author | wenzelm |
Tue, 18 Sep 2012 13:18:45 +0200 (2012-09-18) | |
changeset 49414 | d7b5fb2e9ca2 |
parent 45605 | a89b4bc311a5 |
child 51702 | dcfab8e87621 |
permissions | -rw-r--r-- |
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(* Title: HOL/SET_Protocol/Message_SET.thy |
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Author: Giampaolo Bella |
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Author: Fabio Massacci |
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Author: Lawrence C Paulson |
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*) |
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header{*The Message Theory, Modified for SET*} |
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theory Message_SET |
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imports Main "~~/src/HOL/Library/Nat_Bijection" |
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begin |
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subsection{*General Lemmas*} |
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text{*Needed occasionally with @{text spy_analz_tac}, e.g. in |
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@{text analz_insert_Key_newK}*} |
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lemma Un_absorb3 [simp] : "A \<union> (B \<union> A) = B \<union> A" |
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by blast |
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text{*Collapses redundant cases in the huge protocol proofs*} |
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lemmas disj_simps = disj_comms disj_left_absorb disj_assoc |
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text{*Effective with assumptions like @{term "K \<notin> range pubK"} and |
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@{term "K \<notin> invKey`range pubK"}*} |
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lemma notin_image_iff: "(y \<notin> f`I) = (\<forall>i\<in>I. f i \<noteq> y)" |
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by blast |
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text{*Effective with the assumption @{term "KK \<subseteq> - (range(invKey o pubK))"} *} |
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lemma disjoint_image_iff: "(A <= - (f`I)) = (\<forall>i\<in>I. f i \<notin> A)" |
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by blast |
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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 (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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definition symKeys :: "key set" where |
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"symKeys == {K. invKey K = K}" |
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text{*Agents. We allow any number of certification authorities, cardholders |
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merchants, and payment gateways.*} |
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datatype |
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agent = CA nat | Cardholder nat | Merchant nat | PG nat | Spy |
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text{*Messages*} |
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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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| Pan nat --{*Unguessable Primary Account Numbers (??)*} |
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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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(*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 nat_of_agent :: "agent => nat" where |
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"nat_of_agent == agent_case (curry prod_encode 0) |
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(curry prod_encode 1) |
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(curry prod_encode 2) |
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(curry prod_encode 3) |
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(prod_encode (4,0))" |
