(* Title: HOL/Auth/Event
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1996 University of Cambridge
Theory of events for security protocols
Datatype of events; function "sees"; freshness
*)
open Event;
AddIffs [Spy_in_bad, Server_not_bad];
(**** Function "spies" ****)
(** Simplifying parts (insert X (spies evs))
= parts {X} Un parts (spies evs) -- since general case loops*)
bind_thm ("parts_insert_spies",
parts_insert |>
read_instantiate_sg (sign_of thy) [("H", "spies evs")]);
Goal
"P(event_case sf nf ev) = \
\ ((ALL A B X. ev = Says A B X --> P(sf A B X)) & \
\ (ALL A X. ev = Notes A X --> P(nf A X)))";
by (induct_tac "ev" 1);
by Auto_tac;
qed "expand_event_case";
Goal "spies (Says A B X # evs) = insert X (spies evs)";
by (Simp_tac 1);
qed "spies_Says";
(*The point of letting the Spy see "bad" agents' notes is to prevent
redundant case-splits on whether A=Spy and whether A:bad*)
Goal "spies (Notes A X # evs) = \
\ (if A:bad then insert X (spies evs) else spies evs)";
by (Simp_tac 1);
qed "spies_Notes";
Goal "spies evs <= spies (Says A B X # evs)";
by (simp_tac (simpset() addsimps [subset_insertI]) 1);
qed "spies_subset_spies_Says";
Goal "spies evs <= spies (Notes A X # evs)";
by (Simp_tac 1);
by (Fast_tac 1);
qed "spies_subset_spies_Notes";
(*Spy sees all traffic*)
Goal "Says A B X : set evs --> X : spies evs";
by (induct_tac "evs" 1);
by (ALLGOALS (asm_simp_tac (simpset() addsplits [expand_event_case])));
qed_spec_mp "Says_imp_spies";
(*Spy sees Notes of bad agents*)
Goal "Notes A X : set evs --> A: bad --> X : spies evs";
by (induct_tac "evs" 1);
by (ALLGOALS (asm_simp_tac (simpset() addsplits [expand_event_case])));
qed_spec_mp "Notes_imp_spies";
(*Use with addSEs to derive contradictions from old Says events containing
items known to be fresh*)
val spies_partsEs = make_elim (Says_imp_spies RS parts.Inj) :: partsEs;
Addsimps [spies_Says, spies_Notes];
(**** NOTE REMOVAL--laws above are cleaner, as they don't involve "case" ****)
Delsimps [spies_Cons];
(*** Fresh nonces ***)
Goal "parts (spies evs) <= used evs";
by (induct_tac "evs" 1);
by (ALLGOALS (asm_simp_tac
(simpset() addsimps [parts_insert_spies]
addsplits [expand_event_case])));
by (ALLGOALS Blast_tac);
qed "parts_spies_subset_used";
bind_thm ("usedI", impOfSubs parts_spies_subset_used);
AddIs [usedI];
Goal "parts (initState B) <= used evs";
by (induct_tac "evs" 1);
by (ALLGOALS (asm_simp_tac
(simpset() addsimps [parts_insert_spies]
addsplits [expand_event_case])));
by (ALLGOALS Blast_tac);
bind_thm ("initState_into_used", impOfSubs (result()));
Goal "used (Says A B X # evs) = parts{X} Un used evs";
by (Simp_tac 1);
qed "used_Says";
Addsimps [used_Says];
Goal "used (Notes A X # evs) = parts{X} Un used evs";
by (Simp_tac 1);
qed "used_Notes";
Addsimps [used_Notes];
Goal "used [] <= used evs";
by (Simp_tac 1);
by (blast_tac (claset() addIs [initState_into_used]) 1);
qed "used_nil_subset";
(**** NOTE REMOVAL--laws above are cleaner, as they don't involve "case" ****)
Delsimps [used_Nil, used_Cons];
(*currently unused*)
Goal "used evs <= used (evs@evs')";
by (induct_tac "evs" 1);
by (simp_tac (simpset() addsimps [used_nil_subset]) 1);
by (induct_tac "a" 1);
by (ALLGOALS Asm_simp_tac);
by (ALLGOALS Blast_tac);
qed "used_subset_append";
(*For proving theorems of the form X ~: analz (spies 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 analz.*)
val analz_mono_contra_tac =
let val analz_impI = read_instantiate_sg (sign_of thy)
[("P", "?Y ~: analz (spies ?evs)")] impI;
in
rtac analz_impI THEN'
REPEAT1 o
(dresolve_tac
[spies_subset_spies_Says RS analz_mono RS contra_subsetD,
spies_subset_spies_Notes RS analz_mono RS contra_subsetD])
THEN'
mp_tac
end;