doc-src/TutorialI/CTL/document/PDL.tex
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 nipkow@10123  1 %  nipkow@10123  2 \begin{isabellebody}%  nipkow@10123  3 \def\isabellecontext{PDL}%  nipkow@10123  4 %  nipkow@10178  5 \isamarkupsubsection{Propositional dynamic logic---PDL}  nipkow@10133  6 %  nipkow@10133  7 \begin{isamarkuptext}%  nipkow@10178  8 \index{PDL|(}  nipkow@10133  9 The formulae of PDL are built up from atomic propositions via the customary  nipkow@10133  10 propositional connectives of negation and conjunction and the two temporal  nipkow@10133  11 connectives \isa{AX} and \isa{EF}. Since formulae are essentially  nipkow@10133  12 (syntax) trees, they are naturally modelled as a datatype:%  nipkow@10133  13 \end{isamarkuptext}%  nipkow@10149  14 \isacommand{datatype}\ formula\ {\isacharequal}\ Atom\ atom\isanewline  nipkow@10149  15 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ Neg\ formula\isanewline  nipkow@10149  16 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ And\ formula\ formula\isanewline  nipkow@10149  17 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ AX\ formula\isanewline  nipkow@10149  18 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ EF\ formula%  nipkow@10133  19 \begin{isamarkuptext}%  nipkow@10133  20 \noindent  nipkow@10149  21 This is almost the same as in the boolean expression case study in  nipkow@10149  22 \S\ref{sec:boolex}, except that what used to be called \isa{Var} is now  nipkow@10149  23 called \isa{formula{\isachardot}Atom}.  nipkow@10149  24 nipkow@10133  25 The meaning of these formulae is given by saying which formula is true in  nipkow@10133  26 which state:%  nipkow@10133  27 \end{isamarkuptext}%  nipkow@10187  28 \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}%  nipkow@10149  29 \begin{isamarkuptext}%  nipkow@10149  30 \noindent  nipkow@10149  31 The concrete syntax annotation allows us to write \isa{s\ {\isasymTurnstile}\ f} instead of  nipkow@10149  32 \isa{valid\ s\ f}.  nipkow@10149  33 nipkow@10149  34 The definition of \isa{{\isasymTurnstile}} is by recursion over the syntax:%  nipkow@10149  35 \end{isamarkuptext}%  nipkow@10123  36 \isacommand{primrec}\isanewline  nipkow@10133  37 {\isachardoublequote}s\ {\isasymTurnstile}\ Atom\ a\ \ {\isacharequal}\ {\isacharparenleft}a\ {\isasymin}\ L\ s{\isacharparenright}{\isachardoublequote}\isanewline  nipkow@10149  38 {\isachardoublequote}s\ {\isasymTurnstile}\ Neg\ f\ \ \ {\isacharequal}\ {\isacharparenleft}{\isasymnot}{\isacharparenleft}s\ {\isasymTurnstile}\ f{\isacharparenright}{\isacharparenright}{\isachardoublequote}\isanewline  nipkow@10123  39 {\isachardoublequote}s\ {\isasymTurnstile}\ And\ f\ g\ {\isacharequal}\ {\isacharparenleft}s\ {\isasymTurnstile}\ f\ {\isasymand}\ s\ {\isasymTurnstile}\ g{\isacharparenright}{\isachardoublequote}\isanewline  nipkow@10123  40 {\isachardoublequote}s\ {\isasymTurnstile}\ AX\ f\ \ \ \ {\isacharequal}\ {\isacharparenleft}{\isasymforall}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}t{\isacharparenright}\ {\isasymin}\ M\ {\isasymlongrightarrow}\ t\ {\isasymTurnstile}\ f{\isacharparenright}{\isachardoublequote}\isanewline  nipkow@10133  41 {\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}%  nipkow@10133  42 \begin{isamarkuptext}%  nipkow@10149  43 \noindent  nipkow@10149  44 The first three equations should be self-explanatory. The temporal formula  nipkow@10149  45 \isa{AX\ f} means that \isa{f} is true in all next states whereas  nipkow@10149  46 \isa{EF\ f} means that there exists some future state in which \isa{f} is  nipkow@10149  47 true. The future is expressed via \isa{{\isacharcircum}{\isacharasterisk}}, the transitive reflexive  nipkow@10149  48 closure. Because of reflexivity, the future includes the present.  