Theory Event_SET

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


header{*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} 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"] 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} 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]

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

ML
{*
val analz_mono_contra_tac =
rtac @{thm analz_impI} THEN'
REPEAT1 o (dresolve_tac @{thms analz_mono_contra})
THEN' mp_tac
*}


method_setup analz_mono_contra = {*
Scan.succeed (K (SIMPLE_METHOD (REPEAT_FIRST analz_mono_contra_tac))) *}

"for proving theorems of the form X ∉ analz (knows Spy evs) --> P"

end