--- a/doc-src/TutorialI/Protocol/Message.thy Mon Jul 23 14:30:53 2007 +0200
+++ b/doc-src/TutorialI/Protocol/Message.thy Mon Jul 23 14:31:34 2007 +0200
@@ -5,45 +5,83 @@
Datatypes of agents and messages;
Inductive relations "parts", "analz" and "synth"
-*)
+*)(*<*)
-theory Message imports Main
-uses ("Message_lemmas.ML") begin
+header{*Theory of Agents and Messages for Security Protocols*}
+
+theory Message imports Main uses "../../antiquote_setup.ML" begin
(*Needed occasionally with spy_analz_tac, e.g. in analz_insert_Key_newK*)
-lemma [simp] : "A Un (B Un A) = B Un A"
+lemma [simp] : "A \<union> (B \<union> A) = B \<union> A"
by blast
+(*>*)
+
+section{* Agents and Messages *}
-types
- key = nat
+text {*
+All protocol specifications refer to a syntactic theory of messages.
+Datatype
+@{text agent} introduces the constant @{text Server} (a trusted central
+machine, needed for some protocols), an infinite population of
+friendly agents, and the~@{text Spy}:
+*}
+
+datatype agent = Server | Friend nat | Spy
-consts
- invKey :: "key=>key"
+text {*
+Keys are just natural numbers. Function @{text invKey} maps a public key to
+the matching private key, and vice versa:
+*}
-axioms
- invKey [simp] : "invKey (invKey K) = K"
+types key = nat
+consts invKey :: "key=>key"
+(*<*)
+consts all_symmetric :: bool --{*true if all keys are symmetric*}
- (*The inverse of a symmetric key is itself;
- that of a public key is the private key and vice versa*)
+specification (invKey)
+ invKey [simp]: "invKey (invKey K) = K"
+ invKey_symmetric: "all_symmetric --> invKey = id"
+ by (rule exI [of _ id], auto)
+
+
+text{*The inverse of a symmetric key is itself; that of a public key
+ is the private key and vice versa*}
constdefs
symKeys :: "key set"
"symKeys == {K. invKey K = K}"
+(*>*)
-datatype (*We allow any number of friendly agents*)
- agent = Server | Friend nat | Spy
+text {*
+Datatype
+@{text msg} introduces the message forms, which include agent names, nonces,
+keys, compound messages, and encryptions.
+*}
datatype
- msg = Agent agent (*Agent names*)
- | Number nat (*Ordinary integers, timestamps, ...*)
- | Nonce nat (*Unguessable nonces*)
- | Key key (*Crypto keys*)
- | Hash msg (*Hashing*)
- | MPair msg msg (*Compound messages*)
- | Crypt key msg (*Encryption, public- or shared-key*)
+ msg = Agent agent
+ | Nonce nat
+ | Key key
+ | MPair msg msg
+ | Crypt key msg
+
+text {*
+\noindent
+The notation $\comp{X\sb 1,\ldots X\sb{n-1},X\sb n}$
+abbreviates
+$\isa{MPair}\,X\sb 1\,\ldots\allowbreak(\isa{MPair}\,X\sb{n-1}\,X\sb n)$.
+Since datatype constructors are injective, we have the theorem
+@{thm [display,indent=0] msg.inject(5) [THEN iffD1, of K X K' X']}
+A ciphertext can be decrypted using only one key and
+can yield only one plaintext. In the real world, decryption with the
+wrong key succeeds but yields garbage. Our model of encryption is
+realistic if encryption adds some redundancy to the plaintext, such as a
+checksum, so that garbage can be detected.
+*}
-(*Concrete syntax: messages appear as {|A,B,NA|}, etc...*)
+(*<*)
+text{*Concrete syntax: messages appear as {|A,B,NA|}, etc...*}
syntax
"@MTuple" :: "['a, args] => 'a * 'b" ("(2{|_,/ _|})")
@@ -56,18 +94,15 @@
constdefs
- (*Message Y, paired with a MAC computed with the help of X*)
- HPair :: "[msg,msg] => msg" ("(4Hash[_] /_)" [0, 1000])
- "Hash[X] Y == {| Hash{|X,Y|}, Y|}"
-
- (*Keys useful to decrypt elements of a message set*)
keysFor :: "msg set => key set"
+ --{*Keys useful to decrypt elements of a message set*}
"keysFor H == invKey ` {K. \<exists>X. Crypt K X \<in> H}"
-(** Inductive definition of all "parts" of a message. **)
+
+subsubsection{*Inductive Definition of All Parts" of a Message*}
inductive_set
- parts :: "msg set => msg set"
+ parts :: "msg set => msg set"
for H :: "msg set"
where
Inj [intro]: "X \<in> H ==> X \<in> parts H"
@@ -76,75 +111,825 @@
| Body: "Crypt K X \<in> parts H ==> X \<in> parts H"
-(*Monotonicity*)
-lemma parts_mono: "G<=H ==> parts(G) <= parts(H)"
+text{*Monotonicity*}
+lemma parts_mono: "G \<subseteq> H ==> parts(G) \<subseteq> parts(H)"
apply auto
apply (erule parts.induct)
-apply (auto dest: Fst Snd Body)
