
diff -r 488d35c20949 -r 747c8cf666ca ninems/ninems.tex
--- a/ninems/ninems.tex	Fri Jul 05 21:00:40 2019 +0100
+++ b/ninems/ninems.tex	Fri Jul 05 21:11:21 2019 +0100
@@ -493,10 +493,13 @@
 \]
 
 Now instead of using $nullable$ to give a $yes$, we  call $mkeps$ to construct a parse tree for how $r_3$ matched the string $abc$. $mkeps$ gives the following value $v_3$: \\$Left(Left(Seq(Right(Right(Right(Seq(Empty, Seq(Empty, Empty)))))), Stars []))$\\
-This corresponds to the leftmost term $((0+0+0 + 0 + 1 \cdot 1 \cdot 1) \cdot r^* $ in $r_3$. Note that its leftmost location allows $mkeps$ to choose  it as the first candidate that meets the requirement of being $nullable$. This location is naturally generated by the splitting clause\\ $(r_1 \cdot r_2)\backslash c  (when \, r_1 \, nullable)) \, = (r_1\backslash c) \cdot r_2 \,+\, r_2\backslash c.  \\$. By this clause, we put
-$r_1 \backslash c \cdot r_2 $ at the front and $r_2 \backslash c$ at the back. This allows $mkeps$ to always pick up among two matches the one with a longer prefix. The value \\
+This corresponds to the leftmost term 
+$((0+0+0 + 0 + 1 \cdot 1 \cdot 1) \cdot r^* $\\
+ in $r_3$. Note that its leftmost location allows $mkeps$ to choose  it as the first candidate that meets the requirement of being $nullable$. This location is naturally generated by the splitting clause\\ $(r_1 \cdot r_2)\backslash c  (when \; r_1 \; nullable)) \, = (r_1\backslash c) \cdot r_2 \,+\, r_2\backslash c.  \\$ By this clause, we put
+$r_1 \backslash c \cdot r_2 $ at the $\textit{front}$ and $r_2 \backslash c$ at the $\textit{back}$. This allows $mkeps$ to always pick up among two matches the one with a longer prefix. The value \\
 $Left(Left(Seq(Right(Right(Right(Seq(Empty, Seq(Empty, Empty)))))), Stars []))$\\
-tells us how about the empty string matches the final regular expression after doing all the derivatives: among the regular expressions $(0+0+0 + 0 + 1 \cdot 1 \cdot 1) \cdot r^* + (0+0+0  + 1 + 0) \cdot r*) +((0+1+0  + 0 + 0) \cdot r*+(0+0+0  + 1 + 0) \cdot r* )$ we choose the left most nullable one, which is composed of a sequence of a nested alternative and a folded star that iterates 0 times. In that nested alternative we take the rightmost alternative.
+tells us how about the empty string matches the final regular expression after doing all the derivatives: among the regular expressions in \\$(0+0+0 + 0 + 1 \cdot 1 \cdot 1) \cdot r^* + (0+0+0  + 1 + 0) \cdot r*) +((0+1+0  + 0 + 0) \cdot r*+(0+0+0  + 1 + 0) \cdot r* )$, \\
+we choose the left most nullable one, which is composed of a sequence of an alternative and a star. In that alternative we take the rightmost choice.
 
 Using the value $v_3$, the character c, and the regular expression $r_2$, we can recover how $r_2$ matched the string $[c]$ : we inject $c$ back to $v_3$, and get \\ $v_2 = Left(Seq(Right(Right(Right(Seq(Empty, Seq(Empty, c)))))), Stars [])$, which tells us how $r_2$ matched $c$. After this we inject back the character $b$, and get\\ $v_1 = Seq(Right(Right(Right(Seq(Empty, Seq(b, c)))))), Stars [])$ for how $r_1= (1+0+1 \cdot b + 0 + 1 \cdot b \cdot c) \cdot r*$ matched  the string $bc$ before it split into 2 pieces. Finally, after injecting character a back to $v_1$, we get  the parse tree $v_0= Stars [Right(Right(Right(Seq(a, Seq(b, c)))))]$ for how r matched $abc$.
 We omit the details of injection function, which is provided by Sulzmann and Lu's paper \cite{Sulzmann2014}. 




