--- a/src/HOL/Auth/Message.thy Sat Apr 26 12:38:17 2003 +0200
+++ b/src/HOL/Auth/Message.thy Sat Apr 26 12:38:42 2003 +0200
@@ -7,11 +7,10 @@
Inductive relations "parts", "analz" and "synth"
*)
-theory Message = Main
-files ("Message_lemmas.ML"):
+theory Message = Main:
(*Needed occasionally with spy_analz_tac, e.g. in analz_insert_Key_newK*)
-lemma [simp] : "A Un (B Un A) = B Un A"
+lemma [simp] : "A \<union> (B \<union> A) = B \<union> A"
by blast
types
@@ -83,6 +82,238 @@
done
+(*Equations hold because constructors are injective; cannot prove for all f*)
+lemma Friend_image_eq [simp]: "(Friend x \<in> Friend`A) = (x:A)"
+by auto
+
+lemma Key_image_eq [simp]: "(Key x \<in> Key`A) = (x\<in>A)"
+by auto
+
+lemma Nonce_Key_image_eq [simp]: "(Nonce x \<notin> Key`A)"
+by auto
+
+
+(** Inverse of keys **)
+
+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 \<union> H') = keysFor H \<union> keysFor H'"
+by (unfold keysFor_def, blast)
+
+lemma keysFor_UN [simp]: "keysFor (\<Union>i\<in>A. H i) = (\<Union>i\<in>A. keysFor (H i))"
+by (unfold keysFor_def, blast)
+
+(*Monotonicity*)
+lemma keysFor_mono: "G\<subseteq>H ==> keysFor(G) \<subseteq> keysFor(H)"
+by (unfold keysFor_def, blast)
+
+lemma keysFor_insert_Agent [simp]: "keysFor (insert (Agent A) H) = keysFor H"
+by (unfold keysFor_def, auto)
+
+lemma keysFor_insert_Nonce [simp]: "keysFor (insert (Nonce N) H) = keysFor H"
+by (unfold keysFor_def, auto)
+
+lemma keysFor_insert_Number [simp]: "keysFor (insert (Number N) H) = keysFor H"
+by (unfold keysFor_def, auto)
+
+lemma keysFor_insert_Key [simp]: "keysFor (insert (Key K) H) = keysFor H"
+by (unfold keysFor_def, auto)
+
+lemma keysFor_insert_Hash [simp]: "keysFor (insert (Hash X) H) = keysFor H"
+by (unfold keysFor_def, auto)
+
+lemma keysFor_insert_MPair [simp]: "keysFor (insert {|X,Y|} H) = keysFor H"
+by (unfold keysFor_def, auto)
+
+lemma keysFor_insert_Crypt [simp]:
+ "keysFor (insert (Crypt K X) H) = insert (invKey K) (keysFor H)"
+apply (unfold keysFor_def, auto)
+done
+
+lemma keysFor_image_Key [simp]: "keysFor (Key`E) = {}"
+by (unfold keysFor_def, auto)
+
+lemma Crypt_imp_invKey_keysFor: "Crypt K X \<in> H ==> invKey K \<in> keysFor H"
+by (unfold keysFor_def, blast)
+
+
+subsection{*Inductive relation "parts"*}
+
+lemma MPair_parts:
+ "[| {|X,Y|} \<in> parts H;
+ [| X \<in> parts H; Y \<in> parts H |] ==> P |] ==> P"
+by (blast dest: parts.Fst parts.Snd)
+
+declare MPair_parts [elim!] parts.Body [dest!]
+text{*NB These two rules are UNSAFE in the formal sense, as they discard the
+ compound message. They work well on THIS FILE.
+ @{text MPair_parts} is left as SAFE because it speeds up proofs.
