--- a/src/HOL/SET_Protocol/Event_SET.thy Tue Feb 13 14:24:50 2018 +0100
+++ b/src/HOL/SET_Protocol/Event_SET.thy Thu Feb 15 12:11:00 2018 +0100
@@ -34,11 +34,11 @@
subsection\<open>Agents' Knowledge\<close>
consts (*Initial states of agents -- parameter of the construction*)
- initState :: "agent => msg set"
+ initState :: "agent \<Rightarrow> msg set"
(* Message reception does not extend spy's knowledge because of
reception invariant enforced by Reception rule in protocol definition*)
-primrec knows :: "[agent, event list] => msg set"
+primrec knows :: "[agent, event list] \<Rightarrow> msg set"
where
knows_Nil:
"knows A [] = initState A"
@@ -46,23 +46,23 @@
"knows A (ev # evs) =
(if A = Spy then
(case ev of
- Says A' B X => insert X (knows Spy evs)
- | Gets A' X => knows Spy evs
- | Notes A' X =>
+ Says A' B X \<Rightarrow> insert X (knows Spy evs)
+ | Gets A' X \<Rightarrow> knows Spy evs
+ | Notes A' X \<Rightarrow>
if A' \<in> bad then insert X (knows Spy evs) else knows Spy evs)
else
(case ev of
- Says A' B X =>
+ Says A' B X \<Rightarrow>
if A'=A then insert X (knows A evs) else knows A evs
- | Gets A' X =>
+ | Gets A' X \<Rightarrow>
if A'=A then insert X (knows A evs) else knows A evs
- | Notes A' X =>
+ | Notes A' X \<Rightarrow>
if A'=A then insert X (knows A evs) else knows A evs))"
subsection\<open>Used Messages\<close>
-primrec used :: "event list => msg set"
+primrec used :: "event list \<Rightarrow> msg set"
where
(*Set of items that might be visible to somebody:
complement of the set of fresh items.
@@ -70,9 +70,9 @@
used_Nil: "used [] = (UN B. parts (initState B))"
| used_Cons: "used (ev # evs) =
(case ev of
- Says A B X => parts {X} Un (used evs)
- | Gets A X => used evs
- | Notes A X => parts {X} Un (used evs))"
+ Says A B X \<Rightarrow> parts {X} \<union> (used evs)
+ | Gets A X \<Rightarrow> used evs
+ | Notes A X \<Rightarrow> parts {X} \<union> (used evs))"
@@ -80,7 +80,7 @@
be re-loaded. Addsimps [knows_Cons, used_Nil, *)
(** Simplifying parts (insert X (knows Spy evs))
- = parts {X} Un parts (knows Spy evs) -- since general case loops*)
+ = parts {X} \<union> parts (knows Spy evs) -- since general case loops*)
lemmas parts_insert_knows_A = parts_insert [of _ "knows A evs"] for A evs
@@ -92,33 +92,33 @@
on whether @{term "A=Spy"} and whether @{term "A\<in>bad"}\<close>
lemma knows_Spy_Notes [simp]:
"knows Spy (Notes A X # evs) =
- (if A:bad then insert X (knows Spy evs) else knows Spy evs)"
+ (if A\<in>bad then insert X (knows Spy evs) else knows Spy evs)"
apply auto
done
lemma knows_Spy_Gets [simp]: "knows Spy (Gets A X # evs) = knows Spy evs"
by auto
-lemma initState_subset_knows: "initState A <= knows A evs"
+lemma initState_subset_knows: "initState A \<subseteq> knows A evs"
apply (induct_tac "evs")
apply (auto split: event.split)
done
lemma knows_Spy_subset_knows_Spy_Says:
- "knows Spy evs <= knows Spy (Says A B X # evs)"
+ "knows Spy evs \<subseteq> knows Spy (Says A B X # evs)"
by auto
lemma knows_Spy_subset_knows_Spy_Notes:
- "knows Spy evs <= knows Spy (Notes A X # evs)"
+ "knows Spy evs \<subseteq> knows Spy (Notes A X # evs)"
by auto
lemma knows_Spy_subset_knows_Spy_Gets:
- "knows Spy evs <= knows Spy (Gets A X # evs)"
+ "knows Spy evs \<subseteq> knows Spy (Gets A X # evs)"
by auto
(*Spy sees what is sent on the traffic*)
lemma Says_imp_knows_Spy [rule_format]:
- "Says A B X \<in> set evs --> X \<in> knows Spy evs"
+ "Says A B X \<in> set evs \<longrightarrow> X \<in> knows Spy evs"
apply (induct_tac "evs")
apply (auto split: event.split)
done
@@ -132,24 +132,24 @@
subsection\<open>The Function @{term used}\<close>
-lemma parts_knows_Spy_subset_used: "parts (knows Spy evs) <= used evs"
+lemma parts_knows_Spy_subset_used: "parts (knows Spy evs) \<subseteq> used evs"
apply (induct_tac "evs")
apply (auto simp add: parts_insert_knows_A split: event.split)
done
lemmas usedI = parts_knows_Spy_subset_used [THEN subsetD, intro]
-lemma initState_subset_used: "parts (initState B) <= used evs"
+lemma initState_subset_used: "parts (initState B) \<subseteq> used evs"
apply (induct_tac "evs")
apply (auto split: event.split)
done
lemmas initState_into_used = initState_subset_used [THEN subsetD]
-lemma used_Says [simp]: "used (Says A B X # evs) = parts{X} Un used evs"
+lemma used_Says [simp]: "used (Says A B X # evs) = parts{X} \<union> used evs"
by auto
-lemma used_Notes [simp]: "used (Notes A X # evs) = parts{X} Un used evs"
+lemma used_Notes [simp]: "used (Notes A X # evs) = parts{X} \<union> used evs"
by auto
lemma used_Gets [simp]: "used (Gets A X # evs) = used evs"
@@ -157,7 +157,7 @@
lemma Notes_imp_parts_subset_used [rule_format]:
- "Notes A X \<in> set evs --> parts {X} <= used evs"
+ "Notes A X \<in> set evs \<longrightarrow> parts {X} \<subseteq> used evs"
apply (induct_tac "evs")
apply (rename_tac [2] a evs')
apply (induct_tac [2] "a", auto)
@@ -168,7 +168,7 @@
used_Nil [simp del] used_Cons [simp del]
-text\<open>For proving theorems of the form @{term "X \<notin> analz (knows Spy evs) --> P"}
+text\<open>For proving theorems of the form @{term "X \<notin> analz (knows Spy evs) \<longrightarrow> P"}
New events added by induction to "evs" are discarded. Provided
this information isn't needed, the proof will be much shorter, since
it will omit complicated reasoning about @{term analz}.\<close>
@@ -190,6 +190,6 @@
method_setup analz_mono_contra = \<open>
Scan.succeed (fn ctxt => SIMPLE_METHOD (REPEAT_FIRST (analz_mono_contra_tac ctxt)))\<close>
- "for proving theorems of the form X \<notin> analz (knows Spy evs) --> P"
+ "for proving theorems of the form X \<notin> analz (knows Spy evs) \<longrightarrow> P"
end