(* Title: HOL/Auth/Message
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1996 University of Cambridge
Theory of Shared Keys (common to all symmetric-key protocols)
Server keys; initial states of agents; new nonces and keys; function "sees"
*)
Addsimps [parts_cut_eq];
(*GOALS.ML??*)
fun prlim n = (goals_limit:=n; pr());
(*FUN.ML?? WE NEED A NOTION OF INVERSE IMAGE, OR GRAPH!!*)
goal Set.thy "!!f. B <= range f = (B = f`` {x. f x: B})";
by (fast_tac (!claset addEs [equalityE]) 1);
val subset_range_iff = result();
open Shared;
(*Holds because Friend is injective: thus cannot prove for all f*)
goal thy "(Friend x : Friend``A) = (x:A)";
by (Auto_tac());
qed "Friend_image_eq";
Addsimps [Friend_image_eq];
Addsimps [Un_insert_left, Un_insert_right];
(*By default only o_apply is built-in. But in the presence of eta-expansion
this means that some terms displayed as (f o g) will be rewritten, and others
will not!*)
Addsimps [o_def];
(*** Basic properties of shrK and newK ***)
(* invKey (shrK A) = shrK A *)
bind_thm ("invKey_shrK", rewrite_rule [isSymKey_def] isSym_shrK);
(* invKey (newK evs) = newK evs *)
bind_thm ("invKey_newK", rewrite_rule [isSymKey_def] isSym_newK);
Addsimps [invKey_shrK, invKey_newK];
(*Injectiveness and freshness of new keys and nonces*)
AddIffs [inj_shrK RS inj_eq, inj_newN RS inj_eq, inj_newK RS inj_eq,
fresh_newN, fresh_newK];
(** Rewrites should not refer to initState(Friend i)
-- not in normal form! **)
goal thy "newK evs ~= shrK B";
by (subgoal_tac "newK evs = shrK B --> \
\ Key (newK evs) : parts (initState B)" 1);
by (Fast_tac 1);
by (agent.induct_tac "B" 1);
by (auto_tac (!claset addIs [range_eqI], !simpset addsimps [bad_def]));
qed "newK_neq_shrK";
Addsimps [newK_neq_shrK, newK_neq_shrK RS not_sym];
(*Good for talking about Server's initial state*)
goal thy "!!H. H <= Key``E ==> parts H = H";
by (Auto_tac ());
be parts.induct 1;
by (ALLGOALS (fast_tac (!claset addss (!simpset))));
qed "parts_image_subset";
bind_thm ("parts_image_Key", subset_refl RS parts_image_subset);
goal thy "!!H. H <= Key``E ==> analz H = H";
by (Auto_tac ());
be analz.induct 1;
by (ALLGOALS (fast_tac (!claset addss (!simpset))));
qed "analz_image_subset";
bind_thm ("analz_image_Key", subset_refl RS analz_image_subset);
Addsimps [parts_image_Key, analz_image_Key];
goalw thy [keysFor_def] "keysFor (parts (initState C)) = {}";
by (agent.induct_tac "C" 1);
by (auto_tac (!claset addIs [range_eqI], !simpset));
qed "keysFor_parts_initState";
Addsimps [keysFor_parts_initState];
goalw thy [keysFor_def] "keysFor (Key``E) = {}";
by (Auto_tac ());
qed "keysFor_image_Key";
Addsimps [keysFor_image_Key];
goal thy "shrK A ~: newK``E";
by (agent.induct_tac "A" 1);
by (Auto_tac ());
qed "shrK_notin_image_newK";
Addsimps [shrK_notin_image_newK];
(*Agents see their own shared keys!*)
goal thy "Key (shrK A) : sees A evs";
by (list.induct_tac "evs" 1);
by (agent.induct_tac "A" 1);
by (auto_tac (!claset, !simpset addsimps [bad_def]));
qed "sees_own_shrK";
(*Enemy sees shared keys of bad agents!*)
goal thy "!!i. A: bad ==> Key (shrK A) : sees Enemy evs";
by (list.induct_tac "evs" 1);
by (auto_tac (!claset, !simpset addsimps [bad_def]));
qed "Enemy_sees_bad";
AddSIs [sees_own_shrK, Enemy_sees_bad];
goal thy "Enemy: bad";
by (simp_tac (!simpset addsimps [bad_def]) 1);
qed "Enemy_in_bad";
AddIffs [Enemy_in_bad];
goal thy "!!A. A ~: bad ==> A ~= Enemy";
by (Fast_tac 1);
qed "not_bad_imp_not_Enemy";
AddIffs [Enemy_in_bad];
(** Specialized rewrite rules for (sees A (Says...#evs)) **)
goal thy "sees A (Says A B X # evs) = insert X (sees A evs)";
by (Simp_tac 1);
qed "sees_own";
goal thy "!!A. Server ~= A ==> \
