# HG changeset patch
# User paulson
# Date 1665755868 -3600
# Node ID efc220128637a326e2cae31b4e9800a449423338
# Parent  eb30869e72289fe05186d9f7067f9727d010a9fc# Parent  e7f9e5b3a36a24c84a00e42d88bf06deac4d0f64
merged

diff -r eb30869e7228 -r efc220128637 src/HOL/Auth/NS_Public.thy
--- a/src/HOL/Auth/NS_Public.thy	Fri Oct 14 14:39:52 2022 +0200
+++ b/src/HOL/Auth/NS_Public.thy	Fri Oct 14 14:57:48 2022 +0100
@@ -1,189 +1,194 @@
-(*  Title:      HOL/Auth/NS_Public.thy
+(*  Title:      HOL/Auth/NS_PublicX.thy
     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     Copyright   1996  University of Cambridge
-
-Inductive relation "ns_public" for the Needham-Schroeder Public-Key protocol.
-Version incorporating Lowe's fix (inclusion of B's identity in round 2).
 *)
 
-section\<open>Verifying the Needham-Schroeder-Lowe Public-Key Protocol\<close>
+section\<open>Verifying the Needham-Schroeder Public-Key Protocol\<close>
+
+text \<open>Flawed version, vulnerable to Lowe's attack.
+From Burrows, Abadi and Needham.  A Logic of Authentication.
+  Proc. Royal Soc. 426 (1989), p. 260\<close>
 
 theory NS_Public imports Public begin
 
 inductive_set ns_public :: "event list set"
-  where 
-         (*Initial trace is empty*)
+  where
    Nil:  "[] \<in> ns_public"
-
-         (*The spy MAY say anything he CAN say.  We do not expect him to
-           invent new nonces here, but he can also use NS1.  Common to
-           all similar protocols.*)
+   \<comment> \<open>Initial trace is empty\<close>
  | Fake: "\<lbrakk>evsf \<in> ns_public;  X \<in> synth (analz (spies evsf))\<rbrakk>
           \<Longrightarrow> Says Spy B X  # evsf \<in> ns_public"
-
-         (*Alice initiates a protocol run, sending a nonce to Bob*)
+   \<comment> \<open>The spy can say almost anything.\<close>
  | NS1:  "\<lbrakk>evs1 \<in> ns_public;  Nonce NA \<notin> used evs1\<rbrakk>
           \<Longrightarrow> Says A B (Crypt (pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>)
-                 # evs1  \<in>  ns_public"
-
-         (*Bob responds to Alice's message with a further nonce*)
+                # evs1  \<in>  ns_public"
+   \<comment> \<open>Alice initiates a protocol run, sending a nonce to Bob\<close>
  | NS2:  "\<lbrakk>evs2 \<in> ns_public;  Nonce NB \<notin> used evs2;
            Says A' B (Crypt (pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs2\<rbrakk>
           \<Longrightarrow> Says B A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>)
                 # evs2  \<in>  ns_public"
-
-         (*Alice proves her existence by sending NB back to Bob.*)
+   \<comment> \<open>Bob responds to Alice's message with a further nonce\<close>
  | NS3:  "\<lbrakk>evs3 \<in> ns_public;
            Says A  B (Crypt (pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs3;
-           Says B' A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>)
-              \<in> set evs3\<rbrakk>
+           Says B' A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) \<in> set evs3\<rbrakk>
           \<Longrightarrow> Says A B (Crypt (pubEK B) (Nonce NB)) # evs3 \<in> ns_public"
+   \<comment> \<open>Alice proves her existence by sending @{term NB} back to Bob.\<close>
 
 declare knows_Spy_partsEs [elim]
-declare knows_Spy_partsEs [elim]
 declare analz_into_parts [dest]
 declare Fake_parts_insert_in_Un [dest]
 
-(*A "possibility property": there are traces that reach the end*)
+text \<open>A "possibility property": there are traces that reach the end\<close>
 lemma "\<exists>NB. \<exists>evs \<in> ns_public. Says A B (Crypt (pubEK B) (Nonce NB)) \<in> set evs"
-apply (intro exI bexI)
-apply (rule_tac [2] ns_public.Nil [THEN ns_public.NS1, THEN ns_public.NS2, 
-                                   THEN ns_public.NS3], possibility)
-done
+  apply (intro exI bexI)
+   apply (rule_tac [2] ns_public.Nil [THEN ns_public.NS1, THEN ns_public.NS2, THEN ns_public.NS3])
+  by possibility
+
+
+subsection \<open>Inductive proofs about @{term ns_public}\<close>
 
 (** Theorems of the form X \<notin> parts (spies evs) imply that NOBODY
     sends messages containing X! **)
 
-(*Spy never sees another agent's private key! (unless it's bad at start)*)
-lemma Spy_see_priEK [simp]: 
-      "evs \<in> ns_public \<Longrightarrow> (Key (priEK A) \<in> parts (spies evs)) = (A \<in> bad)"
-by (erule ns_public.induct, auto)
+text \<open>Spy never sees another agent's private key! (unless it's bad at start)\<close>
+lemma Spy_see_priEK [simp]:
+  "evs \<in> ns_public \<Longrightarrow> (Key (priEK A) \<in> parts (spies evs)) = (A \<in> bad)"
+  by (erule ns_public.induct, auto)
+
+lemma Spy_analz_priEK [simp]:
+  "evs \<in> ns_public \<Longrightarrow> (Key (priEK A) \<in> analz (spies evs)) = (A \<in> bad)"
+  by auto
+
 
-lemma Spy_analz_priEK [simp]: 
-      "evs \<in> ns_public \<Longrightarrow> (Key (priEK A) \<in> analz (spies evs)) = (A \<in> bad)"
-by auto
+subsection \<open>Authenticity properties obtained from {term NS1}\<close>
 
-subsection\<open>Authenticity properties obtained from NS2\<close>
+text \<open>It is impossible to re-use a nonce in both {term NS1} and {term NS2}, provided the nonce
+  is secret.  (Honest users generate fresh nonces.)\<close>
+lemma no_nonce_NS1_NS2 [rule_format]:
+      "evs \<in> ns_public
+       \<Longrightarrow> Crypt (pubEK C) \<lbrace>NA', Nonce NA, Agent D\<rbrace> \<in> parts (spies evs) \<longrightarrow>
+           Crypt (pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace> \<in> parts (spies evs) \<longrightarrow>
+           Nonce NA \<in> analz (spies evs)"
+  by (induct rule: ns_public.induct) (auto intro: analz_insertI)
 
 
-(*It is impossible to re-use a nonce in both NS1 and NS2, provided the nonce
-  is secret.  (Honest users generate fresh nonces.)*)
-lemma no_nonce_NS1_NS2 [rule_format]: 
-      "evs \<in> ns_public 
-       \<Longrightarrow> Crypt (pubEK C) \<lbrace>NA', Nonce NA, Agent D\<rbrace> \<in> parts (spies evs) \<longrightarrow>
-           Crypt (pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace> \<in> parts (spies evs) \<longrightarrow>  
-           Nonce NA \<in> analz (spies evs)"
-apply (erule ns_public.induct, simp_all)
-apply (blast intro: analz_insertI)+
-done
-
-(*Unicity for NS1: nonce NA identifies agents A and B*)
-lemma unique_NA: 
-     "\<lbrakk>Crypt(pubEK B)  \<lbrace>Nonce NA, Agent A \<rbrace> \<in> parts(spies evs);  
-       Crypt(pubEK B') \<lbrace>Nonce NA, Agent A'\<rbrace> \<in> parts(spies evs);  
-       Nonce NA \<notin> analz (spies evs); evs \<in> ns_public\<rbrakk>
-      \<Longrightarrow> A=A' \<and> B=B'"
-apply (erule rev_mp, erule rev_mp, erule rev_mp)   
-apply (erule ns_public.induct, simp_all)
-(*Fake, NS1*)
-apply (blast intro: analz_insertI)+
-done
+text \<open>Unicity for {term NS1}: nonce {term NA} identifies agents {term A} and {term B}\<close>
+lemma unique_NA:
+  assumes NA: "Crypt(pubEK B)  \<lbrace>Nonce NA, Agent A \<rbrace> \<in> parts(spies evs)"
+              "Crypt(pubEK B') \<lbrace>Nonce NA, Agent A'\<rbrace> \<in> parts(spies evs)"
+              "Nonce NA \<notin> analz (spies evs)"
+    and evs: "evs \<in> ns_public"
+  shows "A=A' \<and> B=B'"
+  using evs NA
+  by (induction rule: ns_public.induct) (auto intro!: analz_insertI split: if_split_asm)
 