+ The Crypt rule is normally kept UNSAFE to avoid breaking up certificates.*}
+
+lemma parts_increasing: "H \<subseteq> parts(H)"
+by blast
+
+lemmas parts_insertI = subset_insertI [THEN parts_mono, THEN subsetD, standard]
+
+lemma parts_empty [simp]: "parts{} = {}"
+apply safe
+apply (erule parts.induct, blast+)
+done
+
+lemma parts_emptyE [elim!]: "X\<in> parts{} ==> P"
+by simp
+
+(*WARNING: loops if H = {Y}, therefore must not be repeated!*)
+lemma parts_singleton: "X\<in> parts H ==> \<exists>Y\<in>H. X\<in> parts {Y}"
+by (erule parts.induct, blast+)
+
+
+(** Unions **)
+
+lemma parts_Un_subset1: "parts(G) \<union> parts(H) \<subseteq> parts(G \<union> H)"
+by (intro Un_least parts_mono Un_upper1 Un_upper2)
+
+lemma parts_Un_subset2: "parts(G \<union> H) \<subseteq> parts(G) \<union> parts(H)"
+apply (rule subsetI)
+apply (erule parts.induct, blast+)
+done
+
+lemma parts_Un [simp]: "parts(G \<union> H) = parts(G) \<union> parts(H)"
+by (intro equalityI parts_Un_subset1 parts_Un_subset2)
+
+lemma parts_insert: "parts (insert X H) = parts {X} \<union> parts H"
+apply (subst insert_is_Un [of _ H])
+apply (simp only: parts_Un)
+done
+
+(*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} \<union> parts {Y} \<union> parts H"
+apply (simp add: Un_assoc)
+apply (simp add: parts_insert [symmetric])
+done
+
+lemma parts_UN_subset1: "(\<Union>x\<in>A. parts(H x)) \<subseteq> parts(\<Union>x\<in>A. H x)"
+by (intro UN_least parts_mono UN_upper)
+
+lemma parts_UN_subset2: "parts(\<Union>x\<in>A. H x) \<subseteq> (\<Union>x\<in>A. parts(H x))"
+apply (rule subsetI)
+apply (erule parts.induct, blast+)
+done
+
+lemma parts_UN [simp]: "parts(\<Union>x\<in>A. H x) = (\<Union>x\<in>A. parts(H x))"
+by (intro equalityI parts_UN_subset1 parts_UN_subset2)
+
+(*Added to simplify arguments to parts, analz and synth.
+ NOTE: the UN versions are no longer used!*)
+
+
+text{*This allows @{text blast} to simplify occurrences of
+ @{term "parts(G\<union>H)"} in the assumption.*}
+declare parts_Un [THEN equalityD1, THEN subsetD, THEN UnE, elim!]
+
+
+lemma parts_insert_subset: "insert X (parts H) \<subseteq> parts(insert X H)"
+by (blast intro: parts_mono [THEN [2] rev_subsetD])
+
+(** Idempotence and transitivity **)
+
+lemma parts_partsD [dest!]: "X\<in> parts (parts H) ==> X\<in> parts H"
+by (erule parts.induct, blast+)
+
+lemma parts_idem [simp]: "parts (parts H) = parts H"
+by blast
+
+lemma parts_trans: "[| X\<in> parts G; G \<subseteq> parts H |] ==> X\<in> parts H"
+by (drule parts_mono, blast)
+
+(*Cut*)
+lemma parts_cut: "[| Y\<in> parts (insert X G); X\<in> parts H |]
+ ==> Y\<in> parts (G \<union> H)"
+apply (erule parts_trans, auto)
+done
+
+lemma parts_cut_eq [simp]: "X\<in> parts H ==> parts (insert X H) = parts H"
+by (force dest!: parts_cut intro: parts_insertI)
+
+
