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(* Title: HOL/MetisTest/Message.thy
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ID: $Id$
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory
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Testing the metis method
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*)
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theory Message imports Main begin
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(*Needed occasionally with spy_analz_tac, e.g. in analz_insert_Key_newK*)
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lemma strange_Un_eq [simp]: "A \<union> (B \<union> A) = B \<union> A"
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by blast
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types
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key = nat
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consts
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all_symmetric :: bool --{*true if all keys are symmetric*}
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invKey :: "key=>key" --{*inverse of a symmetric key*}
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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 (rule exI [of _ id], auto)
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text{*The inverse of a symmetric key is itself; that of a public key
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is the private key and vice versa*}
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constdefs
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symKeys :: "key set"
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"symKeys == {K. invKey K = K}"
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datatype --{*We allow any number of friendly agents*}
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agent = Server | Friend nat | Spy
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datatype
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msg = Agent agent --{*Agent names*}
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| Number nat --{*Ordinary integers, timestamps, ...*}
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| Nonce nat --{*Unguessable nonces*}
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| Key key --{*Crypto keys*}
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| Hash msg --{*Hashing*}
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| MPair msg msg --{*Compound messages*}
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| Crypt key msg --{*Encryption, public- or shared-key*}
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text{*Concrete syntax: messages appear as {|A,B,NA|}, etc...*}
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syntax
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"@MTuple" :: "['a, args] => 'a * 'b" ("(2{|_,/ _|})")
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syntax (xsymbols)
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"@MTuple" :: "['a, args] => 'a * 'b" ("(2\<lbrace>_,/ _\<rbrace>)")
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translations
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"{|x, y, z|}" == "{|x, {|y, z|}|}"
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"{|x, y|}" == "MPair x y"
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constdefs
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HPair :: "[msg,msg] => msg" ("(4Hash[_] /_)" [0, 1000])
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--{*Message Y paired with a MAC computed with the help of X*}
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"Hash[X] Y == {| Hash{|X,Y|}, Y|}"
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keysFor :: "msg set => key set"
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--{*Keys useful to decrypt elements of a message set*}
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"keysFor H == invKey ` {K. \<exists>X. Crypt K X \<in> H}"
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subsubsection{*Inductive Definition of All Parts" of a Message*}
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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: "{|X,Y|} \<in> parts H ==> X \<in> parts H"
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| Snd: "{|X,Y|} \<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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ML{*ResAtp.problem_name := "Message__parts_mono"*}
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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 Inj set_mp)
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apply (metis Fst)
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apply (metis Snd)
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apply (metis Body)
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done
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text{*Equations hold because constructors are injective.*}
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lemma Friend_image_eq [simp]: "(Friend x \<in> Friend`A) = (x:A)"
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by auto
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lemma Key_image_eq [simp]: "(Key x \<in> Key`A) = (x\<in>A)"
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by auto
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lemma Nonce_Key_image_eq [simp]: "(Nonce x \<notin> Key`A)"
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by auto
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subsubsection{*Inverse of keys *}
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ML{*ResAtp.problem_name := "Message__invKey_eq"*}
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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{*keysFor operator*}
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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{*Monotonicity*}
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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 {|X,Y|} 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{*Inductive relation "parts"*}
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lemma MPair_parts:
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"[| {|X,Y|} \<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{*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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@{text MPair_parts} 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.*}
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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, standard]
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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{*WARNING: loops if H = {Y}, therefore must not be repeated!*}
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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 blast+
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done
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subsubsection{*Unions *}
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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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ML{*ResAtp.problem_name := "Message__parts_insert_two"*}
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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 insert_commute 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{*Added to simplify arguments to parts, analz and synth.
