse of keys\<close>
 
 lemma invKey_eq [simp]: "(invKey K = invKey K') = (K=K')"
 by (metis invKey)
 
 
 subsection\<open>keysFor operator\<close>
 
 lemma keysFor_empty [simp]: "keysFor {} = {}"
 by (unfold keysFor_def, blast)
 
 lemma keysFor_Un [simp]: "keysFor (H \<union> H') = keysFor H \<union> keysFor H'"
 by (unfold keysFor_def, blast)
 
 lemma keysFor_UN [simp]: "keysFor (\<Union>i\<in>A. H i) = (\<Union>i\<in>A. keysFor (H i))"
 by (unfold keysFor_def, blast)
 
 text\<open>Monotonicity\<close>
 lemma keysFor_mono: "G \<subseteq> H \<Longrightarrow> keysFor(G) \<subseteq> keysFor(H)"
 by (unfold keysFor_def, blast)
 
 lemma keysFor_insert_Agent [simp]: "keysFor (insert (Agent A) H) = keysFor H"
 by (unfold keysFor_def, auto)
 
 lemma keysFor_insert_Nonce [simp]: "keysFor (insert (Nonce N) H) = keysFor H"
 by (unfold keysFor_def, auto)
 
 lemma keysFor_insert_Number [simp]: "keysFor (insert (Number N) H) = keysFor H"
 by (unfold keysFor_def, auto)
 
 lemma keysFor_insert_Key [simp]: "keysFor (insert (Key K) H) = keysFor H"
 by (unfold keysFor_def, auto)
 
 lemma keysFor_insert_Hash [simp]: "keysFor (insert (Hash X) H) = keysFor H"
 by (unfold keysFor_def, auto)
 
 lemma keysFor_insert_MPair [simp]: "keysFor (insert \<lbrace>X,Y\<rbrace> H) = keysFor H"
 by (unfold keysFor_def, auto)
 
-lemma keysFor_insert_Crypt [simp]: 
+lemma keysFor_insert_Crypt [simp]:
     "keysFor (insert (Crypt K X) H) = insert (invKey K) (keysFor H)"
 by (unfold keysFor_def, auto)
 
 lemma keysFor_image_Key [simp]: "keysFor (Key`E) = {}"
 by (unfold keysFor_def, auto)
 
 lemma Crypt_imp_invKey_keysFor: "Crypt K X \<in> H \<Longrightarrow> invKey K \<in> keysFor H"
 by (unfold keysFor_def, blast)
 
 
 subsection\<open>Inductive relation "parts"\<close>
 
 lemma MPair_parts:
-     "[| \<lbrace>X,Y\<rbrace> \<in> parts H;        
+     "[| \<lbrace>X,Y\<rbrace> \<in> parts H;
          [| X \<in> parts H; Y \<in> parts H |] ==> P |] ==> P"
-by (blast dest: parts.Fst parts.Snd) 
+by (blast dest: parts.Fst parts.Snd)
 
 declare MPair_parts [elim!]  parts.Body [dest!]
 text\<open>NB These two rules are UNSAFE in the formal sense, as they discard the
-     compound message.  They work well on THIS FILE.  
+     compound message.  They work well on THIS FILE.
   \<open>MPair_parts\<close> is left as SAFE because it speeds up proofs.
   The Crypt rule is normally kept UNSAFE to avoid breaking up certificates.\<close>
 
 lemma parts_increasing: "H \<subseteq> parts(H)"
 by blast
 
 lemmas parts_insertI = subset_insertI [THEN parts_mono, THEN subsetD]
 
 lemma parts_empty [simp]: "parts{} = {}"
 apply safe
 apply (erule parts.induct, blast+)
 done
 
 lemma parts_emptyE [elim!]: "X\<in> parts{} \<Longrightarrow> P"
 by simp
 
 text\<open>WARNING: loops if H = {Y}, therefore must not be repeated!\<close>
 lemma parts_singleton: "X\<in> parts H \<Longrightarrow> \<exists>Y\<in>H. X\<in> parts {Y}"
 by (erule parts.induct, fast+)
 
 
 subsubsection\<open>Unions\<close>
 
 lemma parts_Un_subset1: "parts(G) \<union> parts(H) \<subseteq> parts(G \<union> H)"
 by (intro Un_least parts_mono Un_upper1 Un_upper2)
 
 lemma parts_Un_subset2: "parts(G \<union> H) \<subseteq> parts(G) \<union> parts(H)"
 apply (rule subsetI)
 apply (erule parts.induct, blast+)
 done
 
 lemma parts_Un [simp]: "parts(G \<union> H) = parts(G) \<union> parts(H)"
 by (intro equalityI parts_Un_subset1 parts_Un_subset2)
 
 lemma parts_insert: "parts (insert X H) = parts {X} \<union> parts H"
 by (metis insert_is_Un parts_Un)
 
 text\<open>TWO inserts to avoid looping.  This rewrite is better than nothing.
   Not suitable for Addsimps: its behaviour can be strange.\<close>
 lemma parts_insert2:
      "parts (insert X (insert Y H)) = parts {X} \<union> parts {Y} \<union> parts H"
 by (metis Un_commute Un_empty_right Un_insert_right insert_is_Un parts_Un)
 
 lemma parts_UN_subset1: "(\<Union>x\<in>A. parts(H x)) \<subseteq> parts(\<Union>x\<in>A. H x)"
 by (intro UN_least parts_mono UN_upper)
 
 lemma parts_UN_subset2: "parts(\<Union>x\<in>A. H x) \<subseteq> (\<Union>x\<in>A. parts(H x))"
 apply (rule subsetI)
 apply (erule parts.induct, blast+)
 done
 
 lemma parts_UN [simp]: "parts(\<Union>x\<in>A. H x) = (\<Union>x\<in>A. parts(H x))"
 by (intro equalityI parts_UN_subset1 parts_UN_subset2)
 
 text\<open>Added to simplify arguments to parts, analz and synth.
   NOTE: the UN versions are no longer used!\<close>
 
 
-text\<open>This allows \<open>blast\<close> to simplify occurrences of 
+text\<open>This allows \<open>blast\<close> to simplify occurrences of
   @{term "parts(G\<union>H)"} in the assumption.\<close>
-lemmas in_parts_UnE = parts_Un [THEN equalityD1, THEN subsetD, THEN UnE] 
+lemmas in_parts_UnE = parts_Un [THEN equalityD1, THEN subsetD, THEN UnE]
 declare in_parts_UnE [elim!]
 
 
 lemma parts_insert_subset: "insert X (parts H) \<subseteq> parts(insert X H)"
 by (blast intro: parts_mono [THEN [2] rev_subsetD])
 
 subsubsection\<open>Idempotence and transitivity\<close>
 
 lemma parts_partsD [dest!]: "X \<in> parts (parts H) \<Longrightarrow> X\<in> parts H"
 by (erule parts.induct, blast+)
 
 lemma parts_idem [simp]: "parts (parts H) = parts H"
 by blast
 
 lemma parts_subset_iff [simp]: "(parts G \<subseteq> parts H) = (G \<subseteq> parts H)"
 by (metis parts_idem parts_increasing parts_mono subset_trans)
 
 lemma parts_trans: "[| X \<in> parts G;  G \<subseteq> parts H |] ==> X\<in> parts H"
 by (drule parts_mono, blast)
 
 text\<open>Cut\<close>
 lemma parts_cut:
-     "[| Y\<in> parts (insert X G);  X \<in> parts H |] ==> Y\<in> parts (G \<union> H)" 
-by (blast intro: parts_trans) 
+     "[| Y\<in> parts (insert X G);  X \<in> parts H |] ==> Y\<in> parts (G \<union> H)"
+by (blast intro: parts_trans)
 
 
 lemma parts_cut_eq [simp]: "X \<in> parts H \<Longrightarrow> parts (insert X H) = parts H"
 by (force dest!: parts_cut intro: parts_insertI)
 
 
 subsubsection\<open>Rewrite rules for pulling out atomic messages\<close>
 
 lemmas parts_insert_eq_I = equalityI [OF subsetI parts_insert_subset]
 
 
 lemma parts_insert_Agent [simp]:
      "parts (insert (Agent agt) H) = insert (Agent agt) (parts H)"
-apply (rule parts_insert_eq_I) 
-apply (erule parts.induct, auto) 
+apply (rule parts_insert_eq_I)
+apply (erule parts.induct, auto)
 done
 
 lemma parts_insert_Nonce [simp]:
      "parts (insert (Nonce N) H) = insert (Nonce N) (parts H)"
-apply (rule parts_insert_eq_I) 
-apply (erule parts.induct, auto) 
+apply (rule parts_insert_eq_I)
+apply (erule parts.induct, auto)
 done
 
 lemma parts_insert_Number [simp]:
      "parts (insert (Number N) H) = insert (Number N) (parts H)"
-apply (rule parts_insert_eq_I) 
-apply (erule parts.induct, auto) 
+apply (rule parts_insert_eq_I)
+apply (erule parts.induct, auto)
 done
 
 lemma parts_insert_Key [simp]:
      "parts (insert (Key K) H) = insert (Key K) (parts H)"
-apply (rule parts_insert_eq_I) 
-apply (erule parts.induct, auto) 
+apply (rule parts_insert_eq_I)
+apply (erule parts.induct, auto)
 done
 
 lemma parts_insert_Hash [simp]:
      "parts (insert (Hash X) H) = insert (Hash X) (parts H)"
-apply (rule parts_insert_eq_I) 
-apply (erule parts.induct, auto) 
+apply (rule parts_insert_eq_I)
+apply (erule parts.induct, auto)
 done
 
 lemma parts_insert_Crypt [simp]:
      "parts (insert (Crypt K X) H) = insert (Crypt K X) (parts (insert X H))"
 apply (rule equalityI)
 apply (rule subsetI)
 apply (erule parts.induct, auto)
 apply (blast intro: parts.Body)
 done
 
 lemma parts_insert_MPair [simp]:
-     "parts (insert \<lbrace>X,Y\<rbrace> H) =  
+     "parts (insert \<lbrace>X,Y\<rbrace> H) =
           insert \<lbrace>X,Y\<rbrace> (parts (insert X (insert Y H)))"
 apply (rule equalityI)
 apply (rule subsetI)
 apply (erule parts.induct, auto)
 apply (blast intro: parts.Fst parts.Snd)+
 done
 
 lemma parts_image_Key [simp]: "parts (Key`N) = Key`N"
 apply auto
 apply (erule parts.induct, auto)
 done
 
