Theory Message

theory Message
imports Main
(*  Title:      HOL/Auth/Message.thy
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 ∪ (B ∪ A) = B ∪ A"
by blast

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 (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*}


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{|_,/ _|})")

syntax (xsymbols)
"_MTuple" :: "['a, args] => 'a * 'b" ("(2\<lbrace>_,/ _\<rbrace>)")

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"


text{*Monotonicity*}
lemma parts_mono: "G ⊆ H ==> parts(G) ⊆ 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 ∈ Friend`A) = (x:A)"
by auto

lemma Key_image_eq [simp]: "(Key x ∈ Key`A) = (x∈A)"
by auto

lemma Nonce_Key_image_eq [simp]: "(Nonce x ∉ Key`A)"
by auto


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 (\<Union>i∈A. H i) = (\<Union>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.
@{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 ⊆ 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, 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}"
by (erule parts.induct, fast+)


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"
by (metis insert_is_Un parts_Un)

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} ∪ parts {Y} ∪ parts H"
by (metis Un_commute Un_empty_right Un_insert_right insert_is_Un parts_Un)

lemma parts_UN_subset1: "(\<Union>x∈A. parts(H x)) ⊆ parts(\<Union>x∈A. H x)"
by (intro UN_least parts_mono UN_upper)

lemma parts_UN_subset2: "parts(\<Union>x∈A. H x) ⊆ (\<Union>x∈A. parts(H x))"
apply (rule subsetI)
apply (erule parts.induct, blast+)
done

lemma parts_UN [simp]: "parts(\<Union>x∈A. H x) = (\<Union>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 @{text 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)"
by (metis parts_idem parts_increasing parts_mono subset_trans)

lemma parts_trans: "[| X∈ parts G; G ⊆ parts H |] ==> X∈ parts H"
by (metis parts_subset_iff set_mp)

text{*Cut*}
lemma parts_cut:
"[| Y∈ parts (insert X G); X∈ parts H |] ==> Y∈ parts (G ∪ H)"
by (blast intro: parts_trans)

lemma parts_cut_eq [simp]: "X∈ parts H ==> parts (insert X H) = parts H"
by (metis insert_absorb parts_idem parts_insert)


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: "∃N. ∀n. N≤n --> Nonce n ∉ parts {msg}"
apply (induct msg)
apply (simp_all (no_asm_simp) add: exI parts_insert2)
txt{*Nonce case*}
apply (metis Suc_n_not_le_n)
txt{*MPair case: metis works out the necessary sum itself!*}
apply (metis le_trans nat_le_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"
by (metis analz_increasing analz_subset_parts equalityI parts_mono parts_subset_iff)

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 @{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) ∈ 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) ⊆
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)"
by (metis analz_idem analz_increasing analz_mono subset_trans)

lemma analz_trans: "[| X∈ analz G; G ⊆ analz H |] ==> X∈ analz H"
by (drule analz_mono, blast)

text{*Cut; Lemma 2 of Lowe*}
lemma analz_cut: "[| Y∈ analz (insert X H); X∈ analz H |] ==> Y∈ 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∈ analz H ==> analz (insert X H) = analz H"
by (metis analz_cut analz_insert_eq_I insert_absorb)


text{*A congruence rule for "analz" *}

lemma analz_subset_cong:
"[| analz G ⊆ analz G'; analz H ⊆ analz H' |]
==> analz (G ∪ H) ⊆ analz (G' ∪ H')"

by (metis Un_mono analz_Un analz_subset_iff subset_trans)

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 (\<Union>i∈A. analz (H i)) ==> X∈ analz (\<Union>i∈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∈A. analz (H i)) = analz (\<Union>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 @{text Agent_synth}, as any Agent name can be synthesized.
The same holds for @{term Number}*}


inductive_simps synth_simps [iff]:
"Nonce n ∈ synth H"
"Key K ∈ synth H"
"Hash X ∈ synth H"
"{|X,Y|} ∈ synth H"
"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 (blast intro: synth_mono [THEN [2] rev_subsetD])

subsubsection{*Idempotence and transitivity *}

lemma synth_synthD [dest!]: "X∈ synth (synth H) ==> X∈ synth H"
by (erule synth.induct, auto)

lemma synth_idem: "synth (synth H) = synth H"
by blast

lemma synth_subset_iff [simp]: "(synth G ⊆ synth H) = (G ⊆ synth H)"
by (metis subset_trans synth_idem synth_increasing synth_mono)

lemma synth_trans: "[| X∈ synth G; G ⊆ synth H |] ==> X∈ synth H"
by (drule synth_mono, blast)

text{*Cut; Lemma 2 of Lowe*}
lemma synth_cut: "[| Y∈ synth (insert X H); X∈ synth H |] ==> Y∈ synth H"
by (erule synth_trans, blast)

