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+% Chapter Template
+
+\chapter{A Better Bound and Other Extensions} % Main chapter title
+
+\label{Chapter5} %In Chapter 5\ref{Chapter5} we discuss stronger simplifications to improve the finite bound
+%in Chapter 4 to a polynomial one, and demonstrate how one can extend the
+%algorithm to include constructs such as bounded repetitions and negations.
+ 
+%----------------------------------------------------------------------------------------
+%	SECTION strongsimp
+%----------------------------------------------------------------------------------------
+\section{A Stronger Version of Simplification Inspired by Antimirov}
+%TODO: search for isabelle proofs of algorithms that check equivalence 
+
+
+
+%----------------------------------------------------------------------------------------
+%	SECTION 1
+%----------------------------------------------------------------------------------------
+
+\section{Adding Support for the Negation Construct, and its Correctness Proof}
+We now add support for the negation regular expression:
+\[			r ::=   \ZERO \mid  \ONE
+			 \mid  c  
+			 \mid  r_1 \cdot r_2
+			 \mid  r_1 + r_2   
+			 \mid r^*    
+			 \mid \sim r
+\]
+The $\textit{nullable}$ function's clause for it would be 
+\[
+\textit{nullable}(~r) = \neg \nullable(r)
+\]
+The derivative would be
+\[
+~r \backslash c = ~ (r \backslash c)
+\]
+ 
+The most tricky part of lexing for the $~r$ regex
+ is creating a value for it.
+ For other regular expressions, the value aligns with the 
+ structure of the regex:
+ \[
+ \vdash \Seq(\Char(a), \Char(b)) : a \cdot b
+ \]
+But for the $~r$ regex, $s$ is a member of it if and only if
+$s$ does not belong to $L(r)$. 
+That means when there
+is a match for the not regex, it is not possible to generate how the string $s$ matched
+with $r$. 
+What we can do is preserve the information of how $s$ was not matched by $r$,
+and there are a number of options to do this.
+
+We could give a partial value when there is a partial match for the regex inside
+the $\mathbf{not}$ construct.
+For example, the string $ab$ is not in the language of $(a\cdot b) \cdot c$,
+A value for it could be 
+ \[
+ \vdash \textit{Not}(\Seq(\Char(a), \Char(b))) : ~((a \cdot b ) \cdot c)
+ \]
+ The above example demonstrates what value to construct 
+ when the string $s$ is at most a real prefix
+ of the strings in $L(r)$. When $s$ instead is not a prefix of any strings
+ in $L(r)$, it becomes unclear what to return as a value inside the $\textit{Not}$
+ constructor.
+ 
+ Another option would be to either store the string $s$ that resulted in 
+ a mis-match for $r$ or a dummy value as a placeholder:
+ \[
+ \vdash \textit{Not}(abcd) : ~(a^*)
+ \]
+or
+ \[
+ \vdash \textit{Not}(\textit{Dummy}) : ~(a^*)
+ \] 
+ We choose to implement this as it is most straightforward:
+ \[
+ \mkeps(~(r))  = \textit{if}(\nullable(r)) \; \textit{Error} \; \textit{else} \; \textit{Not}(\textit{Dummy})
+ \]
+ 
+%----------------------------------------------------------------------------------------
+%	SECTION 2
+%----------------------------------------------------------------------------------------
+
+\section{Bounded Repetitions}
+
+