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\section*{Coursework (Strand 2)}

\noindent This coursework is worth 25\% and is due on 12
December at 16:00. You are asked to prove the correctness of a
regular expression matcher from the lectures using the
Isabelle theorem prover. You need to submit a theory file
containing this proof. The Isabelle theorem prover is
available from 

\begin{center}
\url{http://isabelle.in.tum.de}
\end{center}

\noindent This is an interactive theorem prover, meaning that
you can make definitions and state properties, and then help
the system with proving these properties. Sometimes the proofs
are also automatic. There is a shortish user guide for
Isabelle, called ``Programming and Proving in Isabelle/HOL''
at

\begin{center}
\url{http://isabelle.in.tum.de/documentation.html}
\end{center}

\noindent
and also a longer (free) book at

\begin{center}
\url{http://www.concrete-semantics.org}
\end{center}

\noindent The Isabelle theorem prover is operated through the
jEdit IDE, which might not be an editor that is widely known.
JEdit is documented in

\begin{center}
\url{http://isabelle.in.tum.de/dist/Isabelle2014/doc/jedit.pdf}
\end{center}


\noindent If you need more help or you are stuck somewhere,
please feel free to contact me (christian.urban@kcl.ac.uk). I
am a main developer of Isabelle and have used it for
approximately the 14 years. One of the success stories of
Isabelle is the recent verification of a microkernel operating
system by an Australian group, see \url{http://sel4.systems}.
Their operating system is the only one that has been proved
correct according to its specification and is used for
application where high assurance, security and reliability is
needed. 


\subsection*{The Task}

In this coursework you are asked to prove the correctness of
the regular expression matcher from the lectures in Isabelle.
For this you need to first specify what the matcher is
supposed to do and then to implement the algorithm. Finally
you need to prove that the algorithm meets the specification.
The first two parts are relatively easy, because the
definitions in Isabelle will look very similar to the
mathematical definitions from the lectures or the Scala code
that is supplied at KEATS. For example very similar to Scala,
regular expressions are defined in Isabelle as an inductive
datatype:

\begin{lstlisting}[language={},numbers=none]
datatype rexp =
  NULL
| EMPTY
| CHAR char
| SEQ rexp rexp
| ALT rexp rexp
| STAR rexp
\end{lstlisting}

\noindent The meaning of regular expressions is given as 
usual:

\begin{center}
\begin{tabular}{rcl@{\hspace{10mm}}l}
$L(\varnothing)$  & $\dn$ & $\varnothing$   & \pcode{NULL}\\
$L(\epsilon)$     & $\dn$ & $\{[]\}$        & \pcode{EMPTY}\\ 
$L(c)$            & $\dn$ & $\{[c]\}$       & \pcode{CHAR}\\
$L(r_1 + r_2)$     & $\dn$ & $L(r_1) \cup L(r_2)$ & \pcode{ALT}\\
$L(r_1 \cdot r_2)$ & $\dn$ & $L(r_1) \,@\, L(r_2)$ & \pcode{SEQ}\\
$L(r^*)$           & $\dn$ & $(L(r))^*$ & \pcode{STAR}\\
\end{tabular}
\end{center}

\noindent You would need to implement this function in order
to state the theorem about the correctness of the algorithm.
The function $L$ should in Isabelle take a \pcode{rexp} as
input and return a set of strings. Its type is
therefore 

\begin{center}
\pcode{L} \pcode{::} \pcode{rexp} $\Rightarrow$ \pcode{string set}
\end{center}

\noindent Isabelle treats strings as an abbreviation for lists
of characters. This means you can pattern-match strings like
lists. The union operation on sets (for the \pcode{ALT}-case)
is a standard definition in Isabelle, but not the
concatenation operation on sets and also not the
star-operation. You would have to supply these definitions.
The concatenation operation can be defined in terms of the
append function, written \code{_ @ _} in Isabelle, for lists.
The star-operation can be defined as a ``big-union'' of 
powers, like in the lectures, or directly as an inductive set.

