--- a/thys/Paper/Paper.thy	Sun Mar 06 18:40:43 2016 +0000
+++ b/thys/Paper/Paper.thy	Sun Mar 06 20:00:47 2016 +0000
@@ -139,7 +139,7 @@
 The longest initial substring matched by any regular expression is taken as
 next token.\smallskip
 
-\item[$\bullet$] \underline{Rule Priority:}
+\item[$\bullet$] \underline{Priority Rule:}
 
 For a particular longest initial substring, the first regular expression
 that can match determines the token.
@@ -368,7 +368,7 @@
   \noindent Note that no values are associated with the regular expression
   @{term ZERO}, and that the only value associated with the regular
   expression @{term ONE} is @{term Void}, pronounced (if one must) as {\em
-  ``Void''}. It is routine to stablish how values ``inhabiting'' a regular
+  ``Void''}. It is routine to establish how values ``inhabiting'' a regular
   expression correspond to the language of a regular expression, namely
 
   \begin{proposition}
@@ -526,35 +526,24 @@
   \end{tabular}
   \end{center}
 
-
-
-  NOT DONE YET 
-
-  Therefore there are, for example, three
-  cases for sequence regular expressions (for all possible shapes of the
-  value).
- 
-  Again the virtues of this algorithm is that it can be
-  implemented with ease in a functional programming language and
-  also in Isabelle/HOL.
+  \noindent If the regular expression does not match, @{const None} is
+  returned. If the regular expression does match the string, then @{const
+  Some} value is returned. Again the virtues of this algorithm is that it
+  can be implemented with ease in a functional programming language and also
+  in Isabelle/HOL. In the remaining part of this section we prove that
+  this algorithm is correct.
 
-The well-known idea of POSIX lexing is informally defined in (for example)
-\cite{posix}; as correctly argued in \cite{Sulzmann2014}, this needs formal
-specification. The rough idea is that, in contrast to the so-called GREEDY
-algorithm, POSIX lexing chooses to match more deeply and using left choices
-rather than a right choices. For example, note that to match the string 
-@{term "[a, b]"} with the regular expression $(a + \mts)\circ (b+ab)$ the matching
-will return $( Void, Right(ab))$ rather than $(Left\ a, Left\ b)$. [The
-regular expression $ab$ is short for $(Lit\ a) \circ (Lit\ b)$.] Similarly,
-to match {\em ``a''} with $(a+a)$ the leftmost $a$ will be chosen.
+  The well-known idea of POSIX matching is informally defined by the longest
+  match and priority rule; as correctly argued in \cite{Sulzmann2014}, this
+  needs formal specification. 
 
-We use a simple inductive definition to specify this notion, incorporating
-the POSIX-specific choices into the side-conditions for the rules $R tl
-+_2$, $R tl\circ$ and $R tl*$ (as they are now called). By contrast,
-\cite{Sulzmann2014} defines a relation between values and argues that there is a
-maximum value, as given by the derivative-based algorithm yet to be spelt
-out. The relation we define is ternary, relating strings, values and regular
-expressions.
+  We use a simple inductive definition to specify this notion, incorporating
+  the POSIX-specific choices into the side-conditions for the rules $R tl
+  +_2$, $R tl\circ$ and $R tl*$ (as they are now called). By contrast,
+  \cite{Sulzmann2014} defines a relation between values and argues that there is a
+  maximum value, as given by the derivative-based algorithm yet to be spelt
+  out. The relation we define is ternary, relating strings, values and regular
+  expressions.
 
 Our Posix relation @{term "s \<in> r \<rightarrow> v"}