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 (blast intro: synth_increasing [THEN parts_mono, THEN subsetD]
parts.Fst parts.Snd parts.Body)+
done

lemma analz_analz_Un [simp]: "analz (analz G ∪ H) = analz (G ∪ H)"
apply (intro equalityI analz_subset_cong)+
apply simp_all
done

lemma analz_synth_Un [simp]: "analz (synth G ∪ H) = analz (G ∪ H) ∪ 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 ∪ synth H"
by (metis Un_empty_right analz_synth_Un)


subsubsection{*For reasoning about the Fake rule in traces *}

lemma parts_insert_subset_Un: "X∈ G ==> parts(insert X H) ⊆ parts G ∪ parts H"
by (metis UnCI Un_upper2 insert_subset parts_Un parts_mono)

text{*More specifically for Fake. See also @{text Fake_parts_sing} below *}
lemma Fake_parts_insert:
"X ∈ synth (analz H) ==>
parts (insert X H) ⊆ synth (analz H) ∪ parts H"

by (metis Un_commute analz_increasing insert_subset parts_analz parts_mono
parts_synth synth_mono synth_subset_iff)

lemma Fake_parts_insert_in_Un:
"[|Z ∈ parts (insert X H); X: synth (analz H)|]
==> Z ∈ synth (analz H) ∪ parts H"

by (metis Fake_parts_insert set_mp)

text{*@{term H} is sometimes @{term"Key ` KK ∪ spies evs"}, so can't put
@{term "G=H"}.*}

lemma Fake_analz_insert:
"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)", force)
apply (blast intro: analz_mono [THEN [2] rev_subsetD] analz_mono [THEN synth_mono, THEN [2] rev_subsetD])
done

lemma analz_conj_parts [simp]:
"(X ∈ analz H & X ∈ parts H) = (X ∈ analz H)"
by (blast intro: analz_subset_parts [THEN subsetD])

lemma analz_disj_parts [simp]:
"(X ∈ analz H | X ∈ parts H) = (X ∈ 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|} ∈ synth (analz H)) =
(X ∈ synth (analz H) & Y ∈ synth (analz H))"

by blast

lemma Crypt_synth_analz:
"[| Key K ∈ analz H; Key (invKey K) ∈ analz H |]
==> (Crypt K X ∈ synth (analz H)) = (X ∈ synth (analz H))"

by blast


lemma Hash_synth_analz [simp]:
"X ∉ synth (analz H)
==> (Hash{|X,Y|} ∈ synth (analz H)) = (Hash{|X,Y|} ∈ 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 ∉ synth (analz H)
==> (Hash[X] Y ∈ synth (analz H)) =
(Hash {|X, Y|} ∈ analz H & Y ∈ synth (analz H))"

by (auto 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*}


lemmas pushKeys =
insert_commute [of "Key K" "Agent C"]
insert_commute [of "Key K" "Nonce N"]
insert_commute [of "Key K" "Number N"]
insert_commute [of "Key K" "Hash X"]
insert_commute [of "Key K" "MPair X Y"]
insert_commute [of "Key K" "Crypt X K'"]
for K C N X Y K'

lemmas pushCrypts =
insert_commute [of "Crypt X K" "Agent C"]
insert_commute [of "Crypt X K" "Agent C"]
insert_commute [of "Crypt X K" "Nonce N"]
insert_commute [of "Crypt X K" "Number N"]
insert_commute [of "Crypt X K" "Hash X'"]
insert_commute [of "Crypt X K" "MPair X' Y"]
for X K C N X' Y

text{*Cannot be added with @{text "[simp]"} -- messages should not always be
re-ordered. *}

lemmas pushes = pushKeys pushCrypts


subsection{*The set of key-free messages*}

(*Note that even the encryption of a key-free message remains key-free.
This concept is valuable because of the theorem analz_keyfree_into_Un, proved below. *)


inductive_set
keyfree :: "msg set"
where
Agent: "Agent A ∈ keyfree"
| Number: "Number N ∈ keyfree"
| Nonce: "Nonce N ∈ keyfree"
| Hash: "Hash X ∈ keyfree"
| MPair: "[|X ∈ keyfree; Y ∈ keyfree|] ==> {|X,Y|} ∈ keyfree"
| Crypt: "[|X ∈ keyfree|] ==> Crypt K X ∈ keyfree"


declare keyfree.intros [intro]

inductive_cases keyfree_KeyE: "Key K ∈ keyfree"
inductive_cases keyfree_MPairE: "{|X,Y|} ∈ keyfree"
inductive_cases keyfree_CryptE: "Crypt K X ∈ keyfree"

lemma parts_keyfree: "parts (keyfree) ⊆ keyfree"
by (clarify, erule parts.induct, auto elim!: keyfree_KeyE keyfree_MPairE keyfree_CryptE)

(*The key-free part of a set of messages can be removed from the scope of the analz operator.*)
lemma analz_keyfree_into_Un: "[|X ∈ analz (G ∪ H); G ⊆ keyfree|] ==> X ∈ parts G ∪ analz H"
apply (erule analz.induct, auto)
apply (blast dest:parts.Body)
apply (blast dest: parts.Body)
apply (metis Un_absorb2 keyfree_KeyE parts_Un parts_keyfree UnI2)
done

subsection{*Tactics useful for many protocol proofs*}
ML
{*
(*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.
DEPENDS UPON "X" REFERRING TO THE FRADULENT MESSAGE *)

fun impOfSubs th = th RSN (2, @{thm rev_subsetD})

(*Apply rules to break down assumptions of the form
Y ∈ parts(insert X H) and Y ∈ analz(insert X H)
*)
val Fake_insert_tac =
dresolve_tac [impOfSubs @{thm Fake_analz_insert},
impOfSubs @{thm Fake_parts_insert}] THEN'
eresolve_tac [asm_rl, @{thm synth.Inj}];

fun Fake_insert_simp_tac ctxt i =
REPEAT (Fake_insert_tac i) THEN asm_full_simp_tac ctxt i;

fun atomic_spy_analz_tac ctxt =
SELECT_GOAL
(Fake_insert_simp_tac ctxt 1 THEN
IF_UNSOLVED
(Blast.depth_tac
(ctxt addIs [@{thm analz_insertI}, impOfSubs @{thm analz_subset_parts}]) 4 1));

fun spy_analz_tac ctxt i =
DETERM
(SELECT_GOAL
(EVERY
[ (*push in occurrences of X...*)
(REPEAT o CHANGED)
(res_inst_tac ctxt [(("x", 1), "X")] (insert_commute RS ssubst) 1),
(*...allowing further simplifications*)
simp_tac ctxt 1,
REPEAT (FIRSTGOAL (resolve_tac [allI,impI,notI,conjI,iffI])),
DEPTH_SOLVE (atomic_spy_analz_tac ctxt 1)]) 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 ∉ Key ` A"
by auto

lemma Hash_notin_image_Key [simp] :"Hash X ∉ Key ` A"
by auto

lemma synth_analz_mono: "G⊆H ==> synth (analz(G)) ⊆ synth (analz(H))"
by (iprover intro: synth_mono analz_mono)

lemma Fake_analz_eq [simp]:
"X ∈ synth(analz H) ==> synth (analz (insert X H)) = synth (analz H)"
by (metis Fake_analz_insert Un_absorb Un_absorb1 Un_commute
subset_insertI synth_analz_mono synth_increasing synth_subset_iff)

text{*Two generalizations of @{text analz_insert_eq}*}
lemma gen_analz_insert_eq [rule_format]:
"X ∈ analz H ==> ALL G. H ⊆ 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 ∈ synth (analz H)
==> ALL G. H ⊆ G --> (Key K ∈ analz (insert X G)) = (Key K ∈ 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 ∈ synth (analz H) ==> parts{X} ⊆ synth (analz H) ∪ parts H"
by (metis Fake_parts_insert empty_subsetI insert_mono parts_mono subset_trans)

lemmas Fake_parts_sing_imp_Un = Fake_parts_sing [THEN [2] rev_subsetD]

method_setup spy_analz = {*
Scan.succeed (SIMPLE_METHOD' o spy_analz_tac) *}

"for proving the Fake case when analz is involved"

method_setup atomic_spy_analz = {*
Scan.succeed (SIMPLE_METHOD' o atomic_spy_analz_tac) *}

"for debugging spy_analz"

method_setup Fake_insert_simp = {*
Scan.succeed (SIMPLE_METHOD' o Fake_insert_simp_tac) *}

"for debugging spy_analz"

end