(****************************************************************************** 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*} consts pparts :: "msg set => msg set" inductive "pparts H" intros 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*} consts kparts :: "msg set => msg set" inductive "kparts H" intros 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, 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, 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
lemma pparts_is_MPair:
X ∈ pparts H ==> is_MPair X
lemma Crypt_notin_pparts:
Crypt K X ∉ pparts H
lemma Key_notin_pparts:
Key K ∉ pparts H
lemma Nonce_notin_pparts:
Nonce n ∉ pparts H
lemma Number_notin_pparts:
Number n ∉ pparts H
lemma Agent_notin_pparts:
Agent A ∉ pparts H
lemma pparts_empty:
pparts {} = {}
lemma pparts_insertI:
X ∈ pparts H ==> X ∈ pparts (insert Y H)
lemma pparts_sub:
[| X ∈ pparts G; G ⊆ H |] ==> X ∈ pparts H
lemma pparts_insert2:
pparts (insert X (insert Y H)) = pparts {X} ∪ pparts {Y} ∪ pparts H
lemma pparts_insert_MPair:
pparts (insert {|X, Y|} H) = insert {|X, Y|} (pparts ({X, Y} ∪ H))
lemma pparts_insert_Nonce:
pparts (insert (Nonce n) H) = pparts H
lemma pparts_insert_Crypt:
pparts (insert (Crypt K X) H) = pparts H
lemma pparts_insert_Key:
pparts (insert (Key K) H) = pparts H
lemma pparts_insert_Agent:
pparts (insert (Agent A) H) = pparts H
lemma pparts_insert_Number:
pparts (insert (Number n) H) = pparts H
lemma pparts_insert_Hash:
pparts (insert (Hash X) H) = pparts H
lemma pparts_insert:
X ∈ pparts (insert Y H) ==> X ∈ pparts {Y} ∪ pparts H
lemma insert_pparts:
X ∈ pparts {Y} ∪ pparts H ==> X ∈ pparts (insert Y H)
lemma pparts_Un:
pparts (G ∪ H) = pparts G ∪ pparts H
lemma pparts_pparts:
pparts (pparts H) = pparts H
lemma pparts_insert_eq:
pparts (insert X H) = pparts {X} ∪ pparts H
lemmas pparts_insert_substI:
P (pparts {X1} ∪ pparts H1) ==> P (pparts (insert X1 H1))
lemmas pparts_insert_substI:
P (pparts {X1} ∪ pparts H1) ==> P (pparts (insert X1 H1))
lemma in_pparts:
Y ∈ pparts H ==> ∃X. X ∈ H ∧ Y ∈ pparts {X}
lemma pparts_no_Nonce:
[| X ∈ pparts {Y}; Nonce n ∉ parts {Y} |] ==> Nonce n ∉ parts {X}
lemma pparts_analz:
X ∈ pparts H ==> X ∈ analz H
lemma pparts_analz_sub:
[| X ∈ pparts G; G ⊆ H |] ==> X ∈ analz H
lemma kparts_not_MPair:
X ∈ kparts H ==> not_MPair X
lemma kparts_empty:
kparts {} = {}
lemma kparts_insertI:
X ∈ kparts H ==> X ∈ kparts (insert Y H)
lemma kparts_insert2:
kparts (insert X (insert Y H)) = kparts {X} ∪ kparts {Y} ∪ kparts H
lemma kparts_insert_MPair:
kparts (insert {|X, Y|} H) = kparts ({X, Y} ∪ H)
lemma kparts_insert_Nonce:
kparts (insert (Nonce n) H) = insert (Nonce n) (kparts H)
lemma kparts_insert_Crypt:
kparts (insert (Crypt K X) H) = insert (Crypt K X) (kparts H)
lemma kparts_insert_Key:
kparts (insert (Key K) H) = insert (Key K) (kparts H)
lemma kparts_insert_Agent:
kparts (insert (Agent A) H) = insert (Agent A) (kparts H)
lemma kparts_insert_Number:
kparts (insert (Number n) H) = insert (Number n) (kparts H)
lemma kparts_insert_Hash:
kparts (insert (Hash X) H) = insert (Hash X) (kparts H)
lemma kparts_insert:
X ∈ kparts (insert X H) ==> X ∈ kparts {X} ∪ kparts H
lemma kparts_insert_fst:
[| X ∈ kparts (insert Z H); X ∉ kparts H |] ==> X ∈ kparts {Z}
lemma kparts_sub:
[| X ∈ kparts G; G ⊆ H |] ==> X ∈ kparts H
lemma kparts_Un:
kparts (G ∪ H) = kparts G ∪ kparts H
lemma pparts_kparts:
pparts (kparts H) = {}
lemma kparts_kparts:
kparts (kparts H) = kparts H
lemma kparts_insert_eq:
kparts (insert X H) = kparts {X} ∪ kparts H
lemmas kparts_insert_substI:
P (kparts {X1} ∪ kparts H1) ==> P (kparts (insert X1 H1))
lemmas kparts_insert_substI:
P (kparts {X1} ∪ kparts H1) ==> P (kparts (insert X1 H1))
lemma in_kparts:
Y ∈ kparts H ==> ∃X. X ∈ H ∧ Y ∈ kparts {X}
lemma kparts_has_no_pair:
has_no_pair (kparts H)
lemma kparts_no_Nonce:
[| X ∈ kparts {Y}; Nonce n ∉ parts {Y} |] ==> Nonce n ∉ parts {X}
lemma kparts_parts:
X ∈ kparts H ==> X ∈ parts H
lemma parts_kparts:
X ∈ parts (kparts H) ==> X ∈ parts H
lemma Crypt_kparts_Nonce_parts:
[| Crypt K Y ∈ kparts {Z}; Nonce n ∈ parts {Y} |] ==> Nonce n ∈ parts {Z}
lemma kparts_analz:
X ∈ kparts H ==> X ∈ analz H
lemma kparts_analz_sub:
[| X ∈ kparts G; G ⊆ H |] ==> X ∈ analz H
lemma analz_kparts:
[| X ∈ analz H; Y ∈ kparts {X} |] ==> Y ∈ analz H
lemma analz_kparts_analz:
X ∈ analz (kparts H) ==> X ∈ analz H
lemma analz_kparts_insert:
X ∈ analz (kparts (insert Z H)) ==> X ∈ analz (kparts {Z} ∪ kparts H)
lemma Nonce_kparts_synth:
[| Y ∈ synth (analz G); Nonce n ∈ kparts {Y} |] ==> Nonce n ∈ analz G
lemma kparts_insert_synth:
[| Y ∈ parts (insert X G); X ∈ synth (analz G); Nonce n ∈ kparts {Y}; Nonce n ∉ analz G |] ==> Y ∈ parts G
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
lemma analz_pparts_kparts:
X ∈ analz H ==> X ∈ pparts H ∨ X ∈ analz (kparts H)
lemma analz_pparts_kparts_eq:
analz H = pparts H ∪ analz (kparts H)
lemmas analz_pparts_kparts_substI:
P (pparts H1 ∪ analz (kparts H1)) ==> P (analz H1)
lemmas analz_pparts_kparts_substI:
P (pparts H1 ∪ analz (kparts H1)) ==> P (analz H1)
lemmas analz_pparts_kparts_substD:
P (analz H2) ==> P (pparts H2 ∪ analz (kparts H2))
lemmas analz_pparts_kparts_substD:
P (analz H2) ==> P (pparts H2 ∪ analz (kparts H2))