+apply (blast dest: parts.Fst parts.Snd parts.Body)+
+done
+
+
+text{*Equations hold because constructors are injective.*}
+lemma Friend_image_eq [simp]: "(Friend x \<in> Friend`A) = (x:A)"
+by auto
+
+lemma Key_image_eq [simp]: "(Key x \<in> Key`A) = (x\<in>A)"
+by auto
+
+lemma Nonce_Key_image_eq [simp]: "(Nonce x \<notin> Key`A)"
+by auto
+
+
+subsubsection{*Inverse of keys *}
+
+lemma invKey_eq [simp]: "(invKey K = invKey K') = (K=K')"
+apply safe
+apply (drule_tac f = invKey in arg_cong, simp)
+done
+
+
+subsection{*keysFor operator*}
+
+lemma keysFor_empty [simp]: "keysFor {} = {}"
+by (unfold keysFor_def, blast)
+
+lemma keysFor_Un [simp]: "keysFor (H \<union> H') = keysFor H \<union> keysFor H'"
+by (unfold keysFor_def, blast)
+
+lemma keysFor_UN [simp]: "keysFor (\<Union>i\<in>A. H i) = (\<Union>i\<in>A. keysFor (H i))"
+by (unfold keysFor_def, blast)
+
+text{*Monotonicity*}
+lemma keysFor_mono: "G \<subseteq> H ==> keysFor(G) \<subseteq> keysFor(H)"
+by (unfold keysFor_def, blast)
+
+lemma keysFor_insert_Agent [simp]: "keysFor (insert (Agent A) H) = keysFor H"
+by (unfold keysFor_def, auto)
+
+lemma keysFor_insert_Nonce [simp]: "keysFor (insert (Nonce N) H) = keysFor H"
+by (unfold keysFor_def, auto)
+
+lemma keysFor_insert_Key [simp]: "keysFor (insert (Key K) H) = keysFor H"
+by (unfold keysFor_def, auto)
+
+lemma keysFor_insert_MPair [simp]: "keysFor (insert {|X,Y|} H) = keysFor H"
+by (unfold keysFor_def, auto)
+
+lemma keysFor_insert_Crypt [simp]:
+ "keysFor (insert (Crypt K X) H) = insert (invKey K) (keysFor H)"
+by (unfold keysFor_def, auto)
+
+lemma keysFor_image_Key [simp]: "keysFor (Key`E) = {}"
+by (unfold keysFor_def, auto)
+
+lemma Crypt_imp_invKey_keysFor: "Crypt K X \<in> H ==> invKey K \<in> keysFor H"
+by (unfold keysFor_def, blast)
+
+
+subsection{*Inductive relation "parts"*}
+
+lemma MPair_parts:
+ "[| {|X,Y|} \<in> parts H;
+ [| X \<in> parts H; Y \<in> parts H |] ==> P |] ==> P"
+by (blast dest: parts.Fst parts.Snd)
+
+declare MPair_parts [elim!] parts.Body [dest!]
+text{*NB These two rules are UNSAFE in the formal sense, as they discard the
+ compound message. They work well on THIS FILE.
+ @{text MPair_parts} is left as SAFE because it speeds up proofs.
+ The Crypt rule is normally kept UNSAFE to avoid breaking up certificates.*}
+
+lemma parts_increasing: "H \<subseteq> parts(H)"
+by blast
+
+lemmas parts_insertI = subset_insertI [THEN parts_mono, THEN subsetD, standard]
+
+lemma parts_empty [simp]: "parts{} = {}"
+apply safe
+apply (erule parts.induct, blast+)
+done
+
+lemma parts_emptyE [elim!]: "X\<in> parts{} ==> P"
+by simp
+
+text{*WARNING: loops if H = {Y}, therefore must not be repeated!*}
+lemma parts_singleton: "X\<in> parts H ==> \<exists>Y\<in>H. X\<in> parts {Y}"
+by (erule parts.induct, blast+)
+
+
+subsubsection{*Unions *}
+
+lemma parts_Un_subset1: "parts(G) \<union> parts(H) \<subseteq> parts(G \<union> H)"
+by (intro Un_least parts_mono Un_upper1 Un_upper2)
+
+lemma parts_Un_subset2: "parts(G \<union> H) \<subseteq> parts(G) \<union> parts(H)"
+apply (rule subsetI)
+apply (erule parts.induct, blast+)
+done
+
+lemma parts_Un [simp]: "parts(G \<union> H) = parts(G) \<union> parts(H)"
+by (intro equalityI parts_Un_subset1 parts_Un_subset2)
+
+lemma parts_insert: "parts (insert X H) = parts {X} \<union> parts H"
+apply (subst insert_is_Un [of _ H])
+apply (simp only: parts_Un)
+done
+
+text{*TWO inserts to avoid looping. This rewrite is better than nothing.
+ Not suitable for Addsimps: its behaviour can be strange.*}
+lemma parts_insert2:
+ "parts (insert X (insert Y H)) = parts {X} \<union> parts {Y} \<union> parts H"
+apply (simp add: Un_assoc)
+apply (simp add: parts_insert [symmetric])
+done
+
+lemma parts_UN_subset1: "(\<Union>x\<in>A. parts(H x)) \<subseteq> parts(\<Union>x\<in>A. H x)"
+by (intro UN_least parts_mono UN_upper)
+
+lemma parts_UN_subset2: "parts(\<Union>x\<in>A. H x) \<subseteq> (\<Union>x\<in>A. parts(H x))"
+apply (rule subsetI)
+apply (erule parts.induct, blast+)
+done
+
+lemma parts_UN [simp]: "parts(\<Union>x\<in>A. H x) = (\<Union>x\<in>A. parts(H x))"
+by (intro equalityI parts_UN_subset1 parts_UN_subset2)
+
+text{*Added to simplify arguments to parts, analz and synth.
+ NOTE: the UN versions are no longer used!*}
+
+
+text{*This allows @{text blast} to simplify occurrences of
+ @{term "parts(G\<union>H)"} in the assumption.*}
+lemmas in_parts_UnE = parts_Un [THEN equalityD1, THEN subsetD, THEN UnE]
+declare in_parts_UnE [elim!]
+
+
+lemma parts_insert_subset: "insert X (parts H) \<subseteq> parts(insert X H)"
+by (blast intro: parts_mono [THEN [2] rev_subsetD])
+
+subsubsection{*Idempotence and transitivity *}
+
+lemma parts_partsD [dest!]: "X\<in> parts (parts H) ==> X\<in> parts H"
+by (erule parts.induct, blast+)
+
+lemma parts_idem [simp]: "parts (parts H) = parts H"
+by blast
+
+lemma parts_subset_iff [simp]: "(parts G \<subseteq> parts H) = (G \<subseteq> parts H)"
+apply (rule iffI)
+apply (iprover intro: subset_trans parts_increasing)
+apply (frule parts_mono, simp)
+done
+
+lemma parts_trans: "[| X\<in> parts G; G \<subseteq> parts H |] ==> X\<in> parts H"
+by (drule parts_mono, blast)
+
+text{*Cut*}
+lemma parts_cut:
+ "[| Y\<in> parts (insert X G); X\<in> parts H |] ==> Y\<in> parts (G \<union> H)"
+by (blast intro: parts_trans)
+
+
+lemma parts_cut_eq [simp]: "X\<in> parts H ==> parts (insert X H) = parts H"
+by (force dest!: parts_cut intro: parts_insertI)
+
+
+subsubsection{*Rewrite rules for pulling out atomic messages *}
+
+lemmas parts_insert_eq_I = equalityI [OF subsetI parts_insert_subset]
+
+
+lemma parts_insert_Agent [simp]:
+ "parts (insert (Agent agt) H) = insert (Agent agt) (parts H)"
+apply (rule parts_insert_eq_I)
+apply (erule parts.induct, auto)
+done
+
+lemma parts_insert_Nonce [simp]:
+ "parts (insert (Nonce N) H) = insert (Nonce N) (parts H)"
+apply (rule parts_insert_eq_I)
+apply (erule parts.induct, auto)
+done
+
+lemma parts_insert_Key [simp]:
+ "parts (insert (Key K) H) = insert (Key K) (parts H)"
+apply (rule parts_insert_eq_I)
+apply (erule parts.induct, auto)
+done
+
+lemma parts_insert_Crypt [simp]:
+ "parts (insert (Crypt K X) H) = insert (Crypt K X) (parts (insert X H))"
+apply (rule equalityI)
+apply (rule subsetI)
+apply (erule parts.induct, auto)
+apply (blast intro: parts.Body)
+done
+
+lemma parts_insert_MPair [simp]:
+ "parts (insert {|X,Y|} H) =
+ insert {|X,Y|} (parts (insert X (insert Y H)))"
+apply (rule equalityI)
+apply (rule subsetI)
+apply (erule parts.induct, auto)
+apply (blast intro: parts.Fst parts.Snd)+
+done
+
+lemma parts_image_Key [simp]: "parts (Key`N) = Key`N"
+apply auto
+apply (erule parts.induct, auto)
done
-(** Inductive definition of "analz" -- what can be broken down from a set of
- messages, including keys. A form of downward closure. Pairs can
- be taken apart; messages decrypted with known keys. **)
+text{*In any message, there is an upper bound N on its greatest nonce.*}
+lemma msg_Nonce_supply: "\<exists>N. \<forall>n. N\<le>n --> Nonce n \<notin> parts {msg}"
+apply (induct_tac "msg")
+apply (simp_all (no_asm_simp) add: exI parts_insert2)
+ txt{*MPair case: blast works out the necessary sum itself!*}
+ prefer 2 apply auto apply (blast elim!: add_leE)
+txt{*Nonce case*}
+apply (rule_tac x = "N + Suc nat" in exI, auto)
+done
+(*>*)
+
+section{* Modelling the Adversary *}
+
+text {*
+The spy is part of the system and must be built into the model. He is
+a malicious user who does not have to follow the protocol. He
+watches the network and uses any keys he knows to decrypt messages.
+Thus he accumulates additional keys and nonces. These he can use to
+compose new messages, which he may send to anybody.
+
+Two functions enable us to formalize this behaviour: @{text analz} and
+@{text synth}. Each function maps a sets of messages to another set of
+messages. The set @{text "analz H"} formalizes what the adversary can learn
+from the set of messages~$H$. The closure properties of this set are
+defined inductively.
+*}
inductive_set
- analz :: "msg set => msg set"
+ analz :: "msg set \<Rightarrow> msg set"
for H :: "msg set"
where
- Inj [intro,simp] : "X \<in> H ==> X \<in> analz H"
- | Fst: "{|X,Y|} \<in> analz H ==> X \<in> analz H"
- | Snd: "{|X,Y|} \<in> analz H ==> Y \<in> analz H"
+ Inj [intro,simp] : "X \<in> H \<Longrightarrow> X \<in> analz H"
+ | Fst: "\<lbrace>X,Y\<rbrace> \<in> analz H \<Longrightarrow> X \<in> analz H"
+ | Snd: "\<lbrace>X,Y\<rbrace> \<in> analz H \<Longrightarrow> Y \<in> analz H"
| Decrypt [dest]:
- "[|Crypt K X \<in> analz H; Key(invKey K) \<in> analz H|] ==> X \<in> analz H"
+ "\<lbrakk>Crypt K X \<in> analz H; Key(invKey K) \<in> analz H\<rbrakk>
+ \<Longrightarrow> X \<in> analz H"
+(*<*)
+text{*Monotonicity; Lemma 1 of Lowe's paper*}
+lemma analz_mono: "G\<subseteq>H ==> analz(G) \<subseteq> analz(H)"
+apply auto
+apply (erule analz.induct)
+apply (auto dest: analz.Fst analz.Snd)
+done
+
+text{*Making it safe speeds up proofs*}
+lemma MPair_analz [elim!]:
+ "[| {|X,Y|} \<in> analz H;
+ [| X \<in> analz H; Y \<in> analz H |] ==> P
+ |] ==> P"
+by (blast dest: analz.Fst analz.Snd)
+
+lemma analz_increasing: "H \<subseteq> analz(H)"
+by blast
+
+lemma analz_subset_parts: "analz H \<subseteq> parts H"
+apply (rule subsetI)
+apply (erule analz.induct, blast+)
+done
+
+lemmas analz_into_parts = analz_subset_parts [THEN subsetD, standard]
+
+lemmas not_parts_not_analz = analz_subset_parts [THEN contra_subsetD, standard]
-(*Monotonicity; Lemma 1 of Lowe's paper*)
-lemma analz_mono: "G<=H ==> analz(G) <= analz(H)"
+lemma parts_analz [simp]: "parts (analz H) = parts H"
+apply (rule equalityI)
+apply (rule analz_subset_parts [THEN parts_mono, THEN subset_trans], simp)
+apply (blast intro: analz_increasing [THEN parts_mono, THEN subsetD])
+done
+
+lemma analz_parts [simp]: "analz (parts H) = parts H"
apply auto
-apply (erule analz.induct)
-apply (auto dest: Fst Snd)
+apply (erule analz.induct, auto)
+done
+
+lemmas analz_insertI = subset_insertI [THEN analz_mono, THEN [2] rev_subsetD, standard]
+
+subsubsection{*General equational properties *}
+
+lemma analz_empty [simp]: "analz{} = {}"
+apply safe
+apply (erule analz.induct, blast+)
+done
+
+text{*Converse fails: we can analz more from the union than from the
+ separate parts, as a key in one might decrypt a message in the other*}
+lemma analz_Un: "analz(G) \<union> analz(H) \<subseteq> analz(G \<union> H)"
+by (intro Un_least analz_mono Un_upper1 Un_upper2)
+
+lemma analz_insert: "insert X (analz H) \<subseteq> analz(insert X H)"
+by (blast intro: analz_mono [THEN [2] rev_subsetD])
+
+subsubsection{*Rewrite rules for pulling out atomic messages *}
+
+lemmas analz_insert_eq_I = equalityI [OF subsetI analz_insert]
+
+lemma analz_insert_Agent [simp]:
+ "analz (insert (Agent agt) H) = insert (Agent agt) (analz H)"
+apply (rule analz_insert_eq_I)
+apply (erule analz.induct, auto)
+done
+
+lemma analz_insert_Nonce [simp]:
+ "analz (insert (Nonce N) H) = insert (Nonce N) (analz H)"
+apply (rule analz_insert_eq_I)
+apply (erule analz.induct, auto)
+done
+
+text{*Can only pull out Keys if they are not needed to decrypt the rest*}
+lemma analz_insert_Key [simp]:
+ "K \<notin> keysFor (analz H) ==>
+ analz (insert (Key K) H) = insert (Key K) (analz H)"
+apply (unfold keysFor_def)
+apply (rule analz_insert_eq_I)
+apply (erule analz.induct, auto)
+done
+
+lemma analz_insert_MPair [simp]:
+ "analz (insert {|X,Y|} H) =
+ insert {|X,Y|} (analz (insert X (insert Y H)))"
+apply (rule equalityI)
+apply (rule subsetI)
+apply (erule analz.induct, auto)
+apply (erule analz.induct)
+apply (blast intro: analz.Fst analz.Snd)+
+done
+
+text{*Can pull out enCrypted message if the Key is not known*}
+lemma analz_insert_Crypt:
+ "Key (invKey K) \<notin> analz H
+ ==> analz (insert (Crypt K X) H) = insert (Crypt K X) (analz H)"
+apply (rule analz_insert_eq_I)
+apply (erule analz.induct, auto)
+
done
-(** Inductive definition of "synth" -- what can be built up from a set of
- messages. A form of upward closure. Pairs can be built, messages
- encrypted with known keys. Agent names are public domain.
