--- a/src/HOL/SET-Protocol/MessageSET.thy Tue Oct 20 19:52:04 2009 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,957 +0,0 @@
-(* Title: HOL/SET-Protocol/MessageSET.thy
- Author: Giampaolo Bella
- Author: Fabio Massacci
- Author: Lawrence C Paulson
-*)
-
-header{*The Message Theory, Modified for SET*}
-
-theory MessageSET
-imports Main Nat_Int_Bij
-begin
-
-subsection{*General Lemmas*}
-
-text{*Needed occasionally with @{text spy_analz_tac}, e.g. in
- @{text analz_insert_Key_newK}*}
-
-lemma Un_absorb3 [simp] : "A \<union> (B \<union> A) = B \<union> A"
-by blast
-
-text{*Collapses redundant cases in the huge protocol proofs*}
-lemmas disj_simps = disj_comms disj_left_absorb disj_assoc
-
-text{*Effective with assumptions like @{term "K \<notin> range pubK"} and
- @{term "K \<notin> invKey`range pubK"}*}
-lemma notin_image_iff: "(y \<notin> f`I) = (\<forall>i\<in>I. f i \<noteq> y)"
-by blast
-
-text{*Effective with the assumption @{term "KK \<subseteq> - (range(invKey o pubK))"} *}
-lemma disjoint_image_iff: "(A <= - (f`I)) = (\<forall>i\<in>I. f i \<notin> A)"
-by blast
-
-
-
-types
- key = nat
-
-consts
- all_symmetric :: bool --{*true if all keys are symmetric*}
- invKey :: "key=>key" --{*inverse of a symmetric key*}
-
-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}"
-
-text{*Agents. We allow any number of certification authorities, cardholders
- merchants, and payment gateways.*}
-datatype
- agent = CA nat | Cardholder nat | Merchant nat | PG nat | Spy
-
-text{*Messages*}
-datatype
- msg = Agent agent --{*Agent names*}
- | Number nat --{*Ordinary integers, timestamps, ...*}
- | Nonce nat --{*Unguessable nonces*}
- | Pan nat --{*Unguessable Primary Account Numbers (??)*}
- | Key key --{*Crypto keys*}
- | Hash msg --{*Hashing*}
- | MPair msg msg --{*Compound messages*}
- | Crypt key msg --{*Encryption, public- or shared-key*}
-
-
-(*Concrete syntax: messages appear as {|A,B,NA|}, etc...*)
-syntax
- "@MTuple" :: "['a, args] => 'a * 'b" ("(2{|_,/ _|})")
-
-syntax (xsymbols)
- "@MTuple" :: "['a, args] => 'a * 'b" ("(2\<lbrace>_,/ _\<rbrace>)")
-
-translations
- "{|x, y, z|}" == "{|x, {|y, z|}|}"
- "{|x, y|}" == "MPair x y"
-
-
-constdefs
- nat_of_agent :: "agent => nat"
- "nat_of_agent == agent_case (curry nat2_to_nat 0)
- (curry nat2_to_nat 1)
- (curry nat2_to_nat 2)
- (curry nat2_to_nat 3)
- (nat2_to_nat (4,0))"
- --{*maps each agent to a unique natural number, for specifications*}
-
-text{*The function is indeed injective*}
-lemma inj_nat_of_agent: "inj nat_of_agent"
-by (simp add: nat_of_agent_def inj_on_def curry_def
- nat2_to_nat_inj [THEN inj_eq] split: agent.split)
-
-
-constdefs
- (*Keys useful to decrypt elements of a message set*)
- keysFor :: "msg set => key set"
- "keysFor H == invKey ` {K. \<exists>X. Crypt K X \<in> H}"
-
-subsubsection{*Inductive definition of all "parts" of a message.*}
-
-inductive_set
- parts :: "msg set => msg set"
- for H :: "msg set"
- where
- Inj [intro]: "X \<in> H ==> X \<in> parts H"
- | Fst: "{|X,Y|} \<in> parts H ==> X \<in> parts H"
- | Snd: "{|X,Y|} \<in> parts H ==> Y \<in> parts H"
- | Body: "Crypt K X \<in> parts H ==> X \<in> parts H"
-
-
-(*Monotonicity*)
-lemma parts_mono: "G<=H ==> parts(G) <= parts(H)"
-apply auto
-apply (erule parts.induct)
-apply (auto dest: Fst Snd Body)
-done
-
-
-subsubsection{*Inverse of keys*}
-
-(*Equations hold because constructors are injective; cannot prove for all f*)
-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
-
-lemma Cardholder_image_eq [simp]: "(Cardholder x \<in> Cardholder`A) = (x \<in> A)"
-by auto
-
-lemma CA_image_eq [simp]: "(CA x \<in> CA`A) = (x \<in> A)"
-by auto
-
-lemma Pan_image_eq [simp]: "(Pan x \<in> Pan`A) = (x \<in> A)"
-by auto
-
-lemma Pan_Key_image_eq [simp]: "(Pan x \<notin> Key`A)"
-by auto
-
