# Theory EventSET

Up to index of Isabelle/HOL/SET-Protocol

theory EventSET
imports MessageSET
begin

    (*  Title:      HOL/Auth/SET/EventSET
        ID:         $Id: EventSET.thy,v 1.3 2005/06/17 14:13:10 haftmann Exp $
        Authors:     Giampaolo Bella, Fabio Massacci, Lawrence C Paulson
    *)

    header{*Theory of Events for SET*}

    theory EventSET imports MessageSET begin

    text{*The Root Certification Authority*}
    syntax        RCA :: agent
    translations "RCA" == "CA 0"

    text{*Message events*}
    datatype
      event = Says  agent agent msg
            | Gets  agent       msg
            | Notes agent       msg

    text{*compromised agents: keys known, Notes visible*}
    consts bad :: "agent set"

    text{*Spy has access to his own key for spoof messages, but RCA is secure*}
    specification (bad)
      Spy_in_bad     [iff]: "Spy ∈ bad"
      RCA_not_bad [iff]: "RCA ∉ bad"
        by (rule exI [of _ "{Spy}"], simp)

    subsection{*Agents' Knowledge*}

    consts  (*Initial states of agents -- parameter of the construction*)
      initState :: "agent => msg set"
      knows  :: "[agent, event list] => msg set"

    (* Message reception does not extend spy's knowledge because of
       reception invariant enforced by Reception rule in protocol definition*)
    primrec

    knows_Nil:
      "knows A []       = initState A"
    knows_Cons:
        "knows A (ev # evs) =
           (if A = Spy then
            (case ev of
               Says A' B X => insert X (knows Spy evs)
             | Gets A' X => knows Spy evs
             | Notes A' X  =>
                 if A' ∈ bad then insert X (knows Spy evs) else knows Spy evs)
            else
            (case ev of
               Says A' B X =>
                 if A'=A then insert X (knows A evs) else knows A evs
             | Gets A' X    =>
                 if A'=A then insert X (knows A evs) else knows A evs
             | Notes A' X    =>
                 if A'=A then insert X (knows A evs) else knows A evs))"

    subsection{*Used Messages*}

    consts
      (*Set of items that might be visible to somebody:
        complement of the set of fresh items*)
      used :: "event list => msg set"

    (* As above, message reception does extend used items *)
    primrec
      used_Nil:  "used []         = (UN B. parts (initState B))"
      used_Cons: "used (ev # evs) =
                     (case ev of
                        Says A B X => parts {X} Un (used evs)
                      | Gets A X   => used evs
                      | Notes A X  => parts {X} Un (used evs))"

    (* Inserted by default but later removed.  This declaration lets the file
    be re-loaded. Addsimps [knows_Cons, used_Nil, *)

    (** Simplifying   parts (insert X (knows Spy evs))
          = parts {X} Un parts (knows Spy evs) -- since general case loops*)

    lemmas parts_insert_knows_A = parts_insert [of _ "knows A evs", standard]

    lemma knows_Spy_Says [simp]:
         "knows Spy (Says A B X # evs) = insert X (knows Spy evs)"
    by auto

    text{*Letting the Spy see "bad" agents' notes avoids redundant case-splits
          on whether @{term "A=Spy"} and whether @{term "A∈bad"}*}
    lemma knows_Spy_Notes [simp]:
         "knows Spy (Notes A X # evs) =
              (if A:bad then insert X (knows Spy evs) else knows Spy evs)"
    apply auto
    done

    lemma knows_Spy_Gets [simp]: "knows Spy (Gets A X # evs) = knows Spy evs"
    by auto

    lemma initState_subset_knows: "initState A <= knows A evs"
    apply (induct_tac "evs")
    apply (auto split: event.split)
    done

    lemma knows_Spy_subset_knows_Spy_Says:
         "knows Spy evs <= knows Spy (Says A B X # evs)"
    by auto

    lemma knows_Spy_subset_knows_Spy_Notes:
         "knows Spy evs <= knows Spy (Notes A X # evs)"
    by auto

    lemma knows_Spy_subset_knows_Spy_Gets:
         "knows Spy evs <= knows Spy (Gets A X # evs)"
    by auto

    (*Spy sees what is sent on the traffic*)
    lemma Says_imp_knows_Spy [rule_format]:
         "Says A B X ∈ set evs --> X ∈ knows Spy evs"
    apply (induct_tac "evs")
    apply (auto split: event.split)
    done

    (*Use with addSEs to derive contradictions from old Says events containing
      items known to be fresh*)
    lemmas knows_Spy_partsEs =
         Says_imp_knows_Spy [THEN parts.Inj, THEN revcut_rl, standard]
         parts.Body [THEN revcut_rl, standard]

    subsection{*The Function @{term used}*}

    lemma parts_knows_Spy_subset_used: "parts (knows Spy evs) <= used evs"
    apply (induct_tac "evs")
    apply (auto simp add: parts_insert_knows_A split: event.split)
    done

    lemmas usedI = parts_knows_Spy_subset_used [THEN subsetD, intro]

    lemma initState_subset_used: "parts (initState B) <= used evs"
    apply (induct_tac "evs")
    apply (auto split: event.split)
    done

    lemmas initState_into_used = initState_subset_used [THEN subsetD]

    lemma used_Says [simp]: "used (Says A B X # evs) = parts{X} Un used evs"
    by auto

    lemma used_Notes [simp]: "used (Notes A X # evs) = parts{X} Un used evs"
    by auto

    lemma used_Gets [simp]: "used (Gets A X # evs) = used evs"
    by auto

    lemma Notes_imp_parts_subset_used [rule_format]:
         "Notes A X ∈ set evs --> parts {X} <= used evs"
    apply (induct_tac "evs")
    apply (induct_tac [2] "a", auto)
    done

    text{*NOTE REMOVAL--laws above are cleaner, as they don't involve "case"*}
    declare knows_Cons [simp del]
            used_Nil [simp del] used_Cons [simp del]

    text{*For proving theorems of the form @{term "X ∉ analz (knows Spy evs) --> P"}
      New events added by induction to "evs" are discarded.  Provided
      this information isn't needed, the proof will be much shorter, since
      it will omit complicated reasoning about @{term analz}.*}

    lemmas analz_mono_contra =
           knows_Spy_subset_knows_Spy_Says [THEN analz_mono, THEN contra_subsetD]
           knows_Spy_subset_knows_Spy_Notes [THEN analz_mono, THEN contra_subsetD]
           knows_Spy_subset_knows_Spy_Gets [THEN analz_mono, THEN contra_subsetD]
    ML
    {*
    val analz_mono_contra_tac =
      let val analz_impI = inst "P" "?Y ∉ analz (knows Spy ?evs)" impI
      in rtac analz_impI THEN'
         REPEAT1 o (dresolve_tac (thms"analz_mono_contra")) THEN'
         mp_tac
      end
    *}

    method_setup analz_mono_contra = {*
        Method.no_args
          (Method.METHOD (fn facts => REPEAT_FIRST analz_mono_contra_tac)) *}
        "for proving theorems of the form X ∉ analz (knows Spy evs) --> P"

    end

### Agents' Knowledge

### Used Messages

lemmas parts_insert_knows_A:

      parts (insert X (knows A evs)) = parts {X} ∪ parts (knows A evs)

lemmas parts_insert_knows_A:

      parts (insert X (knows A evs)) = parts {X} ∪ parts (knows A evs)

lemma knows_Spy_Says:

      knows Spy (Says A B X # evs) = insert X (knows Spy evs)

lemma knows_Spy_Notes:

      knows Spy (Notes A X # evs) =
      (if A ∈ bad then insert X (knows Spy evs) else knows Spy evs)

lemma knows_Spy_Gets:

      knows Spy (Gets A X # evs) = knows Spy evs

lemma initState_subset_knows:

      initState A ⊆ knows A evs

lemma knows_Spy_subset_knows_Spy_Says:

      knows Spy evs ⊆ knows Spy (Says A B X # evs)

lemma knows_Spy_subset_knows_Spy_Notes:

      knows Spy evs ⊆ knows Spy (Notes A X # evs)

lemma knows_Spy_subset_knows_Spy_Gets:

      knows Spy evs ⊆ knows Spy (Gets A X # evs)

lemma Says_imp_knows_Spy:

      Says A B X ∈ set evs ==> X ∈ knows Spy evs

lemmas knows_Spy_partsEs:

      [| Says A B X ∈ set evs; X ∈ parts (knows Spy evs) ==> PROP W |] ==> PROP W

      [| Crypt K X ∈ parts H; X ∈ parts H ==> PROP W |] ==> PROP W

lemmas knows_Spy_partsEs:

      [| Says A B X ∈ set evs; X ∈ parts (knows Spy evs) ==> PROP W |] ==> PROP W

      [| Crypt K X ∈ parts H; X ∈ parts H ==> PROP W |] ==> PROP W

### The Function @{term used}

lemma parts_knows_Spy_subset_used:

      parts (knows Spy evs) ⊆ used evs

lemmas usedI:

      c ∈ parts (knows Spy evs1) ==> c ∈ used evs1

lemmas usedI:

      c ∈ parts (knows Spy evs1) ==> c ∈ used evs1

lemma initState_subset_used:

      parts (initState B) ⊆ used evs

lemmas initState_into_used:

      c ∈ parts (initState B1) ==> c ∈ used evs1

lemmas initState_into_used:

      c ∈ parts (initState B1) ==> c ∈ used evs1

lemma used_Says:

      used (Says A B X # evs) = parts {X} ∪ used evs

lemma used_Notes:

      used (Notes A X # evs) = parts {X} ∪ used evs

lemma used_Gets:

      used (Gets A X # evs) = used evs

lemma Notes_imp_parts_subset_used:

      Notes A X ∈ set evs ==> parts {X} ⊆ used evs

lemmas analz_mono_contra:

      c ∉ analz (knows Spy (Says A2 B2 X2 # evs2)) ==> c ∉ analz (knows Spy evs2)

      c ∉ analz (knows Spy (Notes A2 X2 # evs2)) ==> c ∉ analz (knows Spy evs2)

      c ∉ analz (knows Spy (Gets A2 X2 # evs2)) ==> c ∉ analz (knows Spy evs2)

lemmas analz_mono_contra:

      c ∉ analz (knows Spy (Says A2 B2 X2 # evs2)) ==> c ∉ analz (knows Spy evs2)

      c ∉ analz (knows Spy (Notes A2 X2 # evs2)) ==> c ∉ analz (knows Spy evs2)

      c ∉ analz (knows Spy (Gets A2 X2 # evs2)) ==> c ∉ analz (knows Spy evs2)