13508
|
1 |
(******************************************************************************
|
|
2 |
date: november 2001
|
|
3 |
author: Frederic Blanqui
|
|
4 |
email: blanqui@lri.fr
|
|
5 |
webpage: http://www.lri.fr/~blanqui/
|
|
6 |
|
|
7 |
University of Cambridge, Computer Laboratory
|
|
8 |
William Gates Building, JJ Thomson Avenue
|
|
9 |
Cambridge CB3 0FD, United Kingdom
|
|
10 |
******************************************************************************)
|
|
11 |
|
|
12 |
header {*Extensions to Standard Theories*}
|
|
13 |
|
|
14 |
theory Extensions = Event:
|
|
15 |
|
|
16 |
|
|
17 |
subsection{*Extensions to Theory @{text Set}*}
|
|
18 |
|
|
19 |
lemma eq: "[| !!x. x:A ==> x:B; !!x. x:B ==> x:A |] ==> A=B"
|
|
20 |
by auto
|
|
21 |
|
|
22 |
lemma Un_eq: "[| A=A'; B=B' |] ==> A Un B = A' Un B'"
|
|
23 |
by auto
|
|
24 |
|
|
25 |
lemma insert_absorb_substI: "[| x:A; P (insert x A) |] ==> P A"
|
|
26 |
by (simp add: insert_absorb)
|
|
27 |
|
|
28 |
lemma insert_Diff_substD: "[| x:A; P A |] ==> P (insert x (A - {x}))"
|
|
29 |
by (simp add: insert_Diff)
|
|
30 |
|
|
31 |
lemma insert_Diff_substI: "[| x:A; P (insert x (A - {x})) |] ==> P A"
|
|
32 |
by (simp add: insert_Diff)
|
|
33 |
|
|
34 |
lemma insert_Un: "P ({x} Un A) ==> P (insert x A)"
|
|
35 |
by simp
|
|
36 |
|
|
37 |
lemma in_sub: "x:A ==> {x}<=A"
|
|
38 |
by auto
|
|
39 |
|
|
40 |
|
|
41 |
subsection{*Extensions to Theory @{text List}*}
|
|
42 |
|
|
43 |
subsubsection{*"minus l x" erase the first element of "l" equal to "x"*}
|
|
44 |
|
|
45 |
consts minus :: "'a list => 'a => 'a list"
|
|
46 |
|
|
47 |
primrec
|
|
48 |
"minus [] y = []"
|
|
49 |
"minus (x#xs) y = (if x=y then xs else x # minus xs y)"
|
|
50 |
|
|
51 |
lemma set_minus: "set (minus l x) <= set l"
|
|
52 |
by (induct l, auto)
|
|
53 |
|
|
54 |
subsection{*Extensions to Theory @{text Message}*}
|
|
55 |
|
|
56 |
subsubsection{*declarations for tactics*}
|
|
57 |
|
|
58 |
declare analz_subset_parts [THEN subsetD, dest]
|
|
59 |
declare image_eq_UN [simp]
|
|
60 |
declare parts_insert2 [simp]
|
|
61 |
declare analz_cut [dest]
|
|
62 |
declare split_if_asm [split]
|
|
63 |
declare analz_insertI [intro]
|
|
64 |
declare Un_Diff [simp]
|
|
65 |
|
|
66 |
subsubsection{*extract the agent number of an Agent message*}
|
|
67 |
|
|
68 |
consts agt_nb :: "msg => agent"
|
|
69 |
|
|
70 |
recdef agt_nb "measure size"
|
|
71 |
"agt_nb (Agent A) = A"
|
|
72 |
|
|
73 |
subsubsection{*messages that are pairs*}
|
|
74 |
|
|
75 |
constdefs is_MPair :: "msg => bool"
|
|
76 |
"is_MPair X == EX Y Z. X = {|Y,Z|}"
|
|
77 |
|
|
78 |
declare is_MPair_def [simp]
|
|
79 |
|
|
80 |
lemma MPair_is_MPair [iff]: "is_MPair {|X,Y|}"
|
|
81 |
by simp
|
|
82 |
|
|
83 |
lemma Agent_isnt_MPair [iff]: "~ is_MPair (Agent A)"
|
|
84 |
by simp
|
|
85 |
|
|
86 |
lemma Number_isnt_MPair [iff]: "~ is_MPair (Number n)"
|
|
87 |
by simp
|
|
88 |
|
|
89 |
lemma Key_isnt_MPair [iff]: "~ is_MPair (Key K)"
|
|
90 |
by simp
|
|
91 |
|
|
92 |
lemma Nonce_isnt_MPair [iff]: "~ is_MPair (Nonce n)"
|
|
93 |
by simp
|
|
94 |
|
|
95 |
lemma Hash_isnt_MPair [iff]: "~ is_MPair (Hash X)"
|
|
96 |
by simp
|
|
97 |
|
|
98 |
lemma Crypt_isnt_MPair [iff]: "~ is_MPair (Crypt K X)"
|
|
99 |
by simp
|
|
100 |
|
|
101 |
syntax not_MPair :: "msg => bool"
|
|
102 |
|
|
103 |
