(******************************************************************************
date: december 2001
author: Frederic Blanqui
email: blanqui@lri.fr
webpage: http://www.lri.fr/~blanqui/
University of Cambridge, Computer Laboratory
William Gates Building, JJ Thomson Avenue
Cambridge CB3 0FD, United Kingdom
******************************************************************************)
header{*Decomposition of Analz into two parts*}
theory Analz imports Extensions begin
text{*decomposition of @{term analz} into two parts:
@{term pparts} (for pairs) and analz of @{term kparts}*}
subsection{*messages that do not contribute to analz*}
inductive_set
pparts :: "msg set => msg set"
for H :: "msg set"
where
Inj [intro]: "[| X:H; is_MPair X |] ==> X:pparts H"
| Fst [dest]: "[| {|X,Y|}:pparts H; is_MPair X |] ==> X:pparts H"
| Snd [dest]: "[| {|X,Y|}:pparts H; is_MPair Y |] ==> Y:pparts H"
subsection{*basic facts about @{term pparts}*}
lemma pparts_is_MPair [dest]: "X:pparts H ==> is_MPair X"
by (erule pparts.induct, auto)
lemma Crypt_notin_pparts [iff]: "Crypt K X ~:pparts H"
by auto
lemma Key_notin_pparts [iff]: "Key K ~:pparts H"
by auto
lemma Nonce_notin_pparts [iff]: "Nonce n ~:pparts H"
by auto
lemma Number_notin_pparts [iff]: "Number n ~:pparts H"
by auto
lemma Agent_notin_pparts [iff]: "Agent A ~:pparts H"
by auto
lemma pparts_empty [iff]: "pparts {} = {}"
by (auto, erule pparts.induct, auto)
lemma pparts_insertI [intro]: "X:pparts H ==> X:pparts (insert Y H)"
by (erule pparts.induct, auto)
lemma pparts_sub: "[| X:pparts G; G<=H |] ==> X:pparts H"
by (erule pparts.induct, auto)
lemma pparts_insert2 [iff]: "pparts (insert X (insert Y H))
= pparts {X} Un pparts {Y} Un pparts H"
by (rule eq, (erule pparts.induct, auto)+)
lemma pparts_insert_MPair [iff]: "pparts (insert {|X,Y|} H)
= insert {|X,Y|} (pparts ({X,Y} Un H))"
apply (rule eq, (erule pparts.induct, auto)+)
apply (rule_tac Y=Y in pparts.Fst, auto)
apply (erule pparts.induct, auto)
by (rule_tac X=X in pparts.Snd, auto)
lemma pparts_insert_Nonce [iff]: "pparts (insert (Nonce n) H) = pparts H"
by (rule eq, erule pparts.induct, auto)
lemma pparts_insert_Crypt [iff]: "pparts (insert (Crypt K X) H) = pparts H"
by (rule eq, erule pparts.induct, auto)
lemma pparts_insert_Key [iff]: "pparts (insert (Key K) H) = pparts H"
by (rule eq, erule pparts.induct, auto)
lemma pparts_insert_Agent [iff]: "pparts (insert (Agent A) H) = pparts H"
by (rule eq, erule pparts.induct, auto)
lemma pparts_insert_Number [iff]: "pparts (insert (Number n) H) = pparts H"
by (rule eq, erule pparts.induct, auto)
lemma pparts_insert_Hash [iff]: "pparts (insert (Hash X) H) = pparts H"
by (rule eq, erule pparts.induct, auto)
lemma pparts_insert: "X:pparts (insert Y H) ==> X:pparts {Y} Un pparts H"
by (erule pparts.induct, blast+)
lemma insert_pparts: "X:pparts {Y} Un pparts H ==> X:pparts (insert Y H)"
by (safe, erule pparts.induct, auto)
lemma pparts_Un [iff]: "pparts (G Un H) = pparts G Un pparts H"
by (rule eq, erule pparts.induct, auto dest: pparts_sub)
lemma pparts_pparts [iff]: "pparts (pparts H) = pparts H"
by (rule eq, erule pparts.induct, auto)
lemma pparts_insert_eq: "pparts (insert X H) = pparts {X} Un pparts H"
by (rule_tac A=H in insert_Un, rule pparts_Un)
lemmas pparts_insert_substI = pparts_insert_eq [THEN ssubst]
lemma in_pparts: "Y:pparts H ==> EX X. X:H & Y:pparts {X}"
by (erule pparts.induct, auto)
subsection{*facts about @{term pparts} and @{term parts}*}
lemma pparts_no_Nonce [dest]: "[| X:pparts {Y}; Nonce n ~:parts {Y} |]
==> Nonce n ~:parts {X}"
by (erule pparts.induct, simp_all)
subsection{*facts about @{term pparts} and @{term analz}*}
lemma pparts_analz: "X:pparts H ==> X:analz H"
by (erule pparts.induct, auto)
lemma pparts_analz_sub: "[| X:pparts G; G<=H |] ==> X:analz H"
by (auto dest: pparts_sub pparts_analz)
subsection{*messages that contribute to analz*}
inductive_set
kparts :: "msg set => msg set"
for H :: "msg set"
where
Inj [intro]: "[| X:H; not_MPair X |] ==> X:kparts H"
| Fst [intro]: "[| {|X,Y|}:pparts H; not_MPair X |] ==> X:kparts H"
| Snd [intro]: "[| {|X,Y|}:pparts H; not_MPair Y |] ==> Y:kparts H"
subsection{*basic facts about @{term kparts}*}
lemma kparts_not_MPair [dest]: "X:kparts H ==> not_MPair X"
by (erule kparts.induct, auto)
lemma kparts_empty [iff]: "kparts {} = {}"
by (rule eq, erule kparts.induct, auto)
lemma kparts_insertI [intro]: "X:kparts H ==> X:kparts (insert Y H)"
by (erule kparts.induct, auto dest: pparts_insertI)
lemma kparts_insert2 [iff]: "kparts (insert X (insert Y H))
= kparts {X} Un kparts {Y} Un kparts H"
by (rule eq, (erule kparts.induct, auto)+)
lemma kparts_insert_MPair [iff]: "kparts (insert {|X,Y|} H)
= kparts ({X,Y} Un H)"
by (rule eq, (erule kparts.induct, auto)+)
lemma kparts_insert_Nonce [iff]: "kparts (insert (Nonce n) H)
= insert (Nonce n) (kparts H)"
by (rule eq, erule kparts.induct, auto)
lemma kparts_insert_Crypt [iff]: "kparts (insert (Crypt K X) H)
= insert (Crypt K X) (kparts H)"
by (rule eq, erule kparts.induct, auto)
lemma kparts_insert_Key [iff]: "kparts (insert (Key K) H)
= insert (Key K) (kparts H)"
by (rule eq, erule kparts.induct, auto)
lemma kparts_insert_Agent [iff]: "kparts (insert (Agent A) H)
= insert (Agent A) (kparts H)"
by (rule eq, erule kparts.induct, auto)
lemma kparts_insert_Number [iff]: "kparts (insert (Number n) H)
= insert (Number n) (kparts H)"
by (rule eq, erule kparts.induct, auto)
lemma kparts_insert_Hash [iff]: "kparts (insert (Hash X) H)
= insert (Hash X) (kparts H)"
by (rule eq, erule kparts.induct, auto)
lemma kparts_insert: "X:kparts (insert X H) ==> X:kparts {X} Un kparts H"
by (erule kparts.induct, (blast dest: pparts_insert)+)
lemma kparts_insert_fst [rule_format,dest]: "X:kparts (insert Z H) ==>
X ~:kparts H --> X:kparts {Z}"
by (erule kparts.induct, (blast dest: pparts_insert)+)
lemma kparts_sub: "[| X:kparts G; G<=H |] ==> X:kparts H"
by (erule kparts.induct, auto dest: pparts_sub)
lemma kparts_Un [iff]: "kparts (G Un H) = kparts G Un kparts H"
by (rule eq, erule kparts.induct, auto dest: kparts_sub)
lemma pparts_kparts [iff]: "pparts (kparts H) = {}"
by (rule eq, erule pparts.induct, auto)
lemma kparts_kparts [iff]: "kparts (kparts H) = kparts H"
by (rule eq, erule kparts.induct, auto)
lemma kparts_insert_eq: "kparts (insert X H) = kparts {X} Un kparts H"
by (rule_tac A=H in insert_Un, rule kparts_Un)
lemmas kparts_insert_substI = kparts_insert_eq [THEN ssubst]
lemma in_kparts: "Y:kparts H ==> EX X. X:H & Y:kparts {X}"
by (erule kparts.induct, auto dest: in_pparts)
lemma kparts_has_no_pair [iff]: "has_no_pair (kparts H)"
by auto
subsection{*facts about @{term kparts} and @{term parts}*}
lemma kparts_no_Nonce [dest]: "[| X:kparts {Y}; Nonce n ~:parts {Y} |]
==> Nonce n ~:parts {X}"
by (erule kparts.induct, auto)
lemma kparts_parts: "X:kparts H ==> X:parts H"
by (erule kparts.induct, auto dest: pparts_analz)
lemma parts_kparts: "X:parts (kparts H) ==> X:parts H"
by (erule parts.induct, auto dest: kparts_parts
intro: parts.Fst parts.Snd parts.Body)
lemma Crypt_kparts_Nonce_parts [dest]: "[| Crypt K Y:kparts {Z};
Nonce n:parts {Y} |] ==> Nonce n:parts {Z}"
by auto
subsection{*facts about @{term kparts} and @{term analz}*}
lemma kparts_analz: "X:kparts H ==> X:analz H"
by (erule kparts.induct, auto dest: pparts_analz)
lemma kparts_analz_sub: "[| X:kparts G; G<=H |] ==> X:analz H"
by (erule kparts.induct, auto dest: pparts_analz_sub)
lemma analz_kparts [rule_format,dest]: "X:analz H ==>
Y:kparts {X} --> Y:analz H"
by (erule analz.induct, auto dest: kparts_analz_sub)
lemma analz_kparts_analz: "X:analz (kparts H) ==> X:analz H"
by (erule analz.induct, auto dest: kparts_analz)
lemma analz_kparts_insert: "X:analz (kparts (insert Z H)) ==>
X:analz (kparts {Z} Un kparts H)"
by (rule analz_sub, auto)
lemma Nonce_kparts_synth [rule_format]: "Y:synth (analz G)
==> Nonce n:kparts {Y} --> Nonce n:analz G"
by (erule synth.induct, auto)
lemma kparts_insert_synth: "[| Y:parts (insert X G); X:synth (analz G);
Nonce n:kparts {Y}; Nonce n ~:analz G |] ==> Y:parts G"
apply (drule parts_insert_substD [where P="%S. Y : S"], clarify)
apply (drule in_sub, drule_tac X=Y in parts_sub, simp)
by (auto dest: Nonce_kparts_synth)
lemma Crypt_insert_synth: "[| Crypt K Y:parts (insert X G); X:synth (analz G);
Nonce n:kparts {Y}; Nonce n ~:analz G |] ==> Crypt K Y:parts G"
apply (drule parts_insert_substD [where P="%S. Crypt K Y : S"], clarify)
apply (drule in_sub, drule_tac X="Crypt K Y" in parts_sub, simp, clarsimp)
apply (ind_cases "Crypt K Y:synth (analz G)")
by (auto dest: Nonce_kparts_synth)
subsection{*analz is pparts + analz of kparts*}
lemma analz_pparts_kparts: "X:analz H ==> X:pparts H | X:analz (kparts H)"
apply (erule analz.induct)
apply (rule_tac X=X in is_MPairE, blast, blast)
apply (erule disjE, rule_tac X=X in is_MPairE, blast, blast, blast)
by (erule disjE, rule_tac X=Y in is_MPairE, blast+)
lemma analz_pparts_kparts_eq: "analz H = pparts H Un analz (kparts H)"
by (rule eq, auto dest: analz_pparts_kparts pparts_analz analz_kparts_analz)
lemmas analz_pparts_kparts_substI = analz_pparts_kparts_eq [THEN ssubst]
lemmas analz_pparts_kparts_substD
= analz_pparts_kparts_eq [THEN sym, THEN ssubst]
end