+(** 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 (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
+
+
+(*In any message, there is an upper bound N on its greatest nonce.*)
+lemma msg_Nonce_supply: "\<exists>N. \<forall>n. N\<le>n --> Nonce n \<notin> 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 (rule_tac x = "N + Suc nat" in exI)
+apply (auto elim!: add_leE)
+done
+
+
+subsection{*Inductive relation "analz"*}
+
(** 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. **)
@@ -104,6 +335,211 @@
apply (auto dest: Fst Snd)
done
+text{*Making it safe speeds up proofs*}
+lemma MPair_analz [elim!]:
+ "[| {|X,Y|} \<in> analz H;
+ [| X \<in> analz H; Y \<in> analz H |] ==> P
+ |] ==> P"
+by (blast dest: analz.Fst analz.Snd)
+
+lemma analz_increasing: "H \<subseteq> analz(H)"
+by blast
+
+lemma analz_subset_parts: "analz H \<subseteq> parts H"
+apply (rule subsetI)
+apply (erule analz.induct, blast+)
+done
+
+lemmas not_parts_not_analz = analz_subset_parts [THEN contra_subsetD, standard]
+
+
+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, standard]
+
+(** 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) \<union> analz(H) \<subseteq> analz(G \<union> H)"
+by (intro Un_least analz_mono Un_upper1 Un_upper2)
+
+lemma analz_insert: "insert X (analz H) \<subseteq> analz(insert X H)"
+by (blast intro: analz_mono [THEN [2] rev_subsetD])
+
+(** 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 \<notin> keysFor (analz H) ==>
+ analz (insert (Key K) H) = insert (Key K) (analz H)"
+apply (unfold keysFor_def)
+apply (rule analz_insert_eq_I)
+apply (erule analz.induct, auto)
+done
+
+lemma analz_insert_MPair [simp]: "analz (insert {|X,Y|} H) =
+ insert {|X,Y|} (analz (insert X (insert Y H)))"
+apply (rule equalityI)
+apply (rule subsetI)
+apply (erule analz.induct, auto)
+apply (erule analz.induct)
+apply (blast intro: analz.Fst analz.Snd)+
+done
+
+(*Can pull out enCrypted message if the Key is not known*)
+lemma analz_insert_Crypt:
+ "Key (invKey K) \<notin> analz H
+ ==> analz (insert (Crypt K X) H) = insert (Crypt K X) (analz H)"
+apply (rule analz_insert_eq_I)
+apply (erule analz.induct, auto)
+
+done
+
+lemma lemma1: "Key (invKey K) \<in> analz H ==>
+ analz (insert (Crypt K X) H) \<subseteq>
+ insert (Crypt K X) (analz (insert X H))"
+apply (rule subsetI)
+apply (erule_tac xa = x in analz.induct, auto)
+done
+
+lemma lemma2: "Key (invKey K) \<in> analz H ==>
+ insert (Crypt K X) (analz (insert X H)) \<subseteq>
+ analz (insert (Crypt K X) H)"
+apply auto
+apply (erule_tac xa = x in analz.induct, auto)
+apply (blast intro: analz_insertI analz.Decrypt)
+done
+
+lemma analz_insert_Decrypt: "Key (invKey K) \<in> analz H ==>
+ analz (insert (Crypt K X) H) =
+ insert (Crypt K X) (analz (insert X H))"
+by (intro equalityI lemma1 lemma2)
+
+(*Case analysis: either the message is secure, or it is not!
+ Effective, but can cause subgoals to blow up!
+ Use with split_if; 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) \<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)
+
+
+(*This rule supposes "for the sake of argument" that we have the key.*)
+lemma analz_insert_Crypt_subset: "analz (insert (Crypt K X) H) \<subseteq>
+ insert (Crypt K X) (analz (insert X H))"
+apply (rule subsetI)
+apply (erule analz.induct, auto)
+done
+
+
+lemma analz_image_Key [simp]: "analz (Key`N) = Key`N"
+apply auto
+apply (erule analz.induct, auto)
+done
+
+
+(** Idempotence and transitivity **)
+
+lemma analz_analzD [dest!]: "X\<in> analz (analz H) ==> X\<in> analz H"
+by (erule analz.induct, blast+)
+
+lemma analz_idem [simp]: "analz (analz H) = analz H"
+by blast
+
+lemma analz_trans: "[| X\<in> analz G; G \<subseteq> analz H |] ==> X\<in> analz H"
+by (drule analz_mono, blast)
+
+(*Cut; Lemma 2 of Lowe*)
+lemma analz_cut: "[| Y\<in> analz (insert X H); X\<in> analz H |] ==> Y\<in> 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\<in> analz H ==> analz (insert X H) = analz H"
+by (blast intro: analz_cut analz_insertI)
+
+
+(** A congruence rule for "analz" **)
+
+lemma analz_subset_cong: "[| analz G \<subseteq> analz G'; analz H \<subseteq> analz H'
+ |] ==> analz (G \<union> H) \<subseteq> analz (G' \<union> 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 \<union> H) = analz (G' \<union> H')"
+apply (intro equalityI analz_subset_cong, simp_all)
+done
+
+
+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: "[| \<forall>X Y. {|X,Y|} \<notin> H; \<forall>X K. Crypt K X \<notin> H |] ==> analz H = H"
+apply safe
+apply (erule analz.induct, blast+)
+done
+
+(*These two are obsolete (with a single Spy) but cost little to prove...*)
+lemma analz_UN_analz_lemma: "X\<in> analz (\<Union>i\<in>A. analz (H i)) ==> X\<in> analz (\<Union>i\<in>A. H i)"
+apply (erule analz.induct)
+apply (blast intro: analz_mono [THEN [2] rev_subsetD])+
+done
+
+lemma analz_UN_analz [simp]: "analz (\<Union>i\<in>A. analz (H i)) = analz (\<Union>i\<in>A. H i)"
+by (blast intro: analz_UN_analz_lemma analz_mono [THEN [2] rev_subsetD])
+
+
+subsection{*Inductive relation "synth"*}
+
(** 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.
@@ -133,7 +569,376 @@
inductive_cases MPair_synth [elim!]: "{|X,Y|} \<in> synth H"
inductive_cases Crypt_synth [elim!]: "Crypt K X \<in> synth H"
-use "Message_lemmas.ML"
+
+lemma synth_increasing: "H \<subseteq> synth(H)"
+by blast
+
+(** 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) \<union> synth(H) \<subseteq> synth(G \<union> H)"
+by (intro Un_least synth_mono Un_upper1 Un_upper2)
+
+lemma synth_insert: "insert X (synth H) \<subseteq> synth(insert X H)"
+by (blast intro: synth_mono [THEN [2] rev_subsetD])
+
+(** Idempotence and transitivity **)
+
+lemma synth_synthD [dest!]: "X\<in> synth (synth H) ==> X\<in> synth H"
+by (erule synth.induct, blast+)
+
+lemma synth_idem: "synth (synth H) = synth H"
+by blast
+
+lemma synth_trans: "[| X\<in> synth G; G \<subseteq> synth H |] ==> X\<in> synth H"
+by (drule synth_mono, blast)
+
+(*Cut; Lemma 2 of Lowe*)
+lemma synth_cut: "[| Y\<in> synth (insert X H); X\<in> synth H |] ==> Y\<in> synth H"
+by (erule synth_trans, blast)
+