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NOTE: the UN versions are no longer used!*}
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text{*This allows @{text blast} to simplify occurrences of
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@{term "parts(G\<union>H)"} in the assumption.*}
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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{*Idempotence and transitivity *}
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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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ML{*ResAtp.problem_name := "Message__parts_subset_iff"*}
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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_Un parts_idem parts_increasing 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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ML{*ResAtp.problem_name := "Message__parts_cut"*}
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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 Un_subset_iff insert_subset parts_increasing parts_trans)
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subsubsection{*Rewrite rules for pulling out atomic messages *}
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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 {|X,Y|} H) =
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insert {|X,Y|} (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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ML{*ResAtp.problem_name := "Message__msg_Nonce_supply"*}
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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{*Inductive relation "analz"*}
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text{*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. *}
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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: "{|X,Y|} \<in> analz H ==> X \<in> analz H"
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| Snd: "{|X,Y|} \<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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text{*Monotonicity; Lemma 1 of Lowe's paper*}
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lemma analz_mono: "G\<subseteq>H ==> analz(G) \<subseteq> analz(H)"
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apply auto
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apply (erule analz.induct)
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apply (auto dest: analz.Fst analz.Snd)
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done
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text{*Making it safe speeds up proofs*}
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lemma MPair_analz [elim!]:
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"[| {|X,Y|} \<in> analz H;
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[| X \<in> analz H; Y \<in> analz H |] ==> P
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|] ==> P"
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by (blast dest: analz.Fst analz.Snd)
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lemma analz_increasing: "H \<subseteq> analz(H)"
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by blast
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lemma analz_subset_parts: "analz H \<subseteq> parts H"
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apply (rule subsetI)
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apply (erule analz.induct, blast+)
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done
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lemmas analz_into_parts = analz_subset_parts [THEN subsetD, standard]
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lemmas not_parts_not_analz = analz_subset_parts [THEN contra_subsetD, standard]
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ML{*ResAtp.problem_name := "Message__parts_analz"*}
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lemma parts_analz [simp]: "parts (analz H) = parts H"
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apply (rule equalityI)
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apply (metis analz_subset_parts parts_subset_iff)
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apply (metis analz_increasing parts_mono)
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done
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|
|
379 |
lemma analz_parts [simp]: "analz (parts H) = parts H"
|
|
380 |
apply auto
|
|
381 |
apply (erule analz.induct, auto)
|
|
382 |
done
|
|
383 |
|
|
384 |
lemmas analz_insertI = subset_insertI [THEN analz_mono, THEN [2] rev_subsetD, standard]
|
|
385 |
|
|
386 |
subsubsection{*General equational properties *}
|
|
387 |
|
|
388 |
lemma analz_empty [simp]: "analz{} = {}"
|
|
389 |
apply safe
|
|
390 |
apply (erule analz.induct, blast+)
|
|
391 |
done
|
|
392 |
|
|
393 |
text{*Converse fails: we can analz more from the union than from the
|
|
394 |
separate parts, as a key in one might decrypt a message in the other*}
|
|
395 |
lemma analz_Un: "analz(G) \<union> analz(H) \<subseteq> analz(G \<union> H)"
|
|
396 |
by (intro Un_least analz_mono Un_upper1 Un_upper2)
|
|
397 |
|
|
398 |
lemma analz_insert: "insert X (analz H) \<subseteq> analz(insert X H)"
|
|
399 |
by (blast intro: analz_mono [THEN [2] rev_subsetD])
|
|
400 |
|
|
401 |
subsubsection{*Rewrite rules for pulling out atomic messages *}
|
|
402 |
|
|
403 |
lemmas analz_insert_eq_I = equalityI [OF subsetI analz_insert]
|
|
404 |
|
|
405 |
lemma analz_insert_Agent [simp]:
|
|
406 |
"analz (insert (Agent agt) H) = insert (Agent agt) (analz H)"
|
|
407 |
apply (rule analz_insert_eq_I)
|
|
408 |
apply (erule analz.induct, auto)
|
|
409 |
done
|
|
410 |
|
|
411 |
lemma analz_insert_Nonce [simp]:
|
|
412 |
"analz (insert (Nonce N) H) = insert (Nonce N) (analz H)"
|
|
413 |
apply (rule analz_insert_eq_I)
|
|
414 |
apply (erule analz.induct, auto)
|
|
415 |
done
|
|
416 |
|
|
417 |
lemma analz_insert_Number [simp]:
|
|
418 |
"analz (insert (Number N) H) = insert (Number N) (analz H)"
|
|
419 |
apply (rule analz_insert_eq_I)
|
|
420 |
apply (erule analz.induct, auto)
|
|
421 |
done
|
|
422 |
|
|
423 |
lemma analz_insert_Hash [simp]:
|
|
424 |
"analz (insert (Hash X) H) = insert (Hash X) (analz H)"
|
|
425 |
apply (rule analz_insert_eq_I)
|
|
426 |
apply (erule analz.induct, auto)
|
|
427 |
done
|
|
428 |
|
|
429 |
text{*Can only pull out Keys if they are not needed to decrypt the rest*}
|
|
430 |
lemma analz_insert_Key [simp]:
|
|
431 |
"K \<notin> keysFor (analz H) ==>
|
|
432 |
analz (insert (Key K) H) = insert (Key K) (analz H)"
|
|
433 |
apply (unfold keysFor_def)
|
|
434 |
apply (rule analz_insert_eq_I)
|
|
435 |
apply (erule analz.induct, auto)
|
|
436 |
done
|
|
437 |
|
|
438 |
lemma analz_insert_MPair [simp]:
|
|
439 |
"analz (insert {|X,Y|} H) =
|
|
440 |
insert {|X,Y|} (analz (insert X (insert Y H)))"
|
|
441 |
apply (rule equalityI)
|
|
442 |
apply (rule subsetI)
|
|
443 |
apply (erule analz.induct, auto)
|
|
444 |
apply (erule analz.induct)
|
|
445 |
apply (blast intro: analz.Fst analz.Snd)+
|
|
446 |
done
|
|
447 |
|
|
448 |
text{*Can pull out enCrypted message if the Key is not known*}
|
|
449 |
lemma analz_insert_Crypt:
|
|
450 |
"Key (invKey K) \<notin> analz H
|
|
451 |
==> analz (insert (Crypt K X) H) = insert (Crypt K X) (analz H)"
|
|
452 |
apply (rule analz_insert_eq_I)
|
|
453 |
apply (erule analz.induct, auto)
|
|
454 |
|
|
455 |
done
|
|
456 |
|
|
457 |
lemma lemma1: "Key (invKey K) \<in> analz H ==>
|
|
458 |
analz (insert (Crypt K X) H) \<subseteq>
|
|
459 |
insert (Crypt K X) (analz (insert X H))"
|
|
460 |
apply (rule subsetI)
|
23755
|
461 |
apply (erule_tac x = x in analz.induct, auto)
|
23449
|
462 |
done
|
|
463 |
|
|
464 |
lemma lemma2: "Key (invKey K) \<in> analz H ==>
|
|
465 |
insert (Crypt K X) (analz (insert X H)) \<subseteq>
|
|
466 |
analz (insert (Crypt K X) H)"
|
|
467 |
apply auto
|
23755
|
468 |
apply (erule_tac x = x in analz.induct, auto)
|
23449
|
469 |
apply (blast intro: analz_insertI analz.Decrypt)
|
|
470 |
done
|
|
471 |
|
|
472 |
lemma analz_insert_Decrypt:
|
|
473 |
"Key (invKey K) \<in> analz H ==>
|
|
474 |
analz (insert (Crypt K X) H) =
|
|
475 |
insert (Crypt K X) (analz (insert X H))"
|
|
476 |
by (intro equalityI lemma1 lemma2)
|
|
477 |
|
|
478 |
text{*Case analysis: either the message is secure, or it is not! Effective,
|
|
479 |
but can cause subgoals to blow up! Use with @{text "split_if"}; apparently
|
|
480 |
@{text "split_tac"} does not cope with patterns such as @{term"analz (insert
|
|
481 |
(Crypt K X) H)"} *}
|
|
482 |
lemma analz_Crypt_if [simp]:
|
|
483 |
"analz (insert (Crypt K X) H) =
|
|
484 |
(if (Key (invKey K) \<in> analz H)
|
|
485 |
then insert (Crypt K X) (analz (insert X H))
|
|
486 |
else insert (Crypt K X) (analz H))"
|
|
487 |
by (simp add: analz_insert_Crypt analz_insert_Decrypt)
|
|
488 |
|
|
489 |
|
|
490 |
text{*This rule supposes "for the sake of argument" that we have the key.*}
|
|
491 |
lemma analz_insert_Crypt_subset:
|
|
492 |
"analz (insert (Crypt K X) H) \<subseteq>
|
|
493 |
insert (Crypt K X) (analz (insert X H))"
|
|
494 |
apply (rule subsetI)
|
|
495 |
apply (erule analz.induct, auto)
|
|
496 |
done
|
|
497 |
|
|
498 |
|
|
499 |
lemma analz_image_Key [simp]: "analz (Key`N) = Key`N"
|
|
500 |
apply auto
|
|
501 |
apply (erule analz.induct, auto)
|
|
502 |
done
|
|
503 |
|
|
504 |
|
|
505 |
subsubsection{*Idempotence and transitivity *}
|
|
506 |
|
|
507 |
lemma analz_analzD [dest!]: "X\<in> analz (analz H) ==> X\<in> analz H"
|
|
508 |
by (erule analz.induct, blast+)
|
|
509 |
|
|
510 |
lemma analz_idem [simp]: "analz (analz H) = analz H"
|
|
511 |
by blast
|
|
512 |
|
|
513 |
lemma analz_subset_iff [simp]: "(analz G \<subseteq> analz H) = (G \<subseteq> analz H)"
|
|
514 |
apply (rule iffI)
|
|
515 |
apply (iprover intro: subset_trans analz_increasing)
|
|
516 |
apply (frule analz_mono, simp)
|
|
517 |
done
|
|
518 |
|
|
519 |
lemma analz_trans: "[| X\<in> analz G; G \<subseteq> analz H |] ==> X\<in> analz H"
|
|
520 |
by (drule analz_mono, blast)
|
|
521 |
|
|
522 |
|
|
523 |
ML{*ResAtp.problem_name := "Message__analz_cut"*}
|
|
524 |
declare analz_trans[intro]
|
|
525 |
lemma analz_cut: "[| Y\<in> analz (insert X H); X\<in> analz H |] ==> Y\<in> analz H"
|
|
526 |
(*TOO SLOW
|
|
527 |
by (metis analz_idem analz_increasing analz_mono insert_absorb insert_mono insert_subset) --{*317s*}
|
|
528 |
??*)
|
|
529 |
by (erule analz_trans, blast)
|
|
530 |
|
|
531 |
|
|
532 |
text{*This rewrite rule helps in the simplification of messages that involve
|
|
533 |
the forwarding of unknown components (X). Without it, removing occurrences
|
|
534 |
of X can be very complicated. *}
|
|
535 |
lemma analz_insert_eq: "X\<in> analz H ==> analz (insert X H) = analz H"
|
|
536 |
by (blast intro: analz_cut analz_insertI)
|
|
537 |
|
|
538 |
|
|
539 |
text{*A congruence rule for "analz" *}
|
|
540 |
|
|
541 |
ML{*ResAtp.problem_name := "Message__analz_subset_cong"*}
|
|
542 |
lemma analz_subset_cong:
|
|
543 |
"[| analz G \<subseteq> analz G'; analz H \<subseteq> analz H' |]
|
|
544 |
==> analz (G \<union> H) \<subseteq> analz (G' \<union> H')"
|
|
545 |
apply simp
|
|
546 |
apply (metis Un_absorb2 Un_commute Un_subset_iff Un_upper1 Un_upper2 analz_mono)
|
|
547 |
done
|
|
548 |
|
|
549 |
|
|
550 |
lemma analz_cong:
|
|
551 |
"[| analz G = analz G'; analz H = analz H'
|
|
552 |
|] ==> analz (G \<union> H) = analz (G' \<union> H')"
|
|
553 |
by (intro equalityI analz_subset_cong, simp_all)
|
|
554 |
|
|
555 |
lemma analz_insert_cong:
|
|
556 |
"analz H = analz H' ==> analz(insert X H) = analz(insert X H')"
|
|
557 |
by (force simp only: insert_def intro!: analz_cong)
|
|
558 |
|
|
559 |
text{*If there are no pairs or encryptions then analz does nothing*}
|
|
560 |
lemma analz_trivial:
|
|
561 |
"[| \<forall>X Y. {|X,Y|} \<notin> H; \<forall>X K. Crypt K X \<notin> H |] ==> analz H = H"
|
|
562 |
apply safe
|
|
563 |
apply (erule analz.induct, blast+)
|
|
564 |
done
|
|
565 |
|
|
566 |
text{*These two are obsolete (with a single Spy) but cost little to prove...*}
|
|
567 |
lemma analz_UN_analz_lemma:
|
|
568 |
"X\<in> analz (\<Union>i\<in>A. analz (H i)) ==> X\<in> analz (\<Union>i\<in>A. H i)"
|
|
569 |
apply (erule analz.induct)
|
|
570 |
apply (blast intro: analz_mono [THEN [2] rev_subsetD])+
|
|
571 |
done
|
|
572 |
|
|
573 |
lemma analz_UN_analz [simp]: "analz (\<Union>i\<in>A. analz (H i)) = analz (\<Union>i\<in>A. H i)"
|
|
574 |
by (blast intro: analz_UN_analz_lemma analz_mono [THEN [2] rev_subsetD])
|
|
575 |
|
|
576 |
|
|
577 |
subsection{*Inductive relation "synth"*}
|
|
578 |
|
|
579 |
text{*Inductive definition of "synth" -- what can be built up from a set of
|
|
580 |
messages. A form of upward closure. Pairs can be built, messages
|
|
581 |
encrypted with known keys. Agent names are public domain.
|
|
582 |
Numbers can be guessed, but Nonces cannot be. *}
|
|
583 |
|
23755
|
584 |
inductive_set
|
|
585 |
synth :: "msg set => msg set"
|
|
586 |
for H :: "msg set"
|
|
587 |
where
|
23449
|
588 |
Inj [intro]: "X \<in> H ==> X \<in> synth H"
|
23755
|
589 |
| Agent [intro]: "Agent agt \<in> synth H"
|
|
590 |
| Number [intro]: "Number n \<in> synth H"
|
|
591 |
| Hash [intro]: "X \<in> synth H ==> Hash X \<in> synth H"
|
|
592 |
| MPair [intro]: "[|X \<in> synth H; Y \<in> synth H|] ==> {|X,Y|} \<in> synth H"
|
|
593 |
| Crypt [intro]: "[|X \<in> synth H; Key(K) \<in> H|] ==> Crypt K X \<in> synth H"
|
23449
|
594 |
|
|
595 |
text{*Monotonicity*}
|
|
596 |
lemma synth_mono: "G\<subseteq>H ==> synth(G) \<subseteq> synth(H)"
|
|
597 |
by (auto, erule synth.induct, auto)
|
|
598 |
|
|
599 |
text{*NO @{text Agent_synth}, as any Agent name can be synthesized.
|
|
600 |
The same holds for @{term Number}*}
|
|
601 |
inductive_cases Nonce_synth [elim!]: "Nonce n \<in> synth H"
|
|
602 |
inductive_cases Key_synth [elim!]: "Key K \<in> synth H"
|
|
603 |
inductive_cases Hash_synth [elim!]: "Hash X \<in> synth H"
|
|
604 |
inductive_cases MPair_synth [elim!]: "{|X,Y|} \<in> synth H"
|
|
605 |
inductive_cases Crypt_synth [elim!]: "Crypt K X \<in> synth H"
|
|
606 |
|
|
607 |
|
|
608 |
lemma synth_increasing: "H \<subseteq> synth(H)"
|
|
609 |
by blast
|
|
610 |
|
|
611 |
subsubsection{*Unions *}
|
|
612 |
|
|
613 |
text{*Converse fails: we can synth more from the union than from the
|
|
614 |
separate parts, building a compound message using elements of each.*}
|
|
615 |
lemma synth_Un: "synth(G) \<union> synth(H) \<subseteq> synth(G \<union> H)"
|
|
616 |
by (intro Un_least synth_mono Un_upper1 Un_upper2)
|
|
617 |
|
|
618 |
|
|
619 |
ML{*ResAtp.problem_name := "Message__synth_insert"*}
|
|
620 |
|
|
621 |
lemma synth_insert: "insert X (synth H) \<subseteq> synth(insert X H)"
|
|
622 |
by (metis insert_iff insert_subset subset_insertI synth.Inj synth_mono)
|
|
623 |
|
|
624 |
subsubsection{*Idempotence and transitivity *}
|
|
625 |
|
|
626 |
lemma synth_synthD [dest!]: "X\<in> synth (synth H) ==> X\<in> synth H"
|
|
627 |
by (erule synth.induct, blast+)
|
|
628 |
|
|
629 |
lemma synth_idem: "synth (synth H) = synth H"
|
|
630 |
by blast
|
|
631 |
|
|
632 |
lemma synth_subset_iff [simp]: "(synth G \<subseteq> synth H) = (G \<subseteq> synth H)"
|
|
633 |
apply (rule iffI)
|
|
634 |
apply (iprover intro: subset_trans synth_increasing)
|
|
635 |
apply (frule synth_mono, simp add: synth_idem)
|
|
636 |
done
|
|
637 |
|
|
638 |
lemma synth_trans: "[| X\<in> synth G; G \<subseteq> synth H |] ==> X\<in> synth H"
|
|
639 |
by (drule synth_mono, blast)
|
|
640 |
|
|
641 |
ML{*ResAtp.problem_name := "Message__synth_cut"*}
|
|
642 |
lemma synth_cut: "[| Y\<in> synth (insert X H); X\<in> synth H |] ==> Y\<in> synth H"
|
|
643 |
(*TOO SLOW
|
|
644 |
by (metis insert_absorb insert_mono insert_subset synth_idem synth_increasing synth_mono)
|
|
645 |
*)
|
|
646 |
by (erule synth_trans, blast)
|
|
647 |
|
|
648 |
|
|
649 |
lemma Agent_synth [simp]: "Agent A \<in> synth H"
|
|
650 |
by blast
|
|
651 |
|
|
652 |
lemma Number_synth [simp]: "Number n \<in> synth H"
|
|
653 |
by blast
|
|
654 |
|
|
655 |
lemma Nonce_synth_eq [simp]: "(Nonce N \<in> synth H) = (Nonce N \<in> H)"
|
|
656 |
by blast
|
|
657 |
|
|
658 |
lemma Key_synth_eq [simp]: "(Key K \<in> synth H) = (Key K \<in> H)"
|
|
659 |
by blast
|
|
660 |
|
|
661 |
lemma Crypt_synth_eq [simp]:
|
|
662 |
"Key K \<notin> H ==> (Crypt K X \<in> synth H) = (Crypt K X \<in> H)"
|
|
663 |
by blast
|
|
664 |
|
|
665 |
|
|
666 |
lemma keysFor_synth [simp]:
|
|
667 |
"keysFor (synth H) = keysFor H \<union> invKey`{K. Key K \<in> H}"
|
|
668 |
by (unfold keysFor_def, blast)
|
|
669 |
|
|
670 |
|
|
671 |
subsubsection{*Combinations of parts, analz and synth *}
|
|
672 |
|
|
673 |
ML{*ResAtp.problem_name := "Message__parts_synth"*}
|
|
674 |
lemma parts_synth [simp]: "parts (synth H) = parts H \<union> synth H"
|
|
675 |
apply (rule equalityI)
|
|
676 |
apply (rule subsetI)
|
|
677 |
apply (erule parts.induct)
|
|
678 |
apply (metis UnCI)
|
|
679 |
apply (metis MPair_synth UnCI UnE insert_absorb insert_subset parts.Fst parts_increasing)
|
|
680 |
apply (metis MPair_synth UnCI UnE insert_absorb insert_subset parts.Snd parts_increasing)
|
|
681 |
apply (metis Body Crypt_synth UnCI UnE insert_absorb insert_subset parts_increasing)
|
|
682 |
apply (metis Un_subset_iff parts_increasing parts_mono synth_increasing)
|
|
683 |
done
|
|
684 |
|
|
685 |
|
|
686 |
|
|
687 |
|
|
688 |
ML{*ResAtp.problem_name := "Message__analz_analz_Un"*}
|
|
689 |
lemma analz_analz_Un [simp]: "analz (analz G \<union> H) = analz (G \<union> H)"
|
|
690 |
apply (rule equalityI);
|
|
691 |
apply (metis analz_idem analz_subset_cong order_eq_refl)
|
|
692 |
apply (metis analz_increasing analz_subset_cong order_eq_refl)
|
|
693 |
done
|
|
694 |
|
|
695 |
ML{*ResAtp.problem_name := "Message__analz_synth_Un"*}
|
|
696 |
declare analz_mono [intro] analz.Fst [intro] analz.Snd [intro] Un_least [intro]
|
|
697 |
lemma analz_synth_Un [simp]: "analz (synth G \<union> H) = analz (G \<union> H) \<union> synth G"
|
|
698 |
apply (rule equalityI)
|
|
699 |
apply (rule subsetI)
|
|
700 |
apply (erule analz.induct)
|
|
701 |
apply (metis UnCI UnE Un_commute analz.Inj)
|
|
702 |
apply (metis MPair_synth UnCI UnE Un_commute Un_upper1 analz.Fst analz_increasing analz_mono insert_absorb insert_subset)
|
|
703 |
apply (metis MPair_synth UnCI UnE Un_commute Un_upper1 analz.Snd analz_increasing analz_mono insert_absorb insert_subset)
