1839
|
1 |
(* Title: HOL/Auth/Message
|
|
2 |
ID: $Id$
|
|
3 |
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
|
|
4 |
Copyright 1996 University of Cambridge
|
|
5 |
|
|
6 |
Datatypes of agents and messages;
|
1913
|
7 |
Inductive relations "parts", "analz" and "synth"
|
1839
|
8 |
*)
|
|
9 |
|
|
10 |
open Message;
|
|
11 |
|
|
12 |
|
|
13 |
(** Inverse of keys **)
|
|
14 |
|
|
15 |
goal thy "!!K K'. (invKey K = invKey K') = (K=K')";
|
|
16 |
by (Step_tac 1);
|
|
17 |
br box_equals 1;
|
|
18 |
by (REPEAT (rtac invKey 2));
|
|
19 |
by (Asm_simp_tac 1);
|
|
20 |
qed "invKey_eq";
|
|
21 |
|
|
22 |
Addsimps [invKey, invKey_eq];
|
|
23 |
|
|
24 |
|
|
25 |
(**** keysFor operator ****)
|
|
26 |
|
|
27 |
goalw thy [keysFor_def] "keysFor {} = {}";
|
|
28 |
by (Fast_tac 1);
|
|
29 |
qed "keysFor_empty";
|
|
30 |
|
|
31 |
goalw thy [keysFor_def] "keysFor (H Un H') = keysFor H Un keysFor H'";
|
|
32 |
by (Fast_tac 1);
|
|
33 |
qed "keysFor_Un";
|
|
34 |
|
|
35 |
goalw thy [keysFor_def] "keysFor (UN i. H i) = (UN i. keysFor (H i))";
|
|
36 |
by (Fast_tac 1);
|
|
37 |
qed "keysFor_UN";
|
|
38 |
|
|
39 |
(*Monotonicity*)
|
|
40 |
goalw thy [keysFor_def] "!!G H. G<=H ==> keysFor(G) <= keysFor(H)";
|
|
41 |
by (Fast_tac 1);
|
|
42 |
qed "keysFor_mono";
|
|
43 |
|
|
44 |
goalw thy [keysFor_def] "keysFor (insert (Agent A) H) = keysFor H";
|
|
45 |
by (fast_tac (!claset addss (!simpset)) 1);
|
|
46 |
qed "keysFor_insert_Agent";
|
|
47 |
|
|
48 |
goalw thy [keysFor_def] "keysFor (insert (Nonce N) H) = keysFor H";
|
|
49 |
by (fast_tac (!claset addss (!simpset)) 1);
|
|
50 |
qed "keysFor_insert_Nonce";
|
|
51 |
|
|
52 |
goalw thy [keysFor_def] "keysFor (insert (Key K) H) = keysFor H";
|
|
53 |
by (fast_tac (!claset addss (!simpset)) 1);
|
|
54 |
qed "keysFor_insert_Key";
|
|
55 |
|
|
56 |
goalw thy [keysFor_def] "keysFor (insert {|X,Y|} H) = keysFor H";
|
|
57 |
by (fast_tac (!claset addss (!simpset)) 1);
|
|
58 |
qed "keysFor_insert_MPair";
|
|
59 |
|
|
60 |
goalw thy [keysFor_def]
|
|
61 |
"keysFor (insert (Crypt X K) H) = insert (invKey K) (keysFor H)";
|
|
62 |
by (Auto_tac());
|
|
63 |
by (fast_tac (!claset addIs [image_eqI]) 1);
|
|
64 |
qed "keysFor_insert_Crypt";
|
|
65 |
|
|
66 |
Addsimps [keysFor_empty, keysFor_Un, keysFor_UN,
|
|
67 |
keysFor_insert_Agent, keysFor_insert_Nonce,
|
|
68 |
keysFor_insert_Key, keysFor_insert_MPair,
|
|
69 |
keysFor_insert_Crypt];
|
|
70 |
|
|
71 |
|
|
72 |
(**** Inductive relation "parts" ****)
|
|
73 |
|
|
74 |
val major::prems =
|
|
75 |
goal thy "[| {|X,Y|} : parts H; \
|
|
76 |
\ [| X : parts H; Y : parts H |] ==> P \
|
|
77 |
\ |] ==> P";
|
|
78 |
by (cut_facts_tac [major] 1);
|
|
79 |
brs prems 1;
|
|
80 |
by (REPEAT (eresolve_tac [asm_rl, parts.Fst, parts.Snd] 1));
|
|
81 |
qed "MPair_parts";
|
|
82 |
|
|
83 |
AddIs [parts.Inj];
|
|
84 |
AddSEs [MPair_parts];
|
|
85 |
AddDs [parts.Body];
|
|
86 |
|
|
87 |
goal thy "H <= parts(H)";
|
|
88 |
by (Fast_tac 1);
|
|
89 |
qed "parts_increasing";
|
|
90 |
|
|
91 |
(*Monotonicity*)
|
|
92 |
goalw thy parts.defs "!!G H. G<=H ==> parts(G) <= parts(H)";
|
|
93 |
by (rtac lfp_mono 1);
|
|
94 |
by (REPEAT (ares_tac basic_monos 1));
|
|
95 |
qed "parts_mono";
|
|
96 |
|
|
97 |
goal thy "parts{} = {}";
|
|
98 |
by (Step_tac 1);
|
|
99 |
be parts.induct 1;
|
|
100 |
by (ALLGOALS Fast_tac);
|
|
101 |
qed "parts_empty";
|
|
102 |
Addsimps [parts_empty];
|
|
103 |
|
|
104 |
goal thy "!!X. X: parts{} ==> P";
|
|
105 |
by (Asm_full_simp_tac 1);
|
|
106 |
qed "parts_emptyE";
|
|
107 |
AddSEs [parts_emptyE];
|
|
108 |
|
1893
|
109 |
(*WARNING: loops if H = {Y}, therefore must not be repeated!*)
|
|
110 |
goal thy "!!H. X: parts H ==> EX Y:H. X: parts {Y}";
|
|
111 |
be parts.induct 1;
|
|
112 |
by (ALLGOALS Fast_tac);
|
|
113 |
qed "parts_singleton";
|
|
114 |
|
1839
|
115 |
|
|
116 |
(** Unions **)
|
|
117 |
|
|
118 |
goal thy "parts(G) Un parts(H) <= parts(G Un H)";
|
|
119 |
by (REPEAT (ares_tac [Un_least, parts_mono, Un_upper1, Un_upper2] 1));
|
|
120 |
val parts_Un_subset1 = result();
|
|
121 |
|
|
122 |
goal thy "parts(G Un H) <= parts(G) Un parts(H)";
|
|
123 |
br subsetI 1;
|
|
124 |
be parts.induct 1;
|
|
125 |
by (ALLGOALS Fast_tac);
|
|
126 |
val parts_Un_subset2 = result();
|
|
127 |
|
|
128 |
goal thy "parts(G Un H) = parts(G) Un parts(H)";
|
|
129 |
by (REPEAT (ares_tac [equalityI, parts_Un_subset1, parts_Un_subset2] 1));
|
|
130 |
qed "parts_Un";
|
|
131 |
|
1852
|
132 |
(*TWO inserts to avoid looping. This rewrite is better than nothing...*)
|
|
133 |
goal thy "parts (insert X (insert Y H)) = parts {X} Un parts {Y} Un parts H";
|
|
134 |
by (stac (read_instantiate [("A","H")] insert_is_Un) 1);
|
|
135 |
by (stac (read_instantiate [("A","{Y} Un H")] insert_is_Un) 1);
|
|
136 |
by (simp_tac (HOL_ss addsimps [parts_Un, Un_assoc]) 1);
|
|
137 |
qed "parts_insert2";
|
|
138 |
|
1839
|
139 |
goal thy "(UN x:A. parts(H x)) <= parts(UN x:A. H x)";
|
|
140 |
by (REPEAT (ares_tac [UN_least, parts_mono, UN_upper] 1));
|
|
141 |
val parts_UN_subset1 = result();
|
|
142 |
|
|
143 |
goal thy "parts(UN x:A. H x) <= (UN x:A. parts(H x))";
|
|
144 |
br subsetI 1;
|
|
145 |
be parts.induct 1;
|
|
146 |
by (ALLGOALS Fast_tac);
|
|
147 |
val parts_UN_subset2 = result();
|
|
148 |
|
|
149 |
goal thy "parts(UN x:A. H x) = (UN x:A. parts(H x))";
|
|
150 |
by (REPEAT (ares_tac [equalityI, parts_UN_subset1, parts_UN_subset2] 1));
|
|
151 |
qed "parts_UN";
|
|
152 |
|
|
153 |
goal thy "parts(UN x. H x) = (UN x. parts(H x))";
|
|
154 |
by (simp_tac (!simpset addsimps [UNION1_def, parts_UN]) 1);
|
|
155 |
qed "parts_UN1";
|
|
156 |
|
1913
|
157 |
(*Added to simplify arguments to parts, analz and synth*)
|
1839
|
158 |
Addsimps [parts_Un, parts_UN, parts_UN1];
|
|
159 |
|
|
160 |
goal thy "insert X (parts H) <= parts(insert X H)";
|
1852
|
161 |
by (fast_tac (!claset addEs [impOfSubs parts_mono]) 1);
|
1839
|
162 |
qed "parts_insert_subset";
|
|
163 |
|
|
164 |
(*Especially for reasoning about the Fake rule in traces*)
|
|
165 |
goal thy "!!Y. X: G ==> parts(insert X H) <= parts G Un parts H";
|
|
166 |
br ([parts_mono, parts_Un_subset2] MRS subset_trans) 1;
|
|
167 |
by (Fast_tac 1);
|
|
168 |
qed "parts_insert_subset_Un";
|
|
169 |
|
|
170 |
(** Idempotence and transitivity **)
|
|
171 |
|
|
172 |
goal thy "!!H. X: parts (parts H) ==> X: parts H";
|
|
173 |
be parts.induct 1;
|
|
174 |
by (ALLGOALS Fast_tac);
|
|
175 |
qed "parts_partsE";
|
|
176 |
AddSEs [parts_partsE];
|
|
177 |
|
|
178 |
goal thy "parts (parts H) = parts H";
|
|
179 |
by (Fast_tac 1);
|
|
180 |
qed "parts_idem";
|
|
181 |
Addsimps [parts_idem];
|
|
182 |
|
|
183 |
goal thy "!!H. [| X: parts G; G <= parts H |] ==> X: parts H";
|
|
184 |
by (dtac parts_mono 1);
|
|
185 |
by (Fast_tac 1);
|
|
186 |
qed "parts_trans";
|
|
187 |
|
|
188 |
(*Cut*)
|
|
189 |
goal thy "!!H. [| X: parts H; Y: parts (insert X H) |] ==> Y: parts H";
|
|
190 |
be parts_trans 1;
|
|
191 |
by (Fast_tac 1);
|
|
192 |
qed "parts_cut";
|
|
193 |
|
|
194 |
|
|
195 |
(** Rewrite rules for pulling out atomic messages **)
|
|
196 |
|
|
197 |
goal thy "parts (insert (Agent agt) H) = insert (Agent agt) (parts H)";
|
|
198 |
by (rtac (parts_insert_subset RSN (2, equalityI)) 1);
|
|
199 |
br subsetI 1;
|
|
200 |
be parts.induct 1;
|
|
201 |
(*Simplification breaks up equalities between messages;
|
|
202 |
how to make it work for fast_tac??*)
|
|
203 |
by (ALLGOALS (fast_tac (!claset addss (!simpset))));
|
|
204 |
qed "parts_insert_Agent";
|
|
205 |
|
|
206 |
goal thy "parts (insert (Nonce N) H) = insert (Nonce N) (parts H)";
|
|
207 |
by (rtac (parts_insert_subset RSN (2, equalityI)) 1);
|
|
208 |
br subsetI 1;
|
|
209 |
be parts.induct 1;
|
|
210 |
by (ALLGOALS (fast_tac (!claset addss (!simpset))));
|
|
211 |
qed "parts_insert_Nonce";
|
|
212 |
|
|
213 |
goal thy "parts (insert (Key K) H) = insert (Key K) (parts H)";
|
|
214 |
by (rtac (parts_insert_subset RSN (2, equalityI)) 1);
|
|
215 |
br subsetI 1;
|
|
216 |
be parts.induct 1;
|
|
217 |
by (ALLGOALS (fast_tac (!claset addss (!simpset))));
|
|
218 |
qed "parts_insert_Key";
|
|
219 |
|
|
220 |
goal thy "parts (insert (Crypt X K) H) = \
|
|
221 |
\ insert (Crypt X K) (parts (insert X H))";
|
|
222 |
br equalityI 1;
|
|
223 |
br subsetI 1;
|
|
224 |
be parts.induct 1;
|
|
225 |
by (Auto_tac());
|
|
226 |
be parts.induct 1;
|
|
227 |
by (ALLGOALS (best_tac (!claset addIs [parts.Body])));
|
|
228 |
qed "parts_insert_Crypt";
|
|
229 |
|
|
230 |
goal thy "parts (insert {|X,Y|} H) = \
|
|
231 |
\ insert {|X,Y|} (parts (insert X (insert Y H)))";
|
|
232 |
br equalityI 1;
|
|
233 |
br subsetI 1;
|
|
234 |
be parts.induct 1;
|
|
235 |
by (Auto_tac());
|
|
236 |
be parts.induct 1;
|
|
237 |
by (ALLGOALS (best_tac (!claset addIs [parts.Fst, parts.Snd])));
|
|
238 |
qed "parts_insert_MPair";
|
|
239 |
|
|
240 |
Addsimps [parts_insert_Agent, parts_insert_Nonce,
|
|
241 |
parts_insert_Key, parts_insert_Crypt, parts_insert_MPair];
|
|
242 |
|
|
243 |
|
1913
|
244 |
(**** Inductive relation "analz" ****)
|
1839
|