\ sees Server (Says A B X # evs) = sees Server evs";
by (Asm_simp_tac 1);
qed "sees_Server";
goal thy "!!A. Friend i ~= A ==> \
\ sees (Friend i) (Says A B X # evs) = sees (Friend i) evs";
by (Asm_simp_tac 1);
qed "sees_Friend";
goal thy "sees Enemy (Says A B X # evs) = insert X (sees Enemy evs)";
by (Simp_tac 1);
qed "sees_Enemy";
goal thy "sees A (Says A' B X # evs) <= insert X (sees A evs)";
by (simp_tac (!simpset setloop split_tac [expand_if]) 1);
by (Fast_tac 1);
qed "sees_Says_subset_insert";
goal thy "sees A evs <= sees A (Says A' B X # evs)";
by (simp_tac (!simpset setloop split_tac [expand_if]) 1);
by (Fast_tac 1);
qed "sees_subset_sees_Says";
(*Pushing Unions into parts; one of the A's equals B, and thus sees Y*)
goal thy "(UN A. parts (sees A (Says B C Y # evs))) = \
\ parts {Y} Un (UN A. parts (sees A evs))";
by (Step_tac 1);
be rev_mp 1; (*for some reason, split_tac does not work on assumptions*)
val ss = (!simpset addsimps [parts_Un, sees_Cons]
setloop split_tac [expand_if]);
by (ALLGOALS (fast_tac (!claset addss ss)));
qed "UN_parts_sees_Says";
goal thy "Says A B X : set_of_list evs --> X : sees Enemy evs";
by (list.induct_tac "evs" 1);
by (Auto_tac ());
qed_spec_mp "Says_imp_sees_Enemy";
Addsimps [Says_imp_sees_Enemy];
AddIs [Says_imp_sees_Enemy];
goal thy "initState C <= Key `` range shrK";
by (agent.induct_tac "C" 1);
by (Auto_tac ());
qed "initState_subset";
goal thy "X : sees C evs --> \
\ (EX A B. Says A B X : set_of_list evs) | \
\ (EX A. X = Key (shrK A))";
by (list.induct_tac "evs" 1);
by (ALLGOALS Asm_simp_tac);
by (fast_tac (!claset addDs [impOfSubs initState_subset]) 1);
br conjI 1;
by (Fast_tac 2);
by (event.induct_tac "a" 1);
by (ALLGOALS (asm_simp_tac (!simpset addsimps [mem_if])));
by (ALLGOALS Fast_tac);
qed_spec_mp "seesD";
Addsimps [UN_parts_sees_Says, sees_own, sees_Server, sees_Friend, sees_Enemy];
Delsimps [sees_Cons]; (**** NOTE REMOVAL -- laws above are cleaner ****)
goal thy "!!K. newK evs = invKey K ==> newK evs = K";
br (invKey_eq RS iffD1) 1;
by (Simp_tac 1);
val newK_invKey = result();
AddSDs [newK_invKey];
(** Rewrites to push in Key and Crypt messages, so that other messages can
be pulled out using the analz_insert rules **)
fun insComm x y = read_instantiate_sg (sign_of thy) [("x",x), ("y",y)]
insert_commute;
val pushKeys = map (insComm "Key ?K")
["Agent ?C", "Nonce ?N", "MPair ?X ?Y", "Crypt ?X ?K'"];
val pushCrypts = map (insComm "Crypt ?X ?K")
["Agent ?C", "Nonce ?N", "MPair ?X' ?Y"];
(*Cannot be added with Addsimps -- we don't always want to re-order messages*)
val pushes = pushKeys@pushCrypts;
val pushKey_newK = insComm "Key (newK ?evs)" "Key (shrK ?C)";
(*No premature instantiation of variables. For proving weak liveness.*)
fun safe_solver prems =
match_tac (TrueI::refl::prems) ORELSE' eq_assume_tac
ORELSE' etac FalseE;
(*Analysis of Fake cases and of messages that forward unknown parts
Abstraction over i is ESSENTIAL: it delays the dereferencing of claset*)
fun enemy_analz_tac i =
SELECT_GOAL
(EVERY [ (*push in occurrences of X...*)
(REPEAT o CHANGED)
(res_inst_tac [("x1","X")] (insert_commute RS ssubst) 1),
(*...allowing further simplifications*)
simp_tac (!simpset setloop split_tac [expand_if]) 1,
REPEAT (resolve_tac [impI,notI] 1),
dtac (impOfSubs Fake_analz_insert) 1,
eresolve_tac [asm_rl, synth.Inj] 1,
Fast_tac 1,
Asm_full_simp_tac 1,
IF_UNSOLVED (deepen_tac (!claset addIs [impOfSubs analz_mono]) 0 1)
]) i;
(** Simplifying parts (insert X (sees A evs))
= parts {X} Un parts (sees A evs) -- since general case loops*)
val parts_insert_sees =
parts_insert |> read_instantiate_sg (sign_of thy) [("H", "sees A evs")]
|> standard;