 
-(*Secrecy: Spy does not see the nonce sent in msg NS1 if A and B are secure
-  The major premise "Says A B ..." makes it a dest-rule, so we use
-  (erule rev_mp) rather than rule_format. *)
-theorem Spy_not_see_NA: 
-      "\<lbrakk>Says A B (Crypt(pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs;
-        A \<notin> bad;  B \<notin> bad;  evs \<in> ns_public\<rbrakk>                     
-       \<Longrightarrow> Nonce NA \<notin> analz (spies evs)"
-apply (erule rev_mp)   
-apply (erule ns_public.induct, simp_all, spy_analz)
-apply (blast dest: unique_NA intro: no_nonce_NS1_NS2)+
-done
+text \<open>Secrecy: Spy does not see the nonce sent in msg {term NS1} if {term A} and {term B} are secure
+  The major premise "Says A B ..." makes it a dest-rule, hence the given assumption order. \<close>
+theorem Spy_not_see_NA:
+  assumes NA: "Says A B (Crypt(pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs"
+              "A \<notin> bad" "B \<notin> bad"
+    and evs: "evs \<in> ns_public"
+  shows "Nonce NA \<notin> analz (spies evs)"
+  using evs NA
+proof (induction rule: ns_public.induct)
+  case (Fake evsf X B)
+  then show ?case
+    by spy_analz
+next
+  case (NS2 evs2 NB A' B NA A)
+  then show ?case
+    by simp (metis Says_imp_analz_Spy analz_into_parts parts.simps unique_NA usedI)
+next
+  case (NS3 evs3 A B NA B' NB)
+  then show ?case
+    by simp (meson Says_imp_analz_Spy analz_into_parts no_nonce_NS1_NS2)
+qed auto
 
 
-(*Authentication for A: if she receives message 2 and has used NA
-  to start a run, then B has sent message 2.*)
-lemma A_trusts_NS2_lemma [rule_format]: 
-   "\<lbrakk>A \<notin> bad;  B \<notin> bad;  evs \<in> ns_public\<rbrakk>                     
+text \<open>Authentication for {term A}: if she receives message 2 and has used {term NA}
+  to start a run, then {term B} has sent message 2.\<close>
+lemma A_trusts_NS2_lemma [rule_format]:
+   "\<lbrakk>A \<notin> bad;  B \<notin> bad;  evs \<in> ns_public\<rbrakk>
     \<Longrightarrow> Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace> \<in> parts (spies evs) \<longrightarrow>
         Says A B (Crypt(pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs \<longrightarrow>
         Says B A (Crypt(pubEK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) \<in> set evs"
-apply (erule ns_public.induct, simp_all)
-(*Fake, NS1*)
-apply (blast dest: Spy_not_see_NA)+
-done
+  by (erule ns_public.induct) (auto dest: Spy_not_see_NA unique_NA)
 
-theorem A_trusts_NS2: 
-     "\<lbrakk>Says A  B (Crypt(pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs;   
+theorem A_trusts_NS2:
+     "\<lbrakk>Says A  B (Crypt(pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs;
        Says B' A (Crypt(pubEK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) \<in> set evs;
-       A \<notin> bad;  B \<notin> bad;  evs \<in> ns_public\<rbrakk>                     
+       A \<notin> bad;  B \<notin> bad;  evs \<in> ns_public\<rbrakk>
       \<Longrightarrow> Says B A (Crypt(pubEK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) \<in> set evs"
-by (blast intro: A_trusts_NS2_lemma)
+  by (blast intro: A_trusts_NS2_lemma)
 
 
-(*If the encrypted message appears then it originated with Alice in NS1*)
+text \<open>If the encrypted message appears then it originated with Alice in {term NS1}\<close>
 lemma B_trusts_NS1 [rule_format]:
-     "evs \<in> ns_public                                         
+     "evs \<in> ns_public
       \<Longrightarrow> Crypt (pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace> \<in> parts (spies evs) \<longrightarrow>
           Nonce NA \<notin> analz (spies evs) \<longrightarrow>
           Says A B (Crypt (pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs"
-apply (erule ns_public.induct, simp_all)
-(*Fake*)
-apply (blast intro!: analz_insertI)
-done
+  by (induction evs rule: ns_public.induct) (use analz_insertI in auto)
+
+
+subsection \<open>Authenticity properties obtained from {term NS2}\<close>
+
+text \<open>Unicity for {term NS2}: nonce {term NB} identifies nonce {term NA} and agent {term A} 
+  [proof closely follows that for @{thm [source] unique_NA}]\<close>
+
+lemma unique_NB [dest]:
+  assumes NB: "Crypt(pubEK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace> \<in> parts(spies evs)"
+              "Crypt(pubEK A') \<lbrace>Nonce NA', Nonce NB, Agent B'\<rbrace> \<in> parts(spies evs)"
+              "Nonce NB \<notin> analz (spies evs)"
+    and evs: "evs \<in> ns_public"
+  shows "A=A' \<and> NA=NA' \<and> B=B'"
+  using evs NB
+  by (induction rule: ns_public.induct) (auto intro!: analz_insertI split: if_split_asm)
 
