Theory KerberosIV

(*  Title:      HOL/Auth/KerberosIV.thy
    Author:     Giampaolo Bella, Cambridge University Computer Laboratory
    Copyright   1998  University of Cambridge
*)

sectionThe Kerberos Protocol, Version IV

theory KerberosIV imports Public begin

textThe "u" prefix indicates theorems referring to an updated version of the protocol. The "r" suffix indicates theorems where the confidentiality assumptions are relaxed by the corresponding arguments.

abbreviation
  Kas :: agent where "Kas == Server"

abbreviation
  Tgs :: agent where "Tgs == Friend 0"


axiomatization where
  Tgs_not_bad [iff]: "Tgs  bad"
   ― ‹Tgs is secure --- we already know that Kas is secure

definition
 (* authKeys are those contained in an authTicket *)
    authKeys :: "event list  key set" where
    "authKeys evs = {authK. A Peer Ta. Says Kas A
                        (Crypt (shrK A) Key authK, Agent Peer, Number Ta,
               (Crypt (shrK Peer) Agent A, Agent Peer, Key authK, Number Ta)
                  )  set evs}"

definition
 (* A is the true creator of X if she has sent X and X never appeared on
    the trace before this event. Recall that traces grow from head. *)
  Issues :: "[agent, agent, msg, event list]  bool"
             ("_ Issues _ with _ on _" [50, 0, 0, 50] 50) where
   "(A Issues B with X on evs) =
      (Y. Says A B Y  set evs  X  parts {Y} 
        X  parts (spies (takeWhile (λz. z  Says A B Y) (rev evs))))"

definition
 (* Yields the subtrace of a given trace from its beginning to a given event *)
  before :: "[event, event list]  event list" ("before _ on _" [0, 50] 50)
  where "(before ev on evs) = takeWhile (λz. z  ev) (rev evs)"

definition
 (* States than an event really appears only once on a trace *)
  Unique :: "[event, event list]  bool" ("Unique _ on _" [0, 50] 50)
  where "(Unique ev on evs) = (ev  set (tl (dropWhile (λz. z  ev) evs)))"


consts
    (*Duration of the authentication key*)
    authKlife   :: nat

    (*Duration of the service key*)
    servKlife   :: nat

    (*Duration of an authenticator*)
    authlife   :: nat

    (*Upper bound on the time of reaction of a server*)
    replylife   :: nat

specification (authKlife)
  authKlife_LB [iff]: "2  authKlife"
    by blast

specification (servKlife)
  servKlife_LB [iff]: "2 + authKlife  servKlife"
    by blast

specification (authlife)
  authlife_LB [iff]: "Suc 0  authlife"
    by blast

specification (replylife)
  replylife_LB [iff]: "Suc 0  replylife"
    by blast

abbreviation
  (*The current time is the length of the trace*)
  CT :: "event list  nat" where
  "CT == length"

abbreviation
  expiredAK :: "[nat, event list]  bool" where
  "expiredAK Ta evs == authKlife + Ta < CT evs"

abbreviation
  expiredSK :: "[nat, event list]  bool" where
  "expiredSK Ts evs == servKlife + Ts < CT evs"

abbreviation
  expiredA :: "[nat, event list]  bool" where
  "expiredA T evs == authlife + T < CT evs"

abbreviation
  valid :: "[nat, nat]  bool" ("valid _ wrt _" [0, 50] 50) where
  "valid T1 wrt T2 == T1  replylife + T2"

(*---------------------------------------------------------------------*)


(* Predicate formalising the association between authKeys and servKeys *)
definition AKcryptSK :: "[key, key, event list]  bool" where
  "AKcryptSK authK servK evs ==
     A B Ts.
       Says Tgs A (Crypt authK
                     Key servK, Agent B, Number Ts,
                       Crypt (shrK B) Agent A, Agent B, Key servK, Number Ts )
          set evs"

inductive_set kerbIV :: "event list set"
  where

   Nil:  "[]  kerbIV"

 | Fake: " evsf  kerbIV;  X  synth (analz (spies evsf)) 
           Says Spy B X  # evsf  kerbIV"

(* FROM the initiator *)
 | K1:   " evs1  kerbIV 
           Says A Kas Agent A, Agent Tgs, Number (CT evs1) # evs1
           kerbIV"

(* Adding the timestamp serves to A in K3 to check that
   she doesn't get a reply too late. This kind of timeouts are ordinary.
   If a server's reply is late, then it is likely to be fake. *)

(*---------------------------------------------------------------------*)

(*FROM Kas *)
 | K2:  " evs2  kerbIV; Key authK  used evs2; authK  symKeys;
            Says A' Kas Agent A, Agent Tgs, Number T1  set evs2 
           Says Kas A
                (Crypt (shrK A) Key authK, Agent Tgs, Number (CT evs2),
                      (Crypt (shrK Tgs) Agent A, Agent Tgs, Key authK,
                          Number (CT evs2))) # evs2  kerbIV"
(*
  The internal encryption builds the authTicket.
  The timestamp doesn't change inside the two encryptions: the external copy
  will be used by the initiator in K3; the one inside the
  authTicket by Tgs in K4.
*)

(*---------------------------------------------------------------------*)

(* FROM the initiator *)
 | K3:  " evs3  kerbIV;
            Says A Kas Agent A, Agent Tgs, Number T1  set evs3;
            Says Kas' A (Crypt (shrK A) Key authK, Agent Tgs, Number Ta,
              authTicket)  set evs3;
            valid Ta wrt T1
         
           Says A Tgs authTicket,
                           (Crypt authK Agent A, Number (CT evs3)),
                           Agent B # evs3  kerbIV"
(*The two events amongst the premises allow A to accept only those authKeys
  that are not issued late. *)

(*---------------------------------------------------------------------*)

(* FROM Tgs *)
(* Note that the last temporal check is not mentioned in the original MIT
   specification. Adding it makes many goals "available" to the peers. 
   Theorems that exploit it have the suffix `_u', which stands for updated 
   protocol.
*)
 | K4:  " evs4  kerbIV; Key servK  used evs4; servK  symKeys;
            B  Tgs;  authK  symKeys;
            Says A' Tgs 
             (Crypt (shrK Tgs) Agent A, Agent Tgs, Key authK,
                                 Number Ta),
             (Crypt authK Agent A, Number T2), Agent B
                 set evs4;
            ¬ expiredAK Ta evs4;
            ¬ expiredA T2 evs4;
            servKlife + (CT evs4)  authKlife + Ta
         
           Says Tgs A
                (Crypt authK Key servK, Agent B, Number (CT evs4),
                               Crypt (shrK B) Agent A, Agent B, Key servK,
                                                Number (CT evs4) )
                # evs4  kerbIV"
(* Tgs creates a new session key per each request for a service, without
   checking if there is still a fresh one for that service.
   The cipher under Tgs' key is the authTicket, the cipher under B's key
   is the servTicket, which is built now.
   NOTE that the last temporal check is not present in the MIT specification.

*)

(*---------------------------------------------------------------------*)

(* FROM the initiator *)
 | K5:  " evs5  kerbIV; authK  symKeys; servK  symKeys;
            Says A Tgs
                authTicket, Crypt authK Agent A, Number T2,
                  Agent B
               set evs5;
            Says Tgs' A
             (Crypt authK Key servK, Agent B, Number Ts, servTicket)
                 set evs5;
            valid Ts wrt T2 
           Says A B servTicket,
                         Crypt servK Agent A, Number (CT evs5) 
               # evs5  kerbIV"
(* Checks similar to those in K3. *)

(*---------------------------------------------------------------------*)

(* FROM the responder*)
  | K6:  " evs6  kerbIV;
            Says A' B 
              (Crypt (shrK B) Agent A, Agent B, Key servK, Number Ts),
              (Crypt servK Agent A, Number T3)
             set evs6;
            ¬ expiredSK Ts evs6;
            ¬ expiredA T3 evs6
         
           Says B A (Crypt servK (Number T3))
               # evs6  kerbIV"
(* Checks similar to those in K4. *)

(*---------------------------------------------------------------------*)

(* Leaking an authK... *)
 | Oops1: " evsO1  kerbIV;  A  Spy;
              Says Kas A
                (Crypt (shrK A) Key authK, Agent Tgs, Number Ta,
                                  authTicket)   set evsO1;
              expiredAK Ta evsO1 
           Says A Spy Agent A, Agent Tgs, Number Ta, Key authK
               # evsO1  kerbIV"

(*---------------------------------------------------------------------*)

(*Leaking a servK... *)
 | Oops2: " evsO2  kerbIV;  A  Spy;
              Says Tgs A
                (Crypt authK Key servK, Agent B, Number Ts, servTicket)
                    set evsO2;
              expiredSK Ts evsO2 
           Says A Spy Agent A, Agent B, Number Ts, Key servK
               # evsO2  kerbIV"

(*---------------------------------------------------------------------*)

declare Says_imp_knows_Spy [THEN parts.Inj, dest]
declare parts.Body [dest]
declare analz_into_parts [dest]
declare Fake_parts_insert_in_Un [dest]


subsectionLemmas about lists, for reasoning about  Issues

lemma spies_Says_rev: "spies (evs @ [Says A B X]) = insert X (spies evs)"
apply (induct_tac "evs")
apply (rename_tac [2] a b)
apply (induct_tac [2] a, auto)
done

lemma spies_Gets_rev: "spies (evs @ [Gets A X]) = spies evs"
apply (induct_tac "evs")
apply (rename_tac [2] a b)
apply (induct_tac [2] a, auto)
done

lemma spies_Notes_rev: "spies (evs @ [Notes A X]) =
          (if Abad then insert X (spies evs) else spies evs)"
apply (induct_tac "evs")
apply (rename_tac [2] a b)
apply (induct_tac [2] a, auto)
done

lemma spies_evs_rev: "spies evs = spies (rev evs)"
apply (induct_tac "evs")
apply (rename_tac [2] a b)
apply (induct_tac [2] a)
apply (simp_all (no_asm_simp) add: spies_Says_rev spies_Gets_rev spies_Notes_rev)
done

lemmas parts_spies_evs_revD2 = spies_evs_rev [THEN equalityD2, THEN parts_mono]

lemma spies_takeWhile: "spies (takeWhile P evs)  spies evs"
apply (induct_tac "evs")
apply (rename_tac [2] a b)
apply (induct_tac [2] "a", auto)
txtResembles used_subset_append› in theory Event.
done

lemmas parts_spies_takeWhile_mono = spies_takeWhile [THEN parts_mono]


subsectionLemmas about termauthKeys

lemma authKeys_empty: "authKeys [] = {}"
apply (unfold authKeys_def)
apply (simp (no_asm))
done

lemma authKeys_not_insert:
 "(A Ta akey Peer.
   ev  Says Kas A (Crypt (shrK A) akey, Agent Peer, Ta,
              (Crypt (shrK Peer) Agent A, Agent Peer, akey, Ta)))
        authKeys (ev # evs) = authKeys evs"
by (unfold authKeys_def, auto)

lemma authKeys_insert:
  "authKeys
     (Says Kas A (Crypt (shrK A) Key K, Agent Peer, Number Ta,
      (Crypt (shrK Peer) Agent A, Agent Peer, Key K, Number Ta)) # evs)
       = insert K (authKeys evs)"
by (unfold authKeys_def, auto)

lemma authKeys_simp:
   "K  authKeys
    (Says Kas A (Crypt (shrK A) Key K', Agent Peer, Number Ta,
     (Crypt (shrK Peer) Agent A, Agent Peer, Key K', Number Ta)) # evs)
         K = K' | K  authKeys evs"
by (unfold authKeys_def, auto)

lemma authKeysI:
   "Says Kas A (Crypt (shrK A) Key K, Agent Tgs, Number Ta,
     (Crypt (shrK Tgs) Agent A, Agent Tgs, Key K, Number Ta))  set evs
         K  authKeys evs"
by (unfold authKeys_def, auto)

lemma authKeys_used: "K  authKeys evs  Key K  used evs"
by (simp add: authKeys_def, blast)


subsectionForwarding Lemmas

text--For reasoning about the encrypted portion of message K3--
lemma K3_msg_in_parts_spies:
     "Says Kas' A (Crypt KeyA authK, Peer, Ta, authTicket)
                set evs  authTicket  parts (spies evs)"
by blast

lemma Oops_range_spies1:
     " Says Kas A (Crypt KeyA Key authK, Peer, Ta, authTicket)
            set evs ;
         evs  kerbIV   authK  range shrK  authK  symKeys"
apply (erule rev_mp)
apply (erule kerbIV.induct, auto)
done

text--For reasoning about the encrypted portion of message K5--
lemma K5_msg_in_parts_spies:
     "Says Tgs' A (Crypt authK servK, Agent B, Ts, servTicket)
                set evs  servTicket  parts (spies evs)"
by blast

lemma Oops_range_spies2:
     " Says Tgs A (Crypt authK Key servK, Agent B, Ts, servTicket)
            set evs ;
         evs  kerbIV   servK  range shrK  servK  symKeys"
apply (erule rev_mp)
apply (erule kerbIV.induct, auto)
done

lemma Says_ticket_parts:
     "Says S A (Crypt K SesKey, B, TimeStamp, Ticket)  set evs
       Ticket  parts (spies evs)"
by blast

(*Spy never sees another agent's shared key! (unless it's lost at start)*)
lemma Spy_see_shrK [simp]:
     "evs  kerbIV  (Key (shrK A)  parts (spies evs)) = (A  bad)"
apply (erule kerbIV.induct)
apply (frule_tac [7] K5_msg_in_parts_spies)
apply (frule_tac [5] K3_msg_in_parts_spies, simp_all)
apply (blast+)
done

lemma Spy_analz_shrK [simp]:
     "evs  kerbIV  (Key (shrK A)  analz (spies evs)) = (A  bad)"
by auto

lemma Spy_see_shrK_D [dest!]:
     " Key (shrK A)  parts (spies evs);  evs  kerbIV   Abad"
by (blast dest: Spy_see_shrK)

lemmas Spy_analz_shrK_D = analz_subset_parts [THEN subsetD, THEN Spy_see_shrK_D, dest!]

textNobody can have used non-existent keys!
lemma new_keys_not_used [simp]:
    "Key K  used evs; K  symKeys; evs  kerbIV
      K  keysFor (parts (spies evs))"
apply (erule rev_mp)
apply (erule kerbIV.induct)
apply (frule_tac [7] K5_msg_in_parts_spies)
apply (frule_tac [5] K3_msg_in_parts_spies, simp_all)
txtFake
apply (force dest!: keysFor_parts_insert)
txtOthers
apply (force dest!: analz_shrK_Decrypt)+
done

(*Earlier, all protocol proofs declared this theorem.
  But few of them actually need it! (Another is Yahalom) *)
lemma new_keys_not_analzd:
 "evs  kerbIV; K  symKeys; Key K  used evs
   K  keysFor (analz (spies evs))"
by (blast dest: new_keys_not_used intro: keysFor_mono [THEN subsetD])



subsectionLemmas for reasoning about predicate "before"

lemma used_Says_rev: "used (evs @ [Says A B X]) = parts {X}  (used evs)"
apply (induct_tac "evs")
apply simp
apply (rename_tac a b)
apply (induct_tac "a")
apply auto
done

lemma used_Notes_rev: "used (evs @ [Notes A X]) = parts {X}  (used evs)"
apply (induct_tac "evs")
apply simp
apply (rename_tac a b)
apply (induct_tac "a")
apply auto
done

lemma used_Gets_rev: "used (evs @ [Gets B X]) = used evs"
apply (induct_tac "evs")
apply simp
apply (rename_tac a b)
apply (induct_tac "a")
apply auto
done

lemma used_evs_rev: "used evs = used (rev evs)"
apply (induct_tac "evs")
apply simp
apply (rename_tac a b)
apply (induct_tac "a")
apply (simp add: used_Says_rev)
apply (simp add: used_Gets_rev)
apply (simp add: used_Notes_rev)
done

lemma used_takeWhile_used [rule_format]: 
      "x  used (takeWhile P X)  x  used X"
apply (induct_tac "X")
apply simp
apply (rename_tac a b)
apply (induct_tac "a")
apply (simp_all add: used_Nil)
apply (blast dest!: initState_into_used)+
done

lemma set_evs_rev: "set evs = set (rev evs)"
by auto

lemma takeWhile_void [rule_format]:
      "x  set evs  takeWhile (λz. z  x) evs = evs"
by auto


subsectionRegularity Lemmas
textThese concern the form of items passed in messages

textDescribes the form of all components sent by Kas
lemma Says_Kas_message_form:
     " Says Kas A (Crypt K Key authK, Agent Peer, Number Ta, authTicket)
            set evs;
         evs  kerbIV    
  K = shrK A  Peer = Tgs 
  authK  range shrK  authK  authKeys evs  authK  symKeys  
  authTicket = (Crypt (shrK Tgs) Agent A, Agent Tgs, Key authK, Number Ta) 
  Key authK  used(before 
           Says Kas A (Crypt K Key authK, Agent Peer, Number Ta, authTicket)
                   on evs) 
  Ta = CT (before 
           Says Kas A (Crypt K Key authK, Agent Peer, Number Ta, authTicket)
           on evs)"
apply (unfold before_def)
apply (erule rev_mp)
apply (erule kerbIV.induct)
apply (simp_all (no_asm) add: authKeys_def authKeys_insert, blast, blast)
txtK2
apply (simp (no_asm) add: takeWhile_tail)
apply (rule conjI)
apply (metis Key_not_used authKeys_used length_rev set_rev takeWhile_void used_evs_rev)
apply blast+
done



(*This lemma is essential for proving Says_Tgs_message_form:

  the session key authK
  supplied by Kas in the authentication ticket
  cannot be a long-term key!

  Generalised to any session keys (both authK and servK).
*)
lemma SesKey_is_session_key:
     " Crypt (shrK Tgs_B) Agent A, Agent Tgs_B, Key SesKey, Number T
             parts (spies evs); Tgs_B  bad;
         evs  kerbIV 
       SesKey  range shrK"
apply (erule rev_mp)
apply (erule kerbIV.induct)
apply (frule_tac [7] K5_msg_in_parts_spies)
apply (frule_tac [5] K3_msg_in_parts_spies, simp_all, blast)
done

lemma authTicket_authentic:
     " Crypt (shrK Tgs) Agent A, Agent Tgs, Key authK, Number Ta
            parts (spies evs);
         evs  kerbIV 
       Says Kas A (Crypt (shrK A) Key authK, Agent Tgs, Number Ta,
                 Crypt (shrK Tgs) Agent A, Agent Tgs, Key authK, Number Ta)
             set evs"
apply (erule rev_mp)
apply (erule kerbIV.induct)
apply (frule_tac [7] K5_msg_in_parts_spies)
apply (frule_tac [5] K3_msg_in_parts_spies, simp_all)
txtFake, K4
apply (blast+)
done

lemma authTicket_crypt_authK:
     " Crypt (shrK Tgs) Agent A, Agent Tgs, Key authK, Number Ta
            parts (spies evs);
         evs  kerbIV 
       authK  authKeys evs"
apply (frule authTicket_authentic, assumption)
apply (simp (no_asm) add: authKeys_def)
apply blast
done

textDescribes the form of servK, servTicket and authK sent by Tgs
lemma Says_Tgs_message_form:
     " Says Tgs A (Crypt authK Key servK, Agent B, Number Ts, servTicket)
            set evs;
         evs  kerbIV 
   B  Tgs  
      authK  range shrK  authK  authKeys evs  authK  symKeys 
      servK  range shrK  servK  authKeys evs  servK  symKeys 
      servTicket = (Crypt (shrK B) Agent A, Agent B, Key servK, Number Ts) 
      Key servK  used (before
        Says Tgs A (Crypt authK Key servK, Agent B, Number Ts, servTicket)
                        on evs) 
      Ts = CT(before 
        Says Tgs A (Crypt authK Key servK, Agent B, Number Ts, servTicket)
              on evs) "
apply (unfold before_def)
apply (erule rev_mp)
apply (erule kerbIV.induct)
apply (simp_all add: authKeys_insert authKeys_not_insert authKeys_empty authKeys_simp, blast)
txtWe need this simplification only for Message 4
apply (simp (no_asm) add: takeWhile_tail)
apply auto
txtFive subcases of Message 4
apply (blast dest!: SesKey_is_session_key)
apply (blast dest: authTicket_crypt_authK)
apply (blast dest!: authKeys_used Says_Kas_message_form)
txtsubcase: used before
apply (metis used_evs_rev used_takeWhile_used)
txtsubcase: CT before
apply (metis length_rev set_evs_rev takeWhile_void)
done

lemma authTicket_form:
     " Crypt (shrK A) Key authK, Agent Tgs, Ta, authTicket
            parts (spies evs);
         A  bad;
         evs  kerbIV 
     authK  range shrK  authK  symKeys  
        authTicket = Crypt (shrK Tgs) Agent A, Agent Tgs, Key authK, Ta"
apply (erule rev_mp)
apply (erule kerbIV.induct)
apply (frule_tac [7] K5_msg_in_parts_spies)
apply (frule_tac [5] K3_msg_in_parts_spies, simp_all)
apply (blast+)
done

textThis form holds also over an authTicket, but is not needed below.
lemma servTicket_form:
     " Crypt authK Key servK, Agent B, Ts, servTicket
               parts (spies evs);
            Key authK  analz (spies evs);
            evs  kerbIV 
          servK  range shrK  servK  symKeys  
    (A. servTicket = Crypt (shrK B) Agent A, Agent B, Key servK, Ts)"
apply (erule rev_mp)
apply (erule rev_mp)
apply (erule kerbIV.induct, analz_mono_contra)
apply (frule_tac [7] K5_msg_in_parts_spies)
apply (frule_tac [5] K3_msg_in_parts_spies, simp_all, blast)
done

textEssentially the same as authTicket_form›
lemma Says_kas_message_form:
     " Says Kas' A (Crypt (shrK A)
              Key authK, Agent Tgs, Ta, authTicket)  set evs;
         evs  kerbIV 
       authK  range shrK  authK  symKeys  
          authTicket =
                  Crypt (shrK Tgs) Agent A, Agent Tgs, Key authK, Ta
          | authTicket  analz (spies evs)"
by (blast dest: analz_shrK_Decrypt authTicket_form
                Says_imp_spies [THEN analz.Inj])

lemma Says_tgs_message_form:
 " Says Tgs' A (Crypt authK Key servK, Agent B, Ts, servTicket)
        set evs;  authK  symKeys;
     evs  kerbIV 
   servK  range shrK 
      (A. servTicket =
              Crypt (shrK B) Agent A, Agent B, Key servK, Ts)
       | servTicket  analz (spies evs)"
by (metis Says_imp_analz_Spy Says_imp_parts_knows_Spy analz.Decrypt analz.Snd invKey_K servTicket_form)


subsectionAuthenticity theorems: confirm origin of sensitive messages

lemma authK_authentic:
     " Crypt (shrK A) Key authK, Peer, Ta, authTicket
            parts (spies evs);
         A  bad;  evs  kerbIV 
       Says Kas A (Crypt (shrK A) Key authK, Peer, Ta, authTicket)
             set evs"
apply (erule rev_mp)
apply (erule kerbIV.induct)
apply (frule_tac [7] K5_msg_in_parts_spies)
apply (frule_tac [5] K3_msg_in_parts_spies, simp_all)
txtFake
apply blast
txtK4
apply (blast dest!: authTicket_authentic [THEN Says_Kas_message_form])
done

textIf a certain encrypted message appears then it originated with Tgs
lemma servK_authentic:
     " Crypt authK Key servK, Agent B, Number Ts, servTicket
            parts (spies evs);
         Key authK  analz (spies evs);
         authK  range shrK;
         evs  kerbIV 
  A. Says Tgs A (Crypt authK Key servK, Agent B, Number Ts, servTicket)
        set evs"
apply (erule rev_mp)
apply (erule rev_mp)
apply (erule kerbIV.induct, analz_mono_contra)
apply (frule_tac [7] K5_msg_in_parts_spies)
apply (frule_tac [5] K3_msg_in_parts_spies, simp_all)
txtFake
apply blast
txtK2
apply blast
txtK4
apply auto
done

lemma servK_authentic_bis:
     " Crypt authK Key servK, Agent B, Number Ts, servTicket
            parts (spies evs);
         Key authK  analz (spies evs);
         B  Tgs;
         evs  kerbIV 
  A. Says Tgs A (Crypt authK Key servK, Agent B, Number Ts, servTicket)
        set evs"
apply (erule rev_mp)
apply (erule rev_mp)
apply (erule kerbIV.induct, analz_mono_contra)
apply (frule_tac [7] K5_msg_in_parts_spies)
apply (frule_tac [5] K3_msg_in_parts_spies, simp_all)
txtFake
apply blast
txtK4
apply blast
done

textAuthenticity of servK for B
lemma servTicket_authentic_Tgs:
     " Crypt (shrK B) Agent A, Agent B, Key servK, Number Ts
            parts (spies evs); B  Tgs;  B  bad;
         evs  kerbIV 
  authK.
       Says Tgs A (Crypt authK Key servK, Agent B, Number Ts,
                   Crypt (shrK B) Agent A, Agent B, Key servK, Number Ts)
        set evs"
apply (erule rev_mp)
apply (erule rev_mp)
apply (erule kerbIV.induct)
apply (frule_tac [7] K5_msg_in_parts_spies)
apply (frule_tac [5] K3_msg_in_parts_spies, simp_all)
apply blast+
done

textAnticipated here from next subsection
lemma K4_imp_K2:
" Says Tgs A (Crypt authK Key servK, Agent B, Number Ts, servTicket)
       set evs;  evs  kerbIV
    Ta. Says Kas A
        (Crypt (shrK A)
         Key authK, Agent Tgs, Number Ta,
           Crypt (shrK Tgs) Agent A, Agent Tgs, Key authK, Number Ta)
         set evs"
apply (erule rev_mp)
apply (erule kerbIV.induct)
apply (frule_tac [7] K5_msg_in_parts_spies)
apply (frule_tac [5] K3_msg_in_parts_spies, simp_all, auto)
apply (blast dest!: Says_imp_spies [THEN parts.Inj, THEN parts.Fst, THEN authTicket_authentic])
done

textAnticipated here from next subsection
lemma u_K4_imp_K2:
" Says Tgs A (Crypt authK Key servK, Agent B, Number Ts, servTicket)
       set evs; evs  kerbIV
    Ta. (Says Kas A (Crypt (shrK A) Key authK, Agent Tgs, Number Ta,
           Crypt (shrK Tgs) Agent A, Agent Tgs, Key authK, Number Ta)
              set evs
           servKlife + Ts  authKlife + Ta)"
apply (erule rev_mp)
apply (erule kerbIV.induct)
apply (frule_tac [7] K5_msg_in_parts_spies)
apply (frule_tac [5] K3_msg_in_parts_spies, simp_all, auto)
apply (blast dest!: Says_imp_spies [THEN parts.Inj, THEN parts.Fst, THEN authTicket_authentic])
done

lemma servTicket_authentic_Kas:
     " Crypt (shrK B) Agent A, Agent B, Key servK, Number Ts
            parts (spies evs);  B  Tgs;  B  bad;
         evs  kerbIV 
   authK Ta.
       Says Kas A
         (Crypt (shrK A) Key authK, Agent Tgs, Number Ta,
            Crypt (shrK Tgs) Agent A, Agent Tgs, Key authK, Number Ta)
         set evs"
by (blast dest!: servTicket_authentic_Tgs K4_imp_K2)

lemma u_servTicket_authentic_Kas:
     " Crypt (shrK B) Agent A, Agent B, Key servK, Number Ts
            parts (spies evs);  B  Tgs;  B  bad;
         evs  kerbIV 
   authK Ta. Says Kas A (Crypt(shrK A) Key authK, Agent Tgs, Number Ta,
           Crypt (shrK Tgs) Agent A, Agent Tgs, Key authK, Number Ta)
              set evs
            servKlife + Ts  authKlife + Ta"
by (blast dest!: servTicket_authentic_Tgs u_K4_imp_K2)

lemma servTicket_authentic:
     " Crypt (shrK B) Agent A, Agent B, Key servK, Number Ts
            parts (spies evs);  B  Tgs;  B  bad;
         evs  kerbIV 
  Ta authK.
     Says Kas A (Crypt (shrK A) Key authK, Agent Tgs, Number Ta,
                   Crypt (shrK Tgs) Agent A, Agent Tgs, Key authK, Number Ta)
        set evs
      Says Tgs A (Crypt authK Key servK, Agent B, Number Ts,
                   Crypt (shrK B) Agent A, Agent B, Key servK, Number Ts)
        set evs"
by (blast dest: servTicket_authentic_Tgs K4_imp_K2)

