--- a/hws/Der.tex	Mon Feb 03 12:35:54 2025 +0000
+++ b/hws/Der.tex	Mon Feb 03 13:25:59 2025 +0000
@@ -2,11 +2,24 @@
 \usepackage{../style}
 \usepackage{../graphics}
 
+\title{Derivative for the NOT-Regular Expression}
+
 \begin{document}
+\maketitle
+\begin{abstract}
+  \noindent
+  This short note explains why the derivative for the NOT-regular
+  expression is defined as
+  \[
+  der\,c(\sim r) \;\dn\; \sim (der\,c\,r)
+  \]
+  The explanation goes via complement sets, the semantic derivative (\textit{Der})
+  and how the derivative relates to the semantic derivative.
+\end{abstract}
 
 \section*{Complement Sets}
 
-Consider the following picture:
+To start with, consider the following picture:
 
 \begin{center}
 \begin{tikzpicture}[fill=gray]
@@ -23,7 +36,7 @@
 
 
 \noindent
-where $\Sigma^*$ is in our case the set of all strings (what follows
+where $\Sigma^*$ is in our case the set of all strings (what follows in this section
 also holds for any kind of ``domain'', like the set of all integers or
 the set of all binary trees, etc). Let us assume $P(s)$ is a property that
 is about strings, for example $P(s)$ could be ``the string $s$ has
@@ -231,7 +244,15 @@
 
 \noindent
 That means we have established the property of derivatives in the $\sim r$-case\dots{}yippee~;o)
+\medskip
 
+\noindent
+The conclusion is: if we want the property $L(der\,c\,r) = Der\,c\,(L(r))$ to hold and the
+semantics of $\sim r$ is defined as $\overline{L(r)}$, then the definition for the derivative
+for the NOT-regular expression must be:
+\[
+der\,c(\sim r) \;\dn\; \sim (der\,c\,r)
+\]  
 \end{document}
 
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