nipkow@10149  49 nipkow@10133  50 Now we come to the model checker itself. It maps a formula into the set of  nipkow@10149  51 states where the formula is true and is defined by recursion over the syntax,  nipkow@10149  52 too:%  nipkow@10133  53 \end{isamarkuptext}%  nipkow@10149  54 \isacommand{consts}\ mc\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}formula\ {\isasymRightarrow}\ state\ set{\isachardoublequote}\isanewline  nipkow@10123  55 \isacommand{primrec}\isanewline  nipkow@10133  56 {\isachardoublequote}mc{\isacharparenleft}Atom\ a{\isacharparenright}\ \ {\isacharequal}\ {\isacharbraceleft}s{\isachardot}\ a\ {\isasymin}\ L\ s{\isacharbraceright}{\isachardoublequote}\isanewline  nipkow@10149  57 {\isachardoublequote}mc{\isacharparenleft}Neg\ f{\isacharparenright}\ \ \ {\isacharequal}\ {\isacharminus}mc\ f{\isachardoublequote}\isanewline  nipkow@10133  58 {\isachardoublequote}mc{\isacharparenleft}And\ f\ g{\isacharparenright}\ {\isacharequal}\ mc\ f\ {\isasyminter}\ mc\ g{\isachardoublequote}\isanewline  nipkow@10123  59 {\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  nipkow@10187  60 {\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}%  nipkow@10133  61 \begin{isamarkuptext}%  nipkow@10149  62 \noindent  nipkow@10149  63 Only the equation for \isa{EF} deserves some comments. Remember that the  nipkow@10187  64 postfix \isa{{\isacharcircum}{\isacharminus}{\isadigit{1}}} and the infix \isa{{\isacharcircum}{\isacharcircum}} are predefined and denote the  nipkow@10149  65 converse of a relation and the application of a relation to a set. Thus  nipkow@10187  66 \isa{M{\isacharcircum}{\isacharminus}{\isadigit{1}}\ {\isacharcircum}{\isacharcircum}\ T} is the set of all predecessors of \isa{T} and the least  nipkow@10187  67 fixpoint (\isa{lfp}) of \isa{{\isasymlambda}T{\isachardot}\ mc\ f\ {\isasymunion}\ M{\isacharcircum}{\isacharminus}{\isadigit{1}}\ {\isacharcircum}{\isacharcircum}\ T} is the least set  nipkow@10149  68 \isa{T} containing \isa{mc\ f} and all predecessors of \isa{T}. If you  nipkow@10149  69 find it hard to see that \isa{mc\ {\isacharparenleft}EF\ f{\isacharparenright}} contains exactly those states from  nipkow@10149  70 which there is a path to a state where \isa{f} is true, do not worry---that  nipkow@10149  71 will be proved in a moment.  nipkow@10149  72 nipkow@10149  73 First we prove monotonicity of the function inside \isa{lfp}%  nipkow@10133  74 \end{isamarkuptext}%  nipkow@10187  75 \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  nipkow@10123  76 \isacommand{apply}{\isacharparenleft}rule\ monoI{\isacharparenright}\isanewline  nipkow@10159  77 \isacommand{apply}\ blast\isanewline  nipkow@10159  78 \isacommand{done}%  nipkow@10149  79 \begin{isamarkuptext}%  nipkow@10149  80 \noindent  nipkow@10149  81 in order to make sure it really has a least fixpoint.  nipkow@10149  82 nipkow@10149  83 Now we can relate model checking and semantics. For the \isa{EF} case we need  nipkow@10149  84 a separate lemma:%  nipkow@10149  85 \end{isamarkuptext}%  nipkow@10149  86 \isacommand{lemma}\ EF{\isacharunderscore}lemma{\isacharcolon}\isanewline  nipkow@10187  87 \ \ {\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}%  nipkow@10149  88 \begin{isamarkuptxt}%  nipkow@10149  89 \noindent  nipkow@10149  90 The equality is proved in the canonical fashion by proving that each set  nipkow@10149  91 contains the other; the containment is shown pointwise:%  nipkow@10149  92 \end{isamarkuptxt}%  nipkow@10123  93 \isacommand{apply}{\isacharparenleft}rule\ equalityI{\isacharparenright}\isanewline  nipkow@10123  94 \ \isacommand{apply}{\isacharparenleft}rule\ subsetI{\isacharparenright}\isanewline  nipkow@10149  95 \ \isacommand{apply}{\isacharparenleft}simp{\isacharparenright}%  nipkow@10149  96 \begin{isamarkuptxt}%  