# Theory Kerberos_BAN

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theory Kerberos_BAN
imports Public
begin

    (*  Title:      HOL/Auth/Kerberos_BAN
        ID:         $Id: Kerberos_BAN.thy,v 1.12 2005/06/17 14:13:06 haftmann Exp $
        Author:     Giampaolo Bella, Cambridge University Computer Laboratory
        Copyright   1998  University of Cambridge

    Tidied and converted to Isar by lcp.
    *)

    header{*The Kerberos Protocol, BAN Version*}

    theory Kerberos_BAN imports Public begin

    text{*From page 251 of
      Burrows, Abadi and Needham (1989).  A Logic of Authentication.
      Proc. Royal Soc. 426

      Confidentiality (secrecy) and authentication properties rely on
      temporal checks: strong guarantees in a little abstracted - but
      very realistic - model.
    *}

    (* Temporal modelization: session keys can be leaked
                              ONLY when they have expired *)

    syntax
        CT :: "event list=>nat"
        Expired :: "[nat, event list] => bool"
        RecentAuth :: "[nat, event list] => bool"

    consts

        (*Duration of the session key*)
        SesKeyLife   :: nat

        (*Duration of the authenticator*)
        AutLife :: nat

    text{*The ticket should remain fresh for two journeys on the network at least*}
    specification (SesKeyLife)
      SesKeyLife_LB [iff]: "2 ≤ SesKeyLife"
        by blast

    text{*The authenticator only for one journey*}
    specification (AutLife)
      AutLife_LB [iff]:    "Suc 0 ≤ AutLife"
        by blast

    translations
       "CT" == "length "

       "Expired T evs" == "SesKeyLife + T < CT evs"

       "RecentAuth T evs" == "CT evs ≤ AutLife + T"

    consts  kerberos_ban   :: "event list set"
    inductive "kerberos_ban"
     intros

       Nil:  "[] ∈ kerberos_ban"

       Fake: "[| evsf ∈ kerberos_ban;  X ∈ synth (analz (spies evsf)) |]
              ==> Says Spy B X # evsf ∈ kerberos_ban"

       Kb1:  "[| evs1 ∈ kerberos_ban |]
              ==> Says A Server {|Agent A, Agent B|} # evs1
                    ∈  kerberos_ban"

       Kb2:  "[| evs2 ∈ kerberos_ban;  Key KAB ∉ used evs2;  KAB ∈ symKeys;
                 Says A' Server {|Agent A, Agent B|} ∈ set evs2 |]
              ==> Says Server A
                    (Crypt (shrK A)
                       {|Number (CT evs2), Agent B, Key KAB,
                        (Crypt (shrK B) {|Number (CT evs2), Agent A, Key KAB|})|})
                    # evs2 ∈ kerberos_ban"

       Kb3:  "[| evs3 ∈ kerberos_ban;
                 Says S A (Crypt (shrK A) {|Number Ts, Agent B, Key K, X|})
                   ∈ set evs3;
                 Says A Server {|Agent A, Agent B|} ∈ set evs3;
                 ~ Expired Ts evs3 |]
              ==> Says A B {|X, Crypt K {|Agent A, Number (CT evs3)|} |}
                   # evs3 ∈ kerberos_ban"

       Kb4:  "[| evs4 ∈ kerberos_ban;
                 Says A' B {|(Crypt (shrK B) {|Number Ts, Agent A, Key K|}),
                             (Crypt K {|Agent A, Number Ta|}) |}: set evs4;
                 ~ Expired Ts evs4;  RecentAuth Ta evs4 |]
              ==> Says B A (Crypt K (Number Ta)) # evs4
                    ∈ kerberos_ban"

            (*Old session keys may become compromised*)
       Oops: "[| evso ∈ kerberos_ban;
                 Says Server A (Crypt (shrK A) {|Number Ts, Agent B, Key K, X|})
                   ∈ set evso;
                 Expired Ts evso |]
              ==> Notes Spy {|Number Ts, Key K|} # evso ∈ kerberos_ban"

