# Theory Message

theory Message
imports Main
```(*  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 ‹Metis Example Featuring Message Authentication›

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        ― ‹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 (metis id_apply)

text‹The inverse of a symmetric key is itself; that of a public key
is the private key and vice versa›

definition symKeys :: "key set" where
"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⦃_,/ _⦄)")
translations
"⦃x, y, z⦄"   == "⦃x, ⦃y, z⦄⦄"
"⦃x, y⦄"      == "CONST MPair x y"

definition HPair :: "[msg,msg] => msg" ("(4Hash[_] /_)" [0, 1000]) where
― ‹Message Y paired with a MAC computed with the help of X›
"Hash[X] Y == ⦃ Hash⦃X,Y⦄, Y⦄"

definition keysFor :: "msg set => key set" where
― ‹Keys useful to decrypt elements of a message set›
"keysFor H == invKey ` {K. ∃X. Crypt K X ∈ 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 ∈ 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 set_rev_mp)
apply (metis parts.Fst)
apply (metis parts.Snd)
by (metis parts.Body)

text‹Equations hold because constructors are injective.›
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‹Inverse of keys›

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 ∪ 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‹Monotonicity›
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‹Inductive relation "parts"›

lemma MPair_parts:
"[| ⦃X,Y⦄ ∈ parts H;
[| X ∈ parts H; Y ∈ 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.
‹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 ⊆ 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‹WARNING: loops if H = {Y}, therefore must not be repeated!›
lemma parts_singleton: "X∈ parts H ==> ∃Y∈H. X∈ parts {Y}"
apply (erule parts.induct)
apply fast+
done

subsubsection‹Unions›

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‹Added to simplify arguments to parts, analz and synth.
NOTE: the UN versions are no longer used!›

text‹This allows ‹blast› to simplify occurrences of
@{term "parts(G∪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) ⊆ parts(insert X H)"
by (blast intro: parts_mono [THEN [2] rev_subsetD])

subsubsection‹Idempotence and transitivity›

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‹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

lemma msg_Nonce_supply: "∃N. ∀n. N≤n --> Nonce n ∉ parts {msg}"
apply (induct_tac "msg")
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 ∈ 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‹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: analz.Fst analz.Snd)
done

text‹Making it safe speeds up proofs›
lemma MPair_analz [elim!]:
"[| ⦃X,Y⦄ ∈ analz H;
[| X ∈ analz H; Y ∈ 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‹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) ∪ 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‹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 ∉ 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) ∉ 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) ⊆
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)) ⊆
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‹Case analysis: either the message is secure, or it is not! Effective,
but can cause subgoals to blow up! Use with ‹if_split›; apparently
‹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) ∈ analz H)
then insert (Crypt K X) (analz (insert X H))
else insert (Crypt K X) (analz H))"

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) ⊆
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∈ 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‹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∈ 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 ⊆ analz G'; analz H ⊆ analz H' |]
==> analz (G ∪ H) ⊆ analz (G' ∪ 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 ∪ H) = analz (G' ∪ 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:
"[| ∀X Y. ⦃X,Y⦄ ∉ H;  ∀X K. Crypt K X ∉ 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∈ 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‹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 ∈ 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‹Monotonicity›
lemma synth_mono: "G⊆H ==> synth(G) ⊆ synth(H)"
by (auto, erule synth.induct, auto)

text‹NO ‹Agent_synth›, as any Agent name can be synthesized.
The same holds for @{term Number}›
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‹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) ∪ 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‹Idempotence and transitivity›

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‹Combinations of parts, analz and synth›

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⇩2 x⇩1. synth x⇩1 ∪ analz (x⇩1 ∪ x⇩2) = analz (synth x⇩1 ∪ x⇩2)" by (metis Un_commute analz_synth_Un)
hence "∀x⇩1. synth x⇩1 ∪ analz x⇩1 = analz (synth x⇩1 ∪ {})" by (metis Un_empty_right)
hence "∀x⇩1. synth x⇩1 ∪ analz x⇩1 = analz (synth x⇩1)" by (metis Un_empty_right)
hence "∀x⇩1. analz x⇩1 ∪ synth x⇩1 = analz (synth x⇩1)" by (metis Un_commute)
thus "analz (synth H) = analz H ∪ synth H" by metis
qed

subsubsection‹For reasoning about the Fake rule in traces›

lemma parts_insert_subset_Un: "X ∈ G ==> parts(insert X H) ⊆ parts G ∪ parts H"
proof -
assume "X ∈ G"
hence "∀x⇩1. G ⊆ x⇩1 ⟶ X ∈ x⇩1 " by auto
hence "∀x⇩1. X ∈ G ∪ x⇩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) ⊆ synth (analz H) ∪ parts H"
proof -
assume A1: "X ∈ synth (analz H)"
have F1: "∀x⇩1. analz x⇩1 ∪ synth (analz x⇩1) = analz (synth (analz x⇩1))"
by (metis analz_idem analz_synth)
have F2: "∀x⇩1. parts x⇩1 ∪ synth (analz x⇩1) = parts (synth (analz x⇩1))"
by (metis parts_analz parts_synth)
have F3: "X ∈ synth (analz H)" using A1 by metis
have "∀x⇩2 x⇩1::msg set. x⇩1 ≤ sup x⇩1 x⇩2" by (metis inf_sup_ord(3))
hence F4: "∀x⇩1. analz x⇩1 ⊆ analz (synth x⇩1)" by (metis analz_synth)
have F5: "X ∈ synth (analz H)" using F3 by metis
have "∀x⇩1. analz x⇩1 ⊆ synth (analz x⇩1)
⟶ analz (synth (analz x⇩1)) = synth (analz x⇩1)"
using F1 by (metis subset_Un_eq)
hence F6: "∀x⇩1. analz (synth (analz x⇩1)) = synth (analz x⇩1)"
by (metis synth_increasing)
have "∀x⇩1. x⇩1 ⊆ analz (synth x⇩1)" using F4 by (metis analz_subset_iff)
hence "∀x⇩1. x⇩1 ⊆ analz (synth (analz x⇩1))" by (metis analz_subset_iff)
hence "∀x⇩1. x⇩1 ⊆ synth (analz x⇩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 ∈  synth (analz H) ∪ 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) ⊆ synth (analz G) ∪ analz (G ∪ 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) ⊆ synth (analz G) ∪ analz (G ∪ 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
```