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+% !TEX program = xelatex
+\documentclass{article}
+\usepackage{../style}
+\usepackage{../langs}
+
+\begin{document}
+
+\section*{Coursework (Strand 2)}
+
+\noindent This coursework is worth 20\% and is due on \cwISABELLE{} at
+18:00. You are asked to prove the correctness of the regular expression
+matcher from the lectures using the Isabelle theorem prover. You need to
+submit a theory file containing this proof and also a document
+describing your 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 completely 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 at kcl.ac.uk). I
+am one of the main developers of Isabelle and have used it for
+approximately 16 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 (like in helicopters which fly over enemy territory).
+
+
+\subsection*{The Task}
+
+In this coursework you are asked to prove the correctness of the
+regular expression matcher from the lectures in Isabelle.  The matcher
+should be able to deal with the usual (basic) regular expressions
+
+\[
+\ZERO,\; \ONE,\; c,\; r_1 + r_2,\; r_1 \cdot r_2,\; r^*
+\]
+
+\noindent
+but also with the following extended regular expressions:
+
+\begin{center}
+\begin{tabular}{ll}
+  $r^{\{n\}}$ & exactly $n$-times\\
+  $r^{\{..m\}}$ & zero or more times $r$ but no more than $m$-times\\
+  $r^{\{n..\}}$ & at least $n$-times $r$\\
+  $r^{\{n..m\}}$ & at least $n$-times $r$ but no more than $m$-times\\
+  $\sim{}r$ & not-regular-expression of $r$\\
+\end{tabular}
+\end{center}
+
+
+\noindent
+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 =
+  ZERO
+| ONE
+| 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}
+$L(\ZERO)$  & $\dn$ & $\varnothing$   & \pcode{ZERO}\\
+$L(\ONE)$     & $\dn$ & $\{[]\}$        & \pcode{ONE}\\ 
+$L(c)$            & $\dn$ & $\{[c]\}$       & \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 ZERO = False"
+| "nullable ONE = 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 ZERO = ZERO"
+| "der c ONE = ZERO"
+| "der c (CHAR d) = (if c = d then ONE else ZERO)"
+| "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 ZERO = True"
+| "zeroable ONE = 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 (ZERO)
+  have "zeroable ZERO" "L ZERO = {}" by simp_all
+  then show "zeroable ZERO $\longleftrightarrow$ (L ZERO = {})" by simp
+next
+  case (ONE)
+  have "$\neg$ zeroable ONE" "L ONE = {[]}" by simp_all
+  then show "zeroable ONE $\longleftrightarrow$ (L ONE = {})" 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}
+
+%%% Local Variables: 
+%%% mode: latex
+%%% TeX-master: t
+%%% End: