Theory Event_SET

(*  Title:      HOL/SET_Protocol/Event_SET.thy
    Author:     Giampaolo Bella
    Author:     Fabio Massacci
    Author:     Lawrence C Paulson
*)

section‹Theory of Events for SET›

theory Event_SET
imports Message_SET
begin

text‹The Root Certification Authority›
abbreviation "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"

(* Message reception does not extend spy's knowledge because of
   reception invariant enforced by Reception rule in protocol definition*)
primrec knows :: "[agent, event list] ⇒ msg set"
where
  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›

primrec used :: "event list ⇒ msg set"
where
  (*Set of items that might be visible to somebody:
    complement of the set of fresh items.
    As above, message reception does extend used items *)
  used_Nil:  "used []         = (UN B. parts (initState B))"
| used_Cons: "used (ev # evs) =
                 (case ev of
                    Says A B X ⇒ parts {X} ∪ (used evs)
                  | Gets A X   ⇒ used evs
                  | Notes A X  ⇒ parts {X} ∪ (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} ∪ parts (knows Spy evs) -- since general case loops*)

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

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, elim_format] 
     parts.Body [elim_format]


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} ∪ used evs"
by auto

lemma used_Notes [simp]: "used (Notes A X # evs) = parts{X} ∪ 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 (rename_tac [2] a 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]

lemmas analz_impI = impI [where P = "Y ∉ analz (knows Spy evs)"] for Y evs

ML
‹
fun analz_mono_contra_tac ctxt = 
  resolve_tac ctxt @{thms analz_impI} THEN' 
  REPEAT1 o (dresolve_tac ctxt @{thms analz_mono_contra})
  THEN' mp_tac ctxt
›

method_setup analz_mono_contra = ‹
    Scan.succeed (fn ctxt => SIMPLE_METHOD (REPEAT_FIRST (analz_mono_contra_tac ctxt)))›
    "for proving theorems of the form X ∉ analz (knows Spy evs) ⟶ P"

end