--- a/doc-src/TutorialI/CTL/document/PDL.tex Tue Aug 16 13:42:21 2005 +0200
+++ b/doc-src/TutorialI/CTL/document/PDL.tex Tue Aug 16 13:42:23 2005 +0200
@@ -1,7 +1,20 @@
%
\begin{isabellebody}%
\def\isabellecontext{PDL}%
-\isamarkupfalse%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+\isamarkuptrue%
%
\isamarkupsubsection{Propositional Dynamic Logic --- PDL%
}
@@ -17,12 +30,12 @@
\cite{HarelKT-DL} looks quite different from ours, but the two are easily
shown to be equivalent.}%
\end{isamarkuptext}%
-\isamarkuptrue%
+\isamarkupfalse%
\isacommand{datatype}\ formula\ {\isacharequal}\ Atom\ atom\isanewline
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ Neg\ formula\isanewline
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ And\ formula\ formula\isanewline
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ AX\ formula\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ EF\ formula\isamarkupfalse%
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ EF\ formula\isamarkuptrue%
%
\begin{isamarkuptext}%
\noindent
@@ -31,8 +44,8 @@
A validity relation between
states and formulae specifies the semantics:%
\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{consts}\ valid\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}state\ {\isasymRightarrow}\ formula\ {\isasymRightarrow}\ bool{\isachardoublequote}\ \ \ {\isacharparenleft}{\isachardoublequote}{\isacharparenleft}{\isacharunderscore}\ {\isasymTurnstile}\ {\isacharunderscore}{\isacharparenright}{\isachardoublequote}\ {\isacharbrackleft}{\isadigit{8}}{\isadigit{0}}{\isacharcomma}{\isadigit{8}}{\isadigit{0}}{\isacharbrackright}\ {\isadigit{8}}{\isadigit{0}}{\isacharparenright}\isamarkupfalse%
+\isamarkupfalse%
+\isacommand{consts}\ valid\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}state\ {\isasymRightarrow}\ formula\ {\isasymRightarrow}\ bool{\isachardoublequote}\ \ \ {\isacharparenleft}{\isachardoublequote}{\isacharparenleft}{\isacharunderscore}\ {\isasymTurnstile}\ {\isacharunderscore}{\isacharparenright}{\isachardoublequote}\ {\isacharbrackleft}{\isadigit{8}}{\isadigit{0}}{\isacharcomma}{\isadigit{8}}{\isadigit{0}}{\isacharbrackright}\ {\isadigit{8}}{\isadigit{0}}{\isacharparenright}\isamarkuptrue%
%
\begin{isamarkuptext}%
\noindent
@@ -40,13 +53,13 @@
\hbox{\isa{valid\ s\ f}}.
The definition of \isa{{\isasymTurnstile}} is by recursion over the syntax:%
\end{isamarkuptext}%
-\isamarkuptrue%
+\isamarkupfalse%
\isacommand{primrec}\isanewline
{\isachardoublequote}s\ {\isasymTurnstile}\ Atom\ a\ \ {\isacharequal}\ {\isacharparenleft}a\ {\isasymin}\ L\ s{\isacharparenright}{\isachardoublequote}\isanewline
{\isachardoublequote}s\ {\isasymTurnstile}\ Neg\ f\ \ \ {\isacharequal}\ {\isacharparenleft}{\isasymnot}{\isacharparenleft}s\ {\isasymTurnstile}\ f{\isacharparenright}{\isacharparenright}{\isachardoublequote}\isanewline
{\isachardoublequote}s\ {\isasymTurnstile}\ And\ f\ g\ {\isacharequal}\ {\isacharparenleft}s\ {\isasymTurnstile}\ f\ {\isasymand}\ s\ {\isasymTurnstile}\ g{\isacharparenright}{\isachardoublequote}\isanewline
{\isachardoublequote}s\ {\isasymTurnstile}\ AX\ f\ \ \ \ {\isacharequal}\ {\isacharparenleft}{\isasymforall}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}t{\isacharparenright}\ {\isasymin}\ M\ {\isasymlongrightarrow}\ t\ {\isasymTurnstile}\ f{\isacharparenright}{\isachardoublequote}\isanewline
