(* Title: HOL/MetisTest/Message.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Testing the metis method
*)
theory Message imports Main begin
(*Needed occasionally with spy_analz_tac, e.g. in analz_insert_Key_newK*)
lemma strange_Un_eq [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*}
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"
ML{*AtpWrapper.problem_name := "Message__parts_mono"*}
lemma parts_mono: "G \<subseteq> H ==> parts(G) \<subseteq> parts(H)"
apply auto
apply (erule parts.induct)
apply (metis Inj set_mp)
apply (metis Fst)
apply (metis Snd)
apply (metis 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 *}
ML{*AtpWrapper.problem_name := "Message__invKey_eq"*}
lemma invKey_eq [simp]: "(invKey K = invKey K') = (K=K')"
by (metis invKey)
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)
apply 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}"
apply (erule parts.induct)
apply fast+
done
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
ML{*AtpWrapper.problem_name := "Message__parts_insert_two"*}
lemma parts_insert2:
"parts (insert X (insert Y H)) = parts {X} \<union> parts {Y} \<union> parts H"
by (metis Un_commute Un_empty_left Un_empty_right Un_insert_left Un_insert_right parts_Un)
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
ML{*AtpWrapper.problem_name := "Message__parts_subset_iff"*}
lemma parts_subset_iff [simp]: "(parts G \<subseteq> parts H) = (G \<subseteq> parts H)"
apply (rule iffI)
apply (metis Un_absorb1 Un_subset_iff parts_Un parts_increasing)
apply (metis parts_idem parts_mono)
done
lemma parts_trans: "[| X\<in> parts G; G \<subseteq> parts H |] ==> X\<in> parts H"
by (blast dest: parts_mono);
ML{*AtpWrapper.problem_name := "Message__parts_cut"*}
lemma parts_cut: "[|Y\<in> parts(insert X G); X\<in> parts H|] ==> Y\<in> parts(G \<union> H)"
by (metis Un_subset_iff insert_subset parts_increasing parts_trans)
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
ML{*AtpWrapper.problem_name := "Message__msg_Nonce_supply"*}
lemma msg_Nonce_supply: "\<exists>N. \<forall>n. N\<le>n --> Nonce n \<notin> parts {msg}"
apply (induct_tac "msg")
apply (simp_all add: parts_insert2)
apply (metis Suc_n_not_le_n)
apply (metis le_trans linorder_linear)
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"
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]
ML{*AtpWrapper.problem_name := "Message__parts_analz"*}
lemma parts_analz [simp]: "parts (analz H) = parts H"
apply (rule equalityI)
apply (metis analz_subset_parts parts_subset_iff)
apply (metis analz_increasing parts_mono)
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 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)
ML{*AtpWrapper.problem_name := "Message__analz_cut"*}
declare analz_trans[intro]
lemma analz_cut: "[| Y\<in> analz (insert X H); X\<in> analz H |] ==> Y\<in> analz H"
(*TOO SLOW
by (metis analz_idem analz_increasing analz_mono insert_absorb insert_mono insert_subset) --{*317s*}
??*)
by (erule analz_trans, blast)
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" *}
ML{*AtpWrapper.problem_name := "Message__analz_subset_cong"*}
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 (metis Un_absorb2 Un_commute Un_subset_iff Un_upper1 Un_upper2 analz_mono)
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. *}
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"
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)
ML{*AtpWrapper.problem_name := "Message__synth_insert"*}
lemma synth_insert: "insert X (synth H) \<subseteq> synth(insert X H)"
by (metis insert_iff insert_subset subset_insertI synth.Inj synth_mono)
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)
ML{*AtpWrapper.problem_name := "Message__synth_cut"*}
lemma synth_cut: "[| Y\<in> synth (insert X H); X\<in> synth H |] ==> Y\<in> synth H"
(*TOO SLOW
by (metis insert_absorb insert_mono insert_subset synth_idem synth_increasing synth_mono)
*)
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 *}
ML{*AtpWrapper.problem_name := "Message__parts_synth"*}
lemma parts_synth [simp]: "parts (synth H) = parts H \<union> synth H"
apply (rule equalityI)
apply (rule subsetI)
apply (erule parts.induct)
apply (metis UnCI)
apply (metis MPair_synth UnCI UnE insert_absorb insert_subset parts.Fst parts_increasing)
apply (metis MPair_synth UnCI UnE insert_absorb insert_subset parts.Snd parts_increasing)
apply (metis Body Crypt_synth UnCI UnE insert_absorb insert_subset parts_increasing)
apply (metis Un_subset_iff parts_increasing parts_mono synth_increasing)
done
ML{*AtpWrapper.problem_name := "Message__analz_analz_Un"*}
lemma analz_analz_Un [simp]: "analz (analz G \<union> H) = analz (G \<union> H)"
apply (rule equalityI);
apply (metis analz_idem analz_subset_cong order_eq_refl)
apply (metis analz_increasing analz_subset_cong order_eq_refl)
done
ML{*AtpWrapper.problem_name := "Message__analz_synth_Un"*}
declare analz_mono [intro] analz.Fst [intro] analz.Snd [intro] Un_least [intro]
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)
apply (metis UnCI UnE Un_commute analz.Inj)
apply (metis MPair_synth UnCI UnE Un_commute Un_upper1 analz.Fst analz_increasing analz_mono insert_absorb insert_subset)
apply (metis MPair_synth UnCI UnE Un_commute Un_upper1 analz.Snd analz_increasing analz_mono insert_absorb insert_subset)
apply (blast intro: analz.Decrypt)
apply blast
done
ML{*AtpWrapper.problem_name := "Message__analz_synth"*}
lemma analz_synth [simp]: "analz (synth H) = analz H \<union> synth H"
proof (neg_clausify)
assume 0: "analz (synth H) \<noteq> analz H \<union> synth H"
have 1: "\<And>X1 X3. sup (analz (sup X3 X1)) (synth X3) = analz (sup (synth X3) X1)"
by (metis analz_synth_Un)
have 2: "sup (analz H) (synth H) \<noteq> analz (synth H)"
by (metis 0)
have 3: "\<And>X1 X3. sup (synth X3) (analz (sup X3 X1)) = analz (sup (synth X3) X1)"
by (metis 1 Un_commute)
have 4: "\<And>X3. sup (synth X3) (analz X3) = analz (sup (synth X3) {})"
by (metis 3 Un_empty_right)
have 5: "\<And>X3. sup (synth X3) (analz X3) = analz (synth X3)"
by (metis 4 Un_empty_right)
have 6: "\<And>X3. sup (analz X3) (synth X3) = analz (synth X3)"
by (metis 5 Un_commute)
show "False"
by (metis 2 6)
qed
subsubsection{*For reasoning about the Fake rule in traces *}
ML{*AtpWrapper.problem_name := "Message__parts_insert_subset_Un"*}
lemma parts_insert_subset_Un: "X\<in> G ==> parts(insert X H) \<subseteq> parts G \<union> parts H"
proof (neg_clausify)
assume 0: "X \<in> G"
assume 1: "\<not> parts (insert X H) \<subseteq> parts G \<union> parts H"
have 2: "\<not> parts (insert X H) \<subseteq> parts (G \<union> H)"
by (metis 1 parts_Un)
have 3: "\<not> insert X H \<subseteq> G \<union> H"
by (metis 2 parts_mono)
have 4: "X \<notin> G \<union> H \<or> \<not> H \<subseteq> G \<union> H"
by (metis 3 insert_subset)
have 5: "X \<notin> G \<union> H"
by (metis 4 Un_upper2)
have 6: "X \<notin> G"
by (metis 5 UnCI)
show "False"
by (metis 6 0)
qed
ML{*AtpWrapper.problem_name := "Message__Fake_parts_insert"*}
lemma Fake_parts_insert:
"X \<in> synth (analz H) ==>
parts (insert X H) \<subseteq> synth (analz H) \<union> parts H"
proof (neg_clausify)
assume 0: "X \<in> synth (analz H)"
assume 1: "\<not> parts (insert X H) \<subseteq> synth (analz H) \<union> parts H"
have 2: "\<And>X3. parts X3 \<union> synth (analz X3) = parts (synth (analz X3))"
by (metis parts_synth parts_analz)
have 3: "\<And>X3. analz X3 \<union> synth (analz X3) = analz (synth (analz X3))"
by (metis analz_synth analz_idem)
have 4: "\<And>X3. analz X3 \<subseteq> analz (synth X3)"
by (metis Un_upper1 analz_synth)
have 5: "\<not> parts (insert X H) \<subseteq> parts H \<union> synth (analz H)"
by (metis 1 Un_commute)
have 6: "\<not> parts (insert X H) \<subseteq> parts (synth (analz H))"
by (metis 5 2)
have 7: "\<not> insert X H \<subseteq> synth (analz H)"
by (metis 6 parts_mono)
have 8: "X \<notin> synth (analz H) \<or> \<not> H \<subseteq> synth (analz H)"
by (metis 7 insert_subset)
have 9: "\<not> H \<subseteq> synth (analz H)"
by (metis 8 0)
have 10: "\<And>X3. X3 \<subseteq> analz (synth X3)"
by (metis analz_subset_iff 4)
have 11: "\<And>X3. X3 \<subseteq> analz (synth (analz X3))"
by (metis analz_subset_iff 10)
have 12: "\<And>X3. analz (synth (analz X3)) = synth (analz X3) \<or>
\<not> analz X3 \<subseteq> synth (analz X3)"
by (metis Un_absorb1 3)
have 13: "\<And>X3. analz (synth (analz X3)) = synth (analz X3)"
by (metis 12 synth_increasing)
have 14: "\<And>X3. X3 \<subseteq> synth (analz X3)"
by (metis 11 13)
show "False"
by (metis 9 14)
qed
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])
ML{*AtpWrapper.problem_name := "Message__Fake_analz_insert"*}
declare analz_mono [intro] synth_mono [intro]
lemma Fake_analz_insert:
"X\<in> synth (analz G) ==>
analz (insert X H) \<subseteq> synth (analz G) \<union> analz (G \<union> H)"
by (metis Un_commute Un_insert_left Un_insert_right Un_upper1 analz_analz_Un analz_mono analz_synth_Un equalityE insert_absorb order_le_less xt1(12))
ML{*AtpWrapper.problem_name := "Message__Fake_analz_insert_simpler"*}
(*simpler problems? BUT METIS CAN'T PROVE
lemma Fake_analz_insert_simpler:
"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) ")
apply (metis Un_commute analz_analz_Un analz_synth_Un)
apply (metis Un_commute Un_upper1 Un_upper2 analz_cut analz_increasing analz_mono insert_absorb insert_mono insert_subset)
done
*)
end