src/HOL/Metis_Examples/Message.thy
author haftmann
Mon Jun 05 15:59:41 2017 +0200 (2017-06-05)
changeset 66010 2f7d39285a1a
parent 63167 0909deb8059b
child 67443 3abf6a722518
permissions -rw-r--r--
executable domain membership checks
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(*  Title:      HOL/Metis_Examples/Message.thy
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    Author:     Lawrence C. Paulson, Cambridge University Computer Laboratory
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    Author:     Jasmin Blanchette, TU Muenchen
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Metis example featuring message authentication.
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*)
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section \<open>Metis Example Featuring Message Authentication\<close>
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theory Message
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imports Main
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begin
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declare [[metis_new_skolem]]
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lemma strange_Un_eq [simp]: "A \<union> (B \<union> A) = B \<union> A"
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by (metis Un_commute Un_left_absorb)
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type_synonym key = nat
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consts
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  all_symmetric :: bool        \<comment>\<open>true if all keys are symmetric\<close>
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  invKey        :: "key=>key"  \<comment>\<open>inverse of a symmetric key\<close>
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specification (invKey)
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  invKey [simp]: "invKey (invKey K) = K"
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  invKey_symmetric: "all_symmetric --> invKey = id"
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by (metis id_apply)
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text\<open>The inverse of a symmetric key is itself; that of a public key
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      is the private key and vice versa\<close>
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definition symKeys :: "key set" where
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  "symKeys == {K. invKey K = K}"
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datatype  \<comment>\<open>We allow any number of friendly agents\<close>
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  agent = Server | Friend nat | Spy
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datatype
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     msg = Agent  agent     \<comment>\<open>Agent names\<close>
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         | Number nat       \<comment>\<open>Ordinary integers, timestamps, ...\<close>
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         | Nonce  nat       \<comment>\<open>Unguessable nonces\<close>
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         | Key    key       \<comment>\<open>Crypto keys\<close>
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         | Hash   msg       \<comment>\<open>Hashing\<close>
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         | MPair  msg msg   \<comment>\<open>Compound messages\<close>
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         | Crypt  key msg   \<comment>\<open>Encryption, public- or shared-key\<close>
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text\<open>Concrete syntax: messages appear as \<open>\<lbrace>A,B,NA\<rbrace>\<close>, etc...\<close>
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syntax
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  "_MTuple"      :: "['a, args] => 'a * 'b"       ("(2\<lbrace>_,/ _\<rbrace>)")
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translations
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  "\<lbrace>x, y, z\<rbrace>"   == "\<lbrace>x, \<lbrace>y, z\<rbrace>\<rbrace>"
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  "\<lbrace>x, y\<rbrace>"      == "CONST MPair x y"
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definition HPair :: "[msg,msg] => msg" ("(4Hash[_] /_)" [0, 1000]) where
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    \<comment>\<open>Message Y paired with a MAC computed with the help of X\<close>
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    "Hash[X] Y == \<lbrace> Hash\<lbrace>X,Y\<rbrace>, Y\<rbrace>"
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definition keysFor :: "msg set => key set" where
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    \<comment>\<open>Keys useful to decrypt elements of a message set\<close>
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  "keysFor H == invKey ` {K. \<exists>X. Crypt K X \<in> H}"
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subsubsection\<open>Inductive Definition of All Parts" of a Message\<close>
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inductive_set
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  parts :: "msg set => msg set"
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  for H :: "msg set"
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  where
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    Inj [intro]:               "X \<in> H ==> X \<in> parts H"
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  | Fst:         "\<lbrace>X,Y\<rbrace>   \<in> parts H ==> X \<in> parts H"
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  | Snd:         "\<lbrace>X,Y\<rbrace>   \<in> parts H ==> Y \<in> parts H"
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  | Body:        "Crypt K X \<in> parts H ==> X \<in> parts H"
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lemma parts_mono: "G \<subseteq> H ==> parts(G) \<subseteq> parts(H)"
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apply auto
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apply (erule parts.induct)
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   apply (metis parts.Inj set_rev_mp)
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  apply (metis parts.Fst)
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 apply (metis parts.Snd)
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by (metis parts.Body)
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text\<open>Equations hold because constructors are injective.\<close>
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lemma Friend_image_eq [simp]: "(Friend x \<in> Friend`A) = (x:A)"
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by (metis agent.inject image_iff)
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lemma Key_image_eq [simp]: "(Key x \<in> Key`A) = (x \<in> A)"
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by (metis image_iff msg.inject(4))
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lemma Nonce_Key_image_eq [simp]: "Nonce x \<notin> Key`A"
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by (metis image_iff msg.distinct(23))
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subsubsection\<open>Inverse of keys\<close>
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lemma invKey_eq [simp]: "(invKey K = invKey K') = (K = K')"
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by (metis invKey)
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subsection\<open>keysFor operator\<close>
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lemma keysFor_empty [simp]: "keysFor {} = {}"
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by (unfold keysFor_def, blast)
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lemma keysFor_Un [simp]: "keysFor (H \<union> H') = keysFor H \<union> keysFor H'"
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by (unfold keysFor_def, blast)
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lemma keysFor_UN [simp]: "keysFor (\<Union>i\<in>A. H i) = (\<Union>i\<in>A. keysFor (H i))"
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by (unfold keysFor_def, blast)
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text\<open>Monotonicity\<close>
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lemma keysFor_mono: "G \<subseteq> H ==> keysFor(G) \<subseteq> keysFor(H)"
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by (unfold keysFor_def, blast)
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lemma keysFor_insert_Agent [simp]: "keysFor (insert (Agent A) H) = keysFor H"
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by (unfold keysFor_def, auto)
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lemma keysFor_insert_Nonce [simp]: "keysFor (insert (Nonce N) H) = keysFor H"
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by (unfold keysFor_def, auto)
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lemma keysFor_insert_Number [simp]: "keysFor (insert (Number N) H) = keysFor H"
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by (unfold keysFor_def, auto)
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lemma keysFor_insert_Key [simp]: "keysFor (insert (Key K) H) = keysFor H"
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by (unfold keysFor_def, auto)
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lemma keysFor_insert_Hash [simp]: "keysFor (insert (Hash X) H) = keysFor H"
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by (unfold keysFor_def, auto)
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lemma keysFor_insert_MPair [simp]: "keysFor (insert \<lbrace>X,Y\<rbrace> H) = keysFor H"
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by (unfold keysFor_def, auto)
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lemma keysFor_insert_Crypt [simp]:
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    "keysFor (insert (Crypt K X) H) = insert (invKey K) (keysFor H)"
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by (unfold keysFor_def, auto)
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lemma keysFor_image_Key [simp]: "keysFor (Key`E) = {}"
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by (unfold keysFor_def, auto)
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lemma Crypt_imp_invKey_keysFor: "Crypt K X \<in> H ==> invKey K \<in> keysFor H"
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by (unfold keysFor_def, blast)
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subsection\<open>Inductive relation "parts"\<close>
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lemma MPair_parts:
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     "[| \<lbrace>X,Y\<rbrace> \<in> parts H;
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         [| X \<in> parts H; Y \<in> parts H |] ==> P |] ==> P"
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by (blast dest: parts.Fst parts.Snd)
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declare MPair_parts [elim!] parts.Body [dest!]
