--- a/thys2/Paper/Paper.thy	Tue Mar 22 10:29:07 2022 +0000
+++ b/thys2/Paper/Paper.thy	Tue Mar 22 11:08:43 2022 +0000
@@ -229,6 +229,7 @@
   not match any string, @{const ONE} for the regular expression that matches
   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
   \emph{language} of a regular expression, written $L$, is defined as usual
   and we omit giving the definition here (see for example \cite{AusafDyckhoffUrban2016}).
@@ -1049,7 +1050,7 @@
      add @{text "f x"} as another ``seen'' element to @{text acc}. We will use
      @{term distinctBy} where @{text f} is the erase function, @{term "erase (DUMMY)"},
      that deletes bitsequences from bitcoded regular expressions.
-     This is clearly a computationally more expensive operation, than @{text nub},
+     This is clearly a computationally more expensive operation than @{text nub},
      but is needed in order to make the removal of unnecessary copies
      to work properly.
 
@@ -1308,9 +1309,9 @@
   \end{center}
   \caption{The rewrite rules that generate simplified regular expressions
   in small steps: @{term "rrewrite r\<^sub>1 r\<^sub>2"} is for bitcoded regular
-  expressions and @{term "rrewrites rs\<^sub>1 rs\<^sub>2"} for \emph{lists} of bitcoded
+  expressions and @{term "srewrite rs\<^sub>1 rs\<^sub>2"} for \emph{lists} of bitcoded
   regular expressions. Interesting is the $LD$ rule that allows copies of regular
-  expressions be removed provided a regular expression earlier in the list can
+  expressions to be removed provided a regular expression earlier in the list can
   match the same strings.}\label{SimpRewrites}
   \end{figure}
 *}
@@ -1422,12 +1423,13 @@
    derivatives in some cases even after aggressive simplification, this
    is a hard to believe fact. A similar claim about a theoretical runtime
    of @{text "O(n\<^sup>2)"} is made for the Verbatim lexer, which calculates
-   tokens according to POSIX rules \cite{verbatim}. For this it uses Brzozowski's
-   derivatives . 
+   tokens according to POSIX rules~\cite{verbatim}. For this it uses Brzozowski's
+   derivatives. 
    They write: ``The results of our empirical tests [..] confirm that Verbatim has
-   $O(n^2)$ time complexity.'' \cite[Section~VII]{verbatim}.
+   @{text "O(n\<^sup>2)"} time complexity.'' \cite[Section~VII]{verbatim}.
    While their correctness proof for Verbatim is formalised in Coq, the claim about
-   the runtime complexity is only supported by some emperical evidence.
+   the runtime complexity is only supported by some emperical evidence obtained
+   by using the code extraction facilities of Coq.
    In the context of our observation with the ``growth problem'' of derivatives,
    we
    tried out their extracted OCaml code with the example

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