
diff -r 2d4c058749a7 -r c00ac928fc6f doc-src/TutorialI/CTL/PDL.thy
--- a/doc-src/TutorialI/CTL/PDL.thy	Sat Jan 06 11:27:09 2001 +0100
+++ b/doc-src/TutorialI/CTL/PDL.thy	Sat Jan 06 12:39:05 2001 +0100
@@ -38,13 +38,13 @@
 "s \<Turnstile> Neg f   = (\<not>(s \<Turnstile> f))"
 "s \<Turnstile> And f g = (s \<Turnstile> f \<and> s \<Turnstile> g)"
 "s \<Turnstile> AX f    = (\<forall>t. (s,t) \<in> M \<longrightarrow> t \<Turnstile> f)"
-"s \<Turnstile> EF f    = (\<exists>t. (s,t) \<in> M^* \<and> t \<Turnstile> f)";
+"s \<Turnstile> EF f    = (\<exists>t. (s,t) \<in> M\<^sup>* \<and> t \<Turnstile> f)";
 
 text{*\noindent
 The first three equations should be self-explanatory. The temporal formula
 @{term"AX f"} means that @{term f} is true in all next states whereas
 @{term"EF f"} means that there exists some future state in which @{term f} is
-true. The future is expressed via @{text"^*"}, the transitive reflexive
+true. The future is expressed via @{text"\<^sup>*"}, the transitive reflexive
 closure. Because of reflexivity, the future includes the present.
 
 Now we come to the model checker itself. It maps a formula into the set of
@@ -58,14 +58,14 @@
 "mc(Neg f)   = -mc f"
 "mc(And f g) = mc f \<inter> mc g"
 "mc(AX f)    = {s. \<forall>t. (s,t) \<in> M  \<longrightarrow> t \<in> mc f}"
-"mc(EF f)    = lfp(\<lambda>T. mc f \<union> M^-1 ``` T)"
+"mc(EF f)    = lfp(\<lambda>T. mc f \<union> M\<inverse> ``` T)"
 
 text{*\noindent
 Only the equation for @{term EF} deserves some comments. Remember that the
-postfix @{text"^-1"} and the infix @{text"```"} are predefined and denote the
+postfix @{text"\<inverse>"} and the infix @{text"```"} are predefined and denote the
 converse of a relation and the application of a relation to a set. Thus
-@{term "M^-1 ``` T"} is the set of all predecessors of @{term T} and the least
-fixed point (@{term lfp}) of @{term"\<lambda>T. mc f \<union> M^-1 ``` T"} is the least set
+@{term "M\<inverse> ``` T"} is the set of all predecessors of @{term T} and the least
+fixed point (@{term lfp}) of @{term"\<lambda>T. mc f \<union> M\<inverse> ``` T"} is the least set
 @{term T} containing @{term"mc f"} and all predecessors of @{term T}. If you
 find it hard to see that @{term"mc(EF f)"} contains exactly those states from
 which there is a path to a state where @{term f} is true, do not worry---that
@@ -74,7 +74,7 @@
 First we prove monotonicity of the function inside @{term lfp}
 *}
 
-lemma mono_ef: "mono(\<lambda>T. A \<union> M^-1 ``` T)"
+lemma mono_ef: "mono(\<lambda>T. A \<union> M\<inverse> ``` T)"
 apply(rule monoI)
 apply blast
 done
@@ -87,7 +87,7 @@
 *}
 
 lemma EF_lemma:
-  "lfp(\<lambda>T. A \<union> M^-1 ``` T) = {s. \<exists>t. (s,t) \<in> M^* \<and> t \<in> A}"
+  "lfp(\<lambda>T. A \<union> M\<inverse> ``` T) = {s. \<exists>t. (s,t) \<in> M\<^sup>* \<and> t \<in> A}"
 
 txt{*\noindent
 The equality is proved in the canonical fashion by proving that each set
@@ -112,11 +112,11 @@
 Having disposed of the monotonicity subgoal,
 simplification leaves us with the following main goal
 \begin{isabelle}
-\ \isadigit{1}{\isachardot}\ {\isasymAnd}s{\isachardot}\ s\ {\isasymin}\ A\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ s\ {\isasymin}\ M{\isacharcircum}{\isacharminus}\isadigit{1}\ {\isacharcircum}{\isacharcircum}\ {\isacharparenleft}lfp\ {\isacharparenleft}{\dots}{\isacharparenright}\ {\isasyminter}\ {\isacharbraceleft}x{\isachardot}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}x{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M{\isacharcircum}{\isacharasterisk}\ {\isasymand}\ t\ {\isasymin}\ A{\isacharbraceright}{\isacharparenright}\isanewline
-\ \ \ \ \ \ \ \ \ {\isasymLongrightarrow}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M{\isacharcircum}{\isacharasterisk}\ {\isasymand}\ t\ {\isasymin}\ A
+\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x{\isachardot}\ x\ {\isasymin}\ A\ {\isasymor}\isanewline
+\ \ \ \ \ \ \ \ \ x\ {\isasymin}\ M{\isasyminverse}\ {\isacharbackquote}{\isacharbackquote}{\isacharbackquote}\ {\isacharparenleft}lfp\ {\isacharparenleft}\dots{\isacharparenright}\ {\isasyminter}\ {\isacharbraceleft}x{\isachardot}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}x{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M\isactrlsup {\isacharasterisk}\ {\isasymand}\ t\ {\isasymin}\ A{\isacharbraceright}{\isacharparenright}\isanewline
+\ \ \ \ \ \ \ \ {\isasymLongrightarrow}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}x{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M\isactrlsup {\isacharasterisk}\ {\isasymand}\ t\ {\isasymin}\ A
 \end{isabelle}
-which is proved by @{text blast} with the help of transitivity of @{text"^*"}:
+which is proved by @{text blast} with the help of transitivity of @{text"\<^sup>*"}:
 *}
 
  apply(blast intro: rtrancl_trans);
@@ -132,13 +132,13 @@
 txt{*\noindent
 After simplification and clarification we are left with
 @{subgoals[display,indent=0,goals_limit=1]}
-This goal is proved by induction on @{term"(s,t)\<in>M^*"}. But since the model
+This goal is proved by induction on @{term"(s,t)\<in>M\<^sup>*"}. But since the model
 checker works backwards (from @{term t} to @{term s}), we cannot use the
 induction theorem @{thm[source]rtrancl_induct} because it works in the
 forward direction. Fortunately the converse induction theorem
 @{thm[source]converse_rtrancl_induct} already exists:
 @{thm[display,margin=60]converse_rtrancl_induct[no_vars]}
-It says that if @{prop"(a,b):r^*"} and we know @{prop"P b"} then we can infer
+It says that if @{prop"(a,b):r\<^sup>*"} and we know @{prop"P b"} then we can infer
 @{prop"P a"} provided each step backwards from a predecessor @{term z} of
 @{term b} preserves @{term P}.
 *}