lemma u_servTicket_authentic:
     " Crypt (shrK B) Agent A, Agent B, Key servK, Number Ts
            parts (spies evs);  B  Tgs;  B  bad;
         evs  kerbIV 
  Ta authK.
     (Says Kas A (Crypt (shrK A) Key authK, Agent Tgs, Number Ta,
                   Crypt (shrK Tgs) Agent A, Agent Tgs, Key authK, Number Ta)
        set evs
      Says Tgs A (Crypt authK Key servK, Agent B, Number Ts,
                   Crypt (shrK B) Agent A, Agent B, Key servK, Number Ts)
        set evs
      servKlife + Ts  authKlife + Ta)"
by (blast dest: servTicket_authentic_Tgs u_K4_imp_K2)

lemma u_NotexpiredSK_NotexpiredAK:
     " ¬ expiredSK Ts evs; servKlife + Ts  authKlife + Ta 
       ¬ expiredAK Ta evs"
  by (metis le_less_trans)


subsectionReliability: friendly agents send something if something else happened

lemma K3_imp_K2:
     " Says A Tgs
             authTicket, Crypt authK Agent A, Number T2, Agent B
            set evs;
         A  bad;  evs  kerbIV 
       Ta. Says Kas A (Crypt (shrK A)
                      Key authK, Agent Tgs, Number Ta, authTicket)
                    set evs"
apply (erule rev_mp)
apply (erule kerbIV.induct)
apply (frule_tac [7] K5_msg_in_parts_spies)
apply (frule_tac [5] K3_msg_in_parts_spies, simp_all, blast, blast)
apply (blast dest: Says_imp_spies [THEN parts.Inj, THEN authK_authentic])
done

textAnticipated here from next subsection. An authK is encrypted by one and only one Shared key. A servK is encrypted by one and only one authK.
lemma Key_unique_SesKey:
     " Crypt K  Key SesKey,  Agent B, T, Ticket
            parts (spies evs);
         Crypt K' Key SesKey,  Agent B', T', Ticket'
            parts (spies evs);  Key SesKey  analz (spies evs);
         evs  kerbIV 
       K=K'  B=B'  T=T'  Ticket=Ticket'"
apply (erule rev_mp)
apply (erule rev_mp)
apply (erule rev_mp)
apply (erule kerbIV.induct, analz_mono_contra)
apply (frule_tac [7] K5_msg_in_parts_spies)
apply (frule_tac [5] K3_msg_in_parts_spies, simp_all)
txtFake, K2, K4
apply (blast+)
done

lemma Tgs_authenticates_A:
  "  Crypt authK Agent A, Number T2  parts (spies evs); 
      Crypt (shrK Tgs) Agent A, Agent Tgs, Key authK, Number Ta
            parts (spies evs);
      Key authK  analz (spies evs); A  bad; evs  kerbIV 
   B. Says A Tgs 
          Crypt (shrK Tgs) Agent A, Agent Tgs, Key authK, Number Ta,
          Crypt authK Agent A, Number T2, Agent B   set evs"  
apply (drule authTicket_authentic, assumption, rotate_tac 4)
apply (erule rev_mp, erule rev_mp, erule rev_mp)
apply (erule kerbIV.induct, analz_mono_contra)
apply (frule_tac [5] Says_ticket_parts)
apply (frule_tac [7] Says_ticket_parts)
apply (simp_all (no_asm_simp) add: all_conj_distrib)
txtFake
apply blast
txtK2
apply (force dest!: Crypt_imp_keysFor)
txtK3
apply (blast dest: Key_unique_SesKey)
txtK5
apply (metis K3_imp_K2 Key_unique_SesKey Spy_see_shrK parts.Body parts.Fst 
             Says_imp_knows_Spy [THEN parts.Inj])
done

lemma Says_K5:
     " Crypt servK Agent A, Number T3  parts (spies evs);
         Says Tgs A (Crypt authK Key servK, Agent B, Number Ts,
                                     servTicket)  set evs;
         Key servK  analz (spies evs);
         A  bad; B  bad; evs  kerbIV 
  Says A B servTicket, Crypt servK Agent A, Number T3  set evs"
apply (erule rev_mp)
apply (erule rev_mp)
apply (erule rev_mp)
apply (erule kerbIV.induct, analz_mono_contra)
apply (frule_tac [5] Says_ticket_parts)
apply (frule_tac [7] Says_ticket_parts)
apply (simp_all (no_asm_simp) add: all_conj_distrib)
apply blast
txtK3
apply (blast dest: authK_authentic Says_Kas_message_form Says_Tgs_message_form)
txtK4
apply (force dest!: Crypt_imp_keysFor)
txtK5
apply (blast dest: Key_unique_SesKey)
done

textAnticipated here from next subsection
lemma unique_CryptKey:
     " Crypt (shrK B)  Agent A,  Agent B,  Key SesKey, T
            parts (spies evs);
         Crypt (shrK B') Agent A', Agent B', Key SesKey, T'
            parts (spies evs);  Key SesKey  analz (spies evs);
         evs  kerbIV 
       A=A'  B=B'  T=T'"
apply (erule rev_mp)
apply (erule rev_mp)
apply (erule rev_mp)
apply (erule kerbIV.induct, analz_mono_contra)
apply (frule_tac [7] K5_msg_in_parts_spies)
apply (frule_tac [5] K3_msg_in_parts_spies, simp_all)
txtFake, K2, K4
apply (blast+)
done

lemma Says_K6:
     " Crypt servK (Number T3)  parts (spies evs);
         Says Tgs A (Crypt authK Key servK, Agent B, Number Ts,
                                     servTicket)  set evs;
         Key servK  analz (spies evs);
         A  bad; B  bad; evs  kerbIV 
       Says B A (Crypt servK (Number T3))  set evs"
apply (erule rev_mp)
apply (erule rev_mp)
apply (erule rev_mp)
apply (erule kerbIV.induct, analz_mono_contra)
apply (frule_tac [5] Says_ticket_parts)
apply (frule_tac [7] Says_ticket_parts)
apply (simp_all (no_asm_simp))
apply blast
apply (metis Crypt_imp_invKey_keysFor invKey_K new_keys_not_used)
apply (clarify)
apply (frule Says_Tgs_message_form, assumption)
apply (metis K3_msg_in_parts_spies parts.Fst Says_imp_knows_Spy [THEN parts.Inj] 
             unique_CryptKey) 
done

textNeeds a unicity theorem, hence moved here
lemma servK_authentic_ter:
 " Says Kas A
    (Crypt (shrK A) Key authK, Agent Tgs, Number Ta, authTicket)  set evs;
     Crypt authK Key servK, Agent B, Number Ts, servTicket
        parts (spies evs);
     Key authK  analz (spies evs);
     evs  kerbIV 
  Says Tgs A (Crypt authK Key servK, Agent B, Number Ts, servTicket)
        set evs"
apply (frule Says_Kas_message_form, assumption)
apply (erule rev_mp)
apply (erule rev_mp)
apply (erule rev_mp)
apply (erule kerbIV.induct, analz_mono_contra)
apply (frule_tac [7] K5_msg_in_parts_spies)
apply (frule_tac [5] K3_msg_in_parts_spies, simp_all, blast)
txtK2
apply (blast dest!: servK_authentic Says_Tgs_message_form authKeys_used)
txtK4 remain
apply (blast dest!: unique_CryptKey)
done


subsectionUnicity Theorems

textThe session key, if secure, uniquely identifies the Ticket
   whether authTicket or servTicket. As a matter of fact, one can read
   also Tgs in the place of B.


(*
  At reception of any message mentioning A, Kas associates shrK A with
  a new authK. Realistic, as the user gets a new authK at each login.
  Similarly, at reception of any message mentioning an authK
  (a legitimate user could make several requests to Tgs - by K3), Tgs
  associates it with a new servK.

  Therefore, a goal like

   "evs ∈ kerbIV
     ⟹ Key Kc ∉ analz (spies evs) ⟶
           (∃K' B' T' Ticket'. ∀K B T Ticket.
            Crypt Kc ⦃Key K, Agent B, T, Ticket⦄
             ∈ parts (spies evs) ⟶ K=K' ∧ B=B' ∧ T=T' ∧ Ticket=Ticket')"

  would fail on the K2 and K4 cases.
*)

lemma unique_authKeys:
     " Says Kas A
              (Crypt Ka Key authK, Agent Tgs, Ta, X)  set evs;
         Says Kas A'
              (Crypt Ka' Key authK, Agent Tgs, Ta', X')  set evs;
         evs  kerbIV   A=A'  Ka=Ka'  Ta=Ta'  X=X'"
apply (erule rev_mp)
apply (erule rev_mp)
apply (erule kerbIV.induct)
apply (frule_tac [7] K5_msg_in_parts_spies)
apply (frule_tac [5] K3_msg_in_parts_spies, simp_all)
txtK2
apply blast
done

textservK uniquely identifies the message from Tgs
lemma unique_servKeys:
     " Says Tgs A
              (Crypt K Key servK, Agent B, Ts, X)  set evs;
         Says Tgs A'
              (Crypt K' Key servK, Agent B', Ts', X')  set evs;
         evs  kerbIV   A=A'  B=B'  K=K'  Ts=Ts'  X=X'"
apply (erule rev_mp)
apply (erule rev_mp)
apply (erule kerbIV.induct)
apply (frule_tac [7] K5_msg_in_parts_spies)
apply (frule_tac [5] K3_msg_in_parts_spies, simp_all)
txtK4
apply blast
done

