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\begin{isabellebody}%

\def\isabellecontext{PDL}%

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\isamarkupsubsection{Propositional dynmic logic---PDL}

%

\begin{isamarkuptext}%

The formulae of PDL are built up from atomic propositions via the customary

propositional connectives of negation and conjunction and the two temporal

connectives \isa{AX} and \isa{EF}. Since formulae are essentially

(syntax) trees, they are naturally modelled as a datatype:%

\end{isamarkuptext}%

\isacommand{datatype}\ formula\ {\isacharequal}\ Atom\ atom\isanewline

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ Neg\ formula\isanewline

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ And\ formula\ formula\isanewline

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ AX\ formula\isanewline

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ EF\ formula%

\begin{isamarkuptext}%

\noindent

This is almost the same as in the boolean expression case study in

\S\ref{sec:boolex}, except that what used to be called \isa{Var} is now

called \isa{formula{\isachardot}Atom}.

The meaning of these formulae is given by saying which formula is true in

which state:%

\end{isamarkuptext}%

\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}%

\begin{isamarkuptext}%

\noindent

The concrete syntax annotation allows us to write \isa{s\ {\isasymTurnstile}\ f} instead of

\isa{valid\ s\ f}.

The definition of \isa{{\isasymTurnstile}} is by recursion over the syntax:%

\end{isamarkuptext}%

\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{\isacharcircum}{\isacharasterisk}\ {\isasymand}\ t\ {\isasymTurnstile}\ f{\isacharparenright}{\isachardoublequote}%

\begin{isamarkuptext}%

\noindent

The first three equations should be self-explanatory. The temporal formula

\isa{AX\ f} means that \isa{f} is true in all next states whereas

\isa{EF\ f} means that there exists some future state in which \isa{f} is

true. The future is expressed via \isa{{\isacharcircum}{\isacharasterisk}}, 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

states where the formula is true and is defined by recursion over the syntax,

too:%

\end{isamarkuptext}%

\isacommand{consts}\ mc\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}formula\ {\isasymRightarrow}\ state\ set{\isachardoublequote}\isanewline

\isacommand{primrec}\isanewline

{\isachardoublequote}mc{\isacharparenleft}Atom\ a{\isacharparenright}\ \ {\isacharequal}\ {\isacharbraceleft}s{\isachardot}\ a\ {\isasymin}\ L\ s{\isacharbraceright}{\isachardoublequote}\isanewline

{\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}\ M{\isacharcircum}{\isacharminus}\isadigit{1}\ {\isacharcircum}{\isacharcircum}\ T{\isacharparenright}{\isachardoublequote}%

\begin{isamarkuptext}%

\noindent

Only the equation for \isa{EF} deserves some comments. Remember that the

postfix \isa{{\isacharcircum}{\isacharminus}\isadigit{1}} and the infix \isa{{\isacharcircum}{\isacharcircum}} are predefined and denote the

converse of a relation and the application of a relation to a set. Thus

\isa{M{\isacharcircum}{\isacharminus}\isadigit{1}\ {\isacharcircum}{\isacharcircum}\ T} is the set of all predecessors of \isa{T} and the least

fixpoint (\isa{lfp}) of \isa{{\isasymlambda}T{\isachardot}\ mc\ f\ {\isasymunion}\ M{\isacharcircum}{\isacharminus}\isadigit{1}\ {\isacharcircum}{\isacharcircum}\ T} is the least set

\isa{T} containing \isa{mc\ f} and all predecessors of \isa{T}. If you

find it hard to see that \isa{mc\ {\isacharparenleft}EF\ f{\isacharparenright}} contains exactly those states from

which there is a path to a state where \isa{f} is true, do not worry---that

will be proved in a moment.

First we prove monotonicity of the function inside \isa{lfp}%

\end{isamarkuptext}%

\isacommand{lemma}\ mono{\isacharunderscore}ef{\isacharcolon}\ {\isachardoublequote}mono{\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ M{\isacharcircum}{\isacharminus}\isadigit{1}\ {\isacharcircum}{\isacharcircum}\ T{\isacharparenright}{\isachardoublequote}\isanewline

\isacommand{apply}{\isacharparenleft}rule\ monoI{\isacharparenright}\isanewline

\isacommand{by}{\isacharparenleft}blast{\isacharparenright}%

\begin{isamarkuptext}%

\noindent

in order to make sure it really has a least fixpoint.

