--- a/cws/main_cw03.tex	Thu Jan 13 12:55:03 2022 +0000
+++ b/cws/main_cw03.tex	Mon Apr 11 23:55:27 2022 +0100
@@ -176,6 +176,7 @@
 5000000: 1.29659 secs.
 \end{lstlisting}%$
 
+
 \subsection*{Preliminaries}
 
 The task is to implement a regular expression matcher that is based on
@@ -241,8 +242,10 @@
 The alternative regular expressions comes in two versions: one is
 binary (+ / \texttt{ALT}) and the other is n-ary ($\sum$ /
 \texttt{ALTs}). The latter takes a list of regular expressions as
-arguments.  In what follows we shall use $rs$ to stand for lists of
-regular expressions.  The binary alternative can be seen as an abbreviation,
+argument.  In what follows we shall use $rs$ to stand for lists of
+regular expressions. When the list is empty, we shall write $\sum\,[]$;
+if it is non-empty, we sometimes write $\sum\,[r_1,..., r_n]$.
+The binary alternative can be seen as an abbreviation,
 that is $r_1 + r_2 \dn \sum\,[r_1, r_2]$. As a result we can ignore the
 binary version and only implement the n-ary alternative.
 
@@ -267,7 +270,7 @@
 
 \item[(2)] Implement a function, called \textit{der}, by recursion over
   regular expressions. It takes a character and a regular expression
-  as arguments and calculates the derivative of a regular expression according
+  as arguments and calculates the \emph{derivative} of a regular expression according
   to the rules:
 
 \begin{center}
@@ -337,8 +340,8 @@
     \end{tabular}  
   \end{center}  
 
-  The first clause just states that empty lists cannot be further
-  flattened. The second removes all $\ZERO$s from the list.
+  The first clause states that empty lists cannot be further
+  flattened. The second removes the first $\ZERO$ from the list and recurses.
   The third is when the first regular expression is an \texttt{ALTs}, then
   the content of this alternative should be spilled out and appended
   with the flattened rest of the list. The last case is for all other
@@ -366,16 +369,16 @@
 expressions $r_1,.. ,r_n$; then flatten the resulting list using
 \texttt{flts}; finally remove all duplicates in this list (this can be
 done in Scala using the function
-\texttt{\_.distinct}). \textcolor{red}{When you perform these
+\texttt{\_.distinct}). When you perform these
   operations, you end up with three cases, namely where the list is
   empty, contains a single element and ``otherwise''. These cases
-  should be processed as follows}
+  should be processed as follows
 \begin{center}
-\textcolor{red}{\begin{tabular}{l@{\hspace{4mm}}c@{\hspace{4mm}}ll}
+\begin{tabular}{l@{\hspace{4mm}}c@{\hspace{4mm}}ll}
 $\sum\;[]$ & $\mapsto$ & $\ZERO$\\ 
 $\sum\;[r]$ & $\mapsto$ & $r$\\ 
 $\sum\;rs$ & $\mapsto$ & $\sum\;rs$ & ``otherwise''\\ 
-\end{tabular}}
+\end{tabular}
 \end{center}
 
   
@@ -515,7 +518,7 @@
 \begin{center}
 \begin{tabular}{@{}cc@{}}
 \multicolumn{2}{c}{Graph: $(a^*)^*\cdot b$ and strings 
-           $\underbrace{a\ldots a}_{n}$}\bigskip\\
+           $\underbrace{a\ldots a}_{n}$}\medskip\\
   
 \begin{tikzpicture}
 \begin{axis}[
@@ -531,7 +534,7 @@
     scaled ticks=false,
     axis lines=left,
     width=6cm,
-    height=5.0cm, 
+    height=5.5cm, 
     legend entries={Python, Java 8, JavaScript, Swift, Dart},  
     legend pos=north west,
     legend cell align=left]
@@ -557,7 +560,7 @@
     scaled ticks=false,
     axis lines=left,
     width=6cm,
-    height=5.0cm, 
+    height=5.5cm, 
     legend entries={Java 9},  
     legend pos=north west]
 \addplot[cyan,mark=*, mark options={fill=white}] table {re-java9.data};