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+% Chapter Template
+
+% Main chapter title
+\chapter{Correctness of Bit-coded Algorithm without Simplification}
+
+\label{Bitcoded1} % Change X to a consecutive number; for referencing this chapter elsewhere, use \ref{ChapterX}
+%Then we illustrate how the algorithm without bitcodes falls short for such aggressive 
+%simplifications and therefore introduce our version of the bitcoded algorithm and 
+%its correctness proof in 
+%Chapter 3\ref{Chapter3}. 
+
+\section*{Bit-coded Algorithm}
+Bits and bitcodes (lists of bits) are defined as:
+
+\begin{center}
+		$b ::=   1 \mid  0 \qquad
+bs ::= [] \mid b::bs    
+$
+\end{center}
+
+\noindent
+The $1$ and $0$ are not in bold in order to avoid 
+confusion with the regular expressions $\ZERO$ and $\ONE$. Bitcodes (or
+bit-lists) can be used to encode values (or potentially incomplete values) in a
+compact form. This can be straightforwardly seen in the following
+coding function from values to bitcodes: 
+
+\begin{center}
+\begin{tabular}{lcl}
+  $\textit{code}(\Empty)$ & $\dn$ & $[]$\\
+  $\textit{code}(\Char\,c)$ & $\dn$ & $[]$\\
+  $\textit{code}(\Left\,v)$ & $\dn$ & $0 :: code(v)$\\
+  $\textit{code}(\Right\,v)$ & $\dn$ & $1 :: code(v)$\\
+  $\textit{code}(\Seq\,v_1\,v_2)$ & $\dn$ & $code(v_1) \,@\, code(v_2)$\\
+  $\textit{code}(\Stars\,[])$ & $\dn$ & $[0]$\\
+  $\textit{code}(\Stars\,(v\!::\!vs))$ & $\dn$ & $1 :: code(v) \;@\;
+                                                 code(\Stars\,vs)$
+\end{tabular}    
+\end{center} 
+
+\noindent
+Here $\textit{code}$ encodes a value into a bitcodes by converting
+$\Left$ into $0$, $\Right$ into $1$, and marks the start of a non-empty
+star iteration by $1$. The border where a local star terminates
+is marked by $0$. This coding is lossy, as it throws away the information about
+characters, and also does not encode the ``boundary'' between two
+sequence values. Moreover, with only the bitcode we cannot even tell
+whether the $1$s and $0$s are for $\Left/\Right$ or $\Stars$. The
+reason for choosing this compact way of storing information is that the
+relatively small size of bits can be easily manipulated and ``moved
+around'' in a regular expression. In order to recover values, we will 
+need the corresponding regular expression as an extra information. This
+means the decoding function is defined as:
+
+
+%\begin{definition}[Bitdecoding of Values]\mbox{}
+\begin{center}
+\begin{tabular}{@{}l@{\hspace{1mm}}c@{\hspace{1mm}}l@{}}
+  $\textit{decode}'\,bs\,(\ONE)$ & $\dn$ & $(\Empty, bs)$\\
+  $\textit{decode}'\,bs\,(c)$ & $\dn$ & $(\Char\,c, bs)$\\
+  $\textit{decode}'\,(0\!::\!bs)\;(r_1 + r_2)$ & $\dn$ &
+     $\textit{let}\,(v, bs_1) = \textit{decode}'\,bs\,r_1\;\textit{in}\;
+       (\Left\,v, bs_1)$\\
+  $\textit{decode}'\,(1\!::\!bs)\;(r_1 + r_2)$ & $\dn$ &
+     $\textit{let}\,(v, bs_1) = \textit{decode}'\,bs\,r_2\;\textit{in}\;
+       (\Right\,v, bs_1)$\\                           
+  $\textit{decode}'\,bs\;(r_1\cdot r_2)$ & $\dn$ &
+        $\textit{let}\,(v_1, bs_1) = \textit{decode}'\,bs\,r_1\;\textit{in}$\\