Now we can relate model checking and semantics. For the \isa{EF} case we need

a separate lemma:%

\end{isamarkuptext}%

\isacommand{lemma}\ EF{\isacharunderscore}lemma{\isacharcolon}\isanewline

\ \ {\isachardoublequote}lfp{\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ M{\isacharcircum}{\isacharminus}\isadigit{1}\ {\isacharcircum}{\isacharcircum}\ T{\isacharparenright}\ {\isacharequal}\ {\isacharbraceleft}s{\isachardot}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}t{\isacharparenright}\ {\isasymin}\ M{\isacharcircum}{\isacharasterisk}\ {\isasymand}\ t\ {\isasymin}\ A{\isacharbraceright}{\isachardoublequote}%

\begin{isamarkuptxt}%

\noindent

The equality is proved in the canonical fashion by proving that each set

contains the other; the containment is shown pointwise:%

\end{isamarkuptxt}%

\isacommand{apply}{\isacharparenleft}rule\ equalityI{\isacharparenright}\isanewline

\ \isacommand{apply}{\isacharparenleft}rule\ subsetI{\isacharparenright}\isanewline

\ \isacommand{apply}{\isacharparenleft}simp{\isacharparenright}%

\begin{isamarkuptxt}%

\noindent

Simplification leaves us with the following first subgoal

\begin{isabelle}

\ \isadigit{1}{\isachardot}\ {\isasymAnd}s{\isachardot}\ s\ {\isasymin}\ lfp\ {\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ M{\isacharcircum}{\isacharminus}\isadigit{1}\ {\isacharcircum}{\isacharcircum}\ T{\isacharparenright}\ {\isasymLongrightarrow}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M{\isacharcircum}{\isacharasterisk}\ {\isasymand}\ t\ {\isasymin}\ A

\end{isabelle}

which is proved by \isa{lfp}-induction:%

\end{isamarkuptxt}%

\ \isacommand{apply}{\isacharparenleft}erule\ Lfp{\isachardot}induct{\isacharparenright}\isanewline

\ \ \isacommand{apply}{\isacharparenleft}rule\ mono{\isacharunderscore}ef{\isacharparenright}\isanewline

\ \isacommand{apply}{\isacharparenleft}simp{\isacharparenright}%

\begin{isamarkuptxt}%

\noindent

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

\end{isabelle}

which is proved by \isa{blast} with the help of a few lemmas about

\isa{{\isacharcircum}{\isacharasterisk}}:%

\end{isamarkuptxt}%

\ \isacommand{apply}{\isacharparenleft}blast\ intro{\isacharcolon}\ r{\isacharunderscore}into{\isacharunderscore}rtrancl\ rtrancl{\isacharunderscore}trans{\isacharparenright}%

\begin{isamarkuptxt}%

We now return to the second set containment subgoal, which is again proved

pointwise:%

\end{isamarkuptxt}%

\isacommand{apply}{\isacharparenleft}rule\ subsetI{\isacharparenright}\isanewline

\isacommand{apply}{\isacharparenleft}simp{\isacharcomma}\ clarify{\isacharparenright}%

\begin{isamarkuptxt}%

\noindent

After simplification and clarification we are left with

\begin{isabelle}

\ \isadigit{1}{\isachardot}\ {\isasymAnd}s\ t{\isachardot}\ {\isasymlbrakk}{\isacharparenleft}s{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M{\isacharcircum}{\isacharasterisk}{\isacharsemicolon}\ t\ {\isasymin}\ A{\isasymrbrakk}\ {\isasymLongrightarrow}\ s\ {\isasymin}\ lfp\ {\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ M{\isacharcircum}{\isacharminus}\isadigit{1}\ {\isacharcircum}{\isacharcircum}\ T{\isacharparenright}

\end{isabelle}

This goal is proved by induction on \isa{{\isacharparenleft}s{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M{\isacharcircum}{\isacharasterisk}}. But since the model

checker works backwards (from \isa{t} to \isa{s}), we cannot use the

induction theorem \isa{rtrancl{\isacharunderscore}induct} because it works in the

forward direction. Fortunately the converse induction theorem

\isa{converse{\isacharunderscore}rtrancl{\isacharunderscore}induct} already exists:

\begin{isabelle}%

\ \ \ \ \ {\isasymlbrakk}{\isacharparenleft}a{\isacharcomma}\ b{\isacharparenright}\ {\isasymin}\ r{\isacharcircum}{\isacharasterisk}{\isacharsemicolon}\ P\ b{\isacharsemicolon}\isanewline

\ \ \ \ \ \ \ \ {\isasymAnd}y\ z{\isachardot}\ {\isasymlbrakk}{\isacharparenleft}y{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharsemicolon}\ {\isacharparenleft}z{\isacharcomma}\ b{\isacharparenright}\ {\isasymin}\ r{\isacharcircum}{\isacharasterisk}{\isacharsemicolon}\ P\ z{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\ y{\isasymrbrakk}\isanewline

\ \ \ \ \ {\isasymLongrightarrow}\ P\ a%

\end{isabelle}

It says that if \isa{{\isacharparenleft}a{\isacharcomma}\ b{\isacharparenright}\ {\isasymin}\ r{\isacharcircum}{\isacharasterisk}} and we know \isa{P\ b} then we can infer

\isa{P\ a} provided each step backwards from a predecessor \isa{z} of

\isa{b} preserves \isa{P}.%

\end{isamarkuptxt}%

\isacommand{apply}{\isacharparenleft}erule\ converse{\isacharunderscore}rtrancl{\isacharunderscore}induct{\isacharparenright}%

\begin{isamarkuptxt}%

\noindent

The base case

\begin{isabelle}

\ \isadigit{1}{\isachardot}\ {\isasymAnd}t{\isachardot}\ t\ {\isasymin}\ A\ {\isasymLongrightarrow}\ t\ {\isasymin}\ lfp\ {\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ M{\isacharcircum}{\isacharminus}\isadigit{1}\ {\isacharcircum}{\isacharcircum}\ T{\isacharparenright}

\end{isabelle}

is solved by unrolling \isa{lfp} once%

\end{isamarkuptxt}%

\ \isacommand{apply}{\isacharparenleft}rule\ ssubst{\isacharbrackleft}OF\ lfp{\isacharunderscore}Tarski{\isacharbrackleft}OF\ mono{\isacharunderscore}ef{\isacharbrackright}{\isacharbrackright}{\isacharparenright}%

\begin{isamarkuptxt}%

\begin{isabelle}

\ \isadigit{1}{\isachardot}\ {\isasymAnd}t{\isachardot}\ t\ {\isasymin}\ A\ {\isasymLongrightarrow}\ t\ {\isasymin}\ A\ {\isasymunion}\ M{\isacharcircum}{\isacharminus}\isadigit{1}\ {\isacharcircum}{\isacharcircum}\ lfp\ {\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ M{\isacharcircum}{\isacharminus}\isadigit{1}\ {\isacharcircum}{\isacharcircum}\ T{\isacharparenright}

\end{isabelle}

and disposing of the resulting trivial subgoal automatically:%

\end{isamarkuptxt}%

\ \isacommand{apply}{\isacharparenleft}blast{\isacharparenright}%

\begin{isamarkuptxt}%

\noindent

The proof of the induction step is identical to the one for the base case:%

\end{isamarkuptxt}%

\isacommand{apply}{\isacharparenleft}rule\ ssubst{\isacharbrackleft}OF\ lfp{\isacharunderscore}Tarski{\isacharbrackleft}OF\ mono{\isacharunderscore}ef{\isacharbrackright}{\isacharbrackright}{\isacharparenright}\isanewline

\isacommand{by}{\isacharparenleft}blast{\isacharparenright}%

\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}%

\isacommand{theorem}\ {\isachardoublequote}mc\ f\ {\isacharequal}\ {\isacharbraceleft}s{\isachardot}\ s\ {\isasymTurnstile}\ f{\isacharbraceright}{\isachardoublequote}\isanewline

\isacommand{apply}{\isacharparenleft}induct{\isacharunderscore}tac\ f{\isacharparenright}\isanewline

\isacommand{by}{\isacharparenleft}auto\ simp\ add{\isacharcolon}EF{\isacharunderscore}lemma{\isacharparenright}\end{isabellebody}%

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