textRevised unicity theorems

lemma Kas_Unique:
     " Says Kas A
              (Crypt Ka Key authK, Agent Tgs, Ta, authTicket)  set evs;
        evs  kerbIV   
   Unique (Says Kas A (Crypt Ka Key authK, Agent Tgs, Ta, authTicket)) 
   on evs"
apply (erule rev_mp, erule kerbIV.induct, simp_all add: Unique_def)
apply blast
done

lemma Tgs_Unique:
     " Says Tgs A
              (Crypt authK Key servK, Agent B, Ts, servTicket)  set evs;
        evs  kerbIV   
  Unique (Says Tgs A (Crypt authK Key servK, Agent B, Ts, servTicket)) 
  on evs"
apply (erule rev_mp, erule kerbIV.induct, simp_all add: Unique_def)
apply blast
done


subsectionLemmas About the Predicate termAKcryptSK

lemma not_AKcryptSK_Nil [iff]: "¬ AKcryptSK authK servK []"
by (simp add: AKcryptSK_def)

lemma AKcryptSKI:
 " Says Tgs A (Crypt authK Key servK, Agent B, Number Ts, X )  set evs;
     evs  kerbIV   AKcryptSK authK servK evs"
apply (unfold AKcryptSK_def)
apply (blast dest: Says_Tgs_message_form)
done

lemma AKcryptSK_Says [simp]:
   "AKcryptSK authK servK (Says S A X # evs) =
     (Tgs = S 
      (B Ts. X = Crypt authK
                Key servK, Agent B, Number Ts,
                  Crypt (shrK B) Agent A, Agent B, Key servK, Number Ts )
     | AKcryptSK authK servK evs)"
by (auto simp add: AKcryptSK_def)


(*A fresh authK cannot be associated with any other
  (with respect to a given trace). *)
lemma Auth_fresh_not_AKcryptSK:
     " Key authK  used evs; evs  kerbIV 
       ¬ AKcryptSK authK servK evs"
apply (unfold AKcryptSK_def)
apply (erule rev_mp)
apply (erule kerbIV.induct)
apply (frule_tac [7] K5_msg_in_parts_spies)
apply (frule_tac [5] K3_msg_in_parts_spies, simp_all, blast)
done

(*A fresh servK cannot be associated with any other
  (with respect to a given trace). *)
lemma Serv_fresh_not_AKcryptSK:
 "Key servK  used evs  ¬ AKcryptSK authK servK evs"
by (unfold AKcryptSK_def, blast)

lemma authK_not_AKcryptSK:
     " Crypt (shrK Tgs) Agent A, Agent Tgs, Key authK, tk
            parts (spies evs);  evs  kerbIV 
       ¬ AKcryptSK K authK evs"
apply (erule rev_mp)
apply (erule kerbIV.induct)
apply (frule_tac [7] K5_msg_in_parts_spies)
apply (frule_tac [5] K3_msg_in_parts_spies, simp_all)
txtFake
apply blast
txtK2: by freshness
apply (simp add: AKcryptSK_def)
txtK4
apply (blast+)
done

textA secure serverkey cannot have been used to encrypt others
lemma servK_not_AKcryptSK:
 " Crypt (shrK B) Agent A, Agent B, Key SK, Number Ts  parts (spies evs);
     Key SK  analz (spies evs);  SK  symKeys;
     B  Tgs;  evs  kerbIV 
   ¬ AKcryptSK SK K evs"
apply (erule rev_mp)
apply (erule rev_mp)
apply (erule kerbIV.induct, analz_mono_contra)
apply (frule_tac [7] K5_msg_in_parts_spies)
apply (frule_tac [5] K3_msg_in_parts_spies, simp_all, blast)
txtK4
apply (metis Auth_fresh_not_AKcryptSK Crypt_imp_keysFor new_keys_not_used parts.Fst parts.Snd Says_imp_knows_Spy [THEN parts.Inj] unique_CryptKey)
done

textLong term keys are not issued as servKeys
lemma shrK_not_AKcryptSK:
     "evs  kerbIV  ¬ AKcryptSK K (shrK A) evs"
apply (unfold AKcryptSK_def)
apply (erule kerbIV.induct)
apply (frule_tac [7] K5_msg_in_parts_spies)
apply (frule_tac [5] K3_msg_in_parts_spies, auto)
done

textThe Tgs message associates servK with authK and therefore not with any
  other key authK.
lemma Says_Tgs_AKcryptSK:
     " Says Tgs A (Crypt authK Key servK, Agent B, Number Ts, X )
            set evs;
         authK'  authK;  evs  kerbIV 
       ¬ AKcryptSK authK' servK evs"
apply (unfold AKcryptSK_def)
apply (blast dest: unique_servKeys)
done

textEquivalently
lemma not_different_AKcryptSK:
     " AKcryptSK authK servK evs;
        authK'  authK;  evs  kerbIV 
       ¬ AKcryptSK authK' servK evs   servK  symKeys"
apply (simp add: AKcryptSK_def)
apply (blast dest: unique_servKeys Says_Tgs_message_form)
done

lemma AKcryptSK_not_AKcryptSK:
     " AKcryptSK authK servK evs;  evs  kerbIV 
       ¬ AKcryptSK servK K evs"
apply (erule rev_mp)
apply (erule kerbIV.induct)
apply (frule_tac [7] K5_msg_in_parts_spies)
apply (frule_tac [5] K3_msg_in_parts_spies, simp_all)
apply (metis Auth_fresh_not_AKcryptSK Says_imp_spies authK_not_AKcryptSK 
             authKeys_used authTicket_crypt_authK parts.Fst parts.Inj)
done

textThe only session keys that can be found with the help of session keys are
  those sent by Tgs in step K4.

textWe take some pains to express the property
  as a logical equivalence so that the simplifier can apply it.
lemma Key_analz_image_Key_lemma:
     "P  (Key K  analz (Key`KK  H))  (KKK | Key K  analz H)
      
      P  (Key K  analz (Key`KK  H)) = (KKK | Key K  analz H)"
by (blast intro: analz_mono [THEN subsetD])


lemma AKcryptSK_analz_insert:
     " AKcryptSK K K' evs; K  symKeys; evs  kerbIV 
       Key K'  analz (insert (Key K) (spies evs))"
apply (simp add: AKcryptSK_def, clarify)
apply (drule Says_imp_spies [THEN analz.Inj, THEN analz_insertI], auto)
done

lemma authKeys_are_not_AKcryptSK:
     " K  authKeys evs  range shrK;  evs  kerbIV 
       SK. ¬ AKcryptSK SK K evs  K  symKeys"
apply (simp add: authKeys_def AKcryptSK_def)
apply (blast dest: Says_Kas_message_form Says_Tgs_message_form)
done

lemma not_authKeys_not_AKcryptSK:
     " K  authKeys evs;
         K  range shrK; evs  kerbIV 
       SK. ¬ AKcryptSK K SK evs"
apply (simp add: AKcryptSK_def)
apply (blast dest: Says_Tgs_message_form)
done


subsectionSecrecy Theorems

textFor the Oops2 case of the next theorem
lemma Oops2_not_AKcryptSK:
     " evs  kerbIV;
         Says Tgs A (Crypt authK
                     Key servK, Agent B, Number Ts, servTicket)
            set evs 
       ¬ AKcryptSK servK SK evs"
by (blast dest: AKcryptSKI AKcryptSK_not_AKcryptSK)
   
textBig simplification law for keys SK that are not crypted by keys in KK
 It helps prove three, otherwise hard, facts about keys. These facts are
 exploited as simplification laws for analz, and also "limit the damage"
 in case of loss of a key to the spy. See ESORICS98.
 [simplified by LCP]
lemma Key_analz_image_Key [rule_format (no_asm)]:
     "evs  kerbIV 
      (SK KK. SK  symKeys  KK  -(range shrK) 
       (K  KK. ¬ AKcryptSK K SK evs)   
       (Key SK  analz (Key`KK  (spies evs))) =
       (SK  KK | Key SK  analz (spies evs)))"
apply (erule kerbIV.induct)
apply (frule_tac [10] Oops_range_spies2)
apply (frule_tac [9] Oops_range_spies1)
apply (frule_tac [7] Says_tgs_message_form)
apply (frule_tac [5] Says_kas_message_form)
apply (safe del: impI intro!: Key_analz_image_Key_lemma [THEN impI])
txtCase-splits for Oops1 and message 5: the negated case simplifies using
 the induction hypothesis
apply (case_tac [11] "AKcryptSK authK SK evsO1")
apply (case_tac [8] "AKcryptSK servK SK evs5")
apply (simp_all del: image_insert
        add: analz_image_freshK_simps AKcryptSK_Says shrK_not_AKcryptSK
             Oops2_not_AKcryptSK Auth_fresh_not_AKcryptSK
       Serv_fresh_not_AKcryptSK Says_Tgs_AKcryptSK Spy_analz_shrK)
txtFake 
apply spy_analz
txtK2
apply blast 
txtK3
apply blast 
txtK4
apply (blast dest!: authK_not_AKcryptSK)
txtK5
apply (case_tac "Key servK  analz (spies evs5) ")
txtIf servK is compromised then the result follows directly...
apply (simp (no_asm_simp) add: analz_insert_eq Un_upper2 [THEN analz_mono, THEN subsetD])
txt...therefore servK is uncompromised.
txtThe AKcryptSK servK SK evs5 case leads to a contradiction.
apply (blast elim!: servK_not_AKcryptSK [THEN [2] rev_notE] del: allE ballE)
txtAnother K5 case
apply blast 
txtOops1
apply simp 
apply (blast dest!: AKcryptSK_analz_insert)
done

textFirst simplification law for analz: no session keys encrypt
authentication keys or shared keys.
lemma analz_insert_freshK1:
     " evs  kerbIV;  K  authKeys evs  range shrK;
        SesKey  range shrK 
       (Key K  analz (insert (Key SesKey) (spies evs))) =
          (K = SesKey | Key K  analz (spies evs))"
apply (frule authKeys_are_not_AKcryptSK, assumption)
apply (simp del: image_insert
            add: analz_image_freshK_simps add: Key_analz_image_Key)
done


textSecond simplification law for analz: no service keys encrypt any other keys.
lemma analz_insert_freshK2:
     " evs  kerbIV;  servK  (authKeys evs); servK  range shrK;
        K  symKeys 
       (Key K  analz (insert (Key servK) (spies evs))) =
          (K = servK | Key K  analz (spies evs))"
apply (frule not_authKeys_not_AKcryptSK, assumption, assumption)
apply (simp del: image_insert
            add: analz_image_freshK_simps add: Key_analz_image_Key)
done


textThird simplification law for analz: only one authentication key encrypts a certain service key.

