(* Title: HOL/Auth/Message
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1996 University of Cambridge
Datatypes of agents and messages;
Inductive relations "parts", "analz" and "synth"
*)
header{*Theory of Agents and Messages for Security Protocols*}
theory Message imports Main begin
(*Needed occasionally with spy_analz_tac, e.g. in analz_insert_Key_newK*)
lemma [simp] : "A \<union> (B \<union> A) = B \<union> 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}"
datatype --{*We allow any number of friendly agents*}
agent = Server | Friend nat | Spy
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*}
text{*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
HPair :: "[msg,msg] => msg" ("(4Hash[_] /_)" [0, 1000])
--{*Message Y paired with a MAC computed with the help of X*}
"Hash[X] Y == {| Hash{|X,Y|}, Y|}"
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}"
subsubsection{*Inductive Definition of All Parts" of a Message*}
consts parts :: "msg set => msg set"
inductive "parts H"
intros
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"
text{*Monotonicity*}
lemma parts_mono: "G \<subseteq> H ==> parts(G) \<subseteq> parts(H)"
apply auto
apply (erule parts.induct)
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_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_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
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_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_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 (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
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 (blast elim!: add_leE)
txt{*Nonce case*}
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. *}
consts analz :: "msg set => msg set"
inductive "analz H"
intros
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"
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]
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
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
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
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
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 xa = 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 xa = 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])
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. *}
consts synth :: "msg set => msg set"
inductive "synth H"
intros
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"
text{*Monotonicity*}
lemma synth_mono: "G\<subseteq>H ==> synth(G) \<subseteq> synth(H)"
by (auto, erule synth.induct, auto)
text{*NO @{text Agent_synth}, as any Agent name can be synthesized.
The same holds for @{term 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"
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 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 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)
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
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
subsection{*HPair: a combination of Hash and MPair*}
subsubsection{*Freeness *}
lemma Agent_neq_HPair: "Agent A ~= Hash[X] Y"
by (unfold HPair_def, simp)
lemma Nonce_neq_HPair: "Nonce N ~= Hash[X] Y"
by (unfold HPair_def, simp)
lemma Number_neq_HPair: "Number N ~= Hash[X] Y"
by (unfold HPair_def, simp)
lemma Key_neq_HPair: "Key K ~= Hash[X] Y"
by (unfold HPair_def, simp)
lemma Hash_neq_HPair: "Hash Z ~= Hash[X] Y"
by (unfold HPair_def, simp)
lemma Crypt_neq_HPair: "Crypt K X' ~= Hash[X] Y"
by (unfold HPair_def, simp)
lemmas HPair_neqs = Agent_neq_HPair Nonce_neq_HPair Number_neq_HPair
Key_neq_HPair Hash_neq_HPair Crypt_neq_HPair
declare HPair_neqs [iff]
declare HPair_neqs [symmetric, iff]
lemma HPair_eq [iff]: "(Hash[X'] Y' = Hash[X] Y) = (X' = X & Y'=Y)"
by (simp add: HPair_def)
lemma MPair_eq_HPair [iff]:
"({|X',Y'|} = Hash[X] Y) = (X' = Hash{|X,Y|} & Y'=Y)"
by (simp add: HPair_def)
lemma HPair_eq_MPair [iff]:
"(Hash[X] Y = {|X',Y'|}) = (X' = Hash{|X,Y|} & Y'=Y)"
by (auto simp add: HPair_def)
subsubsection{*Specialized laws, proved in terms of those for Hash and MPair *}
lemma keysFor_insert_HPair [simp]: "keysFor (insert (Hash[X] Y) H) = keysFor H"
by (simp add: HPair_def)
lemma parts_insert_HPair [simp]:
"parts (insert (Hash[X] Y) H) =
insert (Hash[X] Y) (insert (Hash{|X,Y|}) (parts (insert Y H)))"
by (simp add: HPair_def)
lemma analz_insert_HPair [simp]:
"analz (insert (Hash[X] Y) H) =
insert (Hash[X] Y) (insert (Hash{|X,Y|}) (analz (insert Y H)))"
by (simp add: HPair_def)
lemma HPair_synth_analz [simp]:
"X \<notin> synth (analz H)
==> (Hash[X] Y \<in> synth (analz H)) =
(Hash {|X, Y|} \<in> analz H & Y \<in> synth (analz H))"
by (simp add: HPair_def)
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 HPair_def = thm "HPair_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 [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 Hash_notin_image_Key [simp] :"Hash 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.ctxt_args (fn ctxt =>
Method.METHOD (fn facts =>
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.METHOD (fn facts =>
atomic_spy_analz_tac (local_clasimpset_of ctxt) 1)) *}
"for debugging spy_analz"
method_setup Fake_insert_simp = {*
Method.ctxt_args (fn ctxt =>
Method.METHOD (fn facts =>
Fake_insert_simp_tac (local_simpset_of ctxt) 1)) *}
"for debugging spy_analz"
end