# Theory Analz

Up to index of Isabelle/HOL/Auth

theory Analz
imports Extensions
begin

    (******************************************************************************
    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

### messages that do not contribute to analz

### basic facts about @{term pparts}

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}

### facts about @{term pparts} and @{term parts}

lemma pparts_no_Nonce:

      [| X ∈ pparts {Y}; Nonce n ∉ parts {Y} |] ==> Nonce n ∉ parts {X}

### facts about @{term pparts} and @{term analz}

lemma pparts_analz:

      X ∈ pparts H ==> X ∈ analz H

lemma pparts_analz_sub:

      [| X ∈ pparts G; G ⊆ H |] ==> X ∈ analz H

### messages that contribute to analz

### basic facts about @{term kparts}

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)

### facts about @{term kparts} and @{term parts}

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}

### facts about @{term kparts} and @{term analz}

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

### analz is pparts + analz of kparts

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))