lemma analz_insert_freshK3:
 " AKcryptSK authK servK evs;
    authK'  authK; authK'  range shrK; evs  kerbIV 
         (Key servK  analz (insert (Key authK') (spies evs))) =
                (servK = authK' | Key servK  analz (spies evs))"
apply (drule_tac authK' = authK' in not_different_AKcryptSK, blast, assumption)
apply (simp del: image_insert
            add: analz_image_freshK_simps add: Key_analz_image_Key)
done

lemma analz_insert_freshK3_bis:
 " Says Tgs A
            (Crypt authK Key servK, Agent B, Number Ts, servTicket)
         set evs; 
     authK  authK'; authK'  range shrK; evs  kerbIV 
         (Key servK  analz (insert (Key authK') (spies evs))) =
                (servK = authK' | Key servK  analz (spies evs))"
apply (frule AKcryptSKI, assumption)
apply (simp add: analz_insert_freshK3)
done

texta weakness of the protocol
lemma authK_compromises_servK:
     " Says Tgs A
              (Crypt authK Key servK, Agent B, Number Ts, servTicket)
            set evs;  authK  symKeys;
         Key authK  analz (spies evs); evs  kerbIV 
       Key servK  analz (spies evs)"
  by (metis Says_imp_analz_Spy analz.Fst analz_Decrypt')

lemma servK_notin_authKeysD:
     " Crypt authK Key servK, Agent B, Ts,
                      Crypt (shrK B) Agent A, Agent B, Key servK, Ts
            parts (spies evs);
         Key servK  analz (spies evs);
         B  Tgs; evs  kerbIV 
       servK  authKeys evs"
apply (erule rev_mp)
apply (erule rev_mp)
apply (simp add: authKeys_def)
apply (erule kerbIV.induct, analz_mono_contra)
apply (frule_tac [7] K5_msg_in_parts_spies)
apply (frule_tac [5] K3_msg_in_parts_spies, simp_all)
apply (blast+)
done


textIf Spy sees the Authentication Key sent in msg K2, then
    the Key has expired.
lemma Confidentiality_Kas_lemma [rule_format]:
     " authK  symKeys; A  bad;  evs  kerbIV 
       Says Kas A
               (Crypt (shrK A)
                  Key authK, Agent Tgs, Number Ta,
          Crypt (shrK Tgs) Agent A, Agent Tgs, Key authK, Number Ta)
             set evs 
          Key authK  analz (spies evs) 
          expiredAK Ta evs"
apply (erule kerbIV.induct)
apply (frule_tac [10] Oops_range_spies2)
apply (frule_tac [9] Oops_range_spies1)
apply (frule_tac [7] Says_tgs_message_form)
apply (frule_tac [5] Says_kas_message_form)
apply (safe del: impI conjI impCE)
apply (simp_all (no_asm_simp) add: Says_Kas_message_form less_SucI analz_insert_eq not_parts_not_analz analz_insert_freshK1 pushes)
txtFake
apply spy_analz
txtK2
apply blast
txtK4
apply blast
txtLevel 8: K5
apply (blast dest: servK_notin_authKeysD Says_Kas_message_form intro: less_SucI)
txtOops1
apply (blast dest!: unique_authKeys intro: less_SucI)
txtOops2
apply (blast dest: Says_Tgs_message_form Says_Kas_message_form)
done

lemma Confidentiality_Kas:
     " Says Kas A
              (Crypt Ka Key authK, Agent Tgs, Number Ta, authTicket)
            set evs;
         ¬ expiredAK Ta evs;
         A  bad;  evs  kerbIV 
       Key authK  analz (spies evs)"
by (blast dest: Says_Kas_message_form Confidentiality_Kas_lemma)

textIf Spy sees the Service Key sent in msg K4, then
    the Key has expired.

lemma Confidentiality_lemma [rule_format]:
     " Says Tgs A
            (Crypt authK
               Key servK, Agent B, Number Ts,
                 Crypt (shrK B) Agent A, Agent B, Key servK, Number Ts)
            set evs;
        Key authK  analz (spies evs);
        servK  symKeys;
        A  bad;  B  bad; evs  kerbIV 
       Key servK  analz (spies evs) 
          expiredSK Ts evs"
apply (erule rev_mp)
apply (erule rev_mp)
apply (erule kerbIV.induct)
apply (rule_tac [9] impI)+
  ― ‹The Oops1 case is unusual: must simplify
    termAuthkey  analz (spies (ev#evs)), not letting
   analz_mono_contra› weaken it to
   termAuthkey  analz (spies evs),
  for we then conclude termauthK  authKa.
apply analz_mono_contra
apply (frule_tac [10] Oops_range_spies2)
apply (frule_tac [9] Oops_range_spies1)
apply (frule_tac [7] Says_tgs_message_form)
apply (frule_tac [5] Says_kas_message_form)
apply (safe del: impI conjI impCE)
apply (simp_all add: less_SucI new_keys_not_analzd Says_Kas_message_form Says_Tgs_message_form analz_insert_eq not_parts_not_analz analz_insert_freshK1 analz_insert_freshK2 analz_insert_freshK3_bis pushes)
txtFake
     apply spy_analz
txtK2
    apply (blast intro: parts_insertI less_SucI)
txtK4
   apply (blast dest: authTicket_authentic Confidentiality_Kas)
txtK5
  apply (metis Says_imp_spies Says_ticket_parts Tgs_not_bad analz_insert_freshK2 
             less_SucI parts.Inj servK_notin_authKeysD unique_CryptKey)
txtOops1 
 apply (blast dest: Says_Kas_message_form Says_Tgs_message_form intro: less_SucI)
txtOops2
apply (blast dest: Says_imp_spies [THEN parts.Inj] Key_unique_SesKey intro: less_SucI)
done


textIn the real world Tgs can't check wheter authK is secure!
lemma Confidentiality_Tgs:
     " Says Tgs A
              (Crypt authK Key servK, Agent B, Number Ts, servTicket)
            set evs;
         Key authK  analz (spies evs);
         ¬ expiredSK Ts evs;
         A  bad;  B  bad; evs  kerbIV 
       Key servK  analz (spies evs)"
by (blast dest: Says_Tgs_message_form Confidentiality_lemma)

textIn the real world Tgs CAN check what Kas sends!
lemma Confidentiality_Tgs_bis:
     " Says Kas A
               (Crypt Ka Key authK, Agent Tgs, Number Ta, authTicket)
            set evs;
         Says Tgs A
              (Crypt authK Key servK, Agent B, Number Ts, servTicket)
            set evs;
         ¬ expiredAK Ta evs; ¬ expiredSK Ts evs;
         A  bad;  B  bad; evs  kerbIV 
       Key servK  analz (spies evs)"
by (blast dest!: Confidentiality_Kas Confidentiality_Tgs)

textMost general form
lemmas Confidentiality_Tgs_ter = authTicket_authentic [THEN Confidentiality_Tgs_bis]

lemmas Confidentiality_Auth_A = authK_authentic [THEN Confidentiality_Kas]

textNeeds a confidentiality guarantee, hence moved here.
      Authenticity of servK for A
lemma servK_authentic_bis_r:
     " Crypt (shrK A) Key authK, Agent Tgs, Number Ta, authTicket
            parts (spies evs);
         Crypt authK Key servK, Agent B, Number Ts, servTicket
            parts (spies evs);
         ¬ expiredAK Ta evs; A  bad; evs  kerbIV 
 Says Tgs A (Crypt authK Key servK, Agent B, Number Ts, servTicket)
        set evs"
by (blast dest: authK_authentic Confidentiality_Auth_A servK_authentic_ter)

lemma Confidentiality_Serv_A:
     " Crypt (shrK A) Key authK, Agent Tgs, Number Ta, authTicket
            parts (spies evs);
         Crypt authK Key servK, Agent B, Number Ts, servTicket
            parts (spies evs);
         ¬ expiredAK Ta evs; ¬ expiredSK Ts evs;
         A  bad;  B  bad; evs  kerbIV 
       Key servK  analz (spies evs)"
apply (drule authK_authentic, assumption, assumption)
apply (blast dest: Confidentiality_Kas Says_Kas_message_form servK_authentic_ter Confidentiality_Tgs_bis)
done