245 |
|
|
246 |
val major::prems =
|
1913
|
247 |
goal thy "[| {|X,Y|} : analz H; \
|
|
248 |
\ [| X : analz H; Y : analz H |] ==> P \
|
1839
|
249 |
\ |] ==> P";
|
|
250 |
by (cut_facts_tac [major] 1);
|
|
251 |
brs prems 1;
|
1913
|
252 |
by (REPEAT (eresolve_tac [asm_rl, analz.Fst, analz.Snd] 1));
|
|
253 |
qed "MPair_analz";
|
1839
|
254 |
|
1913
|
255 |
AddIs [analz.Inj];
|
|
256 |
AddSEs [MPair_analz];
|
|
257 |
AddDs [analz.Decrypt];
|
1839
|
258 |
|
1913
|
259 |
Addsimps [analz.Inj];
|
1885
|
260 |
|
1913
|
261 |
goal thy "H <= analz(H)";
|
1839
|
262 |
by (Fast_tac 1);
|
1913
|
263 |
qed "analz_increasing";
|
1839
|
264 |
|
1913
|
265 |
goal thy "analz H <= parts H";
|
1839
|
266 |
by (rtac subsetI 1);
|
1913
|
267 |
be analz.induct 1;
|
1839
|
268 |
by (ALLGOALS Fast_tac);
|
1913
|
269 |
qed "analz_subset_parts";
|
1839
|
270 |
|
1913
|
271 |
bind_thm ("not_parts_not_analz", analz_subset_parts RS contra_subsetD);
|
1839
|
272 |
|
|
273 |
|
1913
|
274 |
goal thy "parts (analz H) = parts H";
|
1839
|
275 |
br equalityI 1;
|
1913
|
276 |
br (analz_subset_parts RS parts_mono RS subset_trans) 1;
|
1839
|
277 |
by (Simp_tac 1);
|
1913
|
278 |
by (fast_tac (!claset addDs [analz_increasing RS parts_mono RS subsetD]) 1);
|
|
279 |
qed "parts_analz";
|
|
280 |
Addsimps [parts_analz];
|
1839
|
281 |
|
1913
|
282 |
goal thy "analz (parts H) = parts H";
|
1885
|
283 |
by (Auto_tac());
|
1913
|
284 |
be analz.induct 1;
|
1885
|
285 |
by (Auto_tac());
|
1913
|
286 |
qed "analz_parts";
|
|
287 |
Addsimps [analz_parts];
|
1885
|
288 |
|
1839
|
289 |
(*Monotonicity; Lemma 1 of Lowe*)
|
1913
|
290 |
goalw thy analz.defs "!!G H. G<=H ==> analz(G) <= analz(H)";
|
1839
|
291 |
by (rtac lfp_mono 1);
|
|
292 |
by (REPEAT (ares_tac basic_monos 1));
|
1913
|
293 |
qed "analz_mono";
|
1839
|
294 |
|
|
295 |
(** General equational properties **)
|
|
296 |
|
1913
|
297 |
goal thy "analz{} = {}";
|
1839
|
298 |
by (Step_tac 1);
|
1913
|
299 |
be analz.induct 1;
|
1839
|
300 |
by (ALLGOALS Fast_tac);
|
1913
|
301 |
qed "analz_empty";
|
|
302 |
Addsimps [analz_empty];
|
1839
|
303 |
|
1913
|
304 |
(*Converse fails: we can analz more from the union than from the
|
1839
|
305 |
separate parts, as a key in one might decrypt a message in the other*)
|
1913
|
306 |
goal thy "analz(G) Un analz(H) <= analz(G Un H)";
|
|
307 |
by (REPEAT (ares_tac [Un_least, analz_mono, Un_upper1, Un_upper2] 1));
|
|
308 |
qed "analz_Un";
|
1839
|
309 |
|
1913
|
310 |
goal thy "insert X (analz H) <= analz(insert X H)";
|
|
311 |
by (fast_tac (!claset addEs [impOfSubs analz_mono]) 1);
|
|
312 |
qed "analz_insert";
|
1839
|
313 |
|
|
314 |
(** Rewrite rules for pulling out atomic messages **)
|
|
315 |
|
1913
|
316 |
goal thy "analz (insert (Agent agt) H) = insert (Agent agt) (analz H)";
|
|
317 |
by (rtac (analz_insert RSN (2, equalityI)) 1);
|
1839
|
318 |
br subsetI 1;
|
1913
|
319 |
be analz.induct 1;
|
1839
|
320 |
(*Simplification breaks up equalities between messages;
|
|
321 |
