diff -r 96e93df60954 -r 823d9b19d21c ChengsongTanPhdThesis/Chapters/ChapterFinite.tex
--- a/ChengsongTanPhdThesis/Chapters/ChapterFinite.tex	Mon May 30 17:24:52 2022 +0100
+++ b/ChengsongTanPhdThesis/Chapters/ChapterFinite.tex	Mon May 30 20:36:15 2022 +0100
@@ -7,67 +7,29 @@
 %of our bitcoded algorithm, that is a finite bound on the size of any 
 %regex's derivatives. 
 
-
-%-----------------------------------
-%	SECTION ?
-%-----------------------------------
-\section{preparatory properties for getting a finiteness bound}
-Before we get to the proof that says the intermediate result of our lexer will
-remain finitely bounded, which is an important efficiency/liveness guarantee,
-we shall first develop a few preparatory properties and definitions to 
-make the process of proving that a breeze.
-
-We define rewriting relations for $\rrexp$s, which allows us to do the 
-same trick as we did for the correctness proof,
-but this time we will have stronger equalities established.
-\subsection{"hrewrite" relation}
-List of 1-step rewrite rules for regular expressions simplification without bitcodes:
-\begin{center}
-\begin{tabular}{lcl}
-$r_1 \cdot \ZERO$ & $\rightarrow$ & $\ZERO$
-\end{tabular}
-\end{center}
-
-And we define an "grewrite" relation that works on lists:
-\begin{center}
-\begin{tabular}{lcl}
-$ \ZERO :: rs$ & $\rightarrow_g$ & $rs$
-\end{tabular}
-\end{center}
-
-
+In this chapter we give a guarantee in terms of time complexity:
+given a regular expression $r$, for any string $s$ 
+our algorithm's internal data structure is finitely bounded.
+To obtain such a proof, we need to 
+\begin{itemize}
+\item
+Define an new datatype for regular expressions that makes it easy
+to reason about the size of an annotated regular expression.
+\item
+A set of equalities for this new datatype that enables one to
+rewrite $\bderssimp{r_1 \cdot r_2}{s}$ and $\bderssimp{r^*}{s}$ etc.
+by their children regexes $r_1$, $r_2$, and $r$.
+\item
+Using those equalities to actually get those rewriting equations, which we call
+"closed forms".
+\item
+Bound the closed forms, thereby bounding the original
+$\blexersimp$'s internal data structures.
+\end{itemize}
 
-With these relations we prove
-\begin{lemma}
-$rs \rightarrow rs'  \implies \RALTS{rs} \rightarrow \RALTS{rs'}$
-\end{lemma}
-which enables us to prove "closed forms" equalities such as
-\begin{lemma}
-$\sflat{(r_1 \cdot r_2) \backslash s} = \RALTS{ (r_1 \backslash s) \cdot r_2 :: (\map (r_2 \backslash \_) (\vsuf{s}{r_1}))}$
-\end{lemma}
+\section{the $\mathbf{r}$-rexp datatype and the size functions}
 
-But the most involved part of the above lemma is proving the following:
-\begin{lemma}
-$\bsimp{\RALTS{rs @ \RALTS{rs_1} @ rs'}} = \bsimp{\RALTS{rs @rs_1 @ rs'}} $ 
-\end{lemma}
-which is needed in proving things like 
-\begin{lemma}
-$r \rightarrow_f r'  \implies \rsimp{r} \rightarrow \rsimp{r'}$
-\end{lemma}
-
-Fortunately this is provable by induction on the list $rs$
-
-
-
-%-----------------------------------
-%	SECTION 2
-%-----------------------------------
-
-\section{Finiteness Property}
-In this section let us describe our argument for why the size of the simplified
-derivatives with the aggressive simplification function is finitely bounded.
- Suppose
-we have a size function for bitcoded regular expressions, written
+We have a size function for bitcoded regular expressions, written
 $\llbracket r\rrbracket$, which counts the number of nodes if we regard $r$ as a tree
 
 \begin{center}
@@ -170,6 +132,65 @@
  regular expression.
 
 
+
+%-----------------------------------
+%	SECTION ?
+%-----------------------------------
+\section{preparatory properties for getting a finiteness bound}
+Before we get to the proof that says the intermediate result of our lexer will
+remain finitely bounded, which is an important efficiency/liveness guarantee,
+we shall first develop a few preparatory properties and definitions to 
+make the process of proving that a breeze.
+
+We define rewriting relations for $\rrexp$s, which allows us to do the 
+same trick as we did for the correctness proof,
+but this time we will have stronger equalities established.
+\subsection{"hrewrite" relation}
+List of 1-step rewrite rules for regular expressions simplification without bitcodes:
+\begin{figure}
+\caption{the "h-rewrite" rules}
+\[
+r_1 \cdot \ZERO \rightarrow_h \ZERO \]
+
+\[
+\ZERO \cdot r_2 \rightarrow \ZERO 
+\]
+\end{figure}
+And we define an "grewrite" relation that works on lists:
+\begin{center}
+\begin{tabular}{lcl}
+$ \ZERO :: rs$ & $\rightarrow_g$ & $rs$
+\end{tabular}
+\end{center}
+
+
+
+With these relations we prove
+\begin{lemma}
+$rs \rightarrow rs'  \implies \RALTS{rs} \rightarrow \RALTS{rs'}$
+\end{lemma}
+which enables us to prove "closed forms" equalities such as
+\begin{lemma}
+$\sflat{(r_1 \cdot r_2) \backslash s} = \RALTS{ (r_1 \backslash s) \cdot r_2 :: (\map (r_2 \backslash \_) (\suffix \; s \; r_1 ))}$
+\end{lemma}
+
+But the most involved part of the above lemma is proving the following:
+\begin{lemma}
+$\bsimp{\RALTS{rs @ \RALTS{rs_1} @ rs'}} = \bsimp{\RALTS{rs @rs_1 @ rs'}} $ 
+\end{lemma}
+which is needed in proving things like 
+\begin{lemma}
+$r \rightarrow_f r'  \implies \rsimp{r} \rightarrow \rsimp{r'}$
+\end{lemma}
+
+Fortunately this is provable by induction on the list $rs$
+
+
+
+%-----------------------------------
+%	SECTION 2
+%-----------------------------------
+
 \begin{theorem}
 For any regex $r$, $\exists N_r. \forall s. \; \llbracket{\bderssimp{r}{s}}\rrbracket \leq N_r$
 \end{theorem}