lemma Confidentiality_B:
     " Crypt (shrK B) Agent A, Agent B, Key servK, Number Ts
            parts (spies evs);
         Crypt authK Key servK, Agent B, Number Ts, servTicket
            parts (spies evs);
         Crypt (shrK A) Key authK, Agent Tgs, Number Ta, authTicket
            parts (spies evs);
         ¬ expiredSK Ts evs; ¬ expiredAK Ta evs;
         A  bad;  B  bad; B  Tgs; evs  kerbIV 
       Key servK  analz (spies evs)"
apply (frule authK_authentic)
apply (frule_tac [3] Confidentiality_Kas)
apply (frule_tac [6] servTicket_authentic, auto)
apply (blast dest!: Confidentiality_Tgs_bis dest: Says_Kas_message_form servK_authentic unique_servKeys unique_authKeys)
done
(*
The proof above is fast.  It can be done in one command in 17 secs:
apply (blast dest: authK_authentic servK_authentic
                               Says_Kas_message_form servTicket_authentic
                               unique_servKeys unique_authKeys
                               Confidentiality_Kas
                               Confidentiality_Tgs_bis)
It is very brittle: we can't use this command partway
through the script above.
*)

lemma u_Confidentiality_B:
     " Crypt (shrK B) Agent A, Agent B, Key servK, Number Ts
            parts (spies evs);
         ¬ expiredSK Ts evs;
         A  bad;  B  bad;  B  Tgs; evs  kerbIV 
       Key servK  analz (spies evs)"
by (blast dest: u_servTicket_authentic u_NotexpiredSK_NotexpiredAK Confidentiality_Tgs_bis)



subsectionParties authentication: each party verifies "the identity of
       another party who generated some data" (quoted from Neuman and Ts'o).

textThese guarantees don't assess whether two parties agree on
         the same session key: sending a message containing a key
         doesn't a priori state knowledge of the key.


textTgs_authenticates_A› can be found above

lemma A_authenticates_Tgs:
 " Says Kas A
    (Crypt (shrK A) Key authK, Agent Tgs, Number Ta, authTicket)  set evs;
     Crypt authK Key servK, Agent B, Number Ts, servTicket
        parts (spies evs);
     Key authK  analz (spies evs);
     evs  kerbIV 
  Says Tgs A (Crypt authK Key servK, Agent B, Number Ts, servTicket)
        set evs"
apply (frule Says_Kas_message_form, assumption)
apply (erule rev_mp)
apply (erule rev_mp)
apply (erule rev_mp)
apply (erule kerbIV.induct, analz_mono_contra)
apply (frule_tac [7] K5_msg_in_parts_spies)
apply (frule_tac [5] K3_msg_in_parts_spies, simp_all, blast)
txtK2
apply (blast dest!: servK_authentic Says_Tgs_message_form authKeys_used)
txtK4
apply (blast dest!: unique_CryptKey)
done


lemma B_authenticates_A:
     " Crypt servK Agent A, Number T3  parts (spies evs);
        Crypt (shrK B) Agent A, Agent B, Key servK, Number Ts
            parts (spies evs);
        Key servK  analz (spies evs);
        A  bad; B  bad; B  Tgs; evs  kerbIV 
  Says A B Crypt (shrK B) Agent A, Agent B, Key servK, Number Ts,
               Crypt servK Agent A, Number T3  set evs"
by (blast dest: servTicket_authentic_Tgs intro: Says_K5)

textThe second assumption tells B what kind of key servK is.
lemma B_authenticates_A_r:
     " Crypt servK Agent A, Number T3  parts (spies evs);
         Crypt (shrK B) Agent A, Agent B, Key servK, Number Ts
            parts (spies evs);
         Crypt authK Key servK, Agent B, Number Ts, servTicket
            parts (spies evs);
         Crypt (shrK A) Key authK, Agent Tgs, Number Ta, authTicket
            parts (spies evs);
         ¬ expiredSK Ts evs; ¬ expiredAK Ta evs;
         B  Tgs; A  bad;  B  bad;  evs  kerbIV 
    Says A B Crypt (shrK B) Agent A, Agent B, Key servK, Number Ts,
                  Crypt servK Agent A, Number T3   set evs"
by (blast intro: Says_K5 dest: Confidentiality_B servTicket_authentic_Tgs)

textu_B_authenticates_A› would be the same as B_authenticates_A› because the servK confidentiality assumption is yet unrelaxed

lemma u_B_authenticates_A_r:
     " Crypt servK Agent A, Number T3  parts (spies evs);
         Crypt (shrK B) Agent A, Agent B, Key servK, Number Ts
            parts (spies evs);
         ¬ expiredSK Ts evs;
         B  Tgs; A  bad;  B  bad;  evs  kerbIV 
    Says A B Crypt (shrK B) Agent A, Agent B, Key servK, Number Ts,
                  Crypt servK Agent A, Number T3   set evs"
by (blast intro: Says_K5 dest: u_Confidentiality_B servTicket_authentic_Tgs)

lemma A_authenticates_B:
     " Crypt servK (Number T3)  parts (spies evs);
         Crypt authK Key servK, Agent B, Number Ts, servTicket
            parts (spies evs);
         Crypt (shrK A) Key authK, Agent Tgs, Number Ta, authTicket
            parts (spies evs);
         Key authK  analz (spies evs); Key servK  analz (spies evs);
         A  bad;  B  bad; evs  kerbIV 
       Says B A (Crypt servK (Number T3))  set evs"
by (blast dest: authK_authentic servK_authentic Says_Kas_message_form Key_unique_SesKey K4_imp_K2 intro: Says_K6)

lemma A_authenticates_B_r:
     " Crypt servK (Number T3)  parts (spies evs);
         Crypt authK Key servK, Agent B, Number Ts, servTicket
            parts (spies evs);
         Crypt (shrK A) Key authK, Agent Tgs, Number Ta, authTicket
            parts (spies evs);
         ¬ expiredAK Ta evs; ¬ expiredSK Ts evs;
         A  bad;  B  bad; evs  kerbIV 
       Says B A (Crypt servK (Number T3))  set evs"
apply (frule authK_authentic)
apply (frule_tac [3] Says_Kas_message_form)
apply (frule_tac [4] Confidentiality_Kas)
apply (frule_tac [7] servK_authentic)
prefer 8 apply blast
apply (erule_tac [9] exE)
apply (frule_tac [9] K4_imp_K2)
apply assumption+
apply (blast dest: Key_unique_SesKey intro!: Says_K6 dest: Confidentiality_Tgs
)
done


subsectionKey distribution guarantees
       An agent knows a session key if he used it to issue a cipher.
       These guarantees also convey a stronger form of 
       authentication - non-injective agreement on the session key


lemma Kas_Issues_A:
   " Says Kas A (Crypt (shrK A) Key authK, Peer, Ta, authTicket)  set evs;
      evs  kerbIV 
   Kas Issues A with (Crypt (shrK A) Key authK, Peer, Ta, authTicket) 
          on evs"
apply (simp (no_asm) add: Issues_def)
apply (rule exI)
apply (rule conjI, assumption)
apply (simp (no_asm))
apply (erule rev_mp)
apply (erule kerbIV.induct)
apply (frule_tac [5] Says_ticket_parts)
apply (frule_tac [7] Says_ticket_parts)
apply (simp_all (no_asm_simp) add: all_conj_distrib)
txtK2
apply (simp add: takeWhile_tail)
apply (blast dest: authK_authentic parts_spies_takeWhile_mono [THEN subsetD] parts_spies_evs_revD2 [THEN subsetD])
done

lemma A_authenticates_and_keydist_to_Kas:
  " Crypt (shrK A) Key authK, Peer, Ta, authTicket  parts (spies evs);
     A  bad; evs  kerbIV 
  Kas Issues A with (Crypt (shrK A) Key authK, Peer, Ta, authTicket) 
          on evs"
by (blast dest: authK_authentic Kas_Issues_A)

lemma honest_never_says_newer_timestamp_in_auth:
     " (CT evs)  T; A  bad; Number T  parts {X}; evs  kerbIV  
       B Y.  Says A B Y, X  set evs"
apply (erule rev_mp)
apply (erule kerbIV.induct)
apply force+
done

lemma honest_never_says_current_timestamp_in_auth:
     " (CT evs) = T; Number T  parts {X}; evs  kerbIV  
       A B Y. A  bad  Says A B Y, X  set evs"
  by (metis eq_imp_le honest_never_says_newer_timestamp_in_auth)

lemma A_trusts_secure_authenticator:
    " Crypt K Agent A, Number T  parts (spies evs);
       Key K  analz (spies evs); evs  kerbIV 
  B X. Says A Tgs X, Crypt K Agent A, Number T, Agent B  set evs  
           Says A B X, Crypt K Agent A, Number T  set evs"
apply (erule rev_mp)
apply (erule rev_mp)
apply (erule kerbIV.induct, analz_mono_contra)
apply (frule_tac [5] Says_ticket_parts)
apply (frule_tac [7] Says_ticket_parts)
apply (simp_all add: all_conj_distrib)
apply blast+
done

lemma A_Issues_Tgs:
  " Says A Tgs authTicket, Crypt authK Agent A, Number T2, Agent B
        set evs; 
     Key authK  analz (spies evs);  
     A  bad; evs  kerbIV 
  A Issues Tgs with (Crypt authK Agent A, Number T2) on evs"
apply (simp (no_asm) add: Issues_def)
apply (rule exI)
apply (rule conjI, assumption)
apply (simp (no_asm))
apply (erule rev_mp)
apply (erule rev_mp)
apply (erule kerbIV.induct, analz_mono_contra)
apply (frule_tac [5] Says_ticket_parts)
apply (frule_tac [7] Says_ticket_parts)
apply (simp_all (no_asm_simp) add: all_conj_distrib)
txtfake
apply blast
txtK3
(*
apply clarify
apply (drule Says_imp_knows_Spy [THEN parts.Inj, THEN authK_authentic, THEN Says_Kas_message_form], assumption, assumption, assumption)
*)
apply (simp add: takeWhile_tail)
apply auto
apply (force dest!: authK_authentic Says_Kas_message_form)
apply (drule parts_spies_takeWhile_mono [THEN subsetD, THEN parts_spies_evs_revD2 [THEN subsetD]])
apply (drule A_trusts_secure_authenticator, assumption, assumption)
apply (simp add: honest_never_says_current_timestamp_in_auth)
done