translations "not_MPair X" == "~ is_MPair X"
|
|
104 |
|
|
105 |
lemma is_MPairE: "[| is_MPair X ==> P; not_MPair X ==> P |] ==> P"
|
|
106 |
by auto
|
|
107 |
|
|
108 |
declare is_MPair_def [simp del]
|
|
109 |
|
|
110 |
constdefs has_no_pair :: "msg set => bool"
|
|
111 |
"has_no_pair H == ALL X Y. {|X,Y|} ~:H"
|
|
112 |
|
|
113 |
declare has_no_pair_def [simp]
|
|
114 |
|
|
115 |
subsubsection{*well-foundedness of messages*}
|
|
116 |
|
|
117 |
lemma wf_Crypt1 [iff]: "Crypt K X ~= X"
|
|
118 |
by (induct X, auto)
|
|
119 |
|
|
120 |
lemma wf_Crypt2 [iff]: "X ~= Crypt K X"
|
|
121 |
by (induct X, auto)
|
|
122 |
|
|
123 |
lemma parts_size: "X:parts {Y} ==> X=Y | size X < size Y"
|
|
124 |
by (erule parts.induct, auto)
|
|
125 |
|
|
126 |
lemma wf_Crypt_parts [iff]: "Crypt K X ~:parts {X}"
|
|
127 |
by (auto dest: parts_size)
|
|
128 |
|
|
129 |
subsubsection{*lemmas on keysFor*}
|
|
130 |
|
|
131 |
constdefs usekeys :: "msg set => key set"
|
|
132 |
"usekeys G == {K. EX Y. Crypt K Y:G}"
|
|
133 |
|
|
134 |
lemma finite_keysFor [intro]: "finite G ==> finite (keysFor G)"
|
|
135 |
apply (simp add: keysFor_def)
|
|
136 |
apply (rule finite_UN_I, auto)
|
|
137 |
apply (erule finite_induct, auto)
|
|
138 |
apply (case_tac "EX K X. x = Crypt K X", clarsimp)
|
|
139 |
apply (subgoal_tac "{Ka. EX Xa. (Ka=K & Xa=X) | Crypt Ka Xa:F}
|
|
140 |
= insert K (usekeys F)", auto simp: usekeys_def)
|
|
141 |
by (subgoal_tac "{K. EX X. Crypt K X = x | Crypt K X:F} = usekeys F",
|
|
142 |
auto simp: usekeys_def)
|
|
143 |
|
|
144 |
subsubsection{*lemmas on parts*}
|
|
145 |
|
|
146 |
lemma parts_sub: "[| X:parts G; G<=H |] ==> X:parts H"
|
|
147 |
by (auto dest: parts_mono)
|
|
148 |
|
|
149 |
lemma parts_Diff [dest]: "X:parts (G - H) ==> X:parts G"
|
|
150 |
by (erule parts_sub, auto)
|
|
151 |
|
|
152 |
lemma parts_Diff_notin: "[| Y ~:H; Nonce n ~:parts (H - {Y}) |]
|
|
153 |
==> Nonce n ~:parts H"
|
|
154 |
by simp
|
|
155 |
|
|
156 |
lemmas parts_insert_substI = parts_insert [THEN ssubst]
|
|
157 |
lemmas parts_insert_substD = parts_insert [THEN sym, THEN ssubst]
|
|
158 |
|
|
159 |
lemma finite_parts_msg [iff]: "finite (parts {X})"
|
|
160 |
by (induct X, auto)
|
|
161 |
|
|
162 |
lemma finite_parts [intro]: "finite H ==> finite (parts H)"
|
|
163 |
apply (erule finite_induct, simp)
|
|
164 |
by (rule parts_insert_substI, simp)
|
|
165 |
|
|
166 |
lemma parts_parts: "[| X:parts {Y}; Y:parts G |] ==> X:parts G"
|
|
167 |
by (drule_tac x=Y in in_sub, drule parts_mono, auto)
|
|
168 |
|
|
169 |
lemma parts_parts_parts: "[| X:parts {Y}; Y:parts {Z}; Z:parts G |] ==> X:parts G"
|
|
170 |
by (auto dest: parts_parts)
|
|
171 |
|
|
172 |
lemma parts_parts_Crypt: "[| Crypt K X:parts G; Nonce n:parts {X} |]
|
|
173 |
==> Nonce n:parts G"
|
|
174 |
by (blast intro: parts.Body dest: parts_parts)
|
|
175 |
|
|
176 |
subsubsection{*lemmas on synth*}
|
|
177 |
|
|
178 |
lemma synth_sub: "[| X:synth G; G<=H |] ==> X:synth H"
|
|
179 |
by (auto dest: synth_mono)
|
|
180 |
|
|
181 |
lemma Crypt_synth [rule_format]: "[| X:synth G; Key K ~:G |] ==>
|
|
182 |
Crypt K Y:parts {X} --> Crypt K Y:parts G"
|
|
183 |
by (erule synth.induct, auto dest: parts_sub)
|
|
184 |
|
|
185 |
subsubsection{*lemmas on analz*}
|
|
186 |
|
|
187 |
lemma analz_UnI1 [intro]: "X:analz G ==> X:analz (G Un H)"
|
|
188 |
by (subgoal_tac "G <= G Un H", auto dest: analz_mono)
|
|
189 |
|
|
190 |
lemma analz_sub: "[| X:analz G; G <= H |] ==> X:analz H"
|
|
191 |
by (auto dest: analz_mono)
|
|
192 |
|
|
193 |
lemma analz_Diff [dest]: "X:analz (G - H) ==> X:analz G"
|
|
194 |
by (erule analz.induct, auto)
|
|
195 |
|
|
196 |
lemmas in_analz_subset_cong = analz_subset_cong [THEN subsetD]
|
|
197 |
|
|
198 |
lemma analz_eq: "A=A' ==> analz A = analz A'"
|
|
199 |
by auto
|
|
200 |
|
|
201 |
lemmas insert_commute_substI = insert_commute [THEN ssubst]
|
|
202 |
|
|
203 |
lemma analz_insertD: "[| Crypt K Y:H; Key (invKey K):H |]
|
|
204 |
==> analz (insert Y H) = analz H"
|
|
205 |
apply (rule_tac x="Crypt K Y" and P="%H. analz (insert Y H) = analz H"
|
|
206 |
in insert_absorb_substI, simp)
|
|
207 |
by (rule_tac insert_commute_substI, simp)
|
|
208 |
|
|
209 |
lemma must_decrypt [rule_format,dest]: "[| X:analz H; has_no_pair H |] ==>
|
|
210 |
X ~:H --> (EX K Y. Crypt K Y:H & Key (invKey K):H)"
|
|
211 |
by (erule analz.induct, auto)
|
|
212 |
|
|
213 |
lemma analz_needs_only_finite: "X:analz H ==> EX G. G <= H & finite G"
|
|
214 |
by (erule analz.induct, auto)
|
|
215 |
|
|
216 |
lemma notin_analz_insert: "X ~:analz (insert Y G) ==> X ~:analz G"
|
|
217 |
by auto
|
|
218 |
|
|
219 |
subsubsection{*lemmas on parts, synth and analz*}
|
|
220 |
|
|
221 |
lemma parts_invKey [rule_format,dest]:"X:parts {Y} ==>
|
|
222 |
X:analz (insert (Crypt K Y) H) --> X ~:analz H --> Key (invKey K):analz H"
|
|
223 |
by (erule parts.induct, auto dest: parts.Fst parts.Snd parts.Body)
|
|
224 |
|
|
225 |
lemma in_analz: "Y:analz H ==> EX X. X:H & Y:parts {X}"
|
|
226 |
by (erule analz.induct, auto intro: parts.Fst parts.Snd parts.Body)
|
|
227 |
|
|
228 |
lemmas in_analz_subset_parts = analz_subset_parts [THEN subsetD]
|
|
229 |
|
|
230 |
lemma Crypt_synth_insert: "[| Crypt K X:parts (insert Y H);
|
|
231 |
Y:synth (analz H); Key K ~:analz H |] ==> Crypt K X:parts H"
|
|
232 |
apply (drule parts_insert_substD, clarify)
|
|
233 |
apply (frule in_sub)
|
|
234 |
apply (frule parts_mono)
|
|
235 |
by auto
|
|
236 |
|
|
237 |
subsubsection{*greatest nonce used in a message*}
|
|
238 |
|
|
239 |
consts greatest_msg :: "msg => nat"
|
|
240 |
|
|
241 |
recdef greatest_msg "measure size"
|
|
242 |
"greatest_msg (Nonce n) = n"
|
|
243 |
"greatest_msg {|X,Y|} = max (greatest_msg X) (greatest_msg Y)"
|
|
244 |
"greatest_msg (Crypt K X) = greatest_msg X"
|
|
245 |
"greatest_msg other = 0"
|
|
246 |
|
|
247 |
lemma greatest_msg_is_greatest: "Nonce n:parts {X} ==> n <= greatest_msg X"
|
|
248 |
by (induct X, auto, arith+)
|
|
249 |
|
|
250 |
subsubsection{*sets of keys*}
|
|
251 |
|
|
252 |
constdefs keyset :: "msg set => bool"
|
|
253 |
"keyset G == ALL X. X:G --> (EX K. X = Key K)"
|
|
254 |
|
|
255 |
lemma keyset_in [dest]: "[| keyset G; X:G |] ==> EX K. X = Key K"
|
|
256 |
by (auto simp: keyset_def)
|
|
257 |
|
|
258 |
lemma MPair_notin_keyset [simp]: "keyset G ==> {|X,Y|} ~:G"
|
|
259 |
by auto
|
|
260 |
|
|
261 |
lemma Crypt_notin_keyset [simp]: "keyset G ==> Crypt K X ~:G"
|
|
262 |
by auto
|
|
263 |
|
|
264 |
lemma Nonce_notin_keyset [simp]: "keyset G ==> Nonce n ~:G"
|
|
265 |
by auto
|
|
266 |
|
|
267 |
lemma parts_keyset [simp]: "keyset G ==> parts G = G"
|
|
268 |
by (auto, erule parts.induct, auto)
|
|
269 |
|
|
270 |
subsubsection{*keys a priori necessary for decrypting the messages of G*}
|
|
271 |
|
|
272 |
constdefs keysfor :: "msg set => msg set"
|
|
273 |
"keysfor G == Key ` keysFor (parts G)"
|
|
274 |
|
|
275 |
lemma keyset_keysfor [iff]: "keyset (keysfor G)"
|
|
276 |
by (simp add: keyset_def keysfor_def, blast)
|
|
277 |
|
|
278 |
lemma keyset_Diff_keysfor [simp]: "keyset H ==> keyset (H - keysfor G)"
|
|
279 |
by (auto simp: keyset_def)
|
|
280 |
|
|
281 |
lemma keysfor_Crypt: "Crypt K X:parts G ==> Key (invKey K):keysfor G"
|
|
282 |
by (auto simp: keysfor_def Crypt_imp_invKey_keysFor)
|
|
283 |
|
|
284 |
lemma no_key_no_Crypt: "Key K ~:keysfor G ==> Crypt (invKey K) X ~:parts G"
|
|
285 |
by (auto dest: keysfor_Crypt)
|
|
286 |
|
|
287 |
lemma finite_keysfor [intro]: "finite G ==> finite (keysfor G)"
|
|
288 |
by (auto simp: keysfor_def intro: finite_UN_I)
|
|
289 |
|
|
290 |
subsubsection{*only the keys necessary for G are useful in analz*}
|
|
291 |
|
|
292 |
lemma analz_keyset: "keyset H ==>
|
|
293 |
analz (G Un H) = H - keysfor G Un (analz (G Un (H Int keysfor G)))"
|
|
294 |
apply (rule eq)
|
|
295 |
apply (erule analz.induct, blast)
|
|
296 |
apply (simp, blast dest: Un_upper1)
|
|
297 |
apply (simp, blast dest: Un_upper2)
|
|
298 |
apply (case_tac "Key (invKey K):H - keysfor G", clarsimp)
|
|
299 |
apply (drule_tac X=X in no_key_no_Crypt)
|
|
300 |
by (auto intro: analz_sub)
|
|
301 |
|
|
302 |
lemmas analz_keyset_substD = analz_keyset [THEN sym, THEN ssubst]
|
|
303 |
|
|
304 |
|
|
305 |
subsection{*Extensions to Theory @{text Event}*}
|
|
306 |
|
|
307 |
|
|
308 |
subsubsection{*general protocol properties*}
|
|
309 |
|
|
310 |
constdefs is_Says :: "event => bool"
|
|
311 |
"is_Says ev == (EX A B X. ev = Says A B X)"
|
|
312 |
|
|
313 |
lemma is_Says_Says [iff]: "is_Says (Says A B X)"
|
|
314 |
by (simp add: is_Says_def)
|
|
315 |
|
|
316 |
(* one could also require that Gets occurs after Says
|
|
317 |
but this is sufficient for our purpose *)
|
|
318 |
constdefs Gets_correct :: "event list set => bool"
|
|
319 |
"Gets_correct p == ALL evs B X. evs:p --> Gets B X:set evs
|
|
320 |
--> (EX A. Says A B X:set evs)"
|
|
321 |
|
|
322 |
lemma Gets_correct_Says: "[| Gets_correct p; Gets B X # evs:p |]
|
|
323 |
==> EX A. Says A B X:set evs"
|
|
324 |
apply (simp add: Gets_correct_def)
|
|
325 |
by (drule_tac x="Gets B X # evs" in spec, auto)
|
|
326 |
|
|
327 |
constdefs one_step :: "event list set => bool"
|
|
328 |
"one_step p == ALL evs ev. ev#evs:p --> evs:p"
|
|
329 |
|
|
330 |
lemma one_step_Cons [dest]: "[| one_step p; ev#evs:p |] ==> evs:p"
|
|
331 |
by (unfold one_step_def, blast)
|
|
332 |
|
|
333 |
lemma one_step_app: "[| evs@evs':p; one_step p; []:p |] ==> evs':p"
|
|
334 |
by (induct evs, auto)
|
|
335 |
|
|
336 |
lemma trunc: "[| evs @ evs':p; one_step p |] ==> evs':p"
|
|
337 |
by (induct evs, auto)
|
|
338 |
|
|
339 |
constdefs has_only_Says :: "event list set => bool"
|
|
340 |
"has_only_Says p == ALL evs ev. evs:p --> ev:set evs
|
|
341 |
--> (EX A B X. ev = Says A B X)"
|
|
342 |
|
|
343 |
lemma has_only_SaysD: "[| ev:set evs; evs:p; has_only_Says p |]
|
|
344 |
==> EX A B X. ev = Says A B X"
|
|
345 |
by (unfold has_only_Says_def, blast)
|
|
346 |
|
|
347 |
lemma in_has_only_Says [dest]: "[| has_only_Says p; evs:p; ev:set evs |]
|
|
348 |
==> EX A B X. ev = Says A B X"
|
|
349 |
by (auto simp: has_only_Says_def)
|
|
350 |
|
|
351 |
lemma has_only_Says_imp_Gets_correct [simp]: "has_only_Says p
|
|
352 |
==> Gets_correct p"
|
|
353 |
by (auto simp: has_only_Says_def Gets_correct_def)
|
|
354 |
|
|
355 |
subsubsection{*lemma on knows*}
|
|
356 |
|
|
357 |
lemma Says_imp_spies2: "Says A B {|X,Y|}:set evs ==> Y:parts (spies evs)"
|
|
358 |
by (drule Says_imp_spies, drule parts.Inj, drule parts.Snd, simp)
|
|
359 |
|
|
360 |
lemma Says_not_parts: "[| Says A B X:set evs; Y ~:parts (spies evs) |]
|
|
361 |
==> Y ~:parts {X}"
|
|
362 |
by (auto dest: Says_imp_spies parts_parts)
|
|
363 |
|
|
364 |
subsubsection{*knows without initState*}
|
|
365 |
|
|
366 |
consts knows' :: "agent => event list => msg set"
|
|
367 |
|
|
368 |
primrec
|
|
369 |
"knows' A [] = {}"
|
|
370 |
"knows' A (ev # evs) = (
|
|
371 |
if A = Spy then (
|
|
372 |
case ev of
|
|
373 |
Says A' B X => insert X (knows' A evs)
|
|
374 |
| Gets A' X => knows' A evs
|
|
375 |
| Notes A' X => if A':bad then insert X (knows' A evs) else knows' A evs
|
|
376 |
) else (
|
|
377 |
case ev of
|
|
378 |
Says A' B X => if A=A' then insert X (knows' A evs) else knows' A evs
|
|
379 |
| Gets A' X => if A=A' then insert X (knows' A evs) else knows' A evs
|
|
380 |
| Notes A' X => if A=A' then insert X (knows' A evs) else knows' A evs
|
|
381 |
))"
|
|
382 |
|
|
383 |
translations "spies" == "knows Spy"
|
|
384 |
|
|
385 |
syntax spies' :: "event list => msg set"
|
|
386 |
|
|
387 |
translations "spies'" == "knows' Spy"
|
|
388 |
|
|
389 |
subsubsection{*decomposition of knows into knows' and initState*}
|
|
390 |
|
|
391 |
lemma knows_decomp: "knows A evs = knows' A evs Un (initState A)"
|
|
392 |
by (induct evs, auto split: event.split simp: knows.simps)
|
|
393 |
|
|
394 |
lemmas knows_decomp_substI = knows_decomp [THEN ssubst]
|
|
395 |
lemmas knows_decomp_substD = knows_decomp [THEN sym, THEN ssubst]
|
|
396 |
|
|
397 |
lemma knows'_sub_knows: "knows' A evs <= knows A evs"
|
|
398 |
by (auto simp: knows_decomp)
|
|
399 |
|
|
400 |
lemma knows'_Cons: "knows' A (ev#evs) = knows' A [ev] Un knows' A evs"
|
|
401 |
by (induct ev, auto)
|
|
402 |
|
|
403 |
lemmas knows'_Cons_substI = knows'_Cons [THEN ssubst]
|
|
404 |
lemmas knows'_Cons_substD = knows'_Cons [THEN sym, THEN ssubst]
|
|
405 |
|
|
406 |
lemma knows_Cons: "knows A (ev#evs) = initState A Un knows' A [ev]
|
|
407 |
Un knows A evs"
|
|
408 |
apply (simp only: knows_decomp)
|
|
409 |
apply (rule_tac s="(knows' A [ev] Un knows' A evs) Un initState A" in trans)
|
|
410 |
by (rule Un_eq, rule knows'_Cons, simp, blast)
|
|
411 |
|
|
412 |
lemmas knows_Cons_substI = knows_Cons [THEN ssubst]
|
|
413 |
lemmas knows_Cons_substD = knows_Cons [THEN sym, THEN ssubst]
|
|
414 |
|
|
415 |
lemma knows'_sub_spies': "[| evs:p; has_only_Says p; one_step p |]
|
|
416 |
==> knows' A evs <= spies' evs"
|
|
417 |
by (induct evs, auto split: event.splits)
|
|
418 |
|
|
419 |
subsubsection{*knows' is finite*}
|
|
420 |
|
|
421 |
lemma finite_knows' [iff]: "finite (knows' A evs)"
|
|
422 |
by (induct evs, auto split: event.split simp: knows.simps)
|
|
423 |
|
|
424 |
subsubsection{*monotonicity of knows*}
|
|
425 |
|
|
426 |
lemma knows_sub_Cons: "knows A evs <= knows A (ev#evs)"
|
13596
|
427 |
by(cases A, induct evs, auto simp: knows.simps split:event.split)
|
13508
|
428 |
|
|
429 |
lemma knows_ConsI: "X:knows A evs ==> X:knows A (ev#evs)"
|
|
430 |
by (auto dest: knows_sub_Cons [THEN subsetD])
|
|
431 |
|
|
432 |
lemma knows_sub_app: "knows A evs <= knows A (evs @ evs')"
|
|
433 |
apply (induct evs, auto)
|
|
434 |
apply (simp add: knows_decomp)
|
|
435 |
by (case_tac a, auto simp: knows.simps)
|
|
436 |
|
|
437 |
subsubsection{*maximum knowledge an agent can have
|
|
438 |
includes messages sent to the agent*}
|
|
439 |
|
|
440 |
consts knows_max' :: "agent => event list => msg set"
|
|
441 |
|
|
442 |
primrec
|
|
443 |
knows_max'_def_Nil: "knows_max' A [] = {}"
|
|
444 |
knows_max'_def_Cons: "knows_max' A (ev # evs) = (
|
|
445 |
if A=Spy then (
|
|
446 |
case ev of
|
|
447 |
Says A' B X => insert X (knows_max' A evs)
|
|
448 |
| Gets A' X => knows_max' A evs
|
|
449 |
| Notes A' X =>
|
|
450 |
if A':bad then insert X (knows_max' A evs) else knows_max' A evs
|
|
451 |
) else (
|
|
452 |
case ev of
|
|
453 |
Says A' B X =>
|
|
454 |
if A=A' | A=B then insert X (knows_max' A evs) else knows_max' A evs
|
|
455 |
| Gets A' X =>
|
|
456 |
if A=A' then insert X (knows_max' A evs) else knows_max' A evs
|
|
457 |
| Notes A' X =>
|
|
458 |
if A=A' then insert X (knows_max' A evs) else knows_max' A evs
|
|
459 |
))"
|
|
460 |
|
|
461 |
constdefs knows_max :: "agent => event list => msg set"
|
|
462 |
"knows_max A evs == knows_max' A evs Un initState A"
|
|
463 |
|
|
464 |
consts spies_max :: "event list => msg set"
|
|
465 |
|
|
466 |
translations "spies_max evs" == "knows_max Spy evs"
|
|
467 |
|
|
468 |
subsubsection{*basic facts about @{term knows_max}*}
|
|
469 |
|
|
470 |
lemma spies_max_spies [iff]: "spies_max evs = spies evs"
|
|
471 |
by (induct evs, auto simp: knows_max_def split: event.splits)
|
|
472 |
|
|
473 |
lemma knows_max'_Cons: "knows_max' A (ev#evs)
|
|
474 |
= knows_max' A [ev] Un knows_max' A evs"
|
|
475 |
by (auto split: event.splits)
|
|
476 |
|
|
477 |
lemmas knows_max'_Cons_substI = knows_max'_Cons [THEN ssubst]
|
|
478 |
lemmas knows_max'_Cons_substD = knows_max'_Cons [THEN sym, THEN ssubst]
|
|
479 |
|
|
480 |
lemma knows_max_Cons: "knows_max A (ev#evs)
|
|
481 |
= knows_max' A [ev] Un knows_max A evs"
|
|
482 |
apply (simp add: knows_max_def del: knows_max'_def_Cons)
|
|
483 |
apply (rule_tac evs1=evs in knows_max'_Cons_substI)
|
|
484 |
by blast
|
|
485 |
|
|
486 |
lemmas knows_max_Cons_substI = knows_max_Cons [THEN ssubst]
|
|
487 |
lemmas knows_max_Cons_substD = knows_max_Cons [THEN sym, THEN ssubst]
|
|
488 |
|
|
489 |
lemma finite_knows_max' [iff]: "finite (knows_max' A evs)"
|
|
490 |
by (induct evs, auto split: event.split)
|
|
491 |
|
|
492 |
lemma knows_max'_sub_spies': "[| evs:p; has_only_Says p; one_step p |]
|
|
493 |
==> knows_max' A evs <= spies' evs"
|
|
494 |
by (induct evs, auto split: event.splits)
|
|
495 |
|
|
496 |
lemma knows_max'_in_spies' [dest]: "[| evs:p; X:knows_max' A evs;
|
|
497 |
has_only_Says p; one_step p |] ==> X:spies' evs"
|
|
498 |
by (rule knows_max'_sub_spies' [THEN subsetD], auto)
|
|
499 |
|
|
500 |
lemma knows_max'_app: "knows_max' A (evs @ evs')
|
|
501 |
= knows_max' A evs Un knows_max' A evs'"
|
|
