--- a/slides/slides04.tex	Wed Oct 21 09:24:32 2020 +0100
+++ b/slides/slides04.tex	Wed Oct 21 14:55:20 2020 +0100
@@ -487,7 +487,7 @@
   \bl{$|Left(v)|$}   & \bl{$\dn$} & \bl{$|v|$}\\
   \bl{$|Right(v)|$}  & \bl{$\dn$} & \bl{$|v|$}\\
   \bl{$|Seq(v_1,v_2)|$}& \bl{$\dn$} & \bl{$|v_1| \,@\, |v_2|$}\\
-  \bl{$|[v_1,\ldots ,v_n]|$} & \bl{$\dn$} & \bl{$|v_1| \,@\ldots @\, |v_n|$}\\
+  \bl{$|Stars\,[v_1,\ldots ,v_n]|$} & \bl{$\dn$} & \bl{$|v_1| \,@\ldots @\, |v_n|$}\\
 \end{tabular}
 \end{center}
 
@@ -604,7 +604,7 @@
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
 
   
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   \begin{frame}[c]
   \frametitle{Inject}
   
@@ -629,7 +629,71 @@
               & result $\mapsto$ a value 
   \end{tabular}
   \end{frame}
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \begin{frame}[t]
+  
+  \begin{center}
+  \begin{tabular}{@{\hspace{-3mm}}l@{\hspace{1mm}}c@{\hspace{1mm}}l@{}}
+    \bl{$inj\,(c)\,c\,(Empty)$} & \bl{$\dn$}  & \bl{$Char\,c$}\\
+  \end{tabular}
+  \end{center}
+
+\end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \begin{frame}[t]
+  
+  \begin{center}
+    \begin{tabular}{@{\hspace{-3mm}}l@{\hspace{1mm}}c@{\hspace{1mm}}l@{}}
+    \bl{$inj\,(r_1 + r_2)\,c\,(Left(v))$} & \bl{$\dn$}  & \bl{$Left(inj\,r_1\,c\,v)$}\\
+    \bl{$inj\,(r_1 + r_2)\,c\,(Right(v))$} & \bl{$\dn$}  & \bl{$Right(inj\,r_2\,c\,v)$}\\
+
+  \end{tabular}
+  \end{center}
+
+\end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \begin{frame}[t]
+  
+  \begin{center}
+    \begin{tabular}{@{\hspace{-8mm}}l@{\hspace{1mm}}c@{\hspace{1mm}}l@{}}
+    \\[-15mm]  
+    \bl{$inj\,(r_1 \cdot r_2)\,c\,(Seq(v_1,v_2))$} & \bl{$\dn$}  & \bl{$Seq(inj\,r_1\,c\,v_1,v_2)$}\\
+    \bl{$inj\,(r_1 \cdot r_2)\,c\,(Left(Seq(v_1,v_2)))$} & \bl{$\dn$}  & \bl{$Seq(inj\,r_1\,c\,v_1,v_2)$}\\
+    \bl{$inj\,(r_1 \cdot r_2)\,c\,(Right(v))$} & \bl{$\dn$}  & \bl{$Seq(mkeps(r_1),inj\,r_2\,c\,v)$}\\  
+  \end{tabular}
+\end{center}
+
+
+\begin{textblock}{13}(1,13.4)
+\begin{bubble}[13cm]
+\small
+\bl{$\der\, c\, (r_1 \cdot r_2)$} \bl{$\dn$} \bl{\textbf{if} $nullable (r_1)$}
+ \bl{\textbf{then} $(\der\,c\,r_1) \cdot r_2 + \der\, c\, r_2$}
+ \bl{\textbf{else} $(\der\, c\, r_1) \cdot r_2$}
+\end{bubble}
+\end{textblock}
+
+\end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \begin{frame}[t]
+  
+  \begin{center}
+  \begin{tabular}{@{\hspace{-3mm}}l@{\hspace{1mm}}c@{\hspace{1mm}}l@{}}
+    \bl{$inj\,(r^*)\,c\,(Seq(v,Stars\,vs))$} & \bl{$\dn$}  & \bl{$Stars\,(inj\,r\,c\,v\,::\,vs)$}
+  \end{tabular}
+  \end{center}
+
+\end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
   
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{frame}[c]
@@ -638,7 +702,7 @@
 \begin{center}
 \begin{tabular}{l@{\hspace{1mm}}c@{\hspace{1mm}}l}
   \bl{$lex\,r\,[]$} & \bl{$\dn$}  & \bl{if $nullable(r)$ then $mkeps(r)$ else $error$}\\
-  \bl{$lex\,r\,c::s$} & \bl{$\dn$}  & \bl{$inj\,r\,c\,lex(der(c,r), s)$}\\
+  \bl{$lex\,r\,a::s$} & \bl{$\dn$}  & \bl{$inj\,r\,a\,lex(der(a,r), s)$}\\
 \end{tabular}
 \end{center}
 
@@ -689,6 +753,17 @@
 \small
 for extracting subpatterns \bl{$(z: ((x:ab) + (y:ba))$}
 
+\begin{textblock}{6}(10,10)
+\begin{bubble}[2cm]
+\small
+\begin{tabular}{l}
+  \bl{$(id : r_{id})$}\\
+  \bl{$(key : r_{key})$}\\
+\end{tabular}
+\end{bubble}
+\end{textblock}
+
+
 \end{frame}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%