+lemma Agent_synth [simp]: "Agent A \<in> synth H"
+by blast
+
+lemma Number_synth [simp]: "Number n \<in> synth H"
+by blast
+
+lemma Nonce_synth_eq [simp]: "(Nonce N \<in> synth H) = (Nonce N \<in> H)"
+by blast
+
+lemma Key_synth_eq [simp]: "(Key K \<in> synth H) = (Key K \<in> H)"
+by blast
+
+lemma Crypt_synth_eq [simp]: "Key K \<notin> H ==> (Crypt K X \<in> synth H) = (Crypt K X \<in> H)"
+by blast
+
+
+lemma keysFor_synth [simp]:
+ "keysFor (synth H) = keysFor H \<union> invKey`{K. Key K \<in> H}"
+apply (unfold keysFor_def, blast)
+done
+
+
+(*** Combinations of parts, analz and synth ***)
+
+lemma parts_synth [simp]: "parts (synth H) = parts H \<union> 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 \<union> H) = analz (G \<union> H)"
+apply (intro equalityI analz_subset_cong)+
+apply simp_all
+done
+
+lemma analz_synth_Un [simp]: "analz (synth G \<union> H) = analz (G \<union> H) \<union> synth G"
+apply (rule equalityI)
+apply (rule subsetI)
+apply (erule analz.induct)
+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 \<union> synth H"
+apply (cut_tac H = "{}" in analz_synth_Un)
+apply (simp (no_asm_use))
+done
+
+
+(** For reasoning about the Fake rule in traces **)
+
+lemma parts_insert_subset_Un: "X\<in> G ==> parts(insert X H) \<subseteq> parts G \<union> 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} \<subseteq> synth (analz H) \<union> parts H *)
+lemma Fake_parts_insert: "X \<in> synth (analz H) ==>
+ parts (insert X H) \<subseteq> synth (analz H) \<union> parts H"
+apply (drule parts_insert_subset_Un)
+apply (simp (no_asm_use))
+apply blast
+done
+
+(*H is sometimes (Key ` KK \<union> spies evs), so can't put G=H*)
+lemma Fake_analz_insert: "X\<in> synth (analz G) ==>
+ analz (insert X H) \<subseteq> synth (analz G) \<union> analz (G \<union> H)"
+apply (rule subsetI)
+apply (subgoal_tac "x \<in> analz (synth (analz G) \<union> H) ")
+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 \<in> analz H & X \<in> parts H) = (X \<in> analz H)"
+by (blast intro: analz_subset_parts [THEN [2] rev_subsetD])
+
+lemma analz_disj_parts [simp]: "(X \<in> analz H | X \<in> parts H) = (X \<in> parts H)"
+by (blast intro: analz_subset_parts [THEN [2] rev_subsetD])
+
+(*Without this equation, other rules for synth and analz would yield
+ redundant cases*)
+lemma MPair_synth_analz [iff]:
+ "({|X,Y|} \<in> synth (analz H)) =
+ (X \<in> synth (analz H) & Y \<in> synth (analz H))"
+by blast
+
+lemma Crypt_synth_analz: "[| Key K \<in> analz H; Key (invKey K) \<in> analz H |]
+ ==> (Crypt K X \<in> synth (analz H)) = (X \<in> synth (analz H))"
+by blast
+
+
+lemma Hash_synth_analz [simp]: "X \<notin> synth (analz H)
+ ==> (Hash{|X,Y|} \<in> synth (analz H)) = (Hash{|X,Y|} \<in> analz H)"
+by blast
+
+
+subsection{*HPair: a combination of Hash and MPair*}
+
+(*** 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)
+
+
+(*** 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 \<notin> synth (analz H)
+ ==> (Hash[X] Y \<in> synth (analz H)) =
+ (Hash {|X, Y|} \<in> analz H & Y \<in> synth (analz H))"
+by (simp add: HPair_def)
+
+
+(*We do NOT want Crypt... messages broken up in protocols!!*)
+declare parts.Body [rule del]
+
+
+ML
+{*