|
|
704 |
apply (blast intro: analz.Decrypt)
|
24759
|
705 |
apply blast
|
23449
|
706 |
done
|
|
707 |
|
|
708 |
|
|
709 |
ML{*ResAtp.problem_name := "Message__analz_synth"*}
|
|
710 |
lemma analz_synth [simp]: "analz (synth H) = analz H \<union> synth H"
|
|
711 |
proof (neg_clausify)
|
|
712 |
assume 0: "analz (synth H) \<noteq> analz H \<union> synth H"
|
|
713 |
have 1: "\<And>X1 X3. sup (analz (sup X3 X1)) (synth X3) = analz (sup (synth X3) X1)"
|
|
714 |
by (metis analz_synth_Un sup_set_eq sup_set_eq sup_set_eq)
|
|
715 |
have 2: "sup (analz H) (synth H) \<noteq> analz (synth H)"
|
|
716 |
by (metis 0 sup_set_eq)
|
|
717 |
have 3: "\<And>X1 X3. sup (synth X3) (analz (sup X3 X1)) = analz (sup (synth X3) X1)"
|
|
718 |
by (metis 1 Un_commute sup_set_eq sup_set_eq)
|
|
719 |
have 4: "\<And>X3. sup (synth X3) (analz X3) = analz (sup (synth X3) {})"
|
|
720 |
by (metis 3 Un_empty_right sup_set_eq)
|
|
721 |
have 5: "\<And>X3. sup (synth X3) (analz X3) = analz (synth X3)"
|
|
722 |
by (metis 4 Un_empty_right sup_set_eq)
|
|
723 |
have 6: "\<And>X3. sup (analz X3) (synth X3) = analz (synth X3)"
|
|
724 |
by (metis 5 Un_commute sup_set_eq sup_set_eq)
|
|
725 |
show "False"
|
|
726 |
by (metis 2 6)
|
|
727 |
qed
|
|
728 |
|
|
729 |
|
|
730 |
subsubsection{*For reasoning about the Fake rule in traces *}
|
|
731 |
|
|
732 |
ML{*ResAtp.problem_name := "Message__parts_insert_subset_Un"*}
|
|
733 |
lemma parts_insert_subset_Un: "X\<in> G ==> parts(insert X H) \<subseteq> parts G \<union> parts H"
|
|
734 |
proof (neg_clausify)
|
|
735 |
assume 0: "X \<in> G"
|
|
736 |
assume 1: "\<not> parts (insert X H) \<subseteq> parts G \<union> parts H"
|
|
737 |
have 2: "\<not> parts (insert X H) \<subseteq> parts (G \<union> H)"
|
|
738 |
by (metis 1 parts_Un)
|
|
739 |
have 3: "\<not> insert X H \<subseteq> G \<union> H"
|
|
740 |
by (metis 2 parts_mono)
|
|
741 |
have 4: "X \<notin> G \<union> H \<or> \<not> H \<subseteq> G \<union> H"
|
|
742 |
by (metis 3 insert_subset)
|
|
743 |
have 5: "X \<notin> G \<union> H"
|
|
744 |
by (metis 4 Un_upper2)
|
|
745 |
have 6: "X \<notin> G"
|
|
746 |
by (metis 5 UnCI)
|
|
747 |
show "False"
|
|
748 |
by (metis 6 0)
|
|
749 |
qed
|
|
750 |
|
|
751 |
ML{*ResAtp.problem_name := "Message__Fake_parts_insert"*}
|
|
752 |
lemma Fake_parts_insert:
|
|
753 |
"X \<in> synth (analz H) ==>
|
|
754 |
parts (insert X H) \<subseteq> synth (analz H) \<union> parts H"
|
|
755 |
proof (neg_clausify)
|
|
756 |
assume 0: "X \<in> synth (analz H)"
|
|
757 |
assume 1: "\<not> parts (insert X H) \<subseteq> synth (analz H) \<union> parts H"
|
|
758 |
have 2: "\<And>X3. parts X3 \<union> synth (analz X3) = parts (synth (analz X3))"
|
|
759 |
by (metis parts_synth parts_analz)
|
|
760 |
have 3: "\<And>X3. analz X3 \<union> synth (analz X3) = analz (synth (analz X3))"
|
|
761 |
by (metis analz_synth analz_idem)
|
|
762 |
have 4: "\<And>X3. analz X3 \<subseteq> analz (synth X3)"
|
|
763 |
by (metis Un_upper1 analz_synth)
|
|
764 |
have 5: "\<not> parts (insert X H) \<subseteq> parts H \<union> synth (analz H)"
|
|
765 |
by (metis 1 Un_commute)
|
|
766 |
have 6: "\<not> parts (insert X H) \<subseteq> parts (synth (analz H))"
|
|
767 |
by (metis 5 2)
|
|
768 |
have 7: "\<not> insert X H \<subseteq> synth (analz H)"
|
|
769 |
by (metis 6 parts_mono)
|
|
770 |
have 8: "X \<notin> synth (analz H) \<or> \<not> H \<subseteq> synth (analz H)"
|
|
771 |
by (metis 7 insert_subset)
|
|
772 |
have 9: "\<not> H \<subseteq> synth (analz H)"
|
|
773 |
by (metis 8 0)
|
|
774 |
have 10: "\<And>X3. X3 \<subseteq> analz (synth X3)"
|
|
775 |
by (metis analz_subset_iff 4)
|
|
776 |
have 11: "\<And>X3. X3 \<subseteq> analz (synth (analz X3))"
|
|
777 |
by (metis analz_subset_iff 10)
|
|
778 |
have 12: "\<And>X3. analz (synth (analz X3)) = synth (analz X3) \<or>
|
|
779 |
\<not> analz X3 \<subseteq> synth (analz X3)"
|
|
780 |
by (metis Un_absorb1 3)
|
|
781 |
have 13: "\<And>X3. analz (synth (analz X3)) = synth (analz X3)"
|
|
782 |
by (metis 12 synth_increasing)
|
|
783 |
have 14: "\<And>X3. X3 \<subseteq> synth (analz X3)"
|
|
784 |
by (metis 11 13)
|
|
785 |
show "False"
|
|
786 |
by (metis 9 14)
|
|
787 |
qed
|
|
788 |
|
|
789 |
lemma Fake_parts_insert_in_Un:
|
|
790 |
"[|Z \<in> parts (insert X H); X: synth (analz H)|]
|
|
791 |
==> Z \<in> synth (analz H) \<union> parts H";
|
|
792 |
by (blast dest: Fake_parts_insert [THEN subsetD, dest])
|
|
793 |
|
|
794 |
ML{*ResAtp.problem_name := "Message__Fake_analz_insert"*}
|
|
795 |
declare analz_mono [intro] synth_mono [intro]
|
|
796 |
lemma Fake_analz_insert:
|
|
797 |
"X\<in> synth (analz G) ==>
|
|
798 |
analz (insert X H) \<subseteq> synth (analz G) \<union> analz (G \<union> H)"
|
|
799 |
by (metis Un_commute Un_insert_left Un_insert_right Un_upper1 analz_analz_Un analz_mono analz_synth_Un equalityE insert_absorb order_le_less xt1(12))
|
|
800 |
|
|
801 |
ML{*ResAtp.problem_name := "Message__Fake_analz_insert_simpler"*}
|
|
802 |
(*simpler problems? BUT METIS CAN'T PROVE
|
|
803 |
lemma Fake_analz_insert_simpler:
|
|
804 |
"X\<in> synth (analz G) ==>
|
|
805 |
analz (insert X H) \<subseteq> synth (analz G) \<union> analz (G \<union> H)"
|
|
806 |
apply (rule subsetI)
|
|
807 |
apply (subgoal_tac "x \<in> analz (synth (analz G) \<union> H) ")
|
|
808 |
apply (metis Un_commute analz_analz_Un analz_synth_Un)
|
|
809 |
apply (metis Un_commute Un_upper1 Un_upper2 analz_cut analz_increasing analz_mono insert_absorb insert_mono insert_subset)
|
|
810 |
done
|
|
811 |
*)
|
|
812 |
|
|
813 |
end
|