 
-subsection\<open>Authenticity properties obtained from NS2\<close>
-
-(*Unicity for NS2: nonce NB identifies nonce NA and agents A, B 
-  [unicity of B makes Lowe's fix work]
-  [proof closely follows that for unique_NA] *)
-
-lemma unique_NB [dest]: 
-     "\<lbrakk>Crypt(pubEK A)  \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace> \<in> parts(spies evs);
-       Crypt(pubEK A') \<lbrace>Nonce NA', Nonce NB, Agent B'\<rbrace> \<in> parts(spies evs);
-       Nonce NB \<notin> analz (spies evs); evs \<in> ns_public\<rbrakk>
-      \<Longrightarrow> A=A' \<and> NA=NA' \<and> B=B'"
-apply (erule rev_mp, erule rev_mp, erule rev_mp)   
-apply (erule ns_public.induct, simp_all)
-(*Fake, NS2*)
-apply (blast intro: analz_insertI)+
-done
+text \<open>{term NB} remains secret\<close>
+theorem Spy_not_see_NB [dest]:
+  assumes NB: "Says B A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) \<in> set evs"
+              "A \<notin> bad" "B \<notin> bad"
+    and evs: "evs \<in> ns_public"
+  shows "Nonce NB \<notin> analz (spies evs)"
+  using evs NB evs
+proof (induction rule: ns_public.induct)
+  case Fake
+  then show ?case by spy_analz
+next
+  case NS2
+  then show ?case
+    by (auto intro!: no_nonce_NS1_NS2)
+qed auto
 
 
-(*Secrecy: Spy does not see the nonce sent in msg NS2 if A and B are secure*)
-theorem Spy_not_see_NB [dest]:
-     "\<lbrakk>Says B A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) \<in> set evs;
-       A \<notin> bad;  B \<notin> bad;  evs \<in> ns_public\<rbrakk>
-      \<Longrightarrow> Nonce NB \<notin> analz (spies evs)"
-apply (erule rev_mp)
-apply (erule ns_public.induct, simp_all, spy_analz)
-apply (blast intro: no_nonce_NS1_NS2)+
-done
-
-
-(*Authentication for B: if he receives message 3 and has used NB
-  in message 2, then A has sent message 3.*)
+text \<open>Authentication for {term B}: if he receives message 3 and has used {term NB}
+  in message 2, then {term A} has sent message 3.\<close>
 lemma B_trusts_NS3_lemma [rule_format]:
-     "\<lbrakk>A \<notin> bad;  B \<notin> bad;  evs \<in> ns_public\<rbrakk> \<Longrightarrow>
-      Crypt (pubEK B) (Nonce NB) \<in> parts (spies evs) \<longrightarrow>
-      Says B A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) \<in> set evs \<longrightarrow>
-      Says A B (Crypt (pubEK B) (Nonce NB)) \<in> set evs"
-by (erule ns_public.induct, auto)
+     "\<lbrakk>evs \<in> ns_public;
+       Crypt (pubEK B) (Nonce NB) \<in> parts (spies evs);
+       Says B A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) \<in> set evs;
+       A \<notin> bad;  B \<notin> bad\<rbrakk>
+      \<Longrightarrow> Says A B (Crypt (pubEK B) (Nonce NB)) \<in> set evs"
+proof (induction rule: ns_public.induct)
+  case (NS3 evs3 A B NA B' NB)
+  then show ?case
+    by simp (blast intro: no_nonce_NS1_NS2)
+qed auto
 
 theorem B_trusts_NS3:
      "\<lbrakk>Says B A  (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) \<in> set evs;
-       Says A' B (Crypt (pubEK B) (Nonce NB)) \<in> set evs;             
-       A \<notin> bad;  B \<notin> bad;  evs \<in> ns_public\<rbrakk>                    
+       Says A' B (Crypt (pubEK B) (Nonce NB)) \<in> set evs;
+       A \<notin> bad;  B \<notin> bad;  evs \<in> ns_public\<rbrakk>
       \<Longrightarrow> Says A B (Crypt (pubEK B) (Nonce NB)) \<in> set evs"
-by (blast intro: B_trusts_NS3_lemma)
+  by (blast intro: B_trusts_NS3_lemma)
 
-subsection\<open>Overall guarantee for B\<close>
+
+subsection\<open>Overall guarantee for {term B}\<close>
 
-(*If NS3 has been sent and the nonce NB agrees with the nonce B joined with
-  NA, then A initiated the run using NA.*)
+text \<open>If NS3 has been sent and the nonce NB agrees with the nonce B joined with
+  NA, then A initiated the run using NA.\<close>
 theorem B_trusts_protocol:
      "\<lbrakk>A \<notin> bad;  B \<notin> bad;  evs \<in> ns_public\<rbrakk> \<Longrightarrow>
       Crypt (pubEK B) (Nonce NB) \<in> parts (spies evs) \<longrightarrow>
diff -r eb30869e7228 -r efc220128637 src/HOL/Auth/NS_Public_Bad.thy
--- a/src/HOL/Auth/NS_Public_Bad.thy	Fri Oct 14 14:39:52 2022 +0200
+++ b/src/HOL/Auth/NS_Public_Bad.thy	Fri Oct 14 14:57:48 2022 +0100
@@ -1,78 +1,68 @@
 (*  Title:      HOL/Auth/NS_Public_Bad.thy
     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     Copyright   1996  University of Cambridge
-
-Inductive relation "ns_public" for the Needham-Schroeder Public-Key protocol.
-Flawed version, vulnerable to Lowe's attack.
-
-From page 260 of
-  Burrows, Abadi and Needham.  A Logic of Authentication.
-  Proc. Royal Soc. 426 (1989)
 *)
 
 section\<open>Verifying the Needham-Schroeder Public-Key Protocol\<close>
 
+text \<open>Flawed version, vulnerable to Lowe's attack.
+From Burrows, Abadi and Needham.  A Logic of Authentication.
+  Proc. Royal Soc. 426 (1989), p. 260\<close>
+
 theory NS_Public_Bad imports Public begin
 
 inductive_set ns_public :: "event list set"
   where
-         (*Initial trace is empty*)
-   Nil:  "[] \<in> ns_public"
-
-         (*The spy MAY say anything he CAN say.  We do not expect him to
-           invent new nonces here, but he can also use NS1.  Common to
-           all similar protocols.*)
+   Nil:  "[] \<in> ns_public" 
+   \<comment> \<open>Initial trace is empty\<close>
  | Fake: "\<lbrakk>evsf \<in> ns_public;  X \<in> synth (analz (spies evsf))\<rbrakk>
           \<Longrightarrow> Says Spy B X  # evsf \<in> ns_public"
-
-         (*Alice initiates a protocol run, sending a nonce to Bob*)
+   \<comment> \<open>The spy can say almost anything.\<close>
  | NS1:  "\<lbrakk>evs1 \<in> ns_public;  Nonce NA \<notin> used evs1\<rbrakk>
           \<Longrightarrow> Says A B (Crypt (pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>)
                 # evs1  \<in>  ns_public"
-
-         (*Bob responds to Alice's message with a further nonce*)
+   \<comment> \<open>Alice initiates a protocol run, sending a nonce to Bob\<close>
  | NS2:  "\<lbrakk>evs2 \<in> ns_public;  Nonce NB \<notin> used evs2;
            Says A' B (Crypt (pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs2\<rbrakk>
           \<Longrightarrow> Says B A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>)
                 # evs2  \<in>  ns_public"
-
-         (*Alice proves her existence by sending NB back to Bob.*)
+   \<comment> \<open>Bob responds to Alice's message with a further nonce\<close>
  | NS3:  "\<lbrakk>evs3 \<in> ns_public;
            Says A  B (Crypt (pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs3;
            Says B' A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>) \<in> set evs3\<rbrakk>
           \<Longrightarrow> Says A B (Crypt (pubEK B) (Nonce NB)) # evs3 \<in> ns_public"
+   \<comment> \<open>Alice proves her existence by sending @{term NB} back to Bob.\<close>
 
 declare knows_Spy_partsEs [elim]
 declare analz_into_parts [dest]
 declare Fake_parts_insert_in_Un [dest]
 
-(*A "possibility property": there are traces that reach the end*)
+text \<open>A "possibility property": there are traces that reach the end\<close>
 lemma "\<exists>NB. \<exists>evs \<in> ns_public. Says A B (Crypt (pubEK B) (Nonce NB)) \<in> set evs"
-apply (intro exI bexI)
-apply (rule_tac [2] ns_public.Nil [THEN ns_public.NS1, THEN ns_public.NS2, 
-                                   THEN ns_public.NS3])
-by possibility
+  apply (intro exI bexI)
+   apply (rule_tac [2] ns_public.Nil [THEN ns_public.NS1, THEN ns_public.NS2, THEN ns_public.NS3])
+  by possibility
 
 
-(**** Inductive proofs about ns_public ****)
+subsection \<open>Inductive proofs about @{term ns_public}\<close>
 
 (** Theorems of the form X \<notin> parts (spies evs) imply that NOBODY
     sends messages containing X! **)
 