how to make it work for fast_tac??*)
|
|
322 |
by (ALLGOALS (fast_tac (!claset addss (!simpset))));
|
1913
|
323 |
qed "analz_insert_Agent";
|
1839
|
324 |
|
1913
|
325 |
goal thy "analz (insert (Nonce N) H) = insert (Nonce N) (analz H)";
|
|
326 |
by (rtac (analz_insert RSN (2, equalityI)) 1);
|
1839
|
327 |
br subsetI 1;
|
1913
|
328 |
be analz.induct 1;
|
1839
|
329 |
by (ALLGOALS (fast_tac (!claset addss (!simpset))));
|
1913
|
330 |
qed "analz_insert_Nonce";
|
1839
|
331 |
|
|
332 |
(*Can only pull out Keys if they are not needed to decrypt the rest*)
|
|
333 |
goalw thy [keysFor_def]
|
1913
|
334 |
"!!K. K ~: keysFor (analz H) ==> \
|
|
335 |
\ analz (insert (Key K) H) = insert (Key K) (analz H)";
|
|
336 |
by (rtac (analz_insert RSN (2, equalityI)) 1);
|
1839
|
337 |
br subsetI 1;
|
1913
|
338 |
be analz.induct 1;
|
1839
|
339 |
by (ALLGOALS (fast_tac (!claset addss (!simpset))));
|
1913
|
340 |
qed "analz_insert_Key";
|
1839
|
341 |
|
1913
|
342 |
goal thy "analz (insert {|X,Y|} H) = \
|
|
343 |
\ insert {|X,Y|} (analz (insert X (insert Y H)))";
|
1885
|
344 |
br equalityI 1;
|
|
345 |
br subsetI 1;
|
1913
|
346 |
be analz.induct 1;
|
1885
|
347 |
by (Auto_tac());
|
1913
|
348 |
be analz.induct 1;
|
|
349 |
by (ALLGOALS (deepen_tac (!claset addIs [analz.Fst, analz.Snd, analz.Decrypt]) 0));
|
|
350 |
qed "analz_insert_MPair";
|
1885
|
351 |
|
|
352 |
(*Can pull out enCrypted message if the Key is not known*)
|
1913
|
353 |
goal thy "!!H. Key (invKey K) ~: analz H ==> \
|
|
354 |
\ analz (insert (Crypt X K) H) = \
|
|
355 |
\ insert (Crypt X K) (analz H)";
|
|
356 |
by (rtac (analz_insert RSN (2, equalityI)) 1);
|
1839
|
357 |
br subsetI 1;
|
1913
|
358 |
be analz.induct 1;
|
1839
|
359 |
by (ALLGOALS (fast_tac (!claset addss (!simpset))));
|
1913
|
360 |
qed "analz_insert_Crypt";
|
1839
|
361 |
|
1913
|
362 |
goal thy "!!H. Key (invKey K) : analz H ==> \
|
|
363 |
\ analz (insert (Crypt X K) H) <= \
|
|
364 |
\ insert (Crypt X K) (analz (insert X H))";
|
1839
|
365 |
br subsetI 1;
|
1913
|
366 |
by (eres_inst_tac [("za","x")] analz.induct 1);
|
1839
|
367 |
by (ALLGOALS (fast_tac (!claset addss (!simpset))));
|
|
368 |
val lemma1 = result();
|
|
369 |
|
1913
|
370 |
goal thy "!!H. Key (invKey K) : analz H ==> \
|
|
371 |
\ insert (Crypt X K) (analz (insert X H)) <= \
|
|
372 |
\ analz (insert (Crypt X K) H)";
|
1839
|
373 |
by (Auto_tac());
|
1913
|
374 |
by (eres_inst_tac [("za","x")] analz.induct 1);
|
1839
|
375 |
by (Auto_tac());
|
1913
|
376 |
by (best_tac (!claset addIs [subset_insertI RS analz_mono RS subsetD,
|
|
377 |
analz.Decrypt]) 1);
|
1839
|
378 |
val lemma2 = result();
|
|
379 |
|
1913
|
380 |
goal thy "!!H. Key (invKey K) : analz H ==> \
|
|
381 |
\ analz (insert (Crypt X K) H) = \
|
|
382 |
\ insert (Crypt X K) (analz (insert X H))";
|
1839
|
383 |
by (REPEAT (ares_tac [equalityI, lemma1, lemma2] 1));
|
1913
|
384 |
qed "analz_insert_Decrypt";
|
1839
|
385 |
|
1885
|
386 |
(*Case analysis: either the message is secure, or it is not!
|
|
387 |
Use with expand_if; apparently split_tac does not cope with patterns
|
1913
|
388 |
such as "analz (insert (Crypt X' K) H)" *)
|
|
389 |
goal thy "analz (insert (Crypt X' K) H) = \
|
|
390 |
\ (if (Key (invKey K) : analz H) then \
|
|
391 |
\ insert (Crypt X' K) (analz (insert X' H)) \
|
|
392 |
\ else insert (Crypt X' K) (analz H))";
|
|
393 |
by (excluded_middle_tac "Key (invKey K) : analz H " 1);
|
|
394 |
by (ALLGOALS (asm_simp_tac (!simpset addsimps [analz_insert_Crypt,
|
|
395 |
analz_insert_Decrypt])));
|
|
396 |
qed "analz_Crypt_if";
|
1885
|
397 |
|
1913
|
398 |
Addsimps [analz_insert_Agent, analz_insert_Nonce,
|
|
399 |
analz_insert_Key, analz_insert_MPair,
|
|
400 |
analz_Crypt_if];
|
1839
|
401 |
|
|
402 |
(*This rule supposes "for the sake of argument" that we have the key.*)
|
1913
|
403 |
goal thy "analz (insert (Crypt X K) H) <= \
|
|
404 |
\ insert (Crypt X K) (analz (insert X H))";
|
1839
|
405 |
br subsetI 1;
|
1913
|
406 |
be analz.induct 1;
|
1839
|
407 |
by (Auto_tac());
|
1913
|
408 |
qed "analz_insert_Crypt_subset";
|
1839
|
409 |
|
|
410 |
|
|
411 |
(** Idempotence and transitivity **)
|
|
412 |
|
1913
|
413 |
goal thy "!!H. X: analz (analz H) ==> X: analz H";
|
|
414 |
be analz.induct 1;
|
1839
|
415 |
by (ALLGOALS Fast_tac);
|
1913
|
416 |
qed "analz_analzE";
|
|
417 |
AddSEs [analz_analzE];
|
1839
|
418 |
|
1913
|
419 |
goal thy "analz (analz H) = analz H";
|
1839
|
420 |
by (Fast_tac 1);
|
1913
|
421 |
qed "analz_idem";
|
|
422 |
Addsimps [analz_idem];
|
1839
|
423 |
|
1913
|
424 |
goal thy "!!H. [| X: analz G; G <= analz H |] ==> X: analz H";
|
|
425 |
by (dtac analz_mono 1);
|
1839
|
426 |
by (Fast_tac 1);
|
1913
|
427 |
qed "analz_trans";
|
1839
|
428 |
|
|
429 |
(*Cut; Lemma 2 of Lowe*)
|
1913
|
430 |
goal thy "!!H. [| X: analz H; Y: analz (insert X H) |] ==> Y: analz H";
|
|
431 |
be analz_trans 1;
|
1839
|
432 |
by (Fast_tac 1);
|
1913
|
433 |
qed "analz_cut";
|
1839
|
434 |
|
|
435 |
(*Cut can be proved easily by induction on
|
1913
|
436 |
"!!H. Y: analz (insert X H) ==> X: analz H --> Y: analz H"
|
1839
|
437 |
*)
|
|
438 |
|
1885
|
439 |
|
1913
|
440 |
(** A congruence rule for "analz" **)
|
1885
|
441 |
|
1913
|
442 |
goal thy "!!H. [| analz G <= analz G'; analz H <= analz H' \
|
|
443 |
\ |] ==> analz (G Un H) <= analz (G' Un H')";
|
1885
|
444 |
by (Step_tac 1);
|
1913
|
445 |
be analz.induct 1;
|
|
446 |
by (ALLGOALS (best_tac (!claset addIs [analz_mono RS subsetD])));
|
|
447 |
qed "analz_subset_cong";
|
1885
|
448 |
|
1913
|
449 |
goal thy "!!H. [| analz G = analz G'; analz H = analz H' \
|
|
450 |
\ |] ==> analz (G Un H) = analz (G' Un H')";
|
|
451 |
by (REPEAT_FIRST (ares_tac [equalityI, analz_subset_cong]
|
1885
|
452 |
ORELSE' etac equalityE));
|
1913
|
453 |
qed "analz_cong";
|
1885
|
454 |
|
|
455 |
|
1913
|
456 |
goal thy "!!H. analz H = analz H' ==> analz(insert X H) = analz(insert X H')";
|