nipkow@10149  97 \noindent  nipkow@10149  98 Simplification leaves us with the following first subgoal  nipkow@10149  99 \begin{isabelle}  nipkow@10149  100 \ \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  nipkow@10149  101 \end{isabelle}  nipkow@10149  102 which is proved by \isa{lfp}-induction:%  nipkow@10149  103 \end{isamarkuptxt}%  wenzelm@10211  104 \ \isacommand{apply}{\isacharparenleft}erule\ lfp{\isacharunderscore}induct{\isacharparenright}\isanewline  nipkow@10149  105 \ \ \isacommand{apply}{\isacharparenleft}rule\ mono{\isacharunderscore}ef{\isacharparenright}\isanewline  nipkow@10149  106 \ \isacommand{apply}{\isacharparenleft}simp{\isacharparenright}%  nipkow@10149  107 \begin{isamarkuptxt}%  nipkow@10149  108 \noindent  nipkow@10149  109 Having disposed of the monotonicity subgoal,  nipkow@10149  110 simplification leaves us with the following main goal  nipkow@10149  111 \begin{isabelle}  nipkow@10149  112 \ \isadigit{1}{\isachardot}\ {\isasymAnd}s{\isachardot}\ s\ {\isasymin}\ A\ {\isasymor}\isanewline  nipkow@10149  113 \ \ \ \ \ \ \ \ \ 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  nipkow@10149  114 \ \ \ \ \ \ \ \ \ {\isasymLongrightarrow}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M{\isacharcircum}{\isacharasterisk}\ {\isasymand}\ t\ {\isasymin}\ A  nipkow@10149  115 \end{isabelle}  nipkow@10212  116 which is proved by \isa{blast} with the help of transitivity of \isa{{\isacharcircum}{\isacharasterisk}}:%  nipkow@10149  117 \end{isamarkuptxt}%  nipkow@10212  118 \ \isacommand{apply}{\isacharparenleft}blast\ intro{\isacharcolon}\ rtrancl{\isacharunderscore}trans{\isacharparenright}%  nipkow@10149  119 \begin{isamarkuptxt}%  nipkow@10149  120 We now return to the second set containment subgoal, which is again proved  nipkow@10149  121 pointwise:%  nipkow@10149  122 \end{isamarkuptxt}%  nipkow@10123  123 \isacommand{apply}{\isacharparenleft}rule\ subsetI{\isacharparenright}\isanewline  nipkow@10149  124 \isacommand{apply}{\isacharparenleft}simp{\isacharcomma}\ clarify{\isacharparenright}%  nipkow@10149  125 \begin{isamarkuptxt}%  nipkow@10149  126 \noindent  nipkow@10149  127 After simplification and clarification we are left with  nipkow@10149  128 \begin{isabelle}  nipkow@10149  129 \ \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}  nipkow@10149  130 \end{isabelle}  nipkow@10149  131 This goal is proved by induction on \isa{{\isacharparenleft}s{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M{\isacharcircum}{\isacharasterisk}}. But since the model  nipkow@10149  132 checker works backwards (from \isa{t} to \isa{s}), we cannot use the  nipkow@10149  133 induction theorem \isa{rtrancl{\isacharunderscore}induct} because it works in the  nipkow@10149  134 forward direction. Fortunately the converse induction theorem  nipkow@10149  135 \isa{converse{\isacharunderscore}rtrancl{\isacharunderscore}induct} already exists:  nipkow@10149  136 \begin{isabelle}%  nipkow@10149  137 \ \ \ \ \ {\isasymlbrakk}{\isacharparenleft}a{\isacharcomma}\ b{\isacharparenright}\ {\isasymin}\ r{\isacharcircum}{\isacharasterisk}{\isacharsemicolon}\ P\ b{\isacharsemicolon}\isanewline  nipkow@10149  138 \ \ \ \ \ \ \ \ {\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  nipkow@10149  139 \ \ \ \ \ {\isasymLongrightarrow}\ P\ a%  nipkow@10149  140 \end{isabelle}  nipkow@10149  141 It says that if \isa{{\isacharparenleft}a{\isacharcomma}\ b{\isacharparenright}\ {\isasymin}\ r{\isacharcircum}{\isacharasterisk}} and we know \isa{P\ b} then we can infer  nipkow@10149  142 \isa{P\ a} provided each step backwards from a predecessor \isa{z} of  nipkow@10149  143 \isa{b} preserves \isa{P}.%  nipkow@10149  144 \end{isamarkuptxt}%  nipkow@10149  145 \isacommand{apply}{\isacharparenleft}erule\ converse{\isacharunderscore}rtrancl{\isacharunderscore}induct{\isacharparenright}%  nipkow@10149  146 \begin{isamarkuptxt}%  nipkow@10149  147 \noindent  nipkow@10149  148 The base case  nipkow@10149  149 \begin{isabelle}  nipkow@10149  150 \ \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}  nipkow@10149  151 \end{isabelle}  nipkow@10149  152 is solved by unrolling \isa{lfp} once%  nipkow@10149  153 \end{isamarkuptxt}%  nipkow@10186  154 \ \isacommand{apply}{\isacharparenleft}rule\ ssubst{\isacharbrackleft}OF\ lfp{\isacharunderscore}unfold{\isacharbrackleft}OF\ mono{\isacharunderscore}ef{\isacharbrackright}{\isacharbrackright}{\isacharparenright}%  nipkow@10149  155 \begin{isamarkuptxt}%  nipkow@10149  156 \begin{isabelle}  nipkow@10149  157 \ \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}  nipkow@10149  158 \end{isabelle}  nipkow@10149  159 and disposing of the resulting trivial subgoal automatically:%  nipkow@10149  160 \end{isamarkuptxt}%  nipkow@10149  161 \ \isacommand{apply}{\isacharparenleft}blast{\isacharparenright}%  nipkow@10149  162 \begin{isamarkuptxt}%  nipkow@10149  163 \noindent  nipkow@10149  164 The proof of the induction step is identical to the one for the base case:%  nipkow@10149  165 \end{isamarkuptxt}%  nipkow@10186  166 \isacommand{apply}{\isacharparenleft}rule\ ssubst{\isacharbrackleft}OF\ lfp{\isacharunderscore}unfold{\isacharbrackleft}OF\ mono{\isacharunderscore}ef{\isacharbrackright}{\isacharbrackright}{\isacharparenright}\isanewline  nipkow@10159  167 \isacommand{apply}{\isacharparenleft}blast{\isacharparenright}\isanewline  nipkow@10159  168 \isacommand{done}%  nipkow@10149  169 \begin{isamarkuptext}%  nipkow@10149  170 The main theorem is proved in the familiar manner: induction followed by  nipkow@10149  171 \isa{auto} augmented with the lemma as a simplification rule.%  nipkow@10149  172 \end{isamarkuptext}%  nipkow@10123  173 \isacommand{theorem}\ {\isachardoublequote}mc\ f\ {\isacharequal}\ {\isacharbraceleft}s{\isachardot}\ s\ {\isasymTurnstile}\ f{\isacharbraceright}{\isachardoublequote}\isanewline  nipkow@10123  174 \isacommand{apply}{\isacharparenleft}induct{\isacharunderscore}tac\ f{\isacharparenright}\isanewline  nipkow@10159  175 \isacommand{apply}{\isacharparenleft}auto\ simp\ add{\isacharcolon}EF{\isacharunderscore}lemma{\isacharparenright}\isanewline  nipkow@10171  176 \isacommand{done}%  nipkow@10171  177 \begin{isamarkuptext}%  nipkow@10171  178 \begin{exercise}  nipkow@10171  179 \isa{AX} has a dual operator \isa{EN}\footnote{We cannot use the customary \isa{EX}  nipkow@10171  180 as that is the ASCII equivalent of \isa{{\isasymexists}}}  nipkow@10171  181 (there exists a next state such that'') with the intended semantics  nipkow@10171  182 \begin{isabelle}%  nipkow@10171  183 \ \ \ \ \ s\ {\isasymTurnstile}\ EN\ f\ {\isacharequal}\ {\isacharparenleft}{\isasymexists}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M\ {\isasymand}\ t\ {\isasymTurnstile}\ f{\isacharparenright}%  nipkow@10171  184 \end{isabelle}  nipkow@10171  185 Fortunately, \isa{EN\ f} can already be expressed as a PDL formula. How?  nipkow@10171  186 nipkow@10171  187 Show that the semantics for \isa{EF} satisfies the following recursion equation:  nipkow@10171  188 \begin{isabelle}%  nipkow@10171  189 \ \ \ \ \ s\ {\isasymTurnstile}\ EF\ f\ {\isacharequal}\ {\isacharparenleft}s\ {\isasymTurnstile}\ f\ {\isasymor}\ s\ {\isasymTurnstile}\ EN\ {\isacharparenleft}EF\ f{\isacharparenright}{\isacharparenright}%  nipkow@10171  190 \end{isabelle}  nipkow@10178  191 \end{exercise}  nipkow@10178  192 \index{PDL|)}%  nipkow@10171  193 \end{isamarkuptext}%  nipkow@10171  194 \end{isabellebody}%  nipkow@10123  195 %%% Local Variables:  nipkow@10123  196 %%% mode: latex  nipkow@10123  197 %%% TeX-master: "root"  nipkow@10123  198 %%% End: