Theory Analz

(*  Title:      HOL/Auth/Guard/Analz.thy
    Author:     Frederic Blanqui, University of Cambridge Computer Laboratory
    Copyright   2001  University of Cambridge
*)

sectionDecomposition of Analz into two parts

theory Analz imports Extensions begin

textdecomposition of termanalz into two parts: 
      termpparts (for pairs) and analz of termkparts

subsectionmessages 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"

subsectionbasic facts about termpparts

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}  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}  pparts H"
by (erule pparts.induct, blast+)

lemma insert_pparts: "X  pparts {Y}  pparts H  X  pparts (insert Y H)"
by (safe, erule pparts.induct, auto)

lemma pparts_Un [iff]: "pparts (G  H) = pparts G  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  X. X  H  Y  pparts {X}"
by (erule pparts.induct, auto)

subsectionfacts about termpparts and termparts

lemma pparts_no_Nonce [dest]: "[| X  pparts {Y}; Nonce n  parts {Y} |]
==> Nonce n  parts {X}"
by (erule pparts.induct, simp_all)

subsectionfacts about termpparts and termanalz

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)

subsectionmessages 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"

subsectionbasic facts about termkparts

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}  kparts {Y}  kparts H"
by (rule eq, (erule kparts.induct, auto)+)

lemma kparts_insert_MPair [iff]: "kparts (insert X,Y H)
= kparts ({X,Y}  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}  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  H) = kparts G  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}  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  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

subsectionfacts about termkparts and termparts

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

subsectionfacts about termkparts and termanalz

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}  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)
apply (auto dest: Nonce_kparts_synth)
done

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"
by (metis Fake_parts_insert_in_Un Nonce_kparts_synth UnE analz_conj_parts synth_simps(5))


subsectionanalz is pparts + analz of kparts

lemma analz_pparts_kparts: "X  analz H  X  pparts H  X  analz (kparts H)"
by (erule analz.induct, auto) 

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