-{\isachardoublequote}s\ {\isasymTurnstile}\ EF\ f\ \ \ \ {\isacharequal}\ {\isacharparenleft}{\isasymexists}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}t{\isacharparenright}\ {\isasymin}\ M\isactrlsup {\isacharasterisk}\ {\isasymand}\ t\ {\isasymTurnstile}\ f{\isacharparenright}{\isachardoublequote}\isamarkupfalse%
+{\isachardoublequote}s\ {\isasymTurnstile}\ EF\ f\ \ \ \ {\isacharequal}\ {\isacharparenleft}{\isasymexists}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}t{\isacharparenright}\ {\isasymin}\ M\isactrlsup {\isacharasterisk}\ {\isasymand}\ t\ {\isasymTurnstile}\ f{\isacharparenright}{\isachardoublequote}\isamarkuptrue%
%
\begin{isamarkuptext}%
\noindent
@@ -59,7 +72,7 @@
Now we come to the model checker itself. It maps a formula into the set of
states where the formula is true. It too is defined by recursion over the syntax:%
\end{isamarkuptext}%
-\isamarkuptrue%
+\isamarkupfalse%
\isacommand{consts}\ mc\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}formula\ {\isasymRightarrow}\ state\ set{\isachardoublequote}\isanewline
\isamarkupfalse%
\isacommand{primrec}\isanewline
@@ -67,7 +80,7 @@
{\isachardoublequote}mc{\isacharparenleft}Neg\ f{\isacharparenright}\ \ \ {\isacharequal}\ {\isacharminus}mc\ f{\isachardoublequote}\isanewline
{\isachardoublequote}mc{\isacharparenleft}And\ f\ g{\isacharparenright}\ {\isacharequal}\ mc\ f\ {\isasyminter}\ mc\ g{\isachardoublequote}\isanewline
{\isachardoublequote}mc{\isacharparenleft}AX\ f{\isacharparenright}\ \ \ \ {\isacharequal}\ {\isacharbraceleft}s{\isachardot}\ {\isasymforall}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}t{\isacharparenright}\ {\isasymin}\ M\ \ {\isasymlongrightarrow}\ t\ {\isasymin}\ mc\ f{\isacharbraceright}{\isachardoublequote}\isanewline
-{\isachardoublequote}mc{\isacharparenleft}EF\ f{\isacharparenright}\ \ \ \ {\isacharequal}\ lfp{\isacharparenleft}{\isasymlambda}T{\isachardot}\ mc\ f\ {\isasymunion}\ {\isacharparenleft}M{\isasyminverse}\ {\isacharbackquote}{\isacharbackquote}\ T{\isacharparenright}{\isacharparenright}{\isachardoublequote}\isamarkupfalse%
+{\isachardoublequote}mc{\isacharparenleft}EF\ f{\isacharparenright}\ \ \ \ {\isacharequal}\ lfp{\isacharparenleft}{\isasymlambda}T{\isachardot}\ mc\ f\ {\isasymunion}\ {\isacharparenleft}M{\isasyminverse}\ {\isacharbackquote}{\isacharbackquote}\ T{\isacharparenright}{\isacharparenright}{\isachardoublequote}\isamarkuptrue%
%
\begin{isamarkuptext}%
\noindent
@@ -84,36 +97,54 @@
First we prove monotonicity of the function inside \isa{lfp}
in order to make sure it really has a least fixed point.%
\end{isamarkuptext}%
-\isamarkuptrue%
+\isamarkupfalse%
\isacommand{lemma}\ mono{\isacharunderscore}ef{\isacharcolon}\ {\isachardoublequote}mono{\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ {\isacharparenleft}M{\isasyminverse}\ {\isacharbackquote}{\isacharbackquote}\ T{\isacharparenright}{\isacharparenright}{\isachardoublequote}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
\isamarkupfalse%
\isacommand{apply}{\isacharparenleft}rule\ monoI{\isacharparenright}\isanewline
\isamarkupfalse%
\isacommand{apply}\ blast\isanewline
\isamarkupfalse%
