--- a/src/HOL/Auth/Guard/Extensions.thy Thu Oct 13 15:49:09 2022 +0100
+++ b/src/HOL/Auth/Guard/Extensions.thy Thu Oct 13 16:00:22 2022 +0100
@@ -306,7 +306,7 @@
"one_step p == \<forall>evs ev. ev#evs \<in> p \<longrightarrow> evs \<in> p"
lemma one_step_Cons [dest]: "\<lbrakk>one_step p; ev#evs \<in> p\<rbrakk> \<Longrightarrow> evs \<in> p"
- unfolding one_step_def by (blast)
+ unfolding one_step_def by blast
lemma one_step_app: "\<lbrakk>evs@evs' \<in> p; one_step p; [] \<in> p\<rbrakk> \<Longrightarrow> evs' \<in> p"
by (induct evs, auto)
@@ -320,7 +320,7 @@
lemma has_only_SaysD: "\<lbrakk>ev \<in> set evs; evs \<in> p; has_only_Says p\<rbrakk>
\<Longrightarrow> \<exists>A B X. ev = Says A B X"
- unfolding has_only_Says_def by (blast)
+ unfolding has_only_Says_def by blast
lemma in_has_only_Says [dest]: "\<lbrakk>has_only_Says p; evs \<in> p; ev \<in> set evs\<rbrakk>
\<Longrightarrow> \<exists>A B X. ev = Says A B X"
--- a/src/HOL/Auth/Guard/Proto.thy Thu Oct 13 15:49:09 2022 +0100
+++ b/src/HOL/Auth/Guard/Proto.thy Thu Oct 13 16:00:22 2022 +0100
@@ -164,12 +164,12 @@
lemma freshD: "fresh p R s n Ks evs \<Longrightarrow> (\<exists>evs1 evs2.
evs = evs2 @ ap' s R # evs1 \<and> Nonce n \<notin> used evs1 \<and> R \<in> p \<and> ok evs1 R s
\<and> Nonce n \<in> parts {apm' s R} \<and> apm' s R \<in> guard n Ks)"
- unfolding fresh_def by (blast)
+ unfolding fresh_def by blast
lemma freshI [intro]: "\<lbrakk>Nonce n \<notin> used evs1; R \<in> p; Nonce n \<in> parts {apm' s R};
ok evs1 R s; apm' s R \<in> guard n Ks\<rbrakk>
\<Longrightarrow> fresh p R s n Ks (list @ ap' s R # evs1)"
- unfolding fresh_def by (blast)
+ unfolding fresh_def by blast
lemma freshI': "\<lbrakk>Nonce n \<notin> used evs1; (l,r) \<in> p;
Nonce n \<in> parts {apm s (msg r)}; ok evs1 (l,r) s; apm s (msg r) \<in> guard n Ks\<rbrakk>
@@ -220,10 +220,10 @@
"safe Ks G \<equiv> \<forall>K. K \<in> Ks \<longrightarrow> Key K \<notin> analz G"
lemma safeD [dest]: "\<lbrakk>safe Ks G; K \<in> Ks\<rbrakk> \<Longrightarrow> Key K \<notin> analz G"
- unfolding safe_def by (blast)
+ unfolding safe_def by blast
lemma safe_insert: "safe Ks (insert X G) \<Longrightarrow> safe Ks G"
- unfolding safe_def by (blast)
+ unfolding safe_def by blast
lemma Guard_safe: "\<lbrakk>Guard n Ks G; safe Ks G\<rbrakk> \<Longrightarrow> Nonce n \<notin> analz G"
by (blast dest: Guard_invKey)
@@ -238,7 +238,7 @@
lemma preservD: "\<lbrakk>preserv p keys n Ks; evs \<in> tr p; Guard n Ks (spies evs);
safe Ks (spies evs); fresh p R' s' n Ks evs; R \<in> p; ok evs R s;
keys R' s' n evs \<subseteq> Ks\<rbrakk> \<Longrightarrow> apm' s R \<in> guard n Ks"
- unfolding preserv_def by (blast)
+ unfolding preserv_def by blast
lemma preservD': "\<lbrakk>preserv p keys n Ks; evs \<in> tr p; Guard n Ks (spies evs);
safe Ks (spies evs); fresh p R' s' n Ks evs; (l,Says A B X) \<in> p;
@@ -253,7 +253,7 @@
lemma monotonD [dest]: "\<lbrakk>keys R' s' n (ev # evs) \<subseteq> Ks; monoton p keys;
ev # evs \<in> tr p\<rbrakk> \<Longrightarrow> keys R' s' n evs \<subseteq> Ks"
- unfolding monoton_def by (blast)
+ unfolding monoton_def by blast
subsection\<open>guardedness theorem\<close>
@@ -328,13 +328,13 @@
secret R n s Ks \<in> parts (spies evs); secret R' n' s' Ks \<in> parts (spies evs);
apm' s R \<in> guard (nonce s n) Ks; apm' s' R' \<in> guard (nonce s n) Ks\<rbrakk> \<Longrightarrow>
secret R n s Ks = secret R' n' s' Ks"
- unfolding uniq_def by (blast)
+ unfolding uniq_def by blast
definition ord :: "proto \<Rightarrow> (rule \<Rightarrow> rule \<Rightarrow> bool) \<Rightarrow> bool" where
"ord p inff \<equiv> \<forall>R R'. R \<in> p \<longrightarrow> R' \<in> p \<longrightarrow> \<not> inff R R' \<longrightarrow> inff R' R"
lemma ordD: "\<lbrakk>ord p inff; \<not> inff R R'; R \<in> p; R' \<in> p\<rbrakk> \<Longrightarrow> inff R' R"
- unfolding ord_def by (blast)
+ unfolding ord_def by blast
definition uniq' :: "proto \<Rightarrow> (rule \<Rightarrow> rule \<Rightarrow> bool) \<Rightarrow> secfun \<Rightarrow> bool" where