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text\<open>NB These two rules are UNSAFE in the formal sense, as they discard the
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     compound message.  They work well on THIS FILE.
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  \<open>MPair_parts\<close> is left as SAFE because it speeds up proofs.
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  The Crypt rule is normally kept UNSAFE to avoid breaking up certificates.\<close>
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lemma parts_increasing: "H \<subseteq> parts(H)"
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by blast
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lemmas parts_insertI = subset_insertI [THEN parts_mono, THEN subsetD]
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lemma parts_empty [simp]: "parts{} = {}"
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apply safe
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apply (erule parts.induct)
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apply blast+
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done
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lemma parts_emptyE [elim!]: "X\<in> parts{} ==> P"
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by simp
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text\<open>WARNING: loops if H = {Y}, therefore must not be repeated!\<close>
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lemma parts_singleton: "X\<in> parts H ==> \<exists>Y\<in>H. X\<in> parts {Y}"
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apply (erule parts.induct)
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apply fast+
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done
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subsubsection\<open>Unions\<close>
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lemma parts_Un_subset1: "parts(G) \<union> parts(H) \<subseteq> parts(G \<union> H)"
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by (intro Un_least parts_mono Un_upper1 Un_upper2)
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lemma parts_Un_subset2: "parts(G \<union> H) \<subseteq> parts(G) \<union> parts(H)"
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apply (rule subsetI)
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apply (erule parts.induct, blast+)
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done
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lemma parts_Un [simp]: "parts(G \<union> H) = parts(G) \<union> parts(H)"
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by (intro equalityI parts_Un_subset1 parts_Un_subset2)
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lemma parts_insert: "parts (insert X H) = parts {X} \<union> parts H"
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apply (subst insert_is_Un [of _ H])
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apply (simp only: parts_Un)
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done
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lemma parts_insert2:
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     "parts (insert X (insert Y H)) = parts {X} \<union> parts {Y} \<union> parts H"
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by (metis Un_commute Un_empty_left Un_empty_right Un_insert_left Un_insert_right parts_Un)
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lemma parts_UN_subset1: "(\<Union>x\<in>A. parts(H x)) \<subseteq> parts(\<Union>x\<in>A. H x)"
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by (intro UN_least parts_mono UN_upper)
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lemma parts_UN_subset2: "parts(\<Union>x\<in>A. H x) \<subseteq> (\<Union>x\<in>A. parts(H x))"
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apply (rule subsetI)
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apply (erule parts.induct, blast+)
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done
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lemma parts_UN [simp]: "parts(\<Union>x\<in>A. H x) = (\<Union>x\<in>A. parts(H x))"
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by (intro equalityI parts_UN_subset1 parts_UN_subset2)
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text\<open>Added to simplify arguments to parts, analz and synth.
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  NOTE: the UN versions are no longer used!\<close>
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text\<open>This allows \<open>blast\<close> to simplify occurrences of
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  @{term "parts(G\<union>H)"} in the assumption.\<close>
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lemmas in_parts_UnE = parts_Un [THEN equalityD1, THEN subsetD, THEN UnE]
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declare in_parts_UnE [elim!]
