author | wenzelm |
Fri, 20 Mar 2009 15:24:18 +0100 | |
changeset 30607 | c3d1590debd8 |
parent 30549 | d2d7874648bd |
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permissions | -rw-r--r-- |
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(* Title: HOL/Auth/Message |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Copyright 1996 University of Cambridge |
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Datatypes of agents and messages; |
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Inductive relations "parts", "analz" and "synth" |
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*) |
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header{*Theory of Agents and Messages for Security Protocols*} |
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theory Message |
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imports Main |
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begin |
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(*Needed occasionally with spy_analz_tac, e.g. in analz_insert_Key_newK*) |
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lemma [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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text{*Monotonicity*} |
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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 (blast dest: parts.Fst parts.Snd parts.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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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, 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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by (erule parts.induct, fast+) |
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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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text{*TWO inserts to avoid looping. This rewrite is better than nothing. |
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Not suitable for Addsimps: its behaviour can be strange.*} |
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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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apply (simp add: Un_assoc) |
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apply (simp add: parts_insert [symmetric]) |
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done |
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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 (iprover intro: subset_trans parts_increasing) |
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apply (frule parts_mono, simp) |
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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 (drule parts_mono, blast) |
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text{*Cut*} |
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lemma parts_cut: |
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"[| Y\<in> parts (insert X G); X\<in> parts H |] ==> Y\<in> parts (G \<union> H)" |
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by (blast intro: parts_trans) |
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lemma parts_cut_eq [simp]: "X\<in> parts H ==> parts (insert X H) = parts H" |
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by (force dest!: parts_cut intro: parts_insertI) |
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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) |
|
296 |
done |
|
297 |
||
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lemma parts_insert_Crypt [simp]: |
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"parts (insert (Crypt K X) H) = insert (Crypt K X) (parts (insert X H))" |
13926 | 300 |
apply (rule equalityI) |
301 |
apply (rule subsetI) |
|
302 |
apply (erule parts.induct, auto) |
|
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303 |
apply (blast intro: parts.Body) |
13926 | 304 |
done |
305 |
||
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lemma parts_insert_MPair [simp]: |
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307 |
"parts (insert {|X,Y|} H) = |
13926 | 308 |
insert {|X,Y|} (parts (insert X (insert Y H)))" |
309 |
apply (rule equalityI) |
|
310 |
apply (rule subsetI) |
|
311 |
apply (erule parts.induct, auto) |
|
312 |
apply (blast intro: parts.Fst parts.Snd)+ |
|
313 |
done |
|
314 |
||
315 |
lemma parts_image_Key [simp]: "parts (Key`N) = Key`N" |
|
316 |
apply auto |
|
317 |
apply (erule parts.induct, auto) |
|
318 |
done |
|
319 |
||
320 |
||
16818 | 321 |
text{*In any message, there is an upper bound N on its greatest nonce.*} |
13926 | 322 |
lemma msg_Nonce_supply: "\<exists>N. \<forall>n. N\<le>n --> Nonce n \<notin> parts {msg}" |
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apply (induct msg) |
13926 | 324 |
apply (simp_all (no_asm_simp) add: exI parts_insert2) |
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txt{*MPair case: blast works out the necessary sum itself!*} |
22424 | 326 |
prefer 2 apply auto apply (blast elim!: add_leE) |
16818 | 327 |
txt{*Nonce case*} |
328 |
apply (rule_tac x = "N + Suc nat" in exI, auto) |
|
13926 | 329 |
done |
330 |
||
331 |
||
332 |
subsection{*Inductive relation "analz"*} |
|
333 |
||
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text{*Inductive definition of "analz" -- what can be broken down from a set of |
1839 | 335 |
messages, including keys. A form of downward closure. Pairs can |
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336 |
be taken apart; messages decrypted with known keys. *} |
1839 | 337 |
|
23746 | 338 |
inductive_set |
339 |
analz :: "msg set => msg set" |
|
340 |
for H :: "msg set" |
|
341 |
where |
|
11192 | 342 |
Inj [intro,simp] : "X \<in> H ==> X \<in> analz H" |
23746 | 343 |
| Fst: "{|X,Y|} \<in> analz H ==> X \<in> analz H" |
344 |
| Snd: "{|X,Y|} \<in> analz H ==> Y \<in> analz H" |
|
345 |
| Decrypt [dest]: |
|
11192 | 346 |
"[|Crypt K X \<in> analz H; Key(invKey K): analz H|] ==> X \<in> analz H" |
1839 | 347 |
|
348 |
||
16818 | 349 |
text{*Monotonicity; Lemma 1 of Lowe's paper*} |
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350 |
lemma analz_mono: "G\<subseteq>H ==> analz(G) \<subseteq> analz(H)" |
11189 | 351 |
apply auto |
352 |
apply (erule analz.induct) |
|
16818 | 353 |
apply (auto dest: analz.Fst analz.Snd) |
11189 | 354 |
done |
355 |
||
13926 | 356 |
text{*Making it safe speeds up proofs*} |
357 |
lemma MPair_analz [elim!]: |
|
358 |
"[| {|X,Y|} \<in> analz H; |
|
359 |
[| X \<in> analz H; Y \<in> analz H |] ==> P |
|
360 |
|] ==> P" |
|
361 |
by (blast dest: analz.Fst analz.Snd) |
|
362 |
||
363 |
lemma analz_increasing: "H \<subseteq> analz(H)" |
|
364 |
by blast |
|
365 |
||
366 |
lemma analz_subset_parts: "analz H \<subseteq> parts H" |
|
367 |
apply (rule subsetI) |
|
368 |
apply (erule analz.induct, blast+) |
|
369 |
done |
|
370 |
||
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lemmas analz_into_parts = analz_subset_parts [THEN subsetD, standard] |
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372 |
|
13926 | 373 |
lemmas not_parts_not_analz = analz_subset_parts [THEN contra_subsetD, standard] |
374 |
||
375 |
||
376 |
lemma parts_analz [simp]: "parts (analz H) = parts H" |
|
377 |
apply (rule equalityI) |
|
378 |
apply (rule analz_subset_parts [THEN parts_mono, THEN subset_trans], simp) |
|
379 |
apply (blast intro: analz_increasing [THEN parts_mono, THEN subsetD]) |
|
380 |
done |
|
381 |
||
382 |
lemma analz_parts [simp]: "analz (parts H) = parts H" |
|
383 |
apply auto |
|
384 |
apply (erule analz.induct, auto) |
|
385 |
done |
|
386 |
||
387 |
lemmas analz_insertI = subset_insertI [THEN analz_mono, THEN [2] rev_subsetD, standard] |
|
388 |
||
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|
389 |
subsubsection{*General equational properties *} |
13926 | 390 |
|
391 |
lemma analz_empty [simp]: "analz{} = {}" |
|
392 |
apply safe |
|
393 |
apply (erule analz.induct, blast+) |
|
394 |
done |
|
395 |
||
16818 | 396 |
text{*Converse fails: we can analz more from the union than from the |
397 |
separate parts, as a key in one might decrypt a message in the other*} |
|
13926 | 398 |
lemma analz_Un: "analz(G) \<union> analz(H) \<subseteq> analz(G \<union> H)" |
399 |
by (intro Un_least analz_mono Un_upper1 Un_upper2) |
|
400 |
||
401 |
lemma analz_insert: "insert X (analz H) \<subseteq> analz(insert X H)" |
|
402 |
by (blast intro: analz_mono [THEN [2] rev_subsetD]) |
|
403 |
||
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|
404 |
subsubsection{*Rewrite rules for pulling out atomic messages *} |
13926 | 405 |
|
406 |
lemmas analz_insert_eq_I = equalityI [OF subsetI analz_insert] |
|
407 |
||
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|
408 |
lemma analz_insert_Agent [simp]: |
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|
409 |
"analz (insert (Agent agt) H) = insert (Agent agt) (analz H)" |
13926 | 410 |
apply (rule analz_insert_eq_I) |
411 |
apply (erule analz.induct, auto) |
|
412 |
done |
|
413 |
||
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|
414 |
lemma analz_insert_Nonce [simp]: |
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|
415 |
"analz (insert (Nonce N) H) = insert (Nonce N) (analz H)" |
13926 | 416 |
apply (rule analz_insert_eq_I) |
417 |
apply (erule analz.induct, auto) |
|
418 |
done |
|
419 |
||
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|
420 |
lemma analz_insert_Number [simp]: |
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|
421 |
"analz (insert (Number N) H) = insert (Number N) (analz H)" |
13926 | 422 |
apply (rule analz_insert_eq_I) |
423 |
apply (erule analz.induct, auto) |
|
424 |
done |
|
425 |
||
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|
426 |
lemma analz_insert_Hash [simp]: |
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|
427 |
"analz (insert (Hash X) H) = insert (Hash X) (analz H)" |
13926 | 428 |
apply (rule analz_insert_eq_I) |
429 |
apply (erule analz.induct, auto) |
|
430 |
done |
|
431 |
||
16818 | 432 |
text{*Can only pull out Keys if they are not needed to decrypt the rest*} |
13926 | 433 |
lemma analz_insert_Key [simp]: |
434 |
"K \<notin> keysFor (analz H) ==> |
|
435 |
analz (insert (Key K) H) = insert (Key K) (analz H)" |
|
436 |
apply (unfold keysFor_def) |
|
437 |
apply (rule analz_insert_eq_I) |
|
438 |
apply (erule analz.induct, auto) |
|
439 |
done |
|
440 |
||
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|
441 |
lemma analz_insert_MPair [simp]: |
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|
442 |
"analz (insert {|X,Y|} H) = |
13926 | 443 |
insert {|X,Y|} (analz (insert X (insert Y H)))" |
444 |
apply (rule equalityI) |
|
445 |
