1.1 --- a/doc-src/TutorialI/CTL/PDL.thy Sat Jan 06 11:27:09 2001 +0100
1.2 +++ b/doc-src/TutorialI/CTL/PDL.thy Sat Jan 06 12:39:05 2001 +0100
1.3 @@ -38,13 +38,13 @@
1.4 "s \<Turnstile> Neg f = (\<not>(s \<Turnstile> f))"
1.5 "s \<Turnstile> And f g = (s \<Turnstile> f \<and> s \<Turnstile> g)"
1.6 "s \<Turnstile> AX f = (\<forall>t. (s,t) \<in> M \<longrightarrow> t \<Turnstile> f)"
1.7 -"s \<Turnstile> EF f = (\<exists>t. (s,t) \<in> M^* \<and> t \<Turnstile> f)";
1.8 +"s \<Turnstile> EF f = (\<exists>t. (s,t) \<in> M\<^sup>* \<and> t \<Turnstile> f)";
1.9
1.10 text{*\noindent
1.11 The first three equations should be self-explanatory. The temporal formula
1.12 @{term"AX f"} means that @{term f} is true in all next states whereas
1.13 @{term"EF f"} means that there exists some future state in which @{term f} is
1.14 -true. The future is expressed via @{text"^*"}, the transitive reflexive
1.15 +true. The future is expressed via @{text"\<^sup>*"}, the transitive reflexive
1.16 closure. Because of reflexivity, the future includes the present.
1.17
1.18 Now we come to the model checker itself. It maps a formula into the set of
1.19 @@ -58,14 +58,14 @@
1.20 "mc(Neg f) = -mc f"
1.21 "mc(And f g) = mc f \<inter> mc g"
1.22 "mc(AX f) = {s. \<forall>t. (s,t) \<in> M \<longrightarrow> t \<in> mc f}"
1.23 -"mc(EF f) = lfp(\<lambda>T. mc f \<union> M^-1 ``` T)"
1.24 +"mc(EF f) = lfp(\<lambda>T. mc f \<union> M\<inverse> ``` T)"
1.25
1.26 text{*\noindent
1.27 Only the equation for @{term EF} deserves some comments. Remember that the
1.28 -postfix @{text"^-1"} and the infix @{text"```"} are predefined and denote the
1.29 +postfix @{text"\<inverse>"} and the infix @{text"```"} are predefined and denote the
1.30 converse of a relation and the application of a relation to a set. Thus
1.31 -@{term "M^-1 ``` T"} is the set of all predecessors of @{term T} and the least
1.32 -fixed point (@{term lfp}) of @{term"\<lambda>T. mc f \<union> M^-1 ``` T"} is the least set
1.33 +@{term "M\<inverse> ``` T"} is the set of all predecessors of @{term T} and the least
1.34 +fixed point (@{term lfp}) of @{term"\<lambda>T. mc f \<union> M\<inverse> ``` T"} is the least set
1.35 @{term T} containing @{term"mc f"} and all predecessors of @{term T}. If you
1.36 find it hard to see that @{term"mc(EF f)"} contains exactly those states from
1.37 which there is a path to a state where @{term f} is true, do not worry---that
1.38 @@ -74,7 +74,7 @@
1.39 First we prove monotonicity of the function inside @{term lfp}
1.40 *}
1.41
1.42 -lemma mono_ef: "mono(\<lambda>T. A \<union> M^-1 ``` T)"
1.43 +lemma mono_ef: "mono(\<lambda>T. A \<union> M\<inverse> ``` T)"
1.44 apply(rule monoI)
1.45 apply blast
1.46 done
1.47 @@ -87,7 +87,7 @@
1.48 *}
1.49
1.50 lemma EF_lemma:
1.51 - "lfp(\<lambda>T. A \<union> M^-1 ``` T) = {s. \<exists>t. (s,t) \<in> M^* \<and> t \<in> A}"
1.52 + "lfp(\<lambda>T. A \<union> M\<inverse> ``` T) = {s. \<exists>t. (s,t) \<in> M\<^sup>* \<and> t \<in> A}"
1.53
1.54 txt{*\noindent
1.55 The equality is proved in the canonical fashion by proving that each set
1.56 @@ -112,11 +112,11 @@
1.57 Having disposed of the monotonicity subgoal,
1.58 simplification leaves us with the following main goal
1.59 \begin{isabelle}
1.60 -\ \isadigit{1}{\isachardot}\ {\isasymAnd}s{\isachardot}\ s\ {\isasymin}\ A\ {\isasymor}\isanewline
1.61 -\ \ \ \ \ \ \ \ \ 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
1.62 -\ \ \ \ \ \ \ \ \ {\isasymLongrightarrow}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M{\isacharcircum}{\isacharasterisk}\ {\isasymand}\ t\ {\isasymin}\ A
1.63 +\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x{\isachardot}\ x\ {\isasymin}\ A\ {\isasymor}\isanewline
1.64 +\ \ \ \ \ \ \ \ \ 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
1.65 +\ \ \ \ \ \ \ \ {\isasymLongrightarrow}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}x{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M\isactrlsup {\isacharasterisk}\ {\isasymand}\ t\ {\isasymin}\ A
1.66 \end{isabelle}
1.67 -which is proved by @{text blast} with the help of transitivity of @{text"^*"}:
1.68 +which is proved by @{text blast} with the help of transitivity of @{text"\<^sup>*"}:
1.69 *}
1.70
1.71 apply(blast intro: rtrancl_trans);
1.72 @@ -132,13 +132,13 @@
1.73 txt{*\noindent
1.74 After simplification and clarification we are left with
1.75 @{subgoals[display,indent=0,goals_limit=1]}
1.76 -This goal is proved by induction on @{term"(s,t)\<in>M^*"}. But since the model
1.77 +This goal is proved by induction on @{term"(s,t)\<in>M\<^sup>*"}. But since the model
1.78 checker works backwards (from @{term t} to @{term s}), we cannot use the
1.79 induction theorem @{thm[source]rtrancl_induct} because it works in the
1.80 forward direction. Fortunately the converse induction theorem
1.81 @{thm[source]converse_rtrancl_induct} already exists:
1.82 @{thm[display,margin=60]converse_rtrancl_induct[no_vars]}
1.83 -It says that if @{prop"(a,b):r^*"} and we know @{prop"P b"} then we can infer
1.84 +It says that if @{prop"(a,b):r\<^sup>*"} and we know @{prop"P b"} then we can infer
1.85 @{prop"P a"} provided each step backwards from a predecessor @{term z} of
1.86 @{term b} preserves @{term P}.
1.87 *}