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lemma parts_insert_subset: "insert X (parts H) \<subseteq> parts(insert X H)"
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by (blast intro: parts_mono [THEN [2] rev_subsetD])
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subsubsection\<open>Idempotence and transitivity\<close>
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lemma parts_partsD [dest!]: "X\<in> parts (parts H) ==> X\<in> parts H"
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by (erule parts.induct, blast+)
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lemma parts_idem [simp]: "parts (parts H) = parts H"
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by blast
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lemma parts_subset_iff [simp]: "(parts G \<subseteq> parts H) = (G \<subseteq> parts H)"
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apply (rule iffI)
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apply (metis Un_absorb1 Un_subset_iff parts_Un parts_increasing)
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apply (metis parts_idem parts_mono)
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done
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lemma parts_trans: "[| X\<in> parts G;  G \<subseteq> parts H |] ==> X\<in> parts H"
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by (blast dest: parts_mono)
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lemma parts_cut: "[|Y\<in> parts (insert X G);  X\<in> parts H|] ==> Y\<in> parts(G \<union> H)"
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by (metis (no_types) Un_insert_left Un_insert_right insert_absorb le_supE
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          parts_Un parts_idem parts_increasing parts_trans)
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subsubsection\<open>Rewrite rules for pulling out atomic messages\<close>
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lemmas parts_insert_eq_I = equalityI [OF subsetI parts_insert_subset]
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lemma parts_insert_Agent [simp]:
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     "parts (insert (Agent agt) H) = insert (Agent agt) (parts H)"
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apply (rule parts_insert_eq_I)
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apply (erule parts.induct, auto)
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done
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lemma parts_insert_Nonce [simp]:
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     "parts (insert (Nonce N) H) = insert (Nonce N) (parts H)"
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apply (rule parts_insert_eq_I)
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apply (erule parts.induct, auto)
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done
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lemma parts_insert_Number [simp]:
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     "parts (insert (Number N) H) = insert (Number N) (parts H)"
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apply (rule parts_insert_eq_I)
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apply (erule parts.induct, auto)
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done
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lemma parts_insert_Key [simp]:
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     "parts (insert (Key K) H) = insert (Key K) (parts H)"
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apply (rule parts_insert_eq_I)
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apply (erule parts.induct, auto)
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done
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lemma parts_insert_Hash [simp]:
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     "parts (insert (Hash X) H) = insert (Hash X) (parts H)"
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apply (rule parts_insert_eq_I)
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apply (erule parts.induct, auto)
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done
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lemma parts_insert_Crypt [simp]:
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     "parts (insert (Crypt K X) H) =
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          insert (Crypt K X) (parts (insert X H))"
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apply (rule equalityI)
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apply (rule subsetI)
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apply (erule parts.induct, auto)
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apply (blast intro: parts.Body)
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done
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lemma parts_insert_MPair [simp]:
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     "parts (insert \<lbrace>X,Y\<rbrace> H) =
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          insert \<lbrace>X,Y\<rbrace> (parts (insert X (insert Y H)))"
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apply (rule equalityI)
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apply (rule subsetI)
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apply (erule parts.induct, auto)
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apply (blast intro: parts.Fst parts.Snd)+
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done
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lemma parts_image_Key [simp]: "parts (Key`N) = Key`N"
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apply auto
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apply (erule parts.induct, auto)
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done
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lemma msg_Nonce_supply: "\<exists>N. \<forall>n. N\<le>n --> Nonce n \<notin> parts {msg}"
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apply (induct_tac "msg")
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apply (simp_all add: parts_insert2)
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apply (metis Suc_n_not_le_n)
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apply (metis le_trans linorder_linear)
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done
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subsection\<open>Inductive relation "analz"\<close>
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text\<open>Inductive definition of "analz" -- what can be broken down from a set of
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    messages, including keys.  A form of downward closure.  Pairs can
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    be taken apart; messages decrypted with known keys.\<close>
