1 (******************************************************************************
3 author: Frederic Blanqui
5 webpage: http://www.lri.fr/~blanqui/
7 University of Cambridge, Computer Laboratory
8 William Gates Building, JJ Thomson Avenue
9 Cambridge CB3 0FD, United Kingdom
10 ******************************************************************************)
12 header{*Decomposition of Analz into two parts*}
14 theory Analz imports Extensions begin
16 text{*decomposition of @{term analz} into two parts:
17 @{term pparts} (for pairs) and analz of @{term kparts}*}
19 subsection{*messages that do not contribute to analz*}
22 pparts :: "msg set => msg set"
25 Inj [intro]: "[| X:H; is_MPair X |] ==> X:pparts H"
26 | Fst [dest]: "[| {|X,Y|}:pparts H; is_MPair X |] ==> X:pparts H"
27 | Snd [dest]: "[| {|X,Y|}:pparts H; is_MPair Y |] ==> Y:pparts H"
29 subsection{*basic facts about @{term pparts}*}
31 lemma pparts_is_MPair [dest]: "X:pparts H ==> is_MPair X"
32 by (erule pparts.induct, auto)
34 lemma Crypt_notin_pparts [iff]: "Crypt K X ~:pparts H"
37 lemma Key_notin_pparts [iff]: "Key K ~:pparts H"
40 lemma Nonce_notin_pparts [iff]: "Nonce n ~:pparts H"
43 lemma Number_notin_pparts [iff]: "Number n ~:pparts H"
46 lemma Agent_notin_pparts [iff]: "Agent A ~:pparts H"
49 lemma pparts_empty [iff]: "pparts {} = {}"
50 by (auto, erule pparts.induct, auto)
52 lemma pparts_insertI [intro]: "X:pparts H ==> X:pparts (insert Y H)"
53 by (erule pparts.induct, auto)
55 lemma pparts_sub: "[| X:pparts G; G<=H |] ==> X:pparts H"
56 by (erule pparts.induct, auto)
58 lemma pparts_insert2 [iff]: "pparts (insert X (insert Y H))
59 = pparts {X} Un pparts {Y} Un pparts H"
60 by (rule eq, (erule pparts.induct, auto)+)
62 lemma pparts_insert_MPair [iff]: "pparts (insert {|X,Y|} H)
63 = insert {|X,Y|} (pparts ({X,Y} Un H))"
64 apply (rule eq, (erule pparts.induct, auto)+)
65 apply (rule_tac Y=Y in pparts.Fst, auto)
66 apply (erule pparts.induct, auto)
67 by (rule_tac X=X in pparts.Snd, auto)
69 lemma pparts_insert_Nonce [iff]: "pparts (insert (Nonce n) H) = pparts H"
70 by (rule eq, erule pparts.induct, auto)
72 lemma pparts_insert_Crypt [iff]: "pparts (insert (Crypt K X) H) = pparts H"
73 by (rule eq, erule pparts.induct, auto)
75 lemma pparts_insert_Key [iff]: "pparts (insert (Key K) H) = pparts H"
76 by (rule eq, erule pparts.induct, auto)
78 lemma pparts_insert_Agent [iff]: "pparts (insert (Agent A) H) = pparts H"
79 by (rule eq, erule pparts.induct, auto)
81 lemma pparts_insert_Number [iff]: "pparts (insert (Number n) H) = pparts H"
82 by (rule eq, erule pparts.induct, auto)
84 lemma pparts_insert_Hash [iff]: "pparts (insert (Hash X) H) = pparts H"
85 by (rule eq, erule pparts.induct, auto)
87 lemma pparts_insert: "X:pparts (insert Y H) ==> X:pparts {Y} Un pparts H"
88 by (erule pparts.induct, blast+)
90 lemma insert_pparts: "X:pparts {Y} Un pparts H ==> X:pparts (insert Y H)"
91 by (safe, erule pparts.induct, auto)
93 lemma pparts_Un [iff]: "pparts (G Un H) = pparts G Un pparts H"
94 by (rule eq, erule pparts.induct, auto dest: pparts_sub)
96 lemma pparts_pparts [iff]: "pparts (pparts H) = pparts H"
