Theory Message

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

  Title:      HOL/Auth/Message
  Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
  Copyright   1996  University of Cambridge

  Datatypes of agents and messages;
  Inductive relations "parts", "analz" and "synth"

********************************************************************************

  Module:  Refinement/Message.thy (Isabelle/HOL 2016-1)
  ID:      $Id: Message.thy 133856 2017-03-20 18:05:54Z csprenge $
  Edited:  Christoph Sprenger <sprenger@inf.ethz.ch>, ETH Zurich

  Integrated and adapted for security protocol refinement

*******************************************************************************)

section ‹Theory of Agents and Messages for Security Protocols›

theory Message imports Keys begin

(*Needed occasionally with spy_analz_tac, e.g. in analz_insert_Key_newK*)
lemma Un_idem_collapse [simp]: "A ∪ (B ∪ A) = B ∪ A"
by blast

datatype
     msg = Agent  agent     ― ‹Agent names›
         | Number nat       ― ‹Ordinary integers, timestamps, ...›
         | Nonce  nonce     ― ‹Unguessable nonces›
         | Key    key       ― ‹Crypto keys›
         | Hash   msg       ― ‹Hashing›
         | MPair  msg msg   ― ‹Compound messages›
         | Crypt  key msg   ― ‹Encryption, public- or shared-key›


text‹Concrete syntax: messages appear as ‹⦃A,B,NA⦄›, etc...›

syntax
  "_MTuple"      :: "['a, args] => 'a * 'b"   (‹(‹indent=2 notation=‹mixfix message tuple››⦃_,/ _⦄)›)
syntax_consts
  "_MTuple"     == MPair
translations
  "⦃x, y, z⦄"   == "⦃x, ⦃y, z⦄⦄"
  "⦃x, y⦄"      == "CONST MPair x y"


definition
  HPair :: "[msg,msg] ⇒ msg"                       (‹(4Hash[_] /_)› [0, 1000])
where
  ― ‹Message Y paired with a MAC computed with the help of X›
  "Hash[X] Y ≡ ⦃Hash⦃X,Y⦄, Y⦄"

definition
  keysFor :: "msg set ⇒ key set"
where
    ― ‹Keys useful to decrypt elements of a message set›
  "keysFor H ≡ invKey ` {K. ∃X. Crypt K X ∈ H}"


subsubsection‹Inductive Definition of All Parts" of a Message›

inductive_set
  parts :: "msg set => msg set"
  for H :: "msg set"
  where
    Inj [intro]:               "X ∈ H ==> X ∈ parts H"
  | Fst:         "⦃X,Y⦄   ∈ parts H ==> X ∈ parts H"
  | Snd:         "⦃X,Y⦄   ∈ parts H ==> Y ∈ parts H"
  | Body:        "Crypt K X ∈ parts H ==> X ∈ parts H"


text‹Monotonicity›
lemma parts_mono: "G ⊆ H ==> parts(G) ⊆ parts(H)"
apply auto
apply (erule parts.induct)
apply (blast dest: parts.Fst parts.Snd parts.Body)+
done


text‹Equations hold because constructors are injective.›
lemma Other_image_eq [simp]: "(Agent x ∈ Agent`A) = (x:A)"
by auto

lemma Key_image_eq [simp]: "(Key x ∈ Key`A) = (x∈A)"
by auto

lemma Nonce_Key_image_eq [simp]: "(Nonce x ∉ Key`A)"
by auto



subsection‹keysFor operator›

lemma keysFor_empty [simp]: "keysFor {} = {}"
by (unfold keysFor_def, blast)

lemma keysFor_Un [simp]: "keysFor (H ∪ H') = keysFor H ∪ keysFor H'"
by (unfold keysFor_def, blast)

lemma keysFor_UN [simp]: "keysFor (⋃i∈A. H i) = (⋃i∈A. keysFor (H i))"
by (unfold keysFor_def, blast)

text‹Monotonicity›
lemma keysFor_mono: "G ⊆ H ==> keysFor(G) ⊆ keysFor(H)"
by (unfold keysFor_def, blast)

lemma keysFor_insert_Agent [simp]: "keysFor (insert (Agent A) H) = keysFor H"
by (unfold keysFor_def, auto)

lemma keysFor_insert_Nonce [simp]: "keysFor (insert (Nonce N) H) = keysFor H"
by (unfold keysFor_def, auto)

lemma keysFor_insert_Number [simp]: "keysFor (insert (Number N) H) = keysFor H"
by (unfold keysFor_def, auto)

lemma keysFor_insert_Key [simp]: "keysFor (insert (Key K) H) = keysFor H"
by (unfold keysFor_def, auto)

lemma keysFor_insert_Hash [simp]: "keysFor (insert (Hash X) H) = keysFor H"
by (unfold keysFor_def, auto)

lemma keysFor_insert_MPair [simp]: "keysFor (insert ⦃X,Y⦄ H) = keysFor H"
by (unfold keysFor_def, auto)

lemma keysFor_insert_Crypt [simp]:
    "keysFor (insert (Crypt K X) H) = insert (invKey K) (keysFor H)"
by (unfold keysFor_def, auto)

lemma keysFor_image_Key [simp]: "keysFor (Key`E) = {}"
by (unfold keysFor_def, auto)

lemma Crypt_imp_invKey_keysFor: "Crypt K X ∈ H ==> invKey K ∈ keysFor H"
by (unfold keysFor_def, blast)


subsection‹Inductive relation "parts"›

lemma MPair_parts:
     "[| ⦃X,Y⦄ ∈ parts H;
         [| X ∈ parts H; Y ∈ parts H |] ==> P |] ==> P"
by (blast dest: parts.Fst parts.Snd)

declare MPair_parts [elim!]  parts.Body [dest!]
text‹NB These two rules are UNSAFE in the formal sense, as they discard the
     compound message.  They work well on THIS FILE.
  ‹MPair_parts› is left as SAFE because it speeds up proofs.
  The Crypt rule is normally kept UNSAFE to avoid breaking up certificates.›

lemma parts_increasing: "H ⊆ parts(H)"
by blast

lemmas parts_insertI = subset_insertI [THEN parts_mono, THEN subsetD]

lemma parts_empty [simp]: "parts{} = {}"
apply safe
apply (erule parts.induct, blast+)
done

lemma parts_emptyE [elim!]: "X∈ parts{} ==> P"
by simp

text‹WARNING: loops if H = {Y}, therefore must not be repeated!›
lemma parts_singleton: "X∈ parts H ==> ∃Y∈H. X∈ parts {Y}"
by (erule parts.induct, fast+)


subsubsection‹Unions›

lemma parts_Un_subset1: "parts(G) ∪ parts(H) ⊆ parts(G ∪ H)"
by (intro Un_least parts_mono Un_upper1 Un_upper2)

lemma parts_Un_subset2: "parts(G ∪ H) ⊆ parts(G) ∪ parts(H)"
apply (rule subsetI)
apply (erule parts.induct, blast+)
done

lemma parts_Un [simp]: "parts(G ∪ H) = parts(G) ∪ parts(H)"
by (intro equalityI parts_Un_subset1 parts_Un_subset2)

lemma parts_insert: "parts (insert X H) = parts {X} ∪ parts H"
apply (subst insert_is_Un [of _ H])
apply (simp only: parts_Un)
done

text‹TWO inserts to avoid looping.  This rewrite is better than nothing.
  Not suitable for Addsimps: its behaviour can be strange.›
lemma parts_insert2:
     "parts (insert X (insert Y H)) = parts {X} ∪ parts {Y} ∪ parts H"
apply (simp add: Un_assoc)
apply (simp add: parts_insert [symmetric])
done

text‹Added to simplify arguments to parts, analz and synth.›


text‹This allows ‹blast› to simplify occurrences of
  @{term "parts(G∪H)"} in the assumption.›
lemmas in_parts_UnE = parts_Un [THEN equalityD1, THEN subsetD, THEN UnE]
declare in_parts_UnE [elim!]