-(*Spy never sees another agent's private key! (unless it's bad at start)*)
+text \<open>Spy never sees another agent's private key! (unless it's bad at start)\<close>
 lemma Spy_see_priEK [simp]: 
-      "evs \<in> ns_public \<Longrightarrow> (Key (priEK A) \<in> parts (spies evs)) = (A \<in> bad)"
-by (erule ns_public.induct, auto)
+  "evs \<in> ns_public \<Longrightarrow> (Key (priEK A) \<in> parts (spies evs)) = (A \<in> bad)"
+  by (erule ns_public.induct, auto)
 
 lemma Spy_analz_priEK [simp]: 
-      "evs \<in> ns_public \<Longrightarrow> (Key (priEK A) \<in> analz (spies evs)) = (A \<in> bad)"
-by auto
+  "evs \<in> ns_public \<Longrightarrow> (Key (priEK A) \<in> analz (spies evs)) = (A \<in> bad)"
+  by auto
 
 
-(*** Authenticity properties obtained from NS2 ***)
+subsection \<open>Authenticity properties obtained from {term NS1}\<close>
 
-(*It is impossible to re-use a nonce in both NS1 and NS2, provided the nonce
-  is secret.  (Honest users generate fresh nonces.)*)
+text \<open>It is impossible to re-use a nonce in both {term NS1} and {term NS2}, provided the nonce
+  is secret.  (Honest users generate fresh nonces.)\<close>
 lemma no_nonce_NS1_NS2 [rule_format]: 
       "evs \<in> ns_public 
        \<Longrightarrow> Crypt (pubEK C) \<lbrace>NA', Nonce NA\<rbrace> \<in> parts (spies evs) \<longrightarrow>
@@ -81,32 +71,42 @@
   by (induct rule: ns_public.induct) (auto intro: analz_insertI)
 
 
-(*Unicity for NS1: nonce NA identifies agents A and B*)
+text \<open>Unicity for {term NS1}: nonce {term NA} identifies agents {term A} and {term B}\<close>
 lemma unique_NA: 
-     "\<lbrakk>Crypt(pubEK B)  \<lbrace>Nonce NA, Agent A \<rbrace> \<in> parts(spies evs);  
-       Crypt(pubEK B') \<lbrace>Nonce NA, Agent A'\<rbrace> \<in> parts(spies evs);  
-       Nonce NA \<notin> analz (spies evs); evs \<in> ns_public\<rbrakk>
-      \<Longrightarrow> A=A' \<and> B=B'"
-apply (erule rev_mp, erule rev_mp, erule rev_mp)   
-apply (erule ns_public.induct, auto intro: analz_insertI)
-done
+  assumes NA: "Crypt(pubEK B)  \<lbrace>Nonce NA, Agent A \<rbrace> \<in> parts(spies evs)"
+              "Crypt(pubEK B') \<lbrace>Nonce NA, Agent A'\<rbrace> \<in> parts(spies evs)"
+              "Nonce NA \<notin> analz (spies evs)"
+    and evs: "evs \<in> ns_public"
+  shows "A=A' \<and> B=B'"
+  using evs NA
+  by (induction rule: ns_public.induct) (auto intro!: analz_insertI split: if_split_asm)
 
 
-(*Secrecy: Spy does not see the nonce sent in msg NS1 if A and B are secure
-  The major premise "Says A B ..." makes it a dest-rule, so we use
-  (erule rev_mp) rather than rule_format. *)
+text \<open>Secrecy: Spy does not see the nonce sent in msg {term NS1} if {term A} and {term B} are secure
+  The major premise "Says A B ..." makes it a dest-rule, hence the given assumption order. \<close>
 theorem Spy_not_see_NA: 
-      "\<lbrakk>Says A B (Crypt(pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs;
-        A \<notin> bad;  B \<notin> bad;  evs \<in> ns_public\<rbrakk>                     
-       \<Longrightarrow> Nonce NA \<notin> analz (spies evs)"
-apply (erule rev_mp)   
-apply (erule ns_public.induct, simp_all, spy_analz)
-apply (blast dest: unique_NA intro: no_nonce_NS1_NS2)+
-done
+  assumes NA: "Says A B (Crypt(pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs"
+              "A \<notin> bad" "B \<notin> bad"
+    and evs: "evs \<in> ns_public"
+  shows "Nonce NA \<notin> analz (spies evs)"
+  using evs NA
+proof (induction rule: ns_public.induct)
+  case (Fake evsf X B)
+  then show ?case
+    by spy_analz
+next
+  case (NS2 evs2 NB A' B NA A)
+  then show ?case
+    by simp (metis Says_imp_analz_Spy analz_into_parts parts.simps unique_NA usedI)
+next
+  case (NS3 evs3 A B NA B' NB)
+  then show ?case
+    by simp (meson Says_imp_analz_Spy analz_into_parts no_nonce_NS1_NS2)
+qed auto
 