97 by (rule eq, erule pparts.induct, auto)
99 lemma pparts_insert_eq: "pparts (insert X H) = pparts {X} Un pparts H"
100 by (rule_tac A=H in insert_Un, rule pparts_Un)
102 lemmas pparts_insert_substI = pparts_insert_eq [THEN ssubst]
104 lemma in_pparts: "Y:pparts H ==> EX X. X:H & Y:pparts {X}"
105 by (erule pparts.induct, auto)
107 subsection{*facts about @{term pparts} and @{term parts}*}
109 lemma pparts_no_Nonce [dest]: "[| X:pparts {Y}; Nonce n ~:parts {Y} |]
110 ==> Nonce n ~:parts {X}"
111 by (erule pparts.induct, simp_all)
113 subsection{*facts about @{term pparts} and @{term analz}*}
115 lemma pparts_analz: "X:pparts H ==> X:analz H"
116 by (erule pparts.induct, auto)
118 lemma pparts_analz_sub: "[| X:pparts G; G<=H |] ==> X:analz H"
119 by (auto dest: pparts_sub pparts_analz)
121 subsection{*messages that contribute to analz*}
124 kparts :: "msg set => msg set"
127 Inj [intro]: "[| X:H; not_MPair X |] ==> X:kparts H"
128 | Fst [intro]: "[| {|X,Y|}:pparts H; not_MPair X |] ==> X:kparts H"
129 | Snd [intro]: "[| {|X,Y|}:pparts H; not_MPair Y |] ==> Y:kparts H"
131 subsection{*basic facts about @{term kparts}*}
133 lemma kparts_not_MPair [dest]: "X:kparts H ==> not_MPair X"
134 by (erule kparts.induct, auto)
136 lemma kparts_empty [iff]: "kparts {} = {}"
137 by (rule eq, erule kparts.induct, auto)
139 lemma kparts_insertI [intro]: "X:kparts H ==> X:kparts (insert Y H)"
140 by (erule kparts.induct, auto dest: pparts_insertI)
142 lemma kparts_insert2 [iff]: "kparts (insert X (insert Y H))
143 = kparts {X} Un kparts {Y} Un kparts H"
144 by (rule eq, (erule kparts.induct, auto)+)
146 lemma kparts_insert_MPair [iff]: "kparts (insert {|X,Y|} H)
147 = kparts ({X,Y} Un H)"
148 by (rule eq, (erule kparts.induct, auto)+)
150 lemma kparts_insert_Nonce [iff]: "kparts (insert (Nonce n) H)
151 = insert (Nonce n) (kparts H)"
152 by (rule eq, erule kparts.induct, auto)
154 lemma kparts_insert_Crypt [iff]: "kparts (insert (Crypt K X) H)
155 = insert (Crypt K X) (kparts H)"
156 by (rule eq, erule kparts.induct, auto)
158 lemma kparts_insert_Key [iff]: "kparts (insert (Key K) H)
159 = insert (Key K) (kparts H)"
160 by (rule eq, erule kparts.induct, auto)
162 lemma kparts_insert_Agent [iff]: "kparts (insert (Agent A) H)
163 = insert (Agent A) (kparts H)"
164 by (rule eq, erule kparts.induct, auto)
166 lemma kparts_insert_Number [iff]: "kparts (insert (Number n) H)
167 = insert (Number n) (kparts H)"
168 by (rule eq, erule kparts.induct, auto)
170 lemma kparts_insert_Hash [iff]: "kparts (insert (Hash X) H)
171 = insert (Hash X) (kparts H)"
172 by (rule eq, erule kparts.induct, auto)
174 lemma kparts_insert: "X:kparts (insert X H) ==> X:kparts {X} Un kparts H"
175 by (erule kparts.induct, (blast dest: pparts_insert)+)
177 lemma kparts_insert_fst [rule_format,dest]: "X:kparts (insert Z H) ==>
178 X ~:kparts H --> X:kparts {Z}"
179 by (erule kparts.induct, (blast dest: pparts_insert)+)
181 lemma kparts_sub: "[| X:kparts G; G<=H |] ==> X:kparts H"
182 by (erule kparts.induct, auto dest: pparts_sub)
184 lemma kparts_Un [iff]: "kparts (G Un H) = kparts G Un kparts H"