- Numbers can be guessed, but Nonces cannot be. **)
+lemma lemma1: "Key (invKey K) \<in> analz H ==>
+ analz (insert (Crypt K X) H) \<subseteq>
+ insert (Crypt K X) (analz (insert X H))"
+apply (rule subsetI)
+apply (erule_tac x = x in analz.induct, auto)
+done
+
+lemma lemma2: "Key (invKey K) \<in> analz H ==>
+ insert (Crypt K X) (analz (insert X H)) \<subseteq>
+ analz (insert (Crypt K X) H)"
+apply auto
+apply (erule_tac x = x in analz.induct, auto)
+apply (blast intro: analz_insertI analz.Decrypt)
+done
+
+lemma analz_insert_Decrypt:
+ "Key (invKey K) \<in> analz H ==>
+ analz (insert (Crypt K X) H) =
+ insert (Crypt K X) (analz (insert X H))"
+by (intro equalityI lemma1 lemma2)
+
+text{*Case analysis: either the message is secure, or it is not! Effective,
+but can cause subgoals to blow up! Use with @{text "split_if"}; apparently
+@{text "split_tac"} does not cope with patterns such as @{term"analz (insert
+(Crypt K X) H)"} *}
+lemma analz_Crypt_if [simp]:
+ "analz (insert (Crypt K X) H) =
+ (if (Key (invKey K) \<in> analz H)
+ then insert (Crypt K X) (analz (insert X H))
+ else insert (Crypt K X) (analz H))"
+by (simp add: analz_insert_Crypt analz_insert_Decrypt)
+
+
+text{*This rule supposes "for the sake of argument" that we have the key.*}
+lemma analz_insert_Crypt_subset:
+ "analz (insert (Crypt K X) H) \<subseteq>
+ insert (Crypt K X) (analz (insert X H))"
+apply (rule subsetI)
+apply (erule analz.induct, auto)
+done
+
+
+lemma analz_image_Key [simp]: "analz (Key`N) = Key`N"
+apply auto
+apply (erule analz.induct, auto)
+done
+
+
+subsubsection{*Idempotence and transitivity *}
+
+lemma analz_analzD [dest!]: "X\<in> analz (analz H) ==> X\<in> analz H"
+by (erule analz.induct, blast+)
+
+lemma analz_idem [simp]: "analz (analz H) = analz H"
+by blast
+
+lemma analz_subset_iff [simp]: "(analz G \<subseteq> analz H) = (G \<subseteq> analz H)"
+apply (rule iffI)
+apply (iprover intro: subset_trans analz_increasing)
+apply (frule analz_mono, simp)
+done
+
+lemma analz_trans: "[| X\<in> analz G; G \<subseteq> analz H |] ==> X\<in> analz H"
+by (drule analz_mono, blast)
+
+text{*Cut; Lemma 2 of Lowe*}
+lemma analz_cut: "[| Y\<in> analz (insert X H); X\<in> analz H |] ==> Y\<in> analz H"
+by (erule analz_trans, blast)
+
+(*Cut can be proved easily by induction on
+ "Y: analz (insert X H) ==> X: analz H --> Y: analz H"
+*)
+
+text{*This rewrite rule helps in the simplification of messages that involve
+ the forwarding of unknown components (X). Without it, removing occurrences
+ of X can be very complicated. *}
+lemma analz_insert_eq: "X\<in> analz H ==> analz (insert X H) = analz H"
+by (blast intro: analz_cut analz_insertI)
+
+
+text{*A congruence rule for "analz" *}
+
+lemma analz_subset_cong:
+ "[| analz G \<subseteq> analz G'; analz H \<subseteq> analz H' |]
+ ==> analz (G \<union> H) \<subseteq> analz (G' \<union> H')"
+apply simp
+apply (iprover intro: conjI subset_trans analz_mono Un_upper1 Un_upper2)
+done
+
+lemma analz_cong:
+ "[| analz G = analz G'; analz H = analz H' |]
+ ==> analz (G \<union> H) = analz (G' \<union> H')"
+by (intro equalityI analz_subset_cong, simp_all)
+
+lemma analz_insert_cong:
+ "analz H = analz H' ==> analz(insert X H) = analz(insert X H')"
+by (force simp only: insert_def intro!: analz_cong)
+
+text{*If there are no pairs or encryptions then analz does nothing*}
+lemma analz_trivial:
+ "[| \<forall>X Y. {|X,Y|} \<notin> H; \<forall>X K. Crypt K X \<notin> H |] ==> analz H = H"
+apply safe
+apply (erule analz.induct, blast+)
+done
+
+text{*These two are obsolete (with a single Spy) but cost little to prove...*}
+lemma analz_UN_analz_lemma:
+ "X\<in> analz (\<Union>i\<in>A. analz (H i)) ==> X\<in> analz (\<Union>i\<in>A. H i)"
+apply (erule analz.induct)
+apply (blast intro: analz_mono [THEN [2] rev_subsetD])+
+done
+
+lemma analz_UN_analz [simp]: "analz (\<Union>i\<in>A. analz (H i)) = analz (\<Union>i\<in>A. H i)"
+by (blast intro: analz_UN_analz_lemma analz_mono [THEN [2] rev_subsetD])
+(*>*)
+text {*
+Note the @{text Decrypt} rule: the spy can decrypt a
+message encrypted with key~$K$ if he has the matching key,~$K^{-1}$.
+Properties proved by rule induction include the following:
+@{named_thms [display,indent=0] analz_mono [no_vars] (analz_mono) analz_idem [no_vars] (analz_idem)}
+
+The set of fake messages that an intruder could invent
+starting from~@{text H} is @{text "synth(analz H)"}, where @{text "synth H"}
+formalizes what the adversary can build from the set of messages~$H$.
+*}
inductive_set
- synth :: "msg set => msg set"
+ synth :: "msg set \<Rightarrow> msg set"
for H :: "msg set"
where
- Inj [intro]: "X \<in> H ==> X \<in> synth H"
- | Agent [intro]: "Agent agt \<in> synth H"
- | Number [intro]: "Number n \<in> synth H"
- | Hash [intro]: "X \<in> synth H ==> Hash X \<in> synth H"
- | MPair [intro]: "[|X \<in> synth H; Y \<in> synth H|] ==> {|X,Y|} \<in> synth H"
- | Crypt [intro]: "[|X \<in> synth H; Key(K) \<in> H|] ==> Crypt K X \<in> synth H"
+ Inj [intro]: "X \<in> H \<Longrightarrow> X \<in> synth H"
+ | Agent [intro]: "Agent agt \<in> synth H"
+ | MPair [intro]:
+ "\<lbrakk>X \<in> synth H; Y \<in> synth H\<rbrakk> \<Longrightarrow> \<lbrace>X,Y\<rbrace> \<in> synth H"
+ | Crypt [intro]:
+ "\<lbrakk>X \<in> synth H; Key(K) \<in> H\<rbrakk> \<Longrightarrow> Crypt K X \<in> synth H"
+(*<*)
+lemma synth_mono: "G\<subseteq>H ==> synth(G) \<subseteq> synth(H)"
+ by (auto, erule synth.induct, auto)
-(*Monotonicity*)
-lemma synth_mono: "G<=H ==> synth(G) <= synth(H)"
-apply auto
-apply (erule synth.induct)
-apply (auto dest: Fst Snd Body)
-done
-
-(*NO Agent_synth, as any Agent name can be synthesized. Ditto for Number*)
-inductive_cases Nonce_synth [elim!]: "Nonce n \<in> synth H"
inductive_cases Key_synth [elim!]: "Key K \<in> synth H"
-inductive_cases Hash_synth [elim!]: "Hash X \<in> synth H"
inductive_cases MPair_synth [elim!]: "{|X,Y|} \<in> synth H"
inductive_cases Crypt_synth [elim!]: "Crypt K X \<in> synth H"
-use "Message_lemmas.ML"
+lemma analz_synth_Un [simp]: "analz (synth G \<union> H) = analz (G \<union> H) \<union> synth G"
+apply (rule equalityI)
+apply (rule subsetI)
+apply (erule analz.induct)
+prefer 5 apply (blast intro: analz_mono [THEN [2] rev_subsetD])
+apply (blast intro: analz.Fst analz.Snd analz.Decrypt)+
+done
+
+lemma analz_synth [simp]: "analz (synth H) = analz H \<union> synth H"
+apply (cut_tac H = "{}" in analz_synth_Un)
+apply (simp (no_asm_use))
+done
+(*>*)
+text {*
+The set includes all agent names. Nonces and keys are assumed to be
+unguessable, so none are included beyond those already in~$H$. Two
+elements of @{term "synth H"} can be combined, and an element can be encrypted
+using a key present in~$H$.
+
+Like @{text analz}, this set operator is monotone and idempotent. It also
+satisfies an interesting equation involving @{text analz}:
+@{named_thms [display,indent=0] analz_synth [no_vars] (analz_synth)}
+Rule inversion plays a major role in reasoning about @{text synth}, through
+declarations such as this one:
+*}
+
+inductive_cases Nonce_synth [elim!]: "Nonce n \<in> synth H"
+
+text {*
+\noindent
+The resulting elimination rule replaces every assumption of the form
+@{term "Nonce n \<in> synth H"} by @{term "Nonce n \<in> H"},
+expressing that a nonce cannot be guessed.
+
+A third operator, @{text parts}, is useful for stating correctness
+properties. The set
+@{term "parts H"} consists of the components of elements of~$H$. This set
+includes~@{text H} and is closed under the projections from a compound
+message to its immediate parts.
+Its definition resembles that of @{text analz} except in the rule
+corresponding to the constructor @{text Crypt}:
+@{thm [display,indent=5] parts.Body [no_vars]}
+The body of an encrypted message is always regarded as part of it. We can
+use @{text parts} to express general well-formedness properties of a protocol,
+for example, that an uncompromised agent's private key will never be
+included as a component of any message.