lemma parts_insert_subset: "insert X (parts H) ⊆ parts(insert X H)"
by (blast intro: parts_mono [THEN [2] rev_subsetD])

subsubsection‹Idempotence and transitivity›

lemma parts_partsD [dest!]: "X∈ parts (parts H) ==> X∈ parts H"
by (erule parts.induct, blast+)

lemma parts_idem [simp]: "parts (parts H) = parts H"
by blast

lemma parts_subset_iff [simp]: "(parts G ⊆ parts H) = (G ⊆ parts H)"
apply (rule iffI)
apply (iprover intro: subset_trans parts_increasing)
apply (frule parts_mono, simp)
done

lemma parts_trans: "[| X∈ parts G;  G ⊆ parts H |] ==> X∈ parts H"
by (drule parts_mono, blast)

text‹Cut›
lemma parts_cut:
     "[| Y∈ parts (insert X G);  X∈ parts H |] ==> Y∈ parts (G ∪ H)"
by (blast intro: parts_trans)


lemma parts_cut_eq [simp]: "X∈ parts H ==> parts (insert X H) = parts H"
by (force dest!: parts_cut intro: parts_insertI)


subsubsection‹Rewrite rules for pulling out atomic messages›

lemmas parts_insert_eq_I = equalityI [OF subsetI parts_insert_subset]


lemma parts_insert_Agent [simp]:
     "parts (insert (Agent agt) H) = insert (Agent agt) (parts H)"
apply (rule parts_insert_eq_I)
apply (erule parts.induct, auto)
done

lemma parts_insert_Nonce [simp]:
     "parts (insert (Nonce N) H) = insert (Nonce N) (parts H)"
apply (rule parts_insert_eq_I)
apply (erule parts.induct, auto)
done

lemma parts_insert_Number [simp]:
     "parts (insert (Number N) H) = insert (Number N) (parts H)"
apply (rule parts_insert_eq_I)
apply (erule parts.induct, auto)
done

lemma parts_insert_Key [simp]:
     "parts (insert (Key K) H) = insert (Key K) (parts H)"
apply (rule parts_insert_eq_I)
apply (erule parts.induct, auto)
done

lemma parts_insert_Hash [simp]:
     "parts (insert (Hash X) H) = insert (Hash X) (parts H)"
apply (rule parts_insert_eq_I)
apply (erule parts.induct, auto)
done

lemma parts_insert_Crypt [simp]:
     "parts (insert (Crypt K X) H) = insert (Crypt K X) (parts (insert X H))"
apply (rule equalityI)
apply (rule subsetI)
apply (erule parts.induct, auto)
apply (blast intro: parts.Body)
done

lemma parts_insert_MPair [simp]:
     "parts (insert ⦃X,Y⦄ H) =
          insert ⦃X,Y⦄ (parts (insert X (insert Y H)))"
apply (rule equalityI)
apply (rule subsetI)
apply (erule parts.induct, auto)
apply (blast intro: parts.Fst parts.Snd)+
done

lemma parts_image_Key [simp]: "parts (Key`N) = Key`N"
apply auto
apply (erule parts.induct, auto)
done


text‹In any message, there is an upper bound N on its greatest nonce.›
(*
lemma msg_Nonce_supply: "∃N. ∀n. N≤n --> Nonce n ∉ parts {msg}"
apply (induct msg)
apply (simp_all (no_asm_simp) add: exI parts_insert2)
 txt{*MPair case: blast works out the necessary sum itself!*}
 prefer 2 apply auto apply (blast elim!: add_leE)
txt{*Nonce case*}
apply (rule_tac x = "N + Suc nat" in exI, auto)
done
*)


subsection‹Inductive relation "analz"›

text‹Inductive definition of "analz" -- what can be broken down from a set of
    messages, including keys.  A form of downward closure.  Pairs can
    be taken apart; messages decrypted with known keys.›

inductive_set
  analz :: "msg set => msg set"
  for H :: "msg set"
  where
    Inj [intro,simp] :    "X ∈ H ==> X ∈ analz H"
  | Fst:     "⦃X,Y⦄ ∈ analz H ==> X ∈ analz H"
  | Snd:     "⦃X,Y⦄ ∈ analz H ==> Y ∈ analz H"
  | Decrypt [dest]:
             "[|Crypt K X ∈ analz H; Key(invKey K): analz H|] ==> X ∈ analz H"


text‹Monotonicity; Lemma 1 of Lowe's paper›
lemma analz_mono: "G⊆H ==> analz(G) ⊆ analz(H)"
apply auto
apply (erule analz.induct)
apply (auto dest: analz.Fst analz.Snd)
done

lemmas analz_monotonic = analz_mono [THEN [2] rev_subsetD]

text‹Making it safe speeds up proofs›
lemma MPair_analz [elim!]:
     "[| ⦃X,Y⦄ ∈ analz H;
             [| X ∈ analz H; Y ∈ analz H |] ==> P
          |] ==> P"
by (blast dest: analz.Fst analz.Snd)