"uniq' p inff secret \<equiv> \<forall>evs R R' n n' Ks s s'. R \<in> p \<longrightarrow> R' \<in> p \<longrightarrow>
--- a/src/HOL/Auth/KerberosIV.thy Thu Oct 13 15:49:09 2022 +0100
+++ b/src/HOL/Auth/KerberosIV.thy Thu Oct 13 16:00:22 2022 +0100
@@ -304,27 +304,27 @@
ev \<noteq> Says Kas A (Crypt (shrK A) \<lbrace>akey, Agent Peer, Ta,
(Crypt (shrK Peer) \<lbrace>Agent A, Agent Peer, akey, Ta\<rbrace>)\<rbrace>))
\<Longrightarrow> authKeys (ev # evs) = authKeys evs"
- unfolding authKeys_def by (auto)
+ unfolding authKeys_def by auto
lemma authKeys_insert:
"authKeys
(Says Kas A (Crypt (shrK A) \<lbrace>Key K, Agent Peer, Number Ta,
(Crypt (shrK Peer) \<lbrace>Agent A, Agent Peer, Key K, Number Ta\<rbrace>)\<rbrace>) # evs)
= insert K (authKeys evs)"
- unfolding authKeys_def by (auto)
+ unfolding authKeys_def by auto
lemma authKeys_simp:
"K \<in> authKeys
(Says Kas A (Crypt (shrK A) \<lbrace>Key K', Agent Peer, Number Ta,
(Crypt (shrK Peer) \<lbrace>Agent A, Agent Peer, Key K', Number Ta\<rbrace>)\<rbrace>) # evs)
\<Longrightarrow> K = K' | K \<in> authKeys evs"
- unfolding authKeys_def by (auto)
+ unfolding authKeys_def by auto
lemma authKeysI:
"Says Kas A (Crypt (shrK A) \<lbrace>Key K, Agent Tgs, Number Ta,
(Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key K, Number Ta\<rbrace>)\<rbrace>) \<in> set evs
\<Longrightarrow> K \<in> authKeys evs"
- unfolding authKeys_def by (auto)
+ unfolding authKeys_def by auto
lemma authKeys_used: "K \<in> authKeys evs \<Longrightarrow> Key K \<in> used evs"
by (simp add: authKeys_def, blast)
@@ -1044,7 +1044,7 @@
(with respect to a given trace). *)
lemma Serv_fresh_not_AKcryptSK:
"Key servK \<notin> used evs \<Longrightarrow> \<not> AKcryptSK authK servK evs"
- unfolding AKcryptSK_def by (blast)
+ unfolding AKcryptSK_def by blast
lemma authK_not_AKcryptSK:
"\<lbrakk> Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, tk\<rbrace>
--- a/src/HOL/Auth/KerberosIV_Gets.thy Thu Oct 13 15:49:09 2022 +0100
+++ b/src/HOL/Auth/KerberosIV_Gets.thy Thu Oct 13 16:00:22 2022 +0100
@@ -270,27 +270,27 @@
ev \<noteq> Says Kas A (Crypt (shrK A) \<lbrace>akey, Agent Peer, Ta,
(Crypt (shrK Peer) \<lbrace>Agent A, Agent Peer, akey, Ta\<rbrace>)\<rbrace>))
\<Longrightarrow> authKeys (ev # evs) = authKeys evs"
- unfolding authKeys_def by (auto)
+ unfolding authKeys_def by auto
lemma authKeys_insert:
"authKeys
(Says Kas A (Crypt (shrK A) \<lbrace>Key K, Agent Peer, Number Ta,
(Crypt (shrK Peer) \<lbrace>Agent A, Agent Peer, Key K, Number Ta\<rbrace>)\<rbrace>) # evs)
= insert K (authKeys evs)"
- unfolding authKeys_def by (auto)
+ unfolding authKeys_def by auto
lemma authKeys_simp:
"K \<in> authKeys
(Says Kas A (Crypt (shrK A) \<lbrace>Key K', Agent Peer, Number Ta,
(Crypt (shrK Peer) \<lbrace>Agent A, Agent Peer, Key K', Number Ta\<rbrace>)\<rbrace>) # evs)
\<Longrightarrow> K = K' | K \<in> authKeys evs"
- unfolding authKeys_def by (auto)
+ unfolding authKeys_def by auto
lemma authKeysI:
"Says Kas A (Crypt (shrK A) \<lbrace>Key K, Agent Tgs, Number Ta,
(Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key K, Number Ta\<rbrace>)\<rbrace>) \<in> set evs
\<Longrightarrow> K \<in> authKeys evs"
- unfolding authKeys_def by (auto)
+ unfolding authKeys_def by auto
lemma authKeys_used: "K \<in> authKeys evs \<Longrightarrow> Key K \<in> used evs"
by (simp add: authKeys_def, blast)
@@ -905,7 +905,7 @@
(with respect to a given trace). *)
lemma Serv_fresh_not_AKcryptSK:
"Key servK \<notin> used evs \<Longrightarrow> \<not> AKcryptSK authK servK evs"
- unfolding AKcryptSK_def by (blast)
+ unfolding AKcryptSK_def by blast
lemma authK_not_AKcryptSK:
"\<lbrakk> Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, tk\<rbrace>
--- a/src/HOL/Auth/Message.thy Thu Oct 13 15:49:09 2022 +0100
+++ b/src/HOL/Auth/Message.thy Thu Oct 13 16:00:22 2022 +0100
@@ -78,7 +78,7 @@
text\<open>Monotonicity\<close>
-lemma parts_mono_aux: "\<lbrakk>G \<subseteq> H; x \<in> parts G\<rbrakk> \<Longrightarrow> x \<in> parts H"
+lemma parts_mono_aux: "\<lbrakk>G \<subseteq> H; X \<in> parts G\<rbrakk> \<Longrightarrow> X \<in> parts H"
by (erule parts.induct) (auto dest: parts.Fst parts.Snd parts.Body)
lemma parts_mono: "G \<subseteq> H \<Longrightarrow> parts(G) \<subseteq> parts(H)"
@@ -105,53 +105,53 @@
subsection\<open>keysFor operator\<close>
lemma keysFor_empty [simp]: "keysFor {} = {}"
- unfolding keysFor_def by (blast)
+ unfolding keysFor_def by blast
lemma keysFor_Un [simp]: "keysFor (H \<union> H') = keysFor H \<union> keysFor H'"
- unfolding keysFor_def by (blast)
+ unfolding keysFor_def by blast
lemma keysFor_UN [simp]: "keysFor (\<Union>i\<in>A. H i) = (\<Union>i\<in>A. keysFor (H i))"
- unfolding keysFor_def by (blast)
+ unfolding keysFor_def by blast
text\<open>Monotonicity\<close>
lemma keysFor_mono: "G \<subseteq> H \<Longrightarrow> keysFor(G) \<subseteq> keysFor(H)"
- unfolding keysFor_def by (blast)
+ unfolding keysFor_def by blast
lemma keysFor_insert_Agent [simp]: "keysFor (insert (Agent A) H) = keysFor H"
- unfolding keysFor_def by (auto)
+ unfolding keysFor_def by auto
lemma keysFor_insert_Nonce [simp]: "keysFor (insert (Nonce N) H) = keysFor H"
- unfolding keysFor_def by (auto)
+ unfolding keysFor_def by auto
lemma keysFor_insert_Number [simp]: "keysFor (insert (Number N) H) = keysFor H"
- unfolding keysFor_def by (auto)
+ unfolding keysFor_def by auto
lemma keysFor_insert_Key [simp]: "keysFor (insert (Key K) H) = keysFor H"
- unfolding keysFor_def by (auto)
+ unfolding keysFor_def by auto
lemma keysFor_insert_Hash [simp]: "keysFor (insert (Hash X) H) = keysFor H"
- unfolding keysFor_def by (auto)
+ unfolding keysFor_def by auto
lemma keysFor_insert_MPair [simp]: "keysFor (insert \<lbrace>X,Y\<rbrace> H) = keysFor H"
- unfolding keysFor_def by (auto)
+ unfolding keysFor_def by auto
lemma keysFor_insert_Crypt [simp]:
"keysFor (insert (Crypt K X) H) = insert (invKey K) (keysFor H)"
- unfolding keysFor_def by (auto)
+ unfolding keysFor_def by auto
lemma keysFor_image_Key [simp]: "keysFor (Key`E) = {}"
- unfolding keysFor_def by (auto)
+ unfolding keysFor_def by auto
lemma Crypt_imp_invKey_keysFor: "Crypt K X \<in> H \<Longrightarrow> invKey K \<in> keysFor H"
- unfolding keysFor_def by (blast)
+ unfolding keysFor_def by blast
subsection\<open>Inductive relation "parts"\<close>
lemma MPair_parts:
- "\<lbrakk>\<lbrace>X,Y\<rbrace> \<in> parts H;
+ "\<lbrakk>\<lbrace>X,Y\<rbrace> \<in> parts H;
\<lbrakk>X \<in> parts H; Y \<in> parts H\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
-by (blast dest: parts.Fst parts.Snd)
+ by (blast dest: parts.Fst parts.Snd)
declare MPair_parts [elim!] parts.Body [dest!]
text\<open>NB These two rules are UNSAFE in the formal sense, as they discard the
@@ -160,52 +160,53 @@
The Crypt rule is normally kept UNSAFE to avoid breaking up certificates.\<close>
lemma parts_increasing: "H \<subseteq> parts(H)"
-by blast
+ by blast
lemmas parts_insertI = subset_insertI [THEN parts_mono, THEN subsetD]
+lemma parts_empty_aux: "X \<in> parts{} \<Longrightarrow> False"
+ by (induction rule: parts.induct) (blast+)
+
lemma parts_empty [simp]: "parts{} = {}"
-apply safe
-apply (erule parts.induct, blast+)
-done
+ using parts_empty_aux by blast
lemma parts_emptyE [elim!]: "X\<in> parts{} \<Longrightarrow> P"
-by simp
+ by simp
text\<open>WARNING: loops if H = {Y}, therefore must not be repeated!\<close>
-lemma parts_singleton: "X\<in> parts H \<Longrightarrow> \<exists>Y\<in>H. X\<in> parts {Y}"
-by (erule parts.induct, fast+)
+lemma parts_singleton: "X \<in> parts H \<Longrightarrow> \<exists>Y\<in>H. X \<in> parts {Y}"
+ by (erule parts.induct, fast+)
subsubsection\<open>Unions\<close>
lemma parts_Un_subset1: "parts(G) \<union> parts(H) \<subseteq> parts(G \<union> H)"
-by (intro Un_least parts_mono Un_upper1 Un_upper2)
+ 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
+ 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)
+ by (intro equalityI parts_Un_subset1 parts_Un_subset2)
lemma parts_insert: "parts (insert X H) = parts {X} \<union> parts H"
-by (metis insert_is_Un parts_Un)
+ by (metis insert_is_Un parts_Un)
text\<open>TWO inserts to avoid looping. This rewrite is better than nothing.
But its behaviour can be strange.\<close>
lemma parts_insert2:
- "parts (insert X (insert Y H)) = parts {X} \<union> parts {Y} \<union> parts H"
-by (metis Un_commute Un_empty_right Un_insert_right insert_is_Un parts_Un)
+ "parts (insert X (insert Y H)) = parts {X} \<union> parts {Y} \<union> parts H"
+ by (metis Un_commute Un_empty_right Un_insert_right insert_is_Un parts_Un)
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)
+ 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
+ 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))"
@@ -214,7 +215,7 @@
lemma parts_image [simp]:
"parts (f ` A) = (\<Union>x\<in>A. parts {f x})"
apply auto
- apply (metis (mono_tags, opaque_lifting) image_iff parts_singleton)
+ apply (metis (mono_tags, opaque_lifting) image_iff parts_singleton)
apply (metis empty_subsetI image_eqI insert_absorb insert_subset parts_mono)
done
@@ -229,29 +230,29 @@
lemma parts_insert_subset: "insert X (parts H) \<subseteq> parts(insert X H)"
-by (blast intro: parts_mono [THEN [2] rev_subsetD])
+ by (blast intro: parts_mono [THEN [2] rev_subsetD])
subsubsection\<open>Idempotence and transitivity\<close>
lemma parts_partsD [dest!]: "X\<in> parts (parts H) \<Longrightarrow> X\<in> parts H"
-by (erule parts.induct, blast+)
+ by (erule parts.induct, blast+)
lemma parts_idem [simp]: "parts (parts H) = parts H"
-by blast
+ by blast
lemma parts_subset_iff [simp]: "(parts G \<subseteq> parts H) = (G \<subseteq> parts H)"
-by (metis parts_idem parts_increasing parts_mono subset_trans)
+ by (metis parts_idem parts_increasing parts_mono subset_trans)
lemma parts_trans: "\<lbrakk>X\<in> parts G; G \<subseteq> parts H\<rbrakk> \<Longrightarrow> X\<in> parts H"
-by (metis parts_subset_iff subsetD)
+ by (metis parts_subset_iff subsetD)
text\<open>Cut\<close>
lemma parts_cut:
- "\<lbrakk>Y\<in> parts (insert X G); X\<in> parts H\<rbrakk> \<Longrightarrow> Y\<in> parts (G \<union> H)"
-by (blast intro: parts_trans)
+ "\<lbrakk>Y\<in> parts (insert X G); X\<in> parts H\<rbrakk> \<Longrightarrow> Y\<in> parts (G \<union> H)"
+ by (blast intro: parts_trans)
lemma parts_cut_eq [simp]: "X\<in> parts H \<Longrightarrow> parts (insert X H) = parts H"
-by (metis insert_absorb parts_idem parts_insert)
+ by (metis insert_absorb parts_idem parts_insert)
subsubsection\<open>Rewrite rules for pulling out atomic messages\<close>
@@ -260,65 +261,65 @@
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
+ "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
+ "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
+ "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
+ "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
+ "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
+ "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 \<lbrace>X,Y\<rbrace> H) =
+ "parts (insert \<lbrace>X,Y\<rbrace> H) =
insert \<lbrace>X,Y\<rbrace> (parts (insert X (insert Y H)))"
-apply (rule equalityI)
-apply (rule subsetI)
-apply (erule parts.induct, auto)
-apply (blast intro: parts.Fst parts.Snd)+
-done
+ 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"
-by auto
+ by auto
text\<open>In any message, there is an upper bound N on its greatest nonce.\<close>
lemma msg_Nonce_supply: "\<exists>N. \<forall>n. N\<le>n \<longrightarrow> Nonce n \<notin> parts {msg}"
proof (induct msg)
case (Nonce n)
- show ?case
- by simp (metis Suc_n_not_le_n)
+ show ?case
+ by simp (metis Suc_n_not_le_n)
next
case (MPair X Y)
- then show ?case \<comment> \<open>metis works out the necessary sum itself!\<close>
- by (simp add: parts_insert2) (metis le_trans nat_le_linear)
+ then show ?case \<comment> \<open>metis works out the necessary sum itself!\<close>
+ by (simp add: parts_insert2) (metis le_trans nat_le_linear)
qed auto
subsection\<open>Inductive relation "analz"\<close>
@@ -335,30 +336,30 @@
| Fst: "\<lbrace>X,Y\<rbrace> \<in> analz H \<Longrightarrow> X \<in> analz H"
| Snd: "\<lbrace>X,Y\<rbrace> \<in> analz H \<Longrightarrow> Y \<in> analz H"
| Decrypt [dest]:
- "\<lbrakk>Crypt K X \<in> analz H; Key(invKey K) \<in> analz H\<rbrakk> \<Longrightarrow> X \<in> analz H"
+ "\<lbrakk>Crypt K X \<in> analz H; Key(invKey K) \<in> analz H\<rbrakk> \<Longrightarrow> X \<in> analz H"
text\<open>Monotonicity; Lemma 1 of Lowe's paper\<close>
lemma analz_mono: "G\<subseteq>H \<Longrightarrow> analz(G) \<subseteq> analz(H)"
-apply auto
-apply (erule analz.induct)
-apply (auto dest: analz.Fst analz.Snd)
-done
+ apply auto
+ apply (erule analz.induct)
+ apply (auto dest: analz.Fst analz.Snd)
+ done
text\<open>Making it safe speeds up proofs\<close>
lemma MPair_analz [elim!]:
- "\<lbrakk>\<lbrace>X,Y\<rbrace> \<in> analz H;
+ "\<lbrakk>\<lbrace>X,Y\<rbrace> \<in> analz H;
\<lbrakk>X \<in> analz H; Y \<in> analz H\<rbrakk> \<Longrightarrow> P
\<rbrakk> \<Longrightarrow> P"
-by (blast dest: analz.Fst analz.Snd)
+ by (blast dest: analz.Fst analz.Snd)
lemma analz_increasing: "H \<subseteq> analz(H)"
-by blast
+ by blast
lemma analz_subset_parts: "analz H \<subseteq> parts H"
-apply (rule subsetI)
-apply (erule analz.induct, blast+)
-done
+ apply (rule subsetI)
+ apply (erule analz.induct, blast+)
+ done
lemmas analz_into_parts = analz_subset_parts [THEN subsetD]
@@ -366,151 +367,151 @@
lemma parts_analz [simp]: "parts (analz H) = parts H"
-by (metis analz_increasing analz_subset_parts equalityI parts_mono parts_subset_iff)
+ 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
+ apply auto
+ apply (erule analz.induct, auto)
+ done
lemmas analz_insertI = subset_insertI [THEN analz_mono, THEN [2] rev_subsetD]
subsubsection\<open>General equational properties\<close>
lemma analz_empty [simp]: "analz{} = {}"
-apply safe
-apply (erule analz.induct, blast+)
-done
+ apply safe
+ apply (erule analz.induct, blast+)
+ done
text\<open>Converse fails: we can analz more from the union than from the
separate parts, as a key in one might decrypt a message in the other\<close>
lemma analz_Un: "analz(G) \<union> analz(H) \<subseteq> analz(G \<union> H)"
-by (intro Un_least analz_mono Un_upper1 Un_upper2)
+ 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])
+ by (blast intro: analz_mono [THEN [2] rev_subsetD])
subsubsection\<open>Rewrite rules for pulling out atomic messages\<close>
lemmas analz_insert_eq_I = equalityI [OF subsetI analz_insert]
lemma analz_insert_Agent [simp]:
- "analz (insert (Agent agt) H) = insert (Agent agt) (analz H)"
-apply (rule analz_insert_eq_I)
-apply (erule analz.induct, auto)
-done
+ "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
+ "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
+ "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
+ "analz (insert (Hash X) H) = insert (Hash X) (analz H)"
+ apply (rule analz_insert_eq_I)
+ apply (erule analz.induct, auto)
+ done
text\<open>Can only pull out Keys if they are not needed to decrypt the rest\<close>
lemma analz_insert_Key [simp]:
- "K \<notin> keysFor (analz H) \<Longrightarrow>
+ "K \<notin> keysFor (analz H) \<Longrightarrow>
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
+ apply (unfold keysFor_def)
+ apply (rule analz_insert_eq_I)
+ apply (erule analz.induct, auto)
+ done
lemma analz_insert_MPair [simp]:
- "analz (insert \<lbrace>X,Y\<rbrace> H) =
+ "analz (insert \<lbrace>X,Y\<rbrace> H) =
insert \<lbrace>X,Y\<rbrace> (analz (insert X (insert Y H)))"
-apply (rule equalityI)
-apply (rule subsetI)
-apply (erule analz.induct, auto)
-apply (erule analz.induct)
-apply (blast intro: analz.Fst analz.Snd)+
-done
+ apply (rule equalityI)
+ apply (rule subsetI)
+ apply (erule analz.induct, auto)
+ apply (erule analz.induct)
+ apply (blast intro: analz.Fst analz.Snd)+
+ done
text\<open>Can pull out enCrypted message if the Key is not known\<close>
lemma analz_insert_Crypt:
- "Key (invKey K) \<notin> analz H
+ "Key (invKey K) \<notin> analz H
\<Longrightarrow> analz (insert (Crypt K X) H) = insert (Crypt K X) (analz H)"
-apply (rule analz_insert_eq_I)
-apply (erule analz.induct, auto)
+ apply (rule analz_insert_eq_I)
+ apply (erule analz.induct, auto)
-done
+ done
lemma lemma1: "Key (invKey K) \<in> analz H \<Longrightarrow>
analz (insert (Crypt K X) H) \<subseteq>
insert (Crypt K X) (analz (insert X H))"
-apply (rule subsetI)
-apply (erule_tac x = x in analz.induct, auto)
-done
+ apply (rule subsetI)
+ apply (erule_tac x = x in analz.induct, auto)
+ done
lemma lemma2: "Key (invKey K) \<in> analz H \<Longrightarrow>
insert (Crypt K X) (analz (insert X H)) \<subseteq>
analz (insert (Crypt K X) H)"
-apply auto
-apply (erule_tac x = x in analz.induct, auto)
-apply (blast intro: analz_insertI analz.Decrypt)
-done
+ 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) \<in> analz H \<Longrightarrow>
+ "Key (invKey K) \<in> analz H \<Longrightarrow>
analz (insert (Crypt K X) H) =
insert (Crypt K X) (analz (insert X H))"
-by (intro equalityI lemma1 lemma2)
+ by (intro equalityI lemma1 lemma2)
text\<open>Case analysis: either the message is secure, or it is not! Effective,
but can cause subgoals to blow up! Use with \<open>if_split\<close>; apparently
\<open>split_tac\<close> does not cope with patterns such as \<^term>\<open>analz (insert
(Crypt K X) H)\<close>\<close>
lemma analz_Crypt_if [simp]:
- "analz (insert (Crypt K X) H) =
+ "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)
+ by (simp add: analz_insert_Crypt analz_insert_Decrypt)
text\<open>This rule supposes "for the sake of argument" that we have the key.\<close>
lemma analz_insert_Crypt_subset:
- "analz (insert (Crypt K X) H) \<subseteq>
+ "analz (insert (Crypt K X) H) \<subseteq>
insert (Crypt K X) (analz (insert X H))"
-apply (rule subsetI)
-apply (erule analz.induct, auto)
-done
+ 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
+ apply auto
+ apply (erule analz.induct, auto)
+ done
subsubsection\<open>Idempotence and transitivity\<close>
lemma analz_analzD [dest!]: "X\<in> analz (analz H) \<Longrightarrow> X\<in> analz H"
-by (erule analz.induct, blast+)
+ by (erule analz.induct, blast+)
lemma analz_idem [simp]: "analz (analz H) = analz H"
-by blast
+ by blast
lemma analz_subset_iff [simp]: "(analz G \<subseteq> analz H) = (G \<subseteq> analz H)"