apply (rule subsetI) |
|
446 |
apply (erule analz.induct, auto) |
|
447 |
apply (erule analz.induct) |
|
448 |
apply (blast intro: analz.Fst analz.Snd)+ |
|
449 |
done |
|
450 |
||
16818 | 451 |
text{*Can pull out enCrypted message if the Key is not known*} |
13926 | 452 |
lemma analz_insert_Crypt: |
453 |
"Key (invKey K) \<notin> analz H |
|
454 |
==> analz (insert (Crypt K X) H) = insert (Crypt K X) (analz H)" |
|
455 |
apply (rule analz_insert_eq_I) |
|
456 |
apply (erule analz.induct, auto) |
|
457 |
||
458 |
done |
|
459 |
||
460 |
lemma lemma1: "Key (invKey K) \<in> analz H ==> |
|
461 |
analz (insert (Crypt K X) H) \<subseteq> |
|
462 |
insert (Crypt K X) (analz (insert X H))" |
|
463 |
apply (rule subsetI) |
|
23746 | 464 |
apply (erule_tac x = x in analz.induct, auto) |
13926 | 465 |
done |
466 |
||
467 |
lemma lemma2: "Key (invKey K) \<in> analz H ==> |
|
468 |
insert (Crypt K X) (analz (insert X H)) \<subseteq> |
|
469 |
analz (insert (Crypt K X) H)" |
|
470 |
apply auto |
|
23746 | 471 |
apply (erule_tac x = x in analz.induct, auto) |
13926 | 472 |
apply (blast intro: analz_insertI analz.Decrypt) |
473 |
done |
|
474 |
||
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|
475 |
lemma analz_insert_Decrypt: |
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|
476 |
"Key (invKey K) \<in> analz H ==> |
13926 | 477 |
analz (insert (Crypt K X) H) = |
478 |
insert (Crypt K X) (analz (insert X H))" |
|
479 |
by (intro equalityI lemma1 lemma2) |
|
480 |
||
16818 | 481 |
text{*Case analysis: either the message is secure, or it is not! Effective, |
482 |
but can cause subgoals to blow up! Use with @{text "split_if"}; apparently |
|
483 |
@{text "split_tac"} does not cope with patterns such as @{term"analz (insert |
|
484 |
(Crypt K X) H)"} *} |
|
13926 | 485 |
lemma analz_Crypt_if [simp]: |
486 |
"analz (insert (Crypt K X) H) = |
|
487 |
(if (Key (invKey K) \<in> analz H) |
|
488 |
then insert (Crypt K X) (analz (insert X H)) |
|
489 |
else insert (Crypt K X) (analz H))" |
|
490 |
by (simp add: analz_insert_Crypt analz_insert_Decrypt) |
|
491 |
||
492 |
||
16818 | 493 |
text{*This rule supposes "for the sake of argument" that we have the key.*} |
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changeset
|
494 |
lemma analz_insert_Crypt_subset: |
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diff
changeset
|
495 |
"analz (insert (Crypt K X) H) \<subseteq> |
13926 | 496 |
insert (Crypt K X) (analz (insert X H))" |
497 |
apply (rule subsetI) |
|
498 |
apply (erule analz.induct, auto) |
|
499 |
done |
|
500 |
||
501 |
||
502 |
lemma analz_image_Key [simp]: "analz (Key`N) = Key`N" |
|
503 |
apply auto |
|
504 |
apply (erule analz.induct, auto) |
|
505 |
done |
|
506 |
||
507 |
||
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changeset
|
508 |
subsubsection{*Idempotence and transitivity *} |
13926 | 509 |
|
510 |
lemma analz_analzD [dest!]: "X\<in> analz (analz H) ==> X\<in> analz H" |
|
511 |
by (erule analz.induct, blast+) |
|
512 |
||
513 |
lemma analz_idem [simp]: "analz (analz H) = analz H" |
|
514 |
by blast |
|
515 |
||
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|
516 |
lemma analz_subset_iff [simp]: "(analz G \<subseteq> analz H) = (G \<subseteq> analz H)" |
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|
517 |
apply (rule iffI) |
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|
518 |
apply (iprover intro: subset_trans analz_increasing) |
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|
519 |
apply (frule analz_mono, simp) |
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changeset
|
520 |
done |
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|
521 |
|
13926 | 522 |
lemma analz_trans: "[| X\<in> analz G; G \<subseteq> analz H |] ==> X\<in> analz H" |
523 |
by (drule analz_mono, blast) |
|
524 |
||
16818 | 525 |
text{*Cut; Lemma 2 of Lowe*} |
13926 | 526 |
lemma analz_cut: "[| Y\<in> analz (insert X H); X\<in> analz H |] ==> Y\<in> analz H" |
527 |
by (erule analz_trans, blast) |
|
528 |
||
529 |
(*Cut can be proved easily by induction on |
|
530 |
"Y: analz (insert X H) ==> X: analz H --> Y: analz H" |
|
531 |
*) |
|
532 |
||
16818 | 533 |
text{*This rewrite rule helps in the simplification of messages that involve |
13926 | 534 |
the forwarding of unknown components (X). Without it, removing occurrences |
16818 | 535 |
of X can be very complicated. *} |
13926 | 536 |
lemma analz_insert_eq: "X\<in> analz H ==> analz (insert X H) = analz H" |
537 |
by (blast intro: analz_cut analz_insertI) |
|
538 |
||
539 |
||
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Removal of the Key_supply axiom (affects many possbility proofs) and minor
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|
540 |
text{*A congruence rule for "analz" *} |
13926 | 541 |
|
14200
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Removal of the Key_supply axiom (affects many possbility proofs) and minor
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diff
changeset
|
542 |
lemma analz_subset_cong: |
17689
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paulson
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diff
changeset
|
543 |
"[| analz G \<subseteq> analz G'; analz H \<subseteq> analz H' |] |
a04b5b43625e
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paulson
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diff
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|
544 |
==> analz (G \<union> H) \<subseteq> analz (G' \<union> H')" |
a04b5b43625e
streamlined theory; conformance to recent publication
paulson
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16818
diff
changeset
|
545 |
apply simp |
a04b5b43625e
streamlined theory; conformance to recent publication
paulson
parents:
16818
diff
changeset
|
546 |
apply (iprover intro: conjI subset_trans analz_mono Un_upper1 Un_upper2) |
13926 | 547 |
done |
548 |
||
14200
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Removal of the Key_supply axiom (affects many possbility proofs) and minor
paulson
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changeset
|
549 |
lemma analz_cong: |
17689
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diff
changeset
|
550 |
"[| analz G = analz G'; analz H = analz H' |] |
a04b5b43625e
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paulson
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|
551 |
==> analz (G \<union> H) = analz (G' \<union> H')" |
14200
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Removal of the Key_supply axiom (affects many possbility proofs) and minor
paulson
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changeset
|
552 |
by (intro equalityI analz_subset_cong, simp_all) |
13926 | 553 |
|
14200
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Removal of the Key_supply axiom (affects many possbility proofs) and minor
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changeset
|
554 |
lemma analz_insert_cong: |
d8598e24f8fa
Removal of the Key_supply axiom (affects many possbility proofs) and minor
paulson
parents:
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diff
changeset
|
555 |
"analz H = analz H' ==> analz(insert X H) = analz(insert X H')" |
13926 | 556 |
by (force simp only: insert_def intro!: analz_cong) |
557 |
||
16818 | 558 |
text{*If there are no pairs or encryptions then analz does nothing*} |
14200
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Removal of the Key_supply axiom (affects many possbility proofs) and minor
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changeset
|
559 |
lemma analz_trivial: |
d8598e24f8fa
Removal of the Key_supply axiom (affects many possbility proofs) and minor
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changeset
|
560 |
"[| \<forall>X Y. {|X,Y|} \<notin> H; \<forall>X K. Crypt K X \<notin> H |] ==> analz H = H" |
13926 | 561 |
apply safe |
562 |
apply (erule analz.induct, blast+) |
|
563 |
done |
|
564 |
||
16818 | 565 |
text{*These two are obsolete (with a single Spy) but cost little to prove...*} |
14200
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Removal of the Key_supply axiom (affects many possbility proofs) and minor
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|
566 |
lemma analz_UN_analz_lemma: |
d8598e24f8fa
Removal of the Key_supply axiom (affects many possbility proofs) and minor
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changeset
|
567 |
"X\<in> analz (\<Union>i\<in>A. analz (H i)) ==> X\<in> analz (\<Union>i\<in>A. H i)" |
13926 | 568 |
apply (erule analz.induct) |
569 |
apply (blast intro: analz_mono [THEN [2] rev_subsetD])+ |
|
570 |
done |
|
571 |
||
572 |
lemma analz_UN_analz [simp]: "analz (\<Union>i\<in>A. analz (H i)) = analz (\<Union>i\<in>A. H i)" |
|
573 |
by (blast intro: analz_UN_analz_lemma analz_mono [THEN [2] rev_subsetD]) |
|
574 |
||
575 |
||
576 |
subsection{*Inductive relation "synth"*} |
|
577 |
||
14200
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diff
changeset
|
578 |
text{*Inductive definition of "synth" -- what can be built up from a set of |
1839 | 579 |
messages. A form of upward closure. Pairs can be built, messages |
3668 | 580 |
encrypted with known keys. Agent names are public domain. |
14200
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Removal of the Key_supply axiom (affects many possbility proofs) and minor
paulson
parents:
14181
diff
changeset
|
581 |
Numbers can be guessed, but Nonces cannot be. *} |
1839 | 582 |
|
23746 | 583 |
inductive_set |
584 |
synth :: "msg set => msg set" |
|
585 |
for H :: "msg set" |
|
586 |
where |
|
11192 | 587 |
Inj [intro]: "X \<in> H ==> X \<in> synth H" |
23746 | 588 |
| Agent [intro]: "Agent agt \<in> synth H" |
589 |
| Number [intro]: "Number n \<in> synth H" |
|
590 |
| Hash [intro]: "X \<in> synth H ==> Hash X \<in> synth H" |
|
591 |
| MPair [intro]: "[|X \<in> synth H; Y \<in> synth H|] ==> {|X,Y|} \<in> synth H" |
|
592 |
| Crypt [intro]: "[|X \<in> synth H; Key(K) \<in> H|] ==> Crypt K X \<in> synth H" |
|
11189 | 593 |
|
16818 | 594 |
text{*Monotonicity*} |
14200
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Removal of the Key_supply axiom (affects many possbility proofs) and minor
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14181
diff
changeset
|
595 |
lemma synth_mono: "G\<subseteq>H ==> synth(G) \<subseteq> synth(H)" |
16818 | 596 |
by (auto, erule synth.induct, auto) |
11189 | 597 |
|
16818 | 598 |
text{*NO @{text Agent_synth}, as any Agent name can be synthesized. |
599 |
The same holds for @{term Number}*} |
|
11192 | 600 |
inductive_cases Nonce_synth [elim!]: "Nonce n \<in> synth H" |
601 |
inductive_cases Key_synth [elim!]: "Key K \<in> synth H" |
|
602 |
inductive_cases Hash_synth [elim!]: "Hash X \<in> synth H" |
|
603 |
inductive_cases MPair_synth [elim!]: "{|X,Y|} \<in> synth H" |
|
604 |
inductive_cases Crypt_synth [elim!]: "Crypt K X \<in> synth H" |
|
11189 | 605 |
|
13926 | 606 |
|
607 |
lemma synth_increasing: "H \<subseteq> synth(H)" |
|
608 |
by blast |
|
609 |
||
14200
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Removal of the Key_supply axiom (affects many possbility proofs) and minor
paulson
parents:
14181
diff
changeset
|
610 |
subsubsection{*Unions *} |
13926 | 611 |
|
16818 | 612 |
text{*Converse fails: we can synth more from the union than from the |
613 |
separate parts, building a compound message using elements of each.*} |
|
13926 | 614 |
lemma synth_Un: "synth(G) \<union> synth(H) \<subseteq> synth(G \<union> H)" |
615 |
by (intro Un_least synth_mono Un_upper1 Un_upper2) |
|
616 |
||
617 |
lemma synth_insert: "insert X (synth H) \<subseteq> synth(insert X H)" |
|
618 |
by (blast intro: synth_mono [THEN [2] rev_subsetD]) |
|
619 |
||
14200
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Removal of the Key_supply axiom (affects many possbility proofs) and minor
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diff
changeset
|
620 |
subsubsection{*Idempotence and transitivity *} |
13926 | 621 |
|
622 |
lemma synth_synthD [dest!]: "X\<in> synth (synth H) ==> X\<in> synth H" |
|
623 |
by (erule synth.induct, blast+) |
|
624 |
||
625 |
lemma synth_idem: "synth (synth H) = synth H" |
|
626 |
by blast |
|
627 |
||
17689
a04b5b43625e
streamlined theory; conformance to recent publication
paulson
parents:
16818
diff
changeset
|
628 |
lemma synth_subset_iff [simp]: "(synth G \<subseteq> synth H) = (G \<subseteq> synth H)" |
a04b5b43625e
streamlined theory; conformance to recent publication
paulson
parents:
16818
diff
changeset
|
629 |
apply (rule iffI) |
a04b5b43625e
streamlined theory; conformance to recent publication
paulson
parents:
16818
diff
changeset
|
630 |
apply (iprover intro: subset_trans synth_increasing) |
a04b5b43625e
streamlined theory; conformance to recent publication
paulson
parents:
16818
diff
changeset
|
631 |
apply (frule synth_mono, simp add: synth_idem) |
a04b5b43625e
streamlined theory; conformance to recent publication
paulson
parents:
16818
diff
changeset
|
632 |
done |
a04b5b43625e
streamlined theory; conformance to recent publication
paulson
parents:
16818
diff
changeset
|
633 |
|
13926 | 634 |
lemma synth_trans: "[| X\<in> synth G; G \<subseteq> synth H |] ==> X\<in> synth H" |
635 |
by (drule synth_mono, blast) |
|
636 |
||
16818 | 637 |
text{*Cut; Lemma 2 of Lowe*} |
13926 | 638 |
lemma synth_cut: "[| Y\<in> synth (insert X H); X\<in> synth H |] ==> Y\<in> synth H" |
639 |
by (erule synth_trans, blast) |
|
640 |
||
641 |
lemma Agent_synth [simp]: "Agent A \<in> synth H" |
|
642 |
by blast |
|
643 |
||
644 |
lemma Number_synth [simp]: "Number n \<in> synth H" |
|
645 |
by blast |
|
646 |
||
647 |
lemma Nonce_synth_eq [simp]: "(Nonce N \<in> synth H) = (Nonce N \<in> H)" |
|
648 |
by blast |
|
649 |
||
650 |
lemma Key_synth_eq [simp]: "(Key K \<in> synth H) = (Key K \<in> H)" |
|
651 |
by blast |
|
652 |
||
14200
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Removal of the Key_supply axiom (affects many possbility proofs) and minor
paulson
parents:
14181
diff
changeset
|
653 |
lemma Crypt_synth_eq [simp]: |
d8598e24f8fa
Removal of the Key_supply axiom (affects many possbility proofs) and minor
paulson
parents:
14181
diff
changeset
|
654 |
"Key K \<notin> H ==> (Crypt K X \<in> synth H) = (Crypt K X \<in> H)" |
13926 | 655 |
by blast |
656 |
||
657 |
||
658 |
lemma keysFor_synth [simp]: |
|
659 |
"keysFor (synth H) = keysFor H \<union> invKey`{K. Key K \<in> H}" |
|
14200
d8598e24f8fa
Removal of the Key_supply axiom (affects many possbility proofs) and minor
paulson
parents:
14181
diff
changeset
|
660 |
by (unfold keysFor_def, blast) |
13926 | 661 |
|
662 |
||
14200
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Removal of the Key_supply axiom (affects many possbility proofs) and minor
paulson
parents:
14181
diff
changeset
|
663 |
subsubsection{*Combinations of parts, analz and synth *} |
13926 | 664 |
|
665 |
lemma parts_synth [simp]: "parts (synth H) = parts H \<union> synth H" |
|
666 |
apply (rule equalityI) |
|
667 |
apply (rule subsetI) |
|
668 |
apply (erule parts.induct) |
|
669 |
apply (blast intro: synth_increasing [THEN parts_mono, THEN subsetD] |
|
670 |
parts.Fst parts.Snd parts.Body)+ |
|
671 |
done |
|
672 |
||
673 |
lemma analz_analz_Un [simp]: "analz (analz G \<union> H) = analz (G \<union> H)" |
|
674 |
apply (intro equalityI analz_subset_cong)+ |
|
675 |
apply simp_all |
|
676 |
done |
|
677 |
||
678 |
lemma analz_synth_Un [simp]: "analz (synth G \<union> H) = analz (G \<union> H) \<union> synth G" |
|
679 |
apply (rule equalityI) |
|
680 |
apply (rule subsetI) |
|
681 |
apply (erule analz.induct) |
|
682 |
prefer 5 apply (blast intro: analz_mono [THEN [2] rev_subsetD]) |
|
683 |
apply (blast intro: analz.Fst analz.Snd analz.Decrypt)+ |
|
684 |
done |
|
685 |
||
686 |
lemma analz_synth [simp]: "analz (synth H) = analz H \<union> synth H" |
|
687 |
apply (cut_tac H = "{}" in analz_synth_Un) |
|
688 |
apply (simp (no_asm_use)) |
|
689 |
done |
|
690 |
||
691 |
||
14200
d8598e24f8fa
Removal of the Key_supply axiom (affects many possbility proofs) and minor
paulson
parents:
14181
diff
changeset
|
692 |
subsubsection{*For reasoning about the Fake rule in traces *} |
13926 | 693 |
|
694 |
lemma parts_insert_subset_Un: "X\<in> G ==> parts(insert X H) \<subseteq> parts G \<union> parts H" |
|
695 |
by (rule subset_trans [OF parts_mono parts_Un_subset2], blast) |
|
696 |
||
16818 | 697 |
text{*More specifically for Fake. Very occasionally we could do with a version |
698 |
of the form @{term"parts{X} \<subseteq> synth (analz H) \<union> parts H"} *} |
|
14200
d8598e24f8fa
Removal of the Key_supply axiom (affects many possbility proofs) and minor
paulson
parents:
14181
diff
changeset
|
699 |
lemma Fake_parts_insert: |
d8598e24f8fa
Removal of the Key_supply axiom (affects many possbility proofs) and minor
paulson
parents:
14181
diff
changeset
|
700 |
"X \<in> synth (analz H) ==> |
13926 | 701 |
parts (insert X H) \<subseteq> synth (analz H) \<union> parts H" |
702 |
apply (drule parts_insert_subset_Un) |
|
703 |
apply (simp (no_asm_use)) |
|
704 |
apply blast |
|
705 |
done |
|
706 |
||
14200
d8598e24f8fa
Removal of the Key_supply axiom (affects many possbility proofs) and minor
paulson
parents:
14181
diff
changeset
|
707 |
lemma Fake_parts_insert_in_Un: |
d8598e24f8fa
Removal of the Key_supply axiom (affects many possbility proofs) and minor
paulson
parents:
14181
diff
changeset
|
708 |
"[|Z \<in> parts (insert X H); X: synth (analz H)|] |
d8598e24f8fa
Removal of the Key_supply axiom (affects many possbility proofs) and minor
paulson
parents:
14181
diff
changeset
|
709 |
==> Z \<in> synth (analz H) \<union> parts H"; |
d8598e24f8fa
Removal of the Key_supply axiom (affects many possbility proofs) and minor
paulson
parents:
14181
diff
changeset
|
710 |
by (blast dest: Fake_parts_insert [THEN subsetD, dest]) |
d8598e24f8fa
Removal of the Key_supply axiom (affects many possbility proofs) and minor
paulson
parents:
14181
diff
changeset
|
711 |
|
16818 | 712 |
text{*@{term H} is sometimes @{term"Key ` KK \<union> spies evs"}, so can't put |
713 |
@{term "G=H"}.*} |
|
14200
d8598e24f8fa
Removal of the Key_supply axiom (affects many possbility proofs) and minor
paulson
parents:
14181
diff
changeset
|
714 |
lemma Fake_analz_insert: |
d8598e24f8fa
Removal of the Key_supply axiom (affects many possbility proofs) and minor
paulson
parents:
14181
diff
changeset
|
715 |
"X\<in> synth (analz G) ==> |
13926 | 716 |
analz (insert X H) \<subseteq> synth (analz G) \<union> analz (G \<union> H)" |
717 |
apply (rule subsetI) |
|
718 |
apply (subgoal_tac "x \<in> analz (synth (analz G) \<union> H) ") |
|
719 |
prefer 2 apply (blast intro: analz_mono [THEN [2] rev_subsetD] analz_mono [THEN synth_mono, THEN [2] rev_subsetD]) |
|
720 |
apply (simp (no_asm_use)) |
|
721 |
apply blast |
|
722 |
done |
|
723 |
||
14200
d8598e24f8fa
Removal of the Key_supply axiom (affects many possbility proofs) and minor
paulson
parents:
14181
diff
changeset
|
724 |
lemma analz_conj_parts [simp]: |
d8598e24f8fa
Removal of the Key_supply axiom (affects many possbility proofs) and minor
paulson
parents:
14181
diff
changeset
|
725 |
"(X \<in> analz H & X \<in> parts H) = (X \<in> analz H)" |
14145
2e31b8cc8788
ZhouGollmann: new example (fair non-repudiation protocol)
paulson
parents:
14126
diff
changeset
|
726 |
by (blast intro: analz_subset_parts [THEN subsetD]) |
13926 | 727 |
|
14200
d8598e24f8fa
Removal of the Key_supply axiom (affects many possbility proofs) and minor
paulson
parents:
14181
diff
changeset
|
728 |
lemma analz_disj_parts [simp]: |
d8598e24f8fa
Removal of the Key_supply axiom (affects many possbility proofs) and minor
paulson
parents:
14181
diff
changeset
|
729 |
"(X \<in> analz H | X \<in> parts H) = (X \<in> parts H)" |
14145
2e31b8cc8788
ZhouGollmann: new example (fair non-repudiation protocol)
paulson
parents:
14126
diff
changeset
|
730 |
by (blast intro: analz_subset_parts [THEN subsetD]) |
13926 | 731 |
|
16818 | 732 |
text{*Without this equation, other rules for synth and analz would yield |
733 |
redundant cases*} |
|
13926 | 734 |
lemma MPair_synth_analz [iff]: |
735 |
"({|X,Y|} \<in> synth (analz H)) = |
|
736 |
(X \<in> synth (analz H) & Y \<in> synth (analz H))" |
|
737 |
by blast |
|
738 |
||
14200
d8598e24f8fa
Removal of the Key_supply axiom (affects many possbility proofs) and minor
paulson
parents:
14181
diff
changeset
|
739 |
lemma Crypt_synth_analz: |
d8598e24f8fa
Removal of the Key_supply axiom (affects many possbility proofs) and minor
paulson
parents:
14181
diff
changeset
|
740 |
"[| Key K \<in> analz H; Key (invKey K) \<in> analz H |] |
13926 | 741 |
==> (Crypt K X \<in> synth (analz H)) = (X \<in> synth (analz H))" |
742 |
by blast |
|
743 |
||
744 |
||
14200
d8598e24f8fa
Removal of the Key_supply axiom (affects many possbility proofs) and minor
paulson
parents:
14181
diff
changeset
|
745 |
lemma Hash_synth_analz [simp]: |
d8598e24f8fa
Removal of the Key_supply axiom (affects many possbility proofs) and minor
paulson
parents:
14181
diff
changeset
|
746 |
"X \<notin> synth (analz H) |
13926 | 747 |
==> (Hash{|X,Y|} \<in> synth (analz H)) = (Hash{|X,Y|} \<in> analz H)" |
748 |
by blast |
|
749 |
||
750 |
||
751 |
subsection{*HPair: a combination of Hash and MPair*} |
|
752 |
||
14200
d8598e24f8fa
Removal of the Key_supply axiom (affects many possbility proofs) and minor
paulson
parents:
14181
diff
changeset
|
753 |
subsubsection{*Freeness *} |
13926 | 754 |