+  & &   $\textit{let}\,(v_2, bs_2) = \textit{decode}'\,bs_1\,r_2$\\
+  & &   \hspace{35mm}$\textit{in}\;(\Seq\,v_1\,v_2, bs_2)$\\
+  $\textit{decode}'\,(0\!::\!bs)\,(r^*)$ & $\dn$ & $(\Stars\,[], bs)$\\
+  $\textit{decode}'\,(1\!::\!bs)\,(r^*)$ & $\dn$ & 
+         $\textit{let}\,(v, bs_1) = \textit{decode}'\,bs\,r\;\textit{in}$\\
+  & &   $\textit{let}\,(\Stars\,vs, bs_2) = \textit{decode}'\,bs_1\,r^*$\\
+  & &   \hspace{35mm}$\textit{in}\;(\Stars\,v\!::\!vs, bs_2)$\bigskip\\
+  
+  $\textit{decode}\,bs\,r$ & $\dn$ &
+     $\textit{let}\,(v, bs') = \textit{decode}'\,bs\,r\;\textit{in}$\\
+  & & $\textit{if}\;bs' = []\;\textit{then}\;\textit{Some}\,v\;
+       \textit{else}\;\textit{None}$                       
+\end{tabular}    
+\end{center}    
+%\end{definition}
+
+Sulzmann and Lu's integrated the bitcodes into regular expressions to
+create annotated regular expressions \cite{Sulzmann2014}.
+\emph{Annotated regular expressions} are defined by the following
+grammar:%\comment{ALTS should have  an $as$ in  the definitions, not  just $a_1$ and $a_2$}
+
+\begin{center}
+\begin{tabular}{lcl}
+  $\textit{a}$ & $::=$  & $\ZERO$\\
+                  & $\mid$ & $_{bs}\ONE$\\
+                  & $\mid$ & $_{bs}{\bf c}$\\
+                  & $\mid$ & $_{bs}\sum\,as$\\
+                  & $\mid$ & $_{bs}a_1\cdot a_2$\\
+                  & $\mid$ & $_{bs}a^*$
+\end{tabular}    
+\end{center}  
+%(in \textit{ALTS})
+
+\noindent
+where $bs$ stands for bitcodes, $a$  for $\mathbf{a}$nnotated regular
+expressions and $as$ for a list of annotated regular expressions.
+The alternative constructor($\sum$) has been generalized to 
+accept a list of annotated regular expressions rather than just 2.
+We will show that these bitcodes encode information about
+the (POSIX) value that should be generated by the Sulzmann and Lu
+algorithm.
+
+
+To do lexing using annotated regular expressions, we shall first
+transform the usual (un-annotated) regular expressions into annotated
+regular expressions. This operation is called \emph{internalisation} and
+defined as follows:
+
+%\begin{definition}
+\begin{center}
+\begin{tabular}{lcl}
+  $(\ZERO)^\uparrow$ & $\dn$ & $\ZERO$\\
+  $(\ONE)^\uparrow$ & $\dn$ & $_{[]}\ONE$\\
+  $(c)^\uparrow$ & $\dn$ & $_{[]}{\bf c}$\\
+  $(r_1 + r_2)^\uparrow$ & $\dn$ &
+  $_{[]}\sum[\textit{fuse}\,[0]\,r_1^\uparrow,\,
+  \textit{fuse}\,[1]\,r_2^\uparrow]$\\
+  $(r_1\cdot r_2)^\uparrow$ & $\dn$ &
+         $_{[]}r_1^\uparrow \cdot r_2^\uparrow$\\
+  $(r^*)^\uparrow$ & $\dn$ &
+         $_{[]}(r^\uparrow)^*$\\
+\end{tabular}    
+\end{center}    
+%\end{definition}
+
+\noindent
+We use up arrows here to indicate that the basic un-annotated regular
+expressions are ``lifted up'' into something slightly more complex. In the
+fourth clause, $\textit{fuse}$ is an auxiliary function that helps to
+attach bits to the front of an annotated regular expression. Its
+definition is as follows:
+
+\begin{center}
+\begin{tabular}{lcl}
+  $\textit{fuse}\;bs \; \ZERO$ & $\dn$ & $\ZERO$\\
+  $\textit{fuse}\;bs\; _{bs'}\ONE$ & $\dn$ &
+     $_{bs @ bs'}\ONE$\\
+  $\textit{fuse}\;bs\;_{bs'}{\bf c}$ & $\dn$ &
+     $_{bs@bs'}{\bf c}$\\
+  $\textit{fuse}\;bs\,_{bs'}\sum\textit{as}$ & $\dn$ &
+     $_{bs@bs'}\sum\textit{as}$\\
+  $\textit{fuse}\;bs\; _{bs'}a_1\cdot a_2$ & $\dn$ &
+     $_{bs@bs'}a_1 \cdot a_2$\\
+  $\textit{fuse}\;bs\,_{bs'}a^*$ & $\dn$ &
+     $_{bs @ bs'}a^*$
+\end{tabular}    
+\end{center}  
+
+\noindent
+After internalising the regular expression, we perform successive
+derivative operations on the annotated regular expressions. This
+derivative operation is the same as what we had previously for the
+basic regular expressions, except that we beed to take care of
+the bitcodes:
+
+
+\iffalse
+ %\begin{definition}{bder}
+\begin{center}
+  \begin{tabular}{@{}lcl@{}}
+  $(\textit{ZERO})\,\backslash c$ & $\dn$ & $\textit{ZERO}$\\  
+  $(\textit{ONE}\;bs)\,\backslash c$ & $\dn$ & $\textit{ZERO}$\\  
+  $(\textit{CHAR}\;bs\,d)\,\backslash c$ & $\dn$ &
+        $\textit{if}\;c=d\; \;\textit{then}\;
+         \textit{ONE}\;bs\;\textit{else}\;\textit{ZERO}$\\  
+  $(\textit{ALTS}\;bs\,as)\,\backslash c$ & $\dn$ &
+  $\textit{ALTS}\;bs\,(map (\backslash c) as)$\\
+  $(\textit{SEQ}\;bs\,a_1\,a_2)\,\backslash c$ & $\dn$ &
+     $\textit{if}\;\textit{bnullable}\,a_1$\\
+					       & &$\textit{then}\;\textit{ALTS}\,bs\,List((\textit{SEQ}\,[]\,(a_1\,\backslash c)\,a_2),$\\
+					       & &$\phantom{\textit{then}\;\textit{ALTS}\,bs\,}(\textit{fuse}\,(\textit{bmkeps}\,a_1)\,(a_2\,\backslash c)))$\\
+  & &$\textit{else}\;\textit{SEQ}\,bs\,(a_1\,\backslash c)\,a_2$\\
+  $(\textit{STAR}\,bs\,a)\,\backslash c$ & $\dn$ &
+      $\textit{SEQ}\;bs\,(\textit{fuse}\, [\Z] (r\,\backslash c))\,
+       (\textit{STAR}\,[]\,r)$
+\end{tabular}    
+\end{center}    
+%\end{definition}
+
+\begin{center}
+  \begin{tabular}{@{}lcl@{}}
+  $(\textit{ZERO})\,\backslash c$ & $\dn$ & $\textit{ZERO}$\\  
+  $(_{bs}\textit{ONE})\,\backslash c$ & $\dn$ & $\textit{ZERO}$\\  
+  $(_{bs}\textit{CHAR}\;d)\,\backslash c$ & $\dn$ &
+        $\textit{if}\;c=d\; \;\textit{then}\;
+         _{bs}\textit{ONE}\;\textit{else}\;\textit{ZERO}$\\  
+  $(_{bs}\textit{ALTS}\;\textit{as})\,\backslash c$ & $\dn$ &
+  $_{bs}\textit{ALTS}\;(\textit{as}.\textit{map}(\backslash c))$\\
+  $(_{bs}\textit{SEQ}\;a_1\,a_2)\,\backslash c$ & $\dn$ &
+     $\textit{if}\;\textit{bnullable}\,a_1$\\
+					       & &$\textit{then}\;_{bs}\textit{ALTS}\,List((_{[]}\textit{SEQ}\,(a_1\,\backslash c)\,a_2),$\\
+					       & &$\phantom{\textit{then}\;_{bs}\textit{ALTS}\,}(\textit{fuse}\,(\textit{bmkeps}\,a_1)\,(a_2\,\backslash c)))$\\
+  & &$\textit{else}\;_{bs}\textit{SEQ}\,(a_1\,\backslash c)\,a_2$\\
+  $(_{bs}\textit{STAR}\,a)\,\backslash c$ & $\dn$ &
+      $_{bs}\textit{SEQ}\;(\textit{fuse}\, [0] \; r\,\backslash c )\,
+       (_{bs}\textit{STAR}\,[]\,r)$
+\end{tabular}    
+\end{center}    
+%\end{definition}
+\fi
+
+\begin{center}
+  \begin{tabular}{@{}lcl@{}}
+  $(\ZERO)\,\backslash c$ & $\dn$ & $\ZERO$\\  
+  $(_{bs}\ONE)\,\backslash c$ & $\dn$ & $\ZERO$\\  
+  $(_{bs}{\bf d})\,\backslash c$ & $\dn$ &
+        $\textit{if}\;c=d\; \;\textit{then}\;
+         _{bs}\ONE\;\textit{else}\;\ZERO$\\  
+  $(_{bs}\sum \;\textit{as})\,\backslash c$ & $\dn$ &
+  $_{bs}\sum\;(\textit{map} (\_\backslash c) as )$\\
+  $(_{bs}\;a_1\cdot a_2)\,\backslash c$ & $\dn$ &
+     $\textit{if}\;\textit{bnullable}\,a_1$\\
+					       & &$\textit{then}\;_{bs}\sum\,[(_{[]}\,(a_1\,\backslash c)\cdot\,a_2),$\\
+					       & &$\phantom{\textit{then},\;_{bs}\sum\,}(\textit{fuse}\,(\textit{bmkeps}\,a_1)\,(a_2\,\backslash c))]$\\
+  & &$\textit{else}\;_{bs}\,(a_1\,\backslash c)\cdot a_2$\\
+  $(_{bs}a^*)\,\backslash c$ & $\dn$ &
+      $_{bs}(\textit{fuse}\, [0] \; r\,\backslash c)\cdot
+       (_{[]}r^*))$
+\end{tabular}    
+\end{center}    
+
+%\end{definition}
+\noindent
+For instance, when we do derivative of  $_{bs}a^*$ with respect to c,
+we need to unfold it into a sequence,
+and attach an additional bit $0$ to the front of $r \backslash c$
+to indicate one more star iteration. Also the sequence clause
+is more subtle---when $a_1$ is $\textit{bnullable}$ (here
+\textit{bnullable} is exactly the same as $\textit{nullable}$, except
+that it is for annotated regular expressions, therefore we omit the
+definition). Assume that $\textit{bmkeps}$ correctly extracts the bitcode for how
+$a_1$ matches the string prior to character $c$ (more on this later),
+then the right branch of alternative, which is $\textit{fuse} \; \bmkeps \;  a_1 (a_2
+\backslash c)$ will collapse the regular expression $a_1$(as it has
+already been fully matched) and store the parsing information at the
+head of the regular expression $a_2 \backslash c$ by fusing to it. The
+bitsequence $\textit{bs}$, which was initially attached to the
+first element of the sequence $a_1 \cdot a_2$, has
+now been elevated to the top-level of $\sum$, as this information will be
+needed whichever way the sequence is matched---no matter whether $c$ belongs
+to $a_1$ or $ a_2$. After building these derivatives and maintaining all
+the lexing information, we complete the lexing by collecting the
+bitcodes using a generalised version of the $\textit{mkeps}$ function
+for annotated regular expressions, called $\textit{bmkeps}$:
+
+
+%\begin{definition}[\textit{bmkeps}]\mbox{}
+\begin{center}
+\begin{tabular}{lcl}
+  $\textit{bmkeps}\,(_{bs}\ONE)$ & $\dn$ & $bs$\\
+  $\textit{bmkeps}\,(_{bs}\sum a::\textit{as})$ & $\dn$ &
+     $\textit{if}\;\textit{bnullable}\,a$\\
+  & &$\textit{then}\;bs\,@\,\textit{bmkeps}\,a$\\
+  & &$\textit{else}\;bs\,@\,\textit{bmkeps}\,(_{bs}\sum \textit{as})$\\
+  $\textit{bmkeps}\,(_{bs} a_1 \cdot a_2)$ & $\dn$ &
+     $bs \,@\,\textit{bmkeps}\,a_1\,@\, \textit{bmkeps}\,a_2$\\
+  $\textit{bmkeps}\,(_{bs}a^*)$ & $\dn$ &
+     $bs \,@\, [0]$
+\end{tabular}    
+\end{center}    
+%\end{definition}
+
+\noindent
+This function completes the value information by travelling along the
+path of the regular expression that corresponds to a POSIX value and
+collecting all the bitcodes, and using $S$ to indicate the end of star
+iterations. If we take the bitcodes produced by $\textit{bmkeps}$ and
+decode them, we get the value we expect. The corresponding lexing
+algorithm looks as follows:
+
+\begin{center}
+\begin{tabular}{lcl}
+  $\textit{blexer}\;r\,s$ & $\dn$ &
+      $\textit{let}\;a = (r^\uparrow)\backslash s\;\textit{in}$\\                
+  & & $\;\;\textit{if}\; \textit{bnullable}(a)$\\
+  & & $\;\;\textit{then}\;\textit{decode}\,(\textit{bmkeps}\,a)\,r$\\
+  & & $\;\;\textit{else}\;\textit{None}$
+\end{tabular}
+\end{center}
+
+\noindent
+In this definition $\_\backslash s$ is the  generalisation  of the derivative
+operation from characters to strings (just like the derivatives for un-annotated
+regular expressions).
+
+Now we introduce the simplifications, which is why we introduce the 
+bitcodes in the first place.
+
+\subsection*{Simplification Rules}
+
+This section introduces aggressive (in terms of size) simplification rules
+on annotated regular expressions
+to keep derivatives small. Such simplifications are promising
+as we have
+generated test data that show
+that a good tight bound can be achieved. We could only
+partially cover the search space as there are infinitely many regular
+expressions and strings. 
+
+One modification we introduced is to allow a list of annotated regular
+expressions in the $\sum$ constructor. This allows us to not just
+delete unnecessary $\ZERO$s and $\ONE$s from regular expressions, but
+also unnecessary ``copies'' of regular expressions (very similar to
+simplifying $r + r$ to just $r$, but in a more general setting). Another
+modification is that we use simplification rules inspired by Antimirov's
+work on partial derivatives. They maintain the idea that only the first
+``copy'' of a regular expression in an alternative contributes to the
+calculation of a POSIX value. All subsequent copies can be pruned away from
+the regular expression. A recursive definition of our  simplification function 
+that looks somewhat similar to our Scala code is given below:
+%\comment{Use $\ZERO$, $\ONE$ and so on. 
+%Is it $ALTS$ or $ALTS$?}\\
+
+\begin{center}
+  \begin{tabular}{@{}lcl@{}}
+   
+  $\textit{simp} \; (_{bs}a_1\cdot a_2)$ & $\dn$ & $ (\textit{simp} \; a_1, \textit{simp}  \; a_2) \; \textit{match} $ \\
+   &&$\quad\textit{case} \; (\ZERO, \_) \Rightarrow  \ZERO$ \\
+   &&$\quad\textit{case} \; (\_, \ZERO) \Rightarrow  \ZERO$ \\
+   &&$\quad\textit{case} \;  (\ONE, a_2') \Rightarrow  \textit{fuse} \; bs \;  a_2'$ \\
+   &&$\quad\textit{case} \; (a_1', \ONE) \Rightarrow  \textit{fuse} \; bs \;  a_1'$ \\
+   &&$\quad\textit{case} \; (a_1', a_2') \Rightarrow   _{bs}a_1' \cdot a_2'$ \\
+
+  $\textit{simp} \; (_{bs}\sum \textit{as})$ & $\dn$ & $\textit{distinct}( \textit{flatten} ( \textit{map} \; simp \; as)) \; \textit{match} $ \\
+  &&$\quad\textit{case} \; [] \Rightarrow  \ZERO$ \\
+   &&$\quad\textit{case} \; a :: [] \Rightarrow  \textit{fuse bs a}$ \\
+   &&$\quad\textit{case} \;  as' \Rightarrow _{bs}\sum \textit{as'}$\\ 
+
+   $\textit{simp} \; a$ & $\dn$ & $\textit{a} \qquad \textit{otherwise}$   
+\end{tabular}    
+\end{center}    
+
+\noindent
+The simplification does a pattern matching on the regular expression.
+When it detected that the regular expression is an alternative or
+sequence, it will try to simplify its child regular expressions
+recursively and then see if one of the children turns into $\ZERO$ or
+$\ONE$, which might trigger further simplification at the current level.
+The most involved part is the $\sum$ clause, where we use two
+auxiliary functions $\textit{flatten}$ and $\textit{distinct}$ to open up nested
+alternatives and reduce as many duplicates as possible. Function
+$\textit{distinct}$  keeps the first occurring copy only and removes all later ones
+when detected duplicates. Function $\textit{flatten}$ opens up nested $\sum$s.
+Its recursive definition is given below:
+
+ \begin{center}
+  \begin{tabular}{@{}lcl@{}}
+  $\textit{flatten} \; (_{bs}\sum \textit{as}) :: \textit{as'}$ & $\dn$ & $(\textit{map} \;
+     (\textit{fuse}\;bs)\; \textit{as}) \; @ \; \textit{flatten} \; as' $ \\
+  $\textit{flatten} \; \ZERO :: as'$ & $\dn$ & $ \textit{flatten} \;  \textit{as'} $ \\
+    $\textit{flatten} \; a :: as'$ & $\dn$ & $a :: \textit{flatten} \; \textit{as'}$ \quad(otherwise) 
+\end{tabular}    
+\end{center}  
+
+\noindent
+Here $\textit{flatten}$ behaves like the traditional functional programming flatten
+function, except that it also removes $\ZERO$s. Or in terms of regular expressions, it
+removes parentheses, for example changing $a+(b+c)$ into $a+b+c$.
+
+Having defined the $\simp$ function,
+we can use the previous notation of  natural
+extension from derivative w.r.t.~character to derivative
+w.r.t.~string:%\comment{simp in  the [] case?}
+
+\begin{center}
+\begin{tabular}{lcl}
+$r \backslash_{simp} (c\!::\!s) $ & $\dn$ & $(r \backslash_{simp}\, c) \backslash_{simp}\, s$ \\
+$r \backslash_{simp} [\,] $ & $\dn$ & $r$
+\end{tabular}
+\end{center}
+
+\noindent
+to obtain an optimised version of the algorithm:
+
+ \begin{center}
+\begin{tabular}{lcl}
+  $\textit{blexer\_simp}\;r\,s$ & $\dn$ &
+      $\textit{let}\;a = (r^\uparrow)\backslash_{simp}\, s\;\textit{in}$\\                
+  & & $\;\;\textit{if}\; \textit{bnullable}(a)$\\
+  & & $\;\;\textit{then}\;\textit{decode}\,(\textit{bmkeps}\,a)\,r$\\
+  & & $\;\;\textit{else}\;\textit{None}$
+\end{tabular}
+\end{center}
+
+\noindent
+This algorithm keeps the regular expression size small, for example,
+with this simplification our previous $(a + aa)^*$ example's 8000 nodes
+will be reduced to just 6 and stays constant, no matter how long the
+input string is.
+
+
+
+
+
+
+
+
+
+
+
+%-----------------------------------
+%	SUBSECTION 1
+%-----------------------------------
+\section{Specifications of Some Helper Functions}
+Here we give some functions' definitions, 
+which we will use later.
+\begin{center}
+\begin{tabular}{ccc}
+$\retrieve \; \ACHAR \, \textit{bs} \, c \; \Char(c) = \textit{bs}$
+\end{tabular}
+\end{center}
+
+
+%----------------------------------------------------------------------------------------
+%	SECTION  correctness proof
+%----------------------------------------------------------------------------------------
+\section{Correctness of Bit-coded Algorithm (Without Simplification)}
+We now give the proof the correctness of the algorithm with bit-codes.