1885
|
457 |
by (asm_simp_tac (!simpset addsimps [insert_def]
|
1913
|
458 |
setloop (rtac analz_cong)) 1);
|
|
459 |
qed "analz_insert_cong";
|
1885
|
460 |
|
1913
|
461 |
(*If there are no pairs or encryptions then analz does nothing*)
|
1839
|
462 |
goal thy "!!H. [| ALL X Y. {|X,Y|} ~: H; ALL X K. Crypt X K ~: H |] ==> \
|
1913
|
463 |
\ analz H = H";
|
1839
|
464 |
by (Step_tac 1);
|
1913
|
465 |
be analz.induct 1;
|
1839
|
466 |
by (ALLGOALS Fast_tac);
|
1913
|
467 |
qed "analz_trivial";
|
1839
|
468 |
|
|
469 |
(*Helps to prove Fake cases*)
|
1913
|
470 |
goal thy "!!X. X: analz (UN i. analz (H i)) ==> X: analz (UN i. H i)";
|
|
471 |
be analz.induct 1;
|
|
472 |
by (ALLGOALS (fast_tac (!claset addEs [impOfSubs analz_mono])));
|
1839
|
473 |
val lemma = result();
|
|
474 |
|
1913
|
475 |
goal thy "analz (UN i. analz (H i)) = analz (UN i. H i)";
|
1839
|
476 |
by (fast_tac (!claset addIs [lemma]
|
1913
|
477 |
addEs [impOfSubs analz_mono]) 1);
|
|
478 |
qed "analz_UN_analz";
|
|
479 |
Addsimps [analz_UN_analz];
|
1839
|
480 |
|
|
481 |
|
1913
|
482 |
(**** Inductive relation "synth" ****)
|
1839
|
483 |
|
1913
|
484 |
AddIs synth.intrs;
|
1839
|
485 |
|
1913
|
486 |
goal thy "H <= synth(H)";
|
1839
|
487 |
by (Fast_tac 1);
|
1913
|
488 |
qed "synth_increasing";
|
1839
|
489 |
|
|
490 |
(*Monotonicity*)
|
1913
|
491 |
goalw thy synth.defs "!!G H. G<=H ==> synth(G) <= synth(H)";
|
1839
|
492 |
by (rtac lfp_mono 1);
|
|
493 |
by (REPEAT (ares_tac basic_monos 1));
|
1913
|
494 |
qed "synth_mono";
|
1839
|
495 |
|
|
496 |
(** Unions **)
|
|
497 |
|
1913
|
498 |
(*Converse fails: we can synth more from the union than from the
|
1839
|
499 |
separate parts, building a compound message using elements of each.*)
|
1913
|
500 |
goal thy "synth(G) Un synth(H) <= synth(G Un H)";
|
|
501 |
by (REPEAT (ares_tac [Un_least, synth_mono, Un_upper1, Un_upper2] 1));
|
|
502 |
qed "synth_Un";
|
1839
|
503 |
|
1913
|
504 |
goal thy "insert X (synth H) <= synth(insert X H)";
|
|
505 |
by (fast_tac (!claset addEs [impOfSubs synth_mono]) 1);
|
|
506 |
qed "synth_insert";
|
1885
|
507 |
|
1839
|
508 |
(** Idempotence and transitivity **)
|
|
509 |
|
1913
|
510 |
goal thy "!!H. X: synth (synth H) ==> X: synth H";
|
|
511 |
be synth.induct 1;
|
1839
|
512 |
by (ALLGOALS Fast_tac);
|
1913
|
513 |
qed "synth_synthE";
|
|
514 |
AddSEs [synth_synthE];
|
1839
|
515 |
|
1913
|
516 |
goal thy "synth (synth H) = synth H";
|
1839
|
517 |
by (Fast_tac 1);
|
1913
|
518 |
qed "synth_idem";
|
1839
|
519 |
|
1913
|
520 |
goal thy "!!H. [| X: synth G; G <= synth H |] ==> X: synth H";
|
|
521 |
by (dtac synth_mono 1);
|
1839
|
522 |
by (Fast_tac 1);
|
1913
|
523 |
qed "synth_trans";
|
1839
|
524 |
|
|
525 |
(*Cut; Lemma 2 of Lowe*)
|
1913
|
526 |
goal thy "!!H. [| X: synth H; Y: synth (insert X H) |] ==> Y: synth H";
|
|
527 |
be synth_trans 1;
|
1839
|
528 |
by (Fast_tac 1);
|
1913
|
529 |