+(*ML bindings for definitions and axioms*)
+
+val invKey = thm "invKey"
+val keysFor_def = thm "keysFor_def"
+val HPair_def = thm "HPair_def"
+val symKeys_def = thm "symKeys_def"
+
+structure parts =
+ struct
+ val induct = thm "parts.induct"
+ val Inj = thm "parts.Inj"
+ val Fst = thm "parts.Fst"
+ val Snd = thm "parts.Snd"
+ val Body = thm "parts.Body"
+ end
+
+structure analz =
+ struct
+ val induct = thm "analz.induct"
+ val Inj = thm "analz.Inj"
+ val Fst = thm "analz.Fst"
+ val Snd = thm "analz.Snd"
+ val Decrypt = thm "analz.Decrypt"
+ end
+
+
+(** Rewrites to push in Key and Crypt messages, so that other messages can
+ be pulled out using the analz_insert rules **)
+
+fun insComm x y = inst "x" x (inst "y" y insert_commute);
+
+bind_thms ("pushKeys",
+ map (insComm "Key ?K")
+ ["Agent ?C", "Nonce ?N", "Number ?N",
+ "Hash ?X", "MPair ?X ?Y", "Crypt ?X ?K'"]);
+
+bind_thms ("pushCrypts",
+ map (insComm "Crypt ?X ?K")
+ ["Agent ?C", "Nonce ?N", "Number ?N",
+ "Hash ?X'", "MPair ?X' ?Y"]);
+*}
+
+text{*Cannot be added with @{text "[simp]"} -- messages should not always be
+ re-ordered. *}
+lemmas pushes = pushKeys pushCrypts
+
+
+subsection{*Tactics useful for many protocol proofs*}
+ML
+{*
+val parts_mono = thm "parts_mono";
+val analz_mono = thm "analz_mono";
+val Key_image_eq = thm "Key_image_eq";
+val Nonce_Key_image_eq = thm "Nonce_Key_image_eq";
+val keysFor_Un = thm "keysFor_Un";
+val keysFor_mono = thm "keysFor_mono";
+val keysFor_image_Key = thm "keysFor_image_Key";
+val Crypt_imp_invKey_keysFor = thm "Crypt_imp_invKey_keysFor";
+val MPair_parts = thm "MPair_parts";
+val parts_increasing = thm "parts_increasing";
+val parts_insertI = thm "parts_insertI";
+val parts_empty = thm "parts_empty";
+val parts_emptyE = thm "parts_emptyE";
+val parts_singleton = thm "parts_singleton";
+val parts_Un_subset1 = thm "parts_Un_subset1";
+val parts_Un_subset2 = thm "parts_Un_subset2";
+val parts_insert = thm "parts_insert";
+val parts_insert2 = thm "parts_insert2";
+val parts_UN_subset1 = thm "parts_UN_subset1";
+val parts_UN_subset2 = thm "parts_UN_subset2";
+val parts_UN = thm "parts_UN";
+val parts_insert_subset = thm "parts_insert_subset";
+val parts_partsD = thm "parts_partsD";
+val parts_trans = thm "parts_trans";
+val parts_cut = thm "parts_cut";
+val parts_cut_eq = thm "parts_cut_eq";
+val parts_insert_eq_I = thm "parts_insert_eq_I";
+val parts_image_Key = thm "parts_image_Key";
+val MPair_analz = thm "MPair_analz";
+val analz_increasing = thm "analz_increasing";
+val analz_subset_parts = thm "analz_subset_parts";
+val not_parts_not_analz = thm "not_parts_not_analz";
+val parts_analz = thm "parts_analz";
+val analz_parts = thm "analz_parts";
+val analz_insertI = thm "analz_insertI";
+val analz_empty = thm "analz_empty";
+val analz_Un = thm "analz_Un";
+val analz_insert_Crypt_subset = thm "analz_insert_Crypt_subset";
+val analz_image_Key = thm "analz_image_Key";
+val analz_analzD = thm "analz_analzD";
+val analz_trans = thm "analz_trans";
+val analz_cut = thm "analz_cut";
+val analz_insert_eq = thm "analz_insert_eq";
+val analz_subset_cong = thm "analz_subset_cong";
+val analz_cong = thm "analz_cong";
+val analz_insert_cong = thm "analz_insert_cong";
