doc-src/TutorialI/CTL/document/PDL.tex
author nipkow
Thu Oct 12 18:38:23 2000 +0200 (2000-10-12)
changeset 10212 33fe2d701ddd
parent 10211 1bece7f35762
child 10242 028f54cd2cc9
permissions -rw-r--r--
*** empty log message ***
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: