--- a/ChengsongTanPhdThesis/Chapters/Chapter2.tex	Tue May 17 01:44:30 2022 +0100
+++ b/ChengsongTanPhdThesis/Chapters/Chapter2.tex	Fri May 20 18:48:34 2022 +0100
@@ -488,8 +488,8 @@
 as suggested by the finiteness bound proof.
 And unfortunately we have concrete examples
 where such regexes grew exponentially large before levelling off:
-$(a ^ * + (a + aa) ^ * + (a + aa + aaa) ^ * + \ldots + 
-(a+ aa + aaa+\ldots \underbrace{a \ldots a}_{\text{n a's}})^*$ will already have a maximum
+$(a ^ * + (aa) ^ * + (aaa) ^ * + \ldots + 
+(\underbrace{a \ldots a}_{\text{n a's}})^*$ will already have a maximum
 size that is  exponential on the number $n$.
 %TODO: give out a graph showing how the size of the regex grows and levels off at a size exponential to the original regex
 
@@ -503,7 +503,7 @@
 %----------------------------------------------------------------------------------------
 %	SECTION 5
 %----------------------------------------------------------------------------------------
-\section{a stronger version of simplification}
+\section{A Stronger Version of Simplification}
 %TODO: search for isabelle proofs of algorithms that check equivalence