--- a/ChengsongTanPhdThesis/Chapters/Finite.tex	Mon Jul 10 01:53:32 2023 +0100
+++ b/ChengsongTanPhdThesis/Chapters/Finite.tex	Mon Jul 10 14:32:48 2023 +0100
@@ -204,8 +204,16 @@
 \subsection{Overview of the Proof}
 \marginpar{trying to make it more intuitive
 and provide more insights into proof}
-The most important intuition is what we call the "closed forms" of
+The most important idea in this chapter %intuition 
+is what we call the "closed forms" of
 regular expression derivatives with respect to strings.
+In short it allows us to express $r \backslash_{rsimps} s$
+as a different recursive function so induction on the size bound can go through.
+A simple induction on $s$ or $r$ fails for $r\backslash_{rsimps} s$, but 
+works for $\textit{ClosedForm}(r,s)$.
+
+
+
 Assume we have a regular expression $r$, be it an alternative,
 a sequence or a star, the idea is if we try to take several derivatives
 of it on paper, we end up getting a list of subexpressions,
@@ -218,11 +226,6 @@
 if we omit the way these regular expressions need to be nested.
 where each $r_i$ ($i \in \{1, \ldots, n\}$) is related to some fragments
 of $r$ and $s$.
-We call the precise formalisation for the shape of 
-\[
- r_1 + r_2 + r_3 + \ldots + r_n
-\]
-"closed form".
 The second important observation is that the list %of regular expressions
 $[r_1, \ldots, r_n]$ %is not
 cannot grow indefinitely because they all come from $r$, and derivatives
@@ -231,6 +234,45 @@
 is powerful enough to counteract the effect of nested structure of alternatives
 and eliminate duplicates
 such that indeed the list in $a\backslash s$ does not grow unbounded.
+We call the precise formalisation for the shape of 
+\[
+ r_1 + r_2 + r_3 + \ldots + r_n
+\]
+"closed form".
+The name was chosen because turning the recursive relation
+\[
+	a \backslash_{bsimps} (c\!::\!s) \dn (\textit{bsimp} \; (a\backslash c)) \backslash_{bsimps} s
+\]
+into some easier-to-estimate forms 
+like
+\[
+	\sum (a_1\backslash s \cdot a_2) :: (\map \; (a_2\backslash\_) \;
+	(\textit{Suffix} \; s \; a_1))
+	%\backslash_{bsimp
+\]
+was reminiscent of 
+%similar to t
+solving recurrence relations like 
+$T \; n = 2 (T \frac{1}{2} n) + n$ to obtain
+their closed forms.
+%$T \; n = n \ln n + (s \; n)$ ($s \; n$ is 
+%some higher-order terms).
+%(for example we know $T$ is $\Theta (n \ln n)$).
+Just like a closed form of a recursive definition makes estimating 
+their growth possible, the closed 
+form of $a \backslash_{bsimps} s$
+allows us to prove the existence of a size bound.
+Note that \ref{eq:approx} is only an approximate
+term to show our point.
+The precise formalised formula (\ref{seqClosedForm})
+needs to wait until all $\textit{rrexp}$-related
+definitions are given, 
+%but for now we can think of the above as "the sequence
+%regular expression $a_1 \cdot a_2$ after derivatives and simplifications
+%w.r.t string $s$ looks like
+%an alternative of giant list of sub-expressions, where each 
+
+
 
 A high-level overview of the main components of the finiteness proof
 is as follows: