--- a/slides/slides02.tex	Tue Sep 29 21:52:52 2020 +0100
+++ b/slides/slides02.tex	Sat Oct 03 00:51:47 2020 +0100
@@ -637,6 +637,118 @@
 \end{frame}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
 
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}[c]
+  \frametitle{$der$ for $n$-times}
+
+Case \bl{$n = 2$} and \bl{$r \cdot r$}: 
+
+\begin{center}
+  \begin{tabular}{lcl}
+    \bl{$der\,c\,(r\cdot r)$} & \bl{$\dn$} & \bl{if \; $nullable(r)$}\\
+                              &   & \bl{then \; \alert<3>{$(der\,c\,r)\cdot r + der\,c\,r$}}\\
+                              &   & \bl{else \; $(der\,c\,r)\cdot r$}\bigskip\pause\\
+    my claim                  &   & \bl{$\equiv\;$} \bl{$(der\,c\,r)\cdot r$}\\
+    (in this case)                                
+  \end{tabular} 
+\end{center}  
+
+\end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}[t]
+
+We know \bl{$nullable(r)$} holds!\pause 
+
+\begin{center}
+  \begin{tabular}{@{}lcl}
+    \bl{$(der\,c\,r)\cdot r + der\,c\,r$}\pause
+    & \bl{$\equiv$} & \bl{$(der\,c\,r)\cdot r + (der\,c\,r) \cdot \ONE$}\medskip\pause\\
+    & \bl{$\equiv$} & \bl{$(der\,c\,r)\cdot (r + \ONE)$}\medskip\pause\\
+    & \bl{$\equiv$} & \bl{$(der\,c\,r)\cdot r$}\\
+    \multicolumn{3}{r}{\small(remember \bl{$r$} is nullable)}
+  \end{tabular} 
+\end{center}\pause
+
+\rule{13cm}{0.8mm}
+
+\begin{textblock}{13}(2,10)
+\only<6>{%
+  \begin{tabular}{lcl}
+    \bl{$der\,c\,(r\cdot r)$} & \bl{$\dn$} & \bl{if \; $nullable(r)$}\\
+                          &   & \bl{then \; $(der\,c\,r)\cdot r + der\,c\,r$}\\
+                          &   & \bl{else \; $(der\,c\,r)\cdot r$}                                
+  \end{tabular}}
+\only<7>{%
+  \begin{tabular}{lcl}
+    \bl{$der\,c\,(r\cdot r)$} & \bl{$\dn$} & \bl{if \; $nullable(r)$}\\
+                          &   & \bl{then \; $(der\,c\,r)\cdot r$}\\
+                          &   & \bl{else \; $(der\,c\,r)\cdot r$}                                
+  \end{tabular}}
+\only<8>{%
+  \begin{tabular}{lcl}
+    \bl{$der\,c\,(r\cdot r)$} & \bl{$\dn$} & \bl{$(der\,c\,r)\cdot r$}                                
+  \end{tabular}}
+\end{textblock}
+
+
+\end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}[c]
+
+\begin{center}
+\begin{tabular}{ll@{\hspace{10mm}}l}
+& \bl{$r\{n\}$} & \bl{$der$}\hspace{20mm}\mbox{}\\\hline\\[-4mm]    
+\bl{$n = 0$} : & \bl{$\ONE$}           & \bl{$\ZERO$}\\
+\bl{$n = 1$} : & \bl{$r$}              & \bl{$(der\,c\,r)$}\\
+\bl{$n = 2$} : & \bl{$r\cdot r$}       & \bl{$(der\,c\,r)\cdot r$}\\
+\bl{$n = 3$} : & \bl{$r\cdot r\cdot r$}& \only<1>{???}\only<2->{\bl{$(der\,c\,r)\cdot r\cdot r$}}\\
+\quad\vdots  
+\end{tabular} 
+\end{center}
+
+\begin{textblock}{13}(1,11)
+\only<3->{%
+  \begin{tabular}{rcl}
+    \bl{$nullable(r\{n\})$} & \bl{$\dn$} &
+                          \bl{if\; $n = 0$\; then\; $true$\; else\; $nullable(r)$}\\
+    \bl{$der\,c\,(r\{n\})$} & \bl{$\dn$} &
+                          \bl{if\; $n = 0$\; then\; $\ZERO$\; else\; $(der\,c\,r)\cdot r\{n - 1\}$}
+  \end{tabular}}
+\end{textblock}
+
+\end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}[c]
+
+\begin{textblock}{13}(1.5,4)
+  \only<1-6>{%
+  \begin{tabular}{@{}lcl@{}}
+    \bl{$der\,c\,(r\cdot r\cdot r)$}
+    & \bl{$\dn$} & \bl{if $nullable(r)$}\medskip\\
+    & & \only<1>{\bl{then\; $(der\,c\,r)\cdot r\cdot r + der\,c\,(r\cdot r)$}}%
+        \only<2>{\bl{then\; $(der\,c\,r)\cdot r\cdot r + (der\,c\,r)\cdot r$}}%
+        \only<3>{\bl{then\; $(der\,c\,r)\cdot (r\cdot r + r)$}}%
+        \only<4>{\bl{then\; $(der\,c\,r)\cdot (r\cdot (r + \ONE))$}}%
+        \only<5>{\bl{then\; $(der\,c\,r)\cdot (r\cdot r)$}}%
+        \only<6>{\bl{then\; $(der\,c\,r)\cdot r\cdot r$}}\medskip\\
+    & & \bl{else\; $(der\,c\,r)\cdot r\cdot r$}\\
+  \end{tabular}}
+  \only<7->{%
+  \begin{tabular}{@{}lcl@{}}
+    \bl{$der\,c\,(r\cdot r\cdot r)$}
+    & \bl{$\dn$} & \bl{$(der\,c\,r)\cdot r\cdot r$}%
+  \end{tabular}} 
+\end{textblock}
+
+\end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{frame}[t]
@@ -869,12 +981,12 @@
 \item extends to most regular expressions, for example
 \bl{$\sim r$} (next slide)
 
-\item is easy to implement in a functional language (slide after)
+\item is easy to implement in a functional language
 
 \item the algorithm is already quite old; there is still
   work to be done to use it as a tokenizer (that is relatively new work)
 
-\item we can prove its correctness\ldots (several slides later)
+\item we can prove its correctness\ldots (another video)
 \end{itemize}
 
 \end{frame}
@@ -889,10 +1001,9 @@
 \item \bl{$L(\sim{}r) \dn UNIV - L(r)$}\medskip
 \item \bl{$nullable (\sim{}r) \dn not\, (nullable(r))$}\medskip
 \item \bl{$der\,c\,(\sim{}r) \dn \;\sim{}(der\,c\,r)$}
-\end{itemize}\bigskip\pause
+\end{itemize}\bigskip\bigskip\medskip\pause
 
 Used often for recognising comments:
-
 \[
 \bl{/ \cdot * \cdot (\sim{}([a\mbox{-}z]^* \cdot * \cdot / \cdot [a\mbox{-}z]^*)) \cdot * \cdot /}
 \]