-by (metis analz_idem analz_increasing analz_mono subset_trans)
+ by (metis analz_idem analz_increasing analz_mono subset_trans)
lemma analz_trans: "\<lbrakk>X\<in> analz G; G \<subseteq> analz H\<rbrakk> \<Longrightarrow> X\<in> analz H"
-by (drule analz_mono, blast)
+ by (drule analz_mono, blast)
text\<open>Cut; Lemma 2 of Lowe\<close>
lemma analz_cut: "\<lbrakk>Y\<in> analz (insert X H); X\<in> analz H\<rbrakk> \<Longrightarrow> Y\<in> analz H"
-by (erule analz_trans, blast)
+ by (erule analz_trans, blast)
(*Cut can be proved easily by induction on
"Y: analz (insert X H) \<Longrightarrow> X: analz H \<longrightarrow> Y: analz H"
@@ -520,41 +521,41 @@
the forwarding of unknown components (X). Without it, removing occurrences
of X can be very complicated.\<close>
lemma analz_insert_eq: "X\<in> analz H \<Longrightarrow> analz (insert X H) = analz H"
-by (metis analz_cut analz_insert_eq_I insert_absorb)
+ by (metis analz_cut analz_insert_eq_I insert_absorb)
text\<open>A congruence rule for "analz"\<close>
lemma analz_subset_cong:
- "\<lbrakk>analz G \<subseteq> analz G'; analz H \<subseteq> analz H'\<rbrakk>
+ "\<lbrakk>analz G \<subseteq> analz G'; analz H \<subseteq> analz H'\<rbrakk>
\<Longrightarrow> analz (G \<union> H) \<subseteq> analz (G' \<union> H')"
-by (metis Un_mono analz_Un analz_subset_iff subset_trans)
+ by (metis Un_mono analz_Un analz_subset_iff subset_trans)
lemma analz_cong:
- "\<lbrakk>analz G = analz G'; analz H = analz H'\<rbrakk>
+ "\<lbrakk>analz G = analz G'; analz H = analz H'\<rbrakk>
\<Longrightarrow> analz (G \<union> H) = analz (G' \<union> H')"
-by (intro equalityI analz_subset_cong, simp_all)
+ by (intro equalityI analz_subset_cong, simp_all)
lemma analz_insert_cong:
- "analz H = analz H' \<Longrightarrow> analz(insert X H) = analz(insert X H')"
-by (force simp only: insert_def intro!: analz_cong)
+ "analz H = analz H' \<Longrightarrow> analz(insert X H) = analz(insert X H')"
+ by (force simp only: insert_def intro!: analz_cong)
text\<open>If there are no pairs or encryptions then analz does nothing\<close>
lemma analz_trivial:
- "\<lbrakk>\<forall>X Y. \<lbrace>X,Y\<rbrace> \<notin> H; \<forall>X K. Crypt K X \<notin> H\<rbrakk> \<Longrightarrow> analz H = H"
-apply safe
-apply (erule analz.induct, blast+)
-done
+ "\<lbrakk>\<forall>X Y. \<lbrace>X,Y\<rbrace> \<notin> H; \<forall>X K. Crypt K X \<notin> H\<rbrakk> \<Longrightarrow> analz H = H"
+ apply safe
+ apply (erule analz.induct, blast+)
+ done
text\<open>These two are obsolete (with a single Spy) but cost little to prove...\<close>
lemma analz_UN_analz_lemma:
- "X\<in> analz (\<Union>i\<in>A. analz (H i)) \<Longrightarrow> X\<in> analz (\<Union>i\<in>A. H i)"
-apply (erule analz.induct)
-apply (blast intro: analz_mono [THEN [2] rev_subsetD])+
-done
+ "X\<in> analz (\<Union>i\<in>A. analz (H i)) \<Longrightarrow> 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])
+ by (blast intro: analz_UN_analz_lemma analz_mono [THEN [2] rev_subsetD])
subsection\<open>Inductive relation "synth"\<close>
@@ -583,132 +584,127 @@
The same holds for \<^term>\<open>Number\<close>\<close>
inductive_simps synth_simps [iff]:
- "Nonce n \<in> synth H"
- "Key K \<in> synth H"
- "Hash X \<in> synth H"
- "\<lbrace>X,Y\<rbrace> \<in> synth H"
- "Crypt K X \<in> synth H"
+ "Nonce n \<in> synth H"
+ "Key K \<in> synth H"
+ "Hash X \<in> synth H"
+ "\<lbrace>X,Y\<rbrace> \<in> synth H"
+ "Crypt K X \<in> synth H"
lemma synth_increasing: "H \<subseteq> synth(H)"
-by blast
+ by blast
subsubsection\<open>Unions\<close>
text\<open>Converse fails: we can synth more from the union than from the
separate parts, building a compound message using elements of each.\<close>
lemma synth_Un: "synth(G) \<union> synth(H) \<subseteq> synth(G \<union> H)"
-by (intro Un_least synth_mono Un_upper1 Un_upper2)
+ 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])
+ by (blast intro: synth_mono [THEN [2] rev_subsetD])
subsubsection\<open>Idempotence and transitivity\<close>
lemma synth_synthD [dest!]: "X\<in> synth (synth H) \<Longrightarrow> X\<in> synth H"
-by (erule synth.induct, auto)
+ by (erule synth.induct, auto)
lemma synth_idem: "synth (synth H) = synth H"
-by blast
+ by blast
lemma synth_subset_iff [simp]: "(synth G \<subseteq> synth H) = (G \<subseteq> synth H)"
-by (metis subset_trans synth_idem synth_increasing synth_mono)
+ by (metis subset_trans synth_idem synth_increasing synth_mono)
lemma synth_trans: "\<lbrakk>X\<in> synth G; G \<subseteq> synth H\<rbrakk> \<Longrightarrow> X\<in> synth H"
-by (drule synth_mono, blast)
+ by (drule synth_mono, blast)
text\<open>Cut; Lemma 2 of Lowe\<close>
lemma synth_cut: "\<lbrakk>Y\<in> synth (insert X H); X\<in> synth H\<rbrakk> \<Longrightarrow> Y\<in> synth H"
-by (erule synth_trans, blast)
+ by (erule synth_trans, blast)
lemma Crypt_synth_eq [simp]:
- "Key K \<notin> H \<Longrightarrow> (Crypt K X \<in> synth H) = (Crypt K X \<in> H)"
-by blast
+ "Key K \<notin> H \<Longrightarrow> (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}"
- unfolding keysFor_def by (blast)