lemma analz_increasing: "H ⊆ analz(H)"
by blast

lemma analz_subset_parts: "analz H ⊆ parts H"
apply (rule subsetI)
apply (erule analz.induct, blast+)
done

lemmas analz_into_parts = analz_subset_parts [THEN subsetD]

lemmas not_parts_not_analz = analz_subset_parts [THEN contra_subsetD]


lemma parts_analz [simp]: "parts (analz H) = parts H"
apply (rule equalityI)
apply (rule analz_subset_parts [THEN parts_mono, THEN subset_trans], simp)
apply (blast intro: analz_increasing [THEN parts_mono, THEN subsetD])
done

lemma analz_parts [simp]: "analz (parts H) = parts H"
apply auto
apply (erule analz.induct, auto)
done

lemmas analz_insertI = subset_insertI [THEN analz_mono, THEN [2] rev_subsetD]

subsubsection‹General equational properties›

lemma analz_empty [simp]: "analz{} = {}"
apply safe
apply (erule analz.induct, blast+)
done

text‹Converse fails: we can analz more from the union than from the
  separate parts, as a key in one might decrypt a message in the other›
lemma analz_Un: "analz(G) ∪ analz(H) ⊆ analz(G ∪ H)"
by (intro Un_least analz_mono Un_upper1 Un_upper2)

lemma analz_insert: "insert X (analz H) ⊆ analz(insert X H)"
by (blast intro: analz_mono [THEN [2] rev_subsetD])

subsubsection‹Rewrite rules for pulling out atomic messages›

lemmas analz_insert_eq_I = equalityI [OF subsetI analz_insert]

lemma analz_insert_Agent [simp]:
     "analz (insert (Agent agt) H) = insert (Agent agt) (analz H)"
apply (rule analz_insert_eq_I)
apply (erule analz.induct, auto)
done

lemma analz_insert_Nonce [simp]:
     "analz (insert (Nonce N) H) = insert (Nonce N) (analz H)"
apply (rule analz_insert_eq_I)
apply (erule analz.induct, auto)
done

lemma analz_insert_Number [simp]:
     "analz (insert (Number N) H) = insert (Number N) (analz H)"
apply (rule analz_insert_eq_I)
apply (erule analz.induct, auto)
done

lemma analz_insert_Hash [simp]:
     "analz (insert (Hash X) H) = insert (Hash X) (analz H)"
apply (rule analz_insert_eq_I)
apply (erule analz.induct, auto)
done

text‹Can only pull out Keys if they are not needed to decrypt the rest›
lemma analz_insert_Key [simp]:
    "K ∉ keysFor (analz H) ==>
          analz (insert (Key K) H) = insert (Key K) (analz H)"
apply (unfold keysFor_def)
apply (rule analz_insert_eq_I)
apply (erule analz.induct, auto)
done

lemma analz_insert_MPair [simp]:
     "analz (insert ⦃X,Y⦄ H) =
          insert ⦃X,Y⦄ (analz (insert X (insert Y H)))"
apply (rule equalityI)
apply (rule subsetI)
apply (erule analz.induct, auto)
apply (erule analz.induct)
apply (blast intro: analz.Fst analz.Snd)+
done

text‹Can pull out enCrypted message if the Key is not known›
lemma analz_insert_Crypt:
     "Key (invKey K) ∉ analz H
      ==> analz (insert (Crypt K X) H) = insert (Crypt K X) (analz H)"
apply (rule analz_insert_eq_I)
apply (erule analz.induct, auto)

done

lemma lemma1: "Key (invKey K) ∈ analz H ==>
               analz (insert (Crypt K X) H) ⊆
               insert (Crypt K X) (analz (insert X H))"
apply (rule subsetI)
apply (erule_tac x = x in analz.induct, auto)
done

lemma lemma2: "Key (invKey K) ∈ analz H ==>
               insert (Crypt K X) (analz (insert X H)) ⊆
               analz (insert (Crypt K X) H)"
apply auto
apply (erule_tac x = x in analz.induct, auto)
apply (blast intro: analz_insertI analz.Decrypt)
done

lemma analz_insert_Decrypt:
     "Key (invKey K) ∈ analz H ==>
               analz (insert (Crypt K X) H) =
               insert (Crypt K X) (analz (insert X H))"
by (intro equalityI lemma1 lemma2)

text‹Case analysis: either the message is secure, or it is not! Effective,
but can cause subgoals to blow up! Use with ‹split_if›; apparently
‹split_tac› does not cope with patterns such as @{term"analz (insert
(Crypt K X) H)"}›
lemma analz_Crypt_if [simp]:
     "analz (insert (Crypt K X) H) =
          (if (Key (invKey K) ∈ analz H)
           then insert (Crypt K X) (analz (insert X H))
           else insert (Crypt K X) (analz H))"
by (simp add: analz_insert_Crypt analz_insert_Decrypt)


text‹This rule supposes "for the sake of argument" that we have the key.›
lemma analz_insert_Crypt_subset:
     "analz (insert (Crypt K X) H) ⊆
           insert (Crypt K X) (analz (insert X H))"
apply (rule subsetI)
apply (erule analz.induct, auto)
done


lemma analz_image_Key [simp]: "analz (Key`N) = Key`N"
apply auto
apply (erule analz.induct, auto)
done


subsubsection‹Idempotence and transitivity›

lemma analz_analzD [dest!]: "X∈ analz (analz H) ==> X∈ analz H"
by (erule analz.induct, blast+)

lemma analz_idem [simp]: "analz (analz H) = analz H"
by blast

lemma analz_subset_iff [simp]: "(analz G ⊆ analz H) = (G ⊆ analz H)"
apply (rule iffI)
apply (iprover intro: subset_trans analz_increasing)
apply (frule analz_mono, simp)
done

lemma analz_trans: "[| X∈ analz G;  G ⊆ analz H |] ==> X∈ analz H"
by (drule analz_mono, blast)

text‹Cut; Lemma 2 of Lowe›
lemma analz_cut: "[| Y∈ analz (insert X H);  X∈ analz H |] ==> Y∈ analz H"
by (erule analz_trans, blast)

(*Cut can be proved easily by induction on
   "Y: analz (insert X H) ==> X: analz H --> Y: analz H"
*)

text‹This rewrite rule helps in the simplification of messages that involve
  the forwarding of unknown components (X).  Without it, removing occurrences
  of X can be very complicated.›
lemma analz_insert_eq: "X∈ analz H ==> analz (insert X H) = analz H"
by (blast intro: analz_cut analz_insertI)


text‹A congruence rule for "analz"›

lemma analz_subset_cong:
     "[| analz G ⊆ analz G'; analz H ⊆ analz H' |]
      ==> analz (G ∪ H) ⊆ analz (G' ∪ H')"
apply simp
apply (iprover intro: conjI subset_trans analz_mono Un_upper1 Un_upper2)
done

lemma analz_cong:
     "[| analz G = analz G'; analz H = analz H' |]
      ==> analz (G ∪ H) = analz (G' ∪ H')"
by (intro equalityI analz_subset_cong, simp_all)

lemma analz_insert_cong:
     "analz H = analz H' ==> analz(insert X H) = analz(insert X H')"
by (force simp only: insert_def intro!: analz_cong)

text‹If there are no pairs or encryptions then analz does nothing›
lemma analz_trivial:
     "[| ∀X Y. ⦃X,Y⦄ ∉ H;  ∀X K. Crypt K X ∉ H |] ==> analz H = H"
apply safe
apply (erule analz.induct, blast+)
done


subsection‹Inductive relation "synth"›

text‹Inductive definition of "synth" -- what can be built up from a set of
    messages.  A form of upward closure.  Pairs can be built, messages
    encrypted with known keys.  Agent names are public domain.
    Numbers can be guessed, but Nonces cannot be.›

inductive_set
  synth :: "msg set => msg set"
  for H :: "msg set"
  where
    Inj    [intro]:   "X ∈ H ==> X ∈ synth H"
  | Agent  [intro]:   "Agent agt ∈ synth H"
  | Number [intro]:   "Number n  ∈ synth H"
  | Hash   [intro]:   "X ∈ synth H ==> Hash X ∈ synth H"
  | MPair  [intro]:   "[|X ∈ synth H;  Y ∈ synth H|] ==> ⦃X,Y⦄ ∈ synth H"
  | Crypt  [intro]:   "[|X ∈ synth H;  Key(K) ∈ H|] ==> Crypt K X ∈ synth H"