+
+Ausaf and Urban cleverly defined an auxiliary function called $\flex$,
+defined as
+\[
+\flex \; r \; f \; [] \; v \; = \; f\; v
+\flex \; r \; f \; c :: s \; v =  \flex r \; \lambda v. \, f (\inj \; r\; c\; v)\; s \; v
+\]
+which accumulates the characters that needs to be injected back, 
+and does the injection in a stack-like manner (last taken derivative first injected).
+$\flex$ is connected to the $\lexer$:
+\begin{lemma}
+$\flex \; r \; \textit{id}\; s \; \mkeps (r\backslash s) = \lexer \; r \; s$
+\end{lemma}
+$\flex$ provides us a bridge between $\lexer$ and $\blexer$.
+What is even better about $\flex$ is that it allows us to 
+directly operate on the value $\mkeps (r\backslash v)$,
+which is pivotal in the definition of  $\lexer $ and $\blexer$, but not visible as an argument.
+When the value created by $\mkeps$ becomes available, one can 
+prove some stepwise properties of lexing nicely:
+\begin{lemma}\label{flexStepwise}
+$\textit{flex} \; r \; f \; s@[c] \; v= \flex \; r \; f\; s \; (\inj \; (r\backslash s) \; c \; v) $
+\end{lemma}
+
+And for $\blexer$ we have a function with stepwise properties like $\flex$ as well,
+called $\retrieve$\ref{retrieveDef}.
+$\retrieve$ takes bit-codes from annotated regular expressions
+guided by a value.
+$\retrieve$ is connected to the $\blexer$ in the following way:
+\begin{lemma}\label{blexer_retrieve}
+$\blexer \; r \; s = \decode  \; (\retrieve \; (\internalise \; r) \; (\mkeps \; (r \backslash s) )) \; r$
+\end{lemma}
+If you take derivative of an annotated regular expression, 
+you can $\retrieve$ the same bit-codes as before the derivative took place,
+provided that you use the corresponding value:
+
+\begin{lemma}\label{retrieveStepwise}
+$\retrieve \; (r \backslash c)  \;  v= \retrieve \; r \; (\inj \; r\; c\; v)$
+\end{lemma}
+The other good thing about $\retrieve$ is that it can be connected to $\flex$:
+%centralLemma1
+\begin{lemma}\label{flex_retrieve}
+$\flex \; r \; \textit{id}\; s\; v = \decode \; (\retrieve \; (r\backslash s )\; v) \; r$
+\end{lemma}
+\begin{proof}
+By induction on $s$. The induction tactic is reverse induction on strings.
+$v$ is allowed to be arbitrary.
+The crucial point is to rewrite 
+\[
+\retrieve \; (r \backslash s@[c]) \; \mkeps (r \backslash s@[c]) 
+\]
+as
+\[
+\retrieve \; (r \backslash s) \; (\inj \; (r \backslash s) \; c\;  \mkeps (r \backslash s@[c]))
+\].
+This enables us to equate 
+\[
+\retrieve \; (r \backslash s@[c]) \; \mkeps (r \backslash s@[c]) 
+\] 
+with 
+\[
+\flex \; r \; \textit{id} \; s \; (\inj \; (r\backslash s) \; c\; (\mkeps (r\backslash s@[c])))
+\],
+which in turn can be rewritten as
+\[
+\flex \; r \; \textit{id} \; s@[c] \;  (\mkeps (r\backslash s@[c]))
+\].
+\end{proof}
+
+With the above lemma we can now link $\flex$ and $\blexer$.
+
+\begin{lemma}\label{flex_blexer}
+$\textit{flex} \; r \; \textit{id} \; s \; \mkeps(r \backslash s)  = \blexer \; r \; s$
+\end{lemma}
+\begin{proof}
+Using two of the above lemmas: \ref{flex_retrieve} and \ref{blexer_retrieve}.
+\end{proof}
+Finally 
+
+
+