-lemma Nonce_Pan_image_eq [simp]: "(Nonce x \<notin> Pan`A)"
-by auto
-
-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)
-
-(*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_Number [simp]: "keysFor (insert (Number 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_Pan [simp]: "keysFor (insert (Pan A) H) = keysFor H"
-by (unfold keysFor_def, auto)
-
-lemma keysFor_insert_Hash [simp]: "keysFor (insert (Hash X) 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
-
-(*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, fast+)
-
-
-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
-
-(*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)
-
-(*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.*}
-declare parts_Un [THEN equalityD1, THEN subsetD, THEN 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_trans: "[| X\<in> parts G; G \<subseteq> parts H |] ==> X\<in> parts H"
-by (drule parts_mono, blast)
-
-(*Cut*)
-lemma parts_cut:
- "[| Y\<in> parts (insert X G); X\<in> parts H |] ==> Y\<in> parts (G \<union> H)"
-by (erule parts_trans, auto)
-
-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_Number [simp]:
- "parts (insert (Number N) H) = insert (Number 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_Pan [simp]:
- "parts (insert (Pan A) H) = insert (Pan A) (parts H)"
-apply (rule parts_insert_eq_I)
-apply (erule parts.induct, auto)
-done
-
-lemma parts_insert_Hash [simp]:
- "parts (insert (Hash X) H) = insert (Hash X) (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 (erule parts.induct)
-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 (erule parts.induct)
-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
-
-lemma parts_image_Pan [simp]: "parts (Pan`A) = Pan`A"
-apply auto
-apply (erule parts.induct, auto)
-done
-
-
-(*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)
-(*MPair case: blast_tac works out the necessary sum itself!*)
-prefer 2 apply (blast elim!: add_leE)
-(*Nonce case*)
-apply (rule_tac x = "N + Suc nat" in exI)
-apply (auto elim!: add_leE)
-done
-
-(* Ditto, for numbers.*)
-lemma msg_Number_supply: "\<exists>N. \<forall>n. N<=n --> Number n \<notin> parts {msg}"
-apply (induct_tac "msg")
-apply (simp_all (no_asm_simp) add: exI parts_insert2)
-prefer 2 apply (blast elim!: add_leE)
-apply (rule_tac x = "N + Suc nat" in exI, auto)
-done
-
-subsection{*Inductive relation "analz"*}
-
-text{*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.*}
-
-inductive_set
- analz :: "msg set => 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"
- | Decrypt [dest]:
- "[|Crypt K X \<in> analz H; Key(invKey K): analz H|] ==> X \<in> analz H"
-
-
-(*Monotonicity; Lemma 1 of Lowe's paper*)
-lemma analz_mono: "G<=H ==> analz(G) <= analz(H)"
-apply auto
-apply (erule analz.induct)
-apply (auto dest: Fst 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]
-
-
-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, 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
-
-(*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
-
-lemma analz_insert_Number [simp]:
- "analz (insert (Number N) H) = insert (Number N) (analz H)"
-apply (rule analz_insert_eq_I)
-apply (erule analz.induct, auto)
-done
-
-lemma analz_insert_Hash [simp]:
- "analz (insert (Hash X) H) = insert (Hash X) (analz H)"
-apply (rule analz_insert_eq_I)
-apply (erule analz.induct, auto)
-done
-
-(*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
-
-(*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
-
-lemma analz_insert_Pan [simp]:
- "analz (insert (Pan A) H) = insert (Pan A) (analz H)"
-apply (rule analz_insert_eq_I)
-apply (erule analz.induct, auto)
-done
-
-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)
-
-(*Case analysis: either the message is secure, or it is not!
- Effective, but can cause subgoals to blow up!