 
 text\<open>In any message, there is an upper bound N on its greatest nonce.\<close>
 lemma msg_Nonce_supply: "\<exists>N. \<forall>n. N\<le>n \<longrightarrow> Nonce n \<notin> parts {msg}"
 apply (induct msg)
 apply (simp_all (no_asm_simp) add: exI parts_insert2)
 txt\<open>Nonce case\<close>
 apply (metis Suc_n_not_le_n)
 txt\<open>MPair case: metis works out the necessary sum itself!\<close>
 apply (metis le_trans nat_le_linear)
 done
 
 
 subsection\<open>Inductive relation "analz"\<close>
 
 text\<open>Inductive definition of "analz" -- what can be broken down from a set of
     messages, including keys.  A form of downward closure.  Pairs can
     be taken apart; messages decrypted with known keys.\<close>
 
 inductive_set
   analz :: "msg set => msg set"
   for H :: "msg set"
   where
     Inj [intro,simp] :    "X \<in> H \<Longrightarrow> X \<in> analz H"
   | Fst:     "\<lbrace>X,Y\<rbrace> \<in> analz H \<Longrightarrow> X \<in> analz H"
   | Snd:     "\<lbrace>X,Y\<rbrace> \<in> analz H \<Longrightarrow> Y \<in> analz H"
-  | Decrypt [dest]: 
+  | Decrypt [dest]:
              "[|Crypt K X \<in> analz H; Key(invKey K): analz H|] ==> X \<in> analz H"
 
 
 text\<open>Monotonicity; Lemma 1 of Lowe's paper\<close>
 lemma analz_mono: "G\<subseteq>H \<Longrightarrow> analz(G) \<subseteq> analz(H)"
 apply auto
-apply (erule analz.induct) 
-apply (auto dest: analz.Fst analz.Snd) 
+apply (erule analz.induct)
+apply (auto dest: analz.Fst analz.Snd)
 done
 
 text\<open>Making it safe speeds up proofs\<close>
 lemma MPair_analz [elim!]:
-     "[| \<lbrace>X,Y\<rbrace> \<in> analz H;        
-             [| X \<in> analz H; Y \<in> analz H |] ==> P   
+     "[| \<lbrace>X,Y\<rbrace> \<in> analz H;
+             [| X \<in> analz H; Y \<in> analz H |] ==> P
           |] ==> P"
 by (blast dest: analz.Fst analz.Snd)
 
 lemma analz_increasing: "H \<subseteq> analz(H)"
 by blast
 
 lemma analz_subset_parts: "analz H \<subseteq> parts H"
 apply (rule subsetI)
 apply (erule analz.induct, blast+)
 done
 
 lemmas analz_into_parts = analz_subset_parts [THEN subsetD]
 
 lemmas not_parts_not_analz = analz_subset_parts [THEN contra_subsetD]
 
 
 lemma parts_analz [simp]: "parts (analz H) = parts H"
 by (metis analz_increasing analz_subset_parts equalityI parts_mono parts_subset_iff)
 
 lemma analz_parts [simp]: "analz (parts H) = parts H"
 apply auto
 apply (erule analz.induct, auto)
 done
 
 lemmas analz_insertI = subset_insertI [THEN analz_mono, THEN [2] rev_subsetD]
 
 subsubsection\<open>General equational properties\<close>
 
 lemma analz_empty [simp]: "analz{} = {}"
 apply safe
 apply (erule analz.induct, blast+)
 done
 
-text\<open>Converse fails: we can analz more from the union than from the 
+text\<open>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\<close>
 lemma analz_Un: "analz(G) \<union> analz(H) \<subseteq> analz(G \<union> H)"
 by (intro Un_least analz_mono Un_upper1 Un_upper2)
 
 lemma analz_insert: "insert X (analz H) \<subseteq> analz(insert X H)"
 by (blast intro: analz_mono [THEN [2] rev_subsetD])
 
 subsubsection\<open>Rewrite rules for pulling out atomic messages\<close>
 
 lemmas analz_insert_eq_I = equalityI [OF subsetI analz_insert]
 
 lemma analz_insert_Agent [simp]:
      "analz (insert (Agent agt) H) = insert (Agent agt) (analz H)"
-apply (rule analz_insert_eq_I) 
-apply (erule analz.induct, auto) 
+apply (rule analz_insert_eq_I)
+apply (erule analz.induct, auto)
 done
 
 lemma analz_insert_Nonce [simp]:
      "analz (insert (Nonce N) H) = insert (Nonce N) (analz H)"
-apply (rule analz_insert_eq_I) 
-apply (erule analz.induct, auto) 
+apply (rule analz_insert_eq_I)
+apply (erule analz.induct, auto)
 done
 
 lemma analz_insert_Number [simp]:
      "analz (insert (Number N) H) = insert (Number N) (analz H)"
-apply (rule analz_insert_eq_I) 
-apply (erule analz.induct, auto) 
+apply (rule analz_insert_eq_I)
+apply (erule analz.induct, auto)
 done
 
 lemma analz_insert_Hash [simp]:
      "analz (insert (Hash X) H) = insert (Hash X) (analz H)"
-apply (rule analz_insert_eq_I) 
-apply (erule analz.induct, auto) 
+apply (rule analz_insert_eq_I)
+apply (erule analz.induct, auto)
 done
 
 text\<open>Can only pull out Keys if they are not needed to decrypt the rest\<close>
-lemma analz_insert_Key [simp]: 
-    "K \<notin> keysFor (analz H) \<Longrightarrow>   
+lemma analz_insert_Key [simp]:
+    "K \<notin> keysFor (analz H) \<Longrightarrow>
           analz (insert (Key K) H) = insert (Key K) (analz H)"
 apply (unfold keysFor_def)
-apply (rule analz_insert_eq_I) 
-apply (erule analz.induct, auto) 
+apply (rule analz_insert_eq_I)
+apply (erule analz.induct, auto)
 done
 
 lemma analz_insert_MPair [simp]:
-     "analz (insert \<lbrace>X,Y\<rbrace> H) =  
+     "analz (insert \<lbrace>X,Y\<rbrace> H) =
           insert \<lbrace>X,Y\<rbrace> (analz (insert X (insert Y H)))"
 apply (rule equalityI)
 apply (rule subsetI)
 apply (erule analz.induct, auto)
 apply (erule analz.induct)
 apply (blast intro: analz.Fst analz.Snd)+
 done
 
 text\<open>Can pull out enCrypted message if the Key is not known\<close>
 lemma analz_insert_Crypt:
-     "Key (invKey K) \<notin> analz H 
+     "Key (invKey K) \<notin> analz H
       \<Longrightarrow> analz (insert (Crypt K X) H) = insert (Crypt K X) (analz H)"
-apply (rule analz_insert_eq_I) 
-apply (erule analz.induct, auto) 
+apply (rule analz_insert_eq_I)
+apply (erule analz.induct, auto)
 
 done
 
-lemma lemma1: "Key (invKey K) \<in> analz H \<Longrightarrow>   
-               analz (insert (Crypt K X) H) \<subseteq>  
+lemma lemma1: "Key (invKey K) \<in> analz H \<Longrightarrow>
+               analz (insert (Crypt K X) H) \<subseteq>
                insert (Crypt K X) (analz (insert X H))"
 apply (rule subsetI)
 apply (erule_tac x = x in analz.induct, auto)
 done
 
-lemma lemma2: "Key (invKey K) \<in> analz H \<Longrightarrow>   
-               insert (Crypt K X) (analz (insert X H)) \<subseteq>  
+lemma lemma2: "Key (invKey K) \<in> analz H \<Longrightarrow>
+               insert (Crypt K X) (analz (insert X H)) \<subseteq>
                analz (insert (Crypt K X) H)"
 apply auto
 apply (erule_tac x = x in analz.induct, auto)
 apply (blast intro: analz_insertI analz.Decrypt)
 done
 
 lemma analz_insert_Decrypt:
-     "Key (invKey K) \<in> analz H \<Longrightarrow>   
-               analz (insert (Crypt K X) H) =  
+     "Key (invKey K) \<in> analz H \<Longrightarrow>
+               analz (insert (Crypt K X) H) =
                insert (Crypt K X) (analz (insert X H))"
 by (intro equalityI lemma1 lemma2)
 
 text\<open>Case analysis: either the message is secure, or it is not! Effective,
 but can cause subgoals to blow up! Use with \<open>if_split\<close>; apparently
 \<open>split_tac\<close> does not cope with patterns such as @{term"analz (insert
-(Crypt K X) H)"}\<close> 
+(Crypt K X) H)"}\<close>
 lemma analz_Crypt_if [simp]:
-     "analz (insert (Crypt K X) H) =                 
-          (if (Key (invKey K) \<in> analz H)                 
-           then insert (Crypt K X) (analz (insert X H))  
+     "analz (insert (Crypt K X) H) =
+          (if (Key (invKey K) \<in> analz H)
+           then insert (Crypt K X) (analz (insert X H))
            else insert (Crypt K X) (analz H))"
 by (simp add: analz_insert_Crypt analz_insert_Decrypt)
 
 
 text\<open>This rule supposes "for the sake of argument" that we have the key.\<close>
 lemma analz_insert_Crypt_subset:
-     "analz (insert (Crypt K X) H) \<subseteq>   
+     "analz (insert (Crypt K X) H) \<subseteq>
            insert (Crypt K X) (analz (insert X H))"
 apply (rule subsetI)
 apply (erule analz.induct, auto)
 done
 
 
 lemma analz_image_Key [simp]: "analz (Key`N) = Key`N"
 apply auto
 apply (erule analz.induct, auto)
 done
 
 
 subsubsection\<open>Idempotence and transitivity\<close>
 
 lemma analz_analzD [dest!]: "X\<in> analz (analz H) \<Longrightarrow> X \<in> analz H"
 by (erule analz.induct, blast+)
 
 lemma analz_idem [simp]: "analz (analz H) = analz H"
 by blast
 
 lemma analz_subset_iff [simp]: "(analz G \<subseteq> analz H) = (G \<subseteq> analz H)"
 by (metis analz_idem analz_increasing analz_mono subset_trans)
 
 lemma analz_trans: "[| X\<in> analz G;  G \<subseteq> analz H |] ==> X\<in> analz H"
 by (drule analz_mono, blast)
 
 text\<open>Cut; Lemma 2 of Lowe\<close>
 lemma analz_cut: "[| Y\<in> analz (insert X H);  X\<in> analz H |] ==> Y\<in> analz H"
 by (erule analz_trans, blast)
 
 (*Cut can be proved easily by induction on
    "Y: analz (insert X H) \<Longrightarrow> X: analz H \<longrightarrow> Y: analz H"
 *)
 
 text\<open>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.\<close>
 lemma analz_insert_eq: "X \<in> analz H \<Longrightarrow> analz (insert X H) = analz H"
 by (blast intro: analz_cut analz_insertI)
 
 
 text\<open>A congruence rule for "analz"\<close>
 
 lemma analz_subset_cong:
-     "[| analz G \<subseteq> analz G'; analz H \<subseteq> analz H' |] 
+     "[| analz G \<subseteq> analz G'; analz H \<subseteq> analz H' |]
       ==> analz (G \<union> H) \<subseteq> analz (G' \<union> H')"
 apply simp
-apply (iprover intro: conjI subset_trans analz_mono Un_upper1 Un_upper2) 
+apply (iprover intro: conjI subset_trans analz_mono Un_upper1 Un_upper2)
 done
 
 lemma analz_cong:
-     "[| analz G = analz G'; analz H = analz H' |] 
+     "[| analz G = analz G'; analz H = analz H' |]
       ==> analz (G \<union> H) = analz (G' \<union> H')"
-by (intro equalityI analz_subset_cong, simp_all) 
+by (intro equalityI analz_subset_cong, simp_all)
 
 lemma analz_insert_cong:
      "analz H = analz H' \<Longrightarrow> analz(insert X H) = analz(insert X H')"
 by (force simp only: insert_def intro!: analz_cong)
 
 text\<open>If there are no pairs or encryptions then analz does nothing\<close>
 lemma analz_trivial:
      "[| \<forall>X Y. \<lbrace>X,Y\<rbrace> \<notin> H;  \<forall>X K. Crypt K X \<notin> H |] ==> analz H = H"
 apply safe
 apply (erule analz.induct, blast+)
 done
 
-text\<open>These two are obsolete but cost little to prove...\<close>
-lemma analz_UN_analz_lemma:
-     "X\<in> analz (\<Union>i\<in>A. analz (H i)) \<Longrightarrow> X\<in> analz (\<Union>i\<in>A. H i)"
-apply (erule analz.induct)
-apply (blast intro: analz_mono [THEN [2] rev_subsetD])+
-done
-
-lemma analz_UN_analz [simp]: "analz (\<Union>i\<in>A. analz (H i)) = analz (\<Union>i\<in>A. H i)"
-by (blast intro: analz_UN_analz_lemma analz_mono [THEN [2] rev_subsetD])
-
 
 subsection\<open>Inductive relation "synth"\<close>
 
 text\<open>Inductive definition of "synth" -- what can be built up from a set of
     messages.  A form of upward closure.  Pairs can be built, messages
     encrypted with known keys.  Agent names are public domain.
     Numbers can be guessed, but Nonces cannot be.\<close>
 
 inductive_set
   synth :: "msg set => msg set"
   for H :: "msg set"
   where
     Inj    [intro]:   "X \<in> H \<Longrightarrow> X \<in> synth H"
   | Agent  [intro]:   "Agent agt \<in> synth H"
   | Number [intro]:   "Number n  \<in> synth H"
   | Hash   [intro]:   "X \<in> synth H \<Longrightarrow> Hash X \<in> synth H"
   | MPair  [intro]:   "[|X \<in> synth H;  Y \<in> synth H|] ==> \<lbrace>X,Y\<rbrace> \<in> synth H"
   | Crypt  [intro]:   "[|X \<in> synth H;  Key(K) \<in> H|] ==> Crypt K X \<in> synth H"
 
 text\<open>Monotonicity\<close>
 lemma synth_mono: "G\<subseteq>H \<Longrightarrow> synth(G) \<subseteq> synth(H)"
-  by (auto, erule synth.induct, auto)  
+  by (auto, erule synth.induct, auto)
 
-text\<open>NO \<open>Agent_synth\<close>, as any Agent name can be synthesized.  
+text\<open>NO \<open>Agent_synth\<close>, as any Agent name can be synthesized.
   The same holds for @{term Number}\<close>
 
 inductive_simps synth_simps [iff]:
  "Nonce n \<in> synth H"
  "Key K \<in> synth H"
  "Hash X \<in> synth H"
  "\<lbrace>X,Y\<rbrace> \<in> synth H"
  "Crypt K X \<in> synth H"
 
 lemma synth_increasing: "H \<subseteq> synth(H)"
 by blast
 
 subsubsection\<open>Unions\<close>
 
-text\<open>Converse fails: we can synth more from the union than from the 
+text\<open>Converse fails: we can synth more from the union than from the
   separate parts, building a compound message using elements of each.\<close>
 lemma synth_Un: "synth(G) \<union> synth(H) \<subseteq> synth(G \<union> H)"
 by (intro Un_least synth_mono Un_upper1 Un_upper2)
 
 lemma synth_insert: "insert X (synth H) \<subseteq> synth(insert X H)"
 by (blast intro: synth_mono [THEN [2] rev_subsetD])
 
 subsubsection\<open>Idempotence and transitivity\<close>
 
 lemma synth_synthD [dest!]: "X \<in> synth (synth H) \<Longrightarrow> X\<in> synth H"
 by (erule synth.induct, auto)
 
 lemma synth_idem: "synth (synth H) = synth H"
 by blast
 
 lemma synth_subset_iff [simp]: "(synth G \<subseteq> synth H) = (G \<subseteq> synth H)"
 by (metis subset_trans synth_idem synth_increasing synth_mono)
 
 lemma synth_trans: "[| X\<in> synth G;  G \<subseteq> synth H |] ==> X\<in> synth H"
 by (drule synth_mono, blast)
 
 text\<open>Cut; Lemma 2 of Lowe\<close>
 lemma synth_cut: "[| Y\<in> synth (insert X H);  X\<in> synth H |] ==> Y\<in> synth H"
 by (erule synth_trans, blast)
 
 lemma Agent_synth [simp]: "Agent A \<in> synth H"
 by blast
 
 lemma Number_synth [simp]: "Number n \<in> synth H"
 by blast
 
 lemma Nonce_synth_eq [simp]: "(Nonce N \<in> synth H) = (Nonce N \<in> H)"
 by blast
 
 lemma Key_synth_eq [simp]: "(Key K \<in> synth H) = (Key K \<in> H)"
 by blast
 
 lemma Crypt_synth_eq [simp]:
      "Key K \<notin> H \<Longrightarrow> (Crypt K X \<in> synth H) = (Crypt K X \<in> H)"
 by blast
 
 
-lemma keysFor_synth [simp]: 
+lemma keysFor_synth [simp]:
     "keysFor (synth H) = keysFor H \<union> invKey`{K. Key K \<in> H}"
 by (unfold keysFor_def, blast)
 
 
 subsubsection\<open>Combinations of parts, analz and synth\<close>
 
 lemma parts_synth [simp]: "parts (synth H) = parts H \<union> synth H"
 apply (rule equalityI)
 apply (rule subsetI)
 apply (erule parts.induct)
-apply (blast intro: synth_increasing [THEN parts_mono, THEN subsetD] 
+apply (blast intro: synth_increasing [THEN parts_mono, THEN subsetD]
                     parts.Fst parts.Snd parts.Body)+
 done
 
 lemma analz_analz_Un [simp]: "analz (analz G \<union> H) = analz (G \<union> H)"
 apply (intro equalityI analz_subset_cong)+
 apply simp_all
 done
 
 lemma analz_synth_Un [simp]: "analz (synth G \<union> H) = analz (G \<union> H) \<union> synth G"
 apply (rule equalityI)
 apply (rule subsetI)
 apply (erule analz.induct)
 prefer 5 apply (blast intro: analz_mono [THEN [2] rev_subsetD])
 apply (blast intro: analz.Fst analz.Snd analz.Decrypt)+
 done
 
 lemma analz_synth [simp]: "analz (synth H) = analz H \<union> synth H"
 by (metis Un_empty_right analz_synth_Un)
 
 
 subsubsection\<open>For reasoning about the Fake rule in traces\<close>
 
 lemma parts_insert_subset_Un: "X \<in> G \<Longrightarrow> parts(insert X H) \<subseteq> parts G \<union> parts H"
 by (metis UnCI Un_upper2 insert_subset parts_Un parts_mono)
 
 text\<open>More specifically for Fake. See also \<open>Fake_parts_sing\<close> below\<close>
 lemma Fake_parts_insert:
-     "X \<in> synth (analz H) \<Longrightarrow>  
+     "X \<in> synth (analz H) \<Longrightarrow>
       parts (insert X H) \<subseteq> synth (analz H) \<union> parts H"
-by (metis Un_commute analz_increasing insert_subset parts_analz parts_mono 
+by (metis Un_commute analz_increasing insert_subset parts_analz parts_mono
           parts_synth synth_mono synth_subset_iff)
 
 lemma Fake_parts_insert_in_Un:
-     "[|Z \<in> parts (insert X H);  X: synth (analz H)|] 
+     "[|Z \<in> parts (insert X H);  X: synth (analz H)|]
       ==> Z \<in>  synth (analz H) \<union> parts H"
 by (metis Fake_parts_insert subsetD)
 
-text\<open>@{term H} is sometimes @{term"Key ` KK \<union> spies evs"}, so can't put 
+text\<open>@{term H} is sometimes @{term"Key ` KK \<union> spies evs"}, so can't put
   @{term "G=H"}.\<close>
 lemma Fake_analz_insert:
-     "X\<in> synth (analz G) \<Longrightarrow>  
+     "X\<in> synth (analz G) \<Longrightarrow>
       analz (insert X H) \<subseteq> synth (analz G) \<union> analz (G \<union> H)"
 apply (rule subsetI)
 apply (subgoal_tac "x \<in> analz (synth (analz G) \<union> H)", force)
 apply (blast intro: analz_mono [THEN [2] rev_subsetD] analz_mono [THEN synth_mono, THEN [2] rev_subsetD])
 done
 
 lemma analz_conj_parts [simp]:
      "(X \<in> analz H \<and> X \<in> parts H) = (X \<in> analz H)"
 by (blast intro: analz_subset_parts [THEN subsetD])
 
 lemma analz_disj_parts [simp]:
      "(X \<in> analz H | X \<in> parts H) = (X \<in> parts H)"
 by (blast intro: analz_subset_parts [THEN subsetD])
 
 text\<open>Without this equation, other rules for synth and analz would yield
   redundant cases\<close>
 lemma MPair_synth_analz [iff]:
-     "(\<lbrace>X,Y\<rbrace> \<in> synth (analz H)) =  
+     "(\<lbrace>X,Y\<rbrace> \<in> synth (analz H)) =
       (X \<in> synth (analz H) \<and> Y \<in> synth (analz H))"
 by blast
 
 lemma Crypt_synth_analz:
-     "[| Key K \<in> analz H;  Key (invKey K) \<in> analz H |]  
+     "[| Key K \<in> analz H;  Key (invKey K) \<in> analz H |]
        ==> (Crypt K X \<in> synth (analz H)) = (X \<in> synth (analz H))"
 by blast
 
 
 lemma Hash_synth_analz [simp]:
-     "X \<notin> synth (analz H)  
+     "X \<notin> synth (analz H)
       \<Longrightarrow> (Hash\<lbrace>X,Y\<rbrace> \<in> synth (analz H)) = (Hash\<lbrace>X,Y\<rbrace> \<in> analz H)"
 by blast
 
 
 subsection\<open>HPair: a combination of Hash and MPair\<close>
 
 subsubsection\<open>Freeness\<close>
 
 lemma Agent_neq_HPair: "Agent A ~= Hash[X] Y"
 by (unfold HPair_def, simp)
 
 lemma Nonce_neq_HPair: "Nonce N ~= Hash[X] Y"
 by (unfold HPair_def, simp)
 
 lemma Number_neq_HPair: "Number N ~= Hash[X] Y"
 by (unfold HPair_def, simp)
 
 lemma Key_neq_HPair: "Key K ~= Hash[X] Y"
 by (unfold HPair_def, simp)
 
 lemma Hash_neq_HPair: "Hash Z ~= Hash[X] Y"
 by (unfold HPair_def, simp)
 
 lemma Crypt_neq_HPair: "Crypt K X' ~= Hash[X] Y"
 by (unfold HPair_def, simp)
 
-lemmas HPair_neqs = Agent_neq_HPair Nonce_neq_HPair Number_neq_HPair 
+lemmas HPair_neqs = Agent_neq_HPair Nonce_neq_HPair Number_neq_HPair
                     Key_neq_HPair Hash_neq_HPair Crypt_neq_HPair
 
 declare HPair_neqs [iff]
 declare HPair_neqs [symmetric, iff]
 
 lemma HPair_eq [iff]: "(Hash[X'] Y' = Hash[X] Y) = (X' = X \<and> Y'=Y)"
 by (simp add: HPair_def)
 
 lemma MPair_eq_HPair [iff]:
      "(\<lbrace>X',Y'\<rbrace> = Hash[X] Y) = (X' = Hash\<lbrace>X,Y\<rbrace> \<and> Y'=Y)"
 by (simp add: HPair_def)
 
 lemma HPair_eq_MPair [iff]:
      "(Hash[X] Y = \<lbrace>X',Y'\<rbrace>) = (X' = Hash\<lbrace>X,Y\<rbrace> \<and> Y'=Y)"
 by (auto simp add: HPair_def)
 
 
 subsubsection\<open>Specialized laws, proved in terms of those for Hash and MPair\<close>
 
 lemma keysFor_insert_HPair [simp]: "keysFor (insert (Hash[X] Y) H) = keysFor H"
 by (simp add: HPair_def)
 
-lemma parts_insert_HPair [simp]: 
-    "parts (insert (Hash[X] Y) H) =  
+lemma parts_insert_HPair [simp]:
+    "parts (insert (Hash[X] Y) H) =
      insert (Hash[X] Y) (insert (Hash\<lbrace>X,Y\<rbrace>) (parts (insert Y H)))"
 by (simp add: HPair_def)
 
-lemma analz_insert_HPair [simp]: 
-    "analz (insert (Hash[X] Y) H) =  
+lemma analz_insert_HPair [simp]:
+    "analz (insert (Hash[X] Y) H) =
      insert (Hash[X] Y) (insert (Hash\<lbrace>X,Y\<rbrace>) (analz (insert Y H)))"
 by (simp add: HPair_def)
 
 lemma HPair_synth_analz [simp]:
-     "X \<notin> synth (analz H)  
-    \<Longrightarrow> (Hash[X] Y \<in> synth (analz H)) =  
+     "X \<notin> synth (analz H)
+    \<Longrightarrow> (Hash[X] Y \<in> synth (analz H)) =
         (Hash \<lbrace>X, Y\<rbrace> \<in> analz H \<and> Y \<in> synth (analz H))"
 by (auto simp add: HPair_def)
 
 
 text\<open>We do NOT want Crypt... messages broken up in protocols!!\<close>
 declare parts.Body [rule del]
 
 
 text\<open>Rewrites to push in Key and Crypt messages, so that other messages can
     be pulled out using the \<open>analz_insert\<close> rules\<close>
 
 lemmas pushKeys =
   insert_commute [of "Key K" "Agent C"]
   insert_commute [of "Key K" "Nonce N"]
   insert_commute [of "Key K" "Number N"]
   insert_commute [of "Key K" "Hash X"]
   insert_commute [of "Key K" "MPair X Y"]
   insert_commute [of "Key K" "Crypt X K'"]
   for K C N X Y K'
 
 lemmas pushCrypts =
   insert_commute [of "Crypt X K" "Agent C"]
   insert_commute [of "Crypt X K" "Agent C"]
   insert_commute [of "Crypt X K" "Nonce N"]
   insert_commute [of "Crypt X K" "Number N"]
   insert_commute [of "Crypt X K" "Hash X'"]
   insert_commute [of "Crypt X K" "MPair X' Y"]
   for X K C N X' Y
 
 text\<open>Cannot be added with \<open>[simp]\<close> -- messages should not always be
   re-ordered.\<close>
 lemmas pushes = pushKeys pushCrypts
 
 
 subsection\<open>The set of key-free messages\<close>
 
 (*Note that even the encryption of a key-free message remains key-free.
   This concept is valuable because of the theorem analz_keyfree_into_Un, proved below. *)
 
 inductive_set
   keyfree :: "msg set"
   where
     Agent:  "Agent A \<in> keyfree"
   | Number: "Number N \<in> keyfree"
   | Nonce:  "Nonce N \<in> keyfree"
   | Hash:   "Hash X \<in> keyfree"
   | MPair:  "[|X \<in> keyfree;  Y \<in> keyfree|] ==> \<lbrace>X,Y\<rbrace> \<in> keyfree"
   | Crypt:  "[|X \<in> keyfree|] ==> Crypt K X \<in> keyfree"
 
 
-declare keyfree.intros [intro] 
+declare keyfree.intros [intro]
 
 inductive_cases keyfree_KeyE: "Key K \<in> keyfree"
 inductive_cases keyfree_MPairE: "\<lbrace>X,Y\<rbrace> \<in> keyfree"
 inductive_cases keyfree_CryptE: "Crypt K X \<in> keyfree"
 
 lemma parts_keyfree: "parts (keyfree) \<subseteq> keyfree"
   by (clarify, erule parts.induct, auto elim!: keyfree_KeyE keyfree_MPairE keyfree_CryptE)
 
 (*The key-free part of a set of messages can be removed from the scope of the analz operator.*)
 lemma analz_keyfree_into_Un: "\<lbrakk>X \<in> analz (G \<union> H); G \<subseteq> keyfree\<rbrakk> \<Longrightarrow> X \<in> parts G \<union> analz H"
 apply (erule analz.induct, auto)
 apply (blast dest:parts.Body)
 apply (blast dest: parts.Body)
 apply (metis Un_absorb2 keyfree_KeyE parts_Un parts_keyfree UnI2)
 done
 
 subsection\<open>Tactics useful for many protocol proofs\<close>
 ML
 \<open>
 (*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.
   DEPENDS UPON "X" REFERRING TO THE FRADULENT MESSAGE *)
 
 fun impOfSubs th = th RSN (2, @{thm rev_subsetD})
 
 (*Apply rules to break down assumptions of the form
   Y \<in> parts(insert X H)  and  Y \<in> analz(insert X H)
 *)
-fun Fake_insert_tac ctxt = 
+fun Fake_insert_tac ctxt =
     dresolve_tac ctxt [impOfSubs @{thm Fake_analz_insert},
                   impOfSubs @{thm Fake_parts_insert}] THEN'
     eresolve_tac ctxt [asm_rl, @{thm synth.Inj}];
 
-fun Fake_insert_simp_tac ctxt i = 
+fun Fake_insert_simp_tac ctxt i =
   REPEAT (Fake_insert_tac ctxt i) THEN asm_full_simp_tac ctxt i;
 
 fun atomic_spy_analz_tac ctxt =
   SELECT_GOAL
    (Fake_insert_simp_tac ctxt 1 THEN
     IF_UNSOLVED
       (Blast.depth_tac
         (ctxt addIs [@{thm analz_insertI}, impOfSubs @{thm analz_subset_parts}]) 4 1));
 
 fun spy_analz_tac ctxt i =
   DETERM
    (SELECT_GOAL
-     (EVERY 
+     (EVERY
       [  (*push in occurrences of X...*)
        (REPEAT o CHANGED)
          (Rule_Insts.res_inst_tac ctxt [((("x", 1), Position.none), "X")] []
           (insert_commute RS ssubst) 1),
        (*...allowing further simplifications*)
        simp_tac ctxt 1,
        REPEAT (FIRSTGOAL (resolve_tac ctxt [allI,impI,notI,conjI,iffI])),
        DEPTH_SOLVE (atomic_spy_analz_tac ctxt 1)]) i);
 \<close>
 
 text\<open>By default only \<open>o_apply\<close> is built-in.  But in the presence of
 eta-expansion this means that some terms displayed as @{term "f o g"} will be
 rewritten, and others will not!\<close>
 declare o_def [simp]
 
 
 lemma Crypt_notin_image_Key [simp]: "Crypt K X \<notin> Key ` A"
 by auto
 
 lemma Hash_notin_image_Key [simp] :"Hash X \<notin> Key ` A"
 by auto
 
 lemma synth_analz_mono: "G\<subseteq>H \<Longrightarrow> synth (analz(G)) \<subseteq> synth (analz(H))"
-by (iprover intro: synth_mono analz_mono) 
+by (iprover intro: synth_mono analz_mono)
 
 lemma Fake_analz_eq [simp]:
      "X \<in> synth(analz H) \<Longrightarrow> synth (analz (insert X H)) = synth (analz H)"
-by (metis Fake_analz_insert Un_absorb Un_absorb1 Un_commute 
+by (metis Fake_analz_insert Un_absorb Un_absorb1 Un_commute
           subset_insertI synth_analz_mono synth_increasing synth_subset_iff)
 
 text\<open>Two generalizations of \<open>analz_insert_eq\<close>\<close>
 lemma gen_analz_insert_eq [rule_format]:
      "X \<in> analz H \<Longrightarrow> \<forall>G. H \<subseteq> G \<longrightarrow> analz (insert X G) = analz G"
 by (blast intro: analz_cut analz_insertI analz_mono [THEN [2] rev_subsetD])
 
 lemma synth_analz_insert_eq [rule_format]:
-     "X \<in> synth (analz H) 
+     "X \<in> synth (analz H)
       \<Longrightarrow> \<forall> G. H \<subseteq> G \<longrightarrow> (Key K \<in> analz (insert X G)) = (Key K \<in> analz G)"
-apply (erule synth.induct) 
-apply (simp_all add: gen_analz_insert_eq subset_trans [OF _ subset_insertI]) 
+apply (erule synth.induct)
+apply (simp_all add: gen_analz_insert_eq subset_trans [OF _ subset_insertI])
 done
 
 lemma Fake_parts_sing:
      "X \<in> synth (analz H) \<Longrightarrow> parts{X} \<subseteq> synth (analz H) \<union> parts H"
 by (metis Fake_parts_insert empty_subsetI insert_mono parts_mono subset_trans)
 
 lemmas Fake_parts_sing_imp_Un = Fake_parts_sing [THEN [2] rev_subsetD]
 
 method_setup spy_analz = \<open>
     Scan.succeed (SIMPLE_METHOD' o spy_analz_tac)\<close>
     "for proving the Fake case when analz is involved"
 
 method_setup atomic_spy_analz = \<open>
     Scan.succeed (SIMPLE_METHOD' o atomic_spy_analz_tac)\<close>
     "for debugging spy_analz"
 
 method_setup Fake_insert_simp = \<open>
     Scan.succeed (SIMPLE_METHOD' o Fake_insert_simp_tac)\<close>
     "for debugging spy_analz"
 
 end
diff --git a/thys/Security_Protocol_Refinement/Refinement/Message.thy b/thys/Security_Protocol_Refinement/Refinement/Message.thy
--- a/thys/Security_Protocol_Refinement/Refinement/Message.thy
+++ b/thys/Security_Protocol_Refinement/Refinement/Message.thy
@@ -1,888 +1,878 @@
 (*******************************************************************************
 
   Title:      HOL/Auth/Message
   Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
   Copyright   1996  University of Cambridge
 
   Datatypes of agents and messages;
   Inductive relations "parts", "analz" and "synth"
 
 ********************************************************************************
 
   Module:  Refinement/Message.thy (Isabelle/HOL 2016-1)
   ID:      $Id: Message.thy 133856 2017-03-20 18:05:54Z csprenge $
   Edited:  Christoph Sprenger <sprenger@inf.ethz.ch>, ETH Zurich
 
   Integrated and adapted for security protocol refinement
 
 *******************************************************************************)
 
 section \<open>Theory of Agents and Messages for Security Protocols\<close>
 
 theory Message imports Keys begin
 
 (*Needed occasionally with spy_analz_tac, e.g. in analz_insert_Key_newK*)
 lemma Un_idem_collapse [simp]: "A \<union> (B \<union> A) = B \<union> A"
 by blast
 
 datatype
      msg = Agent  agent     \<comment> \<open>Agent names\<close>
          | Number nat       \<comment> \<open>Ordinary integers, timestamps, ...\<close>
          | Nonce  nonce     \<comment> \<open>Unguessable nonces\<close>
          | Key    key       \<comment> \<open>Crypto keys\<close>
          | Hash   msg       \<comment> \<open>Hashing\<close>
          | MPair  msg msg   \<comment> \<open>Compound messages\<close>
          | Crypt  key msg   \<comment> \<open>Encryption, public- or shared-key\<close>
 
 
 text\<open>Concrete syntax: messages appear as \<open>\<lbrace>A,B,NA\<rbrace>\<close>, etc...\<close>
 
 syntax
   "_MTuple"      :: "['a, args] => 'a * 'b"       ("(2\<lbrace>_,/ _\<rbrace>)")
 translations
   "\<lbrace>x, y, z\<rbrace>"   == "\<lbrace>x, \<lbrace>y, z\<rbrace>\<rbrace>"
   "\<lbrace>x, y\<rbrace>"      == "CONST MPair x y"
 
 
 definition
   HPair :: "[msg,msg] \<Rightarrow> msg"                       ("(4Hash[_] /_)" [0, 1000])
 where
   \<comment> \<open>Message Y paired with a MAC computed with the help of X\<close>
   "Hash[X] Y \<equiv> \<lbrace>Hash\<lbrace>X,Y\<rbrace>, Y\<rbrace>"
 
 definition
   keysFor :: "msg set \<Rightarrow> key set"
 where
     \<comment> \<open>Keys useful to decrypt elements of a message set\<close>
   "keysFor H \<equiv> invKey ` {K. \<exists>X. Crypt K X \<in> H}"
 
 
 subsubsection\<open>Inductive Definition of All Parts" of a Message\<close>
 
 inductive_set
   parts :: "msg set => msg set"
   for H :: "msg set"
   where
     Inj [intro]:               "X \<in> H ==> X \<in> parts H"
   | Fst:         "\<lbrace>X,Y\<rbrace>   \<in> parts H ==> X \<in> parts H"
   | Snd:         "\<lbrace>X,Y\<rbrace>   \<in> parts H ==> Y \<in> parts H"
   | Body:        "Crypt K X \<in> parts H ==> X \<in> parts H"
 
 
 text\<open>Monotonicity\<close>
 lemma parts_mono: "G \<subseteq> H ==> parts(G) \<subseteq> parts(H)"
 apply auto
 apply (erule parts.induct)
 apply (blast dest: parts.Fst parts.Snd parts.Body)+
 done
 
 
 text\<open>Equations hold because constructors are injective.\<close>
 lemma Other_image_eq [simp]: "(Agent x \<in> Agent`A) = (x:A)"
 by auto
 
 lemma Key_image_eq [simp]: "(Key x \<in> Key`A) = (x\<in>A)"
 by auto
 
 lemma Nonce_Key_image_eq [simp]: "(Nonce x \<notin> Key`A)"
 by auto
 
 
 
 subsection\<open>keysFor operator\<close>
 
 lemma keysFor_empty [simp]: "keysFor {} = {}"
 by (unfold keysFor_def, blast)
 
 lemma keysFor_Un [simp]: "keysFor (H \<union> H') = keysFor H \<union> keysFor H'"
 by (unfold keysFor_def, blast)
 
 lemma keysFor_UN [simp]: "keysFor (\<Union>i\<in>A. H i) = (\<Union>i\<in>A. keysFor (H i))"
 by (unfold keysFor_def, blast)
 
 text\<open>Monotonicity\<close>
 lemma keysFor_mono: "G \<subseteq> H ==> keysFor(G) \<subseteq> keysFor(H)"
 by (unfold keysFor_def, blast)
 
 lemma keysFor_insert_Agent [simp]: "keysFor (insert (Agent A) H) = keysFor H"
 by (unfold keysFor_def, auto)
 
 lemma keysFor_insert_Nonce [simp]: "keysFor (insert (Nonce N) H) = keysFor H"
 by (unfold keysFor_def, auto)
 
 lemma keysFor_insert_Number [simp]: "keysFor (insert (Number N) H) = keysFor H"
 by (unfold keysFor_def, auto)
 
 lemma keysFor_insert_Key [simp]: "keysFor (insert (Key K) H) = keysFor H"
 by (unfold keysFor_def, auto)
 
 lemma keysFor_insert_Hash [simp]: "keysFor (insert (Hash X) H) = keysFor H"
 by (unfold keysFor_def, auto)
 
 lemma keysFor_insert_MPair [simp]: "keysFor (insert \<lbrace>X,Y\<rbrace> H) = keysFor H"
 by (unfold keysFor_def, auto)
 
 lemma keysFor_insert_Crypt [simp]:
     "keysFor (insert (Crypt K X) H) = insert (invKey K) (keysFor H)"
 by (unfold keysFor_def, auto)
 
 lemma keysFor_image_Key [simp]: "keysFor (Key`E) = {}"
 by (unfold keysFor_def, auto)
 
 lemma Crypt_imp_invKey_keysFor: "Crypt K X \<in> H ==> invKey K \<in> keysFor H"
 by (unfold keysFor_def, blast)
 
 
 subsection\<open>Inductive relation "parts"\<close>
 
 lemma MPair_parts:
      "[| \<lbrace>X,Y\<rbrace> \<in> parts H;
          [| X \<in> parts H; Y \<in> parts H |] ==> P |] ==> P"
 by (blast dest: parts.Fst parts.Snd)
 
 declare MPair_parts [elim!]  parts.Body [dest!]
 text\<open>NB These two rules are UNSAFE in the formal sense, as they discard the
      compound message.  They work well on THIS FILE.
   \<open>MPair_parts\<close> is left as SAFE because it speeds up proofs.
   The Crypt rule is normally kept UNSAFE to avoid breaking up certificates.\<close>
 
 lemma parts_increasing: "H \<subseteq> parts(H)"
 by blast
 
 lemmas parts_insertI = subset_insertI [THEN parts_mono, THEN subsetD]
 
 lemma parts_empty [simp]: "parts{} = {}"
 apply safe
 apply (erule parts.induct, blast+)
 done
 
 lemma parts_emptyE [elim!]: "X\<in> parts{} ==> P"
 by simp
 
 text\<open>WARNING: loops if H = {Y}, therefore must not be repeated!\<close>
 lemma parts_singleton: "X\<in> parts H ==> \<exists>Y\<in>H. X\<in> parts {Y}"
 by (erule parts.induct, fast+)
 
 
 subsubsection\<open>Unions\<close>
 
 lemma parts_Un_subset1: "parts(G) \<union> parts(H) \<subseteq> parts(G \<union> H)"
 by (intro Un_least parts_mono Un_upper1 Un_upper2)
 
 lemma parts_Un_subset2: "parts(G \<union> H) \<subseteq> parts(G) \<union> parts(H)"
 apply (rule subsetI)
 apply (erule parts.induct, blast+)
 done
 