    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]

    text{*A "possibility property": there are traces that reach the end.*}
    lemma "[|Key K ∉ used []; K ∈ symKeys|]
           ==> ∃Timestamp. ∃evs ∈ kerberos_ban.
                 Says B A (Crypt K (Number Timestamp))
                      ∈ set evs"
    apply (cut_tac SesKeyLife_LB)
    apply (intro exI bexI)
    apply (rule_tac [2]
               kerberos_ban.Nil [THEN kerberos_ban.Kb1, THEN kerberos_ban.Kb2,
                                 THEN kerberos_ban.Kb3, THEN kerberos_ban.Kb4])
    apply (possibility, simp_all (no_asm_simp) add: used_Cons)
    done

    (**** Inductive proofs about kerberos_ban ****)

    text{*Forwarding Lemma for reasoning about the encrypted portion of message Kb3*}
    lemma Kb3_msg_in_parts_spies:
         "Says S A (Crypt KA {|Timestamp, B, K, X|}) ∈ set evs
          ==> X ∈ parts (spies evs)"
    by blast

    lemma Oops_parts_spies:
         "Says Server A (Crypt (shrK A) {|Timestamp, B, K, X|}) ∈ set evs
          ==> K ∈ parts (spies evs)"
    by blast

    text{*Spy never sees another agent's shared key! (unless it's bad at start)*}
    lemma Spy_see_shrK [simp]:
         "evs ∈ kerberos_ban ==> (Key (shrK A) ∈ parts (spies evs)) = (A ∈ bad)"
    apply (erule kerberos_ban.induct)
    apply (frule_tac [7] Oops_parts_spies)
    apply (frule_tac [5] Kb3_msg_in_parts_spies, simp_all, blast+)
    done

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

    lemma Spy_see_shrK_D [dest!]:
         "[| Key (shrK A) ∈ parts (spies evs);
                    evs ∈ kerberos_ban |] ==> A:bad"
    by (blast dest: Spy_see_shrK)

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

    text{*Nobody can have used non-existent keys!*}
    lemma new_keys_not_used [simp]:
        "[|Key K ∉ used evs; K ∈ symKeys; evs ∈ kerberos_ban|]
         ==> K ∉ keysFor (parts (spies evs))"
    apply (erule rev_mp)
    apply (erule kerberos_ban.induct)
    apply (frule_tac [7] Oops_parts_spies)
    apply (frule_tac [5] Kb3_msg_in_parts_spies, simp_all)
    txt{*Fake*}
    apply (force dest!: keysFor_parts_insert)
    txt{*Kb2, Kb3, Kb4*}
    apply (force dest!: analz_shrK_Decrypt)+
    done

    subsection{* Lemmas concerning the form of items passed in messages *}

    text{*Describes the form of K, X and K' when the Server sends this message.*}
    lemma Says_Server_message_form:
         "[| Says Server A (Crypt K' {|Number Ts, Agent B, Key K, X|})
             ∈ set evs; evs ∈ kerberos_ban |]
          ==> K ∉ range shrK &
              X = (Crypt (shrK B) {|Number Ts, Agent A, Key K|}) &
              K' = shrK A"
    apply (erule rev_mp)
    apply (erule kerberos_ban.induct, auto)
    done

    text{*If the encrypted message appears then it originated with the Server
      PROVIDED that A is NOT compromised!

      This shows implicitly the FRESHNESS OF THE SESSION KEY to A
    *}
    lemma A_trusts_K_by_Kb2:
         "[| Crypt (shrK A) {|Number Ts, Agent B, Key K, X|}
               ∈ parts (spies evs);
             A ∉ bad;  evs ∈ kerberos_ban |]
           ==> Says Server A (Crypt (shrK A) {|Number Ts, Agent B, Key K, X|})
                 ∈ set evs"
    apply (erule rev_mp)
    apply (erule kerberos_ban.induct)
    apply (frule_tac [7] Oops_parts_spies)
    apply (frule_tac [5] Kb3_msg_in_parts_spies, simp_all, blast)
    done

    text{*If the TICKET appears then it originated with the Server*}
    text{*FRESHNESS OF THE SESSION KEY to B*}
    lemma B_trusts_K_by_Kb3:
         "[| Crypt (shrK B) {|Number Ts, Agent A, Key K|} ∈ parts (spies evs);
             B ∉ bad;  evs ∈ kerberos_ban |]
           ==> Says Server A
                (Crypt (shrK A) {|Number Ts, Agent B, Key K,
                              Crypt (shrK B) {|Number Ts, Agent A, Key K|}|})
               ∈ set evs"
    apply (erule rev_mp)
    apply (erule kerberos_ban.induct)
    apply (frule_tac [7] Oops_parts_spies)
    apply (frule_tac [5] Kb3_msg_in_parts_spies, simp_all, blast)
    done

    text{*EITHER describes the form of X when the following message is sent,
      OR     reduces it to the Fake case.
      Use @{text Says_Server_message_form} if applicable.*}
    lemma Says_S_message_form:
         "[| Says S A (Crypt (shrK A) {|Number Ts, Agent B, Key K, X|})
                ∈ set evs;
             evs ∈ kerberos_ban |]
     ==> (K ∉ range shrK & X = (Crypt (shrK B) {|Number Ts, Agent A, Key K|}))
              | X ∈ analz (spies evs)"
    apply (case_tac "A ∈ bad")
    apply (force dest!: Says_imp_spies [THEN analz.Inj])
    apply (frule Says_imp_spies [THEN parts.Inj])
    apply (blast dest!: A_trusts_K_by_Kb2 Says_Server_message_form)
    done

    (****
     The following is to prove theorems of the form

      Key K ∈ analz (insert (Key KAB) (spies evs)) ==>
      Key K ∈ analz (spies evs)

     A more general formula must be proved inductively.

    ****)

    text{* Session keys are not used to encrypt other session keys *}
    lemma analz_image_freshK [rule_format (no_asm)]:
         "evs ∈ kerberos_ban ==>
       ∀K KK. KK ⊆ - (range shrK) -->
              (Key K ∈ analz (Key`KK Un (spies evs))) =
              (K ∈ KK | Key K ∈ analz (spies evs))"
    apply (erule kerberos_ban.induct)
    apply (drule_tac [7] Says_Server_message_form)
    apply (erule_tac [5] Says_S_message_form [THEN disjE], analz_freshK, spy_analz, auto)
    done

    lemma analz_insert_freshK:
         "[| evs ∈ kerberos_ban;  KAB ∉ range shrK |] ==>
          (Key K ∈ analz (insert (Key KAB) (spies evs))) =
          (K = KAB | Key K ∈ analz (spies evs))"
    by (simp only: analz_image_freshK analz_image_freshK_simps)

    text{* The session key K uniquely identifies the message *}
    lemma unique_session_keys:
         "[| Says Server A
               (Crypt (shrK A) {|Number Ts, Agent B, Key K, X|}) ∈ set evs;
             Says Server A'
              (Crypt (shrK A') {|Number Ts', Agent B', Key K, X'|}) ∈ set evs;
             evs ∈ kerberos_ban |] ==> A=A' & Ts=Ts' & B=B' & X = X'"
    apply (erule rev_mp)
    apply (erule rev_mp)
    apply (erule kerberos_ban.induct)
    apply (frule_tac [7] Oops_parts_spies)
    apply (frule_tac [5] Kb3_msg_in_parts_spies, simp_all)
    txt{*Kb2: it can't be a new key*}
    apply blast
    done

    text{* Lemma: the session key sent in msg Kb2 would be EXPIRED
        if the spy could see it! *}

    lemma lemma2 [rule_format (no_asm)]:
         "[| A ∉ bad;  B ∉ bad;  evs ∈ kerberos_ban |]
      ==> Says Server A
              (Crypt (shrK A) {|Number Ts, Agent B, Key K,
                                Crypt (shrK B) {|Number Ts, Agent A, Key K|}|})
             ∈ set evs -->
          Key K ∈ analz (spies evs) --> Expired Ts evs"
    apply (erule kerberos_ban.induct)
    apply (frule_tac [7] Says_Server_message_form)
    apply (frule_tac [5] Says_S_message_form [THEN disjE])
    apply (simp_all (no_asm_simp) add: less_SucI analz_insert_eq analz_insert_freshK pushes)
    txt{*Fake*}
    apply spy_analz
    txt{*Kb2*}
    apply (blast intro: parts_insertI less_SucI)
    txt{*Kb3*}
    apply (case_tac "Aa ∈ bad")
     prefer 2 apply (blast dest: A_trusts_K_by_Kb2 unique_session_keys)
    apply (blast dest: Says_imp_spies [THEN analz.Inj] Crypt_Spy_analz_bad elim!: MPair_analz intro: less_SucI)
    txt{*Oops: PROOF FAILED if addIs below*}
    apply (blast dest: unique_session_keys intro!: less_SucI)
    done

    text{*Confidentiality for the Server: Spy does not see the keys sent in msg Kb2
    as long as they have not expired.*}
    lemma Confidentiality_S:
         "[| Says Server A
              (Crypt K' {|Number T, Agent B, Key K, X|}) ∈ set evs;
             ~ Expired T evs;
             A ∉ bad;  B ∉ bad;  evs ∈ kerberos_ban
          |] ==> Key K ∉ analz (spies evs)"
    apply (frule Says_Server_message_form, assumption)
    apply (blast intro: lemma2)
    done

    (**** THE COUNTERPART OF CONFIDENTIALITY
          [|...; Expired Ts evs; ...|] ==> Key K ∈ analz (spies evs)
          WOULD HOLD ONLY IF AN OOPS OCCURRED! ---> Nothing to prove!   ****)

    text{*Confidentiality for Alice*}
    lemma Confidentiality_A:
         "[| Crypt (shrK A) {|Number T, Agent B, Key K, X|} ∈ parts (spies evs);
             ~ Expired T evs;
             A ∉ bad;  B ∉ bad;  evs ∈ kerberos_ban
          |] ==> Key K ∉ analz (spies evs)"
    by (blast dest!: A_trusts_K_by_Kb2 Confidentiality_S)

    text{*Confidentiality for Bob*}
    lemma Confidentiality_B:
         "[| Crypt (shrK B) {|Number Tk, Agent A, Key K|}
              ∈ parts (spies evs);
            ~ Expired Tk evs;
            A ∉ bad;  B ∉ bad;  evs ∈ kerberos_ban
          |] ==> Key K ∉ analz (spies evs)"
    by (blast dest!: B_trusts_K_by_Kb3 Confidentiality_S)

    lemma lemma_B [rule_format]:
         "[| B ∉ bad;  evs ∈ kerberos_ban |]
          ==> Key K ∉ analz (spies evs) -->
              Says Server A (Crypt (shrK A) {|Number Ts, Agent B, Key K, X|})
              ∈ set evs -->
              Crypt K (Number Ta) ∈ parts (spies evs) -->
              Says B A (Crypt K (Number Ta)) ∈ set evs"
    apply (erule kerberos_ban.induct)
    apply (frule_tac [7] Oops_parts_spies)
    apply (frule_tac [5] Says_S_message_form)
    apply (drule_tac [6] Kb3_msg_in_parts_spies, analz_mono_contra)
    apply (simp_all (no_asm_simp) add: all_conj_distrib)
    txt{*Fake*}
    apply blast
    txt{*Kb2*}
    apply (force dest: Crypt_imp_invKey_keysFor)
    txt{*Kb4*}
    apply (blast dest: B_trusts_K_by_Kb3 unique_session_keys
                       Says_imp_spies [THEN analz.Inj] Crypt_Spy_analz_bad)
    done

    text{*Authentication of B to A*}
    lemma Authentication_B:
         "[| Crypt K (Number Ta) ∈ parts (spies evs);
             Crypt (shrK A) {|Number Ts, Agent B, Key K, X|}
             ∈ parts (spies evs);
             ~ Expired Ts evs;
             A ∉ bad;  B ∉ bad;  evs ∈ kerberos_ban |]
          ==> Says B A (Crypt K (Number Ta)) ∈ set evs"
    by (blast dest!: A_trusts_K_by_Kb2
              intro!: lemma_B elim!: Confidentiality_S [THEN [2] rev_notE])

    lemma lemma_A [rule_format]:
         "[| A ∉ bad; B ∉ bad; evs ∈ kerberos_ban |]
          ==>
             Key K ∉ analz (spies evs) -->
             Says Server A (Crypt (shrK A) {|Number Ts, Agent B, Key K, X|})
             ∈ set evs -->
              Crypt K {|Agent A, Number Ta|} ∈ parts (spies evs) -->
             Says A B {|X, Crypt K {|Agent A, Number Ta|}|}
                 ∈ set evs"
    apply (erule kerberos_ban.induct)
    apply (frule_tac [7] Oops_parts_spies)
    apply (frule_tac [5] Says_S_message_form)
    apply (frule_tac [6] Kb3_msg_in_parts_spies, analz_mono_contra)
    apply (simp_all (no_asm_simp) add: all_conj_distrib)
    txt{*Fake*}
    apply blast
    txt{*Kb2*}
    apply (force dest: Crypt_imp_invKey_keysFor)
    txt{*Kb3*}
    apply (blast dest: A_trusts_K_by_Kb2 unique_session_keys)
    done

    text{*Authentication of A to B*}
    lemma Authentication_A:
         "[| Crypt K {|Agent A, Number Ta|} ∈ parts (spies evs);
             Crypt (shrK B) {|Number Ts, Agent A, Key K|}
             ∈ parts (spies evs);
             ~ Expired Ts evs;
             A ∉ bad;  B ∉ bad;  evs ∈ kerberos_ban |]
          ==> Says A B {|Crypt (shrK B) {|Number Ts, Agent A, Key K|},
                         Crypt K {|Agent A, Number Ta|}|} ∈ set evs"
    by (blast dest!: B_trusts_K_by_Kb3
              intro!: lemma_A
              elim!: Confidentiality_S [THEN [2] rev_notE])

    end

lemma

      [| Key K ∉ used []; K ∈ symKeys |]
      ==> ∃Timestamp.
             ∃evs∈kerberos_ban. Says B A (Crypt K (Number Timestamp)) ∈ set evs

lemma Kb3_msg_in_parts_spies:

      Says S A (Crypt KA {|Timestamp, B, K, X|}) ∈ set evs
      ==> X ∈ parts (knows Spy evs)

lemma Oops_parts_spies:

      Says Server A (Crypt (shrK A) {|Timestamp, B, K, X|}) ∈ set evs
      ==> K ∈ parts (knows Spy evs)

lemma Spy_see_shrK:

      evs ∈ kerberos_ban ==> (Key (shrK A) ∈ parts (knows Spy evs)) = (A ∈ bad)

lemma Spy_analz_shrK:

      evs ∈ kerberos_ban ==> (Key (shrK A) ∈ analz (knows Spy evs)) = (A ∈ bad)

lemma Spy_see_shrK_D:

      [| Key (shrK A) ∈ parts (knows Spy evs); evs ∈ kerberos_ban |] ==> A ∈ bad

lemmas Spy_analz_shrK_D:

      [| Key (shrK A) ∈ analz (knows Spy evs); evs ∈ kerberos_ban |] ==> A ∈ bad

lemmas Spy_analz_shrK_D:

      [| Key (shrK A) ∈ analz (knows Spy evs); evs ∈ kerberos_ban |] ==> A ∈ bad

lemma new_keys_not_used:

      [| Key K ∉ used evs; K ∈ symKeys; evs ∈ kerberos_ban |]
      ==> K ∉ keysFor (parts (knows Spy evs))

### Lemmas concerning the form of items passed in messages

lemma Says_Server_message_form:

      [| Says Server A (Crypt K' {|Number Ts, Agent B, Key K, X|}) ∈ set evs;
         evs ∈ kerberos_ban |]
      ==> K ∉ range shrK ∧
          X = Crypt (shrK B) {|Number Ts, Agent A, Key K|} ∧ K' = shrK A

lemma A_trusts_K_by_Kb2:

      [| Crypt (shrK A) {|Number Ts, Agent B, Key K, X|} ∈ parts (knows Spy evs);
         A ∉ bad; evs ∈ kerberos_ban |]
      ==> Says Server A (Crypt (shrK A) {|Number Ts, Agent B, Key K, X|}) ∈ set evs

lemma B_trusts_K_by_Kb3:

      [| Crypt (shrK B) {|Number Ts, Agent A, Key K|} ∈ parts (knows Spy evs);
         B ∉ bad; evs ∈ kerberos_ban |]
      ==> Says Server A
           (Crypt (shrK A)
             {|Number Ts, Agent B, Key K,
               Crypt (shrK B) {|Number Ts, Agent A, Key K|}|})
          ∈ set evs

lemma Says_S_message_form:

      [| Says S A (Crypt (shrK A) {|Number Ts, Agent B, Key K, X|}) ∈ set evs;
         evs ∈ kerberos_ban |]
      ==> K ∉ range shrK ∧ X = Crypt (shrK B) {|Number Ts, Agent A, Key K|} ∨
          X ∈ analz (knows Spy evs)

lemma analz_image_freshK:

      [| evs ∈ kerberos_ban; KK ⊆ - range shrK |]
      ==> (Key K ∈ analz (Key ` KK ∪ knows Spy evs)) =
          (K ∈ KK ∨ Key K ∈ analz (knows Spy evs))

lemma analz_insert_freshK:

      [| evs ∈ kerberos_ban; KAB ∉ range shrK |]
      ==> (Key K ∈ analz (insert (Key KAB) (knows Spy evs))) =
          (K = KAB ∨ Key K ∈ analz (knows Spy evs))

lemma unique_session_keys:

      [| Says Server A (Crypt (shrK A) {|Number Ts, Agent B, Key K, X|}) ∈ set evs;
         Says Server A' (Crypt (shrK A') {|Number Ts', Agent B', Key K, X'|})
         ∈ set evs;
         evs ∈ kerberos_ban |]
      ==> A = A' ∧ Ts = Ts' ∧ B = B' ∧ X = X'

lemma lemma2:

      [| A ∉ bad; B ∉ bad; evs ∈ kerberos_ban;
         Says Server A
          (Crypt (shrK A)
            {|Number Ts, Agent B, Key K,
              Crypt (shrK B) {|Number Ts, Agent A, Key K|}|})
         ∈ set evs;
         Key K ∈ analz (knows Spy evs) |]
      ==> Expired Ts evs

lemma Confidentiality_S:

      [| Says Server A (Crypt K' {|Number T, Agent B, Key K, X|}) ∈ set evs;
         ¬ Expired T evs; A ∉ bad; B ∉ bad; evs ∈ kerberos_ban |]
      ==> Key K ∉ analz (knows Spy evs)

lemma Confidentiality_A:

      [| Crypt (shrK A) {|Number T, Agent B, Key K, X|} ∈ parts (knows Spy evs);
         ¬ Expired T evs; A ∉ bad; B ∉ bad; evs ∈ kerberos_ban |]
      ==> Key K ∉ analz (knows Spy evs)

lemma Confidentiality_B:

      [| Crypt (shrK B) {|Number Tk, Agent A, Key K|} ∈ parts (knows Spy evs);
         ¬ Expired Tk evs; A ∉ bad; B ∉ bad; evs ∈ kerberos_ban |]
      ==> Key K ∉ analz (knows Spy evs)

lemma lemma_B:

      [| B ∉ bad; evs ∈ kerberos_ban; Key K ∉ analz (knows Spy evs);
         Says Server A (Crypt (shrK A) {|Number Ts, Agent B, Key K, X|}) ∈ set evs;
         Crypt K (Number Ta) ∈ parts (knows Spy evs) |]
      ==> Says B A (Crypt K (Number Ta)) ∈ set evs

lemma Authentication_B:

      [| Crypt K (Number Ta) ∈ parts (knows Spy evs);
         Crypt (shrK A) {|Number Ts, Agent B, Key K, X|} ∈ parts (knows Spy evs);
         ¬ Expired Ts evs; A ∉ bad; B ∉ bad; evs ∈ kerberos_ban |]
      ==> Says B A (Crypt K (Number Ta)) ∈ set evs

lemma lemma_A:

      [| A ∉ bad; B ∉ bad; evs ∈ kerberos_ban; Key K ∉ analz (knows Spy evs);
         Says Server A (Crypt (shrK A) {|Number Ts, Agent B, Key K, X|}) ∈ set evs;
         Crypt K {|Agent A, Number Ta|} ∈ parts (knows Spy evs) |]
      ==> Says A B {|X, Crypt K {|Agent A, Number Ta|}|} ∈ set evs

lemma Authentication_A:

      [| Crypt K {|Agent A, Number Ta|} ∈ parts (knows Spy evs);
         Crypt (shrK B) {|Number Ts, Agent A, Key K|} ∈ parts (knows Spy evs);
         ¬ Expired Ts evs; A ∉ bad; B ∉ bad; evs ∈ kerberos_ban |]
      ==> Says A B
           {|Crypt (shrK B) {|Number Ts, Agent A, Key K|},
             Crypt K {|Agent A, Number Ta|}|}
          ∈ set evs