text‹Monotonicity›
lemma synth_mono: "G⊆H ==> synth(G) ⊆ synth(H)"
  by (auto, erule synth.induct, auto)

text‹NO ‹Agent_synth›, as any Agent name can be synthesized.
  The same holds for @{term Number}›
inductive_cases Nonce_synth [elim!]: "Nonce n ∈ synth H"
inductive_cases Key_synth   [elim!]: "Key K ∈ synth H"
inductive_cases Hash_synth  [elim!]: "Hash X ∈ synth H"
inductive_cases MPair_synth [elim!]: "⦃X,Y⦄ ∈ synth H"
inductive_cases Crypt_synth [elim!]: "Crypt K X ∈ synth H"


lemma synth_increasing: "H ⊆ synth(H)"
by blast

subsubsection‹Unions›

text‹Converse fails: we can synth more from the union than from the
  separate parts, building a compound message using elements of each.›
lemma synth_Un: "synth(G) ∪ synth(H) ⊆ synth(G ∪ H)"
by (intro Un_least synth_mono Un_upper1 Un_upper2)

lemma synth_insert: "insert X (synth H) ⊆ synth(insert X H)"
by (blast intro: synth_mono [THEN [2] rev_subsetD])

subsubsection‹Idempotence and transitivity›

lemma synth_synthD [dest!]: "X∈ synth (synth H) ==> X∈ synth H"
by (erule synth.induct, blast+)

lemma synth_idem: "synth (synth H) = synth H"
by blast

lemma synth_subset_iff [simp]: "(synth G ⊆ synth H) = (G ⊆ synth H)"
apply (rule iffI)
apply (iprover intro: subset_trans synth_increasing)
apply (frule synth_mono, simp add: synth_idem)
done

lemma synth_trans: "[| X∈ synth G;  G ⊆ synth H |] ==> X∈ synth H"
by (drule synth_mono, blast)

text‹Cut; Lemma 2 of Lowe›
lemma synth_cut: "[| Y∈ synth (insert X H);  X∈ synth H |] ==> Y∈ synth H"
by (erule synth_trans, blast)

lemma Agent_synth [simp]: "Agent A ∈ synth H"
by blast

lemma Number_synth [simp]: "Number n ∈ synth H"
by blast

lemma Nonce_synth_eq [simp]: "(Nonce N ∈ synth H) = (Nonce N ∈ H)"
by blast

lemma Key_synth_eq [simp]: "(Key K ∈ synth H) = (Key K ∈ H)"
by blast

lemma Crypt_synth_eq [simp]:
     "Key K ∉ H ==> (Crypt K X ∈ synth H) = (Crypt K X ∈ H)"
by blast


lemma keysFor_synth [simp]:
    "keysFor (synth H) = keysFor H ∪ invKey`{K. Key K ∈ H}"
by (unfold keysFor_def, blast)


subsubsection‹Combinations of parts, analz and synth›

lemma parts_synth [simp]: "parts (synth H) = parts H ∪ synth H"
apply (rule equalityI)
apply (rule subsetI)
apply (erule parts.induct)
apply (blast intro: synth_increasing [THEN parts_mono, THEN subsetD]
                    parts.Fst parts.Snd parts.Body)+
done

lemma analz_analz_Un [simp]: "analz (analz G ∪ H) = analz (G ∪ H)"
apply (intro equalityI analz_subset_cong)+
apply simp_all
done

lemma analz_synth_Un [simp]: "analz (synth G ∪ H) = analz (G ∪ H) ∪ synth G"
apply (rule equalityI)
apply (rule subsetI)
apply (erule analz.induct)
prefer 5 apply (blast intro: analz_mono [THEN [2] rev_subsetD])
apply (blast intro: analz.Fst analz.Snd analz.Decrypt)+
done

lemma analz_synth [simp]: "analz (synth H) = analz H ∪ synth H"
apply (cut_tac H = "{}" in analz_synth_Un)
apply (simp (no_asm_use))
done

text ‹chsp: added›

lemma analz_Un_analz [simp]: "analz (G ∪ analz H) = analz (G ∪ H)"
by (subst Un_commute, auto)+

lemma analz_synth_Un2 [simp]: "analz (G ∪ synth H) = analz (G ∪ H) ∪ synth H"
by (subst Un_commute, auto)+


subsubsection‹For reasoning about the Fake rule in traces›

lemma parts_insert_subset_Un: "X∈ G ==> parts(insert X H) ⊆ parts G ∪ parts H"
by (rule subset_trans [OF parts_mono parts_Un_subset2], blast)

text‹More specifically for Fake.  Very occasionally we could do with a version
  of the form  @{term"parts{X} ⊆ synth (analz H) ∪ parts H"}›
lemma Fake_parts_insert:
     "X ∈ synth (analz H) ==>
      parts (insert X H) ⊆ synth (analz H) ∪ parts H"
apply (drule parts_insert_subset_Un)
apply (simp (no_asm_use))
apply blast
done

lemma Fake_parts_insert_in_Un:
     "[|Z ∈ parts (insert X H);  X ∈ synth (analz H)|]
      ==> Z ∈  synth (analz H) ∪ parts H"
by (blast dest: Fake_parts_insert  [THEN subsetD, dest])

text‹@{term H} is sometimes @{term"Key ` KK ∪ spies evs"}, so can't put
  @{term "G=H"}.›
lemma Fake_analz_insert:
     "X∈ synth (analz G) ==>
      analz (insert X H) ⊆ synth (analz G) ∪ analz (G ∪ H)"
apply (rule subsetI)
apply (subgoal_tac "x ∈ analz (synth (analz G) ∪ H) ")
prefer 2
  apply (blast intro: analz_mono [THEN [2] rev_subsetD]
                      analz_mono [THEN synth_mono, THEN [2] rev_subsetD])
apply (simp (no_asm_use))
apply blast
done

lemma analz_conj_parts [simp]:
     "(X ∈ analz H & X ∈ parts H) = (X ∈ analz H)"
by (blast intro: analz_subset_parts [THEN subsetD])

lemma analz_disj_parts [simp]:
     "(X ∈ analz H | X ∈ parts H) = (X ∈ parts H)"
by (blast intro: analz_subset_parts [THEN subsetD])

text‹Without this equation, other rules for synth and analz would yield
  redundant cases›
lemma MPair_synth_analz [iff]:
     "(⦃X,Y⦄ ∈ synth (analz H)) =
      (X ∈ synth (analz H) & Y ∈ synth (analz H))"
by blast

lemma Crypt_synth_analz:
     "[| Key K ∈ analz H;  Key (invKey K) ∈ analz H |]
       ==> (Crypt K X ∈ synth (analz H)) = (X ∈ synth (analz H))"
by blast


lemma Hash_synth_analz [simp]:
     "X ∉ synth (analz H)
      ==> (Hash⦃X,Y⦄ ∈ synth (analz H)) = (Hash⦃X,Y⦄ ∈ analz H)"
by blast


subsection‹HPair: a combination of Hash and MPair›

subsubsection‹Freeness›

lemma Agent_neq_HPair: "Agent A ~= Hash[X] Y"
by (unfold HPair_def, simp)

lemma Nonce_neq_HPair: "Nonce N ~= Hash[X] Y"
by (unfold HPair_def, simp)

lemma Number_neq_HPair: "Number N ~= Hash[X] Y"
by (unfold HPair_def, simp)

lemma Key_neq_HPair: "Key K ~= Hash[X] Y"
by (unfold HPair_def, simp)

lemma Hash_neq_HPair: "Hash Z ~= Hash[X] Y"
by (unfold HPair_def, simp)

lemma Crypt_neq_HPair: "Crypt K X' ~= Hash[X] Y"
by (unfold HPair_def, simp)

lemmas HPair_neqs = Agent_neq_HPair Nonce_neq_HPair Number_neq_HPair
                    Key_neq_HPair Hash_neq_HPair Crypt_neq_HPair

declare HPair_neqs [iff]
declare HPair_neqs [symmetric, iff]

lemma HPair_eq [iff]: "(Hash[X'] Y' = Hash[X] Y) = (X' = X & Y'=Y)"
by (simp add: HPair_def)

lemma MPair_eq_HPair [iff]:
     "(⦃X',Y'⦄ = Hash[X] Y) = (X' = Hash⦃X,Y⦄ & Y'=Y)"
by (simp add: HPair_def)

lemma HPair_eq_MPair [iff]:
     "(Hash[X] Y = ⦃X',Y'⦄) = (X' = Hash⦃X,Y⦄ & Y'=Y)"
by (auto simp add: HPair_def)


subsubsection‹Specialized laws, proved in terms of those for Hash and MPair›

lemma keysFor_insert_HPair [simp]: "keysFor (insert (Hash[X] Y) H) = keysFor H"
by (simp add: HPair_def)

lemma parts_insert_HPair [simp]:
    "parts (insert (Hash[X] Y) H) =
     insert (Hash[X] Y) (insert (Hash⦃X,Y⦄) (parts (insert Y H)))"
by (simp add: HPair_def)

lemma analz_insert_HPair [simp]:
    "analz (insert (Hash[X] Y) H) =
     insert (Hash[X] Y) (insert (Hash⦃X,Y⦄) (analz (insert Y H)))"
by (simp add: HPair_def)

lemma HPair_synth_analz [simp]:
     "X ∉ synth (analz H)
    ==> (Hash[X] Y ∈ synth (analz H)) =
        (Hash⦃X, Y⦄ ∈ analz H & Y ∈ synth (analz H))"
by (simp add: HPair_def)


text‹We do NOT want Crypt... messages broken up in protocols!!›
declare parts.Body [rule del]


text‹Rewrites to push in Key and Crypt messages, so that other messages can
    be pulled out using the ‹analz_insert› rules›

lemmas pushKeys =
  insert_commute [of "Key K" "Agent C" for K C]
  insert_commute [of "Key K" "Nonce N" for K N]
  insert_commute [of "Key K" "Number N" for K N]
  insert_commute [of "Key K" "Hash X" for K X]
  insert_commute [of "Key K" "MPair X Y" for K X Y]
  insert_commute [of "Key K" "Crypt X K'" for K K' X]

lemmas pushCrypts =
  insert_commute [of "Crypt X K" "Agent C" for X K C]
  insert_commute [of "Crypt X K" "Agent C" for X K C]
  insert_commute [of "Crypt X K" "Nonce N" for X K N]
  insert_commute [of "Crypt X K" "Number N"  for X K N]
  insert_commute [of "Crypt X K" "Hash X'"  for X K X']
  insert_commute [of "Crypt X K" "MPair X' Y"  for X K X' Y]

text‹Cannot be added with ‹[simp]› -- messages should not always be
  re-ordered.›
lemmas pushes = pushKeys pushCrypts


text‹By default only ‹o_apply› is built-in.  But in the presence of
eta-expansion this means that some terms displayed as @{term "f o g"} will be
rewritten, and others will not!›
declare o_def [simp]


lemma Crypt_notin_image_Key [simp]: "Crypt K X ∉ Key ` A"
by auto

lemma Hash_notin_image_Key [simp] :"Hash X ∉ Key ` A"
by auto

lemma synth_analz_mono: "G⊆H ==> synth (analz(G)) ⊆ synth (analz(H))"
by (iprover intro: synth_mono analz_mono)

lemma Fake_analz_eq [simp]:
     "X ∈ synth(analz H) ==> synth (analz (insert X H)) = synth (analz H)"
apply (drule Fake_analz_insert[of _ _ "H"])
apply (simp add: synth_increasing[THEN Un_absorb2])
apply (drule synth_mono)
apply (simp add: synth_idem)
apply (rule equalityI)
apply simp
apply (rule synth_analz_mono, blast)
done

text‹Two generalizations of ‹analz_insert_eq››
lemma gen_analz_insert_eq [rule_format]:
     "X ∈ analz H ==> ALL G. H ⊆ G --> analz (insert X G) = analz G"
by (blast intro: analz_cut analz_insertI analz_mono [THEN [2] rev_subsetD])

lemma synth_analz_insert_eq [rule_format]:
     "X ∈ synth (analz H)
      ==> ALL G. H ⊆ G --> (Key K ∈ analz (insert X G)) = (Key K ∈ analz G)"
apply (erule synth.induct)
apply (simp_all add: gen_analz_insert_eq subset_trans [OF _ subset_insertI])
done

lemma Fake_parts_sing:
     "X ∈ synth (analz H) ==> parts{X} ⊆ synth (analz H) ∪ parts H"
apply (rule subset_trans)
 apply (erule_tac [2] Fake_parts_insert)
apply (rule parts_mono, blast)
done

lemmas Fake_parts_sing_imp_Un = Fake_parts_sing [THEN [2] rev_subsetD]


text‹For some reason, moving this up can make some proofs loop!›
declare invKey_K [simp]


end