lemma Tgs_authenticates_and_keydist_to_A:
  "  Crypt authK Agent A, Number T2  parts (spies evs); 
      Crypt (shrK Tgs) Agent A, Agent Tgs, Key authK, Number Ta
            parts (spies evs);
     Key authK  analz (spies evs);  
     A  bad; evs  kerbIV 
  A Issues Tgs with (Crypt authK Agent A, Number T2) on evs"
by (blast dest: A_Issues_Tgs Tgs_authenticates_A)

lemma Tgs_Issues_A:
    " Says Tgs A (Crypt authK Key servK, Agent B, Number Ts, servTicket )
          set evs; 
       Key authK  analz (spies evs);  evs  kerbIV 
   Tgs Issues A with 
          (Crypt authK Key servK, Agent B, Number Ts, servTicket ) on evs"
apply (simp (no_asm) add: Issues_def)
apply (rule exI)
apply (rule conjI, assumption)
apply (simp (no_asm))
apply (erule rev_mp)
apply (erule rev_mp)
apply (erule kerbIV.induct, analz_mono_contra)
apply (frule_tac [5] Says_ticket_parts)
apply (frule_tac [7] Says_ticket_parts)
apply (simp_all (no_asm_simp) add: all_conj_distrib)
txtK4
apply (simp add: takeWhile_tail)
(*Last two thms installed only to derive authK ∉ range shrK*)
apply (metis knows_Spy_partsEs(2) parts.Fst usedI used_evs_rev used_takeWhile_used)
done

lemma A_authenticates_and_keydist_to_Tgs:
"Crypt authK Key servK, Agent B, Number Ts, servTicket  parts (spies evs);
  Key authK  analz (spies evs); B  Tgs; evs  kerbIV 
  A. Tgs Issues A with 
          (Crypt authK Key servK, Agent B, Number Ts, servTicket ) on evs"
by (blast dest: Tgs_Issues_A servK_authentic_bis)



lemma B_Issues_A:
     " Says B A (Crypt servK (Number T3))  set evs;
         Key servK  analz (spies evs);
         A  bad;  B  bad; B  Tgs; evs  kerbIV 
       B Issues A with (Crypt servK (Number T3)) on evs"
apply (simp (no_asm) add: Issues_def)
apply (rule exI)
apply (rule conjI, assumption)
apply (simp (no_asm))
apply (erule rev_mp)
apply (erule rev_mp)
apply (erule kerbIV.induct, analz_mono_contra)
apply (frule_tac [5] Says_ticket_parts)
apply (frule_tac [7] Says_ticket_parts)
apply (simp_all (no_asm_simp) add: all_conj_distrib)
apply blast
txtK6 requires numerous lemmas
apply (simp add: takeWhile_tail)
apply (blast dest: servTicket_authentic parts_spies_takeWhile_mono [THEN subsetD] parts_spies_evs_revD2 [THEN subsetD] intro: Says_K6)
done

lemma B_Issues_A_r:
     " Says B A (Crypt servK (Number T3))  set evs;
         Crypt (shrK B) Agent A, Agent B, Key servK, Number Ts
             parts (spies evs);
         Crypt authK Key servK, Agent B, Number Ts, servTicket
             parts (spies evs);
         Crypt (shrK A) Key authK, Agent Tgs, Number Ta, authTicket
            parts (spies evs);
         ¬ expiredSK Ts evs; ¬ expiredAK Ta evs;
         A  bad;  B  bad; B  Tgs; evs  kerbIV 
       B Issues A with (Crypt servK (Number T3)) on evs"
by (blast dest!: Confidentiality_B B_Issues_A)

lemma u_B_Issues_A_r:
     " Says B A (Crypt servK (Number T3))  set evs;
         Crypt (shrK B) Agent A, Agent B, Key servK, Number Ts
             parts (spies evs);
         ¬ expiredSK Ts evs;
         A  bad;  B  bad; B  Tgs; evs  kerbIV 
       B Issues A with (Crypt servK (Number T3)) on evs"
by (blast dest!: u_Confidentiality_B B_Issues_A)

lemma A_authenticates_and_keydist_to_B:
     " Crypt servK (Number T3)  parts (spies evs);
         Crypt authK Key servK, Agent B, Number Ts, servTicket
            parts (spies evs);
         Crypt (shrK A) Key authK, Agent Tgs, Number Ta, authTicket
            parts (spies evs);
         Key authK  analz (spies evs); Key servK  analz (spies evs);
         A  bad;  B  bad; B  Tgs; evs  kerbIV 
       B Issues A with (Crypt servK (Number T3)) on evs"
by (blast dest!: A_authenticates_B B_Issues_A)

lemma A_authenticates_and_keydist_to_B_r:
     " Crypt servK (Number T3)  parts (spies evs);
         Crypt authK Key servK, Agent B, Number Ts, servTicket
            parts (spies evs);
         Crypt (shrK A) Key authK, Agent Tgs, Number Ta, authTicket
            parts (spies evs);
         ¬ expiredAK Ta evs; ¬ expiredSK Ts evs;
         A  bad;  B  bad; B  Tgs; evs  kerbIV 
       B Issues A with (Crypt servK (Number T3)) on evs"
by (blast dest!: A_authenticates_B_r Confidentiality_Serv_A B_Issues_A)


lemma A_Issues_B:
     " Says A B servTicket, Crypt servK Agent A, Number T3
            set evs;
         Key servK  analz (spies evs);
         B  Tgs; A  bad;  B  bad;  evs  kerbIV 
    A Issues B with (Crypt servK Agent A, Number T3) on evs"
apply (simp (no_asm) add: Issues_def)
apply (rule exI)
apply (rule conjI, assumption)
apply (simp (no_asm))
apply (erule rev_mp)
apply (erule rev_mp)
apply (erule kerbIV.induct, analz_mono_contra)
apply (frule_tac [5] Says_ticket_parts)
apply (frule_tac [7] Says_ticket_parts)
apply (simp_all (no_asm_simp))
apply clarify
txtK5
apply auto
apply (simp add: takeWhile_tail)
txtLevel 15: case analysis necessary because the assumption doesn't state
  the form of servTicket. The guarantee becomes stronger.
apply (blast dest: Says_imp_spies [THEN analz.Inj, THEN analz_Decrypt']
                   K3_imp_K2 servK_authentic_ter
                   parts_spies_takeWhile_mono [THEN subsetD]
                   parts_spies_evs_revD2 [THEN subsetD]
             intro: Says_K5)
apply (simp add: takeWhile_tail)
done

lemma A_Issues_B_r:
     " Says A B servTicket, Crypt servK Agent A, Number T3
            set evs;
         Crypt (shrK A) Key authK, Agent Tgs, Number Ta, authTicket
            parts (spies evs);
         Crypt authK Key servK, Agent B, Number Ts, servTicket
            parts (spies evs);
         ¬ expiredAK Ta evs; ¬ expiredSK Ts evs;
         B  Tgs; A  bad;  B  bad;  evs  kerbIV 
    A Issues B with (Crypt servK Agent A, Number T3) on evs"
by (blast dest!: Confidentiality_Serv_A A_Issues_B)

lemma B_authenticates_and_keydist_to_A:
     " Crypt servK Agent A, Number T3  parts (spies evs);
         Crypt (shrK B) Agent A, Agent B, Key servK, Number Ts
            parts (spies evs);
         Key servK  analz (spies evs);
         B  Tgs; A  bad;  B  bad;  evs  kerbIV 
    A Issues B with (Crypt servK Agent A, Number T3) on evs"
by (blast dest: B_authenticates_A A_Issues_B)

lemma B_authenticates_and_keydist_to_A_r:
     " Crypt servK Agent A, Number T3  parts (spies evs);
         Crypt (shrK B) Agent A, Agent B, Key servK, Number Ts
            parts (spies evs);
         Crypt authK Key servK, Agent B, Number Ts, servTicket
            parts (spies evs);
         Crypt (shrK A) Key authK, Agent Tgs, Number Ta, authTicket
            parts (spies evs);
         ¬ expiredSK Ts evs; ¬ expiredAK Ta evs;
         B  Tgs; A  bad;  B  bad;  evs  kerbIV 
    A Issues B with (Crypt servK Agent A, Number T3) on evs"
by (blast dest: B_authenticates_A Confidentiality_B A_Issues_B)

textu_B_authenticates_and_keydist_to_A› would be the same as B_authenticates_and_keydist_to_A› because the
 servK confidentiality assumption is yet unrelaxed

lemma u_B_authenticates_and_keydist_to_A_r:
     " Crypt servK Agent A, Number T3  parts (spies evs);
         Crypt (shrK B) Agent A, Agent B, Key servK, Number Ts
            parts (spies evs);
         ¬ expiredSK Ts evs;
         B  Tgs; A  bad;  B  bad;  evs  kerbIV 
    A Issues B with (Crypt servK Agent A, Number T3) on evs"
by (blast dest: u_B_authenticates_A_r u_Confidentiality_B A_Issues_B)

end