# Theory Message_SET

theory Message_SET
imports Nat_Bijection
```(*  Title:      HOL/SET_Protocol/Message_SET.thy
Author:     Giampaolo Bella
Author:     Fabio Massacci
Author:     Lawrence C Paulson
*)

section‹The Message Theory, Modified for SET›

theory Message_SET
imports Main "HOL-Library.Nat_Bijection"
begin

subsection‹General Lemmas›

text‹Needed occasionally with ‹spy_analz_tac›, e.g. in
‹analz_insert_Key_newK››

lemma Un_absorb3 [simp] : "A ∪ (B ∪ A) = B ∪ A"
by blast

text‹Collapses redundant cases in the huge protocol proofs›
lemmas disj_simps = disj_comms disj_left_absorb disj_assoc

text‹Effective with assumptions like @{term "K ∉ range pubK"} and
@{term "K ∉ invKey`range pubK"}›
lemma notin_image_iff: "(y ∉ f`I) = (∀i∈I. f i ≠ y)"
by blast

text‹Effective with the assumption @{term "KK ⊆ - (range(invKey o pubK))"}›
lemma disjoint_image_iff: "(A ⊆ - (f`I)) = (∀i∈I. f i ∉ 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}"

text‹Agents. We allow any number of certification authorities, cardholders
merchants, and payment gateways.›
datatype
agent = CA nat | Cardholder nat | Merchant nat | PG nat | Spy

text‹Messages›
datatype
msg = Agent  agent     ― ‹Agent names›
| Number nat       ― ‹Ordinary integers, timestamps, ...›
| Nonce  nat       ― ‹Unguessable nonces›
| Pan    nat       ― ‹Unguessable Primary Account Numbers (??)›
| Key    key       ― ‹Crypto keys›
| Hash   msg       ― ‹Hashing›
| MPair  msg msg   ― ‹Compound messages›
| Crypt  key msg   ― ‹Encryption, public- or shared-key›

(*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 nat_of_agent :: "agent ⇒ nat" where
"nat_of_agent == case_agent (curry prod_encode 0)
(curry prod_encode 1)
(curry prod_encode 2)
(curry prod_encode 3)
(prod_encode (4,0))"
― ‹maps each agent to a unique natural number, for specifications›

text‹The function is indeed injective›
lemma inj_nat_of_agent: "inj nat_of_agent"
by (simp add: nat_of_agent_def inj_on_def curry_def prod_encode_eq split: agent.split)

definition
(*Keys useful to decrypt elements of a message set*)
keysFor :: "msg set ⇒ key set"
where "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"

(*Monotonicity*)
lemma parts_mono: "G⊆H ==> parts(G) ⊆ parts(H)"
apply auto
apply (erule parts.induct)
apply (auto dest: Fst Snd Body)
done

subsubsection‹Inverse of keys›

(*Equations hold because constructors are injective; cannot prove for all f*)
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

lemma Cardholder_image_eq [simp]: "(Cardholder x ∈ Cardholder`A) = (x ∈ A)"
by auto

lemma CA_image_eq [simp]: "(CA x ∈ CA`A) = (x ∈ A)"
by auto

lemma Pan_image_eq [simp]: "(Pan x ∈ Pan`A) = (x ∈ A)"
by auto

lemma Pan_Key_image_eq [simp]: "(Pan x ∉ Key`A)"
by auto

lemma Nonce_Pan_image_eq [simp]: "(Nonce x ∉ Pan`A)"
by auto

lemma invKey_eq [simp]: "(invKey K = invKey K') = (K=K')"
apply safe
apply (drule_tac f = invKey in arg_cong, simp)
done

subsection‹keysFor operator›

lemma keysFor_empty [simp]: "keysFor {} = {}"
by (unfold keysFor_def, blast)

lemma keysFor_Un [simp]: "keysFor (H ∪ 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)

(*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_Pan [simp]: "keysFor (insert (Pan A) 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, blast+)
done

lemma parts_emptyE [elim!]: "X∈ parts{} ==> P"
by simp

(*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"
apply (subst insert_is_Un [of _ H])
apply (simp only: parts_Un)
done

(*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"
apply (simp add: Un_assoc)
apply (simp add: parts_insert [symmetric])
done

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)

(*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.›
declare parts_Un [THEN equalityD1, THEN subsetD, THEN 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_trans: "[| X∈ parts G;  G ⊆ parts H |] ==> X∈ parts H"
by (drule parts_mono, blast)

(*Cut*)
lemma parts_cut:
"[| Y∈ parts (insert X G);  X∈ parts H |] ==> Y∈ parts (G ∪ H)"
by (erule parts_trans, auto)

lemma parts_cut_eq [simp]: "X∈ parts H ==> parts (insert X H) = parts H"
by (force dest!: parts_cut intro: parts_insertI)

subsubsection‹Rewrite rules for pulling out atomic messages›

lemmas parts_insert_eq_I = equalityI [OF subsetI parts_insert_subset]

lemma parts_insert_Agent [simp]:
"parts (insert (Agent agt) H) = insert (Agent agt) (parts H)"
apply (rule parts_insert_eq_I)
apply (erule parts.induct, auto)
done

lemma parts_insert_Nonce [simp]:
"parts (insert (Nonce N) H) = insert (Nonce N) (parts H)"
apply (rule parts_insert_eq_I)
apply (erule parts.induct, auto)
done

lemma parts_insert_Number [simp]:
"parts (insert (Number N) H) = insert (Number N) (parts H)"
apply (rule parts_insert_eq_I)
apply (erule parts.induct, auto)
done

lemma parts_insert_Key [simp]:
"parts (insert (Key K) H) = insert (Key K) (parts H)"
apply (rule parts_insert_eq_I)
apply (erule parts.induct, auto)
done

lemma parts_insert_Pan [simp]:
"parts (insert (Pan A) H) = insert (Pan A) (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 (erule parts.induct)
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 (erule parts.induct)
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 parts_image_Pan [simp]: "parts (Pan`A) = Pan`A"
apply auto
apply (erule parts.induct, auto)
done

(*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_tac "msg")
apply (simp_all (no_asm_simp) add: exI parts_insert2)
(*MPair case: blast_tac works out the necessary sum itself!*)
prefer 2 apply (blast elim!: add_leE)
(*Nonce case*)
apply (rename_tac nat)
apply (rule_tac x = "N + Suc nat" in exI)
apply (auto elim!: add_leE)
done

(* Ditto, for numbers.*)
lemma msg_Number_supply: "∃N. ∀n. N≤n ⟶ Number n ∉ parts {msg}"
apply (induct_tac "msg")
apply (simp_all (no_asm_simp) add: exI parts_insert2)
prefer 2 apply (blast elim!: add_leE)
apply (rename_tac nat)
apply (rule_tac x = "N + Suc nat" in exI, auto)
done

subsection‹Inductive relation "analz"›

text‹Inductive definition of "analz" -- what can be broken down from a set of
messages, including keys.  A form of downward closure.  Pairs can
be taken apart; messages decrypted with known keys.›

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"

(*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: Fst 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 (rule analz_subset_parts [THEN parts_mono, THEN subset_trans], simp)
apply (blast intro: analz_increasing [THEN parts_mono, THEN subsetD])
done

lemma analz_parts [simp]: "analz (parts H) = parts H"
apply auto
apply (erule analz.induct, auto)
done

lemmas analz_insertI = subset_insertI [THEN analz_mono, THEN [2] rev_subsetD]

subsubsection‹General equational properties›

lemma analz_empty [simp]: "analz{} = {}"
apply safe
apply (erule analz.induct, blast+)
done

(*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

(*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

(*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 analz_insert_Pan [simp]:
"analz (insert (Pan A) H) = insert (Pan A) (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)

(*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 "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)

(*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

lemma analz_image_Pan [simp]: "analz (Pan`A) = Pan`A"
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_trans: "[| X∈ analz G;  G ⊆ analz H |] ==> X∈ analz H"
by (drule analz_mono, blast)

(*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"
*)

(*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 clarify
apply (erule analz.induct)
apply (best intro: analz_mono [THEN subsetD])+
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)

(*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

(*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"

(*Monotonicity*)
lemma synth_mono: "G⊆H ==> synth(G) ⊆ synth(H)"
apply auto
apply (erule synth.induct)
apply (auto dest: Fst Snd Body)
done

(*NO Agent_synth, as any Agent name can be synthesized.  Ditto for 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"
inductive_cases Pan_synth   [elim!]: "Pan A ∈ synth H"

lemma synth_increasing: "H ⊆ synth(H)"
by blast

subsubsection‹Unions›

(*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, blast+)

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

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

(*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 Pan_synth_eq [simp]: "(Pan A ∈ synth H) = (Pan A ∈ 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"
apply (cut_tac H = "{}" in analz_synth_Un)
apply (simp (no_asm_use))
done

subsubsection‹For reasoning about the Fake rule in traces›

lemma parts_insert_subset_Un: "X∈ G ==> parts(insert X H) ⊆ parts G ∪ parts H"
by (rule subset_trans [OF parts_mono parts_Un_subset2], blast)

(*More specifically for Fake.  Very occasionally we could do with a version
of the form  parts{X} ⊆ synth (analz H) ∪ parts H *)
lemma Fake_parts_insert: "X ∈ synth (analz H) ==>
parts (insert X H) ⊆ synth (analz H) ∪ parts H"
apply (drule parts_insert_subset_Un)
apply (simp (no_asm_use))
apply blast
done

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])

(*H is sometimes (Key ` KK ∪ spies evs), so can't put 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) ")
prefer 2 apply (blast intro: analz_mono [THEN [2] rev_subsetD] analz_mono [THEN synth_mono, THEN [2] rev_subsetD])
apply (simp (no_asm_use))
apply blast
done

lemma analz_conj_parts [simp]:
"(X ∈ 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])

(*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

(*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 ‹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" "Pan PAN"]
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 PAN X Y K'

lemmas pushCrypts =
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" "Pan PAN"]
insert_commute [of "Crypt X K" "Hash X'"]
insert_commute [of "Crypt X K" "MPair X' Y"]
for X K C N PAN X' Y

text‹Cannot be added with ‹[simp]› -- messages should not always be
re-ordered.›
lemmas pushes = pushKeys pushCrypts

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)
*)
fun Fake_insert_tac ctxt =
dresolve_tac ctxt [impOfSubs @{thm Fake_analz_insert},
impOfSubs @{thm Fake_parts_insert}] THEN'
eresolve_tac ctxt [asm_rl, @{thm synth.Inj}];

fun Fake_insert_simp_tac ctxt i =
REPEAT (Fake_insert_tac ctxt 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)
(Rule_Insts.res_inst_tac ctxt [((("x", 1), Position.none), "X")] []
(insert_commute RS ssubst) 1),
(*...allowing further simplifications*)
simp_tac ctxt 1,
REPEAT (FIRSTGOAL (resolve_tac ctxt [allI,impI,notI,conjI,iffI])),
DEPTH_SOLVE (atomic_spy_analz_tac ctxt 1)]) i);
›
(*>*)

(*By default only o_apply is built-in.  But in the presence of eta-expansion
this means that some terms displayed as (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 (simp add: synth_mono analz_mono)

lemma Fake_analz_eq [simp]:
"X ∈ synth(analz H) ==> synth (analz (insert X H)) = synth (analz H)"
apply (drule Fake_analz_insert[of _ _ "H"])
apply (simp add: synth_increasing[THEN Un_absorb2])
apply (drule synth_mono)
apply (simp add: synth_idem)
apply (blast intro: synth_analz_mono [THEN [2] rev_subsetD])
done

text‹Two generalizations of ‹analz_insert_eq››
lemma gen_analz_insert_eq [rule_format]:
"X ∈ analz H ==> ∀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)
⟹ ∀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"
apply (rule subset_trans)
apply (erule_tac [2] Fake_parts_insert)
apply (simp add: parts_mono)
done

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
```