- Use with split_if; apparently split_tac does not cope with patterns
- such as "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)
-
-
-(*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
-
-lemma analz_image_Pan [simp]: "analz (Pan`A) = Pan`A"
-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_trans: "[| X\<in> analz G; G \<subseteq> analz H |] ==> X\<in> analz H"
-by (drule analz_mono, blast)
-
-(*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"
-*)
-
-(*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 clarify
-apply (erule analz.induct)
-apply (best intro: analz_mono [THEN subsetD])+
-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)
-
-(*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
-
-(*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])
-
-
-subsection{*Inductive relation "synth"*}
-
-text{*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.*}
-
-inductive_set
- synth :: "msg set => 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"
-
-(*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"
-inductive_cases Pan_synth [elim!]: "Pan A \<in> synth H"
-
-
-lemma synth_increasing: "H \<subseteq> synth(H)"
-by blast
-
-subsubsection{*Unions*}
-
-(*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_trans: "[| X\<in> synth G; G \<subseteq> synth H |] ==> X\<in> synth H"
-by (drule synth_mono, blast)
-
-(*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 Number_synth [simp]: "Number n \<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 Pan_synth_eq [simp]: "(Pan A \<in> synth H) = (Pan A \<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
-
-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
-
-
-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)
-
-(*More specifically for Fake. Very occasionally we could do with a version
- of the form 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])
-
-(*H is sometimes (Key ` KK \<union> spies evs), so can't put 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])
-
-(*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
-
-
-lemma Hash_synth_analz [simp]:
- "X \<notin> synth (analz H)
- ==> (Hash{|X,Y|} \<in> synth (analz H)) = (Hash{|X,Y|} \<in> analz H)"
-by blast
-
-
-(*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*}
-
-lemmas pushKeys [standard] =
- insert_commute [of "Key K" "Agent C"]
- insert_commute [of "Key K" "Nonce N"]
- insert_commute [of "Key K" "Number N"]
- insert_commute [of "Key K" "Pan PAN"]
- insert_commute [of "Key K" "Hash X"]
- insert_commute [of "Key K" "MPair X Y"]
- insert_commute [of "Key K" "Crypt X K'"]
-
-lemmas pushCrypts [standard] =
- insert_commute [of "Crypt X K" "Agent C"]
- insert_commute [of "Crypt X K" "Nonce N"]
- insert_commute [of "Crypt X K" "Number N"]
- insert_commute [of "Crypt X K" "Pan PAN"]
- insert_commute [of "Crypt X K" "Hash X'"]
- insert_commute [of "Crypt X K" "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
-{*
-structure MessageSET =
-struct
-
-(*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 (cs, ss) i =
- force_tac (cs, ss addsimps [@{thm image_eq_UN}]) i THEN
- ALLGOALS (asm_simp_tac ss)
-
-(*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 *)
-
-fun impOfSubs th = th RSN (2, @{thm rev_subsetD})
-
-(*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 @{thm Fake_analz_insert},
- impOfSubs @{thm 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 [@{thm analz_insertI},
- impOfSubs @{thm analz_subset_parts}]) 4 1))
-
-fun spy_analz_tac (cs,ss) i =
- DETERM
- (SELECT_GOAL
- (EVERY
- [ (*push in occurrences of X...*)
- (REPEAT o CHANGED)
- (res_inst_tac (Simplifier.the_context ss)
- [(("x", 1), "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)
-
-end
-*}
-(*>*)
-
-
-(*By default only o_apply is built-in. But in the presence of eta-expansion
- this means that some terms displayed as (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 Hash_notin_image_Key [simp] :"Hash X \<notin> Key ` A"
-by auto
-
-lemma synth_analz_mono: "G<=H ==> synth (analz(G)) <= synth (analz(H))"
-by (simp add: 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 (blast intro: synth_analz_mono [THEN [2] rev_subsetD])
-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 (simp add: parts_mono)
-done
-
-lemmas Fake_parts_sing_imp_Un = Fake_parts_sing [THEN [2] rev_subsetD]
-
-method_setup spy_analz = {*
- Scan.succeed (fn ctxt =>
- SIMPLE_METHOD' (MessageSET.spy_analz_tac (clasimpset_of ctxt))) *}
- "for proving the Fake case when analz is involved"
-
-method_setup atomic_spy_analz = {*
- Scan.succeed (fn ctxt =>
- SIMPLE_METHOD' (MessageSET.atomic_spy_analz_tac (clasimpset_of ctxt))) *}
- "for debugging spy_analz"
-
-method_setup Fake_insert_simp = {*
- Scan.succeed (fn ctxt =>
- SIMPLE_METHOD' (MessageSET.Fake_insert_simp_tac (simpset_of ctxt))) *}
- "for debugging spy_analz"
-
-end