 lemma parts_Un [simp]: "parts(G \<union> H) = parts(G) \<union> parts(H)"
 by (intro equalityI parts_Un_subset1 parts_Un_subset2)
 
 lemma parts_insert: "parts (insert X H) = parts {X} \<union> parts H"
 apply (subst insert_is_Un [of _ H])
 apply (simp only: parts_Un)
 done
 
 text\<open>TWO inserts to avoid looping.  This rewrite is better than nothing.
   Not suitable for Addsimps: its behaviour can be strange.\<close>
 lemma parts_insert2:
      "parts (insert X (insert Y H)) = parts {X} \<union> parts {Y} \<union> parts H"
 apply (simp add: Un_assoc)
 apply (simp add: parts_insert [symmetric])
 done
 
 lemma parts_UN_subset1: "(\<Union>x\<in>A. parts(H x)) \<subseteq> parts(\<Union>x\<in>A. H x)"
 by (intro UN_least parts_mono UN_upper)
 
 lemma parts_UN_subset2: "parts(\<Union>x\<in>A. H x) \<subseteq> (\<Union>x\<in>A. parts(H x))"
 apply (rule subsetI)
 apply (erule parts.induct, blast+)
 done
 
 lemma parts_UN [simp]: "parts(\<Union>x\<in>A. H x) = (\<Union>x\<in>A. parts(H x))"
 by (intro equalityI parts_UN_subset1 parts_UN_subset2)
 
 text\<open>Added to simplify arguments to parts, analz and synth.
   NOTE: the UN versions are no longer used!\<close>
 
 
 text\<open>This allows \<open>blast\<close> to simplify occurrences of
   @{term "parts(G\<union>H)"} in the assumption.\<close>
 lemmas in_parts_UnE = parts_Un [THEN equalityD1, THEN subsetD, THEN UnE]
 declare in_parts_UnE [elim!]
 
 
 lemma parts_insert_subset: "insert X (parts H) \<subseteq> parts(insert X H)"
 by (blast intro: parts_mono [THEN [2] rev_subsetD])
 
 subsubsection\<open>Idempotence and transitivity\<close>
 
 lemma parts_partsD [dest!]: "X\<in> parts (parts H) ==> X\<in> parts H"
 by (erule parts.induct, blast+)
 
 lemma parts_idem [simp]: "parts (parts H) = parts H"
 by blast
 
 lemma parts_subset_iff [simp]: "(parts G \<subseteq> parts H) = (G \<subseteq> parts H)"
 apply (rule iffI)
 apply (iprover intro: subset_trans parts_increasing)
 apply (frule parts_mono, simp)
 done
 
 lemma parts_trans: "[| X\<in> parts G;  G \<subseteq> parts H |] ==> X\<in> parts H"
 by (drule parts_mono, blast)
 
 text\<open>Cut\<close>
 lemma parts_cut:
      "[| Y\<in> parts (insert X G);  X\<in> parts H |] ==> Y\<in> parts (G \<union> H)"
 by (blast intro: parts_trans)
 
 
 lemma parts_cut_eq [simp]: "X\<in> parts H ==> parts (insert X H) = parts H"
 by (force dest!: parts_cut intro: parts_insertI)
 
 
 subsubsection\<open>Rewrite rules for pulling out atomic messages\<close>
 
 lemmas parts_insert_eq_I = equalityI [OF subsetI parts_insert_subset]
 
 
 lemma parts_insert_Agent [simp]:
      "parts (insert (Agent agt) H) = insert (Agent agt) (parts H)"
 apply (rule parts_insert_eq_I)
 apply (erule parts.induct, auto)
 done
 
 lemma parts_insert_Nonce [simp]:
      "parts (insert (Nonce N) H) = insert (Nonce N) (parts H)"
 apply (rule parts_insert_eq_I)
 apply (erule parts.induct, auto)
 done
 
 lemma parts_insert_Number [simp]:
      "parts (insert (Number N) H) = insert (Number N) (parts H)"
 apply (rule parts_insert_eq_I)
 apply (erule parts.induct, auto)
 done
 
 lemma parts_insert_Key [simp]:
      "parts (insert (Key K) H) = insert (Key K) (parts H)"
 apply (rule parts_insert_eq_I)
 apply (erule parts.induct, auto)
 done
 
 lemma parts_insert_Hash [simp]:
      "parts (insert (Hash X) H) = insert (Hash X) (parts H)"
 apply (rule parts_insert_eq_I)
 apply (erule parts.induct, auto)
 done
 
 lemma parts_insert_Crypt [simp]:
      "parts (insert (Crypt K X) H) = insert (Crypt K X) (parts (insert X H))"
 apply (rule equalityI)
 apply (rule subsetI)
 apply (erule parts.induct, auto)
 apply (blast intro: parts.Body)
 done
 
 lemma parts_insert_MPair [simp]:
      "parts (insert \<lbrace>X,Y\<rbrace> H) =
           insert \<lbrace>X,Y\<rbrace> (parts (insert X (insert Y H)))"
 apply (rule equalityI)
 apply (rule subsetI)
 apply (erule parts.induct, auto)
 apply (blast intro: parts.Fst parts.Snd)+
 done
 
 lemma parts_image_Key [simp]: "parts (Key`N) = Key`N"
 apply auto
 apply (erule parts.induct, auto)
 done
 
 
 text\<open>In any message, there is an upper bound N on its greatest nonce.\<close>
 (*
 lemma msg_Nonce_supply: "\<exists>N. \<forall>n. N\<le>n --> Nonce n \<notin> parts {msg}"
 apply (induct msg)
 apply (simp_all (no_asm_simp) add: exI parts_insert2)
  txt{*MPair case: blast works out the necessary sum itself!*}
  prefer 2 apply auto apply (blast elim!: add_leE)
 txt{*Nonce case*}
 apply (rule_tac x = "N + Suc nat" in exI, auto)
 done
 *)
 
 
 subsection\<open>Inductive relation "analz"\<close>
 
 text\<open>Inductive definition of "analz" -- what can be broken down from a set of
     messages, including keys.  A form of downward closure.  Pairs can
     be taken apart; messages decrypted with known keys.\<close>
 
 inductive_set
   analz :: "msg set => msg set"
   for H :: "msg set"
   where
     Inj [intro,simp] :    "X \<in> H ==> X \<in> analz H"
   | Fst:     "\<lbrace>X,Y\<rbrace> \<in> analz H ==> X \<in> analz H"
   | Snd:     "\<lbrace>X,Y\<rbrace> \<in> analz H ==> Y \<in> analz H"
   | Decrypt [dest]:
              "[|Crypt K X \<in> analz H; Key(invKey K): analz H|] ==> X \<in> analz H"
 
 
 text\<open>Monotonicity; Lemma 1 of Lowe's paper\<close>
 lemma analz_mono: "G\<subseteq>H ==> analz(G) \<subseteq> analz(H)"
 apply auto
 apply (erule analz.induct)
 apply (auto dest: analz.Fst analz.Snd)
 done
 
 lemmas analz_monotonic = analz_mono [THEN [2] rev_subsetD]
 
 text\<open>Making it safe speeds up proofs\<close>
 lemma MPair_analz [elim!]:
      "[| \<lbrace>X,Y\<rbrace> \<in> analz H;
              [| X \<in> analz H; Y \<in> analz H |] ==> P
           |] ==> P"
 by (blast dest: analz.Fst analz.Snd)
 
 lemma analz_increasing: "H \<subseteq> analz(H)"
 by blast
 
 lemma analz_subset_parts: "analz H \<subseteq> parts H"
 apply (rule subsetI)
 apply (erule analz.induct, blast+)
 done
 
 lemmas analz_into_parts = analz_subset_parts [THEN subsetD]
 
 lemmas not_parts_not_analz = analz_subset_parts [THEN contra_subsetD]
 
 
 lemma parts_analz [simp]: "parts (analz H) = parts H"
 apply (rule equalityI)
 apply (rule analz_subset_parts [THEN parts_mono, THEN subset_trans], simp)
 apply (blast intro: analz_increasing [THEN parts_mono, THEN subsetD])
 done
 
 lemma analz_parts [simp]: "analz (parts H) = parts H"
 apply auto
 apply (erule analz.induct, auto)
 done
 
 lemmas analz_insertI = subset_insertI [THEN analz_mono, THEN [2] rev_subsetD]
 
 subsubsection\<open>General equational properties\<close>
 
 lemma analz_empty [simp]: "analz{} = {}"
 apply safe
 apply (erule analz.induct, blast+)
 done
 
 text\<open>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\<close>
 lemma analz_Un: "analz(G) \<union> analz(H) \<subseteq> analz(G \<union> H)"
 by (intro Un_least analz_mono Un_upper1 Un_upper2)
 
 lemma analz_insert: "insert X (analz H) \<subseteq> analz(insert X H)"
 by (blast intro: analz_mono [THEN [2] rev_subsetD])
 
 subsubsection\<open>Rewrite rules for pulling out atomic messages\<close>
 
 lemmas analz_insert_eq_I = equalityI [OF subsetI analz_insert]
 
 lemma analz_insert_Agent [simp]:
      "analz (insert (Agent agt) H) = insert (Agent agt) (analz H)"
 apply (rule analz_insert_eq_I)
 apply (erule analz.induct, auto)
 done
 
 lemma analz_insert_Nonce [simp]:
      "analz (insert (Nonce N) H) = insert (Nonce N) (analz H)"
 apply (rule analz_insert_eq_I)
 apply (erule analz.induct, auto)
 done
 
 lemma analz_insert_Number [simp]:
      "analz (insert (Number N) H) = insert (Number N) (analz H)"
 apply (rule analz_insert_eq_I)
 apply (erule analz.induct, auto)
 done
 
 lemma analz_insert_Hash [simp]:
      "analz (insert (Hash X) H) = insert (Hash X) (analz H)"
 apply (rule analz_insert_eq_I)
 apply (erule analz.induct, auto)
 done
 
 text\<open>Can only pull out Keys if they are not needed to decrypt the rest\<close>
 lemma analz_insert_Key [simp]:
     "K \<notin> keysFor (analz H) ==>
           analz (insert (Key K) H) = insert (Key K) (analz H)"
 apply (unfold keysFor_def)
 apply (rule analz_insert_eq_I)
 apply (erule analz.induct, auto)
 done
 
 lemma analz_insert_MPair [simp]:
      "analz (insert \<lbrace>X,Y\<rbrace> H) =
           insert \<lbrace>X,Y\<rbrace> (analz (insert X (insert Y H)))"
 apply (rule equalityI)
 apply (rule subsetI)
 apply (erule analz.induct, auto)
 apply (erule analz.induct)
 apply (blast intro: analz.Fst analz.Snd)+
 done
 
 text\<open>Can pull out enCrypted message if the Key is not known\<close>
 lemma analz_insert_Crypt:
      "Key (invKey K) \<notin> analz H
       ==> analz (insert (Crypt K X) H) = insert (Crypt K X) (analz H)"
 apply (rule analz_insert_eq_I)
 apply (erule analz.induct, auto)
 
 done
 
 lemma lemma1: "Key (invKey K) \<in> analz H ==>
                analz (insert (Crypt K X) H) \<subseteq>
                insert (Crypt K X) (analz (insert X H))"
 apply (rule subsetI)
 apply (erule_tac x = x in analz.induct, auto)
 done
 
 lemma lemma2: "Key (invKey K) \<in> analz H ==>
                insert (Crypt K X) (analz (insert X H)) \<subseteq>
                analz (insert (Crypt K X) H)"
 apply auto
 apply (erule_tac x = x in analz.induct, auto)
 apply (blast intro: analz_insertI analz.Decrypt)
 done
 
 lemma analz_insert_Decrypt:
      "Key (invKey K) \<in> analz H ==>
                analz (insert (Crypt K X) H) =
                insert (Crypt K X) (analz (insert X H))"
 by (intro equalityI lemma1 lemma2)
 
 text\<open>Case analysis: either the message is secure, or it is not! Effective,
 but can cause subgoals to blow up! Use with \<open>split_if\<close>; apparently
 \<open>split_tac\<close> does not cope with patterns such as @{term"analz (insert
 (Crypt K X) H)"}\<close>
 lemma analz_Crypt_if [simp]:
      "analz (insert (Crypt K X) H) =
           (if (Key (invKey K) \<in> analz H)
            then insert (Crypt K X) (analz (insert X H))
            else insert (Crypt K X) (analz H))"
 by (simp add: analz_insert_Crypt analz_insert_Decrypt)
 
 
 text\<open>This rule supposes "for the sake of argument" that we have the key.\<close>
 lemma analz_insert_Crypt_subset:
      "analz (insert (Crypt K X) H) \<subseteq>
            insert (Crypt K X) (analz (insert X H))"
 apply (rule subsetI)
 apply (erule analz.induct, auto)
 done
 
 
 lemma analz_image_Key [simp]: "analz (Key`N) = Key`N"
 apply auto
 apply (erule analz.induct, auto)
 done
 
 
 subsubsection\<open>Idempotence and transitivity\<close>
 
 lemma analz_analzD [dest!]: "X\<in> analz (analz H) ==> X\<in> analz H"
 by (erule analz.induct, blast+)
 
 lemma analz_idem [simp]: "analz (analz H) = analz H"
 by blast
 
 lemma analz_subset_iff [simp]: "(analz G \<subseteq> analz H) = (G \<subseteq> analz H)"
 apply (rule iffI)
 apply (iprover intro: subset_trans analz_increasing)
 apply (frule analz_mono, simp)
 done
 
 lemma analz_trans: "[| X\<in> analz G;  G \<subseteq> analz H |] ==> X\<in> analz H"
 by (drule analz_mono, blast)
 
 text\<open>Cut; Lemma 2 of Lowe\<close>
 lemma analz_cut: "[| Y\<in> analz (insert X H);  X\<in> analz H |] ==> Y\<in> analz H"
 by (erule analz_trans, blast)
 
 (*Cut can be proved easily by induction on
    "Y: analz (insert X H) ==> X: analz H --> Y: analz H"
 *)
 
 text\<open>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.\<close>
 lemma analz_insert_eq: "X\<in> analz H ==> analz (insert X H) = analz H"
 by (blast intro: analz_cut analz_insertI)
 
 
 text\<open>A congruence rule for "analz"\<close>
 
 lemma analz_subset_cong:
      "[| analz G \<subseteq> analz G'; analz H \<subseteq> analz H' |]
       ==> analz (G \<union> H) \<subseteq> analz (G' \<union> H')"
 apply simp
 apply (iprover intro: conjI subset_trans analz_mono Un_upper1 Un_upper2)
 done
 
 lemma analz_cong:
      "[| analz G = analz G'; analz H = analz H' |]
       ==> analz (G \<union> H) = analz (G' \<union> H')"
 by (intro equalityI analz_subset_cong, simp_all)
 
 lemma analz_insert_cong:
      "analz H = analz H' ==> analz(insert X H) = analz(insert X H')"
 by (force simp only: insert_def intro!: analz_cong)
 
 text\<open>If there are no pairs or encryptions then analz does nothing\<close>
 lemma analz_trivial:
      "[| \<forall>X Y. \<lbrace>X,Y\<rbrace> \<notin> H;  \<forall>X K. Crypt K X \<notin> H |] ==> analz H = H"
 apply safe
 apply (erule analz.induct, blast+)
 done
 
-text\<open>These two are obsolete (with a single Spy) but cost little to prove...\<close>
-lemma analz_UN_analz_lemma:
-     "X\<in> analz (\<Union>i\<in>A. analz (H i)) ==> X\<in> analz (\<Union>i\<in>A. H i)"
-apply (erule analz.induct)
-apply (blast intro: analz_mono [THEN [2] rev_subsetD])+
-done
-
-lemma analz_UN_analz [simp]: "analz (\<Union>i\<in>A. analz (H i)) = analz (\<Union>i\<in>A. H i)"
-by (blast intro: analz_UN_analz_lemma analz_mono [THEN [2] rev_subsetD])
-
 
 subsection\<open>Inductive relation "synth"\<close>
 
 text\<open>Inductive definition of "synth" -- what can be built up from a set of
     messages.  A form of upward closure.  Pairs can be built, messages
     encrypted with known keys.  Agent names are public domain.
     Numbers can be guessed, but Nonces cannot be.\<close>
 
 inductive_set
   synth :: "msg set => msg set"
   for H :: "msg set"
   where
     Inj    [intro]:   "X \<in> H ==> X \<in> synth H"
   | Agent  [intro]:   "Agent agt \<in> synth H"
   | Number [intro]:   "Number n  \<in> synth H"
   | Hash   [intro]:   "X \<in> synth H ==> Hash X \<in> synth H"
   | MPair  [intro]:   "[|X \<in> synth H;  Y \<in> synth H|] ==> \<lbrace>X,Y\<rbrace> \<in> synth H"
   | Crypt  [intro]:   "[|X \<in> synth H;  Key(K) \<in> H|] ==> Crypt K X \<in> synth H"
 
 text\<open>Monotonicity\<close>
 lemma synth_mono: "G\<subseteq>H ==> synth(G) \<subseteq> synth(H)"
   by (auto, erule synth.induct, auto)
 
 text\<open>NO \<open>Agent_synth\<close>, as any Agent name can be synthesized.
   The same holds for @{term Number}\<close>
 inductive_cases Nonce_synth [elim!]: "Nonce n \<in> synth H"
 inductive_cases Key_synth   [elim!]: "Key K \<in> synth H"
 inductive_cases Hash_synth  [elim!]: "Hash X \<in> synth H"
 inductive_cases MPair_synth [elim!]: "\<lbrace>X,Y\<rbrace> \<in> synth H"
 inductive_cases Crypt_synth [elim!]: "Crypt K X \<in> synth H"
 
 
 lemma synth_increasing: "H \<subseteq> synth(H)"
 by blast
 
 subsubsection\<open>Unions\<close>
 
 text\<open>Converse fails: we can synth more from the union than from the
   separate parts, building a compound message using elements of each.\<close>
 lemma synth_Un: "synth(G) \<union> synth(H) \<subseteq> synth(G \<union> H)"
 by (intro Un_least synth_mono Un_upper1 Un_upper2)
 
 lemma synth_insert: "insert X (synth H) \<subseteq> synth(insert X H)"
 by (blast intro: synth_mono [THEN [2] rev_subsetD])
 
 subsubsection\<open>Idempotence and transitivity\<close>
 
 lemma synth_synthD [dest!]: "X\<in> synth (synth H) ==> X\<in> synth H"
 by (erule synth.induct, blast+)
 
 lemma synth_idem: "synth (synth H) = synth H"
 by blast
 
 lemma synth_subset_iff [simp]: "(synth G \<subseteq> synth H) = (G \<subseteq> synth H)"
 apply (rule iffI)
 apply (iprover intro: subset_trans synth_increasing)
 apply (frule synth_mono, simp add: synth_idem)
 done
 
 lemma synth_trans: "[| X\<in> synth G;  G \<subseteq> synth H |] ==> X\<in> synth H"
 by (drule synth_mono, blast)
 
 text\<open>Cut; Lemma 2 of Lowe\<close>
 lemma synth_cut: "[| Y\<in> synth (insert X H);  X\<in> synth H |] ==> Y\<in> synth H"
 by (erule synth_trans, blast)
 
 lemma Agent_synth [simp]: "Agent A \<in> synth H"
 by blast
 
 lemma Number_synth [simp]: "Number n \<in> synth H"
 by blast
 
 lemma Nonce_synth_eq [simp]: "(Nonce N \<in> synth H) = (Nonce N \<in> H)"
 by blast
 
 lemma Key_synth_eq [simp]: "(Key K \<in> synth H) = (Key K \<in> H)"
 by blast
 
 lemma Crypt_synth_eq [simp]:
      "Key K \<notin> H ==> (Crypt K X \<in> synth H) = (Crypt K X \<in> H)"
 by blast
 
 
 lemma keysFor_synth [simp]:
     "keysFor (synth H) = keysFor H \<union> invKey`{K. Key K \<in> H}"
 by (unfold keysFor_def, blast)
 
 
 subsubsection\<open>Combinations of parts, analz and synth\<close>
 
 lemma parts_synth [simp]: "parts (synth H) = parts H \<union> synth H"
 apply (rule equalityI)
 apply (rule subsetI)
 apply (erule parts.induct)
 apply (blast intro: synth_increasing [THEN parts_mono, THEN subsetD]
                     parts.Fst parts.Snd parts.Body)+
 done
 
 lemma analz_analz_Un [simp]: "analz (analz G \<union> H) = analz (G \<union> H)"
 apply (intro equalityI analz_subset_cong)+
 apply simp_all
 done
 