-\isacommand{done}\isamarkupfalse%
+\isacommand{done}%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+\isamarkuptrue%
%
\begin{isamarkuptext}%
\noindent
Now we can relate model checking and semantics. For the \isa{EF} case we need
a separate lemma:%
\end{isamarkuptext}%
-\isamarkuptrue%
+\isamarkupfalse%
\isacommand{lemma}\ EF{\isacharunderscore}lemma{\isacharcolon}\isanewline
-\ \ {\isachardoublequote}lfp{\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ {\isacharparenleft}M{\isasyminverse}\ {\isacharbackquote}{\isacharbackquote}\ T{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ {\isacharbraceleft}s{\isachardot}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}t{\isacharparenright}\ {\isasymin}\ M\isactrlsup {\isacharasterisk}\ {\isasymand}\ t\ {\isasymin}\ A{\isacharbraceright}{\isachardoublequote}\isamarkupfalse%
+\ \ {\isachardoublequote}lfp{\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ {\isacharparenleft}M{\isasyminverse}\ {\isacharbackquote}{\isacharbackquote}\ T{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ {\isacharbraceleft}s{\isachardot}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}t{\isacharparenright}\ {\isasymin}\ M\isactrlsup {\isacharasterisk}\ {\isasymand}\ t\ {\isasymin}\ A{\isacharbraceright}{\isachardoublequote}%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isamarkuptrue%
%
\begin{isamarkuptxt}%
\noindent
The equality is proved in the canonical fashion by proving that each set
includes the other; the inclusion is shown pointwise:%
\end{isamarkuptxt}%
-\isamarkuptrue%
+\isamarkupfalse%
\isacommand{apply}{\isacharparenleft}rule\ equalityI{\isacharparenright}\isanewline
\ \isamarkupfalse%
\isacommand{apply}{\isacharparenleft}rule\ subsetI{\isacharparenright}\isanewline
\ \isamarkupfalse%
-\isacommand{apply}{\isacharparenleft}simp{\isacharparenright}\isamarkupfalse%
-\isamarkupfalse%
+\isacommand{apply}{\isacharparenleft}simp{\isacharparenright}\isamarkuptrue%
%
\begin{isamarkuptxt}%
\noindent
@@ -123,12 +154,12 @@
\end{isabelle}
which is proved by \isa{lfp}-induction:%
\end{isamarkuptxt}%
-\ \isamarkuptrue%
+\ \isamarkupfalse%
\isacommand{apply}{\isacharparenleft}erule\ lfp{\isacharunderscore}induct{\isacharparenright}\isanewline
\ \ \isamarkupfalse%
\isacommand{apply}{\isacharparenleft}rule\ mono{\isacharunderscore}ef{\isacharparenright}\isanewline
\ \isamarkupfalse%
-\isacommand{apply}{\isacharparenleft}simp{\isacharparenright}\isamarkupfalse%
+\isacommand{apply}{\isacharparenleft}simp{\isacharparenright}\isamarkuptrue%
%
\begin{isamarkuptxt}%
\noindent
@@ -142,17 +173,17 @@
It is proved by \isa{blast}, using the transitivity of
\isa{M\isactrlsup {\isacharasterisk}}.%
\end{isamarkuptxt}%
-\ \isamarkuptrue%
-\isacommand{apply}{\isacharparenleft}blast\ intro{\isacharcolon}\ rtrancl{\isacharunderscore}trans{\isacharparenright}\isamarkupfalse%
+\ \isamarkupfalse%
+\isacommand{apply}{\isacharparenleft}blast\ intro{\isacharcolon}\ rtrancl{\isacharunderscore}trans{\isacharparenright}\isamarkuptrue%
%
\begin{isamarkuptxt}%
We now return to the second set inclusion subgoal, which is again proved
pointwise:%
\end{isamarkuptxt}%
-\isamarkuptrue%
+\isamarkupfalse%
\isacommand{apply}{\isacharparenleft}rule\ subsetI{\isacharparenright}\isanewline
\isamarkupfalse%
-\isacommand{apply}{\isacharparenleft}simp{\isacharcomma}\ clarify{\isacharparenright}\isamarkupfalse%
+\isacommand{apply}{\isacharparenleft}simp{\isacharcomma}\ clarify{\isacharparenright}\isamarkuptrue%
%
\begin{isamarkuptxt}%
\noindent
@@ -174,8 +205,8 @@
\isa{P\ a} provided each step backwards from a predecessor \isa{z} of
\isa{b} preserves \isa{P}.%
\end{isamarkuptxt}%
-\isamarkuptrue%
-\isacommand{apply}{\isacharparenleft}erule\ converse{\isacharunderscore}rtrancl{\isacharunderscore}induct{\isacharparenright}\isamarkupfalse%
+\isamarkupfalse%
+\isacommand{apply}{\isacharparenleft}erule\ converse{\isacharunderscore}rtrancl{\isacharunderscore}induct{\isacharparenright}\isamarkuptrue%
%
\begin{isamarkuptxt}%
\noindent
@@ -185,8 +216,8 @@
\end{isabelle}
is solved by unrolling \isa{lfp} once%
\end{isamarkuptxt}%
-\ \isamarkuptrue%
-\isacommand{apply}{\isacharparenleft}subst\ lfp{\isacharunderscore}unfold{\isacharbrackleft}OF\ mono{\isacharunderscore}ef{\isacharbrackright}{\isacharparenright}\isamarkupfalse%
+\ \isamarkupfalse%
+\isacommand{apply}{\isacharparenleft}subst\ lfp{\isacharunderscore}unfold{\isacharbrackleft}OF\ mono{\isacharunderscore}ef{\isacharbrackright}{\isacharparenright}\isamarkuptrue%
%
\begin{isamarkuptxt}%
\begin{isabelle}%
@@ -194,32 +225,52 @@
\end{isabelle}
and disposing of the resulting trivial subgoal automatically:%
\end{isamarkuptxt}%
-\ \isamarkuptrue%
-\isacommand{apply}{\isacharparenleft}blast{\isacharparenright}\isamarkupfalse%
+\ \isamarkupfalse%
+\isacommand{apply}{\isacharparenleft}blast{\isacharparenright}\isamarkuptrue%
%
\begin{isamarkuptxt}%
\noindent
The proof of the induction step is identical to the one for the base case:%
\end{isamarkuptxt}%
-\isamarkuptrue%
+\isamarkupfalse%
\isacommand{apply}{\isacharparenleft}subst\ lfp{\isacharunderscore}unfold{\isacharbrackleft}OF\ mono{\isacharunderscore}ef{\isacharbrackright}{\isacharparenright}\isanewline
\isamarkupfalse%
\isacommand{apply}{\isacharparenleft}blast{\isacharparenright}\isanewline
\isamarkupfalse%
-\isacommand{done}\isamarkupfalse%
+\isacommand{done}%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+\isamarkuptrue%
%
\begin{isamarkuptext}%
The main theorem is proved in the familiar manner: induction followed by
\isa{auto} augmented with the lemma as a simplification rule.%
\end{isamarkuptext}%
-\isamarkuptrue%
+\isamarkupfalse%
\isacommand{theorem}\ {\isachardoublequote}mc\ f\ {\isacharequal}\ {\isacharbraceleft}s{\isachardot}\ s\ {\isasymTurnstile}\ f{\isacharbraceright}{\isachardoublequote}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
\isamarkupfalse%
\isacommand{apply}{\isacharparenleft}induct{\isacharunderscore}tac\ f{\isacharparenright}\isanewline
\isamarkupfalse%
\isacommand{apply}{\isacharparenleft}auto\ simp\ add{\isacharcolon}\ EF{\isacharunderscore}lemma{\isacharparenright}\isanewline
\isamarkupfalse%
-\isacommand{done}\isamarkupfalse%
+\isacommand{done}%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+\isamarkuptrue%
%
\begin{isamarkuptext}%
\begin{exercise}
@@ -240,20 +291,58 @@
\end{exercise}
\index{PDL|)}%
\end{isamarkuptext}%
-\isamarkuptrue%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
\end{isabellebody}%
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