(* Title: HOL/Auth/SET/EventSET ID: $Id: EventSET.thy,v 1.5 2007/08/01 19:10:37 wenzelm 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.SIMPLE_METHOD (REPEAT_FIRST analz_mono_contra_tac)) *} "for proving theorems of the form X ∉ analz (knows Spy evs) --> P" end
lemma 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
lemma 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
lemma parts_knows_Spy_subset_used:
parts (knows Spy evs) ⊆ used evs
lemma usedI:
c ∈ parts (knows Spy evs1) ==> c ∈ used evs1
lemma initState_subset_used:
parts (initState B) ⊆ used evs
lemma 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
lemma 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)