--- a/thys3/Paper.thy	Sat Sep 10 15:04:35 2022 +0100
+++ b/thys3/Paper.thy	Sat Sep 10 22:30:22 2022 +0100
@@ -263,7 +263,7 @@
   extend straightforwardly to them too. The importance of the bounded
   regular expressions is that they are often used in practical
   applications, such as Snort (a system for detecting network
-  intrusion) and also in XML Schema definitions. According to Bj\"{o}rklund et
+  intrusions) and also in XML Schema definitions. According to Bj\"{o}rklund et
   al~\cite{BjorklundMartensTimm2015}, bounded regular expressions 
   occur frequently in the latter and can have counters of up to
   ten million.  The problem is that tools based on the classic notion
@@ -718,8 +718,8 @@
   \begin{figure}
   \begin{center}
   \begin{tabular}{@ {}l@ {\hspace{1mm}}c@ {\hspace{1mm}}l@ {}}
-  \multicolumn{3}{@ {}l}{$\textit{decode}'\,bs\,(\ONE)$ $\;\dn\;$ $(\Empty, bs)$ \quad\qquad
-  $\textit{decode}'\,bs\,(c)$ $\;\dn\;$ $(\Char\,c, bs)$}\\
+  $\textit{decode}'\,bs\,(\ONE)$ & $\;\dn\;$ & $(\Empty, bs)$\\
+  $\textit{decode}'\,bs\,(c)$ & $\;\dn\;$ & $(\Char\,c, bs)$\\
   $\textit{decode}'\,(\Z\!::\!bs)\;(r_1 + r_2)$ & $\dn$ &
      $\textit{let}\,(v, bs_1) = \textit{decode}'\,bs\,r_1\;\textit{in}\;
        (\Left\,v, bs_1)$\\
@@ -735,15 +735,11 @@
          $\textit{let}\,(v, bs_1) = \textit{decode}'\,bs\,r\;\textit{in}$\\
   & &   $\textit{let}\,(\Stars\,vs, bs_2) = \textit{decode}'\,bs_1\,r^*$
         \hspace{2mm}$\textit{in}\;(\Stars\,v\!::\!vs, bs_2)$\\
-  $\textit{decode}'\,(\S\!::\!bs)\,(r^{\{n\}})$ & $\dn$ & $(\Stars\,[], bs)$\\
-  $\textit{decode}'\,(\Z\!::\!bs)\,(r^{\{n\}})$ & $\dn$ & 
-         $\textit{let}\,(v, bs_1) = \textit{decode}'\,bs\,r\;\textit{in}$\\
-  & &   $\textit{let}\,(\Stars\,vs, bs_2) = \textit{decode}'\,bs_1\,r^{\{n - 1\}}$
-        \hspace{2mm}$\textit{in}\;(\Stars\,v\!::\!vs, bs_2)$\medskip\\	
-  \multicolumn{3}{@ {}l}{$\textit{decode}\,bs\,r$ $\dn$ 
-     $\textit{let}\,(v, bs') = \textit{decode}'\,bs\,r\;\textit{in}$
-     $\;\,\textit{if}\;bs' = []\;\textit{then}\;\textit{Some}\,v\;
-       \textit{else}\;\textit{None}$}\\[-4mm]   
+  $\textit{decode}'\,bs\,(r^{\{n\}})$ & $\dn$ & $\textit{decode}'\,bs\,r^*$\smallskip\medskip\\
+
+  $\textit{decode}\,bs\,r$ & $\dn$ & 
+     $\textit{let}\,(v, bs') = \textit{decode}'\,bs\,r\;\textit{in}$\\
+  & & $\;\;\;\,\textit{if}\;bs' = []\;\textit{then}\;\textit{Some}\,v\;\textit{else}\;\textit{None}$\\[-4mm]   
   \end{tabular}    
   \end{center}
   \caption{Two functions, called $\textit{decode}'$ and \textit{decode}, for decoding a value from a bitsequence with the help of a regular expression.\\[-5mm]}\label{decode}
@@ -1037,7 +1033,7 @@
 
 
 \begin{theorem}\label{thmone}
-$\textit{lexer}\,r\,s = \textit{blexer}\,r\,s$
+$\textit{blexer}\,r\,s = \textit{lexer}\,r\,s$
 \end{theorem}  
 
 
@@ -1101,7 +1097,7 @@
      de-nest, or spill out, @{text ALTs} as follows
      %
      \[
-     @{term "ALTs bs\<^sub>1 ((ALTs bs\<^sub>2 rs\<^sub>2) # rs\<^sub>1)"}
+     @{term "ALTs bs\<^sub>1 (((ALTs bs\<^sub>2 rs\<^sub>2)) # rs\<^sub>1)"}
      \quad\xrightarrow{bsimp}\quad
      @{term "ALTs bs\<^sub>1 ((map (fuse bs\<^sub>2) rs\<^sub>2) @ rs\<^sub>1)"}
      \]
@@ -1377,10 +1373,10 @@
      \noindent
      With these lemmas in place we can finally establish that @{term "blexer_simp"} and @{term "blexer"}
      generate the same value, and using Theorem~\ref{thmone} from the previous section that this value
-     is indeed the POSIX value.
+     is indeed the POSIX value as generated by \textit{lexer}.
      
      \begin{theorem}
-     @{thm[mode=IfThen] main_blexer_simp}
+     @{thm[mode=IfThen] main_blexer_simp[symmetric]} \; (@{text "= lexer r s"}\; by Thm.~\ref{thmone})
      \end{theorem}
 
      %\begin{proof}
@@ -1543,7 +1539,8 @@
    %point-of-view it is really important to have the formal proofs of
    %the corresponding properties at hand.
 
-   We can of course only make a claim about the correctness and the sizes of the
+   With the results reported here, we can of course only make a claim about the correctness
+   of the algorithm and the sizes of the
    derivatives, not about the efficiency or runtime of our version of
    Sulzman and Lu's algorithm. But we found the size is an important
    first indicator about efficiency: clearly if the derivatives can