 lemma analz_synth_Un [simp]: "analz (synth G \<union> H) = analz (G \<union> H) \<union> synth G"
 apply (rule equalityI)
 apply (rule subsetI)
 apply (erule analz.induct)
 prefer 5 apply (blast intro: analz_mono [THEN [2] rev_subsetD])
 apply (blast intro: analz.Fst analz.Snd analz.Decrypt)+
 done
 
 lemma analz_synth [simp]: "analz (synth H) = analz H \<union> synth H"
 apply (cut_tac H = "{}" in analz_synth_Un)
 apply (simp (no_asm_use))
 done
 
 text \<open>chsp: added\<close>
 
 lemma analz_Un_analz [simp]: "analz (G \<union> analz H) = analz (G \<union> H)"
 by (subst Un_commute, auto)+
 
 lemma analz_synth_Un2 [simp]: "analz (G \<union> synth H) = analz (G \<union> H) \<union> synth H"
 by (subst Un_commute, auto)+
 
 
 subsubsection\<open>For reasoning about the Fake rule in traces\<close>
 
 lemma parts_insert_subset_Un: "X\<in> G ==> parts(insert X H) \<subseteq> parts G \<union> parts H"
 by (rule subset_trans [OF parts_mono parts_Un_subset2], blast)
 
 text\<open>More specifically for Fake.  Very occasionally we could do with a version
   of the form  @{term"parts{X} \<subseteq> synth (analz H) \<union> parts H"}\<close>
 lemma Fake_parts_insert:
      "X \<in> synth (analz H) ==>
       parts (insert X H) \<subseteq> synth (analz H) \<union> parts H"
 apply (drule parts_insert_subset_Un)
 apply (simp (no_asm_use))
 apply blast
 done
 
 lemma Fake_parts_insert_in_Un:
      "[|Z \<in> parts (insert X H);  X \<in> synth (analz H)|]
       ==> Z \<in>  synth (analz H) \<union> parts H"
 by (blast dest: Fake_parts_insert  [THEN subsetD, dest])
 
 text\<open>@{term H} is sometimes @{term"Key ` KK \<union> spies evs"}, so can't put
   @{term "G=H"}.\<close>
 lemma Fake_analz_insert:
      "X\<in> synth (analz G) ==>
       analz (insert X H) \<subseteq> synth (analz G) \<union> analz (G \<union> H)"
 apply (rule subsetI)
 apply (subgoal_tac "x \<in> analz (synth (analz G) \<union> H) ")
 prefer 2
   apply (blast intro: analz_mono [THEN [2] rev_subsetD]
                       analz_mono [THEN synth_mono, THEN [2] rev_subsetD])
 apply (simp (no_asm_use))
 apply blast
 done
 
 lemma analz_conj_parts [simp]:
      "(X \<in> analz H & X \<in> parts H) = (X \<in> analz H)"
 by (blast intro: analz_subset_parts [THEN subsetD])
 
 lemma analz_disj_parts [simp]:
      "(X \<in> analz H | X \<in> parts H) = (X \<in> parts H)"
 by (blast intro: analz_subset_parts [THEN subsetD])
 
 text\<open>Without this equation, other rules for synth and analz would yield
   redundant cases\<close>
 lemma MPair_synth_analz [iff]:
      "(\<lbrace>X,Y\<rbrace> \<in> synth (analz H)) =
       (X \<in> synth (analz H) & Y \<in> synth (analz H))"
 by blast
 
 lemma Crypt_synth_analz:
      "[| Key K \<in> analz H;  Key (invKey K) \<in> analz H |]
        ==> (Crypt K X \<in> synth (analz H)) = (X \<in> synth (analz H))"
 by blast
 
 
 lemma Hash_synth_analz [simp]:
      "X \<notin> synth (analz H)
       ==> (Hash\<lbrace>X,Y\<rbrace> \<in> synth (analz H)) = (Hash\<lbrace>X,Y\<rbrace> \<in> analz H)"
 by blast
 
 
 subsection\<open>HPair: a combination of Hash and MPair\<close>
 
 subsubsection\<open>Freeness\<close>
 
 lemma Agent_neq_HPair: "Agent A ~= Hash[X] Y"
 by (unfold HPair_def, simp)
 
 lemma Nonce_neq_HPair: "Nonce N ~= Hash[X] Y"
 by (unfold HPair_def, simp)
 
 lemma Number_neq_HPair: "Number N ~= Hash[X] Y"
 by (unfold HPair_def, simp)
 
 lemma Key_neq_HPair: "Key K ~= Hash[X] Y"
 by (unfold HPair_def, simp)
 
 lemma Hash_neq_HPair: "Hash Z ~= Hash[X] Y"
 by (unfold HPair_def, simp)
 
 lemma Crypt_neq_HPair: "Crypt K X' ~= Hash[X] Y"
 by (unfold HPair_def, simp)
 
 lemmas HPair_neqs = Agent_neq_HPair Nonce_neq_HPair Number_neq_HPair
                     Key_neq_HPair Hash_neq_HPair Crypt_neq_HPair
 
 declare HPair_neqs [iff]
 declare HPair_neqs [symmetric, iff]
 
 lemma HPair_eq [iff]: "(Hash[X'] Y' = Hash[X] Y) = (X' = X & Y'=Y)"
 by (simp add: HPair_def)
 
 lemma MPair_eq_HPair [iff]:
      "(\<lbrace>X',Y'\<rbrace> = Hash[X] Y) = (X' = Hash\<lbrace>X,Y\<rbrace> & Y'=Y)"
 by (simp add: HPair_def)
 
 lemma HPair_eq_MPair [iff]:
      "(Hash[X] Y = \<lbrace>X',Y'\<rbrace>) = (X' = Hash\<lbrace>X,Y\<rbrace> & Y'=Y)"
 by (auto simp add: HPair_def)
 
 
 subsubsection\<open>Specialized laws, proved in terms of those for Hash and MPair\<close>
 
 lemma keysFor_insert_HPair [simp]: "keysFor (insert (Hash[X] Y) H) = keysFor H"
 by (simp add: HPair_def)
 
 lemma parts_insert_HPair [simp]:
     "parts (insert (Hash[X] Y) H) =
      insert (Hash[X] Y) (insert (Hash\<lbrace>X,Y\<rbrace>) (parts (insert Y H)))"
 by (simp add: HPair_def)
 
 lemma analz_insert_HPair [simp]:
     "analz (insert (Hash[X] Y) H) =
      insert (Hash[X] Y) (insert (Hash\<lbrace>X,Y\<rbrace>) (analz (insert Y H)))"
 by (simp add: HPair_def)
 
 lemma HPair_synth_analz [simp]:
      "X \<notin> synth (analz H)
     ==> (Hash[X] Y \<in> synth (analz H)) =
         (Hash\<lbrace>X, Y\<rbrace> \<in> analz H & Y \<in> synth (analz H))"
 by (simp add: HPair_def)
 
 
 text\<open>We do NOT want Crypt... messages broken up in protocols!!\<close>
 declare parts.Body [rule del]
 
 
 text\<open>Rewrites to push in Key and Crypt messages, so that other messages can
     be pulled out using the \<open>analz_insert\<close> rules\<close>
 
 lemmas pushKeys =
   insert_commute [of "Key K" "Agent C" for K C]
   insert_commute [of "Key K" "Nonce N" for K N]
   insert_commute [of "Key K" "Number N" for K N]
   insert_commute [of "Key K" "Hash X" for K X]
   insert_commute [of "Key K" "MPair X Y" for K X Y]
   insert_commute [of "Key K" "Crypt X K'" for K K' X]
 
 lemmas pushCrypts =
   insert_commute [of "Crypt X K" "Agent C" for X K C]
   insert_commute [of "Crypt X K" "Agent C" for X K C]
   insert_commute [of "Crypt X K" "Nonce N" for X K N]
   insert_commute [of "Crypt X K" "Number N"  for X K N]
   insert_commute [of "Crypt X K" "Hash X'"  for X K X']
   insert_commute [of "Crypt X K" "MPair X' Y"  for X K X' Y]
 
 text\<open>Cannot be added with \<open>[simp]\<close> -- messages should not always be
   re-ordered.\<close>
 lemmas pushes = pushKeys pushCrypts
 
 
 text\<open>By default only \<open>o_apply\<close> is built-in.  But in the presence of
 eta-expansion this means that some terms displayed as @{term "f o g"} will be
 rewritten, and others will not!\<close>
 declare o_def [simp]
 
 
 lemma Crypt_notin_image_Key [simp]: "Crypt K X \<notin> Key ` A"
 by auto
 
 lemma Hash_notin_image_Key [simp] :"Hash X \<notin> Key ` A"
 by auto
 
 lemma synth_analz_mono: "G\<subseteq>H ==> synth (analz(G)) \<subseteq> synth (analz(H))"
 by (iprover intro: synth_mono analz_mono)
 
 lemma Fake_analz_eq [simp]:
      "X \<in> synth(analz H) ==> synth (analz (insert X H)) = synth (analz H)"
 apply (drule Fake_analz_insert[of _ _ "H"])
 apply (simp add: synth_increasing[THEN Un_absorb2])
 apply (drule synth_mono)
 apply (simp add: synth_idem)
 apply (rule equalityI)
 apply simp
 apply (rule synth_analz_mono, blast)
 done
 
 text\<open>Two generalizations of \<open>analz_insert_eq\<close>\<close>
 lemma gen_analz_insert_eq [rule_format]:
      "X \<in> analz H ==> ALL G. H \<subseteq> G --> analz (insert X G) = analz G"
 by (blast intro: analz_cut analz_insertI analz_mono [THEN [2] rev_subsetD])
 
 lemma synth_analz_insert_eq [rule_format]:
      "X \<in> synth (analz H)
       ==> ALL G. H \<subseteq> G --> (Key K \<in> analz (insert X G)) = (Key K \<in> analz G)"
 apply (erule synth.induct)
 apply (simp_all add: gen_analz_insert_eq subset_trans [OF _ subset_insertI])
 done
 
 lemma Fake_parts_sing:
      "X \<in> synth (analz H) ==> parts{X} \<subseteq> synth (analz H) \<union> parts H"
 apply (rule subset_trans)
  apply (erule_tac [2] Fake_parts_insert)
 apply (rule parts_mono, blast)
 done
 
 lemmas Fake_parts_sing_imp_Un = Fake_parts_sing [THEN [2] rev_subsetD]
 
 
 text\<open>For some reason, moving this up can make some proofs loop!\<close>
 declare invKey_K [simp]
 
 
 end