The functions for the matcher are shown in
Figure~\ref{matcher}. The theorem that needs to be proved is

\begin{lstlisting}[numbers=none,language={},keywordstyle=\color{black}\ttfamily,mathescape]
theorem 
  "matches r s $\longleftrightarrow$ s $\in$ L r"
\end{lstlisting}

\noindent which states that the function \emph{matches} is
true if and only if the string is in the language of the
regular expression. A proof for this lemma will need
side-lemmas about \pcode{nullable} and \pcode{der}. An example
proof in Isabelle that will not be relevant for the theorem
above is given in Figure~\ref{proof}.

\begin{figure}[p]
\begin{lstlisting}[language={},keywordstyle=\color{black}\ttfamily,mathescape]
fun 
  nullable :: "rexp $\Rightarrow$ bool"
where
  "nullable NULL = False"
| "nullable EMPTY = True"
| "nullable (CHAR _) = False"
| "nullable (ALT r1 r2) = (nullable(r1) $\vee$ nullable(r2))"
| "nullable (SEQ r1 r2) = (nullable(r1) $\wedge$ nullable(r2))"
| "nullable (STAR _) = True"

fun 
  der :: "char $\Rightarrow$ rexp $\Rightarrow$ rexp"
where
  "der c NULL = NULL"
| "der c EMPTY = NULL"
| "der c (CHAR d) = (if c = d then EMPTY else NULL)"
| "der c (ALT r1 r2) = ALT (der c r1) (der c r2)"
| "der c (SEQ r1 r2) = 
     (if (nullable r1) then ALT (SEQ (der c r1) r2) (der c r2)
                       else SEQ (der c r1) r2)"
| "der c (STAR r) = SEQ (der c r) (STAR r)"

fun 
  ders :: "rexp $\Rightarrow$ string $\Rightarrow$ rexp"
where
  "ders r [] = r"
| "ders r (c # s) = ders (der c r) s"

fun 
  matches :: "rexp $\Rightarrow$ string $\Rightarrow$ bool"
where
  "matches r s = nullable (ders r s)" 
\end{lstlisting}
\caption{The definition of the matcher algorithm in 
Isabelle.\label{matcher}}
\end{figure}

\begin{figure}[p]
\begin{lstlisting}[language={},keywordstyle=\color{black}\ttfamily,mathescape]
fun 
  zeroable :: "rexp $\Rightarrow$ bool"
where
  "zeroable NULL = True"
| "zeroable EMPTY = False"
| "zeroable (CHAR _) = False"
| "zeroable (ALT r1 r2) = (zeroable(r1) $\wedge$ zeroable(r2))"
| "zeroable (SEQ r1 r2) = (zeroable(r1) $\vee$ zeroable(r2))"
| "zeroable (STAR _) = False"

lemma
  "zeroable r $\longleftrightarrow$ L r = {}"
proof (induct)
  case (NULL)
  have "zeroable NULL" "L NULL = {}" by simp_all
  then show "zeroable NULL $\longleftrightarrow$ (L NULL = {})" by simp
next
  case (EMPTY)
  have "$\neg$ zeroable EMPTY" "L EMPTY = {[]}" by simp_all
  then show "zeroable EMPTY $\longleftrightarrow$ (L EMPTY = {})" by simp
next
  case (CHAR c)
  have "$\neg$ zeroable (CHAR c)" "L (CHAR c) = {[c]}" by simp_all
  then show "zeroable (CHAR c) $\longleftrightarrow$ (L (CHAR c) = {})" by simp
next 
  case (ALT r1 r2)
  have ih1: "zeroable r1 $\longleftrightarrow$ L r1 = {}" by fact
  have ih2: "zeroable r2 $\longleftrightarrow$ L r2 = {}" by fact
  show "zeroable (ALT r1 r2) $\longleftrightarrow$ (L (ALT r1 r2) = {})" 
    using ih1 ih2 by simp
next
  case (SEQ r1 r2)
  have ih1: "zeroable r1 $\longleftrightarrow$ L r1 = {}" by fact
  have ih2: "zeroable r2 $\longleftrightarrow$ L r2 = {}" by fact
  show "zeroable (SEQ r1 r2) $\longleftrightarrow$ (L (SEQ r1 r2) = {})" 
    using ih1 ih2 by (auto simp add: Conc_def)
next
  case (STAR r)
  have "$\neg$ zeroable (STAR r)" "[] $\in$ L (r) ^ 0" by simp_all
  then show "zeroable (STAR r) $\longleftrightarrow$ (L (STAR r) = {})" 
    by (simp (no_asm) add: Star_def) blast
qed
\end{lstlisting}
\caption{An Isabelle proof about the function \texttt{zeroable}.\label{proof}}
\end{figure}

\end{document}

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