
diff -r 92ca56c1a199 -r 288637d9dcde Journal/Paper.thy
--- a/Journal/Paper.thy	Thu Sep 12 17:20:48 2013 +0100
+++ b/Journal/Paper.thy	Sat Sep 14 14:08:19 2013 +0100
@@ -341,10 +341,10 @@
   not in the predicate for regularity; and there is no type
   quantification available in HOL.\footnote{Slind already pointed out
   this problem in an email to the HOL4 mailing list on 21st April
-  2005.} Coq, for example, has quantification over types and thus can
-  state such a definition.  This has been recently exploited in a
-  slick formalisation of the Myhill-Nerode theorem in Coq by
-  \citeN{XXX}.
+  2005.} Systems such as Coq permit quantification over types and thus can
+  state such a definition.  This has been recently exploited in an
+  elegant and short formalisation of the Myhill-Nerode theorem in Coq by
+  \citeN{Smolka13}, but as said this is not avaible to us working in Isabelle/HOL.
 
   An alternative, which provides us with a single type for states of automata,
   is to give every state node an identity, for example a natural number, and
@@ -470,6 +470,9 @@
   ``{\it \ldots [b]ut is there a rigorous way to even state that?}'' 
   \end{quote} 
 
+  \noindent
+  We shall give an answer to these questions with our paper.
+
   %Moreover, no side-conditions will be needed for regular expressions,
   %like we need for automata. 
  
@@ -773,14 +776,7 @@
   equivalence classes and only finitely many characters.
   The term @{text "\<lambda>(ONE)"} in the first equation acts as a marker for the initial state, that
   is the equivalence class
-  containing the empty string @{text "[]"}.\footnote{Note that we mark, roughly speaking, the
-  single `initial' state in the equational system, which is different from
-  the method by \citeN{Brzozowski64}, where he marks the
-  `terminal' states. We are forced to set up the equational system in our
-  way, because the Myhill-Nerode Relation determines the `direction' of the
-  transitions---the successor `state' of an equivalence class @{text Y} can
-  be reached by adding a character to the end of @{text Y}. This is also the
-  reason why we have to use our reversed version of Arden's Lemma.}
+  containing the empty string @{text "[]"}.
   In our running example we have the initial equational system
   %
   \begin{equation}\label{exmpcs}
@@ -793,7 +789,17 @@
   \end{tabular}}
   \end{equation}
 
-  \noindent
+  Note that we mark, roughly speaking, the
+  single `initial' state in the equational system, which is different from
+  the method by \citeN{Brzozowski64}, where he marks the
+  `terminal' states. We are forced to set up the equational system in our
+  way, because the Myhill-Nerode Relation determines the `direction' of the
+  transitions---the successor `state' of an equivalence class @{text Y} can
+  be reached by adding a character to the end of @{text Y}. If we had defined
+  our equations the `Brzozowski-way', then our variables do not correspond to 
+  the equivalence classes generated by the Myhill-Nerode relation. This need of reverse marking is also the
+  reason why we have to use our reversed version of Arden's Lemma.
+
   Overloading the function @{text \<calL>} for the two kinds of terms in the
   equational system, we have
   
@@ -818,10 +824,12 @@
   \end{equation}
 
   \noindent
-  holds. The reason for adding the @{text \<lambda>}-marker to our initial equational system is 
+  holds. Again note that the reason for adding the @{text \<lambda>}-marker to our initial equational system is 
   to obtain this equation: it only holds with the marker, since none of 
-  the other terms contain the empty string. The point of the initial equational system is
-  that solving it means we will be able to extract a regular expression for every equivalence class. 
+  the other terms contain the empty string. 
+  The point of the initial equational system is
+  that solving it means we will be able to extract a regular expression for every equivalence class
+  generated by the Myhill-Nerode relation. 
 
   Our representation for the equations in Isabelle/HOL are pairs,
   where the first component is an equivalence class (a set of strings)
@@ -1958,8 +1966,11 @@
   \noindent
   Carrying this idea through, we must not consider the set of all derivatives,
   but the one modulo @{text "ACI"}.  In principle, this can be done formally, 
-  but it is very painful in a theorem prover (since there is no
-  direct characterisation of the set of dissimilar derivatives).
+  but it is very painful in a theorem prover---since there is no
+  direct characterisation of the set of dissimilar derivatives. Therefore
+  \citeN{CoquandSiles12} who follow through this idea had to `stand on their  heads' and
+  first formalise the quite intricate notion of \emph{inductively finite sets} in 
+  order to formalise this property.
 
 
   Fortunately, there is a much simpler approach using \emph{partial
@@ -2668,7 +2679,8 @@
   
   Our future work will focus on formalising the regular expression matchers
   developed by \citeN{Sulzmann12} which generate variable assignments for
-  regular expression submatching.\smallskip
+  regular expression sub-matching. For this they use both, derivatives and
+  partial derivatives of regular expressions.\smallskip
 
   \noindent
   {\bf Acknowledgements:}




