--- a/ChengsongTanPhdThesis/Chapters/Inj.tex	Tue Aug 23 22:59:49 2022 +0100
+++ b/ChengsongTanPhdThesis/Chapters/Inj.tex	Sat Aug 27 00:37:03 2022 +0100
@@ -1181,19 +1181,23 @@
 simplifications arise.
 \section{A Case Requring More Aggressive Simplifications}
 For example, when starting with the regular
-expression $(a^* \cdot a^*)^*$ and building a few successive derivatives (around 10)
+expression $(a^* \cdot a^*)^*$ and building just over
+a dozen successive derivatives 
 w.r.t.~the character $a$, one obtains a derivative regular expression
-with more than 9000 nodes (when viewed as a tree)
-even with simplification.
-\begin{figure}
+with millions of nodes (when viewed as a tree)
+even with simplification, which is not much better compared
+with the naive version without any simplifications:
+\begin{figure}[H]
+	\centering
 \begin{tikzpicture}
 \begin{axis}[
     xlabel={$n$},
     ylabel={size},
-    legend entries={Naive Matcher},  
+    legend entries={Simple-Minded Simp, Naive Matcher},  
     legend pos=north west,
     legend cell align=left]
 \addplot[red,mark=*, mark options={fill=white}] table {BetterWaterloo.data};
+\addplot[blue,mark=*, mark options={fill=white}] table {BetterWaterloo1.data};
 \end{axis}
 \end{tikzpicture} 
 \caption{Size of $(a^*\cdot a^*)^*$ against $\protect\underbrace{aa\ldots a}_\text{n \textit{a}s}$}