+*}
+(*<*)
+lemma synth_increasing: "H \<subseteq> synth(H)"
+by blast
+
+subsubsection{*Unions *}
+
+text{*Converse fails: we can synth more from the union than from the
+ separate parts, building a compound message using elements of each.*}
+lemma synth_Un: "synth(G) \<union> synth(H) \<subseteq> synth(G \<union> H)"
+by (intro Un_least synth_mono Un_upper1 Un_upper2)
+
+lemma synth_insert: "insert X (synth H) \<subseteq> synth(insert X H)"
+by (blast intro: synth_mono [THEN [2] rev_subsetD])
+
+subsubsection{*Idempotence and transitivity *}
+
+lemma synth_synthD [dest!]: "X\<in> synth (synth H) ==> X\<in> synth H"
+by (erule synth.induct, blast+)
+
+lemma synth_idem: "synth (synth H) = synth H"
+by blast
+
+lemma synth_subset_iff [simp]: "(synth G \<subseteq> synth H) = (G \<subseteq> synth H)"
+apply (rule iffI)
+apply (iprover intro: subset_trans synth_increasing)
+apply (frule synth_mono, simp add: synth_idem)
+done
+
+lemma synth_trans: "[| X\<in> synth G; G \<subseteq> synth H |] ==> X\<in> synth H"
+by (drule synth_mono, blast)
+
+text{*Cut; Lemma 2 of Lowe*}
+lemma synth_cut: "[| Y\<in> synth (insert X H); X\<in> synth H |] ==> Y\<in> synth H"
+by (erule synth_trans, blast)
+
+lemma Agent_synth [simp]: "Agent A \<in> synth H"
+by blast
+
+lemma Nonce_synth_eq [simp]: "(Nonce N \<in> synth H) = (Nonce N \<in> H)"
+by blast
+
+lemma Key_synth_eq [simp]: "(Key K \<in> synth H) = (Key K \<in> H)"
+by blast
+
+lemma Crypt_synth_eq [simp]:
+ "Key K \<notin> H ==> (Crypt K X \<in> synth H) = (Crypt K X \<in> H)"
+by blast
+
+
+lemma keysFor_synth [simp]:
+ "keysFor (synth H) = keysFor H \<union> invKey`{K. Key K \<in> H}"
+by (unfold keysFor_def, blast)
+
+
+subsubsection{*Combinations of parts, analz and synth *}
+
+lemma parts_synth [simp]: "parts (synth H) = parts H \<union> synth H"
+apply (rule equalityI)
+apply (rule subsetI)
+apply (erule parts.induct)
+apply (blast intro: synth_increasing [THEN parts_mono, THEN subsetD]
+ parts.Fst parts.Snd parts.Body)+
+done
+
+lemma analz_analz_Un [simp]: "analz (analz G \<union> H) = analz (G \<union> H)"
+apply (intro equalityI analz_subset_cong)+
+apply simp_all
+done
+
+
+subsubsection{*For reasoning about the Fake rule in traces *}
+
+lemma parts_insert_subset_Un: "X\<in> G ==> parts(insert X H) \<subseteq> parts G \<union> parts H"
+by (rule subset_trans [OF parts_mono parts_Un_subset2], blast)
+
+text{*More specifically for Fake. Very occasionally we could do with a version
+ of the form @{term"parts{X} \<subseteq> synth (analz H) \<union> parts H"} *}
+lemma Fake_parts_insert:
+ "X \<in> synth (analz H) ==>
+ parts (insert X H) \<subseteq> synth (analz H) \<union> parts H"
+apply (drule parts_insert_subset_Un)
+apply (simp (no_asm_use))
+apply blast
+done
lemma Fake_parts_insert_in_Un:
"[|Z \<in> parts (insert X H); X: synth (analz H)|]
==> Z \<in> synth (analz H) \<union> parts H";
by (blast dest: Fake_parts_insert [THEN subsetD, dest])
+text{*@{term H} is sometimes @{term"Key ` KK \<union> spies evs"}, so can't put
+ @{term "G=H"}.*}
+lemma Fake_analz_insert:
+ "X\<in> synth (analz G) ==>
+ analz (insert X H) \<subseteq> synth (analz G) \<union> analz (G \<union> H)"
+apply (rule subsetI)
+apply (subgoal_tac "x \<in> analz (synth (analz G) \<union> H) ")
+prefer 2 apply (blast intro: analz_mono [THEN [2] rev_subsetD] analz_mono [THEN synth_mono, THEN [2] rev_subsetD])
+apply (simp (no_asm_use))
+apply blast
+done
+
+lemma analz_conj_parts [simp]:
+ "(X \<in> analz H & X \<in> parts H) = (X \<in> analz H)"
+by (blast intro: analz_subset_parts [THEN subsetD])
+
+lemma analz_disj_parts [simp]:
+ "(X \<in> analz H | X \<in> parts H) = (X \<in> parts H)"
+by (blast intro: analz_subset_parts [THEN subsetD])
+
+text{*Without this equation, other rules for synth and analz would yield
+ redundant cases*}
+lemma MPair_synth_analz [iff]:
+ "({|X,Y|} \<in> synth (analz H)) =
+ (X \<in> synth (analz H) & Y \<in> synth (analz H))"
+by blast
+
+lemma Crypt_synth_analz:
+ "[| Key K \<in> analz H; Key (invKey K) \<in> analz H |]
+ ==> (Crypt K X \<in> synth (analz H)) = (X \<in> synth (analz H))"
+by blast
+
+
+text{*We do NOT want Crypt... messages broken up in protocols!!*}
+declare parts.Body [rule del]
+
+
+text{*Rewrites to push in Key and Crypt messages, so that other messages can
+ be pulled out using the @{text analz_insert} rules*}
+ML
+{*
+fun insComm x y = inst "x" x (inst "y" y insert_commute);
+
+bind_thms ("pushKeys",
+ map (insComm "Key ?K")
+ ["Agent ?C", "Nonce ?N", "Number ?N",
+ "Hash ?X", "MPair ?X ?Y", "Crypt ?X ?K'"]);
+
+bind_thms ("pushCrypts",
+ map (insComm "Crypt ?X ?K")
+ ["Agent ?C", "Nonce ?N", "Number ?N",
+ "Hash ?X'", "MPair ?X' ?Y"]);
+*}
+
+text{*Cannot be added with @{text "[simp]"} -- messages should not always be
+ re-ordered. *}
+lemmas pushes = pushKeys pushCrypts
+
+
+subsection{*Tactics useful for many protocol proofs*}
+ML
+{*
+val invKey = thm "invKey"
+val keysFor_def = thm "keysFor_def"
+val symKeys_def = thm "symKeys_def"
+val parts_mono = thm "parts_mono";
+val analz_mono = thm "analz_mono";
+val synth_mono = thm "synth_mono";
+val analz_increasing = thm "analz_increasing";
+
+val analz_insertI = thm "analz_insertI";
+val analz_subset_parts = thm "analz_subset_parts";
+val Fake_parts_insert = thm "Fake_parts_insert";
+val Fake_analz_insert = thm "Fake_analz_insert";
+val pushes = thms "pushes";
+
+
+(*Prove base case (subgoal i) and simplify others. A typical base case
+ concerns Crypt K X \<notin> Key`shrK`bad and cannot be proved by rewriting
+ alone.*)
+fun prove_simple_subgoals_tac i =
+ force_tac (claset(), simpset() addsimps [@{thm image_eq_UN}]) i THEN
+ ALLGOALS Asm_simp_tac
+
+(*Analysis of Fake cases. Also works for messages that forward unknown parts,
+ but this application is no longer necessary if analz_insert_eq is used.
+ Abstraction over i is ESSENTIAL: it delays the dereferencing of claset
+ DEPENDS UPON "X" REFERRING TO THE FRADULENT MESSAGE *)
+
+(*Apply rules to break down assumptions of the form
+ Y \<in> parts(insert X H) and Y \<in> analz(insert X H)
+*)
+val Fake_insert_tac =
+ dresolve_tac [impOfSubs Fake_analz_insert,
+ impOfSubs Fake_parts_insert] THEN'
+ eresolve_tac [asm_rl, thm"synth.Inj"];
+
+fun Fake_insert_simp_tac ss i =
+ REPEAT (Fake_insert_tac i) THEN asm_full_simp_tac ss i;
+
+fun atomic_spy_analz_tac (cs,ss) = SELECT_GOAL
+ (Fake_insert_simp_tac ss 1
+ THEN
+ IF_UNSOLVED (Blast.depth_tac
+ (cs addIs [analz_insertI,
+ impOfSubs analz_subset_parts]) 4 1))
+
+(*The explicit claset and simpset arguments help it work with Isar*)
+fun gen_spy_analz_tac (cs,ss) i =
+ DETERM
+ (SELECT_GOAL
+ (EVERY
+ [ (*push in occurrences of X...*)
+ (REPEAT o CHANGED)
+ (res_inst_tac [("x1","X")] (insert_commute RS ssubst) 1),
+ (*...allowing further simplifications*)
+ simp_tac ss 1,
+ REPEAT (FIRSTGOAL (resolve_tac [allI,impI,notI,conjI,iffI])),
+ DEPTH_SOLVE (atomic_spy_analz_tac (cs,ss) 1)]) i)
+
+fun spy_analz_tac i = gen_spy_analz_tac (claset(), simpset()) i
+*}
+
+text{*By default only @{text o_apply} is built-in. But in the presence of
+eta-expansion this means that some terms displayed as @{term "f o g"} will be
+rewritten, and others will not!*}
+declare o_def [simp]
+
+
+lemma Crypt_notin_image_Key [simp]: "Crypt K X \<notin> Key ` A"
+by auto
+
+lemma synth_analz_mono: "G\<subseteq>H ==> synth (analz(G)) \<subseteq> synth (analz(H))"
+by (iprover intro: synth_mono analz_mono)
+
+lemma Fake_analz_eq [simp]:
+ "X \<in> synth(analz H) ==> synth (analz (insert X H)) = synth (analz H)"
+apply (drule Fake_analz_insert[of _ _ "H"])
+apply (simp add: synth_increasing[THEN Un_absorb2])
+apply (drule synth_mono)
+apply (simp add: synth_idem)
+apply (rule equalityI)
+apply (simp add: );
+apply (rule synth_analz_mono, blast)
+done
+
+text{*Two generalizations of @{text analz_insert_eq}*}
+lemma gen_analz_insert_eq [rule_format]:
+ "X \<in> analz H ==> ALL G. H \<subseteq> G --> analz (insert X G) = analz G";
+by (blast intro: analz_cut analz_insertI analz_mono [THEN [2] rev_subsetD])
+
+lemma synth_analz_insert_eq [rule_format]:
+ "X \<in> synth (analz H)
+ ==> ALL G. H \<subseteq> G --> (Key K \<in> analz (insert X G)) = (Key K \<in> analz G)";
+apply (erule synth.induct)
+apply (simp_all add: gen_analz_insert_eq subset_trans [OF _ subset_insertI])
+done
+
+lemma Fake_parts_sing:
+ "X \<in> synth (analz H) ==> parts{X} \<subseteq> synth (analz H) \<union> parts H";
+apply (rule subset_trans)
+ apply (erule_tac [2] Fake_parts_insert)
+apply (rule parts_mono, blast)
+done
+
+lemmas Fake_parts_sing_imp_Un = Fake_parts_sing [THEN [2] rev_subsetD]
+
method_setup spy_analz = {*
- Method.no_args (Method.METHOD (fn facts => spy_analz_tac 1)) *}
+ Method.ctxt_args (fn ctxt =>
+ Method.SIMPLE_METHOD (gen_spy_analz_tac (local_clasimpset_of ctxt) 1)) *}
"for proving the Fake case when analz is involved"
+method_setup atomic_spy_analz = {*
+ Method.ctxt_args (fn ctxt =>
+ Method.SIMPLE_METHOD (atomic_spy_analz_tac (local_clasimpset_of ctxt) 1)) *}
+ "for debugging spy_analz"
+
+method_setup Fake_insert_simp = {*
+ Method.ctxt_args (fn ctxt =>
+ Method.SIMPLE_METHOD (Fake_insert_simp_tac (local_simpset_of ctxt) 1)) *}
+ "for debugging spy_analz"
+
+
end
+(*>*)
\ No newline at end of file