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--{*maps each agent to a unique natural number, for specifications*} |
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text{*The function is indeed injective*} |
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lemma inj_nat_of_agent: "inj nat_of_agent" |
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by (simp add: nat_of_agent_def inj_on_def curry_def prod_encode_eq split: agent.split) |
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definition |
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(*Keys useful to decrypt elements of a message set*) |
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keysFor :: "msg set => key set" |
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where "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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(*Monotonicity*) |
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lemma parts_mono: "G<=H ==> parts(G) <= parts(H)" |
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apply auto |
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apply (erule parts.induct) |
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apply (auto dest: Fst Snd Body) |
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done |
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subsubsection{*Inverse of keys*} |
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(*Equations hold because constructors are injective; cannot prove for all f*) |
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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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lemma Cardholder_image_eq [simp]: "(Cardholder x \<in> Cardholder`A) = (x \<in> A)" |
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by auto |
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lemma CA_image_eq [simp]: "(CA x \<in> CA`A) = (x \<in> A)" |
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by auto |
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lemma Pan_image_eq [simp]: "(Pan x \<in> Pan`A) = (x \<in> A)" |
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by auto |
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lemma Pan_Key_image_eq [simp]: "(Pan x \<notin> Key`A)" |
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by auto |
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lemma Nonce_Pan_image_eq [simp]: "(Nonce x \<notin> Pan`A)" |
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by auto |
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lemma invKey_eq [simp]: "(invKey K = invKey K') = (K=K')" |
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apply safe |
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apply (drule_tac f = invKey in arg_cong, simp) |
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done |
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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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(*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_Pan [simp]: "keysFor (insert (Pan A) 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] |
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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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(*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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(*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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(*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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declare parts_Un [THEN equalityD1, THEN subsetD, THEN 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_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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(*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 (erule parts_trans, auto) |
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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_Pan [simp]: |
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"parts (insert (Pan A) H) = insert (Pan A) (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 (erule parts.induct) |
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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 (erule parts.induct) |
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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 parts_image_Pan [simp]: "parts (Pan`A) = Pan`A" |
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apply auto |
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apply (erule parts.induct, auto) |
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done |
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(*In any message, there is an upper bound N on its greatest nonce.*) |
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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 (no_asm_simp) add: exI parts_insert2) |
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(*MPair case: blast_tac works out the necessary sum itself!*) |
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prefer 2 apply (blast elim!: add_leE) |
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(*Nonce case*) |
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apply (rule_tac x = "N + Suc nat" in exI) |
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apply (auto elim!: add_leE) |
|
378 |
done |
|
379 |
||
380 |
(* Ditto, for numbers.*) |
|
381 |
lemma msg_Number_supply: "\<exists>N. \<forall>n. N<=n --> Number n \<notin> parts {msg}" |
|
382 |
apply (induct_tac "msg") |
|
383 |
apply (simp_all (no_asm_simp) add: exI parts_insert2) |
|
384 |
prefer 2 apply (blast elim!: add_leE) |
|
385 |
apply (rule_tac x = "N + Suc nat" in exI, auto) |
|
386 |
done |
|
387 |
||
388 |
subsection{*Inductive relation "analz"*} |
|
389 |
||
390 |
text{*Inductive definition of "analz" -- what can be broken down from a set of |
|
391 |
messages, including keys. A form of downward closure. Pairs can |
|
392 |
be taken apart; messages decrypted with known keys.*} |
|
393 |
||
23755 | 394 |
inductive_set |
395 |
analz :: "msg set => msg set" |
|
396 |
for H :: "msg set" |
|
397 |
where |
|
14199 | 398 |
Inj [intro,simp] : "X \<in> H ==> X \<in> analz H" |
23755 | 399 |
| Fst: "{|X,Y|} \<in> analz H ==> X \<in> analz H" |
400 |
| Snd: "{|X,Y|} \<in> analz H ==> Y \<in> analz H" |
|
401 |
| Decrypt [dest]: |
|
14199 | 402 |
"[|Crypt K X \<in> analz H; Key(invKey K): analz H|] ==> X \<in> analz H" |
403 |
||
404 |
||
405 |
(*Monotonicity; Lemma 1 of Lowe's paper*) |
|
406 |
lemma analz_mono: "G<=H ==> analz(G) <= analz(H)" |
|
407 |
apply auto |
|
408 |
apply (erule analz.induct) |
|
409 |
apply (auto dest: Fst Snd) |
|
410 |
done |
|
411 |
||
412 |
text{*Making it safe speeds up proofs*} |
|
413 |
lemma MPair_analz [elim!]: |
|
414 |
"[| {|X,Y|} \<in> analz H; |
|
415 |
[| X \<in> analz H; Y \<in> analz H |] ==> P |
|
416 |
|] ==> P" |
|
417 |
by (blast dest: analz.Fst analz.Snd) |
|
418 |
||
419 |
lemma analz_increasing: "H \<subseteq> analz(H)" |
|
420 |
by blast |
|
421 |
||
422 |
lemma analz_subset_parts: "analz H \<subseteq> parts H" |
|
423 |
apply (rule subsetI) |
|
424 |
apply (erule analz.induct, blast+) |
|
425 |
done |
|
426 |
||
45605 | 427 |
lemmas analz_into_parts = analz_subset_parts [THEN subsetD] |
14199 | 428 |
|
45605 | 429 |
lemmas not_parts_not_analz = analz_subset_parts [THEN contra_subsetD] |
14199 | 430 |
|
431 |
||
432 |
lemma parts_analz [simp]: "parts (analz H) = parts H" |
|
433 |
apply (rule equalityI) |
|
434 |
apply (rule analz_subset_parts [THEN parts_mono, THEN subset_trans], simp) |
|
435 |
apply (blast intro: analz_increasing [THEN parts_mono, THEN subsetD]) |
|
436 |
done |
|
437 |
||
438 |
lemma analz_parts [simp]: "analz (parts H) = parts H" |
|
439 |
apply auto |
|
440 |
apply (erule analz.induct, auto) |
|
441 |
done |
|
442 |
||
45605 | 443 |
lemmas analz_insertI = subset_insertI [THEN analz_mono, THEN [2] rev_subsetD] |
14199 | 444 |
|
445 |
subsubsection{*General equational properties*} |
|
446 |
||
447 |
lemma analz_empty [simp]: "analz{} = {}" |
|
448 |
apply safe |
|
449 |
apply (erule analz.induct, blast+) |
|
450 |
done |
|
451 |
||
452 |
(*Converse fails: we can analz more from the union than from the |
|
453 |
separate parts, as a key in one might decrypt a message in the other*) |
|
454 |
lemma analz_Un: "analz(G) \<union> analz(H) \<subseteq> analz(G \<union> H)" |
|
455 |
by (intro Un_least analz_mono Un_upper1 Un_upper2) |
|
456 |
||
457 |
lemma analz_insert: "insert X (analz H) \<subseteq> analz(insert X H)" |
|
458 |
by (blast intro: analz_mono [THEN [2] rev_subsetD]) |
|
459 |
||
460 |
subsubsection{*Rewrite rules for pulling out atomic messages*} |
|
461 |
||
462 |
lemmas analz_insert_eq_I = equalityI [OF subsetI analz_insert] |
|
463 |
||
464 |
lemma analz_insert_Agent [simp]: |
|
465 |
"analz (insert (Agent agt) H) = insert (Agent agt) (analz H)" |
|
466 |
apply (rule analz_insert_eq_I) |
|
467 |
apply (erule analz.induct, auto) |
|
468 |
done |
|
469 |
||
470 |
lemma analz_insert_Nonce [simp]: |
|
471 |
"analz (insert (Nonce N) H) = insert (Nonce N) (analz H)" |
|
472 |
apply (rule analz_insert_eq_I) |
|
473 |
apply (erule analz.induct, auto) |
|
474 |
done |
|
475 |
||
476 |
lemma analz_insert_Number [simp]: |
|
477 |
"analz (insert (Number N) H) = insert (Number N) (analz H)" |
|
478 |
apply (rule analz_insert_eq_I) |
|
479 |
apply (erule analz.induct, auto) |
|
480 |
done |
|
481 |
||
482 |
lemma analz_insert_Hash [simp]: |
|
483 |
"analz (insert (Hash X) H) = insert (Hash X) (analz H)" |
|
484 |
apply (rule analz_insert_eq_I) |
|
485 |
apply (erule analz.induct, auto) |
|
486 |
done |
|
487 |
||
488 |
(*Can only pull out Keys if they are not needed to decrypt the rest*) |
|
489 |
lemma analz_insert_Key [simp]: |
|
490 |
"K \<notin> keysFor (analz H) ==> |
|
491 |
analz (insert (Key K) H) = insert (Key K) (analz H)" |
|
492 |
apply (unfold keysFor_def) |
|
493 |
apply (rule analz_insert_eq_I) |
|
494 |
apply (erule analz.induct, auto) |
|
495 |
done |
|
496 |
||
497 |
lemma analz_insert_MPair [simp]: |
|
498 |
"analz (insert {|X,Y|} H) = |
|
499 |
insert {|X,Y|} (analz (insert X (insert Y H)))" |
|
500 |
apply (rule equalityI) |
|
501 |
apply (rule subsetI) |
|
502 |
apply (erule analz.induct, auto) |
|
503 |
apply (erule analz.induct) |
|
504 |
apply (blast intro: analz.Fst analz.Snd)+ |
|
505 |
done |
|
506 |
||
507 |
(*Can pull out enCrypted message if the Key is not known*) |
|
508 |
lemma analz_insert_Crypt: |
|
509 |
"Key (invKey K) \<notin> analz H |
|
510 |
==> analz (insert (Crypt K X) H) = insert (Crypt K X) (analz H)" |
|
511 |
apply (rule analz_insert_eq_I) |
|
512 |
apply (erule analz.induct, auto) |
|
513 |
done |
|
514 |
||
515 |
lemma analz_insert_Pan [simp]: |
|
516 |
"analz (insert (Pan A) H) = insert (Pan A) (analz H)" |
|
517 |
apply (rule analz_insert_eq_I) |
|
518 |
apply (erule analz.induct, auto) |
|
519 |
done |
|
520 |
||
521 |
lemma lemma1: "Key (invKey K) \<in> analz H ==> |
|
522 |
analz (insert (Crypt K X) H) \<subseteq> |
|
523 |
insert (Crypt K X) (analz (insert X H))" |
|
524 |
apply (rule subsetI) |
|
23755 | 525 |
apply (erule_tac x = x in analz.induct, auto) |
14199 | 526 |
done |
527 |
||
528 |
lemma lemma2: "Key (invKey K) \<in> analz H ==> |
|
529 |
insert (Crypt K X) (analz (insert X H)) \<subseteq> |
|
530 |
analz (insert (Crypt K X) H)" |
|
531 |
apply auto |
|
23755 | 532 |
apply (erule_tac x = x in analz.induct, auto) |
14199 | 533 |
apply (blast intro: analz_insertI analz.Decrypt) |
534 |
done |
|
535 |
||
536 |
lemma analz_insert_Decrypt: |
|
537 |
"Key (invKey K) \<in> analz H ==> |
|
538 |
analz (insert (Crypt K X) H) = |
|
539 |
insert (Crypt K X) (analz (insert X H))" |
|
540 |
by (intro equalityI lemma1 lemma2) |
|
541 |
||
542 |
(*Case analysis: either the message is secure, or it is not! |
|
543 |
Effective, but can cause subgoals to blow up! |
|
544 |
Use with split_if; apparently split_tac does not cope with patterns |
|
545 |
such as "analz (insert (Crypt K X) H)" *) |
|
546 |
lemma analz_Crypt_if [simp]: |
|
547 |
"analz (insert (Crypt K X) H) = |
|
548 |
(if (Key (invKey K) \<in> analz H) |
|
549 |
then insert (Crypt K X) (analz (insert X H)) |
|
550 |
else insert (Crypt K X) (analz H))" |
|
551 |
by (simp add: analz_insert_Crypt analz_insert_Decrypt) |
|
552 |
||
553 |
||
554 |
(*This rule supposes "for the sake of argument" that we have the key.*) |
|
555 |
lemma analz_insert_Crypt_subset: |
|
556 |
"analz (insert (Crypt K X) H) \<subseteq> |
|
557 |
insert (Crypt K X) (analz (insert X H))" |
|
558 |
apply (rule subsetI) |
|
559 |
apply (erule analz.induct, auto) |
|
560 |
done |
|
561 |
||
562 |
lemma analz_image_Key [simp]: "analz (Key`N) = Key`N" |
|
563 |
apply auto |
|
564 |
apply (erule analz.induct, auto) |
|
565 |
done |
|
566 |
||
567 |
lemma analz_image_Pan [simp]: "analz (Pan`A) = Pan`A" |
|
568 |
apply auto |
|
569 |
apply (erule analz.induct, auto) |
|
570 |
done |
|
571 |
||
572 |
||
573 |
subsubsection{*Idempotence and transitivity*} |
|
574 |
||
575 |
lemma analz_analzD [dest!]: "X\<in> analz (analz H) ==> X\<in> analz H" |
|
576 |
by (erule analz.induct, blast+) |
|
577 |
||
578 |
lemma analz_idem [simp]: "analz (analz H) = analz H" |
|
579 |
by blast |
|
580 |
||
581 |
lemma analz_trans: "[| X\<in> analz G; G \<subseteq> analz H |] ==> X\<in> analz H" |
|
582 |
by (drule analz_mono, blast) |
|
583 |
||
584 |
(*Cut; Lemma 2 of Lowe*) |
|
585 |
lemma analz_cut: "[| Y\<in> analz (insert X H); X\<in> analz H |] ==> Y\<in> analz H" |
|
586 |
by (erule analz_trans, blast) |
|
587 |
||
588 |
(*Cut can be proved easily by induction on |
|
589 |
"Y: analz (insert X H) ==> X: analz H --> Y: analz H" |
|
590 |
*) |
|
591 |
||
592 |
(*This rewrite rule helps in the simplification of messages that involve |
|
593 |
the forwarding of unknown components (X). Without it, removing occurrences |
|
594 |
of X can be very complicated. *) |
|
595 |
lemma analz_insert_eq: "X\<in> analz H ==> analz (insert X H) = analz H" |
|
596 |
by (blast intro: analz_cut analz_insertI) |
|
597 |
||
598 |
||
599 |
text{*A congruence rule for "analz"*} |
|
600 |
||
601 |
lemma analz_subset_cong: |
|
602 |
"[| analz G \<subseteq> analz G'; analz H \<subseteq> analz H' |
|
603 |
|] ==> analz (G \<union> H) \<subseteq> analz (G' \<union> H')" |
|
604 |
apply clarify |
|
605 |
apply (erule analz.induct) |
|
606 |
apply (best intro: analz_mono [THEN subsetD])+ |
|
607 |
done |
|
608 |
||
609 |
lemma analz_cong: |
|
610 |
"[| analz G = analz G'; analz H = analz H' |
|
611 |
|] ==> analz (G \<union> H) = analz (G' \<union> H')" |
|
612 |
by (intro equalityI analz_subset_cong, simp_all) |
|
613 |
||
614 |
lemma analz_insert_cong: |
|
615 |
"analz H = analz H' ==> analz(insert X H) = analz(insert X H')" |
|
616 |
by (force simp only: insert_def intro!: analz_cong) |
|
617 |
||
618 |
(*If there are no pairs or encryptions then analz does nothing*) |
|
619 |
lemma analz_trivial: |
|
620 |
"[| \<forall>X Y. {|X,Y|} \<notin> H; \<forall>X K. Crypt K X \<notin> H |] ==> analz H = H" |
|
621 |
apply safe |
|
622 |
apply (erule analz.induct, blast+) |
|
623 |
done |
|
624 |
||
625 |
(*These two are obsolete (with a single Spy) but cost little to prove...*) |
|
626 |
lemma analz_UN_analz_lemma: |
|
627 |
"X\<in> analz (\<Union>i\<in>A. analz (H i)) ==> X\<in> analz (\<Union>i\<in>A. H i)" |
|
628 |
apply (erule analz.induct) |
|
629 |
apply (blast intro: analz_mono [THEN [2] rev_subsetD])+ |
|
630 |
done |
|
631 |
||
632 |
lemma analz_UN_analz [simp]: "analz (\<Union>i\<in>A. analz (H i)) = analz (\<Union>i\<in>A. H i)" |
|
633 |
by (blast intro: analz_UN_analz_lemma analz_mono [THEN [2] rev_subsetD]) |
|
634 |
||
635 |
||
636 |
subsection{*Inductive relation "synth"*} |
|
637 |
||
638 |
text{*Inductive definition of "synth" -- what can be built up from a set of |
|
639 |
messages. A form of upward closure. Pairs can be built, messages |
|
640 |
encrypted with known keys. Agent names are public domain. |
|
641 |
Numbers can be guessed, but Nonces cannot be.*} |
|
642 |
||
23755 | 643 |
inductive_set |
644 |
synth :: "msg set => msg set" |
|
645 |
for H :: "msg set" |
|
646 |
where |
|
14199 | 647 |
Inj [intro]: "X \<in> H ==> X \<in> synth H" |
23755 | 648 |
| Agent [intro]: "Agent agt \<in> synth H" |
649 |
| Number [intro]: "Number n \<in> synth H" |
|
650 |
| Hash [intro]: "X \<in> synth H ==> Hash X \<in> synth H" |
|
651 |
