--- a/thys3/Paper.thy	Tue Jul 05 00:42:06 2022 +0100
+++ b/thys3/Paper.thy	Sat Jul 09 14:11:07 2022 +0100
@@ -32,6 +32,7 @@
   ALT ("_ + _" [77,77] 78) and
   SEQ ("_ \<cdot> _" [77,77] 78) and
   STAR ("_\<^sup>*" [79] 78) and
+  NTIMES ("_\<^sup>{\<^sup>_\<^sup>}" [79] 78) and
 
   val.Void ("Empty" 78) and
   val.Char ("Char _" [1000] 78) and
@@ -94,7 +95,13 @@
 Brzozowski's derivatives \cite{Brzozowski1964} is that they are neatly
 expressible in any functional language, and easily definable and
 reasoned about in theorem provers---the definitions just consist of
-inductive datatypes and simple recursive functions.  Derivatives of a
+inductive datatypes and simple recursive functions. Another neat
+feature of derivatives is that they can be easily extended to bounded
+regular expressions, such as @{term "NTIMES r n"}, where numbers or
+intervals specify how many times a regular expression should be used
+during matching.
+
+Derivatives of a
 regular expression, written @{term "der c r"}, give a simple solution
 to the problem of matching a string @{term s} with a regular
 expression @{term r}: if the derivative of @{term r} w.r.t.\ (in
@@ -231,7 +238,8 @@
   @{term "CH c"} $\mid$
   @{term "ALT r\<^sub>1 r\<^sub>2"} $\mid$
   @{term "SEQ r\<^sub>1 r\<^sub>2"} $\mid$
-  @{term "STAR r"} 
+  @{term "STAR r"} $\mid$
+  @{term "NTIMES r n"}
   \end{center}
 
   \noindent where @{const ZERO} stands for the regular expression that does
@@ -239,7 +247,13 @@
   only the empty string and @{term c} for matching a character literal.
   The constructors $+$ and $\cdot$ represent alternatives and sequences, respectively.
   We sometimes omit the $\cdot$ in a sequence regular expression for brevity. 
-  The
+  In our work here we also add to the usual ``basic'' regular expressions 
+  the bounded regular expression @{term "NTIMES r n"} where the @{term n}
+  specifies that @{term r} should match exactly @{term n}-times. For
+  brevity we omit the other bounded regular expressions
+  @{text "r"}$^{\{..n\}}$, @{text "r"}$^{\{n..\}}$ and @{text "r"}$^{\{n..m\}}$
+  which specify an interval for how many times @{text r} should match. Our
+  results extend straightforwardly also to them. The
   \emph{language} of a regular expression, written $L(r)$, is defined as usual
   and we omit giving the definition here (see for example \cite{AusafDyckhoffUrban2016}).