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inductive_set
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  analz :: "msg set => msg set"
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  for H :: "msg set"
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  where
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    Inj [intro,simp] :    "X \<in> H ==> X \<in> analz H"
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  | Fst:     "\<lbrace>X,Y\<rbrace> \<in> analz H ==> X \<in> analz H"
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  | Snd:     "\<lbrace>X,Y\<rbrace> \<in> analz H ==> Y \<in> analz H"
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  | Decrypt [dest]:
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             "[|Crypt K X \<in> analz H; Key(invKey K): analz H|] ==> X \<in> analz H"
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paulson@23449
   329
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   330
text\<open>Monotonicity; Lemma 1 of Lowe's paper\<close>
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   331
lemma analz_mono: "G\<subseteq>H ==> analz(G) \<subseteq> analz(H)"
paulson@23449
   332
apply auto
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   333
apply (erule analz.induct)
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   334
apply (auto dest: analz.Fst analz.Snd)
paulson@23449
   335
done
paulson@23449
   336
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   337
text\<open>Making it safe speeds up proofs\<close>
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   338
lemma MPair_analz [elim!]:
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   339
     "[| \<lbrace>X,Y\<rbrace> \<in> analz H;
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   340
             [| X \<in> analz H; Y \<in> analz H |] ==> P
paulson@23449
   341
          |] ==> P"
paulson@23449
   342
by (blast dest: analz.Fst analz.Snd)
paulson@23449
   343
paulson@23449
   344
lemma analz_increasing: "H \<subseteq> analz(H)"
paulson@23449
   345
by blast
paulson@23449
   346
paulson@23449
   347
lemma analz_subset_parts: "analz H \<subseteq> parts H"
paulson@23449
   348
apply (rule subsetI)
paulson@23449
   349
apply (erule analz.induct, blast+)
paulson@23449
   350
done
paulson@23449
   351
wenzelm@45605
   352
lemmas analz_into_parts = analz_subset_parts [THEN subsetD]
paulson@23449
   353
wenzelm@45605
   354
lemmas not_parts_not_analz = analz_subset_parts [THEN contra_subsetD]
paulson@23449
   355
paulson@23449
   356
lemma parts_analz [simp]: "parts (analz H) = parts H"
paulson@23449
   357
apply (rule equalityI)
paulson@23449
   358
apply (metis analz_subset_parts parts_subset_iff)
paulson@23449
   359
apply (metis analz_increasing parts_mono)
paulson@23449
   360
done
paulson@23449
   361
paulson@23449
   362
paulson@23449
   363
lemma analz_parts [simp]: "analz (parts H) = parts H"
paulson@23449
   364
apply auto
paulson@23449
   365
apply (erule analz.induct, auto)
paulson@23449
   366
done
paulson@23449
   367
wenzelm@45605
   368
lemmas analz_insertI = subset_insertI [THEN analz_mono, THEN [2] rev_subsetD]
paulson@23449
   369
wenzelm@63167
   370
subsubsection\<open>General equational properties\<close>
paulson@23449
   371
paulson@23449
   372
lemma analz_empty [simp]: "analz{} = {}"
paulson@23449
   373
apply safe
paulson@23449
   374
apply (erule analz.induct, blast+)
paulson@23449
   375
done
paulson@23449
   376
wenzelm@63167
   377
text\<open>Converse fails: we can analz more from the union than from the
wenzelm@63167
   378
  separate parts, as a key in one might decrypt a message in the other\<close>
paulson@23449
   379
lemma analz_Un: "analz(G) \<union> analz(H) \<subseteq> analz(G \<union> H)"
paulson@23449
   380
by (intro Un_least analz_mono Un_upper1 Un_upper2)
paulson@23449
   381
paulson@23449
   382
lemma analz_insert: "insert X (analz H) \<subseteq> analz(insert X H)"
paulson@23449
   383
by (blast intro: analz_mono [THEN [2] rev_subsetD])
paulson@23449
   384
wenzelm@63167
   385
subsubsection\<open>Rewrite rules for pulling out atomic messages\<close>
paulson@23449
   386
paulson@23449
   387
lemmas analz_insert_eq_I = equalityI [OF subsetI analz_insert]
paulson@23449
   388
paulson@23449
   389
lemma analz_insert_Agent [simp]:
paulson@23449
   390
     "analz (insert (Agent agt) H) = insert (Agent agt) (analz H)"
blanchet@43197
   391
apply (rule analz_insert_eq_I)
blanchet@43197
   392
apply (erule analz.induct, auto)
paulson@23449
   393
done
paulson@23449
   394
paulson@23449
   395
lemma analz_insert_Nonce [simp]:
paulson@23449
   396
     "analz (insert (Nonce N) H) = insert (Nonce N) (analz H)"
blanchet@43197
   397
apply (rule analz_insert_eq_I)
blanchet@43197
   398
apply (erule analz.induct, auto)
paulson@23449
   399
done
paulson@23449
   400
paulson@23449
   401
lemma analz_insert_Number [simp]:
paulson@23449
   402
     "analz (insert (Number N) H) = insert (Number N) (analz H)"
blanchet@43197
   403
apply (rule analz_insert_eq_I)
blanchet@43197
   404
apply (erule analz.induct, auto)
paulson@23449
   405
done
paulson@23449
   406
paulson@23449
   407
lemma analz_insert_Hash [simp]:
paulson@23449
   408
     "analz (insert (Hash X) H) = insert (Hash X) (analz H)"
blanchet@43197
   409
apply (rule analz_insert_eq_I)
blanchet@43197
   410
apply (erule analz.induct, auto)
paulson@23449
   411
done
paulson@23449
   412
wenzelm@63167
   413
text\<open>Can only pull out Keys if they are not needed to decrypt the rest\<close>
blanchet@43197
   414
lemma analz_insert_Key [simp]:
blanchet@43197
   415
    "K \<notin> keysFor (analz H) ==>
paulson@23449
   416
          analz (insert (Key K) H) = insert (Key K) (analz H)"
paulson@23449
   417
apply (unfold keysFor_def)
blanchet@43197
   418
apply (rule analz_insert_eq_I)
blanchet@43197
   419
apply (erule analz.induct, auto)
paulson@23449
   420
done
paulson@23449
   421
paulson@23449
   422
lemma analz_insert_MPair [simp]:
wenzelm@61984
   423
     "analz (insert \<lbrace>X,Y\<rbrace> H) =
wenzelm@61984
   424
          insert \<lbrace>X,Y\<rbrace> (analz (insert X (insert Y H)))"
paulson@23449
   425
apply (rule equalityI)
paulson@23449
   426
apply (rule subsetI)
paulson@23449
   427
apply (erule analz.induct, auto)
paulson@23449
   428
apply (erule analz.induct)
paulson@23449
   429
apply (blast intro: analz.Fst analz.Snd)+
paulson@23449
   430
done
paulson@23449
   431
wenzelm@63167
   432
text\<open>Can pull out enCrypted message if the Key is not known\<close>
paulson@23449
   433
lemma analz_insert_Crypt:
blanchet@43197
   434
     "Key (invKey K) \<notin> analz H
paulson@23449
   435
      ==> analz (insert (Crypt K X) H) = insert (Crypt K X) (analz H)"
blanchet@43197
   436
apply (rule analz_insert_eq_I)
blanchet@43197
   437
apply (erule analz.induct, auto)
paulson@23449
   438
