(* Title: HOL/Metis_Examples/Message.thy
Author: Lawrence C. Paulson, Cambridge University Computer Laboratory
Author: Jasmin Blanchette, TU Muenchen
Metis example featuring message authentication.
*)
section \<open>Metis Example Featuring Message Authentication\<close>
theory Message
imports Main
begin
declare [[metis_new_skolem]]
lemma strange_Un_eq [simp]: "A \ (B \ A) = B \ A"
by (metis Un_commute Un_left_absorb)
type_synonym key = nat
consts
all_symmetric :: bool \<comment> \<open>true if all keys are symmetric\<close>
invKey :: "key=>key" \<comment> \<open>inverse of a symmetric key\<close>
specification (invKey)
invKey [simp]: "invKey (invKey K) = K"
invKey_symmetric: "all_symmetric --> invKey = id"
by (metis id_apply)
text\<open>The inverse of a symmetric key is itself; that of a public key
is the private key and vice versa\<close>
definition symKeys :: "key set" where
"symKeys == {K. invKey K = K}"
datatype \<comment> \<open>We allow any number of friendly agents\<close>
agent = Server | Friend nat | Spy
datatype
msg = Agent agent \<comment> \<open>Agent names\<close>
| Number nat \<comment> \<open>Ordinary integers, timestamps, ...\<close>
| Nonce nat \<comment> \<open>Unguessable nonces\<close>
| Key key \<comment> \<open>Crypto keys\<close>
| Hash msg \<comment> \<open>Hashing\<close>
| MPair msg msg \<comment> \<open>Compound messages\<close>
| Crypt key msg \<comment> \<open>Encryption, public- or shared-key\<close>
text\<open>Concrete syntax: messages appear as \<open>\<lbrace>A,B,NA\<rbrace>\<close>, etc...\<close>
syntax
"_MTuple" :: "['a, args] => 'a * 'b" ("(2\_,/ _\)")
translations
"\x, y, z\" == "\x, \y, z\\"
"\x, y\" == "CONST MPair x y"
definition HPair :: "[msg,msg] => msg" ("(4Hash[_] /_)" [0, 1000]) where
\<comment> \<open>Message Y paired with a MAC computed with the help of X\<close>
"Hash[X] Y == \ Hash\X,Y\, Y\"
definition keysFor :: "msg set => key set" where
\<comment> \<open>Keys useful to decrypt elements of a message set\<close>
"keysFor H == invKey ` {K. \X. Crypt K X \ H}"
subsubsection\<open>Inductive Definition of All Parts" of a Message\<close>
inductive_set
parts :: "msg set => msg set"
for H :: "msg set"
where
Inj [intro]: "X \ H ==> X \ parts H"
| Fst: "\X,Y\ \ parts H ==> X \ parts H"
| Snd: "\X,Y\ \ parts H ==> Y \ parts H"
| Body: "Crypt K X \ parts H ==> X \ parts H"
lemma parts_mono: "G \ H ==> parts(G) \ parts(H)"
apply auto
apply (erule parts.induct)
apply (metis parts.Inj rev_subsetD)
apply (metis parts.Fst)
apply (metis parts.Snd)
by (metis parts.Body)
text\<open>Equations hold because constructors are injective.\<close>
lemma Friend_image_eq [simp]: "(Friend x \ Friend`A) = (x\A)"
by (metis agent.inject image_iff)
lemma Key_image_eq [simp]: "(Key x \ Key`A) = (x \ A)"
by (metis image_iff msg.inject(4))
lemma Nonce_Key_image_eq [simp]: "Nonce x \ Key`A"
by (metis image_iff msg.distinct(23))
subsubsection\<open>Inverse of keys\<close>
lemma invKey_eq [simp]: "(invKey K = invKey K') = (K = K')"
by (metis invKey)
subsection\<open>keysFor operator\<close>
lemma keysFor_empty [simp]: "keysFor {} = {}"
by (unfold keysFor_def, blast)
lemma keysFor_Un [simp]: "keysFor (H \ H') = keysFor H \ keysFor H'"
by (unfold keysFor_def, blast)
lemma keysFor_UN [simp]: "keysFor (\i\A. H i) = (\i\A. keysFor (H i))"
by (unfold keysFor_def, blast)
text\<open>Monotonicity\<close>
lemma keysFor_mono: "G \ H ==> keysFor(G) \ 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 \ H ==> invKey K \ keysFor H"
by (unfold keysFor_def, blast)
subsection\<open>Inductive relation "parts"\<close>
lemma MPair_parts:
"[| \X,Y\ \ 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\<open>NB These two rules are UNSAFE in the formal sense, as they discard the
compound message. They work well on THIS FILE.
\<open>MPair_parts\<close> is left as SAFE because it speeds up proofs.
The Crypt rule is normally kept UNSAFE to avoid breaking up certificates.\<close>
lemma parts_increasing: "H \ parts(H)"
by blast
lemmas parts_insertI = subset_insertI [THEN parts_mono, THEN subsetD]
lemma parts_empty [simp]: "parts{} = {}"
apply safe
apply (erule parts.induct)
apply blast+
done
lemma parts_emptyE [elim!]: "X\ parts{} ==> P"
by simp
text\<open>WARNING: loops if H = {Y}, therefore must not be repeated!\<close>
lemma parts_singleton: "X\ parts H ==> \Y\H. X\ parts {Y}"
apply (erule parts.induct)
apply fast+
done
subsubsection\<open>Unions\<close>
lemma parts_Un_subset1: "parts(G) \ parts(H) \ parts(G \ H)"
by (intro Un_least parts_mono Un_upper1 Un_upper2)
lemma parts_Un_subset2: "parts(G \ H) \ parts(G) \ parts(H)"
apply (rule subsetI)
apply (erule parts.induct, blast+)
done
lemma parts_Un [simp]: "parts(G \ H) = parts(G) \ parts(H)"
by (intro equalityI parts_Un_subset1 parts_Un_subset2)
lemma parts_insert: "parts (insert X H) = parts {X} \ parts H"
apply (subst insert_is_Un [of _ H])
apply (simp only: parts_Un)
done
lemma parts_insert2:
"parts (insert X (insert Y H)) = parts {X} \ parts {Y} \ parts H"
by (metis Un_commute Un_empty_left Un_empty_right Un_insert_left Un_insert_right parts_Un)
lemma parts_UN_subset1: "(\x\A. parts(H x)) \ parts(\x\A. H x)"
by (intro UN_least parts_mono UN_upper)
lemma parts_UN_subset2: "parts(\x\A. H x) \ (\x\A. parts(H x))"
apply (rule subsetI)
apply (erule parts.induct, blast+)
done
lemma parts_UN [simp]: "parts(\x\A. H x) = (\x\A. parts(H x))"
by (intro equalityI parts_UN_subset1 parts_UN_subset2)
text\<open>Added to simplify arguments to parts, analz and synth.
NOTE: the UN versions are no longer used!\<close>
text\<open>This allows \<open>blast\<close> to simplify occurrences of
\<^term>\<open>parts(G\<union>H)\<close> in the assumption.\<close>
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) \ parts(insert X H)"
by (blast intro: parts_mono [THEN [2] rev_subsetD])
subsubsection\<open>Idempotence and transitivity\<close>
lemma parts_partsD [dest!]: "X\ parts (parts H) ==> X\ 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 \ parts H) = (G \ 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\ parts G; G \ parts H |] ==> X\ parts H"
by (blast dest: parts_mono)
lemma parts_cut: "[|Y\ parts (insert X G); X\ parts H|] ==> Y\ parts(G \ H)"
by (metis (no_types) Un_insert_left Un_insert_right insert_absorb le_supE
parts_Un parts_idem parts_increasing parts_trans)
subsubsection\<open>Rewrite rules for pulling out atomic messages\<close>
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 \<lbrace>X,Y\<rbrace> (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
lemma msg_Nonce_supply: "\N. \n. N\n --> Nonce n \ 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\<open>Inductive relation "analz"\<close>
text\<open>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.\<close>
inductive_set
analz :: "msg set => msg set"
for H :: "msg set"
where
Inj [intro,simp] : "X \ H ==> X \ analz H"
| Fst: "\X,Y\ \ analz H ==> X \ analz H"
| Snd: "\X,Y\ \ analz H ==> Y \ analz H"
| Decrypt [dest]:
"[|Crypt K X \ analz H; Key(invKey K) \ analz H|] ==> X \ analz H"
text\<open>Monotonicity; Lemma 1 of Lowe's paper\<close>
lemma analz_mono: "G\H ==> analz(G) \ analz(H)"
apply auto
apply (erule analz.induct)
apply (auto dest: analz.Fst analz.Snd)
done
text\<open>Making it safe speeds up proofs\<close>
lemma MPair_analz [elim!]:
"[| \X,Y\ \ analz H;
[| X \<in> analz H; Y \<in> analz H |] ==> P
|] ==> P"
by (blast dest: analz.Fst analz.Snd)
lemma analz_increasing: "H \ analz(H)"
by blast
lemma analz_subset_parts: "analz H \ parts H"
apply (rule subsetI)
apply (erule analz.induct, blast+)
done
lemmas analz_into_parts = analz_subset_parts [THEN subsetD]
lemmas not_parts_not_analz = analz_subset_parts [THEN contra_subsetD]
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]
subsubsection\<open>General equational properties\<close>
lemma analz_empty [simp]: "analz{} = {}"
apply safe
apply (erule analz.induct, blast+)
done
text\<open>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\<close>
lemma analz_Un: "analz(G) \ analz(H) \ analz(G \ H)"
by (intro Un_least analz_mono Un_upper1 Un_upper2)
lemma analz_insert: "insert X (analz H) \ analz(insert X H)"
by (blast intro: analz_mono [THEN [2] rev_subsetD])
subsubsection\<open>Rewrite rules for pulling out atomic messages\<close>
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\<open>Can only pull out Keys if they are not needed to decrypt the rest\<close>
lemma analz_insert_Key [simp]:
"K \ 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 \<lbrace>X,Y\<rbrace> (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\<open>Can pull out enCrypted message if the Key is not known\<close>
lemma analz_insert_Crypt:
"Key (invKey K) \ 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) \ 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) \ 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) \ analz H ==>
analz (insert (Crypt K X) H) =
insert (Crypt K X) (analz (insert X H))"
by (intro equalityI lemma1 lemma2)
text\<open>Case analysis: either the message is secure, or it is not! Effective,
but can cause subgoals to blow up! Use with \<open>if_split\<close>; apparently
\<open>split_tac\<close> does not cope with patterns such as \<^term>\<open>analz (insert
(Crypt K X) H)\<close>\<close>
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\<open>This rule supposes "for the sake of argument" that we have the key.\<close>
lemma analz_insert_Crypt_subset:
"analz (insert (Crypt K X) H) \
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\<open>Idempotence and transitivity\<close>
lemma analz_analzD [dest!]: "X\ analz (analz H) ==> X\ 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 \ analz H) = (G \ analz H)"
apply (rule iffI)
apply (iprover intro: subset_trans analz_increasing)
apply (frule analz_mono, simp)
done
lemma analz_trans: "[| X\ analz G; G \ analz H |] ==> X\ analz H"
by (drule analz_mono, blast)
declare analz_trans[intro]
lemma analz_cut: "[| Y\ analz (insert X H); X\ analz H |] ==> Y\ analz H"
by (metis analz_idem analz_increasing analz_mono insert_absorb insert_mono insert_subset)
text\<open>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.\<close>
lemma analz_insert_eq: "X\ analz H ==> analz (insert X H) = analz H"
by (blast intro: analz_cut analz_insertI)
text\<open>A congruence rule for "analz"\<close>
lemma analz_subset_cong:
"[| analz G \ analz G'; analz H \ 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\<open>If there are no pairs or encryptions then analz does nothing\<close>
lemma analz_trivial:
"[| \X Y. \X,Y\ \ H; \X K. Crypt K X \ H |] ==> analz H = H"
apply safe
apply (erule analz.induct, blast+)
done
text\<open>These two are obsolete (with a single Spy) but cost little to prove...\<close>
lemma analz_UN_analz_lemma:
"X\ analz (\i\A. analz (H i)) ==> X\ analz (\i\A. H i)"
apply (erule analz.induct)
apply (blast intro: analz_mono [THEN [2] rev_subsetD])+
done
lemma analz_UN_analz [simp]: "analz (\i\A. analz (H i)) = analz (\i\A. H i)"
by (blast intro: analz_UN_analz_lemma analz_mono [THEN [2] rev_subsetD])
subsection\<open>Inductive relation "synth"\<close>
text\<open>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.\<close>
inductive_set
synth :: "msg set => msg set"
for H :: "msg set"
where
Inj [intro]: "X \ H ==> X \ synth H"
| Agent [intro]: "Agent agt \ synth H"
| Number [intro]: "Number n \ synth H"
| Hash [intro]: "X \ synth H ==> Hash X \ synth H"
| MPair [intro]: "[|X \ synth H; Y \ synth H|] ==> \X,Y\ \ synth H"
| Crypt [intro]: "[|X \ synth H; Key(K) \ H|] ==> Crypt K X \ synth H"
text\<open>Monotonicity\<close>
lemma synth_mono: "G\H ==> synth(G) \ synth(H)"
by (auto, erule synth.induct, auto)
text\<open>NO \<open>Agent_synth\<close>, as any Agent name can be synthesized.
The same holds for \<^term>\<open>Number\<close>\<close>
inductive_cases Nonce_synth [elim!]: "Nonce n \ synth H"
inductive_cases Key_synth [elim!]: "Key K \ synth H"
inductive_cases Hash_synth [elim!]: "Hash X \ synth H"
inductive_cases MPair_synth [elim!]: "\X,Y\ \ synth H"
inductive_cases Crypt_synth [elim!]: "Crypt K X \ synth H"
lemma synth_increasing: "H \ synth(H)"
by blast
subsubsection\<open>Unions\<close>
text\<open>Converse fails: we can synth more from the union than from the
separate parts, building a compound message using elements of each.\<close>
lemma synth_Un: "synth(G) \ synth(H) \ synth(G \ H)"
by (intro Un_least synth_mono Un_upper1 Un_upper2)
lemma synth_insert: "insert X (synth H) \ synth(insert X H)"
by (metis insert_iff insert_subset subset_insertI synth.Inj synth_mono)
subsubsection\<open>Idempotence and transitivity\<close>
lemma synth_synthD [dest!]: "X\ synth (synth H) ==> X\ synth H"
by (erule synth.induct, blast+)
lemma synth_idem: "synth (synth H) = synth H"
by blast
lemma synth_subset_iff [simp]: "(synth G \ synth H) = (G \ 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\ synth G; G \ synth H |] ==> X\ synth H"
by (drule synth_mono, blast)
lemma synth_cut: "[| Y\ synth (insert X H); X\ synth H |] ==> Y\ synth H"
by (metis insert_absorb insert_mono insert_subset synth_idem synth_increasing synth_mono)
lemma Agent_synth [simp]: "Agent A \ synth H"
by blast
lemma Number_synth [simp]: "Number n \ synth H"
by blast
lemma Nonce_synth_eq [simp]: "(Nonce N \ synth H) = (Nonce N \ H)"
by blast
lemma Key_synth_eq [simp]: "(Key K \ synth H) = (Key K \ H)"
by blast
lemma Crypt_synth_eq [simp]:
"Key K \ H ==> (Crypt K X \ synth H) = (Crypt K X \ H)"
by blast
lemma keysFor_synth [simp]:
"keysFor (synth H) = keysFor H \ invKey`{K. Key K \ H}"
by (unfold keysFor_def, blast)
subsubsection\<open>Combinations of parts, analz and synth\<close>
lemma parts_synth [simp]: "parts (synth H) = parts H \ 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
lemma analz_analz_Un [simp]: "analz (analz G \ H) = analz (G \ 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
declare analz_mono [intro] analz.Fst [intro] analz.Snd [intro] Un_least [intro]
lemma analz_synth_Un [simp]: "analz (synth G \ H) = analz (G \ H) \ 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 analz.Fst analz.Inj)
apply (metis MPair_synth UnCI UnE Un_commute analz.Inj analz.Snd)
apply (blast intro: analz.Decrypt)
apply blast
done
lemma analz_synth [simp]: "analz (synth H) = analz H \ synth H"
proof -
have "\x\<^sub>2 x\<^sub>1. synth x\<^sub>1 \ analz (x\<^sub>1 \ x\<^sub>2) = analz (synth x\<^sub>1 \ x\<^sub>2)" by (metis Un_commute analz_synth_Un)
hence "\x\<^sub>1. synth x\<^sub>1 \ analz x\<^sub>1 = analz (synth x\<^sub>1 \ {})" by (metis Un_empty_right)
hence "\x\<^sub>1. synth x\<^sub>1 \ analz x\<^sub>1 = analz (synth x\<^sub>1)" by (metis Un_empty_right)
hence "\x\<^sub>1. analz x\<^sub>1 \ synth x\<^sub>1 = analz (synth x\<^sub>1)" by (metis Un_commute)
thus "analz (synth H) = analz H \ synth H" by metis
qed
subsubsection\<open>For reasoning about the Fake rule in traces\<close>
lemma parts_insert_subset_Un: "X \ G ==> parts(insert X H) \ parts G \ parts H"
proof -
assume "X \ G"
hence "\x\<^sub>1. G \ x\<^sub>1 \ X \ x\<^sub>1 " by auto
hence "\x\<^sub>1. X \ G \ x\<^sub>1" by (metis Un_upper1)
hence "insert X H \ G \ H" by (metis Un_upper2 insert_subset)
hence "parts (insert X H) \ parts (G \ H)" by (metis parts_mono)
thus "parts (insert X H) \ parts G \ parts H" by (metis parts_Un)
qed
lemma Fake_parts_insert:
"X \ synth (analz H) ==>
parts (insert X H) \<subseteq> synth (analz H) \<union> parts H"
proof -
assume A1: "X \ synth (analz H)"
have F1: "\x\<^sub>1. analz x\<^sub>1 \ synth (analz x\<^sub>1) = analz (synth (analz x\<^sub>1))"
by (metis analz_idem analz_synth)
have F2: "\x\<^sub>1. parts x\<^sub>1 \ synth (analz x\<^sub>1) = parts (synth (analz x\<^sub>1))"
by (metis parts_analz parts_synth)
have F3: "X \ synth (analz H)" using A1 by metis
have "\x\<^sub>2 x\<^sub>1::msg set. x\<^sub>1 \ sup x\<^sub>1 x\<^sub>2" by (metis inf_sup_ord(3))
hence F4: "\x\<^sub>1. analz x\<^sub>1 \ analz (synth x\<^sub>1)" by (metis analz_synth)
have F5: "X \ synth (analz H)" using F3 by metis
have "\x\<^sub>1. analz x\<^sub>1 \ synth (analz x\<^sub>1)
\<longrightarrow> analz (synth (analz x\<^sub>1)) = synth (analz x\<^sub>1)"
using F1 by (metis subset_Un_eq)
hence F6: "\x\<^sub>1. analz (synth (analz x\<^sub>1)) = synth (analz x\<^sub>1)"
by (metis synth_increasing)
have "\x\<^sub>1. x\<^sub>1 \ analz (synth x\<^sub>1)" using F4 by (metis analz_subset_iff)
hence "\x\<^sub>1. x\<^sub>1 \ analz (synth (analz x\<^sub>1))" by (metis analz_subset_iff)
hence "\x\<^sub>1. x\<^sub>1 \ synth (analz x\<^sub>1)" using F6 by metis
hence "H \ synth (analz H)" by metis
hence "H \ synth (analz H) \ X \ synth (analz H)" using F5 by metis
hence "insert X H \ synth (analz H)" by (metis insert_subset)
hence "parts (insert X H) \ parts (synth (analz H))" by (metis parts_mono)
hence "parts (insert X H) \ parts H \ synth (analz H)" using F2 by metis
thus "parts (insert X H) \ synth (analz H) \ parts H" by (metis Un_commute)
qed
lemma Fake_parts_insert_in_Un:
"[|Z \ 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])
declare synth_mono [intro]
lemma Fake_analz_insert:
"X \ 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 insert_absorb)
lemma Fake_analz_insert_simpler:
"X \ synth (analz G) ==>
analz (insert X H) \<subseteq> synth (analz G) \<union> analz (G \<union> H)"
apply (rule subsetI)
apply (subgoal_tac "x \ analz (synth (analz G) \ H) ")
apply (metis Un_commute analz_analz_Un analz_synth_Un)
by (metis Un_upper1 Un_upper2 analz_mono insert_absorb insert_subset)
end
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