author | blanchet |
Mon, 06 Jun 2011 20:36:35 +0200 | |
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parent 42463 | f270e3e18be5 |
child 45503 | 44790ec65f70 |
permissions | -rw-r--r-- |
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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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header {* Metis Example Featuring Message Authentication *} |
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theory Message |
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imports Main |
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begin |
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Metis examples use the new Skolemizer to test it
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declare [[metis_new_skolemizer]] |
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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 --{*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 (metis id_apply) |
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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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definition symKeys :: "key set" where |
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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|}" == "CONST MPair x y" |
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definition HPair :: "[msg,msg] => msg" ("(4Hash[_] /_)" [0, 1000]) where |
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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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definition keysFor :: "msg set => key set" where |
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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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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{*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 (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{*Inverse of keys *} |
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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 fast+ |
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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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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{*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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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 Un_insert_left Un_insert_right insert_absorb mem_def parts_Un parts_idem sup1CI) |
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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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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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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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lemma analz_parts [simp]: "analz (parts H) = parts H" |
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apply auto |
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apply (erule analz.induct, auto) |
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done |
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lemmas analz_insertI = subset_insertI [THEN analz_mono, THEN [2] rev_subsetD, standard] |
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subsubsection{*General equational properties *} |
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lemma analz_empty [simp]: "analz{} = {}" |
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apply safe |
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apply (erule analz.induct, blast+) |
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done |
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text{*Converse fails: we can analz more from the union than from the |
23449 | 382 |
separate parts, as a key in one might decrypt a message in the other*} |
383 |
lemma analz_Un: "analz(G) \<union> analz(H) \<subseteq> analz(G \<union> H)" |
|
384 |
by (intro Un_least analz_mono Un_upper1 Un_upper2) |
|
385 |
||
386 |
lemma analz_insert: "insert X (analz H) \<subseteq> analz(insert X H)" |
|
387 |
by (blast intro: analz_mono [THEN [2] rev_subsetD]) |
|
388 |
||
389 |
subsubsection{*Rewrite rules for pulling out atomic messages *} |
|
390 |
||
391 |
lemmas analz_insert_eq_I = equalityI [OF subsetI analz_insert] |
|
392 |
||
393 |
lemma analz_insert_Agent [simp]: |
|
394 |
"analz (insert (Agent agt) H) = insert (Agent agt) (analz H)" |
|
43197 | 395 |
apply (rule analz_insert_eq_I) |
396 |
apply (erule analz.induct, auto) |
|
23449 | 397 |
done |
398 |
||
399 |
lemma analz_insert_Nonce [simp]: |
|
400 |
"analz (insert (Nonce N) H) = insert (Nonce N) (analz H)" |
|
43197 | 401 |
apply (rule analz_insert_eq_I) |
402 |
apply (erule analz.induct, auto) |
|
23449 | 403 |
done |
404 |
||
405 |
lemma analz_insert_Number [simp]: |
|
406 |
"analz (insert (Number N) H) = insert (Number N) (analz H)" |
|
43197 | 407 |
apply (rule analz_insert_eq_I) |
408 |
apply (erule analz.induct, auto) |
|
23449 | 409 |
done |
410 |
||
411 |
lemma analz_insert_Hash [simp]: |
|
412 |
"analz (insert (Hash X) H) = insert (Hash X) (analz H)" |
|
43197 | 413 |
apply (rule analz_insert_eq_I) |
414 |
apply (erule analz.induct, auto) |
|
23449 | 415 |
done |
416 |
||
417 |
text{*Can only pull out Keys if they are not needed to decrypt the rest*} |
|
43197 | 418 |
lemma analz_insert_Key [simp]: |
419 |
"K \<notin> keysFor (analz H) ==> |
|
23449 | 420 |
analz (insert (Key K) H) = insert (Key K) (analz H)" |
421 |
apply (unfold keysFor_def) |
|
43197 | 422 |
apply (rule analz_insert_eq_I) |
423 |
apply (erule analz.induct, auto) |
|
23449 | 424 |
done |
425 |
||
426 |
lemma analz_insert_MPair [simp]: |
|
43197 | 427 |
"analz (insert {|X,Y|} H) = |
23449 | 428 |
insert {|X,Y|} (analz (insert X (insert Y H)))" |
429 |
apply (rule equalityI) |
|
430 |
apply (rule subsetI) |
|
431 |
apply (erule analz.induct, auto) |
|
432 |
apply (erule analz.induct) |
|
433 |
apply (blast intro: analz.Fst analz.Snd)+ |
|
434 |
done |
|
435 |
||
436 |
text{*Can pull out enCrypted message if the Key is not known*} |
|
437 |
lemma analz_insert_Crypt: |
|
43197 | 438 |
"Key (invKey K) \<notin> analz H |
23449 | 439 |
==> analz (insert (Crypt K X) H) = insert (Crypt K X) (analz H)" |
43197 | 440 |
apply (rule analz_insert_eq_I) |
441 |
apply (erule analz.induct, auto) |
|
23449 | 442 |
|
443 |
done |
|
444 |
||
43197 | 445 |
lemma lemma1: "Key (invKey K) \<in> analz H ==> |
446 |
analz (insert (Crypt K X) H) \<subseteq> |
|
447 |
insert (Crypt K X) (analz (insert X H))" |
|
23449 | 448 |
apply (rule subsetI) |
23755 | 449 |
apply (erule_tac x = x in analz.induct, auto) |
23449 | 450 |
done |
451 |
||
43197 | 452 |
lemma lemma2: "Key (invKey K) \<in> analz H ==> |
453 |
insert (Crypt K X) (analz (insert X H)) \<subseteq> |
|
23449 | 454 |
analz (insert (Crypt K X) H)" |
455 |
apply auto |
|
23755 | 456 |
apply (erule_tac x = x in analz.induct, auto) |
23449 | 457 |
apply (blast intro: analz_insertI analz.Decrypt) |
458 |
done |
|
459 |
||
460 |
lemma analz_insert_Decrypt: |
|
43197 | 461 |
"Key (invKey K) \<in> analz H ==> |
462 |
analz (insert (Crypt K X) H) = |
|
23449 | 463 |
insert (Crypt K X) (analz (insert X H))" |
464 |
by (intro equalityI lemma1 lemma2) |
|
465 |
||
466 |
text{*Case analysis: either the message is secure, or it is not! Effective, |
|
467 |
but can cause subgoals to blow up! Use with @{text "split_if"}; apparently |
|
468 |
@{text "split_tac"} does not cope with patterns such as @{term"analz (insert |
|
43197 | 469 |
(Crypt K X) H)"} *} |
23449 | 470 |
lemma analz_Crypt_if [simp]: |
43197 | 471 |
"analz (insert (Crypt K X) H) = |
472 |
(if (Key (invKey K) \<in> analz H) |
|
473 |
then insert (Crypt K X) (analz (insert X H)) |
|
23449 | 474 |
else insert (Crypt K X) (analz H))" |
475 |
by (simp add: analz_insert_Crypt analz_insert_Decrypt) |
|
476 |
||
477 |
||
478 |
text{*This rule supposes "for the sake of argument" that we have the key.*} |
|
479 |
lemma analz_insert_Crypt_subset: |
|
43197 | 480 |
"analz (insert (Crypt K X) H) \<subseteq> |
23449 | 481 |
insert (Crypt K X) (analz (insert X H))" |
482 |
apply (rule subsetI) |
|
483 |
apply (erule analz.induct, auto) |
|
484 |
done |
|
485 |
||
486 |
||
487 |
lemma analz_image_Key [simp]: "analz (Key`N) = Key`N" |
|
488 |
apply auto |
|
489 |
apply (erule analz.induct, auto) |
|
490 |
done |
|
491 |
||
492 |
||
493 |
subsubsection{*Idempotence and transitivity *} |
|
494 |
||
495 |
lemma analz_analzD [dest!]: "X\<in> analz (analz H) ==> X\<in> analz H" |
|
496 |
by (erule analz.induct, blast+) |
|
497 |
||
498 |
lemma analz_idem [simp]: "analz (analz H) = analz H" |
|
499 |
by blast |
|
500 |
||
501 |
lemma analz_subset_iff [simp]: "(analz G \<subseteq> analz H) = (G \<subseteq> analz H)" |
|
502 |
apply (rule iffI) |
|
43197 | 503 |
apply (iprover intro: subset_trans analz_increasing) |
504 |
apply (frule analz_mono, simp) |
|
23449 | 505 |
done |
506 |
||
507 |
lemma analz_trans: "[| X\<in> analz G; G \<subseteq> analz H |] ==> X\<in> analz H" |
|
508 |
by (drule analz_mono, blast) |
|
509 |
||
510 |
||
36553 | 511 |
declare analz_trans[intro] |
512 |
||
23449 | 513 |
lemma analz_cut: "[| Y\<in> analz (insert X H); X\<in> analz H |] ==> Y\<in> analz H" |
514 |
(*TOO SLOW |
|
515 |
by (metis analz_idem analz_increasing analz_mono insert_absorb insert_mono insert_subset) --{*317s*} |
|
516 |
??*) |
|
517 |
by (erule analz_trans, blast) |
|
518 |
||
519 |
||
520 |
text{*This rewrite rule helps in the simplification of messages that involve |
|
521 |
the forwarding of unknown components (X). Without it, removing occurrences |
|
522 |
of X can be very complicated. *} |
|
523 |
lemma analz_insert_eq: "X\<in> analz H ==> analz (insert X H) = analz H" |
|
524 |
by (blast intro: analz_cut analz_insertI) |
|
525 |
||
526 |
||
527 |
text{*A congruence rule for "analz" *} |
|
528 |
||
529 |
lemma analz_subset_cong: |
|
43197 | 530 |
"[| analz G \<subseteq> analz G'; analz H \<subseteq> analz H' |] |
23449 | 531 |
==> analz (G \<union> H) \<subseteq> analz (G' \<union> H')" |
532 |
apply simp |
|
533 |
apply (metis Un_absorb2 Un_commute Un_subset_iff Un_upper1 Un_upper2 analz_mono) |
|
534 |
done |
|
535 |
||
536 |
||
537 |
lemma analz_cong: |
|
43197 | 538 |
"[| analz G = analz G'; analz H = analz H' |
23449 | 539 |
|] ==> analz (G \<union> H) = analz (G' \<union> H')" |
43197 | 540 |
by (intro equalityI analz_subset_cong, simp_all) |
23449 | 541 |
|
542 |
lemma analz_insert_cong: |
|
543 |
"analz H = analz H' ==> analz(insert X H) = analz(insert X H')" |
|
544 |
by (force simp only: insert_def intro!: analz_cong) |
|
545 |
||
546 |
text{*If there are no pairs or encryptions then analz does nothing*} |
|
547 |
lemma analz_trivial: |
|
548 |
"[| \<forall>X Y. {|X,Y|} \<notin> H; \<forall>X K. Crypt K X \<notin> H |] ==> analz H = H" |
|
549 |
apply safe |
|
550 |
apply (erule analz.induct, blast+) |
|
551 |
done |
|
552 |
||
553 |
text{*These two are obsolete (with a single Spy) but cost little to prove...*} |
|
554 |
lemma analz_UN_analz_lemma: |
|
555 |
"X\<in> analz (\<Union>i\<in>A. analz (H i)) ==> X\<in> analz (\<Union>i\<in>A. H i)" |
|
556 |
apply (erule analz.induct) |
|
557 |
apply (blast intro: analz_mono [THEN [2] rev_subsetD])+ |
|
558 |
done |
|
559 |
||
560 |
lemma analz_UN_analz [simp]: "analz (\<Union>i\<in>A. analz (H i)) = analz (\<Union>i\<in>A. H i)" |
|
561 |
by (blast intro: analz_UN_analz_lemma analz_mono [THEN [2] rev_subsetD]) |
|
562 |
||
563 |
||
564 |
subsection{*Inductive relation "synth"*} |
|
565 |
||
566 |
text{*Inductive definition of "synth" -- what can be built up from a set of |
|
567 |
messages. A form of upward closure. Pairs can be built, messages |
|
568 |
encrypted with known keys. Agent names are public domain. |
|
569 |
Numbers can be guessed, but Nonces cannot be. *} |
|
570 |
||
23755 | 571 |
inductive_set |
572 |
synth :: "msg set => msg set" |
|
573 |
for H :: "msg set" |
|
574 |
where |
|
23449 | 575 |
Inj [intro]: "X \<in> H ==> X \<in> synth H" |
23755 | 576 |
| Agent [intro]: "Agent agt \<in> synth H" |
577 |
| Number [intro]: "Number n \<in> synth H" |
|
578 |
| Hash [intro]: "X \<in> synth H ==> Hash X \<in> synth H" |
|
579 |
| MPair [intro]: "[|X \<in> synth H; Y \<in> synth H|] ==> {|X,Y|} \<in> synth H" |
|
580 |
| Crypt [intro]: "[|X \<in> synth H; Key(K) \<in> H|] ==> Crypt K X \<in> synth H" |
|
23449 | 581 |
|
582 |
text{*Monotonicity*} |
|
583 |
lemma synth_mono: "G\<subseteq>H ==> synth(G) \<subseteq> synth(H)" |
|
43197 | 584 |
by (auto, erule synth.induct, auto) |
23449 | 585 |
|
43197 | 586 |
text{*NO @{text Agent_synth}, as any Agent name can be synthesized. |
23449 | 587 |
The same holds for @{term Number}*} |
588 |
inductive_cases Nonce_synth [elim!]: "Nonce n \<in> synth H" |
|
589 |
inductive_cases Key_synth [elim!]: "Key K \<in> synth H" |
|
590 |
inductive_cases Hash_synth [elim!]: "Hash X \<in> synth H" |
|
591 |
inductive_cases MPair_synth [elim!]: "{|X,Y|} \<in> synth H" |
|
592 |
inductive_cases Crypt_synth [elim!]: "Crypt K X \<in> synth H" |
|
593 |
||
594 |
||
595 |
lemma synth_increasing: "H \<subseteq> synth(H)" |
|
596 |
by blast |
|
597 |
||
598 |
subsubsection{*Unions *} |
|
599 |
||
43197 | 600 |
text{*Converse fails: we can synth more from the union than from the |
23449 | 601 |
separate parts, building a compound message using elements of each.*} |
602 |
lemma synth_Un: "synth(G) \<union> synth(H) \<subseteq> synth(G \<union> H)" |
|
603 |
by (intro Un_least synth_mono Un_upper1 Un_upper2) |
|
604 |
||
605 |
lemma synth_insert: "insert X (synth H) \<subseteq> synth(insert X H)" |
|
606 |
by (metis insert_iff insert_subset subset_insertI synth.Inj synth_mono) |
|
607 |
||
608 |
subsubsection{*Idempotence and transitivity *} |
|
609 |
||
610 |
lemma synth_synthD [dest!]: "X\<in> synth (synth H) ==> X\<in> synth H" |
|
611 |
by (erule synth.induct, blast+) |
|
612 |
||
613 |
lemma synth_idem: "synth (synth H) = synth H" |
|
614 |
by blast |
|
615 |
||
616 |
lemma synth_subset_iff [simp]: "(synth G \<subseteq> synth H) = (G \<subseteq> synth H)" |
|
617 |
apply (rule iffI) |
|
43197 | 618 |
apply (iprover intro: subset_trans synth_increasing) |
619 |
apply (frule synth_mono, simp add: synth_idem) |
|
23449 | 620 |
done |
621 |
||
622 |
lemma synth_trans: "[| X\<in> synth G; G \<subseteq> synth H |] ==> X\<in> synth H" |
|
623 |
by (drule synth_mono, blast) |
|
624 |
||
625 |
lemma synth_cut: "[| Y\<in> synth (insert X H); X\<in> synth H |] ==> Y\<in> synth H" |
|
626 |
(*TOO SLOW |
|
627 |
by (metis insert_absorb insert_mono insert_subset synth_idem synth_increasing synth_mono) |
|
628 |
*) |
|
629 |
by (erule synth_trans, blast) |
|
630 |
||
631 |
||
632 |
lemma Agent_synth [simp]: "Agent A \<in> synth H" |
|
633 |
by blast |
|
634 |
||
635 |
lemma Number_synth [simp]: "Number n \<in> synth H" |
|
636 |
by blast |
|
637 |
||
638 |
lemma Nonce_synth_eq [simp]: "(Nonce N \<in> synth H) = (Nonce N \<in> H)" |
|
639 |
by blast |
|
640 |
||
641 |
lemma Key_synth_eq [simp]: "(Key K \<in> synth H) = (Key K \<in> H)" |
|
642 |
by blast |
|
643 |
||
644 |
lemma Crypt_synth_eq [simp]: |
|
645 |
"Key K \<notin> H ==> (Crypt K X \<in> synth H) = (Crypt K X \<in> H)" |
|
646 |
by blast |
|
647 |
||
648 |
||
43197 | 649 |
lemma keysFor_synth [simp]: |
23449 | 650 |
"keysFor (synth H) = keysFor H \<union> invKey`{K. Key K \<in> H}" |
651 |
by (unfold keysFor_def, blast) |
|
652 |
||
653 |
||
654 |
subsubsection{*Combinations of parts, analz and synth *} |
|
655 |
||
656 |
lemma parts_synth [simp]: "parts (synth H) = parts H \<union> synth H" |
|
657 |
apply (rule equalityI) |
|
658 |
apply (rule subsetI) |
|
659 |
apply (erule parts.induct) |
|
660 |
apply (metis UnCI) |
|
661 |
apply (metis MPair_synth UnCI UnE insert_absorb insert_subset parts.Fst parts_increasing) |
|
662 |
apply (metis MPair_synth UnCI UnE insert_absorb insert_subset parts.Snd parts_increasing) |
|
663 |
apply (metis Body Crypt_synth UnCI UnE insert_absorb insert_subset parts_increasing) |
|
664 |
apply (metis Un_subset_iff parts_increasing parts_mono synth_increasing) |
|
665 |
done |
|
666 |
||
667 |
lemma analz_analz_Un [simp]: "analz (analz G \<union> H) = analz (G \<union> H)" |
|
668 |
apply (rule equalityI); |
|
669 |
apply (metis analz_idem analz_subset_cong order_eq_refl) |
|
670 |
apply (metis analz_increasing analz_subset_cong order_eq_refl) |
|
671 |
done |
|
672 |
||
36553 | 673 |
declare analz_mono [intro] analz.Fst [intro] analz.Snd [intro] Un_least [intro] |
674 |
||
23449 | 675 |
lemma analz_synth_Un [simp]: "analz (synth G \<union> H) = analz (G \<union> H) \<union> synth G" |
676 |
apply (rule equalityI) |
|
677 |
apply (rule subsetI) |
|
678 |
apply (erule analz.induct) |
|
679 |
apply (metis UnCI UnE Un_commute analz.Inj) |
|
35095 | 680 |
apply (metis MPair_synth UnCI UnE Un_commute analz.Fst analz.Inj mem_def) |
681 |
apply (metis MPair_synth UnCI UnE Un_commute analz.Inj analz.Snd mem_def) |
|
23449 | 682 |
apply (blast intro: analz.Decrypt) |
24759 | 683 |
apply blast |
23449 | 684 |
done |
685 |
||
686 |
lemma analz_synth [simp]: "analz (synth H) = analz H \<union> synth H" |
|
36553 | 687 |
proof - |
36911 | 688 |
have "\<forall>x\<^isub>2 x\<^isub>1. synth x\<^isub>1 \<union> analz (x\<^isub>1 \<union> x\<^isub>2) = analz (synth x\<^isub>1 \<union> x\<^isub>2)" by (metis Un_commute analz_synth_Un) |
689 |
hence "\<forall>x\<^isub>1. synth x\<^isub>1 \<union> analz x\<^isub>1 = analz (synth x\<^isub>1 \<union> {})" by (metis Un_empty_right) |
|
690 |
hence "\<forall>x\<^isub>1. synth x\<^isub>1 \<union> analz x\<^isub>1 = analz (synth x\<^isub>1)" by (metis Un_empty_right) |
|
691 |
hence "\<forall>x\<^isub>1. analz x\<^isub>1 \<union> synth x\<^isub>1 = analz (synth x\<^isub>1)" by (metis Un_commute) |
|
36553 | 692 |
thus "analz (synth H) = analz H \<union> synth H" by metis |
23449 | 693 |
qed |
694 |
||
695 |
||
696 |
subsubsection{*For reasoning about the Fake rule in traces *} |
|
697 |
||
698 |
lemma parts_insert_subset_Un: "X\<in> G ==> parts(insert X H) \<subseteq> parts G \<union> parts H" |
|
36553 | 699 |
proof - |
700 |
assume "X \<in> G" |
|
36911 | 701 |
hence "G X" by (metis mem_def) |
702 |
hence "\<forall>x\<^isub>1. G \<subseteq> x\<^isub>1 \<longrightarrow> x\<^isub>1 X" by (metis predicate1D) |
|
703 |
hence "\<forall>x\<^isub>1. (G \<union> x\<^isub>1) X" by (metis Un_upper1) |
|
704 |
hence "\<forall>x\<^isub>1. X \<in> G \<union> x\<^isub>1" by (metis mem_def) |
|
705 |
hence "insert X H \<subseteq> G \<union> H" by (metis Un_upper2 insert_subset) |
|
706 |
hence "parts (insert X H) \<subseteq> parts (G \<union> H)" by (metis parts_mono) |
|
707 |
thus "parts (insert X H) \<subseteq> parts G \<union> parts H" by (metis parts_Un) |
|
23449 | 708 |
qed |
709 |
||
710 |
lemma Fake_parts_insert: |
|
43197 | 711 |
"X \<in> synth (analz H) ==> |
23449 | 712 |
parts (insert X H) \<subseteq> synth (analz H) \<union> parts H" |
36553 | 713 |
proof - |
714 |
assume A1: "X \<in> synth (analz H)" |
|
715 |
have F1: "\<forall>x\<^isub>1. analz x\<^isub>1 \<union> synth (analz x\<^isub>1) = analz (synth (analz x\<^isub>1))" |
|
716 |
by (metis analz_idem analz_synth) |
|
717 |
have F2: "\<forall>x\<^isub>1. parts x\<^isub>1 \<union> synth (analz x\<^isub>1) = parts (synth (analz x\<^isub>1))" |
|
718 |
by (metis parts_analz parts_synth) |
|
719 |
have F3: "synth (analz H) X" using A1 by (metis mem_def) |
|
720 |
have "\<forall>x\<^isub>2 x\<^isub>1\<Colon>msg set. x\<^isub>1 \<le> sup x\<^isub>1 x\<^isub>2" by (metis inf_sup_ord(3)) |
|
721 |
hence F4: "\<forall>x\<^isub>1. analz x\<^isub>1 \<subseteq> analz (synth x\<^isub>1)" by (metis analz_synth) |
|
722 |
have F5: "X \<in> synth (analz H)" using F3 by (metis mem_def) |
|
723 |
have "\<forall>x\<^isub>1. analz x\<^isub>1 \<subseteq> synth (analz x\<^isub>1) |
|
724 |
\<longrightarrow> analz (synth (analz x\<^isub>1)) = synth (analz x\<^isub>1)" |
|
725 |
using F1 by (metis subset_Un_eq) |
|
726 |
hence F6: "\<forall>x\<^isub>1. analz (synth (analz x\<^isub>1)) = synth (analz x\<^isub>1)" |
|
727 |
by (metis synth_increasing) |
|
728 |
have "\<forall>x\<^isub>1. x\<^isub>1 \<subseteq> analz (synth x\<^isub>1)" using F4 by (metis analz_subset_iff) |
|
729 |
hence "\<forall>x\<^isub>1. x\<^isub>1 \<subseteq> analz (synth (analz x\<^isub>1))" by (metis analz_subset_iff) |
|
730 |
hence "\<forall>x\<^isub>1. x\<^isub>1 \<subseteq> synth (analz x\<^isub>1)" using F6 by metis |
|
731 |
hence "H \<subseteq> synth (analz H)" by metis |
|
732 |
hence "H \<subseteq> synth (analz H) \<and> X \<in> synth (analz H)" using F5 by metis |
|
733 |
hence "insert X H \<subseteq> synth (analz H)" by (metis insert_subset) |
|
734 |
hence "parts (insert X H) \<subseteq> parts (synth (analz H))" by (metis parts_mono) |
|
735 |
hence "parts (insert X H) \<subseteq> parts H \<union> synth (analz H)" using F2 by metis |
|
736 |
thus "parts (insert X H) \<subseteq> synth (analz H) \<union> parts H" by (metis Un_commute) |
|
23449 | 737 |
qed |
738 |
||
739 |
lemma Fake_parts_insert_in_Un: |
|
43197 | 740 |
"[|Z \<in> parts (insert X H); X: synth (analz H)|] |
23449 | 741 |
==> Z \<in> synth (analz H) \<union> parts H"; |
36553 | 742 |
by (blast dest: Fake_parts_insert [THEN subsetD, dest]) |
23449 | 743 |
|
43197 | 744 |
declare analz_mono [intro] synth_mono [intro] |
36553 | 745 |
|
23449 | 746 |
lemma Fake_analz_insert: |
36553 | 747 |
"X \<in> synth (analz G) ==> |
23449 | 748 |
analz (insert X H) \<subseteq> synth (analz G) \<union> analz (G \<union> H)" |
36553 | 749 |
by (metis Un_commute Un_insert_left Un_insert_right Un_upper1 analz_analz_Un |
750 |
analz_mono analz_synth_Un insert_absorb) |
|
23449 | 751 |
|
752 |
lemma Fake_analz_insert_simpler: |
|
43197 | 753 |
"X \<in> synth (analz G) ==> |
23449 | 754 |
analz (insert X H) \<subseteq> synth (analz G) \<union> analz (G \<union> H)" |
755 |
apply (rule subsetI) |
|
756 |
apply (subgoal_tac "x \<in> analz (synth (analz G) \<union> H) ") |
|
757 |
apply (metis Un_commute analz_analz_Un analz_synth_Un) |
|
39260 | 758 |
by (metis Un_upper1 Un_upper2 analz_mono insert_absorb insert_subset) |
23449 | 759 |
|
760 |
end |