--- a/ChengsongTanPhdThesis/Chapters/Inj.tex	Thu Nov 17 23:13:57 2022 +0000
+++ b/ChengsongTanPhdThesis/Chapters/Inj.tex	Mon Nov 21 23:56:15 2022 +0000
@@ -387,7 +387,8 @@
 returns a new regular expression representing
 the original regular expression's language $L \; r$
 being taken the language derivative with respect to $c$.
-\begin{center}
+\begin{table}
+	\begin{center}
 \begin{tabular}{lcl}
 		$\ZERO \backslash c$ & $\dn$ & $\ZERO$\\  
 		$\ONE \backslash c$  & $\dn$ & $\ZERO$\\
@@ -399,7 +400,10 @@
 	&   & $\mathit{else}\;(r_1\backslash c) \cdot r_2$\\
 	$(r^*)\backslash c$           & $\dn$ & $(r\backslash c) \cdot r^*$\\
 \end{tabular}
-\end{center}
+	\end{center}
+\caption{Derivative on Regular Expressions}
+\label{table:der}
+\end{table}
 \noindent
 The most involved cases are the sequence case
 and the star case.
@@ -694,6 +698,7 @@
 to indicate that a value $v$ could be generated from a lexing algorithm
 with input $r$. They call it the value inhabitation relation,
 defined by the rules.
+\begin{figure}[H]\label{fig:inhab}
 \begin{mathpar}
 	\inferrule{\mbox{}}{\vdash \Char \; c : \mathbf{c}} \hspace{2em}
 
@@ -707,6 +712,8 @@
 
 \inferrule{\forall v \in vs. \vdash v:r \land  |v| \neq []}{\vdash \Stars \; vs : r^*}
 \end{mathpar}
+\caption{The inhabitation relation for values and regular expressions}
+\end{figure}
 \noindent
 The condition $|v| \neq []$ in the premise of star's rule
 is to make sure that for a given pair of regular