|
755 |
lemma Agent_neq_HPair: "Agent A ~= Hash[X] Y" |
|
756 |
by (unfold HPair_def, simp) |
|
757 |
||
758 |
lemma Nonce_neq_HPair: "Nonce N ~= Hash[X] Y" |
|
759 |
by (unfold HPair_def, simp) |
|
760 |
||
761 |
lemma Number_neq_HPair: "Number N ~= Hash[X] Y" |
|
762 |
by (unfold HPair_def, simp) |
|
763 |
||
764 |
lemma Key_neq_HPair: "Key K ~= Hash[X] Y" |
|
765 |
by (unfold HPair_def, simp) |
|
766 |
||
767 |
lemma Hash_neq_HPair: "Hash Z ~= Hash[X] Y" |
|
768 |
by (unfold HPair_def, simp) |
|
769 |
||
770 |
lemma Crypt_neq_HPair: "Crypt K X' ~= Hash[X] Y" |
|
771 |
by (unfold HPair_def, simp) |
|
772 |
||
773 |
lemmas HPair_neqs = Agent_neq_HPair Nonce_neq_HPair Number_neq_HPair |
|
774 |
Key_neq_HPair Hash_neq_HPair Crypt_neq_HPair |
|
775 |
||
776 |
declare HPair_neqs [iff] |
|
777 |
declare HPair_neqs [symmetric, iff] |
|
778 |
||
779 |
lemma HPair_eq [iff]: "(Hash[X'] Y' = Hash[X] Y) = (X' = X & Y'=Y)" |
|
780 |
by (simp add: HPair_def) |
|
781 |
||
14200
d8598e24f8fa
Removal of the Key_supply axiom (affects many possbility proofs) and minor
paulson
parents:
14181
diff
changeset
|
782 |
lemma MPair_eq_HPair [iff]: |
d8598e24f8fa
Removal of the Key_supply axiom (affects many possbility proofs) and minor
paulson
parents:
14181
diff
changeset
|
783 |
"({|X',Y'|} = Hash[X] Y) = (X' = Hash{|X,Y|} & Y'=Y)" |
13926 | 784 |
by (simp add: HPair_def) |
785 |
||
14200
d8598e24f8fa
Removal of the Key_supply axiom (affects many possbility proofs) and minor
paulson
parents:
14181
diff
changeset
|
786 |
lemma HPair_eq_MPair [iff]: |
d8598e24f8fa
Removal of the Key_supply axiom (affects many possbility proofs) and minor
paulson
parents:
14181
diff
changeset
|
787 |
"(Hash[X] Y = {|X',Y'|}) = (X' = Hash{|X,Y|} & Y'=Y)" |
13926 | 788 |
by (auto simp add: HPair_def) |
789 |
||
790 |
||
14200
d8598e24f8fa
Removal of the Key_supply axiom (affects many possbility proofs) and minor
paulson
parents:
14181
diff
changeset
|
791 |
subsubsection{*Specialized laws, proved in terms of those for Hash and MPair *} |
13926 | 792 |
|
793 |
lemma keysFor_insert_HPair [simp]: "keysFor (insert (Hash[X] Y) H) = keysFor H" |
|
794 |
by (simp add: HPair_def) |
|
795 |
||
796 |
lemma parts_insert_HPair [simp]: |
|
797 |
"parts (insert (Hash[X] Y) H) = |
|
798 |
insert (Hash[X] Y) (insert (Hash{|X,Y|}) (parts (insert Y H)))" |
|
799 |
by (simp add: HPair_def) |
|
800 |
||
801 |
lemma analz_insert_HPair [simp]: |
|
802 |
"analz (insert (Hash[X] Y) H) = |
|
803 |
insert (Hash[X] Y) (insert (Hash{|X,Y|}) (analz (insert Y H)))" |
|
804 |
by (simp add: HPair_def) |
|
805 |
||
806 |
lemma HPair_synth_analz [simp]: |
|
807 |
"X \<notin> synth (analz H) |
|
808 |
==> (Hash[X] Y \<in> synth (analz H)) = |
|
809 |
(Hash {|X, Y|} \<in> analz H & Y \<in> synth (analz H))" |
|
810 |
by (simp add: HPair_def) |
|
811 |
||
812 |
||
16818 | 813 |
text{*We do NOT want Crypt... messages broken up in protocols!!*} |
13926 | 814 |
declare parts.Body [rule del] |
815 |
||
816 |
||
14200
d8598e24f8fa
Removal of the Key_supply axiom (affects many possbility proofs) and minor
paulson
parents:
14181
diff
changeset
|
817 |
text{*Rewrites to push in Key and Crypt messages, so that other messages can |
d8598e24f8fa
Removal of the Key_supply axiom (affects many possbility proofs) and minor
paulson
parents:
14181
diff
changeset
|
818 |
be pulled out using the @{text analz_insert} rules*} |
13926 | 819 |
|
27225 | 820 |
lemmas pushKeys [standard] = |
821 |
insert_commute [of "Key K" "Agent C"] |
|
822 |
insert_commute [of "Key K" "Nonce N"] |
|
823 |
insert_commute [of "Key K" "Number N"] |
|
824 |
insert_commute [of "Key K" "Hash X"] |
|
825 |
insert_commute [of "Key K" "MPair X Y"] |
|
826 |
insert_commute [of "Key K" "Crypt X K'"] |
|
13926 | 827 |
|
27225 | 828 |
lemmas pushCrypts [standard] = |
829 |
insert_commute [of "Crypt X K" "Agent C"] |
|
830 |
insert_commute [of "Crypt X K" "Agent C"] |
|
831 |
insert_commute [of "Crypt X K" "Nonce N"] |
|
832 |
insert_commute [of "Crypt X K" "Number N"] |
|
833 |
insert_commute [of "Crypt X K" "Hash X'"] |
|
834 |
insert_commute [of "Crypt X K" "MPair X' Y"] |
|
13926 | 835 |
|
836 |
text{*Cannot be added with @{text "[simp]"} -- messages should not always be |
|
837 |
re-ordered. *} |
|
838 |
lemmas pushes = pushKeys pushCrypts |
|
839 |
||
840 |
||
841 |
subsection{*Tactics useful for many protocol proofs*} |
|
842 |
ML |
|
843 |
{* |
|
24122 | 844 |
structure Message = |
845 |
struct |
|
13926 | 846 |
|
847 |
(*Prove base case (subgoal i) and simplify others. A typical base case |
|
848 |
concerns Crypt K X \<notin> Key`shrK`bad and cannot be proved by rewriting |
|
849 |
alone.*) |
|
30607
c3d1590debd8
eliminated global SIMPSET, CLASET etc. -- refer to explicit context;
wenzelm
parents:
30549
diff
changeset
|
850 |
fun prove_simple_subgoals_tac (cs, ss) i = |
c3d1590debd8
eliminated global SIMPSET, CLASET etc. -- refer to explicit context;
wenzelm
parents:
30549
diff
changeset
|
851 |
force_tac (cs, ss addsimps [@{thm image_eq_UN}]) i THEN |
c3d1590debd8
eliminated global SIMPSET, CLASET etc. -- refer to explicit context;
wenzelm
parents:
30549
diff
changeset
|
852 |
ALLGOALS (asm_simp_tac ss) |
13926 | 853 |
|
854 |
(*Analysis of Fake cases. Also works for messages that forward unknown parts, |
|
855 |
but this application is no longer necessary if analz_insert_eq is used. |
|
856 |
Abstraction over i is ESSENTIAL: it delays the dereferencing of claset |
|
857 |
DEPENDS UPON "X" REFERRING TO THE FRADULENT MESSAGE *) |
|
858 |
||
859 |
(*Apply rules to break down assumptions of the form |
|
860 |
Y \<in> parts(insert X H) and Y \<in> analz(insert X H) |
|
861 |
*) |
|
862 |
val Fake_insert_tac = |
|
24122 | 863 |
dresolve_tac [impOfSubs @{thm Fake_analz_insert}, |
864 |
impOfSubs @{thm Fake_parts_insert}] THEN' |
|
865 |
eresolve_tac [asm_rl, @{thm synth.Inj}]; |
|
13926 | 866 |
|
867 |
fun Fake_insert_simp_tac ss i = |
|
868 |
REPEAT (Fake_insert_tac i) THEN asm_full_simp_tac ss i; |
|
869 |
||
870 |
fun atomic_spy_analz_tac (cs,ss) = SELECT_GOAL |
|
871 |
(Fake_insert_simp_tac ss 1 |
|
872 |
THEN |
|
873 |
IF_UNSOLVED (Blast.depth_tac |
|
24122 | 874 |
(cs addIs [@{thm analz_insertI}, |
875 |
impOfSubs @{thm analz_subset_parts}]) 4 1)) |
|
13926 | 876 |
|
30607
c3d1590debd8
eliminated global SIMPSET, CLASET etc. -- refer to explicit context;
wenzelm
parents:
30549
diff
changeset
|
877 |
fun spy_analz_tac (cs,ss) i = |
13926 | 878 |
DETERM |
879 |
(SELECT_GOAL |
|
880 |
(EVERY |
|
881 |
[ (*push in occurrences of X...*) |
|
882 |
(REPEAT o CHANGED) |
|
27239 | 883 |
(res_inst_tac (Simplifier.the_context ss) [(("x", 1), "X")] (insert_commute RS ssubst) 1), |
13926 | 884 |
(*...allowing further simplifications*) |
885 |
simp_tac ss 1, |
|
886 |
REPEAT (FIRSTGOAL (resolve_tac [allI,impI,notI,conjI,iffI])), |
|
887 |
DEPTH_SOLVE (atomic_spy_analz_tac (cs,ss) 1)]) i) |
|
888 |
||
24122 | 889 |
end |
13926 | 890 |
*} |
891 |
||
16818 | 892 |
text{*By default only @{text o_apply} is built-in. But in the presence of |
893 |
eta-expansion this means that some terms displayed as @{term "f o g"} will be |
|
894 |
rewritten, and others will not!*} |
|
13926 | 895 |
declare o_def [simp] |
896 |
||
11189 | 897 |
|
13922 | 898 |
lemma Crypt_notin_image_Key [simp]: "Crypt K X \<notin> Key ` A" |
899 |
by auto |
|
900 |
||
901 |
lemma Hash_notin_image_Key [simp] :"Hash X \<notin> Key ` A" |
|
902 |
by auto |
|
903 |
||
14200
d8598e24f8fa
Removal of the Key_supply axiom (affects many possbility proofs) and minor
paulson
parents:
14181
diff
changeset
|
904 |
lemma synth_analz_mono: "G\<subseteq>H ==> synth (analz(G)) \<subseteq> synth (analz(H))" |
17689
a04b5b43625e
streamlined theory; conformance to recent publication
paulson
parents:
16818
diff
changeset
|
905 |
by (iprover intro: synth_mono analz_mono) |
13922 | 906 |
|
907 |
lemma Fake_analz_eq [simp]: |
|
908 |
"X \<in> synth(analz H) ==> synth (analz (insert X H)) = synth (analz H)" |
|
909 |
apply (drule Fake_analz_insert[of _ _ "H"]) |
|
910 |
apply (simp add: synth_increasing[THEN Un_absorb2]) |
|
911 |
apply (drule synth_mono) |
|
912 |
apply (simp add: synth_idem) |
|
17689
a04b5b43625e
streamlined theory; conformance to recent publication
paulson
parents:
16818
diff
changeset
|
913 |
apply (rule equalityI) |
a04b5b43625e
streamlined theory; conformance to recent publication
paulson
parents:
16818
diff
changeset
|
914 |
apply (simp add: ); |
a04b5b43625e
streamlined theory; conformance to recent publication
paulson
parents:
16818
diff
changeset
|
915 |
apply (rule synth_analz_mono, blast) |
13922 | 916 |
done |
917 |
||
918 |
text{*Two generalizations of @{text analz_insert_eq}*} |
|
919 |
lemma gen_analz_insert_eq [rule_format]: |
|
920 |
"X \<in> analz H ==> ALL G. H \<subseteq> G --> analz (insert X G) = analz G"; |
|
921 |
by (blast intro: analz_cut analz_insertI analz_mono [THEN [2] rev_subsetD]) |
|
922 |
||
923 |
lemma synth_analz_insert_eq [rule_format]: |
|
924 |
"X \<in> synth (analz H) |
|
925 |
==> ALL G. H \<subseteq> G --> (Key K \<in> analz (insert X G)) = (Key K \<in> analz G)"; |
|
926 |
apply (erule synth.induct) |
|
927 |
apply (simp_all add: gen_analz_insert_eq subset_trans [OF _ subset_insertI]) |
|
928 |
done |
|
929 |
||
930 |
lemma Fake_parts_sing: |
|
13926 | 931 |
"X \<in> synth (analz H) ==> parts{X} \<subseteq> synth (analz H) \<union> parts H"; |
13922 | 932 |
apply (rule subset_trans) |
17689
a04b5b43625e
streamlined theory; conformance to recent publication
paulson
parents:
16818
diff
changeset
|
933 |
apply (erule_tac [2] Fake_parts_insert) |
20648 | 934 |
apply (rule parts_mono, blast) |
13922 | 935 |
done |
936 |
||
14145
2e31b8cc8788
ZhouGollmann: new example (fair non-repudiation protocol)
paulson
parents:
14126
diff
changeset
|
937 |
lemmas Fake_parts_sing_imp_Un = Fake_parts_sing [THEN [2] rev_subsetD] |
2e31b8cc8788
ZhouGollmann: new example (fair non-repudiation protocol)
paulson
parents:
14126
diff
changeset
|
938 |
|
11189 | 939 |
method_setup spy_analz = {* |
30607
c3d1590debd8
eliminated global SIMPSET, CLASET etc. -- refer to explicit context;
wenzelm
parents:
30549
diff
changeset
|
940 |
Scan.succeed (SIMPLE_METHOD' o Message.spy_analz_tac o local_clasimpset_of) *} |
11189 | 941 |
"for proving the Fake case when analz is involved" |
1839 | 942 |
|
11264 | 943 |
method_setup atomic_spy_analz = {* |
30549 | 944 |
Scan.succeed (SIMPLE_METHOD' o Message.atomic_spy_analz_tac o local_clasimpset_of) *} |
11264 | 945 |
"for debugging spy_analz" |
946 |
||
947 |
method_setup Fake_insert_simp = {* |
|
30549 | 948 |
Scan.succeed (SIMPLE_METHOD' o Message.Fake_insert_simp_tac o local_simpset_of) *} |
11264 | 949 |
"for debugging spy_analz" |
950 |
||
1839 | 951 |
end |