(* 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} 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 (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