 
-(*Authentication for A: if she receives message 2 and has used NA
-  to start a run, then B has sent message 2.*)
+text \<open>Authentication for {term A}: if she receives message 2 and has used {term NA}
+  to start a run, then {term B} has sent message 2.\<close>
 lemma A_trusts_NS2_lemma [rule_format]: 
    "\<lbrakk>A \<notin> bad;  B \<notin> bad;  evs \<in> ns_public\<rbrakk>                     
     \<Longrightarrow> Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB\<rbrace> \<in> parts (spies evs) \<longrightarrow>
@@ -119,10 +119,10 @@
        Says B' A (Crypt(pubEK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>) \<in> set evs;
        A \<notin> bad;  B \<notin> bad;  evs \<in> ns_public\<rbrakk>                     
       \<Longrightarrow> Says B A (Crypt(pubEK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>) \<in> set evs"
-by (blast intro: A_trusts_NS2_lemma)
+  by (blast intro: A_trusts_NS2_lemma)
 
 
-(*If the encrypted message appears then it originated with Alice in NS1*)
+text \<open>If the encrypted message appears then it originated with Alice in {term NS1}\<close>
 lemma B_trusts_NS1 [rule_format]:
      "evs \<in> ns_public                                         
       \<Longrightarrow> Crypt (pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace> \<in> parts (spies evs) \<longrightarrow>
@@ -131,65 +131,72 @@
   by (induction evs rule: ns_public.induct) (use analz_insertI in auto)
 
 
-
-(*** Authenticity properties obtained from NS2 ***)
+subsection \<open>Authenticity properties obtained from {term NS2}\<close>
 
-(*Unicity for NS2: nonce NB identifies nonce NA and agent A
-  [proof closely follows that for unique_NA] *)
+text \<open>Unicity for {term NS2}: nonce {term NB} identifies nonce {term NA} and agent {term A} 
+  [proof closely follows that for @{thm [source] unique_NA}]\<close>
+
 lemma unique_NB [dest]: 
-     "\<lbrakk>Crypt(pubEK A)  \<lbrace>Nonce NA, Nonce NB\<rbrace> \<in> parts(spies evs);
-       Crypt(pubEK A') \<lbrace>Nonce NA', Nonce NB\<rbrace> \<in> parts(spies evs);
-       Nonce NB \<notin> analz (spies evs); evs \<in> ns_public\<rbrakk>
-     \<Longrightarrow> A=A' \<and> NA=NA'"
-apply (erule rev_mp, erule rev_mp, erule rev_mp)   
-apply (erule ns_public.induct, simp_all)
-(*Fake, NS2*)
-apply (blast intro!: analz_insertI)+
-done
+  assumes NB: "Crypt(pubEK A) \<lbrace>Nonce NA, Nonce NB\<rbrace> \<in> parts(spies evs)"
+              "Crypt(pubEK A') \<lbrace>Nonce NA', Nonce NB\<rbrace> \<in> parts(spies evs)"
+              "Nonce NB \<notin> analz (spies evs)"
+    and evs: "evs \<in> ns_public"
+  shows "A=A' \<and> NA=NA'"
+  using evs NB 
+  by (induction rule: ns_public.induct) (auto intro!: analz_insertI split: if_split_asm)
 
 
-(*NB remains secret PROVIDED Alice never responds with round 3*)
+text \<open>{term NB} remains secret \emph{provided} Alice never responds with round 3\<close>
 theorem Spy_not_see_NB [dest]:
-     "\<lbrakk>Says B A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>) \<in> set evs;   
-       \<forall>C. Says A C (Crypt (pubEK C) (Nonce NB)) \<notin> set evs;       
-       A \<notin> bad;  B \<notin> bad;  evs \<in> ns_public\<rbrakk>                      
-     \<Longrightarrow> Nonce NB \<notin> analz (spies evs)"
-apply (erule rev_mp, erule rev_mp)
-apply (erule ns_public.induct, simp_all, spy_analz)
-apply (simp_all add: all_conj_distrib) (*speeds up the next step*)
-apply (blast intro: no_nonce_NS1_NS2)+
-done
+  assumes NB: "Says B A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>) \<in> set evs"
+              "\<forall>C. Says A C (Crypt (pubEK C) (Nonce NB)) \<notin> set evs"
+              "A \<notin> bad" "B \<notin> bad"
+    and evs: "evs \<in> ns_public"
+  shows "Nonce NB \<notin> analz (spies evs)"
+  using evs NB evs
+proof (induction rule: ns_public.induct)
+  case Fake
+  then show ?case by spy_analz
+next
+  case NS2
+  then show ?case
+    by (auto intro!: no_nonce_NS1_NS2)
+qed auto
 
 
-(*Authentication for B: if he receives message 3 and has used NB
-  in message 2, then A has sent message 3--to somebody....*)
-
+text \<open>Authentication for {term B}: if he receives message 3 and has used {term NB}
+  in message 2, then {term A} has sent message 3 (to somebody) \<close>
 lemma B_trusts_NS3_lemma [rule_format]:
-     "\<lbrakk>A \<notin> bad;  B \<notin> bad;  evs \<in> ns_public\<rbrakk>                    
-      \<Longrightarrow> Crypt (pubEK B) (Nonce NB) \<in> parts (spies evs) \<longrightarrow>
-          Says B A  (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>) \<in> set evs \<longrightarrow>
-          (\<exists>C. Says A C (Crypt (pubEK C) (Nonce NB)) \<in> set evs)"
-apply (erule ns_public.induct, auto)
-by (blast intro: no_nonce_NS1_NS2)+
+     "\<lbrakk>evs \<in> ns_public; 
+       Crypt (pubEK B) (Nonce NB) \<in> parts (spies evs); 
+       Says B A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>) \<in> set evs; 
+       A \<notin> bad;  B \<notin> bad\<rbrakk>                    
+      \<Longrightarrow> \<exists>C. Says A C (Crypt (pubEK C) (Nonce NB)) \<in> set evs"
+proof (induction rule: ns_public.induct)
+  case (NS3 evs3 A B NA B' NB)
+  then show ?case
+    by simp (blast intro: no_nonce_NS1_NS2)
+qed auto
 
 theorem B_trusts_NS3:
      "\<lbrakk>Says B A  (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>) \<in> set evs;
        Says A' B (Crypt (pubEK B) (Nonce NB)) \<in> set evs;             
        A \<notin> bad;  B \<notin> bad;  evs \<in> ns_public\<rbrakk>                    
       \<Longrightarrow> \<exists>C. Says A C (Crypt (pubEK C) (Nonce NB)) \<in> set evs"
-by (blast intro: B_trusts_NS3_lemma)
+  by (blast intro: B_trusts_NS3_lemma)
 
 
-(*Can we strengthen the secrecy theorem Spy_not_see_NB?  NO*)
-lemma "\<lbrakk>A \<notin> bad;  B \<notin> bad;  evs \<in> ns_public\<rbrakk>            
-       \<Longrightarrow> Says B A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>) \<in> set evs  
-           \<longrightarrow> Nonce NB \<notin> analz (spies evs)"
-apply (erule ns_public.induct, simp_all, spy_analz)
-(*NS1: by freshness*)
+text \<open>Can we strengthen the secrecy theorem @{thm[source]Spy_not_see_NB}?  NO\<close>
+lemma "\<lbrakk>evs \<in> ns_public; 
+        Says B A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>) \<in> set evs; 
+        A \<notin> bad; B \<notin> bad\<rbrakk>            
+       \<Longrightarrow> Nonce NB \<notin> analz (spies evs)"
+apply (induction rule: ns_public.induct, simp_all, spy_analz)
+(*{term NS1}: by freshness*)
 apply blast
-(*NS2: by freshness and unicity of NB*)
+(*{term NS2}: by freshness and unicity of {term NB}*)
 apply (blast intro: no_nonce_NS1_NS2)
-(*NS3: unicity of NB identifies A and NA, but not B*)
+(*{term NS3}: unicity of {term NB} identifies {term A} and {term NA}, but not {term B}*)
 apply clarify
 apply (frule_tac A' = A in 
        Says_imp_knows_Spy [THEN parts.Inj, THEN unique_NB], auto)
diff -r eb30869e7228 -r efc220128637 src/HOL/Auth/OtwayRees.thy
--- a/src/HOL/Auth/OtwayRees.thy	Fri Oct 14 14:39:52 2022 +0200
+++ b/src/HOL/Auth/OtwayRees.thy	Fri Oct 14 14:57:48 2022 +0100
@@ -148,8 +148,8 @@
 
 subsection\<open>Towards Secrecy: Proofs Involving \<^term>\<open>analz\<close>\<close>
 
-(*Describes the form of K and NA when the Server sends this message.  Also
-  for Oops case.*)
+text \<open>Describes the form of K and NA when the Server sends this message.  Also
+  for Oops case.\<close>
 lemma Says_Server_message_form:
      "\<lbrakk>Says Server B \<lbrace>NA, X, Crypt (shrK B) \<lbrace>NB, Key K\<rbrace>\<rbrace> \<in> set evs;
          evs \<in> otway\<rbrakk>