paulson@23449
   439
done
paulson@23449
   440
blanchet@43197
   441
lemma lemma1: "Key (invKey K) \<in> analz H ==>
blanchet@43197
   442
               analz (insert (Crypt K X) H) \<subseteq>
blanchet@43197
   443
               insert (Crypt K X) (analz (insert X H))"
paulson@23449
   444
apply (rule subsetI)
berghofe@23755
   445
apply (erule_tac x = x in analz.induct, auto)
paulson@23449
   446
done
paulson@23449
   447
blanchet@43197
   448
lemma lemma2: "Key (invKey K) \<in> analz H ==>
blanchet@43197
   449
               insert (Crypt K X) (analz (insert X H)) \<subseteq>
paulson@23449
   450
               analz (insert (Crypt K X) H)"
paulson@23449
   451
apply auto
berghofe@23755
   452
apply (erule_tac x = x in analz.induct, auto)
paulson@23449
   453
apply (blast intro: analz_insertI analz.Decrypt)
paulson@23449
   454
done
paulson@23449
   455
paulson@23449
   456
lemma analz_insert_Decrypt:
blanchet@43197
   457
     "Key (invKey K) \<in> analz H ==>
blanchet@43197
   458
               analz (insert (Crypt K X) H) =
paulson@23449
   459
               insert (Crypt K X) (analz (insert X H))"
paulson@23449
   460
by (intro equalityI lemma1 lemma2)
paulson@23449
   461
wenzelm@63167
   462
text\<open>Case analysis: either the message is secure, or it is not! Effective,
wenzelm@63167
   463
but can cause subgoals to blow up! Use with \<open>if_split\<close>; apparently
wenzelm@63167
   464
\<open>split_tac\<close> does not cope with patterns such as @{term"analz (insert
wenzelm@63167
   465
(Crypt K X) H)"}\<close>
paulson@23449
   466
lemma analz_Crypt_if [simp]:
blanchet@43197
   467
     "analz (insert (Crypt K X) H) =
blanchet@43197
   468
          (if (Key (invKey K) \<in> analz H)
blanchet@43197
   469
           then insert (Crypt K X) (analz (insert X H))
paulson@23449
   470
           else insert (Crypt K X) (analz H))"
paulson@23449
   471
by (simp add: analz_insert_Crypt analz_insert_Decrypt)
paulson@23449
   472
paulson@23449
   473
wenzelm@63167
   474
text\<open>This rule supposes "for the sake of argument" that we have the key.\<close>
paulson@23449
   475
lemma analz_insert_Crypt_subset:
blanchet@43197
   476
     "analz (insert (Crypt K X) H) \<subseteq>
paulson@23449
   477
           insert (Crypt K X) (analz (insert X H))"
paulson@23449
   478
apply (rule subsetI)
paulson@23449
   479
apply (erule analz.induct, auto)
paulson@23449
   480
done
paulson@23449
   481
paulson@23449
   482
paulson@23449
   483
lemma analz_image_Key [simp]: "analz (Key`N) = Key`N"
paulson@23449
   484
apply auto
paulson@23449
   485
apply (erule analz.induct, auto)
paulson@23449
   486
done
paulson@23449
   487
paulson@23449
   488
wenzelm@63167
   489
subsubsection\<open>Idempotence and transitivity\<close>
paulson@23449
   490
paulson@23449
   491
lemma analz_analzD [dest!]: "X\<in> analz (analz H) ==> X\<in> analz H"
paulson@23449
   492
by (erule analz.induct, blast+)
paulson@23449
   493
paulson@23449
   494
lemma analz_idem [simp]: "analz (analz H) = analz H"
paulson@23449
   495
by blast
paulson@23449
   496
paulson@23449
   497
lemma analz_subset_iff [simp]: "(analz G \<subseteq> analz H) = (G \<subseteq> analz H)"
paulson@23449
   498
apply (rule iffI)
blanchet@43197
   499
apply (iprover intro: subset_trans analz_increasing)
blanchet@43197
   500
apply (frule analz_mono, simp)
paulson@23449
   501
done
paulson@23449
   502
paulson@23449
   503
lemma analz_trans: "[| X\<in> analz G;  G \<subseteq> analz H |] ==> X\<in> analz H"
paulson@23449
   504
by (drule analz_mono, blast)
paulson@23449
   505
paulson@23449
   506
blanchet@36553
   507
declare analz_trans[intro]
blanchet@36553
   508
paulson@23449
   509
lemma analz_cut: "[| Y\<in> analz (insert X H);  X\<in> analz H |] ==> Y\<in> analz H"
blanchet@46075
   510
by (metis analz_idem analz_increasing analz_mono insert_absorb insert_mono insert_subset)
paulson@23449
   511
wenzelm@63167
   512
text\<open>This rewrite rule helps in the simplification of messages that involve
paulson@23449
   513
  the forwarding of unknown components (X).  Without it, removing occurrences
wenzelm@63167
   514
  of X can be very complicated.\<close>
paulson@23449
   515
lemma analz_insert_eq: "X\<in> analz H ==> analz (insert X H) = analz H"
paulson@23449
   516
by (blast intro: analz_cut analz_insertI)
paulson@23449
   517
paulson@23449
   518
wenzelm@63167
   519
text\<open>A congruence rule for "analz"\<close>
paulson@23449
   520
paulson@23449
   521
lemma analz_subset_cong:
blanchet@43197
   522
     "[| analz G \<subseteq> analz G'; analz H \<subseteq> analz H' |]
paulson@23449
   523
      ==> analz (G \<union> H) \<subseteq> analz (G' \<union> H')"
paulson@23449
   524
apply simp
paulson@23449
   525
apply (metis Un_absorb2 Un_commute Un_subset_iff Un_upper1 Un_upper2 analz_mono)
paulson@23449
   526
done
paulson@23449
   527
paulson@23449
   528
paulson@23449
   529
lemma analz_cong:
blanchet@43197
   530
     "[| analz G = analz G'; analz H = analz H'
paulson@23449
   531
               |] ==> analz (G \<union> H) = analz (G' \<union> H')"
blanchet@43197
   532
by (intro equalityI analz_subset_cong, simp_all)
paulson@23449
   533
paulson@23449
   534
lemma analz_insert_cong:
paulson@23449
   535
     "analz H = analz H' ==> analz(insert X H) = analz(insert X H')"
paulson@23449
   536
by (force simp only: insert_def intro!: analz_cong)
paulson@23449
   537
wenzelm@63167
   538
text\<open>If there are no pairs or encryptions then analz does nothing\<close>
paulson@23449
   539
lemma analz_trivial:
wenzelm@61984
   540
     "[| \<forall>X Y. \<lbrace>X,Y\<rbrace> \<notin> H;  \<forall>X K. Crypt K X \<notin> H |] ==> analz H = H"
paulson@23449
   541
apply safe
paulson@23449
   542
apply (erule analz.induct, blast+)
paulson@23449
   543
done
paulson@23449
   544
wenzelm@63167
   545
text\<open>These two are obsolete (with a single Spy) but cost little to prove...\<close>
paulson@23449
   546
lemma analz_UN_analz_lemma:
paulson@23449
   547
     "X\<in> analz (\<Union>i\<in>A. analz (H i)) ==> X\<in> analz (\<Union>i\<in>A. H i)"
paulson@23449
   548
apply (erule analz.induct)
paulson@23449
   549
apply (blast intro: analz_mono [THEN [2] rev_subsetD])+
paulson@23449
   550
done
paulson@23449
   551
paulson@23449
   552
lemma analz_UN_analz [simp]: "analz (\<Union>i\<in>A. analz (H i)) = analz (\<Union>i\<in>A. H i)"
paulson@23449
   553
by (blast intro: analz_UN_analz_lemma analz_mono [THEN [2] rev_subsetD])
paulson@23449
   554
paulson@23449
   555
wenzelm@63167
   556
subsection\<open>Inductive relation "synth"\<close>
paulson@23449
   557
wenzelm@63167
   558
text\<open>Inductive definition of "synth" -- what can be built up from a set of
paulson@23449
   559
    messages.  A form of upward closure.  Pairs can be built, messages
paulson@23449
   560
    encrypted with known keys.  Agent names are public domain.
wenzelm@63167
   561
    Numbers can be guessed, but Nonces cannot be.\<close>
paulson@23449
   562
berghofe@23755
   563
inductive_set
berghofe@23755
   564
  synth :: "msg set => msg set"
berghofe@23755
   565
  for H :: "msg set"
berghofe@23755
   566
  where
paulson@23449
   567
    Inj    [intro]:   "X \<in> H ==> X \<in> synth H"
berghofe@23755
   568
  | Agent  [intro]:   "Agent agt \<in> synth H"
berghofe@23755
   569
  | Number [intro]:   "Number n  \<in> synth H"
berghofe@23755
   570
  | Hash   [intro]:   "X \<in> synth H ==> Hash X \<in> synth H"
wenzelm@61984
   571
  | MPair  [intro]:   "[|X \<in> synth H;  Y \<in> synth H|] ==> \<lbrace>X,Y\<rbrace> \<in> synth H"
berghofe@23755
   572
  | Crypt  [intro]:   "[|X \<in> synth H;  Key(K) \<in> H|] ==> Crypt K X \<in> synth H"
paulson@23449
   573
wenzelm@63167
   574
text\<open>Monotonicity\<close>
paulson@23449
   575
lemma synth_mono: "G\<subseteq>H ==> synth(G) \<subseteq> synth(H)"
blanchet@43197
   576
  by (auto, erule synth.induct, auto)
paulson@23449
   577
wenzelm@63167
   578
text\<open>NO \<open>Agent_synth\<close>, as any Agent name can be synthesized.
wenzelm@63167
   579
  The same holds for @{term Number}\<close>
paulson@23449
   580
inductive_cases Nonce_synth [elim!]: "Nonce n \<in> synth H"
paulson@23449
   581
inductive_cases Key_synth   [elim!]: "Key K \<in> synth H"
paulson@23449
   582
inductive_cases Hash_synth  [elim!]: "Hash X \<in> synth H"
wenzelm@61984
   583
inductive_cases MPair_synth [elim!]: "\<lbrace>X,Y\<rbrace> \<in> synth H"
paulson@23449
   584
inductive_cases Crypt_synth [elim!]: "Crypt K X \<in> synth H"
paulson@23449
   585
paulson@23449
   586
paulson@23449
   587
lemma synth_increasing: "H \<subseteq> synth(H)"
paulson@23449
   588
by blast
paulson@23449
   589
wenzelm@63167
   590
subsubsection\<open>Unions\<close>
paulson@23449
   591
wenzelm@63167
   592
text\<open>Converse fails: we can synth more from the union than from the
wenzelm@63167
   593
  separate parts, building a compound message using elements of each.\<close>
paulson@23449
   594
lemma synth_Un: "synth(G) \<union> synth(H) \<subseteq> synth(G \<union> H)"
paulson@23449
   595
by (intro Un_least synth_mono Un_upper1 Un_upper2)
paulson@23449
   596
paulson@23449
   597
lemma synth_insert: "insert X (synth H) \<subseteq> synth(insert X H)"
paulson@23449
   598
by (metis insert_iff insert_subset subset_insertI synth.Inj synth_mono)
paulson@23449
   599
wenzelm@63167
   600
subsubsection\<open>Idempotence and transitivity\<close>
paulson@23449
   601
paulson@23449
   602
lemma synth_synthD [dest!]: "X\<in> synth (synth H) ==> X\<in> synth H"
paulson@23449
   603
by (erule synth.induct, blast+)
paulson@23449
   604
paulson@23449
   605
lemma synth_idem: "synth (synth H) = synth H"
paulson@23449
   606
by blast
paulson@23449
   607
paulson@23449
   608
lemma synth_subset_iff [simp]: "(synth G \<subseteq> synth H) = (G \<subseteq> synth H)"
paulson@23449
   609
apply (rule iffI)
blanchet@43197
   610
apply (iprover intro: subset_trans synth_increasing)
blanchet@43197
   611
apply (frule synth_mono, simp add: synth_idem)
paulson@23449
   612
done
paulson@23449
   613
paulson@23449
   614
lemma synth_trans: "[| X\<in> synth G;  G \<subseteq> synth H |] ==> X\<in> synth H"
paulson@23449
   615
by (drule synth_mono, blast)
paulson@23449
   616
paulson@23449
   617
lemma synth_cut: "[| Y\<in> synth (insert X H);  X\<in> synth H |] ==> Y\<in> synth H"
paulson@23449
   618
by (metis insert_absorb insert_mono insert_subset synth_idem synth_increasing synth_mono)
paulson@23449
   619
paulson@23449
   620
lemma Agent_synth [simp]: "Agent A \<in> synth H"
paulson@23449
   621
by blast
paulson@23449
   622
paulson@23449
   623
lemma Number_synth [simp]: "Number n \<in> synth H"
paulson@23449
   624
by blast
paulson@23449
   625
paulson@23449
   626
lemma Nonce_synth_eq [simp]: "(Nonce N \<in> synth H) = (Nonce N \<in> H)"
paulson@23449
   627
by blast
paulson@23449
   628
paulson@23449
   629
lemma Key_synth_eq [simp]: "(Key K \<in> synth H) = (Key K \<in> H)"
paulson@23449
   630
by blast
paulson@23449
   631
paulson@23449
   632
lemma Crypt_synth_eq [simp]:
paulson@23449
   633
     "Key K \<notin> H ==> (Crypt K X \<in> synth H) = (Crypt K X \<in> H)"
paulson@23449
   634
by blast
paulson@23449
   635
paulson@23449
   636
blanchet@43197
   637
lemma keysFor_synth [simp]:
paulson@23449
   638
    "keysFor (synth H) = keysFor H \<union> invKey`{K. Key K \<in> H}"
paulson@23449
   639
by (unfold keysFor_def, blast)
paulson@23449
   640
paulson@23449
   641
wenzelm@63167
   642
subsubsection\<open>Combinations of parts, analz and synth\<close>
paulson@23449
   643
paulson@23449
   644
lemma parts_synth [simp]: "parts (synth H) = parts H \<union> synth H"
paulson@23449
   645
apply (rule equalityI)
paulson@23449
   646
apply (rule subsetI)
paulson@23449
   647
apply (erule parts.induct)
paulson@23449
   648
apply (metis UnCI)
paulson@23449
   649
apply (metis MPair_synth UnCI UnE insert_absorb insert_subset parts.Fst parts_increasing)
paulson@23449
   650
apply (metis MPair_synth UnCI UnE insert_absorb insert_subset parts.Snd parts_increasing)
paulson@23449
   651
apply (metis Body Crypt_synth UnCI UnE insert_absorb insert_subset parts_increasing)
paulson@23449
   652
apply (metis Un_subset_iff parts_increasing parts_mono synth_increasing)
paulson@23449
   653
done
paulson@23449
   654
paulson@23449
   655
lemma analz_analz_Un [simp]: "analz (analz G \<union> H) = analz (G \<union> H)"
huffman@45503
   656
apply (rule equalityI)
paulson@23449
   657
apply (metis analz_idem analz_subset_cong order_eq_refl)
paulson@23449
   658
apply (metis analz_increasing analz_subset_cong order_eq_refl)
paulson@23449
   659
done
paulson@23449
   660
blanchet@36553
   661
declare analz_mono [intro] analz.Fst [intro] analz.Snd [intro] Un_least [intro]
blanchet@36553
   662
paulson@23449
   663
lemma analz_synth_Un [simp]: "analz (synth G \<union> H) = analz (G \<union> H) \<union> synth G"
paulson@23449
   664
apply (rule equalityI)
paulson@23449
   665
apply (rule subsetI)
paulson@23449
   666
apply (erule analz.induct)
paulson@23449
   667
apply (metis UnCI UnE Un_commute analz.Inj)
haftmann@45970
   668
apply (metis MPair_synth UnCI UnE Un_commute analz.Fst analz.Inj)
haftmann@45970
   669
apply (metis MPair_synth UnCI UnE Un_commute analz.Inj analz.Snd)
paulson@23449
   670
apply (blast intro: analz.Decrypt)
wenzelm@24759
   671
apply blast
paulson@23449
   672
done
paulson@23449
   673
paulson@23449
   674
lemma analz_synth [simp]: "analz (synth H) = analz H \<union> synth H"
blanchet@36553
   675
proof -
wenzelm@53015
   676
  have "\<forall>x\<^sub>2 x\<^sub>1. synth x\<^sub>1 \<union> analz (x\<^sub>1 \<union> x\<^sub>2) = analz (synth x\<^sub>1 \<union> x\<^sub>2)" by (metis Un_commute analz_synth_Un)
wenzelm@53015
   677
  hence "\<forall>x\<^sub>1. synth x\<^sub>1 \<union> analz x\<^sub>1 = analz (synth x\<^sub>1 \<union> {})" by (metis Un_empty_right)
wenzelm@53015
   678
  hence "\<forall>x\<^sub>1. synth x\<^sub>1 \<union> analz x\<^sub>1 = analz (synth x\<^sub>1)" by (metis Un_empty_right)
wenzelm@53015
   679
  hence "\<forall>x\<^sub>1. analz x\<^sub>1 \<union> synth x\<^sub>1 = analz (synth x\<^sub>1)" by (metis Un_commute)
blanchet@36553
   680
  thus "analz (synth H) = analz H \<union> synth H" by metis
paulson@23449
   681
qed
paulson@23449
   682
paulson@23449
   683
wenzelm@63167
   684
subsubsection\<open>For reasoning about the Fake rule in traces\<close>
paulson@23449
   685
haftmann@45970
   686
lemma parts_insert_subset_Un: "X \<in> G ==> parts(insert X H) \<subseteq> parts G \<union> parts H"
blanchet@36553
   687
proof -
blanchet@36553
   688
  assume "X \<in> G"
wenzelm@53015
   689
  hence "\<forall>x\<^sub>1. G \<subseteq> x\<^sub>1 \<longrightarrow> X \<in> x\<^sub>1 " by auto
wenzelm@53015
   690
  hence "\<forall>x\<^sub>1. X \<in> G \<union> x\<^sub>1" by (metis Un_upper1)
blanchet@36911
   691
  hence "insert X H \<subseteq> G \<union> H" by (metis Un_upper2 insert_subset)
blanchet@36911
   692
  hence "parts (insert X H) \<subseteq> parts (G \<union> H)" by (metis parts_mono)
blanchet@36911
   693
  thus "parts (insert X H) \<subseteq> parts G \<union> parts H" by (metis parts_Un)
paulson@23449
   694
qed
paulson@23449
   695
paulson@23449
   696
lemma Fake_parts_insert:
blanchet@43197
   697
     "X \<in> synth (analz H) ==>
paulson@23449
   698
      parts (insert X H) \<subseteq> synth (analz H) \<union> parts H"
blanchet@36553
   699
proof -
blanchet@36553
   700
  assume A1: "X \<in> synth (analz H)"
wenzelm@53015
   701
  have F1: "\<forall>x\<^sub>1. analz x\<^sub>1 \<union> synth (analz x\<^sub>1) = analz (synth (analz x\<^sub>1))"
blanchet@36553
   702
    by (metis analz_idem analz_synth)
wenzelm@53015
   703
  have F2: "\<forall>x\<^sub>1. parts x\<^sub>1 \<union> synth (analz x\<^sub>1) = parts (synth (analz x\<^sub>1))"
blanchet@36553
   704
    by (metis parts_analz parts_synth)
haftmann@45970
   705
  have F3: "X \<in> synth (analz H)" using A1 by metis
wenzelm@61076
   706
  have "\<forall>x\<^sub>2 x\<^sub>1::msg set. x\<^sub>1 \<le> sup x\<^sub>1 x\<^sub>2" by (metis inf_sup_ord(3))
wenzelm@53015
   707
  hence F4: "\<forall>x\<^sub>1. analz x\<^sub>1 \<subseteq> analz (synth x\<^sub>1)" by (metis analz_synth)
haftmann@45970
   708
  have F5: "X \<in> synth (analz H)" using F3 by metis
wenzelm@53015
   709
  have "\<forall>x\<^sub>1. analz x\<^sub>1 \<subseteq> synth (analz x\<^sub>1)
wenzelm@53015
   710
         \<longrightarrow> analz (synth (analz x\<^sub>1)) = synth (analz x\<^sub>1)"
blanchet@36553
   711
    using F1 by (metis subset_Un_eq)
wenzelm@53015
   712
  hence F6: "\<forall>x\<^sub>1. analz (synth (analz x\<^sub>1)) = synth (analz x\<^sub>1)"
blanchet@36553
   713
    by (metis synth_increasing)
wenzelm@53015
   714
  have "\<forall>x\<^sub>1. x\<^sub>1 \<subseteq> analz (synth x\<^sub>1)" using F4 by (metis analz_subset_iff)
wenzelm@53015
   715
  hence "\<forall>x\<^sub>1. x\<^sub>1 \<subseteq> analz (synth (analz x\<^sub>1))" by (metis analz_subset_iff)
wenzelm@53015
   716
  hence "\<forall>x\<^sub>1. x\<^sub>1 \<subseteq> synth (analz x\<^sub>1)" using F6 by metis
blanchet@36553
   717
  hence "H \<subseteq> synth (analz H)" by metis
blanchet@36553
   718
  hence "H \<subseteq> synth (analz H) \<and> X \<in> synth (analz H)" using F5 by metis
blanchet@36553
   719
  hence "insert X H \<subseteq> synth (analz H)" by (metis insert_subset)
blanchet@36553
   720
  hence "parts (insert X H) \<subseteq> parts (synth (analz H))" by (metis parts_mono)
blanchet@36553
   721
  hence "parts (insert X H) \<subseteq> parts H \<union> synth (analz H)" using F2 by metis
blanchet@36553
   722
  thus "parts (insert X H) \<subseteq> synth (analz H) \<union> parts H" by (metis Un_commute)
paulson@23449
   723
qed
paulson@23449
   724
paulson@23449
   725
lemma Fake_parts_insert_in_Un:
blanchet@43197
   726
     "[|Z \<in> parts (insert X H);  X: synth (analz H)|]
huffman@45505
   727
      ==> Z \<in>  synth (analz H) \<union> parts H"
blanchet@36553
   728
by (blast dest: Fake_parts_insert [THEN subsetD, dest])
paulson@23449
   729
haftmann@45970
   730
declare synth_mono [intro]
blanchet@36553
   731
paulson@23449
   732
lemma Fake_analz_insert:
blanchet@36553
   733
     "X \<in> synth (analz G) ==>
paulson@23449
   734
      analz (insert X H) \<subseteq> synth (analz G) \<union> analz (G \<union> H)"
blanchet@36553
   735
by (metis Un_commute Un_insert_left Un_insert_right Un_upper1 analz_analz_Un
blanchet@36553
   736
          analz_mono analz_synth_Un insert_absorb)
paulson@23449
   737
paulson@23449
   738
lemma Fake_analz_insert_simpler:
blanchet@43197
   739
     "X \<in> synth (analz G) ==>
paulson@23449
   740
      analz (insert X H) \<subseteq> synth (analz G) \<union> analz (G \<union> H)"
paulson@23449
   741
apply (rule subsetI)
paulson@23449
   742
apply (subgoal_tac "x \<in> analz (synth (analz G) \<union> H) ")
paulson@23449
   743
apply (metis Un_commute analz_analz_Un analz_synth_Un)
blanchet@39260
   744
by (metis Un_upper1 Un_upper2 analz_mono insert_absorb insert_subset)
paulson@23449
   745
paulson@23449
   746
end