qed "synth_cut";
|
1839
|
530 |
|
|
531 |
|
|
532 |
(*Can only produce a nonce or key if it is already known,
|
1913
|
533 |
but can synth a pair or encryption from its components...*)
|
|
534 |
val mk_cases = synth.mk_cases msg.simps;
|
1839
|
535 |
|
1913
|
536 |
(*NO Agent_synth, as any Agent name can be synthd*)
|
|
537 |
val Nonce_synth = mk_cases "Nonce n : synth H";
|
|
538 |
val Key_synth = mk_cases "Key K : synth H";
|
|
539 |
val MPair_synth = mk_cases "{|X,Y|} : synth H";
|
|
540 |
val Crypt_synth = mk_cases "Crypt X K : synth H";
|
1839
|
541 |
|
1913
|
542 |
AddSEs [Nonce_synth, Key_synth, MPair_synth, Crypt_synth];
|
1839
|
543 |
|
1913
|
544 |
goal thy "(Nonce N : synth H) = (Nonce N : H)";
|
1839
|
545 |
by (Fast_tac 1);
|
1913
|
546 |
qed "Nonce_synth_eq";
|
1839
|
547 |
|
1913
|
548 |
goal thy "(Key K : synth H) = (Key K : H)";
|
1839
|
549 |
by (Fast_tac 1);
|
1913
|
550 |
qed "Key_synth_eq";
|
1839
|
551 |
|
1913
|
552 |
Addsimps [Nonce_synth_eq, Key_synth_eq];
|
1839
|
553 |
|
|
554 |
|
|
555 |
goalw thy [keysFor_def]
|
1913
|
556 |
"keysFor (synth H) = keysFor H Un invKey``{K. Key K : H}";
|
1839
|
557 |
by (Fast_tac 1);
|
1913
|
558 |
qed "keysFor_synth";
|
|
559 |
Addsimps [keysFor_synth];
|
1839
|
560 |
|
|
561 |
|
1913
|
562 |
(*** Combinations of parts, analz and synth ***)
|
1839
|
563 |
|
1913
|
564 |
goal thy "parts (synth H) = parts H Un synth H";
|
1839
|
565 |
br equalityI 1;
|
|
566 |
br subsetI 1;
|
|
567 |
be parts.induct 1;
|
|
568 |
by (ALLGOALS
|
1913
|
569 |
(best_tac (!claset addIs ((synth_increasing RS parts_mono RS subsetD)
|
1839
|
570 |
::parts.intrs))));
|
1913
|
571 |
qed "parts_synth";
|
|
572 |
Addsimps [parts_synth];
|
1839
|
573 |
|
1913
|
574 |
goal thy "analz (synth H) = analz H Un synth H";
|
1839
|
575 |
br equalityI 1;
|
|
576 |
br subsetI 1;
|
1913
|
577 |
be analz.induct 1;
|
1839
|
578 |
by (best_tac
|
1913
|
579 |
(!claset addIs [synth_increasing RS analz_mono RS subsetD]) 5);
|
1839
|
580 |
(*Strange that best_tac just can't hack this one...*)
|
1913
|
581 |
by (ALLGOALS (deepen_tac (!claset addIs analz.intrs) 0));
|
|
582 |
qed "analz_synth";
|
|
583 |
Addsimps [analz_synth];
|
1839
|
584 |
|
|
585 |
(*Hard to prove; still needed now that there's only one Enemy?*)
|
1913
|
586 |
goal thy "analz (UN i. synth (H i)) = \
|
|
587 |
\ analz (UN i. H i) Un (UN i. synth (H i))";
|
1839
|
588 |
br equalityI 1;
|
|
589 |
br subsetI 1;
|
1913
|
590 |
be analz.induct 1;
|
1839
|
591 |
by (best_tac
|
1913
|
592 |
(!claset addEs [impOfSubs synth_increasing,
|
|
593 |
impOfSubs analz_mono]) 5);
|
1839
|
594 |
by (Best_tac 1);
|
1913
|
595 |
by (deepen_tac (!claset addIs [analz.Fst]) 0 1);
|
|
596 |
by (deepen_tac (!claset addIs [analz.Snd]) 0 1);
|
|
597 |
by (deepen_tac (!claset addSEs [analz.Decrypt]
|
|
598 |
addIs [analz.Decrypt]) 0 1);
|
|
599 |
qed "analz_UN1_synth";
|
|
600 |
Addsimps [analz_UN1_synth];
|