--- a/ninems/ninems.tex	Fri Jul 05 17:49:54 2019 +0100
+++ b/ninems/ninems.tex	Fri Jul 05 18:01:26 2019 +0100
@@ -440,7 +440,7 @@
 algorithm by a second phase (the first phase being building successive
 derivatives---see \eqref{graph:*}). In this second phase, a POSIX value 
 is generated assuming the regular expression matches  the string. 
-Pictorially, the algorithm as follows:
+Pictorially, the algorithm is as follows:
 
 \begin{center}
 \begin{tikzcd}
@@ -461,7 +461,7 @@
 parse tree incrementally by \emph{injecting} back the characters into
 the values $v_n, \ldots, v_0$. We first call the function
 $\textit{mkeps}$, which builds the parse tree for how the empty string
-is matched the empty regular expression $r_n$. This function is defined
+has matched the empty regular expression $r_n$. This function is defined
 as
 
 	\begin{center}