502 |
by (induct evs, auto split: event.splits)
|
|
503 |
|
|
504 |
lemma Says_to_knows_max': "Says A B X:set evs ==> X:knows_max' B evs"
|
|
505 |
by (simp add: in_set_conv_decomp, clarify, simp add: knows_max'_app)
|
|
506 |
|
|
507 |
lemma Says_from_knows_max': "Says A B X:set evs ==> X:knows_max' A evs"
|
|
508 |
by (simp add: in_set_conv_decomp, clarify, simp add: knows_max'_app)
|
|
509 |
|
|
510 |
subsubsection{*used without initState*}
|
|
511 |
|
|
512 |
consts used' :: "event list => msg set"
|
|
513 |
|
|
514 |
primrec
|
|
515 |
"used' [] = {}"
|
|
516 |
"used' (ev # evs) = (
|
|
517 |
case ev of
|
|
518 |
Says A B X => parts {X} Un used' evs
|
|
519 |
| Gets A X => used' evs
|
|
520 |
| Notes A X => parts {X} Un used' evs
|
|
521 |
)"
|
|
522 |
|
|
523 |
constdefs init :: "msg set"
|
|
524 |
"init == used []"
|
|
525 |
|
|
526 |
lemma used_decomp: "used evs = init Un used' evs"
|
|
527 |
by (induct evs, auto simp: init_def split: event.split)
|
|
528 |
|
|
529 |
lemma used'_sub_app: "used' evs <= used' (evs@evs')"
|
|
530 |
by (induct evs, auto split: event.split)
|
|
531 |
|
|
532 |
lemma used'_parts [rule_format]: "X:used' evs ==> Y:parts {X} --> Y:used' evs"
|
|
533 |
apply (induct evs, simp)
|
|
534 |
apply (case_tac a, simp_all)
|
|
535 |
apply (blast dest: parts_trans)+;
|
|
536 |
done
|
|
537 |
|
|
538 |
subsubsection{*monotonicity of used*}
|
|
539 |
|
|
540 |
lemma used_sub_Cons: "used evs <= used (ev#evs)"
|
|
541 |
by (induct evs, (induct ev, auto)+)
|
|
542 |
|
|
543 |
lemma used_ConsI: "X:used evs ==> X:used (ev#evs)"
|
|
544 |
by (auto dest: used_sub_Cons [THEN subsetD])
|
|
545 |
|
|
546 |
lemma notin_used_ConsD: "X ~:used (ev#evs) ==> X ~:used evs"
|
|
547 |
by (auto dest: used_sub_Cons [THEN subsetD])
|
|
548 |
|
|
549 |
lemma used_appD [dest]: "X:used (evs @ evs') ==> X:used evs | X:used evs'"
|
|
550 |
by (induct evs, auto, case_tac a, auto)
|
|
551 |
|
|
552 |
lemma used_ConsD: "X:used (ev#evs) ==> X:used [ev] | X:used evs"
|
|
553 |
by (case_tac ev, auto)
|
|
554 |
|
|
555 |
lemma used_sub_app: "used evs <= used (evs@evs')"
|
|
556 |
by (auto simp: used_decomp dest: used'_sub_app [THEN subsetD])
|
|
557 |
|
|
558 |
lemma used_appIL: "X:used evs ==> X:used (evs' @ evs)"
|
|
559 |
by (induct evs', auto intro: used_ConsI)
|
|
560 |
|
|
561 |
lemma used_appIR: "X:used evs ==> X:used (evs @ evs')"
|
|
562 |
by (erule used_sub_app [THEN subsetD])
|
|
563 |
|
|
564 |
lemma used_parts: "[| X:parts {Y}; Y:used evs |] ==> X:used evs"
|
|
565 |
apply (auto simp: used_decomp dest: used'_parts)
|
|
566 |
by (auto simp: init_def used_Nil dest: parts_trans)
|
|
567 |
|
|
568 |
lemma parts_Says_used: "[| Says A B X:set evs; Y:parts {X} |] ==> Y:used evs"
|
|
569 |
by (induct evs, simp_all, safe, auto intro: used_ConsI)
|
|
570 |
|
|
571 |
lemma parts_used_app: "X:parts {Y} ==> X:used (evs @ Says A B Y # evs')"
|
|
572 |
apply (drule_tac evs="[Says A B Y]" in used_parts, simp, blast)
|
|
573 |
apply (drule_tac evs'=evs' in used_appIR)
|
|
574 |
apply (drule_tac evs'=evs in used_appIL)
|
|
575 |
by simp
|
|
576 |
|
|
577 |
subsubsection{*lemmas on used and knows*}
|
|
578 |
|
|
579 |
lemma initState_used: "X:parts (initState A) ==> X:used evs"
|
|
580 |
by (induct evs, auto simp: used.simps split: event.split)
|
|
581 |
|
|
582 |
lemma Says_imp_used: "Says A B X:set evs ==> parts {X} <= used evs"
|
|
583 |
by (induct evs, auto intro: used_ConsI)
|
|
584 |
|
|
585 |
lemma not_used_not_spied: "X ~:used evs ==> X ~:parts (spies evs)"
|
|
586 |
by (induct evs, auto simp: used_Nil)
|
|
587 |
|
|
588 |
lemma not_used_not_parts: "[| Y ~:used evs; Says A B X:set evs |]
|
|
589 |
==> Y ~:parts {X}"
|
|
590 |
by (induct evs, auto intro: used_ConsI)
|
|
591 |
|
|
592 |
lemma not_used_parts_false: "[| X ~:used evs; Y:parts (spies evs) |]
|
|
593 |
==> X ~:parts {Y}"
|
|
594 |
by (auto dest: parts_parts)
|
|
595 |
|
|
596 |
lemma known_used [rule_format]: "[| evs:p; Gets_correct p; one_step p |]
|
|
597 |
==> X:parts (knows A evs) --> X:used evs"
|
|
598 |
apply (case_tac "A=Spy", blast dest: parts_knows_Spy_subset_used)
|
|
599 |
apply (induct evs)
|
|
600 |
apply (simp add: used.simps, blast)
|
|
601 |
apply (frule_tac ev=a and evs=list in one_step_Cons, simp, clarify)
|
|
602 |
apply (drule_tac P="%G. X:parts G" in knows_Cons_substD, safe)
|
|
603 |
apply (erule initState_used)
|
|
604 |
apply (case_tac a, auto)
|
|
605 |
apply (drule_tac B=A and X=msg and evs=list in Gets_correct_Says)
|
|
606 |
by (auto dest: Says_imp_used intro: used_ConsI)
|
|
607 |
|
|
608 |
lemma known_max_used [rule_format]: "[| evs:p; Gets_correct p; one_step p |]
|
|
609 |
==> X:parts (knows_max A evs) --> X:used evs"
|
|
610 |
apply (case_tac "A=Spy")
|
|
611 |
apply (simp, blast dest: parts_knows_Spy_subset_used)
|
|
612 |
apply (induct evs)
|
|
613 |
apply (simp add: knows_max_def used.simps, blast)
|
|
614 |
apply (frule_tac ev=a and evs=list in one_step_Cons, simp, clarify)
|
|
615 |
apply (drule_tac P="%G. X:parts G" in knows_max_Cons_substD, safe)
|
|
616 |
apply (case_tac a, auto)
|
|
617 |
apply (drule_tac B=A and X=msg and evs=list in Gets_correct_Says)
|
|
618 |
by (auto simp: knows_max'_Cons dest: Says_imp_used intro: used_ConsI)
|
|
619 |
|
|
620 |
lemma not_used_not_known: "[| evs:p; X ~:used evs;
|
|
621 |
Gets_correct p; one_step p |] ==> X ~:parts (knows A evs)"
|
|
622 |
by (case_tac "A=Spy", auto dest: not_used_not_spied known_used)
|
|
623 |
|
|
624 |
lemma not_used_not_known_max: "[| evs:p; X ~:used evs;
|
|
625 |
Gets_correct p; one_step p |] ==> X ~:parts (knows_max A evs)"
|
|
626 |
by (case_tac "A=Spy", auto dest: not_used_not_spied known_max_used)
|
|
627 |
|
|
628 |
subsubsection{*a nonce or key in a message cannot equal a fresh nonce or key*}
|
|
629 |
|
|
630 |
lemma Nonce_neq [dest]: "[| Nonce n' ~:used evs;
|
|
631 |
Says A B X:set evs; Nonce n:parts {X} |] ==> n ~= n'"
|
|
632 |
by (drule not_used_not_spied, auto dest: Says_imp_knows_Spy parts_sub)
|
|
633 |
|
|
634 |
lemma Key_neq [dest]: "[| Key n' ~:used evs;
|
|
635 |
Says A B X:set evs; Key n:parts {X} |] ==> n ~= n'"
|
|
636 |
by (drule not_used_not_spied, auto dest: Says_imp_knows_Spy parts_sub)
|
|
637 |
|
|
638 |
subsubsection{*message of an event*}
|
|
639 |
|
|
640 |
consts msg :: "event => msg"
|
|
641 |
|
|
642 |
recdef msg "measure size"
|
|
643 |
"msg (Says A B X) = X"
|
|
644 |
"msg (Gets A X) = X"
|
|
645 |
"msg (Notes A X) = X"
|
|
646 |
|
|
647 |
lemma used_sub_parts_used: "X:used (ev # evs) ==> X:parts {msg ev} Un used evs"
|
|
648 |
by (induct ev, auto)
|
|
649 |
|
|
650 |
|
|
651 |
|
|
652 |
end
|