+ "keysFor (synth H) = keysFor H \<union> invKey`{K. Key K \<in> H}"
+ unfolding keysFor_def by blast
subsubsection\<open>Combinations of parts, analz and synth\<close>
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
+ 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
+ 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
+ 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"
-by (metis Un_empty_right analz_synth_Un)
+ by (metis Un_empty_right analz_synth_Un)
subsubsection\<open>For reasoning about the Fake rule in traces\<close>
lemma parts_insert_subset_Un: "X\<in> G \<Longrightarrow> parts(insert X H) \<subseteq> parts G \<union> parts H"
-by (metis UnCI Un_upper2 insert_subset parts_Un parts_mono)
+ by (metis UnCI Un_upper2 insert_subset parts_Un parts_mono)
text\<open>More specifically for Fake. See also \<open>Fake_parts_sing\<close> below\<close>
lemma Fake_parts_insert:
- "X \<in> synth (analz H) \<Longrightarrow>
+ "X \<in> synth (analz H) \<Longrightarrow>
parts (insert X H) \<subseteq> synth (analz H) \<union> parts H"
-by (metis Un_commute analz_increasing insert_subset parts_analz parts_mono
- parts_synth synth_mono synth_subset_iff)
+ 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:
- "\<lbrakk>Z \<in> parts (insert X H); X \<in> synth (analz H)\<rbrakk>
+ "\<lbrakk>Z \<in> parts (insert X H); X \<in> synth (analz H)\<rbrakk>
\<Longrightarrow> Z \<in> synth (analz H) \<union> parts H"
-by (metis Fake_parts_insert subsetD)
+ by (metis Fake_parts_insert subsetD)
text\<open>\<^term>\<open>H\<close> is sometimes \<^term>\<open>Key ` KK \<union> spies evs\<close>, so can't put
\<^term>\<open>G=H\<close>.\<close>
lemma Fake_analz_insert:
- "X\<in> synth (analz G) \<Longrightarrow>
+ "X\<in> synth (analz G) \<Longrightarrow>
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)", force)
-apply (blast intro: analz_mono [THEN [2] rev_subsetD] analz_mono [THEN synth_mono, THEN [2] rev_subsetD])
-done
+ by (metis UnCI Un_commute Un_upper1 analz_analz_Un analz_mono analz_synth_Un insert_subset)
lemma analz_conj_parts [simp]:
- "(X \<in> analz H \<and> X \<in> parts H) = (X \<in> analz H)"
-by (blast intro: analz_subset_parts [THEN subsetD])
+ "(X \<in> analz H \<and> X \<in> parts H) = (X \<in> analz H)"
+ by (blast intro: analz_subset_parts [THEN 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 subsetD])
+ "(X \<in> analz H | X \<in> parts H) = (X \<in> parts H)"
+ by (blast intro: analz_subset_parts [THEN subsetD])
text\<open>Without this equation, other rules for synth and analz would yield
redundant cases\<close>
lemma MPair_synth_analz [iff]:
- "(\<lbrace>X,Y\<rbrace> \<in> synth (analz H)) =
- (X \<in> synth (analz H) \<and> Y \<in> synth (analz H))"
-by blast
+ "\<lbrace>X,Y\<rbrace> \<in> synth (analz H) \<longleftrightarrow> X \<in> synth (analz H) \<and> Y \<in> synth (analz H)"
+ by blast
lemma Crypt_synth_analz:
- "\<lbrakk>Key K \<in> analz H; Key (invKey K) \<in> analz H\<rbrakk>
+ "\<lbrakk>Key K \<in> analz H; Key (invKey K) \<in> analz H\<rbrakk>
\<Longrightarrow> (Crypt K X \<in> synth (analz H)) = (X \<in> synth (analz H))"
-by blast
-
+ by blast
lemma Hash_synth_analz [simp]:
- "X \<notin> synth (analz H)
+ "X \<notin> synth (analz H)
\<Longrightarrow> (Hash\<lbrace>X,Y\<rbrace> \<in> synth (analz H)) = (Hash\<lbrace>X,Y\<rbrace> \<in> analz H)"
-by blast
+ by blast
subsection\<open>HPair: a combination of Hash and MPair\<close>
@@ -734,43 +730,43 @@
unfolding HPair_def by simp
lemmas HPair_neqs = Agent_neq_HPair Nonce_neq_HPair Number_neq_HPair
- Key_neq_HPair Hash_neq_HPair Crypt_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 \<and> Y'=Y)"
-by (simp add: HPair_def)
+ by (simp add: HPair_def)
lemma MPair_eq_HPair [iff]:
- "(\<lbrace>X',Y'\<rbrace> = Hash[X] Y) = (X' = Hash\<lbrace>X,Y\<rbrace> \<and> Y'=Y)"
-by (simp add: HPair_def)
+ "(\<lbrace>X',Y'\<rbrace> = Hash[X] Y) = (X' = Hash\<lbrace>X,Y\<rbrace> \<and> Y'=Y)"
+ by (simp add: HPair_def)
lemma HPair_eq_MPair [iff]:
- "(Hash[X] Y = \<lbrace>X',Y'\<rbrace>) = (X' = Hash\<lbrace>X,Y\<rbrace> \<and> Y'=Y)"
-by (auto simp add: HPair_def)
+ "(Hash[X] Y = \<lbrace>X',Y'\<rbrace>) = (X' = Hash\<lbrace>X,Y\<rbrace> \<and> Y'=Y)"
+ by (auto simp add: HPair_def)
subsubsection\<open>Specialized laws, proved in terms of those for Hash and MPair\<close>
lemma keysFor_insert_HPair [simp]: "keysFor (insert (Hash[X] Y) H) = keysFor H"
-by (simp add: HPair_def)
+ by (simp add: HPair_def)
lemma parts_insert_HPair [simp]:
- "parts (insert (Hash[X] Y) H) =
+ "parts (insert (Hash[X] Y) H) =
insert (Hash[X] Y) (insert (Hash\<lbrace>X,Y\<rbrace>) (parts (insert Y H)))"
-by (simp add: HPair_def)
+ by (simp add: HPair_def)
lemma analz_insert_HPair [simp]:
- "analz (insert (Hash[X] Y) H) =
+ "analz (insert (Hash[X] Y) H) =
insert (Hash[X] Y) (insert (Hash\<lbrace>X,Y\<rbrace>) (analz (insert Y H)))"
-by (simp add: HPair_def)
+ by (simp add: HPair_def)
lemma HPair_synth_analz [simp]:
- "X \<notin> synth (analz H)
+ "X \<notin> synth (analz H)
\<Longrightarrow> (Hash[X] Y \<in> synth (analz H)) =
(Hash \<lbrace>X, Y\<rbrace> \<in> analz H \<and> Y \<in> synth (analz H))"
-by (auto simp add: HPair_def)
+ by (auto simp add: HPair_def)
text\<open>We do NOT want Crypt... messages broken up in protocols!!\<close>
@@ -830,13 +826,13 @@
(*The key-free part of a set of messages can be removed from the scope of the analz operator.*)
lemma analz_keyfree_into_Un: "\<lbrakk>X \<in> analz (G \<union> H); G \<subseteq> keyfree\<rbrakk> \<Longrightarrow> X \<in> parts G \<union> analz H"
-apply (erule analz.induct, auto dest: parts.Body)
-apply (metis Un_absorb2 keyfree_KeyE parts_Un parts_keyfree UnI2)
-done
+ apply (erule analz.induct, auto dest: parts.Body)
+ apply (metis Un_absorb2 keyfree_KeyE parts_Un parts_keyfree UnI2)
+ done
subsection\<open>Tactics useful for many protocol proofs\<close>
ML
-\<open>
+ \<open>
(*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 *)
@@ -882,47 +878,47 @@
lemma Crypt_notin_image_Key [simp]: "Crypt K X \<notin> Key ` A"
-by auto
+ by auto
lemma Hash_notin_image_Key [simp] :"Hash X \<notin> Key ` A"
-by auto
+ by auto
lemma synth_analz_mono: "G\<subseteq>H \<Longrightarrow> synth (analz(G)) \<subseteq> synth (analz(H))"
-by (iprover intro: synth_mono analz_mono)
+ by (iprover intro: synth_mono analz_mono)
lemma Fake_analz_eq [simp]:
- "X \<in> synth(analz H) \<Longrightarrow> 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)
+ "X \<in> synth(analz H) \<Longrightarrow> 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\<open>Two generalizations of \<open>analz_insert_eq\<close>\<close>
lemma gen_analz_insert_eq [rule_format]:
- "X \<in> analz H \<Longrightarrow> \<forall>G. H \<subseteq> G \<longrightarrow> analz (insert X G) = analz G"
-by (blast intro: analz_cut analz_insertI analz_mono [THEN [2] rev_subsetD])
+ "X \<in> analz H \<Longrightarrow> \<forall>G. H \<subseteq> G \<longrightarrow> 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 \<in> synth (analz H)
+ "X \<in> synth (analz H)
\<Longrightarrow> \<forall>G. H \<subseteq> G \<longrightarrow> (Key K \<in> analz (insert X G)) = (Key K \<in> analz G)"
-apply (erule synth.induct)
-apply (simp_all add: gen_analz_insert_eq subset_trans [OF _ subset_insertI])
-done
+ apply (erule synth.induct)
+ apply (simp_all add: gen_analz_insert_eq subset_trans [OF _ subset_insertI])
+ done
lemma Fake_parts_sing:
- "X \<in> synth (analz H) \<Longrightarrow> parts{X} \<subseteq> synth (analz H) \<union> parts H"
-by (metis Fake_parts_insert empty_subsetI insert_mono parts_mono subset_trans)
+ "X \<in> synth (analz H) \<Longrightarrow> parts{X} \<subseteq> synth (analz H) \<union> 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 = \<open>
Scan.succeed (SIMPLE_METHOD' o spy_analz_tac)\<close>
- "for proving the Fake case when analz is involved"
+ "for proving the Fake case when analz is involved"
method_setup atomic_spy_analz = \<open>
Scan.succeed (SIMPLE_METHOD' o atomic_spy_analz_tac)\<close>
- "for debugging spy_analz"
+ "for debugging spy_analz"
method_setup Fake_insert_simp = \<open>
Scan.succeed (SIMPLE_METHOD' o Fake_insert_simp_tac)\<close>
- "for debugging spy_analz"
+ "for debugging spy_analz"
end
--- a/src/HOL/Auth/OtwayReesBella.thy Thu Oct 13 15:49:09 2022 +0100
+++ b/src/HOL/Auth/OtwayReesBella.thy Thu Oct 13 16:00:22 2022 +0100
@@ -185,7 +185,7 @@
A \<notin> bad; evs \<in> orb\<rbrakk>
\<Longrightarrow> Says A B \<lbrace>Nonce M, Agent A, Agent B, Crypt (shrK A) \<lbrace>Nonce Na, Nonce M, Agent A, Agent B\<rbrace>\<rbrace> \<in> set evs"
apply (erule rev_mp, erule orb.induct, parts_explicit, simp_all)
-apply (blast)
+apply blast
done
@@ -312,7 +312,7 @@
txt\<open>Oops\<close>
prefer 4 apply (blast dest: analz_insert_freshCryptK)
txt\<open>OR4 - ii\<close>
-prefer 3 apply (blast)
+prefer 3 apply blast
txt\<open>OR3\<close>
(*adding Gets_imp_ and Says_imp_ for efficiency*)
apply (blast dest:
--- a/src/HOL/Auth/Public.thy Thu Oct 13 15:49:09 2022 +0100
+++ b/src/HOL/Auth/Public.thy Thu Oct 13 16:00:22 2022 +0100
@@ -95,7 +95,7 @@
by blast
lemma symKeys_invKey_iff [iff]: "(invKey K \<in> symKeys) = (K \<in> symKeys)"
- unfolding symKeys_def by (auto)
+ unfolding symKeys_def by auto
lemma analz_symKeys_Decrypt:
"\<lbrakk>Crypt K X \<in> analz H; K \<in> symKeys; Key K \<in> analz H\<rbrakk>
--- a/src/HOL/Auth/Yahalom.thy Thu Oct 13 15:49:09 2022 +0100
+++ b/src/HOL/Auth/Yahalom.thy Thu Oct 13 16:00:22 2022 +0100
@@ -332,7 +332,7 @@
"Says Server A
\<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, na, Nonce NB\<rbrace>, X\<rbrace>
\<in> set evs \<Longrightarrow> KeyWithNonce K NB evs"
- unfolding KeyWithNonce_def by (blast)
+ unfolding KeyWithNonce_def by blast
lemma KeyWithNonce_Says [simp]:
"KeyWithNonce K NB (Says S A X # evs) =
@@ -354,7 +354,7 @@
(with respect to a given trace).\<close>
lemma fresh_not_KeyWithNonce:
"Key K \<notin> used evs \<Longrightarrow> \<not> KeyWithNonce K NB evs"
- unfolding KeyWithNonce_def by (blast)
+ unfolding KeyWithNonce_def by blast
text\<open>The Server message associates K with NB' and therefore not with any
other nonce NB.\<close>