| MPair [intro]: "[|X \<in> synth H; Y \<in> synth H|] ==> {|X,Y|} \<in> synth H" |
|
652 |
| Crypt [intro]: "[|X \<in> synth H; Key(K) \<in> H|] ==> Crypt K X \<in> synth H" |
|
14199 | 653 |
|
654 |
(*Monotonicity*) |
|
655 |
lemma synth_mono: "G<=H ==> synth(G) <= synth(H)" |
|
656 |
apply auto |
|
657 |
apply (erule synth.induct) |
|
658 |
apply (auto dest: Fst Snd Body) |
|
659 |
done |
|
660 |
||
661 |
(*NO Agent_synth, as any Agent name can be synthesized. Ditto for Number*) |
|
662 |
inductive_cases Nonce_synth [elim!]: "Nonce n \<in> synth H" |
|
663 |
inductive_cases Key_synth [elim!]: "Key K \<in> synth H" |
|
664 |
inductive_cases Hash_synth [elim!]: "Hash X \<in> synth H" |
|
665 |
inductive_cases MPair_synth [elim!]: "{|X,Y|} \<in> synth H" |
|
666 |
inductive_cases Crypt_synth [elim!]: "Crypt K X \<in> synth H" |
|
667 |
inductive_cases Pan_synth [elim!]: "Pan A \<in> synth H" |
|
668 |
||
669 |
||
670 |
lemma synth_increasing: "H \<subseteq> synth(H)" |
|
671 |
by blast |
|
672 |
||
673 |
subsubsection{*Unions*} |
|
674 |
||
675 |
(*Converse fails: we can synth more from the union than from the |
|
676 |
separate parts, building a compound message using elements of each.*) |
|
677 |
lemma synth_Un: "synth(G) \<union> synth(H) \<subseteq> synth(G \<union> H)" |
|
678 |
by (intro Un_least synth_mono Un_upper1 Un_upper2) |
|
679 |
||
680 |
lemma synth_insert: "insert X (synth H) \<subseteq> synth(insert X H)" |
|
681 |
by (blast intro: synth_mono [THEN [2] rev_subsetD]) |
|
682 |
||
683 |
subsubsection{*Idempotence and transitivity*} |
|
684 |
||
685 |
lemma synth_synthD [dest!]: "X\<in> synth (synth H) ==> X\<in> synth H" |
|
686 |
by (erule synth.induct, blast+) |
|
687 |
||
688 |
lemma synth_idem: "synth (synth H) = synth H" |
|
689 |
by blast |
|
690 |
||
691 |
lemma synth_trans: "[| X\<in> synth G; G \<subseteq> synth H |] ==> X\<in> synth H" |
|
692 |
by (drule synth_mono, blast) |
|
693 |
||
694 |
(*Cut; Lemma 2 of Lowe*) |
|
695 |
lemma synth_cut: "[| Y\<in> synth (insert X H); X\<in> synth H |] ==> Y\<in> synth H" |
|
696 |
by (erule synth_trans, blast) |
|
697 |
||
698 |
lemma Agent_synth [simp]: "Agent A \<in> synth H" |
|
699 |
by blast |
|
700 |
||
701 |
lemma Number_synth [simp]: "Number n \<in> synth H" |
|
702 |
by blast |
|
703 |
||
704 |
lemma Nonce_synth_eq [simp]: "(Nonce N \<in> synth H) = (Nonce N \<in> H)" |
|
705 |
by blast |
|
706 |
||
707 |
lemma Key_synth_eq [simp]: "(Key K \<in> synth H) = (Key K \<in> H)" |
|
708 |
by blast |
|
709 |
||
710 |
lemma Crypt_synth_eq [simp]: "Key K \<notin> H ==> (Crypt K X \<in> synth H) = (Crypt K X \<in> H)" |
|
711 |
by blast |
|
712 |
||
713 |
lemma Pan_synth_eq [simp]: "(Pan A \<in> synth H) = (Pan A \<in> H)" |
|
714 |
by blast |
|
715 |
||
716 |
lemma keysFor_synth [simp]: |
|
717 |
"keysFor (synth H) = keysFor H \<union> invKey`{K. Key K \<in> H}" |
|
718 |
by (unfold keysFor_def, blast) |
|
719 |
||
720 |
||
721 |
subsubsection{*Combinations of parts, analz and synth*} |
|
722 |
||
723 |
lemma parts_synth [simp]: "parts (synth H) = parts H \<union> synth H" |
|
724 |
apply (rule equalityI) |
|
725 |
apply (rule subsetI) |
|
726 |
apply (erule parts.induct) |
|
727 |
apply (blast intro: synth_increasing [THEN parts_mono, THEN subsetD] |
|
728 |
parts.Fst parts.Snd parts.Body)+ |
|
729 |
done |
|
730 |
||
731 |
lemma analz_analz_Un [simp]: "analz (analz G \<union> H) = analz (G \<union> H)" |
|
732 |
apply (intro equalityI analz_subset_cong)+ |
|
733 |
apply simp_all |
|
734 |
done |
|
735 |
||
736 |
lemma analz_synth_Un [simp]: "analz (synth G \<union> H) = analz (G \<union> H) \<union> synth G" |
|
737 |
apply (rule equalityI) |
|
738 |
apply (rule subsetI) |
|
739 |
apply (erule analz.induct) |
|
740 |
prefer 5 apply (blast intro: analz_mono [THEN [2] rev_subsetD]) |
|
741 |
apply (blast intro: analz.Fst analz.Snd analz.Decrypt)+ |
|
742 |
done |
|
743 |
||
744 |
lemma analz_synth [simp]: "analz (synth H) = analz H \<union> synth H" |
|
745 |
apply (cut_tac H = "{}" in analz_synth_Un) |
|
746 |
apply (simp (no_asm_use)) |
|
747 |
done |
|
748 |
||
749 |
||
750 |
subsubsection{*For reasoning about the Fake rule in traces*} |
|
751 |
||
752 |
lemma parts_insert_subset_Un: "X\<in> G ==> parts(insert X H) \<subseteq> parts G \<union> parts H" |
|
753 |
by (rule subset_trans [OF parts_mono parts_Un_subset2], blast) |
|
754 |
||
755 |
(*More specifically for Fake. Very occasionally we could do with a version |
|
756 |
of the form parts{X} \<subseteq> synth (analz H) \<union> parts H *) |
|
757 |
lemma Fake_parts_insert: "X \<in> synth (analz H) ==> |
|
758 |
parts (insert X H) \<subseteq> synth (analz H) \<union> parts H" |
|
759 |
apply (drule parts_insert_subset_Un) |
|
760 |
apply (simp (no_asm_use)) |
|
761 |
apply blast |
|
762 |
done |
|
763 |
||
764 |
lemma Fake_parts_insert_in_Un: |
|
765 |
"[|Z \<in> parts (insert X H); X: synth (analz H)|] |
|
766 |
==> Z \<in> synth (analz H) \<union> parts H"; |
|
767 |
by (blast dest: Fake_parts_insert [THEN subsetD, dest]) |
|
768 |
||
769 |
(*H is sometimes (Key ` KK \<union> spies evs), so can't put G=H*) |
|
770 |
lemma Fake_analz_insert: |
|
771 |
"X\<in> synth (analz G) ==> |
|
772 |
analz (insert X H) \<subseteq> synth (analz G) \<union> analz (G \<union> H)" |
|
773 |
apply (rule subsetI) |
|
774 |
apply (subgoal_tac "x \<in> analz (synth (analz G) \<union> H) ") |
|
775 |
prefer 2 apply (blast intro: analz_mono [THEN [2] rev_subsetD] analz_mono [THEN synth_mono, THEN [2] rev_subsetD]) |
|
776 |
apply (simp (no_asm_use)) |
|
777 |
apply blast |
|
778 |
done |
|
779 |
||
780 |
lemma analz_conj_parts [simp]: |
|
781 |
"(X \<in> analz H & X \<in> parts H) = (X \<in> analz H)" |
|
782 |
by (blast intro: analz_subset_parts [THEN subsetD]) |
|
783 |
||
784 |
lemma analz_disj_parts [simp]: |
|
785 |
"(X \<in> analz H | X \<in> parts H) = (X \<in> parts H)" |
|
786 |
by (blast intro: analz_subset_parts [THEN subsetD]) |
|
787 |
||
788 |
(*Without this equation, other rules for synth and analz would yield |
|
789 |
redundant cases*) |
|
790 |
lemma MPair_synth_analz [iff]: |
|
791 |
"({|X,Y|} \<in> synth (analz H)) = |
|
792 |
(X \<in> synth (analz H) & Y \<in> synth (analz H))" |
|
793 |
by blast |
|
794 |
||
795 |
lemma Crypt_synth_analz: |
|
796 |
"[| Key K \<in> analz H; Key (invKey K) \<in> analz H |] |
|
797 |
==> (Crypt K X \<in> synth (analz H)) = (X \<in> synth (analz H))" |
|
798 |
by blast |
|
799 |
||
800 |
||
801 |
lemma Hash_synth_analz [simp]: |
|
802 |
"X \<notin> synth (analz H) |
|
803 |
==> (Hash{|X,Y|} \<in> synth (analz H)) = (Hash{|X,Y|} \<in> analz H)" |
|
804 |
by blast |
|
805 |
||
806 |
||
807 |
(*We do NOT want Crypt... messages broken up in protocols!!*) |
|
808 |
declare parts.Body [rule del] |
|
809 |
||
810 |
||
811 |
text{*Rewrites to push in Key and Crypt messages, so that other messages can |
|
812 |
be pulled out using the @{text analz_insert} rules*} |
|
813 |
||
45605 | 814 |
lemmas pushKeys = |
27225 | 815 |
insert_commute [of "Key K" "Agent C"] |
816 |
insert_commute [of "Key K" "Nonce N"] |
|
817 |
insert_commute [of "Key K" "Number N"] |
|
818 |
insert_commute [of "Key K" "Pan PAN"] |
|
819 |
insert_commute [of "Key K" "Hash X"] |
|
820 |
insert_commute [of "Key K" "MPair X Y"] |
|
821 |
insert_commute [of "Key K" "Crypt X K'"] |
|
45605 | 822 |
for K C N PAN X Y K' |
14199 | 823 |
|
45605 | 824 |
lemmas pushCrypts = |
27225 | 825 |
insert_commute [of "Crypt X K" "Agent C"] |
826 |
insert_commute [of "Crypt X K" "Nonce N"] |
|
827 |
insert_commute [of "Crypt X K" "Number N"] |
|
828 |
insert_commute [of "Crypt X K" "Pan PAN"] |
|
829 |
insert_commute [of "Crypt X K" "Hash X'"] |
|
830 |
insert_commute [of "Crypt X K" "MPair X' Y"] |
|
45605 | 831 |
for X K C N PAN X' Y |
14199 | 832 |
|
833 |
text{*Cannot be added with @{text "[simp]"} -- messages should not always be |
|
834 |
re-ordered.*} |
|
835 |
lemmas pushes = pushKeys pushCrypts |
|
836 |
||
837 |
||
838 |
subsection{*Tactics useful for many protocol proofs*} |
|
14218 | 839 |
(*<*) |
14199 | 840 |
ML |
841 |
{* |
|
842 |
(*Analysis of Fake cases. Also works for messages that forward unknown parts, |
|
843 |
but this application is no longer necessary if analz_insert_eq is used. |
|
844 |
Abstraction over i is ESSENTIAL: it delays the dereferencing of claset |
|
845 |
DEPENDS UPON "X" REFERRING TO THE FRADULENT MESSAGE *) |
|
846 |
||
32117
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
30607
diff
changeset
|
847 |
fun impOfSubs th = th RSN (2, @{thm rev_subsetD}) |
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
30607
diff
changeset
|
848 |
|
14199 | 849 |
(*Apply rules to break down assumptions of the form |
850 |
Y \<in> parts(insert X H) and Y \<in> analz(insert X H) |
|
851 |
*) |
|
852 |
val Fake_insert_tac = |
|
24123 | 853 |
dresolve_tac [impOfSubs @{thm Fake_analz_insert}, |
854 |
impOfSubs @{thm Fake_parts_insert}] THEN' |
|
855 |
eresolve_tac [asm_rl, @{thm synth.Inj}]; |
|
14199 | 856 |
|
857 |
fun Fake_insert_simp_tac ss i = |
|
42793 | 858 |
REPEAT (Fake_insert_tac i) THEN asm_full_simp_tac ss i; |
14199 | 859 |
|
42474 | 860 |
fun atomic_spy_analz_tac ctxt = |
42793 | 861 |
SELECT_GOAL |
862 |
(Fake_insert_simp_tac (simpset_of ctxt) 1 THEN |
|
863 |
IF_UNSOLVED |
|
864 |
(Blast.depth_tac (ctxt addIs [@{thm analz_insertI}, |
|
865 |
impOfSubs @{thm analz_subset_parts}]) 4 1)); |
|
14199 | 866 |
|
42474 | 867 |
fun spy_analz_tac ctxt i = |
42793 | 868 |
DETERM |
869 |
(SELECT_GOAL |
|
870 |
(EVERY |
|
871 |
[ (*push in occurrences of X...*) |
|
872 |
(REPEAT o CHANGED) |
|
873 |
(res_inst_tac ctxt [(("x", 1), "X")] (insert_commute RS ssubst) 1), |
|
874 |
(*...allowing further simplifications*) |
|
875 |
simp_tac (simpset_of ctxt) 1, |
|
876 |
REPEAT (FIRSTGOAL (resolve_tac [allI,impI,notI,conjI,iffI])), |
|
877 |
DEPTH_SOLVE (atomic_spy_analz_tac ctxt 1)]) i); |
|
14199 | 878 |
*} |
14218 | 879 |
(*>*) |
880 |
||
14199 | 881 |
|
882 |
(*By default only o_apply is built-in. But in the presence of eta-expansion |
|
883 |
this means that some terms displayed as (f o g) will be rewritten, and others |
|
884 |
will not!*) |
|
885 |
declare o_def [simp] |
|
886 |
||
887 |
||
888 |
lemma Crypt_notin_image_Key [simp]: "Crypt K X \<notin> Key ` A" |
|
889 |
by auto |
|
890 |
||
891 |
lemma Hash_notin_image_Key [simp] :"Hash X \<notin> Key ` A" |
|
892 |
by auto |
|
893 |
||
894 |
lemma synth_analz_mono: "G<=H ==> synth (analz(G)) <= synth (analz(H))" |
|
895 |
by (simp add: synth_mono analz_mono) |
|
896 |
||
897 |
lemma Fake_analz_eq [simp]: |
|
898 |
"X \<in> synth(analz H) ==> synth (analz (insert X H)) = synth (analz H)" |
|
899 |
apply (drule Fake_analz_insert[of _ _ "H"]) |
|
900 |
apply (simp add: synth_increasing[THEN Un_absorb2]) |
|
901 |
apply (drule synth_mono) |
|
902 |
apply (simp add: synth_idem) |
|
903 |
apply (blast intro: synth_analz_mono [THEN [2] rev_subsetD]) |
|
904 |
done |
|
905 |
||
906 |
text{*Two generalizations of @{text analz_insert_eq}*} |
|
907 |
lemma gen_analz_insert_eq [rule_format]: |
|
908 |
"X \<in> analz H ==> ALL G. H \<subseteq> G --> analz (insert X G) = analz G"; |
|
909 |
by (blast intro: analz_cut analz_insertI analz_mono [THEN [2] rev_subsetD]) |
|
910 |
||
911 |
lemma synth_analz_insert_eq [rule_format]: |
|
912 |
"X \<in> synth (analz H) |
|
913 |
==> ALL G. H \<subseteq> G --> (Key K \<in> analz (insert X G)) = (Key K \<in> analz G)"; |
|
914 |
apply (erule synth.induct) |
|
915 |
apply (simp_all add: gen_analz_insert_eq subset_trans [OF _ subset_insertI]) |
|
916 |
done |
|
917 |
||
918 |
lemma Fake_parts_sing: |
|
919 |
"X \<in> synth (analz H) ==> parts{X} \<subseteq> synth (analz H) \<union> parts H"; |
|
920 |
apply (rule subset_trans) |
|
921 |
apply (erule_tac [2] Fake_parts_insert) |
|
922 |
apply (simp add: parts_mono) |
|
923 |
done |
|
924 |
||
925 |
lemmas Fake_parts_sing_imp_Un = Fake_parts_sing [THEN [2] rev_subsetD] |
|
926 |
||
927 |
method_setup spy_analz = {* |
|
42474 | 928 |
Scan.succeed (SIMPLE_METHOD' o spy_analz_tac) *} |
14199 | 929 |
"for proving the Fake case when analz is involved" |
930 |
||
931 |
method_setup atomic_spy_analz = {* |
|
42474 | 932 |
Scan.succeed (SIMPLE_METHOD' o atomic_spy_analz_tac) *} |
14199 | 933 |
"for debugging spy_analz" |
934 |
||
935 |
method_setup Fake_insert_simp = {* |
|
42474 | 936 |
Scan.succeed (SIMPLE_METHOD' o Fake_insert_simp_tac o simpset_of) *} |
14199 | 937 |
"for debugging spy_analz" |
938 |
||
939 |
end |