+val analz_trivial = thm "analz_trivial";
+val analz_UN_analz = thm "analz_UN_analz";
+val synth_mono = thm "synth_mono";
+val synth_increasing = thm "synth_increasing";
+val synth_Un = thm "synth_Un";
+val synth_insert = thm "synth_insert";
+val synth_synthD = thm "synth_synthD";
+val synth_trans = thm "synth_trans";
+val synth_cut = thm "synth_cut";
+val Agent_synth = thm "Agent_synth";
+val Number_synth = thm "Number_synth";
+val Nonce_synth_eq = thm "Nonce_synth_eq";
+val Key_synth_eq = thm "Key_synth_eq";
+val Crypt_synth_eq = thm "Crypt_synth_eq";
+val keysFor_synth = thm "keysFor_synth";
+val parts_synth = thm "parts_synth";
+val analz_analz_Un = thm "analz_analz_Un";
+val analz_synth_Un = thm "analz_synth_Un";
+val analz_synth = thm "analz_synth";
+val parts_insert_subset_Un = thm "parts_insert_subset_Un";
+val Fake_parts_insert = thm "Fake_parts_insert";
+val Fake_analz_insert = thm "Fake_analz_insert";
+val analz_conj_parts = thm "analz_conj_parts";
+val analz_disj_parts = thm "analz_disj_parts";
+val MPair_synth_analz = thm "MPair_synth_analz";
+val Crypt_synth_analz = thm "Crypt_synth_analz";
+val Hash_synth_analz = thm "Hash_synth_analz";
+val pushes = thms "pushes";
+
+
+(*Prove base case (subgoal i) and simplify others. A typical base case
+ concerns Crypt K X \<notin> Key`shrK`bad and cannot be proved by rewriting
+ alone.*)
+fun prove_simple_subgoals_tac i =
+ force_tac (claset(), simpset() addsimps [image_eq_UN]) i THEN
+ ALLGOALS Asm_simp_tac
+
+(*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.
+ Abstraction over i is ESSENTIAL: it delays the dereferencing of claset
+ DEPENDS UPON "X" REFERRING TO THE FRADULENT MESSAGE *)
+
+(*Apply rules to break down assumptions of the form
+ Y \<in> parts(insert X H) and Y \<in> analz(insert X H)
+*)
+val Fake_insert_tac =
+ dresolve_tac [impOfSubs Fake_analz_insert,
+ impOfSubs Fake_parts_insert] THEN'
+ eresolve_tac [asm_rl, thm"synth.Inj"];
+
+fun Fake_insert_simp_tac ss i =
+ REPEAT (Fake_insert_tac i) THEN asm_full_simp_tac ss i;
+
+fun atomic_spy_analz_tac (cs,ss) = SELECT_GOAL
+ (Fake_insert_simp_tac ss 1
+ THEN
+ IF_UNSOLVED (Blast.depth_tac
+ (cs addIs [analz_insertI,
+ impOfSubs analz_subset_parts]) 4 1))
+
+(*The explicit claset and simpset arguments help it work with Isar*)
+fun gen_spy_analz_tac (cs,ss) i =
+ DETERM
+ (SELECT_GOAL
+ (EVERY
+ [ (*push in occurrences of X...*)
+ (REPEAT o CHANGED)
+ (res_inst_tac [("x1","X")] (insert_commute RS ssubst) 1),
+ (*...allowing further simplifications*)
+ simp_tac ss 1,
+ REPEAT (FIRSTGOAL (resolve_tac [allI,impI,notI,conjI,iffI])),
+ DEPTH_SOLVE (atomic_spy_analz_tac (cs,ss) 1)]) i)
+
+fun spy_analz_tac i = gen_spy_analz_tac (claset(), simpset()) 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 \<notin> Key ` A"
by auto
@@ -174,7 +979,7 @@
done
lemma Fake_parts_sing:
- "X \<in> synth (analz H) ==> parts{X} \<subseteq> synth (analz H) Un parts H";
+ "X \<in> synth (analz H) ==> parts{X} \<subseteq> synth (analz H) \<union> parts H";
apply (rule subset_trans)
apply (erule_tac [2] Fake_parts_insert)
apply (simp add: parts_mono)
@@ -200,4 +1005,5 @@
Fake_insert_simp_tac (Simplifier.get_local_simpset ctxt) 1)) *}
"for debugging spy_analz"
+
end