185 by (rule eq, erule kparts.induct, auto dest: kparts_sub)
187 lemma pparts_kparts [iff]: "pparts (kparts H) = {}"
188 by (rule eq, erule pparts.induct, auto)
190 lemma kparts_kparts [iff]: "kparts (kparts H) = kparts H"
191 by (rule eq, erule kparts.induct, auto)
193 lemma kparts_insert_eq: "kparts (insert X H) = kparts {X} Un kparts H"
194 by (rule_tac A=H in insert_Un, rule kparts_Un)
196 lemmas kparts_insert_substI = kparts_insert_eq [THEN ssubst]
198 lemma in_kparts: "Y:kparts H ==> EX X. X:H & Y:kparts {X}"
199 by (erule kparts.induct, auto dest: in_pparts)
201 lemma kparts_has_no_pair [iff]: "has_no_pair (kparts H)"
204 subsection{*facts about @{term kparts} and @{term parts}*}
206 lemma kparts_no_Nonce [dest]: "[| X:kparts {Y}; Nonce n ~:parts {Y} |]
207 ==> Nonce n ~:parts {X}"
208 by (erule kparts.induct, auto)
210 lemma kparts_parts: "X:kparts H ==> X:parts H"
211 by (erule kparts.induct, auto dest: pparts_analz)
213 lemma parts_kparts: "X:parts (kparts H) ==> X:parts H"
214 by (erule parts.induct, auto dest: kparts_parts
215 intro: parts.Fst parts.Snd parts.Body)
217 lemma Crypt_kparts_Nonce_parts [dest]: "[| Crypt K Y:kparts {Z};
218 Nonce n:parts {Y} |] ==> Nonce n:parts {Z}"
221 subsection{*facts about @{term kparts} and @{term analz}*}
223 lemma kparts_analz: "X:kparts H ==> X:analz H"
224 by (erule kparts.induct, auto dest: pparts_analz)
226 lemma kparts_analz_sub: "[| X:kparts G; G<=H |] ==> X:analz H"
227 by (erule kparts.induct, auto dest: pparts_analz_sub)
229 lemma analz_kparts [rule_format,dest]: "X:analz H ==>
230 Y:kparts {X} --> Y:analz H"
231 by (erule analz.induct, auto dest: kparts_analz_sub)
233 lemma analz_kparts_analz: "X:analz (kparts H) ==> X:analz H"
234 by (erule analz.induct, auto dest: kparts_analz)
236 lemma analz_kparts_insert: "X:analz (kparts (insert Z H)) ==> X:analz (kparts {Z} Un kparts H)"
237 by (rule analz_sub, auto)
239 lemma Nonce_kparts_synth [rule_format]: "Y:synth (analz G)
240 ==> Nonce n:kparts {Y} --> Nonce n:analz G"
241 by (erule synth.induct, auto)
243 lemma kparts_insert_synth: "[| Y:parts (insert X G); X:synth (analz G);
244 Nonce n:kparts {Y}; Nonce n ~:analz G |] ==> Y:parts G"
245 apply (drule parts_insert_substD [where P="%S. Y : S"], clarify)
246 apply (drule in_sub, drule_tac X=Y in parts_sub, simp)
247 by (auto dest: Nonce_kparts_synth)
249 lemma Crypt_insert_synth:
250 "[| Crypt K Y:parts (insert X G); X:synth (analz G); Nonce n:kparts {Y}; Nonce n ~:analz G |]
251 ==> Crypt K Y:parts G"
252 by (metis Fake_parts_insert_in_Un Nonce_kparts_synth UnE analz_conj_parts synth_simps(5))
255 subsection{*analz is pparts + analz of kparts*}
257 lemma analz_pparts_kparts: "X:analz H ==> X:pparts H | X:analz (kparts H)"
258 by (erule analz.induct, auto)
260 lemma analz_pparts_kparts_eq: "analz H = pparts H Un analz (kparts H)"
261 by (rule eq, auto dest: analz_pparts_kparts pparts_analz analz_kparts_analz)
263 lemmas analz_pparts_kparts_substI = analz_pparts_kparts_eq [THEN ssubst]
264 lemmas analz_pparts_kparts_substD = analz_pparts_kparts_eq [THEN sym, THEN ssubst]