--- a/slides/slides02.tex	Fri Oct 25 18:54:08 2024 +0100
+++ b/slides/slides02.tex	Sat Nov 09 06:23:35 2024 +0000
@@ -392,6 +392,56 @@
 \end{frame}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
 
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}[c]
+
+\bl{$r ::= \ZERO \,\;|\;\, \ONE \,\;|\;\, c \,\;|\;\, r_1 + r_2 \,\;|\;\, r_1 \cdot r_2 \,\;|\;\, r^{*} \,\;|\;\, r^{\{n\}}$}  
+
+\end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+  
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}[c]
+
+\begin{center}
+\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}}
+\bl{$nullable(\ZERO)$}    & \bl{$\dn$} & \bl{\textit{false}}\\
+\bl{$nullable(\ONE)$}       & \bl{$\dn$} & \bl{\textit{true}}\\
+\bl{$nullable (c)$}             & \bl{$\dn$} & \bl{\textit{false}}\\
+\bl{$nullable (r_1 + r_2)$}     & \bl{$\dn$} & \bl{$nullable(r_1) \vee nullable(r_2)$} \\ 
+\bl{$nullable (r_1 \cdot r_2)$} & \bl{$\dn$} & \bl{$nullable(r_1) \wedge nullable(r_2)$} \\
+\bl{$nullable (r^*)$}           & \bl{$\dn$} & \bl{\textit{true}}\\
+\bl{$nullable (r^{\{n\}})$}      & \bl{$\dn$} & \bl{if $n = 0$ then \textit{true} else $nullable(r)$}\\  
+\end{tabular}
+\end{center}
+
+\end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}[c]
+
+\begin{center}
+\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}}
+  \bl{$\der\, c\, (\ZERO)$}      & \bl{$\dn$} & \bl{$\ZERO$} & \\
+  \bl{$\der\, c\, (\ONE)$}           & \bl{$\dn$} & \bl{$\ZERO$} & \\
+  \bl{$\der\, c\, (d)$}                     & \bl{$\dn$} & \bl{if $c = d$ then $\ONE$ else $\ZERO$} & \\
+  \bl{$\der\, c\, (r_1 + r_2)$}        & \bl{$\dn$} & \bl{$\der\, c\, r_1 + \der\, c\, r_2$} & \\
+  \bl{$\der\, c\, (r_1 \cdot r_2)$}  & \bl{$\dn$}  & \bl{if $nullable (r_1)$}\\
+  & & \bl{then $(\der\,c\,r_1) \cdot r_2 + \der\, c\, r_2$}\\ 
+  & & \bl{else $(\der\, c\, r_1) \cdot r_2$}\\
+  \bl{$\der\, c\, (r^*)$}  & \bl{$\dn$} & \bl{$(\der\,c\,r) \cdot (r^*)$}\\
+
+  \bl{$\der\, c\, (r^{\{n\}})$}  & \bl{$\dn$} & \bl{if $n = 0$ then $\ZERO$ else $(\der\,c\,r) \cdot r^{\{n-1\}}$}\\
+                                                
+  \end{tabular}
+\end{center}
+
+\end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{frame}[c]